Chang, C. E.

Wilcox, W. R.

English

NASA Technical Reports Server

194 
THEXMOCAPILLARY F L O W  I N  A CYLINDRICAL 
* 
LIQUID DROP AT ZERO GRAVITY 
Chong E. Chang and W i l l i a m  R. Wilcox 
Chemical Engineering and Materials Science Departments 
University of Southern California, Los Angeles, California 90007 
ABSTRACT 
Surface tension driven flow i n  the floating zone melting gemtry was 
studied by means of computer solution of the differential equations for heat and 
momMrum transfer. 
top of the zone, and its mirror image i n  the bottom of the zone. 
scanewhat l ike vortex rings (smoke rings). 
crease,secondary cells fo= and the flow probably becomes oscillatory. High 
differences produce turbulence. For silicon, effective heat shields are re- 
quired t o  avoid turbulence, and the flow is not appreciably influenced by zone- 
motion or by earth's gravity. 
on the heat transfer w i t h  good thermal conductors. 
thermal conductors may result in oscillatory flow even with s m a l l  driving 
forces. 
A t  small temperature differences one ce l l  is f o m d  i n  the 
They appear 
As the temperature differences in- 
The laminar cells do not have a strong influence 
Radiant heating of poor 
INTKQDUCTION 
Floating zone melting is used commercially and in research for crystal 
growth and for purification of high melting materials. 
Contact a crucible, as shown schematically i n  Fig. 1. 
tion, by an electron beam, or by induction from a surrounding high frequency 
coil. The zone is moved through the solid either by moving the heater or  by 
moving the solid rods borderfns the zone. 
in  a floating zone: 
zone generated by melting a t  one solid-liquid interface and freezing a t  the other, 
due to movement of the zone through the solid. 
sometimes used t o  maintain a cylindrical zone and growing rod.) 
diff icul t  t o  calculate. 
tional or hccelerational f ie ld  w i t h  density variations i n  the melt. 
along the m e l t  surface. 
The m e l t  does not 
Heating may be by radia- 
There are five sources of convection 
1) When the zone is taken as stationary, there is a flow through the 
2)  Convection due t o  rotation of the two solid rods. 
3) 
4) 
5) 
(Slow rotation is 
With induction heating one obtains electrmagnetic st irring, which is 
Buoyancy-driven natural convection due to interaction of a gravita- 
Surface-tension driven f low due t o  variations i n  surface tension 
* 
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195 
as w e l l ,  through its effect  on the freezing interface shape. 
detexmine the ialuence of surface driven flow on zone m e l t i n g ,  the part ia l  
differential equations for  momentum and heat transfer were considered. The 
momentum eqpations ware expressed i n  f i n i t e  difference &lindrical stream- 
function form, and solved by the Gauss-Seidel i t e r aeon  method (1). 
Raphson method w a s  used to  solve the heat transfer equatfans, w i t h  an Over- 
relaxation parameter of 1.9 for  low Prandtl numbers and an underrelaxation for 
high Prandtl numbers, Solutions were obtained for two situations--a parabolic 
temperature profile along the melt-vapor surface and for  a ring heat source at  
the center of the zone. 
radiant heating, and the ring source corresponds to electron beam heating. 
mass transfer equation coupled with the computed flow fields were @mployed to  
find the impurity concentration fields i n  the m e l t  a t  steady state. 
I n  ordsr t o  
The Newton- 
The parabolic profile corresponds approximately to  
The 
The results of the calculations are expressed i n  dimensionless fonn and 
fo r  the physical properties of silicon w i t h  a 1 an x loa zone, as a concrete 
exampl8. 
0.023, the schtoidt number is ahout f, pf(ay/aT)/1.12 is L4,000, and the eannisaivity 
is ass- to be 0.3 fo r  both the m e l t  and the SOW* 
f ie ld  and ordinary zone travel rates weze found to have negligible influence on 
the convection for silicon. 
For molten silicon the melting point is 14I.O C, the P r a n d t l  number is 
Earth's gravitational 
Dimensional analysis of the momentum equations produced a parameter char- 
actering,the driving force for surface driven flow, M = a(To-Tm)pf(ay/aT)/112, 
where a i s  the radius of the zone, To is the temperature of the surface of the 
m e l t  a t  the center of the zone, Tm is the interface temperature, pf is the density 
of the melt, y is the surface energy or surface temperature, and 1.1 is the 
viscosity. 
S t r ed inas  for  a Parabolic Temperature Profile 
a t  the h e e  Melt Surface (Radiation Heatinq) 
For simplicity, a parabolic temperature profile along the m e l t  surface w a s  
