# Fractal Characteristics of Porosity of Electrospun Nanofiber Membranes

Authors: Ting Wang; Ying Chen; Wenxia Dong; Yong Liu; Luoyi Shi; Rudong Chen; Tiandi Pan
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the method of measuring the porosity of electrostatic nanofiber membrane by VC++ and Matlab is introduced. It is found that the ratio of the calculated porosity to the porosity measured by the mercury intrusion method accords with the famous Feigenbaum constant (α=2.5029078750957⋯). The porosity distribution of nanofiber membranes was studied by VC++ and Matlab based on the image obtained by using a scanning electron microscope. The porosity distribution calculated by using a computer is magnified by eα times which was named as relative porosity distribution. According to the relative porosity distribution, we use the algorithm proposed by Grassberger and Procaccia (briefly referred to as the G-P algorithm) to calculate the correlation fractal dimension. The correlation fractal dimension calculated from the relative porosity distribution series was between 1 and 2, consistent with geometric characteristics of coincidence samples. The fractal meaning of the Feigenbaum constant was verified again. In the end, we obtained the relationship between the associated fractal dimension and the filtration resistance by fitting in accordance with the secondary function relationship and reached the maximum correlation fractal dimension when the filtration resistance was 15–20 pa.


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## Body

## 1. Introduction

Electrospinning is a technology of drawing polymer solution with high viscosity into fibers by electrostatic force. It can produce nanofibers with high porosity, high surface area, fine and uniform fiber, and other characteristics. Electrospinning has the following advantages: continuous fibers can be prepared directly from polymers and composites; it can be applied to a wide range of polymers; the thickness of membrane can be controlled by adjusting the collection time during the electrospun nanofiber; the dimensions and filtration feature of the electrospun nanofiber can be changed by altering the solution properties and process parameters (see [1]).Most of the pores exist between the interior of the electrospun nanofiber membrane and the fibers. These pores will directly affect the properties of the electrospun nanofiber membrane. At present, the methods of measuring the porosity of the electrospun nanofiber membrane include density method, image method, and mercury intrusion method (see [2–4]). However, these methods are time-consuming and costly. In recent years, more and more attention has been paid to the calculation of the porosity of electrospun nanofiber membrane by using computers (see [5, 6]). The powerful image processing ability and matrix computing ability of computers make the calculation of the porosity of electrospun nanofiber membrane simple, easy, and of low cost.Fractal theory is a nonlinear mathematical theory derived in the middle of 1970s. The definition of fractal given by the founder Mandelbrot is that fractal is a form in which parts are in some way similar to the whole. Based on the complexity of fiber structure in electrospinning nanofiber film, fractal can be used as an ideal mathematical research tool.As early as the 80s of the last century, it was found by Krohn and Thompson that pores have fractal characteristic, and fractal dimension is the best description of fractal feature as the quantitative characterization and basic parameter of fractal (see [7–9]). They all look at the fractal characteristics of the pores of rocks and soils. At the beginning of the 20th century, this theory was applied to the study of electrospun nanofiber membrane. The fractal dimensions used were generally box dimensions, as given by Ai et al. (see [10, 11]). However, although the box counting method is simple and effective, it is very difficult to study the pore characteristics of electrospun nanofiber because of their obvious distribution. According to the characteristics of electrospun nanofibers, we choose the correlation fractal dimension method to study. We use the algorithm proposed by Grassberger and Procaccia to calculate its correlation fractal dimension (see [12]).In this paper, we first give a kind of method of calculating the porosity of electrospinning nanofiber membrane by using computers. Then, according to the calculated porosity distribution sequence, using the G-P algorithm, the correlation fractal dimension of the sample is calculated and the mathematical relationship between the fractal dimension and the filtration resistance is finally discussed.


## 2. Preliminaries

The correlation fractal dimension calculated by the G-P algorithm is as follows:Let us consider the system as follows:(1)xk:k=1,2,…,N,which is converted into a M dimensional Euclidean space, as follows:(2)XM,τ=xn,xn+τ,…,xn+M−1τ,n=1,2,…,N.First, the correlation functionCr is defined as(3)CMr=1NN−1∑i≠jθr−Xi−Xj,where θx is the Heaviside step function defined as(4)θx=1,x>0;0,x≤0.where ⋅ represents the distance between state vectors Xi and Xj in Euclidean space.Whenr is small enough, define a constant DM,r relative to M and r (see [10]):(5)DM,r=dlnCMrdlnr.With the proper selection ofM, we can find the correlation fractal dimension: Select the appropriate value of M; when the double log curve lnCMr−lnr has a linear interval long enough, the sample is considered to have a fractal structure. At this point, the linear slope corresponding to the linear interval is D, that is, the correlation fractal dimension of the sample.The introduction of fractal theory and fractal methodology is of great significance in scientific methodology. Fractals are often defined as a rough or fragmented geometry that can be divided into several parts, and each part is (at least approximately) the reduced shape of the whole, i.e., having self-similar properties. Fractal and chaos are closely related; most of them are the self-organizing system, but their meanings are different. Feigenbaum, a physicist in the United States, studied the universal behavior of complex systems in detail and put forward the standard law and universality of chaotic theory (see [13]). The discovery of Feigenbaum constant (δ=4.669201609103⋯,α=1/δ=2.5029078750957⋯) reflects the order of chaotic evolution and opens the research on the universality of chaos (see [13]).In this paper, the porosity of the electrospun nanofiber membrane calculated by Matlab software is compared with that of the electrospun nanofiber membrane obtained by the mercury intrusion method. It is found that the ratio of porosity between the two is close to the Feigenbaum constant (α=2.5029078750957). This feature, in a sense, reflects the fractal characteristics of the porosity distribution of the electrospun nanofiber membrane. When calculating the porosity of the electrospun nanofiber membrane, the porosity distribution is expanded to eα multiple at first, and then, the correlation fractal dimension of the porosity distribution is calculated by using the G-P algorithm. It is found that the dimension is between 1 and 2, which is more in line with the geometric characteristics.


## 3. Experimental

### 3.1. Experimental Materials and Apparatus

#### 3.1.1. Experimental Materials

Polyvinyl alcohol (PVA) with a relative molecule weight (Mn) of 84000∼89000, alcoholysis degree of 86∼89 mol%, and the average degree of polymerization 1700∼1800 was purchased from Changchun Petrochemical Co., Ltd., Taiwan.


