Phase field model, modeling method and electronic device for polycrystalline zirconium-niobium alloy system

Abstract

The present invention discloses a phase field model, a modeling method and an electronic device of a polycrystalline zirconium-niobium alloy system. Based on two elements and equilibrium vacancies in the polycrystalline zirconium-niobium alloy system, the total expression of the Gibbs energy is: The model is used to simulate the parameter changes in the initial sample preparation process for the study of the irradiation effect of the zirconium-niobium alloy cladding material; it is helpful to overcome the problem that the phase field model in the prior art mainly sets the initial simulation parameters through experience, thereby failing to guarantee the accuracy of the subsequent simulation results.

Description

Technical Field

[0001] The invention relates to the technical field of phase field modeling and analysis, and in particular to a phase field model, a modeling method and electronic equipment of a polycrystalline zirconium-niobium alloy system. Background Art

[0002] Reactor fuel elements are the core components of the reactor, mainly composed of fuel pellets and cladding. During the operation of the reactor, nuclear fuel undergoes nuclear reactions to generate heat, which is then conducted away by cooling water. The temperature inside the fuel pellets is very high, and the pellets cannot come into direct contact with the cooling water, so they need to be separated by cladding. The inner surface of the cladding material is in contact with nuclear fission products and is bombarded by high-flux neutrons, which will produce a high concentration of point defects, forming defect clusters such as dislocation rings and voids, which in turn cause swelling, hardening, amorphization and embrittlement of the material.

[0003] At present, zirconium alloys are widely used as cladding materials for pressurized water reactor fuel elements due to their small neutron absorption cross section, good corrosion resistance and good radiation resistance. There are many types of zirconium alloys. In order to explore zirconium alloys with better performance, researchers are still conducting in-pile tests on zirconium cladding.

[0004] At present, the simulation under the working conditions of nuclear power plants is usually carried out in research reactors with high flux at home and abroad. The high neutron flux is used to test whether it is safe and reliable in operation in a short time. However, the high-flux irradiation resources are limited, and the test cost is huge. The post-irradiation inspection is usually accompanied by radioactivity, which has certain harm to the body of the test personnel. With the rapid development of computer technology, the numerical simulation technology of nuclear materials has emerged. Among them, the multi-scale numerical simulation method can simulate the irradiation effect of materials at nano, micro, meso and macro scales. The phase field method is a commonly used mesoscale simulation method. It takes the Ginzburg-Landau theory as the physical basis and reflects the comprehensive effect of diffusion, ordering potential and thermodynamic driving with specific physical mechanisms through differential equations. Compared with other multi-scale simulation methods, this method has many advantages. It takes into account long-range interactions such as elastic fields, and all spatial information of microstructures is included in the field. Compared with atomic simulation, it can consider uniform local long-range fields, such as strain fields generated during the evolution of microstructures.

[0005] Existing sample preparation techniques are time-consuming and costly, and the uncertainty of the samples obtained is very large, which ultimately affects the experimental results and leads to low experimental efficiency. Especially for the preparation of radioactive samples, not only complex shielding measures are required, but also the experimenters need to face the danger of radioactivity. Different from the existing experimental sample preparation technology, simulated sample preparation can greatly reduce the sample preparation time and cost, and the quality of the obtained samples is controllable.

[0006] The phase field model in the prior art mainly sets the initial simulation parameters through experience, which cannot guarantee the accuracy of the subsequent simulation results. The sample preparation method in the present invention can simulate the defects and second phase density parameters generated by the original sample during the processing process, and has a higher degree of restoration of the actual sample, thereby ensuring higher experimental accuracy. Summary of the invention

[0007] The technical problem to be solved by the present invention is that the phase field model in the prior art mainly sets initial simulation parameters through experience, which cannot guarantee the accuracy of subsequent simulation results. The present invention provides a phase field model, modeling method and electronic equipment for a polycrystalline zirconium-niobium alloy system to solve the above problem.

[0008] The present invention is achieved through the following technical solutions:

[0009] A phase field model of a polycrystalline zirconium-niobium alloy system is proposed. Based on the two elements and equilibrium vacancies in the polycrystalline zirconium-niobium alloy system, the total expression of Gibbs energy is:

[0010]

[0011] To simulate the parameter changes during the initial sample preparation process for the study of irradiation effects of zirconium-niobium alloy cladding materials;

[0012] Among them, x Zr is the concentration of zirconium, x Nb is the concentration of niobium, c v To balance the vacancy concentration, is the reference energy of pure zirconium, is the reference energy of pure niobium, is the vacancy formation energy, λ is the energy efficiency due to lattice deformation;

[0013] R is the ideal gas constant, T is the temperature; L Zr,Nb is the interaction coefficient, obtained by least squares fitting of the experimental data; is the interaction energy between the vacancy and the zirconium atom, is the interaction energy between the vacancy and the niobium atom.

