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gas meniscus in the case of edge-defined film-fed growth system
ARTICLE IN PRESS
Journal of Crystal Growth 259 (2003) 121–129
The effect of a periodic movement on the die of the bottom line of the melt/gas meniscus in the case of edge-defined film-fed growth system L. Braescua, A.M. Balintb,*, St. Balintc a
Department of Mathematics, University Polytechnical Timisoara, P-ta: Regina Maria, Nr: 1, 1900 Timisoara, Romania b Department of Physics, University of the West Timisoara, Blv: V. Parvan, Nr. 4, 1900 Timisoara, Romania c Department of Mathematics, University of the West Timisoara, Blv: V. Parvan, Nr. 4, 1900 Timisoara, Romania Received 3 April 2003; accepted 13 July 2003 Communicated by T. Hibiya
Abstract In this paper the usual model which permits to describe the evolution of the radius r ¼ rðtÞ and of the meniscus height h ¼ hðtÞ in the case of filament growth from the melt by edge-defined film-fed growth method is considered. What is specific is that the bottom line of the melt/gas meniscus is movable on the die. The main objective is to show that a periodic movement of the bottom line leads to a periodic change of the crystal radius (as it was observed by practical crystal growers) and to show that this effect can be compensated for example by an adequate periodic change of the pulling rate. r 2003 Elsevier B.V. All rights reserved. PACS: 81.10.h; 81.05.Je; 44.40.+a; 02.70.Dh; 44.90.+c Keywords: A1. Control; A1. Filament; A1. Radius; A2. Edge-defined film-fed growth method
1. Introduction Oxide crystals grown from the melt by edgedefined film-fed growth (EFG) method are used as solid-state laser hosts and materials for acoustoopto-electronic devices. The shape and the quality of the crystal grown by EFG method is determined by the shape of the meniscus (the liquid bridge retained between the crystal and the die) and its behavior during the growth. In the last 20 years *Corresponding author. Tel./fax: +40-56-201-105. E-mail address: [email protected] (A.M. Balint).
many experimental and theoretical studies have been reported regarding this growth process [1–34]. The EFG method is performed to achieve stable crystal growth when the melt/gas meniscus bottom is fixed to the inner or outer edge of the die top. In reality the wetting width of the melt on the die can vary. Machida et al. observed experimentally periodic change of the crystal width during the growth of a rutile (TiO2) single crystal and suggested that the phenomenon is closely related to the wetting of the melt against the die [21]. Kobayashi et al. investigated the effect of wetting,
0022-0248/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0022-0248(03)01595-1
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i.e. the contact angle of the melt against a die, in the EFG crystal growth of oxides [34]. They found that, the increase in melt/die contact angle leads to the decrease of the wetting width on the die and other processing conditions, such as die dimensions, heat power, crystal pulling rate, etc., also affect the wetting width significantly. In this paper we describe the mathematical model reported in Refs. [32,33] for EFG process. We assume that the bottom line of the melt/gas meniscus on the die is movable. We investigate the effect of the movement on the die of the bottom line of the melt/gas meniscus and we show that a periodic movement of the bottom line leads to a periodic change of the crystal radius, we give a model-based numerical proof of the fact that it is possible to compensate the effect of the variation of the wetting width, for example, by an adequate variation of the pulling rate.
2. The mathematical model The system of differential equations which governs the evolution of the filament radius r ¼ rðtÞ and the meniscus height h ¼ hðtÞ is dr ¼ v tg½aðr; h; wÞ a1 ; dt dh 1 ¼v ½ll Gl ðr; hÞ l2 G2 ðr; hÞ: dt Lr2
ð1Þ
For details see Fig. 1 and Refs. [19,20,32,33]. The significance of these quantities and their values for LiNbO3 are given in Table 1. The function aðr; h; wÞ is obtained from the Young–Laplace equation of the capillary surface in equilibrium in the absence of exterior pressure, i.e. p ¼ 0 " " 2 # 2 2 # 2 qz q z qz qz q2 z qz q z 1þ 2 þ 1þ qy qx2 qx qy qxqy qx qy2 " 2 2 #3=2 r1 g qz qz z 1þ ¼ þ : ð2Þ g qx qy For filaments, there are solutionspz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ zðrÞ depending on the considered radius r ¼ x2 þ y2 and for
Fig. 1. Two-dimensional model for filament grown by EFG method.
