J Mol Model (2014) 20:2119 DOI 10.1007/s00894-014-2119-6

ORIGINAL PAPER

Molecular simulation study of PAMAM dendrimer composite membranes Sepideh Amjad-Iranagh & Karim Golzar & Hamid Modarress

Received: 12 September 2013 / Accepted: 14 December 2013 / Published online: 11 February 2014 # Springer-Verlag Berlin Heidelberg 2014

Abstract Pure polysulfone (PSF) and its composites with chitosan (CST), hyaluronic acid (HA), conventional poly(amidoamine), and hydroxyl poly(amidoamine) dendrimers as the membranes for separation of the gases, methane, carbon dioxide, hydrogen sulfide, nitrogen, and oxygen have been studied by molecular dynamics (MD) and grand canonical Monte Carlo (GCMC) simulations. The transport properties (solubility, diffusivity, and permeability) of pure and gas mixtures in the membranes were calculated and the results of the simulations were compared with the available experimental data. The simulated structural properties of the pure and composite PSF membranes including occupied volume, free volume, surface area, fractional free volume (FFV), and radius of gyration (Rg) were evaluated and their effects on the separability of the gases by the membranes were analyzed and interpreted by the obtained results. Keywords Molecular dynamics simulation . Monte Carlo simulation . Transport properties

I kB mi Ni n Pacc p Q rjk ri Rg S T V VS VO VvdW

Scattering intensity curve Boltzmann constant (atm m3 K−1) Mass of atom i (gr mol−1) Current number of component i molecules in the membrane cell Proportionality coefficient Acceptance probability Pressure (atm) Magnitude of the scattering angle (degree Å−1) Distance of atoms j and k in the membrane (Å) Position vector of atom i with respect to the mass center of the molecule (Å) Radius of gyration (Å) Solubility coefficient Temperature (K) Volume of membrane cell (Å3) Specific volume of the polymer chains (Å3) Occupied volume of the polymer chains (Å3) van der Waals volume of the polymer chains (Å3)

Nomenclature C Concentration of gas molecules in the simulation cell (cm3(STP) cm−3 polymer atm−1) E Energy of configuration (kcal mol−1) FFV Fractional free volume fi Fugacity of component i in the gas phase (atm) fk and fj Radial distribution functions

Greek variables Θ Scattering angle (degree) λ X-ray wavelength (Å)

K. Golzar : H. Modarress (*) Department of Chemical Engineering, Amirkabir University of Technology, Tehran, Iran e-mail: [email protected]

HA PSF PSF/CST-HA(x) PSF/CST-HA(x)/C-PAMAM PSF/CST-HA(x)/4OH-PAMAM 4OH-PAMAM

S. Amjad-Iranagh Department of Chemistry, Amirkabir University of Technology, Tehran, Iran

Abbreviations CST C-PAMAM

Chitosan Conventional poly (amidoamine) dendrimer Hyaluronic acid Pure Polysulfone First membrane type Second membrane type Third membrane type Hydroxyl poly (amidoamine) dendrimer

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Introduction Dendritic nanomaterials, which include dendrimers, dendrons, dendigrafts, dendronized polymers, and hyperbranched polymers, are considered as the main building blocks for a variety of nanoscale materials [1–4]. These materials are divided into three subtypes according to their structures. Dendrimers and dendrons are the most controlled structures, followed by dendrigrafts and dendronized polymers, which are semicontrolled, and hyperbranched polymers, which are poorly controlled [5, 6]. Dendrimers are unique synthetic macromolecules with highly branched or tree-like configuration and globular shape and these nanostructural materials are composed of a central core, which correspond to monomer units (interior branch cells) and terminal branch cells [7–9]. The size and structure of dendrimer molecules are highly controllable and their molecular weight distribution is generally very narrow, since the molecular size and generation of dendrimers are increased stepwise via the repetition of a reaction sequence and, because of their compact conformations they do not entangle in spite of their large molecular masses [2, 10–12]. These unique properties of dendrimers are providing new opportunities for this type of polymeric materials, with soft architecture and size in the range of 2–20 nm, to be employed as the hosts for cations, anions and organic/ inorganic solutes, scaffolds and templates for the preparation of metal-encapsulated nanoparticles, scaffolds for bioactive compounds, drug delivery vehicles, gene transfections, medical imaging agents, catalysis and membranes for chemical and separation processes [12–25]. Poly(amidoamine) (PAMAM) dendrimers are the first synthesized and commercialized dendritic polymers family, which are based on an ethylene diamine core and an amidoamine repeat branching structure [26–32]. The size of PAMAM dendrimers and their surface functionality are defined by the number of controlled repetitive additions of monomer units giving rise to different generations and can be synthesized in various well-defined molecular weights [31, 32]. Some characteristics of the PAMAM dendrimers include uniformity, aqueous solubility, easily modified surface chemistry, and controlled size, and have caused the PAMAM dendrimers to be counted as an advantageous material for various applications [2, 15, 16, 27–38]. The interior structure of dendrimers is shown to be capable of encapsulating various hydrophobic or hydrophilic molecules such as drugs inside the nonpolar interior cavities present around the focal core of the dendrimer, therefore, application of dendrimers as drug delivery systems has been of great interest [11, 20, 21, 24, 25, 39–42]. For instance, the fourth generation of PAMAM dendrimer is used as the nanoscale drug delivery unit for controlled release of the drugs [42]. In the last two decades, significant improvements have been made in the performance of the membranes based on

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polymeric materials for the gas separation and attempts have been made to establish meaningful relationships between the structural and transport properties of polymeric membranes [43–46]. Polymeric membranes are more economical than the other conventional membranes in industrial separation of gas mixtures but their application is limited by two essential parameters, gas selectivity and permeability, which highly affect their performance [44, 47]. Membranes based on dendritic materials are used for two reasons: first, dendrimers have an interior structure with cavities that are suitable sites for adsorbing molecules and separate them from their mixtures [44, 47]. It is believed that applying the dendrimers in the membranes causes an increase in the solubility and the permeability of the penetrating gases in the membranes. Second, by changing the end groups of the dendrimers and embedding the most suitable end groups, the interaction energy between the modified surface of the membrane and desired gas molecules to be separated from the mixture will be tuned and this will raise the selectivity of the membrane, significantly [13, 15–18]. Dendrimers are also used as regular or biocides and bacterial membranes either as pure or composite, with other appropriate components such as nanoparticles and common polymer (e.g., poly(vinyl alcohol), chitosan and polysulfone), for chemical and biochemical separation processes [12, 13, 15, 16, 18, 48]. Crespilho et al. [48] developed PAMAM dendrimers with cobalt hexacyanoferrates-modified gold nanoparticles with poly(vinylsulfonic acid) layers on indium tin oxide electrodes as electroactive membranes. Duan et al. [13] prepared the PAMAM membranes which were composed of PSF as a supporting substrate layer and chitosan as a gutter layer and added the hyaluronic acid to the gutter layer for improving the CO2 separation by these composite membranes. Also, they used the hydroxyl PAMAM dendrimer to raise the CO2 selectivity of their produced membranes. The empirical investigation of gas permeation performance of each kind of membrane is difficult, time consuming and costly or needs obtaining special parameters in the critical conditions (such as high temperature and high pressure) [44, 49–59]. On the other hand, by applying the considerable improvements in the computational calculations and using this assumption that the motions of atoms can be described by Newtons’ laws and the interactions among the atoms by the existing empirical potential functions, the molecular dynamics (MD) simulation has provided a general and highly efficient method to simulate various natural processes at the molecular level [54, 60–66]. For instance, Hu et al. [54] investigated the diffusion coefficient and sorption isotherm behavior of the CO2 and CH4 in coal. They calculated the diffusion coefficient of these gases by MD simulation in the order of 10−9 m2/s and obtained the sorption isotherm by applying the grand canonical Monte Carlo (GCMC) method. Pavel and shanks [65] applied the MD simulation technique by utilizing NVE ensemble to evaluate the transport properties of oxygen and

