Metastable phase separation and information retrieval in multicomponent mixtures

TL;DR

The paper develops a general thermodynamic formalism for metastable phase separation in multicomponent liquids and derives precise stationarity and stability conditions for both homogeneous and phase-separated states, showing that metastability of a phase-separated state requires the constituent phases to be individually metastable (except for a soft-mode exception). It then applies this framework to a simple binary model with high-order interactions and to Hopfield liquids, where phase separation can encode and retrieve multiple prescribed compositions via nucleation seeds, with continuum-space simulations confirming the analytical predictions. The results reveal that Hopfield liquids can store and retrieve information through metastable phase separation and that increasing the number of components enhances the retrieval capacity, offering a toy model for cytoplasmic organization and for designing synthetic multicomponent condensates with memory-like behavior. Overall, the work provides a rigorous link between phase behavior in complex mixtures and information processing capabilities, with potential implications for biology and materials design.

Abstract

Liquid mixtures can separate into phases with distinct composition. This phenomenon has recently come back to prominence due to its role in complex biological liquids, such as the cytoplasm, which contain thousands of components. For simple two-component mixtures phase-separated states are global free energy minima. However, local free energy minima, i.e. metastable states, are known to play a dominant role in complex systems with many components. For example, Hopfield neural networks can retrieve information from partial cues via relaxation to metastable states. Under what conditions can phase separated states be metastable, and what are the implications for information processing in multicomponent liquids? In this work we develop the general thermodynamic formalism of metastable phase separation. We then apply this formalism to an illustrative toy example inspired by recent experiments, binary mixtures with high-order interactions. Finally, as core application of the formalism, we study metastability in Hopfield liquids, a class of multicomponent mixtures capable of storing information on the composition of phases. We show that these phases can be retrieved from partial cues via metastable phase separation. Spatial simulations of liquids with a large number of components match our analytical solution. Our work suggests that complex biological mixtures can perform information retrieval through metastable phase separation.

Figures (15)

