Scaling of demixing curves and crossover from critical to tricritical behavior in polymer solutions

Abstract

In this paper we show that the virial expansion up to third order for the osmotic pressure of a dilute polymer solution, including first-order perturbative corrections to the virial coefficients, allows for a scaling description of phase-separation data for polymer solutions in reduced variables. This scaling description provides a method to estimate the Θ-temperature, where demixing occurs in the limit of vanishing polymer volume fraction φ and infinite chain-length N , without explicit assumptions concerning the chain-length dependence of the critical parameters Tc and φc. The scaling incorporates three limiting regimes: the Ising limit asymptotically close to the critical point of phase separation, the pure-solvent limit, and the tricritical limit for the polymer-rich phase asymptotically close to the theta point. We incorporate the effects of critical and tricritical fluctuations on the coexistence curve scaling by using renormalization-group methods. We present a detailed comparison with experimental and simulation data for coexistence-curves and compare our estimates for the Θ-temperatures of several systems with those obtained from different extrapolation schemes.

Vapor–liquid critical properties of multi-component fluid mixture

“…The isomorphism assumption claims that a universal property of a mixture near its critical point is isomorphic to that of a one-component system, provided that certain appropriately chosen thermodynamic variables are kept constant and the correct order parameter is known. Under this assumption, RG theory for a one-component system can be applied directly to mixtures [18,[32][33][34][35]. The results of some RG theories based on the isomorphism approximation are in good agreement with experimental data.…”

Thermodynamics of Fluid Polyamorphism

“…We emphasize that the meaning of the order parameter for the system in which two , the parameter that controls crossover between the singularity-free scenario (finite N and zero ) and polyamorphism with a lambda transition (infinite N and zero ), is [82,83]. If N is finite, the possibility of polyamorphism is always associated with phase separation and requires the existence of nonideality in the Gibbs energy of mixing (finite interaction parameter ).…”

“…avoiding walk singularities of the infinitely long ( N  ) polymer chain [81][82][83]. This makes the infinite-degree of polymerization, at least in the mean-field approximation, to be analogous to the two-state model of superfluidity.…”

“…interconvertible states are controlled by chemical-reaction equilibrium changes from a scalar for all finite N (in the purely symmetric case, 1 N  , this is a fraction of conversion, x) to a specific (zero-component vector) order parameter ψ , with with self-avoiding walk singularities of the infinitely long ( N  ) polymer chain[81][82][83]. This makes the infinite-degree of polymerization, at least in the mean-field approximation, to be analogous to the two-state model of superfluidity.…”

Tricriticality in the ternary system 3-methylpyridine/water/NaBr? The light-scattering intensity

“…In polymer solutions, the expected changes of the critical exponents b, g and n from the Ising values (0.324, 1.24, 0.63) towards the figures at a tricritical point (1.0, 1.0, 0.5) have indeed been observed, when the chain length of the polymer becomes large. [8][9][10][11] For the H 2 O/3MP/NaBr system, the hypothesis of crossover from an Ising to a tricritical point was developed in analogy to the critical properties of polymer solutions. [8][9][10][11] However, in the case of the H 2 O/3MP/NaBr solutions it is not obvious if a parameter is available, which determines the crossover in an analogous way as the chain length in the case of the polymer solutions.…”