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Dynamics of surface directed mesophase formation in block copolymer meltsSevink, G. J. A.; Zvelindovsky, A. V.; van Vlimmeren, B. A. C.; Maurits, N. M.; Fraaije, J. G. E.M.Published in:Journal of Chemical Physics

DOI:10.1063/1.477837

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Citation for published version (APA):Sevink, G. J. A., Zvelindovsky, A. V., van Vlimmeren, B. A. C., Maurits, N. M., & Fraaije, J. G. E. M. (1999).Dynamics of surface directed mesophase formation in block copolymer melts. Journal of Chemical Physics,110(4), 2250-2256. https://doi.org/10.1063/1.477837

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Dynamics of surface directed mesophase formation in block copolymer meltsG. J. A. Sevink, A. V. Zvelindovsky, B. A. C. van Vlimmeren, N. M. Maurits, and J. G. E. M. Fraaije

Citation: J. Chem. Phys. 110, 2250 (1999); doi: 10.1063/1.477837View online: https://doi.org/10.1063/1.477837View Table of Contents: http://aip.scitation.org/toc/jcp/110/4Published by the American Institute of Physics

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Dynamics of surface directed mesophase formation in blockcopolymer melts

G. J. A. Sevink,a) A. V. Zvelindovsky,b) B. A. C. van Vlimmeren, N. M. Maurits,and J. G. E. M. FraaijeFaculty of Mathematics and Natural Sciences, University of Groningen, Nijenborgh 4, 9747 AG Groningen,The Netherlands

~Received 6 August 1998; accepted 23 October 1998!

The dynamic mean-field density functional method is adapted to describe phase separation in thepresence of geometrical constraints. We observe that inclusion of small filler particles~such as rods!already has a dramatic effect on the morphology of polymer melts. The effect is comparable to theeffect of applied simple steady shear. Mesostructures in the presence of large filler particles such asplates are totally governed by the geometry of the particle. Effects of polymer–surface interactionson morphology formation are investigated in detail. ©1999 American Institute of Physics.@S0021-9606~99!70704-X#

I. INTRODUCTION

It is well known that material structures in the mesos-copic domain play an important role in macroscopic materialproperties. Models that describe the dynamics in the mesos-copic domain are of extreme importance for understandingthe relation between processing conditions and particularmesostructures. For instance, a great deal of theoretical andexperimental attention has been given to ordering phenom-ena in block copolymers.1,2 It was found experimentally thatblock copolymers are capable of forming mesoscale struc-tures~such as lamellar mesophases! whose morphology canbe tailored by controlled synthesis. Mesoscopic models thatare used to describe morphology formation at mesoscalelevel form a bridge between models that describe fast mo-lecular kinetics and slow thermodynamic relaxations of mac-roscopic properties.

A method for investigating the dynamic formation ofmesostructures was recently developed by Fraaijeet al.3 Itcombines a~dynamic! mean-field density functional theorywith Gaussian chains as a molecular model. This theory hasbeen validated for specific4 triblock polymer surfactants:Pluronics L64 ~EO!13~PO!30~EO!13 and 4R25~PO!19~EO!33~PO!19. Recently, this theory was adapted fordescribing mesoscale dynamics of block copolymers undershear.5–7 It is well known that flows affect mesostructures,giving rise to global orientation.8,9 As a next step in thispaper, we consider surface effects due to the presence ofsolid objects in a block copolymer melt. These surface ef-fects are known to have great influence on the resulting me-sostructures. Both shear and surface effects are often presentin the experiments. For reasons of simplicity, we only con-sider surface effects resulting from stationary objects. Mov-ing objects, giving rise to much more complex computerimplementations of our parallel algorithms, will be consid-ered in the future.

