Method for calculating vibration free energy based on key length-key stiffness relation

Abstract

The invention relates to the technical field of thermodynamic properties of alloy materials, in particular to a method for calculating vibration free energy based on a bond length-bond stiffness relationship, which comprises the following steps: selecting configurations with the same symmetry according to chemical bond information contained in a crystal structure, uniform stretching or compression deformation is applied to the key body so as to generate key lengths with different lengths; supercells are created for configurations with different key lengths, micro disturbance is applied to atoms with different symmetry in the supercells, and force is calculated; obtaining a force constant matrix between two adjacent atoms according to a force calculation result, and diagonalizing the force constant matrix to obtain tensile bond rigidity and bending bond rigidity; and fitting a key length-key stiffness function according to the tensile key stiffness and the bending key stiffness corresponding to different key lengths to obtain a force constant matrix of all possible key lengths so as to calculate vibration free energy. According to the method, the calculation cost is effectively reduced under the condition that the calculation precision is guaranteed, and a solution is provided for accurately predicting the alloy phase diagram and guiding alloy design.

Description

Technical Field

[0001] The present application relates to the technical field of thermodynamic properties of alloy materials, and in particular to a method for calculating vibrational free energy based on the bond length-bond stiffness relationship. Background Art

[0002] Equilibrium phase diagrams are at the core of current strategies to design new materials with specific microstructures or to optimize existing materials through integrated computational materials engineering. Phase diagrams provide information about the stable phases of a given composition at a given temperature and pressure and are one of the cornerstones of establishing the processing-property-structure of materials science.

[0003] The alloy phase diagram determined by the traditional Calphad method has been reported in the literature. Discrete thermodynamic data are obtained experimentally, and then different thermodynamic models, such as the sublattice model, the substitutional solution model, and the order-disorder model, are used to fit the thermodynamic formula in order to describe the phase equilibrium in a wider concentration range. Nevertheless, the Calphad method is a fitting process rather than a prediction process, which means that it is highly dependent on the accuracy of the experimental data and the rationality of the thermodynamic model, and the Calphad method is expensive and time-consuming. At the same time, the presence of impurities and the accuracy of the measurement will affect the phase diagram, and a long annealing time is required to achieve stability or distinguish between metastable and stable phases because slow kinetics hinder the convergence of thermodynamic equilibrium. Although some efforts have been made, such as replacing some experimental data with single phase quantities calculated by first principles to reduce time and cost, the efforts saved are still limited. In addition, there are often areas in the phase diagram with limited experimental information.

[0004] With the development of integrated computational materials engineering (ICME), there is an urgent need to obtain non-empirical thermodynamic descriptions based on experimental data without conventional optimization. In recent years, methods based on first-principles calculations and statistical physics principles supported by cluster expansion (CE) formalism have been developed. In this method, the mixing enthalpies of different configurations are obtained by density functional theory (DFT) calculations, while the mixing enthalpies on the convex hull in the enthalpy composition space are regarded as the ground state phase at 0K. The thermodynamic properties of each configuration depend on all possible microstates, whose properties can be determined simply and accurately by the CE formalism. Therefore, the Gibbs free energy of different phases, including the contribution of configurational entropy, can be obtained as a function of temperature from Monte Carlo (MC) simulations using statistical mechanics principles.

[0005] Accurate prediction of phase diagrams also requires including vibrational entropy contributions to determine free energies. Assuming the quasi-harmonic approximation holds, the thermal properties of solid materials can be traced back to the thermal properties of the phonon system, whose frequency can be volume dependent. Therefore, phonon calculations must be performed to determine the force constant matrix for each atom, which reflects the energetic response of the lattice vibrations at each temperature. Lattice vibrations are configurational degrees of freedom, i.e., any change in the atomic configuration will make the vibrations different. Therefore, it is not feasible to perform phonon calculations for solid solutions or phases with a certain solubility, as phonon calculations should be performed for all possible microstates. Summary of the invention

[0006] The embodiment of the present application provides a method for calculating vibrational free energy based on the bond length-bond stiffness relationship, and performs phonon calculations on all possible microscopic states through the bond length-bond stiffness relationship, thereby obtaining vibrational entropy and determining its contribution to the free energy of different intermetallic compounds. This method effectively reduces the calculation cost while ensuring the calculation accuracy, and provides a solution for accurately predicting alloy phase diagrams and guiding alloy design.

