Joirnal of Crystal Growth 109 (1991) 127—132 North-Holland
Oscillation phase relations in a Bridgman system David J. Knuteson
Archibald L. Fripp, Glenn A. Woodell, William J. Debnam, Jr.
N~ISA Langley Research Center, Hampton, Virginia 23665-5225, USA
and R inga Narayanan Th partment of Chemical Engineering, University of Florida, Gainesville, Florida 32611, USA
The thermal conditions under which convection in a tin melt undergoes the transition from steady flow to time-dependent flow an investigated. Previously, it has been shown that crystals that are directionally solidified from a time-dependent melt will have a lar ter defect density and increased chance of compositional striations. Thus, it is important for this transition to be well ch. tracterized. The experimental results to be presented were obtained from a two-zone Bridgman furnace with a middle insulation zooe. Thermocouples were placed on the axis and on the outer wall of a cylindrical vitreous silica tube which contained molten tin. Th phase relations between temperature oscillations measured at different positions in the cell are discussed. Fourier transforms are ust d to investigate the increasing complexity of convection as the temperature gradient is increased.
1. Introduction This paper will deal with convection and phase re]ations of temperature oscillations in liquid tin in a vertical Bridgman system. It has been shown [1,2] that the convection regime in the melt of a cr’rstal growth system has a large effect on the co’nposition and defect density of the crystal. In particular, the presence of oscillatory and turbulet t convection causes the crystal quality to decrtase. In order to better understand the transition to oscillatory and then turbulent flow, we have studied the phase relations between temperature oscillations and use FFTs (fast Fourier transfoims) to characterize the flow. Much of the work that has been done on convettive instabilities has been for Rayleigh—Bénard sy~tems. In a Rayleigh—Bénard (R—B) system, a fluid layer is heated from below (or cooled from *
Research Associate in the Department of Materials Science, IJniversity of Virginia.
0022-0248/91/$03.50 © 1991
above) and the resulting buoyancy force causes convection. In a vertical Bridgman configuration, the liquid is usually cooled from below to prevent R—B convection. Even then convection can occur, due to double-diffusive convection and radial ternperature gradients, which are inherently unstable. The main parameters that determine the convective regime in a R—B system are aspect ratio, Rayleigh number, and Prandtl number. For a cylinder, the aspect ratio is the height of the fluid divided by its radius (AR h/R). The Rayleigh number is a dimensionless parameter that depends on physical properties of the fluid, gravity level, fluid height, and temperature gradient. The Rayleigh number (Ra) can be thought of as a dimensionless temperature gradient. The Prandtl number (Pr) is the fluid’s kinematic viscosity divided by its thermal diffusivity. Liquid metals and semiconductors, the materials of interest here, have low Prandtl numbers and are opaque. As a result, the study of temperature behavior is one of the few methods available to
Elsevier Science Publishers By. (North-Holland)
D.J. Knuteson et al.
phase relations in Bridgman system
investigate convective flow in low Pr systems. Several authors [1—8]have done work on unsteady buoyancy-driven convection, but only a few have mentioned the phase relations of temperature oscillations. For instance, Hurle et al. [71measured the phase of surface temperatures in a rectangular boat of liquid gallium located in a horizontal Bridgman furnace. Most of the measurements were taken with length = 3 cm, width = 1.3 cm, and depth = 1.2 cm. They observed “phase vortices” into or out of which 2i~‘s worth of lines of constant phase disappear (or appear). The direction of rotation of the vortices depended on off-axis heat flow through the boat. MUller et al. [21 placed
the TCs are plotted) the thermocouples are very nearly in phase. In addition to studying phase relations, the Rayleigh— Bénard system is also ideal for the study of the onset of turbulence. While many systems become turbulent suddenly (e.g. pipe flow), the Rayleigh— Bénard system approaches the turbulent state in a series of small discreet steps that can be observed and used to explain the underlying mechanism. There are currently three different scenarios that explain the transition from simple oscillatory flow to turbulent flow. These scenarios are by Ruelle and Takens , Feigenbaum , and Pomeau and Manneville . Ruelle and Takens  predicted that chaos
thermocouples around the circumference of their cylindrical cell, with 1.0 ~ h/R ~ 10. Their results
would commence with the appearance of a strange
included characterization of axial and non-axial unsteady flow in liquid gallium, but they reported no phase measurements. Olson and Rosenberger  have measured phase differences with thermocouples at different locations in a cylinder for monocomponent gases. They placed three vertically-spaced thermocouples on one side of the cylinder and three more on the opposite side (180° around the vertical axis) of a cylinder with h/R = 6. Olson et al.’s data show a mirror-image relationship (fig. 13, TC pairs 2—3 and 5—6) for Ra 1620. Interestingly, this coincides with a second oscillatory mode that can be seen in the temperature traces beginning at Ra = 1620. For Ra = 1437 (the lowest Ra at which all
attractor after two or three period-doubling bifurcations. A strange attractor is an orbit into which a solution (velocity, temperature, etc.) settles and from where the solution will exhibit sensitive dependence on initial conditions. Feigenbaum  theorized that chaos occurs after an infinite number of period-doubling bifurcations, that would occur at a finite Rayleigh number. Pomeau and Manneville  pictured chaos as intermittent bursts of turbulent and steady behavior. As the system becomes more chaotic, the turbulent bursts become longer and more frequent. In the current work, the phase relations will be studied for liquid tin in a Rayleigh—Bénard configuration and the relevance of the data to these scenarios will he discussed.
