MOLECULAR SIMULATIONS OF DIFFUSION IN ZEOLITES AND IN AMORPHOUS POLYMERS

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1 MOLECULAR SIMULATIONS OF DIFFUSION IN ZEOLITES AND IN AMORPHOUS POLYMERS George K. Papadopoulos, Doros N. Theodorou Computational Materials Science and Engineering Lab School of Chemical Engineering National Technical University of Athens, Greece

2 APPLICATIONS Zeolites Adsorption separations Industrial catalytic processes Ion exchange Detergents Gas storage Polymers Membrane separations Packaging materials Biomedical devices Performance is governed by sorption and diffusion phenomena which depend on molecular-level structure and interactions.

3 MOLECULAR MODELS Sorbates: Use molecular mechanics force fields that give good predictions for fluid-phase properties united atom representation for alkanes: φ r 1 6 σ σ LJ () r 4ε V = r r θ Lennard-Jones (dispersion attraction and excluded volume interactions): Bond angle bending: V θ 1 ( θ ) = kθ( θ θo) Dihedral angle torsion: 5 k Vφ ( φ) = ck cos ( φ) k = 0 Polar molecules (e.g., CO ): In addition, Coulomb interactions between partial charges: () zize j VC r = 4 r πε o e e e

MOLECULAR MODELS x z y Silicalite Si 96 O 19 Pnma a = 0.07 Å b = 19.91 Å c = 13.

crystallographically known positions. Lennard-Jones and Coulomb interactions with sites on sorbate molecules.

4 MOLECULAR MODELS x z y Silicalite Si 96 O 19 Pnma a = 0.07 Å b = Å c = 13.4 Å ( 4 unit cells shown) x y z Zeolites: Represented as sets of framework atoms and counterions at their crystallographically known positions. Lennard-Jones and Coulomb interactions with sites on sorbate molecules. Partial charges on framework from DFT calculations Rigid framework model allows pretabulation of the zeolite field experienced by sorbate sites: 100 fold savings in CPU time.

5 MOLECULAR MODELS Amorphous Polymers: Represented by same types of force fields as small organic molecules. C 6000 polyethylene 4 polymer chains interaction sites Thoroughly equilibrated using specially designed algorithms, e.g., connectivity-altering Monte Carlo N. Karayiannis, V.G. Mavrantzas, DNT, Phys. Rev. Lett. 88, (00). PVT behavior, packing, segmental dynamics, accessible volume distribution validated against experiments.

6 PREDICTION OF SORPTION ISOTHERMS Grand Canonical Monte Carlo (GCMC) Simulation: Sorbate-zeolite system simulated under constant V s, T, f Elementary moves attempted: Sorbate fugacity displacement removal addition Move acceptance criteria designed so that the sequence of configurations generated samples the equilibrium probability distribution of the grand canonical ensemble at prescribed V s, T, f. <N> = f (V s, T, f ) Equilibrium adsorption isotherm at T

7 PREDICTION OF DIFFUSIVITY: Molecular Dynamics (MD) Simulations Self-Diffusivity D s r(0) r(t) D s = lim t < [ r() t r( 0) ] > 6t (Albert Einstein, 1905) (Adolf Fick, 1855) Transport Diffusivity D t J = D t c flux, mol m - s -1 concentration, mol m -3 sorbate fugacity Corrected Diffusivity D o L.S.Darken, Trans. AIME 1948, 174, 184 D t = D o ln ln f c T lim D ( c) = lim D ( c) = lim D ( c) s t c 0 c 0 c 0 o

8 INCOHERENT QENS SPECTRA FROM MD r r j (t) r j (0) Self-part of the van Hove correlation function: N 1 Gs (,) r t = δ + j(0) j() t N r r r j= 1 Self-part of the intermediate scattering function: 3 Is( Q, t) = Gs( r, t) exp( iq r) d r Incoherent scattering function: + 1 Ss( Q, ω) = Is(, t) exp( iωt) dt π Q Isotropic self-diffusion over large r, t: G 1 r ( r, t) = exp (4 π D t) 4Dst s 3/ s Gaussian in r I Q t D Q t s(, ) = exp( s ) Gaussian in Q, exponential in t S 1 π ω s s ( Q, ω ) = Lorentzian in ω HWHM=D s Q + DQ ( DQ s )

9 DIFFUSION OF METHANE n-butane MIXTURES IN SILICALITE S(Q,ω) CH 4 motion: MD CH 4 : 4 molecules per unit cell T=00 K n-c 4 D 10 : 4 molecules per unit cell Q (Å -1 ) E=ħω (mev) L.N. Gergidis and DNT J. Phys. Chem., 103, (1999) L.N. Gergidis, DNT, and H. Jobic J. Phys. Chem., 104, (000)

