Modeling of molar volumes of solid solutions and compounds in multicomponent alloys and applications

Abstract

This invention discloses a modeling method and application for the molar volume of solid solutions and compounds in multi-component alloys. On one hand, this invention supplements the molar volume-temperature data of linear compounds based on crystallographic data and first-principles calculations; on the other hand, it supplements the molar volume-temperature composition data of single-phase solid solutions based on the thermal expansion data and thermodynamic calculations of multi-component alloys. The molar volume-temperature composition data of compounds and solid solutions are input into the CALPHAD model, which describes their molar volumes as functions related to composition and temperature. This model can not only accurately output the lattice constants, volumes, densities, and other physical quantities of solid solutions and compounds at various components and temperatures in multi-component alloys, but also supplement these physical quantities into multi-component alloy molecular models with sparse data points, increasing the reliability of multi-component alloy molecular models and providing guidance and prediction for the design of novel alloys.

Description

Technical Field

[0001] This invention relates to a model of molar volume, specifically to a model and application of the molar volume of solid solutions and compounds in multi-component alloys, belonging to the technical field of physical modeling of metallic materials. Background Technology

[0002] Molar volume is a fundamental thermophysical property of materials, related to other material physical properties or behaviors such as the coefficient of thermal expansion, lattice constant, density, and hot cracking. Therefore, describing the molar volume of solid solutions and compounds as a function of composition and temperature has been a long-standing goal. However, current molar volume models cannot effectively describe solid solutions and compounds as functions of composition and temperature. Furthermore, to avoid overfitting of model parameters, it is necessary to provide as much molar volume data as possible. Many current models, however, use relatively limited data, obtaining molar volume data solely from experimental crystallographic data for modeling, without considering experimental thermal expansion data or first-order calculated thermal expansion data. Summary of the Invention

[0003] To address the problems existing in the prior art, the first objective of this invention is to provide a model for the molar volume of solid solutions and compounds in multi-component alloys.

[0004] The second objective of this invention is to provide a modeling method for the molar volume of solid solutions and compounds in multi-component alloys. This invention supplements the molar volume-temperature data for linear compounds based on crystallographic data and first-principles calculations. Furthermore, it supplements the molar volume-temperature composition data for single-phase solid solutions based on the thermal expansion data and thermodynamic calculations of multi-component alloys. By inputting the molar volume-temperature composition data of the compounds and solid solutions into the CALPHAD model, and after parameter optimization, the molar volumes of both are described as functions related to composition and temperature. This allows for rapid and accurate calculation and prediction of the lattice constants of solid solution phases in multi-component alloys.

[0005] A third objective of this invention is to provide an application of a model for the molar volume of solid solutions and compounds in multi-component alloys, enabling prediction and supplementation of experimental data for multi-component alloys. Based on the molar volume model for solid solutions and compounds provided by this invention, the molar volume is described as a function of composition and temperature. This model can accurately output physical quantities such as lattice constants, volumes, and densities of solid solutions and compounds in multi-component alloys at various compositions and temperatures, thereby improving the reliability and accuracy of analysis and simulation.

[0006] To achieve the above-mentioned technical objectives, the present invention provides a model for the molar volume of solid solutions and compounds in multi-component alloys, comprising:

[0007] S1: Molar volume-temperature data of linear compounds were obtained based on experimental crystallographic data and first-principles calculations of thermal expansion.

[0008] S2: Based on the thermal expansion coefficient and thermodynamics of the multi-element alloy, the molar volume-temperature data of the multi-element alloy is obtained, and then the molar volume-temperature composition of the single-phase solid solution is obtained by difference calculation.

[0009] S3: Input the molar volume-temperature composition data of the compound and the single-phase solid solution into the CALPHAD model of the multi-element alloy to optimize the model parameters and obtain a binary model of the molar volume of each single-phase solid solution and compound in the multi-element alloy with respect to composition and temperature.

