Nonlocal theory

In this chapter we consider the extension of continuum solvent models to nonlocal theories in the framework of the linear response approximation (LRA). Such an approximation is mainly applicable to electrostatic solute-solvent interactions, which usually obey the LRA with reasonable accuracy. The presentation is confined to this case. 

Generally, e and are tensorial quantities. They reduce to scalars in the case of isotropic media, and then describe the longitudinal polarization effects. Our presentation is devoted to this simple transparent case. Complications introduced by anisotropic phenomena are not considered they do not change the main idea of nonlocal theory only making the notation cumbersome. 

According to the nonlocal theory the vector fields E(r), I)(7) and P(r) in Equations (1.119) can be treated as time dependent and they obey the Maxwell equations [1], Within the LRA, most general expressions are valid ... 

Nevertheless, the concept of spatial dispersion provides a general background for a qualitative understanding of those solvation effects which are beyond the scope of local continuum models. The nonlocal theory creates a bridge between conventional and well developed local approaches and explicit molecular level treatments such as integral equation theory, MC or MD simulations. The future will reveal whether it can survive as a computational tool competitive with these popular and more familiar computational schemes. 

Formulation of a full dynamic nonlocal theory is not practical due to intrinsic difficulties in separation of advective and diffusional fluxes, aggravated in nonlocal theory by impossibility to reduce external forces to boundary integrals. Even in the framework of local theory, computational difficulties of a straightforward approach make it so far impossible to span the entire range from nanoscopic scale of molecular interactions to observable macroscopic scales. 

Research by Kyokong and others [28] lent credibility to Ilcewicz and Wilson s hypothesis. They applied Eringen s nonlocal theory to solid poplar (Populus tremuloides) joints bonded with resorcinol adhesive, substituting the average vessel lumen diameter of aspen (100 pm) as the characteristic dimension. They were able to show that the nonlocal theory using this dimension correlated very closely with the fracture toughness of the joints as determined by classic (local) theory. 

Semi-phenomenological Nonlocal Theory of the Double Layer... 

As space is a notion that is disconnected from the definition of a pole, the Formal Graph approach avoids the problem of transmission speed in having recourse to a nonlocalized theory of influence between entities that will be exposed in Chapter 7. 

In case that all gradients being not too large, we may neglect higher orders of smallness in interplanar spacings at the Taylor expansion. Otherwise, we will arrive at a nonlocal theory of Cahn-Hilliard type. Then elementary mathematics leads to the following expression for the flux... 

Immediately we face the problem of interpreting the square-root operator on the right-hand side in Eq. [46]. Using, for example, a Taylor expansion would lead to an equation containing all powers of the derivative operator and thus to a nonlocal theory. Such theories are very difficult to handle, and they present an unattractive version of the Schrodinger equation with space and time coordinates appearing in an unsymmetrical form. In the interest of mathematical simplicity, we return to Eq. [40], making the transformation to a quantum mechanical operator representation ... 

Optical Response of Nanostructures Microscopic Nonlocal Theory ByK.Cho... 

R. Fuchs and F. Claro, Multipolar response of small metallic spheres nonlocal theory, Phys. Rev. B3 (8), in-nn (i987). 

The basic problem of nonlocal theories is to find an appropriate e(,k,co). Several authors have dealt with this problem for both geometries, a dipole above a surface and a dipole close to a metal nanoparticle.It is certainly beyond the scope of this chapter to go into detail of those theories. However, let us briefly note that the results are miscellaneous. For example, in the case of a dipole close to a metal nanosphere, Leung predicts one to two orders of magnitude less energy transfer to the nanoparticle in the case where the dipolar transition is energetically lower than the particle plasmon resonance. Ekardt and Penzar predict exactly the opposite. 

B. K. Peterson, K. E. Gubbins, G. S. Heffelfinger, U. M. B. Marconi, F. Swol, Lennard-Jones fluids in cylindrical pores Nonlocal theory and computer simulation, J. Chem. Phys. 88 (1988) 6487-6500. 

Eringen s interest in the micromorphic theory continued when he joined the faculty of Princeton University in 1966. During the early Princeton years, he concentrated on the application of this theory to turbulence, liquid crystals, polymers, suspensions, biomechanics, and composite materials. He has always kept a deep interest in questions related to the foundations of continuum mechanics and thermodynamics. In recent years, Eringen has been the most articulate and active proponent of the nonlocal theory of continua with applications to dislocation theory, fracture problems, surface physics, composite materials, and turbulence. 

---