Molecular Dynamics Simulation of Diffusion Behavior of Benzene

Jun 5, 2008 - Diffusion coefficients of benzene and water at “infinite dilution” and saturated condition displayed the same changing tendency, alt...
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Ind. Eng. Chem. Res. 2008, 47, 4440–4447

SEPARATIONS Molecular Dynamics Simulation of Diffusion Behavior of Benzene/Water in PDMS-Calix[4]arene Hybrid Pervaporation Membranes Ben Li, Fusheng Pan, Zhiping Fang, Liang Liu, and Zhongyi Jiang* Key Laboratory for Green Chemical Technology, Ministry of Education of China, School of Chemical Engineering and Technology, Tianjin UniVersity, Tianjin 300072, People’s Republic of China

Molecular dynamics (MD) simulation was employed to investigate diffusion behavior of small penetrants in rubbery-polymer-based hybrid membranes, using pervaporative removal of benzene from its dilute solution by poly(dimethylsiloxane) (PDMS) membranes filled with calix[4]arene (CA) as the model system. In our previous experimental investigation, the normalized permeation rate of benzene (NPRb) and separation factor (benzene/water) through PDMS-CA hybrid membranes did not follow the usual monotonous or single peak/ valley change, but accompanied minimum and maximum values instead. In the present study, nonbonding interaction energy between PDMS and CA, mean-square displacement (MSD), free volume characteristics, and diffusion coefficients of benzene and water in pure PDMS and hybrid membranes were analyzed by molecular dynamics simulation. The simulation results revealed that MSD and fractional free volume (FFV) values were closely dependent on interaction energy. Diffusion coefficients of benzene and water at “infinite dilution” and saturated condition displayed the same changing tendency, although the values at saturated condition were a bit larger. Moreover, it was observed that diffusion coefficients were not only related to FFV but also affected by the interaction between CA and the penetrants. Overall, the MD results agreed well with the experimental results. 1. Introduction Pervaporation (PV) is a separation process in which minor components of a liquid mixture are preferentially transported by partial vaporization through a nonporous permselective membrane. As an emerging and cost-effective technology in environment cleanup operations, especially in the removal of volatile organic compounds (VOCs) from industrial wastewaters or contaminated groundwater, PV has been demonstrated to be a promising alternative to conventional technology such as adsorption and air stripping.1–3 The enhancement of polymer properties by the addition of certain fillers is a combined function of interfacial interactions, interfacial area, and the distribution of interfiller distances. Owing to the improvement in permeability, selectivity, stability, and conductivity, hybrid materials are slated for applications ranging from membranes to fuel cells.4 The existing models describing the transport process through hybrid membranes are either approximate expressions or semiempirical correlations,5 by which it is difficult to accurately predict the performance of hybrid membranes composed of materials with different natures. Molecular dynamics (MD) simulation, in comparison, has been validated to be able to reveal the microstructure of the hybrid membrane and provides the dynamics of the permeating compounds at the molecular level. Although the estimated results, such as diffusion coefficients, were usually within (40-60% of the corresponding experimental values,6 the same changing tendency between experimental and simulated values can be often obtained, which is quite crucial for the rational design of hybrid membranes. Several works have been reported of MD simulation that concerned diffusion behavior of penetrants in hybrid membranes. * To whom correspondence should be addressed. Tel.: +86-2227892143. Fax: +86-22-27892143. E-mail: [email protected].

Zhou and co-workers7 studied the diffusion of gases through poly(1-trimethylsilyl-1-propyne) [PTMSP] membrane filled with silica particles. It was found that the addition of silica particle to PTMSP increased the diffusion coefficients of gases by enhancing the free volume of polymer. Pan and co-workers8,9 simulated the diffusion behavior of benzene/cyclohexane molecules in poly(vinyl alcohol)-graphite hybrid membranes. According to their work, the morphology of the interfacial region that played a crucial role in the separation performance of hybrid materials was systematically studied. The graphite content and different functional groups on the graphite surface exerted considerable influence on the free volume size and distribution, which could be served as strong evidence for elucidating the antitrade-off phenomenon.10 As far as we know, the existing works mainly dealt with glassy-polymer-based hybrid membranes.7–9,11 However, because of its flexible structure and, thus, high permeability, rubbery -polymer-based hybrid membranes were widely used in pervaporative removal of trace amounts of organic compounds from water and separation of organic-organic mixtures. Unfortunately, simulations of penetrants diffusion through rubbery-polymer-based hybrid membranes have not been reported yet. Hofmann and co-workers12 compared the static structure and the dynamic behavior of the free volume in the simulated rubbery polymers with flexible chains and glassy polymers with rigid chains, and the results revealed substantial differences in the diffusion of small molecules in these two kinds of materials. However, it is still rare to investigate the influence of fillers incoprporation on membrane structural morphology and the dynamics of small penetrants in the these hybrid membranes. Poly(dimethylsiloxane) (PDMS) membranes filled with calixarene (CA) were reported for pervaporative removal of benzene