these functions Eq. (2) becomes " 2 #3=2 " 2 # d2 z r1 g dz 1 dz dz z 1þ : ¼ 1þ dr2 g dr r dr dr ð3Þ Eq. (3) is transformed into the system dz ¼ tg a; dr da rg z 1 ¼ 1 tg a dr g cos a r
ð4Þ
for which the following initial values are considered: p zðwÞ ¼ 0; aðwÞ ¼ ac ; ac A 0; : ð5Þ 2 The solution of the initial values Problem (4)–(5) is denoted by z ¼ zðr; ac ; wÞ and a ¼ aðr; ac ; wÞ: The angle ac is expressed from z ¼ zðr; ac ; wÞ as ac ¼ ac ðr; z; wÞ and it is introduced into a ¼ aðr; ac ; wÞ
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Table 1 Material parameters for LiNbO3 Nomenclature
Value
v pulling rate r1 density of the melt r2 density of the crystal l1 thermal conductivity coefficient in the melt l2 thermal conductivity coefficient in the crystal L latent heat ae growth angle (¼ p=2 a1 ) g superficial tension of the liquid c1 heat capacity of the melt c2 heat capacity of the crystal Bi Biot number Tm melting point temperature z coordinate in the pulling direction Ten0 environment temperature at quota z ¼ 0 k environment temperature gradient Ten ðzÞ environment temperature (¼ Ten0 kz) T0 melt temperature at the meniscus basis L length of the seed g gravitational acceleration r0 radius of the die for filaments growth w wetting width w1 thermal diffusivity in the melt (¼ l1 =ðr1 c1 Þ) w2 thermal diffusivity in the crystal (¼ l2 =ðr2 c2 Þ) m1 heat radiation coefficient in the melt ð¼ Bil2 =rÞ m2 heat radiation coefficient in the crystal ð¼ Bi l2 =rÞ aðr; hÞ angle between the Or axis and the tangent to the meniscus in the point ðr; hÞ G1 ðr; hÞ temperature gradient in the melt G2 ðr; hÞ temperature gradient in the crystal
((cm 102)/s) 46 107 (g/(cm 102)3) 46 107 (g/(cm 102)3) 0.0004 (W/(cm 102) K) 0.0005 (W/(cm 102) K) 1000 (J/g) 17 218 107 (N/(cm 102)) 0.633 (J/g K) 0.633 (J/g K) 0.928 1526 (K) 1019 (K) 9.33 (K/(cm 102)) 63.57 (cm 102) 98100 ((cm 102)/s2) 22.5 (cm 102)
obtaining the function a ¼ aðr; z; wÞ: In the above functions r represents the radial coordinate. The temperature gradients G1;2 in the melt and in the crystal, respectively, are given by 1 vkl1 r T0 Ten0 G1 ðr; hÞ ¼ ðb1 ed1 h Þ shðb1 hÞ 2m1 w1 vkl1 r þ Tm Ten0 þ kh 2m1 w1
ðd1 shðb1 hÞ þ b1 chðb1 hÞ k; ð6Þ 1 vkl2 r G2 ðr; hÞ ¼ Tm Ten0 þ kh shðb2 LÞ 2m2 w2 ðd2 shðb2 LÞ b2 chðb2 LÞÞ
vkl2 r d2 L be k; 2m2 w2 2
where di and bi ; i ¼ 1; 2 are defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v v2 2m di ¼ ; bi ¼ þ i for i ¼ 1; 2: 2wi li r 4w2i
The significance and the values of the above quantities are given in Table 1. To obtain these gradients the following boundary value problems for the one-dimensional stationary heat transport equation d2 Ti v dTi 2mi ðTi Ten Þ ¼ 0; wi dz dz2 li r
i ¼ 1; 2;
T1 ð0Þ ¼ T0 ; T1 ðhÞ ¼ T2 ðhÞ ¼ Tm ; T2 ðh þ LÞ ¼ Ten ðh þ LÞ ¼ Ten0 kðh þ LÞ ð7Þ
ð8Þ
ð9Þ
ð10Þ
in the meniscus and in the crystal, respectively, was solved.
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The temperature gradients were obtained as qTi Gi ðr; hÞ ¼ i ¼ 1; 2: ð11Þ qz z¼h
We remark that for any initial temperature distribution in the meniscus and in the crystal the time dependent temperature distributions Ti ðt; z; r; hÞ tend to the stationary distributions Ti ðz; r; hÞ: Ti ðt; z; r; hÞ t
! T ðz; r; hÞ i ¼ 1; 2: þN i
ð12Þ
In order to grow a filament with constant diameter cross-section, the following system: aðr; h; wÞ a1 ¼ 0; 1 ½ll Gl ðr; hÞ l2 G2 ðr; hÞ ¼ 0 v Lr2
totically stable steady-state ðr2 ; h2 Þ will be denoted by ðr ; h Þ: The asymptotically stable steady-state ðr ; h Þ depends on v and T0 and on w: In this model we compute for the 30 different values of the wetting width w in the range 10.5–22.5 (cm 102) the asymptotically steady-states ðr ; h Þ for 280 couples of ðv; T0 Þ: For a fixed couple ðv; T0 Þ we find what is happening if w suddenly changes from the value w1 to the value w2 : Finally we simulate numerically that the effect of a periodic variation of w during the growth can be compensated by an adequate periodic variation of the pulling rate v:
ð13Þ
has to be satisfied. System (13) has two solutions ðr1 ; h1 Þ and ðr2 ; h2 Þ; r1 or2 : The steady-state solution ðr1 ; h1 Þ is not stable and the steady-state solution ðr2 ; h2 Þ is asymptotically stable. In the following the asymp-
3. The determination of the range of the wetting width w for LiNbO3 filament For a die of radius r0 ¼ 22:5 (cm 102), solving Eqs. (13) we find the steady states of Eq. (1) for different values of w and ðv; T0 Þ couples,
Table 2 The filament radii r (cm 102) for the wetting width w ¼ 10:5 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 9.769 9.276 8.306 8.000 r o0 r o0 r o0
r > r0 r > r0 9.759 9.258 8.270 7.957 7.469 6.755 6.538
r > r0 10.306 9.646 9.065 7.885 7.488 6.820 5.242 r o0
10.491 10.285 9.554 8.908 7.551 7.063 6.112 r o0 r o0
10.482 9.907 8.170 6.144 r o0 r o0 r o0 r o0 r o0
10.416 9.366 6.411 r o0 r o0 r o0 r o0 r o0 r o0
Table 3 Filament radii r (cm 102) for the wetting width w ¼ 13 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 12.150 11.583 10.471 r o0 r o0 r o0 r o0
r > r0 r > r0 12.135 11.558 10.422 10.069 9.531 r o0 r o0
12.999 12.766 11.978 11.289 9.895 9.241 8.307 7.440 r o0
12.998 12.734 11.844 11.059 9.125 8.459 7.825 r o0 r o0
12.945 11.957 9.298 r o0 r o0 r o0 r o0 r o0 r o0
12.003 10.516 r o0 r o0 r o0 r o0 r o0 r o0 r o0
