Surface tension

Surface tension, represented by the symbol γ (alternatively σ or T), is measured in force per unit length. Its SI unit is newton per meter but the cgs unit of dyne per centimeter is also used. For example,[6]

Surface tension can be defined in terms of force or energy.

Surface tension γ of a liquid is the force per unit length. In the illustration on the right, the rectangular frame, composed of three unmovable sides (black) that form a "U" shape, and a fourth movable side (blue) that can slide to the right. Surface tension will pull the blue bar to the left; the force F required to hold the movable side is proportional to the length L of the immobile side. Thus the ratio F/L depends only on the intrinsic properties of the liquid (composition, temperature, etc.), not on its geometry. For example, if the frame had a more complicated shape, the ratio F/L, with L the length of the movable side and F the force required to stop it from sliding, is found to be the same for all shapes. We therefore define the surface tension as

The reason for the 1/2 is that the film has two sides (two surfaces), each of which contributes equally to the force; so the force contributed by a single side is γL = F/2.

Surface tension γ of a liquid is the ratio of the change in the energy of the liquid to the change in the surface area of the liquid (that led to the change in energy). This can be easily related to the previous definition in terms of force:[7] if F is the force required to stop the side from starting to slide, then this is also the force that would keep the side in the state of sliding at a constant speed (by Newton's Second Law). But if the side is moving to the right (in the direction the force is applied), then the surface area of the stretched liquid is increasing while the applied force is doing work on the liquid. This means that increasing the surface area increases the energy of the film. The work done by the force F in moving the side by distance Δx is W = FΔx; at the same time the total area of the film increases by ΔA = 2LΔx (the factor of 2 is here because the liquid has two sides, two surfaces). Thus, multiplying both the numerator and the denominator of γ = 1/2F/L by Δx, we get

This work W is, by the usual arguments, interpreted as being stored as potential energy. Consequently, surface tension can be also measured in SI system as joules per square meter and in the cgs system as ergs per cm². Since mechanical systems try to find a state of minimum potential energy, a free droplet of liquid naturally assumes a spherical shape, which has the minimum surface area for a given volume. The equivalence of measurement of energy per unit area to force per unit length can be proven by dimensional analysis.[8]

Several effects of surface tension can be seen with ordinary water:

• A. Water beading on a leaf
• B. Water dripping from a tap
• C. Water striders stay at the top of liquid because of surface tension
• D. Lava lamp with interaction between dissimilar liquids: water and liquid wax
• E. Photo showing the "tears of wine" phenomenon.

Surface tension is visible in other common phenomena, especially when surfactants are used to decrease it:

• Soap bubbles have very large surface areas with very little mass. Bubbles in pure water are unstable. The addition of surfactants, however, can have a stabilizing effect on the bubbles (see Marangoni effect). Surfactants actually reduce the surface tension of water by a factor of three or more.
• Emulsions are a type of colloidal dispersion in which surface tension plays a role. Tiny droplets of oil dispersed in pure water will spontaneously coalesce and phase separate. The addition of surfactants reduces the interfacial tension and allow for the formation of oil droplets in the water medium (or vice versa). The stability of such formed oil droplets depends on many different chemical and environmental factors.

If no force acts normal to a tensioned surface, the surface must remain flat. But if the pressure on one side of the surface differs from pressure on the other side, the pressure difference times surface area results in a normal force. In order for the surface tension forces to cancel the force due to pressure, the surface must be curved. The diagram shows how surface curvature of a tiny patch of surface leads to a net component of surface tension forces acting normal to the center of the patch. When all the forces are balanced, the resulting equation is known as the Young–Laplace equation:[11]

where:

• Δp is the pressure difference, known as the Laplace pressure.[12]
• γ is surface tension.
• R_x and R_y are radii of curvature in each of the axes that are parallel to the surface.

