This article was submitted to Polymer Chemistry, a section of the journal Frontiers in Chemistry
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Polymer microspheres (PMs) are a kind of selfsimilar volume expansion particle, and their fractal dimension varies with hydration swelling. However, there is no unique fractal dimension calculation method for their characteristics. A new model is established in this paper, which is particular to calculate the fractal dimension of PMs. We carried out swelling hydration experiments and scanning electron microscope (SEM) experiments to verify the new model. The new model and the boxcounting model were used to calculate the fractal dimensions of PMs based on the hydration experiment results. Then, a comparison of the calculation results of the two methods was used to verify the validity of the model. Finally, according to the new model calculation results, the fractal dimension characteristics of PMs were analyzed. The research results indicate that the new model successfully correlates the cumulative probability of the PMs dispersed system with the fractal dimension and makes fractal dimension calculation of PMs more accurate and convenient. Based on the experiment results, the new model was used to calculate the fractal dimension of PMs and the boxcounting model, and its findings were all 2.638 at initial state hydration and 2.739 and 2.741 at hydration time as of day 1. This result verifies the correctness of the new model. According to the hydration swelling experiments and the new model calculation results, the fractal dimension is linear correlated to the average particle size of PMs and the standard deviation average particle size. This means the fractal dimension of PMs represents the space occupancy ability and space occupancy effectiveness.
Polymer microspheres (PMs) are spherical polymer composite materials with diameters ranging from nanometers to micrometers, high specific surface area, high reactivity, and other unique physical, chemical, biological properties (
Among its many application fields, oil development is one of the most important application fields (
In 1977, Mandelbrot proposed a fractal dimension theory that described the selfsimilarity, nature fracture, and irregular structures (
In this paper, to calculate PMs’ fractal dimension more accurately and conveniently, a new model was established, which successfully correlates the cumulative probability of the PMs dispersed system with the fractal dimension based on the selfsimilarity theory. We carried out swelling hydration experiments and scanning electron microscope (SEM) experiments to verify the new model and obtained the data needed to calculate the fractal dimension of PMs. Then, the new model and the conventional calculation model (box counting model) were used to calculate the fractal dimension of PMs at different hydration times. The calculation results verify the correctness of the new model. Finally, according to the calculation results of the new model, the fractal dimension characteristics of PMs were analyzed.
There are many popular kinds of fractal dimensions, as stated by definition. According to the fractal geometry theory, for the statistically selfsimilar system of the PMs, an essential relation of fractal scaling law between microsphere accumulated number
Therefore, the slope
The fractal dimension can be calculated using the curve of
From
The negative sign in
The microsphere size distribution is based on dynamic scattering.
We need to find the microsphere diameter corresponding to the microsphere’s maximum numbers to obtain accurate fractal dimension values. We define it as a maximum probability density diameter (MPDD) for the probability distribution of particle size larger than MPDD, obeying a monotonical decreasing rule.
The accumulated number, corresponding to a microsphere size that is less than the MPDD, can be considered integration to replace the original minimum diameter
Replace
The box counting model is one of the commonly used methods to calculate the fractal dimension of the PMs dispersion system. To validate the cumulative probability model, we also use the boxcounting method to calculate the fractal dimension of the same PMs.
The fractal dimension
Bohai Oilfield provides the PMs used in the experiment. Its main component was polymer and synthesized by reverse phase emulsion and reversephase suspension. The physical and chemical properties of this PMs indicated that it was an environmentally friendly PMs. It was nontoxic and noncorrosive and was kept in white oil with an acceptable solid content of 20%. The anhydrous ethanol (CH_{3}CH_{2}OH, purity above 99.5%) was used as the dispersion medium to make a uniform dispersion solution of PMs in the initial state. The simulated formation water was used as the solvent to study the hydration swelling properties of PMs. Bohai Oilfield provides its ion composition, and the total salinity was 9,500 mg/L. The SEM experiment is carried out by using liquid nitrogen to freeze the PMs.
