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equation, it is common to ﬁt the data as t(D) rather than D(t).
Model 3: Grain growth with size-dependent impediment (see
refs 13 and 14 and references therein). This satisﬁes the
À b 3 Dð t Þ
Dð t Þ
21037 dx.doi.org/10.1021/jp207140g |J. Phys. Chem. C 2011, 115, 21034–21040 The Journal of Physical Chemistry C ARTICLE Figure 8. Fits of the relaxation model to 2% Al crystallite size versus
time data for diﬀerent temperatures. Figure 10. Fitted parameters m and D∞ as a function of temperature for
diﬀerent Al contents for the ﬁts shown in Figures 6À9. Figure 9. Fits of the relaxation model to 4% Al crystallite size versus
time data for diﬀerent temperatures. temperatures using the relaxation model (Model 4) are shown
in Figures 6À9.
The ﬁtted parameters m and D∞ are shown for all samples as a
function of temperature in Figure 10. For the experiments with
0% Al, m decreases with increasing temperature while D∞
increases with increasing temperature. These trends are less
pronounced for experiments with nonzero Al content. At
800 °C m is approximately 0.1, which for the generalized
parabolic grain growth (recalling that the two models have
equivalent forms in these conditions) corresponds to a value of
n of 10. Similar values have been observed before in nanocrystalline grain growth studies.15,16
Since D∞ tended toward excessively large numbers at 800 °C
it was not possible to use the D∞ values in an Arrhenius plot.
Instead, the last 10 min of D(t) values were averaged (termed Df),
and a plot of ln Df versus 1/T is given in Figure 11 for samples
of various compositions. These obey Arrhenius relationships
(as indicated by the linear ﬁts) from which the activation energies
can be extracted. The activation energies are shown as a function
of Al content in Figure 12. A value of 24 ( 3 kJ/mol was obtained
for undoped ZnO, while for all of the Al-doped experiments the
values were around 43 ( 4 kJ/mol. Figure 11. Arrhenius plot of Df (average last 10 min) vs 1/T. Linear ﬁts
are shown. ’ DISCUSSION
For all compositions and temperatures studied, the initial
nucleation and grain growth is very fast, with peaks corresponding to crystalline ZnO being observable by the end of the ﬁrst
scan after insertion into the hot air stream, i.e., nucleation occurs
in less than 30 s, with the average crystallite size after 1 min being
larger than 10 nm for most of the temperatures studied. This
implies a low activation energy, which is borne out in Figure 12.
The activation energy for grain growth in undoped ZnO has
been quoted in the literature as being 275 kJ/mol.17 The value we
21038 dx.doi.org/10.1021/jp207140g |J. Phys. Chem. C 2011, 115, 21034–21040 The Journal of Physical Chemistry C Figure 12. Activation energy extracted from the slopes of the ﬁtted lines
in Figure 11 as a function of Al content. obtained, of 24 ( 3 kJ/mol, is less than one-tenth of the
Grain growth activation energies in nanocrystalline oxides
have been o...
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