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Open Physics

formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1


Volume 13 (2015)

A volumetric ablation model of EPDM considering complex physicochemical process in porous structure of char layer

Liu Yang
  • Corresponding author
  • Science and Technology on Combustion, Interior Flow and Thermal-structure Laboratory, Northwestern Polytechnical University, Xi’an 710072, China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Yu Xiao-Jing / Ma Jian-Ming
  • Science and Technology on Combustion, Interior Flow and Thermal-structure Laboratory, Northwestern Polytechnical University, Xi’an 710072, China
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Guan Yi-Wen
  • Science and Technology on Combustion, Interior Flow and Thermal-structure Laboratory, Northwestern Polytechnical University, Xi’an 710072, China
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Li Jiang
  • Science and Technology on Combustion, Interior Flow and Thermal-structure Laboratory, Northwestern Polytechnical University, Xi’an 710072, China
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  • De Gruyter OnlineGoogle Scholar
/ Li Qiang
  • Science and Technology on Combustion, Interior Flow and Thermal-structure Laboratory, Northwestern Polytechnical University, Xi’an 710072, China
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Yang Sa
Published Online: 2017-06-14 | DOI: https://doi.org/10.1515/phys-2017-0038


A volumetric ablation model for EPDM (ethylene- propylene-diene monomer) is established in this paper. This model considers the complex physicochemical process in the porous structure of a char layer. An ablation physics model based on a porous structure of a char layer and another model of heterogeneous volumetric ablation char layer physics are then built. In the model, porosity is used to describe the porous structure of a char layer. Gas diffusion and chemical reactions are introduced to the entire porous structure. Through detailed formation analysis, the causes of the compact or loose structure in the char layer and chemical vapor deposition (CVD) reaction between pyrolysis gas and char layer skeleton are introduced. The Arrhenius formula is adopted to determine the methods for calculating carbon deposition rate C which is the consumption rate caused by thermochemical reactions in the char layer, and porosity evolution. The critical porosity value is used as a criterion for char layer porous structure failure under gas flow and particle erosion. This critical porosity value is obtained by fitting experimental parameters and surface porosity of the char layer. Linear ablation and mass ablation rates are confirmed with the critical porosity value. Results of linear ablation and mass ablation rate calculations generally coincide with experimental results, suggesting that the ablation analysis proposed in this paper can accurately reflect practical situations and that the physics and mathematics models built are accurate and reasonable.

Keywords: Ethylene-propylene-diene monomer; Porous media; Volumetric ablation; Ablation model; Solid Rocket Motor

PACS: 44.30.+v; 47.56.+r

1 Introduction

Owing to the low specific density, high thermal pyrolysis temperature, high char yield, and excellent mechanical properties of ethylene-propylene-diene monomer (EPDM), it is widely used as an interior insulator in solid rocket motors (SRMs) with high temperatures (2500-4000 K), high pressures (30-200 atm), and particle erosion environments [1, 2].

For high-temperature environments, EPDM materials undergo pyrolysis, followed by charring. The entire ablation process forms a three-layer structure comprising a virgin layer, a pyrolysis layer, and a char layer. The char layer is a uniquely loose but interconnected structure, and the density, specific surface area, and interior pores of which can influence the diffusion of oxidative components and thermochemical reactions in the contact area. The char layer falls off when sufficiently loose because it can no longer resist the mechanical damages caused by gas flow and particle erosion. This condition leads to material ablation.

Complex processes occur in the char layer, particularly chemical reactions, head and mass transfer, and material consumption between external gas and substances within materials. The char layer protects insulation materials against ablation. Thus, understanding the principles and mechanisms of the physicochemical changes in the char layer can provide valuable insights into the mechanisms involved in the ablation of insulation materials and facilitate the construction of models that can accurately describe the ablation process.