[0014] Alternatively, the Gibbs free energy density is:

[0015] G(x Zr , x Nb , c v , φ) = G a (x Zr , x Nb , c v ,φ)+q(φ)(2); used to distinguish the second phase;

[0016] The second phase refers to that part of the niobium element exists in the zirconium in the form of the second phase; in order to distinguish the second phase, the order parameter Φ is introduced, and the values ​​are 0 and 1 respectively; G a (x Zr , x Nb , c v , φ) is the approximate Gibbs free energy, q(Φ) is the well potential function, and its relationship with the order parameter is q(φ) = w 0 φ 2 (1-2φ+φ 2 ), w 0 is the potential well depth.

[0017] Alternatively, the Gibbs free energy is used to study the second phase structural transition, as shown in formula (3):

[0018]

[0019] Where c is the concentration of niobium; V m is the molar volume, ε c is the gradient energy of niobium, ε v is the gradient energy of the vacancy, ε Φ is the gradient energy of the required parameter.

[0020] Alternatively, the kinetic process of niobium and vacancies is as follows:

[0021]

[0022]

[0023]

[0024] Among them, J c is the diffusion flux of niobium, J v is the diffusion flux of vacancies, L c is the mobility of niobium, L v is the mobility of the vacancy, M Φ is the kinetic coefficient of the order parameter, ξ c (r, t) is the Gaussian noise of the niobium element, ξ v (r,t) is the Gaussian noise of the vacancy, ξ Φ (r,t) is the Gaussian noise with order parameter.

[0025] A phase field modeling method for a polycrystalline zirconium-niobium alloy system is proposed. Based on two elements and equilibrium vacancies in the polycrystalline zirconium-niobium alloy system, the total expression of the Gibbs energy is constructed using a ternary system:

[0026]

[0027] Among them, xZr is the concentration of zirconium, x Nb is the concentration of niobium, c v To balance the vacancy concentration, is the reference energy of pure zirconium, is the reference energy of pure niobium, is the vacancy formation energy, λ is the energy efficiency due to lattice deformation;

[0028] R is the ideal gas constant, T is the temperature; L Zr,Nb is the interaction coefficient, obtained by least squares fitting of the experimental data; is the interaction energy between the vacancy and the zirconium atom, is the interaction energy between the vacancy and the niobium atom.

[0029] Further optionally, part of the niobium element exists in the form of a second phase in zirconium. In order to distinguish the second phase, an order parameter Φ is introduced, with values ​​of 0 and 1 respectively; the Gibbs free energy density is introduced as:

[0030] G(x Zr , x Nb , c v , φ) = G a (x Zr , x Nb , cv ,φ)+q(φ) (2);

[0031] Among them, G a (x Zr , x Nb , c v , φ) is the approximate Gibbs free energy, q(Φ) is the well potential function, and its relationship with the order parameter is q(φ) = w 0 φ 2 (1-2φ+φ 2 ), w 0 is the potential well depth.

[0032] Alternatively, in order to study the second phase structural transition, only the concentrations of niobium and vacancies are considered. Assuming that the concentration of niobium is c, the Gibbs free energy is:

[0033]

[0034] Where c is the concentration of niobium; V m is the molar volume, ε c is the gradient energy of niobium, ε v is the gradient energy of the vacancy, ε Φ is the gradient energy of the required parameter.

[0035] Further optionally, the kinetic process of niobium and vacancies is designed as shown below:

[0036]

[0037] Among them, J c is the diffusion flux of niobium, J v is the diffusion flux of vacancies, L c is the mobility of niobium, L v is the mobility of the vacancy, M Φ is the kinetic coefficient of the order parameter, ξ c (r,t) is the Gaussian noise of niobium element, ξ v (r,t) is the Gaussian noise of the vacancy, ξ Φ (r,t) is the Gaussian noise with order parameter.

[0038] An application of a phase field model for a polycrystalline zirconium-niobium alloy system comprises the following steps:

[0039] S1: Input parameters include: alloy element type, irradiation temperature, annealing time step, sample size or more;

[0040] S2: Based on solving the phase field model;

[0041] S3: Output results: the evolution of the second phase size with annealing time and / or the evolution of the second phase microstructure in the Zr-Nb alloy system.

