Bridging scales in disordered porous media by mapping molecular dynamics onto intermittent Brownian motion

Role of Adsorption-Induced Deformation on Gas Self-Diffusivity in a Flexible Microporous Coal Matrix

Adsorption-induced deformation has long been underappreciated in gas transport studies of microporous coal, yet it strongly influences pore configurations and diffusive pathways. Here, a hybrid grand canonical Monte Carlo (GCMC)/molecular dynamics (MD) approach and equilibrium MD (EMD) simulations are employed to investigate how matrix flexibility reshapes pore structures and, in turn, impacts CH4 and CO2 self-diffusion in connected pore networks under various gas loadings. The results show that coal matrix deformation enhances adsorption, with CO2 exhibiting greater uptake and volumetric strain than CH4. A universal linear relationship emerges among gas loading, free volume ratio, and self-diffusion coefficients for both rigid and flexible matrices. In flexible matrices, this linearity features a gentler slope, indicating reduced diffusion sensitivity to diminishing free volume with loadings. By comparing geometrical and effective tortuosity, it is revealed that strongly adsorbing CO2 induces significant swelling and complex local rearrangements at elevated loadings, pushing geometrical tortuosity far beyond rigid-matrix levels, whereas CH4─with weaker adsorption─drives smaller, more uniform structural adjustments that only mildly increase geometrical tortuosity. These differences in tortuosity directly reflect changes in path complexity, which in turn governs self-diffusion behavior. Collectively, the findings clarify the dynamic coupling between gas adsorption, matrix deformation, and self-diffusivity in microporous coal, offering critical guidance for enhanced gas recovery and CO2 sequestration strategies that rely on accurate modeling of gas transport in deformable media.

Quantifying anomalous chemical diffusion through disordered porous rock materials

Fickian (normal) diffusion models show limitations in quantifying diffusion-controlled migration of solute species through porous rock structures, as observed in experiments. Anomalous diffusion prevails and can be interpreted using a Continuous Time Random Walk (CTRW) framework with a clear mechanistic underpinning. From the associated fractional diffusion equation we derive solutions over a broad range of anomalous diffusion behaviours, from highly anomalous to nearly Fickian, that yield temporal breakthrough curves and spatial concentration profiles of diffusing solutes. We illustrate that these solutions can be tailored to match realistic experimental conditions and resulting measurements that display anomalous diffusion. In particular, our analysis enables clear differentiation between early-time Fickian and anomalous diffusion, which becomes more pronounced over longer durations. It is shown that recent measurements of diffusion in natural rocks display distinct anomalous behaviour, with significant implications for critical assessment of solute migration in diverse geological and engineering applications.

Probing particle dynamics in a fully opaque porous network using X-ray differential dynamic radiography (XDDR)

Being able to follow in real-time the motion of particles of various nature, shape and size in porous media clearly represents a major research and societal issue. Currently, the leading technique used for tracing particle dynamics in porous media is based on optical microscopy. However, such methods require working with partially transparent samples, which is not the case of many naturally occurring porous media. The latter can indeed be fully opaque, with soil being the most salient example. We propose an alternative method based on recording the time evolution of X-ray radiographs in pure absorption mode. We show that a specific analysis of such a dataset can provide a quantitative determination of the intermediate scattering function (ISF) of these particles in various opaque porous media. The potential of our approach, named X-ray differential dynamic radiography (XDDR), was first checked by simulating random walk dynamics of light colloids inside a porous SiO2 RCP (random close packing) network saturated with water. Potential perturbation induced by Fresnel diffraction is analyzed. Finally, two experiments are performed on the beamline ANATOMIX at the SOLEIL synchrotron, demonstrating the possibility to probe μm SiO2 particle sedimentation either in bulk water or inside a RCP of poly(methyl methacrylate) (PMMA) spheres. XDDR appears to fill a relevant "niche" between DDM (differential dynamics microscopy) and XPCS (X-ray photon correlation spectroscopy), allowing to cover a time scale from 0.2 μs to several minutes and a range (q-range) from 0.1 μm-1 to 5 μm-1.

