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Sorption

Sorption is the broad process by which solutes, such as ions or molecules, are removed from a fluid phase (typically aqueous or gaseous) and accumulated onto or within a solid phase, encompassing both adsorption at the surface and absorption into the bulk material.^[1] This phenomenon is particularly significant in geochemical and environmental contexts, where it governs the retention and mobility of nutrients, pollutants, and contaminants in soils and natural systems.^[2] The term is often employed when the precise mechanism—whether surface-specific or volumetric—is unclear or immaterial to the application.^[1] Adsorption involves the attachment of solutes to the external or internal surfaces of the solid through mechanisms like electrostatic interactions, hydrogen bonding, van der Waals forces, or chemical complexation, while absorption entails the diffusion and incorporation of solutes into the solid's structure.^[2] In soil science, sorption is a key regulator of chemical bioavailability, influencing the uptake of essential elements like potassium and phosphorus by plants, as well as the fate of environmental toxins such as heavy metals and radionuclides.^[2] Factors affecting sorption include solution pH, solute concentration, temperature, and the sorbent's surface properties, such as charge and organic matter content, which can lead to equilibrium states described by isotherms like Freundlich or Langmuir models.^[3] Beyond soils, sorption plays a critical role in water purification, where activated carbon or zeolites sorb organic pollutants and ions to improve water quality, and in nuclear waste management, where clay minerals sorb radioactive species to prevent groundwater contamination.^[3] The process's reversibility—known as desorption—allows for dynamic cycling of substances in ecosystems, but incomplete desorption can result in long-term immobilization of contaminants.^[2] Understanding sorption mechanisms is essential for predicting environmental impacts and designing remediation strategies, with ongoing research focusing on advanced modeling to account for multi-component systems and kinetic effects.^[3]

Definition and Fundamentals

Definition

Sorption is a general term in chemistry and environmental science that encompasses both adsorption and absorption processes, where substances are taken up by a material through surface accumulation or bulk incorporation, respectively.^[4] Adsorption refers to the adhesion of molecules, ions, or atoms to the surface of a sorbent, forming a film on that surface, while absorption involves the penetration and distribution of the sorbate throughout the volume of the absorbing medium.^[1] This broad categorization allows sorption to describe a wide range of interactions, including both physical processes driven by intermolecular forces and chemical processes involving bonding.^[5] In sorption terminology, the sorbate is the substance being accumulated or incorporated, such as a molecule, ion, or gas, while the sorbent is the receiving material, which can be a solid, liquid, or even a gas in certain contexts.^[6] A key quantitative measure of sorption strength is the partition coefficient K_d, defined as the ratio of the equilibrium concentration of the sorbate on the sorbent to its concentration in the solution phase:

K_d = \frac{[\text{sorbed concentration}]}{[\text{solution concentration}]}

This parameter indicates the distribution tendency of the sorbate between phases at equilibrium.^[7] Representative examples illustrate these concepts in simple systems: the sorption of pollutant gases, such as carbon monoxide, onto activated carbon solids in air purification, or the sorption of heavy metal ions, like lead, from aqueous solutions onto soil particles in environmental remediation.^[2]

Historical Development

The concept of sorption traces its roots to early 19th-century investigations into gas interactions with solids and liquids, with notable observations on gas uptake emerging in the mid-century. In 1866, Thomas Graham reported the reversible absorption of hydrogen gas by palladium metal, marking one of the first systematic studies of gas occlusion in solids and highlighting the potential for metals to act as sorbents under varying conditions. These findings built on earlier qualitative reports, such as Carl Wilhelm Scheele's 1773 experiments demonstrating the uptake of air by charcoal, which laid groundwork for understanding surface and bulk interactions in sorption processes.^[8] By the early 20th century, the need for a unified terminology became apparent as distinctions between surface adhesion and bulk penetration blurred in experimental contexts. In 1909, James W. McBain introduced the term "sorption" in his study of hydrogen interactions with charcoal, proposing it as an inclusive descriptor for both adsorption and absorption to avoid ambiguity in mechanisms. This nomenclature facilitated broader application of the concept. A key theoretical milestone followed in 1938 with the development of the Brunauer-Emmett-Teller (BET) theory, which extended earlier monolayer models to describe multilayer gas adsorption on solids, providing a foundational framework for quantifying sorption capacities in porous materials. In the mid-20th century, sorption concepts gained practical traction in soil science and chromatography, bridging fundamental research with applied fields. During the 1940s and 1950s, studies in soil science increasingly applied sorption principles to nutrient retention and contaminant behavior.^[9] Concurrently, advancements in chromatography, such as Archer J.P. Martin and Richard L.M. Synge's 1941 partition method, underscored sorption's role in separation techniques, linking it to efficient analyte isolation in complex mixtures and spurring industrial adoption.

