TheInfoList

Surface tension is the tendency of
liquid A liquid is a nearly incompressible In fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, ...

surfaces at rest to shrink into the minimum
surface area The surface area of a Solid geometry, solid object is a measure of the total area that the Surface (mathematics), surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more ...

possible. Surface tension is what allows objects with a higher density than water such as
razor blade A razor is a bladed tool A tool is an object that can extend an individual's ability to modify features of the surrounding environment. Although many animals use simple tools, only human beings, whose use of stone tools dates back hundre ...

s and insects (e.g. water striders) to float on a water surface without becoming even partly submerged. At liquid–air interfaces, surface tension results from the greater attraction of liquid molecules to each other (due to cohesion) than to the molecules in the air (due to
adhesion Adhesion is the tendency of dissimilar particles In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physic ...

). There are two primary mechanisms in play. One is an inward force on the surface molecules causing the liquid to contract. Second is a tangential force parallel to the surface of the liquid. This ''tangential'' force is generally referred to as the surface tension. The net effect is the liquid behaves as if its surface were covered with a stretched elastic membrane. But this analogy must not be taken too far as the tension in an elastic membrane is dependent on the amount of deformation of the membrane while surface tension is an inherent property of the liquid''–''air or liquid''–''vapour interface. Because of the relatively high attraction of water molecules to each other through a web of
hydrogen bond A hydrogen bond (or H-bond) is a primarily electrostatic Electrostatics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department ...

s, water has a higher surface tension (72.8
millinewton The newton (symbol: N) is the International System of Units (SI) SI derived unit, derived unit of force. It is named after Isaac Newton in recognition of his work on classical mechanics, specifically Newton's laws of motion#Newton's second law, ...
s (mN) per meter at 20 °C) than most other liquids. Surface tension is an important factor in the phenomenon of
capillarity Capillary action (sometimes capillarity, capillary motion, capillary effect, or wicking) is the ability of a liquid to flow in narrow spaces without the assistance of, or even in opposition to, external forces like Gravitation, gravity. The effe ...
. Surface tension has the
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
of
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

per unit
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

, or of
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

per unit
area Area is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in ...

. The two are equivalent, but when referring to energy per unit of area, it is common to use the term
surface energy Surface free energy or interfacial free energy or surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created. In the of s, surfaces must be intrinsically less than the bulk of a material (the molecules ...
, which is a more general term in the sense that it applies also to
solid Solid is one of the four fundamental states of matter 4 (four) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is an ...

s. In
materials science The interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fields like sociology, a ...
, surface tension is used for either
surface stress Surface stress was first defined by Josiah Willard Gibbs Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on th ...
or
surface energy Surface free energy or interfacial free energy or surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created. In the of s, surfaces must be intrinsically less than the bulk of a material (the molecules ...
.

# Causes

Due to the cohesive forces, a molecule located away from the surface is pulled equally in every direction by neighbouring 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 Internal pressure is a measure of how the of a system changes when it expands or contracts at constant . It has the same dimensions as , the of which is the . Internal pressure is usually given the symbol \pi_T. It is defined as a of internal ...
and forces liquid surfaces to contract to the minimum area. There is also a tension parallel to the surface at the liquid-air interface which will resist an external force, due to the cohesive nature of water molecules. The forces of attraction acting between the molecules of same type are called cohesive forces, while those acting between the 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 Wetting is the ability of a liquid to maintain contact with a solid surface science, surface, resulting from intermolecular interactions when the two are brought together. This happens in presence of a gaseous phase or another liquid phase not ...

, the
contact angle The contact angle is the angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Eleme ...

and the shape of 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 (adhesion energy 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. 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 a 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 the smoothest shape it can (mathematical proof that "smooth" shapes minimize surface area relies on use of the
Euler–Lagrange equationIn the calculus of variations, the Euler equation is a second-order partial differential equation whose solutions are the function (mathematics), functions for which a given functional (mathematics), functional is stationary point, stationary. It wa ...
). Since any curvature in the surface shape results in greater area, a higher energy will also result.

