Introduction
Surface tension is the propensity of a liquid surface at equilibrium to contract and thereby minimize area, producing macroscopic behaviour akin to a stretched elastic membrane; this effect permits some objects denser than the liquid (for example razor blades or water‑striding insects) to remain supported without significant immersion. Microscopically, the phenomenon originates at the interface because molecules inside the liquid experience stronger cohesive forces from neighbouring liquid molecules than adhesive forces from molecules in the adjacent gas phase; the imbalance produces a normal (inward) stress that reduces surface area and a tangential stress commonly described as surface tension. For water, strong hydrogen bonding yields a relatively large value of 72.8 mN·m−1 at 20 °C, a reference used widely in physical and engineering calculations.
Dimensionally, surface tension has units of force per unit length, which is equivalent to energy per unit area; the term surface energy is used when emphasizing the latter, while the more general concepts surface stress and surface energy extend to solids. Because it sets the energetic cost of creating or deforming an interface, surface tension is central to capillarity, wetting, adhesion and the shape of menisci, and it governs rise or depression of liquid columns in narrow pores. Droplet formation and breakup reflect the interplay of cohesive surface forces with other influences: Van der Waals interactions at very small scales, contact‑line properties such as hydrophobicity, and hydrodynamic instabilities—most notably the Plateau–Rayleigh instability that fragments cylindrical jets into droplets.
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Surface tension is readily modified experimentally by surfactants (soap films provide classic demonstrations of altered interfacial stresses) and is naturally treated within the continuum mechanics and fluid‑mechanics framework that enforces conservation of mass, momentum and energy and respects thermodynamic constraints such as the Clausius–Duhum (entropy) inequality. Transport in adjacent phases may be described by constitutive relations—for example Fick’s law, J = −D dφ/dx, linking diffusive flux to concentration gradients. In fluid mechanics, surface tension sits alongside adhesion, cohesion and capillarity and interfaces with governing equations (Navier–Stokes, Poiseuille) and material properties (viscosity, buoyancy). Its influence extends into rheology and complex fluids—viscoelastic media, electrorheological and magnetorheological fluids and ferrofluids—where interfacial stresses interact with bulk rheological behaviour measured by rheometers. The theoretical and experimental foundations of these topics build on the work of classical and modern scientists (e.g., Bernoulli, Cauchy, Fick, Navier, Stokes, Truesdell), and analyses typically employ common field variables such as v (velocity), t (time) and e (energy or strain, context dependent).
Causes
Within the liquid interior, molecules experience nearly symmetric cohesive interactions from all sides, producing mechanical equilibrium with no net force. At the liquid–air interface, however, molecules lack neighbors on the vapor side and so are subject to an unbalanced inward attraction; this anisotropy of cohesive forces manifests as a tangential tension along the surface (surface tension) that resists forces tending to stretch or deform the interface. The balance between cohesion (attraction among like molecules) and adhesion (attraction between unlike materials) controls wetting phenomena, contact angle, and the curvature of the meniscus at a solid wall. Energetically, when adhesion is less than half of cohesion, cohesion dominates and wetting is poor, producing a convex meniscus (e.g., mercury on glass); when adhesion exceeds half of cohesion, adhesion prevails and the meniscus is concave (e.g., water on glass). Surface tension also shapes free droplets: by reducing exposed, high‑energy boundary molecules the liquid minimizes surface area, so in the absence of other forces droplets adopt an approximately spherical form that minimizes the required surface tension work; this tendency is quantitatively linked to the pressure jump across a curved interface via Laplace’s relation. Viewed at the molecular level, surface molecules occupy higher potential energy states because they have fewer neighbors, and the system lowers its total energy by decreasing the number of such boundary molecules—hence the smooth, compact interfaces and, in practical situations, the ability of sufficiently high surface tension to inhibit liquid penetration into porous substrates.
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Surface tension (commonly denoted γ, sometimes σ or T) is a scalar interfacial property that can be expressed either as a force per unit length or as an energy per unit area. In SI units it is given as newtons per metre (N·m⁻¹), while older cgs literature typically uses dynes per centimetre (dyn·cm⁻¹). Using base‑unit relations (1 dyn = 10⁻⁵ N, 1 cm = 10⁻² m) yields 1 dyn·cm⁻¹ = (10⁻⁵ N)/(10⁻² m) = 10⁻³ N·m⁻¹ = 0.001 N·m⁻¹.
Because energy and force units are related (1 J = 1 N·m), surface tension may equivalently be reported as energy per unit area: J·m⁻² in SI or erg·cm⁻² in cgs. Converting (1 erg = 10⁻⁷ J, 1 cm² = 10⁻⁴ m²) gives 1 erg·cm⁻² = (10⁻⁷ J)/(10⁻⁴ m²) = 10⁻³ J·m⁻² = 0.001 J·m⁻², demonstrating the dimensional identity N·m⁻¹ ≡ J·m⁻². Consequently, any surface‑tension value can be stated interchangeably as force/length or energy/area (for example, 1 dyn·cm⁻¹ = 0.001 N·m⁻¹ = 0.001 J·m⁻²).
Definition
Surface tension is the property of a liquid surface that resists an increase in area; mechanically it appears as a tangential force distributed along a line in the surface. If a force F acts along a line of length L, the surface tension γ is defined by γ = F/L and is measured in newtons per metre (N·m⁻¹). The magnitude of the required force therefore scales with both the intrinsic surface tension of the material and the length of contact over which the force is applied.
