- Email: [email protected]
Kinetic theory of microstructural evolution in nucleation and growth processes
MATERIALS SCIENCE ENGINEERING
ELSEVI
ER
Materials
Kinetic
theory
Science
and Engineering
A238
(1997)
A
160- 165
of microstructural evolution growth processes
D. Crespo a,*;, T. Pradell b, N. Clavaguera
‘, M.T.
&
in nucleation
and
Clavaguera-Mora
d
a Deparrarnen? de Fkica Apiirudu: Uniwrsitnt Politkzica de Catalm~~n, Campm Nerd UPC, MMul 84, 08034 Bcrr~&na, Spain b Escola Urriversitciriu dlti?ginyrria T&nicu Agriwla, Unicersitnt Polirhica de Carahrnpa, UrgeN 187, 08036 Barceiom, Spain c Grup de Fisim de I%stat Stilid, Departament #EC&f, Fac~dtnr de Fisicn, Lhhersitai de Burcelona, Diagonal 647. 08028 Barcelona, Spain d Grup de Fisica de Maareriafs I, Departamerzt de Fisiru. Fwtrltat de Cihcies, Uiticersitnt Autdnornn de Barcelona, 08193 BeIlatewa, Spain
Abstract
Macroscopic transformation
properties the complete
to tailor properties
of materials knowledge
of technological
are highly of microscopical
dependent on parameters,
their such
microstructure. as mean radius
In materials obtained and grain size distribution,
interest. A kinetic theory is presented, based on the same assumptions
by
a phase is essential
as the KolmogorovP
Johnson, Mel&Avrami (KJMA) formulation; that is to say, randomly distributed active nucleation sites that grow isotropically until collision gives rise to growth stop at the grain boundaries. The theory is an extension of KJMA because it evaluates probabilities of impingement between grains, giving actual grain size populations as a function of time. The formalism allows us to model arbitrary dependencies of the kinetic parameters (nucleation and growth rates) on macroscopic and/or microscopic
variables, and on time. The theory has been tested against Monte Carlo simulations, showing that it is quantitatively exact in its predictions of the grain size populations. Specific dependencies on the kinetic parameters have been studied in order to understand the growth final grain Keyvow’s:
behavior of real materials. In particular, size distribution and related parameters Kinetic
theory;
Macroscopic
properties;
interface and are discussed.
Monte
Carlo
Microstructured materials are becoming a subject of major interest due to their macroscopic properties. The distinctive feature is that their mechanical, magnetic, electric, thermal and superconducting behaviour is essentially controlled by the presence of small particles. There are several methods of developing these type of materials, such as precipitation of a secondary phase from metastable systems under controlled conditions (i.e. temperature, pressure), or vapor deposition. Therefore, the emerging microstructure depends not only on the chemical composition but also on the mode of preparation and the previous history of the materials. Among the possible kinetics of first order phase transition which result in a microstructured material, the nucleation and growth kinetics play a relevant role. Several families of microstructured materials, such as hard magnetic materials, Fe-Nd based, soft magnetic author.
E-mail:
0921.5093/97/517.00 Q 1997 Elsevier PIISO921-5093(97)00446-2
[email protected] Science
S.A. All rights
controlled
growth
are studied,
and
differences
in the
simulations
1. Introduction
* Corresponding
diffusion
0 1997 Elsevier Science S.A.
