A temperature-pressure phase diagram modeling method and device based on a grover model

Abstract

The application relates to the technical field of temperature-pressure phase diagram modeling, and discloses a temperature-pressure phase diagram modeling method based on a Grover model, which comprises the following steps: S1: obtaining the thermal expansion coefficient, volume and isothermal bulk modulus thermodynamic data of a rare earth tantalate system under different temperature and pressure conditions; S2: respectively correcting the molar volume and the isothermal compressibility under standard atmospheric pressure, reducing the influence of excessive sensitivity of the heat capacity on the coupling relationship between the temperature and the pressure, so as to realize the correction of the Grover model; and S3: obtaining the Gibbs free energy change through the corrected Grover model state equation, and linearly superimposing the Gibbs free energy change and the Gibbs free energy under standard atmospheric pressure given by a thermodynamic database to obtain a temperature-pressure phase diagram model. The application can ensure the accuracy of the obtained Gibbs free energy, thereby ensuring stable modeling and ensuring the reliability of the phase diagram.

Description

Technical Field

[0001] This invention relates to the field of temperature and pressure phase diagram modeling technology, specifically to a method and apparatus for temperature and pressure phase diagram modeling based on the Grover model. Background Technology

[0002] Phase diagrams, as a cornerstone tool in physical chemistry and materials science, are essentially geometric diagrams describing the stable phase composition and evolution of multiphase equilibrium systems under different thermodynamic conditions (such as temperature, pressure, and component concentration). By selecting different state functions as coordinate axes, multivariate phase diagrams applicable to various systems can be constructed, providing an indispensable thermodynamic framework for understanding the composition-structure-property relationships of materials.

[0003] In materials science and engineering practice, the vast majority of research objects are in the condensed state (solid and liquid). Therefore, phase diagrams of condensed systems, such as those of metallic alloys and ceramic systems, have the broadest guiding significance. However, phase diagrams only indicate the thermodynamic equilibrium state that a system eventually reaches under given conditions; they are static maps of a "starting point" and an "ending point," but they cannot reveal the "path" from the starting point to the ending point—that is, the phase transformation kinetics. In actual processes, due to limitations such as cooling rate and atomic diffusion capacity, the system often cannot reach the equilibrium state with the lowest global energy. Instead, it forms a series of metastable phases with slightly higher energy that can persist for a long time (such as martensite in steel and diamond in carbon materials). These crucial non-equilibrium structures are precisely what phase diagrams cannot predict, but they are key to the control of material properties. In a narrow sense, a phase diagram specifically refers to a phase equilibrium state diagram, which accurately depicts the phase region distribution, phase boundaries, and characteristic points (such as eutectic points and peritectic points) of a material under different parameters (most commonly temperature and composition). This powerful visualization tool plays a central role in mineralogy, metallurgy, and advanced materials design.

[0004] Among these advancements, the high-temperature, high-pressure phase diagram expands our understanding of matter's behavior, revealing stable phases and phase transitions under extreme conditions (such as the Earth's core, planetary formation processes, or the environments in which artificially synthesized superhard materials are created). For example, under high pressure, graphite transforms into diamond, and ordinary gases can be compressed into a metallic state. These new phases often exhibit unique physicochemical properties unattainable under conventional conditions. Therefore, the study of matter's behavior under high and even ultra-high pressure has become a highly interdisciplinary frontier field, closely linking solid-state physics, deep Earth science, planetary astrophysics, high-pressure chemistry, and the development of new materials.

[0005] Currently, phase diagram modeling methods based on computational materials science have become an important development direction, especially the combination of first-principles calculations and the CALPHAD method. Nevertheless, existing temperature-pressure phase diagram modeling methods still face many prominent problems: the applicability of the equation of state under high pressure conditions is limited, and the commonly used Murnaghan equation of state shows significant deviations in the high-pressure range (especially when the pressure exceeds half of the bulk modulus), making it difficult to accurately describe the volume-pressure relationship of materials under ultra-high pressure; traditional models have not yet taken into account the negative heat capacity and thermal expansion anomalies that occur under high pressure, resulting in inaccurate Gibbs free energy and thus affecting the reliability of phase diagram modeling. Summary of the Invention

[0006] To overcome the shortcomings of the prior art, one of the objectives of this invention is to provide a temperature-pressure phase diagram modeling method based on the Grover model, which eliminates the negative heat capacity and thermal expansion anomaly problems that occur in the traditional equation of state under high pressure, ensuring the accuracy of the obtained Gibbs free energy, thereby ensuring stable modeling and the reliability of the phase diagram.

