# High-pressure, high-temperature phase equilibria with superhard boron-rich compounds of B-C-N-O and B-C-b-Si systems by in situ X-ray diffraction and CALPHAD methodology

Alexandre Courac, Yann Le Godec, Jelena Sjakste, Nathalie Vast, Olivier Rapaud, Vladimir Turkevich

► To cite this version:

Alexandre Courac, Yann Le Godec, Jelena Sjakste, Nathalie Vast, Olivier Rapaud, et al. High-pressure, high-temperature phase equilibria with superhard boron-rich compounds of B-C-N-O and B-C-b-Si systems by in situ X-ray diffraction and CALPHAD methodology. ACS Applied Materials & Interfaces, 2025, 17 (38), pp.53013-53039. 10.1021/acsami.5c08312. hal-05359252

HAL Id: hal-05359252

https://hal.science/hal-05359252v1

Submitted on 11 Nov 2025

**HAL** is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire **HAL**, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# High-Pressure, High-Temperature Phase Equilibria with Superhard Boron-Rich Compounds of B-C-N-O and B-C-Si Systems by In Situ X-Ray Diffraction and CALPHAD Methodology

Alexandre Courac,\*¹,² Yann Le Godec,¹ Jelena Sjakste,³ Nathalie Vast,³ Olivier Rapaud,⁴
Vladimir Turkevich\*⁵

1. Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie (IMPMC), UMR CNRS 7590, Muséum National d'Histoire Naturelle, IRD UMR 206, Sorbonne Université, 75005 Paris, France

2. Institut Universitaire de France (IUF), 75005 Paris, France

3. Laboratoire des Solides Irradiés, Institut Polytechnique de Paris, CEA-DRF-IRAMIS, CNRS UMR 7642, Ecole Polytechnique, F91128 Palaiseau, France

4. Institut de Recherche sur les Céramiques (IRCER), UMR CNRS 7315, Université de Limoges, 87000 Limoges, France

5. National Academy of Sciences of Ukraine, Institute for Superhard Materials (ISM), 04074 Kiev, Ukraine,

## Abstract

Boron-rich compounds within the B–C–N–O and B–C–Si systems exhibit exceptional functional properties, making them highly attractive for industrial applications such as those requiring superhardness, nuclear technologies, or thermoelectricity. This review summarizes recent experimental and theoretical advances on high-pressure, high-temperature (HPHT) phase equilibria up to 20 GPa and 3000 K, focusing on boron carbide (B₄C), boron suboxide (B₆O), boron subnitride (B₁₃N₂), boron silicides and some of their solid solutions. Emphasis is placed on *in situ* x-ray diffraction (XRD), density functional theory (DFT) calculations and CALPHAD thermodynamic modeling. Despite recent progress, significant methodological challenges remain, requiring enhanced experimental accuracy and refined theoretical approaches. Future work should address these gaps to fully leverage the potential of these superhard materials.
# 1. Introduction

Boron-rich compounds find a broad application range, as superhard, nuclear or thermoelectric materials. For efficient explorative research and subsequent application of such materials, *in situ* x-ray diffraction studies, *ab initio* density functional theory calculations (DFT) and the CALPHAD approach make the promising combination that has been efficient in the B–C–N–O and B–C–Si systems (Figure 1), the subject of the present review.

<figure><img src="image_2.png" /><figcaption>**Figure 1** Boron binary compounds of B-C-N-O concentration tetrahedron and B-C-Si concentration triangle.</figcaption></figure>

Recovery high pressure, high temperature (HPHT) experiments are the basis of industrial production of diamond and cubic boron nitride, discovery of new boron-rich solids [1], as well as sintered bindered superhard materials and semiconductors produced by volatile elements (e.g. nitrides like GaN). Generally, the best mechanical and other properties such as optical ones can be achieved in equilibrium synthesis, as it is well known for diamond [2] and boron nitride [3]. Boron carbide produced at HPHT conditions provides another example. However, it is not always clear whether HP phases form at HPHT conditions, or during cooling and/or decompression, nor how it happens, and one needs to extend to these materials the kind of *in situ* studies performed on Si [4, 5]. Indeed, *in situ* methods allow for the observation of phase transformations in solids under HPHT conditions [6, 7], thereby removing the ambiguity that can arise when analyzing the recovered samples a posteriori.

However, *in situ* x-ray diffraction (XRD) of light element compounds [8] is a challenging task, particularly given the low atomic number of the first group of elements. To date, only a limited number of light-element compounds with relatively simple crystal structures has been subjected to such analyses. Observation of x-ray diffraction and scattering of light-element solids and liquids remains a challenging axis of future developments of synchrotron beamlines for geophysics and materials science.

Like diamond that can be obtained by high pressure, high temperature conditions, the synthesis of boron-rich solids at HPHT conditions has attracted much attention in the past
decades [1, 9]. The B-C-N-O concentration tetrahedron and B-C-Si triangle (Figure 1) contain a number of superhard materials that have been extensively studied, and most of them are related to the atomic structure of pristine boron and its compounds. However, the *in situ* HPHT thermodynamic data on boron and of its compounds remains very limited as compared to the graphite-to diamond transformation in various systems (C-Ni-Fe [10], C-Fe [11], Mg-C [12-15], etc.).

In this review we aim to gather all available thermodynamic data for boron-rich solids, with a particular focus on the *p*-*V*-*T* equation of state and enthalpy of formation of HP phases that are lacking in conventional thermodynamic sources. The review is structured as follows: after this Introduction (Section 1) outlining the context and objectives, Section 2 describes the experimental and theoretical methods, highlighting HPHT synthesis protocols, *in situ* X-ray diffraction techniques (including angular dispersive, energy dispersive, and the combined CAESAR method), DFT calculations and CALPHAD thermodynamic modeling. Section 3 examines HPHT phase diagrams of elemental boron, silicon, carbon, and binary boron-rich compounds, addressing their structural characteristics and transformations. Sections 4 and 5 discuss detailed thermodynamics of binary and ternary systems involving boron-rich solids, emphasizing comparative analyses of experimental data with CALPHAD predictions. Section 6 correlates structural characteristics with physical and mechanical properties of boron-rich phases, predicting hardness and evaluating potential industrial applications. Finally, the Conclusions (Section 7) summarize the main findings, highlight critical research gaps, and provide perspectives for future investigations and material applications.
## 2. Review of experimental and theoretical methods

### 2.1 Synthesis and HPHT experiments

In this part we will consider the experimental protocols of ex situ and in situ techniques, HP cells employed and typical operational p-T conditions for each device. In the results presented in this review, three static compression systems were used to obtain reliable and accurate thermodynamic data under pressure: the diamond anvil cell, the multi-anvil press and the portable Paris-Edinburgh press. The Figure 2a presents these three static compression systems used in this review.

<figure><img src="image_3.png" /><figcaption>**Figure 2** Three static compression systems compared at the same scale: 1/ multi-anvil press, 2/ Paris-Edinburgh press, 3/ Diamond anvil cell (DAC) at IMPMC (a). Typical cubic assemblage (b) and X-ray path for in situ probing of reaction volume (c) at MAX80 beamline at DESY, adapted from [16]. Soller slits (d) and diamond anvil cell with Laser heating system and large CCD detector at PSICHE beamline of SOLEIL synchrotron for in situ X-ray diffraction (e).</figcaption></figure>

1/ The diamond anvil cell (DAC) is a hand-sized device employing opposed anvils to create extremely high pressures, reaching up to 600 GPa. A metallic gasket, typically made of rhenium (Re), is positioned between the culetts of two gem-quality diamond anvils. The space enclosed by the anvil culetts and the gasket central hole forms the high-pressure sample chamber, which contains the sample, the pressure transmitting medium (PTM) and the pressure gauge (calibrant). Pressure is generated by squeezing the two diamond anvils together. This is usually accomplished through the utilisation of screws [17] or by inflating a metal membrane [18]. As pressure is defined as the force per unit area, smaller diamond culet sizes allow for higher pressures to be achieved with the same applied force. In addition, the DAC is capable of attaining moderate temperatures ranging up to 1100 K, as well as elevated temperatures reaching as high as 8000 K, by means of either resistive heating or
laser heating, respectively. Pressure within the DAC is typically determined by measuring the visible fluorescence of a known standard, such as ruby. Nitrogen-vacancy centers implemented in one of the diamond anvils also allow detecting magnetic or superconducting transitions [19]. As for temperature, it is assessed using black body radiation measurements, taking advantage of diamond's broad transparency. Generally, for measurements aiming to characterize the *p*-*V*-*T* equation of state of a material, it is crucial to create highly hydrostatic conditions, for which gases, particularly helium or neon, are the preferred pressure-transmitting media. The 300-K EOS data presented in this review were obtained mainly using Neon as the pressure-transmitting medium.

2/ The multi-anvil (MA) press (a brief overview of the principle of operation, including figures, can be found in [8]) is operated using a dual-stage anvil configuration comprising primary and secondary anvils. Uniaxial compression of the press piston drives the six outer steel primary anvils, which in turn compress eight inner secondary anvils (typically composed of WC, c-BN, or sintered diamond) arranged within a cubic volume. The truncated internal corners of these secondary anvils define an octahedral high-pressure assembly cavity, into which a Cr-doped MgO octahedron containing the sample assembly (sample surrounded by a capsule/gasket and a furnace, etc.) is inserted. The magnitude of the generated pressure is inversely correlated with the size of these truncations, enabling typical operating conditions up to 25 GPa and temperatures up to 2500 K. These multi-anvil devices achieve simultaneous high temperature and pressure using internal resistive heating (with materials like graphite, LaCrO₃ chromite, metallic foil, etc.) placed inside the octahedron. Temperature is measured using thermocouples or an electrical power-temperature calibration curve. In *ex situ* experiment, pressure is estimated using pre-established calibration curves that relate sample pressure to applied hydraulic oil pressure. The substantial weight and size of these presses prevent easy relocation to conventional synchrotron diffraction stations. Therefore, their use is generally confined to either fixed laboratory setups for *ex situ* high *p*-*T* synthesis or dedicated high-pressure/high-temperature beamlines at synchrotron facilities (for instance, ID06 at ESRF). The silicon thermodynamic data presented in this review were generally obtained with such a press on ID06 (ESRF synchrotron) and PSICHE (SOLEIL synchrotron) beamlines.

3/ The Paris-Edinburgh (PE) press, with its relatively compact size (around 25x25x25 cm³) and moderate weight (approximately 50 kg in its standard design), presents a different approach compared to larger multi-anvil devices. Originally developed for high-pressure neutron diffraction [20], it has since been adapted for various *in situ* high *p*-*T* measurements, including neutron [21] and x-ray diffractions [22]. Its "two opposed anvils" geometry, featuring a specialized conical high-pressure cavity for the gasket and sample assembly, also makes it a popular choice for *ex situ* synthesis in many academic laboratories. The gasket (usually in amorphous boron epoxy) is defined by its external diameter (ED) and the diameter of its central hole (FS), which corresponds to the flat surface of the high-pressure cavity where the sample assembly is placed. Different ED/FS ratios exist; for example, 12/5, 10/3.5, 7/2.4, and 5/1.5 geometries correspond to maximum achievable pressures of 6, 8, 10, and 17 GPa, respectively, when used with sintered diamond anvils. Similar to multi-anvil presses, the Paris-Edinburgh press achieves heating through Joule heating (or resistive
heating), allowing temperatures to reach values as high as 2500 K. Sample temperature is measured using a coaxial thermocouple, typically without accounting for pressure effect on its electromotive force. If the thermocouple fails, temperature can also be reliably estimated using a pre-determined power-temperature relationship, which is largely influenced by the gasket geometry. For off-line experiments, pressure is estimated using pre-existing calibration curves that correlate sample pressure with the controlled primary (oil) pressure applied to the piston. Some binary boron compounds data presented in this review were generally obtained with such a press on ID27 (ESRF synchrotron) and PSICHE (SOLEIL synchrotron) beamlines.

## 2.2 In situ x-ray diffraction

In this part we will describe the *in situ* characterization techniques, and some specificities for light element study (e.g. low-scattering contrast). The three devices described in the previous section can be combined with conventional x-ray diffraction techniques, either for angular dispersion or energy dispersion (the latter for the Paris-Edinburgh and multi-anvil presses). In the following, a concise overview of these two x-ray diffraction geometries and their merits in the acquisition of high-quality thermodynamic data will be provided. Finally, a recent third hybrid mode associated with x-ray diffraction is presented: the Combined Angle and Energy-dispersive Structural Analysis and Refinement (CAESAR) method.

### 2.2.1. Angular dispersive x-ray diffraction

Angle-dispersive powder x-ray diffraction (ADX) with a monochromatic beam and large area detectors (for example EIGER2 9M at ID27 of ESRF) is the standard XRD technique for high-pressure structural studies due to its large detection area, high resolution, and wide dynamic range. In the DAC experiments typical of this review, the small sample size (around few microns), diamond absorption, and limited DAC aperture necessitate a focused high-energy x-ray beam (synchrotron). This allows for accurate diffraction intensities suitable for the Rietveld refinements. However, a high inelastic background from the diamond anvils (Compton scattering) can degrade data quality, especially for low Z materials, hindering quantitative structural analysis. This issue has been recently addressed using a Soller slits system optimized for DAC geometry, featuring concentric micro-slits (50 and 150 microns) that cover a large solid angle and significantly improve the signal-to-noise ratio [23].

Angle-dispersive x-ray diffraction (ADX) with multi-anvil devices, as implemented at the ID06 beamline of ESRF, presents different challenges compared to DAC experiments. The limited aperture of multi-anvil presses prevents the use of Soller slits for collimating the diffraction signal from each element in the beam path. Therefore, cylindrical amorphous SiBCN x-ray windows and wide amorphous boron epoxy rectangles are inserted into the octahedron and gaskets, respectively, along the beam path. While a monochromatic beam significantly reduces the x-ray flux, these non-crystalline and low-absorbing elements help preserving
high diffraction data quality (see below our experimental data on silicon). However, the
Rietveld refinement under these conditions is typically challenging.

Finally, the compactness and exceptional angular opening (140° for a standard VX3) of the Paris-Edinburgh press have enabled its adaptation for ADX mode with an original Soller slits system [24]. At the ID27 beamline of the European Synchrotron Radiation Facility (ESRF), the diffracted beam can be readily collimated by a translation-mounted Soller slit system, which aligns the central slit with the incident beam. This multichannel collimator consists of two concentric sets of 75 fine slits (WC blades) at 0.8° intervals, covering a total angle of 60°. The slits, positioned at 50 mm and 200 mm from the sample, have slit widths of 0.05 mm and 0.20 mm, respectively, to minimise the detector's viewed volume and maximise signal. The slits are precisely aligned on an Invar base plate, ensuring dimensional stability despite temperature fluctuations. By oscillating the slits at a constant speed during data acquisition, background scattering is effectively eliminated, ensuring the collection of high-quality, homogeneous angle-dispersive data solely from the sample. Using large-area 2D detectors. Consequently, the association of the Paris-Edinburgh press with a multichannel collimator facilitates in situ ADX and the acquisition of comprehensive crystallographic data at extreme pressures and temperatures (see below our data on binary boron compounds).

