Thermodynamic Analysis of the V-Shaped Area of High Pressure and High Temperature in Cubic Boron Nitride Synthesis with Li₃N as a Catalyst

Abstract

The possibilities of different phase transitions to cBN with Li₃N as catalyst at high temperature and high pressure (1600–2200 K, 4.8–6.0 GPa) are analyzed, in the framework of the second law of thermodynamics. The Gibbs free energy (ΔG) of three reactions which may happen in the Li₃N-BN system: hBN + Li₃N→Li₃BN₂, hBN→cBN, and Li₃BN₂→cBN + Li₃N, is calculated, with the influence of high temperature and high pressure on volume included. We show that ΔG of hBN + Li₃N→Li₃BN₂ and hBN→cBN are between −35∼−10 KJ·mol⁻¹ and −25∼−19 KJ·mol⁻¹, respectively. However, ΔG of Li₃BN₂→cBN + Li₃N can be positive or negative. The area formed by the positive data is a V-shaped area, which covers the most part of the cBN growing V-shaped area. It confirms that Li₃BN₂ is stable in the P-T area of cBN synthesis, and cBN is probably transformed directly from hBN. Analysis suggests that Li₃BN₂ promotes the transition from hBN to cBN.

1. Introduction

High static pressure is the most common method to synthesize cubic boron nitride (cBN) with hexagonal boron nitride (hBN) as raw material and with alkaline metals or alkaline-earth metals and their nitrides as catalysts. The cBN synthesized with lithium nitride (Li₃N) is widely used because of its regular crystal shape and suitable particle size. A layer of white powder covers the surface of cBN crystal during the growth. Phase structure characterizations show that the layer is composed of hBN, micro cBN and lithium boron nitrides (Li₃BN₂) [1]. It is very desirable to study their relationships to understand the phase transition mechanism of cBN as they are in direct contact with the cBN single crystals.

Previous studies indicate that no matter what kind of catalyst is adopted, the temperature and pressure forms a V-shaped area [2–4] where cBN can grow stably. Besides there is a short-range ordered or medium-range ordered structure in high temperature and high pressure (HPHT) system [5], namely a solid microstructure existing in the HPHT system. It was proposed that cBN was directly transformed from hBN [6], while [7] believed that cBN was precipitated from solvent. Thermodynamic analysis plays an important role in understanding the synthesis mechanism of diamond [8,9]. Therefore it is possible to investigate the phase transition mechanism of cBN because of the similar structure. However, there are few reports about the thermodynamics of the cBN transition mechanism with catalyst at HPHT due to the absence of thermodynamic data at HPHT and the ambiguity of the mechanism. Bocquillon et al. [3,10] put forward the following reactions in the Li₃N-hBN system:

$${\mathit{Li}}_{3}N+\mathit{hBN}\to {\mathit{Li}}_3{\mathit{BN}}_{\mathit{2}}$$

(1)

$${\mathit{Li}}_{3}B{N}_2\to {\mathit{Li}}_{3}N+\mathit{cBN}$$

(2)

After the reactions, all of the phase are found around the cBN crystals by XRD [1], except for the volatile Li₃N. It is difficult to identify the reactions through experiments because the online inspection is impractical. The synthesis of cBN is an isobaric and isothermal process, and we can apply classical thermodynamics to analyze the reactions in the system by querying the related thermodynamic data. Based on the second law of thermodynamics, we calculate the change of Gibbs free energy in (1) and (2) and analyze the thermodynamic transformation of cBN in V-shaped area of the system.

2. Calculation

The calculation is based on the second law of thermodynamics. The change of Gibbs free energy ΔG is:

$$G_T^P=G_T^P+{\int }_{P^{\theta }}^P\mathit{VdP}$$

(3)

Under the isobaric condition, ΔG can be obtained:

$$\Delta G_T^{\theta }=\Delta H_T^{\theta }-T\Delta S_T^{\theta }={\Delta }_f{H}^{\theta }+{\int }_{T_0}^T\Delta C_{p}dT-T\times ({\Delta }_r{S}^{\theta }+{\int }_{T_0}^T\frac{\Delta C_p}{T}dT)$$

(4)

