Surface stress

The continuum definition of surface free energy is the amount of reversible work dw performed to create new area dA of surface, expressed as: :dw = \gamma dA In this definition the number of atoms at the surface is proportional to the area. Gibbs was the first to define another surface quantity, different from the surface free energy \gamma, that is associated with the reversible work per unit area needed to elastically stretch a pre-existing surface. In a continuum approach one can define a surface stress tensor f_{ij} that relates the work associated with the variation in \gamma A, the total excess free energy of the surface due to a strain tensor e_{ij} :Af_{ij}=d(\gamma A)/de_{ij} = A d\gamma/de_{ij} +\gamma dA/de_{ij} In general there is no change in area for shear, which means that for the second term on the right i=j and dA/de_{ij}=A\delta_{ij}, using the Kronecker delta. Cancelling the area then gives :f_{ij}=d\gamma /de_{ij}+\delta_{ij} \gamma called the Shuttleworth equation. A conventional liquid cannot sustain strains, ==Physical origins of surface stress==

The origin of surface stress is the difference between bonding in the bulk and at a surface. The bulk spacings set the values of the in-plane surface spacings, and consequently the in-plane distance between atoms. However, the atoms at the surface have a different bonding, so would prefer to be at a different spacing, often (but not always) closer together. If they want to be closer, then d\gamma /de_{ij} will be positive—a tensile or expansive strain will increase the surface energy. For many metals the derivative is positive, but in other cases it is negative, for instance solid argon and some semiconductors. The sign can also strongly depend upon molecules adsorbed on the surface. If these want to be further apart that will introduce a negative component. == Surface stress values ==

Theoretical calculations The most common method to calculate the surface stresses is by calculating the surface free energy and its derivative with respect to elastic strain. Different methods have been used such as first principles, atomistic potential calculations and molecular dynamics simulations, with density functional theory most common. A large tabulation of calculated values for metals has been given by Lee et al. Typical values of the surface energies are 1-2 Joule per metre squared (Jm^{-2}), with the trace of the surface stress tensor g_{ij} in the range of -1 to 1 Jm^{-2}. 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). ==Surface stress effects==

Whenever there is a balance between a bulk elastic energy contribution and a surface energy term, surface stresses can be important. Surface contributions are more important at small sizes, so surface stress effects are often important at the nanoscale. Surface structural reconstruction As mentioned above, often the atoms at a surface would like to be either closer together or further apart. 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. The misregistry with the underlying bulk is accommodated by having partial partial dislocations between the first two layers. The silicon (111) is similar, with a 7x7 reconstruction with both more atoms in the plane and some added atoms (called adatoms) on top. Different is the case for anatase (001) surfaces. Here the atoms want to be further apart, so one row "pops out" and sits further from the bulk. Adsorbate-induced changes in the surface stress When atoms or molecules are adsorbed on a surface, two phenomena can lead to a change in the surface stress. One is a change in the electron density of the atoms in the surface, which changes the in-plane bonding and thus the surface stress. A second is due to interactions between the adsorbed atoms or molecules themselves, which may want to be further apart (or closer) than is possible with the atomic spacings in the surface. 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. or x-ray diffraction. This phenomenon has sometimes been written as equivalent to the Laplace pressure, also called the capillary pressure, in both cases with a surface tension. This is not correct since these are terms that apply to liquids. 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. Stabilization of decahedral and icosahedral nanoparticles Also in the area of nanoparticles, surface stress can play a significant role in the stabilization of decahedral nanoparticle and icosahedral twins. In both cases an arrangement of internal twin boundaries leads to lower energy surface energy facets. Balancing this there are nominal angular gaps (disclinations) which are removed by an elastic deformation. While the main energy contributions are the external surface energy and the strain energy, the surface stress couples the two and can have an important role in the overall stability. Deformation and instabilities at surfaces During thin film growth, there can be a balance between surface energy and internal strain, with surface stress a coupling term combining the two. 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. and also a morphological instability in a thin film. ==See also==