Taylor dispersion with adsorption and desorption

From Transient to Stationary Transport in Porous Networks under Various Adsorption Conditions and Kinetics

We investigate the interplay between adsorption and transport in a two-dimensional porous medium by means of an extended Lattice Boltzmann technique within the Two-Relaxation-Time framework. We focus on two canonical adsorption thermodynamics and kinetics formalisms: (1) the Henry model in which the adsorbed amount scales linearly with the free adsorbate concentration and (2) the Langmuir model that accounts for surface saturation upon adsorption. We simulate transport of adsorbing and nonadsorbing particles to investigate the effect of the adsorption/desorption ratio k, initial free adsorbate concentration c₀, surface saturation Γ^∞, and Peclet numbers Pe on their dispersion behavior. In all cases, despite marked differences between the different adsorption models, the three following transport regimes are observed: diffusion-dominated regime, transient regime and Gaussian or nearly Gaussian dispersion regime. On the one hand, at short times, the intermediate transient regime strongly depends on the system's parameters with the shape of the concentration field at a given time being dependent on the amount of particles adsorbed shortly after injection. On the other hand, at longer times, the influence of the initial condition attenuates as particles sample sufficiently the adsorbed and nonadsorbed states. Once such dynamical equilibrium is reached, transport becomes Gaussian (i.e., normal) or nearly Gaussian in the asymptotic regime. Interestingly, the characteristic time scale to reach equilibrium, which varies drastically with the system's parameters, can be much longer than the actual simulation time. In practice, such results reflect many experimental situations such as in water treatment where dispersion is found to remain anomalous (non-Gaussian), even if transport is considered over long macroscopic times.

How Stickiness Can Speed Up Diffusion in Confined Systems

The paradigmatic model for heterogeneous media used in diffusion studies is built from reflecting obstacles and surfaces. It is well known that the crowding effect produced by these reflecting surfaces slows the dispersion of Brownian tracers. Here, using a general adsorption desorption model with surface diffusion, we show analytically that making surfaces or obstacles attractive can accelerate dispersion. In particular, we show that this enhancement of diffusion can exist even when the surface diffusion constant is smaller than that in the bulk. Even more remarkably, this enhancement effect occurs when the effective diffusion constant, when restricted to surfaces only, is lower than the effective diffusivity with purely reflecting boundaries. We give analytical formulas for this intriguing effect in periodic arrays of spheres as well as undulating microchannels. Our results are confirmed by numerical calculations and Monte Carlo simulations.

Lattice Boltzmann method for adsorption under stationary and transient conditions: Interplay between transport and adsorption kinetics in porous media

A numerical method based on the Lattice Boltzmann formalism is presented to capture the effect of adsorption kinetics on transport in porous media. Through the use of a general adsorption operator, canonical models such as Henry and Langmuir adsorption as well as more complex adsorption mechanisms involving collective behavior with lateral interactions and surface aggregation can be investigated using this versatile model. By extending the description of adsorption phenomena to kinetic regimes with any underlying adsorption model, this effective technique allows assessing the coupled dynamics resulting from advection, diffusion, and adsorption in pores not only in stationary conditions but also under transient conditions (i.e., in regimes where the adsorbed amount evolves with time due to diffusion and advection). As illustrated in this paper, the development of such an approach provides a simple tool to determine the reciprocal effect of molecular flow and dispersion on adsorption kinetics. In this context, the use of a Lattice Boltzmann-based approach is important as it allows considering porous media of any morphology and topology. Beyond fundamental implications, this efficient method allows treating real engineering conditions such as pollutant dispersion or surfactant injection in a flowing liquid in soils and porous rocks.

A spatial Markov model for upscaling transport of adsorbing-desorbing solutes

The Spatial Markov Model (SMM) is an upscaled model with a strong track record in predicting upscaled behavior of conservative solute transport across hydrologic systems. Here we propose an SMM that can account for reactive linear adsorption and desorption processes and test it on a simple benchmark problem: flow and transport through an idealized periodic wavy channel. The methodology is built using trajectories that are obtained from a single high resolution random walk simulation of conservative transport across one periodic element. Our approach encodes information about where a particle starts at the inlet, where it leaves at the outlet, how long it takes to cross the domain and one additional piece of information, the number of times a particle strikes the boundary, with the objective of predicting large scale transport with arbitrary linear adsorption and desorption rates. Our benchmark problem demonstrates that predictions made with our proposed SMM agree favorably with results from direct numerical simulations, which resolve the full transport problem.

Transport and adsorption under liquid flow: the role of pore geometry

We study here the interplay between transport and adsorption in porous systems with complex geometries under fluid flow. Using a lattice Boltzmann scheme extended to take into account the adsorption at solid/fluid interfaces, we investigate the influence of pore geometry and internal surface roughness on the efficiency of fluid flow and the adsorption of molecular species inside the pore space. We show how the occurrence of roughness on pore walls acts effectively as a modification of the solid/fluid boundary conditions, introducing slippage at the interface. We then compare three common pore geometries, namely honeycomb pores, inverse opal, and materials produced by spinodal decomposition. Finally, we quantify the influence of those three geometries on fluid transport and tracer adsorption. This opens perspectives for the optimization of materials' geometries for applications in dynamic adsorption under fluid flow.