The quantity in parentheses on the right hand side is in fact (twice) the mean curvature of the surface (depending on normalisation). Solutions to this equation determine the shape of water drops, puddles, menisci, soap bubbles, and all other shapes determined by surface tension (such as the shape of the impressions that a water strider's feet make on the surface of a pond). The table below shows how the internal pressure of a water droplet increases with decreasing radius. For not very small drops the effect is subtle, but the pressure difference becomes enormous when the drop sizes approach the molecular size. (In the limit of a single molecule the concept becomes meaningless.)

When an object is placed on a liquid, its weight F_w depresses the surface, and if surface tension and downward force become equal then it is balanced by the surface tension forces on either side F_s, which are each parallel to the water's surface at the points where it contacts the object. Notice that small movement in the body may cause the object to sink. As the angle of contact decreases, surface tension decreases. The horizontal components of the two F_s arrows point in opposite directions, so they cancel each other, but the vertical components point in the same direction and therefore add up[5] to balance F_w. The object's surface must not be wettable for this to happen, and its weight must be low enough for the surface tension to support it. If m denotes the mass of the needle and g acceleration due to gravity, we have

To find the shape of the minimal surface bounded by some arbitrary shaped frame using strictly mathematical means can be a daunting task. Yet by fashioning the frame out of wire and dipping it in soap-solution, a locally minimal surface will appear in the resulting soap-film within seconds.[8][13]

The reason for this is that the pressure difference across a fluid interface is proportional to the mean curvature, as seen in the Young–Laplace equation. For an open soap film, the pressure difference is zero, hence the mean curvature is zero, and minimal surfaces have the property of zero mean curvature.

The surface of any liquid is an interface between that liquid and some other medium.[note 1] The top surface of a pond, for example, is an interface between the pond water and the air. Surface tension, then, is not a property of the liquid alone, but a property of the liquid's interface with another medium. If a liquid is in a container, then besides the liquid/air interface at its top surface, there is also an interface between the liquid and the walls of the container. The surface tension between the liquid and air is usually different (greater) than its surface tension with the walls of a container. And where the two surfaces meet, their geometry must be such that all forces balance.[8][11]

Where the two surfaces meet, they form a contact angle, θ, which is the angle the tangent to the surface makes with the solid surface. Note that the angle is measured through the liquid, as shown in the diagrams above. The diagram to the right shows two examples. Tension forces are shown for the liquid–air interface, the liquid–solid interface, and the solid–air interface. The example on the left is where the difference between the liquid–solid and solid–air surface tension, γ_ls − γ_sa, is less than the liquid–air surface tension, γ_la, but is nevertheless positive, that is

In the diagram, both the vertical and horizontal forces must cancel exactly at the contact point, known as equilibrium. The horizontal component of f_la is canceled by the adhesive force, f_A.[8]

The more telling balance of forces, though, is in the vertical direction. The vertical component of f_la must exactly cancel the difference of the forces along the solid surface, f_ls − f_sa.[8]

Since the forces are in direct proportion to their respective surface tensions, we also have:[11]

where

• γ_ls is the liquid–solid surface tension,
• γ_la is the liquid–air surface tension,
• γ_sa is the solid–air surface tension,
• θ is the contact angle, where a concave meniscus has contact angle less than 90° and a convex meniscus has contact angle of greater than 90°.[8]

This means that although the difference between the liquid–solid and solid–air surface tension, γ_ls − γ_sa, is difficult to measure directly, it can be inferred from the liquid–air surface tension, γ_la, and the equilibrium contact angle, θ, which is a function of the easily measurable advancing and receding contact angles (see main article contact angle).

This same relationship exists in the diagram on the right. But in this case we see that because the contact angle is less than 90°, the liquid–solid/solid–air surface tension difference must be negative:

Observe that in the special case of a water–silver interface where the contact angle is equal to 90°, the liquid–solid/solid–air surface tension difference is exactly zero.

Another special case is where the contact angle is exactly 180°. Water with specially prepared Teflon approaches this.[11] Contact angle of 180° occurs when the liquid–solid surface tension is exactly equal to the liquid–air surface tension.