reserved
materials, Fe based, and high strength materials, Al based, are the result of this type of kinetics. Since the early 40s a classical nucleation and growth kinetics due to Kolmogorov [I] Johnson and Mehl [2] and Avrami [3] (KJMA) of first order phase transitions exists, and gives an appropriate description of the behavior of an integral variable, namely the transformed volume fraction, which characterizes the process. Although this theory has been generalized to arbitrary grain shapes [4], the resulting microstructure cannot be evaluated from it. Further efforts to provide a better description of the referred systems were performed with the time-cone method [5-71, By using this method it is possible to obtain exact formal expressions which give the time dependence of the spatial two-point correlation function during the phase transition. However, the method has some restrictions. From the practical point of view, the most important are that, first, the growth rate must be a non-increasing function of time; second, the finite radius of the nuclei cannot be considered; third, the
D. Cmpo
cr nl. j’ ,2larerinls
Science
growth protocol cannot modify the surroundings of a growing grain; and, finally, the involved integrals become very intricate for non-trivial kinetics. As a con -sequence, the practical applicability of this method is very restricted. It is worth noting that both theories simultaneously describe very different systems, depending on the behavior of the transformed phase. It is usually defined as a degeneracy parameter, p, which accounts for the number of different, statistically equivalent domains existing at the end of the phase transition. The nondegenerated case, p = 1, corresponds to the situation where the growing domains melt after collision, and thus the final state corresponds to the presence of only one domain in the whole system; as an example of this we can think of magnetic systems (Tsing models) switching from paramagnetic to ferromagnetic state. The completely degenerated case, p-+ co, corresponds to the situation where the growing domains remain distinguishable over the whole transition, as in the case of microstructured materials. In the non-degenerated case the notion of individual growing domain disappears, and thus the spatial correlation function becomes the natural descriptive tool. However, in the completely degenerated system the natural description comes from the grain size distribution, which cannot be calculated from the spatial correlation. Therefore, it is necessary to go further in order to find an accurate description of the transformed phase. The authors have recently presented [8.9] a different approach to this problem, in the case p+ co, based on a mean field theory. This approach can be considered an extension of the KJMA theory, in which discrete populations of grains of a well defined radius by unit volume are defined. Isotropy considerations allow us to define a mean growth rate, and thus equations giving the evolution of the populations are established. The result of the integration of this model was tested against Monte Carlo calculations giving quantitative agreement. The model has been implemented in two situations, namely time-dependent G(r) and radius-dependent G(R) growth rate. In this paper we show the application of the model to two of the most usually considered growth procedures, namely interface controlled and diffusion controlled grain growth. We show the applicability of the above cited model to the two cases and the comparison of the resulting microstructures, discussing its importance in the final macroscopical properties of the materials obtained under these two kinetics.
2. Sketch of the mean field model
In this section we recall the features of the mean
and Etlgitwerirlg
A.238 (1997)
161
160- 165
field model which allows the determination of the microstructure obtained under a given nucleation and growth kinetics. The hypotheses under which the models are developed are: l The nucleation and growth process is studied in an infinite, homogeneous and isotropic volume. l The nucleation rate r(t) is homogeneous. l The growth rate is homogeneous and either time-dependent, G(r), or radius-dependent, G(R), but always isotropic. l The growth is only stopped by impingement, meaning that the classical Avrami expression
(1) where 4 and &(t) account for the transformed and extended volume fractions, is valid. The following parameters are defined: 0 a control volume I/,: l a umt length 11: l the radius of a nuclei after nucleation, R,. Dimensionless variables, shown in lowercase, are built: t:, = V,q3, I’, = R,/ry, and an additional condition is imposed. namely that the growth rate of a nuclei just after nucleation be the unity in the dimensionless scheme; that is .aE> f) = 1
(2)
II=s*+i(i) t,.
which defines a timescale T given by
I
G(R,, t’) dt’
(3)
t
and implies that a nuclei born at time t growing in isolation, reaches radius r6 + 1 at + z(f). This enables us to discretize the time scale, whose integer values are At this time it is necessary to distinguish between time-dependent and radius-dependent grain growth. For time-dependent grain growth G(R, t) becomes G(r) and Eq. (3) is automatically satisfied for all values of R. Thus, a change in the growth rate simply provokes a change in the time step, and the radius scale is always constant. We call this model Mean field constant-radius-scale (MFC) model. For radius-dependent grain growth, however, G(R, t) becomes G(R) and once defined r from Eq. (3) the radius scale becomes defined. This is due to the fact that here R(r) is a universal function for all grains, given by r R(r) = R, +
j
G(t’ - to) dt’
(4)
10
fO being the nucleation time of each grain. Thus we can choose arbitrarily q and obtain the reduced radius scale
from
162
D.
Crmpo
et ul. I Mawrids
Science
crnd
Engineering
A.238
(1997)
160-
I65
r - 10
1 r(t - to) = re = ul s0 p/z s0
G(f)
dt’
r G(t’)dt’=1.(5)-rl’,,
T~+~=Y(~T)
(5)
with the important restriction that now 5 must be constant in order to preserve the radius scale. This model is referred to as the mean field variable-radiusscale (MFV) model. Later on, the dimensionless nucleation rate is defined as f + r(r) I(f) dt’ i(t) = ik = q 3 (6) sI In order to characterize growing grains, a grain of volume ug(rk) is assigned an effective radius li3 A471
Ig=
(7)
u l?(f/J
( 1 and discrete populations & are defined, including all grains such as j - 1 < pg
f
1=
ikvC
+ 1.k + I =
N c,k+
1 =
c1
j
fij,k, -
dN,,,
>
E +
(8) ik(cC
-
Nj+l.k+1=(1-Clk)Nj,1.k+SlkNj,k
L’k) ,j>E
(9)
(10)
which can be solved iteratively, of true populations.
