A crystal phase-field simulation method for coupled microvoid contraction and phase interface evolution

Abstract

The present invention describes a crystal phase field simulation method for coupled microvoid contraction and phase interface evolution. The steps are as follows: using the Helmholtz free energy equation based on density functional theory to describe the thermodynamics of voids and phase interfaces to obtain the vacancy free energy function; based on the Chan-Hilliard diffusion equation, establishing a crystal phase field dynamics equation related to the vacancy free energy function and atomic density; using the semi-implicit Fourier spectrum method to discretize the constructed crystal phase field dynamics equation in space and time to obtain t The atomic density field evolution equation at time +1 is derived; based on this evolution equation, a graph of the morphology of the void and phase interface is derived. Furthermore, by inputting variables, the evolution of void morphology under conditions of different sizes and positions is simulated, and the impact of different conditions on void contraction is analyzed. This invention can clarify the influence of different void sizes and morphologies on void contraction and reveal the influence of different void positions at the phase interface on the contraction rate.

Description

Technical Field

[0001] The present invention relates to the field of crystal simulation technology, in particular to a method for predicting the size of micro-voids and the relative position contraction mechanism and rate with respect to a phase interface based on a crystal phase field method. Background Art

[0002] Metal materials undergo numerous internal defects, such as interfaces, dislocations, and microvoids, through physical processes like high-energy particle irradiation, impact loading (unloading), and manufacturing. These microdefects can significantly affect the mechanical and physical properties of the material. For example, nanoindentation is a method currently used to measure the elastic modulus and hardness of a material's microstructure. During testing, the size and shape of the microvoids affect the indentation load, and the atoms on the microvoid surface also act as a barrier to dislocations. Clearly, the shrinkage mechanism and rate of the microvoids influence the experimental results.

[0003] Traditional experimental research methods can provide the most intuitive information about defects such as microvoids and interfaces in materials. They can understand the configurations of various defects at different spatial and temporal scales and visually observe the defect motion process. However, these detection instruments, such as high-resolution transmission electron microscopy (HRTEM) and scanning probe microscopy (SPM), have different requirements for sample preparation. For example, SPM can easily observe the cross-section of material interfaces, but it requires the experimental sample surface to be atomically smooth. Transmission electron microscopy, combined with an appropriate electron microscope test bench, can provide the ability to observe the dynamic evolution of material interfaces, i.e., in situ observation. However, current in situ observations at the atomic scale are mostly limited to observing samples under vacuum conditions and lack the field effects of real-world application scenarios, such as stress, electromagnetic fields, and temperature fields. Moreover, due to the stringent requirements for the experimental environment, the introduction of external fields can affect the mechanical stability of the sample bench, ultimately affecting the accuracy of the experimental results.

[0004] Molecular dynamics methods are commonly used to study atomic-scale problems, but constructing an appropriate interatomic interaction potential is crucial, leading to significant uncertainty in the calculated results. Currently, molecular dynamics methods can be used to study complex grain boundaries or phase interfaces at the atomic scale, simulating interface evolution and migration processes by applying stress. However, this method's time scale is limited to the atomic vibration timescale, making it difficult to simulate interface migration processes on the diffusion timescale. Consequently, the strain rate used is significantly higher than that used in actual experiments, raising questions about the accuracy of the calculated results. In 2002, Elder et al. proposed an atomic-scale model—the phase field crystal model (PFC). This method combines the advantages of traditional phase field methods and molecular dynamics simulations, enabling simulation of diffusion-timescale physical processes at the atomic scale. As a multiscale simulation method, it can describe atomic-scale crystal nucleation and growth processes. The model incorporates material properties related to crystal symmetry, such as elastic and plastic deformation, multi-orientation, and anisotropy, all of which can be self-consistently described by the PFC. Therefore, the crystal phase field method is a suitable and novel means to study the evolution of microvoids and phase interfaces at the atomic scale.

[0005] Currently, the crystal phase field method is used to study the dynamic evolution of crystal defects, mostly grain boundaries and dislocations, but there is a lack of research on vacancies or microvoids. The crystal phase field method can not only construct multi-oriented interface defects, but also couple microvoids for co-evolution. The crystal phase field method can reveal the mechanism of microvoid contraction and the influence of phase interfaces on microvoid contraction at the atomic scale. Combining the evolution process with the results of nanoindentation experiments can better understand the nanoscopic mechanism affecting the hardness of the material, further reduce the error of the experimental results, and optimize the experimental plan. At the same time, the use of crystal phase field methods can also make up for the long experimental cycle, high cost, high difficulty of sample preparation, and the influence of various unfavorable factors on the experimental results. Therefore, it is very advantageous and necessary to use crystal phase field simulation to study the co-evolution of defects and nanophases under irradiation of different sizes and shapes. Summary of the Invention

[0006] The technical problem solved by the present invention is to provide a crystal phase field simulation method for coupling microvoid contraction and phase interface evolution, so as to solve the problem that existing research cannot predict microvoid contraction and phase interface evolution from the atomic scale.