Previous research on insulation material ablation is mostly based on the concept of SRM nozzle throat-insertablation when the nose-cone of an aircraft re-enters the atmosphere, and thus presumes that the ablation is caused by the thermochemical reactions in the oxidative components, nose-cone, and surface of the throat-insert materials. As the understanding on the mechanisms underlying ablation increases, the ablation model gradually evolved from a surface ablation model [3, 4] to a volumetric ablation model. Based on a porous media perspective, Dimitrienko [5-8] systematically studied the ablation behavior of insulation materials and considered density changes of materials caused by pyrolysis, while adopting Darcy’s porous media permeation principle [9] to analyze the flowing phenomenon within the porous structures of char layers. However, the analysis of Dimitrienko showed that chemical reactions only occur on the char layer, and thus the volumetric ablation concept was not actually realized. Furthermore, he failed to conduct an in-depth discussion and exploration on which mechanism of the char layer is unsuccessful.

Yu [10] built an ablation model using silicone rubber materials in an oxygen-rich environment, considering permeation of oxygen into fuel gas into the char layer interior to achieve an exothermic reaction with materials in the char layer pores, which subsequently consumes the char layer and accelerates ablation. Yu’s research suggested that char layer porosity is an important parameter for the characterization of ablation materials, because it greatly influences the ablation process. In this model, the structure of the char layer is simplified such that the structure can be characterized by porosity, which is a measurable parameter, and the construction of corresponding mathematical models is encouraged.

Curry [11] performed a retrospective analysis on the thermal protection system and considered processes, such as convection radiation, pyrolysis, thermal chocking, vapor deposition, and thermo-chemical ablation, to describe a model for the entire ablation process more accurately. Ayasoufi [12] adopted this model during numerical simulation of ablation for re-entry vehicles. He also described the deposition reaction process of CH4, C2H6, C2H4, and C2H2 gas mixtures.

Based on experimental phenomena, Li and Yang et al. [13-15] built a thermochemical ablation model for heterogeneous char layer structures. This model laid the foundation for future ablation models featuring coupling of particle erosion, gas flow erosion, and thermochemical ablation. A general transportation equation was used to describe all governing equations in order to simplify the numerical process. The discrete equation was deduced by control volume integral method and solved using the SIMPLE (Semi-Implicit Method for Pressure-Linked Equation) algorithm.

For the microporous structures of EPDM char layers, the current paper introduces a porous medium and considers several processes, such as flow, heat transfer, chemical vapor deposition (CVD), and chemical reaction, within the char layer pores for the construction of an accurate EPDM volumetric ablation model, which is validated through thermal testing.

2 Physics ablation model of EPDM insulator

2.1 Analysis of char layer and physical ablation model

A char layer is generated on the surface of insulation materials during ablation processes. The char layer has inflaming-retarding and thermal-insulating functions. After being peeled off, the char layer can remove heat to protect the insulation layer below. Thus, the configurations, structures, and properties of the char layer can directly influence ablation properties of thermal-insulation materials and are closely related to the formulation of insulation materials and state parameters during ablation. Char structure is the basis for the study of the ablation mechanism in insulation materials. Particularly, the properties of the Char layer porous structure form important bases in studying the ablation model. Thus, analysis and characterization of the char layer is important.

The char layer properties are described by four aspects, namely, physical properties, chemical properties, mechanical properties, and structure. The density, porosity, pore diameter and distribution, and specific surface area are essential to ablation mechanism research. Char layer structure and its surface morphology can be analyzed and characterized through scanning electron microscopy (SEM), microscopy, and other similar methods. While, porosity can be measured with a fully-automatic density meter. Specific surface area can be measured through a mercury intrusion method.

Two different SRMs are specially designed (See Figure 1 and Figure 2) to study ablation characteristics of the insulation layer at different gas flow velocities and in different particle erosion environments. This approach allows the comprehensive and objective investigation of the char layer morphology after hot firing test. Figure 3 shows the surface and cross section of SEM images of the EPDM char layer. The char layer generated after the ablation of the insulation materials showed a heterogeneous porous structure. Furthermore, the loosening in the layer gradually decreases from the ablation surface to the interior char layer. After measuring the open porosity within the char layer of the insulation materials, 77.49% out of a 78.09% porosity are open pores. This result indicates that 99.2% of the pores are open pores, which are favorably connected.

