Constitutive equation general molecular theory

Nernst s equation is timeless. Theories of the mechanism of electrode reaction may change as a consequence of the availability of new experimental results and new ideas for interpreting them. However, thermodynamic treatments involve no molecular assumptions. They depend only on the validity of the two great generalizations of experience that constitute the first two laws of thermodynamics. Therefore, conclusions reached by applying them are not expected to change. 

In network models the molecular arguments supply a form for the constitutive equation, but do not provide the detailed connections to molecular structure. As such, they provide a bridge between molecular theories which incorporate specific structural information in rather specific flow situations and continuum models which can generalize such information to arbitrary flows. 

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. 

Doi first proposed the generalized dynamic equations for the concentrated solution of rod-like polymers. Such constitutive equations can be derived from the molecular theory developed by Doi and Edwards (1986). The basis for the molecular theory is the Smoluchowski equation or Fokker-Planck equation in thermodynamics with the mean field approximation of molecular interaction. 

In chemical kinetics the concept of the order of a reaction forms the basis of a kinematics which constitutes a frame for most of the molecular theories of chemical reactions. The fundamental magnitudes of this kinematics are the concentrations and the specific rate constants. In simple cases only the time enters as an independent variable, whereas in a diffusion process both time and space are involved. Diffusion processes are generally described in terms of diffusion coefficients, volume concentrations and thermodynamic potential or activity factors. Partial volume factors and friction coefficients associated with the components of the diffusing mixture are also essential in the description. A feature of the macro-dynamical theory is that it covers any region of concentration. Especially simple equations are connected with the differential diffusion process (diffusion with small concentration differences), for which the different coefficients or factors mentioned above are practically constant. 

Finally, more accurate differential and integral constitutive equations were presented, and their successes and failures in de-st bing experimental data, were discussed. No single nonlinear constitutive equation is best for all purposes, and thus one s choice of an appropriate constitutive equation must be guided by the problem at himd, the accuracy with which one wishes to solve the problem, and the effort one is willing to expend to solve it. Generally differential models of the Maxwell type are easier to implement numerically, and some are available in fluid mechanics codes. Also, some cmistitutive equations are better founded in molecular theory, as discussed in Chiqpter 11. 

New mathematical techniques [22] revealed the structure of the theory and were helpful in several derivations to present the theory in a simple form. The assumption of small transient (elastic) strains and transient relative rotations, employed in the theory, seems to be appropriate for most LCPs, which usually display a small macromolecular flexibility. This assumption has been used in Ref [23] to simplify the theory to symmetric type of anisotropic fluid mechanical constitutive equations for describing the molecular elasticity effects in flows of LCPs. Along with viscoelastic and nematic kinematics, the theory nontrivially combines the de Gennes general form of weakly elastic thermodynamic potential and LEP dissipative type of constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank elasticity in liquid crystalline polymers. It should be mentioned that this theory is suitable only for monodomain molecular nematics. Nevertheless, effects of Frank (orientation) elasticity could also be included in the viscoelastic nematody-namic theory to describe the multidomain effects in flows of LCPs near equilibrium. 

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. 

Material Functions—Linear Viscoelasticity. One of the most important aspects of both the phenomenological and the molecular theories of viscoelasticity is the ability to characterize the material functions. The material functions are the properties that allow one to relate the stress response to a strain (deformation) history and vice versa through a constitutive equation. In linear viscoelasticity theory, generally isotropic descriptions are dealt with that is, the properties are the same in all directions. However, a material may be anisotropic and still have properties that vary with the direction of the test (7). Here only the isotropic case is considered and it is recognized that straight forward extensions can be made to the anisotropic case. In addition, only homogeneous materials, for which the properties are the same at all points within the material, only are discussed. 

In this section, we discuss the assumptions which constitute the foundation of the theory, and derive final equations as rigorously as possible. We also give a theoretical description of the most general case of spin exchange, that in a multi-component system where both intra-and inter-molecular processes take place. The theory presented in this section involves only homonuclear systems of spins. An extension over individual heteronuclear systems should be quite straightforward. 

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