Calculation method of metal grain boundary energy under different temperature conditions

Abstract

The invention discloses a method for calculating metal grain boundary energy under different temperature conditions, and belongs to the field of metal material microstructure analysis. The method focuses on the temperature dependence of the grain boundary energy of aluminum and copper, systematically reveals the quantitative influence rule of the temperature on different types of grain boundary energy in aluminum and copper through molecular dynamics simulation for the first time, and studies the change difference of the grain boundary energy of aluminum and copper at high temperature. The correlation between important grain boundary characteristic parameters of aluminum and copper and grain boundary energy at different temperatures is clarified.

Description

Technical Field

[0001] This invention relates to the field of microstructure analysis of metallic materials, and in particular to a method for calculating the grain boundary energy of metals under different temperature conditions. Background Technology

[0002] Grain boundaries, as one of the most important structural defects in polycrystalline materials, directly determine the material's mechanical behavior, thermal stability, and diffusion-dominated phase transition processes through their energy characteristics. In recent years, grain boundary engineering methods, which customize grain boundary properties by controlling the distribution of grain boundary features, have become a key means of optimizing material properties. However, research on the temperature dependence of grain boundary energy remains significantly lacking, especially in the high-temperature region near the material's melting point, where experimental data is scarce and theoretical predictions are inconsistent. Understanding the evolution of grain boundary energy with temperature is crucial not only for providing a theoretical basis for the design of high-temperature structural materials but also for assessing thermal stability in practical engineering applications such as microelectronic interconnects and nuclear reactor cooling systems.

[0003] Grain boundary energy can be obtained indirectly through experimental measurement or simulation calculation. Currently, experimental determination of grain boundary energy mainly relies on grain boundary etching morphology analysis or equilibrium morphology calculations based on the Herring equation. However, these methods are limited at high temperatures by surface oxidation, atomic diffusion interference, and the difficulty of in-situ observation. In theoretical research, molecular statics (MS) and density functional theory (DFT) are widely used to calculate grain boundary energy at 0K, revealing the close relationship between crystallographic orientation and grain boundary microstructure and grain boundary energy. It is difficult to determine the free energy in atomic-scale simulations, and only a few DFT calculations on temperature-dependent grain boundary energy have been reported. Due to the diversity of grain boundary structures, calculating the contribution of phonons to temperature-dependent grain boundary energy using DFT is extremely challenging. Therefore, it is particularly important to develop computationally efficient methods for calculating grain boundary energy at different temperatures using molecular dynamics simulations.

[0004] Establishing the correlation between grain boundary structure and grain boundary energy is another research hotspot. Recent studies have shown that grain boundary energy may exhibit a nonlinear decreasing trend with increasing temperature, but comparative studies on different metals and different grain boundary structures remain scarce. Currently, many discussions on solute segregation, diffusion, and fracture behavior are based on grain boundary energy at 0 K, and it is unclear whether the relative magnitudes of the energies of different grain boundaries at 0 K are applicable at high temperatures. Furthermore, it is also unclear how grain boundary structural characteristics strongly correlated with grain boundary energy, such as free volume and grain boundary structural order parameters, change with temperature. Summary of the Invention

[0005] The present invention aims to provide a method for calculating the grain boundary energy of metals under different temperature conditions, so as to solve the problems mentioned in the background art.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A method for calculating the grain boundary energy of a metal under different temperature conditions, the method specifically includes:

[0008] Step 1: Calculation of grain boundary energy at different temperatures

[0009] Grain boundary energy is usually defined as the excess energy at the grain boundary, expressed as the total Gibbs free energy G in the bicrystal model. gb Subtract the Gibbs free energy G of a single atom within the crystal bulk Multiply by the number of atoms N in the model and divide by the total grain boundary area A, then calculate according to equation (1):

[0010]

