Relaxation modes

Berezhkovskii A M and Zitserman V Yu 1992 Multidimensional activated rate processes with slowly relaxing mode Physica A 187 519-50... 

The amount of a particular component in a sample can be monitored by examining the height of a spectral absorption peak The reduction of an aldehyde to an alcohol would show up as a decrease in line intensity for the carbonyl and an increase for the hydroxyl peaks in the spectrum. Changes in the relative importance of different relaxation modes in a polymer can also be followed by the corresponding changes in a mechanical spectrum. 

The stress—relaxation process is governed by a number of different molecular motions. To resolve them, the thermally stimulated creep (TSCr) method was developed, which consists of the following steps. (/) The specimen is subjected to a given stress at a temperature T for a time /, both chosen to allow complete orientation of the mobile units that one wishes to consider. (2) The temperature is then lowered to Tq T, where any molecular motion is completely hindered then the stress is removed. (3) The specimen is subsequendy heated at a controlled rate. The mobile units reorient according to the available relaxation modes. The strain, its time derivative, and the temperature are recorded versus time. By mnning a series of experiments at different orientation temperatures and plotting the time derivative of the strain rate observed on heating versus the temperature, various relaxational processes are revealed as peaks (243). 

Turning from chemical exchange to nuclear relaxation time measurements, the field of NMR offers many good examples of chemical information from T, measurements. Recall from Fig. 4-7 that Ti is reciprocally related to Tc, the correlation time, for high-frequency relaxation modes. For small- to medium-size molecules in the liquid phase, T, lies to the left side of the minimum in Fig. 4-7. A larger value of T, is, therefore, associated with a smaller Tc, hence, with a more rapid rate of molecular motion. It is possible to measure Ti for individual carbon atoms in a molecule, and such results provide detailed information on the local motion of atoms or groups of atoms. Levy and Nelson " have reviewed these observations. A few examples are shown here. T, values (in seconds) are noted for individual carbon atoms. 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

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. 

Under these conditions, Eq. (32) indicates the maximum extent to which a particular mode p can reduce S(Q,t) as a function of the momentum transfer Q. Figure 10 presents the Q-dependence of the mode contributions for PE of molecular weights Mw = 2000 and Mw = 4800 used in the experiments to be described later. Vertical lines mark the experimentally examined momentum transfers. Let us begin with the short chain. For the smaller Q the internal modes do not influence the dynamic structure factor. There, only the translational diffusion is observed. With increasing Q, the first mode begins to play a role. If Q is further increased, higher relaxation modes also begin to influence the... 

Figures 13c-e shows how the agreement between experimental data and the calculated structure factor improves if more and more relaxation modes are included. In Fig. 13e, finally, very good agreement between theory and experiment can be noted.

Fig. 14. Relaxation rates Wp for the first four relaxation modes of chains of different molecular mass as a function of the mode number p. The arrows indicate the condition p = N/Ne for each molecular weight. (Reprinted with permission from [52]. Copyright 1993 The American Physical Society, Maryland)...

Exactly at the LST, the material behaves not as a liquid any more and not yet as a solid. The relaxation modes are not independent of each other but are coupled. The coupling is expressed by a power law distribution of relaxation modes [5-7]... 

The glass transition involves additional phenomena which strongly affect the rheology (1) Short-time and long-time relaxation modes were found to shift with different temperature shift factors [93]. (2) The thermally introduced glass transition leads to a non-equilibrium state of the polymer [10]. Because of these, the gelation framework might be too simple to describe the transition behavior. 

Small amplitude oscillatory shear is the method of choice for materials with very broad distributions of relaxation modes, such as materials near LST, and for materials which undergo change during the measurement. The dynamic moduli in Eq. 4-10 are defined by [10]... 

The divergence of the longest relaxation time does not perturb the measurement. In comparison, steady state properties (the steady shear viscosity, for instance) would probe an integral over all relaxation modes and, hence, fail near the gel point. 

This relationship between the relaxation modes and the low shear viscosity is an important one. It indicates that the longest Rouse relaxation time, i.e. the p = 1 mode ... 

The plateau modulus is determined by the region in which these two relaxation modes cross. The plateau modulus, the low shear viscosity and the tube disengagement time are given in Section 6.4.3 as... 

The mode coupling theory [11] has emerged from the hydrodynamics of liquids. This theory is able to explain the splitting of molecular mobility into relaxation modes that are frozen at the glass transition and molecular motion that is still possible below Tg. 

In the low Q-regime RPA describes well the static structure factor for the short chain melt, where the ODT is sufficiently far away (kN 7). In the dynamics we would expect the diblock breathing mode to take over around QRg 2 (Q=0.04 A ). Instead, deviations from Rouse dynamics are already observed at Q values as high as QR =5. At QJ g=3 a crossover to a virtually Q-independent relaxation rate about four to five times faster than the predicted breathing mode is found. This phenomenon is only visible under h-d labelUng. Under single chain contrast (see below) these deviations from RPA are not seen. Thus, the observed fast relaxation mode must be associated with the block contrast. 

The single chain dynamics of one given block or of one chain in a diblock copolymer melt is observed if a matched deuterated diblock is mixed with a small amount of labelled diblocks, where the label could be a protonated a or b block or a protonated chain. In terms of the dynamic RPA such a system is a four-component polymer mixture. It is characterized by four different relaxation modes A1-A4 which - depending on the contrast conditions - appear with... 

Thus, the remaining difference from the Rouse model is a mode-dependent friction coefficient p=(Hpp) for (p>0), which leads to a relaxation mode spectrum with a different mode munber p-dependence. The second term in Hpq is the bead friction with the surrounding me um (solvent), which is the only term present in the Rouse model. The ratio of the diagonal (Rouse-like) friction and the solvent-mediated interaction strength may be expressed by the draining parameter The Rouse model has B=0, whereas the assumption... 

Figure 30.4 Variations of/gp with tube exit relaxation modes. 1 — very fast relaxation 2 — very slow relaxation [11]...

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