Phase field method for crystal growth under diffusion control in molten salt system
By employing a diffusion-controlled phase-field method in a molten salt system, the solid-liquid interface is automatically tracked to simulate the crystal growth process of single-crystal materials grown by the molten salt method. This solves the problems of high computational resource requirements and high mesh generation difficulty in traditional modeling methods, and realizes rapid and accurate crystal growth simulation and directional synthesis scheme design.
Patent Information
- Application Number
- CN202511059641.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-07-30
- Publication Date
- 2025-12-16
AI Technical Summary
Existing technologies struggle to accurately simulate the crystal morphology and size control of salt inclusion compounds synthesized via the molten salt method. Traditional modeling methods require significant computational resources, are difficult to generate meshes, and have poor applicability, making it impossible to accurately predict the microstructure and properties of materials.
By employing the diffusion-controlled phase-field method, a two-dimensional geometric model of crystal growth in molten salt medium is established. This model is combined with the free energy density function and concentration diffusion field to automatically track the solid-liquid interface and simulate the crystal growth process in real time, thereby reducing model complexity and the number of meshes.
It achieves rapid response crystal growth simulation, reduces computational costs, and can accurately predict the concentration distribution and morphology of single crystal materials grown by molten salt method, supporting the design and verification of directional synthesis production schemes.
Smart Images
Figure CN121148545A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The application belongs to the field of numerical simulation of crystal growth in molten salt system, and particularly relates to a phase field method for crystal growth under diffusion control in a molten salt system. BACKGROUND
[0002] At present, the installed capacity of nuclear power generating units is the first in the world, and the development of nuclear power cannot bypass the reprocessing technology. One of the fundamental problems to be solved for the sustainable large-scale development of fissile energy is to reduce the inventory of high-level radioactive waste and achieve safe disposal. Dry reprocessing technology has been widely used internationally, and a large amount of radioactive fission products still exist in the molten salt after electrolytic refining in the dry reprocessing process of spent fuel, which is the focus that needs to be considered in the disposal of waste salt.
[0003] Salt inclusion is a new hierarchical material that can be customized at the molecular or lattice level, and has the potential to fix specific radionuclides in a stable matrix. Compared with traditional glasses, salt inclusion crystal materials have stronger stability. The process of growing salt inclusion by molten salt method is usually to put the reactants into a certain proportion of eutectic salt, heat at high temperature to melt the salt, and then the reactants react with the molten salt to form salt inclusion crystals during the cooling stage. Compared with hydrothermal method and high temperature reaction method, the molten salt method for synthesizing salt inclusion has the advantages of accommodating abnormal valence radionuclides, simple process, less risk of explosion accidents, and easier control of the shape and size of the crystal.
[0004] Crystal morphology and size have an impact on the macroscopic mechanical properties of materials. The mechanical, electrical, magnetic and other properties of materials not only depend on the chemical composition of the material, but also largely depend on the microstructure distribution inside the material. The study of the microstructure of materials not only helps us better understand the relationship between the macroscopic properties and the microstructure of materials, but also helps us improve the performance of materials by optimizing and designing the microstructure.
[0005] Using computer technology to simulate the microstructure evolution process of materials can help us understand the evolution mechanism of the organization and improve the material organization and its mechanical properties, and provide a theoretical basis for optimizing the preparation process and controlling the quality of materials. However, the modeling of crystallization by molten salt method is usually challenging because the basic process of nucleation and growth is very complex, the traditional modeling method requires a large amount of computing resources, the geometric modeling is difficult, the number of grids is large and difficult to generate, and the applicability is poor. At the same time, the material is affected by many factors during the crystallization process, and the intermediate process of crystallization is uncontrollable. Only relying on experiments cannot achieve accurate quantitative analysis. SUMMARY
[0006] The application aims to provide a phase field method for crystal growth under diffusion control in a molten salt system, which can automatically track a solid-liquid interface and simultaneously couple a concentration diffusion field to simulate a crystal growth process under diffusion in real time.