CAP—to & 11
x s-~~ to
a Fig. 1. Thermocouplc placement for capillary cell.
D.J. Knuteson et al.
Oscillation phase relations in Bridgman system
2. Experimental The liquid tin was contained in cylindrical, vi reous silica (quartz) ampoules (ID = 16 mm). The outer walls of the ampoules were instrum ~nted with type K thermocouples (OD = 0.5 mm) ar d insulated with a ceramic blanket. A graphite p1 ig was placed on the top of the liquid tin to help extract heat from the top surface of the tin. Punfi d argon was trickled over the top of the cell to pi rvent oxidation of the tin. The phase measurements were carried out in a ce LI with a quartz capillary (OD = 2 mm) extended th ough the graphite plug and into the tin. The th ~rmocouple (TC) positions can be seen in fig. 1. Three sheathed type K thermocouples (OD = 0.25 m TI) were placed in the capillary and cemented in :0 place. The capillary TCs were positioned to correspond to the vertical placement of the sidew, LIl TCs. The cell was loaded into a Bridgman furnace with two heating zones separated by a ceramic in ;ulation zone. Isothermal liners were placed in b th the top (cold) and bottom (hot) zones to m.tke the gradient in the insulation zone more hr ear. Several experiments were done to ensure th.it the oscillations were due to unsteady convec-
T887. 9 AR7 APIII Ra = 1,05~ 000
3. Results and discussion Representative data from the capillary experiment is shown in fig. 2, for AR 7. The Rayleigh number is greater than 1 X 106, 50 the convective flow in this data has progressed past the purely oscillatory regime which is characterized by a single dominant peak in the FFT. In fig. 2a, it can be seen that the temperatures from the thermocouples that are vertically-aligned, TC8—TC11, stay in phase, despite the high Ra and the erratic behavior of the temperatures. More surprising is the behavior of the thermocouples that are at the same vertical position (see fig. 2b). The capillary thermocouple, CAP-b, is beginning to show some signal, but at lower Ra,
T887. 9 AR7 APIII Ra — 1,05 000
lion in the tin and not due to a furnace effect. In the most definitive experiment, thermocouples were placed next to the furnace blocks, and both inside and outside the insulation blanket that surrounds the cell. When oscillations were observed inside the blanket, none were observed outside the blanket or in the furnace block. More information on the experimental setup is available elsewhere .
TC—8 TC—9 TC—10 TC it
~ ~ ~
Fig. 2. Data from AR7CAPIII, Ra =1,053,000: (a) vertically-aligned thermocouples; (b) thermocouples at the same vertical position. (Note: the y-axis is marked in 1°C increments and the thermocouple traces have been moved to make the phase relations more clear.)
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phase relations in Bridgtnan system
the signal is negligible. The other four TCs, though, are very active. Thermocouples B-b and F-b (as well as TC-10 and S-bO) are almost exact mirror images, or nearly 180° out of phase. This close correlation between B-b and F-lU is unexpected because they are on opposite sides of the ampoule. The two adjacent TCs to any given TC, though, show no particular relationship at high Ra. For example, the TCs adjacent to TC-bO (B-b and
F-b) are sometimes in phase and sometimes out of phase. Also, there is less correlation of peak amplitudes between adjacent TCs than between the opposite pairs. This behavior persisted throughout the upper range of Rayleigh numbers, up to 2.7 x 106. The relations between the thermocouples are more clearly shown with the use of a Lissajou diagram. In a Lissajou diagram, the temperature
TC—10 (°c) T887.09
Fig. 3. Lissajou diagrams for AR7CAPIII, Ra = 1,053,000. (a) Vertically aligned TCs, TC-9 versus TC-10; (b) opposite TCs, S-lU versus TC-10; (c) adjacent TCs, F-b versus TC-10.
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Oscillation phase relations in Bridgman system
from one TC is plotted against the corresponding temperature from another TC, rather than against til ne. A perfect, positive correlation would be repre ;ented by a plot that forms a single line with a p( sitive slope. A perfect, negative (or mirrorin age) correlation would result in a plot of a line
Some of the Lissajou diagrams for the data in fi1. 2 are shown in fig. 3. First, notice the plot of T( -9 versus TC-bO (fig. 3a), a vertical pair of TCs thit are separated by b cm. The plot shows that
sii riultaneously, as do their minimums In other wards these two TCs are in phase The relatively fl~t, or linear, shape of the curve shows that the oscillations of TC-9 and TC-bO remain closely in phase.
