Editing Hydrophobe (section)

===Theory===
In 1805, Thomas Young defined the contact angle ''θ'' by analyzing the forces acting on a fluid droplet resting on a solid surface surrounded by a gas.<ref>{{cite journal |first=T. |last=Young |title=An Essay on the Cohesion of Fluids |journal=[[Philosophical Transactions of the Royal Society|Phil. Trans. R. Soc. Lond.]] |volume=95 |pages=65–87 |year=1805 |doi=10.1098/rstl.1805.0005|s2cid=116124581 |doi-access=free }}</ref>

[[File:Contact angle.svg|thumb|upright=1.15|A liquid droplet rests on a solid surface and is surrounded by gas. The contact angle, ''θ''<sub>C</sub>, is the angle formed by a liquid at the three-phase boundary where the liquid, gas, and solid intersect.]]
[[File:Contact angle microstates.svg|thumb|upright=1.5|A droplet resting on a solid surface and surrounded by a gas forms a characteristic contact angle&nbsp;''θ''. If the solid surface is rough, and the liquid is in intimate contact with the solid asperities, the droplet is in the Wenzel state. If the liquid rests on the tops of the asperities, it is in the Cassie–Baxter state.]]

:<math>\gamma_\text{SG}\ =\gamma_\text{SL}+\gamma_\text{LG}\cos\theta \,</math>
where
:<math>\gamma_\text{SG}\ </math> = [[Interfacial tension]] between the solid and gas
:<math>\gamma_\text{SL}\ </math> = Interfacial tension between the solid and liquid
:<math>\gamma_\text{LG}\ </math> = Interfacial tension between the liquid and gas

''θ'' can be measured using a [[contact angle goniometer]].

Wenzel determined that when the liquid is in intimate contact with a microstructured surface, ''θ'' will change to ''θ''<sub>W*</sub>
:<math>\cos\theta_W* = r \cos\theta \, </math>

where ''r'' is the ratio of the actual area to the projected area.<ref>{{cite journal |first=RN |last=Wenzel |title=Resistance of Solid Surfaces to Wetting by Water |journal= Ind. Eng. Chem. |volume= 28 |pages= 988–994 |year= 1936 |doi= 10.1021/ie50320a024 |issue= 8}}</ref> Wenzel's equation shows that microstructuring a surface amplifies the natural tendency of the surface. A hydrophobic surface (one that has an original contact angle greater than 90°) becomes more hydrophobic when microstructured – its new contact angle becomes greater than the original. However, a hydrophilic surface (one that has an original contact angle less than&nbsp;90°) becomes more hydrophilic when microstructured – its new contact angle becomes less than the original.<ref>{{cite book |first=Pierre-Gilles |last=de Gennes |title=Capillarity and Wetting Phenomena |year=2004 |publisher=Springer |isbn=0-387-00592-7}}</ref>
Cassie and Baxter found that if the liquid is suspended on the tops of microstructures, ''θ'' will change to ''θ''<sub>CB*</sub>:

:<math>\cos\theta_\text{CB}* = \varphi(\cos\theta + 1) - 1 \,</math>

where ''φ'' is the area fraction of the solid that touches the liquid.<ref>{{cite journal |vauthors= Baxter AB, Cassie S |title= Wettability of Porous Surfaces |journal=[[Trans. Faraday Soc.]] |volume=40 |pages=546–551 |year=1944 |doi=10.1039/tf9444000546}}</ref> Liquid in the Cassie–Baxter state is more mobile than in the Wenzel state.{{Citation needed|date=January 2021}}

We can predict whether the Wenzel or Cassie–Baxter state should exist by calculating the new contact angle with both equations. By a minimization of free energy argument, the relation that predicted the smaller new contact angle is the state most likely to exist. Stated in mathematical terms, for the Cassie–Baxter state to exist, the following inequality must be true.<ref>{{cite journal |first=D |last=Quere |title=Non-sticking Drops |journal=[[Reports on Progress in Physics]] |volume=68 |pages=2495–2532 |year=2005 |doi=10.1088/0034-4885/68/11/R01 |issue=11 |bibcode=2005RPPh...68.2495Q|s2cid=121128710 }}</ref>

:<math>\cos\theta<\frac{\varphi-1}{r-\varphi}</math>

A recent alternative criterion for the Cassie–Baxter state asserts that the Cassie–Baxter state exists when the following 2 criteria are met:1) Contact line forces overcome body forces of unsupported droplet weight and 2) The microstructures are tall enough to prevent the liquid that bridges microstructures from touching the base of the microstructures.<ref>{{cite journal |vauthors= Extrand CW |title= Modeling of ultralyophobicity: Suspension of liquid drops by a single asperity |journal= Langmuir |volume= 21 |issue= 23 |pages= 10370–10374 |year= 2005 |pmid= 16262294 |doi= 10.1021/la0513050}}</ref>

A new criterion for the switch between Wenzel and Cassie-Baxter states has been developed recently based on surface roughness and [[surface energy]].<ref>{{cite journal |vauthors= Zhang YL, Sundararajan S |title= Superhydrophobic engineering surfaces with tunable air-trapping ability |journal= Journal of Micromechanics and Microengineering |volume=18 |pages=035024 |year=2008 |doi=10.1088/0960-1317/18/3/035024|issue=3|bibcode= 2008JMiMi..18c5024Z|s2cid= 137395618 }}</ref> The criterion focuses on the air-trapping capability under liquid droplets on rough surfaces, which could tell whether Wenzel's model or Cassie-Baxter's model should be used for certain combination of surface roughness and energy.{{Citation needed|date=January 2021}}

Contact angle is a measure of static hydrophobicity, and [[Contact angle#Contact Angle Hysteresis|contact angle hysteresis]] and slide angle are dynamic measures. Contact angle hysteresis  is a phenomenon that characterizes surface heterogeneity.<ref>{{cite journal |vauthors= Johnson RE, Dettre RH |title=Contact Angle Hysteresis |volume=68 |pages=1744–1750 |year=1964 |journal=[[J. Phys. Chem.]] |doi=10.1021/j100789a012 |issue=7}}</ref> When a pipette injects a liquid onto a solid, the liquid will form some contact angle. As the pipette injects more liquid, the droplet will increase in volume, the contact angle will increase, but its three-phase boundary will remain stationary until it suddenly advances outward. The contact angle the droplet had immediately before advancing outward is termed the advancing contact angle. The receding contact angle is now measured by pumping the liquid back out of the droplet. The droplet will decrease in volume, the contact angle will decrease, but its three-phase boundary will remain stationary until it suddenly recedes inward. The contact angle the droplet had immediately before receding inward is termed the receding contact angle. The difference between advancing and receding contact angles is termed contact angle hysteresis and can be used to characterize surface heterogeneity, roughness, and mobility.<ref>{{Cite web|url=https://blog.biolinscientific.com/how-to-measure-contact-angle-hysteresis|title=How to measure contact angle hysteresis?|last=Laurén|first=Susanna|website=blog.biolinscientific.com|language=en-us|access-date=2019-12-31}}</ref> Surfaces that are not homogeneous will have domains that impede motion of the contact line. The slide angle is another dynamic measure of hydrophobicity and is measured by depositing a droplet on a surface and tilting the surface until the droplet begins to slide. In general, liquids in the Cassie–Baxter state exhibit lower slide angles and [[Contact angle#Contact Angle Hysteresis|contact angle hysteresis]]  than those in the Wenzel state.{{Citation needed|date=January 2021}}