sorption and transport of fluids in zeolites

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

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.

MOLECULAR MODELSSorbates: Use “molecular mechanics” force fields that give good predictions for fluid-phase properties

“united atom” representation for alkanes:

φ θBond angle bending:

21( ) ( )2 okθ θθ θ θ= −V

Dihedral angle torsion:

r

12 6

LJ ( ) 4rr rσ σε

⎡ ⎤⎛ ⎞ ⎛ ⎞= −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

V

Lennard-Jones (dispersion attraction and excluded volume interactions):

Polar molecules (e.g., CO2):

In addition, Coulomb interactions between partial charges:

2

C ( )4

i j

o

z z er

rπε=V

+0.66 e-0.33 e

5

0

( ) cos ( )kk

k

cφ φ φ=

=∑V

-0.33 e

MOLECULAR MODELS

y

z

x

SilicaliteSi96O192

Pnmaa = 20.07 Åb = 19.91 Åc = 13.42 Å

(2 × 2 × 4 unit cells shown)

y

x z

Zeolites: Represented as sets of framework atoms and counterionsat 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.

MOLECULAR MODELSAmorphous Polymers: Represented by same types of force fieldsas small organic molecules.

PVT behavior, packing, segmental dynamics, accessible volume distribution validated against experiments.

Thoroughly equilibrated using specially designed algorithms, e.g., connectivity-altering Monte Carlo N. Karayiannis, V.G. Mavrantzas, DNT, Phys. Rev. Lett. 88, 105503 (2002).

C6000 polyethylene24 polymer chains 192000 interaction sites

PREDICTION OF SORPTION ISOTHERMS

Grand Canonical Monte Carlo (GCMC) Simulation:Sorbate-zeolite system simulated under constant Vs, T, f

Sorbate fugacity

Elementary moves attempted:

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 Vs, T, f.

<N> = f (Vs, T, f ) → Equilibrium adsorption isotherm at T

PREDICTION OF DIFFUSIVITY: Molecular Dynamics (MD) Simulations

Self-Diffusivity Ds

r(0)

r(t)2

s( )[ ]li 0)m

6(

t

tDt→∞

< − >=

r r

(Albert Einstein, 1905)

Transport Diffusivity Dt(Adolf Fick, 1855)

t cD= − ∇Jflux, mol m-2 s-1 concentration, mol m-3

0 0s 0 otlim ( ) lim ( ) lim ( )c c c

D Dc cD c→ → →

= =

sorbate fugacity

Corrected Diffusivity Doot

lnln T

fD Dc

∂=

∂L.S.Darken, Trans. AIME 1948, 174, 184

INCOHERENT QENS SPECTRA FROM MD

Self-part of the van Hove correlation function:

Self-part of the intermediate scattering function:

Incoherent scattering function:

rj(0)

rj(t)

r

s1

1( , ) (0) ( )N

j jj

G t tN

δ=

⎡ ⎤= + −⎣ ⎦∑ rr r r

3s s( , ) ( , ) exp( ) dI t G t i r= ⋅∫Q r Q r

s s1( , ) ( , ) exp( ) d

2S I t i t tω ω

π

+∞

−∞

= −∫Q Q

Isotropic self-diffusion over large r, t:

Gaussian in r

Gaussian in Q, exponential in t

Lorentzian in ωHWHM=DsQ2

2

s 3 / 2s s

1( , ) exp(4 ) 4

rG r tD t D tπ

⎡ ⎤= −⎢ ⎥

⎣ ⎦

2s s( , ) exp( )I Q t D Q t= −

( )2

ss 22 2

s

1( , ) D QSD Q

ωπ ω

=+

Q

DIFFUSION OF METHANE − n-BUTANE MIXTURES IN SILICALITE

L.N. Gergidis and DNT J. Phys. Chem., 103, 3380-3390 (1999) L.N. Gergidis, DNT, and H. Jobic J. Phys. Chem., 104, 5541-5552 (2000)

S(Q,ω)

Q (Å-1)E=ħω (meV)

CH4 motion: MD T=200 KCH4: 4 molecules per unit cell

n-C4D10: 4 molecules per unit cell

SCATTERING FUNCTION LINE SHAPE ANALYSIS: METHANE – n- BUTANE MIXTURES IN SILICALITE

T=200 K

CH4: 2 molecules per unit celln-C4D10: 4 molecules per unit cell

Q2 (Å-2)

