Interface Orientation and Chain Conformation in Simulated Symmetric Diblock Copolymer Lamellar Systems

Interface Orientation and Chain Conformation in

Simulated Symmetric Diblock Copolymer

Lamellar Systems

Jian Yang,1 Mitchell A. Winnik,*1 Tadeusz Pakula*2

1 Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, Ontario, Canada M5S 3H6Fax: (þ1) 416 978-0541; E-mail: [email protected]

2 Max-Planck-Institute fur Polymerforschung, D-55021 Mainz, Postfach 3148, GermanyE-mail: [email protected]

Received: October 14, 2004; Accepted: November 11, 2004; DOI: 10.1002/mats.200400073

Keywords: chain conformation; diblock copolymers; interfaces; junction vector correlations; Monte Carlo simulation

Introduction

The effect of interfaces on the properties of heterogeneous

materials increases with decreasing dimensions of the hete-

rogeneities. In block copolymers, where the heterogeneity

scale is directly related to the sizes of macromolecular

fragments (blocks), it goes down to the level of even few

nanometers, that is, to the size scale of the interface itself. In

such a case, the interfacial fraction becomes predominant in

various aspects. This is particularly interesting in terms of

the detailed information concerning the structure of inter-

faces. Such knowledge can become important for under-

standing and consequently for the prediction of properties

of block copolymer materials.

Both experimental[1–5] and computer simulation[6–10]

methods have been applied to analyze composition profiles

in copolymers of immiscible blocks undergoing microphase

separation to the characteristic morphologies: spherical,

cylindrical, or lamellar, in accordance with their composi-

tion. Most of the attention has been directed towards the

simplest lamellar morphologies for which information at

the molecular level was obtained by means of X-ray and

Summary: We describe the results of Monte Carlo simula-tions, based on the cooperative motion algorithm, of the lame-llar structure generated at finite temperature by a symmetricdiblock copolymer. The (70� 70� 70) simulation box inwhich the polymer chains were embedded for each simula-tion was rotated, based on the interface orientation, to bringthe interfacial planes of the simulated structure into parallel.We found that the interface thickness, as defined by the dis-tribution of the junction points, became narrower at lowertemperature, and that the interface plane was characterizedby a waviness with a maximum peak-to-valley distance of20–30 lattice bonds. Compared with the isotropic state(T/N¼1), chains at lower temperatures were stretched inthe direction perpendicular to the interface; but only modestlycompressed in the direction parallel to the interface. Indivi-dual block chains within the lamellar domains still behavelike random coils. The block copolymer molecules exhibitonly a modest tendency to orient themselves with their end-to-end vector perpendicular to the plane of the lamellar inter-face. Considered as an ensemble average, the results we ob-tained are similar to those reported from small angle neutronscattering measurements for the mean conformation of thePSd blocks of symmetrical PSd-PVP diblock copolymers.

2-D projections onto the X-Z plane of the end beads for theA- and B-chains (gray) and the junction points J (black) atT/N¼ 0.2. The interface plane is oriented parallel to theY-Z plane by rotating the simulation box. The distributionprofiles of junction points and the end beads across the systemin the direction of interface normal are shown in the lowerpart of the figure.

Macromol. Theory Simul. 2005, 14, 9–20 DOI: 10.1002/mats.200400073 � 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Full Paper 9

neutron scattering[1–3] and reflectivity[4,5] measurements.

Early experiments yielded values for the breadth of the

interface higher than predicted from theoretical description

of composition profiles,[4,5] and also found chain exten-

sions in the direction perpendicular to the interface[1–3] that

at the time had not yet been considered theoretically.

Computer simulations of the microphase separated copo-

lymer states offer the possibility to obtain even more details

about the interfacial structures.[6–11] This information in-

cludes: descriptions of chain deformation, orientation

correlations, chain end distributions, and concentration

profiles at the interfaces of the simulated diblock copolymer

systems.

In this paper, we attempt to extend our knowledge about

interfacial structures in symmetric diblock copolymer sys-

tems with alternating lamellae that develop below the

microphase separation temperature. This investigation is

performed in relation to ongoing experimental studies of

block copolymer interfaces by the direct nonradiative

energy transfer (DET) technique.[12,13] Proper interpreta-

tion of these experimental data depends on knowledge

about the orientation of copolymer chains at the interface

and the distribution of junction points between the blocks

within the interface. The simulation method used here is

the cooperative motion algorithm (CMA),[6,7] which has

already been successfully applied for simulation of diblock

copolymer melts under various conditions ranging between

the homogeneous and the strong segregation limits.

Simulation Method

General Conditions and Relation to Former Results

The dynamic Monte Carlo simulations of symmetric di-

block copolymers in bulk were carried out using the coope-

rative motion algorithm, which has been designed for dense

systems.[14–17] Simulations are performed on a face cen-

tered cubic (fcc) lattice (coordination number z¼ 12) with

bond length H2a (a is the half of the fcc lattice constant).

The possible bond angles are 60, 90, 120, and 1808. The

simulations are carried out in a 70a� 70a� 70a simulation

box, and periodic boundary conditions are applied in all

three dimensions in order to reduce boundary effects. The

polymer chain length in the simulation is N¼ 70. The

chains are considered as consisting of two blocks of diffe-

rent monomers A and B in the composition 1:1. The first

35 beads of each chain constitute the block A, and the

second block, B, starts from the 36th bead. All of the lattice

sites are occupied by monomers, and are constrained by the

excluded volume condition (all chains are self- and

mutually-avoiding) so that no lattice sites can be occupied

by more than one monomer. Nonbreakable bonding of

beads into chains and the excluded volume condition cons-

titute the strongest interactions in the systems and are

athermal. The additional interaction between polymer

chains resulting in a partial incompatibility of the A and

B components is governed by the direct interaction

parameters, eij, between neighboring monomers. We

assume eAA¼ eBB¼ 0 and eAB¼ 1. Under such conditions,

the thermodynamic state of the system can be characterized

by the Flory interaction parameter [Equation (1)].

