# Surface tension

In free fall, water takes on a teardrop shape due to air resistance . In weightlessness the drops are spherical.

The surface tension is the phenomenon that occurs in liquids as a result of molecular forces to keep their surface area small. The surface of a liquid behaves like a stretched, elastic film. This effect is what causes water to form droplets , for example , and helps some insects run across the water or allow a razor blade to “float” on water.

A pair of water striders use surface tension

The surface tension (Formula Symbol: , ) is thus a surface tension that occurs between liquid and gas phases. It is measured in the SI units kg / s 2 , which is synonymous with N / m . ${\ displaystyle \ sigma}$${\ displaystyle \ gamma}$

In a broader sense, the near-surface tension on solids is also referred to as surface tension. Eye-catching examples are balloons or toughened glass .

## meaning

Left: Cross-section of a drop of liquid. The direction of action of the surface tension is shown at the curved point. Right: The result of the effect of surface tension. The drop has assumed a spherical shape as a result of the pull on the liquid surface.

Surface tension is a pulling force that is localized on the surface of a liquid and its direction of action is parallel to the surface of the liquid. Accordingly, a liquid surface is always under tension . A liquid surface can thus be compared to a slightly stretched thin film, only that the tension does not depend on the elongation.

The surface tension of the water carries a paper clip

The surface tension gives a liquid surface special properties. Non- wetted objects can float on a water surface as long as their weight is not sufficient to overcome the surface tension. This becomes clear when, for example, a paper clip - made of greasy iron wire - is placed on the surface of water. It is not or only partially wetted, sinks a little below the water level, but takes the surface with it, dents it. The surface tension acts on the paper clip with vertical force components and carries it. This effect is also used by living things such as the water strider in order to be able to walk on a surface of water.

Surface tension is what causes liquids to take on a spherical shape when no other forces act on them. An extreme example of this are drops of liquid in space that are almost not exposed to external forces. After mercury as the front runner among pure substances, water has a particularly high surface tension. This drops significantly as the temperature of the water rises and can be radically reduced by adding even small amounts of surface-active substances ( detergents ).

A practical example are particularly small water droplets. To explain this, think of a liquid whose shape is not spherical. The surface tension acts parallel to the surface of the liquid and compensates for locally deviating curvatures .

When other forces act on a drop of liquid, its shape deviates from the spherical one. An example of this are raindrops with a diameter of more than 1 mm and drops of liquid on a solid surface, where additional attractive forces act between the solid and the liquid ( adhesion ). The shape of the drop deviates more from the spherical and wets the surface of the solid, the higher the adhesion between solid and liquid.

Another example of the effect of surface tension is the hexagonal shape of honeybees' honeycomb cells . The cells are first built around beeswax . However, the material yields (flows) due to the temperatures prevailing in the beehive and forms flat interfaces ( minimal surfaces ) between the individual cells.

## Physical background

Liquid clamped in a bracket. A force pulls parallel to the liquid film and increases its surface (front and back) .${\ displaystyle \ mathrm {d} A = 2L \ mathrm {d} x}$

There are two definitions of surface tension that are consistent. On the one hand there is the mechanical definition, according to which the surface tension is a force per length, and the thermodynamic one , according to which the surface tension is an energy per area.

### Mechanical definition

The mechanical definition can be explained using a bracket with the width in which a liquid film is clamped. If the liquid film by a force parallel to the surface and perpendicular to to is pulled apart, the work on the film is performed and the surface increases by (a factor of 2 because the front and back of the film). The surface tension is the ratio . Accordingly, the surface tension is a force per length that is directed parallel to the surface of the liquid. ${\ displaystyle L}$${\ displaystyle F}$${\ displaystyle L}$${\ displaystyle \ mathrm {d} x}$${\ displaystyle \ mathrm {d} W = F \ mathrm {d} x}$${\ displaystyle \ mathrm {d} A = 2L \ mathrm {d} x}$${\ displaystyle \ gamma = {\ frac {\ mathrm {d} W} {\ mathrm {d} A}} = {\ frac {F} {2L}}}$

The correctness of the idea of ​​surface tension as a force parallel to the surface is shown in numerous measurement methods and effects such as the ironing method, capillarity or the contact angle .

