Length Distributions and the Alignment Transition of Polymers Formed by Linear Reversible Polymerization

The equilibrium distributions of polymer lengths and orientations are calculated for reversible linear polymerization, assuming that the free energy of monomer addition is independent of polymer size. The anisotropy of the equilibrium state at high concentrations has two aspects. (1) In the absence of anisotropy in the interactions between molecules, the system is isotropic with respect to monomer orientation, but the average polymer length is greater along the axis of alignment than orthogonal to it. (2) In the presence of anisotropic interactions between molecules, the orientation of monomers is also anisotropic. The phase diagram differs somewhat from that calculated earlier for an approximation of a reversibly polymerizing system in which all the polymers were constrained to be of identical, albeit reversibly changeable, length. In particular, noninteracting polymers do not form as dense an anisotropic phase as was predicted earlier. However, the temperature dependence of the phase behavior of the present freely reversible system and that of the former constrained reversible system are qualitatively similar to one another and qualitatively distinct from that of irreversibly polymerized systems. In particular, there exists for the reversibly polymerized systems, in contrast to the irreversibly polymerized systems, a temperature below which, for positive enthalpies of polymerization, or above which, for negative enthalpies of polymerization, no phase transition occurs. The maximum concentration of the isotropic phase is found to depend primarily on the free energy of polymerization, to be relatively insensitive to the nature of interparticle interactions, and to have a temperature dependence which is qualitatively similar to that observed for sickle cell hemoglobin. The distribution of polymer lengths in the isotropic phase is significantly different from that observed for sickle cell hemoglobin, presumably due to the relative instability of subnuclear hemoglobin aggregates which is not incorporated into the present model. The minimum concentration of the anisotropic phase is found to depend strongly on both the free energy of polymerization and the interparticle interactions and, in general, has a complicated temperature dependence.