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 becomes under tension from the imbalanced forces, which is
probably where the term "surface tension" came from. Because of the
relatively high attraction of water molecules for each other through a
web of hydrogen bonds, water has a higher surface tension (72.8
millinewtons per meter at 20 °C) compared to that of most other
1 Causes 2 Effects of surface tension
2.1 Water 2.2 Surfactants
3.1 Physical units
4 Methods of measurement 5 Effects
5.1 Liquid in a vertical tube 5.2 Puddles on a surface 5.3 The breakup of streams into drops
6.1 Thermodynamic theories of surface tension
6.2.1 Influence of temperature 6.2.2 Influence of solute concentration 6.2.3 Influence of particle size on vapor pressure
8 Data table 9 Gallery of effects 10 See also 11 Notes 12 References 13 External links
Diagram of the forces on molecules of a liquid
The cohesive forces a molecule is pulled equally in every direction by
neighboring 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 minimal 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. When cohesive forces are
stronger than adhesives forces, the liquid acquires a convex
meniscus(as mercury in a glass container). On the other hand, when
adhesive forces are stronger, the surface of the liquid curves up (as
water in a glass)
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 (not in 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. 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. Consequently, the surface will push back against any curvature in much the same way as a ball pushed uphill will push back to minimize its gravitational potential energy. Effects of surface tension Water Several effects of surface tension can be seen with ordinary water:
Beading of rain water on a waxy surface, such as a leaf.
E. Photo showing the "tears of wine" phenomenon.
Soap bubbles have very large surface areas with very little mass. Bubbles in pure water are unstable. The addition of surfactants, however, can have a stabilizing effect on the bubbles (see Marangoni effect). Note that surfactants actually reduce the surface tension of water by a factor of three or more. Emulsions are a type of colloid in which surface tension plays a role. Tiny fragments of oil suspended in pure water will spontaneously assemble themselves into much larger masses. But the presence of a surfactant provides a decrease in surface tension, which permits stability of minute droplets of oil in the bulk of water (or vice versa).
Physics Physical units Surface tension, usually represented by the symbol σ, 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.
γ = 1
d y n
e r g
displaystyle gamma =1~mathrm frac dyn cm =1~mathrm frac erg cm^ 2 =1~mathrm frac mN m =0.001~mathrm frac N m =0.001~mathrm frac J m^ 2
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This diagram illustrates the force necessary to increase the surface area. This force is proportional to the surface tension.
displaystyle gamma = frac 1 2 frac F L .
The reason for the 1/2 is that the film has two sides, 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, and the change in the surface area of the liquid (that led to the change in energy). This can be easily related to the previous definition in terms of force: if 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
F Δ x
2 L Δ x
displaystyle gamma = frac F 2L = frac FDelta x 2LDelta x = frac W Delta A
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.
Surface curvature and pressure
If no force acts normal to a tensioned surface, the surface must remain flat. But if the pressure on one side of the surface differs from pressure on the other side, the pressure difference times surface area results in a normal force. In order for the surface tension forces to cancel the force due to pressure, the surface must be curved. The diagram shows how surface curvature of a tiny patch of surface leads to a net component of surface tension forces acting normal to the center of the patch. When all the forces are balanced, the resulting equation is known as the Young–Laplace equation:
Δ p = γ
displaystyle Delta p=gamma left( frac 1 R_ x + frac 1 R_ y right)
Δp is the pressure difference, known as the Laplace pressure. γ is surface tension. Rx and Ry are radii of curvature in each of the axes that are parallel to the surface.
The quantity in parentheses on the right hand side is in fact (twice) the mean 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 radius 1 mm 0.1 mm 1 μm 10 nm
Δp (atm) 0.0014 0.0144 1.436 143.6
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 becomes equal than 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 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.
L g = 2 γ L cos θ
displaystyle F_ mathrm w =2F_ mathrm s cos theta quad Leftrightarrow quad rho _ mathrm w A_ mathrm w Lg=2gamma Lcos theta
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. 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 Main article: Contact angle 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.
