Continuum mechanics  
Conservation of mass Conservation of momentum Navier–Stokes equations Tensors


In physics, surface tension is an effect within the surface layer of a liquid that causes that layer to behave as an elastic sheet. This effect allows insects (such as the water strider) to walk on water. It allows small metal objects such as needles, razor blades, or foil fragments to float on the surface of water, and causes capillary action. Interface tension is the name of the same effect when it takes place between two liquids.
Surface tension is caused by the attraction between the molecules of the liquid by various intermolecular forces. In the bulk of the liquid each molecule is pulled equally in all directions by neighboring liquid molecules, resulting in a net force of zero. At the surface of the liquid, the molecules are pulled inwards by other molecules deeper inside the liquid but they are not attracted as intensely by the molecules in the neighboring medium (be it vacuum, air or another liquid). Therefore, all of the molecules at the surface are subject to an inward force of molecular attraction which can be balanced only by the resistance of the liquid to compression. Thus, the liquid squeezes itself together until it has the locally lowest surface area possible.
Another way to think about it is that a molecule in contact with a neighbor is in a lower state of energy than if it weren't in contact with a neighbor. The interior molecules all have as many neighbors as they can possibly have. But the boundary molecules have fewer neighbors than interior molecules and are therefore in a higher state of energy. For the liquid to minimize its energy state, it must minimize its number of boundary molecules and therefore minimize its surface area.^{[1]}
As a result of this minimizing of surface area, the surface will want to assume the smoothest flattest shape it can (rigorous proof that "smooth" shapes minimize surface area relies on use of the EulerLagrange Equation). Since any curvature in the surface shape results in higher area, a higher energy will also result. Consequently, the surface will push back on the disturbing object in much the same way a ball pushed uphill will push back to minimize its gravitational energy.
Some examples of the effects of surface tension seen with ordinary water:
Surface tension has a big influence on other common phenomena, especially when certain substances, surfactants, are used to decrease it:
Surface tension is represented by the symbol σ, γ or T and is defined as the force along a line of unit length where the force is parallel to the surface but perpendicular to the line. One way to picture this is to imagine a flat soap film bounded on one side by a taut thread of length, L. The thread will be pulled toward the interior of the film by a force equal to γL. Surface tension is therefore measured in newtons per meter (N·m^{1}), although the cgs unit of dynes per cm is normally used.^{[3]}
A better definition of surface tension, in order to treat its thermodynamics, is work done per unit area. As such, in order to increase the surface area of a mass of liquid an amount, δA, a quantity of work, γδA, is needed. Since mechanical systems try to find a state of minimum potential energy, a free droplet of liquid naturally assumes a spherical shape. This is because a sphere has the minimum surface area for a given volume. Therefore surface tension can be also measured in joules per square meter (J·m^{2}), or, in the cgs system, ergs per cm^{2}.
The equivalence of both units can be proven by dimensional analysis.
A related quantity is the energy of cohesion, which is the energy released when two bodies of the same liquid become joined by a boundary of unit area. Since this process involves the removal of a unit area of surface from each of the two bodies of liquid, the energy of cohesion is equal to twice the surface energy. A similar concept, the energy of adhesion, applies to two bodies of different liquids. Energy of adhesion is linked to the surface tension of an interface between two liquids. <math>\scriptstyle W_{adh}\ =\ W_{coh}^{\alpha}+W_{coh}^\beta\gamma_{\alpha}^\beta</math>
See also Cassie's law.
The photograph shows water striders standing on the surface of a pond. It is clearly visible that its feet cause indentations in the water's surface. And it is intuitively evident that the surface with indentations has more surface area than a flat surface. If surface tension tends to minimize surface area, how is it that the water striders are increasing the surface area?
