Internal state principle

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

The first term does not contribute to when BC is bound, but it would contribute if it were unbound. Again, one may not put e (or the first term) equal to zero until after scalar products are evaluated. It is in principle possible to solve the above equation for ba, by expanding both sides in the internal states of AB. But the expansion must include continuum (dissociation) states of AB, which has discouraged researchers from attempting this approach so far. However, some efforts are being made in this direction. 

Let us now consider the special case that the reactants are in an equilibrium distribution, both with reference to internal states and also translational states. It is for this special case that we obtain the same result by the method of Section V-C. Before making the assumption for the translational states, however, we stress that Eq. (232) refers to the chemical reaction only, and that we should in principle include terms for collisions without reaction as in Eq. (226). We return to a discussion of this point at the end of this section. 

Since the unperturbed evolution of the internal states a) is known in principle, it is convenient to... 

The computation of internal state densities and partition functions for polyatomic molecules is an essential task in the theoretical treatment of molecular gases. A first principles approach to the statistical thermodynamics of polyatomic gases requires the computation of the internal molecular energy levels based on an ab initio quantum mechanical (QM) determination of portions of the potential energy surface. Likewise, statistical theories of chemical reactions, such as Rice-Ramsberger-KasseUMarcus (RRKM) theory or transition state... 

One of the principal driving forces behind the current intense activity in state-selected chemistry is the desire to find cheaper methods of separating isotopes. Because the internal states of atoms and molecules are not identical for different isotopic forms of the same species (i.e., isotomers), it is frequently possible to excite just one of a mixture of isotomers photochemically. In principle, methods based on selective excitation could lead to complete separation in a single step. In practice, 100% separation is very difficult to achieve (in several important cases, it is not even required), and consequently the word enrichment has been preferred to the word separation in the title to this section. Nevertheless, impressive enrichment factors have been reported... 

In principle, photoactivation allows the selection of a well-defined reactant state. In practice, the quasicontinuous absorption of polyatomic molecules, arising from the high internal state density and the thermal distribution of absorbers over closely packed rovibrational levels, usually frustrate this desirable objective. In single-photon excitation relying on fast internal conversion from an electronically excited state, the distribution is unlikely to be completely random, or to be entirely independent of excitation wavelength, but it is not possible to control the initial nonrandomness. The results of such experiments (see Section 1.4.3) are consistent with rapid energy randomization but no direct tests of the random lifetime assumption have been made. 

The equipartition principle is a classic result which implies continuous energy states. Internal vibrations and to a lesser extent molecular rotations can only be understood in terms of quantized energy states. For the present discussion, this complication can be overlooked, since the sort of vibration a molecule experiences in a cage of other molecules is a sufficiently loose one (compared to internal vibrations) to be adequately approximated by the classic result. 

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

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