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Wetting

Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from the balance of interfacial tensions at the three-phase contact line where the liquid, solid, and vapor meet.^[1] This phenomenon is fundamentally governed by the contact angle formed by the liquid-vapor interface with the solid surface, as first described by Thomas Young in 1805, where a contact angle θ < 90° signifies wetting (hydrophilic behavior, with the liquid spreading), θ = 90° indicates neutral wetting, and θ > 90° denotes non-wetting (hydrophobic behavior, with the liquid beading up).^[2] Young's equation, θ = cos⁻¹[(γ_SV - γ_SL)/γ_LV], relates the equilibrium contact angle to the solid-vapor (γ_SV), solid-liquid (γ_SL), and liquid-vapor (γ_LV) interfacial tensions, providing a thermodynamic foundation for predicting wetting behavior on smooth, homogeneous surfaces.^[3] In practice, wetting is influenced by surface roughness, chemical heterogeneity, and molecular-scale interactions, leading to deviations from ideal models and phenomena such as contact angle hysteresis—the difference between advancing and receding angles during droplet motion—which arises from energy barriers at the contact line.^[1] For rough surfaces, the Wenzel model describes how roughness amplifies wetting (increasing hydrophilicity for θ < 90° or hydrophobicity for θ > 90°), while the Cassie-Baxter model accounts for composite interfaces with trapped air pockets, enabling superhydrophobicity (θ > 150°, low hysteresis) as seen in natural structures like lotus leaves.^[3] These extensions highlight wetting's sensitivity to nanoscale topography and chemistry, critical for engineering surfaces with tailored properties. Wetting phenomena are ubiquitous in nature and technology, underpinning processes from capillary action in plant xylem to industrial applications like coatings, printing, lubrication, and microfluidics.^[1] In enhanced oil recovery, wettability controls fluid displacement in porous media; in biomedicine, it influences cell adhesion on implants; and in self-cleaning materials, superhydrophobic surfaces facilitate water repellency and dirt removal.^[3] Dynamics of wetting, including spreading rates and dewetting transitions, involve hydrodynamic effects at the contact line, often modeled with lubrication approximations or molecular dynamics simulations to address singularities and precursor films.^[1] Ongoing research explores wetting in confined geometries for nanofluidics and phase transitions near critical points, emphasizing its interdisciplinary role in physics, chemistry, and engineering.^[2]

Introduction to Wetting

Definition and Phenomena

Wetting refers to the process by which a liquid spreads across or adheres to a solid surface, driven by the balance of interfacial tensions between the liquid, solid, and surrounding vapor phase.^[4] This phenomenon is fundamental in surface science, characterizing how liquids interact with solids in various environments, from natural settings to industrial applications. In contrast, non-wetting occurs when the liquid minimizes contact with the solid, forming discrete droplets rather than spreading.^[3] Key observable phenomena in wetting include complete wetting, partial wetting, and non-wetting, distinguished by the extent of liquid spreading on the surface. Complete wetting is observed when the liquid fully spreads out to form a thin film, effectively covering the entire solid surface. Partial wetting features a moderate spread, where the liquid forms a droplet with an intermediate shape. Non-wetting, on the other hand, results in the liquid forming a nearly spherical droplet that beads up with minimal adhesion to the solid.^[5] These behaviors are qualitatively assessed through the shape of a deposited liquid drop, reflecting the preferential interaction between the liquid and solid.^[4] Everyday examples illustrate these phenomena clearly: water on clean glass exhibits wetting, as the liquid spreads across the surface to form a flat film. In contrast, water on a lotus leaf demonstrates non-wetting, where droplets bead up and roll off easily without adhering.^[4]^[6] Such observations highlight wetting's role in natural and engineered systems, without invoking underlying mechanisms. A common method for observing wetting is the sessile drop technique, in which a small volume of liquid is placed on the solid surface, and its shape is analyzed to infer spreading behavior.^[7] In dynamic contexts, the advancing contact line refers to the edge of the liquid as it expands across the surface, while the receding contact line describes the edge as the liquid contracts.^[8] These terms capture the motion-dependent aspects of liquid-solid interactions during spreading or retraction.

