Surface Stress - Thermodynamics of Surface Stress

Thermodynamics of Surface Stress

Definition of surface free energy is seemly the amount of reversible work performed to create new area of surface, expressed as:

Gibbs was the first to define another surface quantity, different from, that is associated with the reversible work per unit area needed to elastically stretch a pre-existing surface. Surface stress can be derived from surface free energy as followed:

One can define a surface stress tensor that relates the work associated with the variation in, the total excess free energy of the surface, owing to the strain :

Now let's consider the two reversible paths showed in figure 0. The first path (clockwise), the solid object is cut into two same pieces. Then both pieces are elastically strained. The work associated with the first step (unstrained) is, where and are the excess free energy and area of each of new surfaces. For the second step, work, equals the work needed to elastically deform the total bulk volume and the four (two original and two newly formed) surfaces.

In the second path (counter-clockwise), the subject is first elastically strained and then is cut in two pieces. The work for the first step here, is equal to that needed to deform the bulk volume and the two surfaces. The difference is equal to the excess work needed to elastically deform two surfaces of area to area or:

the work associated with the second step of the second path can be expressed as, so that:

These two paths are completely reversible, or W₂ – W₁ = W₂ – W₁. It means:

Since d(γA) = γdA + Adγ, and dA = Aδ_ijde_ij. Then surface stress can be expressed as:

Where δ_ij is the Kronecker delta and e_ij is elastic strain tensor. Differently to the surface free energy γ, which is a scalar, surface stress f_ij is a second rank tensor. However, for a general surface, set of principle axes that are off-diagonal components are identically zero. Surface that possesses a threefold or higher rotation axis symmetry, diagonal components are equal. Therefore, surface stress can be rewritten as a scalar:

Now it can be easily explained why f and γ are equal in liquid-gas or liquid-liquid interfaces. Due to the chemical structure of liquid surface phase, the term ∂γ/∂e always equals to zero meaning that surface free energy won’t change even if the surface is being stretched. However, ∂γ/∂e is not zero in solid surface due do the fact that surface atomic structure of solid are modified in elastic deformation.