Thermodynamic Analysis About Nucleation and Growth of Cubic Boron Nitride in the System BN-Li3N Under High Pressure and High Temperature

In this paper, cubic boron nitride (cBN) single crystals are synthesized with lithium nitride (Li3N) as catalyst under high pressure and high temperature (HPHT). A lot of nanometer-sized cubic boron nitride nucleuses are found in the near surface layer of cBN crystals by high resolution transmission electron microscopy examinations. Based on the experiment results, the transformation kinetics is described by a nucleation and growth process in the thermodynamic stability region of cBN. The theoretical description is based on the heterogeneous nucleation and layer growth mechanism, and the relevant parameters are estimated and discussed. The results show that critical crystal radius, r * , is increasing with the elevated temperature under the same pressure, and temperature is the main influence on it under lower pressure. At the same time, the results show that crystal growth velocity has different changing tendency with different pressure under proper synthesized scope. The effect of the catalyst is suggested to lower the activation enthalpy for nucleation. These results are well consistent with experimental data.


Introduction
Cubic boron nitride single crystal, as a functional material, has many interesting properties, such as thermal conductivity, high hardness only second to diamond and excellent chemical stability.Furthermore, they are typical III-V group semiconductor materials with a wide energy gap, which can be made easily into both P-and N-type with suitable impurity additions [1][2][3].At present, cBN single crystals can be synthesized by various methods, among which the static catalyst synthetic method under HPHT is commonly used [4,5].According to that method, hexagonal boron nitride (hBN) is used as boron and nitride element sources.Meanwhile, in order to decrease the synthetic temperature and pressure, many catalysts including alkali and alkaline earth nitrides, ammonium salts, and even water were proposed for HPHT synthesis of cBN crystals [6,7].
It may be understood that the key problem in obtaining large crystals of high quality is to control the synthesized pressure and temperature, but the general regulation about synthesized conditions to obtain a well-facetted large single crystal has not been successfully established [8].For the difference of equipments and synthesized methods, researchers maybe draw the different conclusions through their experiments results.On the basis of this background, some work was undertaken to investigate the kinetics of the phase transition and to discuss the mechanism of nucleation and crystal growth.A first version of p, T-diagram for boron nitride was proposed by Bundy and Wentorf [9] in 1963 based on experimental data, and the conversion from hBN into the cubic form under HPHT conditions was depicted in details.Fukunaga had determined the phase boundary between hexagonal and cubic boron nitride by careful observation of the transformation behavior [10].However, the mechanism of nucleation and crystal growth is still a subject for discussion, especially by the thermodynamic calculation as methods.
In this paper, critical crystal radius and crystal growth velocity of cBN under different pressure and temperature was calculated by the existing thermodynamic data.Combined the related experiments results, the mechanism of nucleation and crystal growth were also discussed.

experiments section
The cBN crystals were synthesized by static HPHT method.As starting materials, the hBN (purity 99.9%) were well-mixed with Li 3 N as catalyst in cubic anvil with the weight ratio of 9:1, and the pyrophyllite was used as a transmitting pressure substance.The cell assembly was brought up to a pressure of 4.6-4.8GPaby the high pressure apparatus and then heated to a temperature of approximately 1850K.After keeping the sample at this temperature for 10 min, the temperature was decrease rapidly by turning off the electric power, and the synthesized samples were quenched from high temperature to room temperature and the high pressure was decreased to the ambient pressure.
The morphology of synthesized sample was examined with JSM-6380LA type SEM.The nearsurface region samples were carefully collected from the cBN crystal matrix under the view of an optic microscope, and their phases were determined by means of JEOL JEM-2010F type HRTEM with an operating voltage of 200kV.

calculations
The cBN can be synthesized under HPHT conditions, and this process could be explained by the kinetics of nucleation and growth.The critical crystal radius, r*, is the essential parameter for crystal nucleation, and the linear growth velocity, v, is independent of time for interface-controlled growth.
According to the classical theory of nucleation [11], in the processing of transformation from hBN to cBN, the Gibbs free energy difference, ) (r G  , which includes volume free energy difference and surface free energy difference, is given by Where r is nucleus radius, v G  is the molar volume free energy difference and V m is the molar volume of cBN. is the contact angle between mold wall and crystal nucleus which formed under HPHT with spherical cap shape, and Ls  represents the surface tension coefficient.
The V  , which represents molar volume difference between hBN and cBN, is described by When temperature is constant, v G  can be expressed as Eq. 3 with pressure increases from P 0 to P, According to reference [10], the P-T balancing line between hBN and cBN is given by 0 273 0.79 465 , the critical crystal radius, r*, is given by From Eq. 5, the critical crystal radius could be calculated with different pressure and temperature [12].Fig. 1 shows the growth step on the cBN crystal surface, and this means the growth pattern of cBN is two-dimension nucleation.According to the classical theory of nucleation, the growth velocity, ν, is given by [13] ) exp( ) Where C G is a temperature independent constant, E D is the activation energy for diffusion and γ represents edge free energy of the two-dimensional nucleus.  is the difference of the free enthalpies of both phases and is expanded around the equilibrium point (P 0 , T 0 )  is proportional to the (T 0 -T).The γ represents the interface free energy between crystal and melt.The temperature dependence of γ is less essential and can be regarded as a constant because in the nucleation process there is no roughening transition.