HW

HM

(meV

)

MD:

CH4 motion

4

5 2s,CH 1.8 10 cm / sD −= ×

4 10

6 2s,n-C D 4.1 10 cm / sD −= ×

DIFFUSION OF METHANE – n-BUTANE MIXTURES IN SILICALITE

Distribution in the pores

T=300 K

CH4: 4 molecules per unit cell

n-C4H10: 9 molecules per unit cell

L.N. Gergidis and DNT J. Phys. Chem., 103, 3380-3390 (1999) L.N. Gergidis, DNT, and H. Jobic J. Phys. Chem., 104, 5541-5552 (2000)

Methane self-diffusivity

Τ=200 Κ

4 CH4molecules per unit cell

Dt , Do MEASUREMENT AND SIMULATIONCoherent 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:

2

coh 2 2t

2t( )( , )

( )QS Q DS

QDω

π ω=

+Q

Equilibrium MD Simulation, duration ≤ 10ns

Dt extracted from slope of HWHM vs. Q2 at low Q.

[ ]2cm c

1 1o m

0

1 ( ) (0) lim ( ) (0)3 6

N N

i j ti j

Nddt t tDN dt

∞

→∞= =

= < ⋅ >= −∑∑∫ v v R R

Molecular velocities Green-Kubo

Center of mass of swarm ofN molecules: Einstein

0.0

0.5

1.0

1.5

2.0

2.5

3.0

-0.20.0

0.2

0.40.20

0.250.30

0.350.40

0.45

Inte

nsity

(a.u

.)

E (meV)

Q (Å-1 )

N2/silicaliteT=200K 0.85 molec./u.c.

Inte

nsity

(a.u

.)

Q (Å-1)

G. K. Papadopoulos, H. Jobic, DNT, J. Phys. Chem. B,. 108, 12748

TRANSPORT DIFFUSIVITY OF N2 AND CO2 IN SILICALITE-1: COHERENT QENS MEASUREMENTS AND MD SIMULATIONS

Molecules / u.c.

CO2 MD

0 2 4 6 83.0e-5

3.5e-5

4.0e-5

4.5e-5

5.0e-5

5.5e-5

6.0e-5

6.5e-5

7.0e-5CO2 QENS

0 2 4 6 8 104.0e-5

5.0e-5

6.0e-5

7.0e-5

8.0e-5

9.0e-5

1.0e-4

1.1e-4

N2 MD

0 2 4 6 87.0e-5

8.0e-5

9.0e-5

1.0e-4

1.1e-4

1.2e-4N2 QENS

0 2 4 6 8

Diff

usiv

ity (c

m2 /

s)

4.0e-5

5.0e-5

6.0e-5

7.0e-5

8.0e-5

9.0e-5

1.0e-4

1.1e-4

Dt

D0

Dt

D0

Dt

D0

Dt

D0

200 K

300 K

1000/T2 3 4 5

D

2x10-5

4x10-5

6x10-5

8x10-5

10-4

CO2 at 300 K(2 molecules/u.c.)

Ea = 5.14 kJ/mol

Dt(

cm2 /s

)TRANSPORT OF N2 AND CO2 IN SILICALITE

QENS experiment (H. Jobic)

CO2

(2 molecules/u.c.)-20 ,3

-20 ,1

-19 ,9

-19 ,7

-19 ,5

-19 ,3

-19 ,1

-18 ,9

-18 ,7

-18 ,5

2 2 ,5 3 3 ,5 4 4 ,5 5

1000 /T

ln D

t (m

2 /s)

E a =5 .8 kJ /m o l

MD (Athens)

N2 (1.1 molecules/u.c.)

-19,5

-19,3

-19,1

-18,9

-18,7

-18,5

-18,3

-18,1

-17,9

3 3,5 4 4,5 5 5,5

1000/T

ln D

t (m

2 /s)

Ea=2.7KJ/mol

1000/T3 4 5 6 7

D

2x10-5

4x10-5

6x10-5

8x10-5

10-4

N2 at 200 K(1.1 molecule/u.c.)

