CN104408300A - Microscopic phase-field modeling and analysis method based on age forming/diffusion combination process - Google Patents

Abstract

The invention discloses a microscopic phase-field modeling and analysis method based on an age forming/diffusion combination process, so as to solve the technical problem of poor mechanical performance of material caused by an existing age forming/diffusion combination process. The technical scheme is as follows: constructing a microscopic phase-field model of the age forming/diffusion combination process, and solving out atom occupation information by use of a microscopic atomic diffusion equation coupled with a temperature field, a concentration field and a macro/micro elastic stress field; then converting the atom occupation information into secondary information such as structure morphology evolution, wherein the secondary information is used for analyzing rules in influence of the temperature field, the concentration field and the elastic stress field on the structure morphology evolution and interface structure migration in an age forming/diffusion combination process, the structure morphology evolution expresses the structural changes in the age forming/diffusion combination process, atom occupation can quantitatively characterize the microscopic atomic diffusion mechanisms of structural changes, and structural responsivity of the temperature field, the concentration field and the elastic stress field can be subjected to reverse calculation to obtain process parameters for eliminating oriented coarsening, so that the mechanical performance of the material is improved.

Description

Microscopic Phase Field Modeling and Analysis Method for Age Forming/Diffusion Composite Process

technical field

The invention relates to a microscopic phase field modeling and analysis method, in particular to a microscopic phase field modeling and analysis method of aging forming/diffusion composite technology.

Background technique

Aging forming is a precise forming technology combining creep forming and dynamic precipitation. It realizes precise forming of aluminum alloy components by creep deformation at artificial aging temperature, overcomes the limit problem of cold forming, and obtains the microstructure and properties of dynamic precipitation. Aging-forming large and complex integral components replace welded, riveted, and glued assembled complex panels, reducing mechanical connections such as holes and fasteners, which not only reduces stress concentration, but also significantly reduces weight, and significantly improves service life and reliability. High structural efficiency is an important symbol of the manufacturing technology of large and complex integral components of modern aircraft.

Based on the principle of age forming and the research of aluminum alloy layered composite materials, an integrated technology extending from component aging forming process to composite material preparation and component forming has been developed——diffusion composite/aging forming method, that is, the two alloys are separated at corresponding temperatures After solution treatment, aging forming/diffusion compounding is carried out in the aging forming equipment. Under the action of creep, the contact surface atoms change from the physical distance to the chemical distance, and the physical contact surface is transformed into a chemical bonding surface; during the diffusion compounding stage, the elements on both sides of the interface layer Interdiffusion weakens the tendency to form a continuous precipitation phase at the joint surface; and realizes precise forming, preparing a composite material part with an alloy as the surface layer and an alloy as the center layer, and at the same time realizing the microstructure and performance of deformation aging.

Creep deformation and precipitation strengthening are carried out simultaneously during age forming [International Journal of Machine Tools & Manufacture 51(2011) 1–17]. The creep constitutive equation based on unified theory and aging dynamics is often used to describe stress age hardening, creep-induced precipitation phase evolution, dislocation strengthening, solid solution strengthening and aging strengthening, and multi-stage aging precipitation phase evolution [Ho K.C, Lin J , Dean T.A.J. Mater. Process. Tech. 2004 153-154:122-127]. The constitutive equation provides useful theoretical support for aging forming in describing the evolution of precipitation phases. However, during the aging forming/diffusion recombination process, solute atoms remain in the bonding layer, and the orientation of the precipitation phase is severely coarsened. There is no precipitation zone along the grain, and the structure is not The homogeneity seriously damages the material properties. The constitutive equation cannot observe the atomic-level microstructure inhomogeneity on the spatial scale, and cannot capture the evolution process of precipitation phase nucleation, growth, and orientation coarsening on the time scale.

