Effect of the density-gradient term in the free energy expression on critical exponents

Fluctuations, density gradients, and interfaces near the critical point of one-component fluids

“…From the relationships involved it was possible to infer that10 µ' = y' + 2ß-' (12) It is possible to obtain eq 12 in an entirely different way by consideration of the fluctuations, again by assuming that the boundaries between fluctuations resemble interfaces. We start with the Griffiths inequality11 5 ß( -1) (13) This is believed to be an equality, which in effect states that the behavior of the compressibility is, in the limit, the same along the coexistence curve as along the critical isotherm. From eq 8b the Griffiths inequality may be written v'(2 -n)< ß( -l) which from eq 10 gives…”

“…It does not seem reasonable to suppose that fluctuations in density much greater than the difference in density of two phases which can be in equilibrium could be sustained, so we assume that the equality is correct. Equation 10 can be combined with (13) as an equality…”

Existence of two characteristic lengths in determining the thickness of an interface near the critical point, and the interface profile

“…The central portion will not be an exact straight line, and to connect smoothly on the exponential decay part we will need a different coefficient in the exponential of eq 9 and 10. Let us still approximate the central part of the profile by a straight line, calling the Az defined in this way (i.e., as in Figure 2 and in eq 7 and 8) 2, and we replace eq 12 by b0 = /8 (13) As we shall see, in order to have a consistent picture, and 2 will have to behave slightly differently as the critical point is approached. In general, we may expect the actual curvature of the nearly straight-line midsection of the profile to be such that the slope with which it connects to the exponential drop offs is less than it is at z = 0.…”