MacDiarmid - Grain Growth Kinetics of ZnOAl Nanocrystalline Powders

Fits of various models to crystallite size data shown

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Unformatted text preview: esolution of the data shown in Figure 4, a number of grain growth models were investigated. 21036 dx.doi.org/10.1021/jp207140g |J. Phys. Chem. C 2011, 115, 21034–21040 The Journal of Physical Chemistry C ARTICLE Figure 5. Fits of various models to crystallite size data shown in Figure 2. See text for the model descriptions and formulas. Figure 6. Fits of the relaxation model to 0% Al crystallite size versus time data for different temperatures. These are shown in Figure 5 for the data presented in Figures 1 and 2 (0% Al heated at 600 °C). The models investigated are as follows: Model 1: Generalized parabolic grain growth (see refs 13À16 and references therein). This satisfies the differential equation dDðt Þ a1 ¼ dt Dð t Þ where D(t) is the grain size as a function of time and a1 is a constant dependent on the grain geometry, interfacial energy, and grain boundary mobility. This has the solution Dð t Þ 2 À D0 2 ¼ k 1 t where D0 = D(t = 0) and k1 is the rate constant (k1 = 2a1). It is usually generalized to n Dðt Þ À D0 ¼ k1 t n ð1 Þ Figure 7. Fits of the relaxation model to 1% Al crystallite size versus time data for different temperatures. which has the solution qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dðt Þ ¼ D∞ 2 À ðD∞ 2 À D0 2 ÞexpðÀk3 t Þ where experimentally n has been found to take values up to 10,13À15 although only values between 2 and 4 are considered to have any physical meaning. Model 2: Grain growth with impediment (see refs 13 and 14 and references therein). This satisfies the differential equation ð3Þ where the rate constant k3 is equivalent to k3 = 2b3 = 2a3/D∞2. Model 4: Relaxation model (see refs 15 and 16 and references therein). This has the form dDðt Þ a2 ¼ À b2 dt Dð t Þ Dðt Þ À D0 ¼ 1 À expðÀk4 t m Þ D∞ À D0 where b2 is a growth-retarding term that results in dD(t)/dt = 0 as t f ∞. This has the solution   D0 À Dð t Þ D∞ À D0 k2 t ¼ þ ln ð2 Þ D∞ D∞ À Dð t Þ ð4Þ where k4 is the rate constant and m is the relaxation order.The model that best fits the data in all cases was found to be model 4, the relaxation model. For all cases, fitting D0 (the initial crystallite size) resulted in small values (∼0.1 nm) with uncertainties of greater than 100%; therefore, D0 was set to zero. For the experiments conducted at 800 °C, the fitted parameters tended toward D∞ f ∞; for these D∞ was fixed to 2000 nm to obtain values for k4 and m. In the case where D0 = 0 and D∞ f ∞ it can be shown that eq 4 approaches the form of eq 1, the generalized parabolic grain growth model, with m = 1/n and k4 = k11/n/D∞. See the Supporting Information for this derivation. Fits of all of the data for various Al contents and where D∞ = D(tf∞) and k2 = b22/a2 = a2/D∞2 is the associated rate constant. Because of the unusual form...
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This note was uploaded on 06/15/2013 for the course MSE 101 taught by Professor Sen during the Spring '12 term at Indian Institute of Technology, Kharagpur.

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