Critical relaxation time

Note that the critical relaxation time is predicted to depend upon the square of the particle radius, as observable. This bound-ion-layer model of Schwarz was remarkably successful in several ways. The magnitude of the dielectric increment and its frequency shift with particle radius were reasonably in agreement with observation. Despite this welcome advance in the understandings of dispersions, the Schwarz theory deliberately excluded the possibility of bound counterion exchange with the surrounding medium. It also neglected any contribution to the particle polarizability that arises from the distortion of the outer or surrounding portions of the diffuse double layer. 

Comparison between Cole-Cole plots for the Debye, Cole-Cole and Davidson-Cole equations is made in Fig. 4.2. The arc corresponding to the Cole-Cole equation is symmetrical and forms a portion of a circle, the centre of which is below the 6 -axis. The corresponding distribution function of relaxation times is symmetric, although there is no closed expression for F t) which would give the Cole-Cole equation. The Davidson-Cole arc is a skewed one, and reflects strongly asymmetric distribution of relaxation times. The distribution is peaked at the critical relaxation time Tq, with a decaying tail of shorter relaxation times. There is an exact expression for the autocorrelation function leading to the Davidson-Cole equation. 

Pulsed ENDOR offers several distinct advantages over conventional CW ENDOR spectroscopy. Since there is no MW power during the observation of the ESE, klystron noise is largely eliminated. Furthemiore, there is an additional advantage in that, unlike the case in conventional CW ENDOR spectroscopy, the detection of ENDOR spin echoes does not depend on a critical balance of the RE and MW powers and the various relaxation times. Consequently, the temperature is not such a critical parameter in pulsed ENDOR spectroscopy. Additionally the pulsed teclmique pemiits a study of transient radicals. 

The relaxation time in Eq. (15) and the scaling law Z — 2v+ for the dynamic critical exponent Z are then understood by the condition that the coil is relaxed when its center of mass has diffused over its own size... 

In Fig. 9 the relaxation time shows a very smooth variation with e (analogously smooth data has been obtained for Dj e)). In particular, no evidence for a critical anomaly as e — e, is seen. The straight line in Fig. 9 represents a law = T2 ec)Qxp[ ec — e)/k T], which indicates a simple thermally activated behavior. 

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

Plotting U as a function of L (or equivalently, to the end-to-end distance r of the modeled coil) permits us to predict the coil stretching behavior at different values of the parameter et, where t is the relaxation time of the dumbbell (Fig. 10). When et < 0.15, the only minimum in the potential curve is at r = 0 and all the dumbbell configurations are in the coil state. As et increases (to 0.20 in the Fig. 10), a second minimum appears which corresponds to a stretched state. Since the potential barrier (AU) between the two minima can be large compared to kBT, coiled molecules require a very long time, to the order of t exp (AU/kBT), to diffuse by Brownian motion over the barrier to the stretched state at any stage, there will be a distribution of long-lived metastable states with different chain conformations. With further increases in et, the second minimum deepens. The barrier decreases then disappears at et = 0.5. At this critical strain rate denoted by ecs, the transition from the coiled to the stretched state should occur instantaneously. 

In simple shear flow where vorticity and extensional rate are equal in magnitude (cf. Eq. (79), Sect. 4), the molecular coil rotates in the transverse velocity gradient and interacts successively for a limited time with the elongational and the compressional flow component during each turn. Because of the finite relaxation time (xz) of the chain, it is believed that the macromolecule can no more follow these alternative deformations and remains in a steady deformed state above some critical shear rate (y ) given by [193] (Fig. 65) ... 

It should be stressed that below a critical (Blocking) temperature (Tg) S.P. clusters will have a slow relaxation time (t) at which their net moment will align so-to-speak "parallel" to H, and thus appear to behave as if they had an apparent "bulk-like" ferromagnetic behavior. This aspect will result in a hysteresis or an "apparent" ferromagnetic behavior. Conversely, above the Tg, the hysteresis will disappear and the clusters will show a unique curve with no hysteresis (Fig. 2). 

The longest mode (p=l) should be identical to the motion of the chain. The fundamental correctness of the model for dilute solutions has been shown by Ferry [74], Ferry and co-workers [39,75] have shown that,in concentrated solutions, the formation of a polymeric network leads to a shift of the characteristic relaxation time A,0 (X0=l/ ycrit i.e. the critical shear rate where r becomes a function of y). It has been proposed that this time constant is related to the motion of the polymeric chain between two coupling points. 