Contents of the invention

In order to overcome the shortcomings of poor mechanical properties of materials caused by the existing aging forming/diffusion composite process, the present invention provides a microcosmic phase field modeling and analysis method for the aging forming/diffusion composite process. In this method, the microscopic phase field model of the aging forming/diffusion composite process is constructed, and the atomic occupancy information is solved by the microscopic atomic diffusion equation coupling the temperature field, the concentration field and the macroscopic/microscopic elastic stress field. Then, the atom occupancy information is converted into secondary information such as microstructure evolution, which is used to analyze the influence of temperature field, concentration field and elastic stress field on microstructure evolution and interfacial structure migration in the aging forming/diffusion compounding process. The evolution of tissue morphology expresses the tissue changes in the aging forming/diffusion composite process, the atomic occupancy can quantitatively characterize the microscopic atomic diffusion mechanism of tissue changes, and the tissue responsivity of temperature field, concentration field, and elastic stress field can be reversed to obtain the elimination orientation Coarsening process parameters. This method can reduce experimental trial and error, optimize the aging forming/diffusion composite structure, and improve the mechanical properties of the material.

The technical scheme adopted by the present invention to solve the technical problem: a microscopic phase field modeling and analysis method of aging forming/diffusion composite process, which is characterized in that it includes the following steps:

Step 1: Construct the Langevin form microscopic diffusion equation of the two-dimensional microscopic phase field model.

Onsager-type discrete lattice kinetic equations were used to describe alloy precipitation, and mean-field theory was used to calculate free energy functions.

(a) According to the Onsager diffusion equation, the rate of change of the probability is proportional to the thermodynamic driving force, namely:

$\frac{\mathrm{<span\; class="notranslate">\; dP</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; dt</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; f</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Indicates the probability of A atoms occupying the lattice position r at time t, C ₀ is the average concentration of the matrix, L(rr') is a constant related to the probability of transitioning from the lattice point r' to the unit time; T is the absolute temperature; K _B is the Boltzmann constant; C ₀ is the average concentration of the matrix; F is the total free energy of the system, and the expression of the mean field approximation to the total free energy of the system in the discrete lattice model is:

$\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}\mathrm{<span\; class="notranslate">\; V</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}\mathrm{<span\; class="notranslate">\; T</span>}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; ln</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

V(r-r′) is the effective interaction energy between atoms, which is given by the following formula,

V(rr')=V _AA (rr')+V _BB (rr')-2V _AB (rr') (3)

V _AA is the interaction potential between A atoms, and V _AB is the interaction potential between A atoms and B atoms. A and B are alloy components.

(b) To simplify the calculation, the face-centered cubic three-dimensional space is projected on the [001] orientation, the equation (1) is in the plane projection of the reciprocal space, and the dynamic equation is:

ξ(k, t) is a random noise term, which satisfies the random term of the fluctuation-dissipation theory.

in,

V ₁ and V ₂ are the effective interaction energies between the first and second nearest neighbor atoms, respectively.

Step 2. Macro/micro coupling elastic stress of aging forming/diffusion compounding.

The microelastic expression describing the lattice mismatch energy of the precipitated phase in the microscopic phase field model is,

$\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; e</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 44</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 44</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; [</span>\frac{{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.125</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

Where ε ₀ =(a _p -a ₀ )/a ₀ is the lattice lattice mismatch degree of atomic size mismatch caused by the difference in atomic size, and c _ij is the elastic constant of the cubic lattice.

The elastic strain energy caused by external stress has in Fourier space,

A is a constant, a ₀ is the average distance between atoms when the external force is 0, a is the distance between atoms, and a=a ₀ +Δa, Δa is the deformation caused by external stress, E _<100> , E _<001> are [100 ] and the modulus of elasticity on [001], and △a=σa ₀ /E. σ _x is the component of the external force on the x-axis, and σ _y is the component of the external force on the y-axis.

The elastic strain energy (8) caused by mismatch and external stress is introduced into the free energy expression (2) to calculate the coupling stress effect of aging forming and diffusion composite macroscopic stress and microscopic elastic stress introduced by structural evolution.

Step three, equation solving and data processing.

