General approach for partitioning and phase separation in macromolecular coexisting phases

TL;DR

This paper develops a general classical density functional theory framework to describe phase coexistence and partitioning in multicomponent polymer–colloid mixtures, including two immiscible polymers and a dispersed colloid. The approach combines fundamental measure theory for hard-core effects, free-volume theory for depletion, and an enhanced mean-field term to capture polymer–polymer and colloid–polymer attractions, with a bulk free-energy functional F = F_id + F_exc that incorporates effective densities rho'_i = rho_i / alpha'_i. Coexistence is determined by equal chemical potentials and pressure, using a reduced-densities representation to ensure numerical stability, and the framework is extended to inhomogeneous systems via an AOV-FMT cavity model. Validation against Brownian dynamics shows the EMF formulation improves quantitative agreement and captures trends in partitioning as functions of polymer density, affinity, and size ratios, offering a predictive tool for ATPS, biomolecular condensates, and purification applications.

Abstract

Partitioning of (bio)materials in polymeric mixtures is a key phenomenon both in cellular environments, as well as in industrial applications. In cells, several macromolecules are suspended within different biomolecular phases. On the other hand, the coexistence of polymeric aqueous phases has been exploited for the extraction and purification of (bio)materials suspended in water. Despite its relevance, key physical and chemical properties controlling the phase behavior of these complex systems are still lacking. Here, we have developed a classical density functional theory approach for describing the phase coexistence and partitioning of an arbitrary number of polymers and suspended materials. As a case example, we focus on a binary mixture of phase separating polymers in which a third material is dispersed. We explore the effect of size ratios and affinities between the different materials and address their distribution and coexisting densities, and find optimal conditions for partitioning.

Figures (9)

• Figure 1: (a) Free energy density for a given pair of coexisting densities in the $\rho_{\rm a} \rho_{\rm b}$-plane and (b) in the $\rho_{\rm b} \rho_{\rm c}$-plane, where a and b are two immiscible polymer species and c a colloidal species. The upper and lower triangles correspond to phase-I and phase-II, respectively. The color bar indicates the value of $f$ at each point in the density space, where the two conjugate points represent the same system. All conjugate points represent the same canonical ensemble with $\bar{\rho}_{\rm a} = \bar{\rho}_{\rm b} = 0.18$ and $\bar{\rho}_{\rm c} = 0.09$.
• Figure 2: Density difference between the two phases of the colloidal species, $\Delta \rho_{\rm c} = \rho_{\rm c}^{\rm I} - \rho_{\rm c}^{\rm II}$ are shown using symbols for particle-based Brownian dynamics simulations (similar to those in SCACCHI20251135). The simulation results, for the parameters summarized in Table \ref{['tab:interaction']}, are shown for case-I, case-II and case-III, and are compared with our cDFT using EMF (solid lines), as well as SMF (dashed lines). In panel (a) we show the results while varying the total polymer concentration ($\bar{\rho}_{\rm p} = \bar{\rho}_{\rm a} +\bar{\rho}_{b\rm }$) at fixed colloid concentration $\bar{\rho}_{\rm c} = 0.09$. In panel (b), the results are shown for varying colloid density at fixed polymer concentration $\bar{\rho}_{\rm a} = \bar{\rho}_{\rm b} = 0.27$.
• Figure 3: In (a), binodal curve for the ternary colloid-polymer mixture described by the following parameters: $R_{\rm aa} = R_{\rm bb} = R_{\rm ab} = \sqrt{2} \sigma$, $\epsilon_{\rm aa} = \epsilon_{\rm bb} = 2.0, \epsilon_{\rm ab} = 2.5$, $\sigma_{\rm aa} = \sigma_{\rm bb} = \sigma_{\rm cc}=1$ and $\hat{V}_{\rm ac}=-5.6$. $x$ is the fraction of polymer b with reference to the total polymer densities $\bar{\rho}_{\rm p}$ and $y$ is the fraction of colloids with reference to the total density $\bar{\rho}_{\rm t}$. Here $\bar{\rho}_{\rm c} = 0.09$ is kept fixed. The upper triangles indicate phase-I (polymer a-rich phase), while the lower triangles indicate phase-II (polymer b-rich phase). Coexisting points are connected by isobars, where their color corresponds to the value of $\bar{\rho}_{\rm p}$. The green squares represent the value of $V_{\rm f}$, i.e. the volume ratio of phase-I, while the red squares represent $V_{\rm f}=0.5$. The full circle represents the critical point. In (b), effect of polymer concentration on the coexistence densities of colloids.
• Figure 4: Partitioning of colloids driven for the same system as in Fig. \ref{['fig.phase_diagram']} yet for (a) $\hat{V}_{\rm ac} = -2.8$, (b) $\hat{V}_{\rm ac} = -5.6$ and (c) $\hat{V}_{\rm ac}=-8.4$. The upper triangles indicate phase-I (polymer a-rich phase), while the lower triangles indicate phase-II (polymer b-rich phase). Coexisting points are connected by isobars, where their color corresponds to the value of $\bar{\rho}_{\rm p}$.
• Figure 5: System interacting as in Fig. \ref{['fig.phase_diagram']}. Here $\rho_{\rm r} = 0.27$ and $\bar{\rho}_{\rm c} = 0.09$ are fixed. In (a), phase response, and in (b) partitioning of the system components for varying $\bar{\rho}_{\rm a}$. In (c), phase response, and in (d) partitioning of the system components for varying $\bar{\rho}_{\rm b}$. The upper triangles indicate phase-I (polymer a-rich phase), while the lower triangles indicate phase-II (polymer b-rich phase). Coexisting points are connected by isobars, where their color corresponds to the value of $\bar{\rho}_{\rm a}$ (panel (a)) and $\bar{\rho}_{\rm b}$ (panel (c)). The green squares represent the value of $V_{\rm f}$, i.e. the volume ratio of phase-I, while the red squares represent $V_{\rm f}=0.5$. The differences in the coexisting densities of the colloids, $\Delta \rho_{\rm c}$, are shown in the insets of panel (b) and (d). Dashed horizontal lines shown in panel (b) and (d) represent $N/N_{\rm 0} = 0.5$.