Vapor spinodal

Imre, A.R., Mayer, G., Hazi, G., Rozas, R., and Kraska, T. (2008) Estimation of the liquid-vapor spinodal from interfacial properties obtained from molecular dynamics and lattice Boltzmann simulations, J. Chem. Phys. 128, 114708... 

Here, line ABDFG represents the pressure-volume relationship for a temperature lower than the critical temperature. The liquid phase can exist in a metastable state along line BC and the vapor phase can similarly exist along line FE. The dotted line CDE represents an unstable region. Points C and E, representing the limits of the metastable region, are usually referred to as spinodal points, and these points have loci (for different isotherms) along the lines labeled Liquid Spinodal and Vapor Spinodal in Fig. 15.2. 

Figure 7. Time dependence of the wavelength of the fastest growing density fluctuation (Affiax) during a molecular dynamics simulation of isothermal liquid-vapor spinodal decomposition in the three-dimensional Lennard-Jones fluid kT/e = 0.8 p

FigureZ. PhasedlagramofwaterproposedbySpeedy [18] plotted using the lAPWSEoS [25,26]. The solid curves are equilibrium lines, the dotted curve is the liquid-vapor spinodal, and the dashed curve the LDM. The circles show the experimental determination of the LDM at negative pressure [27]. When the spinodal and the LDM meet, the spinodal pressure reaches a minimum, and (for this EoS) retraces to positive pressure at low temperature. However, note that this would imply an improbable crossing between the spinodal and the metastable liquid-vapor equilibrium (see text for details).

At negative pressure, these scenarios differ with respect to the shapes of the LDM and of the liquid-vapor spinodal curve Ps T) scenario (i) predicts a monotonic LDM and a minimum of Ps as a function of temperature, whereas scenarios (ii) and (iii) predict a turning point in the LDM and a monotonic spinodal. In an experiment, it is difficult to reach the spinodal rather, the liquid will break before by nucleation of vapor bubbles (cavitation). Usually, impurities favor heterogeneous nucleation, and lead to irreproducible results. But for a pristine system, nucleation will occur homogeneously, at a well-defined pressure threshold / cav(T), which is an intrinsic property of the liquid. 

Figure 4. Cavitation pressure as a fimction of temperature for two scenarios for water reentrant spinodal scenario (a) and liquid-liquid critical point scenario (b). These scenarios predict a different temperature behavior for the liquid-vapor spinodal (dotted curve), eitha- with a minimum (a. based on extrapolation of positive pressure data [40]), or monotonic (b. based on molecular dynamics simulations with theTIPSP potential [41]). The solid curve shows the prediction of CNT based on the bulk surface tension of water it becomes unphysical when it goes beyond the liquid-vapor spinodal. The dashed curve is the DPT prediction [40] that correctly remains above the spinodal, and reflects its temperature dependence.

A feed composition in the metastable set is stable to infinitesimal composition disturbances but is unstable to finite ones hence, for such a composition phase splitting can only occur by nucleation, and not simply by Brownian motion (which, at most, supposedly results in infinitesimal composition disturbances). Hence, a metastable composition may be observed as a one-phase system in the laboratory Superheated liquids and subcooled vapors are elementary one-component examples. In contrast with this, one-phase spinodal compositions, by virtue of being unstable to infinitesimal perturbations, will never be observed in the laboratory. 

Fig. 16. Calculated phase diagram of the soft-sphere plus mean-field model, showing the vapor-liquid (VLB), solid-liquid (SI.E ), and solid-vapor (SVE) coexistence loci, the superheated liquid spinodal (s), and the Kauzmann locus (K) in the pressure-temperature plane (P = Pa /e-,T =k T/ ). The Kauzmann locus gives the pressure-dependent temperature at which the entropies of the supercooled lit]uid and the stable crystal are equal. Note the convergence of the Kauzmann and spinodal loci at T = 0. See Debenedetti et al. (1999) for details of this calculation.

