Coexistence lines parameters

In general the width of the coexistence line (Ap, Ax, or AM) is proportional to an order parameter s, and its absolute value may be written as... 

Figure 16.5 Comparison between the self-consistently solved coexistence line and the cloud points (filled triangle) of iPP/EPDM and the melting points (filled circles). The solidus and liquidus lines are virtually overlapped (dots), but the existence of both fines is manifested by the kink in the LCST coexistence line. The phase diagram was calculated using the material parameters, AHufp = 2110calmol , = 162.5°C, rj>p = 1800, repoM = 1000, and = 0.8 at T. ...

Fig. 11.3 Comparisons of phase diagrams of symmetric polymer blends (same chain lengths 32 monomers, only one component crystallizable) obtained from simulations (data point with the same labeled sequences as the solid lines) and from mean-field theory (solid lines for binodals, and dashed lines for liquid-solid coexistence lines) in the cubic lattice space 32. The x-axis is the volume fraction of crystallizable comptment, and the y-axis is the reduced temperature. The data labeled near the solid lines are the reduced eneigy parameter B/Ec, and all the cinves have EpIEc = 1 (Ma et al. 2007) (Reprinted with permission)...

The vapor and liquid density Qi and and the vapor pressure p along the coexistence line have been represented by multi-term functions, comprising 7 adjustable parameters for Qi, 13 for Qv. 9 for Pv (only 6 are defined for the latter by the general form of the vapor pressure equation presented in the orginal reference [2]). Some of the numerical values given between the triple point and the critical point in 5-K intervals in [2] are presented in the following table. 

The notion [136], and accumulating experimental evidence [139], that the the BPni-isotropic coexistence line might terminate in a critical point has stimulated recent theoretical work by Lubensky and Stark [148], Letting Q = Qij denote the customary order parameter tensor, they assume a new, pseudoscalar order parameter formed from the chiral term in the free energy... 

In a conventional helical structure, S q, where S = <(3 cos 0 — l)/2> is the scalar order parameter and q = InlP-, in general, contains information on both the chirality and the order itself This new order parameter is discontinuous across the coexistence line, with the discontinuity decreasing to zero at the critical point. In this sense, the BPIII-isotropic transition is analogous to the liquid gas transition. 

Figure A3.3.5 Tliemiodynamic force as a fiuictioii of the order parameter. Three equilibrium isodiemis (fiill curves) are shown according to a mean field description. For T < J., the isothemi has a van der Waals loop, from which the use of the Maxwell equal area constmction leads to the horizontal dashed line for the equilibrium isothemi. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T < T. The states within the spinodal curve are themiodynaniically unstable, and those between the spinodal and coexistence...

Since industrial separation processes operate in the Li L2 region, it is important to determine how the Margules parameters affect the shape of the coexistence curve and the slope of the tie lines. For any liquid-liquid region to exist, at least one of the binary Margules constants must be greater than 2RT(on y positive values are considered here) this is a consequence of the... 

A simplified parameter space diagram obtained numerically [168] is shown in Fig. 13. The dashed lines bound the region in which both the linear and nonlinear responses of period 1 coexist. The upper line marks the boundary of the linear response, and the lower line marks that for the nonlinear responses. The boundaries of hysteresis for the period 1 resonance are shown by solid lines. The region in which linear response coexists with one or two nonlinear responses of period 2 is bounded by dotted lines. This region is similar to the one bounded by dashed lines. The region of coexistence of the two resonances of period 2 is bounded by the dashed-dotted line. Chaotic states are indicated by small dots. The chaotic state appears as the result of period-doubling bifurcations, and thus corresponds to a nonhyperbolic attractor [167]. Its boundary of attraction Sfl is nonfractal and is formed by the unstable manifold of the saddle cycle of period 1 (SI). 

Figure 13. Phase diagram of the system (35) on the (to, H) plane obtained numerically for the parameter values T = 0.025, coo = 0.597, P = 1, y = 1. See text for a description of the symbols the various lines are guide to the eye. The working point P, with ay = 0.95, h = 0.13, shown by a thick plus, was chosen to lie in the region of coexistence of the period 1 stable limit cycle and of the strange attractor [168],...