10 SCATTERING FUNCTION LINE SHAPE ANALYSIS: METHANE n- BUTANE MIXTURES IN SILICALITE T=00 K CH 4 motion CH 4 : molecules per unit cell n-c 4 D 10 : 4 molecules per unit cell HWHM (mev) MD: 5 D s,ch = cm / s 4 6 D s,n-c D = cm / s 4 10 Q (Å - )

DIFFUSION OF METHANE n-butane MIXTURES IN SILICALITE Distribution in the pores Methane self-diffusivity CH 4 : 4 molecules per unit cell n-c 4 H 10 : 9 molecules per unit

11 DIFFUSION OF METHANE n-butane MIXTURES IN SILICALITE Distribution in the pores Methane self-diffusivity CH 4 : 4 molecules per unit cell n-c 4 H 10 : 9 molecules per unit cell T=300 K Τ=00 Κ 4 CH 4 molecules per unit cell L.N. Gergidis and DNT J. Phys. Chem., 103, (1999) L.N. Gergidis, DNT, and H. Jobic J. Phys. Chem., 104, (000)

DtQ Scoh ( Q, ω) = π ω + ( D Q ) D t extracted from slope of HWHM vs. Q at low Q.

12 D t, D o MEASUREMENT AND SIMULATION Coherent Quasielastic Neutron Scattering (QENS) Dr. Hervé Jobic, Institut de Recherches sur la Catalyse, CNRS, Villeurbanne, France: IN6 spectrometer, ILL, Grenoble Coherent Scattering Function (isotropic motion) at low Q: SQ ( ) DtQ Scoh ( Q, ω) = π ω + ( D Q ) D t extracted from slope of HWHM vs. Q at low Q. t Equilibrium MD Simulation, duration 10ns N N 1 d N Do = dt < vi( t) v j(0) >= lim Rcm( t) Rc m(0) 3N dt t 6 0 i= 1 j= 1 Molecular velocities Green-Kubo Intensity (a.u.) Intensity (a.u.) E (mev) N /silicalite T=00K 0.85 molec./u.c Q (Å -1 ) Q (Å -1 ) [ ] Center of mass of swarm of N molecules: Einstein

13 TRANSPORT DIFFUSIVITY OF N AND CO IN SILICALITE-1: COHERENT QENS MEASUREMENTS AND MD SIMULATIONS G. K. Papadopoulos, H. Jobic, DNT, J. Phys. Chem. B,. 108, e-4 1.0e-4 9.0e-5 8.0e-5 N QENS 00 K D t 1.e-4 1.1e-4 1.0e-4 N MD D t Diffusivity (cm / s) 7.0e-5 9.0e-5 6.0e-5 D 0 5.0e-5 8.0e-5 D 0 4.0e-5 7.0e e-4 7.0e-5 1.0e-4 CO QENS 6.5e K 9.0e-5 6.0e-5 CO MD 8.0e-5 D t 5.5e-5 5.0e-5 7.0e-5 D t 4.5e-5 6.0e-5 4.0e-5 D 5.0e e-5 D 0 4.0e-5 3.0e Molecules / u.c.

14 TRANSPORT OF N AND CO IN SILICALITE -17, Ea=.65 =. kj/mol -18,1 Ea=.7KJ/mol D D t (cm /s) 8x10-5 6x10-5 4x10-5 ln Dt (m /s) -18,3-18,5-18,7-18,9 N at 00 K (1.1 molecule/u.c.) -19,1-19,3 N (1.1 molecules/u.c.) x /T -19,5 3 3,5 4 4,5 5 5,5 1000/T -18, Ea = 5.14 kj/mol -18,7 E a =5.8 kj/mol 8x ,9 D D t (cm /s) 6x10-5 4x10-5 CO at 300 K ( molecules/u.c.) ln Dt (m /s) -19,1-19,3-19,5-19,7-19,9-0,1 CO ( molecules/u.c.) x /T -0,3,5 3 3,5 4 4, /T QENS experiment (H. Jobic) MD (Athens)