10.1021/ie0708935 CCC: $40.75  2008 American Chemical Society Published on Web 06/05/2008

Ind. Eng. Chem. Res., Vol. 47, No. 13, 2008 4441

this study. The developed CA molecule and PDMS-CA were shown in Figure 2. Cell samples were designated as PDMS1CA, PDMS2CA, and PDMS4CA according to the number of CA molecules in the PDMS unit. For the PDMS control membrane, a polymer chain that consisted of 100 repeat units was built. The packing model with a density of 0.95 g/cm3 containing five PDMS chains was developed by the amorphous cell module. A 2000-step energy minimization was carried out at the beginning phase to eliminate local nonequilibrium. As we all know, in experimental operations, it was rather difficult to acquire homogeneous dispersion of the fillers within the polymer matrix. When the fillers became smaller, they tended to agglomerate. In order to simplify the calculation, an idealized hypothesis was often utilized.7,8,10,11 In this study, a single CA molecule (linear size about 14.11 Å) was embedded in the PDMS matrix, rather than a cluster.

Figure 1. Effects of the CA content on separation factor and NPRb.

from its dilute aqueous solution.13,14 Compared with the unfilled PDMS membranes, those membranes filled with CA exhibited higher permselectivity toward benzene. In particular, the unusual change of permeation rate in such membranes was observed.15 With the increased filling amount of CA, the normalized permeation rate of benzene and the separation factor of the hybrid membranes exhibited maximum and minimum values, as shown in Figure 1. NPRb referred to the normalized permeation rate of benzene during pervaporation, which was determined from lWbp (1) tS where l is the thickness of the membrane, Wbp is the weight of benzene in permeate, t is the permeation time, and S is the effective membrane area. In the present study, nonbonding interaction energy between PDMS and CA, mean-square displacement (MSD), free volume characteristics, and diffusion coefficients of benzene and water in pure PDMS and hybrid membranes were analyzed by molecular dynamics simulation to provide the theoretical insight into the unusual change of permeation rate in the hybrid membranes. NPRb )

2. Packing Models and Simulation Details Molecular dynamics simulations in this study were carried out using “discover” and “amorphous cell” module of “Materials Studio”, a powerful workstation developed by Accelrys Software Inc. COMPASS force field was employed in this study. The energy minimization process was conducted using the smart minimizer method that switched from steepest-descent to conjugated-gradient and then to the Newton method as the energy derivatives decreased for the sake of accelerating the computation. For dynamics, the Andersen16 thermostat and Berendsen barostat17 methods were employed to maintain a constant temperature and pressure, respectively. Nonbond cutoff distance was set as 9.5 Å (with a spline width of 1.0 Å and a buffer width of 0.5 Å) to calculate the nonbonding energies. Long-tail corrections to the energy due to cutoff were employed during dynamics simulation. The time step was set as 1 fs for all dynamics runs. Molecular models of CA, PDMS polymers, and mixed matrixes containing different CA loadings were constructed in