Figure 3. Probability distribution of the number N of particles in an LJ fluid. The simulations use the HPT method described in Section IV.B.l. The solid line shows the distribution for a replica whose p—T parameters lie close to coexistence the dashed line (offset) shows the distribution (for the same p-T parameters) obtained by folding in (explicitly) the contributions of all replicas, using multihistogram reweighting. (Taken from Fig. 2 of Ref. 38.)...

We have taken a diffraction pattern of a second sample at 11 K. The degree of contamination by unreacted Cso is slightly less in that sample. We achieved similar R factors in Rietveld fits. The lattice parameters were a = 15.807(1) A, = 12.785(1) A, c = 9.859(1) A, =94.02(1)°. We have also studied a sample prepared from C o dissolved in toluene, described in ref. 5, which had a smaller saturation moment of 0.11/xb per formula unit, It showed a much smaller amount of coexisting f.c.c. Cfto, but essentially the same monoclinic pattern. This shows that this is a simple two-phase system, and hence confirms the validity of ignoring the f.c.c. Qo peaks in the refinement of Fig. 1. Interestingly, the diffraction lines from this sample were not uniformly sharp in particular, lines with h and / both nonzero were so broad as to be virtually undetectable. Evidently, there were solvent molecules present, which acted both to degrade the lattice perfection and the magnetic properties. 

Expressions obtained for the free energy, pressure and chemical potentials can be used to study the thermodynamic properties of the electrolyte solutions, in particular, to describe the phase diagram of ionic fluids. Such a possibility is illustrated in Fig. 7, which shows the effect of ion pairing on liquid-liquid coexistence curve in the ion-dipole model as a function of the ion concentration a = ()%/(pi + ps) and reduced temperature T = (Ms)-1/2, bs = Rpl/Rl The solid line corresponds to the ion-dipole model with the parameter of ion association, B = 10. The dashed line corresponds to the ion-dipole... 

Fig. 16. Projection of the global phase diagram for a compressible mixture of hexadecane, C16H34, and carbon dioxide, CO2, into the temperature-pressure plane for two values of the mixing parameter, = 1 square) and = 0.886 triangle). Simulation results for the liquid-vapor coexistence of the pure components are shown by solid lines and end in critical points that are indicated by arrows. The line of critical points that emerges from the critical point of the less volatile polymer component is indicated by symbols. Prom Virnau et al. [40]...

Fig. lO.a The inset shows the postulated variation of the solubility parameter 8 caused by deuterium labeling (symbols and V correspond to labeled and nonlabeled copolymers, respectively) and due to the change in ethyl ethylene fraction x. The cumulative analysis, described in text, yields the absolute 8 value for deuterated dx (A) and protonated hx (V) copolymers as a function of x at a reference temperature Tref=100 °C determined interaction parameters (as in Fig. 9) allow us to determine two sets of differences AS adjusted here to fit independent PVT data [140,141] measured at 83 °C ( ) and at 121 °C (O). b The interaction parameter, yE/EE, arising from the microstructural difference contribution to the overall effective interaction parameter (hxj/dxpej) in Eq. (19) as a function of the average blend composition (xi+Xj)/2 at a reference temperature of 100 °C.%E/ee values are calculated (see text) from coexistence data ( points correspond to [91,143] and O symbols to [136]) for blend pairs, structurally identical but with swapped labeled component. X marks %e/ee yielded directly [134] for a blend with both components protonated. Solid line is the best fit to data... 

Figure 3. Evolution of isotherms in the P p phase diagram near gas + liquid critical point. Cl -gas + liquid. Red lines (online) are coexistence curves green lines (online) are spinodals. Critical point location tici =0.7949e-3, Zci = 0.0284, ya = 0.0678. Model parameter set Bi, =2.27, Ur,Ua =2,D,=10.29.

Figure 6. Evolution of isochors in the P - 7 phase diagram for the core softened potential with third critical point in metastable region. Cl - gas + liquid, C2 - LDL + LIDL, and C3 - HDL + VHDL critical points. Red lines (online) are coexistence curves. Blue curves (online) are isochors. Critical point location na = 0.0064, Xa = 0.1189, ya =0.0998 nc2 = 0.1423, Xc2 = 0.3856, yc2 = 0.33 Ties = 0.07487, xcs = 0.2398, yes = 0.6856. Model parameter set a = 6.962, bh =2.094, Ur/Ua=3, b,=7.0686.

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