Orientations of the lamellar phase of block copolymer melts under oscillatory shear flow

Abstract

We develop a theory to describe the reorientation phenomena in the lamellar phase of block copolymer melt under reciprocating shear flow. We show that similar to the steady-shear, the oscillating flow anisotropically suppresses fluctuations and gives rise to the →⊥ transition. The experimentally observed high-frequency reverse transition is explained in terms of interaction between the melt and the shear-cell walls.The behaviour of the lamellar phase (a stripped pattern) of block copolymer melts under oscillatory shear flow have attracted attention of numerous experimental studies 1-6 . Shear flow is known to influence the orderdisorder transition (ODT) temperature and the orientation of the lamellae with respect to the shear geometry. Thus, in the vicinity of ODT at low frequencies the lamellae orient with their normal parallel to the shear gradient (the parallel orientation), while at higher frequencies their normal is perpendicular to the velocity and the gradient directions (the perpendicular orientation). Further increase of frequency results in reappearance of the parallel orientation 5 . In this Letter we propose an explanation of this orientation behaviour which is usually referred to as a double-flip phenomena.Earlier theories, which deal with steady shear, emphasize the role of compositional fluctuations 7-9 . The stable orientation is seen as a result of interaction between the shear flow and the fluctuation spectra. In equilibrium fluctuations destroy the long-range correlations and therefore lower the ODT temperature with respect to its mean-field value. Imposition of shear breaks the rotational symmetry and anisotropically suppresses fluctuations. The direction of the strongest suppression will have the highest ODT temperature and the corresponding orientation of lamellae will be selected. We will show that the selected orientation depends on the amplitude and frequency of the flow. We base our analysis on the Fokker-Planck equation for the probability densityHere φ k is a fluctuating scalar field described by the Brazovskii Hamiltonian 10τ is a temperature-controlling parameter, k −1 0 is an intrinsic length-scale of the block-copolymer melt arising from the interplay of interactions and the chain connectivity, µ is an Onsager coefficient which is approximated by µ = µ(k 0 ) and assumed to be frequency independent 11-13 . The last term in Eq.(1) describes a coupling between the shear flow v = A ω cos ωt y e x and the gradient of the order parameter. Here we assume that this form of flow is valid for all ω and A. The Fokker-Planck equation (1) generates the equations for the amplitude of the average order-parameter profile φ k = a(δ k,k0n + δ k,−k0n ) oriented along the unit vector nand for the structure factorwhereThe interaction 14 between fluctuations λ(k, −k, q, −q) = λ 1 − β(k ·q) 2 , withk = k/k, renormalizes the temperature r(k) and makes it anisotropic in the presence of shear.The stability criterion for an orientation is derived from Eq.(3).It has a potential form ∂a/∂t = −(µ/2)∂Φ(a)/∂a, w...

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