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Grain growth in nanocrystalline materials
Feng Liu, Ke Zhang, Mingming Gong
30 Nov 2010
The Saudi International Nanotechnology Conference 2010
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Contents
Experiments
Thermo-kinetics of grain growth
Theoretical background of thermal stability
Introduction
Conclusions
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1 Introduction
1.1 Character and application of nano-material
1.2 Importance of stability of nano-material
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Character Application
Nano-
material
Surface-effect
Size-effect
Quantum-effect
Structure material
Function material
device
1.1 Character and application of nano-material
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1.2 Importance of stability of nano-material
Stability
grain sizeimportant factor
reflecting properties ofnanocrystalline materials
actual use
grain growth
performancedecreasing
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2 Theoretical background ofthermal stability
2.1 Thermodynamic models of grain growth
2.2 Kinetic models of grain growth
2.3 Advantage and disadvantage of previous models2.4 Development of current models
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2.1 Thermodynamic models of grain growth
1. Weissmllers model
---A concept of stabilization of NC solids was first presented.
( ) ( ) { } { } { }0
, , ln
M
dilute M sol sol mix
in M in GB g P T P T M
N
N H H T S N R N
= +
specific excess, , integral molar heats of solution in matrix and GB,
excess entropy of mixing, number of atoms of component in the matrix.N
sol
MinHsol
GBinH
{ }mixS MNNanostruct. Mater., 3(1993) 261
Acta Mater., 50 (2002) 413
2. Kirchheims model
---From a general thermodynamic consideration, an additional analytical
model was derived.
with is the GB energy for pure solvent, the saturated solute excess, the
content of matrix and the enthalpy change.
( )0 0 lnb b g seg R T X H = +
0 0b
segH
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3. Liu and Kirchheims model
Liu and Kirchheims model
Gibbs adsorptionequation
McLeans GBsegregation model
=i
iidSdTd
0 0 03ln b mb b g seg
V R T X H D
= +
Metastable equilibrium grain size D*
( )0 0 0 03
exp ( ) /
b M
b seg b g
VD
X H R T
=
F.Liu et al. J.Cryst.Growth, 264 (2004) 385
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Validation in Ni-P and Pd-Zr alloys [Acta Mater., 48 (2000) 789; Z. Metallkd., 96 (2005) 1134]
as prepared
heat treated, 573K
fits
Relation of grain size and P-contents in NCNi-P alloys at 573K
Relation of grain size and temperature in NCPd-Zr alloys at 573K
It is shown that this model fits well with grain growth in NC Ni-P and Pd-Zr alloys. This further provides evidence
that reduces to zero once grain growth is suppressed with saturated GBs.b
0
(J/ m2)
Gseg
(kJ/ mol)
b
(mol/ m2)
b0
(mol/ m2)
Error
(%)
0.51 55.2 1.751.8510-5 1.8810-5 5
Solute content 0
(J/ m2
)
Gseg
(kJ/ mol)
b0
(mol/ m2
)
Error
(%)Pd90Zr10
Pd85Zr15
Pd80Zr20 (6731400 K)
Pd80Zr20 (11731400K)
0.7
0.7
0.7
0.7
55.9
42.5
59.2
59.2
2.0710-5
3.0710-5
1.7810-5
1.7810-5
2
16
7
2
Fitting parameters and errors of fits are given in Table 1 and 2
Table 1
Table 2
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2.2 Kinetic models of grain growth
1. Burke and Turnbulls model [Prog Met Phys., 3 (1952) 220]
--- A so-called parabolic law applicable to grain growth in highly pure and coarse-grained polycrystalline materials was proposed.
tMDDD
M
dt
dDb
b
== 202
2
with Mas the GB mobility, the initial mean grain size at annealing time t=0 and
the GB energy.
2. Burkes solute drag model [Trans Metall Soc AIME., 175 (1949) 73]
--- Regarding the effect of the solute, the case where a drag term is independent of
the grain size is first considered.