wF ¼ ðz� 2ÞeAB

kBTð1Þ

where T is the temperature, and kB is the Boltzmann

constant, which is assumed here to take a value of 1. For

convenience, we use the ratio of temperature to chain length

(T/N), instead of wFN, to describe the strength of interactions

in the system. At various temperatures, the systems can be

equilibrated by the strictly cooperative dynamics charac-

teristic of the CMA method, according to which in a dense

system (r¼ 1) a segment of one chain can only move if

other segments of different chains move simultaneously.[14]

The motion changes the local energies since the elements

contact new neighbors. An attempt to move a single mono-

mer is defined as one Monte Carlo Step (MCS). We take the

simulation time unit (TU) to be the time when the number of

attempts is equal to the number of beads in the system. The

probability of motion is related to the interaction energy of

the monomer with neighbors in the attempted position. At a

given temperature,T, the Boltzmann factor p¼ exp(-En,final/

kBT) is compared with a random number r (0< r< 1). If

p> r, the motion is favorable and a new attempt is

performed. If p< r, the motion is rejected. Since there is a

repulsive interaction between monomers of different type

(eAB> 0), these monomers tend to separate from each other

at low temperature to reduce the number of A-B contacts,

and consequently, to reduce the energy of the system. All of

the simulations are allowed to equilibrate for at least

2� 107 TU until the averaged interaction energy remains

unchanged for a predetermined length of time. The effective

energy of any specific monomer En, which depends on the

local structure, can be written as [Equation (2)]:

En ¼Xzi¼1

ejkðiÞ ð2Þ

where j and k can be either A or B, depending on the local

configuration.

Figure 1 shows an example of variation of the average

interaction energy and the specific heat with temperature for

a symmetric diblock copolymer system studied previou-

sly.[6] The dependencies allow one to infer that the micro-

phase separation temperature TMS is nearly proportional to

N, which justifies the use of the T/N scale.[6,7] According to

the simulation results, this temperature separates the strong

and weak segregation regimes observed below and above

TMS, respectively. At high temperatures the systems

become homogeneous.

10 J. Yang, M. A. Winnik, T. Pakula

Macromol. Theory Simul. 2005, 14, 9–20 www.mts-journal.de � 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In this paper, we analyze the structure of model systems

at various temperatures below TMS, that is, in the strong

segregation regime. These strongly segregated states are

obtained by quenching systems equilibrated initially under

athermal conditions (eAB¼ 0) to various finite temperatures

(eAB¼ 1) below TMS (indicated by the dashed lines in

Figure 1). In ref.[6] we described the variation of the average

interaction energy with temperature for a similar but shorter

symmetric diblock copolymer system (N¼ 40). The verti-

cal solid line in Figure 1 indicates the position of the

microphase separation transition (TMS) identified for that

system and the dashed lines in Figure 1 indicate the tem-

peratures of systems studied in this work.

Figure 2 shows examples of the final morphologies ob-

tained in the simulated systems quenched from the

homogeneous states to various temperatures (T/N¼ 0.2,

0.3, and 0.4). In each instance, and at each of the three

different temperatures, the diblock chains within the box

evolved to lamellar monodomains. None of the simulations

resulted in a box containing a grain boundary with lamellar

microdomains oriented in different directions.

Determination of the Orientation ofLamellar Interface

Since the simulation is a kind of random sampling tech-

nique, the orientation of the lamellar interface for each

simulated symmetric diblock copolymer system cannot be

controlled unless one prealigns the interface prior to

mixing.[18–21] In our approach, as in most Monte Carlo

methods, the initial system is generated as a homogenous

state. The evolution of the structure, including the lamellar

orientation is fully dependent upon the random numbers

and a sporadic nucleation when the conditions for micro-

phase separation are reached (T< TMS). The orientation of

the lamellae generated in each simulation cannot be

controlled a priori. To analyze the results of multiple simu-

lations in a common framework, it is necessary to have a

method to determine the orientation of the lamellar

interface for all of the simulation runs that generate lamellar

morphologies. From various possibilities we have chosen

the one described below.

Figure 3 shows the schematic illustration of the lamellar

system represented by interfacial planes in the coordinates

related to the simulation box. Geometrically, the interfacial

planes intersect the planes XY, YZ, and ZX of the coordinate

system along parallel lines. The XY, YZ, and ZX planes at

integer values of the corresponding normal coordinates are

regarded asZ-,X-, andY-layers, respectively. Parallel planes

in the simulated lamellar structures should have corres-

ponding parallel lines of intersection on each layer within

the interfaces. Distances between the interfacial parallel

planes of the lamellar systems have the following relation-

ship with the distances between the parallel lines of inter-

section with the layers normal to the system coordinates

[Equation (3)]:

HX-layer ¼H3Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1�sin2a cos2 yp

HY-layer ¼H3Dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1�sin2a sin2yp HZ-layer ¼

H3D

sina2

H23D

¼ 1

H2X-layer

þ 1

H2Y-layer

þ 1

H2Z-layer

ð3Þ

Figure 1. An example of variation of the average interactionenergy and specific heat with temperature for the symmetricdiblock copolymer system (N¼ 40) studied previously.[6] Thevertical solid line indicates the position of the microphaseseparation transition (TMS) and the dashed lines indicate thetemperatures of systems studied in this work.

Figure 2. Three-dimensional equilibrated morphologies for symmetric diblock copolymers (chainlength¼ 70) in a 70� 70� 70 face centered cubic (fcc) lattice box at three different temperatures asindicated in the figure. The different blocks are identified by gray shades and individual bonds are notshown.