Diffusion power spectra as a window into dynamic materials architecture

Chemical recycling of commodity and specialty polymers presents a multifaceted challenge for industrial societies. On one hand, macromolecular architectures must be engineered to yield durable products that, on the other hand, rapidly deconstruct to recyclable monomers under pre-determined conditions. Polymer deconstruction is a chemical process that requires deep understanding of molecular reactivity in heterogeneous media, where porous material architectures evolve in both space and time. To build this understanding, we develop herein experimental and analytical methods describing sets of diffusive eigenmodes that exist within time-varying, non-Euclidean boundary conditions, a situation commonly encountered in the reactive deconstruction of polymers where chain fragments splay, alter their local dynamics, and evolve in their confinement of reacting media. Diffusion power spectra, discerned experimentally by NMR, yield polymer and solvent frequency-domain velocity autocorrelation functions that are analyzed in the context of physical models for chemical reactions parameterized with fractal mathematics. The results connect local motion in polymers to chemical reactivity during acidolysis of circular elastomers.

Dynamics Slowdown Induced by Gas Oversolubility in Nanoconfined Fluids

Oversolubility refers to the observation in nanoconfined liquids of significant gas solubilities that largely surpass the bulk solubility. While this thermodynamic effect is now well-documented, its impact on the dynamics of fluids confined in nanoporous materials has not been explored. Yet, by affecting adsorption and wetting at solid/liquid interfaces, oversolubility is expected to be a key phenomenon in separation and catalysis but also in geological applications such as pollutant migration in soils, carbon capture/storage in natural environments, and underground/atmosphere exchanges. Here, we employ atom-scale simulations and NMR experiments to show that gas oversolubility is expected in hydrated nanoporous materials and that it reduces both water and ion diffusivities [by 10% up to 60% depending on thermodynamic conditions]. Despite the complexity of adsorption/transport coupling in such gas/liquid/solid systems, we establish that diffusivities in the presence of small gases such as CO2, CH4 and H2 can be rationalized by accounting for the increase in the confined fluid viscosity (which is found to be directly linked to the decrease in the free volume accessible to the liquid upon solubilization). Moreover, in agreement with the reported data, by invoking Stokes-Einstein relation between the viscosity and diffusivity, we predict that the dynamics slowdown is identical for the confined water molecules and ionic species. We also show that this oversolubility-induced dynamical effect becomes more pronounced as the strength of the molecular interactions between the solubilized gas and the liquid/solid increases. This approach provides a robust formalism to fluid diffusion in nanoconfined environments subjected to gas solubility and potential oversolubility effects.

Theory and Modeling of Transport for Simple Fluids in Nanoporous Materials: From Microscopic to Coarse-Grained Descriptions

We present the state-of-the-art of theoretical modeling, molecular simulation, and coarse-graining strategies for the transport of gases and liquids in nanoporous materials (pore size 1-100 nm). Special emphasis is placed on the transport of small molecules in zeolites, active carbons, metal-organic frameworks, but also in nanoporous materials with larger pores such as ordered and disordered mesoporous oxides. We present different atomistic and mesoscopic methods as well as available theoretical formalisms to describe such a complex problem. Attention is given to the investigation of different molecular transport coefficients─including the self, collective and transport diffusivities─but also to the determination of free energy barriers and their role in overall adsorption/separation process rates. We further introduce other available approaches such as hierarchical simulations and upscaling strategies. This review focuses on simple fluids in prototypical nanoporous materials. While the phenomena covered here capture the main physical mechanisms in such systems, complex molecules will exhibit additional specific features. For the sake of clarity and brevity, we also omit multicomponent systems (e.g., fluid mixtures, electrolytes, etc.) and electrokinetic effects arising when charged systems are considered (ionic species, charged surfaces, etc.), both of which add to the complexity.

