Metastable states first-order transitions

The phase coexistence observed around the first-order transition in NIPA gels cannot be interpreted by the Flory-Rehner theory because this theory tacitly assumes that the equilibrium state of a gel is always a homogeneous one. Heterogeneous structures such as two-phase coexistence are ruled out from the outset in this theory. Of course, if the observed phase coexistence is a transient phenomenon, it is beyond the thermodynamical theory. However, as will be described below, the result of the detailed experiment strongly indicates that the coexistence of phases is not a transient but rather a stable or metastable equilibrium phenomenon. At any rate, we will focus our attention in this article only on static equilibrium phenomena. 

Fig. 51. Plol of — t)(iiZ)/i)Z z=u versus /j(0) for the cases of a second-order wetting transition (a) and a first-order wetting transition (b). The solution consistent with the boundary condition always is found by intersection of the curve /x2(0) - 1 with the straight line [h + gq.(0)]/y. In case (a) this solution is unique for all choices of k /y (keeping the order parameter g/y fixed). Critical wetting occurs for the case where the solution (denoted by a dot) occurs for y (0) = +1, where then ) i(Z)/dZ z u = I) and hence the interface is an infinite distance away from the surface. For ft] > ftlc the surface is non-wet while for fti > ft C the surface is wet. In case (b) the solution is unique for ft] < ft (only a non-wet state of the surface occurs) and for ft > ft (only a wet state of the surface occurs). For ft > ft] > ft three intersections (denoted by A, B, C in the figure) occur, B being always unstable, while A is stable and C metastable for ft]c > ft > ft and A is metastable and C stable for ft " > ft > ftic. At ftic where the exchange of stability between A and C occurs (i.e., the first-order wetting transition) the shaded areas in fig. 51b are equal. This construction is the surface counterpart of the Maxwell-type construction of the first-order transition in the bulk lsing model (cf. fig. 37). From Schmidt and Binder (1987).

In the low-field range the weak first order transitions are fully reversible, but above 8 teslas we observe the strong hysteresis associated with the formation of a metastable SDW phase. Finally, the direct measurement of the specific heat as a function of the field showed no variation of the density of state at the Fermi level, for the normal metallic state below the FISDW threshold fieldlO... 

Of course, all of this reasoning and arguments have a common presumption M is a valid thermodynamic parameter. In truth, for a first-order transition, the system can present metastable states, and so the measured value of M may not be a good thermodynamic parameter, and also the Maxwell relation is not valid. In the following section, the consequences of using non-equilibrium magnetization data on estimating the MCE is discussed. 

At this stage, we point out a qualitative effect of thermal fluctuations The condensation inside the bubble is a sharp first order transition which ends in a critical point only within mean-field approximation. Since the bubble is only of finite size no true phase transition can occur, because in the vicinity of the transition the free energy difference between the stable and the unstable states is finite. Therefore one can always And the critical nucleus in the metastable state with a small but finite probability, and the transition is rounded. The consideration of fluctuations will replace the sharp transition by a rather rapid but continuous variation of the density inside the bubble. 

FIG. 4 Qualitative phase diagram close to a first-order irreversible phase transition. The solid line shows the dependence of the coverage of A species ( a) on the partial pressure (Ta). Just at the critical point F2a one has a discontinuity in (dashed line) which indicates coexistence between a reactive state with no large A clusters and an A rich phase (hkely a large A cluster). The dotted fine shows a metastability loop where Fas and F s are the upper and lower spinodal points, respectively. Between F2A and Fas the reactive state is unstable and is displaced by the A rich phase. In contrast, between F s and F2A the reactive state displaces the A rich phase. 

Making the flow rate higher or lower will change from stable to metastable the folded or the stretched state, respectively. The effects of hysteresis associated with this first-order discontinuous transition play an important role in the formation of composite crystalline structures. 

I) If A > 0, then h2 < hi, and the reorientation of the second sublattice occurs in a field h = hn by a first-order phase transition. The total magnetization in this field has a jump-like change Am = 2. In the field interval h2 < h < h metastable states exist, and hysteresis is possible (Fig. 15a). 

Figure 13. Schematic phase diagram of water s metastable states. Line (1) indicates the upstroke transition LDA —>HDA —>VHDA discussed in Refs. [173, 174], Line (2) indicates the standard preparation procedure of VHDA (annealing of uHDA to 160 K at 1.1 GPa) as discussed in Ref. [152]. Line (3) indicates the reverse downstroke transition VHDA—>HDA LDA as discussed in Ref. [155]. The thick gray line marked Tx represents the crystallization temperature above which rapid crystallization is observed (adapted from Mishima [153]). The metastability fields for LDA and HDA are delineated by a sharp line, which is the possible extension of a first-order liquid-liquid transition ending in a hypothesized second critical point. The metastability fields for HDA and VHDA are delineated by a broad area, which may either become broader (according to the singularity free scenario [202, 203]) or alternatively become more narrow (in case the transition is limited by kinetics) as the temperature is increased. The question marks indicate that the extrapolation of the abrupt LDA<- HDA and the smeared HDA <-> VHDA transitions at 140 K to higher temperatures are speculative. For simplicity, we average out the hysteresis effect observed during upstroke and downstroke transitions as previously done by Mishima [153], which results in a HDA <-> VHDA transition at T=140K and P 0.70 GPa, which is 0.25 GPa broad and a LDA <-> HDA transition at T = 140 K and P 0.20 GPa, which is at most 0.01 GPa broad (i.e., discontinuous) within the experimental resolution.

In particular, it is well known that, if the macromolecule is supercooled below the 0 temperature, the phase transition isotropic coil-isotropic globule occurs. We emphasize that for the semiflexible macromolecule this is the peculiar phase transition between two metastable states. It should be recalled that the theory of the transition isotropic coil-isotropic globule for the model of beads is formulated in terms of the second and third virial coefficients of the interactions of beads , B and C24). This transition takes place slightly below the 0 point and its type depends on the value of the ratio C1/2/a3 if Cw/a3 I, the coil-globule transition is the first order phase transition with the bound of the macromolecular dimensions, and if C1/2/a3 1, it is a smooth second order phase transition (see24, 25)). 

Immediately ahead of the detonation firont the explosive rests quietly in its metastable state, while to the rear the shocked and reacted material flows at several kilometers per second with a pressure of several hundred thousand atmospheres and temperature of several thousand Kelvins. The rapid compression and heating of matter to these extreme conditions and the associated high velocity flow are properties of detonations that can be shared by strong shockwaves. However, with detonations the heated and compressed flow is selfsustaining. Typically, detonations are maintained by the exothermic chemistry they induce. Detonations driven by first order phase transitions have been envisioned, but have not yet been observed. 

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