CN114678089B - Method for determining appearance of irradiation bubble in nuclear material and influence of irradiation bubble on force and thermal performance - Google Patents
Method for determining appearance of irradiation bubble in nuclear material and influence of irradiation bubble on force and thermal performance Download PDFInfo
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Abstract
The invention discloses a method for determining the appearance of an irradiation bubble in a nuclear material and the influence of the irradiation bubble on force and thermal performance, collecting physical parameters of the nuclear material required in the irradiation process, and obtaining elastic energy density by solving an elastic equilibrium equation; constructing a free energy equation by combining the elastic energy density; establishing and solving a rate theoretical model of irradiation bubble nucleation to obtain initial bubble distribution, and taking the initial bubble distribution as initial conditions of phase field simulation; constructing and solving a phase field evolution equation on the basis of introducing a polycrystalline structure and considering temperature gradient, interfacial energy anisotropy and diffusion coefficient anisotropy; carrying out visual treatment on the numerical solution of the phase field evolution equation to obtain the morphology of bubble nucleation growth and grain growth in the nuclear material irradiation process; counting all bubbles to obtain the size distribution and porosity parameters of the irradiated bubbles in the nuclear material; and obtaining the irradiation swelling and heat conduction rule of the nuclear material by counting the size distribution and the porosity of the irradiation bubbles, thereby realizing the prediction of the service performance of the nuclear material.
Description
Technical Field
The invention relates to the field of nuclear materials, in particular to a method for determining the appearance of irradiation bubbles in a nuclear material and the influence of the irradiation bubbles on force and heat performance.
Background
The nuclear material is relevant to nuclear reactor safety. During operation of a nuclear reactor, the nuclear material is in a harsh operating environment and a significant amount of fission products, including solids and fission gases, are released as a result of the burnup developing fission reactions, wherein the fission gases account for about 25% of the total number of atoms of all fission products. Due to the very large mobility energy and extremely low solubility of the gas atoms, fissile gas generated during the nuclear reaction is often trapped in voids or grain boundaries, forming intra-or inter-crystalline irradiation bubbles. A large number of bubbles are continuously accumulated on the grain boundary, and finally a special through hole structure is formed on the grain boundary, meanwhile, the volume of a fission product is larger than that of a substance before fission, so that the volume of a material element can be increased along with the development of burnup, and the change is called irradiation swelling. The irradiation swelling of the nuclear fuel can cause the interaction of the fuel and the cladding and the pressing of the cladding, causing the radial deformation and the transverse stretching of the cladding tube, and finally causing the cladding to be broken, thereby seriously threatening the operation safety of the nuclear reactor. The effect study of the solid fission products on irradiation swelling is more thorough; however, the behavior of the fission gas is relatively complex, and further research is required in this respect. Therefore, the research on the growth of the irradiated bubble nuclei of the nuclear material and the influence of the nuclear material on irradiation swelling is particularly important. In addition, since the nuclear material is always in high temperature and irradiation extreme service conditions, the generated thermal strain and creep strain can greatly increase the stress in the material, so that the research on the evolution of irradiation bubbles of the material under the action of the stress is particularly important.
Disclosure of Invention
The invention aims to provide a phase field simulation method for researching formation and evolution of intra-crystalline bubbles and material swelling of a nuclear material after the nuclear material is irradiated, namely a method for determining the shape of the irradiated bubbles in the nuclear material and the influence of the irradiated bubbles on force and heat performance.
In order to achieve the above purpose, the invention adopts the following technical scheme:
a method of determining the morphology of irradiated bubbles in a nuclear material and their impact on thermal performance, comprising the steps of:
s1, selecting a phase field variable describing the evolution process of the appearance of the nuclear material irradiation bubble; collecting physical parameters of the nuclear material, including formation energy, diffusion coefficient and gradient coefficient of point defects, and mobility of bubbles and grain boundaries; and calculating mobility of the point defect considering diffusion anisotropy;
s2, calculating elastic modulus distribution related to the position; obtaining initial intrinsic strain distribution by combining Van der Waals equation; establishing and solving an elastic equilibrium equation to obtain irradiation bubble stress strain distribution and elastic energy density;
S3, obtaining a free energy density function of a matrix phase of the nuclear material based on a thermodynamic theory of the material according to the physical parameters of the nuclear material collected in the step S1, obtaining a free energy density function of a bubble phase in the nuclear material through deducing a Van der Waals equation under high temperature and high pressure conditions, and obtaining a total free energy equation by combining interface gradient energy, bubble and grain boundary interaction energy, polycrystal interaction energy and elastic energy density obtained in the step S2;
s4, establishing a rate theoretical model of irradiation bubble nucleation; obtaining bubble size distribution at the initial stage of irradiation bubble nucleation through solving a rate theoretical model, and taking the size distribution as initial conditions of phase field simulation;
s5, constructing a phase field evolution equation according to the total free energy equation obtained in the step S3 and comprehensively considering the factors of temperature gradient, generation of point defects, recombination of the point defects and disappearance of the point defects at defect traps under the irradiation condition; carrying out numerical solution on a phase field evolution equation by adopting a semi-implicit Fourier spectrum method;
s6, carrying out visual treatment on the numerical solution obtained in the step S5 to obtain the morphology of bubble nucleation growth and grain growth in the nuclear material irradiation process; counting all bubbles by adopting a communication area marking method to obtain the size distribution and porosity parameters of the irradiated bubbles in the nuclear material; classifying and counting inter-crystal bubbles and intra-crystal bubbles;
S7, calculating to obtain the effective heat conductivity and the irradiation swelling rule of the nuclear material based on the bubble distribution and the porosity obtained in the step S6.
In one implementation, S1 specifically includes:
s1.1, selecting a phase field variable describing the morphology evolution process of the nuclear material irradiation bubble, wherein the phase field variable representing the concentration of point defects is required to be selected: concentration of gas atoms c g Concentration of vacancies c v And interstitial concentration c i The method comprises the steps of carrying out a first treatment on the surface of the And simultaneously selecting a bubble phase sequence parameter eta for distinguishing a bubble phase and a matrix phase: η=1 when in the bubble phase.0, η=0.0 when in the matrix phase; the multi-order parameter phi needs to be selected i Represents a grain structure, wherein the subscript i represents the ith grain and there is phi within the ith grain i ＝1.0；
S1.2, collecting physical parameters of a nuclear material, including formation energy, diffusion coefficient, mobility and gradient coefficient of gas atoms, vacancies and interstitial atoms and mobility of bubbles and grain boundaries;
s1.3 calculating the mobility of the point defect considering diffusion anisotropy, vacancy mobility M, based on the diffusion coefficients of the gas atoms, vacancies, and interstitial atoms of S1.2 v Has the following expression:
wherein D is v Vacancy diffusion coefficient in tensor form, V m The unit volume of the crystal lattice is R is an ideal gas constant, and T is absolute temperature;
Wherein D is v 0 For the effective diffusion coefficient of the vacancies,and->The components of vacancy bulk diffusion, surface diffusion and grain boundary diffusion, respectively, have the following expressions:
wherein w is b 、w s And w GB Weight coefficients of bulk diffusion, surface diffusion and grain boundary diffusion respectively, I is an identity matrix, T s v And T GB v The surface and grain boundary tensor coefficients, respectively, have the following expression:
in one implementation, S2 specifically includes:
s2.1, calculating the elastic modulus distribution related to the position: in the polycrystalline material, the elastic modulus is related to the orientation of the grains, the elastic modulus of each grain with defined orientation is obtained by coordinate transformation, and the elastic modulus tensor C ijkl The expression (r) is as follows:
wherein,for the modulus of elasticity in the matrix +.>And->Rotation matrix a of p-th crystal grain relative to reference coordinate ij Ia, jb, kc, ld th component of (a), rotating matrix a ij The expression is determined by Euler angles of corresponding grains, and the rotation matrix under the three-dimensional coordinates is as follows:
wherein θ, ζ, and ψ are three euler angles determining the grain orientation;
s2.2 initial intrinsic StrainIs composed of two parts, the first part is intrinsic strain of matrix phase +.>The second part is the intrinsic strain of the bubble phase +.>r is a position vector;
wherein the matrix phase intrinsic strain caused by defect non-uniformity The method comprises the following steps:
wherein ε is g0 Delta is the lattice constant expansion coefficient due to the introduction of gas atoms ij As a function of Kronecker,h (η) =η, the equilibrium concentration of the gas atoms 3 (6η 2 -15 η+10) is a structured interpolation function, satisfying when η=1.0, h (η) =1.0 in the bubble phase; when η=0.0, h (η) =0.0 in the matrix phase; the intrinsic strain of the bubble is very complex because it depends on the internal pressure of the bubbleAnd the size of the bubbles when the internal pressure of the bubbles is greater than 2 gamma s At/r, an outward pressure acts on the lattice around the bubbles, so the characteristic strain of the bubbles is positive, which means that the bubbles are in an overpressure state; when the bubble pressure is equal to 2 gamma s At/r, the intrinsic strain becomes zero; negative intrinsic strain values indicate that the bubble is in an under-pressure state, which results in elastic relaxation of the surrounding lattice of the bubble toward the center of the bubble; gamma ray s The surface energy of the bubble and r is the radius of the bubble;
intrinsic strain of bubble phaseThe description is as follows:
wherein P is g Is the internal pressure of the bubble, 2γ s R is considered as external hydrostatic pressure, C 11 For the principal direction elastic modulus, the gas pressure inside the bubble is always positive, while the intrinsic strain inside the bubble may be negative; this definition shows that intrinsic strain is a function of gas pressure, bubble radius, surface tension and material elastic properties; the gas pressure in the bubble can be obtained using the van der Waals equation:
Wherein Ω is the molecular volume of the corresponding core material, b is the van der Waals constant, k B Is the boltzmann constant;
s2.3, obtaining stress strain distribution by solving an elastic equilibrium equation, wherein the elastic equilibrium equation is as follows:
middle sigma ij Is stress, r j Is the unit displacement along the j direction; epsilon kl (r) is the total strainEquivalent to epsilon ij (r); total strain epsilon ij (r) is the sum of uniform strain and non-uniform strain:
in the middle ofAnd delta epsilon ij (r) uniform and non-uniform strain, respectively; uniform strain->The expression is as follows:
in the method, in the process of the invention,for additional stress, V is the simulated area volume; non-uniform strain delta epsilon ij The expression (r) is as follows:
u in the formula i (r) and u j (r) substituting equation (17) into the elastic equilibrium equation (14) for displacement components in the i and j directions:
and (3) obtaining a displacement field equation after simplifying and finishing:
in the middle ofFor a position-independent elastic modulus, C' ijkl (r) is the position dependent elastic modulus; the Fourier transform is carried out to obtain:
in the middle ofFor a displacement field in fourier space, k is the wave vector in fourier space, k j Is a component of a wave vector; />Is the green tensor; the subscript k indicates that the partial calculation is performed in fourier space; solving formula (20) using perturbation iteration method, the displacement field in the nth iteration is as follows:
The initial value of the iteration is taken asThe displacement field u in real space k (r) byObtaining an inverse Fourier transform, obtaining a displacement field through an iteration solution elastic balance equation, and further obtaining stress-strain distribution through calculation;
the elastic energy density was calculated from the obtained stress strain, and the elastic energy density was expressed as follows:
in one implementation, S3 specifically includes:
s3.1, based on thermodynamic theory deduction, calculating a free energy density function of a nuclear material matrix phase according to the following formula:
wherein in deriving the free energy of the matrix, condition c is always satisfied since the defects are assumed to occupy only perfect lattices g +c v +c i +c m ＝1.0；c m Is the concentration of a perfect lattice in at.%; e (E) g f 、E v f 、E i f The energy of formation of gas atoms, vacancies, and interstitials in the matrix phase;
s3.2 the free energy density function of the bubble phase of the core material is as follows:
wherein A and B are constants,as a function of free energy density within the bubble in relation to the concentration of gas atoms; the gibbs free energy expression G (p) of the bubble is:
wherein p is pressure; v (V) b Is the volume of the bubbles; g (p) 0 ) Is at reference state pressure p 0 The lower gibbs free energy; the van der Waals equation under high pressure is:
p(V b -nb)＝nk B T (26)
wherein n is the number of gas atoms; the gibbs free energy of the non-ideal gas thus obtained is:
The gas atoms in the bubble phase may occupy lattice or non-lattice positions, so the concentration of gas atoms in the bubble is defined as:
wherein V is site Is the volume of a single lattice, from which the pressure can be obtained:
the free energy density function of the bubbles obtained by bringing the above results into the bubbles is:
wherein mu 0 Is the reference state p 0 The chemical potential below;
s3.3, introducing an interpolation function and a double-trap potential function according to the obtained free energy density functions of the matrix phase and the bubble phase to obtain a bulk free energy density function:
f bulk (c g ,c v ,c i ,η,T)＝h(η)f b (c g ,c v ,c i ,T)+[1-h(η)]f m (c g ,c v ,c i ,T)+wg(η) (31)
wherein h (η) =η 3 (6η 2 -15 η+10) is a structured interpolation function, satisfying when η=1.0, h (η) =1.0 in the bubble phase; when η=0.0, h (η) =0.0 in the matrix phase; g (eta) =eta 2 (1-η) 2 As a dual-potential well function, w is the potential well height;
the free energy density function of polycrystalline interactions is expressed as follows:
wherein α, β, γ and δ are image-only parameters.
The combined body free energy density function, elastic energy density, interface gradient energy, bubble and grain boundary interaction energy and polycrystal interaction energy are used for obtaining a total free energy equation:
wherein, kappa g 、κ v 、κ i 、κ η The gradient term coefficients of the gas atom concentration, the vacancy concentration, the interstitial atom concentration and the bubble phase sequence parameter are respectively shown, and the last four terms of the formula represent gradient energy.
In one implementation, S4 specifically includes:
s4.1, combining the physical parameters of the nuclear material obtained in the step S1, and establishing a rate theoretical model of the irradiation bubble nucleation growth behavior: the nuclear material is irradiated with a large amount of gas atoms, and in order to reduce the energy of the nuclear material and minimize strain, the gas atoms tend to be accumulated in a region of low density, and this bubble formation process is called a nucleation process; the change in free energy during nucleation is caused by the core being spherical in radius, the free energy Δf being expressed by the following formula:
wherein Δf is the difference between the free energy of the bubble and the matrix, r is the bubble radius, and σ is the surface energy; whereas a critical core corresponds to the maximum of the free energy, since any larger core will have less energy and will grow spontaneously; thereby obtaining the critical radius R of the nucleus C And activation energy Δf C ：
Randomly introducing bubble cores in a nucleation stage, and ensuring that the average nucleation rate is matched with the expected nucleation rate; assuming that all points in the matrix can become nucleation sites, the probability of forming a core at one atomic position within one nucleation time interval can be calculated by considering the nucleation rate:
wherein Z is Zeldovich correction factor, N is atomic number, beta * The probability of the critical core changing into the supercritical core is represented by tau, and the inoculation time of the core is represented by tau; t is t 0 To calculate a time interval for nucleation; the overall nucleation probability P is thus obtained:
P＝1-exp(-J * t 0 ) (38)
s4.2, solving a formula (34) -a formula (38) to obtain nucleation probabilities corresponding to bubbles with different sizes, namely bubble size distribution in the initial stage of bubble nucleation irradiation; in the rate theory nucleation algorithm, it is necessary to calculate the bubble radius and nucleation probability of each nucleation point, and input the bubble size distribution at the initial stage of irradiation bubble nucleation as an initial condition into the phase field simulation.
