Crossover scaling

N = 32. (b) Crossover scaling description of i g and (c) vs the crossover scaling variable [AjA, where A = e/e — I for different chain lengths. The flat branch of each curve refers to [e] < [ed while the other branch refers to [e] > [ed. Ideally, the slopes of the latter branches should yield the theoretical exponents expected for large A N, 2 u2 — u)/ip 0.55, and -lujip a -2.03, respectively [13]. 

A crossover scaling for and R of a polymer chain in a quenched random medium may be developed if one introduces a correlation length... 

In fact, the variable x /Gi controls the "crossover" from one "universality class" " to the other. I.e., there exists a crossover scaling description where data for various Gi (i.e., various N) can be collapsed on a master curve Evidence for this crossover scaling has been seen both in experiments and in Monte Carlo simulations for the bond fluctuation model of symmetric polymer mixtures, e.g Fig. 1. One expects a scaling of the form... 

Figure 1. Crossover scaling plot for tlie order parameter ( m > = ( ( ia - Bl / (<1>a + B)> of a symmetrical polymer mixture simulated by tlie bond fluctiiatioii model on tlie simple cubic lattice, with a concentration (jiv = 0.5 of vacant sites. Here N " ( m > is plotted vs. N t, and chain lengths from N = 32 to N = 512 are...

We may summarize our discussion by noting that, in the parameter region given by Eqs. (1.18) we find universal behavior. Tt takes the form of crossover scaling laws, in limiting situations reducing to. simple povrer laws. Specific microstructure effects are contained in a few nonuniversal constants. 

The theory as presented here has been called the field theoretic approach , since it relies on powerful results derived in quantum field theory or in statistical field theory of critical phenomena. This aspect, however, will not be important to us. Yet a reader acquainted with critical phenomena will easily see close connections and hopefully may take some interest in the construction of crossover scaling functions as advocated here. 

To summarize, strict e-expansion a priori seems to yield unambiguous results. Closer inspection, however, reveals that in low order calculations considerable ambiguity is hidden in the definition of the physical observables used as variables or chosen to calculate. What is worse, the e-expansion does not incorporate relevant physical ideas predicting the behavior outside the small momentum range or beyond the dilute limit. In particular, it does not give a reasonable form for crossover scaling functions. On the other hand, it can be used to calculate well-defined critical ratios, which are a function of dimensionality only, Even then, however, the precise definition of the ratio matters,... 

A more complete description of the crossover from two- to three-dimensional critical behavior, that incorporates the shift of Tc, Eq. (129), can be written down in terms of crossover scaling functions M for the order parameter M, Eq. (130) and S for the collective scattering intensity Scoll for scattering wavenumber q—>0, which can be accessed by small angle scattering techniques [186]. Defining =1—T/Tc (D->°o) we can write... 

Of particular interest is the behavior near the multicritical point, where again a crossover scaling description applies (Binder and Landau, 1984, 1990)... 

The singular variation of the critical amplitude fi(N) with chain length N is simple seen from the fact that the variable eN which controls the Ginzburg criterion, Eq. (96b), appears as a crossover scaling variable in the order parameter, assuming a smooth crossover between the order parameter in the mean... 

For polymer blends in d = 3 it then turns out that a crossover scaling description applies (74-76), t= — T/Td,... 

Povodyreveta/. (1997) have developedasix-term Landau expansion crossover scaling model to describe the thermodynamic properties of near-critical binary mixtures, based on the same model for pure fluids and the isomorphism principle of the critical phenomena. The model describes densities and concentrations at vapor-liquid equflibrium and isochoric heat capacities in the one-phase region. The description shows crossover from asymptotic Ising-hke critical behavior to classical (mean-field) behavior. This model was applied to aqueous solutions of sodium chloride. 

It is more difficult to predict the critical amplitudes that depend on microscopic details of the system. The essential issue is to predict how the amplitudes depend on the molecular weight of polymer chains. In the absence of detailed renormalization group calculations some crossover scaling arguments have been evoked to suggest that the critical amplitudes vary with molecular weight according to simple power laws e.g.. 

Fig. 1.8 (a) Log-log plot of the reduced mean square gyration radius, / g(0))/(.Rg(O)). vs. rescaled chain lenrth N - l)((/>f ) where f is the root mean-square bond length, and the theoretical value for the exponent i/(i/ = 0.59) is used. All data refer to a bead-spring model with stiff repulsive Lennard-Jones interaction, as described in Section 1.2.1 (Eq. [1.7]). (b) Same as (a) but for the bond fluctuation model. In both (a) and (b) the straight line indicates the asymptotic slope of the crossover scaling function, resulting from eq. (1.15). (From Gerroff et al.)... 

Fig. 7.13 Crossover scaling of characteristic lengths for a polymer mixture. For t Gi, i.e., -logt to the left of -logGi, one has mean-field behavior, with two characteristic lengths mf oi and I

However, combining data for all chain lengths studied in a crossover scaling plot (Fig. 7.14[b]), better evidence for the theoretical exponents can be obtained. 

Fig. 7.27 Semi-log plot of the effective exponent 7efr of the coilective scattering function iScolK - 0) (upper part)j of the order parameter exponent (middle part) and the ratio of critical amplitudes C /Cl (lower part) versus the crossover scaling variable LmjN, which is proportional to the ratio of characteristic lengths m/fcross- L is the mean size of the lattices used in the finite size scaling analysis. Asymptotic limits are shown as dashed straight lines. Open circles refer to the bond fluctuation model which has (.aa = bb = 0, ab 5 0 for effective monomers two lattice spacings apart (Fig. 7.25), full dots to the version of the bond fluctuation model with ab = aa = bb = e and Zefr rj 14 (critical temperatures of this model are included in Fig. 7.4). Curves through the points are guides to the eye only. (From Deutsch and Binder." )...

Unfortunately, a full description of the crossover from tricritical behavior to nonmean-field critical behavior is a difficult theoretical problem [48]. Here, we shall not dwell on recent developments based on the renormalization group approach, since this is outside the scope of the present review, but we only mention the phenomenological extension of the crossover scaling description, Eqs. (24)-(27), to incorporate the Ising behavior [3, 30,49]. There one starts from the observation that the variable appearing in the Ginzburg criterion, r Gi cx is simply proportional to the... 

Finally we mention that also the collective scattering function 5coii(0) and the correlation length f ( crit) must show a crossover scaling... 

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