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Capillary condensation

Capillary condensation is the process by which a vapor condenses into liquid within the narrow pores or crevices of a porous solid at partial pressures below the saturation vapor pressure of the bulk liquid, driven by the curvature of the liquid-vapor meniscus in confined geometries.^[1] This phenomenon occurs preferentially in wetting systems where the contact angle is less than 90°, enabling the formation of stable liquid bridges or menisci that lower the equilibrium vapor pressure required for phase transition.^[2] The underlying mechanism is thermodynamically described by the Kelvin equation, which relates the shift in vapor pressure to the pore radius: \ln(p/p_s) = -\frac{2\gamma V_m \cos \theta}{rRT}, where p is the equilibrium vapor pressure over the curved interface, p_s is the saturation vapor pressure, \gamma is the surface tension, V_m is the molar volume of the liquid, \theta is the contact angle, r is the pore radius, R is the gas constant, and T is the temperature.^[3] Building upon foundational contributions from Thomas Young (1805) on surface tension and Pierre-Simon Laplace (1806) on capillary forces, William Thomson (Lord Kelvin) proposed it in linear form in 1871 to explain vapor equilibrium at curved surfaces, which later developed into the exponential form used today, with practical applications to porous media emerging in the early 20th century, such as Richard Zsigmondy's 1911 studies on silica gels.^[4] A hallmark of capillary condensation is the adsorption-desorption hysteresis observed in isotherms, arising from the metastable states during filling and emptying of pores, which persists across diverse mesoporous materials like metal-organic frameworks and silicas.^[5] Capillary condensation plays a pivotal role in numerous scientific and engineering fields, influencing processes from gas adsorption in catalysis and chromatography to water management in soils and textiles.^[1] In energy applications, it affects shale gas recovery by altering permeability in nanopores and enhances ion transport in supercapacitors.^[6] Emerging uses include atmospheric water harvesting via tailored porous structures and the controlled growth of nanomaterials, such as perovskite nanowires, underscoring its versatility in addressing challenges in sustainability and nanotechnology.^[7]^[8]

Fundamentals

Definition and Mechanism

Capillary condensation is the phenomenon in which a vapor phase transitions to a liquid phase within the narrow confines of pores or capillaries in a porous material, occurring at relative pressures (P_v/P_sat) less than unity, where P_v is the vapor pressure and P_sat is the saturation pressure of the bulk liquid.^[2] This process is fundamentally driven by surface tension at the liquid-vapor interface and the associated Laplace pressure, which arises from the curvature of the interface in confined geometries.^[2] The mechanism initiates with the adsorption of vapor molecules onto the pore walls, forming an initial thin liquid film due to attractive intermolecular forces, primarily van der Waals interactions.^[2] As the relative vapor pressure rises, the film thickens until it reaches an instability point, where nucleation of discrete liquid clusters occurs, often described as a morphological transition from a uniform wetting layer to localized liquid phases.^[9] These clusters develop into curved menisci, whose concave shape in wetting pores generates a capillary pressure that reduces the equilibrium vapor pressure needed for condensation, thereby promoting further liquid formation.^[2] Key physical drivers include surface tension (γ), which dictates the meniscus curvature and pressure differential, and confinement effects that enhance intermolecular forces; in hydrophilic pores, strong wetting favors film stability and rapid filling, whereas hydrophobic pores exhibit delayed or altered condensation dynamics due to reduced adhesion.^[2] The process unfolds step-by-step: vapor adsorption first establishes menisci at pore walls or entrances, creating localized regions of high curvature that draw additional vapor inward via capillary action.^[2] This leads to the growth of liquid bridges connecting opposite pore surfaces, stabilized by the balance of surface tension and disjoining pressures from intermolecular forces.^[9] As pressure increases, these bridges expand and coalesce, ultimately filling the pore volume with liquid condensate.^[2] The quantitative foundation for this pressure reduction is captured by the Kelvin equation, relating vapor pressure to interface curvature.^[10] This effect was first theoretically described in the 19th century by William Thomson (later Lord Kelvin) in the context of atmospheric vapor equilibrium at curved liquid surfaces.^[10]

Thermodynamic Principles

Capillary condensation is governed by thermodynamic principles that minimize the Gibbs free energy of the system, where the phase transition from vapor to liquid in confined geometries reduces the overall free energy through the formation of curved menisci. This process balances the bulk vapor-liquid free energy gain against surface energy contributions from the liquid-vapor interface and the solid-liquid/solid-vapor interactions. In porous media, the curved interface lowers the system's total Gibbs free energy compared to the non-condensed state when the chemical potential difference favors condensation, particularly for wetting surfaces where the spreading coefficient is positive.^[11] The Laplace pressure plays a central role, creating a pressure difference across the curved interface given by \Delta P = \frac{2\gamma \cos\theta}{r}, where \gamma is the liquid-vapor surface tension, \theta is the contact angle, and r is the pore radius. This pressure difference induces supersaturation within the pores, as the liquid phase experiences a lower pressure than the surrounding vapor, stabilizing the condensed phase at vapor pressures below the bulk saturation value. For hydrophilic pores with \theta < 90^\circ, this effect promotes condensation by enhancing the mechanical stability of the liquid bridge.^[11] At equilibrium, the chemical potential of the vapor must equal that of the condensed liquid phase within the confinement, ensuring no net driving force for phase change. This equality shifts the coexistence condition due to the curvature-induced modifications to the interfacial free energy, distinguishing confined systems from bulk behavior. In small pores, nucleation faces significant energy barriers, as the activation energy for forming a critical liquid nucleus scales with the pore size and is modulated by wetting properties; for partial wetting (\theta > 0), higher barriers delay condensation, leading to metastable states.^[11] Unlike bulk condensation, which occurs at the saturation vapor pressure, confinement alters the phase diagram by the Laplace pressure and surface terms, enabling condensation at lower relative pressures and introducing hysteresis due to these energy barriers. This thermodynamic shift is particularly pronounced in nanopores, where surface effects dominate over bulk contributions.^[12]

