**4.1 Calculation of interfacial boundary layer thickness**

One important and effective method for calculation of interfacial boundary layer thickness is developed by the help of crystal rotation. In present high temperature in situ observation system, the rotating growth of oxide crystal has been realized (W. Q. Jin et al., 1998; X. H. Pan et al., 2008). Fig. 10 shows a typical process of BaB2O4 crystal growth with rotation in the center of one toroidal Marangoni flow as depicted in Fig. 7. During the growth, the Marangoni flow may be as a forced convection to rotate the tiny crystal.

Fig. 10. BaB2O4 single crystal growth process with rotation. The dark arrow shows the direction of rotation and rotating velocity ω is 6 rad/s

The arrow in Fig. 10(a) illustrates the rotation direction, and the angular velocity ω is 6 rad/s. The widths of the interfacial momentum, heat and concentration boundary layer are given by following equations respectively. Where ν is the kinematic viscosity, κ is the thermal

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

In crystal growth, interfacial gradients in velocity, concentration or temperature are often summarily referred to as boundary layers. The concept of interfacial boundary layer concerning of crystal growth has been thoroughly expounded by Franz Rosenberger (F. Rosenberger, 1993). The domain of liquid can be subdivided into two regions. Inside the boundary layer, the gradient is high, while out side the boundary layer, the gradient is

One important and effective method for calculation of interfacial boundary layer thickness is developed by the help of crystal rotation. In present high temperature in situ observation system, the rotating growth of oxide crystal has been realized (W. Q. Jin et al., 1998; X. H. Pan et al., 2008). Fig. 10 shows a typical process of BaB2O4 crystal growth with rotation in the center of one toroidal Marangoni flow as depicted in Fig. 7. During the growth, the

Fig. 10. BaB2O4 single crystal growth process with rotation. The dark arrow shows the

The arrow in Fig. 10(a) illustrates the rotation direction, and the angular velocity ω is 6 rad/s. The widths of the interfacial momentum, heat and concentration boundary layer are given by following equations respectively. Where ν is the kinematic viscosity, κ is the thermal

direction of rotation and rotating velocity ω is 6 rad/s

**4. Interfacial boundary layer and microconvection during oxide crystal** 

**4.1 Calculation of interfacial boundary layer thickness** 

Marangoni flow may be as a forced convection to rotate the tiny crystal.

present system.

**growth** 

negligible.

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, respectively.

$$\mathcal{S}\_{\upsilon} = \Im.6 \left( \frac{\nu}{\alpha} \right)^{\mathfrak{Y}^2} \tag{2}$$

$$\mathcal{S}\_{\Gamma} = 1.61 \left( \frac{\nu}{\alpha} \right)^{1/2} P^{-1/3} = 1.61 \kappa^{1/3} \nu^{1/6} \alpha^{-1/2} \tag{3}$$

$$\delta\_{\mathbb{C}} = 1.61 \left( \frac{\nu}{\alpha} \right)^{1/2} S^{-1/3} = 1.61 D^{1/3} \nu^{1/6} \alpha^{-1/2} \tag{4}$$

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

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

Interfacial Mass Transfer and Morphological Instability of Oxide Crystal Growth 541

velocity of mass flow parallel to the solid-liquid interface (transversal flow) and that of flow perpendicular to the interface (normal flow) are depicted as the solid lines in Fig. 14. Here, the origin of coordinate axes is at the center of KNbO3 grain face as indicated in Fig. 13. Since the growth of the grains is restricted to each other, the quenched solution retains their situation in high temperature. And thus the solute concentration near the grain can be

Fig. 13. Growth pattern of KNbO3 solute grains in KNbO3/Li2B4O7 solution, showing the

*n D K KK K* =

1 υ

ρ

 ≈ = ρυ

For the binary system in the solution growth, the solvent is quantitatively rejected at the

 ρ

*K K KK K K*

 υ  υ

ρ υ*i i*

where ρK is component mass density of the KNbO3 solute, ρ is the total mass density, D is binary diffusivity, WK=ρK /ρ is the mass fraction of component KNbO3, and υ is the mass

= +∇*WK* (5)

<sup>=</sup> ∑ (6)

*KK KK* / *W* (7)

= *W nW* = (8)

