Thermodynamics of high polymer solutions. VI. The compatibility of copolymers

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Abstract
It is evident that a copolymeric mixture, if sufficiently heterogeneous in chemical composition and molecular weight, may separate into two or more distinct phases. The conditions for thermodynamic stability of a single‐phase system have been investigated using the modified Flory‐Huggins equations for heterogeneous polymer mixtures. The equation for a “spinodal hyper‐surface” serves as a sufficient, but not always necessary, condition for phase separation. With certain simplifying assumptions, this reduces to the requirement that two or more phases must exist if: where M̄w is the weight average molecular weight, ρ is the density of the high polymer, xk is the mole fraction (or volume fraction) of chains with the same chain‐average chemical composition, represented by the “solubility parameter” δk, and δ is the average δ for the whole mixture. For a typical case (M̄w = 100,000, ρ = 1, T = 300°K), the standard deviation in δ cannot exceed 0.055. Since for a typical copolymer the possible range of δ (from 100% of one monomer to 100% of the other) may be 0.5–1.0 δ‐units, we conclude that most copolymers must be reasonably homogeneous in chemical composition; the over‐all composition of one chain cannot differ greatly from that of another in the same phase.