Static and Dynamic Density Functional Theory and ...called copolymers. Here we consider the class of copolymers called \block copolymers" [7] while there are many kinds of copolymers

$Page 1: Static and Dynamic Density Functional Theory and ...called copolymers. Here we consider the class of copolymers called \block copolymers" [7] while there are many kinds of copolymers$
Static and Dynamic Density Functional Theory

and Simulations for Micellar Structures in Block

Copolymer Systems

Takashi Uneyama


Contents

1 Introduction 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Block Copolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Block Copolymer Micelles and Vesicles . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Theories and Simulations for Block Copolymer Micelles . . . . . . . . . . . 41.3.2 Problems in Previous Continuum Field Studies for Block Copolymer Micelles 61.3.3 Density Functional Approach for Block Copolymer Micelles . . . . . . . . . 7

2 Static Density Functional Theory and Simulations for Micellar Structures inBlock Copolymer Systems 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Hamiltonian and Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 Edwards Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Partition Function and Grand Partition Function . . . . . . . . . . . . . . . 11

2.3 Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Auxiliary Field and Functional Integral Form of Partition Function . . . . . 122.3.2 Functional Taylor Expansion Form with Respect to Auxiliary Field . . . . . 132.3.3 Correlation Functions and Vertex Functions . . . . . . . . . . . . . . . . . . 132.3.4 Random Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.5 Validity of the Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . 16

2.4 Two Point Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.1 Density Functional Integral Form for Grand Partition Function . . . . . . . 172.4.2 Approximations for Saddle Point Equation . . . . . . . . . . . . . . . . . . 192.4.3 Approximations for the Functional Determinant . . . . . . . . . . . . . . . 212.4.4 Explicit Forms for Γ and Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.5 Generalization for Block Copolymers . . . . . . . . . . . . . . . . . . . . . . 232.4.6 Generalization for Block Copolymer Blends . . . . . . . . . . . . . . . . . . 262.4.7 Necessity of Use of the Two Point Density . . . . . . . . . . . . . . . . . . . 26

2.5 Static Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.5.1 Density Functional Theory for Homopolymers . . . . . . . . . . . . . . . . . 282.5.2 Density Functional Theory for Block Copolymers . . . . . . . . . . . . . . . 302.5.3 Density Functional Theory for Block Copolymer Blends . . . . . . . . . . . 312.5.4 Comparison with Other Theories . . . . . . . . . . . . . . . . . . . . . . . . 332.5.5 Physical Properties of Static Density Functional Theory . . . . . . . . . . . 36

2.6 Static Density Functional Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 362.6.1 Static Simulation Method by the Static Density Functional Theory . . . . . 362.6.2 Constraints: Mass Conservation and Incompressibility . . . . . . . . . . . . 372.6.3 Numerical Method and Algorithms for Static Density Functional Simulation 382.6.4 Results of Static Density Functional Simulation . . . . . . . . . . . . . . . . 412.6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.A Self Consistent Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.B Calculation of Coefficients in Approximate Form of Γij . . . . . . . . . . . . . . . . 51

i

ii CONTENTS

2.B.1 Calculation of Monomer-Monomer Two Point Correlation Function . . . . . 522.B.2 Calculation of Aij and Cij . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.C Static Density Functional Theory: Another Derivation . . . . . . . . . . . . . . . . 552.C.1 Intuitive Derivation of the Flory-Huggins-de Gennes-Lifshitz Theory . . . . 552.C.2 Generalization for Block Copolymer Systems . . . . . . . . . . . . . . . . . 552.C.3 Validity of the Heuristic Derivation . . . . . . . . . . . . . . . . . . . . . . . 56

3 Dynamic Density Functional Theory and Simulations for Micellar Structures inBlock Copolymer Systems 593.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2 Time-Dependent Ginzburg-Landau Theory . . . . . . . . . . . . . . . . . . . . . . 59

3.2.1 Equation of Continuity, Fick’s Law, and Mobility . . . . . . . . . . . . . . . 593.2.2 Time-Dependent Ginzburg-Landau Equation . . . . . . . . . . . . . . . . . 603.2.3 Validity of TDGL Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.3 Dynamic Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3.1 Dynamic Density Functional Theory for Colloidal Systems . . . . . . . . . . 613.3.2 Dean Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.3.3 Functional Fokker-Planck Equation for the Dean Equation . . . . . . . . . 643.3.4 Coarse-Graining Method: Archer-Rauscher Approximation . . . . . . . . . 65

3.4 Dynamic Density Functional Theory for Block Copolymer Systems . . . . . . . . . 663.4.1 Approximations for Dynamic Density Functional Equation . . . . . . . . . 663.4.2 ψ-Field Expression of Dynamic Density Functional Equation . . . . . . . . 67

3.5 Discussions on Dynamic Density Functional Equation . . . . . . . . . . . . . . . . 673.5.1 Deterministic and Stochastic Dynamic Density Functional Equations . . . . 673.5.2 Magnitude of Thermal Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 683.5.3 Hydrodynamic Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Dynamic Density Functional Simulation . . . . . . . . . . . . . . . . . . . . . . . . 703.6.1 Incompressible Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.6.2 Numerical Method and Algorithms for Dynamic Density Functional Simulation 713.6.3 Generation Scheme for Thermal Noise Field . . . . . . . . . . . . . . . . . . 733.6.4 Results of Dynamic Density Functional Simulation . . . . . . . . . . . . . . 743.6.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.A Dynamic Density Functional Theory for Rouse Chains . . . . . . . . . . . . . . . . 92

3.A.1 Rouse Model for Single Polymer Chain . . . . . . . . . . . . . . . . . . . . . 923.A.2 Rouse Model for Interacting Many Polymer Chains . . . . . . . . . . . . . . 94

3.B External Potential Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.B.1 Non-Local Mobility Model and Dynamic Equation . . . . . . . . . . . . . . 963.B.2 On Local Mass Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4 Simple Model for Vesicle and Onion Formation Kinetics in Block CopolymerSystems 994.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2 Simple Kinetic Model for Block Copolymer Vesicles and Onions . . . . . . . . . . . 99

4.2.1 Fromherz Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.2.2 Extension of Fromherz Theory for Onion Structures . . . . . . . . . . . . . 1024.2.3 Rough Estimation of ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2.4 Stability of Disk Like Micelles . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.2.5 Onion Formation Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3.1 Vesicle and Onion Formation Mechanisms . . . . . . . . . . . . . . . . . . . 1084.3.2 Difference and Similarity between Vesicles and Onions . . . . . . . . . . . . 108

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.A Two Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.A.1 Fromherz Theory in Two Dimensional Systems . . . . . . . . . . . . . . . . 110

CONTENTS iii

4.A.2 Extension of Fromherz Theory to Onion Formation Kinetics in Two Dimen-sional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5 Summary 1155.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.2.1 On the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.2.2 On the Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116


Chapter 1

Introduction

1.1 Introduction

It is important for engineering to study materials in everyday life. Nowadays there are manyso-called “soft matters” or “soft materials” in everyday life [1]. One of the most important softmaterials is a polymer [2], which has various physical and chemical properties. For example, wecan easily find that polymers such as polyethylene (PE), polypropylene (PP), polystyrene (PS),polyvinyl-chloride (PVC), or polyethylene terephthalate (PET) are used very widely as plastics, orthat polyisoprene (PI) or polybutadiene (PB) are used as rubbers (PI is the natural rubber). Struc-ture formulas for these polymers are shown in Figure 1.1. (For more details see some databases,for example [3].)

Polymers are string like macromolecules consist of units which are called monomers. Because oftheir string like structures, polymers have large internal degrees of freedom, and this makes variousinteresting and useful physical properties of polymers. For example, it is known experimentally andtheoretically that polymers have the viscoelastic properties [4] (polymers behave as viscous liquidsas well as elastic solids, depending on the time scale). Such behaviors are due to the large internaldegrees of freedom, and thus we can change viscoelastic behaviors (for example, the characteristicrelaxation times) by changing the architecture of polymers.

Theoretically it is useful to use coarse-grained units which contains several monomers. Suchunits are called the segments, and by using the concept of segments we can study general propertiesof polymers regardless of their microscopic monomer architectures.

It is well known that blend of polymers composed of different monomer species cause the phaseseparation [5, 6]. This is just like blends of water and oil, and phase separation structures growto the large scale (or macroscopic scale). Therefore such a phase separation is called “macrophaseseparation”.

This is also due to the large internal degrees of freedoms. The phase separation can be under-stood by considering the free energy of the mixing. Here we consider the incompressible binarypolymer blends. For simplicity, we assume that the polymerization indexes (number of segmentsin one polymer molecule) are N for two polymer species. If we write the density of one componentas φ, the free energy can be expressed as the function of φ.

F(φ) = U(φ)− TS(φ) (1.1)

where U is the interaction energy, S is the entropy, and T is the absolute temperature. By usingthe Flory-Huggins mean field theory U and S can be written as follows.

U(φ) ≈ 12kBTχφ(1− φ)V (1.2)

S(φ) ≈ −kB[φ

N

(lnφ

N− 1)

+1− φN

(ln

1− φN− 1)]V (1.3)

1

2 CHAPTER 1. INTRODUCTION

CH2CH

n

CH2CH

nCH3

CH2n

(a) (b) (c)

CH2CH

nCl

(d) (e)

CO

nO

CH2 CH2 O C

O

CH2C

nCH3

(f)

C

H

CH2 CH2C

nH

(g)

C

H

CH2

Figure 1.1: Structure formulas for several popular polymers. (a) polyethylene (PE), (b) polypropy-lene (PP), (c) polystyrene (PS), (d) polyvinyl-chloride (PVC), (e) polyethylene terephthalate(PET), (f) polyisoprene (PI), and (g) polybutadiene (PB).

where kB is the Boltzmann constant, χ is the parameter which represents dimensionless interaction(the Flory-Huggins χ parameter), and V is the volume of the system. Eq (1.2) can be interpretedas the simple mean field type two body interaction. Eq (1.3) can be interpreted as the translationalentropy of each polymer chains [6] (φ/N and (1 − φ)/N correspond to the densities of the centerof mass). From eqs (1.1)-(1.3), the dimensionless free energy density is finally written as follows.

F(φ)kBTV =

12χφ(1− φ) +

φ

N

(lnφ

N− 1)

+1− φN

(ln

1− φN− 1)

=1N

[12

(χN)φ(1− φ) + φ lnφ+ (1− φ) ln(1− φ)]

+ (const)(1.4)

From eq (1.4) it is clear that the thermodynamic behavior is determined by χN . Since N is verylarge for polymers, χN can be large even for small χ.

1.2 Block Copolymers

There are polymers which are composed of two or more monomer species. Such polymers arecalled copolymers. Here we consider the class of copolymers called “block copolymers” [7] whilethere are many kinds of copolymers. Block copolymers consists of two or more subchains (blocks),which are connected chemically each other. Each subchains consists of one monomer species. Weshow schematic image of typical block copolymers in Figure 1.2. The most simple block copolymerspecies is the diblock copolymer (Figure 1.2(a)), which is composed by two subchains (or blocks).If we use three subchains, there are several possible combinations for monomer species or topology.We can construct, for example, ABA linear type copolymers, ABA star type copolymers, ABClinear type copolymers, or ABC star type copolymers (see also Figure 1.2(b)-(d)). If we use fouror more subchains, there are more possible combinations.

Block copolymers in melt states cause phase separation like polymer blends, but because of thechemical bond between subchains, they cannot form macroscopically phase separated structures.Instead they form molecular scale (typically about 10nm – 1µm) phase separation structures called

1.3. BLOCK COPOLYMER MICELLES AND VESICLES 3

(a) (b) (c) (d)

Figure 1.2: Schematic draw of typical block copolymers. Solid light gray line, solid dark gray line,and dashed black line represent subchains of A, B, and C monomers, respectively. (a) AB diblockcopolymer, (b) ABA triblock linear copolymer, (c) ABC triblock linear copolymer , and (d) ABCtriblock star copolymer.

“microphase separation” structures. It is known that there are various self-organized microphaseseparation structures, and the morphologies depend on many parameters such as the strength ofthe interaction (the χ parameters) or the architecture of block copolymers. For example, diblockcopolymers form microphase separation structures such as lamellars (stacked layer like structures),hexagonally packed cylinders, or BCC packed spheres. 1 We show schematic images of microphaseseparation structures formed by diblock copolymers in Figure 1.3.

Lamellars Cylinders SpheresCylindersSpheres

f : small f : largef ∼ 1 / 2

(a) (b) (c) (d) (e)

Figure 1.3: Morphologies of diblock copolymer melts. Light and dark gray represents A and Bsubchains, respectively. f is the block ratio (number fraction) of the A subchain. (a) and (e) BCCspheres, (b) and (d) hexagonal cylinders, and (c) lamellars. It is known that the double gyroidphase is observed in addition to the sphere phase, the cylinder phase, and the lamellar phase.

Microphase separation structures of diblock copolymer melts are studied by several theoreticaland simulation methods. One of the most reliable and accurate result is obtained by using theself consistent field (SCF) theory [8, 12], and is in good agreement with experimental results [9].Triblock copolymers form many complex microphase separation structures [7, 13,14].

Block copolymers which form microphase separation structures are widely used as functionalmaterials (such as plastics [7, 15, 16] or functional gels [17]). For example, rubbers such as poly-isoprene (PI) or polybutadiene (PB) can be toughened by adding block copolymers which containrelatively stiff components such as polystyrene (PS) such as PS-PI diblock copolymers or PS-PBdiblock copolymers.

1.3 Block Copolymer Micelles and Vesicles

Other interesting systems are blends of block copolymers and homopolymers or block copolymersolutions [18]. Amphiphilic molecules such as surfactants or lipids forms various micellar structuresin selective solvents such as water. Block copolymers consists of both the hydrophilic (solvophilic)

1It is know that there are more complicated network type phase separation structures called the double gyroidstructures [8, 9] and the FDDD structures [10,11].


subchain(s) and the hydrophobic (solvophobic) subchain(s) also form micellar structures [19–21]in selective solvents.2

A schematic image of typical micellar structures observed in block copolymers is depicted inFigure 1.4. The typical micellar structures observed are the spherical micelles, the cylindricalmicelles, or the vesicles (closed bilayer structures). These structures are considered to be the sameas ones observed in the systems of low molecular weight amphiphilic molecules. The morphologiesare known to depend various parameters or conditions such as the monomer species of subchains,the architecture of the block copolymer, the temperature, or the solvent quality. Such micellarstructures are considered to be used as functional materials such as microcupsules for the drugdelivery system (DDS) [18,22].

While there are various experimental works on block copolymer micelles, properties of blockcopolymer micelles or mechanisms of the micelle formation are still not well understood.

Spherical micelles

Cylindrical micellesVesicles

(a)

(b)

(c)

Figure 1.4: Schematic draw of micellar structures formed by amphiphilic diblock copolymers. (a)spherical micelles, (b) cylindrical micelles, and (c) vesicles. Light and dark gray subchains representhydrophilic and hydrophobic subchains.

1.3.1 Theories and Simulations for Block Copolymer Micelles

The static and dynamic properties of polymeric micellear systems covers wide time and lengthscales. There are several characteristic scales and simulation methods. We draw the schematicimage of the time and length scales in Figure 1.5. There are some theoretical and simulation modelsfor each scales. At the microscopic (atomic) scale, the combination of the Newton equation andthe Lennard-Jones potential are often used. At the macroscopic scale, continuum field descriptionfor viscoelastic liquids (or solids) works well. At the intermediate, mesoscopic scale, various modelsincluding the coarse-grained particle models and the continuum field models have been used.

So far various theories have been proposed for micellar systems and simulations based on thetheories have been carried out as well. Typical simulation methods are as follows.

• Particle models

– Molecular dynamics (MD) (atomistic models or coarse-grained models)

– Dissipative particle dynamics (DPD)

– Brownian dynamics (BD)

• Lattice models

– Monte Carlo (MC) simulations2In experiments, polar organic solvents are often used instead of water. In this work, for simplicity we use the

words “hydrophilic” and “hydrophobic” as same as the “solvophilic” and “solvophobic” hereafter.


Atomistic MD

Length scale

Tim

e sc

ale

Micro

Meso

Macro

Monomers,Atoms

Polymerchains

Phase separationstructures

Higher orderstructures

Coarse-grained MD

ContinuumField Model

Figure 1.5: Schematic image of the length and time scales in block copolymer systems.

• Continuum field models

– Self consistent field (SCF) theory

– Time-dependent Ginzburg-Landau (TDGL) theory

– Density functional (DF) theory

The most reliable model is the MD simulations. While the MD [23–25] simulations successfullyreproduce vesicle formation dynamics or membrane fusion dynamics for low molecular systems(such as lipids), the MD simulations for block copolymer solutions are not easy because theyrequires large computational costs. The coarse-grained particle simulations, such as the DPD[26,27] or the BD [28] simulations reproduce the kinetics which are qualitatively the same as onesby the MD simulations with reasonable computational costs. Therefore the particle simulationsare widely used to study dynamics of micellar systems from atomistic, fine scales to coarse-grained,mesoscale systems. The lattice Monte Carlo models [29–33] also reproduce static micellar structureswell and they gives several useful suggestions on kinetic pathways.

On the other hand, the continuum field models are not used so much. While the continuum fieldmodels achieved success in static simulations for block copolymer melts or blends (for example,the phase diagrams can be calculated accurately), there are few works for micellar systems.

The first continuum field simulations for block copolymer micelles are performed by He et al [34],for 2 dimensional systems by using the static SCF method. It is worth surprising that the simulationmethod used in their simulations is a standard method [35]. This means that the simulation ofmicellar systems itself is not so difficult nor special as long as the appropriate parameters areused (for example, the block ratio, the volume fraction, or the interaction parameters). TheSCF simulations are also performed for the three dimensional systems [36] or the polydispersesystems [37].

There are few dynamic simulation works based on the continuum field theory. Although thereare several works which aim the dynamic simulations [38], most of them are physically unaccept-able. (We will discuss these works in detail, later.) Most of these previous works are using thephenomenological Ginzburg-Landau (GL) type free energy functional models and the Cahn-Hilliardtype simple phenomenological dynamic equations.

The GL type free energy model is clearly inappropriate for micellar systems because the GLtype free energy is based on the expansion of the order parameter (the density fluctuation) around


the homogeneous state. If the magnitude of the density fluctuation is not small, the validity ofthe GL expansion and the resulting free energy functional are not guaranteed. Unfortunately, inmicellar systems, amphiphilic molecules associate into spatially localized micellar structures. Thisfact means that the density (or the density fluctuation) strongly depends on the position and theGL expansion cannot be justified.3

Together with the GL type free energy model, the Cahn-Hilliard type simple dynamic equation[39] is usually used in previous works. The use of the Cahn-Hilliard type equation is also notinappropriate for micellar systems. As mentioned, the GL expansion is inappropriate for micellarsystems. This means that the Chan-Hilliard equation is also inappropriate because it assumesthe validity of the GL expansion. Besides, as shown by particle simulations, characteristic sizeof micelles are mesoscopic and the effect of the thermal fluctuation is quite large. Thus thedeterministic dynamic equation cannot describe the dynamics correctly.4

1.3.2 Problems in Previous Continuum Field Studies for Block Copoly-mer Micelles

We have shown that there are problems in previous continuum field studies and we need to overcomeseveral difficulties to study block copolymer micelles by using the continuum field model. We cansummarize the problems of previous continuum field studies for block copolymer micelles as follows.

• Micelles are strongly localized in space, but GL expansion type phenomenological free energymodels cannot be applied to such strongly localized structures.

• Cylindrical micelles or vesicles are usually observed in the strong segregation region. How-ever, phenomenological free energy models cannot be applied to the strong segregation field,neither.

• Hydrophilic/hydrophobic interactions are important for the formation and the stabilizationof micellar structures. In most previous studies, the hydrophilic/hydrophobic interaction arenot treated correctly.

• The thermal activation type processes are essential for structure formation dynamics in mi-cellar systems. In most previous studies, this effect is ignored or underestimated.

Physically these problems are rather obvious. What we need for us to study micellar structures inblock copolymer systems is to construct physically acceptable continuum field model.5

In this work, we propose the static and dynamic density functional theory for block copolymersystems, which can be applied for micellar systems unlike previous models. We show that theproblems shown above can be overcome by our density functional theory. We also perform staticand dynamic simulations for micellar systems and show that micellar structures can be formedactually by simulations. We especially emphasis that the dynamic density functional model is thefirst continuum field model which can reproduce various micellar structures including vesicles.

3In most cases, the GL type free energy models are introduced as the “phenomenological models”. However, sincethey never describe the phenomena correctly, it is not good to call them “phenomenological models”. Although theGL theory has some generality and the GL free energy models have some universality, it is not equivalent that wecan apply them to all the systems. Such uses of the GL free energy models will lead qualitatively incorrect results.

4Several improved versions of the Cahn-Hilliard type equation (for example, dynamic equations for stronglysegregated systems or dynamic equations with the thermal noise) are proposed so far. However, as far as the dynamicequation is introduced as the phenomenological model, we have no principle to select the dynamic equation amongthe candidates.

5One may wonder why such simple problems have not been overcome yet. We considers that there are mainlytwo reasons. One is that we can use particle models to study micellar structures. There are various particle modelswhich can be used to study micellar structures, and thus we do not need other unreliable models. Another is thatmacrophase separation or microphase separation dynamics are widely studied by the continuum field models. Thecontinuum field models have achieved successful results, and thus we expect (or believe) that the continuum fieldmodels can be used for micellar systems without any modifications.


1.3.3 Density Functional Approach for Block Copolymer Micelles

In this work, we study block copolymer micelles and vesicles by using the density functional theory.In the density functional theory, the state of the system is expressed only by the density fields. Allthe other informations such as positions of each monomers are not used explicitly.

The word “density functional theory” sometimes means the integral equation type theories (suchas the simple liquid theory [40] or the polymer reference interaction site model (PRISM) [41–43]).However, in this work we use the word “density functional theory” for more general sense. Herewe call the theory which express the thermodynamic function as or the dynamic equation of thesystem by using the functional or the density fields as the density functional theory.

Static Density Functional Theory

The density functional theory can be roughly categorized into two categories. One is the staticdensity functional theory. In the static density functional theory, we express the free energy (orother thermodynamic function) as the functional of the density fields. For example, the free energyof the system F is expressed as

F = F [ρi] (1.5)

where ρi is the density field of the i-th component.6 Since all the thermodynamic properties canbe calculated from the free energy, the main purpose of the static density functional theory is toderive the accurate expression of the free energy functional. It is clear that the expression of thefree energy functional affects all the thermodynamic behaviors qualitatively.

The static density functional theory is generally numerically efficient method to calculate thestatic properties such as phase diagrams.

The difference between our density functional theory and the phenomenological GL type freeenergy models is the use of the expansion. In the GL type models, the free energy functional Fis expanded into the power series of the density fluctuation. As described in the previous section,such an expansion is not valid for micellar systems. This makes our theory qualitatively differentfrom the phenomenological GL type models.

Our density functional theory is rather similar to the SCF theory. The parameters used in ourtheory and the SCF theory is the same, and simulation results agrees qualitatively. The differencebetween our theory and the SCF theory is the accuracy and computational costs for simulations.In the SCF theory, the information about the conformation of polymer chains can be handledcorrectly. This makes SCF simulations accurate, but its computational costs are rather large.In contrast, our density functional theory does not handle the information about conformationscorrectly but use rather rough approximations for it. Thus the DF simulations need rather smallcomputational costs but not so accurate compared with the SCF simulations.

Dynamic Density Functional Theory

Another category is the dynamic density functional theory. In the dynamic density functionaltheory, we express the dynamic equation for the density fields as the closed form of the densityfields.

∂ρi(r, t)∂t

= L[ρi] (1.6)

where L is a functional of ρi and generally depends on other informations such as time or externalforce field and L can be stochastic (eq (1.6) is the generalized Langevin equation if L includes thestochastic term). To describe the time evolution, the free energy functional derived in the staticdensity functional theory is often used. Therefore the accuracy of the dynamic density functionaldepends on both the expression of the free energy functional and one of the dynamic equation.Unfortunately, the statistical mechanical basis of the dynamic functional theory is still not fullyunderstood unlike the static density functional theory. However, the expressions for the free energy

6In polymer physics, the volume fraction field is often used instead of the density field. The two fields coincideif the excluded volume of the segment is unity. We set the excluded volume to unity in this work for the sake ofsimplicity.


functional and the dynamic equation are considered to affect strongly the dynamic behavior of thesystem.

The dynamic density functional theory is numerically not so efficient compared with particlemethods. However, we can use the same parameters used in static density functional simulationsfor dynamic density functional simulations. It means that we can compare or combine the staticand dynamic density functional simulations smoothly. There are many static density functionalsimulation works and we can use the data or knowledge of these works for the dynamic simulations.

Our dynamic density functional theory looks much different from conventional TDGL or dy-namic SCF models. However, this does not mean that our theory is not correct. Oppositely itmeans that conventional TDGL or dynamics SCF models are not correct for micellar systems.This is physically and intuitively clear because our model reproduces the dynamics of micellarsystems (such as the vesicle formation process) qualitatively while the conventional models cannotreproduce the dynamics.7

7However, notice that this does not mean that the conventional models are not correct for all cases. It worksqualitatively well for macrophase separation dynamics, such as the phase separation of polymer blends.

Chapter 2

Static Density Functional Theoryand Simulations for MicellarStructures in Block CopolymerSystems

2.1 Introduction

In this chapter, we consider static density functional theory for block copolymer systems andperform static simulations. The main purpose of this chapter is to derive the free energy functionaltheory which can be applied to micellar systems. After derive the density functional theory weshow that micellar structures including vesicles can be actually formed by simulations.

As mentioned, we use the continuum field model. We derive the thermodynamic function (thefree energy) as the functional of the density fields. Because thermodynamically stable structures(equilibrium or metastable structures) minimize the free energy, we can obtain these structures byminimizing the free energy functional.

However there are large degrees of freedom and resulting structures are generally not simple,thus we minimize the free energy numerically. We show numerical techniques for the minimizationand perform simulations for micellar systems.

2.2 Hamiltonian and Partition Function

2.2.1 Edwards Hamiltonian

First we describe the Hamiltonian of the system. We start from the most simple linear homopoly-mer systems. We assume that polymers are flexible and can be expressed well as the Gaussianchain [4, 6]. For the Gaussian chain, the Hamiltonian can be decomposed into two parts. One isthe interaction energy. The form of the interaction energy is formally just the same as the simplemolecules. Another is the contribution of the conformational entropy of the chain. The form ofthe entropy contribution is the same as the elastic energy of springs, but the spring coefficient isproportional to the temperature. This is the characteristic property of the Gaussian chain.

We consider the case of the non-interacting Gaussian polymer chain. We consider the polymerchain consists of linearly connected N segments and write the position of the j-th segment as Rj .

9

10 CHAPTER 2. STATIC DENSITY FUNCTIONAL THEORY AND SIMULATIONS

For the Gaussian chain, the conformation of the chain is expressed as the random walk.

Rj+1 = Rj +Bj (2.1)〈Bj〉 = 0 (2.2)

〈BjBk〉 = b2δij1 (2.3)

where 〈. . . 〉 means the statistical average, b is the segment size, and 1 is the unit tensor. We canshow that the probability of a conformation Rj can be written as follows.

P (Rj) ∝ exp

− 3

2b2

N−1∑

j=1

|Rj −Rj+1|2 (2.4)

The effective Hamiltonian for the chain can be obtained by inverting eq (2.4).

H0 = −kBT lnP (Rj) (2.5)

where kB is the Boltzmann constant and T is the absolute temperature of the system. Finally theHamiltonian for a polymer of which polymerization degree is N can be expressed as follows.

H0 =3kBT2b2

N−1∑

j=1

|Rj −Rj+1|2 (2.6)

where we have dropped the constant terms.If N is sufficiently large, we can take the continuum limit expression.

H0 =3kBT2b2

∫ N

0

ds

∣∣∣∣∂R(s)∂s

∣∣∣∣2

(2.7)

where R(s) represents the conformation of a polymer chain.If we assume that the Gaussian statistics of the polymer chains in the system is not affected

so much by the interaction between segments, we can describe the Hamiltonian of many polymerchains as follows.

H = H0 + U (2.8)

H0 =3kBT2b2

∑

k

∫ N

0

ds

∣∣∣∣∂Rk(s)∂s

∣∣∣∣2

(2.9)

U =12

∑

k,k′

∫dsds′ v(Rk(s)−Rk′(s′)) (2.10)

where Rk(s) represents the conformation of the k-th polymer chain, and v(r) is the interactionpotential between segments. Eqs (2.8)-(2.10) is called the Edwards Hamiltonian.

Generalization of eqs (2.8)-(2.10) is straightforward. We consider general block copolymersystems here. We distinguish block copolymer species by indices p, q, . . . and subchain species byindices i, j, . . . . Thus one subchain species in the system can be described as double indices like(p, i) or (q, j). For general block copolymer systems, the Edwards Hamiltonian can be describedas follows.

H = H0 + U (2.11)

H0 =3kBT2b2

∑

p,i

∑

k∈p

∫

s∈ids

∣∣∣∣∂Rk(s)∂s

∣∣∣∣2

(2.12)

U =12

∑

p,i,q,j

∑

k∈p,k′∈q

∫

s∈i,s′∈jdsds′ vpi,qj(Rk(s)−Rk′(s′)) (2.13)

2.2. HAMILTONIAN AND PARTITION FUNCTION 11

where k ∈ p means that we take the sum for all polymer chains of which polymer species is p, ands ∈ i means that we take the integral over the i-th subchain. vpi,qj(r) is the interaction betweenthe segments in the subchains (p, i) and (q, j).

The interaction potential is generally not simple form, but in continuum field models, we oftenapproximate it as the local contact interaction.

vpi,qj(r) = (v0 + εpi,qj)δ(r) = (v0 + kBTχpi,qj)δ(r) (2.14)

where v0 is constant and corresponds to the excluded volume parameter. if v0 is sufficiently large,the system is almost incompressible. 1 εpi,qj is the effective contact interaction parameter betweensubchains (p, i) and (q, j). χpi,qj ≡ εpi,qj/kBT is the Flory-Huggins interaction parameter (χparameter) between subchains (p, i) and (q, j). The χ parameter is a dimensionless parameterwhich represents effective strength of interaction between subchains.

The most accurate theory based on the Edwards Hamiltonian (eqs (2.11)-(2.13)) under themean field approximation is the self consistent field (SCF) theory. (We show a brief derivation ofthe SCF in Appendix 2.A.) Other theories such as the Ginzburg-Landau (GL) expansion theory orthe density functional theory shown in the following sections can be interpreted as approximationsfor the SCF.

2.2.2 Partition Function and Grand Partition Function

By using the Edwards Hamiltonian (eqs (2.11)-(2.13)) we can write the partition function for thesystem.

Z =∏p

Zp (2.15)

Zp ≡ 1Mp!

∫

k∈pDRk exp [−βH]

=1Mp!

∫

k∈pDRk exp [−βU ]

(2.16)

where Mp is the number of the p-th polymer chains and β = 1/kBT is the inverse temperature.DRk represents the functional integral over all chain conformations.

∫

k∈pDRk ≡

∫ ∏

k∈pDRk (2.17)

DRk is defined as follows.

DRk ≡ DRk exp

[− 3

2b2

∫ds

∣∣∣∣∂Rk(s)∂s

∣∣∣∣2]

(2.18)

The grand partition function is sometimes more convenient than the canonical partition func-tion. The grand partition function can be expressed as follows.

Ξ =∏p

Ξp (2.19)

Ξp =∞∑

Mp=0

eβµpMp

Mp!

∫

k∈pDRk exp [−βH]

=∞∑

Mp=0

eβµpMp

Mp!

∫

k∈pDRk exp [−βU ]

(2.20)

where µp is the chemical potential for the p-polymer species.1If the incompressible condition is imposed to the system, we can drop the interaction terms which contain v0.


2.3 Ginzburg-Landau Theory

In this work, we do not use the Ginzburg-Landau (GL) expansion type theory.2 However, toconsider the difference between the GL expansion models and our density functional model, weshow the GL theory briefly here.

2.3.1 Auxiliary Field and Functional Integral Form of Partition Function

For simplicity we consider systems consists of one block copolymer species here. We start from thecanonical partition function. (In this section, we ignore the Gibbs factor 1/M ! in eq (2.16).)

Z =∫DRk exp [−βH[Rk]]

=∫DRk exp [−βU [ρi]]

(2.21)

First we introduce the microscopic monomer density (or the monomer density operator) ρi(r)defined as

ρi(r) ≡M∑

k=1

∫

s∈ids δ (r −Rk(s)) (2.22)

By using the identity for the functional delta function, we can transform the partition functioninto the functional integral form over the density field.

1 =∫Dρi

∏

i

δ[ρi − ρi] =∫DρiDWi exp

[i∑

i

ρi ·Wi − i∑

i

ρi ·Wi

](2.23)

where Wi(r) is the auxiliary field and f · g means the functional inner product defined via thefollowing equation.

f · g ≡∫dr f(r)g(r) (2.24)

Using the identity (2.23), eq (2.21) can be rewritten as follows.

Z =∫Dρi exp [−βU [ρi]]

∫DRk

∏

i

δ[ρi − ρi]

=∫DρiDWi exp

[i∑

i

ρi ·Wi − βU [ρi]]∫DRk exp

[−i∑

i

ρi ·Wi

]

=∫DρiDWi exp

[i∑

i

ρi ·Wi − βU [ρi]][∫

DR exp

[−i∑

i

∫

s∈idsWi(R(s))

]]M

=∫DρiDWi exp

[i∑

i

ρi ·Wi − βU [ρi] +M ln Z1[Wi]]

(2.25)

where we defined one-chain partition function Z1[Wi] as follows.

Z1[Wi] ≡∫DR exp

[−i∑

i

∫

s∈idsWi(R(s))

](2.26)

2In this work, we use “Ginzburg-Landau theory” as the functional expansion form free energy functional theory.However, the terminology “Ginzburg-Landau theory” sometimes indicates more general free energy functional theory.Thus one may call the density functional theory in this work as “Ginzburg-Landau theory”.

2.3. GINZBURG-LANDAU THEORY 13

2.3.2 Functional Taylor Expansion Form with Respect to Auxiliary Field

While eq (2.25) is exact, it is practically useless because it is expressed as the double functionalintegral form. We want the free energy functional as the functional of the density field or the densityfluctuation field. To obtain such a free energy functional, we first approximate the functionalintegral in eq (2.25) by the saddle point approximation, and then express it as a functional of thedensity fluctuation field, by using the functional expansion.

We use the saddle point values ρ∗i (r) andW ∗i (r), defined via following equations, to approximateeq (2.25).

δ

δρi(r)

[i∑

i

ρi ·Wi − βU [ρi] +M ln Z1[Wi]]∣∣∣∣∣ρi=ρ∗i ,Wi=W∗i

= 0 (2.27)

δ

δWi(r)

[i∑

i

ρi ·Wi − βU [ρi] +M ln Z1[Wi]]∣∣∣∣∣ρi=ρ∗i ,Wi=W∗i

= 0 (2.28)

Eqs (2.27) and (2.28) can be rewritten as follows.

W ∗i (r) = −i δ

δρ∗i (r)βU [ρ∗i ] (2.29)

ρ∗i (r) = iM

Z1[W ∗i ]δZ1[W ∗i ]δW ∗i (r)

(2.30)

Using the saddle point approximation we have

Z ≈∫DρiDWi

∏

i

δ[ρi − ρ∗i ]δ[Wi −W ∗i ] exp

[i∑

i

ρi ·Wi − βU [ρi] +M ln Z1[Wi]]

= exp

[i∑

i

ρ∗i ·W ∗i − βU [ρ∗i ] +M ln Z1[W ∗i ]]

(2.31)

The free energy F is expressed by using the logarithm of the partition function.

βF [ρ∗i ] = − lnZ

≈ βU [ρ∗i ]− i∑

i

ρ∗i ·W ∗i −M ln Z1[W ∗i ] (2.32)

Eq (2.32) still contains both ρ∗i (r) and W ∗i (r). The next task is to express W ∗i (r) by ρ∗i (r) andwrite down the free energy only by using ρ∗i (r).

