Editing Surface tension (section)

==Effects==

===Water===
Several effects of surface tension can be seen with ordinary water:
{{ordered list|list_style_type=upper-alpha
|Beading of rain water on a waxy surface, such as a leaf. Water [[Hydrophobic effect|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]].
|Formation of [[Drop (liquid)|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.<ref name="MIT5">{{cite web|url=http://web.mit.edu/1.63/www/Lec-notes/Surfacetension/Lecture5.pdf|title=MIT Lecture Notes on Surface Tension, lecture 5|access-date=April 1, 2007|date=May 2004|publisher=Massachusetts Institute of Technology|author=Bush, John W. M.|archive-date=February 26, 2007|archive-url=https://web.archive.org/web/20070226102246/http://web.mit.edu/1.63/www/Lec-notes/Surfacetension/Lecture5.pdf|url-status=live}}</ref>
|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.<ref name="white"/> For example, [[Gerridae|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<ref name="MIT3">{{cite web|url=http://web.mit.edu/1.63/www/Lec-notes/Surfacetension/Lecture3.pdf|title=MIT Lecture Notes on Surface Tension, lecture 3|access-date=April 1, 2007|publisher=Massachusetts Institute of Technology|author=Bush, John W. M.|date=May 2004|archive-date=February 26, 2007|archive-url=https://web.archive.org/web/20070226102227/http://web.mit.edu/1.63/www/Lec-notes/Surfacetension/Lecture3.pdf|url-status=live}}</ref> and tendency of minimization of surface curvature (so area) of the water pushes the insect's feet upward.
|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.
|[[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.
}}
<gallery mode="packed" heights="120">
File:Dew 2.jpg|'''A.''' Water beading on a leaf
File:Water drop animation enhanced small.gif|'''B.''' Water dripping from a tap
File:WaterstriderEnWiki.jpg|'''C.''' [[Water strider]]s stay at the top of liquid because of surface tension
File:1990s Mathmos Astro.jpg|'''D.''' [[Lava lamp]] with interaction between dissimilar liquids: water and liquid wax
File:Wine legs shadow.jpg|'''E.''' Photo showing the "[[tears of wine]]" phenomenon. 
</gallery>

===Surfactants===
Surface tension is visible in other common phenomena, especially when [[surfactant]]s are used to decrease it:

* [[Soap bubble]]s 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]]). Surfactants actually reduce the surface tension of water by a factor of three or more.
* [[Emulsion]]s are a type of colloidal dispersion in which surface tension plays a role. Tiny droplets of oil dispersed in pure water will spontaneously coalesce and phase separate. The addition of surfactants reduces the interfacial tension and allow for the formation of oil droplets in the water medium (or vice versa). The stability of such formed oil droplets depends on many different chemical and environmental factors.{{Clear right}}

===Surface curvature and pressure===
[[Image:CurvedSurfaceTension.png|thumb|right|Surface tension forces acting on a tiny (differential) patch of surface. {{mvar|δθ<sub>x</sub>}} and {{mvar|δθ<sub>y</sub>}} 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]]:<ref name="cwp"/>
<math display="block">\Delta p = \gamma \left( \frac{1}{R_x} + \frac{1}{R_y} \right)</math>
where:
*{{math|Δ''p''}} is the pressure difference, known as the [[Laplace pressure]].<ref name="Physics and Chemistry of Interfaces">{{cite book | title = Physics and Chemistry of Interfaces | author1 = Butt, Hans-Jürgen | author2 = Graf, Karlheinz | author3 = Kappl, Michael | year = 2006 | page = 9|isbn=978-3-527-60640-5|publisher=Wiley}}</ref>
*{{mvar|γ}} is surface tension.
*{{mvar|R<sub>x</sub>}} and {{mvar|R<sub>y</sub>}} are [[Radius of curvature (mathematics)|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.)