[0010] As a preferred embodiment, the formula for converting the experimental crystallographic data into molar volume is:

[0011] Formula 1:

[0012] In Equation 1: V m It is the molar volume, N A N is Avogadro's constant, a, b, and c are the lattice constants of the unit cell, α, β, and γ are the angles between the unit cell axes, and N is the lattice constant of the unit cell. atom It is the number of atoms in the unit cell.

[0013] As a preferred approach, the first principle calculation uses a quasi-harmonic approximation. A unit cell set of the target phase relative to the relaxation volume is constructed based on the Helmholtz free energy equation. The energy of each unit cell at absolute zero is obtained. Then, the force constant and phonon frequency are calculated using density function perturbation theory. Finally, the free energy and unit cell volume are fitted under VinetEOS to obtain the equilibrium molar volume and thermal expansion coefficient of the target phase at finite temperatures. The Helmholtz free energy equation is:

[0014] Formula 2:

[0015] In Equation 2: U0 is the energy of the unit cell at 0K, which is related to the unit cell volume (V); ω q,v,V It is the frequency of the phonon v with respect to the wave vector q and the unit cell volume V, where T is the temperature. and k B These are Planck's constant and Boltzmann's constant, respectively.

[0016] As a preferred embodiment, the process for establishing the molar volume-temperature function of the linear compound is as follows:

[0017] Formula 3:

[0018] In Equation 3: N is the number of experimental crystallographic data. It is the temperature at the i-th experimental point. It is the molar volume at the i-th experimental point. It is relative to temperature T The lattice line expansion is obtained from first-property calculations.

[0019] As a preferred embodiment, the molar volume of the multi-element alloy is the phase fraction-weighted average of the molar volumes of each individual phase, expressed as:

[0020] Equation 4: V α+β =f α V α +f β V β ;

[0021] In Equation 4: f α and f β These are the phase fractions of the α phase and the β phase, respectively, V α and V β These are the molar volumes of the α phase and the β phase, respectively.

[0022] The process for supplementing the molar volume-temperature composition data of the single-phase solid solution is as follows:

[0023] i) Obtain the reference temperature T0 using experiments and relevant databases. and The reference temperature T0 is calculated using Equation 4.

[0024] ii) Using the linear compound molar volume formula in Equation 3, the molar volume of the multi-component alloy at any temperature T is obtained. and the β phase at any temperature T By calculating the difference again using Equation 4, we can obtain the α phase at any temperature T.

[0025] As a preferred embodiment, in the CALPHAD model of the multi-element alloy, the molar volume of the phase with a solid solubility range is divided into the contribution of the end groups and the contribution of the excess molar volume. The relationship between the molar volume of the end groups of the compound and the solid solution and temperature is as follows:

[0026] Equation 5: V T = (A+BT+CT) 2 +DT 3 ) 3 ;

[0027] The molar volume model of the compound is given by (A, B, ...). x (C,D,…) y Taking sub-lattice as an example, it can be represented as:

[0028] Formula 6:

[0029]

[0030] In Equations 5 and 6: T is temperature, A, B, C, and D are parameters to be optimized, and y i ′ and y j "" represents the lattice fraction of element i in the first sublattice and the lattice fraction of element j in the second sublattice, respectively. This represents the molar volume when the first sublattice of the end base is entirely composed of elements i and the second sublattice is entirely composed of elements j. It is the interaction between elements i and k in the first sublattice when the second sublattice consists entirely of elements j; It is the interaction between elements i and l in the first sublattice when the second sublattice consists entirely of elements j.

[0031] As a preferred embodiment, the molar volume model of the solid solution is as follows:

[0032] Formula 7:

[0033] In Equation 7: x i V is the mole fraction of element i. i fcc It is the molar volume of element i under the fcc structure. and These are the parameters to be optimized.

[0034] As a preferred approach, the optimization of interaction parameters in the optimization process of the CALPHAD model of the multi-element alloy must be carried out after the optimization of end-group parameters.

[0035] This invention also provides an application of a model for the molar volume of solid solutions and compounds in multi-component alloys, which is used to predict and supplement experimental data of multi-component alloys.