2.3.3 Correlation Functions and Vertex Functions

To obtain the free energy as an explicit functional of ρ∗i (r), we consider to expand the free energyaround the reference state.3

We have to determine the reference state here. We employ the homogeneous state, whichis realized for the non-interacting ideal case, as the reference state. The homogeneous state isreproduced by setting

ρ∗i (r) = ρi (2.33)

where ρi is the spatial average of ρi(r). In this case, from eq (2.29) we have

W ∗i (r) = −i δ

δρ∗i (r)βU [ρ∗i ]

∣∣∣∣ρ∗i=ρi

= Wi (2.34)

3In the SCF, the saddle point value W ∗i (r) can be calculated from ρ∗i (r) without any further approximationsuch as expansion method used in this section. See Appendix 2.A for detail.


where Wi is constant. Thus both ρ∗i (r) and W ∗i (r) are constant. We write the free energy for thereference state as F0.

βF0 ≡ βF [ρi] = βU [ρi]− i∑

i

ρi · Wi −M ln Z1[Wi] (2.35)

To expand the free energy, we introduce the fluctuation field around the reference state.

δρi(r) ≡ ρ∗i (r)− ρi (2.36)δWi(r) ≡W ∗i (r)− Wi (2.37)

The free energy F [ρi] cab be expanded around the reference state, as follows.

βF [ρ∗i ] = βF0 +12

∑

i,j

∫drdr′

δ2FδW ∗i (r)δW ∗j (r′)

∣∣∣∣∣W∗i =Wi

δWi(r)δWj(r′) + · · ·

= βF0 − M

2

∑

i,j

∫drdr′

[1

Z1[W ∗i ]δ2Z1[W ∗i ]

δW ∗i (r)δW ∗j (r′)

∣∣∣∣∣W∗i =Wi

− 1(Z1[W ∗i ])2

δZ1[W ∗i ]δW ∗i (r)

δZ1[W ∗i ]δW ∗j (r′)

∣∣∣∣∣W∗i =Wi

]δWi(r)δWj(r′) + · · ·

= βF0 − 12

∑

ij

∫drdr′ S(2)

ij (r − r′)δWi(r)δWj(r′) + · · ·

(2.38)

where S(2)ij (r−r′) is the two point correlation function. (Higher order terms, which are not shown

here, include higher order correlation functions.)Next we expand δWi(r) into the power series of δρi(r). From eq (2.30),

δρi(r) = iM

Z1[W ∗i ]δZ1[W ∗i ]δW ∗i (r)

− ρi

= iM∑

j

∫dr′[

1Z1[W ∗i ]

δ2Z1[W ∗i ]δW ∗i (r)δW ∗j (r′)

∣∣∣∣∣W∗i =Wi

− 1(Z1[W ∗i ])2

δZ1[W ∗i ]δW ∗i (r)

δZ1[W ∗i ]δW ∗j (r′)

∣∣∣∣∣W∗i =Wi

]δWj(r′) + · · ·

= i∑

j

∫dr′ S(2)(r − r′)δWj(r′) + · · ·

(2.39)

Inverting eq (2.39) and we have

δWi(r) = −i∑

j

∫dr′ Γ(2)

ij (r − r′)δρj(r′) + . . . (2.40)

where Γ(2)ij (r − r′) is the inverse of S(2)(r − r′) defined via the following equation.

∑

j

∫dr′ S(2)

ij (r − r′)Γ(2)jk (r′ − r′′) = δikδ(r − r′′) (2.41)

Γ(2)ij (r − r′) is called the second order vertex function.

2.3. GINZBURG-LANDAU THEORY 15

By using the vertex function, the free energy is expressed as the functional of δρi(r).

βδF [δρi] ≡ βF [ρ∗i ]− βF0

= −12

∑

ij

∫drdr′ S(2)

ij (r − r′)

× (−1)∑

kl

∫dr′′dr′′′ Γ(2)

ik (r − r′′)Γ(2)jl (r′ − r′′′)δρk(r′′)δρl(r′′′) + · · ·

=12

∑

ij

∫drdr′ Γ(2)

ij (r − r′)δρi(r)δρj(r′) + · · ·

(2.42)

Eq (2.53) is the so-called GL free energy functional. While we have derived only the lowest orderterm (the second order temr), higher order terms can be derived systematically [44,45].

2.3.4 Random Phase Approximation

To get the explicit form of the GL expansion free energy, we have to get the explicit form of thevertex function. Here we calculate it by using the random phase approximation (RPA).

By using the RPA, eq (2.39) can be approximated as follows.

δρi(r) = i∑

j

∫dr′ S(2)

ij (r − r′)δWj(r′) + · · ·

≈ i∑

j

∫dr′ S(2)

ij (r − r′)[iδWj(r′)−W (RPA)

j (r′)]

+ · · ·(2.43)

where S(2)(r−r′) is the correlation function for the ideal system. W (RPA)i (r) is the RPA potential

which is assumed to be linear in δρi(r). This is true for the case where the interaction can beexpressed as the bilinear form of δρi(r). For example, the Flory-Huggins type interaction can beexpressed as this form.

W(RPA)i (r) ≈

∑

j

χijδρj(r) (2.44)

where χij is the χ parameter. Using eqs (2.44) and (2.44), we have

δWi(r) = −i∑

j

∫dr′ Γ(2)

ij (r − r′)δρj(r′)− iW (RPA)i (r)

= −i∑

j

∫dr′

[Γ(2)ij (r − r′) + χijδ(r − r′)

]δρj(r′)

(2.45)

where Γ(2)(r − r′) is the vertex function for the ideal system.

∑

j

∫dr′ S(2)

ij (r − r′)Γ(2)jk (r′ − r′′) = δikδ(r − r′′) (2.46)

Finally we have the explicit form of the vertex function as follows.

Γ(2)ij (r − r′) ≈ Γ(2)

ij (r − r′) + χijδ(r − r′) (2.47)

Substituting eq (2.47) into (2.42) gives the explicit expression for the GL free energy.

βδF [δρi] ≈ 12

∑

ij

∫drdr′ Γ(2)

ij (r − r)δρi(r)δρj(r′) +12

∑

ij

∫dr χijδρi(r)δρj(r) + · · · (2.48)

We have to evaluate the term which contains the vertex function Γ(2)(r − r′) in numericalsimulations. There are roughly two methods to evaluate the vertex function. One is to calculate it


directly in the Fourier space [46, 47]. This is most accurate method under the RPA, but requiressome computational costs. Another is to use the approximate form of the vertex function. Forexample, Brazovskii-Leibler-Fredrickson-Helfand type free energy model [44, 48, 49] or the Ohta-Kawasaki type model [50–52] are often used. While such approximate forms require less numericalcosts, they are less accurate compared with the direct evaluation method.

Generalization for multicomponent systems are straightforward [6, 47,52].

βδF [δρpi] ≈ 12

∑

p,ij

∫drdr′ Γ(2)

p,ij(r − r)δρpi(r)δρpj(r′) +12

∑

pi,qj

∫dr χpi,qjδρpi(r)δρqj(r) + · · ·

(2.49)where δρpi(r) is the density fluctuation field for the (p, i) subchain, and Γ(2)

p,ij(r− r′) is the vertexfunction for the p-th polymer at the ideal state.

At the end of this section, we note that the same free energy function form as eq (2.48) canbe obtained by just to approximate the vertex function by one for the ideal system and add theinteraction term to the free energy. That is,

Γ(2)ij (r − r′) ≈ Γ(2)

ij (r − r′) (2.50)

βδF [δρi] ≈ 12

∑

ij

∫drdr′ Γ(2)

ij δρi(r)δρj(r′) + βδU [δρi] + · · · (2.51)

βδU [δρi] ≡ 12

∫dr χijδρi(r)δρj(r) (2.52)

Such an approximation is correct only for the second order terms compared with the RPA. Weshould use the RPA to calculate higher order terms accurately [44,53]. However, in most practicalcases, higher order vertex functions are not calculated explicitly but assumed to be simple forms(for example, non-local coupling effects are often ignored) [49] and thus this simple approximationwill be sufficient.

2.3.5 Validity of the Ginzburg-Landau Theory

Now we ask ourselves whether the GL expansion theory is generally valid. For example, wheterthe GL expantion free energy can describe the micellar structures. As mentioned above, the GLexpansion is justified only for the cases where the density fluctuations are sufficiently small. Thiscondition can be written explicitly as follows.

∣∣∣∣δρi(r)ρi

∣∣∣∣ 1 (2.53)

From eq (2.53), we immediately know that

• For strongly segregated systems, |δρi(r)| ∼ 1 and the GL expansion is no longer justified.

• For dilute components, 1/ρi 1 and thus the GL expansion is not justified unless the densityfluctuation is extremely small.

Thus the validity of the GL expansion is guaranteed only for weakly segregated and nearly homo-geneous systems.

So far, theoretical works and simulations works have been done for block copolymer systems ormicellar systems based on the GL expansion theory. However, we know that in many cases the GLexpansion theory can be qualitatively inaccurate or physically unacceptable. We should be carefulto study block copolymer systems by using the GL theory, or we will have unphysical results whichare inconsistent with other theories or experiments. While this fact is based on the very simpleargument, it seems to be ignored or not to be considered seriously in most of previous studied.

To overcome these problems and study strongly segregated and/or strongly localized systems,in this work we use non-GL expansion type free energy functional.

2.4. TWO POINT DENSITY FUNCTIONAL THEORY 17

2.4 Two Point Density Functional Theory

To overcome the difficulties associated with the GL expansion theory, we need another free energyfunctional theory. To derive such a theory, we formulate a new theory which is not based on theconventional GL type theory.

In this section we derive the two point density functional theory, which is used as the base formore coarse-grained free energy functional theories. In the two point density functional theory,the free energy is expressed as a functional of two point density. While it cannot be used directlyin numerical simulations because of the large computational costs, it can be used to derive the onepoint density functional theory which can be used in actual simulations.

2.4.1 Density Functional Integral Form for Grand Partition Function

We start from the one component linear homopolymer systems. The grand partition function (eqs(2.19) and (2.20)) is expressed by using the Edwards Hamiltonian (eqs (2.8)-(2.10)).

By introducing the the microscopic monomer density ρ(r) defined as

ρ(r) ≡M∑

k=1

∫ N

0

ds δ (r −Rk(s)) (2.54)

the grand partition function for the system Ξ is expressed as follows.

Ξ =∞∑

M=0

eβµM

M !

∫ M∏

k=1

DRk exp [−βU [ρ]] (2.55)

where U [ρ] is the functional expression of the interaction energy.

U [ρ] =∫dr1dr2

12v(r1 − r2)ρ(r1)ρ(r2) (2.56)

The standard way to obtain the free energy as the functional of the density field is to use thefollowing identity.

1 =∫Dρ δ[ρ− ρ] =

∫DρDW exp [iρ ·W − iρ ·W ] (2.57)

where we used the expression for the inner product of functions

ρ ·W ≡∫drρ(r)W (r) (2.58)

and the Fourier transform of the δ functional. Using eq (2.57), we can rewrite the grand partitionfunction as follows.

Ξ =∞∑

M=0

eβµM

M !

∫ M∏

k=1

DRk

∫Dρ δ[ρ− ρ] exp [−βU [ρ]]

=∞∑

M=0

eβµM

M !

∫ M∏

k=1

DRk

∫DρDW exp [−βU [ρ] + iρ ·W − iρ ·W ]

=∫DρDW exp

[−βU [ρ] + iρ ·W + eβµ

∫DR exp

[−i∫ N

0

dsW (R(s))

]](2.59)

However, this formulation is not suitable to consider strongly localized systems since there areno information about the individual polymer chains. We cannot use the monomer density as theinformation about the each polymer chains, and clearly we need other density fields. Pagonabarragaand Cates [54] proposed to use the center of mass density of polymer chains and Frusawa [55]


formulated the density functional integral theory by using two density fields; the monomer densityand the center of mass density.

In this work, we also use the information about the center of mass density. However, unlikethe Frusawa theory, here we propose to use the monomer - center of mass two point density (ortwo point distribution function) instead of two density fields (monomer density and center of massdensity). The microscopic monomer - center of mass two point density (the monomer - center ofmass two point density operator) is defined as

ω(r1; r0) ≡M∑

k=1

∫ N

0

ds δ(r1 −Rk(s))δ(r0 −RCM,k) (2.60)

where RCM,k is the position of the center of mass of the k-th polymer chain.

RCM,k ≡ 1N

∫ N

0

dsRk(s) (2.61)

It is straightforward to calculate the monomer density or the center of mass density from themonomer - center of mass two point density.

ρ(r1) =∫dr0 ω(r1; r0) (2.62)

c(r0) =1N

∫dr1 ω(r1; r0) (2.63)

where c(r0) is the microscopic center of mass density. The merit of the use of the monomer - centerof mass two point density is that we can avoid the problems associated with the conversion (or therelation) between the monomer density field ρ(r1) and the center of mass density field c(r0). Wecan derive both ρ(r1) and c(r0) straightforwardly from ω(r1; r0).

We use the following identity.

1 =∫Dω δ[ω − ω] =

∫DωDV exp [iω : V − iω : V ] (2.64)

where we used the expression for the inner product of functions.

ω : V ≡∫dr1dr0 ω(r1; r0)V (r1; r0) (2.65)

By using (2.64) we can rewrite the grand partition function as

Ξ =∞∑

M=0

eβµM

M !

∫ M∏

k=1

DRk

∫Dω δ[ω − ω] exp [−βU [ρ]]

=∞∑

M=0

eβµM

M !

∫ M∏

k=1

DRk

∫DωDV exp [−βU [ρ] + iω : V − iω : V ]

=∫DωDV exp

[−βU [ρ] + iω : V + eβµZ[V ]]

(2.66)

where

Z[V ] ≡∫DR exp

[−i∫ N

0

ds V (R(s);RCM )

](2.67)

is the one chain partition function.The saddle point value of V (r1; r0), V ∗(r1; r0) satisfies following equation.

δ

δV (r1; r0)[−βU [ρ] + iω : V + eβµZ[V ]

]∣∣∣∣V=V ∗

= 0 (2.68)


Eq (2.68) can be written as

ω(r1; r0) = ieβµδZ[V ]

δV (r1; r0)

∣∣∣∣V=V ∗

(2.69)

We expand the exponent of the integrand up to the second order in ∆V ≡ V −V ∗ and integrateover ∆V [55].

Ξ ≈∫DωD∆V exp

[− βU [ρ] + iω : V ∗ + eβµZ[V ∗]

−∫dr0dr1dr2A(r1, r2; r0)∆V (r1; r0)∆V (r2; r0)

]

=∫Dω det

[(πA−1

)1/2]exp

[−βU [ρ] + iω : V ∗ + eβµZ[V ∗]]

(2.70)

where

A(r1, r2; r0) ≡ −12eβµ

δ2Z[V ]δV (r1; r0)δV (r2; r0)

∣∣∣∣V=V ∗

(2.71)

We introduce the new order parameter defined as

σ(r1; r0) ≡√ω(r1; r0) (2.72)

By using the identity

1 =∫Dσ2 δ[σ2 − ω] =

∫Dσ det [2σ] δ[σ2 − ω] (2.73)

the grand partition function can be expressed as

Ξ ≈∫Dω

∫Dσ det [2σ] δ[σ2 − ω] det

[(πA−1

)1/2]exp

[−βU [ρ] + iω : V ∗ + eβµZ[V ∗]]

=∫Dσ det [2σ] det

[(πA−1

)1/2]exp

[−βU [ρ] + iσ2 : V ∗ + eβµZ[V ∗]]

=∫Dσ det [2σ] det

[(πA−1

)1/2]exp

[−βF [σ] +

βµ

N

∫dr0dr1 σ

2(r1; r0)]

(2.74)

where we defined the free energy functional F [σ] as follows.

F [σ] ≡ U [ρ] + kBT[−iσ2 : V ∗ − eβµZ[V ∗]

]+µ

N

∫dr0dr1 σ

2(r1; r0)

= U [ρ] + kBT

∫dr0dr1

σ2(r1; r0)N

[ln eβµ−iNV

∗(r1;r0) − 1] (2.75)

Introducing the entropy functional S[σ], the free energy functional is finally expressed as

F [σ] = U [ρ]− TS[σ] (2.76)

S[σ] ≡ −kB∫dr0dr1

σ2(r1; r0)N

[ln eβµ−iNV

∗(r1;r0) − 1]

(2.77)

2.4.2 Approximations for Saddle Point Equation

To obtain the free energy functional as the explicit functional of the monomer - center of masstwo point density, we have to solve the saddle point equation (2.69). However, it is impossible tosolve the saddle point equation exactly and we need some approximations. The standard way tosolve the saddle point equation is to use power series expansion which is used in the GL expansiontheory. But the power series expansion limits the validity of the theory and therefore we do not


want to use it. Here we seek a non power series expansion approximation to solve the saddle pointequation.

First we rewrite the saddle point equation (2.69) by using the normalized microscopic one chainmonomer density ρ(1) and the microscopic one chain center of mass density c(1) which are definedas follows.

ρ(1)(r1) ≡ 1N

∫ N

0

ds δ(r −R(s)) (2.78)

c(1)(r0) ≡ δ(r0 −RCM ) (2.79)

Eq (2.69) can be rewritten as

σ2(r1; r0) = eβµ∫DRNρ(1)(r1)c(1)(r0) exp

[−i∫ N

0

ds V ∗(R(s);RCM )

]

= N

∫DR ρ(1)(r1)c(1)(r0) exp

[βµ− iN

∫dr2 ρ

(1)(r2)V ∗(r2; r0)] (2.80)

The exponential factor in eq (2.80) is similar to the exponential factor in the entropy functional(eq (2.77)). Therefore we consider that we can get the approximate form of the exponential factor,instead of the approximate form for V ∗ itself. A possible most simple approximation form for eq(2.80) will be

σ2(r1; r0) ≈ N∫DR ρ(1)(r1)c(1)(r0) exp [βµ− iV ∗(r1; r0)]

= NΩ(r1 − r0) exp [βµ− iV ∗(r1; r0)](2.81)

where Ω is the monomer - center of mass correlation function [55–57].

Ω(r1 − r0) ≡∫DR ρ(1)(r1)c(1)(r0)

≈(

9πNb2

)3/2

exp[− 9Nb2

(r1 − r0)] (2.82)

From eq (2.81) the entropy functional (eq (2.77)) can be expressed as follows.

S[σ] ≈ −kB∫dr0dr1

σ2(r1; r0)N

[ln

σ2(r1; r0)NΩ(r1 − r0)

− 1]

(2.83)

This is the most simple approximation form for the entropy functional. The form of eq (2.83) issimilar to the standard Flory-Huggins entropy, and in fact, it reduces to the Flory-Huggins entropyfor the case of the homogeneous ideal systems.

We expect that eq (2.81) can be the reference state of the approximation, instead of the homo-geneous state. Next we seek the higher order approximation form. We propose to approximate eq(2.80) as follows.

σ2(r1; r0) ≈ N∫dr2dr3

[∫DR ρ(1)(r1)ρ(1)(r2)ρ(1)(r3)c(1)(r0)

]

× exp[βµ− iN

2V ∗(r2; r0)− iN

2V ∗(r3; r0)

]

= N

∫dr2dr3 S

(3,1)(r1, r2, r3; r0) exp[βµ− iN

2V ∗(r2; r0)− iN

2V ∗(r3; r0)

](2.84)

where S(3,1) is the monomer - monomer - monomer - center of mass four point correlation function.

S(3,1)(r1, r2, r3; r0) ≡∫DR ρ(1)(r1)ρ(1)(r2)ρ(1)(r3)c(1)(r0) (2.85)


Note that if we perform power series expansion for eq (2.84) with respect to V ∗, we can see thateq (2.84) is correct up to the first order.

Unfortunately S(3,1) is not simple function and thus we need an approximation further. Weemploy the decoupling approximation here.

S(3,1)(r1, r2, r3; r0) ≈ S(2)(r1 − r2)S(2)(r1 − r3)√

Ω(r2 − r0)Ω(r3 − r0) (2.86)

S(2)(r1 − r2) ≡∫DR ρ(1)(r1)ρ(1)(r2) (2.87)

S(2) is the monomer - monomer two point density correlation function. Ω corresponds to themonomer - center of mass two point correlation function, but we do not give the explicit definitionfor Ω here. We determine it later so that the monomer - center of mass two point density recoversthe exact form for homogeneous systems. Clearly there are many possible approximate forms forS(3,1). However, not all the approximate forms are analytically tractable. Besides, among theanalytically tractable candidates, eq (2.86) gives physically most reasonable results. Thus in thiswork we use eq (2.86) as the approximation form for S(3,1).

By using these approximations, we can solve the saddle point equation.

exp[βµ

2− iN

2V ∗(r2; r0)

]=

1√N Ω(r1 − r0)

∫dr2 Γ(r1 − r2)σ(r2; r0) (2.88)

where Γ is the inverse of S(2).∫dr2 S

(2)(r1 − r2)Γ(r2 − r3) = δ(r1 − r3) (2.89)

The entropy functional (eq (2.77)) can be expressed as follows.

S[σ] = −kB∫dr0dr1

σ2(r1; r0)N

2 ln

1√

N Ω(r1 − r0)

∫dr2 Γ(r1 − r2)σ(r2; r0)

− 1

≈ −kB∫dr0dr1

σ2(r1; r0)N

[ln

σ2(r1; r0)N Ω(r1 − r0)

− 1]

− kB∫dr0dr1dr2

2Nσ(r1; r0)

[Γ(r1 − r2)− δ(r1 − r2)

]σ(r2; r0)

(2.90)

2.4.3 Approximations for the Functional Determinant

We do not have the explicit form of the functional determinant in the grand partition function (eq(2.74)). To calculate the determinant, we need to approximate A which is defined by eq (2.71).Eq (2.71) has the similar form as the saddle point equation (2.69) and thus we approximate it inthe similar way to the approximations for the saddle point equation.

A(r1, r2; r0) =N2

2

∫DR ρ(1)(r1)ρ(1)(r2)c(1)(r0) exp

[βµ− iN

∫dr3 ρ

(1)(r3)V ∗(r3; r0)]

≈ N2

2

∫dr3dr4 S

(4,1)(r1, r2, r3, r4; r0) exp[βµ− iN

2V ∗(r3; r0)− iN

2V ∗(r4; r0)

]

(2.91)

where S(4,1) is the five point correlation function and we approximate it as follows.

S(4,1)(r1, r2, r3, r4; r0) ≡∫DR ρ(1)(r1)ρ(1)(r2)ρ(1)(r3)ρ(1)(r4)c(1)(r0)

≈ S(2)(r1 − r2)S(2)(r1 − r3)S(2)(r2 − r4)√

Ω(r3 − r0)Ω(r4 − r0)(2.92)


Then eq (2.91) can be written as the following form, by using σ.

A(r1, r2; r0) ≈ N

2S(2)(r1 − r2)σ(r1; r0)σ(r2; r0) (2.93)

By using eq (2.93), the determinant in the grand partition function (eq (2.74)) can be written as

det [2σ] det[(πA−1

)1/2]= det [2σ] det

[((NS(2)/2π

)−1/2

σ

)−1]

= det

[(8πN

Γ)1/2

](2.94)

The last form of the determinant in eq (2.94) is independent of σ as expected, and thus we donot have anomaly. Here it should be emphasized that it is essential to use the order parameterσ ≡ √ω to remove the anomaly. This fact justifies the use of the order parameter which is definedas the square root of the density field.

Finally, we have the following approximate form for the grand partition function.

Ξ ≈∫Dσ det

[(8πN

Γ)1/2

]exp

[−βF [σ] +

βµ

N

∫dr0dr1 σ

2(r1; r0)]

(2.95)

Notice that the determinant in the functional integral of eq (2.95) is independent of σ and thereforemost of the thermodynamic quantities such as the free energy are not affected by this factor.

2.4.4 Explicit Forms for Γ and Ω

Here we describe the explicit forms for Γ and Ω. First, Γ can be obtained easily by using theapproximation using the asymptotic forms [52].

Γ(r1 − r2) ≈ N[

1Nδ(r1 − r2)− b2

12∇2

1δ(r1 − r2)]

(2.96)

where ∇1 ≡ ∂/∂r1. Substituting eq (2.96) into eq (2.90), we have the explicit form for the entropyfunctional.

S[σ] ≈ −kB∫dr0dr1

σ2(r1; r0)N

[ln

σ2(r1; r0)N Ω(r1 − r0)

− 1]

− kB∫dr0dr1

b2

6|∇1σ(r1; r0)|2

(2.97)

Next we derive the explicit form for Ω. We require Ω to reproduce the correct monomer - centerof mass two point density function for homogeneous state. The equilibrium two point density fieldis given as the field which minimize the grand potential. Thus we can write the equation whichthe equilibrium density field σ(eq) satisfies.

δ

δσ(r1; r0)

[F [σ]− µ

N

∫dr0dr1 σ

2(r1; r0)]∣∣∣∣σ=σ(eq)

= 0 (2.98)

For homogeneous state, σ(eq) should satisfy the following equation.

σ(eq)(r1; r0) =√ρΩ(r1 − r0) (2.99)

For simplicity here we assume that U = 0. Then from eqs (2.98), (2.99) and (2.97) we obtain thefollowing equation for Ω.

1N

lnρΩ(r1 − r0)N Ω(r1 − r0)

− b2

6∇2

1

√Ω(r1 − r0)√

Ω(r1 − r0)− βµ

N= 0 (2.100)


From eq (2.100) we can write

Ω(r1 − r0) =(

452πNb2

)3/2

exp[− 45

2Nb2(r1 − r0)2

](2.101)

µ = kBT

[92

+32

ln25

+ lnρ

N

](2.102)

Notice that Both Ω and Ω (eqs (2.101) and (2.82)) are Gaussian but the numerical factors aredifferent. This is due to the approximation for the monomer - monomer - monomer - center ofmass correlation function S(3,1).

2.4.5 Generalization for Block Copolymers

We have derived the two point density functional theory for homopolymers. In this section, wegeneralize the two point density functional theory for block copolymers. We show the generalizationfor block copolymer melts which consists of one block copolymer species in this section and, showthe generalization for block copolymer blends in the next section.

We consider block copolymers with arbitrary architecture [52]. We index each subchains inblock copolymers as i, j, . . . . The Edwards Hamiltonian and the interaction is given by eqs (2.11)-(2.13) (here we consider only one block copolymer species and thus drop the polymer species indexp).

By introducing ρi(r), the microscopic density for the i-th subchain, and using the functionalexpression for the interaction energy, we can formulate the two point density functional theory forblock copolymers.

ρi(r) ≡M∑

k=1

∫

s∈ids δ (r −Rk(s)) (2.103)

U [ρi] =∑

ij

∫dr1dr2

12vij(r1 − r2)ρi(r1)ρj(r2) (2.104)

After the straight forward calculations, we have the following approximate form of the partitionfunction and the saddle point equation.

Ξ ≈∫Dσi det [2σi] det

[(π

Aij

)1/2]

exp

[−βF [σi] +

βµ

N

∑

i

∫dr0dr1 σ

2i (r1; r0)

](2.105)

F [σi] ≡ U [ρi]− TS[σi] (2.106)

S[σi] ≡ −kB∑

i

∫dr0dr1

σ2i (r1; r0)N

[ln eβµ−iNV

∗i (r1;r0) − 1

](2.107)

σ2i (r1; r0) = ieβµ

δZ[Vi]δVi(r1; r0)

∣∣∣∣V=V ∗

(2.108)

Aij(r1, r2; r0) ≡ −12eβµ

δ2Z[Vi]δVi(r1; r0)δVj(r2; r0)

∣∣∣∣V=V ∗

(2.109)

Z[Vi] ≡∫DR exp

[−i∫

s∈ids Vi(R(s);RCM )

](2.110)

where σi(r1; r0) is the square root of the monomer (in the i-th subchain)-center of mass two pointdensity.


The saddle point equation (2.108) is expressed as

σ2i (r1; r0) = eβµ

∫DRNfiρ

(1)i (r1)c(1)(r0) exp

−i

∑

j

∫

s∈jds V ∗j (R(s);RCM )

= Nfi

∫DR ρ

(1)i (r1)c(1)(r0) exp

βµ− iN

∑

j

fj

∫dr2 ρ

(1)j (r2)V ∗j (r2; r0)

(2.111)

where fi is the block ratio of the i-th subchain and ρ(1)i is one chain monomer density for the i-th

subchain. They are defined as follows.

fi ≡ 1N

∫

s∈ids (2.112)

ρ(1)i (r1) ≡ 1

Nfi

∫

s∈ids δ(r1 −R(s)) (2.113)

We approximate the saddle point equation as follows, in the same way as the homopolymer case.

σ2i (r1; r0) ≈ Nfi

∑

jk

∫dr2dr3

[∫DR ρ

(1)i (r1)ρ(1)

j (r2)ρ(1)k (r3)c(1)(r0)

]

× fjfk exp[βµ− iN

2V ∗j (r2; r0)− iN

2V ∗k (r3; r0)

]

= Nfi

∫dr2dr3 S

(3,1)ijk (r1, r2, r3; r0)fjfk exp

[βµ− iN

2V ∗j (r2; r0)− iN

2V ∗k (r3; r0)

]

(2.114)

where S(3,1)ijk is the monomer - monomer - monomer - center of mass four point correlation function.

We approximate S(3,1)ijk as follows.

S(3,1)ijk (r1, r2, r3; r0) ≡

∫DR ρ

(1)i (r1)ρ(1)

j (r2)ρ(1)k (r3)c(1)(r0)

≈ S(2)ij (r1 − r2)S(2)

ik (r1 − r3)√

Ωj(r2 − r0)Ωk(r3 − r0)(2.115)

S(2)ij (r1 − r2) ≡

∫DR ρ

(1)i (r1)ρ(1)

j (r2) (2.116)

Then we can write

exp[βµ

2− iN

2V ∗i (r2; r0)

]=

1√NfiΩi(r1 − r0)

∑

j

∫dr2

Γij(r1 − r2)√fifj

σj(r2; r0) (2.117)

where Γij is the inverse of S(2)ij .

∑

j

∫dr2 S

(2)ij (r1 − r2)Γjk(r2 − r3) = δikδ(r1 − r3) (2.118)


Finally the entropy functional can be expressed as follows.

S[σi] = −kB∑

i

∫dr0dr1

σ2i (r1; r0)N

×2 ln

1√

NfiΩi(r1 − r0)

∑

j

∫dr2

Γij(r1 − r2)√fifj

σj(r2; r0)

− 1

≈ −kB∑

i

∫dr0dr1

σ2i (r1; r0)N

[ln

σ2i (r1; r0)

NfiΩi(r1 − r0)− 1]

− kB∑

ij

∫dr0dr1dr2

2Nσi(r1; r0)

[Γij(r1 − r2)√

fifj− δijδ(r1 − r2)

]σj(r2; r0)

(2.119)

Eq (2.109) can be approximated similarly.

Aij(r1, r2; r0) ≈ 12N√fifjS

(2)ij (r1 − r2)σi(r1)σj(r2) (2.120)

By substituting eq (2.120) into eq (2.105) we can write the grand partition function as follows.

Ξ ≈∫Dσi det

(

8πN

Γij√fifj

)1/2 exp

[−βF [σi] +

βµ

N

∑

i

∫dr0dr1 σ

2i (r1; r0)

](2.121)

To obtain the explicit forms of the entropy functional and the free energy functional, we needthe explicit forms for Γij and Ωi. Γ can be approximated as follows.

Γij(r1 − r2) ≈ Nfifj[AijG(r1 − r2) + Cijδ(r1 − r2)− b2δij

12fi∇2

1δ(r1 − r2)]

(2.122)

where Aij and Cij are determined from N, fi, b and the architecture of the block copolymer. Theexplicit form of Aij and Cij is shown in Ref [52] or Appendix 2.B). G(r1−r2) is the Green functionwhich satisfies

−∇21G(r1 − r2) = δ(r1 − r2) (2.123)

Ωi can be approximated as the case of the homopolymer systems. However, it gives somehow com-plicated form. Here we roughly approximate Ωi by using the approximate form for homopolymers((2.101)).

Ωi(r1 − r0) ≈(

452πNb2

)3/2

exp[− 45

2Nb2(r1 − r0)2

](2.124)

Eq (2.119) together with eqs (2.122) and (2.124) gives the explicit entropy functional.By using eq (2.122), finally the entropy functional (eq (2.119)) can be expressed as follows.

S[σi] ≈ −kB∑

i

∫dr0dr1

σ2i (r1; r0)N

[ln

σ2i (r1; r0)

NfiΩi(r1 − r0)− 1]

− kB∑

ij

∫dr0dr1dr2 2

√fifjAijG(r1 − r2)σi(r1; r0)σj(r2; r0)

− kB∑

ij

∫dr0dr1

[2√fifjCij − 2

Nδij

]σi(r1; r0)σj(r1; r0)

− kB∑

ij

∫dr0dr1

b2

6|∇1σi(r1; r0)|2

(2.125)

Unlike the homopolymer case (eq (2.97)), the entropy functional for block copolymer systems con-tains non-local term (the second term in the right hand side of eq (2.125)). This is the characteristicproperty of the entropy functional for block copolymers. Intuitively we can interpret that suchlong range entropic interaction is the effect of the chemical bonds which tether subchains. Thislong range interaction reproduces the microphase separations.


2.4.6 Generalization for Block Copolymer Blends

The generalization for the blend systems is straight forward. In general block copolymer blend sys-tems, there are multiple block copolymer species. We index the block copolymer species as p, q, . . . ,and index the subchains in each block copolymer species as i, j, . . . . Thus each subchains can bedistinguished by using double index such as pi, qj (or (p, i), (q, j)). The Edwards Hamiltonian canbe expressed by eqs (2.11)-(2.13).

After the straight forward calculations, we have the following approximate form for the grandpartition function.

Ξ ≈∫Dσpi det

(

8πNp

Γp,ij√fpifpj

)1/2 exp

−βF [σpi] +

∑

p,i

βµpNp

∫dr0dr1 σ

2pi(r1; r0)

(2.126)The free energy functional in eq (2.126) is expressed as follows.

F [σpi] ≡ U [ρpi]− T∑p

Sp[σpi] (2.127)

U [ρpi] =∑

pi,qj

∫dr1dr2

12vpi,qj(r1 − r2)ρpi(r1)ρqj(r2) (2.128)

Sp[σpi] ≈ −kB∑

i

∫dr0dr1

σ2pi(r1; r0)N

[ln

σ2pi(r1; r0)

NpfpiΩpi(r1 − r0)− 1

]

− kB∑

ij

∫dr0dr1dr2

2Nσpi(r1; r0)

[Γp,ij(r1 − r2)√

fpifpj− δijδ(r1 − r2)

]σpj(r2; r0)

(2.129)

where Γp,ij and Ωpi represent Γij and Ωi for the p-th polymer species respectively, and fpi is theblock ratio of the (p, i) subchain.

Eq (2.129) can be modified by using eq (2.122).

Sp[σpi] ≈ −kB∑

i

∫dr0dr1

σ2pi(r1; r0)N

[ln

σ2pi(r1; r0)

NfiΩpi(r1 − r0)− 1

]

− kB∑

ij

∫dr0dr1dr2 2

√fpifpjAp,ijG(r1 − r2)σpi(r1; r0)σpj(r2; r0)

− kB∑

ij

∫dr0dr1

[2√fpifpjCp,ij − 2

Npδij

]σpi(r1; r0)σpj(r1; r0)

− kB∑

ij

∫dr0dr1

b2

6|∇1σpi(r1; r0)|2

(2.130)

2.4.7 Necessity of Use of the Two Point Density

It is possible to formulate the non GL expansion type static density functional theory using onlythe monomer density field ρi(r). Then the entropy functional is expressed as the functional ofρi(r), and the form of the entropy functional is similar to one of the two point density functionaltheory. (We derive the entropy functional which is expressed only by the monomer density fieldin the following section.) Here we discuss about the necessity of the use of the two point densityωi(r1; r0) (or σi(r1; r0)).

