Conformational behaviors of a charged-neutral star micelle

in salt-free solution

Mingge Deng,ab Ying Jiang,ab Xuejin Li,*ab Lei Wangab and Haojun Liang*ab

Received 18th November 2009, Accepted 5th March 2010

First published as an Advance Article on the web 20th April 2010

DOI: 10.1039/b924281c

The conformational behaviors of charged brushes on a micelle self-assembled by charged-neutral

diblock copolymers in salt-free solution are extensively analyzed using a coarse-grained dissipative

particle dynamic (DPD) simulation. When only monovalent counterions exist, the brush

conformation of the corona in the micelle is exactly consistent with the predictions from the

blob-scaling theory based on the spherical polyelectrolyte brush model, which differentiates the

system into three distinct regimes: (I) quasi-neutral regime, (II) ‘‘Pincus’’ regime, and (III) osmotic

regime. For multivalent counterions such as divalence and trivalence, however, the strong

electrostatic correlations lead the micelle structures to deviate obviously from those of scaling

predictions. The collapse of the brush appears to be due to the drop in the osmotic pressure

inside the corona region of the micelle.

1 Introduction

The conformations and physical properties of polyelectrolyte

brushes are of great interest because of their fundamental

significance and potential applications in the field of bio- and

nanotechnology.1–3 In particular, spherical polyelectrolyte

brushes (SPBs) with long polyelectrolyte chains grafted onto

a solid core in the size of colloidal dimensions have been

widely utilized as novel carrier particles for functional bio-

molecules.4–6 Generally, photo-emulsion polymerization7 and

controlled radical polymerization8 are two normal methods of

manufacturing the SPB. Recently, an efficient and easy ap-

proach has been developed by simply dissolving charged-

neutral diblock copolymer in appropriate solvents,9 whereby

micelles with neutral cores and charged hairs, so-called

charged-neutral star micelles, were produced. The distinctively

responsive properties of this type of polymer brush and its

potential applications in industry have been extensively

investigated previously in terms of transition mechanisms

in a controlled environment.10 It is very likely that the

comprehensive investigations on the SPBs, in comparison with

neutral polymeric brushes, may offer us an opportunity to

understand deeply the manner of conformational transformation

of a polymeric brush. In the past decades, there has been active

interest in this field, and it has attracted the attention of many

experimental and theoretical scientists.11–20 However, owing

to fact that the performances of the charged chains are

governed mutually by multiple parameters such as ionic

strength, electrostatic interaction, and valence of counterions,

the system is expected to respond in a complicated manner

during stimulations emanating from the circumstance. Up to this

date, an understanding of this sort of system is far from

complete, and many problems are still left to challenge us. To

comprehend the conformational behaviors of this brush system,

we studied the conformational transitions of charged-neutral star

micelles built with charged-neutral diblock copolymers using a

dissipative particle dynamics (DPD) simulation.

2 Model and method

2.1 Dissipative particle dynamics formulation

We study the conformational transitions of charged-neutral

star micelles in salt-free solution with the help of the DPD

simulation technique. DPD is a simple but intrinsically

promising simulation method that allows the study of the

conformational behaviors of charged-neutral block copolymers.21

In DPD simulation, a particle represents the center of mass of

a cluster of atoms, and the position and momentum of the

particle is updated in a continuous phase but spaced at discrete

time steps. Particles i and j at positions ri and rj interact with

each other via a pairwise additive force, consisting of three

components: (i) a conservative force, FCij ; (ii) a dissipative

force, FDij ; and (iii) a random force, FR

ij . All forces are

non-zero within a cut-off radius rc. Hence, the total force on

particle i is given by

Fi ¼Xiaj

FCij þ FD

ij þ FRij ð1Þ

where the sum acts over all particles within rc. Specifically, in

our simulations

Fi ¼Xiaj

aijoðrijÞnij � go2ðrijÞðnij � vijÞnij þ soðrijÞzijDt�1=2nij

ð2Þ

where aij is a maximum repulsion between particles i and j, rij is

the distance between them, with the corresponding unit vector

a CAS Key Laboratory of Soft Matter Chemistry, Department ofPolymer Science and Engineering, University of Science andTechnology of China, Hefei, Anhui 230026, People’s Republic ofChina. E-mail: xjli7@ustc.edu, hjliang@ustc.edu.cn

bHefei National Laboratory for Physical Sciences at Microscale,University of Science and Technology of China, Hefei,Anhui 230026, People’s Republic of China

This journal is �c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 6135–6139 | 6135

PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics

nij, vij is the difference between the two velocities, zij is a

random number with zero mean and unit variance, and gand s are parameters coupled by s2 = 2gkBT. The weight

function o(rij) is given by

oðrijÞ ¼1� rij=rc rijorc0 rij � rc

�ð3Þ

The standard values s=3.0 and g=4.5 are used in our study.