Interface Orientation and Chain Conformation in Simulated Symmetric Diblock Copolymer Lamellar Systems 11


where HZ-layer, HY-layer, and HX-layer are the distances

between the parallel lines of intersection of the interfacial

planes with the Z-layer, the Y-layer, and the X-layer,

respectively. Here, a is the angle between the interface

normal and the Z-axis, and y is the angle between the pro-

jection of interface normal on the XY plane and the

X-axis. All of the parameters in Equation (3) are illustrated

in Figure 3.

According to Equation (3), we can determine the orien-

tation of the lamellar interface normal (a, y) based on the

known lamellar spacing in three dimensions and known

spacings between intersection lines on all layers in the

simulated diblock copolymer system. All H distances have

been determined on the basis of correlation functions

describing distributions of junction points between the

blocks of the copolymer chains, defined as follows

[Equation (4)]:

rðrÞ ¼ 1

nk

Xnki¼1

1

nðriÞXnðriÞj¼1

cðri; jÞ ð4Þ

where nk is the total number of polymer chains (4 500) in the

system and c(ri,j) refers to the correlation density of

junction points j at a distance ri from the chosen junction

point i. n(ri) is the total number of junction point pair (i,j) at

the distance ri. The midpoint of each chain is regarded as the

junction point, a virtual point in the lattice model located at

the A-B-bond, halfway between beads 35 and 36 of the

chain. The correlation curves were summed for every

junction point considering all other junction points around

it within a sphere (or a circle in two dimensions) with a

radius up to 70 (the box dimension). Figure 4 shows an

example of correlation curves for the simulated system with

T/N¼ 0.2. For the system at finite temperatures, one can

Figure 3. Schematic representation of the relationship between the orientation of thelamellar structure and the X-, Y-, and Z-axes of the laboratory coordinate system. In (a), H3D

refers to the lamellar spacing as defined within the axis system of the simulated lamellae. Theterms HX-layer, HY-layer, HZ-layer refer, respectively, to the distances between adjacent parallellines of intersection of the lamellar planes of the simulated structure with the Y-Z,X-Z, andX-Y planes in the Cartesian laboratory axis system. The interface normal points in the samedirection as theH3D vector a is the angle between the interface normal and the Z-axis, while yis the angle between the projection of interface normal on theX-Y plane and theX-axis. In (b),we show planes corresponding to a series of adjacent interfaces in the lamellar structure andtheir orientation within the Cartesian laboratory axis system. The distances corresponding tothe terms HX-layer, HY-layer, HZ-layer are indicated in the figure.

Figure 4. Values (solid points) of the correlation functionrðrÞ asdefined in Equation (4), for the entire system in three dimensionsand within the individual X-, Y-, and Z-layers, for the simulateddiblock copolymer lamellae at T/N¼ 0.2. The simulated morphol-ogy is also shown in the figure. The dashed line in gray in theupper-most figure is the curve showing the absence of correlationsobtained from the simulation of the homogeneous state atT/N¼1. The solid lines are the fits to the correlation curvesusing Equation (5).



observe periodic correlations, whereas there is almost no

correlation in the homogeneous state.

In order to precisely determine the correlation period (H),

a periodically decaying function [Equation (5)]

rðrÞ ¼ P1

rP5cosð2pP2r þ P3Þ þ P4

H ¼ffiffiffi2

p

P2

ð5Þ

has been used in which the adjustable parameters Pi (i¼ 1–

5) were obtained by fitting the correlation curves obtained

for the simulated lamellar systems. The parameters P2

correspond to the characteristic periods Hi (in units of bond

length) of the simulated diblock copolymer lamella. The

orientation of the interface can be then determined using

Equation (3) with the Hi values obtained from the fitting

performed for the three layer orientations, giving HX, HY,

and HZ, as well as, for the global correlation giving H3D.

As examples, we present in Table 1 the lamellar spacings

and orientations obtained for three simulated systems at

various temperatures. In all of the orientation determina-

tions, the orientations of interfaces (a, y) have errors not

exceeding 38.The results presented in Table 1 show that the lamellar

spacings that we obtained do not vary monotonically with

temperature. This result is probably related to the fact that

we impose a large temperature jump upon the system in

cooling from the homogenous state. One expects the lamel-

lar spacing to decrease as the temperature is increased, and

previous simulations involving many small temperature

steps show these changes.[6] We also performed parallel

simulations with smaller temperature jumps in a smaller

simulation box (30a� 30a� 40a, polymer chain length:

N¼ 40) and observed this variation. Although the simula-

tions with small temperature jumps are essential for study-

ing the kinetics of the system in response to a change in

temperature, such as changes in the vicinity of the order–

disorder transition (ODT), the large temperature-step

procedure that we employ here prevents the lamellar phase

from adjusting to the continuously changing equilibrium

period.[22,23] Once the lamellar morphology is formed at a

temperature lower than ODT, it is very difficult for a system

to adjust its orientation and its lamellar spacing to another

temperatures because of the high potential barrier of mov-

ing an entire polymer from one lamella to another or to

reorient the whole domain. Since we are interested in the

equilibrium state of polymer chains with respect to the

junction plane of individual lamellae, we can accept

the limitations of nonideal lamellar spacing. We let the

temperature change directly from infinity (the homogenous

state) to the desired temperature, and allow the polymer

chains to equilibrate for a sufficiently long period of time

(typically 2� 107 TU) to ensure that the system locally

reaches equilibrium within each lamella.

Results and Discussion

Interface Characterization

Having information about the orientation of the parallel

planes of a lamellar structure with respect to the systems

coordinates, we can analyze the lamellar systems in the pro-

jection on planes perpendicular to the interfaces. Figure 5

illustrates such results. Figure 5a–c shows examples of the

projections of the junction points and the end beads of each

block for all three temperatures. Figure 5d shows an ex-

panded view at T/N¼ 0.2 of the B-block domain, in which

we show the ensemble-averaged distributions of the end-

beads. Here, we distinguish chain ends for chains emanat-

ing from different adjacent interfaces, and we show the

summed end-bead distribution as well. We see that the two

subclasses of B chains having the block junction points in

different interfaces interpenetrate each other over the entire

B-block region.