[0042] An electronic device includes a display, a memory, a processor, and one or more modules; the one or more modules are stored in the memory and configured to be executed by the processor; the one or more modules include:

[0043] The parameter input module is used to input parameters including one or more of alloy element type, irradiation temperature, annealing time step, and sample size;

[0044] The solution module is used to solve the model constructed based on the phase field modeling method of a polycrystalline zirconium-niobium alloy system mentioned above;

[0045] The output parameter module is used to output the evolution process of the second phase size in the zirconium-niobium alloy system with annealing time and / or the evolution process of the second phase microstructure.

[0046] The present invention has the following advantages and beneficial effects:

[0047] The phase field model and modeling method in the present invention can simulate the defects and second phase density parameters generated in the zirconium-niobium alloy system during the heat treatment process, simulate the evolution of the microstructure in the zirconium-niobium alloy with time during sample preparation, and obtain a zirconium-niobium alloy organizational state that is closer to the actual situation, thereby providing more accurate initial samples and input parameters for the numerical simulation calculation of irradiation defect evolution or mechanical properties in the alloy.

[0048] 1. The sample preparation method of the present invention can simulate the defects and second phase density parameters generated by the original sample during the processing, and has a higher degree of restoration of the actual sample, thereby ensuring higher experimental accuracy.

[0049] 2. The main content of the present invention is a phase field modeling method for a polycrystalline zirconium-niobium alloy system. The phase field simulation method is used to establish a model, which can simulate the evolution of the microstructure in the zirconium-niobium alloy over time during sample preparation, and obtain the spatial distribution of the two elements and the initial vacancy concentration field in alloys with different compositions, thereby providing more accurate initial samples and input parameters for numerical simulation of irradiation defect evolution or mechanical properties in the alloy.

[0050] 3. The main content of the present invention is a phase field modeling method for a polycrystalline zirconium-niobium alloy system, which performs simulated annealing on a simulated sample by a numerical simulation method, thereby obtaining an initial state of the simulated sample that is closer to the actual situation. Using this method and the model system for simulation calculations, the element distribution and initial distribution of point defects of zirconium-niobium alloys with different alloy components under different temperature fields can be obtained. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] The drawings described herein are used to provide a further understanding of the embodiments of the present invention, constitute a part of this application, and do not constitute a limitation of the embodiments of the present invention. In the drawings:

[0052] Figure 1 This is the evolution process of the second phase size with annealing time in Example 3.

[0053] Figure 2 This is the evolution process of the second phase microstructure of the zirconium-niobium alloy system in Example 3. DETAILED DESCRIPTION

[0054] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with embodiments and drawings. The exemplary embodiments of the present invention and their description are only used to explain the present invention and are not intended to limit the present invention.

[0055] Throughout the specification, references to "one embodiment," "an embodiment," "an example," or "an example" mean that a particular feature, structure, or characteristic described in conjunction with the embodiment or example is included in at least one embodiment of the present invention. Therefore, the phrases "one embodiment," "an embodiment," "an example," or "an example" appearing in various places throughout the specification do not necessarily all refer to the same embodiment or example. In addition, particular features, structures, or characteristics may be combined in one or more embodiments or examples in any suitable combination and / or sub-combination. In addition, it will be appreciated by those of ordinary skill in the art that the figures provided herein are for illustrative purposes and that the figures are not necessarily drawn to scale. The term "and / or" as used herein includes any and all combinations of one or more of the associated listed items.

[0056] Example 1

[0057] This embodiment provides a phase field modeling method for a polycrystalline zirconium-niobium alloy system, which is specifically as follows:

[0058] Considering that there are two elements and equilibrium vacancies in the Zr-Nb alloy, a ternary system is used here. Interstitial atoms are not considered here because their concentration is very small.

[0059] Assume that the Gibbs free energy of the system is G = G ref +G id +G ex Among them, G ref is the reference Gibbs energy, G id represents the ideal Gibbs energy contribution due to random mixing of atoms, G ex is an additional term that defines the deviation from the ideal value.

[0060] The reference Gibbs energy is is the reference energy of pure zirconium, is the reference energy of pure niobium, is the vacancy formation energy, x Zr is the concentration of zirconium, x Nb is the concentration of niobium, c v is the equilibrium vacancy concentration.

[0061] The ideal Gibbs energy is G id =RT[x Zr x r +x Nb ln Nb +c v lnc v ], where R is the ideal gas constant and T is the temperature.