ρ υ ρυ ρ

In the absence of temperature gradient for a binary system in the solution growth, it is rational to assume that the solvent is quantitatively rejected at the interface and the velocity of solvent flow is zero (W. Q. Jin et al., 2005, 2006). Then the mass transport is mainly by diffusive flux. Diffusion is in general understood as the component flux with respect to an average velocity of the system. The total mass flux of solute KNbO3 in the binary system can

examined microscopically by the electron-microprobe analyses.

existence of microconvection near the solid-liquid interface

interface and the velocity of solvent flow is zero. So one obtain

υ

ρ

υρ

be expressed as

average velocity

So the "convective" term in (5) is

and the coolest point A across the heater. The values of ω with various ∆T have been experimentally measured as shown in Fig. 12(a). With a larger value of ∆T, the crystal rotates faster and then a smaller width of concentration boundary layer is obtained. It should be noted that present calculating method for boundary layer thickness is unsuitable for the cases with too large or too small values of ∆T. In the former case the rotation is unsteady, while in the latter case no obvious convection is visible due to the viscous force of the melt.

Fig. 12. The dependence of rotating angular velocity on temperature (a) and the variation of concentration boundary layer thickness δC versus Marangoni number Ma (b)

In general, the strength of Marangoni convection can be characterized by dimensionless Marangoni number *M Td a T* = Δ σ /ηκ . Where σ σ *<sup>T</sup>* = *d dT* / is the temperature coefficient of surface tension σ, d is the characteristic length of the liquid, η is dynamic viscosity and κ is the thermal diffusivity. By taking d = 0.8 mm, σT = -0.06 dyn/cm•K, η = 0.16 Pa•s, and κ = 10-2cm2/s, then the calculated concentration boundary layer thickness δC versus Ma is plotted in Fig. 12(b). The thickness of concentration boundary layer is in inverse proportion to the value of Marangoni number. It suggests that the concentration boundary layer is suppressed by the enhancement of Marangoni convection. This is due to the fact that heat and mass transfer is enhanced and therefore the interfacial diffusion field is weakened.

#### **4.2 Microconvection within the boundary layer**

Since in the central portion of the loop, the melt/solution temperature gradient along the radial direction can be negligible, this has an advantage for studying the "pure diffusive" mass flow. Fig. 13 shows the KNbO3 grains and the surrounding flow structure during growth in the center of KNbO3/Li2B4O7 solution. One significant flow cell is observed around each grain. Since such flow is usually restricted only in a narrow region of several tens of microns near the solid-liquid interface, that is the size scale of grain, cell or dendrite, it can be called the microconvection as described by Sahm and Tensi (P. R. Sahm and H. M. Tensi, 1981).

The growth of the grains is restricted to each other. As a result, the growing rate of the grains is much low which is helpful for velocity measurement of microconvection. The

and the coolest point A across the heater. The values of ω with various ∆T have been experimentally measured as shown in Fig. 12(a). With a larger value of ∆T, the crystal rotates faster and then a smaller width of concentration boundary layer is obtained. It should be noted that present calculating method for boundary layer thickness is unsuitable for the cases with too large or too small values of ∆T. In the former case the rotation is unsteady, while in the latter case no obvious convection is visible due to the viscous force of

Fig. 12. The dependence of rotating angular velocity on temperature (a) and the variation of

In general, the strength of Marangoni convection can be characterized by dimensionless

surface tension σ, d is the characteristic length of the liquid, η is dynamic viscosity and κ is the thermal diffusivity. By taking d = 0.8 mm, σT = -0.06 dyn/cm•K, η = 0.16 Pa•s, and κ = 10-2cm2/s, then the calculated concentration boundary layer thickness δC versus Ma is plotted in Fig. 12(b). The thickness of concentration boundary layer is in inverse proportion to the value of Marangoni number. It suggests that the concentration boundary layer is suppressed by the enhancement of Marangoni convection. This is due to the fact that heat and mass transfer is enhanced and therefore the interfacial diffusion field is weakened.

Since in the central portion of the loop, the melt/solution temperature gradient along the radial direction can be negligible, this has an advantage for studying the "pure diffusive" mass flow. Fig. 13 shows the KNbO3 grains and the surrounding flow structure during growth in the center of KNbO3/Li2B4O7 solution. One significant flow cell is observed around each grain. Since such flow is usually restricted only in a narrow region of several tens of microns near the solid-liquid interface, that is the size scale of grain, cell or dendrite, it can be called the microconvection as described by Sahm and Tensi (P. R. Sahm and H. M.

