__fElII

20 November

J-a1995

PHYSICS

EISEVIER

LETTERS

APhysics Letters A 208 (1995) 108-112

Scattering function of the disordered phase of block copolymers
under shear flow
Shigeyuki Komura a,1,Jun-ichi Fukuda b,2
a Department of Mechanical System Engineering, Kyushu Institute of Technology, Iizuka 820, Japanb Department of Physics, Faculty of Science, Kyoto University, Kyoto 606, Japan
Received 26 July 1995; accepted for publication 13 September
Communicated by A.R. Bishop

1995

Abstract
The scattering function of the disordered phase of a block copolymer (BCP) melt under a steady shear flow isinvestigated by the cell dynamical system (CDS) approach. Agreement with previous theoretical calculations indicates theapplicability of the CDS approach to the disordered phase as well as the ordered phase.
PACS: 05.40 + j; 61.25.Hq

Block copolymers (BCP) are linear chains typically composed of two homopolymer
subchains
grafted covalently at one end. Their properties haveattracted great scientific and technological interest. It
is well-known that the BCP system exhibits various
fascinating periodic structures as well as a homogeneous disordered phase depending on the temperature or the molecular composition. The mesoscopic
description of these equilibriumphases has beenrather well established according to the theory byLeibler for the weak segregation regime [l], or thatby Ohta and Kawasaki for the strong segregation
regime [21.
In this Letter, using the cell dynamical system(CDS) approach [3,4], we investigate the effect of a’ E-mail: komura@post.isci.kyutech.ac.jp.
’ E-mail: fukuda@ton.scphys.kyoto-u.ac.jp.
03759601/95/$09.50
0 1995 Elsevier Science B.V. All rights reserved
SSDI 0375-9601(95)00728-8

steady shear flow on the scattering function of thedisordered phase in a BCP system. The present
computer simulation is motivated by the previous
mean field calculation of Fredrickson, who showedthat the shear flow affects the scattering function
highly anisotropically,
whereas the peak intensity is
considerably attenuated [51. In our computer simulation, we especially pay attention to the scaling analysis of the attenuation rate in the peak intensity. So far
the CDS approaches have been used only to simulate
the ordered phase (low temperature phase) in thespinodal decomposition or in the micro-phase separation of the BCP system. Our result shows that theCDS method is also applicable to investigating thehigh temperature disordered phase at least in thelatter case.
According to a previous theory [l], the orderparameter of the BCP system can be introduced in
the following way. Let 4,,(r) and &,(r) denote theS. Komura, J. Fukuda /Physics Letters A 208 (I 995) 108-l 12

local volume fraction of segments a and b; N, andNb represent the degrees of polymerization of blocksa and b, respectively. Under the condition that themolten phase is incompressible (4,(r) + +b(~) = l),
the system can be described by a single order parameter $(r) = 4&r) - &(r).
The spatial average ofG(r) is determined solely by the relation 3 = 2f- 1,
with f= N,/(N, + N,), where f is the block ratio.
For the dynamics of the BCP system, Oono andShiwa first proposed an equation of motion for thetime dependent order parameter $(r, t) 141. Recently
Ohta et al. generalized this equation to the case inwhich a macroscopic flow u is present [6],-B(ICI-Icl).

(1)

In the above, B, D, T and u are positive phenomenological parameters and the last term is due tothe long range interaction. (The transport coefficient
has been eliminated in Eq. (1) after an appropriatescaling.) As for the macroscopic flow in Eq. (l), weconsider the following shear flow,ux(r)= +y,uy = u, = 0,(2)where the shear rate i/ is the time derivative of theshear strain y. In this case, the convective term inEq. (1) becomes ~y(@/ax)
and we are allowed toperform our simulation in two-dimensional
space [6].
The CDS model for spinodal
decomposition
proved to be an efficient algorithm for numericalsimulations[3]. As a slight modificationof thismodel, Oono and Shiwa proposed the corresponding
CDS model of Eq. (1) [4], and a subsequent detailed
investigation
has been performed by Bahiana andOono [7]. In the present simulation, we employ thesame CDS model as in Ref. [6] where the rheologicalproperties of the hexagonal phase under an applied
shear flow are reported. The difference equation forthe order parameter #(n, t) is given byrCl(% t+ 1)= $(n,t) -@-;?++,+1,+ ((I(%rlr(n,t) - $1

ny, t)-@(n,-Lt)>> -I(%t) +Q(n,ny’ t)]t),

(3)