[0007] The technical solution adopted in this invention is as follows: A temperature-pressure phase diagram modeling method based on the Grover model, comprising the following steps:

[0008] S1: Obtain the initial crystal structure of the rare earth tantalate system, optimize the initial crystal structure of the rare earth tantalate system based on first-principles calculations, and obtain the cold energy, electronic density of states, and Fermi level of the rare earth tantalate system. Obtain the thermal expansion coefficient, volume, and isothermal modulus thermodynamic data of the rare earth tantalate system under different temperature and pressure conditions through the cold energy, electronic density of states, and Fermi level.

[0009] S2: Correct the molar volume and isothermal compressibility under standard atmospheric pressure respectively to reduce the excessive sensitivity of heat capacity to the coupling relationship between temperature and pressure, thereby correcting the state equation of the Grover model.

[0010] S3: The change in Gibbs free energy is obtained through the modified Grover model equation of state. The change in Gibbs free energy is linearly superimposed with the Gibbs free energy under standard atmospheric pressure given by the thermodynamic database to obtain the temperature-pressure phase diagram model, as shown in formulas (1) and (2).

[0011] (1),

[0012] (2),

[0013] In formula (1), This represents the change in Gibbs free energy. For parameters to be optimized, The isothermal compressibility coefficient under standard atmospheric pressure. The molar volume under standard atmospheric pressure. The molar volume includes temperature and pressure;

[0014] In formula (2), For Gibbs free energy, Let Gibbs free energy be the energy along the isobars under atmospheric pressure. This represents the change in Gibbs free energy.

[0015] In a preferred embodiment of the present invention, in S2, an exponential decay factor is introduced into the temperature dependence term in the isothermal compressibility coefficient expression to correct the molar volume under standard atmospheric pressure, thereby suppressing the excessive contribution of the molar volume under standard atmospheric pressure to thermal expansion, as shown in formulas (3) and (4).

[0016] (3),

[0017] (4),

[0018] In formulas (3) and (4), To obtain from reference temperature up to the current temperature The cumulative thermal expansion effect, For temperature, For pressure, To cut off the pressure, , , These are the polynomial fitting coefficients. The molar volume under standard atmospheric pressure. For 1 bar and reference temperature molar volume below It is an exponential decay factor.

[0019] In a preferred embodiment of the present invention, in S2, in order to suppress the generation of negative heat capacity under high pressure, an exponential decay factor is introduced into the temperature dependence term in the expression of the isothermal compressibility coefficient to correct the isothermal compressibility coefficient, thereby correcting the temperature dependence of the isothermal compressibility coefficient, as shown in formulas (5) and (6).

[0020] (5),

[0021] (6),

[0022] In formulas (5) and (6), This is the corrected isothermal compressibility coefficient. , , , These are the fitting coefficients in the polynomial. For temperature, For pressure, This is the cutoff pressure.

[0023] The second objective of this invention is to provide a temperature-pressure phase diagram modeling device based on the Grover model, comprising a thermodynamic data calculation unit and a temperature-pressure phase diagram modeling unit. The thermodynamic data calculation unit is used to virtually calculate the thermodynamic data of each phase of the rare earth tantalate system. The temperature-pressure phase diagram modeling unit is used to substitute the calculated thermodynamic data as input into the temperature-pressure phase diagram formula with temperature and pressure as independent variables to obtain the Gibbs free energy and establish the temperature-pressure phase diagram model, as shown in formulas (1) and (2).

[0024] (1),

[0025] (2),

[0026] In formula (1), This represents the change in Gibbs free energy. For parameters to be optimized, The isothermal compressibility coefficient under standard atmospheric pressure. The molar volume under standard atmospheric pressure. The molar volume includes temperature and pressure;

[0027] In formula (2), For Gibbs free energy, Let Gibbs free energy be the energy along the isobars under atmospheric pressure. This represents the change in Gibbs free energy.

[0028] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0029] This invention combines the double-shielded Coulomb correction method with MFP mean field potential calculation to construct an automated, cross-scale modeling system from first-principles calculations to temperature-pressure phase diagram output. By introducing Grover's equation of state, this method addresses the anomalous coupling problem between thermal expansion and compressibility under high pressure. It introduces a pressure-dependent exponential decay term into the polynomials of thermal expansion coefficient and compressibility, thereby achieving physical coupling between temperature and pressure. This successfully eliminates the negative heat capacity and thermal expansion anomaly problems that occur in traditional equations of state under high pressure, ensuring the accuracy of the obtained Gibbs free energy, thus ensuring stable modeling and the reliability of the phase diagram. Attached Figure Description

[0030] Figure 1 This is a flowchart of the temperature-pressure phase diagram modeling method based on the Grover model of the present invention;