Historical Development

The study of wetting phenomena originated in the Renaissance era, with Leonardo da Vinci recording detailed observations of liquid spreading on surfaces and capillary action in his notebooks around 1500, attributing such behaviors to cohesive forces within fluids.^[9] These early insights laid informal groundwork for understanding interfacial behaviors, though they remained qualitative without quantitative models.^[10] Advancements accelerated in the 19th century, marked by Thomas Young's 1805 essay "On the Cohesion of Fluids," which provided the first qualitative description of the contact angle as the equilibrium angle formed by a liquid drop on a solid surface, serving as a precursor to later quantitative interfacial energy models.^[11] Concurrently, Pierre-Simon Laplace advanced the theoretical framework of capillarity in his 1806 treatise, integrating molecular attraction to explain capillary rise and meniscus formation in tubes, building on Newtonian principles to quantify pressure differences across curved interfaces.^[12] These contributions shifted wetting from empirical observation to a mechanics-based science, influencing subsequent derivations of interfacial tensions.^[13] In the 20th century, focus turned to surface topography's role in wetting, with Robert Wenzel introducing a model in 1936 to describe how roughness enhances or diminishes wettability on homogeneous solids, based on experiments with water and mercury drops.^[14] This was extended in 1944 by A.B.D. Cassie and S. Baxter, who proposed a theory for composite interfaces where air pockets trap beneath liquid drops on heterogeneous surfaces, again using simple drop-based assays with water and non-wetting liquids like mercury.^[15] Early experiments relied on visual inspection of sessile drops, but by the 1950s, precise goniometry emerged as a standard technique, enabling accurate measurement of contact angles through optical protractors and improved microscopy, as pioneered in studies of polymer and metal surfaces.^[16] The late 20th century brought biological inspirations to wetting research, highlighted by Wilhelm Barthlott's 1977 discovery of the "lotus effect" on Nelumbo nucifera leaves, where micro- and nanostructures create superhydrophobic self-cleaning surfaces by minimizing liquid-solid contact.^[17] This observation spurred integration of wetting principles with nanotechnology from the 1990s onward, enabling engineered biomimetic surfaces for applications in coatings and microfluidics.^[18]

Fundamental Principles

Contact Angle and Interfacial Energies

The contact angle, denoted as θ, is defined as the angle formed between the tangent to the liquid-vapor interface and the solid surface at the three-phase contact line, measured through the liquid phase.^[19] This angle serves as a primary quantitative indicator of wettability, where θ < 90° indicates partial wetting (hydrophilic behavior for water), θ > 90° suggests non-wetting (hydrophobic), and θ = 0° corresponds to complete spreading.^[20] The concept was first articulated by Thomas Young in his 1805 essay on fluid cohesion, where he described the angle as arising from the balance at the contact line.^[21] Wetting phenomena are governed by interfacial energies, which represent the excess free energy per unit area at the boundaries between phases. These include the solid-vapor interfacial energy (γ_sv), the solid-liquid interfacial energy (γ_sl), and the liquid-vapor surface tension (γ_lv, often simply called the liquid surface tension).^[5] The values of these energies dictate the tendency of the liquid to spread or bead up on the solid; for instance, high γ_sv relative to γ_sl favors wetting by reducing the system's total energy.^[22] These energies are typically on the order of 20–70 mJ/m² for common liquids and solids, with γ_lv for water around 72 mJ/m² at room temperature, providing a scale for the driving forces in wetting processes.^[22] In equilibrium conditions, the contact angle θ represents a static configuration where the interfacial energies are balanced, minimizing the total free energy of the system at the contact line.^[23] However, real systems often exhibit dynamic contact angles due to motion of the contact line, leading to hysteresis: the advancing contact angle (θ_adv) is larger than the receding contact angle (θ_rec) because of surface pinning effects from roughness, chemical heterogeneities, or molecular adsorption. Hysteresis, typically 10–20° for smooth surfaces but up to 90° on rough ones, quantifies energy dissipation during wetting and dewetting, with θ_adv measured as the liquid front advances and θ_rec as it recedes.^[24] Contact angles and interfacial energies are measured using several established techniques. The sessile drop method involves placing a liquid droplet on a solid surface and imaging the profile to fit the tangent at the contact line, yielding θ directly; it is widely used for static and advancing/receding angles on flat samples. The pendant drop technique suspends a droplet from a needle and analyzes its shape under gravity to determine γ_lv, which can be combined with contact angle data for solid interfacial energies. The Wilhelmy plate method immerses a thin plate into the liquid and measures the force to compute both γ_lv and dynamic contact angles via the wetting force balance, particularly useful for hysteresis on fibers or plates.^[25] Thermodynamically, wetting occurs through minimization of the system's Helmholtz free energy, where the equilibrium contact angle configuration reduces the total interfacial energy without external work.^[23] This principle underlies the balance of the three interfacial energies, as later formalized in Young's equation relating θ to γ_sv, γ_sl, and γ_lv.^[5]

Young's Equation and Derivations

Young's equation provides the fundamental relationship for the equilibrium contact angle \theta at the three-phase contact line where a liquid-vapor interface meets a solid surface, expressed as

