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{{Short description|Ability of a liquid to maintain contact with a solid surface}}
{{Redirect|Wetness|the biological secretion|subpreputial wetness}}
{{other uses}}
{{other uses of|Wet}}
[[File:A water droplet DWR-coated surface2 edit1.jpg|thumb|alt=Close-up of a drop of water (almost spherical) on blue fabric, with a shadow under it|Water bead on a fabric that has been made non-wetting by chemical treatment.]]

'''Wetting''' is the ability of a [[liquid]] to displace gas to maintain contact with a [[solid]] [[surface science|surface]], resulting from [[intermolecular]] interactions when the two are brought together.<ref>{{Cite journal |last1=Carroll |first1=Gregory T. |last2=Turro |first2=Nicholas J. |last3=Mammana |first3=Angela |last4=Koberstein |first4=Jeffrey T. |date=2017 |title=Photochemical Immobilization of Polymers on a Surface: Controlling Film Thickness and Wettability |url=https://onlinelibrary.wiley.com/doi/10.1111/php.12751 |journal=Photochemistry and Photobiology |language=en |volume=93 |issue=5 |pages=1165–1169 |doi=10.1111/php.12751 |pmid=28295380 |issn=0031-8655}}</ref> These interactions occur in the presence of either a gaseous phase or another liquid phase not [[miscible]] with the wetting liquid. The degree of wetting (wettability) is determined by a force balance between [[adhesion|adhesive]] and [[cohesion (chemistry)|cohesive]] [[force]]s. There are two types of wetting: non-reactive wetting and reactive wetting.<ref>{{cite journal|doi=10.1007/s10853-009-4128-x|title=Fundamental issues of reactive wetting by liquid metals|journal=Journal of Materials Science|volume=45|issue=16|pages=4256–4264|year=2010|last1=Dezellus|first1=O.|last2=Eustathopoulos|first2=N.|bibcode=2010JMatS..45.4256D|s2cid=4512480|url=https://hal.archives-ouvertes.fr/hal-00516332/file/Euromat2009_Dezellus_HAL.pdf}}</ref><ref>{{cite journal|doi= 10.1039/C3CP51848E|pmid= 23949186|title= The effect of oxygen vacancies on water wettability of a ZnO surface|journal= Physical Chemistry Chemical Physics|volume= 15|issue= 39|pages= 16557–65|year= 2013|last1= Hu|first1= Han|last2= Ji|first2= Hai-Feng|last3= Sun|first3= Ying|author3-link=Ying Sun (mechanical engineer)|bibcode= 2013PCCP...1516557H|s2cid= 205850095}}</ref>

Wetting is important in the [[Chemical bond|bonding]] or [[adhesion|adherence]] of two materials.<ref>{{Cite book|url=https://books.google.com/books?id=dvsODgAAQBAJ&pg=PA74|title=Bio-aggregates Based Building Materials: State-of-the-Art Report of the RILEM Technical Committee 236-BBM|last1=Amziane|first1=Sofiane|last2=Collet|first2=Florence|date=2017-03-05|publisher=Springer|isbn=9789402410310}}</ref> The wetting power of a liquid, and surface forces which control wetting, are also responsible for related effects, including [[capillary action|capillary]] effects. [[Surfactant]]s can be used to increase the wetting power of liquids such as water.

Wetting has gained increasing attention in nanotechnology and nanoscience research, following the development of nanomaterials over the past two decades (i.e., [[graphene]],<ref>{{cite journal|pmid=22266468|year=2012|last1=Rafiee|first1=J.|title=Wetting transparency of graphene |url=https://homepages.rpi.edu/~koratn/resources/publications/nmat3228.pdf|archive-url=https://web.archive.org/web/20171115234915/https://homepages.rpi.edu/~koratn/resources/publications/nmat3228.pdf|url-status=dead|archive-date=2017-11-15|journal=Nature Materials|volume=11|issue=3|pages=217–22|last2=Mi|first2=X.|last3=Gullapalli|first3=H.|last4=Thomas|first4=A. V.|last5=Yavari|first5=F.|last6=Shi|first6=Y.|last7=Ajayan|first7=P. M.|last8=Koratkar|first8=N. A.|doi=10.1038/nmat3228|bibcode=2012NatMa..11..217R}}</ref> [[Carbon nano tube|carbon nanotube]], boron nitride [[nanomesh]]<ref>{{cite journal|url=https://www.physik.uzh.ch/~greber/greber185.pdf |archive-url=https://web.archive.org/web/20190411120409/https://www.physik.uzh.ch/~greber/greber185.pdf |url-status=dead |archive-date=2019-04-11 |doi=10.1038/nature18275|pmid=27357755|title=Switching stiction and adhesion of a liquid on a solid|journal=Nature|volume=534|issue=7609|pages=676–679|year=2016|last1=Mertens|first1=Stijn F. L.|last2=Hemmi|first2=Adrian|last3=Muff|first3=Stefan|last4=Gröning|first4=Oliver|last5=De Feyter|first5=Steven|last6=Osterwalder|first6=Jürg|last7=Greber|first7=Thomas|bibcode=2016Natur.534..676M|s2cid=205249367 }}</ref>).

==Explanation==
[[File:Contact angle schematic.svg|thumb|right|Figure 1: Contact angle for a liquid droplet on a solid surface]]

Wetting of a [[solid]] material with a [[liquid]] substance occurs when [[adhesion|adhesive]] forces cause the liquid (as a droplet) to spread across the surface of the solid at the ''solid-liquid'' interface. However, [[cohesion (chemistry)|cohesive]] forces acting on the liquid - at the ''liquid-vapor'' interface - counteract the adhesive forces to prevent the droplet from making full contact with the surface. 

{| class="wikitable floatright" style="text-align:center;"
|-
! rowspan=2 | Fig. 2
! rowspan=2 | Contact angle
! rowspan=2 | Degree of <br />wetting
! colspan=2 | Interaction strength
|-
! Solid–liquid
! Liquid–liquid
|-
| S || θ = 0
| Perfect wetting || Strong || Weak
|-
| rowspan=2 | C
| rowspan=2 | 0 < θ < 90°
| rowspan=2 | High wettability
| Strong || Strong
|-
| Weak   || Weak
|-
| B
| 90° ≤ θ < 180°
| Low wettability
| Weak || Strong
|-
| A
| θ = 180°
| Non-wetting
| Weak || Strong
|}

[[File:Surface tension.svg|thumb|upright=1.25|Figure 2: Wetting of different fluids: ''A'' shows a [[fluid]] with very little wetting, while ''C'' shows a fluid with more wetting. ''A'' has a large contact angle, and ''C'' has a small contact angle.]]

The [[contact angle]] (θ), as seen in Figure 1, is the angle at which the liquid–vapor interface meets the solid–liquid interface, and is determined by the balance between adhesive and cohesive forces. As the tendency of a drop to spread out over a flat, solid surface increases, the contact angle decreases. Thus, the contact angle is used as an inverse measure of wettability.<ref name="sharfin">{{cite journal|first1 = E.|last1 = Sharfrin|title = Constitutive relations in the wetting of low energy surfaces and the theory of the retraction method of preparing monolayers|journal = The Journal of Physical Chemistry|volume = 64|issue= 5|pages = 519–524|year = 1960|doi = 10.1021/j100834a002|last2 = Zisman|first2 = William A.}}</ref><ref name="mantanis et al.">{{cite journal | last1=Mantanis | first1=G. I. | last2=Young | first2=R. A. | title=Wetting of wood | journal=Wood Science and Technology | volume=31 | issue=5 | date=1997 | issn=0043-7719 | doi=10.1007/BF01159153 | pages=339–353}}</ref>

A contact angle less than 90° (low contact angle) usually indicates that wetting of the surface is very favorable, and the fluid will spread over a large area of the surface. Contact angles greater than 90° (high contact angle) generally mean that wetting of the surface is unfavorable, so the fluid will minimize contact with the surface and form a compact liquid droplet.