0D b
tD
k
DD
DD
D
DD
max2
max
0max
max
0ln =
+
with as the maximum size and kinetic parameter.bk M=maxD
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3. Michelss model [Acta Mater., 47 (1999) 2143 ]
--- A grain-size-dependent drag term is introduced to restrain the grain growth andto stabilize the grain size of nano-material.
( )2
1
max2
2
0max2
max2 2exp
=D
ktDDDD
4. Rabkins model [Scripta Mater., 42 (2000) 1199 ]
--- From Cahn, the drag term acts as function of both GB concentration and V,Rabkin then shows that,
( ) ( ) tMDDMDD b
=+3
0
32
0
2
32
1
with , =const.
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2.3 Advantage and disadvantage of models
Thermodynamic model favors a state or tendency,
whereas, cannot describe the evolution of grain size
Weissmllersmodel
Liu et alsmodel
Kirchheimsmodel
Thermodynamics
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Constant GB energy b is assumed and stabilized grain
size cannot be determined. It points to a real process.
Burke andTurnbulls
RabkinsBurkes Michelss
Kinetics
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2.4 Development of current models
Advantage and
disadvantage of
thermodynamic
and kinetic models
Establish an intact nano-graingrowth model considering mixed
effect of thermodynamicsand kinetics
Factors affecting
grain growth
Until now, thermodynamics and kinetics of grain growth have been studied onlyindependently of one another for NC materials.
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3 Thermo-kinetics of grain growth
3.1 Validity of thermo-kinetic models
3.2 State criterion of initial GB segregation3.2.1 Thermodynamic state with saturated GB segregation
3.2.2 Evolution of thermodynamic factors upon grain growth with unsaturated
GB segregation
3.3 Thermo-kinetic models of grain growth3.3.1 Thermo-kinetic models with saturated GB segregation
3.3.2 Thermo-kinetic models with unsaturated GB segregation
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3.1 Validity of thermo-kinetic models
Kinetic process Thermodynamicstate
Lius GB energymodel
Both Kinetic process and Thermodynamic state can lead to the same GB energy model.
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The coupling of GB diffusion, bulk diffusion (kinetics) and GB energy (thermodynamics) is reasonable.
Kinetic process
2
0 0
ln ln
b
l
b a a
DkT Dm ma a
=
( )
( )
( )
( )
( )
( )
( )
0 0
0 0 0 0
0 0 0
0 0
0 0 0
b b b
fb mb bb g l b l b
fb fb f f l l l
fb mb bb g l l l b
b b g mb mb m mb b b
f m b g l b l b
bb bb b b
l l l f m b g
S S S R H H H H
S S S R R T H H H H
S S S R H H H H
S S S R
+ + + + + = + + + + + + + + + +
ln X segH
0 0 lnb b g seg R T X H = +
b l bG G =
Borisov semi-empirical equation[Fiz. Metal. Metall., 17 (1964) 80]
The term ofentropy change
The term ofenthalpy change
Lius GB energy model
F.Liu et al. Acta Mater., 57 (2009) 1466
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Both kinetic process and thermodynamic state lead to the same GB energy model. This further proves that
incorporation of GB energy decreasing with GB segregation into the parabolic kinetics is physically practicable.
R. Kirchheim, Acta Mater., 50 (2002) 413
F.Liu et al. J.Cryst.Growth, 264 (2004) 385
Thermodynamic state
Kirchheims model:The segregation enthalpy movesolute atoms from the GBs into thegrains. The changes of theconfigurational entropy is( ). Assuming further
that the GBs are always saturatedwith solute atoms ( ) leads to Lius GB energy model
+=
seg
g
bbH
XTR
ln00
McLeans GB segregation model
= TR
H
X
X
XX
X
g
seg
GBGB
GB
exp10
==
dX
d
TR
X
d
d
g
b
Gibbs adsorption equation
0 0ln
b b g seg R T X H = +
segH
( ) lnb gda R T X
0b b =
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3.2 State criterion of initial GB segregation
Upon grain growth, the evolution of GB energy and GB excess amount with grain size must depend on whetherGB segregation is saturated or not. So it is necessary to analyze the initial GB excess amount and evaluate thestate of initial GB segregation.
