Material microstructure digital design method based on non-local model

Abstract

The invention provides a material microstructure digital design method based on a non-local model. The material microstructure digital design method can be used for controlling the microstructure and performance of a material. The method comprises the following steps: establishing a fractional order Cahn-Hilliard equation model for describing diffusion and phase separation processes of components in a material; selecting non-local parameters to control the phase distribution characteristics of the material; the fractional order equation is solved through numerical values, and material component space distribution under different non-local parameter values is obtained; constructing a material atom model based on the obtained component distribution; and analyzing the performance of the constructed model in combination with molecular dynamics simulation. By regulating and controlling non-local parameters, materials with different microstructures can be obtained under the same component proportion, accurate control of the microstructures and collaborative optimization of multiple properties such as strength, toughness and conductivity are achieved, and the method is suitable for material design in the fields of electronic packaging, high-performance conductors, aerospace and the like.

Description

Technical Field

[0001] This invention belongs to the field of materials science and engineering technology, and relates to the design and performance control technology of material microstructure, specifically to a digital design method for material microstructure based on a nonlocal model. Background Technology

[0002] With the rapid development of modern industrial technology, the demand for high-performance materials is increasing, especially in key fields such as electronic packaging, aerospace, and new energy, which place higher demands on the precise control of material microstructure and performance optimization. Traditional material design methods mainly rely on experience and trial and error, which are difficult to meet the needs of modern engineering for accurate prediction of material properties and customized design.

[0003] Taking Cu / Ag alloys as an example, in actual production, companies typically need to repeatedly adjust parameters such as melting temperature, cooling rate, and heat treatment processes to optimize the silver phase distribution. The development cycle often lasts for months, resulting in high R&D costs. More importantly, due to a lack of theoretical guidance, even if an ideal microstructure is obtained, it is difficult to guarantee batch-to-batch consistency. This "trial and error" R&D model can no longer meet the rapidly changing market demands and increasingly stringent quality requirements.

[0004] Differential equations, as an important mathematical tool for describing physical processes in materials, are widely used in theoretical modeling and simulation analysis in the field of materials science. While traditional integer-order differential equations are effective in describing the local behavior of materials, they have significant limitations in dealing with complex interactions in material systems, particularly in accurately describing long-range effects and memory properties in materials.

[0005] Fractional differential equations, as a generalization of integer differential equations, can more accurately describe the nonlocal properties and complex diffusion phenomena in materials. Compared with traditional integer-order models, fractional-order models, with their pronounced nonlocal characteristics, can better capture long-range interactions within materials, providing a new theoretical tool for accurately describing the behavior of complex materials. More importantly, fractional-order models can achieve precise control over the microstructure of materials through the adjustment of a single parameter, transforming traditional multi-parameter optimization problems into predictable mathematical designs, significantly improving the efficiency of materials development.

[0006] In recent years, the rapid development of atomic-level manufacturing technology has provided new technical means to transform the microstructures predicted by theoretical models into actual materials. Advanced fabrication technologies such as atomic layer deposition (ALD), molecular beam epitaxy (MBE), and scanning tunneling microscopy manipulation can achieve precise control of material structures at the atomic scale, making the technology chain from theoretical prediction to material preparation more complete.

[0007] There are currently some relevant technical solutions for the control of material microstructure and properties.

[0008] Patent CN116646035A discloses a method and system for modeling two-phase alloys based on a controllable second phase using molecular dynamics. This method creates a three-dimensional array based on a pre-set three-dimensional model. Based on the volume percentage and particle number of the second phase, it uses a random coordinate generation method and a breadth-first search algorithm to determine the distribution of the second phase. Then, it combines the matrix material with the reinforcement geometry model to generate a composite material model. This method can effectively control the content, morphology, and distribution of the second phase, making the model closer to the actual microstructure. However, this technology mainly focuses on static structural design and lacks the ability to predict the dynamic evolution of materials under service conditions. Furthermore, it still requires extensive experiments to verify and optimize design parameters, limiting its engineering implementation efficiency.

[0009] Patent CN117275621A proposes a method for obtaining the thermo-elastic-diffusion behavior of hollow cylindrical structures. This method establishes a fractional-order generalized thermo-elastic-diffusion model, a kinematic model, and a constitutive model. By solving the governing equations through Laplace transform and the eigenvalue method, the distribution patterns of the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained. This method considers the change in thermal conductivity with temperature, providing a more accurate description of the thermo-elastic-diffusion coupling problem. However, this technique mainly targets the material response under specific geometries, lacking guidance value for the design of general material microstructures, and its research on the correlation between the concentration of internal defects and dynamic relaxation behavior is insufficient.

