PHYSICAL REVIEW E 85, 051402 (2012) Pressure and density scaling for colloid-polymer systems in the protein limit Nathan A. Mahynski, Thomas Laﬁtte, and Athanassios Z. Panagiotopoulos * Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08540, USA (Received 22 February 2012; published 11 May 2012) Grand canonical Monte Carlo and histogram reweighting techniques are used to study the ﬂuid-phase behavior of an athermal system of colloids and nonadsorbing polymers on a ﬁne lattice in the “protein limit,” where polymer dimensions exceed those of the colloids. The main parameters are the chains’ radius of gyration, R g , the diameter of the colloids, σ c , and the monomer diameter, σ s . The phase behavior is controlled by the macroscopic size ratio, q r = 2R g /σ c , and the microscopic size ratio, d = σ s /σ c . The latter ratio is found to play a signiﬁcant role in determining the critical monomer concentration for q r  4 and the critical colloid density for all chain lengths. However, the critical (osmotic) pressure is independent of the microscopic size ratio at all macroscopic size ratios studied. Quantitative agreement is observed between our simulation results and experimental data. We scale our results based on the polymer correlation length, which has previously been suggested to universally collapse these binodals [Bolhuis et al., Phys. Rev. Lett. 90, 068304 (2003); Fleer and Tuinier, Phys. Rev. E 76, 041802 (2007)]. While the density binodals exhibit universal characteristics along the low-colloid-density branch, such features are not present in the corresponding high-density phase. However, pressure binodals do collapse nicely under such a scaling, even far from the critical point, which allows us to produce a binodal curve whose shape is independent of either size ratio. DOI: 10.1103/PhysRevE.85.051402 PACS number(s): 82.70.−y, 64.75.Gh, 64.70.pv, 64.70.km I. INTRODUCTION The addition of nonadsorbing polymers to a suspension of colloids has long been known to give rise to an effect known as depletion. This phenomenon, ﬁrst described theoretically by Asakura and Oosawa [1,2], arises as nearby colloids mutually exclude polymers from within their overlapping depletion radii, creating an osmotic pressure imbalance between the locally polymer depleted region and the bulk. This creates an effective attraction between the suspended colloids which can induce phase separation into colloidal liquid-like and vapor- like phases. This is a highly advantageous system to study not only for its applications in industrial [3] and biological [4] settings, but also because it represents a controllable model for ﬂuids. While molecular ﬂuids have intermolecular forces ﬁxed by their structural and electronic properties, colloid-polymer systems allow the experimentalist to tune these interactions for a wide array of technological applications [5]. As originally surmised by Asakura and Oosawa, these effects also play an important role in biochemical systems, though their signiﬁcance is only now beginning to be fully ap- preciated [6]. While a single depletion interaction between two colloids may only be of the order of one k B T , many such inter- actions may work in concert, with a total energy comparable to that of a hydrogen bond in a protein. For instance, it has been demonstrated that depletion forces play a role in actin dimer- ization and bundling, and the addition of noninteracting crowd- ing agents, such as polysaccharides, has been shown to increase the refolding rates of proteins, such as hen lysozyme, two to ﬁve times times over its uncrowded state [7]. In fact, the com- mon practice of adding large macromolecules to assist in the crystallization of smaller proteins has given the “protein limit” its moniker [8]. Due to the ubiquity of depletion, the need to understand the underlying physics becomes readily apparent. * azp@princeton.edu Typically, colloid-polymer systems are characterized by the ratio q r = 2R g /σ c , where R g is the polymer’s radius of gyra- tion, generally estimated in its pure or reservoir state, and σ c is the colloid diameter. In the “colloid limit” (q r < 1), polymer dimensions are smaller than those of the colloid and their internal degrees of freedom are often negligible. As a result, coarse-graining the polymers has proven to be a successful technique in this limit. In the less explored “protein limit” (q r > 1), where polymer dimensions exceed those of the col- loid, neglecting these degrees of freedom is less appropriate. Investigations in this limit can be rather tedious as monomer interactions, solvent quality, and many-body effects become increasingly important [9–14]. Nonetheless, this limit is at least as important as the colloid limit for many applications. Early scaling analysis of the protein limit led many to believe widespread miscibility would be observed [15,16], until it was later shown that such mixtures do, in fact, phase separate [17,18]. In the protein limit, phase separation is observed at much higher polymer concentrations than in the colloid limit. Due to increased polymer-polymer interactions at these concentrations, the characteristic interaction length scale (depletion radius) is reduced from the polymer’s radius of gyration to its correlation length, ξ [15]. As a result, the phase behavior shifts from being governed by the chain length (in the colloid limit) to the polymer volume fraction (in the protein limit). With sufﬁciently short-ranged attractions or high enough packing densities, ﬂuid-ﬂuid coexistence becomes metastable with respect to a ﬂuid-solid transition [12,19–21]. Estimates of this limit in terms of q r range from roughly 0.15 for purely pairwise interactions [19] to approximately 0.32 for perturbation [20] and free volume theories [21] and up to 0.45 for lattice simulations [22]. In the protein limit q r is in sufﬁcient excess of these estimates, so we restrict our considerations in this report to ﬂuid-ﬂuid phase behavior only. Many theories have been proposed to describe the phase behavior of colloid-polymer systems, including 051402-1 1539-3755/2012/85(5)/051402(9) ©2012 American Physical Society