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P. Binkele, S. Schmauder: An atomistic Monte Carlo simulation of precipitation in a binary system
Peter Binkele, Siegfried Schmauder
Staatliche Materialprüfungsanstalt (MPA), University of Stuttgart, Germany
An atomistic Monte Carlo simulation of
precipitation in a binary system
Dedicated to Professor Dr. Dr. h. c. Manfred R'uhle on the occasion of his 65th birthday
2. Monte Carlo simulation
An atomistic Monte Carlo simulation of coherent precipitation on a body-centred cubic crystallattice is presented. A
binary system with atom types A and B is considered, while
the 'diffusion' of the atoms is realized via a vacancy mechanism. Starting with a random distribution of 91 % A
atoms and 9 % B atoms, the formation and growth of precipitates is simulated at a constant temperature of 773 K. As a
result of the simulation the precipitate radii distributions at
different states of the system and the time evolution of the
precipitate mean radii have been calculated. The results of
the simulation are compared with the predictions of the
classical Landau-SlyozovWagner (LSW) theory. At the
beginning of precipitation the mean radius R(t) grows proportional to tO.180, while at later states the growth exponent
approaches the classical value of 1/3. In order to reveal
the growth exponent's exact value, an even larger simulation is required, which remains as an interesting task for
the future.
Keywords: Coarsening; Precipitation; Thermal ageing;
Landau-SlyozovWagner theory; Computer simulation
2.1. Model
In the present simulation, a fixed body-centred cubic (bcc)
crystal lattice is used with periodic boundary conditions
and a fixed side length of L = 128 lattice constants. Thus,
N = 2L3 lattice sites are available. The lattice is occupied
with NA atoms of type A, NB atomes of type B and one vacancy (Nv = 1). For the total number of lattice sites the following equation holds: N = NA + NB + Nv. The chemical
binding between atoms has been described 1;>yfirst- and
second-neighbour
pair interactions e~~,e~~,e~~ with
i E {I, 2}, denoting the ith neighbours. The binding between atoms and the vacancy has been described by firstneighbour interaction energies e~~ and e~I~. The movement
of the atoms occurs by change of the vacancy with a nearest
neighbour atom. Such a change is thermally activated and
the transition rates TA V for an A atom and TB V for a B
atom are given by:
'
,
TA,v
= vAexp ( --,;:r!J.EA'V)
(1)
TB,v
=
--,;:r!J.EB,v)
(2)
1. Introduction
The classical theory of precipitation was developed by Lifshitz, Slyozov [61Lif] and Wagner [61Wag] (LSW theory).
Using the assumption of an infinite dilution of the second
phase, the coarsening of precipitates occurs by shrinking
of small precipitates and growth of large precipitates. This
results in an increase of the mean radius of the precipitates
and a concomitant decrease of the number of precipitates.
The LSW theory predicts a growth exponent CI: = 1/3 of
the mean radius of precipitates and provides a precipitate
radii distribution function.
Nowadays, atomistic computer simulations provide a detailed insight in the process of precipitation. In early simulations the growth exponent a was found in the range
0.17 -0.25 [91Wag]. In the Monte Carlo simulations of
Soisson [96Soi] on the Fe-Cu system with a low copper
content growth exponents very close to 1/3 were found,
while the precipitate size distribution strongly differed from
the LSW distribution function. In these simulations the initial stages of precipitation were analysed.
In the present simulation the process of precipitation is
studied with a good statistical accuracy in an extended
range with respect to the simulation volume and particle
sizes.
Z. Metallkd. 94 (2003) 8
© Carl Hanser Verlag, München
VBexp
(
Here, VA and VB represent the attempt frequencies for an A
and B atom, respectively and kT has its usual meaning.
The activation energy !J.Ex,v,X E {A,B}, is the energy difference between the stable position and the saddle point position of a diffusing atom A or B, which is situated next to
the vacancy (Fig. 1). The activation energies depend on the
local atom configuration, and a simple model is applied to
calculate them, which is presented in the following for atom
type A: The activation energies depend on the saddle point
energy ESp,Aand the interatomic energies of the first and
second-nearest neighbours of the A atom, and on the interaction energies of the nearest neighbours of the vacancy.