Theoretical Framework

Kelvin Equation

The Kelvin equation is derived from the thermodynamic requirement that the chemical potentials of the vapor and liquid phases are equal at equilibrium during capillary condensation. This builds on the principle of chemical potential balance between coexisting phases.^[13] For the vapor phase, assumed to behave as an ideal gas, the chemical potential is expressed as
\mu_v = \mu^0(T) + RT \ln\left(\frac{P_v}{P_\mathrm{sat}}\right),
where \mu^0(T) is the standard chemical potential at temperature T, R is the universal gas constant, P_v is the equilibrium vapor pressure in the pore, and P_\mathrm{sat} is the saturation vapor pressure over a flat liquid-vapor interface.^[13] For the liquid phase in the bulk at saturation, \mu_l^\mathrm{bulk} = \mu^0(T). In a capillary pore, the liquid experiences a pressure reduction due to the Laplace pressure across the curved meniscus at the liquid-vapor interface, given by \Delta P = \frac{2\gamma \cos\theta}{r} for a cylindrical geometry, where \gamma is the liquid-vapor surface tension, \theta is the contact angle, and r is the pore radius. The pressure in the liquid is thus P_l = P_v - \Delta P. Assuming the liquid is incompressible with constant molar volume V_m, the chemical potential of the confined liquid becomes
\mu_l = \mu_l^\mathrm{bulk} + V_m (P_l - P_\mathrm{sat}) = \mu^0(T) + V_m (P_v - \Delta P - P_\mathrm{sat}). ^[13] Setting \mu_v = \mu_l at equilibrium yields
RT \ln\left(\frac{P_v}{P_\mathrm{sat}}\right) = V_m (P_v - P_\mathrm{sat} - \Delta P).
Given that P_v \approx P_\mathrm{sat} for modest curvatures (small \Delta P), the term V_m (P_v - P_\mathrm{sat}) is second-order small and can be neglected, resulting in the Kelvin equation:
\ln\left(\frac{P_v}{P_\mathrm{sat}}\right) = -\frac{2 \gamma V_m \cos\theta}{r RT}.
This form indicates that capillary condensation occurs at a relative pressure P_v / P_\mathrm{sat} < 1, lower than the bulk saturation pressure, due to the concave curvature stabilizing the liquid phase in the pore. The equation originates from the work of William Thomson (Lord Kelvin) in 1871, who first related vapor equilibrium to curved surfaces.^[14]^[13] The derivation relies on several key assumptions: ideal cylindrical pores with smooth walls; partial to complete wetting (\theta \leq 90^\circ); incompressible liquid with bulk thermodynamic properties; ideal gas behavior for the vapor; and negligible gas adsorption or multilayer film effects on the pore walls. For complete wetting (\theta = 0), the equation simplifies to
\ln\left(\frac{P_v}{P_\mathrm{sat}}\right) = -\frac{2 \gamma V_m}{r RT}. ^[13]^[15] In spherical pore geometry, the meniscus approximates a hemisphere, leading to a mean curvature of $2/r and \Delta P = 2\gamma / r (assuming \theta = 0), which yields the same simplified form as the cylindrical case with complete wetting:
\ln\left(\frac{P_v}{P_\mathrm{sat}}\right) = -\frac{2 \gamma V_m}{r RT}. ^[15] The Kelvin equation holds reliably for mesopores, defined as having widths between 2 nm and 50 nm, where continuum thermodynamics applies and surface curvature dominates without significant molecular-scale deviations. It breaks down in micropores (widths < 2 nm), where enhanced wall-fluid interactions, molecular discreteness, and non-bulk liquid densities invalidate the assumptions, necessitating approaches like density functional theory.^[16]^[13]

