Thermodynamically Consistent Incorporation of the Langmuir Adsorption Model into Compressible Fluctuating Hydrodynamics

TL;DR

This work develops a thermodynamically consistent framework to couple Langmuir adsorption with compressible fluctuating hydrodynamics at a gas–solid interface. By deriving a mass–energy update scheme and a thermodynamically consistent Langmuir (TCR) model, the authors show that adsorption/desorption events can be integrated without violating equilibrium statistics, identifying an essential internal-energy correction term of $-rac{1}{2} k_B T$ per molecule. Analytical stochastic analysis and equilibrium simulations validate that the update scheme reproduces correct variances and structure factors, while thermodynamic inconsistency (e.g., using mean rates or omitting the energy correction) leads to significant deviations and nonzero correlations. The methodology lays a principled foundation for future thermodynamically consistent FHD–KMC coupling and may extend to more complex reactive interfacial systems, including flows.

Abstract

For a gas-solid interfacial system where chemical species undergo reversible adsorption, we develop a mesoscopic stochastic modeling method that simulates both gas-phase hydrodynamics and surface coverage dynamics by coupling the Langmuir adsorption model with compressible fluctuating hydrodynamics. To this end, we derive a thermodynamically consistent mass-energy update scheme that accounts for how the mass and energy variables in the gas and surface subsystems should be updated according to the changes in the number of molecules of each species in each subsystem due to adsorption and desorption events. By performing a stochastic analysis for the ideal Langmuir model and the full hydrodynamic system, we analytically confirm that our mass-energy update scheme captures thermodynamic equilibrium predicted by equilibrium statistical mechanics. We find that an internal energy correction term is needed, which is attributed to the difference in the mean kinetic energy of gas molecules colliding with the surface from that computed from the Maxwell-Boltzmann distribution. By performing an equilibrium simulation study for an ideal gas mixture of CO and Ar with CO undergoing reversible adsorption, we validate our overall simulation method and implementation.

Figures (8)

• Figure 1: Ideal Langmuir model undergoing reversible molecular (i.e., one-site) adsorption. It consists of the gas subsystem containing an ideal gas mixture and the solid subsystem containing an ideal adsorbent. We assume that the ideal gas mixture consists of a chemical species ($\mathrm{A}$, orange) undergoing reversible adsorption \ref{['eq:revads']} and a nonreactive species ($\mathrm{B}$, green).
• Figure 2: For a gas molecule colliding with a wall normal to the $z$ axis (blue) and a gas molecule far from the wall (red), the probability density functions of (a) the normal velocity component, $v_z$, and (b) the corresponding kinetic energy, $\frac{1}{2} m v_z^2$, are compared. In panel (a), velocity is normalized by $\sqrt{k_B T/m}$. The blue curve depicts the Rayleigh distribution, whereas the red curve shows the Maxwell--Boltzmann distribution. In panel (b), kinetic energy is normalized by $k_B T$. The vertical dashed lines indicate the mean kinetic energies: $k_B T$ for the Rayleigh distribution (blue) and $\frac{1}{2} k_B T$ for the Maxwell--Boltzmann distribution (red).
• Figure 3: An illustration of the spatially discretized system. It consists of $N_{cell}$ gas cells and an adsorbent surface. The states of gas cells are described by $\delta\mathbf{Q}^{(i)}$, $i=1,2,\dots,N_{cell}$ (and collectively by $\delta\mathbf{Q}$), and the state of the surface is described by $\delta\theta$. It is also shown which variables the augmented variables $\delta\tilde{\mathbf{Q}}^{(1)}$ and $\delta\tilde{\mathbf{Q}}$ contain.
• Figure 4: Cell variances of (a) $\rho_\mathrm{CO}$ and (b) $\rho_\mathrm{Ar}$ obtained using our update scheme for $\bar{T}=800K.$ The normalized cell variances $C_\phi(z)/C_{\phi,eq}$ are plotted as a function of $z$, where $z = (i-0.5)\Delta z$ is the distance of the $i$th layer ($i=1,\dots,16)$ from the adsorbent surface. Error bars show 95% confidence intervals.
• Figure 5: Structure factor spectra obtained using our update scheme for $\bar{T}=800K$. The results for the bottom layer contacting with the adsorbent surface are shown: (a) total mass density, (b) $x$-velocity component, (c) temperature, (d) mass density of $\mathrm{CO}$, (e) mass density of $\mathrm{Ar}$, and (f) surface coverage. The normalized structure factors $S_\phi(\kappa)/S_{\phi,eq}$ are plotted as a function of $\kappa=\sqrt{\kappa_x^2+\kappa_y^2}$, where $\kappa_\alpha=k_\alpha(2\pi/L_\alpha)^{-1}$ is the wave index in the $\alpha$-direction ($\alpha=x,y$).