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, γls − γsa, is less than the liquid–air surface tension, γla, but is nevertheless positive, that is[dubious – discuss]
displaystyle gamma _ mathrm la >gamma _ mathrm ls -gamma _ mathrm sa >0
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.
displaystyle f_ mathrm A =f_ mathrm la sin theta
The more telling balance of forces, though, is in the vertical direction. The vertical component of fla must exactly cancel the force, fls.
displaystyle f_ mathrm ls -f_ mathrm sa =-f_ mathrm la cos theta
Liquid Solid Contact angle
water paraffin wax 107°
methyl iodide soda-lime glass 29°
lead glass 30°
fused quartz 33°
mercury soda-lime glass 140°
Some liquid–solid contact angles
Since the forces are in direct proportion to their respective surface tensions, we also have:
displaystyle gamma _ mathrm ls -gamma _ mathrm sa =-gamma _ mathrm la cos theta
γls is the liquid–solid surface tension, γla is the liquid–air surface tension, γsa is the solid–air surface tension, θ is the contact angle, where a concave meniscus has contact angle less than 90° and a convex meniscus has contact angle of greater than 90°.
This means that although the difference between the liquid–solid and solid–air surface tension, γls − γsa, 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:
> 0 >
displaystyle gamma _ mathrm la >0>gamma _ mathrm ls -gamma _ mathrm sa
displaystyle gamma _ mathrm la =gamma _ mathrm ls -gamma _ mathrm sa >0qquad theta =180^ circ
Methods of measurement
Because surface tension manifests itself in various effects, it offers a number of paths to its measurement. Which method is optimal depends upon the nature of the liquid being measured, the conditions under which its tension is to be measured, and the stability of its surface when it is deformed.
Du Noüy ring method: The traditional method used to measure surface
or interfacial tension.
Effects Liquid in a vertical tube Main article: Capillary action
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 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:
ρ g r
displaystyle h= frac 2gamma _ mathrm la cos theta rho gr
h is the height the liquid is lifted, γla is the liquid–air surface tension, ρ is the density of the liquid, r is the radius of the capillary, g is the acceleration due to gravity, θ is the angle of contact described above. If θ is greater than 90°, as with mercury in a glass container, the liquid will be depressed rather than lifted.
Puddles on a surface
Profile curve of the edge of a puddle where the contact angle is 180°. The curve is given by the formula:
displaystyle x-x_ 0 = frac 1 2 Hcosh ^ -1 left( frac H h right)-H sqrt 1- frac h^ 2 H^ 2
H = 2
displaystyle H=2 sqrt frac gamma grho
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.
h = 2
displaystyle h=2 sqrt frac gamma grho
h is the depth of the puddle in centimeters or meters. γ is the surface tension of the liquid in dynes per centimeter or newtons per meter. g is the acceleration due to gravity and is equal to 980 cm/s2 or 9.8 m/s2 ρ is the density of the liquid in grams per cubic centimeter or kilograms per cubic meter
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:
1 − cos θ
displaystyle h= sqrt frac 2gamma _ mathrm la left(1-cos theta right) grho .
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
Intermediate stage of a jet breaking into drops. Radii of curvature in the axial direction are shown. Equation for the radius of the stream is R(z) = R0 + Ak cos (kz), where R0 is the radius of the unperturbed stream, Ak is the amplitude of the perturbation, z is distance along the axis of the stream, and k is the wave number
Main article: Plateau–Rayleigh instability
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, 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. He introduced and studied
thermodynamics of two-dimensional 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
T , P , n
displaystyle gamma =left( frac partial G partial A right)_ T,P,n
where G is
Gibbs free energy
A , P
displaystyle left( frac partial gamma partial T right)_ A,P =-S^ A
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.
= γ − T
displaystyle H^ A =gamma -Tleft( frac partial gamma partial T right)_ P
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. He added to the energy density the term
c ( ∇ ρ
displaystyle c(nabla rho )^ 2 ,
where c is the capillarity coefficient and ρ is the density. For the
multiphase equilibria, the results of the van der Waals approach
practically coincide with the Gibbs formulae, but for modelling of the
dynamics of phase transitions the van der Waals approach is much more
convenient. The van der Waals capillarity energy is now widely
used in the phase field models of multiphase flows. Such terms are
also discovered in the dynamics of non-equilibrium gases.
d F = − P
d V + γ
displaystyle dF=-P,dV+gamma ,dA
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,
d V = γ
displaystyle P,dV=gamma ,dA
For a spherical bubble, the volume and surface area are given simply by
d V ≈ 4 π
d R ,
displaystyle V= tfrac 4 3 pi R^ 3 quad rightarrow quad dVapprox 4pi R^ 2 ,dR,
A = 4 π
d A ≈ 8 π R
d R .
displaystyle A=4pi R^ 2 quad rightarrow quad dAapprox 8pi R,dR.