In one implementation, S5 specifically includes:
s5.1, considering the evolution equation of the temperature gradient, the generation of point defects, the recombination of the point defects and the disappearance process of the point defects at defect traps under the irradiation condition, wherein the evolution equation is as follows:
evolution equation of gas atomic concentration:
evolution equation of vacancy atom concentration:
evolution equation of interstitial concentration:
evolution equation of bubble phase field variables:
evolution equation of the poly-crystalline phase field variable:
wherein t is the simulation time; f is total free energy; l is the free interface mobility; m is M g 、M v 、M i Atomic mobilities of gas atoms, vacancy atoms, and interstitial atoms, respectively;thermal fluctuation terms of gas atoms, vacancy atoms, interstitial atoms and bubble phases respectively; / >The generation rate of gas atoms and point defects under irradiation conditions are respectively; />For the recombination rate of point defect vacancies and interstitial atoms, < >>Is an absorption term for grain boundary point defects. The influence of temperature gradient on bubble migration and grain growth is considered in an evolution equation, wherein Q is heat transfer efficiency, and C is a constant;
and->The production rates of gas atoms, vacancies and interstitial gas atoms under irradiation conditions are respectively; the gas atom generation rate is as follows:
wherein f r Is the rate of cracking; ran is a random number between 0 and 1; Λ is a constant;
the generation of interstitial and vacancy uses the following expression:
wherein R is 1 And R is 2 Two random numbers which are randomly generated on each time step and each space lattice point and are uniformly distributed between 0 and 1; e is a biased constant that can be varied to represent unequal numbers of vacancies and interstitials created by the dislocation damage; parameter P casc Representing probability of cascade collision in unit time and unit volume, V G Representing a maximum increase in vacancies due to a cascade collision event; in eta<The 0.8 condition ensures that cascade collisions occur only in the matrix phase, not in the bubble phase;
when point defect vacancies and interstitials meet, they recombine to form a perfect crystal lattice, expressed in terms of recombination rate:
In the formula, v r The recombination rate is: v (v) r ＝v b +η 2 v s ，v b And v s Recombination rates of point defects at bulk phase and interface, respectively; v b ＝4πr iv (D i +D v )/Ω，r iv Is the composite volume radius; d (D) i And D v Diffusion coefficients of interstitial and vacancy, respectively; omega is the lattice volume;the absorption term for grain boundary-to-point defects is expressed as follows:
wherein the method comprises the steps ofIs a grain boundary absorption factor, representing the strength of grain boundary to point defect absorption; phi = Σphi i 2 As a function of the position of the grains, there is Φ=1.0 in the grains, while Φ < 1.0 at the grain boundaries, +.>Is the equilibrium concentration of gas atoms, vacancies, interstitial atoms;
s5.2, solving a phase field evolution equation by using a semi-implicit Fourier spectrum method, and solving evolution equations of gas atom concentration, vacancy concentration and interstitial atom concentration, wherein the solution is shown as follows:
and solving an evolution equation of the bubble phase field variable and the polycrystal phase field variable, wherein the evolution equation is shown as the following formula:
wherein the superscript n denotes the value of the portion in the nth time step, the free energy f is the sum of the bulk free energy density, the polycrystalline free energy density and the elastic free energy density, k= (k) 1 ,k 2 ) Vector coordinates in fourier space, Δt being the simulated time step;
in one implementation, S6 specifically includes:
s6.1, carrying out visualization processing on the numerical solution result in the step S5 by using visualization software to obtain the morphology of bubble nucleation growth and grain growth in the nuclear material irradiation process:
And (3) characterizing visual variables in the irradiation bubble evolution simulation process:
in the method, in the process of the invention,visual variables for definition; the characteristics of the bonded diffusion interface are as follows: inside the grain->At the grain boundary +.>In the bubble->Therefore, the grain, grain boundary and bubble are distinguished by distinguishing the visual variable values at each point in space;
s6.2, counting the size distribution of bubbles in the output result by using an algorithm of the communication area marking, wherein the method comprises the following steps of:
in the connected domain marking, scanning from left to right and from top to bottom during the first marking, each effective pixel is set with a label value, and the judgment rule is as follows:
(1) When the left adjacent pixel and the upper adjacent pixel of the pixel are invalid values, a new label value is set for the pixel, and label is +1;
(2) When one of the left adjacent pixel or the upper adjacent pixel of the pixel is an effective value, assigning the label of the effective value pixel to the label value of the pixel;
(3) When the left adjacent pixel and the upper adjacent pixel of the pixel are both effective values, selecting a smaller label value from the effective values and assigning the smaller label value to the pixel;
finally, counting the area of the region contained in each label value to obtain the size distribution and porosity parameters of the bubbles;
s6.3, counting inter-crystal bubbles, and realizing the method by using the following algorithm:
(1) Numbering each bubble on the basis of S6.2 statistics, and recording the position of each bubble;
(2) Judging whether a bubble is an inter-crystalline bubble or not, wherein the method comprises the following steps: taking a range larger than the bubble as a calculation area to calculate a multi-order parameter phi i A sum Γ within this range i Finally multiplying it to obtainThe bubbles are within the grains if λ=0.0; if λ+.0.0 then the bubble is at the grain boundary;
(3) Judging whether all bubbles are inter-crystal bubbles in sequence, respectively obtaining inter-crystal bubble distribution and intra-crystal bubble distribution, and obtaining inter-crystal bubble coverage rate by using the area of the inter-crystal bubbles to be compared with the total area of the grain boundary;
in one implementation, S7 specifically includes:
s7.1, calculating the effective heat conductivity of the nuclear material, wherein a heat transfer equation needs to be solved:
k in local For localized thermal conductivity, related to the thermal transfer properties of the spatial location, k when located inside the grain local ＝k bulk ，k bulk Is the thermal conductivity within the grain; k when located on grain boundary local ＝k GB ，k GB Is the thermal conductivity at the grain boundaries; k when located in the bubble local ＝k bubble ，k bubble Is the thermal conductivity in the bubble; solving a heat transfer equation (56) using a finite difference method to obtain an equilibrium temperature field and an effective thermal conductivity;
s7.2, combining the bubble size distribution and the porosity calculated in the step S6 to obtain an irradiation swelling rule of the nuclear material irradiation bubbles, wherein the fuel swelling caused by the irradiation bubbles in the nuclear material is estimated by the following formula:
Where ΔV is the fuel volume change, V 0 For initial fuel volume, V f To include the final fuel volume where the bubble induced swelling, the fuel swelling was calculated in a two-dimensional simulation using the ratio of the bubble area to the total area.
Compared with the prior art, the invention has the following advantages:
according to the invention, the simulation precision of the irradiation bubble evolution process under the irradiation condition is greatly improved; the influence of various objective conditions on the evolution of the irradiation bubble is fully considered, and the application range of the phase field model is expanded;
further, by introducing and solving an elastic equilibrium equation, the bubble internal pressure and elastic energy are fully considered, so that the simulation process is more similar to the actual physical process;
further, the free energy density of the bubble phase is deduced from the Van der Waals equation, so that the defect of using an empirical function in the traditional phase field simulation is avoided, and the simulation precision of the bubble evolution process is improved;
further, a communication area marking method is adopted to accurately count and classify the irradiation bubbles in the evolution process, so that the accuracy of a simulation result is improved;
further, factors such as grain size, external stress, anisotropic diffusion coefficient and the like are fully considered in simulation, so that the irradiation bubble nucleation and growth process and the influence rule of the irradiation bubble nucleation and growth process on force and heat performance are fully considered, and the application range of the phase field model is expanded.
Furthermore, the invention is implemented by self programming of Fortran language, can also be implemented by Matlab and other software, and has good flexibility and expansibility.
Drawings
Fig. 1 is a flow chart of the method of the present invention.
FIGS. 2a, 2b, 2c are stress components sigma of a single irradiated bubble, respectively xx Distribution, stress component sigma yy Distribution, stress component sigma xy Distribution.
FIG. 3 is a graph of average bubble diameter as a function of irradiation intensity and comparison with experimental results.
FIGS. 4a, 4b and 4c show the distribution of irradiation bubbles generated by polycrystalline tungsten having a grain size of 181nm, 45nm and 32nm, respectively.
Fig. 5 is a graph of porosity of polycrystalline tungsten as a function of grain size.
Fig. 6 is a graph of irradiated bubble density of polycrystalline tungsten as a function of grain size.
Detailed Description
The invention will be described in further detail below with reference to the drawings and the specific examples.
The main conception of the invention is as follows:
as shown in fig. 1, obtaining nuclear material physical parameters required in the irradiation bubble evolution process, and solving elastic energy in a coupling way and constructing a free energy equation; establishing a phase field model of irradiation bubble morphology evolution on the basis of introducing a polycrystalline structure and considering temperature gradient, interfacial energy anisotropy and diffusion coefficient anisotropy, and constructing a phase field evolution equation; establishing and solving a rate theoretical model of irradiation bubble nucleation to obtain initial bubble distribution, and taking the initial bubble distribution as initial conditions of phase field simulation; and solving a phase field evolution equation by using a semi-implicit Fourier spectrum method to obtain the rule of influence of grain size, external stress and anisotropic diffusion coefficient factors on the irradiation bubble nucleation growth process and on force and thermal properties. And the irradiation swelling and heat conduction rule of the nuclear material are obtained by counting the parameters such as the size, the density, the porosity and the like of the irradiation bubbles and combining the visual result, so that the service performance of the nuclear material is predicted.
In order to make the object technical scheme and advantages of the present invention more clear, the present invention will be further described in detail below with reference to the accompanying drawings and examples thereof. It should be understood that the specific embodiments described herein are for purposes of illustration only and are not limiting the invention.