ΔG between different temperatures and pressures can be expressed as [11]:

$$\begin{array}{l}\Delta G_T^P=(\Delta H_T^0-T\Delta S_T^0)+({\int }_{T_0}^T\Delta C_P-T{\int }_{T_0}^T\frac{\Delta C_P}{T}dT)+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\int }_{P0}^P[{(V_0+\Delta V_T+\Delta V_P)}_{\text{Product}}-{(V_0+\Delta V_T+\Delta V_P)}_{\text{Reactant}}]dP\end{array}$$

(5)

Equation (5) takes the influence of temperature and pressure to volume into consideration. $\Delta H_T^0$ and $\Delta S_T^0$ are enthalpy and entropy of crystalline phase in the system, respectively. V₀ is the mole volume at normal temperature and pressure. ΔV_T and ΔV_P are the variation of mole volume with constant temperature and pressure, respectively. According to [12,13], the related thermodynamic quantities can be acquired. Similarly, we get ΔG of Li₃BN₂ from [14]:

$$\begin{array}{r}\hfill \Delta G^{\theta }(T,P^{\theta })=-425.42206+0.07512T-1.778\times {10}^{-4}{T}^2+\\ \hfill +4.144\times {10}^{-8}{T}^3+\frac{27.9}{T}-9.306\times {10}^{-3}T\text{ln}T\end{array}$$

(6)

298 K is chosen as the start temperature. ΔV_T and ΔV_P are calculated in the following.

2.1. ΔV_P

Based on the state equation of crystal under high pressure by Sung [15], we use the Birch-Murnaghan equation of crystal state to calculate the variation of volume under high pressure:

$$P=\frac{3}{2}{(\frac{V_0}{V_0+\Delta V_P})}^{\frac{5}{3}}[{(\frac{V_0}{V_0+\Delta V_P})}^{\frac{2}{3}}=1]\hspace{0.17em}\{1+\frac{3}{4}(B_0^{\text{'}}-4)[{(\frac{V_0}{V_0+\Delta V_P})}^{\frac{2}{3}}-1]\}$$

(7)

In Equation (7)B₀ is bulk modulus, and B₀′ is the first derivative of the pressure with respect to bulk modulus. From Equation (7), the relation between ΔV_P and pressure can be obtained.

Table 1. Physical parameters of the materials participating in the reactions.

Material	V0/cm3·mol−1	B0/GPa	B0′/GPa
cBN	7.1150	398.6 [16]	3.85 [16]
hBN	10.8820	36.7 [17]	5.6 [17]
Li3N	27.2031	98 [18]	4.04 [18]
Li3BN2	34.0850	-	-

shows the physical parameters of materials in reactions in the Li₃N-BN HPHT system.

In Table 1, the B₀ and B₀′ are not found in references. According to [11], they can be deduced from Vegard’s law from the B₀ and B₀′ of Li₃N and cBN:

$$B_0=x_1{B}_1+x_2{B}_2=B_{{\mathit{Li}}_{3}N}+B_{\mathit{cBN}}$$

(8)

In Equation (8)B₀ and B₁ and B₂ are the bulk modulus of Li₃BN₂, Li₃N and cBN, respectively. x₁, x₂ are the percentage of Li₃N and cBN, with x₂ = 1 − x₁. B₀′ could be calculated by the same method. The B₀ of Li₃BN₂ is 248.3 and B₀′ is 3.945.

2.2. ΔV_T

ΔV_T can be obtained on the basis of the thermal expansion properties of crystals:.

$$\Delta V_T=V_0\beta \Delta T=V_0\beta (T-T_0)$$

(9)

In Equation (9), β refers to the volume expansion coefficient, and its value of hBN, cBN and Li₃N can be obtained from [19,20], but there has been no report on the value β of Li₃BN₂. It can be calculated as follows:

$$C_p-C_v=\frac{V_{0}T{\beta }^2}{\kappa }$$

(10)

where C_v is the molar heat capacity at constant volume and κ is the isothermal compressibility with κ = 1/B. According to the definition of bulk modulus, the Equation (10) can be written as

$$C_P-C_V=(B_0+B_0^{\text{'}}\times P)V_{0}T{\beta }^2$$

(11)

