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The effect of interfacial kinetics on the morphological stability of a spherical particle
Journal of Crystal Growth 362 (2013) 20–23
Contents lists available at SciVerse ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
The effect of interfacial kinetics on the morphological stability of a spherical particle Ming-Wen Chen a, Xin-Feng Wang b,n, Fei Wang a, Guo-Biao Lin c, Zi-Zong Wang c,n a
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China Basic Course Department, Beijing Union University, Beijing 100101, China c School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China b
a r t i c l e i n f o
a b s t r a c t
Available online 23 July 2012
This paper studies the effect of the temperature-depended interface kinetics on the morphological stability of a spherical particle under the non-equilibrium solidification condition. The result shows that as the interfacial kinetics parameter increases, the growth velocity of the spherical particle decreases. Compared with the case of neglecting interfacial kinetics, the temperature-depended interfacial kinetics has the significant stable effect on the morphological stability of the spherical crystal and the critical stability radius increases as the interfacial kinetics parameter increases. & 2012 Published by Elsevier B.V.
Keywords: A1. Interfaces A1. Nucleation A1. Crystal morphology A1. Morphological stability A2. Growth from melt
1. Introduction
2. The theoretical model
Interfacial kinetics has an important effect on the interfacial morphologies during solidification[1]. For typical metals, the growth velocity of a spherical particle during solidification attains several hundred meters per second, this implies that the number of atoms attached to the interface from the liquid phase is far larger than that escaped from the interface and solidification at the interface significantly departs the equilibrium condition. At the interface, the total undercooling consists of the thermal undercooling, the constitutional undercooling, the curvature undercooling and the kinetic undercooling. The kinetic undercooling relates to both growth velocity of the interface and interfacial temperature. Previous studies mainly took the linear relation between the kinetic undercooling and the growth velocity [2–6]. Actually, the kinetic undercooling is generally a function of interfacial temperature [7], which gives rise to local change in interfacial temperature. Li et al. [8] demonstrated that the dependence of the interfacial kinetics coefficient on interfacial temperature significantly influences the morphological stability of the planar interface during directional solidification. In the paper, the temperature-depended interfacial kinetics coefficient under the non-equilibrium solidification condition is considered in the model of the growth of a spherical particle. By the asymptotic analysis method [9–10], we analyze the effect of the temperature-depended interface kinetic coefficient on the morphological stability of the spherical particle.
Nucleation occurs instantaneously when a group of atoms form a critical nucleus. As more atoms attach on the interface and further extraction of heat from the nucleus causes the nucleus to grow into a spherical crystal. It is assumed that the initial radius of the sphere is r0, the far-field temperature is TN. TL and TS denote the temperatures in the liquid phase and solid phase, respectively, and R¼R(y,j,t) denotes the crystal-melt interface. The heat transfer processes in the liquid phase after nucleation are governed by the temperature fields,
n
Corresponding authors. Tel.: þ 86 10 62333749; fax: þ 86 10 62333152. E-mail addresses: [email protected] (X.-F. Wang), [email protected] (Z.-Z. Wang). 0022-0248/$ - see front matter & 2012 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.jcrysgro.2012.02.042