Figure 6 shows the ensemble-averaged distributions of

the block junction points for the three temperatures. Accord-

ing to Helfand–Tagami theory (HTT),[24] the junction

profile across an interface can be described by the expres-

sion [Equation (6)]:

PðxÞ ¼ 2pd

sech2x

d

� �ð6Þ

where x is the distance across the interface (x¼ 0 at the

interface), and d is the interface thickness. We fitted the

curves obtained in the simulation by Equation (6) and

obtained the ratios of interface thickness to the half-

lamellar spacing (d/(H/2)) as 0.21, 0.33, and 0.62 for T/

N¼ 0.2, 0.3, and 0.4, respectively.

Geisinger et al.,[10] as well as Vassiliev and Matsen,[22]

have established that the interface thickness obtained in a

Monte Carlo simulation is much broader than that charac-

terized by mean field theory, since the mean field ignores

capillary wave fluctuations.[25] In order to see this effect in

our simulations, we have chosen one of the interfaces

(T/N¼ 0.2) and plotted it in the form of a surface profile, as

Table 1. Lamellar spacings (in units of bond length) for thesimulated structure for three temperatures.

T/N H3Da) HX-layer

b) HY-layerb) HZ-layer

b) ac) yc)

8 8

0.2 20.6 22.3 22.9 36.2 35 470.3 17.7 24.5 25.6 17.7 90 460.4 18.5 19.5 24.5 25.7 46 65

a) Distance between adjacent planes in the as-generated lamellarstructure.

b) Distances between parallel lines of intersection of the lamellarplanes with theXY, YZ, andZX planes as defined in Equation (3).

c) a is the angle between the interface normal and the Z-axis, and yis the angle between the projection of interface normal on theXYplane and the X-axis.



shown in Figure 7a. The waviness of the interface can be

observed in the figure, but there are no clear-cut methods

available to characterize this waviness. Figure 7b shows a

projection of the interface shown in Figure 7a onto the

perpendicular plane. We calculated the standard deviation

of the junction position across the interface to be�1.9a (a is

the half of the lattice constant in fcc). From an estimate of

the span of the interface waviness on the bases of the profile

shown in Figure 7a, we obtain a value of the order of

20a�30a.[26]

Conformation Analysis

In Table 2, we collect values for a variety of mean-squared

distances that characterize the block copolymer chains at

different temperatures. With a decrease of temperature,

the mean-squared end-end distance (hre-e2 i), the mean-

squared radius of gyration of polymer chains (hrg2i), and the

mean-squared distance between the center of mass of the

A- and B-blocks within the copolymer (hrAB2 i), all increase.

This result indicates that the polymer chains are stretched

Figure 5. For simulation boxes rotated so that the interface plane is parallel to the Y-Zplane, (a)–(c) present for various temperatures two-dimensional projections onto the X-Zplane of the end beads for the A- and B-chains (gray) and the junction points J (black). Thedistribution profiles of junction points and the end beads across the system in the direction ofinterface normal are also shown in the lower part of each figure. In (d) we show the ensembleaveraged B-chain end-bead distributions along the direction normal to the interface atT/N¼ 0.2. The solid gray circles refer to chains whose junction is anchored in the right-handinterface and the open circles refer to chains emanating from the left interface. The solidsquares describe the overall end-bead distribution of the B-block. The solid lines are thecorresponding smoothed distributions.

Figure 6. Ensemble averaged interface profiles for all threetemperatures as shown in the figure. The profile curves were norma-lized by the area of the individual peaks. The solid lines are the fits tothe distributions using Helfand–Tagami theory [Equation (6)].



and separated as the temperature is decreased. But the

mean-squared radius of gyration of the individual blocks

(hrgA2 i) and (hrgB

2 i) did not change very much, suggesting

that the blocks are not very perturbed in their mean

dimensions in these well-defined lamellae. Vassiliev and

Matsen[22] found similar results in their Monte Carlo simu-

lations of diblock copolymers, but they also argued that a

change in the radius of gyration is not sufficient to charac-

terize the perturbation of a chain in a lamellar structure,

since it is not uniquely related to the shapes of the individual

blocks.

In order to delve further into the issue of chain confor-

mation in the simulated diblock copolymer systems we

considered two correlation functions (frr-ee and frr-AB),

which characterize the chain stretching and orientation

inside the system [Equation (7)]:

frr ¼1

23 cos2yij� �

� 1� �

ð7Þ

In this expression, frr is the orientation correlation para-

meter.[6] yij is the angle between the end-to-end vectors

(directed from the end of the A-block to the end of the B-

block) for frr-ee, or between the center of mass vectors (rAB�!)

for frr-AB. Table 3 shows the correlation parameters at the

three different temperatures in comparison with the ather-

mal state. From both quantities, we see that the polymer

chains gradually become more aligned with decreasing

temperature. The orientation correlation related to rAB�!

reaches higher values because it does not include effects of

chain end freedom. The small values of the correlation

parameters indicate that the polymer chains are not strongly

oriented in the phase-separated state.

Since we have already determined the orientation of the

interface (Table 1), we can calculate the average angle

between the end-to-end vector and the interface (yee-inter),

the average angle between rAB�!and the interface (yAB-inter),

and finally, the ‘‘bending’’ angle, yA-j-B (the angle between

two vectors emanating from the block junctions and poin-

ting to the centers of mass (CM) of the two blocks). In order

to make a comparison with the homogenous system

(T/N¼1), we arbitrarily chose three virtual interface

orientations as shown in Table 4. The identical angle values

for all three different virtual orientations indicate isotropic

properties in the homogeneous state. With the decrease of

temperature, the angles increase as a result of chain

Table 2. Values of characteristic mean-square distances for thesimulated structure at the athermal state and the three finitetemperatures.