Equivalently, surface tension can be understood energetically as the incremental energy needed to create additional surface area. In this formulation γ = dE/dA, where dE is the work required to produce an incremental area dA; the corresponding unit is joules per square metre (J·m⁻²). The two definitions are consistent because their units are dimensionally identical (1 N·m⁻¹ = 1 J·m⁻²), and the work required to enlarge a surface by ΔA is W = γ·ΔA.
In practical systems the mechanical and energetic descriptions coincide: a pulling force must overcome surface tension acting over the relevant contact length, and when a film has two exposed interfaces (for example a soap film) both surfaces contribute, effectively doubling the force and the energetic cost to increase the film’s area.
Surface tension, denoted γ, is the lateral force per unit length acting along a liquid surface and is determined by intrinsic factors such as composition and temperature. In a standard experiment a liquid film spans a rectangular frame with three fixed sides and a fourth side free to slide; the film meets this movable side on both faces. The film’s surface tension produces a horizontal pull on the movable side, so an external opposing force F must be applied to maintain equilibrium. The magnitude of F is proportional to the length L of the movable edge, and the ratio F/L is characteristic of the liquid alone—independent of the overall frame geometry. Because the film presents two free surfaces that each contribute equally to the lateral pull, the surface-tension coefficient is given quantitatively by γ = F/(2L); equivalently, each individual surface exerts a force γL = F/2 on the movable side.
Surface tension γ quantifies the increase in a liquid’s interfacial energy per unit increase of surface area, defined thermodynamically as γ = ΔE/ΔA. The same quantity can be derived mechanically for a thin liquid film: if a film edge of length L is displaced by Δx under a lateral force F, the work done is W = FΔx while the film’s exposed area increases by ΔA = 2LΔx (the factor 2 accounts for the two free surfaces). Substituting gives the identity
γ = W/ΔA = F/(2L),
so γ may be viewed equivalently as energy per unit area or as force per unit length acting along the surface. The force F represents the magnitude required either to prevent the edge from beginning to move or, by Newton’s second law, to sustain motion at constant speed; when the edge moves in the force direction the applied work is stored as potential energy of the interface. Because mechanical systems seek to minimize stored potential energy, an unconstrained liquid adopts a shape (a sphere) that minimizes surface area for a given volume. In SI units γ is expressed as joules per square metre (J m−2) and equivalently as newtons per metre (N m−1); dimensional analysis shows that energy per area and force per length are consistent units, supporting the dual interpretation. The explicit factor of 2 in γ = F/(2L) pertains to thin films with two interfaces; more generally, the principle that creating additional interfacial area requires work proportional to that area applies across films, droplets and other interfacial phenomena, with numerical prefactors determined by the number and geometry of interfaces.
Water
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Surface tension in water arises from strong cohesive forces among molecules that favor configurations minimizing exposed surface area. On hydrophobic substrates—such as waxy plant leaves—weak adhesion to the solid combined with strong cohesion within the liquid causes water to retract into discrete, nearly spherical drops; the sphere minimizes surface area for a given volume, so surface tension drives the beading morphology.
When liquid accumulates at a constriction (for example, a faucet) the droplet grows until gravitational force exceeds the tensile restoring force provided by surface tension, at which point the neck pinches off and the detached drop relaxes toward a spherical shape. A continuous vertical jet undergoes an analogous capillary (Rayleigh–Plateau) instability: stretching by gravity and surface-tension-driven necking cause the stream to fragment into individual droplets during free fall.
A nonwetting object can be supported by a liquid surface even if its bulk density exceeds that of the liquid because the contact prevents molecular attraction and the surface deforms elastically. A water-strider’s hydrophobic legs depress the surface, increasing local curvature and area; the resulting surface-tension forces that act to restore minimal curvature produce an upward component that can balance the insect’s weight provided the surface-tension forces are sufficient.
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At boundaries between two immiscible liquids (e.g., water and liquid wax) an interfacial tension exists that is physically analogous to liquid–gas surface tension. This interface tension promotes macroscopic phase separation and discrete domains of each liquid, as seen in devices such as lava lamps where immiscible components remain distinct.
Spatial variations in surface tension within mixed liquids drive flow: for instance, ethanol lowers the surface tension of water and evaporates more rapidly, producing surface-tension gradients that induce Marangoni flows. In a wine glass this process concentrates liquid into upward-moving rivulets that eventually form the characteristic “tears of wine” drops and streaks.
Representative visual examples: (A) beaded water on a waxy leaf illustrating hydrophobic adhesion versus cohesion; (B) a droplet forming and detaching from a tap showing mass accumulation and capillary breakup; (C) water striders on a pond surface demonstrating nonwettability and elastic-film behavior of the liquid surface; (D) a lava lamp displaying interfacial tension and phase separation between immiscible liquids; (E) the “tears of wine” pattern in an alcoholic mixture, evidencing surface-tension gradients and Marangoni-driven flows.
Surfactants
Surfactants adsorb to liquid–air and liquid–liquid interfaces and thereby govern the mechanical behavior of thin films and dispersed droplets by altering interfacial energetics. In aqueous systems they can lower surface tension by more than a factor of three and impart surface viscoelasticity (elasticity and surface viscosity), which slows film drainage, modifies rupture kinetics, and increases the interface’s resistance to deformation. Without such agents, thin spherical films (bubbles) with very large surface-area-to-mass ratios are intrinsically unstable: unmitigated surface tension drives thinning and rapid rupture.
