Stability, conditions mechanical

The molecular dynamics constraint technique presented in the previous section is designed to simulate steady solutions of the Euler equations but there is no guarantee that all of the simulated solutions are physical. Some steady solutions are characterized by unboimded volume expansion, and others may not be the particular shock wave solutions desired. This section defines mechanical stability conditions that characterize shock waves and then shows that the molecular dynamics constraint technique naturally takes the system through states that satisfy these stability conditions. 

Mechanical Stability. Condition (29-123 ) requires that an > 0. Hence, pressure must decrease at larger volume along a. p-V isotherm otherwise, the system splits into two different states of matter. 

Every configurative point inside the spinodal implies the violation of the mechanical stability conditions for the one-phase state, and this region of the state diagram is characterized as absolutely unstable to the one-phase existence. As a result, the system... 

One can derive the critical phase equation in a much simpler form in many specific cases using Equations 1.1.2--30... 34 as the stability conditions rather than Equation 1.1.2-12. The subsequent line of argument should remain unchanged. The molar Helmholtz potential F is convenient to use for a one-component system with Equations 1.1.2-31 and 1,1.2-32 to express the thermal and mechanical stability conditions, respectively. As the second inequality is stronger, one can confine oneself with it. In this case, the determinant D/r degenerates into one term... 

Formula 21 readily illustrates the mechanical stability condition (Equation 10) the system is in stable equilibrium if volume fluctuations are small and decrease substantially with increasing AV. This is redized when the exponent in Equation 21 is negative. As (AV), k, T are always positive, then... 

The upper portion of Fig. 4.1 shows plots of the universal van der Waals equation for three different values of t (solid lines). Of course Eq. (4.12) always deviates from the ideal gas law at low v. In fact we did not plot the pressure for v-values below the singularity at w = 1 /3, because there the molecules overlap. But we also notice that the universal van der Waals equation exhibits strange behavior iff < 1.0. There is a v-range in which the pressure rises even though the volume increases. Here we find an isothermal compressibility cr < 0—in clear violation of the mechanical stability condition in Eq. (3.16) Had we plotted Eq. (4.12) for even smaller f-values, we would have obtained negative pressures in addition. All in all, for certain v and t, the van der Waals equation does describe states which cannot be equilibrium states. It turns out however that we can fix this problem, and at the same time we may describe a new phenomenon—the phase transformation between gas and liquid. 

Composite aperture-spanning membranes resist electron beam under low-dose conditions A significant increase in the mechanical stability of solid—supported lipid membranes is observed... 

Equation (5.2) also implies that a crystalline solid becomes mechanically unstable when an elastic constant vanishes. Explicitly, for a three-dimensional cubic solid the stability conditions can be expressed in terms of the elastic stiffness coefficients of the substance [9] as... 

However, a more realistic model for the phase transition between baryonic and quark phase inside the star is the Glendenning construction [16], which determines the range of baryon density where both phases coexist. The essential point of this procedure is that both the hadron and the quark phase are allowed to be separately charged, still preserving the total charge neutrality. This implies that neutron star matter can be treated as a two-component system, and therefore can be parametrized by two chemical potentials like electron and baryon chemical potentials [if. and iin. The pressure is the same in the two phases to ensure mechanical stability, while the chemical potentials of the different species are related to each other satisfying chemical and beta stability. The Gibbs condition for mechanical and chemical equilibrium at zero temperature between both phases reads... 

A quantum-mechanical interpretation of Miedema s parameters has already been proposed by Chelikowsky and Phillips (1978). Extensions of the model to complex alloy systems have been considered. As an interesting application we may mention the discussion on the stabilities of ternary compounds presented by de Boer et al. (1988). In the case of the Heusler-type alloys XY2Z, for instance, the stability conditions with respect to mechanical mixtures of the same nominal composition (XY2+Z, X+Y2Z, XY+YZ, etc.) have been systematically examined and presented by means of diagrams. The Miedema s parameters, A t>, A ws1/3, moreover, have been used as variables for the construction of structural maps of intermetallic phases (Zunger 1981, Rajasekharan and Girgis 1983). 

Position C does not correspond to the lowest minimum of the energy following a small displacement, the block will return to the initial position whereas large displacements will move the block to the more stable position A. In A there is an (absolutely) stable equilibrium and in C a metastable equilibrium. For this mechanical system the stability conditions and the trends of spontaneous (natural) processes are related to minima (relative or absolute) of the gravitational potential energy. 

The statistical mechanical interpretation of the stability condition is quite simple. From Eq. (2.1.3) we obtain by differentiation... 

The YBG equation is a two point boundary value problem requiring the equilibrium liquid and vapor densities which in the canonical ensemble are uniquely defined by the number of atoms, N, volume, V, and temperature, T. If we accept the applicability of macroscopic thermodynamics to droplets of molecular dimensions, then these densities are dependent upon the interfacial contribution to the free energy, through the condition of mechanical stability, and consequently, the droplet size dependence of the surface tension must be obtained. 

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