comparison between kinetic and thermodynamic effects on grain growth
TRANSCRIPT
www.elsevier.com/locate/tsf
Thin Solid Films 466 (2004) 108–113
Comparison between kinetic and thermodynamic effects
on grain growth
Feng Liua,b,*, Reiner Kirchheimb
aMax-Planck-Institute for Metals Research, Heisenbergstr. 3, Stuttgart 70569, Germanyb Institute for Materialphysik, University of Gottingen, Tammannstr. 1, Gottingen 37077, Germany
Received 14 August 2003; received in revised form 21 January 2004; accepted 2 March 2004
Available online 27 April 2004
Abstract
A detailed comparison between the kinetic and the thermodynamic effects on grain growth was given, and the corresponding models were
fitted to the experimental data, respectively. It was found that both models can explain the derivation from the normal parabolic growth under
ideal condition. According to the kinetic model, a single isothermal grain growth can be understood in terms of a single, thermally-activated
rate process with constant Q and grain boundary (GB) energy, rb; impurity atoms accumulated in the GBs might exert a retarding force on
GB migration, but do not change the grain growth activation energy, Q. On the other hand, it is necessary to invoke variable Q and rbaccording to the thermodynamic model, where the probably existing impurities and/or surface oxide seem to block surface and/or GB
diffusion path, thus increasing Q and reducing rb.D 2004 Elsevier B.V. All rights reserved.
Keywords: Nanostructures; Grain boundary; Segregation; Diffusion
1. Introduction
The stability of nanocrystalline (NC) alloy with res-
pect to grain growth is discussed controversially. Some
authors [1–3] claim that solute drag by the alloying or
impurity atoms reduces the mobility of the grain bound-
aries (GBs), whereas others attribute it to a vanishing
driving force [4–6]. For NC materials, the large change
in total GB area accompanying grain growth would
greatly affect the grain growth kinetics. As GB area
diminishes, the concentration of solute or impurity atoms
segregated to the GBs is expected to increase rapidly and
introduces a grain-size-dependent retarding force on GB
migration. Considering this grain-size-dependent drag
force, Michels et al. [2] obtained the following grain
growth equation:
Dt ¼ D2max � ðD2
max � D0Þexp � 2kt
D2max
� �� �1=2
ð1Þ
0040-6090/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.tsf.2004.03.018
* Corresponding author. Max-Planck-Institute for Metals Research,
Heisenbergstr. 3, Stuttgart 70569, Germany.
E-mail address: [email protected] (F. Liu).
where Dt is the final grain size, D0 the initial grain size,
Dmax the maximum grain size, t the annealing time and k
the temperature-dependent rate constant. It is implied in
the last equation that the grain growth activation energy
Q and the GB energy rb are assumed to be constant. For
NC alloy with a large fraction of interfacial volume,
however, rb, as the driving force for grain growth, cannot
be kept constant throughout the whole process.
According to the Gibbs adsorption equation [7,8] and the
thermodynamic treatment of Weissmuller [4] and Kirch-
heim [6] based on this concept, rb reduces with solute
segregation.
rb ¼ r0 � Cb0 tRT lnX0 þ DHseg b ð2Þ
where r0 is the GB energy for pure solvent, X0 the bulk
content, Cb0 the solute excess of GB monolayer available
for segregation at saturation and DHseg the enthalpy change
of segregation per mole of solute. Whenever rb is positive,grain growth will decrease the free energy of the system.
Since systems with rb<0 are not thermodynamically stable,
the only case where grain growth can be suppressed is
where rb=0 (see Section 2).
Fig. 1. Fits of Eq. (5) with s=2 (solid lines) and s=3 (dash-dotted lines) to
the grain sizes calculated from Eq. (1) at different temperatures.