# Effects of surface tension

## Water

Several effects of surface tension can be seen with ordinary 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 strider The Gerridae are a family of insects in the order Hemiptera, commonly known as water striders, water skeeters, water scooters, water bugs, pond skaters, water skippers, Jesus bugs, or water skimmers. Consistent with the classification of the Ge ...

s stay at the top of liquid because of surface tension File:1990s Mathmos Astro.jpg, D.
Lava lamp A lava lamp is a decorative lamp, invented in 1963 by British entrepreneur Edward Craven Walker, the founder of the lighting company Mathmos. The lamp consists of a bolus of a special coloured wax mixture inside a glass vessel, the remainder o ...

with interaction between dissimilar liquids: water and liquid wax File:Wine legs shadow.jpg, E. Photo showing the " tears of wine" phenomenon.

## Surfactants

Surface tension is visible in other common phenomena, especially when
surfactant Surfactants are compounds that lower the surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than wate ...

s are used to decrease it: *
Soap bubble A soap bubble is an extremely thin film A film, also called a movie, motion picture or moving picture, is a work of visual art The visual arts are art forms such as painting Painting is the practice of applying paint ...
s 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 The Marangoni effect (also called the Gibbs–Marangoni effect) is the mass transfer along an Interface (chemistry), interface between two fluids due to a gradient of the surface tension. In the case of temperature dependence, this phenomenon may be ...
). Note that surfactants actually reduce the surface tension of water by a factor of three or more. *
Emulsion An emulsion is a mixture In chemistry, a mixture is a material made up of two or more different chemical substances which are not chemically combined. A mixture is the physical combination of two or more substances in which the identities are ...

s are a type of colloid in which surface tension plays a role. Tiny fragments of oil suspended in pure water will spontaneously assemble themselves into much larger masses. But the presence of a surfactant provides a decrease in surface tension, which permits stability of minute droplets of oil in the bulk of water (or vice versa).

# Physics

## Physical units

Surface tension, represented by the symbol '''' (alternatively '''' or ''''), is measured in
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

per
unit length Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit ...

. Its SI unit is
newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...
per meter but the unit of
dyne The dyne (symbol dyn, from grc, δύναμις, dynamis, power, force) is a derived unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of acti ...
per centimeter is also used. For example, :$\gamma = 1 ~\mathrm = 1 ~\mathrm = 1~\mathrm = 0.001~\mathrm = 0.001~\mathrm.$

## Surface area growth

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 :$\gamma=\frac\frac.$ 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 :$\gamma=\frac=\frac=\frac$. 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
erg The erg is a unit of energy equal to 10−7joule The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an objec ...
s 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 In engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range ...
.

## Surface curvature and pressure

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 In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...
: :$\Delta p = \gamma \left\left( \frac + \frac \right\right)$ where: * is the pressure difference, known as the
Laplace pressure The Laplace pressure is the pressure Pressure (symbol: ''p'' or ''P'') is the force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , ...
. * 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 curvatureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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 The Gerridae are a family of insects in the order Hemiptera, commonly known as water striders, water skeeters, water scooters, water bugs, pond skaters, water skippers, Jesus bugs, or water skimmers. Consistent with the classification of the Ge ...

'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.)

## Floating objects

When an object is placed on a liquid, its weight depresses the surface, and if surface tension and downward force becomes equal than 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 to balance . 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 denotes the mass of the needle and acceleration due to gravity, we have :$F_\mathrm = 2 F_\mathrm \sin \theta \quad\Leftrightarrow\quad m g = 2 \gamma L \sin \theta$

## Liquid surface

To find the shape of the
minimal surface 180px, A helicoid minimal surface formed by a soap film on a helical frame In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...

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. The reason for this is that the pressure difference across a fluid interface is proportional to the
mean curvatureIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
, as seen in the
Young–Laplace equation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...
. 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.In a
mercury barometer A barometer is a scientific instrument that is used to measure air pressure Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth File:Atmosphere gas proportions.svg ...
, the upper liquid surface is an interface between the liquid and a vacuum containing some molecules of evaporated liquid.
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. Where the two surfaces meet, they form a
contact angle The contact angle is the angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Eleme ...