2.2.2. Energy dispersive x-ray diffraction

Unlike angular diffraction, which varies the diffraction angle, energy-dispersive diffraction keeps the angle constant and varies the photon energy. Energy-dispersive experiments, therefore, utilise the white beam produced by a bending magnet or wiggler at a synchrotron source. The white beam is diffracted by the sample, and the energy of the diffracted photons is analysed and counted by a solid-state detector (germanium detector). When combined with large volume presses such as PE and multi-anvil (at PSICHE beamline of SOLEIL Figure 3a,b and MAX80 at Hasylab [6], Figure 2b), this x-ray configuration provides several advantages, including high flux at the sample, rapid acquisition of the entire energy spectrum, and effective limitation of the diffracting volume, with the double slits acting as Soller slits to minimize parasitic scattering from the sample environment. However, it also has limitations: the detector sees only a limited portion of reciprocal space, making the technique sensitive to preferred orientation and poor grain statistics, and obtaining reliable peak intensities for the Rietveld refinement is very challenging.

2.2.3. Combined Angle and Energy-dispersive Structural Analysis and Refinement (CAESAR)
method

In order to combine the advantages of both energy- and angle-dispersive diffraction, the CAESAR technique was recently developed for HPHT experiments at the PSICHE beamline of the SOLEIL synchrotron [25, 26]. This system employs a germanium detector mounted on a two-θ-axis arm. The arm is scanned in theta, and at each angular position, an energy-dispersive spectrum is recorded. This results in a three-dimensional spectrum, with axes
representing theta, energy, and intensity. This facilitates data analysis in multiple modes (Figure 3): at a fixed energy (equivalent to angle-dispersive diffraction), at a fixed angle (equivalent to energy-dispersive diffraction), or as a function of d-spacing, which incorporates the information from all energies and angles. The CAESAR setup is particularly advantageous for use with large volume multi-anvil and PE presses and yields spectra of sufficient quality for the Rietveld refinement. Some of the thermodynamic data presented in this review were acquired with this mode.

<figure><img src="image_4.png" /><figcaption>**Figure 3** In situ data (a) obtained at HPHT with Combined Angle and Energy-dispersive Structural Analysis and Refinement (CAESAR) method (b) at beamline PSYCHE of SOLEIL synchrotron. 1-D detector at ID06-LVP (c) and typical in situ XRD sequence (d).</figcaption></figure>

## 2.3 Density functional theory calculations

DFT calculations enable the prediction of stable atomic structures, phase transitions and *ab initio* convex hulls by the computation of enthalpy and vibrational entropy in the quasi-harmonic approximation [27]. To this end, the total energy E(P), volume V (P), phonon frequencies ω(P), phonon DOS g(ω(P)), zero-point energy E<sub>ZP</sub>(P), and vibrational entropy S<sub>ivb</sub>(P) have been computed with DFT and density functional perturbation theory (DFPT) for various pressures. As the volume expansion is however not negligible at moderate to high temperatures, the quasi-harmonic approximation is used to obtain the temperature change of the lattice parameters by minimizing the free enthalpy E+PV-TS<sub>vib</sub> [28]. Both the local density approximation (LDA) and the generalized gradient approximation (GGA) in the PW91 or in the PBE form have been used for the exchange and correlation potential. Typically, absolute energy changes are of the order of 31 meV/at. between DFT-LDA and DFT-GGA-
PW91 for B₁₃C₂, and 27 meV/at. for B₄C. Such differences cancel out to a large extent in the energy difference which defines the formation enthalpy [27]. Lattice parameters (resp. phonons frequencies) are generally more accurately described with respect to experiment in DFT-GGA (resp. DFT LDA). The pressure at which the formation enthalpy goes from negative to positive is lowered by as much as 16 GPa in B₄C between DFT-LDA and DFT-GGA-PW91.

DFT has been used to provide theoretical pressure and stoichiometry ranges in which some of the computed phases are found to be thermodynamically stable for the B-C [[27] in DFT-GGA-PW91, [29-31] in DFT-GGA-PBE], B-Si [[32, 33], both DFT-GGA-PBE] and for B-C-N compounds [[34] in DFT-PBE]. Stoichiometry changes and insertion of carbon and/or nitrogen atoms are guided by the fulfilment of Wade's rule for the electron counting, and by charge compensation [27, 35, 36].

DFT calculations also help characterizing samples synthesized or transformed at HPHT by providing spectroscopy fingerprints allowing to understand the atomic structure and some point defects observed in XRD [[37], DFT-GGA-PW91 and DFT-LDA], Raman [[38, 39] in DFT-LDA] or NMR [[40] in DFT-PBE] experiments. The atomic structures as determined by DFT are also of importance in the choice of the CALPHAD structural model for the solid solution under pressure, as described in the next section.

Finally, depending on the size of the unit-cell, such DFT investigations require intensive *ab initio* calculations and a high performance computing (HPC) environment [41].

## 2.4 Thermodynamic modeling (CALPHAD)

Principles of the CALPHAD method for boron-based systems are quite general, and use a number of standard integrated models. Most relevant to our case are compound energy formalism (CEF) [42] using sublattices and interaction parameters between end-members, the use of associates in sublattices, and may be extended to ionic models for liquid and solid phases. Often for HPHT simulations are available only ideal stoichiometric solids, subregular liquid solutions, equations of state of Murnaghan [43] and build-in Lu-Selleby-Sundman model [44], or simply noted as Lu's EOS in this review. High pressure was included via the equations of state (EOS) of individual solids, i.e. crystallographic density principally based on *in situ* experimental data. Input parameters (either experimental or *ab initio* thermodynamic data in some cases) will be described in the chapters devoted to various systems. Here we will underline the procedure by which EOS can be extracted from limited HPHT data and how it can be integrated into the Thermo-Calc database. The validity of the Murnaghan EOS to 20 GPa, i.e. in the range of our interest, is generally accepted [43]. Lu's EOS [44] allows getting some improvements and higher flexibility (e.g. in the case of Si with phases with close atomic volumes and distinct compressibility), however it is also limited in pressure and requires some adjustments to obtain the physical behavior of second derivatives (such as the heat capacity) at high pressure [45]. All gaseous phases were typically suspended in order not only to simplify the diagrams, but also to easily compare them to HP counterparts. Phase diagrams at ambient pressure are not discussed in this review.

### 2.4.1 Gibbs energy at HPHT by p-V-T equation of state

The Gibbs energy at HPHT conditions can be estimated using *p-V-T* EOS according to equation
$$G(p, T) = G^{\circ}(T) + \int_{0}^{p} V(p', T) dp' \qquad (1)$$

The Anderson-Grüneisen equation [46] takes into account the pressure dependence of the thermal expansion through the volume change, i.e.

$$\alpha(p, T) = \alpha(0, T) \left[ \frac{V(p, T)}{V(0, T)} \right]^{\delta_T}, \qquad (2)$$

and can be integrated (under the assumption that $\delta_T$ is constant) to the Kurakevych-Solozhenko equation [47]

$$V(p, T) = [V(0, T)^{-\delta_T} + V(p, 300)^{-\delta_T} - V(0, 300)^{-\delta_T}]^{-1/\delta_T}, \qquad (3)$$

where the thermal expansion (i.e. $V(0,T)$ at 0.1 MPa) and isothermal compression (i.e. $V(p,300)$ at 300 K) can be presented in any analytical form, typically polynomial

$$V(0,T) = V(0,300) [1 + \alpha (T-300) + b (T-300)^2] \qquad (4)$$

and the Murnaghan [43] (or, in principle, any other) equation of state

$$V(p, 300) = V(0, 300) (1 + B'_{0} p / B_{0})^{-1/B_{0}}. \qquad (5)$$

The parameters describing the p-V-T EOS of phase are, therefore, $\rho_0$ and related $V_0 \equiv V(1$ MPa, 300 K) $\equiv M / \rho_0$, $B_0$, $B_0'$, $a$, $b$ and $\delta_T$ (usually ~5-6 [48]). Such a form of the EOS is useful to approximate $V(p,T)$ in the p-T domain of its formation and stability, which is often the principal domain of interest in terms of the HPHT thermodynamics). Our evaluation of accuracy of $\delta_T$ is of ~0.5, so $\delta_T = 5.5$ was chosen as a value for phases with limited or non-reliable data. This seems to be a reasonable estimate since for $B_6O$ this value is ~6 obtained by fitting p-V-T data [49]. The values selected in this review were principally estimated in situ using XRD (the 300-K EOS can be measured on decompression and recovery).

In some other cases, the knowledge of phase equilibrium curves may allow evaluating the HPHT EOS of the unknown phase through the known one(s). For example, we succeeded to fit the experimental melting curves (p-T coordinates) of $\alpha$- and $\beta$-B$_2$O$_3$ and to find out the parameters determining the p-V-T EOS for liquid B$_2$O$_3$ [50] or Mg [51]. The melting was observed experimentally, while the bulk modulus and thermal expansion parameters were adjusted to fit the theoretical curve to the experimental points. This approach is limited, however, to the case of the absence of liquid polymorphism.

### 2.4.2 EOS within Thermo-Calc using the CALPHAD method

The CALPHAD p-V-T blocks of databases to calculate high pressure equilibria was developed in the past for most geologically important systems and a number of procedures to introduce this data into calculation of Gibbs free energy was developed.

For HP materials science, diamond and boron-rich systems are of particular interest, and our major goal consists in the explanation, with reasonable precision, of the phase equilibria with quite complicated chemistry, rather than through extrapolation to very high pressures. One important particularity is the narrowness of the p-T range of the existence of some compounds, and thus the number of points in the EOS is very limited. Only EOS with relatively low and physically meaningful parameters are acceptable. One of those is the Kurakevych-Solozhenko p-V-T EOS easily adaptable for solids and liquids, containing only basic parameters that can be obtained experimentally, empirically and through ab initio
calculations. This EOS has the form that can be easily used for fitting the p-T-V data with reduced number of reliable parameters with direct physical sense (e.g. B₀, thermal expansion coefficient a; and especially δₜ and B₀' with “typical” values of 5.5 and 4 for unknown phases), even if the data is limited to four reliable experimental points. Compressibility of liquid can also be derived from the experimental melting curve.

In Thermo-Calc, Lu-Selleby-Sundman EOS was chosen [44], one can determine parameters of EOS using the mentioned above Anderson-Grüneisen parameter δₜ as:

$$V0= V_0=M / \rho_0 \text{ (linear extrapolation to 0 K)} \qquad (6)$$

$$VA = a \times T + b \times T^2 \qquad (7)$$

$$VC= V_0 / B'_0 \qquad (8)$$

$$VK= k_0+k_1 \times T+k_2 \times T^2 \qquad (9)$$

where

$$k_0 = 1/B_0; k_1 = a \times k_0 \times \delta_T; k_2 = b \times k_0 \times \delta_T \qquad (10a-c)$$

Using this equation, the pressure change of Gibbs energy is calculated automatically. An alternative use of the Murnaghan equation [43] with the temperature dependence of the compressibility is possible and provides a reasonable approximation to 15-20 GPa, i.e. in the range of interest. In this case the parameters of the Kurakevych-Solozhenko equation [49] are linked via the expression of k(T) at ambient pressure, i.e. k₀+k₁×T+k₂×T².
### 3. Boron and boron-rich compounds at HPHT conditions

#### 3.1 HPHT phase diagrams of simple substances

In this section we will present a summary of observed phase transformations in simple elements with a particular focus on boron and silicon, which have been the subject of the most experimental studies as compared to other simple substances and compounds included in this review. We will then compare the literature data with results of the CALPHAD models and thus evaluate the phase stability as function of pressure and temperature. In the case of boron, principally *ex situ* data is available, while in the case of silicon, the analyzed data consist of *in situ* x-ray diffraction.

##### 3.1.1 Boron

At least five generally recognized icosahedral allotropes of boron can be recovered from high pressure experiments at ambient conditions. They are trigonal α-B₁₂ (or α-B, also may be denoted as α-rh B), trigonal β-B₁₀₆+x (β-rh B or simply β-B in this review), orthorhombic γ-B₂₈ (or γ-B [52, 53]), tetragonal t-B₅₂-x (δ-B, [54], or also known as β-tetragonal or II-t in old papers, while mentioned as α-t in some recent papers [55]), and trigonal ε-B₁₅ (ε-B, [54]). For β-B and δ-B the number of atoms per cell may change as a function of *p*-T conditions. δ-B is presumed to be metastable at any *p*-T condition by some authors [55], stabilized by impurities [56] or to be stable at high temperature and high pressure by others [57]. At present time all mentioned icosahedral allotropes can be obtained in the form of single crystals by high pressure, high temperature synthesis. The α-Ga non-icosahedral structure of B becomes stable at high pressures, it has been predicted [52] and observed *in situ* [58], however it cannot be recovered at ambient conditions.

High pressure synthesis of single crystals of α-boron is feasible when employing a Pt/B or Cu/B contact [59] (Figure 4b). The HPHT crystallization of α-B can be achieved in the presence of carbon [60]. α-B has a simple unit cell of rhombohedral symmetry with twelve boron atoms organized as icosahedra with the so-called 3-center electron-deficient bonds where the electron charge is accumulated at the center of a triangle formed by three adjacent atoms.

γ-B₂₈ is a high-pressure phase of boron predicted by *ab initio* calculations and confirmed by experiments. Its structure consists of icosahedral B₁₂ clusters and B₂ dumbbells in a NaCl-type arrangement (B₂)⁶⁺(B₁₂)⁶⁻ and displays a significant charge transfer δ~0.5-0.6. This is a textbook example of recovery at ambient condition of element form with exotic chemical bondings, typical for high pressures and clearly illustrated by LO-TO phonon splitting, confirming partially “ionic” interactions similar to III-V semiconductors (or significant charge transfer) between various structural groups of the same element [52, 53]. The discovery of this phase proved essential for the understanding and construction of the phase diagram of boron [52]. This phase has the highest bulk modulus and hardness of boron-rich solids (see section 6) [61]. Also, it is a transparent phase, which is not typical for boron-rich solids. At the same time, *in situ* studies of synthesis and phase transformations of this phase has not been reported so far.
Thermodynamic stability and fundamental state of boron were previously discussed quite a lot, and the prevailing consensus that β-B is thermodynamically stable (also called as a fundamental state) at ambient conditions has been achieved through the use of low-temperature calorimetry on high-purity samples available, and measurements of the α-to-β transformation enthalpy [62]. Direct irreversible transformation of α-to-β boron has been observed in situ by XRD at quite high temperature [50]. From the theoretical point of view, the main controversy arises from the fact that allotrope β-B represents a substantial challenge in terms of understanding its crystal structure [63] and stability.