{| class="wikitable" style="float:center; clear:right;"
|+ {{math|Δ''p''}} for water drops of different radii at [[standard conditions for temperature and pressure|STP]]
|-
! style="width:120px;" | Droplet radius
| style="width:120px;" | 1&nbsp;mm
| style="width:120px;" | 0.1&nbsp;mm
| style="width:120px;" | 1&nbsp;[[micrometre|μm]]
| style="width:120px;" | 10&nbsp;[[nanometer|nm]]
|-
! {{math|Δ''p''}} ([[Atmosphere (unit)|atm]])
| 0.0014
| 0.0144
| 1.436
| 143.6
|}
{{Clear right}}

=== Floating objects ===
{{See also|Flotation of flexible objects}}[[Image:Surface Tension Diagram.svg|thumb|Cross-section of a needle floating on the surface of water. {{math|''F''<sub>w</sub>}} is the weight and {{math|''F''<sub>s</sub>}} are surface tension resultant forces.]]

When an object is placed on a liquid, its weight {{math|''F''<sub>w</sub>}} depresses the surface, and if surface tension and downward force become equal then it is balanced by the surface tension forces on either side {{math|''F''<sub>s</sub>}}, 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 {{math|''F''<sub>s</sub>}} arrows point in opposite directions, so they cancel each other, but the vertical components point in the same direction and therefore add up<ref name="white"/> to balance {{math|''F''<sub>w</sub>}}. 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 {{math|''m''}} denotes the mass of the needle and {{math|''g''}} acceleration due to gravity, we have

<math display="block"> F_\mathrm{w} = 2 F_\mathrm{s} \sin \theta \quad\Leftrightarrow\quad m g = 2 \gamma L \sin \theta </math>

===Liquid surface===
[[Image:Površinska napetost milnica.jpg|thumb|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.<ref name="s_z"/><ref>Aaronson, Scott (March 2005) [https://arxiv.org/abs/quant-ph/0502072 NP-complete Problems and Physical Reality] {{Webarchive|url=https://web.archive.org/web/20180223065653/https://arxiv.org/abs/quant-ph/0502072 |date=2018-02-23 }}. ''ACM SIGACT News''</ref>

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|Contact angle}}
The surface of any liquid is an interface between that liquid and some other medium.<ref group=note>In a [[mercury barometer]], the upper liquid surface is an interface between the liquid and a vacuum containing some molecules of evaporated liquid.</ref> 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.<ref name="s_z"/><ref name="cwp"/>

{| style="float:right;"
|-
|[[Image:SurfTensionContactAngle.png|thumb|255px|right|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]], {{mvar|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, {{math|''γ''<sub>ls</sub> − ''γ''<sub>sa</sub>}}, is less than the liquid–air surface tension, {{math|''γ''<sub>la</sub>}}, but is nevertheless positive, that is

<math display="block">\gamma_\mathrm{la} > \gamma_\mathrm{ls} - \gamma_\mathrm{sa} > 0</math>

In the diagram, both the vertical and horizontal forces must cancel exactly at the contact point, known as [[Mechanical equilibrium|equilibrium]]. The horizontal component of {{math|''f''<sub>la</sub>}} is canceled by the adhesive force, {{math|''f''<sub>A</sub>}}.<ref name="s_z"/>

<math display="block">f_\mathrm{A} = f_\mathrm{la} \sin \theta</math>

The more telling balance of forces, though, is in the vertical direction. The vertical component of {{math|''f''<sub>la</sub>}} must exactly cancel the difference of the forces along the solid surface, {{math|''f''<sub>ls</sub> − ''f''<sub>sa</sub>}}.<ref name="s_z"/>

<math display="block">f_\mathrm{ls} - f_\mathrm{sa} = -f_\mathrm{la} \cos \theta</math>