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carbon dioxide as small molecule penetrants in models polyester blends of bulk amorphous poly(ethylene terephthalate) and aromatic polyesters in a temperature range of (300, 500, and 600 K). Lee and Larson [67] simulated the 0 %, 50 %, and 100 % acetylated third and fifth generation of PAMAM dendrimers in dipalmitoylphosphatidylcholine (DPPC) bilayers with explicit water by utilizing the coarse-grained (CG) model and evaluated the effect of dendrimer size, extent of terminal acetylation, temperature, and salt concentration on dendrimer-lipid bilayer interactions and also tested the ability of CG models to predict the experimentally measured dendrimer-bilayer properties. In another study of this group, they simulated the size and internal configuration of the PAMAM dendrimer which was grafted with arginine and histidine [68]. In this work, three distinguished types of polysulfone (PSF) composite membranes were simulated by Materials Studio software (http://accelrys.com/products/materialsstudio). All membranes contained the PSF as a supporting substrate and the differences between the simulated membranes were in their gutter layers. In the first type, the gutter layer contained chitosan (CST). This layer for the second type was prepared byadding the conventional poly(amidoamine) dendrimer (C-PAMAM) to chitosan. In the third type, to study the role of the end groups on each characteristic of dendritic composite membrane, the novel hydroxyl poly(amidoamine) dendrimer (4OH-PAMAM) was embedded into the gutter layers of membranes instead of conventional poly(amidoamine) dendrimer. The hyaluronic acid (HA) concentration, in the three membrane types, was changed from 0 to 50 %. Some properties including radius of gyration for PSF, fraction free volume, X-ray patterns, and solubilites of some gases including CH4, CO2, H2S, N2, and O2, which are commonly used in chemical industry, were calculated and for scrutinizing the accuracy of the obtained simulation results, a comparison between simulation results and the available experimental data has been made. The first, second and third types of the membranes are denoted by PSF/ CST-HA(x), PSF/CST-HA(x)/C-PAMAM, and PSF/CSTHA(x)/4OH-PAMAM, respectively. In these abbreviations, x refers to the weight percent of HA molecules in each membrane.

Molecular simulation methodology Force field It is possible to simulate a system accurately by applying the quantum mechanical techniques, but these techniques are often too time consuming and are usually feasible just for systems that include a few hundred interacting particles. Fortunately, the system bulk properties are primarily controlled

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by the atomic nuclei location and knowledge of the electronic structure of atoms which are provided by the quantum mechanical techniques is not required. By utilizing a reasonable, physically based approximation of the force field an insight into the physical and mechanical behaviors of a system can be obtained. For this purpose, a force field that generates a set of system configurations which are statistically consistent with a fully quantum mechanical description is used [69–71]. Based on the above mentioned assumptions the force field must be capable of calculating all the essential interaction energies of the particles in the system and should be able to produce the calculated results consistent with the experimental data, therefore, in atomistic simulation study of multi particle systems the choice of the appropriate force field is a crucial point. The suitable force field must calculate only the important interactions and ignore the weak interactions among the atoms and thereby reducing the time of calculations. Therefore, the force field should be capable of fast and accurate calculation of various kinds of important interaction energies, according to the atomic structure of the material under study. All the simulations in this study have been conducted by utilizing Materials Studio 4.3 developed by Accelrys Software Inc (http://accelrys.com/products/materials-studio). For calculating the interaction energies between PSF, CST, HA, C-PAMAM, 4OH-PAMAM dendrimers and gas molecules condensed-phase optimized molecular potential for atomistic simulation studies (COMPASS) [71] force field was utilized. It is necessary to note that the applied simulation software (http://accelrys.com/products/materials-studio) includes COMPASS, Dreiding and PCFF force fields. The density of pure PSF membrane, calculated in this work, by COMPASS force field is in close agreement with the experimental result represent in ref. [72] in comparison to the calculated density of pure PSF membrane by Dreiding and PCFF force fields. Therefore, this force field is selected for all calculations in simulation process. The detailed information about this force field and also the equations for calculating the attractive and repulsive interaction energies between the molecules of the simulated system can be found in the literature [71, 73, 74]. Construction of amorphous cells To evaluate the solubility and solubility-selectivity of the gases including CH4, CO2, H2S, N2, and O2 into the composite membranes based on PSF, CST, HA, C-PAMAM, and 4OH-PAMAM dendrimers, by employing the molecular dynamics (MD) simulation, the repeating unit of these components as shown in Fig. 1a, d, g, j, and l, respectively, were drawn by Materials Studio 6.0 software (http://accelrys.com/ products/materials-studio). The initial monomer structures were located in the Forcite and Discover modules of Materials Studio software and to perform the geometry optimization and energy minimization, COMPASS force

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J Mol Model (2014) 20:2119 O

O S

CH3 O

C

O

CH3

A OH

OH

O

O

HO

HO

O

B

C

E

F

H

I

O NHCOCH3

NH2

n

m

D COOH

OH

HO O

O

HO

O

O

OH

NHCOCH3 n

G H2N

O N H

O

N H

N

H2N

NH2

N

H N

H N

O O

NH2

J

K

O

H2N HO

O

N H

N H

N

NH2 HO

OH H2N

N

H N

H N

O O

OH NH2

L

M

Fig. 1 Monomer structures

field was used along with the smart minimizer method. This method is a part of the Amorphous Cell module of Materials Studio software and combines the steepest descent, conjugate gradient and Newton methods in a cascading manner to minimize the energy of the molecular configurations of each material. Then the resulted configurations of the monomers

were neutralized (see Fig. 1b, e, h, k, and m). To construct the polymer chains for PSF, CST, and HA, the Amorphous Cell module of the software was utilized. This software is capable of producing the initial configurations and then folding these chains at all densities by using their mirror images [72]. In this simulation study, all PSF, CST, and HA chains had

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respectively 10, 10, and 20 repeating units (see Fig. 1c, f, and i). Before embedding the chains in the amorphous cells, it must be subjected to the geometry optimization and energy minimization by performing the same procedure as used for their monomers. To simulate the pure PSF membrane (without gutter layer), one PSF chain (with ten repeating units) were folded in the amorphous cell under periodic boundary condition at initial density of 0.868 g/cm3, which corresponds to the 70 % of a well-accepted PSF experimental density (1.24 g/cm3) [72, 75]. The initial constructed atomistic configuration was subsequently optimized and minimized by the following procedure. To eliminate the undesirable contacts including overlapping and close contacts, a 20,000-step energy minimization was adopted. An annealing procedure was performed by using the temperature cycle protocol in the Forcite module and the system was heated at intervals of 20 ˚C from 80 to 280 °C, well above the glass transition temperature (Tg) of the polymer and then it was cooled back at intervals of 20 ˚C and at each step, 100 ps MD simulation on the NPT ensemble was applied to achieve the equilibrium structure, that is the lowest conformation energy of the polymer chain and is similar to the real configuration of polymer chain in amorphous cell. After this stage the equilibrium density of the amorphous cell was evaluated by using the 2 ns NPT simulation process. In the NPT ensemble the number of molecules, N, pressure, P, and the temperature, T, of the system are kept constant. To control pressure (at 1 atm) and temperature (at 313.15 K) of the system, the Andersen barostat and Nose thermostat were utilized [72, 76–78]. The equations of motion were integrated by the velocity Verlet algorithm with a time step of 1 fs for all simulation runs. The electrostatic interaction were calculated by Ewald summation method [78] with the accuracy 0.001 kcal mol−1 and the van der Waals interaction were approximated by the Lennard–Jones 6–12 potential with a cut-off distance of 9–18 Å (with a spline width of 0.1 nm and a buffer width of 0.05 nm). This range of cut-off distance was chosen because, in this study, the variation range of simulation cell length is approximately 19–37 Å and the cutoff distance must be less than half of the cell length. On the endpoint of the NPT run, to confirm that the equilibrium molecular structure was achieved, an additional 500 ps NVT run was performed, and the atomic trajectories were recorded every 1 ps for the subsequent analysis. At the end of this minimization and optimization procedure, the relaxed configuration (as shown in Fig. 1c) with the cell length of 19.33±0.03 Å and the equilibrium density of 1.17±0.08 g/ cm3 (the experimental density of PSF is 1.24 g/cm3) was obtained. In a previous study by Wang et al. [78] the amorphous cell contained one chain of PSF with 20 repeating units and the simulation cell length was around 18.1 Å which is in agreement with the cell length used in this study.