Unexpected coupling between flow and adsorption in porous media

We study the interplay between transport and adsorption in porous systems under a fluid flow, based on a lattice Boltzmann scheme extended to account for adsorption. We performed simulations on well-controlled geometries with slit and grooved pores, investigating the influence of adsorption and flow on dispersion coefficient and adsorbed density. In particular, we present a counterintuitive effect where fluid flow induces heterogeneity in the adsorbate, displacing the adsorption equilibrium towards downstream adsorption sites in grooves. We also present an improvement of the adsorption-extended lattice Boltzmann scheme by introducing the possibility for saturating Langmuir-like adsorption, while earlier work focused on linear adsorption phenomena. We then highlight the impact of this change in situations of high concentration of adsorbate.

Biased diffusion in three-dimensional comb-like structures

In this paper, we study biased diffusion of point Brownian particles in a three-dimensional comb-like structure formed by a main cylindrical tube with identical periodic cylindrical dead ends. It is assumed that the dead ends are thin cylinders whose radius is much smaller than both the radius of the main tube and the distance between neighboring dead ends. It is also assumed that in the main tube, the particle, in addition to its regular diffusion, moves with a uniform constant drift velocity. For such a system, we develop a formalism that allows us to derive analytical expressions for the Laplace transforms of the first two moments of the particle displacement along the main tube axis. Inverting these Laplace transforms numerically, one can find the time dependences of the two moments for arbitrary values of both the drift velocity and the dead-end length, including the limiting case of infinitely long dead ends, where the unbiased diffusion becomes anomalous at sufficiently long times. The expressions for the Laplace transforms are used to find the effective drift velocity and diffusivity of the particle as functions of its drift velocity in the main tube and the tube geometric parameters. As might be expected from common-sense arguments, the effective drift velocity monotonically decreases from the initial drift velocity to zero as the dead-end length increases from zero to infinity. The effective diffusivity is a more complex, non-monotonic function of the dead-end length. As this length increases from zero to infinity, the effective diffusivity first decreases, reaches a minimum, and then increases approaching a plateau value which is proportional to the square of the particle drift velocity in the main tube.

Aris-Taylor dispersion in tubes with dead ends

This paper deals with transport of point Brownian particles in a cylindrical tube with dead ends in the presence of laminar flow of viscous fluid in the cylindrical part of the tube (Poiseuille flow). It is assumed that the dead ends are identical and are formed by spherical cavities connected to the cylindrical part of the tube by narrow necks. The focus is on the effective velocity and diffusivity of the particles as functions of the mean flow velocity and geometric parameter of the tube. Entering a dead end, the particle interrupts its propagation along the tube axis. Later it returns, and the axial motion continues. From the axial propagation point of view, the particle entry into a dead end and its successive return to the flow is equivalent to the particle reversible binding to the tube wall. The effect of reversible binding on the transport parameters has been previously studied assuming that the particle survival probability in the bound state decays as a single exponential. However, this is not the case when the particle enters a dead end, since escape from the dead end is a non-Markovian process. Our analysis of the problem consists of two steps: First, we derive expressions for the effective transport parameters in the general case of non-Markovian binding. Second, we find the effective velocity and diffusivity by substituting into these expressions known results for the moments of the particle lifetime in the dead end [L. Dagdug, A. M. Berezhkovskii, Yu. A. Makhnovskii, and V. Yu. Zitserman, J. Chem. Phys. 127, 224712 (2007)]. To check the accuracy of our theory, we compare its predictions with the values of the effective velocity and diffusivity obtained from Brownian dynamics simulations. The comparison shows excellent agreement between the theoretical predictions and numerical results.

Aris-Taylor dispersion with drift and diffusion of particles on the tube wall

A laminar stationary flow of viscous fluid in a cylindrical tube enhances the rate of diffusion of Brownian particles along the tube axis. This so-called Aris-Taylor dispersion is due to the fact that cumulative times, spent by a diffusing particle in layers of the fluid moving with different velocities, are random variables which depend on the realization of the particle stochastic trajectory in the radial direction. Conceptually similar increase of the diffusivity occurs when the particle randomly jumps between two states with different drift velocities. Here we develop a theory that contains both phenomena as special limiting cases. It is assumed (i) that the particle in the flow can reversibly bind to the tube wall, where it moves with a given drift velocity and diffusivity, and (ii) that the radial and longitudinal diffusivities of the particle in the flow may be different. We derive analytical expressions for the effective drift velocity and diffusivity of the particle, which show how these quantities depend on the geometric and kinetic parameters of the model.

Accounting for adsorption and desorption in lattice Boltzmann simulations

We report a Lattice-Boltzmann scheme that accounts for adsorption and desorption in the calculation of mesoscale dynamical properties of tracers in media of arbitrary complexity. Lattice Boltzmann simulations made it possible to solve numerically the coupled Navier-Stokes equations of fluid dynamics and Nernst-Planck equations of electrokinetics in complex, heterogeneous media. With the moment propagation scheme, it became possible to extract the effective diffusion and dispersion coefficients of tracers, or solutes, of any charge, e.g., in porous media. Nevertheless, the dynamical properties of tracers depend on the tracer-surface affinity, which is not purely electrostatic and also includes a species-specific contribution. In order to capture this important feature, we introduce specific adsorption and desorption processes in a lattice Boltzmann scheme through a modified moment propagation algorithm, in which tracers may adsorb and desorb from surfaces through kinetic reaction rates. The method is validated on exact results for pure diffusion and diffusion-advection in Poiseuille flows in a simple geometry. We finally illustrate the importance of taking such processes into account in the time-dependent diffusion coefficient in a more complex porous medium.