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respectively. The corresponding radius r obtained for 54 couples ðv; T0 Þ for 9 different values of w are presented, as being representative, in Tables 2–10. Computations show that r becomes less than zero when T0 is relatively high and the pulling rate v decreases. This is because in these conditions
125
h > hmax —admissible, where hmax —admissible represents the maximum with respect to r of the function z ¼ zðr; a1 ; wÞ obtained expressing z from the equation a1 ¼ aðr; z; wÞ: These computations show also that it is possible to obtain an asymptotically stable steady-state
Table 4 Filament radii r (cm 102) for the wetting width w ¼ 13:5 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 12.584 11.960 10.725 r o0 r o0 r o0 r o0
r > r0 r > r0 12.567 11.930 10.666 10.264 9.635 r o0 r o0
r > r0 13.248 12.379 11.607 10.010 9.445 8.490 r o0 r o0
13.492 13.211 12.215 11.325 9.387 8.629 r o0 r o0 r o0
13.401 12.094 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
Table 5 Filament radii r (cm 102) for the wetting width w ¼ 14 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 13.091 12.483 11.292 r o0 r o0 r o0 r o0
r > r0 r > r0 13.074 12.454 11.236 10.856 10.274 r o0 r o0
r > r0 13.746 12.891 12.140 10.618 10.114 9.284 r o0 r o0
13.995 13.709 12.730 11.866 10.048 9.394 r o0 r o0 r o0
r > r0 12.605 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
Table 6 Filament radii r (cm 102) for the wetting width w ¼ 14:5 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 13.523 12.858 11.536 r o0 r o0 r o0 r o0
r > r0 r > r0 13.504 12.824 11.469 11.036 10.358 r o0 r o0
14.494 14.227 13.287 12.450 10.704 10.314 9.384 r o0 r o0
14.493 14.183 13.093 12.116 10.251 9.569 r o0 r o0 r o0
14.181 12.610 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
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Table 7 Filament radii r (cm 102) for the wetting width w ¼ 14:9 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 13.930 13.245 11.989 r o0 r o0 r o0 r o0
r > r0 r > r0 13.911 13.242 11.924 11.510 10.871 r o0 r o0
r > r0 14.628 13.697 12.875 11.193 10.624 9.657 r o0 r o0
14.893 14.584 13.505 12.549 10.491 9.711 r o0 r o0 r o0
r > r0 13.018 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
Table 8 Filament radii r (cm 102) for the wetting width w ¼ 15 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 14.032 13.381 12.102 r o0 r o0 r o0 r o0
r > r0 r > r0 14.012 13.347 12.037 11.627 10.996 r o0 r o0
14.995 14.727 13.800 12.983 11.313 10.754 9.812 r o0 r o0
14.893 14.584 13.505 12.549 10.491 9.711 r o0 r o0 r o0
14.673 13.122 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
1.6
5.5
7
18.491 18.060 16.553 15.168 11.772 9.627 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
Table 9 Filament radii r (cm 102) for the wetting width w ¼ 18:5 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 17.288 16.455 r o0 r o0 r o0 r o0 r o0
r > r0 r > r0 17.257 16.402 14.662 14.096 r o0 r o0 r o0
ðr ; h Þ with r ¼ 10 (cm 102) (as considered by Kobayashi et al. in Ref. [34]) if the wetting width w is in the range 10.5–18.5 (cm 102). This fact implies that it is not possible to obtain a LiNbO3 filament with r ¼ 10 (cm 102) if the meniscus bottom is fixed to the outer edge of the die i.e. w ¼ r0 ¼ 22:5 (cm 102) (see Table 10).
1 18.493 18.140 16.900 15.778 13.331 12.418 10.520 r o0 r o0
In the following we will show what happens if we change suddenly w from w1 to w2 : For this purpose the unsteady state equations Eq. (1) are solved numerically. We assume that during the growth, ðv; T0 Þ ¼ ð1; 2027Þ ((cm 102)/s; K) and we start with the initial conditions ðr0 ; h0 ; w0 Þ ¼ ð10; 5; 13:5Þ: At the
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Table 10 Filament radii r (cm 102) for the wetting width w ¼ 22:5 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 r > r0 20.049 r o0 r o0 r o0 r o0 r o0
r > r0 r > r0 21.014 19.970 17.800 17.080 r o0 r o0 r o0
22.490 22.048 20.471 19.005 15.625 14.208 r o0 r o0 r o0
22.488 21.915 19.890 17.962 11.690 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
Fig. 2. Evolution of the crystal radius due to the following movements of the bottom line: w0 ¼ 13:5-w1 ¼ 14-w2 ¼ 13:5-w3 ¼ 14:5-w4 ¼ 13:5-w5 ¼ 14:9-w6 ¼ 13:5-w7 ¼ 15-w8 ¼ 13:5 (cm 102).
moments t1 ¼ 59; t2 ¼ 118; t3 ¼ 179; t4 ¼ 238; t5 ¼ 301; t6 ¼ 360; t7 ¼ 421; t8 ¼ 480 (s), during the growth we change the wetting width as follows: w1 ¼ 14-w2 ¼ 13:5-w3 ¼ 14:5-w4 ¼ 13:5-w5 ¼ 14:9-w6 ¼ 13:5-w7 ¼ 15-w8 ¼ 13:5 (cm 102). The computed filament radius is plotted in Fig. 2. This simulation shows that if jw1 w2 j is less than 1.5 (cm 102) the filament radius r ðw1 Þ evolves in time to r ðw2 Þ while if jw1 w2 j ¼ 1:5 (cm 102) then r ðw1 Þ does not evolve to r ðw2 Þ; in the last change w7 ¼ 15-w8 ¼ 13:5; r ð15Þ ¼ 11:303 evolves to the value 12.600 which is different from r ð13:5Þ ¼ 10:0005: This behavior of r ðw1 Þ is due to the fact that the steady state ðr ðw1 Þ; h ðw1 ÞÞ; in which we are at the moment when w switches from w1 to w2 ; does not belong to the region of attraction of the steady state ðr ðw2 Þ; h ðw2 Þ). For nonlinear systems the numerically determination of the region of attraction of an asymptotically stable steady-state is an open problem in mathematics. What we know today is
to give numerically an accurate estimation of the region of attraction [35].