[0007] The application achieves the goal by the following technical solutions.
[0008] The application achieves the goal by the following technical solutions.
[0009] Step 1: establishing a two-dimensional geometric model of crystal growth in a molten salt medium;
[0010] Step 2: determining the width of the solid-liquid interface, and dividing the two-dimensional geometric model of crystal growth in the molten salt medium into a grid meeting the simulation of computational fluid dynamics;
[0011] Step 3: constructing a free energy density function describing the crystal growth process of a single crystal material in the molten salt medium;
[0012] Step 4: varying the free energy density function with respect to an order parameter to obtain an order parameter evolution equation;
[0013] Step 5: varying the free energy density function of the system with respect to a concentration parameter, and combining a source term generated by the discontinuity of the solute concentration gradient on the interface and a net flux generated by the discontinuity of the solute concentration on the interface to obtain a concentration parameter evolution equation;
[0014] Step 6: obtaining physical property parameters and phase field parameters, substituting them into the order parameter evolution equation and the concentration parameter evolution equation to solve, obtaining convergent solutions of the order parameter and the concentration, and finally performing visual processing.
[0015] Further, the simulation domain of the two-dimensional geometric model of crystal growth in the molten salt medium in step 1 is larger than the crystal size, and a no-flux boundary condition is set on the crucible wall surface of the simulation domain.
[0016] Further, the grid size meeting the simulation of computational fluid dynamics in step 2 is one-tenth of the width of the solid-liquid interface.
[0017] Further, the free energy density function in step 3 is:
[0018] f(φ,c)=f1(φ)+λf2(φ)c
[0019] Where f1(φ) and f2(φ) are double-potential well interpolation functions, φ is an order parameter, that is, a quantity of a solid-liquid state of a substance; c is a concentration in a dimensionless form; and λ is a coupling coefficient.
[0020]
[0021] When φ in f1(φ) and f2(φ) takes values of -1 and +1, the free energy density function takes minimum values respectively.
[0022] Further, the step 4 is the order parameter evolution equation:
[0023]
[0024] Wherein, τ is a time constant, φ is an order parameter; t is time; ε is an interface width constant; ▽ is a Hamiltonian operator; c is a dimensionless concentration; γ is a gradient term coefficient; λ is a coupling coefficient; κ is an interface curvature.
[0025] Further, the step 5 is a source term generated by the discontinuity of solute concentration gradient on the interface And net flux The concentration evolution equation is obtained:
[0026]
[0027] Wherein, α is a dimensionless constant, φ is an order parameter; t is time; c is a dimensionless concentration; D is a diffusion coefficient; ▽ is a Hamiltonian operator.
[0028] Further, the step 6 is that the phase field parameters include: interface mobility, time constant, coupling coefficient; physical property parameters include: single crystal material density, interface energy, equilibrium concentration.
[0029] Further, the dimensionless concentration c=(C-C eq ) / C eq , wherein C is the ion concentration of a single crystal in a molten salt, and C eq is the equilibrium concentration of a single crystal in a molten salt.
[0030] Further, the time constant τ=1 / (Lω), wherein L is phase field mobility, and ω is a trap height of a double potential well function; the interface width constant Curvature term Wherein, ▽ is a Hamiltonian operator, is a unit normal vector of a solid-liquid interface, and points to a liquid phase.
[0031] Further, the dimensionless constant Wherein ρ is density; C eq is the equilibrium concentration of a single crystal in a molten salt, and C is the ion concentration of a single crystal in a molten salt.