Many models that are currently used to describe the be-

havior of polymers are based on traditional free energy ex-pansion methods~Cahn–Hilliard,10 Oono–Puri,11 Flory–Huggins–de Gennes12!. A disadvantage of the use of thesemodels for studies of specific processes is that they containonly the basic physics of phase separation13 and are not wellsuited for specific application to different complex industrialand biological liquids. In contrast to these phenomenologicaltheories we do not truncate the free energy expansion, butrather retain the full polymer path integral by a numericalprocedure.13–19 Very recently a similar approach has beenalso started to use by other authors.20 The benefit of such anapproach is that it allows for the description of the mesos-copic dynamics ofspecificcomplex polymer liquids.4

The morphologies of complex liquids in confined geom-etries has been the subject of numerous studies both by ex-perimentalists and theoreticians. Several aspects have beenconsidered. Mesostructures in diblock and triblock copoly-mer thin films are studied because1,2,21–27the surface prop-erties of these materials, such as adhesion, friction, and wet-ting, are controlled by the mesostructures near the surface.For instance, atomic force microscope~AFM! experiments2

show interesting ordering phenomena in spin-coated films ofa commercial triblock copolymer. Cylinders with two kindsof orientation~parallel and perpendicular! are found that arepacked in large islands. Related studies have been carried outfor mesostructure orientation of block copolymer films whenconfined between parallel flat plates.28–30 Hydrodynamic ef-fects on the surface directed phase separation have beenstudied31 as well as the effect of patterned surfaces ondiblock-copolymer melts.32 More general studies with re-spect to several aspects of interaction phenomena of copoly-mers at surfaces and interfaces are very numerous.33–42 Forexample, the dynamics of polymer absorption at a wall hasrecently been addressed in Ref. 42 using self-consistent fieldtheory. In this article we extend the dynamic density func-tional method14,15 to simple surface interactions with station-ary filler particles. The advantage of this method with respectto the previously mentioned~phenomenological! methods isthat the mesostructure formation inspecificpolymer systems

a!Electronic mail: sevink@chem.rug.nlb!Electronic mail: andrei@chem.rug.nl

JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 4 22 JANUARY 1999

22500021-9606/99/110(4)/2250/7/$15.00 © 1999 American Institute of Physics

with specificsurface interactions can be investigated. More-over, no simulations of polymer morphology formation withboundary interactions in three dimensions have been per-formed until this moment. This step is important since inmany cases such as the AFM measurements, one wants torelate measurements of the surface morphologies@two-dimensional~2D!# of specific polymers to bulk morphologies~3D!. With the help of this extension we can investigate allkinds of interesting phenomena: dependency of the orienta-tion of mesostructures~lamellar, hexagonal, etc! on the sur-face interactions and confinement of melts between flatplates and filler particles combined with shear. More impor-tantly, the time evolution of the mesophase structures can befollowed depending on geometry of filler particles. The fillerparticles constitute regions where the densities of the differ-ent bead types are zero by exclusion. This exclusion effectcan be treated by introducing so-called mask fields. In ourcalculations, the positions of the surface~the positions of themask field! are bound to the grid elements. For the moment,all simulations are carried out for a symmetric diblock co-polymer melt. More specific applications, such as the influ-ence of surface interactions on mesostructures in PL64, willbe considered in the future.

II. METHOD

The polymer melt is modeled as a compressible system,consisting of Gaussian chain molecules in a mean-field en-vironment. The free energy functional for copolymer meltshas a form that is similar to the free energy that is used inRefs. 4, 15, and 16:

F@$r%#52kT lnFn

n!2(

IE

VUI~r !r I~r !dr

11

2 (I ,J

EV2

e IJ~r2r 8!r I~r !rJ~r 8!dr dr 8

11

2 (IE

V2e IM ~r2r 8!r I~r !rM~r 8!dr dr 8

1kH

2 EVS (

In I@r I~r !2r I

0# D 2

dr , ~1!

