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Morphological instability in a float zone
Journal of Crystal Growth 100 (1990) 31—50 North-Holland
31
MORPHOLOGICAL INSTABILITY IN A FLOAT ZONE Laura B. HUMPHREYS, John A. HEMINGER and Gerald W. YOUNG Department of Mathematical Sciences, University of Akron, Akron, Ohio 44325, USA
Received 12 January 1989; manuscript received in final form 7 September 1989
Morphological instability in a crystal sheet, float-zone geometry for which the width and thickness of the sheet are much larger than the height of the liquid zone is examined. The heat transfer across the boundaries of the system is assumed to be large so that the temperature profiles in the melt and solids are identical with the heater profile. A piecewise linear profile, applicable to systems with slow velocities of solidification and for which the thermal diffusivities are larger than the solutal diffusivities. is imposed. The height of the zone is calculated consistent with these assumptions. Linear and weakly nonlinear analyses in the limit of small segregation coefficient determine that for shorter zone configurations the system is less susceptible to long wavelength morphological instabilities. Solute redistribution leading to flatter axial concentration profiles is the proposed mechanism for the increased stability as compared to unidirectional solidification. A nonlinear evolution equation governing the dynamics of the solidification front and which includes surface free energy variation with interfacial concentration is also developed. Bifurcation analysis and numerical simulations of this equation predict transitions from planar to three-dimensional hexagonal node structures or two-dimensional band structures.
1. Infroduction When a heat source is passed along a crystal rod, a travelling liquid zone is created. The zone is shaped by the melting interface at the feed crystal, the melt-gas free surface, and the solidifying interface at the growing crystal. This process, shown in fig 1, is known as the floating zone technique. This method is suggested for use in space for crystal growth of electronic materials, One concern of crystal growers is the shape and stability of the float zone, both of which affect the quality of the grown crystal. Various theoretical studies of the float zone process have been performed. Plateau [1], Rayleigh [2], and Heywang and Ziegler [3] determine that for static stability of a cylindrtcal zone melt, the maximum zone height, L, must be less than 27rR, where 2R is the zone thickness. The melt motion is exammed, for examplc, by Chang and Wilcox [4,5], Clark and Wilcox [61, Sen and Davis [7], and Lai, Ostrach, and Kamotani [8]. Coriell, Hardy, and Cordes [91investigate instability in the shape of the zone as a function of the contact angle. The most realistic 0022-0248/90/803.50 © Elsevier Science Publishers B.V. (North-Holland)
model of the float zone process is that of Duranceau and Brown [TO],who determine the interfacial positions consistent with a conduction dominated heat transfer model. Young and Chait [11] use a coupled analytical/numerical procedure to determine the zone shape in the limit R << L. In this work we examine morphological instability of the freezing interface of the float zone for a sheet geometry as shown in fig. 2. This interface ultimately determines the concentration profile in
\
Slope
\
G~j
_~._
I
J
“
Temp
LOSS
/
~r’ysi~iØ/7/~/~ ‘///////////
//A
I
Fig. 1. Float zone with a heater source of magnitude TH and linear temperature profiles applied in the feed and growing crystal regions.
32
L. II, Jiimiphreii ci a!.
~
S/ar~thiilagua! instainli i in float zone
~
the effect of buoyancy in the system. Novick— (‘ohen and Sivashinsky [161 include latenl heat in the evolution equation. Young, Davis. and l3ratt— kus [171 analyze anisotropic interface kinetics. The first to analyze morphological instability considering a finite melt height is Young and Davis [18]: however, their work does not consider a float—zone process. hut rather a melt hounded by an upper solid wall. All these analyses have the disadvantage that the evolution equations are not Ufli— form for large time. Kurtze [1~1and Hvman et al. [201 suggest means of stabilizing the equalion by including higher order terms [191. or retaining full expressions for the interfacial curvature and unex-
Fig. 2. Schematic of a float zone in a crv~talsheet.
the solid which in turn determines the material and electronic properties of the crystal. It is well known that morphological instability occurs when the temperature gradient in the liquid is shallow leading to constitutional supercooling. A shallow gradient is desirable, though, to prevent buckling and dislocations in the solid crystal which result from nonlinearities in the temperature profile. Hence, there is a need to optimize the temperature profile so as to minimize the morphological and structural instabilities. It is our purpose to establish criterion determining the onset of interfacial instability in a float zone system. The majority of morphological instability analyses have been done for a directional solidification geometry no upper boundary - rather than for the float zone. Mullins and Sekerka [121 complete the first analysis of morphological inslahility using a linear analysis. Woilkind and Segel [13] use bifurcation theory to conduct a weakly nonlinear analysis. Sivashinsky [14] develops a nonlinear evolution equation governing morphological instability at the interface in the limit of small segregation coefficient, k. and 0(1) surface free energy. This evolution equation retains much of the physics of the problem, hut involves the analysis of a single equation as opposed to the more complicated, coupled system. Il has also been used by Young and Davis [15] to examine
panded exponential concentration profilcs [20] Brattkus and Davis [21] develop an evolution equation in the limit of absolute stability where the surface free energy is large. For this condition with k near unity. they predict that Ihe planar interface bifurcates to a stable three-dimensional hexagonal node solution. Riley and Davis [22] tie together the Sivashinsky and Brattkus and Davis limits by developing an evolution equation for Ic ~ I and large surface free energy. Numerical integrations of the absolute stability limit evolution equation do stabilize to steady state solutions [211. In our approach to the float zone conliguration. we assume a rectangular geometry as shown in fig. 2. with W>~R >~L. and neglect the presence of the melt-gas interface and crystal rod boundaries. In effect, our liquid zone is finite in height and infinite in width and thickness. The height of the zone is determined consistent with an imposed temperature profile in the melt and solid rods. The transfer of heat from the heater to the system. across the surface of the system is assumed to be large. i.e. the Biot number. B, = hR/k, >~ 1. where h, is the heat transfer coefficient and k, is the thermal conductivity in the feed rod, melt, or growing crystal. Hence, the imposed temperature profile in each region is identical to the heater temperature as a consequence of the large surface heat transfer and 11) approximation (infinite extent of the system in the lateral directions s and i). Further, we neglect any melt motion. Together with the “frozen temperature” approximation [231 discussed in section
LB. Humphrevs et al.
/
33
Morphological instability in float zone
3, these assumptions lead to nearly planar interfacial shapes and x, y-independent, linear temperature profiles as shown in figs. I and 2. This is clearly an extreme simplification of the process, neglecting the practical importance of radially (ydirection) distributed (parabolic) temperature profiles which lead to structural instabilities, However, it allows for calculation of the zone height consistent with the temperature gradients and incorporation of concentration boundary layers at the solid interface. In section 2 we formulate the problem. We find the basic state solution and introduce scalings in section 3 ; we analyze the linear stability of the system in section 4, looking for criterion for a long wavelength instability. We find that the float zone
tion of large R (applicable to thin sheets) one would require the solid/liquid frdnts in the v direction to be everywhere perpendicular to the direction of growth, i.e. the contact angle 7T/2. Boundary layer corrections due to the presence of the melt—gas interface are not considered here. These are discussed by Young and Chait [11]. The coordinate system translates upwards with the heating element at a constant speed V. with z 0 representing the position of the heater. We assume there is no density change upon solidification and neglect melt motion. The latter is rcasonable in a microgravity environment for systems with small Marangoni number so that surface tension driven flows are not present. We define L~ and L1~,to be the planar posi-
configuration is more stable than the directional solidification geometry. In section 5 we develop long-wave evolution equations in the Sivashinsky limit for the shapes of the solidifying and melting interfaces. We include the possibility that the surface free energy varies with interfacial concentration [24]. We discuss stability characteristics in section 6 including a bifurcation analysis of the evolution equation for the solidifying interface of the float zone together with numerical simulations of the same equation. The assumption of the surface free energy varying with concentration leads to supercritical bifurcation of two-dimensional band structures and a range of stable three-dimensional hexagonal node structures.
tions of the freezing and melting interfaces. respectively. T,, Tm, T, C.,, C.,,, and C are the temperatures and concentrations in the growing crystal, feed crystal, and melt, respectively. i~, D. Ks, and D5 denote the thermal and solutal diffusivities in the liquid and solid phases. The equations representing the thermal and concentration fields in the three zones are:
=
=
—
.
(i)
Solidifying crystal
K5(7,,, + 7/)
—
=
(—
~
—
L. + hf(x. t))
VT.,z + L,,.
(2.1)
where h1(x, t) denotes deviations from the planar solidifying front, The neglect of solute diffusion in the solidifying crystal while considering such in the feed crystal will be discussed shortly. (ii) Feed crystal (Lrn + h5~(x,t)
2. Formulation i~s(Tm.,.,+ ~
The floating zone technique is applied to a crystal as shown in fig. 2. For simplicity, we consider a two-dimensional coordinate system for the derivation and extend the results to three dimensions, We neglect the presence of the melt— gas interface and crystal rod boundaries by assuming W>> R >> L where W is the width of the feed and growing crystal sheets, L is the height of the liquid zone, and 2R is the thickness, This leads to a liquid zone which is infinite in the x and 3’ directions. Thus leading order expressions for the temperature and solute fields will be one-dimensional (functions of z only). To relax the assump-
D5 ( Cm,, + C~,- -)
V7~~+ T5,,.
—
=
=
—
VC’m - +
(2.2)
cm,,
(2.3)
-
where hrn(X, t) denotes deviations from the planar melting front, (ni) Melt (—L1
+ h~(x, t) <: < L
+h Ti
K(T.,., + T.)
=
—
VT + T,.
(2.4)
D(C+C~)=—VC+C. --
(2.5)
-
At the freezing interface z — L1 assume continuity of temperature =
T’
1/
(x t)) 111
+ hf(x. I)
we
(2.6a)
34
L. B. Huniphrer.s et a!.
/1
Morphological instuhilitr in I/oar zoo,’
and thermodynamic equilibrium T = mC + Tv
1 I + yIi1,.,/~( I + h~,)3
~]
(2.ob)
Here T51 is the pure substance melting point. y is the surface free energy. m is the slope of the liquidus line and £° is the latent heat per unit volume. y is taken to be isotropic hut we allow the possibility that it varies with interfacial concentration [241 by considering
J -
L .
(2.6hh) I
where y~has units of energy per unit area and is constant, ~.t is constant, and c0 is the uniform concentration of the feed crystal. There is also conservation of solute
—
(hf,+ V)C(k~ 1)
IT
=D(C—C.,h~,)
(2.6c)
-
..
.
Fig. 3. Basic state concentration and temperature profiles in
where k is the segregation coefficient such that (/=kC.
the feed crystal, melt, and growing crystal regions.
(2.6d)
and conservation of energy k5(T
—
T h~ )
—
k1 (I-
—
Tip, ‘
‘
)
=~(v+h,. (7 —.
)
c
At the melting interface = L + it (x, 1). the conditions are similar. Eqs. (2.6a) and (2.6b) are now T= 7/,
(2.7a) ~.2
T= mC + T~1[i + yh51,,~~(1+ h;,)
the growing The redistribution argument for ofthissolute approach is thatcrystal. significant in the feed rod upon melting is expected so that the product of the feed concentration gradient with the feed diffusion coefficient D5 (DSC,,,T in eq. (2.7c)) is comparable to this same product (DC) in the liquid phase at the melting interface. Here we expect C, ~ C, (flatter melt concentration profile away from the solidification front as shown in fig. 3) so that the effect of D5 ~< D is
L (2.7b)
cancelled. On the other hand, this will not he true at the solidification front. Here C. >~C so that the product D5C., is negligible in eq. (2.6c). This -
Conservation of solute requires (h
,
+ V )C( k
—
1)
=D(C— C.,h5,.,) C
=
kC
results in the inclusion of diffusion of solute in the —
D5(C~— C,,,.,h,,,J, (2.7c) (2 7d)
‘ii
Implicit in the neglect of a governing expression for concentration C, in the solidifying crystal and in the formulation of eqs. (2.6c) and (2.7c) is the assumption that solute diffusion in the feed (melting) crystal is significant but that it is negligible in
feed crystal at the melting front as discussed by Woodruff [25], Chen and Jackson [26j, and Wollkind and Raissi [27]. Finally, the equation of energy conservation at
the melting front is
k5(T~,,— 7/.,h51,) ~
V + h,,,,).
—
kL(T
—
T~h.,) (2.7e)
LB. Humphreys et a!.
/
Morphological instability in float zone
Conservation of mass also requires that far from the freezing and melting interfaces
Cm=Co
as
~
(2.8)
as
~
(2.9)
where C0 is the concentration of the feed crystal, which is assumed to be uniform. At these distances, we control the temperature gradient so that 7/
—s
G~
Tm~~GM
as as
z —~ ~
—
~,
(2.10) (2.11)
as in fig. 1 where G~and GM must satisfy the compatibility requirements to be discussed in eqs. (3.6a) and (3.6b).
35
fig. 1. The temperature profile in the melt will be represented as the piecewise linear profile (TUTu+Guz, )0
T(z)
(3.2a)
t),
=
TL i.,
=
(3.2b)
TL + GLz,
L1 + h1 (x, t)
where Ti,, G~,and GL are to be determined. We require the temperature to be continuous and equal to the maximum heater temperature, at 7/,
7/,
z0, Tu=TL=TH.