Schematic diagram of hot firing test SRM-I
Figure 1

Schematic diagram of hot firing test SRM-I

Schematic diagram of hot firing test SRM-II
Figure 2

Schematic diagram of hot firing test SRM-II

Microstructure of EPDM char layer surface and cross-section after firing test
Figure 3

Microstructure of EPDM char layer surface and cross-section after firing test

The char layer is prepared through different methods, such as tube furnace, laser ablation, oxygen acetylene torch, and hot firing test SRM, to examine char layer properties of the EPDM insulation materials. A fully-automatic density meter is then used to measure the prepared char layers. Table 1 shows the density, open porosity, and obturator rate of the char layer under different test conditions. Wide applications of ablation physics model are considered and EPDM char layer in different formulation systems is measured. Measurement results show that the porosity of the char layer is large (65%-80%). The char layer pores are mostly open pores, and the obturator rate is low. According to these characteristics, the physical model of EPDM insulation materials in the char layer is described as follows:

  1. After analyzing the char layers of several insulation materials, results confirmed that the char layer is a heterogeneous loose material, the structural properties of which are in line with the definition of porous media. Its interior flow paths can be regarded as interconnected and thus can be defined as a porous medium for the convenience of describing gas flow and chemical reactions within char layer.

  2. The characteristic sizes of the pores are far larger than those in the average free range of gas molecules. The interior gas flow forms a continuous medium. The oxidative components in pyrolysis gas and main fuel gas undergo heterogeneous reactions with char in the skeleton surface of the char layer. Affected by the char layer skeleton, the gas fuel flow within the char layer turns into an incompressible layer flow. Meanwhile, owing to the slow velocity of the pyrolysis gas and main fuel gas, the model often assumes that the char layer skeleton and local gas are in a thermal equilibrium state and the heat transfer within the char layer is mainly completed through gas phase relative flow and solid phase heat conduction.

Table 1

Char layer porous configuration under different test conditions

2.2 Analysis of compact/loose structure in char layer

The porous structure of the char layer under different ablation conditions have a heterogeneous distribution along ablation direction, and thus show an evident compact/loose structure (Figure 4). The compact structure is a contrast to the loose structure and has a porosity as high as 80%. Heterogeneous compact/loose structures are ubiquitous, and their structural characteristics are closely related to experimental conditions. Under different external erosion conditions, the porosity, formation position, and thickness of the compact structure within the char layer vary. Under conditions of laser ablation and oxygen acetylene torch, the compact layer is often located on the surface. Under hot firing test SRM condition, in which the fuel gas erosion velocity is slow [See Figure 4(a)], a compact structure with certain degree of strength, minor porosity, and thickness, forms on the char layer surface. Along with the acceleration of erosion velocity, the thickness of the compact structure decreases. At a particle two-phase flow erosion condition, the compact structure shown in Figure 4(b) forms within the char layer. As erosion velocity increases, the compact structure moves farther away from char layer surface (Figure 4(c)). Under extreme conditions, the entire char layer structure is compact.

Cross-section of the micro-structure of the EPDM char layer after the firing test
Figure 4

Cross-section of the micro-structure of the EPDM char layer after the firing test

The presence of compact structures impedes the overflow of the pyrolysis gas generated in the pyrolysis layer, increases the local pyrolysis gas concentration, and slows the hydrolysis of the insulation materials. These compact structures hinder the inward diffusion of oxidative components in the fuel gas and alleviate the chemical ablation process. Compared with loose structures, compact structures have higher strength and more significant mechanical properties and thus can protect the char layer against mechanical and particle erosions.

During debugging of the preliminary ablation model, when the compact structure is not considered, the calculated mass ablation rate is approximately three times higher than that obtained through the experiment. Conversely, when the compact structure is artificially added to the model, the error is only 10%. Therefore, constructing a reasonable ablation model is critical to the identification of the mechanism involved in compact structure formation.

Micrometer computed tomography is conducted on a char layer with a compact structure. Figure 5 shows the porosity distribution and average pore diameter along a thick char layer. Item 1 stands for the char layer surface. Porosity is approximately 58.5% near the char layer surface and 89% at the back, and the gap between the two ratios is large. The changes in the average pore diameter are also dramatic. Approximately 3/4 of the surface area of the char layer has an average pore diameter smaller than 2 μm. The other areas on the back increase sharply to approximately 10.8 μm. This result validates the uneven distribution of pores along the thick char layer and the presence of a loose/compact structures.

