Relaxation, normal-mode

All in all, the Rouse model provides a reasonable description of polymer dynamics when the hydrodynamic interactions, excluded volume effects and entanglement effects can be neglected a classical example of its applicability is short-chain polymer melts. Since the Rouse model is exactly solvable for polymer chains, it represents a basic reference frame for comparison with more involved models of polymer dynamics. In particular, the decouphng of the dynamics of the Rouse chain into a set of independently relaxing normal modes is fundamental and plays an important role in other cases, such as more complex objects of study, or in other models, such as the Zimm model. 

FIGURE 24.7 Local segmental (upper panel) and normal mode (lower panel) relaxation times for 1,4-polyisoprene M = 11 kg/mol) [90], plotted versus specific volume. 

With the boundary conditions that the chain ends are free of forces, Eq. (13) is readily solved by cos-Fourier transformation, resulting in a spectrum of normal modes. Such solutions are similar, e.g. to the transverse vibrational modes of a linear chain except that relaxation motions are involved here instead of periodic vibrations. 

How can one hope to extract the contributions of the different normal modes from the relaxation behavior of the dynamic structure factor The capability of neutron scattering to directly observe molecular motions on their natural time and length scale enables the determination of the mode contributions to the relaxation of S(Q, t). Different relaxation modes influence the scattering function in different Q-ranges. Since the dynamic structure factor is not simply broken down into a sum or product of more contributions, the Q-dependence is not easy to represent. In order to make the effects more transparent, we consider the maximum possible contribution of a given mode p to the relaxation of the dynamic structure factor. This maximum contribution is reached when the correlator in Eq. (32) has fallen to zero. For simplicity, we retain all the other relaxation modes = 1 for s p. 

In a conventional relaxation kinetics experiment in a closed reaction system, because of mass conservation, the system can be described in a single equation, e.g., SCc(t) = SCc(0)e Rt where R = ((Ca) + ( C b)) + kh- The forward and reverse rate constants are k and k t, respectively. In an open system A, B, and C, can change independently and so three equations, one each for A, B, and C, are required, each equation having contributions from both diffusion and reaction. Consequently, three normal modes rather than one will be required to describe the fluctuation dynamics. Despite this complexity, some general comments about FCS measurements of reaction kinetics are useful. 

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. 

When a chain has lost the memory of its initial state, rubbery flow sets in. The associated characteristic relaxation time is displayed in Fig. 1.3 in terms of the normal mode (polyisoprene displays an electric dipole moment in the direction of the chain) and thus dielectric spectroscopy is able to measure the relaxation of the end-to-end vector of a given chain. The rubbery flow passes over to liquid flow, which is characterized by the translational diffusion coefficient of the chain. Depending on the molecular weight, the characteristic length scales from the motion of a single bond to the overall chain diffusion may cover about three orders of magnitude, while the associated time scales easily may be stretched over ten or more orders. 

Fig. 4.7 Temperature dependence of the mean relaxation time (r) divided by the rheological shift factor for the dielectric normal mode (plus) the dielectric segmental mode (cross) and NSE at Qinax=l-44 A (empty circle) and Q=1.92 A (empty square) [7] (Reprinted with permission from [8]. Copyright 1992 Elsevier)...

Section 5.3 The vibrations of CO adsorbed on the Cu(100) surface were examined using an asymmetric slab model of 4 layers with the atoms in the two bottommost layers fixed at bulk positions and all remaining atoms allowed to relax previous to the calculation of the normal modes. The supercell had c(2 x 2) surface symmetry, containing 2 metal atoms... 

If the M-H-M array is bent, the selection rules are relaxed from the linear case all three bands are allowed in the ir and Raman. Also, the form of the normal modes changes, in that the two symmetric modes (those associated with and V2) are mixed. 

Configurational relaxation in the absence of flow is governed by the normal modes and their corresponding relaxation times. The autocorrelation function for the end-to-end vector is a measure of the configurational memory ... 

The configurational response to flow depends upon which of the normal modes interact frictionally with the flow field. In simple shear the distribution envelope in the flow direction alone is altered, and only the N normal modes associated with the flow direction are active. The polymer contribution to the shear relaxation modulus for a system with v chains per unit volume is ... 

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

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