F. Liu, R. Kirchheim / Thin Solid Films 466 (2004) 108–113 109
Grain growth in NC alloy is accompanied by a reduc-
tion in diffusion, i.e. an increase of activation enthalpy of
diffusion [9]. Upon grain growth, Q should be increased,
continuously, in contrast with a reduction of rb. Only if
grain growth stops with saturated GBs, rb could be zero
[6]. Approximately, a single process can be separated as
several domains, and Q and rb are kept constant within
each domain,
Z Ds
D0
DdD ¼Z D1
D0
DdDþ . . .þZ Ds
Ds�1
DdD
¼Z t1
t0
½A1rb1�exp � Qb1
RT
� �dt þ . . .
þZ ts
ts�1
Asrbsexp � Qbs
RT
� �� �ð3Þ
where A1,. . .,As is constant and rb1>rb2>. . .>rbs and Qb1<
Qb2<. . .<Qbs with s as the number of the domains. Inte-
grating Eq. (3) gives
D2s � D2
0 ¼ k1t1 þ k2ðt2 � t1Þ þ . . .þ ksðts � ts�1Þ ð4Þ
Table 1
Values of the rate constants and the relative errors obtained from fitting Eq. (5
temperatures
Different T Eq. (5), s=2 Eq. (5), s=3
k1 (nm2/s) k2 (nm
2/s) k1 (nm2/s) k2 (nm
2/s)
1 0.1595 0.005769 0.1442 0.01671
2 0.8861 0.04459 0.8042 0.1275
3 14.279 0.08361 13.061 0.4204
with k1>k2>. . .>ks. In general, the last equation can be
rewritten as
D2t � D2
0 ¼
k1t tVt1
k1t1 þ k2ðt � t1Þ t2 > t > t1
. . .
. . . ts�1 > t > ts�2
k1t1 þ k2ðt2 � t1Þ þ . . .þ ksðt � ts�1Þ ts > t > ts�1
8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:
9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;
:
ð5Þ
Apparently higher value of s gives better fits to the
experimental data (see Section 3). Generally, s can be
chosen as 2 or 3.
By fitting both models (Eqs. (1), (3), (4) and (5)) to the
experimental data obtained in crystalline growth in nano-
nickel samples with different oxygen amounts at 673 K
[10] and in normal grain growth in silver thin film at
different temperatures [11], a detailed comparison between
the kinetic and the thermodynamic effects on grain growth
inhibition has been given in the present paper.
2. Comparison between kinetic effect and
thermodynamic effect
According to Eq. (1), Dt, at constant annealing temper-
ature T, is determined by D0, Dmax and k, and can be
described as a function of t (see Fig. 1). Eq. (1) is derived
by assuming a linear velocity/driving force (V/rb) relation
where the pinning force, assumed to be dependent on grain
size, is subtracted directly from the capillary force term
[1], whereas it is necessary to invoke variable Q and rbaccording to Eq. (5). Fitting Eq. (5) to Dt calculated from
Eq. (1) gives the values of k1 and k2 with s=2 or k1, k2 and
k3 with s=3, as well as the relative error of the fit, as
gathered in Table 1, respectively. Obviously, good fits are
obtained, and fit corresponding to s=3 is better than that
corresponding to s=2. This further indicates that it is
reasonable to assume variable Q and rb.Furthermore, a relation kik1
s=2ik1s=3>k2
s=2ik2s=3>k3
s=3,
deduced from Table 1, implies that higher T strengthens the
) with s=2 and s=3 to the grain sizes calculated from Eq. (1) at different
Eq. (1) Error of fit (%)
k3 (nm2/s) Dmax (nm) k (nm2/s) s=2 s=3
0.001384 15 0.1 2.9 2.1
0.002173 35 0.5 4.0 1.6
0.01173 120 10 3.1 1.5
Table 2
Values of the rate constants and the relative errors obtained from fitting Eqs.
(1) and (5) (with s=2) to the grain sizes measured in nickel thin film with
oxygen contents between 956 and 6039 wt. ppm at 673 K [10]
T=673 K Eq. (5), s=2 Eq. (1) Error of fit (%)
ppm O2 k1(nm2/s)
k2(nm2/s)
Dmax
(nm)
k
(nm2/s)
Eq.