The Euler iterative method was used to solve the microscopic diffusion equation, and the atomic occupancy information was calculated. Using the interatomic potential, thermal fluctuations, projected lattice information, elastic constants, etc. as input parameters, solve the phase field equation in the reciprocal space to obtain the atomic occupancy probability value, and then transform it to the real space to draw the atomic evolution diagram . In the process of solving the diffusion equation by computer, if the occupancy probability is greater than 1 or less than 0, the program terminates; if the occupancy probability is a value between 0 and 1, the program continues. After the calculation is completed, the data processing process saves the atomic occupancy probability in the real space in the form of a four-dimensional matrix, and is used to draw the morphology evolution diagram and subsequent atomic occupancy analysis. By analyzing the morphology evolution diagram, the ordering process of the solid solution state, the nucleation, growth and coarsening process of the precipitate phase, and the influence of time, composition, and coupling stress on the microstructure of the precipitate are obtained; further analysis of the morphology evolution diagram , to obtain the interface structure information of the homogeneous and heterogeneous precipitated phases; use the atomic occupancy to quantitatively characterize the atomic diffusion law of the evolution of the precipitated phase, and the atomic composition, distribution and orientation diffusion law at the interface.

The beneficial effect of the present invention is: the method solves the atomic occupancy information by constructing the microcosmic phase field model of the aging forming/diffusion composite process with the microcosmic atomic diffusion equation coupling the temperature field, the concentration field and the macroscopic/microscopic elastic stress field. Then, the atom occupancy information is converted into secondary information such as microstructure evolution, which is used to analyze the influence of temperature field, concentration field and elastic stress field on microstructure evolution and interfacial structure migration in the aging forming/diffusion compounding process. The evolution of tissue morphology expresses the tissue changes in the aging forming/diffusion composite process, the atomic occupancy can quantitatively characterize the microscopic atomic diffusion mechanism of tissue changes, and the tissue responsivity of temperature field, concentration field, and elastic stress field can be reversed to obtain the elimination orientation Coarsening process parameters. This method can reduce experimental trial and error, optimize aging forming/diffusion composite structure, and improve the mechanical properties of the material.

The present invention will be described in detail below in conjunction with the accompanying drawings and specific embodiments.

Description of drawings

Figure 1 is a flow chart of the method of the present invention.

Fig. 2 is a microstructure evolution diagram of the aging forming/diffusion compounding process in the embodiment of the present invention.

Fig. 3 is a graph showing the influence of aging forming/diffusion composite temperature on the microstructure morphology in the embodiment of the present invention.

Fig. 4 is an orientation growth diagram of different compositional structures under the action of macro/micro coupling stress of aging forming/diffusion compounding in an embodiment of the present invention.

Fig. 5 is a structure diagram of two different precipitation phase interfaces of the face-centered cubic structure in the embodiment of the present invention.

Fig. 6 is a diagram showing the distribution and evolution of alloying elements at the heterogeneous domain boundary and its two sides in the embodiment of the present invention.

Detailed ways

The present invention will be described in detail below with reference to FIGS. 1-6.

Step 1. The Langevin form micro-diffusion equation of the two-dimensional micro-phase field model is constructed.

Onsager-type discrete lattice kinetic equations were used to describe alloy precipitation, and mean-field theory was used to calculate free energy functions. It can quantitatively describe the evolution of atomic scale microstructure.

(a) According to the Onsager diffusion equation, the rate of change of the probability is proportional to the thermodynamic driving force, namely:

$\frac{\mathrm{<span\; class="notranslate">\; dP</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; dt</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}\mathrm{<span\; class="notranslate">\; T</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>\frac{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; f</span>}}{<span\; class="notranslate">\; \&PartialD;</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>$

Indicates the probability of A atoms occupying the lattice position r at time t, C ₀ is the average concentration of the matrix, L(rr') is a constant related to the probability of transitioning from the lattice point r' to the unit time; T is the absolute temperature; K _B is the Boltzmann constant; C ₀ is the average concentration of the matrix; F is the total free energy of the system, and the expression of the mean field approximation to the total free energy of the system in the discrete lattice model is:

$\mathrm{<span\; class="notranslate">\; f</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}\mathrm{<span\; class="notranslate">\; V</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; K</span>}}_{\mathrm{<span\; class="notranslate">\; B</span>}}\mathrm{<span\; class="notranslate">\; T</span>}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; ln</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; P</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

V(r-r′) is the effective interaction energy between atoms, which is given by the following formula,

V(rr')=V _AA (rr')+V _BB (rr')-2V _AB (rr') (3)

V _AA is the interaction potential between A atoms, and V _AB is the interaction potential between A atoms and B atoms.

