Surface tension

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Rain water flux from a canopy. Among the forces that govern drop formation: surface tension, cohesion, Van der Waals force, Plateau-Rayleigh instability. RainDrops1.jpg
Rain water flux from a canopy. Among the forces that govern drop formation: surface tension, cohesion, Van der Waals force, Plateau–Rayleigh instability.
Surface tension and hydrophobicity interact in this attempt to cut a water droplet.
Surface tension experimental demonstration with soap

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 water such as razor blades 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). [1]

There are two primary mechanisms in play. One is an inward force on the surface molecules causing the liquid to contract. [2] [3] Second is a tangential force parallel to the surface of the liquid. [3] 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 liquidair or liquidvapour interface. [4]

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 (mN) 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. [4] 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.


Diagram of the cohesive forces on molecules of a liquid WassermolekuleInTropfchen.svg
Diagram of the cohesive forces on molecules of a liquid

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 and forces liquid surfaces to contract to the minimum area. [2]

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. [2] [3]

The forces of attraction acting between molecules of the same type are called cohesive forces, while those acting between 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.

Water droplet lying on a damask. Surface tension is high enough to prevent seeping through the textile Water droplet lying on a damask.jpg
Water droplet lying on a damask. Surface tension is high enough to prevent seeping through the textile

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. [5] 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.

Effects of surface tension


Several effects of surface tension can be seen with ordinary water:

  1. Beading of rain water on a waxy surface, such as a leaf. Water adheres weakly to wax and strongly to itself, so water clusters into drops. Surface tension gives them their near-spherical shape, because a sphere has the smallest possible surface area to volume ratio.
  2. Formation of drops occurs when a mass of liquid is stretched. The animation (below) shows water adhering to the faucet gaining mass until it is stretched to a point where the surface tension can no longer keep the drop linked to the faucet. It then separates and surface tension forms the drop into a sphere. If a stream of water were running from the faucet, the stream would break up into drops during its fall. Gravity stretches the stream, then surface tension pinches it into spheres. [6]
  3. Flotation of objects denser than water occurs when the object is nonwettable and its weight is small enough to be borne by the forces arising from surface tension. [5] For example, water striders use surface tension to walk on the surface of a pond in the following way. The nonwettability of the water strider's leg means there is no attraction between molecules of the leg and molecules of the water, so when the leg pushes down on the water, the surface tension of the water only tries to recover its flatness from its deformation due to the leg. This behavior of the water pushes the water strider upward so it can stand on the surface of the water as long as its mass is small enough that the water can support it. The surface of the water behaves like an elastic film: the insect's feet cause indentations in the water's surface, increasing its surface area [7] and tendency of minimization of surface curvature (so area) of the water pushes the insect's feet upward.
  4. Separation of oil and water (in this case, water and liquid wax) is caused by a tension in the surface between dissimilar liquids. This type of surface tension is called "interface tension", but its chemistry is the same.
  5. Tears of wine is the formation of drops and rivulets on the side of a glass containing an alcoholic beverage. Its cause is a complex interaction between the differing surface tensions of water and ethanol; it is induced by a combination of surface tension modification of water by ethanol together with ethanol evaporating faster than water.


Surface tension is visible in other common phenomena, especially when surfactants are used to decrease it:


Physical units

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, [8]

Surface area growth

This diagram illustrates the force necessary to increase the surface area. This force is proportional to the surface tension. Surface growing.png
This diagram illustrates the force necessary to increase the surface area. This force is proportional to the surface tension.

Surface tension can be defined in terms of force or energy.

In terms of force

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.

In terms of energy

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: [9] 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 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. [10]

Surface curvature and pressure

Surface tension forces acting on a tiny (differential) patch of surface. dthx and dthy indicate the amount of bend over the dimensions of the patch. Balancing the tension forces with pressure leads to the Young-Laplace equation CurvedSurfaceTension.png
Surface tension forces acting on a tiny (differential) patch of surface. δθx and δθy indicate the amount of bend over the dimensions of the patch. Balancing the tension forces with pressure leads to the Young–Laplace equation

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: [11]


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.)