and gives the evolution
3. Application to interface controlled and diffusion controlled grain growth Interface controlled and diffusion controlled grain growth are two of the most usually considered grain growth protocols in true nucleation and growth processes. Interface controlled grain growth is well represented by the classical PJM (p degenerated
Johnson-Mehl) model where the nucleation and growth rates are considered constant and p -+ cc1is the number of grains. In this situation it is supposed that the growth of an isolated grain does not affect its surroundings, and that the growth is only stopped by impingement with neighbor grains. In a more general situation the growth rate could be a function of time, and thus the corresponding expression for the growth rate is
$ =G(r)
(12)
Diffusion controlled grain growth appears naturally in partitioning transitions, i.e. those where the transformed phase (the growing grains) has a composition different from that of the original phase. In this situation some element must be expelled and/or included into the growing grains, and then the kinetics of the process is controlled by the diffusion of the slowest moving species. The diffusion process around an isolated growing grain is well established; see for instance Kurtz and Fisher [lo]. In the steady state the growth rate is given by dR -=df
D R
(13)
where D is diffusion coefficient of the slowest moving species under the considered external conditions. Interface controlled grain growth can easily be modelled with the MFC model, while diffusion controlled grain growth can only be modelled, under steady external conditions, with the MFV model. In order to compare the influence in the microstructure of both growth protocols, we will study two processes performed under steady external conditions, thus assuming that G(r) = G,, constant, in the interface controlled grain growth as in the pJM model. For comparison purposes we have chosen the same value of the nucleation rate I,, constant in both processes. and a value of the diffusion coefficient D given by D/R, = Go. This means that the initial growing rate of a nuclei in both processes is the same. Before looking at the microstructure, it is interesting to compare the evolution of the transformed fraction with time in both situations. Fig. 1 shows this for both processes and, for comparison, the transformed volume fraction in two interface controlled processes with slower nucleation rates, namely &,/IO0 and &,/lOOO. It is seen that diffusion controlled growth delays the transformation in a different way than reducing the nucleation i-ate. Differences in the driving mechanisms lead to different transformation kinetics, suggesting that the developed microstructure can also have different features. The evolution of the microstructure versus transformed fraction is shown in Fig. 2(a). It shows the
D. Crespo
et al. ,‘Materials
Science
md Engineering
f (arb. units) Fig. 1. Transfonned fraction vs. time in three interface controlled growth processes of growth rates G,, and nucleation rates I,, I,!100 and I,/lOOO, compared with a diffusion controlled grain growth process with D/R, = G,. and nucleation rate I,.
volume grain size distribution, which gives the number of grains of a given size by unit volume, at several transformed volume fractions. Four striking features are readily observable. First of all, the grain size distribution of the interface controlled process is more or less flat, while the diffusion controlled one is highly peaked; secondly, the shape of this peak is very asymmetric;
A-738 (1997)
163
160-165
third, the total number of grains, which corresponds to the total area, is much higher in the diffusion controlled process than in the interface controlled; and, lastly, the maximum grain size is heavily reduced by the diffusion mechanism. This can be explained as follows: although both interface controlled and diffusion controlled grains stop their growth by impingement, the growing speed of diffusion controlled grains decreases as their radius increases, and therefore grains accumulate at smaller radius sizes. The asymmetry of the distribution can be accounted for in the same way, because it is easier for a grain to grow while its radius is still small. Furthermore, the higher number of grains born in the diffusion controlled grain growth is due to the fact that the microstructure is displayed for the same transformed fractions but not at the same transformation time. Therefore, as diffusion controlled grains are, on average, smaller than interface controlled grains, more grains are needed to occupy the same volume. Although the volume grain size distribution gives an accurate description of the microstructure, it is not easy to characterize it from the experimental point of view. In fact, it may be determined by direct inspection by Lopttcal microscopy (OM), scanning and transmission electron microscopy (SEM and TEM), or by scattering techniques such as small angle X-ray and neutron scat-
Grain Size Distributions Volume
Surface = -
hiFf(D~ffukm MFC (Interiace
Transformed Fraction controlled) Controlledi
[ = 0.60
6 = 0.80
Fig. 2. Microstructure
evolution during interface controlled and diffusion controlled grain growth. See the text for details.
