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On the formation of the banded structure in rapid solidification
Materials Science and Engineering, A178 ( 1994 ) 153-157
153
On the formation of the banded structure in rapid solidification Armand Sarkissian and Alain Karma
Physics Department, Northeastern University, Boston, MA 02115 (USA)
Abstract Rapid solidification experiments on metallic alloys over the last decade have provided widespread observations of a novel banded structure. We present the results of numerical and analytical studies of the non-linear dynamics of the solidifying interface which provide insights into the mechanism of formation of this structure. These studies are based on a model of directional solidification which includes non-equilibrium effects and consists of a one-dimensional boundary integral equation for the temperature field combined with a boundary-layer approximation for the solute field. Three main ingredients of the banding mechanism emerge from these studies: the presence of non-linear oscillations of the solidification rate driven by solute trapping, the dramatic effect of latent heat diffusion on the character of these oscillations, and the propagation of"optical growth fronts" connecting collision-controlledand diffusion-controlled inteffacial regions.
1. Introduction
Perhaps one of the most novel and surprising observations which has emerged from rapid solidification experiments over the last decade is that of the socalled banded structure. First observed as an isolated phenomenon [1], this structure has subsequently been found to occur in a broad class of metallic alloys solidified under growth conditions which approach absolute stability (for a complete list of references on banding see ref. 2). The most puzzling aspect of the banded structure, from a theoretical standpoint, is the presence of alternating light and dark bands which lie parallel to the solidification front. Detailed experimental studies [2-5] have revealed that the dark bands have a precipitate structure, either cellular-dendritic or eutectic, which is similar to the precipitate structure observed at lower growth rate just preceding the appearance of the banded structure. In contrast, the light bands have a microsegregation-free structure similar to that observed at the larger growth rate just following the disappearance of this structure. An attractive explanation for the origin of banding was set forth by Carard, Gremaud, Zimmermann, and Kurz (CGZK) [6]. These authors remarked that, owing to non-equilibrium effects, the Tss(V) curve relating the steady-state temperature and growth rate of the solidification front has an S-shape characteristic of bistable systems when the growth rate is in the vicinity of the absolute stability limit (Fig. 1). On the basis of this observation, they proposed a classic "relaxationoscillator" picture of banding in which periodic changes in growth rate cause the solidification front to 0921-5093/94/$7.00 SSI)I 0921-5093(93)04528-P
make abrupt transitions between two steady-state branches of this S-shaped Tss(V) curve: a low-velocity branch corresponding to a cellular-dendritic growth morphology, and a high-velocity branch corresponding to a planar morphology (thereby accounting for the observed alternation of structure). This model has provided a valuable first step towards an understanding of banding but, at the same time, set forth the difficult task of elucidating whether the basic dynamical equations of solidification, including phenomenological nonequilibrium corrections, actually produce such nonlinear oscillations.
91 0
,,
Vo=1.0 m/s
/,,t
906 =
/:'l
~O .
F I"'""',
,=o
\
..................' > ............
,2
E 902
i
8 9 8
,
0.1
,,:
....
,
. . . . . . . .