<figure><img src="image_5.png" /><figcaption>**Figure 4** (a) In situ XRD patterns collected at 5 GPa and temperatures up to boron melting [57]. α-B transforms into β-B above 2000 K and melts above 2400 K. (b) Phase diagram of boron based on in situ melting and ex situ phase transformation data. Stability domain of α-B from Thermo-Calc simulations (c) vs ex situ single-crystal growth (d) and the latest ambient-pressure calorimetry e. Triple points α-B/β-B/γ-B are shown as circles, and is expected to be situated at ~1700(200) K and 8(1) GPa. Experimental points from ref. [64].</figcaption></figure>
A recrystallization - in the sense that the Raman spectrum shows sharper peaks- has been observed in β boron treated with HPHT, typically 5 GPa [60, 64, 65]. It was also observed that the stability of the crystal structure can be altered by changes in temperature or pressure, which in turn affects the occupancy of the crystal sites [55]. α-B becomes stable at high pressure, but only at low and moderate temperatures, since its heat capacity increase stronger than that of β-B. Experimental pressure-temperature phase diagram of boron has been also studied in the presence of Pt and Cu metals, and the single crystal growth data [64] is in reasonable agreement with theoretical diagrams of pristine boron on the values of boundaries and triple point between α/β/γ phases [57].

**Table 1** presents the full data set (experimental data and estimates) available for boron allotropes. The experimental melting curve allowed us obtaining an estimate of the bulk modulus of liquid boron, a value to be confirmed from future *in situ* studies of the liquid boron structure.

<table><caption>*Table 1 Parameters of the p-V-T equation of state data for boron-rich solids.*</caption><thead><tr><th></th><th>ρ<sub>0</sub></th><th>B<sub>0</sub></th><th>B'<sub>0</sub></th><th>a</th><th>b</th><th>δ<sub>T</sub></th><th>Refs.</th></tr></thead><tbody><tr><td>α-B<sub>12</sub></td><td>2.447</td><td>224</td><td>3.0</td><td>18.3</td><td>0</td><td>5.5*</td><td>[66, 67]</td></tr><tr><td>β-B</td><td>2.365</td><td>210</td><td>2.23</td><td>19.2</td><td>0</td><td>5.5*</td><td>[68-70]</td></tr><tr><td>γ-B<sub>28</sub></td><td>2.544</td><td>237</td><td>2.7</td><td>6</td><td>0</td><td>5.5*</td><td>[71]</td></tr><tr><td>t-B<sub>52</sub></td><td>2.493</td><td>232</td><td>2.7</td><td>6</td><td>0</td><td>5.5*</td><td>[47]</td></tr><tr><td>L-B</td><td></td><td>210*</td><td>2.2*</td><td>19.2*</td><td>0</td><td>5.5*</td><td></td></tr><tr><td>B<sub>6</sub>O</td><td>2.601</td><td>180</td><td>6</td><td>14</td><td>5</td><td>6</td><td>[49, 72]</td></tr><tr><td>B<sub>13</sub>N<sub>2</sub></td><td>2.666</td><td>200</td><td>4</td><td>21.3</td><td>0</td><td>1.5</td><td>[73-75]</td></tr><tr><td>HP-B<sub>50</sub>N<sub>2</sub></td><td>2.408</td><td>167</td><td>4</td><td>15.7</td><td>0</td><td>5.5*</td><td>[74, 75]</td></tr><tr><td>B<sub>4</sub>C</td><td></td><td>221</td><td>3.3</td><td>17.2</td><td>0</td><td>5.5*</td><td>[76]</td></tr></tbody></table>

Units: ρ₀ in g/cm³; B₀ in GPa; B'₀ is dimensionless; a in 10⁶ K⁻¹; b in 10⁹ K⁻²; δT is dimensionless.

Experimental phase diagrams have been proposed by different groups [50, 64], at the same time the crystallization of metastable phases cannot be precluded, as well as the crystal growth conditions cannot be unambiguously attributed to the highest pressure and
temperature values achieved in recovery experiments. The best estimate of up-to-date experimental phase equilibria can be presented as a self-consistent set of equations that agree with available thermodynamic data (Table 2):

$$T_{L/β} = 2360 + 15.5 p \qquad (6)$$

$$T_{V/β} = -4534 + 707.8 p \qquad (7)$$

$$T_{t/V} = 2176 + 5.1 p \qquad (8)$$

$$T_{t/β} = 3756 - 160.2 p \qquad (9)$$

In general case, one can consider such simplified equations as a form of thermodynamic equation

$$T_{eq} = \Delta H/\Delta S + p \Delta V/\Delta S \qquad (10)$$

with constant ΔH, ΔS and ΔV.

<table><caption>Table 2 Change of thermodynamic functions in phase transformations in boron at HPHT.</caption><thead><tr><th>Phase transition Y/X</th><th>ΔH<sub>Y/X</sub> (kJ/mol), ΔS<sub>Y/X</sub> (J/mol K) or ΔV<sub>Y/X</sub> (cm<sup>3</sup>/mol)</th><th>In situ / ex situ data</th><th>Reference data</th></tr></thead><tbody><tr><td>α/β</td><td>ΔH = 0.3; ΔS = -1.7 +0.02*LN(T)</td><td>Present analysis</td><td>ΔV<sub>300K, 1MPa</sub> =-0.15</td></tr><tr><td>L/β</td><td>ΔH = 50; ΔV = 0.32; ΔS = 21</td><td>Experimental [77-84]</td><td>ΔH<sub>melt</sub> = 50.2 -48.93; ΔV = 0.22 to 0.46</td></tr><tr><td>γ/β</td><td>ΔH = 2.5; ΔV = -0.35; ΔS = -0.55</td><td>Ab initio &amp; experimental [52, 53, 80]</td><td>ΔH<sub>0K</sub> = 2.5; ΔV<sub>300K, 1MPa</sub> = -0.32</td></tr><tr><td>t/β</td><td>ΔH = 8.8; ΔV = -0.38; ΔS = 2.3</td><td>Experimental &amp; fitting [80, 85]</td><td>ΔV<sub>300K, 1MPa</sub> =-0.30</td></tr><tr><td>t/γ</td><td>ΔH = 6.3; ΔV = 0.01; ΔS = 2.9</td><td>Experimental &amp; fitting [52, 53, 85]</td><td>ΔV<sub>300K, 1MPa</sub> = 0.02</td></tr><tr><td>L/t</td><td>ΔH = 41.2; ΔV = 0.7; ΔS = 19</td><td>Thermodynamic consistency (fitting) [57]</td><td></td></tr></tbody></table>

* “L” stands for liquid.

Ab initio phase diagrams of boron are interesting to compare with described above experimental observations. The recent simulations, taking into account different possibilities of interstitial site occupancies, indicate the instability, at high temperature, of any of the two known tetragonal phase, either (α-)t B<sub>192</sub> or (β-)t B<sub>52</sub>, at high temperature. The actual
extreme-condition challenges are (i) the *in situ* melting of gamma boron and (ii) the *in situ* observation, nature and melting of the HPHT tetragonal phase, whether it exists and whether it is stable. It should be noted here that t-B₁₉₂ phase, known only by chemical route synthesis, was also once identified in HPHT experiments above 10 GPa after heating of initial β-B₂ to 2200 K in diamond anvil cell [86]; however, this finding was not replicated in subsequent studies by other groups. In fact, both known tetragonal phases were stable for a long time believed to be stabilized by impurities [56]. Some *ab initio* simulations suggest its HPHT thermodynamic stability [52] and others not [55]. Various t-B₅₂ species of pure boron were observed experimentally at such HPHT conditions [57, 85, 87].

### 3.1.2 Silicon & carbon

The Thermo-Calc phase diagram of Si constructed using available data on HPHT transformations is presented at Figure 5. *In situ* data about melting curve have been used to fit the solid/liquid transition. Liquid polymorphism, reported above 15 GPa [88] was neglected in these simulations. The EOS parameters satisfactorily describe the available 300K *in situ* data on crystallographic density (Table 3). Figure 5 shows the best fit using Lu's EOS [44]. Although this model has some limitations at very high pressures, it is reliable at the domain of our interest up to 20 GPa and below 3000 K.

Carbon has a controversy phase diagram above 3000 K [89-91], however, this is out of the *p*-T domain of our interest. Here we used conventional version with carbon melting in the 4500-5000 K interval up to 30 GPa (Figure 6) that show reasonable agreement with experimental graphite-to-diamond transition.

<figure><img src="image_6.png" /><figcaption>**Figure 5** The phase diagram of Si simulated by Thermo-Calc in comparison with *in situ* XRD data [92]. HPHT contribution to the Gibbs energy was estimated using Lu's EOS [44].</figcaption></figure>
<table><caption>Table 3 Parameters of p-V-T equation of state data for Si and C.</caption><thead><tr><th></th><th>ρ<sub>0</sub></th><th>B<sub>0</sub></th><th>B'<sub>0</sub></th><th>a</th><th>b</th><th>δ<sub>T</sub></th><th>Refs.</th></tr></thead><tbody><tr><td>Si-I (A4)</td><td>2.33</td><td>101</td><td>3.4</td><td>2.6</td><td>0*</td><td>5.5*</td><td>[93]</td></tr><tr><td>Si-II (A5)</td><td>2.58*</td><td>80-150</td><td>4*</td><td>12**</td><td>0*</td><td>5.5*</td><td></td></tr><tr><td>Si-XI</td><td>2.30*</td><td>45</td><td>4.0</td><td>12**</td><td>0*</td><td>5.5*</td><td>[93]</td></tr><tr><td>Si-V (A<sub>f</sub>)</td><td>2.40*</td><td>95</td><td>4.6</td><td>12**</td><td>0*</td><td>5.5*</td><td>[93]</td></tr><tr><td>L-Si</td><td>2.72***</td><td>25</td><td>4</td><td>60</td><td>20</td><td>5.5*</td><td></td></tr><tr><td>C graphite</td><td>2.245</td><td>33.8</td><td>8.9</td><td>30</td><td>0*</td><td>5.5*</td><td>[94, 95]</td></tr><tr><td>C diamond</td><td>3.516</td><td>445</td><td>4.2</td><td>0.7</td><td>0*</td><td>5.5*</td><td>[96]</td></tr><tr><td>L-C</td><td>1.573***</td><td>6.25</td><td>2</td><td>23.2</td><td>5.7</td><td>5.5*</td><td>[91]</td></tr><tr><td>SiC</td><td>3.220</td><td>218</td><td>4.19</td><td>6.2 (3.1)</td><td>0* (2.64)</td><td>5.5*</td><td>[97, 98]</td></tr></tbody></table>

Units: ρ₀ in g/cm³; B₀ in GPa; B'₀ is dimensionless; a in 10⁶ K⁻¹; b in 10⁹ K⁻²; δT is dimensionless.

* Fixed to common values. **Here we have used the same ratio as for A5/A4 (metal/semiconductor) thermal expansions of tin allotropes, ~4.8 [99]. *** Values extrapolated to OK.

<figure><img src="image_7.png" /></figure>
**Figure 6** The phase diagram of carbon by Thermo-Calc software optimized by CALPHAD assessed database [91].
## 3.2 Binary boron-rich compounds at HPHT

In this section, we will present a summary of phase transformations observed in binary compounds. We will then compare the literature data with the results of CALPHAD simulations and evaluate phase stability as a function of pressure and temperature. The *p*-T phase diagram are often more complicated as compared to simple elements and may include the domains of phase decomposition, typically observed in *in situ* x-ray diffraction or electrical probing experiments (a method described in ref. [100]).

### 3.2.1 Boron-rich compounds of C, N and O: crystal structures and melting

Most allotropes of boron, except the gamma phase, give rise to a family of compounds with similar structures, sometimes called as related to α-rh B or simply α- B₁₂ (B₆O, B₁₃N₂ and B₄C, Figure 7); β-rh B or simply β- B₁₀₆+ₓ (see ref. [62] and section 3.2.2), β -t or t-B₅₂ (B₅₀C₂, B₅₀N₂ [101]) and α-t B or t-B₁₉₂ (α-AlB₁₂ [102]). Crystal structures were studied in the past, and are characterized by a high degree of structural disorder for some of the allotropes (e.g. β-rh B [63]). At the same time, it was previously reported and seems to be correct that high pressure synthesis allows obtaining equilibrium structures with the highest order in the atomic distribution, both in pristine boron and in related compounds [73, 103].

<figure><img src="image_8.png" /><figcaption>**Figure 7** Lattice parameters from ICSD database and bibliography (a) and simplified crystal structure model (b) of α- B type compounds and solid solutions. Oxygen can be replaced with Nitrogen. Carbon occupancy may be zero, in the case of suboxide and subnitride (N replaces O). In the case of X = Y = B, the structure becomes ε-B₁₅.</figcaption></figure>

During past years, a number of reports has appeared on the high-pressure synthesis of the tetragonal allotrope of boron, the t-B₅₂ (δ-B) phase. Although detailed characterization of the crystal structure has been performed in some simple cases, there remains remarkable
variation of the a/c lattice-parameter ratio that raises strong doubts about its uniqueness. The Rietveld refinement (Figure 8b) or single crystal structure solution remain unique. For example, the crystal structure of the high pressure – high temperature boron phase namely “pc-B$_{52}$” synthesized by direct solid-state transformation of rhombohedral β-B$_{106}$ at 20 GPa and 2500 K belongs to the t-B$_{52}$ type, and its structure can be considered as pseudo-cubic (pc) with a/c ratio of √2 [85].

At Figure 8, the solid circle (●) represents pc-B$_{52}$ phase synthesized by us, open circles (o) show the lattice parameters of phases obtained at ambient or low pressure (compounds B$_{50}$C$_{2}$ and B$_{50}$B$_{2}$) [104], triangles correspond to t-B$_{52}$ phases synthesized at high pressures from β-B$_{106}$ (∇) [104], and borane (Δ) [105]. Among these phases, only B$_{50}$N$_{2}$ and, to a high probability, t-B$_{52}$ participate in equilibria at HPHT conditions. Only these tetragonal phases have been integrated into CALPHAD simulations so far.

