Continuum mechanics  

Laws


Surface tension is the tendency of liquid surfaces to shrink into the minimum surface area possible. Surface tension allows insects (e.g. water striders), usually denser than water, to float and slide on a water surface.
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). The net effect is an inward force at its surface that causes the liquid to behave as if its surface were covered with a stretched elastic membrane. Thus, the surface comes under tension from the imbalanced forces, which is probably where the term "surface tension" came from.^{[1]} Because of the relatively high attraction of water molecules to each other through a web of hydrogen bonds, water has a higher surface tension (72.8 millinewtons per meter at 20 °C) than most other liquids. Surface tension is an important factor in the phenomenon of capillarity.
Surface tension has the dimension of force per unit length, or of energy per unit area. The two are equivalent, but when referring to energy per unit of area, it is common to use the term surface energy, which is a more general term in the sense that it applies also to solids.
In materials science, surface tension is used for either surface stress or surface energy.
Due to the cohesive forces a molecule 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 and forces liquid surfaces to contract to the minimum area. 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, the contact angle 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 higher state of energy than if it were alone (with some contact with a neighbor). 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.^{[2]} 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 equation). Since any curvature in the surface shape results in greater area, a higher energy will also result.
Several effects of surface tension can be seen with ordinary water:
C. Water striders stay atop the liquid because of surface tension
D. Lava lamp with interaction between dissimilar liquids: water and liquid wax
E. Photo showing the "tears of wine" phenomenon.
Surface tension is visible in other common phenomena, especially when surfactants are used to decrease it:
Surface tension, represented by the symbol γ (alternatively σ or T), is measured in force per unit length. Its SI unit is newton per meter but the cgs unit of dyne per centimeter is also used. For example,^{[5]}
Surface tension can be defined in terms of force or energy.
Surface tension γ 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 F required to hold the movable side is proportional to the length L of the immobile side. Thus the ratio F/L 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 F/L, with L the length of the movable side and F the force required to stop it from sliding, is found to be the same for all shapes. We therefore define the surface tension as
The reason for the 1/2 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 γL = F/2.
Surface tension γ 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:^{[6]} if F 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 F in moving the side by distance Δx is W = FΔx; at the same time the total area of the film increases by ΔA = 2LΔx (the factor of 2 is here because the liquid has two sides, two surfaces). Thus, multiplying both the numerator and the denominator of γ = 1/2F/L by Δx, we get
This work W is, by the usual arguments, interpreted as being stored as potential energy. Consequently, surface tension can be also measured in SI system as joules per square meter and in the cgs system as ergs per cm^{2}. 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.^{[7]}
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:^{[8]}
where:
The quantity in parentheses on the right hand side is in fact (twice) the mean curvature of the surface (depending on normalisation). Solutions to this equation determine the shape of water drops, puddles, menisci, soap bubbles, and all other shapes determined by surface tension (such as the shape of the impressions that a water strider's feet make on the surface of a pond). The table below shows how the internal pressure of a water droplet increases with decreasing radius. For not very small drops the effect is subtle, but the pressure difference becomes enormous when the drop sizes approach the molecular size. (In the limit of a single molecule the concept becomes meaningless.)
Droplet radius  1 mm  0.1 mm  1 μm  10 nm 

Δp (atm)  0.0014  0.0144  1.436  143.6 
When an object is placed on a liquid, its weight F_{w} depresses the surface, and if surface tension and downward force becomes equal than is balanced by the surface tension forces on either side F_{s}, 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 F_{s} arrows point in opposite directions, so they cancel each other, but the vertical components point in the same direction and therefore add up^{[2]} to balance F_{w}. 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.
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, a locally minimal surface will appear in the resulting soapfilm within seconds.^{[7]}^{[10]}
The reason for this is that the pressure difference across a fluid interface is proportional to the mean curvature, as seen in the Young–Laplace equation. For an open soap film, the pressure difference is zero, hence the mean curvature is zero, and minimal surfaces have the property of zero mean curvature.
The surface of any liquid is an interface between that liquid and some other medium.^{[note 1]} 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.^{[7]}^{[8]}
Where the two surfaces meet, they form a contact angle, θ, 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, γ_{ls} − γ_{sa}, is less than the liquid–air surface tension, γ_{la}, but is nevertheless positive, that is
In the diagram, both the vertical and horizontal forces must cancel exactly at the contact point, known as equilibrium. The horizontal component of f_{la} is canceled by the adhesive force, f_{A}.^{[7]}
The more telling balance of forces, though, is in the vertical direction. The vertical component of f_{la} must exactly cancel the difference of the forces along the solid surface, f_{ls} − f_{sa}.^{[7]}
Liquid  Solid  Contact angle  