,
1.0 10.0 Velocity (m/s)
,,~_ 100.0
Fig. 1. Comparison of the CGZK banding cycle ( 1-2-3-4-1 ) and the dynamic transients of the planar interface without (L = 0) and with (L ¢ 0)latent heat diffusion for V0 = 1.0 m s J. © 1994 - Elsevier Sequoia. All rights reserved
A. Sarkissian, A. Karma / Formationof the banded structure
154 2. Model
al. [12] to study dendritic growth. In the present context, this approximation leads to the set of equations:
To investigate this question we study a model of rapid directional solidification which takes into account both heat and solute diffusion and nonequilibrium corrections. The latter are included using a thermodynamically consistent interface condition
dh =vn(C~o-kCL)-vn~ch+Dc
L
1 -- k e
(2)
J
developed by Boettinger and Coriell [7] (this condition also corresponds to the special case of solute trapping without solute drag of a more general interface condition developed recently [8]) and a velocity dependent form of partition coefficient
k
-
h
(1)
v*
1 - k(1 -ln[k/k~])l m =In e
(c -c f
CL(1 -- k)vn = Dc
RTM 2 V n Ti= TM + m Q - - F r - - -
ke + Vn/Vo 1 + Vn/V d
CL-C~o OS]
(5) (6)
where the variables h(S, t) denotes the total impurity content in the solute boundary-layer ahead of the interface and d/dtln denotes the time derivative along the normal to the interface. The dynamics of the interface in our model is then completely specified by eqns. (1)-(6) together with the usual interface kinematic relations = --
K2-.{-
(7)
Vn
(3) dS
developed by Aziz and Kaplan [9]. We focus here on a computationally tractable description of the diffusion fields which allows study of both planar and nonplanar interface dynamics, thereby extending our previous study of planar dynamics (only) that was based on a boundary integral formulation of fully nonlocal models [10, 11]. The temperature field is represented here by the one-dimensional boundary integral equation
Ti(S,t ) - To=G(~(S,t)- Vot ) L i + -CP x
(I L
o
dt' [4~DT(t_t,)]l/2
s;
Lx o
dS'Vn(S' ,
t')-
(4)
which was derived previously [10, 11] and shown to provide a quantitatively accurate approximation of the full boundary integral equation of the non-local symmetric model in the small Lewis number limit (Dc/ D T "* 1 ). In this model, both phases are assumed to have the same diffusivity and specific heat. Here ~j(S,t) denotes the vertical displacement of the interface parallel to the temperature gradient, vn(S, t) the normal velocity of the interface, S the arclength coordinate along the interface, L x the lateral dimension of the system, V0 the isotherm velocity which is the main control parameter of our model, and TO the steadystate temperature of the planar interface at velocity V0. The solute field is described using a boundary-layer approximation introduced originally by Ben-Jacob et
,
=
,
,
dS 7¢(S,t)vn(S ,t)
(8)
o
where r(S, r) denotes the interface curvature.
3. Results
3.1. Planar dynamics, latent heat, and period doubling All the calculations presented here were performed for an A1-2wt.%Fe alloy using material parameters defined in refs. 10 and 11. This choice was made to allow a direct comparison with experiment [5] and the CGZK model [6]. The results obtained for the nonlinear oscillations of the planar interface were found to be in good quantitative agreement with those previously obtained using a fully non-local model [10, 11] (which indicates that using a boundary-layer approximation for the present purposes should be adequate). We emphasize that the dynamic transients and nonlinear oscillations of the planar interface, obtained by artificially constraining the latter to remain planar, only provide an indication of the general form of the overall banding cycle in the temperature-velocity (TV) plane since they do not take into account the cellular morphology forming during the low velocity portion of the cycle. Three main conclusions can be drawn from numerical studies of planar dynamics [10, 11]. As we shall see below, at least the first two carry over to nonplanar dynamics and, hence, to banding. (i) Without latent heat diffusion (which corresponds to setting L = 0 in eqn. (4)), banding should be relatively well described by the CGZK model as indicated by the dynamic transient of the planar interface shown
A. Sarkissian, A. Karma
/
in Fig. 1. The planar interface leaves the unstable middle branch of the Tss(V) curve and follows closely the cycle 1 - 2 - 3 - 4 of CGZK. The portion 4-1 of the cycle is not described by our numerics but is expected to follow a cellular-dendritic steady-state branch. (Details of the instability of the up-sloping part of the planar steady-state branch, in the absence of latent heat diffusion can be found in ref. 13. For details in the presence of latent heat diffusion see ref. 11.) (ii) With latent heat diffusion, the dynamic transient of the planar interface is altered significantly (Fig. 1). This difference can be understood [10, 11] by rewriting eqn. (4) in the form T~ - To = A TG + A T L
155
Formation of the banded structure 910
_
'
'
......
I
':, (a)
. . . . . . . .
I
. . . . . . . .
<
m/s
~ 906
1
E 902
891 0.1
,,
11
. . . .
1.0
10.0
100.0
Velocity (m/s)
(9) . . . .