4. The effect of a periodic movement on the die of the bottom line of the melt/gas meniscus We will show that a periodic movement of the bottom line leads to a periodic variation of the filament radius. We assume that during the growth ðv; T0 Þ ¼ ð1; 2027Þ ((cm 102)/s; K), the initial conditions are ðr0 ; h0 ; w0 Þ ¼ ð10; 5; 13:5Þ and during the process, at the moments t1 ¼ 59; t2 ¼ 118; t3 ¼ 179; t4 ¼ 238; t5 ¼ 299 (s), the wetting width changes periodically as follows: w1 ¼ 14-w2 ¼ 13-w3 ¼ 14-w4 ¼ 13-w5 ¼ 14 (cm 102). The computed evolution of the filament radius is represented in Fig. 3. This figure shows that there are six stationary phases and six transition phases, respectively, in this growth process. The first transition phase
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Fig. 3. Periodic evolution of the crystal radius due to the following periodic movements of the bottom line: w0 ¼ 13:5-w1 ¼ 14-w2 ¼ 13-w3 ¼ 14-w4 ¼ 13-w5 ¼ 14 (cm 102).
t1 ¼ 9 (s) is due to the fact that the initial condition with which we start the growth process is not stationary. The other five phases are due to the five movements on the die of the bottom line of the melt/gas meniscus and the relaxation times corresponding to these transition processes are t2 ¼ 9; t3 ¼ 11; t4 ¼ 9; t5 ¼ 11; t6 ¼ 9 (s). This computation shows that the crystal radius varies periodically around the average value r ¼ 10 (cm 102).
5. Modeling the compensation of the effect of periodically wetting width changes by an adequate changes of the pulling rate We will show how we can compensate the effect of a given periodical variation of the wetting width w by an adequate variation the pulling rate v; in order to obtain a filament with constant radius r ¼ 10 (cm 102) (as considered by Kobayashi et al. in Ref. [34]). For that we assume that during the growth at the moments t1 ¼ 59; t2 ¼ 119; t3 ¼ 181; t4 ¼ 241; t5 ¼ 303 (s) there are the following changes of the wetting width: w1 ¼ 14-w2 ¼ 13-w3 ¼ 14-w4 ¼ 13-w5 ¼ 14 (cm 102). The adequate pulling rate changes are: v1 ¼ 1:632-v2 ¼ 0:83-v3 ¼ 1:632-v4 ¼ 0:83-v5 ¼ 1:632 ((cm 102)/s). The computed shape of the filament radius, due to the above w and v changes, is plotted in Fig. 4. The relaxation times of the transition processes are t1 ¼ 9; t2 ¼ 10; t3 ¼ 12; t4 ¼ 10; t5 ¼ 12; t6 ¼ 10 (s). In these transition periods only the fourth digit of the crystal radius varies.
Fig. 4. Evolution of the crystal radius due to the following periodic movements of the bottom line: w0 ¼ 13:5-w1 ¼ 14-w2 ¼ 13-w3 ¼ 14-w4 ¼ 13-w5 ¼ 14 (cm 102) and the adequate pulling rate changes: v ¼ 1-v1 ¼ 1:632-v2 ¼ 0:83-v3 ¼ 1:632-v4 ¼ 0:83-v5 ¼ 1:632 ((cm 102)/s).
6. Conclusions (i) If the meniscus bottom is fixed to the outer edge of the die, i.e. w ¼ r0 ¼ 22:5 (cm 102) (see Table 10) then we cannot obtain a LiNbO3 filament with radius r ¼ 10 (cm 102). We have to admit that the bottom line of the melt/gas meniscus is movable. (ii) The range of jw1 w2 j for which r ðw1 Þ evolves in time to r ðw2 Þ is 0–1.49 (cm 102). If jw1 w2 j is equal to 1.5 (cm 102), then the evolution of r ðw1 Þ to r ðw2 Þ is not sure. (iii) A periodic movement on the die of the bottom line of the melt/gas meniscus leads to a periodic variation of the crystal radius which can be compensated by an adequate periodic variation of the pulling rate.
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[22] P.I. Antonov, S.I. Bakholdin, E.A. Tropp, V.S. Yuferev, J. Crystal Growth 50 (1980) 62. [23] V.S. Yuferev, M.G. Vasil’ev, J. Crystal Growth 82 (1987) 31. [24] V.S. Yuferev, M.G. Vasil’ev, L.A. Stefanova, E.N. Kolesnikova, J. Crystal Growth 128 (1993) 144. [25] M.F. Modest, Radiative Heat Transfer, McGraw-Hill, New York, 1993. [26] J.L. Duranceanu, R.A. Brown, J. Crystal Growth 75 (1986) 367. [27] T. Tsukada, H. Hozawa, N. Imaishi, J. Chem. Eng. Japan 27 (1994) 25. [28] M. Kobayashi, T. Hagino, T. Tsukada, M. Hozawa, J. Crystal Growth 235 (2002) 258. [29] C.M. Spuckler, R. Siegel, J. Thermophys. Heat Transfer 6 (1992) 596. [30] C.M. Spuckler, R. Siegel, J. Thermophys. Heat Transfer 8 (1994) 193. [31] H. Machida, K. Hoshikawa, T. Fukuda, Jpn. J. Appl. Phys. 31 (Part 2) (1992) L974. [32] V.A. Tatarchenko, in: D.T.J. Hurle (Ed.), Handbook of Crystal Growth, North-Holland, Amsterdam, 1994, pp. 1017–1048. [33] V.A. Tatarchenko, in: Shaped Crystal Growth, Kluwer Academic Publishers, Dordecht, 1993, pp. 19–145. [34] M. Kobayashi, T. Tsukada, M. Hozawa, J. Crystal Growth 249 (1–2) (2003) 230. [35] E. Kaslik, A.M. Balint, St. Balint, Nonlinear Stud. 10 (1) (2003) 67.