0
exp1
segGB
GB GB g
HX X
X X X R T
=
McLeans GBsegregation modelConservationlaw Conservationlaw
0 3 /b M X V D = b GBX =
( )03
0
1
exp1
seg
g
b M
b
Hb R T
V
DX
+
( ) ( )0
0
0
3
00
1 1
exp exp1 1
seg seg
g g
b M
b
H Hb R T R T
V
DXX
< +++
>>+
=
1112211
121211
11
2
0
2
... nnnn
t
tttttkttktk
tttttktk
tttk
DDLL
1 Lius thermo-kinetic model
with k1>k2>>kn and n can be chosen as 2 or 3.
F.Liu et al. Thermochimica Acta, 443 (2006) 212
Lius model
---Liu et al introduced variable activation energy Q and into the parabolic growthlaw.
b
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Validation in nano-RuAl alloy [Acta Mater., 49 (2001) 395]
The grain size, D, of nano-RuAl produced bymechanical alloying, was evaluated as a
function of annealing time, t
n = 2
n = 3
The present model can describe the experiment results well (n=2 and n=3).
( )12
2 2 2
max max 0 2
max
2exp
kt D D D D
D
=
Michels,
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2 Thermo-kinetic numerical model
Lius modelParabolic
equation
Numerical model
+=
seg
g
bbH
XTR
ln00
Numerical model of thermo-kinetics
0 0 0
3ln b mb g seg
V R T X H
DdDM
dt D
+ =
( )
( )
33
0 3
330
0 3
exp
1 1
b
segb
b b gb
D DXHD
R TD DX
D
= +
McLeans GB
segregation model
=
TR
H
X
X
XX
X
g
seg
GBGB
GB exp10
Conservation
law
( )fX
fXx GB
+=
1
0
F.Liu et al. J. Alloys Compd., 475 (2009) 893
Two-stage (i.e., thermo-kinetic) fittingis applied,
1. Determination of thermodynamicparameters: , , and
2. Determination of kineticparameters: M
0 b 0b segH
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Validation in nano-CGO [Acta Mater., 54 (2006) 1721]
Average grain size of the CGO spray pyrolysis films as afunction of annealing time and temperature
The generalized parabolic growth model cannot predict well, whereas a good fit to the experimental data is
obtained using numerical model.
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Effect of GB energy
obviously tends to its saturated value , a metastable equilibrium, i.e. GB energy tends to zero, results and
grain growth stops.
b 0b
The GB energy vs. grain size with the temperaturesA plot of solute excess vs. grain size fordifferent annealing temperatures
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Regarding the effect of the solute segregation, Q=2.00.3 eV>Qsurface=1.30.1 eV.
The consumed annealing time before grain growth stops should be determined by the GB mobility which is
described by annealing temperature and atom diffusion activation energy.
= TR
QMM
g
exp0
Effect of interface mobility
The GB energy vs. annealing time with the temperatures Arrhenius plot of the GB mobility Magainst
the reciprocal annealing temperature
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0
0 0
bGB
V V X X X V V
= + ( ) XXGBGBb =
Conservationequation
( ) 06
b GB
D X X X
= +
( )00
6
6
seg
b
H X D
+=
GB excess
bsegb H = 0
Krills model
Simplified GB energy model
It is consistent with the phase-field approach in which the GB energy reduces linearly;
Only one parameter is used.seg
3 Thermo-kinetic analytical model
F.Liu et al. Acta Mater., 57 (2009) 1466
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In comparison with Rabkins and Kirchheims models, the present model is a relative intact one in that it couples
the effects due to thermodynamics and kinetics.
