Constitutive equations molecular modeling approach

Continuum mechanics is the approach most frequently used to develop constitutive equations and predict material functions. This approach to constitutive modeling has established the framework for all constitutive modeling, including molecular modeling approaches and thermodynamic and stochastic methods. 

The proposed mechanisms of models to explain the drag reduction phenomenon are based on either a molecular approach or fluid dynamical continuum considerations, but these models are mainly empirical or semi-empirical in nature. Models constructed from the equations of motion (or energy) and from the constitutive equations of the dilute polymer solutions are generally not suitable for use in engineering applications due to the difficulty of placing numerical values on all the parameters. In the absence of a more generally accurate model, semi-empirical ones remain the most useful for applications. 

Time constants are related to the relaxation times and can be found in equations based on mechanical models (phenomenological approaches), in constitutive equations (empirical or semiempirical) for viscoelastic fluids that are based on either molecular theories or continuum mechanics. Equations based on mechanical models are covered in later sections, particularly in the treatment of creep-compliance studies while the Bird-Leider relationship is an example of an empirical relationship for viscoelastic fluids. 

There are several approaches leading to the formulation of these non-linear relationships. Unfortunately it is not possible to summarize them here because the field is very rich in the variety of systems and behaviors. Consequently, such a summary hes beyond the scope of this study. An excellent and detailed review is given in Ref. (Beris Edwards, 1994). However, let us mention that many attempts have been made to bring together continuum mechanics theories and molecular models in order to formulate appropriate constitutive equations for... 

For the study of excited states and molecular properties, the EOM-CC model constitutes a conceptually simple approach closely related to the Cl model, with the emphasis shifted away fiom excitations from determinants towards excitations from a more general state. However, the EOM-CC approach is somewhat restricted in the sense that it can be applied only to the standard coupled-cluster mcxlels such as CCSD and CCSDT. For the calculation of molecular properties and excitation energies, the response-fuiKtion approach - based on the solution of the time-dependent Schrodinger equation - constitutes a more general framework, also encompassing, for example, the quadratic Cl model of Section 13.8.2 and the iterative hybrid CC2 and CC3 models of Section 14.6.2 [20,21]. 

The DE Constitutive Equations. The DE model (52-56) made a major breakthrough in polymer viscoelasticity in that it provided an important new molecular physics based constitutive relation (between the stress and the applied deformation history). This section outlines the DE approach that built on the reptation-tube model developed above and gave a nonlinear constitutive equation, which in one simplified form gives the K-BKZ equation (70,71). The model also inspired a significant amount of experimental work. One should begin by... 

Master equations have been used to describe relaxation and kinetics of clusters. The first approaches were extremely approximate, and served primarily as proof-of-principle. ° Master equations had been used to describe relaxation in models of proteins somewhat earlier and continue to be used in that context. " More elaborate master-equation descriptions of cluster behavior have now appeared. These have focused on how accurate the rate coefficients must be in order that the master equation s solutions reproduce the results of molecular dynamics simulations and then on what constitutes a robust statistical sample of a large master equation system, again based on both agreement with molecular dynamics simulations and on the results of a full master equation.These are only indications now of how master equations may be used in the future as a way to describe and even control the behavior of clusters and nanoscale systems of great complexity. ... 

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