Cubic coexistence curve

Fig. 10.7. Phase diagram for a homopolymer of chain length r = 8onal0xl0xl0 simple cubic lattice of coordination number z = 6. Filled circles give the reduced temperature, T and mean volume fraction, () of the three runs performed. Arrows from the run points indicate the range of densities sampled for each simulation. The thick continuous line is the estimated phase coexistence curve. Reprinted by permission from [6], 2000IOP Publishing Ltd...

Figure 9 shows coexistence curves for polymers of 100, 600, 1000, and 2000 sites. The lines are the results of this work, and the open symbols are simulation data from the literature [25]. For n = 100 and n = 600, our results are in good agreement with literature reports. Note, however, that with the new method, we are able to explore the phase behavior of long polymer chains down to fairly low temperatures. The computational demands of the new method are relatively modest. For example, calculation of the full phase diagram for polymer chains of length 2000 required less than 5 days on a workstation. It is important to emphasize that, for the cubic lattice model adopted here, chains of 2000 segments correspond to polystyrene solutions... 

Nearly all experimental coexistence curves, whether from liquid-gas equilibrium, liquid mixtures, order-disorder in alloys, or in ferromagnetic materials, are far from parabolic, and more nearly cubic, even far below the critical temperature. This was known for fluid systems, at least to some experimentalists, more than one hundred years ago. Verschaffelt (1900), from a careful analysis of data (pressure-volume and densities) on isopentane, concluded that the best fit was with p = 0.34 and 5 = 4.26, far from the classical values. Van Laar apparently rejected this conclusion, believing that, at least very close to the critical temperature, the coexistence curve must become parabolic. Even earlier, van der Waals, who had derived a classical theory of capillarity with a surface-tension exponent of 3/2, found (1893)... 

In 1945 Guggenheim [10]. as part of an extensive discussion of the law of corresponding states, showed that, when plotted as reduced temperature 7], versus reduced density p., all the coexistence-curve measurements on three inert gases (Ar, Kr, Xe) fell on a single curve, and that Ne, N2, O2, CO and CH also fit the same curve very closely. Moreover he either rediscovered or re-emphasized the fact that the curve was unequivocally cubic (i.e. p = 1/3) over the entire range of experimental temperatures, writing for p. ... 

For simple fluids Nq is estimated to be about 0.01, and Kostrowicka Wyczalkowska et al [29] have used this to apply crossover theory to the van der Waals equation with interesting results. The critical temperature is reduced by 11% and the coexistence curve is of course flattened to a cubic. The critical density p. is almost unchanged (by 2%), but the critical pressure p is reduced greatly by 38%. These changes reduce the critical... 

Assertions that a quadratic coexistence curve has actually been found are rare, - for either liquids or liquid mixtures. Flat-topped curves have been foimd, cubic curves have been found, and it has definitely been established that w = 9 for a square-lattice gas, > but the coefficient may not have the classical significance given in Eq. (4). 

The effects of gravity on the one-component coexistence curve are severe. > > - In long vertical tubes it is to be expected that large density gradients will occur in the critical region. The effect of these gradients is to cause a flat-topped coexistence ciuve if short tubes are used for the measurements, the curves become rounded and, in fact, cubic. 

Attempts have been made to relate the cubic coexistence curve to Eq. (4) with w = 4 and p — Pe replaced by its absolute value.i >i It cannot be said at the moment that this procedure is incorrect, but it sacrifices the differentiability of P with respect to p at the critical point without, it appears to the author, a corresponding compensation. 

If Gm(T, p, x), the appropriate thermodynamic function for a binary mixture, is analytic, the deductions about the behaviour of the various thermodynamic properties are entirely analogous to those for the one-component fluid. The same conclusions arise from any general Taylor series expansion in which all the coefficients are non-zero except those two required to define the critical point [equations (6a, b)]. In particular, the coexistence curve (T vs. x at constant p) should be parabolic, the critical isotherm vs. x at constant T and p) should be cubic, and the molar heat capacity C, ,m should be everywhere finite. 

