Surface Charge Relaxation Controls the Lifetime of Out-of-Equilibrium Colloidal Crystals

TL;DR

This work addresses the stability of out-of-equilibrium charged colloidal crystals under density-dependent electrostatic interactions. It combines Poisson-Boltzmann cell theory to obtain renormalized charge $Z^*$ and screening length $\tilde{\kappa}$ as functions of packing fraction, with Brownian dynamics that evolve these quantities in time via a relaxation timescale $\tau$. The key finding is that slow surface-charge relaxation can significantly prolong crystal lifetimes by creating a dynamic, density-regulated interaction landscape, providing a mechanism for metastable, long-lived structures without invoking true like-charge attractions. The results highlight the importance of electrostatic feedback and charging kinetics in nonequilibrium colloidal systems and suggest avenues for tuning stability through controlled charge-regulation dynamics and system geometry.

Abstract

Interactions between charged colloidal particles are profoundly influenced by charge regulation and charge renormalization, rendering the effective potential highly sensitive to local particle density. In this work, we investigate how a dynamically evolving, density-dependent Yukawa interaction affects the stability of out-of-equilibrium colloidal structures. Motivated by a series of experiments where unexpectedly long-lived colloidal crystals have suggested the presence of like-charged attractions, we systematically explore the role of charge regulation and charge renormalization. Using Poisson-Boltzmann cell theory, we compute the effective colloidal charge and screening length as a function of packing fraction. These results are subsequently incorporated into Brownian dynamics simulations that dynamically resolve the evolving colloid charge as a function of time and local density. In the case of slow relaxation dynamics, our results show that incorporating these charging effects significantly prolongs the lifetimes of out-of-equilibrium colloidal crystals, providing an explanation for the experimental observation of long-lived crystals. These findings demonstrate that the interplay of surface charge dynamics and colloidal interactions can give rise to complex and rich nonequilibrium behavior in charged colloidal suspensions, opening new pathways for tuning colloidal stability through electrostatic feedback mechanisms.

Figures (5)

• Figure 1: Effect of Charge Regulation and Charge Renormalization on Colloidal Interactions. (a) Bare charge $Z$ (dark blue) and renormalized charge $Z^*$ (light blue) as functions of packing fraction $\eta$. (b) Renormalized screening parameter $\tilde{\kappa} a$ as a function of $\eta$. (c) Total interaction potential $\beta U(r)$ between two colloids as a function of center-of-mass distance $r$ for two packing fractions, $\eta=0.0006$ (blue) and 0.145 (red), representative of the initial gas and crystal phase, respectively. (d) Initial configuration used in the simulations consisting of a cubic crystallite surrounded by a dilute fluid phase. (e) Slice of thickness $5\sigma$ through the center of the simulation box, showing the renormalized charge $Z^*$ of individual particles (see colorbar for color coding). (f) Same slice as in (e), but with particle radii scaled by the local screening length according to $5/\kappa \sigma+0.7$. Panels (e) and (f) correspond to time $t=2\tau_d$. The parameters used in the calculations are adopted from Ref. larsen-1997.
• Figure 2: Melting Behavior of a Cubic Crystal with an FCC structure. For clarity, only slices of thickness $6\sigma$ through the center of the simulation box are shown. The colors of the particles represent their renormalized charge $Z^*$, as indicated by the color bar. (a) Slice of the initial configuration used in the melting simulations. (b) Typical configurations from the melting simulation with relaxation time $\tau = 0.04 \tau_d$. (c) Time evolution of the fraction of particles identified as crystalline by our machine learning approach. (d) Typical configurations from the melting simulation with $\tau = 4 \tau_d$.
• Figure 3: Melting Behavior of Cubic Crystallites. Fraction of crystalline particles as a function of time $t/\tau_d$ for varying relaxation times $\tau$ and different initial packing fractions $\eta_i$ of the crystal and global packing fraction $\eta_g$: (a) $(\eta_i,\eta_g) = (0.057806,0.001047)$ from simulations $B_1, B_2, B_3$, (b) $(\eta_i,\eta_g) = (0.115613,0.002094)$ from simulations $B_5, B_6, B_7$, and (c) $(\eta_i,\eta_g) = (0.404644,0.004189)$ from simulations $B_{13}, B_{14}, B_{15}$. In this dilute regime, the small variations in $\eta_g$ between simulations have a negligible impact on the crystal lifetime.
• Figure 4: Melting Behavior of a Crystal with an FCC structure in a Slab Geometry. (a) Initial configuration with particles colored according to their renormalized charge $Z^*$, as shown by the color bar. The crystal has an initial packing fraction of $\eta_i=0.089215$, and a global packing fraction of $\eta_g = 0.009$. The relaxation time is set to $\tau = 0.04 \tau_d$. This corresponds to simulation $P_3$ of Table \ref{['tab:plates']}. (b) Time evolution of the system during expansion of the crystal. The particle colors represent the probability of belonging to the crystal phase. The images show slices of thickness $10\sigma$, centered along the height of the simulation box with dimensions of approximately $72\sigma \times 147\sigma$. For visual clarity, all particle radii are set to $0.7\sigma$.
• Figure 5: Expansion of a crystal in a slab geometry at a global packing fraction $\eta_g=0.02$, corresponding to simulation $P_7$ of Table \ref{['tab:plates']}. (a) The first four frames show the evolution of the crystalline regions during expansion of the crystal. (b) Packing fraction gradient across the simulation box at time $t=64 \tau_d$, with the color bar indicating the local packing fraction $\eta$.