109

with

I( n, t) = -Atanh +( n, t) + @(n,-D[((rCl(n,t)t)>> - rCr(n, t)],(4)

where 12= (n,, ny) assigns the two-dimensionallattice point. The operator ((X))
defines the isotropic
spatial average of the quantity X, and is defined on
the squarelatticeby ((X))= iCX(nearestneighbor cells) + &CX(next-nearest-neighborcells)[3]. The last term on the r.h.s. of Eq. (3) represents
the noise term, C being the noise amplitude andrl(n, t) = q&r, + 1, IIY, t) - 77x(nx, IzY, t) +77y(lt*, nY + 1, t> - rl,(n,, nyr t>, where 77, and 77y
are random numbers uniformly
distributed in theinterval [ - 1, l] [7]. In the present simulation, wehave fixed the parameters to the values B = 0.02,
C = 0.01, D = 0.5, 6 = 0, whereas the shear rate has
been changed for the values i, = 0, 1 X 10m4, 3 X
10-4, 5 X 10m4 and 1 X 10m3.
The linear stability analysis based on Eq. (1)without any flow (o = 0) shows that the spinodalline is determined by (T- 3~$‘)~ - 4BD = 0 [8].
Since the line of microphase-separation
temperature(MST) and the spinodal line coincides for $ = 0(symmetric BCP), the transition temperature between
the lamellar phase and the disordered phase is givenby rc = 2fi= 0.2 for above values of parameters.
Hence the corresponding critical value of A in Eq.
(4) should be A, = 1 + TV= 1.2. We have chosen
several values of A to lie between 1.190 and 1.199
representing just above the transition temperature.
(In general the flow changes the locus of the spinodal and MST, but we do not consider this problem
here.)
In our simulation we have studied a system on asquare lattice of size L XL = 64 X 64 and the initial
distribution of 4(n) is specified by a random uniform distribution in the range [ -0.025, 0, 0251. In
the presence of the shear flow given by Eq. (2), therequiredboundarycondition
shouldsatisfy*(n,, nY, t) = +(n, + iL + yjL, ny + jL, t> for arbitrary integers i and j [6]. The Fourier components$(k, t) under this sheared boundary condition canbe calculated by performing the affine transforma-

S. Komura, J. Fukuda /Physics Letters A 208 (1995) 108-I 12

110tion of $(n,t) as

X+(x, Y, t) exp[i(k,x+ kyy)],(5)
wherex1=xyyandk = (k,,k 1 =(277-/LXm, n - rm) with m and n integers [6 I. The
scattering functions S(k) = ( 1t)(k) 12> have been
computed after 5000 time steps, which is long enoughto observe the steady state behavior of the system.
We have also checked that S(k) does not change
appreciably for longer runs up to 10000 time steps.
In order to improve the statistics, the average has
been taken over 200 sample patterns for each param-eter set. It is important to add the noise in thissimulation since we are close to the transition point.
For a fixed value of A = 1.198, two-dimensional
scattering patterns S(k) are presented for (a) i/ = 0,(b) 3 X 10p4, (c) 5 X low4 and (d) 1 X 10e3 in Fig.
1. The dark area denotes the region of higher scattering intensity. When there is no applied shear flow
(i, = O), the scattering pattern is isotropic and theradius of the circular ring corresponds to the absolutevalue of the most unstable wave vector with which
the lamellar pattern will be developed below thetransition temperature. As the shear rate is increased,
the pattern becomes more and more anisotropic.
Depending on the direction in the scattering patterns,
the peak positions in S(k) are shifted towards larger
or smaller value of k = I k I as compared to the case

(a)

(b)

(cl(d)

Fig. 1. Two-dimensional
scattering patterns for A = 1.198 and (a) 9 = 0, (b) 3 X 10w4, (c) 5 X 10e4 and (d) 1 X lo-‘.
denotes the region of higher scattering intensity.

