PHYSICAL REVIEW E

VOLUME 53, NUMBER 6

JUNE 1996

Interface dynamics in a block copolymer melt and the effect of noise
Shigeyuki Komura*
Department of Mechanical System Engineering, Kyushu Institute of Technology, Iizuka 820, Japan

Jun-ichi Fukuda†
Department of Physics, Faculty of Science, Kyoto University, Kyoto 606, Japan

Glenn C. Paquette‡
Media Dynamics Corp., Ltd., 781 Honryusui-cho, Nishinotoin-dori, Bukkoji-sagaru, Shimogyo-ku, Kyoto 600, Japan~Received 20 February 1996!
The role of fluctuations in the disorder-lamellar transition in a block copolymer melt is investigated using acell dynamical system simulation by measuring the propagation velocity of the interface between ordered anddisordered regions. Our results strongly suggest that near the transition temperature, in the absence of noise, thevelocity increases with quench depth as v ; At @t 5(T c 2T)/T c is the reduced temperature measured from thetransition temperature T c #, while in the presence of noise, the velocity increases as v ; t . These results lead us
to conclude that the addition of noise causes the disorder-lamellar transition to change from second order tofirst order. This conclusion is consistent with the prediction of Brazovskii @Sov. Phys. JETP 41, 85 ~1975!#.
@S1063-651X~96!50606-9#
PACS number~s!: 61.25.Hq, 05.40.1j

Block copolymers ~BCP’s! are linear polymer chains,
typically composed of two homopolymer subchains grafted
covalently at one end. Their properties have attracted a great
deal of interest, both scientific and technological. Depending
on temperature and polymer composition, BCP systems havebeen found to exhibit many fascinating periodic structures aswell as a homogeneous disordered phase. The mean field
theories of Leibler @1#, in the weak segregation regime, andOhta and Kawasaki @2#, in the strong segregation regime,
yield phase diagrams which reproduce many features ofphase diagrams obtained experimentally. According to these
theories, the order parameter of the BCP system is introduced in the following way. Let f a (r) and f b (r) denote thelocal volume fractions of a and b segments, and N a andN b represent the degrees of polymerization of a and b
blocks. Under the condition that the molten phase is incompressible @ f a (r)1 f b (r)51#, the system can be describedby a single order parameter: c (r)5 f a (r)2 f b (r). The spatial average of c~r! is given by c̄ 52 f 21, where f is theblock ratio, f 5N a /(N a 1N b ).
According to Leibler’s mean field theory, for a symmetricchain @ N a 5N b # , the transition from a disordered system toan ordered system is second order, and the resulting orderedphase consists of alternating ‘‘stripes’’ ~lamellae! of a-richand b-rich regions. Systems which exhibit such a transition
between a homogeneous isotropic disordered phase and alamellar phase belong to the ‘‘Brazovskii universality class’’@3#. For such systems, the transition to the ordered state,predicted to be second order by mean field theory, is expected to become first order due to the effect of fluctuations.
Treating fluctuations within the Hartree approximation, Fre*Electronic address: komura@iizuka.isc.kyutech.ac.jp†‡

Electronic address: fukuda@ton.scphys.kyoto-u.ac.jp
Electronic address: VYE03010@niftyserve.or.jp

