# Transport Properties of Thermoplastic R-BAPB Polyimide: Molecular Dynamics Simulations and Experiment

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{2}), nitrogen (N

_{2}), and methane (CH

_{4}). The validity of the results obtained was confirmed by studying the correlation of the experimental solubilities and diffusion coefficients of He, O

_{2}, and N

_{2}in R-BAPB, with their critical temperatures and the effective sizes of the gas molecules, respectively. The solubilities obtained in the molecular dynamics simulations are in good quantitative agreement with the experimental data. A good qualitative relationship between the simulation results and the experimental data is also observed when comparing the diffusion coefficients of the gases. Analysis of the Robeson plots shows that R-BAPB has high selectivity for He, N

_{2}, and CO

_{2}separation from CH

_{4}, which makes it a promising polymer for developing gas-separation membranes. From this point of view, the simulation models developed and validated in the present work may be put to effective use for further investigations into the transport properties of R-BAPB polyimide and nanocomposites based on it.

## 1. Introduction

_{2}) and oxygen (O

_{2}), the processing of natural gas to produce helium (He), and methane extraction (CH

_{4}) during the comprehensive preparation of associated petroleum gas [1,2,3]. The efficient performance of these tasks opens up significant opportunities for the reduction of mass greenhouse gas emissions into the atmosphere, the optimization of the processes involved in petrochemical production, and the rational and efficient use of hydrocarbon resources. In recent years, polymer membrane technology was increasingly used [1,2] to address these issues. Currently, polymer membranes are being successfully used for the removal of nitrogen from the air [2], for processing the associated petroleum gas, and for drying natural gas and extracting helium and carbon dioxide from it [3,4].

_{4}, O

_{2}/N

_{2}, CO

_{2}/CH

_{4}, He/N

_{2}, N

_{2}/CH

_{4}, CO

_{2}/N

_{2}, and He/CO

_{2}). Moreover, in order to provide comprehensive descriptions, both at macroscopic and microscopic levels, we will supplement our simulations with actual experimental investigations. These important and preliminary investigations aimed at evaluating the transport properties of the polymer itself and developing simulation models will encourage further investigations of nanocomposite membranes based on R-BAPB polyimide, which have a heterogeneous morphology associated with nanoparticle inclusion, and will be performed in the future.

## 2. Materials and Methods

#### 2.1. Materials and Methods

#### 2.2. Film Preparation

#### 2.3. Characterization of Transport Properties

^{3}, and may be increased up to 312.24 cm

^{3}to achieve a stable state in the case of high flows or prolonged experiments. Up to eight gas cylinders can be connected to the apparatus simultaneously. The gas pressure above the membrane can be adjusted from 0 to 1.3 bar, while the actual value is detected with a resolution of up to 0.1 mbar. The permeate pressure is measured in the range from 0 to 10 mbar, with a resolution of 0.001 mbar. The diameter of the polymer membrane is 76 mm, but the effective area can be reduced by using appropriate masks. The system is fully automated and computer controlled. The pressure of the gas supply is set by pneumatic valves, and the gases can be alternated automatically.

^{2}(d = 4.19 cm). The measurements were carried out at 303 K, with a partial gas pressure of 0.6 bar.

^{−7}bar, using a Pfeiffer Vacuum HiCube 80 Eco turbomolecular pump, before the first measurement, to remove dissolved gases or vapors from the sample and rubber seals. Between the two subsequent gases, the system was also evacuated, purged with a second gas, and again evacuated to ensure the complete removal of the previous gas.

^{2}/s), and S is the solubility coefficient (cm

^{3}(STP)/(cm

^{3}∙bar)).

_{0}) [49]:

#### 2.4. Molecular Dynamics Simulations

#### 2.4.1. Simulation Details

_{2}, O

_{2}, and CH

_{4}). Therefore, in accordance with common practice, we have used additional parameters, which are calibrated for the thermodynamic, structural, and dynamic properties of the gases in question [28,60,61,62,63]. The parameters for He were proposed by Lee et al. [28,60]; for N

_{2}by Fischer and Lago [61]; for O

_{2}by Cheung and Powles [62]; and for CH

_{4}by Yin and MacKerell [63], and are summarized in Table 2. Finally, non-bonded parameters for “gas-polymer” interactions were calculated using standard geometric combination rules implemented in the Gromos53a6 force field [53,54]. We show that implementation of the parameters for gas molecules into Gromos53a6 allows us to reproduce quantitative trends and correlations between solubility and diffusivity of the gases observed in the experiment.

