"Zero" and "Off-Zero" Critical Concentrations in Solutions of Polydisperse Polymers with Very High Molar Masses

Abstract

Polymer solutions with a concentration-dependent interaction parameter g(ϕ) are known to have sometimes critical polymer concentrations ϕc converging to a non-zero value (a so-called "off-zero" limiting critical point (CP)), as the chain length, m, grows to infinity, rather than to zero as usual (a "zero" limiting CP). In this report the criteria for the existence of both types, known for binary solutions with a linear g = g0 + g1ϕ, are extended to cover polydisperse polymers with a quadratic interaction function g(ϕ). Its coefficients g2 and ∆g2 = g2 - g1 determine the number and type of limiting CPs. Accordingly, the plane g2, ∆g2 is divided into the regions I (a zero CP), II (an off-zero CP), and III (a zero + two off-zero CPs). The region II is restricted to the half-plane with ∆g2 < -1/6, whereas the other half-plane with ∆g2 > -1/6 is shared by I and III. By varying the interactions, two limiting CPs may be brought together and merged in a heterogeneous double limiting CP. Such instances define the boundaries between the regions: at the I/III line, two off-zero CPs merge, whereas at the II/III line an off-zero CP coincides with a zero CP. A first-order perturbation theory of the latter double CPs, and a second-order perturbation theory of single "zero" CPs are developed, enabling meaningful extrapolations of data on polymers with high but finite molar masses. The latter theory yields extrapolation formulas for determination of Η-temperature, taking into account the polymer polydispersity and the concentration dependence of g. Solutions of polyisobutene in diphenyl ether and, possibly, in benzene appear to present experimental examples of off-zero limiting critical concentrations.