[0007] To solve the above technical problems, an embodiment of the present application provides a method for calculating vibration free energy based on the bond length-bond stiffness relationship, comprising the following steps: first, selecting a configuration with the same symmetry according to the chemical bond information contained in the crystal structure, and applying uniform tensile or compressive deformation thereto to produce bond lengths of different lengths, thereby obtaining a configuration with bond lengths of different lengths; then, creating a supercell for the configuration with different bond lengths, applying small perturbations to atoms with different symmetries in the supercell to produce atomic displacements, and calculating the force to obtain a force calculation result; next, obtaining a force constant matrix between two adjacent atoms according to the force calculation result, and diagonalizing it to obtain a tensile bond stiffness and a bending bond stiffness; finally, according to the tensile bond stiffness and the bending bond stiffness corresponding to different bond lengths, fitting the bond length-bond stiffness function to obtain a force constant matrix for all possible bond lengths, thereby calculating the vibration free energy.

[0008] In some exemplary embodiments, a configuration with the same symmetry is selected according to the chemical bond information contained in the crystal structure, and uniform tensile or compressive deformation is applied thereto to produce bond lengths of different lengths, thereby obtaining configurations with bond lengths of different lengths, including: step one, selecting a configuration with the same symmetry according to the chemical bond information contained in the crystal structure, and optimizing each configuration; step two, applying a uniform compressive strain or tensile strain of 10% to the optimized configuration with a strain interval of 2% to obtain 5 configurations of different volumes; step three, relaxing the configurations of different volumes, and this process only allows the atomic position to change while fixing the crystal volume to eliminate internal stress and obtain energy at 0K.

[0009] In some exemplary embodiments, in step 1, the electron exchange correlation is described by the generalized gradient approximation with the Perdew-Burke-Ernzerhof exchange correlation function and the ultrasoft pseudopotential, and the plane wave cutoff energy of the system is set by the convergence test, and the Monkhorst-Pack scheme is used with the k-point density as The Brillouin zone is sampled; the convergence criterion is: the energy convergence reaches 10 -7 Ry / cell, force convergence reaches 10 -6 Ry / Bohr, self-consistent convergence criterion reaches 10 -8 Ry / cell.

[0010] In some exemplary embodiments, in step three, the same convergence criterion as step one is adopted.

[0011] In some exemplary embodiments, a supercell is created for a configuration with different bond lengths, a small perturbation is applied to atoms with different symmetries in the supercell to generate atomic displacement, and a force is calculated to obtain a force calculation result, including: using a periodic boundary condition calculation in a first-principles calculation method to construct a reasonable supercell size to effectively eliminate the dipole effect; according to the symmetry information of the crystal structure, each atom with different symmetry is given a The displacement is used to generate disturbances; static calculations are performed using the first-principles method to obtain the forces on each atom in the x, y, and z directions.

[0012] In some exemplary embodiments, a reasonable supercell size contains 60-100 atoms.

[0013] In some exemplary embodiments, a force constant matrix between two adjacent atoms is obtained based on the force calculation results, and the force constant matrix is ​​diagonalized to obtain the tensile bond stiffness and the bending bond stiffness, including the following steps: based on the force calculation results, the force constant matrix between two atomic pairs is calculated; the force constant matrix is ​​subjected to Fourier transformation to obtain a dynamics matrix to calculate the phonon state density, obtain the vibration entropy, and thus calculate the vibration free energy.

[0014] In some exemplary embodiments, the force constant matrix between two atomic pairs is calculated as:

[0015]

[0016] Where Ψ(i, j) represents the force constant matrix between atoms i and j; E represents the total energy of the configuration; u i and u j represents the displacement of atoms i and j; the force constant matrix is ​​simplified to obtain a diagonal matrix related only to the tensile bond stiffness and the bending bond stiffness;

[0017]

[0018] Where s is the tensile bond stiffness and b is the bending bond stiffness.

[0019] In some exemplary embodiments, the vibration entropy is obtained by integrating the phonon frequency, and the vibration entropy is calculated as follows:

[0020]

[0021] Among them, S vib (V, T) represents vibration entropy; ω is the phonon frequency; is the reduced Planck constant, k B is the Boltzmann constant, g(ω) is the phonon state density; the vibrational free energy is obtained by multiplying the temperature by the vibrational entropy, and its calculation formula is:

[0022] F vib (V, T) = -TS vib (V, T)

[0023] Among them, F vib (V, T) represents vibrational free energy; T is temperature.

[0024] In some exemplary embodiments, the bond length-bond stiffness function is fitted according to the tensile bond stiffness and bending bond stiffness corresponding to different bond lengths to obtain the force constant matrix of all possible bond lengths, thereby calculating the vibration free energy, including: drawing a graph according to the tensile bond stiffness and bending bond stiffness corresponding to different bond lengths, and for all bond types in the same lattice, fitting their bond stiffnesses to obtain the bond length-bond stiffness function relationship; for a given configuration, obtaining the force constant matrix between each two atoms according to the bond length-bond stiffness function relationship, and calculating the vibration entropy and vibration free energy of the configuration.

