Glassy states and microphase separation in cross-linked homopolymer blends

– The physical properties of blends of distinct homopolymers, cross-linked beyond the gelation point, are addressed via a Landau approach involving a pair of coupled order-parameter fields: one describing vulcanisation, the other describing local phase separation. Thermal concentration fluctuations, present at the time of cross-linking, are frozen in by cross-linking, and the structure of the resulting glassy fluctuations is analysed at the Gaussian level in various regimes, determined by the relative values of certain physical length scales. The enhancement, due to gelation, of the stability of the blend with respect to demixing is also analysed. Beyond the corresponding stability limit, gelation prevents complete demixing, replacing it by microphase separation, which occurs up to a length scale set by the rigidity of the network, as a simple variational scheme reveals. Introduction. – Molten blends of distinct homopolymers have a strong tendency to phase separate, compared with unpolymerised mixtures, due to the fact that their entropy of mixing is reduced by a factor of the degree of polymerisation. Random permanent cross-linking hampers the demixing process, and completely inhibits macroscopic phase separation, provided enough cross-links are introduced to cause the blend to undergo a transition to a gel. de Gennes [1] was the first to point out that the region of compatibility (i.e. the region in which the melt remains mixed) is substantially increased by cross-linking, and that instead of macroscopic phase separation, the gel undergoes microphase separation (MPS) with a domain size comparable to the typical mesh size of the random macromolecular network. de Gennes’ predictions were subsequently confirmed in scattering experiments [2]. However, a discrepancy remained, concerning the scattering intensity at small wave numbers. Whereas de Gennes predicted a vanishing intensity at zero wave number, via an analogy with the polarisation of a dielectric medium, the experiments showed a non-vanishing intensity. Several modifications of the de Gennes theory have been proposed. Benmouna et al. [3] suggest that the discrepancy arises through the neglect of concentration fluctuations, which are present during the cross-linking process and are “frozen in” by the cross-linking process. These authors introduce a screening length, self-consistently, by assuming that the scattering intensity at zero wave number is not changed by cross-linking. Read et al. [4] instead suggest a microscopic model for the gel, in which each chain is anchored at its ends to fixed random points in space. By using reasonable assumptions for the distribution of the quenched end-to-end vectors, they c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2004-10522-9 844 EUROPHYSICS LETTERS arrive at an intensity similar to that of ref. [3], and also investigate effects of applied strain. Simulations by Lay et al. [5] reveal rather large cooperative rearrangements of the network upon MPS, allowing for domain sizes larger than the radius of gyration of the network strands. In this letter we focus on “charge” fluctuations in gels that have been prepared from blends in a homogeneously mixed state, characterised by a sufficiently small Flory incompatibility parameter. By “charge” we mean the difference between the local densities of the two species of homopolymer. We concentrate on two themes: – To what extent do the charge fluctuations present during cross-linking become frozen-in in the gel phase? This is particularly interesting if the cross-linking is performed close to phase separation, so that fluctuations are present up to a very large length scale, which can be either smaller or larger than the localisation length scale characterising the gel. – How do frozen-in charge fluctuations affect the scattering intensity and MPS? To what extent does gelation enhance the stability of the mixed state, and to what extent can one characterise the MPS state that emerges at large enough incompatibility parameter? Our starting point is a Landau expansion for the free energy in terms of two order parameter fields: the local charge fluctuations associated with phase separation, and the local static density fluctuations associated with the gelation transition. This free energy can be derived from a microscopic model [6], extending previous work on the gelation transition [7] to include the incompatibility of homopolymer blends. In this letter we do not dwell upon the derivation of the Landau free energy, but rather work out its consequences. As we shall see, the phase diagram is controlled by three parameters: the cross-link density control parameter μ, and the incompatibilities at cross-linking χp and at measurement χm (p stands for preparation, and m for measurement). The Landau free energy allows us to compute the frozen-in charge fluctuations without any ad hoc assumptions. We show that there are competing length scales: in a strong gel, the fluctuations are frozen-in almost completely, and hence preserve the native length scale at cross-linking; in a weak gel, by contrast, the charge fluctuations are only partially frozen in, limited by the length scale characterising localisation in the gel. We calculate the modification of the range of compatibility due to cross-linking, in terms of the gel order parameter, and discuss the scattering intensity in the gel, including the region in which MPS is approached. With the aim of developing a rough picture of the MPS state itself, we also analyse a lamellarstate Ansatz for the MPS state beyond, but close to, the transition. Model. – We consider a blend of two incompatible homopolymer species, “A” and “B”. The two species are modelled identically as Gaussian chains of length L, and the blend is taken to contain N/2 chains of each type. The mutual repulsion is modelled by the interaction