T/N hre-e2 ia) hrg

2ib) hrgA2 ic) hrgB

2 ic) hrAB2 id)

1 90.7 15.4 7.5 7.5 31.60.4 112.2 18.2 7.5 7.4 43.10.3 121.5 19.4 7.6 7.6 47.20.2 136.3 21.3 7.7 7.8 54.2

a) Mean squared end-to-end distance.b) Mean squared radius of gyration of the entire chain.c) Mean squared radius of gyration of the A- and B-blocks,

respectively.d) Mean squared distance between the centers of mass (CM) of the

A- and B-blocks.

Figure 7. (a) Three-dimensional interface plot representing thepositions of beads from A-block that are in contact with at least onebead from B-block, (denoted as ‘‘interface beads’’) for one of theinterfaces in the simulated system at T/N¼ 0.2. The gray scale barrepresenting the interface-normal direction is also shown. (b) Theprojection onto the X-Z plane of interface beads of A-block asshown in Figure 7a. The solid line is the averaged position(X¼ 46.9) and the dotted lines refer to one standard deviation(�1.9a).

Table 3. Magnitude of the correlation parameters describingchain orientation at the athermal state and at the three temperaturesbelow the TMS.

T/N frr-ee frr-AB

1 2.3e-4 �3.6e-40.4 0.033 0.0790.3 0.073 0.140.2 0.11 0.22



extension. The yA-j-B angle has values deviating signifi-

cantly from 1808, which could be regarded as a bending of

the chain, but in fact it results from the nature of bead

distribution within the ellipsoidal statistical chain in which

a displacement of the junction point away from the axis

connecting the centers of mass of the blocks is predicted.[27]

Distributions of the angle (yee-inter) normalized by

considering the volume occupied by chains oriented at

different angles, using the expression [Equation (8)]:

Pðyee-interÞ ¼nðyee-interÞ

2pr sin yee-interdyee-inter

ð8Þ

where n(yee-inter) is the number of chains, which have the

same value of yee-inter are shown in Figure 8. We can see that

the normalized distributions (P(yee-inter)) for systems at low

temperatures have very broad maxima at around 908 and

minima at 08, whereas P(yee-inter) at T/N¼1 shows a

nearly constant value.

Another interesting parameter for examining chain stret-

ching along the polymer backbone is the correlation of vec-

tors representing the backbone orientation at the junction

point between the blocks (the ‘‘junction vector’’) as a

function of the spatial separation of the junctions belonging

to different chains. This parameter conveys information

about how the polymer backbones influence one another.

We define the junction vector for each diblock copolymer

chain in terms of the four beads that span the junction point,

two on the A side and two on the B side as shown in

Figure 9,[28] where the A-block is shown in gray and the

B-block is shown in black. The junction vector is directed

from the midpoint between beads 34 and 35 of the A-block,

immediately adjacent to the A/B junction, to the midpoint

between beads 36 and 37 of the B-block.

We define the following correlation function [Equa-

tion (9)]:

cjvðrÞ ¼1

23 cos2 yðrijÞ� �

� 1� �

cos2 yðrijÞ� �

¼ 1

nk

XNk

i¼1

1

nðriÞXnðriÞj¼1

rjv!ðiÞ � rjv!ðri; jÞrjv!ðiÞ � rjv

!ðri; jÞ

!2

ð9Þ

where, nk is the total number of polymer chains in the

system, n(ri) is the total number of junction vector pairs (i,j)

at distance ri, rjv!ðiÞ is the junction vector of one selected

polymer chain, and rjv!ðri; jÞ is j-th junction vector located at

a distance ri from rjv!ðiÞ. The calculation of the junction

vector correlation was carried out for an ensemble over all

of the junction vectors within a sphere of radius: ri (up

to R¼ 35) centered at rjv!ðiÞ, and then is summed for all of

the blocks junction vectors. The results are presented in

Figure 10 for the athermal case and the three temperatures

below TMS. There are no correlations for the athermal case,

but weak correlations at finite temperatures that increase

in magnitude at lower temperatures. All the above results

Table 4. Angles in the simulated systems at four temperatures.

T/N Orientation of interfacea) yee-interb) yCM-inter

c) yCM-j-CMd)

8 8 8

1 a¼ 1508; y¼ 268 32 33a¼ 608; y¼ 368 34 33 94a¼ 368; y¼ 3008 33 33

0.4 a¼ 468; y¼ 658 41 46 1110.3 a¼ 908; y¼ 468 46 51 1150.2 a¼ 358; y¼ 478 49 55 120

a) ForT/N¼1, we arbitrarily selected three different planes as thevirtual interface for calculations.

b) Mean orientation of the end-to-end vector with respect to theplane of the interface.

c) Mean orientation of the vectors connecting the CMs of theA- and B-blocks with respect to the plane of the interface.

d) Mean value of bending angles for the diblock copolymer chains.The bending angle for one polymer chain is formed by twovectors originating from the junction point, one directed to theCM of the A-block, the other to the CM of the B-block.

Figure 8. Temperature dependences of P(yee-inter), shown with-out error bars for clarity. The solid lines are the correspondingsmoothed curves (Table 4).