Journal of Crystal Growth 259 (2003) 121–129
The effect of a periodic movement on the die of the bottom line of the melt/gas meniscus in the case of edge-defined film-fed growth system L. Braescua, A.M. Balintb,*, St. Balintc a
Department of Mathematics, University Polytechnical Timisoara, P-ta: Regina Maria, Nr: 1, 1900 Timisoara, Romania b Department of Physics, University of the West Timisoara, Blv: V. Parvan, Nr. 4, 1900 Timisoara, Romania c Department of Mathematics, University of the West Timisoara, Blv: V. Parvan, Nr. 4, 1900 Timisoara, Romania Received 3 April 2003; accepted 13 July 2003 Communicated by T. Hibiya
Abstract In this paper the usual model which permits to describe the evolution of the radius r ¼ rðtÞ and of the meniscus height h ¼ hðtÞ in the case of filament growth from the melt by edge-defined film-fed growth method is considered. What is specific is that the bottom line of the melt/gas meniscus is movable on the die. The main objective is to show that a periodic movement of the bottom line leads to a periodic change of the crystal radius (as it was observed by practical crystal growers) and to show that this effect can be compensated for example by an adequate periodic change of the pulling rate. r 2003 Elsevier B.V. All rights reserved. PACS: 81.10.h; 81.05.Je; 44.40.+a; 02.70.Dh; 44.90.+c Keywords: A1. Control; A1. Filament; A1. Radius; A2. Edge-defined film-fed growth method
1. Introduction Oxide crystals grown from the melt by edgedefined film-fed growth (EFG) method are used as solid-state laser hosts and materials for acoustoopto-electronic devices. The shape and the quality of the crystal grown by EFG method is determined by the shape of the meniscus (the liquid bridge retained between the crystal and the die) and its behavior during the growth. In the last 20 years *Corresponding author. Tel./fax: +40-56-201-105. E-mail address: [email protected] (A.M. Balint).
many experimental and theoretical studies have been reported regarding this growth process [1–34]. The EFG method is performed to achieve stable crystal growth when the melt/gas meniscus bottom is fixed to the inner or outer edge of the die top. In reality the wetting width of the melt on the die can vary. Machida et al. observed experimentally periodic change of the crystal width during the growth of a rutile (TiO2) single crystal and suggested that the phenomenon is closely related to the wetting of the melt against the die [21]. Kobayashi et al. investigated the effect of wetting,
0022-0248/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0022-0248(03)01595-1
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i.e. the contact angle of the melt against a die, in the EFG crystal growth of oxides [34]. They found that, the increase in melt/die contact angle leads to the decrease of the wetting width on the die and other processing conditions, such as die dimensions, heat power, crystal pulling rate, etc., also affect the wetting width significantly. In this paper we describe the mathematical model reported in Refs. [32,33] for EFG process. We assume that the bottom line of the melt/gas meniscus on the die is movable. We investigate the effect of the movement on the die of the bottom line of the melt/gas meniscus and we show that a periodic movement of the bottom line leads to a periodic change of the crystal radius, we give a model-based numerical proof of the fact that it is possible to compensate the effect of the variation of the wetting width, for example, by an adequate variation of the pulling rate.
2. The mathematical model The system of differential equations which governs the evolution of the filament radius r ¼ rðtÞ and the meniscus height h ¼ hðtÞ is dr ¼ v tg½aðr; h; wÞ a1 ; dt dh 1 ¼v ½ll Gl ðr; hÞ l2 G2 ðr; hÞ: dt Lr2
ð1Þ
For details see Fig. 1 and Refs. [19,20,32,33]. The significance of these quantities and their values for LiNbO3 are given in Table 1. The function aðr; h; wÞ is obtained from the Young–Laplace equation of the capillary surface in equilibrium in the absence of exterior pressure, i.e. p ¼ 0 " " 2 # 2 2 # 2 qz q z qz qz q2 z qz q z 1þ 2 þ 1þ qy qx2 qx qy qxqy qx qy2 " 2 2 #3=2 r1 g qz qz z 1þ ¼ þ : ð2Þ g qx qy For filaments, there are solutionspz ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ zðrÞ depending on the considered radius r ¼ x2 þ y2 and for
Fig. 1. Two-dimensional model for filament grown by EFG method.