Physical interpretation of the use of the information of the center of mass is that it can be used todistinguish each polymer. First we consider to split the entropy functional into two contributions;the translational entropy and the conformational entropy.

S = S(trans) + S(conf) (2.131)

2.5. STATIC DENSITY FUNCTIONAL THEORY 27

Under the mean field approximation, the total conformational entropy is simply expressed as asum of the conformational entropy S(conf) for each polymer chain.

Thus we can write

S(conf) =M∑

k=1

S(conf),(1)[ρ(k)i ] (2.132)

where M is the number of polymer chains in the system, S(conf),(1) is the conformation entropyfor one polymer chain, and ρ

(k)i (r) is the density of the i-th subchain of the k-th polymer chain.

ρi(r) can be calculated from ρ(k)i (r) as

ρi(r) =M∑

k=1

ρ(k)i (r) (2.133)

It is not easy to handle eq (2.132) in our formalism. Instead of considering each polymer chainsexplicitly, we use other information which can be used to extract the information about one polymerchain. The position of the center of mass is one of the candidate. We may rewrite eq (2.132) as

S(conf) =∫dr0 S(conf),(1)[ωi(·; r0)] (2.134)

where S(conf),(1)[ωi(·; r0)] is the conformational entropy of polymer chain of which center of massis at r0. It is clear that the form of eq (2.134) is consistent with eq (2.119). Although there willbe some other candidates for such information. Among tried candidates, the center of mass is themost easy to use and behaves well. Thus we use the information of the center of mass in this work.

There is another reason why we need the two point density. The main difference between thetheory with the two density and one with the monomer density is that the form of the nonlocal,long range term. In the two point density functional theory, the long range entropic interaction(the second term of the right hand side of (2.125)) is expressed as

S(long-range)[σi] = −kB∑

ij

∫dr0dr1dr2 2

√fifjAijG(r1 − r2)σi(r1; r0)σj(r2; r0) (2.135)

while if we use only the density field ρ(r) it will be expressed as follows.

S(long-range)[ρi] = −kB∑

ij

∫dr1dr2 2

√fifjAijG(r1 − r2)

√ρi(r1)ρj(r2) (2.136)

(See also Appendix 2.C.) While eq (2.135) and eq (2.136) are similar, there are serious difference. Ingeneral S(long-range)[ρi] diverges while S(long-range)[σi] does not diverge. One possible methodto remove this divergence is to introduce the cutoff length to the kernel G(r1− r2). However, sucha method is ad-hoc and artificial. We do not need such a method if we use of the two point density.

2.5 Static Density Functional Theory

We have derived the free energy as the functional of the monomer - center of mass two pointdensity in the previous section. This form (we refer it as the “σ-form”) is however not suitable forsimulations, since it needs the two point density field as the input. In most cases, we are interestedin only the one point density field such as the monomer density field or the center of mass densityfield. In this section, we derive the approximate form of the free energy as the one point densityfunctional. There are two one point density fields; ρ(r1) and c(r0).

ρpi(r1) ≡∫dr0 σ

2pi(r1; r0) (2.137)

cp(r0) ≡∑

i

1Np

∫dr1 σ

2pi(r1; r0) (2.138)


Thus we derive the free energy functional as the monomer density functional form (we refer thisform as the “ρ-form” or the “ψ-form”) and the center of mass density functional form (we referthis form as the “c-form”).

2.5.1 Density Functional Theory for Homopolymers

We start from the homopolymer systems. What we want to do here is to reduce the σ-form freeenergy functional (eq (2.75) or eq (2.76)) into the functional of the center of mass density or thefunctional of the monomer density.

The Colloid Density Field Form (c-Form) Expression

The c-form expression can be obtained as the same way as the ψ-form. To obtain the c-form, weset the monomer - center of mass two point density field as follows.

σ(r1; r0) =√c(r0)

√Ng0(r1; r0) (2.139)

where g0(r1; r0) is the function which satisfies∫dr1 g0(r1; r0) = 1 (2.140)

While eq (2.139) is exact, but we cannot obtain g0(r1; r0) as an explicit functional form ofc(r0). For homogeneous state g0(r1; r0) should be reduced to Ω(r1 − r0), but in general it is notsimple function.

Physically g0(r1; r0) should be determined to minimize the free energy under the given monomerdensity profile. However it is practically impossible (especially for three dimensional systems)and thus we need some approximations. One of the most natural way is to calculate g0(r1; r0)analytically under some approximations and use it (for example, by using approximation methodsused in the simple liquid theory [40]). Such an approach will be physically natural.

Unfortunately, it is quite difficult to find analytically tractable approximation method in thiscase, because the system is inhomogeneous and thus it is not translational invariant. Instead ofusing the approximate method to determine g0(r1; r0), we attempt to use an approximate form ofg0(r1; r0) directly. Of course such an approach is quite rough but we expect that this approach isqualitatively still acceptable4.

Here we employ the following approximation form for g0(r1; r0).

g0(r1; r0) ≈ g(r1 − r0) ≡(

1πλ2

)3/2

exp[− 1λ2

(r1 − r0)2

](2.141)

where λ is the characteristic length of polymer chains. Notice that if we set λ2 = Nb2/9, eq (2.148)reduces to the exact form of g0(r1; r0) for homogeneous systems. Intuitively this approximationcorresponds to taking the spatial average and the spherical average.

From eq (2.141) the entropy functional (eq (2.97)) can be written as the following c-form.

S[c] ≈ −kB∫dr0 c(r0) [ln c(r0)− 1]−NJ

∫dr0 c(r0) (2.142)

J ≡ kB∫dr

[g(r)N

lng(r)Ω(r)

+b2

6|∇g(r)|2

](2.143)

J is considered to be a sort of the external field or the chemical potential. We can neglect the termwhich contains J because the constant external field does not affect the thermodynamic propertiesas long as its value is constant, and J can be canceled by shifting the chemical potential µ. Thec-form entropy functional (eq (2.142)) is just the same form as the translational entropy of colloids.

4The free energy functional derived by using this approximation is not sensitive to the approximation form. Atleast, the parameter λ in eq (2.141) does not affect the thermodynamic behavior such as morphologies so much [58].


Using eqs (2.76), (2.56), and (2.142) the free energy functional in the c-form can be expressedas follows.

F [c] ≈∫dr0dr

′0

12v(r0 − r′0)c(r0)c(r′0) + kBT

∫dr0 c(r0) [ln c(r0)− 1] (2.144)

where v(r0 − r′0) is the effective interaction potential defined as

v(r0 − r′0) ≡ N2

∫dr1dr2 v(r1 − r2)g(r1 − r0)g(r2 − r′0) (2.145)

This form of the effective interaction potential is consistent with the Frusawa’s theory [55]. Theinteraction potential between centers of mass v(r0−r′0) has the same form as the Flory-Krigbaumtype interaction [55, 59, 60]. Note that the free energy functional in the c-form (eq (2.144)) is thesame form as the free energy for interacting colloid particles under the mean field approximation.

The Monomer Density Field Form (ψ-Form) Expression

To obtain the ψ-form, we set the monomer - center of mass two point density field as follows.

σ(r1; r0) = ψ(r1)√g1(r1; r0) (2.146)

where ψ(r1) ≡√ρ(r1) and g1(r1; r0) is the function which satisfies

∫dr0 g1(r1; r0) = 1 (2.147)

Eq (2.146) is similar to (2.139), and we expect the functions g1(r1; r0) and g0(r1; r0) are alsosimilar. Thus we assume that g0(r1; r0) is approximately the same form as the approximate formfor g1(r1; r0) and approximate g1(r1; r0) as

g1(r1; r0) ≈ g(r1 − r0) ≡(

1πλ2

)3/2

exp[− 1λ2

(r1 − r0)2

](2.148)

Using eq (2.148) we can write the entropy functional (2.97) as the functional of ψ.

S[ψ] ≈ −kB∫dr1

ψ2(r1)N

[lnψ2(r1)N

− 1]− kB

∫dr1

b2

6|∇1ψ(r1)|2 − J

∫dr1 ψ

2(r1) (2.149)

where J is given by eq (2.143). Since the term which contains J in eq (2.149) is linear in ψ2, wecan eliminate it by changing the chemical potential µ, as in the case of the c-form. Thus hereafterwe neglect the term which contains J .

It will be convenient to express the entropy functional as

S[ψ] = S(trans)[ρ] + S(conf)[ψ] (2.150)

where S(trans) and S(conf) are the translational entropy functional and the conformational entropyfunctional.

S(trans)[ρ] = −kB∫dr1

ρ(r1)N

[lnρ(r1)N− 1]

(2.151)

S(conf)[ψ] ≈ −kB∫dr1

b2

6|∇1ψ(r1)|2 (2.152)

Eq (2.151) is the Flory-Huggins type translational entropy of the polymer chains, and eq (2.152)is the Lifshitz conformational entropy [61–63].


From eqs (2.76), (2.56), (2.151) and (2.152), finally we have the following ψ-form free energyfunctional.

F [ψ] = U [ρ]− TS(trans)[ρ]− TS(conf)[ψ]

≈∫dr1dr2

12v(r1 − r2)ψ2(r1)ψ2(r2)

+ kBT

∫dr1

ψ2(r1)N

[lnψ2(r1)N

− 1]

+ kBT

∫dr1

b2

6|∇1ψ(r1)|2

(2.153)

Eq (2.153) is the Flory-Huggins-de Gennes-Lifshitz type free energy functional. Thus it is shownthat the σ-form free energy functional can be reduced to the standard Flory-Huggins-de Gennes-Lifshitz free energy functional.

Thus the σ-form free energy functional is shown to be reduced to both the Flory-Huggins-deGennes type free energy and the free energy for interacting colloid particles.

2.5.2 Density Functional Theory for Block Copolymers

In this section, we generalize the formulation in the previous section to general block copolymersystems. First we generalize the theory to block copolymer melts, and then we generalize it toblock copolymer blend systems. The generalization is rather straight forward and the result issimilar to our previous work [52] which is based on the intuitive, physical argument (see Appendix2.C).

The c-Form Expression

To obtain the c-form expression for the free energy functional, we approximate σi as follows.

σi(r1; r0) ≈√c(r0)

√Nfig(r0 − r1) (2.154)

where g(r0−r1) is defined by eq (2.141). Then the free energy functional can be written as follows.

F [c] ≈∫dr0dr

′0

12v(r0 − r′0)c(r0)c(r′0) + kBT

∫dr0 c(r0) [ln c(r0)− 1] (2.155)

where v(r0 − r′0) is the effective interaction potential defined as

v(r0 − r′0) ≡∑

ij

N2fifj

∫dr1dr2 vij(r1 − r2)g(r1 − r0)g(r2 − r′0) (2.156)

Eq (2.155) has the same form as eq (2.144) because we approximated density field as simple functionof c(r0) (eq (2.154)). Thus the c-form is not suitable to study microphase separation structures.

The ψ-Form Expression

The ψ-form free energy functional can be obtained by expressing σi approximately as follows.

σi(r1; r0) ≈ ψi(r1)√g(r1 − r0) (2.157)

where ψi(r1) ≡√ρi(r1) and g(r0 − r1) is defined by eq (2.148).

For microphase separated states, generally polymer chains are stretched and thus the value ofλ in eq (2.148) will be larger than the value at the ideal state,

√Nb2/9. Because the characteristic

size is of polymer chains is of the order of the size of the microphase separation structures, weexpect λ ∝ N2/3b [58] for strongly segregated cases 5.

5In this case we can write λ = αN2/3b where α is constant. It will be difficult to determine the numerical coeffi-cient α theoretically, and we will need to perform SCF simulations or MD simulations to determine it numerically.


The free energy functional can be expressed as follows.

F [ψi] = U [ρi]− TS(trans)[ρi]− TS(conf)[ψi] (2.158)

S(trans)[ρi] = −kB∑

i

∫dr1

ρi(r1)N

[lnρi(r1)Nfi

− 1]

(2.159)

S(conf)[ψi] ≈ −kB∑

ij

∫dr1dr2 2

√fifjAij G(r1 − r2)ψi(r1)ψj(r2)

− kB∑

ij

∫dr1 2

[√fifjCij − δij

N

]ψi(r1)ψj(r1)

− kB∑

i

∫dr1

b2

6|∇1ψi(r1)|2

(2.160)

where G(r1 − r2) is the effective long range interaction kernel defined as

G(r1 − r2) ≡ G(r1 − r2)∫dr0

√g(r1 − r0)g(r2 − r0)

= G(r1 − r2) exp[− 1

4λ2(r1 − r2)2

] (2.161)

The effective long range interaction decays rapidly (exponentially) as |r1 − r2| → ∞. Eq (2.161)means that the original long range interaction kernel (eq (2.123)) which has the infinite interactionrange is “screened” by the kernel g, and the effective interaction range is finite (more strictly,the interaction range is about 2λ). The necessity of the screening is first pointed by Tang andFreed [64], based on the physical argument. Eq (2.161) justifies the use of the screened kernelwithout rough physical arguments. The physical interpretation of eq (2.161) is rather straightforward. The effective long range interaction is the product of the bare long range interaction andthe probability that monomers at positions r1 and r2 share the same polymer chain.6

2.5.3 Density Functional Theory for Block Copolymer Blends

The c-Form Expression

The c-form expression for blend systems is almost the same as the previous two cases.

σpi(r1; r0) ≈√cp(r0)

√Npfpigp(r0 − r1) (2.162)

where gp(r1 − r0) is defined as follows.

gp(r1 − r0) ≡(

1πλ2

p

)exp

[− 1λ2p

(r1 − r0)2

](2.163)

λp is the characteristic length of the p-th polymer species. Notice that λp depends on the poly-mer species (for example, λp can be written as λp = Npb

2/9 for linear polymer systems at thehomogeneous state and generally larger than

√Npb2/9).

The free energy functional can be written as follows.

F [cp] ≈∑pq

∫dr0dr

′0

12vpq(r0 − r′0)cp(r0)cq(r′0) +

∑p

kBT

∫dr0 cp(r0) [ln cp(r0)− 1] (2.164)

where vpq(r0 − r′0) is the effective interaction potential defined as

vpq(r0 − r′0) ≡∑

i,j

NpNqfpifqj

∫dr1dr2 vpi,qj(r1 − r2)gp(r1 − r0)gq(r2 − r′0) (2.165)

6The screening length can be changed by changing λ. Currently we have not shown any theoretical method tocalculate the appropriate value of λ. It will be natural to expect that λ becomes large in the strong segregationregion. However, it is not clear whether this form of the effective long range interaction is really valid in the strongsegregation region.


The ψ-Form Expression

For block copolymer blend systems, the ψ-form free energy functional can be obtained in the sameway as the case of the block copolymer melt case.

σpi(r1; r0) ≈ ψpi(r1)√gp(r1 − r0) (2.166)

where ψpi(r1) ≡√ρpi(r1).The ψ-form free energy functional can be expressed as follows.

F [ψpi] = U [ρpi]− T∑p

S(trans)p [ρpi]− T

∑p

S(conf)p [ψpi] (2.167)

S(trans)p [ρpi] = −kB

∑

i

∫dr1

ρpi(r1)Np

[lnρpi(r1)Npfpi

− 1]

(2.168)

S(conf)p [ψpi] ≈ −kB

∑

ij

∫dr1dr2 2

√fpifpjAp,ij Gp(r1 − r2)ψpi(r1)ψpj(r2)

− kB∑

ij

∫dr1 2

[√fpifpjCp,ij − δij

N

]ψpi(r1)ψpj(r1)

− kB∑

i

∫dr1

b2

6|∇1ψpi(r1)|2

(2.169)

where Gp(r1 − r2) is the effective long range interaction kernel for the p-th polymer species.

Gp(r1 − r2) ≡ G(r1 − r2)∫dr0

√gp(r1 − r0)gp(r2 − r0)

≈ G(r1 − r2) exp[− 1

4λ2p

(r1 − r2)2

] (2.170)

Substituting eqs (2.128), (2.168), and (2.169) into eq (2.167) we have the following ψ-formexpression for the free energy functional.

F [ψpi]kBT

≈∑

pi,qj

∫dr1dr2

12vpi,qj(r1 − r2)

kBTψ2pi(r1)ψ2

qj(r2)

+∑

p,i

∫dr1

2ψ2pi(r1)Np

[lnψpi(r1)√Npfi

− 1

]

+∑

p,ij

∫dr1dr2 2


+∑

p,ij

∫dr1 2

[√fpifpjCp,ij − δij

N

]ψpi(r1)ψpj(r1) +

∑

p,i

∫dr1

b2

6|∇1ψpi(r1)|2

(2.171)

This is one of the main results in this chapter. The ψ-form free energy functional (eq (2.167)) issimilar to the free energy functional in the previous work [52], but there are several differences.One difference is the coefficient of the Flory-Huggins type entropy (translational entropy term, thesecond term in eq (2.171)). In the previous theory, the coefficient is 2fpiCp,ii while in this theorythe coefficient is 2/Np. However, since Cp,ii is proportional to 1/Np this difference is qualitativelynot so important. (Especially for homopolymer systems, Cp,ii can be expressed as Cp = 1/Npand thus coefficients in two theories are exactly the same.) Another difference is the form of thelong range interaction kernel. In the previous theory, the long range interaction kernel G(r1 − r2)(eq (2.123)) has the infinite interaction range. In this theory the long range interaction kernelGp(r1 − r2) (eq (2.170)) has the finite interaction range.


While there are several differences between our theory and the previous theory, and there arestill difficulty about the accuracy, it should be noted here that our free energy functional (eq(2.171)) is derived without expansions in terms of the density fluctuation. Thus we expect that eq(2.171) works qualitatively well for strongly segregated and/or strongly localized systems includingmicellar systems.

We further modify eq (2.171) slightly as follows.

F [ψpi]kBT

≈∫dr1dr2

12v0

kBT

∑

pi

ψ2pi(r1)

2

+∑

pi,qj

∫dr1

12χpi,qjψ

2qj(r1)

+∑

p,i

∫dr1 fpiCp,iiψ

2pi(r1)

[lnψpi(r1)√Npfi

− 1

]

+∑

p,ij

∫dr1dr2 2


+∑

p,i 6=j

∫dr1 2

√fpifpjCp,ijψpi(r1)ψpj(r1) +

∑

p,i

∫dr1

b2

6|∇1ψpi(r1)|2

(2.172)

To derive eq (2.172) from eq (2.171), we applied two approximations. One is the contact interactionapproximation for the interaction potential (eq (2.14)). Another is the approximation (or the mod-ification) of the coefficient of the Flory-Huggins entropy. The former is widely used approximationand useful especially for numerical simulations, because we can express the interaction only by theχ parameter and the excluded volume parameter7. The latter one is qualitatively not importantand introduced just for the consistency with the previous theory.

If v0 is sufficiently large, the system is nearly incompressible. Then, by imposing the incom-pressible condition, we can ignore the first term in the right hand side of eq (2.172). It can beapproximately expressed as follows.

∫dr1dr2

12v0

kBT

∑

pi

ψ2pi(r1)

2

→∫dr1 P (r1)

∑

pi

ψ2pi(r1)− 1

(2.173)

where P (r1) is the Lagrange multiplier and determined to satisfy the incompressible condition(physically P (r1) corresponds to the inhomogeneous pressure field).8

2.5.4 Comparison with Other Theories

There are many theoretical and simulations works for block copolymer systems or micellar systemsbased on the continuum field model. Even if we consider only the theories which have explicitfunctional expressions for the free energy, there are still several theories. In this section we compareour DF theory with other theories and discuss about the differences.

Ginzburg-Landau Expansion Theory

The GL expansion type free energy functional is one of the most widely used free energy modelsfor block copolymer systems. The most large difference between our free energy functional andthe GL expansion for of the free energy (eq (2.49)) is that our free energy functional is expressedby using the ψ-field while the GL expansion model is expressed by the density fluctuation field.

7It should be noted here that we assume that the interaction between monomers can be approximated by thecontact interaction and the χ parameter represent the effective strength of the interaction. In generally, the longrange interaction such as electrostatic interaction cannot be approximated by a simple contact interaction and wewill need more realistic interaction potential model for such cases.

8In some cases, it seems that for numerical simulations, the Lagrange multiplier method works better than thesmall compressibility method.


As shown in Appendix 2.C, we can derive our density functional model from the GL expansionmodel with rather rough extrapolation method. Therefore, our free energy functional contains theGL expansion model as a special case. We can obtain the GL expansion form easily by expanding(2.171) into the power series of δρpi(r) ≡ ρpi(r) − fpiρp (ρp is the spatial average density of thep-th polymer species, and thus fpiρp is the spatial average of ρpi(r)). Thus we can interpret thatour density functional theory is more general model than the GL expansion model.

Of course, our DF theory is not limited to the weak segregation region nor limited to the nearlyhomogeneous state. Or in other words, there is no condition like the validity for the GL expansion(eq (2.53)) in our DF theory. As mentioned, such a difference leads qualitative differences inmicellar systems. This is empirically clear because so far no GL expansion type models reproducethe micellar structures successfully.

General Random Phase Approximation Theory

Next we consider free energy functional models which is not the standard GL expansion typemodel.

Bohbot-Raviv and Wang [46] proposed to combine the Flory-Huggins entropy and the secondorder terms calculated by the RPA, for block copolymer melts.

βFBRW[ρi] =∑

i

∫dr

1Nfi

ρi(r) ln ρi(r)

+12

∑

i,j

∫drdr′

[1ρS−1ij (r − r′) +

vij(r − r′)kBT

− δijNfiρ

]∆ρi(r)∆ρj(r)

(2.174)

where ∆ρi(r) ≡ ρi(r) − fiρ and S−1ij (r − r′) is the inverse of the correlation function for ideal

systems (without any further approximations). Eq (2.174) is exact up to the second order in∆ρi(r) under the mean field approximation. This kind of combined models are also proposed byKawakatsu [65] for homopolymer / diblock copolymer blends.

Honda and Kawakatsu [47] proposed generalization of eq (2.174) for block copolymer blends(generalized RPA, or GRPA).

βFGRPA[ρpi] =∑

pi

∫dr

1Npfpi

ρpi(r) ln ρpi(r)

+12

∑

pi,qj

∫drdr′

[δpqρpS−1p,ij(r − r) +

vpi,qj(r − r′)kBT

− δpqδijNfpiρp

]∆ρpi(r)∆ρqj(r)

(2.175)

where ∆ρpi(r) ≡ ρpi(r)− fpiρp. Eqs (2.174) and (2.175) are rather straightforward generalizationof the conventional GL expansion free energy functional.

We can interpret that higher order terms (more than the third order) in ∆ρpi(r) are approxi-mated by the Flory-Huggins entropy term (this can be seen easily by expanding the Flory-Hugginsentropy into the power series of ∆ρpi(r)). The purpose of such an approximation is to include thehigher order terms in ∆ρpi(r) without calculating the higher order vertex functions. Thus we cansay our theory and the GRPA theory are similar in a sense.

However, there are several differences. There are mainly two differences. One is that our freeenergy contains higher order non local coupling terms and gradient terms while the GRPA doesnot, and another is that our free energy functional uses the approximation form for the vertexfunction (or the correlation function) while the GRPA does not (in the GRPA simulations, thevertex function is calculated exactly in the Fourier space). The first point is especially importantfor micellar systems, because in micellar systems, power series expansion with respect to thedensity fluctuation is not appropriate. Thus we expect that for strongly segregated and/or stronglylocalized structures, the GRPA will be inaccurate. In fact, the GRPA produces wrong densityprofiles for micellar systems [47]. The second point is important when the we consider the accuracy.Because of the approximation for the vertex function, our free energy functional cannot reproduce


exact behavior around the critical point (or in the weak segregation region). Even in the strongsegregation region, our free energy seems to overestimate the long range interaction.

Phenomenological GL Expansion Free Energy Functional Model

There are several phenomenological GL expansion models for amphiphilic systems [38] (see alsoreferences in [38]). Roughly speaking, the phenomenological models can be described as follows.

Fphenomenological[φi] =∑

i

∫dr

[12a

(2)i φ2

i (r) +13a

(3)i φ3

i (r) +14a

(4)i φ4

i (r) + · · ·]

+∑

i

∫dr

[12b(1)i |∇φi(r)|2 +

12b(2)i

(∇2φi(r))2

+ · · ·]

+∑

i

∫dr

[12c(2)ij φi(r)φj(r) +

13c(3)ij φi(r)φ2

j (r)

+14c(4)ij φ

2i (r)φ2

j (r) + · · ·]

(2.176)

where φi(r) is the order parameter field such as the density fluctuation, and a(n)i , b

(n)i and c(n)

ij aresome constants. Eq (2.176) is usually truncated at the fourth or sixth order in the order parameter.The parameters a(n)

i , b(n)i and c(n)

ij are often treated as the fitting parameters and determined basedon some physical arguments.

It is clear that eq (2.176) is much different from our density functional theory. One may considerthat eq (2.176) can be used for any systems by fitting parameters (a(n)

i , b(n)i and c

(n)ij ) properly.

However, it is not a simple task. For multicomponent systems, there are considerably many fittingparameters and practically it is impossible to give appropriate sets of parameters. Besides, asmentioned earlier, in micellar systems, the GL expansion is not valid. Thus we can conclude eq(2.176) is not even a qualitative model for micellar systems.9

The Soft Colloid Picture

The ψ-form free energy functional can be interpreted as the free energy for soft colloids, accordingto the soft colloid picture [54,55,66].

If we employ the soft colloid picture, the free energy functional for homopolymer systems canbe expressed as follows [54,55].

F = U [ρ]− TS[c] (2.177)

where U [ρ] is the interaction functional as the functional of the monomer density field ρ(r), andS[c] is the entropy functional as the functional of the center of mass density field c(r). Eq (2.177)is a sort of hybrid form which uses both ρ(r) and c(r). If we want to express the free energyfunctional F only by ρ(r), we have to use some approximation form for S[c]. On the other hand,if we want to express F only by c(r), we need the approximation form for U [ρ]. Such a behavior issomehow similar to the uncertainty principle or the complementarity in quantum mechanics (Thisis first pointed by Pagonabarraga and Cates [54]).

We can interpret that the c-form free energy functional and the ψ-form free energy functionalare the approximation forms for eq (2.177). Thus we consider that our density functional theoryis consistent with the soft colloid picture. However, notice that the approximations we employedin our calculations are rough compared with other works based on the soft colloid picture, andtherefore our theory is considered to be quantitatively inaccurate. The accuracy may be improvedby considering more fine and accurate calculation methods which is used in the soft colloid works[66–70].

9One may feel such a claim is not correct. However, considering that the phenomenological free energy functionalmodels do not reproduce any practical phenomena in micellar systems, this claim is at least empirically correct.Of course we does not claim that the phenomenological models are useless for micellar systems. If one construct amodel so that it reproduces phenomena in micellar systems well, we believe that it is “a phenomenological model”.


2.5.5 Physical Properties of Static Density Functional Theory

Here we compare some physical properties of our static DF theory with ones of other theories. Asmentioned, there are several properties which is essential for micellar structures but not consideredwell in previous theories.

Reference State for Expansion

In many continuum field models the free energy functional is expressed by using some expansionmethod. We need the reference state to calculate the expansion form of the free energy, and thereference state affect the thermodynamic behaviors qualitatively.

The most popular reference state is the homogeneous state (ρpi(r) = fpiρp), which is widelyused in the GL expansion theory, the GRPA theory, or the phenomenological model. Whilethe expansion around the homogeneous state is well studied and useful in some cases, it is notappropriate for micellar systems. In the micellar systems, the density profiles are much differentfrom the homogeneous state. Thus the expansion around the homogeneous state is consideredto give qualitatively inaccurate result. In fact, even the GRPA theory (which is one of the mostaccurate expansion model around the homogeneous state) cannot reproduce micellar structuresqualitatively well.

On the other hand, our DF and the soft colloid model uses the inhomogeneous state as the ref-erence state. This makes the expansion complicated and difficult, compared with the homogeneousreference state case, but the resulting expansion form can be valid for micellar systems.

Extensivity of Conformational Entropy

The conformational entropy functional satisfies the following equation under the mean field ap-proximation.

S(conf)[αρi] = αS(conf)[ρi] (2.178)

where α is a positive constant. Eq (2.178) corresponds to the extensivity of the conformationalentropy.

It is easy to show that our conformational entropy functional satisfies eq (2.178). The confor-mational entropy in the SCF theory and one in the soft colloid model also satisfy eq (2.178).

The GL expansion theory and the GRPA theory also satisfies eq (2.178). It should be noticedhere that the reference state is changed as ρp → αρp by the change of the density field rescalingρpi(r) → αρpi(r). Since these theories are using the expansion around the homogeneous state,several coefficients in the free energy functional (for example, the coefficient of the square gradientterm) is changed by the density field rescaling. Such a behavior is not intuitive and this will notbe good for micellar systems.

The phenomenological free energy model does not satisfy eq (2.178) unlike other theories. Thisimplies that the phenomenological model is qualitatively inconsistent with other theories.

2.6 Static Density Functional Simulation

In this section, we perform static density functional simulations by minimizing the free energyfunctional numerically. We first describe the numerical method and then show the simulationresults. Finally we compare our DF simulation results and experimental results.

2.6.1 Static Simulation Method by the Static Density Functional Theory

The steady state density field ρ∗pi(r) minimizes the free energy functional F [ψpi] (eq (2.172)).Thus ρ∗pi(r) satisfies the following equation.

δF [√ρpi]δρpi(r)

∣∣∣∣ρpi(r)=ρ∗pi(r)

= 0 (2.179)

2.6. STATIC DENSITY FUNCTIONAL SIMULATION 37

Since F [√ρpi] is nonlinear functional of ρpi(r), the equation for ρ∗pi(r) is also nonlinear. Whilethere are several numerical methods to solve the nonlinear equation, we cannot use methods whichrequire large computational costs. For example, the Newton-Raphson method, which is widelyused to solve nonlinear equations, requires the evaluation of Jacobian matrix and not suitable forour current problems.

In analogy to usual minimization methods for the free energy functional, we first consider touse the density field ρpi(r) by the steepest-descent method. It is the most simple way to obtain thesteady state structures by the iteration. The density field is iteratively updated by the followingequation.

ρ(new)pi (r) = ρpi(r)− ωδF [√ρpi]

δρpi(r)(2.180)

where ω is a sufficiently small positive constant. Roughly speaking, eq (2.180) is equivalent toassume the following artificial dynamic equation and simulate the artificial dynamics.

∂ρpi(r, t)∂t

=δF [√ρpi]δρpi(r)

(2.181)

While the steepest-descent method is not so efficient, we can obtain static (equilibrium or metastable) structures if it works.

However, this steepest-descent method for the density field does not work because of the nu-merical instability. We need other method to avoid this instability. One simple, but practicalway is to minimize the ψ-field directly, instead of the density field. This is rather natural becausethe free energy is expressed as the functional of the ψ-field. The steepest-descent method is thenwritten as

ψ(new)pi (r) = ψpi(r)− ωδF [ψpi]

δψpi(r)(2.182)

unlike eq (2.180), eq (2.182) is not numerically unstable. This means that the variable transfor-mation from ρpi(r) to ψpi(r) removes singularity. The artificial dynamic equation corresponds toeq (2.182) is

∂ψpi(r, t)∂t

= −δF [ψpi]δψpi(r)

(2.183)

Eq (2.183) is equivalent to the following artificial dynamic equation for ρpi(r).

∂ρpi(r, t)∂t

= −4ρpi(r)δF [√ρpi]δρpi(r)

(2.184)

The difference between eq (2.181) and eq (2.184) is the factor 4ρpi(r), and this factor removessingularity which causes numerical instability.

2.6.2 Constraints: Mass Conservation and Incompressibility

We have to consider constraints of the system when performing numerical simulations. We are nowconsidering canonical systems and the total mass must be conserved. This constraint is written as

∫dr ρpi(r) = fpiρpV (2.185)

where V is the volume of the system. We also impose the incompressible condition.∑

pi

ρpi(r) = 1 (2.186)

The free energy should be minimized under these constraints.


To satisfy the constraints (2.185) and (2.186), we use the Lagrange multiplier method. Wemodify the free energy functional as follows.

F [ψpi]→ F [ψpi] +12

∑

pi

∫dr [λpi + κ(r)]

[ψ2pi(r)− fpiρp

](2.187)

where λpi and κ(r) are the Lagrange multipliers which correspond to eqs (2.185) and (2.186),respectively. (Because the Lagrange multiplier for the incompressible condition is used, we set theexcluded volume interaction term in (2.172), which contains v0, to 0.)

2.6.3 Numerical Method and Algorithms for Static Density FunctionalSimulation

We have to minimize the free energy numerically by solving eq (2.182) numerically. Becausewe want to perform simulations in real space, using regular meshes, we have to discretize thefields. We have to consider the numerical scheme for the update, too. In this section, we showthe discretization method and update scheme for fields. For simplicity, here we assume that allthe simulations are performed in three dimensional systems. The generalization for one or twodimensional systems is straightforward.

Discretization

We use the regular mesh in the following simulations. All the fields are defined only on the latticepoints. The position of the lattice point, G, is represented as follows.

G(nα) = nxhx + nyhy + nzhz (2.188)

where nα = 0,±1,±2, . . . is the index of the lattice point and hα corresponds to the lattice vector.ψpi(r) or other fields are defined only on this lattice.

The differential operator (Laplacian) or the integral are replaced by the filter. For example,

∂2

∂x2ψpi(r)→ 1

|hx|2 [ψpi(r − hx)− 2ψpi(r) + ψpi(r + hx)] (2.189)

∇2ψpi(r)→∑α

1|hα|2 [ψpi(r − hα)− 2ψpi(r) + ψpi(r + hα)] (2.190)

∫dr ψ2

pi(r)→∑

nαψ2pi(r)|hxhyhz| (2.191)

Hereafter we does not show such replacements explicitly for simplicity, but all the operators orintegrals are replaced appropriately.

The terms which contain Green function G(r − r′) can be calculated similarly, but it requireslarge numerical costs. To avoid it, we calculated it by using the fast Fourier transform (FFT). TheGreen function terms can be evaluated by using the Fourier transform as follows.10

∫dr′ G(r − r′)ψpi(r′) =

1(2π)3

∫dq e−iq·r

∫dr eiq·r

′′∫dr′ G(r′′ − r′)ψpi(r′)

=1

(2π)3

∫dq e−iq·r

[1q2ψpi(q)

] (2.192)

where ψpi(q) is the Fourier transform of ψpi(r). The numerical scheme is as follows.10Strictly speaking, the calculation of the Green function term based on eq (2.192) is inconsistent with the

discretization of the Laplacian. However, in most practical cases, the difference between the scheme shown hereand the scheme based on the exact discretization is rather small. Especially considering that there are very roughapproximations in the ADI type free energy minimization scheme, such a small difference will not be importantpractically.


1. Calculate the Fourier transform of ψpi(r) by the FFT.

2. Divide ψpi(q) by q2. To avoid divergence, the zero wave number component q = 0 is set to0.

ψpi(q)→

1q2ψpi(q) (q 6= 0)

0 (q = 0)(2.193)

3. Calculate the inverse Fourier transform of ψpi(q) by the inverse FFT.

We use the FFTW library [71] for the FFT and the inverse FFT11.

Update Scheme

Now we can solve eq (2.182) numerically by using the discretization scheme introduced above.

µpi(r) =∑

j

∫dr′ 4

√fpifpjAp,ijG(r − r′)ψpj(r′) + 2fpiCp,ii [ψpi(r) + 2ψpi(r) lnψpi(r)]

+∑

j(j 6=i)4√fpifpjCp,ijψpj(r)− b2

3∇2ψpi(r) +

∑

qj

χpi,qjψ2qj(r)

(2.194)

ψ(new)pi (r) = ψpi(r)− ω [µpi(r) + [κ(r) + λpi]ψpi(r)] (2.195)

λ(new)pi = λpi − ω

2

∫dr[ψ2pi(r)− fpiρp

](2.196)

κ(new)(r) = κ(r)− ω

2

∑

pi

ψ2pi(r)− 1

(2.197)

By iterating the update using eqs (2.194)-(2.197), the ψ-field for steady state which minimizes thefree energy functional can be obtained.

