The theoretical description of contact angle arises from the consideration of a
thermodynamic equilibrium between the three
phases: the
liquid phase (L), the
solid phase (S), and the gas or
vapor phase (G) (which could be a mixture of ambient atmosphere and an equilibrium concentration of the liquid vapor). (The "gaseous" phase could be replaced by another
immiscible liquid phase.) If the solid–vapor
interfacial energy is denoted by , the solid–liquid interfacial energy by , and the liquid–vapor interfacial energy (i.e. the
surface tension) by , then the equilibrium contact angle is determined from these quantities by the
Young equation: \gamma_{\rm SG} - \gamma_{\rm SL} - \gamma_{\rm LG} \cos \theta_{\rm C}=0 \, The contact angle can also be related to the work of
adhesion via the
Young–Dupré equation: \gamma_{\rm LG} (1 + \cos \theta_{\rm C} )= \Delta W_{\rm SLG} \, where \Delta W_{\rm SLG} is the solid – liquid adhesion energy per unit area when in the medium G.
Modified Young’s equation The earliest study on the relationship between contact angle and surface tensions for sessile droplets on flat surfaces was reported by Thomas Young in 1805. A century later Gibbs proposed a modification to Young's equation to account for the volumetric dependence of the contact angle. Gibbs postulated the existence of a line tension, which acts at the three-phase boundary and accounts for the excess energy at the confluence of the solid-liquid-gas phase interface, and is given as: \cos\theta = \frac{\gamma_{\rm SG}-\gamma_{\rm SL}}{\gamma_{\rm LG}} + \frac{\kappa}{\gamma_{\rm LG}} \frac{1}{a} where is the line tension in
Newtons and is the droplet radius in meters. Although experimental data validates an affine relationship between the cosine of the contact angle and the inverse line radius, it does not account for the correct sign of and overestimates its value by several orders of magnitude.
Contact angle prediction while accounting for line tension and Laplace pressure With improvements in measuring techniques such as
atomic force microscopy,
confocal microscopy, and
scanning electron microscope, 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 confirmed that the modified Young's equation does not hold at the micro-nano scales. Jasper The advancing contact angle can be described as a measure of the liquid-solid cohesion while the receding contact angle is a measure of liquid-solid adhesion. The advancing and receding contact angles can be measured directly using different methods and can also be calculated from other wetting measurements such as force tensiometry (aka
Wilhemy-Plate method). Advancing and receding contact angles can be measured directly from the same measurement if drops are moved linearly on a surface. For example, a drop of liquid will adopt a given contact angle when static, but when the surface is tilted the drop will initially deform so that the contact area between the drop and surface remains constant. The "downhill" side of the drop will adopt a higher contact angle while the "uphill" side of the drop will adopt a lower contact angle. As the tilt angle increases the contact angles will continue to change but the contact area between the drop and surface will remain constant. At a given surface tilt angle, the advancing and receding contact angles will be met and the drop will move on the surface. In practice, the measurement can be influenced by shear forces and momentum if the tilt velocity is high. The measurement method can also be challenging in practice for systems with high (>30 degrees) or low ( \theta_{\rm c} = \arccos\left(\frac{r_{\rm A}\cos\theta_{\rm A} + r_{\rm R}\cos\theta_{\rm R}}{r_{\rm A}+r_{\rm R}}\right) where \begin{align} r_{\rm A} &= \sqrt[3]{ \frac{\sin^3\theta_{\rm A}}{2-3\cos\theta_{\rm A} + \cos^3 \theta_{\rm A}} } \\[4pt] r_{\rm R} &= \sqrt[3]{\frac{\sin^3\theta_{\rm R}}{2-3\cos\theta_{\rm R} + \cos^3 \theta_{\rm R}} } \end{align} Equations for the advancing and receding contact angles have been derived also based on a purely thermodynamic theory On a surface that is rough or contaminated, there will also be contact angle hysteresis, but now the local equilibrium contact angle (the Young equation is now only locally valid) may vary from place to place on the surface. According to the Young–Dupré equation, this means that the adhesion energy varies locally – thus, the liquid has to overcome local energy barriers in order to wet the surface. One consequence of these barriers is contact angle
hysteresis: the extent of wetting, and therefore the observed contact angle (averaged along the contact line), depends on whether the liquid is advancing or receding on the surface. Because liquid advances over previously dry surface but recedes from previously wet surface, contact angle hysteresis can also arise if the solid has been altered due to its previous contact with the liquid (e.g., by a
chemical reaction, or absorption). Such alterations, if slow, can also produce measurably time-dependent contact angles.
Effect of roughness to contact angles Surface roughness has a strong effect on the contact angle and wettability of a surface. The effect of roughness depends on if the droplet will wet the surface grooves or if air pockets will be left between the droplet and the surface. If the surface is wetted homogeneously, the droplet is in Wenzel state. In Wenzel state, adding surface roughness will enhance the wettability caused by the chemistry of the surface. The Wenzel correlation can be written as \cos\theta_m = r\cos\theta_Y where is the measured contact angle, is the Young contact angle and is the roughness ratio. The roughness ratio is defined as the ratio between the actual and projected solid surface area. If the surface is wetted heterogeneously, the droplet is in Cassie-Baxter state. The most stable contact angle can be connected to the Young contact angle. The contact angles calculated from the Wenzel and Cassie-Baxter equations have been found to be good approximations of the most stable contact angles with real surfaces.
Dynamic contact angles For liquid moving quickly over a surface, the contact angle can be altered from its value at rest. The advancing contact angle will increase with speed, and the receding contact angle will decrease. The discrepancies between static and dynamic contact angles are closely proportional to the
capillary number, noted Ca. ==Contact angle curvature==