Quantitative simulation method for binary polycrystal solidification process

Abstract

The invention discloses a binary polycrystal solidification process quantitative simulation method which comprises the following steps: establishing a unified huge potential functional expression capable of uniformly describing a solid phase, a liquid phase, a solid-liquid interface and a grain boundary according to thermodynamic description of a binary polycrystal system; constructing a kinetic equation of the non-conservative field sequence parameters according to an energy dissipation theory and a Cahn-Harliad formula; constructing a chemical potential kinetic equation according to the Fick's second law and progressive analysis; obtaining a calculation formula of solute concentration through thermodynamic definition of chemical potential; coupling thermal noise; materials and physical parameters learned by phase field simulation are obtained through literatures, thermodynamic software and experiments; carrying out discrete solution on the kinetic equation by using methods of finite difference, finite volume or finite element and the like; and post-processing and visualizing the numerical solution process to complete the process. According to the method, the dilute solution limitation and polycrystal limitation of a phase field model based on free energy functional are broken through, and quantitative calculation of the grain boundary energy in the binary polycrystal solidification process is realized.

Description

Technical Field

[0001] The invention belongs to the field of material metallurgical process simulation, and in particular relates to a quantitative simulation method for a binary polycrystalline solidification process. Background Art

[0002] The performance of metal materials (alloys) is mainly affected by their microstructure and structural defects. For many alloys, the solidification process is a major step in the alloy production process, and the solidification process even directly determines the main microstructure of its service process, such as aluminum alloys. In addition, many important process defects are also related to the solidification process, such as hot cracks, pores, segregation and inclusions.

[0003] The evolution of microstructures and the generation of structural defects during the solidification process are closely related to the composition and solidification conditions of the alloy. Usually, the solidification process inside metal alloy products cannot be directly observed. At the same time, experimental research generally faces disadvantages such as long cycles, high costs, and difficulty in achieving specific conditions. Numerical simulation methods can easily solve the above difficulties. At present, the simulation methods for microstructure transformation during alloy solidification processes mainly include Monte Carlo method, cellular automaton method and phase field method. The phase field method is a deterministic method because its model construction is based on the physical mechanism of material phase change. Under certain conditions, quantitative simulation of the phase change process can be achieved.

[0004] For binary alloys, Yang (L. Yang, J. Yang, F. Han, et al. Hot cracking susceptibility prediction from quantitative multi-phase field simulations with grain boundary effects [J]. Acta Materialia, 2023, 250 (5): 118821.) constructed a phase field model based on free energy functional that can quantitatively simulate the growth and solidification process of multiple branches in binary dilute solution systems. The solidification process is the process in which two or more freely growing branches begin to interact and form grain boundaries. The dilute solution approximation generally assumes that the solute concentration is low and the solidus and liquidus are straight lines, that is, the equilibrium solute distribution coefficient is a constant at any temperature during the solidification process. In addition, Yang used this model to quantitatively simulate the quantitative solidification process of twin crystals and extracted the solidification path information from it. Combined with the RDG model, the effect of solute concentration on hot crack susceptibility under different grain boundary energy conditions was studied. However, with the gradual increase of solute concentration, the deviation of the results simulated by the phase field model also gradually increased. Therefore, how to accurately and quantitatively simulate the solidification process of multiple dendrites under high solute concentration is an important issue that needs to be solved. Summary of the invention

[0005] In order to solve the problem of quantitatively simulating the microstructural evolution of multiple dendrite growth and solidification processes of binary alloys, the present invention constructs a quantitative phase field model of binary polycrystalline solidification and solidification processes, and provides corresponding parameter acquisition methods and numerical solution methods.

[0006] The purpose of the present invention is achieved through the following technical solutions:

[0007] A method for quantitatively simulating a binary polycrystalline solidification process of the present invention, the specific steps of the method are as follows:

[0008] A quantitative phase field model for binary multi-dendrite growth and solidification process based on giant potential functional, and a quantitative model for binary alloy multi-dendrite growth and solidification process based on the model, including the following steps:

[0009] Step 1: Based on the thermodynamic description of the binary polycrystalline system, a functional expression of the giant potential describing the solid phase, liquid phase, solid-liquid interface, and grain boundary is established, and its expression is:

[0010]

[0011] Where ω is the giant potential density, μ is the chemical potential, T is the temperature, r is the space vector, α represents and traverses all solid phase grains, l represents the liquid phase, g(φ) represents the interpolation function, and the giant potential density of each phase is interpolated through the order parameter φ. int is the interface giant potential density;

[0012] An expression covering the solid-liquid interface and grain boundary is established, and its expression is:

[0013]

