Normal modes of relaxation

It is useful at this point to introduce the compact notation that (, ) denotes the scalar product of two vectors thus (S,-,Sj) = 0, and (S,-,Sj) = 1. In this style, equation (2.12) becomes 

Equation (2.12) may be separated into a series of independent components, one for each value of j. We write 

If we sum (2.16) over all j, we get equation (2.12), for each i. It is easy to see that 2,(ft )j=0 for all j= 0, since (Sq =0 this means that each of these independent processes simply shuffles molecules among the energy levels, each with its own rate constant A [76.P2], and that the evolution of the population of any individual state is given by 

3 One could arrive at the same conclusion by taking a Markov chain approach to the problem of calculating the evolution, see e.g. [79.R3]. 

The of equations (2.16) and (2.17) are dependent on the starting distribution n(0), and the larger the numerical magnitude of the elements of any the greater the contribution of the process with rate constant Xj to the overall relaxation. I have written equation (2.16) as a product of two terms (M ), x Cj, the first of which is the ith element of the th normal mode, and the second of which is the flux for that mode, i.e. 

---

Furthermore, it may be seen that for all the normal modes of relaxation, including the most rapid, the freely jointed chain model and the Rouse model are identical if we set n = N + 1 that is, the relaxation time xp of the pth normal mode of a freely-jointed chain is the same as that of a Rouse marcromolecule composed of N + 1 subchains, each of mean square end-to-end length b2. Moreover, for the special choice a = 0, Eq. (10) is true for arbitrarily large departures from equilibrium. We thus seem to have confirmed analytically the discovery of Verdier24 that quite short chains executing a stochastic process described by Eqs. (1) and (3) on a simple cubic lattice display Rouse relaxation behavior. Of course, Verdier s Monte Carlo technique permits study of excluded volume effects, quite beyond the range of our present efforts. 

I have previously used the term normal modes of reaction in this context [71.M] the same term has been used in rather different contexts on other occasions [68.B2 69.H] for the time being, at least until their properties have been investigated in more detail, it is perhaps preferable to be more cautious and to refer to the quantities (SoliCFy), as simply perturbed normal modes (of relaxation). 

The wavelength of the torsion normal mode with relaxation time r = 1 ns is A >50 bp for a >3.8x10 12 dyn-cm [from Eq. (4.34)]. Thus, the shortest torsion normal modes resolved in the FPA have wavelengths extending over about five full turns of the helix. The rms angular displacement of a base pair around its helix axis is about 18° at t= 1 ns and increases without bound as t goes to infinity. 

In this respect, another insufficiency of Lodge s treatment is more serious, viz. the lack of specification of the relaxation times, which occur in his equations. In this connection, it is hoped that the present paper can contribute to a proper valuation of the ideas of Bueche (13), Ferry (14), and Peticolas (13). These authors adapted the dilute solution theory of Rouse (16) by introducing effective parameters, viz. an effective friction factor or an effective friction coefficient. The advantage of such a treatment is evident The set of relaxation times, explicitly given for the normal modes of motion of separate molecules in dilute solution, is also used for concentrated systems after the application of some modification. Experimental evidence for the validity of this procedure can, in principle, be obtained by comparing dynamic measurements, as obtained on dilute and concentrated systems. In the present report, flow birefringence measurements are used for the same purpose. 

In order to show that this procedure leads to acceptable results, reference is briefly made to the normal coordinate transformation mentioned at the end of Section 2.2. By this transformation the set of coordinates of junction points is transformed into a set of normal coordinates. These coordinates describe the normal modes of motion of the model chain. It can be proved that the lowest modes, in which large parts of the chain move simultaneously, are virtually uninfluenced by the chosen length of the subchains. This statement remains valid even when the subchains are chosen so short that their end-to-end distances no longer display a Gaussian distribution in a stationary system [cf. a proof given in the appendix of a paper by Ham (75)]. As a consequence, the first (longest or terminal) relaxation time and some of the following relaxation times will be quite insensitive for the details of the chain... 

The time evolution of a system may also be characterized according to the degree of perturbation from its equilibrium state. Linear theories hold if local equilibrium prevails, that is, each volume element of the non-equilibrium system can still be unambiguously defined by the usual set of (local) thermodynamic state variables. Often, a crystal is in (partial) equilibrium with respect to externally predetermined P and 7j but not with external component chemical potentials pik. Although P, T, and nk are all intensive functions of state, AP relaxes with sound velocity, A7 by heat conduction, and A/ik by matter transport. In solids, matter transport is normally much slower than the other modes of relaxation. 

Recent numerical experiments by the method of molecular dynamics have shown that, for a chain model consisting of particles joined by ideally rigid bonds, the Van der Waals interactions of chain units cause only a little change in the dependence of relaxation times on the wave vector of normal modes of motions, i.e. in the character and shape of the relaxation spectrum. It was found that for the model chain the important relationship... 

Figure 11. Relaxation, 4>(r), of the center of energy is plotted for wave packets propagated by the normal modes of cytochrome c hydrated by 400 water molecules (circles) and myoglobin (squares). Curve is a stretched exponential, Eq. (33), with p = 2v = 0.52, the value fit to the computed energy diffusion data for cytochrome c plotted in Fig. 10, and time constant, t — 11 ps.

Rotation and relaxation. If we move the hydrogen atom towards Silicon without symmetry constraint, we obtain the transition state TS2. The bridged hydrogen atom is in a plane at an angle of 25° with respect to the plane of SiNH. This structure has an imaginary frequency, the associated normal mode of which leads to the planar HSiNH species. 

Studies have been made on the terminal relaxation, chain diffusion, and chain normal modes of the fast component in HAPB by simulations (Brodeck et al., 2010 Diddens et al., 2011 Moreno and Colmenero, 2008), neutron scattering (Niedzwiedz et al., 2007), and dielectric relaxation experiments (Arrese-Igor et al., 2012). Several remarkable properties of the chain dynamics of the fast component have been found by these studies. 

In view of the remarkable changes of dispersion and molecular weight dependence of normal modes of unentangled chains of the fast component in HAPB found by MD simulations and dielectric relaxation experiment, such chain normal modes are highlighted as Property (iii), which is summarized as follows. 

The chain normal modes of unentangled fast component polymer in HAPB have dynamics totally changed from the Rouse dynamics seen in the pure polymer, especially at low temperatures where there is large disparity in mobility of the two components. The relaxation time, fi = tr, becomes much longer... 

---