[0031] Figure 2 YTaO4 obtained by the method of this invention Figure showing the fitting effect of the volumetric thermal expansion coefficient under the phase. Figure of integral thermal expansion fitting effect, and figures of fitting residual and relative error analysis;

[0032] Figure 3 YTaO4 obtained by the method of this invention Figure showing the fitting effect of the volumetric thermal expansion coefficient under the phase. Figure of integral thermal expansion fitting effect, and figures of fitting residual and relative error analysis;

[0033] Figure 4 YTaO4 obtained by the method of this invention Figure showing the fitting effect of the volumetric thermal expansion coefficient under the phase. Figure of integral thermal expansion fitting effect, and figures of fitting residual and relative error analysis;

[0034] Figure 5 YTaO4 obtained by the method of this invention Look down The graph shows the fitting results of parameters (isothermal compressibility coefficient), the comparison of fitting methods (quadratic, linear, exponential), the comparison of R2 (coefficient of determination), the comparison of RMSE (root mean square error), the best fit display graph, and the residual analysis graph.

[0035] Figure 6 YTaO4 obtained by the method of this invention Look down The graph shows the fitting results of parameters (isothermal compressibility coefficient), the comparison of fitting methods (quadratic, linear, exponential), the comparison of R2 (coefficient of determination), the comparison of RMSE (root mean square error), the best fit display graph, and the residual analysis graph.

[0036] Figure 7 YTaO4 obtained by the method of this invention Look down The graph shows the fitting results of parameters (isothermal compressibility coefficient), the comparison of fitting methods (quadratic, linear, exponential), the comparison of R2 (coefficient of determination), the comparison of RMSE (root mean square error), the best fit display graph, and the residual analysis graph.

[0037] Figure 8 YTaO4 obtained by the method of this invention is respectively in , Gibbs free energy at 0-50 GPa;

[0038] Figure 9 YTaO4 obtained by the method of this invention is respectively in , The volumetric thermal expansion coefficient at 0-50 GPa;

[0039] Figure 10 YTaO4 obtained by the method of this invention is respectively in , Isothermal bulk modulus at 0-50 GPa;

[0040] Figure 11 This is a temperature-pressure phase diagram of YTaO4 obtained by the method of this invention at temperatures of 0-2000K and pressures of 0-50GPa. Detailed Implementation

[0041] Typical embodiments embodying the features and advantages of the present invention will be specifically described in the following description. It should be understood that the present invention can have various variations in different embodiments without departing from the scope of the present invention, and the descriptions and illustrations herein are for illustrative purposes only and not intended to limit the present invention.

[0042] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.

[0043] This embodiment discloses a temperature-pressure phase diagram modeling method based on the Grover model, such as... Figure 1 As shown, it includes the following steps:

[0044] S1: Obtain the initial crystal structure of the rare earth tantalate system, optimize the initial crystal structure of the rare earth tantalate system based on first-principles calculations, and obtain the cold energy, electronic density of states, and Fermi level of the rare earth tantalate system. Obtain the thermal expansion coefficient, volume, and isothermal modulus thermodynamic data of the rare earth tantalate system under different temperature and pressure conditions through the cold energy, electronic density of states, and Fermi level.

[0045] Specifically, S1 also includes the following steps:

[0046] S11: Obtain the initial crystal structure of the rare earth tantalate system from the Materials Project database, then preprocess the initial crystal structure of the rare earth tantalate system using Vesta software. The preprocessed initial crystal structure of the rare earth tantalate system is used to establish an atomic structure model of the rare earth tantalate system using a special quasi-random structure method. The atomic structure model of the rare earth tantalate system is then input into VASP software to perform structural optimization and convergence of the atomic structure of the rare earth tantalate system.

[0047] In this embodiment, the Materials Project database generates crystal structure information based on first-principles calculations (DFT), including unit cell parameters, atomic coordinates, and space group symmetry. First-principles calculations are a method based on quantum mechanics, which directly predicts the microscopic structure and macroscopic properties of matter by solving the Schrödinger equation. VASP (Vienna Ab-initioSimulation Package) is one of the core tools for performing first-principles calculations.

[0048] S12: The atomic structure of the rare earth tantalate system after structural optimization convergence was obtained by applying the double-shielded coulomb correction method to rare earth elements. electronic or The electrons are corrected to make the atomic structure of the rare earth tantalate system more stable at the electronic level. Then, under the first principle calculation framework, the cell volume parameter of the rare earth tantalate system is changed. The structural relaxation calculation of the atomic structure of the rare earth tantalate system under different volumes is performed by static calculation until complete convergence, thereby obtaining a dataset of the system energy changing with the cell volume. The energy-cell volume data obtained is fitted based on the fourth-order Birch-Murnaghan equation of state, as shown in formulas (8), (9), and (10), to obtain the energy-volume curve of the rare earth tantalate system.

