Mechanical stability criterion

IUPAC defines the lower limit of mesopores as 2 nm [1] which was considered as the limit below which the adsorption will occur by volume filling. However, in our recent article, based on the tensile stress hypothesis, we have shown that this limit is different than IUPAC limit. Using the mechanical stability criterion for the cylindrical meniscus (during adsorption), the critical size is obtained from... 

Note that it appears to be possible for (8.1.30) to be satisfied by having both Cj, < 0 and K.J < 0 however, this is only a mathematical possibility that cannot actually occur. In fact, we expect that the mechanical stability limit will be violated before the thermal limit, because the mechanical limit represents a response of higher-order than the thermal limit [3] higher-order terms approach zero before lower-order terms. This expectation is confirmed experimentally whenever an initially stable system is driven into an unstable region of its phase diagram, the mechanical stability limit is always violated before the thermal limit. In other words, a state may be mechanically unstable but remain thermally stable, because Kj appears only in (8.1.31) and not in (8.1.23). The mechanical stability criterion (8.1.31) is a stronger test than the thermal stability criterion (8.1.23). 

A conventional way to address the criteria (a) and (b) is to employ a volumetric equation of state of the form P(T, v) that applies to all fluid phases of our pure substance. The Redlich-Kwong equation (8.2.1) is an example. Any properly constructed model for a volumetric equation of state should satisfy the thermal stability criterion C > 0), and as far as we are aware, all cubic equations of state having constant parameters (a and b) do so. Consequently, thermal stability only needs to be tested when we construct complicated equations of state, such as those that are high-order polynomials in v or that have temperature-dependent parameters. Moreover, as we noted under (8.1.31), the mechanical stability criterion is a stronger test, so we do not consider thermal stability further here. 

At this point we have eliminated all roots that fail to satisfy the mechanical stability criterion, but we do not yet have a unique root that is stable. To select from among the remaining alternatives, we apply criterion (b), cited at the start of this section. That criterion is a consequence of the equilibrium conditions developed in 7.1.5 the stable equilibrium state will have a lower value of the Gibbs energy than any other state that might exist at the specified T and P. 

If the pressure is reduced to 30 bar, below the mechanical critical point, the loop in the fugacity becomes more pronounced. Here violates the diffusional stability criterion dfildxi > 0 for stability) over only a small range of Xj but the mechanical stability criterion (k > 0) is also violated at states between the extrema in fp Finally, at the lowest pressure (10 bar), the loop in has completely closed, dividing into two parts a vapor part that is linear and obeys the ideal-gas law, and a fluid part that includes stable and metastable liquid states at small x values plus mechanically unstable fluid states at higher x values. The broken horizontal lines in Figure 10.1 are vapor-liquid tie lines, computed by solving the phi-phi equations (10.1.3) simultaneously for both components. 

For the hydrodynamic instability model, Lienhard and Dhir (1973b) extended the Zuber model to the CHF on finite bodies of several kinds (see Sec. 2.3.1, Fig. 2.18). Lienhard and Hasan (1979) proposed a mechanical energy stability criterion The vapor-escape wake system in a boiling process remains stable as long as the net mechanical energy transfer to the system is negative. They concluded that there is no contradiction between this criterion and the hydrodynamic instability model. 

Lienhard, J. H., and M. M. Hasan, 1979, On Predicting Boiling Burnout with the Mechanical Energy Stability Criterion, Trans ASME, J. Heat Transfer 707 276 (2)... 

Actually an additional stability criterion is needed, see M.E. Fisher, Archives Rat. Mech. Anal. 17, 377 (1964) D. Ruelle, Statistical Mechanics, Rigorous Results (Benjamin, New York 1969). A collection of point particles with mutual gravitational attraction is an example where this criterion is not satisfied, and for which therefore no statistical mechanics exists. 

There are two criteria for the mechanical stability of a shock wave. [9] The first criterion requires v > c , where c is the speed of sound in the pre-shock material. The second criterion requires Uj +c, > v, where the subscript 1 denotes the post-shock state. 

However, unlike the case for the pure fluid, this inflection point is not the real mi.xture critical point. The mixture critical point is the point of intersection of the dew point and bubble point curves, and this must be determined from phase equilibrium calculations, more complicated mixture stability conditions, or experiment, not simply from the criterion for mechanical stability as for a pure fluid. 

The criterion of mechanical stability reveals that (dV/dp)T, aii v, < 0, as discussed... 

Spinodal points represent the boundary between positive and negative curvature of A-V isotherms. An equilibrium state on the spinodal curve is defined by (9p/9V)7 ,au JV = 0. Regions between the spinodal points are intrinsically unstable and violate the criterion of mechanical stability. 

It has been found that prestrain can significantly improve the actuation performance of dielectric elastomer devices [143, 165]. The observed improvements have been largely attributed to an increase in the breakdown strength [168-169], which has been explained via a thermodynamic stability criterion [170]. Prestrain has the additional benefits of improving the mechanical efficiency [171] and response speed of most dielectric elastomers while causing a marginal decrease in the dielectric... 

This is the criterion for mechanical stability for a thermally stable system to also be mechanically stable, the system volume must always decrease in response to any isothermal fluctuation that increases the pressure. 

More problematic are those situations in which the equation of state provides multiple roots for v at the given T and P. Which of these are observable To decide, we first eliminate any p-roots that fail to satisfy the differential criterion for mechanical stability (8.1.31). That criterion can be written in several forms, but it may be more helpful here to state it as a criterion for instabilities. 

For states at which the equation of state provides only one real root for v, then Kj > 0 and the simplification (8.3.14) is legitimate. But when the equation of state has bifurcated, producing multiple roots for v, then we must exercise care when using (8.3.14) in place of (8.3.13). Some of those volume roots will have Kj < 0 and therefore will be mechanically unstable, even if those roots also have Gn > 0, so they satisfy (8.3.14). Consequently, those fluids are diffusionally unstable because (8.3.13) is violated. For cubic equations of state, it is the "middle" root for v that has Gn > 0, but Kj < 0, as illustrated in Figure 8.11. Equations of state that are higher-order pol)momials in v will have additional roots that behave as in Figure 8.11. So when we test for the observability of proposed states and we do not know where that state lies on a phase diagram, we should apply the complete stability criterion (8.3.13), rather than the abbreviated form (8.3.14). 

The criterion (8.3.13) implies that if a mixture is mechanically unstable kj < 0), then it is also diffusionally unstable, just as (8.1.30) implies that if a fluid is thermally unstable (Q < 0), then it is also mechanically unstable. But a fluid may be diffusion-ally unstable while remaining mechanically and thermally stable. In fact, whenever a stable mixture is driven into an unstable region of its phase diagram, the diffusional stability limit is always violated before the mechanical or thermal limits are violated, because higher-order terms approach zero before lower-order terms [3]. This can be seen in Figure 8.11. This means that the diffusional stability criterion (8.3.13) is a stronger test for thermodynamic stability than the mechanical criterion and (as noted in 8.1.2) the mechanical criterion, in turn, is a stronger test than the thermal criterion. 

The middle envelope is the spinodal the set of states that separate metastable states from unstable states. Recall from 8.3 that one-phase mixtures become diffusionally unstable before becoming mechanically unstable. Therefore, the mixture spinodal is the locus of points at which the diffusional stability criterion (8.3.14) is first violated that is, it is the locus of points having... 

Figure 8. Bom criteria for mechanical stability. Three calculations are illustrated. The black dots are from a pseudopotential calculation. The open circles and squares are from interatomic potential calculations. Note how the third criterion, is violated at high pressures.

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