except for an extra fourth term that contributes only in thedirect vicinity of the filler particles. This accounts for theinteraction of a polymer melt with surfaces. In this equation,n is the number of polymer molecules,F is the intramolecu-lar partition function for ideal Gaussian chains in an externalfield U, I is a component index,r I are the density fields ofthe different bead typesI, andV is the system volume. Insidethe filler particles, the densitiesr I of the different bead typesare equal to zero. Since the densityr is present in all inte-grals in the definition of the free energy@Eq. ~1!#, integrals~except ‘‘surface’’ term! over the entire volumeV are equalto the integrals restricted toV/V0, standing for the total vol-umeV with the exception of the volume taken by the fillerparticles, denoted asV0. The filler particles considered hereare constrained to the condition of stationary position intime. The constant density fieldrM ~where M represents

beads of the filler particle type! that appears in Eq.~1! isdefined asrM(r )51 for rPV0 andrM(r )50 for rPV/V0.The average concentration isr I

0 and n I is the particle vol-ume. The cohesive interactions have kernelse IJ $e IJ(ur2r 8u)5e IJ

0 (3/2pa2)3/2exp@2(3/(2a2))(r2r 8)2#%. The sur-face interactions have kernelse IM . The Helfand compress-ibility parameter iskH .16

The external potentialsUI are conjugate to the densitiesr I via the Gaussian chain density functional.15 For our sys-tem, Eq.~1!, the statistical distribution function of a chain ofN beads, in a certain conformation specified by the coordi-nates of the beads$R1 ,¯ ,RN% is3

C~R1 ,¯ ,RN!51

Fexp2S 3

2a2 (s52

N

~Rs2Rs21!2

1 (s851

N

Us8~Rs8!D , ~2!

where the external fieldsUs8 are in unitskT and a is theGaussian bond strength parameter. The ensemble averageparticle densityrs(r ) of a certain beads at position r inspace is

rs@U#~r !5CM~r !EVN

C~R1 ,¯ ,RN!

3d~r2Rs!dR1 ...dRN , ~3!

whereC is a normalization constant~see Ref. 3! and a maskfieldM~r ! is used that is defined as

M~r !5 H01

rPV0

rPV/V0.

The density functional can be calculated via Green propaga-tors ~see Ref. 19 for details!

rs~r !}Gs~r !s@Gs11inv #~r !. ~4!

The set of once integrated Greens functionsGs(r ) andGs11

inv (r ) are related by the recurrence relations

Gs~r !5M~r !e2Us~r !s@Gs21#~r !,~5!

Gsinv~r !5M~r !e2Us~r !s@Gs11

inv #~r !,

where G0(r )5GN11inv (r )51. The linkage operators

5s@ f #(r ) is defined as a convolution with a Gaussian kernel

s@ f #~r !5S 3

2pa2D 2/3EVe2„3/~2a2!…~r2r8!2

f ~r 8!dr 8. ~6!

The time evolution of the density fieldr I(r ) can be de-scribed by a time dependent Landau–Ginzburg typeequation8,43

]r I

]t5MI¹•r I¹m I1h I . ~7!

Herem I5dF/dr I is the intrinsic chemical potential,MI is amobility, and h I is a stochastic noise which is distributedaccording to the fluctuation–dissipation theorem.18

The boundary conditions that are used on the simulationbox are periodic boundary conditions. For the diffusion flux

2251J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 Sevink et al.

in the vicinity of the filler particles, rigid-wall boundary con-ditions are used. A simple way to implement these boundaryconditions in accordance with the conservation law is to al-low no flux through the filler particle surfaces, i.e.,

¹m I•n50, ~8!

wheren is the normal pointing towards the filler particle. Itis easy to show that with these boundary conditions on thefiller particles, the free energy decreases with time~neglect-ing the contribution of the noise!:

]F

]t5E

V/V0(I

dF

dr I

]r I

]tdV

5EV/V0(I

m I

]r I

]tdV5E

V/V0(Im IM I¹r I¹m IdV

5(I

M IE]~V/V0!

m Ir I¹m In dS

2(I

M IEV/V0

r I~¹m I !2dV<0, ~9!

where we have used Gauss’ theorem. Since we use a 27 pointstencil for the calculation of derivatives, the presence of fillerparticles affects only the grid elements one cell away fromthe rigid boundaries.19

The same boundary conditions apply to the noiseh I .The noise has a Gaussian distribution with moments dictatedby the fluctuation–dissipation theorem, and is equal to18 ~inthe notation of that paper!

h I~r ,t !5A2

b¹ r•AMIr~r !wK~r ,t !, ~10!

where wK is a completely decorrelated random field withGaussian distribution

^wIc~r ,t !&50,

~11!^wI

c~r ,t !wJc8~r 8,t8!&5d IJdcc8d~r 2r 8!d~ t2t8!.

