1. Introduction
Owing to their excellent self-lubricating properties, oil-containing bearings are widely used in aerospace, microelectromechanical systems, deep-sea exploration, and so on. The self-lubrication of oil-containing bearings is achieved by the oil flow in and out of the porous media. As a kind of oil-containing bearing, bearings with a porous cage realize its self-lubrication through the oil flow circulation inside the micron and submicron pores of the porous cage. As the rotational speed rises, the lubricating oil stored in the porous cage continuously seeps out under the joint effect of centrifugal force, thermal effect, and capillary force to play a lubricating role on the friction surface. When the speed of the cage decreases, the lubricant on the cage surface is drawn back into the porous matrix under the capillary force of the micro-porous channel. In this way, bearing self-lubrication is realized. The pore structure of the porous cage is a key factor affecting the internal fluid properties, which further affects the bearing lubrication characteristics. Therefore, the study of the flow characteristics and flow mechanism inside the porous structure of the cage is a prerequisite for the study of its seepage process and self-lubrication performance.
At present, the lubrication performance of cages is mainly studied by various experimental means. For example, the Lanzhou Institute of Chemical Physics prepared porous polyimide (PI) materials with different pore parameters by the cold-pressing sintering technique and studied their performance of oil storage and tribological behavior [
1]. In addition, the Henan University of Science and Technology [
2], Nanjing University [
3], Inha University in South Korea [
4], and NASA in the United States [
5] have also conducted relevant experimental studies on this topic. The above studies focused on the lubrication characteristics of the material by observing and summarizing the macroscopic patterns of the experiments. Unfortunately, it is difficult to directly obtain the flow process inside porous materials by experimental means, which leads to the inability to explain the lubrication mechanism. The pore structure has a significant impact on the flow characteristics and lubrication performance of the oil inside the bearing cage. Unfortunately, the pore structure inside the cage has a micron, or even submicron, scale and is highly complex, random, and disordered, making the set-up of the simulation model and the flow behavior analysis very difficult. Therefore, the key to investigating the self-lubrication mechanism of porous materials lies in the numerical modeling of the microscopic pore structure and the flow analysis of the internal pores, especially for porous oil-containing cage bearings, whose pore structure is characterized by tortuous, complex, low porosity, and small average pore size. The establishment of a highly restored numerical model similar to the real pore structure has become a prerequisite for the analysis of the lubrication mechanism of the cage.
Scholars have proposed various modeling methods for the microscopic pore structure of porous materials. In general, modeling methods for porous materials can be broadly classified into digital core methods and pore network models. The pore network model is an abstract and simplified model for the complex pore internal structure inside porous materials, which can be regarded as a combination of simple geometric shapes and features. Its basic principle is to divide the pore structure into some combinations of units with different functions. Among them, a throat is used to characterize the narrow pore space, a pore body is used to represent the larger pore space, and a regular geometry is used to characterize the pore body and throat. The structure of the model constructed by this method is simple and can visually represent the parameters of the pore structure (such as pore size and coordination number) inside the material. The disadvantages of the pore network model are also obvious: it differs greatly from the real pore structure, resulting in a low degree of similarity. In addition, it cannot quantitatively analyze the degree of irregularity in porous materials. When the pore morphology of porous materials is complex, the accuracy of the model is greatly reduced, which can easily cause large deviations. In summary, the pore network model provides a means to extract the structural parameters of pores and to study the flow behavior of pore-scale fluids but lacks an effective method to quantitatively describe the degree of irregularity of porous materials. The common modeling methods for pore network models include the maximal ball algorithm (MB), the watershed algorithm (WA), and the medial axis algorithm. In 2019, Todor et al. used the maximal ball algorithm and the watershed algorithm to extract the pore network model [
6]. In addition, Wang combined the maximal ball algorithm and the medial axis algorithm to extract the pore network model from 2D SEM images in 2020 [
7]. In addition, the ball-and-stick model structure employed in the PNM approach is also applied in the field of molecular dynamics analysis, thus simplifying and analyzing the particle simulation for convenience. Similar approaches were utilized in the literature [
8,
9] for the modeling and simulation analysis of a nanoparticle in oligomeric poly(methyl methacrylate) and triblock Janus particles, respectively. Given the complexity of the pore structure with submicronic scale inside the porous cage material, the existing pore network model is not suitable for the modeling of porous cages. However, as the model can easily extract the structural parameters of the porous material, the pore network model can be used as an auxiliary means during pore structure characteristic analysis.
