Surface stress tensor

Here 4 are the shear stresses in the phases a and b, respectively, with the interface of tension y. By introducing the shear stress s through the Newton s equation of liquid motion and by considering a uniform dilatation = y = 0, the surface stress tensor becomes scalar and Eq. (3.39) can be rewritten as... 

Surface Rheology. Surface rheology deals with the functional relationships that link the dynamic behavior of a surface to the stress that is placed on the surface. The complex nature of these relationships is often expressed in the form of a surface stress tensor, P8. Both elastic and viscous resistances oppose the expansion and deformation of surface films. The isotropic (diagonal) components of this stress tensor describe the di-latational behavior of the surface element. The components that are off-diagonal relate the resistance to changing the shape of the surface element to the applied shear stresses. Equation 7 demonstrates the general form of the surface stress tensor. 

To demonstrate the details of the surface stress tensor, Ps, a simplified surface geometry is presumed. For a soap film that is stretched on a wire frame parallel to the x—y plane, the surface stress tensor, confined to this geometry (Figure 10b), reduces to... 

A model that pictures the interface as a thin layer with high viscosity (the Bous-sinesq model [2]) has gained widespread acceptance. For such a layer, one has to write down a special phenomenological correlation that connects the surface stress tensor with the rate of surface strain tensor [25). Paper [26] uses this model to examine the problem of the influence of surfactants on the dynamics of a free interface. 

Angle between surface tension vectors Surface coverage Surface coverage, ad-atoms Surface stress tensor Degree of dissociation... 

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. 

Coefficients aij T) depend explicitly on temperature T. Coefficients afj, aijki, are supposed to be temperature independent, constants giju and Vijkimn determine the magnitude of the gradient energy. Tensors gijki, Oijki and positively defined. Tensor is the surface excess elastic moduli, p, p is the surface stress tensor [81,82], is the surface piezoelectric or piezomagnetic tensor [67], qijki are the bulk striction coefficients Ciju are components of elastic stiffness tensor [83]. 

To demonstrate spontaneous flexoeffect contribution to the nanoferroic properties, we hereafter neglect the surface excess elastic moduli, surface stress tensor and surface piezoelectric effect contributions into the surface energy (4.11). We consider mechanically free nanoparticles without misfit dislocations, which should lead to the external flexoeffect only. The contribution of misfit dislocations into the flexoelectric effect in thin films has been considered in details by Catalan et al. [72, 73]. 

However, in nanowires and nanorods (Fig. 4.41a), the intrinsic surface stress exists spontaneously due to the surface curvature and typically does not relax. Surface stress is inversely proportional to the wire radius and directly proportional to the surface stress tensor (similar to Laplace surface tension). The intrinsic surface stress should depend both on the growth conditions and the surface termination morphology [136, 137]. Although surface tension appears even for the case of non-reconstructed geometrical surfaces due to the surface curvature [137], surface reconstruction should affect the surface tension value or even be responsible for the appearance of surface stresses [138,139]. 

Xy is the intrinsic surface stress tensor that determines the excess pressure exerted on the solid under the curved surface [137]. We used the isotropic approximation (X . = (x8y, where x is the scalar coefficient. Due to the lack of experimental measurements of the x value for EuTiOs, we select an experimentally reasonable value based on the data for surface tension coefficients measured in ferroelectric ABO3 perovskites. The values reported for other ABOs-type perovskites vary in the range 3-30 N/m 36.6 N/m for PbTiOs [142] (or even 50N/m [143]), 2.6-10 N/m for PbTiOs and BaTiOs nanowires [144], and 9.4 N/m for Pb(Zr,Ti)03 [145]. Here we use averaged (x value 10 N/m, which is close to that extracted recently from Pb(Zr,Ti)03 sponges tetragonality temperature dependence. For comparison, we characterize the effect of (x = 30 N/m (higher end of the reported values) on the multiferroic phase transition. 

The components of the surface stress tensor depend upon the extent and the rate of surface deformation, in a relationship involving the resistance of the surface to both changes in area and shape. Either of these two types of resistance can be expressed in a modulus which combines an elastic with a viscous term. This leaves us with four formal rheological coefficients which suffice for a description of the surface stress. Two of these, viz., the surface dilatational elasticity, and viscosity, measure the surface resistance to changes in area, the other two, viz., the surface shear elasticity, e, and viscosity, r describe the... 

For two-phase flow, the conservation of mass and momentum requires corresponding transmission conditions at the interface, the so-called jump condi-ti(Mis. At this point, we assume a non-contaminated and fully mobile interface. The former corresponds to negligible surface mass densities, i.e., no occurrence of adsorbed species, while the latter means constant surface tension, i.e., no Marangoni stresses due to surface gradient of the surface tension, as well as zero surface viscosities. In this case, the surface stress tensor reduces to with constant a. Since no phase change is considered, the additional jump condi-ticHis read... 

Calcnlations of the surface stress tensor and surface energy of the (111) snrfaces of iridium, platinum and gold, R. J. Needs and M. Mansfield, J. Phys Condensed Matter, 1989,1, 7555. 

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