Relative Pressure Dependence

The Kelvin equation predicts that the relative vapor pressure P_v / P_{\text{sat}} at which capillary condensation occurs in a pore of radius r follows an exponential dependence given by \ln(P_v / P_{\text{sat}}) \propto -1/r, such that smaller pores exhibit condensation at significantly lower relative pressures than the bulk saturation value of P_v / P_{\text{sat}} = 1. This shift arises from the enhanced curvature of the liquid-vapor meniscus in confined spaces, which lowers the equilibrium vapor pressure required for phase transition.^[17] Graphically, this dependence is evident in type IV adsorption isotherms typical of mesoporous materials, where the capillary condensation manifests as a steep uptake step at P_v / P_{\text{sat}} < 1, with the position of the step inversely related to the pore radius; for instance, narrower pores show the transition at lower relative pressures, while bulk-like behavior approaches P_v / P_{\text{sat}} = 1 for very large pores. These isotherms allow visual comparison of confined versus bulk condensation, highlighting how pore confinement stabilizes the liquid phase at sub-saturation conditions. Temperature influences the relative pressure shift through the RT term in the Kelvin equation, where higher temperatures decrease the magnitude of \ln(P_v / P_{\text{sat}}), bringing the condensation closer to bulk saturation due to the dominance of thermal energy over interfacial effects. For quantitative examples, in a 10 nm radius cylindrical pore with water at 25°C (assuming complete wetting), P_v / P_{\text{sat}} \approx 0.90; similarly, for nitrogen at 77 K in the same pore size, P_v / P_{\text{sat}} \approx 0.90, and for argon at 87 K, values are comparably close to 0.90 under standard conditions.^[17] This pressure-radius relationship underpins the predictive modeling of adsorption isotherms in porous materials, forming the basis for techniques like the Barrett-Joyner-Halenda (BJH) method to derive pore size distributions from experimental data.

Curvature and Contact Angle Effects

In capillary condensation within cylindrical pores, the curvature of the liquid-vapor meniscus plays a critical role in determining the condensation pressure, with the effective pore radius accounting for the pre-existing adsorbate layer thickness t. Specifically, the effective radius is given by r_{\text{eff}} = r - t, where r is the nominal pore radius, allowing the Kelvin equation to predict the onset of condensation more accurately by considering the reduced space available for the meniscus formation.^[18] During the filling process, the meniscus transitions from a cylindrical shape, which dominates at lower relative pressures and features a single principal radius of curvature, to a spherical shape as the pore fills, reflecting the evolving geometry and increasing the mean curvature, which shifts the equilibrium vapor pressure.^[19] The contact angle \theta, which quantifies surface wettability, modulates the pressure required for condensation through its influence on the meniscus geometry. For hydrophilic surfaces where \theta < 90^\circ, the cosine term \cos \theta > 0 enhances the capillary forces, promoting condensation at lower relative pressures compared to flat interfaces. In contrast, hydrophobic surfaces with \theta > 90^\circ yield \cos \theta < 0, which inhibits condensation by requiring higher pressures or preventing it altogether in some cases.^[20] The standard Kelvin equation is modified to incorporate non-zero contact angles, yielding the form for a spherical meniscus:

\ln\left(\frac{P_v}{P_{\text{sat}}}\right) = -\frac{2 \gamma V_m \cos \theta}{r R T}

where \gamma is the surface tension, V_m is the molar volume of the liquid, R is the gas constant, and T is the temperature; for cylindrical menisci during initial filling, the factor of 2 is replaced by 1 to reflect the geometry:

\ln\left(\frac{P_v}{P_{\text{sat}}}\right) = -\frac{ \gamma V_m \cos \theta}{r R T}.

In real materials, \theta varies significantly: hydrophilic silica typically exhibits \theta \approx 0^\circ, facilitating condensation in pores as small as 2-5 nm at relative pressures near 0.4, while hydrophobic carbon surfaces show \theta \approx 80^\circ - 90^\circ, shifting thresholds to higher pressures and altering pore-filling dynamics.^[21]^[22] Experimentally, the contact angle is often determined by fitting adsorption isotherms to modified Kelvin models, where deviations in pore-filling thresholds from ideal predictions reveal \theta values; for instance, nitrogen adsorption on silica gels at 77 K allows extraction of \theta by comparing observed condensation steps to calculated pressures, confirming hydrophilic behavior that lowers filling thresholds by up to 20% relative to assumed \theta = 0^\circ. This approach validates how \theta directly impacts the relative humidity or pressure at which pores fill, with higher \theta delaying condensation in controlled mesoporous samples.^[20] In boundary cases, complete wetting (\theta = 0^\circ) leads to unbounded multilayer adsorption prior to bulk condensation, forming thick precursor films that effectively reduce the pore radius and enhance capillary forces, as seen in systems like water on clean silica where film thicknesses exceed 1 nm at sub-saturation pressures. Partial wetting ($0^\circ < \theta < 90^\circ) limits multilayer growth to finite thicknesses, resulting in sharper condensation transitions without extensive pre-wetting layers, which has implications for adsorption capacity in porous materials by altering the balance between surface coverage and capillary filling.^[23]