[0049] (8),

[0050] (9),

[0051] (10)

[0052] In formulas (8), (9), and (10), For a volume of Energy at that time In reference volume The corresponding energy under zero pressure, The bulk elastic modulus under zero pressure. For the volume under zero pressure, For a volume of The pressure they endured at that time , These are parameters related to the bulk elastic modulus. This is a dimensionless parameter related to volume.

[0053] Static calculation is a basic calculation mode in first-principles calculations (such as VASP, Quantum ESPRESSO, etc.). Its core objective is to calculate the ground-state energy, electronic structure, and other properties of a system under the condition of fixed atomic positions and unit cell parameters.

[0054] S13: The crystal structure and corresponding energy-volume of the rare earth tantalate system obtained by static calculation under different unit cell volumes are fitted to the average field potential function, as shown in formula (11). After generating representative structure samples through the average field potential function, the self-consistent electronic structure is further calculated to obtain the electronic state density, cold energy and Fermi level of the rare earth tantalate system.

[0055] (11),

[0056] In formula (11), The average field potential function is a function of the ion deviation distance r and the unit cell volume V. It is used to describe the potential energy of lattice ions in rare earth tantalate systems when they deviate from their equilibrium positions at different unit cell volumes. This represents the distance by which a lattice ion deviates from its equilibrium position. is the lattice constant. The unit cell volume, For parameters, For a specific location ( It can be the energy value corresponding to a certain energy-volume relationship at different points (such as R+r, R−r, or R).

[0057] S14: Cold energy, electronic density of states, and Fermi level are calculated using MFP commands to generate input files in vec.dat, vfermi.dat, and dos.dat formats, respectively. Then, the corresponding input files are called through the control file and combined with the physical model to obtain the thermal expansion coefficient, volume, and isothermal modulus thermodynamic data of the rare earth tantalate system under different temperatures and pressures.

[0058] In this embodiment, the physical model includes a lattice dynamics model, an electronic structure model, and a thermodynamic statistical model.

[0059] In this implementation, the coefficient of thermal expansion is the initial value. The volume is the initial volume. (1 bar (approximately standard atmospheric pressure at low pressure) and reference temperature) The molar volume at that point), and the isothermal modulus, which is the initial value. The reciprocal of.

[0060] In this embodiment, the control file, also known as the control script, controls the MFP2 program. It primarily calculates the contributions of ion thermal vibrations and electronic thermal excitations to free energy, internal energy, and specific heat using the Mean Field Potential (MFP) model and the Debye model. It supports calculations using various thermodynamic paths, including isotherms, Hugonius lines, isentropic lines, phase diagrams, and high-pressure melting line predictions. Users need to provide data files such as Fermi energy, electronic density of states, and cold energy, and select the model (e.g., ion vibration model, Debye temperature generation method, electronic excitation model) and set the computational grid (temperature, pressure) through the input file. During use, it is important to ensure sufficient cold energy data range, avoid low-temperature numerical overflow, and select appropriate model combinations based on the calculation temperature range (e.g., MFP for high temperatures, Debye with quantum correction for low temperatures) to ensure the reasonableness and stability of the results.

[0061] In this embodiment, the thermal expansion coefficient, heat capacity, and volumetric thermodynamic data of the rare earth tantalate system in the literature, as well as the thermal expansion coefficient, heat capacity, and volumetric thermodynamic data of the rare earth tantalate system in the experimental data, and the thermal expansion coefficient, heat capacity, and volume calculated above are integrated as input data for the temperature-pressure phase diagram model.

[0062] S2: Correct the molar volume and isothermal compressibility under standard atmospheric pressure respectively to reduce the excessive sensitivity of heat capacity to the coupling relationship between temperature and pressure, thereby correcting the state equation of the Grover model.

[0063] Specifically, in S2, in order to prevent non-physical anomalies from occurring in the Grover model in the high-temperature and high-pressure coupling region and to keep thermal expansion constant under high pressure, an exponential decay factor is introduced into the temperature dependence term in the isothermal compressibility coefficient expression to correct the molar volume under standard atmospheric pressure, thereby suppressing the excessive contribution of the molar volume under standard atmospheric pressure to thermal expansion, as shown in equations (3) and (4).