A and B are alloy components.

(b) To simplify the calculation, the face-centered cubic three-dimensional space is projected on the [001] orientation, the equation (1) is in the plane projection of the reciprocal space, and the dynamic equation is:

ξ(k, t) is a random noise term, which satisfies the random term of the fluctuation-dissipation theory.

in,

V ₁ and V ₂ are the effective interaction energies between the first and second nearest neighbor atoms, respectively.

Step 2. Macro/micro coupling elastic stress of aging forming/diffusion compounding.

The microelastic expression describing the lattice mismatch energy of the precipitated phase in the microscopic phase field model is,

$\mathrm{<span\; class="notranslate">\; B</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; e</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 44</span>}}<span\; class="notranslate">\; )</span>}{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 44</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; [</span>\frac{{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; h</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 0.125</span>}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

Where ε ₀ =(a _p -a ₀ )/a ₀ is the lattice lattice mismatch degree of atomic size mismatch caused by the difference in atomic size, and c _ij is the elastic constant of the cubic lattice.

The elastic strain energy caused by external stress has in Fourier space,

A is a constant, a ₀ is the average distance between atoms when the external force is 0, a is the distance between atoms, and a=a ₀ +Δa, Δa is the deformation caused by external stress, E _<100> , E _<001> are [100 ] and the modulus of elasticity on [001], and △a=σa ₀ /E. σ _x is the component of the external force on the x-axis, and σ _y is the component of the external force on the y-axis.

The elastic strain energy (8) caused by mismatch and external stress is introduced into the free energy expression (2) to calculate the coupling stress effect of aging forming and diffusion composite macroscopic stress and microscopic elastic stress introduced by structural evolution.

Step three, equation solving and data processing.

The Euler iterative method was used to solve the microscopic diffusion equation, and the atomic occupancy information was calculated. Using the interatomic potential, thermal fluctuations, projected lattice information, elastic constants, etc. as input parameters, solve the phase field equation in the reciprocal space to obtain the atomic occupancy probability value, and then transform it to the real space to draw the atomic evolution diagram . In the process of solving the diffusion equation by computer, if the occupancy probability is greater than 1 or less than 0, the program terminates; if the occupancy probability is a value between 0 and 1, the program continues. After the calculation is completed, the data processing process saves the atomic occupancy probability in the real space in the form of a four-dimensional matrix, and is used to draw the morphology evolution diagram and subsequent atomic occupancy analysis. By analyzing the morphology evolution diagram, the ordering process of the solid solution state, the nucleation, growth and coarsening process of the precipitate phase, and the influence of time, composition, and coupling stress on the microstructure of the precipitate can be obtained; further analysis of the morphology evolution The interface structure information of the homogeneous and heterogeneous precipitation phase can be obtained; the atomic diffusion law of the evolution of the precipitation phase is quantitatively characterized by atomic occupancy, and the atomic composition, distribution and orientation diffusion law at the interface.

(1) Construction of the microscopic phase field equation: establish a microscopic phase field equation that couples the macroscopic deformation stress and the microscopic elastic mismatch stress, and solve the equation in the reciprocal space. The solution of the equation is the atomic occupancy information in the reciprocal space. The atomic occupancy information in the reciprocal space is Fourier transformed to obtain the atomic occupancy probability value in the real space. The occupancy probability is used to draw the morphology evolution diagram, the atomic occupancy evolution diagram curve, the atomic diffusion, and the raw data of the precipitation phase composition analysis.