[0011] The Gibbs free energy is calculated using the thermodynamic integration method of the Frenkel-Ladd path. The Frenkel-Ladd method is based on constructing a reversible path to a known free energy state; in which the Einstein crystal model is used with harmonic springs in all three dimensions; since the Einstein solid is structurally identical to the initial solid, this path is likely to be reversible without phase transition; starting from the initial solid, the reference state is slowly reached by opening the harmonic spring forces centered on the atomic positions while closing the interatomic potentials of the initial solid, which is controlled by the parameter λ, and the parameter Hamiltonian H(λ) can be written as Equation (2):

[0012] H(λ)=λH E +(1-λ)H0 (2)

[0013] Where H0 and H E Let the initial and final Hamiltonians be represented. The Hamiltonian of the initial state is given by equation (3):

[0014]

[0015] The Hamiltonian of Einstein solids can be written as equation (4):

[0016]

[0017] Where m is the mass of particle i, p is the momentum, U is the interatomic potential of the applied force field, ω is the frequency of the harmonic oscillator, and r i and r i 0 The update and initial lattice position of particle i are represented; λ = 0 and 1 in formula (1) represent the initial and final states, respectively; therefore, when the average free energy of the integral ensemble changes along the path, the free energy difference between the two states can be calculated:

[0018]

[0019] This reversible work involves two processes: a forward process 0→E and a reverse process E→0; the forward irreversible work... and reverse irreversible work Determined by molecular dynamics simulations:

[0020]

[0021] Where Γ is the scaling time, the reversible work can be expressed as the irreversible work:

[0022]

[0023] The free energy of a solid, as defined by Einstein, is called:

[0024]

[0025] Therefore, the Gibbs free energy of interest can be obtained through the following formula:

[0026] G0 = G E +W rev (9)

[0027] Compared with the prior art, the present invention has the following beneficial effects:

[0028] This invention focuses on the temperature dependence of grain boundary energy in aluminum and copper. For the first time, through molecular dynamics simulations, it systematically reveals the quantitative influence of temperature on the energy of different types of grain boundaries in Al / Cu, studies the relative contributions of atomic vibrational entropy and electronic entropy to the temperature coefficient of grain boundary energy, and clarifies the differentiation of grain boundary energy evolution paths between Al and Cu due to differences in electronic structure. Attached Figure Description

[0029] Figure 1 The grain boundary energy in the bicrystal model of aluminum varies with temperature;

[0030] Figure 2 The grain boundary energy in the copper bicrystal model varies with temperature;

[0031] Figure 3 The grain boundary structure of aluminum and copper Σ5(310) varies with temperature;

[0032] Figure 4 The relationship between grain boundary energy and grain boundary characteristics for aluminum and copper. Detailed Implementation

[0033] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments:

[0034] The method specifically includes:

[0035] Step 1: Calculation of the Gibbs free energy of the block

[0036] The calculations were performed in the large-scale atomic and molecular parallel simulator LAMMPS, using the potential functions of Al and Cu developed by Mishin et al. These two potential functions were fitted with a large amount of experimental and ab initio data, and were able to accurately reproduce the fundamental equilibrium and energy properties of the metals. First, a single-crystal model of aluminum and a single-crystal model of copper with a volume of 10×10×10 lattice were constructed, and the number of atoms in the model contained 4000 atoms. The Gibbs free energy was calculated using the thermodynamic integral method of the Frenkel-Ladd path. The Frenkel-Ladd method is based on constructing a reversible path to a known free energy state; in which the Einstein crystal model is used with all three-dimensional harmonic springs; since the Einstein solid is structurally identical to the initial solid, this path is likely to be reversible without phase transition; starting from the initial solid, the reference state is slowly reached by opening the harmonic spring force centered on the atomic position while closing the interatomic potential of the initial solid, which is controlled by the parameter λ, and the parameter Hamiltonian H(λ) can be written as Equation (2):

[0037] H(λ)=λH E +(1-λ)H0 (1)

[0038] Where H0 and H E Let the initial and final Hamiltonians be represented. The Hamiltonian of the initial state is given by equation (3):

