A method for determining the percolation threshold of a particle composite material comprising hard particles, soft inclusions and particles of interphase

Abstract

This invention discloses a method for determining the percolation threshold of particle-type composite materials applicable to hard particles, softened inclusions, and those containing mesophases. The method includes: determining the percolation threshold of hard particle systems based on mesophase thickness reduction; determining the percolation threshold of particle systems containing thick mesophases based on Maxwell homogenization, volume-averaged depolarization factor, and scattering tensor; constructing an interpolation function based on the correlation ratio, considering the correlation effects of short-range repulsive fields and long-range scattering fields; and determining the percolation threshold of non-spherical particle systems with mesophases of arbitrary thickness based on the relative thickness parameter of the mesophase and the constructed interpolation function. This invention overcomes the limitations of existing theoretical techniques, which cannot cover a wide range of non-spherical particle systems and cannot quantify the percolation characteristics of materials under the influence of mesophase evolution. It also overcomes the technical bottlenecks of low efficiency and difficulty in guaranteeing accuracy in numerical simulation, enabling the effective determination of percolation phase transitions in drastically different particle-type composite materials.

Description

Technical Field

[0001] This invention belongs to the field of composite material micromechanics and materials engineering technology, and includes theoretical methods such as Bethe lattice percolation model, continuous percolation model, scale theory, Maxwell homogenization, and colloidal interaction field. It relates to the evolution method of hard, softened, and mesophase-containing particle composite materials, and specifically relates to a method for determining the percolation threshold of particle composite materials applicable to hard particles, softened inclusions, and mesophase-containing particles. Background Technology

[0002] In recent years, materials with excellent stiffness, strength, thermal conductivity, and electrical conductivity, such as particle-reinforced materials, carbon fiber materials, ceramic matrix materials, and graphene materials, have been widely used. Therefore, characterizing the structural features of materials and their properties is crucial. Quantifying the correlation between the two experimentally is expensive and time-consuming; thus, reasonable theoretical prediction and explanation are essential. Some theories employ Maxwell's equivalent field, Mori-Tanaka model, differential effective medium theory, and generalized self-consistent models to characterize the mechanical and transport properties of materials. However, these early theories often have certain limitations, such as the assumption of low density, regular inclusion morphology, and perfect integration of inclusions into the matrix. Furthermore, some theories cannot represent the behavior of percolation within the material's internal structure leading to abrupt changes in its properties. In nature, percolation of some hard particles and fibers may be accompanied by the emergence of rigid load-bearing capacity, while percolation of some soft inclusions and colloids may lead to gelation, deposition, and the propagation of pores and cracks. However, the types of percolation in particulate systems are numerous, and experimental and numerical studies often require separate consideration of different systems, making it impossible to summarize them into a unified framework. Therefore, developing a universal method for accurately predicting the percolation threshold of a material's internal structure is a key factor in material design. On the one hand, particles may belong to different types, such as hard, soft, and hard-core-soft-shell parallel structures with intermediate phases; on the other hand, there are uncontrollable interactions between particles.

[0003] Current methods for predicting percolation thresholds neglect the continuity of the spatial configuration evolution of the mesophase and lack a robust and universally applicable method for determining percolation characteristics that can cover both the classical hard-core-soft-shell model and the two extreme cases of hard particle systems and softened inclusion systems. Therefore, it is urgent to establish a conceptually clear, easy-to-operate, and widely applicable method to quantify the mesophase structure information, encompass different types of non-spherical particle composite materials, and determine the percolation threshold of corresponding material types. Summary of the Invention

[0004] Purpose of the invention: In order to overcome the shortcomings of the existing technology, this invention provides a method for determining the percolation threshold of particle-type composite materials with hard particles, softened inclusions and intermediate phases. The method is conceptually clear and easy to operate, and is of great significance for its application and promotion in the field of composite materials.

[0005] Technical Solution: To achieve the above objectives, this invention provides a method for determining the percolation threshold of particle-type composite materials containing hard particles, softened inclusions, and mesophases, comprising the following steps:

[0006] S1: Based on the reduction of the thickness of the intermediate phase, a method for determining the percolation threshold of the hard particle system is given, and the implicit function of the percolation threshold of hard particles in two-phase particle composite materials is obtained.