Nonuniform surfactant distributions produce surface-tension gradients that drive Marangoni flows; these surface-driven circulations counter local thinning, re-distribute liquid to thickened regions, and are a principal mechanism by which soap films and bubbles gain longevity. In emulsions—colloidal dispersions of one liquid in another—interfacial tension controls droplet coalescence and phase separation because high interfacial energy favors reduction of total interfacial area. By reducing interfacial tension, surfactants enable the formation and persistence of oil-in-water or water-in-oil droplet ensembles.
The stability and microstructure of surfactant-stabilized emulsions are determined by a complex interplay of chemical and physical factors: surfactant identity and concentration, ionic strength and pH, temperature, shear history, droplet-size distribution, viscosities of the continuous and dispersed phases, and the rheology of the interface, together with kinetic processes such as coalescence, flocculation and Ostwald ripening. Thus, long-term stability reflects a balance between lowered interfacial energy afforded by surfactants and the various destabilizing kinetic pathways that remain active.
Surface curvature and pressure
Local bending of a fluid interface produces tangential surface-tension forces whose non-cancelling components act normal to the surface patch; if a net normal force is present the surface must deform until equilibrium is restored. In mechanical equilibrium a pressure discontinuity across the interface is balanced by the normal components of surface tension, so a nonzero pressure jump requires curvature of the surface. Balancing these forces yields the Young–Laplace relation,
Δp = γ (1/Rx + 1/Ry),
where Δp is the pressure difference across the interface (Laplace pressure), γ is the surface tension, and Rx and Ry are the principal radii of curvature in two orthogonal tangent directions. The combination (1/Rx + 1/Ry) is proportional to the mean curvature of the surface (in the common convention it equals twice the mean curvature), so the Young–Laplace equation directly links Laplace pressure with mean curvature and surface tension.
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Solutions of the Young–Laplace equation determine the equilibrium shapes and internal pressures of capillary-controlled systems such as drops, puddles, menisci at contact lines, soap bubbles and localized surface deformations (e.g., impressions from an insect’s foot). The Laplace pressure grows as curvature increases (radius decreases): for water at STP one finds Δp ≈ 0.0014 atm for a 1 mm droplet, ≈ 0.0144 atm for 0.1 mm, ≈ 1.436 atm for 1 μm, and ≈ 143.6 atm for 10 nm. Thus the effect is modest for macroscopic droplets but becomes dominant at submicron scales; in the molecular limit the continuum concepts of surface tension and the Young–Laplace relation lose validity.
A slender needle resting on a liquid surface depresses the interface at two symmetric contact lines; at each contact the surface-tension force acts tangentially to the deformed interface and is inclined by an angle θ to the horizontal. Equilibrium of vertical forces requires the needle’s weight F_w to be balanced by the sum of the upward vertical components of these two surface‑tension resultants, so F_w = 2 F_s sin θ. Since the magnitude of each resultant equals the surface tension γ times the contact length L (F_s = γL), the equilibrium condition can be written m g = 2 γ L sin θ (with m the needle mass and g gravitational acceleration). The horizontal components of the two surface‑tension forces are equal and opposite and therefore cancel, so only the vertical components contribute to supporting the weight.
The ability of surface tension to sustain the needle therefore depends on the contact geometry and the object’s mass: a sufficiently large γL sin θ is required so that 2 γ L sin θ ≥ m g. As θ decreases (the interface flattens at the contact), each vertical component F_s sin θ diminishes and the total support falls, making the floating state potentially marginally stable—small perturbations that reduce θ can lead to sinking. In practice the object’s surface must also resist wetting (so the interface retains the upward‑curved geometry at the contact) and the mass must be small enough that surface tension, possibly aided by buoyancy, can prevent immersion.
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Liquid surface
Determining the exact geometry of a minimal surface spanning an arbitrarily shaped fixed frame is a challenging boundary‑value problem; nevertheless, a simple laboratory procedure—shaping a wire frame and immersing it in soap solution—produces a stable, locally minimal film within seconds, offering an immediate empirical realization of the mathematical solution. Physically, the soap film’s form is governed by the Young–Laplace relation, which ties the jump in pressure across a fluid interface to the surface’s mean curvature; this relation converts the variational task of area minimization into a balance between pressure and curvature. In the case of an open film spanning a frame the pressure difference is effectively zero, so the mean curvature must vanish and the film attains a minimal surface. More generally, minimal surfaces are those whose mean curvature is zero at every smooth point; soap films exemplify such locally area‑minimizing configurations constrained by boundary curves, with the imposed frame selecting a particular (not necessarily global) minimizer.
Contact angles
Surface tension is an interfacial property and therefore depends on which two phases form the interface; a liquid will generally have different surface‑tension coefficients at liquid–air and liquid–solid boundaries. Where three phases meet (solid, liquid, gas) the geometry of the liquid surface at the contact line is described by the contact angle θ, defined as the angle between the tangent to the liquid surface and the solid, measured through the liquid. This angle determines the meniscus curvature: θ < 90° produces a concave meniscus (liquid wets the solid), whereas θ > 90° yields a convex meniscus (poor wetting).
Mechanical equilibrium at the three‑phase contact line requires that the surface tensions acting along the three interfaces balance their horizontal and vertical components. Horizontally, the adhesive force along the solid balances the horizontal component of the liquid–air tension (per unit length): fA = f_la sin θ. Vertically, the balance of forces along the solid surface yields f_ls − f_sa = − f_la cos θ. Because force per unit length is equal to the corresponding surface‑tension coefficient, this vertical balance is commonly written in terms of surface tensions (Young’s relation):
γ_ls − γ_sa = − γ_la cos θ,
or equivalently γ_sa = γ_ls + γ_la cos θ.
