Comparing Penetrants Transport in Composite Poly (4-methyl-2

Article pubs.acs.org/IECR

Comparing Penetrants Transport in Composite Poly (4-methyl-2pentyne) and Nanoparticles of Cristobalite Silica and Faujasite Silica through Molecular Dynamics Simulation Quan Yang,*,‡ Luke E. Achenie,‡ and Weibin Cai† ‡

Department of Chemical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, United States Honghao Qingyuan Technology Corp. Ltd., Beijing, 102218, China

†

ABSTRACT: In industry, membrane separation of gas mixtures is of considerable significance. For example, the composite poly (4-methyl-2-pentyne) (PMP) and silica nanoparticle is used to separate C4H10 (n-butane) from mixtures of C4H10 and CH4, H2, etc. Silica has different crystalline forms. The transport properties of penetrants in the composite PMP and nanoparticles of cristobalite silica (PMPC) and the composite PMP and nanoparticles of faujasite silica (PMPF) are different. It is essential to explore this difference from the molecular level in order to have an in-depth knowledge and aid the design of the membrane based on composite PMP and silica nanoparticles. In this work, the molecular dynamics (MD) method was employed to explore the transport of different penetrants in the composites of PMP and nanoparticles of two forms of silica. The simulation results show that both types of composites have high permeability compared with PMP for the same penetrants and PMPF has higher permeability than PMPC for the same penetrants. The selectivity of n-C4H10 over CH4 in PMPF, on the contrary, is lower than in PMPC. Finally, the influence of weight concentration of nanoparticle in PMPC on selectivity of n-C4H10 over CH4 was explored.

1. INTRODUCTION In industry, the separation of methane from higher hydrocarbons, organic monomers from nitrogen, and others are important processes. In the production of natural gas, raw gas is treated to separate butane and higher hydrocarbons from methane in order to bring the heating value and the dew point to pipeline specification and to recover the valuable higher hydrocarbons as chemical feedstock. Similarly, approximately 1% of the 30 billion lb/year of monomer used in polyethylene and polypropylene production is lost in nitrogen vent streams from resin purge operation. Recovery of these monomers would save US producers about $100 million/year.1 In these processes, membranes made of composite material2,3 are used due to their better performance than membranes consisting of polymer solely. For example, the composite poly (4-methyl-2-pentyne) (PMP) and silica nanoparticle is used to separate C4H10 (n-butane) from mixtures of C4H10 and CH4, H2, etc. Silica has different crystalline forms. In cristobalite, the Si and O atoms are so densely packed that there are probably no pores through which penetrants can pass, while in faujasite crystalline form, pores exist that are probably large enough to allow penetrants to pass through. Amorphous silica may contain a mixture of small and large pores. The transport properties of penetrants in the composite PMP and nanoparticles of cristobalite silica (PMPC) and the composite PMP and nanoparticles of faujasite silica (PMPF) are different. It is essential to explore the reasons that lead to the difference; this knowledge would aid the design of membrane made of composite PMP and silica nanoparticles. In the research, the transport of different penetrants in PMPC and PMPF was simulated and compared. © 2013 American Chemical Society

Molecular dynamics (MD) simulation is extensively employed to explore the transport properties of penetrants in organic polymers and inorganic materials. To our knowledge, only rather limited research4 has been done to explore the transport property of penetrants in composites using MD simulation. The structure of the composites of a polymer and nanoparticles is generally complex. In the work, the PMPC and PMPF structures were created and relaxed and gas permeation simulation was done via MD simulation. We employed the logarithmic plot of mean square displacement averaged over different time origin versus time to determine if the transport is in the Fickian diffusive regime.5−7 Subsequently, values of the diffusivity were determined through the slope of the line obtained from a least-squares fit of mean square displacement. Finally, the solubilities and permeabilities of different penetrants, including H2, O2, CH4, Ar, and C4H10 in PMPC and PMPF were obtained. The paper is organized as follows: in Section 2, the computational method of the simulation is presented. The results and discussion are presented in Section 3, and in Section 4, the conclusions are drawn.