For PDMS-CA membranes, the CA molecules were embedded symmetrically in the model of the PDMS control membrane. PDMS1CA was put in the cubic cell center; PDMS2CA and PDMS4CA were put on the diagonal lines of the cubic cell symmetrically. Mass fractions of CA molecules in the membranes were calculated by their molecule weight (Figure 3). Taking PDMS2CA as example, the molecular weight of CA is (11 × 12.011 + 15.9994 + 14 × 1.0079) × 4 ) 648.924; the molecular weight of the PDMS chain of 100 repeat units is [28.0855 + 15.9994 + (12.011 + 3 × 1.0079) × 2] × 100 + 2 × 1.0079 + 15.9994 ) 7433.4452; and the CA content in PDMS2CA is 648.924 × 2/7433.4452 × 5 + 648.924 × 2 ≈ 3.3%. Accordingly, mass fractions of CA molecules in the membranes were about 1.5, 3, and 6 wt % for PDMS1CA, PDMS2CA, and PDMS4CA, respectively. The resulting atomistic structures were subsequently optimized by a 5000-step energy minimization. NPT-NVT ensembles were utilized for calculating the diffusivities. When constructing an amorphous cell, the molecules may be not equally distributed throughout the cell, creating areas of vacuum. To correct this, a short time of molecular dynamics simulation has to be run to equilibrate the cell. This procedure of minimization and molecular dynamics is known as structure relaxing and should be carried out whenever an amorphous cell is constructed. There are different types of molecular dynamics simulations, and these are usually classified by the ensemble names. Taking NVT and NPT as examples, N refers to constant number of moles, V refers to constant volume, T refers to constant temperature, and P refers to constant pressure. For equilibrating a cell in preparaing for a diffusivity calculation, the NPT ensemble is the best candidate. Subject to the NPT ensemble, the system could swell sufficiently and achieve the equilibrium state. Hereafter, the NVT ensemble was utilized for collecting data because it ran much faster than the NPT ensemble. With regard to this study, a 200 ps MD equilibration run on the membrane was performed in the NPT (T ) 323 K, P ) 1.01 × 105 Pa) ensemble to further equilibrate the models prior to data collection. The length of the final periodic boundary cubic cell varied from 38.60 to 40.08 Å depending on the embedded CA content. The specific parameter of the cells was listed in Table 1. An additional 100 ps NVT (T ) 323 K) dynamics was performed on the end point of the NPT run to obtain equilibrium molecular structures, and the atomic trajectory was recorded every picosecond for the subsequent analysis.

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Figure 2. Molecule models of PDMS control membrane and PDMS-CA hybrid membranes: (a) PDMS, (b) PDMS1CA, (c) PDMS2CA, and (d) PDMS4CA. CA molecules were highlighted by white frames. Table 2. Interaction Energies between PDMS and CA (kJ/mol)

PDMS1CA PDMS2CA PDMS4CA

Enonbond

EvdW

Eelectrostatic

Evalence

-54.0397 13.01248 -92.6578

-17.7055 -91.4112 -71.3612

-56.3338495 84.42370102 -41.296198

216.6851 -81.0183 653.5025

kJ/mol), which quantitatively conveys the consistency between CA molecules and PDMS polymers, can be calculated as follows: ∆E ) EPDMS-CA - (ECA + EPDMS)

Figure 3. Structures of CA (a) and PDMS (b). Table 1. Specific Parameters of the Cells

3

volume (Å ) density (g/m3) cell length (Å)

PDMS

PDMS1CA

PDMS2CA

PDMS4CA

58161.68 1.0589 38.74

57504.03 1.0897 38.60

64681.68 0.9854 40.08

57764.03 1.1408 38.66

3. Results and Discussions 3.1. Interaction between Filler and Polymer. For hybrid materials, it is important to investigate the interfacial regions between the filler and the bulk polymer matrix, which greatly influence the separation performance of the membrane.18 When small-size fillers are well-dispersed in the membranes, interfacial interaction between fillers and polymer becomes very crucial to the properties of the hybrid membranes. According to thermodynamics theory, the interaction energy ∆E (quoted in

(2)

where EPDMS-CAis the potential energy of the PDMS-CA hybrid membrane and ECA and EPDMS are the energies of optimized CA molecule and PDMS polymer, respectively. In the PDMS-CA membranes, the fillers and the polymer were just physically blended, no chemical reaction energies were involved, and the nonbonding energies were exclusively taken into account. The nonbonding interaction energies between CA and PDMS polymer were calculated by COMPASS force field and the results were listed in Table 2. According to eq 2, ∆E is the difference between the potential energy of the hybrid membrane and the sum of each component; thus, the negative and positive values of interaction energy, respectively, correspond to the attractive force and the repulsive force between CA and PDMS polymers. Within the simulation time in this study, the interaction energy was negative for model PDMS1CA and PDMS4CA but positive for PDMS2CA. The phenomenon that different loadings of CA led to different interaction energies may be attributed to the synergistic effects such as the inherent natures of PDMS and CA, the filler size and distribution, and the polymer chain packing. The calculation results indicated

