Crystallization stability during capillary shaping

Crystallization stability during capillary shaping

Journal of Crystal Growth 50 (1980) 45—50 © North-Holland Publishing Company CRYSTALLIZATION STABILITY DURING CAPILLARY SHAPING II. Capillary stabili...

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Journal of Crystal Growth 50 (1980) 45—50 © North-Holland Publishing Company

CRYSTALLIZATION STABILITY DURING CAPILLARY SHAPING II. Capillary stability for arbitrary small perturbations G.I. BABKIN, E.A. BRENER and V.A. TATARCHENKO The Solid State Physics Institute of the Academy of Sciences of the USSR, Chernogolovka 142432, Moscow district, USSR Received 3 October 1978; manuscript received in final form 1 June 1979

The problem of stability of the crystal shape grown from the melt is treated. The contour restricting the cross section of a crystal is subjected to an arbitrary small perturbation that can be expanded in a series over a corresponding set of orthogonal functions. Solution of a linearized two-dimensional capillary equation determines the time dependence of the expansion coefficients. The shape of a cross section is stable if the coefficients of all the expansion terms decrease with time. The analysis shows that, in case of the zero-harmonic stability, the higher harmonics are stable too. A possibility is indicated for the development of a first harmonic instability during the growth of thin fibers. Such an instability, resulting in the crystal bending, has been observed during the growth of sapphire fibers, 300 ~imin diameter.

1. Introduction The growth of shaped single and polycrystalline material by the capillary shaping technique is a problem of particular interest from both scientific and technological points of view. The best-known technological variants of this technique are the Stepanov method [1] and EFG [21. Recently, some other methods have appeared [3,41.The capillary shaping technique involves the presence, in the melt, of a solid-state body which affects with its walls or sharp edges the process of crystaffization. However, the walls and edges alone do not define the dimensions of the crystallizing sample. The shape is set by the conditions of the crystallization process. Therefore, it is very important to choose schemes for shaping and growth conditions that result in a stable process. Previously, we investigated the problem of stability of the crystallization front, and the stability of a characteristic cross section of the growing crystal profIle [5,6]. The cross section of the crystal was subjected to a homogeneous perturbation (a change in the cylinder radius or in the plate thickness). Such an

In this paper, an attempt is made to investigate in detail both the dimensional and the shape stability of the growing crystal. For this purpose, a contour which defines the crystal cross section is subjected to an arbitrary small perturbation, which can be expanded in a series of a corresponding set of orthogonal functions depending on the shape of the unperturbed contour. Next, the time dependence of the expansion coefficients is calculated. The shape of the cross section is stable if the coefficients of all terms in the expansion decrease with time. The first term of the expansion corresponds to the homogeneous perturbation, that is, to a change of the contour dimension. In order to calculate the dependence of contour shape on time, it is necessary to know the deviation of the angle between the horizontal line and the tangent to the melt surface near the crystallization front from the equilibrium value s~(see fig. 1) [6]. This deviation can be found in the following manner. To the unperturbed crystal contour, there corresponds the unperturbed shape of the meniscus, described by the function Z0, which satisfies the cap-

approach suggests that the characteristic profile dimension is stable, but it is still unclear how to maintam the shape of the growing crystal.

ifiary equation with the corresponding boundary conditions and the equilibrium angle For a small change in the crystal contour, there wifi correspond ~.

45

46

G.I. Babkin et at

the solution Z

/ Crystallization stability during capillary shaping. II

Z

0 +Z1 satisfying the capillary equation linearized with respect to a small correction, Z1 and a corresponding angle a = a~+ Sa differing from a0 by the amount Sa. Since all equations are linear with respect to Z, one can analyze the behaviour of a small and sufficiently smooth perturbation by considering the superposition of expansion components over the corresponding set of orthogonal functions. In this paper, we consider the problem of stability of the crystal shape in the growth of plates and of circular cylinders.

L

Zoixa0

(3)

.

Here 2x0 is the size of the shaper, 2a0 is the plate thickness, r~= pg/y, d = P/pg, P is the pressure of the melt, p is the density of the melt,g is the acceleration of gravity, and y is the melt surface tension. The angle between the horizontal and the tangent to the meniscus at the crystaffization front is equal to the equilibrium value a0. Then tan a0 zZ~_ao ~‘~i’c=a0 P0 hence 4 P. The solution of the perturbed meniscus shape is assumed to be of the form:

2. The growth of plates

Z

The equation for the unperturbed shape of the meniscus, in the coordinate system of fig. I, has the form [7]:

where Z0(x) satisfies eqs. (l)—(3). By substituting this solution into the two-dimensional capillary equation, and retaining the first-order terms of Z1 only, we obtain a linearized equation for Z1(x,y):

(1) At the shaper, the boundary condition of “catching” [81 is realized:

Z

=

Z0(x) + Z1(x, y),

2z a 1 az13z0p — —

~-~-

(1 +

~

Z

2)3t2

+

2) = 0,

(4)

~ +p

1~(1 + p

0I~~0=d.