, , 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, , is less than the liquid–air surface tension, , but is nevertheless positive, that is :$\gamma_\mathrm > \gamma_\mathrm - \gamma_\mathrm > 0$ In the diagram, both the vertical and horizontal forces must cancel exactly at the contact point, known as
equilibrium List of types of equilibrium, the condition of a system in which all competing influences are balanced, in a wide variety of contexts. Equilibrium may also refer to: Film and television * Equilibrium (film), ''Equilibrium'' (film), a 2002 scien ...
. The horizontal component of is canceled by the adhesive force, . :$f_\mathrm = f_\mathrm \sin \theta$ The more telling balance of forces, though, is in the vertical direction. The vertical component of must exactly cancel the difference of the forces along the solid surface, . :$f_\mathrm - f_\mathrm = -f_\mathrm \cos \theta$ Since the forces are in direct proportion to their respective surface tensions, we also have: :$\gamma_\mathrm - \gamma_\mathrm = -\gamma_\mathrm \cos \theta$ where * is the liquid–solid surface tension, * is the liquid–air surface tension, * is the solid–air surface tension, * 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°.Sears, Francis Weston; Zemanski, Mark W. (1955) ''University Physics 2nd ed''. Addison Wesley 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 The contact angle is the angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Eleme ...

). 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: :$\gamma_\mathrm > 0 > \gamma_\mathrm - \gamma_\mathrm$

### 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 Polytetrafluoroethylene (PTFE) is a synthetic fluoropolymer A fluoropolymer is a fluorocarbon Fluorocarbons, sometimes referred to as perfluorocarbons or PFCs, are organofluorine B: isoflurane Isoflurane, sold under the brand name Foran ...

approaches this. Contact angle of 180° occurs when the liquid–solid surface tension is exactly equal to the liquid–air surface tension. :$\gamma_\mathrm = \gamma_\mathrm - \gamma_\mathrm > 0\qquad \theta = 180^\circ$

# Methods of measurement

Because surface tension manifests itself in various effects, it offers a number of paths to its measurement. Which method is optimal depends upon the nature of the liquid being measured, the conditions under which its tension is to be measured, and the stability of its surface when it is deformed. An instrument that measures surface tension is called tensiometer. * : The traditional method used to measure surface or interfacial tension. Wetting properties of the surface or interface have little influence on this measuring technique. Maximum pull exerted on the ring by the surface is measured. *: A universal method especially suited to check surface tension over long time intervals. A vertical plate of known perimeter is attached to a balance, and the force due to wetting is measured. * Spinning drop method: This technique is ideal for measuring low interfacial tensions. The diameter of a drop within a heavy phase is measured while both are rotated. * Pendant drop method: Surface and interfacial tension can be measured by this technique, even at elevated temperatures and pressures. Geometry of a drop is analyzed optically. For pendant drops the maximum diameter and the ratio between this parameter and the diameter at the distance of the maximum diameter from the drop apex has been used to evaluate the size and shape parameters in order to determine surface tension. * Bubble pressure method (Jaeger's method): A measurement technique for determining surface tension at short surface ages. Maximum pressure of each bubble is measured. * Drop volume method: A method for determining interfacial tension as a function of interface age. Liquid of one density is pumped into a second liquid of a different density and time between drops produced is measured. * Capillary rise method: The end of a capillary is immersed into the solution. The height at which the solution reaches inside the capillary is related to the surface tension by the equation discussed below. *
Stalagmometric method The stalagmometric method ( grc, στάλαγμα, stálagma, drop) is one of the most common methods for measuring surface tension Surface tension is the tendency of liquid A liquid is a nearly incompressible In fluid mechanics ...
: A method of weighting and reading a drop of liquid. * Sessile drop method: A method for determining surface tension and
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

by placing a drop on a substrate and measuring the
contact angle The contact angle is the angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Eleme ...

(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 Buoyancy (), or upthrust, is an upward exerted by a that opposes the of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pressure at the bo ...