@T L @T S 2 2 ¼ kL r T L , ¼ kS r T S , @t @t
ð2:1Þ
where kL and kS are the thermal diffusivities in the liquid phase and solid phase respectively. The temperature at the interface is governed by the heat equilibrium condition and the Gibbs–Thomson condition, TL ¼ TS ¼ TI,
T I ¼ T M ð1 þ
ð2:2Þ
gK DH
ÞDT K ,
ð2:3Þ
where TI is the interface temperature, K is the twice local mean curvature at the interface, g is the isotropic surface energy, DH is the latent heat per unit volume, DTK is the kinetic undercooling which is necessary for the attachment of atoms to the interface [7,8], rRT T rRT TU U DT K ¼ L g M I ln 1 I L g M I I , ð2:4Þ DHM0 V0 DHM0 V 0
M.-W. Chen et al. / Journal of Crystal Growth 362 (2013) 20–23
in which, V0 is the sound speed, R0 is the gas constant, rL is the density in the melt, M0 is the mole mass, and UI is the local growth velocity. Moreover, at the crystal-melt interface, the difference of the heat flux in and out that transports the heat to melt the crystal equals the heat gained from the latent heat to flow away from the interface, at the interface the energy conservation condition holds, ðDH þ gKÞU I ¼ nUrðkS T S kL T L Þ,
ð2:5Þ
where kL and kS are the heat conduction coefficients in the liquid and solid phase, respectively. Finally, the initial conditions are given beforehand. In the present paper it is assumed that the densities in the liquid phase and solid phase are equal, and the buoyancy effects are neglected. We choose the initial radius of the spherical crystal r0 as the length scale, and r0/V as the time scale, where V is the characteristic velocity of the interface. The scales of the temperature fields are taken as the undercooling DT¼TM TN (TM is the equilibrium melting temperature, TM 4TN. We introduce the non-dimensional physical quantities T L T M T S T M r t , T S, r- , tr0 r 0 =V DT DT
21
in which each order term is expanded into spherical harmonics series. Substituting (3.1) into Eqs. (2.2)–(2.10) and expanding the interface curvature in the interface conditions (2.9) and (2.10) into 2 e 1 @ @ 1 @2 K ¼ þ þ 2 ðL þ 2ÞRB1 þ , L ¼ sin y RB0 RB0 sin y @y @y sin2 y @j2 equating the terms of like powers of e, we have the each order terms in (3.1) [4–6] T L0 ¼ 1 þ
T S0 ¼
RB0 2Gð1EÞbE1 MRB0 R0B0 1 , r 1þ bMR0B0
ð3:2Þ
2GbE1 MRB0 R0B0 , RB0 þ bMRB0 R0B0
ð3:3Þ
and RB0 satisfies the ordinary differential equation that satisfies and the initial condition RB0(0)¼1, RB0 R0B0 ¼
RB0 2G ðRB0 2GEÞð1 þ bMR0B0 Þ þ bðE1 MMÞ
,
ð3:4Þ
where the prime ‘‘0 ’’ denotes the derivative with respect to t.
ð2:6Þ
T L1 ¼
W1 W3 r db0 ðtÞ , g ðtÞ þ þ W 2 r 0,0 6kW 2 r 2 dt
ð3:5Þ
In the spherical coordinates whose origin is at the center of the sphere, the heat transfer processes in the liquid phase after nucleation are governed by the energy equation
T S1 ¼
W4 W5 r 2 d b0 ðtÞ g 0,0 ðtÞ þ þ , W2 6kW 2 6k dt RB0
ð3:6Þ
T L-
e
@T L @T S ¼ r2 T L , e ¼ kr2 T S @t @t
ð2:7Þ
RB1 ¼ g 0,0 ðtÞ, where
where e ¼ DT/(DH/cprL). At the interface, the heat equilibrium condition and Gibbs–Thomson condition are transferred into T L ¼ T S,
ð2:8Þ
T S ¼ GKbMU I T I bE1 MU I ,
ð2:9Þ
ð1þ GEKÞU I ¼ ðkrT S rT L ÞUn,
ð2:10Þ
g 0,0 ðtÞ ¼
b0 ðtÞ ¼
W1 ¼
where
G¼
ð3:7Þ
R t W 6 ðtÞ Z t R W 6 ðtÞ 1 W 7 ðtÞ t W dt dt e 0 2 ðtÞ dt, e 0 W 2 ðtÞ 3k W ð t Þ 2 0
ð3:8Þ
RB0 2GðE1 MMÞbRB0 R0B0 , 1 þ bMR0B0
ð3:9Þ
2bMW 8 b0 ðtÞ þ GERB0 R0B0 þ EW 9 b0 ðtÞ þ 2GEðRB0 2GEÞ, RB0
W 2 ¼ bMW 8 þ RB0 EW 9 ,
r Rg T M gT M kS kS DT V , M¼ L , k¼ , k¼ , E¼ , b¼ TM V0 r 0 DT DH DHM0 kL kL
Here, we call M as the interfacial kinetics parameter. The lefthand side in (2.10) includes the interfacial stretching energy in the heat release from the interface during solidification [11,12].