Types and Mechanisms

Adsorption

Adsorption is a surface phenomenon in which molecules or atoms (adsorbate) adhere to the surface of a solid or liquid (adsorbent), forming a film of varying thickness on the surface without entering the bulk of the adsorbent material.^[10] This process is distinct from absorption, which involves penetration into the bulk phase. Adsorption can be classified into two primary types based on the nature of the interaction forces: physisorption and chemisorption. Physisorption, also known as physical adsorption, involves weak van der Waals forces between the adsorbate and the adsorbent surface, making it reversible and typically occurring at lower temperatures.^[11] The heat of adsorption for physisorption is low, ranging from 5 to 40 kJ/mol, reflecting the non-specific and multilayer nature of the bonding.^[12] In contrast, chemisorption, or chemical adsorption, entails the formation of strong chemical bonds, often involving electron sharing or transfer, which renders the process largely irreversible and activated, requiring higher temperatures for initiation.^[11] The heat of adsorption in chemisorption is significantly higher, typically between 40 and 800 kJ/mol, comparable to the energies of chemical reactions.^[12] Surface coverage in adsorption can be described by models that differentiate between monolayer and multilayer formations. Monolayer adsorption assumes a single layer of adsorbate molecules covering the surface sites uniformly, with no lateral interactions beyond site occupancy, as idealized in the Langmuir model.^[13] Multilayer adsorption, prevalent in physisorption, allows for multiple layers to form, with subsequent layers interacting via weaker forces similar to condensation, extending beyond the initial surface monolayer.^[14] A fundamental model for monolayer adsorption is the Langmuir isotherm, which describes the equilibrium fractional surface coverage \theta as a function of the adsorbate pressure p and the equilibrium constant K:

\theta = \frac{K p}{1 + K p}

This equation assumes homogeneous surface sites, no adsorbate-adsorbate interactions, and dynamic equilibrium between adsorption and desorption rates.^[13] Practical examples of adsorption include the removal of gases such as volatile organic compounds from air streams using activated carbon, where the high surface area of the carbon facilitates physisorption to capture pollutants efficiently.^[15] Similarly, activated carbon is widely employed for dye removal from wastewater, leveraging both physisorption and chemisorption mechanisms to adsorb organic dyes onto its porous surface, achieving high removal efficiencies in aqueous environments.^[16]

Absorption

Absorption refers to the process in which a sorbate, or absorbate, is incorporated into the bulk volume of a sorbent, or absorbent, through diffusion into its interior structure, often governed by the solubility of the sorbate within the sorbent material.^[1] This mechanism contrasts with surface-bound processes, as the absorbate disperses throughout the sorbent's matrix rather than accumulating at its exterior. In sorption contexts, absorption typically involves the sorbate transitioning from a fluid phase into the solid or liquid bulk phase of the sorbent, resembling a dissolution process that requires minimal energy compared to surface interactions.^[1] For gas absorption, a fundamental relationship is described by Henry's law, which states that the solubility of a gas in a liquid is proportional to its partial pressure above the liquid at equilibrium and constant temperature. The law is expressed as p = K_H \cdot x, where p is the partial pressure of the gas, x is the mole fraction of the gas in the liquid phase, and K_H is Henry's constant, which quantifies the gas's solubility under those conditions.^[17] This principle applies primarily to dilute solutions where no chemical reactions occur between the gas and solvent. Representative examples of absorption include the uptake of water in polymeric materials, where moisture diffuses into the polymer matrix, leading to swelling and influenced by the polymer's solubility parameters and temperature-dependent diffusion rates. Another key application is the absorption of CO₂ in liquid solvents for carbon capture, where CO₂ dissolves into amine-based or other solvents to separate it from flue gases.^[18] Absorption can be classified as physical or reactive (chemical). Physical absorption involves non-covalent dissolution of the sorbate into the sorbent's bulk via weak intermolecular forces, such as van der Waals interactions, resulting in reversible uptake without altering the sorbate's chemical structure.^[18] In contrast, reactive absorption entails a chemical reaction between the sorbate and sorbent within the bulk phase, forming new compounds like carbamates in CO₂-amine systems, which enhances capacity but requires energy for regeneration.^[18]