(3.3)
Furthermore, the temperature gradients in the liquid give rise to the jump condition TU(z
=
0~)— TL(z
0)
=
=
G~ GL <0. (3.4) —
3. Basic state and scaling
We assume that TH, G~,and GM are known. Hence the heater profile of fig. 1 is physically
We now seek a steady basic state solution to the governing equations which is x independent and for which h 0. The temperature profiles in the feed crystal, the liquid zone, and in the growing crystal are taken to be identical to the established heater temperature profile. This assumes a large Biot number approximation so that there is large heat transfer across the boundaries of the system. Further the temperature profiles of the system (or without loss of generality, the heater) are not arbitrary but are chosen consistent with the assumption that K and KS are much larger than the solutal diffusivities, D and D~,i.e. ic >> D. Under this assumption and consistent with the scales to be defined in eq. (3.19), the energy equations (2.1), (2.2), and (2.4) effectively reduce to the Laplace equation, L~T=0, in the solidifying and feed crystals and melt. These simplifications are relevant to systems where the velocity of solidification V is slow. The temperature fields will then be represented by piecewise combinations of theform
equivalent to linear profiles far from the melt region and a line heat source in the melt region as shown in fig. 2, and mathematically represented by the Dirac delta function. The assumption B,>> 1 transfers this source term from the heat transfer boundary condition at the liquid/gas interface to the bulk liquid. Integration of eq. (2.4) with respect to z in the neighborhood of z 0 (the line source location) will determine the magnitude of the source. The constants in eq. (3.2) are determined from (2.6e), (2.7e) and (3.3) so that the temperature of the melt zone is (k~ ~PV
=
T=T*+Gz,
=
(3.1)
where T * is to be determined and G corresponds to the different temperature gradients as shown in
=
\
=
T TL
kL GM — —
—
~~~__)Z L
+ TH,
(3.5a)
\
(ks 2?V k L G~— ~ L z +
)
7/.
(3.5b)
Using eq. (3.2), we have G G M~’ L
(3.6a)
GL=~-~.Gs-—--~-—, ~2~V
(3.6b)
which serve as compatibility requirements upon
L. B. Humphrei’.i
36
Ct a!.
/ Morphological
the temperature gradients and velocity of solidification. These requirements are the energy balanees across the melting and solidifying interfaces, respectively. The solid temperature profiles are determined using (2.6a), (2.7a), (2.10). and (2.11). We find =
T
=
=
TH
—
L1G1,
+ T11 +
—
(3 .7h)
LmGii.
(3.8)
The concentrations in the feed crystal and melt are found by solving (2.3) and (2.5). subject to (2.6c. d), (2.7c, d), and (2.9) so that
—C C,
+
+ 1—k k
D)( 6,
t.,~
1]
+ k—
}
(3.9a) (3-. 9i~\ /
e -_/Dgt,,+z)j ~.
FTc
— —
G5)
—
<
KL
-
—
C,
—
—
T=
— —
C,
(3.14)
from eq. (3.6h) and U1 > 0. Note that eqs. (3.11) and (3.12) indicate that the height of the liquid zone varies directly with the maximum heater temperature. T , and inversely with the temperaH
ture gradients. U1 and G1. Increasing the magnitude of the gradients leads to a shorter zone height L = L1 + iS,,,. (3.15) . .increasing the maximum temperature results while in a longer zone. Further as V increases in eq. (3.12), L decreases toward 0 so that the location of the heat source C, = 0) is closer to the melting interface, Only in the case V = 0. G~ = ---
(symmetric heater profile) is the source midway between the freezing and melting fronts. These
(3.10)
results agree with the findings in refs. [101 and [111. We define the solute gradient at the interface, G~.to he
This measures the strength of the line heat source so that the energy in the feed-melt-growing cyrstal system balances the energy transferred from the surroundings. The planar freezing front position L1 is determined by replacing T and C in (2.6h) with (3.5b) and (3.9b) to give
G
=
-~
D
C1(k —1) k
—
(3 16) ‘
-
so that our concentration equations (3.9a) and (3.9h) can be written
2(i (3.11) The planar melting front position is determined from the solution of Lf=(__~~~~+TH_TM)GLt.
=
(3.13)
1 )J.
~—
—
-
L,,
T11.
G5 >~V/k5
,
Upon integration of the energy equation (2.4) in the liquid in the neighborhood of = 0 and using eqs. (3.4) and (3.6a, b), one finds the value k5
7/
This is reasonable since the temperature of the heating element must be greater than the equilihrium melting point. We restrict
(tI),SLz)
e x [(I — k )e
K~(G~1
(3.9a). The above can easily he solved using fixed point methods. The basic state gives rise to a few restrictions. For L 1 >0. eq. (3.11) leads to
(3.7a)
c~
—
which results from eq. (2.7b) using (3.5a) and
mC’,/k +
G5(: + L1) + Gvt( L,,)
The concentration in the solidifying crystal is constant. C,
in.viahilitt’ in float zone
—e (I
~
(3.17)
C~+
c
=
C,,
+ kG~D (I
T~+ c 0m
x (i + 1
—
h
x [1 (1)5/., + 1))]
G~ (3.12)
—
e
(t~ /)iitz
/1],
(3.18)
We non-dimensionalize the system 2. by thescaling characteristic tinieD/V, for diffusion, concentrations lengths with time withand D/V
L. B. Humphrevs et a!.
/ Morphological instability in float zone
with G~D/V. The scaled variables, denoted by primes, are D
x
=
h
-~
=
x’,
z
‘~h’ V as’
D =
-~
=
h
z’, 2
V
-~--~‘
D =
-~
h
37
pendent of the solute concentration, and unaffected by perturbations at the fronts. This might be termed the frozen temperature approximation [23] as applied to the float-zone configuration. Without eq. (3.25a), the latent heat at the interadjust the linear profiles as shown in refs. faces is from significant and the temperature fields here
~,
‘
C C—
0 —i--
=
G.D C —~----C’, ~
C0 —
-i--
=
G.D —~C,,.
[16,18]. (3 19)
We substitute eq. (3.19) into the remaining governing equations (2.3), (2.5), (2.6b, c), (2.7b, c, d), and (2.9). If we drop the primes and denote
As a result of these scalings, we define the following non-dimensionalized groups: 2, (3.20) 1= (TMy/~mGC)(V/D)
(3.21) measuring the effects of the surface free energy and its variation with interfacial concentration,
V
V
and £~/ = ~‘ñ~Lm, the scaled governing equations are
(3.27)
C,—C=C.,.,+C.
(3.28)
~‘f
=
+ Cm==)
~(Cmxx
~Cm= +
=
ç,,,
(3.29)
.
MI=mGC/GL.
(3.22)
subject to the boundary conditions at z—5~
MU=mGC/GLJ,
(3.23)
Cm=0,
the morphological parameters indicating criterion for supercooling, and
at z
=
(3.30a) +
—~
h I
—
D
DS/D,
=
(3.24)
C
M~j1h
—
1+
a ratio of solid versus melt solutal diffusivity. In the stability analysis to follow, we make the fur-
3/2
0,
=
(3.30b)
(i +h~.,)
(1
+ h11)[1 + —~ + h
ther assumptions
(k
1)C]
—
=
C,
C.,h1.,,
—
(3.30c)
1h+1 ~V/kLGL<
(3.25a)
5=k~.
(3.25b)
k
C——e~+M + ~
(3 30d
~cC)hm..,x
3~2 From equations (3.6a, b), it follows that G~ G~ and G~ GM. =
=
(i (3.26)
This allows for replacing the temperatures in the
governing equations (2.1)—(2.11) with the linear basic state temperature fields shown together with the basic state concentration field in fig. 3. Henceforth, eq. (3.5a) will be identical to eq. (3.7b) and similarly, eq. (3.5b) and (3.7a) are equivalent. As a result of the small latent heat assumption, eq. (3.25a), ic >> D, and k 5 = kL, the heat (created/required) at the interfaces (freezing/ melting) may be neglected. The temperature field will thus be fixed, linearly as in fig. 3, inde-
Cm
+
‘
h~.,)
C,
=
(3.30e)
(I + hm,)[1 + (k =
—
h
~
—
ms
—
l)C]
kD~C
—
m
k
C
h
(3 30f
mx m v)~
S
The scaled basic state is -
C~ 1e ~
(~ —
—
M
h
0
I
hm
=
~
e
—
=
~
0.
[s”,.,, Z
3 31
—
exPi~
—
,
38
LB. Hurnphrevs ci al.
/
Morphological instability in float zone
4. Linear stability analysis We now allow disturbances to the basic state: h 1=0+eH, = 0 + H,,,.
(4.la)
hrn C
1
=
C,,,
[1
=
(4.lh)
e _ c~’+> + cC *
—
e~”~]
—
If we solve the linearized thermodynamic equation (4.Sh) for H we obtain 2F+ M~‘—1). (4.6) H= C’/(a
(4.1 c)
e©
‘~“~>
+
~
(4.ld)
We substitute eq. (4.1) into the scaled equations (3.28)_-(3.30), linearize with respect toe, and drop the asterisk from the perturbed quantities. Further, we assume for all dependent variables a separation of variables form qi(x, z, t) e°’e”’F(z), (4.2) =
where the growth rate, a, determines the stability and a represents the disturbance wave number.
The governing equations become C+C—(a+a)C=0.
(4.3) (4,4)
2~)c,, 0. =
+ c,,, (a + a These equations are subject to the following boundary conditions: at z
~l’M~—° 0, then H —s 0. This is consistent with eq. (3.22), since there would he no morphological instability when there is zero undercooling. Substituting eq. (4.6) into eq. (4.5c). mass conservation of solute at the solidifying front becomes 21+ +Mk )C 11t _1 +(k—l)C—C=0. (~
(4.7)
a
with the first term representing a perturbation in the concentration field due to the change in the shape of the interface. As M1 approaches I, the critical value for instability, we see in eq. (4.7) that the first term is proportional to 1/u. If we assume ~2 small, then this term would dominate the diffusional term C. We take a~1.
(4.8)
so that the wavelength 3’.
—
3’
2~D ~ ~‘
=
(4.9)
—~
satisfies C’,,,
0,
=
(4.5a) (4.10)
at z = C ± H(1 (a +
—
M~t)
k )H + (k
—
—
Fa2H
l)C
—
0,
=
C
=
0.
(4.sb)
implying long wavelength instabilities.
(4.5c)
tems with small k so that
From eq. (4.8) we rescale the problem for sys-
atz=2/
k=a2k, a=a2&
M~J’J+ F*a2H~,
C+ ~
(4.lla)
(4.llh)
(4.5d) C
C,,,
—
+ H,,,k1
—
~
i)
e_1~,c>~,,,) + D
0, (4.5e) =
(k—1)C—C±kDC,,,
with the first term of eq. (4.7) retained as an order one quantity near ML = 1. If we likewise solve eq. (4.5d) for H,, we obtain H,
C
=
=aH,,,[(k—l)e~’~~’~kI
(4.12)
M_e~’~’,_a)F*
-
and substituting into eq. (4.Se) we have (4.5f) where =
C’ —
—
~~(1
—
e~~)j.
(4.5g)
x [(1
+ —
Mt
—
—
Dt) e~©’C~)
+ D
=
0.
(4.13)
LB. Humphreys eta!.
/
Morphological instability in float zone
The governing equations are now
with B an arbitrary constant of integration and (4.14)
2(6+1)C=0.
C+C—a
DCm== + Cm=
—
a2(ä
+ ~i5)Cm
=
0,
B*=Be’_©
(4.15)
(1
F
xli
L
subject to conditions (4.5a) as z —s
39
e~1~m>+ (h_i)
~_1)
—
+
ML~1 e —
—
~m)
~
—
a2 F *
and
~,
1 ] (4.20)
At order a2 we obtain at z
=
a2(~+ff)C
2k)C—
a2F+M~—1
C+(1—a
at C
(4.16a)
C 1 =D* + Fe~—(6+ 1)B e’(l +z), Cmi~~F*exp[z/D]
c
Cm +
-
~
x
[(1
—
~
+
(1
—
=
=
0,
(4.16b)
—
—
a2F*
/
~ e~’’)
—1) e~’”’~
M~1 e~~”> —
—
k/D]
0. (4.16c)
—
a2F* (4.24)
=
We assume a power series expansion for all dependent variables in the problem, C=C
(6+ 1)(1 +~m)I, (4.23)
1) e’~””’~
+ ~(D_i
—
D*=Be~’,,
e’~2”~
[6(a2~
e~”+ F)
+Bi(D*
a2~~)C
a2C M~t —
(4.22)
Satisfying the boundary conditions, we find F* B* [(~ + 6)(~ +~m)
a2F*
e1m)+~i]
_~1)
+
x
—
(4.21)
exp[—z/71.
_(6+f~)B*[h+z}
Z=~flm
M~t —
C
=0,
with F an arbitrary constant of integration. We obtain an expression for the linear growth rate, a, by substituting C 2 term 0 and C1 into the order a of eq. (4.16a). Unscaling ~ and 6, we find
2C 0+a
1+..., 2Cmt +
...,
(4.17a) (4.17b)
a=((1_e_~n~>)[(1_M~)a2_a4FI} x{(a2F+ M/1)
Cm= Cmo+a
and substitute these into eqs. (4.14)—(4.16). The goal is to find an expression for the growth rate, a, valid in the long-wavelength limit. Alternative to the expansion procedure presented here, one could also accomplish the above by exactly solving the eigenvalue problem, eqs. (4.3)—(4.5), and then applying the long-wave limit to the resulting growth rate expression. At leading order the solution is C B e~, Cm~=B*e_T,~, 0
=
(4.18) (4.19)
_l’~i±.2”m>M*(2p+Mi —
k —
1
—
e’~’-’~M*N *
(4.25)
where M
*
=
M M~’— 13 ~
2F* —
a
—
(4.26) a2F*
2F + M/1 — 1)/(a2F + M~1). (4.27) N* = (a Examining eq. (4.25) in the limits 2’~—s0 and
LB. Humphrecs ci a!.