164
D. Crespo
er al. :’ Mutevinls
Science rind Engineering
tering (SAXS and SANS). However, none of them give the volume grain size distribution; they offer a different view of the microstructure which can be obtained from the volume grain size distribution. The microstructure determined by OM and SEM is a surface grain size distribution, i.e. the number of grains of a given radius by unit surface. Assuming that the observed surface is a cross-section of the sample, the observed radius of a grain hardly ever corresponds to its true radius. The track of a grain in the surface depends on the distance between the cutting plane and the center of the grain. By using classical stereology [I l] it is possible to demonstrate that
where NsJ,~ is the surface grain size distribution of radius j at time tk. This distribution is shown in Fig. 2(b) for interface and diffusion controlled growth. Both surface distributions become more rounded than the volume distributions. This result arises from the f&t that, on one hand, the probability of cutting a grain exactly by its center is very low, which reduces the number of observable large grains; and, on the other hand, the probability of cutting a small grain is small. Furthermore, the asymmetry of the diffusion controlled distribution diminishes because large grains contribute to all the sizes smaller than their own, while small grains cannot contribute to larger sizes. The particular characteristics of TEM virtually prevent comparison of experimental data with theoretical calculations. By TEM it is possible to observe the grains embedded in the sample and the tracks of cut grains on the surface. The relative proportion of both volume and surface distributions depends on the sample thickness. In the case of SANS/SAXS, the information obtainable may be related to the transformed volume fraction distribution as a function of the grain radius [12]; namely 4 && = - nrj3IQ 3
AZ38
:
(1997)
160- 165
I’I 4’ K I;
11,’ I
/
t (arb. units) Fig. 3. Time dependence of the grain density for interfacecontrolled growth with growth rate G,, and nucleation rates I,> and 1,,/100, compared to diffusion controlled growth with D/R, = G, and nucleation rate I,. The arrows indicate the end of the transformation in each case.
the nucleation rate in interface controlled growth. The effect in grain density is the opposite: while diffusion controlled growth dramatically increases the grain density, reduction of the nucleation rate reduces it. As a consequence, the average grain radius shows an opposite behavior: it diminishes for a diffusion controlled growth and increases for interpace controlled growth with smaller nucleation rate.
4. Conclusions
Interface controlled and diffusion controlled grain growth are very common growing mechanisms in first order phase transitions. Therefore, complete knowledge of their influence in the developed microstructure is of great theoretical and practical interest. In this paper, we have applied a mean-field extension of the Kolmogorov-Johnson, Mehl-Avrami kinetics, which has
(15)
This distribution is shown in Fig. 2(c). The shape is similar for interface and diffusion controlled growth, and is more sharply peaked than the volume grain size distributions. Only the peak position and the maximum grain size enable us to distinguish between both mechanisms. Detailed information can also be obtained from the knowledge of the grain size distributions. Time evolution of several macroscopic parameters of interest, such as grain density and average grain radius, can be easily computed, and are shown in Figs. 3 and 4. As previously seen, diffusion controlled grain growth delays the transformation in a different way than the reduction of
a F .M cd G
f (arb. units) Fig. 4. Time dependence of the average grain radius; see Fig. 3.
D. Crespo
et al. /Materials
Scierlce
allowed us to characterize the developed microstructures. It has been shown that, under similar conditions, a diffusion controlled kinetics induces a larger grain density than the interface controlled kinetics. The maximum grain size is heavily reduced by the diffusion mechanism and, as a consequence, the grain size distribution becomes much more peaked, and thus much more homogeneous than by the interface kinetics. Comparison of the presented data with experimental measurements is made easy by computing the theoretical surface and transformed fraction grain size distributions. Beyond the particular kinetics here studied, we have also shown that the mean-field method here employed can determine the microstructure of nucleation and growth kinetics characterized either by time-dependent growth rates G(r) or radius-dependent growth rates G(R). However, a general formulation, able to deal with the general time and radius-dependent growth rate G(R, t) induced kinetics, has still not, to our knowledge, been found.
Acknowledgements This work was financed by CICYT,
grants PB94-1209
and Etlgitzeemg
AZ38
(1997)
160-165~
165
and MAT96-0692, UPC, grant PR9505, and Generalitat de Catalunya, grant 1995SGR 00514. D. Crespo acknowledges also the financial support received from ETSECCPB and Departament de Politica Territorial i Obres Ptibliques de la Generalitat de Catalunya.