<figure><img src="image_9.png" /><figcaption>Figure 8 Lattice parameters (a) of various phases with structure related to tetragonal boron t-B$_{52}$ and best quality reported ex situ DRX (b). Insert: crystal structure of low-pressure B$_{50}$C$_{2}$ (ICSD 444).</figcaption></figure>

Melting (i.e. congruent one without decomposition of incongruent one with decomposition) of some boron-rich compounds has been studied experimentally, in situ by electrical measurements or/and XRD. The pressure dependence of melting curves of B$_{4}$C [106], B$_{13}$N$_{2}$ [107] and B$_{6}$O [108] have the following analytical expressions in the 2.7 -7.7 GPa pressure range:

$$B_4C (19.5 at\% B): T_m = 2720(50) - 13(6) \times p \qquad (11)$$

$$B_{13}N_{2}: T_{m} = 2430(20) + 31(3) \times p \qquad (12)$$

$$B_6O: T_m = 2350 + 60(10) \times p \qquad (13)$$

Only B$_{6}$O [108] melts congruently in the mentioned pressure range with positive slope; while melting temperature of B$_{4}$C decrease with pressure. B$_{13}$N$_{2}$ melts with decomposition (Figure 9).
<figure><img src="image_10.png" /><figcaption>Figure 9 The p-T section of the binary B-BN phase diagram: incongruent melting by decomposition of B₁₃N₂ (a) and of B₅₀N₂ (b). Insets show crystal structures of B₁₃N₂ (rhombohedral setting) and B₅₀N₂. (c,d) The p-T section of B-C system with compositions of carbon-rich boron carbides. Thick red dashed lines show equations (12) for B₁₃N₂ and (11) for B₄C.</figcaption></figure>

### 3.2.2 Boron-rich compounds of Si

All of the Si borides reported so far belongs to a family of boron-rich solids with unknown EOS parameters. Some tentative values of EOS are presented at Table 4, based on ab initio estimations [33, 109, 110]. Thermal stability of B₆Si has been studied at HPHT conditions using in situ electrical measurements [111], that indicated the corresponding temperatures of melt formation and decomposition (both for B:Si = 1:6):

$$B_6Si: T_m = 2150 - 31(2) \times p \qquad (14)$$

$$B_nSi (12 < n < 50, +Si): T_m = 2684 - 92(3) \times p \qquad (15)$$

The simulated p-T section of binary B-Si phase diagram is presented at Figure 10. The general tendency of negative melting slope is reproduced, as well as melting temperature range.

Reliable EOS data at HPHT conditions, once available, are expected to improve the agreement between simulated and experimental phase-diagram sections.
<figure><img src="image_11.png" /><figcaption>**Figure 10** The tentative p-T section of boron silicide $B_6Si$ by Thermo-Calc simulations. Thick red dashed line represent equation (14).</figcaption></figure>

<table><caption>**Table 4** Tentative parameters of p-V-T equation of state for boron silicides.</caption><thead><tr><th></th><th>ρ<sub>0</sub></th><th>B<sub>0</sub></th><th>B'<sub>0</sub></th><th>a</th><th>b</th><th>δ<sub>T</sub></th><th>Refs</th></tr></thead><tbody><tr><td>SiB<sub>n</sub></td><td></td><td>200*</td><td>2.2*</td><td>6*</td><td>0*</td><td>5.5</td><td>β-B</td></tr><tr><td>α-SiB<sub>6</sub></td><td></td><td>153</td><td>4*</td><td>4.3*</td><td>0*</td><td>5.5</td><td>R-3m [110]</td></tr><tr><td>β-SiB<sub>6</sub></td><td></td><td>147</td><td>4*</td><td>4.3*</td><td>0*</td><td>5.5</td><td>Cmce* [110]</td></tr><tr><td>γ-SiB<sub>6</sub></td><td></td><td>146</td><td>4*</td><td>4.3*</td><td>0*</td><td>5.5</td><td>Cubic [110]</td></tr><tr><td>α-SiB<sub>3-x</sub></td><td>2.478</td><td>100*</td><td>4*</td><td>2.6*</td><td>0*</td><td>5.5</td><td>R-3m [33]</td></tr><tr><td>β-SiB<sub>3</sub></td><td>2.455</td><td>100*</td><td>4*</td><td>2.6*</td><td>0*</td><td>5.5</td><td>Imma [33]</td></tr></tbody></table>

Units: ρ₀ in g/cm³; B₀ in GPa; B'₀ is dimensionless; a in 10⁶ K⁻¹; b in 10⁹ K⁻²; δT is dimensionless. * Orthorhombic structure predicted to be stable.

### 3.2.3 Boron nitride polymorphs

Known allotropes of boron nitride with parameters that allow computing the pressure term in the Gibbs energy are listed in Table 5 [112]. Their crystal structures are relatively simple and they are presented here merely for the completeness of our thermodynamic consideration of the B-BN system.

Phase diagram to quite high pressure (~25 GPa) contains only three phases, hBN, cBN and liquid [113]. Some thermodynamic data, obtained in situ, includes the p-V-T EOS [113] and
some phase equilibria [114]. Experimental phase diagram and typical *in situ* XRD sequence of cBN synthesis are shown at Figure 11.

<figure><img src="image_12.png" /><figcaption>**Figure 11** The phase diagram of cBN by Thermo-Calc compared to ref. [113] (Insert) (a) with *in situ* observation of cBN crystallization (b) from ref. [114] (Inserts show crystal structures of hBN and cBN, on the left and on the right, respectively).</figcaption></figure>

<table><caption>**Table 5** Parameters of the p-V-T equation of state for some boron nitride polymorphs.</caption><thead><tr><th></th><th>ρ<sub>0</sub></th><th>B<sub>0</sub></th><th>B'<sub>0</sub></th><th>a</th><th>b</th><th>δ<sub>T</sub></th><th>Refs</th></tr></thead><tbody><tr><td>cBN</td><td>3.489</td><td>390</td><td>3.35</td><td>1</td><td>5</td><td>3</td><td>[112]</td></tr><tr><td>wBN</td><td>3.524</td><td>375</td><td>2.3</td><td>1</td><td>5</td><td>5.5</td><td>[112]</td></tr><tr><td>hBN</td><td>2.279</td><td>36.7</td><td>5.6</td><td>41</td><td>0</td><td>2.5</td><td>[112]</td></tr><tr><td>tBN</td><td>2.212</td><td>17.2</td><td>11.4</td><td>41</td><td>0</td><td>2.5</td><td>[112]</td></tr><tr><td>L-BN</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Nano cBN</td><td>3.489</td><td>375</td><td>2.3</td><td>1</td><td>5</td><td>3</td><td>[115]</td></tr></tbody></table>

Units: ρ₀ in g/cm³; B₀ in GPa; B'₀ is dimensionless; a in 10⁶ K⁻¹; b in 10⁹ K⁻²; δₜ is dimensionless.

### 3.2.4 Boron oxide polymorphs
Boron oxide is not a boron-rich compound, but it participates into HPHT equilibria that are important for the synthesis of superhard boron suboxide. Boron oxide exists in two allotropes, namely α-B₂O₃ and the β-B₂O₃ high-pressure phase. Crystal structures were previously resolved [116] and can be easily identified by *in situ* XRD diffraction.

High pressure thermodynamics has been studied *in situ* in our previous work [50], including such an important point as reversibility of the phase transformation, to avoid the overestimation of equilibrium pressure which is often attributable to hysteresis. Boron oxide is a glass-forming material, and its crystalline forms (including the ambient-pressure form) can be obtained principally by HP techniques. One of possible explanation of such behavior, supported by *ab initio* simulations, consists in hidden polymorphism and competing nucleation in boron oxide [117, 118]. It is also important to note that at the pressure of 0.1 MPa, metastable melting of the HP β-B₂O₃ phase occurs at a higher temperature than that of α-B₂O₃. The set of thermodynamic parameters has been established (Table 6), so that the calculates phase equilibria correspond to the experimental data (Figure 12a). Satisfactory agreement between experimental and calculated triple point at ~5 GPa and 1500 K is achieved by CALPHAD simulations, however, the data base improvement is required to improve the agreement between experimental data and simulated equilibrium curves (Figure 12b).

<figure><img src="image_13.png" /><figcaption>Figure 12 (a) The phase diagram of B₂O₃ with experimental *in situ* data [50]; and (b) the CALPHAD-assessed *p*-T section of boron oxide B₂O₃ composition.</figcaption></figure>

<table><caption>Table 6 The parameters of the *p*-V-T equation of state for boron oxide phases.</caption><thead><tr><th></th><th>ρ<sub>0</sub></th><th>B<sub>0</sub></th><th>B<sub>0</sub>'</th><th>a</th><th>b</th><th>δ<sub>T</sub></th><th>Refs.</th></tr></thead><tbody><tr><td>α-B<sub>2</sub>O<sub>3</sub></td><td>2.554</td><td>40</td><td>5.5</td><td>30</td><td>80</td><td>6</td><td>[50]</td></tr><tr><td>β-B<sub>2</sub>O<sub>3</sub></td><td>3.11</td><td>170</td><td>5</td><td>15</td><td>11</td><td>5.5</td><td>[50, 72]</td></tr></tbody></table>
<table><tr><td>L-B<sub>2</sub>O<sub>3</sub></td><td>2.00 *</td><td>16</td><td>8</td><td>620</td><td>-400*</td><td>7.5</td><td>[50]</td></tr></table>

Units: ρ₀ in g/cm³; B₀ in GPa; B'₀ is dimensionless; a in 10⁶ K⁻¹; b in 10⁹ K⁻²; δT is dimensionless. * Extrapolated to 300 K.
# 4. HPHT thermodynamics of the binary systems with boron-rich and regular borides

## 4.1. Boron suboxide and the B-B₂O₃ system

Boron suboxide [119] is usually believed to be of the B₆O stoichiometry with typical oxygen deficiency when produced under conditions that do not involve high pressure [120]. High pressure allows getting highly stoichiometric species [121, 122], including single crystals. Reported crystal structures are typical for B₆O (Figure 7b) and similar to other chalcogenides. At higher pressure ~10 GPa, though, higher boron content in the unit cell is suggested, and the α-c lattice parameter evolution follows tendency similar to the case of boron carbide with high boron content (B-doped B₄C from the thermodynamic point of view, see section 4.3).

Theoretical phase diagrams at ambient and high pressure can be speculated by taking into account the stability of the orthorhombic β-B₆O polymorph, predicted by the evolutionary ab initio USPEX algorithm [123], which is produced by alternative arrangement of B₁₂ icosahedra. This phase has not been observed experimentally, to the best of our knowledge.

The B₆O phase and its synthesis have been extensively studied in situ [49, 108, 124] using energy-dispersive diffraction. The thermoelastic equation of state of B₆O has been reported based on numerous HPHT data obtained during the study of phase transformations.

The experimental melting curve of B₆O is shown at Figure 13a [108]. From low pressures, up to several GPa, B₆O decomposes into boron-rich and oxygen-rich fluids (e.g. detected B₂O₂ species in gaseous phase [120]), and fast quenching allows recovering the decomposition products. At the same time, the HP melting (if not overheated) is congruent and the recovery of well-crystallized B₆O ingots is possible. Low-pressure decomposition (i.e. oxygen loss [120]) cannot be efficiently simulated using the available CALPHAD assessed data (Figure 14a) with only stoichiometric B₆O, and ambient pressure data should be completed. Also, above 10 GPa the phase should be considered as B₁₃O₂, probably similar to B₁₃N₂, although the possibility of simplified version of boron carbide solutions are not precluded.

In situ data up to 5 GPa allowed to get general features of the B-B₂O₃ phase diagram [124], including the melting temperature at a given concentration, the corresponding phase diagram has been calculated using available thermodynamic data for simple substances and compounds at 0.1 MPa, p-V-T EOS and adjusted parameters for the subregular solution model that allows describing the interaction in the liquid phase.
<figure><img src="image_14.png" /><figcaption>**Figure 13** The experimental melting curve of B<sub>6</sub>O (a) and in situ energy-dispersive XRD patterns in the course of the B<sub>6</sub>O melting in the B - B<sub>2</sub>O<sub>3</sub> system [124] obtained at MAX80(b).</figcaption></figure>

The HPHT transformations and chemical reactions generally take place in the presence of the liquid phase, and under pressure the diffusion process is quite fast to reach the thermodynamic equilibrium. Formation of the liquid phase, in some cases, may be explained by metastable eutectic melting and is often useful to the phase diagram. It has not been observed in the B-B<sub>2</sub>O<sub>3</sub> system at 5 GPa.

The phase diagram of the B-B₂O₃ system has been calculated up to 20 GPa in recent works using the Thermo-Calc software [125]. Two diagrams at ambient pressure and at 5 GPa are shown at Figure 14.

<figure><img src="image_15.png" /><figcaption>**Figure 14** Simulated B-B<sub>2</sub>O<sub>3</sub> phase diagrams at 0.1 MPa (exceptionally for this review, with gaseous phase) (a) and 5 GPa (b) using Thermo-Calc software with database from ref. [125].</figcaption></figure>

## 4.2 Boron subnitrides and B-BN system
Long debates on the stoichiometry and crystal structure of boron subnitrides and on the possibility of their solid-state HPHT synthesis [73, 126, 127] led to general admittance of the $B_{13}N_2$ structural model (Figure 7b) which is different from both $B_6O$ and $B_4C$ structures, and can be considered as an ordered $B_{13}C_2$-like structural type. Theoretical calculations also predict thermodynamic stability of $B_{13}N_2$ on the B-N convex hull ($B_{12.67}N_2$ exact stoichiometry) [34] rather than the $B_6N$ structure. The thermal expansion of the boron subnitride data [74] combined with compressibility data [128] and two experimental points at HPHT [129] (Figure 15a) gives the dimensionless Anderson Grüneisen parameter $\delta_T$ to be close to 1.5 [130].

The chemical interaction and phase relations in the B-BN system have been studied in situ at 5 GPa and temperatures up to 2800 K using x-ray diffraction with synchrotron radiation [129]. Thus, the in situ thermodynamic data were obtained only for the “low” temperature part of phase diagram at ~2400 K. The equilibrium and metastable phase diagrams of the B-BN system were proposed of the system at 5 GPa (Figure 15b). The only thermodynamically stable boron subnitride, $B_{13}N_2$, melts incongruently at 2600 K and forms eutectic equilibrium with boron at 2300 K and 4 at. % of nitrogen.

Subsequent studies at high pressure indicated the existence of an additional thermodynamically stable compounds, $B_{50}N_2$ as individual phase [74], and the phase diagram of the B-BN system has been constructed with its stability domain up to 20 GPa (Figure 16). Ambient pressure phase diagram (Figure 16a), excluding gaseous phases, shows metastable equilibria and cannot act as a guide for efficient synthesis of this phase below ~2 GPa.