[0007] S2: Based on Maxwell homogenization scheme, volume-average depolarization factor, and scattering tensor, the self-consistent relationship of particle composite materials is derived by far-field matching. Based on the percolation limit ratio of scattering tensor, a method for determining the percolation threshold of particle systems containing thick mesophase is given, and the implicit function of the percolation threshold of particle systems containing thick mesophase is obtained.

[0008] S3: Consider the correlation between short-range repulsive field and long-range scattering field, and construct an interpolation function based on the correlation ratio;

[0009] S4: Based on the relative thickness parameter of the mesophase and the constructed interpolation function, the interpolation correlation field of the non-spherical particle system is designed, and a method for determining the percolation threshold of the non-spherical particle system with an intermediate phase of arbitrary thickness is given, and the percolation threshold of the non-spherical particle system with an intermediate phase of arbitrary thickness is obtained.

[0010] Further, step S1 specifically includes:

[0011] By utilizing the limit of the intermediate phase thickness t→0 (approaching 0), the average contact number Z of hard particles is obtained. MF , represented as:

[0012]

[0013] Where ρ is the particle number density, g² is the radial distribution function for the hard-core-soft-shell parallel structure, and t is the thickness of the intermediate phase. It is the particle volume fraction. It is the volume fraction of the intermediate phase, M f It is the shape factor of non-spherical particles with smooth and sharp surfaces, g CS For congruent hard spheres, the radial distribution function is given by the classic Carnahan-Starling (CS) model, V. ex ITZ The volume of the intermediate phase in the hard-core-soft-shell parallel structure is affected by repulsion. D is the average curvature diameter of the particle. eq The equivalent diameter is defined as the diameter of an equivalent sphere with the same volume as the non-spherical particle, ss is the sphericity of the non-spherical particle, and γ is the critical index of lattice percolation.

[0014] By utilizing the lattice percolation critical condition, the percolation threshold of hard particles in two-phase particle composite materials was obtained. The implicit function is expressed as:

[0015]

[0016] in, It is the critical occupancy probability of hard particles, which is the critical volume fraction of hard particles under the mean-field model.

[0017] Further, step S2 specifically includes:

[0018] Introducing a three-phase Maxwell far-field equivalent scheme, and based on Maxwell far-field matching, the critical volume fraction of a hard-core-soft-shell parallel structure with a relatively thick mesophase is given. Represented as:

[0019]

[0020] Among them, S c 11 S c 22 ,Sc 33 These are the volume-average depolarization factors along the three principal axes of the non-spherical hard core-soft shell parallel structure.

[0021] The volume fraction of the mesophase Expanding on this, we derive the percolation threshold of the particulate system containing a thick mesophase. The implicit function is expressed as:

[0022]

[0023] Where λ=t / D eq .

[0024] Furthermore, in step S3, a dimensionless parameter d is introduced to construct an interpolation function, specifically expressed as follows:

[0025]

[0026] Where d = t / D eqThe relative thickness of the intermediate phase is the criterion for particle association distance, and u1 and u2 are potential functions. Furthermore, the association between particles also depends on anisotropy; non-spherical particles with an aspect ratio κ far deviating from 1 exhibit more pronounced repulsive effects. The logarithmic form (|lgκ|) is chosen to characterize the influence of κ on anisotropy and repulsive effects. The larger |lgκ| is, the greater the contact probability between particles, and the more pronounced the repulsive association, leading to changes in the interpolation function I. f (e f(κ,d) )Increase.

[0027] Furthermore, the process for obtaining the percolation threshold of the non-spherical particle system with an intermediate phase of arbitrary thickness in step S4 is as follows:

[0028] Based on the relative thickness parameter and interpolation function of the mesophase, an interpolation correlation field is designed for the non-spherical particle system to determine the percolation threshold of the particle system containing a thin mesophase. Percolation threshold of particulate systems containing thick mesophase Interpolation is performed to ultimately give the percolation threshold of a granular system with an intermediate phase of arbitrary thickness. Represented as:

[0029]

[0030] This invention starts with a hard-core-soft-shell structure, and based on the gradual evolution of the shrinkage and expansion of the intermediate phase to two limits: hard particle systems and soft inclusion systems, it reveals the potential correlation between percolation phase transitions and blockage phase transitions, and summarizes the percolation phase transition problem of different particle systems into a generalized model framework. This invention is of great significance for predicting the microstructure characteristics and macroscopic properties of non-spherical particle composite materials.