(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. *By smartphone: Some smartphones can be used to measure the surface tension of a transparent liquid. The method is based on measuring the wavelength of capillary waves of known frequency. The smartphone is placed on top of a cup with the liquid. Then smartphone's vibro-motor excites (through the cup) capillary ripples on the surface of the liquid, which are captured by smartphone's camera.

# Effects

## Liquid in a vertical tube

An old style
mercury Mercury usually refers to: * Mercury (planet) Mercury is the smallest planet in the Solar System and the closest to the Sun. Its orbit around the Sun takes 87.97 Earth days, the shortest of all the Sun's planets. It is named after the Roman g ...

barometer A barometer is a scientific instrument that is used to measure air pressure in a certain environment. Pressure tendency can forecast short term changes in the weather. Many measurements of air pressure are used within surface weather analysis to ...

consists of a vertical glass tube about 1 cm in diameter partially filled with mercury, and with a vacuum (called '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 Capillary action (sometimes capillarity, capillary motion, capillary effect, or wicking) is the ability of a liquid A liquid is a nearly incompressible In fluid mechanics Fluid mechanics is the branch of physics concerned with the ...
. The height to which the column is lifted is given by Jurin's law: :$h = \frac$ where * is the height the liquid is lifted, * is the liquid–air surface tension, * is the density of the liquid, * is the radius of the capillary, * is the acceleration due to gravity, * is the angle of contact described above. If is greater than 90°, as with mercury in a glass container, the liquid will be depressed rather than lifted.

## Puddles on a surface

Pouring mercury onto a horizontal flat sheet of glass results in a
puddle A puddle is a small accumulation of liquid, usually water, on a surface. It can form either by pooling in a depression on the surface, or by surface tension Surface tension is the tendency of liquid A liquid is a nearly incompressi ...

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: :$h = 2 \sqrt$ 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 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: :$h = \sqrt.$ For mercury on glass, = 487 dyn/cm, = 13.5 g/cm3 and = 140°, which gives = 0.36 cm. For water on paraffin at 25 °C, = 72 dyn/cm, = 1.0 g/cm3, and = 107° which gives = 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.

## The breakup of streams into drops

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 The Plateau–Rayleigh instability, often just called the Rayleigh instability, explains why and how a falling stream of fluid breaks up into smaller packets with the same volume but less surface area. It is related to the Rayleigh–Taylor insta ...
, 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 A sine wave or sinusoid is any of certain mathematical curves that describe a smooth periodic oscillation Oscillation is the repetitive variation, typically in time Time is the indefinite continued sequence, progress of existence and ev ...

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.

# Thermodynamics

## Thermodynamic theories of surface tension

developed the thermodynamic theory of capillarity based on the idea of surfaces of discontinuity. 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 $U - TS - \mu_1 N_1 - \mu_2 N_2 \cdots$, a quantity later named as the
grand potential The grand potential is a quantity used in statistical mechanics, especially for irreversible processes in Open system (systems theory), open systems. The grand potential is the characteristic state function for the grand canonical ensemble. Definit ...
and given the symbol $\Omega$. Considering a given subvolume $V$ containing a surface of discontinuity, the volume is divided by the mathematical surface into two parts A and B, with volumes $V_$ and $V_$, with $V = V_ + V_$ exactly. Now, if the two parts A and B were homogeneous fluids (with pressures $p_$, $p_$) and remained perfectly homogeneous right up to the mathematical boundary, without any surface effects, the total grand potential of this volume would be simply $-p_V_-p_V_$. The surface effects of interest are a modification to this, and they can be all collected into a surface free energy term $\Omega_$ so the total grand potential of the volume becomes: :$\Omega = -p_V_-p_V_ + \Omega_.$ For sufficiently macroscopic and gently curved surfaces, the surface free energy must simply be proportional to the surface area: :$\Omega_ = \gamma A,$ for surface tension $\gamma$ and surface area $A$. 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 $\Omega$, 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 Spontaneous may refer to: * Miscarriage, Spontaneous abortion * Spontaneous bacterial peritonitis * Spontaneous combustion * Excited utterance, Spontaneous declaration * Spontaneous emission * Spontaneous fission * Spontaneous generation * Spontane ...