[0007] The technical solutions provided to achieve the purpose of the present invention are as follows:

[0008] A crystal phase-field simulation method for coupled microvoid contraction and phase interface evolution:

[0009] (1) Using the Helmholtz free energy equation based on density functional theory to describe the thermodynamics of the void and the phase interface, the vacancy free energy function is obtained;

[0010] (2) Based on the Chan-Hilliard diffusion equation, the crystal phase field dynamics equation related to the vacancy free energy function and atomic density is established;

[0011] (3) The constructed crystal phase field dynamics equation is discretized in space and time by using the semi-implicit Fourier spectrum method to obtain the atomic density field evolution equation at time t+1;

[0012] (4) According to the evolution equation of step (3), the evolution diagram of the morphology of the void and the phase interface is obtained; and the evolution of the void morphology under different size and position conditions is simulated by inputting variables, and the influence of different conditions on the void shrinkage is analyzed.

[0013] Furthermore, in step (1), the vacancy free energy function based on density functional theory is used to describe the thermodynamics of microvoids and phase interfaces. The specific process is as follows: the periodic atomic density order parameter is introduced into the vacancy free energy equation, and the vacancy free energy term is coupled to obtain the vacancy free energy function:

[0014]

[0015] Where F is the total free energy of the system, f vac =H(|ρ| n -ρ n ) represents the expression of the vacancy free energy term. For the convenience and stability of numerical calculation, H = 1500, n = 3, a, λ and g are phenomenological parameters related to the material properties, a is related to the supercooling temperature, and a smaller absolute value represents a higher temperature. g is determined by the density wave amplitude in the solid phase, and its specific value is determined by the two-point correlation function of the material. q0 is the modulus of the nearest neighbor reciprocal lattice vector in the crystal lattice, the interplanar spacing is 2π / q0, F is the free energy, ρ is the atomic density of the system, is the Laplace operator.

[0016] Furthermore, based on the Chan-Hilliard diffusion equation, a crystal phase field dynamics equation related to the vacancy free energy function and atomic density is established. The crystal phase field dynamics equation is as follows:

[0017]

[0018] Where ε is a parameter representing the system temperature. The larger its value, the lower the temperature and the greater the supercooling. Δt is the time step and ρ is the atomic density.

[0019] Furthermore, the expression of atomic density ρ is as follows:

[0020]

[0021] in, is the average atomic density, A is the amplitude, q = 2π / a, where a is the lattice constant of the hexagonal phase,

[0022] Furthermore, in step (3), the constructed crystal phase field dynamics equation is discretized in space and time by the semi-implicit Fourier spectrum method to obtain the atomic density field evolution equation at time t+1, wherein the atomic density field evolution equation at time t+1 is specifically as follows:

[0023]

[0024] in,{·} k is the Fourier transform of the atomic density order parameter, Δt is the time step, where {ρ} k =∫dre ik·r ρ(r,t), represents the reciprocal space wave vector, k=(k1,k2), represents the reciprocal space wave vector, its magnitude is

[0025] Compared with the prior art, the present invention has the following significant advantages:

[0026] This method incorporates a vacancy crystal phase-field model and exploits the near-zero atomic density of microvoids to couple the coexistence of microvoids with phase interfaces. Compared to experimental methods, this method can predict and analyze the coevolution of microvoids and phase interfaces at the atomic scale, reducing experimental cost and time. Comparative molecular dynamics methods can also be used to study the dynamic evolution of microvoids or interfaces. The vacancy crystal phase-field model can describe the microvoid contraction mechanism and the influence of phase interfaces on the microvoid contraction rate on a larger diffusion timescale. BRIEF DESCRIPTION OF THE DRAWINGS

[0027] Figure 1 The evolution process of a circular microvoid with radius R = 15Δx and a phase boundary, with an average atomic density of ρ = 0.2 and a supercooling parameter ε = -0.9.

[0028] Figure 2 The time-varying equivalent radius of a circular microvoid with radius R = 15Δx on and off the phase boundary. DETAILED DESCRIPTION

[0029] In order to more clearly illustrate the specific implementation of the present invention, it is further described through specific embodiments shown in the accompanying drawings.

