Propagator, first-order

In this chapter the regimes of mechanical response nonlinear elastic compression stress tensors the Hugoniot elastic limit elastic-plastic deformation hydrodynamic flow phase transformation release waves other mechanical aspects of shock propagation first-order and second-order behaviors. 

First order stimulated Stokes scattering experiences an exponential gain in intensity as the fields propagate tlirough the scattering medium. This is given by the expression [75]... 

Forward Analysis In this type of analysis, we are interested in the propagation of initial perturbations Sxq along the flow of (1), i.e., in the growth of the perturbations 5x t xo) = (xo -h Sxq) — xq. The condition number K,(t) may be defined as the worst case error propagation factor (cf. textbook [4]), so that, in first order perturbation analysis and with a suitable norm j ... 

A parameter such as a rate constant is usually obtained as a consequence of various arithmetic manipulations, and in order to estimate the uncertainly (error) in the parameter we must know how this error is related to the uncertainties in the quantities that contribute to the parameter. For example, Eq. (2-33) for a pseudo-first-order reaction defines k, which can be determined by a semilogarithmic plot according to Eq. (2-6). By a method to be described later in this section the uncertainty in itobs (expressed as its variance associated with cb. Thus, we need to know how the errors in fcobs and cb are propagated into the rate constant k. 

It frequently happens that we plot or analyze data in terms of quantities that are transformed from the raw experimental variables. The discussion of the propagation of error leads us to ask about the distribution of error in the transformed variables. Consider the first-order rate equation as an important example ... 

The propagation reaction itself is of the first order with respect to the monomer concentration. This was demonstrated by measuring the propagation rate constants Kp at different monomer concentrations (98). 

As for the dependence of the polymerization rate V on the monomer concentration some authors have also found first-order kinetics (84, 90, 96, 99), but sometimes deviations from the first order were observed (38, 51, 88) that may be connected with a change in the number of propagation centers with monomer concentration. 

The propagation rate constant did not depend on the monomer concentration which corresponds to the first-order propagation step. The activation energy of the propagation calculated according to the variation of Kp with temperature was found to be 6.5 0.5 kcal/mole. 

For all one-component catalysts the first order of the propagation rate on the monomer concentration is observed. It can be consistent with two cases ... 

This expression suggests a rate-controlling step in which RM reacts with an intermediate. If so, [Int] °c [RM] /2. To be consistent with this, the initiation step should be first-order in [RM] and the termination step second-order in [Int]. Since O2 is not involved in the one propagation step deduced, it must appear in the other, because it is consumed in the overall stoichiometry. On the other hand, given that one RM is consumed by reaction with the intermediate, another cannot be introduced in the second propagation step, since the stoichiometry [Eq. (8-3)] would disallow that. Further, we know that the initiation and propagation steps are not the reverse of one another, since the system is not well-behaved. From this logic we write this skeleton ... 

Biernath et al. concluded that phenolic novolac and epoxidized cresol novolac cure reactions using triphenylphosphine as the catalyst had a short initiation period wherein the concentration of phenolate ion increased, followed by a (steady-state) propagation regime where the number of reactive phenolate species was constant.85 The epoxy ring opening was reportedly first order in the steady-state regime. 

The exceptionally low propagation constants of t-butyl and of phenyl methacrylate are notable. The polymerization of the former monomer was thoroughly examinedS5). At temperatures even as high as 25 °C this reaction, when performed in THF in the presence of salts depressing dissociation of ion-pairs, yields polymers of highly uniform size. The reaction is strictly first order in growing polymers and in monomer, and no... 

Table 3. Pseudo-first order rate constant kapp of propagation of lithium o-methoxy styrene in toluene at 20 °C. The values collected in the first two columns are taken from Table 2 of Ref. 69)...

However, the mechanisms by which the initiation and propagation reactions occur are far more complex. Dimeric association of polystyryllithium is reported by Morton, al. ( ) and it is generally accepted that the reactions are first order with respect to monomer concentration. Unfortunately, the existence of associated complexes of initiator and polystyryllithium as well as possible cross association between the two species have negated the determination of the exact polymerization mechanisms (, 10, 11, 12, 13). It is this high degree of complexity which necessitates the use of empirical rate equations. One such empirical rate expression for the auto-catalytic initiation reaction for the anionic polymerization of styrene in benzene solvent as reported by Tanlak (14) is given by ... 

Here Uj and Uj are Cartesian unit vectors, a) and j3) are localized orbitals that are doubly occupied in the HF ground state, jm) and n) are virtual orbitals. Rq is the position vector of the local gauge origin assigned to orbital a) and = (r — R ) x p is the angular momentum relative to Re- Superscript 1 denotes terms to first order in the fluctuation potential, and = [A — is the principal propagator at the zero energy... 

The correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... 

---