The sign of γ_ls − γ_sa (or the sign of cos θ) determines wetting behaviour: if γ_ls − γ_sa > 0 the contact angle exceeds 90° (non‑wetting); if γ_ls − γ_sa < 0 the contact angle is less than 90° (wetting). In practice γ_ls − γ_sa is difficult to measure directly; instead the liquid–air surface tension γ_la (readily measured) and the equilibrium contact angle θ (obtained experimentally from advancing and receding contact angles) are used with Young’s relation to infer the solid–related terms.
Selected empirical examples illustrate the range of contact angles: water on soda‑lime glass, lead glass or fused quartz → θ ≈ 0° (complete wetting); methyl iodide on soda‑lime glass, lead glass, fused quartz → θ ≈ 29°, 30°, 33° respectively; water on paraffin wax → θ ≈ 107°; silver (as a solid substrate) → θ ≈ 90°; mercury on soda‑lime glass → θ ≈ 140°. Other low‑angle cases include common organic liquids such as ethanol, diethyl ether, carbon tetrachloride, glycerol and acetic acid on wetting substrates.
In the context of contact angles at a three‑phase (liquid–solid–air) boundary, wetting behaviour is governed by the balance of the three interfacial tensions: γ_la (liquid–air), γ_ls (liquid–solid) and γ_sa (solid–air). A contact angle of θ = 90° corresponds to a neutral wetting state in which the liquid–solid and solid–air tensions are equal in magnitude; their difference vanishes and the free‑surface meets the solid at a right angle. At the opposite extreme, θ → 180° represents maximal non‑wetting: the liquid virtually detaches from the substrate and forms an almost spherical cap, as can be approached experimentally with water on appropriately treated Teflon. In this limit the interfacial tensions satisfy γ_la = γ_ls − γ_sa (with γ_la > 0), reflecting the constraint on γ_ls and γ_sa that produces the largest possible contact angle. Thus θ = 90° signals a tension balance (neutral wetting), whereas θ = 180° indicates conditions under which liquid–solid adhesion is effectively negligible and the drop attains maximal non‑wetting geometry.
A traditional mercury barometer consists of a vertical glass tube (≈1 cm diameter, r ≈ 0.5 cm) containing a mercury column capped by a vacuum (Torricelli’s vacuum). The exposed mercury surface in such a tube adopts a convex, dome-like meniscus: this shape simultaneously reduces the liquid’s exposed surface area and shifts the column’s center of mass slightly downward relative to a flat top. Those competing geometric and energetic effects—surface-energy reduction versus gravitational potential—are resolved by the meniscus shape that minimizes the total potential energy of the column.
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The convex form reflects mercury’s non‑wetting behavior on glass: mercury does not adhere to glass, so the liquid–air surface tension acts over the full exposed surface including regions adjacent to the wall, producing a raised center. By contrast, when mercury strongly adheres to the container (e.g., copper), the meniscus is concave because the system lowers its energy by increasing liquid–solid contact; this is sometimes described qualitatively as an effective negative contribution from the liquid–container interface, whereby enlarging the contact area outweighs the gravitational cost of raising fluid near the walls.
When capillary dimensions are small and adhesion is significant, surface tension drives capillary rise or depression. Jurin’s law gives the equilibrium height h of the liquid column:
h = (2 γ_la cos θ) / (ρ g r),
where γ_la is the liquid–air surface tension, θ the contact angle at the wall, ρ the liquid density, g gravitational acceleration, and r the capillary radius. The sign and magnitude of cos θ determine whether the liquid is drawn up (θ < 90°, cos θ > 0) or depressed (θ > 90°, cos θ < 0); mercury on glass is a classic non‑wetting example that yields a negative h. Jurin’s law therefore implies h ∝ γ_la cos θ and h ∝ 1/(ρ g r): increasing surface tension, improving wetting (larger cos θ), or using a narrower capillary increases rise, whereas greater density, stronger gravity, or a larger radius reduce it.
A liquid puddle on a smooth, horizontal, non‑adhesive substrate attains a finite equilibrium thickness because gravity, which favors spreading to reduce potential energy, is opposed by surface tension, which resists increases in surface area. The relevant capillary–gravity length scale is
H = 2 sqrt(γ/(g ρ)),
where γ is the liquid–air surface tension, g is gravitational acceleration and ρ is liquid density (use a consistent cm–g–s or m–kg–s unit set). For the ideal non‑wetting limit (contact angle θ = 180°) the equilibrium depth equals this length,
h(θ = 180°) = 2 sqrt(γ/(g ρ)) = H,
and the lateral meniscus profile measured from a reference x0 is given implicitly by
x − x0 = (1/2) H arcosh(H/h) − H sqrt(1 − h^2/H^2),
which describes the puddle edge when the liquid does not wet the substrate.
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For realistic contact angles (θ < 180°) the equilibrium depth is reduced according to
h = sqrt[ 2 γ (1 − cos θ) / (g ρ) ],
so increased wettability (smaller θ) lowers h, with h → 0 as θ → 0° (complete wetting and formation of a micro‑thin film). Because most real surfaces are not perfectly non‑wetting, observed puddle thicknesses are typically somewhat less than the θ = 180° prediction; the relations above quantify how γ, ρ, g and θ set both the puddle depth and the meniscus shape.