References [I] A.N. Kolmogorov, Bull. Acad. Sci., USSR Phys. Ser. 1 (1937) 355. [2] W.A. Johnson, P.A. Mehl, Trans. Am. Inst. Mining Metall. Eng. 135 (1939) 316. [3] M. Avrami, J. Chem. Phys. 7 (1939) 1103; 8 (1940) 212; 9 (1941). [4] K. Sekimoto, Phys. Lett. 105A (1984) 177. [5] J.D. Axe. Y. Yamada, Phys. Rev. B 34 (1986) 1599. [6] S. Ohta. T. Ohta, K. Kawasaki. Physica 140A (1987) 478. [7] J.W. Cahn. Thermodynamics and Kinetics of Phase Transfotmations, MRS Symp. Proc. No. 398, Materials Research Society, Pittsburgh, PA, 1996, p, 425. [8] D. Cresps T. Pradell. Phys. Rev. B 54 (1996)3101. [9] D. Crespo, T. Pradell. M.T. Clavaguera-Mora, N. Clavaguera, Phys. Rev. B 55 (1997) 3435. [lo] W. Kurtz, D.J. Fisher, Fundamentals of Solidification, Trans Tech, Aedermannsdorf, 1992. [l I] S.A. Saltykov, Stereology, Springer-Verlag. New York, 1963. 1121 J. Kohlbrecher, A.~~iedenmann, H. Wollenberaer, Physica B, 213:214 (1995) 579.
ELSEVI
ER
Materials
Kinetic
theory
Science
and Engineering
A238
(1997)
A
160- 165
of microstructural evolution growth processes
D. Crespo a,*;, T. Pradell b, N. Clavaguera
‘, M.T.
&
in nucleation
and
Clavaguera-Mora
d
a Deparrarnen? de Fkica Apiirudu: Uniwrsitnt Politkzica de Catalm~~n, Campm Nerd UPC, MMul 84, 08034 Bcrr~&na, Spain b Escola Urriversitciriu dlti?ginyrria T&nicu Agriwla, Unicersitnt Polirhica de Carahrnpa, UrgeN 187, 08036 Barceiom, Spain c Grup de Fisim de I%stat Stilid, Departament #EC&f, Fac~dtnr de Fisicn, Lhhersitai de Burcelona, Diagonal 647. 08028 Barcelona, Spain d Grup de Fisica de Maareriafs I, Departamerzt de Fisiru. Fwtrltat de Cihcies, Uiticersitnt Autdnornn de Barcelona, 08193 BeIlatewa, Spain
Abstract
Macroscopic transformation
properties the complete
to tailor properties
of materials knowledge
of technological
are highly of microscopical
dependent on parameters,
their such
microstructure. as mean radius
In materials obtained and grain size distribution,
interest. A kinetic theory is presented, based on the same assumptions
by
a phase is essential
as the KolmogorovP
Johnson, Mel&Avrami (KJMA) formulation; that is to say, randomly distributed active nucleation sites that grow isotropically until collision gives rise to growth stop at the grain boundaries. The theory is an extension of KJMA because it evaluates probabilities of impingement between grains, giving actual grain size populations as a function of time. The formalism allows us to model arbitrary dependencies of the kinetic parameters (nucleation and growth rates) on macroscopic and/or microscopic
variables, and on time. The theory has been tested against Monte Carlo simulations, showing that it is quantitatively exact in its predictions of the grain size populations. Specific dependencies on the kinetic parameters have been studied in order to understand the growth final grain Keyvow’s:
behavior of real materials. In particular, size distribution and related parameters Kinetic
theory;
Macroscopic
properties;
interface and are discussed.
Monte
Carlo
Microstructured materials are becoming a subject of major interest due to their macroscopic properties. The distinctive feature is that their mechanical, magnetic, electric, thermal and superconducting behaviour is essentially controlled by the presence of small particles. There are several methods of developing these type of materials, such as precipitation of a secondary phase from metastable systems under controlled conditions (i.e. temperature, pressure), or vapor deposition. Therefore, the emerging microstructure depends not only on the chemical composition but also on the mode of preparation and the previous history of the materials. Among the possible kinetics of first order phase transition which result in a microstructured material, the nucleation and growth kinetics play a relevant role. Several families of microstructured materials, such as hard magnetic materials, Fe-Nd based, soft magnetic author.
E-mail:
0921.5093/97/517.00 Q 1997 Elsevier PIISO921-5093(97)00446-2
[email protected] Science
S.A. All rights
controlled
growth
are studied,
and
differences
in the
simulations
1. Introduction
* Corresponding
diffusion
0 1997 Elsevier Science S.A.
