Multiple relaxation modes

In some systems, especially when studied with more modem instrumentation, of diffusing probes reveals several distinct relaxational modes. These have been studied most systematically in hydroxypropylcellulose water, as treated in a later section. 

Delfino, et al. used QELSS to study probes in aqueous solutions of car-boxymethylcellulose, nominal molecular weight 700 kDa, at concentrations 0.2-11.7 g/l(38). The probe particles were 14, 47, and 102 nm radius polystyrene spheres. Ubbelohde viscometers and a concentric-cylinder Couette viscometer were used to determine rheological properties of the solutions. From the intrinsic 

Phillies, et al. report three studies of probe diffusion in polyelectrolyte solu-tions(41 3). Two examine probes in dilute and concentrated solutions of high(41) 

Phillies, et al. examined polystyrene spheres, radius 7, 34, and 95 nm, diffusing through aqueous polystyrene sulfonate(43). The experiments determined the initial slope a of Dp against polymer c for various polymer M, and confirmed the model prediction of Phillies and Kirkitelos(44). Polymers had seven molecular 

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However, Eq. (3-24) can be extended to allow multiple relaxation modes simply by setting mtf — t equal to a sum of exponentials (Lodge 1956.1968 Yamamoto 1956.1957. 

The relaxation time r is a fundamental dynamic property of all viscoelastic liquids. Polymer liquids have multiple relaxation modes, each with its own relaxation time. Any stress relaxation modulus can be described by a series combination of Maxwell elements. 

This relaxation time for the Maxwell model is an average relaxation time [see Eq. 7.136] whenever a material has multiple relaxation modes. 

The Rouse model (Rouse, 1953) extends these theories to multiple beads and springs (or multiple-relaxation modes). Here the expression for the viscosity becomes... 

B.8 Predictions of the McLeish-Larson Constitutive Equation for Branched Polymers (McLeish and Larson, 1998). The stress tensor for a branched polymer, which is given in Table 3.7, is a function of the dynamic variable S, which describes the average backbone orientation, and A, which describes the average backbone stretch. The model described in the table is written for multiple relaxation modes and represents the simplest model for a branched polymer. Dynamic expressions for S and X for each relaxation mode are given in Eqs. 2 and 3 in Table 3.7. Tb is the /th mode of the backbone relaxation time and Ts, is the backbone stretch orientation time, v is taken as Hq, where q is the number of branch arms associated with a given Ts. 

Recent ultrasonic measurements on S02 by Bass, Winter, and Evans [258] confirms the earlier observations of multiple relaxation [259,260] and extends the temperature range to 1090°K. As T increases, the relaxation time for the bending mode increases with increasing temperature, while that for the stretching modes decreases in the normal manner. Shields and Anderson [261], using an improved absorption apparatus, have studied pure S02 and its mixtures with argon. Under careful inspection, the results are shown to... 

Hermans and Van Beek626 have recently used the new model of polymer molecules suggested by Rouse.46 At high frequencies the whole molecule cannot follow the field so it is divided into a number of submolecules small enough to follow the field and yet sufficiently large to have a Gaussian distribution. Dielectric relaxation for the case of dipoles parallel to the chain has been calculated by Founder sum transforms. The distribution of relaxation modes appears though the multiplicity of the mathematical solution for the diffusion equation. 

Brown and collaborators interpret their spectra as showing that S q, t) in some systems has multiple slow modes, some being -independent while others scale as (43,48). Brown, et a/. s interpretation potentially explains all slow mode behaviors, namely in different circumstances the slow mode is dominated by a -dependent or by a -independent component. A spectrum whose modes have -dependent shapes might in some cases also be described as a mixture of and °-dependent relaxations whose relative amplitudes are not constant. The relationship between the spectral analyses of Brown and Stepanek(29), who interpret their spectra via a regularized Laplace transform method, and the work of Phillies and collaborators(87), who interpret S q,t) as a sum of stretched exponentials whose parameters depend on q and c, has not been completely analyzed. The latter interpretation has the virtue of supplying quantitative parameters for further analysis. 

Traditionally, polymer relaxation is characterized by a spectnun of multiple relaxation times corresponding to various modes of relaxation a polymer chain can undergo. The longest relaxation time corresponds to the relaxation of a whole chain, while the shorter ones correspond to the relaxation of short parts of the macromolecules. In this paper, we are particularly interested in the relaxation behavior at molecular chain level. Dynamic rheometers are the most used means for providing relaxation time spectrum of polymer melts [4-6]. However, since they only generate small scale melt deformation dming measurements, the relaxation time at macromolecular level... 

Kr. In the B-emitting states, a slower stepwise relaxation was observed. Figure C3.5.5 shows the possible modes of relaxation for B-emitting XeF and some experimentally detennined time constants. Although a diatomic in an atomic lattice seems to be a simple system, these vibronic relaxation experiments are rather complicated to interiDret, because of multiple electronic states which are involved due to energy transfer between B and C sites. 

The purpose of these comparisons is simply to point out how complete the parallel is between the Rouse molecular model and the mechanical models we discussed earlier. While the summations in the stress relaxation and creep expressions were included to give better agreement with experiment, the summations in the Rouse theory arise naturally from a consideration of different modes of vibration. It should be noted that all of these modes are overtones of the same fundamental and do not arise from considering different relaxation processes. As we have noted before, different types of encumbrance have different effects on the displacement of the molecules. The mechanical models correct for this in a way the simple Rouse model does not. Allowing for more than one value of f, along the lines of Example 3.7, is one of the ways the Rouse theory has been modified to generate two sets of Tp values. The results of this development are comparable to summing multiple effects in the mechanical models. In all cases the more elaborate expressions describe experimental results better. 

It should be noted that the decomposition shown in Eq. 3.7.2 is not necessarily a subdivision of separate sets of spins, as all spins in general are subject to both relaxation and diffusion. Rather, it is a classification of different components of the overall decay according to their time constant. In particular cases, the spectrum of amplitudes an represents the populations of a set of pore types, each encoded with a modulation determined by its internal gradient. However, in the case of stronger encoding, the initial magnetization distribution within a single pore type may contain multiple modes (j)n. In this case the interpretation could become more complex [49]. 

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