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 N 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 =Aφ 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 φq ...

Structure of phase-separated athermal colloid-polymer systems in the protein limit

“…In the "protein" limit, q r > 1, such approaches begin to fail as the overall free energy of the system becomes sensitive to details of the polymer. A previous study of the phase behavior of lattice colloids and linear chains demonstrated * azp@princeton.edu that proposed rescaling arguments involving the polymer correlation length, ξ, qualitatively failed to collapse the colloid-rich portion of the binodal in the protein limit [16]; this observation had not been previously captured in simulation or analytical models which suggested such a collapse should be possible [17][18][19], however it is consistent with recent experimental findings [20].…”

Coarse-graining polymer solutions: A critical appraisal of single- and multi-site models

“…In particular, we focus on colloidal dispersions comprising large particles, colloids, usually modeled as hard spheres, and polymers in an implicit solvent. These systems are particularly interesting as they show a complex phase diagram which depends crucially on the polymer-to-colloid size ratio: for small ratios, only fluid-solid coexistence is observed, while for larger values an additional fluid-fluid transition is present [79][80][81][82][83][84]. Even in the absence of an explicit solvent, the computation of the full phase diagram is quite difficult, especially if one is interested in polymers with a large degree of polymerization.…”

Fluid–fluid coexistence in an athermal colloid–polymer mixture: thermodynamic perturbation theory and continuum molecular-dynamics simulation

“…This value still exceeds the theoretical threshold below which the liquid phase is expected to be metastable, and corresponds to a depletion thickness to colloid radius ratio of q s ∼ 0.3-0.4. The phase diagram for a system with this value of q s may be expected to correspond roughly to that provided in figure 1.1 of reference [34], which features 'gas-liquid', 'gas-solid' and 'fluid-solid' phase boundaries. It is noticeable that the solid phase is characterised by η c 0.54, corresponding to the well-known coexistence packing fraction for pure HS, and significantly in excess of the total packing fraction (polymers + colloids) of the denser of the two phases we find in our simulations.…”

“…To date, the phase behaviour of colloid-polymer mixtures has been addressed in a number of theoretical studies, in addition to those already mentioned; see, for example, references [34,43,[47][48][49][50][51][52][53][54][55][56][57]. Among these, we highlight the studies [43,[53][54][55]58] based on Wertheim's first-order thermodynamic perturbation theory (TPT1) [59,60]; these deal with the polymer at the monomer segment level [61], in contrast to studies based on the AO model or the AO approximation.…”

“…Thorough explanations of the depletion interaction are provided in, for example, references [1,[31][32][33][34][35]; depletion forces have been the focus of a number of studies, including [36][37][38][39][40][41][42][43]. In particular, Binder et al [35] have provided a detailed appraisal of the Asakura-Oosawa (AO) model, and its scope.…”