3. The basic state solution and stability analysis For the dimensionless parameter e ¼ DT/(DH/cprL), if the undercoolings are low, the value of DT is small, then the dimensionless parameter e is small; if the value of DT is not small but the value of DH/cprL is relatively large, then the dimensionless parameter e is also small. For the typical metals, the parameter e is small. For example, in the experiment of fabrication of a nanocomposite from in situ iron nanoparticle reinforced copper alloy, the melting temperature of Fe particles in the solid phase is TM ¼1728 K, the melting temperature in Cu matrix of liquid phase [13] is TM ¼1356 K, the undercooling DT¼373 K is formed during the solidification of Fe particles. The latent heat of Fe per unit volume is DH¼2.404 109 J m 3, the specific heat and surface tension of Fe are cp ¼477.3 J/(kg K), rL ¼7874 kg/m3kg/m, so we calculate the parameter e ¼0.535 which is still taken as a small parameter. We expand the physical quantities of the solidification system into the asymptotic series in powers of e as e-0, T LB ¼ T L0 þ eT L1 þ ,T SB ¼ T S0 þ eT S1 þ ,RB ¼ RB0 þ eRB1 þ ð3:1Þ
db0 ðtÞ d b0 ðtÞ db0 ðtÞ 3 2kRB0 RB0 , 3kEW 9 W 3 ¼ bMR2B0 W 8 3k dt dt RB0 dt W4 ¼
bMW 8 ðb0 ðtÞ þ 2GERB0 R0B0 Þ R2B0
þ
2GEðRB0 2GEÞ , RB0
db0 ðtÞ d b0 ðtÞ ð1 þ 2kÞRB0 W 5 ¼ bMRB0 W 8 6k dt dt RB0 b0 ðtÞ 3 d W 9 ERB0 , dt RB0 W6 ¼
2GEEðb0 ðtÞ þ2GERB0 R0B0 Þð1 þ bMR0B0 Þ , RB0
d b0 ðtÞ db0 ðtÞ , W 7 ¼ ER2B0 ð1 þ bMR0B0 Þ kRB0 3k dt RB0 dt in which W8 and W9 are two abbreviations W 8 ¼ RB0 ERB0 þEb0 ðtÞ,W 9 ¼ ðRB0 2GEÞð1 þ bMR0B0 Þ ˙ Therefore, we obtain the regular expansion solution in (3.1) up to the first order term. The dimensionless local growth rate interface velocity is dRB dRB0 dRB1 ¼ þe þ dt dt dt
ð3:10Þ
Fig. 1 shows the relationship of the radius of the spherical crystal with the time under the influence of interface kinetics.
22
M.-W. Chen et al. / Journal of Crystal Growth 362 (2013) 20–23 0
0.4
dRB /dt
0
d (t)/d(t)o0. d (t)/d(t) ¼0 corresponds to the critical state. Fig. 2 0 shows the perturbation growth rate s0 ¼ d (t)/d(t) versus different
0.5
0.3
0.2
0.1 0
2
4
t
6
8
10
As the interfacial kinetics parameter M increases, the growth velocity of the spherical crystal decreases. For the stability analysis, we superpose the perturbation term on the basic state solution,
0.003
~ T L ¼ T LB þ T~ L ,T S ¼ T SB þ T~ S ,R ¼ RB þ R:
t
0.001
' t
Fig. 1. The variations of the growth velocity of the spherical particle with the time for different interfacial kinetics parameters M ¼ 0.0, 0.5, 1.0, 2.0 (from top to bottom) where b ¼ 1.0, k¼ 1.0, k ¼ 1.0, G ¼ 0.3, E ¼0.3, and e ¼ 0.1.
radii of the basic state of the sphere RB0 (or equivalently for different times t) for different modes n. It is seen that compared with the case that neglects interfacial kinetics in Ref. [2], the mode n ¼1 is still neutrally stable, the first mode that determines instability is still the mode n ¼2. Fig. 3 shows the variations of the perturbation growth rate corresponding to the mode n¼2 with different radii of the basic state RB0 for different interfacial kinetics parameter M. As the interfacial kinetics parameter M increases the perturbation growth rate decreases. It implies that the growth of the spherical particle tends to more stable and the critical stability radius increases as the interfacial kinetics parameter M increases. If the interfacial kinetics is neglected, i.e. the interfacial kinetics parameter M¼0, our result reduces to the result obtained by Mullins and Sekerka [2]. As we know, up to now, the available experimental data for the temperaturedepended interfacial kinetics has not been reported. Maybe it is due to the difficulty of experimental measurement. We consider
0.000
ð3:11Þ
Further, we expand the perturbation terms in (3.11) into the following form T~ L ¼ T~ L0 þ eT~ L1 þ , T~ S ¼ T~ S0 þ eT~ S1 þ , R~ ¼ R~ 0 þ eR~ 1 þ
r T~ L0 ¼ 0, r2 T~ S0 ¼ 0
ð3:12Þ
4
n
5
1 n
-0.001
0
-0.002 -0.003 10
20
30
40
50
RB0
ð3:14Þ
and the interface conditions
0
@T L0 ~ T~ L0 ¼ T~ S0 R0 @r
ð3:15Þ
G @R~ 0 T~ S0 ¼ 2 ðL þ2ÞR~ 0 bMR0B0 T~ S0 bðE1 M þ MT S0 Þ , @t RB0
ð3:16Þ
GE @R~ 0 GER0B0 @T~ S0 @T~ L0 @2 T L0 ~ R0 þ Þ ðL þ 2ÞR~ 0 ¼ k RB0 @t @r @r @r 2 R2B0
Fig. 2. The perturbation growth rate s0 ¼ d (t)/d(t) versus different radii of the basic state of the sphere RB0 under the influence of interfacial kinetics for the modes n¼ 0, 1, 2, 3, 4, and 5, where M ¼1.0, b ¼1.0, k¼ 1.0, k ¼1.0, G ¼0.15, and E¼ 0.3.