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Figure 4. MSD of PDMS in PDMS control membrane and PDMS-CA hybrid membranes.

that, within the time limit, polymer chains in PDMS2CA were under repulsion by CA fillers and tended to move apart. In membranes PDMS1CA and PDMS4CA, however, polymer chains were attracted or “seized” by CA molecules. These different interactions could be supported by the cell length or volume listed in Table 1. The volume of PDMS2CA (64681.679 Å3) was obviously larger than that of PDMS1CA (57504.03 Å3) and PDMS2CA (57764.03 Å3). In other words, polymer chains were more closely packed in membranes PDMS1CA and PDMS4CA than in membrane PDMS2CA. It was, thus, reasonably assumed that the difference of interaction values existing between the above three hybrid membranes would lead to different structures of the interfacial region, which will be discussed later in our paper. 3.2. Mobility of Polymer Chains. To design a suitable separation membrane, it is crucial to consider the flexibility of polymer chains. It is well-known that penetrants first adsorb into and then diffuse through polymers. Subnanometer random “jumps” of penetrants occur when a sufficient large transient gap is formed in the polymer next to the penetrant due to thermally induced motions of the polymer chain segments.6,19 As a result, the mobility or flexibility of polymer chains considerably affects the permeation rate of small penetrants and, consequently, the membrane separation performance. Polymers with higher flexibility win more opportunities and frequency for the penetrant molecules to jump from one void to another. However, the penetrant molecules in polymers with high rigidity often exhibit low diffusion rate. Moreover, polymer chains mobility are closely related to the interaction between fillers and polymers in hybrid membranes. Therefore, it is important to investigate how the mobility of PDMS-CA changes with different CA loadings. Polymer chain mobility can be analyzed by mean-square displacements (MSDs). g(t) ) 〈(ri(t) - ri(0))2 〉

(3)

where ri(t) and ri(0) are the position of atom i at time t and 0, respectively. The bracket denotes the ensemble average, which is obtained from averaging over all penetrants and all time origining from t ) 0. The MSDs of polymer chains in PDMS control membrane and PDMS-CA hybrid membranes in this study were shown in Figure 4. The larger slope of the MSD curve reflected the higher chains mobility. The slopes of the dotted MSD of PDMS control membranes and the hybrid membranes with different CA loadings studied above followed the order of PDMS2CA

Figure 5. Simulated X-ray diffraction patterns of PDMS control membrane and PDMS/CA hybrid membranes.

(3 wt %) > PDMS > PDMS4CA (6 wt %) > PDMS1CA (1.5 wt %). It is rather difficult to accurately predict the influence on mobility of polymer backbones after fillers are incorporated into the polymer matrix,20 which strongly depends on the interaction between fillers and the backbone of polymers. Eisenberg et al.21 have proposed a model that suggested that the mobility of a polymer chain in the neighborhood of “multiplet-clusters” could become more restricted relative to the chains in the bulk phase. They regarded the clusters with anchored polymer chains as effective cross-linkers in the bulk polymer to increase the Tg of the material. Because of the negative interaction energy, CA molecules in PDMS1CA and PDMS4CA acted as cross-linkers, which substantially confined the mobility of polymer chains within the interface region. In PDMS2CA, however, because of the positive interaction energy, CA enhanced the mobility of polymer chains within the interface region. This could be supported by the DSC result.22 The Tg of PDMS-3 wt % CA hybrid membrane was around -111.9 °C, even below that of the PDMS control membrane (-109.9 °C), which indicated that polymer chain segments in PDMS1CA became more flexible. It seems that MD methods enable the prediction or validation of the complicated influence of different fillers on the mobility of the backbone of polymers. On the basis of the experimental results in Figure 1, PDMS2CA with the most flexible polymer segments exhibited the largest permeation flux, but PDMS1CA with the most rigid polymer segments ranked the lowest. PDMS and PDMS4CA lied just in between. 3.3. Simulated X-ray Diffraction Pattern of the Membranes. The X-ray diffraction pattern of the membranes was obtained using the “discover” module, and the result was shown in Figure 5. It was shown that PDMS control membrane and PDMS–CA hybrid membranes generally existed in the amorphous state. Moreover, the peak intensity of PDMS2CA is stronger than those of the other three, which indicated that PDMS2CA possessed more amorphous area that was favorable for the transport of small penetrants. 3.4. Free Volume of the Membranes. There are two phases in polymer membranes: a solid phase occupied by polymer chains and a space phase named as free volume. Free volume size and distribution serve as the most convenient and direct parameters to evaluate the quality of dense membranes and have the potential to connect the microscopic membrane morphology with its macroscopic separation performances. Positron annihilation lifetime spectroscopy (PALs) is the prevalently adopted experimental method to determine the free volume

