Critical exponent values

With the relations given in Table 2.2-1 and the critical exponent values given in Table 2.2-2, the thermodynamic behaviour of a pure component close to the critical point can be described exactly, however further away from the critical point also the mean field contributions have to be taken into account. A theory which is in principle capable to describe... 

Several paradoxes have become apparent from modern descriptions of phase transitions, and these have driven much of the research activity in this field. The intermolecular interactions that are responsible for the phase transition are relatively short-ranged, yet they serve to create very long-range order at the transition temperature. The quantum mechanical details of the interactions governing various transitions are very different, and the length scales over which they operate vary considerably, yet the observation of scaling laws and the equivalences of a given critical exponent value within a fixed dimensionality of the order parameter show that some additional principle not described by quantum mechanics must also be at work. Also, the partition... 

Other physical properties like correlation lengths and percolation probabilities follow as well power laws in the vicinity of pc, however with different critical exponents. Values of the critical exponent /r in Equation 3.102 are known in 2D and 3D from computer simulations (Isichenko, 1992). For lattice percolation in 2D, it is u. 1.3... 

In contrast to critical exponents, values of percolation thresholds depend on the lattice topology and the type of the percolation problem considered. Percolation thresholds for a few known lattice types are listed in Table 3.2. In ID, it is trivially Pc = 1. In 2D, values forpc are known exactly for specific lattice types. In 3D, values of Pc can only be found with the help of computer simulations. 

The results of the conductivity measurements are given in Figure 4.10 [/) along with the linear fittings of the equation given in Table 4.1. The determined percolation thresholds, ultimate conductivities, and critical exponent values are summarized in Table 4.3. 

From the knowledge of g and the tangent of p g ) and the definition of the critical exponents, values of the critical exponents can be obtained. 

J.V. Sengers, J.G. Shanks, Experimental critical exponent values for fluids. J. Stat. Phys. 137, 857 (2009)... 

Table A2.3.4 simnnarizes the values of these critical exponents m two and tliree dimensions and the predictions of mean field theory.

This implies that the critical exponent y = 1, whether the critical temperature is approached from above or below, but the amplitudes are different by a factor of 2, as seen in our earlier discussion of mean-field theory. The critical exponents are the classical values a = 0, p = 1/2, 5 = 3 and y = 1. 

The scientific shidies of the early 1970s are fiill of concern whether the critical exponents detemiined experimentally, particularly those for fluids, could be reconciled with the calculated values, and at times it appeared that they could not be. However, not only were the theoretical values more uncertain (before RG calculations) than first believed, but also there were serious problems with the analysis of the experiments, in addition to those associated with the Wegner... 

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. 

Another interesting version of the MM model considers a variable excluded-volume interaction between same species particles [92]. In the absence of interactions the system is mapped on the standard MM model which has a first-order IPT between A- and B-saturated phases. On increasing the strength of the interaction the first-order transition line, observed for weak interactions, terminates at a tricritical point where two second-order transitions meet. These transitions, which separate the A-saturated, reactive, and B-saturated phases, belong to the same universality class as directed percolation, as follows from the value of critical exponents calculated by means of time-dependent Monte Carlo simulations and series expansions [92]. 

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. 

Comparing to equation 7.12, we thus see that the mean-field value for the critical exponent (3 exists and is given by... 

In both equations, k and k are proportionality constants and 0 is a constant known as the critical exponent. Experimental measurements have shown that 0 has the same value for both equations and for all gases. Analytic8 equations of state, such as the Van der Waals equation, predict that 0 should have a value of i. Careful experimental measurement, however, gives a value of 0 = 0.32 0.01.h Thus, near the critical point, p or Vm varies more nearly as the cube root of temperature than as the square root predicted from classical equations of state. 

The form of equations (8.11) and (8.12) turns out to be general for properties near a critical point. In the vicinity of this point, the value of many thermodynamic properties at T becomes proportional to some power of (Tc - T). The exponents which appear in equations such as (8.11) and (8.12) are referred to as critical exponents. The exponent 6 = 0.32 0.01 describes the temperature behavior of molar volume and density as well as other properties, while other properties such as heat capacity and isothermal compressibility are described by other critical exponents. A significant scientific achievement of the 20th century was the observation of the nonanalytic behavior of thermodynamic properties near the critical point and the recognition that the various critical exponents are related to one another ... 

A is a constant and p is the critical exponent which adopts values from 0.3 to 0.5. Values around p = 0.5 are observed for long-range interactions between the particles for short-range interactions (e.g. magnetic interactions) the critical exponent is closer to p 0.33. As shown in the typical curve diagram in Fig. 4.2, the order parameter experiences its most relevant changes close to the critical temperature the curve runs vertical at Tc. 

Only the value of the relaxation exponent is needed. The critical exponent a of the longest relaxation time (compare Eqs. 1-6 and 1-7) is therefore on an equal footing with the critical exponent of the viscosity ... 

Our first two critical gels had an exponent value of n as 0.5, which made us believe initially that this would be the only possible value... 

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. 

Similar methods produce the following values of the critical exponents ... 

Scaling laws provide an improved estimate of critical exponents without a scheme for calculating their absolute values or elucidating the physical changes that occur in the critical region. 

Further reading)). Monte Carlo simulations have thus played, for example, an important role in developing an understanding of behaviour approaching critical points, and provided valuable insights, for instance, into the fundamental physics responsible for the values of critical exponents. 

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