### Thermodynamic definition

The thermodynamic conception of surface tension as energy per area comes from the picture that the symmetry of the liquid molecules on the liquid surface is disturbed. The lack of liquid molecules vertical to the liquid surface and the resulting “missing” binding energy must be compensated for by positive energy . In order to increase the surface of a liquid one needs energy, whereby the surface tension is defined as the energy that is needed to enlarge the liquid surface by a unit area. So it follows ${\ displaystyle E}$${\ displaystyle \ gamma}$

${\ displaystyle \ gamma = {\ frac {\ mathrm {d} E} {\ mathrm {d} A}} = {\ frac {\ mathrm {d} W} {\ mathrm {d} A}} = {\ frac {F} {2L}}}$,

This shows the analogy between the idea of ​​“missing binding energy” and the mechanical definition.

However, this clear interpretation is not yet sufficient to define the surface tension thermodynamically. To do this one goes from the change in the Gibbs free energy at a constant temperature and constant pressure which is described by equation (1), wherein the enthalpy , the temperature and the entropy features. ${\ displaystyle G}$${\ displaystyle T}$${\ displaystyle p}$${\ displaystyle H}$${\ displaystyle T}$${\ displaystyle S}$

${\ displaystyle \ mathrm {d} G = \ mathrm {d} HT \ mathrm {d} S \ qquad (1)}$

One can rewrite this equation by inserting the definition of enthalpy and taking into account that it holds. ${\ displaystyle H = U + pV}$${\ displaystyle \ mathrm {d} p = 0}$

The surface tension causes the drink to bulge over the glass.
${\ displaystyle \ mathrm {d} G = \ mathrm {d} U + p \ mathrm {d} VT \ mathrm {d} S \ qquad (2)}$

The change in internal energy is used, where stands for the work done. The following applies to the amount of heat . It follows: ${\ displaystyle \ mathrm {d} U = \ mathrm {d} q + \ mathrm {d} W}$${\ displaystyle \ mathrm {d} W}$${\ displaystyle \ mathrm {d} q = T \ mathrm {d} S}$

${\ displaystyle \ mathrm {d} G = \ mathrm {d} W + p \ mathrm {d} V \ qquad (3)}$

The expression for work can be broken down into a term for volume work and non-expansive work . ${\ displaystyle \ mathrm {d} W _ {\ mathrm {vol}} = - p \ mathrm {d} V}$${\ displaystyle \ mathrm {d} W_ {ne}}$

${\ displaystyle \ mathrm {d} G = \ mathrm {d} W_ {ne} -p \ mathrm {d} V + p \ mathrm {d} V = \ mathrm {d} W_ {ne} \ quad (4) }$

At constant temperature and constant pressure, the change in free enthalpy corresponds to the non-expansive work. This expression can now be related to surface tension . If only work is used to enlarge the surface of a liquid, this corresponds to the expression . Since the surface tension is now defined as work per unit area, the surface of the liquid must still be taken into account. So it follows: ${\ displaystyle \ gamma}$${\ displaystyle dW_ {ne}}$${\ displaystyle A}$

${\ displaystyle \ gamma = \ left ({\ frac {\ partial G} {\ partial A}} \ right) _ {T, p} = \ left ({\ frac {\ partial W_ {ne}} {\ partial A}} \ right) _ {T, p} \ qquad (5)}$

The surface tension is therefore thermodynamically defined as the partial derivative of the free enthalpy according to the surface at constant temperature and constant pressure.

### Molecular Theory of Surface Tension

Schematic representation of the Lennard-Jones potential as a function of the distance r .