By joining consecutive particles with a spring force, we

can construct coarse-grained models of polymers.22,23 The

harmonic spring force with a spring constant ks = 10.0 and

an equilibrium bond length a0 = 0.86 in our simulations has

the form,

FSij = ks(1 � rij/a0)nij. (4)

The total force can also have an electrostatic contribution,

which is derived from the electrostatic field solved locally on a

grid. For the dimensionless Poissons equation which is scaled

with DPD length and energy,

r(p(r)rc(r)) = �lBr(r), (5)

where lB = e2/(kBTe) is the Bjerrum length that measures the

distance at which two charged particles interact with each

other with thermal energy kBT, and p(r) is the dielectric

permittivity related to the value in pure water. With an

iterating algorithm,24,25 we can solve the field equation

successfully, then the electrostatic force on charged particle i

is given by

FEi ¼ �qi

Xj

rcðrjÞ1� jrj � rij=RePj0 ð1� jrj0 � rij=ReÞ

" #( )ð6Þ

where Re is the smearing radius and rj is the grid position for

smearing out this point charge. Details on the grid method

that we used are available elsewhere.25

The simulations are performed using a modified version of the

DPD code named MYDPD.26,27 Time integration of motion

equations is calculated by a modified velocity–Verlet algorithm22

with l = 0.65 and time step Dt = 0.04.

2.2 Mesoscopic model for charged-neutral block copolymers

Within the DPD approach, some molecules of the system are

coarse-grained by a set of particles. In our simulations,

the polyelectrolyte is modeled as a block copolymer with

section of hydrophilic and hydrophobic blocks. Specifically,

we considered a subsystem containing m diblock copolymer

chains in a salt-free solution, each with a hydrophilic block A

built with NA charged monomers and a hydrophobic block B

with NB neutral monomers. Each of the charged monomers

carries one unit of positive charge q = +1, and the total

charges carried by block copolymers are Q = mNAq. The

valence n of counterions is constrained to n = Q/NC (where

NC is the number of counterions) with the electro-neutrality

requirement in this presumed salt-free system. In our specific

case, polyelectrolyte chains each having twelve charged

monomers and four neutral monomers are encapsulated into

a cuboid cell (subsystem) of length d = 40. These diblock

polyelectrolytes self-aggregate into a spherical micelle with a

neutral core radius of RB and a charged corona thickness of

RA, as shown in Fig. 1.

We used the simple model to characterize the dilute micelle

solution with volume fraction f ¼ 43pðRA þ RBÞ3=d3. The

scaling approaches based on a simple SPB model5,11 are briefly

elucidated in the following. Following Shusharuna et al.,5 the

elastic force per chain related to the conformational entropy

losses in the stretched chain is,

Felast

kBT� R

3=2A

N3=2A a

5=20

ð7Þ

In our simple SPB model, the neutral monomers (hydrophobic

particles) are collapsed into the core of the charged-neutral

micelle, the elastic force of this part can be balanced by the

hydrophobic interactions between the hydrophobic particles

and the hydrophilic particles/solvent particle. Thus, in our

opinion, it can be ignored in the simple SPB model. In our

simulations, the charged monomer A and neutral monomer B

are set to have the same bond length. When most of the

counterions are outside the micelle, the elastic force Felast is

balanced mainly by the unscreened electrostatic force

Felec

kBT� lBQ

2

R2Am

ð8Þ

This is referred to as the ‘‘Pincus’’ regime identified by

Shusharina.5 Then, the equilibrium thickness of the corona is

RA B NA3/7au2/7m�2/7Q4/7, (9)

where u= lB/a is the relative electrostatic interaction strength.

As we have stated earlier, the total charges carried by the block

copolymers are Q = mNAq, thus, the equilibrium thickness of

the corona can be rewritten as:

RA B NAau2/7m2/7. (10)

In the osmotic regime,28 most of the counterions are inside the

corona, and the elastic force Felast is balanced mostly by the

osmotic pressure of the counterions inside the corona

Fosm

kBT� NA

nRAð11Þ

Fig. 1 Schematic representation of a charged-neutral star micelle. d is

the length of the simulation box, RA is the width of the corona formed

by the charged blocks, and RB is the radius of the core formed by

neutral blocks.

6136 | Phys. Chem. Chem. Phys., 2010, 12, 6135–6139 This journal is �c the Owner Societies 2010

And the equilibrium thickness of the corona is

RA B n�2/5NAa. (12)

Together with the scaling approaches, we carried out DPD

simulations to study the conformational behaviors of a

charged-neutral star micelle in salt-free solution.

