Scaling of the surface tension of phase-separated polymer solutions

Abstract

The tension of the interface between the equilibrium phases of a phase-separated polymer solution is obtained in the simplest mean-field approximation from the functional equation for the composition profile of the interface. For temperaturesT near the critical solution temperatureT _c, i.e., for Flory parameterχ nearχ _c, and for high degrees of polymerizationN, the profile and tension scale withχ=N ^1/2(χ−χ _c), just as do the compositions of the coexisting phases in mean-field approximation. The surface tensionσ in the asymptotic limitN→∞,χ→χ _c at fixedx, is found to be given bya ²σ/kT _c ∼(2c'/c)^1/2 N ^-5/4∑(x), wherea is the lattice spacing of an underlying lattice (or, roughly, the length of a monomer),c′ andc are the vertical and total coordination numbers of the lattice, andΣ(x) is a scaling function, known for allx, with the asymptotic behavior$\Sigma (x) \sim 4\sqrt 2 x^{3/2}$ asx→0 and$\Sigma (x) \sim (6\sqrt {2/5} ) x^{3/2}$ asx→∞. The latter implies thatσ becomes independent ofN asN→∞ at fixedT nearT _c; the former implies thatσ becomes proportional toN ^−1/2(1−T/T _c)^3/2 asT→T _c at fixedN≫1, as found previously.