For water, a wettable surface may also be termed [[hydrophilic]] and a nonwettable surface [[hydrophobic]]. Superhydrophobic surfaces have contact angles greater than 150°, showing almost no contact between the liquid drop and the surface. This is sometimes referred to as the "[[Lotus effect]]". The table describes varying contact angles and their corresponding solid/liquid and liquid/liquid interactions.<ref name="Eustat">{{cite book|first = N.| last = Eustathopoulos |author2=Nicholas, M.G. |author3=Drevet B.|title = Wettability at high temperatures|publisher = Pergamon|location = Oxford, UK|year = 1999|isbn = 978-0-08-042146-9}}</ref> For nonwater liquids, the term lyophilic is used for low contact angle conditions and lyophobic is used when higher contact angles result. Similarly, the terms omniphobic and omniphilic apply to both [[chemical polarity|polar]] and apolar liquids.

==High-energy vs. low-energy surfaces==

Liquids can interact with two main types of solid surfaces. Traditionally, solid surfaces have been divided into high-[[energy]] and low-energy solids. The relative energy of a solid has to do with the bulk nature of the solid itself. Solids such as metals, [[glass]]es, and [[ceramic]]s are known as 'hard solids' because the [[chemical bond]]s that hold them together (e.g., [[Covalent bond|covalent]], [[Ionic bonding|ionic]], or [[Metallic bonding|metallic]]) are very strong. Thus, it takes a large amount of energy to break these solids (alternatively, a large amount of energy is required to cut the bulk and make two separate surfaces), so they are termed "high-energy". Most molecular liquids achieve complete wetting with high-energy surfaces.

The other type of solid is weak molecular crystals (e.g., [[fluorocarbon]]s, [[hydrocarbon]]s, etc.) where the molecules are held together essentially by physical forces (e.g., [[van der Waals force]]s and [[hydrogen bond]]s). Since these solids are held together by weak forces, a very low amount of energy is required to break them, thus they are termed "low-energy". Depending on the type of liquid chosen, low-energy surfaces can permit either complete or partial wetting.<ref>{{cite book|first = M.E| last = Schrader|author2=Loeb, G.I. |title = Modern Approaches to Wettability. Theory and Applications| publisher = Plenum Press|location = New York|year = 1992|isbn = 978-0-306-43985-8}}</ref><ref name="de Gennes">{{cite journal|first = P.G.|last = de Gennes|title = Wetting: statics and dynamics |url = https://www.physics.uci.edu/~taborek/publications/other/deGennesWettingReview.pdf|archive-url = https://web.archive.org/web/20160910030323/https://www.physics.uci.edu/~taborek/publications/other/deGennesWettingReview.pdf|url-status = dead|archive-date = 2016-09-10| journal = Reviews of Modern Physics| volume = 57|issue= 3|pages = 827–863|year = 1985|doi = 10.1103/RevModPhys.57.827|bibcode=1985RvMP...57..827D}}</ref>

Dynamic surfaces have been reported that undergo changes in [[surface energy]] upon the application of an appropriate stimuli. For example, a surface presenting photon-driven [[molecular motor]]s was shown to undergo changes in water contact angle when switched between [[Bistable structure|bistable]] conformations of differing surface energies.<ref>{{Cite journal |doi = 10.1021/ja412110t|title = Control of Surface Wettability Using Tripodal Light-Activated Molecular Motors|journal = Journal of the American Chemical Society|volume = 136|issue = 8|pages = 3219–3224|year = 2014|last1 = Chen|first1 = Kuang-Yen|last2 = Ivashenko|first2 = Oleksii|last3 = Carroll|first3 = Gregory T.|last4 = Robertus|first4 = Jort|last5 = Kistemaker|first5 = Jos C. M.|last6 = London|first6 = Gábor|last7 = Browne|first7 = Wesley R.|last8 = Rudolf|first8 = Petra|last9 = Feringa|first9 = Ben L.|pmid = 24490770| s2cid=207106544 }}</ref>

===Wetting of low-energy surfaces===
Low-energy surfaces primarily interact with liquids through dispersive ([[Van der Waals force|van der Waals]]) forces. [[William Zisman]] produced several key findings:<ref name="Johnson"/>

Zisman observed that cos θ increases linearly as the [[surface tension]] (γ<sub>LV</sub>) of the liquid decreased. Thus, he was able to establish a [[linear function]] between cos θ and the surface tension (γ<sub>LV</sub>) for various [[organic compound|organic]] liquids.

A surface is more wettable when γ<sub>LV</sub> and θ is low. Zisman termed the intercept of these lines when cos θ = 1 as the [[critical surface tension]] (γ<sub>c</sub>) of that surface. This critical surface tension is an important parameter because it is a characteristic of only the solid.

Knowing the critical surface tension of a solid, it is possible to predict the wettability of the surface.<ref name="sharfin"/>
The wettability of a surface is determined by the outermost chemical groups of the solid.
Differences in wettability between surfaces that are similar in structure are due to differences in the packing of the atoms. For instance, if a surface has branched chains, it will have poorer packing than a surface with straight chains.
Lower critical surface tension means a less wettable material surface.

==Ideal solid surfaces==
An [[ideal surface]] is flat, rigid, perfectly smooth, chemically homogeneous, and has zero [[Contact angle#Contact Angle Hysteresis|contact angle hysteresis]]. Zero [[hysteresis]] implies the advancing and receding contact angles are equal. In other words, only one thermodynamically stable contact angle exists. When a drop of liquid is placed on such a surface, the characteristic contact angle is formed as depicted in Figure 1. Furthermore, on an ideal surface, the drop will return to its original shape if it is disturbed.<ref name="Eustat"/><ref name="Johnson">Johnson, Rulon E. (1993) in ''Wettability'' Ed. Berg, John. C. New York, NY: Marcel Dekker, Inc. {{ISBN|0-8247-9046-4}}</ref> The following derivations apply only to ideal solid surfaces; they are only valid for the state in which the interfaces are not moving and the phase boundary line exists in equilibrium.

===Minimization of energy, three phases===
[[File:3PhaseCoexistence.svg|thumb|Figure 3: Coexistence of three fluid phases in mutual contact; here, α, β, and θ each indicate both a phase and its contact angle.]]
[[File:NeumannTriangle.svg|thumb|Figure 4: Neumann's triangle relating the surface energies and contact angles of three fluid phases coexisting in static equilibrium, as depicted in Figure&nbsp;3]]

Figure 3 shows the line of contact where three phases meet. In [[Thermodynamic equilibrium|equilibrium]], the net [[force]] per unit length acting along the boundary line between the three phases must be zero. The components of net force in the direction along each of the interfaces are given by:

:<math>\begin{align}
  \gamma_{\alpha\theta}                        + \gamma_{\theta\beta}\cos\left(\theta\right) + \gamma_{\alpha\beta}\cos\left(\alpha\right) &= 0 \\
  \gamma_{\alpha\theta}\cos\left(\theta\right) + \gamma_{\theta\beta}                        + \gamma_{\alpha\beta}\cos\left(\beta\right)  &= 0 \\
  \gamma_{\alpha\theta}\cos\left(\alpha\right) + \gamma_{\theta\beta}\cos\left(\beta\right)  + \gamma_{\alpha\beta}                        &= 0
\end{align}</math>

where α, β, and θ are the angles shown and γ<sub>ij</sub> is the surface energy between the two indicated phases. These relations can also be expressed by an analog to a triangle known as Neumann's triangle, shown in Figure 4. Neumann's triangle is consistent with the geometrical restriction that <math>\alpha + \beta + \theta = 2\pi</math>, and applying the law of sines and law of cosines to it produce relations that describe how the interfacial angles depend on the ratios of surface energies.<ref>{{cite book|first = J.S.| last = Rowlinson|author2=Widom, B. |title = Molecular Theory of Capillarity|publisher = Clarendon Press|location = Oxford, UK|year = 1982|isbn = 978-0-19-855642-8}}</ref>

Because these three surface energies form the sides of a [[triangle]], they are constrained by the triangle inequalities, γ<sub>ij</sub> < γ<sub>jk</sub> + γ<sub>ik</sub> meaning that not one of the surface tensions can exceed the sum of the other two. If three fluids with surface energies that do not follow these inequalities are brought into contact, no equilibrium configuration consistent with Figure 3 will exist.