Other Forms

Ion exchange represents a specialized sorption process characterized by the selective uptake of ions through reversible electrostatic replacement between a solution and a solid sorbent, such as synthetic resins or natural soil components. In ion-exchange resins, which are typically cross-linked polymers with fixed charged groups, target ions in the aqueous phase displace counter-ions on the resin via stoichiometric exchange, enabling high selectivity based on ion charge and size. This mechanism extends to soils, where clay minerals and organic matter facilitate similar ionic substitutions, contributing to nutrient retention and contaminant immobilization. Unlike pure adsorption, which primarily involves surface accumulation, or absorption, which entails bulk dissolution, ion exchange integrates both surface and volumetric interactions through charged site occupancy.^[19]^[20]^[21] A prominent application of ion exchange in sorption is the removal of heavy metals from aqueous environments using natural zeolites, aluminosilicate minerals with a porous, cage-like structure that supports cation exchange. For instance, clinoptilolite, a common natural zeolite, effectively sorbs heavy metals such as lead (Pb²⁺), cadmium (Cd²⁺), and zinc (Zn²⁺) from mine drainage or industrial wastewater by exchanging these toxic ions for less harmful ones like sodium or calcium within its framework, achieving removal efficiencies often exceeding 90% under optimized conditions. This process leverages the zeolite's high cation-exchange capacity, typically ranging from 1 to 3 meq/g, making it a cost-effective sorbent for environmental cleanup without requiring extensive pretreatment.^[22]^[23] Partitioning, another hybrid sorption form, describes the equilibrium distribution of solutes—particularly nonionic organic compounds—between immiscible phases, such as the aqueous phase and soil organic matter in soil-water systems. This process is governed by the solute's hydrophobicity and the sorbent's organic carbon content, often quantified by the soil-water partition coefficient (K_d) or organic carbon-normalized partition coefficient (K_oc), which reflect the compound's preference for partitioning into the solid organic phase over dissolution in water. In contrast to adsorption's surface-specific binding or absorption's uniform bulk uptake, partitioning emphasizes phase partitioning akin to liquid-liquid extraction, frequently combining diffusive incorporation into organic matrices with minor surface effects.^[6]^[24]^[1] In environmental remediation, partitioning sorption is critical for assessing and mitigating the fate of organic contaminants like polycyclic aromatic hydrocarbons (PAHs) or pesticides in contaminated sites, where high K_oc values (e.g., >10⁴ L/kg for hydrophobic compounds) indicate strong retention in soil, reducing leaching to groundwater. Techniques such as partitioning tracer tests exploit this behavior to map subsurface immiscible fluid distributions, informing strategies like in situ bioremediation or pump-and-treat systems by predicting contaminant mobility and bioavailability.^[25]^[26]

Kinetics and Thermodynamics

Sorption Rate

The sorption rate refers to the speed at which a sorbate is taken up by a sorbent, typically governed by sequential processes involving diffusion across the boundary layer and within the sorbent particles, as well as the attachment of the sorbate to active sites.^[27] This rate determines how quickly the system approaches equilibrium, where the net uptake of sorbate ceases.^[28] A widely used model for describing sorption kinetics is the pseudo-first-order equation, originally proposed by Lagergren in 1898 for the adsorption of organic acids onto charcoal.^[28] The model assumes that the rate of sorption is proportional to the number of available sites on the sorbent and is expressed as:

