Corresponding author: Vladimir S. Berdnikov ( berdnikov@itp.nsc.ru ) © 2019 Vladimir S. Berdnikov.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Berdnikov VS (2019) Hydrodynamics and heat exchange of crystal pulling from melts. Part I: Experimental studies of free convection mode. Modern Electronic Materials 5(3): 91100. https://doi.org/10.3897/j.moem.5.3.46647

This work is a brief overview of experimental study results for hydrodynamics and convective heat exchange in thermal gravity capillary convection modes for the classic Czochralski technique setup obtained at the Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences. The experiments have been carried out at test benches which simulated the physics of the Czochralski technique for 80 and 295 mm diameter crucibles. Melt simulating fluids with Prandtl numbers Pr = 0.05, 16, 45.6 and 2700 have been used. Experiments with transparent fluids have been used for comparing the evolution of flow structure from laminar mode to welldeveloped turbulent mode. Advanced visualization and measurement methods have been used. The regularities of local and integral convective heat exchange in the crucible/melt/crystal system have been studied. The experiments have shown that there are threshold Grashof and Marangoni numbers at which the structure of the thermal gravity capillary flow undergoes qualitative changes and hence the regularities of heat exchange in the melt change. The effect of melt hydrodynamics on the crystallization front shape has been studied for Pr = 45.6. Crystallization front shapes have been determined for the 1 × 10^{5} to 1.9 × 10^{5} range of Grashof numbers. We show that the crystallization front shape depends largely on the spatial flow pattern and the temperature distribution in the melt.
crystal growth, Czochralski technique, heat exchange, thermal gravity capillary convection, boundary layers, laminarturbulent transition, local heat flows, crystallization front shape
Solving the practical task to improve the quality of crystals and hence optimize single crystal melt growth process modes requires fundamental studies of melt hydrodynamics and complex conjugate heat and mass transfer in growth units of general type [
Below we will present the first part of this fourpart overview of experimental and numerical results for convection in various fluids simulating various setups of the Czochralski technique obtained at the Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences. By now the most complete hydrodynamic and convective heat exchange simulation data have been obtained for the classic fixed crucible setup of the Czochralski technique. The aim of the present series of studies is to simulate the crystal growth conditions at different process stages and develop a fundamental basis for improving the methods of controlling the melt hydrodynamics and the heat exchange conditions. When designing the benches for the simulation of physical models of various directional crystallization techniques, we used proprietary methods for the visualization and parameter measurement of velocity and temperature fields developed at the IT SB RAS for the fundamental investigations of nearwall turbulence [42]. The experiments dealt with laminarturbulent transition (LTT) processes in freeconvention boundary layers and heat exchange at free liquid surfaces [
Under real process conditions, melt hydrodynamics are determined by the cooperative action of a set of bulk forces (buoyancy, centrifugal, Coriolis, shear and electromagnetic forces) and surface forces (thermo and concentrationcapillary, Laplacian and friction forces). If the effect of the buoyancy forces is significant, the temperature and velocity fields are selfconsistent. The presence of the thermocapillary effect further increases the hydrodynamics/heat exchange feedback. The development of hydrodynamic control methods requires understanding of the specific influence mechanisms (or the specific expression regularities) and the relative contributions of each of the forces in question to the formation of the melt flow structure and analysis of the results of their nonlinear interaction. This problem cannot be solved using solely experimental tools since the buoyancy forces which are inevitable in Earth conditions and other bulk forces act jointly with surface forces. Controlling their relative contributions or separately studying their individual action under the conditions of a physical experiment is an extremely complex or even impossible task. Therefore there is a need for combined complementary experimental and numerical investigations. Results of numerical investigations into free convection flow modes will be presented in the second part of this overview, and experimental and numerical results for the effect of the abovementioned forces on the convective processes in the melt during crystal and crucible rotation will be discussed in the third and the fourth parts.