[0014] where σ α represents the solid-liquid interface energy coefficient, H α represents the nucleation energy barrier between the solid and liquid phases, α αβ represents the solid-solid interface interaction coefficient, f DW (φ α ) is a double potential well function, and its expression is:

[0015] f DW (φ α )=φ α 2 (1-φα) 2

[0016] In the giant potential calculation, the free energy function adopts the parabola approximation, and its expression is:

[0017]

[0018] The bulk giant potential density can be expressed as:

[0019]

[0020] The chemical potential is:

[0021]

[0022] Step 2: Define supersaturation:

[0023]

[0024] In the formula, χ l(eq) ≡1 / A l , ΔB α ≡B l -B α , χ α(eq) ≡1 / A α , k α(eff) ≡χ α(eq) / x l(eq) ;

[0025] Step 3: Construct the dynamic equation of non-conservative field order parameters based on energy dissipation theory and Cahn-Halliard formula:

[0026]

[0027] Where W α is the characteristic width of the solid-liquid interface, τ α is the solid-liquid interface phase field relaxation time, λ α is the coupling coefficient between order parameter and solute field, ω αβ is the solid-solid interface interaction coefficient, is an interpolation function that satisfies

[0028] in, is the solid-liquid interface interpolation function:

[0029]

[0030] Step 4: Construct the chemical potential kinetic equation based on Fick's second law and asymptotic analysis:

[0031]

[0032] in:

[0033]

[0034] Where ξ is the ratio of the solid phase diffusion coefficient to the liquid phase diffusion coefficient, is the equilibrium concentration difference between the liquid and solid phases, and the interpolation function

[0035] Step 5: Obtain the calculation formula for solute concentration through the thermodynamic definition of chemical potential:

[0036]

[0037] Step 6: Coupled thermal noise:

[0038]

[0039] Step 7: Obtain the materials and physical parameters learned from phase field simulation through thermodynamic software and experiments;

[0040] Step 8: Discretely solve the dynamic equations using finite difference, finite volume or finite element methods.

[0041] Beneficial Effects

[0042] The method of the present invention can quantitatively simulate the solidification and coagulation process of binary alloys of any concentration, and is no longer limited by the dilute solution when constructing similar models before. The model can be extended to quantitatively simulate the solidification and coagulation process of binary alloys of any concentration under rapid solidification conditions. Combined with conventional nucleation methods, the model can be extended to quantitatively simulate the entire solidification process of multi-dendrite nucleation, dendrite growth and dendrite coagulation of binary alloys of any concentration. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 is a graph showing the relationship between the grain boundary energy of the Al-Cu alloy obtained in Example 1 and the interaction coefficient;

[0044] Figure 2 This is a simulated morphology diagram of the Al-Cu alloy molten liquid obtained in Example 1 in the late stage of solidification. DETAILED DESCRIPTION

[0045] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0046] Example 1

[0047] This embodiment takes an Al-Cu alloy with a solute concentration of 8.0wt.% as an example. In this embodiment, a method for quantitatively simulating the solidification process of binary polycrystals is implemented by the following steps:

[0048] Step 1: Based on the thermodynamic description of the binary polycrystalline system, a unified giant potential functional expression is established to uniformly describe the solid phase, liquid phase, solid-liquid interface, and grain boundary:

[0049]

[0050] Where ω is the giant potential density, μ is the chemical potential, T is the temperature, r is the space vector, α represents and traverses all solid phase grains, l represents the liquid phase, g(φ) represents the interpolation function, and the giant potential density of each phase is interpolated through the order parameter φ. int is the interface giant potential density, covering the solid-liquid interface and grain boundary, and its expression is:

[0051]

[0052] where σ α represents the solid-liquid interface energy coefficient, H α represents the nucleation energy barrier between the solid and liquid phases, α αβ represents the solid-solid interface interaction coefficient, f DW (φ α ) is a double potential well function, and its expression is:

[0053] f DW (φ α )=φ α 2 (1-φ α 2 )

[0054] And the free energy function adopts parabolic approximation in giant potential calculation:

[0055]

[0056] The bulk giant potential density can be expressed as:

[0057]

[0058] The chemical potential is

[0059]

[0060] Step 2: Define supersaturation:

[0061]

[0062] In the formula, χ l(eq) ≡1 / A l , ΔB α ≡B l -B α , χ α(eq) ≡1 / Aα , k α(eff) ≡ α(eq) / x l(eq) ;

[0063] Step 3: Construct the dynamic equation of non-conservative field order parameters based on energy dissipation theory and Cahn-Halliard formula:

[0064]