During the rapid acceleration of the interface, the change in interface temperature due to the interface advancing relative to the moving isotherm is easily seen to scale as A TG - GDc/Vo, while the change due to the extra amount of latent heat produced at the interface scales as A T L - L / c p ( D c / D T ) 1/2. For material parameters characteristic of metallic alloys and growth conditions where banding occurs, A TL>>ATG, which implies that interfacial temperature changes are dominated by latent heat diffusion. An important consequence of this effect is that the total band spacing should depend only weakly on the strength of the temperature gradient, in contrast to the case without latent heat where the latter is predicted to be inversely proportional to G [6]. (iii) The oscillatory dynamics of the planar interface in the presence of latent heat diffusion is richer than anticipated and exhibits a classic period doubling sequence of bifurcation to chaos with decreasing V0. An example of chaotic oscillation for V0 = 1 m s-~ is shown in Fig. 2. An interesting analogy exists between these oscillations and thermal oscillations observed during the rapid crystallization of metastable amorphous films which exhibit a similar bifurcation sequence [14]. 3.2. Non-planar dynamics, cells, "optical growth fronts", and banding
Results of simulation for non-planar dynamics are shown in Figs. 3, 4 and 5 which correspond to the same run with V0 = 1 m s- ~. Figure 3 shows a comparison of the initial part of the planar (P) cycle corresponding to Fig. 2 and the non-planar (NP) cycle obtained by starting the interface with a small sinusoidal perturbation. Figure 4 provides a direct visualization of the sequence of interface deformation which leads to the alteration of growth morphology. The concentration profile corresponding to one of the interfaces in this sequence is shown in Fig. 5. Two principal features stand out
I ' ' ' ' l
. . . .
(b)
lo.o
o >
1.0
0
Vo=1.0 m/s 0.1
,,, . . . . . . . . . . . 0
, .... , ....
4 8 I n t e r f a c e Position ( # m )
2
Fig. 2. Characteristic chaotic non-linear oscillation of the planar interface displayed (a) in the TI/plane and (b) with the instantaneous interface velocity plotted as a function of the instantaneous interface position.
,0If '
. . . . .
I
= 9ool-/t,"
.
.
.
.
.
.
.
.
I
B
>,/',
895
890 1
10 Velocity (m/s)
Fig. 3. Comparison of planar (P) and non-planar (NP) cycles for I/0 = 1.0 m s- 1.The two cycles were started from the same initial condition, a small sinusoidal perturbation being added to the interface in the NP case. For the NP case, the tip temperature and velocity are used to draw the cycle.
156
A. Sarkissian, A. Karma
3 4 0 ~
0
' i ' '
20
~
I
C
40 60 x (nm)
80
Fig. 4. Sequence of interfacial deformation corresponding to the NP-cycle of Fig. 3. The letters A-C indicate the same stage of the cycle in Fig. 3 with tA= 323 ns, tB= 392 ns, and tc = 404 ns. The interfaces are equally spaced in time with At = 8.62 ns between A and B, and At = 1.33 ns between B and C. The light band spacing, corresponding to the total interface displacement up to the formation of the cellular morphology is about 250 nm, while the dark band spacing, corresponding to the subsequent interface displacement up to the collision of the optical fronts is about 80 nm. These predictions agree reasonably well with observed spacings for this alloy (A1-2wt.%Fe) [5] at a growth rate of lms-I
15
'
I
'
'
'
I
'
'
'
I
'
'
'
I
.~_
Formation of the banded structure
P-cycle. This part of the cycle is conceptually similar to the CGZK model with the main difference that the non-planar morphology is cellular instead of dendritic. This difference is due indirectly to the effect of latent heat diffusion on the cycles. (ii) After intersecting this curve, the interface starts accelerating towards the high-velocity branch of the planar Tss(V) steady-state curve (see section A-B in Figs. 3 and 4). The feature, at first surprising, is that although the interface is accelerating towards a morphologically stable domain, its deformation increases rather than decreases. This effect can be understood by noting that lateral diffusion is inefficient for restabilizing the interface on the short timescale where the accelerating takes place. In particular, the latter scales as Dc/Vo 2 while the timescale necessary for restabilization by diffusion scales as 22/Dc ,>Dc/ V02, where 2 is the cell spacing. Consequently, the interface is restabilized by a different mechanism which involves the lateral propagation of high-velocity collision-controlled fronts, initiated at the cell tips, into diffusion-controlled low velocity regions of the interface (see section B-C in Figs. 3 and 4). In the short time interval between their formation and their collision, these fronts propagate as circular growth fronts which closely resemble the circular wave fronts emanating from a point source in geometrical optics (the name "optical growth fronts" was inspired here by this analogy). Since the normal velocity is nearly constant along these fronts, and k is close to unity, the dynamics of the interface is governed essentially by the time rate of change of its radius of curvature Ro(t) which is given by the interface boundary condition:
dRo(t) dt "~RTM2
o
8 5
TM-- Ti(t)+mC~
r i).