reserved
materials, Fe based, and high strength materials, Al based, are the result of this type of kinetics. Since the early 40s a classical nucleation and growth kinetics due to Kolmogorov [I] Johnson and Mehl [2] and Avrami [3] (KJMA) of first order phase transitions exists, and gives an appropriate description of the behavior of an integral variable, namely the transformed volume fraction, which characterizes the process. Although this theory has been generalized to arbitrary grain shapes [4], the resulting microstructure cannot be evaluated from it. Further efforts to provide a better description of the referred systems were performed with the time-cone method [5-71, By using this method it is possible to obtain exact formal expressions which give the time dependence of the spatial two-point correlation function during the phase transition. However, the method has some restrictions. From the practical point of view, the most important are that, first, the growth rate must be a non-increasing function of time; second, the finite radius of the nuclei cannot be considered; third, the
D. Cmpo
cr nl. j’ ,2larerinls
Science
growth protocol cannot modify the surroundings of a growing grain; and, finally, the involved integrals become very intricate for non-trivial kinetics. As a con -sequence, the practical applicability of this method is very restricted. It is worth noting that both theories simultaneously describe very different systems, depending on the behavior of the transformed phase. It is usually defined as a degeneracy parameter, p, which accounts for the number of different, statistically equivalent domains existing at the end of the phase transition. The nondegenerated case, p = 1, corresponds to the situation where the growing domains melt after collision, and thus the final state corresponds to the presence of only one domain in the whole system; as an example of this we can think of magnetic systems (Tsing models) switching from paramagnetic to ferromagnetic state. The completely degenerated case, p-+ co, corresponds to the situation where the growing domains remain distinguishable over the whole transition, as in the case of microstructured materials. In the non-degenerated case the notion of individual growing domain disappears, and thus the spatial correlation function becomes the natural descriptive tool. However, in the completely degenerated system the natural description comes from the grain size distribution, which cannot be calculated from the spatial correlation. Therefore, it is necessary to go further in order to find an accurate description of the transformed phase. The authors have recently presented [8.9] a different approach to this problem, in the case p+ co, based on a mean field theory. This approach can be considered an extension of the KJMA theory, in which discrete populations of grains of a well defined radius by unit volume are defined. Isotropy considerations allow us to define a mean growth rate, and thus equations giving the evolution of the populations are established. The result of the integration of this model was tested against Monte Carlo calculations giving quantitative agreement. The model has been implemented in two situations, namely time-dependent G(r) and radius-dependent G(R) growth rate. In this paper we show the application of the model to two of the most usually considered growth procedures, namely interface controlled and diffusion controlled grain growth. We show the applicability of the above cited model to the two cases and the comparison of the resulting microstructures, discussing its importance in the final macroscopical properties of the materials obtained under these two kinetics.
2. Sketch of the mean field model
In this section we recall the features of the mean
and Etlgitwerirlg
A.238 (1997)
161
160- 165
field model which allows the determination of the microstructure obtained under a given nucleation and growth kinetics. The hypotheses under which the models are developed are: l The nucleation and growth process is studied in an infinite, homogeneous and isotropic volume. l The nucleation rate r(t) is homogeneous. l The growth rate is homogeneous and either time-dependent, G(r), or radius-dependent, G(R), but always isotropic. l The growth is only stopped by impingement, meaning that the classical Avrami expression
(1) where 4 and &(t) account for the transformed and extended volume fractions, is valid. The following parameters are defined: 0 a control volume I/,: l a umt length 11: l the radius of a nuclei after nucleation, R,. Dimensionless variables, shown in lowercase, are built: t:, = V,q3, I’, = R,/ry, and an additional condition is imposed. namely that the growth rate of a nuclei just after nucleation be the unity in the dimensionless scheme; that is .aE> f) = 1
(2)
II=s*+i(i) t,.
which defines a timescale T given by
I
G(R,, t’) dt’
(3)
t
and implies that a nuclei born at time t growing in isolation, reaches radius r6 + 1 at + z(f). This enables us to discretize the time scale, whose integer values are At this time it is necessary to distinguish between time-dependent and radius-dependent grain growth. For time-dependent grain growth G(R, t) becomes G(r) and Eq. (3) is automatically satisfied for all values of R. Thus, a change in the growth rate simply provokes a change in the time step, and the radius scale is always constant. We call this model Mean field constant-radius-scale (MFC) model. For radius-dependent grain growth, however, G(R, t) becomes G(R) and once defined r from Eq. (3) the radius scale becomes defined. This is due to the fact that here R(r) is a universal function for all grains, given by r R(r) = R, +
j
G(t’ - to) dt’
(4)
10
fO being the nucleation time of each grain. Thus we can choose arbitrarily q and obtain the reduced radius scale
from
162
D.
