Relaxations short time mode

Figure 3-19. Calculations of the viscosity from the stress relaxation master curve for NBS PIB at 25 °C (see Figure 3-12) via equation (3-89), showing the important contribution to the integral from the long-time data and the negligible contribution from the short-time modes in the glassy region.

The short time mode corresponds to the glass transition. In polymers like polystyrene, a narrow distribution is observed. Ihe width of the distribution reflects the width of the distribution of the order parameter it is increased after mechanical orientation by addition of a dopant or additive, or under special glass forming conditions (hydrostatic pressure or rheomolding). The distributed relaxation times obey a compensation law, they are reduced to a single time at the compensation temperature T. The departure of from the glass transition is related to the kinetic aspect of the transition. Thermodynamic models are based on the linear relationship between the activation enthalpy and the activation entropy. 

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. 

The relaxation rate R t) described by Eqs. (4.49)-(4.51) embodies our universal recipe for dynamically controlled relaxation [10, 21], which has the following merits (i) it holds for any bath and any type of interventions, that is, coherent modulations and incoherent interruptions/measurements alike (ii) it shows that in order to suppress relaxation, we need to minimize the spectral overlap of G( ), given to us by nature, and Ffo)), which we may design to some extent (iii) most importantly, it shows that in the short-time domain, only broad (coarse-grained) spectral features of G( ) and Ffa>) are important. The latter implies that, in contrast to the claim that correlations of the system with each individual bath mode must be accounted for, if we are to preserve coherence in the system, we actually only need to characterize and suppress (by means of Ffco)) the broad spectral features of G( ), the bath response function. The universality of Eqs. (4.49)-(4.51) will be elucidated in what follows, by focusing on several limits. 

The processes observed in the depolarized Rayleigh spectrum correspond to internal modes of motion. Thus, they may have relaxation times which substantially exceed those obtained from the longitudinal or bulk relaxation alone. Nevertheless they are a part of the a relaxation process as it is normally observed in the creep compliance. All processes with the same shift factors make up the full a relaxation. In liquids with substantial depolarized Rayleigh scattering the slowly relaxing part of the W scattering is also dominated by the orientation fluctuations associated with the internal modes of motion. Each internal mode contributes some intensity, but it is believed that fairly short wavelength modes dominate the scattered intensity. 

The behavior of VACF and of D in one-dimensional systems are, therefore, of special interest. The transverse current mode of course does not exist here, and the decay of the longitudinal current mode (related to the dynamic structure factor by a trivial time differentiation) is sufficiently fast to preclude the existence of any "dangerous" long-time tail. Actually, Jepsen [181] was the first to derive die closed-form expression for the VACF and the diffusion coeffident for hard rods. His study showed that in the long time VACF decays as 1/f3, in contrast to the t d 2 dependence reported for the two and three dimensions. Lebowitz and Percus [182] studied the short-time behavior of VACF and made an exponential approximation for VACF [i.e, Cv(f) = e 2 ], for the short times. Haus and Raveche [183] carried out the extensive molecular dynamic simulations to study relaxation of an initially ordered array in one dimension. This study also investigated the 1/f3 behavior of VACF. However, none of the above studies provides a physical explanation of the 1/f3 dependence of VACF at long times, of the type that exists for two and three dimensions. 

A more accurate evaluation of the first modes with the open-chain transform [83] leads to a coefiicient 0.467 instead of 0.425.] The correlation function B k, t) cannot be easily evaluated in closed form for the Zimm limit. However, it may be seen from Eqs. (3.1.18) that for sufficiently short times, B(k, t)/Ai is a universal function of the two variables k/N and t/r ax- In particular, the time dependence of the self-correlation function B(0, t) is again a simple power law if t is comprised between the two extremal relaxation times. We get, from (3.1.18) and (3.2.12) [90],... 

For high frequencies uj > 1 /tq, there are no relaxation modes in the Rouse model. The storage modulus becomes independent of frequency, and equal to the short time stress relaxation modulus, which is kT per monomer G uj) (pkTjb. This high-frequency saturation is not included in Eqs (8.49) and (8.50). At low frequencies a < 1/tr, the storage modulus is proportional to the square of frequency and the loss modulus is pro-portional to frequency, as is the case for the terminal response of any viscoelastic liquid. 

The use of hannonic baths to model the thennal enviromnent of molecular systems does not rest only on such timescale arguments. We have seen in Chapter 3 that the radiation field constitutes an harmonic environment that determines the radiative relaxation properties of all material systems. We will see in Chapters 15 and 16 that dielectric solvents can be modeled as harmonic environments in which the harmonic modes are motions of the polarization field. In the latter case the harmonic environment picture does not rely on a short time approximation, but rather stems from the long wavelength nature of the motions involved that makes it possible to view the solvent as a continuum dielectric. 

Analysis of a typical evolution of the Fano factors F 2, such as presented in Fig. la, leads to the conclusion that after initial short-time (gt < 1) relaxations in both modes, a strongly super-Poissonian (F 2 ) This behavior occurs for the majority of initial coherent states ai) and cq) except a certain set of initial... 

Because relaxing solute normal mode frequencies are often much higher than typical solvent translational-rotational frequencies, the following short time picture of solute VER emerges [22,24]. In zeroth order the relaxing mode executes conservative harmonic notion in a hypothetical solvent that is nonresponsive to this motion. The velocity autocorrelation function /(f) for this zeroth order motion is... 

In order to stndy the short time vibrational energy transfer behavior of a vibra-tionally excited system, we employ a non-Markovian time-dependent perturbation theory [83]. Onr approach builds on the successful application of Markovian time-dependent pertnrbation theory by Leitner and coworkers to explore heat flow in proteins and glasses, and Tokmakoff, Payer, and others, in modeling vibrational population relaxation of selected modes in larger molecules. In a separate chapter in this volnme, Leitner provides an overview of the development of normal mode-based methods, snch as the one employed here, for the study of energy flow in solids and larger molecnlar systems. 

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