(3)
Here, n~~(n~) is the number of AA bonds (AB bonds) to
the ith neighbours of the A atom (i E {I, 2} ), and
n~~(n~l~) is the number of AV 'bonds' (BV 'bonds') with
the nearest neighbours of the vacancy.
1
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P. Binkele, S. Schmauder: An atomistie Monte Carlo simulation of preeipitation in a binary system
Distance
•
Aatom
Batom
Vacancy
Fig. 1. Schematic representation of activation energies and saddle
point energies.
Vacancy
•
Aatom
Batorn
For the B atoms an analogous eonsideration provides the
following equation:
=
I1EB,Y
(I)
(I)
ESp,B - nABeAB
(2)
(2)
-nBBeBB
(I)
(I)
(I)
- nBBeBB
(I)
- nAyeAy
(I)
(2)
(2)
- nABeAB
(I)
- nByeBy
(4)
The energies eCfl and e~k, i E {I, 2}, were estimated from
the eohesive energies of the pure metals, using the assumptions e;e. = e~li2 and e~~ = e~V2. Herewith, the pair interaetion energies with the second-nearest neighbours are
half of the pair interaetion energies with the first-nearest
neighbours. This assumption is justified by the shape of
atom pair potentials. With Zl = 8 first-nearest neighbour
lattiee sites, and Z2 = 6 seeond-nearest neighbour lattice
sites on a bee lattiee, the following equations hold:
(5)
Ecoh,B
(I)
Z2 (2)
= Zl"2 eBB
+ "2
eBB
(
6)
The energies e~k, i E {I, 2}, are related to the mixing energy WAB, which is defined as:
(7)
For mixing energies WAB < 0 the system shows a tendeney
to form precipitates, for WAB = 0 A and B atoms are ideally
solvable, and for WAB > 0 a tendeney to form superstruetures exists [96Soi~.
The energies e~y and e~l~ are related to the vaeaney formation energies as:
F
Ey,A
= ZleAy + Ecoh,A
(I)
F
Ey,B
=
(1)
ZI eBy
Thus,
+ Ecoh,B
Fig. 2. Interaction energies in the bcc crystallattice.
(8)
(9)
with the knowledge of the eohesive energies
the mixing energy WAB and the vaeaney formation energies E~ A' E~ B' th~ simulatjon parameters for
.
'.
' e AA'
(I)
(I)
(,)
t h·
e mteraetlOn
energles
eBB,
e AB'
I. EI,{
2 } and
(I)
(1)
.
eAy, eBy ean be calculated, see Fig. 2.
In order to use 'realistie' material data, an approaeh to
the Fe-Cu system was used, see [02Seh]. The kinetie parameters were adjusted to diffusion data, assuming an Arrhenius law.
D
= Doexp(
~i)
with
Q
= E~ + E~
(10)
The attempt frequencies were estimated using the Debye
frequeneies VD of pure metals: VA = VB = VD. The saddle
point energies determine the vaeaney migration energies in
metals and the following equations were used for the case
of dissolved B atoms [98Soi]:
ESp,A
=
ESp,B
M + ZleBy
(1) + ( ZI
= Ey,B
M
Ey,A
+
(I)
ZleAy
+
(ZI -
-
I)
(I)
eAA
(2)
+ Z2eAA
(
1)
(I)
eAB
(2)
+ Z2eAB
(12)
11 )
where EtJ A and EtJ B are the vaeaney migration energies of
atom types A and 13.
It is weIl known from experiment that small copper precipitates with radii smaller than 2-3 nm are coherent and
possess the bee structure of iron [940th, 90Piz]. As the cohesive energy of bee cop per is unknown, a symmetrie al
m.odel was applied in the present simulation, i. e.,
e~~ = e~k (i = 1,2). An attempt to use the eohesive energy
of fee eopper in Eq. ~6) lea,ds to an unrealistie high asymmetrie energy u = 2:i=1 (e~~ - e~k) and to unrealistie aetivati on energies I1E. This shows the limits of the model presen ted where simple broken bonds are assumed (Fig. 2) and
indicates that some energetie eontributions are not eonsidered in the theoretical model. The applied material data are
listed in Table 1, and the simulation parameters in Table 2.