(2) Overflow judgment: convert the atomic occupancy information in the reciprocal space into the occupancy information in the real space at intervals of a certain number of time steps (the interval can be omitted, or any integer can be selected to calculate the time steps, generally 1000 steps are selected), The bit probability value performs an overflow judgment. The basis for the judgment is that the occupancy of an atom at a certain grid point cannot be greater than 1 or less than 0. Therefore, when the calculation result is between 0 and 1, the calculation is in line with the actual situation, and the calculation can continue. , and when it is outside the range of 0 to 1, the program terminates, and the parameters need to be modified, and then a new round of calculation is performed.

(3) The value of occupancy probability converted to real space is stored in a four-dimensional matrix of (n, n, t, x), and n, t, x are divided into tables to calculate the spatial scale, time scale, and number of components. Among them, t is positively correlated with time, but there is no specific corresponding relationship. t represents the number of states from the beginning to the end of the calculation, and the number of storage can be selected according to the demand. For example: the total number of calculation steps is 10 0000 steps, if you choose to store data once every calculation step, then t=10 0000, the advantage is that you can observe the slight changes in each calculation step, the disadvantage is that the storage capacity is too large for computer storage Quantity is a huge challenge; if you choose to store data every 1000 steps, then t=10 0000/1000, that is, t=100, which means that this calculation stores data every 1000 steps, and stores 100 sets of data in total. Take the occupancy probability of a certain component in the (n, n, t, x) four-dimensional matrix, draw the shape evolution diagram, and obtain a set of shape evolution diagrams with a size of n×n and a total of t pairs. By analyzing the evolution information of tissue morphology, information such as orientation arrangement, directional coarsening, interface structure and component diffusion can be obtained.

The specific results are as follows:

a. Tissue morphology.

Referring to the time-, temperature-, and composition-related morphology evolution diagrams in Figures 2, 3, and 4, the spatial scale of the evolution diagram is a two-dimensional lattice of 128×128 grid points (256×256 can also be selected). Figure 2 is the evolution diagram of the microstructure over time during the aging forming/diffusion compounding process, and Figure 2(a) is the morphology diagram of the initial stage of aging forming/diffusion compounding, which shows the large-area concentration fluctuation and structure fluctuation at the initial stage, although the structure is not yet clear Recognizable. During the aging forming process, under the action of heat and coupling stress, the atomic movement is violent and the diffusion speed is fast. It is observed that a precipitate phase is precipitated in Figure 2(b), which is represented by A. The face-centered cubic structure of the A precipitate phase is clearly distinguishable . With the prolongation of the aging forming/diffusion compounding process, the A precipitation phase grows and coarsens obviously. In Fig. 2(d), a red precipitate phase begins to precipitate at the boundary of the A precipitate phase, denoted by B, which engulfs the A precipitate phase at the boundary and gradually grows and becomes coarser. There are two kinds of A and B face centers in the system Cubic precipitated phase.

Figure 3 shows the effect of aging forming/diffusion composite temperature on the microstructure morphology when the temperature changes, and Figure 3(a)-(f) are the morphology diagrams when the temperature gradually increases. At low temperature, the precipitation phases with two structures, A and B, fill the whole system, and the particles of the two precipitation phases are fine and dense, and they are independent and dispersed. As the temperature increases, the number of precipitated phases decreases and the particle radius increases. The precipitated phase gradually transitions from single particles at low temperature to interconnected particles at high temperature. The lower the temperature, the greater the undercooling and the more nucleation, forming a dense two-phase region; the higher the temperature, the smaller the undercooling of the crystal nuclei, but long-range diffusion dominates, and the potential for nucleation to overcome The barrier increases with the increase of temperature, and the crystal nuclei are difficult to form at this time. When the temperature is higher, a small amount of disordered phase exists, and the higher the temperature, the larger the proportion of disordered phase.

Figure 4 is a diagram of the orientation growth morphology of different compositional structures under the action of macro/micro coupling stress. With the adjustment of alloy composition, the ratio of A and B precipitated phases changes, and with the change of the proportion of precipitated phases, the orientation and arrangement of precipitated phases also change. When the B precipitates dominate, the A precipitates are unidirectionally oriented, and the orientation is obvious, and the B precipitates are arranged in rafts between the A precipitates with orientation distribution, and the coarsening is severe. As the proportion of A precipitation phase increases, the orientation distribution still exists, but the directionality of the orientation is not obvious, and the orientation growth direction changes from unidirectional growth to [001] and [100] two directions.

b. Interface and placeholder.

