Johnson-Mehl

In an amorphous material, the aUoy, when heated to a constant isothermal temperature and maintained there, shows a dsc trace as in Figure 10 (74). This trace is not a characteristic of microcrystalline growth, but rather can be well described by an isothermal nucleation and growth process based on the Johnson-Mehl-Avrami (JMA) transformation theory (75). The transformed volume fraction at time /can be written as... 

L. E. Levine, K. Lakshmi Narayan, K. F. Kelton. Finite size corrections for the Johnson-Mehl-Avrami-Kolmogorov equation. J Mater Res 72 124, 1997. 

Table 9.7 shows the results of the calculations of average parameters of PBU/P for isotropic DRP, fulfilled by Serra [134] and Meijering [152], Serra used VD-method while Meijering used the Johnson-Mehl s (JM) statistical model [150] of simultaneous growth of crystals until the total filling of the whole free space was accomplished. The parameter Nv in the table is the number of PBUs in a unit of system volume, thus Nv 1 is the mean volume of a single PBU, which is related to the relative density of the packing (1—e) with an interrelation... 

Table 9.7 Characteristics of PBU/Ps by V (Voronoi), JM (Johnson-Mehl), as well as TKH and RDH...

The hydrogen absorption/desorption kinetics are usually analyzed by applying the JMAK (Johnson-Mehl-Avrami-Kolmogorov) theory of phase transformations, which is based on nucleation and growth events [166-168] where a is the fraction transformed at time t or alternatively for hydrides the fraction absorbed... 

Karty et al. [21] pointed out that the value of the reaction order r and the dependence of k on pressure and temperature in the JMAK (Johnson-Mehl-Avrami-Kolmogorov) equation (Sect. 1.4.1.2), and perhaps on other variables such as particle size, are what define the rate-limiting process. Table 2.3 shows the summary of the dependence of p on growth dimensionality, rate-limiting process, and nucleation behavior as reported by Karty et al. [21]. 

One of the earliest attempts to explicitly combine thermodynamics and kinetics in rapid solidification was by Saunders et al. (1985). They examined the equations derived by Davies (1976) and Uhlmann (1972) for predicting TTT diagrams. These were based on Johnson-Mehl-Avrami kinetics for predicting glass formation during rapid solidification where the ruling equation could be given as... 

Equations (6.15) and (6.17) phenomenologically describe the overall growth kinetics after the initial nucleation took place and further nucleation is still occurring. Indeed, the sigmoidal form of the X(t) curve represents a wide variety of transformation reactions. Equation (6.13) is named after Johnson, Mehl, and Avrami [W. A. Johnson, R. E Mehl (1939) M. Avrami (1939)]. Let us finally mention two points. 1) Plotting Vin (1 -X) vs. t should give a straight line with slope km. 2) The time ty of the inflection point (d2X/dt2 = 0) on X(t) is suitable to derive either m or km, namely... 

The theory of the kinetics of concurrent nucleation and growth reactions has a rich history that includes work by Kolmogorov [1], Johnson and Mehl [2], Avrami [3-5], Jackson [6], and Cahn [7]. Cahn s time-cone method for treating a class of these problems is the most general of these, with the most transparent assumptions, and is presented here. The method of Johnson, Mehl, and Avrami is covered in Section 4 of Christian s text [8]. 

The function ((t) in Eq. 21.12 has a characteristic sigmoidal shape with a maximum rate of transformation at intermediate times. Examples are shown in Fig. 21.2. The d = 3 form of Eq. 21.12 is commonly known as the Johnson-Mehl-Avrami equation. 

However, this is valid only for f 1. At longer times the fraction of the volume that is available for growth and nucleation is (1 — /). When this term is included, the Johnson-Mehl equation becomes... 

Nucleation and Growth Johnson-Mehl-Avrami Equation... 

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],... 

The different combinations of nucleation, growth, and impingement processes give rise to the Johnson-Mehl-Avrami kinetic model [4], which results in the following equation... 

Nucleation and growth processes such as glass crystallization generally follow the Johnson-Mehl-Avrami (JMA) model ... 

Most reconstructive transitions take place by processes of formation and growth of nuclei of the product phase and the kinetics are often described by the model developed by Johnson, Mehl, Avrami and Erofeev (see Chapter 3). These reactions are, therefore, identical with other decompositions and represent simple chemical changes. Studies are often relatively difficult and greater interest has been directed towards the experimentally more accessible reactions that jdeld gaseous products. 

An example of an intensively studied set of polymorphs whose decompositions are of great theoretical and practical importance (see Chapter 12) is CaCOj which may exist (in order of decreasing thermodynamic stability) as calcite, aragonite or vaterite [18]. Vaterite can be prepared by precipitation from aqueous solutions under carefully controlled conditions. A DTA curve for the vaterite calcite transition is shown in Figure 2.3. The transition is exothermic AH = -34.3 J g ) with onset at 704 K. Isothermal extent of conversion against time curves were described [18] by the Johnson, Mehl, Avrami, Erofeev model (see Chapter 3) with n = 2. The measured Arrhenius parameters were F, = 210 kJ mol and A = 1.15x10 min. The decomposition of vaterite and its concurrent transformation to calcite under various conditions were compared [18] with the decomposition of calcite xmder the same conditions (see Chapter 12). 

Equation (3.2) is often referred to as the Avrami-Erofeev (A-E) equation, or more fittingly, on account of the substantial contributions from other workers, especially Johnson and Mehl [26] in the field of metallurgy, as the Johnson-Mehl-Avrami-Erofeev-Kholmogorov (JMAEK) equation. The values of n obtained from kinetic... 

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