Multiscale simulation of liquid chromatography: Effective diffusion in macro-mesoporous beds and the B-term of the plate height equation

We performed multiscale simulations of analyte sorption and diffusion in hierarchical porosity models of monolithic silica columns for reversed-phase liquid chromatography to investigate how the mean mesopore size of the chromatographic bed and the analyte-specific interaction with the chromatographic interface influence the analyte diffusivity at various length scales. The reproduced experimental conditions comprised the retention of six analyte compounds of low to moderate solute polarity on a silica-based, endcapped, C18 stationary phase with water‒acetonitrile and water-methanol mobile phases whose elution strength was varied via the volumetric solvent ratio. Detailed information about the analyte-specific interfacial dynamics received from molecular dynamics simulations was incorporated through appropriate linker schemes into Brownian dynamics diffusion simulations in three hierarchical porosity models received from physical reconstructions of silica monoliths with a mean macropore size of 1.23 µm and mean mesopore sizes of 12.3, 21.3, or 25.7 nm. The mean mesopore size was found to have a similar influence on the effective mesopore diffusivity as the analyte polarity and the mobile-phase elution strength, which together determine the analyte residence time on a column. A smaller mesopore size attenuated the increase of the effective mesopore diffusivity with increasing mobile-phase elution strength significantly. The effective bed diffusivity was limited by the analyte residence time rather than by morphological details of the mesopore space. The stronger an analyte was retained by the chromatographic interface inside the mesopores, the slower became the mass transfer between the pore space hierarchies and the lower was the effective bed diffusivity. The B-terms of the plate height equation were finally generated with the bed diffusivities and phase-based retention factors derived from the hierarchical porosity models using additional information about the stationary-phase limit obtained from the analysis of analyte-bonded phase contacts. The B-terms highlight analyte- and mobile phase-specific behavior relevant to isocratic and gradient elution conditions in chromatographic practice. In particular, U-shaped B-term curves are observed due to the dominating contribution of the retention factor and the bed diffusivity to the B-term at low and high elution strength of the mobile phase, respectively.

Mind the Gap: The Role of Mass Transfer in Shaped Nanoporous Adsorbents for Carbon Dioxide Capture

Adsorptive separations by nanoporous materials are major industrial processes. The industrial importance of solid adsorbents is only expected to grow due to the increased focus on carbon dioxide capture technology and energy-efficient separations. To evaluate the performance of an adsorbent and design a separation process, the adsorption thermodynamics and kinetics must be known. However, although diffusion kinetics determine the maximum production rate in any adsorption-based separation, this aspect has received less attention due to the challenges associated with conducting diffusion measurements. These challenges are exacerbated in the study of shaped adsorbents due to the presence of porosity at different length scales. As a result, adsorbent selection typically relies mainly on adsorption properties at equilibrium, i.e., uptake capacity, selectivity and adsorption enthalpy. In this Perspective, based on an extensive literature review on mass transfer of CO2 in nanoporous adsorbents, we discuss the importance and limitations of measuring diffusion in nanoporous materials, from the powder form to the adsorption bed, considering the nature of the process, i.e., equilibrium-based or kinetic-based separations. By highlighting the lack of and discrepancies between published diffusivity data in the context of CO2 capture, we discuss future challenges and opportunities in studying mass transfer across scales in adsorption-based separations.