(1) Selecting phase field variables and collecting physical parameters of nuclear materials
Selecting a phase field variable describing the evolution process of the nuclear material irradiation bubble morphology, wherein the phase field variable representing the point defect concentration is required to be selected: concentration of gas atoms c g Concentration of vacancies c v And interstitial concentration c i The method comprises the steps of carrying out a first treatment on the surface of the And simultaneously selecting a bubble phase sequence parameter eta for distinguishing a bubble phase and a matrix phase: η=1.0 when in the bubble phase and η=0.0 when in the matrix phase; the multi-order parameter phi needs to be selected i Represents a grain structure, wherein the subscript i represents the ith grain and there is phi within the ith grain i ＝1.0；
Collecting physical parameters of the nuclear material, including formation energy, diffusion coefficient, mobility and gradient coefficient of gas atoms, vacancies and interstitial atoms, and mobility of bubbles and grain boundaries;
calculating the diffusion coefficient of gas atoms, vacancies and interstitial atoms according to S1.2, and taking the diffusion into considerationBulk anisotropic point defect mobility, vacancy mobility M v Has the following expression:
wherein D is v Vacancy diffusion coefficient in tensor form, V m The unit volume of the crystal lattice is R is an ideal gas constant, and T is absolute temperature;
wherein D is v 0 For the effective diffusion coefficient of the vacancies,and->The components of vacancy bulk diffusion, surface diffusion and grain boundary diffusion, respectively, have the following expressions:
wherein w is b 、w s And w GB Weight coefficients of bulk diffusion, surface diffusion and grain boundary diffusion respectively, I is an identity matrix, T s v And T GB v The surface and grain boundary tensor coefficients, respectively, have the following expression:
/>
(2) Solving the elastic equilibrium equation to obtain elastic energy density
Calculating a position dependent elastic modulus distribution: in the polycrystalline material, the elastic modulus is related to the orientation of the grains, the elastic modulus of each grain with defined orientation is obtained by coordinate transformation, and the elastic modulus tensor C ijkl The expression (r) is as follows:
wherein,for the modulus of elasticity in the matrix +.>And->Rotation matrix a of p-th crystal grain relative to reference coordinate ij Ia, jb, kc, ld th component of (a), rotating matrix a ij The expression is determined by Euler angles of corresponding grains, and the rotation matrix under the three-dimensional coordinates is as follows:
wherein θ, ζ, and ψ are three euler angles determining the grain orientation;
initial intrinsic strainIs composed of two parts, the first part is the intrinsic response of the matrix phase Change->The second part is the intrinsic strain of the bubble phase +.>r is a position vector;
wherein the matrix phase intrinsic strain caused by defect non-uniformityThe method comprises the following steps:
wherein ε is g0 Delta is the lattice constant expansion coefficient due to the introduction of gas atoms ij As a function of Kronecker,h (η) =η, the equilibrium concentration of the gas atoms 3 (6η 2 -15 η+10) is a structured interpolation function, satisfying when η=1.0, h (η) =1.0 in the bubble phase; when η=0.0, h (η) =0.0 in the matrix phase; the intrinsic strain of the bubble is very complex because it depends on the internal pressure of the bubble and the size of the bubble when the internal pressure of the bubble is greater than 2γ s At/r, an outward pressure acts on the lattice around the bubbles, so the characteristic strain of the bubbles is positive, which means that the bubbles are in an overpressure state; when the bubble pressure is equal to 2 gamma s At/r, the intrinsic strain becomes zero; negative intrinsic strain values indicate that the bubble is in an under-pressure state, which results in elastic relaxation of the surrounding lattice of the bubble toward the center of the bubble; gamma ray s The surface energy of the bubble and r is the radius of the bubble;
intrinsic strain of bubble phaseThe description is as follows:
wherein P is g Is the internal pressure of the bubble, 2γ s R is considered as external hydrostatic pressure, C 11 For the principal direction elastic modulus, the gas pressure inside the bubble is always positive, while the intrinsic strain inside the bubble may be negative; this definition shows that intrinsic strain is a function of gas pressure, bubble radius, surface tension and material elastic properties; the gas pressure in the bubble can be obtained using the van der Waals equation:
Wherein Ω is the molecular volume of the corresponding core material, b is the van der Waals constant, k B Is the boltzmann constant;
the stress strain distribution is obtained by solving an elastic equilibrium equation, which is as follows:
middle sigma ij Is stress, r j Is the unit displacement along the j direction; epsilon kl (r) is the total strain equivalent to ε ij (r)；ε ij (r) is the sum of uniform strain and non-uniform strain:
in the middle ofAnd delta epsilon ij (r) uniform and non-uniform strain, respectively; uniform stressChange->The expression is as follows:
in the method, in the process of the invention,for additional stress, V is the simulated area volume; non-uniform strain delta epsilon ij The expression (r) is as follows:
u in the formula i (r) and u j (r) substituting equation (17) into the elastic equilibrium equation (14) for displacement components in the i and j directions:
and (3) obtaining a displacement field equation after simplifying and finishing:
in the middle ofFor a position-independent elastic modulus, C' ijkl (r) is the position dependent elastic modulus; the Fourier transform is carried out to obtain:
in the middle ofFor a displacement field in fourier space, k is the wave vector in fourier space, k j Is a component of a wave vector; />Is the green tensor; the subscript k indicates that the partial calculation is performed in fourier space; solving formula (20) using perturbation iteration method, the displacement field in the nth iteration is as follows:
/>
The initial value of the iteration is taken asThe displacement field u in real space k (r) by->Obtaining an inverse Fourier transform, obtaining a displacement field through an iteration solution elastic balance equation, and further obtaining stress-strain distribution through calculation;
the elastic energy density was calculated from the obtained stress strain, and the elastic energy density was expressed as follows:
(3) Construction of free energy equation
Based on thermodynamic theory deduction, the free energy density function of the matrix phase of the nuclear material is calculated according to the following formula:
wherein in deriving the free energy of the matrix, condition c is always satisfied since the defects are assumed to occupy only perfect lattices g +c v +c i +c m ＝1.0；c m Is the concentration of a perfect lattice in at.%; e (E) g f 、E v f 、E i f The energy of formation of gas atoms, vacancies, and interstitials in the matrix phase;
the free energy density function of the bubble phase of the core material is as follows:
wherein A and B are constants,as a function of free energy density within the bubble in relation to the concentration of gas atoms; the gibbs free energy expression G (p) of the bubble is:
wherein p is pressure; v (V) b Is the volume of the bubbles; g (p) 0 ) Is at reference state pressure p 0 The lower gibbs free energy; the van der Waals equation under high pressure is:
p(V b -nb)＝nk B T (26)
wherein n is the number of gas atoms; the gibbs free energy of the non-ideal gas thus obtained is:
The gas atoms in the bubble phase may occupy lattice or non-lattice positions, so the concentration of gas atoms in the bubble is defined as:
wherein V is site Is the volume of a single lattice, thereby obtaining the pressureForce:
the free energy density function of the bubbles obtained by bringing the above results into the bubbles is:
wherein mu 0 Is the reference state p 0 The chemical potential below;
according to the obtained free energy density functions of the matrix phase and the bubble phase, introducing an interpolation function and a double-well potential function to obtain a free energy density function of the body:
f bulk (c g ,c v ,c i ,η,T)＝h(η)f b (c g ,c v ,c i ,T)+[1-h(η)]f m (c g ,c v ,c i ,T)+wg(η) (31)
wherein h (η) =η 3 (6η 2 -15 η+10) is a structured interpolation function, satisfying when η=1.0, h (η) =1.0 in the bubble phase; when η=0.0, h (η) =0.0 in the matrix phase; g (eta) =eta 2 (1-η) 2 As a dual-potential well function, w is the potential well height;
the free energy density function of polycrystalline interactions is expressed as follows:
wherein α, β, γ and δ are image-only parameters.
The combined body free energy density function, elastic energy density, interface gradient energy, bubble and grain boundary interaction energy and polycrystal interaction energy are used for obtaining a total free energy equation:
wherein, kappa g 、κ v 、κ i 、κ η The gradient term coefficients of the gas atom concentration, the vacancy concentration, the interstitial atom concentration and the bubble phase sequence parameter are respectively shown, and the last four terms of the formula represent gradient energy.
(4) Acquiring initial bubble size distribution by combining velocity theory
Establishing a rate theoretical model of irradiation bubble nucleation growth behavior: the nuclear material is irradiated with a large amount of gas atoms, and in order to reduce the energy of the nuclear material and minimize strain, the gas atoms tend to be accumulated in a region of low density, and this bubble formation process is called a nucleation process; the change in free energy during nucleation is caused by the core being spherical in radius, the free energy Δf being expressed by the following formula:
wherein Δf is the difference between the free energy of the bubble and the matrix, r is the bubble radius, and σ is the surface energy; whereas a critical core corresponds to the maximum of the free energy, since any larger core will have less energy and will grow spontaneously; thereby obtaining the critical radius R of the nucleus C And activation energy Δf C ：
Randomly introducing bubble cores in a nucleation stage, and ensuring that the average nucleation rate is matched with the expected nucleation rate; assuming that all points in the matrix can become nucleation sites, the probability of forming a core at one atomic position within one nucleation time interval can be calculated by considering the nucleation rate:
wherein Z is Zeldovich correction factor, N is atomic number, beta * The probability of the critical core changing into the supercritical core is represented by tau, and the inoculation time of the core is represented by tau; t is t 0 To calculate a time interval for nucleation; the overall nucleation probability P is thus obtained:
P＝1-exp(-J * t 0 ) (38)
solving the formula (34) -the formula (38) to obtain nucleation probabilities corresponding to bubbles with different sizes, namely bubble size distribution at the initial stage of bubble nucleation by irradiation; in the rate theory nucleation algorithm, it is necessary to calculate the bubble radius and nucleation probability of each nucleation point, and input the bubble size distribution at the initial stage of irradiation bubble nucleation as an initial condition into the phase field simulation.
