Superposition principle, molecular models

In principle, the time evolution of a particular linear superposition on the molecular base states will reflect a chemical process via the changes shown by the amplitudes. This represents a complete quantum mechanical representation of the chemical processes in Hilbert space. The problem is that the separability cannot be achieved in a complete and exact manner. One way to introduce a model that is able to keep as much as possible of the linear superposition principle is to use generalized electronic diabatic base functions. 

Theories of reactions between two molecules A and B, where the kinetics are practically diffusion controlled have been the subject of many studies which used continuous models (Fick s law or similar molecular systems [1-4]). In a general way, if we accept the notion of "molecular chaos" (fluctuations of forces applied on a molecule are only correlated during a very short period in front of displacement time of molecules) and if we use a superposition principle, the introduction of a space dependent chemical rate content, k(r), leads, for a simple reaction such as A + B C, to ... 

In Chapter 4, it was noted that linear viscoelastic behavior is observed only in deformations that are very small or very slow. The response of a polymer to large, rapid deformations is nonlinear, which means that the stress depends on the magnitude, the rate and the kinematics of the deformation. Thus, the Boltzmann superposition principle is no longer valid, and nonlinear viscoelastic behavior cannot be predicted from linear properties. There exists no general model, i.e., no universal constitutive equation or rheological equation of state that describes all nonlinear behavior. The constitutive equations that have been developed are of two basic types empirical continuum models, and those based on a molecular theory. We will briefly describe several examples of each type in this chapter, but since our primary objective is to relate rheological behavior to molecular structure, we will be most interested in models based on molecular phenomena. The most successful molecular models to date are those based on the concept of a molecule in a tube, which was introduced in Chapter 6. We therefore begin this chapter with a brief exposition of how nonlinear phenomena are represented in tube models. A much more complete discussion of these models will be provided in Chapter 11. 

R. M. Guedes. A viscoelastic model for a biomedical ultra-high molecular weight polyethylene using the time-temperature superposition principle. Polymer Testing 30 (3), 294-302 (2011). 

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