Isochore critical

The specific heat along the critical isochore hence has the same smgularity as (5 P /5T )p for a lattice gas. 

Finally, we consider the isothennal compressibility = hi V/dp)y = d hi p/5p) j, along tlie coexistence curve. A consideration of Figure A2.5.6 shows that the compressibility is finite and positive at every point in the one-phase region except at tlie critical point. Differentiation of equation (A2.5.2) yields the compressibility along the critical isochore ... 

Figure 8,6 (A) P-T regions of H2O stability. (B) Extended P-T field showing location of the critical region. The critical isochore is p = 0.322778 g/cm. From Johnson and Norton (1991), American Journal of Science, 291, 541-648. Reprinted with permission of American Journal of Science.

In spite of the constant density of the gel, the friction of the poly(N-isopropylacrylamide) gel reversibly decreases by three orders of magnitude and appears to diminish as the gel approaches a certain temperature. This phenomenon should be universal and may be observed in any gel under optimal experimental conditions of the solvent composition and the temperature because the unique parameter describing the friction is the correlation length which tends to diverge in the vicinity of the volume phase transition point of gels. The exponent v for the correlation length obtained from the frictional experiment is far from the theoretical value. It will, therefore, be important to study a poly(N-isopropylacrylamide) gel prepared at the critical isochore where the frictional property of gel may be governed by the critical density fluctuations of the gel. 

The order-parameter fluctuations are temperature- and system-dependent and their decay rate is related to the transport coefficients (5) Usually the magnitude of the fluctuations are characterized by a correlation length . Along a critical isochore or isopleth, the correlation length diverges as... 

Correlation function measurements were made along four critical isochores for each of the three systems CO -n-heptane,... 

Plot of the measured decay rates and the calculated critical and background contributions along a critical isochore of the.benzene-CO system at 97.9 mol % CO and Pc = 0.546 g/cm. ... 

On the basis of this brief summary of RPM criticality, one might be tempted to conclude that the problem has been solved all finite-size scaling analysis point towards the Ising universality class. There is, however, one critical phenomenon which does not seem to have been demonstrated unambiguously in the RPM. This is the critical divergence of the constant-volume heat capacity, Cy. Recall that on the critical isochore and close to the critical temperature where the parameter t = (T — Tc)/Tc is small,... 

On the basis that the asymptotic interactions between CHDs are dipolar, and that polar fluids are expected to exhibit short-range criticality [42], one might expect C along the critical isochore to exhibit a strong peak in the vicinity of Tc. In Fig. 3 we show Cy along the critical isochore from NVT... 

Figure 3. Constant-volume heat capacity, Cy, for the CHD fluid on the critical isochore as obtained from NVT MC simulations fluctuation formula (6) (points) from a [5,5] Pade approximant (9) fitted to the energy (solid line). The kinetic contribution is not included.

We now turn to measurements of the constant-volume heat capacity, Cy, along the critical isochore. In Fig. 5 we show Cy as measured in NVT and pVT simulations of a system with a = 6 and L = lOer. The bulk critical... 

Figure 5. Constant-volume heat capacity, Cy, of the AHS fluid with a = 6 as obtained from NVT (filled circles and solid line) and pVT (open circles and dashed line) MC simulations of systems with L = 10a along the bulk critical isochore. The kinetic contribution (equal to SNIcb/Z) is included.

Mixed-field finite-size scaling studies of the CHD fluid - in which cation-anion pairs are fused together - are consistent with Ising criticality. When calculated in NVT MC simulations, Cy along the critical isochore shows no sign of a near-critical peak. 

For a lattice gas on its critical isochore, (4.68) is exact, and it is for this reason that Stell and Htzfye chose to use it in their study of the critical behavior of S2, which we summarize in the next section. 

Here 2 is the exponent for the heat capacity measured along the critical isochore (i.e. in the two-phase region) below the critical temperature, while is the exponent for the isothermal compressibility measured in the one-phase region at the edge of the coexistence curve. These inequalities say nothing about the exponents a and y in the one-phase region above the critical temperature. 

In our experience, a necessary but insufficient condition for a well-behaved critical isotherm is that, at the critical point, the slope of the critical isochore from the equation of state be equal to the slope of the vapor-pressure equation, 6P/6T = dP /dT. This constraint always is applied in the following work via the least-squares program (7). 

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