Ea = 2.2 kJ/mol

Dt(

cm2 /s

)

Ea=2.65 kJ/mol

ANALYTICAL MODEL FOR D0(θ )

G K. Papadopoulos, H. Jobic, D. N. Theodorou, J. Phys. Chem. B 2004, 108, 12748

• 3D-lattice of uniform z• fraction of occupied sites θ

Langmuir: 0

0

( )1

(0)DD

θθ= −

Γ (θ) Γ (j) Ρ(j)

Individual jump rate

Quasichemical isotherm

nearest-neighbor interactions w

Effective jump rate of molecules surrounded by j neighbors in the lattice

∝

Reed D.A , Ehrlich G., Surface Sci. 1981,102, 588

θ0 .0 0 .2 0 .4 0 .6 0 .8 1 .0

D0(θ

) / D

0(0)

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

1 .2

1 .4

1 .6

M D N 2M D C O 2βw = -0 .1βw = 0 .7βw = 1 .15βw = -0 .9

f (atm)0.01 0.1 1 10

θ

0.0

0.2

0.4

0.6

0.8

1.0

GCMC N2GCMC CO2

βw = -0.1, z = 8βw = 0.7, z = 8

0

0

1( ) ( , ) 1 ( , ) 1 21 exp(2 / )(0) 2 2 2 2

D w wz zD

z zzwθ ζ ζ θ β

θ θ

− −+ − +⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

QC:

RATIOS OF PURE GAS PERMEABILITIES THROUGH MFI MEMBRANES AT 300 K

Experiments: Supported Membrane PermeationY. Yan, M.E. Davis, G.R. Gavalas, Ind. Eng. Chem. Res., 34, 1652 (1995).

Simulations: Equilibrium NVE MD and GCMCK. Makrodimitris, G.K. Papadopoulos, DNT, J. Phys. Chem. B, 777, 105 (2001).

Sorbates Experiments Simulations

CH4/N2 1.2 1.6

CO2/CH4 2.3 2.2

CO2/N2 2.8 3.4

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.

-1 ,0 -0,5 0 ,0 0 ,5 1,0- 1

0

1

2

3

4

5

6

7

8

Re

lativ

e P

ote

ntia

l En

erg

y (k

cal/m

ol)

R elative R eac tio n C o ord in ate (Å )

0: Local minimum ofV(r)

1: Local minimum of V(r)

†: Saddle point of V(r)ELEMENTARY

DIFFUSIONAL JUMP

CO2 in PAI, 300 K

N. Vergadou

Transition-State Theory provides an estimate of the rate constant k0→1 in terms of the energies and eigenfrequencies at 0, †.

Diffusion: Sequence (Poisson process) of uncorrelated elementary jumps in a disordered network of sorption sites

14.14 nm

Self-diffusivity Ds:

M.L. Greenfield and DNT, Macromolecules 34, 8561 (2001)

2

s( )[ ]li 0)m

6(

t

tDt→∞

< − >=

r r

Kinetic Monte Carlo simulation

ORIGIN OF ANOMALOUS DIFFUSION IN GLASSY POLYMERS2 , 1nr t n∝ <

Penetrant motion in the glassy matrix exhibits great dynamic heterogeneity

Distribution of rate constants for elementary jumps.

Distribution of elementary jump lengths.

CH4/glassy atactic polypropyleneM.L. Greenfield and DNT, Macromolecules 31, 7068 (1998)

SIMPLE INVESTIGATION OF THE EFFECTS OF DYNAMIC HETEROGENEITY

Simple cubic lattice of sites.

Random assignment of intersite transition rate constants ki→j from a distribution.

Calculation of Ds through Kinetic Monte Carlo simulation (KMC) and Effective Medium Approximation (EMA).

-25 -20 -15 -10 -5 0 50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 sd(ki->j)=0.100vo

sd(ki->j)=0.163vo

sd(ki->j)=0.220vo

sd(ki->j)=0.232vo

prob

abili

ty d

ensi

ty

ln(ki->j/vo)ln(ki→j/ νo)

sd(ki→j)=0.100 νo



sd(ki→j)=0.232 νopr

obab

ility

den

sity

ki→j Distribution Functions:

• Same mean, νo/6

• Various standard deviations, sd(ki→j)

• Truncated Gaussian distribution of barrier heights.