40 —* ~,
/
Morphological invtahilitt’ in float zn/i,’
which recover a directional solidification
geometry, we find that eq. (4.25) reduces to a= (I
—k.
‘ri
(4.28)
This expression is equivalent to that found by Young and Davis [15] with their R = 0 (no convection). Considering the limit M 1 1 with a ~ I. we find eq. (4.25) reduces tO 41 j — k. a = (I — e ‘~‘> )[(1 — ~-t )a — a (4.29)
‘~
‘~.9
[1
--
/2
LT
.
.
Hg. 4. Phase diagram plot of the directional solidification I I)S) and float zone (FZ) concentration profiles for identical material and s’sstem paranseiers.
fig. 4. For identical material properties and 1cmSince a> 0 implies instability, then (1
—
M
1 ‘) > 0 is a destabilizing term. Therefore, when M1 > I we have ntG~/G~ > I. which is a criterion for undercooling. On the other hand, increasing k and the presence of the surface free energy. F. are stabilizing effects. From eq. (4.29) the magnitude of the undercooling (1 — M~) decreases by the factor (1 — e ~‘1~ which indicates the height of the liquid zone. There is no instability when ~ =~,, = 0. In this case we have all solid and no melt. On the other hand, the most unstable configuration is .~-= 0. .~/ -~ ~, which is directional solidification. Hence. a float zone configuration is always more stable to long-wave morphological disturbances than a directional solidification geometry. Further. the shorter the melt height. the more stable is the zone. To understand the mechanism responsible for the increased stabilization we examine the concentration profile in the liquid, Because of the presence of the melting interface of the float zone, solute diffusion in the axial direction is hindered. This leads to a higher solute concentration in the liquid zone in comparison ~o the directional solidification system This can he seen from eq. (4.21) where a term of the form
perature gradient G1 . the float zone system has a flatter axial concentration gradient at the solidifying front. Hence, the float zone is niore stable. Using eq. (3.11) we see that L —sO, as U1 increases. As G1 increases, M1 decreases and the system is more stable. Likewise. ~,, —~ 0 as U1 becomes large, as seen in eq. (3.12). Hence, steep gradients are necessary for morphological stahility, which is no surprise. In practice, though, steep gradients are to he avoided due to buckling and structural concerns. Eqs. (3.11), (3.12). and (4.29) suggest balancing the above steep versus non-steep issue so as to keep the zone height small by decreasing the maximum heater temperature. T11. which gives the same effect as increasing the temperature gradients. As an example, consider the data listed in table I. Using the assumption that U1 = — U1 . fig. 5 shows the neutral curve (a = 0 in eq. (4.29)) as a function of the temperature gradient U1 and max-
D*(l — e ~ ) (4.30) will appear since F is arbitrary. The source of this term is the boundary conditions applied at the melting front, in contrast to the directional solidification system where we require C 1 —f 0 as c ty~. Note D* 0 as -~ ~. The effect of the term is best illustrated by the phase diagram shown in
D V A
Table I
M iterial and s~siem~ iranleters for
(Jc(J I
Parameter
Value
T
1213K 4cni~s 4x1() l.3>< 10 4cmz,/s
,~,
1)1 ._4 K~t~
~,
0.5 wi/
Is,~
ft39 W/crn K
-
L. B. Humphreys et a!. (‘K 1
/ Morphological instability
STSBL
F
5. Weakly nonlinear analysis
=5
54~-
41
as fig. 5 for systems of interest would show desirable heater gradients and temperatures which would not give rise to morphological instability.
~
G.,,~.,.
in float zone
____
In order to obtain the weakly nonlinear behavior of the system, and investigate instabilities of
_____________
-
‘
the planar front, we examine near the critical point
JIISTIBLE
52
ML 1 + , for c ~< 1. We rescale eqs. (3.28)—(3.31) as =
~
I
[
I
I
1260
1280
1300
~I
xe1”2 k,X. c2tz =
1200
1220
1240
1320
T,,(’K)
Fig. 5. Neutral stability curve for the directional solidification (DS) and float zone configurations corresponding to the data of table 1.
=
= =
Z, T,
hf = ci], hm = C = C, C,,, = Cm.
(5.1)
(5.2a)
These scalings are determined by eq. (5.1) and the results of section 4. Additionally, since we have that k O(2), then ~ as defined in eq. (3.21) will be large. We take =
imum heater temperature, TH. This curve is obtained by solving for the planar positions, Lf and Lm, using eqs. (3.11) and (3.12) and then using a Newton iteration to find the value of GL which gives a 0 for a fixed We note that eqs. (3.11) and (3.12) are valid, independent of eq. (3.25), provided eq. (3.6) is satisfied, and thus give a =
7/.
means with of determining theline melt zone height consistent the proposed heat source. Fig. 5 demonstrates that for the data chosen, the morphological stability characteristics of the float zone are similar to the directional solidification geometry for heater temperatures about 25 K above the critical value set by eq. (3.13). For these temperatures, the melt height is sufficiently large that the presence of the melting boundary does not influence the stability criteria. Note that as the zone height is smaller for lower values of TH, the critical temperature gradient for stability decreases, so that the system is more stable. One possible application of these results would be to the hot-wall float zone technique. Here the temperature gradients are kept small and the heater temperature is kept low near the melting point. The reason for this design is to reduce thermally induced imperfections and possibly reduce the level of surface-tension driven flows. A result such
~
(5.2h)
~/.
=
that ~.c 0(e) and the surface free energy varies slightly with the interfacial concentration. The so
=
governing equations become (2CT~ C
7=C~+ C77, 2CmT, + Cm7z) = — C,,,7 + E
~(C
(5.3) (54)
subject to conditions at z —~
Cm
0,
=
at Z C
=
(5 .5a) —~
+ cH
eM~tH + 2F(1
—
—
~ii~C/c)
x H~~(1 + c~H,~)3”2 = 0. (5.5b) (3H~+ i)[i + (2~ — i)C] = C~.— c2C~H~, (5.5c) at Z
=
£~=‘m+ c H,,,
C
—
e”~”~ + M~1Hm+ 1 + c2(F*
x
Hmxx(1 + c~H~~)3”2,
=
C,,,
=
C,
—
(5.5d) (5 Se)
LB. Hurnphreys et al.
42
2k
(1
±c~H,,,T)[i + C(c
—
1)1
C
=
Morphological in.stahilitt in float zoni’
2C~H,,,~ 7
——
/
—
c
Consequently, from eq. (SlOe) we find the dcviation from the planar melting front.
(5Sf) H
We seek solutions in powers of c as follows: C
I
=
e>
—
+ cC
~‘~1
2C +
H
(1— et.~~”s)) exp[(.~ 2C,i, .... +cC~,1±c
=
H
=
2H +
—
:)/D1
. ..,
=
~(M~
c> “—I)
Note the melting front is more stable than the freezing front as H,,, 11 = 0. The concentration in
(5.6h)
the melting crystal is found by solving eq. (5.9) subject to eqs. — (5.lOa) and (5.lOd).
(5.6c)
C,,,
~
=
1
M
+
1~ eJ’1
0 + H1 + c H,,,
H,,,0 ± H,,,1 + cH,,7 ± ....
=
(5.6d)
‘~
At leading order the system is identically satisfied by the basic state. At 0(c) we find C
H 0=0.
=
DC,,,,77 +
(/77
=
0.
unknown, so we consider the 0(c’) problem (only
(5.8)
C’,//±C’)/=aV\e
(5.9)
at Z
those equations needed to complete the analysis are listed):
at Z=~,1
=
+
(/7
—
+ C,
=
kH0
(5.16)
+ H01.
C37+C7=O.
(5.17)
Eqs. (5.15) and (5.16) give
~
C7 ± H1 at Z
(5.lOa)
(5.15)
~
=
at Z—* ~
z
(5.14)
).
The freezing front interfacial position. /-1~.is still
C,7
at
-
D
subject to conditions
(/7=0.
I
(57)
The O(2) system is C77+C7=O.
e
I
—
xexp(
=
(5.13)
.
(5.6a)
1 ±c C
-I
—
C
~H1~+ F[1
—
~~H0IH0 ~
0, (SlOb)
=
0.
=
(SlOe)
~
C’~ kH0 + H07 =
~
—
±~
~
I/i
±a5.~Z e >‘~, (5.18) 3) termsconstant of eq. (5.Sb). (5.18) andfrom (5.17) The /3 can Eqs. he determined thelead 0(c to
C,,, 7
—
C
+ H,,,1 [(D
—
1) e
—
~
an evolution equation for the interfacial shape. (5.lOd)
C= H,,,t(M~t
C>>’I+.Y’Ill)).
(/7+C,=O. -
-
(SlOe) (5.lOf)
=
a e
-
±kH1+H1~=0.
-
We find the concentration in the melt by solving eq. (5.8) subject to eqs. (S.IOb). (SlOe) and (S.lOf) to give C
[—H0H0., ).~+ Ho,\v+FHO.\.\.v,v F~.(H11H11~~) ~~](1 —e
~
We rescale eq. (5.19) ,to obtain a two parameter evolution equation using the following transformations:
(5.11)
~
H11=F, where a
=
H~— H11
T=
T
1 —
(5.19)
FH() xx + F~ H11 H0,~~.
(5.12)
,
X=Ft
3~
—
C, -
LB. Hutnphreys et a!. / Morphological instability in float zone
Eq. (5.19) is rewritten as
43
ities of the planar front, F= 0. We linearize (6.1) about F 0, expand in the normal modes F= F* ~ (6 2 =
[(1
—
F)F~]~+
—
)~+ KF+
~
=
(5.21) and find where (6.3a) K=kF/c11
—e’~”~
(5.22)
.
Eq. (5.21) is of the same form (with ii,,
0) as the
=
one parameter evolution equation found by Sivashinsky [141, and Young and Davis [15].
where ~2
+
=
(6.3b)
~2
‘I
Mathematically, eq. (5.21) will behave the same as in refs. [14.15] with increasing K corresponding to stabilization. However, the physical interpretation is different. K as defined by eq. (5.22) is effectively larger for the float zone system, due to the
Eq. (6.3a) is the scaled form of the growth rate expression eq. (4.29). The critical wave mode for the most unstable disturbance is w,, l/v~ and the critical parameter for the onset of instability is K,, ~, with K < corresponding to instability. To follow the weakly nonlinear development of
1
=
=
-~
e~”~”~ term, than the directional solidifi-
instability we examine the response of the planar
cation system for equal material and system parameters, k, F, and c. Thus, the float zone will be more stable. However, in the limit ~I1,, 0, these evolution equations are known to predict subcritical bifurcation to two-dimensional band structures and exhibit unbounded numerical simulations as i- —s oc [15,17]. To investigate the effects of interfacial surface free energy variation with concentra-
interface for a special class of perturbations with critical wave mode to.,,. In particular after Segel and Stuart [28] and Sriranganathan [24] we assume that
tion (ji,, ~ 0) we now examine the bifurcation behavior of eq. (5.21).
(6.4)
—
=
cos(co,,fl + B(T*) cos(w,,~/2)
F= 6[A(T*)
~1
cos(~w,,31/2)1+ 8F2
x
+ 6F3 +
where 2T, we have introduced the slow time scale T* “~6
6. Stability characteristics An evolution equation for a three-dimensional solidification front can be determined by extending eq. (5.21) to find
6 ~‘sz1. Substituting into eq. (6.1) to third order we find that the amplitudes A and B satisfy the equations 2
~1
1
A (T*)=a,,A —aB —A(atA~+a 7B).
(6.Sa)
B’(7*)=a,B_4aAB —B[2asA —
v
(FvF)
=
0,
where =
and with
± (a1 +
2a2)B/4],
(6.1)
(6.5b)
where 2a,, = 6 a = (—ji,,
+ y
scaling to ~ as x scales to
~.
This
+
1)/16.
—
ji,,)(2
(6.6b)
a 1
equation governs the nonlinear behavior of the interface and can be used to investigate instabil-
(6.6a)
K.
—
=
—(1
—
5jiJ/16(4 K). 2fL,, )/32( ~ K). —
—
a2
=
—
3(1
— ji,,
)(1
—
-
—
(6.6c) (6 .6d)
L. B. Hun,phretn ci a!.
44
Morpholo,g,cal ,n,rtahilii~in float .zoiie
2D bands
hexagonal nodes I
II~I~1II
—f—r—r—’~rr—1~ ~III~F I
liii~
—
—~
/ I IN/ISO I:
11:111 IIIII.I11110
NO
‘.1.501.1
0105 1100
I.N’.0711H
11000,,
.1
~ IIJI
N~~LI/lOIN
1.
1/001
,.~. 2 I
i
2
_______
LIII
N
,.‘ I
II
I
I
I
II
I
Fig. 6. Regions in the p.~— K parameter space yielding stable 2D band structures. Neither stable nor unstable 2D bands exist in the ‘~no solution regions.