Real-System Variations

Pore Geometry Influences

In real porous materials, deviations from ideal cylindrical pore geometries significantly alter the dynamics of capillary condensation by influencing the shape and stability of the liquid-vapor meniscus. For instance, ink-bottle pores, characterized by a wide body connected to a narrow neck, lead to condensation initiating in the neck at a higher relative pressure corresponding to the smaller neck radius, followed by rapid filling of the larger body once the meniscus advances, resulting in distinct steps in adsorption isotherms.^[24] Slit-like pores, in contrast to cylindrical ones, promote condensation at lower relative pressures due to the attraction between opposing walls forming a flatter meniscus, whereas spherical or more closed geometries require higher pressures for meniscus formation because of increased curvature constraints. These shape-dependent meniscus configurations shift the equilibrium condensation pressure away from predictions based on uniform cylinders, as demonstrated in simulations of linear mesopores where slit-shaped pores exhibit earlier filling than cylindrical or spherical counterparts. Cylindrical pores with conical variations or tapered geometries further modify the filling sequence, with condensation progressing along the pore axis in a pressure-dependent manner that creates multiple plateaus in isotherms, reflecting sequential meniscus advancements from wider to narrower sections.^[25] In such non-uniform structures, the local radius variation along the pore axis, denoted as r(z), dictates the progression of the liquid bridge, leading to delayed or accelerated filling compared to straight cylindrical pores of equivalent average size.^[24] Experimental isotherms from materials like MCM-41 (cylindrical) versus disordered silicas (conical or irregular) reveal these effects through inflection points indicating geometry-driven transitions. Pre-adsorbed multilayer films on pore walls reduce the effective pore size, shifting the condensation threshold to higher relative pressures by effectively decreasing the radius available for meniscus formation; this multilayer thickness, often derived from t-curves, can be 0.3–0.5 nm for nitrogen at 77 K, altering the apparent geometry.^[24] Modeling these influences typically involves modifications to the Kelvin equation to account for varying r(z), such as integrating local curvature terms along the pore length, though classical approaches like BJH assume cylindrical symmetry and thus underestimate sizes in irregular pores by up to 20–30%. Numerical simulations using non-local density functional theory (NLDFT) or grand canonical Monte Carlo (GCMC) provide more accurate predictions for complex shapes, capturing multilayer effects and meniscus evolution in slit-like or ink-bottle geometries without relying on idealized assumptions. Experimental characterization via methods like BET surface area analysis or BJH pore size distribution extracts geometry insights from isotherm steps; for example, sharp plateaus in nitrogen adsorption at 77 K on ordered mesoporous silicas indicate uniform cylindrical pores, while broader or multi-step features signal irregular shapes like slits or bottles, validated against TEM imaging. These techniques, refined in seminal works, highlight how pore geometry controls condensation accessibility and thus material uptake capacities.

Hysteresis Behavior

Hysteresis in capillary condensation manifests as a discrepancy between the adsorption (condensation) and desorption (evaporation) branches of the isotherm, where liquid condenses in pores at a higher relative pressure than the pressure at which it evaporates. This behavior arises from the advancing meniscus during adsorption, which forms at higher pressures due to favorable wetting conditions, versus the receding meniscus during desorption, which persists at lower pressures owing to metastable configurations.^[26] The primary causes of this hysteresis include pore blocking in constricted geometries, such as ink-bottle pores with narrow necks, where adsorbed liquid in larger cavities blocks evaporation from the wide body until the meniscus in the neck reaches a critical curvature.^[27] Metastable menisci also contribute, as the liquid-vapor interface remains stable in non-equilibrium states during desorption, delaying evaporation.^[28] Additionally, cavitation in wider pores during desorption triggers sudden vapor bubble formation when the tensile stress in the metastable liquid exceeds its mechanical stability, leading to abrupt emptying independent of pore connectivity.^[29] Hysteresis loops exhibit distinct characteristics depending on the underlying mechanism: Type A loops, associated with pore blocking in neck-dominated structures, show a steep desorption branch at lower pressures with gradual adsorption, often observed in materials with uniform narrow openings.^[28] In contrast, Type B loops arise from cavitation in expansive pore bodies, featuring parallel adsorption and desorption branches separated by a wide pressure gap, particularly in disconnected or wide-pore systems.^[28] The width of these loops is influenced by the pore size distribution, with broader distributions yielding wider hysteresis due to sequential filling and emptying across varied pore dimensions. Theoretical modeling of hysteresis often employs the independent domain theory, originally developed by Everett for parallel, non-interacting pores, which treats the porous material as an assembly of independent domains that fill or empty at specific threshold pressures, enabling prediction of loop shapes and reversibility. This framework extends to scanning curves in partial adsorption-desorption cycles, where intermediate paths trace reversible filling within domains between primary loop boundaries, providing insights into metastable state distributions.^[26] Experimentally, hysteresis is measured through nitrogen physisorption isotherms at 77 K, where the adsorption branch reflects equilibrium capillary condensation via the Kelvin equation, while the desorption branch reveals irreversible processes like blocking or cavitation, allowing differentiation between reversible multilayer adsorption and irreversible pore filling. These observations have implications for assessing pore network accessibility, as narrow hysteresis indicates reversible filling suitable for reversible storage applications, whereas wide loops signal irreversible trapping that affects material performance in catalysis or filtration.^[30]