We divided the experiment into two parts. One part of the experiment studied the hydration swelling properties of PMs. The main equipment used in this experiment was Ultrasonic Instrument (produced by Tianjin Autoscience Instrument Co., Ltd., China.) and Nanoparticle Size Analyzer (produced by Beckman Coulter, USA). Ultrasonic Instrument was primarily used to disperse the PMs solution. Its ultrasonic frequency and rated powers were 40 kHz and 120 W, respectively. Nanoparticle Size Analyzer was mainly used to measure the diameter of PMs and its measurement range is 0.6 nm–7 μm. Other equipment, such as thermostats, quartz cuvettes (10 ml), and electromagnetic stirrers, were also used in the experiment. The other experiment was to obtain the morphology and particle size distribution of PMs. The main equipment used in this experiment was a light microscope (BX41) manufactured by Olympus Corporation in Japan and a scanning electron microscope (Quanta 200F) manufactured by FEI in the United States.
The main purpose of the hydration swelling experiment is to measure the particle size distribution of the PMs dispersion system. The experiment was carried out at 65 C, and the experiment data was measured at 0, 1, 3, 5,10, and 30 days. The experimental procedures of hydration swelling properties are summarized as follows:
1) Take 100 ml of anhydrous ethanol into a beaker and add an appropriate amount of PMs. Stir it to get the test sample. Rinse the sample with anhydrous ethanol more than three times.
2) Prepare the 5% microsphere solution using anhydrous ethanol as solvent. Then, measure the particle size distribution at room temperature and calculate the average particle size as the data at the initial state.
3) Prepare the 5% microsphere solution using the simulated formation water as solvent. Then, put it into the thermostat and record the time as the initial state. Set the temperature of the thermostat to 65 °C. Measure the particle size distribution and calculate the average particle size at 1, 3, 5, 10, and 30 days.
We carried out the SEM experiment and light microscope experiment to find out the morphology and particle size distribution of PMs. As these two experiments can represent the PMs as pictures, we were able to observe the microscopic morphology of PMs and measure the particle size from the image. The experimental procedures are summarized as follows:
1) SEM experiment
a. After immersing the cover glass in the washing solution for 12 h, rinse it repeatedly with tap water and deionized water and put it in the ultraclean working platform to dry naturally.
b. Absorb the microsphere dispersion system with a clean dropper, put one or two drops of the solution on the cover glass. Then, dry it naturally in the ultraclean working platform to obtain a dry film of the polymer microsphere dispersion system.
c. Fix the dry film of the dispersion system on a glass slide. Take an observation of a wide area and select typical samples with a light microscope.
d. Fix the selected dry film of the polymer microsphere dispersion system on the template. Spray gold on the surface to make the sample conductive to avoid charge accumulation. After drying, put it into the SEM sample chamber, which should be preheated for 30 min. Pump the sample Vacuum and cool with liquid nitrogen for 30 min. We could observe the sample and select a specific region to take pictures.
2) Light microscope
a. Stir the microsphere dispersion system evenly.
b. Put a small amount of microsphere dispersion system on a glass slide with a glass rod and dye with methylene blue.
c. Place the glass slide on the microscope stage for observation. Then, set the microscope magnification to 400 times, and select an appropriate area to observe and take pictures.
The particle size distribution of PMs dispersion system measured in the swelling hydration experiment at 1, 3, 5, 10, and 30 days are shown in
Size probability and the cumulative probability distribution of PMs versus different hydration time:
The diameter of PMs at different hydration times.
Hydration time/day  λave/μm 


0  18.12  30.25 
1  25.31  45.47 
3  34.8  55.12 
5  36.37  55.83 
10  39.35  60.35 
30  39.81  60.51 
The PMs expanse in the process of hydration can be observed, which leads to the increase of average particle size (
The images of PMs dispersion system are taken with SEM and light microscopes separately at the initial state and 1 day (
Shape of PMs at different hydration times:
PMs distribution at different hydration times after processing with ImageJ:
The fractal dimensions were calculated by fitting the size distribution of PMs measured in the swelling hydration experiment at differents hydrations times.
Comparison of size distribution with cumulative probability method of PMs at the different hydration time:
Based on the results shown in
The PMs fractal dimensions D and fitted parameters at different hydration times.
Hydration time/day  λcut/μm  D/f  R2/f 

0  15  2.638  0.992 
1  26.4  2.739  0.996 
3  26.81  2.853  0.993 
5  27.84  2.866  0.995 
10  36.77  2.914  0.996 
30  28.9  2.918  0.997 
Fractal dimensions calculated with fractal cumulative probability method versus hydration time.