#### 3.1.2. Experimental Apparatus

The experimental apparatuses used were as follows: DXES-01 automatic electrospinning machine, produced by Shanghai Dongxiang Nano Technology Co., Ltd.; TSI8130 Automatic Filtration Tester, from Tsai Sai Central Instrument Trade Co., Ltd. (Beijing); TM3030 Desktop Scanning Electron Microscopy, from Japan Hitachi Agency, Hitachi High-Technologies Nagano Office; and AutoPore IV 9510 high-performance automatic mercury presser, from Micromeritics Instrument Co., Ltd. (Shanghai).


### 3.2. Spinning Solution Preparation

Polyvinyl alcohol was dissolved in distilled water and formulated into a polyvinyl alcohol solution with a concentration of 12% (to avoid excessive parameters, we fixed the concentration). The resulting mixed solution was stirred in a water bath at 80 C for 1 h until a homogeneous solution was formed, and then, it was statically defoamed.


### 3.3. Sample Preparation

At room temperature, the prepared spinning solution was poured into four 5-mL syringes with a needle diameter of 1.2 mm. From previous studies, there are five process parameters for the manufacture of electrospun nanofiber membrane as follows: 90 minutes for the spinning time; 12% for the solution concentration; the needle-to-collector distance (cm) taken as 11, 13, 15, 17, and 19; applied voltage (kV) taken as 15, 18, 20, 23, and 26; flow rate (mL/h) taken as 0.5, 0.7, 1, 1.2, and 1.5. To ensure the generality of the experiment and to avoid too many experiments, we adopted an orthogonal experiment idea and added five more experiments based on orthogonal experiment, as shown in Table1.Table 1
Experiment table.
Number of samples
Needle-to-collector distance (cm)
Applied voltage (kV)
Volume flow rate (mL/h)
The filtration resistance (pa)
Number of samples
Needle-to-collector distance (cm)
Applied voltage (kV)
Volume flow rate (mL/h)
The filtration resistance (pa)
1
11
15
0.5
11.1
16
15
20
1.5
43.1
2
13
15
0.7
22.1
17
17
20
0.5
8.5
3
15
15
1
21
18
19
20
0.7
4.9
4
17
15
1.2
11.1
19
11
23
1.2
30.1
5
19
15
1
8.9
20
13
23
1.5
29
6
19
15
1.5
6.1
21
15
23
0.5
7
7
11
18
0.5
11.5
22
17
23
0.7
5.8
8
11
18
0.7
19.3
23
17
23
1.5
7.3
9
13
18
1
15.6
24
19
23
1
9
10
15
18
1.2
19.8
25
11
26
1.5
15.7
11
17
18
1.5
9
26
13
26
0.5
9.9
12
19
18
0.5
8.1
27
13
26
1
25.9
13
11
20
1
31.2
28
15
26
0.7
8
14
13
20
1.2
27.5
29
17
26
1
5.7
15
15
20
0.7
9
30
19
26
1.2
4.3The filtration resistance of each sample is tested by the TSI8310 automatic filter tester (see Table1).


### 3.4. Morphological Characterization

We used the TM-3030 scanning electron microscope (SEM) of the Technology Research Center of Tianjin Polytechnic University to obtain a SEM image of the gold-sprayed sample.


## 3.1. Experimental Materials and Apparatus

### 3.1.1. Experimental Materials

Polyvinyl alcohol (PVA) with a relative molecule weight (Mn) of 84000∼89000, alcoholysis degree of 86∼89 mol%, and the average degree of polymerization 1700∼1800 was purchased from Changchun Petrochemical Co., Ltd., Taiwan.


### 3.1.2. Experimental Apparatus

The experimental apparatuses used were as follows: DXES-01 automatic electrospinning machine, produced by Shanghai Dongxiang Nano Technology Co., Ltd.; TSI8130 Automatic Filtration Tester, from Tsai Sai Central Instrument Trade Co., Ltd. (Beijing); TM3030 Desktop Scanning Electron Microscopy, from Japan Hitachi Agency, Hitachi High-Technologies Nagano Office; and AutoPore IV 9510 high-performance automatic mercury presser, from Micromeritics Instrument Co., Ltd. (Shanghai).


## 3.1.1. Experimental Materials

Polyvinyl alcohol (PVA) with a relative molecule weight (Mn) of 84000∼89000, alcoholysis degree of 86∼89 mol%, and the average degree of polymerization 1700∼1800 was purchased from Changchun Petrochemical Co., Ltd., Taiwan.


## 3.1.2. Experimental Apparatus

The experimental apparatuses used were as follows: DXES-01 automatic electrospinning machine, produced by Shanghai Dongxiang Nano Technology Co., Ltd.; TSI8130 Automatic Filtration Tester, from Tsai Sai Central Instrument Trade Co., Ltd. (Beijing); TM3030 Desktop Scanning Electron Microscopy, from Japan Hitachi Agency, Hitachi High-Technologies Nagano Office; and AutoPore IV 9510 high-performance automatic mercury presser, from Micromeritics Instrument Co., Ltd. (Shanghai).


## 3.2. Spinning Solution Preparation

Polyvinyl alcohol was dissolved in distilled water and formulated into a polyvinyl alcohol solution with a concentration of 12% (to avoid excessive parameters, we fixed the concentration). The resulting mixed solution was stirred in a water bath at 80 C for 1 h until a homogeneous solution was formed, and then, it was statically defoamed.


## 3.3. Sample Preparation

At room temperature, the prepared spinning solution was poured into four 5-mL syringes with a needle diameter of 1.2 mm. From previous studies, there are five process parameters for the manufacture of electrospun nanofiber membrane as follows: 90 minutes for the spinning time; 12% for the solution concentration; the needle-to-collector distance (cm) taken as 11, 13, 15, 17, and 19; applied voltage (kV) taken as 15, 18, 20, 23, and 26; flow rate (mL/h) taken as 0.5, 0.7, 1, 1.2, and 1.5. To ensure the generality of the experiment and to avoid too many experiments, we adopted an orthogonal experiment idea and added five more experiments based on orthogonal experiment, as shown in Table1.Table 1
Experiment table.
Number of samples
Needle-to-collector distance (cm)
Applied voltage (kV)
Volume flow rate (mL/h)
The filtration resistance (pa)
Number of samples
Needle-to-collector distance (cm)
Applied voltage (kV)
Volume flow rate (mL/h)
The filtration resistance (pa)
1
11
15
0.5
11.1
16
15
20
1.5
43.1
2
13
15
0.7
22.1
17
17
20
0.5
8.5
3
15
15
1
21
18
19
20
0.7
4.9
4
17
15
1.2
11.1
19
11
23
1.2
30.1
5
19
15
1
8.9
20
13
23
1.5
29
6
19
15
1.5
6.1
21
15
23
0.5
7
7
11
18
0.5
11.5
22
17
23
0.7
5.8
8
11
18
0.7
19.3
23
17
23
1.5
7.3
9
13
18
1
15.6
24
19
23
1
9
10
15
18
1.2
19.8
25
11
26
1.5
15.7
11
17
18
1.5
9
26
13
26
0.5
9.9
12
19
18
0.5
8.1
27
13
26
1
25.9
13
11
20
1
31.2
28
15
26
0.7
8
14
13
20
1.2
27.5
29
17
26
1
5.7
15
15
20
0.7
9
30
19
26
1.2
4.3The filtration resistance of each sample is tested by the TSI8310 automatic filter tester (see Table1).


