Unstable equilibrium state

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

The plate buckling equations inherently cannot be derived from the equilibrium of a differential element. Instead, the buckling problem represents the departure from the equilibrium state when that state becomes unstable because the in-plane load is too high. The departure from the equilibrium state is accompanied by waves or buckles in the surface of the plate. That is, the plate cannot remain flat when the... 

Truss Stress Analysis The computation of member forces in an arbitrary plane truss is now examined. There exist some simple counting tests that may determine if a given truss is unstable. Failing that, one must attempt to compute the equilibrium state given some external forces in the process, one obtains values for all member forces. In this example, all truss members are identical in terms of material and area, grown in a developmental space where units are measured in meters EA is set to 1.57 x 104 N, corresponding to a modulus of elasticity for steel and a cylindrical member of diameter 1 cm. Consider a general truss with n joints and m beams external forces are applied at joints and the member forces are computed. Let the structure forces be... 

These multiple branches are not equivalent. From the contour lines in Figure 7.8, we can deduce which branch is stable, and which branch is not. The middle branch (C-C1) lies in a range where, for a given value of p, F increases with u. The derivative of F with respect to u is therefore positive which is just the criterion we found for an unstable equilibrium. Any fluctuation of u at constant p will drive the system away from the branch C-C. The opposite holds for the upper and lower branches A-C and A -C that lie in a range where F decreases when u increases. The derivative of F with respect to u is therefore negative and any concentration fluctuation around an equilibrium state along these branches dies out rapidly. The branches A-C and A -C are stable steady-states. 

Stable, metastable and unstable states a simple analogy. A simple mechanical model is shown in Fig. 2.37 a block on a stand may be in different equilibrium states. In A and C the centre of gravity (G) of the block is lower than... 

Figure 2.37. A simple mechanical system and its equilibrium states. Different positions of a block on a stand and the corresponding values of the gravitation potential energy are shown. Point G is the centre of gravity of the block. In A there is stable equilibrium, in C metastable, in B unstable.

From Figure 7 it is deduced that the number of the equilibrium states depends on the number of points where the straight line yo = constant intersects with the curve defined by Eq.(13). With a value of yo 0.025, there are three equilibrium points Pi, P2, P3, being P stable, P2 unstable and P3 can be stable or unstable depending on the real part of the eigenvalues of the linearized system at this point. When the line yo = constant is tangent to the curve yo = fiy ) (be. point M) a new behavior of the reactor appears, which can be characterized from dyo/dy = 0 in Eq.(13) as follows ... 

An unstable equilibrium state (or configuration) that is at maximal potential energy. A metastable state is at equilibrium, and its potential energy [written here as U(x)] is such that any displacement (dx) from Xequiiibnum will result in the loss of potential energy. 

If the composition of a phase falls inside the spinode, the phase would undergo spontaneous decomposition into two phases. That is, heterogeneity would arise from homogeneity. From an energetic point of view, the homogeneous phase is at an unstable equilibrium state, meaning that without disturbance at all, the equilibrium could persist, but the tiniest microscopic perturbation (there would always be perturbation due to thermal motion) would grow to produce hetero-... 

Fig. la, h. The elastic part (ne) and the negative of the mixing part (— Jtm) of the osmotic pressure as functions of polymer concentration < >. The intercepts of ae and — nm correspond to the equilibrium state of neutral gels. Numbers besides each curve of — represent Xi> which increases with temperature, (a) x2 = 0. Only one root at all temperatures, (b) Xi = 0.56. Three roots appear in the intermediate temperature range (around Xi = 0.465), which correspond to stable, unstable, and metastable states, respectively. (Reproduced with permission from Ref. 20)... 

When dealing with general thermodynamic systems, the fact that entropy tends to a maximum in the trend toward equilibrium of a natural process generalizes the above mechanical consideration with respect to stability. An equilibrium state can be characterized as a stable equilibrium when the entropy is a maximum neutral equilibrium when displacement from one equilibrium state lo another does not involve changing entropy and unstable equilibrium when entropy is a minimum. Any slight disturbance from an unstable equilibrium state or a system will lead to transition to another state of equilibrium. 