(5)
Eq.
(1)
956 53.561 0.1370 142 68.843 2.5 0.8
1805 9.1921 0.04928 62.2 9.6801 3 3.2
6309 1.7336 0.03181 33.6 1.8827 0.4 1.8
F. Liu, R. Kirchheim / Thin Solid Films 466 (2004) 108–113110
effect of solute segregation, and thus increases Q, e.g.
activation enthalpy for GB diffusion. Regarding both kinetic
and thermodynamic effects, which one controls the grain
growth inhibition is really a difficult problem.
Another mechanism for grain growth inhibition was
proposed considering the correlation between GB struc-
ture and grain growth [12]. The GBs at low temperature
tend to be ordered (facet or straight) with mobility much
lower than that of disordered (defacet or curved) GBs at
high temperature. When grain growth occurs at relatively
low temperature, the fraction of high energy GBs
decreases while that of low energy GBs increases. When
most of GBs are facet or straight, the mobility becomes
extreme low and the grain growth can practically stop.
Therefore, this interpretation indicates that, although rb
decreases with segregation to a degree, it is difficult to
accept that rb decreases eventually to zero, where grain
growth stops. In reality, ‘‘zero’’ value according to the
thermodynamic interpretation should be considered as the
ideally reachable limit for rb. This can be supported by
the fact that the {111}, {110} and {211} coherent twist
boundary in Cu possesses rb=0 if the inclination angles
/111, /110 and /211 are close to zero [13]. As for NC
alloy, however, experiment has shown that the effect of
solute atoms on rb can be substantial [14], and theoretical
considerations predict that driving rb to zero is possible
in alloy with high segregation energy [6,14].
According to Ref. [12], grain growth will re-start
because the GBs undergo a transition from ordered
to disordered structure with increasing T. This kind
of strong temperature-dependence can be also deduced
from the kinetic effect (Eq. (1)). Without considering this
correlation between GB structure and grain growth,
potentially more effective would be the thermodynamic
Fig. 2. Fits of Eq. (1) (dash-dotted lines) and Eq. (5) with s=2 (solid lines)
to the grain sizes measured in nickel thin film with oxygen contents
between 956 and 6039 wt. ppm at 673 K (data points were taken from [10];
note that not all the data were taken).
effect, which manifests only weak temperature-depen-
dence [15].
3. Application of the kinetic and the thermodynamic
models
3.1. Grain growth in nanocrystalline nickel doped with
nickeloxide
According to Ref. [10], a pulse reverse technique was
used to deposit nanocrystalline nickel (19 nm) doped with
nickel oxide. The effect of GB nickel oxide on the
thermal stability of NC Nickel was studied on samples
with oxygen contents between 956 and 6039 wt. ppm at
673 K. Fig. 2 shows different fits to the measured data
points using Eqs. (1) and (5), respectively. The corres-
ponding values for the fitting parameters are gathered in
Table 2.
From Fig. 2 and Table 2, both models give good fits to
the experimental data. According to the kinetic mechanism,
constant Q and linear V/rb relation prevail throughout a
single process with definite oxygen content, and increasing
Fig. 3. Fits of Eq. (1) (dashed lines) and Eq. (5) with s=2 (dash-doted lines)
and s=3 (solid lines) to the grain sizes measured in silver thin film at
different annealing temperatures [11].
