Phase transformation first order

After the investigation of hydrodynamics and mass transfer, the next step is the examination of the reactor model. For example, let us consider here the two-phase model with plug flow of gas in both bubble and emulsion phase and first-order reaction (see Section 3.8.3). The first step at this stage is to transform its equations to dimensionless forms. 

Sheu S-Y, Mou C-Y and Lovett R 1995 How a solid can be turned into a gas without passing through a first-order phase transformation Phys. Rev. E 51 R3795-8... 

Tolbert S H and Aiivisatos A P 1994 Size dependence of a first order solid-solid phase transition the wurtzite to rock salt transformation in CdSe nanocrystais Science 265 373... 

In this chapter the regimes of mechanical response nonlinear elastic compression stress tensors the Hugoniot elastic limit elastic-plastic deformation hydrodynamic flow phase transformation release waves other mechanical aspects of shock propagation first-order and second-order behaviors. 

It has been a persistent characteristic of shock-compression science that the first-order picture of the processes yields readily to solution whereas second-order descriptions fail to confirm material models. For example, the high-pressure, pressure-volume relations and equation-of-state data yield pressure values close to that expected at a given volume compression. Mechanical yielding behavior is observed to follow behaviors that can be modeled on concepts developed to describe solids under less severe loadings. Phase transformations are observed to occur at pressures reasonably close to those obtained in static compression. 

In this chapter studies of physical effects within the elastic deformation range were extended into stress regions where there are substantial contributions to physical processes from both elastic and inelastic deformation. Those studies include the piezoelectric responses of the piezoelectric crystals, quartz and lithium niobate, similar work on the piezoelectric polymer PVDF, ferroelectric solids, and ferromagnetic alloys which exhibit second- and first-order phase transformations. The resistance of metals has been investigated along with the distinctive shock phenomenon, shock-induced polarization. 

T. Pusztai, L. Granasy. Monte Carlo simulation of first-order phase transformations with mutual blocking of anisotropically growing particles up to all relevant orders. Phys Rev B 57 14110, 1998. 

Many metals and metallic alloys show martensitic transformations at temperatures below the melting point. Martensitic transformations are structural phase changes of first order which belong to the broader class of diffusion js solid-state phase transformations. These are structural transformations of the crystal lattice, which do not involve long-range atomic movements. A recent review of the properties and the classification of diffusionless transformations has been given by Delayed... 

A. Kmmhansl and Y. Yamada, Some new aspects of first-order displacive phase transformations ... 

Following three phase transformations [951] (>298 K), NH C decomposition begins [915] in the solid phase at 423 K but only becomes extensive well above the melting point ( 440 K). Decomposition with the evolution of N20 and H20 from the melt is first order [952,953] (E = 153—163 kJ mole-1), the mechanism suggested involving intermediate nitramide formation. Other proposed schemes have identified NOj [954] or the radical NH2NO [955] (<473 K) as possible participants. Studies [956,957] have been made of the influence of additives on NH C decomposition. 

In the previous chapter we examined cellular automata simulations of first-order reactions. Because these reactions involved just transformations of individual ingredients, the simulations were relatively simple and straightforward to set up. Second-order cellular automata simulations require more instructions than do the first-order models described earlier. First of all, since movement is involved and ingredients can only move into vacant spaces on the grid, one must allow a suitable number of vacant cells on the grid for movement to take place in a sensible manner. For a gas-phase reaction one might wish to allow at least 5-10 vacant cells for each ingredient, so that on a 100 x 100 = 10,000... 

Early studies of ET dynamics at externally biased interfaces were based on conventional cyclic voltammetry employing four-electrode potentiostats [62,67 70,79]. The formal pseudo-first-order electron-transfer rate constants [ket(cms )] were measured on the basis of the Nicholson method [99] and convolution potential sweep voltammetry [79,100] in the presence of an excess of one of the reactant species. The constant composition approximation allows expression of the ET rate constant with the same units as in heterogeneous reaction on solid electrodes. However, any comparison with the expression described in Section II.B requires the transformation to bimolecular units, i.e., M cms . Values of of the order of 1-2 x lO cms (0.05 to O.IM cms ) were reported for Fe(CN)g in the aqueous phase and the redox species Lu(PC)2, Sn(PC)2, TCNQ, and RuTPP(Py)2 in DCE [62,70]. Despite the fact that large potential perturbations across the interface introduce interferences in kinetic analysis [101], these early estimations allowed some preliminary comparisons to established ET models in heterogeneous media. 

First-order phase transitions exhibit hysteresis, i.e. the transition takes place some time after the temperature or pressure change giving rise to it. How fast the transformation proceeds also depends on the formation or presence of sites of nucleation. The phase transition can proceed at an extremely slow rate. For this reason many thermodynamically unstable modifications are well known and can be studied in conditions under which they should already have been transformed. 

Cp is the specific heat at constant pressure, k is the compressibility at constant temperature. The conversion process of a second-order phase transition can extend over a certain temperature range. If it is linked with a change of the structure (which usually is the case), this is a continuous structural change. There is no hysteresis and no metastable phases occur. A transformation that almost proceeds in a second-order manner (very small discontinuity of volume or entropy) is sometimes called weakly first order . 

MnAs exhibits this behavior. It has the NiAs structure at temperatures exceeding 125 °C. When cooled, a second-order phase transition takes place at 125 °C, resulting in the MnP type (cf. Fig. 18.4, p. 218). This is a normal behavior, as shown by many other substances. Unusual, however, is the reappearance of the higher symmetrical NiAs structure at lower temperatures after a second phase transition has taken place at 45 °C. This second transformation is of first order, with a discontinuous volume change AV and with enthalpy of transformation AH. In addition, a reorientation of the electronic spins occurs from a low-spin to a high-spin state. The high-spin structure (< 45°C) is ferromagnetic,... 

It has been found that both the anhydrous Form III and dihydrate phases of carbamazepine exhibit fluorescence in the solid state [78]. The fluorescence intensity associated with the dihydrate phase was determined to be significantly more intense than that associated with the anhydrate phase, and this difference was exploited to develop a method for study of the kinetics of the aqueous solution-mediated phase transformation between these forms. Studies were conducted at temperatures over the range of 18 40 °C, and it was found that the phase transformation was adequately characterized by first-order reaction kinetics. The temperature dependence in the calculated rate constants was used to calculate activation energy of 11.2 kCal/ mol (47.4 cal/g) for the anhydrate-to-dihydrate phase conversion. 

Several years ago Baer proposed the use of a matrix A, that transforms the adiabatic electronic set to a diabatic one [72], (For a special twofold set this was discussed in [286,287].) Computations performed with the diabatic set are much simpler than those with the adiabatic set. Subject to certain conditions, A is the solution of a set of first order partial differential equations. A is unitary and has the form of a path-ordered phase factor, in which the phase can be formally written as... 

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