[0036] Compared with the prior art, the beneficial technical effects of the technical solution of the present invention are as follows:

[0037] 1) The molar volume data of solid solutions and compounds in multi-component alloys provided by this invention supplements the molar volume-temperature data of linear compounds based on crystallographic data and first-principles calculation methods. Furthermore, it supplements the molar volume-temperature composition data of single-phase solid solutions based on the thermal expansion data and thermodynamic calculations of multi-component alloys. By inputting the molar volume-temperature composition data of compounds and solid solutions into the CALPHAD model, and after parameter optimization, the molar volumes of both are described as functions related to composition and temperature. This allows for rapid and accurate calculation and prediction of the lattice constants of solid solution phases in multi-component alloys.

[0038] 2) In the technical solution provided by the present invention, based on the molar volume model of solid solutions and compounds provided by the present invention, the molar volume is described as a function of composition and temperature. The model can accurately output the lattice constant, volume, density and other physical quantities of solid solutions and compounds of each component and at each temperature in multi-component alloys, thereby improving the reliability and accuracy of analysis and simulation.

[0039] 3) The technical solution provided by this invention can accurately calculate and predict the thermal expansion coefficient of alloy materials, make precise calculations on the thermal expansion behavior between different phases inside the material, guide alloy design, and thus avoid hot cracking of the alloy. Attached Figure Description

[0040] Figure 1 A flowchart illustrating the molar volume modeling method for compounds and solid solutions provided in this invention;

[0041] Figure 2 A schematic diagram illustrating the verification and application of methods for supplementing molar volume data for linear compounds;

[0042] in, Figure 2 (a) and Figure 2 (b) This section compares the molar volume obtained by combining experimental crystallographic data within 50 K above room temperature with the experimental values, in order to verify the reliability of this supplementary molar volume data method. Figure 2 (c) and Figure 2 (d) To compare the molar volume obtained by combining experimental crystallographic data with first-order thermal expansion over the entire temperature range with the experimental value, the method used to indicate that this supplementary molar volume data can be applied to linear compounds with limited molar volume data.

[0043] Figure 3 Flowchart of a method for supplementing molar volume data for solid solutions;

[0044] Figure 4 The graph shows the relationship between the molar volume of some compounds and solid solutions and temperature composition after the CALPHAD model has been optimized.

[0045] in, Figure 4 (a) is a schematic diagram of the θ-Al2Cu compound as a function of composition and temperature; Figure 4 (b) is a schematic diagram of Al-Cu solid solution as a function of composition and temperature; Figure 4 (c) is a schematic diagram of Al-Mg solid solution as a function of composition and temperature; Figure 4 (d) is a schematic diagram of Al-Ag solid solution as a function of composition and temperature; Figure 4 (e) is a schematic diagram of Al-Zn solid solution as a function of composition and temperature; Figure 4(f) is a schematic diagram of Al-Si solid solution as a function of composition and temperature. Detailed Implementation

[0046] The following comparative examples and embodiments are intended to further illustrate the present invention and not to limit the invention.

[0047] Example 1

[0048] The specific process of the modeling method for the molar volume of solid solutions and compounds in multi-component alloys provided by this invention is as follows:

[0049] 1. Molar volume data of linear compounds

[0050] 1.1. Obtain experimental crystallographic data of the target linear compound through X-ray diffraction experiments or literature review. Then, convert the experimental crystallographic data into molar volume using the formula:

[0051]

[0052] In the formula, V m It is the molar volume, N A N is Avogadro's constant, a, b, and c are the lattice constants of the unit cell, α, β, and γ are the angles between the unit cell axes, and N is the lattice constant of the unit cell. atom This refers to the number of atoms within the unit cell. Furthermore, the temperature corresponding to the crystallographic data must be determined in the literature;

[0053] 1.2. First-principles calculations commonly used for high-temperature properties employ the quasi-harmonic approximation (QHA). In QHA, the Helmholtz free energy (F) is a function of the cell volume (V) and temperature, expressed as:

[0054]