2. COMPUTATIONAL METHOD 2.1. Composite PMP and Silica Nanoparticle. Like Tamai et al., 5 the PMP sample is modeled as H(CCH3CC3H7)31H. The methyl (including the methyl in Received: Revised: Accepted: Published: 6462

February April 18, April 19, April 19,

17, 2013 2013 2013 2013

dx.doi.org/10.1021/ie400524k | Ind. Eng. Chem. Res. 2013, 52, 6462−6469

Industrial & Engineering Chemistry Research

Article

transport, other systems composed of a different number of chains and nanoparticles were simulated. MD simulation has also been done to simulate systems composed of 40 chains and 8 nanoparticles evenly distributed in the simulation cell. The mass ratio of both scales of systems between PMP chains and nanoparticles is the same as the system was composed of 5 chains and 1 nanoparticle. The simulation results of diffusivity and permeability of both systems are in agreement. 2.2. Simulation Parameters. The DLPOLY12 software package was employed here. The Verlet algorithm12 was used to solve equations of motion with time step of 2.5 fs under constant pressure and temperature condition. Time step of 1.5 fs was also tried. The simulation results are in agreement with that corresponding to time step of 2.5 fs, except that much more computer simulation time is required. 5000 steps of energy minimization were performed using the steepest descent algorithm to obtain a reasonable starting configuration. Subsequently, the structure was equilibrated with 500 ps MD simulation in the NPT ensemble. The weak coupling technique12 was used to modulate the T and P with relaxation time of 0.1 and 0.5 ps, respectively. The equilibration procedure was followed by 3 ns MD production run in NVT evans12 ensemble. The temperature was set to 300 K. The pressure was 1 bar. The VDW interaction potentials were cut off at 1.2 nm. The electrostatic interaction potential cut off distance was also 1.2 nm. The time for production run was set as 3 ns for diffusion in PMP and composite PMP and silica nanoparticle or set according to eq 1. The production run time can be estimated approximately from the distance traveled by the penetrant in one diffusive jump (ljump, equal to 0.5−1 nm) according to the following equation:

propyl) groups are taken as united groups, while H and C are treated as individual units. The density of PMP at 300 K was used to determine the cell size. The glass transition temperature of PMP is over 523 K.8 As a result, all the systems simulated are in glassy status. The molecular weight of one chain is 2606. The computational details about PMP and the penetrants are described in Appendix A. The diameter of the nanoparticle is 2.5 nm. Most researchers9−11 use nanoparticles of similar size in their MD simulation of composites. To facilitate comparison, simulation cells corresponding to composite PMPC and PMPF are composed of the same number of PMP chains and the same size of nanoparticle. The structures of both the cristobalite silica and the faujasite silica are presented in Figure 1. The structures

t=

ljump2 (1)

6×D

where ljump is the distance traveled by the penetrant in one diffusive jump, equal to 0.5−1 nm. When the time for the production run was set as 10 ns, the results coincided with that with production run time of 3 ns. 2.3. Determination of Diffusivity. When the penetrants enter the Fickian diffusive regime, the mean square displacement of penetrants averaged over different time origin can be employed to calculate diffusivity with the following equation:5,13,14 D = lim

Δt →∞

6 × Δt

(2)

where R(t) is vector position of penetrants at time t and < > means an ensemble average. According to eq 2, diffusivity may be determined with leastsquares fit under the condition of large Δt. Furthermore, the displacement must occur in the Fickian diffusive zone. 2.4. Calculation of Solubility. In mass transport process, the permeability is widely used to compare transport properties of different penetrants.2 The equation to calculate permeability is as follows:15−19

Figure 1. Structures of the cristobalite (a) silica and the faujasite (b) silica.

differ significantly. The silica of the faujasite form has large pores that probably allow penetrants to easily pass through. The Si and O atoms of cristobalite silica are packed densely compared with the faujasite silica. The simulated 2.5 nm nanoparticle of faujasite silica contains 92 SiO2 units, while the nanoparticle of cristobalite silica is made up of 172 SiO2 units. In the composite, the nanoparticle is placed in the center of the simulation cell, among the PMP chains. The simulated systems were composed of 5 chains and 1 nanoparticle. Finally, to explore the influence of nanoparticle mass fraction on