Digital cores are digital models used to characterize the microstructure of porous materials. The current method of modeling the digital cores can be divided into two main categories: image reconstruction and numerical reconstruction. The method of image reconstruction mainly acquires a series of two-dimensional images of materials by imaging devices such as scanning electron microscopy and CT tomography and then constructs three-dimensional models using reconstruction methods such as the slice method. In 1997, Zhu built up a three-dimensional microstructure reconstruction system using stereo technology of a scanning electron microscope [
10]. This method can visually reproduce the material structure, but it is time-consuming and requires high equipment requirements. The principle of the numerical reconstruction method is to construct an image model using mathematical algorithms based on the statistical information of the material image. It mainly includes the simulated annealing method, Gaussian random field method, multi-point statistics (MPS) method, the random seed method, the quartet structure generation set (QSGS) method, etc. For example, Hidajat et al. combined the Gaussian field method and simulated annealing method to construct a porous material model, which extremely improved the modeling efficiency [
11]. However, the Gaussian field method and the simulated annealing method only perform some calculations of flow simulation directly on the reconstructed image and do not extract the available porous material model [
12]. Hajizadeh et al. used the MPS method to achieve the reconstruction of the 3D pore space from 2D training images [
13], which avoided the need for 3D images in the traditional MPS method and thus improved the computational speed. Renmin et al. used the QSGS method to construct a two-dimensional microstructure model of clay and investigated the groundwater percolation in the clay [
14]. The construction process of this method has similarity with the generation process of porous materials, but the models constructed by this method are mostly two-dimensional images, and the original three-dimensional models generated are point data. It is difficult to use this method to build solid models, and it cannot be directly applied to the numerical analysis of multi-physics field coupling. Recently, in the pore-scale modeling of porous oil-containing cages, Yin et al. proposed a modeling method for the 2D/3D structure of porous cages based on the random seed method. A pore structure model, which has a certain degree of similarity in terms of quantitative indicators (porosity and pore size) and qualitative indicators (pore morphology and pore distribution), was obtained [
15]. However, due to the Boolean operation between seeds, cusps appear at the pore edges of the 3D model constructed by Yin et al., which lead to distortion of the pore structure. More importantly, the cusps in the model make the meshing process difficult, resulting in the inability to perform simulation calculations in software. Due to this, the simulation of the single-phase flow inside porous materials mainly uses the lattice Boltzmann (LBM) approach and mainly focuses on the acquisition of absolute permeability [
16].
In summary, the current modeling method is mainly based on a two-dimensional model, and the three-dimensional model has a low model similarity and easy distortion, which is difficult to apply to multi-physics field-coupled finite element simulation software. The analysis of the 3D model is mainly based on the LBM method, and many models and flow field types cannot be analyzed and calculated. For example, the color-model is one common LB model for simulating multiphase flow in porous media, while it is limited to density-matched fluids, and numerical stability issues arise for large viscosity ratios and/or small capillary numbers [
17]. Besides, Fakhari et al. proposed a conservative phase-field lattice-Boltzmann model for ternary fluids, which has shown promising results. However, it reduces the fidelity of the simulations compared with the FD-based approach [
18].
It can be seen that it is necessary to improve the digital core-based modeling method for better meshing and simulation analysis of 3D models. This is the key to researching porous materials. To this end, in this study, according to the manufacturing process of porous materials, a modeling method that combines the QSGS method and the slice method was proposed. The pore boundary structure by the median filtering method was optimized and the pore surface was generated and interpolated by Avizo software using the slicing method. In this way, the tips and distortions of the random seed method were avoided, which made the constructed pore structure closer to the actual morphology of porous materials. According to this method, the meshing of the 3D model of the pore structure and the simulation analysis within the finite element software were realized. On this basis, the pore network model was used to extract multiple sets of parameters for the porous material model, and the generated 3D model was compared and analyzed with the prepared specimens. Finally, the simulation study of the oil dumping by centrifugal force and the capillary oil absorption of porous materials was carried out according to the constructed porous material model. The simulation results were verified with experimental data and then analyzed in combination with the relevant pore structure parameters. The model meshing processing, calculation efficiency, and simulation results reflect the advantages of the modeling method by this paper.
3. Modeling Process
Based on the above modeling idea, slices were extracted from the output of the QSGS method, processed by image filtering and segmentation, and reused as the input of the slice method. Subsequently, MATLAB and Avizo 3D reconstruction software were used to construct the 3D solid model of the pore structure of porous materials. The modeling flow is shown in
Figure 2.