( )00
6
6
seg
b
H X D
+=
Simplified GB energymodel
Kinetic equation withsolute drag
with1 0 0
2 0
,
6
seg
seg
H X
H X
= =
( ) ( )[ ] ( )
t
D
D
M
DDM
DD
=
+
+
++
021
21
3
2
2
1
2
2
1
02
2
1
2
2
021
2
213
2
ln
21
2
Analytical thermo-kinetic model of grain growth
Analytical thermo-kinetic model
F.Liu et al. Acta Mater., 57 (2009) 1466
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Analytical thermo-
kinetic model of
grain growth
b
Pure thermodynamic model
(is zero)
( ) ( ) tMDDMDD b
=+ 3032
0
2
32
1
MtDD
D
D=
+
2
0
021
21
2
2
1 ln
( ) ( )[ ]
( ) MtD
DDD
DD
=
++
++
021
21
2
2
21
2
2
10
2
1
2
2
021
2
212
2
ln
21
2
1
Pure kinetic model( is constant)
Mixed model (= /M)2
NC Ni-O alloys annealed at 673K
Validation in NC Ni-O alloys [Scripta Mater., 44 (2001) 2321]
Pure thermodynamic model and mixed model can describe the experimental data well, whereas, the pure kinetic
model cannot bring into agreement.
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4 Experiments
4.1 Preparation and stability of NC Fe-C alloys
4.2 Thermal stability of as-spun nano-Fe-B alloys
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XRD patterns of Fe-C alloy powders and grain size
in them corresponding to different milling time
SEM pictures of Fe-C alloy powders
subjected to different milling time
4.1 Preparation and stability of NC Fe-C alloys
XRD
Nano-crystalline Fe-1at%C alloy powdershave been synthesized by high energyball milling.
The morphologies of powders have beenrevealed by SEM.
The average grain size evolution withmilling time have been obtained from XRDpatterns by applying Scherrer formula.
Final grain size is about 8nm.
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Isothermal and isochronal DSC curves of NC Fe-1at%C solid solution
The nano-scale growth (stability) can be studied experimentally by DSC method. The grainsize evolution with annealing time can be indirectly reflected by the obtained DSC curves.
Isochronal DSC experiments subjected to different heating rates have been performed to reveal the temperaturerange where grain growth happens.
Isothermal DSC experiments at annealing temperatures of 250, 270, 320, 350 and 450 oC have been conducted.
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The GB energy (Krills GB energy model), the total GB enthalpy of the nano-multicrystalline system are
incorporated into the kinetic equation for grain growth. And the equations used to describe the total energy
change during grain growth are obtained.
and
( )211n gb gb
gb mn n nm
dH MH
H gV dt g V = +
1
2
m
gb m
gVD
H gV
=
+
1 0 0
2 0
,
6seg
seg
H XH X
= =
with the enthalpy of GB,
the shape factor,
the molar volume
gbH
g
mV
1n
dD M
dt D
=
0
0
0
1 2
( 6 )
6
b seg
seg
H
H x D
D
= +
=
=
mgb
g VH
D
=
Krills GB energy model
Kinetic equation
Total GB enthalpy
The Isothermal DSC curves can be analytically fitted by the following model which
describes the total energy change during isothermal grain growth.
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3.115.02.003.710-18142450
2.512.12.177.310-20165350
4.411.02.221.910-20176320
Error %D*(nm)nMHseg (kJ mol-1)Temperature()
Fitting parameters and errors of fits are given in the following table
The reasons for the much larger values ofHsegare:
1. All the other factors that stabilize the grain size (e.g. entropy) have been incorporated into the thermodynamic effect (i.e. GB
energy decreasing with grain growth);2. The present alloy system is a little bit different from an ideal dilute solution, which is the basis of the thermodynamic model used
here.
Fitted results of DSC curves for isothermal anneal of NC Fe-C powders
With increasing annealing temperature
T, GB mobility Mincreases due to the
improvement of atom activity.The metastable equilibrium grain size
D* increases with increasing T.
Manuscript is in preparation
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4.2 Thermal stability of as-spun nano-Fe-B alloys
XRD profile for the melt spun Fe-10at.%B alloy
(7000rpm) annealed at 700
for different time
Bright field TEM image for Fe-10at.%B nano-grain(7000rpm) annealed at 700 oC for 0.5 h
For Fe-8at.%B, Fe-10at.%B, Fe-12at.%Band Fe-14at.%B alloys, the melt spinningwith 7000 rpm was performed, andsubsequently, isothermal annealing of theas-spun nano-grain at 700oC wasconducted for different time.