itssmed fo r  our in i t ia l  calculations. This enabled us t o  sol- the momentum 
equations without simultaneously solving the heat transfer equations. 
bolic profile, w i t h  a maximum a t  the center, seems reasonable for radiant heat- 
ing. 
the zone a and the temperature variation along the m e l t  surface increases. 
Figures 2 through 4, the streamlines for parabolic temperature profiles are 
Shawn to i l lustrate  the effect of increased values of M (350, 3500 and 7000). 
Donut-shaped vortex cells w e r e  formed. With M = 35 and 350 only two e l l s  are 
generated, and the centers  of the vortices move closer toward the liquid/solid 
interfaces as M increases. W i t h  M = 3500, however, which corresponds to a 
A para- 
The dimensionless surface tension parameter M increases as th% radius of 
In  
196 
condition of a = 0.5 cm and (To-%) * 0.5OC fo r  silicon,* secondary VOW cells 
were induced behind the primary vortex, as shown' i n  Figure 3. 
M was further increased up to 7000, third and fourth vortices were produced, as 
shown in  Figure 4. 
lations and incipient turbulence, which cannot be calculated i n  a steady state 
analysis (oscillations are frequently found with free convection in enclosed 
cavities between laminar and fully turbulent flow). The maximum velocity of 
the m e l t  f o r  1 cm diameter silicon w i t h  (To-Tm) = l.O°C is calculated to be 2 
Wsec. 
As the value of 
These multiple vortices are probably indicative of osJ1- 
Fluid Flow Coupled with H e a t  Transfer 
(Electron Beam Heating) 
Electron beam heating is commonly used i n  floating zone melting and w a s  con- 
Rather than specify the power input, the circlrmferential temperature To 
sidered as another heating mode. 
assumed. 
a t  +&e center of t h e  zone was specified for  convenience i n  analysis. 
cedure used to solve the coupled heat-momentum transport problem w a s  as follows: 
t ive heat transfer. 
first approximate solution of fluid flow. 
A ring heat source at  the center of the zone was 
The pro- 
' 1)- The temperature f ie ld  for conduction w a s  canputed by neglecting convec- 
2) The surface temperature profile from 1) w a s  utilized to calculate the 
3) The temperature f i e ld  w a s  recalculated using the f low fields from 2).  
4) Steps 2) and 3) were repeated until the temperature and the fluid flow 
The resulting streamlines w i t h  M * 350** are drawn i n  Figure 5 i n  which w e  
f ields no longer changed appreciably. 
took values for  silicon w i t h  the surroundings a t  the melting point, i.e., the 
heat shielding about the zone i s  extremely effective. 
respond exactly with r ea l i t y ,  it does show the correct features. Comparing w i t h  
Figure 2, we see that the vortex centers are shifted nearer t o  the heat source 
f rom the liquid/solid interfaces. 
gradient is a t  the'center of the zone. The max imum velocity for  silicon w a s  0.55 
cm/sec. 
M = 350 with those for pure conduction, there w a s  no significant change except fo r  
a sl ight one near the center of the zone. 
greater than convective heat transfer as would be expected for the s m a l l  Prandtl 
number for silicon. 
maximum vorticity and its location for various conditions are also summarized i n  
Table I. 
While this does not cor- 
This is because the steepest temperature 
comparing the temperature fields w i t h  surface tension driven flow for  
This indicates that  conduction is much 
The The vorticity fields for  PI = 350 are shown i n  Figure 6 .  
._ - .  - __ - -  
Influence of Gravity on Flow 
In a vertical silicon m e l t  w i t h  M = 350 a t  earth's gravity the flow and 
*Without heat shielding, we estimate the temperature difference (To-Tm) would be 
on the order of 10 to  2OoC, and the f l o w  would be turbulent. 
For electron beam heating of 1 cm diameter silicon, the Marangoni nusaber M a  is 
about 1/50 of the value of M. 
about 60,000. 
** 
Without heat shielding Ma is estimated t o  be 
197 
temperature f ie lds  i n  tLe floating. zone do not change appreciably from those a t  
zero g.* men the acceiezation is increased to ten times the earth's gravity, 
the lower vortex contracts while the upper one 5s expanded, p a r t i c u l t r l - a t -  the 
center of the zone where the  f l aw f ie ld  is relat ively weak. 
F i w e  7 and may be canpared with Figwe 5 a t  zero g. T h i s  means tha t  surface 
driven flow predominates even on earth for  many materials for  radiant and 
electron beam heating. It may even be important when induction heating is em- 
ployed and may account fo r  sgme of the caapositional inhomogeneities observed. 
