We formulate a microscopic, force-level statistical mechanical theory for the activated diffusion of dilute penetrants in dense liquids, colloidal suspensions, and glasses. The approach explicitly and self-consistently accounts for coupling between penetrant hopping and matrix dynamic displacements that actively facilitate the hopping event. The key new ideas involve two mechanistically (at a stochastic trajectory level) coupled dynamic free energy functions for the matrix and spherical penetrant particles. A single dynamic coupling parameter quantifies how much the matrix displaces relative to the penetrant when the latter reaches its transition state which is determined via the enforcement of a temporal causality or coincidence condition. The theory is implemented for dilute penetrants smaller than the matrix particles, with or without penetrant-matrix attractive forces. Model calculations reveal a rich dependence of the penetrant diffusion constant and degree of dynamic coupling on size ratio, volume fraction, and attraction strength. In the absence of attractions, a near exponential decrease of penetrant diffusivity with size ratio over an intermediate range is predicted, in contrast to the much steeper, non-exponential variation if one assumes local matrix dynamical fluctuations are not correlated with penetrant motion. For sticky penetrants, the relative and absolute influence of caging versus physical bond formation is studied. The conditions for a dynamic crossover from the case where a time scale separation between penetrant and matrix activated hopping exists to a “slaved” or “constraint release” fully coupled regime are determined. The particle mixture model is mapped to treat experimental thermal systems and applied to make predictions for the diffusivity of water, toluene, methanol, and oxygen in polyvinylacetate liquids and glasses. The theory agrees well with experiment with values of the penetrant-matrix size ratio close to their chemically intuitive values.

1.
J. S.
Vrentas
and
C. M.
Vrentas
,
Diffusion and Mass Transfer
(
CRC Press
,
Boca Raton
,
2013
).
2.
S. C.
George
and
S.
Thomas
,
Prog. Polym. Sci.
26
,
985
(
2001
).
3.
J.
Guan
,
B.
Wang
, and
S.
Granick
,
ACS Nano
8
,
3331
(
2014
).
4.
M. T.
Cicerone
,
F. R.
Blackburn
, and
M. D.
Ediger
,
J. Chem. Phys.
102
,
471
(
1995
).
5.
K.
Paeng
,
H.
Park
,
D. T.
Hoang
, and
L. J.
Kaufman
,
Proc. Natl. Acad. Sci. U. S. A.
112
,
4952
(
2015
).
6.
D. F.
Sanders
,
Z. P.
Smith
,
R.
Guo
,
L. M.
Robeson
,
J. E.
Mcgrath
,
D. R.
Paul
, and
B. D.
Freeman
,
Polymer
54
,
4729
(
2013
).
7.
G. M.
Geise
,
H.-S.
Lee
,
D. J.
Miller
,
B. D.
Freeman
,
J. E.
Mcgrath
, and
D. R.
Paul
,
J. Polym. Sci., Part B: Polym. Phys.
48
,
1685
(
2010
).
8.
B. J.
Blaiszik
,
S. L. B.
Kramer
,
S. C.
Olugebefola
,
J. S.
Moore
,
N. R.
Sottos
, and
S. R.
White
,
Annu. Rev. Mater. Res.
40
,
179
(
2010
).
9.
C. E.
Diesendruck
,
N. R.
Sottos
,
J. S.
Moore
, and
S. R.
White
,
Angew. Chem., Int. Ed.
54
,
10428
(
2015
).
10.
J. F.
Patrick
,
M. J.
Robb
,
N. R.
Sottos
,
J. S.
Moore
, and
S. R.
White
,
Nature
540
,
363
(
2016
).
11.
G. M.
Geise
,
D. R.
Paul
, and
B. D.
Freeman
,
Prog. Polym. Sci.
39
,
1
(
2014
).
12.
J. N.
Ryan
and
M.
Elimelech
,
Colloids Surf., A
107
,
1
(
1996
).
13.
P. R.
Johnson
,
N.
Sun
, and
M.
Elimelech
,
Environ. Sci. Technol.
30
,
3284
(
1996
).
14.
E.
Zaccarelli
,
J. Phys.: Condens. Matter
19
,
323101
(
2007
).
15.
E.
Zaccarelli
and
W. C. K.
Poon
,
Proc. Natl. Acad. Sci. U. S. A.
106
,
15203
(
2009
).
16.
C. R.
Wilke
and
P.
Chang
,
AICHE J.
1
,
264
(
1955
).
17.
J. C.
Bosma
and
J. A.
Wesselingh
,
Chem. Eng. Res. Des.
77
,
325
(
1999
).
18.
S.