Fig. 7. Regions in the /L~- K parameter space yielding stable 31) node structures. Neither stable nor unstable 3D nodes exist in the ‘‘no solution ‘‘ regions.
For the chosen values of ji~,a as defined by eq.
solutions (A <0). which project into the solid, exist subcritically (K> ~) for the range 0.48
(6.6h) is small in magnitude allowing the quadratic terms of eq. (6.5) to be valid to third order. The critical points of eq. (6.5) corresponding to A’(T*) B’(r*) 0 are classified as [24,28] =
<0.93. We find no regions in which hexagonal cell solutions (A >0) (projecting into the melt) are
=
I: A
B
=
0. planar interface.
=
(6.7a)
‘~
‘~1I’
05959
II: A III: A
=
=
±,/~- B=O. 2D bands.
v
,
I~N -
(6.7h)
a
1
i.~ 11
2 ± a,,(a
—2a ±~4a
1 ± 407)
~,-
-
--
a1±4a7 B= ±2A. 3D hexagonal solutions.
/
(6.7c)
We do not examine the Class V solutions listed in ref. [24]. To determine whether or not the structures corresponding to the solutions listed in eq. (6.7) are realistic we investigate their stability by linearizing eq. (6.5) about each. The Class I planar interface analysis reveals stability for a,, < 0. as expected. The Class II and III results are shown in
figs. 6 and 7. Fig. 6 demonstrates that stable 2D bands exist supercritically (K < ~) for the range 0.4 < 1. Fig. 7 shows that hexagonal node
1’
I
/
-
/
.
-
.
7
,,,:,
..
I
. Fig. 8. Bifurcation diagramI obtained from the nonlinear analy= 0.5959. Solid lines denote stable solutions: dashed
sis for ~
denote unstable.
L. B. Humphreys et a!.
/
Morphological instability in float zone
stable. These results are consistent with the bifurcation analysis performed in ref. [24] on the full problem for k <1. There
a’y/aC> 0
(6.8)
[eq. (2.6bb)] is necessary for stability. Since we find stable solutions for jii,, > 0, then p. <0 by eq.
(3.21). Hence,
45
points found in eq. (6.7). The stable regions are denoted by solid lines. Numerical simulations of the full equation (6.1) (to be examined below) reveal critical values of K below which the 3D and 2D structures are unstable, while the linear analysis predicts stability. To examine the nonlinear development of these
solutions we perform a numerical integration [29] of eq. (6.1). The domain for all integrations is
ay/aC>o is also required in this analysis. Further, for the
case Ii,, 0, fig. 6 reveals that 2D bands bifurcate subcritically and are unstable for this case, which agrees with the results of the bifurcation analysis in ref. [17]. Thus the inclusion of 0 yields stable solutions, not found in the small k, F 0(1), Sivashinsky [14] type evolution equations. Fig. 8 shows a typical bifurcation diagram for the steady state amplitude, A + B for the critical
(~
~)
E
~
=
ji,,
‘‘
=
,
x 0, Vito,,
~,, =
[0 p1 x [0 Q] ‘
J
(6 9
I
which corresponds to 3 wavelengths in the ~ and ij directions for the leading order terms of eq. (6.4). Periodic boundary conditions are applied at the edges of the rectangular domain. An explicit finite difference scheme with second order approxima-
~ (~urntsof ~f~’
~
O~53.315
0
i~ 30.781
-1.474 < F < 1.038
Fig. 9. Two-dimensional band structures resulting from the numerical simulation of eq. (6.1) for 4,,
=
0.5959 and K
=
0.21.
1.. B. Huinphrei
46
11
i
ci a!.
8.Ioi’pholog, a! in ctab,l,ti ii, float zol,s’
~
~H
F~~un1tsof~..,/~
O~53.315
0
<
F < 0.883
Fig. 10. Fhree-dirnensional node struclureN resulting from the ni,n,crical N,mulation of eq. (6 1) for ji,
((5751) and ,~,
0.21
tions in space and first order forward differencing in time is employed. A SO X 50 grid is used for all calculations to follow with several runs on a 100 >< 100 grtd as a check on the stability of the scheme, Initial conditions similar to the form of the leading term in eq. (6.4) with various magnitudes for A and B (typically of the order 0.05 to 0.1) as well
minimum root of the steady structures. The presence of jii,, ~ 0 stabilizes the runaway ohsersed in previous numerical integrations of the Sivashinsk\ type evolution equations. Fig. 13 shows the hifurcation diagram obtained by the numerical integration for a fixed ji,, as K varies. The amplitude plotted is tip minus root. Here dotted lines corre-
as the initial conditions A* cos(2~/P) cos(2~si /Q) are input. Typical results yielding steady 2D
spond to unstable solutions. This diagram differs somewhat in detail than fig. 8. Here a different
hand solutions and 3D hexagonal node solutions are shown in figs.. 9 and 10, Whether one obtains
amplitude is plotted which corresponds to arhitrary amplitude solutions of eq. (6.1). while fig. 8
hands or nodes depends upon the initial condi-
is restricted to small amplitudes. Nevertheless the subcritical (3D) and supercritical (2D) trends are
tions in those (K. p.,,) regions where the linear analysis predicts both are stable. Figs. 11 and 12 show typical tracking of the maximum tip and
qualitatively the same in each. The general trend is that as velocity of solidification V increases (cor-
Tip
L. B. Huinphreys ci a!.
/ Morphological instability
in float :one
~i,,i~irri
I
I
47
~E 1.1
I
I
I
I
I
I
I
I
I
I
I
I
I I
I
I
I
I
I
I
1.0
0 I
0
ii,,,
50
Ii,
tOO
II,
t50
(1
i,i,i,,,i
200
—
II
250
interface undergoes a transition to small amplitude hexagonal node solutions and then to bands,
finally losing stability to large amplitude structures which are not predicted by the scalings introduced in eq. (5.2).
7. Conclusion We examine morphological instability in a crystal sheet, float-zone geometry for which the height of the liquid zone, L L1 + Lm, is smaller than the width and thickness of the sheet. Here L1 and Lm denote the planar positions of the solidifying and melting interfaces. We calculate these positions using equations (3.6), (3.11) and =
—
350
400
e~r~1>)
Fig. 11. Maximum cell tip height development with respect to time (2D structures,
responding to increasing the magnitude of c in eq. (5.1) yielding smaller K, eq. (5.22)) an initially flat
II
300
4,,
=
0.5959. K
=
0.21).
(3.12). The controllable experimental parameters are the temperature gradients, G 5 and GM, in the growing and feed crystals, and the maximum ternperature, TH, of the heat source. As discussed by
Young and Chait [11] for large heat transfer across the boundaries of the system, the melt temperature profile is identical to the heater profile. The above result is applicable to this situation and gives a simple means of estimating the zone height. We calculate basic state concentration profiles for a planar interfacial system. A linear stability analysis of this system, in the limit of small segregation coefficient and limited to the frozen ternperature profile shown in fig. 3, shows that the
growth rate of disturbances, eq. (4.25). is a modified version of that for a directional solidification system. We find that shorter melt zones are more stable due to solute redistribution caused by the
48
L. B. lluniphreis ci a!.
/
VIorpholog,cal ,nstabil,tt in float zo,,i’
sot Depth (~
—.2
IIIIiIIIIIIII,IIIIIiIIIIiIIIIiIIIIiIIII
.10
I
I
I
I 50 I
I
I~IIIIIII
I
10))
15))
( ~ ~2
I
I
20))
F (1
I
I
25))
I
I
I
I
I
I
319)
I
I
350
I
I
I
4)9)
DI
e~’/V)
—
Fig. (2. Minimum root depth development with respect to time (2D structures.
4~
=
0.5959. K
=
0.21).
presence of the melting front. The shorter zones have flatter axial concentration profiles making them less susceptible to undercooling. Thus our results indicate that to suppress morphological instability in a float zone system. one needs to decrease the liquid zone height which reduces the axial concentration gradient. This can be accomplished by increasing the temperature gradients in the feed and growing crystal or decreasing the maximum heater temperature. On the macroscopic level, this temperature profile tailoring would need to be done in conjunction with efforts to keep thermally induced stresses to a minimum.
We perform a weakly nonlinear analysis near the critical conditions for instability. The result is an evolution equation giving the departure of the melting and solidifying interfaces from a purely sinusoidal shape due to nonlinear effects. This equation is similar to those developed in previous small segregation coefficient, 0(1) surface free energy, long wavelength analyses hut has the added feature of surface free energy variation with interfacial concentration. Both bifurcation analysis and numerical simulation of the equation reveal transition from a planar front to 2D hand or 3D hexagonal node structures. Qualitatively, this
Hence, a lower heater temperature with shallow
agrees with results obtained for a non-simplified
thermal gradients in the feed and growing crystal is the most desirable situation. Our results, fig. S. suggest this can be accomplished while keeping the solidification front morphologically stable.
system [24]. The point being that evolution equations, while developed under severe restrictions, yield simpler problems which retain much of the physics of solidifying systems. The results derived
L.B. Humphreys ci a!.
/
49
Morphological instability in float :one
I ‘
I
I
I
I
I I
I
I I
7c=o.5959
I
I
I
3-0 NODES
I
‘I’ip—Rt:t I
_.~~.~—~-;i(
D~
l~,E~’)
-
LIN
2-DBANDS
in
X
-
-~ in
I
-
)
-
IN
I -
I —
-in
I
\ ‘I
-
S S l~
in
I in
.0
I w
in
ro
IN
C’i
I
I I
I I
I
in
in
iO
.0
01
iN
‘-
,-
I w I
I
(N
I in
IR~
K Fig. 13. Bifurcation diagram obtained from numerical simulation of eq. (6.1) for 7,, = 0.5959. Solid lines denote stable solutions and dashed denote unstable: X corresponds to calculated values.
from this approach predict solidification structures which are determined by the interplay of surface energy, thermal and solutal diffusion, and interface attachment kinetics. These results are suggestive of the behavior of these systems even though there may be no materials which exactly satisfy each of the restrictions.
Donald Wakefield for his assistance with the numerical computations and graphics. References [1] J.A.F. Plateau, Staiique Expérimentak ci Théorique des Liquides Soumis aux Seules Forces Moléculaires (Gauthier-Villars. Paris, 1873). [21J.W.S. Rayleigh, Proc. Roy. Soc. (London) 29 (1879) 71. 131 W. Heywang and G. Ziegler, Z. Naturforsch. 9a (1954)
Acknowledgements
561.
[4] CE. Chang and W.R. Wilcox. J. Crystal Growth 28 (1975)
This work is supported by NSF Grants DMS-
8604047 and DMS-89-57534, and NASA Grant NCC-3-104. The authors also wish to thank
[5]
8.
CE. Chang and W.R. Wilcox. Inter. J. Heat Mass Trans-
(1976) 335. [6] fer P.A.19Clark and W R. Wilcox. J. Crystal Growth 50 (1980) 461.
50
LB. Hllfllphrev.c ci a!. / Morphological in.oalnlity in float co/Ic
[7] AK. Sen and S.H. Davis. J. Fluid Mechanics 121 (1982) [8] [9] [101 [11] [12]
163. CL. Lai. S. Ostrach and Y. Karnotani. 1985 US—Japan Heat Transfer Jo1nt Seminar, 1985. SR. Coriell. S.C. Hardy and MR. (.ordes. J. (olloid Interface Sci. 60 (1977) 126. J.L. Duranceau and R.A. Brown. J. Crystal Growth 75 (1986) 367. G.W. Young and A. Chait. J. Crystal Growth 96 (1989) 65. W.W. Mullins and R.F. Sekerka. J. AppI. Phys. 35(1964)
[18] G.W. Young and SF1. Da’is. SIAM J. App). Math. 49 (1989) 152. 19] D.A. Kurtze. Phys. Rev. B57 (1988) 370. [2).)] J.M. H’sman. A. Noxick-Cohen and P. Rosenau. Ph~s. Rev. B37 (1988) 76).)3. [21] K. Brattkus and S.H. Davis. Ph~s.Rev. B38 (1988)11452. [221 D.S. Riley and S.H. Davis. SIAM .J. AppI. Math., submuted. [23] J.S. Langer. Rev. Mod. Phys. 52 (1980) 1. [24] R. Sriranganathan. l).J. Wollkind and D.B. ()ulion. J. Crsstal Growth 62 (1983) 265.
444. [13] D.J. Wollkind and L.A. Scgel. Phil. Trans. Roy. Soc.
[25] D.P. Woodruff, Phil. Mag. 17 (1968) 283. [261 H.S. Chen and K.A. Jackson, J. Crystal Growth 8 (1971)
London .A268 (1970) 351. [14] G.I. Sisashinsky, Phvsica 8D (1983) 3388. [15] G.W. Young and S.H. Davis. Phys. Rex. B34 (1986) 237. [16] A. Novick-Cohcn and G.I. Sivashinsky, Phvsic~i 201) (1986) 237.
181). [27] D.J. Wollkind and S. Raissi. J. C’r~sialGrowth 2h (1974) 277. [28] L.A. Segel and J.T. Stuart. J. Fluid Mech. 13 (1962) 289. [29] J.,A. Hcminger. Tech. Report ~ 88-07. I)epartnient of
[17] G.W. Young. S.H. Davis and K. Brattkus. J. C’rvstal Growth 83 (1987) 560.