Figure 5 is the calculated heterogeneous interface structure of A and B precipitation phases. Figure 5 contains five heterogeneous interfaces, namely (100) _B // (200) _A (shown by arrows A and B), (100) _B //(200) _A 1/2[001] _A , (002) _B //(002) _A 1/2[100] _A and (002) _B //(002) _A (arrows C, D and E shown), (002) _B // (001) _A (shown by arrow F). In the aging forming process, the A and B precipitated phases are coarsened respectively, and the ratio of the two phases changes. The migration law of the heterogeneous interface in this process is that except for (002) _B //(001) _A , the other four ordered domain boundaries Migration can occur. (100) _B //(200) _A and (100) _B //(200) _A 1/2[001] _A remains unchanged before and after migration, while (002) _B //(002 ) _A and (100) _B //(200) _A ·1/2[001] _A appear alternately during the migration process.

Figure 6 is the distribution and evolution diagram of alloying elements on the ordered domain boundary and its two sides. (a), (b), (c), (d) are respectively (100) _B //(200) _A , (100) _B //(200) _A 1/2[001] _A , (002) _B //(002) _A 1/2[100] _A and (002) _B //(002) _A , (002) _B //(001) _A (corresponding to arrows A, C, E, G refers to the ordered domain boundary) and the distribution of alloying elements on both sides. The solid line parallel to the ordinate in the figure indicates the initial position of the ordered domain boundary, and the dotted line indicates the termination position of the ordered domain boundary after migration. As time goes on, although the ordered domain boundary migrates, the segregation and depletion tendency of alloying elements at the ordered domain boundary does not change.

Claims (1)

1. A microscopic phase field modeling and analyzing method of an age forming/diffusion composite process is characterized by comprising the following steps:

step one, constructing a Langevin form microscopic diffusion equation of a two-dimensional microscopic phase field model;

describing alloy precipitation by adopting an Onsager type discrete lattice point form kinetic equation, and calculating a free energy function by adopting an average field theory;

(a) according to the Onsager diffusion equation, the rate of change of probability is proportional to the thermodynamic driving force, namely:

<math> <mrow> <mfrac> <mrow> <mi>dP</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> <mi>dt</mi> </mfrac> <mo>=</mo> <mfrac> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>C</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>K</mi> <mi>B</mi> </msub> <mi>T</mi> </mrow> </mfrac> <msub> <mi>&Sigma;</mi> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> </msub> <mi>L</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>-</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mfrac> <mrow> <mo>&PartialD;</mo> <mi>F</mi> </mrow> <mrow> <mo>&PartialD;</mo> <mi>P</mi> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>,</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </math>

denotes the probability of the A atom occupying lattice position r at time t, C₀L (r-r ') is a constant related to the probability of transition from the lattice point r' per unit time, which is the average concentration of the matrix; t is the absolute temperature; k_BBoltzmann constant; c₀Is the average concentration of the matrix; f is the total free energy of the system, averagedThe field approximation is represented by the total free energy of the system in the discrete lattice point model as:

<math> <mrow> <mi>F</mi> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>&Sigma;</mi> <mi>r</mi> </msub> <msub> <mi>&Sigma;</mi> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> </msub> <mi>V</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>-</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mi>P</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mi>P</mi> <mrow> <mo>(</mo> <msup> <mi>r</mi> <mo>&prime;</mo> </msup> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>K</mi> <mi>B</mi> </msub> <mi>T</mi> <msub> <mi>&Sigma;</mi> <mi>r</mi> </msub> <mo>[</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mi>ln</mi> <mrow> <mo>(</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <mi>P</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>)</mo> </mrow> <mi>ln</mi> <mi>P</mi> <mrow> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> <mo>]</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </math>

v (r-r') is an interatomic effective energy represented by the following formula,

V(r-r′)＝V_AA(r-r′)+V_BB(r-r′)-2V_AB(r-r′) (3)