[0031] This invention systematically solves the percolation phase transition problem under different particle systems, overcomes the limitations of existing theoretical techniques that cannot cover a wide range of non-spherical particle systems and cannot quantify the percolation characteristics of materials under the influence of intermediate phase evolution, and overcomes the technical bottlenecks of low efficiency and difficulty in guaranteeing accuracy in numerical simulation, enabling the percolation phase transition of drastically different particle composite materials to be effectively measured.

[0032] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0033] 1. This invention overcomes the limitations of existing theoretical techniques in fully characterizing the geometric features of the intermediate phase and the percolation characteristics during the spatial configuration evolution process. It provides a continuous, robust, and universal method for determining the percolation threshold for the transition from the classic hard-core-soft-shell model to the two extreme cases of hard particle systems and softened inclusion systems.

[0034] 2. This invention overcomes the technical bottlenecks of low efficiency and difficulty in guaranteeing accuracy in numerical simulation, making the calculation method of global percolation threshold of particle-reinforced materials in different systems more universal and representative. Attached Figure Description

[0035] Figure 1 This is a flowchart of the method of the present invention;

[0036] Figure 2 The graph shows the interpolation function as a function of the relative intermediate phase thickness and aspect ratio.

[0037] Figure 3 Theoretical prediction of the permeation threshold for a completely impenetrable hard particle system;

[0038] Figure 4 Theoretical prediction diagram of percolation threshold for particle systems with different relative mesophase thicknesses (d);

[0039] Figure 5 A graph comparing the relative errors of different theoretical methods for determining the percolation threshold;

[0040] Figure 6 Theoretical prediction diagram for the percolation threshold of softened particulate systems. Detailed Implementation

[0041] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading this invention, any modifications of the invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0042] This invention provides a method for determining the percolation threshold of particle-type composite materials applicable to hard particles, softened inclusions, and particle-type composite materials containing mesophases, such as... Figure 1 As shown, it includes the following steps:

[0043] S1: Based on the reduction of the mesophase thickness, a method for determining the percolation threshold of hard particle systems is presented, obtaining the implicit function of the percolation threshold of hard particles in two-phase particle composite materials:

[0044] By utilizing the limit of the intermediate phase thickness t→0 (approaching 0), the average contact number Z of hard particles is obtained. MF , represented as:

[0045]

[0046] Where ρ is the particle number density, g² is the radial distribution function for the hard-core-soft-shell parallel structure, and t is the thickness of the intermediate phase. It is the particle volume fraction. It is the volume fraction of the intermediate phase, M fIt is the shape factor of non-spherical particles with smooth and sharp surfaces, g CS For congruent hard spheres, the radial distribution function is given by the classic Carnahan-Starling (CS) model, V. ex ITZ is the volume of the intermediate phase repulsion region in the hard-core-soft-shell parallel structure, ī is the average curvature diameter of the particle, and D is the average curvature diameter of the particle. eq The equivalent diameter is defined as the diameter of an equivalent sphere with the same volume as the non-spherical particle, ss is the sphericity of the non-spherical particle, and γ is the critical index of lattice percolation.

[0047] By utilizing the lattice percolation critical condition, the percolation threshold of hard particles in two-phase particle composite materials was obtained. The implicit function is expressed as:

[0048]

[0049] in, It is the critical occupancy probability of hard particles, which is the critical volume fraction of hard particles under the mean-field model.

[0050] S2: Based on the Maxwell homogenization scheme, volume-averaged depolarization factor, and scattering tensor, the self-consistent relationship of particle-type composite materials is derived through far-field matching. Based on the percolation limit ratio of the scattering tensor, a method for determining the percolation threshold of particle systems containing thick mesophases is given, and the implicit function of the percolation threshold of particle systems containing thick mesophases is obtained.

[0051] Introducing a three-phase Maxwell far-field equivalent scheme, and based on Maxwell far-field matching, the critical volume fraction of a hard-core-soft-shell parallel structure with a relatively thick mesophase is given. Represented as:

[0052]

[0053] Among them, Sc 11 ,Sc 22 S c 33 These are the volume-average depolarization factors along the three principal axes of the non-spherical hard core-soft shell parallel structure.

