Distinct roughness dependence of self- and transport
diffusion in mesoporous materials
Kourosh Malek, Marc-Olivier Coppens
Physical Chemistry and Molecular Thermodynamics, DelftChemTech, Delft University of Technology,
Julianalaan 136, 2628 BL Delft, The Netherlands
________________________________________________________________________________
Introduction
The geometrical disorder of disordered nanoporous systems strongly
influences the dynamic and thermodynamic processes that take place
inside pore networks.
Molecular motion in the Knudsen regime, which
involves ballistic motions between points on the
surface, could be influenced by roughness.
Goal: Investigating the effect of pore surface
roughness on Knudsen diffusion in
nanoporous media by dynamic Monte-Carlo
simulations in three-dimensional rough fractal
pores.
Disordered nanoporous media have a fractal rough internal surface over a finite scaling range
What is the conceptual difference
between the roughness dependence
of diffusion in the presence of a
concentration gradient (transport
diffusion) and gradientless (self- or
tracer) diffusion?
Fractal pore models
Roughness factor:
2D → ξ =
First passage time:
Increases from t0 to t
3D → ξ =
DK 0
DK =
1 + α [1 − (δ ') β ]
fractal
fjord tree
α ( Dads , p0 ),
u
β ( Dads , p0 )
Dads : Fractal adsorption dimension
P0 : Return probability, depends on
accessibility distribution
δ
Result and discussion
•
Various 2-D and 3-D fractal pore models (Fig.1 & 2)
•
Self- and transport diffusivities vs. roughness
•
Comparison to the analytical results, Fig.3
Calculation of
diffusivities:
5
3
ξ = ( )n
1
2
D self = lim
r (t ) − r (t 0 )
t → ∞ 6t
fC u
flux
flux = t A → Dtrans = − G
∝ ft
4
∇C A
ξ = (
Sn
S0
13 n
)
9
Fig.1. Segment of a 3-D fractal pore
n=2
n=1
n=0
Fig.2. Typical part of a trajectory in 2-D as calculated by computer
simulations
1
DK/DK0
Analytical calculations
Pn
P0
Simulation
Analytical
0.9
0.8
0.7
1.2
0.6
D K/DK0
1
0.5
Transport diffusivity
Different, like
for zeolites !
0.8
0.6
0.4
0.4
0.3
0.2
0.1
Self-diffusivity
0.2
0
1
0
1
•
2
3
4
5
Roughness Factor (ξ )
6
7
Anisotropic roughness of the walls, Fig. 4
n=0
Dt=0.0142
Ds=0.0141
n=1
Dt=0.0143
Ds=0.0142
n=2
Dt=0.0142
Ds=0.0143
Fig.4. No effect of anisotropic roughness on diffusivity
2
3
4
ξ
5
6
7
Fig.3. Comparison of MC and analytical results in 3-D
Conclusion
Self- (or tracer) diffusivities and transport diffusivities of gases in
nanoporous materials may differ in the Knudsen regime, when the pore
surface is rough on molecular scales. Self-diffusion is roughness
dependent; transport diffusion is not.
References
Sahimi, M., 1993, Rev. Mod. Phys. 65, 1393.
Coppens, M.-O., 1999, Catalysis Today 53, 225-243.
Malek, K. and Coppens, M.-O., 2001, Phys. Rev. Lett. 87,125505.
Malek, K. and Coppens, M.-O., 2002, Coll. Surf. A. 206, 335.
Malek, K. and Coppens, M.-O., 2003, J. Chem. Phys. 119, 2801.
Coppens, M.-O. and Malek, K., 2003, Chem. Eng. Sci. 58, 4787.
Acknowledgement
Delft University of Technology