[0064] (3),

[0065] (4),

[0066] In formulas (3) and (4), To obtain from reference temperature up to the current temperature The cumulative thermal expansion effect (volume thermal expansion coefficient). For temperature, For pressure, The cutoff pressure is 109. , , , These are the polynomial fitting coefficients. The molar volume under standard atmospheric pressure. 1 bar (approximate to standard atmospheric pressure at low pressure) and reference temperature Molar volume below;

[0067] In this embodiment, the thermal expansion behavior described by the cumulative thermal expansion effect in the Grover model is set to be only related to temperature and independent of pressure. This simplification leads to non-physical anomalies in the high-temperature and high-pressure coupling region. That is, as the temperature rises, thermal expansion not only continues to increase, but its rate of increase (second derivative) may also become excessively high. This abnormal thermal expansion behavior is transmitted through thermodynamic relationships, leading to non-physical drastic changes or even negative values ​​in the isobaric heat capacity, ultimately causing distortion in the temperature-pressure phase diagram prediction based on the Grover model.

[0068] To maintain accurate fitting of experimental data in low-pressure regions while ensuring reasonable thermodynamic behavior (i.e., thermal expansion stabilization) of the model in high-pressure regions, this invention modifies the Grover model. In the temperature polynomial expression for the cumulative thermal expansion effect, higher-order temperature terms (such as...) are modified. , (These are the main factors leading to abnormal growth) Introducing an exponential decay factor related to pressure. The function of this factor is to, at low pressure ( ≪ The value is approximately 1, which does not affect the fitting of the original data; under high pressure ( Close to or greater than The thermal expansion will significantly decrease, thereby suppressing the excessive growth of thermal expansion under high pressure and causing it to tend to a stable value.

[0069] Specifically, in S2, in order to suppress the generation of negative heat capacity under high pressure, an exponential decay factor is introduced into the temperature dependence term in the expression of the isothermal compressibility coefficient to correct the isothermal compressibility coefficient, thereby correcting the dependence of the isothermal compressibility coefficient on temperature, as shown in formulas (5) and (6).

[0070] (5),

[0071] (6),

[0072] In formulas (5) and (6), This is the corrected isothermal compressibility coefficient. , , , These are the fitting coefficients in the polynomial. For temperature, For pressure, This is the cutoff pressure.

[0073] According to thermodynamic laws, the isobaric heat capacity can be obtained. With constant volume heat The difference satisfies ,in The coefficient of volumetric thermal expansion is 1. Absolute temperature For molar volume, This is the isothermal compressibility coefficient. Therefore, it can be seen that the isothermal compressibility coefficient's dependence on temperature directly affects... The value, if It increases significantly with increasing temperature, and will lead to [further consequences] under high pressure. Excessive compression could even lead to non-physical anomalies in negative heat capacity. Therefore, to ensure the thermodynamic consistency of the model across the entire temperature and pressure space, it is necessary to... The temperature dependence is constrained.

[0074] In this embodiment, by simultaneously introducing pressure-dependent exponential decay corrections to the molar volume and isothermal compressibility under standard atmospheric pressure reference conditions, their excessive sensitivity to temperature under high pressure can be effectively suppressed. This synergistic correction ensures that the thermal expansion behavior and equation of state derived from them remain reasonable under extreme conditions, thereby fundamentally preventing the occurrence of negative heat capacity anomalies. This results in reliable prediction accuracy and physical consistency of the temperature-pressure phase diagram based on the Grover model over a wide range (especially in the high-pressure region).

[0075] In this embodiment, the initial coefficient of thermal expansion The reciprocal of the initial isothermal modulus is .

[0076] S3: The change in Gibbs free energy is obtained through the modified Grover model equation of state. The change in Gibbs free energy is linearly superimposed with the Gibbs free energy at standard atmospheric pressure given by the thermodynamic database or CALPHAD assessment to obtain the temperature-pressure phase diagram model, as shown in formulas (1) and (2).

[0077] (1),

[0078] (2),

[0079] In formula (1), This represents the change in Gibbs free energy. For parameters to be optimized, The isothermal compressibility coefficient under standard atmospheric pressure. The molar volume under standard atmospheric pressure. The molar volume includes temperature and pressure;

[0080] In formula (2), For Gibbs free energy, Let Gibbs free energy be the energy along the isobars under atmospheric pressure. This represents the change in Gibbs free energy.

[0081] Specifically:

[0082] S31: Using the modified Grover model equation of state, combined with the definition of isothermal bulk modulus and thermodynamic relationships, a complete expression describing the molar volume and isothermal compressibility coefficient of the rare earth tantalate system at arbitrary temperature and pressure is established, as shown in formula (7).

[0083] (7),

[0084] In formula (7), The molar volume includes temperature and pressure. The molar volume under standard atmospheric pressure. For parameters to be optimized, The isothermal compressibility coefficient under standard atmospheric pressure. For pressure The isothermal compressibility coefficient at this temperature;

[0085] S32: By integrating the corrected molar volume containing temperature and pressure along the pressure, the change of Gibbs free energy with pressure can be obtained, as shown in formula (2).