Figure 9. The junction vector for each diblock copolymer chainin terms of the four beads that span the junction point, here withtwo on the A side and two on the B side (A-block: gray; B-blockblack).



show that the junction points of polymer chains confined to

well-defined diblock lamellae are not strongly correlated

nor are the chains highly oriented. This behavior can be

regarded as resulting from the rather weak correlations of

local orientations of chain fragments with the global chain

position. When the whole chain orientations observed in the

lamellar system are not very strong (maximum values of

frr-ee are not higher than 0.1), the orientation correlation on a

smaller scale can be lost almost completely (the maximal

values of cjv do not exceed 0.02). It may also be worthwhile

to notice that the temperatures at which these extremal

values were observed in simulation are not usually acces-

sible for experiments. The observation of negligibly small

orientation of the junction vectors will help us to analyze

experimental data in our studies of energy transfer in di-

block copolymer systems.[29]

In order to obtain a deeper insight into the conformation

and orientation of the chains within the lamellar structure,

we consider a single simulation box at each temperature and

translate all of the polymer chains to a common point by

shifting the virtual junction point for each polymer chain to

the origin of our coordinate system. This kind of translation

does not change the dimensions or orientation of any chain.

We then project the coordinates of the beads of the ensem-

ble of chains onto the XZ plane. (The interface layers lie in

Y-Z planes.) The results of this procedure are shown in

Figure 11. For T/N¼ 0.4, 0.3, and 0.2, the values of a and yshown in the figure correspond to the box rotation necessary

to align the lamellae with the laboratory coordinate system.

Two images are shown for T/N¼1, representing arbitrary

(virtual) orientations of the box for chains in the homo-

genous state. To simplify the representation, we designate

as A-chains those whose centers of mass fall at lower X

values than the centers of mass of the other half of the chain

(i.e., XCMA<XCMB). On the left side of the figure, the

images describe the distribution of beads of the A-block,

shown in green, and the distribution of the B-block, shown

in red. The apparent segregation seen in the top image on

the left-hand side is a consequence of the choice of

XCMA<XCMB to specify the direction of individual chains.

That this segregation is apparent and not real is indicated by

an essentially identical distribution of colors for the same

box rotated as indicated by the second example forT/N¼1in Figure 11.

The images on the right side of Figure 11 depict the

distribution of the centers of mass of the A-block in green,

of the B-block in red, and of the entire chain in blue. As the

temperature is lowered, the blue domain becomes more

concentrated in the center, and the green and red domains

become strongly segregated to opposite side of the Z-axis.

Figure 10. Correlation of the vectors representing the localbackbone orientation, calculated using Equation (9), as function ofthe distance between junctions.

Figure 11. Structures in which the lamellar structures areoriented so that the interface is parallel to the Y-Z plane. Leftside: Projections onto the Z-X plane of chain bead distributions foreach temperature. All chains were translated by setting thejunction point as the origin, and to organize the blocks, weassumed that XCMA<XCMB. The interface normal is along X-axis.Beads from A-block are shown in green, while beads from B-blockare in red. Right side: X-Z plane projections of the centers of mass(CM) of the entire polymer chain (CM, blue), A-block (CMA,green), and B-block (CMB, red). The upper and lower lines forT/N¼ 0.4, 0.3, and 0.2 indicate the lamellar spacings at eachtemperature.



To put these results in perspective, and to facilitate compa-

rison of the centers of mass distribution in the right-hand

column with the segment distribution shown in the left-

hand column, we drew horizontal lines for T/N¼ 0.4, 0.3,

and 0.2 in gray. The upper and lower lines indicate the

lamellar spacing, and the middle line represents the plane of

the interface. On the left side of the figure, one can see that

as the temperature is lowered from T/N¼ 0.4 to 0.2, the

segregation of A and B segments increases, and fewer

chains protrude beyond the limits of the lamellar spacing.

In summary, with the decrease in temperature, the two

blocks tend to separate and the shape of the conformation

ensemble changed from spherical to dumbbell-like. The

change in spatial distribution of the centers of mass is more

obvious than the change in the distribution of beads. A

sketch illustrating the shape change for the segment distri-

bution is shown in Figure 12. The different gray shades

represent the different blocks. The reason for the shape

change is that the blocks separate and the individual chains

become more confined to the interface normal direction.

Nevertheless, these changes are not large. There is still

substantial interpenetration of the two blocks across the

interface even at the lowest temperature considered here.

Finally, it should be noted that the representation of

polymer behavior at the interface such as presented in

Figure 11 makes qualitatively a slightly different impres-

sion than the diagrams of this kind reported in earlier

publications.[6,30,31] Nevertheless, these results are not

in contradiction. The difference results from different

choice of reference points. In this paper, the block junction

points and, in the earlier reported cases, the centers of

mass were used as the reference. The natural mean dis-

placement of the junction point from the center of mass of

chains,[27] mentioned earlier in this paper, resulting from a

distribution of the junction around the mass center explains

the effect.

Since the spatial distribution for individual chains has a

different concentration profile for chains oriented at dif-

ferent angles, which cannot be seen in this plane projection,

we cannot say if individual blocks compressed along

the interface normal. As shown in Table 4, the mean

orientation angle for the end-to-end vector of the chains

(yee-inter) changed from about 338 (expected for random

orientation) for T/N¼1 to 498 at T/N¼ 0.2, and that for

the vector connecting the centers of mass of the

A- and B-blocks (yAB-inter) increased from 33 to 558.While this is evidence that, with a decrease of temperature

the chains move closer to the interface normal direction,

the chains still behave like random coils within the well-

defined lamellae.

Some information about chain conformation in block

copolymer lamellae is available through experiments.

Matsushita et al.[3] used small angle neutron scattering

(SANS) to measure radii of gyration (rg) in poly(deuterated

styrene-block-2-vinylpyridine) (PSd-PVP) diblock copoly-

mer lamellae, where the polystyrene block was deuterated,

and rg refers to an experimentally derived value. Based on

the values of rg along three different directions (rgk, in the

YZ plane, parallel to the interface; rg\1, in the XZ plane;

rg\2, in the XY plane, both are perpendicular to the inter-

face), they inferred that the polystyrene block was extended

along the interface normal direction, while it had contracted

along the parallel direction to keep the volume occupied by

the block chain constant. We know that the radius of

gyration obtained from SANS is a statistical-averaged value

for all of the polymer chains in the sample. With the infor-

mation available from the experiment, one cannot say

anything about the shape of individual block copolymer

chains in the lamellae.