\cos \theta = \frac{\gamma_{SV} - \gamma_{SL}}{\gamma_{LV}},

where \gamma_{SV}, \gamma_{SL}, and \gamma_{LV} are the solid-vapor, solid-liquid, and liquid-vapor interfacial tensions, respectively.^[26] This equation was first proposed by Thomas Young in 1805 based on observations of capillary action and surface cohesion.^[26] The mechanical derivation of Young's equation arises from the condition of horizontal force balance at the contact line in equilibrium, where the system experiences no net tangential force along the solid surface.^[27] The liquid-vapor interfacial tension \gamma_{LV} acts tangentially to the liquid surface at angle \theta to the solid, contributing a horizontal component \gamma_{LV} \cos \theta, while the solid-liquid tension \gamma_{SL} pulls directly along the solid; for balance, this must equal the solid-vapor tension \gamma_{SV}, yielding \gamma_{SV} = \gamma_{SL} + \gamma_{LV} \cos \theta.^[27] This force equilibrium assumes the interfaces meet at a point and ignores vertical components, which are balanced by the solid's rigidity.^[27] An alternative variational derivation obtains the same relation by minimizing the total interfacial free energy of a sessile liquid drop on a planar solid surface.^[28] The total energy E is given by E = \gamma_{SV} A_{SV} + \gamma_{SL} A_{SL} + \gamma_{LV} A_{LV}, where A_{ij} are the respective interfacial areas; for a fixed drop volume, varying the contact angle to minimize E leads to the condition that the derivative \frac{\partial E}{\partial \theta} = 0, resulting in Young's equation after accounting for the geometric dependence of areas on \theta.^[28] This approach emphasizes the thermodynamic equilibrium and is equivalent to the force balance under the same assumptions.^[28] Young's equation relies on several key assumptions, including an ideal smooth, homogeneous, and chemically inert solid surface that is rigid and molecularly flat, with no adsorption or precursor films at the contact line.^[29] It also neglects gravitational deformation, inertial effects, and thermal fluctuations, applying primarily to sessile drops where the contact line is macroscopic and the liquid is incompressible.^[29] These conditions ensure the interfacial tensions are well-defined and isotropic near the contact line.^[29] The equation holds reliably at macroscopic scales but exhibits limitations for nanoscale drops, where line tension effects and molecular discreteness cause deviations from the predicted contact angle, as the contact line energy becomes significant relative to interfacial contributions.^[30] In such cases, a modified form incorporating line tension \tau is needed: \cos \theta = \frac{\gamma_{SV} - \gamma_{SL}}{\gamma_{LV}} - \frac{\tau}{\gamma_{LV} r}, where r is the base radius, highlighting the breakdown for drops smaller than tens of nanometers.^[30]

Wetting on Smooth Surfaces

High-Energy and Low-Energy Surfaces

Solid surfaces are classified as high-energy or low-energy based on their solid-vapor interfacial energy (γ_sv), which governs the wetting behavior of liquids on smooth, ideal surfaces. High-energy surfaces, typically featuring clean metals or metal oxides, exhibit γ_sv values typically greater than 50 mJ/m² (often 70-1000 mJ/m² or higher for metals), promoting strong liquid adhesion and near-complete wetting with equilibrium contact angles (θ) approaching 0° for many liquids.^[31] For instance, water on clean glass (a silica-based oxide) spreads extensively, yielding θ < 30°, due to favorable interactions between the liquid and surface.^[32] At the molecular level, high-energy surfaces arise from polar functional groups, such as hydroxyl (-OH) moieties on oxides, or unsaturated dangling bonds on clean metals, which enable strong dipole-dipole and hydrogen-bonding interactions with polar liquids like water.^[33] This contrasts with low-energy surfaces, common in polymers or fluorinated coatings, where γ_sv is below 50 mJ/m², resulting in weak adhesion and non-wetting behavior with θ > 90° for water.^[34] Examples include paraffin wax (γ_sv ≈ 25 mJ/m², θ ≈ 110° for water) and Teflon (polytetrafluoroethylene, γ_sv ≈ 18 mJ/m², θ ≈ 110°), where water forms beads to minimize contact.^[35]^[36] The molecular origins of low surface energy stem from non-polar terminal groups, such as methyl (-CH3) in hydrocarbons or trifluoromethyl (-CF3) in fluoropolymers, which primarily engage in weak van der Waals (dispersive) forces, reducing adhesion to polar liquids.^[37] These classifications align with Young's equation, which relates θ to the balance of interfacial energies (γ_sv, γ_sl, γ_lv), underscoring how high γ_sv favors low θ while low γ_sv elevates it.^[38] Surfaces with θ < 90° are deemed hydrophilic (water-attracting) on high-energy materials, facilitating spreading for applications like coatings, whereas θ > 90° indicates hydrophobicity (water-repelling) on low-energy surfaces, useful in non-stick or self-cleaning technologies.^[34] For example, clean glass demonstrates hydrophilicity as water spreads into a thin film, while Teflon exhibits hydrophobicity, causing water to bead and roll off.^[4]

Spreading Coefficient and Wetting Regimes

The spreading coefficient S, a fundamental thermodynamic quantity in wetting theory, quantifies the driving force for a liquid to spread over a solid surface in the presence of vapor. It is defined as

