Surface tension

thumb|upright=1.2|Rain water flux from a canopy. Among the forces that govern drop formation: surface tension by [[Cohesion (chemistry)|cohesion, Van der Waals force, Plateau–Rayleigh instability.]]

thumb|Surface tension and [[hydrophobicity interact in this attempt to cut a water droplet.]]

thumb|Surface tension experimental demonstration with soap

Surface tension is the energy per unit area due to having a surface in a liquid. It has the dimension of force per unit length, or energy per unit area. The two are equivalent, but when referring to energy per unit of area, it is common to use the term surface energy, which is a more general term in the sense that it applies also to solids. Surface tension is used for liquids, while surface stress and surface energy are more commonly used for solids.

An example of its relevance is the tendency of liquid surfaces at rest to shrink to the minimum surface area possible. Because of the relatively high attraction of water molecules to each other through a web of hydrogen bonds, water has a higher surface tension (72.8 millinewtons (mN) per meter at 20 °C) than most other liquids. This allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to float on a water surface without becoming even partly submerged.

The magnitude of the surface tension is connected to the forces between the molecules at the surface. Therefore surfactants are often used to reduce it so there is more contact between the liquid and another material, for instance detergents. It can also lead to pressure inside water bubbles, as well as many other phenomena; it is a classic, well-studied property common to all liquids.

Causes

thumb|left|Diagram of the cohesive forces on molecules of a liquid

Due to the cohesive forces, a molecule located away from the surface is pulled equally in every direction by neighboring liquid molecules, resulting in a net force of zero. The molecules at the surface do not have the same molecules on all sides of them and therefore are pulled inward. This creates some internal pressure and forces liquid surfaces to contract to the minimum area.

The forces of attraction acting between molecules of the same type are called cohesive forces, while those acting between molecules of different types are called adhesive forces. The balance between the cohesion of the liquid and its adhesion to the material of the container determines the degree of wetting, the contact angle, and the shape of the meniscus. When cohesion dominates (specifically, adhesion energy is less than half of cohesion energy) the wetting is low and the meniscus is convex at a vertical wall (as for mercury in a glass container). On the other hand, when adhesion dominates (when adhesion energy is more than half of cohesion energy) the wetting is high and the similar meniscus is concave (as in water in a glass).

Surface tension is responsible for the shape of liquid droplets. Although easily deformed, droplets of water tend to be pulled into a spherical shape by the imbalance in cohesive forces of the surface layer. In the absence of other forces, drops of virtually all liquids would be approximately spherical. The spherical shape minimizes the necessary "wall tension" of the surface layer according to Laplace's law.

thumb|Water droplet lying on a [[damask. Surface tension is high enough to prevent seeping through the textile]]

Another way to view surface tension is in terms of energy. A molecule in contact with a neighbor is in a lower state of energy than if it were alone. The interior molecules have as many neighbors as they can possibly have, but the boundary molecules are missing neighbors (compared to interior molecules) and therefore have higher energy. For the liquid to minimize its energy state, the number of higher energy boundary molecules must be minimized. The minimized number of boundary molecules results in a minimal surface area.

As a result of surface area minimization, a surface will assume a smooth shape.

Physics

Physical units

Surface tension, represented by the symbol γ (alternatively σ or T), is measured in force per unit length. Its SI unit is newton per metre but the cgs unit of dyne per centimetre is also used, particularly in the older literature. For example,

<math display="block"> \gamma = 1 ~\mathrm{\frac{dyn}{cm = 1 ~\mathrm{\frac{erg}{cm^2 = 1~\mathrm{\frac{10^{-7}\,m\cdot N}{10^{-4}\, m^2 = 0.001~\mathrm{\frac{N}{m = 0.001~\mathrm{\frac{J}{m^2.</math>

Definition

thumb|right|This diagram illustrates the force necessary to increase the surface area. This force is proportional to the surface tension.

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

In terms of force

Surface tension 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 required to hold the movable side is proportional to the length of the immobile side. Thus the ratio 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 , with the length of the movable side and the force required to stop it from sliding, is found to be the same for all shapes. We therefore define the surface tension as<math display="block">\gamma=\frac{F}{2L}.</math>

The reason for the 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 .

In terms of energy

Surface tension 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: if 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 in moving the side by distance is ; at the same time the total area of the film increases by (the factor of 2 is here because the liquid has two sides, two surfaces). Thus, multiplying both the numerator and the denominator of by , we get

<math display="block">\gamma=\frac{F}{2L}=\frac{F \Delta x}{2 L \Delta x}=\frac{W}{\Delta A} .</math>

This work 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 as first pointed out by Gibbs.