The growth of the grains is restricted to each other. As a result, the growing rate of the grains is much low which is helpful for velocity measurement of microconvection. The

σ

 σ

*<sup>T</sup>* = *d dT* / is the temperature coefficient of

(a) (b)

ηκ

concentration boundary layer thickness δC versus Marangoni number Ma (b)

. Where

the melt.

Marangoni number *M Td a T* = Δ

Tensi, 1981).

σ/

**4.2 Microconvection within the boundary layer** 

velocity of mass flow parallel to the solid-liquid interface (transversal flow) and that of flow perpendicular to the interface (normal flow) are depicted as the solid lines in Fig. 14. Here, the origin of coordinate axes is at the center of KNbO3 grain face as indicated in Fig. 13. Since the growth of the grains is restricted to each other, the quenched solution retains their situation in high temperature. And thus the solute concentration near the grain can be examined microscopically by the electron-microprobe analyses.

Fig. 13. Growth pattern of KNbO3 solute grains in KNbO3/Li2B4O7 solution, showing the existence of microconvection near the solid-liquid interface

In the absence of temperature gradient for a binary system in the solution growth, it is rational to assume that the solvent is quantitatively rejected at the interface and the velocity of solvent flow is zero (W. Q. Jin et al., 2005, 2006). Then the mass transport is mainly by diffusive flux. Diffusion is in general understood as the component flux with respect to an average velocity of the system. The total mass flux of solute KNbO3 in the binary system can be expressed as

$$
\rho n\_{\rm K} = \rho\_{\rm K} \upsilon\_{\rm K} = \rho\_{\rm K} \upsilon + \rho D \nabla W\_{\rm K} \tag{5}
$$

where ρK is component mass density of the KNbO3 solute, ρ is the total mass density, D is binary diffusivity, WK=ρK /ρ is the mass fraction of component KNbO3, and υ is the mass average velocity

$$
\omega = \frac{1}{\rho} \sum \rho\_i \nu\_i \tag{6}
$$

For the binary system in the solution growth, the solvent is quantitatively rejected at the interface and the velocity of solvent flow is zero. So one obtain

$$
\Delta \boldsymbol{\nu} \approx \rho\_{\mathcal{K}} \boldsymbol{\nu}\_{\mathcal{K}} \;/\; \rho = \mathsf{W}\_{\mathcal{K}} \boldsymbol{\nu}\_{\mathcal{K}} \tag{7}
$$

So the "convective" term in (5) is

$$
\rho\_K \boldsymbol{\nu} = \rho\_K \mathbf{V} \mathbf{V}\_K \boldsymbol{\nu}\_K = \mathbf{n}\_K \mathbf{V} \mathbf{V}\_K \tag{8}
$$

Interfacial Mass Transfer and Morphological Instability of Oxide Crystal Growth 543

If only one single seed crystal is formed in the centre of the loop, it can grow larger because no surrounding grains restrain its growth. At the initial stage of growth, the crystal size is comparatively small. In this case, the growing solid-liquid interface is in the "pure" diffusion region and the interfacial mass transport visualized by Schlieren technique is shown in Fig. 15(a). However, the interfacial mass transport becomes different when the solid-liquid interface enters the diffusion-convective region near the melt margin as indicated in Fig. 15 (b). Here the mass transport is governed by the diffusive-convective flow

**5. Effect of mass transport on interfacial kinetics and morphological** 

Fig. 15. Morphology of the interfacial fluid flow for one single crystal growth

Fig. 16. The data plotted as V~ΔT for (100) face of KNbO3 in the pure diffusive region and in

To examine the influence of the mass transport on the interface growth kinetics, the growing rate data plotted as V~ΔT are shown in Fig. 16 for two different flow states. The growing rate in pure diffusive region can be described as V=1.1exp(-4.5×104/TΔT), while that in the diffusive-convective region is expressed as V=0.59exp(-8.4×104/TΔT). These exponential

**instability** 

due to the significant temperature gradient.