====Simplification to planar geometry, Young's relation====
[[File:Contact angle.svg|thumb|upright=1.25|Figure 5: Contact angle of a liquid droplet wetted to a rigid solid surface]]
If the β phase is replaced by a flat rigid surface, as shown in Figure 5, then β = π, and the second net force equation simplifies to the Young equation,<ref>{{cite journal|first = T.|last = Young|title = An Essay on the Cohesion of Fluids |url= https://upload.wikimedia.org/wikipedia/commons/8/85/Thomas_Young-An_Essay_on_the_Cohesion_of_Fluids.pdf |journal = Philosophical Transactions of the Royal Society of London|volume = 95|pages = 65–87|year = 1805|doi = 10.1098/rstl.1805.0005|s2cid = 116124581}}</ref>

:<math>\gamma_{SG} = \gamma_{SL} + \gamma_{LG}\cos\left(\theta\right)</math><ref>{{cite journal|author = T. S. Chow|title = Wetting of rough surfaces|year = 1998|journal = [[Journal of Physics: Condensed Matter]]|volume = 10|issue = 27|pages = L445–L451|doi = 10.1088/0953-8984/10/27/001|bibcode = 1998JPCM...10L.445C | s2cid=250874015 }}</ref>

which relates the surface tensions between the three phases: [[solid]], [[liquid]] and [[gas]]. Subsequently, this predicts the contact angle of a liquid [[droplet]] on a solid surface from knowledge of the three surface energies involved. This equation also applies if the "gas" phase is another liquid, [[immiscible]] with the droplet of the first "liquid" phase.

====Simplification to planar geometry, Young's relation derived from variational computation====
Consider the interface as a curve <math> y(x) </math> for <math> x\in I=[0,L] </math> where <math> L </math> is a free parameter. The free energy to be minimized is
:<math> {\cal F}[y,L] = \int_0^L \left( \gamma_{LG}\sqrt{1+y'^2} + (\gamma_{SL}-\gamma_{SG}) \right) dx </math>
with the constraints <math> y(0)=y(L)=0 </math> which we can write as <math>\int_I y'dx=0 </math> and fixed volume <math>\int_I ydx= A </math>.

The modified Lagrangian, taking into account the constraints is therefore
:<math> {\cal L} =  \gamma_{LG}\sqrt{1+y'^2} + (\gamma_{SL}-\gamma_{SG}) -\lambda_1 y' - \lambda_2 y </math>
where <math> \lambda_i </math> are Lagrange multipliers. By definition, the momentum <math> p=\partial_{y'}{\cal L} </math> and the Hamiltonian <math> {\cal H} = p y' - {\cal L} </math> which is computed to be:
:<math> {\cal H} =  -\gamma_{LG}\frac{1}{\sqrt{1+y'^2}} -
(\gamma_{SL}-\gamma_{SG}) + \lambda_2 y </math>

Now, we recall that the boundary is free in the <math>x </math> direction and <math> L </math> is a free parameter. Therefore, we must have:
:<math> \frac{\partial{\cal F}}{\partial L} = -{\cal H} = 0 </math>
At the boundary <math> y(L)=0 </math> and <math>(1+y'^2)^{-1/2} = \cos\theta</math>, therefore we recover the Young equation.

====Non-ideal smooth surfaces and the Young contact angle====
The Young equation assumes a perfectly flat and rigid surface often referred to as an [[ideal surface]]. In many cases, surfaces are far from this ideal situation, and two are considered here: the case of rough surfaces <!-- (see Non-ideal rough solid surfaces) --> and the case of smooth surfaces that are still real (finitely rigid). Even in a perfectly smooth surface, a drop will assume a wide spectrum of contact angles ranging from the so-called advancing contact angle, <math>\theta_\mathrm{A}</math>, to the so-called receding contact angle, <math>\theta_\mathrm{R}</math>. The equilibrium contact angle (<math>\theta_\mathrm{c}</math>) can be calculated from <math>\theta_\mathrm{A}</math> and <math>\theta_\mathrm{R}</math> as was shown by Tadmor<ref name="Tadm">{{cite journal|title=Line energy and the relation between advancing, receding and Young contact angles|doi=10.1021/la049410h|year=2004|last1=Tadmor|first1=Rafael|journal=Langmuir|volume=20|pmid=15323516|issue=18|pages=7659–64}}</ref> as,

:<math>\theta_\mathrm{c} = \arccos\left(
  \frac{r_\mathrm{A}\cos\left(\theta_\mathrm{A}\right) + r_\mathrm{R}\cos\left(\theta_\mathrm{R}\right)}
       {r_\mathrm{A} + r_\mathrm{R}}
\right)</math>

where
:<math>
  r_\mathrm{A} = \left(
    \frac{\sin^3\left(\theta_\mathrm{A}\right)}{2 - 3\cos\left(\theta_\mathrm{A}\right) + \cos^3\left(\theta_\mathrm{A}\right)}
  \right)^\frac{1}{3}
~;~~
  r_\mathrm{R} = \left(
    \frac{\sin^3\left(\theta_\mathrm{R}\right)}{2 - 3\cos\left(\theta_\mathrm{R}\right) + \cos^3\left(\theta_\mathrm{R}\right)}
  \right)^\frac{1}{3}
</math>

====The Young–Dupré equation and spreading coefficient====
The Young–Dupré equation ([[Thomas Young (scientist)|Thomas Young]] 1805; Anthanase Dupré and Paul Dupré 1869) dictates that neither γ<sub>SG</sub> nor γ<sub>SL</sub> can be larger than the sum of the other two surface energies.<ref>{{Cite journal |last=Schrader |first=Malcolm E. |date=1995 |title=Young-Dupre Revisited |journal=Langmuir |language=EN |volume=11 |issue=9 |pages=3585–3589 |doi=10.1021/la00009a049}}</ref><ref>{{Cite book|title = Théorie mécanique de la chaleur|url = https://archive.org/details/thoriemcaniqued00duprgoog|publisher = Gauthier-Villars|date = 1869-01-01|language = French|first = Paul Dupré|last = Athanase M. Dupré}}</ref> The consequence of this restriction is the prediction of complete wetting when γ<sub>SG</sub> > γ<sub>SL</sub> + γ<sub>LG</sub> and zero wetting when γ<sub>SL</sub> > γ<sub>SG</sub> + γ<sub>LG</sub>. The lack of a solution to the Young–Dupré equation is an indicator that there is no equilibrium configuration with a contact angle between 0 and 180° for those situations.<ref>{{cite news|title=Contact Angle Spreading Coefficient|first=Carl|last=Clegg|url=http://www.ramehart.com/newsletters/2016-01_news.htm|work=www.ramehart.com|publisher=ramé-hart|date=2016|access-date=6 January 2016}}</ref>

A useful parameter for gauging wetting is the ''spreading parameter S'',
:<math>S = \gamma_{SG} - \left(\gamma_{SL} + \gamma_{LG}\right)</math>

When ''S'' > 0, the liquid wets the surface completely (complete wetting). When ''S'' < 0, partial wetting occurs.

