Note: the following is taken extensively from reference [4].
Gibbs was among the first to study the thermodynamics of surfaces. Our interest in this is to understand the formation of surfaces and how they can change under the application of forces. Gibbs, of course, took an energy approach, developing the concept of the surface free energy, g. We will now follow the development that leads to an understanding that this parameter is the same as the surface tension (or stress) that we've introduced earlier. g is also the work per unit area required to create a planar surface of area A, a process which is thermodynamically reversible.
g = dW/dA
The SI units of g would be J/m2. Alternatively, the units are N/m, which we recognize as the units of a stress.
Gibbs also pointed out that for solids there is another important quantity to be considered. It is the work per unit area needed to elastically stretch (or compress) a pre-existing surface. A surface strain tensor eij (where i,j = 1,2) is used to express the elastic deformation of the solid surface. An infinitesimal elastic strain deij is then introduced in Gibbs' approach to cause a small variation in the surface area. A surface stress tensor fi,j is then defined that relates the work producing a variation in the total excess surface free energy, gA, resulting from the strain dei,j. The relationship is
(9)
d(gA) = Afi,j dei,j
where the repeated indicies represent a summation over the indicies. To understand equation (9), consider the following figure.
Figure 7. A cube is stretched and split in half. There are two reversible paths one can take to go from the initial state (top left) to the final state (lower right). In one case, the cube is stretched by an amount dx (top right) and then split. In the other case (lower left), the cube is split and then the two pieces are stretched.
Although the surface stress f is a second-rank tensor, for most surfaces it refers to a set of principle axes with the off-diagonal components of the tensor equal to zero. In the case of high-symmetry surfaces, the diagonal components are all equal, i.e. the surface stress is isotropic. In this case, the surface stress can be viewed as a scalar quantity (14)
f = g + dg/de.
This indicates that the difference between the surface free energy and the surface stress is the change in the free energy per unit change in the elastic strain of the surface.
Both f and g are referred to in the literature as "surface tension." All it does is lead to confusion. A better way is to consider f as the force per unit length (or work per unit area) exerted on a surface during elastic deformation; and to consider g as the force per unit length (or work per unit area) exerted on a surface during plastic deformation.
The physical origin of surface stress lies in understanding the chemical bonding of atoms at the surface. The atomic configuration of atoms at the surface is somewhat different than it is in the bulk. Surface atoms therefore have a different arrangement than if they were constrained to remain in the same positions as if they were in the interior of the solid. Therefore, the interior atoms are viewed as exerting a stress on the surface (moving them out of the positions they would otherwise occupy).
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