Mathematical Sciences. The University of Akron.
31
MORPHOLOGICAL INSTABILITY IN A FLOAT ZONE Laura B. HUMPHREYS, John A. HEMINGER and Gerald W. YOUNG Department of Mathematical Sciences, University of Akron, Akron, Ohio 44325, USA
Received 12 January 1989; manuscript received in final form 7 September 1989
Morphological instability in a crystal sheet, float-zone geometry for which the width and thickness of the sheet are much larger than the height of the liquid zone is examined. The heat transfer across the boundaries of the system is assumed to be large so that the temperature profiles in the melt and solids are identical with the heater profile. A piecewise linear profile, applicable to systems with slow velocities of solidification and for which the thermal diffusivities are larger than the solutal diffusivities. is imposed. The height of the zone is calculated consistent with these assumptions. Linear and weakly nonlinear analyses in the limit of small segregation coefficient determine that for shorter zone configurations the system is less susceptible to long wavelength morphological instabilities. Solute redistribution leading to flatter axial concentration profiles is the proposed mechanism for the increased stability as compared to unidirectional solidification. A nonlinear evolution equation governing the dynamics of the solidification front and which includes surface free energy variation with interfacial concentration is also developed. Bifurcation analysis and numerical simulations of this equation predict transitions from planar to three-dimensional hexagonal node structures or two-dimensional band structures.
1. Infroduction When a heat source is passed along a crystal rod, a travelling liquid zone is created. The zone is shaped by the melting interface at the feed crystal, the melt-gas free surface, and the solidifying interface at the growing crystal. This process, shown in fig 1, is known as the floating zone technique. This method is suggested for use in space for crystal growth of electronic materials, One concern of crystal growers is the shape and stability of the float zone, both of which affect the quality of the grown crystal. Various theoretical studies of the float zone process have been performed. Plateau [1], Rayleigh [2], and Heywang and Ziegler [3] determine that for static stability of a cylindrtcal zone melt, the maximum zone height, L, must be less than 27rR, where 2R is the zone thickness. The melt motion is exammed, for examplc, by Chang and Wilcox [4,5], Clark and Wilcox [61, Sen and Davis [7], and Lai, Ostrach, and Kamotani [8]. Coriell, Hardy, and Cordes [91investigate instability in the shape of the zone as a function of the contact angle. The most realistic 0022-0248/90/803.50 © Elsevier Science Publishers B.V. (North-Holland)
model of the float zone process is that of Duranceau and Brown [TO],who determine the interfacial positions consistent with a conduction dominated heat transfer model. Young and Chait [11] use a coupled analytical/numerical procedure to determine the zone shape in the limit R << L. In this work we examine morphological instability of the freezing interface of the float zone for a sheet geometry as shown in fig. 2. This interface ultimately determines the concentration profile in
\
Slope
\
G~j
_~._
I
J
“
Temp
LOSS
/
~r’ysi~iØ/7/~/~ ‘///////////
//A
I
Fig. 1. Float zone with a heater source of magnitude TH and linear temperature profiles applied in the feed and growing crystal regions.
32
L. II, Jiimiphreii ci a!.
~
S/ar~thiilagua! instainli i in float zone
~
the effect of buoyancy in the system. Novick— (‘ohen and Sivashinsky [161 include latenl heat in the evolution equation. Young, Davis. and l3ratt— kus [171 analyze anisotropic interface kinetics. The first to analyze morphological instability considering a finite melt height is Young and Davis [18]: however, their work does not consider a float—zone process. hut rather a melt hounded by an upper solid wall. All these analyses have the disadvantage that the evolution equations are not Ufli— form for large time. Kurtze [1~1and Hvman et al. [201 suggest means of stabilizing the equalion by including higher order terms [191. or retaining full expressions for the interfacial curvature and unex-
Fig. 2. Schematic of a float zone in a crv~talsheet.
the solid which in turn determines the material and electronic properties of the crystal. It is well known that morphological instability occurs when the temperature gradient in the liquid is shallow leading to constitutional supercooling. A shallow gradient is desirable, though, to prevent buckling and dislocations in the solid crystal which result from nonlinearities in the temperature profile. Hence, there is a need to optimize the temperature profile so as to minimize the morphological and structural instabilities. It is our purpose to establish criterion determining the onset of interfacial instability in a float zone system. The majority of morphological instability analyses have been done for a directional solidification geometry no upper boundary - rather than for the float zone. Mullins and Sekerka [121 complete the first analysis of morphological inslahility using a linear analysis. Woilkind and Segel [13] use bifurcation theory to conduct a weakly nonlinear analysis. Sivashinsky [14] develops a nonlinear evolution equation governing morphological instability at the interface in the limit of small segregation coefficient, k. and 0(1) surface free energy. This evolution equation retains much of the physics of the problem, hut involves the analysis of a single equation as opposed to the more complicated, coupled system. Il has also been used by Young and Davis [15] to examine
panded exponential concentration profilcs [20] Brattkus and Davis [21] develop an evolution equation in the limit of absolute stability where the surface free energy is large. For this condition with k near unity. they predict that Ihe planar interface bifurcates to a stable three-dimensional hexagonal node solution. Riley and Davis [22] tie together the Sivashinsky and Brattkus and Davis limits by developing an evolution equation for Ic ~ I and large surface free energy. Numerical integrations of the absolute stability limit evolution equation do stabilize to steady state solutions [211. In our approach to the float zone conliguration. we assume a rectangular geometry as shown in fig. 2. with W>~R >~L. and neglect the presence of the melt-gas interface and crystal rod boundaries. In effect, our liquid zone is finite in height and infinite in width and thickness. The height of the zone is determined consistent with an imposed temperature profile in the melt and solid rods. The transfer of heat from the heater to the system. across the surface of the system is assumed to be large. i.e. the Biot number. B, = hR/k, >~ 1. where h, is the heat transfer coefficient and k, is the thermal conductivity in the feed rod, melt, or growing crystal. Hence, the imposed temperature profile in each region is identical to the heater temperature as a consequence of the large surface heat transfer and 11) approximation (infinite extent of the system in the lateral directions s and i). Further, we neglect any melt motion. Together with the “frozen temperature” approximation [231 discussed in section
LB. Humphrevs et al.
/
33
Morphological instability in float zone
3, these assumptions lead to nearly planar interfacial shapes and x, y-independent, linear temperature profiles as shown in figs. I and 2. This is clearly an extreme simplification of the process, neglecting the practical importance of radially (ydirection) distributed (parabolic) temperature profiles which lead to structural instabilities, However, it allows for calculation of the zone height consistent with the temperature gradients and incorporation of concentration boundary layers at the solid interface. In section 2 we formulate the problem. We find the basic state solution and introduce scalings in section 3 ; we analyze the linear stability of the system in section 4, looking for criterion for a long wavelength instability. We find that the float zone
tion of large R (applicable to thin sheets) one would require the solid/liquid frdnts in the v direction to be everywhere perpendicular to the direction of growth, i.e. the contact angle 7T/2. Boundary layer corrections due to the presence of the melt—gas interface are not considered here. These are discussed by Young and Chait [11]. The coordinate system translates upwards with the heating element at a constant speed V. with z 0 representing the position of the heater. We assume there is no density change upon solidification and neglect melt motion. The latter is rcasonable in a microgravity environment for systems with small Marangoni number so that surface tension driven flows are not present. We define L~ and L1~,to be the planar posi-
configuration is more stable than the directional solidification geometry. In section 5 we develop long-wave evolution equations in the Sivashinsky limit for the shapes of the solidifying and melting interfaces. We include the possibility that the surface free energy varies with interfacial concentration [24]. We discuss stability characteristics in section 6 including a bifurcation analysis of the evolution equation for the solidifying interface of the float zone together with numerical simulations of the same equation. The assumption of the surface free energy varying with concentration leads to supercritical bifurcation of two-dimensional band structures and a range of stable three-dimensional hexagonal node structures.
tions of the freezing and melting interfaces. respectively. T,, Tm, T, C.,, C.,,, and C are the temperatures and concentrations in the growing crystal, feed crystal, and melt, respectively. i~, D. Ks, and D5 denote the thermal and solutal diffusivities in the liquid and solid phases. The equations representing the thermal and concentration fields in the three zones are:
=
=
—
.
(i)
Solidifying crystal
K5(7,,, + 7/)
—
=
(—
~
—
L. + hf(x. t))
VT.,z + L,,.
(2.1)
where h1(x, t) denotes deviations from the planar solidifying front, The neglect of solute diffusion in the solidifying crystal while considering such in the feed crystal will be discussed shortly. (ii) Feed crystal (Lrn + h5~(x,t)
2. Formulation i~s(Tm.,.,+ ~
The floating zone technique is applied to a crystal as shown in fig. 2. For simplicity, we consider a two-dimensional coordinate system for the derivation and extend the results to three dimensions, We neglect the presence of the melt— gas interface and crystal rod boundaries by assuming W>> R >> L where W is the width of the feed and growing crystal sheets, L is the height of the liquid zone, and 2R is the thickness, This leads to a liquid zone which is infinite in the x and 3’ directions. Thus leading order expressions for the temperature and solute fields will be one-dimensional (functions of z only). To relax the assump-
D5 ( Cm,, + C~,- -)
V7~~+ T5,,.
—
=
=
—
VC’m - +
(2.2)
cm,,
(2.3)
-
where hrn(X, t) denotes deviations from the planar melting front, (ni) Melt (—L1
+ h~(x, t) <: < L
+h Ti
K(T.,., + T.)
=
—
VT + T,.
(2.4)
D(C+C~)=—VC+C. --
(2.5)
-
At the freezing interface z — L1 assume continuity of temperature =
T’
1/
(x t)) 111
+ hf(x. I)
we
(2.6a)
34
L. B. Huniphrer.s et a!.
/1
Morphological instuhilitr in I/oar zoo,’
and thermodynamic equilibrium T = mC + Tv
1 I + yIi1,.,/~( I + h~,)3
~]
(2.ob)
Here T51 is the pure substance melting point. y is the surface free energy. m is the slope of the liquidus line and £° is the latent heat per unit volume. y is taken to be isotropic hut we allow the possibility that it varies with interfacial concentration [241 by considering
J -
L .
(2.6hh) I
where y~has units of energy per unit area and is constant, ~.t is constant, and c0 is the uniform concentration of the feed crystal. There is also conservation of solute
—
(hf,+ V)C(k~ 1)
IT
=D(C—C.,h~,)
(2.6c)
-
..
.
Fig. 3. Basic state concentration and temperature profiles in
where k is the segregation coefficient such that (/=kC.
the feed crystal, melt, and growing crystal regions.
(2.6d)
and conservation of energy k5(T
—
T h~ )
—
k1 (I-
—
Tip, ‘
‘
)
=~(v+h,. (7 —.
)
c
At the melting interface = L + it (x, 1). the conditions are similar. Eqs. (2.6a) and (2.6b) are now T= 7/,
(2.7a) ~.2
T= mC + T~1[i + yh51,,~~(1+ h;,)
the growing The redistribution argument for ofthissolute approach is thatcrystal. significant in the feed rod upon melting is expected so that the product of the feed concentration gradient with the feed diffusion coefficient D5 (DSC,,,T in eq. (2.7c)) is comparable to this same product (DC) in the liquid phase at the melting interface. Here we expect C, ~ C, (flatter melt concentration profile away from the solidification front as shown in fig. 3) so that the effect of D5 ~< D is
L (2.7b)
cancelled. On the other hand, this will not he true at the solidification front. Here C. >~C so that the product D5C., is negligible in eq. (2.6c). This -
Conservation of solute requires (h
,
+ V )C( k
—
1)
=D(C— C.,h5,.,) C
=
kC
results in the inclusion of diffusion of solute in the —
D5(C~— C,,,.,h,,,J, (2.7c) (2 7d)
‘ii
Implicit in the neglect of a governing expression for concentration C, in the solidifying crystal and in the formulation of eqs. (2.6c) and (2.7c) is the assumption that solute diffusion in the feed (melting) crystal is significant but that it is negligible in
feed crystal at the melting front as discussed by Woodruff [25], Chen and Jackson [26j, and Wollkind and Raissi [27]. Finally, the equation of energy conservation at
the melting front is
k5(T~,,— 7/.,h51,) ~
V + h,,,,).
—
kL(T
—
T~h.,) (2.7e)
LB. Humphreys et a!.
/
Morphological instability in float zone
Conservation of mass also requires that far from the freezing and melting interfaces
Cm=Co
as
~
(2.8)
as
~
(2.9)
where C0 is the concentration of the feed crystal, which is assumed to be uniform. At these distances, we control the temperature gradient so that 7/
—s
G~
Tm~~GM
as as
z —~ ~
—
~,
(2.10) (2.11)
as in fig. 1 where G~and GM must satisfy the compatibility requirements to be discussed in eqs. (3.6a) and (3.6b).
35
fig. 1. The temperature profile in the melt will be represented as the piecewise linear profile (TUTu+Guz, )0
T(z)
(3.2a)
t),
=
TL i.,
=
(3.2b)
TL + GLz,
L1 + h1 (x, t)
where Ti,, G~,and GL are to be determined. We require the temperature to be continuous and equal to the maximum heater temperature, at 7/,
7/,
z0, Tu=TL=TH.