## 3.4. Morphological Characterization

We used the TM-3030 scanning electron microscope (SEM) of the Technology Research Center of Tianjin Polytechnic University to obtain a SEM image of the gold-sprayed sample.


## 4. Results and Discussion

### 4.1. Porosity Measurement

Current methods for measuring material porosity include nitrogen adsorption (see [14]), mercury intrusion (see [4]), impregnation (see [15]), and mass density (see [16]). The pore size of materials with pore size less than 50 nm can be measured by this nitrogen adsorption method, but it is only suitable for ultrafine powder materials. The impregnation method is a suitable test porosity method for liquid permeable nonwovens with low surface tension. However, the PVA used experimentally in this paper is a water-soluble material, so the impregnation method is not suitable for electrospinning nanofibers. Both the mercury intrusion porosimetry method and the mass density method are suitable for testing the porosity of electrospun nanofiber membrane. We tested the porosity with the mercury intrusion porosimetry at the Yanhao material laboratory in Beijing, but the test was difficult and costly, with a sample of 550 CNY. As a result, we tested only five samples: 5, 11, 19, 20, and 28. To solve the problems of high cost testing and difficult testing, a new calculation method is proposed. This method makes use of the powerful image processing ability of VC++ software and the matrix computing ability of Matlab software to process the image of electrospun nanofiber membrane. In this way, we calculated the porosity of electrospun nanofibrous membranes by an economical, simple method.


#### 4.1.1. Threshold Segmentation

The threshold command converts gray or color images to high-contrast black and white images by specify a certain color level as the threshold. All pixels brighter than the threshold are converted to white, and all pixels darker than the threshold are converted to black. Scanning electron microscope image of the electrospun nanofiber membrane was read by VC++ software. Firstly, the image was transformed into a gray image, and the threshold value was adjusted according to the average grayscale value of the image to black and white images (binary images). The specific steps are as follows: running the VC++ program, preprocessing the pictures, image gray processing, binary processing, and obtaining binary black and white image, where the black part represents the nonpore part and the white part represents the pore part.


#### 4.1.2. Calculation Using Matlab Software

For the binary black and white image after segmentation, the pixel value of each position is calculated using Matlab software and the table is generated. The porosity is the ratio of the number of white pixels to the total number of pixels, and the porosity can be calculated from the data from the table. If the porosity of the electrospun nanofiber membrane is recorded asp, then(6)p%=nN×100,where n is the number of white pixels and N is the total number of pixels.


#### 4.1.3. Determination of Threshold

The threshold is actually a “critical point,” meaning that greater than this critical point, the threshold, is white, and the one that is less than this critical point, the threshold, is black, so there are only black and white two cases in the image. The selection of threshold is the key to the conversion of grayscale image to binary image. In this paper, we take85% times of the average gray value of the whole image, rounded, as the image threshold, to the image binarization. The diagram before and after processing is shown in Figure 1.Figure 1
Image before and after processing.
(a)
(b)


#### 4.1.4. Porosity Test Results

The measured results of the porosity of electrospun nanofiber membranes calculated by the mercury intrusion method and image processing are shown in Table2. Table 2 shows that the ratio of porosity calculated by the mercury intrusion method and image processing method is very close to the Feigenbaum constant (α=2.5029078750957⋯).Table 2
Results of porosity of the electrospun nanofiber membrane.
Number of samples
Threshold
Porosity (%)
Ratio
Whole-image average gray value
Image binarization selected threshold
Mercury intrusion method
Image processing
Image selection threshold/whole-image average gray value
Mercury intrusion method/image processing
5
116.357832
98
83.0
33.9032
0.8508
2.4461
11
119.866425
100
66.1
28.2820
0.8509
2.3126
19
106.131277
80
83.5
33.7828
0.8480
2.4717
20
127.836432
115
85
27.9989
0.8527
2.8793
28
115.542275
98
78.8
30.1811
0.8482
2.6109
Average
0.8501
2.5432Of course, to further determine this conclusion, we also selected some samples 1, 6, 7, 21, and 2, with large differences in mean gray values for empirical evidence. When the image threshold is selected, about85% of the average gray value of the entire image is still selected. The experimental and calculated results are shown in Table 3. The set a1 represents the porosity of the electrospun nanofibrous membrane obtained by mercury invasion, and the set a2 represents the porosity of the electrospun nanofibrous membrane calculated by image processing. Then, the ratio of a1 to a2 remains essentially consistent with the Feigenbaum constant.Table 3
Porosity of electrospun nanofiber membrane in contrast experiment.
Number of samples
Threshold
Porosity (%)
Ratio
Whole-image average gray value
Image binarization selected threshold
Mercury intrusion method
Image processing
Image selection threshold/whole-image average gray value
Mercury intrusion method/image processing
1
121.414597
104
83.043
34.0404
0.8565
2.4539
6
126.814396
109
86.123
33.6947
0.8595
2.5560
7
119.745632
102
85.519
35.4594
0.8518
2.4936
21
133.345346
113
86.606
37.5705
0.8549
2.3584
29
131.217394
113
89.107
33.4530
0.8535
2.6636
Average
0.8552
2.5028We believe that the measurement of porosity by the mercury intrusion method and the calculation of porosity by the image processing method should satisfy a certain relationship. The experimental results confirm that the ratio of the two is a Feigenbaum constant, which should be derived from the intrinsic structure of the electrospinning nanofibers with fractal properties, thus making their ratio more directly related to the chaotic constants. It also makes it more convenient for us to calculate the porosity of electrospun nanofibers with computer image technology and Matlab.It is known that with chaos, there are fractals. Only chaos emphasizes dynamic process, while fractal emphasizes geometric structure. In fact, the chaotic attractor is a fractal structure in the phase space, the fractal structure is the product of chaotic dynamics process, and the geometric convergence constant of the attractor is also a fractal dimension. Therefore, finding the physics content of universal constant of chaotic attractor is essentially the same problem as studying physical meaning of fractal dimension. Therefore, after discovering the relationship between the electrospun nanofiber membrane and the Feigenbaum constant, we further use this constant to expand the porosity distribution to the originaleα times (called it the relative porosity distribution) and to study the fractal characteristics of the structure of electrospun nanofiber membrane.