3.1. The QSGS Method
In this paper, the bonding of solid particles during the preparation of porous materials was considered as the growth of the growth phase in the spatial range. Based on this, the quartet structure generation set (QSGS) method was used to generate the three-dimensional scatter data needed to construct a three-dimensional model of porous materials. The QSGS method was first proposed by Wang [
19], which adjusts the growth process of particles of porous materials by introducing the concept of growth probability. Thus, a growth model for porous materials with pores and matrix solid particles was generated.
In this paper, the pore was considered as the nongrowth phase and the matrix solid-phase as the growth phase. The specific process is as follows.
- (1)
Set the dimensions (lx, ly, lz) and porosity n of the microscopic model, spread all the meshes within the set 3D porous model dimensions, and generate the growth phase nodes randomly, with each node representing a solid phase particle.
- (2)
Set the growth probability and growth direction of the growing phase nodes so that the solid particles grow randomly in the three-dimensional space. The growth directions include 6 principal directions (Pd1–Pd6) and 12 angular directions (Pd7–Pd18). Due to the isotropic nature of the porous material, set Pd1–Pd6 to be equal and Pd7–Pd18 to be equal.
- (3)
Iterate over the node coordinates of all growth phases, and assign a random number between 0 and 1 in each growth direction of each node. If this random number is less than the corresponding probability Pdi, add a growth phase node along the corresponding direction.
- (4)
Count the number of the nodes of the growth phase, and calculate the porosity reached during growth to see if the porosity defined is reached; if not, repeat step (3).
- (5)
The growth mechanism of the QSGS method leads to a lower growth probability of the grid edge than that of the internal growth phase. Therefore, remove the model boundary for a specific length range from the original pore model.
Based on the above process, the program for the QSGS method was written in MATLAB, and 3D scattering maps representing the pores and the matrix solid particles were generated. The scatter diagram of the generated pore model is shown in
Figure 3. The data points in the diagram represent the matrix solid particles of the material, and space not covered by the points in the cubic modeling space constitutes the pore structure of the porous material model.
3.2. Slice Extraction and Median Filtering
According to the generated scatter data graphs, these scatter data can be considered as a stack of planar data images along the Z-axis. Based on this idea, the pore structure data were extracted and the slices for subsequent 3D modeling were reorganized using the slice method. The slice method is a popular method used in 3D reconstruction software. The principle is to generate a sufficient number of slices of continuous 3D structural cross-sections by grinding, cutting, and image acquisition of experimental specimens. After that, the intermediate surfaces between the slices are generated by sequential superposition and image processing techniques, finally constructing the resulting 3D structure images. This method requires high accuracy for the relevant experimental equipment. In this paper, the X and Y coordinates of all growth phases in each layer were extracted along the Z-axis, and the binary slice maps were generated based on the obtained coordinate data. The continuity of the pore and matrix solid phases within the generated set of slice maps can be ensured on a three-dimensional scale because the scatter maps are randomly grown in space from the growth phase. Besides, as the object of slicing is not an entity but a binary data point map, the requirement for experimental equipment is avoided. The obtained slice image is shown in
Figure 4a. Furthermore, as can be seen from
Figure 4a, many growth phase nodes failed to grow sufficiently during the generation of the continuous pore structure, resulting in many noise points inside the images obtained by slice extraction. Moreover, the pore slice image in MATLAB was made by stacking particles of the growth phase, and each growth phase was a pixel point. Due to the limitation of the computational speed of the computer, the size of the modeled space set was small and the pixel points constituting the pore space were relatively large. These pixel points were connected, resulting in a stepped pore structure boundary.
Based on the microscopic morphology of porous materials and considering the simplification of calculations in subsequent simulations, tips should be avoided as much as possible when generating meshes for porous models. In this paper, the nodes of the growth phase that fail to grow sufficiently in the image were treated as speckle noise, they were filtered by the median filtering method, and the pore boundaries were smoothed.
Median filtering is a nonlinear signal processing technique developed based on the statistical theory of order, which was first proposed by Turky in 1971. This method is widely used because of its simple algorithm structure, good noise-reduction effect, and better protection of the edge details of the image from blurring. The steps of median filtering are as follows: To begin with, divide an odd-length sampling window in a digital image or sequence. Then, sort the samples in the window by grayscale value. Next, generate the monotonically increasing or monotonically decreasing data sequence. Finally, replace the value of the window centroid with the grayscale value of the middle sample of the sequence. In this way, the filtering of noise is achieved. The median filtering of two-dimensional digital images is specified as follows.