An old style mercury barometer consists of a vertical glass tube about 1 cm in diameter partially filled with mercury, and with a vacuum (called Torricelli's vacuum) in the unfilled volume (see diagram to the right). Notice that the mercury level at the center of the tube is higher than at the edges, making the upper surface of the mercury dome-shaped. The center of mass of the entire column of mercury would be slightly lower if the top surface of the mercury were flat over the entire cross-section of the tube. But the dome-shaped top gives slightly less surface area to the entire mass of mercury. Again the two effects combine to minimize the total potential energy. Such a surface shape is known as a convex meniscus.

We consider the surface area of the entire mass of mercury, including the part of the surface that is in contact with the glass, because mercury does not adhere to glass at all. So the surface tension of the mercury acts over its entire surface area, including where it is in contact with the glass. If instead of glass, the tube was made out of copper, the situation would be very different. Mercury aggressively adheres to copper. So in a copper tube, the level of mercury at the center of the tube will be lower than at the edges (that is, it would be a concave meniscus). In a situation where the liquid adheres to the walls of its container, we consider the part of the fluid's surface area that is in contact with the container to have negative surface tension. The fluid then works to maximize the contact surface area. So in this case increasing the area in contact with the container decreases rather than increases the potential energy. That decrease is enough to compensate for the increased potential energy associated with lifting the fluid near the walls of the container.

If a tube is sufficiently narrow and the liquid adhesion to its walls is sufficiently strong, surface tension can draw liquid up the tube in a phenomenon known as capillary action. The height to which the column is lifted is given by Jurin's law:[8]

where

• h is the height the liquid is lifted,
• γ_la is the liquid–air surface tension,
• ρ is the density of the liquid,
• r is the radius of the capillary,
• g is the acceleration due to gravity,
• θ is the angle of contact described above. If θ is greater than 90°, as with mercury in a glass container, the liquid will be depressed rather than lifted.

Pouring mercury onto a horizontal flat sheet of glass results in a puddle that has a perceptible thickness. The puddle will spread out only to the point where it is a little under half a centimetre thick, and no thinner. Again this is due to the action of mercury's strong surface tension. The liquid mass flattens out because that brings as much of the mercury to as low a level as possible, but the surface tension, at the same time, is acting to reduce the total surface area. The result of the compromise is a puddle of a nearly fixed thickness.

The same surface tension demonstration can be done with water, lime water or even saline, but only on a surface made of a substance to which water does not adhere. Wax is such a substance. Water poured onto a smooth, flat, horizontal wax surface, say a waxed sheet of glass, will behave similarly to the mercury poured onto glass.

The thickness of a puddle of liquid on a surface whose contact angle is 180° is given by:[11]

where

• h is the depth of the puddle in centimeters or meters.
• γ is the surface tension of the liquid in dynes per centimeter or newtons per meter.
• g is the acceleration due to gravity and is equal to 980 cm/s² or 9.8 m/s²
• ρ is the density of the liquid in grams per cubic centimeter or kilograms per cubic meter

In reality, the thicknesses of the puddles will be slightly less than what is predicted by the above formula because very few surfaces have a contact angle of 180° with any liquid. When the contact angle is less than 180°, the thickness is given by:[11]

For mercury on glass, γ_Hg = 487 dyn/cm, ρ_Hg = 13.5 g/cm³ and θ = 140°, which gives h_Hg = 0.36 cm. For water on paraffin at 25 °C, γ = 72 dyn/cm, ρ = 1.0 g/cm³, and θ = 107° which gives h_H2O = 0.44 cm.

The formula also predicts that when the contact angle is 0°, the liquid will spread out into a micro-thin layer over the surface. Such a surface is said to be fully wettable by the liquid.

In day-to-day life all of us observe that a stream of water emerging from a faucet will break up into droplets, no matter how smoothly the stream is emitted from the faucet. This is due to a phenomenon called the Plateau–Rayleigh instability,[11] which is entirely a consequence of the effects of surface tension.

