Contribution of Surface Tension to Nanoferroics Properties

The surface energy and surface tension in particular is one of the most essential factors defining the nanoferroics properties. The following considerations confirm this statement. 

We write Gibbs free energy for multicomponent order parameter t) (which are the components of magnetization, polarization or strain) for the case of monodomain ferroic. The latter is valid for small enough sizes ( 100 nm), where the domain structure disappears [1-3]. For the materials with inversion center in paraphrase, the bulk (Gy) and surface (Gs) parts of free energy can be written as [4]  

Here Oy are the components of stress tensor, the upper index S in Eq. (3.1b) shows, that the coefficients describe the surface energy. 

The form of tensor L is determined by the nanoparticle shape (e.g. spherical, ellipsoidal or cylindrical) and mechanical boundary conditions. It should be noted that surface stress tensor components depend on the chemical properties of the nanoparticle ambient material and the presence of oxide interface layer [6]. In the case of chemically clean surface under the thermodynamic equilibrium with environment the diagonal components have to be positive like the surface tension for liquids, although in general case may have both positive and negative sign. 

For nanoparticles of spherical, cylindrical or ellipsoidal shapes with diameter less than 100 nm, the stress ct can be considered as homogeneous and represented in the above form with L tensor. Collecting the squares of corresponding order parameters in the Eq. (3.1), one can rewrite the coefficient URij (renormalized by the striction) in the form 

---