[0039]

[0040] The Hamiltonian of Einstein solids can be written as equation (4):

[0041]

[0042] Where m is the mass of particle i, p is the momentum, U is the interatomic potential of the applied force field, ω is the frequency of the harmonic oscillator, and r i and r i 0 The update and initial lattice position of particle i are represented; λ = 0 and 1 in formula (1) represent the initial and final states, respectively; therefore, when the average free energy of the integral ensemble changes along the path, the free energy difference between the two states can be calculated:

[0043]

[0044] This reversible work involves two processes: a forward process 0→E and a reverse process E→0; the forward irreversible work... and reverse irreversible work Determined by molecular dynamics simulations:

[0045]

[0046] Where Γ is the scaling time, the reversible work can be expressed as the irreversible work:

[0047]

[0048] The free energy of a solid, as defined by Einstein, is called:

[0049]

[0050] Therefore, the Gibbs free energy of the block can be obtained by the following formula:

[0051] G0 = G E +W rev (8)

[0052] For each temperature, the system was first calculated using an NPT ensemble equilibrium MD simulation at the desired temperature and pressure (P=0) for 40 ps until the model size reached equilibrium. The time step for all MD simulations was 0.002 ps. The Frenkel Ladd path was performed in the NVE ensemble. The equations of motion were integrated using a Langevin thermostat. In the Frenkel Ladd path MD simulation, the λ parameter varied with the scaling time τ according to the following equation (9):

[0053] λ(τ)=τ 5 (70τ 4 -315τ 3 +540τ 2 -420τ+126) (9)

[0054] The simulation time for both forward and backward λ-integrals was 10 ps. To maximize the efficiency of the Frenkel-Ladd path, the spring constant k of the harmonic oscillator in the Einstein solid was chosen to minimize the difference between the initial Hamiltonian H0 and the Einstein solid Hamiltonian HE. This was achieved by measuring the mean square displacement <(Δr) of the atoms in the system in a separate MD simulation at the desired temperature within 10 ps. 2 > and k is obtained using the equal-partition theorem (10).

[0055]

[0056] Where k B is the Boltzmann constant, and T is the temperature. The parameters used to calculate the grain boundary energy of different types of bicrystal models using the above method are shown in Table 1.

[0057] Table 1. Parameters used to calculate grain boundary energy at different temperatures in this study.

[0058]

[0059]

[0060] Step 2: Calculation of grain boundary energy at different temperatures

[0061] A series of symmetrical tilts of aluminum and copper were constructed

[001] , and Grain boundary model. To ensure the convergence of free energy and system size, the models all have more than 40,000 atoms, and the size of the bicrystalline model satisfies equations (10) and (11). The specific parameters of the model are shown in Table 1:

[0062]

[0063] Where H is the height of the single crystal, W is the width of the grain boundary, h, k and l are the crystal orientation indices in each direction, and m and n are scaling factors related to the lattice constant.

[0064] After relaxing the grain boundary model using the same parameters as the single-crystal model at different temperatures, the total Gibbs free energy of the model is calculated using the thermodynamic integration method of the Frenkel-Ladd path. Then, according to equation (13), the total Gibbs free energy G of the bicrystalline model is used. gb Subtract the Gibbs free energy G of a single atom within the crystal bulk Multiply by the number of atoms N in the model and divide by the total grain boundary area A:

[0065]

[0066] Step 3: Calculation of grain boundary eigenvalues

[0067] Five grain boundary characteristics were calculated based on the relaxed models of each grain boundary at different temperatures: orientation difference angle (θ), interface normal (h kl), free volume (FV), orientation order (Q), and centrosymmetry parameter (CSP). The orientation difference angle was corrected according to the spatial rotational symmetry relationship and converted into a sine value (sinθ), and the interface normal was converted into the interface spacing (dGB). The interface spacing was calculated as shown in Equation (14), where a represents the lattice constant.

