Avrami—Erofeev equation

For a solid-state reaction, one of the solutions of Equation 3.1 is the Avrami-Erofeev equation [3], The phase transition model that derives this equation supposes that the germ nuclei of the new phase are distributed randomly within the solid following a nucleation event, grains grow throughout the old phase until the transformation is complete. Then, the Avrami-Erofeev equation is [3]... 

Rickett equation [49], The sigmoid form of the nr-time curve expressed by equation (3.6) is similar in shape to those derived from the Avrami-Erofeev equation. 

Incorporation of a diffusion term in nucleation and growth reaction models has been proposed by Hulbert [68]. Interface advance is assiuned to fit the parabolic law and is proportional to but the nucleation step is uninhibited. The overall rate expressions have the same form as the Avrami-Erofeev equation ... 

The kinetics of many solid state reactions have been reported as being satisfactorily represented by the first-order rate equation [70] (which is also one form of the Avrami-Erofeev equation (n = 1)). Such kinetic behaviour may be expected in decompositions of fine powders if particle nucleation occurs on a random basis and growth does not advance beyond the individual crystallite nucleated. 

Care should be taken in defining the procedure for calculating values of k fi om the experimental data. There is always the possibihty that the apparent is a compound term containing several individual rate coefficients for separable processes (such as nucleation and growth). It is important that the dimensions of k (and hence of A) should be (time). For example, the power law (Table 3.3.) should be written as = kt and not as ar = k t. Similarly the Avrami-Erofeev equation (An) is [-ln(l - a)Y = kt. The use of k in place of A in the Arrhenius equation will produce an apparent activation energy /i, which is n times the conventional activation energy obtained using k. 

For those rate equations containing exponents, e.g the Avrami-Erofeev equations (An), it would appear that plots of ln[- ln(l - a)] against In (t - Q would provide the most direct method for the determination of the value of the exponent, n. Such plots are, however, notoriously insensitive and errors in t, together with contributions from initial rate processes, can influence the apparent value of n. Non-integer values of n have been reported [12],... 

Ammonium copper(II) sulfate hexahydrate [64] loses 4H2O between 339 and 393 K with no influence of water vapour pressure on reaction rate below 172 N m . The Avrami-Erofeev equation n = 2 fitted the data below 350 K with E, = 170 kJ mol. Above this temperature the contracting area equation applied and E, = 82 kJ mol. ... 

The dehydration and rehydration reactions of calcium sulfate dihydrate (gypsiun) are of considerable technological importance and have been the subject of many studies. On heating, CaS04.2H20 may yield the hemihydrate or the anhydrous salt and both the product formed and the kinetics of the reaction are markedly dependent upon the temperature and the water vapour pressure. At low temperatures (i.e. < 383 K) the process fits the Avrami-Erofeev equation (n = 2) [75]. The apparent activation energy for nucleation varies between 250 and 140 kJ mol in 4.6 and 17.0 Torr water v our pressure, respectively. Reactions yielding the anhydrous salt (< 10 Torr) and the hemihydrate ( (HjO) >17 Torr) proceeded by an interface mechanism, for which the values of E, were 80 to 90 kJ mol. At temperatures > 383 K the reaction was controlled by diffusion with E, = 40 to 50 kJ mol. ... 

A recent kinetic study [108] of the overall dehydration rates of KA1(S04)2.12H20 and KCr(S04)2.12H20 showed that measured ur-time data were well described by the Avrami-Erofeev equation with = 2. This was not consistent with expectation for the growth of three dimensional nuclei for which A = 3 and = 0 or 1, respectively. In accordance with the reaction models described above, there must be limited water losses from all surfaces together with an overall reaction controlled by product recrystallization that confers the apparent topotacticity on the overall... 

The thermal dehyi-ation of Na2C03.H20 between 336 and 400 K fits the Avrami-Erofeev equation with = 2 (E, = 71.5 kJ mol and., 4 = 2.2 x 10 s [110]). The apparent reduction in rate resulting from an increase of /r(H20) is ascribed to competition from the rehydration reaction. Electron micrographs confirm the nucleation and growth mechanism indicated by the kinetic behaviour, nucleation develops from circular defects that may be occluded solution. 

The kinetics of dehydration [128] of Na2S203.5H20 were difficult to interpret because the course of the reaction was markedly influenced by the perfection of the initial reactant surface and the reaction conditions. No reliable Arrhenius parameters could be obtained. The mechanism proposed to account for behaviour was the initial formation of a thin superficial layer of the anhydrous salt which later reorganized to form dihydrate. The first step in the reaction pentahydrate - dihydrate was satisfactorily represented by the contracting area (0.08 < or, < 0.80) expression. The second reaction, giving the anhydrous salt, fitted the Avrami-Erofeev equation (n = 2) between 0.05 < 2< 0.8. The product layer offers no impedance to product water vapoiu escape and no evidence of diffusion control was obtained. The mechanistic discussions are supported by microscopic observations of the distributions and development of nuclei as reaction proceeds. 

The final stages of the above reaction overlapped with the onset of the nucleation and growth process that continued to complete the dehydration. Growth of three dimensional nuclei was confirmed microscopically. This second rate process was well described by the Avrami-Erofeev equation with = 2 and E, for crystals was 175 30 kJ mol (with a considerable scatter of data) below 460 K and a more reproducible reaction rate, with E, = 153 10 kJ mol, for powder. Above about 450 K there were some indications of intracrystalline melting of single crystals and the value of , increased markedly to 350 50 kJ mol (again with significant scatter of data). 