Nanoscale Corrections

At nanoscale dimensions, particularly for pore radii below 2 nm, the classical Kelvin model for capillary condensation breaks down due to the overlapping molecular potentials from opposing pore walls, which enhance adsorption beyond continuum predictions. This overlap creates a strong potential field gradient that favors fluid accumulation even at very low relative pressures, shifting the process from meniscus-driven condensation to cooperative filling. Additionally, disjoining pressure arises from wall-fluid interactions, such as van der Waals forces, which stabilize thin adsorbed films and modify the effective pressure difference across the interface, leading to deviations from macroscopic capillarity assumptions.^[15]^[31] To address these limitations, corrections incorporate the curvature dependence of surface tension, parameterized by the Tolman length, which quantifies the shift between the equimolar surface and the surface of tension. For water, the Tolman length is approximately -0.07 nm (-0.71 Å), enabling the Kelvin equation to remain valid down to curvatures of about 0.5 nm when this dependence is included, as demonstrated in molecular simulations of condensation in carbon nanotubes.^[32] Furthermore, the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory integrates van der Waals attractions and electrostatic repulsions to model disjoining pressures in thin films, providing a framework for predicting enhanced adsorption in small pores where these long-range forces dominate over Laplace pressure.^[33] Advanced models like density functional theory (DFT) offer a molecular-level description for micropores, accounting for inhomogeneous density profiles and fluid-solid interactions that the Kelvin equation overlooks. Non-local DFT kernels accurately predict adsorption isotherms and phase transitions in pores as small as 1-2 nm, bridging simulations and macroscopic thermodynamics by incorporating disjoining effects and potential overlaps. Potential theory complements this by quantifying enhanced adsorption through wall potential gradients, revealing filling mechanisms in confined geometries.^[34] In materials like zeolites and metal-organic frameworks (MOFs) with sub-2 nm pores, these corrections manifest as condensation—or more precisely, volume filling—at relative pressures (P_v/P_sat) approaching zero, driven by overlapping field gradients that lower the free energy barrier for fluid ingress. For instance, in hydrophilic zeolites, initial adsorption occurs at low humidities due to strong site interactions, transitioning to pore filling without a distinct capillary step. Similarly, certain MOFs exhibit steep uptake at P_v/P_sat < 0.1, enabling applications in water harvesting.^[35]^[36] Recent advancements as of 2025 include refined Kelvin theory models that describe capillary condensation features in sub-nanometer pores, accounting for extreme confinement and bridging classical thermodynamics with molecular simulations.^[37] A key limitation of these nanoscale corrections is the transition from the capillary condensation regime to a volume filling regime in ultrasmall pores (D < 4σ, where σ is the molecular diameter), where interfaces blur, and the process becomes a continuous, second-order density increase rather than a first-order phase jump. This crossover, governed by a pore-specific critical temperature, eliminates hysteresis and aligns with enhanced potential effects, as validated by simulations and zeolite experiments.^[31]

Associated Phenomena

Capillary Adhesion

Capillary adhesion refers to the attractive force between two solid surfaces mediated by liquid bridges formed via capillary condensation. This phenomenon occurs when vapor condenses in the narrow gap between surfaces, creating a meniscus with a curved interface that generates a pressure difference according to the Laplace equation. The resulting Laplace pressure deficit within the liquid bridge pulls the surfaces together, while the surface tension acts along the three-phase contact line to enhance the attraction. For small liquid bridges where the bridge height is comparable to the neck radius, the adhesion force can be approximated as F \approx 2\pi r \gamma \cos\theta, with r the radius of the bridge neck, \gamma the liquid-vapor surface tension, and \theta the equilibrium contact angle.^[38] Liquid bridges form through capillary condensation in gaps between particles or flat surfaces, particularly for hydrophilic materials exposed to ambient vapors like water. This condensation initiates at relative humidities around 50-70% RH for rough hydrophilic surfaces with small gaps, depending on gap size and surface wettability, as the vapor pressure required is reduced in confined geometries per the Kelvin equation.^[39] Once formed, the bridge volume and shape evolve with humidity, leading to stronger adhesion as the meniscus curvature adjusts.^[40] The magnitude of the capillary adhesion force typically ranges from 10 to 100 nN for micron-scale bridges, scaling with bridge volume, separation distance (decreasing sharply beyond a few micrometers), and relative humidity. At fixed separation, higher humidity increases the effective bridge radius and reduces the mean curvature, amplifying the force until saturation near 100% RH. This force dominates van der Waals interactions in humid conditions and can exceed 10 times the dry adhesion for rough or deformable surfaces.^[40] In static conditions, the adhesion force represents the equilibrium pull-off force required to break the bridge at constant humidity, often exhibiting hysteresis due to pinning at the contact line. Dynamically, during surface separation, the force evolves through stages of neck elongation, instability, and rupture, with the maximum force occurring just before catastrophic breakup; rupture dynamics can generate transient peaks up to twice the static value.^[41] Even in ostensibly dry environments with trace moisture (e.g., <20% RH), capillary adhesion can cause unexpected sticking between closely apposed surfaces, as minute amounts of condensed vapor form nanoscale bridges that persist due to kinetic barriers to evaporation. This effect is particularly relevant in microscale devices and powder handling, where it contributes to stiction or agglomeration despite low ambient humidity.^[40]