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quantitatively. However, presently the experimental technique is often inaccurate and cannot clearly give the details about the morphology of the free volume voids. The MD method employs nanometer molecules as probes to characterize the free volume properties of dense membranes and often obtains satisfying results. The simulated atomistic models allow an accurate determination of geometrical quantities characterizing the structure. The fractional free volumes (FFVs) of the equilibrated PDMS control membranes and PDMS-CA membranes were determined by a hard spherical probe. The atoms composing the membranes are represented by hard spheres with van der Waals radius (C, 1.55 Å; H, 1.10 Å; O, 1.35 Å; Si, 2.10 Å). The penetrant molecules, benzene and water, were selected as the probe molecules, which were modeled by spheres with radii 2.63 and 1.20 Å (collision radius), respectively. The Connolly surface was calculated when the probe molecule with the radius Rp rolled over the van der Waals surface, and free volume is defined as the volume on the side of the Connolly surface without atoms. The fractional free volume was determined by the ratio of free volume to total volume of the model. It should be mentioned that the free volume obtained by this method excluded the volume that was unaccessible for the probes. The FFV obtained by a hard spherical probe depended strongly on the size of the probe. The pictures of the free volume morphology of each hybrid membrane focusing on the interface are shown in Figure 6. The FFV values using molecule probes with different radii Rp are listed in Table 3. It clearly indicated that the FFV decreased rapidly as the probe size increased. It could be seen that the FFV of PDMS2CA (3 wt %) was obviously larger than those of the other three membranes no matter which probe was employed, and the FFV of the hybrid membranes followed the order of PDMS2CA (3 wt %) > PDMS > PDMS1CA (1.5 wt %) > PDMS4CA (6 wt %). The simulation exhibited the same changing tendency as the PALs results.22 The free volume of a hybrid membrane is commonly a comprehensive outcome of many contributions. In this study, it was noted that the FFV in the interfacial region was closely dependent on the interaction energy between CA and polymers. As mentioned in Section 3.2, polymer chains in PDMS1CA and PDMS4CA were seriously restricted by CA fillers, which decreased the polymer segmental mobility. Polymer segmental mobility of PDMS2CA, however, was enhanced. FFV is the comprehensive outcome of thermal motion of polymer chains; the change in segmental mobility of polymers will, therefore, lead to the change of the FFV value. Moreover, the FFV of PDMS4CA with lower interaction energy was smaller than that of PDMS1CA with higher interaction energy. Through the above analysis, it could be derived that the FFV was strongly related to interaction energy in this hybrid membranes. Membranes with higher interaction energy exhibited higher FFV values. 3.5. Diffusion of Penetrants through the Control Membrane and the Hybrid Membranes. The diffusion coefficients of benzene and water in the membrane were calculated from the slope of their mean-square displacements for long time by the Einstein relation:6 Na

∑

d 1 Di ) lim 〈(r (t) - ri(0))2 〉 6 tf∞ dt i)1 i

(4)

where Di is the diffusion coefficient, and 〈(ri(t) - ri(0))2〉 is the mean-square displacement of the i penetrant. In this study, two conditions were taken into consideration, i.e., infinite-dilution and saturated (feed-membrane thermodynamics balance state) conditions, which could be regarded as

Figure 6. Simulated morphology of free-volume voids (shown in highlight) in equilibrated PDMS control membrane and PDMS-CA hybrid membrane models probed by benzene and water. Table 3. Fractional Free Volume (FFV) and the FFV Ratio of PDMS Control Membrane to PDMS-CA Hybrid Membranes Obtained Geometrically for Each Probe Molecule