3 Results and discussion

For simplicity, the charged-neutral micelle in monovalent-

counterion solution is firstly chosen as our model. Three

typical regions corresponding to the conformations of brush

block as a function of relative electrostatic interactions u are

displayed in Fig. 2 and Fig. 3. Within the region of the small

value of u, most of the counterions distribute into the solution

without entering into the interior of the corona region because

of the extremely weaker attractions of charged segments on

counterions, which are incapable of overcoming the large

entropy aroused by fluctuation of the ions (see Fig. 2a).

The conformation of the charged blocks on the micelles is

essentially dominated by the elastic force similar to those in the

neutral micelles made of the amphiphilic diblock copolymers.

The swelling of the PE corona of the charged-neutral micelle is

not remarkable in comparison with the neutral corona. The

region is referred to as the quasi-neutral regime. The remarkable

character of this quasi-neutral regime is that the thickness of

corona RA is independent of u. Our simulation results soundly

demonstrated this point, as indicated in u o 0.01 in Fig. 3.

In the intermediate regimes, i.e., the ‘‘Pincus’’ regime, with

the enhancement of attractive force on the counterions by

charged segments, a part of the ions are adsorbed into the

micelles, while a large number of ions still remain in solution

due to the entropy effect. Owing to the lack of sufficient

quantities of counterions within the corona region of the

micelles, only a small part of the charges on the brush block

is screened, and the majority of charges remain unscreened.

The conformation of the block in this circumstance is balanced

by two factors: repulsion among charges on the block and

recoiling force with the requirement of maximum conforma-

tional entropy of chain. The scaling theory indicates the

exponent relation of RA B u0.28 (eqn (10)) for the growth of

corona thickness with u value. Our present calculation results

are consistent with the scaling theoretical prediction (Fig. 3).

As for the system having larger values of u, more ions are

absorbed into the corona of the micelle, which results in the

rise of osmotic pressure and screening of the charges on the

polyelectrolyte. The two effects are mutually responsible for

the conformation of charged corona blocks. As indicated in

Fig. 3, the growth of the thickness of the corona starts to

deviate from the ‘‘Pincus’’ regime elucidated by eqn (10).

When the fluctuation entropy of counterions are effectively

hurdled, the ions involved in the micelle inside of the micelle

reach saturation, and most parts of the charges on the polyion

(polyelectrolyte) are screened by counterions. The charged

blocks gain a maximum extension in length and then remain

constant (Fig. 3), coinciding with the scaling prediction in

eqn (12) This region is designated in terms of osmotic regime

(Fig. 2c).

Despite agreement of scaling theory with our calculations in

the case of the monovalent counterion, the behavior of RA for

divalent and trivalent counterion is perceived to deviate from

the scaling predication in both ‘‘Pincus’’ and osmotic regimes.

For instance, the corona thickness of the charged blocks in

a di- or trivalent counterion circumstance exhibits a tiny

collapse in osmotic regime for very large u value, unlike in the

monovalent counterion where the thickness remains steady. A

detailed explanation of this phenomenon will be given later on.

Next, the details on the conformation and behavior of the

micellar corona are elucidated in Fig. 4 via a calculation of the

Fig. 2 Density profile of polyions (black line), counterions (red line)

and net charges (blue line) as a function of the radius from the

center of the micelle R with different relative electrostatic interaction

(a) u = 0.001 ,(b) u = 0.1 and (c) u = 2.5, respectively. The

morphologies on the right (solvent are omitted for clarity; Block A,

green; Block B, blue; Counterions, red) correspond to each density

profile.

Fig. 3 Micelle corona thickness RA as a function of relative electro-

static interaction u.

This journal is �c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 6135–6139 | 6137

pair correlation functions, g(r), between charged segments and

the counterions of mono-, di-, and tri-valence, respectively. In

a simulation, it is straightforward to measure g(r), which is the

ratio between the average particle number density r(r) at a

distance r from a reference particle and the density at a

distance r from a particle in an ideal gas at the same overall

density. It has been previously proposed that the following

equation can be used to calculate the g(r),29

gðrÞ ¼ rðrÞVN¼ nðrÞV

4pr2DrNð13Þ

where n(r) is the number of particles that are a distance

between r and r + Dr away from the reference particle, and

N and V are the total number of particles in the system and the

volume of the system, respectively. The relative higher peak

of g(r) for trivalent counterions reveals the existence of a

relatively strong correlation between charged segments and

the ions with multiple valences. Furthermore, an obvious

appearance of the second peak in the cases of di- and

trivalence indicates the establishment of somewhat ordered

structures of the ions around the charged segments, implying a

strong condensation of multivalent counterions yielded by the

strong electrostatic correlations. The drop of osmotic pressure

in the system can be attributed primarily to the condensation

of the counterions.15–17 Moreover, the reduction in the

number of multivalent ions with the charged-neutral requirement

in the system is another subordinate reason for the drop of

osmotic pressure. This clarifies why the brushes formed by

these charged blocks collapse in multivalent ion solution for

the larger value of u (Fig. 3).