Bridging Mechanisms

Capillary condensation facilitates the formation of liquid bridges between closely spaced particles or surface features, where vapor condenses into menisci that connect adjacent elements, often resulting in necklace-like chains of these menisci along particle alignments. These bridges arise when the relative humidity reaches levels sufficient to promote condensation in the narrow gaps, typically on the order of micrometers for micron-sized particles, leading to attractive forces that promote aggregation. The process is driven by the minimization of surface free energy, with the liquid phase wetting the particle surfaces and forming concave menisci that exert both surface tension and Laplace pressure components. The strength and stability of bridges also depend on the contact angle, with better wetting (lower θ) promoting stronger forces. The stability of such capillary bridges is highly sensitive to the separation distance between particles, with stable configurations generally limited to critical separations of approximately 1-5 times the particle radius r, beyond which the bridge ruptures due to insufficient capillary pressure to sustain the meniscus shape. For separations closer than this threshold, the bridges exhibit robust connectivity, enabling the linking of multiple particles into extended structures. In multi-particle systems, such as loosely packed powders, these bridges induce coalescence by forming networks that enhance overall material cohesion, where the effective strength at each contact point scales with the number of individual bridges formed per particle pair, often increasing nonlinearly with liquid content.^[42]^[43] Environmental factors, particularly relative humidity, dictate the onset of bridging, with thresholds typically at high relative humidities of 70-95% RH or higher for micron-sized particles with smooth surfaces, and lower (down to ~50%) for rough particles or smaller sizes due to nanoscale crevices, depending on particle size, surface chemistry, and geometry; below these thresholds, insufficient vapor pressure prevents condensation.^[44] Upon decreasing humidity, these liquid bridges evaporate; however, in the presence of dissolved impurities, drying can leave solid residues that form permanent adhesive bonds persisting even at low humidity. Analytical models approximate the capillary force F between two spherical particles as F = 2\pi \gamma r^2 / d, where \gamma is the liquid-vapor surface tension and d is the center-to-center separation, capturing the dominant hydrostatic contribution for small bridges; more detailed derivations incorporate contact angle effects for precision.^[44]^[38] Numerical simulations of random particle packings reveal that bridge distribution follows the local void geometry, with forces varying spatially and leading to anisotropic stress fields that stabilize loose assemblies against shear. In granular materials, these mechanisms underpin cohesion under ambient humidity, enabling phenomena like self-supporting arches in sands or enhanced flow resistance in powders without full saturation.^[45]

Practical Applications

Pore Size Analysis

Capillary condensation serves as a foundational principle in pore size analysis, enabling the determination of pore size distributions in porous materials through the analysis of gas adsorption-desorption isotherms. The process relies on the Kelvin equation, which predicts the relative pressure at which vapor condenses in pores of varying radii due to enhanced capillary forces. This approach traces its origins to William Thomson (Lord Kelvin)'s 1871 derivation of the relationship between vapor pressure and the curvature of the liquid-vapor interface in capillaries. Over time, advancements have refined these concepts into standardized methods for characterizing porous structures, culminating in the International Union of Pure and Applied Chemistry (IUPAC) recommendations of 1985, which classify pores as micropores (width < 2 nm), mesopores (2–50 nm), and macropores (> 50 nm) based on adsorption behavior.^[46]^[47] The seminal Barrett-Joyner-Halenda (BJH) method, developed in 1951, applies the Kelvin equation to the desorption branch of nitrogen adsorption isotherms to calculate pore size distributions, assuming cylindrical pores and stepwise multilayer adsorption followed by capillary evaporation. In this model, the volume of gas desorbed at each relative pressure corresponds to the emptying of pores with a specific radius, derived from the Kelvin radius adjusted for the thickness of the adsorbed multilayer film. The procedure begins with obtaining the adsorption-desorption isotherm, typically using nitrogen at 77 K, followed by correction for multilayer thickness using statistical thickness curves such as the de Boer t-curve or those derived from Brunauer-Emmett-Teller (BET) theory, which models multilayer adsorption on non-porous reference surfaces. The t-plot method further refines this by plotting adsorbed volume against statistical thickness to isolate micropore filling from multilayer adsorption on external surfaces and mesopore walls.^[48]^[48]^[49]^[50] The BJH method excels in analyzing mesopores, providing reliable distributions for materials with pore sizes between 2 and 50 nm, but it has notable limitations for micropores, where enhanced adsorbate-adsorbent interactions and pore filling mechanisms deviate from classical capillary condensation assumptions, often leading to underestimation of pore volumes. For microporous materials, more advanced techniques like density functional theory (DFT) are preferred, as they account for fluid-wall potentials and non-uniform density profiles without relying on macroscopic Kelvin thermodynamics. Hysteresis observed in isotherms, arising from differences in adsorption and desorption mechanisms, is typically addressed by focusing on the desorption branch in BJH analysis to approximate equilibrium conditions.^[48]^[51]^[51] In practical applications, pore size analysis via capillary condensation is essential for characterizing catalysts and adsorbents, where mesopore distributions influence reactant diffusion and selectivity. For instance, BJH analysis of activated carbon reveals broad mesopore networks that enhance gas storage capacity for applications like CO2 capture, while in silica gels, it identifies uniform mesopore sizes around 2–10 nm that optimize moisture adsorption in drying processes. These insights guide material design, ensuring tailored porosity for industrial performance.^[52]^[53]