The equivalence of measurement of energy per unit area to force per unit length can be proven by dimensional analysis.

|Flotation of objects denser than water occurs when the object is nonwettable and its weight is small enough to be borne by the forces arising from surface tension. and tendency of minimization of surface curvature (so area) of the water pushes the insect's feet upward.

|Separation of oil and water (such as water and liquid wax in a lava lamp) is caused by a tension in the surface between dissimilar liquids. This type of surface tension is called "interface tension", but its chemistry is the same.

|Tears of wine is the formation of drops and rivulets on the side of a glass containing an alcoholic beverage. Its cause is a complex interaction between the differing surface tensions of water and ethanol; it is induced by a combination of surface tension modification of water by ethanol together with ethanol evaporating faster than water.

File:Dew 2.jpg|A. Water beading on a leaf

File:Water drop animation enhanced small.gif|B. Water dripping from a tap

File:WaterstriderEnWiki.jpg|C. Water striders stay at the top of liquid because of surface tension

File:1990s Mathmos Astro.jpg|D. Lava lamp with interaction between dissimilar liquids: water and liquid wax

File:Wine legs shadow.jpg|E. Photo showing the "tears of wine" phenomenon.

</gallery>

Surfactants

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.

Surface curvature and pressure

thumb|right|Surface tension forces acting on a tiny (differential) patch of surface. and indicate the amount of bend over the dimensions of the patch. Balancing the tension forces with pressure leads to the [[Young–Laplace equation]]

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:

is surface tension.
and 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.)

{| class="wikitable" style="float:center; clear:right;"

|+ for water drops of different radii at STP

! style="width:120px;" | Droplet radius

| style="width:120px;" | 1 mm

| style="width:120px;" | 0.1 mm

| style="width:120px;" | 1 μm

| style="width:120px;" | 10 nm

! (atm)

| 0.0014

| 0.0144

| 1.436

| 143.6

Floating objects

thumb|Cross-section of a needle floating on the surface of water. is the weight and are surface tension resultant forces.

When an object is placed on a liquid, its weight depresses the surface, and if surface tension and downward force become equal then it is balanced by the surface tension forces on either side , 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 arrows point in opposite directions, so they cancel each other, but the vertical components point in the same direction and therefore add up

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.

Contact angles

The surface of any liquid is an interface between that liquid and some other medium. 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.

This means that although the difference between the liquid–solid and solid–air surface tension, , is difficult to measure directly, it can be inferred from the liquid–air surface tension, , and the equilibrium contact angle, , 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:

<math display="block">\gamma_\mathrm{la} > 0 > \gamma_\mathrm{ls} - \gamma_\mathrm{sa}</math>

Special contact angles

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.

<math display="block">h = 2 \sqrt{\frac{\gamma} {g\rho</math>

where

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

thumb|Illustration of how lower contact angle leads to reduction of puddle depth

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: 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 and chemical potentials ), 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 <math>U - TS - \mu_1 N_1 - \mu_2 N_2 \cdots </math>, a quantity later named as the grand potential and given the symbol <math>\Omega</math>.

thumb|Gibbs' placement of a precise mathematical surface in a fuzzy physical interface.

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

<math display="block">\Omega = -p_\text{A} V_\text{A} - p_\text{B} V_\text{B} + \Omega_\text{S}.</math>

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

<math display="block">\Omega_\text{S} = \gamma A,</math>

for surface tension <math>\gamma</math> and surface area <math>A</math>.

As stated above, this implies the mechanical work needed to increase a surface area A is , 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 <math>\Omega</math>, 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. 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 ; however, it does have a very strong effect on the values of the surface entropy, surface excess mass densities, and surface internal energy, which are the partial derivatives of the surface tension function <math>\gamma(T, \mu_1, \mu_2, \cdots)</math>.

Gibbs emphasized that for solids, the surface free energy may be completely different from surface stress (what he called surface tension): 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. He added to the energy density the term <math>c (\nabla \rho)^2,</math> 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. 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.

Thermodynamics of bubbles

The pressure inside an ideal spherical bubble can be derived from thermodynamic free energy considerations. <math display="block">\gamma V^{2/3} = k(T_\mathrm{C}-T) .</math> Here is the molar volume of a substance, is the critical temperature and is a constant valid for almost all substances. A variant on Eötvös is described by Ramay and Shields: <math display="block">\gamma V^{2/3} = k \left(T_\mathrm{C} - T - 6\,\mathrm{K}\right)</math> where the temperature offset of 6 K provides the formula with a better fit to reality at lower temperatures.