Two illustrative calculations: for mercury on glass (γ ≈ 487 dyn/cm, ρ ≈ 13.5 g/cm3, θ ≈ 140°) the formula yields h ≈ 0.36 cm, consistent with puddles that remain several millimetres thick rather than spreading arbitrarily thin. For water on paraffin at 25 °C (γ ≈ 72 dyn/cm, ρ ≈ 1.0 g/cm3, θ ≈ 107°) the same expression gives h ≈ 0.44 cm, showing that non‑wetting substrates produce perceptible puddle thicknesses even for low‑surface‑tension liquids.
Breakup of streams into drops
The fragmentation of a continuous liquid jet into separate droplets is governed by the Plateau–Rayleigh instability, a process driven solely by surface tension acting on an initially cylindrical column of fluid. Any real jet contains arbitrarily small shape perturbations; these can be represented as a superposition of sinusoidal (Fourier) modes, each specified by a wave number that quantifies its spatial frequency along the column.
Each Fourier component evolves independently: depending on its wave number and on the jet’s radius a given mode either decays or grows, and the growth/decay rates are determined entirely by those two parameters. Because different unstable modes amplify at different rates, a subset of modes becomes dominant, producing a characteristic pattern of necking and pinch-off that sets the spacing and timing of resulting droplets. Consequently, even a visually smooth jet will inevitably disintegrate into droplets as surface-tension-driven mode selection amplifies unavoidable perturbations.
Gallery
At small length scales the mechanics of liquid interfaces are dominated by surface tension, which acts along the free surface to minimize area and resist deformation. In practical situations this line tension couples with gravity, inertia and internal pneumatic pressure to set the shape of menisci, films and free-surface geometries and to determine whether an object is supported or submerges.
When a moving sheet of water strikes and rebounds from a spoon, inertial stretching and momentum tend to fling the sheet outward while capillary forces work to contract the surface. The transient result is an expanding thin sheet that may develop a thickened rim and, as capillary instabilities overcome cohesion, break into ligaments and droplets. Similarly, a continuous sheet of flow that adheres to a human hand is sustained by a combination of surface tension and liquid–solid adhesion: the contact line and local contact angle control how the free surface meets skin and how the film rises and wraps around edges.
Thin liquid films, such as soap bubbles, owe their stability to a balance between surface tension acting along the curved film and the pressure difference between the bubble interior and the surrounding air; curvature generates the Young–Laplace pressure jump that inflates the bubble and counters film tension. Objects denser than water can nonetheless be held aloft when the liquid surface is deformed so that the net upward force (including contributions from the raised meniscus) offsets weight. An aluminium coin on water at 10 °C exemplifies this marginal equilibrium: small additional loads destroy the balance and cause sinking. A daisy positioned entirely below the undisturbed free surface illustrates capillary containment, where surface tension draws the liquid smoothly around petal edges and prevents water from intruding into inter‑petal air spaces, thereby preserving an internal air pocket.
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A fine wire object such as a metal paper clip floats because the liquid surface is depressed around it, producing an upward resultant at the contact line; several clips can be added until the cumulative deformation and weight exceed the surface-tension capacity and the interface ruptures. The subtle interface shapes produced by such loads can be visualized by placing a grille before a light source: contour-line reflections reveal the meniscus profile and local slopes of the deformed surface.
Across these examples the same physical elements recur: surface tension, wetting (contact angle and adhesion), curvature (and the associated Young–Laplace pressure), hydrostatic pressure and external forces (weight and inertia). The observed small-scale geometries — whether a sheet remains coherent or fragments, an object floats or sinks, or a bubble persists — reflect the delicate balance among these factors.
Thermodynamic theories of surface tension
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Gibbs established a rigorous thermodynamic description of capillarity by introducing an arbitrary mathematical dividing surface inside the microscopically diffuse physical interface and by treating that surface as a subsystem with possible excess energy, entropy and particle content. The natural thermodynamic potential for such a subsystem is the grand potential, Ω = U − T S − Σi μi Ni. For a control volume V that contains the dividing surface, V is partitioned into subvolumes A and B (V = VA + VB) and the total grand potential can be written as the bulk contributions −pA VA − pB VB plus a surface term ΩS, so that Ω = −pA VA − pB VB + ΩS. In the macroscopic, gently curved limit the surface contribution is proportional to area, ΩS = γA, where γ is the surface free energy (surface tension); consequently the reversible mechanical work to increase the interfacial area at fixed adjacent volumes is dW = γ dA. At constant temperature and chemical potentials spontaneous processes lower Ω (equivalently increase total entropy when exchanges with the bulk are accounted for), which explains the tendency of isolated liquid masses to minimize surface area unless countervailing energy couplings exist.
Because the mathematical dividing surface may be placed arbitrarily at the microscopic scale, the numerical values of surface excess quantities (surface entropy, excess mass densities, surface internal energy) depend on that placement and appear as partial derivatives of γ(T, μ1, μ2, …). By contrast, for macroscopic or planar interfaces the scalar value of γ is essentially independent of the chosen surface. For highly curved, microscopic interfaces the approximation of a size-independent γ fails: curvature-dependent corrections (e.g., those associated with the Tolman length) must be introduced and surface thermodynamics modified for small radii of curvature. Gibbs also distinguished scalar surface free energy (energy cost to create new area) from tensorial surface stress (work required to stretch an existing surface); for fluid–fluid interfaces formation and stretching coincide, but for solids stretching alters the surface state and surface stress is direction-dependent.
An alternative, developed by van der Waals about fifteen years after Gibbs, replaces the sharp dividing surface by a continuous density field and augments the local energy density with a gradient term c(∇ρ)2 (c the capillarity coefficient). The continuous-density (square-gradient) theory reproduces Gibbs’s equilibrium results while providing a natural framework for interfacial structure and the dynamics of phase transitions; the c(∇ρ)2 term is widely employed in phase-field models of multiphase flows and in non‑equilibrium descriptions of gas dynamics.