Sintering and Materials Processing

In the sintering process, capillary condensation plays a crucial role by enabling the formation of liquid bridges from ambient vapor at the contact points or grain boundaries of powder particles, particularly under low-temperature conditions. This phenomenon occurs when the relative humidity exceeds the threshold dictated by the Kelvin equation, causing water vapor to condense into menisci between particles, thereby generating attractive capillary forces that draw grains together and initiate neck formation. These forces promote surface diffusion and material transport, facilitating initial bonding without requiring high temperatures, as demonstrated in studies of pendular liquid bridges between spherical particles.^[54] In ceramics such as yttria-stabilized zirconia (YSZ) and barium titanate (BaTiO₃), this mechanism enhances particle rearrangement and densification at temperatures below 200°C, often in conjunction with applied pressure.^[55] The sintering process unfolds in distinct stages influenced by these liquid bridges. During the initial stage, capillary forces from the condensed bridges act to compact particles and smooth asperities, increasing green body strength and enabling handling prior to full densification; this is particularly evident in humid environments where bridge volumes can reach up to 10% of particle separation distances for submicron grains. In the intermediate stage, the capillary pressure—governed by the Young-Laplace equation, ΔP = 2σ/r (where σ is surface tension and r is meniscus radius)—drives dissolution-precipitation mechanisms that fill interparticle voids, reducing porosity and promoting grain boundary diffusion for neck growth. Models of bridge geometry, such as the toroidal approximation, link bridge volume V to shrinkage rates, showing that smaller particle radii (e.g., 25 nm) yield higher forces (up to several nN) at 85% relative humidity and 85°C, accelerating densification by 10-20% compared to dry conditions.^[54]^[55] Examples include cold sintering of barium titanate, where controlled vapor exposure achieves 92-95% relative density.^[55] This approach is applied under conditions of ambient or controlled humidity (e.g., 50-85% relative humidity) and low pressures (350-520 MPa), making it suitable for energy-efficient processing of ceramics and metals like silver nanoparticles, where vapor-assisted bridges lower activation energies for mass transport. Benefits include enhanced green strength for improved handling of fragile compacts and reduced thermal stresses, enabling compatibility with temperature-sensitive additives or substrates in materials like porous ceramics. However, issues arise from uneven evaporation of the condensed liquid, potentially leading to localized stresses and cracking during subsequent drying or heating phases, which can be mitigated by gradual humidity reduction. Quantitative assessments indicate that capillary pressure contributions can boost shrinkage rates by linking bridge volumes directly to porosity reduction, with models predicting up to 15% volume shrinkage in early stages for humid-assisted processes.^[54]^[55]

Nanotechnology Devices

Capillary condensation poses significant challenges in the fabrication and operation of microelectromechanical systems (MEMS) and similar nanoscale devices, primarily through the formation of liquid bridges that cause stiction, where suspended structures adhere irreversibly to underlying surfaces. During wet etching processes, residual rinse liquids evaporate and form menisci in narrow gaps, generating attractive forces that exceed the restoring forces of microstructures, leading to collapse and yield losses. In humid environments, vapor-induced capillary bridges during operation can similarly pin moving parts, compromising functionality. Early surface-micromachined silicon devices often experienced failure rates exceeding 50% due to this in-process stiction, highlighting its role as a dominant yield limiter in initial polysilicon-based designs. To mitigate these issues, dry release techniques such as supercritical CO2 drying have been widely adopted, as the supercritical fluid eliminates surface tension during the transition from liquid to gas, preventing meniscus formation and enabling high-yield release of delicate structures like cantilever beams and gears. Additionally, applying hydrophobic coatings with contact angles greater than 90°, such as self-assembled monolayers of perfluorinated silanes on silicon surfaces, reduces wettability and minimizes capillary bridge strength, thereby enhancing device reliability in ambient humidity. These strategies have improved fabrication yields to over 90% in modern MEMS processes. Beyond challenges, capillary condensation enables beneficial applications in nanotechnology devices, particularly for humidity-responsive actuation in sensors. In these designs, controlled formation and rupture of water bridges between electrodes or compliant elements alter capacitance or mechanical deflection in response to relative humidity changes, providing sensitive detection without external power. For instance, carbon nanotube-enhanced capacitive sensors exploit enhanced capillary condensation in nanopores to achieve rapid response times under 1 second for humidity variations from 30% to 90% RH. In energy storage, capillary condensation in nanoporous electrodes enhances ion transport in supercapacitors by facilitating electrolyte distribution and reducing resistance. In resource extraction, it influences shale gas recovery by altering permeability in nanopores through phase transitions that affect fluid flow. Emerging applications as of 2025 include atmospheric water harvesting using amphiphilic nanoporous films that condense undersaturated vapor for sustainable water collection, and controlled growth of perovskite nanowire arrays for photodetectors via capillary-driven crystallization on flexible substrates.^[56]^[8] Representative examples include silicon microstructures in accelerometers and RF switches, where sub-micron gaps amplify capillary forces relative to elastic restoring forces, following inverse scaling laws that make adhesion dominant at nanoscale separations—often orders of magnitude stronger than van der Waals interactions alone. These forces, akin to those in capillary adhesion phenomena, can reach several micro-Newtons in 100 nm gaps at 50% RH, underscoring the need for precise gap engineering. In the 2020s, advances in integrating vapor barriers, such as atomic layer deposition (ALD) of alumina or parylene thin films, have further bolstered reliability by achieving water vapor transmission rates below 10^{-5} g/m²/day, effectively shielding internal structures from ambient moisture and reducing humidity-induced stiction in implantable and harsh-environment MEMS devices like pressure sensors.