[0025] The technical solution provided by the embodiments of the present application has at least the following advantages:

[0026] An embodiment of the present application provides a method for calculating vibration free energy based on the bond length-bond stiffness relationship, comprising the following steps: first, selecting a configuration with the same symmetry based on the chemical bond information contained in the crystal structure, and applying uniform tensile or compressive deformation thereto to produce bond lengths of different lengths, thereby obtaining configurations with bond lengths of different lengths; then, creating a supercell for the configurations with bond lengths of different lengths, applying small perturbations to atoms with different symmetries in the supercell to produce atomic displacements, and calculating the force to obtain a force calculation result; next, obtaining a force constant matrix between two adjacent atoms based on the force calculation result, and diagonalizing it to obtain a tensile bond stiffness and a bending bond stiffness; finally, fitting a bond length-bond stiffness function based on the tensile bond stiffness and the bending bond stiffness corresponding to different bond lengths to obtain a force constant matrix for all possible bond lengths, thereby calculating the vibration free energy.

[0027] The present application provides a bond length-bond stiffness relationship for phonon calculation of all possible microscopic states, thereby obtaining vibrational entropy and determining its contribution to the free energy of different intermetallic compounds. This method effectively reduces the computational cost while ensuring the accuracy of the calculation, and provides a solution for accurately predicting alloy phase diagrams and guiding alloys. The thermodynamic properties of each configuration depend on all possible microscopic states, and it is not feasible to calculate the configurational entropy and vibrational entropy of each configuration using the first principles and traditional supercell methods. The proposal based on cluster expansion (CE) successfully overcomes this problem, but it requires the energy and vibrational entropy of hundreds of configurations. The problem of calculating vibrational entropy is usually simplified to the phonon problem. As we all know, the calculation of phonons is time-consuming and expensive. The proposal of the bond length-bond stiffness relationship effectively solves this problem, and it only needs to fit the bond length-bond stiffness relationship of all bond types contained in the system to obtain the vibrational entropy of all configurations based on the same lattice type. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] One or more embodiments are exemplarily described by pictures in the corresponding drawings, and these exemplifications do not constitute limitations on the embodiments. Unless otherwise stated, the pictures in the drawings do not constitute proportional limitations.

[0029] Figure 1 A schematic flow chart of a method for calculating vibrational free energy based on the bond length-bond stiffness relationship provided in one embodiment of the present application.

[0030] Figure 2 A schematic diagram of the crystal structure of an ordered fcc Al-Li configuration provided in one embodiment of the present application.

[0031] Figure 3 A schematic diagram of the bond length-bond stiffness relationship in an fcc lattice provided for one embodiment of the present application.

[0032] Figure 4 A schematic diagram of the crystal structure of an ordered bcc Al-Li configuration provided in one embodiment of the present application.

[0033] Figure 5 A schematic diagram of the bond length-bond stiffness relationship in a bcc lattice provided for one embodiment of the present application.

[0034] Figure 6 A schematic diagram of the vibrational free energy of different stable phases in the Al-Li system provided in one embodiment of the present application.

[0035] Figure 7 A schematic diagram of the crystal structure of an ordered bcc Ni-Al configuration provided in one embodiment of the present application.

[0036] Figure 8A schematic diagram of the bond length-bond stiffness relationship in a bcc lattice provided for one embodiment of the present application.

[0037] Fig. 9 A schematic diagram of the crystal structure of an ordered fcc Ni-Al configuration provided in one embodiment of the present application.

[0038] Fig.10 A schematic diagram of the bond length-bond stiffness relationship in an fcc lattice provided for one embodiment of the present application.

[0039] Fig.11 A schematic diagram of the vibrational free energy of different stable phases in a Ni-Al system provided in one embodiment of the present application. DETAILED DESCRIPTION

[0040] As can be seen from the background technology, the traditional Calphad method is highly dependent on the accuracy of experimental data and the rationality of the thermodynamic model, and the Calphad method is expensive and time-consuming. Accurate prediction of phase diagrams requires the inclusion of vibrational entropy contributions to determine free energy, and phonon calculations must be performed to determine the force constant matrix for each atom. Since any change in the atomic configuration will make the vibration different, phonon calculations should be performed for all possible microstates, resulting in increased computational costs.