Combining the spreading parameter definition with the Young relation yields the Young–Dupré equation:
:<math>S = \gamma_{LG}\left(\cos\left(\theta\right) - 1\right)</math>

which only has physical solutions for θ when S < 0.

==== A generalized model for the contact angle of droplets on flat and curved surfaces ====
With improvements in measuring techniques such as AFM, confocal microscopy and SEM, researchers were able to produce and image droplets at ever smaller scales. With the reduction in droplet size came new experimental observations of wetting. These observations confirm that the modified Young's equation<ref name=":0">{{Cite book |last=Gibbs |first=Josiah Willard Auteur du texte |url=https://gallica.bnf.fr/ark:/12148/bpt6k95192s |title=The collected works / of J. W. Gibbs,... |date=1928 |language=EN}}</ref> does not hold at the micro-nano scales. In addition the sign of the line tension is not maintained through the modified Young's equation.<ref name=":1">{{Cite journal |last1=Wciślik |first1=Sylwia |last2=Mukherjee |first2=Sayantan |date=June 2022 |title=Evaluation of three methods of static contact angle measurements for TiO 2 nanofluid droplets during evaporation |journal=Physics of Fluids |language=en |volume=34 |issue=6 |pages=062006 |doi=10.1063/5.0096644 |bibcode=2022PhFl...34f2006W |s2cid=249715372 |issn=1070-6631|doi-access=free }}</ref>

For a sessile droplet, the free energy of the three phase system can be expressed as:<ref>{{Cite journal |last=Marmur |first=Abraham |date=February 1997 |title=Line Tension and the Intrinsic Contact Angle in Solid–Liquid–Fluid Systems |url=https://linkinghub.elsevier.com/retrieve/pii/S0021979796946664 |journal=Journal of Colloid and Interface Science |language=en |volume=186 |issue=2 |pages=462–466 |doi=10.1006/jcis.1996.4666|pmid=9056376 |bibcode=1997JCIS..186..462M }}</ref><ref>{{Cite journal |last=Tadmor |first=Rafael |date=2004-08-01 |title=Line Energy and the Relation between Advancing, Receding, and Young Contact Angles |url=https://pubs.acs.org/doi/10.1021/la049410h |journal=Langmuir |language=en |volume=20 |issue=18 |pages=7659–7664 |doi=10.1021/la049410h |pmid=15323516 |issn=0743-7463}}</ref>

:<math>\delta w = \gamma_{LV}dA_{LV}+\gamma_{SL}dA_{SL}+\gamma_{SV}dA_{SV}-\kappa dL-PdV-VdP</math>

At constant volume in thermodynamic equilibrium, this reduces to:<ref>{{Cite journal |last1=Jasper |first1=Warren J. |last2=Rasipuram |first2=Srinivasan |date=December 2017 |title=Relationship between contact angle and contact line radius for micro to atto [10−6 to 10−18] liter size oil droplets |url=https://linkinghub.elsevier.com/retrieve/pii/S0167732217345683 |journal=Journal of Molecular Liquids |language=en |volume=248 |pages=920–926 |doi=10.1016/j.molliq.2017.10.134}}</ref>

:<math>0= \frac{dA_{LG}}{dA_{SL}} + \frac{\gamma_{SL}-\gamma_{SG}}{\gamma_{LG}} - \frac{\kappa}{\gamma_{LG}}\frac{dL}{dA_{SL}}-\frac{V}{\gamma_{LG}} \frac{dP}{dA_{SL}}</math>

Usually, the ''VdP'' term has been neglected for large droplets, however, ''VdP'' work becomes significant at small scales. The variation in pressure at constant volume at the free liquid-vapor boundary is due to the [[Laplace pressure]], which is proportional to the mean curvature of the droplet, and is non zero. Solving the above equation for both convex and concave surfaces yields:<ref name=":2">{{Cite journal |last1=Jasper |first1=Warren J. |last2=Anand |first2=Nadish |date=May 2019 |title=A generalized variational approach for predicting contact angles of sessile nano-droplets on both flat and curved surfaces |url=https://linkinghub.elsevier.com/retrieve/pii/S0167732218360628 |journal=Journal of Molecular Liquids |language=en |volume=281 |pages=196–203 |doi=10.1016/j.molliq.2019.02.039|s2cid=104412970 }}</ref>

:<math>\cos(\theta\mp\alpha)=A+B\frac{\cos(\alpha)}{a}\pm C\sin(\theta\mp\alpha)(\cos(\theta)+1)^2\biggl(\frac{\sin(\alpha)(\cos(\alpha)+2)}{(\cos(\alpha)+1)^2}\mp\frac{\sin(\theta)(\cos(\theta)+2)}{(\cos(\theta)+1)^2}\biggr)</math>

Where the constant parameters A, B, and C are defined as:
[[File:Diagram droplets.tif|thumb|Schematic Diagrams for droplets on flat (a) concave (b) and convex (c) surfaces<ref name=":2" />]]
:<math>A = \frac{\gamma_{SG}-\gamma_{SL}}{\gamma_{LG}}</math>, <math>B = \frac{\kappa}{\gamma_{LG}}</math> and <math>C = \frac{\gamma}{3\gamma_{LG}}</math>

This equation relates the contact angle <math>\theta</math>, a geometric property of a sessile droplet to the bulk thermodynamics, the energy at the three phase contact boundary, and the curvature of the surface α. For the special case of a sessile droplet on a flat surface (α=0),

:<math>\cos(\theta) = \frac{\gamma_{SG}-\gamma_{SL}}{\gamma_{LG}} + \frac{\kappa}{\gamma_{LG}} \frac{1}{a} -\frac{\gamma}{3\gamma_{LG}}(2+\cos(\theta)-2\cos^2(\theta)-\cos^3(\theta))</math>

The first two terms are the modified Young's equation,<ref name=":0" /> while the third term is due to the Laplace pressure. This nonlinear equation correctly predicts the sign and magnitude of κ, the flattening of the contact angle at very small scales, and contact angle hysteresis.<ref name=":2" /><ref name=":1" />

===Computational prediction of wetting===
For many surface/adsorbate configurations, surface energy data and experimental observations are unavailable. As wetting interactions are of great importance in various applications, it is often desired to predict and compare the wetting behavior of various material surfaces with particular crystallographic orientations, with relation to water or other adsorbates. This can be done from an atomistic perspective with tools including [[molecular dynamics]] and [[density functional theory]].<ref>{{Cite journal|last=Sun|first=Xuegui | year=2017 |title=Molecular dynamics simulation of wetting behaviors of Li on W surfaces|journal=[[Fusion Engineering and Design]]|volume=117|pages=188–193|doi=10.1016/j.fusengdes.2016.06.037}}</ref><ref>{{Cite journal|last=Costa|first=D | year=2017 |title=Edge wetting effects of γ-Al2O3 and anatase-TiO2 supports by MoS2 and CoMoS active phases: A DFT study|journal=Journal of Catalysis|volume=246 |issue=2 |pages= 325–334|doi=10.1016/j.jcat.2006.12.007 }}</ref> In the theoretical prediction of wetting by ''ab initio'' approaches such as DFT, ice is commonly substituted for water. This is because DFT calculations are generally conducted assuming conditions of zero thermal movement of atoms, essentially meaning the simulation is conducted at [[absolute zero]]. This simplification nevertheless yields results that are relevant for the adsorption of water under realistic conditions and the use of ice for the theoretical simulation of wetting is commonplace.<ref>Hydrophobicity of low index CeO2 planes, [https://hal.archives-ouvertes.fr/hal-02308396/document/#page=5 Applied Surface Science, Elsevier, 2019, 478, pp.68-74.] in HAL archives ouvertes</ref>