Δp for water drops of different radii at STP
Droplet radius1 mm0.1 mm1  μm 10  nm
Δp (atm)0.00140.01441.436143.6

Floating objects

Cross-section of a needle floating on the surface of water. Fw is the weight and Fs are surface tension resultant forces. Surface Tension Diagram.svg
Cross-section of a needle floating on the surface of water. Fw is the weight and Fs are surface tension resultant forces.

When an object is placed on a liquid, its weight Fw depresses the surface, and if surface tension and downward force become equal then it is balanced by the surface tension forces on either side Fs, 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 Fs arrows point in opposite directions, so they cancel each other, but the vertical components point in the same direction and therefore add up [5] to balance Fw. 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 m denotes the mass of the needle and g acceleration due to gravity, we have

Liquid surface

Minimal surface Povrsinska napetost milnica.jpg
Minimal surface

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 soap-solution, a locally minimal surface will appear in the resulting soap-film within seconds. [10] [13]

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.

Contact angles

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. [10] [11]

Forces at contact point shown for contact angle greater than 90deg (left) and less than 90deg (right) SurfTensionContactAngle.png
Forces at contact point shown for contact angle greater than 90° (left) and less than 90° (right)

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, γlsγsa, is less than the liquid–air surface tension, γla, 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 fla is canceled by the adhesive force, fA. [10]

The more telling balance of forces, though, is in the vertical direction. The vertical component of fla must exactly cancel the difference of the forces along the solid surface, flsfsa. [10]

Some liquid–solid contact angles [10]
soda-lime glass
lead glass
fused quartz
diethyl ether
carbon tetrachloride
acetic acid
water paraffin wax107°
methyl iodide soda-lime glass29°
lead glass30°
fused quartz33°
mercury soda-lime glass140°

Since the forces are in direct proportion to their respective surface tensions, we also have: [11]


This means that although the difference between the liquid–solid and solid–air surface tension, γlsγsa, is difficult to measure directly, it can be inferred from the liquid–air surface tension, γla, 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:

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 approaches this. [11] Contact angle of 180° occurs when the liquid–solid surface tension is exactly equal to the liquid–air surface tension.

Methods of measurement

Force tensiometer is using Du Nouy ring method and Wilhelmy plate method. Sigma tensiometer.jpg
Force tensiometer is using Du Noüy ring method and Wilhelmy plate method.

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.


Liquid in a vertical tube

Diagram of a mercury barometer HgBarometer.gif
Diagram of a mercury barometer

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 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.

Illustration of capillary rise and fall. Red=contact angle less than 90deg; blue=contact angle greater than 90deg CapillaryAction.svg
Illustration of capillary rise and fall. Red=contact angle less than 90°; blue=contact angle greater than 90°

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: [10]


Puddles on a surface

Profile curve of the edge of a puddle where the contact angle is 180deg. The curve is given by the formula:
{\displaystyle x-x_{0}={\frac {1}{2}}H\cosh ^{-1}\left({\frac {H}{h}}\right)-H{\sqrt {1-{\frac {h^{2}}{H^{2}}}}}}
{\textstyle H=2{\sqrt {{\gamma }/{g\rho }}}} SurfTensionEdgeOfPool.png
Profile curve of the edge of a puddle where the contact angle is 180°. The curve is given by the formula:
Small puddles of water on a smooth clean surface have perceptible thickness. Exploring new continents 1200728.JPG
Small puddles of water on a smooth clean surface have perceptible thickness.

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: [11]


Illustration of how lower contact angle leads to reduction of puddle depth Surface tension.svg
Illustration of how lower contact angle leads to reduction of puddle depth

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: [11]

For mercury on glass, γHg = 487 dyn/cm, ρHg = 13.5 g/cm3 and θ = 140°, which gives hHg = 0.36 cm. For water on paraffin at 25 °C, γ = 72 dyn/cm, ρ = 1.0 g/cm3, and θ = 107° which gives hH2O = 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

Breakup of an elongated stream of water into droplets due to surface tension. Dripping faucet 2.jpg
Breakup of an elongated stream of water into droplets due to surface tension.

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, [11] 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.


Thermodynamic theories of surface tension

J.W. Gibbs developed the thermodynamic theory of capillarity based on the idea of surfaces of discontinuity. [25] 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 T and chemical potentials μi), 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 , a quantity later named as the grand potential and given the symbol .

