Lagrangian tricritical theory of polymer chain solutions near the θ-point

P. G. de Gennes has shown that the θ-point of very long polymer chains is a tricritical point. We consider here a Lagrangian model of continuous chains, of Brownian area S, with local two-body and three-body interactions, with respective coefficients go and w. At the theta-point, go (< 0) and w (> 0) compensate each other. The physical cut-off s0 is a minimal Brownian area between two interaction points along a chain. We use the identity of the grand canonical ensemble of chains and of the Landau-Ginzburg-Wilson ϕ field theory, with n = 0 components, and interactions g0(ϕ 2)2 and w(ϕ2)3. For d = 3, the tricritical logarithmic divergences are studied with the renormalization group equation as a function of the physical cut-off s0. This allows a complete calculation of the tricritical logarithmic laws for polymer solutions, including all the prefactors depending on w. We calculate the square radius R2 of a single chain, and its specific heat at θ : Cv = w/20(2 π)3 S/ s0 (11/60(2 π)2 w ln S/s 0)3/11, where the exponent 3/11 agrees with that of de Gennes. For chains in dilute or semi-dilute solutions at concentration C, we calculate the universal tricritical laws for the mean square radius R2 and the osmotic pressure Π. The extension of the tricritical θ domain is studied in the plane { C, g = (T — θ)/θ }. The coexistence curve of infinite chains, which forms the lower border of the tricritical domain, is found to follow the universal equation $$ where C 2 s0