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

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 due to the significant temperature gradient.

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 the diffusive-convective region, respectively

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

Interfacial Mass Transfer and Morphological Instability of Oxide Crystal Growth 545

Fig. 17. Cellular interface during BaB2O4 crystal growth indicating the occurence of

Fig. 18. Skeletal morphology of BaB2O4 single crystal with high growth rate

growth of BaB2O4 single crystal from high-temperature solutions (X. H. Pan et al., 2006, 2007, 2009), where skeletal shape of growing interface is obtained due to the reduction of convective regime by rapid growth. Fig. 18 shows a typical microscopic morphology of BaB2O4 single crystal. It is obvious that the interface is deformed and the crystal presents a shape of snowflake. This kind of interface with the centre region depressed is the so called skeletal shape. The formation of skeletal shape is the result of the extremely non-uniform supersaturation in front of the interface owing to reduction of convective effect. The concentration of solute can be examined by element detection with electro-microprobe analysis when the solution is quenched as described in section 4.2. Unfortunately, the quantitative data about interfacial concentration of BaB2O4 in the solution is impossible primarily due to the existence of light element boron that can not be examined effectively

morphological instability

even by electro-microprobe technique.

functions indicate a growth mechanism by two-dimension nucleation. The experimental data coincide well with the dotted curves predicted by the theory of two-dimension nucleation. It can be seen that, at the lower supercoolings two-dimensional nucleation growth has been obtained irrespective of the state of mass transport in the melt. However, at the same supercooling, the discrepancy between the growth rates for two different states of convection may be assigned to the buoyancy driven convection of the interfacial mass flow. The best model to describe the growth kinetics of two-dimensional nucleation is the birth and spread model (J. W. Cahn et al., 1964). According to this model, many nuclei occur on a flat crystal surface, and the steps annihilate when they spread and impinge. The growth rate V is given by

$$V = V\_{\alpha} \left(\frac{L\Delta T}{RT\_m}\right)^{1/6} e^{\Lambda G^\*/3KT} \tag{13}$$

where V∝ is the velocity of a straight step, L is the latent heat of fusion per mole, Tm is the melting temperature, and ΔG\* is the thermodynamic potential barrier for two dimensional nucleation.

The energy barrier ΔG\* for this is given by the following equation (W. Q. Jin, 1983)

$$
\Delta \mathbf{G}^\* = \frac{\pi \mathbf{z}^2 T\_m}{L \Delta T \, ^\*} \tag{14}
$$

in which ε is the free energy per unit length of a step and ΔT \* is the threshold supercooling for the growth of a flat surface. The values of ε and ΔG\* have been calculated from the experimental data for two different states of mass transport. The various results are summarized in Table.1. On comparing these experimental results with the theoretical kinetic equations (13) and (14), it is implicated that the growth kinetics is dependent upon the convection flow through the two quantities ε and V∝.


Table 1. Data of ε and ΔG\* of KNbO3 crystal growth for the (001) face in different mass transport state

The mass transport in the melt may also have great effect on the morphological stability of the growing crystal. In general, the solid-liquid interface during oxide crystal growth from high temperature melt-solution is flat and smooth for its high melting entropy, and the shape of single crystal is usually polygonal in two dimensions. However, the instability of solid-liquid interface may occur when the mass transport becomes unsteady. This is especially for the case of unsteady convection or rapid growth. Fig. 17 shows a typical unsteady growth of BaB2O4 melt where significant cellular shape is observed for the solidliquid interface. Cellular growth and the resulted striations are usually found accompanied with appearance of oscillatory or turbulent flows.

In case of high cooling rate the morphology of solid-liquid interface may also become unsteady. The interfacial morphology instability has been in situ observed in the rapid

functions indicate a growth mechanism by two-dimension nucleation. The experimental data coincide well with the dotted curves predicted by the theory of two-dimension nucleation. It can be seen that, at the lower supercoolings two-dimensional nucleation growth has been obtained irrespective of the state of mass transport in the melt. However, at the same supercooling, the discrepancy between the growth rates for two different states of convection may be assigned to the buoyancy driven convection of the interfacial mass flow. The best model to describe the growth kinetics of two-dimensional nucleation is the birth and spread model (J. W. Cahn et al., 1964). According to this model, many nuclei occur on a flat crystal surface, and the steps annihilate when they spread and impinge. The growth rate

1/6

2 \* \* *Tm <sup>G</sup> L T* πεΔ =

in which ε is the free energy per unit length of a step and ΔT \* is the threshold supercooling for the growth of a flat surface. The values of ε and ΔG\* have been calculated from the experimental data for two different states of mass transport. The various results are summarized in Table.1. On comparing these experimental results with the theoretical kinetic equations (13) and (14), it is implicated that the growth kinetics is dependent upon the

mass transport state ΔT\* (oC) ε(J/m) ΔG\* (J) pure diffusive 3.0 1.2×10-11 6.3×10-19 diffusive-convective 5.0 1.6×10-11 7.0×10-19