[0062] The additional term is G ex =xZr x Nb L Zr , N b , L Zr，Nb is the interaction coefficient, which is obtained by least squares fitting of the experimental data.

[0063] Additional Gibbs Energy in is the interaction energy between the vacancy and the zirconium atom, is the interaction energy between the vacancy and the niobium atom.

[0064]

[0065]

[0066] in, is the cohesive energy in zirconium, is the cohesive energy in niobium, is the vacancy formation energy in zirconium, is the vacancy formation energy in niobium, and Z represents the coordination number.

[0067] In summary, the overall expression of Gibbs energy is:

[0068]

[0069] Part of the niobium element exists in the form of a second phase in zirconium, that is, zirconium is α phase and niobium is β phase. In order to distinguish the second phase, the order parameter φ is introduced, with values ​​of 0 and 1 respectively. The Gibbs free energy density is introduced below:

[0070] G(x Zr , x Nb , c v , φ) = G a (x Zr , x Nb , c v ,φ)+q(φ);

[0071] Among them G a (x Zr , x Nb , c v , φ) is the approximate Gibbs free energy, q(Φ) is the well potential function, and its relationship with the order parameter is q(φ) = w 0 φ 2 (1-2φ+φ 2 ), w 0 is the potential well depth.

[0072] In order to study the second phase structural transition, we only consider the concentration of niobium and vacancies. Assuming that the concentration of niobium is c, the Gibbs free energy is:

[0073]

[0074] Among them, V m is the molar volume, ε c is the gradient energy of niobium, ε v is the gradient energy of the vacancy, ε φ is the gradient energy of the required parameter.

[0075] The kinetics of niobium and vacancies are shown below:

[0076]

[0077]

[0078]

[0079] Among them, J c is the diffusion flux of niobium, J v is the diffusion flux of vacancies, L c is the mobility of niobium, L v is the mobility of the vacancy, MΦ is the kinetic coefficient of the order parameter, ξ c (r,t) is the Gaussian noise of niobium element, ξ v (r,t) is the Gaussian noise of the vacancy, ξ Φ (r,t) is the Gaussian noise with order parameter.

[0080] Example 2

[0081] This embodiment provides an application of a phase field model of a polycrystalline zirconium-niobium alloy system, as shown below:

[0082] S1: Input parameters: alloying elements, irradiation temperature, annealing time step and sample size;

[0083] S2: Based on solving the phase field model provided in Example 1;

[0084] S3: Output parameters: the evolution of the second phase size with annealing time and the evolution of the second phase microstructure in the Zr-Nb alloy system.

[0085] Example 3

[0086] Based on the method provided in Example 2, an application case is given.

[0087] Assuming the content of alloying element niobium is 10% and the temperature is 550K, the evolution of the second phase size with annealing time is obtained, as shown in Figure 1 As shown; the evolution process of the second phase microstructure of the zirconium-niobium alloy system is obtained, as shown in Figure 2 shown.

[0088] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A phase field model of a polycrystalline zirconium-niobium alloy system, characterized in that: Based on the two elements and equilibrium vacancies in the polycrystalline zirconium-niobium alloy system, the overall expression of the Gibbs energy is: To simulate the parameter changes during the initial sample preparation process for the study of irradiation effects of zirconium-niobium alloy cladding materials; Among them, x Zr is the concentration of zirconium, x Nb is the concentration of niobium, c v To balance the vacancy concentration, is the reference energy of pure zirconium, is the reference energy of pure niobium, is the vacancy formation energy, λ is the energy efficiency due to lattice deformation; R is the ideal gas constant, T is the temperature; L Zr,Nb is the interaction coefficient, obtained by least squares fitting of the experimental data; is the interaction energy between the vacancy and the zirconium atom, is the interaction energy between the vacancy and the niobium atom; The Gibbs free energy density is: G(x Zr ,x Nb ,c υ ,φ)=G a (x Zr ,x Nb ,c υ ,φ)+q(φ) (2); Used to distinguish the second phase; The second phase refers to that part of the niobium element exists in the zirconium in the form of the second phase; in order to distinguish the second phase, the order parameter Φ is introduced, and the values ​​are 0 and 1 respectively; G a (x Zr , x Nb , c υ , φ) is the approximate Gibbs free energy, q(Φ) is the well potential function, and its relationship with the order parameter is q(φ) = w0φ 2 (1-2φ+φ 2 ), w0 is the potential well depth.