While the explicit scheme is easy to implement, we expect that it is numerically not efficient touse such a fully explicit scheme. This is similar to the case of the simple diffusion equation [72].12

From eqs (2.194) and (2.195), we find that there is the Laplacian term in µpi(r), and this Laplacianterm leads numerical instability for large ω.

It is known that the the diffusion equation can be solved stalely by using the implicit scheme suchas the fully implicit scheme or the Crank-Nicholson scheme. Therefore, we expect that the implicitscheme will improve stability in our simulations. There are several numerical methods to solveimplicit diffusion type equations. For example, the successive over relaxation (SOR) scheme (thisis rather classical) [72], the preconditioned conjugate gradient scheme (CG) (especially incompleteCholesky conjugate gradient scheme (ICCG)) [73–75], or the multigrid scheme [76] are often used.In this work we employ the alternating direction implicit method (ADI) [72]. the ADI method isapproximate method to solve implicit diffusion type equations and its not so accurate.

However, our evolution equation is nonlinear and it is not clear how to design the implicitscheme.13 The accuracy strongly depends on the scheme, and several implicit schemes may givenot so accurate results. Besides, several implicit schemes requires numerically inefficient relaxationmethod (such as the multi-dimensional Newton-Raphson method) and total computational costmay be considerably large.

11The FFTW library is based on the automated empirical optimization of software (AEOS) technique, andautomatically tuned at runtime. Therefore the calculation of the Green function by the FFTW is sufficiently fast inmost cases. (Other schemes such as the ICCG scheme or the multigrid scheme may improve the numerical efficiency.)

12In most cases, the fully explicit schemes are numerically not stable for the large time step (in this case, for largeω) and we have to set the time step sufficiently small.

13There are several works on unconditionally stable schemes for nonlinear systems, called the discrete variationalscheme [77, 78]. The discrete variational scheme works well for relatively simple systems such as the Cahn-Hillardequation or the Korteweg-de Vries (KdV) equation.


Thus we employ relatively simple ADI scheme in this work. To make things simpler, we employthe partial implicit ADI type scheme only for the Laplacian term. The computational cost of theADI method for the Laplacian term is rather small and it is constant (the computational costs forthe other relaxation methods generally depend on several conditions or parameters, and difficultto estimate).14 Thus we expect that the ADI method is sufficient for our purpose.

The ADI scheme for our evolution equation (eq (2.182)) can be described as follows.

µ(n)pi (r) =

∑

j

∫dr′ 4

√fpifpjAp,ijG(r − r′)ψ(n)

pj (r′) +∑

j(j 6=i)4√fpifpjCp,ijψ

(n)pj (r)

+ 2fpiCp,ii[ψ

(n)pi (r) + 2ψ(n)

pi (r) lnψ(n)pi (r)

]− b2

3∇2ψ

(n)pi (r) +

∑

qj

χpi,qj

[ψ

(n)qj (r)

]2 (2.198)

ψ(0)pi (r) = ψpi(r) (2.199)

ψ(1)pi (r) = ψ

(0)pi (r)− ω

3

[[µ

(0)pi (r) +

b2

3∂2

∂x2ψ

(0)pi (r)

]− b2

3∂2

∂x2ψ

(1)pi (r)

](2.200)

ψ(2)pi (r) = ψ

(1)pi (r)− ω

3

[[µ

(1)pi (r) +

b2

3∂2

∂y2ψ

(1)pi (r)

]− b2

3∂2

∂y2ψ

(2)pi (r)

](2.201)

ψ(3)pi (r) = ψ

(2)pi (r)− ω

3

[[µ

(2)pi (r) +

b2

3∂2

∂z2ψ

(2)pi (r)

]− b2

3∂2

∂z2ψ

(3)pi (r)

](2.202)

ψ(4)pi (r) = ψ

(3)pi (r)− ωκ(r)ψ(3)

pi (r) (2.203)

ψ(new)pi (r) =

fpiρpV∫dr[ψ

(4)pi (r)

]2

1/2

ψ(4)pi (r) (2.204)

κ(new)(r) = κ(r)− ω

2

∑

pi

ψ2pi(r)− 1

(2.205)

Note that we have changed the update schemes for λpi and κ(r). λpi is not updated explicitly inthe new scheme. Since the Lagrange multiplier λpi is used for the mass conservation constraint,we don’t need to update λpi if we update ψpi(r) by eq (2.204), which ensures the global massconservation. This is equivalent to update λpi and ψ

(new)pi (r) as

λpi =1ω− 1ω

fpiρpV∫dr[ψ

(4)pi (r)

]2

1/2

(2.206)

ψ(new)pi (r) = ψ

(4)pi (r)− ωλpiψ(4)

pi (r) (2.207)

instead of eq (2.204).The ADI scheme (eqs (2.198)-(2.205)) is known to be more stable than the explicit scheme (eqs

(2.194)-(2.197)) [79]. We use the ADI scheme to perform simulations in the following section. Allthe algorithms shown in this section is implemented to the simulator which is named “drops”, andthe simulation results in the following section is obtained by this simulator. 15

14For the diffusion equation in periodic systems, the ADI scheme can be reduced to solve the inverse of the cyclictridiagonal matrix. The cyclic tridiagonal matrix can be solved easily by using the Sherman-Morrison formula [72].The Sherman-Morrison formula is direct method and thus the computational costs for each evolution steps areconstant.

15The simulator “drops” is distributed as a free software on the author’s web site, under the GNU General PublicLicense (GPL):

http://www.ton.scphys.kyoto-u.ac.jp/~uneyama/drops.html (this URL will be changed in the future).


2.6.4 Results of Static Density Functional Simulation

In this section, we show the results of the static density functional simulations. All the simulationsshown in this section are performed in three dimensional systems, and the boundary condition isthe periodic boundary condition. We also set the segment size b = 1 and λ = ∞ (no cutoff forthe long range interaction) in all the simulations. We note that the value of λ does not affect themorphologies so much unless it is too small [58]. Simulations with finite value of λ will give thesimilar results.

The relaxation parameter ω is selected so that the simulations efficiently achieve the steadystate. Most of the simulations are started from homogeneous initial state with small noise, andsteady state structures are efficiently produced. However, in several cases (the systems with smallvolume fraction of the block copolymer and/or small χ parameter) the numerical scheme doesnot works well and no structures are produced. In such cases, we start the simulations from thenearly homogeneous state by using rather large χ parameter and/or large volume fraction. Forsuch parameters large density fluctuations grow easily after several hundreds iteration steps. Wethen switch the χ parameter and/or the volume fraction into the final target value and continuethe iterations to get the steady state.

Although such a procedure may look arbitrary and artificial, this procedure is needed in order toget the micellar structure for several cases. In fact, without the procedure, we could not reproducethe phase separation in the region where the χ parameter or the volume fraction are small. Besides,our static simulation scheme is the artificial dynamics which is designed to minimize the free energyfunctional and we do not need to follow the scheme strictly. Therefore we use this procedure forsmall χ parameter and/or small volume fraction cases.

AB Diblock Copolymer / C Homopolymer Blends

It is known that the blends of diblock copolymers and homopolymers show various interestingstructures other than micellar structures (but somehow similar to the micellar structures). Here weperform the simulations for AB diblock copolymer / C homopolymer blends, as an demonstration.

We set χ parameters so that AB diblock copolymers and C homopolymers cause macro phaseseparation. It is expected that there will be diblock copolymer rich regions and homopolymer richregions, and diblock copolymers form microphase separation structures in the diblock copolymerrich regions.

Figure 2.1 shows the result of the simulations. The parameters for the symmetric case (Figure2.1(a)) are as follows: the volume fractions ρAB = 0.2, ρC = 0.8, the degrees of polymerizationNAB = 10, NC = 20, the block ratios fA = fB = 0.5, and the χ parameters χAB = 1.2, χBC =1, χCA = 0.5. The parameters for the asymmetric case (Figure 2.1(b)) are as follows: the volumefractions ρAB = 0.3, ρC = 0.7, the degrees of polymerization NAB = 10, NC = 20, the block ratiosfA = 0.35, fB = 0.65, and the χ parameters χAB = 1.75, χBC = 1, χCA = 0.5. The system sizeand the lattice points are 40b × 40b × 40b and 64 × 64 × 64, respectively, and common for bothcases.

As expected, we can observe diblock copolymer rich regions with microphase separated struc-tures in homopolymer rich regions. It should be noted here that diffuse A rich layers are formedat the interface between AB diblock copolymer rich regions and C homopolymer rich regions (thiscannot be observed from Figure 2.1 since ρA(r) < 0.5 and ρB(r) < 0.5 in these layers). The multi-layered structure in Figure 2.1(a) is called the onion structure, and the onion structure is actuallyobserved by the experiment [80]. While the experimental system is PS-PI diblock copolymer / PShomopolymer, we expect that the onion structures observed in Figure 2.1(a) will be qualitativelythe same as the onion structures observed in experiments.

AB Diblock Copolymer Solutions

Here we perform simulations for amphiphilic AB diblock copolymer solutions (AB diblock copoly-mer / S solvent blends). This will be the most simple system in which block copolymer micellesare formed.


Figure 2.1: Steady state structures for AB diblock copolymer / C homopolymer blends. (a)symmetric diblock copolymer case, and (b) asymmetric diblock copolymer case. Light and darkgray surfaces show the isosurfaces for ρA(r) = 0.5 and ρB(r) = 0.5, respectively. Notice that weinserted the “cutting plane” to see the structures inside block copolymer rich region.

Results for AB diblock copolymer solutions with various χ parameters are shown in Figure2.2. The parameters are as follows: the volume fractions ρAB = 0.1, ρS = 0.9, the degrees ofpolymerization NAB = 20, NC = 1, the block ratios fA = 1/3, fB = 2/3, and the χ parametersχAB = 1, χBS = 1.75, χAS = 0.5, 0,−0.175. The system size is 48b× 48b× 48b (96× 96× 96 latticepoints).

In the simulations, we changed the χ parameter between the hydrophilic A subchain and theS solvent. Figure 2.2(a) corresponds to relatively hydrophobic case and Figure 2.2(c) correspondsto relatively hydrophilic case. Figure 2.2(b) is the neutral case (there is no effective energeticinteraction between the A subchain and the S solvent) As the A subchain becomes hydrophilic, weobserve that relatively long cylindrical micelles, bilayers, or vesicles are formed.

Results for AB diblock copolymer solutions with various block ratios are shown in Figure 2.3.The parameters are as follows: the volume fractions ρAB = 0.1, ρS = 0.9, the degrees of polymer-ization NAB = 20, NC = 1, the block ratios fA = 1−fB , fB = 0.3, 1/3, 0.4, 0.5, 0.6, 2/3, 0.7, 0.8, 0.9,and the χ parameters χAB = 1, χBS = 1.75, χAS = −0.175. The system size is 32b × 32b × 32b(64 × 64 × 64 lattice points). (Most of parameters are the same as the case of Figure 2.2(c), butthe simulation box is slightly small.)

We can observe that as fB , the block ratio of the B subchain, increases the morphologies ofmicellar structures changed from spherical micelles (Figure 2.3(b) and (c)) to cylindrical micelles(Figure 2.3(c) and (d)), then to vesicles (Figure 2.3(e) and (f)). It is also observed that if fB issufficiently small, no structures are formed and the system is homogeneous (Figure 2.3(a)), and iffB is sufficiently large, the system undergoes the macrophase separation and block copolymer richdroplets are formed instead of micellar structures (Figure 2.3(g)-(i)).

AB Diblock Copolymer / A Homopolymer Blends

Next we perform simulations for AB diblock copolymer / A homopolymer blends. Micellar struc-tures are also formed in these blends and we expect that they can be used as the model systemsof diblock copolymer solutions. Strictly speaking, there are several differences between the AB di-block copolymer solution case and AB diblock copolymer / A homopolymer blend case. However,we use blends here because the resulting micellar structures are qualitatively almost the same forboth cases, and it is easy to get strongly segregated structures in blend systems.

Results of simulations for AB diblock copolymers / A homopolymers are shown in Figure 2.4.The parameters are: the volume fractions ρAB = 0.1, ρA = 0.9, the degrees of polymerizationNAB = 20, NA = 10, the block ratios fA = 1/3, fB2/3, and the χ parameter χAB = 1. The system


Figure 2.2: Steady state structures for AB diblock copolymer solutions with various χAS . (a)χAS = 0.5, (b) χAS = 0, and (c) χAS = −0.175. Gray surfaces show the isosurfaces for ρB(r) = 0.5(B subchains form cores of micellar structures).

size is 48b × 48b × 48b (96 × 96 × 96 lattice points). Figure 2.4(a)-(d) are results for differentrandom seeds. We can observe that several vesicles with formed, and there are several sizes ormorphologies. For example, an elongated vesicle is formed in Figure 2.4(a) or fused vesicles areformed in Figure 2.4(d). However, most of the produced structures are spherical vesicles. Thus weconsider that the spherical vesicles are formed in this set of parameters.

To confirm this conclusion more clearly, we perform simulations for larger systems. Theparameters are: the volume fractions ρAB = 0.1, ρA = 0.9, the degrees of polymerizationNAB = 20, NA = 10, the block ratios fA = 1/3, fB2/3, and the χ parameters χAB = 1. Thesystem size is 64b×64b×64b (128×128×128 lattice points). (Only the box size and the number oflattice points are different.) Results of simulations for AB diblock copolymers / A homopolymersare shown in Figure 2.5. We can observe vesicles are formed in this case, too. While there areseveral non-spherical vesicles, we consider spherical vesicles will be the most stable.

At the end of this section, we scan the χAB-ρAB parameter space and draw the phase diagram.To perform many simulations with various parameters, we use rather small simulation box here.Thus the result may not be accurate, but we believe it is still qualitatively acceptable. Theparameters used for the simulations are as follows: the degrees of polymerization NAB = 20, NA =10, the block ratios fA = 1/3, fB = 2/3. The system size is 32b × 32b × 32b and the number oflattice points is 64× 64× 64. The volume fraction of the A homopolymer is set to ρA = 1− ρAB ,and we vary ρAB and χAB .

The result is shown in Figure 2.6. Symbols in Figure 2.6, such as circles, squares or triangles


Figure 2.3: Steady state structures for AB diblock copolymer solutions with various block ratio fB .(a) fB = 0.3, (b) fB = 1/3, (c) fB = 0.4, (d) fB = 0.5, (e) fB = 0.6, (f) fB = 2/3, (g) fB = 0.7,(h) fB = 0.8, (i) fB = 0.9, Gray surfaces show the isosurfaces for ρB(r) = 0.5 (B subchains formcores of micellar structures).

represent the results of each simulations. The dotted lines in Figure 2.6 shows the boundary ofphases (notice that, however, these lines are drawn very roughly). The resulting phase diagramis qualitatively in agreement with experimental phase diagrams [20, 81, 82] while the simulationsystems and the experimental systems are different.

2.6.5 Discussion

Morphologies of Micellar Structures by Simulations

We have observed several micellar structures (spherical micelles, cylindrical micelles, open bilay-ers, and vesicles) by the static DF simulations. All the observed morphologies are non periodicstructures and periodic structures which is often observed in block copolymer melt systems (suchas lamellars, hexagonally packed cylinders, BCC spheres) are not observed.

However, the periodic structures can be formed in amphiphilic block copolymer systems. Forexample, Matsen [83, 84] applied the SCF spectrum method for AB diblock copolymer / A ho-mopolymer systems and calculated the phase diagrams. However there are several differences


Figure 2.4: Steady state structures for AB diblock copolymer / A homopolymer blends. Parametersare the same for (a)-(d) except for the initial random seed. Gray surfaces show the isosurfaces forρB(r) = 0.5 (B subchains form cores of micellar structures).

between our simulation and the SCF simulation.One is the assumption about the morphologies. In the SCF spectrum simulation, the symme-

tries of morphologies are given as an input. Thus we cannot study non symmetric (non periodic)structures by the spectrum method. Another is that we performed simulations for rather dilutesolutions. Periodic structures are mainly observed in rather dense solutions [38], and this will bethe reason why we have only the non periodic structures. We will be able to reproduce severalperiodic structures if we perform simulations for dense solutions.

It should be also noted that our simulation method generally gives the metastable state, notthe most stable (equilibrium) state. This is the limitation of the numerical method, thus in somesimulations the most stable state will be the periodic structures. However, the structures obtainedby our simulations are sufficiently stable in most cases, and thus we believe the simulation resultsare physically valid.

Dependence of Morphologies of Micellar Structures on Parameters

We have performed many simulations with various parameter sets. Here we consider the depen-dence of the morphologies of micellar structures on simulation parameters.

First we consider the effect of the block ratio. As shown in the simulations for AB diblockcopolymer solutions (Figure 2.3), micellar structures are not formed if fB (the block ratio of thehydrophobic subchain) is too small or too large. If fB is too small, the diblock copolymers are


Figure 2.5: Steady state structures for AB diblock copolymer / A homopolymer blends with thelarge simulation box. Parameters are the same for (a) and (b) except for the initial random seed.Gray surfaces show the isosurfaces for ρB(r) = 0.5 (B subchains form cores of micellar structures).

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1

ρAB

χAB

: Vesicles: Bilayers: Cylindrical micelles: Spherical micelles: Disordered

Vesicles

Cylindricalmicelles

Sphericalmicelles

Disordered

Figure 2.6: Phase diagram for AB diblock copolymer / A homopolymer blends determined fromstatic density functional simulations. Grey symbols represent the results of simulations, and dottedlines are drawn to indicate the phase boundary lines roughly.

soluble to the solvent and no structures are formed. On the other hand, if fB is too large, thediblock copolymers behave as B homopolymers and cause usual macrophase separations. Thus we


need some intermediate fB for micellar structures.In the case of the intermediate fB , morphologies of the formed micellar structures depends on

fB . From Figure 2.3 we know that we have spherical micelles for relatively small fB (so-called the“hairy block copolymers”), vesicles for relatively large fB (so-called the “crew-cut block copoly-mers”), and cylindrical micelles for intermediate fB . This is consistent with physical intuition andexperimental results.

We have to notice that vesicles are “not” formed for symmetric block copolymer case (fB = 0.5).If one consider the classical packing theory for low molecule surfactants, one may expect that blockcopolymer vesicles are formed for nearly symmetric block copolymers. But this is not correct.Because the hydrophilic subchains become swollen by solvents, their exclusive volumes becomelarger. Thus one should not simply apply the packing theory for block copolymer micelles.16

Next we consider the effects of the volume fraction and the χ parameter. These effects are shownin Figure 2.6, for the AB diblock copolymer / A homopolymer blend case. If we increase the χparameter without varying the volume fraction, the morphologies of micelles changes from sphericalmicelles to cylindrical micelles, and then to vesicles. Eisenberg and coworkers [20,81,82] performedexperiments for the mixture of polyacrylicacid-polystyrene (PAA-PS) diblock copolymers, dioxianeand water with various block ratios, polymerization degree, or volume fractions. Because dioxianeis polar organic solvent and water-soluble, we expect that the system can be regarded as themixture of amphiphilic diblock copolymers and the single component effective solvents. Changingthe volume fraction of water without changing the volume fraction of the block copolymer will thencorrespond to changing the χ parameters. It is reported that by adding water the morphologiesof micelles change from spherical micelles to cylindrical micelles, and to vesicles. And it is alsoreported that by adding dioxiane, the morphologies change with the reverse order. We consider oursimulation result are consistent with the experimental results or molecular dynamics simulationresults [25].

If we increase the volume fraction of block copolymers without changing the χ parameter, weobserve the similar morphological change. By increasing the volume fraction, spherical micelleschange into cylindrical micelles, and cylindrical micelles change into vesicles. Such a tendency isalso consistent with experimental results.

Interpretation Based on Simple Model

Here we try to interpret the simulation results by a simple theory. We consider the effects ofthe χ parameter and the block ratio. For the sake of simplicity, we neglect the contribution ofthe translational entropy (this is why we do not consider the effect of the volume fraction here).Further we assume that some micellar structures are formed and stabilized.

We express the free energy of the system as the sum of two contributions; the free energyof hydrophilic subchains (corona) and the interfacial energy between the hydrophobic subchains(core), and the solvents.

F ≈ F (corona) + F (interface) (2.208)

We approximate the free energy of the corona by the free energy of the swollen polymer brush[6, 85, 86]. The free energy of the swollen polymer brush is proportional to the polymerizationindex and the two third power of the of the graft density. If we assume that the graft density isinversely proportional to the interfacial area, we have

F (corona) ∝ fAS−2/3A (2.209)

where SA is the total interfacial area between the corona regions and the outside solvent regions.The interfacial energy can be expressed as follows, by using the Helfand-Tagami type interfacialtension [87,88].

F (interface) ∝ χ1/2SB (2.210)16If we use some kind of effective volume of the segment instead of the original segment volume, the packing

theory will work well. However, the estimation of the effective volume of the segment will not be easy because itdepends on the solvent quality (the χ parameter).


where SB is the total interfacial area between the core regions and the corona regions, and χis the χ parameter between the hydrophobic subchain and the solvent (χBS for the AB diblockcopolymer solution case and χAB for the AB diblock copolymer / A homopolymer blend case).

If we assume SA ∝ SB , then we can write the free energy as the function of SB .

F(SB) = α1fAS−2/3B + α2χ

1/2SB (2.211)

where α1 and α2 are positive constants. Differentiating eq (2.211) and we have

∂F(SB)∂SB

= −23α1fAS

−5/3B + α2χ

1/2 (2.212)

Thus the free energy (eq (2.211)) takes minima at SB = S∗B .

S∗B ≡(

2α1fA3α2χ1/2

)3/5

(2.213)

We expect that large value of SB corresponds to spherical micelles, the intermediate value corre-sponds to cylindrical micelles, and the small value corresponds to vesicles. Therefore we find thefollowings, from eq (2.213).

• The shape of micelles change from spheres to cylinders, and then vesicles, by increasing theχ parameter.

• The shape of micelles change from vesicles to cylinders and to spheres, by increasing theblock ratio of the hydrophilic subchain (or, equivalently, by decreasing the block ratio of thehydrophobic subchain).

These trends are qualitatively consistent with simulations, and we can interpret that this is theeffect of the parameters to morphologies.

Comparison with Experiments

As discussed in the previous sections, our simulation results are qualitatively consistent with ex-periments. Here we compare our simulation χ parameters with real experimental χ parameters.

The most important parameters in static simulations are the Flory-Huggins χ parameters.The χ parameters are experimentally measured or calculated from the solubility parameters. Theχ parameter between the hydrophilic subchain and the hydrophobic subchain is in most casessufficiently large to cause the microphase separation. For example, Bhargava et al [89] calculatedthe χ parameter between polystyrene (PS), which can be used as the hydrophobic monomer [20],and water as χPS,water = 6.27. Dormidontova [90] measured the χ parameter between poly(ethyleneoxide) (PEO) and water and reported that the χ parameter can be expressed as χPEO,water =−0.0615 + 70/T (T is the absolute temperature). Lam and Goldbeck-Wood [91] also measuredχPEO,water and obtained χPEO,water = 1.35. Xu et al [92] measured the interaction between PS andPEO and reported χPS,PEO = 0.02 ∼ 0.03. Zhu et al [93] also measured χPS,PEO and and obtainedχPS,PEO = −0.00705 + 21.3/T . The polymerization index of PS-PEO diblock copolymer used inthe experiments [89] is typically ' 1000, thus we expect that χPS,PEON is sufficiently larger thanthe critical value χcN = 10.495 [12]. Note that the polyelectrolytes such as poly(acrylic acid)(PAA) is often used as the hydrophilic subchain [81, 82]. Such polyelectrolytes can be dissolvedinto water easily, and thus we expect the χ parameter between the polyelectrolytes and the wateris negative.

Thus we can say the diblock copolymers are in strong segregation region, and the effect of wateraddition (the solvent quality change) is expected to be especially large for hydrophobic subchains,and not so large for hydrophilic subchains. While we used AB diblock copolymer / A homopolymerblend to scan the phase diagram, we believe that our simulation is qualitatively applicable as therealistic model of micellar systems. In fact, our phase diagram is qualitatively similar to the

2.7. SUMMARY 49

experimental phase diagram [79]. Thus we consider that our simulation is qualitatively consistentwith these experiments.

Notice that, however, it is quite difficult or practically impossible to compare the χ param-eters used in the simulations and ones measured by experiments directly because the value ofthe χ parameters depends on the definition of the segment, and our free energy functional is notquantitatively accurate as the SCF theory.

We also note that most experiments are carried out in the very strong segregation region inwhich it is quite hard to perform continuum field simulations. (It should be noted that evenfor the SCF simulations it is numerically quite difficult to handle the very strong segregationregion.) While it is difficult to compare our simulations with real experiments, our simulations areexpected to give qualitatively correct physical process since our simulations take the crucial physicalproperties correctly; the hydrophobic interaction χBS is sufficiently large and the hydrophilicinteraction χAS is sufficiently small or negative. These properties are not considered well in previousphenomenological GL expansion models.

Comparison with SCF Simulations

There are many SCF simulation works for amphiphilic block copolymer solutions [34, 36, 37]. Be-cause of the numerical costs for SCF simulations, the investigated parameter range is not so wide.However, so far, the DF simulation results and SCF simulation results seem to be qualitativelyconsistent. We consider that this is rather trivial, because both the DF simulations and the SCFsimulations use the similar parameters (the χ parameters or the block ratio). Even if the models aredifferent, simulation results should be qualitatively the same as long as the models are applicableand the input parameter are the same.

In these SCF simulations, standard static SCF simulation method by Drolet and Fredrickson[35] is used. While the Drolet-Fredrickson method is not a steepest-descent method, our DFsimulations and SCF simulations give qualitatively the same evolution pathway [34, 79]. (Wediscuss the vesicle formation pathway in detail, in the next chapter.) This is not trivial but thelocally non-conserved evolution scheme may favor the similar pathways.

2.7 Summary

We have derived the static density functional theory for block copolymers and have performedstatic simulations for diblock copolymer solutions and diblock copolymer / homopolymer blendsin this chapter.

In our density functional theory, the free energy is expressed as the functional of the ψ-field (thefield of the square root of the density). Our free energy functional can be used for strongly segre-gated structures while conventional GL expansion free energy models do not work for such cases.This means that our DF theory can be applied to micellar systems where the block copolymersare strongly segregated and strongly localized.

By performing numerical simulations we have shown that our free energy actually reproducemicellar structures (spherical micelles, cylindrical micelles, and vesicles). Our simulation result isconsidered to be qualitatively consistent with experimental results for amphiphilic block copolymersolutions or previous SCF simulation results. The DF simulations require less computationalcosts than the SCF simulations and considered to be a useful tool to study micellar structures inamphiphilic block copolymer systems.

Appendix

2.A Self Consistent Field Theory

In this appendix, we show the brief derivation of the self consistent field (SCF) theory [94,95]. Forsimplicity we consider homopolymer systems here. Generalization for block copolymer melts and


blends are straight forward [6].The SCF theory can be used sorely as an accurate simulation method as well as the base to

derive the explicit free energy functional form [63,96].We start from the grand partition function.

Ξ =∫DρDW exp

[−βU [ρ] + iρ ·W + eβµ

∫DR exp

[−i∫ N

0

dsW (R(s))

]](2.214)

We use the saddle point approximation just like the GL expansion theory. The saddle pointequations can written as follows.

− δ

δρ∗(r)βU [ρ∗] + iW ∗(r) = 0 (2.215)

iρ∗(r)− ieβµ∫DR

[∫ N

0

ds δ(r −R(s))

]exp

[−i∫ N

0

dsW ∗(R(s))

]= 0 (2.216)

To solve the saddle point equations, we introduce the path integral field (or the partial statisticalweight field).

q(r, s) ≡∫DR δ(r −R(s)) exp

[−i∫ s

0

ds′W ∗(R(s′))]

(2.217)

q†(r, s) ≡∫DR δ(r −R(s)) exp

[−i∫ N

s

ds′W ∗(R(s′))

](2.218)

Using eqs (2.217) and (2.218), eq (2.216) can be rewritten as follows.

ρ∗(r) = eβµ∫ N

0

ds q†(r, s)q(r, s) (2.219)

From eqs (2.217) and (2.218), we can get the following diffusion type equation for the path integralfield.

∂q(r, s)∂s

=b2

6∇2q(r, s)− iW ∗(r)q(r, s) (2.220)

∂q†(r, s)∂(−s) =

b2

6∇2q†(r, s)− iW ∗(r)q†(r, s) (2.221)

Eq (2.220) (or eq (2.221)) is called the Edwards equation. The initial conditions for q(r, s) andq†(r, s) are as follows.

q(r, 0) = 1 (2.222)

q†(r, N) = 1 (2.223)

These equations (eqs (2.215), and (2.219) - (2.221)) should be solved self consistently. Wewrite them down again here. For simplicity, we write ρ(r) = ρ∗(r) and V (r) = iW ∗(r). The selfconsistent set of equations are:

∂q(r, s)∂s

=b2

6∇2q(r, s)− V (r)q(r, s) (2.224)

∂q†(r, s)∂(−s) =

b2

6∇2q†(r, s)− V (r)q†(r, s) (2.225)

q(r, 0) = 1 (2.226)

q†(r, N) = 1 (2.227)

ρ(r) = eβµ∫ N

0

ds q†(r, s)q(r, s) (2.228)

V (r) =δ

δρ(r)βU [ρ] (2.229)

2.B. CALCULATION OF COEFFICIENTS IN APPROXIMATE FORM OF ΓIJ 51

The partition function is then written as follows.

Ξ ≈ exp[−βU [ρ] + ρ · V +

∫dr ρ(r)

](2.230)

And the grand potential functional J can be expressed as

J ≡ −kBT ln Ξ

≈ U [ρ]− kBTρ · V − kBT∫dr ρ(r)

(2.231)

The free energy functional for the canonical ensemble can be obtained by using the Legendretransform. The number of polymers in the system can be expressed as follows.

M =1N

∫dr ρ(r) (2.232)

From eqs (2.229) and (2.232) we have

µ = −kBT ln

[1

MN

∫dr

∫ N

0

ds q†(r, s)q(r, s)

](2.233)

Thus the free energy functional F can be expressed as follows.

F ≡ J + µM

≈ U [ρ]− kBTρ · V + kBTM

[lnM − 1− ln

[∫dr

1N

∫ N

0

ds q†(r, s)q(r, s)

]](2.234)

In most cases, the self consistent set of equations (2.224)-(2.229) cannot be solved analytically,17

and we should solve them numerically [8,12]. Numerical simulations for eqs (2.224)-(2.229) requireslarge computational costs and this makes it difficult to perform large scale simulations. There areseveral standard algorithms to solve the self consistent set of equations in real space [35,95,97] andin the eigenfunction space [8] (this is sometimes called the spectrum method).

While the numerical cost is rather large, the SCF simulations are known to be very accurate.Thus the SCF simulations are good reference for the GL or DF simulations.18 That is, we canverify our DF simulations or estimate the accuracy of the DF simulations by comparing them withthe SCF simulations.

It should be notice that there are various improved numerical methods for the SCF calculations.For example, the narrow interface approximation (or the narrow interphase approximation, NIA)[84, 98–100], the external potential dynamics (EPD) method [101], or the use of the complexfield [102,103] are known to improve numerical efficiency.

The self consistent set of equations shown here can be used to calculate the free energy for agiven density profile [6, 104]. In such cases, only eqs (2.224)-(2.228) are used. (Because eq (2.229)corresponds to the condition for the steady state distribution.) Usually V (r) which reproduces thegiven ρ(r) is calculated by using some relaxation method. The expressions for the grand potentialfunctional (eq (2.231)) or the free energy functional (eq (2.234)) are unchanged.

2.B Calculation of Coefficients in Approximate Form of Γij

In this appendix, we calculate the coefficients in the approximate form of Γij (eq (2.122)).17There are a few analytical solutions such as interfaces of strongly segregated polymer blends [87,88] or grafted

polymer brush [85,86].18Honda and Kawakatsu developed the hybrid method which combines the SCF theory and the free energy

functional theory. Such a method will be useful to perform numerically efficient and accurate simulations


2.B.1 Calculation of Monomer-Monomer Two Point Correlation Func-tion

Because Γij is defined as the inverse of S(2)ij (eq (2.118)), first we calculate the explicit expression

for S(2)ij .

From eqs (2.113) and (2.116), S(2)ij can be written as follows.

S(2)ij (r1 − r2) =

1N2fifj

∫DR

∫

s∈ids

∫

s∈jds′ δ(r1 −R(s))δ(r2 −R(s′))

=1

N2fifj

∫

s∈ids

∫

s∈jds′ exp

[−3|r1 − r2|2

2|s− s′|b2] (2.235)

where |s−s′|b2 represents the average distance along with the polymer chain (the chemical distance)between R(s) and R(s′). It is convenient to transform eq (2.235) into the Fourier space.

S(2)ij (q) ≡

∫d(r1 − r2) eiq·(r1−r2)S

(2)ij (r1 − r2)

=1

N2fifj

∫

s∈ids

∫

s∈jds′ exp

[−|s− s

′|b26

q2

] (2.236)

For the case of i = j, S(2)ij (q) reduces to the Debye function [4, 6].

S(2)ii (q) =

1N2f2

i

∫ Nfi

0

ds

∫ Nfi

0

ds′ exp[−|s− s′|b

2

6q2

]

=72

N2f2i b

4(q2)2

[exp

[−1

6b2Nfiq

2

]− 1 +

16Nfib

2q2

] (2.237)

For the case of i 6= j, S(2)ij (q) can be expressed as follows.

S(2)ij (q) =

1N2f2

i

∫ Nfi

0

ds

∫ Nfj

0

ds′ exp

[−(s+

l2ijb2

+ s′)b2

6q2

]

=36

N2fifjb4(q2)2

[1− exp

[−1

6Nfib

2q2

]] [1− exp

[−1

6Nfjb

2q2

]]exp

[−1

6l2ijq

2

]

(2.238)

where lij is the chemical distance between the i-th subchain and the j-th subchain.From eqs (2.237) and (2.238), we have

S(2)ij (q) =

2ξ2i

[e−ξi − 1 + ξi

](i = j)

1ξiξj

[e−ξi − 1

] [e−ξj − 1

]exp

[−1

6l2ijq

2

](i 6= j)

(2.239)

where we definedξi ≡ 1

6Nfiq

2 (2.240)

2.B.2 Calculation of Aij and Cij

It is convenient to introduce the following two functions.

hij(q) ≡ NfifjS(2)ij (q) (2.241)

gij(q) ≡ 1Nfifj

Γij(r) (2.242)

2.B. CALCULATION OF COEFFICIENTS IN APPROXIMATE FORM OF ΓIJ 53

It is also convenient to use matrix and vector notations (for example, we write hij(q) as a matrixh(q) or fi as a column vector f). From (2.118) we have

h(q) · g(q) = E (2.243)

where E is the unit matrix.First we consider the low wave number limit in the Fourier space. At the limit of |q| → 0, h(q)

can be expanded as follows.h(q) = Nff t −Hq2 +O(q4) (2.244)

where f t is the transposed vector of f and the matrix H is defined as follows.

Hij ≡

118N2f3

i b2 (i = j)

112Nfifj

[N(fi + fj)b2 + 2l2ij

](i 6= j)

(2.245)

We also expand g(q) into the power series of q2.

g(q) = A1q2

+B +O(q2) (2.246)

Substituting eqs (2.244) and (2.246) into eq (2.243) gives

(Nff t ·A) 1

q2+(−A ·H +NB · ff t)+O(q2) = E (2.247)

To satisfy eq (2.247) for arbitrary q, we require that the following set of equations holds (up tothe O(q0)).

ff t ·A = 0 (2.248)

−H ·A+Nff t ·B = E (2.249)

Multiply H−1 from the left to eq (2.249) and we have

−A+NH−1 · ff t ·B = H−1 (2.250)

Then multiplying ff t from the left to eq (2.250) and using (2.248) gives

−ff t ·A+Nff t ·H−1 · ff t ·B = ff t ·H−1

N(f t ·H−1 · f)ff t ·B = ff t ·H−1

Nff t ·B =ff t ·H−1

f t ·H−1 · f

(2.251)

Substituting eq (2.251) into eq (2.250) finally we have

A = −H−1 +H−1 · ff t ·H−1

f t ·H−1 · f (2.252)

Next we consider the high wave number limit. At the limit of |q| → ∞, h(q) can be expressedas follows.

h(q) = K1q2

+L1q4

+O(1/q6) (2.253)

where matrices K and L are defined as

Kij ≡ 12δijfib2

(2.254)

Lij ≡

− 72Nb4

(i = j)

− 36Nb4

(i 6= j, l2ij = 0)

0 (i 6= j, l2ij 6= 0)

(2.255)


As before, we expand g(q) into the power series of q2.

g(q) = Dq2 + F +O(1/q2) (2.256)

From eqs (2.243), (2.253), and (2.256) we have

K ·D = E (2.257)K · F +L ·D = 0 (2.258)

From eqs (2.257) and (2.258) we have the following expressions for D and F .