\frac{dq_t}{dt} = k_1 (q_e - q_t)

where q_t is the amount of sorbate sorbed at time t, q_e is the amount sorbed at equilibrium, and k_1 is the pseudo-first-order rate constant (typically in units of time^{-1}).^[28] Integrating this differential equation with the initial condition q_t = 0 at t = 0 yields the linear form \log(q_e - q_t) = \log q_e - \frac{k_1 t}{2.303}, which is fitted to experimental data to determine k_1 and validate the model's applicability.^[29] This model is empirical and simplifies the complex interplay of diffusion and reaction, often providing a good fit for initial sorption stages but less accurately for later phases where equilibrium is neared.^[28] The sorption process generally proceeds through three main stages: external mass transfer, where the sorbate diffuses from the bulk solution through the liquid film to the sorbent surface; intraparticle diffusion, involving transport of the sorbate into the pores of the sorbent; and attachment, where the sorbate binds to internal sites via physical or chemical interactions.^[27] The intraparticle diffusion stage is often analyzed using the Weber-Morris model, which posits that the sorption rate is proportional to the square root of time, expressed as q_t = k_{id} t^{0.5} + C, where k_{id} is the intraparticle diffusion rate constant and C reflects the boundary layer effect. These stages highlight that the overall rate is limited by the slowest step, frequently intraparticle diffusion in porous sorbents.^[30] Sorption rates are commonly measured using batch experiments, in which a fixed mass of sorbent is contacted with a sorbate solution of known initial concentration, and the residual sorbate concentration is monitored over time via techniques such as spectrophotometry or chromatography.^[31] Samples are typically withdrawn at regular intervals, filtered to separate the sorbent, and analyzed to calculate q_t from the mass balance q_t = \frac{(C_0 - C_t) V}{m}, where C_0 and C_t are initial and time-dependent concentrations, V is the solution volume, and m is the sorbent mass.^[31] This approach allows for the construction of kinetic curves and the fitting of models like pseudo-first-order to quantify the rate constant, providing insights into the dynamic behavior leading toward equilibrium.^[31]

Equilibrium and Isotherms

In sorption processes, equilibrium is the state where the rate of sorbate uptake onto the sorbent equals the rate of sorbate release from the sorbent, leading to a constant amount sorbed over time. This dynamic balance governs the distribution of the sorbate between the sorbed phase and the bulk fluid phase, typically achieved under constant temperature and pressure conditions. Sorption isotherms mathematically describe this equilibrium by relating the equilibrium sorbed amount (q_e) to the equilibrium concentration in the fluid phase (C_e). The Freundlich isotherm is an empirical model particularly applicable to sorption on heterogeneous surfaces, where adsorption sites vary in affinity for the sorbate. It is expressed as

q_e = K_f C_e^{1/n}

where K_f represents the Freundlich capacity factor, indicative of the sorbent's sorption potential, and $1/n (typically between 0 and 1) reflects the surface heterogeneity and favorability of sorption, with lower values signaling stronger nonlinearity and higher affinity at low concentrations. Developed empirically in the early 20th century, this model effectively captures multilayer sorption and non-ideal behaviors on irregular surfaces, such as activated carbons or soils, without assuming a finite maximum capacity.^[32]^[33] The Brunauer-Emmett-Teller (BET) isotherm extends monolayer sorption theory to account for multilayer formation, assuming localized adsorption in the first layer followed by non-localized layers with weaker interactions. Proposed in 1938, it builds on the Langmuir model by incorporating successive layers, where the first layer forms with a specific heat of adsorption and subsequent layers resemble liquid condensation. This extension is crucial for systems involving porous materials or high relative pressures, enabling the estimation of monolayer capacity as a foundational metric for total sorption potential.^[34]^[35] Thermodynamic parameters provide insight into the driving forces of sorption at equilibrium. The Gibbs free energy change (\Delta G) is calculated as \Delta G = -RT \ln K, where K is the equilibrium constant obtained from isotherm fitting (e.g., Langmuir or distribution coefficient), R is the gas constant, and T is temperature in Kelvin; negative \Delta G values confirm spontaneity. Enthalpy change (\Delta H) distinguishes physisorption (exothermic, |\Delta H| < 40 kJ/mol) from chemisorption (more exothermic), while entropy change (\Delta S) reflects the disorder reduction upon sorbate immobilization, often negative for adsorption. These parameters, derived via van't Hoff analysis of temperature-dependent isotherms, elucidate process feasibility and energy requirements.^[36]^[37] Isotherm models like Freundlich and BET are instrumental in predicting maximum sorption capacity, which informs sorbent selection and process optimization in environmental and industrial applications. For instance, BET-derived monolayer capacity quantifies surface availability, while Freundlich parameters estimate uptake under varying conditions, allowing extrapolation to real-world scenarios without exhaustive experimentation.^[38]