1063-651X/96/53~6!/5588~4!/$10.00

53

drickson and Helfand calculated the phase diagram for the
BCP system and obtained results significantly different fromthose obtained within the mean field theory @4#. Some of
their predictions have been confirmed experimentally @5#.
In this article, we investigate the effects of fluctuations on
a symmetric BCP system through computer simulation by
studying the propagation velocity of the interface between
the lamellar phase and the disordered phase near the transition temperature. It should be noted that the system we study
presents special technical problems. The only work of which
we are aware concerning the identification of a fluctuationinduced first order transition in a numerical simulation wasrecently reported by Shiwa et al. @6#. Using a cell dynamicalsystem ~CDS! approach @7# to study the Swift-Hohenberg
equation, they found a double-plateau evolution of the Nusselt number in the presence of noise.
Perhaps the most direct method to distinguish a first ordertransition from a second order transition is to consider theevolution of droplets of the equilibrium phase in the parent
phase, that is, to determine if sufficiently small droplets will
shrink and eventually disappear rather than grow. ~Of course,
in this case, it is never possible to make a certain identification of a second order transition.! For the present case, however, such a study is not practically feasible. The system westudy is very close to the critical temperature. In general, if
both phases appear in such a system, their interface will extend over a very long distance. ~The thickness of the interface is essentially the correlation length of the system.! If weplace a single small droplet of the equilibrium phase in asystem prepared in the parent phase, its dynamics will initially be dominated by diffusion, and it will quickly spreadand its amplitude shrink. In a noiseless system, this would
not prevent us from determining the fate of this droplet. Itsevolution could be followed indefinitely, and its amplitude
would eventually be seen to everywhere approach zero or its
equilibrium value. However, to investigate the behavior in
R5588© 1996 The American Physical Society

53

INTERFACE DYNAMICS IN A BLOCK COPOLYMER MELT AND . . .

which we are interested, we must add noise to the system. In
this case, when we place a localized droplet of the orderedphase in the disordered phase, its amplitude quickly decays,
and it becomes ‘‘lost’’ in the noise. Its subsequent evolution
cannot be distinguished from the evolution induced by thenoise. The time development of the droplet thus cannot be
isolated, and it cannot be said to either ultimately shrink anddisappear or to grow.
Even in the system we study, it is possible in principle tostudy the evolution of droplets, but in order to do so, theymust extend over a region whose size is on the order of thecorrelation length. With the parameter values used in thepresent numerical study, and using results appearing in Ref.
@8#, this length was found to be between approximately 1000
and 2000 cells. It may be thought that in such a situation, it
would be possible to simply coarse-grain the system so thatthe correlation length corresponds to any desired reasonably
small number of cells. However, in the present case, thiscannot be done since coarse-graining beyond a certain point
will result in the period of the ordered structure becoming
smaller than the cell size. For the parameter values we consider here, the period of the ordered structure is approximately 25 to 50 times shorter than the correlation length.
Thus to honestly study the dynamics of droplets for thepresent purpose would require a system size of perhaps atleast 200032000 ~approximately 60 times larger than thesystem used in our investigation! and a somewhat tedious
preparation of the initial droplets. In a system such as this, ametastable disordered state cannot be distinguished from an
unstable disordered state through a study of local dynamics.
In this sense, we can say that locally there is no difference
between the two. Only when we consider more macroscopic
behavior can the distinction between the two be made, and in
this sense, only here does the distinction have meaning.
Another method which could be used to identify a firstorder phase transition is to measure the scattering function of
the equilibrium pattern in the neighborhood of the transition
temperature and look for a discontinuity in the scatteringpeak height. We attempted this, but were unable to detect
any such discontinuity. Evidently, the precision of our numerical computation was not sufficient.
The method we use in this study relies on the fact that thetemperature dependence of the steady state propagation velocity of an interface separating the disordered and orderedphases directly reflects the order of the transition @9,10#. In
the case of a second order transition, the steady-state velocityof an interface separating the stable ordered state and theunstable disordered state depends on the velocity in accordance with the relation v ; At @t 5(T c 2T)/T c , T c is thetransition temperature#, provided fluctuations and initial inhomogeneities are sufficiently small @11#. In the case of afirst order transition, the interface velocity in the steady state
depends linearly on the temperature near T c @9#. By performing a CDS simulation for the BCP system, we have carriedout a dynamical test of the BCP system to determine how
fluctuations affect the nature of the phase transition. In fact,
an earlier investigation of the type of phenomena in which
we are interested was carried out experimentally in a liquid
crystal system near the nematic–smectic-A transition by Cladis et al., whose results strongly suggest that this transition is
weakly first order @9,10#. A similar experiment for BCP sys-