[0041] In order to solve the above technical problems, the embodiment of the present application provides a method for calculating vibration free energy based on the bond length-bond stiffness relationship, comprising the following steps: first, according to the chemical bond information contained in the crystal structure, a configuration with the same symmetry is selected, and uniform tensile or compressive deformation is applied to it to produce bond lengths of different lengths, so as to obtain configurations with bond lengths of different lengths; then, a supercell is created for the configurations with different bond lengths, and a small perturbation is applied to atoms with different symmetries in the supercell to produce atomic displacements, and the force is calculated to obtain the calculation result of the force; next, according to the calculation result of the force, the force constant matrix between two adjacent atoms is obtained, and it is diagonalized to obtain the tensile bond stiffness and the bending bond stiffness; finally, according to the tensile bond stiffness and the bending bond stiffness corresponding to different bond lengths, the bond length-bond stiffness function is fitted to obtain the force constant matrix of all possible bond lengths, so as to calculate the vibration free energy. The present application provides a method for calculating vibration free energy based on the bond length-bond stiffness relationship, and phonon calculation is performed on all possible microscopic states through the bond length-bond stiffness relationship, so as to obtain vibration entropy and determine its contribution to the free energy of different intermetallic compounds. This method effectively reduces the computational cost while ensuring the computational accuracy, and provides a solution for accurately predicting alloy phase diagrams and guiding alloy design.

[0042] The following will describe the various embodiments of the present application in detail with reference to the accompanying drawings. However, it will be appreciated by those skilled in the art that in the various embodiments of the present application, many technical details are provided in order to enable the reader to better understand the present application. However, even without these technical details and various changes and modifications based on the following embodiments, the technical solution claimed in the present application can be implemented.

[0043] See also Figure 1 , the embodiment of the present application provides a method for calculating vibration free energy based on the bond length-bond stiffness relationship, comprising the following steps:

[0044] Step S1, selecting a configuration with the same symmetry according to the chemical bond information contained in the crystal structure, and applying uniform tensile or compressive deformation to it to produce bond lengths of different lengths, thereby obtaining configurations with bond lengths of different lengths.

[0045] Step S2, creating a supercell for configurations with different bond lengths, applying small perturbations to atoms with different symmetries in the supercell to generate atomic displacements, and calculating the force to obtain the force calculation results.

[0046] Step S3: Obtain a force constant matrix between two adjacent atoms according to the force calculation result, and diagonalize it to obtain the tensile bond stiffness and the bending bond stiffness.

[0047] Step S4 fits the bond length-bond stiffness function according to the tensile bond stiffness and bending bond stiffness corresponding to different bond lengths to obtain the force constant matrix of all possible bond lengths, thereby calculating the vibration free energy.

[0048] In some embodiments, in step S1, a configuration with the same symmetry is selected according to the chemical bond information contained in the crystal structure, and uniform tensile or compressive deformation is applied thereto to produce bond lengths of different lengths, thereby obtaining configurations with bond lengths of different lengths, comprising the following steps:

[0049] Step 1: Select configurations with the same symmetry based on the chemical bond information contained in the crystal structure and optimize each configuration.

[0050] Step 2: Apply 10% uniform compressive strain or tensile strain to the optimized configuration with a strain interval of 2% to obtain 5 configurations with different volumes.

[0051] Step 3: Relax the configurations of different volumes. This process only allows the atomic positions to change while keeping the crystal volume unchanged to eliminate internal stress and obtain energy at 0K.

[0052] In some embodiments, in step 1, the electron exchange correlation is described by the generalized gradient approximation with the Perdew-Burke-Ernzerhof exchange correlation function and the ultrasoft pseudopotential, and the plane wave cutoff energy of the system is set by the convergence test, and the Monkhorst-Pack scheme is used with the k-point density as The Brillouin zone is sampled; the convergence criterion is: the energy convergence reaches 10 -7 Ry / cell, force convergence reaches 10 -6 Ry / Bohr, self-consistent convergence criterion reaches 10 -8 Ry / cell.

[0053] In some embodiments, in step three, the same convergence criteria as step one are used.

[0054] In some embodiments, in step S2, creating a supercell for configurations with different bond lengths, applying small perturbations to atoms with different symmetries in the supercell to generate atomic displacements, and calculating forces to obtain force calculation results includes the following steps:

[0055] Step S201: using periodic boundary condition calculation in the first principle calculation method to construct a reasonable supercell size to effectively eliminate the dipole effect.

[0056] Step S202: According to the symmetry information of the crystal structure, each atom with different symmetry is given displacement to produce disturbance.

[0057] Step S203: Perform static calculations using a first-principles method to obtain the force conditions of each atom in the x, y, and z directions.

[0058] In some embodiments, a reasonable supercell size in step S201 includes 60 to 100 atoms.

[0059] In some embodiments, in step 3, the force constant matrix between two adjacent atoms is obtained according to the force calculation result, and the force constant matrix is ​​diagonalized to obtain the tensile bond stiffness and the bending bond stiffness, comprising the following steps:

[0060] Step S301: Calculate the force constant matrix between two atomic pairs according to the force calculation result.

[0061] Step S302: The force constant matrix is ​​subjected to Fourier transformation to obtain a dynamics matrix to calculate the phonon state density, obtain the vibration entropy, and thus calculate the vibration free energy.

