Crossover function

Extension of the classical Landau-Ginzburg expansion to incorporate nonclassical critical fluctuations and to yield detailed crossover functions were first presented by Nicoll and coworkers [313, 314] and later extended by Chen et al. [315, 316]. These extensions match Ginzburg theory to RG theory, and thus interpolate between the lower-order terms of the Wegner expansion at T -C Afa and mean-field behavior at f Nci-... 

Jn the excluded volume limit similar scaling behavior is found for other observables like the scattering functions, and it turns out that all nonuni versal temperature and chemistry dependence can be absorbed into the single parameter B. Since the sealing functions typically interpolate among asymptotic limits (,s —+ 0 or s — oo, for instance), they also are known as crossover functions ... 

An essential aspect, we want to demonstrate, is the fundamental importance of a (qualitatively) correct choice of the uncritical manifold. It is by this choice that our theory generates smooth crossover functions, which interpolate among asymptotic power law behavior as expected from scaling theory. It is most important that the correct behavior is found even in lowest order approximation. Otherwise higher order corrections, trying to reconstruct the correct asymptotics, must blow up. Then a one loop calculation cannot be reliable quantitatively... 

Such doubly logarithmic plots are a common tool in visualizing crossover functions. They however give a strongly distorted picture of the physical behavior. Consider for instance the crossover from the dilute to the semidilute excluded volume behavior, which for our example takes place in the range 10-4 < s < 0.1, Figure 14.1b shows this crossover in a direct plot. Clearly... 

The very broad crossover has important implications for an experimental test of the theory. Indeed, to illustrate clearly the effects, in the figures of this chapter we have driven the parameters into extreme regions. No experiment, for instance, will cover six decades of overlap, nor will wre reach a range of z sufficient to cover both the 0- and the excluded volume limits. Any physical experiment or any simulation will see only some small section of the crossover function, the full function being tested only by combining results for several chemical systems. In particular we have no chance to see a double crossover as typically exhibited here, in a single experiment. 

Numerically the crossover function for Ja(q) i to be calculated from Eq. (14,5) with the help of the full RG mapping (13.27). Figure 14.2 shows a doubly logarithmic plot of vs where the isolated chain... 

A more phenomenological approach to describe crossover critical phenomena in simple fluids has been developed by Kiselev and coworkers [76-79]. This approach starts from the asymptotic power-law expansion including the leading correction-to-scaling terms which is then multiplied by an empirical crossover functions so that the equation becomes analytic far away from the critical point. A comparison of this approach with the crossover theory based on a Landau expansion has been discussed in earlier publications [13, 78]. One principal difference is that in the application of the results of the RG theory to the Landau expansion the leading correction to asymptotic scaling law is incorporated in the crossover function and recovered upon expanding the crossover function [18]. 

Nicoll, J.F. and Albright, P.C. (1985) Crossover functions by renormalization-group matching Three-loop results, Phys. Rev. B 31, 4576-4589. 

Kiselev, S.B. (1988) Universal crossover function for the free energy of singlecomponent and two-component fluids in the critical region. High Temp. 28, 42-47. 

The scaling concept was applied for the analysis of chain conformations and static properties of the semidilute polymer solutions. The unique characteristic length scale in dilute solution imposes a unique characteristic concentration of the solution, which coincides with the intramolecular concentration c in an isolated coil. All the properties of the semidilute solution can be derived from those of the dilute solution by scaling procedure with the aid of proper crossover functions of a single dimensionless variable c/c. These crossover functions are universal, that is, independent of any details of chemical stmcture of the chains, and exhibit power-law asymptotic behavior at c/c 1. 

A more rigorous approach to crossover between the polymer and critical regimes shows that the crossover variable x itself changes from xoc(np) lATI in the polymer regime to xoc( p) Af (with 2v=1.2) in the critical regime. " This feature makes crossover expressions implicit and more complex, without, however, significantly changing the shape of the crossover functions. 

The approach of Kiselev, based on the work of Sengers and co-workers and Kiselev and co-workers, " utilizes a renormalized Landau expansion that smoothly transforms the classical Helmholtz energy density into an equation that incorporates the fluctuation-induced singular scaling laws near the critical point, and reduces to the classical expression far from the critical point. The Helmholtz energy density is separated into ideal and residual terms, and the crossover function applied to the critical part of the Helmholtz energy Aa(AT, Av), where Aa(AT, Av) = a(T, v) — a, g(T, v) and the background contribution abg(T, v) is expressed as. 

Equations 10.85 to 10.88 define what has been called a two-term crossover Landau model (CLM). In the classical limit (A/ic 1) the crossover function Y approaches unity and one recovers from eq 10.85 the classical expansion of eq 10.83. In the critical region (A/k 1) the crossover function approaches zero as Y x k/uA) and one recovers from eq 10.85 the power-law expansions specified in Table 10.5 with expressions for the critical amplitudes listed in Table 10.8. The values for the critical-amplitude ratios implied by the crossover Landau model are included in Table 10.3. The nonasymptotic critical behaviour is governed by u and A/cJ or, equivalently by u and by Nq, known... 

In this expression T, are and are universal functions with Pq and Pi representing an asymmetric and a first correction-to-scaling function, respectively, while A, are universal exponents with A2 = 2Ai and A3 = A4 = y-fjS — 1. In eqs 10.97 and 10.99 R q) plays the role of a crossover function defined... 

In these equations k, a, ct, d ( = Pu in eq 10.29), and g are system-dependent coefficients with g being related to the inverse of the Ginzburg number AIq. Slightly different versions for the crossover function R q) have also been used. In the critical limit 0 one recovers the linear-model parametric equation in Section 10.2.2 with coefficients a and k. In the classical limit q-rcc. Ad becomes an analytic function of AT and Ap. For a comparison of this phenomenological parametric crossover equation with the crossover Landau models the reader is referred to some previous publications. " " ... 

Fig. 5 Renormalized susceptibility So = Soa XQ as a function of renormalized temperature f = t/tq in accordance with the crossover function Eq. (8) in water-in-oil droplet systems of both AOT/D20/n-decane (LCST) and BHDC/water/benzene (UCST) near the phase boundary. The vertical dotted line shows a boundary f = 1 below which the critical theory applies. The curved dotted line shows the calculated mean-field behavior...

To evaluate the crossover functions 2 and H, not only the specific heat capacities cp and cv are needed, but also the correlation length f. From equations (6.11) and (6.12) it follows that at yo = />c asymptotically close to the critical temperamre... 

Several theoretical attempts have been made recently to give a more complete description of the transport coefficients of mixtures in the critical region. Folk Moser (1993) performed dynamic renormalization-group calculations for binary mixtures near plait points and obtained nonasymptotic expressions for the kinetic coefficients. Kiselev Kulikov (1994) derived phenomenological crossover functions for the transport coefficients by factorizing the Kubo formulas for the transport coefficients a, p and y. This approach is referred to as the decoupled-mode approximation (Ferrell 1970). Their calculations yield for the thermal conductivity the expected finite enhancement in the asymptotic critical region and also a smooth crossover to the background far away from... 

The derivation of the correlation for the excess viscosity A (p, T) from experimental data in general is much easier, since for the viscosity the critical enhancement is just restricted to a very narrow region around the critical point. After the correlations for the excess thermal conductivity and the excess viscosity have been determined, the coincidence of the theoretical results calculated for the thermal conductivity in the critical region with those obtained experimentally could be optimized by adjusting the value of the parameter a microscopic cutoff distance in the crossover function... 

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