### 4.2. Calculation of the Correlation Fractal Dimension of Porosity Distribution

#### 4.2.1. Calculation of Porosity Distribution

First, the image is processed with gray and binary using VC++ software. Second, the black and white images after binary processing are imported into the Matlab software to calculate the number of gray level of each position of the image and to generate the table. For the image of each sample, the porosity distribution of the image is obtained by measuring the proportion of porosity in the transverse coordinates of the pixel every 10 pixels (or the proportion of points whose gray value is greater than85% of the average gray value on the line). Taking sample 3 as an example, the original image is processed, and the porosity distribution is calculated by Matlab, as shown in Figure 2: 20.8117 26.8470 29.0323 29.5525 28.5120 30.2810 28.6160 26.6389 29.7607 29.7607 29.0323 29.0323 27.5754 27.3673 28.7201 28.3039 26.5349 21.6441 23.3091 28.3039 25.7024 23.9334 29.8647 26.5349 22.2685 28.1998 28.0957 24.6618 34.1311 23.1009 27.3673 26.2227 29.1363 28.8241 25.3902 25.8065 27.9917 26.4308 24.4537 22.7888 30.1769 28.5120 28.1998 27.6795 29.6566 27.7836 29.3444 27.1592 22.1644 26.9511 24.2456 23.6212 26.9511 27.6795 30.6972 29.5525 30.2810 28.6160 19.5630 31.6337 28.3039 33.1946 28.5120 24.4537 31.7378 27.3673 28.8241 29.7607 26.3267 27.7836 29.1363 23.1009 28.5120 29.1363 28.0957 27.1592 29.2404 27.5754 30.5931 27.8876 27.5754 25.1821 26.3267 35.1717 31.9459 29.8647 33.7149 32.0499 27.4714 29.8647 28.5120 36.0042 32.9865 30.6972 33.1946 37.4610 32.7784 31.4256 33.1946 37.2529 35.4839 28.7201 35.3798 35.0676 32.9865 32.4662 32.0499 32.8824 31.2175 34.6514 35.2758 34.4433 32.9865 32.3621 27.5754 30.2810 30.3850 29.6566 28.7201 27.0552 31.0094 37.1488 15.8169 29.3444 24.0375 21.9563 31.6337 30.9053.Figure 2
(a) Grayscale image. (b) Binary image.
(a)
(b)


#### 4.2.2. Calculation of the Correlation Fractal Dimension

After the porosity distribution is calculated, the correlation fractal dimension of porosity distribution is calculated by the G-P algorithm. However, we are not simply using porosity distribution data to calculate correlation fractal dimensions. In the previous section, we introduce a Feigenbaum constant associated with chaos. So, when we calculate the correlation fractal dimension of the porosity distribution, the porosity distribution data are enlarged byeα times and the correlation fractal dimension obtained in this way accords with its geometric characteristics between 1 and 2 (see Table 4).Table 4
Correlation fractal dimension of the porosity distribution.
Number of samples
Correlation fractal dimension
Straight line intercept
Correlation coefficient
F test
Collinear interval left-end point
Collinear interval right-end point
M value
1
1.56139686172
−9.8763242714585
0.97572039
10247.6407
301
557
10
2
1.38839671912
−8.3838786693477
0.96888635
3705.6886
264
384
36
3
1.44812324781
−8.945075864163
0.98810103
15777.7817
278
469
35
4
1.4315907248
−8.3576487968196
0.95253119
1986.5803
261
361
34
5
1.4846835618
−9.4652715877316
0.97047590
6837.0920
387
596
14
6
1.58022407525
−10.491951920014
0.95075924
2934.8740
623
776
23
7
1.33696289787
−8.4468109385468
0.95156398
3536.2426
393
574
15
8
1.55601201630
−9.3009103192696
0.92082875
1407.3325
289
411
17
9
1.62172150467
−9.797105817880
0.94754834
1842.6480
339
442
45
10
1.69049555394
−10.436913933398
0.96850045
7163.9328
239
473
16
11
1.39038706342
−8.1930399219638
0.92011168
1209.3348
280
386
26
12
1.43218883519
−8.8340433166209
0.97796717
7678.9222
242
416
12
13
1.35997149018
−8.2296202827107
0.92136516
2648.0441
249
476
20
14
1.61659512319
−9.391379672001
0.90790099
995.6459
249
351
22
15
1.35311753723
−8.0186328299234
0.94924793
2225.7320
267
387
13
16
1.02169015267
−6.1646549547954
0.94651734
1840.5557
338
443
40
17
1.50027056401
−8.8875693433643
0.92285285
992.8661
308
392
20
18
1.4409252032
−9.0887640173833
0.97236387
7177.6416
330
535
10
19
1.54486251839
−9.1404089326107
0.93143693
1181.9048
302
390
33
20
1.50943945985
−8.8179988551138
0.90004084
819.3718
257
349
20
21
1.39264037832
−9.1939612974754
0.96590318
10339.8107
380
746
11
22
1.32247337693
−8.0590456568434
0.95292453
3906.8002
269
463
10
23
1.41210304077
−8.5260610755409
0.95869219
3202.7729
208
347
9
24
1.55990236576
−9.6005829927966
0.95863387
3267.5858
343
485
15
25
1.66255509007
−10.142526043924
0.94636998
2099.9066
341
461
21
26
1.41933998306
−9.2910837497103
0.97327046
6590.5368
304
486
12
27
1.55758791438
−9.4910847598210
0.98852764
15251.3988
239
417
9
28
1.44789169077
−8.7761030513857
0.9733818
3876.2389
253
360
18
29
1.59986009698
−10.023238276336
0.95830634
4321.0791
356
545
11
30
1.42842868531
−8.9589363493934
0.93127739
1395.7789
453
557
36


### 4.3. Filtration Resistance Test

The TSI8310 automatic filter tester is used to test the filtration resistance of each sample (see Table1).