[0030] The present invention is a phase field simulation method for the coordinated evolution of irradiation defects and nanophases, the steps of which are:

[0031] (1) Establish a vacancy free energy function based on density functional theory to achieve a thermodynamic description of microvoids and phase interfaces;

[0032] According to the crystal phase field simulation method of coupled microvoid contraction and phase interface evolution, the vacancy free energy function based on density functional theory is as follows:

[0033]

[0034] Where F is the total free energy of the system, f vac =H(|ρ| n -ρ n ) represents the vacancy free energy expression. For the convenience and stability of numerical calculations, H = 1500 and n = 3. a, λ, and g are phenomenological parameters related to material properties. a is related to the supercooling temperature; a smaller absolute value of a represents a higher temperature. g is determined by the density wave amplitude in the solid phase, and its specific value is determined by the two-point correlation function of the material. q0 is the modulus of the nearest neighbor reciprocal vector in the crystal lattice, the interplanar spacing is 2π / q0, and ρ is the atomic density of the system. is the Laplace operator.

[0035] (2) Based on the Chan-Hilliard diffusion equation, the crystal phase field dynamics equation related to the vacancy free energy function and atomic density is established;

[0036] The atomic density field changes dramatically at the atomic position. If the density field is expanded in the form of density wave superposition, there must be many terms in the expansion. The general expression of atomic density ρ is:

[0037]

[0038] where a i,j is the amplitude coefficient, r is the vector in the positive space, G is the vector in the reciprocal space, and ρ0 is the average atomic density. The classical density functional theory uses real physical quantities to construct the density field and describe the material properties as accurately as possible. The crystal phase field model uses a simplified density functional, which fundamentally limits the sum of the atomic density sum terms and takes only a small set of reciprocal lattice vectors in order to simulate the evolution of field variables on the largest possible spatial and temporal scales. Studies have shown that although the simplified crystal phase field model ignores many real data compared to actual conditions, the model can quantitatively reproduce some important properties of the crystal that affect the evolution of its microstructure, such as crystal bulk modulus and grain boundary energy. Assume that the basis vectors of the hexagonal lattice in the positive space are a=(1,0)a0, According to the following transformation relationship between positive space and reciprocal space:

[0039]

[0040] Where c = (0, 0, 1), so the reciprocal vector expression of the hexagonal phase is as follows:

[0041]

[0042] Where q = 2π / a0. In order to be consistent with the single-mode crystal phase field, the single-mode approximation is also used when approximating the analytical form of the hexagonal phase, that is, only the first group of series is retained - all lengths are equal to the basis vector a * According to the relationship between the vectors of the positive space and the reciprocal space of the hexagonal lattice, there are six groups of reciprocal vectors G that meet this requirement, including (i, j) = (±1, 0), (±1, 0), (1, -1) and (-1, 1), where G = ia * +jb * In addition, for the amplitude coefficient a of formula (2.4) i,j There are two requirements. First, because the atomic density is a real function, the coefficients must satisfy the relationship a i,j =a -i,j =a i,-j Moreover, due to the symmetry of the hexagonal phase, the relationship between the coefficients can be deduced as follows: 0,±1 =a ±1,0 =a 1,-1 =a -1,1 , and set it to A. Substitute it into the general expression of atomic density ρ appropriately, and then according to the Euler equation e ix =cosx+i·sinx, then the expression for atomic density ρ is:

[0043]

[0044] Where, is the average atomic density, A is the amplitude, q = 2π / a, where a is the lattice constant of the hexagonal phase. Then the obtained single-mode approximate solution is substituted into the free energy function to obtain the following hexagonal phase free energy density function f ρ The formula is:

[0045]

[0046] In order to obtain a stable solution of the system, the free energy density function f is required according to the definition of the stable state. ρ Find the partial derivatives of A and q respectively, that is So we get

[0047] The crystal phase field dynamics equation in step (2) of the present invention is as follows:

[0048]

[0049] Where ε is a parameter representing the system temperature. The larger its value is, the lower the temperature is and the greater the supercooling is.

[0050] (3) The constructed crystal phase field dynamics equation is discretized in space and time by using the semi-implicit Fourier spectrum method to obtain the atomic density field evolution equation at time t+1;

[0051] The semi-implicit Fourier spectral method described therein solves the crystal phase field kinetic equations as follows:

[0052]

[0053] in{·} k is the Fourier transform of the atomic density order parameter, where {ρ} k =∫dre ik·r ρ(r,t), k=(k1,k2), represents the reciprocal space wave vector, its magnitude is

[0054] By sorting out, we get the atomic density field evolution equation at time t+1:

[0055]

[0056] (4) According to the evolution equation of step (3), the evolution of microvoid morphology under different void sizes and positions is simulated by inputting variables, and the evolution diagram of microvoid and phase interface morphology is obtained; the influence of different void sizes and positions on microvoid shrinkage is analyzed. Specifically, the average atomic density of the hexagonal lattice is set to ρ = 0.2, and the supercooling parameter ε = -0.9. The initial position and size of the microvoid are set, and the calculation grid size is 256Δx×256Δy, where Δx = Δy = 0.78 is the spatial step size. In addition, the flexible multiple orientation of the crystal phase field is used to apply the rotation matrix algorithm to the atomic density expression to construct an interface with a certain orientation difference. Then, the evolution process of the circular microvoid coupled phase interface is simulated, and the change of the equivalent radius of the circular microvoid with and without the phase boundary is calculated to reveal the influence of the circular microvoid with and without the phase boundary on the shrinkage rate.