D = K−1 (2.259)

F = −K−1 ·L ·K−1 (2.260)

Eqs (2.259) and (2.260) can be rewritten as follows.

Dij ≡ b2δij12fi

(2.261)

Fij ≡

12f2i N

(i = j)

− 14fifjN

(i 6= j, l2ij = 0)

0 (i 6= j, l2ij 6= 0)

(2.262)

Interpolating two limits, gij(q) can be approximated as follows.

gij(q) ≈ Aij 1q2

+ Cij +b2δij12fi

q2 (2.263)

The remaining task is to determine Cij in eq (2.263). Here we determine Cij so that it repro-duces the original value of gij(q) at the minima. If we write the wave number where gij(q) takethe minima as q∗, from eq (2.263) we have

1(q∗)2

=

√b2

12fiAii(i = j)

0 (i 6= j)

(2.264)

Thus we fit Cij as follows.

Cij =

h−1ij (q∗)− 2

√b2Aii12fi

(i = j)

Fij = − 14fifjN

(i 6= j, l2ij = 0)

Fij = 0 (i 6= j, l2ij 6= 0)

(2.265)

where h−1ij (q∗) is the inverse matrix of hij(q∗).

From these results we have the explicit expression for Γij(q) as

Γij(q) ≈ Nfifj[Aij

1q2

+ Cij +b2δij12fi

q2

](2.266)

where Aij and Cij are given by eqs (2.252) and (2.265). The inverse Fourier transform of eq (2.266)gives eq (2.122).

2.C. STATIC DENSITY FUNCTIONAL THEORY: ANOTHER DERIVATION 55

2.C Static Density Functional Theory: Another Derivation

In this appendix, we show another derivation for the static density functional theory. Strictlyspeaking, the derivation shown in this appendix is the original derivation [52]. Although thisderivation contains intuitive approximations (or a sort of extrapolations), but it will be useful toshow it here and compare it with the derivation in main sections.

2.C.1 Intuitive Derivation of the Flory-Huggins-de Gennes-Lifshitz The-ory

As mentioned, our density functional theory reduces to the Flory-Huggins-de Gennes-Lifshitz the-ory for homopolymer cases. There are several derivation for the Flory-Huggins-de Gennes-Lifshitztheory [54,55,63], here we show one of the most simple (but mathematically not sound) derivation.

For simplicity, we consider one component homopolymer systems. We start from the GLexpansion free energy functional.

F [δρ]kBT

=F0

kBT+

12

∫drdr′ Γ(r − r′)δρ(r)δρ(r′) + · · ·

≈ F0

kBT+∫dr

[1

2Nρδρ2(r) +

b2

24ρ|∇δρ(r)|2

]+ · · ·

(2.267)

The GL expansion form is formally correct, but as long as we truncate higher order terms, it canbe applied only for the weak segregation limit.

This is because the free energy functional (2.267) is expressed by using the density fluctuationδρ(r) and it contains the spatial average of the density ρ. We consider then, whether it is possibleto remove these factors by some manipulations. If we recall that the GL expansion is based on thefunctional Taylor expansion, we may interpret eq (2.267) as the following variational expression.

1kBT

δ2F [ρ]∣∣ρ=ρ

=1

2Nρδρ2(r) +

b2

24ρ|∇δρ(r)|2 (2.268)

If we accept eq (2.268), what we have to do is to find the true free energy functional F [ρ] of whichsecond order variational satisfies eq (2.268).

Clearly there are many possible candidates. Thus we modify eq (2.268) further as follows.

1kBT

δ2F [ρ] ≈ 12Nρ(r)

δρ2(r) +b2

24ρ(r)|∇δρ(r)|2 (2.269)

Of course this is very rough approximation and mathematically not exact, but here we employ(2.269) to proceed. It is now straightforward to show that the free energy functional can bedescribed as follows.

F [ρ]kBT

=∫dr

[1Nρ(r) ln ρ(r) +

b2

6

∣∣∣∇√ρ(r)

∣∣∣2]

(2.270)

Eq (2.270) is nothing but the Flory-Huggins-de Gennes-Lifshitz free energy. It is also straightfor-ward to show that eq (2.270) satisfies eq (2.268) or reduces to eq (2.267).

2.C.2 Generalization for Block Copolymer Systems

The derivation of the Flory-Huggins-de Gennes-Lifshitz theory shown in the previous section canbe regarded as the method to extrapolate a GL expansion free energy into the non-GL expansionfree energy.

In this section we apply this extrapolation method to block copolymer systems, and derive thedensity functional theory as the generalized Flory-Huggins-de Gennes-Lifshitz theory.


The GL expansion free energy can be written as follows.

F [δρpi]kBT

=F0

kBT+

12

∑

pi,qj

∫drdr′ [δpqΓp,ij(r − r′) + χpi,qj ] δρ(r)δρpi(r′)δρqj(r) + · · ·

≈ F0

kBT+∑p

∑

i,j

∫drdr′

Ap,ij2ρpG(r − r′)δρpi(r)δρpj(r′)

+∑p

∑

i,j

∫dr

[Cp,ij2ρp

δρpi(r)δρpj(r) +δijb

2

24fpiρpi|∇δρi(r)|2

]

+∑

pi,qj

∫dr

12χpi,qjδρpi(r)δρqj(r) + · · ·

(2.271)

The main differences between the homopolymer case and the block copolymer case are that thevertex function Γp,ij(r) contains the nonlocal kernel and that multiple density components mustbe considered simultaneously. Although there are differences, the approximation and extrapolationcan be used similarly.

1kBT

δ2F [ρpi]∣∣ρpi=ρpi

=Ap,ij2ρpG(r − r′)δρpi(r)δρpj(r′) +

Cp,ij2ρp

δρpi(r)δρpj(r)

+δijb

2

24fpiρp|∇δρpi(r)|2 +

12χpi,qjδρpi(r)δρqj(r)

(2.272)

As before, we modify eq (2.272) further.

1kBT

δ2F [ρpi] =

√fpifpjAp,ij

2√ρpi(r)ρpj(r′)

G(r − r′)δρpi(r)δρpj(r′) +

√fpifpjCp,ij

2√ρpi(r)ρpj(r)

δρpi(r)δρpj(r)

+δijb

2

24ρpi(r)|∇δρpi(r)|2 +

12χpi,qjδρpi(r)δρqj(r)

(2.273)

where we used the following replacement for ρp.

ρp →√ρpi(r)ρpj(r′)

fpifpj(2.274)

Finally we have the following expression for the free energy functional.

F [ρpi]kBT

=∑

p,ij

∫drdr′ 2

√fpifpjAp,ijG(r − r′)√ρpi(r)ρpj(r′)

+∑

p,i

∫dr fpiCp,iiρpi(r) ln ρpi(r) +

∑

p,i 6=j

∫dr 2

√fpifpjCp,ij

√ρpi(r)ρpj(r)

+∑

p,i

∫dr

b2

6

∣∣∣∇√ρpi(r)

∣∣∣2

+∑

pi,qj

∫dr

12χpi,qjρpi(r)ρqj(r)

(2.275)

2.C.3 Validity of the Heuristic Derivation

While the derivation shown in this section is rough, there is an attempt similar to this derivation,in the density functional theory for liquids by Ebner, Saam and Stroud [105].19

Ebner, Saam and Stroud approximated ρ in the density functional theory for simple fluids [40]by several mean values such as [ρ(r) + ρ(r′)]/2,

√ρ(r)ρ(r′), or ρ((r + r′)/2). They calculated

19This is first pointed by Wu and Li [106].

2.C. STATIC DENSITY FUNCTIONAL THEORY: ANOTHER DERIVATION 57

the surface tension by using these approximate expressions and reported that [ρ(r) + ρ(r′)]/2and ρ((r + r′)/2) gives almost the same result and

√ρ(r)ρ(r′) slightly overestimates the surface

tension. However, the difference among the three expressions are about the order of 1%, and so weconsider that the detail expression of the mean will not be important qualitatively. Thus we expectthat the rough approximation used in this appendix will be applicable to our current purpose.

It is also noted that most of the coefficients derived in this section and ones derived in mainsections is the same (although there are several differences). For example, the coefficients suchas√fpifpjAp,ij or b2/6 are consistent with the density functional theory in the main section.

This fact may justify the use of the replacement (2.274) as a rough but physically not so badapproximation.


Chapter 3

Dynamic Density FunctionalTheory and Simulations forMicellar Structures in BlockCopolymer Systems

3.1 Introduction

In the previous chapter we have shown that the DF simulation reproduce various micellar struc-tures including vesicles as steady state (equilibrium or metastable) structures. While the staticsimulations are useful to study thermodynamic behaviors, we cannot study dynamic behaviorsfrom static simulations.

The vesicle formation process or morphological transition process of micellar structures areinteresting phenomena and important for application purposes. While there are several simulationworks on micelle or vesicle formation processes [23, 27, 28, 107], the vesicle formation process arestill not well understood.

In this chapter, we extend the static DF theory to dynamics and perform dynamics simulationsfor micellar systems. We show that our DF simulations successfully reproduce physically naturalvesicle formation dynamics and morphological transition dynamics. We consider the vesicle for-mation mechanism or the morphological transition mechanism from the simulation results. Thesimulation results are also compared with other simulation results.

3.2 Time-Dependent Ginzburg-Landau Theory

To study dynamics of block copolymer systems based on continuum field models, the time-dependent Ginzburg-Landau (TDGL) theory [108,109] is widely used [39,104].

However in this work we do not use the TDGL theory, just like we have not used the GLexpansion free energy in the previous chapter. In this section, we show the standard TDGL theorybriefly and show that it is not suitable for micellar systems.

3.2.1 Equation of Continuity, Fick’s Law, and Mobility

Here we consider the density fluctuation field δρi(r) as the order parameter field. We start fromthe equation of continuity for the order parameter field. It is clear that the equation of continuityholds since the order parameter field is locally conserved (the local mass conservation). Thus wehave

∂

∂tδρi(r, t) +∇ · ji(r, t) = 0 (3.1)

59

60 CHAPTER 3. DYNAMIC DENSITY FUNCTIONAL THEORY AND SIMULATIONS

where ji(r, t) is the flux of the density fluctuation. What we have to do here is to calculate ji(r, t)as the function which is closed in δφi(r, t).

We expect that the flow of the order parameter field obeys Fick’s law if the chemical potentialgradient is not so large.

ji(r, t) = −∑

j

∫dr′ Lij [δρi](r, r′)∇′ δF [δρi]

δ(δρj(r′))(3.2)

where Lij [δρi](r, r′) represents the coupling strength of the chemical potential and the flow, andis called the mobility. Lij [δρi](r, r′) depends the position r, species i, and the order parameterδρi(r).

We have to determine the form of the mobility to get explicit form of the dynamic equation.Clearly this is not easy because the form of the mobility strongly depends on the underlyingparticles’ microscopic dynamics. The most popular model for the mobility will be the constantmodel. That is, we approximate the mobility as

Lij [δρi](r, r′) ≈Miδijδ(r − r′) (3.3)

where Mi is a positive constant. This model gives a dynamic equation which seems to be simple.However, as we see in following sections, this approximation form of mobility is not derived insystematic way and thus physically not acceptable. Another popular model for the mobility isso-called the local coupling model.

Lij [δρi](r, r′) ≈Miδijρi(r)δ(r − r′) = Mi (δρi(r) + ρi) δ(r − r′) (3.4)

In dynamic SCF simulations, this model is widely employed. We show that this form of mobilitycan be derived from microscopic dynamics.

While other mobility models are proposed [101, 110, 111], such mobility models are non-localand complicated. Thus here we do not consider such models. (We discuss the external potentialdynamics which uses the nonlocal mobility model in Appendix 3.B.)

3.2.2 Time-Dependent Ginzburg-Landau Equation

We can obtain the dynamic equation for the density fluctuation by substituting eqs (3.2) and (3.3)(or (3.4)) into eq (3.1). If we employ the constant mobility model (eq (3.3)) we have

∂

∂tδρi(r, t) = ∇ ·

∑

j


δ(δρj(r′))

≈Mi∇2 δF [δρi]δ(δρi(r))

(3.5)

Eq (3.5) is called the time dependent Ginzburg-Landau (TDGL) equation. If we employ the localcoupling type mobility (eq (3.4)) we have the following TDGL equation instead.

∂

∂tδρi(r, t) ≈Mi∇ ·

[(δρi(r) + ρi)∇δF [δρi]

δ(δρi(r))

](3.6)

It can be easily shown that the free energy functional monotonically decays as the time evolves.Taking the differential of the free energy functional with respect to time, we have

∂F [δρi]∂t

=∑

i

∫dr

δF [δρi]δ(δρi(r))

∂

∂tδρi(r)

=∑

i

∫dr

δF [δρi]δ(δρi(r))

∇ ·∑

j


δ(δρj(r′))

= −∑

i,j

∫drdr′ Lij [δρi](r, r′)

[∇δF [δρi]δ(δρi(r))

]·[∇′ δF [δρi]δ(δρj(r′))

](3.7)

3.3. DYNAMIC DENSITY FUNCTIONAL THEORY 61

Eq (3.7) is bilinear form, and if the mobility is positive definite the right hand side of (3.7) ispositive. For constant and local coupling models, the mobility is clearly positive definite and thuswe have

∂F [δρi]∂t

≤ 0 (3.8)

3.2.3 Validity of TDGL Equation

It should be noticed that the validity of the TDGL equation is generally not guaranteed. Whilethe free energy functional monotonically decreases as the time evolves by the TDGL equation, thisis not sufficient. The main problems are:

• The TDGL theory itself cannot give the explicit form of the mobility. Clearly the dynamicsdepends strongly the form of the mobility, and thus inappropriate mobility model may leadqualitatively incorrect and unphysical results.

• In the mesoscopic systems, we cannot ignore the effect of the thermal fluctuation. In otherwords, the dynamic equation must include the random thermal noise term. However, theTDGL equation (3.4) (or (3.6)) is deterministic.

These problems are not so important for macrophase separation cases (see for example, [112]). Inthat case, we can derive the mobility from microscopic dynamics (see, for example [4]), and thefluctuation effect is negligibly small in macroscopic scale.

However, for microphase separation case, the explicit form of the mobility is generally not wellknown and the thermal fluctuation is very important. If there are free energy barriers, the thermalnoise to overcome barriers is essential, and we know that there are many free energy barriers inmicrophase separated systems. Therefore we can say that the TDGL equation is not appropriatefor microphase separation dynamics. We need another approach other than the TDGL theory, tostudy the dynamics of block copolymer systems.

The thermal fluctuation is also quite important for micellar systems. For example, there are freeenergy barriers in the fusion or fission processes of micelles. The deterministic dynamic equationcannot reproduce such processes.

3.3 Dynamic Density Functional Theory

The approaches to dynamics based on the TDGL equation have been proposed and simulationshave been performed in previous works. However, so far no previous works achieved to reproducethe vesicle formation process which is one of the most interesting processes in micellar systems.

In this section we derive the new dynamic density functional theory for block copolymers whichcan be applied for micellar systems. Then we can perform dynamic density functional simulationsfor micellar systems such as the vesicle formation process or the morphological transition process.

3.3.1 Dynamic Density Functional Theory for Colloidal Systems

In this section, we derive the dynamic equation of the density field (the dynamic density functionalequation) for colloidal systems. For simplicity, here we consider the single component colloidalsystems. (It is straight forward to generalize the theory for multicomponent colloidal systems.)The interaction energy for the system is given by

U(Ri) =∑

i,j

12v(Ri −Rj) (3.9)


whereRi is the position of the i-th colloid particle and v(r) is the interaction potential. We assumethat the colloid particles obey the following overdamped Langevin equation.1

ζdRi(t)dt

= −∂U(Ri)∂Ri

+ ηi(t)

= − ∂

∂Ri

∑

j

v(Ri −Rj) + ηi(t)(3.10)

where Ri(t) represents the position of the i-th colloid particle at the time t, ζ is the frictioncoefficient, U(Ri) is the interaction potential, and ηi(t) is the thermal noise (the Gaussian whitenoise) which satisfies the following fluctuation-dissipation relations.

〈ηi(t)〉 = 0 (3.11)〈ηi(t)ηj(t′)〉 = 2ζkBTδijδ(t− t′)1 (3.12)

where 〈. . . 〉 means statistical average over the noise. It can be written as follows by using thefunctional integral over ηi(t).

〈. . . 〉 ≡∫Dηi (. . . ) exp

[− 1

4ζkBT

∑

i

∫dtη2

i (t)

](3.13)

3.3.2 Dean Equation

Dean [113] has shown that we can derive the exact dynamic equation for the microscopic densityfield ρ(r, t) from the Langevin equation (3.10). The microscopic density field (or the densityoperator) is defined as follows.

ρ(r, t) ≡∑

i

δ(r −Ri(t)) (3.14)

It should be noticed that the microscopic density field defined by eq (3.14) is not a “density field”in the usual (or physical) sense. ρ(r, t) is expressed as a sum of δ functions and thus it is notsmooth function. We need to define averaged or coarse grained density field as a “density field”.(So far, use of the statistically averaged density field [114, 115] or the temporally coarse graineddensity field [116] is proposed.) We perform a coarse graining in the following sections to obtainthe dynamic density functional equation for an usual smooth density field.

Here we derive the dynamic density functional equation based on the Dean’s theory. Using theIto calculus, we obtain the following partial differential equation.

∂ρ(r, t)∂t

=∑

i

∂

∂tδ(r −Ri(t))

= −∇ ·∑

i

δ(r −Ri(t))1ζ

−∇

∑

j

v(r −Rj) + ηi(t)

+kBT

ζ∇2∑

i

δ(r −Ri(t))

=∇ ·[ρ(r)ζ∇[kBT ln ρ(r) +

∫dr′ v(r − r′)ρ(r′)

]]−∇ ·

[1ζ

∑

i

δ(r −Ri(t))ηi(t)

]

(3.15)

Eq (3.15) is closed form in ρ(r, t), except for the last term (the random noise term),

ξ(r, t) ≡ −∇ ·[

1ζ

∑

i


](3.16)

1Strictly speaking, the use of eq (3.10) is not always justified. For example, there are hydrodynamic interactionsbetween colloid particles immersed in solvents, or the inertial effect is not always negligible. One way to overcomethese difficulties is to use the DPD dynamic equation instead of the overdamped Langevin equation (3.10).


If ξ(r, t) can be expressed as the closed form, we have the closed dynamic equation for ρ(r, t). Thefirst and second order moments of ξ(r, t) can be derived as follows.

⟨ξ(r, t)

⟩= −1

ζ

⟨∇ ·[∑

i


]⟩= −1

ζ∇ ·[∑

i

〈δ(r −Ri(t))〉〈ηi(t)〉]

= 0 (3.17)

⟨ξ(r, t)ξ(r′, t′)

⟩=

1ζ2

⟨∇ ·[∑

i


]∇′ ·

∑

j

δ(r′ −Rj(t′))ηj(t′)

⟩

=1ζ2∇∇′ :

∑

i,j

〈δ(r −Ri(t))δ(r′ −Rj(t′))〉〈ηi(t)ηj(t′)〉

=2kBTζ∇ · ∇′ [〈ρ(r, t)〉 δ(r − r′)δ(t− t′)]

= −2kBT∇ ·[ 〈ρ(r, t)〉

ζ∇δ(r − r′)

]δ(t− t′)

(3.18)

From eqs (3.17) and (3.18), the moments of ξ(r, t) is closed in ρ(r, t). Thus we can write eq (3.16)as the closed form. The explicit expression is

ξ(r, t) = ∇ ·[√

ρ(r, t)η(r, t)]

(3.19)

where η(r, t) is the thermal noise field which satisfies the following fluctuation-dissipation relations.

〈η(r, t)〉 = 0 (3.20)

〈η(r, t)η(r, t′)〉 =2kBTζ

δ(r − r′)δ(t− t′)1 (3.21)

Finally the dynamic equation (3.15) can be written as follows.

∂ρ(r, t)∂t

= ∇ ·[ρ(r)ζ∇ δH[ρ]δρ(r, t)

]+∇ ·

[√ρ(r, t)η(r, t)

](3.22)

where H[ρ] is the Hamiltonian functional defined as

H[ρ] ≡∫dr kBT ρ(r) [ln ρ(r)− 1] +

∫drdr′

12v(r − r′)ρ(r)ρ(r′) (3.23)

The first term in the right hand side of eq (3.23) corresponds to the translational entropy ofparticles. Eq (3.22) is called as Dean equation. (A similar but slightly different dynamic densityfunctional equation2 has been previously derived by Kawasaki [117], and thus eq (3.22) is alsocalled the Kawasaki-Dean equation).

As mentioned, ρ(r, t) is not a smooth density field (it is rather an operator). Thus, while eq(3.22) is derived by a simple calculation based on the Ito calculus, it is not the dynamic densityfunctional equation in the usual sense. We cannot use eq (3.22) as a dynamic density functionalequation in the usual sense, without coarse graining or averaging.

It should be noticed here that the noise term in the Dean equation (3.22) is the Ito type, notthe Stratonovich type. An intuitive explanation for this fact is that the Hamiltonian functional(eq (3.23)) is a sort of the free energy rather mechanical Hamiltonian because it contains theentropic term. If the noise is the Stratonovich type, there should not be the entropic term in theHamiltonian, thus the noise term should be the Ito type.

2The potential v(r) is replaced by −kBTc(2)(r). where c(2)(r) is the direct correlation function [40]. However,under the mean field approximation there are no difference between two expressions because −kBTc(2)(r) = v(r).


We can modify eq (3.22) as follows

∂ρ(r, t)∂t

= ∇ ·[ρ(r)∇δ(H[ρ]/kBT )

δρ(r)

]+∇ ·

[√2ρ(r)w(r, t)

](3.24)

where t ≡ tζ/kBT , w is the Gaussian white noise field which satisfies⟨w(r, t)

⟩= 0 (3.25)⟨

w(r, t)w(r, t′)⟩

= δ(r − r′)δ(t− t′)1 (3.26)

Eq (3.24) means that the magnitude of the thermal noise is independent of the temperature T aslong as the dimensionless Hamiltonian H[ρ]/kBT is used. That is, the magnitude of the thermalnoise is no longer the control parameter.3

3.3.3 Functional Fokker-Planck Equation for the Dean Equation

The Fokker-Planck equations are sometimes useful to analyze Langevin equations. Here we derivethe Fokker-Planck equation for the dynamic density functional equation (3.22) [118]. Becauseeq (3.22) is the stochastic differential equation for the function, the corresponding Fokker-Planckequation is the functional Fokker-Planck equation. To derive the functional Fokker-Planck equationwe first introduce the following distribution functional.

P [ρ](t) ≡ 〈δ[ρ(r)− ρ(r, t)]〉 (3.27)

The dynamic of P [ρ](t) can be obtained as follows.

∂P [ρ](t)∂t

=⟨∂

∂tδ[ρ(r)− ρ(r, t)]

⟩

= −∫dr

δ

δρ(r)

[P [ρ]∇ ·

[ρ(r)ζ∇δH[ρ]δρ(r)

]]

− 12

∫drdr′

δ2

δρ(r)δρ(r′)

[2kBTP [ρ]∇ ·

[ρ(r)ζ∇δ(r − r′)

]]

= −∫dr

δ

δρ(r)

[[∇ ·[ρ(r)ζ∇δH[ρ]δρ(r)

]]P [ρ] + kBT∇ ·

[ρ(r)ζ∇δP [ρ]δρ(r)

]]

(3.28)

Finally we have the following functional Fokker-Planck equation [116,118].

∂P [ρ](t)∂t

= −∫dr

δ

δρ(r)∇ ·[ρ(r)ζ∇[kBT

δ

δρ(r)+δH[ρ]δρ(r)

]P [ρ]

](3.29)

We can modify eq (3.29) as follows.

∂P [ρ](t)∂t

= −∫dr

δ

δρ(r)∇ ·[ρ(r)ζ∇[P [ρ]

δ

δρ(r)[kBT lnP [ρ] +H[ρ]]

]](3.30)

The equilibrium distribution for the density field, P (eq)[ρ], can be obtained as follows.

P (eq)[ρ] ∝ exp[−H[ρ]kBT

](3.31)

Eq (3.31) is rather trivial but it implies that this form of the equilibrium distribution cannotbe reproduced without the thermal noise. We can interpret this result as follows.

3There are some works which assume the magnitude of the thermal noise can be treated as the control parameterand controls them freely. However, as long as the Langevin equation is valid, they are physically incorrect. It maywork as the kind of energy minimizer (the simulated annealing method), but it does not correspond to the realdynamics.


First, the microscopic density field ρ(r) is fluctuating even at the equilibrium state. This meansthat, the density distribution which minimizes the Hamiltonian functional is not the equilibriumstate, but the most probable state of the density field. Thus we cannot obtain the true equilib-rium density distribution by the minimization of the Hamiltonian functional (or the free energyfunctional, in the case of the previous sections). We may need the renormalization theory to takethe fluctuation effect correctly.4

Second, the dynamic density functional equation should be stochastic. While the deterministicdynamic equation for the density field is widely used for mesoscale field simulations, most ofthem are not based on the microscopic motion of particles. We need to use the stochastic densityfunctional equation based on the mesoscopic dynamics to reproduce qualitatively collect dynamics.

3.3.4 Coarse-Graining Method: Archer-Rauscher Approximation

We have derived the exact dynamic equation for the microscopic density field, but the dynamicequation cannot be used for simulations directly. The main reasons are

• The microscopic density field is sum of δ functions (spikes) and thus it is not expressed bystandard finite difference representation for the continuum fields.

• The magnitude of the thermal noise is so large that we cannot perform numerical simulationsstably (or even if we can perform simulations no structures are formed).

To avoid difficulties, we need to perform a coarse-graining for the microscopic density field andobtain the smoothed density field. Archer and Rauscher [116] proposed to use the temporallycoarse-grained density field.

ρ(r, t) ≡∫dtKτ (t− t′)ρ(r, t′) (3.32)

where Kτ (t) is a kernel function which has the resolution τ and decay rapidly as |t| → ∞.The dynamic equation for ρ(r, t) then becomes

∂ρ(r, t)∂t

=kBT

ζ∇2ρ(r) +∇ ·

[1ζ

∫dr′ [∇v(r − r′)]

∫dt′Kτ (t− t′)ρ(r, t′)ρ(r′, t′)

]

+∇ ·[∫

dt′Kτ (t− t′)√ρ(r, t′)η(r, t′)

]

=∇ ·[ρ(r)ζ∇δFτ [ρ, ρ(2)]

δρ(r)

]+∇ ·

[∫dt′Kτ (t− t′)

√ρ(r, t′)η(r, t′)

](3.33)

where Fτ [ρ, ρ(2)] is the coarse-grained free energy functional (or the coarse-grained Hamiltonianfunctional)

Fτ [ρ, ρ(2)] ≡∫dr kBT ρ(r) [ln ρ(r)− 1] +

∫drdr′

12v(r − r′)ρ(2)(r, r′) (3.34)

and ρ(2)(r, r′) is the coarse-grained two point density.

ρ(2)(r, r′) ≡∫dt′Kτ (t− t′)ρ(r, t′)ρ(r′, t′) (3.35)

Unfortunately, the free energy functional (3.34) is not closed in ρ(r). We need some approximationsto get the closed equation. For example, by replacing the two point density by the equilibrium twopoint density, we can get the closed form of the free energy functional.

Besides, the noise term

ξ(r, t) ≡ ∇ ·[∫

dt′K(t− t′)√ρ(r, t′)η(r, t′)

](3.36)

4However, it is not clear whether the standard renormalization theory can be applied for our free energy functional.Since the free energy is expressed as the functional of the ψ-field (not the density fluctuation field) and it containsnon-polynomial terms and long-range interactions, the standard procedure may not be used.


is also not closed in ρ(r). Archer and Rauscher proposed to approximate eq (3.36) as follows.

ξ(r, t) ≈ ∇ ·[√

ρ(r, t)√τ

τ0η(r, t)

](3.37)

where τ0 is the characteristic time scale for the microscopic equation of motion. Although thisapproximation is hand-weaving and rather rough, it keeps one of the most important properties;the coarse-graining makes the magnitude of the noise smaller.

By using such approximations, the dynamic equation (3.33) can be approximately expressed asthe closed form in ρ(r)

3.4 Dynamic Density Functional Theory for Block Copoly-mer Systems

3.4.1 Approximations for Dynamic Density Functional Equation

Unlike the case of colloids, we cannot derive the exact and closed density functional theory fordensity fields (see also Appendix 3.A). Therefore we need several approximations to obtain thedynamic density functional theory which can be used for numerical simulations.

Here we introduce two approximations. The first one is the Archer-Rauscher type temporalcoarse-graining approximation. For simplicity we write the temporal coarse-grained density asρi(r) here after. The second one is the approximation which replaces the effective free energyfunctional as the equilibrium free energy functional under the mean field approximation.

Fτ [ρi, ρ(2)ij ] ≈ F [ρi] (3.38)

where F [ρi] is the free energy functional calculated for the equilibrium state (eq (2.172)). Eq(3.38) means that we approximate the coarse-grained free energy functional by the equilibriumfree energy functional. This approximation can be roughly interpreted as assuming that all therelaxation modes are completely relaxed.

The resulting dynamic density functional equation can be described as follows.

∂ρi(r, t)∂t

= ∇ ·[ρi(r)∇δ(F [ρi]/kBT )

δρi(r)

]+∇ ·

[√ρi(r)

√2β−1wi(r, t)

](3.39)

where we used the rescaled time t and the noise magnitude parameter β−1 ≡ τ0/τ . The fluctuation-dissipation relation for the thermal noise wi(r, t) is as follows.

⟨wi(r, t)

⟩= 0 (3.40)⟨

wi(r, t)wj(r, t′)⟩

= δijδ(r − r′)δ(t− t′)1 (3.41)

The approximations employed here are very rough and thus the resulting dynamic densityfunctional equation (3.39) is also very rough approximation. Nevertheless, eq (3.39) has physicallyimportant properties compared with phenomenological dynamic equations.

• The form of the mobility is obtained from the microscopic model. The mobility is sometimesapproximated as constant, but the constant mobility leads unphysical behavior.5 For exam-ple, the dynamic equation does not reduce to the simple diffusion equation for ideal case ifwe approximate the mobility as constant.

• The dynamic equation is stochastic. In most cases, however, the deterministic dynamicequations are assumed. Even if there are thermal noise term, the magnitude of the thermalnoise is assumed to be very small. Clearly this is unphysical for mesoscopic systems.

• The density field is temporally coarse grained density field. Thus the magnitude of thethermal noise is smaller than one for the microscopic density field.

5As shown by Dean [113], the dynamic density equation does not reduce to the widely used Cahn-Hilliard typeequation which employs the constant mobility.

3.5. DISCUSSIONS ON DYNAMIC DENSITY FUNCTIONAL EQUATION 67

3.4.2 ψ-Field Expression of Dynamic Density Functional Equation

In the static density functional theory and simulations, we introduced the ψ-field, which is definedas the square root of the density field. We expect that the ψ-field is also useful for the dynamicdensity functional theory and simulations. As the static density functional theory, we define theψ-field via the following equation.

ψi(r, t) ≡√ρi(r, t) (3.42)

where ρi(r, t) is the temporally coarse grained density field introduced in the previous section(ρi(r, t) = ρi(r)).

Here we rewrite eq (3.39) by using the ψ-field [58].

∂ρi(r, t)∂t

= ∇ ·[ψ2i (r)∇

[δ(F/kBT )δψi(r)

∂ψi(r)∂ψ2

i (r)

]]+∇ ·

[ψi(r)

√2β−1wi(r, t)

]

=12∇ ·[ψ2i (r)∇µi(r)

ψi(r)

]+∇ ·

[ψi(r)

√2β−1wi(r, t)

]

=12∇ · [ψi(r)∇µi(r)− µi(r)∇ψi(r)] +∇ ·

[ψi(r)

√2β−1wi(r, t)

]

=12[ψi(r)∇2µi(r)− µi(r)∇2ψi(r)

]+∇ ·

[ψi(r)

√2β−1wi(r, t)

]

=12

[ψi(r)∇2µi(r)− µi(r)∇2ψi(r) +∇ ·

[ψi(r)

√4β−1wi(r, t/2)

]]

(3.43)

where we used the definition of the chemical potential for the ψ-field, µi(r) ≡ δ(F/kBT )/δψi(r).By introducing the new rescaled time t′ ≡ t/2, finally we have

∂ρi(r, t′)∂t′

= ψi(r)∇2µi(r)− µi(r)∇2ψi(r) +∇ ·[ψi(r)

√4β−1wi(r, t′)

](3.44)

For simplicity, we write t′ = t hereafter.

∂ρi(r, t)∂t

= ψi(r)∇2µi(r)− µi(r)∇2ψi(r) +∇ ·[ψi(r)

√4β−1wi(r, t)

](3.45)

Eq (3.45) is the main result of this section and it is also actually used in simulations in followingsections.

3.5 Discussions on Dynamic Density Functional Equation

3.5.1 Deterministic and Stochastic Dynamic Density Functional Equa-tions

We have derived the stochastic density functional equation for block copolymer systems. However,the deterministic dynamic density functional equations are widely used to study block copoly-mer systems. In this section we discuss about the deterministic and stochastic dynamic densityfunctional equations.

First of all, we have to emphasize that the Dean’s stochastic dynamic density functional isformally exact. The Dean equation is derived from the Langevin equation for colloid particles,without any approximations. However, it should be also emphasized that the density field used inthe Dean equation is the microscopic density field, which is expressed as the sum of delta functions.Thus we can say that the Dean equation is exact but it is not for the smooth density field (thedensity field in the usual sense).

As mentioned, to obtain the dynamic equation for the smoothed density field, we need somecoarse-grainings. One possible definition of the smoothed (coarse-grained) density is the averageover the noise distribution [113–115].

ρ(r, t) ≡ 〈ρ(r, t)〉 (3.46)


Taking the average over eq (3.22), we have the following dynamic density functional equation forρ(r, t).

∂ρ(r, t)∂t

=kBT

ζ∇2ρ(r) +∇ ·

[1ζ

∫dr′ [∇v(r − r′)] 〈ρ(r)ρ(r′)〉

](3.47)

Clearly, eq (3.47) is not the closed in ρ(r, t) but contains the noise averaged two point density〈ρ(r, t)ρ(r′, t)〉. The dynamic equation for 〈ρ(r, t)ρ(r′, t)〉 is also not closed, and the three pointnoise averaged densities are needed. Thus the BBGKY like hierarchy set of equations appears[113,116,118]. This BBGKY like hierarchy is exact, thus we can say that the deterministic dynamicdensity functional theory is also correct. However, the BBGKY like hierarchy cannot be handledexactly without some cutting approximations. The most simple approximation is to replace thenoise averaged density by the equilibrium two point density [116]. Such an approximation will bevalid near the equilibrium state, but generally not good.6

We draw the schematic image of the dynamics of the deterministic dynamic density functionaltheory and the dynamics of the stochastic dynamic density functional theory in Figure 3.1. Thedynamics of the deterministic density functional theory is trapped at the local minimum whichis the nearest to the initial state while the dynamics of the stochastic density functional theorymoves around by overcoming the free energy barrier by the thermal fluctuation. The effect of thethermal fluctuation is not important if there are no free energy barriers, or there are only few freeenergy barriers. However we know that there are many free energy barriers in the cases of micellarsystems, and thus for micellar systems only the stochastic density functional theory works well.

global minimum

initial state

deterministicstochastic

F[ρ]

ρ

Figure 3.1: Schematic draw of deterministic and stochastic dynamics in the dynamic densityfunctional theory. The dashed light gray line represents free energy landscape F [ρi]. The soliddark gray and dotted black lines represent deterministic (without noise) and stochastic (withnoise) dynamics, respectively. If we employ the stochastic dynamics, the state can overcome thefree energy barrier and fluctuate in the phase space. If we employ the deterministic dynamics, thestate will be trapped at the nearest local minimum and frozen.