Natural convection was experimentally studied at different stages from laminar mode to welldeveloped turbulent flow mode [
Below we present the results of studies conducted using two physical models of the classic Czhochralski technique. These models had almost similar designs of the working zones, but the crucibles differed considerably in size: the crucible diameter was D_{c} = 80 mm for the first section and D_{c} = 295 mm for the second section. The working media were gallium/indium/tin melt (Prandtl number Pr = 0.05), ethyl alcohol (96 %, Pr = 16), PES5 siloxane fluid (Pr = 2700), and saturated hydrocarbons, i.e., hexadecane and heptadecane (Pr = 45.6). The two latter fluids have the solidification points at 18 and 22 °C, respectively, and allow studying crystallization processes for a transparent fluid. Transparent fluids were visualized using specially selected fraction of scaleshaped aluminum particles 10 to 15 µm in size. Fluid movement was observed in the central section (the r–z plane). The observation area was continuously illuminated with 1 to 2 mm thick planar light beams. The observation results were also photographed and video recorded in the r–z plane at specific preset levels relative to the bottom. In these cases the working zones had transparent bottoms and were observed through a rotating mirror installed underneath at a 45 arc deg angle to the bottom plane. The thermocouples for temperature measurement at the crucible surface and at the model crystals were made from 80 µm diam. copper and 60 µm diam. constantan wires. The temperature distribution in the working fluid volume was measured with the thermocouple probes. The measurement increment was controlled accurate to 0.01 mm along both coordinates. The nichromeconstantan probe thermocouples (0.1 mm wire diameter) were electrochemically etched off to 10–25 µm diam. at the working solders. The solders of the three thermocouple probes were arranged one under another in the same vertical plane so the temperature distribution could be measured in the CF boundary layers, at the crucible side wall and at the free melt surface. The two top thermocouple probes were installed with a constant spacing (δz ≤ 0.3 mm); this arrangement allowed measuring local (instantaneous and average) heat flows in the fluid for differential switching of the thermocouple probes and in simultaneous signal recording mode. The thermocouple sensitivity was 42 µV/K. The accuracy of the measurements with Sch300 or F30 microvoltmeter/ammeters was 1 µV.
Thermal gravity convection caused by the temperature difference between the CF and the crucible walls in the Czochralski technique is fundamentally unavoidable and can hardly be controlled, and this is the case for any nonisothermal system in the gravity field. Its physical nature, contribution and intensity are determined by the temperature dependence of density and by the effect of buoyancy (Archimedes) forces and are described by the Grashof number Gr = (βg/ν^{2})ΔTR_{sc}^{3} or the Rayleigh number Ra = GrPr = (βg/aν)ΔTR_{sc}^{3}, where g is the gravity acceleration, β is the volumetric expansion coefficient, ν = µ/ρ is the kinematic viscosity coefficient, µ is the dynamic viscosity coefficient, ρ is the density, a = λ/ρC_{p} is the thermal diffusivity coefficient, λ is the thermal conductivity coefficient, C_{p} is the constant pressure heat capacity and R_{sc} is the crystal radius. The Prandtl number Pr = ν/a describes the thermophysical properties of the melt. The presence of a free melt surface portion and the temperature drop ΔT along the free surface between the CF and the crucible walls leads to a combined effect of the buoyancy forces and the thermocapillary effect. The thermocapillary effect originates from the temperature dependence of the surface tension coefficient σ and the forces acting along the melt surface in radial directions. As a rule the surface tension coefficient decreases with an increase in temperature. The resultant tangential force directed from the heated melt surface portion to the cold one causes thermocapillary convection. The intensity of thermocapillary convection is described by the Marangoni number Ma = (–δσ/δT)ΔTR_{sc}/aµ.
Thermal gravity capillary convection (TGCC) in the melt is always the initial mode for the search of optimum crystal growth conditions, or the working mode for the growth of some oxide crystals. Depending on the free surface state and the conditions of heat emission to the ambience, this initial mode may have variable relative contributions of the buoyancy forces and the thermocapillary effect. The two limit cases are thermal gravity convection mode for which the surface tension coefficient depends but slightly on temperature and gravity capillary convection mode with a free adiabatic boundary for which the contribution of the thermocapillary effect is the greatest in the presence of a radial temperature gradient.
For free convection which always has a gravity capillary origin in a physical experiment, the flow arrangement is as follows. In the center (Fig.
Figure
Spatial flow pattern in buoyancy and thermocapillary driven convection mode for Pr = 16: (a) H/R_{c} = 1.5, R_{c}/R_{sc} = 6.68, Gr = 1.05 × 10^{4}; (b) H/R_{c} = 1.5, R_{c}/R_{sc} = 1.29, Gr = 2.05 × 10^{6}; (c) H/R_{c} = 0.9, R_{c}/R_{sc} = 2.75, Gr = 1.35 × 10^{6}. H is the melt layer height; R_{c} is the crucible radius.
