Critical exponents measurements

On the whole, n the sol —> gel transition, we are dealing with a cross-linking of preexisting long chains if these chains are rather concentrated, the above discussion may apply, and we expect a trend towards mean field behavior. Thus many practical cases may be intermediatB between percolation and mean field this may explain (at least partly) the large discrepancies between critical exponents measured in different systems. 

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. 

Figure C2.10.1. Potential dependence of the scattering intensity of tire (1,0) reflection measured in situ from Ag (100)/0.05 M NaBr after a background correction (dots). The solid line represents tire fit of tire experimental data witli a two dimensional Ising model witli a critical exponent of 1/8. Model stmctures derived from tire experiments are depicted in tire insets for potentials below (left) and above (right) tire critical potential (from [15]).

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. 

Above the transition temperature cf> = 0 and below the critical temperature 4> > 0. At T = 0, = 1. Any order parameter defined to have a magnitude of unity at zero temperature is said to be normalized. Criteria for testing the theory include a number of critical exponents, which are accurately known from experimental measurement. 

Figure 4 Deviations of mean field theory (dashed lines) from experimental measurements described by critical exponents (solid lines) near the critical point. Tmf is the critical temperature predicted by mean field theory.

Close to the gel point, in the range AX/X <0.1, the static modulus cannot be measured. Strong relaxation effects are present even at the lowest frequency which could be used, to be consistent with the kinetics (one period of oscillation = 67s). Beyond this range for AX/X >0.1, G (0,015Hz) corresponds to the static relaxed modulus. A critical exponent for the relaxed modulus can be determined by using the equation X - X... 

Figure 6. The log-log plot of the shear modulus G measured at f 0.015 Hz, versus (X - X )/X. The slopes of the continuous lines give the critical exponents t.

Additional comment deserve magnetostriction measurements near the ordering temperature 7c reflecting critical phenomena. Few data for critical expansion is available, such as have been reported by Dolejsi and Swenson (1981) for the case of Gd metal. The thermal expansion coefficient in the critical region should assume the form 1(7 — Tc)/Tc °-The critical exponent or should be the same as for the specific heat and depend only on the universality class (dimensionality, No. of degrees of freedom) of the system. For Gd metal this universality class has been determined by Frey et al. (1997). 

The kinetic equations [1-3, 12] were rewritten in [15] for a special choice of the recombination law cr(r), adequate to the NAN model, and solved numerically for d— 1. The general conclusion was drawn that the Kirkwood approximation is quite correct but leads to the error of the order of 10% for the critical exponent a in the asymptotic decay law n(R) oc R a. This quantity (10%) was suggested to be used as a measure of the accuracy of the Kirkwood approximation in the kinetics of the bimolecular reaction A + B -> 0. 

Oleinikova and Bonetti [133] extended this type of measurements to Bu4NPic dissolved in 1-dodecanol, in 1,4-butanediol, and in mixtures of these solvents. From all systems employed in conductance measurements, Bu4NPic + 1-dodecanol may come closest to a Coulombic fluid. The anomaly becomes notable near T = 10 2. The anomalous contribution never exceeded a few percent of the background contribution, which makes any decisive determination of the critical exponent extremely difficult. If correction exponents were included, the fits gave the best results for a (1 - a) anomaly. No essential differences between results for 1-dodecanol and 1,4-butanediole systems seem to be present. 

Not only do the thermodynamic properties follow similar power laws near the critical temperatures, but the exponents measured for a given property, such as heat capacity or the order parameter, are found to be the same within experimental error in a wide variety of substances. This can be seen in Table 13.3. It has been shown that the same set of exponents (a, (3, 7, v, etc.) are obtained for phase transitions that have the same spatial (d) and order parameter (n) dimensionalities. For example, (order + disorder) transitions, magnetic transitions with a single axis about which the magnetization orients, and the (liquid + gas) critical point have d= 3 and n — 1, and all have the same values for the critical exponents. Superconductors and the superfluid transition in 4He have d= 3 and n = 2, and they show different values for the set of exponents. Phase transitions are said to belong to different universality classes when their critical exponents belong to different sets. 

Salje EKH, Wrack B (1983) Specific-heat measurements and critical exponents of the ferroelastic phase transiton in Pb3(P04)2 and Pb3(Pi xAsx04)2. Phys Rev B 28 6510-6518 Salje EKH, Devarajan V, Bismayer, U, Guimaraes DMC (1983) Phase transitions in Pb3(Pi.xAsx04)2 influence of the central peak and flip mode on the Raman scattering of hard modes. J Phys C 16 5233-52343... 

A fit of the data for anorthite to Equations 7 and 8 describes the change in chemical shift with temperature in the PI phase (Fig. 18) and yields a value for the critical exponent, P = 0.27( 0.04), that is consistent with measurements using techniques sensitive to muchjonger length scales, such as X-ray diffraction (Redfem et al. 1987), that indicate the PI -71 transition in Si,Al ordered anorthite is tricritical. 

Measurements of the liquid gas transition of salts, e.g. of NH4CI at 1150 K [12] that yielded the mean field value of 3 can hardly achieve the mK accuracy required in work on critical phenomena. Therefore, measurements of the liquid-liquid coexistence occurring near ambient conditions are more apt for measuring critical exponents. 

To test this theory, the room temperature conductivity of "Nafion" perfluorinated resins was measured as a function of electrolyte uptake by a standard a.c. technique for liquid electrolytes (15). The data obey the percolation prediction very well. Figure 9 is a log-log plot of the measured conductivity against the excell volume fraction of electrolyte (c-c ). The principal experimental uncertainty was in the determination of c as shown by the horizontal error bars. The dashed line is a non-linear least square law to the data points. The best fit value for the threshold c is 10% which is less than the ideal value of 15% for a completely random system. This observation is consistent with a bimodal cluster distribution required by the cluster-network model. In accord with the theoretical prediction, the critical exponent n as determined from the slope of... 

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