[0086] (2),

[0087] In formula (2), This represents the change in Gibbs free energy. For parameters to be optimized, The isothermal compressibility coefficient under standard atmospheric pressure. The molar volume under standard atmospheric pressure. The molar volume includes temperature and pressure;

[0088] S33: Then, the change in Gibbs free energy is directly added to the Gibbs free energy on the standard atmospheric pressure isobar obtained from thermodynamic databases (such as SGTE) or CALPHAD assessments to construct a complete thermodynamic phase diagram model with temperature and pressure as variables, as shown in formula (1).

[0089] (1)

[0090] In formula (1), For Gibbs free energy, Let Gibbs free energy be the energy along the isobars under atmospheric pressure. This represents the change in Gibbs free energy.

[0091] In this embodiment, It is a key fitting parameter in high-pressure thermodynamic models, used to describe the sensitivity of a material's volume to pressure changes, and quantifies the material's ability to resist compression. A higher value indicates that the material is easier to compress. The smaller the value, the more difficult it is to compress. This is obtained by fitting high-pressure experimental data. (Right now The parameters can accurately predict the volumetric behavior of materials under high pressure.

[0092] In this embodiment, Gibbs free energy is the core thermodynamic quantity for constructing and interpreting the phase diagram, while the phase diagram is a visual representation of how Gibbs free energy changes with temperature and pressure.

[0093] In this embodiment, the rare earth tantalate system has , , Three phases, The phase is the most common stable monoclinic phase. The phase is the most important functional phase, and it has excellent luminescent properties as a scintillator. The phase is a high-temperature stable tetragonal phase;

[0094] Rare earth tantalate system Phase temperature-pressure phase diagram model:

[0095]

[0096] Rare earth tantalate system Phase temperature-pressure phase diagram model:

[0097]

[0098] Rare earth tantalate system Phase temperature-pressure phase diagram model:

[0099]

[0100] The temperature-pressure phase diagram modeling method disclosed in this embodiment is particularly suitable for describing the phase behavior of solids under extreme conditions.

[0101] This embodiment discloses a temperature-pressure phase diagram modeling device based on the Grover model, including a thermodynamic data calculation unit and a temperature-pressure phase diagram modeling unit. The thermodynamic data calculation unit is used to virtually calculate the thermodynamic data of each phase of the rare earth tantalate system. The temperature-pressure phase diagram modeling unit is used to substitute the calculated thermodynamic data as input into the temperature-pressure phase diagram formula with temperature and pressure as independent variables to obtain the Gibbs free energy and establish the temperature-pressure phase diagram model, as shown in formulas (1) and (2).

[0102] (1),

[0103] (2),

[0104] In formula (1), This represents the change in Gibbs free energy. For parameters to be optimized, The isothermal compressibility coefficient under standard atmospheric pressure. The molar volume under standard atmospheric pressure. The molar volume includes temperature and pressure;

[0105] In formula (2), For Gibbs free energy, Let Gibbs free energy be the energy along the isobars under atmospheric pressure. This represents the change in Gibbs free energy.

[0106] First, you need to prepare a thermodynamic database (TDB file) containing Grover model parameters, which must include four key parameters: , , and These parameters are typically obtained through first-principles calculations or fitting experimental data. Before the calculations, we add these parameter definitions to the pycalphad parser using a Python script to ensure correct reading of the TDB file.

[0107] The calculation process begins by defining the grid range for pressure (P) and temperature (T), typically covering pressures from 0 to tens of GPa and temperatures from room temperature to thousands of K. The program first loads the TDB file and identifies the phases within it (such as BCC, HCP, etc.). For each phase, pycalphad is used to calculate its Gibbs free energy G(T,P0) at a reference pressure P0 (typically 1 bar). Simultaneously, four volumetric parameters for that phase are extracted from the database.

[0108] Subsequently, the molar volume under arbitrary pressure P was calculated using Grover's equation of state. This equation, based on the exponential integral function and its inverse function, uses numerical solutions to calculate the volume under the reference pressure. The volume change is converted to the target pressure. The calculated volume change is used to determine the pressure-induced Gibbs free energy correction. This correction term is expressed by the formula. Calculation. Finally, the total Gibbs free energy of each phase at a given (T, P) (temperature, pressure) is: .

[0109] After calculating the Gibbs free energy at all grid points, the stable phase is determined by comparing the Gibbs free energy values ​​of each phase: at each (T, P) point, the phase with the lowest Gibbs free energy is the most stable phase. The program automatically identifies phase boundaries (i.e., points where the Gibbs free energies of two phases are equal) and generates a temperature-pressure phase diagram. The phase diagram is displayed as a two-dimensional curve, where different regions represent the stable existence range of different phases, and the phase boundary lines represent the equilibrium conditions between the two phases.

