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Zeitschrift
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2444-8656
Erstveröffentlichung
01 Jan 2016
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# Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics

###### Akzeptiert: 28 Mar 2022
Zeitschriftendaten
Format
Zeitschrift
eISSN
2444-8656
Erstveröffentlichung
01 Jan 2016
Erscheinungsweise
2 Hefte pro Jahr
Sprachen
Englisch
Introduction

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.

Modeling method
Gaussian Random Field and Migration Set Theory

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 f(x) that satisfies the standard normal distribution, the covariance equation C(h) of two random points in the field can be defined by the correlation length: $C(h)=exp(−h2Lc2)$ C\left(h \right) = \exp \left({- {{{h^2}} \over {L_c^2}}} \right)

h represents the spatial distance between two points. Lc stands for correlation length [3]. The correlation length represents the minimum distance that makes the correlation between any two points in the field. Therefore, the larger the correlation length of the random field, the larger the range of points with correlation. As shown in Figure 1, the large correlation length makes the field shape smooth, and the small correlation length makes the field shape undulating.

We set a threshold t for the random field f (x) in the range K. The offset set Es is definedas the part of f (x) greater than t: $EsΔ__{x∈K|f(x)≥t}$ {E_s}\underline{\underline \Delta} \left\{{x \in K|f\left(x \right) \ge t} \right\}

If the one-dimensional case is extended to three-dimensional and the offset set Es in the random field is regarded as one phase, the other part except Es is regarded as another phase. After thresholding, the three-dimensional continuous field is transformed into a discrete two-phase field.

The size of the defined threshold t controls the relative proportion of the two phases to play an important role in the shape of the two-phase field after thresholding. Assume that the offset set Es represents the pores in the porous medium, and the rest of the random field represents the matrix [4]. The decrease in the threshold t corresponds to the increase in the proportion of Es and the increase in porosity. Conversely, an increase in the threshold t reduces the porosity. The relationship between pore φ and threshold t can be expressed as: $t=2erfinv(1−2φ)$ t = \sqrt 2 erfinv\left({1 - 2\varphi} \right)

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.

Union of offset sets

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 Lc. The selected threshold only determines the volume fraction of pores in this range [6]. To build a model that satisfies the material porosity and pore size distribution, we need to combine multiple offset sets with different correlation lengths and thresholds. That is, n(n > 1) independent offset sets are combined to form a collection $Es∪$ E_s^ \cup expressed as: $Es∪=∪k−1nEsk$ E_s^ \cup = \bigcup\nolimits_{k - 1}^n {E_s^k}

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): $φ(Es∪)=φ(Es1∪Es2)=φ(Es1)+φ(Es2)−φ(Es1∩Es2)$ \varphi \left({E_s^ \cup} \right) = \varphi \left({E_s^1\bigcup {E_s^2}} \right) = \varphi \left({E_s^1} \right) + \varphi \left({E_s^2} \right) - \varphi \left({E_s^1\bigcap {E_s^2}} \right)

Assume that the pore size of a medium with total porosity of φ1 is mainly distributed in three ranges from large to small. The volume fractions are φ1 = 25%φt, φ2 = 30%φt and φ3 = 45%φt respectively. Due to the overlap effect in the collaborative process, part of the small-sized pores will merge into the large-sized pores [7]. This makes the volume fraction of pores in a part of the size range in the collection different from actual ones. Therefore, the volume fraction of part of the size range needs to be corrected. The corrected values $φ1new$ \varphi _1^{new} , $φ2new$ \varphi _2^{new} and $φ3new$ \varphi _3^{new} are as follows: $φ1new=φ1=25%φt$ \varphi _1^{new} = {\varphi _1} = 25\% {\varphi _t} $φ2new=φ21−φ1=30%φt1−25%φt$ \varphi _2^{new} = {{{\varphi _2}} \over {1 - {\varphi _1}}} = {{30\% {\varphi _t}} \over {1 - 25\% {\varphi _t}}} $φ3new=φ31−φ1−φ2=45%φt1−55%φt$ \varphi _3^{new} = {{{\varphi _3}} \over {1 - {\varphi _1} - {\varphi _2}}} = {{45\% {\varphi _t}} \over {1 - 55\% {\varphi _t}}}

Modeling and verification
Material selection

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%.

Modeling

The model is built in a cube with a side length 3 μm. The grid is divided into 300×300×300, and the minimum element size is 0.01 μm. Therefore, the main pore portion (0.02–0.20 μm) of the silicate slurry of W / C = 0.64 can be present in the cube. We analyze the cumulative pore size distribution curve in Figure 3. 6 correlation lengths are 0.02μm, 0.04μm, 0.06μm, 0.08μm, 0.10μm, 0.20μm respectively. The porosity set in each field is 2.0%, 4.0%, 2.5%, 3.5%, 3.0%, 1.1%, respectively. The total porosity is 16.1%. The porosities of the generated fields are 2.29%, 4.47%, 2.59%, 3.63%, 3.18%, and 1.34%, respectively. The combined model has a total porosity of 16.3%. The relevant parameters are shown in Table 1, and the generated model is shown in Figure 4.

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
Construction method and model verification

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 d. The image to be processed is shown in the irregular shape part in Figure 5(a). The interaction between the image and the structural elements in the erosion operation can be simply understood as shrinking the image along the boundary by the same size as the width of d / 2. The part of the original image with a size smaller than d will be eliminated (Figure 5(b). The image expansion after the erosion process can be understood that the image is extended by the width of d / 2 along the boundary of the same size, and the part that is reduced and retained in the erosion operation can be reconstructed (Figure 5(c)). The process of opening operation can be defined as: $M∘ E=(M⊗E)⊗E$ M \circ \,E = \left({M \otimes E} \right) \otimes E

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 μm of the main pore size of the model, several structural element sizes are selected in order from small to large for opening calculation. The model's cumulative pore size distribution curve can be obtained by calculating the residual porosity after each calculation [11]. We compare it with the experimental data (Figure 6). The comparative analysis shows that the constructed three-dimensional pore structure model of W / C = 0.64 silicate slurry can reduce the pore size distribution.

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.

Conclusion

This paper uses Gaussian random field migration set theory and mathematical morphology image processing method to complete and verify W / C = 0.64 the three-dimensional pore structure model of silicate slurry. This method has better control of the pore size distribution to ensure that the porosity of the model is consistent with the true value. At the same time, this method can present the randomness of the pore network's three-dimensional shape and spatial distribution. The combination of finer meshes or more two-phase fields with different correlation lengths can make the pore size distribution of the model more accurately meet the test pore size distribution curve. This method is also suitable for modeling other types of porous media that do not consider natural defects in the material.

#### 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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