15 1.0 GCMC N GCMC CO βw = -0.1, z = 8 βw = 0.7, z = 8 ANALYTICAL MODEL FOR D 0 (θ ) Reed D.A, Ehrlich G., Surface Sci. 1981,10, θ D-lattice of uniform z fraction of occupied sites θ f (atm) 1.6 MD N MD CO β w = -0.1 β w = 0.7 β w = 1.15 β w = -0.9 Effective jump rate of molecules surrounded by j neighbors in the lattice Γ (θ) Γ (j) Ρ(j) Individual jump rate Quasichemical isotherm nearest-neighbor interactions w D 0 (θ) / D 0 (0) θ QC: z z D0 ( θ) ζ( w, z) + 1 ζ( w, z) 1+ θ = 1 exp( βw/ z) D (0) + θ θ 0 D0 ( θ ) Langmuir: D (0) G K. Papadopoulos, H. Jobic, D. N. Theodorou, J. Phys. Chem. B 004, 108, = 1 θ 1

16 RATIOS OF PURE GAS PERMEABILITIES THROUGH MFI MEMBRANES AT 300 K Experiments: Supported Membrane Permeation Y. Yan, M.E. Davis, G.R. Gavalas, Ind. Eng. Chem. Res., 34, 165 (1995). Simulations: Equilibrium NVE MD and GCMC K. Makrodimitris, G.K. Papadopoulos, DNT, J. Phys. Chem. B, 777, 105 (001). Sorbates Experiments Simulations CH 4 /N CO /CH 4.3. CO /N.8 3.4

17 GAS PERMEABILITY OF AMORPHOUS POLYMERS Basic question: How do the solubility and diffusivity of small molecules in polymers depend on chemical constitution and temperature? Common diffusion mechanism: Sequence of elementary jumps between accessible volume clusters within the polymer. Jumps are infrequent events.

0: Local minimum of V(r) Relative Potential Energy (kcal/m ol) 8 7 6 5 4 3 1 0-1 -1,0-0,5

18 ELEMENTARY DIFFUSIONAL JUMP Transition-State Theory provides an estimate of the rate constant k 0 1 in terms of the energies and eigenfrequencies at 0,. 0: Local minimum of V(r) Relative Potential Energy (kcal/m ol) ,0-0,5 0,0 0,5 1,0 Relative Reaction Coordinate (Å) : Saddle point of V(r) 1: Local minimum of V(r) CO in PAI, 300 K N. Vergadou

Carlo simulation Self-diffusivity D s : 14.14 nm M.L.

19 Diffusion: Sequence (Poisson process) of uncorrelated elementary jumps in a disordered network of sorption sites Kinetic Monte Carlo simulation Self-diffusivity D s : nm M.L. Greenfield and DNT, Macromolecules 34, 8561 (001) D s = lim t < [ r() t r( 0) ] > 6t

20 ORIGIN OF ANOMALOUS DIFFUSION IN GLASSY POLYMERS n r t n, < 1 Penetrant motion in the glassy matrix exhibits great dynamic heterogeneity Distribution of rate constants for elementary jumps. CH 4 /glassy atactic polypropylene Distribution of elementary jump lengths. M.L. Greenfield and DNT, Macromolecules 31, 7068 (1998)

21 SIMPLE INVESTIGATION OF THE EFFECTS OF DYNAMIC HETEROGENEITY Simple cubic lattice of sites. Random assignment of intersite transition rate constants k i j from a distribution. Calculation of D s through Kinetic Monte Carlo simulation (KMC) and Effective Medium Approximation (EMA). k i j Distribution Functions: Same mean, ν o /6 Various standard deviations, sd(k i j ) Truncated Gaussian distribution of barrier heights. probability density probability density 0.8 sd(k i->j )=0.100v o sd(k i->j )=0.163v o 0.7 sd(k i->j )=0.0v o sd(k i->j )=0.3v o sd(k i j )=0.100 ν o sd(k i j )=0.163 ν o sd(k i j )=0.0 ν o sd(k i j )=0.3 ν o ln(k i->j /v o ) ln(k i j / ν o ) N. Ch. Karayiannis, V. G. Mavrantzas and D. N. Theodorou, Chem. Eng. Sci. 56, 789 (001)

22 Diffusion coefficient as a function of the width of the jump-rate constant distribution Mean square displacement (msd) as a function of time (KMC& ΕΜΑ) self-diffusivity Diffusivity D/ (í D o l s ) / (ν o l ) 0.18 kinetic MC EM T of ( k i-> j / v o ) standard deviation of k i j / ν o <r >/l <r >/l MC sd(k i->j )=0.100 EMA sd(k i->j )=0.100 MC sd(k i->j )=0.163 EMA sd(k i->j )=0.163 MC sd(k i->j )=0.0 EMA sd(k i->j )=0.0 MC sd(k i->j )=0.3 EMA sd(k i->j )=0.3 slope= Dimensionless t v o time t ν o Time-dependent Effective Medium Approximation (EMA): K.W. Kehr, K. Mussawisade, T. Wichmann, in J. Kärger, P. Heitjans, and R. Haberlandt (Eds.) Diffusion in Condensed Matter, Vieweg: Wiesbaden (1998), p Long time limit: S. Kirkpatrick, Phys.Rev.Lett., 7, (1971).