<figure><img src="image_16.png" /></figure>

**Figure 15** The typical in situ XRD patterns collected at the MAX80 beamline: observation of chemical reaction B + BN → $B_{13}N_2$ at 5 GPa (a) and original experimental phase diagram of the B-BN system at 5 GPa: metastable (blue dashed lines) and equilibrium (black solid lines) (b).
<figure><img src="image_17.png" /><figcaption>**Figure 16** The most recent phase diagrams of the B-BN system with thermodynamically stable B<sub>50</sub>N<sub>2</sub> and B<sub>13</sub>N<sub>2</sub> at ambient conditions, previously believed to be metastable. Simulated B-BN phase diagrams at 1 MPa (a) 5 GPa (b) and 10 GPa (c) using ThermoCalc software and related database from ref. [74].</figcaption></figure>

## 4.3 Boron carbides and the B-C system

Boron carbides adopt an atomic structure of so-called α-B<sub>12</sub> family (in fact, it is closer to exotic HPHT ε-B<sub>15</sub> boron allotrope) which adopts “on average” trigonal syngony and is experimentally known to be stable over a wide range of atomic carbon concentrations (at% C). Metastable graphite-like and diamond-like phases have been reported both theoretically and experimentally, and some phase relationships between them are formally similar to the graphite-to-diamond transformation.

### 4.3.1 Stable B-C phases with boron-related crystal structures

Icosahedral boron carbide can be produced by various methods, such as common hot pressing method [131], magnesiothermy, carbothermal reduction, pyrolysis [132], including high pressure synthesis from the elements [133, 134]. At low pressure, the formation of boron carbide from the elements typically occurs around 1333°C in hot pressing experiments by an instantaneous formation of boron rich boron carbide, with high solid-state diffusion of boron and carbon atoms, and finishes at about 1773 K [135]. At high pressure however, the formation temperature increases considerably, e.g. by more than 250 K at 2 GPa, compared to the low pressure synthesis temperature [136].

The B-C system contains only one thermodynamically stable compound, typically considered by experimentalists as a unique solid solution in which the carbon concentration varies from ~18.8 at% down to ~9 at% [137]. In this single-phase regime, the compound mainly consists of (B<sub>12</sub>) and (B<sub>11</sub>C) icosahedra and of triatomic C-B-C chains [138]. This picture holds at elevated temperature, typically beyond 1273 K, where phase diagrams have been measured [137, 139] and configurational disorder is expected to be important.

At low temperatures however, a number of different locally stable arrangements has been predicted by *ab initio* calculations [140], and some configurations having periodic order have proven to be stable theoretically [27]. Firstly, the B<sub>4</sub>C phase with 15 atoms per elementary unit-cell, one (B<sub>11</sub>C) icosahedron and one C-B-C chain (see Figure 7b) has the lowest formation enthalpy per atom [137]. Secondly, intericosahedral blocks can consist of four atoms, either quasi linear or forming a B<sub>4</sub> rhombus also called a <B<sub>B</sub><sup>B</sup>B> pantograph as in
**Figure 17a.** Then, two new phases containing (B₁₂) icosahedra and, instead of only C-B-C chains, one third of either pantographs (first defined in Ref. [141], see Figure 17a), or one third of B-C-B-C chains (Figure 17b) were found stable, so that it became possible to define the theoretical convex hull between 8.7 at% and 20 at% at vanishing temperature [27]. The phase with one third of B₂C₂ pantographs, was found quasi-stable [142] in DFT-LDA, competing with the OPO₂ phase with one third of B-C-B-C chains but less stable than the OP₂ phase in DFT-GGA-PW91 (figure S2 of ref. [126]).

<figure><img src="image_18.png" /></figure>

**Figure 17** B₄ pantographs (a) or four-atom chains (b) replacing one third of the C-B-C chains in boron-rich boron carbide allow to define B₁₀.₅C and B₆.₇C boron carbides, with respectively 8.7 at% and 13.0 at% carbon compositions. Icosahedra are (B₁₂), two thirds of the chains are C-B-C and one third of the chains is B₄ (a) or B-C-B-C (b), leading to the formulae (B₁₂)[C-B-C]₂/₃ [B₄]₁/₃ (OPO₁) or (B₁₂)[C-B-C]₂/₃ [B-C-B-C]₁/₃ (OPO₂) at vanishing temperatures. Red atoms are atoms at interstitial sites of the averaged rhombohedral 15 atom unit-cell of B₄C, leading to the concept of "ordered partial occupations" (OPO) and to the crystal names OPO₁ and OPO₂ that are stable with 414 atoms/unit cell [27]. (c) Lattice parameters (hexagonal setting) of B₄C and α-B obtained in the B-C system at HPHT (solid symbols) as compared to pressureless synthesis (pluses for pressureless B₄C, open circles for B₆O, open stars for B₁₃N₂ and open triangles for pressureless α-B).

While there is a long standing consensus that the crystal structure of B₄C near 20 at% is mainly (B₁₁C)C-B-C, with (B₁₁C) icosahedra interconnected with C-B-C chains [138] (Figure 7b), boron-rich crystal structures have been debated for a long time [27, 141, 143, 144]. In particular, the stoichiometric compounds B₁₃C₂ having the same structure as B₄C has been widely studied experimentally with the proposed structure (B₁₂)C-B-C [27, 103, 140, 141, 143-146] which, with an odd nominal number of electrons, is not compatible with the semiconducting property observed in boron-rich solids, see ref. [138] and references herein. Moreover, (B₁₂)C-B-C is found dynamically instable beyond 35 GPa in DFT-GGA (soft phonons, [27]). While the latter has been interpreted as an electronic charge transfer in the chains of B₁₃C₂ [103], the recently proposed theoretical model OPO₂, of formula (B12)[C-B-C]₂/₃ [B-C-B-C]₁/₃, where one third of the chains contains four atoms, has the advantages of respecting Wade's rules, being semiconducting and lying on the theoretical convex hull [27]
(Figure 17b). Moreover, the presence of one B-C-B-C chain and three (B₁₂) in a supercell of B₄C remarkably explains the behavior of the low frequency vibrational band observed when increasing pressure [38, 147], strong evidence that four atom chains do exist at least as defects in B₄C. All these issues can have some concerns with phenomenological thermodynamic simulations, like CALPHAD, for example the free electron terms in heat capacity (absent for dielectrics and for known boron carbides) or, especially, the number of atoms per unit cell, quite different in ambient-pressure and high-pressure equilibrium phases.

The average model with 15 atoms used for the crystal structure refinement (Figure 7b) seems to be suitable for samples at high carbon concentration and at HPHT, as in these conditions the structural disorder is typically less pronounced [103]. Similar models have been developed for thermodynamic simulations at ambient pressure, e.g. using the CALPHAD method [137]. At the same time, the predictions of such models should be taken with some precaution when dealing with HPHT conditions, as they rely on the assumption that the high pressure is generally believed to stabilize the regular atomic distribution with all crystallographic sites occupied (no vacancies noted as VA in the ambient-pressure CALPHAD assessed databases or as symbol ☐ in Figure 7b) [103]. We note however that the recently found OPO₁ and OPO₂ theoretical crystal structures have been studied under pressure and they are found to be on the theoretical convex hull up to ~20 GPa at vanishing temperature in DFT-GGA-PW91 [27].

Turning to the EOS and to the comparison of the compressibility of the volume of the icosahedra with respect to the volume of the unit-cell, the EOS studies of B₄C at room temperature with NaCl as a pressure-transmitting medium first reported an ‘inverted molecular-like’ solid, in the sense that the former was experimentally found more compressible than the latter [148]. This has then been disputed by *ab initio* calculations on B₄C and α-B₁₂ [149] and by experiments on B₄C performed with neon as the pressure transmitting medium [150]: icosahedra were shown to have a higher bulk modulus than the unit-cell, despite the fact that the covalent intericosahedral bonds in B₄C are expected to be stronger than the three-center bonds in the icosahedra. Last but not least, the experimental EOS study of B₁₃C₂ with neon as the pressure transmitting medium did not reveal a significant difference of the bulk moduli of the icosahedra and of the unit cell [145]. Finally, HP compression at ambient temperature of B₄C and B₁₃C₂ did not reveal any structural phase transition in DFT-GGA up to respectively 70 GPa [150] and 68 GPa [145], as also predicted by *ab initio* calculations for B₄C in the absence of vacancy-based defects in the chains: 70 GPa in DFT-LDA [151], 54 GPa in DFT-GGA-PW91 [27]. These issues are therefore not relevant to the reliability of construction of CALPHAD databases, at least up to 20 GPa.

These vacancies are a part of B₄C and related solid solution models in available CALPHAD databases for Thermocalc. One can expect that at high pressure their concentration rapidly decreases to zero, which can be a good test for CALPHAD assessed databases that consider high pressure.

Thermodynamic *in situ* data is very limited [106], and the actual experimental phase diagrams represent principally the data of recovery experiments at ambient conditions. *In situ* data up to 5 GPa allowed to get general features of the B-C phase diagram, like melting temperature of B₄C [106] that decrease with pressure (in contrast with most other known
compounds), the corresponding phase diagram has been calculated using available thermodynamic data for simple substances and compounds at 0.1 MPa, p-V-T EOS and adjusted parameters for the subregular solution model that allows describing the interaction in liquid phase in the framework of CALPHAD method. Phase diagram of the B-C system can be calculated up to 20 GPa using ThermoCalc software and database reported in ref. [152] (Figure 18).

<figure><img src="image_19.png" /><figcaption>**Figure 18** The phase diagrams of B-C system at 6 GPa (a) and 20 GPa (b) simulated using Thermo-Calc [152]. (c,d) Two p-T sections of phase diagram show boron carbide stability domains in boron-rich part (12 and 15 at% of C).</figcaption></figure>

### 4.3.2 Metastable B-C phases with carbon-related crystal structures

Some other metastable icosahedral phases, principally related to t-B₅₂ structural family, tetragonal (B₁₂)₄B₂C₂, (B₁₂)₄B₃C, (B₁₂)₄BC₃, and orthorhombic [(B₁₂)₄C₆]₈, as well as amorphous boron carbide, have been obtained by CVD [132, 146, 153, 154]. The same method, CVD allows also obtain highly boron-substituted carbons with a graphite-like structure [155].

The B-C graphite-like and diamond-like phases are not formally boron-rich compounds with crystal structures produced by boron clusters, but boron concentration can reach 50 at% [155]. They are generally considered as metastable and *ab initio* calculations predict their heavily positive formation enthalpy (at OK), while their metastability is typically explained by
the high strength of C-C, B-C and B-B covalent bonds and dynamic stability. They often decompose into boron (rich) carbides and boron-doped diamond at high enough pressure. Strictly speaking, they are not electronic analogues of carbon, but can be considered similarly to a certain extent in both theoretical [156] and experimental [71] research.

The 300-K equations of state of graphite-like B-C phases with boron content from 0 to 50 at% have been studied using powder x-ray diffraction with synchrotron radiation up to 35 GPa [157-159]. The compression axis moduli show the non-monotonic dependence on boron content and exhibit a strong correlation with lattice parameters. The values of bulk moduli $B_0$ range from 17.5 GPa (t-BC$_4$) to 30.7 GPa (t-BC$_{1.5}$), while for turbostratic graphite $B_0 = 29.5$ GPa.

For diamond-like c-BC$_5$ [71, 128] the parameters for the calculation of the high-pressure term in the Gibbs energy are: $\rho_0 = 3.267 \text{ gcm}^{-3}$, $B_0 = 335 \text{ GPa}$, $B_0' = 4.5$, $a = 13 \text{ K}^{-1}$, $b = 0 \text{ K}^{-2}$ and $\delta_T = 1.5$ [130]. The last parameter value has been refined using the only HPHT observation in DAC [71]. The parameters of the 300 K EOS of graphite-like phases [157-159] can be found in Table 7.

<table><caption>Table 7 300 K EOS data of turbostratic (t-BC$_x$) heavily boron-doped graphitic carbons and turbostratic graphite t-C.</caption><thead><tr><th>Phase</th><th>EOS parameters: ρ<sub>0</sub> (g/cm<sup>3</sup>), B<sub>0</sub> (GPa), B'<sub>0</sub></th></tr></thead><tbody><tr><td>t-BC</td><td>ρ<sub>0</sub> = 2.141, B<sub>0</sub>=23, B'<sub>0</sub>=8.0</td></tr><tr><td>t-BC<sub>1.5</sub></td><td>ρ<sub>0</sub> = 2.120, B<sub>0</sub>=30.7, B'<sub>0</sub>=8.6</td></tr><tr><td>t-BC<sub>3</sub></td><td>ρ<sub>0</sub> = 2.125, B<sub>0</sub>=24.8, B'<sub>0</sub>=12.5</td></tr><tr><td>t-BC<sub>4</sub></td><td>ρ<sub>0</sub> = 2.159, B<sub>0</sub>=17.5, B'<sub>0</sub>=13.9</td></tr><tr><td>t-C</td><td>ρ<sub>0</sub> = 2.249, B<sub>0</sub>=29.5, B'<sub>0</sub>=9.4</td></tr></tbody></table>

## 4.4 B-Si system

B-Si system contains only boron-rich solids with related crystal structure, and thus we will consider them in this chapter. All boron rich compounds of Si are known to be non-stoichiometric and easily form solid solutions [33]. Two compounds, SiB$_3$ and SiB$_6$ and one Si solid solution in β-B (SiB$_n$) are well characterized and have their stability domain (Figure 19). The employed CALPHAD assessed database include simplified models of both, SiB$_3$ and SiB$_6$, considered as stoichiometric compounds [160]. Application of high pressure is expected to increase the Si solubility in β-B (at ~5 GPa) and suppress B$_6$Si stability at high temperatures (at ~10 GPa). In situ and ex situ experiments are required to test these predictions.
<figure><img src="image_20.png" /><figcaption>Figure 19 (a-c) Crystal structures of SiB₃ (ICSD 12651), SiB₆ and Si / β-B solid solution (ICSD 413978), respectively; and (d-e) phase diagrams of B-Si system at 0.1 MPa, 5 GPa and 10 GPa, respectively.</figcaption></figure>
## 5 Phase diagrams of ternary systems B-X-Y systems at HPHT with boron-rich and regular borides

A number of ternary boron-rich solid solution of the B-C-N-O concentration tetrahedron has been attempted in the past, using only one boron-rich solid (amorphous or crystalline boron, B₄C or B₆O) as starting material [127, 161, 162]. The thus obtained phases, from crystallographic point of view, are found to be very close to known binary boron-rich compounds.

### 5.1 B-C-O system: boron-rich solutions

A number of boron-rich B-C-O solid solutions has been reported in previous HPHT studies by McMillan group [127]. Crystal structures of reported so far B-C-O boron-rich solids of α-B structural family can be understood in the framework of described above simplified model (Figure 7b). Lattice parameters of these HPHT O-doped B₄C phases are presented at Figure 20a; they are very close to boron carbide structural family, and O in the inter-icosahedral chains is expected to be responsible for expansion of the unit cell along c-axis (up to 1%), just like the B atoms do. The evolution of α-parameter is more pronounced (±10%) and more complex, again, similar to low-carbon B₄C phases.