==Non-ideal rough solid surfaces==

[[File:Cahysteresis(crop).jpg|thumb|upright=1.2|left|Figure 6: Schematic of advancing and receding contact angles]]

Unlike ideal surfaces, real surfaces do not have perfect smoothness, rigidity, or chemical homogeneity. Such deviations from ideality result in phenomenon called [[Contact angle#Contact Angle Hysteresis|contact angle hysteresis]], which is defined as the difference between the advancing (θ<sub>a</sub>) and receding (θ<sub>r</sub>) contact angles<ref>{{cite journal|author = Robert J. Good|title = Contact angle, wetting, and adhesion: a critical review|year = 1992|journal = Journal of Adhesion Science and Technology|volume = 6|issue = 12|pages = 1269–1302|doi = 10.1163/156856192X00629}}</ref>
:<math>\text{H} = \,\theta_{a} - \,\theta_{r}</math>

When the contact angle is between the advancing and receding cases, the contact line is considered to be pinned and hysteretic behaviour can be observed, namely [[Contact angle#Contact Angle Hysteresis|contact angle hysteresis]]. When these values are exceeded, the displacement of the contact line, such as the one in Figure 3, will take place by either expansion or retraction of the droplet.<ref name="CAH">{{cite journal|last1= Shi| first1= Z. | display-authors=etal | title= Dynamic contact angle hysteresis in liquid bridges | journal= Colloids and Surfaces A  | year=2018 | volume=555  | pages= 365–371| doi=10.1016/j.colsurfa.2018.07.004| arxiv= 1712.04703 | s2cid= 51916594 }}</ref> Figure 6 depicts the advancing and receding contact angles. The advancing contact angle is the maximum stable angle, whereas the receding contact angle is the minimum stable angle. Contact angle hysteresis occurs because many different thermodynamically stable contact angles are found on a nonideal solid. These varying thermodynamically stable contact angles are known as metastable states.<ref name = "Johnson"/>

Such motion of a phase boundary, involving advancing and receding contact angles, is known as dynamic wetting. The difference between dynamic and static wetting angles is proportional to the [[capillary number]], '''<math>Ca</math>''', When a contact line advances, covering more of the surface with liquid, the contact angle is increased and is generally related to the velocity of the contact line.<ref name="CAH" /><ref>{{cite book|first = P. G.|last = De Gennes|title = Soft Interfaces|year = 1994|publisher = Cambridge University Press|location = Cambridge, UK|isbn = 978-0-521-56417-5}}</ref> If the velocity of a contact line is increased without bound, the contact angle increases, and as it approaches 180°, the gas phase will become entrained in a thin layer between the liquid and solid. This is a kinetic nonequilibrium effect which results from the contact line moving at such a high speed that complete wetting cannot occur.

A well-known departure from ideal conditions is when the surface of interest has a rough texture. The rough texture of a surface can fall into one of two categories: homogeneous or heterogeneous. A homogeneous wetting regime is where the liquid fills in the grooves of a rough surface. A heterogeneous wetting regime, though, is where the surface is a composite of two types of patches. An important example of such a composite surface is one composed of patches of both air and solid. Such surfaces have varied effects on the contact angles of wetting liquids. Cassie–Baxter and Wenzel are the two main models that attempt to describe the wetting of textured surfaces. However, these equations only apply when the drop size is sufficiently large compared with the surface roughness scale.<ref name = "abraham">{{cite journal|author = Abraham Marmur|title = Wetting of Hydrophobic Rough Surfaces: To be heterogeneous or not to be| year = 2003|journal = Langmuir|volume = 19|pages = 8343–8348|doi = 10.1021/la0344682|issue = 20}}</ref> When the droplet size is comparable to that of the underlying pillars, the effect of line tension should be considered.<ref>{{cite journal|doi=10.1103/PhysRevLett.109.116101|pmid=23005650|title=Evaporation of Droplets on Superhydrophobic Surfaces: Surface Roughness and Small Droplet Size Effects |url=https://pdfs.semanticscholar.org/a5ae/b08b7f2816c9c5322d43e5122ee3fdb15d40.pdf|archive-url=https://web.archive.org/web/20190411123804/https://pdfs.semanticscholar.org/a5ae/b08b7f2816c9c5322d43e5122ee3fdb15d40.pdf|url-status=dead|archive-date=2019-04-11|journal=Physical Review Letters|volume=109|issue=11|pages=116101(1–6)|year=2012|last1=Chen|first1=Xuemei|last2=Ma|first2=Ruiyuan|last3=Li|first3=Jintao|last4=Hao|first4=Chonglei|last5=Guo|first5=Wei|last6=Luk|first6=B. L.|last7=Li|first7=Shuai Cheng|last8=Yao|first8=Shuhuai|last9=Wang|first9=Zuankai|bibcode=2012PhRvL.109k6101C|s2cid=29794436}}</ref>

===Wenzel's model===
[[File:Wenzel.png|thumb|Figure 7: Wenzel model]]
The Wenzel model <ref>Wenzel, Robert N. "Resistance of solid surfaces to wetting by water." Industrial & engineering chemistry 28.8 (1936): 988-994.</ref> describes the homogeneous wetting regime, as seen in Figure 7, and is defined by the following equation for the contact angle on a rough surface:<ref name="degennesbook"> de Gennes, Pierre-Gilles; Brochard-Wyart, Françoise; Quéré, David (2004). Capillarity and Wetting Phenomena. Springer New York. doi:10.1007/978-0-387-21656-0. ISBN 978-1-4419-1833-8. S2CID 137894832 </ref>
:<math>\cos\,\left(\theta^*\right) = r \cos\,\left(\theta\right)</math>

where <math>\theta^*</math> is the apparent contact angle which corresponds to the stable equilibrium state (i.e. minimum free energy state for the system). The roughness ratio, r, is a measure of how surface roughness affects a homogeneous surface. The roughness ratio is defined as the ratio of true area of the solid surface to the apparent area.

θ is the [[Contact_angle#Modified_Young’s_equation|contact angle]] for a system in thermodynamic equilibrium, defined for a perfectly flat surface. Although Wenzel's equation demonstrates the contact angle of a rough surface is different from the intrinsic contact angle, it does not describe contact angle [[Hysteresis#Contact angle hysteresis|hysteresis]].<ref name="degennesbook"/>

===Cassie–Baxter model===
[[File:Cassie-Baxter.png|thumb|left|Figure 8: Cassie–Baxter model]]

When dealing with a heterogeneous surface, the Wenzel model is not sufficient. A more complex model is needed to measure how the apparent contact angle changes when various materials are involved. This heterogeneous surface, like that seen in Figure 8, is explained using the Cassie–Baxter equation ([[Cassie's law]]):<ref name="abraham"/>

:<math>\cos\,\left(\theta^*\right) = r_f\,f \,\cos\,\left(\theta_\text{Y}\right) + f - 1</math>

Here the ''r<sub>f</sub>'' is the roughness ratio of the wet surface area and ''f'' is the fraction of solid surface area wet by the liquid. When ''f'' = 1 and ''r<sub>f</sub> = r'', the Cassie–Baxter equations becomes the Wenzel equation. On the other hand, when there are many different fractions of surface roughness, each fraction of the total surface area is denoted by <math>f_i</math>.