N. Ch. Karayiannis, V. G. Mavrantzas and D. N. Theodorou, Chem. Eng. Sci. 56, 2789 (2001)

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)

0.00 0.05 0.10 0.15 0.20 0.25 0.300.06

0.08

0.10

0.12

0.14

0.16

0.18 k inet ic M C EM T

Dif

fusi

vity

D/(

í ol2)

standard deviat ion of ( k i-> j/ vo)standard deviation of ki→j/ νo

self-

diff

usiv

ity D

s/ (ν

ol2 )

10 100 1000

10

100

1000

<r2 >/

l2

t vo

MC sd(ki->j)=0.100 EMA sd(ki->j)=0.100 MC sd(ki->j)=0.163 EMA sd(ki->j)=0.163 MC sd(ki->j)=0.220 EMA sd(ki->j)=0.220 MC sd(ki->j)=0.232 EMA sd(ki->j)=0.232 slope=1

<r2 >

/l2

Dimensionless time t νo

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 265-305. Long time limit: S. Kirkpatrick, Phys.Rev.Lett., 27, 1722-1725 (1971).

APPLICATION: Ο2 DIFFUSIVITY IN PET, PEI

OO

OO

OO

OO

PET PEIPolyester ρ

(imposed in simulation) (g/cm3)

ρ(experimental) (g/cm3)

Ds(calculated)(10-9 cm2/sec)

Ds, (experimental) (10-9 cm2/sec)

ΡΕΤ 1.335 1.336 (1.333) 9.6 9.5 (6.5)

ΡΕI 1.356 1.356 (1.346) 5.4 2.7

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, 2978 (2004).

Gusev-Suter, 1993

DYNAMICS OF PHENYL RINGS IN PET, PEI

û

( ) ( ) (0)M t t=< ⋅ >u u u

Segmental mobility higher in PET.

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.

ACKNOWLEDGEMENTS

Dr. Leonidas GergidisDr. Nikos KarayiannisDr. Konstantinos MakrodimitrisMs. Niki Vergadou

Prof. Michael L. Greenfield, University of Rhode Island, USA

Dr. Hervé Jobic, Institut de Recherches sur la Catalyse, CNRS, Villeurbanne, France

DG 12 of the European Commission, programme TROCAT (G5RD-CT-2001-00520), J. Kärger and S. Vasenkov, coordinators

1. Construction of 3D lattice with periodic boundary conditions. Each site represents a macrostate.

2. Assignment of the energy (Εi) and calculation of the occupancy probability (pieq) for each site.

3. Assignment of the barrier energy (Εij), calculation of the activation energy and the rate constant

for each transition.

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:

3.Α Monte Carlo Algorithm

⎥⎦

⎤⎢⎣

⎡ −−=→ Tk

EEvk

b

iijoji exp

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

=∞→ t

rtrD

t 6)0()(

lim2

3.Γ ResultsN. Ch. Karayiannis, V. G. Mavrantzas and D. N. Theodorou, Chem. Eng. Sci. 56, 2789 (2001)

Energy distributions

8 12 16 20 24 28 32 360.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8 F.C. Ei F.C. s=0.5 F.C. s=3.0 F.C. s=6.0

<Ei>=10.23kbT, sd(Ei)=0.250kbT

<Eij>=12.18kbT, sd(Eij)=0.500kbT



prob

abili

ty d

ensi

ty

E/(kbT)

Jump Rate constants distributions

-25 -20 -15 -10 -5 0 50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8 sd(ki->j)=0.100vo

sd(ki->j)=0.163vo

sd(ki->j

)=0.220vo

sd(ki->j)=0.232vo

prob

abili

ty d

ensi

tyln(ki->j/vo)

Rate constant distribution of elementary CΗ4 jumps in atactic ΡΡ[Greenfield & Theodorou, 1998]

Duration of the “anomalous diffusion” phenomenon as a function of the width of the jump rate constant distribution

0.00 0.05 0.10 0.15 0.20 0.250

100

200

300

400cr

osso

ver t

ime

(vot)

standard deviation of (ki->j/vo)

kinetic MC EMT

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]

[ ]0

1,021

max

min

=−

−−

−∫ ijij

k

k

eff

effij

effij dkk

kkk

sGsz

kk)ρ(

(s)(s)

)(

(s)

The effective rate constant (Laplace space) is calculated from:effk

where:ρ(kij) : probability distribution function of kij,z : lattice coordination number (z = 6),

: initial site occupancy probability [Horner et al., 1995]:),0( sG

∫ ∫ ∫π

π−

π

π−

π

π− −−−+π=

321

3213

cos2cos2cos26

121,0

wwwks

dwdwdwk

sG

eff

eff

)()(

Mean Square Displacement (Laplace space) is calculated using: zsl

p(s)k

(s)r eqav

eff2

22 =><

Diffusivity is calculated from: (s)kk effeff lim 0s→

=2

.

