Phase transitions, first order

TMDSC of Poly(ethylene Terephthalate) (initially quenched to be amorphous) 

DSC Heatinq Trace of (Slassy DDA-12 with Liquid-crystalline (5rder 

The phase area 4 in Fig. 6.1 is less likely for liquid crystals than for condis crystals since the small increase in order that is needed to change a mobile melt into a mobile liquid crystal occurs fast and is usually completely reversible as seen in the reversible transition shown in TMDSC of Fig. 5.144 for a small molecule and the upper curves of Fig. 5.154 for a macromolecule. Once in phase area 2, the polymeric liquid crystals will, however, not crystallize completely on cooling, but rather become a semicrystalline material. In the present scheme, they will produce metastable phase areas 9, followed by 10. 

On crystallization of polyethylene at atmospheric pres sure, the structure typical for a semicrystalline polymer results. It usually consists of nanophase-separated crystalline/amorphous phase structures, as described in Chap. 5, and is represented by phase areas 7 and 8 of Fig. 6.1. A zero-entropy production path on heating, discussed in Sect. 2.4, permits to evaluate the free enthalpy distribution in areas 7 and 8, as shown in Fig. 6.3 [4] (see also schematics of G in Figs. 2.88 and 2.120). 

The glass transition of the amorphous phase can be used to estimate the strain introduced into the metastable, amorphous nanophases by the interfaces with the crystals. The local strain manifests itself in an increase of the glass-transition temperature and the possible presence of a rigid amorphous fraction that does not show a contribution to the increase in heat capacity at T, as is discussed in Sect 6.1.3. Most bulk-crystallized macromolecules which do not exhibit intervening mesophases are described by the phase areas 7 and 8 of Fig. 6.1. 

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Stigter and Dill [98] studied phospholipid monolayers at the n-heptane-water interface and were able to treat the second and third virial coefficients (see Eq. XV-1) in terms of electrostatic, including dipole, interactions. At higher film pressures, Pethica and co-workers [99] observed quasi-first-order phase transitions, that is, a much flatter plateau region than shown in Fig. XV-6. 

For both first-order and continuous phase transitions, finite size shifts the transition and rounds it in some way. The shift for first-order transitions arises, crudely, because the chemical potential, like most other properties, has a finite-size correction p(A)-p(oo) C (l/A). An approximate expression for this was derived by Siepmann et al [134]. Therefore, the line of intersection of two chemical potential surfaces Pj(T,P) and pjj T,P) will shift, in general, by an amount 0 IN). The rounding is expected because the partition fiinction only has singularities (and hence produces discontinuous or divergent properties) in tlie limit i—>oo otherwise, it is analytic, so for finite Vthe discontinuities must be smoothed out in some way. The shift for continuous transitions arises because the transition happens when L for the finite system, but when i oo m the infinite system. The rounding happens for the same reason as it does for first-order phase transitions whatever the nature of the divergence in thennodynamic properties (described, typically, by critical exponents) it will be limited by the finite size of the system. 

Consider simulating a system m the canonical ensemble, close to a first-order phase transition. In one phase, is essentially a Gaussian centred around a value j, while in the other phase tlie peak is around Ejj. 

In the microcanonical ensemble, the signature of a first-order phase transition is the appearance of a van der Waals loop m the equation of state, now written as T(E) or P( ). The P( ) curve switches over from one... 

Lovett R 1995 Can a solid be turned into a gas without passing through a first order phase transition Observation, Prediction and Simuiation of Phase Transitions in Compiex Fiuids vol 460 NATO ASi Series O ed M Baus, L F Rull and J-P Ryckaert (Dordrecht Kluwer) pp 641-54... 

Frenkel D 1986 Free-energy computation and first-order phase transitions Moiecuiar Dynamics Simuiation of Statisticai Mechanicai Systems ed G Ciccotti and W G Hoover (Amsterdam North-Holland) pp 151-88... 

Huckaby D A and Blum L 1991 A model for sequential first-order phase transitions occuring in the underpotential deposition of metals J. Eiectroanai. Chem. 315 255-61... 

Prenkel, D. Pree energy computation and first order phase transitions. In Molecular Dynamic Simulation of Statistical Mechanical Systems, Enrico Fermi Summer School, Varenna 1985, G. Ciccotti and W. Hoover, eds. North Holland, Amsterdam (1986) 43-65. 

The indicated transition pressure of 15 GPa is in agreement with the published data with shock-wave structure measurements on a 3% silicon-iron alloy, the nominal composition of Silectron. A mixed phase region from 15 to 22.5 GPa appears quite reasonable based on shock pressure-volume data. Thus, the direct measure of magnetization appears to offer a sensitive measure of characteristics of shock-induced, first-order phase transitions involving a change in magnetization. 