A summation of all <math>f_i</math> equals 1 or the total surface. Cassie–Baxter can also be recast in the following equation:<ref name=Whyman>{{cite journal|first1 = G.|last1 = Whyman| title = The rigorous derivation of Young, Cassie–Baxter and Wenzel equations and the analysis of the contact angle hysteresis phenomenon |url= https://www.researchgate.net/publication/239161424 | journal = Chemical Physics Letters| volume = 450| pages = 355–359|year = 2008|doi = 10.1016/j.cplett.2007.11.033|last2 = Bormashenko|first2 = Edward|last3 = Stein|first3 = Tamir|bibcode = 2008CPL...450..355W|issue = 4–6 }}</ref>
:<math>\gamma\cos\,\left(\theta^*\right) = \sum_{n=1}^N f_i\left(\gamma_\text{i,sv} - \gamma_\text{i,sl}\right)</math>

Here <math>\gamma</math> is the Cassie–Baxter surface tension between liquid and vapor, <math>\gamma_{i,sv}</math> is the solid vapor surface tension of every component, and <math>\gamma_{i,sl}</math> is the solid liquid surface tension of every component. A case that is worth mentioning is when the liquid drop is placed on the substrate and creates small air pockets underneath it. This case for a two-component system is denoted by:<ref name="Whyman"/>
:<math>\gamma\cos\,\left(\theta^*\right) = f_1\left(\gamma_\text{1,sv} - \gamma_\text{1,sl}\right) - \left(1 - f_1\right)\gamma</math>

Here the key difference to notice is that there is no surface tension between the solid and the vapor for the second surface tension component. This is because of the assumption that the surface of air that is exposed is under the droplet and is the only other substrate in the system. Subsequently, the equation is then expressed as (1 – ''f''). Therefore, the Cassie equation can be easily derived from the Cassie–Baxter equation. Experimental results regarding the surface properties of Wenzel versus Cassie–Baxter systems showed the effect of pinning for a Young angle of 180 to 90°, a region classified under the Cassie–Baxter model. This liquid/air composite system is largely hydrophobic. After that point, a sharp transition to the Wenzel regime was found where the drop wets the surface, but no further than the edges of the drop. Actually, the Young, Wenzel and Cassie-Baxter equations represent the transversality conditions of the variational problem of wetting.<ref>{{Cite journal|last=Bormashenko|first=Edward|date=2009-08-05|title=Young, Boruvka–Neumann, Wenzel and Cassie–Baxter equations as the transversality conditions for the variational problem of wetting|url=http://www.sciencedirect.com/science/article/pii/S0927775709002805|journal=Colloids and Surfaces A: Physicochemical and Engineering Aspects|language=en|volume=345|issue=1|pages=163–165|doi=10.1016/j.colsurfa.2009.04.054|issn=0927-7757}}</ref><ref>{{Cite journal|last=Bormashenko|first=Edward|date=2020-01-17|title=Variational framework for defining contact angles: a general thermodynamic approach|journal=Journal of Adhesion Science and Technology|volume=34|issue=2|pages=219–230|doi=10.1080/01694243.2019.1663030|s2cid=203537637|issn=0169-4243}}</ref>

====Precursor film====

With the advent of high resolution imaging, researchers have started to obtain experimental data which have led them to question the assumptions of the Cassie–Baxter equation when calculating the apparent contact angle. These groups{{who|date=August 2020}} believe the apparent contact angle is largely dependent on the triple line. The triple line, which is in contact with the heterogeneous surface, cannot rest on the heterogeneous surface like the rest of the drop. In theory, it should follow the surface imperfection. This bending in the triple line is unfavorable and is not seen in real-world situations. A theory that preserves the Cassie–Baxter equation while at the same time explaining the presence of the minimized energy state of the triple line hinges on the idea of a precursor film. This film of submicrometer thickness advances ahead of the motion of the droplet and is found around the triple line. Furthermore, this precursor film allows the triple line to bend and take different conformations that were originally considered unfavorable. This precursor fluid has been observed using [[Environmental scanning electron microscope|environmental scanning electron microscopy]] (ESEM) in surfaces with pores formed in the bulk. With the introduction of the precursor film concept, the triple line can follow energetically feasible conformations, thereby correctly explaining the Cassie–Baxter model.<ref name=Bormashenko>{{cite journal|first = E.|last = Bormashenko| title = Why does the Cassie–Baxter equation apply? |url= https://vdocuments.mx/why-does-the-cassiebaxter-equation-apply.html | journal = Colloids and Surfaces A | volume = 324|issue = 1–3| pages = 47–50|year = 2008|doi = 10.1016/j.colsurfa.2008.03.025 }}</ref>

===="Petal effect" vs. "lotus effect"====<!--[[Petal effect]] redirects to this section-->
[[File:Petal lotus.JPG|thumb|upright=1.35|Figure 9: "Petal effect" vs. "lotus effect"]]
The intrinsic [[Hydrophobe|hydrophobicity]] of a surface can be enhanced by being textured with different length scales of [[Rugosity|roughness]]. The red [[rose]] takes advantage of this by using a hierarchy of micro- and nanostructures on each petal to provide sufficient roughness for superhydrophobicity. More specifically, each rose petal has a collection of micropapillae on the surface and each papilla, in turn, has many nanofolds. The term "[[petal effect]]" describes the fact that a water droplet on the surface of a rose petal is spherical in shape, but cannot roll off even if the petal is turned upside down. The water drops maintain their spherical shape due to the superhydrophobicity of the petal (contact angle of about 152.4°), but do not roll off because the petal surface has a high adhesive force with water.<ref>{{cite journal|first1 = F.|last1 = Lin|title = Petal Effect: A Superhydrophobic State with High Adhesive Force |journal = Langmuir| volume = 24|issue= 8|pages = 4114–4119|year = 2008|doi = 10.1021/la703821h|pmid = 18312016|last2 = Zhang|first2 = Y|last3 = Xi|first3 = J|last4 = Zhu|first4 = Y|last5 = Wang|first5 = N|last6 = Xia|first6 = F|last7 = Jiang|first7 = L}}</ref>

When comparing the "[[petal effect]]" to the "[[lotus effect]]", it is important to note some striking differences. The surface structure of the lotus leaf and the rose petal, as seen in Figure 9, can be used to explain the two different effects.

The lotus leaf has a randomly rough surface and low [[contact angle]] hysteresis, which means the water droplet is not able to wet the microstructure spaces between the spikes. This allows air to remain inside the texture, causing a heterogeneous surface composed of both air and solid. As a result, the adhesive force between the water and the solid surface is extremely low, allowing the water to roll off easily (i.e. "[[Self-cleaning surfaces|self-cleaning]]" phenomenon).

The rose petal's micro- and nanostructures are larger in scale than those of the lotus leaf, which allows the liquid film to impregnate the texture. However, as seen in Figure 9, the liquid can enter the larger-scale grooves, but it cannot enter into the smaller grooves. This is known as the Cassie impregnating wetting regime. Since the liquid can wet the larger-scale grooves, the adhesive force between the water and solid is very high. This explains why the water droplet will not fall off even if the petal is tilted at an angle or turned upside down. This effect will fail if the droplet has a volume larger than 10 μL because the balance between weight and surface tension is surpassed.<ref>{{cite journal|doi=10.1016/j.apsusc.2017.11.261|title=Duty cycle dependent chemical structure and wettability of RF pulsed plasma copolymers of acrylic acid and octafluorocyclobutane|journal=Applied Surface Science|volume=436|pages=411–418|year=2018|last1=Muzammil|first1=I.|last2=Li|first2=Y.P.|last3=Li|first3=X.Y.|last4=Lei|first4=M.K.|bibcode=2018ApSS..436..411M}}</ref>

