Document 1755580

Diffusion of He in Frozen Polypropylene Structures:
Polymer Density & Simulation Size Effects
1 email:
Norman J. Wagner and Raul F. Lobo
Abstract
Analysis of Molecular Dynamics Results
1e+05
r 2 = limt →∞ 6Dt
100
t (ps)
1000
Total 10ns of
simulation
10000
1.0
0.5
0.0
0.1
1
10
100
1000
0.03
15.9% .0216432000
38.2% .1645020000
39.1% .4730790000
0.02
0.00
20
40
60
80
100
0
20
40
60
80
100
120
0.05
0.030
1000 ps
0.04
Corr.
1.0%
4.4%
13.4%
40.4%
40.6%
0.03
0.02
2500 ps
0.025
= 0.997505
.0001255440
.0010416100
.0151168000
.1294690000
.4167750000
• Van Hove function represents the
probability of a penetrant moving a
distance r after time t.
.6%
2.6%
8.8%
41.2%
46.5%
0.015
0.010
.0000420861
.0003487790
.0048572900
.0711180000
.3632690000
• Limited form (ideal) is a Gaussian
distribution around an average r
0.005
0.00
0.000
0
20
40
0.030
60
80
100
120
Corr. = 0.976022
.3% .0000230149
1.4% .0001758470
8.6% .0068740600
45.4% .0638795000
43.9% .3972920000
0.010
50
0.020
5000 ps
0.025
0
100
• Van Hove function can be deconvoluted to underlying modes:
150
7500 ps
Corr.
.7%
2.3%
21.3%
41.2%
34.3%
0.015
0.010
= 0.958600
.0000193984
.0001476900
.0185312000
.0956370000
.4446190000
−3 / 2 −r 2 / 4Di t
Gs ( r ,t ) = ∑ xi (4πDi t )
0.005
xi - Average fraction of populatio
diffusing with a diffusivity of Di
6.0
g(r)
0.60
0
1
2
3
4
5
0
3
9
12
3.50
2.5
ex
3.0
3.5
4.0
4.5
5.0
Probability
Probability
20
2.50
1.00
0.10
0.05
1.05
1.10
-1
q (Å )
Specific Volume (cc/g)
1.15
1.20
0.00
0.5
1.0
1.5
2.0
2.5
3.0
Hard Sphere Diameter (Å)
• S(q) agrees semi-quantitatively with experimental data
• Solubility agrees with previous studies
• He hard sphere insertion probability used to calculate
porosity: 1.3 → 5.5%
38
Å
*
Maeda, T. Master’s Thesis, U. Penn., 1987
6
4
2
1.000
0.8
0.7
0.6
0.5
van Hove Analysis
MSD Analysis
0.4
0.3
0
50
100
150
200
350
300
250
1.050
1.100
300
200
1.150
1.200
60
200
Simulation Box Size
50
40
150
100
Turnover distance limited by
simulation box size
30
20
25
• Diffusion no longer by means of a
hopping mechanis
• Define mean-free path as: average
distance traveled in the time require
for velocity autocorrelation function to
pass through zero
Mean-free
path
Lc = <v>tc
Kinetic
Diffusivity
DK = 1/3 Lc<v>
Tortuosity
τ = ε DK/D
Solubility
calculations show
smaller structures
permit only a
limited distribution
of sorption sites
27.145 Å
38.003 Å
48.754 Å
59.719 Å
250
150
30
35
40
45
50
55
50
60
65
0
2.40
2.50
2.60
2.70
2.80
ex
µ (xHe→ 0) kJ/mol
Box Dimension (Å)
µ (xHe→ 0) (kJ/mol)
3.00
15
200
Kinetic diffusivity - diffusion mechanism
4.0
6
150
r (Å)
2.0
0.40
100
0.4
Conclusions
0.872 g/cc
0.892 g/cc
0.910 g/cc
0.934 g/cc
0.955 g/cc
0.971 g/cc
0.2
acf
r (Å)
50
<v >
0.80
0
0.872 g/cc
0.892 g/cc
0.910 g/cc
0.934 g/cc
0.955 g/cc
0.971 g/cc
0.15
0
2.0
0.000
0.20
40
ex
1.00
µ (x He → 0) (kJ/mol)
Experimental Maede 1987
1.20
S(q)
4.00
0.000
e
i
0.005
*
8
• Significant size dependence shown for:
– Diffusivity
– Crossover from anomalous to
fickian diffusion
• Crossover is artificially induced by
averaging penetrant pathway over
small simulation box
0.9
Corr. = 0.994937
0.020
0.01
0.015
1.40
2.25
2.00
1.0
2
Corr. = 0.998088
1.3% .0002258520
5.4% .0018583600
0.04
Crossover Time (ps)
4π
πr2 Gs(r,t)
The graphs show Gs(r,t) for a 0.872
g/cc structure, box dimension 38 Å,
these results are representative
600 ps
0.01
0.020
60
1.2
Results: Simulation Size Effects
-4
Corr. = 0.997890
1.7% .0004596500
6.8% .0038064100
19.2% .0381631000
32.9% .2213970000
39.3% .6085870000
0
Cavity Size Distribution
1.15
• Kinetic diffusivity order of
magnitude larger than longtime diffusivity
• Tortuosity factors: 2.5 → 12
Reasonable values considering
the structures
D (10 cm /s)
0.05
300 ps
0.01
Generation method: See Ref. [3]
1.1
1.75
Crossover Distance (Å)
200ps at 1000K,
300ps @ 233K
80
1.05
2.50
Specific Volume (cc/g)
0.00
4.50
1
1/ρ (cc/g)
Time (ps)
0.02
(for 0.872 g/cc structures)
2.75
τ
10
2
1
dlog<r >/dlog(t)
1
0.1
• MSD results show absence of a
short time diffusivity - slope
goes from 2 (ballistic motion)
to 0.5 (anomalous diffusion)
• Dynamics in the anomalous
regime appear similar to that
of a reptating polymer chain
0.872 g/cc
0.934 g/cc