, 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 $\gamma\left(T, \mu_1, \mu_2, \cdots\right)$. 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, developed the theory of capillarity effects based on the hypothesis of a continuous variation of density. He added to the energy density the term $c \left(\nabla \rho\right)^2,$ 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. The van der Waals capillarity energy is now widely used in the
phase field modelsA phase-field model is a mathematical model for solving interfacial problems. It has mainly been applied to solidification dynamics, but it has also been applied to other situations such as viscous fingering, fracture mechanics, hydrogen embrittleme ...
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. The above free energy can be written as: :$\Omega = -\Delta P V_ - p_ V + \gamma A$ where $\Delta P = p_ - p_$ is the pressure difference between the inside (A) and outside (B) of the bubble, and $V_$ is the bubble volume. In equilibrium, , and so, :$\Delta P\,dV_ = \gamma\, dA$. For a spherical bubble, the volume and surface area are given simply by :$V_ = \tfrac43\pi R^3 \quad\rightarrow\quad dV_ = 4\pi R^2 \,dR,$ and :$A = 4\pi R^2 \quad\rightarrow\quad dA = 8\pi R\, dR.$ Substituting these relations into the previous expression, we find :$\Delta P = \frac\gamma,$ which is equivalent to the
Young–Laplace equation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Ph ...
when .

### Influence of temperature

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 Critical or Critically may refer to: *Critical, or critical but stable, medical state Medical state is a term used to describe a hospital A hospital is a health care institution providing patient treatment with specialized medical and nursing ...
. For further details see
Eötvös ruleImage:SFT-benzene.png, Temperature dependence of the surface tension of benzene The Eötvös rule, named after the Hungarian physicist Loránd Eötvös, Loránd (Roland) Eötvös (1848–1919) enables the prediction of the surface tension of an arb ...
. There are only empirical equations to relate surface tension and temperature: * Eötvös: :: $\gamma V^ = k\left(T_\mathrm-T\right) .$ : Here is the molar volume of a substance, is the
critical temperature Critical or Critically may refer to: *Critical, or critical but stable, medical state Medical state is a term used to describe a hospital A hospital is a health care institution providing patient treatment with specialized medical and nursing ...
and is a constant valid for almost all substances. A typical value is = . For water one can further use = 18 ml/mol and = 647 K (374 °C). : A variant on Eötvös is described by Ramay and Shields: :: $\gamma V^ = k\left(thrm - T - 6\right)$ : where the temperature offset of 6 K provides the formula with a better fit to reality at lower temperatures. * Guggenheim–Katayama: :: $\gamma = \gamma^\circ \left\left( 1-\frac \right\right)^n$ : is a constant for each liquid and is an empirical factor, whose value is for organic liquids. This equation was also proposed by , who further proposed that could be given by the expression ::$K_2 T^_\mathrm P^_\mathrm,$ : where is a universal constant for all liquids, and is the
critical pressure In thermodynamics, a critical point (or critical state) is the end point of a phase Equilibrium (thermodynamics), equilibrium curve. The most prominent example is the liquid–vapor critical point, the end point of the pressure–temperature cur ...
of the liquid (although later experiments found to vary to some degree from one liquid to another). 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.

### Influence of solute concentration

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 Sugar is the generic name for sweet-tasting, soluble carbohydrate is a disaccharide A disaccharide (also called a double sugar or ''biose'') is the sugar formed when two monosaccharides are joined by glycosidic linkage. Like monosacc ...

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 In chemistry, alcohol is an organic compound that carries at least one hydroxyl functional group (−OH) bound to a Saturated and unsaturated compounds, saturated carbon atom. The term alcohol originally referred to the primary alcohol ethan ...

at water, air * Decrease surface tension until certain critical concentration, and no effect afterwards:
surfactants Surfactants are compounds that lower the surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than wate ...
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 The Gibbs adsorption isotherm for multicomponent systems is an equation used to relate the changes in concentration of a component in contact with a surface with changes in the surface tension Surface tension is the tendency of liquid ...
states that: :$\Gamma = - \frac \left\left( \frac \right\right)_$ * 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 * is the concentration of the substance in the bulk solution. * is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant The Boltzmann constant ( or ) is the proportionality f ...
and the
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