2.2. Residenee time algorithm
Ecoh,A, Ecoh,B,
2
In the simulations a rejection-free residenee time algorithm
is applied, whieh shows significant ealculation time advantages in eomparison to a Metropolis algorithm [96Soi].
Z. MetaIlkd. 94 (2003) 8
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P. Binkele, S. Schmauder: An atomistic Monte Carlo simulation of precipitation in a binary system
Tab1e 1. Fe-Cu material data.
3. Simulation results
0.287
nm
-0.48eV
Cohesive
-4.28
energy
eV
Fe
-4.28
aDFe
Deu
1.60
0.90
eV
[76Kit]
2.01
2.16.10-4
8.70·
1.20
.1.60
10-4
Calculated
Calculated
Calculated
1012
eV
[eV
9ILan,1
m2/s
I/s
with
with
with
91Lan2]
WFeC::u
[91Lan2]
[97Sch]
[97Liu]
Symmetrie
Assumption
model
[81Lan]
Calculated
[91Lan2,
Ecoh,Fe
with
83Smi]
[97Sch]
Ecoh,cu
V:;~bye
EC,Fe
EC,eu
E'4,Fe
E'4,eu
Diffusion constant Fe
3.1. Simulation für T= 773 K
Eight nearest neighbours are surrounding the vacancy. A
jump rate is now ca1culated for each of these eight jump
candidates, depending on their first and second neighbours.
This provides eight independent jump frequencies
I, 2, ... , 8. One of these eight possibilities is now selected on the basis of its prob ability by a random number
and the site change is then performed. The total number of
performed vacancy jumps, also denoted as Monte Carlo
steps (MCS), during this simulation is 5.0 . 1011.
In order to define a time scale, the averaged residence
time was used which is given by:
r r
tMC
r
= ( 2:r;
1=1
8 )-1
(13)
However, the different vacancy concentrations in the simulation [CV,sim= 1/(2· L3)] and in reality [we used the adaphave still to be corretation CV,real= 280· exp(E~/kT)]
lated. In order to obtain the time t, the Monte Carlo time
tMC is multiplied by a time adjusting factor as foIlows:
t
=
The aim of the presented simulation is
to ca1culate the precipitate size distribution with a preferably good accuracy
at different states of the system and to
get information about the spatial distribution of the precipitates.
The underlying bcc lattice with periodic boundary conditions posseses a
side length of L = 128 lattice constants
which corresponds to an absolute value
of 36.8 nm. It consists of N =
2 . L3 = 4 194 304 lattice sites which
are occupied by 3816816 A atoms
(91%),377487
B atoms (9%) and
one vacancy. Für this simulation an elevated temperature of 773 K was used which is 1/3 of the critical temperature Tc of the simulated system, given by the
relation kTc/lwABI = 2.45. At this temperature it is expected that diffusional processes can be analysed quite weIl.
At the beginning of the simulation at t = 0, aIl atoms and
the vacancy are randomly distributed on the crystallattice.
Nearly aIl B atoms are dissolved in the matrix, which
means, that no other B atom is present among the first
neighbours of a given B atom. In this state the system is
completely disordered, see Fig. 3.
Figure 4 shows four snapshots of the system at the times
tl = 9 h, t2 = 6 d, t3 = 18 d and t4 = 45 d, where each dot
represents aB atom. At tl a large number of smaIl precipitates have been formed. In the course of the simulation the
number of precipitates reduces, while the mean size of the
precipitates increases, due to minimisation of energy. Considered geometricaIly, the simulated precipitates are polyhedrons. Next, a method has to be developed to define the
radii of the simulated precipitates. We ca1culated the radius
of the sphere having the same volume as the precipitate. An
elementary ceIl of a body-centred cubic lattice inc1udes two
atoms and posesses the volume Vec = a3• As the number of
(20)
(Cv,sim)
CV,real . tMC
The thus ca1culated time scale is sensitively dependent on
the used energies and attempt frequencies. Therefore, the
ca1culated time periods for the simulation results have to
be considered with some care. In any case, the thus ca1culated time is directly proportional to real time.
