Hancock and Sharpe

A much better way is to follow the method of Hancock and Sharp ( 1948). If one can measure x, the amount formed in time, t, then the... 

The above equations summarize the three (3) diffusion mechanisms, one of which usually predominates in any given case (see 3.1.74.). But these equations contain quantities that are hard to measure, or even estimate. A much better way is to follow the method of Hancock and Sharp ( 1938). If one can measure x, the amount of reactant formed in time, t, then the following equation applies ... 

Barium titanate powders were produced using either an amorphous hydrous Ti gel or anatase precursor in a barium hydroxide (Ba(OH)2) solution via a hydrothermal technique in order to discern the nucleation and formation mechanisms of BaTi03 as a function of Ti precursor characteristics. Isothermal reaction of the amorphous Ti hydrous gel and Ba(OH)2 suspension is believed to be limited by a phase boundary chemicd interaction. In contrast, the proposed BaTi03 formation mechanism from the anatase and Ba(OH>2 mixture entails a dissolution and recrystallization process. BaTi03 crystallite nucleation, studied using high resolution transmission electron microscopy, was observed at relatively low temperatures (38°C) in the amorphous hydrous Ti gel and Ba(OH)2 mixture. Additional solution conditions required to form phase pure crystallites include a C02-free environment, temperature >70°C and solution pH >13.4. Analysis of reaction kinetics at 75°C was performed using Hancock and Sharp s modification of the Johnson-Mehl-Avrami approach to compare observed microstructural evolution by transmission electron microscopy (7). 

BaTiOj Kinetic Analysis. Utilization of the Hancock and Sharp modification of the Johnson-Mehl-Avrami analysis permits formation mechanism evaluation of an otherwise closed hydrothermal system(7,79). Through application of the following equation,... 

Figure 2. Hancock and Sharp kinetic analysis of BaTiO formation mechanisms for the (a) amorphous Ti gel and (b) anatase precursor. The m is a constant determined by the system geometry and r is the correlation coefficient.

The amorphous gel reaction produced relatively spherical 5nm to lOnm BaTi03 nuclei which grew within the porous amorphous gel matrix. Based on the Hancock and Sharp kinetic analysis, the proposed BaTi03 formation mechanism entails a phase boundaiy limited reaction where the chemical interaction at the gel surface is the rate controlling step. In the case of BaTi03 formation from the anatase precursor, lOnm size crystallites which nucleated atop the aggregated anatase were observed. It is believed... 

J. S. Hancock and J. H. Sharp, Method of comparing solid-state kinetic data and its application to the decomposition ofkaolinite, brucite, and BaCO, J. Am. Ceram. Soc. 55,74-77(1972). 

The most reliable way to find n is to use the method of Sharp and Hancock. This consists of taking logarithms of the Avrami-Erofe ev equation, to give... 

The gradient of this graph therefore permits the determination of n, and the intercept allows k to be calculated. The advantage of using a Sharp-Hancock plot rather than a least squares fitting process with the Avrami equation is that if Avrami kinetics are not applicable, this can be seen in the former plot, and hence other kinetic models may be investigated. Purely diffusion controlled processes can be identified using a Sharp-Hancock plot n is foimd to be 0.5 in such cases. 

Reactions were performed over a range of temperatures. Sharp-Hancock plots were used to determine the values of n and k for each temperature. The plots are given in Eig. 6 and the parameters extracted in Table 3. 

Fig. 6 Sharp-Hancock plots for the temperature dependence of the intercalation of 10 M LiCl into gihbsite at 140 °C ( ) 120 °C (A), 100 °C (o), 80 °C (V) and 60 °C (0). Reproduced with permission from Chem Mater (1999) 11 1771-1775...

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