### 4.4. Relationship between the Correlation Fractal Dimension of Porosity Distribution and the Filtration Resistance of Electrospun Nanofiber Membrane

The porosity of electrospun nanofiber membrane has an extremely important influence on the filtration performance of electrospun nanofiber membrane (see [17]). The porosity distribution of electrospun nanofiber membrane has fractal characteristics, and generally speaking, the complicated microstructure can be studied by fractal theory. So, we had researched that the correlation fractal dimension of porosity is related to the filtration performance. We fitted the experimental data (Table 1) and the calculated correlation fractal dimension (Table 4) by Matlab. It was found that there was a quadratic function relationship between the correlation fractal dimension of porosity distribution and the filtration resistance. Here, the quadratic function relation is(7)y=0.000834818733652098x2+0.0313732097764176x+1.26579722242935.F test calculated that F=12.4165091855022; look at the F test table and know that F0.992,27=6.48851 (see Table 5). Therefore, the quadratic function relationship between the correlation fractal dimension of porosity distribution and the filtration resistance is extremely significant.Table 5
F test table.
Source of variance
Sum of squares
Degrees of freedom
Square
F
Conspicuousness
Regression
SSR
2
SSR/2
12.4165092
Extremely significant
Residual error
SSE
27
SSE/27
Sum
Lyy
29The fitting diagram is shown in Figure3. From Figure 3, we can see that when the filtration resistance is 15–20 pa, the correlation fractal dimension of porosity distribution is maximum.Figure 3
Relation between the correlation fractal dimension and the filtration resistance.


## 4.1. Porosity Measurement

Current methods for measuring material porosity include nitrogen adsorption (see [14]), mercury intrusion (see [4]), impregnation (see [15]), and mass density (see [16]). The pore size of materials with pore size less than 50 nm can be measured by this nitrogen adsorption method, but it is only suitable for ultrafine powder materials. The impregnation method is a suitable test porosity method for liquid permeable nonwovens with low surface tension. However, the PVA used experimentally in this paper is a water-soluble material, so the impregnation method is not suitable for electrospinning nanofibers. Both the mercury intrusion porosimetry method and the mass density method are suitable for testing the porosity of electrospun nanofiber membrane. We tested the porosity with the mercury intrusion porosimetry at the Yanhao material laboratory in Beijing, but the test was difficult and costly, with a sample of 550 CNY. As a result, we tested only five samples: 5, 11, 19, 20, and 28. To solve the problems of high cost testing and difficult testing, a new calculation method is proposed. This method makes use of the powerful image processing ability of VC++ software and the matrix computing ability of Matlab software to process the image of electrospun nanofiber membrane. In this way, we calculated the porosity of electrospun nanofibrous membranes by an economical, simple method.


### 4.1.1. Threshold Segmentation

The threshold command converts gray or color images to high-contrast black and white images by specify a certain color level as the threshold. All pixels brighter than the threshold are converted to white, and all pixels darker than the threshold are converted to black. Scanning electron microscope image of the electrospun nanofiber membrane was read by VC++ software. Firstly, the image was transformed into a gray image, and the threshold value was adjusted according to the average grayscale value of the image to black and white images (binary images). The specific steps are as follows: running the VC++ program, preprocessing the pictures, image gray processing, binary processing, and obtaining binary black and white image, where the black part represents the nonpore part and the white part represents the pore part.


### 4.1.2. Calculation Using Matlab Software

For the binary black and white image after segmentation, the pixel value of each position is calculated using Matlab software and the table is generated. The porosity is the ratio of the number of white pixels to the total number of pixels, and the porosity can be calculated from the data from the table. If the porosity of the electrospun nanofiber membrane is recorded asp, then(6)p%=nN×100,where n is the number of white pixels and N is the total number of pixels.


### 4.1.3. Determination of Threshold

The threshold is actually a “critical point,” meaning that greater than this critical point, the threshold, is white, and the one that is less than this critical point, the threshold, is black, so there are only black and white two cases in the image. The selection of threshold is the key to the conversion of grayscale image to binary image. In this paper, we take85% times of the average gray value of the whole image, rounded, as the image threshold, to the image binarization. The diagram before and after processing is shown in Figure 1.Figure 1
Image before and after processing.
(a)
(b)


### 4.1.4. Porosity Test Results

The measured results of the porosity of electrospun nanofiber membranes calculated by the mercury intrusion method and image processing are shown in Table2. Table 2 shows that the ratio of porosity calculated by the mercury intrusion method and image processing method is very close to the Feigenbaum constant (α=2.5029078750957⋯).Table 2
Results of porosity of the electrospun nanofiber membrane.
Number of samples
Threshold
Porosity (%)
Ratio
Whole-image average gray value
Image binarization selected threshold
Mercury intrusion method
Image processing
Image selection threshold/whole-image average gray value
Mercury intrusion method/image processing
5
116.357832
98
83.0
33.9032
0.8508
2.4461
11
119.866425
100
66.1
28.2820
0.8509
2.3126
19
106.131277
80
83.5
33.7828
0.8480
2.4717
20
127.836432
115
85
27.9989
0.8527
2.8793
28
115.542275
98
78.8
30.1811
0.8482
2.6109
Average
0.8501
2.5432Of course, to further determine this conclusion, we also selected some samples 1, 6, 7, 21, and 2, with large differences in mean gray values for empirical evidence. When the image threshold is selected, about85% of the average gray value of the entire image is still selected. The experimental and calculated results are shown in Table 3. The set a1 represents the porosity of the electrospun nanofibrous membrane obtained by mercury invasion, and the set a2 represents the porosity of the electrospun nanofibrous membrane calculated by image processing. Then, the ratio of a1 to a2 remains essentially consistent with the Feigenbaum constant.Table 3
Porosity of electrospun nanofiber membrane in contrast experiment.
Number of samples
Threshold
Porosity (%)
Ratio
Whole-image average gray value
Image binarization selected threshold
Mercury intrusion method
Image processing
Image selection threshold/whole-image average gray value
Mercury intrusion method/image processing
1
121.414597
104
83.043
34.0404
0.8565
2.4539
6
126.814396
109
86.123
33.6947
0.8595
2.5560
7
119.745632
102
85.519
35.4594
0.8518
2.4936
21
133.345346
113
86.606
37.5705
0.8549
2.3584
29
131.217394
113
89.107
33.4530
0.8535
2.6636
Average
0.8552
2.5028We believe that the measurement of porosity by the mercury intrusion method and the calculation of porosity by the image processing method should satisfy a certain relationship. The experimental results confirm that the ratio of the two is a Feigenbaum constant, which should be derived from the intrinsic structure of the electrospinning nanofibers with fractal properties, thus making their ratio more directly related to the chaotic constants. It also makes it more convenient for us to calculate the porosity of electrospun nanofibers with computer image technology and Matlab.It is known that with chaos, there are fractals. Only chaos emphasizes dynamic process, while fractal emphasizes geometric structure. In fact, the chaotic attractor is a fractal structure in the phase space, the fractal structure is the product of chaotic dynamics process, and the geometric convergence constant of the attractor is also a fractal dimension. Therefore, finding the physics content of universal constant of chaotic attractor is essentially the same problem as studying physical meaning of fractal dimension. Therefore, after discovering the relationship between the electrospun nanofiber membrane and the Feigenbaum constant, we further use this constant to expand the porosity distribution to the originaleα times (called it the relative porosity distribution) and to study the fractal characteristics of the structure of electrospun nanofiber membrane.