The dark area

S. Komura, J. Fukuda/Physics

111

Letters A 208 (1995) 108-112

perature approaches the transition temperature, thepeak intensity is more attenuated, even if the shear
rate is kept unchanged. In order to summarize thesebehaviors, we look at the attenuation rate defined bywhere Z(A, y, 4) is the maximum value of S(k)depending on A, i, and 4. Recalling the lowest
order calculation of the maximum peak intensity by
Fredrickson [5], we anticipate the following scaling
form,T(A,Fig. 2. S(k)( k>* as a function of the scaled wavevectors k/(k)for A = 1.198 and (a) q5 = 0, (b) $T, (c) f~ and (d) :T. (k) isthe location of the peak in the absence of shear flow for A = 1.198.
The solid, dotted and dot-dashed lines represent t = 0, 3 X low4
and 1 X 10-j, respectively.

of Jo= 0. To specify the orientation of k in thek,-k,
plane, we let 4 denote the polar angle measured relative to the k,-axis. For the same fixedvalue of A = 1.198, the shape of the scatteringfunctions for + = 0 (solid), 3 X 10m4 (dotted) and1 X 10e3 (dot-dashed) are presented in each graph ofFig. 2 corresponding
to the direction of (a) 4 = 0,(b) an, (c) $n and (d) tn. In these four graphs, thedimensionlessquantities S(k)(k)2
are plotted as afunction of the normalized wavevector k/(k),where(k) is the averaged peak position for A = 1.198 and
9 = 0. Considering
the symmetry of Eq. (21, thescattering function should not be changed by theshear flow for 4 = in (k, = 0) and this requirement
is well reproduced in Fig. 2c, where we see noappreciable difference in the scattering function fordifferent values of T. As was predicted by Fredrickson [5], the scattering peak should be shifted tosmaller values of k for 4 = 0 and f7r whereas thepeak is shifted to larger k for 4 = irr. Moreover thepeak height is significantly
attenuated in all directions except for 4 = in. One can also observe thatthe larger the shear rate +, the larger both the peakshift and the attenuation are.
For different values of A, the effect of the shear
flow turns out to give rise to the same effect as
described above for A = 1.198. However as the tem-?, 4) =f(x,4),with x= y2/(l.2-A)3.

(7)

In the limit of x + 0, f(x) should behave as f(x)+ 1. Fig. 3 is the log-linear plot of f(x,4 = i rr ) asa function of X. Although the data collapse is notsatisfactory due to the large error bars, the above
described qualitative behavior is well reproduced.
Below the transition temperature (A > 1.2) and in
the absence of the shear flow, the above CDS model
does not provide a well-definedlamellar structure
but rather a complicated bicontinous pattern in contrast to the real system. Bahiana and Oono tried to
avoid this difficulty by performing such as three-dimensional simulation, adding sufficiently large thermal noise, imposing a bending penalty or taking into
account the hydrodynamicinteractions, but none ofthese attempts could change the pattern significantly

-71

1.0

Fo.900if@00

0080+? 0.8 0%
00.7

00.6

““Iloo*“,,,,,’

10-l“““’IO”“““’‘-IO’

10’x

Fig. 3. Log-linear

plot of f(x,C#J= $T) as a function of x.

112

S. Komura, _I. Fukuda /Physics[7]. In spite of these difficulties, we found that awell-ordered lamellar structure can be obtained by
imposing a macroscopic shear flow given by Eq. (2).
The scattering pattern of such a lamellar structure
exhibits two sharp spots on k, = 0 with much larger
intensity than that of disordered phase.
In summary, we investigated the scattering function of the disordered phase of a BCP melt under asteady shear flow through the CDS approach. Theresults indicate that the CDS approach is also applicable to the disordered phase.
This work is supported by a Grant-in-Aidfor
Scientific Research, Ministry of Education, Scienceand Culture, Japan (06740320).

Letters A 208 (1995) 108-l 12References
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