Editing Surface tension (section)

==Thermodynamics==
===Thermodynamic theories of surface tension===

[[Josiah Willard Gibbs|J.W. Gibbs]] developed the thermodynamic theory of capillarity based
on the idea of surfaces of discontinuity.<ref name=gibbseq>{{Citation
| last    = Gibbs
| first   = J.W.
| year     = 2002
| orig-year =  1876–1878
| title    = The Scientific Papers of J. Willard Gibbs
| chapter  = [[On the Equilibrium of Heterogeneous Substances]]
| volume  = 1
| publisher = Ox Bow Press
| place    = Woodbridge, CT
| pages = 55–354
| editor-last   = Bumstead
| editor-first  = H.A.
| editor2-last  = Van Nameeds
| editor2-first = R.G.
| isbn =  978-0918024770
}}</ref> 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 {{mvar|T}} and chemical potentials {{math|''μ''<sub>i</sub>}}), 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 <math>U - TS - \mu_1 N_1 - \mu_2 N_2 \cdots </math>, a quantity later named as the [[grand potential]] and given the symbol <math>\Omega</math>.

[[File:Gibbs Model.tif|thumb|Gibbs' placement of a precise mathematical surface in a fuzzy physical interface.]]

Considering a given subvolume <math>V</math> containing a surface of discontinuity, the volume is divided by the mathematical surface into two parts A and B, with volumes <math>V_\text{A}</math> and <math>V_\text{B}</math>, with <math>V = V_\text{A} + V_\text{B}</math> exactly. Now, if the two parts A and B were homogeneous fluids (with pressures <math>p_\text{A}</math>, <math>p_\text{B}</math>) and remained perfectly homogeneous right up to the mathematical boundary, without any surface effects, the total grand potential of this volume would be simply <math>-p_\text{A} V_\text{A} - p_\text{B} V_\text{B}</math>. The surface effects of interest are a modification to this, and they can be all collected into a surface free energy term <math>\Omega_\text{S}</math> so the total grand potential of the volume becomes:
<math display="block">\Omega = -p_\text{A} V_\text{A} - p_\text{B} V_\text{B} + \Omega_\text{S}.</math>

For sufficiently macroscopic and gently curved surfaces, the surface free energy must simply be proportional to the surface area:<ref name=gibbseq/><ref name=landaulifshitz>{{cite book |last1=Landau |last2=Lifshitz |title=Course of Theoretical Physics Volume 5: Statistical Physics I |date=1980 |publisher=Pergamon |pages=517–537 |edition=3}}</ref>
<math display="block">\Omega_\text{S} = \gamma A,</math>
for surface tension <math>\gamma</math> and surface area <math>A</math>.

As stated above, this implies the mechanical work needed to increase a surface area ''A'' is {{math|''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 <math>\Omega</math>, 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 process|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.<ref name="Rusanov2005">{{cite journal|last1=Rusanov|first1=A|title=Surface thermodynamics revisited|journal=Surface Science Reports|volume=58|issue=5–8|year=2005|pages=111–239|issn=0167-5729|doi=10.1016/j.surfrep.2005.08.002|bibcode=2005SurSR..58..111R}}</ref> 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 {{mvar|γ}}; however, it does have a very strong effect on the values of the surface entropy, surface excess mass densities, and surface internal energy,{{r|gibbseq|p=237}} which are the partial derivatives of the surface tension function <math>\gamma(T, \mu_1, \mu_2, \cdots)</math>.

Gibbs emphasized that for solids, the surface free energy may be completely different from surface stress (what he called surface tension):{{r|gibbseq|p=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 [[Cauchy stress tensor|stress tensor]]) while surface energy is scalar.