P = S·D = K ·

C0 ·D p0

(3)

where P and S represent permeability and solubility of penetrants, respectively, p0 is the standard pressure, 1 bar, C0 is the ideal gas concentration at standard condition, and K is 6463



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the solubility coefficient with its unit being 1, which is computed with eq 4:

∫−∞

N (r ) Nt ·Δr

(10)

−ΔE / RT

+∞

K=

p(r ) =

d(ΔE)ρ(ΔE)e

(4)

Here, ΔE is the potential energy difference due to the insertion of penetrants and ρ(ΔE) is the possibility density of the energy difference being ΔE. The integration in eq 4 gives the ensemble average of e −ΔE/RT . It may be discretized as Widom presented.15,20,21 The Widom test particle insertion method20,22 is employed to calculate the solubility coefficients. It is assumed that the particle is inserted into the system randomly and the potential change due to the penetrant insertion is ΔEk. If N times of insertions are done and N is large enough, then the solubility coefficient can be determined with eq 5. N

K=

∑k = 1 e−ΔEk / RT (5)

N

According to Appendix B, it is observed that, when the penetrants composition in matrices reaches Cm, the experimental values and the calculated values of permeability satisfy eq 6: Pcal = Pexp·γ(Cm)

(6)

When the composition of penetrants in matrices increases, the activity coefficient will differ more from one and the calculated values of permeability will differ more from the experimental values. If two types of penetrants, 1 and 2, were transported in matrices, the calculated selectivity of penetrant 1 over 2, without considering activity coefficient, would be

Selcal =

Figure 2. Cavity size distribution in PMP, PMPC, and PMPF; p(r) is the probability density of the cavity radius being r. (a) Shows the cavity radius distribution when the size of the matrix units is ignored (regarded as zero). (b) Shows the cavity radius distribution when the radius of the matrix units is considered as one-half of the LennardJones size parameters of the corresponding units.

Pcal,1 Pcal,2

(7)

To get a clear picture of the real cavity size distribution in PMP and the composite, two types of cavity size distribution were evaluated according to the trajectory files. When the size of the units of the matrices is ignored, we get cavity radius distribution in Figure 2a. When the radius of the matrix units is considered as one-half of the Lennard-Jones size parameters of the corresponding units, we obtain Figure 2b. From Figure 2, it is observed that, in the composite PMP and silica nanoparticle, the fraction of large cavities is higher than in pure PMP. Therefore, due to the existence of the silica nanoparticle, the cavity size distribution varies, which lead to higher diffusivity in the composite. Furthermore, the fraction of large cavities in the composite PMPF is higher than that in the composite PMPC. The pores in the faujasite silica nanoparticle contribute to the difference. 3.2. Diffusivity Calculation. At different times, the distances of CH4 to the simulation cell center in PMPC and PMPF are presented in Figure 3. The radius of the silica nanoparticle is 1.25 nm. From Figure 3a, it is seen that CH4 is only transported to an area further than 1.25 nm from the cell center. That is, the penetrant definitely cannot pass through the nanoparticle of the cristobalite silica. On the contrary, Figure 3b shows that, in the composite PMPF, CH4 penetrants can be transported in areas closer than 1.25 nm from the cell center. The penetrant can definitely pass through the nanoparticle of faujasite silica.

while the experimental value of selectivity of penetrant 1 over 2, according to eq 6, should be Selexp =

Pcal,1 Pcal,2·γ1

(8)

This assumes that the solubility of penetrant 2 is small and the corresponding activity coefficient can be regarded as 1. γ1 is the activity coefficient of penetrant 1 solvated in the composite matrices. Assume the solubility of penetrant 1 in PMPF is higher than in PMPC. The corresponding activity coefficient in PMPF should be higher. If both types of penetrants diffuse in two types of matrices, PMPC and PMPF, according to eqs 7 and 8, eq 9 would be obtained: Selcal,PMPC Selcal,PMPF

=

Selexp ,PMPC γ1,PMPC Selexp ,PMP − C · < Selexp ,PMPF γ1,PMPF Selexp ,PMP − F

(9)

3. RESULTS AND DISCSSION 3.1. Cavity Size Distribution. The number of cavities with radius in certain narrow ranges (r, r + Δr) was counted first. The counted cavities number N(r) divided by the total number of cavities over the whole range, Nt, and radius range width, Δr, will give the probability density, p(r), in Figure 2. That is: 6464



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Figure 4. MSD distribution for methane in PMPC for Δt = 0.1 ns ; p(d) is the probability density of the MSD being d.