[0062] In some embodiments, the calculation formula of the force constant matrix between two atomic pairs in step S301 is:

[0063]

[0064] Where Ψ(i, j) represents the force constant matrix between atoms i and j; E represents the total energy of the configuration; u i and u j represents the displacement of atoms i and j.

[0065] Studies have shown that the interaction between the nearest neighbor atoms can accurately calculate the vibration entropy of the configuration. Therefore, only the force constants between the nearest neighbor atoms are considered in this method. The force constant matrix Ψ(i, j) can be simplified to a diagonal matrix related only to the tensile bond stiffness and the bending bond stiffness;

[0066]

[0067] Where s is the tensile bond stiffness and b is the bending bond stiffness.

[0068] In some exemplary embodiments, the vibration entropy in step S302 is obtained by integrating the phonon frequency, and the calculation formula of the vibration entropy is:

[0069]

[0070] Among them, S vib (V, T) represents vibration entropy; ω is the phonon frequency; is the reduced Planck constant, k B is the Boltzmann constant, g(ω) is the phonon state density; the vibrational free energy is obtained by multiplying the temperature by the vibrational entropy, and its calculation formula is:

[0071] F vib (VT)=-TS vib (V, T)

[0072] Among them, F vib (V, T) represents vibrational free energy; T is temperature.

[0073] In some embodiments, in step S4, according to the tensile bond stiffness and bending bond stiffness corresponding to different bond lengths, fitting the bond length-bond stiffness function to obtain the force constant matrix of all possible bond lengths, thereby calculating the vibration free energy, comprises the following steps:

[0074] Step S401: Draw a graph based on the tensile bond stiffness and bending bond stiffness corresponding to different bond lengths, and for all bond types in the same lattice, fit their bond stiffnesses to obtain a bond length-bond stiffness function relationship.

[0075] Step S402: For a given configuration, the force constant matrix between every two atoms is obtained according to the bond length-bond stiffness function relationship, and the vibration entropy and vibration free energy of the configuration are calculated.

[0076] Specifically, in step S401, a graph is drawn according to the bond stiffness corresponding to different bond lengths. For all bond types in the same lattice, their bond stiffness can always be fitted into a first-order or second-order polynomial or even a higher-order polynomial related to the bond length. Therefore, in the same lattice, the bond stiffness corresponding to any bond length can be obtained through this functional relationship.

[0077] In step 402, for a given configuration, the force constant matrix between each two atoms is obtained according to the bond length-bond stiffness relationship, and the vibration entropy and vibration free energy of the configuration can be calculated by executing step 302. In order to evaluate the accuracy of the bond length-bond stiffness relationship in predicting the vibration free energy results, the traditional supercell method is also used to calculate the vibration free energy of the unit cell, which is implemented in the Phonopy software.

[0078] The method for calculating vibration free energy based on the bond length-bond stiffness relationship provided in the present application is described in detail below through specific examples.

[0079] Example 1

[0080] This embodiment is described by taking Al-Li alloy as an example. The method for calculating vibration free energy based on the bond length-bond stiffness relationship is as follows:

[0081] Step 1: First, select a configuration with the same symmetry based on the chemical bond information contained in the crystal structure, and apply uniform tensile or compressive deformation to it to produce bond lengths of different lengths. Based on the Al-Li binary alloy phase diagram, it is found that all phases except those belonging to the fcc lattice are bcc lattices. First, four ordered configurations belonging to the fcc lattice are selected ( Figure 2 ) and three ordered configurations belonging to the bcc lattice ( Figure 4 ) The first-principles calculation method was used to completely relax these seven configurations to obtain the optimal structure. The plane wave cutoff energy was set to 114Ry, and the Monkhorst-Pack scheme was used with a k-point density of The Brillouin zone is sampled. The energy convergence reaches 10 -7 Ry / cell, force convergence reaches 10 -6 Ry / Bohr, self-consistent convergence criterion reaches 10 -6 Ry / cell. Then, unit cells with different bond lengths are created by uniformly compressing and expanding 10% of the volume. For each volume of the unit cell, relaxation is performed. This relaxation process only allows the atomic positions to change while the unit cell volume is fixed to obtain the energy of the corresponding unit cell at 0 K, using the same energy and force convergence criteria as the complete relaxation.

[0082] Step 2: Create a supercell for the unit cell generated in step 1, and apply displacement. Without wasting computing resources and ensuring computing accuracy, the number of atoms in the created supercell is between 60 and 100. Static calculations are performed on all supercells with displacements to obtain forces. The energy and force convergence criteria set in this process are the same as those in step 1. The force constant matrix between the two atoms is obtained by solving the first-order partial derivative of the force with respect to the atomic displacement. Then, by diagonalizing the force constant matrix, two bond stiffnesses related to the bond length are obtained, namely the tensile bond stiffness (s) and the bending bond stiffness (b). The Al-Li system contains three different chemical bonds: Al-Al, Al-Li and Li-Li bonds. The bond stiffnesses corresponding to the nearest neighbor bond lengths of different bond types are as follows: Figure 3 and Figure 5 As shown. Whether it is fcc( Figure 3 ) or bcc( Figure 5 ) crystal structure, the tensile bond stiffness of the three bonds (Al-Al, Al-Li and Li-Li bonds) can be fitted into a quadratic polynomial related to the bond length, while their bending bond stiffness can be fitted into a linear polynomial.