Recall that what nature really tries to minimize is potential energy. By increasing the surface area of the water, the water striders have increased the potential energy of that surface. But note also that the water striders' center of mass is lower than it would be if they were standing on a flat surface. So their potential energy is decreased. Indeed when you combine the two effects, the net potential energy is minimized. If the water striders depressed the surface any more, the increased surface energy would more than cancel the decreased energy of lowering the insects' center of mass. If they depressed the surface any less, their higher center of mass would more than cancel the reduction in surface energy.^{[4]}
The photo of the water striders also illustrates the notion of surface tension being like having an elastic film over the surface of the liquid. In the surface depressions at their feet it is easy to see that the reaction of that imagined elastic film is exactly countering the weight of the insects.
An old style mercury barometer consists of a vertical glass tube about 1 cm in diameter partially filled with mercury, and with a 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 domeshaped. 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 crosssection of the tube. But the domeshaped 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.
The reason people consider the surface area of the entire mass of mercury, including the part of the surface that is in contact with the glass, is because mercury does not adhere at all to glass. 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 were 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 rather than higher 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.
The angle of contact of the surface of the liquid with the wall of the container can be used to determine the surface tension of the liquidsolid interface provided that the surface tension of the liquidair interface is known. The relationship is given by:
where
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 the column is lifted to is given by:^{[5]}
where
Pouring mercury onto a horizontal flat sheet of glass results in a puddle that has a perceptible thickness (do not try this except under a fume hood. Mercury vapor is a toxic hazard). The puddle will spread out only to the point where it is a little under half a centimeter 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 is the compromise of a puddle of a nearly fixed thickness.
The same surface tension demonstration can be done with water, but only on a surface made of a substance that the water does not adhere to. 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 nonadhesive horizontal surface is given by
where
<math>\scriptstyle h</math> is the depth of the puddle in centimeters or meters. 
<math>\scriptstyle \gamma</math> is the surface tension of the liquid in dynes per centimeter or newtons per meter. 
<math>\scriptstyle g</math> is the acceleration due to gravity and is equal to 980 cm/s^{2} or 9.8 m/s^{2} 
<math>\scriptstyle \rho</math> is the density of the liquid in grams per cubic centimeter or kilograms per cubic meter 
For mercury, <math>\scriptstyle \gamma_\mathrm{Hg}\ =\ 487\ \mathrm{\frac{dyn}{cm}}</math> and <math>\scriptstyle \rho_\mathrm{Hg}\ =\ 13.5\ \mathrm{\frac{g}{cm^3}}</math>, which gives <math>\scriptstyle h_\mathrm{Hg}\ =\ 0.38\ \mathrm{cm}</math>. For water at 25 °C, <math>\scriptstyle \gamma_\mathrm{H_2O}\ =\ 72\ \mathrm{\frac{dyn}{cm}}</math> and <math>\scriptstyle \rho_\mathrm{H_2O}\ = 1.0\ \mathrm{\frac{g}{cm^3}}</math>, which gives <math>\scriptstyle h_\mathrm{H_2O}\ =\ 0.54\ \mathrm{cm}</math>.
In reality, the thicknesses of the puddles will be slightly less than these calculated values. This is due to the fact that surface tension of the mercuryglass interface is slightly less than that of the mercuryair interface. Likewise, the surface tension of the waterwax interface is less than that of the waterair interface. The contact angle, as described in the previous subsection, determines by how much the puddle thickness is reduced from the theoretical.
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 soapsolution, an approximately minimal surface will appear in the resulting soapfilm within seconds. Without a single calculation, the soapfilm arrives at a solution to a complex minimization equation on its own.^{[5]} ^{[6]}
As stated above, the mechanical work needed to increase a surface is <math>\scriptstyle dW = \gamma dA</math>. For a reversible process, <math>\scriptstyle dG = VdP + \gamma dA  SdT</math>, therefore at constant temperature and pressure, surface tension equals Gibbs free energy per surface area:
<math>\gamma = \left( \frac{\partial G}{\partial A} \right)_{P,T}</math>, where <math>\scriptstyle G</math> is Gibbs free energy and <math>\scriptstyle A</math> is the area.
Surface tension depends 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. There are only empirical equations to relate surface tension and temperature.
