Surface stress

Surface stress was first defined by Josiah Willard Gibbs[1] (1839–1903) as the amount of the reversible work per unit area needed to elastically stretch a pre-existing surface.

Depending upon the convention used, the area is either the original, unstretched one which represents a constant number of atoms, or sometimes is the final area; these are atomistic versus continuum definitions.

Some care is needed to ensure that the definition used is also consistent with the elastic strain energy, and misinterpretations and disagreements have occurred in the literature.

Both terms represent an energy per unit area, equivalent to a force per unit length, so are sometimes referred to as "surface tension", which contributes further to the confusion in the literature.

The continuum definition of surface free energy is the amount of reversible work

, that is associated with the reversible work per unit area needed to elastically stretch a pre-existing surface.

, the total excess free energy of the surface due to a strain tensor

[5][2] An alternative approach is an atomistic one, which defines all quantities in terms of the number of atoms, not continuum measures such as areas.

For many metals the derivative is positive, but in other cases it is negative, for instance solid argon and some semiconductors.

Different methods have been used such as first principles, atomistic potential calculations and molecular dynamics simulations, with density functional theory most common.

Some metals such as aluminum are calculated to have fairly high, positive values (e.g. 0.82) indicating a strong propensity to contract, whereas others such as calcium are quite negative at -1.25, and others are close to zero such as cesium (-0.02).

Countering this, the atoms below (substrate) have a fixed in-plane spacing onto which the surface has to register.

One way to reduce the total energy is to have extra atoms in the surface, or remove some.

Note that since adsorption often depends strongly upon the environment, for instance gas pressure and temperature, the surface stress tensor will show a similar dependence.

Therefore surface contributions to the energy can become important at small sizes in nanoparticles.

If the energy of the surface atoms is lower when they are closer, this can be accomplished by shrinking the whole particle.

Combined these lead to a change in the lattice parameter that scales inversely with size.

One complication is that the changes in lattice parameter lead to more involved forms for nanoparticles with more complex shapes or when surface segregation can occur.

[24] Balancing this there are nominal angular gaps (disclinations) which are removed by an elastic deformation.

Instead of growing as a continuous thin film, a morphological instability can occur and the film can start to become very uneven, in many cases due to a breakdown of a balance between elastic and surface energies.

[27][28][4] The surface stress can lead to comparable wrinkling in nanowires,[29] and also a morphological instability in a thin film.