Gibbs' placement of a precise mathematical surface in a fuzzy physical interface. Gibbs Model.tif
Gibbs' placement of a precise mathematical surface in a fuzzy physical interface.

Considering a given subvolume containing a surface of discontinuity, the volume is divided by the mathematical surface into two parts A and B, with volumes and , with exactly. Now, if the two parts A and B were homogeneous fluids (with pressures , ) and remained perfectly homogeneous right up to the mathematical boundary, without any surface effects, the total grand potential of this volume would be simply . The surface effects of interest are a modification to this, and they can be all collected into a surface free energy term so the total grand potential of the volume becomes:

For sufficiently macroscopic and gently curved surfaces, the surface free energy must simply be proportional to the surface area: [25] [26]

for surface tension and surface area .

As stated above, this implies the mechanical work needed to increase a surface area A is dW = γ dA, 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 , 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, 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. [27] 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 γ however it does have a very strong effect on the values of the surface entropy, surface excess mass densities, and surface internal energy, [25] :237 which are the partial derivatives of the surface tension function .

Gibbs emphasized that for solids, the surface free energy may be completely different from surface stress (what he called surface tension): [25] :315 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, J.D. van der Waals developed the theory of capillarity effects based on the hypothesis of a continuous variation of density. [28] 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. [29] [30] 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 non-equilibrium gases. [31]

Thermodynamics of bubbles

The pressure inside an ideal spherical bubble can be derived from thermodynamic free energy considerations. [26] The above free energy can be written as:

where is the pressure difference between the inside (A) and outside (B) of the bubble, and is the bubble volume. In equilibrium, dΩ = 0, and so,

For a spherical bubble, the volume and surface area are given simply by


Substituting these relations into the previous expression, we find

which is equivalent to the Young–Laplace equation when Rx = Ry.

Influence of temperature

Temperature dependence of the surface tension between the liquid and vapor phases of pure water Temperature dependence surface tension of water.svg
Temperature dependence of the surface tension between the liquid and vapor phases of pure water
Temperature dependency of the surface tension of benzene SFT-benzene.png
Temperature dependency of the surface tension of benzene

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:

  • Eötvös: [14] [32] [33]
    Here V is the molar volume of a substance, TC is the critical temperature and k is a constant valid for almost all substances. [14] A typical value is k = 2.1×10−7 J K−1 mol23. [14] [33] For water one can further use V = 18 ml/mol and TC = 647 K (374 °C). [34] A variant on Eötvös is described by Ramay and Shields: [35]
    where the temperature offset of 6 K provides the formula with a better fit to reality at lower temperatures.
  • Guggenheim–Katayama: [32]
    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 K2 is a universal constant for all liquids, and PC is the critical pressure of the liquid (although later experiments found K2 to vary to some degree from one liquid to another). [32]

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 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 at water|air
  • Decrease surface tension until certain critical concentration, and no effect afterwards: surfactants 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 states that:

  • Γ 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

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 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. [35]

Molecules on the surface of a tiny droplet (left) have, on average, fewer neighbors than those on a flat surface (right). Hence they are bound more weakly to the droplet than are flat-surface molecules. TinyDropletMolecules.png
Molecules on the surface of a tiny droplet (left) have, on average, fewer neighbors than those on a flat surface (right). Hence they are bound more weakly to the droplet than are flat-surface molecules.
  • Pv° is the standard vapor pressure for that liquid at that temperature and pressure.
  • V is the molar volume.
  • R is the gas constant
  • rk is the Kelvin radius, the radius of the droplets.

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. [35]

This equation is also used in catalyst chemistry to assess mesoporosity for solids. [36]

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/P0 for water drops of different radii at STP [32]
Droplet radius (nm)1000100101

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.

Surface tension of water and of seawater

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.

Surface tension of water

The surface tension of pure liquid water in contact with its vapor has been given by IAPWS [37] as

where both T and the critical temperature TC = 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. [37] For temperatures below 100 °C, the uncertainty is ±0.5%.