[0083] Step 3: Use the polynomial function fitted in step 2 to predict the force constant matrix of any fcc and bcc Al-Li configuration. The force constant matrix of each configuration is Fourier transformed to obtain the dynamical matrix, which can be used to calculate the phonon state density. Then, the vibration entropy (S) is obtained by integrating the phonon frequency ω. vib ):

[0084]

[0085] in, is the reduced Planck constant, k B is the Boltzmann constant, g(ω) represents the phonon state density. The final vibration free energy F vib (V, T) can be obtained by multiplying temperature and vibration entropy: F vib (V, T) = E-TS vib (V, T), where E represents the total energy of the configuration at 0 K.

[0086] Step 4: In order to evaluate the accuracy of the bond length-bond stiffness relationship in predicting the vibrational free energy, the traditional supercell method was used to calculate the vibrational free energy of the ground state phases of Al, Al3Li, AlLi, Al2Li3, Al4Li9, and Li. The calculation results of the two methods are shown in Figure 4. Figure 6 As shown in the figure, the bond length-bond stiffness relationship predicts the phonon state density (DOS) and vibrational free energy (F vib (V, T)) are consistent with the results calculated by the traditional supercell method, indicating that under the same lattice type, the bond length-bond stiffness relationship can accurately predict the vibrational free energy of the configuration.

[0087] Example 2

[0088] This embodiment is described by taking Ni-Al alloy as an example. The method for calculating vibration free energy based on the bond length-bond stiffness relationship is as follows:

[0089] Step 1, first select a configuration with the same symmetry according to the chemical bond information contained in the crystal structure, and apply uniform tensile or compressive deformation to it to produce bond lengths of different lengths. Based on the Ni-Al binary alloy phase diagram, the stable phases include Al, Al3Ni, Al3Ni2, AlNi, Al3Ni5, AlNi3 and Ni. According to their crystal structure information, all phases in the Ni-Al phase diagram (except the Al3Ni phase) can be regarded as configurations on the fcc or bcc lattice, and their atomic sites accommodate Al and Ni atoms. Al3Ni2 can also be regarded as a configuration on the bcc lattice. The Ni sublattice site of AlNi (CsCl-type B2 structure) accommodates Ni atoms and vacancies, while the Al sublattice site of AlNi only accommodates Al atoms. In summary, four ordered configurations belonging to the fcc lattice were first selected ( Figure 7 ) and four ordered configurations belonging to the bcc lattice ( Fig. 9 )The first-principles calculation method is used to fully relax these eight configurations to obtain the optimal structure. Figure 8 Schematic diagram of the bond length-bond stiffness relationship in the bcc lattice. The plane wave cutoff energy is set to 60Ry, the Monkhorst-Pack scheme is used, and the k-point density is The Brillouin zone is sampled. The energy convergence reaches 10 -7 Ry / cell, force convergence reaches 10 -6 Ry / Bohr, self-consistent convergence criterion reaches 10 -6 Ry / cell. Then, unit cells with different bond lengths are created by uniformly expanding the volume by 10%. For each volume of the unit cell, relaxation is performed. This relaxation process only allows the atomic positions to change while the unit cell volume is fixed to obtain the energy of the corresponding unit cell at 0 K, using the same energy and force convergence criteria as the complete relaxation.

[0090] Step 2: Create a supercell for the unit cell generated in step 1, and apply displacement. Without wasting computing resources and ensuring computing accuracy, the number of atoms in the created supercell is between 60 and 100. Static calculations are performed on all supercells with displacements to obtain forces. The energy and force convergence criteria set in this process are the same as those in step 1. The force constant matrix between the two atoms is obtained by solving the first-order partial derivative of the force with respect to the atomic displacement. Then, by diagonalizing the force constant matrix, two bond stiffnesses related to the bond length are obtained, namely the tensile bond stiffness (s) and the bending bond stiffness (b). The Ni-Al system contains three different chemical bonds: Al-Al, Al-Ni and Ni-Ni bonds. The bond stiffnesses corresponding to the nearest neighbor bond lengths of different bond types are as follows: Figure 7 and Fig. 9 shown. Fig.10 is a schematic diagram of the bond length-bond stiffness relationship in the fcc lattice. Figure 7 ) or bcc( Fig. 9 ) crystal structure, the tensile bond stiffness and bending bond stiffness of the three bonds (Al-Al, Al-Ni and Ni-Ni bonds) can be fitted into a quadratic polynomial related to the bond length.