R5589

tems is also important not only to determine the order of thetransition but also to test the various properties of front
propagation in such systems and compare these with the corresponding theoretical predictions.
Oono and Shiwa proposed the following partial differential equation to describe the dynamics of microphase separation in a BCP system ~with no macroscopic flow! @12#:]c
5¹ 2 @ 2D¹ 2 c 2a c 1u c 3 # 2B @ c 2 c̄ # ,]t~1!

where ¹ 2 is the Laplacian, and B, D, a and u are positive
phenomenological parameters. The term proportional to B
reflects the long range interaction. On the basis of this equation, the problem of front propagation in a BCP system wasstudied numerically by Liu and Goldenfeld for a symmetric
one-dimensional system @13#. The value of the selected front
velocity they determined agrees with that derived using themarginal stability conjecture @11#. Later, employing the reductive perturbation method, one of the present authors~G.P.! studied the invasion of lamellar, triangular and bcc
ordered phases into the disordered region and calculated
propagation velocities of front-envelope profiles for various
invasion processes near the spinodal line @8#.
A CDS model corresponding to Eq. ~1! in the presence ofnoise was also introduced by Oono and Shiwa @12# as aslight modification of the CDS model describing spinodal
decomposition. A detailed investigation of this system was
subsequently performed by Bahiana and Oono @14#. We employ the same CDS model, where the difference equation forthe order parameter c is given byc ~ n,t11 ! 5 c ~ n,t ! 1 ^^ I~ n,t ! && 2I~ n,t ! 2B@ c ~ n,t ! 2 c̄ #1Ch ~ n,t ! ,

~2!

with
I~ n,t ! 52D@ ^^ c ~ n,t ! && 2 c ~ n,t !# 2Atanhc ~ n,t ! 1 c ~ n,t ! ,~3!
where n5(n x ,n y ) designates the two-dimensional lattice
point (n x and n y are integers!. We choose the units of lengthand time in a such way that the lattice constant and thetransport coefficient do not appear in the above equations.
The quantity ^^ X && is an isotropic spatial average, definedon the square lattice by ^^ X && 5(1/6) ( X~nearest-neighborcells! 1~1/12!( X~next-nearest-neighbor cells!. Hence thecorrespondence between the coefficients in Eq. ~1! andthe above equations is a5A21, B53B, D5D/3. Time t
is also scaled by factor of 3. The last term in Eq. ~2! represents the noise; C is the noise amplitude, taken here as anindependent parameter @15# and h (n,t)5 h x (n x 11,n y ,t)
2 h x (n x ,n y ,t)1 h y (n x ,n y 11,t)2 h y (n x ,n y ,t), where h x
and h y are random numbers uniformly distributed in theinterval @ 21,1# . In our simulations, we fixed the parameters
as B50.02, C50 or 0.01, D50.5, c̄ 50, and varied A
around the transition temperature Ac .
Linear stability analysis of Eq. ~1! shows that the spinodalline is determined by (a23u c̄ 2 ) 2 24BD50 @8#. Since themicrophase-separation temperature ~MST! and the spinodal
temperature coincide for the symmetric, c̄ 50 case, the tran-

R5590

KOMURA, FUKUDA, AND PAQUETTE

53

FIG. 1. Schematic representation of the invasion of a lamellar
phase into a homogeneous disordered phase.