0.03
ð3:17Þ
Eq. (3.13) have normal-mode solutions
0.02
ð3:18Þ
where Yn,m are spherical harmonics that satisfy LYn,m ¼ n(n þ1) Yn,m. We write R~ 0 ¼ dðtÞY n,m and substitute (3.18) into the inter-
M
0.0
t
Y n,m ðy, jÞ, T~ L0 ¼ B0 r Y n,m ðy, jÞ n
M ' t
T~ L0 ¼ A0 r
n
ð3:13Þ
T~ L0 -0 as r-1
n1
3
n
in which T~ L0 and T~ S0 are subject to the far-field condition
ð1
n
0.002
then the temperature equations for the leading order terms are quasi-steady, 2
n 2
0.2
0.01 M
0.8
face conditions (3.15)–(3.17) to derive that
d0 ðtÞ ðn1Þ ¼ RðRB0 ,nÞ, dðtÞ R2B0
0.00 2.0
-0.01
where RðRB0 , nÞ ¼
M
ð3:19Þ
ð1 þn þknÞðn þ 2ÞGð1 þ bMR0B0 Þ½ðn þ 2ÞGERB0 R0B0 b0 ðtÞ
bMð1þ n þ knÞðE1 þ T S0 ÞR2B0 þ ðRB0 2GEÞð1 þ bMR0B0 Þ
Eq. (3.19) reveals the relation between the perturbation growth rate and different radii of the basic state and the modes n under the influence of interfacial kinetics. The growth of 0 the spherical particle is unstable as d (t)/d(t)40 and is stable as
2
4
6
8
10
RB0 Fig. 3. The variation of the perturbation growth rate corresponding to the mode n¼2 with different radii of the basic state RB0 for different interfacial kinetics parameters M ¼0.0, 0.2, 0.8, and 2.0, whereb ¼ 1.0, k¼1.0, k ¼1.0,G ¼ 0.1, E¼ 0.3, and e ¼ 0.1.
M.-W. Chen et al. / Journal of Crystal Growth 362 (2013) 20–23
that the theoretical analysis can circumvent the difficulty. Our analytical result is expected to be verified by the experiment.
4. Conclusion We consider the temperature-depended interface kinetics under non-equilibrium solidification condition for the growth of a spherical particle. By the asymptotic analysis method, we analyze the influence of the temperature-depended interfacial kinetics on the morphological stability of the spherical particle. The results show that as the interfacial kinetics parameter increases, the growth velocity of the spherical particle decreases. Compared with the case of neglecting interface kinetics, the temperature-depended interfacial kinetics significantly decreases the growth velocity of the spherical particle, the growth of the spherical particle tends to more stable and the critical stability radius increases.
Acknowledgments This work was financially supported by the National Natural Science Foundation of China (Grant nos. 10972030and 50904005), the 863 Project of China (Grant no. 2007AA03Z108), the Fundamental
23
Research Funds for the Central University, the Metallurgy Research Fund of State Key Lab of Advanced Metals and Materials no. 2009Z-01, Overseas Distinguished Scholar program by the Ministry of Chinese Education.