3.5.2 Magnitude of Thermal Noise

To use the stochastic density functional theory, we have to discuss about the magnitude of thethermal noise (or the effective temperature of the field). It is known that we cannot performsimulations if we assume that the noise field satisfies the fluctuation-dissipation relation with thereal temperature T .

We have to use a sort of effective, reduced temperature T , or equivalently, we have to introducethe noise scaling factor T /T . The problem here is how to determine the effective temperature.There are roughly two approaches. One is to use the spatial coarse-graining. In this approach, the

6Besides, if we use the mean field approximation, the information about the equilibrium two point density isdiscarded. We usually use the mean field approximation for polymer systems, and thus this type of dynamic equationcan be incorrect even near the equilibrium state.

3.5. DISCUSSIONS ON DYNAMIC DENSITY FUNCTIONAL EQUATION 69

degree of freedom of the simulation cell is reduced by the factor 1/Ω, where Ω is the number ofthe degree of freedom in the cell. Another is to use the temporal coarse-graining. This approachis introduced in the Archer-Rauscher theory, and the temperature is reduced by the factor β−1 ≡τ/τ0. Therefore, two approaches give the similar results, and we can map one onto another [58].

Ω =|hx||hy||hz|

β−1(3.48)

where hα is the lattice vector. Unfortunately, we do not have any systematic way to calculatethe factor Ω or β−1. Thus what we can to do here is to just estimate the factor roughly, andfor such an estimation the temporal coarse-graining will be useful. Thus we employ the temporalcoarse-graining approach here. We estimate the factor and discuss about it later.

3.5.3 Hydrodynamic Interaction

In our dynamic density functional equation (3.39) (or eq (3.45)) does not contain the hydrody-namic interaction. The hydrodynamic interaction for continuum field models is taken by using theOseen tensor [4] or by coupling the Navier-Stokes equation for the velocity fields with the dynamicequation for order parameters (model H) [108,109].

To include the hydrodynamic equation, here we consider to use nonlocal mobility which isexpressed by the Oseen tensor. The Langevin equation for colloid particles (eq (3.10) ) can bemodified as follows.

dRi(t)dt

=∑

j

Hij(Ri −Rj) ·[−∂U(Ri)

∂Ri

]+ ηi(t)

=∑

j

Hij(Ri −Rj) ·− ∂

∂Ri

∑

j

v(Ri −Rj)

+ ηi(t)

(3.49)

where Hij(Ri −Rj) is the mobility tensor.

Hij(Ri −Rj) ≡ δijζ

1 + (1− δij)Ω(Ri −Rj) (3.50)

Ω(r) ≡ 18πηr

[1 +

rr

r2

](3.51)

ζ is the effective friction coefficient of colloid particles and η is the effective viscosity of the fluid.Ω(r) is the Oseen tensor which describes hydrodynamic interaction between different particles.To avoid the self hydrodynamic interaction we assume Ω(0) = 0 [119]. The thermal noise ηi(t)satisfies the following fluctuation-dissipation relation.

〈ηi(t)〉 = 0 (3.52)〈ηi(t)ηj(t′)〉 = 2kBTHij(Ri −Rj) (3.53)

The dynamic density functional equation for the microscopic density field is then written as

∂ρ(r, t)∂t

= ∇ ·[ρ(r)ζ

∫dr′ [δ(r − r′)1 + ζΩ(r − r′)ρ(r′)] · ∇′ δH[ρ]

δρ(r′)

]

+∇ ·[√

ρ(r)ζ

∫dr′

[δ(r − r′)1 +

√ζΩ1/2(r − r′)

]· η(r′, t)

] (3.54)

where Ω1/2(r) is the Cholesky decomposition of the Oseen tensor Ω(r) defined via the followingequation ∫

dr′ (Ω1/2)t(r − r′) ·Ω1/2(r′ − r′′) = Ω(r − r′′) (3.55)


((Ω1/2)tαβ(r − r′) = Ω1/2βα (r′ − r) is the transpose of Ω1/2) and η(r, t) is the thermal noise field

which satisfies eqs (3.20) and (3.21).Eq (3.54) can be further modified.

∂ρ(r, t)∂t

= ∇ ·[ρ(r)ζ∇δH[ρ]δρ(r)

]+∇ ·

[√ρ(r)η(r, t)

]

+∇ ·[ζ

∫dr′ ρ(r)Ω(r − r′)ρ(r′) · ∇ δH[ρ]

δρ(r′)

]

+∇ ·[√

ζ

∫dr′ ρ(r)Ω1/2(r − r′) · η(r′, t)

](3.56)

We can interpret the right hand side of eq (3.56) as the sum of the right hand side of eq (3.22)(dynamic density functional equation without hydrodynamic interaction) and the correction termby hydrodynamic interaction. The coarse-grained dynamic density functional equation will be thesimilar form (as the case without hydrodynamic interaction).

The importance of the hydrodynamic interaction is still not fully understood. There are severalsimulation works which investigate the importance of the hydrodynamic interaction. For example,Groot et al [120] compared the DPD simulations and BD simulations and concluded that thehydrodynamic interaction is important. On the other hand, Horsch et al [121] compared the MD,BD, and DPD simulations and concluded that the hydrodynamic interaction is not important.

As long as the functional Fokker-Planck equation holds and the time evolution is ergodic, thecorrect Boltzmann type equilibrium distribution is recovered. This property is independent ofthe existence of the hydrodynamic interaction. Therefore, we consider that the hydrodynamicinteraction is not so important for our dynamic density functional theory. However, we have tonotice that the ergodicity of the system is not guaranteed. For example, Zhang and Wang [122]have shown that the glass transition temperature can be very closed to the order-disorder transitiontemperature for block copolymers.7 This implies that phase separated block copolymer systemsare essentially glassy and non-ergodic. Nevertheless, the hydrodynamic interaction is not the mostimportant effect. If the system is non-ergodic, what the most important is the thermal fluctuationeffect. Without thermal fluctuation, the system will be completely trapped at a local minima andfrozen.

Thus we consider the thermal fluctuation effect is the most dominant in the kinetic pathwaysof block copolymer systems, and expect that the hydrodynamic interaction does not affect thekinetic pathways so much. (The characteristic time scale or growth exponent will be changed bythe hydrodynamic interaction, as the usual phase separation dynamics.) In this work we neglectthe hydrodynamic interaction (drop the terms which contains the Oseen tensor in (3.56)).

3.6 Dynamic Density Functional Simulation

We show the numerical simulation method and results of dynamics simulations in this section.While the used free energy functional is the same, static and dynamic simulations are different,and thus we need different algorithms to simulate dynamics. The simulator “drops”, which isoriginally developed for the static simulation, is extended to dynamics. 8

While we have derived the approximate form of the dynamic equation, eq (3.45), the dynamicsimulations is numerically difficult like the case of the static simulations. We show the algorithmsto integrate eq (3.45) in this section.

7They used the Fredrickson-Helfand [49] theory. In the Fredrickson-Helfand theory, the thermal fluctuation effectis included by the renormalization method (with the self consistent type approximation). Thus their result meansdiblock copolymers can be glass even if we consider the thermal fluctuation effect.

8The simulator “drops” is distributed as a free software on the author’s web site, under the GNU General PublicLicense (GPL):

http://www.ton.scphys.kyoto-u.ac.jp/~uneyama/drops.html (this URL will be changed in the future).

3.6. DYNAMIC DENSITY FUNCTIONAL SIMULATION 71

3.6.1 Incompressible Condition

We first consider the constraint for the system. In static simulations, there are two constraints;the (global) mass conservation and the incompressibility. In dynamic simulations, as long asusing the realistic dynamic equation which can be expressed as the continuity equation, the localmass conservation condition is automatically satisfied. Therefore there is only the incompressibleconstraint.

As before, we modify the free energy as follows to satisfy this constraint.

F [ψi]→ F [ψi] +12

∫dr κ(r)

[∑

i

ψ2i (r)− 1

](3.57)

where κ(r) is the Lagrange multiplier field which corresponds to the pressure field. κ(r) is deter-mined so that the total density field becomes 1 at each positions.

3.6.2 Numerical Method and Algorithms for Dynamic Density Func-tional Simulation

In this section, we show the numerical scheme for the integration of the dynamic equation (3.45).The most simple scheme to integrate the dynamic equation is the explicit Euler scheme.

ρi(r, t+ ∆t) = ρi(r, t) + ∆t[ψi(r, t)∇2µi(r, t)− µi(r, t)∇2ψi(r, t)

+∇ ·[ψi(r, t)

√4β−1wi(r, t)

] ] (3.58)

where ∆t is the size of the time step. The density field ρi(r, t) is defined only on the discretetime-space, just like the static simulation case. While the explicit Euler scheme (eq (3.58)) is easyto implement, it is unstable for large ∆t and thus numerically inefficient in most cases. (This issimilar to the case of the static simulations.)

In analogy to the static simulation schemes, we expect that the use of an implicit schemeimproves the numerical stability and the efficiency. To use the implicit scheme, we notice that theright hand side of eq (3.45) contains the Laplacian (or the diffusion) term.

ψi(r)∇2µi(r)− µi(r)∇2ψi(r)

= ψi(r)∇2[Ciψi(r) [4 lnψi(r) + 2]

]−[Ciψi(r) [4 lnψi(r) + 2]

]∇2ψi(r) + · · ·

= 4Ci[|∇ψi(r)|2 + ψi(r)∇2ψi(r)

]+ · · ·

= 2Ci∇2ρi(r) + · · ·

(3.59)

where Ci is constant determined from the free energy functional. Using eq (3.59), we can expresseq (3.45) as follows.

∂ρi(r, t)∂t

= 2Ci∇2ρi(r) + ψi(r)∇2[µi(r)− Ciψi(r) [4 lnψi(r) + 2]

]

−[µi(r)− Ciψi(r) [4 lnψi(r) + 2]

]∇2ψi(r) +∇ ·

[ψi(r)

√4β−1wi(r, t)

] (3.60)


From eq (3.60) we can construct the ADI type implicit scheme to update ρi(r, t) into ρi(r,∆t).

ρ(0)i (r) = ρi(r, t) (3.61)[1− 2∆t

3Ci

∂2

∂x2

]ρ

(1)i (r) = ρ

(0)i (r) +

∆t3

[− 2Ci

∂2

∂x2ρ

(0)i (r) + ψ

(0)i (r)∇2µ

(0)i (r)

− µ(0)i (r)∇2ψ

(0)i (r) + ξ

(0)i (r, t)

] (3.62)

[1− 2∆t

3Ci

∂2

∂y2

]ρ

(2)i (r) = ρ

(1)i (r) +

∆t3

[− 2Ci

∂2

∂y2ρ

(1)i (r) + ψ

(1)i (r)∇2µ

(1)i (r)

− µ(1)i (r)∇2ψ

(1)i (r) + ξ

(1)i (r, t)

] (3.63)

[1− 2∆t

3Ci

∂2

∂z2

]ρ

(3)i (r) = ρ

(2)i (r) +

∆t3

[− 2Ci

∂2

∂z2ρ

(2)i (r) + ψ

(2)i (r)∇2µ

(2)i (r)

− µ(2)i (r)∇2ψ

(2)i (r) + ξ

(2)i (r, t)

] (3.64)

ρi(r, t+ ∆t) = ρ(3)i (r) (3.65)

where ρ(n)i (r), µ(n)

i (r), and ξ(n)i (r) are the ψ field, the chemical potential field, and the thermal

noise field for ρ(n)i (r) at the n-th ADI step. We have to update the ψ field at the each ADI steps.

We employ the following scheme to calculate ψ(n)i (r) form ρ

(n)i (r).

ψ(n)i (r) =

0(ρ

(n)i (r) < 0

)

1(ρ

(n)i (r) > 1

)√ρ

(n)i (r) (otherwise)

(3.66)

Eq (3.66) is rather artificial but it avoids numerical difficulty associated with negative ρ(n)i (r).

µ(n)i (r) can be calculated easily by using ψ(n)

i (r) calculated by eq (3.66).The Lagrange multiplier κ(r) is also updated at each ADI steps. Because of the large thermal

noise, the incompressible condition is not always satisfied exactly.9 Therefore, we consider that it isnot important for simulations to calculate κ(r) accurately. In this work, we employ the followingrather rough approximation scheme to calculate κ(r). If we ignore the terms in the chemicalpotential except for the Lagrange multiplier term, the ADI update scheme can be approximatelywritten as

ρ(n+1)i (r) ≈ ρ(n)

i (r) +∆t3∇ ·[ρ

(n)i (r)∇κ(r)

](3.67)

From eq (3.67) we have

∑

i

ρ(n+1)i (r) ≈

∑

i

ρ(n)i (r) +

∆t3∇ ·[∑

i

ρ(n)i (r)∇κ(r)

]

1∆t/3

[∑

i

ρ(n+1)i (r)−

∑

i

ρ(n)i (r)

]≈ ∇2κ(r)−∇ ·

[[1−

∑

i

ρ(n)i (r)

]∇κ(r)

] (3.68)

If we assume that the incompressible condition is approximately satisfied for ρ(n)(r), we can write|1−∑i ρ

(n)(r)| 1. Using this condition and the constraint∑i ρ

(n+1)i (r) = 1 finally we have

∇2κ(r) ≈ 1(∆t/3)

[1−

∑

i

ρ(n)i (r)

](3.69)

9This is rather natural if we consider the case of the particle simulations. In particle simulations, the densityfields are strongly fluctuating and the incompressible is generally not satisfied. The incompressible condition issatisfied only for the some kind of averaged density fields.


In this work we calculate κ(r) from eq (3.69) with FFTW3 [71], just like the calculation of thelong range interaction terms.

3.6.3 Generation Scheme for Thermal Noise Field

The Gaussian white random noise is generated by the Mersenne Twister [123] and the standardBox-Muller method [72]. The thermal noise field generation method has been proposed by vanVlimmaren and Fraaije [124] for general mobility model. In this work, to make the form of thethermal noise consistent with the dynamic equation, we do not use the original van Vlimmaren-Fraaije method directly but derive a modified version of the noise generation method.

We recall that our dynamic equation, eq (3.45), is expressed by using the ψ-field. We write thethermal noise term in eq (3.45) as ξi(r, t).

ξi(r, t) ≡ ∇ ·[ψi(r)

√4β−1wi(r, t)

](3.70)

We derive the numerical scheme for generating random noise field which satisfies the fluctuationdissipation relation (eqs (3.40) and (3.41)). The fluctuation dissipation relation for ξi(r, t) can bewritten as follows.

〈ξi(r, t)〉 = 0 (3.71)

〈ξi(r, t)ξj(r, t′)〉 = −4β−1δij∇ ·[⟨ψ2i (r, t)

⟩∇δ(r − r′)] δ(t− t′) (3.72)

To generate the noise numerically, we need the discretized version of eq (3.70). While eq(3.70) contains the gradient operator, the discretized ψ-field ψi(r, t) and the discretized noise fieldξi(r, t) is defined only on the lattice points. As before, we express the position of the lattice pointas r = nxhx + nyhy + nzhz where hα (α = x, y, z) is the lattice vector and nα is the integer. Thetime t (strictly speaking, the rescaled time t) is also discretized as t = nt∆t where nt is the integer.The Laplacian operator is replaced by the standard center finite difference operator as follows.

∇2f(r)→∑

α=x,y,z

1|hα|2 [f(r + hα)− f(r) + f(r − hα)] (3.73)

where f(r) is a discretized field defined on the lattice points. The delta functions are replaced asfollows

δ(r − r′)→ δrr′

|hx||hy||hz| (3.74)

δ(t− t′)→ δtt′

∆t(3.75)

where δrr′ = δnxn′xδnyn′yδnzn′z and δtt = δntn′t .The fluctuation dissipation relation for the second order moment (eq (3.72)) can be rewritten

as

〈ξi(r, t)ξj(r′, t′)〉 = −4β−1δij∇ ·[⟨ψ2i (r)

⟩∇δ(r − r′)] δ(t− t′)

= −4β−1δij∇ ·[⟨ψ2i (r)∇ψi(r)δ(r − r′)

ψi(r)

⟩]δ(t− t′)

= −4β−1δij[⟨ψi(r)∇2 [ψi(r)δ(r − r′)]⟩− ⟨[ψi(r)δ(r − r′)]∇2ψi(r)

⟩]δ(t− t′)

(3.76)

Using eqs (3.73) - (3.75), we get the discretized version of the fluctuation dissipation relation (eq(3.76)).

〈ξi(r, t)ξj(r′, t′)〉 ≈ − 4β−1δij|hx||hy||hz|∆t

∑α=x,y,z

1|hα|2

[〈ψi(r + hα)ψi(r)〉 [δ(r+hα)r′ − δrr′

]

− 〈ψi(r)ψi(r − hα)〉 [δrr′ − δ(r−hα)r′] ]δtt′

(3.77)


In this work, we employ the following discretized equation to generate the noise field.

ξi(r, t) ≈ 2√β−1

∑α=x,y,z

1|hα|

[√ψi(r + hα)ψi(r) ω(α)

i (r + hα/2, t)

−√ψi(r)ψi(r − hα) ω(α)

i (r − hα/2, t)] (3.78)

where ω(α)i (r, t) is the α element of ωi(r, t). ωi(r, t) is the discretized Gaussian white noise field

which satisfies

〈ωi(r, t)〉 = 0 (3.79)

〈ωi(r, t)ωj(r′, t′)〉 =δijδrr′δtt′

|hx||hy||hz|∆t1 (3.80)

Notice that ωi(r, t) is defined on the staggered lattice. The position of the staggered lattice pointis r = (nx + 1/2)hx + nyhy + nzhz, nxhx + (ny + 1/2)hy + nzhz, nxhx + nyhy + (nz + 1/2)hzwhere nα is the integer. It is easy to show that the noise field generated by eq (3.78) satisfies eq(3.77).

Eq (3.78) can be obtained by replacing the discrete nabla operator as

∇f(r)→∑

α=x,y,z

1|hα| [f(r + hα/2)− f(r − hα/2)]

hα|hα| (3.81)

and defining the value of function on the staggered lattice as

f(r + hα/2)→√f(r)f(r + hα) (3.82)

Thus we can directly interpret that eq (3.78) as the discrete version of eq (3.70). (However, weshould remember that such a discrete version of the nabla operator is not always valid.)

The noise generation scheme is finally written as follows.

1. Generate the Gaussian white normal distribution vector noise field ωi(r, t) on the staggeredlattice points. The noise is generated by using the Mersenne twister pseudo-random numbergenerator [123] and the standard Box-Muller method [72].

2. Calculate the noise field ξi(r, t) from ψi(r) and ωi(r, t) by using eq (3.78).

The scheme shown here is numerically efficient because we do not need to use the numericallyinefficient gradient or divergence operators). What we need to use is only the simple operatorswhich can be implemented easily.

3.6.4 Results of Dynamic Density Functional Simulation

In this section, we perform dynamics simulations for amphiphilic AB diblock copolymer solutionsand show the results. We also consider the case of the blend of hydrophobic AB diblock copolymerand C homopolymer. As the case of the static simulations, all the simulations are performed inthree dimensional systems with the periodic boundary condition, and the segment size is set tob = 1. The screening length for the long range interaction kernel (λp in eq (2.170)) are set so thatthe microphase separation structures are correctly reproduced. It is known empirically that theresults are qualitatively not so sensitive to the value of λAB [58].

Unlike the statics simulations, the dynamics simulations employ the dynamic density functionalequation which is based on the real dynamics. Thus we do not use any additional artificial schemessuch as one used for the small volume fraction and/or the small χ parameter cases in staticsimulations.


Micelle and Vesicle Formation Dynamics

First we perform simulations for the structure formation dynamics from homogeneous initial statefor amphiphilic diblock copolymer solutions (mixtures of AB diblock copolymers and S solvents).As shown in static simulations, we can control the resulting morphologies by controlling the χparameters. Here we vary χBS , the χ parameter between the hydrophobic B subchain and thesolvent.

The parameters for the simulation are as follows: the volume fractions ρAB = 0.2, ρS = 0.8, thedegrees of polymerization NAB = 10, NS = 1, the screening length for the AB diblock copolymersλAB = 2.5, the block ratios fA = 1/3, fB = 2/3, and the χ parameters χAB = 2.5, χAS =−0.5, χBS = 2.5, 3, 5, the effective temperature β−1 = 0.0390625, and the time step ∆t = 0.0025.The initial condition is the homogeneous state and the system size is 32b × 32b × 32b and thenumber of the lattice points is 64× 64× 64.

The snapshots of the simulations are shown in Figures 3.2-3.4. We can observe that sphericalmicelles (Figure 3.2), cylindrical micelles (Figure 3.3), or vesicles / bilayers (Figure 3.4) are formedspontaneously from the homogeneous initial state.

For the small χ parameter (χBS = 2.5), spherical micelles are formed. In this case, smallmicelles are formed rapidly from the homogeneous initial state, then they collide and aggregateinto larger micelles. The shapes of micelles are nearly spherical. If micelles grow into its equilibriumsize, they do not grow any more. From snapshots, we can observe that characteristic size of sphericalmicelles are nearly the constant about from t = 312.5 (Figure 3.2(c)) to t = 3125 (Figure 3.2(f))Thus we consider that the formation process of the spherical micelles are relatively fast process.

For the intermediate value of χBS (χBS = 3), cylindrical micelles are formed. The cylindri-cal micelle formation process is similar to the spherical micelle formation process. At the firststage, small spherical structures are formed rapidly and then grow into larger spherical micelles bycollision. In this case, spherical micelles further grow into cylindrical micelles by collision. The for-mation process of the cylindrical micelles are relatively slower than the spherical micelle formationprocess.

Vesicles and bilayers (not closed bilayers) are formed for the large χ parameter case (χBS = 5).The structural formation process for this case is the most complicated in three processes obtainedhere. First small spherical micellar structures are formed rapidly from the initial homogeneousstate. Then micellar structures grow into spherical micelles and cylindrical micelles, by collisionand aggregation. The cylindrical micelles grow further into open bilayer like micelles. Finally someopen bilayers close spontaneously to form vesicles (some open bilayers still remain). The first stageis similar to the case of spherical micelles or cylindrical micelles (Figures 3.4(a) and (b)). At theintermediate stage, cylindrical micelles and open bilayer micelles are formed (Figures 3.4(c)-(e)).The final closure process is observed in Figures 3.4(e)-(g). The closure process takes very long time(about 3000 in dimensionless simulation time unit).

This vesicle formation process is qualitatively the same as ones observed in previous particlemodel simulations [23, 27, 28]. Since the vesicle formation process consist of several stages, thisprocess is very slow compared with the spherical micelle formation process or the cylindrical micelleformation process.

There are common mechanism among the three structure formation processes. The first stageof structure formation processes are almost the same for all the cases. Small spherical micellarstructures are formed rapidly from the homogeneous initial state. This is somehow similar tothe spinodal decomposition in usual macrophase separation dynamics. At the next stage, smallstructures collide and aggregate into larger spherical micelles. So far, the three processes arecommon, but for small χBS (χBS = 2.5), the structure formation process stops here. For othertwo cases spherical micelles grow into cylindrical micelles, and for intermediate χBS (χBS = 3), theformation process stops at this stage. Only for large χBS (χBS = 5), cylindrical micelles furthergrow into open bilayer structures. Some open bilayers structures spontaneously close and formvesicles finally (this process takes long time).


Figure 3.2: Snapshots of a spherical micelle formation dynamics simulation for the AB diblockcopolymer solution for χBS = 2.5. Gray surfaces show the isosurfaces for ρB(r) = 0.5. (a)t = 6.25, (b) t = 62.5, (c) t = 312.5, (d) t = 625, (e) t = 1562.5, and (f) t = 3125.


Figure 3.3: Snapshots of a cylindrical micelle formation dynamics simulation for the AB diblockcopolymer solution for χBS = 3. Gray surfaces show the isosurfaces for ρB(r) = 0.5. (a) t = 6.25,(b) t = 62.5, (c) t = 312.5, (d) t = 625, (e) t = 1562.5, and (f) t = 3125.


Figure 3.4: Snapshots of a vesicle formation dynamics simulation for the AB diblock copolymersolution for χBS = 5. Gray surfaces show the isosurfaces for ρB(r) = 0.5. (a) t = 6.25, (b)t = 62.5, (c) t = 312.5, (d) t = 625, (e) t = 1562.5,(f) t = 3125, (g) t = 4687.5, and (h) t = 6250.


Morphological Transition Dynamics of Micellar Structures

Next we perform morphological transition dynamics simulations. In experiments [20, 81, 82, 89],it is shown that the morphologies of micellar structures can be controlled thermodynamically, bychanging the solvent quality. Strictly speaking, what changed in experiments are volume fractionsof mixed solvents (such as dioxiane / water mixtures or DMF / water mixtures). But for simplicitywe use one component solvents and mimic the solvent quality change by changing the χ parameter.

To simulate the morphological transition dynamics, we use the final states of the previousstructure formation dynamics (t = 3125 for χBS = 2.5 and 3, and t = 6250 for χBS = 5) as theinitial states of the simulations.

The results are shown in Figures 3.5-3.10. Several morphological transition processes are similarto the structural formation dynamics. The transition from spherical micelles to cylindrical micelles(Figure 3.5), from spherical micelles to vesicles / bilayers (Figure 3.6), and from cylindrical mi-celles to bilayers (Figure 3.8) are quite similar to the morphological transition dynamics from thehomogeneous state. On the other hand, the transitions from cylindrical micelles to spherical mi-celles (Figure 3.7), from vesicles to spherical micelles (Figure 3.9), and from vesicles to cylindricalmicelles (Figure 3.10) are not similar to the formation dynamics from the homogeneous state.

In the former case, the initial micellar structures grow by collision, and the process is relativelyslow. In the latter case, the initial micellar structures become unstable and fracture into smallerpieces. This process is relatively fast.

Onion Formation Dynamics

At the end of this section, we show the result of the simulation for AB diblock copolymer / Chomopolymer blends. In this system, the onion structures are formed instead of the micelles orvesicles. The result is shown in Figure 3.11.

The parameters for the simulation are as follows: the volume fractions ρAB = 0.2, ρC = 0.8, thedegrees of polymerization NAB = 10, NC = 10, the screening length for the AB diblock copolymersλAB = 2.5, the block ratios fA = fB = 0.5, and the χ parameters χAB = 1.75, χCA = 0.5, χBC =2.5, the effective temperature β−1 = 0.0390625, and the time step ∆t = 0.0025. The initialcondition is the homogeneous state and the system size is 24b× 24b× 24b and the number of thelattice points is 48×48×48 (the system size is smaller than simulations for AB diblock copolymersolutions, but sufficient for the onion formation dynamics in this system).

At the early stage of the structural formation dynamics (Figure 3.11(a) and (b)) the snapshotsseem similar to ones for the vesicle formation dynamics. However, in this case, cylindrical structuresor bilayer structures are not formed. The structures are always rather spherical and like droplets inmacrophase separation dynamics. At the late stage, structures grow by collision and rearrangementof microphase separation structures inside droplets. This is qualitatively different from the vesicleformation dynamics.

3.6.5 Discussion

We have performed various dynamics simulations for amphiphilic AB diblock copolymer solutionsystems in the previous section. In this section, we discuss about the simulation results andcompare them with other simulation studies or experiments.

We consider the mechanisms of the micellar structure formation process or the morphologicaltransition process. We also consider the mechanism of the onion formation process and discuss thedifference between the vesicle formation process and the onion formation process.

Vesicle Formation Mechanism

The vesicle formation mechanism in lipid systems have been studied theoretically [125, 126] andexperimentally [127–130], and the mechanism of the vesicle formation has been understood grad-ually.

The scenario is as follows [126]. First, the small flake like (or small disk like) common bilayerstructures are formed. The common structures are prepared by, for example, sonification or growth


Figure 3.5: Snapshots of a morphological transition dynamics simulation (from spherical micellesto cylindrical micelles) for the AB diblock copolymer solution. Gray surfaces show the isosurfacesfor ρB(r) = 0.5. (a) t = 3131.25, (b) t = 3187.5, (c) t = 3437.5, (d) t = 3750, (e) t = 4687.5, and(f) t = 6250.


Figure 3.6: Snapshots of a morphological transition dynamics simulation (from spherical micellesto bilayers and vesicles) for the AB diblock copolymer solution. Gray surfaces show the isosurfacesfor ρB(r) = 0.5. (a) t = 3131.25, (b) t = 3187.5, (c) t = 3437.5, (d) t = 3750, (e) t = 4687.5, and(f) t = 6250.


Figure 3.7: Snapshots of a morphological transition dynamics simulation (from cylindrical micellesto spherical micelles) for the AB diblock copolymer solution. Gray surfaces show the isosurfacesfor ρB(r) = 0.5. (a) t = 3131.25, (b) t = 3187.5, (c) t = 3437.5, (d) t = 3750, (e) t = 4687.5, and(f) t = 6250.


Figure 3.8: Snapshots of a morphological transition dynamics simulation (from cylindrical micellesto bilayers) for the AB diblock copolymer solution. Gray surfaces show the isosurfaces for ρB(r) =0.5. (a) t = 3131.25, (b) t = 3187.5, (c) t = 3437.5, (d) t = 3750, (e) t = 4687.5, and (f) t = 6250.


Figure 3.9: Snapshots of a morphological transition dynamics simulation (from vesicles to sphericalmicelles) for the AB diblock copolymer solution. Gray surfaces show the isosurfaces for ρB(r) = 0.5.(a) t = 6256.25, (b) t = 6312.5, (c) t = 6562.5, (d) t = 6875, (e) t = 7812.5, and (f) t = 9375.


Figure 3.10: Snapshots of a morphological transition dynamics simulation (from vesicles to cylin-drical micelles) for the AB diblock copolymer solution. Gray surfaces show the isosurfaces forρB(r) = 0.5. (a) t = 6256.25, (b) t = 6312.5, (c) t = 6562.5, (d) t = 6875, (e) t = 7812.5, and (f)t = 9375.


Figure 3.11: Snapshots of an onion formation dynamics simulation for the AB diblock copolymer/ C homopolymer blend. Dark transparent gray and light gray surfaces show the isosurfaces forρA(r) = 0.5 and ρB(r) = 0.5, respectively. (a) t = 6.25, (b) t = 62.5, (c) t = 312.5, (d) t = 625,(e) t = 1562.5, and (f) t = 3125.


induced by the solvent quality change. Second, the small common structures collide and fuse toform larger disk like structures. Finally grown large disk like structures close spontaneously [125]and thus vesicles are formed.

In our simulations, the similar mechanism is observed. First, small spherical micelles areformed. Then spherical micelles grow into cylindrical micelles by the collision and coalescence typemechanism. The cylindrical micelles further grow into bilayer sheet like micelles, and finally theyclose spontaneously. Here we call such a mechanism as the “mechanism I”. The mechanism I isalso supported by various simulations such as the MD [23,24]10, the BD [28], the DPD [27], or theMonte Carlo [32,33] simulations.11

It is known that, however, there is another vesicle formation mechanism observed in simulations[34,79,107]. In the static SCF, the static DF, and the EPD simulations, the vesicles are formed asfollows. First, small spherical micelles appears. These spherical micelles are similar to ones in themechanism I, but they does not grow into cylindrical micelles. The spherical micelles grow intolarger spherical micellar structures instead, by the evaporation and condensation type mechanism.If the micelles grow sufficiently large multilayer structures, they take solvents inside them becausethey are energetically unfavorable. By taking solvents inside, the large spherical micelles finallychange into vesicles. We call this mechanism as the “mechanism II”.

The EPD, which is proposed to simulate the dynamics with the non-local mobility, reproducesmechanism II, not mechanism I. One may feel this is strange because the mechanism II is mainlyobserved in static simulations. We discuss why the EPD reproduce the mechanism II in Appendix3.B.

(a) Mechanism I

(b) Mechanism II

Figure 3.12: Schematic draw of two vesicle formation mechanisms. Grey and Black colors representhydrophilic A subchain and hydrophobic B subchain, respectively. (a) the mechanism I, and (b)the mechanism II.

The schematic draw of two vesicle formation mechanisms are shown in Figure 3.12. The mecha-nism I is considered to be a physically natural mechanism. Unlike the mechanism I, the mechanismII is physically not natural. In fact, it is observed only in static simulations or dynamic simulationsbased on the unrealistic assumption. Especially considering that the mechanism II is not supportedby realistic dynamics simulations, we conclude that the mechanism I is the real vesicle formationmechanism.

It should be noted here that our dynamic DF model supports mechanism I while other simula-tions based on continuum field models do not support mechanism I. This implies that the continuum

10The bilayer formation process is also studied in detail by the MD simulation [131].11Strictly speaking, the Monte Carlo simulations based on the Metropolis algorithm [32, 33] are “not” dynamics

simulations, so one may not consider the Monte Carlo simulations supports the mechanism I. However, it is empiri-cally known that the Monte Carlo simulations sometimes give qualitatively appropriate kinetic pathway (of course,this is not always true). Thus here we consider they supports the mechanism I.


field models except for our model cannot reproduce physically natural results and therefore theyare inconsistent with other simulation methods or experiments. As mentioned, previous continuumfield models ignore several physically important properties. Our dynamic DF model reproduce thecorrect result is because we consider these properties at least qualitatively correctly.

Morphological Transition Mechanism

The morphological transition mechanism is roughly categorized into to groups. One is the casewhen we increase the χ parameter between the hydrophobic subchain and the solvent. In thiscase, the resulting mechanism is similar to the mechanism I. The initial micellar structures growby the collision and the coalescence. The spherical micelles grow into cylindrical micelles, and thecylindrical micelles grow into bilayers or vesicles.

Another is the case when we decrease the χ parameter between the hydrophobic subchain andthe solvent. In this case, initial micellar structures become unstable. The fluctuations of thestructures grow and finally the initial structures are broken into smaller pieces. This process isconsidered to be a sort of the fracture type or the rupture type mechanisms.

(a) Collision-Coalescence

(b) Fracture / Rupture

Figure 3.13: Schematic draw of morphological transition mechanisms. Grey and Black colorsrepresent hydrophilic A subchain and hydrophobic B subchain, respectively. (a) the collision-coalescence type mechanism which is observed in the cases of the spheres to cylinders, spheres tobilayers/vesicles, or cylinders to bilayers/vesicles. (b) the fracture or the rupture type mechanismwhich is observed in the cases of the cylinders to spheres, vesicles to spheres, or vesicles to cylinders.

Onion Formation Mechanism

Here we consider about the similarity and difference between the vesicle formation dynamics andthe onion formation dynamics. We have shown that onion structures are formed in the AB diblockcopolymer / C homopolymer blend system. More generally, we have onions in the case where bothA and B monomers are hydrophobic [132,133].

In such systems, first small spherical micelle like structures are formed. Then small structuresaggregate and grow into larger structures. As the size of structures become large, they bent andfinally form closed multilayer onion structures.

One may consider this is the mechanism I. However, there are severe difference. Because bothsubchains are hydrophobic, there are no solvent rich cavity inside onions. This means that onionsare essentially the diblock copolymer rich droplets, and multilayer structures are just the sameas the usual microphase separation structure (although they may be slightly swollen by solvents).Thus we should distinguish the onion formation mechanism from the vesicle formation mechanism.Here we call such a mechanism as the “mechanism III”. We show the schematic draw of themechanism III in Figure 3.14.