Colloidal Templating in Catalyst Design for Thermocatalysis

Conventional catalyst preparative methods commonly entail the impregnation, precipitation, and/or immobilization of nanoparticles on their supports. While convenient, such methods do not readily afford the ability to control collective ensemble-like nanoparticle properties, such as nanoparticle proximity, placement, and compartmentalization. In this Perspective, we illustrate how incorporating colloidal templating into catalyst design for thermocatalysis confers synthetic advantages to facilitate new catalytic investigations and augment catalytic performance, focusing on three colloid-templated catalyst structures: 3D macroporous structures, hierarchical macro-mesoporous structures, and discrete hollow nanoreactors. We outline how colloidal templating decouples the nanoparticle and support formation steps to devise modular catalyst platforms that can be flexibly tuned at different length scales. Of particular interest is the raspberry colloid templating (RCT) method which confers high thermomechanical stability by partially embedding nanoparticles within its support, while retaining high levels of reactant accessibility. We illustrate how the high modularity of the RCT approach allows one to independently control collective nanoparticle properties, such as nanoparticle proximity and localization, without concomitant changes to other catalytic descriptors that would otherwise confound analyses of their catalytic performance. We next discuss how colloidal templating can be employed to achieve spatially disparate active site functionalization while directing reactant transport within the catalyst structure to enhance selectivity in multistep catalytic cascades. Throughout this Perspective, we highlight developments in advanced characterization that interrogate transport phenomena and/or derive new insights into these catalyst structures. Finally, we offer our outlook on the future roles, applications, and challenges of colloidal templating in catalyst design for thermocatalysis.

Fast Rare Events in Exit Times Distributions of Jump Processes

Rare events in the first-passage distributions of jump processes are capable of triggering anomalous reactions or series of events. Estimating their probability is particularly important when the jump probabilities have broad-tailed distributions, and rare events are therefore not so rare. We formulate a general approach for estimating the contribution of fast rare events to the exit probabilities in the presence of fat-tailed distributions. Using this approach, we study three jump processes that are used to model a wide class of phenomena ranging from biology to transport in disordered systems, ecology, and finance: discrete time random walks, Lévy walks, and the Lévy-Lorentz gas. We determine the exact form of the scaling function for the probability distribution of fast rare events, in which the jump process exits from an interval in a very short time at a large distance opposite to the starting point. In particular, we show that events occurring on timescales orders of magnitude smaller than the typical timescale of the process can make a significant contribution to the exit probability. Our results are confirmed by extensive numerical simulations.

On De Gennes narrowing of fluids confined at the molecular scale in nanoporous materials

Beyond well-documented confinement and surface effects arising from the large internal surface and severely confining porosity of nanoporous hosts, the transport of nanoconfined fluids remains puzzling in many aspects. With striking examples such as memory, i.e., non-viscous effects, intermittent dynamics, and surface barriers, the dynamics of fluids in nanoconfinement challenge classical formalisms (e.g., random walk, viscous/advective transport)-especially for molecular pore sizes. In this context, while molecular frameworks such as intermittent Brownian motion, free volume theory, and surface diffusion are available to describe the self-diffusion of a molecularly confined fluid, a microscopic theory for collective diffusion (i.e., permeability), which characterizes the flow induced by a thermodynamic gradient, is lacking. Here, to fill this knowledge gap, we invoke the concept of "De Gennes narrowing," which relates the wavevector-dependent collective diffusivity D0(q) to the fluid structure factor S(q). First, using molecular simulation for a simple yet representative fluid confined in a prototypical solid (zeolite), we unravel an essential coupling between the wavevector-dependent collective diffusivity and the structural ordering imposed on the fluid by the crystalline nanoporous host. Second, despite this complex interplay with marked Bragg peaks in the fluid structure, the fluid collective dynamics is shown to be accurately described through De Gennes narrowing. Moreover, in contrast to the bulk fluid, the departure from De Gennes narrowing for the confined fluid in the macroscopic limit remains small as the fluid/solid interactions in severe confinement screen collective effects and, hence, weaken the wavevector dependence of collective transport.