Factors Influencing Sorption

Environmental Factors

Environmental factors play a crucial role in modulating sorption processes in natural environments, where external conditions like pH, temperature, and ionic strength can significantly alter the interaction between sorbates and sorbents. These variables often fluctuate due to natural events such as precipitation or seasonal changes, influencing the availability and mobility of contaminants in ecosystems like soils and water bodies.^[2] The pH of the surrounding medium profoundly affects sorption by governing the protonation and deprotonation of both sorbate molecules and sorbent surfaces, which in turn modifies surface charge and electrostatic interactions. At lower pH levels, protonation of functional groups on sorbents like metal oxides or organic matter increases positive surface charge, enhancing attraction to anionic sorbates but repelling cations. Conversely, higher pH promotes deprotonation, leading to negative surface charges that favor cationic sorption while hindering anions through repulsion. This pH-dependent speciation shifts sorption affinity, with optimal pH often aligning near the point of zero charge for the sorbent-sorbate pair.^[2]^[39]^[40] Temperature influences sorption kinetics and equilibrium, typically accelerating the rate through enhanced molecular diffusion and reduced solution viscosity, thereby facilitating faster approach to equilibrium. However, for many exothermic sorption processes—common in physical adsorption and ion exchange—rising temperature decreases equilibrium capacity as the endothermic desorption becomes more favorable, weakening sorbate-sorbent bonds. This trade-off is evident in systems like organic pollutant uptake by soils, where moderate warming boosts initial uptake rates but limits total sorbed amounts at higher temperatures.^[41]^[42]^[43] Ionic strength, determined by the concentration of background electrolytes in solution, generally reduces sorption efficiency through competitive effects, where coexisting ions vie for binding sites on the sorbent surface. Higher ionic strength compresses the electrical double layer around charged sorbents, diminishing electrostatic attraction for target sorbates, particularly in outer-sphere complexation mechanisms. Divalent cations like Ca²⁺ often exert stronger competition than monovalent ones such as Na⁺, further suppressing sorption of heavy metals or nutrients in saline environments.^[44]^[45]^[46] In natural settings, these factors interact dynamically; for instance, rainfall-induced acidification of soils lowers pH, which can enhance sorption of cationic pollutants like heavy metals by increasing positive surface charges on mineral surfaces, while mobilizing anions through reduced affinity. Such pH shifts from acid rain events have been observed to alter the retention of agrochemicals in agricultural soils, affecting their leaching potential and environmental persistence. These external influences contrast with intrinsic material properties of sorbents, which are addressed separately.^[47]^[48]^[49]