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

### Influence of particle size on vapor pressure

The
Clausius–Clapeyron relation The Clausius–Clapeyron relation, named after Rudolf Clausius and Benoît Paul Émile Clapeyron, specifies the temperature dependence of pressure, most importantly vapor pressure, at a discontinuous phase transition between two phases of matter o ...
leads to another equation also attributed to Kelvin, as the
Kelvin equation The Kelvin equation describes the change in vapour pressure due to a curved liquid–vapor interface, such as the surface of a droplet. The vapor pressure at a convex curved surface is higher than that at a flat surface. The Kelvin equation is ...
. It explains why, because of surface tension, the
vapor pressure Vapor pressure (or vapour pressure in British English British English (BrE) is the standard dialect A standard language (also standard variety, standard dialect, and standard) is a language variety that has undergone substantial cod ...

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. :$P_\mathrm^\mathrm=P_\mathrm^\circ e^$ * is the standard vapor pressure for that liquid at that temperature and pressure. * is the molar volume. * is the
gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant The Boltzmann constant ( or ) is the proportionality f ...
* is the Kelvin radius, the radius of the droplets. The effect explains
supersaturation Supersaturation occurs with a chemical solution when the concentration of a solute exceeds the concentration specified by the value equilibrium solubility Solubility is the property of a solid, liquid or gaseous chemical substance called ''s ...
of vapors. In the absence of
nucleation Nucleation is the first step in the formation of either a new thermodynamic phase or a new structure via self-assembly File:Iron oxide nanocube.jpg, upright=1.2, Transmission electron microscopy image of an iron oxide nanoparticle. Regularly ar ...

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 In chemistry Chemistry is the scientific Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is an occurrence in ...
point. This equation is also used in
catalyst Catalysis () is the process of increasing the rate of a chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substance A chemical substance is a form of matter In cla ...

chemistry to assess mesoporosity for solids.Ertl, G.; Knözinger, H. and Weitkamp, J. (1997) ''Handbook of heterogeneous catalysis'', Vol. 2, p. 430. Wiley-VCH, Weinheim. 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 Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
analysis.

# Surface tension of water and of seawater

The two most abundant liquids on the Earth are fresh
water Water (chemical formula H2O) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known li ...

and
seawater Seawater, or salt water, is water Water (chemical formula H2O) is an , transparent, tasteless, odorless, and , which is the main constituent of 's and the s of all known living organisms (in which it acts as a ). It is vital for al ...

. This section gives correlations of reference data for the surface tension of both.

## Surface tension of water

The surface tension of pure liquid water in contact with its vapor has been given by IAPWS as : where both and the critical temperature = 647.096 K are expressed in
kelvin The kelvin is the base unit of temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal en ...

s. 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. For temperatures below 100 °C, the uncertainty is ±0.5%.

## Surface tension of seawater

Nayar et al. 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 Desalination is a process that takes away mineral components from saline water Saline water (more commonly known as salt water) is water Water (chemical formula H2O) is an Inorganic compound, inorganic, transparent, tasteless, odor ...
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 :$\gamma_\mathrm = \gamma_\mathrm \left\left( 1+ 3.766\times 10^ S +2.347\times 10^ S t\right\right)$ 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.

# Gallery of effects

Image:UnstableLiquidSheet.jpg, Breakup of a moving sheet of water bouncing off of a spoon. Image:SurfaceTension.jpg, Photo of flowing water adhering to a hand. Surface tension creates the sheet of water between the flow and the hand. Image:Ggb in soap bubble 1.jpg, A soap bubble balances surface tension forces against internal
pneumatic Pneumatics (from Greek ‘wind, breath’) is a branch of engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and building ...
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

. Image:2006-01-15 coin on water.jpg, 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 Buoyancy (), or upthrust, is an upward exerted by a that opposes the of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pressure at the bo ...

to balance mass. Image:3_Moeda_(5).jpg, An aluminium coin floats on the surface of the water at 10 °C. Any extra weight would drop the coin to the bottom. Image:Dscn3156-daisy-water 1200x900.jpg, 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 filling the air between the petals and possibly submerging the flower. Image:Surface Tension 01.jpg, A metal paper clip floats on water. Several can usually be carefully added without overflow of water. Image:Paperclip floating on water (with 'contour lines').jpg, 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.