Figure 3. Distance, r, of CH4 penetrant to the simulation cell center in PMPC (a) and PMPF (b) at different times.

The shape of the loci in Figure 3a,b is different. In Figure 3a, it is seen that the transport is composed of large jumps separated by a quiescence of long periods, while in Figure 3b, the penetrant motion contains frequent diffusive jumps and lacks quiescent periods. Therefore, in the PMPC, the penetrant hops after long periods of localization in voids in the PMP structure, while in PMPF, the penetrant is no longer trapped in voids when it diffuses in the nanoparticle of faujasite silica and the view of the penetrant motion as that of hopping between voids is no longer applicable. However, the penetrant jumps back and forth between PMP portion and the nanoparticle portion; therefore, the diffusivity in PMPF will not be much higher than that in PMPC. The statistics of the displacement of the penetrants follow the Gaussian behavior. We analyzed the mean square displacement (MSD) with different time origin for time period Δt = 0.1 ns. The number of MSD in certain narrow ranges (d, d + Δd) was counted first. The counted number N(d)divided by the total number of MSD computed for the certain time period, Nt, and radius range width, Δd, will give the probability density. That is:

Figure 5. Relationship between the mean square displacement averaged over different time origin and time when CH4 diffuses in PMP, PMPC, and PMPF. R2 is the square of correlation coefficient.

results of diffusivity of different penetrants in PMP, PMPC, and PMPF are presented in Table 1. Table 1. Diffusivity of Penetrants in PMP, PMPC, and PMPF (D × 109 m2/s)a diffusivity

a

penetrants

DPMP

DPMPC

DPMPF

H2 O2 Ar CH4 n-C4H10

47.5(16.1) 0.610(0.17) 0.166(0.07) 0.092(0.05) 0.030(0.02)

31.7(8.9) 4.31(1.23) 0.732(0.327) 0.112(0.051) 0.108(0.054)

136.1(39.2) 7.52(3.21) 1.053(0.345) 0.338(0.171) 0.115(0.063)

The data in the parentheses are the standard deviations.

From Table 1, it is observed that, from H2, O2, CH4, and Ar to n-C4H10, the corresponding diffusivity in PMP decreases. This trend matches the changing trend of the diffusivity of these penetrants in polyethylene.2,17,23,24 3.3. Computation of Solubility Coefficients and Permeability. At different production time, the values of the calculated solubility coefficients would fluctuate, because during the production run, the PMP structures and the composite structures have minor variation due to minor oscillation of atoms around their equilibrium positions. As a result, to get suitable results of the solubility coefficients, the values of the solubility coefficients corresponding to structures at different times with certain interval were calculated first. Then, the average value of all the coefficients obtained would be regarded as the solubility coefficients of the corresponding penetrants in

N (d ) p(d ) = Nt ·Δd

The calculated results are presented in Figure 4. It is observed that the displacement of the penetrants follow the Gaussian behavior. The logarithmic plot of mean square displacement averaged over different time origin versus time was used to determine if the transport is in the Fickian diffusive regime. Subsequently, the values of the diffusivity were determined via the slope of the line obtained from a least-squares fit. Figure 5 shows the relationship between the mean square displacement averaged over different time origin and the time when CH4 diffuses in PMP, PMPC, and PMPF. The simulation 6465



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that material. The solubility coefficients of CH4 in PMP, PMPC, and PMPF at different times are presented in Figure 6.

Table 2. Solubility Coefficients Calculation Results of Different Penetrants in PMP and Two Types of Compositesa penetrant matrix H2 O2 Ar CH4 nC4H10

PMP 0.250(0.0631) (0.0000248) 0.269(0.117) (0.000790) 0.838(0.216) (0.00307) 1.654(0.323) (0.00243) 107.304(24.167) (0.363)

PMPC 0.320(0.074) (0.0000318) 0.381(0.134) (0.00112) 3.235(0.647) (0.0119) 7.539(1.398) (0.0117) 305.376(142.566) (0.584)

PMPF 0.725(0.120) 0.0000720) 1.345(0.245) (0.00213) 12.567(3.283) (0.0251) 65.231(11.024) (0.0517) 6002.36(2192.75) (0.898)

The data in the first parentheses of each cell are the standard deviation; the data in the second parentheses of each cell are weight concentration of the penetrants. The units are all 1. a

Figure 6. Solubility coefficients of CH4 in PMP (a), PMPC (b), and PMPF (c) at different times.