[0110] In addition to being displayed graphically, the calculation results are automatically output as structured data files. Gibbs free energy data are saved as Excel files, with each worksheet corresponding to one phase, and rows and columns representing temperature and pressure values, facilitating subsequent analysis and plotting. Simultaneously, the raw data is also saved in text format for use by other software. The entire process is highly automated; users only need to provide the correct TDB file and set the calculation range to obtain complete phase diagrams and related thermodynamic data, providing an important reference for research on high-pressure, high-temperature materials.

[0111] Figure 11 The temperature-pressure phase diagram shows that YTaO4 exists in a monoclinic phase in the low-temperature region (0–500 K) almost throughout the entire pressure range studied. The phase is stable, but changes occur under low to moderate pressure (<20 GPa) conditions as the temperature increases. →M structural phase transition. Upon further heating, the material transforms from a monoclinic phase to a tetragonal T phase in the high-temperature region (>2000 K), a transformation requiring the lowest temperature under moderate pressure. After entering the high-pressure region (>20 GPa), the pressure stabilizes again. The phase causes a significant contraction in the stable region of the M phase, and passes through at medium and high temperatures. →The T phase transition leads to a tetragonal structure. Overall, low temperature and high pressure favor low-symmetry monoclinic phases ( The tetragonal T phase is stable under high temperature conditions, driven by entropy effects, while the M phase exists around 23 GPa and 2000 K. The three-phase region with phase T.

[0112] The above embodiments are merely preferred embodiments of the present invention and should not be construed as limiting the scope of protection of the present invention. Any non-substantial changes and substitutions made by those skilled in the art based on the present invention shall fall within the scope of protection claimed by the present invention.

Claims

1. A method for modeling temperature-pressure phase diagrams based on the Grover model, characterized in that: Includes the following steps: S1: Obtain the initial crystal structure of the rare earth tantalate system, optimize the initial crystal structure of the rare earth tantalate system based on first-principles calculations, and obtain the cold energy, electronic density of states, and Fermi level of the rare earth tantalate system. Obtain the thermal expansion coefficient, volume, and isothermal modulus thermodynamic data of the rare earth tantalate system under different temperature and pressure conditions through the cold energy, electronic density of states, and Fermi level. S2: Correct the molar volume and isothermal compressibility under standard atmospheric pressure respectively to reduce the excessive sensitivity of heat capacity to the coupling relationship between temperature and pressure, thereby correcting the state equation of the Grover model. S3: The change in Gibbs free energy is obtained through the modified Grover model equation of state. The change in Gibbs free energy is linearly superimposed with the Gibbs free energy under standard atmospheric pressure given by the thermodynamic database to obtain the temperature-pressure phase diagram model, as shown in formulas (1) and (2). （1）， （2）， In formula (1), This represents the change in Gibbs free energy. For parameters to be optimized, The isothermal compressibility coefficient under standard atmospheric pressure. The molar volume under standard atmospheric pressure. The molar volume includes temperature and pressure; In formula (2), For Gibbs free energy, This represents the Gibbs free energy of isobars under standard atmospheric pressure. This represents the change in Gibbs free energy.

2. The temperature and pressure phase diagram modeling method based on the Grover model according to claim 1, characterized in that: In S2, the molar volume under standard atmospheric pressure is corrected by introducing an exponential decay factor into the temperature dependence term in the expression for the isothermal compressibility coefficient, thereby suppressing the excessive contribution of the molar volume under standard atmospheric pressure to thermal expansion, as shown in formulas (3) and (4). （3）， （4）， In formulas (3) and (4), To obtain from reference temperature up to the current temperature The cumulative thermal expansion effect, For temperature, For pressure, To cut off the pressure, , , These are the polynomial fitting coefficients. The molar volume under standard atmospheric pressure. For 1 bar and reference temperature molar volume below It is an exponential decay factor.

3. The temperature and pressure phase diagram modeling method based on the Grover model according to claim 1, characterized in that: In S2, in order to suppress the generation of negative heat capacity under high pressure, an exponential decay factor is introduced into the temperature dependence term in the expression of the isothermal compressibility coefficient to correct the isothermal compressibility coefficient, thereby correcting the dependence of the isothermal compressibility coefficient on temperature, as shown in formulas (5) and (6). （5）， （6）， In formulas (5) and (6), This is the corrected isothermal compressibility coefficient. , , , These are the fitting coefficients in the polynomial. For temperature, For pressure, This is the cutoff pressure.

