Crossover Regime

When the conditions 720 7io vvith a driving force for phase 1 being larger than for phase 2 are satisfied, that is, 

It is now of interest to study, through a practical example, whether or not a metastable state, at the sample level, can be attained owing to a very high nudeation frequency, and how such a metastable state may compete with the formation of the stable phase. 

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The crossover regime occurs at obstacle density C bs when the size of the random polymer coil becomes equal to the size of the pores, Rg =, obs-From Eqs. (46) and (47) in this regime follows... 

Thus, with only two parameters, the values of which are both close to expectations, the Hess model allows a complete description of all experimental spectra. In the complex crossover regime from Rouse motion to entanglement controlled behavior, this very good agreement confirms the significant success of this theory. 

A quantitative analysis of scattering data, originating from the crossover regime between short-time Rouse motion and local reptation, needs explicit consideration of the initial Rouse motion neglected by de Gennes. Ronca [50] proposed an effective medium approach, where he describes the time-dependent... 

For PS/d-cyclohexane, the only system where comparable static and dynamic data from the crossover regime are available, the static and dynamic crossover points Q (x) = l/c/x) do not coincide... 

A (empty square), 0.3 A (filled diamond) and 0.4 A (empty triangle). The solid line represents r (t))-f -. The dashed lines mark the characteristic times and lengths discussed in the text. The thick solid line in the upper right corner displays the mean squared displacement in the Rouse regime. The two dotted lines extrapolate into the crossover regime. (Reprinted with permission from [39]. Copyright 2004 EDP Sciences)... 

Sanchez and DiMarzio identified [41] a crossover regime 11 [Fig. 1.15(c)], where g is more rapid than in 1 and less than in 111. On the basis of the LH model, crystallization kinetics in these three regimes are obtained as follows. 

The most interesting result of Refs. [58, 59] concerns the crossover regime between dilute and semidilute regions of polymer 0-solution. The author shows that in this crossover regime there exists the critical concentration c corresponding to the appearance of an infinite cluster of entangled with each other macromolecules. It is also shown that near this critical concentration the relative viscosity t r of the 0-solution has a scaling form ... 

Figure 6. A schematic phase diagram for the cuprates. The Tc line is determined by the pairing line (7pair), decreasing with x, and the coherence line f/COii), increasing with x. Broken lines should not be regarded as sharp lines (except when T 0), but as crossover regimes. The MIT point is where a metal-insulator transition occurs at T = 0 when SC is suppressed.

The noninteracting gas corresponds to Kp 1 and density waves or pairing correlations are divergent at low temperature for Kp < 1 and > 1, respectively. The observation of power law dependences for the response functions in an extended domain of temperatures should thus be taken as the signature of one-dimensionality. One-dimensionality is indeed limited at low temperature by the crossover regime toward a physics at higher dimensionality. 

For spheres, [t)] = 2.5. An exponential dependence on has been found for the relaxation time of flexible polymer molecules in the crossover regime from dilute to concentrated solutions (Amelar et al. 1991). [For polymers, the intrinsic viscosity is a dimensional quantity see Eq. (3-5).]... 

Crossover. Generally, crossover from an Ising-like asymptotic behavior to mean-field behavior further away from the critical point [86, 87] may be expected. Such a behavior is also expected for nonionic fluids, but occurs so far away, that conditions close to mean-field behavior are never reached. Reports about crossover [88] and the finding of mean field criticality [14—16] suggest that in ionic systems the temperature distance of the crossover regime from the... 

Applications are then presented in Section IV. These examples should served as a guide as to what kinds of problems can be studied with these techniques and the limitations and possibilities for these methods. We present three examples (1) a dynamical test of the centroid quantum transition-state theory for electron transfer (ET) reactions in the crossover regime between adiabatic and nonadiabatic electron transfer, (2) the primary electron transfer reaction in bacterial photosynthesis, and (3) the diffusion kinetics of a Brownian particle in a periodic potential. Finally, Section V offers an outlook and a perspective of the current status of the field from our vantage point. 

It is well known that a long flexible polymer in a good solvent can form a self-similar adsorbed layer near an attractive wall at the critical temperature T . Using the correspondence between an cidsorbed polymer chain and the model of ferromagnets with n-vector spins in the limit n — 0 with a free surface, it hcis been shown that the adsorption point Ta corresponds to a tricritical point and in its proximity a crossover regime is observed. In particular, the mean number of monomers, M, at the surface is shown to behave as [47]... 

These predictions for different regimes for melt brushes are summarised in figure 6.12 (Aubouy et al. 1995). It should be stressed that in reality the sharp boundaries between regimes will be replaced by broader regions of crossover indeed, if the degree of polymerisation of the brush chains is modest, most of the accessible parameter space will be in one or other of these crossover regimes and it will be difficult to establish the as5miptotic power law behaviour. 

For a thermodynamic description of this intermediate, crossover regime, see Refs. [107] and [108]. 

Polymer solutions also display many interesting features in the crossover regimes from dilute to semidilute and from semidilute to concentrated solutions. The observations and the concepts necessary to comprehend these phenomena are discussed in Section 6.5. 

Figure 13.13 Gibbs energy dependence on the depletion of the matrix in the crossover regime. Point S is the point of crossover of energy advantage.

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