(5) Establishing and solving a phase field evolution equation
The evolution equation of the process of considering the temperature gradient under the irradiation condition, the generation of the point defect, the recombination of the point defect and the disappearance of the point defect at the defect trap is as follows:
evolution equation of gas atomic concentration:
evolution equation of vacancy atom concentration:
evolution equation of interstitial concentration:
evolution equation of bubble phase field variables:
evolution equation of the poly-crystalline phase field variable:
wherein t is the simulation time; f is total free energy; l is the free interface mobility; m is M g 、M v 、M i Atomic mobilities of gas atoms, vacancy atoms, and interstitial atoms, respectively;thermal fluctuation terms of gas atoms, vacancy atoms, interstitial atoms and bubble phases respectively; />The generation rate of gas atoms and point defects under irradiation conditions are respectively; / >For the recombination rate of point defect vacancies and interstitial atoms, < >>Is an absorption term for grain boundary point defects. The influence of temperature gradient on bubble migration and grain growth is considered in an evolution equation, wherein Q is heat transfer efficiency, and C is a constant;
and->The production rates of gas atoms, vacancies and interstitial gas atoms under irradiation conditions are respectively; the gas atom generation rate is as follows: />
Wherein f r Is the rate of cracking; ran is a random number between 0 and 1; Λ is a constant;
the generation of interstitial and vacancy uses the following expression:
wherein R is 1 And R is 2 Two random numbers which are randomly generated on each time step and each space lattice point and are uniformly distributed between 0 and 1; e is a biased constant that can be varied to represent unequal numbers of vacancies and interstitials created by the dislocation damage; parameter P casc Representing probability of cascade collision in unit time and unit volume, V G Representing a maximum increase in vacancies due to a cascade collision event; in eta<The 0.8 condition ensures that cascade collisions occur only in the matrix phase, not in the bubble phase;
when point defect vacancies and interstitials meet, they recombine to form a perfect crystal lattice, expressed in terms of recombination rate:
In the formula, v r The recombination rate is: v (v) r ＝v b +η 2 v s ，v b And v s Recombination rates of point defects at bulk phase and interface, respectively; v b ＝4πr iv (D i +D v )/Ω，r iv Is the composite volume radius; d (D) i And D v Diffusion coefficients of interstitial and vacancy, respectively; omega is the lattice volume;the absorption term for grain boundary-to-point defects is expressed as follows:
wherein the method comprises the steps ofIs a grain boundary absorption factor, representing the strength of grain boundary to point defect absorption; phi = Σphi i 2 As a function of the position of the grains, there is Φ=1.0 in the grains, while Φ < 1.0 at the grain boundaries, +.>Is the equilibrium concentration of gas atoms, vacancies, interstitial atoms;
solving a phase field evolution equation by using a semi-implicit Fourier spectrum method, and solving evolution equations of the gas atom concentration, the vacancy concentration and the interstitial atom concentration, wherein the solution is shown as follows:
/>
and solving an evolution equation of the bubble phase field variable and the polycrystal phase field variable, wherein the evolution equation is shown as the following formula:
wherein the superscript n denotes the value of the portion in the nth time step, the free energy f is the sum of the bulk free energy density, the polycrystalline free energy density and the elastic free energy density, k= (k) 1 ,k 2 ) Vector coordinates in fourier space, Δt being the simulated time step;
(6) Visualization processing and simulation result statistics
Carrying out visualization processing on a numerical solution result of a phase field evolution equation by using visualization software to obtain the morphology of bubble nucleation growth and grain growth in the nuclear material irradiation process:
And (3) characterizing visual variables in the irradiation bubble evolution simulation process:
in the method, in the process of the invention,visual variables for definition; the characteristics of the bonded diffusion interface are as follows: inside the grain->At the grain boundary +.>In the bubble->Therefore, the grain, grain boundary and bubble are distinguished by distinguishing the visual variable values at each point in space;
the bubble size distribution in the output result is counted by using an algorithm of the communication area marking, and the method comprises the following steps:
in the connected domain marking, scanning from left to right and from top to bottom during the first marking, each effective pixel is set with a label value, and the judgment rule is as follows:
(1) When the left adjacent pixel and the upper adjacent pixel of the pixel are invalid values, a new label value is set for the pixel, and label is +1;
(2) When one of the left adjacent pixel or the upper adjacent pixel of the pixel is an effective value, assigning the label of the effective value pixel to the label value of the pixel;
(3) When the left adjacent pixel and the upper adjacent pixel of the pixel are both effective values, selecting a smaller label value from the effective values and assigning the smaller label value to the pixel;
finally, counting the area of the region contained in each label value to obtain the size distribution and porosity parameters of the bubbles;
the inter-crystalline bubbles were counted and implemented using the following algorithm:
(1) Numbering each bubble on the basis of S6.2 statistics, and recording the position of each bubble;
(2) Judging whether a bubble is an inter-crystalline bubble or not, wherein the method comprises the following steps: taking a range larger than the bubble as a calculation area to calculate a multi-order parameter phi i A sum Γ within this range i Finally multiplying it to obtainThe bubbles are within the grains if λ=0.0; if λ+.0.0 then the bubble is at the grain boundary;
(3) Judging whether all bubbles are inter-crystal bubbles in sequence, respectively obtaining inter-crystal bubble distribution and intra-crystal bubble distribution, and obtaining inter-crystal bubble coverage rate by using the area of the inter-crystal bubbles to be compared with the total area of the grain boundary;
(7) Calculating effective thermal conductivity and irradiation swelling
Calculating the effective thermal conductivity of the nuclear material requires solving a heat transfer equation:
k in local For localized thermal conductivity, related to the thermal transfer properties of the spatial location, k when located inside the grain local ＝k bulk ，k bulk Is the thermal conductivity within the grain; k when located on grain boundary local ＝k GB ，k GB Is the thermal conductivity at the grain boundaries; k when located in the bubble local ＝k bubble ，k bubble Is the thermal conductivity in the bubble; solving a heat transfer equation (56) using a finite difference method to obtain an equilibrium temperature field and an effective thermal conductivity;
according to the size distribution and the porosity of the bubbles, an irradiation swelling rule of the nuclear material irradiation bubbles is obtained, and the fuel swelling caused by the irradiation bubbles in the nuclear material is estimated by the following formula:
Where ΔV is the fuel volume change, V 0 For initial fuel volume, V f To include the final fuel volume where the bubble induced swelling, the fuel swelling was calculated in a two-dimensional simulation using the ratio of the bubble area to the total area. And obtaining the irradiation swelling and heat conduction rule of the nuclear material through calculation, so as to realize the prediction of the service performance of the nuclear material.
Some specific examples are provided below.
Example 1
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be further described in detail with reference to the accompanying drawings and examples.
The invention provides a simulation method for the evolution process of nuclear material irradiation bubbles, which is characterized in that according to the flow in a specific implementation mode, a phase field model is utilized to carry out numerical simulation on the evolution process of pure tungsten irradiation bubbles at 1273K, and simulation using parameters are shown in table 1:
TABLE 1 simulation parameters for pure tungsten
Firstly, coupling and solving elastic energy, wherein the elastic energy density is obtained by solving a mechanical equilibrium equation, and the expression is as follows:
wherein C is ijkl (r) is the elastic constant tensor, ε ij 0 Is the initial intrinsic strain, used to represent the inelastic strained portion,for uniform strain, delta epsilon ij (r) is non-uniform strain.
Reconstructing a free energy equation, deducing based on thermodynamic theory, and calculating a free energy density function of a nuclear material matrix phase according to the following formula:
Wherein in deriving the free energy of the matrix, condition c is always satisfied since the defects are assumed to occupy only perfect lattices g +c v +c i +c m ＝1.0；c m Is the concentration of a perfect lattice in at.%; k (k) B Is the boltzmann constant; t is absolute temperature; e (E) g f 、E v f 、E i f The energy of formation of gas atoms, vacancies, and interstitials in the matrix phase;
the free energy density function of the bubble phase of the core material is as follows:
wherein A and B are constants,is self-contained in bubbles which are related to the atomic concentration of gasFrom the energy density function, the ++is calculated by Van der Waals equation at high pressure>/>
Wherein V is site Mu, the volume of a single lattice 0 And p 0 The chemical potential and the pressure in the reference state are shown, and b is a van der Waals constant;
according to the obtained free energy density functions of the matrix phase and the bubble phase, introducing an interpolation function and a double-well potential function to obtain a free energy density function of the body:
f bulk (c g ,c v ,c i ,η,T)＝h(η)f b (c g ,c v ,c i ,T)+[1-h(η)]f m (c g ,c v ,c i ,T)+wg(η) (31)
wherein h (η) =η 3 (6η 2 -15 η+10) is a structured interpolation function, satisfying when η=1.0, h (η) =1.0 in the bubble phase; when η=0.0, h (η) =0.0 in the matrix phase; g (eta) =eta 2 (1-η) 2 As a dual-potential well function, w is the potential well height;
the free energy density function of polycrystalline interactions is expressed as follows:
wherein α, β, γ and δ are image-only parameters.