Mechanism III

Figure 3.14: Schematic draw of the onion formation mechanism. Grey and Black colors representthe A and B subchains, respectively. the B subchain is strongly hydrophobic while the A subchainis not so hydrophobic.

If we look only the much hydrophobic B subchain, the onion formation mechanism is similarto the mechanism I. But what important in micellar systems is that the diblock copolymers areamphiphilic. So we have to look the A subchain. If we look only the A subchain (in the onionformation case it is hydrophobic, while in the vesicle formation case it is hydrophilic), the differenceis rather clear. It should be also noted that the size of the monolayer onions are almost the samewhile the size of the monolayer vesicles are not the same (the vesicle size strongly depends on thekinetics). This also implies that onions are not formed by the vesicle formation type mechanism(the mechanism I). We will consider this more detail in the next chapter.

Magnitude of Thermal Noise (Effective Temperature)

To reproduce the structural formation dynamics or morphological transition dynamics in micellarsystems, the effect of the thermal noise is essential. However, currently there are no standardmethod to calculate the the thermal noise correctly in continuum field simulations. We haveintroduced the dimensionless effective temperature β−1 to represent the effective magnitude of thethermal noise, based on the rather rough physical argument.

Here we consider the value of β−1 used in our simulations. As discussed, the effective tem-perature is directly related to the level of the coarse-graining. Unfortunately we do not know theexplicit form of F for polymer systems, but we know the free energy functional F . Thus we wantto approximate F as F ≈ F . Here we estimate the microscopic time scale τ0, the coarse grainedtime scale τ , and the effective temperature, which is appropriate for the approximation F ≈ F .

Consider the field is splitted into cells, of which edge length is b0. We can split a polymer chaininto segments, of which size is b0 (Figure 3.15(a) and (b)). We assume that each cells are in localequilibrium state after the Rouse relaxation time [4] of the segment, τR = ζ0b

20/3π

2kBT (ζ0 is thefriction coefficient for the segment). We cannot observe dynamics at any time scales smaller thanthis time scale on the cells, and thus this Rouse relaxation time is considered to be the microscopictime scale, so we have τ0 ≈ τR.

The coarse grained time scale τ should be determined to justify the approximation F ≈ F .Since the effective free energy F in eq (3.45) is used only for the calculation of the thermodynamicforce (the gradient of the chemical potential field, ∇(δF /δρ(r))), we expect that the approximationF ≈ F is justified if the thermodynamic force is approximated well by the thermodynamic forcefor the equilibrium state. The thermodynamic force between cells are calculated as the differenceof the chemical potential between the cells. Thus if the neighboring two cells (or a supercell) arein local equilibrium, we can replace the effective free energy by the equilibrium free energy. Theneighboring two cells are expected to be in local equilibrium state after the Rouse relaxation timefor the partial chain which contains several segments (Figure 3.15(c)). In this work we employ theRouse relaxation time for the partial chain of which average size is 2b0 (this partial chain contains4 segments) as the coarse grained time scale τ , thus we have τ ≈ ζ0b2042/3π2kBT = 16τR.

Now the factor for the effective temperature of the noise for the coarse grained continuumdensity field (Figure 3.15(d)) is estimated as

β−1 =τ0τ≈ 1

16= 0.0625 (3.83)


It is noted that β−1 is independent of the segment size (or the cell edge length) b0. Althoughthe value of the β−1 estimated here is rough, nevertheless it is a good reference value for us todetermine the value of β−1. In our simulations we employed β−1 = 0.0390625. This value is of theorder of 0.0625 and thus our value of β−1 is considered to be appropriate. Of course the estimationperformed here is quite rough, but we believe that the order of the effective temperature is not sononsense.

∆t ~ τ0

∆t ~ τ

(c)

(b)(a)

(d)

∆t ~ τ

Figure 3.15: Schematic image of the coarse graining for the time scale in diblock copolymers.The square mesh represents the grid size used in simulations. (a) Gaussian polymer chain (blackand gray curves represent A and B subchains, respectively), (b) coarse-grained polymer chain atthe time scale ∆t ∼ τ0 (black and gray circles represent A and B segments), (c) more coarse-grained polymer chain at the time scale ∆t ∼ τ0, and (d) continuum field representation for thecoarse-grained polymer chains.

Comparison with Experiments

Next we discuss about the relation between the simulation results and the experiments. We usedmany parameters for the simulations and they should be related to the real experimental parame-ters. Unfortunately, because the accuracy of our theory is not high and many approximations areinvolved, it is difficult to calculate the real parameters from our simulation parameters.

The Flory-Huggins χ parameters (or the product of the χ parameter and the degree of poly-merization, χN) are estimated in the previous chapter. We consider that the χ parameters used inour simulations are qualitatively not bad, but it is difficult to compare them with the experimentalχ parameters.

Other interesting properties are the size of the vesicles, the content of solvents inside vesicles,or the higher order structures. These properties cannot be discussed from our current results,since the system size is not large. We have only two vesicles in the simulation box (Figures 3.4)and there may be the finite size effect. Nevertheless, we consider our model and simulation resultsare still valuable. These properties of vesicles should be observed if we perform the simulations


for large systems while it is too difficult to perform large scale simulations by the current models,algorithms and CPU power.

The characteristic time scales are also interesting and important property. As mentioned above,the real time scale can be estimated from the diffusion data. Here we estimate the time scale byusing the empirical equation for diffusion coefficient of short polystyrene melts (at Tg + 125C) byWatanabe and Kotaka [134].

D ' 6.3× 10−5M−1cm2/s (3.84)

where M is the molecular weight.We have to determine the number of monomers in one segment used in our simulations. Since

the polymerization index of typical amphiphilic block copolymers used in experiments [82, 89] areabout M ' 50 ∼ 1000, we consider that each segment used in simulations contains about 10monomer units. In our simulations, we used dimensionless segment size (b = 1). Thus we alsoneed the size of the segment. The segment size of the styrene monomer is about 7A (it canbe calculated from the experimental data of the radius of gyration [3, 135]). So if we assumethe Gaussian statistics for monomers in the segment, size of the segment can be expressed asb0 '

√10× 7A.

Now we can estimate the characteristic time scale for the system. If we assume that thepolymer chains are not entangled and obey the Rouse dynamics, the characteristic time scale τcan be calculated as follows.

τ =2ζb20kBT

=2b20ND

(3.85)

b0 is the size of the segment and N is the number of segments in one polymer chain (polymerizationindex), and D is the diffusion coefficient. N and M can be related as M = 104× 10×N and thuswe have

τ ' 2× 1040× (√10× 7× 10−10m)2

6.5× 10−9m2/s' 1.6× 10−6s (3.86)

Thus we know that the characteristic time scale in our simulations are about 1 microsecond, andthe time scale of the vesicle formation is about 10 milliseconds. Of course the time scale estimatedhere is quite rough and cannot be compared with real experiments directly, but the time scale ofour simulation is considered to be much smaller than one of experiments. The possible reasons arethat the accuracy of the dynamic equation used for the simulation is not so good, and that thesizes of the formed vesicles are small since the system size is small.

Generality of the Vesicle Formation Mechanism

At the end of this section, we briefly comment on the similar structural formation mechanism isfound in nano-scale systems. Irle [136] and coworkers studied fullerene formation dynamics byusing the DFTB (density functional tight binding) based QM/MD hybrid simulation method, andreproduced the fullerene formation dynamics after the temperature quench. Fullerenes are shell-like molecules composed of carbon atoms. Although the characteristic scale of vesicles and one offullerenes are quite differ (fullerenes are of the order of nm), their structures look similar. In theirsimulations, first graphite like small structures grows by collision. The graphite like structureshave legs, and the legs catches carbon dimers or origomers. Because the graphite like structuresare bent (they have non-zero spontaneous curvature), finally closed structures are formed. Afterthe structures are closed, excess legs or carbons are cut and thrown away.

This fullerene formation mechanism is similar to our vesicle formation process, the mechanismI. While there are several differences (for example, in our vesicle formation process, very long leglike structures are not observed), it is interesting that there are such similar mechanisms in differentscale, in the nature. There are other interesting nano structures formed by carbon atom such ascarbon nanotubes or carbon nanohorns. The studies on nano carbon structures may be useful tounderstand mesoscale amphiphilic block copolymer systems, and vice versa.


3.7 Summary

In this chapter, we have derived the dynamic density functional theory for block copolymer systemsand have performed dynamics simulations for block copolymer micellar systems.

By using the dynamic density functional theory for colloidal particles and by employing thetemporal coarse-graining method, we derived the dynamic density functional equation. The dy-namic density functional equation can be expressed by using the psi-field, and it makes numericalsimulations easily. Unlike previous continuum field models for dynamics, our theory is based onthe microscopic dynamics and includes the thermal fluctuation effect correctly.

We have succeeded in reproducing the structural formation dynamics or morphological transi-tion dynamics in amphiphilic diblock copolymer solutions. Our simulation results are physicallynatural and consistent with previous particle model simulations. From the dynamics DF simu-lation results we discussed about the mechanisms of the structure formation dynamics and themorphological transition dynamics. We also discussed about the onion formation dynamics.

It is worth noted that so far our dynamic DF simulation method is only the continuum fieldmethod which can reproduce the correct dynamics in micellar systems.

Appendix

3.A Dynamic Density Functional Theory for Rouse Chains

In this appendix, we derive the dynamic density functional theory from the Rouse model [4]. Whilethe dynamic density functional theory shown in this appendix is formally exact (as long as theRouse model is appropriate), it is practically useless especially for numerical simulations (becauseit is impossible to solve it numerically).

It will be possible to derive the dynamic density functional used in main sections from thedynamic density functional theory shown in this appendix with some approximations, but thereare many difficulties (we show them and discuss later) and it will be the future work.

3.A.1 Rouse Model for Single Polymer Chain

We first consider one Rouse chain. The state of the system is described by the conformation of aRouse chain R(s) (0 ≤ s ≤ N).

Here we describe the Rouse model [4] briefly. We start from the following Hamiltonian for oneGaussian chain.

H[R] =3kBT2b2

∫ds

∣∣∣∣∂R(s)∂s

∣∣∣∣2

(3.87)

The dynamic equation for the conformation of the chain is described by the following overdampedLangevin equation.

ζ∂R(s, t)∂t

= −∂H[R]∂R(s)

+ η(s, t)

=3kBTb2

∂2R(s, t)∂s2

+ η(s, t)

(3.88)

where ζ is the friction coefficient and η(s, t) is the thermal noise which satisfies

〈η(s, t)〉 = 0 (3.89)〈η(s, t)η(s′, t′)〉 = 2ζkBTδ(s− s′)δ(t− t′)1 (3.90)

The boundary condition for R(s, t) is given as follows.

∂R(s, t)∂s

∣∣∣∣s=0

=∂R(s, t)∂s

∣∣∣∣s=N

= 0 (3.91)

3.A. DYNAMIC DENSITY FUNCTIONAL THEORY FOR ROUSE CHAINS 93

Since eq (3.88) is linear in R(s, t), it is convenient to decompose it into Fourier modes (Rousemodes).

Xj(t) ≡ 1N

∫ N

0

dsR(s, t) cos(sjπ

N

)(j = 0, 1, 2, . . . ) (3.92)

ζ∂Xj(t)∂t

= −3π2j2kBT

N2b2Xj(t) + Yj(t) (3.93)

where Yj(t) is the Fourier transform of the noise η(s, t).

Yj(t) ≡ 1N

∫ N

0

dsη(s, t) cos(sjπ

N

)(j = 0, 1, 2, . . . ) (3.94)

The fluctuation-dissipation relation for Yj(t) becomes as follows.

〈Yj(t)〉 = 0 (3.95)

〈Yj(t)Yk(t′)〉 =

2ζkBTδjkδ(t− t′)1 (j = 0)ζkBTδjkδ(t− t′)1 (j 6= 0)

(3.96)

From eq (3.93) the relaxation time for the j-th mode is expressed as

τj =ζN2b2

3π2j2kBT=τRj2

(3.97)

where τR ≡ τ1 = ζN2b2/3π2kBT is the Rouse time. From eq (3.97) the relaxation times have verywide distribution. It will be clear that it is quite difficult to reproduce such relaxation modes onlyfrom the dynamic density functional equation which is closed in the density field.

Here we try to derive the dynamic density functional equation for the Rouse model. Themicroscopic density can be defined as follows.

ρ(r, t) ≡∫ds δ(r −R(s, t)) (3.98)

The dynamic equation can be written as follows.

∂ρ(r, t)∂t

=∫ds

∂

∂tδ(r −R(s))

= −∇ ·[∫

dsδ(r −R(s))

ζ

[3kBTb2

∂2R(s, t)∂s2

∣∣∣∣R(s)=r

+ η(s, t)

]]

+kBT

ζ∇2

∫ds δ(r −R(s))

=kBT

ζ∇2ρ(r)−∇ ·

[∫dsδ(r −R(s))

ζ

3kBTb2

∂2R(s, t)∂s2

∣∣∣∣R(s)=r

]

+∇ ·[√

ρ(r)η(r, t)]

(3.99)

Unlike the case of colloid particles, the dynamic equation is not closed in ρ(r, t). Besides, theexpression of the interaction term is very complicated and thus the approximation for the dynamicdensity functional equation will be difficult. Thus we know that it will be quite difficult to constructthe dynamic density functional model which reproduces the Rouse type viscoelastic behavior.12

This is as expected from the distribution of the relaxation times (eq (3.97)).What we can calculate exactly in this case is the dynamic equation for the microscopic distri-

bution function defined via the following equation.

Ψ[R](t) ≡ δ[R(s)−R(s, t)] (3.100)12There are some attempts to derive the dynamic equations from the Rouse model to using the projection operator

method [137,138].


From eq (3.100), we have the following dynamic density functional equation.

∂Ψ[R](t)∂t

= −∫ds

δ

δR(s)·[δ[R−R]

ζ

∂R(s)∂t

∣∣∣∣R=R

]

+12

∫dsds′

δ

δR(s)δ

δR(s′):[δ[R−R]

2kBTζ

δ(s− s′)1]

=∫ds

δ

δR(s)·[kBT

ζ

δΨ[R]δR(s)

− 3kBTζb2

∂2R(s)∂s2

Ψ[R]

]

−∫ds

δ

δR(s)·[

Ψ[R]ζη(s, t)

]

(3.101)

while eq (3.101) is exact and closed in Ψ[R], it is practically not useful especially for simulations,because it is the distribution in the function space.

∂Ψ[R](t)∂t

=∫ds

δ

δR(s)·[

Ψ[R]ζ

δH[Ψ]δR(s)

]+∫ds

δ

δR(s)·[√

2kBTζ

Ψ[R]W [R](t)

](3.102)

where H[Ψ] is the Hamiltonian as a functional of distribution functional, and W [R](t) is thethermal noise in the functional space. H[Ψ] is defined as follows

H[Ψ] =∫DR

kBT Ψ[R]

[ln Ψ[R]− 1

]+

∫ds

3kBT2b2

∣∣∣∣∣∂R(s)∂s

∣∣∣∣∣

2 Ψ[R]

(3.103)

and W [R](t) satisfies the following fluctuation dissipation relations.⟨W [R](t)

⟩= 0 (3.104)

⟨W [R](t)W [R′](t′)

⟩= δ[R− R′]δ(t− t′)1 (3.105)

The functional Fokker-Planck equation for eq (3.102) is as follows.

∂P [Ψ](t)∂t

= −∫DR δ

δΨ[R]

∫ds

δ

δR(s)·[

Ψ[R]ζ

δ

δR(s)

[kBT

δ

δΨ[R]+δH[Ψ]δΨ[R]

]P [Ψ]

](3.106)

where P [Ψ] is the average distribution functional of the functional Ψ[R].

P [Ψ] ≡⟨δ[Ψ− Ψ]

⟩(3.107)

It is clear that eq (3.106) is useless for numerical simulations because P [Ψ] is the functional of thefunctional, and it is impossible to handle it numerically.

3.A.2 Rouse Model for Interacting Many Polymer Chains

Next we consider the system consists of many interacting Rouse chains. For simplicity, we considerinteracting many homopolymer chains here. (The generalization for block copolymers or blends isstraightforward.) The Hamiltonian for the system is described as follows.

H[Ri] =3kBT2b2

∑

i

∫ds

∣∣∣∣∂Ri(s)∂s

∣∣∣∣2

+12

∑

i,j

∫dsds′ v(Ri(s)−Rj(s′)) (3.108)

3.A. DYNAMIC DENSITY FUNCTIONAL THEORY FOR ROUSE CHAINS 95

where v(r) is the interaction potential between monomers. The Langevin equation is written as

ζ∂Ri(s, t)

∂t= −∂H[Ri]

∂Ri(s)+ ηi(s, t)

=3kBTb2

∂2Ri(s, t)∂s2

− ∂

∂Rk(s)

∑

j

∫ds′ v(Ri(s)−Rj(s′)) + ηi(s, t)

(3.109)

where ηi(s, t) is the thermal noise which satisfies following fluctuation-dissipation relations.

〈ηi(s, t)〉 = 0 (3.110)〈ηi(s, t)ηj(s′, t′)〉 = 2ζkBTδijδ(s− s′)δ(t− t′)1 (3.111)

From eq (3.109) we can derive the dynamic density functional equation for the microscopic distri-bution function.

Ψ[R](t) ≡∑

i

δ[R(s)−Ri(s, t)] (3.112)

The dynamic density functional equation can be written as

∂Ψ[R](t)∂t

= −∑

i

∫ds

δ

δR(s)·[δ[R−Ri]

ζ

∂Ri(s)∂t

∣∣∣∣Ri=R

]

+12

∑

i,j

∫dsds′

δ

δR(s)δ

δR(s′):[δijδ[R−Ri]

2kBTζ

δ(s− s′)1]

=∫ds

δ

δR(s)·[kBT

ζ

δΨ[R]δR(s)

− 3kBTζb2

∂2R(s)∂s2

Ψ[R]

]

+∫ds

δ

δR(s)·[

Ψ[R]ζ

δ

δR(s)

∫DR

[∫ds v(R(s)− R′(s′))Ψ[R′]

]]

−∑

i

∫ds

δ

δR(s)·[δ[R(s)−Ri(s)]

ζηi(s, t)

]

(3.113)

We can write the dynamic density functional equation in the closed form of Ψ[R].

∂Ψ[R](t)∂t

=∫ds

δ

δR(s)·[

Ψ[R]ζ

δH[Ψ]δR(s)

]+∫ds

δ

δR(s)·[√

2kBTζ

Ψ[R]W [R](t)

](3.114)

where H[Ψ] is the Hamiltonian functional defined as follows.

H[Ψ] =∫DR

kBT Ψ[R]

[ln Ψ[R]− 1

]+

∫ds

3kBT2b2

∣∣∣∣∣∂R(s)∂s

∣∣∣∣∣

2 Ψ[R]

+∫DRDR′

[12

∫dsds′ v(R(s)− R′(s′))Ψ[R]Ψ[R′]

] (3.115)

W [R](t) is the thermal noise field which satisfies the following fluctuation dissipation relations.⟨W [R](t)

⟩= 0 (3.116)

⟨W [R](t)W [R′](t′)

⟩= δ[R− R′]δ(t− t′)1 (3.117)

Eq (3.114) is derived from eq (3.109) without any approximations. Thus eq (3.114) reproduceviscoelastic properties correctly. However, as the case of the dynamic density functional equationfor a single Rouse chain, eq (3.114) is practically useless for numerical simulations. (We may needsome approximations and coarse-grainings to get dynamic density functional which is closed in thedensity field.)


3.B External Potential Dynamics

In this appendix, we discuss about the external potential dynamics (EPD) model [95, 101]. TheEPD uses non-local mobility model, and it is known to be computationally efficient. However, forthe vesicle formation dynamics, the EPD simulations reproduce the mechanism II while our DF orother particle simulations reproduce the mechanism I. We show that the EPD dynamic equationdoes not satisfy the local mass conservation and this will cause serious problems including themechanism II in the vesicle formation case.

3.B.1 Non-Local Mobility Model and Dynamic Equation

We start from introducing the non-local mobility based on the Rouse type polymer chain motion[101,110].

Pij(r, r′) ≡∫

s∈ids

∫

s′∈jds′ 〈δ(r −R(s))δ(r′ −R(s′))〉 (3.118)

Notice that, Pij(r, r′) is generally not transitionally invariant. If we employ the non-local mobility(eq (3.118)), the TDGL equation can be written as follows.

∂ρi(r)∂t

= ∇ ·∑

j

∫dr′ Pij(r, r′)∇′F [ρi]

δρj(r′)(3.119)

For simplicity, we have dropped the thermal noise term here. Eq (3.119) is numerically not efficientbecause we have to calculate the non-local mobility Pij(r, r′) and calculate the integral over r′.Thus to use such a non-local mobility model, we need to modify the dynamic equation further.

Eq (3.118) has the same form as the correlation function. Thus we can write

δρi(r)δVj(r′)

= −Pij(r, r′) (3.120)

Using eq (3.120), we can write the time derivative of ρi(r) by using the time derivative of Vi(r).

∂ρi(r)∂t

=∑

j

∫dr′

δρi(r)δVj(r′)

∂Vj(r′)∂t

=∑

j

∫dr′ Pij(r, r′)∂Vj(r

′)∂t

(3.121)

From eqs (3.119) and (3.121), we get the following dynamic equation for Vi(r).

∑

j


′)∂t

= ∇ ·∑

j


δρj(r′)(3.122)

Both the right hand side and the left hand side of eq (3.122) contains the non-local mobilityPij(r, r′). It is expected that if we can modify the equation further and cancel out Pij(r, r′), theresulting dynamic equation will be a simple form.

To obtain a simple dynamic equation, here we assume the following commutation.

∇Pij(r, r′) ≈ −∇′Pij(r, r′) (3.123)

Eq (3.123) is correct if Pij(r, r′) is transitionally invariant, but generally eq (3.123) is just anapproximation (or an assumption). If we use eq (3.123), the right hand side of eq (3.122) can bemodified as

∇ ·∑

j


δρj(r′)≈∑

j

∫dr′ (−∇′) ·

[Pij(r, r′)∇′F [ρi]

δρj(r′)

]

= −∑

j

∫dr′ Pij(r, r′)

[∇′2F [ρi]

δρj(r′)

] (3.124)

3.B. EXTERNAL POTENTIAL DYNAMICS 97

Finally we have a simple dynamic equation, the EPD dynamic equation.

∂Vi(r)∂t

= −∇2F [ρi]δρi(r)

(3.125)

Eq (3.125) does not contain the non-local mobility Pij(r, r′), and is known to be numericallyefficient.

3.B.2 On Local Mass Conservation

He and Schmid [107, 139] employed the EPD to study the vesicle formation process in diblockcopolymer solutions. While they succeeded in obtaining vesicles, their results are inconsistent withour results or results of particle simulations. Here we consider why there are difference betweenthe EPD simulations and other simulations.

The main difference is that the EPD dynamic equation (3.125) does not satisfy the local massconservation. That is, we cannot write the EPD dynamic equation as a continuity equation.

∂ρi(r)∂t

+∇ · ji(r) = 0 (3.126)

It can be shown as follows, by using eq (3.122).

∂ρi(r)∂t

= −∑

j


′)∂t

= −∑

j

∫dr′ Pij(r, r′)

[∂Vj(r′)∂t

+∇′2F [ρi]δρj(r′)

]

−∑

j

∫dr′ [∇′Pij(r, r′)] · ∇′F [ρi]

δρj(r′)

(3.127)

If we use the EPD equation of motion (eq (3.125)), eq (3.127) can be rewritten as follows.

∂ρi(r)∂t

= −∑

j

∫dr′ [∇′Pij(r, r′)] · ∇′F [ρi]

δρj(r′)

=∑

j

∫dr′ Pij(r, r′)∇′2F [ρi]

δρj(r′)

(3.128)

It is clear that eq (3.128) does not satisfy the local mass conservation, and therefore eq (3.125)also does not satisfy the local mass conservation. By integrating eq (3.128) we find

∂

∂t

∫dr ρi(r) =

∑

j

∫dr′

[∫drPij(r, r′)

]∇′2F [ρi]

δρj(r′)

= Nfi∑

j

∫dr′

∫

s∈jds 〈δ(r′ −R(s))〉∇′2F [ρi]

δρj(r′)

= Nfi∑

j

∫dr′ ρj(r′)∇′2F [ρi]

δρj(r′)

(3.129)

where N is the polymerization degree, fi is the block ratio of the i-th subchain. From eq (3.129),in general we have

∂

∂t

∫dr ρi(r) 6= 0 (3.130)


Eq (3.129) means that eq (3.128) does not satisfy the global mass conservation. Notice that, eq(3.128) is modified if we calculate ρi(r) by using standard method for the canonical ensemble as

ρ(r, t) =

V ρ∫

s∈ids

∫DR δ(r −R(s)) exp

[−∑

j

∫

s′∈jds′ Vj(R(s′), t)

]

N

∫DR exp

[−∑

j

∫

s′∈jds′ Vj(R(s′), t)

] (3.131)

where V is the volume of the system and ρ is the spatial average density of the block copolymer.By using eq (3.131), eq (3.128) is modified as follows.

∂ρi(r)∂t

=∑

j

∫dr′ Pij(r, r′)∇′2F [ρi]

δρj(r′)− Nρi(r)

V ρ∑

j

∫dr′ ρj(r′)∇′2F [ρi]

δρj(r′)

=∑

j

∫dr′

[Pij(r, r′)− N

V ρ ρi(r)ρj(r′)]∇′2F [ρi]

δρj(r′)

(3.132)

It is easy to show that eq (3.132) satisfies the global mass conservation unlike eq (3.128).Because EPD simulations are usually performed for canonical ensemble, eq (3.132) is what is

really calculated in EPD simulations. Thus we know that the approximation introduced to derivethe EPD dynamic equation (eq (3.123)) is inconsistent with the local mass conservation.

This fact means that the dynamic simulations based on the EPD equation of motion maylead unphysical mass transport. Thus the kinetic pathway will be rather similar to the evolutionpathway of the standard static free energy minimization scheme, which also does not satisfy thelocal mass conservation. This will be the reason why the mechanism II is observed in the EPDsimulations. This is consistent with that the mechanism II is observed in static SCF simulations andstatic DF simulations (the static simulations does not satisfy the local mass conservation). Whilethe EPD dynamics or its variants [103] are numerically efficient to find equilibrium structures, wehave to be careful to use them to study dynamics.

Chapter 4

Simple Model for Vesicle andOnion Formation Kinetics inBlock Copolymer Systems

4.1 Introduction

In the previous chapter we have shown that diblock copolymers dissolved into solvents can formmicellar structures spontaneously from the homogeneous state.

The onions [132, 133] and the vesicles [34, 58, 79] are the most interesting structures in diblockcopolymer systems. While we have observed that the formation dynamics of vesicles can be re-produced by the dynamic density functional theory, the dynamic density functional model is notsimple and it is difficult to understand why vesicles are spontaneously formed.

In this chapter, we consider the kinetic pathways of the formation processes of diblock copolymervesicles and onions. We show a simple theory based on the Fromherz’s theory [125] for the vesicleformation kinetics and the onion formation kinetics. We have observed the qualitative differencesbetween the vesicle formation mechanism and the onion formation dynamics. By using the theoryintroduced in this chapter, we show these differences clearly and simply.

4.2 Simple Kinetic Model for Block Copolymer Vesicles andOnions

4.2.1 Fromherz Theory

First we introduce the Fromherz’s original theory of vesicle formation kinetics [125]. Fromherzproposed a simple theory for the vesicle formation process by assuming that the shape of a micelleis a cup-like spherical shell. Consider that there is a disk-like micelle and it deforms to form avesicle. We describe the micelle by using the curvature radius R and the angle θ, just as depictedin Figure 4.1.

Fromherz employed Helfrich type free energy.

F = F (bend) + F (edge) (4.1)

where F (bend) is Helfrich bending free energy [140] and F (edge) is the edge free energy (the linetension). For spherical shell, two principle curvature are the constant. Thus we can write

F (bend) =

[12κ

(2R

)2

+ κ

(1R

)2]A = 2κ

A

R2(4.2)

99

100 CHAPTER 4. SIMPLE MODEL FOR VESICLE AND ONION FORMATION KINETICS

radius: R

angle: θ

surface area: Aedge length: l

Figure 4.1: Parameters for a vesicle used in the Fromherz theory. We consider a cup-like micellewhich is a part of a spherical shell.

where κ, κ are splay and saddle-splay moduli and κ ≡ κ + κ/2 is the effective modulus. A is thesurface area of the micelle. The edge free energy is assumed to be the following form.

F (edge) = γl (4.3)

where γ is the line tension and l is the edge length.The surface area of the micelle A and the edge length l can be written as follows.

A = 2πR2(1− cos θ) (4.4)l = 2πR sin θ (4.5)

Eq (4.5) can be modified further by using eq (4.4).

l = 2πR√

1− cos2 θ =

√4πA

(1− A

4πR2

)(4.6)

Using eqs (4.2), (4.2) and (4.6), eq (4.1) can be rewritten as

F = 2κA

R2+ γ

√4πA

(1− A

4πR2

)

= 8πκ

[A

4πR2+

γ

2κ

√A

4π

(1− A

4πR2

)]

= 8πκ[Ω2 + VF

√1− Ω2

]

(4.7)

where we introduced the shape parameter Ω and the vesiculation index VF defined via the followingequations.

Ω ≡√

A

4πR2(4.8)

VF ≡ γ

2κ

√A

4π(4.9)

The shape parameter Ω represents the shape of the micelle. Ω = 0 corresponds to a flat disk-likemicelle and Ω = 1 corresponds to a vesicle (see also Figure 4.2). The vesiculation index VF can beregarded as an index whether a vesicle is formed spontaneously or not.

The free energy (eq (4.7)) is now expressed as a function of only two variables Ω and VF . Thuswe can study the vesicle formation kinetics simply and easily.

First we calculate the free energy for a disk and one for a vesicle.

F(0) = 8πκ, F(1) = 8πκVF (4.10)

4.2. SIMPLE KINETIC MODEL FOR BLOCK COPOLYMER VESICLES AND ONIONS 101

(Disk) (Vesicle)Ω=0 Ω=0.707Ω=0.383 Ω=0.924 Ω=1

Figure 4.2: Schematic draw of the Fromherz’s shape parameter Ω. At Ω = 0, a sheet is a flat disk,and at Ω = 1, a sheet is a closed vesicle. 0 < Ω < 1 corresponds to intermediate structures.

Thus a vesicle is more stable than a disk for VF ≥ 1.Next we consider the stability of the structure. From eq (4.7) we can calculate Ω∗ such that

the free energy has a maxima at Ω = Ω∗.

∂F∂Ω

∣∣∣∣Ω∗

= 8πκ

[2Ω∗ − VF Ω∗√

1− (Ω∗)2

]= 0 (4.11)

Considering that 0 ≤ Ω ≤ 1, we have

Ω∗ =

√1−

(VF2

)2

(VF < 2)

0 (VF ≥ 2)

(4.12)

We can calculate the free energy barrier for the disk-to-vesicle transition and one for the vesicle-to-disk transition.

∆Fdisk→vesicle = F(Ω∗)−F(0) =

8πκ(

1− VF2

)2

(VF < 2)

0 (VF ≥ 2)(4.13)

∆Fvesicle→disk = F(Ω∗)−F(1) =

8πκ(VF2

)2

(VF < 2)

8πκ(VF − 1) (VF ≥ 2)(4.14)

From eqs (4.13) and (4.14) the stability can be analyzed. For the case of VF < 1, a disk is morestable than a vesicle. The transition from a vesicle to a disk is the thermal activation process.For the case of 1 ≤ VF < 2, a vesicle is more stable than a disk. The transition process is athermal activation process just like the case of VF < 1. For the case of VF ≥ 2, F is monotonicallydecreasing function of Ω (∂F/∂Ω ≤ 0). Thus a disk-like micelle (Ω = 0) spontaneously changes itsshape into a vesicle (Ω = 1) without the effect of the thermal noise.

The Fromherz theory shown in this section indicate simple but important physics of the vesicleformation kinetics. Vesicles are spontaneously formed if the following condition is satisfied.

VF =γ

2κ

√A

4π≥ 2 (4.15)

Thus vesicles tend to be formed for large γ, small κ, or large A. If γ and κ are constant, from eq(4.15) we know that a vesicle is spontaneously formed if the size of a disk A becomes larger thanthe critical value A∗.

A∗ ≡ 4π(

4κγ

)2

(4.16)


Usually the disk-like micelles grow by collision, and if the disk size becomes sufficiently large(A ≥ A∗), the disk forms a vesicle [126]. Since the collision process is driven by the thermal noise,we know that the thermal noise is essential for the vesicle formation kinetics.

4.2.2 Extension of Fromherz Theory for Onion Structures

The onion structures observed in simulations [132, 133] look similar to vesicles in a sense. Themain difference between the onion and the vesicle is that the onion does not contain cavity regionsof solvents inside it.

Sevink and Zvelindvsky [133] reported that sheet like structures spontaneously close to formonions. In this section we consider whether this onion formation mechanism is the same as thevesicle formation mechanism. To consider this problem, we extend the Fromherz theory slightly.

Before we consider the extension, here we notice that the arguments in this section is quiterough compared with the original Fromherz theory. Nevertheless, it will be useful to understandthe formation mechanisms of vesicles and onions, especially if we concern on the difference betweenthe onion formation mechanism and the vesicle formation mechanism.

In the original Fromherz theory, the shape of the disk is assumed to be a part of a sphericalshell, and this makes it easy to calculate the free energy. To calculate the free energy of onions weintroduce several assumptions. While the assumptions are rather rough, our theory will be usefulto understand what is physically essential for the onion formation kinetics.

We assume that the disk-like small micellar structures are formed in the middle stage of thestructure formation, just like the vesicle formation case. Then we can describe the onion formationkinetics in the similar way to the vesicle formation kinetics.

We also assume that the shape of the micellar droplet is a capped disk, and the shape of thecore is a part of a spherical shell. The shape of the core is similar to the original theory, but thethickness of the core is finite. The shape of the micellar droplet (or the outer corona) is not bentdisk. This is due to the fact that the corona forming subchain is hydrophobic, and the small corona/ solvent interfacial area is preferred.

We further assume that the volume of the micellar droplet is constant in the morphologicaltransition process considered here. We also assume that the area and the thickness of the sheetlike structure, DA, are approximately constant1. We write the volume of the droplet and the areaof the sheet as V and A, respectively. The schematic draw of droplet shapes are shown in Figure4.3.

We assume that the free energy is expressed as the sum of the three contributions; the bendingfree energy F (bend), the edge free energy F (edge), and the surface free energy F (surf).2

F = F (bend) + F (edge) + F (surf) (4.17)

F (bend) and F (edge) is essentially the same as the vesicle formation case.3 F (surf) is not consideredin the vesicle formation case. It is important only for the onion formation case. (We will showthat F (surf) is irrelevant and negligible in the vesicle formation case.)

The surface free energy F (surf) is given as follows.

F (surf) = εS (4.18)

where ε is the surface energy per unit surface area and S is the total surface area of the droplet(the total area of the corona / solvent interface).

1We do not use the value of DA. What important here is that hydrophobic subchains form disk-like structures,and the thickness DA is qualitatively not important. If we consider onion formation kinetics quantitatively, DA willbe important. For example, the moduli κ and κ will depends on DA.

2It is not clear whether this assumption is applicable for relatively small structures. If the ratio of the inverse ofthe curvature and the membrane thickness is sufficiently large, the free energy is reduced to the Helfrich type [141].In this work, the ratio is not large (it is of the order of 1) and thus we will need some correction terms. However,we believe that the result is not changed qualitatively even if we ignore the correction terms.

3Notice that, strictly speaking, the Helfrich free energy cannot be used for this case, because the thickness of thedisk like structure is comparable to its radius. We believe that any free energy model which qualitatively reproducethe bending free energy can be used for current purpose and thus employ the Helfrich free energy, which is one ofthe most simple models.