From [14] C_p of Li₃BN₂ can be obtained as follows:

$$C_P=9.306+355.6\times {10}^{-3}T-5.58\times {10}^4{T}^{-2}-24.86\times {10}^{-5}{T}^2$$

(12)

and the β of Li₃BN₂ can thus be calculated:

$$\beta =1.68324\times {10}^{-4}+1.21253\times {10}^{-8}T-8.96867\times {10}^{-11}{T}^2+1.30857\times {10}^{-14}{T}^3$$

(13)

3. Results and Discussion

Different P-T areas of cBN growth in the Li₃N-hBN system [2–4] have been proposed, which may be related to the purity and degree of order of the hBN and the synthesis process used. The lowest synthesis pressure is 4.8 GPa and the minimum synthesis temperature is 1690 K at 5.0 GPa in Li₃N-hBN system [4], which approximates the V-shaped area proposed in [3]. Ko [7] found that the lowest temperature was 1620 K at 5.3 GPa, lower than the lowest temperature (1723 K) reported by Bocquillon [3]. Considering the industrial application and the reliability of the numerical calculation is performed in an expanded region of 1600–2200 K and 4.8–6.0 GPa.

Table 2. ΔG of hBN + Li₃N→Li₃BN₂ at different temperatures and pressures.

T/K	ΔG /(KJ·mol−1)
	4.8 GPa	5.0 GPa	5.2 GPa	5.4 GPa	5.6 GPa	5.8 GPa	6.0 GPa
1600	−16.4319	−17.9111	−19.3804	−20.8399	−22.2898	−23.7301	−25.1611
1650	−17.3670	−18.9879	−20.5988	−22.2000	−23.7916	−25.3736	−26.9463
1700	−18.1161	−19.8872	−21.6483	−23.3996	−25.1413	−26.8735	−28.5963
1750	−18.6561	−20.5857	−22.5054	−24.4154	−26.3156	−28.2064	−30.0878
1800	−18.9630	−21.0597	−23.1464	−25.2233	−27.2905	−29.3483	−31.3967
1850	−19.0122	−21.2842	−23.5463	−25.7985	−28.0411	−30.2743	−32.4980
1900	−18.7784	−21.2340	−23.6797	−26.1156	−28.5418	−30.9586	−33.3660
1950	−18.2357	−20.8831	−23.5206	−26.1482	−28.7663	−31.3749	−33.9740
2000	−17.3573	−20.2046	−23.0420	−25.8696	−28.6875	−31.4959	−34.2951
2050	−16.1162	−19.1714	−22.2166	−25.2521	−28.2779	−31.2942	−34.3011
2100	−14.4845	−17.7554	−21.0163	−24.2674	−27.5089	−30.7409	−33.9634
2150	−12.4339	−15.9281	−19.4124	−22.8869	−26.3518	−29.8071	−33.2531
2200	−9.9353	−13.6604	−17.3756	−21.0809	−24.7767	−28.4629	−32.1398

shows the change of Gibbs free energy of Li₃N + hBN→Li₃BN₂ in the HPHT area (1600–2200 K, 4.8–6.0 GPa) of cBN synthesis in the Li₃N-hBN system. The values of ΔG are between −35 and −10 KJ·mol⁻¹. This illustrates that Li₃N can transform into Li₃BN₂ in this P-T area, in coincidence with the XRD results of the catalyst after synthesis [1].

ΔG varies regularly with the increase of temperature and pressure. It decreases firstly and then rises with the increasing temperature under constant pressure. ΔG decreases monotonically with the increasing pressure when temperature keeps constant. The higher temperature and pressure, the more negative ΔG, it is easier to generate Li₃BN₂.The possibility of the reaction exists, although the absolute value of ΔG is very small.

Table 3. ΔG of hBN→cBN at different temperatures and pressures.