PDMS PDMS1CA PDMS2CA PDMS4CA

FFV%benzene

FFV%water

FFVbenzene/water

4.18 3.08 6.56 1.28

17.42 14.58 21.59 11.43

0.24 0.21 0.30 0.11

the conditions in downstream phase and upstream phase of the membranes, respectively. It would be more convincing if these two conditions displayed the same changing tendency. For the infinite-dilution case, penetrant molecules (four benzene molecules and four water molecules) were inserted into the models of the optimized membranes in such a way that the distance between any two molecules is at least one-half of the cell length, in order to ensure that the interactions could be neglected. The models were equilibrated using the same

Ind. Eng. Chem. Res., Vol. 47, No. 13, 2008 4445 Table 4. Simulation Details Corresonding to the Saturated Condition

PDMS PDMS1CA PDMS2CA PDMS4CA

PDMS

water

benzene

5 × 100repeatunit 5 × 100repeatunit 5 × 100repeatunit 5 × 100repeatunit

6 5 8 10

86 72 142 213

Table 5. (a) Diffusion Coefficients of Benzene and Water and Diffusion Selectivity in PDMS Control Membrane and PDMS-CA Hybrid Membranes at Infinite Dilution Condition; (b) Diffusion Coefficients of Benzene and Water and Diffusion Selectivity in PDMS Control Membrane and PDMS-CA Hybrid Membranes at Saturated Condition Dwater/ (10-12 m2/s)

Dbenzene/ (10-12 m2/s)

diffusion selectivity (benzene/water)

(a) PDMS PDMS1CA PDMS2CA PDMS4CA

119.74 99.25 125.10 128.00

4.83 1.42 6.83 5.30

0.0403 0.0143 0.0546 0.0414

7.58 2.45 13.90 9.92

0.05314 0.02125 0.0777 0.05221

Figure 7. Diffusion selectivity for PDMS control membrane and PDMS-CA hybrid membranes.

(b) PDMS PDMS1CA PDMS2CA PDMS4CA

142.70 115.51 179.25 190.36

Table 6. Comparison of Benzene Diffusion Coefficients in PVA-graphite Membranes and in PDMS-CA Hybrid Membranes membranes

Dbenzene/(10-12 m2/s)

membranes

Dbenzene/(10-12 m2/s)

PVA CG1/PVA CG2/PVA CG3/PVA

1.05 1.75 4.00 4.41

PDMS PDMS1CA PDMS2CA PDMS4CA

4.83 1.42 6.83 5.30

procedure as mentioned in Section 2. The diffusion runs were performed under the NVT (T ) 323 K) conditions for 2 ns. The diffusion coefficient was an averaged value. For the saturated case, the number of benzene and water molecules added into the cell was determined by experimental outcome.22 Given concentration of feed (0.15 wt % benzene in water), sorption selectivity, and swelling degree, the number of benzene and water molecules added into each hybrid membrane was calculated and is listed in Table 4. For the packing models containing PDMS chains, benzene and water were developed by amorphous cell module. A 5000-step energy minimization was carried out to eliminate local nonequilibrium. After that, the CA molecules were embedded in the model of the PDMS control membrane. Then, the models were equilibrated using the same procedure as mentioned in Section 2. The diffusion runs were performed under the NVT (T ) 323 K) conditions for 500 ps. The diffusion coefficient was an averaged value. The diffusion coefficients of benzene and water in the control membrane and the hybrid membranes were listed in Table 5. Compared with glassy PVA-graphite and PVA-silica hybrid membranes (Table 6),8,9 benzene diffusion coefficients in PDMS hybrid membranes were much bigger, which resulted from the higher flexibility of the polymer chains and the larger FFV of the PDMS membranes. This could be ascribed to the biggest difference between rubbery and glassy polymers. Moreover, we found that, at saturated conditions, both water and benzene diffusion coefficients increased with their concentrations. Generally speaking, the diffusion coefficient of a single component has a strong relationship with its concentration in the membrane and follows an exponential equation, Di ) Di0 exp(AiCmi)