The increase of the bulk pressure at a small increment after achievement of the equihbrium density distribution allows obtaining the adsorption branch of the isotherm. If the pore is wide enough, the capillary condensation will occur, with the pressure of the condensation being corresponded to the vapor-like spinodal point. Similarly, desorption branch of the isotherm will be obtained at the decrease of pressure. In this case, the capillary evaporation will occur at a hquid-like spinodal point. The equilibrium transition pressure is obtained by comparing the grand thermodynamic potentials corresponding to the adsorption and the desorption branches of the isotherm. It corresponds to the equality of these values of the grand thermodynamic potential. 

Abstract The virtual terms binodal and spinodal are equivalent to the experimental terms eoexistence curve (CXC) and metastability limit (ML), respectively, within an inherent accuracy of any semi-empirical EOS at the description of a real fluid behavior. Any predicted location of mechanical spinodal at positive pressures Psp(T)>0 merits verification because the Maxwell rule is a model (EOS)-dependent method based on the non-measurable values of chemical potential for both phases. It is not a reliable tool of CXC- and ML-prediction especially at low temperatures between the triple and normal boiling ones [Tt,Ti where the actual vapor pressures Ps(T) > 0 are quite small while... 

Indicated in Fig. 9 are temperature ranges of supercooled, stable and superheated water at atmospherie pressure. Ibidem one can see curves representing the temperature dependenee of the logarithm of the homogeneous nucleation rate for crystallization (curve 1) and boiling-up (curve 2). The maximum rate of formation of vapor nuclei is attained at the approach of the spinodal determined by condition (3). Fig. 9 also shows how the inverse isothermal eompressibility =-v(5p/5v) changes with temperature (curve 3). An arrow shows the temperature of the spinodal of superheated water. 

Figure 5 depicts the liquid spinodal curves Sp(L) in a pressure-temperature diagram for fixed CO2 compositions. The region of negative pressures, which is of interest for describing the capillary properties of CO2 aqueous solutions, has been also included. Interestingly, it can be noted that spinodal Sp(L) isopleths present a pressure-temperature trend, which looks similar to the liquid spinodal curve of pure water.At low temperatures, the Sp(L) isopleths are decreasing steeply before to reach a pressure minimum. Then at subcritical temperatures, isopleths are less spaced and sloped, and they finish to meet the H2O-CO2 critical curve. The temperature appears as a determining parameter in the explosivity control of CO2 aqueous solutions. Like for water, the easiest way to generate an explosive vaporization is a sudden depressurization in the superspinodal domain, where spinodal curves have a gentle slope in a P-T diagram (Fig. 5). This superspinodal field can be estimated theoretically irom the PRSV equation of...

Isotherms on an A-V diagram are useful to illustrate the origin of p-V diagrams, superheated liquids and subcooled vapors in metastable states, unstable states between the spinodal points, critical points, and the requirement of mechanical stability. It is reasonable to assume that the slope of A vs. V at constant temperamre and composition is negative, because... 

Binodal points represent the points of contact of a common tangent to A vs. V at constant temperature and composition when a region of negative curvature exists between two regions of positive curvature. The locus of binodal points, known as the binodal curve or two-phase envelope, represents the experimentally observed phase boundary under normal conditions. For example, saturated liquid and saturated vapor represent states on the binodal curve. The binodal region exists between the binodal and spinodal curves, where p/ V)T,aa jv < 0. 

Along any pure-fluid, subcritical isotherm, the spinodal separates unstable states from metastable states. At the other end of an isotherm s metastable range, metastable states are separated from stable states by the points at which vapor-Uquid, phase-equilibrium criteria are satisfied. Those criteria were stated in 7.3.5 the two-phase situation must exhibit thermal equilibrium, mechanical equilibrium, and diffusional equilibrium. Since we are on an isotherm, the temperatures in the two phases must be the same, and the thermal equilibrium criterion is satisfied. 