{| class="toccolours" border="1" style="float: right; clear: right; margin: 0 0 1em 1em; border-collapse: collapse;"
|+'''Some liquid–solid contact angles'''<ref name="s_z"/>
|- style="text-align:center; background:#c0c0f0;"
! Liquid
! Solid
! Contact<br>angle
|-
| [[water]]
| rowspan="6" |
{| cellpadding="0" cellspacing="0" border="0"
|- style="background:#f8f8f8;"
|soda-lime glass
|- style="background:#f8f8f8;"
| lead glass
|- style="background:#f8f8f8;"
| [[fused quartz]]
|}
| rowspan="6" style="text-align:center;"|0°
|-
| [[ethanol]]
|-
| [[diethyl ether]]
|-
| [[carbon tetrachloride]]
|-
| [[glycerol]]
|-
| [[acetic acid]]
|-
| rowspan="2"|[[water]]
| paraffin wax
| style="text-align:center;"|107°
|-
| silver
| style="text-align:center;"|90°
|-
| rowspan="3"| [[iodomethane|methyl iodide]]
| soda-lime glass
| style="text-align:center;"|29°
|-
| lead glass
| style="text-align:center;"|30°
|-
| fused quartz
| style="text-align:center;"|33°
|-
| [[mercury (element)|mercury]]
| soda-lime glass
| style="text-align:center;"|140°
|}

Since the forces are in direct proportion to their respective surface tensions, we also have:<ref name="cwp"/>

<math display="block">\gamma_\mathrm{ls} - \gamma_\mathrm{sa} = -\gamma_\mathrm{la} \cos \theta</math>

where
* {{math|''γ''<sub>ls</sub>}} is the liquid–solid surface tension,
* {{math|''γ''<sub>la</sub>}} is the liquid–air surface tension,
* {{math|''γ''<sub>sa</sub>}} is the solid–air surface tension,
* {{mvar|θ}} is the contact angle, where a concave [[Meniscus (liquid)|meniscus]] has contact angle less than 90° and a convex meniscus has contact angle of greater than 90°.<ref name="s_z">Sears, Francis Weston; Zemanski, Mark W. (1955) ''University Physics 2nd ed''. Addison Wesley</ref>

This means that although the difference between the liquid–solid and solid–air surface tension, {{math|''γ''<sub>ls</sub> − ''γ''<sub>sa</sub>}}, is difficult to measure directly, it can be inferred from the liquid–air surface tension, {{math|''γ''<sub>la</sub>}}, and the equilibrium contact angle, {{mvar|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:

<math display="block">\gamma_\mathrm{la} > 0 > \gamma_\mathrm{ls} - \gamma_\mathrm{sa}</math>

====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 [[polytetrafluoroethylene|Teflon]] approaches this.<ref name="cwp"/> Contact angle of 180° occurs when the liquid–solid surface tension is exactly equal to the liquid–air surface tension.

<math display="block">\gamma_\mathrm{la} = \gamma_\mathrm{ls} - \gamma_\mathrm{sa} > 0\qquad \theta = 180^\circ</math>
{{Clear right}}

===Liquid in a vertical tube===
{{Main|Capillary action}}
[[Image:HgBarometer.gif|thumb|upright=0.6|Diagram of a [[mercury (element)|mercury]] [[barometer]]]]

An old style [[mercury (element)|mercury]] [[barometer]] consists of a vertical glass tube about 1&nbsp;cm in diameter partially filled with mercury, and with a vacuum (called [[Evangelista Torricelli|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.

[[Image:CapillaryAction.svg|thumb|upright=0.6|Illustration of capillary rise and fall. Red=contact angle less than 90°; blue=contact angle greater than 90°]]
<!-- [[Image:Dscn3156-daisy-water 1200x900.jpg|thumb|Surface tension prevents this flower from sinking]] -->

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]]:<ref name="s_z"/>

<math display="block">h = \frac {2\gamma_\mathrm{la} \cos\theta}{\rho g r}</math>

where

* {{mvar|h}} is the height the liquid is lifted,
* {{math|''γ''<sub>la</sub>}} is the liquid–air surface tension,
* {{mvar|ρ}} is the density of the liquid,
* {{mvar|r}} is the radius of the capillary,
* {{mvar|g}} is the acceleration due to gravity,
* {{mvar|θ}} is the angle of contact described above. If {{mvar|θ}} is greater than 90°, as with mercury in a glass container, the liquid will be depressed rather than lifted.
{{Clear}}