The mass transport in the melt may also have great effect on the morphological stability of the growing crystal. In general, the solid-liquid interface during oxide crystal growth from high temperature melt-solution is flat and smooth for its high melting entropy, and the shape of single crystal is usually polygonal in two dimensions. However, the instability of solid-liquid interface may occur when the mass transport becomes unsteady. This is especially for the case of unsteady convection or rapid growth. Fig. 17 shows a typical unsteady growth of BaB2O4 melt where significant cellular shape is observed for the solidliquid interface. Cellular growth and the resulted striations are usually found accompanied

In case of high cooling rate the morphology of solid-liquid interface may also become unsteady. The interfacial morphology instability has been in situ observed in the rapid

Table 1. Data of ε and ΔG\* of KNbO3 crystal growth for the (001) face in different mass

2

where V∝ is the velocity of a straight step, L is the latent heat of fusion per mole, Tm is the melting temperature, and ΔG\* is the thermodynamic potential barrier for two dimensional

*m L T VV e RT*

∞ ⎛ ⎞ <sup>Δ</sup> <sup>=</sup> ⎜ ⎟ ⎝ ⎠

The energy barrier ΔG\* for this is given by the following equation (W. Q. Jin, 1983)

convection flow through the two quantities ε and V∝.

with appearance of oscillatory or turbulent flows.

\*/3

Δ

*G KT*

(13)

<sup>Δ</sup> (14)

V is given by

nucleation.

transport state

Fig. 17. Cellular interface during BaB2O4 crystal growth indicating the occurence of morphological instability

growth of BaB2O4 single crystal from high-temperature solutions (X. H. Pan et al., 2006, 2007, 2009), where skeletal shape of growing interface is obtained due to the reduction of convective regime by rapid growth. Fig. 18 shows a typical microscopic morphology of BaB2O4 single crystal. It is obvious that the interface is deformed and the crystal presents a shape of snowflake. This kind of interface with the centre region depressed is the so called skeletal shape. The formation of skeletal shape is the result of the extremely non-uniform supersaturation in front of the interface owing to reduction of convective effect. The concentration of solute can be examined by element detection with electro-microprobe analysis when the solution is quenched as described in section 4.2. Unfortunately, the quantitative data about interfacial concentration of BaB2O4 in the solution is impossible primarily due to the existence of light element boron that can not be examined effectively even by electro-microprobe technique.

Fig. 18. Skeletal morphology of BaB2O4 single crystal with high growth rate

Interfacial Mass Transfer and Morphological Instability of Oxide Crystal Growth 547

It is obvious that the solute concentration along the interface is uneven both for pure diffusive and diffusive-convective regions which is in related with the specie supplies. In terms of the Berg effect, the apexes of a polyhedral crystal are the best supplied regions and the concentration along the crystal surface varies, being the lowest at the face centre. This is the shape destabilizing factor. The shape stabilizing factor is connected with the anisotropy in the surface growth kinetics. If the kinetic coefficient with its anisotropy is not sufficient to compensate for the inhomogeneity in the concentration distribution over the growing crystal face, the appearance of morphological instability in the crystal faces takes place. The curves in Fig.20 indicate that not only the concentration difference between the centre and corner of a KNbO3 grain face but also the concentration gradient along the face in the convective region are lower than that in the diffusive region. This is attributed to the enhanced mass transport and thus the improved homogeneity of solute concentration by convection. As a result, morphological instability occurs more easily in the diffusive region where dendrites are found whereas crystal in the convective region conserves a polyhedral shape. This result indicates that, even underlying a very high cooling rate, flat interface may

It should be emphasized from above results that, only steady convection is helpful for the maintaining of flat interface during oxide crystal growth. Mass transfer governed by pure diffusion or unsteady convection may leads to morphological instability of growing

In crystal growth, steady convection is always desired because it is helpful for mass transfer and thus provides an enhanced renewal of the melt/solution in the region near the crystallization interface. However, when the temperature gradient gets larger enough, the convective flow may become oscillatory or even turbulent, which inevitably gives rise to generation of striations. In some cases, the unsteady flow may be suppressed by external force such as rotation, vibration or magnetic field. In this part, some experimental results will be given about the effect of external forces on mass transport during oxide crystal growth.

