Surface stress tensor general form

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. 

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. 

The measurement of the work needed to increase the surface area of a solid material (e.g., an electrode metal) is more difficult. The work required to form unit area of new surface by stretching under equilibrium conditions is the surface stress (g1 ) which is a tensor because it is generally anisotropic. For an isotropic solid the work, the generalized surface parameter , or specific surface energy (ys) is the sum of two contributions ... 

The expression —gwxw y (SI units N/m2) is an averaged momentum flow per unit area, and so comparable to a shear stress A force in the direction of the y-axis acts at a surface perpendicular to the a -axis. Terms of the general form —gwf-wij are called Reynolds stresses or turbulent stresses. They are symmetrical tensors. In a corresponding manner, the energy equation (3.135), contains a turbulent heat flux of the form... 

The minus sign in this equation is a matter of convention t(n) is considered positive when it acts inward on a surface whereas n is the outwardly directed normal, andp is taken as always positive. The fact that the magnitude of the pressure (or surface force) is independent of n is self-evident from its molecular origin but also can be proven on purely continuum mechanical grounds, because otherwise the principle of stress equilibrium, (2 25), cannot be satisfied for an arbitrary material volume element in the fluid. The form for the stress tensor T in a stationary fluid follows immediately from (2 59) and the general relationship (2-29) between the stress vector and the stress tensor ... 

General Extensions. - Bader applied ideas of AIM to the atomic force microscope (AFM). In a quantum system, the force exerted on the tip is the Ehrenfest force, a force that is balanced by the pressure exerted on every element of its surface, as determined by the quantum stress tensor. The surface separating the tip from the sample is an IAS. Thus the force measured in the AFM is exerted on a surface determined by the boundaries separating the atoms in the tip from those in the sample, and its response is a consequence of the atomic form of matter. This approach is contrasted with literature results that equate it to the Hellmann-Feynman forces exerted on the nuclei of the atoms in the tip. 

X10. The next three rows present the viscosity rj, the surface tension, and its tenqterature dependence, in the liquid state. The next properties are the coefficient of linear thermal expansion a and the sound velocity, both in the solid and in the liquid state. A number of quantities are tabulated for the presentation of the elastic properties. For isotropic materials, we list the volume compressihility k = —(l/V)(dV/dP), and in some cases also its reciprocal value, the bulk modulus (or compression modulus) the elastic modulus (or Young s modulus) E the shear modulus G and the Poisson number (or Poisson s ratio) fj,. Hooke s law, which expresses the linear relation between the strain s and the stress a in terms of Young s modulus, reads a = Ee. For monocrystalline materials, the components of the elastic compliance tensor s and the components of the elastic stiffness tensor c are given. The elastic compliance tensor s and the elastic stiffness tensor c are both defined by the generalized forms of Hooke s law, a = ce and e = sa. At the end of the list, the tensile strength, the Vickers hardness, and the Mohs hardness are given for some elements. 

To evaluate the normal component of the stress balance, we must first evaluate V n. This is slightly more subtle than it may seem. Because we are calculating everything in general vector/tensor form, we note that the surface gradient operator in (8-61) can be expressed in the form... 

Two approaches, mechanical and thermodynamical, exist for the theoretical description of general curved interfaces and membranes. The first approach originates from the classical theory of shells and plates, reviewed in Refs. 202 and 204. The surface is regarded as a two-dimensional continuum whose deformation is described in terms of the rate-of-strain tensor and the tensor of curvature. In addition, the forces and the force moments acting in the interface are expressed by the tensors of the interfacial stresses, , and moments (torques), M. Figure 11 illustrates the physical meaning of the components of the latter two tensors. Usually, they are expressed in the form... 

---