[0065] Where W α is the characteristic width of the solid-liquid interface, τ α is the solid-liquid interface phase field relaxation time, λ α is the coupling coefficient between order parameter and solute field, ω αβ is the solid-solid interface interaction coefficient, is an interpolation function that satisfies:

[0066]

[0067] in, is the solid-liquid interface interpolation function:

[0068]

[0069] Step 4: Construct the chemical potential kinetic equation based on Fick's second law and asymptotic analysis:

[0070]

[0071] in:

[0072]

[0073] Where ξ is the ratio of the solid phase diffusion coefficient to the liquid phase diffusion coefficient, is the equilibrium concentration difference between the liquid and solid phases, and the interpolation function

[0074] Step 5: Obtain the calculation formula for solute concentration through the thermodynamic definition of chemical potential:

[0075]

[0076] Step 6: Coupled thermal noise:

[0077]

[0078] Step 7: Obtain the materials and physical parameters learned from phase field simulation through literature, thermodynamic software, and experiments.

[0079] Step 8: Discretely solve the dynamic equations using finite difference, finite volume or finite element methods.

[0080] Step 9: Post-process and visualize the numerical solution process.

[0081] In this embodiment, various physical parameters involved in the above formulas are obtained through experiments, literature, calculations, etc.

[0082] Figure 1 The relationship between the grain boundary energy of the Al-Cu alloy and the interaction coefficient obtained in this embodiment shows that this embodiment realizes the quantitative calculation of the grain boundary energy, and the grain boundary energy shows a trend of first rising and then stabilizing with the increase of the interaction coefficient, which meets the metallurgical principles.

[0083] Figure 2 The simulated morphology of the late directional solidification of the Al-Cu alloy molten liquid obtained in this embodiment shows that the molten liquid solidification morphology obtained by a method for quantitatively simulating the binary polycrystalline solidification process involved in this embodiment highly reproduces the actual molten liquid morphology under high solute concentration. The grain boundaries are generated by the solidification process of adjacent grains and are well preserved under large grain boundary supercooling. The cellular-dendrite transformation of the grains near the solid-liquid interface is also well reproduced.

[0084] The above embodiments are merely examples for the purpose of clear explanation and are not intended to limit the implementation methods. For those skilled in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to list all the implementation methods here. The obvious changes or modifications derived therefrom are still within the scope of protection of the invention.

Claims

1. A method for quantitative simulation of binary polycrystalline solidification process, characterized in that: The specific steps of this method are as follows: Step 1: Based on the thermodynamic description of the binary polycrystalline system, a functional expression of the giant potential describing the solid phase, liquid phase, solid-liquid interface, and grain boundary is established, and its expression is: Where ω is the giant potential density, μ is the chemical potential, T is the temperature, r is the space vector, α represents and traverses all solid phase grains, l represents the liquid phase, g(φ) represents the interpolation function, and the giant potential density of each phase is interpolated through the order parameter φ. int is the interface giant potential density; An expression covering the solid-liquid interface and grain boundary is established, and its expression is: where σ α represents the solid-liquid interface energy coefficient, H α represents the nucleation energy barrier between the solid and liquid phases, α αβ represents the solid-solid interface interaction coefficient, f DW (φ α ) is a double potential well function, and its expression is: f DW (f α )=φ α 2 (1-φ α ) 2 In the giant potential calculation, the free energy function adopts the parabola approximation, and its expression is: The bulk giant potential density can be expressed as: The chemical potential is: Step 2: Define supersaturation: wherein, ΔB α ≡B l -B α , χ α(eq) ≡1 / A α , k α(eff) ≡χ α(eq) / χ l(eq) ; Step 3: Construct the dynamic equation of non-conservative field order parameters based on energy dissipation theory and Cahn-Halliard formula: Where W α is the characteristic width of the solid-liquid interface, τ α is the solid-liquid interface phase field relaxation time, λ α is the coupling coefficient between order parameter and solute field, ω αβ is the solid-solid interface interaction coefficient, is an interpolation function that satisfies in, is the solid-liquid interface interpolation function: Step 4: Construct the chemical potential kinetic equation based on Fick's second law and asymptotic analysis: in: Where ξ is the ratio of the solid phase diffusion coefficient to the liquid phase diffusion coefficient, is the equilibrium concentration difference between the liquid and solid phases, and the interpolation function Step 5: Obtain the calculation formula for solute concentration through the thermodynamic definition of chemical potential: Step 6: Coupled thermal noise: Step 7: Obtain the materials and physical parameters learned from phase field simulation through thermodynamic software and experiments; Step 8: Discretely solve the dynamic equations using finite difference, finite volume or finite element methods.

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