===Cassie–Baxter to Wenzel transition===
[[File:Cassie transition.png|thumb|right|Figure 10: Mushroom state]]
In the [[Cassie's law|Cassie–Baxter model]], the drop sits on top of the textured surface with trapped air underneath. During the [[wetting transition]] from the Cassie state to the Wenzel state, the air pockets are no longer thermodynamically stable and liquid begins to nucleate from the middle of the drop, creating a "mushroom state" as seen in Figure 10.<ref name=Ishino>{{Cite journal|first1 = C.|last1 = Ishino|title = Wetting transitions on textured hydrophilic surfaces |url = http://phys.ocha.ac.jp/okumuralab/h16/inside/papers/IshinoEPJE.pdf|archive-url = https://web.archive.org/web/20190411125530/http://phys.ocha.ac.jp/okumuralab/h16/inside/papers/IshinoEPJE.pdf|url-status = dead|archive-date = 2019-04-11|journal = European Physical Journal|volume = 25|pages = 415–424|year = 2008|doi = 10.1140/epje/i2007-10308-y|pmid = 18431542|last2 = Okumura|first2 = K|issue = 4|bibcode = 2008EPJE...25..415I |s2cid = 35973585}}</ref> The penetration condition is given by:
:<math>\cos\,\left(\theta_\text{C}\right)= \frac{\phi - 1}{r - \phi}</math>

where
:* ''θ<sub>C</sub>'' is the critical contact angle
:* ''Φ'' is the fraction of solid/liquid interface where drop is in contact with surface
:* ''r'' is solid roughness (for flat surface, r = 1)

[[File:Wenzel film.png|thumb|left|Figure 11: Penetration front spreads beyond drop]]

The penetration front propagates to minimize the surface energy until it reaches the edges of the drop, thus arriving at the Wenzel state. Since the solid can be considered an absorptive material due to its surface roughness, this phenomenon of spreading and imbibition is called hemiwicking. The contact angles at which spreading/imbibition occurs are between 0 and π/2.<ref>{{cite journal|first1 = D.|last1 = Quere|title = Wetting of Textured Surfaces|journal = Colloids and Surfaces A|pages = 41–46|year = 2008|doi = 10.1016/S0927-7757(02)00061-4|volume = 206|issue = 1–3|url = http://www.pmmh.espci.fr/~jbico/bico02b.pdf|last2 = Thiele|first2 = Uwe|last3 = Quéré|first3 = David|access-date = 2011-12-17|archive-date = 2012-05-27|archive-url = https://web.archive.org/web/20120527014400/http://www.pmmh.espci.fr/~jbico/bico02b.pdf|url-status = dead}}</ref>

The Wenzel model is valid between θ<sub>C</sub> and π/2. If the contact angle is less than Θ<sub>''C''</sub>, the penetration front spreads beyond the drop and a liquid film forms over the surface. Figure 11 depicts the transition from the Wenzel state to the surface film state. The film smoothes the surface roughness and the Wenzel model no longer applies. In this state, the equilibrium condition and Young's relation yields:
:<math>\cos\,\left(\theta^*\right) = \phi\cos\,\left(\theta_C\right) + \left(1 - \phi\right)</math><ref name="Ishino"/>

By fine-tuning the surface roughness, it is possible to achieve a transition between both superhydrophobic and superhydrophilic regions. Generally, the rougher the surface, the more hydrophobic it is.

==Spreading dynamics==

If a drop is placed on a smooth, horizontal surface, it is generally not in the equilibrium state. Hence, it spreads until an equilibrium contact radius is reached (partial wetting). While taking into account capillary, gravitational, and viscous contributions, the drop radius as a function of time can be expressed as<ref>{{Cite journal | doi=10.1002/macp.201100631| title=Simple Approach for Spreading Dynamics of Polymeric Fluids| journal=Macromolecular Chemistry and Physics| volume=213| issue=6| pages=654–665| year=2012| last1=Härth| first1=Michael| last2=Schubert| first2=Dirk W.}}</ref>

: <math>r(t) = r_e\left[
    1 - \exp\left(-\left(\frac{2\gamma_{LG}}{r^{12}_e} + \frac{\rho g}{9r^{10}_e}\right)\frac{24\lambda V^4 \left(t + t_0\right)}{\pi^2\eta}\right)
\right]^\frac{1}{6}.</math>

For the complete wetting situation, the drop radius at any time during the spreading process is given by

: <math>r(t) = \left[
    \left(\gamma_{LG} \frac{96\lambda V^4}{\pi^2 \eta} \left(t + t_0\right)\right)^\frac{1}{2} +
    \left(\frac{\lambda(t + t_0)}{\eta}\right)^\frac{2}{3} \frac{24\rho g V^\frac{8}{3}}{7 \cdot 96^\frac{1}{3} \pi^\frac{4}{3} \gamma_{LG}^\frac{1}{3}}
  \right]^\frac{1}{6},
</math>

where
:*''γ''<sub>LG</sub> is [[surface tension]] of the fluid
:*''V'' is drop volume
:*''η'' is [[viscosity]] of the fluid
:*''ρ'' is [[density]] of the fluid
:*''g'' is [[gravitational constant]]
:*''λ'' is shape factor, 37.1{{nbsp}}m<sup>−1</sup>
:*''t''<sub>0</sub> is experimental delay time
:*''r''<sub>e</sub> is drop radius in equilibrium

==Modifying wetting properties==
===Surfactants===
Many technological processes require control of liquid spreading over solid surfaces. When a drop is placed on a surface, it can completely wet, partially wet, or not wet the surface. By reducing the surface tension with [[surfactants]], a nonwetting material can be made to become partially or completely wetting. The excess free energy (σ) of a drop on a solid surface is:<ref name="Lee"/>
:<math>\sigma = \gamma S + PV + \pi\,R^2 \left(\gamma_\text{SL} - \gamma_\text{SV}\right)</math>

:* ''γ'' is the liquid–vapor interfacial tension
:* γ<sub>SL</sub> is the solid–liquid interfacial tension
:* γ<sub>SV</sub> is the solid–vapor interfacial tension
:* ''S'' is the area of liquid–vapor interface
:* ''P'' is the excess pressure inside liquid
:* ''R'' is the radius of droplet base

Based on this equation, the excess free energy is minimized when γ decreases, γ<sub>SL</sub> decreases, or γ<sub>SV</sub> increases. Surfactants are absorbed onto the liquid–vapor, solid–liquid, and solid–vapor interfaces, which modify the wetting behavior of hydrophobic materials to reduce the free energy. When surfactants are absorbed onto a hydrophobic surface, the polar head groups face into the solution with the tail pointing outward. In more hydrophobic surfaces, surfactants may form a bilayer on the solid, causing it to become more hydrophilic. The dynamic drop radius can be characterized as the drop begins to spread. Thus, the contact angle changes based on the following equation:<ref name=Lee>{{cite journal|first1 = K. S.|last1 = Lee|title = Kinetics of wetting and spreading by aqueous surfactant solutions|year = 2008|journal = Advances in Colloid and Interface Science|pages = 54–65|doi = 10.1016/j.cis.2008.08.005|volume = 144|pmid = 18834966|last2 = Ivanova|first2 = N.|last3 = Starov|first3 = V. M.|last4 = Hilal|first4 = N.|last5 = Dutschk|first5 = V.|issue = 1–2}}</ref>
:<math>\cos\,\left(\theta(t)\right) = \cos\,\left(\theta_0\right) + \left(\cos\,\left(\theta_\infty\right) - \cos\,\left(\theta_0\right)\right)\left(1 - \mathrm{e}^{-\frac{t}{\tau}}\right)</math>

:* ''θ''<sub>0</sub> is initial contact angle
:* ''θ''<sub>∞</sub> is final contact angle
:* ''τ'' is the surfactant transfer time scale

As the surfactants are absorbed, the solid–vapor surface tension increases and the edges of the drop become hydrophilic. As a result, the drop spreads.

