Surface tension

In physics, surface tension is an effect within the surface layer of a liquid that causes the layer to behave as an elastic sheet. It is the effect that allows insects (such as the water strider) to walk on water, and causes capillary action, for example.

Surface tension is caused by the attraction between the molecules of the liquid, due to 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 there are no liquid molecules on the outside to balance these forces. (There may also be a small outward attraction caused by air molecules, but as air is much less dense than the liquid, this force is negligible.) All of the molecules at the surface are therefore 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 lowest surface area possible.

Surface tension, measured in newtons per meter (N·m^-1), is represented by the symbol σ or γ or T and is defined as the force along a line of unit length perpendicular to the surface, or work done per unit area.

Dimensional analysis and the work-energy theorem show that the units of surface tension (N·m^-1) are equivalent to joules per square metre (J·m^-2). This means that surface tension can also be considered as surface energy. If a surface with surface tension σ is expanded by a unit area, then the increase in the surface's stored energy is also equal to σ.

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. $W_{adh}=W_{coh}^{\alpha}+W_{coh}^\beta-\gamma_{\alpha}^\beta$

Measuring methods

Du Noüy Ring method: 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.

Wilhelmy Plate method: 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.

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.

Surface tension and thermodynamics

Thermodynamic Definition

From a thermodynamic point of view surface tension is defined as

$\gamma=\left( \frac{\partial G}{\partial A} \right)_{P,T}$ ,

where $G$ is Gibbs free energy and $A$ is the area.

From that definition it is possible to deduce Kelvin Equation I which gives the surface enthalpy (or surface energy in most cases) $H^A=\gamma - T \left( \frac {\partial \gamma}{\partial T} \right)_P$

Influence of temperature on surface tension

There are only empirical equations:

Eötvös:

$\gamma V^{2/3}=k(T_C-T)\,\!$

$V$ is the molar volume of that substance

$T_C$ is the critical temperature

$k$ is a constant for each substance.

For example for water k = 1.03 erg/°C, V= 18 ml/mol and T_C= 374°C.

Guggenheim-Katayama:

$\gamma = \gamma^o \left( 1-\frac{T}{T_C} \right)^n$

$\gamma^o$ is a constant for each liquid and n is an empirical factor, whose value is 11/9 for organic liquids

Influence of solute concentration on surface tension

Solutes can have different effects on surface tension depending on their structure:

No effect, for example sugar
Increase of surface tension, inorganic salts
Decrease surface tension progressively, alcohols
Decrease surface tension and, once a minimum is reached, no more effect: surfactants

Josiah Willard Gibbs proved that, $\Gamma = - \frac{1}{RT} \left( \frac{\partial \gamma}{\partial \ln C} \right)_{T,P}$

$\Gamma$ is known as surface concentration, it represents excess of solute per unit area of the surface over what would be present if the bulk concentration prevailed all the way to the surface. It has units of mol/m²

C is the concentration of the substance in the bulk solution.

R is the gas constant and T the temperature

Pressure jump across a curved surface

If viscous forces are absent, the pressure jump across a curved surface is given by the Young-Laplace Equation, which relates pressure inside a liquid with the pressure outside it, the surface tension and the geometry of the surface.

$\Delta P=\gamma \frac{dA}{dV}$ .

This equation can be applied to any surface:

For a flat surface $\frac{dA}{dV}=0$ so the pressure inside is the same as the pressure outside.
For a spherical surface $P_I=P_O+\frac{2 \gamma}{R}$
For a toroidal surface $P_I=P_O+\gamma \left( \frac{1}{R} + \frac{1}{r} \right)$ , where r and R are the radii of the toroid.

Influence of particle size on vapour pressure

Starting from Clausius-Clapeyron 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.

$P_v^{fog}=P_v^o e^{\frac{V 2\gamma}{RT r}}$

$P_v^o$ is the standard vapor pressure for that liquid at that temperature and pressure.

$V$ is the molar volume.

$R$ is the gas constant

External links

MIT Lecture Notes on Surface Tension

Fluid mechanics | Physical chemistry

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