Below a critical concentration, c, in a thermodynamically good solvent, r 0 can be standardised against the overlap parameter c [r)]. However, for c>c, and in the case of a 0-solvent for parameter c-[r ]>0.7, r 0 is a function of the Bueche parameter, cMw The critical concentration c is found to be Mw and solvent independent, as predicted by Graessley. In the case of semi-dilute polymer solutions the relaxation time and slope in the linear region of the flow are found to be strongly influenced by the nature of polymer-solvent interactions. Taking this into account, it is possible to predict the shear viscosity and the critical shear rate at which shear-induced degradation occurs as a function of Mw c and the solvent power. 

The first possibility is that the attractive potential associated with the solid surface leads to an increased gaseous molecular number density and molecular velocity. The resulting increase in both gas-gas and gas-wall collision frequencies increases the T1. The second possibility is that although the measurements were obtained at a temperature significantly above the critical temperature of the bulk CF4 gas, it is possible that gas molecules are adsorbed onto the surface of the silica. The surface relaxation is expected to be very slow compared with spin-rotation interactions in the gas phase. We can therefore account for the effect of adsorption by assuming that relaxation effectively stops while the gas molecules adhere to the wall, which will then act to increase the relaxation time by the fraction of molecules on the surface. Both models are in accord with a measurable increase in density above that of the bulk gas. 

As shown in Figure 3.5.3, the relaxation time versus pressure curves are dramatically different from those obtained using CF4 at a temperature well above its critical point. Indeed, while the overall form of the Tx curves for CF4 in fumed silica was similar to that of the bulk gas, the shape of the Ti plots for c-C4F8 in Vycor more closely resembles that of an adsorption isotherm (Ta of CF4 in Vycor is largely invariant with pressure, as gas-wall collisions in this material are more frequent than gas-gas collisions). This is not surprising given that we expect the behavior of this gas at 291 K to be shifted towards the adsorbed phase. The highest pressure... 

CC- critical exponent for longest relaxation time before LST... 

With increasing distance from the gel point, the simplicity of the critical state will be lost gradually. However, there is a region near the gel point in which the spectrum still is very closely related to the spectrum at the gel point itself, H(A,pc). The most important difference is the finite longest relaxation time which cuts off the spectrum. Specific cut-off functions have been proposed by Martin et al. [13] for the spectrum and by Martin et al. [13], Friedrich et al. [14], and Adolf and Martin [15] for the relaxation function G(t,pc). Sufficiently close to the gel point, p — pc <4 1, the specific cut-off function of the spectrum is of minor importance. The problem becomes interesting further away from the gel point. More experimental data are needed for testing these relations. 

Only the value of the relaxation exponent is needed. The critical exponent a of the longest relaxation time (compare Eqs. 1-6 and 1-7) is therefore on an equal footing with the critical exponent of the viscosity ... 

For the relaxation of the solid near the gel point, the critical gel may serve as a reference state. The long time asymptote of G(t) of the nearly critical gel, the equilibrium modulus Ge, intersects the G(t) = St n of the critical gel at a characteristic time (Fig. 6) which we will define as the longest relaxation time of the nearly critical gel [18]... 

Fig. 6. Evaluation of the longest relaxation time for a sample beyond the gel point, p > pc intersect of horizontal line for Ge with the power law of the critical gel, St n...

Fig. 11. Schematic of relaxation time spectrum of the critical gel of PBD 44 (Mw = 44 000). The entanglement and glass transition is governed by the precursor s BSW-spectrum, while the CW spectrum describes the longer modes due to the crosslinking [60]. denotes the longest relaxation time of PBD44 before crosslinking...

This most simple model for the relaxation time spectrum of materials near the liquid-solid transition is good for relating critical exponents (see Eq. 1-9), but it cannot be considered quantitatively correct. A detailed study of the evolution of the relaxation time spectrum from liquid to solid state is in progress [70], Preliminary results on vulcanizing polybutadienes indicate that the relaxation spectrum near the gel point is more complex than the simple spectrum presented in Eq. 3-6. In particular, the relation exponent n is not independent of the extent of reaction but decreases with increasing p. 

Predictions using the observed relaxation time spectrum at the gel point are consistent with further experimental observations. Such predictions require a constitutive equation, which now is available. Insertion of the CW spectrum, Eq. 1-5, into the equation for the stress, Eq. 3-1, results in the linear viscoelastic constitutive equation of critical gels, called the critical gel equation ... 

The transient viscosity f] = T2i(t)/y0 diverges gradually without ever reaching steady shear flow conditions. This clarifies the type of singularity which the viscosity exhibits at the LST The steady shear viscosity is undefined at LST, since the infinitely long relaxation time of the critical gel would require an infinitely long start-up time. 

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