R~t
(10)
0
o
0
20
40 60 x (nm)
80
Fig. 5. Interfacial concentration CL as a function of position for the interface profile corresponding to the arrow in Fig. 4. Collision-controlled and diffusion-controlled regions of the interface are connected by sharp spatial variations in composition.
from the present results, the second novel and somewhat unexpected. (i) The P and NP cycles follow initially the same trajectory but start to diverge when a cellular morphology is formed during the low velocity portion of the cycle (Figs. 3 and 4). As a consequence, the NP cycle intersects the unstable branch of the planar Tss(V) steady-state curve at a higher temperature than the
It is worth noting that optical growth fronts generate an excess amount of latent heat by increasing interfacial area. Thus, although only present during a very short portion of the overall banding cycle, they affect the whole cycle by subsequent diffusion of this excess heat. What happens after these fronts collide? Unavoidably, the interface will pinch-off, leaving behind impurity (Fe) rich liquid droplets, and the newly formed high-curvature regions will be smoothed out by surface tension on a very short timescale, thereby causing the interface to be partially restabilized. Subsequently, the interface will start slowing down again and form a cellular morphology after which the whole cycle will be repeated. The present results provide a detailed view of the sequence of growth morphology (summarized in Fig. 6) which should give rise to banding. It should be noted, however, that our present numerics only describe the non-linear evolution of a
A. Sarkissian, A. Karma
/
(iv)
~ ~ ~ ~ ¢ ~ ( i i i )
Formation of the banded structure
157
morphology giving rise to banding is mediated by the rapid lateral spreading of collision-controlled optical growth fronts, emanating at the cell tips, into diffusioncontrolled regions of the interface. Finally, predicted light and dark band spacings (Fig. 4) are in reasonably good agreement with experiment for the AI-2wt.%Fe alloy studied here.
(ii) i
~
I
I
-
i
i
i
Acknowledgment
(i)
Fig. 6. Schematic summary of the banding cycle consisting of four main stages. In stage (i) the interface is completely diffusion controlled and a cellular structure is formed as a consequence of morphological instability. In stage (ii) the cell tips start accelerating rapidly and emanate collision-controlled optical growth fronts which propagate into diffusion-controlled regions. The borders between these two regions are indicated by vertical dashed lines. In stage (iii), which is expected on theoretical grounds but not simulated, the optical fronts collide and the growth of the interface becomes predominantly collision controlled. In stage (iv) the interface is nearly completely restabilized after rapid smoothing of the cusps by surface tension; stage (i) is then repeated.
spatially periodic deformation of the interface. In experiment, it is likely that optical growth fronts do not trigger cell tips synchronously, but that the lateral spreading of a single front is responsible for the formation of a light band. In conclusion, we have performed a numerical study of the non-linear dynamics of the solid-liquid interface in the vicinity of the absolute stability limit using a model of rapid solidification which includes both solute and heat diffusion, and non-equilibrium effects. Solute trapping gives rise to relaxation oscillations of the solidification front which are strongly influenced by latent heat diffusion. T h e alternation of growth
This research was supported by US D O E grant No. D E - F G 0 2 - 9 2 E R 4 5 4 7 1 and has benefited from supercomputer time allocation at NERSC.
