Sharp phase transitions

Numerical simulations of the k = oo case reveal a sharp phase transition at Ac = 0.27 [wootters]. Simulations also suggest that the spread in values of entropy decreases with increasing k, and that the width of the transition region probably goes as k f [woot90]. 

Although CMC is not a sharp phase transition and the degree of amplification is not phenomenal, a clear non-linear change in surface tension against the amount of photochemical change is observed as shown in Figure 2. 

Phospholipid vesicles (and bilayers) composed of phospholipids with well-defined fatty acid side chains undergo a sharp transition from a crystallinelike state to an amorphous state as the temperature is raised.107 The transition temperature depends on the nature of the fatty acid side chains. For example, for C12 saturated fatty acid chains on lecithin the transition temperature is 0° and for C18 saturated fatty acid chains it is 58°C for unsaturated lecithins the transition temperature is below zero.107 For real membranes sharp phase transitions are not observed, because of the heterogeneous composition of the membrane. In the case of /3 hydroxybutyrate dehydrogenase, the enzymic activity apparently is not influenced by this phase transition as judged by the temperature dependence of the reaction rate. However, for some membrane-bound proteins, a plot of the reaction rate versus the reciprocal temperature... 

Figure 1.16 (a) Change of the phase transition tempreratures for different dopants and (b) change of a sharp phase transition (BaTiC ) to a diffuse phase transition observed in a broadening of the dielectric peak... 

When nucleation is highly unfavorable (i.e., a l) the polymer system exhibits a biphasic behavior depending on the total monomer concentration A0. In this case there is a sharp phase transition between the all-monomer state for A0 < 1 /K, where l/K is the critical monomer concentration. When A0 exceeds 1 /K the free monomer concentration stays fixed at [A eq = l/K. This type of nonsmooth behavior at x = lforcr = 1 is called a transcritical bifurcation in non-linear dynamics [191]. It is also widely known as phase transition in physics. Figure 10.5 shows that for a less than unity, the transition is smooth. Hence we see that the... 

Figure 2. Schematic isotherms for small and infinite system. Note sharp phase transition in infinite system and smoothed-out transition in finite system. (From Hill. )...

The curve in Figure 4 is a DTA spectrum of acetohexamide (Lilly Reference Standard, Lot No. 2KT47) as produced by a DuPont 900 D.T. analyzer. The spectrum shows a sharp phase transition at 192°C. 

Analyzer using 2mm microtubes and N2 atmosphere is used to obtain this curve. Two sharp phase transitions are shown, the first at l69°C and the second at 233°C. Melting is associated with the first transition, and decomposition with second. The operating parameters are glass beads for... 

Let us briefly review the well-studied subject of phase transitions and critical phenomena [39], Examples of critical points include a magnet at the onset of ordering, a liquid-vapor system at the critical temperature and pressure, and a binary liquid system that is about to phase-separate. The key point is that the fluctuations in a system at its critical point occur at all scales, and the system is exquisitely sensitive to tiny perturbations. Even though sharp phase transitions can occur only in infinitely large systems, behavior akin to that at a phase transition is observed for systems of finite size as well. Indeed, for a system near a critical point, the largest scale over which fluctuations occur is determined either by how far away one is from the critical point or by the finite size of the system. 

At non-zero temperatures, of course there is no sharp phase transition in an one-dimensional mode with short-range interactions. One finds that already in the commensurate region (5 < 5C) kink and antikink excitations appear via spontaneous thermal fluctuations (Brazovskii et al., 1977 Burkov and Talapov, 1980). 

This situation should dearly be distinguished from the weak random potential limit for example -CH substitution in the TCNQ /Bechgaard et al to be published/, where only a smearing of an otherwise sharp phase transition is observed. 

This results in the sharp phase transition at 54 K from the metal to a semiconducting or insulating phase, the so-called Peierls transition. Here, the conductivity decreases by more than an order of magnitude within a very small temperature interval (c Fig. 9.5). 

ELPs are macromolecules composed of the monomeric pentapeptide (VPGXG), where X can be any amino acid residue except proline (18,19). Through variation in the length of the pol3uner and the residue in the X position, numerous polymers have been synthesized that exhibit sharp phase transitions, due to hydrophobic collapse and aggregation, when the temperature is raised above a temperatm-e termed the... 

The terms Pmax and Pmax denote the limiting capacities of the inspiratory muscles, and n is an efficiency index. The optimal Pmus(t) output is found by minimization of / subjects to the constraints set by the chemical and mechanical plants. Equation 11.1 and Equation 11.9. Because Pmusif) is generally a continuous time function with sharp phase transitions, this amounts to solving a difficult dynamic nonlinear optimal control problem with piecewise smooth trajectories. An alternative approach adopted by Poon and coworkers [1992] is to model Pmus t) as a biphasic function... 

In order as the conformational changes induced by the external stimuli that occur at the molecular level to become macroscopic perceptible, the composition of the environmental medium is in such manner adjusted that the smart polymer to be in the proximity of phase transition. In this way small changes of stimuli will cause sharp phase transitions, which acts as an amplifier of the external signal [114]. Two typical examples of such phase transitions for soluble polymers and hydrogels are illustrated in Figure 4.24. The most frequent responses of to the stimuli action are listed in Table 4.6. 

Hydrophobically end-capped poly(EO-co-PO) Tcp is in the range of 18-71°C depending on the end group, c = 0.5 wt% sharp phase transition within 3°C and small hysteresis two liquid phases above is Unearly... 

In contrast to classic ferroelectrics, with a sharp phase transition at Tc and along the MPB, respectively relaxor ceramics are characterized by a diffuse OD phase transition without any change in macroscopic symmetry, a slim hysteresis loop (as opposed to the wide hysteresis in classical ferroelectrics see Figures 8.5 and 8.10), and associated small coercive fields Eq as well as small remanent (Pr) and spontaneous polarizations (Ps). Most importantly, the polarization of relaxor ceramics does not vanish at Tc, but rather retains finite values up to a higher temperature, termed the Bums temperature (Tg), as shown schematically in Figure 8.17. The dielectric permittivity of relaxor ceramics attains a maximum value at a temperature for a particular frequency and, as this frequency increases, r, also increases. The temperature dependence of Er does not obey the Curie-Weiss law just above r ax, but only beyond Tr where Tr > Tm . In contrast to the displacive phase transformation in classical ferroelectrics, the difiuse transition in relaxor ceramics does not involve any change in macroscopic symmetry, and is of neither first nor second order. 

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