Substituting these relations into the previous expression, we find
displaystyle P= frac 2 R gamma ,
which is equivalent to the
= k (
− T ) .
displaystyle gamma V^ frac 2 3 =k(T_ mathrm C -T).
Here V is the molar volume of a substance, TC is the critical temperature and k is a constant valid for almost all substances. A typical value is k = 6993210000000000000♠2.1×10−7 J K−1 mol−2⁄3. For water one can further use V = 18 ml/mol and TC = 647 K (374 °C). A variant on Eötvös is described by Ramay and Shields:
− T − 6
displaystyle gamma V^ frac 2 3 =kleft(T_ mathrm C -T-6right)
where the temperature offset of 6 kelvins provides the formula with a better fit to reality at lower temperatures.
displaystyle gamma =gamma ^ circ left(1- frac T T_ mathrm C right)^ n
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
displaystyle K_ 2 T_ mathrm C ^ frac 1 3 P_ mathrm C ^ frac 2 3 ,
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). 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 waterair, most organic compounds at oilair Increase surface tension, most inorganic salts at waterair Non-monotonic change, most inorganic acids at waterair Decrease surface tension progressively, as with most amphiphiles, e.g., alcohols at waterair 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.
Γ = −
∂ ln C
T , P
displaystyle Gamma =- frac 1 RT left( frac partial gamma partial ln C right)_ T,P
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
Influence of particle size on vapor pressure
See also: Gibbs–Thomson effect
f o g
V 2 γ
displaystyle P_ mathrm v ^ mathrm fog =P_ mathrm v ^ circ e^ frac V2gamma RTr_ mathrm k
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
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. This equation is also used in catalyst chemistry to assess mesoporosity for solids. 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
Droplet radius (nm) 1000 100 10 1
P/P0 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.
1 − 0.625
displaystyle gamma _ text w =235.8left(1- frac T T_ text C right)^ 1.256 left[1-0.625left(1- frac T T_ text C right)right]~ text mN/m ,
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
1 + 3.766 ×
S + 2.347 ×
displaystyle gamma _ mathrm sw =gamma _ mathrm w left(1+3.766times 10^ -4 S+2.347times 10^ -6 Stright)
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 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 range of temperature and salinity encompasses both the oceanographic range and the range of conditions encountered in thermal desalination technologies. Data table
Acetic acid 20 27.60
Acetone 20 23.70
Diethyl ether 20 17.00
Ethanol 20 22.27
Glycerol 20 63.00
n-Hexane 20 18.40
Isopropanol 20 21.70
Liquid helium II −273 0.37
Liquid nitrogen −196 8.85
Mercury 15 487.00
Methanol 20 22.60
n-Octane 20 21.80
Water 0 75.64
Water 25 71.97
Water 50 67.91
Water 100 58.85
Toluene 25 27.73
Gallery of effects
Breakup of a moving sheet of water bouncing off of a spoon.
Photo of flowing water adhering to a hand.
A soap bubble balances surface tension forces against internal pneumatic pressure.
A daisy. The entirety of the flower lies below the level of the
(undisturbed) free surface. The water rises smoothly around its edge.
A metal paper clip floats on water. Several can usually be carefully added without overflow of water.
An aluminium coin floats on the surface of the water at 10 °C. Any extra weight would drop the coin to the bottom.
A metal paperclip floating on water. A grille in front of the light has created the 'contour lines' which show the deformation in the water surface caused by the metal paper clip.
Underwater diving portal
Bond number or Eötvös number Capillary number Marangoni number Weber number
Dortmund Data Bank
^ In a mercury barometer, the upper liquid surface is an interface between the liquid and a vacuum containing some molecules of evaporated liquid.
^ "Surface Tension (
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On surface tension and interesting real-world cases
Surface Tensions of Various Liquids
Calculation of temperature-dependent surface tensions for some common
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