[0032] The present application has the beneficial effects that:
[0033] The present application can avoid the direct tracking process of solid-liquid interface. Therefore, the method of the present application reduces the complexity of the model and the number of grids and the time calculation cost, and the calculation response is rapid, which meets the demand of rapid response of calculation of key components in other fields such as crystal growth; and at the same time, the supersaturation concentration of the crystal can be directly calculated from the diffusion equation, and the ion concentration distribution in the molten salt body can be calculated. The present application can be applied to the process monitoring and prediction of single crystal materials grown by the molten salt method, so as to realize the design and verification of the production scheme of directional synthesis of single crystal materials. BRIEF DESCRIPTION OF DRAWINGS
[0034] Figure 1 The flow chart of the numerical simulation method of the present application is shown in the figure;
[0035] Figure 2 The initial concentration field map is shown in the figure;
[0036] Figure 3 The initial order parameter field map is shown in the figure;
[0037] Figure 4 The grid division map is shown in the figure;
[0038] Figure 5 The concentration field and order parameter evolution result map is shown in the figure. DETAILED DESCRIPTION
[0039] The technical solutions in the present application will be clearly and completely described in combination with the accompanying drawings of the present application. Obviously, the described embodiments are part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor are within the scope of the present application.
[0040] In the description of the present application, the functions and steps marked in the flow chart and block diagram of the present application can occur in different order from that marked in the accompanying drawings without conflict. The present application will be further described in combination with the accompanying drawings of the embodiments of the present application:
[0041] As shown in the figure, Figure 1 The present application relates to a phase field model establishment method suitable for crystal growth under diffusion control in a molten salt system, which comprises the following steps:
[0042] Step 1: a two-dimensional geometric model of crystal growth in a molten salt medium is established by a geometric model building software; the two-dimensional geometric model takes an approximately infinite simulation domain, i.e. the simulation domain is much larger than the size of the crystal, to ensure the sustainability of the simulation; mainly including the crucible wall and the molten salt medium. On the boundary of the simulation domain, i.e. the crucible wall, a no-flux boundary condition is set to ensure no additional source flow.
[0043] A rectangular unit that can represent the crystal morphology properties and characteristics is selected as the simulation domain for the molten salt medium and the crucible wall, the complex internal environment is equivalent to a single medium homogenization flow domain, and the rectangular simulation domain boundary is the crucible wall surface without considering the specific structure of the crucible inside. As shown in Figure 2 , the simulation domain is square, an initial circular crystal nucleus is arranged in the middle, the sequence parameter value in the crystal nucleus is -1, which is solid phase, the outside of the crystal nucleus is molten salt medium, the sequence parameter value is 1, the solid-liquid boundary is set as a step function, and the sequence parameter value is -1 to 1. The simulation domain boundary is the crucible wall surface, and the no-flux boundary condition is adopted, and the specific structure of the crucible inside is not considered. As shown in Figure 3 , the initial dimensionless concentration of the concentration field is set to 0.97, that is, c=(C-C eq ) / C eq ; wherein c is the dimensionless concentration, C is the ion concentration of the single crystal in the molten salt, and C eq is the equilibrium concentration of the single crystal in the molten salt. The ion concentration in the crystal nucleus is set to 0, which is pure solid phase.
[0044] Step 2: The two-dimensional geometric model of crystal growth in the molten salt medium is divided into a grid for computational fluid dynamics simulation by a grid division software; the grid size should generally be less than one-tenth of the selected solid-liquid interface width to ensure the authenticity and effectiveness of the simulation.
[0045] As shown in Figure 4 , a free quadrilateral mapping grid is adopted; a grid that is completely consistent and coincides in space is established by grid replication, and the sequence parameter field and the concentration field are independently divided into calculation domains.