[0091] Step 3: Use the polynomial function fitted in step 2 to predict the force constant matrix of any fcc and bcc Al-Ni configuration. The force constant matrix of each configuration is Fourier transformed to obtain the dynamical matrix, which can be used to calculate the phonon state density. Then, the vibration entropy (S) is obtained by integrating the phonon frequency ω. vib ):

[0092]

[0093] in, is the reduced Planck constant, k B is the Boltzmann constant, g(ω) represents the phonon state density. The final vibration free energy F vib (V, T) can be obtained by multiplying temperature and vibration entropy: F vib (V, T) = -TS vib (V, T).

[0094] Step 4: In order to evaluate the accuracy of the bond length-bond stiffness relationship in predicting the vibrational free energy results, the traditional supercell method was used to calculate the vibrational free energy of the ground state phases of Al, Al3Ni2, AlNi, Al3Ni5, AlNi3 and Ni. The calculation results of the two methods are shown in Figure 4. Fig.11 As shown in the figure, the bond length-bond stiffness relationship predicts the phonon state density (DOS) and vibrational free energy (F vib (V, T)) are consistent with the results calculated by the traditional supercell method, indicating that under the same lattice type, the bond length-bond stiffness relationship can accurately predict the vibrational free energy of the configuration.

[0095] Based on the above technical scheme, an embodiment of the present application provides a method for calculating vibration free energy based on the bond length-bond stiffness relationship, comprising the following steps: first, according to the chemical bond information contained in the crystal structure, a configuration with the same symmetry is selected, and uniform tensile or compressive deformation is applied thereto to produce bond lengths of different lengths, thereby obtaining configurations with bond lengths of different lengths; then, a supercell is created for the configurations with different bond lengths, and a small perturbation is applied to atoms with different symmetries in the supercell to produce atomic displacement, and the force is calculated to obtain the force calculation result; next, according to the force calculation result, a force constant matrix between two adjacent atoms is obtained, and it is diagonalized to obtain the stretching bond stiffness and the bending bond stiffness; finally, according to the stretching bond stiffness and the bending bond stiffness corresponding to different bond lengths, the bond length-bond stiffness function is fitted to obtain the force constant matrix of all possible bond lengths, thereby calculating the vibration free energy.

[0096] The present invention provides a method for predicting vibrational free energy based on a bond length-bond stiffness relationship, which is applicable to a precipitated phase in any alloy phase diagram and has a wide range of applications. The present application proposes to perform phonon calculations on all possible microscopic states through a bond length-bond stiffness relationship, thereby obtaining vibrational entropy and determining its contribution to the free energy of different intermetallic compounds. This method effectively reduces the computational cost while ensuring the computational accuracy, and provides a solution for accurately predicting alloy phase diagrams and guiding alloy design. The thermodynamic properties of each configuration depend on all possible microscopic states, and it is not feasible to calculate the configurational entropy and vibrational entropy of each configuration using the first principles and traditional supercell methods. The proposal based on cluster expansion (CE) successfully overcomes this problem, but it requires the energy and vibrational entropy of hundreds of configurations. The problem of calculating vibrational entropy is usually simplified to a phonon problem. As is well known, the calculation of phonons is time-consuming and expensive. The proposal of a bond length-bond stiffness relationship effectively solves this problem, and it only needs to fit the bond length-bond stiffness relationship of all bond types contained in the system to obtain the vibrational entropy of all configurations based on the same lattice type.

[0097] Those skilled in the art can understand that the above-mentioned embodiments are specific examples for implementing the present application, and in practical applications, various changes can be made to them in form and detail without departing from the spirit and scope of the present application. Any person skilled in the art can make their own changes and modifications without departing from the spirit and scope of the present application, so the scope of protection of the present application shall be based on the scope defined in the claims.

Claims

1. A method for calculating vibrational free energy based on the bond length-bond stiffness relationship, characterized in that: The following steps are involved: According to the chemical bond information contained in the crystal structure, a configuration with the same symmetry is selected, and uniform tensile or compressive deformation is applied thereto to produce bond lengths of different lengths, thereby obtaining configurations with bond lengths of different lengths; Create supercells for configurations with different bond lengths, apply small perturbations to atoms with different symmetries in the supercells to generate atomic displacements, and calculate the forces to obtain the calculated force results; The force constant matrix between two adjacent atoms is obtained based on the force calculation results, and the tensile bond stiffness and the bending bond stiffness are obtained by diagonalizing it; According to the tensile bond stiffness and bending bond stiffness corresponding to different bond lengths, the bond length-bond stiffness function is fitted to obtain the force constant matrix of all possible bond lengths, thereby calculating the vibrational free energy.