For the numerical integration of Eq.~7! a Crank–Nicolsonscheme is used. Starting configurations for the time integra-tion are homogeneous density distributions and the externalfield is equal to zero.

III. RESULTS AND DISCUSSION

As an example, we simulate the behavior of a modelpolymer which is represented by anA8B8 Gaussian chain.For all simulations, a cubic grid of dimensionL3L ~2D! orL3L3L ~3D! is used. The dimensionless parameters thatare used in the numerics are chosen to be similar to the onesused in Ref. 5. They are:15,4 the mesh sizeh of the physicalgrid equal toh50.7 nm, the dimensionless time stepDt5b21Mh22Dt50.2 (Dt510 ns), and a bond lengthagiven by the optimal value19 of the grid parameterd5ah2151.1543. The noise scaling parameterV5n21h3

5100, with n the bead volume which in our case is takenequal for all bead species~see also Refs. 3 and 4 for moredetails!. The exchange parameter is chosen to be equal tobeABn2151.0 (eAA5eBB50) and the compressibility pa-

rameterk85bkHn512.0. The diffusion coefficientD that isrelated to the mobilityM by DI5b21MI is chosen equal toDI510211m2/s for both bead species. The surface relatedinteraction parametersbeAMn21 and beBMn21 are chosento be different in the different simulations. Taking into ac-count noise and compressibility is important for creating aphysically realistic model. We have discussed their effect onphase separation in detail earlier.16,18

First we consider a simulation in two dimensions (L5128) ~Fig. 1!. The filler part of the melt consist of fourblock-shaped particles of 12312 grid cells each. The subse-quent interaction parameters are chosen equal tobeAMn21

54.0 andbeBMn2150.0. Note that in the incompressiblecase the surface contribution in the free energy, Eq.~1!, re-duces to a constant, if these two parameters are equal.

In Fig. 1~a! the results are shown at an early stage (t5250). One clearly observes the phase separation that startsto take place. Around the filler particles surface directed mi-crophase separation can be seen. Closed lamellae parallel tothe filler particles surface start to form. At a later stage thelamellae become more distinct and grow in length until mostof them span the whole box. In Fig. 1~b! we show the finalstage att52000. It can be observed that the orientation ofthe lamellae is determined to a great extent by the orientationof the filler particles.

After having observed the process of formation and ori-entation of lamellae in two dimensions, we now look at a 3Dcase. In all three-dimensional examples considered, the boxparameterL532. First we consider two cases~displayed inFigs. 2 and 4! where the filler particles are almost identical.In Fig. 2~a!, the fieldM~r ! is shown for a filler particleconsisting of a plane (3033031 grid cells! in the Cartesianx andy directions. The interaction parameters are chosen tobe equal tobeAMn21521.0 andbeBMn2151.0. These pa-rameters will be used in all the following simulations~whereinteraction between filler particle and polymer melt ispresent!. This plane almost spans the box, leaving slots ateither sides of the plane of one grid cell wide. Az projectionof a part of the total three-dimensional space is shown in Fig.2~b!. The slots between the planes can clearly be observed.In this case, the pattern periodicity of the surface is 30. Incontrast, in Fig. 4~a!, the mask field is shown for a fillerparticle consisting of a plane (3233231) that spans thewhole box.

As can be seen from Figs. 2~c! (t5500) and 2~d! (t

FIG. 1. Two dimensional simulation ofA8B8 block copolymer melt in1283128 box in the presence of four filler particles:~a! left: early stage att5250, ~b! right: final stage att52000.

2252 J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 Sevink et al.

52000), the presence of the filler particle results in lamellaeparallel to the filler particles surface. Parallel lamellae start toform close to the filler particle and spread through the melt.When closely examined, the slots result is a slight bending ofthe lamellae at the edges of the lamellae@Fig. 2~d!#. Oneshould note that the time scales on which the lamellae areformed are much shorter than the time scales of lamellaeformation in the absence of this filler particle.5 Thus, thepresence of surfaces speeds up the formation of mesophaseformation enormously.

In the absence of surface interactions, we may observethe influence of confinement~by the filler particle! on theresulting mesostructures of the polymer material. At thesame time, we get a feeling of the influence of surface inter-actions on the orientation of the final morphologies. For thispurpose, we consider the same filler particle as before, themask field of which is shown in Fig. 2~a!, and takeeAM

5eBM50.0. Given the mask field of Fig. 3~e! @the same as inFig. 2~a! but different observation angle# we see in Fig. 3~f!that at t5500 a structure appears without apparent globalorientation. If we compare this with the structure shown inFig. 2~c! at the same stage, we conclude that surface inter-action is, especially in the early stages of phase separation,apparently responsible for speeding up the mesophase forma-tion. At t54000 @Fig. 3~g!#, where the previous examplewith boundary interaction already reached the stage of per-fect ~slightly bended! lamellae parallel to the boundary sur-face, we see larger structures but still no general global ori-entation. Only att510 000 @Fig. 3~h!# do we observe aglobal orientationperpendicularto the surface of the fillerparticle. Although such orientation requires more investiga-tion, we note that the system is confined between two plates~due to the box periodicity! that act as a strong regularization

of the lammelar spacing and orientation. For the hexagonalcylindrical phase perpendicular orientation has been ob-served in experiments with thin polymer films.2 We concludethat introducing surface interactions induces fast mesostruc-ture formation, with an orientation mainly directed by theinteraction parameters.

In the case of a melt between two planes that are infinitein two Cartesian dimensions~Fig. 4! and the sameeAM andeBM as before in Fig. 2, we observe fast convergence to astable morphology. Lamellae start to form immediately afterthe quench@Fig. 4~b! for t5250#, their number increasingwith time and stable in time. Att51000 @Fig. 4~c!# we ob-

FIG. 2. Lamellar formation of anA8B8 copolymer melt in the presence ofsquare plates of one grid cell thickness. The interaction of polymer blockswith the surface isbeAMn21521.0 andbeBMn2151.0: ~a! view of fillerparticle in simulation box,~b! space filled with filler particles~the slotsbetween filler particles are drawn as white lines!, ~c! morphology ofA beads~isolevel nrA50.5) in one simulation box att5500, ~d! same for t52000.

FIG. 3. Lamellar formation of anA8B8 copolymer melt in the presence ofthe same filler particle as the previous figure. Moreover, there is no interac-tion between the polymer beads and boundaries of the filler particles:~e!view of filler particle in simulation box,~f! morphology ofA beads~isolevelnrA50.5) in one simulation box att5500, ~g! same fort54000,~h! finalmorphology att510 000.

FIG. 4. Lamellar formation of anA8B8 copolymer melt in the presence ofinfinite planes of one grid cell thickness:~a! view of the filler particle in thesimulation box,~b! morphology ofA beads~isolevelnrA50.5) att5250,~c! same fort51000.

2253J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 Sevink et al.

serve five lamellae of slightly different thickness, especiallytwo very thin ones at the filler particle boundaries. When weconsider only a different thickness of the filler particle~aplane of 3233234 grid cells!, we observe the same numberof lamellae at the same time@Fig. 5~a! at t5250 and Fig.5~b! at t51500# because the width of the lamellae hasdecreased.