To date, this ternary system has not been studied *in situ*, and only thermodynamic CALPHAD simulations using interaction parameters of binary systems [125, 152] can be made. High-temperature isothermal sections of this diagram are shown at Figure 20b, which is the preliminary estimate of the phase stability domains prior to getting reliable experimental data at ambient and high pressures.

<figure><img src="image_21.png" /><figcaption>Figure 20 (a) Lattice parameters of O-doped boron carbides (solid triangles) obtained at HPHT conditions as compared to other phases (pluses for pressureless B₄C, open circles for B₆O and stars for B₁₃N₂). (b) The tentative 2400 K isothermal sections of the B-C-B₂O₃ phase diagram at 0.1 MPa (b) and 8 GPa (c).</figcaption></figure>
## 5.2 B-N-O system: boron-rich solid solutions

Crystal structures of B₆(O,N,B) solid solutions reported so far [127] well agree with proposed above simplified model (Figure 7b). Lattice parameters of such phases have not been reported so far, also oxidized boron subnitride samples (amorphous boron + BN) indicate the possibility of 50% replacement of N by oxygen in boron subnitride phase, B₁₃N₂ [161].

Typical combined *in situ* and *ex situ* data for thermodynamic analysis of B-N-O system [162] is shown at Figure 21. At first step, the concentration of eutectic in the B₆O-BN binary section was established *ex situ* in quench and recovery experiments at 5 GPa, by optical observation of microstructure of polished samples (sample 2 at Figure 21a). After that, the melting temperature of preformed eutectic alloy has been *in situ* measured at MAX80 at the same pressure. The adjustment of parameters of subregular liquid solution model was made in order to match this experimental melting temperature; and the ternary phase diagram was constructed using this model. The CALPHAD database for this system is to be constructed.

<figure><img src="image_22.png" /></figure>

**Figure 21** Typical combined *in situ* and *ex situ* data for thermodynamic analysis of B-N-O system: (a1,2,3) concentration of eutectic (optical microscopy *ex situ*), (b) binary section B₆O-BN with melting temperature of eutectic (*in situ* at MAX80) and (c) B-B₂O₃-BN phase diagram without ternary phases. (d) Composition of O-doped boron subnitrides observed by EELS [161].
### 5.3 B-C-Si system: boron-rich solid solutions

Thermo-Calc simulation of the B-C-Si system at high pressure has been first performed in this work by combination ambient-pressure database [160] and previously reported CALPHAD-assessed HPHT databases for B-C system [152] and the EOS parameters (experimental, ab initio or empirical described in previous chapters). No ternary interaction parameter for liquid phase was introduced. The sequence of isopleth sections of B₄C – B₆Si is shown at Figure 22. With pressure increase, we predict the stabilization of B₃Si in this system, and reduction of SiC stability domain. *In situ* HPHT experiments are required to confirm our approach or to refine the CALPHAD model parameters.

<figure><img src="image_23.png" /><figcaption>Figure 22. Isopleth sections of phase diagram of ternary B-C-Si system at 0.1 MPa (a), 5 GPa (b) and 10 GPa (c). Thermo-Calc simulations were performed using CALPHAD assessed database for ternary system at ambient conditions [160] in combination with HP databases for binary subsystems of B-C-Si triangle.</figcaption></figure>
## 6. Physical and mechanical properties of boron-rich HPHT phases

Hardness is a complex property that reflect the ability of solid's surface to resist the penetration or scratching. It is a non-trivial function of thermodynamic parameters like composition (for solid solutions), temperature and pressure. The latter two are of particular significance in the domains of high-speed cutting tools and the study of deep Earth and planetary science. Theoretical models of hardness often refer to thermodynamic properties of solids and their crystal chemistry characteristics. Such models can be purely empirical [163], semiempirical *ab initio* [164] or purely *ab initio* taking into account stress-strain relationships in different crystallographic directions [165] or dislocation propagation energy [166]. Generally, all of them provide the 10% accuracy, both at ambient conditions and at high temperatures (Figure 23a). We should also note that the highest, or ideal values of crystalline solids are typically attributed to single crystals. The polycrystalline samples can reach these values mainly via high-pressure sintering, although alternative “pressureless” methods of processing are possible [167].

Theoretical hardness scale for boron-rich solids has been proposed [168]. Hardness as a function of thermodynamic parameters can be also predicted using so-called “thermodynamic” model of hardness [169] and well agrees with experimental data, at least for concentration and temperature dependences (Figure 23b,c). A number of *ab initio* predictions have been made to highlight how mechanical properties can be improved via crystallographic and thermodynamic consideration. Experimentally, the best mechanical properties of boron carbide ceramics require high pressure sintering. Nanostructuring is also an efficient tool for boron-rich solids [40], similar to cubic boron nitride [170].

<figure><img src="image_24.png" /><figcaption>**Figure 23** (a) Theoretical harness scaled vs experimental Vickers hardness. Inset shows B₁₃N₂ single crystal. (b,c) Temperature and concentration dependence of Vickers hardness of boron carbide: experimental values (symbols) vs predictions of thermodynamic model of hardness (curves).</figcaption></figure>
At Figure 24 theoretical (o) and experimental (x) values of Vickers hardness of boron-
containing binary compounds are presented as a function of the atomic fraction of second
element. The approximate boundaries of Si-hard (H<sub>v</sub> > 10 GPa), superhard (H<sub>v</sub> > 40 GPa) and
boron-rich (boron content ≥ 80 at%) phases are given by horizontal and vertical lines. The
highest values of hardness can be observed only for pure elements, i.e. H<sub>v</sub> = 60 GPa in the
case of γ-B, around 60% of diamond, the hardest known phase.

<figure><img src="image_25.png" /><figcaption>Atomic fraction of foreign elements</figcaption></figure>

**Figure 24** Vickers hardness of boron-rich compounds filling the gap between Si and diamond. Red solid circles show superhard boron-rich solids, solid dark-yellow circles – silicon borides, black open circles – boron-rich compounds of group III and IV elements (left section of the diagram) and regular borides (right section of the diagram). Black crosses show upper and low limits for some phases. Black solid stars show hardness values for diamond Si (Si-I) and diamond C (or simply diamond).

Silicon (Si-I) has an order of magnitude lower hardness than diamond, yet it can nevertheless be regarded as a hard material. Hardness of compounds from the Si-B system is remarkably lower than that of other boron-rich solids [171], however, we should notice that reference values of hardness were typically chosen as those of single crystals, which is not the case for silicon borides. Anyway, systematic increase of hardness was observed while increasing boron content (dark yellow symbols, Figure 24). Microscopic mechanism of hardness is quite complex, and has been considered mainly for Si with simple diamond structure [166]. This theory considers the molecular orbitals of diamond-like structures and link the hardness to the indirect bandgap value of Si, which rise some contradiction with Si-III allotrope [5], that has narrow bandgap of 30 meV [172], but higher hardness than Si-I.
Si is also considered as a potential doping-element with the capacity to enhance the
mechanical properties of boron carbide particularly with regard to the fracture toughness
[173]. At the same time, efficient production of best species of such materials requires high
temperatures for kinetic activation of diffusion and high pressure to assure stoichiometry in
the case of complicated chemistry with volatile phases. This fact renders the try-and-test
method costly for exploring new advances materials, and reliable simulation techniques, like
those described in this review (CALPHAD, *ab initio*), are highly desirable.
7. Conclusions

In this review, we systematically summarized and analyzed the current experimental,
theoretical, and thermodynamic understanding of phase equilibria and transformations in
boron-rich compounds within the binary and ternary of B–C–N–O and B–C–Si systems at
high-pressure and high-temperature (HPHT) conditions. Employing advanced *in situ* x-ray
diffraction techniques combined with density functional theory calculations and the
CALPHAD thermodynamic modeling methodology, we provided comprehensive phase
diagrams, detailed *p*-*V*-*T* equations of state, and thermodynamic properties essential for
characterizing these superhard materials.

Our review highlights significant progress in resolving long-standing uncertainties related to the thermodynamic stability, crystallographic structures, and phase transitions of various boron-rich phases. Notably, the CALPHAD approach demonstrated remarkable efficiency in integrating experimental *in situ* XRD data with theoretical predictions, significantly improving our understanding of complex binary and ternary systems.

Nevertheless, several key areas require further work: (1) Enhanced precision of *in situ* x-ray diffraction experiments to better characterize low-Z element phases under extreme conditions; (2) Improvement and validation of thermodynamic databases, specifically refining *p*-*V*-*T* equations of state parameters and high-temperature enthalpies; (3) Expansion of CALPHAD modeling to address limited or uncertain experimental data, particularly for high-pressure boron-rich compounds; and (4) Exploration of previously unexplored or poorly understood phases, especially metastable and transient states observed during synthesis and cooling processes.

Future studies should also focus on the correlation between crystallographic disorder and thermodynamic stability under extreme conditions, aiming to clarify the role of atomic arrangements on mechanical properties. Continued development in this interdisciplinary field will significantly enhance theoretical insights and practical applications, particularly in industries utilizing superhard cutting tools, wear-resistant coatings, nuclear materials, and thermoelectrics. Integrating precise experimental techniques with advanced computational modeling will be essential in discovering novel materials with superior mechanical and physical properties.
Graphical abstract
**Supporting information**

N/A

**Author statement**

The paper has been written by the contribution of all coauthors.

**Declaration of competing interest**

No competing interest to declare.

**Data availability**

Data will be available on request.

**Acknowledgements**

AC, YLG, JS, NV and OR acknowledge the ANR - FRANCE (French National Research Agency) for the financial support of the BCSi project number ANR-21-CE08-0018. Calculation resources are granted by the French HPC centers GENCI-IDRIS, GENCI-CINES and GENCI-TGCC (Project 2210) and by the Institut Polytechnique de Paris (École Polytechnique) through the 3Labs-HPC project. VZT work was supported by project No. 2023.05/0007 financed by the National Research Fund of Ukraine.

**References**

1. Kurakevych, O.O., Superhard phases of simple substances and binary compounds of the B-C-N-O system: from diamond to the latest results (a Review). Journal of Superhard Materials, 2009. **31**(3): p. 139-157.

2. Novikov, N.V., New trends in high-pressure synthesis of diamond. Diamond Relat. Mater., 1999. **8**: p. 1427-1432.

3. Taniguchi, T. and S. Yamaoka, Spontaneous nucleation of cubic boron nitride single crystal by temperature gradient method under high pressure. Journal of Crystal Growth, 2001. **222**(3): p. 549-557.

4. Kurakevych, O.O., et al., Synthesis of Bulk BC8 Silicon Allotrope by Direct Transformation and Reduced-Pressure Chemical Pathways (v6ol 55, pg 8943, 2016). Inorganic Chemistry, 2016. **55**(19): p. 9949-9949.

5. Kurakevych, O.O., et al., Synthesis of Bulk BC8 Silicon Allotrope by Direct Transformation and Reduced-Pressure Chemical Pathways. Inorganic Chemistry, 2016. **55**(17): p. 8943-8950.

6. Shimomura, O., Current Activity of MAX80 at the Photon Factory. Physica B+C, 1986. **139-140**: p. 292-300.

7. Guignard, J. and W.A. Crichton, The large volume press facility at ID06 beamline of the European synchrotron radiation facility as a High Pressure-High Temperature deformation apparatus. Rev. Sci. Instrum., 2015. **86**(8): p. 085112.

8. Le Godec, Y. and A. Courac, In Situ High-Pressure Synthesis of New Outstanding Light-Element Materials under Industrial P-T Range. Materials, 2021. **14**(15): p. 4245.

9. Le Godec, Y., A. Courac, and V.L. Solozhenko, High-pressure synthesis of superhard and ultrahard materials. Journal of Applied Physics, 2019. **126**(15).

10. Solozhenko, V.L., et al., Kinetics of diamond crystallization from the melt of the Fe-Ni-C system. Journal of Physical Chemistry B, 2002. **106**(26): p. 6634-6637.

11. Fei, Y. and E. Brosh, Experimental study and thermodynamic calculations of phase relations in the Fe-C system at high pressure. Earth and Planetary Science Letters, 2014. **408**: p. 155-162.
12. Turkevich, V.Z., et al., Mg–C System up to 20 GPa: Its Phase Diagram and Stable Magnesium Carbides. The Journal of Physical Chemistry C, 2023. **127**(4): p. 1965-1972.

13. Strobel, T.A., et al., Synthesis of beta-Mg2C3: A Monoclinic High-Pressure Polymorph of Magnesium Sesquicarbide. Inorganic Chemistry, 2014. **53**(13): p. 7020-7027.

14. Kurakevych, O.O., et al., High-Pressure and High-Temperature Stability of Antifluorite Mg2C by in Situ X-ray Diffraction and ab Initio Calculations. Journal of Physical Chemistry C, 2014. **118**(15): p. 8128-8133.

15. Kurakevych, O.O., et al., Synthesis of Mg2C: A Magnesium Methanide. Angewandte Chemie-International Edition, 2013. **52**(34): p. 8930-8933.

16. Mueller, H.J., F.R. Schilling, and C. Lathe, Multianvil techniques in conjunction with synchrotron radiation at Deutsches ElektronenSYnchrotron (DESY) – HAmburger SYnchrotron LABor (HASYLAB), in Advances in High-Pressure Mineralogy, E. Ohtani, Editor. 2007, Geological Society of America. p. 0.

17. Kantor, I., et al., BX90: A new diamond anvil cell design for X-ray diffraction and optical measurements. Review of Scientific Instruments, 2012. **83**(12).

18. Letoullec, R., J. Pinceaux, and P. Loubeyre, The membrane diamond anvil cell: a new device for generating continuous pressure and temperature variations. International Journal of High Pressure Research, 1988. **1**(1): p. 77-90.

19. Dailledouze, C., et al., Imaging the Meissner Effect and Flux Trapping of Superconductors under High Pressure using N-V Centers. arXiv, 2025. https://arxiv.org/abs/2501.14504.

20. Besson, J., et al., Neutron powder diffraction above 10 GPa. Physica B: Condensed Matter, 1992. **180**: p. 907-910.

21. Le Godec, Y., et al., A new high PT cell for neutron diffraction up to 7 GPa and 2000 K with measurement of temperature by neutron radiography. International Journal of High Pressure Research, 2001. **21**(5): p. 263-280.

22. Mezouar, M., et al., Paris–Edinburgh large-volume cell coupled with a fast imaging-plate system for structural investigation at high pressure and high temperature. Journal of Synchrotron Radiation, 1999. **6**(6): p. 1115-1119.