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In this work, to construct the simulation cell for the other studied membranes which included gutter layer, by using the detail information about the number of each component in each membrane as presented in Tables 1, 2, and 3, all simulation cells were made and then by following the above mentioned procedure step by step for minimizing and optimizing their structures, the real configurations or the relaxed architectures were obtained. For example, to fabricate the special membrane that includes the PSF and chitosan as gutter layer with 20 % HA (in this study, this type of membranes is denoted as PSF/CST-HA(x), where x is the weight percent of HA in the membrane), the number of PSF, CST and HA chains were 1, 3, and 2, respectively. This type of membranes did not contain C-PAMAM or 4OH-PAMAM dendrimers, but for manufacturing the membrane that its gutter layer consisted of CST, 20 % HA and 5 % C-PAMAM, respectively 3, 2, 7 chains of each components were embedded. It is necessary to note that in this simulation study, the amount of HA in each type of membranes was changed in range of 0–50 %, but the percentage of C-PAMAM and 4OHPAMAM in the second and third types was fixed and was kept at 5 %. In the last column of Table 1, the exact amounts of HA for each membrane in the first type of membranes are presented. For example, to make the simulation cell for PSF/CSTHA(50), membrane which contains 50 % HA, the exact percentage of HA is 49.08 %. The exquisite values of HA, CPAMAM and 4OH-PAMAM for the second and third membrane types have been listed in the last two columns of Tables 2 and 3, respectively.

Gas solubility of membranes In the simulation process, by applying the grand canonical Monte Carlo method (GCMC) the solubility was calculated [79, 80]. For this purpose, gas molecules were randomly embedded in the membrane models and by using the Sorption module in Materials Studio software (http://accelrys.com/ products/materials-studio) the solubility of gases in membranes were calculated [75]. Non-bonded energy of all Table 1 The number chain(s) of each component in the first type of simulated membrane No.

Sample

PSF

CST

HA

HA/(CST+HA)

1 2 3 4 5 6 7

PSF PSF/CST-HA(0) PSF/CST-HA(10) PSF/CST-HA(20) PSF/CST-HA(30) PSF/CST-HA(40) PSF/CST-HA(50)

1 1 1 1 1 1 1

0 1 7 3 4 4 3

0 0 2 2 4 7 7

0.00 0.00 10.55 21.59 29.23 41.95 49.08

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Table 2 The number chain(s) of each component in the second type of simulated membrane No.

Sample

PSF

CST

HA

1 2 3 4

PSF/C-PAMAM PSF/CST-HA(0)/C-PAMAM PSF/CST-HA(10)/C-PAMAM PSF/CST-HA(20)/C-PAMAM

1 1 1 1

0 1 3 3

0 0 1 2

5 6 7

PSF/CST-HA(30)/C-PAMAM PSF/CST-HA(40)/C-PAMAM PSF/CST-HA(50)/C-PAMAM

1 1 1

4 4 3

4 7 8

molecules in the interface of the model molecule was calculated by the COMPASS force field [70, 73]. In this force field the sorption process is simulated by using both van der Waals and Coulombic forces. In the GCMC method, the concentration probability of the penetrant gas molecules based on the energy change between new configuration and previous configurations, ΔE, is determined [80]. In this procedure, the well-known Metropolis algorithm is used to accept or reject a configurational move of a penetrant gas molecule. To simulate the absorption in the process of gas penetration into the membrane, the sorbate molecules can be translated, rotated, created and destroyed at random probabilities. By displacing randomly a penetrate gas molecule in the x, y, or z directions, the translational move with an acceptance probability (Pacc), as expressed by the following equation, was calculated [77, 80]: 
 ΔE Pacc ðold→newÞ ¼ min 1; exp − ; kBT

ð1Þ

where ΔE can be calculated from the non-bonded potential terms (van der Waals and Coulombic interaction) which are obtained from the energy change between new and previous configurations of the sorbate molecules, kB is the Boltzmann constant and T is the Kelvin temperature. For the rotation movement, a random sorbate molecule in the membrane cell is chosen. After choosing the rotation axis for this movement,

PAMAM

HA/(CST+HA+ PAMAM)

PAMAM/(CST+ HA+PAMAM)

1 2 7 7

0.00 0.00 11.44 20.54

100 5.27 5.40 4.84

11 13 12

27.73 39.86 49.77

5.13 4.99 5.03

randomly, the molecule is rotated by a random amount. The energy of the new configuration is calculated by the same probability formula as expressed by Eq. (1). The acceptance probability of creation and destruction are respectively expressed by the following equations [77, 80]: 




ΔE ðN i þ 1Þk B T Pacc ðold→newÞ ¼ min 1; exp − −ln kBT fiV

 ð2Þ


 ΔE N ikBT  ln Pacc ðold→newÞ ¼ min 1; exp − ; kBT f iV

ð3Þ

where fi is the fugacity of component i in the gas phase, and Ni is the current number of molecule i and V is the membrane cell volume. By performing the simulation over a range of pressures from 0 to 1 atm, where at this low pressure the fugacity, fi, in Eqs. 2 and 3, can be replaced by pressure, the solubility can be calculated in the form of a plot of the concentration of the penetrant gas C, versus pressure (or fugacity) at constant temperature. At each pressure, 1,000,000 steps of GCMC calculations were performed with the initial equilibration period of 100,000 steps. The solubility coefficient, S, is then

Table 3 The number chain(s) of each component in the third type of simulated membrane No.

