Relaxation spectra rate equations

Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors ( modes ) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules that it is possible to solve the kinetic equation for every module in the explicit form. All such solvable networks are described. The obtained multiscale approximations, that we call dominant systems are... 

Which solvent motions represented in the bend force spectrum [see Equation (6)] are important in inducing the rate limiting transition from the OH stretch into the HOD bend For the same D2O model, Marti et al. (69) have found that the D20 librational spectrum is peaked at 400 cm 1 with a FWHM of 300 cm 1. Thus, the dominant motions in the bend force spectrum at 530 cm 1 responsible for the calculated relaxation are the solvent librations. 

The treatment presented thus far applies to systems where only one independent variable is subject to relaxation. Frequently, however, m > 1) such variables are needed to describe the relaxation properties of interest. Under these circumstances, a set of m relaxation equations of the type given in Eq. [4] can be established. Accordingly, m relaxation times are determined and in a specific relaxation process each relaxation time will contribute its share to the overall effect in proportion to a corresponding amplitude. The ensemble of relaxation times and amplitudes is called the relaxation spectrum of the process under consideration. It reflects the underlying molecular rate mechanism. Thus, in principle, experimental relaxation spectrometry offers a way to elucidate kinetic mechanisms. 

Understanding the structure and function of biomolecules requires insight into both thermodynamic and kinetic properties. Unfortunately, many of the dynamical processes of interest occur too slowly for standard molecular dynamics (MD) simulations to gather meaningful statistics. This problem is not confined to biomolecular systems, and the development of methods to treat such rare events is currently an active field of research. - If the kinetic system can be represented in terms of linear rate equations between a set of M states, then the complete spectrum of M relaxation timescales can be obtained in principle by solving a memoryless master equation. This approach was used in the last century for a number of studies involving atomic... 

In contrast to simple elastic solids and viscous liquids, the situation with polymeric fluids is somewhat more complicated. Polymer melts (and most adhesives are composed of polymers) display elements of both Newtonian fluid behavior and elastic solid behavior, depending on the temperature and the rate at which deformation takes place. One therefore characterizes polymers as viscoelastic materials. Furthermore, if either the total strain or the rate of strain is low, the behavior may be described as one of linear or infinitesimal viscoelasticity. In such a case, the stress-deformation relationship (the constitutive equation) involves not just a single time-independent constant but a set of constants called the relaxation spectrum,(2) and this, too, may be determined from a single stress relaxation experiment, or an experiment involving small-amplitude oscillatory motion. 

Most chemical processes including micellar kinetics involve several steps and are characterized by several relaxation times (relaxation spectrum). The maximum number of observable relaxation times is equal to the number of independent rate equations that can be written for the system investigated. This number is equal to that of chemical species minus the number of mass-balance equations. > ... 

An exact derivation and implementation of the constitutive equations needed for this multimode model is not given here the interested reader is refeued to References 13 and 67. What is important to note is that the method just desctihed allows the determination of the relaxation spectrum from a simple set of uniaxial tensile or compression experiments at different strain rates and does not require lengthy relaxation experiments. The nonlinearity in stress is the same as used in the flow stress itself... 

The long-range coupling via the flow field which only decreases with 1/r leads to a qualitatively different behavior from that of the Rouse model. Equation (75) is approximately solved by transformation to Rouse normal coordinates. Its solution [6,91] leads to the spectrum of relaxation rates... 

In Fig. 15.27, the transient extensional viscosity of a low-density polyethylene, measured at 150 °C for various extensional rates of strain, is plotted against time (Munstedt and Laun, 1979). Qualitatively this figure resembles the results of the Lodge model for a Maxwell model in Fig. 15.26. For small extensional rates of strain (qe < 0.001 s ) 77+(f) is almost three times rj+ t). For qe > 0. 01 s 1 r/+ (f) increases fast, but not to infinite values, as is the case in the Lodge model. The drawn line was estimated by substitution of a spectrum of relaxation times of the polymer (calculated from the dynamic shear moduli, G and G") in Lodge s constitutive equation. The resulting viscosities are shown in Fig. 15.28 after a constant value at small extensional rates of strain the viscosity increases to a maximum value, followed by a decrease to values below the zero extension viscosity. 

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