The explanation of this instability begins with the existence of tiny perturbations in the stream. These are always present, no matter how smooth the stream is. If the perturbations are resolved into sinusoidal components, we find that some components grow with time while others decay with time. Among those that grow with time, some grow at faster rates than others. Whether a component decays or grows, and how fast it grows is entirely a function of its wave number (a measure of how many peaks and troughs per centimeter) and the radii of the original cylindrical stream.

• Breakup of a moving sheet of water bouncing off of a spoon.
• Photo of flowing water adhering to a hand. Surface tension creates the sheet of water between the flow and the hand.
• A soap bubble balances surface tension forces against internal pneumatic pressure.
• Surface tension prevents a coin from sinking: the coin is indisputably denser than water, so it must be displacing a volume greater than its own for buoyancy to balance mass.
• An aluminium coin floats on the surface of the water at 10 °C. Any extra weight would drop the coin to the bottom.
• A daisy. The entirety of the flower lies below the level of the (undisturbed) free surface. The water rises smoothly around its edge. Surface tension prevents water from displacing the air between the petals and possibly submerging the flower.
• A metal paper clip floats on water. Several can usually be carefully added without overflow of water.
• A metal paperclip floating on water. A grille in front of the light has created the 'contour lines' which show the deformation in the water surface caused by the metal paper clip.

J.W. Gibbs developed the thermodynamic theory of capillarity based on the idea of surfaces of discontinuity.[14] Gibbs considered the case of a sharp mathematical surface being placed somewhere within the microscopically fuzzy physical interface that exists between two homogeneous substances. Realizing that the exact choice of the surface's location was somewhat arbitrary, he left it flexible. Since the interface exists in thermal and chemical equilibrium with the substances around it (having temperature T and chemical potentials μ_i), Gibbs considered the case where the surface may have excess energy, excess entropy, and excess particles, finding the natural free energy function in this case to be ${\displaystyle U-TS-\mu _{1}N_{1}-\mu _{2}N_{2}\cdots }$, a quantity later named as the grand potential and given the symbol ${\displaystyle \Omega }$.

Considering a given subvolume ${\displaystyle V}$ containing a surface of discontinuity, the volume is divided by the mathematical surface into two parts A and B, with volumes ${\displaystyle V_{\text{A}}}$ and ${\displaystyle V_{\text{B}}}$, with ${\displaystyle V=V_{\text{A}}+V_{\text{B}}}$ exactly. Now, if the two parts A and B were homogeneous fluids (with pressures ${\displaystyle p_{\text{A}}}$, ${\displaystyle p_{\text{B}}}$) and remained perfectly homogeneous right up to the mathematical boundary, without any surface effects, the total grand potential of this volume would be simply ${\displaystyle -p_{\text{A}}V_{\text{A}}-p_{\text{B}}V_{\text{B}}}$. The surface effects of interest are a modification to this, and they can be all collected into a surface free energy term ${\displaystyle \Omega _{\text{S}}}$ so the total grand potential of the volume becomes:

For sufficiently macroscopic and gently curved surfaces, the surface free energy must simply be proportional to the surface area:[14][15]

for surface tension ${\displaystyle \gamma }$ and surface area ${\displaystyle A}$.

As stated above, this implies the mechanical work needed to increase a surface area A is dW = γ dA, assuming the volumes on each side do not change. Thermodynamics requires that for systems held at constant chemical potential and temperature, all spontaneous changes of state are accompanied by a decrease in this free energy ${\displaystyle \Omega }$, that is, an increase in total entropy taking into account the possible movement of energy and particles from the surface into the surrounding fluids. From this it is easy to understand why decreasing the surface area of a mass of liquid is always spontaneous, provided it is not coupled to any other energy changes. It follows that in order to increase surface area, a certain amount of energy must be added.