S = \gamma_{SV} - \gamma_{SL} - \gamma_{LV},

where \gamma_{SV}, \gamma_{SL}, and \gamma_{LV} represent the solid-vapor, solid-liquid, and liquid-vapor interfacial tensions, respectively. This parameter originates from early thermodynamic analyses of interfacial energies, with its practical significance for spreading processes first highlighted in studies of liquid sprays on solids.^[38] The sign of S determines the wetting regime on smooth surfaces. If S > 0, complete wetting occurs, characterized by a zero equilibrium contact angle (\theta = 0^\circ), where the liquid forms a thin film covering the entire substrate to minimize the total interfacial energy. In contrast, when S < 0, partial wetting prevails, with the equilibrium contact angle satisfying $0^\circ < \theta < 180^\circ, resulting in a finite drop shape rather than full spreading. This framework connects directly to Young's equation, \cos \theta = \frac{\gamma_{SV} - \gamma_{SL}}{\gamma_{LV}}, which balances interfacial tensions at the three-phase contact line. For partial wetting (S < 0), substituting the definition of S yields the relation

\cos \theta = 1 + \frac{S}{\gamma_{LV}},

demonstrating how a negative S limits spreading and establishes a nonzero \theta. Complementing this, the Young-Dupré equation provides the reversible work of adhesion per unit area,

W_a = \gamma_{LV} (1 + \cos \theta),

which measures the energy gained when a unit area of solid-vapor interface is replaced by solid-liquid and liquid-vapor interfaces; this expression, combining Young's law with Dupré's thermodynamic work term, underscores the energetic favorability of wetting. Within the partial wetting regime, finer distinctions arise based on the magnitude of S. Pseudopartial wetting emerges when S is slightly negative, leading to the formation of ultrathin precursor films (typically nanometers thick) ahead of the macroscopic drop due to long-range van der Waals attractions, which effectively modify the local interfacial energies despite the overall negative S. At the opposite extreme, when S \ll 0, non-spreading behavior dominates, with \theta approaching $180^\circ, minimizing the liquid-solid contact area and resulting in highly spherical droplets that barely interact with the substrate. These wetting regimes, governed by S, have broad practical implications, particularly on low-energy surfaces where non-spreading is common. For instance, a favorable S (near zero or positive) is essential for efficient spreading in painting and coating processes, ensuring uniform adhesion and coverage, while negative values can lead to defects like dewetting or poor ink transfer in printing applications.

Wetting on Rough Surfaces

Wenzel's Roughness Model

The Wenzel's roughness model, proposed in 1936, describes how microscopic surface roughness on chemically homogeneous solids modifies the apparent contact angle of a liquid droplet by increasing the effective interfacial area. This model builds on Young's equation for smooth surfaces, where the intrinsic contact angle θ is determined by the balance of interfacial tensions. The core of the model is expressed by Wenzel's equation:

\cos \theta^* = r \cos \theta

where θ* is the apparent contact angle on the rough surface, θ is the intrinsic contact angle on a smooth counterpart, and r is the roughness factor defined as the ratio of the actual surface area to the projected flat area (r > 1). The mechanism relies on the amplification of the solid-liquid interfacial energy due to the expanded contact area; for hydrophilic surfaces (θ < 90°), this drives θ* lower than θ, promoting greater spreading, while for hydrophobic surfaces (θ > 90°), θ* increases, enhancing non-wetting behavior.^[39] Key assumptions include full penetration of the liquid into the roughness features, ensuring homogeneous wetting without air entrapment, and applicability primarily to microscale roughness where capillary forces dominate over gravitational effects.^[40]^[41] The model holds for surfaces with isotropic or anisotropic roughness patterns, such as grooves or pillars, as long as the liquid wets the entire topography.^[28] Experimental validation has been achieved through techniques like chemical etching and mechanical sandblasting to introduce controlled roughness. For instance, on intrinsically hydrophilic silicon surfaces roughened by plasma etching, the model predicts and observes superhydrophilic behavior with θ* approaching 0°, as the increased r amplifies spreading.^[42] Similarly, sandblasted metal surfaces with moderate hydrophobicity (θ ≈ 100°) exhibit θ* up to 140°, confirming the model's enhancement of intrinsic properties without air involvement.^[43] These results align with contact angle goniometry measurements on textured substrates, where r is quantified via atomic force microscopy or profilometry.^[44] The model has limitations when the assumption of complete liquid penetration fails, such as on surfaces with features promoting air pocket formation, leading to deviations from predicted θ*.^[45] It also assumes chemical homogeneity and may not fully capture nanoscale effects or dynamic wetting scenarios.^[46]

Cassie-Baxter Composite Model

The Cassie-Baxter model describes the wetting behavior on rough or chemically heterogeneous surfaces where the liquid does not fully penetrate the surface texture, instead forming a composite interface that includes both solid and air (or vapor) phases.^[47] In this regime, the apparent contact angle \theta^* is given by the equation