## 4.1.1. Threshold Segmentation

The threshold command converts gray or color images to high-contrast black and white images by specify a certain color level as the threshold. All pixels brighter than the threshold are converted to white, and all pixels darker than the threshold are converted to black. Scanning electron microscope image of the electrospun nanofiber membrane was read by VC++ software. Firstly, the image was transformed into a gray image, and the threshold value was adjusted according to the average grayscale value of the image to black and white images (binary images). The specific steps are as follows: running the VC++ program, preprocessing the pictures, image gray processing, binary processing, and obtaining binary black and white image, where the black part represents the nonpore part and the white part represents the pore part.


## 4.1.2. Calculation Using Matlab Software

For the binary black and white image after segmentation, the pixel value of each position is calculated using Matlab software and the table is generated. The porosity is the ratio of the number of white pixels to the total number of pixels, and the porosity can be calculated from the data from the table. If the porosity of the electrospun nanofiber membrane is recorded asp, then(6)p%=nN×100,where n is the number of white pixels and N is the total number of pixels.


## 4.1.3. Determination of Threshold

The threshold is actually a “critical point,” meaning that greater than this critical point, the threshold, is white, and the one that is less than this critical point, the threshold, is black, so there are only black and white two cases in the image. The selection of threshold is the key to the conversion of grayscale image to binary image. In this paper, we take85% times of the average gray value of the whole image, rounded, as the image threshold, to the image binarization. The diagram before and after processing is shown in Figure 1.Figure 1
Image before and after processing.
(a)
(b)


## 4.1.4. Porosity Test Results

The measured results of the porosity of electrospun nanofiber membranes calculated by the mercury intrusion method and image processing are shown in Table2. Table 2 shows that the ratio of porosity calculated by the mercury intrusion method and image processing method is very close to the Feigenbaum constant (α=2.5029078750957⋯).Table 2
Results of porosity of the electrospun nanofiber membrane.
Number of samples
Threshold
Porosity (%)
Ratio
Whole-image average gray value
Image binarization selected threshold
Mercury intrusion method
Image processing
Image selection threshold/whole-image average gray value
Mercury intrusion method/image processing
5
116.357832
98
83.0
33.9032
0.8508
2.4461
11
119.866425
100
66.1
28.2820
0.8509
2.3126
19
106.131277
80
83.5
33.7828
0.8480
2.4717
20
127.836432
115
85
27.9989
0.8527
2.8793
28
115.542275
98
78.8
30.1811
0.8482
2.6109
Average
0.8501
2.5432Of course, to further determine this conclusion, we also selected some samples 1, 6, 7, 21, and 2, with large differences in mean gray values for empirical evidence. When the image threshold is selected, about85% of the average gray value of the entire image is still selected. The experimental and calculated results are shown in Table 3. The set a1 represents the porosity of the electrospun nanofibrous membrane obtained by mercury invasion, and the set a2 represents the porosity of the electrospun nanofibrous membrane calculated by image processing. Then, the ratio of a1 to a2 remains essentially consistent with the Feigenbaum constant.Table 3
Porosity of electrospun nanofiber membrane in contrast experiment.
Number of samples
Threshold
Porosity (%)
Ratio
Whole-image average gray value
Image binarization selected threshold
Mercury intrusion method
Image processing
Image selection threshold/whole-image average gray value
Mercury intrusion method/image processing
1
121.414597
104
83.043
34.0404
0.8565
2.4539
6
126.814396
109
86.123
33.6947
0.8595
2.5560
7
119.745632
102
85.519
35.4594
0.8518
2.4936
21
133.345346
113
86.606
37.5705
0.8549
2.3584
29
131.217394
113
89.107
33.4530
0.8535
2.6636
Average
0.8552
2.5028We believe that the measurement of porosity by the mercury intrusion method and the calculation of porosity by the image processing method should satisfy a certain relationship. The experimental results confirm that the ratio of the two is a Feigenbaum constant, which should be derived from the intrinsic structure of the electrospinning nanofibers with fractal properties, thus making their ratio more directly related to the chaotic constants. It also makes it more convenient for us to calculate the porosity of electrospun nanofibers with computer image technology and Matlab.It is known that with chaos, there are fractals. Only chaos emphasizes dynamic process, while fractal emphasizes geometric structure. In fact, the chaotic attractor is a fractal structure in the phase space, the fractal structure is the product of chaotic dynamics process, and the geometric convergence constant of the attractor is also a fractal dimension. Therefore, finding the physics content of universal constant of chaotic attractor is essentially the same problem as studying physical meaning of fractal dimension. Therefore, after discovering the relationship between the electrospun nanofiber membrane and the Feigenbaum constant, we further use this constant to expand the porosity distribution to the originaleα times (called it the relative porosity distribution) and to study the fractal characteristics of the structure of electrospun nanofiber membrane.