23. Itié, J.-P., et al., Crystallography under high pressure using synchrotron radiation. Journal of Physics D: Applied Physics, 2015. **48**(50): p. 504007.

24. Morard, G., et al., High efficiency multichannel collimator for structural studies of liquids and low-Z materials at high pressures and temperatures. Review of Scientific Instruments, 2011. **82**(2): p. 023904.

25. King, A., et al., Advanced diffraction techniques on the PSICHE beamline, Synchrotron SOLEIL. Acta Crystallographica Section A: Foundations and Advances [2014-...], 2022. **78**(a2): p. e802-e802.

26. King, A., et al., Combined angular and energy dispersive diffraction: optimized data acquisition, normalization and reduction. Journal of Applied Crystallography, 2022. **55**(2): p. 218-227.

27. Jay, A., et al., Theoretical phase diagram of boron carbide from ambient to high pressure and temperature. Journal of Applied Physics, 2019. **125**(18).

28. Giannozzi, P., et al., Advanced capabilities for materials modelling with Quantum ESPRESSO. Journal of Physics: Condensed Matter, 2017. **29**(46): p. 465901.

29. Widom, M. and W.P. Huhn, Prediction of orientational phase transition in boron carbide. Solid State Sciences, 2012. **14**(11): p. 1648-1652.

30. Huhn, W., Thermodynamics from first principles: Prediction of phase diagrams and materials properties using density functional theory. Carnegie Mellon University: Mellon

31. Yao, S., W.P. Huhn, and M. Widom, Phase transitions of boron carbide: Pair interaction model of high carbon limit. Solid State Sciences, 2015. **47**: p. 21-26.

32. Liang, X., et al., Prediction of superconductivity in pressure-induced new silicon boride phases. Physical Review B, 2020. **101**(1): p. 014112.
33. Eklöf, D., et al., *Mysterious SiB3: Identifying the Relation between α- and β-SiB3*. ACS Omega, 2019. **4**(20): p. 18741-18759.

34. Zhang, H., S. Yao, and M. Widom, *Predicted phase diagram of boron-carbon-nitrogen*. Physical Review B, 2016. **93**(14): p. 144107.

35. Wade, K., *The structural significance of the number of skeletal bonding electron-pairs in carboranes, the higher boranes and borane anions, and various transition-metal carbonyl cluster compounds*. Journal of the Chemical Society D: Chemical Communications, 1971(15): p. 792-793.

36. Welch, A.J., *The significance and impact of Wade's rules*. Chemical Communications, 2013. **49**(35): p. 3615-3616.

37. Chakraborti, A., et al., *Boron carbide under torsional deformation: Evidence of the formation of chain vacancies in the plastic regime*. Acta Materialia, 2022. **226**: p. 117553.

38. Jay, A., et al., *Theoretical Raman spectrum of boron carbide B4.3C under pressure*. Acta Materialia, 2023. **255**: p. 119085.

39. Roma, G., et al., *Reply to "Comment on 'Understanding first-order Raman spectra of boron carbides across the homogeneity range'"*. Physical Review Materials, 2022. **6**(1): p. 016602.

40. Igoa Saldaña, F., et al., *Transforming Nanocrystals into Superhard Boron Carbide Nanostructures*. ACS Nano, 2024. **18**(44): p. 30473-30483.

41. Carnimeo, I., et al., *Quantum ESPRESSO: One Further Step toward the Exascale*. Journal of Chemical Theory and Computation, 2023. **19**(20): p. 6992-7006.

42. Hillert, M., *The compound energy formalism*. Journal of Alloys and Compounds, 2001. **320**(2): p. 161-176.

43. Murnaghan, F.D., *The compressibility of media under extreme pressures*. Proc. Nation. Acad. Sci. USA, 1944. **30**(9): p. 244-247.

44. Lu, X.-G., M. Selleby, and B. Sundman, *Implementation of a new model for pressure dependence of condensed phases in Thermo-Calc*. Calphad, 2005. **29**(1): p. 49-55.

45. Joubert, J.-M., J.-C. Crivello, and G. Deffrennes, *Modification of Lu's (2005) high pressure model for improved high pressure/high temperature extrapolations*. Part I: Modeling of platinum at high pressure/high temperature. Calphad, 2021. **74**: p. 102304.

46. Anderson, O.L. and D.G. Isaak, *The dependence of the Anderson-Grueneisen parameter $\delta_T$ upon compression at extreme conditions*. J. Phys. Chem. Solids, 1993. **54**(2): p. 221-227.

47. Solozhenko, V.L., K.A. Cherednichenko, and O.O. Kurakevych, *Thermoelastic Equation of State of Boron Subphosphide B12P2*. Journal of Superhard Materials, 2017. **39**(1): p. 71-74.

48. Chopelas, A. and R. Boehler, *Thermal expansion measurements at very high pressure, systematics, and a case for a chemically homogeneous mantle*. Geophysical Research Letters, 1989. **16**(11): p. 1347-1350.

49. Kurakevych, O.O. and V.L. Solozhenko, *Thermoelastic equation of state of boron suboxide B6O up to 6 GPa and 2700 K: Simplified Anderson-Gruneisen model and thermodynamic consistency*. Journal of Superhard Materials, 2014. **36**(4): p. 270-278.

50. Solozhenko, V.L., et al., *Thermodynamically Consistent p-T Phase Diagram of Boron Oxide B2O3 by in Situ Probing and Thermodynamic Analysis*. Journal of Physical Chemistry C, 2015. **119**(35): p. 20600-20605.

51. Courac, A., et al., *Thermoelastic equation of state and melting of Mg metal at high pressure and high temperature*. Journal of Applied Physics, 2020. **127**(5).

52. Oganov, A.R., et al., *Ionic high-pressure form of elemental boron*. Nature, 2009. **457**(7231): p. 863-867.

53. Oganov, A.R., et al., *Ionic high-pressure form of elemental boron (vol 457, pg 863, 2009)*. Nature, 2009. **460**(7252): p. 292-292.

54. Parakhonskiy, G., et al., *High pressure synthesis and investigation of single crystals of metastable boron phases*. High Pressure Research, 2013. **33**(3): p. 673-683.

55. Hayami, W., et al., *Thermodynamic stability of elemental boron allotropes with varying numbers of interstitial atoms*. Journal of Solid State Chemistry, 2024. **329**: p. 124407.
56. Amberger, E., W. Stumpf, and K. Buschbeck, *Boron compounds, elemental boron and boron carbides*, in *Gmelin Handbook of Inorganic Chemistry*. 1981, Springer-Verlag: Berlin.

57. Solozhenko, V.L. and O.O. Kurakevych, *Equilibrium p-T Phase Diagram of Boron: Experimental Study and Thermodynamic Analysis*. SCIENTIFIC REPORTS, 2013. **3**.

58. Chuvashova, I., et al., *Nonicosahedral boron allotrope synthesized at high pressure and high temperature*. Physical Review B, 2017. **95**(18): p. 180102.

59. Parakhonskiy, G., et al., *High pressure synthesis of single crystals of α-boron*. Journal of Crystal Growth, 2011. **321**(1): p. 162-166.

60. Chakraborti, A., et al., *When carbon impurities trigger the synthesis of alpha boron at high pressure and high temperature*. Acta Materialia, 2023. **249**: p. 118820.

61. Solozhenko, V.L., O.O. Kurakevych, and A.R. Oganov, *On the hardness of a new boron phase, orthorhombic gamma-B-28*. Journal of Superhard Materials, 2008. **30**(6): p. 428-429.

62. Albert, B. and H. Hillebrecht, *Boron: Elementary Challenge for Experimenters and Theoreticians*. Angewandte Chemie International Edition, 2009. **48**(46): p. 8640-8668.

63. Ogitsu, T., E. Schwegler, and G. Galli, *β-Rhombohedral Boron: At the Crossroads of the Chemistry of Boron and the Physics of Frustration*. Chemical Reviews, 2013. **113**(5): p. 3425-3449.

64. Parakhonskiy, G., et al., *Experimental pressure-temperature phase diagram of boron: resolving the long-standing enigma*. Scientific Reports, 2011. **1**(1): p. 96.

65. Parakhonskiy, G., *Synthesis and Investigation of Boron Phases at High Pressures and Temperatures* 2012, University of Bayreuth: Bayreuth, Germany.

66. Chuvashova, I., et al., *High-pressure behavior of α-boron studied on single crystals by X-ray diffraction, Raman and IR spectroscopy*. Journal of Solid State Chemistry, 2017. **245**: p. 50-60.

67. Cherednichenko, K.A. and V.L. Solozhenko, *Thermal expansion of α-boron and some boron-rich pnictides*. Solid State Communications, 2019. **303-304**: p. 113735.

68. Nelmes, R.J., et al., *Neutron- and x-ray-diffraction measurements of the bulk modulus of boron*. Physical Review B, 1993. **47**(13): p. 7668-7673.

69. Lundström, T., B. Lönnberg, and J. Bauer, *Thermal expansion of β-rhombohedral boron*. Journal of Alloys and Compounds, 1998. **267**(1): p. 54-58.

70. Sanz, D.N., P. Loubeyre, and M. Mezouar, *Equation of State and Pressure Induced Amorphization of β-Boron from X-Ray Measurements up to 100 GPa*. Physical Review Letters, 2002. **89**(24): p. 245501.

71. Le Godec, Y., et al., *Equation of state of orthorhombic boron, gamma-B-28*. Solid State Communications, 2009. **149**(33-34): p. 1356-1358.

72. Nieto-Sanz, D., et al., *X-ray study of the synthesis of boron oxides at high pressure: phase diagram and equation of state*. Physical Review B, 2004. **70**: p. 214108 1-6.

73. Kurakevych, O.O. and V.L. Solozhenko, *Rhombohedral boron subnitride, B13N2, by X-ray powder diffraction*. Acta Crystallographica Section C-Crystal Structure Communications, 2007. **63**: p. I80-I82.

74. Cherednichenko, K.A., L. Gigli, and V.L. Solozhenko, *Thermal expansion of boron subnitrides*. SOLID STATE COMMUNICATIONS, 2018. **275**: p. 21-24.

75. Cherednichenko, K.A. and V.L. Solozhenko, *Structure and equation of state of tetragonal boron subnitride B₅₀N₂*. JOURNAL OF APPLIED PHYSICS, 2017. **122**(15).

76. Somayazulu, M., et al., *P-V-T equation of state of boron carbide*. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2023. **381**(2258): p. 20220331.

77. Brazhkin, V.V., et al., *Fabrication of β-boron by chemical-reaction and melt-quenching methods at high pressures*. J. Mater. Res., 2004. **19**(6): p. 1643-1648.

78. Chase, M.W.J., *NIST-JANAF Thermochemical Tables, 4th Edition*. J. Phys. Chem. Ref. Data. Vol. Monograph 9. 1998, Melville, NY: AIP Publishing. 1952.

79. Stout, N.D., R.W. Mar, and W.O. Boo, *High-temperature enthalpy and enthalpy of fusion of boron by drop calorimetry*. High Temp. Sci., 1973. **5**: p. 241.
80. Hoard, J.L., et al., *The structure analysis of β-rhombohedral boron*. J. Solid State Chem., 1970. **1**(2): p. 268-277.

81. Krishnan, S., et al., *Optical-Properties and Melting-Point of Pure Boron*. High Temp. Sci., 1991. **31**(1): p. 45-57.

82. Lundstrom, T., B. Lonnberg, and J. Bauer, *Thermal expansion of beta-rhombohedral boron*. J. Alloy. Comp., 1998. **267**(1-2): p. 54-58.

83. Tsagareishvili, D.S. and G.V. Tsagareishvili, *Some thermoelastic properties of liquid boron at melting point*. Journal of the Less Common Metals, 1979. **67**(2): p. 541-550.

84. Tsagareishvili, G.V., D.S. Tsagareishvili, and A.G. Khvedelidze, *Estimation of some thermoelastic properties of [beta] rhombohedral boron in wide ranges of temperature and pressure*. Journal of the Less Common Metals, 1980. **75**(1): p. 141-145.

85. Kurakevych, O.O. and V.L. Solozhenko, *Crystal structure of dense pseudo-cubic boron allotrope, pc-B-52, by powder X-ray diffraction*. Journal of Superhard Materials, 2013. **35**(1): p. 60-63.

86. Ma, Y., et al., *High-pressure high-temperature x-ray diffraction of β-boron to 30 GPa*. Physical Review B, 2003. **67**(17): p. 174116.

87. Qin, J., et al., *Phase relations in boron at pressures up to 18 GPa and temperatures up to 2200 °C*. Physical Review B, 2012. **85**(1): p. 014107.

88. Funamori, N. and K. Tsuji, *Pressure-Induced Structural Change of Liquid Silicon*. Physical Review Letters, 2002. **88**(25): p. 255508.

89. Yang, L., et al., *Melting diamond in the diamond cell by laser-flash heating*. High Pressure Research, 2023. **43**(1): p. 1-14.

90. Ghiringhelli, L.M., et al., *Modeling the Phase Diagram of Carbon*. Physical Review Letters, 2005. **94**(14): p. 145701.

91. Gustafson, P., AN EVALUATION OF THE THERMODYNAMIC PROPERTIES AND THE P-PHASE, T-PHASE DIAGRAM OF CARBON. Carbon, 1986. **24**(2): p. 169-176.

92. Kubo, A., et al., *Melting curve of silicon to 15 GPa determined by two-dimensional angle-dispersive diffraction using a Kawai-type apparatus with X-ray transparent sintered diamond anvils*. Journal of Physics and Chemistry of Solids, 2008. **69**(9): p. 2255-2260.

93. Anzellini, S., et al., *Quasi-hydrostatic equation of state of silicon up to 1 megabar at ambient temperature*. Scientific Reports, 2019. **9**(1): p. 15537.

94. Hanfland, M., H. Beister, and K. Syassen, *Graphite under pressure: Equation of state and first-order Raman modes*. Physical Review B, 1989. **39**(17): p. 12598-12603.

95. Tsang, D.K.L., et al., *Graphite thermal expansion relationship for different temperature ranges*. Carbon, 2005. **43**(14): p. 2902-2906.

96. Dewaele, A., et al., *High pressure--high temperature equations of state of neon and diamond*. Physical Review B, 2008. **77**(9): p. 094106.

97. Daviau, K. and K.K.M. Lee, *High-Pressure, High-Temperature Behavior of Silicon Carbide: A Review*. Crystals, 2018. **8**(5): p. 217.

98. Miozzi, F., et al., *Equation of State of SiC at Extreme Conditions: New Insight Into the Interior of Carbon-Rich Exoplanets*. Journal of Geophysical Research: Planets, 2018. **123**(9): p. 2295-2309.