For an ideal spherical bubble let p_A and p_B denote the internal and external pressures, respectively, ΔP = p_A − p_B the pressure jump across the interface, V_A the bubble volume, A its surface area, γ the surface tension, and Ω the thermodynamic free energy. One convenient form of the free energy is Ω = −ΔP V_A − p_B V + γ A, and mechanical equilibrium of the bubble is obtained from the stationarity condition dΩ = 0. This variational condition yields the differential balance ΔP dV_A = γ dA, which relates infinitesimal changes of the bubble volume and surface area to the pressure difference and surface tension.
For a sphere of radius R, V_A = (4/3)πR^3 and dV_A = 4πR^2 dR, while A = 4πR^2 and dA = 8πR dR. Substituting these differentials into ΔP dV_A = γ dA and cancelling the common factor 4π dR gives the algebraic relation ΔP = 2γ/R. Physically, this Young–Laplace result for a sphere implies that smaller bubbles require a higher internal pressure relative to the exterior to balance surface tension. The spherical form is a special case of the general Young–Laplace law ΔP = γ(1/R_x + 1/R_y), obtained when the two principal radii of curvature are equal (R_x = R_y = R), which reduces to ΔP = 2γ/R.
Influence of temperature
Surface tension at a liquid–vapor interface is a strong function of temperature: γ typically falls as temperature T rises and vanishes at the critical temperature T_C, where liquid and vapor become indistinguishable. For this reason any reported value of γ must state the temperature of measurement. Because no single exact theoretical law links γ and T across all fluids and conditions, practical work relies on empirical relations chosen to reproduce the observed decrease of γ and, for physically consistent forms, the limit γ → 0 at T = T_C.
A longstanding macroscopic rule is the Eötvös relation, written γ V^{2/3} = k (T_C − T), with V the molar volume and k an approximately universal constant (k ≈ 2.1×10^−7 J K^−1 mol^−2/3). Applied to water, for example, one may use V ≈ 18 mL·mol^−1 and T_C ≈ 647 K to estimate γ. A modified form proposed by Ramay and Shields inserts a 6 K offset, γ V^{2/3} = k (T_C − T − 6 K), which typically improves agreement with experimental values at lower temperatures but no longer enforces γ = 0 at T = T_C.
An alternative and widely used empirical scaling is the Guggenheim–Katayama form, γ = γ° (1 − T/T_C)^n, in which γ° and the exponent n are fitted for each liquid; this form explicitly satisfies the critical-point endpoint. For many organic liquids n ≈ 11/9 has been found to give good agreement. Van der Waals suggested estimating the scale parameter from critical properties, γ° = K_2 T_C^{1/3} P_C^{2/3} (with P_C the critical pressure), but experimental work shows K_2 is not strictly universal and varies among substances.
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In practice the choice among these empirical forms balances the required accuracy across a given temperature range against the availability of molar and critical data: both the original Eötvös relation and the Guggenheim–Katayama law respect the γ → 0 endpoint, whereas the Ramay–Shields variant may be preferred for low‑temperature fits despite its incorrect critical‑point behavior.
Solute-induced changes in interfacial tension fall into distinct empirical patterns determined by both the solute identity and the specific interface. Some solutes produce negligible change (for example, sugars at the water–air interface and many organics at oil–air), whereas most inorganic salts raise the water–air surface tension. Certain solutes (notably many inorganic acids) induce non‑monotonic variation with concentration. Amphiphilic molecules, such as short-chain alcohols, typically reduce surface tension progressively as their bulk concentration increases, while classical surfactants lower γ up to a characteristic concentration beyond which further bulk addition has little effect (the critical micelle concentration).
Linking bulk composition to the interfacial state requires recognizing that the surface composition need not equal the bulk composition; a solute can be either enriched or depleted at the interface relative to the bulk, and the sign and magnitude of this surface excess depend on the specific solute–solvent and solute–interface interactions. Quantitatively, for a dilute two‑component solution the Gibbs adsorption isotherm relates the surface excess Γ to changes in surface tension γ via
Γ = −(1/RT) (∂γ/∂ln C)_{T,P},
where Γ is the excess amount of solute per unit interfacial area (units mol m−2), C is the bulk concentration, R is the gas constant, and T is the absolute temperature. Because Γ is proportional to −(∂γ/∂ln C), a negative gradient of γ with respect to ln C indicates net adsorption (Γ > 0) and corresponds to a decrease in γ with increasing bulk concentration; a positive gradient indicates net depletion (Γ < 0) and an increase in γ.
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This simple Gibbs relation applies under the assumptions used in its derivation—essentially ideal, very dilute, two‑component systems without interfacial chemical reactions. Multicomponent solutions, nonideal behavior at higher concentrations, and reactive or structurally complex interfaces require extended or alternative theoretical treatments.