(3.3)
Furthermore, the temperature gradients in the liquid give rise to the jump condition TU(z
=
0~)— TL(z
0)
=
=
G~ GL <0. (3.4) —
3. Basic state and scaling
We assume that TH, G~,and GM are known. Hence the heater profile of fig. 1 is physically
We now seek a steady basic state solution to the governing equations which is x independent and for which h 0. The temperature profiles in the feed crystal, the liquid zone, and in the growing crystal are taken to be identical to the established heater temperature profile. This assumes a large Biot number approximation so that there is large heat transfer across the boundaries of the system. Further the temperature profiles of the system (or without loss of generality, the heater) are not arbitrary but are chosen consistent with the assumption that K and KS are much larger than the solutal diffusivities, D and D~,i.e. ic >> D. Under this assumption and consistent with the scales to be defined in eq. (3.19), the energy equations (2.1), (2.2), and (2.4) effectively reduce to the Laplace equation, L~T=0, in the solidifying and feed crystals and melt. These simplifications are relevant to systems where the velocity of solidification V is slow. The temperature fields will then be represented by piecewise combinations of theform
equivalent to linear profiles far from the melt region and a line heat source in the melt region as shown in fig. 2, and mathematically represented by the Dirac delta function. The assumption B,>> 1 transfers this source term from the heat transfer boundary condition at the liquid/gas interface to the bulk liquid. Integration of eq. (2.4) with respect to z in the neighborhood of z 0 (the line source location) will determine the magnitude of the source. The constants in eq. (3.2) are determined from (2.6e), (2.7e) and (3.3) so that the temperature of the melt zone is (k~ ~PV
=
T=T*+Gz,
=
(3.1)
where T * is to be determined and G corresponds to the different temperature gradients as shown in
=
\
=
T TL
kL GM — —
—
~~~__)Z L
+ TH,
(3.5a)
\
(ks 2?V k L G~— ~ L z +
)
7/.
(3.5b)
Using eq. (3.2), we have G G M~’ L
(3.6a)
GL=~-~.Gs-—--~-—, ~2~V
(3.6b)
which serve as compatibility requirements upon
L. B. Humphrei’.i
36
Ct a!.
/ Morphological
the temperature gradients and velocity of solidification. These requirements are the energy balanees across the melting and solidifying interfaces, respectively. The solid temperature profiles are determined using (2.6a), (2.7a), (2.10). and (2.11). We find =
T
=
=
TH
—
L1G1,
+ T11 +
—
(3 .7h)
LmGii.
(3.8)
The concentrations in the feed crystal and melt are found by solving (2.3) and (2.5). subject to (2.6c. d), (2.7c, d), and (2.9) so that
—C C,
+
+ 1—k k
D)( 6,
t.,~
1]
+ k—
}
(3.9a) (3-. 9i~\ /
e -_/Dgt,,+z)j ~.
FTc
— —
G5)
—
<
KL
-
—
C,
—
—
T=
— —
C,
(3.14)
from eq. (3.6h) and U1 > 0. Note that eqs. (3.11) and (3.12) indicate that the height of the liquid zone varies directly with the maximum heater temperature. T , and inversely with the temperaH
ture gradients. U1 and G1. Increasing the magnitude of the gradients leads to a shorter zone height L = L1 + iS,,,. (3.15) . .increasing the maximum temperature results while in a longer zone. Further as V increases in eq. (3.12), L decreases toward 0 so that the location of the heat source C, = 0) is closer to the melting interface, Only in the case V = 0. G~ = ---
(symmetric heater profile) is the source midway between the freezing and melting fronts. These
(3.10)
results agree with the findings in refs. [101 and [111. We define the solute gradient at the interface, G~.to he
This measures the strength of the line heat source so that the energy in the feed-melt-growing cyrstal system balances the energy transferred from the surroundings. The planar freezing front position L1 is determined by replacing T and C in (2.6h) with (3.5b) and (3.9b) to give
G
=
-~
D
C1(k —1) k
—
(3 16) ‘
-
so that our concentration equations (3.9a) and (3.9h) can be written
2(i (3.11) The planar melting front position is determined from the solution of Lf=(__~~~~+TH_TM)GLt.
=
(3.13)
1 )J.
~—
—
-
L,,
T11.
G5 >~V/k5
,
Upon integration of the energy equation (2.4) in the liquid in the neighborhood of = 0 and using eqs. (3.4) and (3.6a, b), one finds the value k5
7/
This is reasonable since the temperature of the heating element must be greater than the equilihrium melting point. We restrict
(tI),SLz)
e x [(I — k )e
K~(G~1
(3.9a). The above can easily he solved using fixed point methods. The basic state gives rise to a few restrictions. For L 1 >0. eq. (3.11) leads to
(3.7a)
c~
—
which results from eq. (2.7b) using (3.5a) and
mC’,/k +
G5(: + L1) + Gvt( L,,)
The concentration in the solidifying crystal is constant. C,
in.viahilitt’ in float zone
—e (I
~
(3.17)
C~+
c
=
C,,
+ kG~D (I
T~+ c 0m
x (i + 1
—
h
x [1 (1)5/., + 1))]
G~ (3.12)
—
e
(t~ /)iitz
/1],
(3.18)
We non-dimensionalize the system 2. by thescaling characteristic tinieD/V, for diffusion, concentrations lengths with time withand D/V
L. B. Humphrevs et a!.
/ Morphological instability in float zone
with G~D/V. The scaled variables, denoted by primes, are D
x
=
h
-~
=
x’,
z
‘~h’ V as’
D =
-~
=
h
z’, 2
V
-~--~‘
D =
-~
h
37
pendent of the solute concentration, and unaffected by perturbations at the fronts. This might be termed the frozen temperature approximation [23] as applied to the float-zone configuration. Without eq. (3.25a), the latent heat at the interadjust the linear profiles as shown in refs. faces is from significant and the temperature fields here
~,
‘
C C—
0 —i--
=
G.D C —~----C’, ~
C0 —
-i--
=
G.D —~C,,.
[16,18]. (3 19)
We substitute eq. (3.19) into the remaining governing equations (2.3), (2.5), (2.6b, c), (2.7b, c, d), and (2.9). If we drop the primes and denote
As a result of these scalings, we define the following non-dimensionalized groups: 2, (3.20) 1= (TMy/~mGC)(V/D)
(3.21) measuring the effects of the surface free energy and its variation with interfacial concentration,
V
V
and £~/ = ~‘ñ~Lm, the scaled governing equations are
(3.27)
C,—C=C.,.,+C.
(3.28)
~‘f
=
+ Cm==)
~(Cmxx
~Cm= +
=
ç,,,
(3.29)
.
MI=mGC/GL.
(3.22)
subject to the boundary conditions at z—5~
MU=mGC/GLJ,
(3.23)
Cm=0,
the morphological parameters indicating criterion for supercooling, and
at z
=
(3.30a) +
—~
h I
—
D
DS/D,
=
(3.24)
C
M~j1h
—
1+
a ratio of solid versus melt solutal diffusivity. In the stability analysis to follow, we make the fur-
3/2
0,
=
(3.30b)
(i +h~.,)
(1
+ h11)[1 + —~ + h
ther assumptions
(k
1)C]
—
=
C,
C.,h1.,,
—
(3.30c)
1h+1 ~V/kLGL<
(3.25a)
5=k~.
(3.25b)
k
C——e~+M + ~
(3 30d
~cC)hm..,x
3~2 From equations (3.6a, b), it follows that G~ G~ and G~ GM. =
=
(i (3.26)
This allows for replacing the temperatures in the
governing equations (2.1)—(2.11) with the linear basic state temperature fields shown together with the basic state concentration field in fig. 3. Henceforth, eq. (3.5a) will be identical to eq. (3.7b) and similarly, eq. (3.5b) and (3.7a) are equivalent. As a result of the small latent heat assumption, eq. (3.25a), ic >> D, and k 5 = kL, the heat (created/required) at the interfaces (freezing/ melting) may be neglected. The temperature field will thus be fixed, linearly as in fig. 3, inde-
Cm
+
‘
h~.,)
C,
=
(3.30e)
(I + hm,)[1 + (k =
—
h
~
—
ms
—
l)C]
kD~C
—
m
k
C
h
(3 30f
mx m v)~
S
The scaled basic state is -
C~ 1e ~
(~ —
—
M
h
0
I
hm
=
~
e
—
=
~
0.
[s”,.,, Z
3 31
—
exPi~
—
,
38
LB. Hurnphrevs ci al.
/
Morphological instability in float zone
4. Linear stability analysis We now allow disturbances to the basic state: h 1=0+eH, = 0 + H,,,.
(4.la)
hrn C
1
=
C,,,
[1
=
(4.lh)
e _ c~’+> + cC *
—
e~”~]
—
If we solve the linearized thermodynamic equation (4.Sh) for H we obtain 2F+ M~‘—1). (4.6) H= C’/(a
(4.1 c)
e©
‘~“~>
+
~
(4.ld)
We substitute eq. (4.1) into the scaled equations (3.28)_-(3.30), linearize with respect toe, and drop the asterisk from the perturbed quantities. Further, we assume for all dependent variables a separation of variables form qi(x, z, t) e°’e”’F(z), (4.2) =
where the growth rate, a, determines the stability and a represents the disturbance wave number.
The governing equations become C+C—(a+a)C=0.
(4.3) (4,4)
2~)c,, 0. =
+ c,,, (a + a These equations are subject to the following boundary conditions: at z
~l’M~—° 0, then H —s 0. This is consistent with eq. (3.22), since there would he no morphological instability when there is zero undercooling. Substituting eq. (4.6) into eq. (4.5c). mass conservation of solute at the solidifying front becomes 21+ +Mk )C 11t _1 +(k—l)C—C=0. (~
(4.7)
a
with the first term representing a perturbation in the concentration field due to the change in the shape of the interface. As M1 approaches I, the critical value for instability, we see in eq. (4.7) that the first term is proportional to 1/u. If we assume ~2 small, then this term would dominate the diffusional term C. We take a~1.
(4.8)
so that the wavelength 3’.
—
3’
2~D ~ ~‘
=
(4.9)
—~
satisfies C’,,,
0,
=
(4.5a) (4.10)
at z = C ± H(1 (a +
—
M~t)
k )H + (k
—
—
Fa2H
l)C
—
0,
=
C
=
0.
(4.sb)
implying long wavelength instabilities.
(4.5c)
tems with small k so that
From eq. (4.8) we rescale the problem for sys-
atz=2/
k=a2k, a=a2&
M~J’J+ F*a2H~,
C+ ~
(4.lla)
(4.llh)
(4.5d) C
C,,,
—
+ H,,,k1
—
~
i)
e_1~,c>~,,,) + D
0, (4.5e) =
(k—1)C—C±kDC,,,
with the first term of eq. (4.7) retained as an order one quantity near ML = 1. If we likewise solve eq. (4.5d) for H,, we obtain H,
C
=
=aH,,,[(k—l)e~’~~’~kI
(4.12)
M_e~’~’,_a)F*
-
and substituting into eq. (4.Se) we have (4.5f) where =
C’ —
—
~~(1
—
e~~)j.
(4.5g)
x [(1
+ —
Mt
—
—
Dt) e~©’C~)
+ D
=
0.
(4.13)
LB. Humphreys eta!.
/
Morphological instability in float zone
The governing equations are now
with B an arbitrary constant of integration and (4.14)
2(6+1)C=0.
C+C—a
DCm== + Cm=
—
a2(ä
+ ~i5)Cm
=
0,
B*=Be’_©
(4.15)
(1
F
xli
L
subject to conditions (4.5a) as z —s
39
e~1~m>+ (h_i)
~_1)
—
+
ML~1 e —
—
~m)
~
—
a2 F *
and
~,
1 ] (4.20)
At order a2 we obtain at z
=
a2(~+ff)C
2k)C—
a2F+M~—1
C+(1—a
at C
(4.16a)
C 1 =D* + Fe~—(6+ 1)B e’(l +z), Cmi~~F*exp[z/D]
c
Cm +
-
~
x
[(1
—
~
+
(1
—
=
=
0,
(4.16b)
—
—
a2F*
/
~ e~’’)
—1) e~’”’~
M~1 e~~”> —
—
k/D]
0. (4.16c)
—
a2F* (4.24)
=
We assume a power series expansion for all dependent variables in the problem, C=C
(6+ 1)(1 +~m)I, (4.23)
1) e’~””’~
+ ~(D_i
—
D*=Be~’,,
e’~2”~
[6(a2~
e~”+ F)
+Bi(D*
a2~~)C
a2C M~t —
(4.22)
Satisfying the boundary conditions, we find F* B* [(~ + 6)(~ +~m)
a2F*
e1m)+~i]
_~1)
+
x
—
(4.21)
exp[—z/71.
_(6+f~)B*[h+z}
Z=~flm
M~t —
C
=0,
with F an arbitrary constant of integration. We obtain an expression for the linear growth rate, a, by substituting C 2 term 0 and C1 into the order a of eq. (4.16a). Unscaling ~ and 6, we find
2C 0+a
1+..., 2Cmt +
...,
(4.17a) (4.17b)
a=((1_e_~n~>)[(1_M~)a2_a4FI} x{(a2F+ M/1)
Cm= Cmo+a
and substitute these into eqs. (4.14)—(4.16). The goal is to find an expression for the growth rate, a, valid in the long-wavelength limit. Alternative to the expansion procedure presented here, one could also accomplish the above by exactly solving the eigenvalue problem, eqs. (4.3)—(4.5), and then applying the long-wave limit to the resulting growth rate expression. At leading order the solution is C B e~, Cm~=B*e_T,~, 0
=
(4.18) (4.19)
_l’~i±.2”m>M*(2p+Mi —
k —
1
—
e’~’-’~M*N *
(4.25)
where M
*
=
M M~’— 13 ~
2F* —
a
—
(4.26) a2F*
2F + M/1 — 1)/(a2F + M~1). (4.27) N* = (a Examining eq. (4.25) in the limits 2’~—s0 and
LB. Humphrecs ci a!.