_{T}= 0.5 ps and τ

_{p}= 2.5 ps, respectively. Equations of motion were integrated every 2 fs for temperatures T < 430 K and 1 fs for temperatures T > 430 K. The cut-off radius of the non-bonded interactions was taken to be 1 nm.

^{11}K/min. This technique is widely used to obtain systems in molecular dynamics simulations to study the thermophysical and mechanical properties of polymers [70], including thermoplastic polyimides [44,46,57,59].

#### 2.4.2. Solubility Calculations

_{ex}) of the test particle insertion into the system is calculated from the ∆U, according to the following formula:

_{mol}is the molar volume of the gas at STP. Thus, the solubility unit is $\frac{c{m}^{3}\left(STP\right)}{c{m}^{3}\ast bar}$.

^{5}. Further increase in the number of test insertions does not lead to a significant change in the solubility values within the error, as the additional analysis shows (see Supporting Information, Figure S2), while fewer insertions lead to an overestimation of the solubility coefficients.

_{p}is the volume of the simulation cell, and P is the pressure.

#### 2.4.3. Diffusion Coefficient Calculations

_{0}is the pre-exponential factor, and E

_{D}is the diffusion activation energy.

## 3. Results and Discussion

#### 3.1. Solubility

_{4}), 37.5% (for N

_{2}), 47% (for O

_{2}), and 47% (for He). However, we could conclude about the good agreement between the results, since the average values of the solubility coefficients are close to each other, taking into account the error bars. This agreement proves the adequacy of the parameters describing interactions between the gas molecules and the polymer. Moreover, the solubilities of the gases display a linear dependence on the gases’ respective critical temperatures, Figure 4b, in accordance with the literature data [52], which further indicates the reliability of the results obtained.

#### 3.2. Diffusion Coefficients

_{4}) molecules. In the case of He, this is due to its low solubility in R-BAPB, which means that He molecules are practically absent in the simulation cell in question. Given such small values for the equilibrium concentration of the gases, as well as the fact that gas solubility decreases with increasing temperature, the question arises about the correct choice of the number of particles for studying the gas molecules’ mobility, especially at elevated temperatures.

_{4}) at 470 K. An analysis of the MSD curves of the gases (given in Supporting Information, Figure S4) showed that gas dynamics tend to increase upon increasing the concentration of the gas molecules, especially when the number of molecules in the system is above 20. This may be a consequence of polymer swelling; thus, only 10 gas molecules were used to determine their diffusion coefficients in R-BAPB, in order to negate the possible influence of polymer swelling.

_{g}= 485 K) on timescales from 500 to 1000 ns, depending on the temperature considered. Then, the MSD functions of the gas molecules were calculated (see Figure 5).

_{4}is approximately 2–3 times larger than the corresponding experimental values, while the diffusion coefficients of O

_{2}and N

_{2}, on the contrary, are approximately 2–3 times lower. Only for He the desirable consistency is observed. Therefore, in spite of the observed qualitative and quantitative agreement between the solubilities of the gases in the experiment and simulations, the situation with the diffusion coefficients turns out to be more complicated. Nevertheless, a quantitative conclusion could still be drawn about the transport properties of R-BAPB with respect to He, O

_{2}, and N

_{2}, since the experimentally relationship between their diffusion coefficients (D

_{He}> D

_{O2}> D

_{N2}) is preserved in the simulations.

_{4}being slightly larger than that of N

_{2}could be discerned by analyzing the dependence of the activation energies of gas diffusion on the effective diameter of the gas molecules (see Figure 8a). As can be seen, for He, N

_{2}, and O

_{2}, diffusion activation energies depend linearly on the effective diameter of the gases involved, as is typically observed in the literature [52]. However, the activation energy for CH

_{4}lies below the linear dependence. The same deviation is observed when considering the dependence of the diffusion coefficients of gases on the squared effective diameter of the gas molecules (see Figure 8b). A possible reason for this deviation may be related to the absence of partial charges on CH

_{4}atoms in the simulations. Verification of this assumption will be the subject of our further investigations.

#### 3.3. Permeability and Selectivity

_{4}, for which the R-BAPB permeability is significantly lower than the corresponding experimental value, which in turn is due to an underestimation of the diffusion coefficient in the simulations discussed above.

