Associative and Segregative Liquid-Liquid Phase Separation in Macromolecular Solutions

TL;DR

This work contrasts associative (ALLPS) and segregative (SLLPS) liquid–liquid phase separation in symmetric binary polymer–solvent mixtures using Flory–Huggins mean-field theory, validated by self-consistent field (SCF) lattice computations. It derives analytic critical-point and binodal expressions, clarifying that ALLPS depends on the solvent quality via $\Delta\chi$ and yields solvent-independent CPs, while SLLPS is solvent-sensitive with CP at $\chi_{12}^{CP,\mathrm{segr}} = 2/N$. The study reveals universal interfacial scaling: $\gamma \sim \left(1-\frac{\chi_{12}^{CP}}{\chi_{12}}\right)^{3/2}(1-\phi_3)$ and $w_0 \sim \sqrt{\frac{\chi_{12}}{\chi_{12}-\chi_{12}^{CP}}}$, with ALLPS producing higher interfacial tension and thinner interfaces. These results distinguish the two LLPS modes and offer design principles for materials and insights into intracellular phase behavior.

Abstract

We investigate liquid-liquid phase separation (LLPS) and interfacial properties of two LLPS modes: associative (ALLPS) and segregative (SLLPS). Analytical expressions for the critical point (CP) and binodal boundaries are derived and show excellent agreement with self-consistent field (SCF) lattice computations. Distinct thermodynamic features differentiate ALLPS from SLLPS: (1) in ALLPS, polymers co-concentrate within a single dense phase coexisting with a solvent-rich phase, whereas in SLLPS each polymer forms a separate phase; (2) the attractive interaction per monomer in ALLPS is strongly dependent on solvent quality, but solvent-independent in SLLPS; and (3) ALLPS binodals exhibit near-universal behavior, largely independent of solvent content. SCF results further show that interfacial tension increases and interfacial width decreases with distance from the CP. We provide scaling relations for both quantities are provided. Compared with SLLPS, ALLPS displays higher interfacial tension and a thinner interface, reflecting distinct molecular organization at the liquid-liquid boundary.

Figures (9)