23 APPLICATION: Ο O O O PET DIFFUSIVITY IN PET, PEI O O O O O PEI Polyester ρ (imposed in simulation) (g/cm 3 ) ρ (experimental) (g/cm 3 ) D s (calculated) (10-9 cm /sec) Gusev-Suter, 1993 D s, (experimental) (10-9 cm /sec) ΡΕΤ (1.333) (6.5) ΡΕI (1.346) Lower permeability of PEI is mainly due to its smaller segmental mobility. The latter is attributable to the different mode of connection of phenyl groups within the chains. N. Ch. Karayiannis, V.G. Mavrantzas, and DNT, Macromolecules 37, 978 (004).

24 DYNAMICS OF PHENYL RINGS IN PET, PEI û Mu() t =< u() t u(0) > Segmental mobility higher in PET.

25 CONCLUSIONS Molecular simulations are useful in elucidating mechanisms and predicting rates of diffusion in separation materials. Molecular simulations aid in the interpretation of diffusion measurements. Broad spectra of length and time scales present in technologically important separation materials necessitate the development of hierarchical modelling and simulation approaches.

26 ACKNOWLEDGEMENTS Dr. Leonidas Gergidis Dr. Nikos Karayiannis Dr. Konstantinos Makrodimitris Ms. Niki Vergadou Prof. Michael L. Greenfield, University of Rhode Island, USA Dr. Hervé Jobic, Institut de Recherches sur la Catalyse, CNRS, Villeurbanne, France DG 1 of the European Commission, programme TROCAT (G5RD-CT ), J. Kärger and S. Vasenkov, coordinators

27 3.Α Monte Carlo Algorithm 1. Construction of 3D lattice with periodic boundary conditions. Each site represents a macrostate.. Assignment of the energy (Ε i ) and calculation of the occupancy probability (p eq i ) for each site. 3. Assignment of the barrier energy (Ε ij ), calculation of the activation energy and the rate constant for each transition. k i j Eij Ei = voexp kbt 4. Distribution of a large number of penetrant walkers on the lattice sites. There are no interactions between the walkers. 5. Track down the trajectories for all walkers for a long period of time. 6. Calculate penetrant diffusivity (D) according to Einstein s relation: D = lim r( t) r(0) t 6 t

28 3.Γ Results N. Ch. Karayiannis, V. G. Mavrantzas and D. N. Theodorou, Chem. Eng. Sci. 56, 789 (001) Energy distributions Jump Rate constants distributions probability density F.C. Ei <E i >=10.3k b T, sd(e F.C. s=0.5 i )=0.50k b T <E ij >=1.18k b T, sd(e F.C. s=3.0 ij )=0.500k b T <E ij >=13.31k b T, F.C. sd(es=6.0 ij )=.064k b T <E ij >=13.9k b T, sd(e ij )=.919k b T probability density 0.8 sd(k i->j )=0.100v o sd(k i->j )=0.163v o 0.7 sd(k i->j )=0.0v o sd(k i->j )=0.3v o E/(k b T) ln(k i->j /v o ) Rate constant distribution of elementary CΗ 4 jumps in atactic ΡΡ [Greenfield & Theodorou, 1998]

29 Duration of the anomalous diffusion phenomenon as a function of the width of the jump rate constant distribution 400 kinetic MC EMT 300 crossover time (v o t) standard deviation of (k i->j /v o )

30 3.Β Time dependent Effective Medium Theory Application of t-d E.M.T. to calculate mean square displacement in time. [Argyrakis et al., 1995] The effective rate constant k eff (Laplace space) is calculated from: k max k ij k eff [ sg ( 0, s) 1] (s) k k min ij k 1 z keff where: ρ(k ij ) : probability distribution function of k ij, z : lattice coordination number (z= 6), eff (s) ρ( k (s) ij ) dk G ( 0, s) : initial site occupancy probability [Horner et al., 1995]: 1 G ( 0, s) = ( ) π 3 1 k eff π π π π π π s k eff ij cos w 1 = 0 dw dw dw 3 cos w cos w 3 Mean Square Displacement (Laplace space) is calculated using: < r > (s) = k (s) eff eq pav l s z Diffusivity is calculated from: D k eff = l, eq pav. k eff = lim s 0 k eff (s)

