Thermodynamic consistency and fluctuations in mesoscopic stochastic simulations of reactive gas mixtures

TL;DR

The paper develops a thermodynamically-consistent reaction (TCR) model for mesoscopic stochastic simulations of reactive gas mixtures within a compressible fluctuating hydrodynamics (FHD) framework based on chemical Langevin equations. It derives how forward and reverse rate constants must relate under equilibrium constraints and shows that reaction rates must be evaluated at the instantaneous temperature to correctly capture fluctuations. Using NO$_2$/N$_2$O$_4$ dimerization as a test, the authors demonstrate that TCR-FHD reproduces the correct equilibrium structure factors and interior temperature variance in nonequilibrium steady states, while thermodynamically inconsistent treatments distort long-wavelength fluctuation spectra. The work provides a practical methodology to couple thermodynamics with stochastic chemistry in mesoscopic reactive hydrodynamics and lays groundwork for incorporating detailed temperature-dependent thermochemistry data.

Abstract

It is essential that mesoscopic simulations of reactive systems reproduce the correct statistical distributions at thermodynamic equilibrium. By considering a compressible fluctuating hydrodynamics (FHD) simulation method of ideal gas mixtures undergoing reversible reactions described by the chemical Langevin equations, we show that thermodynamic consistency in reaction rates and the use of instantaneous temperatures for the evaluation of reaction rates is required for fluctuations for the overall system to be correct. We then formulate the required properties of a thermodynamically-consistent reaction (TCR) model. As noted in the literature, while reactions are often discussed in terms of forward and reverse rates, these rates should not be modeled independently because they must be compatible with thermodynamic equilibrium for the system. Using a simple TCR model where each chemical species has constant heat capacity, we derive the explicit condition that the forward and reverse reaction rate constants must satisfy in order for the system to be thermodynamically consistent. We perform equilibrium and non-equilibrium simulations of ideal gas mixtures undergoing a reversible dimerization reaction to measure the fluctuational behavior of the system numerically. We confirm that FHD simulations with the TCR model give the correct static structure factor of equilibrium fluctuations. For the statistically steady simulation of a gas mixture between two isothermal walls with different temperatures, we show using the TCR model that the temperature variance agrees with the corresponding thermodynamic-equilibrium temperature variance in the interior of the system, whereas noticeable deviations are present in regions near walls, where chemistry is far from equilibrium.

Figures (6)

• Figure 1: In panel (a), the values of the equilibrium constant $K(T)$ obtained from our constant-heat-capacity TCR model are compared with those computed by the Shomate equation in the temperature range $\hbox{300\K}\le T\le\hbox{400\K}$. Their relative differences, $(K_\mathrm{model}-K_\mathrm{Shomate})/K_\mathrm{Shomate}$, are shown in the inset. In panel (b), the values of the forward and reverse reaction rate constants, $k^+ (T)$ and $k^-(T)$, are plotted versus $T$. Two $y$-axes are used; the left one is for $k^+(T)$ (depicted by the blue solid line), whereas the right one is for $k^-(T)$ (red dashed line).
• Figure 2: Equilibrium structure factor spectra obtained from reactive FHD based on our thermodynamically-consistent chemistry formulation. For various field variables, the structure factor values $S(\mathbf{k})$ are normalized by the theoretical values $S_{eq}$ (see the color bar) and plotted versus $\kappa=\sqrt{\kappa_x^2 + \kappa_y^2 + \kappa_z^2}$, where $\kappa_i = k_i \left(2\pi/L\right)^{-1}$ is the integer wave index in the $i$-direction. Panels show scatter plots for the structure factors of (a) $\rho$, (b) $\rho u_x$, (c) $\rho E$, (d) $\rho_1$, (e) $\rho_2$, and (f) $T$. The horizontal black lines at unity show the theoretically expected value.
• Figure 3: Temperature equilibrium structure factor spectra obtained from (a) the temperature-independent rate constants case and (b) our TCR model are compared. Note that the same data are shown in panel (b) of this figure and panel (f) of Figure \ref{['fig:sr_all_fields']} but different vertical scales are used.
• Figure 4: For various partial pressure values of $\mathrm{N_2}$, the temperature structure factor spectra obtained from the temperature-independent rate constants case are compared. For visual clarity, rather than plotting all data points, representative values of $S(\kappa)/S_{eq}$ obtained by averaging within each subinterval of $\kappa$ are shown.
• Figure 5: For the steady state of an $\mathrm{NO_2/N_2O_4}$ mixture placed between two isothermal walls, (a) mean temperature $\langle T\rangle_z$, (b) variance $\langle \delta T^2 \rangle_z$, and (c) the ratio of $\langle \delta T^2 \rangle_z$ to the corresponding local-equilibrium values $\langle \delta T^2 \rangle_{z,eq}$ are plotted as a function of $z$ as red circles. Results obtained from the mean-profile rate-constant model are depicted by blue crosses. As reference, corresponding results for the deterministic simulation and the local-equilibrium fluctuations are shown in panels (a) and (b), respectively. The temperature profile obtained from the nonreactive FHD simulation is also shown for comparison in panel (a).