Isothermal phase transformation

R.F. Sekerka. Application of the time-dependent theory of interface stability to an isothermal phase transformation. J. Chem. Phys. Solids, 28(6) 983—994, 1967. 

Fusion is an isothermal phase transformation and hence the enthalpy change of M increases without accompanying a temperature change. AH, represents die heat of fusion of M. 

The kinetic theory presented by Johnson and Mehl [2] and Avrami [3] predicts the volume fraction transformed, as a function of time, t, during an isothermal phase transformation. The derivation of the Johnson-Mehl-Avrami kinetics is based on the grouping of the three individual partial processes, that is, nucleation, growth, and impingement of growing particles [5],... 

Figure 4 presents the isothermal phase transformation diagram of the template-free syntheses in which the SiCL/Alo ratio and the time t of crytallization are varied. The Siof/Nafcr and l O/SiCL ratios are 10 and 30, respectively. The pentasil phase could only be synthesized for n = SiC /A O, =30-50 and t = 36 - 72 h. Outside of this area amorphous material, mordenite, sheet structures similar to kenyaite, quartz and crystobalite can be found. For values of n less than 25 the crytalline product is mordenite. For 30 < n < 50 a yield of 95% (related to the SiC content) ZSM-5 type, which was proved by X-ray diffraction pattern, could be found. Depending on n and the crystallization time, t, a more or less large amount of amorphous material is produced. This is shown in Figure 5. A long crystallization time causes recrystallization and is harmful to the yield of ZSM-5 products.

Figure 4. Isothermal phase transformation diagram of the template free syntheses. SS denotes sheet structures.

Avrami Analysis The Avrami equation, a general approach for description of isothermal phase transformation kinetics originally developed for polymers (46), is often used for describing nucleation and crystal growth in fats. The Avrami equation is given as... 

For describing phase equilibrium, the entropy change AS in phase transformations is an important parameter. Therefore, we shall set up an expression to calculate AS for isothermal phase transformations in systems of matter. Normally, a phase transformation will be connected with absorption or release of heat this is known, for example, in the form of the heat of fusion of ice and the heat of evaporation of water. This enthalpy change - the transformation enthalpy ArHr - is a direct measure of the entropy change by the phase transformation concerned. 

The entropy increase AS by an isothermal phase transformation at constant pressure p is... 

Solution. We consider a reversible, isothermal phase transformation where water is transformed into water vapour by the process... 

The isotherms represented in Fig. 1 give a general idea of the equilibria in the Pd-H system under different p-T conditions. Most experimental evidence shows, however, that the equilibrium pressure over a + /3 coexisting phases depends on the direction of the phase transformation process p a-p > pp-a (T, H/Pd constant). This hysteresis effect at 100°... 

When the heat exchange between the inner vessel and its surroundings, maintained at a constant temperature T0, occurs at an infinitely large rate isothermal calorimeter, 2 in Fig. 1), the temperature of the inner vessel also remains constant. The heat produced or absorbed is generally evaluated from the intensity of a physical modification occuring at a constant temperature in the surrounding medium (phase transformation). 

Structural polymorphism has been already reported as a peculiar solid-solid phase transition with a large spectral shift in the cast film of CgAzoCioN+ Br (chapter 4). The type 1 spectrum was thermally transformed to the type VI spectrum and then backed to the type I by the isothermal moisture treatment. The reversible spectral change between the type I and VI is a good experimental evidence of Okuyama s prediction on the molecular packing. Since the type VI state is assumed to be a metastable state, the isothermal phase transition to the type I state is expected to be induced by some external stimuli. Water molecules adsorbed to cast bilayer films might act as an accelerator of the phase transition. 

An XRPD system equipped with a heatable sample holder has been described, which permitted highly defined heating up to 250°C [55]. The system was used to study the phase transformation of phenan-threne, and the dehydration of caffeine hydrate. An analysis scheme was developed for the data that permitted one to extract activation parameters for these solid-state reactions from a single non-isothermal study run at a constant heating rate. 

Most industrially relevant transformation processes are not isothermal and even in a controlled laboratory environment, it is difficult to perform experiments that are completely isothermal. The kinetics of nonisothermal phase transformations are more complex, of course, but there are some useful relationships that have been developed that allow for the evaluation of kinetic parameters under nonisothermal conditions. One such equation takes into account the heating rate, (p usually in K/min, used in the experiment [4] ... 

The temperature of maximum transformation rate is easily determined using either of two similar techniques called differential scanning calorimetry (DSC) or differential thermal analysis (DTA). These techniques are extremely useful in the kinetic study of both isothermal and nonisothermal phase transformations. 

By a change of temperature or pressure, it is often possible to cross the phase limits of a homogeneous crystal. It supersaturates with respect to one or several of its components, and the supersaturated components eventually precipitate. This is an additive reaction. It occurs either externally at the surfaces, or in the crystal bulk by nucleation and growth. Reactions of this kind from initially homogeneous and supersaturated solid solutions will be discussed in Chapter 12 on phase transformations. Internal reactions in the sense of the present chapter occur after crystal A has been brought into contact with reactant B, and the product AB forms isothermally in the interior of A or B. Point defect fluxes are responsible for the matter transport during internal reactions, and local equilibrium is often established throughout. 

The foregoing classification is not without ambiguity. For example, it is common practice to call the reaction A - B +C° (see Fig. 6-1) induced by decreasing the temperature a phase transformation. The similar (peritectoid) reaction C = a+fi (Fig. 12-2) induced by a temperature increase, however, is named a decomposition reaction. In addition, the isothermal reaction AO = A+j02, which occurs if the intensive variable fio2 is decreased so that AO decomposes, is called a metal oxide reduction. It is thus categorized as a genuine heterogeneous solid state reaction (the... 

Useful insights into the kinetics of a phase transformation that proceeds by nucle-ation and growth can be obtained by observing the fraction transformed, , under isothermal conditions at a series of different temperatures. This is usually done by undercooling rapidly to a fixed temperature and then observing the resulting isothermal transformation. The kinetics generally follows the typical C-shaped behavior described in Exercise 18.4. If a series of such curves is obtained at different temperatures, the time required to achieve, for example, ( = 0.01, 0.50, and... 

Fig. 6. Specific volume pressure curves for the l.c. polymer shown in Fig. 5. Thin dashed lines pressure dependence of the phase transformation temperatures l.c. to isotropic, Tc, and the glass transition temperatures, T , full line specific volume-temperature cut at 2000 bar (isothermal measurements)...

An isothermal change in phase (phase transformation) of a substance produced by input of energy as heat always leads to an increase in the entropy of the substance. 

It is significant that the second calorimetric peak and the associated isotherm sub-step were detectable only if the graphitized thermal black had been heated at temperatures above 1700°C. These results suggest that the 2-D phase transformation is very sensitive to the perfection of the surface basal planes and this is a further indication that the phase change leads to the development of a commensurate structure. 

In Eq. (2) dp and dTrtitx to infinitesimal changes in pand Tfor an equihbrium system composed of a pure substance with both phases always present A5 and A Prefer to the change in S and V when one phase transforms to the other at constant p and T. Since the change in state (1) is isothermal and AG is zero, AAmay be replaced by AMZ The result is... 

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