Shuttleworth equation

If a surface area is increased by stretching, the increase in free energy (which in that case is TdA) is given by the surface tension plus the change of the surface tension with the elastic strain. The Shuttleworth equation indicates that, in order to know the surface stress, we need to know the surface tension but also the dependence of 7 on the elastic strain ee. 

Surface stress, (7, and surface free energy, j, can be related using the Shuttleworth equation [20],... 

The cantilever deflection results from a change in the surface stress as the electrode potential is varied. The surface stress is related to the surface energy by the Shuttleworth equation ... 

The question of how to describe a solid surface is still being developed. To give some examples, Grafov recently derived a generalized Shuttleworth equation. The present concept was critically discussed by Lang and Heusler and Guidelli. Experimental efforts to verify the thermodynamic approach were published, e.g., by Haiss, Nichols, Sass, and Charle. ... 

Because the effects become less difficult to predict than the more realistic case of constant heating rate sintaing, we shall limit our discussion to isothermal sintering. The linear densification rate of the free matrix (i.e., the glass) can be found from the models for viscous sintaing described in Chapter 8. The Mackenzie and Shuttleworth equation is used because of its simple form, giving... 

The quantity Y is also called surface stress. With this formulation of the Shuttleworth equation, the surface stress is given as both the sum of surface tension and the change of surface tension y with the elastic strain Eehstk-... 

For solids, crystal face, there is a unique value of asv (which is a scalar) while ySv depends also on the orientation along the face. Moreover, ctsv is always a positive quantity (breaking bonds needs work to be done) while ysv can be either positive or negative (Nolfi and Johnson 1972). For high symmetry surfaces, the surface tension is related to the surface energy by the equation (Shuttleworth 1950) ... 

For the rate of densification of Bingham s solid containing n pores of the same size per unit volume, Mackenzie and Shuttleworth derived the following equation ... 

The significance of the latter parameters can be realised if one considers the initial drop impact. In a fibrous structure such as paper a liquid is unable to discern differences in pore depth until entering the bulk structure. Prior to penetration, following dissipation of mechanical forces, drop wetting and spreading of an aqueous ink jet drop will be primarily determined by the combined effects of chemical heterogenity and physical roughness. The combination of these effects can be approximated (24) in terms of Cassie and Baxter (25) and Shuttleworth and Bailey s (26) equations ... 

Equation (593) is called Shuttleworth s equation and shows how we can calculate the surface stress. In theory, if we want to calculate the Gibbs free energy per unit area from surface tension and surface stress we may write... 

In accordance with Equations 1.1.1 23,-33, the basic thermodynamic parameters are calculated (Moore and Shuttleworth, 1963 Golovin and Lotmintsev, 1981) ... 

In Eqs. 23 and 24, Xg and yg are initial values in Eq. 25, a = (8V2) . Equation 24 is plotted as the dashed curve in Fig. 7. In view of the profound difference in the structure, the densification curve is surprisingly close to that found by Mackenzie and Shuttleworth [24], As the cylinder model densifies, the cylinders become shorter and thicker and, when the relative density reaches p 0.94, neighboring cylinders touch. At that point, the unit cell contains an isolated pore, so the remainder of the densification is described by the MS model. ... 

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