In order to take a closer look at the influence of rough-ness, we now consider a filler particle that simulates a‘‘rough’’ plate, shown in Fig. 6~a!. The mask fieldM~r !consists of a plane (3233231) spanning the whole box,

with a smaller plane (1831831) on top. From a compari-son of the result att5250 @Fig. 6~b!# of this simulation andthe result att5250 of a ‘‘smooth’’ filler particle@Fig. 4~b!#we see that the influence of the roughness in the early stagesis not clearly visible. In Fig. 6~c!, the result is shown att5500. The morphology is clearly different from the mor-phologies in Fig. 4: the formation of parallel lamellae takesplace at a much smaller rate as in the ‘‘smooth’’ case. Att52500 @Fig. 6~d!#, one can clearly observe a final state ofalmost parallel lamellae. The process of parallelization caneven be considered more closely in Figs. 6~e! and 6~f!, whereorthogonal slices halfway through the box of the same mor-phologies@t5500 in Fig. 6~e! andt52500, Fig. 6~f!# can beseen.

Another type of filler particle that has been considered isa rod. In Fig. 7~a! the mask fieldM~r ! is shown for oneinfinite rod. At t5500 @Fig. 7~b!# large structures withoutapparent global orientation are present. At later timest52000 @Fig. 7~c!# andt54000 @Fig. 7~d!# larger structuresare formed with an orientation that is apparently parallel tothe rod. However, close examination of the resulting struc-tures ~not shown here! shows clusters of different orienta-tions ~parallel, perpendicular!.

The configuration of Fig. 7~d! can be used as a startingconfiguration for the application of simple steady shear.5 Al-though the theory considered here is only applicable to sta-tionary objects, we can apply shear in the direction of the roddue to the infinite length of the rod in the direction of theflow. The dimensionless shear rate is chosen equal tog̃50.001 ~for details see Ref. 5!. In Fig. 8~a!, the configura-tion is shown att55000. A global orientation starts to de-velop. This orientation becomes more apparent at later times@Fig. 8~b! at t57500 and Fig. 8~c! at t510 000#. The finalglobal orientation is one of almost parallel lamellae that are

FIG. 5. Same as in Fig. 4 but for thicker infinite planes~four grid cellsthick!: ~a! morphology ofA beads~isolevelnrA50.5) att5250, ~b! samefor t51500.

FIG. 6. Lamellar formation ofA8B8 copolymer melt in the presence of a‘‘rough’’ plane: ~a! view of the filler particle in the simulation box,~b!morphology of A beads~isolevel nrA50.5) at t5250, ~c! same fort5500, ~d! final stage att52500, ~e! orthoslice through the middle of thebox of the morphology shown in~c!, ~f! orthoslice for~d!.

FIG. 7. Mesoscale formation ofA8B8 copolymer melt in the presence of oneinfinite rod. The interaction of polymer blocks with the surface isbeAMn21521.0 andbeBMn2151.0: ~a! view of the filler particle in thesimulation box,~b! morphology ofA beads~isolevelnrA50.5) att5500,~c! same fort52000,~d! final stage att54000.

2254 J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 Sevink et al.

under a 45° angle with the shearing direction. From the or-thogonal projections in Figs. 8~d!, 8~e! and 8~f!, it can beseen that the lamellae are slightly bent. However, we havefound from previous experiments that the global orientationis in this case much affected by the rather small box size(L532). Tilted lamellae are known to be governed by thesize of the box. Given the influence of one rod~the volumefraction of rodsV0/V is about 0.1%! we will now considerthe influence of a few~in our case, six! randomly positionedrods ~the volume fraction is about 0.6%!. In Fig. 9~a!, theposition of the rods can be observed. Again, shear is appliedat t54000 in the direction of the rods. The same shear rateis used as in the previous example. From Figs. 9~b! and 9~d!,where the morphology and the orthogonal projection in thedirection of the rods is shown fort54000, we conclude thatparallel lamellae are formed with a clear global orientation.This orientation is to a large extent governed by the presence

of the rods, since the rods can be observed to be in themiddle of A rich regions, and therefore can be consideredresponsible for the particular bending of the lamellae. Theordering is similar to the ordering in Fig. 8~e!, so that weconclude that the process istwice as fast, not even takingshear into account. At the final stage att55750 @shown inFigs. 9~c! and 9~e!# almost perfect lamellae are formed. De-fects are removed by the shear, but the local bending of thelamellae is not affected, and it can be concluded to be causedby the presence of the rods.