These conditions identify both vapor-liquid and liquid-liquid critical points. For vapor-liquid equilibria, they are satisfied when the spinodal coincides with the vapor-liquid saturation curve. However, that point need not occur either at the maximum in the saturation envelope or at the maximum in the spinodal see Figure 8.12. Along a spinodal the one-phase metastable system is balanced on the brink of an instability at a critical point that balance coincides with a two-phase situation and the resulting fluctuations cause critical opalescence, just as they do at pure-fluid critical points. 

Figure 9.3 Along subcritical isotherms for pure fluids, the fugacity passes through stable, metastable, and unstable regions just as does the pressure. Here we have plotted the subcritical isotherm TlT = 0.863 for a van der Waals fluid. Each point (a-f) on the fugacity plot corresponds to the point of the same label on the Pv diagram. Points b and e have the same fugacity and pressure (P /= 0.539) and therefore locate the vapor-liquid equUibrium state. Points c and d are on the spinodal. Line segment be locates metastable liquid states segment de locates metastable vapor states segment cd locates unstable states.

The intersection of those two one-phase lines satisfies (9.2.2) and therefore identifies the vapor-liquid saturation point. The lines for one-phase liquids terminate at the spinodal—they become unstable— and the unstable portion of the van der Waals loop is represented by the broken line on the fP plot. 

These mean that an isothermal-isobaric plot of /j vs. x passes through a point of inflection at the critical point, as was illustrated in Figure 8.13. Points that satisfy only (9.3.23) locate the spinodal, and when the spinodal coincides with the vapor-liquid saturation curve, then both (9.3.23) and (9.3.24) are satisfied and a vapor-liquid critical point occurs. 

When the liquid state is close to point A, — dP/dV)f j. becomes close to zero and according to (10.2) the liquid is very close to unstable cmidition. At this stage any perturbation can shift the liquid to another phase. Point A is where — dP/dV)i j becomes zero and it is called the limit of intrinsic instability or spinodal limit. From point A to point B — dP/dV)j j is positive and the material cannot physically be stable. Notice between these two points the liquid is supposed to expand when the pressure is increased. Any slight expansion due to the local density fluctuation generates more pressure itself and hence larger volume. This effect continues until the specific volume exceeds point B. A similar process can be described to decrease the specific volume below point A. Therefore, the material cannot stay at any point between A and B. The locus of the limit of intrinsic stability in the vapor dome is called spinodal curve. In order to find the spinodal curve equation, dPjdv is set equal to zero in the van der Waals equation. 

Figure 4. Portion of subsonic adiabats for n-octane with upstream states on selected spinodal points and subsonic downstream states. Initial states are labeled by the reduced temperature = T/Tc. Downstream states are mixtures for initial points of Tr = 0.95 and 0.93 the adiabat crosses the saturation curve for % = 0.917 the adiabat lies completely in the vapor region for = 0.9108.

Numerical solution of the jump conditions for octane [13] verifies this simple estimate. The subsonic-subsonic portion of the adiabats are shown in Fig. 4 for four upstream states on the spinodal. As the upstream pressure is decreased, the adia-bat moves toward and across the satinration boundary. At an initial pressure of one atmosphere (T = 0.91 Tc), the subsonic adiabat lies completely in the vapor state. Computations [13] with lower hydrocarbons in the alkane series, butane and pentane, show only mixture downstream states. 

Figure 5. Schematic of the postulated similarity fiowfield for the steady growth of a liquid-vapor mixture bubble within a superheated liquid. Radial variation of pressure is shown for a bubble radial velocity of 147.1 m/s in water superheated to the spinodal point of 600 K and 2.89 MPa. The bubble velocity corresponds to an evaporation wave velocity of 72.85 m/s, slightly above the CJ velocity of 67.7 m/s but below the maximum velocity of 78.5 m/s.

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