Machine Learning-integrated Multiscale Simulation Framework: Bridging Scales in Associative Polymer-Colloid Suspensions

TL;DR

A novel, unified multiscale simulation framework for telechelic polymer-colloid suspensions integrating explicit-chain Brownian dynamics resolving polymer-particle association kinetics; active learning metamodels compressing kinetics into efficient surrogates; and Population Balance-Brownian Dynamics (Pop-BD) computing network-scale dynamics from metamodel predictions, enabling simulations of larger systems over longer timescales.

Abstract

Predicting the rheological behavior of associative polymers bridging colloidal particles into transient networks is fundamentally challenging because the coupled spatiotemporal scales prevent efficient molecular-fidelity modeling. We address this through a novel, unified multiscale simulation framework for telechelic polymer-colloid suspensions integrating: explicit-chain Brownian dynamics resolving polymer-particle association kinetics; active learning metamodels compressing kinetics into efficient surrogates; and Population Balance-Brownian Dynamics (Pop-BD) computing network-scale dynamics from metamodel predictions. Validated against explicit-chain Brownian dynamics, our framework accurately reproduces time-and frequency-dependent stress relaxation moduli, enabling simulations of larger systems over longer timescales. Systematic investigations reveal that network connectivity exhibits critical transitions at specific chain-to-particle ratios, with bond density and lifetime correlating to enhanced relaxation times and moduli. Higher particle volume fractions yield more persistent bonds and slower relaxation. This framework connects chain-level dynamics to macroscopic rheology, enabling computationally efficient rational design of associative colloidal materials for waterborne coatings and soft-matter applications.

Figures (15)

• Figure 1: Novel ML-enabled multiscale simulation framework integrating fine-grained explicit chain Brownain dynamics (BD) simulations with Population Balance-Brownian dynamics (Pop-BD) through active learning metamodeling. The workflow begins with system parametrization defining the five-dimensional parameter space (chain length $N_k$, particle radius $R_p$, sticker binding energy $\varepsilon _s$, polymer surface density $n_{pol}/n_{col}$, and particle volume fraction $\phi$, which can be converted to normalized gap between particles, $H/N_kb_k$, in explicit chain BD simulations). Explicit-chain Brownian dynamics simulations resolve individual polymer chains and capture stochastic association/dissociation events, providing transition times and bridge fractions. These data inform Gaussian process-based active learning metamodels (previously developed, Ref. abdolahi2025interpretable) that efficiently and intelligently map the parameter space with an order of magnitude less computation compared to conventional exhaustive search methods. The resulting metamodels serve as input for coarse-grained Pop-BD simulations, which track bridge/loop populations without explicitly resolving polymer chains, enabling access to larger timescales. Rheological analysis validates predictions against explicit-chain simulations and explores structure-property relationships. Left top: Schematic visualization of the polymer-bridged colloidal network structure in explicit-chain BD simulations. Left bottom: Pop-BD representation where explicit chains are replaced with dynamical bonds, enabling simulation of larger systems and timescales.
• Figure 2: The stress relaxation modulus, $G(t)$, for a single polymer chain with a non-zero rest length $r_0$ (represented by blue squares), a zero rest length ($r_0 = 0$ and $k = 3k_BT/b_k^2$, shown by orange stars), and a bridged configuration is analysed. The stress relaxation modulus $G(t)$ for a single chain with a resting length of $r_0 = 1 b_k$ and $k = 400k_BT/b_k^2$ demonstrates a double decay pattern. This pattern is attributed to rapid bond-level relaxation followed by slower chain conformational changes. In contrast, when the resting length $r_0$ is zero, $G(t)$ exhibits a single slower decay, which is due to chain reconfiguration dynamics. The bridged chain with $r_0 = 1 b_k$ and $k = 400k_BT/b_k^2$ displays a stiffer modulus and a slightly longer relaxation time compared to the free chain. This behaviour results from the restricted motion of the chain and its enhanced load-bearing capability.
• Figure 3: The stress relaxation modulus $G(t)$ is analysed for systems with a fixed particle at the centre of a $60 \, b_k \times 60 \, b_k \times 60 \, b_k$ simulation box, containing a chain of $N_k = 20$. The chains have parameters $k = 400 \, k_B T/b_k^2$ and $r_0 = 1.0 \, b_k$, with a sticker attraction strength of $\varepsilon_s = 20 \, k_B T$. The longest relaxation time increases with particle radius, leading to longer loop relaxation times for larger particles. The other two relaxation times are the similar to those shown in Figure \ref{['f601']}. Each simulation was repeated five times with different random seeds, with error bars showing mean values and standard deviations.
• Figure 4: Gap-dependent transition rates and equilibrium bridge fraction (EBF) for a system with $\varepsilon_s$ = 8 $k_B T$, $N_k =$10, $R_p/b_k$= 10, and $n_{pol}/n_{col}$ = 10. The bridge-to-loop transition rate $M$ (blue dashed line) and loop-to-bridge transition rate $L$ (pink dash-dot line) from active learning metamodels are compared with the adjusted loop-to-bridge rate $L_{adj}$ (purple solid line) that asymptotically matches the analytical expression (Equation \ref{['eq_utility_601']}) at large separations. The EBF (black dash-dot line, right axis) obtained from the metamodel decreases rapidly with increasing gap distance.
• Figure 5: Comparison of the stress relaxation modulus $G(t)$ from Pop-BD simulations using active learning transition rates and bridge fractions shows excellent agreement with fully explicit-chain BD results. All simulations were conducted for a system of eight colloidal particles at a particle volume fraction of $\phi = 20\%$, with $\varepsilon_s = 8 k_B T$, $N_k = 10$, $R_p/b_k = 10$, and $n_{pol}/n_{col} = 10$. Each simulation was repeated five times with different random seeds, with error bars showing mean values and standard deviations.