[0046] Step 3: Establish a free energy density function for describing the crystal growth process of single crystal materials in the molten salt medium; select a suitable double-well interpolation function, and combine the concentration parameter and the sequence parameter into the local free energy density function by using the coupling coefficient, and add the gradient terms of the sequence parameter and the concentration parameter to form the free energy density functional of the system. The selection of the double-well interpolation function in the free energy density function should ensure that the free energy density function takes the minimum value when the sequence parameter takes the value of -1 and +1. Therefore, the free energy density function can be selected as f(φ,c)=f1(φ)+λf2(φ)c; wherein: f is the free energy density function; a simple double-well interpolation function can be written as φ is the sequence parameter, which is a quantity describing the solid-liquid state of the substance; λ is the coupling coefficient; the free energy density function can be represented as
[0047] Step 4: Variation of the free energy density function of the system to establish the sequence parameter evolution equation; the sequence parameter evolution equation can be represented as wherein L is the phase field mobility; is the Hamiltonian operator. The evolution equation of the sequence parameter can be written as where γ is the gradient term coefficient, ω is the trap height of the double potential well function, and κ is the interface curvature. Define the time constant τ = 1 / (Lω) and the interface width constant The evolution equation of the order parameter can be rewritten as where τ is the time constant; t is the time. The curvature term κ is defined as where is the unit normal vector of the solid-liquid interface, pointing to the liquid phase.
[0048] Step 5: The free energy density function of the system is varied with respect to the concentration parameter, and the evolution equation of the concentration parameter is established; the evolution equation of the concentration can be expressed as where D is the diffusion coefficient. The evolution equation of the concentration can be written as where α is a dimensionless constant, which can be calculated by where ρ is the density. The second term in the above equation is the source term generated by the discontinuity of the solute concentration gradient, and the third term is the net flux generated by the discontinuity of the solute concentration on the interface, thus obtaining the evolution equation of the concentration parameter.
[0049] Step 6: The physical parameters of the selected material (single crystal material density, interface energy, equilibrium concentration) and the phase field parameters (interface mobility, time constant, coupling coefficient) are substituted into the evolution equations of the order parameter and the concentration parameter to solve, and the converged solutions of the order parameter and the concentration are obtained, and the evolution distribution diagram of the concentration of the single crystal material in the molten salt with time and space and the crystal morphology evolution process diagram at each time are obtained.
[0050] As Figure 5 The upper graph in the figure represents the evolution of the concentration field, and the lower graph represents the evolution of the order parameter field. In this embodiment, the crystal has a four-fold symmetry structure. For the convenience of visualization processing and analysis, the order parameter φ is expressed as i.e. in the result graph or 1 represents φ = 1 or -1, respectively. In the initial state, the crystal material is molten in the molten salt in the concentration field, and the dimensionless form of the concentration is 0.97, wherein the circular crystal nucleus represents the nucleated crystal, and the concentration is 0, and the concentration c at the interface varies between 0 and 1; in the order parameter field, the liquid part is φ` = 0, the circular crystal nucleus is solid, that is, φ` = 1, and the interface φ` varies between 0 and 1. Under the driving of supersaturation, the crystal is precipitated from the molten salt and gradually grows, and the newly grown crystal undergoes a process from liquid to solid, that is, the dimensionless form of the concentration c at the new phase changes to 0, and the order parameter field changes from φ` = 0 to φ` = 1. At the same time, under the diffusion effect, the concentration of the crystal molten in the molten salt also changes under the driving of the concentration gradient, and as the time increases, the crystal in the molten salt is precipitated, the dimensionless form of the concentration c in the concentration field near the precipitated solid decreases to 0, the φ` near the precipitated solid in the order parameter field increases to 1, and under the action of crystallographic orientation, it grows to the equilibrium shape. The spatial distribution diagram of the dimensionless form of the concentration c and the order parameter field φ` at different times can obtain the morphology diagram of the crystal growth.
[0051] The application further provides a phase field system for crystal growth under diffusion control in a molten salt system, which comprises a memory, a processor and a computer program stored in the memory, and the processor executes the computer program to realize the steps of the phase field method for crystal growth under diffusion control in a molten salt system, which will not be repeated here.
[0052] The above merely describes the preferred embodiments of the present application and is not used to limit the present application. For those skilled in the art, the present application can have various modifications and changes. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application shall be included in the protection scope of the present application.