(2)

The position of crystallization front is determined by

with the boundary conditions (5)

a

Z(Zo+Zl)ixa(y)L,

where x = a(y) is the perturbed meniscus contour. The solution of this equation can be given in the form:

/

<~ /

-~

Z1(x, y) =

/

~ I -

7

Rx~x)cos(Xy) dx.

(7)

An arbitrary small perturbation of the contour can be presented in the form of a Fourier integral:

2

-

(6)

aQ’)—ao

/

H ~ /L (~.\ ~

7

fl~cos(Xy)dX.

(8)

-

a

/

—-

The small perturbation condition corresponds to: ~/(x0—a0)~l, ~ (9)

/

-

a~

x.

—4--x



Fig. 1. Schematic of shaped crystal pulling from the melt. (1) crystal, (2) melt, (3) shaper, (4) level of the free melt surface.

where /l~is the expansion coefficient, and 2ir/X is the perturbation wavelength. 3k, i.e. Obviously, the values of R~ depend linearly on / R~= Ox.f(~). (10)

G.I. Babkin et al. / Crystallization stability during capillary shaping. II

As a result, an equation forf(A) can be derived:

+

f~—f~3iiZop(l+p2)112 2(l + p2) —

+ ~(l

r?(l +p2)312]

=

47

2X(l +p2)f~,

(19)

F~~x + p2)312]

=

0 ,

(11)

0= 0, F~jxao = 0. From eq. (19), it follows that whenfx >0, the func-

fx [X

fxI~=~ 0 =

0,

(12)

(13) To calculate tan a0, it is necessary to find the derivative of Z along the normal to the contour. But the direction of the normal differs infinitesimally from the direction of the x-axis; therefore one can assume that

fJx=a0=Po.

tan a = aZ/ax

x=a(y) .

For the small difference ~a

=

a



a0, we obtain

and remain positive. Thus, we come to the conclusion that, if the sys-

ff~Ix_ao +z”Ix~o

öa

3x

0

cos(Xy)

Previously, we 1 have +p~shown1that [6]

dA

.

tion F~<0, since where this function cannot have a maximum in the region it is positive. Moreover, since F?jx=ao=O, we have F~I~ = ao <0, and inequality (18) proves to be true. We now must show that fx > 0. Since Po <0 [7], from eq. (11) and boundary conditions (12) and (13), it follows that f~>0 and f. < 0. This is accounted for by the fact that function fx cannot have maxima in the region where it is positive, and minima where it is negative. Hence, it must decrease monotonically

(14)

tem is stable with respect to a homogeneous perturbation turbations (X = as 0),well. thenInit the is stable same against way, one all other can show perthat inequality (18) is fulfilled under the boundary conditions of “wetting” at the shaper [8].

[a(y) a —

0]

=

V5a ,

(15)

where V is the velocity of pulling, and t is the time. It follows therefore that:

f 13x cos(Xy) dX J’ ~ =

V

0

0

+Z

0 ‘x=ap ~

1 +p~

X cos(Xy) dX ,

(16)

the dot denotes differentiation with respect to time. Finally, we obtain:

where

It has already been shown [6] that, under the boundary condition of catching, shaped crystal growth is stable with respect to a homogeneous perturbation (X = 0). This analysis shows that it is also stable against all other perturbations. As an example, we consider crystal growth at zero pressure, with gravitational effects neglected. It is sufficient to assume that g = 0 and d = 0. From eqs. (1)—(3), it follows that dZo/dX —=P Po is independent of x. By integrating eq. (11), with allowance for eqs. (12) and (13), we obtain 112~x

~=

Vx_~b0ZO~=ao

(17)

l+p~

It can be shown that the ratio Ox!13x decreases with increasing X. For this purpose, it is sufficient to show that

Ox ~=—VIp0I

coth[X(l +p~)

1+p~

~0

<~

(20)

for all values of X.

3. The growth of circular cylinders (18)

~U~x=a0)<0.

First, we differentiate eq. (11) with respect to X, and introduce the function Fx = 5f?.J~X.Then, we have the following equation and the boundary conditions for Fx: F~—3F~Z

2)—Fx[X2(l +p2) 0p(1+p

The equation and boundary conditions for the unperturbed meniscus contour, in the cylindrical coordinate system (Z, r, p) shown in fig. 2, has the form [7]: 1Z~(1 +Z~2) ~Z 2)312= 0, (21) Z~+r ZoIrrrrod, ZOIrR0(l +Z~ 0L , —

48

G.I. Babkin et al.