4. The temperature and pressure phase diagram modeling method based on the Grover model according to claim 1, characterized in that: In S3, the Gibbs free energy change is obtained through the modified Grover model state equation, including the following steps: S31: Using the modified Grover model equation of state, combined with the definition of isothermal bulk modulus and thermodynamic relationships, an expression describing the molar volume and isothermal compressibility coefficient of the rare earth tantalate system at arbitrary temperature and pressure is established, as shown in formula (7). （7）， In formula (7), The molar volume includes temperature and pressure. The molar volume under standard atmospheric pressure. For parameters to be optimized, The isothermal compressibility coefficient under standard atmospheric pressure. For pressure The isothermal compressibility coefficient at this temperature; S32: By integrating the corrected molar volume containing temperature and pressure along the pressure, the change of Gibbs free energy with pressure can be obtained, as shown in formula (2). （2）， In formula (2), This represents the change in Gibbs free energy. For parameters to be optimized, The isothermal compressibility coefficient under standard atmospheric pressure. The molar volume under standard atmospheric pressure. This is the molar volume, which includes temperature and pressure.

5. The temperature and pressure phase diagram modeling method based on the Grover model according to claim 1, characterized in that: In S1, the initial crystal structure of the rare earth tantalate system is obtained from the Materials Project database. Then, the initial crystal structure of the rare earth tantalate system is preprocessed using Vesta software. The preprocessed initial crystal structure of the rare earth tantalate system is used to establish an atomic structure model of the rare earth tantalate system using a special quasi-random structure method. The atomic structure model of the rare earth tantalate system is then input into VASP software to perform structural optimization and convergence of the atomic structure of the rare earth tantalate system.

6. The temperature and pressure phase diagram modeling method based on the Grover model according to claim 1, characterized in that: In S1, the atomic structure of the rare earth tantalate system after structural optimization convergence is obtained by applying the double-shielded coulomb correction method to the rare earth elements. electronic or The electrons are corrected to make the atomic structure of the rare earth tantalate system more stable at the electronic level. Then, under the first principle calculation framework, the cell volume parameter of the rare earth tantalate system is changed. The structural relaxation calculation of the atomic structure of the rare earth tantalate system under different volumes is performed by static calculation until complete convergence, thereby obtaining a dataset of the system energy changing with the cell volume. The energy-cell volume data obtained is fitted based on the fourth-order Birch-Murnaghan equation of state, as shown in formulas (8), (9), and (10), to obtain the energy-volume curve of the rare earth tantalate system. （8）， （9）， （10）， In formulas (8), (9), and (10), For a volume of Energy at that time In reference volume The corresponding energy under zero pressure, The bulk elastic modulus under zero pressure. For the volume under zero pressure, For a volume of The pressure they endured at that time , These are parameters related to the bulk elastic modulus. A dimensionless parameter related to volume; The crystal structure and corresponding energy-volume of the rare earth tantalate system obtained by static calculation under different unit cell volumes are all fitted to the average field potential function, as shown in formula (11). After generating representative structure samples through the average field potential function, the self-consistent electronic structure is further calculated to obtain the electronic state density, cold energy and Fermi level of the rare earth tantalate system. （11）， In formula (11), This represents the average potential function. This represents the distance by which a lattice ion deviates from its equilibrium position. is the lattice constant. The unit cell volume, For parameters, For a specific location ( It can be the energy value corresponding to a certain energy-volume relationship at different points (such as R+r, R−r, or R).

7. The temperature and pressure phase diagram modeling method based on the Grover model according to claim 1, characterized in that: Rare earth tantalate system Phase temperature-pressure phase diagram model: Rare earth tantalate system Phase temperature-pressure phase diagram model: Rare earth tantalate system Phase temperature-pressure phase diagram model: 。 8. A Grover-based The system includes a thermodynamic data calculation unit and a temperature-pressure phase diagram modeling unit. The thermodynamic data calculation unit is used to virtually calculate the thermodynamic data of each phase of the rare earth tantalate system. The temperature-pressure phase diagram modeling unit is used to input the calculated thermodynamic data into the temperature-pressure phase diagram formula with temperature and pressure as independent variables to obtain the Gibbs free energy and establish a temperature-pressure phase diagram model. As shown in formulas (1) and (2), （1）， （2）， In formula (1), This represents the change in Gibbs free energy. For parameters to be optimized, The isothermal compressibility coefficient under standard atmospheric pressure. The molar volume under standard atmospheric pressure. The molar volume includes temperature and pressure; In formula (2), For Gibbs free energy, Let Gibbs free energy be the energy along the isobars under atmospheric pressure. This represents the change in Gibbs free energy.

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