[0010] Patent CN119943169A discloses a method for predicting the defect concentration of amorphous alloys through dynamic relaxation analysis. This method uses a fractional-order viscoelastic model to describe the dynamic relaxation behavior of amorphous alloys, verifies the model through dynamic relaxation experiments, and obtains the evolution of defect concentration. The advantage of this method is that it requires fewer parameters and can effectively reflect the changes in defect concentration during the dynamic relaxation process of amorphous alloys. However, this technology focuses on the macroscopic mechanical behavior analysis of amorphous alloys, lacks systematic guidance for the microstructural control of different types of materials, and is mainly aimed at the later-stage evaluation of material properties, failing to provide forward-looking structural optimization schemes in the design phase.

[0011] Patent CN116189816A discloses a molecular dynamics modeling method for multi-alloy element alloy steel. This method first establishes an iron matrix model, then adds a first type of alloying element through atomic substitution, and a second type of alloying element through random insertion, forming an initial molecular dynamics model. Further, gaps and microstructural cracks in the model are eliminated through energy minimization, simulated melting, cooling, and room temperature aging processes. Finally, nanoindentation simulation is used to verify the model's accuracy. This method can quickly and accurately construct alloy steel models containing alloying elements of arbitrary mass fraction and evaluate their physical and mechanical properties. However, its random atomic insertion method can only guarantee macroscopic uniformity of distribution, lacking precise control over microstructural parameters such as phase interface morphology, phase region size, and connectivity, making it difficult to achieve customized design of the material's microstructure.

[0012] Based on the analysis of existing technologies, current research on the microstructure and performance control of materials still needs improvement in the following aspects:

[0013] First, existing methods for controlling the microstructure of materials lack theoretical predictive capabilities. Traditional methods for material preparation and performance control mainly rely on empirical adjustments of process parameters, such as melting temperature, cooling rate, and heat treatment processes, lacking the ability to theoretically predict and precisely control the microstructure formation process. This "trial and error" R&D approach is time-consuming, costly, and makes it difficult to achieve systematic research and quantitative prediction of the relationship between microstructure and macroscopic properties, resulting in low efficiency in the development of high-performance materials.

[0014] Second, traditional integer-order phase-field models have limitations in describing nonlocal effects and long-range interactions in material systems. While integer-order differential equations are effective in describing local behavior, they struggle to accurately capture the complex phase evolution processes and interfacial behaviors in alloys when dealing with complex interactions within material systems. These limitations lead to insufficient understanding of the formation mechanisms of material microstructures and a lack of effective microstructure design tools, making it difficult to obtain diverse microstructure features while maintaining similar component proportions.

[0015] Third, existing technologies lack effective methods for continuous control of microstructure through a single parameter. Current microstructure control technologies typically require the simultaneous adjustment of multiple process parameters, which are coupled together, making optimization complex and hindering precise and continuous control of the microstructure. In particular, the technical means to obtain different microstructure morphologies under the same composition conditions are limited, and there is a lack of simple and efficient structural design methods, which restricts the customized design and consistent control of material microstructures in mass production. Summary of the Invention

[0016] The purpose of this invention is to propose a digital design method for the microstructure of materials based on a nonlocal model. This method aims to overcome the shortcomings of existing materials preparation methods in terms of precise control over microstructure and to break through the limitations of traditional integer-order models in describing nonlocal effects. It achieves an effective combination of microstructure control and macroscopic performance prediction. This invention is applicable to material systems such as binary metal alloys, and by extending the corresponding model parameters, it can be extended to other material types. It provides new theoretical methods and technical approaches for functional material design, and has broad application value in fields such as electronic packaging, conductive contacts, and aerospace.

[0017] To achieve the above objectives, this invention provides a digital design method for material microstructures based on a nonlocal model, comprising the following steps:

[0018] (1) Establish the fractional Cahn-Hilliard equations for the nonlocal model:

[0019]

[0020] Where u represents the normalized component concentration, t represents time, α is a parameter controlling the nonlocality of the material system, ε is the interface thickness parameter, and f(u) = u 3 -u is the derivative of the double-well energy potential function;

[0021] (2) Select the nonlocality parameter α based on the material microstructure design objectives;

[0022] (3) The three-dimensional spatial distribution of material components is obtained by using a numerical solution method based on the selected nonlocality parameter α;

[0023] (4) Based on the component distribution data, construct an atomic model.