===Surface changes===
[[File:Wettability of a silica surface with a bound ferrocene-substituted polymer.jpg|thumb|upright=2.2|Strands of an uncharged ferrocene-substituted polymer are tethered to a [[hydrophobic]] [[silica]] surface. Oxidation of the ferrocenyl groups produces a [[hydrophilic]] surface due to electrostatic attractions between the resulting charges and the polar solvent.<ref name = Pietschnig />]]
[[Ferrocene]] is a [[redox]]-active organometallic compound<ref>
{{cite journal
 | first1 = N. G. | last1 = Connelly
 | first2 = W. E. | last2 = Geiger
 | title = Chemical Redox Agents for Organometallic Chemistry
 | url = https://www.chem.uci.edu/~jennyyy/uploads/2/1/6/1/21612644/chemical_redox_agents.pdf | archive-url = https://web.archive.org/web/20160122152345/https://www.chem.uci.edu/~jennyyy/uploads/2/1/6/1/21612644/chemical_redox_agents.pdf | url-status = dead | archive-date = 2016-01-22 | journal = Chemical Reviews
 | year = 1996 | volume = 96 | issue = 2 | pages = 877–910
 | doi = 10.1021/cr940053x | pmid = 11848774
}}</ref> which can be incorporated into various [[monomer]]s and used to make [[polymer]]s which can be tethered onto a surface.<ref name = Pietschnig /> [[Vinylferrocene]] (ferroceneylethene) can be prepared by a [[Wittig reaction]]<ref>
{{cite journal
 | last1 = Liu | first1 = W.-Y.
 | last2 = Xu | first2 = Q.-H.
 | last3 = Ma | first3 = Y.-X.
 | last4 = Liang | first4 = Y.-M.
 | last5 = Dong | first5 = N.-L.
 | last6 = Guan | first6 = D.-P.
 | journal =  Journal of Organometallic Chemistry
 | year = 2001 | volume = 625 | pages = 128–132
 | doi = 10.1016/S0022-328X(00)00927-X
 | title = Solvent-free synthesis of ferrocenylethene derivatives
}}</ref> and then polymerized to form polyvinylferrocene (PVFc), an [[Structural analog|analog]] of [[polystyrene]]. Another polymer which can be formed is poly({{nowrap|2-(methacryloyloxy)ethyl}} ferrocenecarboxylate), PFcMA. Both PVFc and PFcMA have been tethered onto [[silica]] wafers and the wettability measured when the polymer chains are uncharged and when the ferrocene [[moiety (chemistry)|moieties]] are oxidised to produce positively charged groups, as illustrated at right.<ref name = Pietschnig>
{{cite journal
 | first = R. | last = Pietschnig
 | title = Polymers with pendant ferrocenes
 | journal = Chemical Society Reviews
 | year = 2016 | volume = 45 | issue = 19
 | pages = 5216–5231
 | doi = 10.1039/C6CS00196C
| pmid = 27156979
 | doi-access = free
 }}</ref> The contact angle with water on the PFcMA-coated wafers was 70° smaller following oxidation, while in the case of PVFc the decrease was 30°, and the switching of wettability has been shown to be reversible. In the PFcMA case, the effect of longer chains with more ferrocene groups (and also greater [[molar mass]]) has been investigated, and it was found that longer chains produce significantly larger contact angle reductions.<ref name = Pietschnig /><ref>
{{cite journal
 | first1 = J. | last1 = Elbert
 | first2 = M. | last2 = Gallei
 | first3 = C. | last3 = Rüttiger
 | first4 = A. | last4 = Brunsen
 | first5 = H. | last5 = Didzoleit
 | first6 = B. | last6 = Stühn
 | first7 = M. | last7 = Rehahn
 | journal = Organometallics
 | year = 2013 | volume = 32 | issue = 20 | pages = 5873–5878
 | title = Ferrocene Polymers for Switchable Surface Wettability
 | doi = 10.1021/om400468p
}}</ref>

===Oxygen vacancies===
Rare earth oxides exhibit intrinsic hydrophobicity, and hence can be used in thermally stable [[heat exchangers]] and other applications involving high-temperature hydrophobicity.<ref>{{cite journal|url=https://cen.acs.org/articles/91/i4/Rare-Earth-Oxides-Naturally-Hydrophobic.html |journal= Chemical & Engineering News |volume =91 |issue =4 |date=28 January 2013|author=Kemsley, Jyllian|title=Rare-Earth Oxides Are Naturally Hydrophobic }}</ref> The presence of oxygen vacancies at surfaces of ceria or other rare earth oxides is instrumental in governing surface wettability. [[Adsorption]] of water at oxide surfaces can occur as molecular adsorption, in which H<sub>2</sub>O molecules remain intact at the terminated surface, or as [[dissociative adsorption]], in which OH and H are adsorbed separately<ref>{{cite journal|doi=10.1016/j.apsusc.2019.01.208|arxiv=1902.02662|title=Theoretical insights into the hydrophobicity of low index CeO<sub>2</sub> surfaces|journal=Applied Surface Science|volume=478|pages=68–74|year=2019|last1=Fronzi|first1=Marco|last2=Assadi|first2=M. Hussein N.|last3=Hanaor|first3=Dorian A.H.|last4=Hanaor|first4=Dorian A. H.|last5=Gan|first5=Yixiang|bibcode=2019ApSS..478...68F|s2cid=118895100}}</ref> at solid surfaces. The presence of oxygen vacancies is generally found to enhance hydrophobicity while promoting dissociative adsorption.<ref>{{cite journal|doi=10.1021/acs.jpcc.6b08991 |title=Role of Oxygen Vacancy on the Hydrophobic Behavior of TiO<sub>2</sub> Nanorods on Chemically Etched Si Pyramids |journal=The Journal of Physical Chemistry C |volume=121 |pages=278–283 |year=2017 |last1=Saini |first1=C. P. |last2=Barman |first2=A. |last3=Das |first3=D. |last4=Satpati |first4=B. |last5=Bhattacharyya |first5=S. R. |last6=Kanjilal |first6=D. |last7=Ponomaryov |first7=A. |last8=Zvyagin |first8=S. |last9=Kanjilal |first9=A. }}</ref>

==See also==
{{colbegin}}
* {{annotated link|Adsorption}}
* {{annotated link|Amott test}}
* {{annotated link|Anti-fog}}
* {{annotated link|Dewetting}}
* {{annotated link|Electrowetting}}
* {{annotated link|Flotation (buoyancy)|Flotation}}
* {{annotated link|Sessile drop technique}}
* {{annotated link|Soap bubble}}
* {{annotated link|Rise in core}}
* {{annotated link|Wetting current}}
* {{annotated link|Wetting voltage}}
* {{annotated link|Lak wettability index}}
* {{annotated link|USBM wettability index}}
{{colend}}

==References==
{{Reflist|30em}}

==Further reading==
* {{cite book|last1=de Gennes|first1=Pierre-Gilles|last2=Brochard-Wyart|first2=Françoise|last3=Quéré|first3=David|title=Capillarity and Wetting Phenomena|year=2004|doi=10.1007/978-0-387-21656-0|isbn=978-1-4419-1833-8|publisher=Springer New York|s2cid=137894832}}
* {{cite book|author1=Victor M. Starov|author2=Manuel G. Velarde|author2-link=Manuel García Velarde|author3=Clayton J. Radke|title=Wetting and Spreading Dynamics|url=https://books.google.com/books?id=NT6KPxETpPMC|date=2 April 2007|publisher=CRC Press|isbn=978-1-4200-1617-8}}

==External links==
* [https://blog.biolinscientific.com/what-is-wettability What is wettability?]
* {{Commons category-inline|Wetting}}

{{good article}}
{{Authority control}}

[[Category:Fluid mechanics]]
[[Category:Surface science]]
[[Category:Hysteresis]]