Next, notice the plot of S-bO and TC-10 in fig. 3b. These thermocouples are located on opposite 511 es of the cell. The plot looks much like the plot fo TC-9 and TC-bO except that the maximum vaues of 5-bO correspond to the minimum values for TC-lO and vice versa. This indicates that the twD TCs are b80° out of phase. Furthermore, sir ce the curve is even flatter than the plot for T(’-9 and TC-bO, these oscillation peaks are more ck sely correlated than the peaks of TCs that lie on a vertical line. Finally, look at the adjacent TCs that lie 90° ap irt on the outside of the cell. The pair of F-bO an I TC-l0 is plotted in fig. 3c. There is no domina~tphase relationship (in phase or out of phase) foi this pair. The poor correlation of these two th rmocouples is shown by the broad pattern that appears in the Lissajou diagram. The relationships th~t are represented in fig. 3, are typical for these experiments and continued up to Ra 2.7 X 106, th upper range of these experiments. Furthermc re, the cell was rotated 90° and more data we -e collected. The relationships shown in figs. 2 and 3 were still observed, ~ lower Ra, where the temperature response is nearly sinusoidal, the effect is less striking. The opposite pairs of TCs remain out of phase at low Ra The adjacent TCs now also appear to be closely correlated, but this could be due to the sin plc nature of the temperature response. For an’ given TC, one adjacent TC will be in phase
~io ,r,,ss _____
Fig. 4. FFT of Tb312.07 (TC-1O), AR7CAPIII, showing period-doubling bifurcations. (The vertical lines are drawn at bifurcation frequencies.)
and the other out of phase. This behavior by the adjacent TCs begins to break down at Ra 585,000. The capillary thermocouples appear to have an increase in amplitude near the time the sinusoidal oscillations break down and the adjacent TC correlations weaken. It was mentioned earlier that FFTs can be used to characterize convective flow. An FFT for ternperature data taken for AR 7, is shown in fig. 4. The primary frequency at 0.0086 Hz will be denoted by f0. It can be seen that two or three period-doubling bifurcations have occurred, as shown by the peaks at f0/2, f0/4, 3f~/4,etc. Such bifurcations were often observed in the FFTs, but resolving each new bifurcation required the time for an experimental run to double. Since the oscillations occur at low frequencies, observing a large number of bifurcations would take a very long time with this system. Still, the period-doubling behavior shown agrees well with Feigenbaum’s [bO] theory for the onset of turbulence.
4. Conclusions Oscillation phase relationships have been measured for liquid tin, heated from below, in a verti-
D.J. Knuteson ci at.
cal Bridgman furnace. It was found that temperalure traces from thermocouples that were aligned vertically remained in phase from the onset of oscillatory flow to fairly turbulent conditions (Ra = 2.7 x 106). Thermocouples that were placed on opposite sides of the ampoule developed a mirrorimage relationship, or were 180° out of phase. Again, this behavior was observed from the onset of oscillatory flow to very large Rayleigh numbers. Oddly, the thermocouples on opposite walls were
phase relations in Bridgman system
from experimental data, and intermittent behavior is by nature random and unpredictable. Therefore, not surprisingly, no confirmation was found for these two scenarios. So, while these results support
Feigenbaum’s theory, none of the results contradict the theories of Ruelle and Takens or Pomeau and Manneville.
the most closely correlated thermocouples on the cell. Finally, the thermocouples that were adjacent
 K.M. Kim, A.F. Witt and H.C. Gatos, J. Electrochem.
to each other had an unpredictable relationship. At low Ra, the adjacent thermocouples were either
[21 G. MUller, G. Neumann
Soc. 119 (1972) 1218. and W. Weber. J. Crystal Growth
perfectly in phase or out of phase. At higher Ra,
70 (1984) 78.  H.T. Rossby, J. Fluid Mech. 36 (1969) 309.  J.D. Verhoeven. Phys. Fluids 12 (1969) 1733.
these relations broke down and the adjacent ther-
 R. Krishnamurti, J. Fluid Mech. 60 (1973) 285.
mocouples appeared to have a nearly random
 J.M. Olson and F. Rosenberger, J. Fluid Mech. 92 (1979)
association. These findings agree with the results of Olson and Rosenberger  for monocomponent gases in a vertical cylinder. With the use of Fourier transforms, perioddoubling bifurcations were observed in this systern. This supports the conjecture of Feigenbaum  concerning the onset of turbulence. No cvidence was found to support the conjectures of Ruelle and Takens  or Pomeau and Manneville . It is difficult to detect a strange attractor
D.T.J. Hurle, F. Jakeman and C.P. Johnson. J. Mech. 64 (1974) 565.  F.H. Busse, J. Fluid Mech. 52 (1972) 97. 
 D. Ruelle and F. Takens. Commun. Math. Phys. 20(1971) 167.  M.J. Feigenbaum, Los Alamos Sci. 1(1981) 4. ~
Y. Pomeau and P. Manneville, Commun. Math. Phys. 74 (1980) 189.  D.J. Knuteson, Experimental study of convection in tin in a vertical Bridgman configuration. Doctoral Thesis. iJniversity of Florida (1989).