The idea of ​​the missing liquid molecules on the surface intuitively leads to the assumption that the surface tension is a force vertical to the liquid surface. However, this does not agree with the mechanical definition of surface tension. In order to bring the mechanical definition into line with the thermodynamic one, one has to consider that within a liquid both attractive and repulsive forces act on a molecule. While in a solid body either attractive or repulsive forces act locally, because the particles are in fixed places, the molecules in a liquid are mobile. The distances between the liquid molecules can change and thus repulsive and attractive forces can act on a liquid particle. This fact can also be illustrated in a Lennard-Jones potential . This generally describes the potential between two uncharged particles as a function of their distance. If the particles come into contact at short distances, they repel each other, while at greater distances they attract. While the distance between two particles is fixed in a solid body, it can change in a liquid due to the thermal movement , which enables attractive and repulsive forces on a liquid molecule. The picture on the right shows a schematic representation of a Lennard-Jones potential, which explains the forces between liquid molecules. If the liquid molecules are in contact, they repel each other (orange area), while at great distances they attract (blue area). In a liquid, the distances between the particles change constantly due to the movement of heat, which is shown by the black double arrow in the figure. This means that both attractive and repulsive forces can act on a liquid molecule.

One can interpret the repulsive forces as contact forces. Because of this, their effect in space can be viewed as direction-independent, i.e. isotropic. The attractive forces within a liquid act at further distances, are conditioned by the structure of the molecules and can be viewed as direction-dependent in space, i.e. anisotropic.

Representation of the forces that act on a molecule on the surface of the liquid and in the interior of the liquid.

At the phase interface between liquid and gas phase, the density of the liquid changes abruptly in the range of a few molecule lengths until it remains constant at the value of the inside of the liquid. This has the effect that the repulsive forces in the liquid also increase abruptly on the surface until they reach the constant value of the interior of the liquid, this increase being the same in all spatial directions due to the isotropic nature of the repulsive forces. The picture on the right serves as a further explanation, in which the forces on a liquid molecule on the surface and inside are illustrated. The symmetry is disturbed on the surface of the liquid, i.e. the molecules there have no neighboring molecules in the vertical direction. Thus, in the vertical direction, only repulsive forces act on the molecules from below (gray arrow). In order to maintain the balance of forces , the repulsive forces are balanced in the vertical direction by attractive forces (orange arrow). This is not necessary in the horizontal direction, i.e. parallel to the surface, since the symmetry is not disturbed. This means that in the horizontal direction repulsive forces act on the liquid molecules on the surface from all sides. In addition to the repulsive forces, attractive forces also act in the horizontal direction. However, these are not necessary to maintain the balance of forces and, because of their anisotropic nature, can therefore be greater in magnitude than the repulsive forces. This means that on the surface of the liquid in the horizontal direction the attractive forces on the liquid molecules are greater than the repulsive forces. In the interior of the liquid, the attractive and repulsive forces on a molecule are equal.

Illustration of the surface tension as a force parallel to the liquid surface.
Schematic representation of the change in density and surface tension on a liquid surface.

In order to further understand the surface tension as a force parallel to the surface, it is clear to divide the liquid into two halves, as shown in the picture on the right. There you can see a dotted and a non-dotted half, whereby these only serve to mark the two parts. Consider the forces exerted by the non-dotted part on the dotted part of the liquid. a.) First, the dividing line between the two halves of the liquid is laid parallel to the surface of the liquid. The density increases towards the inside of the liquid, therefore the repulsive forces (gray) on the dotted part also increase. These are balanced by attractive forces (orange). b.) If you now lay the dividing line between the halves in a vertical direction, you can again draw in the repulsive forces that act on the dotted part. Due to their isotropic nature, these are of the same magnitude as in the vertical direction. However, the attractive forces on the dotted part are not isotropic in nature and their magnitude can be greater than the repulsive forces. One can also see that this difference decreases the further one progresses into the interior of the liquid. Already after a few molecule lengths, attractive and repulsive forces balance each other in the horizontal direction, as the density increases towards the inside of the liquid. c.) The non-dotted part of the liquid exerts an attractive force on the dotted part which decreases in the direction of the interior of the liquid.