Material Properties

Material properties of sorbents play a pivotal role in determining their efficiency in sorption processes, as these intrinsic characteristics dictate the availability of binding sites and the ease of molecular interactions. Key attributes include surface area, porosity, functional groups, and particle size, each contributing uniquely to the capacity and selectivity of sorption. These properties are engineered or naturally occurring in materials like activated carbons, polymers, and clays to optimize performance in removing contaminants from aqueous or gaseous phases. Surface area and porosity are fundamental to adsorption-dominated sorption, where higher values directly enhance capacity by providing more sites for adsorbate accumulation. For instance, activated carbons typically exhibit specific surface areas exceeding 1000 m²/g due to their highly developed microporous and mesoporous structures, allowing for extensive monolayer or multilayer adsorption. Porosity, characterized by pore volume and distribution, further influences accessibility; micropores (less than 2 nm) favor selective gas sorption, while mesopores (2-50 nm) facilitate faster diffusion in liquid-phase applications.^[50]^[51] Functional groups on the sorbent surface enable selective binding through chemical interactions such as hydrogen bonding, electrostatic attraction, or complexation, tailoring sorption to specific analytes. Hydroxyl (-OH) and amine (-NH₂) groups, often introduced via surface modification, are particularly effective for heavy metal ions or organic pollutants, as they provide nucleophilic sites that coordinate with electrophilic species. In carbon-based sorbents, oxygen-containing groups like carboxyl and phenolic moieties increase hydrophilicity and ion-exchange potential, thereby improving selectivity over non-polar adsorbates.^[52]^[53] Particle size significantly affects sorption kinetics by altering intraparticle diffusion paths; smaller particles reduce the distance solutes must travel to reach internal sites, thereby accelerating the overall rate. In granular sorbents, reducing size from millimeters to micrometers can decrease diffusion time by orders of magnitude, though it may increase pressure drop in fixed-bed systems. This property is crucial in designing sorbents for rapid-response applications, such as water treatment filters.^[54]^[55] Smectite clays, such as montmorillonite, exemplify sorbents with high cation exchange capacity (CEC) of 80 to 150 meq/100g, while kaolinite has a lower CEC of 3 to 15 meq/100g. The CEC governs ion sorption through electrostatic replacement of interlayer cations in high-CEC clays. This property arises from the negatively charged aluminosilicate layers, enabling efficient binding of divalent metals like Pb²⁺ or Cu²⁺ in soil remediation. The layered structure also contributes to swelling and interlayer accessibility, enhancing overall sorption versatility.^[56]^[57]^[58]

Applications and Modeling

Practical Applications

Sorption plays a pivotal role in water purification processes, particularly through the use of activated carbon to remove organic contaminants. Activated carbon adsorbs a wide range of organic pollutants, such as pesticides, volatile organic compounds, and disinfection byproducts, from drinking water and wastewater, achieving removal efficiencies often exceeding 90% under optimal conditions.^[59] This method is widely employed in granular activated carbon (GAC) filters, where the high surface area of the carbon—typically 500–1500 m²/g—facilitates physical adsorption via van der Waals forces and hydrophobic interactions.^[60] In municipal treatment plants, GAC systems have been integrated to meet regulatory standards for safe potable water, demonstrating long-term efficacy in reducing taste, odor, and health risks associated with organics.^[61] In gas separation applications, zeolites serve as selective sorbents for purifying air and processing natural gas. These crystalline aluminosilicates exhibit molecular sieving and ion-exchange properties, enabling the separation of gases like CO₂, N₂, and CH₄ based on differences in kinetic diameters and affinities.^[62] For air purification, zeolite-based pressure swing adsorption (PSA) units remove water vapor and CO₂ to produce high-purity oxygen or nitrogen, with industrial systems achieving purities above 99%.^[63] In natural gas processing, zeolites such as 5A or 13X are used to dehydrate and sweeten raw gas by adsorbing H₂O and H₂S, improving pipeline quality and reducing corrosion risks.^[64] Soil remediation benefits from sorption barriers, often implemented as permeable reactive barriers (PRBs), to immobilize heavy metals at contaminated sites. These in situ structures, filled with sorbents like zero-valent iron, apatite, or organoclays, intercept groundwater plumes and promote adsorption, precipitation, or complexation of metals such as Pb, Cd, and Cr.^[65] PRBs have demonstrated metal retention rates of 80–95% over years of operation, preventing migration to aquifers while minimizing excavation needs.^[66] For instance, at industrial brownfield sites, such barriers have successfully stabilized arsenic and chromium, restoring environmental safety without generating secondary waste.^[67] In nuclear waste management, sorption is essential for containing radionuclides and preventing their release into the environment. Clay minerals, such as bentonite, are widely used as engineered barriers in deep geological repositories due to their high sorption capacity for actinides and fission products through ion exchange and surface complexation. Natural and modified clays effectively immobilize species like uranium, plutonium, and cesium, with studies showing distribution coefficients (Kd) ranging from 10 to 10^5 mL/g depending on the mineral and conditions.^[68] This application supports long-term safety assessments for repositories, with recent research as of 2023 emphasizing functionalized clays for enhanced performance. In biomedical contexts, sorbent polymers enable advanced drug delivery systems by controlling the loading and release of therapeutics through adsorption-desorption mechanisms. Zwitterionic or pH-responsive polymers, such as polybetaines, sorb drugs like antibiotics onto their charged surfaces, allowing sustained release in targeted physiological environments.^[69] These systems enhance bioavailability and reduce dosing frequency; for example, sorption-based microparticles have shown controlled doxycycline release over 24–48 hours, minimizing systemic toxicity.^[70] Applications include implantable matrices for localized cancer therapy or oral formulations for chronic conditions, where polymer sorption ensures precise pharmacokinetics.^[71]