T/K	ΔG/(KJ·mol−1)
	4.8 GPa	5.0 GPa	5.2 GPa	5.4 GPa	5.6 GPa	5.8 GPa	6.0 GPa
1600	−19.7724	−20.4187	−21.0596	−21.6952	−22.3255	−22.9507	−23.5707
1650	−19.7917	−20.4391	−21.0810	−21.7176	−22.3490	−22.9752	−23.5963
1700	−19.8295	−20.4774	−21.1198	−21.7569	−22.3887	−23.0154	−23.6370
1750	−19.8857	−20.5334	−21.1758	−21.8127	−22.4445	−23.0710	−23.6925
1800	−19.9600	−20.6070	−21.2485	−21.8847	−22.5156	−23.1414	−23.7620
1850	−20.0523	−20.6977	−21.3377	−21.9723	−22.6016	−23.2258	−23.8450
1900	−20.1622	−20.8052	−21.4427	−22.0750	−22.7019	−23.3238	−23.9405
1950	−20.2893	−20.9290	−21.5633	−22.1922	−22.8158	−23.4343	−24.0477
2000	−20.4333	−21.0687	−21.6986	−22.3232	−22.9425	−23.5567	−24.1657
2050	−20.5936	−21.2235	−21.8480	−22.4672	−23.0811	−23.6898	−24.2935
2100	−20.7697	−21.3930	−22.0109	−22.6234	−23.2307	−23.8328	−24.4299
2150	−20.9609	−21.5763	−22.1863	−22.7909	−23.3903	−23.9845	−24.5736
2200	−21.1665	−21.7726	−22.3733	−22.9686	−23.5587	−24.1436	−24.7235

shows ΔG of transition hBN-cBN in HPHT area (1600–2200 K, 4.8–6.0 GPa) of cBN synthesis. In consideration of the influence of temperature and pressure on phase volume, relatively precise calculation results are obtained. ΔG of the transition is between −25 and −19 KJ·mol⁻¹. ΔG is negative and its absolute value increases with the increasing temperature and pressure, and it is more and more probable for the transition to occur. From Table 3, we find that ΔG changes more sharply with the pressure than with the temperature. Therefore, the transition of hBN to cBN is probable in the P-T area, and pressure is greater impact than temperature.

We show the results in

Table 4. ΔG of Li3BN₂→cBN + Li₃N at different temperatures and pressures.

T/K	ΔG/(KJ·mol−1)
	4.8 GPa	5.0 GPa	5.2 GPa	5.4 GPa	5.6 GPa	5.8 GPa	6.0 GPa
	1600	−3.3405	−2.4757	−1.6792	−0.8553	−0.0358	0.7795	1.5904
1650	−2.4247	−1.4193	−0.4822	0.4824	1.4426	2.3984	3.3500
1700	−1.7134	−0.5583	0.5285	1.6427	2.7525	3.8580	4.9592
1750	−1.2296	0.0842	1.3297	2.6026	3.8712	5.1354	6.3953
1800	−0.9970	0.4846	1.8979	3.3386	4.7750	6.2070	7.6347
1850	−1.0401	0.5865	2.2086	3.8263	5.4395	7.0484	8.6530
1900	−1.3837	0.4289	2.2370	4.0406	5.8399	7.6348	9.4254
1950	−2.0537	−0.0459	1.9573	3.9561	5.9505	7.9405	9.9263
2000	−3.0760	−0.8640	1.3434	3.5464	5.7451	7.9393	10.1293
2050	−4.4774	−2.0521	0.3686	2.7849	5.1968	7.6043	10.0076
2100	−6.2851	−3.6376	−0.9945	1.6440	4.2783	6.9081	9.5337
2150	−8.5270	−5.6482	−2.7739	0.0960	2.9615	5.8226	8.6795
2200	−11.2312	−8.1122	−4.9977	−1.8877	1.2180	4.3193	7.4164

supposing that Li₃BN₂ could decompose into cBN. We find that when the pressure is as low as 4.8 GPa, ΔG is negative and ΔG increases from negative to positive with increasing temperature and pressure. ΔG increases from negative to positive with increasing pressure when the temperature is constant. However, ΔG increases first and then decreases with the increasing temperature under constant pressure, and ΔG of reaction Li₃BN₂→cBN + Li₃N can be negative or positive when the pressure is between 4.8 and 6.0 GPa. It can be obtained from Table 4 that the boundary of negative and positive values forms an approximate V-shaped area (the upper area of curve A in Figure 1), in which ΔG of reaction Li₃BN₂→cBN + Li₃N is positive and this reaction does not occur. Therefore Li₃BN₂ can exist stably, and cBN does not come from the decomposition of Li₃BN₂ in the V-shaped area. Upper area of Curve B in Figure 1 is the V-shaped area from [8] where cBN can grow stably. Figure 1 indicates that the V-shaped area formed in Table 4 covers the most part of the V-shaped area of cBN growth from the references. In other words, the reaction Li₃BN₂→cBN + Li₃N cannot occur in the V-shaped area where the macroaggregated and high-quality cBN crystals can be obtained. That is to say, cBN cannot be produced by the treatment of only Li₃BN₂ in the V-shaped area obtained from Table 4. However, it can be seen from Table 3 that in the same area, ΔG of the hBN-cBN transition is negative, and it becomes more and more negative with increasing temperature and pressure. We speculate that in the V-shaped area, cBN is probably transformed directly from hBN.

