Thermodynamics of Crystallization in High Polymers. IV. A Theory of Crystalline States and Fusion in Polymers, Copolymers, and Their Mixtures with Diluents

Abstract

The configurational entropy of a mixture of solvent and polymer molecules composed of segments connected by flexible bonds has been derived for the case in which the lattice they occupy contains limited regions of perfect order (i.e., the crystallites), which are reserved for occupancy by polymer segments exclusively. Although the derivation is carried out primarily for polymer chains of uniform length, modifications required for the treatment of heterogeneous polymers are included. An expression for the free energy of fusion ΔF_f of the semicrystalline mixture of polymer and solvent is derived assuming a vanLaar heat of mixing term. The equilibrium (average) crystallite length ζ_e, derived by differentiation of ΔF_f, depends principally on the ``nucleation parameter'' D which, though it evades explicit evaluation from theory, represents the relative ease with which the first layer of a crystallite tends to form; ζ<sub>e</sub> may depend to a lesser extent on the degree of crystallinity. The equilibrium degree of crystallinity, similarly deduced, depends primarily on the temperature. Fusion is predicted to occur over a range of temperatures, but the temperature T<sub>m</sub> (the ``melting point'') at which crystallinity totally disappears is sharply defined. The melting range is predicted to be narrower the greater the chain length. If the degree of polymerization is greater than 100, the range (depending on the heat of fusion) for equilibrium melting should be quite narrow, as much as 75 percent of the fusion occurring within a few degrees of T_m. The dependence of the melting point on chain length is expressed approximately by a proportionality between the temperature depression given by ΔT_m=T_m⁰—T_m, where T_m⁰ is the melting point for infinite chain length and the reciprocal of the degree of polymerization. Incorporation of a diluent should markedly broaden the melting range. If the degree of polymerization is very great, ΔT_m should be proportional to the number of moles of diluent per unit volume. Extension of the treatment to random copolymers yields $${\text{1/T}}_m{\text{−1/T}}_m^0{\mathrm{=(R/h}}_u\mathrm{)[-}\ln X_A{\mathrm{-\mu }}^{\prime }{\text{(1−v}}_A{)}^2\mathrm{],}$$ where h_u is the heat of fusion, μ′ is the heat of mixing parameter, and X_A and v_A are the ``mole'' fraction and volume fraction, respectively, of the crystallizing structural unit in the copolymer. This expression is the exact analog of the freezing point relationship for ``regular'' binary mixtures of monomeric substances. Crystallinity in an ordered copolymer (the partial analog of an ``irregular'' solution) in which the structural units of a given type are arranged in long sequences should disappear at a temperature only slightly below T<sub>m</sub> for the pure polymer. The phase rule cannot be applied to the crystal‐amorphous transformation in polymers since the free energy per unit amount of the amorphous ``phase'' is not a unique function of the composition. Copolymers should melt over a wider temperature range and their crystallinity‐temperature relationship is predicted to be sigmoid in character.