This paper applies the Gaussian random field to the mathematical modeling of new-type silicate glass-ceramic trachoma detection. The article established a three-dimensional numerical model of trachoma structure based on the anisotropic random field. Then, the open and closed operations in mathematical morphology are used to obtain the strongly connected boundaries of the new-type silicate glass-ceramic trachoma image. At the same time, the connected domain detection in binary morphology is used to remove the noise to obtain the target image of the silicate glass-ceramic pore. The study found that this method can better meet the requirements of silicate glass-ceramic trachoma measurement than the classic edge detection operator.

#### Keywords

- Silicate glass-ceramics
- Porous media
- Material modeling
- Ceramic trachoma
- Mathematical morphology

#### MSC 2010

- 60G15

The mechanical properties of silicate glass-ceramics are closely related to the pore structure characteristics. The matrix and pore network characteristics at the micro or nanoscale determine the mechanical response of the material at the macro scale. Therefore, studying the three-dimensional pore structure of silicate glass ceramics helps to understand its characteristics more thoroughly. This research focuses on constructing a three-dimensional pore structure model of silicate glass ceramics that meets the material pore characteristics (porosity, pore size distribution) and has a random morphology [1]. The article adopts the migration set theory to transform the continuously distributed Gaussian random field into a two-phase field with random shapes to represent the matrix and pores of the material. The model is constructed jointly by multiple two-phase fields. At the same time, we use mathematical morphology image processing methods for analysis. The results verified that the method constructed a pore structure that satisfies the pore size distribution curve of the material test.

We set a threshold for the continuously distributed Gaussian random field. After the two-phase treatment, it can simulate the pore network and matrix part with random morphology in the porous medium [2]. For a Gaussian random field

_{c}

We set a threshold _{s}

If the one-dimensional case is extended to three-dimensional and the offset set _{s}_{s}

The size of the defined threshold _{s}_{s}

The correlation length can control the fluctuation of the random field pattern. Therefore, when the correlation length is reduced, the range of points with correlation in the field becomes smaller. The random field's fluctuation increases and each phase's shape becomes narrow after thresholding [5]. Different pore structure models can be generated by adjusting the values of the two parameters, the threshold and the appropriate length (Figure 2). The larger correlation length and threshold make the pores larger and fewer, and the smaller correlation length and threshold make the pores smaller and more.

The pore size distribution range of silicate glass ceramics is large, and the span can be from the nanometer scale to the centimeter scale. After thresholding, a single Gaussian random field can only simulate the pores within a limited range near the relative length _{c}

Due to the overlap of the pores after the union, the total integral number of the collection is not equal to the sum of the volume fractions of the single offset set (compared to small):

Assume that the pore size of a medium with total porosity of _{1} is mainly distributed in three ranges from large to small. The volume fractions are _{1} = 25%_{t}_{2} = 30%_{t}_{3} = 45%_{t}

There have been many research results on the porosity characteristics of silicate slurry, such as porosity and pore size distribution. The mercury intrusion method can only study the pore size distribution of open pores, and the characteristics of closed pores rely on direct scanning technology [8]. The pore size distribution of the silicate slurry used in the modeling in this study is shown in Figure 3. The pore size of our silicate slurry is mainly distributed in the range of 0.02~0.20μm. The porosity is 16.1%.

The model is built in a cube with a side length 3

Related parameters of silicate slurry pore structure modeling (

NO. | Correlation length Lc/μm | Set porosity φ/% | Threshold t | Model porosity φ’/ % |
---|---|---|---|---|

1 | 0. 02 | 2. 0 | 1. 991 | 2.29 |

2 | 0.04 | 4 | 1. 701 | 4. 47 |

3 | 0.06 | 2.5 | 1. 926 | 2.59 |

4 | 0.08 | 3.5 | 1.793 | 3.63 |

5 | 0. 10 | 3 | 1.876 | 3. 18 |

6 | 0.2 | 1.1 | 2. 289 | 1. 34 |

Under the premise of satisfying the porosity, the pore size distribution of the model needs to be discussed. In this study, the mathematical morphology image processing method was used to screen the model of pore network size.

Mathematical morphology is the theory and technology of analyzing and processing geometric structures based on lattice theory and topology. It is mainly used for image processing. The calculation process transforms the initial image into a new image by interacting with a “probe” of a certain shape and size. The size of the structure element controls the result of the operation. There is no theoretical limit to the choice of the shape of the structural element. A space surface forms a network of pores. The structural element is defined as an octahedron similar to a sphere. The basic operations of mathematical morphology include corrosion, expansion, opening operations, and closing operations [9]. In this study, the pore size selection used the open operation.

The opening operation is a combination of the two basic steps of corrosion and expansion. Define a structure element of size

M is the image to be processed. E is a structural element. Open operation. ⊗ is the corrosion operation. ⊕ is the expansion operation. The original image whose size is less than or equal to the structural element will be eliminated after the open operation is processed. We regard the pore network in the built silicate slurry pore structure model as the image to be processed. We perform a series of opening operations with gradually increasing sizes of structural elements, and the remaining pores in the model will gradually decrease [10]. After we perform the open operation of the structural element of 50Hill on the generated model, the pores with a size less than or equal to 50nm will be eliminated. Only the pores with a size larger than 110nm are retained after the opening operation of the structural element of 110nm is performed on the generated model.

In the range of 0.02~0.20

We took out a certain 5000nmx5000nm square unit from the FIB/SEM image and compared it with a certain two-dimensional cross-section of 3000nmx3000HE intercepted in the model (Figure 7). The pore network shape and distribution of the model have strong randomness. The characteristics of the random field determine this. If you take a two-dimensional cross-section arbitrarily from the model, the pore distribution in different cross-sections is quite different. Some two-dimensional cross-sections cut to sections with fewer large pores will appear as large pores [12]. Some two-dimensional cross-sections are not intercepted with fewer large pores but are dominated by small pores. Therefore, the pore size distribution in the two-dimensional section cannot correspond to the test pore size distribution curve. Still, the overall three-dimensional pore size satisfies the test pore size distribution after morphological calculation with octahedron as the structural unit. In addition, because this method randomizes the shape and distribution of the pore network, the inversion of the pore distribution curve can not show some natural characteristics of the material pores and natural micro-cracks. This affects the degree of similarity between the model and the real pore network to a certain extent.

This paper uses Gaussian random field migration set theory and mathematical morphology image processing method to complete and verify

#### Related parameters of silicate slurry pore structure modeling (W / C = 0.64).

NO. | Correlation length Lc/μm | Set porosity φ/% | Threshold t | Model porosity φ’/ % |
---|---|---|---|---|

1 | 0. 02 | 2. 0 | 1. 991 | 2.29 |

2 | 0.04 | 4 | 1. 701 | 4. 47 |

3 | 0.06 | 2.5 | 1. 926 | 2.59 |

4 | 0.08 | 3.5 | 1.793 | 3.63 |

5 | 0. 10 | 3 | 1.876 | 3. 18 |

6 | 0.2 | 1.1 | 2. 289 | 1. 34 |

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