Sample

PSF

CTS

HA

4OHPAMAM

HA/(CST+HA+ 4OH-PAMAM)

4OH-PAMAM/ (CST+HA+4OHPAMAM)

1 2 3 4 5 6 7

PSF/4OH-PAMAM PSF/CST-HA(0)/4OH-PAMAM PSF/CST-HA(10)/4OH-PAMAM PSF/CST-HA(20)/4OH-PAMAM PSF/CST-HA(30)/4OH-PAMAM PSF/CST-HA(40)/4OH-PAMAM PSF/CST-HA(50)/4OH-PAMAM

1 1 1 1 1 1 1

0 1 3 3 2 3 2

0 0 1 2 2 5 5

1 2 6 7 5 9 7

0.00 0.00 11.47 20.42 27.70 38.62 48.20

100 5.88 5.21 5.41 5.24 5.26 5.11

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evaluated from the limiting slope of the solubility at zero pressure in the following form [81]:
 C S ¼ lim ; ð4Þ p→0 p where C is in the units of cm3(STP)/cm3 polymer, and p is the pressure. Gas diffusivity of membranes In MD simulation, diffusion coefficient for the penetrant is calculated by means of the Einstein equation [82–85]. D¼

N E 1 d X D lim ri ðt Þ−ri ð0Þ2  ; 6N t→∞ dt t¼1

ð5Þ

where N is the number of diffusing atoms i, ri is the position vector of atom i, ± represents the ensemble average of the mean square displacement (MSD) of the inserted gas molecule trajectories; ri(t) and ri(0) are the final and initial positions of the center of mass of the gas molecules over the time interval t. The diffusion coefficient can also be calculated from the velocity autocorrelation function that was developed by simulation software. The Einstein relationship assumes a random walk for the diffusing particles. For particles that diffuse slowly, anomalous diffusion is sometimes observed and is characterized by [82, 83]: hjri ðt Þ−ri ð0Þj∝t n i;

ð6Þ

complex system such as amorphous polymeric cell is fully equilibrated but by checking some criteria such as potential energy and temperature, it is possible to ensure the simulated amorphous cell has reached the fully thermodynamic equilibrium state. In this simulation study, in order to verity cell equilibration, two well-known equilibrium parameters including potential energy and temperature were monitored during the MD simulations. As can be seen from Fig. 2, both the total potential energy and temperature stays unchanged with negligible fluctuations through-out the 2 ns production phase of the MD simulations for the membranes of the first type (similar to the two other types of the simulated membranes) therefore, it is presumed that all simulated systems reached the equilibrium state and the evaluated properties are reliable. Another indication of the equilibrium state for the simulated membranes that contain dendrimer molecule, is to obtain an acceptable relationship between the radius of gyration (Rg) and the number of atoms, N, in the dendrimer molecule (Rg ∝ Nn) [86]. In this study, the values of n, for C-PAMAM and 4OH-PAMAM were calculated as 0.33 and 0.35, respectively which are very close to the experimental value (0.31), therefore, it can be concluded that the final structures of the simulation cells and the calculated properties, were in the equilibrium state.

X-ray diffraction

where n for this regime is less than one, whereas, if the simulation time is more than the hydrodynamic limit and sufficiently long, a transition from anomalous to Einstein diffusion may be observed. For this regime, a linear relationship between MSD and t may be observed, so n in Eq. (6) is equal to one.

The simulated scattering intensity curve, I(Q), for each membrane can be related through a Fourier transform operation to the radial distribution functions fk and fj by the following equation [87]:   X X f j f k sinQ rjk I ðQ Þ ¼ ; ð7Þ Q rjk j k

Gas permeability of membranes

where the magnitude of the scattering angle, Q, is given by:

Since the permeability depends on the solubility and diffusivity, after predicting the diffusivity and solubility of each nanocomposite membranes, the corresponding value of permeability of the membrane can be calculated by employing PA=DA ×SA as has been used successfully by other researchers [68, 74, 80].

Q¼

Results and discussion Amorphous cell equilibrium One of the crucial steps in the MD method is fully equilibrating the simulated amorphous cell. In this context, it is necessary to notice that there is no guarantee that a simulated

4 πsinθ : λ

ð8Þ

In this equation, θ is the scattering angle, λ is the X-ray wavelength and rjk is the distance of atoms j and k in the simulation cell of membrane. The indices j and k range over all the atoms in the molecule [87]. Generally, the maximum peak of the diffraction pattern is noteworthy, because by using it along with the Bragg equation, d=λ/2sinθ, the corresponding d-spacing values that are representative of intersegmental distances between polymer backbones can be calculated. By allowing the real-time coupling of the structure modeling to the experimental results and applying the Forcite modules of the simulation software (http://accelrys.com/products/ materials-studio), the X-ray diffraction pattern of each membrane was evaluated. The angle of diffraction was varied from

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Fig. 2 Total potential energy and temperature

0° to 30° using a step size of 0.05˚. The X-ray scattering pattern for the crystalline structure consists of a series of sharp peaks, but for amorphous materials it has a broad background signal. Therefore, the X-ray scattering can be used to determine the crystallinity by comparing the integrated intensity of the background pattern to that of the sharp peaks. The polymeric materials show a semi ordered crystal structure due to the folding of the polymer chain [77]. The simulated XRD patterns of the final structures for the simulated membranes in the first membrane type which are presented in Fig. 3 show a sharp peak at 2θ=0–5° (like the second and third types). Therefore, the membranes exhibit a semi-crystalline structure. In addition, it can be seen that the values of 2θ angle decrease by raising the amount of HA, so, the d-spacing which has direct influence on transport properties of composite membranes (particularly solubility), becomes larger. For example, the maximum and minimum of 2θ for the first type are obtained for PSF/CST-HA(0) and PSF/CST-HA(50), respectively, and the same behavior is seen for the other membrane types. Therefore, as expected and as will be confirmed by the calculated results which will be presented later in this text, the solubility properties of the membranes, for the gases, with 50 % HA (e.g., PSF/CST-HA(50), PSF/CST-HA(50)/CPAMAM, and PSF/CST-HA(50)/4OH-PAMAM) is higher than other membranes, because these composites have the highest amount of d-spacing and as a result the highest ability to dissolve gases in themselves. The X-ray patterns also show that addition of C-PAMAM and 4OH-PAMAM molecules to the simulation cells of the membranes have no significant effect.

phase which is occupied by polymer chains and an empty space phase known as free volume are determined [77]. Molecular diffusion through a polymer membrane depends strongly on its morphology and free volume. The free volume is the sum of the static voids which are created by chain packing or transient gaps generated by thermally induced chain rearrangement and provides the diffusing molecules a low-resistance path for their transport [88]. The larger and the more numerous free volume elements, would allow larger numbers of gas molecules to migrate through a polymer based membrane. The free volume usually has a direct influence on the gas solubility in the polymers, that is the solubility increases with increasing polymer free volume, but this influence has much weaker effect on the diffusivity [88]. The permeability is resulted from two contributions; the solubility and diffusivity, so the effect of free volume on the permeability is deduced from the influence of free volume on these parameters.

Free volume characteristic In considering the configuration of a membrane based on polymeric materials, two distinct phases including a solid

Fig. 3 Simulated XRD patterns of the final structures for the simulated membranes in the first membrane type

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To investigate the transport and selectivity characteristics of any membranes, the fractional free volume (FFV) is calculated as the most important quantity, because the FFV has a direct influence on the transport properties of membranes. The FFV is commonly employed to characterize the efficiency of chain packing and the amount of free space in a polymer matrix [89, 90]. Generally, the FFV parameter can be evaluated from Bondi group contribution methods [91] as the ratio of the difference between the polymer bulk specific volume and the volume occupied by the polymer chains. The occupied volume of the polymer chains is expressed as 1.3 times the van der Waals volume of the polymer chains, and the free volume is the outlines of the van der Waals surface of polymer chains [91]. The well-known definition of FFV is given as: FFV ¼