Guggenheim–Katayama:

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:

{| class="toccolours" border="1" style="float: center; margin: 0 0 1em 1em; border-collapse: collapse;"

! style="text-align:center; background:#c0c0f0;" colspan="5"| for water drops of different radii at STP

Stalagmometric method: A method of weighting and reading a drop of liquid.
Sessile drop method: A method for determining surface tension and density by placing a drop on a substrate and measuring the contact angle (see Sessile drop technique).
Du Noüy–Padday method: A minimized version of Du Noüy method uses a small diameter metal needle instead of a ring, in combination with a high sensitivity microbalance to record maximum pull. The advantage of this method is that very small sample volumes (down to few tens of microliters) can be measured with very high precision, without the need to correct for buoyancy (for a needle or rather, rod, with proper geometry). Further, the measurement can be performed very quickly, minimally in about 20 seconds.
Vibrational frequency of levitated drops: The natural frequency of vibrational oscillations of magnetically levitated drops has been used to measure the surface tension of superfluid 4He. This value is estimated to be 0.375 dyn/cm at = 0 K.
Resonant oscillations of spherical and hemispherical liquid drop: The technique is based on measuring the resonant frequency of spherical and hemispherical pendant droplets driven in oscillations by a modulated electric field. The surface tension and viscosity can be evaluated from the obtained resonant curves.
Drop-bounce method: This method is based on aerodynamic levitation with a split-able nozzle design. After dropping a stably levitated droplet onto a platform, the sample deforms and bounces back, oscillating in mid-air as it tries to minimize its surface area. Through this oscillation behavior, the liquid's surface tension and viscosity can be measured.

Values

Data table

{| class="wikitable sortable" style="margin:0; text-align: right;"

|+ style="background:#C0C0F0; border: 1px solid #AAA" |Surface tension of various liquids in dyn/cm against air Mixture compositions denoted "%" are by mass dyn/cm is equivalent to the SI units of mN/m (millinewton per meter)

! Liquid !! Temperature (°C)!! Surface tension,

| Acetic acid || 20 || 27.60

| Acetic acid (45.1%) + Water || 30 || 40.68

| Acetic acid (10.0%) + Water || 30 || 54.56

| Acetone || 20 || 23.70

| Benzene || 20 || 28.88

| Blood || 22 || 55.89

| Butyl acetate || 20 || 25.09

| Butyric acid || 20 || 26.51

| Carbon tetrachloride || 25 || 26.43

| Chloroform || 25 || 26.67

| Diethyl ether || 20 || 17.00

| Diethylene glycol || 20 || 30.09

| Dimethyl sulfoxide || 20|| 43.54

| Ethanol || 20 || 22.27

| Ethanol (40%) + Water || 25 || 29.63

| Ethanol (11.1%) + Water || 25 || 46.03

| Ethylene glycol ||| 25 || 47.3

| Glycerol || 20 || 63.00

| Heptane || 20 || 20.14

| n-Hexane || 20 || 18.40

| Hydrochloric acid 17.7 M aqueous solution || 20 || 65.95

| Isopropanol || 20 || 21.70

| Liquid helium II || −273 || 0.37

| Mercury || 20 || 486.5

| Liquid nitrogen || −196 || 8.85

| Nonane || 20 || 22.85

| Liquid oxygen || −182 || 13.2

| Mercury || 15 || 487.00

| Methanol || 20 || 22.60

| Methylene iodide || 20 || 67.00

| Molten Silver chloride || 650 || 163

| Molten Sodium chloride/Calcium chloride (47/53 mole %) || 650 || 139

| n-Octane || 20 || 21.62

| Propionic acid || 20 || 26.69

| Propylene carbonate || 20 || 41.1

| Sodium chloride 6.0 M aqueous solution || 20 || 82.55

| Sodium chloride (molten) || 1073 || 115

| Sucrose (55%) + water || 20 || 76.45

| Toluene || 25 || 27.73

| Water || 0 || 75.64

| Water || 25 || 71.97

| Water || 50 || 67.91

| Water || 100 || 58.85

Surface tension of water

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

<math display="block">\gamma_\text{w} = 235.8\left(1 - \frac{T}{T_\text{C\right)^{1.256} \left[1 - 0.625\left(1 - \frac{T}{T_\text{C\right)\right]~\text{mN/m},</math>

where both and the critical temperature = 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. published reference data for the surface tension of seawater over the salinity range of and a temperature range of 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

<math display="block"> \gamma_\mathrm{sw} = \gamma_\mathrm{w} \left( 1+ 3.766\times 10^{-4} S +2.347\times 10^{-6} S t\right)</math>

where is the surface tension of seawater in mN/m, is the surface tension of water in mN/m, is the reference salinity in g/kg, and 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.