The steady-state solution that is an extension of the equilibrium state, called the thermodynamic branch, is stable until the parameter A reaches the critical value A,. For values larger than A<, there appear two new branches (61) and (62). Each of the new branches is stable, but the extrapolation of the thermodynamic branch (a ) is unstable. Using the mathematical methods of bifurcation theory, one can determine the point A, and also obtain the new solution, (i.e., the dissipative structures) in the vicinity of A, as a function of (A - A,.). One must emphasize that... 

By using external reservoirs, some of these parameters can be kept at values different from those of thermodynamic equilibrium, / /, j = 1, , m < n. As a result, a non-equilibrium state arises, which is characterized by nonvanishing values of some fluxes /,-, i = 1, s < r and of the corresponding forces Xj. Examples of such processes are diffusion and related effects, Peltier effect, etc.45,46. Such a state can either be stationary or time-dependent, stable or unstable. 

Equilibrium thermodynamics is the most important, most tangible result of classical thermodynamics. It is a monumental collection of relations between state properties such as temperature, pressure, composition, volume, internal energy, and so forth. It has impressed, maybe more so overwhelmed, many to the extent that most were left confused and hesitant, if not to say paralyzed, to apply its main results. The most characteristic thing that can be said about equilibrium thermodynamics is that it deals with transitions between well-defined states, equilibrium states, while there is a strict absence of macroscopic flows of energy and mass and of driving forces, potential differences, such as difference in pressure, temperature, or chemical potential. It allows, however, for nonequilibrium situations that are inherently unstable, out of equilibrium, but kinetically inhibited to change. The driving force is there, but the flow is effectively zero. 

In colloid science, the terms thermodynamically stable and metastable mean that a system is in a state of equilibrium corresponding to a local minimum of free energy (Ref. [978]). If several states of energy are accessible, the lowest is referred to as the stable state and the others are referred to as metastable states unstable states are not at a local minimum. Most colloidal systems are metastable or unstable with respect to the separate bulk phases. See also Colloid Stability, Kinetic Stability. 

While the above system is an example where two-dimensional phase separation in the sense of Fig. la,b (or Fig. 5) occurs, there exist also good examples where no lateral phase separation exists in equilibrium, and the system forms a single interface parallel to the surfaces (Fig. Id). However, if one chooses the initial state such that the phase preferred by air is close to the substrate and the phase preferred by the substrate is next to the air surface [77], the system is unstable and surface phase inversion takes place. A laterally inhomogeneous state then occurs only as a transient phenomenon necessary to trigger the inversion kinetics, but not as an equilibrium state [77]. 

The solid white form really is only in a state of false equilibrium, being unstable with respect to the polymerised forms at all realisable temperatures. There are also the other forms—red, scarlet and black phosphorus—the behaviour of which under definite conditions of pressure and temperature cannot be stated with any certainty. Further, the melting-point even of the well-crystallised white phosphorus can be made fcto vary under certain conditions (see p. 15). In fact, all the condensed phases, liquid and solid, behave as mixtures rather than as single pure substances. 

A system may be in a stable, metastable, unstable, or neutral equilibrium state. In a stable system, a perturbation causes small departures from the original conditions, which are restorable. In an unstable equilibrium, even a small perturbation causes large irreversible changes. A metastable system may be stable or unstable according to the level and direction of perturbation. All thermodynamic equilibria are stable or metastable, but not unstable. This means that all natural processes evolve toward an equilibrium state, which is a global attractor. 

Fig. 1.6. Shape of contact area of gas bubbles (a)- equilibrium state of three bubbles (b)- unstable...

The second law ensures that there will be a thermodynamic driving force in the direction of equilibrium. The equilibrium state is finally attained when there is no change with time in any of the system s macroscopic properties. Observing changes, unfortunately, is not a useful test to determine whether a system has reached equilibrium, because there is no standard length scale for time. From a practical standpoint, therefore, three possible types of states can be envisioned stable, metastable, or unstable, all of which were previously defined. 

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