2. The method for calculating vibration free energy based on the bond length-bond stiffness relationship according to claim 1, characterized in that: According to the chemical bond information contained in the crystal structure, a configuration with the same symmetry is selected, and uniform tensile or compressive deformation is applied to it to produce bond lengths of different lengths, thereby obtaining configurations with different bond lengths, including: Step 1: Select configurations with the same symmetry according to the chemical bond information contained in the crystal structure, and optimize each configuration; Step 2: Apply 10% uniform compressive strain or tensile strain to the optimized configuration, with a strain interval of 2%, to obtain 5 configurations with different volumes; Step 3: Relax the configurations of different volumes. This process only allows the atomic positions to change while keeping the crystal volume unchanged to eliminate internal stress and obtain energy at 0K.

3. The method for calculating vibration free energy based on the bond length-bond stiffness relationship according to claim 2, characterized in that: In step 1, the electron exchange correlation is described by the generalized gradient approximation with the Perdew-Burke-Ernzerhof exchange-correlation function and ultrasoft pseudopotentials. The plane wave cutoff energy of the system is set by convergence test, and the Monkhorst-Pack scheme is used with the k-point density as The Brillouin zone is sampled; the convergence criterion is: the energy convergence reaches 10 -7 Ry / cell, force convergence reaches 10 -6 Ry / Bohr, self-consistent convergence criterion reaches 10 -8 Ry / cell.

4. The method for calculating vibration free energy based on the bond length-bond stiffness relationship according to claim 3, characterized in that: In step three, the same convergence criteria as in step one are used.

5. The method for calculating vibrational free energy based on the bond length-bond stiffness relationship according to claim 1, characterized in that: Create supercells for configurations with different bond lengths, apply small perturbations to atoms with different symmetries in the supercell to generate atomic displacements, and calculate the forces to obtain the force calculation results, including: In the first-principles calculation method, periodic boundary conditions are used to construct a reasonable supercell size to effectively eliminate the dipole effect; According to the symmetry information of the crystal structure, each atom with different symmetry is given displacement to produce disturbance; Static calculations are performed using the first-principles method to obtain the forces on each atom in the x, y, and z directions.

6. The method for calculating vibrational free energy based on the bond length-bond stiffness relationship according to claim 5, characterized in that: A reasonable supercell size contains 60 to 100 atoms.

7. The method for calculating vibrational free energy based on the bond length-bond stiffness relationship according to claim 1, characterized in that: The force constant matrix between two adjacent atoms is obtained according to the force calculation results, and the tensile bond stiffness and the bending bond stiffness are obtained by diagonalizing the matrix, including the following steps: Based on the force calculation results, calculate the force constant matrix between the two atomic pairs; The force constant matrix is ​​transformed into a dynamical matrix through Fourier transformation to calculate the phonon state density, obtain the vibration entropy, and then calculate the vibration free energy.

8. The method for calculating vibrational free energy based on the bond length-bond stiffness relationship according to claim 7, characterized in that: The force constant matrix between two atomic pairs is calculated as: Where Ψ(i, j) represents the force constant matrix between atoms i and j; E represents the total energy of the configuration; u i and u j represents the displacement of atoms i and j; The force constant matrix is ​​simplified to obtain a diagonal matrix related only to the tensile bond stiffness and the bending bond stiffness; Where s is the tensile bond stiffness and b is the bending bond stiffness.

9. The method for calculating vibrational free energy based on the bond length-bond stiffness relationship according to claim 7, characterized in that: The vibration entropy is obtained by integrating the phonon frequency, and the calculation formula of the vibration entropy is: Among them, S vib (V, T) represents vibration entropy; ω is the phonon frequency; is the reduced Planck constant, k B is the Boltzmann constant, g(ω) is the phonon state density; The vibrational free energy is obtained by multiplying the temperature by the vibrational entropy, and the calculation formula is: F vib (V,T)=-TS vib (V,T) Among them, F vib (V, T) represents vibrational free energy; T is temperature.

10. The method for calculating vibration free energy based on the bond length-bond stiffness relationship according to claim 1, characterized in that: According to the tensile bond stiffness and bending bond stiffness corresponding to different bond lengths, the bond length-bond stiffness function is fitted to obtain the force constant matrix of all possible bond lengths, thereby calculating the vibration free energy, including: Draw a graph based on the tensile bond stiffness and bending bond stiffness corresponding to different bond lengths. For all bond types in the same lattice, fit their bond stiffness to obtain the bond length-bond stiffness function relationship. For a given configuration, the force constant matrix between every two atoms is obtained according to the bond length-bond stiffness function relationship, and the vibration entropy and vibration free energy of the configuration are calculated.

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