The Kelvin equation, obtained by combining surface-tension effects with the Clausius–Clapeyron relation, predicts that a curved liquid interface exhibits a higher equilibrium vapor pressure than a planar one: the increased internal pressure of a small droplet shifts the vapor–liquid equilibrium toward greater vapor concentration. In quantitative form,
P_v^{fog} = P_v^{\circ} \exp!\left(\frac{2\gamma V}{R T r_k}\right),
where P_v^{\circ} is the standard vapor pressure at the given temperature, P_v^{fog} the equilibrium vapor pressure in the presence of a droplet, γ the liquid surface tension, V the molar volume, R the gas constant, T the absolute temperature, and r_k the droplet (Kelvin) radius. Microscopically, curvature reduces the average number of nearest neighbors for surface molecules and thus weakens cohesive binding, which raises the propensity for molecules to enter the vapor phase and therefore increases equilibrium vapor pressure for small radii. As a consequence, homogeneous droplet formation in pure vapor requires supersaturation: vapor pressures must exceed the flat-interface vapor pressure by an amount set by the Kelvin relation before nanoscopic droplets become thermodynamically stable and can subsequently grow at lower vapor pressures. Numerical values for water at standard conditions illustrate the steep size dependence: P/P_0 ≈ 1.001 at r = 1000 nm, 1.011 at 100 nm, 1.114 at 10 nm, and about 2.95 at 1 nm, so the effect is negligible at micrometer scales but becomes dominant in the nanometer regime. This curvature-dependent vapor pressure underpins practical techniques such as pore-size characterization by capillary condensation in mesoporous catalysts and other porous solids. Finally, when droplet radii approach ∼1 nm (containing on the order of 100 molecules), continuum thermodynamics ceases to be reliable and molecular- or quantum-level descriptions are required for accurate thermodynamic predictions.
Methods of measurement
Selection of a tensiometer and an appropriate measurement protocol depends on the liquid pair’s physical properties, the temporal and thermal regime of interest, and the mechanical stability of the interface under deformation. Measurement techniques exploit distinct observables—equilibrium forces, capillary heights, shape geometries, pressure transients, or oscillation frequencies—and differ in their sensitivity to wetting, sample volume, interfacial age and tension magnitude.
Force-based techniques infer tension from a measured vertical force at an interface. The classical Du Noüy ring method records the maximum upward pull on a horizontal ring as it is withdrawn through an interface; because it is relatively insensitive to wetting details, it remains widely used for equilibrium tensions. The Wilhelmy plate method suspends a vertical plate from a balance and converts the wetting-generated vertical force into surface tension; its continuous measurement capability makes it well suited to long-duration monitoring of surface tension. A miniaturized Du Noüy–Padday variant replaces the ring with a small-diameter needle or rod and a microbalance, enabling high-precision measurements on microliter samples and rapid determinations without large buoyancy corrections for appropriately shaped probes.
Shape-analysis methods determine tension by comparing observed drop geometries to theoretical capillary shapes. Pendant-drop analysis images a hanging droplet and applies shape-to-tension relations to its profile; it is adaptable to elevated temperatures and pressures and yields reliable interfacial tensions from a single droplet. The sessile drop technique measures the contact angle and shape of a droplet on a solid substrate to extract surface tension (and often density), but results are sensitive to solid–liquid wetting behavior. The drop-volume and stalagmometric approaches are gravimetric or time-based shape methods: drop-volume measures the volume or timing between successive detachments to infer interfacial tension as the interface ages, while stalagmometry uses mass or drop counts per detachment to derive tension.
Methods tailored to specific regimes include the spinning-drop technique for very low interfacial tensions, in which a heavy phase in a rotating capillary elongates a lighter drop; the equilibrium diameter under rotation is used to calculate extremely small tensions. The capillary-rise method is a static, classical approach in which the equilibrium height of liquid in a tube gives surface tension via the capillary equation, explicitly accounting for tube radius, density, contact angle and gravity.
Dynamic and short-age measurements capture instantaneous or time-dependent interfacial properties. The bubble-pressure (Jaeger) method generates bubbles at a capillary and relates the peak formation pressure of each bubble to the instantaneous surface tension of the newly forming interface; it is therefore useful for surfactant dynamics at short surface ages. Drop-volume measurements, when timed or volumetrically resolved, likewise provide tension as a function of interface age during detachment events.
Oscillatory and levitation-based methods derive tension from natural or driven resonances of free droplets. Resonant oscillation techniques drive spherical or hemispherical pendant drops with a modulated electric field and analyze frequency spectra to extract both surface tension and viscosity. Magnetically levitated droplet methods determine tension from the vibrational eigenfrequencies of levitated drops; this approach has been applied to extreme systems such as superfluid 4He (reported ≈0.375 dyn cm−1 at T = 0 K). Aerodynamic levitation with controlled drop impact (drop-bounce) records transient mid-air oscillations as a levitated droplet deforms and recoils, permitting simultaneous estimation of surface tension and viscosity from the decay and frequency of the oscillations.
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In practice, method choice balances sample volume, required temporal resolution, temperature/pressure constraints, sensitivity to wetting and buoyancy artifacts, and the magnitude of the tension to be measured. Combining complementary techniques is common when a single method cannot satisfy all experimental constraints.
Surface-tension data summary
Surface tension values are reported in dyn/cm (equivalent to mN/m); composition percentages denote mass percent of the components. The data span cryogenic liquids, common molecular solvents, aqueous mixtures, polyhydric and aprotic oxygenated liquids, halogenated and dense elemental fluids, molten ionic salts, and liquid metals, permitting direct comparison of interfacial cohesion across very different chemical classes and temperatures.
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Pure water provides a useful baseline: γ decreases monotonically with temperature, from 75.64 dyn/cm at 0 °C to 71.97 dyn/cm at 25 °C, 67.91 dyn/cm at 50 °C and 58.85 dyn/cm at 100 °C. Dissolved salts, strong acids and concentrated sugars can shift γ relative to pure water; for example, 6.0 M NaCl (20 °C) yields 82.55 dyn/cm, a 55% sucrose solution (20 °C) gives 76.45 dyn/cm, and concentrated HCl (17.7 M, 20 °C) is 65.95 dyn/cm, demonstrating composition-dependent increases or decreases toward the water value.