In highly viscous media with Pr = 2700 the thermocapillary effect has but a little effect on the spatial pattern of the laminar flow, and no superficial swirl forms. In nontransparent gallium/indium/tin melt with Pr = 0.05 the temperature distributions normal to the CF were measured with microprobe thermocouples. The temperature profiles were similar to those observed in media with Pr = 16 ÷ 2700 having clearly delimited thermal boundary layer portions and a radial distribution of the local heat fluxes that is typical for free laminar convection.
In TGCC mode an increase in the temperature gradient and hence a growth of Gr and Ma cause an unstable and nonsteadystate flow which transits gradually to random turbulent mode. The qualitative parameters of the transition to turbulent flow mode depend largely on the relative and absolute dimensions of the thermohydrodynamic system in question. For a 80 mm crucible diameter the volume of fluid in turbulent mode is filled with variable size swirls (Fig.
Laminar fluid convection mode with Pr = 16 shown in Fig.
RayleighBenard instability develops under the CF in largediameter crystals: cells forming in the boundary layer are entrained by the main flow from the crystal edge toward the center.
Figure
TGCC mode temperature oscillation power spectra for Region IIa (Fig.
Once threshold conditions are reached (Region IIb between Curves 3 and 4) when spaceordered secondary flows emerge, the temperature oscillations under the CF at a certain distance from the front edge are regular, the respective spectra becoming discrete. Oscillation power spectra typical of this region are shown in Fig.
TGCC mode temperature oscillation power spectra for Region IIb (Fig.
TGCC mode temperature oscillation power spectra for Region IIc (Fig.
An increase in Gr causes a shrinkage of the ordered secondary flow existence region and noise contamination of the temperature oscillation spectra. The outer edge of the boundary layer exhibits irregular outbursts of the melt cooled under the CF to far beyond the boundary layer. In Region IIc (Fig.
The temperature oscillation behavior features noted above correlate with the regularities of heat boundary layer development at the CF and the local heat emission regularities. Figure
Radial local heat flow distributions at CF for different Grashof numbers Gr, 10^{6}: (a) (1) 0.0062; (2) 0.041; (3) 0.068; (4) 0.149; (5) 0.269; (6) 0.313; (b): (1) 0.813; (2) 1.645; (3) 3.174; (4) 5.175; (5) 7.259; (c): (1) 0.558; (2) 3.173.
For a largescale model with a 295 mm diameter crucible the laminarturbulent transition processes in the boundary layer at the CF occur at as small ΔT as several tenths of a Kelvin. Typical q (r) curves for this case are shown in Fig.
Practically important feature of free convective flow modes is that the abovementioned CF boundary layer evolution stages correlate with the bends in the curve of the integral heat flow Q at the CF as a function of temperature or Gr. This feature is typical of flow modes in various – scale setups of the classic Czochralski technique with significantly different absolute dimensions of the crucible/melt/crystal system. This integral heat flux vs temperature difference curve pattern is typical of convection in horizontal fluid layers heated from below [
Another important problem is the dependence of melt flow structure and heat exchange regularities at the CF on the CF shape. Free convection modes can be naturally expected to produce a close to conical front shape. This is confirmed by a real hexadecane crystallization experiment (Pr = 45.6). Figure
CF shape as a function of free convection intensity: (а) Gr = 1.9 · 10^{5}; (b) 1.8 · 10^{5}; (c) 1.3 · 10^{5}; (d) 1.0 · 10^{5}.
One more problem is associated with the effect of temperature boundary conditions at the crucible bottom. Figure
Buoyancy and thermocapillary driven convection was studied experimentally for melt simulating fluids in thermal hydrodynamic systems similar to the classic Czochralski technique, for laminar to welldeveloped turbulent flow modes, with 80 and 295 mm crucible diameter models. The experiments proved that an increase in the Grashof and Marangoni numbers induces multiple stepwise qualitative changes in the structure of the initial buoyancy and thermocapillary driven flow. Once the next threshold temperature difference (or the threshold Gr and Ma numbers) is reached the spacetime flow arrangement becomes more complex (new secondary swirl flow develops against the background of the main flow or a new oscillation harmonic emerges). The heat exchange regularities change accordingly. The scale factor proved to affect the hydrodynamics, laminarturbulent transition and heat exchange regularities, yet further studies for fullscale models are required for practical purposes. TGCC modes are distinguished by a significant radial inhomogeneity of local heat flows in laminar, transition and turbulent modes. Model fluid experiments (Pr = 45.6) with real crystallization showed that the CF shape depends largely on the spatial melt flow structure and confirm the possible crystallization front shape hypotheses made on the basis of singlephase convection studies.
This research was realized under the project III.18.2.5, number of state registration ААААА171170228500213.