Porous ratio and average porosity distributions
Figure 5

Porous ratio and average porosity distributions

The SEM images show that the back of the char layer features a loose structure under different ablation types and lengths. The structure is validated by the micrometer CT. This loose structure at the back of the char layer is the preliminary layer formed after pyrolysis. Furthermore, the compact structure mainly concentrates on the surface or in the middle and thus the compact layer is not formed immediately after pyrolysis. Given that the interior part of char layer is interconnected, the fuel and pyrolysis gases are both likely to undergo deposition reaction in the porous structure. Laser ablation surface forms an obvious compact layer. Therefore, the deposition of pyrolysis gas, rather than fuel gas, contributes to compact layer formation.

The results of the pyrolysis experiment on the EPDM insulation materials indicate that pyrolysis gas consists of several components, such as methane, ethylene, and propylene. Methane and ethylene are commonly used to prepare C/C composite materials through CVD. Furthermore, the porous structure of the char layer is loose and interconnected and has a large specific surface area. These characteristics facilitate the diffusion of gas molecules and guarantee a large reaction area and the subsequent chemical deposition reaction. Meanwhile, the results of the ablation heat transfer calculation show that the temperature within the char layer under hot firing test SRM condition ranges from 800 K to 3,000 K from the bottom to the surface. However, the general temperature range of CVD reaction is 1000-1500 K. This result indicates that interior temperature distribution of the char layer met the CVD conditions.

2.3 The description of EPDM insulator physical ablation model

Figure 6 shows the physical ablation model of EPDM insulator. The mode considers the porous structure of the char layer.

Diagram of EPDM volumetric ablation model
Figure 6

Diagram of EPDM volumetric ablation model

Figure 6 shows that virgin layer only undergoes heat conduction. High polymers are pyrolysed in the pyrolysis layer and subsequently generate hydrolysis gas, which then flows to the char layer through seepage. The hydrolysis gas is generated at 450-550 and 550-650°C, which correspond to EPDM rubber and aramid fiber, respectively. The equilibrium pyrolysis gaseous product at 450-550° C is mainly composed of CH4 and C6H6, whereas that at 450-550°C are mainly composed of CO, CH4, C6H6, H2, and N2. Thus, in a pyrolysis layer, high polymer pyrolysis, pyrolysis gas seepage, and heat conduction occurs. The char layer is a loose and complex layered structure mainly composed of carbon. Meanwhile, oxidative components flow into the pyrolysis gas, and oxidative components in incoming fuel gas enter through diffusion. In the char layer pores, the oxidative gases undergo thermochemical reaction with C in the char layer skeleton. The char layer porosity increases when C is consumed. Under certain temperature conditions, pyrolysis gas undergoes heterogeneous deposition reaction with the char layer skeleton. In char layer generated through ablation, the surface porosity is larger than the clinical porosity and the structural strength of the former is extremely small. After subjecting to fuel gas and two-phase flow erosion, the char layer falls off. Thus, a critical value is measured under different experimental conditions.

In the entire ablation process of the EPDM insulation materials, the thermochemical process continuously consumes the char layer, which becomes loose and tends to fall off. The thickness of the char layer must be decreased through gas flow erosion and particle erosion to influence heat transfer and diffusion of components. The heat generated by particle erosion should increase the heat flux on insulation layer to influence pyrolysis and thermochemical ablation.

3 EPDM insulator numerical ablation model

3.1 Pyrolysis model

The pyrolysis principles regarding EPDM insulation materials and major raw materials can be obtained on the basis of a thermal analysis experiment. The Ozawa method [16] is adopted to reduce the pyrolysis activation energy, and the Šatava method is used to deduce the mechanism function form. Through calculation and analysis, the pyrolysis kinetic equation can be obtained: dαdT=1β×9.413×1013×exp229540RT×1α, where α is the reaction fraction to be solved, β is the heating rate, R is the perfect gas constant; and T is the material temperature.

