Machine-Learned Many-Body Potentials for Charged Colloids reveal Gas-Liquid Spinodal Instabilities only in the strong-coupling regime of Primitive Models

TL;DR

This work addresses whether like-charge attractions and gas–liquid spinodal instabilities in highly charged colloidal suspensions persist beyond Poisson–Boltzmann predictions. It builds force-matching, linear-regression ML potentials from primitive-model simulations to obtain a colloids-only potential $U^{ML}$, enabling large-scale MD studies across $Z$, $\\sigma/\\lambda_B$, and salt strength. The key finding is that like-charge attractions and spinodal phase behavior emerge only in the strongly coupled regime, with the melting line scaling as $\\sigma/\\lambda_B \\approx (Z/13.7)^{1/2}$ at high $Z$, and PB-based extrapolations failing in this domain; adding salt enhances attractions but does not broadly extend the coexistence region. Overall, the ML framework allows efficient exploration of PM physics in large colloidal systems and clarifies that correlated counterion fluctuations drive cohesion in the strong-coupling regime, not volume-term effects alone.

Abstract

Past experimental observations of gas-liquid and gas-crystal coexistence in low-salinity suspensions of highly charged colloids have suggested the existence of like charge attraction. Evidence for this phenomenon was also observed in primitive-model simulations of (asymmetric) electrolytes and of low-charge nanoparticle dispersions. These results from low-valency simulations have often been extrapolated to experimental parameter regimes of high colloid valency where like-charge attraction between colloids has been reported. However, direct simulations of highly charged colloids remain computationally demanding. To circumvent slow equilibration, we employ a machine-learning (ML) framework to construct ML potentials that accurately describe the effective colloid interactions. Our ML potentials enable fast simulations of dispersions and successfully reproduce the gas-liquid and gas-solid phase separation observed in primitive-model simulations at low charge numbers. Extending the ML-based simulations to higher valencies, where primitive-model simulations become prohibitively slow, also reveals like-charge attractions and gas-liquid spinodal instabilities, however only in the regime of strongly coupled electrostatic interactions and not in the weakly coupled Poisson-Boltzmann regime of the experimental observations of colloidal like-charge attractions.

Figures (9)

• Figure 1: (a) Illustration of the three species in the primitive model, representing colloids, counterions, and coions with charges $Zq$, $-q$, and $+q$, and diameters $\sigma$, $\sigma_i$, and $\sigma_i$, respectively, where $Z>0$ is the colloid valency, $q$ the elementary charge, and $\sigma_i=\sigma/20$ the diameter of the co- and counter-ions. (b) Configuration of a simulation of $N = 32$ colloids and 3854 ions, used in generating ML potentials. This configuration is representative of a system in the strong-coupling regime.
• Figure 2: (a) Parity plot comparing the Cartesian components of the effective many-body ML forces $\mathbf{F}^{ML}_{i,\alpha}$ (in units of $k_B T/\sigma$) predicted by the ML model with the corresponding PM forces $\mathbf{F}^{PM}_{i,\alpha}$ measured in primitive-model simulations for the same configurations. The ML potential describes a salt-free system of colloids with valency $Z=100$ and monovalent counterions at effective temperature $\sigma/\lambda_B = 2.5$, with ion to colloid ratio $\sigma_i/\sigma = 0.05$. The ML model was trained on 240 configurations of 32 colloids at packing fractions in the range $\eta \in \lbrack0.001, 0.45\rbrack$. (b) The effective two-body ML potential $U_2^{ML}(R)$ for effective temperature $\sigma/\lambda_B \in \lbrack2.0, 3.5 \rbrack$, trained using the same procedure as the potential in plot (a).
• Figure 3: Radial distribution function $g(R)$ as obtained from primitive-model simulations (lines) and from coarse-grained simulations using ML potentials (triangles) for a salt-free colloidal dispersion with $Z=100$ and ion-to-colloid size ratio $\sigma_i/\sigma=0.05$ as described in Section \ref{['sec:assym100']} at several packing fractions $\eta$ (see labels), for effective temperatures (a) $\sigma/\lambda_B = 2.5$ and (b) $\sigma/\lambda_B = 3.0$. The ML model is trained on configurations with packing fractions in the range of $\eta \in [0.001, 0.45]$.
• Figure 4: Typical configurations from primitive-model simulations (top row) and coarse-grained simulations using ML potentials (bottom row) for a salt-free colloidal suspension ($\exp{[\beta\mu/2]} \rightarrow0$) with $Z = 100$ and ion-to-colloid size ratio $\sigma_i/\sigma=0.05$. (a) Initial configuration in all simulations is a face-centered cubic crystal phase adjacent to a vacuum. (b), (c), and (d) Final configurations after equilibration from primitive-model simulations at effective temperatures $\sigma/\lambda_B = 2.0, 2.5,$ and $3.0$, respectively. (e), (f), and (g) Final configurations after equilibration from simulations using the ML potentials at the same temperatures.
• Figure 5: Double logarithmic representation of the colloid valency $Z$ and effective temperature $\sigma/\lambda_B$ plane for a binary (salt-free) colloid-counterion mixture at ion-to-colloid size ratio $\sigma_i/\sigma=0.05$. The full plane in (a) shows the highest temperatures at which our ML simulations exhibit gas-crystal coexistence (blue dots) and includes literature results (black symbols, lines), and the zoom for $Z\in[100,1000]$ in (b) shows the presence (blue dots), with black rings indicating spontaneous phase separation, and absence (red dots) of gas-crystal phase coexistence as obtained from simulations using the ML potential. The crosses in (a) correspond to critical temperatures found from simulations of asymmetric electrolytes for $Z = 2,3$Panagiotopoulos2002 and for $Z = 10, 20, 40, 80$,Linse2000 while the triangles indicate experimentally observed coexistence or like-charge attraction in colloidal suspensions at low salinity with increasing $Z$ (Refs. Monovoukas1989, Tata1997, Larsen1997, Gomez2009). The two lines are literature estimates of the "critical" effective temperature separating regimes with and without gas-crystal coexistence, the strong-coupling result (blue) given by $\sigma/\lambda_B=(Z/13.7)^{0.5}$ from MC simulations,Hynninen2007 and the weak-coupling Poisson-Boltzmann prediction $\sigma/\lambda_B=Z/12$ (black dashed).Zoetekouw2006prl