References 1 G. V. S. Sastry and C. Suryanarayana, Mater. Sci. Eng., 47 (1981)193. 2 W. Kurz and R. Trivedi, Acta MetalL, 38 (1990) 1. 3 W. J. Boettinger, D. Shechtman, R. J. Schaefer and F. S. Biancaniello, Metall. Trans. A, 15 (1984) 55. 4 M. Zimmermann, M. Carrard and W. Kurz, Acta Metall., 37 (1989) 3305. 5 M. Gremaud, M. Carrard and W. Kurz, Acta Metall., 39 (1991)1431. 6 M. Carrard, M. Gremaud, M. Zimmermann and W. Kurz, Acta Metall., 40 (1992) 983. 7 W. J. Boettinger and S. R. Coriell, in R. P. Sahm, H. Jones and C. M. Adam (eds.), Science and Technology of the Undercooled Melt, NATO, Washington, DC, 1986. 8 M.J. Aziz and W. J. Boettinger, Acta Metall., 42 (1994) 527. 9 M.J. Aziz and T. Kaplan, Acta Metall., 36 (1988) 2335. 10 A. Karma and A. Sarkissian, Phys. Rev. Lett., 27 (1992) 2616. 11 A. Karma and A. Sarkissian, Phys. Rev. E, 47 (1993) 513. 12 E. Ben-Jacob, N. Goldenfeld, J. S. Langer and G. Schon, Phys. Rev. A, 29 (1984) 330. 13 G. J. Merchant and S. H. Davis, Acta Metall., 38 (1990) 2683. 14 W. van Saarloos and J. D. Weeks, Phys. Rev. Lea., 51 (1983) 1046; Physica D, 12 (1984) 279.
153
On the formation of the banded structure in rapid solidification Armand Sarkissian and Alain Karma
Physics Department, Northeastern University, Boston, MA 02115 (USA)
Abstract Rapid solidification experiments on metallic alloys over the last decade have provided widespread observations of a novel banded structure. We present the results of numerical and analytical studies of the non-linear dynamics of the solidifying interface which provide insights into the mechanism of formation of this structure. These studies are based on a model of directional solidification which includes non-equilibrium effects and consists of a one-dimensional boundary integral equation for the temperature field combined with a boundary-layer approximation for the solute field. Three main ingredients of the banding mechanism emerge from these studies: the presence of non-linear oscillations of the solidification rate driven by solute trapping, the dramatic effect of latent heat diffusion on the character of these oscillations, and the propagation of"optical growth fronts" connecting collision-controlledand diffusion-controlled inteffacial regions.
1. Introduction
Perhaps one of the most novel and surprising observations which has emerged from rapid solidification experiments over the last decade is that of the socalled banded structure. First observed as an isolated phenomenon [1], this structure has subsequently been found to occur in a broad class of metallic alloys solidified under growth conditions which approach absolute stability (for a complete list of references on banding see ref. 2). The most puzzling aspect of the banded structure, from a theoretical standpoint, is the presence of alternating light and dark bands which lie parallel to the solidification front. Detailed experimental studies [2-5] have revealed that the dark bands have a precipitate structure, either cellular-dendritic or eutectic, which is similar to the precipitate structure observed at lower growth rate just preceding the appearance of the banded structure. In contrast, the light bands have a microsegregation-free structure similar to that observed at the larger growth rate just following the disappearance of this structure. An attractive explanation for the origin of banding was set forth by Carard, Gremaud, Zimmermann, and Kurz (CGZK) [6]. These authors remarked that, owing to non-equilibrium effects, the Tss(V) curve relating the steady-state temperature and growth rate of the solidification front has an S-shape characteristic of bistable systems when the growth rate is in the vicinity of the absolute stability limit (Fig. 1). On the basis of this observation, they proposed a classic "relaxationoscillator" picture of banding in which periodic changes in growth rate cause the solidification front to 0921-5093/94/$7.00 SSI)I 0921-5093(93)04528-P
make abrupt transitions between two steady-state branches of this S-shaped Tss(V) curve: a low-velocity branch corresponding to a cellular-dendritic growth morphology, and a high-velocity branch corresponding to a planar morphology (thereby accounting for the observed alternation of structure). This model has provided a valuable first step towards an understanding of banding but, at the same time, set forth the difficult task of elucidating whether the basic dynamical equations of solidification, including phenomenological nonequilibrium corrections, actually produce such nonlinear oscillations.
91 0
,,
Vo=1.0 m/s
/,,t
906 =
/:'l
~O .
F I"'""',
,=o
\
..................' > ............
,2
E 902
i
8 9 8
,
0.1
,,:
....
,
. . . . . . . .
