Theory Of Microphase Separation Research Articles

Theory of Microphase Separation in Concentrated Solutions of Sequence-Specific Charged Heteropolymers

Theory of Microphase Separation in Concentrated Solutions of Sequence-Specific Charged Heteropolymers

Mesophase formation in two-component cylindrical bottlebrush polymers

When two types of side chains (A,B) are densely grafted to a (stiff) backbone and the resulting bottlebrush polymer is in a solution under poor solvent conditions, an incompatibility between A and B leads to microphase separation in the resulting cylindrical brush. The possible types of ordering are reminiscent of the ordering of block copolymers in cylindrical confinement. Starting from this analogy, Leibler's theory of microphase separation in block copolymer melts is generalized to derive a description of the system in the weak segregation limit. Also molecular dynamics simulation results of a corresponding coarse-grained bead-spring model are presented. Using side chain lengths up to N = 50 effective monomers, the ratio of the Lennard-Jones energy parameter between unlike monomers (ε(AB)) and monomers of the same kind (ε(AA) = ε(BB)) is varied. Various correlation functions are analyzed to study the conditions when (local) Janus-cylinder-type ordering and when (local) microphase separation in the direction along the cylinder axis occurs. Both the analytical theory and the simulations give evidence for short-range order due to a tendency toward microphase separation in the axial direction, with a wavelength proportional to the side chain gyration radius, irrespective of temperature and grafting density, for a wide range of these parameters.

Theory of microphase separation in charged crosslinked polymer blends with arbitrary charge distribution

In this paper, we are interested in the phase behavior and scattering properties of charged crosslinked polymer blends in solution. The system undergoes a microphase separation, below some critical temperature. To study such a transition, use is made of the standard de Gennes theory based on an analogy with a dielectric medium. This analogy is extended to include the effects of the initial composition fluctuations in order to improve its agreement with experimental data in the small wave vector range. The excluded-volume interactions are explicitly introduced through the blob model. The charge effects on the phase behavior are examined, for any charge distribution of polyions and for any salt concentration. This completes a previous study which was concerned with the situation where only one species is charged. The early kinetics of microphase separation is discussed, and the charges contribution to the growth rate is also evaluated.

Theory of microphase separation in multiple segment-type statistical multiblock copolymers with arbitrary block molecular weight distributions

A Landau free energy is derived for the weak segregation regime (WSR) of melts belonging to a very general class of statistical multiblock copolymers, referred to as “multiple segment-type statistical multiblock copolymers.” Copolymer chains in this class consist of sequences of up to M⩾2 chemically different types of segments, organized into sequences of blocks of varying lengths (molecular weights). The possible sequences of blocks that are encountered in the copolymer chains, as far as their type is concerned, are described by a first-order Markov process, while the block molecular weight distributions of these M types of blocks are completely arbitrary. The number of blocks per chain is assumed to be large. This class of copolymers is sufficiently general to encompass all industrial relevant bulk statistical multiblock copolymers, such as all known thermoplastic elastomers. The particular free energy considered is just one realization of an even more general Landau free energy which is applicable to the WSR of melts of all conceivable copolymers, including homopolymers and all possible blends. The derivation of this Landau free energy is given in Appendix A.

Effects of critical fluctuation in the theory of microphase separation in symmetric systems

We consider the effects of critical fluctuations in systems, which are undergoing microphase separations, such as diblock copolymers and interpenetrating networks. We predict the critical regime which, governed by the elastic constant C0, prevents the system from global separation. We obtain that in two dimensions there is no narrowing of the critical fluctuations for large degrees of polymerization studied within the field theoretic renormalization group. The scattering function S(p) in the homogeneous phase is obtained to behave as (∂[p2S−1(p)]/∂p2)p→0 ≂m0 p*(γ−1/ν), where γ and ν are the critical exponents and p*=2π/λ with λ being the dimension of the microphase.

Kinetic theory of microphase separation in block copolymers

A semiphenomenological theory that has the advantage of taking into account nonlinear and nonlocal contributions in the free energy for microphase separation of block copolymers is proposed. A kinetic nonlinear equation defining the process of structure formation from a melt is obtained, and its analytical solution at the melt-structure transition temperature is examined. In this region, the structure formation proceeds in two stages. The first one is characterized by damping of all but stable Fourier-components of density distribution and the second, by stabilization of the amplitude of the distribution. Characteristic times of these processes are estimated. The applied approach allows a comparatively simple definition of lamellar, hexagonal and body-centered cubic structures near Ts. Equilibrium structures at T ~ Ts are described as well.

Theory of microphase separation in block copolymer solutions

Microphase separation in semidilute solutions of block copolymers in nonselective good solvent is analyzed. The swelling due to the presence of solvent is taken into account. The solvent effects on the thermodynamics in the disordered region and in the ordered region in the weak segregation limit are discussed. It is found that the concentration of block copolymer at the microphase separation transition scales as N−0.62 in good solvents and as N−1/2 in Φ solvents. In the asymptotic semidilute regime in good solvents, the coexistence curve at the microphase separation transition is negligible, and the transition scales as (χ′AB φ1.61N)t ∼constant, where the constant is determined by the relative composition of the blocks in the block copolymer and χAB is proportional to the usual Flory interaction parameter.

Fluctuation effects in the theory of microphase separation in block copolymers

The effect of composition fluctuations on the microphase separation transition in diblock copolymers is investigated. Such fluctuation corrections, which were neglected in the mean field treatment of Leibler, are found to be significant for the molecular weights usually encountered. The analysis is facilitated by reducing the block copolymer Hamiltonian to a form previously studied by Brazovskii. Our principal results are the following: (i) A symmetric diblock copolymer is predicted to undergo a first order phase transition at a larger value of χN than the second order transition found by Leibler. The Flory interaction parameter is denoted χ and N is the number of statistical segments per chain. The location of the transition is predicted to be at (χN)t=10.495+41.022 N−1/3, where the peak in the scattering function attains its maximum value of 0.12328 N1/3. (ii) We find windows in composition, with finite width, through which it is possible to pass from the disordered phase to each of the ordered microphases (lamellar, hexagonal, and body-centered-cubic) by changing temperature. (iii) All the results of Leibler are recovered in the limit of block copolymers with infinite molecular weight.

Theory of Microphase Separation in Block Copolymers

Theory of Microphase Separation in Block Copolymers