[0057] Figure 1 The evolution of a circular microvoid with radius R = 15Δx and a phase boundary. The average atomic density ρ = 0.2, the undercooling parameter ε = -0.9, and Grain 1 and Grain 2 represent two grains with different orientations. The figure shows that, on the one hand, the movement of atoms on the surface of the circular microvoid causes localized migration at the phase boundary. On the other hand, the collapse of the microvoid generates dislocation entanglement and lattice distortion. Compared to the contraction of microvoids on the phase boundary, circular microvoids not on the phase boundary take longer to shrink and collapse, indicating that the phase boundary promotes the absorption of microvoids.

[0058] Figure 2 is the change of the equivalent radius of a circular micro-void with radius R = 15Δx on and off the phase boundary with time, K a is the average slope of change. The figure shows that the shrinkage process of the two circular microvoids is not continuous; there are also periods where the equivalent radius stops decreasing over time. This indicates that the movement of atoms on the microvoid surface is not continuous toward the microvoid center, but rather that the microvoid shrinks sequentially over time. Furthermore, the shrinkage rate of circular microvoids at the phase boundary is faster, which is consistent with the evolution of circular microvoids and the phase boundary.

[0059] In general, the relative position of microvoids and the phase interface affects the shrinkage rate. Circular microvoids at the phase interface shrink at a higher rate. Furthermore, after a microvoid shrinks and collapses, atoms rearrange at the original microvoid location, creating dislocation entanglement or stacking faults. Studying the microvoid shrinkage process at the atomic scale can serve as a theoretical basis for nanoindentation testing and further understand the microscopic mechanisms underlying changes in material mechanical properties.

[0060] The above is a detailed description of the preferred implementation cases and principles of the present invention. For technicians in this field based on the ideas provided by the present invention, differences in specific implementation will be considered as the scope of protection of the present invention.

Claims

1. A crystal phase field simulation method for coupling microvoid contraction and phase interface evolution, characterized in that: The following steps are involved: (1) Using the Helmholtz free energy equation based on density functional theory to describe the thermodynamics of the void and the phase interface, the vacancy free energy function is obtained; (2) Based on the Cahn-Hilliard diffusion equation, the crystal phase field dynamics equation related to the vacancy free energy function and atomic density is established; (3) The constructed crystal phase field dynamics equation is discretized in space and time by using the semi-implicit Fourier spectrum method to obtain the atomic density field evolution equation at time t+1; (4) According to the evolution equation of step (3), the evolution diagram of the morphology of the void and the phase interface is obtained; and the evolution of the void morphology under different size and position conditions is simulated by inputting variables, and the influence of different conditions on the void shrinkage is analyzed; in: In step (1), the vacancy free energy function based on density functional theory is used to describe the thermodynamics of microvoids and phase interfaces. The specific process is as follows: the periodic atomic density order parameter is introduced into the vacancy free energy equation, and the vacancy free energy term is coupled to obtain the vacancy free energy function: Where F is the total free energy of the system, f vac =H(|ρ| n -ρ n ) represents the expression of the vacancy free energy term, where H = 1500, n = 3, a, λ and g are phenomenological parameters related to the material properties, a is related to the supercooling temperature, and the smaller the absolute value of a, the higher the temperature. g is determined by the density wave amplitude in the solid phase, and the specific value is determined by the two-point correlation function of the material. q0 is the modulus of the nearest neighbor reciprocal lattice vector in the crystal lattice, the interplanar spacing is 2π / q0, F is the free energy, and ρ is the atomic density of the system. is the Laplace operator; In step (2), based on the Cahn-Hilliard diffusion equation, a crystal phase field dynamics equation related to the vacancy free energy function and the atomic density is established. The crystal phase field dynamics equation is as follows: Where ε is a parameter representing the system temperature. The larger its value, the lower the temperature and the greater the supercooling. Δt is the time step, and ρ is the atomic density. The expression of atomic density ρ is as follows: Where x and y are two-dimensional space coordinates. is the average atomic density, A is the amplitude, q = 2π / a, where a is the lattice constant of the hexagonal phase, In step (3), the constructed crystal phase field dynamics equation is discretized in space and time by the semi-implicit Fourier spectrum method to obtain the atomic density field evolution equation at time t+1, wherein the atomic density field evolution equation at time t+1 is specifically as follows: Where, {·} k is the Fourier transform of the atomic density order parameter, Δt is the time step, where {ρ} k =∫dre ik·r ρ(r,t), represents the reciprocal space wave vector, k=(k1,k2), and its magnitude is

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