Galwey and Hood [160] showed that NajCOj.l.SHjOj decomposed in vacuum (360 to 410 K) to produce Na COj + l.SHjO + O.TSOj. ar-time curves were sigmoidal and the kinetics could be described by the Avrami-Erofeev equation with = 2 or 3. The activation energy was 112 8 kJ mol. The reaction rate between 313 and 343 K was significantly increased by the presence of small amounts of liquid water. This deceleratory reaction was fitted by the first-order equation (E, = 80 10 kJ mol ) and it was concluded that breakdown of hydrogen peroxide proceeded in the liquid water, possibly with trace amounts of impurity transition-metal ions acting as catalysts. 

Although the rates of decomposition of different preparations of lithium azide differed markedly [34], reproducible behaviour was observed for salt which had been crushed and pelleted. Such pretreatment was believed to produce a uniform concentration of defects within the reactant assemblages. The sigmoid nr-time curves fitted the Avrami-Erofeev equation with = 3 between 0.02 < or < 0.58 and the contracting volume expression across the wider interval 0.05 < a< 0.95, the value of was 119 kJ mol". Reaction involved the three-dimensional growth of a constant number of nuclei and it was suggested that acceleration of rate following preirradiation resulted from an increase in the number of such nuclei. 

Tanaka [45] used mass loss measurements to study the isothermal decomposition of NaHCOj between 383 and 397 K. Results fitted the Avrami-Erofeev equation, = 1.41. The value of , was 109 5 kJ mol, which was greater than the reaction enthalpy of 71.8 0.8 kJ mol. It was concluded that this is a nucleation and growth process. 

The thermal decomposition of cobalt carbonate fitted the Avrami-Erofeev equation with n = 1.37 at 493 K and 2.54 at 543 K [74]. These data were later reanalyzed by Engberg [75] who allowed for an initial reaction, 0.02 and found , = 95 kJ mol, which is close to the Co-O bond dissociation energy (91 kJ mol ). 

The products of decomposition of KIO4 and NaI04 are unusual in undergoing efficient diliusion so that each partly reacted periodate crystal becomes surrounded by a fi agile layer of almost pure iodate [20]. ar-time curves fit the Avrami-Erofeev equation and values of n increase with temperature =1 to 2 for KIO4 and n = 2 to 3 for NaI04. Further work is required to elucidate these complicated reactions. 

Decomposition of CsBrOj proceeds [30] in the molten or semi-molten state of a eutectic formed with the CsBr product. Kinetics were fitted by the Prout-Tompkins and Avrami-Erofeev equations. The reaction rate (673 K) was accelerated significantly both by y-irradiation damage (which leads to rupture of Br-0 bonds) and by the presence of added Ba " ions which introduce local strain into the crystal and thereby promote Br03 ion breakdown. 

A complete analysis of the reaction would require measurements of the variations with time of all the phases participating. The product giving unusual textures identified by Brown et al. [18] may, perhaps, be K3(Mn04)2. The fit of kinetic data to the Avrami-Erofeev equation (n = 2) [18], together with the appearance of nuclei, illustrated in Figure 14.1., can be regarded now [17] as only an incomplete representation of this more complicated reaction. 

Nickel permanganate decomposed [44] (- NiMnjOj + I.5O2) between 356 and 400 K. The sigmoid ar-time curves were well expressed by the Avrami-Erofeev equation ( = 2). An initial electron transfer step was identified as rate controlling, with = 100 5 kJ mol . The rate of the first half of reaction a < 0.5) was decreased by the presence of water vapour. The rate of this autocatalytic reaction also proceeded more rapidly in the solid state than the comparable reaction in aqueous solution. 

De Waal et al. [98] used Raman spectroscopy to measure the decomposition kinetics from the isothermal time-dependence of the totally symmetric Cr - 0 vibration mode in (NH4)2Cr04 between 343 and 363 K, The results were fitted to the Avrami-Erofeev equation with n = 2. for microcrystals was 97 10 kJ mol and 49 1 kJ mol for powdered samples. 

The decomposition of iron(II) oxalate in hydrogen is an autocatalytic reaction which fits the Avrami-Erofeev equation with n = 2 (0.02 < nr < 0.25) and = 3 (0.3... 

Iron(III) oxalate decomposed between 410 and 450 K to give CO and iron(II) oxalate which retained about 10% of the iron(lll) salt [57]. Melting was not detected. The sigmoid a - time curves were identified as being due to a nucleation and growth process. The first half of the reaction was well represented by the Avrami - Erofeev equation (n = 2) and the latter half by the contracting volume equation. Values of were relatively low, 107 to 120 kJ mol , and rate control was ascribed to either electron transfer or C - C bond rupture. 

The more stable iron(II) oxalate decomposed [58] between 596 and 638 K to yield FeO, as the initial solid product, which then disproportionated to Fe and Fej04, together with COj and CO (in a 3 2 ratio), a - time data again fitted the Avrami -Erofeev equation (n = 2) with E = 175 kJ mol . The reaction was proposed to proceed by a nucleation and growth process without melting. The behaviour was comparable with the decompositions of other metal oxalates. 

---