*
Anti-fog Anti-fog agents, also known as anti-fogging agents and treatments, are chemicals that prevent the condensation Condensation is the change of the state of matter In physics, a state of matter is one of the distinct forms in which matter can ...
*
Capillary wave A capillary wave is a wave In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the relate ...
— short waves on a water surface, governed by surface tension and inertia * Cheerio effect — the tendency for small wettable floating objects to attract one another. * Cohesion * Dimensionless numbers ** Bond number or Eötvös number ** Capillary number ** Marangoni number ** Weber number * Dortmund Data Bank — contains experimental temperature-dependent surface tensions * Electrodipping force * Electrowetting * Electrocapillarity *
Eötvös ruleImage:SFT-benzene.png, Temperature dependence of the surface tension of benzene The Eötvös rule, named after the Hungarian physicist Loránd Eötvös, Loránd (Roland) Eötvös (1848–1919) enables the prediction of the surface tension of an arb ...
— a rule for predicting surface tension dependent on temperature * Fluid pipe * Hydrostatic equilibrium—the effect of gravity pulling matter into a round shape * Interface (chemistry) * Meniscus (liquid), Meniscus — surface curvature formed by a liquid in a container * Mercury beating heart — a consequence of inhomogeneous surface tension * Microfluidics * Sessile drop technique * Sow-Hsin Chen * Specific surface energy — same as surface tension in isotropic materials. * Spinning drop method *
Stalagmometric method The stalagmometric method ( grc, στάλαγμα, stálagma, drop) is one of the most common methods for measuring surface tension Surface tension is the tendency of liquid A liquid is a nearly incompressible In fluid mechanics ...
* Surface pressure#Physical chemistry, Surface pressure * Surface science * Surface tension biomimetics * Surface tension values * Surfactants — substances which reduce surface tension. * Szyszkowski equation — calculating surface tension of aqueous solutions * Tears of wine — the surface tension induced phenomenon seen on the sides of glasses containing alcoholic beverages. * Tolman length — leading term in correcting the surface tension for curved surfaces. * Wetting and dewetting

# References

What is surface tension?"Why is surface tension parallel to the interface?"
''Physics Stack Exchange''. Retrieved 2021-03-19. *Michael Berry (physicist), Berry, M V (1971-03-01). "The molecular mechanism of surface tension". ''Physics Education''. 6 (2): 79–84. Doi (identifier), doi:10.1088/0031-9120/6/2/001. ISSN (identifier), ISSN 0031-9120. *Marchand, Antonin; Weijs, Joost H.; Snoeijer, Jacco H.; Andreotti, Bruno (2011-09-26). "Why is surface tension a force parallel to the interface?". ''American Journal of Physics''. 79 (10): 999–1008. Doi (identifier), doi:10.1119/1.3619866. ISSN (identifier), ISSN 0002-9505. arXiv: https://arxiv.org/abs/1211.3854
On surface tension and interesting real-world cases

Calculation of temperature-dependent surface tensions for some common components

Surface tension calculator for aqueous solutions
containing the ions H+, , Na+, K+, Mg2+, Ca2+, , , Cl, , Br and OH. * T. Proctor Hall (1893
New methods of measuring surface tension in liquids
Philosophical Magazine (series 5, 36: 385–415), link from Biodiversity Heritage Library.
The Bubble Wall
(Audio slideshow from the National High Magnetic Field Laboratory explaining cohesion, surface tension and hydrogen bonds)
C. Pfister: Interface Free Energy. Scholarpedia 2010
(from first principles of statistical mechanics)
Fundamentals of surface and interfacial tension

Surface and Interfacial Tension
* {{DEFAULTSORT:Surface Tension Fluid dynamics Surface science Articles containing video clips Intermolecular forces