In the fitting process, the cut value is essential to obtain the best accurate fractal dimension. The variation of this value could change the acceptable value of the fractal dimension. When calculating the cumulative probability with different cut values at the same fractal dimension, the error calculating equation can be expressed as follows:
From
The logarithmic plot is the number of boxes in various box sizes obtained from Fractalyse, as shown in
The number of boxes in various box sizes at the different hydration times in loglog coordinate:
The fractal dimension calculated results of the PMs dispersion system with the boxcounting method and cumulative probability model are shown in
PMs fractal dimensions calculated by cumulative probability model and boxcounting method.
Hydration time/day  Db2/f  Db3/f  D/f 

0  1.638  2.638  2.638 
1  1.741  2.741  2.739 
The relationship between the average diameters of PMs and hydration time was plotted based on the hydration swelling experiment results, as shown in
The average size of PMs versus hydration time.
Fractal dimension versus the average size of PMs.
The fractal dimension reflects and represents the space occupancy ability of the PMs dispersion system. Furthermore, the larger the PMs average particle size is, the larger the fractal dimension is, and the stronger the ability to occupy space. At the initial hydration stage of PMs, the spaceoccupying rate quickly increases with the PMs size rapidly increasing, and so does the fractal dimension. When the swelling equilibrium is reached, the particle size of PMs reaches the largest. Meanwhile, the spaceoccupying rate of the PMs dispersion system reaches the maximum, and so does the fractal dimension. After that, the fractal dimension is barely varied because the space occupancy rate is unchanged when the particle size of PMs ceases to increase.
The relationship between the standard deviations of the PMs’ average size and hydration time was shown in
The average size of PMs with different hydration times.
The relationship between fractal dimension and the standard deviations of the PMs’ average size was shown in
Fractal dimension versus the average size of PMs.
The standard deviation of the PMs’ average size represents the space occupancy effectiveness of the PMs dispersion system. In a real reservoir, the size of the pore and throat is a discrete distribution. Therefore, the higher the dispersion of the PMs average particle size, the greater the probability of occupying the reservoir space. As the hydration swells, the particle size distribution of PMs becomes more dispersive. The standard deviation of PMs’ average size increases, and the fractal dimension tends to do so. This change is that the fractal dimension is a measure of the irregularity of the complex system. The more eccentric the system is, the larger the fractal dimension can be. That means the fractal dimension will be larger if the size of the PMs differs considerably. The continuity of the particle size distribution is relatively poor as the fractal dimension represents the uniformity of particle size composition.
This paper established a new model to calculate the fractal dimension of PMs. Hydration swelling experiments, SEM experiments, and light microscope experiments were carried out to verify this model. Then, the new model and the box counting model were used to calculate the fractal dimension of PMs based on the results of the experiment. The correctness of the new model is verified by comparing the calculation results of the two models. Finally, according to the calculation results of the new model, the fractal dimension characteristics of PMs are analyzed. The major conclusions that can be drawn from this study are as follows:
1) Based on the selfsimilarity theory, a fractal dimension calculation model that is particular for PMs was established. This model successfully correlates the cumulative probability of the PMs dispersed system with the fractal dimension and makes fractal dimension calculation of PMs more accurately and conveniently.
2) The fractal dimension of PMs calculated by the new model and the box counting model are all 2.638 at the initial state and 2.739 and 2.741 at hydration time as 1 day based on the experiment results. Comparing the calculation results of the two models indicates that the new model can be used to calculate the fractal dimension of PMs. In addition, the hydration swelling results indicate that the PMs reach the hydration equilibrium after 10 days of hydration. Meanwhile, the fractal dimension of PMs increases from 2.638 to 2.918 based on the new model calculation results.
3) The fractal dimension of PMs calculated by the new model indicates that the fractal dimension is linear correlated to the average particle size of PMs and the standard deviation average particle size during the hydration process. Specifically, with the increase of fractal dimension, the average particle size of PMs increases from 18.12 to 39.35 μm, and the standard deviation of average particle size increases from 30.25 to 60.51 μm. That means that the fractal dimension of PMs represents the space occupancy ability and space occupancy effectiveness.
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
WZ made a draft of this paper. GH and JY searched and collected all references. TL and JH helped in critically assessing this paper. RL and PD helped in analyzing experimental data.
This research was funded by National Science and Technology Major Projects (grant number 2017ZX05009004).
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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