[0068]

[0069] The free volume is equal to the atomic volume at the grain boundary (ΣV) GB Subtract the atomic volume (V) of the same number of atomic blocks bulk ), and then divide by the grain boundary area (A), as shown in equation (15).

[0070] FV=(∑V GB -nV bulk ) / A (15)

[0071] The orientational order degree (Q) is first calculated by defining the local order parameter as the average of the spherical harmonics of each neighbor according to Equation (16). These are complex components of a three-dimensional simulation of the two-dimensional order parameter, implemented as a LAMMPS calculation of the sixth order per atom. The summation is performed over the 12 nearest neighbors of the central atom. The angle summation is the standard spherical polar angle defining the direction of the bond vector. Then, all the degree components are summed according to Equation (17).

[0072]

[0073] The centrosymmetric parameter CSP is obtained according to equation (18), where R i and R i+N / 2 It is a vector from the central atom to a specific pair of nearest neighbor atoms.

[0074]

[0075] The calculation of the three eigenvalues—free volume (FV), orientational order (Q), and central symmetry parameter (CSP)—was performed in LAMMPS. A script written in Python read these three eigenvalues ​​from different models at all temperatures in batches from the log file and copied them to the output file for correlation analysis.

[0076] The grain boundary energy in the bicrystal model of aluminum varies with temperature as follows: Figure 1 As shown. Among them Figure 1 (a) shows the relationship between the grain boundary energy and orientation difference angle of the symmetrical tilted

[001] grain boundary at different temperatures. At 0 K, the energy curve of the

[001] grain boundary shows a trend of first increasing and then decreasing as the orientation difference angle gradually increases. The grain boundary energy reaches its maximum value when the orientation difference angle slightly exceeds 45°, showing a slight asymmetry. In addition, the grain boundary energy-orientation difference angle curve shows a slight decrease at Σ5(310) and Σ5(210). As the temperature increases, the grain boundary energy of different orientation difference angles decreases to different degrees. For example, the grain boundary energy of Σ5(310) changes less with temperature than that of Σ13(510). The change of the grain boundary energy-orientation difference angle curve is similar to that at 0K in the temperature range of 0.2-0.6 times the melting point, but the grain boundary energy reaches its maximum value when the temperature reaches 0.8 times the melting point and the orientation difference is close to 45°. In addition, the energy sharpness phenomenon at the Σ5(310) and Σ5(210) grain boundaries becomes very small, and the symmetry and continuity of the energy curve are significantly improved. Figure 1 (b) demonstrates symmetrical tilt The relationship between grain boundary energy and orientation difference angle. The energy curves of the symmetrical tilted grain boundaries obtained by rotation show a significant drop at the Σ11(113) and Σ3(111) grain boundaries. Furthermore... The grain boundary energies of symmetrical grain boundaries exhibit more significant asymmetry compared to

[001] grain boundaries. For example, the grain boundary energies of Σ3(111) and Σ11(113) are much lower than those of other grain boundaries, while the grain boundary energies of Σ3(112) and Σ11(332) are much higher than those of other grain boundaries. With increasing temperature, the grain boundary energies of Σ3(111) and Σ11(113) change very little, while those of other grain boundaries... The grain boundary energy of symmetrical grain boundaries is significantly reduced. Symmetrical tilt... The grain boundary energy changes as follows Figure 1 As shown in (c), as the orientation difference angle gradually increases up to 32°, the grain boundary energy at 0 K gradually increases and reaches a maximum value, with a sudden drop in energy at the Σ7(123) grain boundary followed by stabilization. At higher temperatures, the grain boundary energy gradually decreases, except at temperatures 0.8 times the melting point. The grain boundary energy is not very continuous with the change in orientation difference, but it basically exhibits a similar variation pattern to that at 0K at other temperatures. Figure 1 (d) Demonstrates symmetrical tilt

[001] , and The average value of all grain boundary energies varies with temperature. Generally speaking, the change of grain boundary energy in aluminum with temperature can be divided into two stages: the grain boundary energy changes relatively slowly in the temperature range of 0 K to 0.2 times the melting point, while above 0.2 times the melting point, the grain boundary energy decreases more rapidly and exhibits an approximately linear change.