,
1.0 10.0 Velocity (m/s)
,,~_ 100.0
Fig. 1. Comparison of the CGZK banding cycle ( 1-2-3-4-1 ) and the dynamic transients of the planar interface without (L = 0) and with (L ¢ 0)latent heat diffusion for V0 = 1.0 m s J. © 1994 - Elsevier Sequoia. All rights reserved
A. Sarkissian, A. Karma / Formationof the banded structure
154 2. Model
al. [12] to study dendritic growth. In the present context, this approximation leads to the set of equations:
To investigate this question we study a model of rapid directional solidification which takes into account both heat and solute diffusion and nonequilibrium corrections. The latter are included using a thermodynamically consistent interface condition
dh =vn(C~o-kCL)-vn~ch+Dc
L
1 -- k e
(2)
J
developed by Boettinger and Coriell [7] (this condition also corresponds to the special case of solute trapping without solute drag of a more general interface condition developed recently [8]) and a velocity dependent form of partition coefficient
k
-
h
(1)
v*
1 - k(1 -ln[k/k~])l m =In e
(c -c f
CL(1 -- k)vn = Dc
RTM 2 V n Ti= TM + m Q - - F r - - -
ke + Vn/Vo 1 + Vn/V d
CL-C~o OS]
(5) (6)
where the variables h(S, t) denotes the total impurity content in the solute boundary-layer ahead of the interface and d/dtln denotes the time derivative along the normal to the interface. The dynamics of the interface in our model is then completely specified by eqns. (1)-(6) together with the usual interface kinematic relations = --
K2-.{-
(7)
Vn
(3) dS
developed by Aziz and Kaplan [9]. We focus here on a computationally tractable description of the diffusion fields which allows study of both planar and nonplanar interface dynamics, thereby extending our previous study of planar dynamics (only) that was based on a boundary integral formulation of fully nonlocal models [10, 11]. The temperature field is represented here by the one-dimensional boundary integral equation
Ti(S,t ) - To=G(~(S,t)- Vot ) L i + -CP x
(I L
o
dt' [4~DT(t_t,)]l/2
s;
Lx o
dS'Vn(S' ,
t')-
(4)
which was derived previously [10, 11] and shown to provide a quantitatively accurate approximation of the full boundary integral equation of the non-local symmetric model in the small Lewis number limit (Dc/ D T "* 1 ). In this model, both phases are assumed to have the same diffusivity and specific heat. Here ~j(S,t) denotes the vertical displacement of the interface parallel to the temperature gradient, vn(S, t) the normal velocity of the interface, S the arclength coordinate along the interface, L x the lateral dimension of the system, V0 the isotherm velocity which is the main control parameter of our model, and TO the steadystate temperature of the planar interface at velocity V0. The solute field is described using a boundary-layer approximation introduced originally by Ben-Jacob et
,
=
,
,
dS 7¢(S,t)vn(S ,t)
(8)
o
where r(S, r) denotes the interface curvature.
3. Results
3.1. Planar dynamics, latent heat, and period doubling All the calculations presented here were performed for an A1-2wt.%Fe alloy using material parameters defined in refs. 10 and 11. This choice was made to allow a direct comparison with experiment [5] and the CGZK model [6]. The results obtained for the nonlinear oscillations of the planar interface were found to be in good quantitative agreement with those previously obtained using a fully non-local model [10, 11] (which indicates that using a boundary-layer approximation for the present purposes should be adequate). We emphasize that the dynamic transients and nonlinear oscillations of the planar interface, obtained by artificially constraining the latter to remain planar, only provide an indication of the general form of the overall banding cycle in the temperature-velocity (TV) plane since they do not take into account the cellular morphology forming during the low velocity portion of the cycle. Three main conclusions can be drawn from numerical studies of planar dynamics [10, 11]. As we shall see below, at least the first two carry over to nonplanar dynamics and, hence, to banding. (i) Without latent heat diffusion (which corresponds to setting L = 0 in eqn. (4)), banding should be relatively well described by the CGZK model as indicated by the dynamic transient of the planar interface shown
A. Sarkissian, A. Karma
/
in Fig. 1. The planar interface leaves the unstable middle branch of the Tss(V) curve and follows closely the cycle 1 - 2 - 3 - 4 of CGZK. The portion 4-1 of the cycle is not described by our numerics but is expected to follow a cellular-dendritic steady-state branch. (Details of the instability of the up-sloping part of the planar steady-state branch, in the absence of latent heat diffusion can be found in ref. 13. For details in the presence of latent heat diffusion see ref. 11.) (ii) With latent heat diffusion, the dynamic transient of the planar interface is altered significantly (Fig. 1). This difference can be understood [10, 11] by rewriting eqn. (4) in the form T~ - To = A TG + A T L
155
Formation of the banded structure 910
_
'
'
......