Fifteen years after Gibbs, [[Johannes Diderik van der Waals|J.D. van der Waals]] developed the theory of capillarity effects based on the hypothesis of a continuous variation of density.<ref>{{cite journal |last1=Rowlinson |first1=J. S. |title=Translation of J. D. van der Waals' ?The thermodynamik theory of capillarity under the hypothesis of a continuous variation of density? |journal=Journal of Statistical Physics |date=February 1979 |volume=20 |issue=2 |pages=197–200 |doi=10.1007/BF01011513 }}</ref> He added to the energy density the term <math>c (\nabla \rho)^2,</math> 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.<ref>{{Citation
| last1    = Cahn
| first1   = J.W.
| last2   = Hilliard
| first2  = J.E.
| year     = 1958
| title    = Free energy of a nonuniform system. I. Interfacial free energy
| journal = J. Chem. Phys.
| pages = 258–266
| doi = 10.1063/1.1744102 
| bibcode= 1958JChPh..28..258C
| volume=28
| issue   = 2
}}</ref><ref>{{Citation
| last1    = Langer
| first1   = J.S.
| last2   = Bar-On
| first2  = M.
| last3   = Miller
| first3  = H.D.
| year     = 1975
| title    = New computational method in the theory of spinodal decomposition
| journal = Phys. Rev. A 
| pages = 1417–1429
| doi = 10.1103/PhysRevA.11.1417
| bibcode= 1975PhRvA..11.1417L
| volume=11| issue   = 4
}}</ref> 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.<ref>
{{Citation
| last1    = Gorban
| first1   = A.N.
| last2   = Karlin
| first2  = I. V.
| year     = 2016
| title    = Beyond Navier–Stokes equations: capillarity of ideal gas
| type    = Review article
| journal = Contemporary Physics
| doi = 10.1080/00107514.2016.1256123 
| arxiv= 1702.00831
| bibcode= 2017ConPh..58...70G
| volume=58
| issue = 1
| pages=70–90
}}</ref>

===Thermodynamics of bubbles===
The pressure inside an ideal spherical bubble can be derived from thermodynamic free energy considerations.<ref name=landaulifshitz/> The above free energy can be written as:
<math display="block">\Omega = -\Delta P V_\text{A} - p_\text{B} V + \gamma A</math>
where <math>\Delta P = p_\text{A} - p_\text{B}</math> is the pressure difference between the inside (A) and outside (B) of the bubble, and <math>V_\text{A}</math> is the bubble volume. In equilibrium, {{math|1=''d''Ω = 0}}, and so,
<math display="block">\Delta P\,dV_\text{A} = \gamma\, dA.</math>

For a spherical bubble, the volume and surface area are given simply by
<math display="block">V_\text{A} = \tfrac43\pi R^3 \quad\rightarrow\quad dV_\text{A} = 4\pi R^2 \,dR,</math>
and
<math display="block">A = 4\pi R^2 \quad\rightarrow\quad dA = 8\pi R\, dR.</math>

Substituting these relations into the previous expression, we find
<math display="block">\Delta P = \frac{2}{R}\gamma,</math>
which is equivalent to the [[Young–Laplace equation]] when {{math|''R<sub>x</sub>'' {{=}} ''R<sub>y</sub>''}}.

====Influence of temperature====
[[Image:Temperature dependence surface tension of water.svg|thumb|upright=1.3|Temperature dependence of the surface tension between the liquid and vapor phases of pure water]]
[[Image:SFT-benzene.png|thumb|upright=1.3|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:<ref name="phywe">{{cite web|url=http://www.nikhef.nl/~h73/kn1c/praktikum/phywe/LEP/Experim/1_4_05.pdf|title=Surface Tension by the Ring Method (Du Nouy Method)|access-date=2007-09-08|publisher=PHYWE|archive-date=2007-09-27|archive-url=https://web.archive.org/web/20070927010256/http://www.nikhef.nl/~h73/kn1c/praktikum/phywe/LEP/Experim/1_4_05.pdf|url-status=live}}</ref><ref name="adam">{{Cite book|title=The Physics and Chemistry of Surfaces, 3rd ed|author=Adam, Neil Kensington|publisher=Oxford University Press|year=1941}}</ref><ref name="Physical Properties Sources Index (PPSI)">{{cite web|url=http://www.ppsi.ethz.ch/fmi/xsl/eqi/eqi_property_details_en.xsl?node_id=1113|title=Physical Properties Sources Index: Eötvös Constant|access-date=2008-11-16|url-status=dead|archive-url=https://web.archive.org/web/20110706231759/http://www.ppsi.ethz.ch/fmi/xsl/eqi/eqi_property_details_en.xsl?node_id=1113|archive-date=2011-07-06}}</ref> <math display="block">\gamma V^{2/3} = k(T_\mathrm{C}-T) .</math> Here {{mvar|V}} is the molar volume of a substance, {{math|''T''<sub>C</sub>}} is the [[critical temperature]] and {{mvar|k}} is a constant valid for almost all substances.<ref name="phywe"/> A typical value is {{mvar|k}} = {{val|2.1|e=-7|u=J K<sup>−1</sup> mol<sup>−{{2/3}}</sup>}}.<ref name="phywe"/><ref name="Physical Properties Sources Index (PPSI)"/> For water one can further use {{mvar|V}} = 18&nbsp;ml/mol and {{math|''T''<sub>C</sub>}} = 647&nbsp;K (374&nbsp;°C).<ref>{{cite journal|doi=10.1063/1.555688|url=https://www.nist.gov/data/PDFfiles/jpcrd231.pdf|title=International Tables of the Surface Tension of Water|journal=Journal of Physical and Chemical Reference Data|volume=12|issue=3|pages=817|year=1983|last1=Vargaftik|first1=N. B.|last2=Volkov|first2=B. N.|last3=Voljak|first3=L. D.|bibcode=1983JPCRD..12..817V|access-date=2017-07-13|archive-date=2016-12-21|archive-url=https://web.archive.org/web/20161221094427/http://nist.gov/data/PDFfiles/jpcrd231.pdf|url-status=dead}}</ref> A variant on Eötvös is described by Ramay and Shields:<ref name="moore">{{Cite book|title=Physical Chemistry, 3rd ed|author=Moore, Walter J.|publisher=Prentice Hall|year=1962}}</ref> <math display="block">\gamma V^{2/3} = k \left(T_\mathrm{C} - T - 6\,\mathrm{K}\right)</math> where the temperature offset of 6 K provides the formula with a better fit to reality at lower temperatures.
* Guggenheim–Katayama:<ref name="adam"/> <math display="block">\gamma = \gamma^\circ \left( 1-\frac{T}{T_\mathrm C} \right)^n </math> {{math|''γ''°}} is a constant for each liquid and {{mvar|n}} is an empirical factor, whose value is {{sfrac|11|9}} for organic liquids. This equation was also proposed by [[Johannes Diderik van der Waals|van der Waals]], who further proposed that {{math|''γ''°}} could be given by the expression <math display="block">K_2 T^{1/3}_\mathrm{C} P^{2/3}_\mathrm{C},</math> where {{math|''K''<sub>2</sub>}} is a universal constant for all liquids, and {{math|''P''<sub>C</sub>}} is the [[critical pressure]] of the liquid (although later experiments found {{math|''K''<sub>2</sub>}} to vary to some degree from one liquid to another).<ref name="adam"/>