• Figure 1: Metastable states in multicomponent mixtures.A. Schematic of a homogeneous state in a multicomponent mixture (solutes represented by colored circles, solvent by blue background). B. Schematic of a phase separated state in a multicomponent mixture. Several droplets of diverse sizes and compositions can coexist in such states. C. Representation of the free energy landscape for a mixture with many components at fixed state parameters. At high temperature, there is only one stable state, which is the homogeneous state. D. Same as C, but at lower temperatures. In this case, the homogeneous state is only one of many possible metastable states. All other metastable states are phase separated, including the global minimum corresponding with the thermodynamic equilibrium.
• Figure 2: Schematic illustration of the geometry underlying metastable phase separation.A. Standard binary mixtures admit a single phase separated state, which can be found by the common tangent construction (orange line connecting two gray circles). The only metastable states are homogeneous states (dashed green line) B. The geometry of the state space of a binary mixture is a straight line. The phase separated state is stable at intermediate solute volume fractions, with high and low volume fractions corresponding to stable homogeneous phases. In addition, there are two metastable regions corresponding to homogeneous phases. The square images show schematics of 2D spatial simulations of the system. C. In presence of non-linear interactions, binary mixtures may admit multiple phase separated states, corresponding to multiple common tangent constructions (three in this example, corresponding to the orange solid line and the two orange dashed lines). D Besides globally stable phase separated states and homogeneous metastable states, the system in C also admits two metastable phase separated states. Gray box contains color legend. E. For a ternary mixture the state space geometry is a triangle. Due to the increase in dimensionality the system now admits a large variety of common tangent lines and planes (three examples are shown, corresponding to the violet planes and the red point). Of these, only those belonging to the convex hull correspond to stable equilibrium states. F. Examples of a stable phase separated state, a metastable phase separated state, and a metastable homogenenous state. The three states correspond to the same thermodynamic parameters, as can be seen by the corresponding points (green, blue, red) that appear in panel E. This figure is analogous to Fig. \ref{['fig:scheme']}D. However, now we have two species, $N=2$, and thus we need to specify the spatial distribution of both $\phi_1$ and $\phi_2$.
• Figure 3: Free energy and stability diagram for a binary mixture with quartic potential. A. Free energy density $f(\phi)$, in units of $\nu_0^{-1},$ from Eq. \ref{['eq:free_bin']} with the energy function in Eq. \ref{['eq:ubin']} for $c=120$. Common tangent lines correspond to phase separated states, solid lines are stable and dashed metastable (see SI \ref{['SI:quartic']} for more details). B. Same as in A, but for $c=135$. Stability is reversed: metastable states in A become stable, and the stable state in A is now metastable. C. Stability diagram in the $(c^{-1},\phi)$ parameter space, with y $c^{-1}$ the inverse interaction energy and $\phi$ the volume fraction of the solute. The spinodal line of the homogeneous state is given by Eq. \ref{['eq:spinquart']} and plotted in pink. There are three additional lines delineating the regions where the three phase separated are stable: high and low volume fraction (red - family I), low and intermediate volume fraction (green - family II), and high and intermediate volume fraction (blue - family III). The corresponding binodal lines can be found in SI Fig. \ref{['fig:si_saddlev1']}. Tie lines are given by horizontal lines connecting coexisting volume fractions in the phase separated states, three examples are shown: one for $c = 71$ (blue dashed line), one for $c =83$ (green dashed line) and another for $c=500$ (red dashed line). A phase diagram can be constructed by selecting the phase of lowest free-energy for each parameter, see SI Fig. \ref{['fig:si_fig3']}
• Figure 4: Phase separation dynamics for a binary mixture with quartic potential. Each row shows the evolution of the system towards a phase separated state belonging to one of the three families. A. Phase separation into a family III state from an initial homogeneous state with $\phi = 0.751$ (see SI Video 1 for the full time evolution), represented in the black cross on top of the common tangent line. B. Phase separation into a family II state from an initial homogeneous state with $\phi = 0.259$ (see SI Video 2). C. Relaxation into the globally stable state of family I, starting from a non-homogeneous initial condition where solute is concentrated in two regions with $\phi = 0.264$ (see SI \ref{['SI:numerics']} for more details and SI Video 3). Common to A, B and C: the left panels show the free energy density $f(\phi)$, in units of $\nu_0^{-1}$, with the final state indicated on the common tangent; the right top series corresponds to four snapshot of the spatial distribution of solute at different times; the right bottom series shows histograms of the solute volume fraction. In the histograms, the y-axis provides an estimate of the volume of compartments, $w^{(c)}$. The points in black correspond to the analytical predictions. The y error-bars correspond to half of the length of the interface between domains and the x error bars to half the bin width. In all panels we used $c = 135$, a lateral length of the box $L = 64/\sqrt{6}$, and units of time and length that equal $k\nu_0/\ell$ and $\sqrt{k\nu_0}$, respectively (see SI \ref{['si:nondim']} for the non-dimensional Cahn-Hilliard equation). See SI \ref{['SI:numerics']} for details of the numerical implementation.
• Figure 5: Stationarity and stability diagrams of the liquid Hopfield model.A. Overlap $a_\ast$ as a function of the volume fraction $\phi$, for different values of the cubic interaction $v_3$ and at fixed $\ v_2 = 4$. The onset of non-zero overlap indicates the emergence of stationary target states. The quantity $a_\ast$ is obtained by solving Eq. \ref{['eq:easy1']}. B. Spinodal lines in the $(v_2, \phi)$-plane, delimiting regions where either the homogeneous state (violet) or all retrieval phases (green) are stable and for fixed $v_3=3$. In the orange region neither the homogeneous nor the retrieval states are stable. The curves drawn correspond to solving the equations obtained by replacing the inequalities in Eqs. \ref{['eq:stab_targets']} and \ref{['eq:phiv2v3']} by equalities. C. Same as B. but in the $(v_3, \phi)$ plane for fixed $v_2 = 4$. D. Schematic representation of the stability diagram in the $v_2, v_3, \phi$ space. For clarity, the small region between the homogeneous and retrieval region was omitted.