References [1] M. Asta, C. Beckermann, A. Karma, W. Kurz, R. Napolitano, M. Plapp, G. Purdy, M. Rappaz, R. Trivedi, Acta Materialia 57 (2009) 941. [2] W.W. Mullins, R.F. Sekerka, Journal of Applied Physics 34 (1963) 323. [3] C.W. Lan, Journal of Crystal Growth 274 (2005) 379–386. [4] M.W. Chen, R.J. Sun, Z.D. Wang, Acta Physica Sinica 56 (2007) 1819. [5] M.W. Chen, Z.D. Wang, J.J. Xu, Science in China Series E 51 (2008) 225. [6] M.W. Chen, Z.D. Wang, J.X. Xie, J.J. Xu, Acta Mechanica Sinica 24 (2008) 681. [7] D. Turnbull, Metallurgical Transactions 12A (1981) 695. [8] J. Li, G. Yang, Y. Zhou, Materials Research Bulletin 35 (2000) 1775. [9] M.W. Chen, X.F. Wang, Z.D. Wang, J.X. Xie, Journal of Crystal Growth 310 (2008) 655. [10] M.W. Chen, H. Jiang, Y.L. Wang, Q.S. Wang, Z.D. Wang, Journal of Crystal Growth 318 (2011) 145. [11] S.H. Davis, Theory of solidification, Cambridge University Press, United Kingdom, 2001. [12] M.W. Chen, Y.L. Wang, H. Zhang, L.Y. Wu, Z.D. Wang, Journal of Applied Physics 109 (2011) 103517. [13] Z.D. Wang, X.W. Wang, Q.S. Wang, I. Shih, J.J. Xu, Nanotechnology 20 (2009) 075605.
Contents lists available at SciVerse ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
The effect of interfacial kinetics on the morphological stability of a spherical particle Ming-Wen Chen a, Xin-Feng Wang b,n, Fei Wang a, Guo-Biao Lin c, Zi-Zong Wang c,n a
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China Basic Course Department, Beijing Union University, Beijing 100101, China c School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China b
a r t i c l e i n f o
a b s t r a c t
Available online 23 July 2012
This paper studies the effect of the temperature-depended interface kinetics on the morphological stability of a spherical particle under the non-equilibrium solidification condition. The result shows that as the interfacial kinetics parameter increases, the growth velocity of the spherical particle decreases. Compared with the case of neglecting interfacial kinetics, the temperature-depended interfacial kinetics has the significant stable effect on the morphological stability of the spherical crystal and the critical stability radius increases as the interfacial kinetics parameter increases. & 2012 Published by Elsevier B.V.
Keywords: A1. Interfaces A1. Nucleation A1. Crystal morphology A1. Morphological stability A2. Growth from melt
1. Introduction
2. The theoretical model
Interfacial kinetics has an important effect on the interfacial morphologies during solidification[1]. For typical metals, the growth velocity of a spherical particle during solidification attains several hundred meters per second, this implies that the number of atoms attached to the interface from the liquid phase is far larger than that escaped from the interface and solidification at the interface significantly departs the equilibrium condition. At the interface, the total undercooling consists of the thermal undercooling, the constitutional undercooling, the curvature undercooling and the kinetic undercooling. The kinetic undercooling relates to both growth velocity of the interface and interfacial temperature. Previous studies mainly took the linear relation between the kinetic undercooling and the growth velocity [2–6]. Actually, the kinetic undercooling is generally a function of interfacial temperature [7], which gives rise to local change in interfacial temperature. Li et al. [8] demonstrated that the dependence of the interfacial kinetics coefficient on interfacial temperature significantly influences the morphological stability of the planar interface during directional solidification. In the paper, the temperature-depended interfacial kinetics coefficient under the non-equilibrium solidification condition is considered in the model of the growth of a spherical particle. By the asymptotic analysis method [9–10], we analyze the effect of the temperature-depended interface kinetic coefficient on the morphological stability of the spherical particle.