Gibbs and other scientists have wrestled with the arbitrariness in the exact microscopic placement of the surface.[16] For microscopic surfaces with very tight curvatures, it is not correct to assume the surface tension is independent of size, and topics like the Tolman length come into play. For a macroscopic-sized surface (and planar surfaces), the surface placement does not have a significant effect on γ; however, it does have a very strong effect on the values of the surface entropy, surface excess mass densities, and surface internal energy,[14]^: 237 which are the partial derivatives of the surface tension function ${\displaystyle \gamma (T,\mu _{1},\mu _{2},\cdots )}$.

Gibbs emphasized that for solids, the surface free energy may be completely different from surface stress (what he called surface tension):[14]^: 315 the surface free energy is the work required to form the surface, while surface stress is the work required to stretch the surface. In the case of a two-fluid interface, there is no distinction between forming and stretching because the fluids and the surface completely replenish their nature when the surface is stretched. For a solid, stretching the surface, even elastically, results in a fundamentally changed surface. Further, the surface stress on a solid is a directional quantity (a stress tensor) while surface energy is scalar.

Fifteen years after Gibbs, J.D. van der Waals developed the theory of capillarity effects based on the hypothesis of a continuous variation of density.[17] He added to the energy density the term ${\displaystyle c(\nabla \rho )^{2},}$ where c is the capillarity coefficient and ρ is the density. For the multiphase equilibria, the results of the van der Waals approach practically coincide with the Gibbs formulae, but for modelling of the dynamics of phase transitions the van der Waals approach is much more convenient.[18][19] The van der Waals capillarity energy is now widely used in the phase field models of multiphase flows. Such terms are also discovered in the dynamics of non-equilibrium gases.[20]

The pressure inside an ideal spherical bubble can be derived from thermodynamic free energy considerations.[15] The above free energy can be written as:

where ${\displaystyle \Delta P=p_{\text{A}}-p_{\text{B}}}$ is the pressure difference between the inside (A) and outside (B) of the bubble, and ${\displaystyle V_{\text{A}}}$ is the bubble volume. In equilibrium, dΩ = 0, and so,

For a spherical bubble, the volume and surface area are given simply by

and

Substituting these relations into the previous expression, we find

which is equivalent to the Young–Laplace equation when R_x = R_y.

Surface tension is dependent on temperature. For that reason, when a value is given for the surface tension of an interface, temperature must be explicitly stated. The general trend is that surface tension decreases with the increase of temperature, reaching a value of 0 at the critical temperature. For further details see Eötvös rule. There are only empirical equations to relate surface tension and temperature:

• Eötvös:[21][22][23]
  $${\displaystyle \gamma V^{2/3}=k(T_{\mathrm {C} }-T).}$$
  Here V is the molar volume of a substance, T_C is the critical temperature and k is a constant valid for almost all substances.[21] A typical value is k = 2.1×10⁻⁷ J K⁻¹ mol^−2⁄3.[21][23] For water one can further use V = 18 ml/mol and T_C = 647 K (374 °C).[24] A variant on Eötvös is described by Ramay and Shields:[25]
  $${\displaystyle \gamma V^{2/3}=k\left(T_{\mathrm {C} }-T-6\,\mathrm {K} \right)}$$
  where the temperature offset of 6 K provides the formula with a better fit to reality at lower temperatures.
• Guggenheim–Katayama:[22]
  $${\displaystyle \gamma =\gamma ^{\circ }\left(1-{\frac {T}{T_{\mathrm {C} }}}\right)^{n}}$$
  γ° is a constant for each liquid and n is an empirical factor, whose value is 11/9 for organic liquids. This equation was also proposed by van der Waals, who further proposed that γ° could be given by the expression
  $${\displaystyle K_{2}T_{\mathrm {C} }^{1/3}P_{\mathrm {C} }^{2/3},}$$
  where K₂ is a universal constant for all liquids, and P_C is the critical pressure of the liquid (although later experiments found K₂ to vary to some degree from one liquid to another).[22]

Both Guggenheim–Katayama and Eötvös take into account the fact that surface tension reaches 0 at the critical temperature, whereas Ramay and Shields fails to match reality at this endpoint.