In summary, it can be said that in the range of a few molecule lengths the density on the surface of the liquid (red curve in the picture on the right) changes until it reaches the constant value inside the liquid. As a result, a pulling force acts on the surface of the liquid in the horizontal direction. The blue curve in the picture on the right describes the difference between attractive and repulsive forces that are exerted by the non-dotted part of the liquid on the dotted part in the horizontal direction. It corresponds to the surface tension and is localized on the surface in the area of ​​a few molecular diameters.

## Dependencies

Temperature dependence of surface tension using the example of water
Temperature dependence of the surface tension using the example of benzene
${\ displaystyle \ sigma = 0 {,} 07275 \ cdot \ left (1-0 {,} 002 \ cdot \ left ({\ frac {T} {\ mathrm {K}}} - 291 \ right) \ right) \ qquad (6)}$ (in SI unit: N / m)
• In a drop of liquid there is an increased pressure due to the surface tension, just like inside a soap bubble . The pressure increase in the liquid drop is described by the Young-Laplace equation .
• When liquid particles form on condensation nuclei , the curvature effect occurs. It can be seen that a higher saturation vapor pressure occurs over the curved surfaces of the resulting liquid droplets than in comparison to a flat water surface.
• Surface-active substances such as surfactants reduce the surface tension. Their effect can be described by a lateral pressure opposite to the surface tension . is not a pressure, but has the same unit as the surface tension.${\ displaystyle \ pi}$${\ displaystyle \ pi}$
• The adjacent air layer is saturated with the vapor of the liquid. The penetration of other vapors from outside can significantly change the surface tension.
• The surface tension is temperature dependent and generally decreases with increasing temperature. At the critical point it is zero. The temperature dependence is described by Eötvös' rule ; the equation given above is a special case of this rule that applies to water.

## values

Table of values ​​for the surface tension at 20 ° C
liquid Surface tension

in mN / m = 10 −3  N / m

n -pentane 16.00
n -hexane 18.40
Ethanol 22.55
Methanol 22.60
acetone 23.30
benzene 28.90
Ethylene glycol 48.4
Water at 80 ° C 62.6
Glycerin 63.4
Water at 50 ° C 67.9
Water at 20 ° C 72.75
Mercury at 18 ° C 471.00
Mercury at 20 ° C 476.00
Galinstan at 20 ° C 718.0

So water has a comparatively high surface tension (see also water pressure tables in WikiBooks).

## Measurement

The surface tension can be measured, for example, with the help of the ring (from Lecomte De Noüy ), plate (from Wilhelmy) or bracket method (from Lenard), with a tensiometer or by the capillary effect.

The lying or hanging drop can also be measured using an optical evaluation and thus the surface tension of the liquid can be determined.

### Ironing method

Measurement of surface tension using the ironing method
Strap in the liquid

With the ironing method (also known as the tear-off method), a hanger with an extremely thin wire (usually made of platinum ) soldered into it is hung in the liquid so that it is just immersed in the liquid and is wetted by it. The tensile force on the bracket is then gradually increased using a precision spring balance. The wire is then pulled out of the liquid and pulls a film of liquid with it. At a certain point this film breaks down.

By pulling on the bracket, work is done against the surface tension. The surface tension can then be calculated from the maximum possible tensile force on the bracket before the liquid film breaks, the dimensions of the bracket and the density of the liquid.

For liquids such as ethanol and wire lengths of 2–3 cm with a radius of 0.1 mm, the expected value for the mass is in the two to three-digit milligram range. So very precise scales are required. With a measurement uncertainty of the balance of 5 mg and a measurement of the wire with an accuracy of 1 µm, the largest error in the final result is already 8 to 12%.

### Measurement with the capillary effect

Capillary in the measuring liquid

This measurement method makes use of the capillary effect, i.e. liquids rise up in thin tubes. You need a vessel ( e.g. a cuvette ) and the thinnest possible capillary . This is then simply placed in the liquid and the height of rise is measured.