[0077] The grain boundary energy in the bicrystal model of copper varies with temperature as follows: Figure 2 As shown. Figure 2 (a)-(c) show that although the grain boundary energy of copper grain boundaries with the same orientation is much higher than that of aluminum, the grain boundary energy of symmetrically tilted grain boundaries in copper at 0K varies with the orientation difference angle similarly to that of aluminum. For example, sharp energy angles appear at certain specific orientations, and at

[001] grain boundaries and... The energy observed at grain boundaries is not entirely symmetrical about the orientation difference angle. Similarly, the grain boundary energy of copper shows a decreasing trend as the temperature gradually increases. By statistically averaging the grain boundary energies of these three types of grain boundaries in copper at different temperatures, we obtained... Figure 2 (d) shows the trend of grain boundary energy in copper with temperature. The grain boundary energy of symmetrical tilted grain boundaries in copper shows a non-linear decreasing trend with temperature, exhibiting a linear decrease in the range of 0K-0.6 times the melting point. However, the grain boundary energy decreases more significantly when the temperature reaches 0.8 times the melting point, which is different from the trend shown by aluminum.

[0078] To explain the difference in grain boundary energy between aluminum and copper with temperature, the structural evolution of the Σ5(310) grain boundary at different temperatures was selected as an example, such as... Figure 3 As shown. Figure 3(a) shows the grain boundary structures of Σ5(310) grain boundaries in aluminum and copper within a temperature range of 0.2 to 0.8 times their respective melting points. As the temperature gradually increases, the number of grain boundary atoms gradually increases, and the atomic arrangement becomes more disordered. At a temperature of 0.8 times the melting point, the number of unknown atoms in the grain boundary region of copper increases significantly compared to aluminum. Figure 3 (b) The changes in the number of unknown atoms with temperature were quantitatively analyzed. At low temperatures, the change in the number of unknown atoms was not significant. For both aluminum and copper, the number of unknown atoms began to increase significantly at 0.6 times their melting point, and the proportion increased rapidly at 0.8 times their melting point. This indicates that pre-melting occurred in the grain boundary region at high temperatures. The degree of pre-melting in copper was significantly higher than that in aluminum, which is why the grain boundary energy of copper is significantly lower than that of aluminum in the high-temperature region at 0.8 times the melting point. Figure 3 (c) The radial distribution functions of the grain boundary region at different temperatures further confirm this hypothesis. At 0 K, the peaks of the radial distribution functions of aluminum and copper in the grain boundary region are relatively sharp. As the temperature increases, the peaks broaden, the peak height decreases, and some peaks disappear. The peak intensity of copper is weaker than that of aluminum, and at 0.8 times its melting point, all peaks except the nearest neighbor peak disappear. This indicates that the grain boundary structure gradually transitions from the solid phase to the liquid phase, i.e., pre-melting occurs at the grain boundaries.