I
':, (a)
. . . . . . . .
I
. . . . . . . .
<
m/s
~ 906
1
E 902
891 0.1
,,
11
. . . .
1.0
10.0
100.0
Velocity (m/s)
(9) . . . .
During the rapid acceleration of the interface, the change in interface temperature due to the interface advancing relative to the moving isotherm is easily seen to scale as A TG - GDc/Vo, while the change due to the extra amount of latent heat produced at the interface scales as A T L - L / c p ( D c / D T ) 1/2. For material parameters characteristic of metallic alloys and growth conditions where banding occurs, A TL>>ATG, which implies that interfacial temperature changes are dominated by latent heat diffusion. An important consequence of this effect is that the total band spacing should depend only weakly on the strength of the temperature gradient, in contrast to the case without latent heat where the latter is predicted to be inversely proportional to G [6]. (iii) The oscillatory dynamics of the planar interface in the presence of latent heat diffusion is richer than anticipated and exhibits a classic period doubling sequence of bifurcation to chaos with decreasing V0. An example of chaotic oscillation for V0 = 1 m s-~ is shown in Fig. 2. An interesting analogy exists between these oscillations and thermal oscillations observed during the rapid crystallization of metastable amorphous films which exhibit a similar bifurcation sequence [14]. 3.2. Non-planar dynamics, cells, "optical growth fronts", and banding
Results of simulation for non-planar dynamics are shown in Figs. 3, 4 and 5 which correspond to the same run with V0 = 1 m s- ~. Figure 3 shows a comparison of the initial part of the planar (P) cycle corresponding to Fig. 2 and the non-planar (NP) cycle obtained by starting the interface with a small sinusoidal perturbation. Figure 4 provides a direct visualization of the sequence of interface deformation which leads to the alteration of growth morphology. The concentration profile corresponding to one of the interfaces in this sequence is shown in Fig. 5. Two principal features stand out
I ' ' ' ' l
. . . .
(b)
lo.o
o >
1.0
0
Vo=1.0 m/s 0.1
,,, . . . . . . . . . . . 0
, .... , ....
4 8 I n t e r f a c e Position ( # m )
2
Fig. 2. Characteristic chaotic non-linear oscillation of the planar interface displayed (a) in the TI/plane and (b) with the instantaneous interface velocity plotted as a function of the instantaneous interface position.
,0If '
. . . . .
I
= 9ool-/t,"
.
.
.
.
.
.
.
.
I
B
>,/',
895
890 1
10 Velocity (m/s)
Fig. 3. Comparison of planar (P) and non-planar (NP) cycles for I/0 = 1.0 m s- 1.The two cycles were started from the same initial condition, a small sinusoidal perturbation being added to the interface in the NP case. For the NP case, the tip temperature and velocity are used to draw the cycle.
156
A. Sarkissian, A. Karma
3 4 0 ~
0
' i ' '
20
~
I
C
40 60 x (nm)
80
Fig. 4. Sequence of interfacial deformation corresponding to the NP-cycle of Fig. 3. The letters A-C indicate the same stage of the cycle in Fig. 3 with tA= 323 ns, tB= 392 ns, and tc = 404 ns. The interfaces are equally spaced in time with At = 8.62 ns between A and B, and At = 1.33 ns between B and C. The light band spacing, corresponding to the total interface displacement up to the formation of the cellular morphology is about 250 nm, while the dark band spacing, corresponding to the subsequent interface displacement up to the collision of the optical fronts is about 80 nm. These predictions agree reasonably well with observed spacings for this alloy (A1-2wt.%Fe) [5] at a growth rate of lms-I
15
'
I
'
'
'
I
'
'
'
I
'
'
'
I
.~_
Formation of the banded structure
P-cycle. This part of the cycle is conceptually similar to the CGZK model with the main difference that the non-planar morphology is cellular instead of dendritic. This difference is due indirectly to the effect of latent heat diffusion on the cycles. (ii) After intersecting this curve, the interface starts accelerating towards the high-velocity branch of the planar Tss(V) steady-state curve (see section A-B in Figs. 3 and 4). The feature, at first surprising, is that although the interface is accelerating towards a morphologically stable domain, its deformation increases rather than decreases. This effect can be understood by noting that lateral diffusion is inefficient for restabilizing the interface on the short timescale where the accelerating takes place. In particular, the latter scales as Dc/Vo 2 while the timescale necessary for restabilization by diffusion scales as 22/Dc ,>Dc/ V02, where 2 is the cell spacing. Consequently, the interface is restabilized by a different mechanism which involves the lateral propagation of high-velocity collision-controlled fronts, initiated at the cell tips, into diffusion-controlled low velocity regions of the interface (see section B-C in Figs. 3 and 4). In the short time interval between their formation and their collision, these fronts propagate as circular growth fronts which closely resemble the circular wave fronts emanating from a point source in geometrical optics (the name "optical growth fronts" was inspired here by this analogy). Since the normal velocity is nearly constant along these fronts, and k is close to unity, the dynamics of the interface is governed essentially by the time rate of change of its radius of curvature Ro(t) which is given by the interface boundary condition:
dRo(t) dt "~RTM2
o
8 5
TM-- Ti(t)+mC~
r i).
