Surface stress

Surface stress was first defined by Josiah Willard Gibbs (1839-1903) as the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface. A suggestion is surface stress define as association with the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface instead of up definition. A similar term called “surface free energy”, which represents the excess free energy per unit area needed to create a new surface, is easily confused with “surface stress”. Although surface stress and surface free energy of liquid–gas or liquid–liquid interface are the same, they are very different in solid–gas or solid–solid interface, which will be discussed in details later. Since both terms represent a force per unit length, they have been referred to as “surface tension”, which contributes further to the confusion in the literature. Surface stress was first defined by Josiah Willard Gibbs (1839-1903) as the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface. A suggestion is surface stress define as association with the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface instead of up definition. A similar term called “surface free energy”, which represents the excess free energy per unit area needed to create a new surface, is easily confused with “surface stress”. Although surface stress and surface free energy of liquid–gas or liquid–liquid interface are the same, they are very different in solid–gas or solid–solid interface, which will be discussed in details later. Since both terms represent a force per unit length, they have been referred to as “surface tension”, which contributes further to the confusion in the literature. Definition of surface free energy is seemly the amount of reversible work d w {displaystyle dw} performed to create new area d A {displaystyle dA} of surface, expressed as: Gibbs was the first to define another surface quantity, different from the surface tension γ {displaystyle gamma } , 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 f i j {displaystyle f_{ij}} that relates the work associated with the variation in γ A {displaystyle gamma A} , the total excess free energy of the surface, owing to the strain d e i j {displaystyle de_{ij}} : Now 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 W 1 = 2 γ 0 A 0 {displaystyle W_{1}=2gamma _{0}A_{0}} , where γ 0 {displaystyle gamma _{0}} and A 0 {displaystyle A_{0}} are the excess free energy and area of each of new surfaces. For the second step, work ( w 2 {displaystyle w_{2}} ), 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, w 1 {displaystyle w_{1}} is equal to that needed to deform the bulk volume and the two surfaces. The difference w 2 − w 1 {displaystyle w_{2}-w_{1}} is equal to the excess work needed to elastically deform two surfaces of area A 0 {displaystyle A_{0}} to area A ( e i j ) {displaystyle A(e_{ij})} or: the work associated with the second step of the second path can be expressed as W 2 = 2 γ ( e i j ) A ( e i j ) {displaystyle W_{2}=2gamma (e_{ij})A(e_{ij})} , so that: These two paths are completely reversible, or W2 – W1 = W2 – W1. It means: Since d(γA) = γdA + Adγ, and dA = Aδijdeij. Then surface stress can be expressed as: