**3.2 Marangoni convection**

534 Mass Transfer - Advanced Aspects

Buoyancy and thermal capillary convections (namely Marangoni convection) are the main styles of mass transfer in melt especially for high temperature condition. In this section, the typical steady buoyancy and Marangoni convections in the oxide melt/solution suspended on the loop heater will be shown. The unsteady convective flows will be illustrated in the

Generally, buoyancy driven convection is in close correlation to the temperature distribution in the liquid phase. Fig. 4 shows the typical temperature distribution in the horizontal direction of the thin liquid film suspended by Pt loop. In the central portion of the loop, the melt temperature gradient can be negligible. This region is called pure diffusion region as indicated by sign A in Fig. 4. The situation has an advantage for studying interfacial kinetics process with pure diffusion transport. But in the marginal portion of the heater, the horizontal thermal gradient is significant, and this portion is called diffusion-convective region as hinted by the sign B in Fig. 4. In this region, the growth may be controlled by the buoyancy driven convection due to the higher temperature gradient. The width of diffusionconvective region depends on the loop diameter as well as on the thermophysical

> 0 0.2 0.4 0.6 0.8 1.0 1.2 *D*/*mm*

Fig. 4. Typical radial temperature distribution in the horizontal direction of the thin liquid film suspended by Pt loop. D is the distance from the loop margin. Here the temperature

Buoyancy driven convection can be visualized since Schlieren technique has been introduced. However, quantitative measurement of flow velocity needs the help of some tracing particles. Since some tiny crystals can be nucleated when the melt becomes undercooled, the buoyancy driven convection can be indicated by observing the movement of tiny crystals as described in the reference (W. Q. Jin, et al., 1993). As shown in Fig. 5, the tiny crystals move from the margin to the center of the heater. This is typical buoyancy

**B A**

0

*D* 1.25

parameters of oxide melt, such as the density, thermal diffusivity and viscosity.

**3. Buoyancy and Marangoni convection in oxide melt** 

section 6 of this chapter.

**3.1 Buoyancy driven convection** 

1020

1010

1000

*T*/*°C*

990

980

970

distribution profile is supplied for KNbO3 melt

Marangoni convection is driven by the variation of surface tension along the free surface. The temperature distribution along the azimuthal coordinate of the loop-like heater is measured by our developed non-contact method (X. A. Liang et al., 2000), which is based on the fact that dissolvability is one-valued function of the temperature. The result is shown in Fig. 6. Here ΔTx=TB-TA. Clearly, the temperature along the free surface is not uniform, which is caused by existence of thermocouple and other attachment. The side with thermocouple on has a lower temperature.

Fig. 6. Temperature difference along the azimuthal coordinate of the heater

Interfacial Mass Transfer and Morphological Instability of Oxide Crystal Growth 537

where C is a geometry-dependent factor, σ is the surface tension,C<sup>σ</sup> is specific heat at constant surface tension, b is a constant independent of temperature, and R is the gas constant.

Fig. 8. Helix flow in the loop (a), the corresponding schematic figure (b) and schematic

(a) (b)

shaped heater ( • Ved , Vsd , S (Ved-Vsd ) , ∇ Vg )

Fig. 9. (a) Comparison between experimental data Veθ and theoretical calculation Vsθ of surface tension convection along the azimuthal direction of the loop-shaped heater; (b) The velocities of buoyancy and surface tension convection along the radial direction of the loop-

By the similar calculation process mentioned above, the convection velocity Vd, which results from the temperature difference along radial direction, can be calculated and profiled in Fig. 9(b). It is obvious that the calculated surface tension flow velocity Vsd is less than Ved obtained by experimental data. Based on an axial symmetry solution of Navier-Stokes equation, the flow velocity Vg that is driven by buoyancy convection has also theoretically

For oxide melt, by using typical values such as σ ~ 0.2 g/s2 , C<sup>σ</sup> = 1.73×109 g⋅cm/s2⋅K and the experimental data, the constants can be determined as C = 2.5×10-5 cm⋅s/g and b/R = 1/K. Thus, from the temperature data measured, one can get the theoretical velocity along the azimuthal loop direction (Vsθ) profile as shown in Fig. 9(a), which agrees with the experimental data (Veθ). This means that the flow along the azimuthal loop is mainly driven

figure of the coordinates (c)

by Marangoni convection.

Fig. 7 shows the streamlines of the steady Marangoni convection in the BaB2O4- Li2B4O7 melt-solution. Compared with the buoyancy driven flow in the oxide melt (Fig.5), the type of Marangoni streamlines is quite different. The streamlines in Fig. 7 are concentrated at the surface. This is observed by changing the focus plane of the object lens. As one can see, the flow structure of Marangoni convection is of two symmetrical vortices, with one in the one half of the loop and its mirror image in the other half. The flow direction is from the hot point C to the cooler point A as shown in the inset of Fig.6. This is a typical pattern for Marangoni flow induced by surface tension driven convection. Such convective pattern has also been observed in KNbO3 melt film (W. Q. Jin et al., 1999). It should be mentioned that if the temperature difference between the hot point C and the cooler point A exceeds some critical value, the Marangoni convection may become unsteady, which has been proved in our previous work (Y. Hong et al., 2004, 2005). Those cases are always undesired in crystal growth because growth rate fluctuation as well as microscopic defects may occur, and consequently the quality of the as-grown crystal is deteriorated.

Fig. 7. Steady Marangoni convection in the BaB2O4- Li2B4O7 melt-solution. The arrows indicate the flow direction

#### **3.3 Coupling of buoyancy and Marangoni convection**

The relation between buoyancy and surface tension convection is investigated in the convective region of the loop melt, since only within this region both convections exist simultaneously. In this region, a typical helix flow can be observed as shown in Fig. 8(a) and its schematic figure is drawn in Fig. 8(b). The velocities of the flow are divided into the cross section through the diameter of the loop heater (Vd) and azimuthal direction along the loop heater (Vθ), as shown in Fig. 8(c).

The flow velocity Vθ along the azimuthal direction of the loop due to the surface tension effect may be estimated by balancing the surface tension force with viscous force (F. Ai et al., 2009). This gives

$$V\_{\theta} = \text{C}\Lambda T \left(\frac{\sigma}{T} - \text{C}\_{\sigma}\right) \exp\left(-\frac{b}{RT}\right) \tag{1}$$

Fig. 7 shows the streamlines of the steady Marangoni convection in the BaB2O4- Li2B4O7 melt-solution. Compared with the buoyancy driven flow in the oxide melt (Fig.5), the type of Marangoni streamlines is quite different. The streamlines in Fig. 7 are concentrated at the surface. This is observed by changing the focus plane of the object lens. As one can see, the flow structure of Marangoni convection is of two symmetrical vortices, with one in the one half of the loop and its mirror image in the other half. The flow direction is from the hot point C to the cooler point A as shown in the inset of Fig.6. This is a typical pattern for Marangoni flow induced by surface tension driven convection. Such convective pattern has also been observed in KNbO3 melt film (W. Q. Jin et al., 1999). It should be mentioned that if the temperature difference between the hot point C and the cooler point A exceeds some critical value, the Marangoni convection may become unsteady, which has been proved in our previous work (Y. Hong et al., 2004, 2005). Those cases are always undesired in crystal growth because growth rate fluctuation as well as microscopic defects may occur, and

Fig. 7. Steady Marangoni convection in the BaB2O4- Li2B4O7 melt-solution. The arrows

The relation between buoyancy and surface tension convection is investigated in the convective region of the loop melt, since only within this region both convections exist simultaneously. In this region, a typical helix flow can be observed as shown in Fig. 8(a) and its schematic figure is drawn in Fig. 8(b). The velocities of the flow are divided into the cross section through the diameter of the loop heater (Vd) and azimuthal direction along the loop

The flow velocity Vθ along the azimuthal direction of the loop due to the surface tension effect may be estimated by balancing the surface tension force with viscous force (F. Ai et al.,

*<sup>b</sup> V CT C*

σ

 σ

=Δ − − ⎜ ⎟⎜ ⎟

exp

⎝ ⎠⎝ ⎠ (1)

*T RT*

⎛ ⎞⎛ ⎞

consequently the quality of the as-grown crystal is deteriorated.

**3.3 Coupling of buoyancy and Marangoni convection** 

θ

indicate the flow direction

heater (Vθ), as shown in Fig. 8(c).

2009). This gives

where C is a geometry-dependent factor, σ is the surface tension,C<sup>σ</sup> is specific heat at constant surface tension, b is a constant independent of temperature, and R is the gas constant.

Fig. 8. Helix flow in the loop (a), the corresponding schematic figure (b) and schematic figure of the coordinates (c)

For oxide melt, by using typical values such as σ ~ 0.2 g/s2 , C<sup>σ</sup> = 1.73×109 g⋅cm/s2⋅K and the experimental data, the constants can be determined as C = 2.5×10-5 cm⋅s/g and b/R = 1/K. Thus, from the temperature data measured, one can get the theoretical velocity along the azimuthal loop direction (Vsθ) profile as shown in Fig. 9(a), which agrees with the experimental data (Veθ). This means that the flow along the azimuthal loop is mainly driven by Marangoni convection.

Fig. 9. (a) Comparison between experimental data Veθ and theoretical calculation Vsθ of surface tension convection along the azimuthal direction of the loop-shaped heater; (b) The velocities of buoyancy and surface tension convection along the radial direction of the loopshaped heater ( • Ved , Vsd , S (Ved-Vsd ) , ∇ Vg )

By the similar calculation process mentioned above, the convection velocity Vd, which results from the temperature difference along radial direction, can be calculated and profiled in Fig. 9(b). It is obvious that the calculated surface tension flow velocity Vsd is less than Ved obtained by experimental data. Based on an axial symmetry solution of Navier-Stokes equation, the flow velocity Vg that is driven by buoyancy convection has also theoretically

Interfacial Mass Transfer and Morphological Instability of Oxide Crystal Growth 539

diffusivity, D is the mass diffusivity, *P* is the Prantdl number and S is the Schmidt number. With ν = 40 mm2/s , ω = 6 rad/s, and the estimation of κ = 10-2cm2/s and D = 10-5cm2/s for oxide melt (W. Q. Jin, 1997), one then obtain δυ = 9.2 mm, δT = 1.25 mm and δC = 0.12mm,

> 3.6 υ

⎛ ⎞ <sup>=</sup> ⎜ ⎟ ⎝ ⎠

13 13 16 12 1.61 1.61 *<sup>T</sup> <sup>P</sup>*

13 13 16 12 1.61 1.61 *<sup>C</sup> S D*

⎛ ⎞ − − <sup>=</sup> <sup>=</sup> ⎜ ⎟ ⎝ ⎠

Fig. 11 shows the schematic comparison among concentration, velocity and temperature distribution at solid-liquid interface of oxide crystal growth. It is obvious that there is a big difference between the diffusivities for momentum, heat and species in the solution. The magnitudes of δT and δυ are at least one order larger than that of δC. This indicates that the thermal and momentum transition zone extends further into the fluid than the concentration transitional region. Thus, in the concentration boundary layer, the flow velocity and conductive heat flux are reduced to relatively small values and it is obviously meaningless

⎛ ⎞ − − <sup>=</sup> <sup>=</sup> ⎜ ⎟ ⎝ ⎠

δ

1 2

1 2

ν

ω

ν

ω

to discuss interfacial heat and velocity transport in oxide crystal growth.

**0 0.1 1 10 100**

Fig. 11. Schematic comparison between δυ , δT and δC for BaB2O4 crystal rotated at 6 rad/s One important parameter determining the thickness of interfacial boundary layer is the rotating rate ω as indicates from equations (2) to (4). Since the driving force of crystal rotation is the surface tension-driven flow caused by the horizontal temperature difference ∆T along the heater, the value of ω is consequently correlated with the strength of the Marangoni convection. Here, ∆T is the temperature difference between the hottest point C

δ<sup>T</sup> **=1.25mm**

δ

δ

δC **=0.12mm**

1 2

 κ ν ω

> ν ω

> > **C**

δυ**=9.2mm**

 **T**

**∞**

 **V**

**∞**

**∞**

(2)

(3)

(4)

ν

ω

respectively.

**0**

**C T 0**

**C T V**

calculated and is also illustrated in Fig. 9(b). For comparison, the profile of (Ved-Vsd) is simultaneously illustrated. Obviously, Vg agrees well with the profile of (Ved-Vsd). That is, the difference between Ved and Vsd indeed results from buoyancy convection effect in present system.
