Lammelar phase of diblock copolymer melt under shear: kineticsand conformational analysis.A.V. Zvelindovsky, G.J.A. Sevink, B.A.C. van Vlimmeren, N.M. Maurits, J.G.E.M. FraaijeDept. Biophysical Chemistry, University of Groningen,Nijenborgh 4, 9747 AG Groningen, The NetherlandsShort title: Lammelar phase of copolymer melt under shear.Key words: shear, polymer, lammelar, density functional.Address for proofs:Dr A V Zvelindovsky,Dept. Biophys. Chem.,University of Groningen,Nijenborgh 4, 9747 AG Groningen,The Netherlands

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Abstract3D computer simulations of lammelar phase of a diblock copolymer meltunder simple steady shear are performed. The polymer system is describedby the density functional as a collection of Gaussian chains in a mean �eldenvironment. The 3D mesophase morphology is obtained as a solution of adi�usion-convection equation with random noise �eld. Here, an analysis ofpolymer chain conformations in lamellae is considered.

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Because of the wide area of industrial applications the behaviour of complex polymersystems in shear ows is intensively studied both experimentally and theoretically [1{3].The stability of morphology structures and phase transitions in complex polymer liquids inshear have been investigated by many authors [4{7]. During the last years the kinetics ofmorphologies of complex uids in applied external shear has also been studied by computersimulation techniques using time dependent Landau-Ginzburg models [8{13]. These modelsare based on traditional free energy expansion methods [14{16] which contain only the basicphysics of phase separation [17]. In contrast to these phenomenological theories we do nottruncate the free energy at a certain level, but rather retain the full polymer path integralby a numerical procedure [17{23]. Recently Kawakatsu and Doi started to use a similarapproach [24,25].Calculation of polymer path integrals is computationally very intensive and requirese�cient numerical techniques. However it allows us to describe mesoscopic dynamics of aspeci�c complex polymer liquid.We model the polymer melt as a compressible system, consisting of collection of Gaussianchains in a mean �eld environment. The free energy functional for copolymer melts has theform [19,20]: F [f�g] = �kT ln �nn! �XI ZV UI(r)�I(r)dr+12XI;J ZV 2 �IJ(r� r0)�I(r)�J(r0)drdr0 +�H2 ZV XI �I(�I(r)� �0I)!2 dr (1)where n is the number of polymer molecules, � is the intra-molecular partition function forideal Gaussian chains in an external �eld U , I is a component index and V is the systemvolume. The external potentials UI are conjugate to the densities �I via de Gaussian chaindensity functional. The average concentration is �0I and �I is the molecular volume. Thecohesive interactions have kernels �IJ . The Helfand compressibility parameter is �H [20].For our system (1) the statistical distribution function of a copolymer chain of N beads, in3

a certain conformation speci�ed by the coordinates of the beads fR1; � � � ;RNg, is (R1; � � � ;RN) = e�[ 32a2 PNs=2(Rs�Rs�1)2+PNs0=1 Us0(Rs0 )] (2)where the external �elds Us0 are in units kT and a is the Gaussian bond strength parameter.The ensemble average particle density �s(r) of a certain bead s at position r in space is�s[U ](r) / ZV N (R1; � � � ;RN) �(r�Rs)dR1 � � �dRN (3)Since the sequence space is discrete, the density functional can be calculated via Greenpropagators [23] �s(r) / Gs(r)��[Ginvs+1](r) (4)The set of (once integrated) Green functions Gs(r) and Ginvs (r) are related by the recurrenceequations Gs(r) = e�Us(r)�[Gs�1](r) (5)Ginvs (r) = e�Us(r)�[Ginvs+1](r)with G0(r) = GinvN+1(r) = 1. The linkage operator � is de�ned as a convolution with aGaussian kernel: � [f ] (r) � � 32�a2� 32 ZV e� 32a2 (r�r0)2f(r0)dr0: (6)As for the dynamic model, the time evolution of the density �eld �I(r) under simpleshear ow, vx = _ y; vy = vz = 0; can be described by the di�usion{convection equationwith a stochastic term [8,12]_�I = MIr � �Ir �F��I � _ yrx�I + �I (7)MI is a mobility parameter, _ is the shear rate (the time derivative of the strain ), �Iis a thermal noise �eld which is distributed according to the uctuation-dissipation theorem4

[22]. For a L� L� L cubic grid we use a sheared periodic boundary condition [26]:�(x; y; z; t) = �(x + iL + jL; y + jL; z + kL; t). Here we consider an approximation in thesense that the form of the chain distribution function (2) is not changed in shear.The density functional technique allows us also to perform conformation analysis ofpolymer chains. For example, the probability of �nding bead s at coordinate r while thebead s0 of the same molecule �xed at r0 is�s[U ](r) / ZV N (R1; � � � ;RN) (8)� �(r�Rs)�(r0 �Rs0)dR1 � � �dRNIn the Green propagator algorithm (5) this leads to replacement of exp(�Us0(r)) by �(r�r0).We simulate the behaviour of a model polymer which is represented by an A8B8 Gaussianchain. The dimensionless parameters that are used in the numerics are chosen as (see [19]for details): the dimensionless time � = ��1Mh�2t, the grid parameter d = ah�1 = 1:1543,the noise scaling parameter equal to 100, the exchange parameter �AB = ��AB��1 = 1:0(�AA = �BB = 0) and the compressibility parameter �0 = ��H� = 12:0. The new dimen-sionless parameter e_ for the shear is chosen equal to e_ = �t _ = 10�3. In our simulations��1Mh�2�t = 1=2.FIG. 1 and 2 hereWe performed a 3D simulation (L=64) starting with the initial structure at � = 500(Fig.1) where no lamellar structure can be observed. During the application of steady sim-ple shear between � = 500 and � = 7500 a global lamellar orientation appears. The structureessentially di�ers from one without shear, where only local lamellar clusters with many de-fects were observed. Thus application of shear speeds up lamellar formation in a diblockpolymer melt. The observed alignment is so called \perpendicular" lamellae (Fig.2). Fromexperiments and stability analysis this orientation is well known to be the most stable oneat high shear rates [3]. 5

