States absolutely unstable state

There is a region of absolutely unstable states between two coexisting phases situated as closely to the critical point as one likes. Thus, the critical point must belong to both the spinodal and binodal, it is the point of their being tangent. 

The spinodal represents a hypersurface within the space of external parameters where the homogeneous state of an equilibrium system becomes thermodynamically absolutely unstable. The loss of this stability can occur with respect to the density fluctuations with wave vector either equal to zero or distinct from it. These two possibilities correspond, respectively, to trivial and nontrivial branches of a spinodal. The Lifshitz points are located on the hyperline common for both branches. 

Equilibrium is defined as the state of absolute rest from which the system has no tendency to depart such stable systems are based on true thermodynamic equilibrium and are the subject of this book. This is to be distinguished from unstable states where processes may be imperceptibly slow such systems are sometimes called inert, unreactive, or unstable (Pitzer, 1995) and are not the subject of this book. Models based on equilibrium thermodynamics (e.g., FREZCHEM) predict stable states. In the real world, unstable states may persist indefinitely. 

Stable A term describing a system in a state of equilibrium corresponding to a local minimum of the appropriate thermodynamic potential for the specified constraints on the system. Stability cannot be defined in an absolute sense, but if several states are in principle accessible to the system under given conditions, that with the lowest potential is called the stable state, while the other states are described as metastable. Unstable states are not at a local minimum. Transitions between metastable and stable states occur at rates that depend on the magnitude of the appropriate activation energy barriers that separate them. 

The Differential Equation of State 1 provides not only a good qualitative description of isothermal behavior at subcritical temperatures T < 1, but also yields accurate quantitative representations of experimentally measured data. It describes not only the stable vapor and liquid branches, but also the two-phase transition region, additionally yielding information on the nature of metastable and absolutely unstable phases. A complete and simple description of the vapor-liquid-phase transition and the critical point also is provided by the differential equation of state. 

Transition state theory or absolute reaction rate theory is built upon these ideas of a potential energy surface and reaction coordinate to account for reactivity. The theory seeks to understand and appreciate reactivity in terms of the structure and behaviour of reaction transition states. A transition state is dehned as a transient, unstable species that is found... 

Dividing blobs, chemical flowers and patterned islands Different modes of propagation of the patterned state have been observed in CDIMA reaction (chlorine dioxide-iodine-malonic acid - PVA in a one-sided-fed reactor) in an absolutely unstable uniform state, among which are a spot division and finger printing mode [55]. 

There may be situations when pjquations 7 and 9 are realized while Equations 8 and 10 are not realized, i.c. the system is stable toward infinitely small perturbations while being unstable toward finite ones (a local maximum of entropy or a minimum of internal energy with at least one additional extremum). In such cases it is generally agreed to speak of a metastable equilibrium (metastable state) of the system. Conditions 8 and 10 define a stable equilibrium. When simultaneously breaking conditions 7 10, the system proves to be absolutely unstable. 

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... 

The system evolution upon transfer of the configurative point from outside the binodal to inside its dome is considered in a special division of phase transition science, namely, phase transition kinetics. The mechanism of evolution will essentially depend on where the configurative point is in the metrtstable or in the absolutely unstable region of the state diagram (even under the same pressure when the final two-phase states will be equal). 

If the system is in the symmetrical phase, the dependence G Q), when T > T" i has the only minimum at Qo = 0. At T = T, there appear points of inflection on the dependence AG = /(Q), and on further cooling (T < T"), extrema appear (Figure 1.28)-The AG minima at Qos and Q04 correspond to the metastable asymmetric phase. The system is capable to keep its initial (more symmetrietj) phase until the minimum at Qo = 0 disappears (this will happen when T = T ). Then, the symmetrical phase will prove to be absolutely unstable, and the system will abruptly change to a state with... 

Ec to unwind the helical structure in the fingerprint state. If the field is decreased from e q, the free energy of the planar state becomes lower than that of the homeotropic state, but the energy barrier persists. The energy barrier becomes lower with decreasing field. When the field is sufficiently low, the energy barrier decreases to zero, and the homeotropic state will become absolutely unstable. The critical field ehp = J /K i can be obtained from the equation... 

Most systems used in material science are nonequilibrium ones aging (supersaturated) alloys dissociate by initiation and coarsening of decay products. Grains start growing in nano- and polycrystaUine materials, amorphous alloys crystallize, interdiffusion takes place in protective coatings and powder alloys, metals oxidize in the atmosphere irreversibly, and so on. All materials listed above are considered to be either metastable or absolutely unstable ones and it is just a matter of the time period required for relaxation to equilibrium, or, more commonly, to a less nonequilibrium state. The production of those materials following the chemical reactions, thermal treatment or mechanical operation is accompanied thus, by irreversible nonquasistatic processes. 

Fig. 4 —Illustration of the asymmetry in the Landau free energy density when a non-zero third-order term is included. For T> Tc, S 0 s the absolute minimum. As T approaches Tc from above a second minimum appears and for T - the physical states corresponding to the two minima have the same free energy density and are separated by a barrier of height h. for Tc < T< TcVne S 0 minimum represents the stable state and the S = 0 minimum represents the metastable (supercooled) state. For temperatures below Tc the metastable state becomes unstable.

Therefore the globally stable pattern corresponds, for given fi, to the absolute minimum of C[A,A ], whereas the relative minima represent metastable structures that arise when multistability is present. The maxima naturally represent unstable states. 

Fig. 11. Absolutely and convectively unstable conditions for the generation of a ID Turing structure in the presence of a simple flow of velocity field v(r) = volx In the represented space-time plots (time running down vertically) the initial perturbation is at the center of the reactor, (a) vq = 0 development in time of the normal Turing structure (b) Vo < critical the pattern develops in the whole reactor as the left front is able to proceed against the flow because of the advection one however has a traveling Turing pattern (waves) (c) vo > identical during its growth the pattern is advected away by the flow in a finite reactor the developing pattern eventually leaves beyond a convectively unstable state very sensitive to imperfections (noise).

Recently [7] we constructed an example showing that interfacial flexibility can cause instability of the uniform state. Two elastic capacitors, C and C2, were connected in parallel. The total charge was fixed, but it was allowed to redistribute between C and C2. It was shown that if the interface was absolutely soft , i.e., contraction of the two gaps was not coupled, the uniform distribution became unstable at precisely the point where the dimensionless charge density s reached the critical value, = (2/3). In other words, the uniform distribution became unstable at the point where, under a control,... 

Absolute Rate Theory(also known as Transition State or Activated Complex Theory). A theory of reaction rates based on the postulate that molecules form, before undergoing reaction, an activated complex which is in equilibrium with the reactants. The rate of reaction is controlled by the concn of the complex present at any instant. In general, the complex is unstable and has a very brief existance(See also Collision Theoty of Reaction)... 

Of the two stable equilibrium points, D may be further classified as absolutely stable (deepest minimum) and A as metastable (local minimum). The unstable point B is also called a transition state. Critical point C is a marginally stable state that may be further classified as stable or unstable according to the sign of the lowest nonvanishing derivative. 

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