V S −V O V S −1:3 V vdW ¼ ; VS VS

ð9Þ

where VS, VO and VvdW are the specific volume, occupied volume and the van der Waals volume of the polymer chains, respectively. Table 4 presents the cell length, occupied volume, free volume, surface area and FFV for all studied membranes. The cell length for the pure PSF membrane is calculated as 19.33±0.03 Å, whereas by embedding the 10 % HA into this

membrane, increases to 23.98±0.08 Å. The cell length variation in each membrane type shows an increasing trend. Addition of 0-50 % HA molecules to the membranes raises the cell length of the first, second and third types from 19.33±0.03 to 35.74±0.01 Å, 19.15±0.07 to 37.04±0.09 Å, and 20.20±0.05 to 36.96±0.10 Å, respectively. For the occupied volume, free volume and surface area parameters, the same increasing trend can be observed. By comparing the exact amounts of each parameter for the three membrane types, it can be understood that by embedding the C-PAMAM into the simulation cells, the FFV decreases slightly, but by adding the 4OH-PAMAM dendrimer, these parameters change to the greater values. For example, the FFV values for PSF/CST-HA(40), PSF/CSTHA(40)/C-PAMAM, and PSF/CST-HA(40)/4OH-PAMAM are 0.4022±0.02, 0.3862±0.03, and 0.4169±0.07, respectively. Therefore, the order of FFV values for the three membrane types is PSF/CST-HA(x)/4OH-PAMAM>PSF/CST-HA(x)> PSF/CST-HA(x)/C-PAMAM. The range of increase in FFV values for all the simulated membranes is less than 0.42 and the actual ranges for the first, second, and third type of membranes are 0.3263±0.03–0.4124±0.05, 0.2044±0.01– 0.3901±0.07, and 0.3648±0.06–0.4135±0.05, respectively. In Fig. 4, the snapshots of all simulation cells for the studied membranes after 2 ns simulation time are presented. By comparing the snapshots of each membrane type, it can be

Table 4 Some volume characteristics of simulation cells of all membranes and the radius of gyration of the PSF chains in each type of simulated membranes No.

Sample

Cell length (Å3)

Occupied volume (Å3)

Free volume (Å3)

Surface area (Å2)

Fractional free volume

Radius of gyration (Å)

1 2 3 4 5 6 7 8 9 10 11 12 13

PSF PSF/CST-HA(0) PSF/CST-HA(10) PSF/CST-HA(20) PSF/CST-HA(30) PSF/CST-HA(40) PSF/CST-HA(50) PSF/C-PAMAM PSF/CST-HA(0)/C-PAMAM PSF/CST-HA(10)/C-PAMAM PSF/CST-HA(20)/C-PAMAM PSF/CST-HA(30)/C-PAMAM PSF/CST-HA(40)/C-PAMAM

19.33±0.03 23.98±0.08 41.54±0.03 34.04±0.05 31.00±0.01 34.01±0.04 35.74±0.01 19.15±0.07 24.93±0.02 30.94±0.09 35.23±0.07 32.14±0.03 35.20±0.02

4870.93±0.12 8660.62±0.16 42779.26±0.11 23834.89±0.43 17897.35±0.98 23524.78±0.28 26847.37±0.48 5594.02±0.10 9937.26±0.35 18403.56±0.87 26869.87±0.54 20491.15±0.21 26788.58±0.78

2359.54±0.53 5135.24±0.88 28921.70±0.26 15633.38±0.63 11901.14±0.99 15829.53±0.18 18839.91±0.47 1437.41±0.21 5574.02±0.66 11230.44±0.90 16886.86±0.87 12737.96±0.75 16854.26±0.98

2831.6±0.89 3876.41±0.78 18660.54±0.99 10617.60±0.43 8826.81±0.83 10925.06±0.56 12329.84±0.20 2174.82±0.45 4425.53±0.76 7799.34±0.91 11173.16±0.87 8993.74±0.43 9912.71±0.56

0.3263±0.03 0.3722±0.04 0.4034±0.06 0.3961±0.03 0.3994±0.08 0.4022±0.02 0.4124±0.05 0.2044±0.01 0.3594±0.03 0.3726±0.09 0.3859±0.03 0.3833±0.05 0.3862±0.03

11.09±0.03 16.02±0.05 17.00±0.07 17.52±0.01 17.59±0.03 22.51±0.05 26.48±0.04 13.48±0.04 19.94±0.06 21.01±0.06 22.71±0.07 25.57±0.06 26.32±0.07

14 15 16 17 18 19 20 21

PSF/CST-HA(50)/C-PAMAM PSF/4OH-PAMAM PSF/CST-HA(0)/4OH-PAMAM PSF/CST-HA(10)/4OH-PAMAM PSF/CST-HA(20)/4OH-PAMAM PSF/CST-HA(30)/4OH-PAMAM PSF/CST-HA(40)/4OH-PAMAM PSF/CST-HA(50)/4OH-PAMAM

37.04±0.09 20.20±0.05 25.05±0.05 31.07±0.01 35.38±0.04 32.28±0.08 35.35±0.02 36.96±0.10

31005.19±0.92 5236.78±0.33 9824.48±0.68 18269.23±0.21 26713.99±0.58 20300.59±0.81 25759.87±0.37 29621.31±0.14

19828.32±0.67 3007.94±0.23 5899.17±0.28 11736.51±0.54 17573.85±0.12 13353.57±0.51 18414.29±0.74 20885.63±0.19

12366.82±0.18 2579.64±0.76 4440.83±0.17 8378.80±0.43 12316.77±0.65 9046.95±0.71 11062.91±0.88 13617.85±0.95

0.3901±0.07 0.3648±0.06 0.3752±0.04 0.3865±0.03 0.3968±0.07 0.3968±0.01 0.4169±0.07 0.4135±0.05

26.98±0.02 17.35±0.06 18.11±0.02 18.25±0.07 18.60±0.08 19.90±0.04 26.95±0.07 25.31±0.01

2119, Page 10 of 20

J Mol Model (2014) 20:2119

First Type

PSF

PSF/CST-HA(0)

PSF/CST-HA(10)

PSF/CST-HA(20)

PSF/CST-HA(30)

PSF/CST-HA(40)

PSF/CST-HA(50)

PSF/CST-HA(30)/C-PAMAM

PSF/CST-HA(40)/C-PAMAM

PSF/CST-HA(50)/C-PAMAM

PSF/CST-HA(30)/4OH-PAMAM

PSF/CST-HA(40)/4OH-PAMAM

PSF/CST-HA(50)/4OH-PAMAM

Second Type

PSF/C-PAMAM

PSF/CST-HA(0)/C-PAMAM

PSF/CST-HA(10)/C-PAMAM

PSF/CST-HA(20)/C-PAMAM

Third Type

PSF/4OH-PAMAM

PSF/CST-HA(0)/4OH-PAMAM

PSF/CST-HA(10)/4OH-PAMAM

PSF/CST-HA(20)/4OH-PAMAM

Fig. 4 Snapshots of all simulation cells for the studied membranes after 2 ns simulation time

seen that the blue or white areas of the figures which are related to the free volumes of membranes were raised significantly. The gray area, in Fig. 4, shows the occupied volume of simulation cell, so, the FFV increases by adding HA molecules to the membranes. Therefore, the solubility of penetrating gas molecules in the membranes which have a higher amount of HA or contains C-PAMAM and 4OH-PAMAM molecules is greater than other membranes. As a result, adding C-PAMAM and 4OH-PAMAM dendrimers or increasing the percentage HA molecules into the gutter layer of conventional membranes, cause an increase in the FFVand create more suitable sites to adsorb the penetrating gas molecules and enhancing their solubility in the dendritic composite membranes. Radius of gyration The radius of gyration (Rg) is used to describe the dimensions of a polymer chain [77] and is calculated as follows: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u0 X ðj r i j Þ2 m i u uB i C Rg ¼ u A; t@ X mi