Table 3
Values of the rate constants and the relative errors obtained by fitting Eqs. (1) and (5) (with s=2 and s=3) to the grain sizes measured in silver thin film at
different annealing temperatures [11]
T Eq. (5), s=2 Eq. (5), s=3 Eq. (1) Error of fit (%)
(K)k1 (nm
2/s) k2 (nm2/s) k1 (nm
2/s) k2 (nm2/s) k3 (nm
2/s) Dmax (nm) k (nm2/s) Eq. (1) Eq. (5)
s=3 s=2
523 15.136 1.3132 14.060 7.0260 5.1427 94.9 13.180 2.4 2.9 4
473 5.4695 0.7227 6.4965 1.9641 0.4576 69.3 5.6039 3.1 2.8 3.5
448 1.3528 0.1944 2.6034 0.7351 0.1088 53.1 4.9807 0.4 0.33 2.8
F. Liu, R. Kirchheim / Thin Solid Films 466 (2004) 108–113 111
the oxygen content enhances Q (see Table 2). So the
stagnation of grain growth after longer annealing is due to
the segregated impurities and/or second phase precipitates
to stabilize GB migration. According to Eq. (5), however, a
single process consists of several domains—Q for the later
domain is much higher than that for the earlier domain and
a linear V/rb relation does not hold within the whole
process. Furthermore, k1 and k2 are reduced with increasing
the oxygen content, but k2 differs in smaller order of
magnitude than k1 does, inferring that Q in the second
domain does not change a lot with the oxygen content.
Therefore, it can be concluded that if grain growth has
(almost) stopped with saturated GBs, variations of Q and rb
Fig. 4. Arrhenius plot of the parameters (a) k (in Eq. (1)) and (b) k1, k2 and
k3 (in Eq. (5) with s=3) against the reciprocal annealing temperature.
should be negligible, independent of the overall solute
content [10].
3.2. Normal grain growth in silver thin film
Normal grain growth in 80-nm-thick sputter-deposited
Ag films was studied via in situ heating stage transmission
electron microscopy [11]. A grain growth exponent n=1/3
from the law D1/n�D01/n=k(T)t was calculated by minimiz-
ing the deviation in the fitting function to the experimental
data, and Q=0.53 eV was found, which is close to surface
diffusion [16]. According to Ref. [11], n=1/3 indicates that
some impurities and/or surface oxide might play a role in
inhibiting GB migration by segregation in normal grain
growth in silver thin film. Here, re-fitting Eqs. (1) and (5)
that assume n=0.5 to the experimental data [11] gives
good results, as shown in Fig. 3 and Table 3. Comparing
Table 3 with Table 1 demonstrates that the relation,
kik1s=2ik1
s=3>k2s=2ik2
s=3>k3s=3 is also deduced, and fit
by Eq. (5) with s=3 is better than that with s=2.
Whether both the kinetic and the thermodynamic models
can explain the deviation from the ideal parabolic manner
for normal grain growth needs further consideration of
the revolution of Q with time or temperature. When n
deviates from the ideal value of 0.5, Q is generally diffi-
cult to assess particularly. Since n=0.5 can be guaranteed
when modelling by using Eqs. (1) and (5), Q can be
subsequently obtained from the slope of a plot of ln k against
1/T. It is assumed that the rate constant k has an Arrhenius
relation with temperature [1]:
k ¼ k0 exp � Q
RT
� �ð6Þ
where R is the gas constant and k0 a constant. The results are
shown in Fig. 4 and Table 4.
Table 4
Grain growth activation energies obtained from Arrhenius plot of the
parameters (a) k (in Eq. (1)) and (b) k1, k2 and k3 (in Eq. (5) with s=3)
against the reciprocal annealing temperature
Temperature Eq. (1) Eq. (5), s=3
range (K)Q (eV) Q1 (eV) Q2 (eV) Q3 (eV)
448–523 0.31 0.4 0.6 1
F. Liu, R. Kirchheim / Thin Solid Films 466 (2004) 108–113112
4. Discussion
The present experimental data strongly suggests that the
probably existing impurities and/or surface oxide segregated
to GBs are the cause of the kinetic or thermodynamic
mechanism that suppresses grain growth in silver thin film
[11]. The difficult question is how or what is the mechanism
by which the impurities and/or surface oxide acts.