these functions Eq. (2) becomes " 2 #3=2 " 2 # d2 z r1 g dz 1 dz dz z 1þ : ¼ 1þ dr2 g dr r dr dr ð3Þ Eq. (3) is transformed into the system dz ¼ tg a; dr da rg z 1 ¼ 1 tg a dr g cos a r
ð4Þ
for which the following initial values are considered: p zðwÞ ¼ 0; aðwÞ ¼ ac ; ac A 0; : ð5Þ 2 The solution of the initial values Problem (4)–(5) is denoted by z ¼ zðr; ac ; wÞ and a ¼ aðr; ac ; wÞ: The angle ac is expressed from z ¼ zðr; ac ; wÞ as ac ¼ ac ðr; z; wÞ and it is introduced into a ¼ aðr; ac ; wÞ
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Table 1 Material parameters for LiNbO3 Nomenclature
Value
v pulling rate r1 density of the melt r2 density of the crystal l1 thermal conductivity coefficient in the melt l2 thermal conductivity coefficient in the crystal L latent heat ae growth angle (¼ p=2 a1 ) g superficial tension of the liquid c1 heat capacity of the melt c2 heat capacity of the crystal Bi Biot number Tm melting point temperature z coordinate in the pulling direction Ten0 environment temperature at quota z ¼ 0 k environment temperature gradient Ten ðzÞ environment temperature (¼ Ten0 kz) T0 melt temperature at the meniscus basis L length of the seed g gravitational acceleration r0 radius of the die for filaments growth w wetting width w1 thermal diffusivity in the melt (¼ l1 =ðr1 c1 Þ) w2 thermal diffusivity in the crystal (¼ l2 =ðr2 c2 Þ) m1 heat radiation coefficient in the melt ð¼ Bil2 =rÞ m2 heat radiation coefficient in the crystal ð¼ Bi l2 =rÞ aðr; hÞ angle between the Or axis and the tangent to the meniscus in the point ðr; hÞ G1 ðr; hÞ temperature gradient in the melt G2 ðr; hÞ temperature gradient in the crystal
((cm 102)/s) 46 107 (g/(cm 102)3) 46 107 (g/(cm 102)3) 0.0004 (W/(cm 102) K) 0.0005 (W/(cm 102) K) 1000 (J/g) 17 218 107 (N/(cm 102)) 0.633 (J/g K) 0.633 (J/g K) 0.928 1526 (K) 1019 (K) 9.33 (K/(cm 102)) 63.57 (cm 102) 98100 ((cm 102)/s2) 22.5 (cm 102)
obtaining the function a ¼ aðr; z; wÞ: In the above functions r represents the radial coordinate. The temperature gradients G1;2 in the melt and in the crystal, respectively, are given by 1 vkl1 r T0 Ten0 G1 ðr; hÞ ¼ ðb1 ed1 h Þ shðb1 hÞ 2m1 w1 vkl1 r þ Tm Ten0 þ kh 2m1 w1
ðd1 shðb1 hÞ þ b1 chðb1 hÞ k; ð6Þ 1 vkl2 r G2 ðr; hÞ ¼ Tm Ten0 þ kh shðb2 LÞ 2m2 w2 ðd2 shðb2 LÞ b2 chðb2 LÞÞ
vkl2 r d2 L be k; 2m2 w2 2
where di and bi ; i ¼ 1; 2 are defined by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v v2 2m di ¼ ; bi ¼ þ i for i ¼ 1; 2: 2wi li r 4w2i
The significance and the values of the above quantities are given in Table 1. To obtain these gradients the following boundary value problems for the one-dimensional stationary heat transport equation d2 Ti v dTi 2mi ðTi Ten Þ ¼ 0; wi dz dz2 li r
i ¼ 1; 2;
T1 ð0Þ ¼ T0 ; T1 ðhÞ ¼ T2 ðhÞ ¼ Tm ; T2 ðh þ LÞ ¼ Ten ðh þ LÞ ¼ Ten0 kðh þ LÞ ð7Þ
ð8Þ
ð9Þ
ð10Þ
in the meniscus and in the crystal, respectively, was solved.
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The temperature gradients were obtained as qTi Gi ðr; hÞ ¼ i ¼ 1; 2: ð11Þ qz z¼h
We remark that for any initial temperature distribution in the meniscus and in the crystal the time dependent temperature distributions Ti ðt; z; r; hÞ tend to the stationary distributions Ti ðz; r; hÞ: Ti ðt; z; r; hÞ t
! T ðz; r; hÞ i ¼ 1; 2: þN i
ð12Þ
In order to grow a filament with constant diameter cross-section, the following system: aðr; h; wÞ a1 ¼ 0; 1 ½ll Gl ðr; hÞ l2 G2 ðr; hÞ ¼ 0 v Lr2
totically stable steady-state ðr2 ; h2 Þ will be denoted by ðr ; h Þ: The asymptotically stable steady-state ðr ; h Þ depends on v and T0 and on w: In this model we compute for the 30 different values of the wetting width w in the range 10.5–22.5 (cm 102) the asymptotically steady-states ðr ; h Þ for 280 couples of ðv; T0 Þ: For a fixed couple ðv; T0 Þ we find what is happening if w suddenly changes from the value w1 to the value w2 : Finally we simulate numerically that the effect of a periodic variation of w during the growth can be compensated by an adequate periodic variation of the pulling rate v:
ð13Þ
has to be satisfied. System (13) has two solutions ðr1 ; h1 Þ and ðr2 ; h2 Þ; r1 or2 : The steady-state solution ðr1 ; h1 Þ is not stable and the steady-state solution ðr2 ; h2 Þ is asymptotically stable. In the following the asymp-
3. The determination of the range of the wetting width w for LiNbO3 filament For a die of radius r0 ¼ 22:5 (cm 102), solving Eqs. (13) we find the steady states of Eq. (1) for different values of w and ðv; T0 Þ couples,
Table 2 The filament radii r (cm 102) for the wetting width w ¼ 10:5 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 9.769 9.276 8.306 8.000 r o0 r o0 r o0
r > r0 r > r0 9.759 9.258 8.270 7.957 7.469 6.755 6.538
r > r0 10.306 9.646 9.065 7.885 7.488 6.820 5.242 r o0
10.491 10.285 9.554 8.908 7.551 7.063 6.112 r o0 r o0
10.482 9.907 8.170 6.144 r o0 r o0 r o0 r o0 r o0