In our simulations, we can also calculate the ensemble-

averaged parallel and perpendicular mean-squared radii of

gyration and we use the notation hrg2i to refer to values

obtained through simulation. These values are collected in

Table 5. We use the same definitions of rg, rgk, rg\1, and

rg\2 as those in ref.[3] which were described in the previous

paragraph. In the homogenous state (T/N¼1) with diffe-

rent virtual orientations, the radii of gyration of the whole

polymer chain, as well as of the individual blocks, are iden-

tical because of the isotropic nature of the system.

With a decrease of temperature, the values of hrg\2 i in-

crease dramatically due to chain stretching, while hrgk2 ivalues decrease slightly. The changes in the radii of gyration

for the individual blocks are similar to those of the entire

chains, but less obvious due to the small values of the radii.

Values of the two projections hrg(A,B)\2 i at the same temper-

ature are identical, both for the whole polymer chain and for

the individual blocks. We observe that the values of

hrg(A,B)\2 i at finite temperature are larger than the correspon-

ding values of hrg(A,B)k2 i in the lamellar system, although the

differences are quite small. Thus our results, based on simu-

lations, are similar to the results obtained by Matsushita

et al.[3] from SANS experiments. We can infer that in

well-defined lamellae at low temperatures, the polymer

chains are stretched along the interface normal direction

and not much compressed in the plane parallel to the

interface. We also know that the chain backbone is not well

oriented perpendicular to the interface, and that individual

chains within each lamellar domain behave like a random

coil similar to their behavior in the homogenous state.Figure 12. A sketch illustrating the shape change for thesegment distribution.



Densities of Junction Beads A (or B) at the Interface

In the previous section, we concluded that the polymer

chains in a lamellar structure are stretched in the direction

perpendicular to the interface and that they have some

tendency to orient towards normal to the interface. In this

section, we look at the composition of the interface region

and investigate which beads of the chain are present in the

interface. We begin by selecting beads of the A-block that

have contacts with the beads of the B-block and calculate

the density of contacts for each bead. The density is

expressed as the mean number of AB contacts per A-bead,

normalized by the total number of A/B contacts, for each

bead in the sequence of the A-block. The results of the

density distributions are shown in Figure 13. In the homo-

genous state (T/N¼1), all of the beads experience a

similar environment. With the decrease in temperature, the

contact densities become more asymmetric, with the

highest density of A/B contacts experienced by beads 30–

35 along the chain, that is, those with the smallest contour

distance from the junction point. The densities decrease for

beads being further away from the junction and increase

again for beads near the chain end.

The increased contact probability for the beads at the

chain end is connected to the greater conformational flexi-

bility of the chain ends. Chain end contacts can occur for

chains, which loop back into the interface containing the A/

B junction, or they can occur for chains which span a

lamellar period and connect with the adjacent interfaces.

The data in Figure 4d provide evidence for the penetration

of chains bound to one interface deep into the brush of

chains anchored in the adjacent interface. In Figure 11 we

see that for some chains, the dimensions along the interface

normal exceed the value of lamellar spacing, which means

that those chains have contact with the two adjacent inter-

faces. In Figure 13, we can also see that at a lower temper-

ature the beads close to the junction have a higher contact

probability than at a higher temperature, and the situation is

reversed for the chain ends. This result provides further

evidence that the chains are more highly stretched at lower

values of T/N.

Conclusion

We described the results of simulations based on the coope-

rative motion algorithm performed for the lamellar structu-

res in symmetric diblock copolymer systems at various

Table 5. Mean squared radii of gyration for the block copolymer fragments at the athermal state and at the three finite temperatures.

T/N Orientation of interfacea) hrg2i hrgA

2 i hrgB2 i hrgk

2 i hrg\12 i hrg\2

2 i

1 a¼ 1508; y¼ 268 15.4 7.5 7.5 10.3 10.3 10.3a¼ 608; y¼ 368 15.4 7.5 7.5 10.3 10.3 10.3

0.4 a¼ 468; y¼ 658 18.2 7.5 7.4 10.3 13.1 13.10.3 a¼ 908; y¼ 468 19.4 7.6 7.6 10.1 14.5 14.20.2 a¼ 358; y¼ 478 21.3 7.7 7.8 9.9 16.3 16.4

hrgAk2 i hrgA\1

2 i hrgA\22 i hrgBk

2 i hrgB\12 i hrgB\2

2 i1 a¼ 1508; y¼ 268 5.0 5.0 5.0 5.0 5.0 5.1

a¼ 608; y¼ 368 5.0 5.0 5.0 5.0 5.0 5.00.4 a¼ 468; y¼ 658 4.9 5.0 5.0 4.9 5.0 5.00.3 a¼ 908; y¼ 468 5.0 5.1 5.0 4.9 5.1 5.10.2 a¼ 358; y¼ 478 5.0 5.2 5.3 4.9 5.3 5.4

a) Calculated in units of (bond length)2 from the expression, hr2gi ¼ 1

2NkNw

PNk

i¼1

PNw

j¼1 rij, where rij is the distance between individual beads:

rij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2i þ y2

i þ z2i

pfor hr2

gi; rij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiy2i þ z2

i

pfor hr2

gjji; rij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2i þ z2

i

pfor hr2

g?1i; and rij ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2i þ y2

i

pfor hr2

g?2i; xi, yi, and zi are the

coordinates for bead i. The variables with the subscript A are the squared radii for the A-block, while those with the subscript B are thesquared radii for the B-block. All others are for the whole polymer chain. The definitions of the radii of gyration are the same as thosegiven in ref.[3] see text for details.