[0024] The method also includes analyzing material properties through molecular dynamics simulations, verifying the effectiveness of the design scheme by comparing the material microstructure design target with the material performance analysis results; if the material performance analysis results do not meet the requirements, the nonlocality parameter α is adjusted and the material atomic model is reconstructed.

[0025] Preferably, the numerical solution method employs the Fourier spectrum method to handle the fractional Laplace operator in spatial discretization and the convex splitting technique to handle nonlinear terms in time discretization.

[0026] Preferably, the nonlocality parameter α is selected from 0.5 to 1.

[0027] Preferably, the nonlocality parameter α is selected from 0.65 to 0.85.

[0028] Preferably, the interface thickness parameter ε ranges from 0.05 to 0.8.

[0029] Preferably, the material is a binary metal alloy.

[0030] Preferably, the method for constructing the atomic model is as follows: when the component concentration u is greater than a preset threshold, a first type of atom is placed; when the component concentration u is less than the preset threshold, a second type of atom is placed, wherein the component concentration u represents the concentration of the first component; the threshold is determined based on the target mass fraction of the second component.

[0031] Preferably, the mass fraction of the second component is 5% to 20%.

[0032] Preferably, the binary metal alloy is selected from any one of Cu / Ag alloy, Al / Cu alloy, and Fe / Ni alloy.

[0033] The beneficial effects of this invention compared to the prior art are as follows:

[0034] (1) Theoretical Prediction and Precise Control of Microstructure: This invention establishes a quantitative design method for the microstructure of materials based on the fractional-order Cahn-Hilliard equation, overcoming the limitations of traditional methods that rely on experience and trial and error in existing technologies. Compared with the static structure design using random coordinates generated in patent CN116646035A, this invention realizes the mathematical prediction of the microstructure formation process through theoretical modeling, transforming the traditional multi-process parameter optimization problem into a predictable digital design, thus shortening the material development cycle.

[0035] (2) Accurate Description of Nonlocal Effects and Modeling Effects of Long-Range Interactions: This invention introduces fractional-order differential operators, overcoming the limitations of traditional integer-order phase-field models in describing nonlocal effects in material systems. Compared to the traditional integer-order thermo-elastic-diffusion model used in patent CN117275621A, the nonlocal model better describes long-range interactions and memory effects in materials. By adjusting the nonlocal parameters, continuously tunable microstructures ranging from uniform distribution to highly aggregated structures can be achieved, expanding the predictive capabilities of traditional models.

[0036] (3) Single-parameter continuous control and simplified structural design: This invention achieves precise control of the material's microstructure through a single nonlocal parameter, solving the complexity of multi-parameter coupled control in existing technologies. Compared to methods such as patent CN116189816A that require simultaneous adjustment of multiple atom insertion and distribution parameters, this invention, by adjusting a single nonlocal parameter, can obtain materials with different microstructural characteristics under the same component ratio, simplifying the traditional multi-parameter optimization problem into single-parameter continuous control, thus improving the controllability of microstructure design and the convenience of engineering implementation.

[0037] In summary, this invention solves the problems mentioned in the background art, such as insufficient prediction of existing material preparation theories, limitations in describing nonlocal effects of traditional models, and complexity of multi-parameter control, through three technical innovations: establishing a fractional-order phase field theoretical model, realizing precise control of microstructure, and establishing a single-parameter design method. It provides materials science with a universal and efficient digital design method and technical tool. Attached Figure Description

[0038] Figure 1 This is a flowchart illustrating the overall technical process of the present invention, showing the technical path from establishing fractional-order Cahn-Hilliard equations, numerical solution, atomic model construction to result analysis.

[0039] Figure 2 This is a schematic diagram of the numerical solution method for the fractional-order Cahn-Hilliard equation of the present invention, which shows the spatial and temporal discretization scheme and its solution process;

[0040] Figure 3 The diagrams show a comparison of the microstructures of Cu / Ag alloys with different nonlocal parameters α according to the present invention, illustrating the three-dimensional atomic distribution under three typical parameter values ​​of α = 0.55, 0.75, and 0.90.

[0041] Figure 4 This is a distribution map of silver atoms after removing copper atoms in one embodiment of the present invention (α=0.55), showing a highly dispersed uniform distribution.