[0079] To investigate the relationship between grain boundary structure and grain boundary energy (GBE), five eigenvalues ​​were selected: orientation difference angle (sinθ), interface normal (dGB), free volume (FV), orientation order (Q), and centrosymmetry parameter (CSP), because previous research results have shown that these eigenvalues ​​are key parameters that determine grain boundary energy. Figure 4 The correlation between grain boundary energy and grain boundary characteristics of aluminum and copper varies with temperature. Figure 4 (a) shows the correlation analysis between the grain boundary energy of aluminum at 0 K and five grain boundary characteristics. Except for the orientation degree of freedom and the centrosymmetry parameter, which show a strong negative correlation, the other selected characteristic values ​​exhibit weak correlations, indicating strong independence among the selected parameters. Among these five characteristic values, the orientation difference angle and interface normal show very little correlation with the grain boundary energy, while the free volume shows the strongest positive correlation. Furthermore, the orientation degree of freedom exhibits a strong negative correlation with the grain boundary energy, and the centrosymmetry parameter also shows some correlation with the grain boundary energy, but it is smaller than that of the free volume. Figure 4 (b) At 0 K, the relationship between the grain boundary characteristics and grain boundary energy of copper is similar to that of aluminum. However, the correlation between the free volume and orientation order of copper and the grain boundary energy is stronger than that of aluminum, and the free volume and orientation order show a strong correlation. Figure 4(c) shows that with increasing temperature, the orientation difference angle and interface normal in aluminum still do not show a significant correlation with grain boundary energy, while the correlation coefficients of free volume, orientation order, and centrosymmetry parameter with grain boundary energy gradually increase. At 0.8 times the melting point, these three characteristic values ​​all show a strong correlation with grain boundary energy. The results for copper are as follows... Figure 4 As shown in (c), the free volume exhibits a strong correlation with the grain boundary energy at 0 K, and the correlation coefficient fluctuates only slightly with increasing temperature. The correlation coefficients between orientational order and centrosymmetry parameters and the grain boundary energy steadily increase and slightly exceed those of the free volume at 0.8 times the melting point.

[0080] The above descriptions are merely embodiments of the present invention, and common knowledge such as specific technical solutions and / or characteristics are not described in detail here. It should be noted that those skilled in the art can make various modifications and improvements without departing from the technical solutions of the present invention, and these should also be considered within the scope of protection of the present invention. These modifications and improvements will not affect the effectiveness of the implementation of the present invention or the practicality of the patent. The scope of protection claimed in this application should be determined by the content of its claims, and the specific embodiments described in the specification can be used to interpret the content of the claims.

Claims

1. A method for calculating the grain boundary energy of a metal under different temperature conditions, characterized in that: The method specifically includes: Step 1: Calculation of grain boundary energy at different temperatures Grain boundary energy is usually defined as the excess energy at the grain boundary, expressed as the total Gibbs free energy G in the bicrystal model. gb Subtract the Gibbs free energy G of a single atom within the crystal bulk Multiply by the number of atoms N in the model and divide by the total grain boundary area A, then calculate according to equation (1): The Gibbs free energy is calculated using the thermodynamic integration method of the Frenkel-Ladd path. The Frenkel-Ladd method is based on constructing a reversible path to a known free energy state; in which the Einstein crystal model is used with harmonic springs in all three dimensions; since the Einstein solid is structurally identical to the initial solid, this path is likely to be reversible without phase transition; starting from the initial solid, the reference state is slowly reached by opening the harmonic spring forces centered on the atomic positions while closing the interatomic potentials of the initial solid, which is controlled by the parameter λ, and the parameter Hamiltonian H(λ) can be written as Equation (2): H(λ)=λH E +(1-λ)H0 (2) Where H0 and H E Let the initial and final Hamiltonians be represented. The Hamiltonian of the initial state is given by equation (3): The Hamiltonian of Einstein solids can be written as equation (4): Where m is the mass of particle i, p is the momentum, U is the interatomic potential of the applied force field, ω is the frequency of the harmonic oscillator, and r i and r i 0 The update and initial lattice position of particle i are represented; λ = 0 and 1 in formula (1) represent the initial and final states, respectively; therefore, when the average free energy of the integral ensemble changes along the path, the free energy difference between the two states can be calculated: This reversible work involves two processes: a forward process 0→E and a reverse process E→0; the forward irreversible work... and reverse irreversible work Determined by molecular dynamics simulations: Where Γ is the scaling time, the reversible work can be expressed as the irreversible work: The free energy of a solid, as defined by Einstein, is called: Therefore, the Gibbs free energy of interest can be obtained through the following formula: G0=G E +W rev (9)

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