Nonpolar hydrocarbons and simple volatile oxygenates occupy the low end of the room-temperature molecular liquids: linear alkanes (n‑hexane → n‑nonane) range ≈18–23 dyn/cm, aromatic solvents (benzene, toluene) ≈28 dyn/cm, and small ethers/ketones (diethyl ether, acetone) ≈17–24 dyn/cm. Short-chain monohydric alcohols have similarly low γ (~22 dyn/cm), while adding water raises mixture γ toward that of pure water (e.g., ethanol–water mixtures: 40% ethanol ≈29.6 dyn/cm at 25 °C, 11.1% ethanol ≈46.0 dyn/cm at 25 °C).
Liquids with stronger dipolar interactions or greater hydrogen-bonding capacity show substantially higher surface tensions: ethylene glycol (25 °C) ≈47.3 dyn/cm, glycerol (20 °C) ≈63.0 dyn/cm, and aprotic but highly polar solvents such as propylene carbonate (20 °C) ≈41.1 dyn/cm and DMSO (20 °C) ≈43.5 dyn/cm. Heavy, highly polarizable halogenated liquids can give variable, sometimes elevated γ (methylene iodide ≈67.0 dyn/cm; chloroform, CCl4 ≈26–27 dyn/cm).
Aqueous carboxylic-acid mixtures illustrate the approach toward water with dilution: pure acetic acid ≈27.6 dyn/cm (20 °C), whereas aqueous acetic acid mixtures increase to ≈40.7 dyn/cm (45.1% at 30 °C) and ≈54.6 dyn/cm (10.0% at 30 °C), depending on mass fraction. Complex biological fluids such as blood (22 °C) have intermediate γ (≈55.9 dyn/cm), reflecting the combined effects of proteins, lipids and solutes on the interface.
Molten ionic salts and liquid metals display very high surface tensions, consistent with strong ionic or metallic cohesion at the liquid–vapor boundary: molten NaCl (1073 °C) ≈115 dyn/cm, a NaCl–CaCl2 eutectic (650 °C) ≈139 dyn/cm, molten AgCl (650 °C) ≈163 dyn/cm, while mercury shows exceptionally large γ ≈486–487 dyn/cm near room temperature.
Cryogenic fluids exhibit the smallest γ values: liquid nitrogen (−196 °C) ≈8.85 dyn/cm, liquid oxygen (−182 °C) ≈13.2 dyn/cm, and superfluid helium II (≈−273 °C) an exceptionally low ≈0.37 dyn/cm, reflecting very weak interfacial forces in these low-temperature states.
Overall trend: the lowest surface tensions (~0.4–29 dyn/cm) are found for cryogenic fluids, nonpolar hydrocarbons and small volatile organics; intermediate-to-high γ (≈40–82 dyn/cm) characterizes polar, hydrogen-bonding liquids and concentrated aqueous solutions; and the highest values (>100 dyn/cm, up to several hundred dyn/cm) occur for molten ionic salts and liquid metals, where strong electrostatic or metallic bonding dominates interfacial energy.
Surface tension of water
The surface tension of pure liquid water in equilibrium with its vapor is described empirically by the relation
γw = 235.8 (1 − T/TC)^1.256 [1 − 0.625 (1 − T/TC)] mN·m^−1,
where T and the critical temperature TC are given in kelvins. The numerical prefactor (235.8), the exponent (1.256) and the secondary coefficient (0.625) are fixed constants in this formulation, and the combination (1 − T/TC) (the reduced temperature) governs the temperature dependence of γw. The critical temperature used is TC = 647.096 K.
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This expression was developed to represent surface tension along the liquid–vapor saturation curve, and it is intended to be valid from the triple point of water (0.01 °C) up to the critical point (TC), i.e., across the coexistence line between liquid and vapor. When extrapolated beyond the equilibrium freezing point it gives physically reasonable values under metastable (supercooled) conditions and has been shown to remain useful down to about −25 °C for practical estimates.
The formulation has been codified by the International Association for the Properties of Water and Steam (IAPWS): it was first adopted in 1976 and updated in 1994 to accord with the International Temperature Scale of 1990 (ITS‑90), ensuring consistency with contemporary temperature traceability. Reported uncertainties accompany the recommendation: for temperatures below 100 °C the relative uncertainty is about ±0.5%, indicating the expected accuracy of γw values obtained from this equation in the common low‑temperature range.
Surface-tension measurements of seawater were performed at atmospheric pressure across salinities of 20–131 g kg−1 and temperatures of 1–92 °C, a range chosen to encompass both natural oceanographic conditions and environments encountered in thermal desalination. Individual measurement uncertainties lay between 0.18 and 0.37 mN m−1 (mean 0.22 mN m−1). The data are represented by the empirical relation
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γ_sw = γ_w [1 + (3.766 × 10−4) S + (2.347 × 10−6) S t],
where γ_sw and γ_w are seawater and pure-water surface tensions in mN m−1, S is reference salinity in g kg−1, and t is temperature in °C; the first numerical term expresses the linear dependence on salinity and the second term captures the salinity–temperature interaction. This formulation reproduces the measurements with high fidelity (mean absolute percentage deviation 0.19%, maximum 0.60%) and has been endorsed by the International Association for the Properties of Water and Steam (IAPWS) as a standard guideline for both oceanographic research and engineering applications such as thermal desalination.