[0042] Figure 5 This is a cross-sectional view of the alloy model center (α = 0.55) in one embodiment of the present invention, showing the two-dimensional projection distribution of the zy plane, zx plane and xy plane;

[0043] Figure 6 This is a distribution map of silver atoms after removing copper atoms in one embodiment of the present invention (α=0.75), showing a moderately aggregated network-like distribution characteristic;

[0044] Figure 7 This is a cross-sectional view of the alloy model center section (α = 0.75) in one embodiment of the present invention, showing the two-dimensional projection distribution of the zy plane, zx plane and xy plane;

[0045] Figure 8 This is a distribution map of silver atoms after removing copper atoms in one embodiment of the present invention (α = 0.90), showing the distribution characteristics of a highly clustered connected network;

[0046] Figure 9 The center section of the alloy model is displayed (α = 0.90), showing the two-dimensional projection distribution of the zy, zx, and xy planes. Detailed Implementation

[0047] The following describes specific embodiments and appendices. Figure 1-9 The technical solution of the present invention will be further described in detail below.

[0048] This invention proposes a digital design method for material microstructures based on nonlocal models, comprising the following steps, forming a technical path from theoretical modeling to performance prediction:

[0049] (1) Establish the fractional Cahn-Hilliard equations for the nonlocal model:

[0050]

[0051] Where u represents the normalized component concentration, α is a nonlocal parameter controlling the nonlocality of the material system, ε is the interface thickness parameter, and f(u) = u 3 -u is the derivative of the double-well energy potential function; the fractional operator (-Δ) is... α It has long-range correlation and can describe the nonlocal characteristics of component diffusion in materials.

[0052] (2) Based on the design objectives of the material's microstructure, select the nonlocal parameter α, with a value ranging from 0.5 to 1. Choose the α value according to the desired phase distribution characteristics: a smaller value (0.55-0.65) is selected when a uniform dispersion is required; a medium value (0.65-0.85) is selected when a network-like connected structure is required; and a larger value (0.85-0.95) is selected when significant phase separation is required. By precisely adjusting the α value, continuous and controllable adjustment of the material's microstructure can be achieved.

[0053] (3) The three-dimensional spatial distribution of material components is obtained by numerical solution method. A three-dimensional computational domain is set, a spatial grid is established, and the initial condition is set as a component distribution with random perturbation. The solution process continues until the system reaches a steady state.

[0054] (4) Based on the component distribution data, construct an atomic model. The continuous component concentration field is converted into a discrete atomic distribution. In the binary alloy system, by adjusting the mapping parameters, the mass fraction of the second component is precisely controlled, generating an atomic model containing atomic type and three-dimensional coordinate information.

[0055] (5) Analyze material properties using molecular dynamics simulation. Minimize the energy of the atomic model, then achieve thermal equilibrium at a set temperature, and finally perform mechanical loading simulation to obtain the material's performance parameters and establish the relationship between microstructure and properties.

[0056] The effectiveness of the design scheme is verified by comparing the design objectives with the simulation results. If the results do not meet the requirements, the nonlocal parameter α can be adjusted and the design redesigned, forming a design-verification-optimization process. The method of this invention achieves precise control of the material's microstructure through a single parameter α, which simplifies the design process and improves the predictability of the design compared to traditional multi-parameter control methods.

[0057] In this embodiment, the numerical solution method employs the Fourier spectral method spatially and the convex splitting technique temporally. This method offers spectral accuracy, accurately handles the long-range correlation characteristics of fractional operators, and boasts advantages in both accuracy and computational efficiency, making it suitable for high-precision numerical solutions of fractional partial differential equations.

[0058] In this embodiment, the preferred range of the nonlocal parameter α is 0.65 to 0.85. Within this range, the material microstructure exhibits better phase distribution characteristics, the phase regions have moderate sizes, and the formed phase distribution structure has a certain degree of connectivity, which is beneficial for load transfer and electron transport, achieving a balanced optimization of mechanical and electrical properties.

[0059] In this embodiment, the interface thickness parameter ε ranges from 0.05 to 0.8. The ε value controls the sharpness of the phase interface and the thickness of the transition region: a smaller ε value results in a sharper phase interface, suitable for studying material systems with clear phase boundaries; a larger ε value results in a more diffuse phase interface, suitable for material systems with continuous compositional changes. The choice of ε value should match the inherent interface width of the material to ensure the physical rationality of the model.