V_AAis between A atomsInteraction potential, V_ABIs the interaction potential between the A atom and the B atom; A. b is an alloy component;

(b) to simplify the calculation, the three-dimensional space of the face-centered cubic is projected in the [001] orientation, equation (1) is in the planar projection of the reciprocal space, and the kinetic equation is:

zeta (k, t) is a random noise term which satisfies the fluctuation-dissipation theory;

wherein,

V₁，V₂respectively the effective interaction energy between the atoms of the first adjacent neighbor and the next adjacent neighbor;

step two, aging forming/diffusion composite macro/micro coupling elastic stress;

the microscopic elastic expression describing the lattice misfit energy of the precipitated phase in the microscopic phase field model is,

<math> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>e</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>-</mo> <mfrac> <mrow> <mo>-</mo> <mn>4</mn> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mn>11</mn> </msub> <mo>+</mo> <mn>2</mn> <msub> <mi>C</mi> <mn>12</mn> </msub> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>C</mi> <mn>11</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mn>11</mn> </msub> <mo>+</mo> <msub> <mi>C</mi> <mn>12</mn> </msub> <mo>+</mo> <msub> <mrow> <mn>2</mn> <mi>C</mi> </mrow> <mn>44</mn> </msub> <mo>)</mo> </mrow> </mrow> </mfrac> <msubsup> <mi>&epsiv;</mi> <mn>0</mn> <mn>2</mn> </msubsup> <mrow> <mo>(</mo> <msub> <mi>C</mi> <mn>11</mn> </msub> <mo>-</mo> <msub> <mi>C</mi> <mn>12</mn> </msub> <mo>-</mo> <msub> <mrow> <mn>2</mn> <mi>C</mi> </mrow> <mn>44</mn> </msub> <mo>)</mo> </mrow> <mo>[</mo> <mfrac> <mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mrow> <msup> <mi>h</mi> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mo>-</mo> <mn>0.125</mn> <mo>]</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow> </math>

wherein₀＝(a_p-a₀)/a₀Lattice misfit of degree of mismatch of atomic size due to difference of atomic size, c_ijIs the elastic constant of the cubic lattice;

the elastic strain energy caused by external stress is in fourier space,

a is a constant number, a₀Is the average spacing of atoms when the external force is 0, a is the atomic spacing, and a ═ a₀+ Δ a, Δ a being the amount of deformation caused by external stress, E_<100>、E_<001>Is [100]]And [001]]And Δ a ═ σ a₀/E；σ_xIs the fraction of the external force on the x-axisQuantity, σ_yIs the component of the external force in the y-axis;

introducing an elastic strain energy (8) formula caused by mismatching and external stress into a free energy expression (2), and calculating the coupling stress action of the macroscopic stress of age forming and diffusion compounding and the microscopic elastic stress introduced by structure evolution;

solving an equation and processing data;

solving a microscopic diffusion equation by using an Euler iteration method, and calculating to obtain atom occupation information; taking interatomic action potential, thermal fluctuation, projected lattice information, elastic constants and the like as input parameters, solving a phase-field equation in a reciprocal space to obtain an atom occupation rate value, and then transforming the atom occupation rate value to a real space for drawing an atom evolution diagram; solving a diffusion equation process in a computer, and if the occupancy probability is greater than 1 or less than 0, terminating the program; if the occupancy probability is a value between 0 and 1, the process continues; after calculation, the atom occupation probability of the real space is stored in a four-dimensional matrix form in the data processing process and is used for drawing a morphology evolution diagram and subsequent atom occupation analysis; by analyzing the appearance evolution diagram, the solid solution state ordering process, the nucleation, growth and coarsening processes of the precipitation phase and the influence of time, components and coupling stress on the appearance of the micro precipitation structure are obtained; further analyzing the morphology evolution diagram to obtain the interface structure information of the in-phase and out-phase precipitated phases; and (3) quantitatively representing the atomic diffusion rule of precipitated phase evolution, and the atomic composition, distribution and orientation diffusion rule at the interface by using atomic occupation.