R~t
(10)
0
o
0
20
40 60 x (nm)
80
Fig. 5. Interfacial concentration CL as a function of position for the interface profile corresponding to the arrow in Fig. 4. Collision-controlled and diffusion-controlled regions of the interface are connected by sharp spatial variations in composition.
from the present results, the second novel and somewhat unexpected. (i) The P and NP cycles follow initially the same trajectory but start to diverge when a cellular morphology is formed during the low velocity portion of the cycle (Figs. 3 and 4). As a consequence, the NP cycle intersects the unstable branch of the planar Tss(V) steady-state curve at a higher temperature than the
It is worth noting that optical growth fronts generate an excess amount of latent heat by increasing interfacial area. Thus, although only present during a very short portion of the overall banding cycle, they affect the whole cycle by subsequent diffusion of this excess heat. What happens after these fronts collide? Unavoidably, the interface will pinch-off, leaving behind impurity (Fe) rich liquid droplets, and the newly formed high-curvature regions will be smoothed out by surface tension on a very short timescale, thereby causing the interface to be partially restabilized. Subsequently, the interface will start slowing down again and form a cellular morphology after which the whole cycle will be repeated. The present results provide a detailed view of the sequence of growth morphology (summarized in Fig. 6) which should give rise to banding. It should be noted, however, that our present numerics only describe the non-linear evolution of a
A. Sarkissian, A. Karma
/
(iv)
~ ~ ~ ~ ¢ ~ ( i i i )
Formation of the banded structure
157
morphology giving rise to banding is mediated by the rapid lateral spreading of collision-controlled optical growth fronts, emanating at the cell tips, into diffusioncontrolled regions of the interface. Finally, predicted light and dark band spacings (Fig. 4) are in reasonably good agreement with experiment for the AI-2wt.%Fe alloy studied here.
(ii) i
~
I
I
-
i
i
i
Acknowledgment
(i)
Fig. 6. Schematic summary of the banding cycle consisting of four main stages. In stage (i) the interface is completely diffusion controlled and a cellular structure is formed as a consequence of morphological instability. In stage (ii) the cell tips start accelerating rapidly and emanate collision-controlled optical growth fronts which propagate into diffusion-controlled regions. The borders between these two regions are indicated by vertical dashed lines. In stage (iii), which is expected on theoretical grounds but not simulated, the optical fronts collide and the growth of the interface becomes predominantly collision controlled. In stage (iv) the interface is nearly completely restabilized after rapid smoothing of the cusps by surface tension; stage (i) is then repeated.
spatially periodic deformation of the interface. In experiment, it is likely that optical growth fronts do not trigger cell tips synchronously, but that the lateral spreading of a single front is responsible for the formation of a light band. In conclusion, we have performed a numerical study of the non-linear dynamics of the solid-liquid interface in the vicinity of the absolute stability limit using a model of rapid solidification which includes both solute and heat diffusion, and non-equilibrium effects. Solute trapping gives rise to relaxation oscillations of the solidification front which are strongly influenced by latent heat diffusion. T h e alternation of growth
This research was supported by US D O E grant No. D E - F G 0 2 - 9 2 E R 4 5 4 7 1 and has benefited from supercomputer time allocation at NERSC.
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