40 —* ~,
/
Morphological invtahilitt’ in float zn/i,’
which recover a directional solidification
geometry, we find that eq. (4.25) reduces to a= (I
—k.
‘ri
(4.28)
This expression is equivalent to that found by Young and Davis [15] with their R = 0 (no convection). Considering the limit M 1 1 with a ~ I. we find eq. (4.25) reduces tO 41 j — k. a = (I — e ‘~‘> )[(1 — ~-t )a — a (4.29)
‘~
‘~.9
[1
--
/2
LT
.
.
Hg. 4. Phase diagram plot of the directional solidification I I)S) and float zone (FZ) concentration profiles for identical material and s’sstem paranseiers.
fig. 4. For identical material properties and 1cmSince a> 0 implies instability, then (1
—
M
1 ‘) > 0 is a destabilizing term. Therefore, when M1 > I we have ntG~/G~ > I. which is a criterion for undercooling. On the other hand, increasing k and the presence of the surface free energy. F. are stabilizing effects. From eq. (4.29) the magnitude of the undercooling (1 — M~) decreases by the factor (1 — e ~‘1~ which indicates the height of the liquid zone. There is no instability when ~ =~,, = 0. In this case we have all solid and no melt. On the other hand, the most unstable configuration is .~-= 0. .~/ -~ ~, which is directional solidification. Hence. a float zone configuration is always more stable to long-wave morphological disturbances than a directional solidification geometry. Further. the shorter the melt height. the more stable is the zone. To understand the mechanism responsible for the increased stabilization we examine the concentration profile in the liquid, Because of the presence of the melting interface of the float zone, solute diffusion in the axial direction is hindered. This leads to a higher solute concentration in the liquid zone in comparison ~o the directional solidification system This can he seen from eq. (4.21) where a term of the form
perature gradient G1 . the float zone system has a flatter axial concentration gradient at the solidifying front. Hence, the float zone is niore stable. Using eq. (3.11) we see that L —sO, as U1 increases. As G1 increases, M1 decreases and the system is more stable. Likewise. ~,, —~ 0 as U1 becomes large, as seen in eq. (3.12). Hence, steep gradients are necessary for morphological stahility, which is no surprise. In practice, though, steep gradients are to he avoided due to buckling and structural concerns. Eqs. (3.11), (3.12). and (4.29) suggest balancing the above steep versus non-steep issue so as to keep the zone height small by decreasing the maximum heater temperature. T11. which gives the same effect as increasing the temperature gradients. As an example, consider the data listed in table I. Using the assumption that U1 = — U1 . fig. 5 shows the neutral curve (a = 0 in eq. (4.29)) as a function of the temperature gradient U1 and max-
D*(l — e ~ ) (4.30) will appear since F is arbitrary. The source of this term is the boundary conditions applied at the melting front, in contrast to the directional solidification system where we require C 1 —f 0 as c ty~. Note D* 0 as -~ ~. The effect of the term is best illustrated by the phase diagram shown in
D V A
Table I
M iterial and s~siem~ iranleters for
(Jc(J I
Parameter
Value
T
1213K 4cni~s 4x1() l.3>< 10 4cmz,/s
,~,
1)1 ._4 K~t~
~,
0.5 wi/
Is,~
ft39 W/crn K
-
L. B. Humphreys et a!. (‘K 1
/ Morphological instability
STSBL
F
5. Weakly nonlinear analysis
=5
54~-
41
as fig. 5 for systems of interest would show desirable heater gradients and temperatures which would not give rise to morphological instability.
~
G.,,~.,.
in float zone
____
In order to obtain the weakly nonlinear behavior of the system, and investigate instabilities of
_____________
-
‘
the planar front, we examine near the critical point
JIISTIBLE
52
ML 1 + , for c ~< 1. We rescale eqs. (3.28)—(3.31) as =
~
I
[
I
I
1260
1280
1300
~I
xe1”2 k,X. c2tz =
1200
1220
1240
1320
T,,(’K)
Fig. 5. Neutral stability curve for the directional solidification (DS) and float zone configurations corresponding to the data of table 1.
=
= =
Z, T,
hf = ci], hm = C = C, C,,, = Cm.
(5.1)
(5.2a)
These scalings are determined by eq. (5.1) and the results of section 4. Additionally, since we have that k O(2), then ~ as defined in eq. (3.21) will be large. We take =
imum heater temperature, TH. This curve is obtained by solving for the planar positions, Lf and Lm, using eqs. (3.11) and (3.12) and then using a Newton iteration to find the value of GL which gives a 0 for a fixed We note that eqs. (3.11) and (3.12) are valid, independent of eq. (3.25), provided eq. (3.6) is satisfied, and thus give a =
7/.
means with of determining theline melt zone height consistent the proposed heat source. Fig. 5 demonstrates that for the data chosen, the morphological stability characteristics of the float zone are similar to the directional solidification geometry for heater temperatures about 25 K above the critical value set by eq. (3.13). For these temperatures, the melt height is sufficiently large that the presence of the melting boundary does not influence the stability criteria. Note that as the zone height is smaller for lower values of TH, the critical temperature gradient for stability decreases, so that the system is more stable. One possible application of these results would be to the hot-wall float zone technique. Here the temperature gradients are kept small and the heater temperature is kept low near the melting point. The reason for this design is to reduce thermally induced imperfections and possibly reduce the level of surface-tension driven flows. A result such
~
(5.2h)
~/.
=
that ~.c 0(e) and the surface free energy varies slightly with the interfacial concentration. The so
=
governing equations become (2CT~ C
7=C~+ C77, 2CmT, + Cm7z) = — C,,,7 + E
~(C
(5.3) (54)
subject to conditions at z —~
Cm
0,
=
at Z C
=
(5 .5a) —~
+ cH
eM~tH + 2F(1
—
—
~ii~C/c)
x H~~(1 + c~H,~)3”2 = 0. (5.5b) (3H~+ i)[i + (2~ — i)C] = C~.— c2C~H~, (5.5c) at Z
=
£~=‘m+ c H,,,
C
—
e”~”~ + M~1Hm+ 1 + c2(F*
x
Hmxx(1 + c~H~~)3”2,
=
C,,,
=
C,
—
(5.5d) (5 Se)
LB. Hurnphreys et al.
42
2k
(1
±c~H,,,T)[i + C(c
—
1)1
C
=
Morphological in.stahilitt in float zoni’
2C~H,,,~ 7
——
/
—
c
Consequently, from eq. (SlOe) we find the dcviation from the planar melting front.
(5Sf) H
We seek solutions in powers of c as follows: C
I
=
e>
—
+ cC
~‘~1
2C +
H
(1— et.~~”s)) exp[(.~ 2C,i, .... +cC~,1±c
=
H
=
2H +
—
:)/D1
. ..,
=
~(M~
c> “—I)
Note the melting front is more stable than the freezing front as H,,, 11 = 0. The concentration in
(5.6h)
the melting crystal is found by solving eq. (5.9) subject to eqs. — (5.lOa) and (5.lOd).
(5.6c)
C,,,
~
=
1
M
+
1~ eJ’1
0 + H1 + c H,,,
H,,,0 ± H,,,1 + cH,,7 ± ....
=
(5.6d)
‘~
At leading order the system is identically satisfied by the basic state. At 0(c) we find C
H 0=0.
=
DC,,,,77 +
(/77
=
0.
unknown, so we consider the 0(c’) problem (only
(5.8)
C’,//±C’)/=aV\e
(5.9)
at Z
those equations needed to complete the analysis are listed):
at Z=~,1
=
+
(/7
—
+ C,
=
kH0
(5.16)
+ H01.
C37+C7=O.
(5.17)
Eqs. (5.15) and (5.16) give
~
C7 ± H1 at Z
(5.lOa)
(5.15)
~
=
at Z—* ~
z
(5.14)
).
The freezing front interfacial position. /-1~.is still
C,7
at
-
D
subject to conditions
(/7=0.
I
(57)
The O(2) system is C77+C7=O.
e
I
—
xexp(
=
(5.13)
.
(5.6a)
1 ±c C
-I
—
C
~H1~+ F[1
—
~~H0IH0 ~
0, (SlOb)
=
0.
=
(SlOe)
~
C’~ kH0 + H07 =
~
—
±~
~
I/i
±a5.~Z e >‘~, (5.18) 3) termsconstant of eq. (5.Sb). (5.18) andfrom (5.17) The /3 can Eqs. he determined thelead 0(c to
C,,, 7
—
C
+ H,,,1 [(D
—
1) e
—
~
an evolution equation for the interfacial shape. (5.lOd)
C= H,,,t(M~t
C>>’I+.Y’Ill)).
(/7+C,=O. -
-
(SlOe) (5.lOf)
=
a e
-
±kH1+H1~=0.
-
We find the concentration in the melt by solving eq. (5.8) subject to eqs. (S.IOb). (SlOe) and (S.lOf) to give C
[—H0H0., ).~+ Ho,\v+FHO.\.\.v,v F~.(H11H11~~) ~~](1 —e
~
We rescale eq. (5.19) ,to obtain a two parameter evolution equation using the following transformations:
(5.11)
~
H11=F, where a
=
H~— H11
T=
T
1 —
(5.19)
FH() xx + F~ H11 H0,~~.
(5.12)
,
X=Ft
3~
—
C, -
LB. Hutnphreys et a!. / Morphological instability in float zone
Eq. (5.19) is rewritten as
43
ities of the planar front, F= 0. We linearize (6.1) about F 0, expand in the normal modes F= F* ~ (6 2 =
[(1
—
F)F~]~+
—
)~+ KF+
~
=
(5.21) and find where (6.3a) K=kF/c11
—e’~”~
(5.22)
.
Eq. (5.21) is of the same form (with ii,,
0) as the
=
one parameter evolution equation found by Sivashinsky [141, and Young and Davis [15].
where ~2
+
=
(6.3b)
~2
‘I
Mathematically, eq. (5.21) will behave the same as in refs. [14.15] with increasing K corresponding to stabilization. However, the physical interpretation is different. K as defined by eq. (5.22) is effectively larger for the float zone system, due to the
Eq. (6.3a) is the scaled form of the growth rate expression eq. (4.29). The critical wave mode for the most unstable disturbance is w,, l/v~ and the critical parameter for the onset of instability is K,, ~, with K < corresponding to instability. To follow the weakly nonlinear development of
1
=
=
-~
e~”~”~ term, than the directional solidifi-
instability we examine the response of the planar
cation system for equal material and system parameters, k, F, and c. Thus, the float zone will be more stable. However, in the limit ~I1,, 0, these evolution equations are known to predict subcritical bifurcation to two-dimensional band structures and exhibit unbounded numerical simulations as i- —s oc [15,17]. To investigate the effects of interfacial surface free energy variation with concentra-
interface for a special class of perturbations with critical wave mode to.,,. In particular after Segel and Stuart [28] and Sriranganathan [24] we assume that
tion (ji,, ~ 0) we now examine the bifurcation behavior of eq. (5.21).
(6.4)
—
=
cos(co,,fl + B(T*) cos(w,,~/2)
F= 6[A(T*)
~1
cos(~w,,31/2)1+ 8F2
x
+ 6F3 +
where 2T, we have introduced the slow time scale T* “~6
6. Stability characteristics An evolution equation for a three-dimensional solidification front can be determined by extending eq. (5.21) to find
6 ~‘sz1. Substituting into eq. (6.1) to third order we find that the amplitudes A and B satisfy the equations 2
~1
1
A (T*)=a,,A —aB —A(atA~+a 7B).
(6.Sa)
B’(7*)=a,B_4aAB —B[2asA —
v
(FvF)
=
0,
where =
and with
± (a1 +
2a2)B/4],
(6.1)
(6.5b)
where 2a,, = 6 a = (—ji,,
+ y
scaling to ~ as x scales to
~.
This
+
1)/16.
—
ji,,)(2
(6.6b)
a 1
equation governs the nonlinear behavior of the interface and can be used to investigate instabil-
(6.6a)
K.
—
=
—(1
—
5jiJ/16(4 K). 2fL,, )/32( ~ K). —
—
a2
=
—
3(1
— ji,,
)(1
—
-
—
(6.6c) (6 .6d)
L. B. Hun,phretn ci a!.
44
Morpholo,g,cal ,n,rtahilii~in float .zoiie
2D bands
hexagonal nodes I
II~I~1II
—f—r—r—’~rr—1~ ~III~F I
liii~
—
—~
/ I IN/ISO I:
11:111 IIIII.I11110
NO
‘.1.501.1
0105 1100
I.N’.0711H
11000,,
.1
~ IIJI
N~~LI/lOIN
1.
1/001
,.~. 2 I
i
2
_______
LIII
N
,.‘ I
II
I
I
I
II
I
Fig. 6. Regions in the p.~— K parameter space yielding stable 2D band structures. Neither stable nor unstable 2D bands exist in the ‘~no solution regions.