_{4}selectivity (around 700), which is also confirmed in the experiment (see Figure 9b). From Figure 9b, it can also be seen that simulations are capable of reproducing the qualitative difference in the R-BAPB selectivities observed in the experiment. At the same time, the quantitative difference between the selectivity values in the experiment and simulations owes more to the kinetic component of the selectivity than its thermodynamic aspect, which could be attributed to the difference in the structure of the free volume of the R-BAPB samples in molecular dynamics simulations and the experiment, and will be investigated further.

_{2}, and CO

_{2}, (see Figure 10a,c,e), since they are comparable to the values for other commercial PIs (such as Kapton, ULTEM, Upilex-R, Matrimid 5218, P84, and 6FDA-durene) [13,14,15,16,17,18,19,20], which are typically used to develop efficient gas separation membranes [6]. In the next stage of our research, we will evaluate transport properties of R-BAPB with respect to gas mixtures.

## 4. Conclusions

_{2}, N

_{2}, and CH

_{4}, were estimated. The simulation results on solubility coefficients of gases were found to be in good quantitative and qualitative agreement with the corresponding experimental data. Computer simulations also reproduced qualitatively the relationship between the diffusion coefficient values for He, O

_{2}, and N

_{2}observed in the experiment. The reliability of the results was also confirmed while examining the dependencies of the activation energies and diffusion coefficients of the gases on the effective diameters of the gas molecules. Nevertheless, the quantitative difference between the simulation and experimental results calls for further investigations aimed at elucidating possible routes for its improvement by simultaneous experimental and simulation studies.

_{2}, CO

_{2}, and He from CH

_{4}. This should stimulate further optimization of polymer gas separation membranes based on R-BAPB polyimide and the development of mixed-matrix membranes based on it, which may be potentially competitive to other commercial membranes.

## Supplementary Materials

_{4}, O

_{2}, N

_{2}, or He at different temperatures after energy minimization. Figure S2: Dependence of the solubility coefficient of CH

_{4}, O

_{2}, N

_{2}, or He at STP conditions calculated from a single trajectory obtained at T = 300 K on the number of the test particle insertions. Figure S3: Time dependence of mean squared displacement of He or CH

_{4}gas molecules in R-BAPB at different concentrations (number of gas molecules) at T = 470 K, calculated over 100 ns. Figure S4: Time dependence of the slope of the mean squared displacement of CH

_{4}, O

_{2}, N

_{2}, or He molecules in R-BAPB at different temperatures.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Schematic diagram of the “HZG G&V Permeability Test Unit” apparatus used to determine the gas transport properties of flat membranes.

**Figure 4.**(

**a**) Solubilities of gases in R-BAPB under STP conditions obtained in molecular dynamics simulations (shaded columns) and in experiment (solid columns). (

**b**) Solubilities of gases in R-BAPB versus their critical temperatures obtained in molecular dynamics simulations (blank red circles) and in experiment (solid black circles). Semi-logarithmic axes. The error bars denote standard deviation.

**Figure 5.**Time dependence of the mean squared displacement (MSD) of (

**a**) CH

_{4}, (

**b**) N

_{2}, (

**c**) O

_{2}, and (

**d**) He in R-BAPB, at various temperatures. Double-logarithmic axes.

**Figure 6.**Dependence of the diffusion coefficients of (

**a**) CH

_{4}, (

**b**) N

_{2}, (

**c**) O

_{2}, and (

**d**) He in R-BAPB on the reciprocal temperature obtained in molecular dynamics simulations. Semi-logarithmic axes. The red dotted line indicates the glass transition temperature of R-BAPB. The black dashed line corresponds to T = 300 K. The solid black symbols indicate the diffusion coefficients at T = 300 K, obtained by extrapolating the diffusion coefficients at higher temperatures, using the Arrhenius Law. The error bars are of the size of the symbols.

**Figure 7.**Diffusion coefficients of gases in R-BAPB at temperature T = 300 K, obtained in molecular dynamics simulations (shaded columns) and in the experiment (solid columns). Semi-logarithmic axes. The error bars denote standard deviation (see main text for details). The diffusion coefficient of N

_{2}lies in the range (0.007 ÷ 0.09) × 10

^{−3}nm

^{2}/ns, with 0.02 nm

^{2}/ns being the value given by the approximation line.