Nucleation occurs instantaneously when a group of atoms form a critical nucleus. As more atoms attach on the interface and further extraction of heat from the nucleus causes the nucleus to grow into a spherical crystal. It is assumed that the initial radius of the sphere is r0, the far-field temperature is TN. TL and TS denote the temperatures in the liquid phase and solid phase, respectively, and R¼R(y,j,t) denotes the crystal-melt interface. The heat transfer processes in the liquid phase after nucleation are governed by the temperature fields,
n
Corresponding authors. Tel.: þ 86 10 62333749; fax: þ 86 10 62333152. E-mail addresses: [email protected] (X.-F. Wang), [email protected] (Z.-Z. Wang). 0022-0248/$ - see front matter & 2012 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.jcrysgro.2012.02.042
@T L @T S 2 2 ¼ kL r T L , ¼ kS r T S , @t @t
ð2:1Þ
where kL and kS are the thermal diffusivities in the liquid phase and solid phase respectively. The temperature at the interface is governed by the heat equilibrium condition and the Gibbs–Thomson condition, TL ¼ TS ¼ TI,
T I ¼ T M ð1 þ
ð2:2Þ
gK DH
ÞDT K ,
ð2:3Þ
where TI is the interface temperature, K is the twice local mean curvature at the interface, g is the isotropic surface energy, DH is the latent heat per unit volume, DTK is the kinetic undercooling which is necessary for the attachment of atoms to the interface [7,8], rRT T rRT TU U DT K ¼ L g M I ln 1 I L g M I I , ð2:4Þ DHM0 V0 DHM0 V 0
M.-W. Chen et al. / Journal of Crystal Growth 362 (2013) 20–23
in which, V0 is the sound speed, R0 is the gas constant, rL is the density in the melt, M0 is the mole mass, and UI is the local growth velocity. Moreover, at the crystal-melt interface, the difference of the heat flux in and out that transports the heat to melt the crystal equals the heat gained from the latent heat to flow away from the interface, at the interface the energy conservation condition holds, ðDH þ gKÞU I ¼ nUrðkS T S kL T L Þ,
ð2:5Þ
where kL and kS are the heat conduction coefficients in the liquid and solid phase, respectively. Finally, the initial conditions are given beforehand. In the present paper it is assumed that the densities in the liquid phase and solid phase are equal, and the buoyancy effects are neglected. We choose the initial radius of the spherical crystal r0 as the length scale, and r0/V as the time scale, where V is the characteristic velocity of the interface. The scales of the temperature fields are taken as the undercooling DT¼TM TN (TM is the equilibrium melting temperature, TM 4TN. We introduce the non-dimensional physical quantities T L T M T S T M r t , T S, r- , tr0 r 0 =V DT DT
21
in which each order term is expanded into spherical harmonics series. Substituting (3.1) into Eqs. (2.2)–(2.10) and expanding the interface curvature in the interface conditions (2.9) and (2.10) into 2 e 1 @ @ 1 @2 K ¼ þ þ 2 ðL þ 2ÞRB1 þ , L ¼ sin y RB0 RB0 sin y @y @y sin2 y @j2 equating the terms of like powers of e, we have the each order terms in (3.1) [4–6] T L0 ¼ 1 þ
T S0 ¼
RB0 2Gð1EÞbE1 MRB0 R0B0 1 , r 1þ bMR0B0
ð3:2Þ
2GbE1 MRB0 R0B0 , RB0 þ bMRB0 R0B0
ð3:3Þ
and RB0 satisfies the ordinary differential equation that satisfies and the initial condition RB0(0)¼1, RB0 R0B0 ¼
RB0 2G ðRB0 2GEÞð1 þ bMR0B0 Þ þ bðE1 MMÞ
,
ð3:4Þ
where the prime ‘‘0 ’’ denotes the derivative with respect to t.
ð2:6Þ
T L1 ¼
W1 W3 r db0 ðtÞ , g ðtÞ þ þ W 2 r 0,0 6kW 2 r 2 dt
ð3:5Þ
In the spherical coordinates whose origin is at the center of the sphere, the heat transfer processes in the liquid phase after nucleation are governed by the energy equation
T S1 ¼
W4 W5 r 2 d b0 ðtÞ g 0,0 ðtÞ þ þ , W2 6kW 2 6k dt RB0
ð3:6Þ
T L-
e
@T L @T S ¼ r2 T L , e ¼ kr2 T S @t @t
ð2:7Þ
RB1 ¼ g 0,0 ðtÞ, where
where e ¼ DT/(DH/cprL). At the interface, the heat equilibrium condition and Gibbs–Thomson condition are transferred into T L ¼ T S,
ð2:8Þ
T S ¼ GKbMU I T I bE1 MU I ,
ð2:9Þ
ð1þ GEKÞU I ¼ ðkrT S rT L ÞUn,
ð2:10Þ
g 0,0 ðtÞ ¼
b0 ðtÞ ¼
W1 ¼
where
G¼
ð3:7Þ
R t W 6 ðtÞ Z t R W 6 ðtÞ 1 W 7 ðtÞ t W dt dt e 0 2 ðtÞ dt, e 0 W 2 ðtÞ 3k W ð t Þ 2 0
ð3:8Þ
RB0 2GðE1 MMÞbRB0 R0B0 , 1 þ bMR0B0
ð3:9Þ
2bMW 8 b0 ðtÞ þ GERB0 R0B0 þ EW 9 b0 ðtÞ þ 2GEðRB0 2GEÞ, RB0
W 2 ¼ bMW 8 þ RB0 EW 9 ,
r Rg T M gT M kS kS DT V , M¼ L , k¼ , k¼ , E¼ , b¼ TM V0 r 0 DT DH DHM0 kL kL
Here, we call M as the interfacial kinetics parameter. The lefthand side in (2.10) includes the interfacial stretching energy in the heat release from the interface during solidification [11,12].