[0060] In this embodiment, the atomic model in the binary alloy system is constructed as follows: when u is greater than a set threshold, a first type of atom is placed; when u is less than the set threshold, a second type of atom is placed. The threshold is determined based on the target mass fraction of the second component. By adjusting the threshold parameter, the volume fraction of each component can be precisely controlled. The generated atomic model data format is compatible with mainstream molecular dynamics software, facilitating subsequent performance calculations.

[0061] In this embodiment, the material is a binary metal alloy, and the mass fraction of the second component is 5% to 20%. The binary metal alloy is selected from any one of Cu / Ag alloy, Al / Cu alloy, and Fe / Ni alloy. By adjusting the corresponding thermodynamic parameters and interfacial energy parameters, the method of the present invention can be effectively applied to different alloy systems.

[0062] When the binary metal alloy is a Cu / Ag alloy, different silver phase distribution morphologies can be obtained by adjusting the nonlocal parameter α. When α = 0.55-0.65, silver atoms are relatively uniformly dispersed in the copper matrix, forming a solid solution strengthening effect; when α = 0.70-0.80, the silver phase forms a network structure with connectivity, achieving a balance between strength and conductivity; when α = 0.85-0.95, the silver phase exhibits an aggregated and separated distribution, forming larger silver-enriched regions, which is beneficial to improving conductivity.

[0063] The material designed using the above method has a microstructure based on a nonlocal model, and the characteristic size of the phase region is tunable in the range of 5 nm to 100 nm. When the nonlocal parameter α is between 0.65 and 0.85, the second phase exhibits a network-like distribution regulated by nonlocal effects.

[0064] This invention is applicable to the microstructure design and performance control of binary metal alloys and other material systems. In existing technologies, the microstructure of binary metal alloys is mainly controlled by process parameters such as melting temperature and cooling rate, making it difficult to accurately predict and control the distribution of the second phase. Often, a large number of experiments are required to obtain the ideal microstructure. This invention establishes a digital design method for the microstructure of materials based on a nonlocal model. By adjusting the nonlocal parameter α, the microstructure can be continuously and controllably modulated, overcoming the limitations of existing technologies that rely on experience and trial and error. The implementation process of the technical solution is illustrated below using a Cu / Ag alloy as the main embodiment, but the application scope of this invention is not limited to this.

[0065] Example 1: A Digital Method for the Microstructure of Cu / Ag Alloys Based on Nonlocal Models

[0066] like Figure 1 As shown, the overall technical process of this invention includes five main steps: establishing a fractional-order model, parameter selection, numerical solution, microstructure construction, and performance analysis, forming a complete technical path from theory to practical application.

[0067] This invention employs the fractional-order Cahn-Hilliard equation to describe the diffusion and phase separation processes of components in Cu / Ag alloys. The equation is expressed as:

[0068]

[0069] Where u represents the normalized component concentration (for Cu / Ag alloys, u = 1 represents pure silver, u = -1 represents pure copper), α is a nonlocal parameter that controls the degree of nonlocality of the system, ε is the interface width parameter that determines the thickness of the phase interface, and f(u) = u 3 -u is the derivative of the double-well energy potential function.

[0070] The degree of nonlocality and memory effect of the Cu / Ag alloy system can be tuned by varying the nonlocality α within the range of (0.5,1). When α approaches 1 / 2, the system exhibits stronger nonlocality and memory effect, the phase interface is more blurred, and the composition distribution is more uniform. When α approaches 1, the equation approximates the traditional integer-order Cahn-Hilliard equation, and the system exhibits more obvious phase separation, forming a larger single-phase region.

[0071] This invention addresses the solution of fractional-order Cahn-Hilliard equations by employing the Fourier spectral method combined with convex splitting techniques for spatial discretization, and utilizes a semi-implicit time discretization scheme to ensure computational stability and accuracy. Figure 2 As shown, the computational process of the Fourier spectral method includes transforming the initial component distribution to the frequency domain using a fast Fourier transform, calculating the fractional Laplace term in the frequency domain, returning to the spatial domain using an inverse transform, calculating the nonlinear term, and updating the solution.

[0072] First, using the Fourier decomposition method, we define the expansion of u(x) in the Fourier spectral space:

[0073]

[0074] in, These are the Fourier coefficients of u, and

[0075] i is the imaginary unit, x = (x1, x2, x3) is the coordinate variable in the three-dimensional physical space, and k, l, m correspond to the wave number in the x1, x2, and x3 directions, respectively.