Solutes can have different effects on surface tension depending on their structure:
If viscous forces are absent, the pressure jump across a curved surface is given by the YoungLaplace Equation, which relates pressure inside a liquid with the pressure outside it, the surface tension and the geometry of the surface.
This equation can be applied to any surface:
The table shows an example of how the pressure increases, showing that for not very small drops the effect is subtle but the pressure difference becomes enormous when the drop sizes approach the molecular size (a drop with a 1 nm radius contains approximately 100 water molecules), this can be attributed to the fact that at a very small scale the laws of continuum physics cannot be applied anymore.
ΔP for water drops of different radii at STP  

Droplet radius  1 mm  0.1 mm  1 μm  10 nm 
ΔP (atm)  0.0014  0.0144  1.436  143.6 
Starting from ClausiusClapeyron relation Kelvin Equation II can be obtained; it explains that because of surface tension, 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 concentration of vapor of that liquid in the surroundings is greater, this is due to the fact that the pressure inside the droplet is greater than outside.
<math>\scriptstyle P_v^o</math> is the standard vapor pressure for that liquid at that temperature and pressure.
<math>\scriptstyle V</math> is the molar volume.
<math>\scriptstyle R</math> is the gas constant
<math>r_k</math> is the Kelvin radius, the radius of the droplets.
This equation is used in catalyst chemistry to assess mesoporosity for solids.^{[8]}
The table shows some calculated values of this effect for water at different drop sizes:
P/P_{0} for water drops of different radii at STP  

Droplet radius (nm)  1000  100  10  1 
P/P_{0}  1.0011  1.0106  1.1115  2.8778 
The effect becomes clear for very low drop sizes, as a drop on 1 nm radius has about 100 molecules inside, which is a quantity small enough to require a quantum mechanics analysis.
Surface tension values for some interfaces  

Interface  Temperature  γ in (mN·m^{–1}) 
Water  air  20º C  72.86±0.05^{[9]} 
Water  air  21.5º C  72.75 
Water  air  25º C  71.99±0.05^{[9]} 
Methylene iodide  air  20º C  67.00 
Methylene iodide  air  21.5º C  63.11 
Ethylene glycol  air  25º C  47.3 
Ethylene glycol  air  40º C  46.3 
Dimethyl sulfoxide  air  20º C  43.54 
Propylene carbonate  air  20º C  41.1 
Benzene  air  20º C  28.88 
Benzene  air  30º C  27.56 
Toluene  air  20º C  28.52 
Chloroform  air  25º C  26.67 
Propionic acid  air  20º C  26.69 
Butyric acid  air  20º C  26.51 
Carbon tetrachloride  air  25º C  26.43 
Butyl acetate  air  20º C  25.09 
Diethylene Glycol  air  20º C  30.09 
Nonane  air  20º C  22.85 
Methanol  air  20º C  22.50 
Ethanol  air  20º C  22.39 
Ethanol  air  30º C  21.55 
Octane  air  20º C  21.62 
Heptane  air  20º C  20.14 
Ether  air  25º C  20.14 
Mercury  air  20º C  486.5 
Mercury  air  25º C  485.5 
Mercury  air  30º C  484.5 
NaCl  air  1073º C  115 
KClO3  air  20º C  81 
Water  1Butanol  20º C  1.8 
Water  Ethyl acetate  20º C  6.8 
Water  Heptanoic acid  20º C  7.0 
Water  Benzaldehyde  20º C  15.5 
Water  Mercury  20º C  415 
Ethanol  Mercury  20º C  389 
Surface tension values^{[10]} for some interfaces at the indicated temperatures. Note that the SI units millinewtons per meter (mN·m^{–1}) are equivalent to the cgs units, dynes per centimeter (dyn·cm^{–1}).
All links retrieved November 14, 2008.
General subfields within physics  
Atomic, molecular, and optical physics  Classical mechanics  Condensed matter physics  Continuum mechanics  Electromagnetism  General relativity  Particle physics  Quantum field theory  Quantum mechanics  Special relativity  Statistical mechanics  Thermodynamics 
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