3. The basic state solution and stability analysis For the dimensionless parameter e ¼ DT/(DH/cprL), if the undercoolings are low, the value of DT is small, then the dimensionless parameter e is small; if the value of DT is not small but the value of DH/cprL is relatively large, then the dimensionless parameter e is also small. For the typical metals, the parameter e is small. For example, in the experiment of fabrication of a nanocomposite from in situ iron nanoparticle reinforced copper alloy, the melting temperature of Fe particles in the solid phase is TM ¼1728 K, the melting temperature in Cu matrix of liquid phase [13] is TM ¼1356 K, the undercooling DT¼373 K is formed during the solidification of Fe particles. The latent heat of Fe per unit volume is DH¼2.404 109 J m 3, the specific heat and surface tension of Fe are cp ¼477.3 J/(kg K), rL ¼7874 kg/m3kg/m, so we calculate the parameter e ¼0.535 which is still taken as a small parameter. We expand the physical quantities of the solidification system into the asymptotic series in powers of e as e-0, T LB ¼ T L0 þ eT L1 þ ,T SB ¼ T S0 þ eT S1 þ ,RB ¼ RB0 þ eRB1 þ ð3:1Þ
db0 ðtÞ d b0 ðtÞ db0 ðtÞ 3 2kRB0 RB0 , 3kEW 9 W 3 ¼ bMR2B0 W 8 3k dt dt RB0 dt W4 ¼
bMW 8 ðb0 ðtÞ þ 2GERB0 R0B0 Þ R2B0
þ
2GEðRB0 2GEÞ , RB0
db0 ðtÞ d b0 ðtÞ ð1 þ 2kÞRB0 W 5 ¼ bMRB0 W 8 6k dt dt RB0 b0 ðtÞ 3 d W 9 ERB0 , dt RB0 W6 ¼
2GEEðb0 ðtÞ þ2GERB0 R0B0 Þð1 þ bMR0B0 Þ , RB0
d b0 ðtÞ db0 ðtÞ , W 7 ¼ ER2B0 ð1 þ bMR0B0 Þ kRB0 3k dt RB0 dt in which W8 and W9 are two abbreviations W 8 ¼ RB0 ERB0 þEb0 ðtÞ,W 9 ¼ ðRB0 2GEÞð1 þ bMR0B0 Þ ˙ Therefore, we obtain the regular expansion solution in (3.1) up to the first order term. The dimensionless local growth rate interface velocity is dRB dRB0 dRB1 ¼ þe þ dt dt dt
ð3:10Þ
Fig. 1 shows the relationship of the radius of the spherical crystal with the time under the influence of interface kinetics.
22
M.-W. Chen et al. / Journal of Crystal Growth 362 (2013) 20–23 0
0.4
dRB /dt
0
d (t)/d(t)o0. d (t)/d(t) ¼0 corresponds to the critical state. Fig. 2 0 shows the perturbation growth rate s0 ¼ d (t)/d(t) versus different
0.5
0.3
0.2
0.1 0
2
4
t
6
8
10
As the interfacial kinetics parameter M increases, the growth velocity of the spherical crystal decreases. For the stability analysis, we superpose the perturbation term on the basic state solution,
0.003
~ T L ¼ T LB þ T~ L ,T S ¼ T SB þ T~ S ,R ¼ RB þ R:
t
0.001
' t
Fig. 1. The variations of the growth velocity of the spherical particle with the time for different interfacial kinetics parameters M ¼ 0.0, 0.5, 1.0, 2.0 (from top to bottom) where b ¼ 1.0, k¼ 1.0, k ¼ 1.0, G ¼ 0.3, E ¼0.3, and e ¼ 0.1.