Although Eqs. (33), (34), and especially (35), are useful they have a problem. They all predict that the hard sphere system is a fluid until = 1. This is beyond close packing and quite impossible. In fact, hard spheres undergo a first order phase transition to a solid phase at around pd 0.9. This has been estabhshed by simulations [3-5]. To a point, the BGY approximation has the advantage here. As is seen in Fig. 1, the BGY equation does predict that dp dp)j = 0 at high densities. However, the location of the transition is quite wrong. Another problem with the PY theory is that it can lead to negative values of g(r). This is a result of the linearization of y(r) - 1 that... 

FIG. 5 Schematic representation of adsorption isotherms in the region of the first-order phase transition on a homogeneous (solid line) and heterogeneous (filled circles) surface. 

R. M. Zif, B. J. Brosilow. Investigation of the first-order phase transition in the A-B2 reaction model using a constant-coverage kinetic ensemble. Phys Rev A 46 4630-4633, 1992. 

J. D. Gunton, M. San Miguel, P. S. Sahni. The dynamics of first order phase transitions. In C. Domb, ed. Phase Transitions and Critical Phenomena. London, Academic Press, 1987, Vol. 8, pp. 261-A61. 

An intrinsic surface is built up between both phases in coexistence at a first-order phase transition. For the hard sphere crystal-melt interface [51] density, pressure and stress profiles were calculated, showing that the transition from crystal to fluid occurs over a narrow range of only two to three crystal layers. Crystal growth rate constants of a Lennard-Jones (100) surface [52] were calculated from the fluctuations of interfaces. There is evidence for bcc ordering at the surface of a critical fee nucleus [53]. 

Before trying to solve the master equation for growth processes by direct stochastic simulation it is usually advisable to first try some analytical approximation. The mean-field approximation often gives very good results for questions of first-order phase transitions, and at least it provides a qualitative understanding for the interplay of the various model parameters. 

In equilibrium, this describes the coexistence of two different phases (solid and liquid), just as in the case of the Ising model ( hising) with the up and down magnetization phases. When h 0, one of these two phases has a priority. Therefore, a sign change of h -h induces a first-order phase transition. (Note that for modeling reasons h(T) may be assumed to depend on temperature.)... 

The martensite - austenite transition temperatures we find are for all systems in accordance with the previously published ones . Some minor deviations can be attributed to the fact that we are simulating an overheated first order phase transition. Therefore, for our limited system sizes, one cannot expect a definite transition temperature. 

Observe how in each of these four events, H is zero until, at some critical Ac (which is different for different cases), H abruptly jumps to some higher value and thereafter proceeds relatively smoothly to its final maximum value i max = log2(8) = 3 at A = 7/8. In statistical physics, such abrupt, discontinuous changes in entropy are representative of first-order phase transitions. Interestingly, an examination of a large number of such transition events reveals that there is a small percentage of smooth transitions, which are associated with a second-order phase transition [li90a]. 

Table 7.3 lists the four rules in this minimally-diluted rule-family, along with their corresponding iterative maps. Notice that since rules R, R2 and R3 do not have a linear term, / (p = 0) = 0 and mean-field-theory predicts a first-order phase transition. By first order we mean that the phase transition is discontinuous there is an abrupt, discontinuous change at a well defined critical probability Pc, at which the system suddenly goes from having poo = 0 as the only stable fixed point to having an asymptotic density Poo 7 0 as the only stable fixed point (see below). 

The boundary layers, or interphases as they are also called, form the mesophase with properties different from those of the bulk matrix and result from the long-range effects of the solid phase on the ambient matrix regions. Even for low-molecular liquids the effects of this kind spread to liquid layers as thick as tens or hundreds or Angstrom [57, 58], As a result the liquid layers at interphases acquire properties different from properties in the bulk, e.g., higher shear strength, modified thermophysical characteristics, etc. [58, 59], The transition from the properties prevalent in the boundary layers to those in the bulk may be sharp enough and very similar in a way to the first-order phase transition [59]. 

First-Order Phase Transitions The phase transitions we have described are examples of first-order phase transitions. In a phase transition represented as... 

In a first-order phase transition, the second derivatives of AG are also not zero. That is... 

We will be looking at first-order phase transitions in a mixture so that the Clapeyron equation, as well as the Gibbs phase rule, apply. We will describe mostly binary systems so that C = 2 and the phase rule becomes... 

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