water 

0°  
ethanol  
diethyl ether  
carbon tetrachloride  
glycerol  
acetic acid  
water  paraffin wax  107°  
silver  90°  
methyl iodide  sodalime glass  29°  
lead glass  30°  
fused quartz  33°  
mercury  sodalime glass  140°  
Some liquid–solid contact angles^{[7]} 
Since the forces are in direct proportion to their respective surface tensions, we also have:^{[8]}
where
This means that although the difference between the liquid–solid and solid–air surface tension, γ_{ls} − γ_{sa}, is difficult to measure directly, it can be inferred from the liquid–air surface tension, γ_{la}, and the equilibrium contact angle, θ, which is a function of the easily measurable advancing and receding contact angles (see main article contact angle).
This same relationship exists in the diagram on the right. But in this case we see that because the contact angle is less than 90°, the liquid–solid/solid–air surface tension difference must be negative:
Observe that in the special case of a water–silver interface where the contact angle is equal to 90°, the liquid–solid/solid–air surface tension difference is exactly zero.
Another special case is where the contact angle is exactly 180°. Water with specially prepared Teflon approaches this.^{[8]} Contact angle of 180° occurs when the liquid–solid surface tension is exactly equal to the liquid–air surface tension.
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.
An old style mercury barometer consists of a vertical glass tube about 1 cm in diameter partially filled with mercury, and with a vacuum (called Torricelli'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 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.
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. The height to which the column is lifted is given by Jurin's law:^{[7]}
where
Pouring mercury onto a horizontal flat sheet of glass results in a puddle 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:^{[8]}
where
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:^{[8]}
For mercury on glass, γ_{Hg} = 487 dyn/cm, ρ_{Hg} = 13.5 g/cm^{3} and θ = 140°, which gives h_{Hg} = 0.36 cm. For water on paraffin at 25 °C, γ = 72 dyn/cm, ρ = 1.0 g/cm^{3}, and θ = 107° which gives h_{H2O} = 0.44 cm.
The formula also predicts that when the contact angle is 0°, the liquid will spread out into a microthin layer over the surface. Such a surface is said to be fully wettable by the liquid.
In daytoday 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,^{[8]} 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 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.
J.W. Gibbs developed the thermodynamic theory of capillarity based on the idea of surfaces of discontinuity.^{[20]} He introduced and studied thermodynamics of twodimensional objects – surfaces. These surfaces have area, mass, entropy, energy and free energy. As stated above, the mechanical work needed to increase a surface area A is dW = γ dA. Hence at constant temperature and pressure, surface tension equals Gibbs free energy per surface area:^{[8]}
where G is Gibbs free energy and A is the area.
Thermodynamics requires that for systems held at constant pressure and temperature all spontaneous changes of state are accompanied by a decrease in Gibbs free energy.
From this it is easy to understand why decreasing the surface area of a mass of liquid is always spontaneous (G < 0), 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 free energy is defined by the equation^{[21]} G = H − TS, where H is enthalpy and S is entropy. Based upon this and the fact that surface tension is Gibbs free energy per unit area, it is possible to obtain the following expression for entropy per unit area:
Kelvin's equation for surfaces arises by rearranging the previous equations. It states that surface enthalpy or surface energy (different from surface free energy) depends both on surface tension and its derivative with temperature at constant pressure by the relationship.^{[22]}
Fifteen years after Gibbs, J.D. van der Waals developed the theory of capillarity effects based on the hypothesis of a continuous variation of density.^{[23]} He added to the energy density the term 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.^{[24]}^{[25]} The van der Waals capillarity energy is now widely used in the phase field models of multiphase flows. Such terms are also discovered in the dynamics of nonequilibrium gases.^{[26]}
The pressure inside an ideal (one surface) soap bubble can be derived from thermodynamic free energy considerations. At constant temperature and particle number, dT = dN = 0, the differential Helmholtz energy is given by
where P is the difference in pressure inside and outside of the bubble, and γ is the surface tension. In equilibrium, dF = 0, and so,
For a spherical bubble, the volume and surface area are given simply by
and
Substituting these relations into the previous expression, we find
which is equivalent to the Young–Laplace equation when R_{x} = R_{y}. For real soap bubbles, the pressure is doubled due to the presence of two interfaces, one inside and one outside.
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. For further details see Eötvös rule. There are only empirical equations to relate surface tension and temperature:
Here V is the molar volume of a substance, T_{C} is the critical temperature and k is a constant valid for almost all substances.^{[11]} A typical value is k = 2.1×10^{−7} J K^{−1} mol^{−2⁄3}.^{[11]}^{[27]} For water one can further use V = 18 ml/mol and T_{C} = 647 K (374 °C).^{[28]}
A variant on Eötvös is described by Ramay and Shields:^{[21]}
where the temperature offset of 6 K provides the formula with a better fit to reality at lower temperatures.
γ° is a constant for each liquid and n is an empirical factor, whose value is 11/9 for organic liquids. This equation was also proposed by van der Waals, who further proposed that γ° could be given by the expression
where K_{2} is a universal constant for all liquids, and P_{C} is the critical pressure of the liquid (although later experiments found K_{2} to vary to some degree from one liquid to another).^{[22]}
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.
Solutes can have different effects on surface tension depending on the nature of the surface and the solute:
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 states that:
Certain assumptions are taken in its deduction, therefore Gibbs isotherm can only be applied to ideal (very dilute) solutions with two components.
The Clausius–Clapeyron relation leads to another equation also attributed to Kelvin, as the Kelvin equation. It explains why, because of surface tension, the 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 equilibrium concentration of its vapor in its surroundings is greater. This arises because the pressure inside the droplet is greater than outside.^{[21]}
The effect explains supersaturation of vapors. In the absence of nucleation 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 point.^{[21]}
This equation is also used in catalyst chemistry to assess mesoporosity for solids.^{[29]}
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:
P/P_{0} for water drops of different radii at STP^{[22]}  

