Mean Field and Monte Carlo Modeling of Multiblock Copolymers
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Mean Field and Monte Carlo Modeling of Multiblock Copolymers K. . Rasmussen, T. D. Sewell, T. Lookman, and A. Saxena Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545
Abstract We compare for multiblock copolymers the results of mean eld calculations with those from Monte Carlo simulations based on the bond uctuation method and experimental results from scattering data. The application of Leibler's [1] theory for copolymers and the results of Monte Carlo simulations indicate that the microphase separation transition occurs at larger N as the number of blocks is increased beyond two (i.e., beyond diblock), and that the characteristic length scale of the emerging morphology decreases as the number of blocks increases. The latter is in qualitative agreement with published experimental results [2] for model multiblock poly(styrene-isoprene) systems and recent results [3] for a segmented poly(ester-urethane).
Introduction The elastic scattering of light, x-rays, and neutrons due to thermal concentration uctuations in homopolymer melts is one of the more interesting aspects of polymer physics because the
uctuations re ect microscopic properties of single chains such as the radius of gyration, as well as thermodynamic interactions between monomers such as the Flory-Huggins interaction parameter . In the case of blends, copolymers, etc., analysis of the uctuations leads to information about, and improved understanding of, the microphase separation transition (MST) in multicomponent systems. One of the most general and widely applied theories for the MST in diblock systems was developed by Leibler [1] in the context of the random phase approximation [4]. In the present report we describe extensions of the theory to treat multiblock systems [A B ] , where (m + n)p = N is the total chain length. Of particular interest are (1) the N -dependence of the MST and (2) the variation of characteristic length scale for concentration uctuations based on peak positions in the structure factor, both as functions of the block number p and the ratio = m=(m + n). Although the theory has previously been applied to multiblock copolymers [5], our objective is to compare the results of this approach to Monte Carlo simulations and small angle x-ray and neutron scattering data. The mean eld predictions for the structure factors are compared in selected cases to the results of lattice Monte Carlo realizations performed using the bond uctuation algorithm with identical values of N , , and . This provides for an internal consistency check between the two approaches and sets the stage for more thorough studies of the e ects of \aging" on chain-level details of polymer morphology and other properties. m
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Mean Field Model In this section we apply a formalism initially developed by Leibler [1] and Ohta and Kawasaki [6] for diblock copolymer melts. The method begins with the microscopic Edwards Hamiltonian [7] for a two component system consisting of N polymer chains of length N composed of m monomers o
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