Crmpo
et ul. I Mawrids
Science
crnd
Engineering
A.238
(1997)
160-
I65
r - 10
1 r(t - to) = re = ul s0 p/z s0
G(f)
dt’
r G(t’)dt’=1.(5)-rl’,,
T~+~=Y(~T)
(5)
with the important restriction that now 5 must be constant in order to preserve the radius scale. This model is referred to as the mean field variable-radiusscale (MFV) model. Later on, the dimensionless nucleation rate is defined as f + r(r) I(f) dt’ i(t) = ik = q 3 (6) sI In order to characterize growing grains, a grain of volume ug(rk) is assigned an effective radius li3 A471
Ig=
(7)
u l?(f/J
( 1 and discrete populations & are defined, including all grains such as j - 1 < pg
f
1=
ikvC
+ 1.k + I =
N c,k+
1 =
c1
j
fij,k, -
dN,,,
>
E +
(8) ik(cC
-
Nj+l.k+1=(1-Clk)Nj,1.k+SlkNj,k
L’k) ,j>E
(9)
(10)
which can be solved iteratively, of true populations.
and gives the evolution
3. Application to interface controlled and diffusion controlled grain growth Interface controlled and diffusion controlled grain growth are two of the most usually considered grain growth protocols in true nucleation and growth processes. Interface controlled grain growth is well represented by the classical PJM (p degenerated
Johnson-Mehl) model where the nucleation and growth rates are considered constant and p -+ cc1is the number of grains. In this situation it is supposed that the growth of an isolated grain does not affect its surroundings, and that the growth is only stopped by impingement with neighbor grains. In a more general situation the growth rate could be a function of time, and thus the corresponding expression for the growth rate is
$ =G(r)
(12)
Diffusion controlled grain growth appears naturally in partitioning transitions, i.e. those where the transformed phase (the growing grains) has a composition different from that of the original phase. In this situation some element must be expelled and/or included into the growing grains, and then the kinetics of the process is controlled by the diffusion of the slowest moving species. The diffusion process around an isolated growing grain is well established; see for instance Kurtz and Fisher [lo]. In the steady state the growth rate is given by dR -=df
D R
(13)
where D is diffusion coefficient of the slowest moving species under the considered external conditions. Interface controlled grain growth can easily be modelled with the MFC model, while diffusion controlled grain growth can only be modelled, under steady external conditions, with the MFV model. In order to compare the influence in the microstructure of both growth protocols, we will study two processes performed under steady external conditions, thus assuming that G(r) = G,, constant, in the interface controlled grain growth as in the pJM model. For comparison purposes we have chosen the same value of the nucleation rate I,, constant in both processes. and a value of the diffusion coefficient D given by D/R, = Go. This means that the initial growing rate of a nuclei in both processes is the same. Before looking at the microstructure, it is interesting to compare the evolution of the transformed fraction with time in both situations. Fig. 1 shows this for both processes and, for comparison, the transformed volume fraction in two interface controlled processes with slower nucleation rates, namely &,/IO0 and &,/lOOO. It is seen that diffusion controlled growth delays the transformation in a different way than reducing the nucleation i-ate. Differences in the driving mechanisms lead to different transformation kinetics, suggesting that the developed microstructure can also have different features. The evolution of the microstructure versus transformed fraction is shown in Fig. 2(a). It shows the
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f (arb. units) Fig. 1. Transfonned fraction vs. time in three interface controlled growth processes of growth rates G,, and nucleation rates I,, I,!100 and I,/lOOO, compared with a diffusion controlled grain growth process with D/R, = G,. and nucleation rate I,.
volume grain size distribution, which gives the number of grains of a given size by unit volume, at several transformed volume fractions. Four striking features are readily observable. First of all, the grain size distribution of the interface controlled process is more or less flat, while the diffusion controlled one is highly peaked; secondly, the shape of this peak is very asymmetric;
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third, the total number of grains, which corresponds to the total area, is much higher in the diffusion controlled process than in the interface controlled; and, lastly, the maximum grain size is heavily reduced by the diffusion mechanism. This can be explained as follows: although both interface controlled and diffusion controlled grains stop their growth by impingement, the growing speed of diffusion controlled grains decreases as their radius increases, and therefore grains accumulate at smaller radius sizes. The asymmetry of the distribution can be accounted for in the same way, because it is easier for a grain to grow while its radius is still small. Furthermore, the higher number of grains born in the diffusion controlled grain growth is due to the fact that the microstructure is displayed for the same transformed fractions but not at the same transformation time. Therefore, as diffusion controlled grains are, on average, smaller than interface controlled grains, more grains are needed to occupy the same volume. Although the volume grain size distribution gives an accurate description of the microstructure, it is not easy to characterize it from the experimental point of view. In fact, it may be determined by direct inspection by Lopttcal microscopy (OM), scanning and transmission electron microscopy (SEM and TEM), or by scattering techniques such as small angle X-ray and neutron scat-
Grain Size Distributions Volume
Surface = -
hiFf(D~ffukm MFC (Interiace
Transformed Fraction controlled) Controlledi
[ = 0.60
6 = 0.80
Fig. 2. Microstructure
evolution during interface controlled and diffusion controlled grain growth. See the text for details.