\cos \theta^* = f (\cos \theta + 1) - 1,

where \theta is the equilibrium contact angle on a smooth surface of the same material, and f (0 < f < 1) represents the fraction of the projected area under the droplet that is in contact with the solid.^[47] This equation arises from minimizing the free energy of the system, accounting for the parallel contributions of the solid-liquid and liquid-vapor interfaces, with the vapor phase effectively contributing a contact angle of 180° due to its non-wetting nature.^[48] In the Cassie-Baxter state, the liquid droplet rests atop the roughness features (such as peaks or protrusions), with air pockets trapped in the valleys below, which significantly reduces the actual solid-liquid contact area compared to a homogeneous wetting scenario.^[47] This configuration lowers the overall interfacial energy, promoting higher apparent contact angles and enabling superhydrophobicity, typically defined as \theta^* > 150^\circ with low contact angle hysteresis.^[49] Unlike the Wenzel model, which assumes complete liquid impregnation of the roughness, the Cassie-Baxter state relies on this air entrapment to enhance repellency.^[47] A thin precursor film of liquid may sometimes advance ahead of the main droplet on such surfaces, potentially influencing the establishment of the full Cassie-Baxter state, though it is not invariably present and depends on the system's dynamics and surface energy.^[48] Natural examples of the Cassie-Baxter model include the leaves of the lotus plant (Nelumbo nucifera), where hierarchical micro- and nanostructures covered with hydrophobic wax create air-trapping surfaces, resulting in \theta^* \approx 160^\circ and self-cleaning properties known as the "lotus effect."^[50] Artificial implementations often involve micropillar arrays on silicon or polymer substrates, chemically modified to be hydrophobic, which mimic this air-pocket mechanism to achieve stable superhydrophobic states.^[51] Within superhydrophobic Cassie-Baxter surfaces, variations in adhesion lead to distinct behaviors: the "lotus effect" features low droplet adhesion and easy roll-off for self-cleaning, while the "petal effect," observed on rose petals with densely packed micropapillae, exhibits high \theta^* > 150^\circ but strong pinning (high roll-off angles >10°), enabling sticky superhydrophobicity useful for applications like droplet transport.^[52]

Transitions and Dynamics

State Transitions Between Models

On rough surfaces, the Wenzel and Cassie-Baxter states represent distinct wetting configurations separated by energy barriers, where the Cassie-Baxter state often serves as a metastable configuration due to trapped air pockets, while the Wenzel state involves complete liquid penetration into the surface texture. Transitions between these states are governed by the relative free energies described in the respective models, with the direction and likelihood depending on surface design and external conditions. Overcoming these barriers typically requires external stimuli to alter the interfacial energies or apply mechanical forces. The Cassie-to-Wenzel transition is more common and occurs when external forces displace the air beneath the droplet, allowing liquid to impregnate the roughness features. This can be induced by hydrostatic pressure, where the Laplace pressure within the droplet, ΔP = 2γ_lv sin θ / R (with γ_lv as liquid-vapor surface tension, θ as the equilibrium contact angle, and R as the base radius of the drop), exceeds the critical value needed to deform the air-liquid interface into the texture.^[53] Vibration also facilitates this transition by providing mechanical energy to break the air pockets, with the required amplitude scaling with surface roughness and increasing hysteresis observed as the droplet pins at texture edges during the process.^[54] Factors influencing the transition include drop size, where smaller droplets favor the Cassie state due to reduced gravitational sagging, surface geometry such as wider pillar spacing that lowers the energy barrier for penetration, and higher vibration amplitudes that accelerate air displacement.^[55] In contrast, the Wenzel-to-Cassie transition is rarer and typically requires active intervention to promote dewetting and air re-entry into the texture. Electrowetting achieves this by applying an electric field to reduce the solid-liquid interfacial energy, enabling the liquid to retract from the grooves and restore the composite interface, particularly on surfaces designed with re-entrant geometries to prevent re-penetration.^[56] Superhydrophobic designs with overhanging structures can also stabilize the Cassie state post-transition, minimizing hysteresis. These transitions are often irreversible without energy input, as droplets become pinned in metastable states, exhibiting increased contact angle hysteresis that reflects the energy landscape's multiple minima.