ð10Þ

i

where mi is the mass of atom i and ri is the position vector of atom i with respect to the mass center of the molecule. The Rg for the PSF chain in each simulation cell of the membrane has been tabulated in Table 4. By analyzing the obtained values of Rg for the three types of the studied membranes two results can be

obtained: (i) in the first type of the simulated membranes, by adding the HA to the pure PSF the Rg rises from 11.09±0.03 Å to 16.02±0.05 Å and by adding the HA to the gutter layers of the membranes from 10 to 50 %, Rg increases to 26.48±0.04 Å. The interaction forces of CST and HA molecules with the PSF chain in each membrane cause an increase in the Rg values. The same increasing trend can be observed in the simulation results of Rg for the other two membrane types. For example, in the second types as shown in Table 4, the Rg for PSF/CST-HA(0)/PAMAM is 13.48±0.04 Å and shifts to 26.98±0.02 Å for PSF/CSTHA(50)/PAMAM. In the third type, by adding the HA molecules (from 0 to 50 %) to the gutter layer, the Rg of the membranes increases from 17.35±0.06 Å to 25.31±0.01 Å. (ii) for the second and third simulated types which have C-PAMAM and 4OH-PAMAM molecules in their simulation cells Rg is greater than the corresponding membranes in the first type that contain the same percentage of HA. The increase in Rg is due to the extra interaction between PSF chain and C-PAMAM molecules in the second membrane type and PSF chain and 4OH-PAMAM molecules in the third membrane type, therefore, in these membranes, the PSF chain becomes rigid and the segmental motions is hindered and Rg increases. Gas transport behaviors Gas solubility of membranes The solubility of the pure gases (CH4, CO2, H2S, N2, and O2) and their binary mixtures (CO2/CH4, CO2/H2S, CO2/N2, and

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CO2/O2) in the three types of the simulated membranes were evaluated by using the grand canonical Monte Carlo (GCMC) simulation [54, 77]. The slope of the concentration curve versus pressure at zero pressure limits is known as the solubility [77]. The solubility results for pure CH4, CO2, H2S, N2, and O2 gases into the three type membranes at 313.15 K and in 0–1 atm pressure are presented in Table 5. As is seen from this table, the solubility for the PSF/CST-HA(0) is higher than the corresponding values in the pure PSF membrane. For example, the CO2 solubility in the PSF/CST-HA(0) is 0.1869± 0.08 cm3(STP)/cm3atm, whereas this value for the pure PSF is 0.1292±0.04 cm3(STP)/cm3atm. These results, in Table 5, indicate that, only adding CST molecules as the gutter layer in the simulation cell of the pure PSF membrane, increase the solubility to 69.12 %, because the adsorption of PSF/CSTHA(0) membrane, due to the presence of the active side groups in chemical structure of CST, is larger than PSF. This enhancement in the gas solubility for CH4, H2S, N2, and O2 are 130.70, 163.24, 110.06, and 108.07 %, respectively. The maximum and minimum effects on the solubility, by adding CST molecules to the pure PSF simulation cell, as is seen in Table 5, are observed for H2S and CO2, respectively. Therefore, the PSF membrane which was composited with CST is appropriate for separation of both H2S and CO2 from gas mixture. By adding the CST to the PSF/C-PAMAM and PSF/4OH-PAMAM membranes, the same increasing trends

for gas solubility are observed. Also as Table 5 shows, by adding the CST to the PSF/C-PAMAM membrane, only the H2S solubility is significantly increased (from 0.1091±0.01 to 0.4664±0.05 cm3(STP)/cm3atm), but for other gases this enhancement is not significant. Therefore, according to the obtained simulation results, PSF/CST-HA(0)/C-PAMAM membrane was recommended for H2S removing any gas separation and purification process such as sweetening natural gas. In the second type of the simulated membranes, only CPAMAM dendrimer was added to the simulation cell of the pure PSF. The results (Table 5) for this membrane show an increase in the solubility of CO2 and N2 compared with the pure PSF membrane, whereas for CH4, H2S, and O2 the results show a decrease. However, by adding 5 % 4OHPAMAM dendrimer to the PSF (third type of the simulated membranes) the solubility of the gases is increased. This increase can be attributed to the effective interactions of CPAMAM and 4OH-PAMAM with the gas molecules. By comparing the simulation results of the solubility for each membrane type (Table 5), it is observed that: (i) the solubility of H2S in the membranes is greater than other gases, because it has stronger interaction with the membrane and the solubility of gases is in the order of H2S > CO2 > CH4 > O2 > N2. (ii) By adding C-PAMAM and 4OH-PAMAM dendrimers to the gutter layer of membranes, the solubility of the gases in

Table 5 The solubility (cm3(STP)/cm3 atm) results of all pure gases in three simulated membrane types No.

Sample

CH4

CO2

H2S

N2

O2

1 2 3 4 5 6 7 8 9 10 11 12 13

PSF PSF/CST-HA(0) PSF/CST-HA(10) PSF/CST-HA(20) PSF/CST-HA(30) PSF/CST-HA(40) PSF/CST-HA(50) PSF/C-PAMAM PSF/CST-HA(0)/C-PAMAM PSF/CST-HA(10)/C-PAMAM PSF/CST-HA(20)/C-PAMAM PSF/CST-HA(30)/C-PAMAM PSF/CST-HA(40)/C-PAMAM

0.0381±0.004 0.0498±0.006 0.0529±0.008 0.0558±0.001 0.0663±0.005 0.0679±0.008 0.0765±0.006 0.0143±0.007 0.0434±0.009 0.0666±0.002 0.0698±0.005 0.0781±0.004 0.0837±0.008

0.1292±0.004 0.1869±0.008 0.1998±0.001 0.2119±0.007 0.2201±0.003 0.2493±0.005 0.2582±0.005 0.1351±0.001 0.1849±0.006 0.1838±0.008 0.1828±0.003 0.2371±0.004 0.2499±0.005

0.2054±0.002 0.3353±0.005 0.3492±0.001 0.3822±0.005 0.4504±0.005 0.4553±0.007 0.5092±0.008 0.1091±0.001 0.4664±0.005 0.4238±0.002 0.4812±0.006 0.5553±0.007 0.5931±0.002

0.0149±0.003 0.0164±0.004 0.0203±0.001 0.0209±0.006 0.0225±0.005 0.0227±0.006 0.0233±0.007 0.0151±0.005 0.0166±0.009 0.0178±0.008 0.0219±0.001 0.0213±0.003 0.0366±0.004

0.0161±0.002 0.0174±0.007 0.0206±0.003 0.0209±0.006 0.0235±0.002 0.0239±0.005 0.0235±0.006 0.0148±0.001 0.0178±0.005 0.0193±0.001 0.0208±0.004 0.0229±0.009 0.0236±0.003

14 15 16 17 18 19 20 21

PSF/CST-HA(50)/C-PAMAM PSF/4OH-PAMAM PSF/CST-HA(0)/4OH-PAMAM PSF/CST-HA(10)/4OH-PAMAM PSF/CST-HA(20)/4OH-PAMAM PSF/CST-HA(30)/4OH-PAMAM PSF/CST-HA(40)/4OH-PAMAM PSF/CST-HA(50)/4OH-PAMAM

0.0812±0.003 0.0466±0.006 0.0695±0.005 0.0777±0. 006 0.0859±0.010 0.0971±0.009 0.0646±0.007 0.0827±0.002

0.2965±0.001 0.2998±0.006 0.3191±0.003 0.4067±0.004 0.4044±0.007 0.4691±0.008 0.4863±0.001 0.4974±0.005

0.6116±0.001 0.5985±0.007 0.6493±0.001 0.6618±0.006 0.6743±0.006 0.4493±0.008 0.6106±0.008 0.7806±0.009

0.0498±0.003 0.0375±0.001 0.0397±0.003 0.0367±0.006 0.0397±0.004 0.0435±0.001 0.0456±0.008 0.0507±0.005

0.0249±0.007 0.0269±0.006 0.0371±0.001 0.0384±0.008 0.0397±0.004 0.0341±0.003 0.0357±0.004 0.0414±0.007