Surface tension of seawater

Nayar et al. [38] 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 γsw is the surface tension of seawater in mN/m, γw is the surface tension of water in mN/m, S is the reference salinity [39] 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. [40]

Data table

Surface tension of various liquids in dyn/cm against air [41]
Mixture compositions denoted "%" are by mass
dyn/cm is equivalent to the SI units of mN/m (millinewton per meter)
LiquidTemperature (°C)Surface tension, γ
Acetic acid 2027.60
Acetic acid (45.1%) + Water3040.68
Acetic acid (10.0%) + Water3054.56
Acetone 2023.70
Blood 2255.89
Diethyl ether 2017.00
Ethanol 2022.27
Ethanol (40%) + Water2529.63
Ethanol (11.1%) + Water2546.03
Glycerol 2063.00
n-Hexane 2018.40
Hydrochloric acid 17.7  M aqueous solution2065.95
Isopropanol 2021.70
Liquid helium II −2730.37 [42]
Liquid nitrogen −1968.85
Liquid oxygen −18213.2
Mercury 15487.00
Methanol 2022.60
Molten Silver chloride 650163 [43]
Molten Sodium chloride/Calcium chloride (47/53 mole %)650139 [44]
n-Octane 2021.80
Sodium chloride 6.0  M aqueous solution2082.55
Sucrose (55%) + water2076.45
Water 075.64
Toluene 2527.73

See also

Explanatory notes

  1. In a mercury barometer, the upper liquid surface is an interface between the liquid and a vacuum containing some molecules of evaporated liquid.

Related Research Articles

<span class="mw-page-title-main">Hydrophobe</span> Molecule or surface that has no attraction to water

In chemistry, hydrophobicity is the physical property of a molecule that is seemingly repelled from a mass of water. In contrast, hydrophiles are attracted to water.

<span class="mw-page-title-main">Surface energy</span> Quantifies the disruption of intermolecular bonds that occurs when a surface is created

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 physics of solids, surfaces must be intrinsically less energetically favorable than the bulk of a material, otherwise there would be a driving force for surfaces to be created, removing the bulk of the material. The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk, or it is the work required to build an area of a particular surface. Another way to view the surface energy is to relate it to the work required to cut a bulk sample, creating two surfaces. There is "excess energy" as a result of the now-incomplete, unrealized bonding at the two surfaces.

Electrowetting is the modification of the wetting properties of a surface with an applied electric field.

<span class="mw-page-title-main">Wetting</span> Ability of a liquid to maintain contact with a solid surface

Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. This happens in presence of a gaseous phase or another liquid phase not miscible with the first one. The degree of wetting (wettability) is determined by a force balance between adhesive and cohesive forces.

<span class="mw-page-title-main">Dewetting</span>

In fluid mechanics, dewetting is one of the processes that can occur at a solid–liquid, solid-solid or liquid–liquid interface. Generally, dewetting describes the process of retraction of a fluid from a non-wettable surface it was forced to cover. The opposite process—spreading of a liquid on a substrate—is called wetting. The factor determining the spontaneous spreading and dewetting for a drop of liquid placed on a solid substrate with ambient gas, is the so-called spreading coefficient S:

<span class="mw-page-title-main">Langmuir–Blodgett trough</span> Laboratory equipment

A Langmuir–Blodgett trough is a laboratory apparatus that is used to compress monolayers of molecules on the surface of a given subphase and measures surface phenomena due to this compression. It can also be used to deposit single or multiple monolayers on a solid substrate.

<span class="mw-page-title-main">Contact angle</span> The angle between a liquid–vapor interface and a solid surface

The contact angle is the angle, conventionally measured through the liquid, where a liquid–vapor interface meets a solid surface. It quantifies the wettability of a solid surface by a liquid via the Young equation. A given system of solid, liquid, and vapor at a given temperature and pressure has a unique equilibrium contact angle. However, in practice a dynamic phenomenon of contact angle hysteresis is often observed, ranging from the advancing (maximal) contact angle to the receding (minimal) contact angle. The equilibrium contact is within those values, and can be calculated from them. The equilibrium contact angle reflects the relative strength of the liquid, solid, and vapour molecular interaction.

<span class="mw-page-title-main">Cassie's law</span>

Cassie's law, or the Cassie equation, describes the effective contact angle θc for a liquid on a chemically heterogeneous surface, i.e. the surface of a composite material consisting of different chemistries, that is non uniform throughout. Contact angles are important as they quantify a surface's wettability, the nature of solid-fluid intermolecular interactions. Cassie's law is reserved for when a liquid completely covers both smooth and rough heterogeneous surfaces.