Fig. 7. Regions in the /L~- K parameter space yielding stable 31) node structures. Neither stable nor unstable 3D nodes exist in the ‘‘no solution ‘‘ regions.
For the chosen values of ji~,a as defined by eq.
solutions (A <0). which project into the solid, exist subcritically (K> ~) for the range 0.48
(6.6h) is small in magnitude allowing the quadratic terms of eq. (6.5) to be valid to third order. The critical points of eq. (6.5) corresponding to A’(T*) B’(r*) 0 are classified as [24,28] =
<0.93. We find no regions in which hexagonal cell solutions (A >0) (projecting into the melt) are
=
I: A
B
=
0. planar interface.
=
(6.7a)
‘~
‘~1I’
05959
II: A III: A
=
=
±,/~- B=O. 2D bands.
v
,
I~N -
(6.7h)
a
1
i.~ 11
2 ± a,,(a
—2a ±~4a
1 ± 407)
~,-
-
--
a1±4a7 B= ±2A. 3D hexagonal solutions.
/
(6.7c)
We do not examine the Class V solutions listed in ref. [24]. To determine whether or not the structures corresponding to the solutions listed in eq. (6.7) are realistic we investigate their stability by linearizing eq. (6.5) about each. The Class I planar interface analysis reveals stability for a,, < 0. as expected. The Class II and III results are shown in
figs. 6 and 7. Fig. 6 demonstrates that stable 2D bands exist supercritically (K < ~) for the range 0.4 < 1. Fig. 7 shows that hexagonal node
1’
I
/
-
/
.
-
.
7
,,,:,
..
I
. Fig. 8. Bifurcation diagramI obtained from the nonlinear analy= 0.5959. Solid lines denote stable solutions: dashed
sis for ~
denote unstable.
L. B. Humphreys et a!.
/
Morphological instability in float zone
stable. These results are consistent with the bifurcation analysis performed in ref. [24] on the full problem for k <1. There
a’y/aC> 0
(6.8)
[eq. (2.6bb)] is necessary for stability. Since we find stable solutions for jii,, > 0, then p. <0 by eq.
(3.21). Hence,
45
points found in eq. (6.7). The stable regions are denoted by solid lines. Numerical simulations of the full equation (6.1) (to be examined below) reveal critical values of K below which the 3D and 2D structures are unstable, while the linear analysis predicts stability. To examine the nonlinear development of these
solutions we perform a numerical integration [29] of eq. (6.1). The domain for all integrations is
ay/aC>o is also required in this analysis. Further, for the
case Ii,, 0, fig. 6 reveals that 2D bands bifurcate subcritically and are unstable for this case, which agrees with the results of the bifurcation analysis in ref. [17]. Thus the inclusion of 0 yields stable solutions, not found in the small k, F 0(1), Sivashinsky [14] type evolution equations. Fig. 8 shows a typical bifurcation diagram for the steady state amplitude, A + B for the critical
(~
~)
E
~
=
ji,,
‘‘
=
,
x 0, Vito,,
~,, =
[0 p1 x [0 Q] ‘
J
(6 9
I
which corresponds to 3 wavelengths in the ~ and ij directions for the leading order terms of eq. (6.4). Periodic boundary conditions are applied at the edges of the rectangular domain. An explicit finite difference scheme with second order approxima-
~ (~urntsof ~f~’
~
O~53.315
0
i~ 30.781
-1.474 < F < 1.038
Fig. 9. Two-dimensional band structures resulting from the numerical simulation of eq. (6.1) for 4,,
=
0.5959 and K
=
0.21.
1.. B. Huinphrei
46
11
i
ci a!.
8.Ioi’pholog, a! in ctab,l,ti ii, float zol,s’
~
~H
F~~un1tsof~..,/~
O~53.315
0
<
F < 0.883
Fig. 10. Fhree-dirnensional node struclureN resulting from the ni,n,crical N,mulation of eq. (6 1) for ji,
((5751) and ,~,
0.21
tions in space and first order forward differencing in time is employed. A SO X 50 grid is used for all calculations to follow with several runs on a 100 >< 100 grtd as a check on the stability of the scheme, Initial conditions similar to the form of the leading term in eq. (6.4) with various magnitudes for A and B (typically of the order 0.05 to 0.1) as well
minimum root of the steady structures. The presence of jii,, ~ 0 stabilizes the runaway ohsersed in previous numerical integrations of the Sivashinsk\ type evolution equations. Fig. 13 shows the hifurcation diagram obtained by the numerical integration for a fixed ji,, as K varies. The amplitude plotted is tip minus root. Here dotted lines corre-
as the initial conditions A* cos(2~/P) cos(2~si /Q) are input. Typical results yielding steady 2D
spond to unstable solutions. This diagram differs somewhat in detail than fig. 8. Here a different
hand solutions and 3D hexagonal node solutions are shown in figs.. 9 and 10, Whether one obtains
amplitude is plotted which corresponds to arhitrary amplitude solutions of eq. (6.1). while fig. 8
hands or nodes depends upon the initial condi-
is restricted to small amplitudes. Nevertheless the subcritical (3D) and supercritical (2D) trends are
tions in those (K. p.,,) regions where the linear analysis predicts both are stable. Figs. 11 and 12 show typical tracking of the maximum tip and
qualitatively the same in each. The general trend is that as velocity of solidification V increases (cor-
Tip
L. B. Huinphreys ci a!.
/ Morphological instability
in float :one
~i,,i~irri
I
I
47
~E 1.1
I
I
I
I
I
I
I
I
I
I
I
I
I I
I
I
I
I
I
I
1.0
0 I
0
ii,,,
50
Ii,
tOO
II,
t50
(1
i,i,i,,,i
200
—
II
250
interface undergoes a transition to small amplitude hexagonal node solutions and then to bands,
finally losing stability to large amplitude structures which are not predicted by the scalings introduced in eq. (5.2).
7. Conclusion We examine morphological instability in a crystal sheet, float-zone geometry for which the height of the liquid zone, L L1 + Lm, is smaller than the width and thickness of the sheet. Here L1 and Lm denote the planar positions of the solidifying and melting interfaces. We calculate these positions using equations (3.6), (3.11) and =
—
350
400
e~r~1>)
Fig. 11. Maximum cell tip height development with respect to time (2D structures,
responding to increasing the magnitude of c in eq. (5.1) yielding smaller K, eq. (5.22)) an initially flat
II
300
4,,
=
0.5959. K
=
0.21).
(3.12). The controllable experimental parameters are the temperature gradients, G 5 and GM, in the growing and feed crystals, and the maximum ternperature, TH, of the heat source. As discussed by
Young and Chait [11] for large heat transfer across the boundaries of the system, the melt temperature profile is identical to the heater profile. The above result is applicable to this situation and gives a simple means of estimating the zone height. We calculate basic state concentration profiles for a planar interfacial system. A linear stability analysis of this system, in the limit of small segregation coefficient and limited to the frozen ternperature profile shown in fig. 3, shows that the
growth rate of disturbances, eq. (4.25). is a modified version of that for a directional solidification system. We find that shorter melt zones are more stable due to solute redistribution caused by the
48
L. B. lluniphreis ci a!.
/
VIorpholog,cal ,nstabil,tt in float zo,,i’
sot Depth (~
—.2
IIIIiIIIIIIII,IIIIIiIIIIiIIIIiIIIIiIIII
.10
I
I
I
I 50 I
I
I~IIIIIII
I
10))
15))
( ~ ~2
I
I
20))
F (1
I
I
25))
I
I
I
I
I
I
319)
I
I
350
I
I
I
4)9)
DI
e~’/V)
—
Fig. (2. Minimum root depth development with respect to time (2D structures.
4~
=
0.5959. K
=
0.21).
presence of the melting front. The shorter zones have flatter axial concentration profiles making them less susceptible to undercooling. Thus our results indicate that to suppress morphological instability in a float zone system. one needs to decrease the liquid zone height which reduces the axial concentration gradient. This can be accomplished by increasing the temperature gradients in the feed and growing crystal or decreasing the maximum heater temperature. On the macroscopic level, this temperature profile tailoring would need to be done in conjunction with efforts to keep thermally induced stresses to a minimum.
We perform a weakly nonlinear analysis near the critical conditions for instability. The result is an evolution equation giving the departure of the melting and solidifying interfaces from a purely sinusoidal shape due to nonlinear effects. This equation is similar to those developed in previous small segregation coefficient, 0(1) surface free energy, long wavelength analyses hut has the added feature of surface free energy variation with interfacial concentration. Both bifurcation analysis and numerical simulation of the equation reveal transition from a planar front to 2D hand or 3D hexagonal node structures. Qualitatively, this
Hence, a lower heater temperature with shallow
agrees with results obtained for a non-simplified
thermal gradients in the feed and growing crystal is the most desirable situation. Our results, fig. S. suggest this can be accomplished while keeping the solidification front morphologically stable.
system [24]. The point being that evolution equations, while developed under severe restrictions, yield simpler problems which retain much of the physics of solidifying systems. The results derived
L.B. Humphreys ci a!.
/
49
Morphological instability in float :one
I ‘
I
I
I
I
I I
I
I I
7c=o.5959
I
I
I
3-0 NODES
I
‘I’ip—Rt:t I
_.~~.~—~-;i(
D~
l~,E~’)
-
LIN
2-DBANDS
in
X
-
-~ in
I
-
)
-
IN
I -
I —
-in
I
\ ‘I
-
S S l~
in
I in
.0
I w
in
ro
IN
C’i
I
I I
I I
I
in
in
iO
.0
01
iN
‘-
,-
I w I
I
(N
I in
IR~
K Fig. 13. Bifurcation diagram obtained from numerical simulation of eq. (6.1) for 7,, = 0.5959. Solid lines denote stable solutions and dashed denote unstable: X corresponds to calculated values.
from this approach predict solidification structures which are determined by the interplay of surface energy, thermal and solutal diffusion, and interface attachment kinetics. These results are suggestive of the behavior of these systems even though there may be no materials which exactly satisfy each of the restrictions.
Donald Wakefield for his assistance with the numerical computations and graphics. References [1] J.A.F. Plateau, Staiique Expérimentak ci Théorique des Liquides Soumis aux Seules Forces Moléculaires (Gauthier-Villars. Paris, 1873). [21J.W.S. Rayleigh, Proc. Roy. Soc. (London) 29 (1879) 71. 131 W. Heywang and G. Ziegler, Z. Naturforsch. 9a (1954)
Acknowledgements
561.
[4] CE. Chang and W.R. Wilcox. J. Crystal Growth 28 (1975)
This work is supported by NSF Grants DMS-
8604047 and DMS-89-57534, and NASA Grant NCC-3-104. The authors also wish to thank
[5]
8.
CE. Chang and W.R. Wilcox. Inter. J. Heat Mass Trans-
(1976) 335. [6] fer P.A.19Clark and W R. Wilcox. J. Crystal Growth 50 (1980) 461.
50
LB. Hllfllphrev.c ci a!. / Morphological in.oalnlity in float co/Ic
[7] AK. Sen and S.H. Davis. J. Fluid Mechanics 121 (1982) [8] [9] [101 [11] [12]
163. CL. Lai. S. Ostrach and Y. Karnotani. 1985 US—Japan Heat Transfer Jo1nt Seminar, 1985. SR. Coriell. S.C. Hardy and MR. (.ordes. J. (olloid Interface Sci. 60 (1977) 126. J.L. Duranceau and R.A. Brown. J. Crystal Growth 75 (1986) 367. G.W. Young and A. Chait. J. Crystal Growth 96 (1989) 65. W.W. Mullins and R.F. Sekerka. J. AppI. Phys. 35(1964)
[18] G.W. Young and SF1. Da’is. SIAM J. App). Math. 49 (1989) 152. 19] D.A. Kurtze. Phys. Rev. B57 (1988) 370. [2).)] J.M. H’sman. A. Noxick-Cohen and P. Rosenau. Ph~s. Rev. B37 (1988) 76).)3. [21] K. Brattkus and S.H. Davis. Ph~s.Rev. B38 (1988)11452. [221 D.S. Riley and S.H. Davis. SIAM .J. AppI. Math., submuted. [23] J.S. Langer. Rev. Mod. Phys. 52 (1980) 1. [24] R. Sriranganathan. l).J. Wollkind and D.B. ()ulion. J. Crsstal Growth 62 (1983) 265.
444. [13] D.J. Wollkind and L.A. Scgel. Phil. Trans. Roy. Soc.
[25] D.P. Woodruff, Phil. Mag. 17 (1968) 283. [261 H.S. Chen and K.A. Jackson, J. Crystal Growth 8 (1971)
London .A268 (1970) 351. [14] G.I. Sisashinsky, Phvsica 8D (1983) 3388. [15] G.W. Young and S.H. Davis. Phys. Rex. B34 (1986) 237. [16] A. Novick-Cohcn and G.I. Sivashinsky, Phvsic~i 201) (1986) 237.
181). [27] D.J. Wollkind and S. Raissi. J. C’r~sialGrowth 2h (1974) 277. [28] L.A. Segel and J.T. Stuart. J. Fluid Mech. 13 (1962) 289. [29] J.,A. Hcminger. Tech. Report ~ 88-07. I)epartnient of
[17] G.W. Young. S.H. Davis and K. Brattkus. J. C’rvstal Growth 83 (1987) 560.
Mathematical Sciences. The University of Akron.