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tering (SAXS and SANS). However, none of them give the volume grain size distribution; they offer a different view of the microstructure which can be obtained from the volume grain size distribution. The microstructure determined by OM and SEM is a surface grain size distribution, i.e. the number of grains of a given radius by unit surface. Assuming that the observed surface is a cross-section of the sample, the observed radius of a grain hardly ever corresponds to its true radius. The track of a grain in the surface depends on the distance between the cutting plane and the center of the grain. By using classical stereology [I l] it is possible to demonstrate that
where NsJ,~ is the surface grain size distribution of radius j at time tk. This distribution is shown in Fig. 2(b) for interface and diffusion controlled growth. Both surface distributions become more rounded than the volume distributions. This result arises from the f&t that, on one hand, the probability of cutting a grain exactly by its center is very low, which reduces the number of observable large grains; and, on the other hand, the probability of cutting a small grain is small. Furthermore, the asymmetry of the diffusion controlled distribution diminishes because large grains contribute to all the sizes smaller than their own, while small grains cannot contribute to larger sizes. The particular characteristics of TEM virtually prevent comparison of experimental data with theoretical calculations. By TEM it is possible to observe the grains embedded in the sample and the tracks of cut grains on the surface. The relative proportion of both volume and surface distributions depends on the sample thickness. In the case of SANS/SAXS, the information obtainable may be related to the transformed volume fraction distribution as a function of the grain radius [12]; namely 4 && = - nrj3IQ 3
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I’I 4’ K I;
11,’ I
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t (arb. units) Fig. 3. Time dependence of the grain density for interfacecontrolled growth with growth rate G,, and nucleation rates I,> and 1,,/100, compared to diffusion controlled growth with D/R, = G, and nucleation rate I,. The arrows indicate the end of the transformation in each case.
the nucleation rate in interface controlled growth. The effect in grain density is the opposite: while diffusion controlled growth dramatically increases the grain density, reduction of the nucleation rate reduces it. As a consequence, the average grain radius shows an opposite behavior: it diminishes for a diffusion controlled growth and increases for interpace controlled growth with smaller nucleation rate.
4. Conclusions
Interface controlled and diffusion controlled grain growth are very common growing mechanisms in first order phase transitions. Therefore, complete knowledge of their influence in the developed microstructure is of great theoretical and practical interest. In this paper, we have applied a mean-field extension of the Kolmogorov-Johnson, Mehl-Avrami kinetics, which has
(15)
This distribution is shown in Fig. 2(c). The shape is similar for interface and diffusion controlled growth, and is more sharply peaked than the volume grain size distributions. Only the peak position and the maximum grain size enable us to distinguish between both mechanisms. Detailed information can also be obtained from the knowledge of the grain size distributions. Time evolution of several macroscopic parameters of interest, such as grain density and average grain radius, can be easily computed, and are shown in Figs. 3 and 4. As previously seen, diffusion controlled grain growth delays the transformation in a different way than the reduction of
a F .M cd G
f (arb. units) Fig. 4. Time dependence of the average grain radius; see Fig. 3.
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allowed us to characterize the developed microstructures. It has been shown that, under similar conditions, a diffusion controlled kinetics induces a larger grain density than the interface controlled kinetics. The maximum grain size is heavily reduced by the diffusion mechanism and, as a consequence, the grain size distribution becomes much more peaked, and thus much more homogeneous than by the interface kinetics. Comparison of the presented data with experimental measurements is made easy by computing the theoretical surface and transformed fraction grain size distributions. Beyond the particular kinetics here studied, we have also shown that the mean-field method here employed can determine the microstructure of nucleation and growth kinetics characterized either by time-dependent growth rates G(r) or radius-dependent growth rates G(R). However, a general formulation, able to deal with the general time and radius-dependent growth rate G(R, t) induced kinetics, has still not, to our knowledge, been found.
Acknowledgements This work was financed by CICYT,
grants PB94-1209
and Etlgitzeemg
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and MAT96-0692, UPC, grant PR9505, and Generalitat de Catalunya, grant 1995SGR 00514. D. Crespo acknowledges also the financial support received from ETSECCPB and Departament de Politica Territorial i Obres Ptibliques de la Generalitat de Catalunya.
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