[0055] In the formula, U0 is the energy of the unit cell at 0K, which is related to the unit cell volume (V); ω q,v,V It is the frequency of the phonon v with respect to the wave vector q and the unit cell volume V, where T is the temperature. and k BThese are Planck's constant and Boltzmann's constant, respectively. This method requires constructing unit cells of different volumes and calculating the free energy and phonon modes of each cell at 0 K. Based on the Helmholtz free energy formula, relaxation calculations are first performed on the unit cells of the target phase. Then, a set of unit cells with volumes ranging from -5% to +5% relative to the relaxation volumes, with a step size of 0.5%, is constructed. Static calculations are then performed on each unit cell to obtain its energy at 0 K. Density function perturbation theory (DFPT) is then used to calculate each unit cell sequentially to obtain the force constants for calculating phonon frequencies. The number of atoms in the supercell participating in the DFPT calculation is kept as close as possible to 100 to balance accuracy and time. The free energy and unit cell volume are fitted by VinetEOS at 10 K intervals between 0 K and 1000 K, thus obtaining the equilibrium molar volume and thermal expansion at finite temperatures.

[0056] 1.3. For linear compounds, their molar volume and temperature satisfy the following condition:

[0057]

[0058] In the formula, This refers to the linear thermal expansion at temperature T relative to a reference temperature T0, where the linear expansion is zero at the reference temperature T0. V T and These are the molar volumes at T and T0, respectively. For linear compounds with known crystallographic data at temperature T0, It can be obtained from crystallographic data in the literature; This is obtained through QHA calculations. Therefore, the linear lattice expansion calculated by QHA can be used to determine the lattice expansion. and experimental molar volume Substituting these values ​​into the above equation yields the molar volume of the compound. If crystallographic data at different temperatures is available, all such data should be taken into account. Finally, V T The molar volume at temperature T can be obtained from the following formula.

[0059]

[0060] Where N is the number of experimental crystallographic data. It is the temperature at the i-th experimental point. It is the molar volume at the i-th experimental point. It is relative to temperature T The lattice lines expand. Figure 2 To verify and apply the molar volume model of linear compounds, among which, Figure 2 (a) and Figure 2In (b), only experimental crystallographic data within 50 K above room temperature were used to combine with the thermal expansion calculated by QHA to obtain the molar volume. From room temperature to 1000 K, the molar volume obtained by supplementing the data was in very good agreement with the experimental value, which verified the reliability of the method. Figure 2 (c) and Figure 2 (d) Provides a large amount of accurate molar volume data for phases with scarce experimental crystallographic data in different multi-component alloys, thereby participating in the optimization of molecular model parameters of multi-component alloys and increasing the reliability of multi-component alloy models.

[0061] 2. Molar volume data of solid solutions

[0062] 2.1. Obtain the experimental thermal expansion data of the two-phase region (α+β) containing the solid solution phase through dilatometer measurement or literature search. In subsequent steps, the solid solution will be treated as the α phase. If sufficient crystallographic data is available for the β phase but insufficient data is available for the α phase, the molar volume data of the α phase can be obtained from these crystallographic data, experimental thermal expansion, and thermodynamic calculations. The core formula lies in the molar volume V of the two-phase region containing the α and β phases. α+β The phase fraction weighted average, equal to the molar volume of each individual phase, is expressed as follows:

[0063] V α+β =f α V α +f β V β

[0064] In the formula, f α and f β These are the phase fractions of the α phase and the β phase, respectively, V α and V β These are the molar volumes of the α and β phases, respectively. A flowchart illustrating how combining thermal expansion data of the two phase regions with thermodynamic calculations from a thermodynamic database to provide molar volume data for solid solution phases is shown below. Figure 3 .

[0065] 2.2. First, data preprocessing is performed, using the thermal expansion of the two-phase region in the experiment to obtain... T0 is the thermal expansion reference temperature; the β-phase molar volume is described as a function of composition and temperature, used to provide... (V at temperature T0) β )and (V at temperature T) β Phase fraction and (f at temperature T0) α and f β )and and (f at temperature T) α and fβ This is obtained through thermodynamic calculations using a thermodynamic database; the solid solution phase has the highest equilibrium solute concentration in the two-phase region, whereas it is usually at equilibrium composition. (V at temperature T0) α (This information is unavailable.) Therefore, experimental data on nearby components and temperatures can be used to pre-optimize the model parameters to estimate...