31 ZEOLITES STRUCTURE framework geometry framework charge distribution size, shape, flexibility, and charge distribution of sorbed molecules MATERIAL PROPERTIES Sorption thermodynamics Henry s law constants Sorption isotherms Intracrystalline transport Self-diffusivities Transport diffusivities MICROSCOPIC ASPECTS Siting of sorbate molecules Conformational changes upon sorption Modes and characteristic times of sorbate motion PERFORMANCE Selectivity and rates in sorption separation and catalytic applications

32 MODEL AND POTENTIAL ENERGY FUNCTION Rigid zeolite framework: O-atoms = Lennard-Jones spheres at positions determined by X-ray diffraction in Pnma-HZSM-5 (D.H. Olson et al. J. Phys. Chem. 85, 38 (1981). V = V (positions of all sites constituting the sorbate molecules) = Conformational energy of individual sorbate molecules + Zeolite/sorbate Lennard-Jones interactions + Zeolite/sorbate Coulomb interactions + Sorbate/sorbate Lennard-Jones interactions + Sorbate/sorbate Coulomb interactions flexible alkanes aromatic sorbates, N, CO

33 DIFFUSIVITY FROM RATE CONSTANTS: KINETIC MONTE CARLO z y Methane/Silicalite sorption states x straight channel zig-zag channel intersection Poisson process: Succession of uncorrelated jumps between sorption states, with rate constants known from atomistic analysis. r(t) < [ r() t r( 0) ] > Self-Diffusivity Ds = lim t 6t r(0) (Einstein, 1905)

ELEMENTARY DIFFUSIONAL JUMP 6 : Saddle point of V(r) Potential Energy (kcal/mol) 4 0-80 -60-40 -0 0 0 40 60 0: Local minimum of V(r) Reaction Coordinate (Å) 1: Local minimum of V(r).

34 ELEMENTARY DIFFUSIONAL JUMP 6 : Saddle point of V(r) Potential Energy (kcal/mol) : Local minimum of V(r) Reaction Coordinate (Å) 1: Local minimum of V(r).8 nm Transition-State Theory provides an estimate of the rate constant k 0 1 in terms of the energies and eigenfrequencies at 0,. M.L. Greenfield and DNT, Macromolecules 31, 7068 (1998)

35 QUASICHEMICAL MODEL OF SORPTION Regular lattice of sites z coordination number w nearest-neighbor interactions Quasichemical approximation (QC) 1.0 GCMC N GCMC CO βw = -0.1, z = 8 βw = 0.7, z = 8 w AA QC isotherm: bf w zw AA = θ θ = 1 θ ζ( w, z) + 1 θ z 0.8 θ 0.6 { } ζ ( w, z) = 1 4 θ(1 θ)[1 exp( βw/ z)] 1/ Langmuir isotherm: bf = θ 1 θ f (atm)

36 ANALYTICAL MODEL FOR D 0 (θ ) D.A. Reed and G. Ehrlich. Surface Sci., 10, (1981) 588. Lattice model of uniform z MD N MD CO βw = -0.1 βw = 0.7 βw = 1.15 βw = -0.9 Molecular jumps with fixed attempt rate. Attempt to jump into occupied site unsuccessful. Overall effective jump rate is a function of the individual jump rates of a molecule and the probabilities of being surrounded by i neighbours. D 0 (θ) / D 0 (0) QC: z z D0 ( θ) ζ( w, z) + 1 ζ( w, z) 1+ θ = 1 exp( βw/ z) D (0) + θ θ 0 Langmuir: D D 0 0 ( θ ) (0) = 1 θ θ

lattice 10000 1000 Random 6x5 3x10 x15 slope=1 <r

37 Comparison between random and spatially correlated lattices Spatially correlated lattice Random lattice Random 6x5 3x10 x15 slope=1 <r >/l sd(k i->j ) =0.0v o t v o

38 INCOHERENT QENS SPECTRA FROM MD dσ dωde σ S ( Q, ω) + σ S ( Q, ω) inc inc coh coh r r j (t) r j (0) N 1 Gs (,) r t = δ + j(0) j() t N r r r j= 1 inc 1 S( Q, ω) = d dt exp { i( - ωt) } G(, t) π r Q r r coh N N 1 G(,) r t = δ + i(0) j() t N r r r i= 1 j= 1 S 1 π ω DQ s s( Q, ω) = + ( DQ s )

Permeation of Hexane Isomers across ZSM-5 Zeolite Membranes

2618 Ind. Eng. Chem. Res. 2000, 39, 2618-2622 Permeation of Hexane Isomers across ZSM-5 Zeolite Membranes Rajamani Krishna* and Dietmar Paschek Department of Chemical Engineering, University of Amsterdam,