0.955 g/cc
0.971 g/cc
1.5
0.04
Box Dimensions:
27 → 60 Å
3.00
-4
0.0
10
0.03
0.872 g/cc
0.892 g/cc
0.910 g/cc
0.934 g/cc
0.955 g/cc
0.971 g/cc
0.2
2.0
Structure Creation & Characterization
1.60
3.25
2
2
2
100
3.50
cm2/s
• D ~
• For lower density structures - ~1000ps in this
regime
1000
0.4
Dk(10 cm /s)
5 ·10-5
0.05
He Solubility
0.6
2
2
-4
10000
(Box.Dim)
D (10 cm /s)
• Slope within 10% of 1.0: fickian diffusion:
0.872 g/cc
0.892 g/cc
0.910 g/cc
0.934 g/cc
0.955 g/cc
0.971 g/cc
Simulation Model
X-Ray Scattering Factor
[4]
Cuthbert et al.' 99
Van Hove function analysis
MSD Analysis
Self part of van Hove space-time autocorrelation function
Density:
0.872 → 0.971 g/cc
• Order of magnitude difference in
diffusivity between mobile and
immobilized matrices
• Polymer motion aids diffusion in
the dynamic formation of cavities
- hopping mechanism
1.0
0.8
[1] Gusev, A. A.; Mueller-Plathe, F.; van Gunsteren, W.F.; Suter, U.W., Adv. Polym. Sci. 1994, 116, 209.
[2] Weber, H.; Paul, W.; Phys. Rev. E 1996, 54, 3999.
[3] Kotelyanskii, M.J.; Wagner, N.J.; Paulaitis, M.E. Macrom. 1996, 29, 8497.
[4] Cuthbert, T.R.; Wagner, N.J.; Paulaitis, M.E.; Murgia, G., D’Aguanno, B. Macrom. 1999, 32, 5017.
Gaussian chain annealed:
Results: Diffusion Mechanism
Mean Squared Displacement
<r > (Å )
Molecular simulations of penetrant diffusion in glassy polymers show a regime of anomalous diffusion
between ballistic motion and fickian diffusion. Various investigations attribute this effect to the tortuous
diffusion pathway topology and correlated motion of the polymeric structure in which the penetrants
diffuse [1]. The turnover from anomalous to Fickian diffusion is also known to depend on the simulation
box size [2], which can lead to erroneous calculations of the diffusivity. This work aims at decoupling the
effect of the polymer topology and motion on the molecular diffusion of small molecule penetrants in
atactic, glassy polypropylene. United atom polypropylene structures of different sizes and densities are
created using the Gaussian Lattice algorithm of Kotelyanskii et al. [3]. Penetrant gas diffusion is studied by
direct molecular dynamics simulations in frozen polymer structures and compared to results of simulations
in which the polymer is allowed thermal motion [4]. The diffusion mechanism and simulations size effects
are studied by analysis of the self-part of the Van-Hove space-time autocorrelation function for penetrant
diffusion. For the frozen matrix, the diffusion is no longer an activated process. Comparisons with fully
mobile matrix simulations demonstrate the influence of polymer motion on the mechanism of molecular
diffusion in glassy polymers and are in qualitative agreement with previous studies [1]. Furthermore, the
anomalous diffusion regime is identified to be a consequence of the tortuosity in the percolated diffusion
pathways in the polymer structure, which itself is connected to the correlation length in the polymer glass.
System size effects are studied by simulating polymeric structures with central linear dimensions ranging
from approximately 25 to 60 Å. The turnover from anomalous to fickian diffusion shows a marked
simulation size dependence not seen in the results in which the polymer is allowed thermal motion.
[email protected] http://che.udel.edu
Probability
Jan H.D. Boshoff
1,
Center for Molecular and Engineering Thermodynamics
University of Delaware
Department of Chemical Engineering
Newark, DE 19716
0.0
Average time for the
velocity vector to turn
90º due to collisions
with the rigid porous
walls
-0.2
-0.4
0
tc
1
2
Time (ps)
3
4
• We have shown that removing polymer motion from the system changes the
mechanism of diffusion from an activated process to kinetic diffusion in a rigi
matrix
• The van Hove function can be de-convoluted into underlying independent modes of
diffusion
• Anomalous diffusion are due to the tortuosity in the polymer structures - becomes a
reptatio -like diffusion along percolated paths
• We have confirmed that simulation size effects influence the thermodynamics an
molecular motion of penetrants in frozen polymer matrices
Acknowledgements:
(# EEC-0085461)
Computational Chemistry Facility