3.2 Deposition reaction model

Hydrocarbon deposition, which involves gas phase pyrolysis, gas phase reaction kinetic process, and gas-solid heterogeneous phase chemical reaction kinetic process, is extremely complex. The current paper refers to deposition reaction adopted from literature [11,12] to illustrate the deposition reaction of the pyrolysis gas in the char layer: C6H6(g)6C(s)+3H2(g)CH4(g)0.5C2H6(g)+0.5H2(g)C2H6(g)C2H4(g)+H2(g)C2H4(g)C2H2(g)+H2(g)C2H4(g)2C(s)+H2(g)

During the calculation process of CH4 and C2H4, the intermediate reaction is omitted to simplify the deposition reaction into a single-step reaction. According to Arrhenius formula, the deposition rate of C can be obtained, as follows: m˙d=Mc6A1P1expE1/RTMC6H6+A2P2expE2/RTMCH4+2A3P3expE3/RTMC2H4.

3.3 Thermo-chemical reaction model

Based on the description of char layer above, the char layer thermochemical ablation model features volumetric ablation. In other words, the consumption of C in char layer by the gas phase components in the oxidation layer is mainly realized through heterogeneous chemical reaction on the solid skeleton surface. The following traditional three-step chemical reaction model is thus adopted: C(s)+O(g)CO(g),C(s)+CO2(g)2CO(g), and C(s)+H2O(g)CO(g)+H2(g), where chemical reaction rate is controlled by chemical kinetics. According to Arrhenius formula, the consumption rate of different gas phase components and consumption rate of C, c, can be obtained as follows: m˙c=McA1P1expE1/RTMO+A2P2expE2/RTMCO2+A3P3expE3/RTMH2O, where A1, A2, A3 and E1, E2, E3 are the pre-exponential factor and reaction activation energy of the three reaction equations, respectively, Pi is the partial pressure of the gas phase component, i, within the pores of the char layer. εn+1=11εn1m˙cfcΩ where fc is the mass percent composition of C in the char layer skeleton of the control volumetric, and Ω is the specific surface area of the char layer skeleton.

3.4 Gas erosion model

The influence of gas velocity on ablation is mainly reflected in two aspects. Firstly, the heat transfer between the high-temperature fuel gas and insulation layer surface is enhanced when the gas velocity increases. Second, high-velocity gas flow erodes the char layer surface and thus peels off loose and weak char layer, subsequently lowering the heat boundary and accelerating the ablation process.

The heat transfer model can be made on the basis of Bazz formula when the gas flows through the char layer surface. Meanwhile, the mechanical damage of char layer can be modeled by connecting the surface shear stress and surface porosity. εc=fσgas On the basis of the cold flow experiment, the shear stress of a gas flow on the char layer can be measured, and the formula for the calculation of the shear stress on the char layer surface is obtained through the following fitting: τ=0.092ρulμ0.26ρu2

3.5 Particle erosion model

During insulation layer ablation process, the erosion of the condensed phase particles has thermal and mechanical damage effects. The thermal effect occurs when the heat transfer triggered by heat conduction and kinetic energy changes to insulation layer samples instantly when the particles erode on the surface of insulation layer surface. Mechanical damage effect mainly refers to the mechanical damage and consumption of insulation samples caused by fast particle erosion.

Considering the relationship between porosity and char layer strength, this paper selects an average porosity of char layer surface obtained through experiment to represent the critical intensity of mechanical damage caused by particle erosion. The porosity of char layer surface is measured through a series of experiments to express the relationship between the critical porosity on the char layer surface and particle erosion into the following equations: εc=0.777×np0.196×vp0.199×(cosα)0.376 when below the critical velocity (Vp < 32 m/s) and εc=0.306×np0.638×vp0.546×(cosα)0.177 when above the critical velocity (Vp < 32 m/s). In these equations, ɛc is the critical porosity on char layer surface, np is the mass flux of particle erosion, kg/m2s, Vp is the particle erosion velocity, m/s, and α is the erosion angle, °.

The heat-gain model of particle erosion is shown below [17]: q˙=94076np0.436vp0.854 where is the heat gain of particle erosion, J/m2s.

The equation above can be adopted to determine the heat gained at different particle erosion conditions and can be combined with heat transfer heat flux and heat radiation heat flux to form the heat boundary for heattransfer calculation.

