Relaxation maps

In order to represent the temperature dependence of the process frequencies corresponding to the various transitions, it is usual to consider a relaxation map, as it is called, in which the logarithm of the frequency is plotted as a function of 1/T, where T is the absolute temperature. A typical example is shown in Fig. 2 for a polymer exhibiting two solid state transitions (/I and y), in addition to the a transition. It is worth pointing out that the lower the transition temperature, the smaller the activation energy. 

The relaxation map in Fig. 2 clearly shows that for a measurement performed at a low frequency,/i, the various transitions appear at temperatures well separated from each other. At a higher frequency,, the transitions appear, firstly, at higher temperatures and, secondly, closer to each other, in such a way that at high frequency an overlapping (merging) of the transitions occurs. 

It is worth pointing out that the relaxation map is quite useful for comparing results obtained by techniques operating at various frequencies, like dynamic mechanical measurements, dielectric relaxation, NMR, etc. 

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In a similar fashion. Thermally Stimulated Current spectrometry (TSC) makes use of an appHed d-c potential that acts as the stress to orient dipoles. The temperature is then lowered to trap these dipoles, and small electrical currents are measured during heating as the dipoles relax. The resulting relaxation maps have been related to G and G" curves obtained by dynamic mechanical analysis (244—246). This technique, long carried out only in laboratory-built instmments, is available as a commercial TSC spectrometer from Thermold Partners L.P., formerly Solomat Instmments (247). 

In Fig. 3 all conformers are plotted within 3 kcal/mol above, the global minimum found by CICADA for each of the glycosidic torsions, superimposed on the corresponding disaccharide relaxed map. Fig. 4 clearly indicates that most of the local minima present in the corresponding disaccharide fragments are also explored along the PES of the pentasaccharide by CICADA... 

Fig. 22. Relaxation map for PBLG side chain motion. Experiments except 2H NMR measurements are open symbols, and PBLG-Kdi (filled circle) and PBLG-fd2 (filled triangle).

The jump rates obtained by the line shape simulations are plotted on the relaxation map in Fig. 22 together with values obtained by other experimental methods. The points of the mechanical and dielectric relaxations correspond to the process of the large-scale side chain motions refered to as the -process and follow the WLF equation very well above Jg,. 11 It should be noted that the present 2FI NMR results are located on the curve obtained by other relaxation experiments. This fact shows that... 

Fig. 1.3 Relaxation map of polyisoprene results from dielectric spectroscopy (inverse of maximum loss frequency/w// symbols), rheological shift factors (solid line) [7], and neutron scattering pair correlation ((r(Q=1.44 A )) empty square) [8] and self correlation ((t(Q=0.88 A" )) empty circle) [9],methyl group rotation (empty triangle) [10]. The shadowed area indicates the time scales corresponding to the so-called fast dynamics [11]...

Fig. 4.10 a Characteristic relaxation times determined from dielectric measurements [137] (diamonds), and from NSE spectra at (triangles) for triol (open symbols) and PU (solid symbols). The full lines correspond to Vogel-Fulcher and the dotted lines to Arrhenius descriptions, b Relaxation times from NSE spectra have been arbitrarily multiplied by a factor 6 for triol and 40 for PU to build a normalized relaxation map. (Reprinted with permission from [127]. Copyright 2002 Elsevier)... 

Several recent papers coiqpare the two types of analysis (32-34) (see also the paper herein by Tran and Brady). Brant and Christ coiqpare the abilities of the two approaches to predict experimental behavior in their chapter herein. Two strategies for constructing relaxed maps are discussed in the chapters by Tran and Brady and by French, Tran and P6rez. 

French has recently presented comparisons of rigid and relaxed conformational maps for cellobiose and maltose obtained with the MMP2(1985), which includes ano-meric effects. The fully relaxed maps show interesting details. 

Figure 5a. An example of a partial energy map, the local relaxed map for the S4 family of conformations. Contours are indicated at 4, 6, and 8 kcal/mol above the global SI minimum, which does not appear on this map. The dashed lines surround the different inter-residue hydrogen bond domains (with a cutoff criterion of 2.05A for the O. .. H distance), with the tic marks on the d hes pointing toward the region where the given hydrogen bond is allowed.

Without using any motional model, the temperature positions of T and Tip minima can be assigned an appropriate frequency 90 MHz at 120 °C from Ti and 43 kHz at - 34 °C from T r These two results fit quite well on the relaxation map of BPA-PC obtained from dynamic mechanical and dielectric relaxation. They support the fact that phenyl ring motions are involved in the /3 relaxation of BPA-PC. Furthermore, the Ti and T f> data can be simulated by considering the Williams-Watts fractional correlation function [33] ... 

The relaxation map of PMMA obtained from E" and s" is shown in Fig. 111. It shows a good agreement between the two types of investigations, in agreement with the fact that the same motional groups are involved in the response of both techniques. 

Fig. Ill Relaxation map of PMMA derived from dielectric e" (, O) and mechanical E" ( ) data (from [75])...

The 13 C powder NMR line shapes of the carboxyl group chemical shift tensor are shown Fig. 115 as a function of temperature. At 27 °C, a nearly regular powder spectrum is found, with o = 268 ppm, <722 = 150 ppm and (733 = 112 ppm. As temperature rises, increasingly pronounced line-shape changes are observed, which are indicative of large motions with rates exceeding 10 kHz. The motional rates estimated at the various temperatures fit quite well on the relaxation map of PMMA obtained from mechanical and dielectric measurements (Fig. 111). Thus, the motions of the carboxyl groups observed by NMR are directly related to the f3 transition. 

From the relaxation map of the copolymer and the frequency range involved in the NMR experiment, the motional processes involved in the p transition should be detected by NMR in the temperature range 20-140 °C. 

C chemical shift anisotropy NMR measurements deal with frequencies of the order of or higher than 104 Hz and, consequently, the temperature range at which the motions involved in the p transition contribute to the NMR response are considerably shifted towards higher temperatures. The dielectric relaxation map of the p transition allows one to determine the following temperature ranges for the considered MGIMx copolymers ... 

Figure 11.2 Shape of relaxation maps (coordinates of transitions a, p, and y in a graph In (frequency) - reciprocal temperature). Left polymers having their a and p transitions well separated (example polycarbonate, amine-crosslinked epoxy). Right polymers with close a and p transitions (example polystyrene, unsaturated polyester).

As shown in Chapter 10, every peak corresponds to a type of molecular motion. From the coordinates (co, T) of the peaks, a relaxation map can be built (see Fig. 11.2). The following observations can be made ... 

A typical relaxation map (relaxation time or frequency at the maximum of dissipation peaks against reciprocal absolute temperature) is shown in Fig. 11.2. The general characteristics of relaxation maps are the following ... 

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