Droplet radius (nm)  1000  100  10  1 
P/P_{0}  1.001  1.011  1.114  2.95 
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 analysis.
The two most abundant liquids on the Earth are fresh water and seawater. This section gives correlations of reference data for the surface tension of both.
The surface tension of pure liquid water in contact with its vapor has been given by IAPWS^{[30]} as
where both T and the critical temperature T_{C} = 647.096 K are expressed in kelvins. The region of validity the entire vapor–liquid saturation curve, from the triple point (0.01 °C) to the critical point. It also provides reasonable results when extrapolated to metastable (supercooled) conditions, down to at least −25 °C. This formulation was originally adopted by IAPWS in 1976 and was adjusted in 1994 to conform to the International Temperature Scale of 1990.
The uncertainty of this formulation is given over the full range of temperature by IAPWS.^{[30]} For temperatures below 100 °C, the uncertainty is ±0.5%.
Nayar et al.^{[31]} published reference data for the surface tension of seawater over the salinity range of 20 ≤ S ≤ 131 g/kg and a temperature range of 1 ≤ t ≤ 92 °C at atmospheric pressure. The range of temperature and salinity encompasses both the oceanographic range and the range of conditions encountered in thermal desalination technologies. The uncertainty of the measurements varied from 0.18 to 0.37 mN/m with the average uncertainty being 0.22 mN/m.
Nayar et al. correlated the data with the following equation
where γ_{sw} is the surface tension of seawater in mN/m, γ_{w} is the surface tension of water in mN/m, S is the reference salinity^{[32]} in g/kg, and t 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^{[33]}.
Liquid  Temperature (°C)  Surface tension, γ 

Acetic acid  20  27.60 
Acetic acid (45.1%) + Water  30  40.68 
Acetic acid (10.0%) + Water  30  54.56 
Acetone  20  23.70 
Blood  22  55.89 
Diethyl ether  20  17.00 
Ethanol  20  22.27 
Ethanol (40%) + Water  25  29.63 
Ethanol (11.1%) + Water  25  46.03 
Glycerol  20  63.00 
nHexane  20  18.40 
Hydrochloric acid 17.7 M aqueous solution  20  65.95 
Isopropanol  20  21.70 
Liquid helium II  −273  0.37^{[35]} 
Liquid nitrogen  −196  8.85 
Liquid oxygen  −182  13.2 
Mercury  15  487.00 
Methanol  20  22.60 
Molten Silver chloride  650  163^{[36]} 
Molten Sodium chloride/Calcium chloride (47/53 mole %)  650  139^{[37]} 
nOctane  20  21.80 
Sodium chloride 6.0 M aqueous solution  20  82.55 
Sucrose (55%) + water  20  76.45 
Water  0  75.64 
Water  25  71.97 
Water  50  67.91 
Water  100  58.85 
Toluene  25  27.73 
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 to balance mass.
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