Bhattacharyya
and
B.
Bagchi
,
J. Chem. Phys.
106
,
1757
(
1997
).
19.
J.
Budzien
,
J. D.
McCoy
,
D.
Rottach
, and
J. G.
Curro
,
Polymer
45
,
3923
(
2004
).
20.
D. N.
Theodorou
, “
Principles of molecular simulation of gas transport in polymers
,” in
Materials Science of Membranes for Gas and Vapor Separation
, edited by
Yu.
Yampolskii
,
I.
Pinnau
, and
B. D.
Freeman
(
Wiley
,
Hoboken, NJ
,
2006
), pp.
49
94
.
21.
L.
Xi
,
M.
Shah
, and
B. L.
Trout
,
J. Phys. Chem. B
117
,
3634
(
2013
).
22.
J. S.
Vrentas
and
J. L.
Duda
,
J. Polym. Sci., Part B: Polym. Phys.
15
,
403
(
1977
).
23.
J. S.
Vrentas
and
J. L.
Duda
,
J. Polym. Sci., Part B: Polym. Phys.
15
,
417
(
1977
).
24.
N.
Ramesh
,
P. K.
Davis
,
J. M.
Zielinski
,
R. P.
Danner
, and
J. L.
Duda
,
J. Polym. Sci., Part B: Polym. Phys.
49
,
1629
(
2011
).
25.
R.
Zhang
and
K. S.
Schweizer
,
J. Chem. Phys.
143
,
144906
(
2015
).
26.
R.
Zhang
and
K. S.
Schweizer
,
Macromolecules
49
,
5727
(
2016
).
27.
S.
Mirigian
and
K. S.
Schweizer
,
J. Phys. Chem. Lett.
4
,
3648
(
2013
).
28.
S.
Mirigian
and
K. S.
Schweizer
,
J. Chem. Phys.
140
,
194506
(
2014
).
29.
D.
Emin
,
Polarons
(
Cambridge University Press
,
Cambridge
,
2013
).
30.
T. R.
Kirkpatrick
and
P. G.
Wolynes
,
Phys. Rev. A
35
,
3072
(
1987
).
31.
J.-P.
Hansen
and
I. R.
McDonald
,
Theory of Simple Liquids
(
Academic Press
,
London
,
1986
).
32.
K. S.
Schweizer
and
E. J.
Saltzman
,
J. Chem. Phys.
119
,
1181
(
2003
).
33.
K. S.
Schweizer
,
J. Chem. Phys.
123
,
244501
(
2005
);
[PubMed]
K. S.
Schweizer
and
G.
Yatsenko
,
ibid.
127
,
164505
(
2007
).
34.
R.
Zwanzig
,
Nonequilibrium Statistical Mechanics
(
Oxford University Press
,
Oxford
,
2001
).
35.
V. N.
Novikov
,
K. S.
Schweizer
, and
A. P.
Sokolov
,
J. Chem. Phys.
138
,
164508
(
2013
).
36.
J.
Colmenero
,
F.
Alvarez
,
Y.
Khairy
, and
A.
Arbe
,
J. Chem. Phys.
139
,
044906
(
2013
).
37.
D. C.
Viehman
and
K. S.
Schweizer
,
J. Chem. Phys.
128
,
084509
(
2008
).
38.
D. C.
Viehman
and
K. S.
Schweizer
,
Phys. Rev. E
78
,
051404
(
2008
).
39.
D. M.
Sussman
and
K. S.
Schweizer
,
J. Chem. Phys.
134
,
064516
(
2011
).
40.
D.
Chandler
and
J. P.
Garrahan
,
Annu. Rev. Phys. Chem.
61
,
191
(
2010
).
41.
S.
Mirigian
and
K. S.
Schweizer
,
J. Chem. Phys.
140
,
194507
(
2014
).
42.
S.
Mirigian
and
K. S.
Schweizer
,
Macromolecules
48
,
1901
(
2015
).
43.
G. L.
Hunter
and
E. R.
Weeks
,
Rep. Prog. Phys.
75
,
066501
(
2012
).
44.
D.
Meng
and
S. K.
Kumar
, “
Simulation of spherical penetrant diffusion in bead-spring polymer model glasses
” (unpublished).
45.
S.-J.
Xie
and
K. S.
Schweizer
,
Macromolecules
49
,
9655
(
2016
).
46.
R. P.
Danner
,
Fluid Phase Equilib.
362
,
19
(
2014
).
47.
B.
Flaconnèche
,
J.
Martin
, and
M. H.
Klopffer
,
Oil Gas Sci. Technol.
56
,
261
(
2001
).
You do not currently have access to this content.