This is sham i n  
rnfluence of P rand t l  Nuraber 
The Prand t l  number WM increased from 0.023 t o  0.3 and 2.0 by increasing 
the specific heat, and keeping the viscosity and the thermal conductivity of the 
m e l t  constant. As the Prandtl number increases, the temperature gradients along 
the free liquid surface (for fixed To-Tm) increase near the heat source and also 
neap the liquid/solid interfaces, but decrease significantly i n  between. Since 
the convective heat t ransfer  becomes more significant as a result of increasing 
the P rand t l  number, the isotherms are increasingly distorted from those of pure 
conduction. T h i s ,  in turn, causes changes i n  the flow f ie ld-  For example, the 
center of the vortex cell near the liquid/solid interface shifts closer t o  the 
interface as the Prandtl nuuiber increases, a6 shown i n  Figure 8. 
Influence of Zone Travel on Hydrodynamics 
The effect of zone mt ion  on the flow f i e ld  w a s  found to be negligible in 
all of our calculated streamlines for  zone t rave l  ra tes  of up t o  5 cm/hout i n  
silicon.** However, the effect becomes significant when the zone t ravel  ra te  
becomes comparable with the velocity of surface-driven flow. 
The flow f i e ld  for  electron beam heating with M = 35 is  taken as a m o d e l  
case i n  order t o  show the influence of zone t rave l  on hydrodynamics i n  the 
floating zone. The maximum velocity of the m e l t  for  M = 35 is 0.07 cm/sec (or 
250 cm/hour), and the average velocity is about 70 cm/hour. 
streamlines i n  floating zone melting of s i l icon a t  zero g are shown for  a 
freezing rate of 30 cmfiour. 
interface and its s ize  is reduced as the zone travel rate increases. 
In Figure 9, the 
The lower vortex cell f loa ts  away from the b o t t m  
CONCLUSION 
For si l icon w e  have seen tha t  thermacapillaq flow i s  very vigorous, and 
Withmaderate heat shielding, oscillatory convection is likely, 
i s  turbulent when heat shields are not employed. 
i n  comparison. 
but was  not studied here. 
convection unless the temperature gradients along the m e l t  surface are made very 
small by effective heat shielding. With laminar flow, tha convection has only a 
small  influence on heat transfer a t  small Prandtl numb- (Pr << 11, but a large 
influence for  P r  > 1. 
The buoyancy flow is negligible 
Zone m a o n  does not have an appreciable effect  on the 
, 
* 
** Only temperature variations w e r e  considered. 
A zone t rave l  rate of 5 cm/hour is typical for  growth of single crystals, w h i l e  
faster rates are  employed for  vacuum outgassing and lower rates for  zone refining. 
198 
REFEICENCES 
I. B. W. hrden and K. N. A s t i l l ,  "Numerical  Alg0ri-r 
Applications," Addison-Wesley, Reading, Mass. (1970) 
Origin and 
199 
GAS OR 
VACUUM 
SOLID 
I 
LIQUID 
* :  
! 
I 
I 
I 
SOLID 
vz 
GAS OR 
2 =-1 
200 
Fig. 2. Canputad d%~aena&onless streamlines $ for  surface tension driven flow in 
a floating zone at zero gravity with a parabolic temperature profile on 
the free liquid surface with M - 350 and vc - 0 .  
CT,-Td = 0.OS0C and a - 0.5 cm. For silicon with 
201 
SOLI 0 
-17 
- I6 
-14 
“12 
VACUUM 
Fig. 3. Coaputed d ~ i o n I , e 8 8  streamlines Fp for surface drivan flow in a 
floating zone a t  zero gravity with a parabolic temperature profile 
oa the free liquid -face with M * 3500 and v, - 0. Silicon with 
Q o - ~ )  * 0.50C rpd a 0.5 a. 
202 
-0.92 
SOLID 
-12 
‘IO 
‘ 0  
VACUUM 
Fig. 4. computed dimmnaionlesa rtreamlbes 9 for surface tension flaw in a 
floating 2one at zero gravity with a parabolic temperature profile 
on the free liquid surface with M = 7000 and vc - 0. Silicon with 
(To=&) l .O°C and 8 0.5 em. 
203 
UM 
Fig. 5 Isotherms 8 and streamlines JI at  zero gravity for electron beam heating 
of silicon with (To-T,,J = 0.050G, T, - Tm. e, = 0.3, a - 0.5 cm, M * 350, 
Pr 0.023, VC * 0. 
204 
SOLID 
Fig. 6. Dimensionless vorticity f i e l d  w corresponding to  the streamline f i e l d  
shown in Figure 4. 
20s 
I .( 
-- 
SOLI 0 
SOLID 
VACUUM 
Fig .  7. Streamlines and isotherms for combinad surfacedriven and buoyancy- 
driven flat with G r h  - 775. Other conditions are the same 88 Figure 4. 
206 
SOLID 
Fig. 8. Isotherma 8 and streamliDne J, for the same conditlonr, as .in Figure 4, 
except pt 2. 
207 
Fig. 9. Streamlines 11 at a 2098 traval rate of 30 c d h m  with M = 35. P r  = 0.023. 
For cl i l fcolr ,  a = 0.5 cm, (To+) = 0.005°C, T, - %, es 0.3. 


Thermocapillary flow in a cylindrical liquid drop at zero gravity

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