Claims
1. A phase-field method for diffusion-controlled crystal growth in a molten salt system, characterized in that: Specifically, the following steps are included: Step 1: Establish a two-dimensional geometric model for crystal growth in molten salt medium; Step 2: Determine the width of the solid-liquid interface, and divide the two-dimensional geometric model of crystal growth in the molten salt medium into a mesh that satisfies the computational fluid dynamics simulation. Step 3: Construct the free energy density function describing the crystal growth process of single-crystal materials in molten salt medium; Step 4: Perform variational analysis on the free energy density function with respect to the order parameter to obtain the order parameter evolution equation; Step 5: Perform variational analysis on the system's free energy density function with respect to the concentration parameter, and combine the source term generated by the discontinuity of the solute concentration gradient at the interface with the net flux generated by the discontinuity of the solute concentration at the interface to obtain the concentration parameter evolution equation. Step 6: Obtain the physical property parameters and phase field parameters, substitute them into the order parameter evolution equation and concentration parameter evolution equation to solve for the convergent solutions of the order parameter and concentration.
2. The phase-field method for diffusion-controlled crystal growth in a molten salt system according to claim 1, characterized in that: In step 1, the simulation domain of the two-dimensional geometric model for crystal growth in the molten salt medium is larger than the crystal size, and a flux-free boundary condition is set on the crucible wall of the simulation domain.
3. The phase-field method for diffusion-controlled crystal growth in a molten salt system according to claim 1, characterized in that: In step 2, the mesh size for computational fluid dynamics simulation is one-tenth of the width of the solid-liquid interface.
4. The phase-field method for diffusion-controlled crystal growth in a molten salt system according to claim 1, characterized in that: The free energy density function in step 3 is: f(φ,c)=f1(φ)+λf2(φ)c Where f1(φ) and f2(φ) are double-potential-well interpolation functions, φ is the order parameter, i.e. the quantity of the substance in solid-liquid state; c is the dimensionless concentration; and λ is the coupling coefficient. When φ in f1(φ) and f2(φ) takes the values of -1 and +1, the free energy density function takes the minimum value.
5. The phase-field method for diffusion-controlled crystal growth in a molten salt system according to claim 1, characterized in that: The order parameter evolution equation in step 4 is as follows: Where τ is the time constant, φ is the order parameter, t is time, and ε is the interface width constant; λ is the Hamiltonian operator; c is the dimensionless concentration; γ is the gradient coefficient; λ is the coupling coefficient; and κ is the interface curvature.
6. The phase-field method for diffusion-controlled crystal growth in a molten salt system according to claim 1, characterized in that: The source term of the discontinuity of the solute concentration gradient at the interface in step 5. and net flux The concentration evolution equation is obtained as follows: Where α is a dimensionless constant, φ is an order parameter, t is time, c is the dimensionless concentration, and D is the diffusion coefficient; It is a Hamiltonian operator.
7. The phase-field method for diffusion-controlled crystal growth in a molten salt system according to claim 1, characterized in that: The phase-field parameters in step 6 include: interface mobility, time constant, and coupling coefficient; the physical property parameters include: single crystal material density, interface energy, and equilibrium concentration.
8. The phase-field method for diffusion-controlled crystal growth in a molten salt system according to claim 4, characterized in that: The dimensionless concentration c = (CC eq ) / C eq Where C is the ion concentration of the single crystal in the molten salt, C eq This represents the equilibrium concentration of single crystals in the molten salt.
9. The phase-field method for diffusion-controlled crystal growth in a molten salt system according to claim 5, characterized in that: The time constant τ = 1 / (Lω), where L is the phase mobility and ω is the trap height of the double-well function; the interface width constant... curvature term in, For Hamiltonian operators, This is the unit normal vector of the solid-liquid interface, pointing towards the liquid phase.
10. The phase-field method for diffusion-controlled crystal growth in a molten salt system according to claim 6, characterized in that: The dimensionless constant Where ρ is density; C eq Let be the equilibrium concentration of the single crystal in the molten salt, and C be the ion concentration of the single crystal in the molten salt.