[0066] 2.3. Next, using the core formula, and Able to calculate (V at temperature T0) α+β Then, using the previously mentioned formulas for thermal expansion and molar volume, and Able to calculate (V at temperature T) α+β Finally, using the core formula, and It is possible to calculate The molar volumes of (Al) solid solutions calculated from the thermal expansion data of Al-Cu, Al-Si, and Al-Mg binary alloys are shown in Table 1. This method allows us to obtain the molar volumes of solid solutions at different temperatures and with different compositions using thermal expansion data from the two-phase region. These data can be used to optimize the parameters of multi-element alloy molecular models, increasing the reliability of these models.

[0067] 3. Optimization of CALPHAD models for solid solutions and compounds

[0068] 3.1 In the CALPHAD model, the molar volume of a phase with a solid solubility range is divided into two parts: the contribution of the end groups and the contribution of the excess molar volume. The molar volume of the end groups of both compounds and solid solutions is related to temperature as follows:

[0069] V T = (A+BT+CT) 2 +DT 3 ) 3

[0070] Where T is temperature, and A, B, C, and D are parameters to be optimized. The parameters can be optimized using experimental crystallographic data of end-group composition or data supplementation of molar volume data.

[0071] A general molar volume model for compounds is based on (A, B, ...). x (C,D,…) y Taking sub-lattice as an example, it can be represented as:

[0072]

[0073] Among them, y i ′ and y j "" represents the lattice fraction of element i in the first sublattice and the lattice fraction of element j in the second sublattice, respectively. This represents the molar volume when the first sublattice of the end base is entirely composed of elements i and the second sublattice is entirely composed of elements j. When the second sublattice consists entirely of elements j, the interaction between elements i and k in the first sublattice is the parameter to be optimized. This refers to the interaction between elements i and l in the first sublattice when the second sublattice consists entirely of element j. Optimization of the compound interaction parameters must be performed after the optimization of the end-group parameters. The interaction parameters can be optimized by incorporating molar volume data of components located within the solid solution region.

[0074] To verify the universal molar volume model of the above-mentioned compounds, this invention also analyzed and verified θ-Al₂Cu, and the results are as follows: Figure 4 As shown in (a).

[0075] 3.2 The general molar volume model for solid solutions can be expressed as:

[0076]

[0077] Where, x i V is the mole fraction of element i. i fcc It is the molar volume of element i under the fcc structure. and These are the parameters to be optimized. Optimization of solid solution interaction parameters must be performed after the optimization of end-group parameters. The interaction parameters can be optimized by incorporating molar volume data of components within the solid solution range. Figure 4 (b)~ Figure 4 (f) describes the relationship between the molar volume and composition temperature of Al-Cu, Al-Mg, Al-Ag, Al-Zn, and Al-Si solid solutions. Furthermore, multiple elemental solutes may exist within the solid solution. This invention also conducted experimental and simulation calculations on the solid solutions of Al-Mg-Zn ternary alloys and commercial aluminum alloys at different heat treatment temperatures. The comparison between the experimental and model-calculated values ​​of the phase lattice constants is shown in Table 2. The lattice constants are obtained by converting molar volumes. As can be seen from Table 2, the calculated values ​​obtained using the model provided by this invention have a very small error compared to the experimental values.