It is reasonable to conceive that the strong correlation

between the charged segments and the counterions can

account for not only the localization of ions in the space

around the charged segments but the suppression of the

mobility of these ions in the solution as well. The self-diffusion

coefficient D0 of the ions is calculated based on the function of

the time dependence of average mean-square displacement

(MSD) hDr2i = 6D0t (inset of Fig. 4). Compared with the

mono- and di-valence cases, the mobility of trivalence is

significantly hindered, implying the ‘‘freeze’’ effect of ions

inside the corona of the charged-neutral micelles.

Based on the above analysis, conformations of the charged

blocks, stretched out or collapsed on the spherical core built in

hydrophobic chains, is intimately correlative to the strength of

electrostatic interaction of two charged segments and that

of ions and segments. Upon the increase in the relative

electrostatic interaction u, the thickness of brush RA increases

in an S style as the system goes from the quasi-neutral to the

‘‘Pincus’’ regime and then to osmotic regime (Fig. 3). To

provide a deeper insight into the process, the dependence of

electrostatic potential energy E on the relative electrostatic

interaction u is presented.

E ¼Z

rðrÞcðrÞ � pðrÞ8plB

jrcðrÞj2� �

dr: ð14Þ

To calculate the electrostatic potential energy, we follow the

iterating algorithm of Beckers et al.24 and the grid method of

Groot,25 where the electrostatic field is solved on a grid. This

method has been used to evaluate the electrostatic force FE

in our simulations, it is consistently used to evaluate the

electrostatic potential energy E. Calculation strategies adopted

here permit the consideration of the inhomogeneity of the

electrostatic permittivity in the system.

In the quasi-neutral regime, as the majority of the counter-

ions are distributed in the solution without being contained

inside of the micellar corona, positive and negative charges

do not screen each other at all, and the electrostatic potential

energy is expected as E p u, as shown in Fig. 5. In the

‘‘Pincus’’ regime, although the electrostatic potential energy

still increases with the relative electrostatic interactions u,

it deviates from the linear relationship of E p u, due to

the screening effect produced by the counterions adsorbed

inside of micelles. In the osmotic regime, more counterions

intrude into the coronal part of the micelle due to strong

relative electrostatic interactions. The screening effects are

also strengthened heavily, especially on the multivalent

counterions.30 Thus, we can observe an obvious decline of

Fig. 4 Pair correlation function g(r) between charged segments and

charged segments in osmotic regime at u = 2.5. Inset: MSD (hDr2i) ofcounterions as a function of reduced time at u=2.5. The self-diffusion

coefficient D0(n=+1) = 0.83, D0(n=+2) = 0.46 and D0(n=+3) =

0.21. (The black, red and blue lines represent the valence of the counterions

as +1, +2 and +3, respectively.)

Fig. 5 Electrostatic potential energy E as a function of relative

electrostatic interaction u.

6138 | Phys. Chem. Chem. Phys., 2010, 12, 6135–6139 This journal is �c the Owner Societies 2010

electrostatic potential energy E, as shown in the inset of Fig. 5,

and the transition from ‘‘Pincus’’ regime to osmotic regime

appears.

4 Conclusion

In this paper, we have made a systematical analysis of

the conformation transition of spherical polyelectrolyte

brushes in salt-free solution of charged-neutral micelles under

different valent counterions using the dissipative particle

dynamic simulation. Our calculation results indicate that the

scaling predictions can well match our simulation results for

the case of monovalent counterions in the systems but deviate

from those for multiple valence counterions. The deviation

implies that the scaling analysis fails for the treatment of

complex circumstances such as those existing in the strong

correlation between the charged segments and counterions in

the multiple valence counterions system. In our simulation, we

found that the trivalent counterions can condense to the

charged segments when electrostatic interactions are extremely

strong. This condensation may suppress the osmotic activity

of the trivalent counterions inside the micelle corona and lead

to the collapse of the corona. Moreover, the transitions

from quasi-neutral to ‘‘Pincus’’ regime and from ‘‘Pincus’’ to

osmotic regime are clearly understood in terms of electrostatic

potential energy.

Acknowledgements

We would like to thank the two anonymous referees whose

critical comments helped us in improving the quality of our

manuscript. We are grateful for the financial support provided

by the Program of the National Natural Science Foundation

of China (Nos. 20934004, 20874094 and 50773072), NBRPC

(Nos. 2005CB623800 and 2010CB934500). X. Li would like to

acknowledge the financial support provided by China

Postdoctoral Science Foundation (No. 20090460729) the

Fundamental Research Funds for the Central Universities,

the K. C. Wong Education Foundation, Hong Kong. Parts of

the simulations were carried out at the Shanghai Super-

computer Center.

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