[0076] The fractional Laplace operator is defined as (-Δ). α u:

[0077]

[0078] The computational domain is set as a cube with side length L = 4π, interface width ε = 0.5, and spatially discretized into a 120 × 120 × 120 grid, with each grid cell having a length of L / N. The total simulation time is T, the time step is τ = 0.01, and the physical time t corresponding to the nth time step is... n =nτ, (time step number n = 1, 2, ..., T / τ),

[0079] In terms of time discretization, let ω = ε 2 (-Δ)αu+f(u), using a semi-implicit scheme to improve computational efficiency and stability:

[0080]

[0081] Using the convex splitting technique, the nonlinear term Λ(u) is obtained.n+1 The semi-implicit format of )

[0082]

[0083] In the Fourier spectral space, the discrete form of the equation is:

[0084]

[0085] Here and They are u n+1 and Λ(u n+1 The Fourier coefficients in the frequency domain Fourier space.

[0086] The nonlocal parameter α plays a crucial role in the fractional-order Cahn-Hilliard equations, representing not only the initial state of the system but also determining the evolution path of phase separation dynamics. In nonlocal models, the influence of α is amplified and complicated due to the long-range transfer and memory effects of information during system evolution caused by the nonlocal derivative. Even small changes in the value of α can lead to significant differences in the final state of the system in nonlocal models; this high sensitivity is not present in integer-order models.

[0087] To investigate the effect of parameter α on the microstructure and mechanical properties of the alloy, a series of different parameters α were set. The initial condition was set as: u(x,y,z,0)=u0+η(x,y,z), where u(x,y,z,0) is the velocity at the initial time t=0, u0 is a constant, and η(x,y,z) is a Gaussian random perturbation with a mean of 0 and a standard deviation of 0.01, used to trigger the phase separation process. To accurately obtain the effect of phase distribution on alloy properties, the percentage of Ag atoms in the Cu / Ag alloy was set to 10%. This non-monotonic mapping relationship stems from the nonlinear characteristics of the fractional-order Cahn-Hilliard equation, resulting in a complex correspondence between the initial parameter α and the final composition distribution after phase separation.

[0088] By numerically solving the fractional-order Cahn-Hilliard equations, the spatial distributions of alloy components under different α values ​​were obtained. These distributions were then used to construct an atomic model of the Cu / Ag alloy. Specifically, the continuous component distribution u(x,y,z) was mapped to a discrete atomic type distribution.

[0089] Using this method, three Cu / Ag alloy models corresponding to different α values ​​were constructed. Each model contains approximately 500,000 atoms and has a size of approximately [missing information]. like Figure 3As shown, Cu / Ag alloys with three different α values ​​(0.55, 0.75, and 0.90) are represented, all with an Ag atom content of 10%. The figure reveals significantly different microstructural features as the α value changes.

[0090] (1) When α=0.55, the phase interface is relatively blurred, and Ag atoms are distributed in fine particles in the Cu matrix;

[0091] (3) When α=0.75, a network-like interconnected phase distribution structure is formed, the phase region size and distribution reach a better balance, and Ag atoms form an interconnected network structure;

[0092] (5) When α=0.90, the phase region is highly aggregated, forming a large single-phase region with a clear phase interface. Ag atoms form separate clusters with almost no connection between the clusters.

[0093] This invention achieves the technical effect of obtaining different microstructural characteristics under the same component ratio by adjusting the fractional-order parameter α. This is because the nonlocal derivative alters the kinetic behavior of atomic diffusion in the alloy. Lower α values ​​(0.55-0.65) result in a shorter "memory" of the system for previous states, promoting random atomic diffusion and mixing; higher α values ​​(0.85-0.95) enhance the "memory" effect of the system for historical states, promoting the aggregation of similar atoms and phase separation; while moderate α values ​​(0.70-0.80) form a network-like interconnected structure, achieving optimal balance between phases. This method of precisely controlling the microstructure of materials through fractional-order parameters overcomes the limitations of traditional integer-order models in describing the nonlocal effects of materials.