99. Jochym, P.T., et al., *Influence of anharmonicity on the negative thermal expansion of $\ ensuremath{\alpha} \text{\ensuremath{-}} \mathrm{Sn} \ensuremath{\}$*. Physical Review Materials, 2022. **6**(11): p. 113601.

100. Courac, A., et al., *High-Pressure Melting Curve of Zintl Sodium Silicide Na4Si4 by In Situ Electrical Measurements*. Inorganic Chemistry, 2019. **58**(16): p. 10822-10828.

101. Ploog, K., et al., $B_{48}B_2C_2$ and $B_{48}B_2N_2$, two non-metal borides with structure of so-called I-tetragonal boron. Journal of Less-Common Metals, 1972. **29**: p. 161-169.

102. Vlasse, M., et al., *Crystal structure of tetragonal boron related to α-AlB₁₂*. Journal of Solid State Chemistry, 1979. **28**(3): p. 289-301.
103. Mondal, S., et al., *Disorder and defects are not intrinsic to boron carbide*. Scientific Reports, 2016. **6**(1): p. 19330.

104. Kurakevych, O.O., et al., *Comparison of solid-state crystallization of boron polymorphs at ambient and high pressures*. High Pressure Research, 2012. **32**(1): p. 30-38.

105. Ekimov, E.A., I.P. Zibrov, and A.V. Zoteev, *Preparation of Boron Microcrystals via High-Pressure, High-Temperature Pyrolysis of Decaborane, B₁₀H₁₄*. Inorg. Mater., 2011. **47**(11): p. 1194-1198.

106. Mukhanov, V.A., P.S. Sokolov, and V.L. Solozhenko, *On melting of B4C boron carbide under pressure*. Journal of Superhard Materials, 2012. **34**(3): p. 211-213.

107. Solozhenko, V.L. and V.A. Mukhanov, *On melting of boron subnitride B13N2 under pressure*. Journal of Superhard Materials, 2017. **39**(2): p. 147-148.

108. Kurakevych, O.O. and V.L. Solozhenko, *Experimental Study and Critical Review of Structural, Thermodynamic and Mechanical Properties of Superhard Refractory Boron Suboxide B6O*. Journal of Superhard Materials, 2011. **33**(6): p. 421-428.

109. Karacaoğlan, A.Ö.Ç. and M. Durandurdu, *Possible boron-rich amorphous silicon borides from ab initio simulations*. Journal of Molecular Modeling, 2023. **29**(4): p. 92.

110. Škundrić, T., et al., *Structure Prediction and Mechanical Properties of Silicon Hexaboride on Ab Initio Level*. Materials (Basel), 2021. **14**(24).

111. Solozhenko, V.L., M.V. A., and V.V. and Brazhkin, *Melting and decomposition of orthorhombic B6Si under high pressure*. High Pressure Research, 2020. **40**(4): p. 488-494.

112. Kurakevych, O.O. and V.L. Solozhenko, *High-Pressure Design of Advanced BN-Based Materials*. Molecules, 2016. **21**(10).

113. Solozhenko, V.L., V.Z. Turkevich, and W.B. Holzapfel, *Refined Phase Diagram of Boron Nitride*. J. Phys. Chem. B, 1999. **103**(15): p. 2903-2905.

114. Solozhenko, V.L., et al., *In situ studies of boron nitride crystallization from BN solutions in supercritical N-H fluid at high pressures and temperatures*. Physical Chemistry Chemical Physics, 2002. **4**(21): p. 5386-5393.

115. Le Godec, Y., et al., *Effect of nanostructuration on compressibility of cubic BN*. Journal of Superhard Materials, 2012. **34**(5): p. 336-338.

116. Prewitt, C.T. and R.D. Shannon, *Crystal structure of a high-pressure form of B₂O₃*. Acta Crystallographica B, 1968. **24**: p. 869-874.

117. Baroni, A., et al., *Many-body effects at the origin of structural transitions in B₂O₃*. The Journal of Chemical Physics, 2019. **151**(22).

118. Ferlat, G., et al., *Hidden polymorphs drive vitrification in B₂O₃*. Nature Materials, 2012. **11**(11): p. 925-929.

119. Rizzo, H.F., W.C. Simmons, and H.O. Bielstein, *The Existence and Formation of the Solid B₆O*. Journal of the Electrochemical Society, 1962. **109**(11): p. 1079-1082.

120. Olofsson, M. and T. Lundstrom, *Synthesis and structure of non-stoichiometric B₆O*. Journal of Alloys Compounds, 1997. **257**: p. 91-95.

121. Hubert, H., et al., *High-pressure, high-temperature synthesis and characterization of boron suboxide (B₆O). Chemistry of Materials*, 1998. **10**(6): p. 1530-1537.

122. Solozhenko, V.L., O.O. Kurakevych, and P. Bouvier, *First and second-order Raman scattering of B6O*. Journal of Raman Spectroscopy, 2009. **40**(8): p. 1078-1081.

123. Dong, H., et al., *Prediction of a new ground state of superhard compound B6O at ambient conditions*. Scientific Reports, 2016. **6**(1): p. 31288.

124. Solozhenko, V.L., et al., *Phase diagram of the B-B₂O₃ system at 5 GPa: Experimental and theoretical studies*. Journal of Physical Chemistry B, 2008. **112**(21): p. 6683-6687.

125. Turkevich, V.Z., D.V. Turkevich, and V.L. Solozhenko, *Phase diagram of the B–B2O3 system at pressures to 24 GPa*. Journal of Superhard Materials, 2016. **38**(3): p. 216-218.

126. Solozhenko, V.L., Y. Le Godec, and O.O. Kurakevych, *Solid-state synthesis of boron subnitride, B6N: myth or reality?*. Comptes Rendus Chimie, 2006. **9**(11-12): p. 1472-1475.
127. Hubert, H., et al., *High-pressure, high-temperature syntheses in the B–C–N–O system. I. Preparation and Characterization*. Journal of Solid State Chemistry, 1997. **133**: p. 356-364.

128. Kurakevych, O.O. and V.L. Solozhenko, *300-K equation of state of rhombohedral boron subnitride*, B13N2. Solid State Communications, 2009. **149**(47-48): p. 2169-2171.

129. Solozhenko, V.L., et al., *Phase Diagram of the B-BN System at 5 GPa*. JOURNAL OF PHYSICAL CHEMISTRY B, 2010. **114**(17): p. 5819-5822.

130. Courac, A., V. Turkevich, and Y. Le Godec, *Thermodynamics and crystallography of hard light-element boron-rich compounds by in situ X-ray diffraction under high-pressure, high-temperature conditions*. Solid State Sciences, 2025. **163**: p. 107909.

131. Werheit, H. and S. Shalamberidze, *Advanced microstructure of boron carbide*. Journal of Physics: Condensed Matter, 2012. **24**(38): p. 385406.

132. Pallier, C., et al., *Structure of an Amorphous Boron Carbide Film: An Experimental and Computational Approach*. Chemistry of Materials, 2013. **25**(13): p. 2618-2629.

133. Chakraborti, A., N. Vast, and Y. Le Godec, *Synthesis of boron carbide from its elements at high pressures and high temperatures*. Solid State Sciences, 2020. **104**: p. 106265.

134. Iwan, S., et al., *High-pressure high-temperature synthesis and characterization of B10C*. AIP Advances, 2024. **14**(6).

135. Roszeitis, S., et al., *Reactive sintering process and thermoelectric properties of boron rich boron carbides*. Journal of the European Ceramic Society, 2014. **34**(2): p. 327-336.

136. Chakraborti, A., et al., *Synthesis of boron carbide from its elements up to 13 GPa*. Journal of Physics and Chemistry of Solids, 2021. **159**: p. 110253.

137. Rogl, P.F., et al., *The B-rich side of the B–C phase diagram*. Calphad, 2014. **44**: p. 3-9.

138. Domnich, Vladislav, et al., *Boron Carbide: Structure, Properties, and Stability under Stress*. Journal of the American Ceramic Society, 2011. **94**(11): p. 3605-3628.

139. Okamoto, H., *B-C (Boron-Carbon) phase diagram*. J. Phase Equilib., 2012. **13**: p. 436.

140. Rasim, K., et al., *Local Atomic Arrangements and Band Structure of Boron Carbide*. Angewandte Chemie International Edition, 2018. **57**(21): p. 6130-6135.

141. Shirai, K., K. Sakuma, and N. Uemura, *Theoretical study of the structure of boron carbide B13C2*. Physical Review B, 2014. **90**(6): p. 064109.

142. Roma, G., et al., *Understanding first-order Raman spectra of boron carbides across the homogeneity range*. Physical Review Materials, 2021. **5**(6): p. 063601.

143. Yang, X., W.A. Goddard, III, and Q. An, *Structure and Properties of Boron-Very-Rich Boron Carbides: B12 Icosahedra Linked through Bent CBB Chains*. The Journal of Physical Chemistry C, 2018. **122**(4): p. 2448-2453.

144. Clark, H.K. and J.L. Hoard, *The Crystal Structure of Boron Carbide*. Journal of the American Chemical Society, 1943. **65**(11): p. 2115-2119.

145. Chuvashova, I., et al., *Structural stability and mechanism of compression of stoichiometric B13C2 up to 68GPa*. Scientific Reports, 2017. **7**(1): p. 8969.

146. Ploog, K., *Composition and structure of boron carbides prepared by CVD*. Journal of Crystal Growth, 1974. **24-25**: p. 197-204.

147. Hushur, A., et al., *High-pressure phase transition makes B4.3C boron carbide a wide-gap semiconductor*. Journal of Physics: Condensed Matter, 2016. **28**(4): p. 045403.

148. Nelmes, R.J., et al., *Observation of Inverted-Molecular Compression in Boron Carbide*. Physical Review Letters, 1995. **74**(12): p. 2268-2271.

149. Lazzari, R., et al., *Atomic Structure and Vibrational Properties of Icosahedral ${\Sigma}B_{4}$(4)C$^{5}$ Boron Carbide*. Physical Review Letters, 1999. **83**(16): p. 3230-3233.

150. Dera, P., et al., *New insights into the enigma of boron carbide inverse molecular behavior*. Journal of Solid State Chemistry, 2014. **215**: p. 85-93.

151. Betranhandy, E., N. Vast, and J. Sjakste, *Ab initio study of defective chains in icosahedral boron carbide B4C*. Solid State Sciences, 2012. **14**(11): p. 1683-1687.

152. Turkevich, V.Z. and V.L. Solozhenko, *Thermodynamic calculation of the B-C system at pressures to 24 GPa*. Journal of Superhard Materials, 2014. **36**(5): p. 358-360.
153. Ekbom, L.B. and C.-O. Amundin, *Microstructural evaluation of sintered boron carbides with different compositions*. Science of Ceramics, 1981. **1**: p. 237.

154. Pallier, C., *Évolution structural des céramiques (Si)-B-C sous sollications thermomécaniques*. 2012, Université Bordeaux I.

155. Shirasaki, T., et al., *Synthesis and characterization of boron-substituted carbons*. Carbon, 2000. **38**(10): p. 1461-1467.

156. Lowther, J.E., *Potential super-hard phases and the stability of diamond-like boron-carbon structures*. Journal of Physics: Condensed Matter, 2005. **17**(21): p. 3221.

157. Kurakevych, O.O., T. Chauveau, and V.L. Solozhenko, *On Crystal Lattice Parameters of Graphite-Like Phases of the B-C System*. Journal of Superhard Materials, 2010. **32**(4): p. 231-235.

158. Kurakevych, O.O., E.G. Solozhenko, and V.L. Solozhenko, *High-pressure study of graphite-like B-C phases*. High Pressure Research, 2009. **29**(4): p. 605-611.

159. Solozhenko, V.L., et al., *Equation of state of graphite-like BC*. Solid State Communications, 2006. **137**(5): p. 268-271.

160. Chen, H.M., et al., *Thermodynamic assessment of the B–C–Si system*. Journal of Alloys and Compounds, 2009. **481**(1): p. 182-189.

161. Garvie, L.A.J., et al., *High-Pressure, High-Temperature Syntheses in the B–C–N–O System*. Journal of Solid State Chemistry, 1997. **133**(2): p. 365-371.

162. Solozhenko, V.L., et al., *Phase Equilibria in the B-BN-B2O3 System at 5 GPa*. Journal of Physical Chemistry C, 2013. **117**(36): p. 18642-18647.

163. Mukhanov, V.A., O.O. Kurakevych, and V.L. Solozhenko, *The interrelation between hardness and compressibility of substances and their structure and thermodynamic properties*. Journal of Superhard Materials, 2008. **30**(6): p. 368-378.

164. Mazhnik, E. and A.R. Oganov, *A model of hardness and fracture toughness of solids*. Journal of Applied Physics, 2019. **126**(12).

165. Veprek, S., R.F. Zhang, and A.S. Argon, *Mechanical properties and hardness of boron and boron-rich solids*. Journal of Superhard Materials, 2011. **33**(6): p. 409-420.

166. Gilman, J.J., *Why Silicon Is Hard*. Science, 1993. **261**(5127): p. 1436-1439.

167. Demirskyi, D. and O. Vasylkiv, *Microstructure and mechanical properties of boron suboxide ceramics prepared by pressureless microwave sintering*. Ceramics International, 2016. **42**(12): p. 14282-14286.

168. Mukhanov, V.A., O.O. Kurakevych, and V.L. Solozhenko, *Thermodynamic Model of Hardness: Particular Case of Boron-Rich Solids*. Journal of Superhard Materials, 2010. **32**(3): p. 167-176.

169. Mukhanov, V.A., O.O. Kurakevych, and V.L. Solozhenko, *Hardness of materials at high temperature and high pressure*. Philosophical Magazine, 2009. **89**(25): p. 2117-2127.

170. Solozhenko, V.L., O.O. Kurakevych, and Y. Le Godec, *Creation of Nanostuctures by Extreme Conditions: High-Pressure Synthesis of Ultrahard Nanocrystalline Cubic Boron Nitride*. Advanced Materials, 2012. **24**(12): p. 1540-1544.

171. Polkowski, W., et al., *Microstructural characteristics and mechanical properties of Si-B alloys for functional and structural ultra-high temperature applications*. Journal of Alloys and Compounds, 2023. **935**: p. 167-172.

172. Zhang, H.D., et al., *BC8 Silicon (Si-III) is a Narrow-Gap Semiconductor*. Physical Review Letters, 2017. **118**(14): p. 146601.

173. Ayguzer Yasar, Z., K. Christian, and R.A. Haber, *The effect of silicon and boron doping on the microstructure of arc melted boron carbide*. Materialia, 2025. **39**: p. 1023-1035.
<figure><img src="image_26.png" /></figure>