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the second and third membrane types, as listed in Table 5, is enhanced. The results in Table 5 shows that by embedding CPAMAM and 4OH-PAMAM dendrimers to the PSF/CSTHA(40) composite membrane, the N2 solubilities increases and the influence of 4OH-PAMAM dendrimer is more than CPAMAM dendrimer. Therefore, by applying a proper structural change in the dendrimers, it is possible to increase the separation ability of the conventional membranes. (iii) In each type of membrane, adding HA to the gutter layer of the membranes, increases solubilities of all gases. The maximum effect of this addition, on the solubility was observed for CH4 solubility in the PSF/CST-HA(x)/C-PAMAM membranes. The solubility of binary mixtures CO2/CH4, CO2/H2S, CO2/N2, and CO2/O2 in four different pure and composite membranes including pure PSF, PSF/CST-HA(0), PSF/CPAMAM, and PSF/4OH-PAMAM, were investigated by the GCMC simulation method. It was found that addition of CST, C-PAMAM, and 4OH-PAMAM dendrimers affects the separation characteristics of PSF based membranes. By considering the fact that the removal of CO2 from gas mixtures is a global environmental concern [13, 15, 16, 92], in all binary gas mixtures of this study, CO2 was used. As is observed from Table 6, the solubility-selectivity of membranes for CO2/CH4 to capture the CO2 molecules is higher than CH4. However, the selectivity of membranes consisting of pure PSF and composite with CST for this gas mixture is close to each other. By adding 5 % C-PAMAM or 4OH-PAMAM dendrimers to the PSF membrane, without changing any operating conditions of separation process, the CO2 solubility selectivities were greatly increased. The solubility-selectivity value of PSF/4OH-PAMAM indicates that this membrane is the best choice for removing CO2 from CH4. Since the mole fraction of CH4 in the natural gas is more than 90 %, it is plausible to assume that the natural gas behavior is approximately similar to the pure CH4, so on the basis of the calculation results the PSF/4OH-PAMAM membrane can be considered as an appropriate composite membrane for CO2 capturing from natural gas and gas sweetening. Therefore, PSF/CST-HA(0)/CPAMAM and PSF/4OH-PAMAM membranes are respectively recommended for removing two acidic gases, H2S and CO2, from natural gas. Figure 5a, b, c, and d depicts the density distribution of binary mixture of CO2/H2S in pure and composite PSF membranes. The red and green points in these figures imply respectively the density distribution of CO2 and H2S in the Table 6 The solubility-selectivity results of all mixed gases in three simulated membrane types

simulation cells. As can be seen from these figures, the density of red points in the PSF and PSF/CST-HA(0) is greater than the green points. However, in Fig. 5a and b, their density ratio is almost the same. This means that the ability of pure PSF and PSF/CST-HA(0) for removing CO2 from CO2/H2S mixture is almost the same, but their affinity to adsorb CO2 is more than H2S. In Fig. 5c and d, the density of red points is significantly higher than green points, so, it can be concluded that the suitable sites for adsorbing CO2 in PSF/C-PAMAM and PSF/4OH-PAMAM membranes are more than H2S. The affinity of pure PSF and PSF/CST-HA(0) to adsorb H2S is higher than CO2, but by adding dendrimes to the pure PSF membranes, the affinity is changed. Generally, unlike the PSF/ 4OH-PAMAM membrane, the affinity of pure PSF, PSF/CSTHA(0), and PSF/C-PAMAM membranes to capture CO2 and H2S is very close to each other, so, for capturing CO2 form CO2/H2S binary mixture, the simulated results of this study propose PSF/4OH-PAMAM as the proper membrane. The results also show a high solubility-selectivity of this membrane to capture CO2 from CO2/CH4, CO2/N2, and CO2/O2 mixtures. Therefore, in this case, all membranes are appropriate. But addition of CST, C-PAMAM and 4OH-PAMAM, increases the CO2 solubility selectivities of the membranes are increased. Figure 6a, b, c, and d shows the solubility energy distribution of the studied gases in the membranes. It is seen in Fig. 6a and b, that the maximum solubility peaks with higher energy are for solubility of H2S in pure PSF and PSF/CST-HA(0), therefore, as expected the solubility of H2S in these membranes is higher than the others. The maximum solubility peaks for the energy distribution of the gases, as seen in Fig. 6, are in the order H2S > CO2 > CH4 > O2 > N2. Thus, the solubility-selectivity of CO2/H2S must be lower than unity. However, this parameter for CO2/CH4, CO2/N2, and CO2/O2 is higher than unity and this fact is in agreement with the obtained simulation results (Table 6). By adding the CPAMAM and 4OH-PAMAM dendrimers to the pure PSF membrane, the order of the maximum solubility peak energy distribution of the gases, as seen in Fig. 6c and d, changes to CO2 > H2S > N2 > O2 > CH4 and CO2 > H2S > O2 > CH4 > N2, respectively, and this indicates that the solubilityselectivities of the binary gas mixtures (CO2/CH4, CO2/H2S, CO2/N2, and CO2/O2) must be greater than unity and this is confirmed by the simulation results for PSF/C-PAMAM and PSF/4OH-PAMAM membranes.

No.

Sample

CO2/CH4

CO2/H2S

CO2/N2

CO2/O2

1 2 3 4

PSF PSF/CST-HA(0) PSF/C-PAMAM PSF/4OH-PAMAM

3.34±0.01 3.68±0.03 9.19±0.05 16.43±0.08

0.49±0.09 0.55±0.02 1.24±0.05 5.50±0.02

8.67±0.07 11.39±0.03 8.94±0.01 17.99±0.05

8.02±0.03 10.74±0.05 9.12±0.07 15.14±0.06

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Fig. 5 Density distribution of binary mixture of CO2/H2S in pure and composite PSF membranes

(A) PSF

(B) PSF/CST-HA(0)

(C) PSF/C-PAMAM

(D) PSF/4OH-PAMAM

Gas diffusivity of membranes To calculate the diffusivity of each gas in the simulated composite membranes, firstly, the three gas molecules are embedded into the final structures of each membrane. The reason for embedding the three gas molecules simultaneously is to obtain the mean squared displacement (MSD) from the trajectories of the three gas molecule due to penetration in each membrane and calculate their average diffusion coefficients. The average of these three calculated diffusion coefficients is considered as diffusion coefficient of the studied system. Diffusion coefficients of CH4, CO2, H2S, N2, and O2 molecules in the equilibrated cells was determined by 2.5 ns NVT run at 308.15 K. For this purpose, after embedding the three molecules of each gas in the optimum amorphous cell of each membrane, the new amorphous cells were relaxed by 500 ps NPT at 308.15 K and 1 atm. Then a NVT was subsequently performed at the density of the model membrane systems for 2.5 ns to study the detailed motion of the gas

molecules in the each cell. From the trajectories recorded at 1 ps intervals, the consecutive positions of the penetrant molecules diffusing in the nanocomposite membrane matrixes were computed as a function of time and the diffusion coefficients for the penetrants was calculated by means of the Einstein equation [82–85]. As mentioned earlier (Eq. (5)), to calculate the diffusion coefficient by Einstein equation, the simulation time must be sufficiently long to pass the anomalous diffusion regime and reach the condition of Einstein diffusion regime. During this transition, the parameter n in Eq. (6), changes from nA PSF/CST-HA(x)/C-PAMAM > PSF/CST-HA(x)/4OH-PAMAM. By comparing diffusion coefficients for the studied gases, it can be seen that the diffusion coefficients of O2 and N2 in the composite membranes are approximately the same but greater than other gases (CH4, CO2 and H2S). From physical viewpoint, O2 and N2 molecules have almost the same size but are smaller than CH4, CO2, and H2S molecules and in electrostatic viewpoint, these molecules are non-polar, so the interaction forces between them and other compounds are much smaller than other gases, so, O2 and N2 molecules displace more easily in membrane and subsequently have higher diffusivity. Among the other three gases, the size of CH4 is almost the same as CO2 and H2S, but this gas is non-polar, so based on physical viewpoint the mobility of CH4 in the membrane is higher than CO2 and H2S. Generally, the order of diffusivity for studies gases is O2 ≈ N2 > CH4 > CO2 ≈ H2S.

nanocomposite membranes, it is interesting to evaluate the corresponding permeability of the membranes. The predicted and available experimental permeabilities related to each gas in the membranes are tabulated in Table 9. As is seen from this table, the variation trend of permeability of the studied gases in the three membrane types is similar to the variation of the diffusion coefficients. Therefore, entire explanations which were presented for the diffusion coefficient is true for permeability results. For the sake of comparison, the only available experimental data as presented in Table 9 are considered and are compared with the corresponding simulation results. From the comparison, it can be concluded that the obtained results are in agreement with the experimental data and confirm the reliability of MD and MC simulation for predicting the transport properties of composite membrane.