The combined body free energy density function, elastic energy density, interface gradient energy, bubble and grain boundary interaction energy and polycrystal interaction energy are used for obtaining a total free energy equation:
Wherein, kappa g 、κ v 、κ i 、κ η The gradient term coefficients of the gas atom concentration, the vacancy concentration, the interstitial atom concentration and the bubble phase sequence parameter are respectively, and the last four terms of the formula represent gradient energy;
constructing a phase field evolution equation, and considering a temperature gradient under irradiation conditions, generation of point defects, recombination of the point defects and disappearance process evolution equation of the point defects at defect traps, wherein the evolution equation comprises the following steps:
evolution equation of gas atomic concentration:
evolution equation of vacancy atom concentration:
evolution equation of interstitial concentration:
evolution equation of bubble phase field variables:
evolution equation of the poly-crystalline phase field variable:
wherein t is the simulation time; f is total free energy; l is the free interface mobility; m is M g 、M v 、M i Atomic mobilities of gas atoms, vacancy atoms, and interstitial atoms, respectively;thermal fluctuation terms of gas atoms, vacancy atoms, interstitial atoms and bubble phases respectively; />The generation rate of gas atoms and point defects under irradiation conditions are respectively; />For the recombination rate of point defect vacancies and interstitial atoms, < >>Is an absorption term for grain boundary point defects. The influence of temperature gradient on bubble migration and grain growth is considered in an evolution equation, wherein Q is heat transfer efficiency, and C is a constant;
solving a phase field evolution equation by a semi-implicit Fourier spectrum method, and carrying out visualization processing on a numerical result by using Paraview, wherein the stress distribution of a single bubble in the simulation result is shown in fig. 2a, 2b and 2 c: FIG. 2a shows the stress component sigma xx Distribution, FIG. 2b shows stress component sigma yy Distribution, FIG. 2c shows stress component sigma xy Distribution. The elastic field distribution generated by a single bubble is obtained through the simulation calculation of the first embodiment, so that the accuracy of the coupling elastic energy solution is ensured.
Example two
In the example, on the basis of the first embodiment, a phase field simulation study is performed on the evolution of bubbles under the irradiation condition of single crystal tungsten. Initial bubble distribution was obtained using rate theory: establishing a rate theoretical model of irradiation bubble nucleation growth behavior: the nuclear material is irradiated with a large amount of gas atoms, which tend to accumulate in the areas of low density in order to reduce the energy of the system and minimize the strain, a process of forming bubbles is called a nucleation process; the change in free energy during nucleation is caused by the core being spherical in radius, the free energy Δf being expressed by the following formula:
wherein Δf is the difference between the free energy of the bubble and the matrix, r is the bubble radius, and σ is the surface energy; whereas a critical core corresponds to the maximum of the free energy, since any larger core will have less energy and will grow spontaneously; thereby obtaining the critical radius R of the nucleus C And activation energy Δf C ：
Randomly introducing bubble cores in a nucleation stage, and ensuring that the average nucleation rate is matched with the expected nucleation rate; assuming that all points in the matrix can become nucleation sites, the probability of forming a core at one atomic position within one nucleation time interval can be calculated by considering the nucleation rate:
Wherein Z is Zeldovich correction factor, N is atomic number, beta * The probability of the critical core changing into the supercritical core is represented by tau, and the inoculation time of the core is represented by tau; t is t 0 To calculate a time interval for nucleation; the overall nucleation probability P is thus obtained:
P＝1-exp(-J * t 0 ) (38)
solving the formula (34) -the formula (38) to obtain nucleation probabilities corresponding to bubbles with different sizes, namely bubble size distribution at the initial stage of bubble nucleation by irradiation; in a rate theory nucleation algorithm, the bubble radius and nucleation probability of each nucleation point are required to be calculated, and the bubble size distribution at the initial stage of irradiation bubble nucleation is used as an initial condition to be input into a phase field simulation;
constructing a phase field evolution equation, and considering a temperature gradient under irradiation conditions, generation of point defects, recombination of the point defects and disappearance process evolution equation of the point defects at defect traps, wherein the evolution equation comprises the following steps:
evolution equation of gas atomic concentration:
evolution equation of vacancy atom concentration:
evolution equation of interstitial concentration:
evolution equation of bubble phase field variables:
evolution equation of the poly-crystalline phase field variable:
wherein L is the mobility of a free interface; m is M g 、M v 、M i Atomic mobilities of gas atoms, vacancy atoms, and interstitial atoms, respectively;thermal fluctuation terms of gas atoms, vacancy atoms, interstitial atoms and bubble phases respectively; The generation rate of gas atoms and point defects under irradiation conditions are respectively; />For the recombination rate of point defect vacancies and interstitial atoms, < >>Is an absorption term for grain boundary point defects. The influence of temperature gradient on bubble migration and grain growth is considered in an evolution equation, wherein Q is heat transfer efficiency, and C is a constant;
and->The production rates of gas atoms, vacancies and interstitial gas atoms under irradiation conditions are respectively; the gas atom generation rate is as follows:
wherein f r Is the rate of cracking; ran is a random number between 0 and 1; Λ is a constant;
the generation of interstitial and vacancy uses the following expression:
wherein R is 1 And R is 2 Two random numbers which are randomly generated on each time step and each space lattice point and are uniformly distributed between 0 and 1; e is a biased constant that can be varied to represent unequal numbers of vacancies and interstitials created by the dislocation damage; parameter P casc Representing probability of cascade collision in unit time and unit volume, V G Representing a maximum increase in vacancies due to a cascade collision event; in eta<The 0.8 condition ensures that cascade collisions occur only in the matrix phase, not in the bubble phase;
the phase field evolution equation is solved through a semi-implicit Fourier spectrum method, the logarithmic result is subjected to visual processing by using Paraview, the average bubble diameters of different irradiation intensities and the comparison between the average bubble diameters and experimental results are shown in figure 3, and the average bubble diameters increase along with the increase of the irradiation intensities. The calculation result of the phase field simulation is basically consistent with the experimental result.
Example III
In the embodiment, on the basis of the second embodiment, a phase field simulation study is performed on the evolution of bubbles under the irradiation condition of polycrystalline tungsten, polycrystalline structures with different sizes are introduced into the embodiment, and the evolution equation of the process of considering the generation of point defects, the recombination of the point defects, the disappearance of the point defects at defect traps and the like under the irradiation condition is obtained as follows:
evolution equation of gas atomic concentration field:
vacancy atomic concentration field evolution equation:
evolution equation of interstitial concentration field:
evolution equation of bubble phase sequence parameters:
evolution equation of the multicrystalline order parameter:
wherein L is the mobility of a free interface; m is M g 、M v 、M i Atomic mobilities of gas atoms, vacancy atoms, and interstitial atoms, respectively;thermal fluctuation terms of gas atoms, vacancy atoms, interstitial atoms and bubble phases respectively;the generation rate of gas atoms and point defects under irradiation conditions are respectively; />Is the recombination rate of point defect vacancies and interstitial atoms. />The absorption term for grain boundary-to-point defects is expressed as follows:
wherein the method comprises the steps ofThe grain boundary absorption factor represents the strength of grain boundary to point defect absorption. Phi = Σphi i 2 As a function of the position of the grains, there is Φ=1.0 in the grains, while Φ < 1.0 at the grain boundaries.
Solving an evolution equation by a Fourier spectrum method, and carrying out visualization treatment on a numerical result by using Paraview, wherein the irradiation bubble distribution generated by polycrystalline tungsten with different grain sizes is shown in fig. 4a, 4b and 4 c: both bubble size and density decrease as the grain size becomes smaller. This is because smaller size grains have a larger grain boundary area, and the grain boundary absorbs a lot of surrounding point defects, resulting in smaller irradiation swelling of the smaller size grains, which also provides a possible idea for improving the irradiation resistance of pure tungsten. The average bubble size at the different grain sizes is shown in fig. 5, the bubble density at the different grain sizes is shown in fig. 6, and the average bubble size and the bubble density at the different grain sizes change substantially in accordance with experimental observations. Through simulation research of the embodiment, the influence of the grain size on the irradiation bubble evolution is quantitatively analyzed, and theoretical explanation is provided for experimental phenomena.
Claims (8)
1. A method for determining the appearance of irradiation bubbles in a nuclear material and the influence of the irradiation bubbles on force and heat performance is characterized by comprising the following steps: the method comprises the following steps:
s1, selecting a phase field variable describing the evolution process of the appearance of the nuclear material irradiation bubble; collecting physical parameters of the nuclear material, including formation energy, diffusion coefficient and gradient coefficient of point defects, and mobility of bubbles and grain boundaries; and calculating mobility of the point defect considering diffusion anisotropy;
s2, calculating elastic modulus distribution related to the position; obtaining initial intrinsic strain distribution by combining Van der Waals equation; establishing and solving an elastic equilibrium equation to obtain irradiation bubble stress strain distribution and elastic energy density;
s3, obtaining a free energy density function of a matrix phase of the nuclear material based on a thermodynamic theory of the material according to the physical parameters of the nuclear material collected in the step S1, obtaining a free energy density function of a bubble phase in the nuclear material through deducing a Van der Waals equation under high temperature and high pressure conditions, and obtaining a total free energy equation by combining interface gradient energy, bubble and grain boundary interaction energy, polycrystal interaction energy and elastic energy density obtained in the step S2;
s4, establishing a rate theoretical model of irradiation bubble nucleation; obtaining bubble size distribution at the initial stage of irradiation bubble nucleation through solving a rate theoretical model, and taking the size distribution as initial conditions of phase field simulation;
S5, constructing a phase field evolution equation according to the total free energy equation obtained in the step S3 and comprehensively considering the factors of temperature gradient, generation of point defects, recombination of the point defects and disappearance of the point defects at defect traps under the irradiation condition; carrying out numerical solution on a phase field evolution equation by adopting a semi-implicit Fourier spectrum method;
s6, carrying out visual treatment on the numerical solution obtained in the step S5 to obtain the morphology of bubble nucleation growth and grain growth in the nuclear material irradiation process; counting all bubbles by adopting a communication area marking method to obtain the size distribution and porosity parameters of the irradiated bubbles in the nuclear material; classifying and counting inter-crystal bubbles and intra-crystal bubbles;
s7, calculating to obtain the effective heat conductivity and the irradiation swelling rule of the nuclear material based on the bubble distribution and the porosity obtained in the step S6.