FIG. 3 and 4 hereConformational analysis provides the detailed view on the density map of polymer chainsin observed morphologies. Two examples are shown on �gures 3 and 4. In the �rst case we�xed one bead from a central part of a chain at the surface of perfect lamella. The symmet-ric shape of the density probability isosurface corresponds to almost undeformed polymerchains. This is not the case in the vicinity of defects, like inter-lamellar necks (Fig. 4).FIG. 5 hereNote that lamellar structure is a stable con�guration of a symmetric diblock copolymermelt. After reaching almost perfect lamellar structure we have applied alternative simplesteady shear ow with a velocity gradient perpendicular to lamellar plane (Fig.5). Thesystem has not been destroyed, which indicates that the structure is enough stable. Thenumber of defects did not increase as well.Our approach can be used for the investigation of shear induced ordering in speci�cpolymer systems. The simulation of morphology evolution of Pluronic (lamellar, hexagonal,micellar, bicontinious phases) under shear is in progress.

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REFERENCES[1] Bird R B, Hassager O, and Armstrong R C (1987) Dynamics of polimeric liquids. Wiley,New York.[2] Schmittmann B, Zia R K P (1994) In: Domb C, Lebowitz J (eds) Phase Transitionsand Critical Phenomena. Academic, London.[3] Chen Z R, Korn�eld J A, Smith S D, Grothaus J T, Satkowski M M (1997) Science277:1248{1253[4] Fredrickson G H (1994) J Rheol 38:1045{1067[5] Goulian M, Milner S T (1995) Physical Review Letters 74:1775{1778[6] Cates M E, Milner S T (1989) Physical Review Letters 62:1856{1859[7] Balsara N P, Dai H J (1996) J Chem Phys 105:2942{2945[8] Kodama H, Doi M (1996) Macromolecules 29:2652{2658[9] Qiwei He D, Nauman E B (1997) Chemical Engineering Science 52:481{496[10] Gonnella G, Orlandini E, Yeomans J M (1997) Physical Review Letters 78:1695{1698[11] P�atzold G, Dawson K (1996) J Chem Phys 104:5932{5941[12] Kodama H, Komura S (1997) J Phys II France 7:7{14[13] Ohta T, Nozaki H, Doi M (1990) Physics Letter A 145:304{308[14] Cahn J W, Hilliard J E (1958) J Chem Phys 28:258{267[15] Oono Y, Puri S (1987) Phys Rev Let 58:836{839[16] de Gennes P G 91980) J Chem Phys 72:4756{4763[17] Maurits N M, Fraaije J G E M (1997) J Chem Phys 106:6730{6743[18] Fraaije J G E M (1993) J Chem Phys 99:9202{92127

[19] Fraaije J G E M, van Vlimmeren B A C, Maurits N M, Postma M, Evers O A, Ho�mannC, Altevogt P, Goldbeck-Wood G (1997) J Chem Phys 106:4260{4269[20] Maurits N M, van Vlimmeren B A C, Fraaije J G E M (1997) Phys Rev E 56:816{825[21] Maurits N M, Fraaije J G E M (1997) J Chem Phys 107:5879{5889[22] van Vlimmeren B A C, Fraaije J G E M (1996) Comp Phys Comm 99:21{28[23] Maurits N M, Altevogt P, O A Evers O A, Fraaije J G E M (1996) Comp Pol Sci 6:1{8[24] Kawakatsu T (1997) Phys Rev E 56:3240{3250[25] Hasegawa R, Doi M (1997) Macromolecule 30:3086{3089[26] Doi M, Chen D (1989) J Chem Phys 90:5271{5279Address:Dr. A. V. ZvelindovskyDept. Biophys. ChemistryUniversity of GroningenNijenborgh 4, 9747 AG Groningen, The Netherlands

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FIGURE LEGENDSFIG. 1Isosurface representation of the A8B8 melt at � = 500 with di�usion only. The isolevel is�A = ��A = 0:3. This structure is used as an initial structure for shearing.FIG. 2Isosurface representation of the A8B8 melt at � = 7500 and shear ow vx = _ y; vy = vz = 0,using the structure at � = 500 (Fig.1) as a starting structure. The isolevel is �A = ��A = 0:3.One can clearly observe the global lamellar orientation. The x and y-axes are indicated.FIG. 3Conformational analysis of A8B8 molecule while bead An=7 �xed at the surface of perfectlamella. This position indicated by small gray square on �gure a. Figure b shows the iso-surface of the density probability (8) of �nding A� (light gray) and B� (dark grey) tails ofthe chain in space.FIG. 4Conformational analysis of A8B8 molecule while bead An=7 �xed at the surface of an inter-lamellae neck. This position indicated by small square on �gures a for two rotations of thesame 3D picture. Figures b show the isosurface of the density probability (8) of �ndingA� (light gray) and B� (dark) tails of the chain in space. The density map has clearlynon-symmetric shape.FIG. 5Isosurface representation of the A8B8 melt in alternating shear ow of the same shear rateas in Fig. 2. The structure of the Fig.2 has been taken as initial one after rotating aroundx-axis over �=2. The shear has been applied during time � = 1325.15