2. The phase field model of a polycrystalline zirconium-niobium alloy system according to claim 1, characterized in that: Gibbs free energy is used to study the second phase structural transition, as shown in formula (3): Where c is the concentration of niobium; V m is the molar volume, ε c is the gradient energy of niobium, ε v is the gradient energy of the vacancy, ε Φ is the gradient energy of the order parameter; G(c, c v , φ; T) is the Gibbs free energy of the component, represents the spatial derivative of niobium concentration, represents the spatial derivative of the vacancy concentration, represents the spatial derivative of the order parameter.

3. The phase field model of a polycrystalline zirconium-niobium alloy system according to claim 1, characterized in that: The kinetics of niobium and vacancies are shown below: Among them, J c is the diffusion flux of niobium, J v is the diffusion flux of vacancies, L c is the mobility of niobium, L v is the mobility of the vacancy, M Φ is the kinetic coefficient of the order parameter, ξ c (r,t) is the Gaussian noise of niobium element, ξ v (r,t) is the Gaussian noise of the vacancy, ξ Φ (r,t) is the Gaussian noise with order parameter; The calculation method is to first take the derivative of the concentration and then take the derivative with respect to the space.

4. A phase field modeling method for a polycrystalline zirconium-niobium alloy system, characterized in that: Based on the two elements and the equilibrium vacancies in the polycrystalline zirconium-niobium alloy system, the total expression of the Gibbs energy is constructed using the ternary system: Among them, x Zr is the concentration of zirconium, x Nb is the concentration of niobium, c v To balance the vacancy concentration, is the reference energy of pure zirconium, is the reference energy of pure niobium, is the vacancy formation energy, λ is the energy efficiency due to lattice deformation; R is the ideal gas constant, T is the temperature; L Zr,Nb is the interaction coefficient, obtained by least squares fitting of the experimental data; is the interaction energy between the vacancy and the zirconium atom, is the interaction energy between the vacancy and the niobium atom; Part of the niobium element exists in the form of a second phase in zirconium. In order to distinguish the second phase, the order parameter Φ is introduced, with values ​​of 0 and 1 respectively; the Gibbs free energy density is introduced as: G(x Zr ,x Nb ,c υ , φ)=Ga(x Zr ,x Nb ,c υ , φ)+q(φ) (2); Among them, G a (x Zr , x Nb , c υ , φ) is the approximate Gibbs free energy, q(Φ) is the well potential function, and its relationship with the order parameter is q(φ) = w0φ 2 (1-2φ+φ 2 ), w0 is the potential well depth.

5. According to the phase field modeling method of a polycrystalline zirconium-niobium alloy system as claimed in claim 4, in order to study the second phase structural transformation, only the concentrations of niobium and vacancies are considered, and it is assumed that the concentration of niobium element is c, then the Gibbs free energy is: in, c is the concentration of niobium; V m is the molar volume, ε c is the gradient energy of niobium, ε v is the gradient energy of the vacancy, ε Φ is the gradient energy of the required parameter.

6. The phase field modeling method of a polycrystalline zirconium-niobium alloy system according to claim 4, characterized in that: The kinetics of designing niobium and vacancies are shown below: Among them, J c is the diffusion flux of niobium, J v is the diffusion flux of vacancies, L c is the mobility of niobium, L v is the mobility of the vacancy, M Φ is the kinetic coefficient of the order parameter, ξ c (r,t) is the Gaussian noise of niobium element, ξ v (r,t) is the Gaussian noise of the vacancy, ξ Φ (r,t) is the order parameter of Gaussian noise.

7. An application of a phase field model for a polycrystalline zirconium-niobium alloy system, characterized in that: S1: Input parameters include: alloy element type, irradiation temperature, annealing time step, sample size or more; S2: A phase field model constructed based on solving the phase field model of any one of claims 1 to 3 or the modeling method of any one of claims 4 to 6; S3: Output results: the evolution of the second phase size with annealing time and / or the evolution of the second phase microstructure in the Zr-Nb alloy system.

8. An electronic device, characterized in that: The device comprises a display, a memory, a processor and one or more modules; the one or more modules are stored in the memory and configured to be executed by the processor; the one or more modules comprise a parameter input module, a solution module, and an output parameter module; The parameter input module is used to input parameters including one or more of alloy element type, irradiation temperature, annealing time step, and sample size; The solution module is used to solve based on the phase field model of any one of claims 1 to 3 or the modeling method of any one of claims 4 to 6; The output parameter module is used to output the evolution process of the second phase size in the zirconium-niobium alloy system with annealing time and / or the evolution process of the second phase microstructure.

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