Table 2. Simulation parameters.
I
(1)
GAA
= -0.778
eV
G~~
= -0.778
eV
= -0.735
G~~ = -0.335
ESp,A = -9.262
eV
G~~
VA
= 8.70
eV
eV
. 1012 S-1
Z. MetaIlkd. 94 (2003) 8
(2)
= -0.389
GBB = -0.389
(2)
GAB = -0.367
(1)
GBV = -0.335
ESp,B = -9.125
VB = 8.70.1012
GAA
(2)
eV
eV
eV
eV
eV
S-1
o MCS
(10
0)
Fig. 3. Initial state of the simulation. Random distribution of
3816816 A atoms (91 %), 377 487 B atoms (9 %) and 1 vacancy on
2· 1283 = 4194304 bcc lattice sites. For visibility reasons B atoms
are shown only.
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P. Binkele, S. Schmauder: An atomistic Monte Carlo simulation of precipitation in a binary system
(a) 5· 109 MCS (tl = 9 h)
(b) 70· 109 MCS (t2 = 6 d)
Fig. 4. Monte Carlo simulation with
91 % A atoms and 9% B atoms. Time
(c) 210· 109 MCS (t3 = 18 d)
(d) 500· 109 MCS (t4 = 45 d)
atoms N in each precipitate is known, the following equation holds for large values of N(N > 20).
4
3
N 3
(15)
VSphere = '3nR = 2'a
This equation provides the precipitate radius R
R= -N
8n
(3a3
)
(16)
1/3
which was used to calculate the following precipitate size
distributions. In order to recognize and characterize the precipitates, the number of atoms in the precipitates, their center of masses and radii [according to Eq. (16)], the mean radii and the radii distributions were calculated at several
states of the simulated system. As a result the precipitate radii distributions are shown in Fig. 5. At time tl there are
1732 precipitates with the mean radius R = 0.81 nm, at t2
476 precipitates with R = 1.24 nm, at t3 244 precipitates
with R = 1.56 nm, and at t4 151 precipitates with
R = 1.83 nm.
The precipitate radii distributions can be compared with
the LSW radii distribution function [97Nem, 61Wag,
61Lif], given by (in normalized form):
gLSW(P)
-P
{4 o
4
=
--
9
3
2(
---
exp
1.5 - P
3 )7/3(
1.5 ~)11/3
+P
evolution of precipitates during thermal ageing at T = 773 K. For visibility
reasons B atoms are shown only.
---
G
P<
1.5
P
)
1.5
1.5
(17)
R is the radius of an individual precipitate, R the mean radius, and p = R/R is the reduced precipitate radius. gLSW
is a self-similar function when scaled with the time-dependent mean radius R, and it possesses a cut-off radius at
p = 1.5. With knowledge ofR from the simulation, the corresponding LSW radius distributions in Fig. 5 were calculated with the condition
100gSim(R)
dR
= 100gLsw(R)
dR
(18)
in order to achieve the same integral values ofboth distributions.
Compared with the LSW distributions, the radii distrubutions fram the simulation show a more Gauss-like
shape, especially at the beginning of the precipitation.
Also it is found that precipitates exist with radii larger
than the LSW cut-off radius. The mean values for the precipitate radii of the simulation are always below the maxima predicted by the theory due to interaction of precipitates in the simulation in contrast to the assumptions of
the LSW theory.
A main point of this simulation is the calculation of the
time evolution of the precipitate mean radii. The mean radius was calculated at 17 different states of the simulation
and is plotted versus time in Fig. 6. Assuming a time law
ofthe form
P 2:
R(t)
= K·
t'"
(19)
Z. Metallkd. 94 (2003) 8
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P. Binkele, S. Schmauder: An atomistic Monte Carlo simulation of precipitation in a binary system
::l
1§
400
300
'"
'"
Z
.0
E 0 200
100
'ü
Ci.
