Critical point exponents

Remarkably different molar mass dependencies are obtained with randomly branched or randomly crosshnked macromolecules. Often, below the critical point exponents v in are found which are close to v=0.5, and sometimes... 

Many physical properties diverge near (i.e. show large values as is approached from either side). Interestingly, divergences of similar quantities in different phase transitions are strikingly similar as shown by the typical Cp-temperature curves in Fig. 4.8. These divergences can be quantified in terms of the so-called critical exponents. A critical point exponent is given by... 

These equations may be regarded as the definitions of the critical point exponents a, jS, y and S. 

While 9.2 is a necessary digression on tiie theory of tiie oorrdation function near the critical point, 9.3 is an equally necessary digression on critical-point exponents what their non-daisical (non-mean-field-theory) values are, and how, according to the ideas of critical-point scaling and homogeneity, they must be related to eadi other. Then in 9.4 we shall see how one incorporates non-dassical exponents into the framework of the van der Waals theory of the interface. 

The exponent t), along with the exponents we already took note of in 9.1 and others that we shall introduce, describe the analytic form of thermodynamic functions and correlation functions near the critical point, and, in particular, index the critical-point singularities of those functions. In 9.3 we shall see how the many critical-point exponents are related to each other, and what their values are, both in the classical, mean-field theories and in reality. 

In 9.1 and 9.2 we saw thermodynamic functions and parameters in correlation functions vanishing or diverging at a critical point proportionally to some power of the distance—often measured as (T-T )—from that point. Those powers arc the critical-point exponents, central to any discussion of critical phenomena. Here we define and discuss those that are most frequently referred to, and to which we shall ourselves refer in the remaining sections of this chapter. 

We list in Table 9.1 the values of the critical-point exponents a, y, 8, V, T, and il. In the first column are the classical, mean-field-theory values, in the second column the values for the two-dimensional Ising model (lattice gas), which are also known exactly, and in the third column those for real, three-dimensional fluids as determined from present-day critical-point theory, " from experiment, or from both. As theoretical values the latter are probably correct to 0-01, but as experimental values their uncertainties may be two or three times that. 

There are further relations among critical-point exponents in whidi the dimensionality d appears explicitly. The equilibrium fluctuations of the density about its average value are coherent (that is, are of one sign) over distances of order These fluctuations that thus occur spontaneously in regions of linear dimension are the elementary density fluctuations with each of which is associated a free energy of order kT — k H. The free-energy density associated with equilibrium density (or composition) fluctuations near a critical point is thus of order But... 

We turn next to the question of how to modify the mean-field theory of the near-critical interface that was outlined in 8 9.1, so as to incorporate in it the correct, non-classical values of the critical-point exponents. 

The analytical forms of the thermodynamic functions at a critical point are universal, the same for all substances. That includes the critical-point exponents and the functional forms, up to multiplicative constants, of functions such as u(x) or T°-r(p) whfle ffie locations of the critical points (the values of T°, p , etc.) and the amplitudes of the scaling functions (the multiplicative constants, such as that in (9.51)) are non-universal, varying from substance to substance, llie principle of universality goes beyond the principle of corresponding states, for it does not require a universal form of the intermolecular potentials but it would be implied by the principle of corresponding states and is in practice difficult to distinguish from it. 

H. K. Kim, M. H. W. Chan, Experimental determination of a two-dimensional liquid-vapor critical-point exponent, Phys. Rev. Lett. 53(1984)170-173. 

This is the well known equal areas mle derived by Maxwell [3], who enthusiastically publicized van der Waal s equation (see figure A2.3.3. The critical exponents for van der Waals equation are typical mean-field exponents a 0, p = 1/2, y = 1 and 8 = 3. This follows from the assumption, connnon to van der Waals equation and other mean-field theories, that the critical point is an analytic point about which the free energy and other themiodynamic properties can be expanded in a Taylor series. 

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. 

The assumption that the free energy is analytic at the critical point leads to classical exponents. Deviations from this require tiiat this assumption be abandoned. In mean-field theory. 

An exponent a governs the limiting slope of the molar heat capacity, variously y, ( or along a line tln-ongh the critical point,... 

A third exponent y, usually called the susceptibility exponent from its application to the magnetic susceptibility x in magnetic systems, governs what m pure-fluid systems is the isothennal compressibility k, and what in mixtures is the osmotic compressibility, and detennines how fast these quantities diverge as the critical point is approached (i.e. as > 1). 

It is curious that he never conuuented on the failure to fit the analytic theory even though that treatment—with the quadratic fonn of the coexistence curve—was presented in great detail in it Statistical Thermodynamics (Fowler and Guggenlieim, 1939). The paper does not discuss any of the other critical exponents, except to fit the vanishing of the surface tension a at the critical point to an equation... 

However, the discovery in 1962 by Voronel and coworkers [H] that the constant-volume heat capacity of argon showed a weak divergence at the critical point, had a major impact on uniting fluid criticality widi that of other systems. They thought the divergence was logaritlnnic, but it is not quite that weak, satisfying equation (A2.5.21) with an exponent a now known to be about 0.11. The equation applies both above and... 

The brackets symbolize fiinction of, not multiplication.) Smce there are only two parameters, and a, in this expression, the homogeneity assumption means that all four exponents a, p, y and S must be fiinctions of these two hence the inequalities in section A2.5.4.5(e) must be equalities. Equations for the various other thennodynamic quantities, in particular the singidar part of the heat capacity Cy and the isothemial compressibility Kp may be derived from this equation for p. The behaviour of these quantities as tire critical point is approached can be satisfied only if... 

If the finite size of the system is ignored (after all, A is probably 10 or greater), the compressibility is essentially infinite at the critical point, and then so are the fluctuations. In reality, however, the compressibility diverges more sharply than classical theory allows (the exponent y is significantly greater dian 1), and thus so do the fluctuations. 

Moreover, well away from the critical point, the range of correlations is much smaller, and when this range is of the order of the range of the intenuolecular forces, analytic treatments should be appropriate, and the exponents should be classical . The need to reconcile the nonanalytic region with tlie classical region has led to attempts to solve the crossover problem, to be discussed in section A2.5.7.2. 

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. 

Alone among all known physical phenomena, the transition in low-temperature (T < 25 K) superconducting materials (mainly metals and alloys) retains its classical behaviour right up to the critical point thus the exponents are the analytic ones. Unlike the situation in other systems, such superconducting interactions are tndy long range and thus... 

P is the critical exponent and t denotes the reduced distance from the critical temperature. In the vicinity of the critical point, the free energy can be expanded in tenns of powers and gradients of the local order parameter m (r) = AW - I bW ... 

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