[0054] The volume fraction of the mesophase Expanding on this, we derive the percolation threshold of the particulate system containing a thick mesophase. The implicit function is expressed as:

[0055]

[0056] Where λ=t / D eq .

[0057] S3: Considering the correlation between the short-range repulsive field and the long-range scattering field, construct an interpolation function based on the correlation ratio:

[0058] Introducing a dimensionless parameter d, we construct an interpolation function, specifically expressed as:

[0059]

[0060] Where d = t / D eq The relative thickness of the intermediate phase is the criterion for the correlation distance between particles, and u1 and u2 are potential functions. In addition, the correlation between particles also depends on the influence of anisotropy. Non-spherical particles with an aspect ratio κ that deviates far from 1 have a more obvious repulsive effect.

[0061] like Figure 2 As shown, the logarithmic form (|lgκ|) is chosen to characterize the influence of κ on anisotropy and repulsion. When |lgκ| is larger, the contact probability between particles is greater, the repulsive correlation is more obvious, and the interpolation function I is affected. f (e f(κ,d) )Increase.

[0062] S4: Based on the relative thickness parameter of the mesophase and the constructed interpolation function, an interpolation correlation field for the non-spherical particle system is designed. A method for determining the percolation threshold of the non-spherical particle system with an arbitrary thickness mesophase is given, and the percolation threshold of the non-spherical particle system with an arbitrary thickness mesophase is obtained.

[0063] Based on the relative thickness parameter and interpolation function of the mesophase, an interpolation correlation field is designed for the non-spherical particle system to determine the percolation threshold of the particle system containing a thin mesophase. Percolation threshold of particulate systems containing thick mesophase Interpolation is performed to ultimately give the percolation threshold of a granular system with an intermediate phase of arbitrary thickness. Represented as:

[0064]

[0065] S5: Based on the evolution of mesophase expansion, the percolation threshold of the non-spherical softening particle system is derived:

[0066] Using the limit of d→∞ (approaching infinity), the percolation threshold of softening particle-type composite materials is derived. The function is represented as:

[0067]

[0068] Based on the above scheme, in order to verify the effectiveness of the method of the present invention, a comparative experiment was conducted in this embodiment, and the following simulation data was obtained:

[0069] Figure 3The theoretical value of the percolation threshold of the hard particle system derived by the method of this invention is shown, and compared with existing experimental and numerical results.

[0070] Figure 4 The theoretical values ​​of the percolation threshold of particulate systems with different relative mesophase thicknesses derived by the method of this invention are presented and compared with existing numerical results.

[0071] Figure 6 The theoretical value of the percolation threshold of the softened particulate system derived by the method of this invention is presented and compared with existing numerical results.

[0072] By comparing the theoretical and numerical results for various intermediate phase thicknesses, and combining... Figure 5 The relative errors of various theoretical predictions for different relative mesophase thicknesses shown demonstrate that the method of this invention has high accuracy. It also encompasses different types of non-spherical particle composite materials, addressing the urgent need to determine the percolation threshold of corresponding material types.

[0073] This embodiment also provides a percolation threshold determination system applicable to hard particles, softened inclusions, and particle-type composite materials containing intermediate phases. The system includes a network interface, a memory, and a processor. The network interface is used to receive and send signals during information exchange with other external network elements. The memory is used to store computer program instructions that can run on the processor. The processor is used to execute the steps of the consensus method described above when running the computer program instructions.

[0074] This embodiment also provides a computer storage medium storing a computer program that, when executed by a processor, can implement the methods described above. The computer-readable medium can be considered tangible and non-transitory. Non-limiting examples of non-transitory tangible computer-readable media include non-volatile memory circuitry (e.g., flash memory circuitry, erasable programmable read-only memory circuitry, or masked read-only memory circuitry), volatile memory circuitry (e.g., static random access memory circuitry or dynamic random access memory circuitry), magnetic storage media (e.g., analog or digital magnetic tape or hard disk drive), and optical storage media (e.g., CD, DVD, or Blu-ray disc). The computer program includes processor-executable instructions stored on at least one non-transitory tangible computer-readable medium. The computer program may also include or depend on stored data. The computer program may include a basic input / output system (BIOS) for interacting with the hardware of a dedicated computer, device drivers for interacting with specific devices of the dedicated computer, one or more operating systems, user applications, background services, background applications, etc.