10.416 9.366 6.411 r o0 r o0 r o0 r o0 r o0 r o0
Table 3 Filament radii r (cm 102) for the wetting width w ¼ 13 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 12.150 11.583 10.471 r o0 r o0 r o0 r o0
r > r0 r > r0 12.135 11.558 10.422 10.069 9.531 r o0 r o0
12.999 12.766 11.978 11.289 9.895 9.241 8.307 7.440 r o0
12.998 12.734 11.844 11.059 9.125 8.459 7.825 r o0 r o0
12.945 11.957 9.298 r o0 r o0 r o0 r o0 r o0 r o0
12.003 10.516 r o0 r o0 r o0 r o0 r o0 r o0 r o0
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respectively. The corresponding radius r obtained for 54 couples ðv; T0 Þ for 9 different values of w are presented, as being representative, in Tables 2–10. Computations show that r becomes less than zero when T0 is relatively high and the pulling rate v decreases. This is because in these conditions
125
h > hmax —admissible, where hmax —admissible represents the maximum with respect to r of the function z ¼ zðr; a1 ; wÞ obtained expressing z from the equation a1 ¼ aðr; z; wÞ: These computations show also that it is possible to obtain an asymptotically stable steady-state
Table 4 Filament radii r (cm 102) for the wetting width w ¼ 13:5 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 12.584 11.960 10.725 r o0 r o0 r o0 r o0
r > r0 r > r0 12.567 11.930 10.666 10.264 9.635 r o0 r o0
r > r0 13.248 12.379 11.607 10.010 9.445 8.490 r o0 r o0
13.492 13.211 12.215 11.325 9.387 8.629 r o0 r o0 r o0
13.401 12.094 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
Table 5 Filament radii r (cm 102) for the wetting width w ¼ 14 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 13.091 12.483 11.292 r o0 r o0 r o0 r o0
r > r0 r > r0 13.074 12.454 11.236 10.856 10.274 r o0 r o0
r > r0 13.746 12.891 12.140 10.618 10.114 9.284 r o0 r o0
13.995 13.709 12.730 11.866 10.048 9.394 r o0 r o0 r o0
r > r0 12.605 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
Table 6 Filament radii r (cm 102) for the wetting width w ¼ 14:5 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 13.523 12.858 11.536 r o0 r o0 r o0 r o0
r > r0 r > r0 13.504 12.824 11.469 11.036 10.358 r o0 r o0
14.494 14.227 13.287 12.450 10.704 10.314 9.384 r o0 r o0
14.493 14.183 13.093 12.116 10.251 9.569 r o0 r o0 r o0
14.181 12.610 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
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Table 7 Filament radii r (cm 102) for the wetting width w ¼ 14:9 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 13.930 13.245 11.989 r o0 r o0 r o0 r o0
r > r0 r > r0 13.911 13.242 11.924 11.510 10.871 r o0 r o0
r > r0 14.628 13.697 12.875 11.193 10.624 9.657 r o0 r o0
14.893 14.584 13.505 12.549 10.491 9.711 r o0 r o0 r o0
r > r0 13.018 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
Table 8 Filament radii r (cm 102) for the wetting width w ¼ 15 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 14.032 13.381 12.102 r o0 r o0 r o0 r o0
r > r0 r > r0 14.012 13.347 12.037 11.627 10.996 r o0 r o0
14.995 14.727 13.800 12.983 11.313 10.754 9.812 r o0 r o0
14.893 14.584 13.505 12.549 10.491 9.711 r o0 r o0 r o0
14.673 13.122 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
1.6
5.5
7
18.491 18.060 16.553 15.168 11.772 9.627 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
Table 9 Filament radii r (cm 102) for the wetting width w ¼ 18:5 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 17.288 16.455 r o0 r o0 r o0 r o0 r o0
r > r0 r > r0 17.257 16.402 14.662 14.096 r o0 r o0 r o0
ðr ; h Þ with r ¼ 10 (cm 102) (as considered by Kobayashi et al. in Ref. [34]) if the wetting width w is in the range 10.5–18.5 (cm 102). This fact implies that it is not possible to obtain a LiNbO3 filament with r ¼ 10 (cm 102) if the meniscus bottom is fixed to the outer edge of the die i.e. w ¼ r0 ¼ 22:5 (cm 102) (see Table 10).
1 18.493 18.140 16.900 15.778 13.331 12.418 10.520 r o0 r o0
In the following we will show what happens if we change suddenly w from w1 to w2 : For this purpose the unsteady state equations Eq. (1) are solved numerically. We assume that during the growth, ðv; T0 Þ ¼ ð1; 2027Þ ((cm 102)/s; K) and we start with the initial conditions ðr0 ; h0 ; w0 Þ ¼ ð10; 5; 13:5Þ: At the
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Table 10 Filament radii r (cm 102) for the wetting width w ¼ 22:5 (cm 102) v ((cm 102)/s) T0 (K)
0.001
0.1
1
1.6
5.5
7
1527 1576 1726 1839 2026 2076 2145 2226 2246
r > r0 r > r0 r > r0 20.049 r o0 r o0 r o0 r o0 r o0
r > r0 r > r0 21.014 19.970 17.800 17.080 r o0 r o0 r o0
22.490 22.048 20.471 19.005 15.625 14.208 r o0 r o0 r o0
22.488 21.915 19.890 17.962 11.690 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0 r o0
Fig. 2. Evolution of the crystal radius due to the following movements of the bottom line: w0 ¼ 13:5-w1 ¼ 14-w2 ¼ 13:5-w3 ¼ 14:5-w4 ¼ 13:5-w5 ¼ 14:9-w6 ¼ 13:5-w7 ¼ 15-w8 ¼ 13:5 (cm 102).