'0.
Ci
'0
Mean radius: 2.81 a = 0.81 10
nm
'"
.0
Z
::l
Ci.
E
'ü
1§
40
80
20
60
100 0
'0.
Ci
'0
Mean radius: 4.33 a = 1.24 nm
Maximum radius: 4.90 a = 1.40 nm
0
<n
<n
Number of precipitates:
1732
0
Maximum radius: 6.90 a = 1.98 nm
Number of precipitates:
2
6
4
8
100
Mean radius: 5.43 a = 1.56 nm
0.8
§
<n
40
80
60
20
Maximum radius: 7.90 a = 2.27 nm '"
.0
10
Number of precipitates: 244
0
Mean radius: 6.37 a = 1.83 nm
§
Z::l
E
Ci.
'ü
40
80
60<n
20
Maximum radius: 9.70 a = 2.78 nm
0
'0.
Ci
'0
Number of precipilates:
4
6
8
2
Radius (in units of a = 0.287 nm)
(c) 210· 109 MCS (t3 = 18 d)
= (R\O) + KLSW'
C/J
••..•..
(20)
t)I/3
R(t) = (R (O)+KLswt)
DA
10.0
6
8
10
m ____m ••
2.0 1.6
1.2
~
E:J
~
U
Fig.5. Calculated precipitate radii distributions
at
t3
t1
=
=
9 h,
t2
t4
=
6 d,
d.
= 45
18 d,
Histogram: simulation.
curve: LSW distribution.
(d) 500· 109 MCS (t4 = 45 d)
we find the coarsening rate constant to be KLSW =
1.287.10-6 nm and R(O) = 1.181 nm, see Fig. 6. In order
to answer the question about the exact value of the growth
exponent, an even larger simulation is required, which remains as an interesting task for the future.
8 c: 25
431
Ia:
:2
c:
~ 3
4
Radius (in units 01 a = 0.287 nm)
we find that the mean radii from the simulation are fitted
weIl with a growth exponent a = 0.180 (:::d 1/5.5), see
Fig. 7. Assuming the classical time law
-
151
0
2
R(t)
10
(b) 70· 109 MCS (tz = 6 d)
100
.0
Z::l
E
Ci.
'ü
~0
'0.
'0
8
Radius (in units of a = 0.287 nm)
(a) 5· 109 MCS (tl = 9 h)
'"
6
4
2
Radius (in units 01 a = 0.287 nm)
476
In order to get information about the spatial distribution
of the precipitates, the radial distribution function (RDF)
was calculated für the state of the system at tl = 9 h, see
Fig. 7. The RDF is defined as the ratio of the number of precipitates per volume the centers of which lie within a radial
shell of radius rand the thickness r + dr surrounding a
precipitate, to the number of precipitates per volume of the
entire system.
RDF(r)
=
Np in a radial shell of radius r to r
+ dr
Np
(21)
where Np is the number of precipitates per volume and Np
the overall precipitate density. For a random particle distri-
(j)
<ll
7
2.0
0
~
Fig. 6. R versus time t. The R values from the simulation are plotted as
dots. Assuming a time law of tbe form R(t) = K . t'" a very good fit
witb a growtb exponent lf = 0.180 is found. A fit assuming tbe classical time law R(t) = (R (0) + KLSW . t)1/3 is plotted as dashed line
and .e.rovidestbe coarsening rate constant KLSW = 1.287· 10-6 nm3/s
and R(O) = 1.181 nm.
Z. Metallkd. 94 (2003) 8
LC
a:
1.5
1.0
0.5
0.0
o
10
20
30
40
50
Distance r / a
Fig.7. Calculated radial distribution function (RDF) for the state of
tbe system at t1 = 9 h.
