Starting from the microscopic description of random block copolymer melts sug-gested by Fredrickson et al. [17], we have derived an effective free energy for this system in two different formulations, averaged over the quenched randomness by means of the replica method: First, in terms of component densities, classifying all chains with respect to their overall composition, disregarding their sequence and their conformation and thus yielding a coarse-grained description. Second, in terms of the local density of monomers and the charge density, i.e. the local difference in the concentrations ofA and B monomers.
For incompressible melts, we have shown the multi-component free energy to be equivalent to the free energy of ref. [15] usually obtained with a Flory-Huggins approach, and we have recovered the results of Nesarikar et al.[16] for the coexis-tence of multiple homogeneous phases. The phase equilibria have been computed using the algorithm proposed in [15]. Our derivation establishes the link to the alternative formulation in terms of the particle and charge density obtained from the same model. Like in [17], this non-coarse-grained formulation was the basis for a Landau expansion approach capable of describing both fundamental mech-anisms of phase separation in a random copolymer system: the separation into macroscopic phases and the formation of a microscopically structured phase. Al-though focusing on the incompressible case, our derivation initially retains effects of compressibility. For the incompressible case we obtained a free energy similar, but not identical, to that of ref. [17]. Substantial differences arise from certain approximations made in [17]: aiming at the case of a high number Q of blocks per chain and a large block size M, Fredrickson et al. neglected terms in the free energy appearing subdominant in 1/Q. The potential gain in the free energy asso-ciated with microphase separation is, however, of the order 1/Q, as well, hence the neglect of these terms is unjustified. With an analytical approximation based on the full free energy expansion, we found that microphases set in with finite wave-length, in contrast to the prediction in [17]. The analytic approximation has been complemented with data obtained by numerical minimisation of the free energy expansion. For very small “blocks” of just a single monomer,i.e.M = 1, the theory predicts the phase behaviour to be highly sensitive to the sequence correlation: the melt either remains homogenous or microphase separates on the smallest possible length-scale, which is bounded from below by the diameter of a monomer.
In the limit of infinite block and chain lengths, we recover the results of Fredrick-son et al. [17], apart from the microphase wavelength at the transition. In the case of symmetric chains and uncorrelated sequences, the spinodal temperature is proportional to the block length, and the temperature range of two-phase co-existence is of the order 1/Q. The microphases set in with large domain spacing and become narrower on going deeper into the microphase separated regime. In polyurethanes with suppressed crystallisation, Ryan et al. [31] measured the do-main spacing as a function of temperature for two different chain lengths by means of small-angle X-ray scattering. Due to a considerable polydispersity of block and
chain lengths, the microphase transition was stretched over a finite temperature in-terval. Ryan et al.observed that the domain spacing, after an increase during the transition attributed to chain stretching, decreases with decreasing temperature in agreement with the theory. Moreover, the transition temperature was found to increase with the molecular weight. However, the prepolymer blocks used in ref. [31] reveal a considerable amount of polydispersity, and the two components have molecular weights differing by nearly a factor of 20; thus it is difficult to compare the results directly to the ideal case of the theory discussed here.
Coarse-graining the theory in terms of monomer and charge densities, which allows for using the exact free energy instead of an expansion, we have shown the equivalence to the multi-component theory of ref. [16] for multiphase equilibria.
The multi-component theory is correct to all orders in the local composition, hence leads to the proper mean-field description of the coexistence of two (or more) phases. By comparison we found, in accordance with ref. [16], that the Landau theory based on the fourth-order expansion accurately describes the two-phase coexistence to leading order for the symmetric case, yet fails for asymmetric melts, where the transition is of first order. In the latter case, the range of validity of the Landau expansion is exceeded because of the finite jump of the order parameter.
Monte-Carlo simulations by Houdayer and M¨uller suggest the possibility of the coexistence of homogeneous and microstructured phases, dependent on the frac-tionation of the chains according to their sequence: homopolymeric chains pre-fer macrophases (rich in A or B), copolymeric chains prefer the microstructured phase [21]. A Landau theory of the three-phase state, however without explicit con-sideration of fractionation, was given by Subbotin and Semenov [22]. To study said three-phase coexistence, especially the effects of fractionation, we have developed and employed a caricature of the three-phase state, in which the homopolymeric chains were allowed to “choose” between micro- and macrophases. Because of the rapid decrease in the statistical weight of the strictly homopolymeric chains with the number Qof blocks per chain, this simplified picture is limited to small values of Q. We have shown that fractionation is sufficient to enable microphase separa-tion, since the homopolymeric chains hampering microphase separation are sorted out into the macrophases, thus increasing the affinity to microdomain formation in the remainder. In particular, we have shown that this mechanism is an alternative to the theory of Fredrickson et al. [17], in which a subtle balance of the spatial dependence of the quadratic and fourth-order terms in the Landau expansion is required for microphase separation.