moments t1 ¼ 59; t2 ¼ 118; t3 ¼ 179; t4 ¼ 238; t5 ¼ 301; t6 ¼ 360; t7 ¼ 421; t8 ¼ 480 (s), during the growth we change the wetting width as follows: w1 ¼ 14-w2 ¼ 13:5-w3 ¼ 14:5-w4 ¼ 13:5-w5 ¼ 14:9-w6 ¼ 13:5-w7 ¼ 15-w8 ¼ 13:5 (cm 102). The computed filament radius is plotted in Fig. 2. This simulation shows that if jw1 w2 j is less than 1.5 (cm 102) the filament radius r ðw1 Þ evolves in time to r ðw2 Þ while if jw1 w2 j ¼ 1:5 (cm 102) then r ðw1 Þ does not evolve to r ðw2 Þ; in the last change w7 ¼ 15-w8 ¼ 13:5; r ð15Þ ¼ 11:303 evolves to the value 12.600 which is different from r ð13:5Þ ¼ 10:0005: This behavior of r ðw1 Þ is due to the fact that the steady state ðr ðw1 Þ; h ðw1 ÞÞ; in which we are at the moment when w switches from w1 to w2 ; does not belong to the region of attraction of the steady state ðr ðw2 Þ; h ðw2 Þ). For nonlinear systems the numerically determination of the region of attraction of an asymptotically stable steady-state is an open problem in mathematics. What we know today is
to give numerically an accurate estimation of the region of attraction [35].
4. The effect of a periodic movement on the die of the bottom line of the melt/gas meniscus We will show that a periodic movement of the bottom line leads to a periodic variation of the filament radius. We assume that during the growth ðv; T0 Þ ¼ ð1; 2027Þ ((cm 102)/s; K), the initial conditions are ðr0 ; h0 ; w0 Þ ¼ ð10; 5; 13:5Þ and during the process, at the moments t1 ¼ 59; t2 ¼ 118; t3 ¼ 179; t4 ¼ 238; t5 ¼ 299 (s), the wetting width changes periodically as follows: w1 ¼ 14-w2 ¼ 13-w3 ¼ 14-w4 ¼ 13-w5 ¼ 14 (cm 102). The computed evolution of the filament radius is represented in Fig. 3. This figure shows that there are six stationary phases and six transition phases, respectively, in this growth process. The first transition phase
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Fig. 3. Periodic evolution of the crystal radius due to the following periodic movements of the bottom line: w0 ¼ 13:5-w1 ¼ 14-w2 ¼ 13-w3 ¼ 14-w4 ¼ 13-w5 ¼ 14 (cm 102).
t1 ¼ 9 (s) is due to the fact that the initial condition with which we start the growth process is not stationary. The other five phases are due to the five movements on the die of the bottom line of the melt/gas meniscus and the relaxation times corresponding to these transition processes are t2 ¼ 9; t3 ¼ 11; t4 ¼ 9; t5 ¼ 11; t6 ¼ 9 (s). This computation shows that the crystal radius varies periodically around the average value r ¼ 10 (cm 102).
5. Modeling the compensation of the effect of periodically wetting width changes by an adequate changes of the pulling rate We will show how we can compensate the effect of a given periodical variation of the wetting width w by an adequate variation the pulling rate v; in order to obtain a filament with constant radius r ¼ 10 (cm 102) (as considered by Kobayashi et al. in Ref. [34]). For that we assume that during the growth at the moments t1 ¼ 59; t2 ¼ 119; t3 ¼ 181; t4 ¼ 241; t5 ¼ 303 (s) there are the following changes of the wetting width: w1 ¼ 14-w2 ¼ 13-w3 ¼ 14-w4 ¼ 13-w5 ¼ 14 (cm 102). The adequate pulling rate changes are: v1 ¼ 1:632-v2 ¼ 0:83-v3 ¼ 1:632-v4 ¼ 0:83-v5 ¼ 1:632 ((cm 102)/s). The computed shape of the filament radius, due to the above w and v changes, is plotted in Fig. 4. The relaxation times of the transition processes are t1 ¼ 9; t2 ¼ 10; t3 ¼ 12; t4 ¼ 10; t5 ¼ 12; t6 ¼ 10 (s). In these transition periods only the fourth digit of the crystal radius varies.
Fig. 4. Evolution of the crystal radius due to the following periodic movements of the bottom line: w0 ¼ 13:5-w1 ¼ 14-w2 ¼ 13-w3 ¼ 14-w4 ¼ 13-w5 ¼ 14 (cm 102) and the adequate pulling rate changes: v ¼ 1-v1 ¼ 1:632-v2 ¼ 0:83-v3 ¼ 1:632-v4 ¼ 0:83-v5 ¼ 1:632 ((cm 102)/s).
6. Conclusions (i) If the meniscus bottom is fixed to the outer edge of the die, i.e. w ¼ r0 ¼ 22:5 (cm 102) (see Table 10) then we cannot obtain a LiNbO3 filament with radius r ¼ 10 (cm 102). We have to admit that the bottom line of the melt/gas meniscus is movable. (ii) The range of jw1 w2 j for which r ðw1 Þ evolves in time to r ðw2 Þ is 0–1.49 (cm 102). If jw1 w2 j is equal to 1.5 (cm 102), then the evolution of r ðw1 Þ to r ðw2 Þ is not sure. (iii) A periodic movement on the die of the bottom line of the melt/gas meniscus leads to a periodic variation of the crystal radius which can be compensated by an adequate periodic variation of the pulling rate.
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