the diffusive-convective region, respectively

The flow in (8) is the so-called diffusion-induced bulk flow as described by Franz Rosenberger (F. Rosenberger, 1983). Substitution of (8) in to (5) yields for the "purely diffusive" component mass flux towards the growing crystal

$$m\_K = \frac{\rho D}{1 - W\_K} \bullet \nabla W\_K \tag{9}$$

rather than the widely used

$$m\_K = \rho D \bullet \nabla \mathcal{W}\_K \tag{10}$$

Furthermore, substitution of υ*K K* =υ/*W* into (9), one obtains

$$
\omega = \frac{D}{1 - W\_{\mathcal{K}}} \bullet \nabla W\_{\mathcal{K}} \tag{11}
$$

Then the mass average velocity of present system in two-dimensional treatment is

$$\begin{split} \vec{\nu}(\mathbf{x}, y) &= \frac{D}{1 - \mathcal{W}\_{\mathbf{K}}(\mathbf{x}, y)} \left( \frac{\partial}{\partial \mathbf{x}} \mathcal{W}\_{\mathbf{K}}(\mathbf{x}, y) \vec{i} + \frac{\partial}{\partial y} \mathcal{W}\_{\mathbf{K}}(\mathbf{x}, y) \vec{j} \right) \\ &= \nu\_1(\mathbf{x}, y) \vec{i} + \nu\_2(\mathbf{x}, y) \vec{j} \end{split} \tag{12}$$

By the experimentally measured KNbO3 concentration WK along the growing interface and normal to the interface, respectively, the authors have got the calculated profiles of velocities both along and normal to the solid-liquid interface during oxide crystal growth (W. Q. Jin et al., 2006). This is indicated by the dotted line as 1( ,0) *<sup>t</sup>* υ *x* and 2 (0, ) *<sup>t</sup>* υ *y* in Fig. 14, which agrees with the experimental profile data 1 ( ,0) *<sup>e</sup>* υ *<sup>x</sup>* and 2(0, ) *<sup>e</sup>* υ *y* , respectively. It means that on segregation at the interface, the solute diffusion-induced bulk flow is exactly nonzero and can be detected experimentally.

Fig. 14. Comparison between experimental velocities of KNbO3 microconvection flow (solid line) and theoretical calculation velocities of KNbO3 diffusion-induced bulk flow (dotted lines): (a) parallel flow to the interface 1 υ ( ,0) *x* ; (b) normal flow to the interface 2 υ(0, ) *y*

The flow in (8) is the so-called diffusion-induced bulk flow as described by Franz Rosenberger (F. Rosenberger, 1983). Substitution of (8) in to (5) yields for the "purely diffusive" component

> 1 *K K K <sup>D</sup> n W W* ρ

> > *n DW K K* = ρ

/*W* into (9), one obtains

1 *<sup>K</sup> K <sup>D</sup> <sup>W</sup> W*

(,) (,) (,) 1 (,)

G G G

*x y W xyi W xyj W xy x <sup>y</sup>*

By the experimentally measured KNbO3 concentration WK along the growing interface and normal to the interface, respectively, the authors have got the calculated profiles of velocities both along and normal to the solid-liquid interface during oxide crystal growth (W. Q. Jin et

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

> *t* υ

> > *e* υ

*x* and 2(0, )

segregation at the interface, the solute diffusion-induced bulk flow is exactly nonzero and

Fig. 14. Comparison between experimental velocities of KNbO3 microconvection flow (solid line) and theoretical calculation velocities of KNbO3 diffusion-induced bulk flow (dotted

υ

*K K*

 *x* and 2 (0, ) *t* υ

( ,0) *x* ; (b) normal flow to the interface 2

G G (12)

= •∇ <sup>−</sup> (9)

• ∇ (10)

*y* in Fig. 14, which agrees

υ(0, ) *y*

*y* , respectively. It means that on

= •∇ <sup>−</sup> (11)

mass flux towards the growing crystal

υ

1 2

≡ +

al., 2006). This is indicated by the dotted line as 1( ,0)

υ

with the experimental profile data 1 ( ,0)

lines): (a) parallel flow to the interface 1

can be detected experimentally.

υ

(,) (,)

 υ

*xyi xy j*

*K D*

> *e* υ

*K K* =υ

υ

Then the mass average velocity of present system in two-dimensional treatment is

rather than the widely used

Furthermore, substitution of