2. The method of determining the morphology of irradiated bubbles in a nuclear material and their effect on thermal performance of force according to claim 1, wherein: said step S1 comprises the sub-steps of:
s1.1, selecting a phase field variable describing the morphology evolution process of the nuclear material irradiation bubble, wherein the phase field variable representing the concentration of point defects is required to be selected: concentration of gas atoms c g Concentration of vacancies c v And interstitial concentration c i The method comprises the steps of carrying out a first treatment on the surface of the And simultaneously selecting a bubble phase sequence parameter eta for distinguishing a bubble phase and a matrix phase: η=1.0 when in the bubble phase and η=0.0 when in the matrix phase; the multi-order parameter phi needs to be selected i Represents a grain structure, wherein the subscript i represents the ith grain and there is phi within the ith grain i ＝1.0；
S1.2, collecting physical parameters of a nuclear material, including formation energy, diffusion coefficient, mobility and gradient coefficient of gas atoms, vacancies and interstitial atoms and mobility of bubbles and grain boundaries;
s1.3 calculating the mobility of the point defect considering diffusion anisotropy, vacancy mobility M, based on the diffusion coefficients of the gas atoms, vacancies, and interstitial atoms of S1.2 v Has the following expression:
wherein D is v Vacancy diffusion coefficient in tensor form, V m The unit volume of the crystal lattice is R is an ideal gas constant, and T is absolute temperature;
wherein D is v 0 For the effective diffusion coefficient of the vacancies,and->The components of vacancy bulk diffusion, surface diffusion and grain boundary diffusion, respectively, have the following expressions:
wherein w is b 、w s And w GB Weight coefficients of bulk diffusion, surface diffusion and grain boundary diffusion respectively, I is an identity matrix, T s v And T GB v The surface and grain boundary tensor coefficients, respectively, have the following expression:
3. The method of determining the morphology of irradiated bubbles in a nuclear material and their effect on thermal performance of force according to claim 1, wherein: said step S2 comprises the sub-steps of:
s2.1, calculating the elastic modulus distribution related to the position: in the polycrystalline material, the elastic modulus is related to the orientation of the grains, the elastic modulus of each grain with defined orientation is obtained by coordinate transformation, and the elastic modulus tensor C ijkl The expression (r) is as follows:
wherein,for the modulus of elasticity in the matrix +.>And->Rotation matrix a of p-th crystal grain relative to reference coordinate ij Ia, jb, kc, ld th component of (a), rotating matrix a ij The expression is determined by Euler angles of corresponding grains, and the rotation matrix under the three-dimensional coordinates is as follows:
wherein θ, ζ, and ψ are three euler angles determining the grain orientation;
s2.2 initial intrinsic StrainIs composed of two partsThe first part is the intrinsic strain of the matrix phase>The second part is the intrinsic strain of the bubble phase +.>r is a position vector;
wherein the matrix phase intrinsic strain caused by defect non-uniformityThe method comprises the following steps:
wherein ε is g0 Delta is the lattice constant expansion coefficient due to the introduction of gas atoms ij As a function of Kronecker,h (η) =η, the equilibrium concentration of the gas atoms 3 (6η 2 -15 η+10) is a structured interpolation function, satisfying when η=1.0, h (η) =1.0 in the bubble phase; when η=0.0, h (η) =0.0 in the matrix phase; the intrinsic strain of the bubble is very complex because it depends on the internal pressure of the bubble and the size of the bubble when the internal pressure of the bubble is greater than 2γ s At/r, an outward pressure acts on the lattice around the bubbles, so the characteristic strain of the bubbles is positive, which means that the bubbles are in an overpressure state; when the bubble pressure is equal to 2 gamma s At/r, the intrinsic strain becomes zero; negative intrinsic strain values indicate that the bubble is in an under-pressure state, which results in elastic relaxation of the surrounding lattice of the bubble toward the center of the bubble; gamma ray s The surface energy of the bubble and r is the radius of the bubble;
intrinsic strain of bubble phaseThe description is as follows:
wherein P is g Is the internal pressure of the bubble, 2γ s R is considered as external hydrostatic pressure, C 11 For the principal direction elastic modulus, the gas pressure inside the bubble is always positive, while the intrinsic strain inside the bubble may be negative; this definition shows that intrinsic strain is a function of gas pressure, bubble radius, surface tension and material elastic properties; the gas pressure in the bubble can be obtained using the van der Waals equation:
wherein Ω is the molecular volume of the corresponding core material, b is the van der Waals constant, k B Is the boltzmann constant;
s2.3, obtaining stress strain distribution by solving an elastic equilibrium equation, wherein the elastic equilibrium equation is as follows:
middle sigma ij Is stress, r j Is the unit displacement along the j direction; epsilon kl (r) is the total strain equivalent to ε ij (r); total strain epsilon ij (r) is the sum of uniform strain and non-uniform strain:
in the middle ofAnd delta epsilon ij (r) respectivelyIs uniform strain and non-uniform strain; uniform strain->The expression is as follows:
in the method, in the process of the invention,for additional stress, V is the simulated area volume; non-uniform strain delta epsilon ij The expression (r) is as follows:
u in the formula i (r) and u j (r) substituting equation (17) into the elastic equilibrium equation (14) for displacement components in the i and j directions:
and (3) obtaining a displacement field equation after simplifying and finishing:
in the middle ofFor a position-independent elastic modulus, C' ijkl (r) is the position dependent elastic modulus; the Fourier transform is carried out to obtain:
in the middle ofFor a displacement field in fourier space, k is the wave vector in fourier space, k j Is a component of a wave vector;is the green tensor; the subscript k indicates that the partial calculation is performed in fourier space; solving formula (20) using perturbation iteration method, the displacement field in the nth iteration is as follows:
the initial value of the iteration is taken asThe displacement field u in real space k (r) by- >Obtaining an inverse Fourier transform, obtaining a displacement field through an iteration solution elastic balance equation, and further obtaining stress-strain distribution through calculation;
the elastic energy density was calculated from the obtained stress strain, and the elastic energy density was expressed as follows:
4. the method of determining the morphology of irradiated bubbles in a nuclear material and their effect on thermal performance of force according to claim 1, wherein: said step S3 comprises the sub-steps of:
s3.1, based on thermodynamic theory deduction, calculating a free energy density function of a nuclear material matrix phase according to the following formula:
wherein in deriving the free energy of the matrix, condition c is always satisfied since the defects are assumed to occupy only perfect lattices g +c v +c i +c m ＝1.0；c m Is the concentration of a perfect lattice in at.%;the energy of formation of gas atoms, vacancies, and interstitials in the matrix phase;
s3.2 the free energy density function of the bubble phase of the core material is as follows:
wherein A and B are constants,as a function of free energy density within the bubble in relation to the concentration of gas atoms; the gibbs free energy expression G (p) of the bubble is:
wherein p is pressure; v (V) b Is the volume of the bubbles; g (p) 0 ) Is at reference state pressure p 0 The lower gibbs free energy; the van der Waals equation under high pressure is:
p(V b -nb)＝nk B T (26)
Wherein n is the number of gas atoms; the gibbs free energy of the non-ideal gas thus obtained is:
the gas atoms in the bubble phase may occupy lattice or non-lattice positions, so the concentration of gas atoms in the bubble is defined as:
wherein V is site Is the volume of a single lattice, from which the pressure can be obtained:
the free energy density function of the bubbles obtained by bringing the above results into the bubbles is:
wherein mu 0 Is the reference state p 0 The chemical potential below;
s3.3, introducing an interpolation function and a double-trap potential function according to the obtained free energy density functions of the matrix phase and the bubble phase to obtain a bulk free energy density function:
f bulk (c g ,c v ,c i ,η,T)＝h(η)f b (c g ,c v ,c i ,T)+[1-h(η)]f m (c g ,c v ,c i ,T)+wg(η)(31)
wherein h (η) =η 3 (6η 2 -15 η+10) is a structured interpolation function, satisfying when η=1.0, h (η) =1.0 in the bubble phase; when η=0.0, h (η) =0.0 in the matrix phase; g (eta) =eta 2 (1-η) 2 As a dual-potential well function, w is the potential well height;
the free energy density function of polycrystalline interactions is expressed as follows:
wherein α, β, γ and δ are image-only parameters;
the combined body free energy density function, elastic energy density, interface gradient energy, bubble and grain boundary interaction energy and polycrystal interaction energy are used for obtaining a total free energy equation:
wherein, kappa g 、κ v 、κ i 、κ η The gradient term coefficients of the gas atom concentration, the vacancy concentration, the interstitial atom concentration and the bubble phase sequence parameter are respectively shown, and the last four terms of the formula represent gradient energy.