Typical bright field morphologies (a) anddiffraction spot (b) of the melt spun Fe-10at.%B
SSSS with rotational speed as 7000 rpm.
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Pure kinetic, pure thermodynamic, and thermo-kinetic models have been adopted to fit the experimental data.Obviously, pure kinetic model with solute drag cannot be brought into agreement with the experimental data.However, the thermo-kinetic model can be adopted to describe well the experimental data due to the introduction
of GB energy effect.
Evolution of the average grain size with the annealing timefor as-quenched Fe-B nano-grain (7000rpm) annealed at
700 oC, The symbols are the experimental data; the dotted,dashed, solid lines are calculated using pure kinetic, purethermodynamic and thermo-kinetic models, respectively.
Pure thermodynamic model(is zero)
( ) ( ) tMDDMDD b
=+ 3032
0
2
32
1
MtDD
D
D=
+
2
0
021
21
2
2
1 ln
( ) ( )[ ]( ) Mt
D
DDD
DD
=
++
++
021
21
2
2
2
1
2
2
10
2
1
2
2
021
2
2122
ln
21
2
1
Pure kinetic model
( is constant)
thermo-kinetic model (= /M)
Thermo-kinetic
model at
unsaturated GB
segregation
b
2
F.Liu et al. J. Crystal Growth, 2010 (313): 81-93
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Effect of GB energy
decreases with grain growth, but it approaches only infinitely zero, indicating that the effect of GB energy plays
an dominated role in inhibiting grain growth in Fe-B alloys.b
GB energy equationMtb
b 2
20
0
1 ln
=+
F.Liu et al. J. Crystal Growth, 2010 (313): 81-93
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Conclusions
1. Based on Borisovs equation, a qualitative description between self-diffusionin the lattice and along the GBs and the GB energy is provided. It is shownthat the incorporation of decreasing GB energy into the grain growth kineticsis physically practicable.
2. A state criterion of initial GB segregation has been proposed, which can beused to evaluate whether the GBs are saturated or not at the initial stage.
3. Based on the state criterion for initial GB segregation, the stop of graingrowth corresponds to the saturated GB, but the saturated GB does notnecessarily correspond to the stop of grain growth.
4. Thermo-kinetic model with saturated GB segregation has been proposed andsuccessfully applied in NC Ni-O alloy with oxygen content as 6039ppm.
5. Thermo-kinetic models with unsaturated GB segregation have also beenderived and successfully applied in NC RuAl, NC CGO and NC Ni-O alloyswith oxygen contents as 956 and 1805 ppm.
6. NC Fe-1at%C alloy powders have been prepared by high energy ball millingand their thermal stability has been investigated by DSC. Isothermalannealing of as-spun Fe-B NC grain at 700oC was conducted. It is due to thereduction of GB energy, but not the kinetic factor, e.g. solute drag, thatdominantly controls the thermal stability of NC materials.
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3.2.1 Thermodynamic state with saturated GB segregation
For the condition that the
initial reaches its
saturated value , the
bulk concentration will
approach the ideal bulkconcentration at
metastable equilibrium
b
0b
X
0X
3 2 2 Evolution of thermodynamic factors upon grain growth with
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3.2.2 Evolution of thermodynamic factors upon grain growth with
unsaturated GB segregation
Grain growth is a kinetic
process controlled by
thermodynamic factor.
, withD and finally GB energy
approaches 0, grain growth
stops
0b b
0X X
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( ) ( )0 00
0
0
ln
3
ln
1 ln
gb
segm
g seg
g seg
D X X R T XHV
R T X H
R T X H
= +
+
= +
For strong segregated alloys, ,and decrease upon growth
For weak segregated alloys, ,and increases upon growth
Large makes close to 0;
while small ( ) leadsto deviating substantially from zeroat the state of
0X X >X
0X