## 4.2. Calculation of the Correlation Fractal Dimension of Porosity Distribution

### 4.2.1. Calculation of Porosity Distribution

First, the image is processed with gray and binary using VC++ software. Second, the black and white images after binary processing are imported into the Matlab software to calculate the number of gray level of each position of the image and to generate the table. For the image of each sample, the porosity distribution of the image is obtained by measuring the proportion of porosity in the transverse coordinates of the pixel every 10 pixels (or the proportion of points whose gray value is greater than85% of the average gray value on the line). Taking sample 3 as an example, the original image is processed, and the porosity distribution is calculated by Matlab, as shown in Figure 2: 20.8117 26.8470 29.0323 29.5525 28.5120 30.2810 28.6160 26.6389 29.7607 29.7607 29.0323 29.0323 27.5754 27.3673 28.7201 28.3039 26.5349 21.6441 23.3091 28.3039 25.7024 23.9334 29.8647 26.5349 22.2685 28.1998 28.0957 24.6618 34.1311 23.1009 27.3673 26.2227 29.1363 28.8241 25.3902 25.8065 27.9917 26.4308 24.4537 22.7888 30.1769 28.5120 28.1998 27.6795 29.6566 27.7836 29.3444 27.1592 22.1644 26.9511 24.2456 23.6212 26.9511 27.6795 30.6972 29.5525 30.2810 28.6160 19.5630 31.6337 28.3039 33.1946 28.5120 24.4537 31.7378 27.3673 28.8241 29.7607 26.3267 27.7836 29.1363 23.1009 28.5120 29.1363 28.0957 27.1592 29.2404 27.5754 30.5931 27.8876 27.5754 25.1821 26.3267 35.1717 31.9459 29.8647 33.7149 32.0499 27.4714 29.8647 28.5120 36.0042 32.9865 30.6972 33.1946 37.4610 32.7784 31.4256 33.1946 37.2529 35.4839 28.7201 35.3798 35.0676 32.9865 32.4662 32.0499 32.8824 31.2175 34.6514 35.2758 34.4433 32.9865 32.3621 27.5754 30.2810 30.3850 29.6566 28.7201 27.0552 31.0094 37.1488 15.8169 29.3444 24.0375 21.9563 31.6337 30.9053.Figure 2
(a) Grayscale image. (b) Binary image.
(a)
(b)


### 4.2.2. Calculation of the Correlation Fractal Dimension

After the porosity distribution is calculated, the correlation fractal dimension of porosity distribution is calculated by the G-P algorithm. However, we are not simply using porosity distribution data to calculate correlation fractal dimensions. In the previous section, we introduce a Feigenbaum constant associated with chaos. So, when we calculate the correlation fractal dimension of the porosity distribution, the porosity distribution data are enlarged byeα times and the correlation fractal dimension obtained in this way accords with its geometric characteristics between 1 and 2 (see Table 4).Table 4
Correlation fractal dimension of the porosity distribution.
Number of samples
Correlation fractal dimension
Straight line intercept
Correlation coefficient
F test
Collinear interval left-end point
Collinear interval right-end point
M value
1
1.56139686172
−9.8763242714585
0.97572039
10247.6407
301
557
10
2
1.38839671912
−8.3838786693477
0.96888635
3705.6886
264
384
36
3
1.44812324781
−8.945075864163
0.98810103
15777.7817
278
469
35
4
1.4315907248
−8.3576487968196
0.95253119
1986.5803
261
361
34
5
1.4846835618
−9.4652715877316
0.97047590
6837.0920
387
596
14
6
1.58022407525
−10.491951920014
0.95075924
2934.8740
623
776
23
7
1.33696289787
−8.4468109385468
0.95156398
3536.2426
393
574
15
8
1.55601201630
−9.3009103192696
0.92082875
1407.3325
289
411
17
9
1.62172150467
−9.797105817880
0.94754834
1842.6480
339
442
45
10
1.69049555394
−10.436913933398
0.96850045
7163.9328
239
473
16
11
1.39038706342
−8.1930399219638
0.92011168
1209.3348
280
386
26
12
1.43218883519
−8.8340433166209
0.97796717
7678.9222
242
416
12
13
1.35997149018
−8.2296202827107
0.92136516
2648.0441
249
476
20
14
1.61659512319
−9.391379672001
0.90790099
995.6459
249
351
22
15
1.35311753723
−8.0186328299234
0.94924793
2225.7320
267
387
13
16
1.02169015267
−6.1646549547954
0.94651734
1840.5557
338
443
40
17
1.50027056401
−8.8875693433643
0.92285285
992.8661
308
392
20
18
1.4409252032
−9.0887640173833
0.97236387
7177.6416
330
535
10
19
1.54486251839
−9.1404089326107
0.93143693
1181.9048
302
390
33
20
1.50943945985
−8.8179988551138
0.90004084
819.3718
257
349
20
21
1.39264037832
−9.1939612974754
0.96590318
10339.8107
380
746
11
22
1.32247337693
−8.0590456568434
0.95292453
3906.8002
269
463
10
23
1.41210304077
−8.5260610755409
0.95869219
3202.7729
208
347
9
24
1.55990236576
−9.6005829927966
0.95863387
3267.5858
343
485
15
25
1.66255509007
−10.142526043924
0.94636998
2099.9066
341
461
21
26
1.41933998306
−9.2910837497103
0.97327046
6590.5368
304
486
12
27
1.55758791438
−9.4910847598210
0.98852764
15251.3988
239
417
9
28
1.44789169077
−8.7761030513857
0.9733818
3876.2389
253
360
18
29
1.59986009698
−10.023238276336
0.95830634
4321.0791
356
545
11
30
1.42842868531
−8.9589363493934
0.93127739
1395.7789
453
557
36


## 4.2.1. Calculation of Porosity Distribution

First, the image is processed with gray and binary using VC++ software. Second, the black and white images after binary processing are imported into the Matlab software to calculate the number of gray level of each position of the image and to generate the table. For the image of each sample, the porosity distribution of the image is obtained by measuring the proportion of porosity in the transverse coordinates of the pixel every 10 pixels (or the proportion of points whose gray value is greater than85% of the average gray value on the line). Taking sample 3 as an example, the original image is processed, and the porosity distribution is calculated by Matlab, as shown in Figure 2: 20.8117 26.8470 29.0323 29.5525 28.5120 30.2810 28.6160 26.6389 29.7607 29.7607 29.0323 29.0323 27.5754 27.3673 28.7201 28.3039 26.5349 21.6441 23.3091 28.3039 25.7024 23.9334 29.8647 26.5349 22.2685 28.1998 28.0957 24.6618 34.1311 23.1009 27.3673 26.2227 29.1363 28.8241 25.3902 25.8065 27.9917 26.4308 24.4537 22.7888 30.1769 28.5120 28.1998 27.6795 29.6566 27.7836 29.3444 27.1592 22.1644 26.9511 24.2456 23.6212 26.9511 27.6795 30.6972 29.5525 30.2810 28.6160 19.5630 31.6337 28.3039 33.1946 28.5120 24.4537 31.7378 27.3673 28.8241 29.7607 26.3267 27.7836 29.1363 23.1009 28.5120 29.1363 28.0957 27.1592 29.2404 27.5754 30.5931 27.8876 27.5754 25.1821 26.3267 35.1717 31.9459 29.8647 33.7149 32.0499 27.4714 29.8647 28.5120 36.0042 32.9865 30.6972 33.1946 37.4610 32.7784 31.4256 33.1946 37.2529 35.4839 28.7201 35.3798 35.0676 32.9865 32.4662 32.0499 32.8824 31.2175 34.6514 35.2758 34.4433 32.9865 32.3621 27.5754 30.2810 30.3850 29.6566 28.7201 27.0552 31.0094 37.1488 15.8169 29.3444 24.0375 21.9563 31.6337 30.9053.Figure 2
(a) Grayscale image. (b) Binary image.
(a)
(b)


