Continuum level description

Up to now, our equations have been continuum-level descriptions of mass flow. As with the other transport properties discussed in this chapter, however, the primary objective here is to examine the microscopic, or atomistic, descriptions, a topic that is now taken up. The transport of matter through a solid is a good example of a phenomenon mediated by point defects. Diffusion is the result of a concentration gradient of solute atoms, vacancies (unoccupied lattice, or solvent atom, sites), or interstitials (atoms residing between lattice sites). An equilibrium concentration of vacancies and interstitials are introduced into a lattice by thermal vibrations, for it is known from the theory of specific heat, atoms in a crystal oscillate around their equilibrium positions. Nonequilibrium concentrations can be introduced by materials processing (e.g. rapid quenching or irradiation treatment). 

Here the situation is very similar to that encountered in connection with the need for continuum (constitutive) models for the molecular transport processes in that a derivation of appropriate boundary conditions from the more fundamental, molecular description has not been accomphshed to date. In both cases, the knowledge that we have of constitutive models and boundary conditions that are appropriate for the continuum-level description is largely empirical in nature. In effect, we make an educated guess for both constitutive equations and boundary conditions and then normally judge the success of our choices by the resulting comparison between predicted and experimentally measured continuum velocity or temperature fields. Models derived from molecular theories, with the exception of kinetic theory for gases, are generally not available for comparison with the empirically proposed models. We discuss some of these matters in more detail later in this chapter, where specific choices will be proposed for both the constitutive equations and boundary conditions. 

The computations required for accurate modeling and simulation of large-scale systems with atomistic resolution involve a hierarchy of levels of theory quantum mechanics (QM) to determine the electronic states force fields to average the electronics states and to obtain atom based forces (FF), molecular dynamics (MD) based on such an FF mesoscale or coarse grain descriptions that average or homogenize atomic motions and finally continuum level descriptions (see Fig. 1). 

Since in this problem not only the limit but also the character of convergence matters we conclude that consistent homogenization of the micromodel should lead to a description in a broader functional space than is currently accepted. One interesting observation is that the concave part of the energy is relevant only in the region with zero measure where the singular, measure valued contribution to the solution is nontrivial (different from point mass). We remark that the situation is similar in fracture mechanics where a problem of closure at the continuum level can be addressed through the analysis of a discrete lattice (e.g. Truskinovsky, 1996). 

Solids undergoing martensitic phase transformations are currently a subject of intense interest in mechanics. In spite of recent progress in understanding the absolute stability of elastic phases under applied loads, the presence of metastable configurations remains a major puzzle. In this overview we presented the simplest possible discussion of nucleation and growth phenomena in the framework of the dynamical theory of elastic rods. We argue that the resolution of an apparent nonuniqueness at the continuum level requires "dehomogenization" of the main system of equations and the detailed description of the processes at micro scale. 

Future research in the modeling of surface-tension-driven flows can be directed towards the development of integrated multi-scale models that can more effectively represent the sub-continuum (essentially, molecular)-level transport features in the framework of continuum-based descriptions of the surface-tension-aided microfluidic transport. 

QM/MM methods -which enable one to simulate the core region quantum mechanically, to model bond formation or breaking, coupled with an atomistic level description of the surrounding regions, which include microstructural features and nanoarchitecture, together with continuum methods that effectively extend the material to infinity -while at time of writing are in their infancy, span hierarchical length scales from the electronic structure to a continuum and have the potential to truly Simulate the dirt ... 

With the advent of sophisticated simulation techniques, the physics of the flow-enhanced nucleation process at the molecular level are gradually being unraveled (see Chapter 6). The results of such investigations can serve to validate and/or improve continuum-level FIC models. Some of the most advanced of these are compared here in terms of the formulation of flow-enhanced nucleation kinetics. A description of flow-induced oriented structure formation and application to IM are discussed in Section 14.4.2 and Section 14.4.3, respectively. We focus on models that calculate the number density and dimensions of nuclei since this is necessary to predict morphological features beyond merely the degree of crystallization or the volume fraction of semicrystalline material. Therefore, approaches based on a (modified) Nakamura equation are left out of consideration. 

Simplified Descriptions Most existing continuum-level FlC models make use of the assumption that aU FIPs are active, n = 0. The activation terms in Equation (14.16) and Equation (14.18) then vanish since pf n = 0, /) = 0 infinitely thin precursors are unstable. The superscript/indicates that the distribution contains FIPs only pf= p - p ). The rate of change of the number density of FIPs can thus be written as... 

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