**Figure 8.**(

**a**) Dependence of the activation energy of diffusion of the gases in R-BAPB on their effective diameter [51], obtained in molecular dynamics simulations. (

**b**) The values of the diffusion coefficients of gases in R-BAPB at T = 300 K at their effective diameters [51], obtained in molecular dynamics simulations (blank red circles) and the experiment (solid black circles). Semi-logarithmic axes. The error bars denote the standard deviation (see main text for details). The gray lines are to guide the eye.

**Figure 9.**(

**a**) Permeability of R-BAPB with respect to CH

_{4}, N

_{2}, O

_{2}, and He at T = 300 K. (

**b**) Selectivity of R-BAPB for various gas pairs at T = 300 K, obtained in molecular dynamics simulations and in experiment. The results obtained in molecular dynamics simulations are shown by the shaded columns, and experimental data are given in the solid columns. Semi-logarithmic axes.

**Figure 10.**Robeson plots for (

**a**) He/CH

_{4}, (

**b**) O

_{2}/N

_{2}, (

**c**) CO

_{2}/CH

_{4}, (

**d**) He/N

_{2}, (

**e**) N

_{2}/CH

_{4}, (

**f**) CO

_{2}/N

_{2}, and (

**g**) He/CO

_{2}. Solid black dots indicate data for commercial PIs available in the literature [13,14,15,16,17,18,19,20]. Open red symbols correspond to the experimental results on the transport properties of R-BAPB obtained in the present work. Solid lines indicate the Robeson upper bound [8].

**Table 1.**Properties of R-BAPB and commercially available polyimides used in membrane gas separation.

R-BAPB | Kapton [13] | ULTEM [20] | Upilex-R [13] | Matrimid 5218 [15] | P84 [18] | 6FDA-Durene [19] | PBI [16] | |
---|---|---|---|---|---|---|---|---|

T_{g}, K | 485 | 693 | 488 | 543 | 583 | 573 | 697 | 690 |

ρ, kg/m^{3} | 1324 | 1395 | 1280–1300 | 1366 | 1.172 | 1336 | 1333 | 1311 |

FFV,% | 13.9 | 12.9 | 10.8–11.5 | 12.1 | 26.8 | 20.3 | 18.0 | 11.6 |

Gas | Atom | Interaction Potential Parameters | Reference | |
---|---|---|---|---|

ε/k_{B}, K | σ, Å | |||

He | He | 6.030 | 2.6282 | [28,60] |

N_{2} | N | 37.296 | 3.31 | [61] |

O_{2} | O | 44.6 | 3.09 | [62] |

CH_{4} | C | 47.8 | 3.7595 | [63] |

H | 8.5546 | 2.3876 |

**Table 3.**The number of gas molecules in the simulation cell at T = 300 K, calculated on the basis of the excess chemical potentials obtained in molecular dynamics simulations.

Gas | Number of Gas Molecules |
---|---|

CH_{4} | 4 |

N_{2} | 1 |

O_{2} | 3 |

He | 0.1 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Volgin, I.V.; Andreeva, M.V.; Larin, S.V.; Didenko, A.L.; Vaganov, G.V.; Borisov, I.L.; Volkov, A.V.; Klushin, L.I.; Lyulin, S.V. Transport Properties of Thermoplastic R-BAPB Polyimide: Molecular Dynamics Simulations and Experiment. *Polymers* **2019**, *11*, 1775.
https://doi.org/10.3390/polym11111775

**AMA Style**

Volgin IV, Andreeva MV, Larin SV, Didenko AL, Vaganov GV, Borisov IL, Volkov AV, Klushin LI, Lyulin SV. Transport Properties of Thermoplastic R-BAPB Polyimide: Molecular Dynamics Simulations and Experiment. *Polymers*. 2019; 11(11):1775.
https://doi.org/10.3390/polym11111775

**Chicago/Turabian Style**

Volgin, Igor V., Maria V. Andreeva, Sergey V. Larin, Andrey L. Didenko, Gleb V. Vaganov, Ilya L. Borisov, Alexey V. Volkov, Leonid I. Klushin, and Sergey V. Lyulin. 2019. "Transport Properties of Thermoplastic R-BAPB Polyimide: Molecular Dynamics Simulations and Experiment" *Polymers* 11, no. 11: 1775.
https://doi.org/10.3390/polym11111775