Spreading and Dynamic Contact Angles

Spreading dynamics describe the time evolution of a liquid drop's contact area on a solid surface following deposition, transitioning from an initial inertial phase to a slower viscous regime. In the early stage, the drop's kinetic energy upon impact drives rapid expansion, governed primarily by inertia and surface tension, before viscous forces dominate and dissipate energy. For complete wetting conditions, where the equilibrium contact angle approaches zero as per Young's equation, the subsequent spreading follows Tanner's law, with the drop radius scaling as r \sim t^{1/10}, reflecting the balance between capillary driving forces and viscous resistance in thin precursor films near the contact line. This power-law behavior was experimentally observed and theoretically derived for low-viscosity silicone oils on smooth, wettable surfaces.^[57] Dynamic contact angles deviate from the static equilibrium value, increasing for advancing contact lines (θ_dyn > θ) due to the liquid's forward motion and decreasing for receding lines (θ_dyn < θ) as the liquid retracts. These deviations arise from unbalanced forces at the moving three-phase contact line, where viscous shear and molecular-scale processes alter the apparent angle. In the molecular-kinetic theory, contact line motion is modeled as a thermally activated process at the molecular level, drawing from Eyring's transition state theory, with the velocity v proportional to \exp(-E_a / kT), where E_a is the activation energy for molecular jumps across the solid-liquid interface, k is Boltzmann's constant, and T is temperature. This framework, originally applied to liquid-liquid displacement, emphasizes adsorption-desorption kinetics over bulk hydrodynamics for low-speed regimes.^[58] Hydrodynamic models complement the molecular-kinetic approach by focusing on macroscopic flow effects at higher speeds, where the Cox-Voinov relation predicts the dynamic contact angle through \theta_\mathrm{dyn}^3 \propto \mathrm{Ca} \ln(1/\mathrm{Ca}), with the capillary number \mathrm{Ca} = \eta v / \gamma_{lv} quantifying the ratio of viscous to capillary forces (\eta is viscosity, v is velocity, and \gamma_{lv} is liquid-vapor surface tension). Developed for viscous flow near the contact line with slip boundary conditions, this relation resolves the stress singularity in classical hydrodynamics and holds for small Ca, linking microscopic slip lengths to observable angles. The original analysis addressed immiscible liquid displacement but extends to solid wetting.^[59]^[60] On partially wetting surfaces, spreading often involves precursor films—ultrathin liquid layers (typically 10–100 nm thick) that advance ahead of the macroscopic contact line, driven by long-range van der Waals forces. These films, first theoretically described for "dry" solids, facilitate smooth motion by reducing pinning and enabling disjoining pressure gradients that pull the bulk liquid forward. Their presence reconciles partial wetting with observed complete film formation over time scales longer than direct capillary spreading.^[61] In applications like inkjet printing, understanding these dynamics is crucial for controlling drop impact, where high-speed deposition (up to m/s velocities) leads to initial inertial spreading followed by viscous relaxation and precursor film evolution, determining print resolution and ink-substrate adhesion. Experimental studies highlight how dynamic angles and spreading rates influence dot formation, with deviations from Tanner's law occurring due to substrate heterogeneity and evaporation.

Modification and Control

Chemical Modifications and Surfactants

Surfactants are amphiphilic molecules that reduce the liquid-vapor interfacial tension (γ_lv) of water, typically from 72 mN/m to around 30 mN/m, thereby promoting the spreading of liquids on otherwise hydrophobic surfaces.^[62] For instance, sodium dodecyl sulfate (SDS), an anionic surfactant, lowers water's surface tension to approximately 30-40 mN/m at concentrations near its critical micelle concentration (CMC).^[63] This reduction facilitates better wetting by decreasing the energy barrier for liquid-solid contact. The primary mechanism by which surfactants enhance wetting involves their adsorption at the liquid-vapor, solid-liquid, and solid-vapor interfaces, with a greater reduction in the solid-liquid interfacial tension (γ_sl) compared to the solid-vapor tension (γ_sv).^[64] This selective adsorption alters the balance in Young's equation, cos θ = (γ_sv - γ_sl)/γ_lv, shifting the equilibrium contact angle θ toward 0° and enabling partial or complete wetting on low-energy surfaces.^[64] However, the effect is limited by the CMC, above which additional surfactant forms micelles in the bulk solution rather than further adsorbing at interfaces, capping the reduction in interfacial tensions.^[65] Surfactants are classified into types based on their polar head groups, each suited to specific wetting applications. Ionic surfactants, such as the cationic cetyltrimethylammonium bromide (CTAB), effectively alter wettability in charged systems like carbonate rocks, reducing oil-water interfacial tension to as low as 1.1 mN/m at CMC and promoting water-wet conditions in enhanced oil recovery.^[66] Non-ionic surfactants, like Triton X-100, provide neutral wetting enhancement without sensitivity to pH or ionic strength, commonly used in detergents for improved spreading on fabrics and in coatings to ensure uniform film formation.^[67] Beyond surfactants, other chemical modifications involve silanes and thiols that form self-assembled monolayers (SAMs) on surfaces, creating low-energy coatings with water contact angles exceeding 110°.^[68] For example, alkylsilane SAMs on silica yield hydrophobic surfaces with advancing contact angles up to 115°, ideal for anti-wetting applications in microfluidics.^[69] These modifications achieve interfacial tension reductions of 20-40 mN/m, driving transitions from non-wetting to partial wetting regimes depending on the underlying substrate energy.