However, just as is the case for many other kinds of critical phenomena (e.g. the one-component fluid, magnetism, order-disorder transitions in solids, etc.) such predictions do not agree either with the results of careful experimental measurements or with simple theoretical models that can be treated nearly exactly. The coexistence curve is more nearly cubic than parabolic, the critical isotherm is of distinctly higher order than cubic, and the heat capacity Cp,x,m diverges at the critical point. 

Critical exponents indicate how fluid properties behave near a critical point. They arc defined by Sengers and Anisimov, these proceedings. By inspecting Fig. lb, one would guess that the simplest forms the special curves could take would be a parabola for the density or volume dilTerence along the coexistence curve, and a cubic for the pressure along the critical isotherm as function of volume or density ... 

Fig. 7.30 (a) Normalized mean-square gyration radius at constant — 4>v) = 0.9 plotted vs. inverse temperature for the model of Fig. 7.3 on the mple cubic lattice, choosing N = 32,4>v = 0.6, (.ab = 0, iAA = BB = —f Tbe crosses refer to the majority component (A), the circles to the minority component (B). The largest value of e/kgT shown corresponds to a state at the coexistence curve. (From Sariban and Binder. ) (b) Mean-square gyration radius of an isolated B chain in an A-rich matrix, for the same model as in (a) but three chain lengths. Note that the inverse critical temperatures for these mixtures are e/fcaTss 0.1 (A = 16),0.092(1V = 32), and 0.049(A = 64), respectively. (From Sariban and Binder. ")... 

Fig. 27. Phase diagram of an adsorbed film in- the simple cubic lattice from mean-fleld calculations (full curves - flrst-order transitions, broken curves -second-order transitions) and from a Monte Carlo calculation (dash-dotted curve - only the transition of the first layer is shown). Phases shown are the lattice gas (G), the ordered (2x1) phase in the first layer, lattice fluid in the first layer F(l) and in the bulk F(a>). For the sake of clarity, layering transitions in layers higher than the second layer (which nearly coincide with the layering of the second layer and merge at 7 (2), are not shown. The chemical potential at gas-liquid coexistence is denoted as ttg, and 7 / is the mean-field bulk critical temperature. While the layering transition of the second layer ends in a critical point Tj(2), mean-field theory predicts two tricritical points 7 (1), 7 (1) in the first layer. Parameters of this calculation are R = —0.75, e = 2.5p, 112 = Mi/ = d/2, D = 20, and L varied from 6 to 24. (From Wagner and Binder .)...

The function p(v) at fixed temperature (the isotherm) is shown in Fig. 5.1. The curves 1, 2, 3 correspond to different temperatures. The curve 3 corresponds to a temperature above the critical temperature (T > T ). In this state the curve changes smoothly, pressure falls with increase of o, and the substance can be in equilibrium only in the gaseous form. The second curve corresponds to the critical temperature It is the highest temperature at which liquid and vapor states can coexist in balance with each other. At temperature T < Ti (curve 1) the dependence p o) is non-monotonous. To the left of the point B (line AB) the substance is in the mono-phase liquid state, to the right of point G (line GH) the substance is in the mono-phase vapor state. The region between points B and G corresponds to the equilibrium the bi-phase state liquid - vapor. In accordance with the Maxwell s rule, squares of areas BDE and EFG are equal. From the form of isotherms it follows that in pre-critical area (T < Tc) the cubic equation... 

Fig. 14 Top Profiles of the orientational order parameter S(z) s-shaped curves), and of the volume fraction of lattice sites taken by monomeric units, (z) thick lines), for a bond fluctuation model (see Sect. 2.2) on the 80 x 80 x 1000 simple cubic lattice, and chain length N = 20, with A/" = 1,600 chains in the system, choosing parameters/ = 8.0 (9)and2g = 0.01. Squares indicate the density values at coexistence and open circles indicate the order parameter at the transition. These values were extracted from a bulk grand canonical simulation [123]. The inset shows an enlarged plot of the transition region. Bottom Two-dimensional xz-map of the coarse-grained order parameter profile for a system snapshot corresponding to the graph. From Ivanov et al. [123]...

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