<span class="mw-page-title-main">Sessile drop technique</span> Method used for the characterization of solid surface energies

The sessile drop technique is a method used for the characterization of solid surface energies, and in some cases, aspects of liquid surface energies. The main premise of the method is that by placing a droplet of liquid with a known surface energy, the shape of the drop, specifically the contact angle, and the known surface energy of the liquid are the parameters which can be used to calculate the surface energy of the solid sample. The liquid used for such experiments is referred to as the probe liquid, and the use of several different probe liquids is required.

In fluid statics, capillary pressure is the pressure between two immiscible fluids in a thin tube, resulting from the interactions of forces between the fluids and solid walls of the tube. Capillary pressure can serve as both an opposing or driving force for fluid transport and is a significant property for research and industrial purposes. It is also observed in natural phenomena.

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, which results in a corresponding change in surface energy. For a binary system, the Gibbs adsorption equation in terms of surface excess is:

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 dependent upon thermodynamic principles and does not allude to special properties of materials. It is also used for determination of pore size distribution of a porous medium using adsorption porosimetry. The equation is named in honor of William Thomson, also known as Lord Kelvin.

<span class="mw-page-title-main">Young–Laplace equation</span>

In physics, the Young–Laplace equation is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although use of the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It's a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface :

<span class="mw-page-title-main">Capillary length</span>

The capillary length or capillary constant, is a length scaling factor that relates gravity and surface tension. It is a fundamental physical property that governs the behavior of menisci, and is found when body forces (gravity) and surface forces are in equilibrium.

In fluid mechanics and mathematics, a capillary surface is a surface that represents the interface between two different fluids. As a consequence of being a surface, a capillary surface has no thickness in slight contrast with most real fluid interfaces.

<span class="mw-page-title-main">Vapor–liquid–solid method</span> Mechanism to grow nano wires

The vapor–liquid–solid method (VLS) is a mechanism for the growth of one-dimensional structures, such as nanowires, from chemical vapor deposition. The growth of a crystal through direct adsorption of a gas phase on to a solid surface is generally very slow. The VLS mechanism circumvents this by introducing a catalytic liquid alloy phase which can rapidly adsorb a vapor to supersaturation levels, and from which crystal growth can subsequently occur from nucleated seeds at the liquid–solid interface. The physical characteristics of nanowires grown in this manner depend, in a controllable way, upon the size and physical properties of the liquid alloy.

Disjoining pressure, in surface chemistry, according to an IUPAC definition, arises from an attractive interaction between two surfaces. For two flat and parallel surfaces, the value of the disjoining pressure can be calculated as the derivative of the Gibbs energy of interaction per unit area in respect to distance. There is also a related concept of disjoining force, which can be viewed as disjoining pressure times the surface area of the interacting surfaces.

Adsorption is the adhesion of ions or molecules onto the surface of another phase. Adsorption may occur via physisorption and chemisorption. Ions and molecules can adsorb to many types of surfaces including polymer surfaces. A polymer is a large molecule composed of repeating subunits bound together by covalent bonds. The adsorption of ions and molecules to polymer surfaces plays a role in many applications including: biomedical, structural, coatings, environmental and petroleum.

<span class="mw-page-title-main">Elasto-capillarity</span> Physical phenomenon

Elasto-capillarity is the ability of capillary force to deform an elastic material. From the viewpoint of mechanics, elastocapillarity phenomena essentially involve competition between the elastic strain energy in the bulk and the energy on the surfaces/interfaces. In the modeling of these phenomena, some challenging issues are, among others, the exact characterization of energies at the micro scale, the solution of strongly nonlinear problems of structures with large deformation and moving boundary conditions, and instability of either solid structures or droplets/films.The capillary forces are generally negligible in the analysis of macroscopic structures but often play a significant role in many phenomena at small scales.

<span class="mw-page-title-main">Ideal surface</span>

An ideal solid surface is flat, rigid, perfectly smooth, and chemically homogeneous, and has zero contact angle hysteresis. Zero hysteresis implies the advancing and receding contact angles are equal.


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