[0094] Example 2: Design of Dispersed Microstructures with Nonlocal Parameter α = 0.55

[0095] Using the basic method of Example 1, the nonlocal parameter is set to α = 0.55, a value corresponding to a strong nonlocal effect state. For example... Figure 4 As shown, when α = 0.55, silver atoms exhibit a highly dispersed distribution in the copper matrix. The distribution of silver atoms after removing copper atoms shows that silver atoms mainly exist as single atoms or small clusters of several atoms, with cluster sizes less than 1 nanometer, and are uniformly distributed in three-dimensional space. Figure 5 The three orthogonal cross-sectional images shown demonstrate that the silver atom distribution exhibits a clear isotropic characteristic, achieving a relatively uniform dispersion at the 10-nanometer scale.

[0096] To achieve this dispersed microstructure, the following theoretical framework guides the design of the corresponding process: Based on the characteristics of the Cu-Ag phase diagram, the high solid solubility of silver in copper at high temperatures is utilized, employing a solution-controlled precipitation process. Specifically, homogenization treatment can be performed at temperatures above the Cu-Ag eutectic temperature to ensure sufficient silver atom dissolution; subsequently, a supersaturated solid solution is obtained through appropriate cooling processes to avoid significant phase separation during cooling; finally, low-temperature aging treatment is used to control precipitation kinetics, allowing silver atoms to precipitate in a fine, dispersed form. The core of this process lies in utilizing the metastable characteristics of the supersaturated solid solution, achieving refined dispersion of the silver phase through precise control of aging temperature and time.

[0097] Based on theoretical analysis of the dispersed microstructure, this structure is expected to have the following performance characteristics: Due to the highly dispersed distribution of silver atoms, it mainly functions through solid solution strengthening and grain refinement mechanisms, resulting in high yield strength and good plastic deformation capacity. In terms of electrical conductivity, although the silver atoms are relatively dispersed, considering the excellent conductivity of silver, it can still improve the conductivity of the matrix to a certain extent. This microstructure, through a uniform strengthening effect, is particularly suitable for structural material applications that require a balance between strength and toughness. In the aerospace field, it can be used to manufacture conductive structural components that require high strength and good formability.

[0098] Example 3: Design of a network-type microstructure with nonlocal parameter α = 0.75

[0099] Setting the nonlocal parameter to α = 0.75, which falls within the preferred range of 0.65 to 0.85 for nonlocal parameters, corresponds to the intermediate state where diffusion and phase separation reach equilibrium. For example... Figure 6 As shown, silver atoms form a medium-sized interconnected network structure, and the silver phase exhibits a network distribution regulated by nonlocal effects. Compared to the dispersed state at α = 0.55, silver atoms begin to aggregate to form clusters with sizes ranging from 3 to 8 nanometers, and connections appear between these clusters to form a three-dimensional network. Figure 7 The cross-sectional view shown indicates that the silver phase has a certain degree of connectivity in all directions, forming conductive channels.

[0100] To achieve this network-like microstructure, a process design principle based on phase separation kinetics can be adopted: appropriate cooling conditions are selected to allow silver atoms sufficient diffusion time to form medium-sized second-phase particles; the holding temperature should be chosen based on the variation of silver solid solubility with temperature in the Cu-Ag phase diagram, promoting silver phase precipitation while controlling its growth rate within this temperature range; the holding time needs to consider the diffusion kinetics of silver atoms, finding a balance between promoting moderate second-phase growth and avoiding excessive coarsening. The key to this process lies in the rational matching of heat treatment parameters to enable the second phase to form a three-dimensional network structure during precipitation, growth, and bonding.

[0101] Based on the characteristic analysis of the network-type microstructure, this structure is expected to have the following performance advantages: the network-like silver phase distribution can provide a good strengthening effect through the dispersion strengthening mechanism; the interfacial strengthening effect between the second phase and the matrix helps to improve the material strength; at the same time, the continuity of the network structure ensures good plastic deformation capacity, achieving a good balance between strength and toughness; the three-dimensional connectivity of the silver phase is conducive to the formation of effective conductive pathways, and it is expected to achieve synergistic optimization of strength, toughness, and conductivity; this microstructure achieves a unity of structural and functional performance, and is suitable for applications that require both certain mechanical strength and good conductivity. In the field of electronic packaging, it can be used to manufacture conductive substrates for high-density integrated circuits, and in high-performance conductor applications, it can be used to manufacture conductive components for high current transmission.