In most cases, the activation energy of grain growth is
close to that of GB diffusion. Due to the limited temper-
ature range for grain growth in silver thin film, constant
value for Q could be obtained by fitting the data points
derived from k (Eqs. (1) and (6)) with a straight line (see
Fig. 4a). On this basis, Q=0.31 eV is obtained (Table 4),
which is significantly less than that of GB diffusion in
silver, 0.95 eV, but close to that measured for the
abnormal grain growth, 0.274 eV [17]. This value is
consistent with a surface diffusion process, reported to
be 0.3 eV in electro-migration pore formation experiments
[17]. So the enhanced thermal stability due to the pres-
ence of impurities can be probably explained on the basis
of GB pinning by an oxide phase that seems to be
present in the silver sample. These impurities accumulated
in GBs, as an external agent, might exert a retarding force
on GB migration, but do not change the activation energy
for grain growth, i.e. surface diffusion (see Fig. 4a and
Table 4).
As solute segregation proceeds upon grain growth, GB
concentration reaches the saturated value [18,19], thus
eliminating the GB vacancies, altering the GB dislocation
structure, and reducing the free volume by simply attach-
ing itself to the areas of poor fits [20]. This will increase
the activation energy for GB diffusion, DGb, as grain
growth relies upon diffusion while diffusion is concerned
with the number and type of neighbouring atoms. As
shown in Refs. [18,19], grain growth in nano-grained
FexY1�x (0<x<0.3) produced by gas condensation was
apparently influenced by solute segregation of Fe to the
Y GBs. When GBs are saturated with Fe atoms, the
grains and GBs reach a metastable state and grain growth
stops; grain size keeps stable and grain growth will not
occur until precipitation happens. This kind of phenome-
non was also observed in GB self-diffusion in a Cu
polycrystal of different purity showing that increasing
the GB impurity concentration leads to a decrease in
GB energy, while enhancing the activation enthalpy for
GB self-diffusion [21].
By fitting the data points derived from k1, k2 and k3 (Eq.
(5) with s=3 and Eq. (6)), Q, kept constant within each
domain, is declared to increase dramatically with annealing
time (Fig. 4b and Table 4). This shows that isothermal
annealing results in equilibrium grain size with saturated
GBs, while Q is increased with solute segregation and grain
growth. As given in Table 4, Q1, Q2 and Q3 for the first, the
second and the third domains are obtained as 0.4, 0.6 and 1
eV, respectively. Therefore, it should be due to the reduced
GB energy resulting from solute segregation that suppresses
grain growth.
The atomic mechanism involved in the motion of the
GBs consists of jumps across the interface from one
crystallite to another. This is expected to occur with Q
between 0.75 and 1.25 eV [11,17], more consistent with
GB diffusion in silver. For a thin film specimen, GB
motion and grain growth can be inhibited by the formation
of thermal groove, which pins the GBs at the film surface
[17]. The thermal groove forms by surface diffusion, and
surface diffusion should have activation energy, 0.3 eV
[17]. As given in Table 4, Q=0.4 eV is initially close to
that of surface diffusion, but upon grain growth, the
probably existing impurities and/or surface oxide seem to
block surface and/or GB diffusion path, thus increasing Q
(=1 eV) and reducing rb.
5. Conclusion
A detailed comparison between the kinetic and the
thermodynamic effects on grain growth was given. Both
mechanisms provide good interpretation for derivation
from the normal parabolic growth under ideal conditions.
It was found that, both models originate from segregation
of solute and/or impurity atoms at GBs, present different
explanations for grain growth inhibition, but give analo-
gous results. According to the thermodynamic effect, the
probably existing impurities and/or surface oxide seems to
block surface and/or GB diffusion path, thus enhances the
activation energy, and in turn, reduces GB energy. Accord-
ing to the kinetic effect, however, impurity atoms accu-
mulated in GBs, as an external agent, might exert a
retarding force on GB migration, but does not change
the activation energy for grain growth.
Acknowledgements
This research is supported by the Alexander von
Humboldt Foundation.
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