/ Crystallization

a /

get certain proportions between the size of the crystal and the shaper which result in the appearance of an instability with respect to homogeneous perturbations (m = 0) [6]. Therefore, to find the regions of stable and unstable growth at zero and higher harmonics, one should solve eqs. (23). As an example, we consider the case of pulling of a thin fiber at zero pressure. For r~r1~ 1, one can neglect the weight of the melt column. Eqs. (21) and (23), in this case, are of the form:

/

T -

L

~



2 3

\-

+ r’Z~(l + 42)

4

______ 0

stability during capillary shaping. II

r.

R.

// —

I

r

0,

(24)



I

ZoIr=r

-

00

0

Fig. 2. Coordinate system in the case of shaped cylindrical crystal growth. Designations ito 4 are the same as in fig. 1.

where r0 is the shaper radius, and R0 is the crystal radius. By performing an analysis analogous to that presented above for the case of plates, we obtain the expression for nJ~m (22)

0

(25)

+3p =

0,

fm r=R0

=

Po•

In ref. [6], it is shown that in this case 4 = p <0 I being > IP I. Eq. (25), in the general case, has no solutions expressed in the form of quadratures. We consider with Po

some particular cases below: (1) Assume m = 0. In this

case, eq. (25) can be

integrated, and the stability boundary is determined from the following relationships: for I~o~ 1 (small angle of growth),

2

ln[q

V~~RO +ZoIr=R

, +p2)fm=0, R0
ZOIrR0L 2)—r2m2(l

,

~+f~r~(l fm ‘r=r

-

=

=

+

q

(q

+ 1)1/21

=

q(q2



l)h/2

r0/Ro;

(26)

l÷p~

and for p0 ~ 1 (large angle ofgrowth), The equation and boundary conditions forfm are: 2) 3I7Zop(l +p2)112j J~+f~[r~(l + 3p frn [m2(l + p2) r2 + si(l + p2)3/2] = 0 , (23) —



fmlrro

=

0,

fmlrR0

=

Po•

(23a)

An essential difference from the case of plates is that, due to rotational symmetry, the value m can assume only whole numbers. As in the case of plates, one can show that fm+1 rR 0
In the intermediate case, the expressions are (27) too lnql. lengthy, and their derivation will not be given here. Eqs. (26) and (27) determine q 1. For q q~,it is unstable. The values of q~depend on Po and are within the limits 1.9 ~ 2.7. These results agree with the previous analysis [6]. (2) Assume IPo I ~ 21.can Then p I < Ipo I In ~ this 1, and in be neglected. case, eq. (25) the terms in p eq. (25) can be integrated to obtain an expression for ~mI~m.

both for the boundary condition of catching and for the boundary condition of wetting at the shaper. This condition implies that the existence of zero-harmonic stability indicates the stability of higher harmonics. However, in growing cylindrical crystals, one can

=

—v ‘P0 I rm coth(m ln q) l+p~

_l].

For m = 0, the stability boundary is 1, which agrees with eq. (27). For m

by ln q~= 1, the stability

given

G.I. Babkin et at / Crystallization stability during capillary shaping. II

boundary is determined by the solution of the equation coth(ln q)

(28) t2p’ 1. In other words, instability at i.e., q~. 2~ the first harmonic for ~ I ‘~ 1 is possible only at very =

1 +

~‘

small crystal radii. The remaining harmonics (m> for ~ I ~vi1 are stable. (3) Assume ~ I 1. In this case, we introduce the new notations ~

49

By matching functions fm and the derivatives f,~, given by eqs. (32) and (36), at the pointx = 21/2, and using the boundary conditions in eq. (31), one can get the expression for 13n’113m and determine the region of stable and unstable growth. For m = 1, q~satisfies an approximate equation 2 \ l.3q~—4.4 (1.8 ~, + 0.8’ (37) exp —

~‘

x

r/bo

=

and

b~= R~p~/(1 + Po)~ 2’

(x2

=



(29)

and eq. (25) becomes x2+2

6. For q >q~,the



i)_~,

f~(x2l)+fm

I

growth is unstable. For m > 1, instability can arise only in the case of ~ I 1, where our method of evaluation is unsuit-

Then from eq. (24) it follows that p2

which has solutions for ~

—m2fmO,

able. But, for m pa I, one can easily perform an evaluation. In this case, one can neglect the second term in eq. (30). The equation is then integrated, and ~‘

(30)

forllm/fimwehave: ~

(31)

Thus, for both ~

X fin x=r 0/b0

=

f~‘x=R0/b0

0,

=

Po,

1
arise only in the zeroth and first harmonics; the first-

From eq. (29), we find that ~I = 1 forx = 21/2, and, for x ~‘ 21/2, I~ I ~ 1. In the latter case, the equation can be integrated (as in point 2 above), and the solution can be presented in the form: tm~2 m (m0) (32) f~1x 2~values of x are within the limits Forx ~ ~ the 1 <1

+

p~2
21/2

~

.