[0102] Example 4: Design of Aggregated Microstructures with Nonlocal Parameter α = 0.90

[0103] Setting the nonlocal parameter to α = 0.90 corresponds to the state where phase separation mechanism dominates. For example... Figure 8 As shown, silver atoms form a highly interconnected three-dimensional network structure, with the size of the silver phase region significantly increasing to 10 to 20 nanometers, forming a continuous sheet-like or band-like distribution. Figure 9 The cross-sectional view shown indicates that the silver phase is continuously distributed in all directions, demonstrating the microstructure control capability of this invention at the micrometer scale.

[0104] To achieve this aggregated microstructure, a corresponding process can be designed based on phase coarsening theory: a heat treatment process favorable to second-phase growth is employed, providing sufficient diffusion time for silver atoms through a relatively slow cooling process; a long-term holding treatment is performed within an appropriate temperature range, determined based on the precipitation and stability characteristics of the silver phase in the Cu-Ag phase diagram, utilizing the Ostwald ripening mechanism to promote the coarsening and growth of the second phase; by extending the holding time and increasing the holding temperature, the diffusion ability of silver atoms is enhanced, promoting the dissolution of small particles and the growth of large particles, ultimately forming a large and highly interconnected silver phase network. The core of this process is to utilize the synergistic effect of thermodynamic driving force and diffusion kinetics to achieve directional coarsening of the second phase.

[0105] Based on theoretical analysis of the aggregated microstructure, this structure is expected to exhibit the following performance characteristics: the highly interconnected silver phase network can form continuous three-dimensional conductive channels, significantly improving the conductivity of the material; the interface strengthening effect of the second phase can still provide a certain mechanical strengthening effect. Although the overall strength may be relatively low, due to the good connectivity of the second phase, the material can maintain good integrity during deformation and exhibit good toughness; this microstructure exhibits outstanding conductivity and is suitable for applications such as electronic packaging with high conductivity requirements. In the field of high-frequency electronic devices, it can be used to manufacture conductive connection components, and in high-power electronic devices, it can be used to manufacture heat dissipation conductive components.

Claims

1. A digital design method for material microstructures based on a nonlocal model, characterized in that, Includes the following steps: (1) Establish the fractional Cahn-Hilliard equations for the nonlocal model: Where u represents the normalized component concentration, t represents time, α is a parameter controlling the nonlocality of the material system, ε is the interface thickness parameter, and f(u) = u 3 -u is the derivative of the double-well energy potential function; (2) Select the nonlocality parameter α based on the material microstructure design objectives; (3) The three-dimensional spatial distribution of material components is obtained by using a numerical solution method based on the selected nonlocality parameter α; (4) Based on the component distribution data, construct an atomic model.

2. The digital design method for material microstructure based on a nonlocal model according to claim 1, characterized in that, The method also includes analyzing material properties through molecular dynamics simulations, verifying the effectiveness of the design scheme by comparing the material microstructure design target with the material performance analysis results; if the material performance analysis results do not meet the requirements, the nonlocality parameter α is adjusted and the material atomic model is reconstructed.

3. The digital design method for material microstructure based on a nonlocal model according to claim 1, characterized in that, The numerical solution method employs the Fourier spectrum method to handle the fractional Laplace operator in spatial discretization and the convex splitting technique to handle nonlinear terms in time discretization.

4. The digital design method for material microstructure based on a nonlocal model according to claim 1, characterized in that, The nonlocality parameter α is selected from 0.5 to 1.

5. The digital design method for material microstructure based on a nonlocal model according to claim 1, characterized in that, The nonlocality parameter α is selected from 0.65 to 0.

85.

6. The digital design method for material microstructure based on a nonlocal model according to claim 1, characterized in that, The interface thickness parameter ε ranges from 0.05 to 0.

8.

7. The digital design method for material microstructure based on a nonlocal model according to any one of claims 1-6, characterized in that, The material is a binary metal alloy.

8. The digital design method for material microstructure based on a nonlocal model according to claim 7, characterized in that, The method for constructing the atomic model is as follows: when the component concentration u is greater than a preset threshold, a first type of atom is placed; when the component concentration u is less than the preset threshold, a second type of atom is placed, wherein the component concentration u represents the concentration of the first component; the threshold is determined based on the target mass fraction of the second component.

9. The digital design method for material microstructure based on a nonlocal model according to claim 7, characterized in that, The mass fraction of the second group of elements is 5% to 20%.

10. The digital design method for material microstructure based on a nonlocal model according to claim 9, characterized in that, The binary metal alloy is selected from any one of Cu / Ag alloy, Al / Cu alloy, and Fe / Ni alloy.

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