The relaxation time

In the chemical relaxation methods for following fast reactions, the rate coefficients are determined from a parameter called the relaxation time, x. The physical significance of x will be considered in terms of the step-function model, but the same quantity is measured by the other main group of relaxation methods, the stationary methods. 

The full line in Fig. 5 represents the concentration of species i at times before and after the application of the step-function (dashed line). c° is a time-independent reference concentration, while Cj is the corresponding (final) equilibrium value which might be time-dependent. Then 

The relationship between the reciprocal relaxation time and the rate coefficients for equilibrium (1), k 2 and derived quite simply. 

This illustrates an unusual feature about relaxation times their expressions always contain the sum of a contribution from the forward and the backward reaction. This has the convenient result that x is independent of the sign of the perturbation, i.e., it is immaterial for a determination of x whether the disturbance causes the equilibrium to move to the left or to the right. 

RELAXATION TIMES IN TERMS OF EQUILIBRIUM CONCENTRATIONS AND RATE COEFFICIENTS FOR SINGLE-STEP REACTIONS cf. Ref. 33) 

---

It is important to recognize the approximations made here the electric field is supposed to be sulficiently small so that the equilibrium distribution of velocities of the ions is essentially undisturbed. We are also assuming that the we can use the relaxation approximation, and that the relaxation time r is independent of the ionic concentration and velocity. We shall see below that these approximations break down at higher ionic concentrations a primary reason for this is that ion-ion interactions begin to affect both x and F, as we shall see in more detail below. However, in very dilute solutions, the ion scattering will be dominated by solvent molecules, and in this limiting region A2.4.31 will be an adequate description. 

This is the famous Lorentzian fimction which is very often found for spectra of radicals in solution. In order to detenuiue the relaxation times and T2, a series of EPR spectra is recorded with the MW power varying... 

The design of a pulsed EPR spectrometer depends heavily on tlie required pulse lengdi and pulse power which in turn are mainly dictated by the relaxation times of tlie paramagnetic species to be studied, but also by the type of experiment perfomied. When pulses of the order of a few nanoseconds are required (either to compete... 

When relaxation of the internal motion during the collision is fast compared with the slow collision speed v, or when the relaxation time is short compared with the collision time, the kinetic energy operator... 

The simulations also revealed that flapping motions of one of the loops of the avidin monomer play a crucial role in the mechanism of the unbinding of biotin. The fluctuation time for this loop as well as the relaxation time for many of the processes in proteins can be on the order of microseconds and longer (Eaton et al., 1997). The loop has enough time to fluctuate into an open state on experimental time scales (1 ms), but the fluctuation time is too long for this event to take place on the nanosecond time scale of simulations. To facilitate the exit of biotin from its binding pocket, the conformation of this loop was altered (Izrailev et al., 1997) using the interactive molecular dynamics features of MDScope (Nelson et al., 1995 Nelson et al., 1996 Humphrey et al., 1996). 

We assume that the unbinding reaction takes place on a time scale long ( ompared to the relaxation times of all other degrees of freedom of the system, so that the friction coefficient can be considered independent of time. This condition is difficult to satisfy on the time scales achievable in MD simulations. It is, however, the most favorable case for the reconstruction of energy landscapes without the assumption of thermodynamic reversibility, which is central in the majority of established methods for calculating free energies from simulations (McCammon and Harvey, 1987 Elber, 1996) (for applications and discussion of free energy calculation methods see also the chapters by Helms and McCammon, Hermans et al., and Mark et al. in this volume). 

T(f) corresponds to the actual temperature at the time t, At is the integration time step, and the relaxation time represents the strength of the coupling (smaller values mean stronger coupling to the bafli). If the coupling is too strong (r smaller... 

Of the adjustable parameters in the Eyring viscosity equation, kj is the most important. In Sec. 2.4 we discussed the desirability of having some sort of natural rate compared to which rates of shear could be described as large or small. This natural standard is provided by kj. The parameter kj entered our theory as the factor which described the frequency with which molecules passed from one equilibrium position to another in a flowing liquid. At this point we will find it more convenient to talk in terms of the period of this vibration rather than its frequency. We shall use r to symbolize this period and define it as the reciprocal of kj. In addition, we shall refer to this characteristic period as the relaxation time for the polymer. As its name implies, r measures the time over which the system relieves the applied stress by the relative slippage of the molecules past one another. In summary. 

Suppose we divide the flow segments into classes according to relaxation times and index the various states by the subscript i. Thus the relaxation time and the component of shear stress borne by the segments in class i are and Fj, respectively. The applied shear force is related to the Fj s through... 

This simple derivation gives us the desired result, a relationship between the relaxation time and the degree of polymerization ... 

In connection with a discussion of the Eyring theory, we remarked that Newtonian viscosity is proportional to the relaxation time [Eqs. (2.29) and (2.31)]. What is needed, therefore, is an examination of the nature of the proportionality between the two. At least the molecular weight dependence of that proportionality must be examined to reach a conclusion as to the prediction of the reptation model of the molecular weight dependence of viscosity. 

Aside from the side chains, the movement of the backbone along the main reptation tube is still given by Eq. (2.67). With the side chains taken into account, the diffusion velocity must be decreased by multiplying by the probability of the side-chain relocation. Since the diffusion velocity is inversely proportional to r, Eq. (2.67) must be divided by Eq. (2.69) to give the relaxation time for a chain of degree of polymerization n carrying side chains of degree of polymerization n ... 

The Newtonian viscosity is given by the product of the relaxation time and the Hookean modulus. This result was anticipated in the discussion of Eqs. (2.29) and (2.31). 

Viscosity is considerably more sensitive to temperature than elasticity. By varying the temperature, the relaxation time of the polymer will be changed. Hence different mechanical response might be expected on a fixed laboratory time scale for samples examined at different temperatures. 

If we combine Eqs. (3.98) and (3.94), we can eliminate from the latter to obtain an expression for the relaxation time of mode p which is free of any reference to the subchain ... 

Now the relaxation times for all higher modes of vibration can be expressed relative to n ... 

Evaluate the relaxation time associated with each of these molecular weights and verify that the molecular weight dependence of r corresponds to the value given in Sec. 2.13. 

The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

The relaxation times, and desctibe the times required to reheve stress on the cessation of strain and to reheve strain on the cessation of stress, respectively. The fuU Oldroyd tensor requires knowledge of eight material properties. 

The relaxation time (arbitrarily defined as the time taken for the stress to relax to half its original value) can be calculated from the power-law creep data as follows. Consider a bolt which is tightened onto a rigid component so that the initial stress in its shank is CTj. In this geometry (Fig. 17.3(c)) the length of the shank must remain constant - that is, the total strain in the shank e must remain constant. But creep strain can rqjiace elastic strain e - , causing the stress to relax. At any time t... 

Accordingly, the relaxation time of a C atom will increase the fewer hydrogen atoms it bonds to and the faster the motion of the molecule or molecular fragment in which it is located. From this, it can be deduced that the spin-lattice relaxation time of C nuclei provides information concerning four molecular characteristics ... 

The number of bonded H atoms if all parts within a molecule move at the same rate (the same Tc for all C atoms), the relaxation times Tj decrease from C//via C// to C//j in the ratio given by ... 

Knowing and p (or k), the relaxation time (r) can be estimated. The determination of relaxation time allows the rate of charge dissipation to be calculated for any process obeying Ohm s law. 

The tank potential is independent of the relaxation time of the liquid, since the tank acts as a Faraday pail (3-5.1.2). 

---