[0075] Those skilled in the art will understand that embodiments of this application can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application can take the form of a computer program product embodied on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0076] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart... Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0077] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0078] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

Claims

1. A method for determining the percolation threshold of particle-type composite materials containing hard particles, softened inclusions, and mesophases, characterized in that, Includes the following steps: S1: Based on the reduction of the thickness of the intermediate phase, a method for determining the percolation threshold of the hard particle system is given, and the implicit function of the percolation threshold of hard particles in two-phase particle composite materials is obtained. S2: Based on Maxwell homogenization scheme, volume-average depolarization factor, and scattering tensor, the self-consistent relationship of particle composite materials is derived by far-field matching. Based on the percolation limit ratio of scattering tensor, a method for determining the percolation threshold of particle systems containing thick mesophase is given, and the implicit function of the percolation threshold of particle systems containing thick mesophase is obtained. S3: Consider the correlation between short-range repulsive field and long-range scattering field, and construct an interpolation function based on the correlation ratio; S4: Based on the relative thickness parameter of the mesophase and the constructed interpolation function, the interpolation correlation field of the non-spherical particle system is designed, and a method for determining the percolation threshold of the non-spherical particle system with an intermediate phase of arbitrary thickness is given, and the percolation threshold of the non-spherical particle system with an intermediate phase of arbitrary thickness is obtained. Step S1 is as follows: Utilizing the thickness of the intermediate phase The limit is to obtain the average number of contacts of hard particles. , represented as: (1)； in, It is the number density of particles. This is the radial distribution function for a hard-core-soft-shell parallel structure, where t is the thickness of the intermediate phase. It is the particle volume fraction. It is the volume fraction of the intermediate phase. It is the shape factor of non-spherical particles with smooth and sharp surfaces. It is the radial distribution function for congruent hard spheres. The volume of the intermediate phase in the hard-core-soft-shell parallel structure is affected by repulsion. It is the average curvature diameter of the particle. The equivalent diameter is defined as the diameter of an equivalent sphere with the same volume as a non-spherical particle. The sphericity of non-spherical particles. It is the critical index for lattice percolation; Using the lattice percolation critical condition, the percolation threshold φ of hard particles in two-phase particle composite materials is obtained. h The implicit function is expressed as: (2)； in, It is the critical occupancy probability of hard particles, which is the critical volume fraction of hard particles under the mean field model. Step S2 is as follows: Introducing a three-phase Maxwell far-field equivalent scheme, and based on Maxwell far-field matching, the critical volume fraction of a hard-core-soft-shell parallel structure with a relatively thick mesophase is given. , represented as: (3)； in, , , These are the volume-average depolarization factors along the three principal axes of the non-spherical hard core-soft shell parallel structure. The volume fraction of the mesophase Expanding on this, we derive the percolation threshold of the particulate system containing a thick mesophase. The implicit function is expressed as: (4)； in, ; In step S3, a dimensionless parameter d is introduced, and an interpolation function is constructed, specifically expressed as: (5)； in, It refers to the relative thickness of the intermediate phase, which is a criterion for the correlation distance between particles. and It is a potential function; choose logarithmic form ( ) characterization The influence of anisotropy and repulsion on the law of action; The process for obtaining the percolation threshold of the non-spherical particle system with an intermediate phase of arbitrary thickness in step S4 is as follows: Based on the relative thickness parameter and interpolation function of the mesophase, an interpolation correlation field is designed for the non-spherical particle system to determine the percolation threshold of the particle system containing a thin mesophase. Percolation threshold of particulate systems containing thick mesophase Interpolation is performed to ultimately give the percolation threshold of a granular system with an intermediate phase of arbitrary thickness. , represented as: (6)。 2. The method for determining the percolation threshold of particle-type composite materials applicable to hard particles, softened inclusions, and mesophases according to claim 1, characterized in that, In step S4, based on formula (6), the following is used: The limit of softening particle-type composite materials is derived. The function is represented as: （7）。

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