As a final example, we consider the effect of a plane~2332332 grid cells! combined with an almost sphericalparticle ~with a radius of five grid cells!. The box-size ischosen to be noncubic: 64332332. In Fig. 10~a!, the maskfieldM~r ! is shown. Att5250 @Fig. 10~b!# lamellae paral-lel to the plane start to form. Att52500 @Fig. 10~c!# weobserve that these lamellae span the whole box, apart from avery tiny region where the spherical particle is present. Froman orthogonal slice att52500 @shown in Fig. 10~d!# weconclude that the parallel orientation is due to the presenceof the plate. This orientation is too strong to be affected bythe presence of the spherical particle.

IV. CONCLUDING REMARKS

Here we have extended the dynamic density functionalmethod formulated before for unconfined block copolymermelts3 to the same systems in the presence of different sur-faces. The free energy functional includes an extra termwhich describes interactions of polymer beads with a sur-face. In the present paper we focus on the surface directedphase separation itself and limit ourselves to the case of im-mobile surfaces. With this limitation, we are still able toinvestigate a large number of interesting phenomena. The

FIG. 8. Lamellar formation of the melt of Fig. 7 in the presence of shear inthe direction of the rod. The conformation of Fig. 7~d! is taken as a startingstructure:~a! morphology ofA beads~isolevel nrA50.5) at t55000, ~b!Same fort57500,~c! final stage att510 000,~d! projection in the direc-tion of shear of the morphology shown in~a!, ~e! projection of ~b!, ~f!,projection of~c!.

FIG. 9. Lamellae formation ofA8B8 copolymer melt in the presence of sixinfinite rods:~a! view of the filler particles in the simulation box,~b! mor-phology ofA beads~isolevelnrA50.5) att54000 @comparable in time toFig. 7~d!#. At t54000, shear is applied with the same shear rate as in Fig.8, ~c! final structure fort55750,~d! projection in the direction of shear ofthe morphology shown in~b!, ~e! projection of~c!.

FIG. 10. Lamellae formation ofA8B8 copolymer melt in the presence of aplane and a spherical particle. The interaction of polymer blocks with thesurface isbeAMn21521.0 andbeBMn2151.0: ~a! view of the filler par-ticles in the simulation box,~b! morphology of A beads~isolevel nrA

50.5) at t5250, ~c! final stage att52500, ~d! orthoslice through themiddle of the box of the morphology shown in~c!.

2255J. Chem. Phys., Vol. 110, No. 4, 22 January 1999 Sevink et al.

analysis of moving colloidal particles in the polymer envi-ronment, implying moving boundary conditions, is left tofuture publications. Time evolution of mesoscopic block co-polymer morphologies is studied using time dependentLandau–Ginzburg type equations for the densities of poly-mer blocks. Noise and compressibility of the system are ex-plicitly taken into account. Simulations of different systemsare performed in 2D and 3D. They include random posi-tioned square-like filler particles in 2D, infinite and finiteplane boundaries in 3D, rough surfaces, cylindrical rods, andcombined geometries like a sphere near a wall. A simplecase of a system under shear is considered as well. We ob-serve a large influence of small filler particles such as infiniterods. These filler particle limit the degrees of freedom inwhich mesostructures can orient themselves, and thereforelead to perfect lamellae already at a very early stage. Shearand filler particles have similar influences on the conforma-tional behavior, as can be seen from a copolymer melt withone and six rods. More generally, inserting a surface into adiblock copolymer melt induces order in the system andspeeds up mesophase separation enormously. The analysis ofother polymer systems in the presence of surfaces is inprogress.

ACKNOWLEDGMENTS

Two of the authors~G.J.A.S. and A.V.Z.! acknowledgethe support of the MesoDyn project ESPRIT No. EP22685 ofthe European Community. Discussions with Henk Huininkare gratefully appreciated.

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