===Puddles on a surface===
[[Image:SurfTensionEdgeOfPool.png|thumb|Profile curve of the edge of a puddle where the contact angle is 180°. The curve is given by the formula:<ref name="cwp"/> <math display="block">x - x_0 = \frac 1 2 H \cosh^{-1}\left(\frac {H}{h}\right) - H \sqrt{1 - \frac{h^2} {H^2}}</math> where <math display="inline">H = 2 \sqrt{ {\gamma} / {g \rho}}</math>]]
[[Image:Exploring new continents 1200728.JPG|thumb|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#Physics|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:<ref name="cwp">{{Cite book|title=Capillarity and Wetting Phenomena—Drops, Bubbles, Pearls, Waves|author1=Pierre-Gilles de Gennes|author-link1=Pierre-Gilles de Gennes|author2=Françoise Brochard-Wyart|author3=David Quéré|publisher=Springer|year=2002|isbn=978-0-387-00592-8|others=Alex Reisinger}}</ref>

<math display="block">h = 2 \sqrt{\frac{\gamma} {g\rho}}</math>

where

* {{mvar|h}} is the depth of the puddle in centimeters or meters.
* {{mvar|γ}} is the surface tension of the liquid in dynes per centimeter or newtons per meter.
* {{mvar|g}} is the acceleration due to gravity and is equal to 980&nbsp;cm/s<sup>2</sup> or 9.8&nbsp;m/s<sup>2</sup>
* {{mvar|ρ}} is the density of the liquid in grams per cubic centimeter or kilograms per cubic meter

[[Image:Surface tension.svg|thumb|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:<ref name="cwp"/>

<math display="block">h = \sqrt{\frac{2\gamma_\mathrm{la}\left( 1 - \cos \theta \right)} {g\rho}}.</math>

For mercury on glass, {{math|''γ''<sub>Hg</sub>}} = 487 dyn/cm, {{math|''ρ''<sub>Hg</sub>}} = 13.5 g/cm<sup>3</sup> and {{mvar|θ}} = 140°, which gives {{math|''h''<sub>Hg</sub>}} = 0.36&nbsp;cm. For water on paraffin at 25&nbsp;°C, {{mvar|γ}} = 72 dyn/cm, {{mvar|ρ}} = 1.0 g/cm<sup>3</sup>, and {{mvar|θ}} = 107° which gives {{math|''h''<sub>H<sub>2</sub>O</sub>}} = 0.44&nbsp;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.{{Clear right}}

===Breakup of streams into drops===
[[File:Dripping faucet 2.jpg|thumb|upright|Breakup of an elongated stream of water into droplets due to surface tension.]]

{{Main|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]],<ref name="cwp"/> 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 [[Sine wave|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.

===Gallery===
<gallery widths="160px" heights="120px" class="center">
Image:UnstableLiquidSheet.jpg|Breakup of a moving sheet of water bouncing off of a spoon.
Image:SurfaceTension.jpg|Photo of flowing water adhering to a hand. Surface tension creates the sheet of water between the flow and the hand.
Image:Ggb in soap bubble 1.jpg|A soap bubble balances surface tension forces against internal [[pneumatic]] [[pressure]].
Image:2006-01-15 coin on water.jpg|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.
Image:3_Moeda_(5).jpg|An aluminium coin floats on the surface of the water at 10&nbsp;°C. Any extra weight would drop the coin to the bottom.
Image:Dscn3156-daisy-water 1200x900.jpg|A daisy. The entirety of the flower lies below the level of the (undisturbed) free surface. The water rises smoothly around its edge. Surface tension prevents water from displacing the air between the petals and possibly submerging the flower.
Image:Surface Tension 01.jpg|A metal paper clip floats on water. Several can usually be carefully added without overflow of water.
Image:Paperclip floating on water (with 'contour lines').jpg|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.
</gallery>