Interface Stresses in Nanostructured Multilayered Materials
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(1)
The total work needed to create a planar surface of area A is equal to YA. Gibbs [1] was the first to point out that there is another type of solid surface quantity that is associated with the reversible work per unit area needed to elastically stretch a pre-existing surface. The relationship between this quantity and y can be derived in the following manner. The elastic deformation of a solid surface can be expressed in terms of a surface elastic strain tensor eij, where ij = 1,2. Consider a reversible process that causes a small variation in the area of the surface through an infinitesimal elastic strain deij. One can define a surface stress tensor fij that relates the work associated with the variation in yA, the total excess free energy of the surface, to the strain dcij: d(yA) = A f1i deij.
(2)
497 Mat. Res. Soc. Symp. Proc. Vol. 405 01996 Materials Research Society
Equation (2) was first given by Shuttleworth [2]. Since d(yA) = y dA + A dy, and dA = A 8ij deij (where 8ij is the Kronecker delta), the surface stress can be expressed as (3)
fij = 7 8ij + aY/aij.
In contrast to the excess surface free energy y, which is a scalar, the surface stress fij is a second rank tensor. For a general surface, it can be referred to a set of principal axes such that the off-diagonal components are identically zero. Furthermore, the diagonal components are equal for a surface possessing a three-fold or higher rotation axis symmetry. This means that the surface stress for high symmetry surfaces is isotropic and can be taken as a scalar f = y + ay/ae.
(4)
INTERFACE STRESS As with the free solid surface, there is a stress associated with a interface between to solid phases. In fact, as pointed out by Brooks [3], a general interface has associated with it two interface stresses corresponding to the two phases that are separated by the interface that can be strained independently. For simplicity, the following discussion assumes that these interface stresses can be taken as scalars. Following Cahn and Larch6 [4], one can define an interface stress g corresponding to the reversible work per unit area needed to strain one of the phases relative to the other, and another interface stress h associated with the reversible work per unit area needed to equally stretch both phases. For an interface between two crystals, the strain associated with the interface stress g results in a change of the interface structure. As an example, consider an epitaxial thin film on a semi-infinite rigid substrate. Changing the misfit dislocation density at the interface allows the lattice parameter of the film to be varied while keeping the lattice parameter of the substrate fixed. The interface stress g can be interpreted as the specific surface work associated with changing the dislocation density. A simple model for this type of interface stress is now given. Consider as the reference state a noncoherent interface where both the film and substrate have their bulk lattice parameters. Let em be defined as the misfit strain equal to (as - af)
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