Machine learning many-body potentials for charged colloids in primitive 1:1 electrolytes

TL;DR

This work addresses the failure of DLVO in strongly coupled electrolytes by learning density-dependent, many-body coarse-grained potentials U^{ML} that reproduce ion-averaged forces from primitive-model simulations. Using two- and three-body symmetry-function descriptors within a cutoff, the authors fit linear-weighted models to force data via force-matching, obtaining U^{ML} and its gradient to drive colloids-only simulations with significantly reduced computational cost. The approach is validated across several regimes, including low-polar solvents, salt-free strong coupling, and aqueous 1:1 electrolytes, where the ML potentials capture essential features such as like-charge attractions and the correct structure via g(R) and PM-like two- and three-body interactions; however, density dependence and long-range Coulomb physics at dilute densities remain challenges. Overall, the framework enables scalable exploration of phase behavior and interfacial phenomena in charged colloids and electrolytes, with potential extensions to density-aware or neural-network descriptors and improved long-range handling.

Abstract

Effective interactions between charged particles dispersed in an electrolyte are most commonly modeled using the Derjaguin-Landau-Verwey-Overbeek (DLVO) potential, where the ions in the suspension are coarse-grained out at mean-field level. However, several experiments point to shortcomings of this theory, as the distribution of ions surrounding colloids is governed by nontrivial correlations in regimes of strong Coulomb coupling (e.g. low temperature, low dielectric constant, high ion valency, high surface charge). Insight can be gained by explicitly including the ions in simulations of these colloidal suspensions, even though direct simulations of dispersions of highly charged spheres are computationally demanding. To circumvent slow equilibration, we employ a machine-learning (ML) framework to generate density-dependent ML potentials that accurately describe the effective colloid interactions at given system parameters. These ML potentials enable fast simulations and make large-scale simulations of charged colloids in suspension possible, opening the possibility for a systematic study of their phase behaviour, in particular gas-liquid and fluid-solid coexistence.

Figures (7)

• Figure 1: (a) Primitive model of a colloidal dispersion in a 1:1 electrolyte, represented as a three-component mixture of charged colloids, counterions, and coions with charges $Zq$, $-q$, and $+q$, and diameters $\sigma$, $\sigma_i$, and $\sigma_i$, respectively. (b) Typical configuration from a primitive model simulation with $N_c = 32$ colloidal particles of valency $Z=90$, $N_+=2551$ coions, $N_-=5431$ counterions, and size ratio $\sigma_i/\sigma=0.05$. This snapshot is representative of the simulations used to generate the training data.
• Figure 2: The inverse screening length $\kappa\sigma$ as a function of ion activity $\exp{\left[\beta \mu/2 \right]}$ with $\mu$ the chemical potential of salt pairs for a 1:1 electrolyte at temperature $\sigma_i/\lambda_B=9.85$ as obtained from Grand Canonical Monte Carlo simulations (blue line) along with the ideal gas prediction for comparison (dashed black line).
• Figure 3: 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 system consists of colloids with $Z =90$, effective temperature $\sigma/\lambda_B = 197.0$, inverse screening length $\kappa \sigma = 1.56$, and ion-to-colloid size ratio $\sigma_i/\sigma=0.05$. The ML model was trained on configurations with packing fractions in the range (a) $\eta \in \lbrack 0.001, 0.45 \rbrack$ and (b) $\eta \in \lbrack 0.001, 0.1 \rbrack$, where the scale difference of the axes between (a) and (b) should be noted.
• Figure 4: The effective pair potential $U^{ML}_2(R)$ predicted by our ML models (blue and red solid lines) and the potential of mean force $U_2^{PM}(R)$ measured in primitive model (PM) simulations (yellow crosses) for a system consisting of colloids with charge $Z=90$, effective temperature $\sigma/\lambda_B = 197$, inverse screening length $\kappa\sigma = 1.56$, and ion-to-colloid size ratio $\sigma_i/\sigma = 0.05$. Inset: effective three-body ML potential $U^{ML}_3(R)$ and the potential of mean force $U^{PM}_3(R)$ for three colloids placed in an equilateral triangle with edge length $R$ for the same system parameters. ML models are trained on configurations with packing fractions in the range $\eta \in \lbrack0.001,0.45\rbrack$ (blue) and on dilute configurations with $\eta \in \lbrack0.001,0.1\rbrack$ (red).
• Figure 5: Radial distribution function $g(R)$ for the colloidal dispersion described in Section \ref{['sec:colsys']} at several packing fractions $\eta$ (see labels) as obtained from primitive model (PM) simulations (lines) and from coarse-grained simulations using ML potentials (triangles), in (a) trained on configurations with packing fractions in the range $\eta \in [0.001, 0.45]$, with the curves shifted upwards for $\eta = 0.1$, $\eta = 0.2$, and $\eta = 0.3$ by $+1$, $+2$, and $+3$, respectively, and in (b) on dilute configurations with $\eta \in \lbrack 0.001, 0.1 \rbrack$, with the curves shifted upwards for $\eta = 0.05$ and $\eta = 0.1$ by $+1$ and $+2$, respectively