D = D0DA

C = 0(a)

(b) (c)

D / 2

D

C-1

DA

D / 2

C-1

D

DA

RB

RBRB=0

Figure 4.3: Shapes and parameters considered in the extended Fromherz theory for onion formationkinetics. (a) a flat disk like structure, (b) a thick disk like structure with a bent disk like core, and(c) an onion. C = 1/R is the curvature of the core, DA is the thickness of the core (constant), Dis the thickness of the whole droplet, and RB is the radius of the droplet (except for the cap).

S can be determined from the following set of equations.

S = 2πR2B + π2RBD + πD2 (4.19)

V = πR2BD +

π2

4RBD

2 +π

6D3 (4.20)

where RB is the radius of the capped disk. We can write S as the function of D and V from eqs(4.19) and (4.20). The explicit expression of S is a bit complicated, but after the straightforwardcalculation, we have

S = π

2π

V

D+

23D2 +

(π4

)2

D2

−1 +

√1−

(4π

)3(π

6− V

D3

) (4.21)

It is easy to show that D ≤ (6V/π)1/3 because the shape of droplet becomes sphere at V = πD3/6.Combining this with D ≥ D0 (where D0 is the value of D for the flat disk, and is the lower limitfor D) gives

D0 ≤ D ≤(

6Vπ

)1/3

(4.22)

Substituting eq (4.21) into eq (4.18) gives

F (surf) = πε

2π

V

D+

23D2 +

(π4

)2

D2

−1 +

√1−

(4π

)3(π

6− V

D3

) (4.23)

4.2.3 Rough Estimation of ε

Here we estimate ε roughly. If the corona chain is hydrophobic, we expect that there is rathersharp interface between the corona phase and the solvent phase. As an extreme case, we employthe Helfand-Tagami [87, 88] type interfacial density profile. The interfacial free energy per unitarea is then given as

ε = kBTb

√χAS

6(4.24)


where χAS is the χ parameter between the corona and the solvent. Strictly speaking, eq (4.24)holds only for the strong segregation limit. But as far as considering qualitative behavior, weexpect it is not so inaccurate to use eq (4.24) as the interfacial energy for the hydrophobic coronasubchains, including weak segregation cases. However, notice that, clearly eq (4.24) cannot beused for χAS < 0. This means that we need to estimate another form of ε for hydrophilic coronasubchain cases.

For hydrophilic corona subchain cases, we expect that there is no sharp interface but the diffuseinterface. In such cases, the corona is expected to be similar to the swollen polymer brush [6,85,86].The energy per unit interfacial area, of the swollen brush grafted at the wall, is roughly estimatedas follows.

ε ≈ (12)5/3π2/3v2/30 σ

5/30 NA

240b2/3(4.25)

where v0 is the excluded volume parameter, σ0 is the graft density, and NA is the polymerizationindex of the corona forming subchain. if we assume the graft density is inversely proportional tothe surface area,

σ0 ≈ M

S(4.26)

where M is the constant which corresponds to the number of polymer chains in the droplet. Fromeqs (4.25) and (4.26), we have

ε ≈ (12)5/3π2/3v2/30 NAM

5/3

240b2/31

S5/3

= ε1

S5/3

(4.27)

where we defined

ε =(12)5/3π2/3v

2/30 NAM

5/3

240b2/3(4.28)

Eq (4.27) means that ε depends on S in the case of the hydrophilic subchains, unlike the case ofthe hydrophobic subchains.

Of course the estimations shown in this section are very rough and we may need more accurateestimation. Nevertheless it reproduces properties of hydrophilic corona subhchains qualitatively.What important here is the qualitative behavior of F (surf), and eqs (4.24) and (4.27) are sufficientfor this purpose.

4.2.4 Stability of Disk Like Micelles

In this section, we consider the stability of the disk like micelles. First we consider the hydrophobicsubchain case. To express the free energy as a function of D, we have to express the surface freeenergy and the edge free energy as functions of D. For this purpose, we simply assume that thecurvature of the core, C, and the thickness of the droplet, D, is related via the following relation.

D = D0 +AC

2π(4.29)

Using eqs (4.2), (4.3), (4.23), and (4.29) the free energy can be rewritten as follows.

F (bend) = 8π2κV 2/3

A(X −X0)2 (4.30)

F (edge) = γ

√4πA

[1− πV 2/3

A(X −X0)2

](4.31)

F (surf) = πεV 2/3

2π

1X

+23X2 +

(π4

)2

X2

−1 +

√1−

(4π

)3(π

6− 1X3

) (4.32)


where ε is defined by eq (4.24), and X and X0 are defined as follows.

X ≡ D

V 1/3(4.33)

X0 ≡ D0

V 1/3(4.34)


X0 ≤ X ≤(

6π

)1/3

(4.35)

as a condition for X. Using eqs (4.30)-(4.32), finally we have the explicit form of the free energyas the following function of X.

F(X) = 8π2κV 2/3

A(X −X0)2

+ γ

√4πA

(1− πV 2/3

A(X −X0)2

)

+ πεV 2/3

2π

1X

+23X2 +

(π4

)2

X2

−1 +

√1−

(4π

)3(π

6− 1X3

)

(4.36)

We analyze the stability of the disk-like micelles by using the free energy derived in the previoussection.

∂F(X)∂X

∣∣∣∣X=X0

=∂F (surf)

∂X

∣∣∣∣X=X0

= πεV 2/3 df(X)dX

∣∣∣∣X=X0

(4.37)

where we defined the function f(X) as follows.

f(X) ≡ 2π

1X

+23X2 +

(π4

)2

X2

−1 +

√1−

(4π

)3(π

6− 1X3

) (4.38)

Then the surface free energy can be written as F (surf) = πεV 2/3f(X). Differentiating eq (4.38)and we have

df(X)dX

≡ − 2π

1X2

+43X + 2

(π4

)2

X

−1 +

√1−

(4π

)3(π

6− 1X3

)

− 12π

1X2

1√1−

(4π

)3(π

6− 1X3

)(4.39)

The graphs of f(X) and df(X)/dX are shown in Figure 4.4. From Figure 4.4(b), df(X)/dX isalways negative. Thus the sign of ∂F/∂X depends only on the sign of ε. From eq (4.24) clearlyε > 0. Therefore we have

∂F(X)∂X

∣∣∣∣X=X0

< 0 (4.40)

Thus the disk-like micelles are unstable and spontaneously changes their form into onions, forhydrophobic corona subchain cases.

Next we consider the hydrophilic subchain case. In this case, the surface free energy is expressedas follows.

F (surf) =ε

π2/3V 4/9

2π

1X

+23X2 +

(π4

)2

X2

−1 +

√1−

(4π

)3(π

6− 1X3

)−2/3

=ε

π2/3V 4/9

1f2/3(X)

(4.41)


0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1 1.2

f(X

)

X

(a)

-200

-150

-100

-50

0

0 0.2 0.4 0.6 0.8 1 1.2

df(X

) / d

X

X

(b)

Figure 4.4: Graphs of functions f(X) and df(X)/dX. (a) f(X), and (b) df(X)/dX. We canobserve that df(X)/dX < 0 and thus f(X) are the monotonically decreasing function of X.

where ε is defined by eq (4.28). Then the total free energy can be expressed as

F(X) = 8π2κV 2/3

A(X −X0)2 + γ

√4πA

(1− πV 2/3

A(X −X0)2

)+

ε

π2/3V 4/9

1f2/3(X)

(4.42)

Since ε > 0 from eq (4.28), we have

∂F(X)∂X

∣∣∣∣X=X0

=∂F (surf)

∂X

∣∣∣∣X=X0

= − 2ε3π2/3V 4/9

1f5/3(X)

df(X)dX

∣∣∣∣X=X0

< 0 (4.43)

Unlike the hydrophobic corona subchain cases, the disk-like micelles are stable for hydrophiliccorona subchain cases.

We expect that the disk-like micelles will grow larger by collision processes, and then closespontaneously by the Fromherz type mechanism. If the size of the disk-like micelles are sufficientlylarge, the surface free energy is not affected so much by the bending. Under such a circumstancewe may neglect the contribution of the surface free energy to the total free energy. Then the theoryis reduced to the original Fromherz theory.

Notice that, the extended theory is not reduced to the original Fromherz theory for the hy-drophobic corona subchain cases, even if we consider large disk like structures4. If the coronaforming subchain is hydrophobic, the property of the free energy functional is qualitatively muchdifferent form the hydrophilic corona subchain case. This makes the onion formation kineticsqualitatively different from the vesicle formation kinetics. It will be shown in the next section.

4.2.5 Onion Formation Kinetics

In this section we consider the case of the hydrophobic corona subchains. As shown in the previoussection, the disk-like micellar droplets are unstable in this case. Thus we consider that the dropletchange its form from disk-like shape into other shape before it grows by the collision-coalescencelike process.

We expect the final structure is the onion structure (a droplet which consists of layered struc-tures) in this case.

For further analysis, we approximate the function f(X) which has relatively complicated form(eq (4.38)) by the simple form.

f(X) ≈ B

X+B′ (4.44)

4Strictly speaking, we have to introduce other deformation modes if we consider large structures. Currently welimit the shape of core to a part of a spherical shell, and thus only one deformation mode is allowed. Clearly weneed more complicated deformation modes which have relatively high wave number to describe the kinetics of largedroplets.


where B and B′ are constant and determined from the fitting. Here we use B ≈ 0.59358 andB′ ≈ 0.82038.

It will be convenient to introduce the Fromherz’s shape parameter Ω just like the originalFromherz theory.

Ω ≡√π

AV 1/3(X −X0) (4.45)

The surface free energy can be approximately expressed as

F (surf) ≈ ε√π3

AV

[B

Ω + Ω0+

√A

π

B′

V 1/3

](4.46)

where Ω0 ≡√π/AV 1/3X0. The total free energy can be then written as the function of Ω

F(Ω) = 8πκΩ2 + γ√

4πA√

1− Ω2 + ε

√π3

AV

[B

Ω + Ω0

]+ (const)

= 8πκ[Ω2 + VF

√1− Ω2 +

K

Ω + Ω0

]+ (const)

(4.47)

where VF is the vesiculation index defined by eq (4.9). K is a new index defined by the followingequation.

K ≡ εV B

8κ

√π

A(4.48)

We can calculate the free energy for the flat sheet case and for the closed sheet case as follows.

F(Ω = 0) = 8πκ(VF +

K

Ω0

)(4.49)

F(Ω = 1) = 8πκ(

1 +K

1 + Ω0

)(4.50)

Thus the closed sheet structure is more stable than the flat sheet structure in the case of

VF +K

Ω0(1 + Ω0)≥ 1 (4.51)

Unfortunately, we cannot proceed analytic calculations further even if we use the approximateform (eq (4.47)). To proceed the calculation, we consider a special case in which the contributionof the edge free energy is negligible (VF = 0).

F(Ω) ≈ 8πκ[Ω2 +

K

Ω + Ω0

]+ (const) (4.52)

Differentiating eq (4.52) gives

∂F(Ω)∂Ω

= 8πκ[2Ω− K

(Ω + Ω0)2

](4.53)

and∂F(Ω)∂Ω

∣∣∣∣Ω=0

= −8πκK

Ω20

< 0 (4.54)

∂F(Ω)∂Ω

∣∣∣∣Ω=1

= 8πκ[2− K

(1 + Ω0)2

](4.55)

Since ∂F(Ω)/∂Ω is monotonically increasing function, ∂F(Ω)/∂Ω is always negative if the followingcondition is satisfied.

K

(1 + Ω0)2> 2 (4.56)

In this case, there are no free energy barrier between the flat sheet and the closed sheet and theclosed sheet structure is spontaneously formed. On the other hand, in the case of K/(1+Ω0)2 < 2,there are free energy barrier and therefore the formation process will be the thermal activationtype process. From eq (4.24), this condition is satisfied for sufficiently large χAS .


4.3 Discussion

4.3.1 Vesicle and Onion Formation Mechanisms

We summarize the formation mechanisms of vesicles and onions in Table 4.3.1.

Corona Driving Resisting Stability ofSubchain Force Force Small Disks

Vesicles Hydrophilic Edge Tension Bending Energy StableOnions Hydrophobic Surface Energy Bending Energy Unstable

Table 4.1: Summary of the formation mechanisms of vesicles and onions.

There are several similarities between the vesicle formation process and the onion formationprocess considered here. The similarities are that the kinetics of the disk like cores is mostly thesame for two cases. The disk like cores have positive bending modulus, and thus they favor theflat shapes. This is the main thermodynamic force which resists the vesicle or onion formation.

On the other hand, the driving thermodynamic forces for the structure formation are quitedifferent. In the vesicle formation case, the driving force is the line tension of the edge. In the onionformation case, on contrary, the line tension is not important but the surface energy is important.It should be noted that due to the surface energy, flat disk like shapes are unconditionally unstablein onion forming systems while they can be stable in vesicle forming systems. Besides, because ofthe difference of the driving force, there are significant differences in two dimensional systems (SeeAppendix 4.A).5

Combining these mechanism with the Lasic’s model [126] and the results of the dynamic DFsimulations, we can draw the schematic image of the kinetic pathways for the vesicle formationand the onion formation (Figure 4.5). At the early stage, both the vesicle formation dynamics andthe onion formation dynamics are very similar. Small spherical structures are formed rapidly fromthe initial homogeneous state, and then they grow into small disk like structures by collision andcoalescence type process. However, there are are differences at the middle stage. For hydrophiliccorona subchain case, disk like structures are stable and grow to larger disks by collision [128,129,142]. On the other hand, for hydrophobic corona subchain case, small disk like structures areunstable and they change into monolayer onions spontaneously. The kinetics are different at thelast stage, too. For hydrophilic corona subchain case, it is expected that if the disk like structuresgrow sufficiently large, they close spontaneously by the Fromherz type mechanism.

For onion formation cases, the final structures observed in simulations are multilayered struc-tures. Thus the onion structure given by the extended Fromherz theory (we may call such astructure the monolayer onion) is not considered to be the final structure. A possible pathway tothe multilayer onion is as follows; monolayer onions collide and then aggregate into larger droplets.The forms of droplets spontaneously change into spheres (to reduce the surface free energy). Therearrangement of sheets (lamellars) occurs and the final multilayer onion structures are formed.This process will be thermal activation type process because there will be the free energy barrier.Such a kinetic pathway can be observed in the dynamic DF simulations. We do not considerthe details here, but the Fromherz type simple analysis may be possible if the kinetic pathway isdescribed by simple parameters such as the shape parameter Ω.

4.3.2 Difference and Similarity between Vesicles and Onions

We claim that the vesicles and the onions are qualitatively different.6 The driving force for thevesicle formation is the line tension of the edge while one for the onion formation is the interfacial

5While the kinetics in two dimensional systems may be unphysical, it is true that there are clear differencesbetween two cases.

6Unfortunately, this difference is sometimes ignored or misunderstood. As shown previously, the vesicles cannotbe obtained by conventional phenomenological simulation models. To understand the dynamics of vesicles, thedistinction between the vesicles and the onions are essential. Without correct distinction, one may arrive unphysicaland incorrect conclusions.

4.4. SUMMARY 109

Vesicle

OnionUnstable disks

Stable disks

(hydrophilic)

(hydrophobic)

(collision) (closure)

(collision)(closure)

Figure 4.5: Schematic draw of the kinetic pathways for the vesicle formation (the upper route, thehydrophilic subchain case) and the onion formation (the lower route, the hydrophobic subchaincase), from the homogeneous initial state. Grey and Black colors represent corona subchain andcore subchain, respectively.

tension between the hydrophobic corona subchains and the solvents. Thus we must not regardthe vesicle formation processes are the same as the onion formation processes while there areseveral similarities. Clearly this is consistent with our dynamic DF simulation results. The vesicleformation process is reproduced only if the proper parameters and dynamic equations are given, andin previous continuum field simulation works such conditions are not satisfied. On the other hand,the onion formation process can be reproduced without such strict conditions. Neither the effectof the thermal noise nor the hydrophilic interaction is needed. Only the hydrophobic interactionwhich causes usual phase separation dynamics and small selectivity (or a kind of wettability) isrequired. This is the reason why the onions can be formed easily in simulations, compared withvesicles.

As discussed in the previous section, the final structures in the onion formation cases arethe multilayer onions. The multilayer onions are considered to be formed by the collision andrearrangement type process. However, if the segregation is not so large, they can grow by theevaporation-condensation type process. The evaporation-condensation type growth is observed inthe TDGL simulations or the dynamic SCF simulations in the weak segregation region [132, 133].In contrast, the final structures observed in vesicle formation cases are normally monolayeredstructures, and there are no realistic growth mechanism based on the evaporation-condensationtype. This is why we need the large thermal noise in our dynamic DF simulations.

4.4 Summary

We have formulated a simple kinetic theory to describe the vesicle and onion formation dynamics,in this chapter. Our theory is an extension of the Fromherz’s theory of vesicle formation to moregeneral systems.

By using the extended theory, we have shown that the vesicle formation dynamics and theonion formation dynamics in block copolymer systems are qualitatively different. In the vesicleformation case, the corona forming subchain is hydrophilic, and the kinetics is described roughlyby the competition between the Helfrich bending free energy and the edge free energy. On theother hand, in the the onion formation case, the corona forming subchain is hydrophobic, and thecompetition between the bending free energy and the interfacial energy describe the kinetics. Wealso found that the effect of the thermal noise is not important for the onion formation case whileit is quite important for the vesicle formation case.

These results are consistent with our dynamic DF simulations, and thus consistent with other


simulation results based on particle models.

Appendix

4.A Two Dimensional Systems

In this appendix, we consider vesicles and onions in two dimensional systems. So far, no realisticvesicle formation dynamics simulations based on the continuum field model are performed in twodimensional systems. By comparing the two dimensional case and the three dimensional case, wediscuss why vesicles are not formed in two dimensional systems. On the other hand, there are onionformation dynamics simulations in two dimensional systems [132]. We also discuss why onions canbe formed in two dimensional systems unlike vesicles.

4.A.1 Fromherz Theory in Two Dimensional Systems

In this section we consider the two dimensional version of the Fromherz theory. We consider aribbon like micelle of which shape is a part of cylindrical surface (see Figure 4.6). We consider herethe free energy per unit length. Using the surface are per unit length A and the edge length perunit area l, the Fromherz free energy can be used for two dimensional systems with minor changes.

radius: R

angle: θsurface area perunit length: A

edge length perunit length: l

Figure 4.6: Parameters for a vesicle used in the two dimensional version of the Fromherz theory.We consider the cut cylinder like shell structure.

The two principle curvatures of cylindrical surfaces are 0 and 1/R (where R is the radius of thecylinder). Thus the Helfrich bending energy per unit area can be expressed as

F (bend) =12κA

(1R

)2

(4.57)

The expression for the edge free energy per unit area is the same as eq (4.3).The surface area per unit length A and the edge length per unit area l can be expressed as

follows, by using the curvature radius R and the angle θ.

A = 2Rθ (4.58)

l =

2 (0 ≤ θ < π)0 (θ = π)

(4.59)

Eq (4.59) can be rewritten as follows.

l =

2 (0 ≤ A/2R < π)0 (A/2R = π)

(4.60)

The Fromherz free energy can be written as

F =

12κA

R2+ 2γ (0 ≤ A/2R < π)

12κA

R2(A/2R = π)

(4.61)

4.A. TWO DIMENSIONAL SYSTEMS 111

It will be convenient to introduce the shape parameter and the vesiculation index just like thethree dimensional case.

Ω ≡ A

2πR(4.62)

VF ≡ Aγ

π2κ(4.63)

Finally eq (4.61) can be rewritten as follows.

F =

2π2κ

A

[Ω2 + VF

](0 ≤ Ω < 1)

2π2κ

AΩ2 (Ω = 1)

(4.64)

The form of eq (4.64) is different from eq (4.7), and it means that there are qualitative differencebetween a two dimensional vesicle and a three dimensional vesicle. Unlike eq (4.7), eq (4.64) isnon continuous at Ω = 1. Thus we have to be careful to consider the stability.

We first consider the free energy for a flat ribbon (Ω = 0) and one for a closed tube (Ω = 1).

F(0) =2π2κ

AVF F(1) =

2π2κ

A(4.65)

Thus a flat ribbon is more stable for VF < 1 and a closed tube is more stable for VF ≥ 1. This isthe same result as the three dimensional case.

We then calculate the free energy barrier for transitions to consider the stability. From eq(4.64), for 0 ≤ Ω < 1 we have

∂F∂Ω

=4π2κ

AΩ ≥ 0 (4.66)

Thus the free energy is the monotonically increasing function of Ω (except for Ω = 1). The freeenergy barrier can be expressed as follows.

∆Fribbon→tube = limΩ→1−0

F(Ω)−F(0) =2π2κ

A(4.67)

∆Ftube→ribbon = limΩ→1−0

F(Ω)−F(1) =2π2κ

AVF (4.68)

Unlike the three dimensional case, the free energy barrier from a ribbon to a tube is independentof VF . Thus we know that all the transition process is thermal activation type even for large VF .

This result indicates that the vesicle formation kinetics strongly depends on the dimensionality.It will be quite difficult to reproduce vesicles in two dimensional systems, and this is consistentwith dynamic simulation results [58].7

4.A.2 Extension of Fromherz Theory to Onion Formation Kinetics inTwo Dimensional Systems

Next we consider the extension of the Fromherz theory for the onion formation kinetics in twodimensional systems. As before, we consider the free energy is expressed as the sum of threecontributions; the bending free energy, the edge free energy, and the surface free energy. Thebending free energy and the edge free energy are the same as the vesicle formation case, just likethe case of the three dimensional systems. What we have to calculate here is the surface freeenergy.

7Vesicles can be formed easily even in two dimensional systems by static the SCF simulations [34] or the EPDsimulations [107]. It will be natural to consider that this is due to the dynamics (or pseudo dynamics) is not basedon the locally conserved realistic dynamics. This fact also supports our discussion about the EPD in the previouschapter.


We consider the two dimensional version of Figure 4.3. The surface area per unit length, S,and the volume per unit length, V , can be expressed as follows in the two dimensional system.

S = 2RB + πD (4.69)

V = RBD +π

4D2 (4.70)

From eqs (4.69) and (4.70), we have the following equation.

S = 2V

D+π

2D (4.71)

Finally the surface free energy can be written as

F (surf) = εV 1/2

[2V 1/2

D+π

2D

V 1/2

](4.72)

By considering that πD2/4 ≤ V , we can show the area of D becomes

D0 ≤ D ≤(

4Vπ

)1/2

(4.73)

where D0 is the minimum value of D.To express the free energy as a function of D, we have to get the relation between the curvature

C and the thickness D. In two dimensional case we assume the following relation.

D = D0 +1C

[1− cos

(AC

2

)](4.74)

Especially for the case of C 1, eq (4.74) can be reduced to the simple form.

D ≈ D0 +A2C

8(4.75)

We consider the stability of the flat ribbon like micelles. The three contributions can beexpressed as follows for D ≈ D0.

F (bend) =32V κA3

(X −X0)2 (4.76)

F (edge) = 2γ (4.77)

F (surf) = 2εV 1/2

[1X

+π

4X

](4.78)

where X and X0 are defined via the following equations.

X ≡ D

V 1/2(4.79)

X0 ≡ D0

V 1/2(4.80)

Stability of Ribbon Like Micelles

Here we consider the stability of the ribbon like micelles. In the case of the hydrophobic coronasubchains, ε is independent of S, and thus ε is independent of X. From eqs (4.76)-(4.78) we have

F(X) =32V κA3

(X −X0)2 + 2γ + 2εV 1/2

[1X

+π

4X

](4.81)

and∂F(X)∂X

∣∣∣∣X=X0

= 2εV 1/2

[− 1X2

0

+π

4

]≤ 0 (4.82)

4.A. TWO DIMENSIONAL SYSTEMS 113

where we used X0 ≤ (4/π)1/2. Therefore, the ribbon like micelles are unstable. This means thateven in two dimensional systems, onions (in this case, multilayer tube like structures) can be formedspontaneously.

In the case of the hydrophilic corona subchains, ε depends on S. Using the result of the threedimensional systems, we have

F (surf) =ε

22/3V 1/3

[1X

+π

4X

]−2/3

(4.83)

From eqs (4.76),(4.77), and (4.83) we have

F(X) =32V κA3

(X −X0)2 + 2γ +ε

22/3V 1/3

[1X

+π

4X

]−2/3

(4.84)

and∂F(X)∂X

∣∣∣∣X=X0

= − 21/3ε

3V 1/3

[1X0

+π

4X0

]−5/3 [− 1X0

+π

4

]≥ 0 (4.85)

Thus the ribbon like micelles are stable.These results are qualitatively the same as the case of the three dimensional systems. This

implies that the first stages of the onion formation kinetics are common for the two and threedimensional systems.

Onion Formation Kinetics

To discuss the onion formation kinetics in two dimensional systems, it is convenient to express thefree energy as the function of the shape parameter Ω. What we have to do is to get an approximateexpression for the following function f(X), as the function of Ω.

f(X) ≡ 1X

+π

4X (4.86)

We attempt to approximate f(X) as same as the case of the three dimensional systems.

f(X) ≈ 1X

+π

4(4.87)

Ω and X can be related by using eqs (4.74) and (4.79).

X = X0 +(

A

2πV 1/2

)1Ω

[1− cos (πΩ)] (4.88)

Because the form of eq (4.88) is not simple, we approximate it as follows.

X ≈ X0 +(

A

πV 1/2

)Ω[Ω +

π2

4(1− Ω)

]

=(

πA

4V 1/2

)[Ω0 + Ω−

(1− 4

π2

)Ω2

] (4.89)

where

Ω0 =4V 1/2X0

πA(4.90)

By substituting eq (4.89) into eq (4.87) we get

f(Ω) ≈(

4V 1/2

πA

)1

Ω0 + Ω− (1− 4/π2)Ω2+√π

4(4.91)

and

F (surf)(Ω) =(

8εVπA

)1

Ω0 + Ω− (1− 4/π2)Ω2+ (const) (4.92)


Using the Fromherz free energy in two dimensional systems (eq (4.64)), the total free energycan be expressed as follows.

F(Ω) =

2π2κ

A

[Ω2 +

K

Ω0 + Ω− (1− 4/π2)Ω2+ VF

]+ (const) (0 ≤ Ω < 1)

2π2κ

A

[Ω2 +

K

Ω0 + Ω− (1− 4/π2)Ω2

]+ (const) (Ω = 1)

(4.93)

where we defined the index K asK ≡ 4εV

π3κ(4.94)


F(0) =2π2κ

A

[K

Ω0+ VF

](4.95)

F(1) =2π2κ

A

[1 +

K

Ω0 + 4/π2

](4.96)

where we have dropped the constant terms. Therefore, onions are stable if the following conditionis satisfied.

VF +K

Ω0(1 + (π2/4)Ω0)≥ 1 (4.97)

Differentiating eq (4.93) gives

∂F(Ω)∂Ω

=2π2κ

A

[2Ω− K[1− (2− 8/π2)Ω]

[Ω0 + Ω− (1− 4/π2)Ω2]2

](0 ≤ Ω < 1) (4.98)

and

∂F(Ω)∂Ω

∣∣∣∣Ω=0

= −2π2κ

A

K

Ω0< 0 (4.99)

limΩ→1−0

∂F(Ω)∂Ω

=2π2κ

A

[2− K(8/π2 − 1)

(1 + Ω0)2

]> 0 (4.100)

Unlike three dimensional systems, onion formation kinetics in two dimensional systems is alwaysthe thermal activation type process. However it should be noticed that we have introduced manyapproximations to simplify the free energy, and thus this conclusion may be changed by consideringother approximation method. Nevertheless, the current conclusion is useful to understand twodimensional systems. In vesicle formation cases, the flat ribbon like micelles are stable for almostall the value of VF . In the onion formation case, on contrary, the flat ribbon like micelles areunstable and bend spontaneously. Thus we can claim that onions can be formed spontaneously intwo dimensional systems while vesicles cannot be formed.

Chapter 5

Summary

5.1 Conclusion

In this work, we have studied statics and dynamics of micellar structures in diblock copolymersolutions. We used the density functional theory for simulations and it has reproduced variousmicellar structures qualitatively well. The state of the system is represented by the density fieldsfor each subchains and solvents in the density functional theory. We derived the free energy as thefunctional of the density fields, and the dynamic density functional equation for the density fields.

In static simulations, we performed simulations with various parameter sets. The steady statestructures were obtained numerically by finding the density fields which minimize the free energyfunctional. The static simulations reproduce micellar structures for amphiphilic AB diblock copoly-mer solutions (spherical micelles, cylindrical micelles, and vesicles) and the results are qualitativelyconsistent with experiments.

We investigated, for example, the effect of the χ parameters, the block ratio, or the volumefraction. It has been shown that by controlling the solvent quality or the block ratio, we can controlthe morphologies of the micellar structures. Roughly speaking, if the hydrophilic interaction and/orthe hydrophobic interaction is large, vesicles are formed and if the hydrophilic interaction and/orthe hydrophobic interaction is small, spherical micelles are formed. At the intermediate cases,cylindrical micelles are formed. Crew-cut copolymers (the block ratio of the hydrophilic subchainsmall) form vesicles and hairy copolymers (the block ratio of the hydrophilic subchain is large)form spherical micelles. These results will be useful to design amphiphilic block copolymers orcontrol the morphology of micellar structures.

The parameters used in the static DF simulations are the same as the parameters used in thestatic SCF simulations. While the accuracy of the static DF simulations are not so high, it requiresless numerical costs compared with the SCF simulations. Therefore the static DF simulations willbe useful to scan some parameter spaces roughly.

In dynamic simulations, we performed simulations for structural formation processes or mor-phological transition processes. The dynamics simulations were performed by integrating thedynamic density functional equation. Especially it is worth noted that our simulation is the firstrealistic vesicle formation dynamics simulation based on the continuum field model. The dynamicssimulation results are qualitatively consistent with experiments and previous simulations based onparticle models. Results of the morphological transition dynamics are also qualitatively consistentwith theories or experiments.

The numerical cost for the dynamic DF simulations are much large compared with the staticsimulations. However, so far this is the only the continuum field model which reproduces the dy-namics micellar structures. Thus the dynamic DF will be useful to compare the statics simulationswith dynamics simulations, or to study the effect of several parameters which are used in staticcontinuum field model simulations.

We also performed static and dynamic simulations for onion structures. We observed that thereare qualitative difference between the onions and the vesicles.

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116 CHAPTER 5. SUMMARY

Results of the dynamic simulations for the vesicle formation dynamics are compared with theFromherz type simple kinetic model. We have found that our simulation results are consistent withthe kinetic model, and that there are qualitative difference between the vesicle formation dynamicsand the onion formation dynamics. The mechanism of the onion formation is also consistent withthe dynamic DF simulation result.

5.2 Future Works

5.2.1 On the Theory

The static and dynamic density functional theory shown and used in this work is still theoreticallyambiguous or using too rough approximations. Clearly we need more clear and reliable formula-tion. As shown before, so called the phenomenological approaches are not appropriate for micellarsystems. We must construct our theory systematically from microscopic model. Thus we can saythat what needed is the systematic coarse-graining theory.

Besides, we ignored several effects which will be important in real systems. For example, we as-sumed that all the monomers are electrostatically neutral. However, many water-soluble monomersare polyelectrolytes. In this work, we assumed that all the interactions between monomers can beexpressed by the simple Flory-Huggins type interaction with the χ parameter. If the electrostaticinteraction is important, our assumption does not work. In fact, there are several reports on theeffect of added salts to morphologies of amphiphilic block copolymer micelles [143]. To studysuch systems, we need to modify our DF theory to include the electrostatic interactions correctly.We also assumed that viscoelasticity and the hydrodynamic interaction are not important. Theseeffects may be important in several situations, and then we will need to consider these effects, too.

Recently multi-scale or multi-physics simulations which use several different scales or models areconsidered to be very important. In multi-scale simulations, the systematic coarse-graining theoryis still under construction and demanding. We will need to construct general coarse-graining theory,which can be applied to various models (such as particle models or continuum field models) andvarious scales (from molecular scale to macroscopic scale). Then we will be able to connect differentmodels for different scales and perform numerically efficient large scale simulations.

It is empirically known that particle models are sometimes very robust and suitable for simu-lations where continuum models are numerically instable. However, several particle models suchas the DPD are introduced intuitively and theoretical back ground is still ambiguous. 1 Thereforeit will be interesting to combine continuum field models and particle models theoretically. As longas the basis and the target phenomena are the same, any models should give the same result.2

It is also known that simple theories or models which have physically essential properties issometimes quite useful to understand experimental data or simulation data. By introducing someassumptions or simplifications to our theory, it will be possible to derive simple models. Forexample, if we calculate the free energy of spherical micelles, cylindrical micelles, or vesicles, bysimple model, we will understand the physics of micellar structures well. It will be useful to studyexperiments or to design block copolymers which form required micellar structures.

5.2.2 On the Simulations

We have performed many simulations for micellar structures, but there are many parameters evenfor the simple diblock copolymer solutions, and thus there are very wide unexplored parameterregions. Besides, there will be very slow events in dynamics. For example, we expect that thefusion or fission of vesicles [19, 28] are slower process than the vesicle formation. It is known that

1There are several theoretical approaches to derive coarse-grained particle models from microscopic dynamics.For example, Kinjo and Hyodo [144] showed that the BD or the DPD dynamic equation can be obtained frommicroscopic Hamiltonian dynamics, by using the projection operator method.

2Of course this is rather trivial. However, most of phenomenological continuum field models of micellar systemsare inconsistent with other models or experiments. There must be no such inconsistency models which are derivedsystematically.

5.2. FUTURE WORKS 117

there are similar problems in the dynamics simulations for block copolymer melts (the growth oflarge scale higher order structures).

Unfortunately, it is quite difficult (or nearly impossible) to explore all the parameter regions orperform very large scale simulations by current computational resources. We need new approachesfor these problems.

There will be mainly two approaches. One is to improve numerical algorithms and then useparallel computers. The numerical algorithms used in our simulations are not well considered andwe can improve them to improve the performance. For example, we used the steepest descentmethod to minimize the free energy functional. However, the steepest descent method is known tobe inefficient compared with improved algorithms such as the conjugate gradient method. Thereare many works on the minimization algorithms, and the use of improved minimization algorithmswill make the simulations efficient. We used the partially implicit integration scheme for dynamicssimulations, but this is also not so efficient. Improved version of integration schemes will make thedynamics simulation efficient. Parallelizing the simulator will also improve the performance.

Another is to construct a new theory based on the current theory. If we construct a morecoarse-grained (but still physically valid) theory, the numerical costs of simulations will be reduced.This will be able to be achieved if we have systematic coarse-graining theory. This is the mostnatural way in the view point of the theory. Besides, constructing a new theory will be needed formulti-scale simulations which uses consistent several simulation models.

Acknowledgment

The author first thanks Prof. T. Ohta, for valuable discussions and comments, and for giving theauthor the opportunities to study dynamics of block copolymers and non-equilibrium statisticalmechanics.

The author also thanks the following people for helpful comments and discussions: Prof. M.Doi, Prof. T. Kawakatsu, Dr. H. Morita, Dr. T. Honda, Prof. H. Frusawa, Dr. J. Fukuda, Prof.A. Onuki, Prof. K. Yoshikawa, Prof. H. Hayakawa, Prof. R. Yamamoto, Prof. T. Taniguchi, Prof.Y. Masubuchi, Dr. S. Yamamoto, Prof. J. Takimoto, Prof. Z.-G. Wang, Prof. A. Zvelindovsky,Dr. A. Sevink, Prof. S. Irle, the ex-members of JCII Doi-project and Doi-laboratory at NagoyaUniversity, and the members of Ohta-laboratory.

This work is done by using many free / open-source softwares (FOSS) such as Linux or GNUsoftwares. These softwares helped the author very much. The author thanks all the creators ofthe FOSS used in this work.

This work is supported by the Research Fellowships of the Japan Society for the Promotion ofScience for Young Scientists.

Parts of this work were initially started while the author was studying at the master course ofComputational Science and Engineering, Graduate School of Engineering, Nagoya University, andis partially supported by JST (CREST Bio-Rheo project).

118

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Static and Dynamic Density Functional Theory and ...called copolymers. Here we consider the class of copolymers called \block copolymers" [7] while there are many kinds of copolymers