Figure 1. The growth step of cBN crystal
The activation energy for diffusion, E D , is described as 00 ( ) ( ) ( ) When the P=P 0 , 00 By above method, the growth velocity under different pressure and temperature could be calculated.

Results and Discussion
The structures and morphologies of the near-surface region in cBN are closely associated with the crystal growth under HPHT, and it may be of great significance to explain the process of cBN growth [14,15].When the synthetic process finished and the synthetic cell assembly was cooled rapidly, much information about cBN growth under HPHT could be recorded.
Fig. 2 shows the HRTEM image of sample which collected from the near-surface region of cBN crystals.According to the two dimension lattice stripe photograph, the phase in selected area is inferred as nanometer-sized cBN nucleus with a reflection of g = 110.Those particles have no obvious crystal shape and cannot be distinguished by optical microscope, and they are distributed all over the sample.
These nanometer-sized cBN nucleuses could be transformed by hBN with catalysis of Li 3 BN 2 , and its size and growth velocity are closely related to synthesized pressure and temperature.When the pressure is constant, with the increase of temperature, the critical crystal radium is increasing.
Under lower pressure (such as P=4.5GPa shown as Fig. 3), the critical crystal radium increases rapidly, so the temperature is the main affecting factor in this nucleation process.However, under high pressure (such as P=6.0GPa shown as Fig. 3), the critical crystal radium has a tendency to increase, but the tendency is not obvious, and it means temperature has little influence on the nucleation of cBN under high pressure.Similarly, the critical crystal radium is increasing with the higher of pressure when the temperature is constant, and this changing trend is obvious under low pressure.The calculated results show the cBN crystals would have larger critical radium under lower pressure and at higher temperature, and cBN crystals with coarse grain structure could be obtained under these conditions.From above results, it can be inferred that the critical radium is closely related to synthesized pressure and temperature.The coarse crystals could be obtained under lower pressure and higher temperature, for the critical radium depends more on temperature under lower pressure.However, the larger critical radium maybe makes the phase transition probability decreased, and makes the nucleation more difficult.This result is consistent to our synthesized experiments, and the crystals which are synthesized under lower pressure and lower temperature have poor quality in reality.
Fig. 4 shows the relationships between crystal growth velocity and temperature under different pressures.According to the previous synthesized experiments, we limit the temperature scope is 1600-2200K, and the pressure scope is 4.5-6.0GPa.It can be obviously observed that growth velocity exhibits parabolic trend with the increasing temperature under different pressures.At lower temperature, the crystal growth velocities with different pressures change a little, but it can be influenced obviously with the increasing of temperature.When the synthesized pressure reaches to 5.5GPa, the growth velocity of cBN crystals has the maximum compared to other pressures.For the phases have short and medium range orders under HPHT conditions, so they could exist with certain flowability.Once the nanometer-sized cBN nucleuses were formed in the HPHT melt, it could diffuse towards to the larger crystal nucleuses to decrease the free energy of system.The whole diffusion velocity is controlled by the synthesized pressure and temperature.When the pressure gets higher, the activation energy of diffusion gets higher, so the diffusion velocity gets slower.On the other hand, the diffusion coefficient gets higher with the increasing of temperature.Considering the effect of pressure and temperature, the cBN has smaller nucleation radius and lower nucleation energy under 5.5GPa pressure, and the nucleation process would become easier.The diffusing process gets easier with the concentration of cBN particles gradually become higher, so the cBN crystals would have the higher growth velocity under this condition.is short-range ordered under HTHP, and it can be dissolved into well-ordered hBN [18].Without catalyst, higher temperature than 3000K and higher pressure than 13GPa are needed to obtain cBN crystal [19].It could be explained that the role of catalyst is to consist in the promotion of nucleation and to reduce the corresponding activation enthalpy by above thermodynamic calculations.

Conclusions
In this paper, the cBN crystals are synthesized by Li 3 N as catalyst under HPHT conditions, and the nanometer-sized cBN nucleuses are found in the near surface region.The experimental results are well described by the classical theory of nucleation and growth.
From the systematic investigation on the nucleation and growth velocity in the cBN synthesis under HPHT conditions, it is found that critical crystal radium is influenced by temperature and pressure obviously.When the pressure is constant, with the increase of temperature, the critical crystal radium is increasing, and it has a gentle tendency to increase under higher pressure.On the other hand, when the temperature is constant, critical crystal radium is increasing with the pressure gets higher, and pressure has the main influence on the nucleation when the temperatures above a certain threshold.
Also, it can be obvious observed that growth velocity exhibits parabolic trend with the increasing temperature under different pressures.Under 5.5GPa, the growth velocity is higher than other pressure.
As to the role of catalyst, it could be explained to consist in the promotion of nucleation and to reduce the corresponding activation enthalpy.

V
 are the heat of transition and volume change, respectively, in the point (P 0 , T 0 ).When the pressure is constant (P=P 0 ), 

Figure 2 .
Figure 2. HRTEM image of cBN particle in the near-surface region of cBN crystal

Figure 3 .
Figure 3.The relationships between critical crystal radius r * and temperature under different pressures

Figure 4 .
Figure 4.The relationships between crystal growth velocity and temperature under different pressures