sition temperature between the lamellar and disorderedphases is a c 52 ABD52 ABD50.2 for the above parametervalues. Hence the corresponding critical value of A in Eq.
~3! for c̄ 50 is Ac 5110.251.2.
In order to examine the velocity of the propagating interfaces, we first produced sufficiently developed ~30 000 time
steps! ‘‘seed’’ patterns on a square lattice of size 1283128
for various values of T,T c (A.Ac ) near the transition
point. Then the ‘‘seed’’ patterns were mounted in the middleof lattice of size 1283512 with periodic boundary conditions. Having prepared such an initial state, we allowed theinterface to propagate with the same value of A as that withwhich the ‘‘seed’’ pattern was produced. In contrast to thereal system, however, the above CDS model with c̄ 50 does
not produce a well-defined lamellar structure, but rather acomplicated bicontinuous pattern. Bahiana and Oono attempted to avoid this difficulty by performing a threedimensional simulation, adding thermal noise, imposing abending penalty, and taking into account hydrodynamic interactions, but none of these attempts changed the patternsignificantly @14#. We have found that a well-ordered lamellar structure can be obtained by imposing a macroscopic
shear flow, the velocity of which is given by v x (r)5 ġ y andv y 5 v z 50. In this case, the right-hand side of Eq. ~1! acquires the convective term 2 ġ y( ] c / ] x), and a corresponding discretized term appears in Eq. ~2!. It should be stressed,
however, the shear flow was applied only during the preparation of the ‘‘seed’’ patterns and was turned off when wemeasured the interface propagation velocity.
Recently, the nucleation of a lamellar phase in afluctuation-induced first order transition was discussed by
Hohenberg and Swift @16#. They showed that the critical
droplet shape is anisotropic in general. In their terminology,
the subject of the present investigation is the propagationof a ‘‘longitudinal’’ interface, as shown in Fig. 1. In thecalculation of the interface velocity, the front position
was specified for every ten time steps by measuring the total
length of the ordered region. The length l(t) of the orderedregion was estimated by the value l(t)52 ( c (n,t) 2 /L c 2max , where c max is the maximum value of c (n), and L is
as shown in Fig. 1. In the presence of noise, an average overmore than ten propagations was taken. While there are obvious problems with this method of calculating the velocity~especially in the presence of noise!, for the present problem,
we feel it is probably the most precise method which can berealized using a relatively simple numerical procedure, aslong as lk 0 is not too large, where k 0 is the wave number ofthe equilibrium ordered pattern. Fortunately, we were able toobtain meaningful results using a small enough number oftime steps that lk 0 never became larger than approximately
10. For values of A very close to Ac 51.2 ~approximately
1.202 and less!, the propagation velocity was not large

FIG. 2. Propagation velocity for the invasion of a lamellar ordered region into a disordered region in the absence and presence ofnoise.

enough to obtain meaningful results using our numerical procedure.
We first discuss the noiseless system, i.e., C50. In Ref.
@8#, the propagation velocity v is calculated using the reductive perturbation method asv 5 @ 2a ~ a 2 24BD ! /D # 1/2.

~4!

If we consider the temperature just below T c , we put
a5a c 1 t 52 ABD1 t (a c @ t , t .0), and hence v ; At tolowest order in t . In terms of A, we have v '0.327
3(A21.2) 1/2 for the above parameter values. In Fig. 2, thepropagation velocity v determined through our numericalsimulations is plotted as a function of A. As shown, the data
are fit well by the curve v 50.3453(A21.20) 0.54060.005. If
the point at A51.210 is removed from the fit, we obtainv 50.3063(A21.20) 0.51360.006 . For a range of values of
A infinitesimally close to Ac , the exponent appearing here
should be exactly 1/2. As the maximum value of A increases, however, this value should also increase @as seen by
expanding Eq. ~4! to second order#. Our results are consistent
with this behavior.
We now study the effect of the noise for the case
C50.01. This value was also used in Ref. @6#. In this case,the lamellar pattern develops spontaneously in the region
ahead of the front. To minimize the error introduced by thepresence of this ‘‘undesired’’ order, in the calculation ofl(t) we removed the contribution of c (n) 2 for values of n
for which the deviation from zero had not been caused by thefront. Of course, this procedure itself is inherently errorridden, but it is perhaps the best alternative. After a sufficiently long time, the propagation of the front is impeded
significantly by the spontaneously developed ordered phasegrowing before it. We determined the propagation velocity
before this ‘‘collision’’ occurs, and the result is also shownin Fig. 2. The data are fit well with the curvev 52.273(A21.20) 0.93860.015 given in the figure. Again if
we remove the point at A51.210 from the fit, we obtainv 53.053(A21.20) 1.00560.013. As in the noise-free system,
we expect that the observed deviation from 1 in the former

53

INTERFACE DYNAMICS IN A BLOCK COPOLYMER MELT AND . . .