Physical and Structural Changes

Physical methods to control wetting often involve altering surface topography or introducing defects that amplify intrinsic wetting properties, leading to either enhanced hydrophilicity or superhydrophobicity. Surface texturing techniques, such as lithography and anodization, create hierarchical roughness scales that promote the Cassie-Baxter state for water repellency. For instance, black silicon surfaces fabricated via reactive ion etching exhibit apparent contact angles exceeding 170°, enabling robust superhydrophobicity without requiring low-surface-energy coatings.^[70] These structures trap air pockets beneath liquid droplets, minimizing contact area and facilitating self-cleaning by reducing adhesion of contaminants.^[71] Defect engineering, particularly the introduction of oxygen vacancies in metal oxides like TiO₂, significantly enhances surface hydrophilicity under specific conditions. These vacancies act as active sites that lower the energy barrier for water adsorption, causing the contact angle to drop from approximately 20° to near 0° upon ultraviolet (UV) irradiation. This photo-induced superhydrophilicity is reversible; in the absence of UV light, the surface reverts to its moderately hydrophilic state as vacancies are passivated by hydroxyl groups.^[72] The mechanism involves photocatalytic generation of electron-hole pairs that dissociate water molecules, forming a continuous hydrophilic layer.^[73] Plasma etching and laser ablation provide versatile routes to roughen surfaces, tailoring wetting behavior according to the Wenzel or Cassie-Baxter regimes. Plasma etching on hydrophilic substrates increases surface area, amplifying wettability to achieve complete spreading in the Wenzel state, while on hydrophobic materials, it fosters micro-nano protrusions that stabilize the Cassie state with contact angles up to 160°.^[74] Similarly, femtosecond laser ablation creates re-entrant geometries that enhance hydrophobicity, with treated surfaces showing low hysteresis and roll-off angles below 5°.^[75] These techniques allow precise control over feature sizes, from micrometers to nanometers, to optimize wetting transitions.^[76] Environmental factors like temperature and humidity also influence wetting through physical interactions at the interface. For most liquid-solid systems, the contact angle decreases with rising temperature due to reduced liquid-vapor surface tension and increased molecular mobility, promoting spreading; this effect is pronounced above 50°C where thermal energy overcomes adhesion barriers.^[77] Humidity modulates precursor films—ultrathin liquid layers that precede macroscopic droplets—by altering adsorption rates; high humidity thickens these films on partially wetting surfaces, facilitating easier droplet nucleation and reducing effective contact angles.^[78] In practical applications, such as UV-activated TiO₂-coated tiles for self-cleaning windows, photo-induced superhydrophilicity ensures rapid water sheeting that rinses away dirt without manual intervention.^[79]

Advanced Models and Predictions

Non-Ideal and Curved Surfaces

Non-ideal smooth surfaces introduce chemical heterogeneities, such as patches of varying surface energy, which disrupt the uniform contact line assumed in ideal models. These heterogeneities cause the contact line to pin at high-energy sites during advancement or receding, leading to contact angle hysteresis where the advancing angle exceeds the receding angle by up to several tens of degrees. For instance, isolated nanometric defects on otherwise homogeneous surfaces can induce hysteresis through local energy barriers that the contact line must overcome, resulting in stick-slip dynamics.^[80]^[81] Line tension, an additional energy per unit length associated with the three-phase contact line, provides a correction to Young's equation for these non-ideal cases, particularly when the contact line curvature is significant. The generalized form incorporates this term as \cos \theta = \frac{\gamma_{SV} - \gamma_{SL}}{\gamma_{LV}} - \frac{\Gamma}{\gamma_{LV} R \sin \theta}, where \Gamma is the line tension (typically on the order of $10^{-11} to $10^{-9} J/m), R is the radius of curvature of the contact line, and the other terms follow the standard notation from Young's equation for flat, ideal surfaces. This correction becomes negligible for macroscopic drops but alters the apparent contact angle by several degrees for nanoscale features.^[82] On curved surfaces, such as cylinders or spheres, the planar assumptions of Young's equation require generalization to account for the substrate's geometry, adjusting the force balance at the contact line by the curvature radius. For a cylindrical fiber, the equilibrium contact angle satisfies a modified balance where \cos \theta = \frac{\gamma_{SV} - \gamma_{SL}}{\gamma_{LV}}, but the overall drop shape deviates from spherical caps due to the azimuthal spreading, with the maximum cross-sectional radius scaling as r \propto V^{1/3} (1 - \cos \theta)^{1/3} for partial wetting, where V is the drop volume. On nanofibers (radii below 100 nm), this leads to distinct morphologies: complete wetting results in axisymmetric wrapping films that coat the fiber uniformly, while partial wetting forms beaded structures where liquid segments bridge or pearl along the fiber, influenced by capillary and gravitational forces.^[83] For spherical nanoparticles, partial wetting stabilizes Pickering emulsions, where particles adsorb at the oil-water interface with a contact angle near 90° relative to the oil phase, forming a jammed layer that prevents coalescence and enhances emulsion stability for months. The attachment energy, proportional to the particle radius squared and \sin^2 \theta, favors irreversible binding for radii above 10 nm, with convex curvature reducing the effective interfacial area covered compared to flat surfaces.^[84] Precursor films, thin liquid layers ahead of the apparent contact line, form on curved convex surfaces due to long-range van der Waals forces quantified by disjoining pressure \Pi(h) = A/(6\pi h^3) (for non-retarded Hamaker interactions, where A is the Hamaker constant and h is film thickness), driving complete wetting even on partially wetting substrates by minimizing the grand potential. On nanofibers or nanoparticles, this promotes initial film spreading over the curvature, altering macroscopic dynamics. Planar models like Young's equation introduce errors exceeding 10% in predicted contact angles for substrate radii below 1 μm, as line tension and curvature effects dominate the energy balance, necessitating these generalizations for accurate predictions in micro- and nanoscale applications.^[82]