Surface Characterization Techniques

Atomic force microscopy (AFM) leverages capillary condensation to probe nanoscale surface properties by measuring tip-sample interactions under controlled humidity conditions. In AFM setups, capillary forces arise from water menisci forming between the sharp tip (typically with a radius of 10-50 nm) and the sample surface due to ambient vapor condensation, dominating adhesion at relative humidities above 20-30%. These forces are quantified through cantilever deflection, where the tip approaches the surface, and the onset of condensation is detected as a sudden increase in attractive force when the tip-sample separation reaches a critical distance of approximately 9-10 nm.^[57] Force-distance curves, obtained by ramping the tip toward and away from the surface at speeds of 25-325 nm/s, reveal this condensation event as an abrupt jump in force, allowing precise mapping of local surface energetics without physical contact instability.^[57] This approach provides insights into surface chemistry and topography, as the meniscus force scales with the tip-sample geometry and vapor pressure.^[58] The resolution of AFM in detecting capillary bridges enables characterization down to sub-nanometer scales, with vertical sensitivity below 0.1 nm and lateral resolution approaching 1 nm for bridge formation. Studies have visualized and quantified bridges as small as 0.5-1 nm in radius, particularly in confined geometries, by analyzing force jumps and retraction hysteresis in the curves.^[59] Wettability mapping is achieved by correlating adhesion force variations with contact angle θ, where higher forces indicate more hydrophilic regions (θ < 30°) and lower forces hydrophobic ones (θ > 90°); for instance, on natural fibers, treated surfaces show up to 2-3 times greater adhesion due to enhanced polarity.^[60] This technique maps heterogeneous wettability across rough or fibrous substrates, resolving θ variations over areas of 10-100 nm², which is crucial for understanding biomaterial or polymer interfaces.^[60] Beyond AFM, environmental scanning electron microscopy (ESEM) facilitates in-situ observation of capillary bridge dynamics by maintaining vapor pressures up to 100% relative humidity within the chamber, allowing real-time imaging of meniscus formation and evaporation on surfaces. In ESEM, bridges between particles or features are visualized at magnifications up to 100,000×, revealing growth kinetics and stability on superhydrophobic substrates where bridges persist at separations of 1-10 μm.^[61] Quartz crystal microbalance (QCM) complements these by monitoring adsorption kinetics of water vapor leading to capillary condensation, measuring mass uptake on thin films or nanoparticle layers with picogram sensitivity. For SiO₂ nanoparticles, QCM detects rapid sorption-desorption cycles, with frequency shifts indicating bridge formation times on the order of seconds under humidity steps from 10% to 80%.^[62] Combined QCM-ESEM setups have confirmed that capillary bridges enhance particle cohesion, with kinetics following Langmuir-type isotherms modified for multilayer adsorption.^[62] Data analysis in these techniques often involves fitting experimental force or mass data to the Kelvin equation, which relates the condensation relative humidity to the local meniscus radius r via \ln(RH) = -\frac{2\gamma V_m}{r RT}, where \gamma is surface tension, V_m molar volume, R gas constant, and T temperature; this yields local r values of 0.5-5 nm and \gamma variations up to 20% on heterogeneous surfaces.^[63] Humidity control, achieved through sealed chambers with saturated salt solutions or gas flow regulators maintaining RH to ±1%, isolates capillary contributions by minimizing van der Waals forces, which dominate below 10% RH and contribute <10% of total adhesion at higher levels.^[64] For example, at 50% RH, capillary forces account for 80-90% of pull-off adhesion on silica, separable via RH-dependent extrapolation.^[58] Recent advances in the 2020s have pushed AFM toward single-molecule sensitivity for capillary studies on heterogeneous surfaces, incorporating qPlus sensors and low-temperature operation to achieve sub-Ångstrom force resolution and atomic-scale imaging of interfacial water layers. In 3D nanomenisci (80-250 nm³), these setups map hydration structures with signal-to-noise ratios exceeding 100, detecting individual water molecule rearrangements during condensation on insulators like graphite.^[59] Such capabilities reveal curvature-dependent \gamma down to molecular scales, enabling characterization of biomolecular adsorption sites where capillary forces modulate binding energies by 10-50 kT.^[65]

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