Theoretical Modeling

Molecular dynamics (MD) simulations offer a powerful atomistic approach to model sorbate-sorbent interactions at the molecular level, capturing dynamic processes such as binding, diffusion, and conformational changes during sorption. By solving Newton's equations of motion for interacting particles, MD reveals detailed mechanisms, including van der Waals forces, hydrogen bonding, and electrostatic interactions that govern selectivity and capacity in porous materials like metal-organic frameworks (MOFs) or activated carbons. For instance, MD has been applied to probe the adsorption of small organic compounds on diverse sorbents, demonstrating how molecular flexibility influences uptake pathways and equilibrium configurations.^[72] In CO2 capture studies, MD simulations quantify sorbate mobility and interaction energies within zeolite pores, aiding the design of high-performance sorbents by predicting diffusion coefficients and binding affinities under varying conditions.^[73] Pore diffusion models adapt Fick's laws to describe intraparticle mass transport in sorbents, accounting for the concentration gradients driving sorbate movement through liquid- or gas-filled pores. These models typically solve the diffusion equation in spherical coordinates for porous particles, expressed as

\frac{\partial q}{\partial t} = D_p \left( \frac{\partial^2 q}{\partial r^2} + \frac{2}{r} \frac{\partial q}{\partial r} \right),

where q is the sorbed concentration, t is time, r is the radial position, and D_p is the pore diffusivity, often coupled with external film resistance for comprehensive kinetics. This framework has been used to simulate phenol sorption on activated carbon, highlighting how pore tortuosity and size distribution limit intraparticle transport rates.^[74] In broader applications, such models integrate surface diffusion to predict overall uptake in hierarchical pore structures, improving accuracy for industrial-scale predictions.^[75] Machine learning (ML) approaches enable predictive modeling of sorption isotherms directly from sorbent characteristics, such as surface area, pore volume, and functional groups, alongside sorbate properties like molecular weight and polarity. Supervised algorithms, including gradient boosted trees and neural networks, trained on large datasets of experimental adsorption data, achieve high accuracy in forecasting equilibrium uptake without time-intensive simulations. For example, ML models have successfully predicted water adsorption isotherms on MOFs for atmospheric water harvesting, incorporating features like metal node type and ligand chemistry to estimate performance metrics.^[76] Similarly, sorbate-sorbent-based ML frameworks have been developed to anticipate organic pollutant sorption on microplastics, using descriptors from quantum chemical calculations to capture nonlinear dependencies.^[77] Basic isotherms like Freundlich and Langmuir serve as foundational components when integrated into larger environmental models to simulate sorption-driven contaminant fate and transport. In groundwater systems, these isotherms are incorporated into numerical solvers such as MODFLOW's Mobile Solute Transport package, which handles nonlinear sorption to predict plume migration and retardation under advective-dispersive conditions.^[78] This integration allows for scenario analysis of pollutant dispersion, where Langmuir parameters model saturation effects in aquifer remediation designs.^[79]

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Predicting aqueous sorption of organic pollutants on microplastics ...
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[PDF] Documentation for the MODFLOW 6 Groundwater Transport Model
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Learning Groundwater Contaminant Diffusion‐Sorption Processes ...
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