Surface Energy - Measuring The Surface Energy of A Solid

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). In that case, in order to increase the surface area of a mass of liquid by an amount, δA, a quantity of work, γδA, is needed (where γ is the surface energy density of the liquid). 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 γ is the surface energy density of a cylindrical rod of radius and length at high temperature and a constant uniaxial tension, 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 is the Gibbs free energy and is the surface area of the rod:

$A = 2\pi r^2 + 2\pi r l \qquad \implies \qquad \delta A = 4\pi r\delta r + 2\pi l\delta r + 2\pi r\delta l$

Also, since the volume of the rod remains constant, the variation of the volume is zero, i.e.,

$V = \pi r^2 l = \text{constant} \qquad \implies \qquad \delta V = 2\pi r l \delta r + \pi r^2 \delta l = 0 \implies \delta r = -\cfrac{r}{2l}\delta l ~.$

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, and at equilibrium.

This method is valid only if the solid is isotropic, meaning the surface energy is the same for all crystallographic orientations. While this is only strictly true for amorphous solids (glass) and liquids, isotropy is a good approximation for many other materials. In particular, if the sample is polygranular (most metals) or made by powder sintering (most ceramics) this is a good approximation.

In the case of single-crystal materials, such as natural gemstones, anisotropy in the surface energy leads to faceting. The shape of the crystal (assuming equilibrium growth conditions) is related to the surface energy by the Wulff construction. The surface energy of the facets can thus be found to within a scaling constant by measuring the relative sizes of the facets.

Read more about this topic: Surface Energy

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