Smoothed Dissipative Particle Dynamics for Mesoscale Advection-Diffusion-Reaction Problems

TL;DR

The paper develops a compositional Smoothed Dissipative Particle Dynamics (SDPD) framework for advection–diffusion–reaction (ADR) problems, integrating transport of reactive species with fluctuating hydrodynamics in a thermodynamically consistent, particle-based scheme. Implemented in LAMMPS, the ADR-SDPD model enables direct control of species diffusivities and reaction kinetics, while using GENERIC-inspired discretization and kernel-based momentum and concentration updates. It validates the method across diffusion-dominated, reaction-dominated, and coupled ADR regimes, including Turing-pattern formation, and shows that the mean diffusion is approximately additive between deterministic and stochastic contributions though spatial patterns can be more sensitive to fluctuations than the mean diffusivity suggests. The results indicate broad applicability to biology, chemistry, materials science, and environmental engineering, offering a versatile tool for mesoscale ADR simulations and insights into fluctuation-driven pattern formation.

Abstract

Smoothed dissipative particle dynamics (SDPD) is a widely used particle-based method for modelling soft matter systems at mesoscopic and macroscopic scales, offering thermodynamic consistency and direct control over the fluid's transport properties. Here, we present an SDPD model that incorporates the transport of reactants on scales smaller than the discretising particles, including the evolution of compositional fields. The proposed methodology is well-suited for modelling complex systems governed by advection-diffusion-reaction (ADR) dynamics. Implemented in LAMMPS, the model is validated using a range of benchmark problems spanning diffusion-dominated, reaction-dominated, and coupled ADR regimes. Our simulation results demonstrate that the implemented SDPD model effectively captures complex behaviours, such as Turing pattern formation. The proposed model holds promise for applications across various fields, including biology, chemistry, materials science, and environmental engineering.

Figures (9)

• Figure 1: Temporal evolution of the concentration profile $C(x,t)$ at $t = [5, 20, 100, 300, 500]\,\tau$ for (a) Dirichlet and (b) Neumann boundary condition tests. Numerical results are compared against analytical solutions (see Appendix \ref{['app:Difftest1D']}). Insets depict the respective channel configurations. Simulation parameters are listed in Table \ref{['tab:sdpd-params']}.
• Figure 2: Computational validation for 2D diffusion. Panel (a) illustrates the three distinct simulation schemes, each with a large gray sphere representing an SDPD particle and a smaller blue sphere inside to symbolize a molecule of the chemical substance. These schemes investigate deterministic ($D_{\mathrm{det}}$) and stochastic ($D_{\mathrm{stoch}}$) diffusion, with temperature $k_BT$ as a variable. Subfigure (f) displays Gaussian profiles with estimated parameters --mean ($\mu$) and standard deviation ($\sigma$)-- along with the corresponding $R^2$ values, captured at times $t$=0, $t$=0.5, $t$=2 and $t$=5. On top of (f), the scatter plots (b)-(e) represent the concentration distribution at these time points. Subfigure (b) also highlights the periodic boundary conditions applied to the 2D computational domain. $k_BT = 1$, $D^{\alpha} = 2$.
• Figure 3: Estimated effective diffusion coefficient ($D_{\text{eff}}$) for different input values of $D$, plotted across a range of temperatures ($k_BT = 0, 1, 3, 5$). The results illustrate the transition from stochastic to deterministic diffusion, with $D_{\text{eff}}$ approaching the deterministic limit as $k_BT \to 0$. Line colors indicate input $D$ values.
• Figure 4: Mean concentration profiles $C^{\alpha}$ of each chemical species $\alpha$ over dimensionless time ($t^* = t/\tau_{\mathrm{r}}$), with concentrations in Molar units. Solid lines show ODE solutions; circles indicate SDPD results. (a) Sulfur trioxide formation: initial equilibrium state (thick lines, filled circles) and evolution toward a new equilibrium after perturbation (thin lines, open circles). (b) Temporal evolution of species in the complex reaction system originally designed for a plug-flow reactor (PFR). (c) Enzyme-substrate dynamics according to the Michaelis–Menten model.
• Figure 5: Top: sketch of the channel geometry for the Poiseuille flow configuration, depicting the characteristic parabolic velocity profile and the boundary conditions for the concentration. Bottom: comparison of analytical concentration contours and simulation results for $C^\alpha$. Simulation parameters: $u_0 = 0.24$, $D = 0.22 l^2/\tau$, with $L_y/2=H$ and $H=5$. $L_x=40$.