Since the liquid theoretically needs infinitely long to reach its final level, the liquid is first drawn up in the capillary (e.g. with a syringe) and then allowed to sink again. The surface tension can then be read off directly from the height of rise if the density of the liquid and the capillary radius are known. Since the measurement is quite difficult, it is believed disposable micro- pipettes and measures their length. Since the volume is known, the inner radius can be calculated in this way.

In capillaries with a radius of 0.2 mm, water rises up to 7 cm. A cathetometer , for example, is suitable for measuring the height of rise as precisely as possible . If the density of the liquid is known exactly and the height of rise can be read with an accuracy of 0.1 mm, the error is in the lower single-digit percentage range.

### Other methods

• Du Noüy ring method: classic method for measuring interfacial tension and surface tension. Uncritical even with difficult wetting conditions. The force of a liquid lamella pulled up by the ring is measured.
• Wilhelmy plate method : Universal method, especially suitable for surface tension measurements over a longer period of time. The force that results from wetting the vertically suspended plate is measured.
• Contact angle measurement : Provides information about the wettability of a substance. Thesurface tension can be calculated from the cosine of the contact angle using Young's equation .
• Spinning-drop method : to determine interfacial tensions. Particularly suitable for low to extremely low measuring ranges. The diameter of a rotating drop in the heavy phase is measured.
• Pendant drop method : suitable for interfacial and surface tension measurements. Measurement options even at extreme pressures and temperatures. Optical detection of the drop geometry. The size of the drops that drip from a capillary is proportional to the surface tension.
• Sessile drop method : Determination of interfacial and surface tensions from the profile of a drop resting on a substrate. In the past, popular method for measuring liquid metals and alloys, since the measurement under high temperatures and / or extreme pressures can be carried out relatively easily with this method.
• Bubble pressure method : suitable for measuring the dynamic surface tension (measurement depending on the surface age). Common measurement methods are the maximum pressure method and the differential pressure method.
• Drop volume method : superior method for dynamic measurement of interfacial tensions. The number of drops into which a given volume of liquid is divided is measured.
• Test ink method : a test used in industry (e.g. when gluing self-adhesive films) on plastics. A colored liquid (“ink”) with a defined surface tension is applied to the surface to be tested using a brush. If the surface is wetted by the ink (i.e. the brushstroke persists for> 3 seconds without contracting), the surface tension of the tested surface is equal to or greater than that of the test ink. If, on the other hand, the brushstroke contracts within 3 seconds, the surface tension of the tested surface is lower than that of the test ink.
• Expanding / Oscillating-Drop-Method (EDM / ODM) : Method for recording the surface rheological properties of liquids. It describes the dependence of the surface tension on the degree and the speed of the areal expansion of a drop, which is either expanded quickly and then stands still (EDM) or is subject to a sinoidally oscillating oscillation (ODM). With the help of this measurement technique, the foam stability and the emulsion stability can be described.
• Method with a mixture of ethylene glycol monoethyl ether and formamide . Both liquids are mixed together in a certain ratio. This gives you a defined Dyn value for determining the surface tension. The mixture is produced using a tensiometer.
• The stalagmometer method is also based on the drop shape.

## Historical

The term surface tension was first used by Niccolò Cabeo in 1629 and clarified by Johann Andreas von Segner in 1751 . Valuable contributions were made to the theory in 1805 by Thomas Young , in 1806 by Pierre-Simon Laplace , in 1830 by Siméon Denis Poisson (see also Young-Laplace equation , Young's equation ) and from 1842 to 1868 by Joseph Plateau .

• Dortmund database and DETHERM : Collection of experimentally determined surface tensions
• DIPPR 801: Parameters for calculating surface tension (mostly by polynomial)

## literature

• Cyril Isenberg: The Science of Soap Films and Soap Bubbles. Tieto, Clevedon 1978, ISBN 0-905028-02-3 .