R5591

case is due to the presence of a non-first-order contribution.
~Note in this case, the second-order effect should cause theexponent to decrease.! We believe that our result reflects thefirst order nature of the transition due to the Brazovskii effect. We point out that our result is consistent with the calculation of the growth kinetics of a BCP lamellar droplet byFredrickson and Binder in the sense that v is proportional tot @17#. As mentioned above @16#, however, their assumptionof a spherical droplet is problematic since, as discussed inRef. @8#, the propagation velocity of the front is highly anisotropic. An investigation using other values of the noise
amplitude is presently underway, and the details of our re-sults will be published elsewhere. The predicted effects ofshear flow on the Brazovskii type transition @18# will also be
investigated in the future.
In summary, within the CDS approach, we found evidence of a fluctuation-induced first order transition in a symmetric BCP system by measuring the velocity of the interface between the ordered and disordered phases.

@1# L. Leibler, Macromolecules 13, 1602 ~1980!.
@2# T. Ohta and K. Kawasaki, Macromolecules 19, 2621 ~1986!.
@3# S. A. Brazovskii, Zh. Éksp. Teor. Fiz. 68, 175 ~1975! @Sov.
Phys. JETP 41, 85 ~1975!#.
@4# G. H. Fredrickson and E. Helfand, J. Chem. Phys. 87, 697~1987!.
@5# F. S. Bates, J. H. Rosedale, G. H. Fredrickson and C. J. Glinka,
Phys. Rev. Lett. 61, 2229 ~1988!; F. S. Bates, J. H. Rosedale
and G. H. Fredrickson, J. Chem. Phys. 92, 6255 ~1990!.
@6# Y. Shiwa, H. Matsunaga, M. Yoshikawa, and H. Yoshitomi,
Phys. Rev. E 49, 2082 ~1994!.
@7# Y. Oono and S. Puri, Phys. Rev. Lett. 58, 836 ~1987!; Phys.
Rev. A 38, 434 ~1988!.
@8# G. C. Paquette, Phys. Rev. A 44, 677 ~1991!; Phys. Rev. E 51,
3280 ~1995!.
@9# P. E. Cladis, W. van Saarloos, D. A. Huse, J. S. Patel, J. W.

Goodby, and P. L. Finn, Phys. Rev. Lett. 62, 1764 ~1989!.
@10# M. A. Anisimov, P. E. Cladis, E. E. Gorodetskii, D. A. Huse,
V. E. Podneks, V. G. Taratuta, W. van Saarloos, and V. P.
Voronov, Phys. Rev. A 41, 6749 ~1990!.
@11# W. van Saarloos, Phys. Rev. A 37, 211 ~1988! and references
therein.
@12# Y. Oono and Y. Shiwa, Mod. Phys. Lett. B 1, 49 ~1987!.
@13# F. Liu and N. Goldenfeld, Phys. Rev. A 39, 4805 ~1989!.
@14# M Bahiana and Y. Oono, Phys. Rev. A 41, 6763 ~1990!.
@15# S. Puri and Y. Oono, J. Phys. A 21, L755 ~1988!.
@16# P. C. Hohenberg and J. B. Swift, Phys. Rev. E 52, 1828
~1995!.
@17# G. H. Fredrickson and K. Binder, J. Chem. Phys. 91, 7265
~1989!.
@18# M. E. Cates and S. T. Milner, Phys. Rev. Lett. 62, 1856
~1989!; M. Goulian and S. T. Milner, ibid. 74, 1775 ~1995!.

We would like to thank Professor Y. Shiwa and Dr. H.
Kodama for their helpful discussions. This work is supported
by a Grant-in-Aid for Scientific Research, Ministry of Education, Science and Culture, Japan ~06740320!.