## 4.2.2. Calculation of the Correlation Fractal Dimension

After the porosity distribution is calculated, the correlation fractal dimension of porosity distribution is calculated by the G-P algorithm. However, we are not simply using porosity distribution data to calculate correlation fractal dimensions. In the previous section, we introduce a Feigenbaum constant associated with chaos. So, when we calculate the correlation fractal dimension of the porosity distribution, the porosity distribution data are enlarged byeα times and the correlation fractal dimension obtained in this way accords with its geometric characteristics between 1 and 2 (see Table 4).Table 4
Correlation fractal dimension of the porosity distribution.
Number of samples
Correlation fractal dimension
Straight line intercept
Correlation coefficient
F test
Collinear interval left-end point
Collinear interval right-end point
M value
1
1.56139686172
−9.8763242714585
0.97572039
10247.6407
301
557
10
2
1.38839671912
−8.3838786693477
0.96888635
3705.6886
264
384
36
3
1.44812324781
−8.945075864163
0.98810103
15777.7817
278
469
35
4
1.4315907248
−8.3576487968196
0.95253119
1986.5803
261
361
34
5
1.4846835618
−9.4652715877316
0.97047590
6837.0920
387
596
14
6
1.58022407525
−10.491951920014
0.95075924
2934.8740
623
776
23
7
1.33696289787
−8.4468109385468
0.95156398
3536.2426
393
574
15
8
1.55601201630
−9.3009103192696
0.92082875
1407.3325
289
411
17
9
1.62172150467
−9.797105817880
0.94754834
1842.6480
339
442
45
10
1.69049555394
−10.436913933398
0.96850045
7163.9328
239
473
16
11
1.39038706342
−8.1930399219638
0.92011168
1209.3348
280
386
26
12
1.43218883519
−8.8340433166209
0.97796717
7678.9222
242
416
12
13
1.35997149018
−8.2296202827107
0.92136516
2648.0441
249
476
20
14
1.61659512319
−9.391379672001
0.90790099
995.6459
249
351
22
15
1.35311753723
−8.0186328299234
0.94924793
2225.7320
267
387
13
16
1.02169015267
−6.1646549547954
0.94651734
1840.5557
338
443
40
17
1.50027056401
−8.8875693433643
0.92285285
992.8661
308
392
20
18
1.4409252032
−9.0887640173833
0.97236387
7177.6416
330
535
10
19
1.54486251839
−9.1404089326107
0.93143693
1181.9048
302
390
33
20
1.50943945985
−8.8179988551138
0.90004084
819.3718
257
349
20
21
1.39264037832
−9.1939612974754
0.96590318
10339.8107
380
746
11
22
1.32247337693
−8.0590456568434
0.95292453
3906.8002
269
463
10
23
1.41210304077
−8.5260610755409
0.95869219
3202.7729
208
347
9
24
1.55990236576
−9.6005829927966
0.95863387
3267.5858
343
485
15
25
1.66255509007
−10.142526043924
0.94636998
2099.9066
341
461
21
26
1.41933998306
−9.2910837497103
0.97327046
6590.5368
304
486
12
27
1.55758791438
−9.4910847598210
0.98852764
15251.3988
239
417
9
28
1.44789169077
−8.7761030513857
0.9733818
3876.2389
253
360
18
29
1.59986009698
−10.023238276336
0.95830634
4321.0791
356
545
11
30
1.42842868531
−8.9589363493934
0.93127739
1395.7789
453
557
36


## 4.3. Filtration Resistance Test

The TSI8310 automatic filter tester is used to test the filtration resistance of each sample (see Table1).


## 4.4. Relationship between the Correlation Fractal Dimension of Porosity Distribution and the Filtration Resistance of Electrospun Nanofiber Membrane

The porosity of electrospun nanofiber membrane has an extremely important influence on the filtration performance of electrospun nanofiber membrane (see [17]). The porosity distribution of electrospun nanofiber membrane has fractal characteristics, and generally speaking, the complicated microstructure can be studied by fractal theory. So, we had researched that the correlation fractal dimension of porosity is related to the filtration performance. We fitted the experimental data (Table 1) and the calculated correlation fractal dimension (Table 4) by Matlab. It was found that there was a quadratic function relationship between the correlation fractal dimension of porosity distribution and the filtration resistance. Here, the quadratic function relation is(7)y=0.000834818733652098x2+0.0313732097764176x+1.26579722242935.F test calculated that F=12.4165091855022; look at the F test table and know that F0.992,27=6.48851 (see Table 5). Therefore, the quadratic function relationship between the correlation fractal dimension of porosity distribution and the filtration resistance is extremely significant.Table 5
F test table.
Source of variance
Sum of squares
Degrees of freedom
Square
F
Conspicuousness
Regression
SSR
2
SSR/2
12.4165092
Extremely significant
Residual error
SSE
27
SSE/27
Sum
Lyy
29The fitting diagram is shown in Figure3. From Figure 3, we can see that when the filtration resistance is 15–20 pa, the correlation fractal dimension of porosity distribution is maximum.Figure 3
Relation between the correlation fractal dimension and the filtration resistance.


## 5. Conclusion

In this paper, the porosity using the mercury intrusion method and that calculated using the image processing method are compared. It was found that the ratio of porosity between the two was very close to the Feigenbaum constant (α=2.5029078750957⋯ ). Secondly, the correlation fractal dimension of the electrospun nanofibrous membrane treated with the Feigenbaum constant (expansion e times) is between 1 and 2 (more consistent with its geometric characteristics). Finally, through regression analysis, we use Matlab software to prove the relationship between the related fractal dimension and the filtration resistance. The quadratic function relationship between them is very significant. It is known that the correlation fractal dimension is maximum when the filtration resistance is 15–20 pa.Note. This result is similar to the relationship between fractal dimension and the filtration resistance of the spatial distribution of electrospinning nanofibers membrane in reference [18].



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*Source: 2503154-2020-02-24.xml*