lpk

D eqav

eff= ,

ZEOLITES

STRUCTURE• framework

geometry• framework

chargedistribution

• size, shape, flexibility,and chargedistribution ofsorbed molecules

MICROSCOPIC ASPECTS• Siting of sorbate molecules• Conformational changes

upon sorption• Modes and characteristic

times of sorbate motion

MATERIAL PROPERTIESSorption thermodynamics• Henry’s law constants• Sorption isotherms

Intracrystalline transport• Self-diffusivities• Transport diffusivities

PERFORMANCESelectivity and rates in sorption separation and catalytic applications

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, 2238 (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, N2, CO2

DIFFUSIVITY FROM RATE CONSTANTS: KINETIC MONTE CARLO

x

y

z

Methane/Silicalitesorption states

straight channelzig-zag channelintersection

Poisson process: Succession of uncorrelated jumps between sorption states, with rate constants known from atomistic analysis.

Self-Diffusivity

(Einstein, 1905)

2

s( )[ ]li 0)m

6(

t

tDt→∞

< − >=

r r

r(0)

r(t)

ELEMENTARY DIFFUSIONAL JUMP

-80 -60 -40 -20 0 20 40 60

0

2

4

6

Transition-State Theory provides an estimate of the rate constant k0→1 in terms of the energies and eigenfrequencies at 0, †.

Pot

entia

l Ene

rgy

(kca

l/mol

)

Reaction Coordinate (Å)

0: Local minimum ofV(r)

1: Local minimum of V(r)

†: Saddle point of V(r)

2.28 nm

M.L. Greenfield and DNT, Macromolecules 31, 7068 (1998)

wAA

• Regular lattice of sites

• z coordination number

• w nearest-neighbor interactions• Quasichemical approximation (QC)

2 21 ( , ) 1 2

bfz

zwθ θθ ζ θ⎛ ⎞−

= ⎜ ⎟− + −⎝ ⎠

QC isotherm:

1/ 2( , ) 1 4 (1 )[1 exp( 2 / )]{ }w wz zζ θ θ β= − − − −

2AAw zw

=

QUASICHEMICAL MODEL OF SORPTION

f (atm)0.01 0.1 1 10

θ

0.0

0.2

0.4

0.6

0.8

1.0

GCMC N2GCMC CO2

βw = -0.1, z = 8βw = 0.7, z = 8

1bf θ

θ=

−Langmuir isotherm:

ANALYTICAL MODEL FOR D0(θ )

D.A. Reed and G. Ehrlich . Surface Sci., 102, (1981) 588.

Lattice model of uniform z. 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.

θ0.0 0.2 0.4 0.6 0.8 1.0

D0(θ )

/ D

0(0)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

MD N2MD CO2βw = -0.1βw = 0.7βw = 1.15βw = -0.9

0

0

1( ) ( , ) 1 ( , ) 1 21 exp(2 / )(0) 2 2 2 2

D w wz zD

z zzwθ ζ ζ θ β

θ θ

− −+ − +⎡ ⎤ ⎡ ⎤= +⎢ ⎥ ⎢ ⎥− −⎣ ⎦ ⎣ ⎦

QC:

0

0

( )1

(0)DD

θθ= −Langmuir:

Comparison between random and spatially correlated latticesSpatially correlated lattice Random lattice

1 10 100 1000 10000

1

10

100

1000

10000

sd(ki->j) =0.220vo

<r2 >/

l2

t vo

Random 6x5 3x10 2x15 slope=1

INCOHERENT QENS SPECTRA FROM MD2

( , ) ( , )inc inc coh cohd S Q S Q

d dEσ σ ω σ ω∝ +Ω

rj(0)

rj(t)r

s1

1( , ) (0) ( )N

j jj

G t tN

δ=

⎡ ⎤= + −⎣ ⎦∑ rr r r

{ }1( , ) exp ( - ) ( , )2

S d dt i t G tω ωπ

= ⋅∫∫Q r Q r r

inc

coh

1 1

1( , ) (0) ( )N N

i ji j

G t tN

δ= =

⎡ ⎤= + −⎣ ⎦∑∑ r rrr

( )2

ss 22 2

s

1( , ) D QSD Q

ωπ ω

=+

Q

sorption and transport of fluids in zeolites

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