The solubility coefficients calculation results of different penetrants in PMP and composite are listed in Table 2. The weight concentration of penetrants in PMP is (mpenetrants)/ (mpenetrants + mPMP), while that in composite is (mpenetrants)/ (mpenetrants + mPMP + mNP). From Table 2, it is observed that, from H2, O2, Ar, and CH4 to n-C4H10, the corresponding solubility coefficients decreases. This trend matches the changing trend of the solubility values of these penetrants in polyethylene.15 To explore the space distribution of potential difference in the simulation cell due to the penetrant insertion, 10 000 successive random insertions were done in the matrix structure corresponding to 2.5 ns in Figure 6b,c. According to eq 5, the contribution of each insertion is proportional to e−ΔE/RT. Calculation results show that, in both PMPC and PMPF, the 25 insertions with the 25 lowest potential change account for 99.6% of the contributions of all the insertions. Figure 7a,b shows the values of the 25 lowest potential differences and the corresponding position of the insertions in PMPC and PMPF, respectively. The label number i means that the corresponding potential difference is the ith lowest potential differences due to insertion. From Figure 7, it is observed that, in PMPC, the positions of the 25 insertions that contribute most are further

Figure 7. (a, b) Shows the values of the 25 lowest potential differences and the corresponding position of the insertions in PMPC and PMPF, respectively. The label number i means that the corresponding potential difference is the ith lowest.

from the cell center than 1.25 nm. On the contrary, in PMPF, the 25 insertions that contribute most locate inside the nanoparticle of faujasite silica, so in PMPF, the nanoparticle solvates most penetrants; it is the large pores in the nanoparticle of faujasite silica and the strong interaction between the penetrants and the Si and O atoms of the nanoparticle that lead to the high solubility of different penetrants in PMPF. Then, with the values of diffusivity and solubility coefficients, the permeability values of different penetrants in PMP and composite PMP and silica nanoparticle were calculated and listed in Table 3. Table 3 shows that in PMP and PMPC the calculated permeability of CH4 is closer to the experimental 6466



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Table 3. Calculated Permeability of Different Penetrants in PMP and Compositesa penetrants PMP

a

3

matrix

P (10 barrer)

H2 O2 Ar CH4 n-C4H10

15.623 0.216 (0.1852,24) 0.183 0.203 (0.19125) 4.236 (2.73525)

PMPC γ

PMPF γ

P (103 barrer)

1.275 1.942

129.853 13.311 17.415 29.015 908.397

3

P (10 barrer) 13.349 2.161 3.116 1.113 (0.87325) 43.403 (22.35725)

1.167 1.068 1.549

The data in parentheses are the corresponding experimental results.