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 compounds|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:
<math display="block">\Gamma = - \frac{1}{RT} \left( \frac{\partial \gamma}{\partial \ln C} \right)_{T,P} </math>

* {{mvar|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/m<sup>2</sup>
* {{mvar|C}} is the concentration of the substance in the bulk solution.
* {{mvar|R}} is the [[gas constant]] and {{mvar|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====
{{See also|Gibbs–Thomson effect}}
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.<ref name="moore"/>

<math display="block">P_\mathrm{v}^\mathrm{fog}=P_\mathrm{v}^\circ e^{V 2\gamma/(RT r_\mathrm{k})}</math>

[[Image:TinyDropletMolecules.png|thumb|[[Molecule]]s 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.]]

*{{math|''P''<sub>v</sub>°}} is the standard vapor pressure for that liquid at that temperature and pressure.
*{{mvar|V}} is the molar volume.
*{{mvar|R}} is the [[gas constant]]
*{{math|''r''<sub>k</sub>}} 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.<ref name="moore"/>

This equation is also used in [[catalyst]] chemistry to assess [[Mesoporous material|mesoporosity]] for solids.<ref name="Handbook">Ertl, G.; Knözinger, H. and Weitkamp, J. (1997). ''Handbook of heterogeneous catalysis'', Vol. 2, p. 430. Wiley-VCH, Weinheim. {{ISBN|3-527-31241-2}}</ref>

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:

{| class="toccolours" border="1" style="float: center; margin: 0 0 1em 1em; border-collapse: collapse;"
|-
! style="text-align:center; background:#c0c0f0;" colspan="5"|{{math|{{sfrac|''P''|''P''<sub>0</sub>}}}} for water drops of different radii at [[Standard temperature and pressure|STP]]<ref name="adam"/>
|- style="text-align:center;" 
| style="width:120px; "|Droplet radius (nm)
| style="width:120px; "|1000
| style="width:120px; "|100
| style="width:120px; "|10
| style="width:120px; "|1
|- style="text-align:center;" 
|| {{math|{{sfrac|''P''|''P''<sub>0</sub>}}}}|| style="text-align:center;"| 1.001|| style="text-align:center;"| 1.011|| style="text-align:center;"|1.114|| style="text-align:center;"| 2.95
|}

The effect becomes clear for very small drop sizes, as a drop of 1&nbsp;nm radius has about 100 molecules inside, which is a quantity small enough to require a [[quantum mechanics]] analysis.