Trajectory-Extending Kinetic Monte Carlo Simulations to Evaluate Pure and Gas Mixture Diffusivities through a Dense Polymeric Membrane

With renewed interest in CO2 separations, carbon molecular sieving (CMS) membrane performance evaluation requires diffusion coefficients as inputs to have a reliable estimate of the permeability. An optimal material is desired to have both high selectivity and permeability. Gases diffusing through dense CMS and polymeric membranes experience extended subdiffusive regimes, which hinders reliable extraction of diffusion coefficients from mean squared displacement data. We improve the sampling of the diffusive landscape by implementing the trajectory-extending kinetic Monte Carlo (TEKMC) technique to efficiently extend molecular dynamics (MD) trajectories from ns to μs time scales. The obtained self-diffusion coefficient of pure CO2 in CMS membranes derived from a 6FDA/BPDA-DAM precursor polymer melt is found to linearly increase from 0.8-1.3 × 10-6 cm2 s-1 in the pressure range of 1-20 bar, which supports previous experimental findings. We also extended the TEKMC algorithm to evaluate the mixture diffusivities in binary mixtures to determine the permselectivity of CO2 in CH4 and N2 mixtures. The mixture diffusion coefficient of CO2 ranges from 1.3-7 × 10-6 cm2 s-1 in the binary mixture CO2/CH4, which is significantly higher than the pure gas diffusion coefficient. Robeson plot comparisons show that the permselectivity obtained from pure gas diffusion data is significantly lower than that predicted using mixture diffusivity data. Specifically, in the case of the CO2/N2 mixture, we find that using mixture diffusivities led to permselectivities lying above the Robeson limit highlighting the importance of using mixture diffusivity data for an accurate evaluation of the membrane performance. Combined with gas solubilities obtained from grand canonical Monte Carlo (GCMC) simulations, our work shows that simulations with the TEKMC method can be used to reliably evaluate the performance of materials for gas separations.

Analysis of Vibrational Spectra of Tetrafluoroethane Glasses Deposited by Physical Vapor Deposition

This paper presents the results obtained in the study of structural phase transitions in thin films of R134A. The samples were condensed on a substrate by physical deposition of R134A molecules from the gas phase. Structural phase transformations in samples were investigated by observing the changes in characteristic frequencies of Freon molecules in the mid-infrared range with the help of Fourier-transform infrared spectroscopy. The experiments were carried out in the temperature range from 12 to 90 K. A number of structural phase states, including glassy forms, were detected. The changes in thermogram curves at fixed frequencies of half-widths of absorption bands of R134A molecules were revealed. These changes indicate a large bathochromic shift of these bands at frequencies of ν = 842 cm^-1, ν = 965 cm^-1, and ν = 958 cm^-1 and a hypsochromic shift of the bands at frequencies of ν = 1055 cm^-1, ν = 1170 cm^-1, and ν = 1280 cm^-1 at temperatures from T = 80 K to T = 84 K. These shifts are related to the structural phase transformations in the samples.

Sorption-Deformation-Percolation Model for Diffusion in Nanoporous Media

Diffusion of molecules in porous media is a critical process that is fundamental to numerous chemical, physical, and biological applications. The prevailing theoretical frameworks are challenged when explaining the complex dynamics resulting from the highly tortuous host structure and strong guest-host interactions, especially when the pore size approximates the size of diffusing molecule. This study, using molecular dynamics, formulates a semiempirical model based on theoretical considerations and factorization that offer an alternative view of diffusion and its link with the structure and behavior (sorption and deformation) of material. By analyzing the intermittent dynamics of water, microscopic self-diffusion coefficients are predicted. The apparent tortuosity, defined as the ratio of the bulk to the confined self-diffusion coefficients, is found to depend quantitatively on a limited set of material parameters: heat of adsorption, elastic modulus, and percolation probability, all of which are experimentally accessible. The proposed sorption-deformation-percolation model provides guidance on the understanding and fine-tuning of diffusion.