Computational Approaches

Computational approaches to wetting leverage numerical simulations to predict and analyze behaviors that analytical models, such as those by Wenzel and Cassie-Baxter, cannot fully capture in complex geometries or under dynamic conditions. These methods span scales from quantum to mesoscale, enabling the study of interfacial energies, droplet spreading, and transitions on nanostructured surfaces. By integrating atomistic details with continuum fluid dynamics, simulations address limitations in experimental resolution, particularly for nanoscale effects and non-equilibrium processes.^[85] Molecular dynamics (MD) simulations provide atomistic insights into wetting by modeling the interactions of liquid molecules with solid surfaces using classical force fields. For instance, MD has been used to predict the contact angle of water droplets on graphene, yielding values around 120° through explicit simulations of droplet equilibrium shapes. Common force fields include TIP4P for water, which accurately reproduces liquid properties and interfacial tensions in such systems. These simulations reveal how molecular-scale interactions, like van der Waals forces, influence macroscopic wetting angles.^[86] Phase-field models offer a diffuse-interface approach to simulate dynamic wetting phenomena, particularly on rough surfaces, by solving the Cahn-Hilliard equation coupled with Navier-Stokes equations. The Cahn-Hilliard equation governs the evolution of the phase field variable \phi, representing the liquid-vapor interface:

\frac{\partial \phi}{\partial t} = \nabla \cdot \left( M \nabla \mu \right),

where M is the mobility, and the chemical potential \mu = -\epsilon \nabla^2 \phi + f'(\phi) balances surface tension \epsilon and the double-well potential f(\phi). This framework captures spreading dynamics and contact line motion without explicitly tracking the interface, making it suitable for irregular topographies. Applications include predicting droplet imbibition and hysteresis on microstructured substrates.^[87]^[88] The lattice Boltzmann method (LBM) serves as a mesoscale technique for simulating fluid flow and wetting transitions, approximating the Navier-Stokes equations on a discrete lattice while incorporating boundary conditions for contact angles. In LBM, the equilibrium contact angle is enforced via wall interaction forces that align the fluid density distribution with the desired \theta. This method excels in modeling multi-phase flows, such as droplet impingement and invasion of porous media, where it resolves Cassie-to-Wenzel transitions under external forcings like vibration. For example, LBM simulations demonstrate how oscillatory vibrations can depin droplets from composite states, facilitating switches between wetting regimes.^[89]^[90] Density functional theory (DFT) enables quantum-level predictions of solid-liquid interfacial free energies (\gamma_{sl}), crucial for designing wetting properties in materials. Calculations often employ DFT+U corrections for transition metals to account for electron localization. On TiO₂ surfaces, DFT reveals that oxygen vacancies reduce \gamma_{sl} by altering electronic structure and surface reactivity, with formation energies decreasing from ~3 eV on stoichiometric sites to lower values near defects, promoting hydrophilicity. These insights guide the engineering of photocatalytic or self-cleaning surfaces.^[91] Validation of these methods confirms their reliability against benchmarks like Young's equation. MD simulations reproduce equilibrium contact angles within 5° of experimental values for simple planar systems, such as water on silica, by averaging droplet profiles over equilibrated trajectories. Similarly, LBM accurately predicts Cassie-Wenzel transitions under vibrational forcing, matching experimental pinning and depinning thresholds within 10% for micropillar arrays. These accuracies stem from refined boundary implementations and force field parametrizations.^[92] Computational approaches particularly excel in addressing gaps in analytical models, such as nanoscale defect-induced hysteresis, where MD quantifies pinning energies from rugged free-energy landscapes. For dynamic hysteresis, phase-field and LBM models simulate contact line friction in non-equilibrium spreading, revealing velocity-dependent angles beyond Tanner-Voinov predictions. In multi-component fluids, like oil-water mixtures on heterogeneous surfaces, LBM and MD capture phase separation and selective wetting, informing applications in microfluidics and enhanced oil recovery. These post-2010 advancements surpass earlier continuum assumptions by incorporating molecular discreteness and thermal fluctuations.^[80]^[85]

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