Solutes can have different effects on surface tension depending on the nature of the surface and the solute:

• Little or no effect, for example sugar at water|air, most organic compounds at oil/air
• Increase surface tension, most inorganic salts at water|air
• Non-monotonic change, most inorganic acids at water|air
• Decrease surface tension progressively, as with most amphiphiles, e.g., alcohols at water|air
• Decrease surface tension until certain critical concentration, and no effect afterwards: surfactants that form micelles

What complicates the effect is that a solute can exist in a different concentration at the surface of a solvent than in its bulk. This difference varies from one solute–solvent combination to another.

Gibbs isotherm states that:

• Γ is known as surface concentration, it represents excess of solute per unit area of the surface over what would be present if the bulk concentration prevailed all the way to the surface. It has units of mol/m²
• C is the concentration of the substance in the bulk solution.
• R is the gas constant and T the temperature

Certain assumptions are taken in its deduction, therefore Gibbs isotherm can only be applied to ideal (very dilute) solutions with two components.

The Clausius–Clapeyron relation leads to another equation also attributed to Kelvin, as the Kelvin equation. It explains why, because of surface tension, the vapor pressure for small droplets of liquid in suspension is greater than standard vapor pressure of that same liquid when the interface is flat. That is to say that when a liquid is forming small droplets, the equilibrium concentration of its vapor in its surroundings is greater. This arises because the pressure inside the droplet is greater than outside.[25]

• P_v° is the standard vapor pressure for that liquid at that temperature and pressure.
• V is the molar volume.
• R is the gas constant
• r_k is the Kelvin radius, the radius of the droplets.

The effect explains supersaturation of vapors. In the absence of nucleation sites, tiny droplets must form before they can evolve into larger droplets. This requires a vapor pressure many times the vapor pressure at the phase transition point.[25]

This equation is also used in catalyst chemistry to assess mesoporosity for solids.[26]

The effect can be viewed in terms of the average number of molecular neighbors of surface molecules (see diagram).

The table shows some calculated values of this effect for water at different drop sizes:

The effect becomes clear for very small drop sizes, as a drop of 1 nm radius has about 100 molecules inside, which is a quantity small enough to require a quantum mechanics analysis.

The surface tension of pure liquid water in contact with its vapor has been given by IAPWS[38] as

where both T and the critical temperature T_C = 647.096 K are expressed in kelvins. The region of validity the entire vapor–liquid saturation curve, from the triple point (0.01 °C) to the critical point. It also provides reasonable results when extrapolated to metastable (supercooled) conditions, down to at least −25 °C. This formulation was originally adopted by IAPWS in 1976 and was adjusted in 1994 to conform to the International Temperature Scale of 1990.

The uncertainty of this formulation is given over the full range of temperature by IAPWS.[38] For temperatures below 100 °C, the uncertainty is ±0.5%.

Nayar et al.[39] published reference data for the surface tension of seawater over the salinity range of 20 ≤ S ≤ 131 g/kg and a temperature range of 1 ≤ t ≤ 92 °C at atmospheric pressure. The range of temperature and salinity encompasses both the oceanographic range and the range of conditions encountered in thermal desalination technologies. The uncertainty of the measurements varied from 0.18 to 0.37 mN/m with the average uncertainty being 0.22 mN/m.

Nayar et al. correlated the data with the following equation

where γ_sw is the surface tension of seawater in mN/m, γ_w is the surface tension of water in mN/m, S is the reference salinity[40] in g/kg, and t is temperature in degrees Celsius. The average absolute percentage deviation between measurements and the correlation was 0.19% while the maximum deviation is 0.60%.

The International Association for the Properties of Water and Steam (IAPWS) has adopted this correlation as an international standard guideline.[41]