(5)

where Di0 is the diffusion coefficient of component i at dilute condition; Cmi is the concentration of component i in the membrane; and Ai is a constant number that represents the interaction between component i and the polymers. The diffusion coefficients of benzene and water, in each rubbery hybrid membrane, followed the order of PDMS1CA < PDMS < PDMS4CA < PDMS2CA. Generally speaking, the more fractional free volume of separation membrane, the higher is the diffusivity of membrane that will be obtained. Higher FFV, which means more accessible free volume cavities in the membranes or a shorter path being needed for the penetrant hopping to the nearby cavities, will certainly facilitate the penetrant diffusion. In this study, however, the diffusion coefficients of benzene and water in PDMS4CA appeared larger than those in PDMS and PDMS1CA, whereas the FFV of PDMS4CA was smaller than those of PDMS and PDMS1CA. This peculiar phenomenon was also reported elsewhere,7 and it might be attributed to the “facilitated transport” effect in this hybrid membranes. It is well-known that the π-rich cavity in calixarene is suitable for the inclusion of a neutral guest of complementary size including benzene, owing to the supramolecular reversible interaction. Therefore, when a sufficient amount of CA is added into the polymer, CA serves as “carriers” that help or accelerate the transport of benzene molecules. Such interaction cannot be reflected by free volume and polymer segment mobility characteristics and will be further investigated in our future study. From the diffusion simulation result, it was easy to explain the unusual permeation rate in these hybrid membranes. The permeation rate is a synergistic outcome of adsorption and diffusion processes. Mathematically, the permeability of penetrant A, PA, is defined as the product of the average diffusion, DA, and sorption, SA, coefficients in the membrane: PA ) DA × SA 22

(6)

In a previous study, it was found that the adsorption selectivity (benzene over water) was almost linear with CA content varying from 0 to 6 wt %,22 so the unusual permeation rate mainly results from the diffusion process instead of the solution process. The diffusion selectivity (benzene/water) was obtained by the ratio of the diffusion coefficients of benzene and water. The collision radius of the benzene molecule is much larger than that of water; the diffusion coefficient of benzene is, thus, much smaller, and correspondingly, the diffusion selectivity of each membrane was PDMS > PDMS4CA (6 wt %) > PDMS1CA (1.5 wt %). The FFV values of the membranes followed the order of PDMS2CA (3 wt %) > PDMS > PDMS1CA (1.5 wt %) > PDMS4CA (6 wt %). The diffusion coefficients at infinite dilution and saturated conditions were respectively calculated. The diffusion rate of the penetrants was usually proportional to the free volume of the hybrid membranes. PDMS4CA exhibited higher diffusion rate although its FFV was smaller than that of PDMS control membrane and PDMS1CA membrane. This unusual change might be attributed to the inherent interaction between benzene and CA. At saturated conditions, both water and benzene diffusion coefficients increased with their concentrations. The simulation results displayed the same order of magnitude and changing tendency as the experimental results, although obvious deviations were found. Overall, the MD approach in this study was able to roughly predict and elucidate the diffusion behavior and separation performance in rubberypolymer-based hybrid membranes. Acknowledgment The authors are grateful for the financial support from the Cross-Century Talent Raising Program of Ministry of Education of China and the Program for Changjiang Scholars and

NPRb ) normalized permeation rate of benzene Di ) diffusion coefficient of self-diffusivity of species i, m2/s Enonbond ) nonbonding energy, kJ/mol Evalence ) valence energy, kJ/mol EvdW ) van der Waal energy, kJ/mol EElectrostatic ) electrostatic energy, kJ/mol ECA-PDMS ) energy of the CA/PDMS hybrid membrane, kJ/mol ECA ) energy of the optimized CA molecule, kJ/mol EPDMS ) energy of the optimized PDMS polymer, kJ/mol FFV ) fraction of the free volume, % g(t) ) mean-square displacement, Å2 P ) pressure, Pa Rp ) radius of the probe molecular, Å ri(t) ) positions of atom i at time t ri(0) ) positions of atom i at time 0 T ) temperature, K Tg ) glass transition temperature t ) time, ps ∆E ) interaction energy, kJ/mol

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ReceiVed for reView June 29, 2007 ReVised manuscript receiVed April 22, 2008 Accepted April 29, 2008 IE0708935