Lattice model of living polymerization. III. Evidence for particle clustering from phase separation properties and “rounding” of the dynamical clustering transition

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Abstract
Equilibrium polymerization is studied here as a prototype for clustering transitions that commonly occur in systems of interacting particles at equilibrium. These transitions are often difficult to locate because of transition “rounding” associated with a limited extent of cluster growth, competing association or dissociation processes that initiate or inhibit clustering, and other constraints on the particle clustering dynamics. Instead of singularities in thermodynamic and transport properties, more subtle property changes signal the onset of particle clustering, explaining why clustering transitions are often overlooked or misinterpreted. We utilize a Flory–Huggins model for the equilibrium (“living”) polymerization of linear polymer chains to identify experimental signatures (features in the osmotic pressure, osmotic compressibility, and specific heat) that can be used to locate and quantify the transition “rounding” in general clustering transitions. The computation of a flattening in the concentration dependence of the osmotic pressure in the one-phase region motivates our consideration of the temperature dependence of the second virial coefficient and the variation of the theta temperature Tθ with “sticking energy” Δh as possible important indicators of particle clustering. The ratio of the critical temperature Tc for phase separation to Tθ, along with other “critical constant” ratios, such as the osmotic compressibility factor Zc, are also calculated and discussed in connection with establishing criteria for identifying particle clustering transitions and for quantifying the relative “strength” of these transitions.