value than the computed permeability of n-C4H10. According to Table 2, the solubility of n-C4H10 is high in PMP and PMPC compared with O2 and CH4. The size of n-C4H10 is larger than O2 and CH4. The insertion of these n-C4H10 molecules will cause more changes of matrices structures. The activity coefficient of n-C4H10 in PMP and PMPC will differ more from one than O2 and CH4. Therefore, the calculated permeability of n-C4H10 in PMP and PMPC will differ more from the corresponding experimental values than O2 and CH4. From Table 3, it is observed that these two types of composites both have higher permeability than PMP for the same penetrants. As observed in Figure 7, it is the large pores in the nanoparticle of faujasite silica (so cavities play most significant roles) and the strong interaction between the penetrants and the Si and O atoms of the nanoparticles that lead to the high solubility of different penetrants in PMPF. The results of the in-depth analysis show that, though PMPF has higher solubility because there are more cavities with higher radius in PMPF than PMPC, it is also these cavities that lead to lower selectivity of n-butane over methane in PMPF than PMPC. It is just like the performance of the sieves we use daily. Especially for n-C4H10, the insertion of silica nanoparticle in PMP significantly increases permeability. The selectivity of CH4 over O2 does not differ much from one in both PMP and composites, as the molecule sizes and VDW interaction of CH4 with matrix atoms are close to O2. The selectivity of n-C4H10 over CH4 increases from 20 to 31 due to the insertion of nanoparticle of faujasite silica and to 38 because of the insertion of nanoparticle of cristobalite silica. The ratio of the computed value of selectivity of n-C4H10 over CH4 in PMPC to that in PMPF is therefore 38/31. When two types of penetrants, n-C4H10 (1) and CH4 (2), are transported in matrices, the solubility of n-C4H10 in PMPF is significantly higher than in PMPC and the corresponding activity coefficient will be higher. Therefore, according to eq 8, the ratio of the experimental value of selectivity of n-C4H10 over CH4 in PMPC to that in PMPF would be higher than 38/31. The composite PMP and nanoparticle of cristobalite has better performance and is therefore widely employed to separate the mixture of n-C4H10 with CH4 in industry.2,23 3.4. Influence of Mass Fraction of Nanoparticle on Selectivity. The CH4 and n-C4H10 transport in other systems listed in Table 4 has been simulated with the MD method as well. Figure 8 gives the approximate relationship between the selectivity of n-C4H10 over CH4 and the weight concentration of nanoparticle in composite. The selectivity increases with the increase of weight concentration of nanoparticle. When the weight concentration is smaller than 0.264, the selectivity also decreases with the decrease of weight concentration until the selectivity reaches a minimum when composite becomes pure

Table 4. Number of Nanoparticles and Chains, the Corresponding Mass Fraction of Nanoparticles, and the Cell Volume for Different Simulation Systems of Composite PMP and Nanoparticle of Cristobalite Silica mass fraction of nanoparticles

number of chains

number of nanoparticles

cell volume (nm3)

0.264 0.305 0.360 0.441 0.497

11 9 7 5 8

1 1 1 1 2

65.643 55.196 44.748 34.300 58.152

Figure 8. Relationship between the selectivity of n-C4H10 over CH4 and the weight concentration of nanoparticle in composite PMP and nanoparticle of cristobalite silica.

PMP. Figure 8 can be employed to guide composite material and membrane design.

4. CONCLUSION In industry, the processes to separate gas mixtures are rather significant. The composite PMP and silica nanoparticle is used to separate C4H10 (n-butane) from mixtures of C4H10 and CH4, H2, etc. Silica has different crystalline forms. The transport properties of penetrants in the composite PMP and nanoparticles of these two forms of silica are different. It is essential to explore the real reasons that lead to the difference from molecular level in order to aid the design of membrane made of composite PMP and silica nanoparticles. The molecular dynamics method was employed in the work to explore the transport of different penetrants, including H2, O2, CH4, Ar, and C4H10 in PMP and the composites of PMP and two forms of silica nanoparticles, the cristobalite form and the faujasite form. The complicated structures of PMPC and PMPF were established and relaxed. With the structure, the cavity size distribution was analyzed and it is observed that composite PMP and silica nanoparticle has more large cavities than pure PMP, while PMPF has more large cavities than PMPC. The results show that insertion of nanoparticles changes the 6467



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polymer packing significantly. Cavity size distribution is the molecular level of intrinsic factor that determines the magnitude of diffusivity and solubility of different penetrants in matrices. It is the difference in the cavity distribution that leads to the differences in diffusivity, solubility, and permeability of penetrants in different materials. The diffusivity of different penetrants was determined through least-squares fit of the data of mean square displacement in Fickian diffusive regime averaged over different time origin. The results show that, from H2, O2, CH4, and Ar to n-C4H10, the corresponding diffusivity in PMP decreases. This trend matches the changing trend of the diffusivity of these penetrants in polyethylene. The solubility coefficients and the permeability of different penetrants in PMP and the composite were calculated. The calculated results show that these two types of composites both have high permeability compared with PMP for the same penetrants and PMPF has higher permeability than PMPC for the same penetrants. The selectivity of n-C4H10 over CH4 increases from 20 to 31 due to the insertion of nanoparticle of faujasite silica and to 38 because of the insertion of nanoparticle of cristobalite silica. The ratio of the real selectivity of n-C4H10 over CH4 in PMPC to that in PMPF is higher than 38/31. PMPC has better performance than PMPF and is widely employed to separate the mixture of n-C4H10 with other gas molecules, like CH4, in industry. Finally, the influence of weight concentration of nanoparticles on penetrants transport was explored. According to the simulation results, the selectivity of n-C4H10 over CH4 increases with the increase of mass fraction of nanoparticle.