5. The method of determining the morphology of irradiated bubbles in a nuclear material and their effect on thermal performance of force according to claim 1, wherein: said step S4 comprises the sub-steps of:
s4.1, combining the physical parameters of the nuclear material obtained in the step S1, and establishing a rate theoretical model of the irradiation bubble nucleation growth behavior: the nuclear material is irradiated with a large amount of gas atoms, and in order to reduce the energy of the nuclear material and minimize strain, the gas atoms tend to be accumulated in a region of low density, and this bubble formation process is called a nucleation process; the change in free energy during nucleation is caused by the core being spherical in radius, the free energy Δf being expressed by the following formula:
wherein Δf is the difference between the free energy of the bubble and the matrix, r is the bubble radius, and σ is the surface energy; whereas a critical core corresponds to the maximum of the free energy, since any larger core will have less energy and will grow spontaneously; thereby obtaining the critical radius R of the nucleus C And activation energy Δf C ：
Randomly introducing bubble cores in a nucleation stage, and ensuring that the average nucleation rate is matched with the expected nucleation rate; assuming that all points in the matrix can become nucleation sites, the probability of forming a core at one atomic position within one nucleation time interval can be calculated by considering the nucleation rate:
Wherein Z is Zeldovich correction factor, N is atomic number, beta * The probability of the critical core changing into the supercritical core is represented by tau, and the inoculation time of the core is represented by tau; t is t 0 To calculate a time interval for nucleation; the overall nucleation probability P is thus obtained:
P＝1-exp(-J * t 0 ) (38)
s4.2, solving a formula (34) -a formula (38) to obtain nucleation probabilities corresponding to bubbles with different sizes, namely bubble size distribution in the initial stage of bubble nucleation irradiation; in the rate theory nucleation algorithm, it is necessary to calculate the bubble radius and nucleation probability of each nucleation point, and input the bubble size distribution at the initial stage of irradiation bubble nucleation as an initial condition into the phase field simulation.
6. The method of determining the morphology of irradiated bubbles in a nuclear material and their effect on thermal performance of force according to claim 1, wherein: said step S5 comprises the sub-steps of:
s5.1, considering the evolution equation of the temperature gradient, the generation of point defects, the recombination of the point defects and the disappearance process of the point defects at defect traps under the irradiation condition, wherein the evolution equation is as follows:
evolution equation of gas atomic concentration:
evolution equation of vacancy atom concentration:
evolution equation of interstitial concentration:
evolution equation of bubble phase field variables:
evolution equation of the poly-crystalline phase field variable:
wherein t is the simulation time; f is total free energy; l is the free interface mobility; m is M g 、M v 、M i Atomic mobilities of gas atoms, vacancy atoms, and interstitial atoms, respectively;thermal fluctuation terms of gas atoms, vacancy atoms, interstitial atoms and bubble phases respectively; />The generation rate of gas atoms and point defects under irradiation conditions are respectively; />For the recombination rate of point defect vacancies and interstitial atoms, < >>As an absorption term of the grain boundary to point defects, the influence of temperature gradient on bubble migration and grain growth is considered in an evolution equation, wherein Q is heat transferEfficiency of delivery, C is a constant;
and->The production rates of gas atoms, vacancies and interstitial gas atoms under irradiation conditions are respectively; the gas atom generation rate is as follows:
wherein f r Is the rate of cracking; ran is a random number between 0 and 1; Λ is a constant;
the generation of interstitial and vacancy uses the following expression:
wherein R is 1 And R is 2 Two random numbers which are randomly generated on each time step and each space lattice point and are uniformly distributed between 0 and 1; e is a biased constant that can be varied to represent unequal numbers of vacancies and interstitials created by the dislocation damage; parameter P casc Representing probability of cascade collision in unit time and unit volume, V G Representing a maximum increase in vacancies due to a cascade collision event; in eta <The 0.8 condition ensures that cascade collisions occur only in the matrix phase, not in the bubble phase;
when point defect vacancies and interstitials meet, they recombine to form a perfect crystal lattice, expressed in terms of recombination rate:
in the formula, v r The recombination rate is: v (v) r ＝v b +η 2 v s ，v b And v s Recombination rates of point defects at bulk phase and interface, respectively; v b ＝4πr iv (D i +D v )/Ω，r iv Is the composite volume radius; d (D) i And D v Diffusion coefficients of interstitial and vacancy, respectively; omega is the lattice volume;the absorption term for grain boundary-to-point defects is expressed as follows:
wherein the method comprises the steps ofIs a grain boundary absorption factor, representing the strength of grain boundary to point defect absorption; phi = Σphi i 2 As a function of the position of the grains, there is Φ=1.0 in the grains, while Φ < 1.0 at the grain boundaries, +.>Is the equilibrium concentration of gas atoms, vacancies, interstitial atoms;
s5.2, solving a phase field evolution equation by using a semi-implicit Fourier spectrum method, and solving evolution equations of gas atom concentration, vacancy concentration and interstitial atom concentration, wherein the solution is shown as follows:
and solving an evolution equation of the bubble phase field variable and the polycrystal phase field variable, wherein the evolution equation is shown as the following formula:
wherein the superscript n denotes the value of the portion in the nth time step, the free energy f is the sum of the bulk free energy density, the polycrystalline free energy density and the elastic free energy density, k= (k) 1 ,k 2 ) For vector coordinates in fourier space, Δt is the simulated time step.
7. The method of determining the morphology of irradiated bubbles in a nuclear material and their effect on thermal performance of force according to claim 1, wherein: said step S6 comprises the sub-steps of:
s6.1, carrying out visualization processing on the numerical solution result in the step S5 by using visualization software to obtain the morphology of bubble nucleation growth and grain growth in the nuclear material irradiation process:
and (3) characterizing visual variables in the irradiation bubble evolution simulation process:
in the method, in the process of the invention,visual variables for definition; the characteristics of the bonded diffusion interface are as follows: inside the grain->At the grain boundary +.>In the bubble->Therefore, the grain, grain boundary and bubble are distinguished by distinguishing the visual variable values at each point in space;
s6.2, counting the size distribution of bubbles in the output result by using an algorithm of the communication area marking, wherein the method comprises the following steps of:
in the connected domain marking, scanning from left to right and from top to bottom during the first marking, each effective pixel is set with a label value, and the judgment rule is as follows:
(1) When the left adjacent pixel and the upper adjacent pixel of the pixel are invalid values, a new label value is set for the pixel, and label is +1;
(2) When one of the left adjacent pixel or the upper adjacent pixel of the pixel is an effective value, assigning the label of the effective value pixel to the label value of the pixel;
(3) When the left adjacent pixel and the upper adjacent pixel of the pixel are both effective values, selecting a smaller label value from the effective values and assigning the smaller label value to the pixel;
finally, counting the area of the region contained in each label value to obtain the size distribution and porosity parameters of the bubbles;
s6.3, counting inter-crystal bubbles, and realizing the method by using the following algorithm:
(1) Numbering each bubble on the basis of S6.2 statistics, and recording the position of each bubble;
(2) Judging whether a bubble is an inter-crystalline bubble or not, wherein the method comprises the following steps: taking a range larger than the bubble as a calculation area to calculate a multi-order parameter phi i A sum Γ within this range i Finally multiplying it to obtainThe bubbles are within the grains if λ=0.0; if λ+.0.0 then the bubble is at the grain boundary;
(3) And judging whether all the bubbles are inter-crystal bubbles in sequence, respectively obtaining inter-crystal bubble distribution and intra-crystal bubble distribution, and obtaining the coverage rate of the inter-crystal bubbles by using the area of the inter-crystal bubbles to the total area of the grain boundary.
8. The method of determining the morphology of irradiated bubbles in a nuclear material and their effect on thermal performance of force according to claim 1, wherein: said step S7 comprises the sub-steps of:
s7.1, calculating the effective heat conductivity of the nuclear material, wherein a heat transfer equation needs to be solved:
▽·(k local ▽T)＝0 (56)
K in local For localized thermal conductivity, related to the thermal transfer properties of the spatial location, k when located inside the grain local ＝k bulk ，k bulk Is the thermal conductivity within the grain; k when located on grain boundary local ＝k GB ，k GB Is the thermal conductivity at the grain boundaries; k when located in the bubble local ＝k bubble ，k bubble Is the thermal conductivity in the bubble; solving a heat transfer equation (56) using a finite difference method to obtain an equilibrium temperature field and an effective thermal conductivity;
s7.2, combining the bubble size distribution and the porosity calculated in the step S6 to obtain an irradiation swelling rule of the nuclear material irradiation bubbles, wherein the fuel swelling caused by the irradiation bubbles in the nuclear material is estimated by the following formula:
where ΔV is the fuel volume change, V 0 For initial fuel volume, V f To include the final fuel volume where the bubble induced swelling, the fuel swelling was calculated in a two-dimensional simulation using the ratio of the bubble area to the total area.
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