Single-chain relaxation time

The rheology of lamellar phases has attracted considerable attention. For a quenched lamellar phase it has been observed that where G = G" both scale as to112 for to < wc, where determined operationally as being approximately equal to 0.1 r1, where r is a single-chain relaxation time defined as the frequency where G and G" cross (Bates et al. 1987 Rosedale and Bates 1990). Similar dynamic moduli scaling was found with PS-PI-PS and PS-PB-PS triblocks (here... 

For a quenched lamellar phase it has been observed that G = G"scales as a>m for tv < tvQ. where tvc is defined operationally as being approximately equal to 0.1t and r is a single-chain relaxation time defined as the frequency where G and G" cross (Bates et al. 1990 Rosedale and Bates 1990). This behaviour has been accounted for theoretically by Kawasaki and Onuki (1990). For a PEP-PEE diblock that was presheared to create two distinct orientations (see Fig. 2.7(c)), Koppi et al. (1992) observed a substantial difference in G for quenched samples and parallel and perpendicular lamellae. In particular, G[ and the viscosity rjj for a perpendicular lamellar phase sheared in the plane of the lamellae were observed to exhibit near-terminal behaviour (G tv2, tj a/), which is consistent with the behaviour of an oriented lamellar phase which flows in two dimensions. These results highlight the fact that the linear viscoelastic behaviour of the lamellar phase is sensitive to the state of sample orientation. 

If one end of an active chain dissociates from a junction due to thermal motion, or a tension caused by the external force, the chain becomes dangling and relaxes to an equilibrium state after the single-chain relaxation time r, which is of the order of the Rouse relaxation time tr = /()7t k T)n in the unentangled regime. We... 

From the graph, we can also obtain a rough idea of the relaxation time of the expanded ensemble. With the ratio of 100 1 between SMC moves and attempts to switch between neighboring states of the expanded ensemble, structural and thermodynamic quantities relax on the same order of timescales. Note that the relaxation time of the morphology is significantly larger than the single-chain relaxation time, Rio/D. 

The proton T relaxation time was determined as a function of temperature for samples with varying content of hydrophilic Aerosil (300 m g ) [7]. Diie to H spin-diffusion, only a single h relaxation time is usually measured in heterogeneous polymers [23]. The presence of Aerosil in PDMS suppresses the T minimum at 195 K, ascribed to the chain motion (a-relaxation) in unfilled PDMS, and leads to the appearance of a minimum at higher temperature, in the vicinity of 280 K as shown in Fig. 6. 

It turns out that a rather simple description of this nonlinear relaxation in terms of a single relaxation time,, depending on the final average chain length Loo, is suggested by a scaling plot of L t) for different L o, as shown in Fig. 18 for an initial exponential MWD. It is evident from Fig. 18 that the response curves, L o — L t), for different L o may be collapsed onto a single master curve, 1 - L t)lLoo = /(V Loo) measured in units of a... 

Single chains confined between two parallel purely repulsive walls with = 0 show in the simulations the crossover from three- to two-dimensional behavior more clearly than in the case of adsorption (Sec. Ill), where we saw that the scaling exponents for the diffusion constant and the relaxation time slightly exceeded their theoretical values of 1 and 2.5, respectively. In sufficiently narrow slits, D density profile in the perpendicular direction (z) across the film that the monomers are localized in the mid-plane z = Djl so that a two-dimensional SAW, cf. Eq. (24), is easily established [15] i.e., the scaling of the longitudinal component of the mean gyration radius and also the relaxation times exhibit nicely the 2 /-exponent = 3/4 (Fig. 13). 

Sikorsky and Romiszowski [172,173] have recently presented a dynamic MC study of a three-arm star chain on a simple cubic lattice. The quadratic displacement of single beads was analyzed in this investigation. It essentially agrees with the predictions of the Rouse theory [21], with an initial t scale, followed by a broad crossover and a subsequent t dependence. The center of masses displacement yields the self-diffusion coefficient, compatible with the Rouse behavior, Eqs. (27) and (36). The time-correlation function of the end-to-end vector follows the expected dependence with chain length in the EV regime without HI consistent with the simulation model, i.e., the relaxation time is proportional to l i+2v The same scaling law is obtained for the correlation of the angle formed by two arms. Therefore, the model seems to reproduce adequately the main features for the dynamics of star chains, as expected from the Rouse theory. A sim-... 

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. 4.20 Temperature dependence of the average relaxation times of PIB results from rheological measurements [34] dashed-dotted line), the structural relaxation as measured by NSE at Qmax (empty circle [125] and empty square), the collective time at 0.4 A empty triangle), the time corresponding to the self-motion at Q ax empty diamond),NMR dotted line [136]), and the application of the Allegra and Ganazzoli model to the single chain dynamic structure factor in the bulk (filled triangle) and in solution (filled diamond) [186]. Solid lines show Arrhenius fitting curves. Dashed line is the extrapolation of the Arrhenius-like dependence of the -relaxation as observed by dielectric spectroscopy [125]. (Reprinted with permission from [187]. Copyright 2003 Elsevier)...

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. 

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