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P. Binkele, S. Schmauder: An atomistic Monte Carlo simulation of precipitation in a binary system
bution, RDF equals the value 1.0 for all distances. The calculated RDF equals zero for very small distances, shows a
small peak value at a distance of 2 lattice constants which
is not analysed here, and an absolute maximum at lilattice
constants, representing the first coordination shell of neighbouring precipitates. The RDF converges against 1 for large
distances, a long range order does not exist.
Thus, with the knowledge ofthe RDF, which provides information about the spatial distribution of the precipitates,
and the ca1culated radius distribution, a given state of the
simulated precipitates is characterized clearly.
4. Summary
An atomistic Monte Carlo simulation of coherent precipitation was presented, where the diffusion of atoms was realized via a vacancy mechanism. The process of precipitation
was analysed with good statistical accuracy. At the beginning of the simulation the mean radius of the precipitates
R(t) grows proportional to tO.180, while at later states the
growth exponent approaches the classical value of 1/3.
The obtained precipitate radii distributions of the simulation are comparable to the classical LSW radii distribution
function, but they are not identical. The observeddeviations
can be explained by the simplified assumptions, which are
used in the classical LSW theory.
[8 I Lan]
Landolt-Börnstein:
Numerical Data and Functional Relationships in Science and Technology,
New Series I1I,
Vol. Ba, Metals: Phonon States, Electron States and Fermi
Surfaces, Springer-Verlag, Heidelberg (1981).
[83Smi] CJ. Smithell: Smithells Metals Reference Book, 6th edition,
Ed. E.A. Brandes, Butterworths, London (1983).
[90Piz]
S. Pizzini, KJ. Roberts, WJ. Phythian, CA Englisch,
G.N. Greaves: Phil. Mag. Lett. 61 (1990) 223.
[9ILanl] Landolt-Börnstein:
Numerical Data and Functional Relationships in Science and Technology,
New Series I1I,
Vol. 25, Atomic Defects in Metals, Springer-Verlag, Heidelberg (1991).
[9ILan2] Landolt-Börnstein:
Numerical Data and Functional Relationships in Science and Technology,
New Series I1I,
Vol. 26, Diffusion in Solids, Metals and Alloys, SpringerVerlag, Heidelberg (1991).
[91 Wag] R. Wagner, R. Kampmann, in: Phase Transformations in Materials, Ed. P. Haasen, VCH, Weinheim (1991).
[940th]
PJ. Othens, M.L. Jenkins, G.D.W. Smith: Phil. Mag. A 70
(1994) I.
[96Soi]
F. Soisson, A. Barbu, G. Martin: Acta Mater. 44 (1996) 3789.
[97Liu]
c.L. Liu, G.R. Odette, B.D. Wirth, G.E. Lucas: Mater. Sci.
Eng. A 238 (1997) 202.
[97Nem] E. Nembach: Particle Strenghtening of Metals and Alloys,
Wiley, New York (1997).
[97Sch] M. Schick, 1. Wiedemann, D. Willer: Technischer Bericht,
MPA Stuttgart (1997).
[98Soi]
F. Soisson: Private communication (1998).
[02Sch] S. Schmauder, P. Binkele: Comp. Mater. Sci. 24 (2002) 42.
(Received September 1,2002)
The authors would like to thank the ministry of education, science, research and technology (BMBF) for financial support of this work under
contract number BMBF 1501029 and Dr. F. Soisson and Dr. G. Martin
from CEA, Saclay for the support in the development of the Monte
Carlo software.
Correspondence
References
Staatliche Materialprüfungsanstalt
(MPA)
Pfaffenwaldring 32, D-70569 Stuttgart, Germany
Tel.: +49711 685 •
Fax: +49711 6852635
[6ILif]
I.M. Lifshitz, V.V. Slyozov: J. Phys. Chem. Solids 19 (1961)
35.
E-mail: [email protected]
[6IWag]
[76Kit]
C. Wagner: Z. Elektrochern. 65 (1961) 581.
C. Kittel: lntroduction to Solid State Physics,
York (1976).
6
address
Prof. Dr. S. Schmauder
Wiley, New
Z. Metallkd. 94 (2003) 8