~+

~

Thus, pa2 <~ ~ ~



1 <

(34)

cross-sectional shape, in the first approximation, does not result in the change of perimeter and cross sectional area and it therefore does not change the heat flow problem considered in ref. [6]. Therefore, the relative stabilization discussed in that analysis is absent for these harmonics. For stable growth in this case, it is necessary to operate with parameters for which the stability criteria in ref. [6] are satisfied and ~J13~<0 for m * 0. As shown in this paper, in the case of thermal and capillary stability with respect to homogeneous perturbations (i3~/j3~ <0), there exists capillary stability for all harmonics (j3~~Jf3~ <0), and

.

~r’m2f~

0. Integrating eq. (35), we find forx ~ 2~’~ =

fm

=



exp {m [2(x

=

(35)

1)] 1/2)



C 3

(x



1)1~’2

exp{—m[2(x l)]h/’2) (x 1)1/2 —

+

4. Discussion The analysis presented here allows one to generalize the results of stability investigations [6] with respect to perturbations of the growing crystal cross section, and the position of the crystallization front in the case of arbitrary perturbations. For all harmonics with m * 0, the alteration of the

By substituting eq. (34) into eq. (30), and retaining only the terms linear in ~, we obtain: C

harmonic instability arises for q ~ 1, i.e. for small radii of the crystal.

(33)

Now we introduce a change of variable: =

I ~ 1 and ~ I ~ 1, the instabilities

C4



(36)

the conclusions of ref. [6] give the correct answer to the growth stability. capillary In thisinstability paper, wefor have first-harmonic indicated a perturbations possibility of

G.I. Babkin et al. / Crystallization stability during capillary shaping. II

50

rr

(~J)

b

Fig. 3. First-harmonic perturbation for a round cylindrical crystal grown in the Ufl

rod (a), and example of a sapphire stable regime (b).

Fig. 4. Cross of a plate ~2a.showing first-harmonic perturbations at thesection plate edges.

two different regions. Far from the ends, under the boundary condition of catching, the plate growth is stable. A change in the plate width without thickness variation, or a bend in the plane of the plate (with respect to the unperturbed plane) can be represented as a first-harmonic perturbation of the semi-cylindrical edge regions. Therefore, for stable growth of a small-thickness plate, it is necessary that the condilions of first-harmonic stability for a fiber of corresponding diameter be satisfied.

Acknowledgements during the growth of thin fibers. It should be noted that the conclusions [6] about general stability in the region of capillary instability for homogeneous perturbations (m = 0) are still valid. However, the problem of the limit of the stable growth needs to be resolved in the following way. The minimal radius of a fiber that can be steadily grown is determined by the greater of two boundary radii; namely, the boundary radius from ref. [6], which determines the boundary of the region of general stability against homogeneous perturbations, and the boundary radius defming the region of capillary stability against second-harmonic perturbations, as determined in this paper. The first harmonic perturbation corresponds to a small shift (fig. 3a), and the instability which arises results in the crystal bending. Such an instability has been observed during growth of sapphire fibers of about 300 jim in diameter (fig. 3b). In the growth of plates of some predetermined width (fig. 4), we need to consider the stability in

The authors wish to express their deep gratitude to A.A. Chernov for fruitful discussions and to G.A. Satunkin and T.N. Yalovets for the photograph of the crystal. References [1] A.V. Stepanov,Zh. Tekhn. liz. 29 (1959) 381. [2] B. Chalmers, H.E. Labelle and A.l. Mlavsky, J. Crystal Growth 13/14 (1972) 84. [3] G.H. Schwuttke, Phys. Status Solidi (a) 43 (1977) 43. [41J. Electron. Eng. 113 (1976) 46. [5] V.A. Tatarchenko, Fiz. i Khim. Obrabotki Mater. 6 (1973) 47. [6] V.A. Tatarchenko and E.A. Brener, lzv. Akad. Nauk USSR, Ser. Fiz. 40 (1976) 1456; V.A. Tatarchenko and E.A. Brener, J. Crystal Growth 50 (1980) 33. [7] V.A. Tatarchenko, J. Crystal Growth 37 (1977) 272. [8] V.A. Tatarchenko, A.I. Saet and A.V. Stepanov, Izv. Akad. Nauk SSSR, Ser. Fiz. 33 (1969) 1954.