Set the window as a two-dimensional template of size n × n and the grayscale value of the original image as
f(
x,
y), and define the filter window as
A and the grayscale value of the median filtered image as
g(
x,
y); then, the median-filtered output of the filter window is
The window used for filtering varies in its applicable shape depending on the processing object. In this paper, a square window was used considering the shape characteristics of the pore structure. The effect of noise reduction using median filtering is shown in
Figure 4b.
3.3. Watershed Segmentation and Pore Size Control
Taking the filtered slices of the porous material model as samples, the watershed segmentation algorithm was used for pore size segmentation to obtain the average pore size of the holes in the slice images, which facilitated the subsequent resizing of the model to obtain the desired pore size.
The watershed segmentation algorithm is a region-based method whose concept was proposed by Digabel et al. based on mathematical morphology and further developed by Vincent and Soille [
20]. In this paper, the watershed algorithm was used for the segmentation of binary images. Its principle is based on the elevation gradient of natural topography in topography. When rain falls from different locations on the surface, it converges at the local minimum surface of the terrain, forming a connected area called a “catchment basin,” and the junction line of the basin not covered by rain is called a “watershed.” Based on this idea, the watershed algorithm converts the grayscale image into a gradient map and then extracts the boundaries and catchment basins to achieve the segmentation of the image [
21].
The watershed segmentation algorithm can be described as:
(1) The image is processed by the gradient operation to obtain the corresponding gradient map, and let
h(
x,
y) be the processed gradient image; then:
where
grad represents the function of gradient operation.
(2) Let
M1,
M2, …
MR be the local minima of the gradient image,
C(
Mi) be the set of coordinates adjacent to
Mi, and
T(
n) denote the set of coordinates with gradient
h less than
n. Take
Cn(
Mi) as the set of
T(
n) adjacent to the minima point
Mi, i.e.,
Take
C[
n] as the union of each
Cn(
Mi) in the image, representing the connected catchment area covered by rainwater in each minimal area (valley) in the gradient map.
In this way, the process of covering rainwater from low to high in the valley is realized, thus achieving the segmentation process of the image.
The segmented image is shown in
Figure 5.
The area of each segmented region was counted, and the average pore size was calculated by considering them as circular pores:
where
n is the total number of pores segmented in the slice image, Re
solution is the spatial resolution in microns/pixel, and
Si is the area of the
ith pore.
In this paper, the obtained average pore size was compared with the preset value and, thus, the model size that should be set for the subsequent 3D reconstruction was calculated. In this way, we can adjust the average pore size of the model of porous materials. Subsequently, the pixel size of the image was resized using the imresize function to ensure that the pixel of the subsequent 3D model was cubic. This process caused a loss of sharpness in the 2D image, but benefitted the slice method to interpolate the model surface in the subsequent 3D reconstruction, thus optimizing the surface and making it smoother.
3.4. Model Reconstruction and Meshing
Finally, the processed set of pore model slices was imported into Avizo software to generate the intermediate surfaces by the slice method to construct the porous material model. Afterward, the model surface was interpolated to construct the desired porous material model. The pore structure and material matrix structure of the obtained 3D solid model are shown in
Figure 6.
In this paper, the model established in this study was compared with the 3D model constructed by Yin et al. using the random seed method, and the details of both models were enlarged for analysis and comparison, as shown in
Figure 7.
As shown in
Figure 7, the traditional random seed modeling method distributed the spherical pores randomly in the modeling space and constructed the pore model by the Boolean operation between balls. As can be seen from the enlarged details, the pore structure of the random seed model differed from the actual morphology of the porous material, and the intersection of the balls led to the existence of sharp angles in the model, which did not match the actual structure of the porous material and was not conducive to the model meshing and simulation analysis, so the simulation of the 3D model of the porous material was not successfully carried out based on this model. In contrast, the present modeling method adopted the median filtering algorithm and the slice method to optimize the intermediate surface, which successfully avoided the spiking phenomenon caused by the Boolean operation between the spherical pore models. From the figure, we can see that the morphology of the model was more similar to the actual material, the surface was smoother, and the transition of the pores was more natural, which was conducive to the 3D meshing and simulation analysis of the solid model.
The mesh model of the three-dimensional model of porous material was successfully constructed by this method, as shown in
Figure 8. The mesh division was more uniform, and the similarity with the actual shape was higher, which was conducive to the subsequent import of software for simulation analysis.