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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 growthretarding 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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