It is known from [2] that cBN does not form when Li₃BN₂ melts or decomposes, and only when excess hBN is added can cBN be found. Li₃BN₂ exists stably in the HPHT area of cBN growth, and Li₃BN₂ appears to be in equilibrium with cBN in the Li₃N-hBN system. We infer that the catalyst promotes the transition of hBN to cBN. The melting point of hBN is over 3000 K at ambient pressure. hBN has a stable structure with a sp² hybridization state in the hexagonal ring layer and Van der Waals’ force between layers. The hybridized orbital of B atom is sp² + 2p⁰, and N atom is 2s² + 2p² in hBN. However, the hybridized orbital of B and N atom is sp³ in cBN, and the chemical bond between B and N atom is a stable σ-bond, so the most important driving force for the transition from hBN to cBN is to transfer an electron from N atom to B atom. Part of Li⁺ in the short-range ordered structure of Li₃BN₂ absorbs electrons from N atom at the high active state. Because of its instability, Li cannot exist stably in the high temperature and high pressure system. B atoms with empty orbitals absorb the electrons of Li, and then complete the electronic transfer from N to B atoms [21], so both the electrical structures of the B and N atoms are sp² + 2p¹. Electrons in the s orbital of B and N atoms at the high temperature are excited into the empty p orbitals. The B and N atom can form a sp³ hybridization state. B and N atoms in this sp³ hybridization state form σ covalent bonds in a head-to-head mode with energy fluctuations, and generate the microstructure of cBN at high pressure and high temperature.

Li₃BN₂ is short-range ordered at molten-like state under HTHP, and it can be dissolved into well-ordered hBN. BN₂³⁻ ion invades hBN and affects the van der Waals forces between the layers, causing inter-layer slippage or breakage. hBN can be disintegrated into two dimensional B-N dusts with low polymerization degree. At this time, the long-range ordered structure disappears, rendering a short-range ordered structure. Without catalyst, higher temperatures than 3000 K and higher pressures than 13 GPa are needed to obtain cBN crystals [4]. With the catalyst, a short-range or medium-range ordered molten-like structure [5] forms to greatly reduce the temperature and pressure of cBN synthesis. From the point of crystal nucleation, the B-N dusts produced by hBN degradation can transform into cBN microstructures by heterogeneous nucleation with energy fluctuation [22]. Li₃BN₂ could be the matrix of cBN nucleation, because of the similar crystal structure and larger lattice constant [23]. The process only forms the microstructure of cBN crystals. In order to obtain cBN crystals, it is essential for the tiny crystal nucleus to grow gradually [24].

4. Conclusions

(1) In the HPHT area (1600∼2200 K, 4.8∼6.0 GPa) of cBN growth, the reaction hBN + Li₃N→Li₃BN₂ can occur, and the ΔG of the reaction is between −35 and −10 KJ·mol⁻¹. However, in the same P-T area, ΔG of transition hBN→cBN is between −25 and −19 KJ·mol⁻¹, indicating a more probable transition. The ΔG of decomposition Li₃BN₂→cBN + Li₃N is almost positive, thus Li₃BN₂ can exist stably.

(2) The area of the positive ΔG of Li₃BN₂ decomposition forms an approximate V-shaped area, which covers the most part of the V-shaped area of cBN crystal growth. Therefore, cBN does not come from the decomposition of Li₃BN₂ and cBN is probably transformed from hBN. However, analysis shows that Li₃BN₂ promotes the transfer of an electron from a N atom to a B atom in order to form the micro cBN nuclei.