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Table A1. Lennard-Jones Parameters of Different Penetrants and Silica Atoms penetrant

LJ ε (kcal/mol)

LJ σ (nm)

H213 O213 CH44,13,26 Ar11 CH3(C4H10)5,27,29 CH2(C4H10)5,30 Si5,28 O5,28

0.0752 0.0885 0.294 0.237 0.175 0.117 0.584 0.203

0.232 0.309 0.373 0.341 0.391 0.391 0.338 0.296

■

APPENDIX B Equation 3 holds only when the composition of penetrants in matrices is low. When the composition of penetrants in matrices is high, the nonideal property of penetrants should be considered. Assuming the gas phase is the ideal gas at standard condition and the fugacity of penetrants in matrices can be computed according to Henry’s theory, the following equations can be obtained: fg = p0 = R ·T ·C0 fm = H ·x·γ = H ·

(B.1)

Cm ·γ Ct

(B.2)

where Cm stands for penetrants composition in matrices and Ct represents total molar composition of the matrix phase including penetrants. Ct can be regarded as constant even when penetrants compositions change. At equilibrium, the fugacity of penetrants in the gas phase and the matrix phase should be equal, so eq B.3 will be obtained:

APPENDIX A

A.1. PMP

Like Tamai et. al.,5 the PMP sample is modeled as H(CCH3CC3H7)31H. In the simulation, the AMBER/OPLS force field26,27 was used. The electrostatic interaction was computed using the Ewald sum12 algorithm. The VDW interaction was computed with the Lennard-Jones equation.12 The generally employed self-avoid random walk algorithm28 was employed to create the initial structure of PMP. The chains were folded when the periodic boundary condition was employed. 5000 steps of energy minimization were performed according to steepest descent algorithm to obtain a reasonable starting configuration. This was followed by a 3 ns MD production run.

K=

R·T ·Ct Cm = C0 H ·γ

(B.3)

The activity coefficient γ is one when the solution of penetrants in matrices is infinitely dilute. When the Widom test particle insertion method is employed to calculate solubility coefficient, the composition of penetrants is very low, so the corresponding γ can be regarded as one and eq B.4 becomes: Kcal =

R·T ·Ct Cm = C0 H

(B.4)

The experimental results are obtained when the penetrants compositions in matrices reaches Cm, so according to eqs 3, B.3, and B.4, the experimental values and the calculated values of permeability satisfy eq B.5:

A.2. Penetrants

The united-atom model is employed to describe the gas molecules. H2, O2, CH4, and Ar are all treated as single groups. n-C4H10 is regarded as consisting of two methyl groups and two methylene groups. The Lennard-Jones equation is employed to calculate VDW interactions. The Lennard-Jones parameters are listed in Table A1.5,13,15,27−31 For interactions between different types of units, the Lorentz−Berthelot mixing rule is used. It should be noted that the gas molecules should be inserted before energy minimization was performed. Otherwise, if the penetrants are inserted after the energy minimization run of the PMP or composite structure, because gas molecules cause the system lose relaxation again due to the additional interaction between the gas molecules and the matrix, a second time of energy minimization must be done. Seven molecules are inserted in every simulation.

Pcal = Pexp·γ(Cm)

(B.5)

When the composition of penetrants in matrices increases, the activity coefficient will differ more from one, and as a result, the calculated values of permeability will differ more from the experimental values.

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AUTHOR INFORMATION

Corresponding Author

*Tel: 540-235-2971. Fax: 540-231-5022. E-mail: quany@vt. edu. Notes

The authors declare no competing financial interest. 6468



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Comparing Penetrants Transport in Composite Poly (4-methyl-2

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