Confinement Effects on Distribution and Transport of Neutral Solutes in a Small Hydrophobic Nanopore

Molecular dynamics simulations are used to study confinement effects in small cylindrical silica pores with extended hydrophobic surface functionalization as realized, for example, in reversed-phase liquid chromatography (RPLC) columns. In particular, we use a 6 nm cylindrical and a 10 nm slit pore bearing the same C₁₈ stationary phase to compare the conditions inside the smaller-than-average pores within an RPLC column to column-averaged properties. Two small, neutral, apolar to moderately polar solutes are used to assess the consequences of spatial confinement for typical RPLC analytes with water (W)-acetonitrile (ACN) mobile phases at W/ACN ratios between 70/30 and 10/90 (v/v). The simulated data show that true bulk liquid behavior, as observed over an extended center region in the 10 nm slit pore, is not recovered within the 6 nm cylindrical pore. Instead, the ACN-enriched solvent layer around the C₁₈ chain ends (the ACN ditch), a general feature of hydrophobic interfaces equilibrated with aqueous-organic liquids, extends over the entire pore lumen of the small cylindrical pore. This renders the entire pore a highly hydrophobic environment, where, contrary to column-averaged behavior, neither the local nor the pore-averaged sorption and diffusion of analytes scales directly with the W/ACN ratio of the mobile phase. Additionally, the solute polarity-related discrimination between analytes is enhanced. The consequences of local ACN ditch overlap in RPLC columns are reminiscent of ion transport in porous media with charged surfaces, where electrical double-layer overlap occurring locally in smaller pores leads to discrimination between co- and counterionic species.

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.

Multiscale Modeling of Broadband Perfect Absorbers Based on Gold Metallic Molecules

The modeling of functional photonic devices that rely on molecular materials continues to be a major contemporary challenge. It is a challenge because, in the Maxwell equations, which govern the light-matter interaction, material properties are primarily introduced on phenomenological grounds and not from first principles. To overcome such limitations, we outline a multiscale modeling approach that bridges multiple length scales. We can predict with our approach the optical response of a photonic device that exploits in its design molecular materials whose properties were determined using time-dependent density functional theory. The specifically considered device is a broadband perfect absorber that uses in part a thin film comprising gold molecules made from 144 atoms. Our methodology discloses various chemical and physical effects that define such a device's response. Our methodology is versatile, and a larger number of applications will profit from this development.

Surface Protolysis and Its Kinetics Impact the Electrical Double Layer

Surface conductivity in the electrical double layer (EDL) is known to be affected by proton hopping and diffusion at solid-liquid interfaces. Yet, the role of surface protolysis and its kinetics on the thermodynamic and transport properties of the EDL are usually ignored as physical models consider static surfaces. Here, using a novel molecular dynamics method mimicking surface protolysis, we unveil the impact of such chemical events on the system's response. Protolysis is found to strongly affect the EDL and electrokinetic aspects with major changes in ζ potential and electro-osmotic flow.

Heterogeneous Microscopic Dynamics of Intruded Water in a Superhydrophobic Nanoconfinement: Neutron Scattering and Molecular Modeling

With their strong confining porosity and versatile surface chemistry, zeolitic imidazolate frameworks-including the prototypical ZIF-8-display exceptional properties for various applications. In particular, the forced intrusion of water at high pressure (∼25 MPa) into ZIF-8 nanopores is of interest for energy storage. Such a system reveals also ideal to study experimentally water dynamics and thermodynamics in an ultrahydrophobic confinement. Here, we report on neutron scattering experiments to probe the molecular dynamics of water within ZIF-8 nanopores under high pressure up to 38 MPa. In addition to an overall confinement-induced slowing down, we provide evidence for strong dynamical heterogeneities with different underlying molecular dynamics. Using complementary molecular simulations, these heterogeneities are found to correspond to different microscopic mechanisms inherent to vicinal molecules located in strongly adsorbing sites (ligands) and other molecules nanoconfined in the cavity center. These findings unveil a complex microscopic dynamics, which results from the combination of surface residence times and exchanges between the cavity surface and center.

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.