Critical Exponents and Scaling

Computer simulation in space takes into account spatial correlations of any range which result in Intramolecular reaction. The lattice percolation was mostly used. It was based on random connections of lattice points of rigid lattice. The main Interest was focused on the critical region at the gel point, l.e., on critical exponents and scaling laws between them. These exponents were found to differ from the so-called classical ones corresponding to Markovian systems irrespective of whether cycllzatlon was approximated by the spanning-tree... 

Tang, C. and Bak, P., Critical exponents and scaling relations for self organized critical phenomena, Physical Review Letters, Vol. 60, 1988, pp. 2347-2350. 

In this chapter we provide a heuristic introduction to scaling procedures that characterize properties of systems close to their critical point. The objective is to provide some general insights and to convey the flavor of the methodology. Much of the treatment of critical phenomena falls into the province of statistical mechanics, well outside the confines of classical thermodynamics. However, certain illuminating aspects that can be discussed without having to resort to statistical approaches will be taken up below. We center the discussion on critical exponents and the Landau theory of critical phenomena. For a proper exposition of the subject the reader is referred to several sources in the literature. ... 

We have thereby specified the scaling exponents in terms of several experimentally observed critical exponents and have succeeded in providing a thermodynamic description of the properties of materials near their critical points. 

In view of the anomalous critical behavior of the correlation length and the osmotic compressibility, it appeared of interest to characterize the behavior of other properties. Bell-ocq and Gazeau investigated how the interfacial tension between the coexisting phases on the one hand and the difference of density of these phases on the other hand vanished at various points of the critical line P (Fig. 25) [152]. The aim of the experiments was to determine the associated critical exponents and and check whether the scaling laws that relate v,p, and f.i were valid all along the critical line. Data obtained for two critical points defined by Xc = 1.55 and Xc = 1.207 indicate that the values of the critical exponents and )U show an X dependence similar to that found for v and y. Furthermore, within the experimental accuracy, the obtained values of v, y, (i, and are in reasonable agreement with the theoretical predictions v = y 2/ = 3v (Table 2). 

It should be mentioned here that the RK equation of state and most others used are analytical, resulting in so-called classical critical exponents, which is in contradiction to non-classical behavior found in very accurate experiments. Since normally no effects extremely near to critical states have to be considered, it is not necessary to use non-analytical relations that would be in accordance with the accurate critical exponents and the scaling laws [52]. 

Scaling Fields, Critical Exponents, and Critical Amplitndes... 

Here a,b > 0 are dynamic critical exponents and F is a suitable scaling function (3 is some temperature-like parameter, and /3e is the critical point. Now suppose that F is continuous and strictly positive, with F x) decaying rapidly (e.g., exponentially) as x —> oo. Then it is not hard to see that... 

The equation of state detemiined by Z N, V, T ) is not known in the sense that it cannot be written down as a simple expression. However, the critical parameters depend on e and a, and a test of the law of corresponding states is to use the reduced variables T, and as the scaled variables in the equation of state. Figure A2.3.5 bl illustrates this for the liquid-gas coexistence curves of several substances. As first shown by Guggenlieim [19], the curvature near the critical pomt is consistent with a critical exponent (3 closer to 1/3 rather than the 1/2 predicted by van der Waals equation. This provides additional evidence that the law of corresponding states obeyed is not the fomi associated with van der Waals equation. Figure A2.3.5 (b) shows tliat PIpkT is approximately the same fiinction of the reduced variables and... 

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

Many of the earlier uncertainties arose from apparent disagreements between the theoretical values and experimental detemiinations of the critical exponents. These were resolved in part by better calculations, but mainly by measurements closer and closer to the critical point. The analysis of earlier measurements assumed incorrectly that the measurements were close enough. (Van der Waals and van Laar were right that one needed to get closer to the critical point, but were wrong in expectmg that the classical exponents would then appear.) As was shown in section A2.5.6.7. there are additional contributions from extended scaling. 

In Eq. (15) the second term reflects the gain in entropy when a chain breaks so that the two new ends can explore a volume Entropy is increased because the excluded volume repulsion on scales less than is reduced by breaking the chain this effect is accounted for by the additional exponent 9 = y — )/v where 7 > 1 is a standard critical exponent, the value of 7 being larger in 2 dimensions than in 3 dimensions 72 = 43/32 1.34, 73j 1.17. In MFA 7 = 1, = 0, and Eq. (15) simplifies to Eq. (9), where correlations, brought about by mutual avoidance of chains, i.e., excluded volume, are ignored. 

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... 

Thus, z and pc can be approximated by finding the intersection of the functions Y N + 1) and Y N) the iV —) 00 result is obtained by extrapolation. The other critical exponents may be obtained through scaling of the corresponding partial derivatives of t and the usual scaling relations. 

Precise knowledge of the critical point is not required to determine k by this method because the scaling relation holds over a finite range of p at intermediate frequency. The exponent k has been evaluated for each of the experiments of Scanlan and Winter [122]. Within the limits of experimental error, the experiments indicate that k takes on a universal value. The average value from 30 experiments on the PDMS system with various stoichiometry, chain length, and concentration is k = 0.214 + 0.017. Exponent k has a value of about 0.2 for all the systems which we have studied so far. Colby et al. [38] reported a value of 0.24 for their polyester system. It seems to be insensitive to molecular detail. We expect the dynamic critical exponent k to be related to the other critical exponents. The frequency range of the above observations has to be explored further. 

It was shown by Wilson [131] that the Kadanoff procedure, combined with the Landau model, may be used to identify the critical point, verify the scaling law and determine the critical exponents without obtaining an exact solution, or specifying the nature of fluctuations near the critical point. The Hamiltonian for a set of Ising spins is written in suitable units, as before... 

This expression has the form of the scaling hypothesis (5) and the critical exponents may be written down in terms of / and g using (6). 

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