radii of the basic state of the sphere RB0 (or equivalently for different times t) for different modes n. It is seen that compared with the case that neglects interfacial kinetics in Ref. [2], the mode n ¼1 is still neutrally stable, the first mode that determines instability is still the mode n ¼2. Fig. 3 shows the variations of the perturbation growth rate corresponding to the mode n¼2 with different radii of the basic state RB0 for different interfacial kinetics parameter M. As the interfacial kinetics parameter M increases the perturbation growth rate decreases. It implies that the growth of the spherical particle tends to more stable and the critical stability radius increases as the interfacial kinetics parameter M increases. If the interfacial kinetics is neglected, i.e. the interfacial kinetics parameter M¼0, our result reduces to the result obtained by Mullins and Sekerka [2]. As we know, up to now, the available experimental data for the temperaturedepended interfacial kinetics has not been reported. Maybe it is due to the difficulty of experimental measurement. We consider
0.000
ð3:11Þ
Further, we expand the perturbation terms in (3.11) into the following form T~ L ¼ T~ L0 þ eT~ L1 þ , T~ S ¼ T~ S0 þ eT~ S1 þ , R~ ¼ R~ 0 þ eR~ 1 þ
r T~ L0 ¼ 0, r2 T~ S0 ¼ 0
ð3:12Þ
4
n
5
1 n
-0.001
0
-0.002 -0.003 10
20
30
40
50
RB0
ð3:14Þ
and the interface conditions
0
@T L0 ~ T~ L0 ¼ T~ S0 R0 @r
ð3:15Þ
G @R~ 0 T~ S0 ¼ 2 ðL þ2ÞR~ 0 bMR0B0 T~ S0 bðE1 M þ MT S0 Þ , @t RB0
ð3:16Þ
GE @R~ 0 GER0B0 @T~ S0 @T~ L0 @2 T L0 ~ R0 þ Þ ðL þ 2ÞR~ 0 ¼ k RB0 @t @r @r @r 2 R2B0
Fig. 2. The perturbation growth rate s0 ¼ d (t)/d(t) versus different radii of the basic state of the sphere RB0 under the influence of interfacial kinetics for the modes n¼ 0, 1, 2, 3, 4, and 5, where M ¼1.0, b ¼1.0, k¼ 1.0, k ¼1.0, G ¼0.15, and E¼ 0.3.
0.03
ð3:17Þ
Eq. (3.13) have normal-mode solutions
0.02
ð3:18Þ
where Yn,m are spherical harmonics that satisfy LYn,m ¼ n(n þ1) Yn,m. We write R~ 0 ¼ dðtÞY n,m and substitute (3.18) into the inter-
M
0.0
t
Y n,m ðy, jÞ, T~ L0 ¼ B0 r Y n,m ðy, jÞ n
M ' t
T~ L0 ¼ A0 r
n
ð3:13Þ
T~ L0 -0 as r-1
n1
3
n
in which T~ L0 and T~ S0 are subject to the far-field condition
ð1
n
0.002
then the temperature equations for the leading order terms are quasi-steady, 2
n 2
0.2
0.01 M
0.8
face conditions (3.15)–(3.17) to derive that
d0 ðtÞ ðn1Þ ¼ RðRB0 ,nÞ, dðtÞ R2B0
0.00 2.0
-0.01
where RðRB0 , nÞ ¼
M
ð3:19Þ
ð1 þn þknÞðn þ 2ÞGð1 þ bMR0B0 Þ½ðn þ 2ÞGERB0 R0B0 b0 ðtÞ
bMð1þ n þ knÞðE1 þ T S0 ÞR2B0 þ ðRB0 2GEÞð1 þ bMR0B0 Þ
Eq. (3.19) reveals the relation between the perturbation growth rate and different radii of the basic state and the modes n under the influence of interfacial kinetics. The growth of 0 the spherical particle is unstable as d (t)/d(t)40 and is stable as
2
4
6
8
10
RB0 Fig. 3. The variation of the perturbation growth rate corresponding to the mode n¼2 with different radii of the basic state RB0 for different interfacial kinetics parameters M ¼0.0, 0.2, 0.8, and 2.0, whereb ¼ 1.0, k¼1.0, k ¼1.0,G ¼ 0.1, E¼ 0.3, and e ¼ 0.1.
M.-W. Chen et al. / Journal of Crystal Growth 362 (2013) 20–23
that the theoretical analysis can circumvent the difficulty. Our analytical result is expected to be verified by the experiment.
4. Conclusion We consider the temperature-depended interface kinetics under non-equilibrium solidification condition for the growth of a spherical particle. By the asymptotic analysis method, we analyze the influence of the temperature-depended interfacial kinetics on the morphological stability of the spherical particle. The results show that as the interfacial kinetics parameter increases, the growth velocity of the spherical particle decreases. Compared with the case of neglecting interface kinetics, the temperature-depended interfacial kinetics significantly decreases the growth velocity of the spherical particle, the growth of the spherical particle tends to more stable and the critical stability radius increases.
Acknowledgments This work was financially supported by the National Natural Science Foundation of China (Grant nos. 10972030and 50904005), the 863 Project of China (Grant no. 2007AA03Z108), the Fundamental
23
Research Funds for the Central University, the Metallurgy Research Fund of State Key Lab of Advanced Metals and Materials no. 2009Z-01, Overseas Distinguished Scholar program by the Ministry of Chinese Education.
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