Figure 13. Relationship between the density in the interface forA-chains and bead number. Bead 1 is the chain end of the A-blockand bead 35 is the bead adjacent to the junction and connected to aB-chain bead. For each type of bead in the chain, we counted thenumber of beads in contact with beads from B-block, and thendivided this contact number by the total number A/B contacts inthe system. The total density for each curve sums to unity.



finite temperatures within the range of strong segregation.

We examined many features of the systems with the aim to

characterize the behavior of copolymer chains at the inter-

face. We found that the interface thickness defined by the

distribution of the junction points becomes narrower at

lower temperatures and that the interface exhibits a wavi-

ness with a maximum peak-to-valley distance of 20–30

lattice bonds. The most interesting results concern the

dimensions and orientation of the block copolymer chains

in the lamellar structure. Compared with the isotropic state

(T/N¼1), chains at lower temperatures are stretched in the

direction perpendicular to the interface and only weakly

compressed in the direction parallel to the interface. The

block copolymer molecules exhibit a modest tendency to

orient themselves with their end-to-end vector perpendi-

cular to the plane of the lamellar interface. This orientation

is, however, not transmitted to the size scale of polymer

segments so that the backbone fragment considered as the

junction between the blocks shows only negligible orienta-

tion correlation with the interface. The results we obtained

are similar to those reported from SANS measurements for

the mean conformation of the PSd blocks of symmetrical

PSd-PVP diblock copolymers.[3]

Acknowledgements: The authors thank NSERC Canada fortheir support of this research.

[1] Y. Matsushita, K. Mori, R. Saguchi, Y. Nakao, I. Noda,M. Nagasawa, Macromolecules 1990, 23, 4313.

[2] H. Hasegawa, T. Hashimoto, H. Kawai, T. P. Lodge, E. J.Amis, C. J. Glinka, C. C. Han, Macromolecules 1985, 18,67.

[3] Y. Matsushita, K. Mori, Y. Mogi, R. Saguchi, I. Noda,M. Nagasawa, T. Chang, C. J. Glinka, C. C. Han,Macromolecules 1990, 23, 4317.

[4] M. Mayes, R. D. Johnson, T. P. Russell, S. D. Smith,S. K. Satija, C. F. Majkrzak, Macromolecules 1993, 26,1047.

[5] N. Torikai, I. Noda, A. Karim, S. K. Satija, C. C. Han,Y. Matsushita, Macromolecules 1997, 30, 2907.

[6] T. Pakula, K. Karatassos, S. H. Anastasiadis, G. Fytas,Macromolecules 1997, 30, 8463.

[7] T. Pakula, G. Floudas, ‘‘Block Copolymers’’, in: F. J. BaltaCalleja, Z. Roslaniec, Eds., Marcel Dekker, New York 2000,p. 123.

[8] M. Banaszak, J. H. R. Clarke, Phys. Rev. E 1999, 60, 5753.[9] T. Geisinger, M. Muller, K. Binder, J. Chem. Phys. 1999,

111, 5241.[10] T. Geisinger, M. Muller, K. Binder, J. Chem. Phys. 1999,

111, 5251.[11] U. Micka, K. Binder,Macromol. Theory Simul. 1995, 4, 419.[12] J. Yang, J. Lu, Y. Rharbi, L. Cao, M. A. Winnik, Y. Zhang,

U. B. Wiesner, Macromolecules 2003, 36, 4485.[13] J. Yang, R. S. Roller, M. A. Winnik, Y. Zhang, T. Pakula,

Macromolecules 2005, in press.[14] T. Pakula, ‘‘Simulations on the Completely Occupied

Lattice’’, in: ‘‘Simulation Methods for Polymers’’, M. J.Kotelyanskii, D. N. Theodorou, Eds., Marcel-Dekker, 2004,Chapter 5.

[15] T. Pakula, Macromolecules 1987, 20, 679.[16] T. Pakula, S. Geyler, Macromolecules 1987, 20, 2909.[17] T. Pakula, S. Geyler, T. Edling, D. Boese, Rheol. Acta 1996,

35, 631.[18] F. Olaj, T. Petrik, G. Zifferer, J. Chem.Phys. 1998, 108, 8214.[19] Q. Wang, P. F. Nealey, J. J. de Pablo, Macromolecules 2002,

35, 9563.[20] S. H. Kim, W. H. Jo, J. Chem. Phys. 1999, 110, 12193.[21] A. Werner, F. Schmid, K. Binder, M. Muller, Macromo-

lecules 1996, 29, 8241.[22] N. Vassiliev, M. W. Matsen, J. Chem. Phys. 2003, 118, 7700.[23] M. Banaszak, S. Woloszczuk, S. Jurga, T. Pakula, J. Chem.

Phys. 2003, 119, 11451.[24] [24a] E. Helfand, Y. Tagami, J. Chem. Phys. 1972, 56, 3592;

[24b] E. Helfand, Macromolecules 1975, 8, 552; [24c] E.Helfand, Z. R. Wasserman, Macromolecules 1980, 13, 994.

[25] N. Semenov, Macromolecules 1993, 26, 6617.[26] This was estimated as a distance between the highest peak

strip and the center of an adjacent bottom strip.[27] H. Yamakawa, ‘‘Modern Theory of Polymer Solutions’’,

Harper & Row Publishers, New York 1971.[28] This definition allows us to overcome a limitation of the

lattice model which would underestimate any backboneorientation if we were to define this vector in terms of the twobeads immediately adjacent to the junction.

[29] J. Yang, M. A. Winnik, T. Pakula, unpublished results.[30] T. Pakula, Macromol. Chem. Theory Simul. 1993, 2, 239.[31] T. Pakula, E. B. Zhulina, J. Chem. Phys. 1991, 95, 4691.



Documents

Interface Orientation and Chain Conformation in Simulated Symmetric Diblock Copolymer Lamellar Systems