Local density gradients

The inability to analyze flow-related material defects is a serious drawback shear and normal stresses associated with the foaming process can be responsible for a variety of defects such as cell structure irregularities, large voids, excessive curvature or even splitting of the foam-free surface, local density gradients and leakages from molds. 

As a result of long-range fluctuations, the local density will vary with position in the classical Landan-Ginzburg theory of fluctuations this introdnces a gradient tenn. A Ginzburg number N is defined (for a... 

One of the most usefiil applications of the mean free path concept occurs in the theory of transport processes in systems where there exist gradients of average but local density, local temperature, and/or local velocity. The existence of such gradients causes a transfer of particles, energy or momentum, respectively, from one region of the system to another. 

I he function/(r) is usually dependent upon other well-defined functions. A simple example 1)1 j functional would be the area under a curve, which takes a function/(r) defining the curve between two points and returns a number (the area, in this case). In the case of ni l the function depends upon the electron density, which would make Q a functional of p(r) in the simplest case/(r) would be equivalent to the density (i.e./(r) = p(r)). If the function /(r) were to depend in some way upon the gradients (or higher derivatives) of p(r) then the functional is referred to as being non-local, or gradient-corrected. By lonlrast, a local functional would only have a simple dependence upon p(r). In DFT the eiK igy functional is written as a sum of two terms ... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

The heat flow density q of a material depends on the local temperature gradient, according to Fourier s law ... 

Finally, we relate the gradient of the local density ViPq(fi) to the pair correlation functions. For this purpose we take the gradient of the expansion (5)... 

Nonlocal density gradient corrections (GC)-local spin density (LDA) approximation. 

The local density approximation is highly successful and has been used in density functional calculations for many years now. There were several difficulties in implementing better approximations, but in 1991 Perdew et al. successfully parametrised a potential known as the generalised gradient approximation (GGA) which expresses the exchange and correlation potential as a function of both the local density and its gradient ... 

This discrepancy arises primarily from the fact that spontaneous liquid flows will always develop in any hquid even without artificial stirring (e.g., under the action of density gradients caused by local temperature or concentration fluctuations). This phenomenon has been termed natural convection. Electrochemical reactions reinforce natural convection, since the concentrations of substances involved in the reaction will change near the electrode surfaces, and also since heat is evolved. Gas evolution attending the reactions has a particularly strong effect on naturaf convection. 

Bray, M. R., Deeth, R. J., Paget, V. J., Sheen, P. D., 1996, The Relative Performance of the Local Density Approximation and Gradient Corrected Density Functional Theory for Computing Metal-Ligand Distances in Werner-Type and Organometallic Complexes , Int. J. Quant. Chem., 61, 85. 

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