Conclusions Gas permeability of membranes Permeability is another important transport property of the membranes which can be considered as a property composed of contributions from solubility and diffusivity. Therefore, after predicting the diffusivity and solubility of each

Molecular dynamics (MD) and Monte Carlo (MC) simulations have been extensively applied to investigate the properties of these types of polymer composite membranes for separation of gases CH4, CO2, H2S, N2. In the first membrane type, pure polysulfone (PSF) and its composite with chitosan (CST) and with various concentration of hyaluronic acid (HA)

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Table 9 The permiability (×10−10 m3(STP)/m2 s Pa) results of all pure gases in three simulated membrane types No.

Sample

CH4

CO2

H2S

N2

O2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

PSF PSF/CST-HA(0) PSF/CST-HA(10) PSF/CST-HA(20) PSF/CST-HA(30) PSF/CST-HA(40) PSF/CST-HA(50) PSF/C-PAMAM PSF/CST-HA(0)/C-PAMAM PSF/CST-HA(10)/C-PAMAM PSF/CST-HA(20)/C-PAMAM PSF/CST-HA(30)/C-PAMAM PSF/CST-HA(40)/C-PAMAM PSF/CST-HA(50)/C-PAMAM PSF/4OH-PAMAM PSF/CST-HA(0)/4OH-PAMAM PSF/CST-HA(10)/4OH-PAMAM

11.4±0.21 14.8±0.45 14.5±0.76 29.4±0.89 44.7±0.91 55.8±0.34 70.2±0.56 8.5±0.77 12.5±0.89 16.9±0.23 25.6±0.55 38.3±0.78 47.9±0.43 60.1±0.67 6.8±0.98 11.0±0.43 15.9±0.67

9.8±0.34 12.4±0.78 18.3±0.71 24.1±0.67 39.9±0.98 45.8±0.34 60.3±0.67 5.9±0.16 6.1±0. 18(2.1)a 15.4±0.67 15.9±0. 98(10.0) 25.7±0.65 39.4±0.54 45.6±0.78 4.6±0.43 6.1±0.22 14.0±0.37

7.2±0.76 10.8±0.87 15.6±0.98 22.9±0.34 31.5±0.56 49.4±0.59 55.1±0.43 3.8±0.55 5.2±0.69 9.9±0.77 12.8±0.11 21.7±0.59 39.3±0.54 44.0±0.78 1.6±0.86 3.8±0.43 11.2±0.78

19.3±0.65 29.4±0.43 45.9±0.23 60.1±0.45 66.5±0.67 83.7±0.43 98.4±0.67 15.4±0.87 21.7±0.66 44.6±0.52 58.2±0.45 61.1±0.67 78.4±0.58 89.2±0.17 10.2±0.54 17.4±0.67 40.8±0.89

18.3±0.43 25.7±0.56 49.9±0.88 55.1±0.87 70.9±0.12 79.2±0.34 105.3±0.56 11.6±0.78 29.3±0.76 58.0±0.54 65.2±0.32 73.4±0.65 89.3±0.67 92.9±0.80 5.1±0.37 5.4±0.68 23.4±0.67

18 19 20 21

PSF/CST-HA(20)/4OH-PAMAM PSF/CST-HA(30)/4OH-PAMAM PSF/CST-HA(40)/4OH-PAMAM PSF/CST-HA(50)/4OH-PAMAM

19.8±0.27 29.7±0.38 32.8±0.58 45.2±0.93

15.1±0.54(8.8) 19.8±0.47 27.9±0.77 30.4±0.79

13.0±0.91 22.5±0.16 21.4±0.68 28.1±0.91

55.8±0.43 59.4±0.45 60.2±0.68 68.1±0.23

45.1±0.36 66.8±0.78 73.8±0.79 75.2±0.32

a

The number in brackets is the experimental value

were investigated. In second and third membrane types, respectively, the conventional poly(amidoamine) (C-PAMAM) and hydroxyl poly(amidoamine) (4OH-PAMAM) dendrimers were added to the first membrane type. The main goal of this research was to explore the effect of added chemicals and their concentrations on the transport behaviors of gases in the pure and composite PSF membranes. For this purpose, by using MD method, the simulation cells of the membranes containing PSF, CST, HA, C-PAMAM, and 4OH-PAMAM dendrimers were constructed (in three distinguished types). After obtaining the fully thermodynamic equilibrium state for each simulation cell and by using 2 ns NPT runs for each membrane, the properties of the simulated membranes including cell length, occupied volume, free volume, surface area, fractional free volume (FFV), radius of gyration (Rg), and X-ray patterns were calculated. By comparing the simulation results for the membranes, it was observed all above mentioned properties increased by adding HA, CST, C-PAMAM, and 4OH-PAMAM to the pure PSF membrane. By using MC method, the solubility of pure gases including CH4, CO2, H2S, N2, and O2 and the solubility-selectivity of their binary mixtures CO2/CH4, CO2/H2S, CO2/N2, and CO2/O2 in all simulation membrane cells were calculated. The obtained results revealed that the solubility of H2S in all membranes was higher than other gases. The gas solubility in the three membrane types

was in the order of H2S > CO2 > CH4 > O2 > N2. In each membrane type, higher percentage of HA increased the gas solubility of the membranes and this parameter for the membranes which contained C-PAMAM or 4OH-PAMAM dendrimer in their simulation cells (second and third types) was higher than corresponding membranes of first type which contained the same HA percentage. The solubility-selectivity of gases in pure PSF, PSF/CST-HA(0), PSF/C-PAMAM, and PSF/4OHPAMAM were calculated. The results indicated that the PSF/ 4OH-PAMAM membrane has the high ability for capturing CO2 from any binary gas mixtures. By performing 2.5 ns NVT run the amount of diffusion coefficient of all gases in three simulated membrane types were calculated. The results indicated that adding CST and HA to the pure PSF increased the gas diffusion coefficients, whereas embedding C-PAMAM and 4OHPAMAM dendrimers into the membranes decreased the diffusion coefficient. In addition, according to the size and polarity of gas molecules the order of gas diffusion coefficients for studied gases is O2 ≈ N2 > CH4 > CO2 ≈ H2S. After calculating the solubility (S) and diffusivity (D) for gases in each membrane type, the permeability was evaluated. Considering the permeability results of the membranes in this study, reveals that the variation trend of gas permeability is similar to the gas diffusivity and the influence of diffusivity on permeability is more than that of solubility.

2119, Page 18 of 20 Acknowledgments The services of the High Performance Computing cluster of Amirkabir University of Technology are gratefully acknowledged for provision of facilities to perform the simulation runs in this study.

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