# Surface energy

Editor-In-Chief: C. Michael Gibson, M.S., M.D. 

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Contact angle measurements can be used to determine the surface energy of a material. Here, a drop of water on glass.

Surface energy quantifies the disruption of intermolecular bonds that occurs when a surface is created. In the physics of solids, surfaces must be intrinsically less energetically favourable than the bulk of a material; otherwise there would be a driving force for surfaces to be created, and surface is all there would be (see sublimation (physics)). The surface energy may therefore be defined as the excess energy at the surface of a material compared to the bulk.

For a liquid, the surface tension (force per unit length) and the surface energy density are identical. Water has a surface energy density of 0.08 J/m2 and a surface tension of 0.08 N/m. However, in general, the surface energy of a solid is not equal to its surface tension.

Cutting a solid body into pieces disrupts its bonds, and therefore consumes energy. If the cutting is done reversibly (see reversible), then conservation of energy means that the energy consumed by the cutting process will be equal to the energy inherent in the two new surfaces created. The unit surface energy of a material would therefore be half of its energy of cohesion, all other things being equal; in practice, this is true only for a surface freshly prepared in vacuum. Surfaces often change their form away from the simple "cleaved bond" model just implied above. They are found to be highly dynamic regions, which readily rearrange or react, so that energy is often reduced by such processes as passivation or adsorption.

## Measuring the surface energy of a liquid

As first described by Thomas Young in 1805 in the Philosophical Transactions of the Royal Society of London, it is the interaction between the forces of cohesion and the forces of adhesion which determines whether or not wetting, the spreading of a liquid over a surface, occurs. If complete wetting does not occur, then a bead of liquid will form, with a contact angle which is a function of the surface energies of the system.

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Contact angle and surface energy measurements can be made using a contact angle goniometer.

Surface energy is most commonly quantified using a contact angle goniometer and a number of different methods.

Thomas Young described surface energy as the interaction between the forces of cohesion and the forces of adhesion which, in turn, dictate if wetting occurs. If wetting occurs, the drop will spread out flat. In most cases, however, the drop will bead to some extent and by measuring the contact angle formed where the drop makes contact with the solid the surface energies of the system can be measured.

### Young's equation

Young established the well-regarded Young's Equation which defines the balances of forces caused by a wet drop on a dry surface. If the surface is hydrophobic then the contact angle of a drop of water will be larger. Hydrophilicity is indicated by smaller contact angles and higher surface energy. (Water has rather high surface energy by nature; it is polar and forms hydrogen bonds). The Young equation gives the following relation,

$\gamma _{\mathrm {SL} }+\gamma _{\mathrm {LV} }\cos {\theta _{\mathrm {c} }}=\gamma _{\mathrm {SV} }\,$ where $\gamma _{\mathrm {SL} }$ , $\gamma _{\mathrm {LV} }$ , and $\gamma _{\mathrm {SV} }$ are the interfacial tensions between the solid and the liquid, the liquid and the vapor, and the solid and the vapor, respectively. The equilibrium contact angle that the drop makes with the surface is denoted by $\theta _{\mathrm {c} }$ . To derive the Young equation, normally the interfacial tensions are described as forces per unit length and from the one-dimensional force balance along the $x$ axis Young equation is obtained.

The Young equation assumes a perfectly flat surface, and in many cases surface roughness and impurities cause a deviation in the equilibrium contact angle from the contact angle predicted by Young's equation. Even in a perfectly smooth surface a drop will assume a wide spectrum of contact angles ranging from the so called advancing contact angle, $\theta _{\mathrm {A} }$ , to the so called receding contact angle, $\theta _{\mathrm {R} }$ . The equilibrium contact angle ($\theta _{\mathrm {c} }$ ) can be calculated from $\theta _{\mathrm {A} }$ and $\theta _{\mathrm {R} }$ as was shown by Tadmor  as,

$\theta _{\mathrm {c} }=\arccos \left({\frac {r_{\mathrm {A} }\cos {\theta _{\mathrm {A} }}+r_{\mathrm {R} }\cos {\theta _{\mathrm {R} }}}{r_{\mathrm {A} }+r_{\mathrm {R} }}}\right)$ where

$r_{\mathrm {A} }=\left({\frac {\sin ^{3}{\theta _{\mathrm {A} }}}{2-3\cos {\theta _{\mathrm {A} }}+\cos ^{3}{\theta _{\mathrm {A} }}}}\right)^{1/3}~;~~r_{\mathrm {R} }=\left({\frac {\sin ^{3}{\theta _{\mathrm {R} }}}{2-3\cos {\theta _{\mathrm {R} }}+\cos ^{3}{\theta _{\mathrm {R} }}}}\right)^{1/3}$ In the case of "dry wetting", one can use the Young-Dupré equation which is expressed by the work of adhesion. This method accounts for the surface pressure of the liquid vapor which can be significant. Pierre-Gilles De Gennes, a Nobel Prize Laureate in Physics, describes wet and dry wetting and how the difference between the two relate to whether or not the vapor is saturated .

## Measuring the surface energy of a solid

The surface energy of a liquid may be measured by stretching a liquid membrane (which increases the surface area and hence the surface energy density). However, such a method cannot be used to measure the surface energy of a solid because stretching of a solid membrane induces elastic energy in the bulk in addition to increasing the surface energy.

The surface energy of a solid is usually measured at high temperatures. At such temperatures the solid creeps and even though the surface area changes, the volume remains approximately constant. If $\gamma$ is the surface energy density of a cylindrical rod of radius $r$ and length $l$ at high temperature and a constant uniaxial tension $P$ , then at equilibrium, the variation of the total Gibbs free energy vanishes and we have

$\delta G=-P~\delta l+\gamma ~\delta A=0\qquad \implies \qquad \gamma =P{\cfrac {\delta l}{\delta A}}$ where $G$ is the Gibbs free energy and $A$ is the surface area of the rod:

$A=2\pi r^{2}+2\pi rl\qquad \implies \qquad \delta A=4\pi r\delta r+2\pi l\delta r+2\pi r\delta l$ Also, since the volume ($V$ ) of the rod remains constant, the variation ($\delta V$ ) of the volume is zero, i.e.,

$V=\pi r^{2}l={\text{constant}}\qquad \implies \qquad \delta V=2\pi rl\delta r+\pi r^{2}\delta l=0\implies \delta r=-{\cfrac {r}{2l}}\delta l~.$ Therefore, the surface energy density can be expressed as

$\gamma ={\cfrac {Pl}{\pi r(l-2r)}}~.$ The surface energy density of the solid can be computed by measuring $P$ , $r$ , and $l$ at equilibrium.