[0078] Table 1

[0079]

[0080]

[0081] Table 2

[0082]

Claims

1. A method of modeling molar volumes of solid solutions and compounds in multicomponent alloys, characterized in that, The application relates to a method for establishing a molar volume-temperature function of a multi-component alloy. S1: obtaining linear compound molar volume-temperature data according to experimental crystallographic data and first principle calculation thermal expansion results; S2: obtaining multi-component alloy molar volume-temperature data according to the thermal expansion rate of the multi-component alloy and thermodynamic calculation, and then obtaining single-phase solid solution molar volume-temperature composition data through difference calculation; S3: inputting the molar volume-temperature composition data of the compound and the single-phase solid solution into a multi-component alloy CALPHAD model to perform model parameter optimization, and obtaining a binary model of the molar volume of each single-phase solid solution and compound in the multi-component alloy with respect to composition and temperature; In the multi-component alloy CALPHAD model, the molar volume of a phase with a solid solubility range is divided into the contribution of end groups and the contribution of excess molar volume, wherein the relationship between the molar volume of the end groups of the compound and the solid solution and temperature is as follows: Formula 5: ; The linear compound can be regarded as the end group of the compound The molar volume model of the compound is (A, B,...) x (C, D,...) y The sublattice can be represented as follows: Formula 6: ； In Equations 5 and 6: Let A be temperature, and let A, B, C, and D be the parameters to be optimized. and Representing elements respectively In the first sub-lattice lattice fraction, the element The lattice fraction in the second sublattice; This indicates that the first sublattice of the end base is all... Elements, the second sub-matrix is ​​all Molar volume of elements; It is the second sub-matrix, all of which are When elements, and The interaction of elements in the first sublattice; It is the second sub-matrix, all of which are When elements, and The interaction of elements in the first sublattice; The molar volume model of the solid solution is as follows: Formula 7: ; In formula 7: is the molar fraction of the element , is the molar volume of the element in the fcc structure, and are the parameters to be optimized.

2. The method of modeling the molar volume of solid solutions and compounds in a multi-component alloy according to claim 1, characterized in that: The formula for converting the experimental crystallographic data into the molar volume is as follows: Formula 1: ; In formula 1: is the molar volume, is the Avogadro constant, , and are the lattice constants of the unit cell, , and are the angles between the axes of the unit cell, is the number of atoms in the unit cell.

3. The method of modeling the molar volume of solid solutions and compounds in a multi-component alloy of claim 1, wherein: The first principle calculation is quasi-harmonic approximation, a cell group of a volume target phase with respect to a relaxation volume is constructed according to a Helmholtz free energy equation, the energy of each cell at absolute zero is obtained, force constants and phonon frequencies are obtained according to a density functional perturbation theory, and the free energy and the cell volume are fitted under Vinet EOS to obtain the equilibrium molar volume and the thermal expansion coefficient of the target phase at a finite temperature, wherein the Helmholtz free energy equation is as follows: Formula 2: ; In formula 2: is the energy of a unit cell of OK, related to the unit cell volume (V); is the phonon at the wave vector and the unit cell volume at the frequency is the temperature, and are the Planck and Boltzmann constants, respectively.

4. The method of modeling the molar volume of solid solutions and compounds in a multi-component alloy of claim 1, wherein: The establishment process of the molar volume-temperature function of the linear compound is as follows: Formula 3: ; In formula 3: N is the number of experimental crystallographic data, is the temperature of the first experimental point, is the molar volume of the first experimental point, is the lattice expansion relative to at the temperature , obtained from first principles calculations.

5. The method of modeling the molar volume of solid solutions and compounds in a multi-component alloy according to claim 4, characterized in that: The molar volume of the multi-component alloy is a phase fraction weighted average value of the molar volumes of each single phase, and is expressed as follows: Formula 4: ; in formula 4: and are respectively the phase and the phase fraction of the phase, and are respectively the phase and the molar volume of the phase; The establishment process of the molar volume-temperature function of the single-phase solid solution is as follows: i) obtaining the values of the reference temperature , , , and at the reference temperature using experiments and databases of correlation by means of formula 4 ; ii) Using the linear compound molar volume formula in equation 3, the phase of the multicomponent alloy at any temperature and at any temperature is obtained again by taking the difference in equation 4 to obtain at any temperature at any temperature .​​ 6. The method of modeling the molar volume of solid solutions and compounds in a multi-component alloy of claim 1, wherein: In the optimization process of the multi-component alloy CALPHAD model, the optimization of the interaction parameters must be performed after the optimization of the end group parameters.

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