• Figure 1: From associative (left) to homogeneous (middle) to segregative (right) Liquid--Liquid Phase Separation (LLPS) state diagram of two symmetric polymers $1$ (blue) and $2$ (red) with chain lengths $N$ in a common solvent (gray). In the segregative case ($\chi_{12} N (1-\phi_3) \geq 2$), one phase (say, $\alpha$) is polymer $1$-rich, whereas the other phase ($\beta$) is polymer $2$-rich. In the associative scenario ($\chi_{12} \leq 4 \chi_{13}-2(1+1/\sqrt{N})^2$), one phase is solvent-rich ($\alpha$) and polymer-lean and both associated polymers concentrate in the other phase ($\beta$). The phase boundaries are indicated in terms of the (scaled) critical polymer--polymer interaction $\chi_{12}^\text{CP}$.
• Figure 2: LLPS binodals in the $\chi_{12}$-$\phi_i$ phase space for $N_1=N_2=N=200$ and $N_3=1$ in a $\theta-$solvent ($\chi_{13}=\chi_{23}=0.5$): (a) polymer 1 coexistence volume fractions, $\phi_1$ (from which $\phi_2$ directly follows), (b) solvent coexistence volume fractions, $\phi_3$. Solid curves are calculated FH-binodals, and symbols correspond to SCF-computed binodals. Solvent volume fractions ($\phi_3$) are indicated in the (common) legend. Gray dashed curve correspond to the polymer blend case ($\phi_3 \rightarrow 0$). Black dashed curves correspond to the $\phi_3$-dependent critical points.
• Figure 3: ALLPS phase coexistence curves in terms of the effective interaction $\Delta \chi= \chi_{12}-4\chi_{13}$ versus $\phi_i$ for various polymer chain lengths $N$ and solvencies. Solid curves are the theoretical FH binodals. Solid gray dashed curves are the $N$-dependent critical points. SCF results are indicated with colored symbols, with a set of solvency conditions $\chi_{13} = \{0, 0.1, 0.2, 0.3, 0.4, 0.5\}$. The three-pointed gray star symbol represents the critical endpoint in the long-chain limit of the association binodal critical points in common solvent conditions. Different symbols are used for different solvency conditions: $\chi_{13} = 0.5$ (up triangles), $\chi_{13} = 0.4$ (down triangles), $\chi_{13} = 0.3$ (diamonds), $\chi_{13} = 0.2$ (four-pointed stars), $\chi_{13} = 0.1$ (up triangle truncated), $\chi_{13}=0$ (disks).
• Figure 4: Polymer segment (a-c) and solvent (d-f) density profiles across the interface of demixed polymer 1 - polymer 2 - solvent mixtures for SLLPS (a,b,d,e) and ALLPS (c,f). Curves are fits to SCF computations using Eqs. (\ref{['tanh']}) and (\ref{['solventads']}). Examples correspond to selected phase diagrams in Fig. \ref{['fig:SCFTheoComparison_N200']} ($N=200$), so for the $\theta$-solvent condition $\chi_{13}=\chi_{23}=0.5$. In panels (d-f) the solvent concentration is plotted for SLLPS. In the segregative scenario (a,b,d,e), the polymer--polymer interaction parameter $\chi_{12}$ decreases from $0.3$ to the corresponding critical value: for $\phi_3 = 0.5$ (segr.), $\chi_{12} = \{0.3, 0.296,...,0.032,0.028\}$ for $\phi_3 = 0.95$ (segr.), $\chi_{12} = \{0.3, 0.297,...,0.204,0.201\}$ ; (black to light gray as approaching the CP). Blue dashed lines correspond to $1-\phi_3$. In the association case (c,f) $\phi_3 = 0.95$ (assoc.), $\chi_{12} = \{-3.0,-2.9,...,-0.3\}$ (approaching the CP from black to light gray). A systematic comparison of these fitted profiles with the SCF-generated profiles is provided in the SI.
• Figure 5: Interfacial tension $\gamma$ (a) and interfacial width $w_0$ (b) for polymer 1 - polymer 2 - solvent mixtures with $N_1=N_2=200$. Numerical SCF lattice computations (data) for both SLLPS (gray symbols) and ALLPS (colored symbols) follow the scaling laws of Eq. (\ref{['gammascaling']}) (a) and Eq. (\ref{['width']}) (b). Star symbols correspond to the ALLPS interfacial properties; other symbols correspond to SLLPS for various solvent concentrations: $\phi_3=0.95$ (circles), $\phi_3=0.9$ (up triangles), $\phi_3=0.8$ (down triangles), $\phi_3=0.7$ (left triangles) $\phi_3=0.6$ (diamonds), and $\phi_3=0.5$ (up truncated triangles). Scaling relations are indicated on the abscissa. The boundaries of scaling ranges for the surface tension (a) in terms of the system parameters are $\gamma = (0.2-5.5) \left(1-{\chi_{12}^\text{CP}}/{\chi_{12}}\right)^{3/2}(1-\phi_3)$ (association, increasing with decreasing $\chi_{13}$), and $\gamma = (0.05-0.1) \left(1-{\chi_{12}^\text{CP}}/{\chi_{12}}\right)^{3/2}(1-\phi_3)$ (segregation, practically independent of $\chi_{13}$). For the interfacial widths (b), the boundaries in terms of the scaling relation are $w_0 = (4.5-1.5)\sqrt{\chi_{12}/(\chi_{12}-\chi_{12}^\text{CP})}$ (association, increasing with increasing $\chi_{13}$) and $w_0 = (5-10)\sqrt{\chi_{12}/(\chi_{12}-\chi_{12}^\text{CP})}$ (segregation).