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The Shuttleworth equation, or Shuttleworth effect is the relation first derived by R. Shuttleworth that relates surface energy to surface tension. The Shuttleworth equation reads

$$\Upsilon = \gamma + \frac{d \gamma}{d \epsilon}$$

with $$\Upsilon$$ the surface tension in N/m, $$\gamma$$ the surface energy in J/m^2 and $$\epsilon$$ the linear surface strain. This suggest that the surface tension does not nessesarly needs to be equal to the surface energy when the surface energy is a function of the surface strain. (Source)

The source of this equation is that the surface tension is defined as the linear scalar to change in Helmholtz free energy by stretching a surface $$dF = \Upsilon dA$$ (note: the surface energy is defined the energy per surface it cost to create new surface by cutting) and into when stretching a surface both energy need to be put into creating new surface area $$ \gamma dA $$ and stretching the already existing surface $$ \frac{d\gamma}{d\epsilon} dA$$. This can be shown formally by applying the chain rule on change in Helmholtz free energy is and simplifying $$ dF = d(\gamma A) = A d\gamma + \gamma dA = (\frac{d\gamma}{d\epsilon} + \gamma) dA $$. Thus by using this result with the definition of the surface tension the Shuttleworth equation is acquired.

Because the surface energy for fluids is not strain dependent (source), the surface tension and the surface energy can be used interchangeably. But for condensed solids (source) and soft solids (source) the second term is generally non-zero (source). Observed effects include
 * Nagative surface tension in small beats
 * Absorption
 * Self assembly
 * jump in the surface tension on the contact line resulting in a tangential force on the solid.
 * A positive relation between the surface tension and the surface strain.

Other forms of the Shuttleworth equation
The last derivation of the Shuttleworth equation can be done more generally to include the possibility of direction depended strain surface energy. Doing this gives a tensor expression for the surface tension. (source)

$$\Upsilon_{ij} = \gamma \delta_{ij} + \delta \gamma /\delta \epsilon_{ij}$$

$$\mathbf{\Upsilon} = \gamma \mathbf{I} + \delta \gamma /\delta \mathbf{\epsilon}$$

Another way to express the surface tension that is more appropriate in the context of finite deformable solids. By defining two surface energies $$\gamma_R$$ energy per surface area of the reference configuration and $$\gamma_C$$ energy per surface area of the current configuration (hence $$\gamma_C J^s = \gamma_R$$) another expression of the shuttleworth equation can be derived (Source). This is that the surface tension is

$$\mathbf{\Upsilon} = \frac{\delta \gamma_R}{\delta \mathbf{\epsilon}} = \frac{\delta (\gamma_C J^s)}{\delta \mathbf{\epsilon}} $$

$$ = \gamma_C \frac{\delta J^s}{\delta \mathbf{\epsilon}} + J^s \frac{\delta \gamma_C}{\delta \mathbf{\epsilon}} $$

When the current configuration is equal to the reference configuration ($$J^s = 1$$,$$\delta J^s / \delta \epsilon=1$$ and $$\gamma_R = \gamma_C$$) the original expression for the Shuttleworth equation is regained.

Role of Shuttleworth equation in condensed solid
nagavtive surface tension elastic instabilities [37], surface segregation [38], surface adsorption [39,40], surface reconstruction [41,42], nanostructuration [43] self-assembly [44,45]

Role of Shuttleworth equation in soft solids
jump in the surface tension young laplace