Analytical phase diagrams for colloids and non-adsorbing polymer

Abstract

We review the free-volume theory (FVT) of Lekkerkerker et al. [ Europhys. Lett. 20 (1992) 559] for the phase behavior of colloids in the presence of non-adsorbing polymer and we extend this theory in several aspects: (i) We take the solvent into account as a separate component and show that the natural thermodynamic parameter for the polymer properties is the insertion work Πv, where Π is the osmotic pressure of the (external) polymer solution and v the volume of a colloid particle. (ii) Curvature effects are included along the lines of Aarts et al. [ J. Phys.: Condens. Matt. 14 (2002) 7551] but we find accurate simple power laws which simplify the mathematical procedure considerably. (iii) We find analytical forms for the first, second, and third derivatives of the grand potential, needed for the calculation of the colloid chemical potential, the pressure, gas–liquid critical points and the critical endpoint (cep), where the (stable) critical line ends and then coincides with the triple point. This cep determines the boundary condition for a stable liquid. We first apply these modifications to the so-called colloid limit, where the size ratio q R = R/ a between the radius of gyration R of the polymer and the particle radius a is small. In this limit the binodal polymer concentrations are below overlap: the depletion thickness δ is nearly equal to R, and Π can be approximated by the ideal (van 't Hoff) law Π = Π 0 = φ/ N, where φ is the polymer volume fraction and N the number of segments per chain. The results are close to those of the original Lekkerkerker theory. However, our analysis enables very simple analytical expressions for the polymer and colloid concentrations in the critical and triple points and along the binodals as a function of q R . Also the position of the cep is found analytically. In order to make the model applicable to higher size ratio's q R (including the so-called protein limit where q R > 1) further extensions are needed. We introduce the size ratio q = δ/ a, where the depletion thickness δ is no longer of order R. In the protein limit the binodal concentrations are above overlap. In such semidilute solutions δ ≈ ξ, where the De Gennes blob size (correlation length) ξ scales as ξ ∼ φ − γ , with γ = 0.77 for good solvents and γ = 1 for a theta solvent. In this limit Π = Π sd ∼ φ 3γ. We now apply the following additional modifications: (iv) Π = Π 0 + Π sd, where Π sd = A φ 3γ; the prefactor A is known from renormalization group theory. This simple additivity describes the crossover for the osmotic pressure from the dilute limit to the semidilute limit excellently. (v) δ − 2 = δ 0 − 2 + ξ − 2 , where δ 0 ≈ R is the dilute limit for the depletion thickness δ. This equation describes the crossover in length scales from δ 0 (dilute) to ξ (semidilute). With these latter two modifications we obtain again a fully analytical model with simple equations for critical and triple points as a function of q R . In the protein limit the binodal polymer concentrations scale as q R 1/γ, and phase diagrams φ q R − 1/γ versus the colloid concentration η become universal (i.e., independent of the size ratio q R ). The predictions of this generalized free-volume theory (GFVT) are in excellent agreement with experiment and with computer simulations, not only for the colloid limit but also for the protein limit (and the crossover between these limits). The q R 1/γ scaling is accurately reproduced by both simulations and other theoretical models. The liquid window is the region between φ c (critical point) and φ t (triple point). In terms of the ratio φ t / φ c the liquid window extends from 1 in the cep (here φ t − φ c = 0) to 2.2 in the protein limit. Hence, the liquid window is narrow: it covers at most a factor 2.2 in (external) polymer concentration.