4 Calculating flow and experimental validation

The calculation of the ablation model includes flow field calculation, thermal chemical reaction, and heat transfer calculation. This paper also includes the diffusion process calculation. These calculation processes are interconnected. Figure 7 shows the overall calculation flow as follows:

 Ablation model calculation flow chart
Figure 7

Ablation model calculation flow chart

Firstly, the fuel gas components and transport parameters can be obtained through thermodynamic calculation. The molar concentrations of various fuel gas components and physical parameters of the fuel gas outside the boundary layer are also obtained through this calculation. Second, through the calculation modules of the boundary layer, the oxidative components distribution and heat transfer coefficient on the boundary layer are acquired. The heat flux of the preliminary heat transfer and heat radiation on the upper surface are obtained through these results. Third, the gas phase and pyrolysis gas parameters of the boundary layer are placed into the char layer diffusion module to determine the density distribution of the oxidative components within the char layer. Fourth, the heat transfer and heat radiation conditions of the boundary layer are placed into the heat transfer calculation module to obtain the temperature distribution. Finally, the distributions of the char and pyrolysis layers are evaluated on the basis of temperature. Meanwhile, the CVD and thermochemical ablation calculations are conducted within the char layer to rectify the porosity distribution, obtain mass ablation rate, and confirm the energy equation source. The char layer physical parameters are then rectified under a new porous structure and compared with critical porosity in the gas flow corrosion and particle erosion models successively. At critical value, the porosity becomes 1, that is, the char layer falls off. Then, mass ablation and linear erosion rates are confirmed.

The ablation calculation flow is adopted to calculate the charring ablation rate under five operating cases of SMRs. The calculation results are shown in Table 2. The comparison between experimental and calculation results shows that error is within 11%. This result indicates that the coupling calculation flow constructed in this paper is fairly accurate.

Table 2

Particle erosion parameters and ablation rate

The calculation results of the char layer structure are compared with the experimental results to validate the feasibility of the coupling model. In this study, Case 3 is adopted as an example. The SEM images of the char layer cross-section are shown in Figure 4(b). Figure 8 shows porosity distribution calculated through Case 3. The compact layer in the middle was observed. From the surface to the back, the char layer structure changes from compact to loose to compact structure. This observation co-incides with the structure shown in the char layer cross-section in Figure 4(b). Figure 9 compares the porosity distribution of the char layer and micrometer CT test results. The comparison shows good coincidence and has a maximum relative error of 15.3%. Thus, the calculation model proposed in this paper can guarantee coincidence not only between the calculation results of the macroscopic parameters, such as ablation rate and experimental results, but also between the calculation results of the microscopic parameters, such as porosity distribution and experimental results. The reasonability and accuracy of the model are further validated. In this paper, the porous structure evolution, caused by thermal stress, in the char layer is not considered. Thus, a difference between calculated and experimental results occurs.

Porosity distribution based on calculation results (Case 3)
Figure 8

Porosity distribution based on calculation results (Case 3)

Comparison of porosity distribution in char layer
Figure 9

Comparison of porosity distribution in char layer

5 Conclusion

An EPDM volumetric ablation model that considers the complex physicochemical process in the porous structure of char layer has been established. The model also consider the CVD and thermochemical ablation process of the pyrolysis and fuel gases in the char layer loose structure and erosion process of char layer by fuel gas and particles. Subsequently, the volumetric ablation mechanism is revealed and is in line with the practical ablation process.

Several important conclusions are obtained as follows:

  1. The pyrolysis gas deposition reaction in the char layer of EPDM insulation materials depends on local temperature. The formation and evolution of the char layer porous structure is a continuous dynamic process. The effect of oxidation reaction on the increasing porosity and deposition reaction at decreasing porosity has competitive relationship. Compact layer can easily appear in area dominated by deposition reaction along the thick char layer.

  2. The calculation results on the porous media volumetric ablation model built in this paper shows good agreement with experimental results in terms of ablation rate and char layer porous structure distribution. Thus, the model can accurately simulate evolution process of char layer porous structure and ablation process of insulation layer.


This research was supported by the National Natural Science Foundation of China (Grant No. 51276150 and 51576165) and the Fundamental Research Funds for the Central Universities (Grant No. 3102016ZB046).


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About the article


Received: 2016-10-01

Accepted: 2017-03-02

Published Online: 2017-06-14

Citation Information: Open Physics, Volume 15, Issue 1, Pages 344–353, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0038.

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