Johnson and Mehl

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]. 

Johnson and Mehl analyzed the kinetics of pearlite formation by assuming that the growth rates, G, in three dimensions and the nucleation rate, N, are constant. With a constant growth rate, the volume, V, of a spherical particle nucleated at a time, r, at a time, t, is... 

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],... 

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... 

The theoretical basis for use of the DSC or DTA for study of crystallization rates was developed independently by Johnson and Mehl and by Avrami. The volume fraction of a sample crystallized, x, as a function of time, t, is expressed in terms of the nucleation rate per unit volume, /, and the crystal growth rate, u, via the equation ... 

By use of the Poison distribution, Avrami derived the famous Avrami phenomenological equation to treat a kinetic process (Avrami 1939, 1940, 1941). Kolmogorov first discussed the formulation of this equation (Kolmogorov 1937). Johnson and Mehl also made similar derivation independently (Johnson and Mehl 1939). Evans proposed a very concise derivation as introduced below (Evans 1945). 

Several authors (Leslie et al., 1961 Scheucher, 1969a,b Murphy and Ball, 1972 Hausmann, 1987) tried to explain this recrystallization behavior by the Avrami and Johnson-Mehl relation. This relation was originally derived by Avrami to explain the kinetics of diffusive phase transitions and later applied by Johnson and Mehl to recrystallization. Accordingly the quantity 1 — R, where R is normalized yield stress, should behave as a function of annealing time like the growth of the volume of a new phase as a function of time. The experimental curve can in fact be fitted by desired relation... 

Weinberg (1992a), Weinberg et al. (1997), and Zanotto (1997), reported in detail on transformation kinetics via nucleation and crystal growth. The standard theory of this type of phase transformation kinetics was developed by Johnson and Mehl and Avrami and Kolmogorov (see Weinberg et al., 1997). Therefore, this theory is called the JMAK theory. The JMAK equation (Eq. 1-6) is universal and applicable to glass-ceramics. 

The general Avrami equation is applicable to any type of crystallization. It is not restricted to polymers. It describes the time evolution of the overall crystallinity. The pioneer work was conducted during the 1930s and 1940s by Evans, Kolmogoroff, Johnson and Mehl, and Avrami. Wunderlich (1978) concludes that without the parallel knowledge of the microscopic, independently proven mechanism, the macroscopic, experimentally derived Avrami equation and the Avrami parameters are only a convenient means to represent empirical data of crystallization. However, interest in the Avrami equation has been... 

Measurements of the overall crystallization rate involve often the macroscopic determination of crystallinity as a function of time. The first effort to describe quantitatively macroscopic development of crystallinity in term of nucleation and linear crystal growth was made by Kolmogoroff [132], Johnson and Mehl [133], and Avrami [134],... 

Many mathematical models have bees advanced relating nucleation and nuclei growth rates to the overall kinetics of phase transformation, such as Johnson and Mehl [427], Avrami [428], Yerofyeyev [429], Kolmogorov [430] as well as Jacobs-Tompkins [431] or Mampel [432] and were agreeably suinmarized elsewhere [1,3,413,144, 421,422,423,426,43 I ]. 

Similar reasoning, apphed by Johnson and Mehl [6] to a problem of nucleation and radial growth of spherical entities, resulted in Equation (7.5), with E expressed by Equation (7.10). The derivations of Tobin [27-29] were also based on a similar principle. However, over-simphfied reasoning, for instance, the incorrect assumption of the proportionality of an average spherulite volume (instead of a volume increment) to the unconverted fraction, led Tobin to the erroneous result [30]. 

In powdered samples, the shapes and sizes of particles are of great importance for the kinetic or rate laws. Johnson and Mehl [JOH 39], Manpel [MAM 40], Avrami [AVR 39], and Delmon [DEL 69] analyzed these phenomena and enabled... 

The concept of fictitious fractional extent was introduced by Johnson and Mehl [JOH 39]. Its purpose is to simultaneously take into account the mutual covering of growing nucleus and the variation of free volume for further nucleatioa... 

Larsen and Livesay (1980) chose SmCoj as a representative compound on which to study the intrinsic hydriding kinetics of compounds of the RCoj family. The compound LaNij and several pseudobinaries were studied by Belkbir et al. (1980, 1981). All these authors analysed their data in terms of the Johnson-Mehl equation (Johnson and Mehl, 1939 Avrami, 1940)... 

Equation (5.45) represents the main theoretical formula resulting from the Kolmogoroff s treatment [5.12], Later it has been obtained in practically the same form by Johnson and Mehl [5.13], Avrami [5.14-5.16] and Evans [5.17]. 

W.A. Johnson and R.F. Mehl. Reaction kinetics in processes of nucleation and growth. Trans. AIME, 135 416-442, 1939. See also discussion on pp. 442-458. 

Hu WB (2005) Molecular segregation in polymer melt crystallization simulation evidence and unified-scheme interpretation. Macromolecules 38 8712-8718 Hu WB, Cai T (2008) Regime transitions of polymer crystal growth rates molecular simulations and interpretation beyond Lauritzen-Hoffman model. Macromolecules 41 2049-2061 Jeziomy A (1971) Parameters characterizing the kinetics of the non-isothermal crystallization of poly(ethylene terephthalate) determined by DSC. Polymer 12 150-158 Johnson WA, Mehl RT (1939) Reaction kinetics in processes of nucleation and growth. Trans Am Inst Min Pet Eng 135 416-441... 

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... 

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... 

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