**6.1 Suppression of oscillatory flow by transverse magnetic field in NaBi(WO4)2 melt**  For oxide melt, oscillatory convection can be observed if the temperature gradient along the loop heater gets large enough. Fig. 21(a) shows a typical unsteady flow pattern of NaBi(WO4)2 melt. This pattern comprises one main trunk and the branches. The main trunk oscillates with time, and the arrows I, II represent the range of oscillation. Fig. 21(b) shows the schematic diagram of oscillatory pattern. The main trunk oscillates around the position A with the amplitude as shown by the bi-directional arrow 1. The oscillatory frequency reached about 10 Hz, and the amplitude was about 500 μm. Similar convective oscillations have also been observed in KNbO3, BaB2O4 and Bi12SiO20 melts or solutions suspended on a

When a 60 mT transverse static magnetic field is applied, the distinct attenuation of the oscillation is observed and arrows III, IV represent the range of oscillation as shown in Fig. 22(a). The main trunk oscillates around the position A with the amplitude (~200 μm) as shown by the bi-directional arrow 2 in Fig. 22(b), which is smaller than that as shown by arrow 1. So the oscillatory amplitude of main trunk decreased when the magnetic field is applied. The frequency of oscillation is measured to be about 4 Hz. This means that the

loop heater (Z. H. Liu et al., 1998; W. Q. Jin et al., 2004; Y. Hong et al., 2006).

instability of convective flow has been effectively reduced.

still keep its shape in case of the existence of certain convection.

**6. Mass transfer with external forces in oxide crystal growth** 

interface, and thus deteriorate the crystal quality.

Morphological instability has also been obtained for KNbO3 grains (X. H. Pan et al., 2009). Fig. 19 shows the microscopic morphologies of quenched KNbO3 grains observed by electromicroprobe, which are achieved under the condition of rapid solidification. Dendrite grows in the central region of the melt indicating the appearance of morphological instability, whereas crystal with smooth surface is observed in the quenched melt adjacent to the periphery of the loop heater.

Fig. 19. Morphologies of KNbO3 grains in (a) the central region and (b) the marginal region of the melt-solution observed by electron-microprobe

The instability of interfacial morphology is close by related to the solute distribution in front of the solid-liquid interface. Fig.20 shows the concentrations of solute KNbO3 in the quenched melt–solution near KNbO3 grains examined by electron-microprobe analysis. Here, WKN is the experimentally measured mass fraction of the solute distribution in the solution and r refers to the distance in the direction parallel to the interface with origin of xcoordinate axe being at the face centre. The point "▲" is for the grain obtained in the central region while point "●" is for the grain obtained in the marginal region of the melt.

Fig. 20. Solute distribution along the solid-liquid interface of KNbO3 grains in solution

Morphological instability has also been obtained for KNbO3 grains (X. H. Pan et al., 2009). Fig. 19 shows the microscopic morphologies of quenched KNbO3 grains observed by electromicroprobe, which are achieved under the condition of rapid solidification. Dendrite grows in the central region of the melt indicating the appearance of morphological instability, whereas crystal with smooth surface is observed in the quenched melt adjacent to the

Fig. 19. Morphologies of KNbO3 grains in (a) the central region and (b) the marginal region

The instability of interfacial morphology is close by related to the solute distribution in front of the solid-liquid interface. Fig.20 shows the concentrations of solute KNbO3 in the quenched melt–solution near KNbO3 grains examined by electron-microprobe analysis. Here, WKN is the experimentally measured mass fraction of the solute distribution in the solution and r refers to the distance in the direction parallel to the interface with origin of xcoordinate axe being at the face centre. The point "▲" is for the grain obtained in the central


Fig. 20. Solute distribution along the solid-liquid interface of KNbO3 grains in solution

r(μm)

region while point "●" is for the grain obtained in the marginal region of the melt.

central region marginal region

of the melt-solution observed by electron-microprobe

60

55

50

45

WKN(wt%)

40

35

30

periphery of the loop heater.

It is obvious that the solute concentration along the interface is uneven both for pure diffusive and diffusive-convective regions which is in related with the specie supplies. In terms of the Berg effect, the apexes of a polyhedral crystal are the best supplied regions and the concentration along the crystal surface varies, being the lowest at the face centre. This is the shape destabilizing factor. The shape stabilizing factor is connected with the anisotropy in the surface growth kinetics. If the kinetic coefficient with its anisotropy is not sufficient to compensate for the inhomogeneity in the concentration distribution over the growing crystal face, the appearance of morphological instability in the crystal faces takes place. The curves in Fig.20 indicate that not only the concentration difference between the centre and corner of a KNbO3 grain face but also the concentration gradient along the face in the convective region are lower than that in the diffusive region. This is attributed to the enhanced mass transport and thus the improved homogeneity of solute concentration by convection. As a result, morphological instability occurs more easily in the diffusive region where dendrites are found whereas crystal in the convective region conserves a polyhedral shape. This result indicates that, even underlying a very high cooling rate, flat interface may still keep its shape in case of the existence of certain convection.

It should be emphasized from above results that, only steady convection is helpful for the maintaining of flat interface during oxide crystal growth. Mass transfer governed by pure diffusion or unsteady convection may leads to morphological instability of growing interface, and thus deteriorate the crystal quality.
