Chapter 15
Theory and Computer Simulation of Structure, Transport, and Flow of Fluid in Micropores
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H. T. Davis, I. Bitsanis, T. K. Vanderlick, and M. V. Tirrell Chemical Engineering and Materials Science Department, University of Minnesota, Minneapolis, MN 55455 An overview is given of recent progress made in our labo ratory on this topic. The density profiles of fluid in micro pores are found by solving numerically an approximate Yvon-Born-Green equation. A related local average density model (LADM) allows prediction of transport and flow in inhomogeneous fluids from density profiles. A rigorous extension of the Enskog theory of transport is also outlined. Simple results of this general approach for the tracer diffusion and Couette flow between planar micropore walls are presented. Equilibrium and flow (molecular dynamics) simulations are compared with the theoretical predictions. Simulated density profiles of the micropore fluid exhibit sub stantial fluid layering. The number and sharpness of fluid layers depend sensitively on the pore width. The solvation force and the pore average density and diffusivity are oscil lating functions of the pore width. The theoretical predic tions for these quantities agree qualitatively with the simula tion results. The flow simulations indicate that the flow does not affect the fluid structure and diffusivity even at extremely high shear rates (10 s ). The fluid structure induces large deviations of the shear stress and the effective viscosity from the bulk fluid values. The flow velocity profiles are correlated with the density profiles and differ from those of a bulk fluid. The LADM and extended Enskog theory predictions for the velocity profiles and the pore average diffusivity agree very well with each other and with the simulation results. The LADM predictions for the shear stress and the effective viscosity agrees fairly well with the simulation results. 10
-1
E x a m p l e s o f fluids c o n f i n e d In pores a n d spaces o f m o l e c u l a r o r n a n o m e ter d i m e n s i o n s a b o u n d In t e c h n o l o g i c a l a n d natural products a n d processes. These Include w e t t i n g a n d l u b r i c a t i o n , zeolite s u p p o r t e d catalysis, s i l i c a gel based c h r o m a t r o g r a p h l c separations, d r y i n g o f paper
0097-6156/87/0353-0257$07.25/0 Φ1987 American Chemical Society
In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
SUPERCOMPUTER RESEARCH
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products a n d clay dispersions, aggregation o f c o l l o i d s , p e r m e a t i o n o f V l c o r a n d o t h e r s i n t e r e d glasses, the f o r m a t i o n o f soap films, f o a m s a n d e m u l s i o n s , a n d water o r o i l rich zones In l y o t r o p i c l i q u i d crystals a n d v e s i c u l a r b i l a y e r structures. I n s u c h c o n f i n e m e n t the fluids can be s t r o n g l y l n h o m o g e n e o u s and so the u s u a l theories o f fluid structure a n d d y n a m i c s m a y n o t be applicable. O w i n g to the m o l e c u l a r d i m e n s i o n s I n v o l v e d , e x p e r i m e n t a l characterization o f fluid In m i c r o p o r e s Is also difficult. T h u s , c o m p u t e r s i m u l a t i o n o n m o d e l systems b e c o m e s an I m p o r t a n t t o o l to test Ideas a n d s u p p l e m e n t e x p e r i m e n t s o n real systems In t r y i n g to u n d e r s t a n d the b e h a v i o r o f fluids c o n f i n e d o n the n a n o m e t e r scale. In this paper, we report recent progress made i n o u r l a b o r a t o r y In u s i n g m o l e c u l a r t h e o r y a n d c o m p u t e r s i m u l a t i o n to u n d e r s t a n d the struc t u r e , flow and transport o f fluids c o n f i n e d b y planar s o l i d walls separated by a few m o l e c u l a r diameters. M o l e c u l a r T h e o r y o f Structure and T r a n s p o r t E q u i l i b r i u m T h e o r y of F l u i d Structure. In all the theoretical work r e p o r t e d h e r e i n , we assume that the particles Interact w i t h pair additive forces whose pair potentials can be a p p r o x i m a t e d b y u(s) = u ( s ) + u ( s ) R
(1)
A
where
U
R( ) x
=
=
0
° , s
σ
(2)
a n d u ( s ) Is the c o n t i n u o u s , attractive part o f the pair p o t e n t i a l . T h e pore walls c o n f i n i n g the fluid w i l l be represented b y the c o n s e r v a t i v e potential u ( r ) . A t e q u i l i b r i u m the density n ( r ) o f the fluid obeys the Y v o n - B o r n G r e e n ( Y B G ) equation A
e
k Tvn + nyu B
e
-
3
n / n(r+s)g(r, r + s ) ^ - u ' ( s ) d s A
2
2
+ n k T / n ( r + ak)g(r,r 4- T(x)n (x)dx// n°(x)dx o o 0
(28)
D ( x ) , the local dlffuslvlty parallel to the pore w a l l s , Is g i v e n b y T
1
(^Τ/ππι) / DT(X)
=
2
F
(29)
2
4 σ / g°(a,5°(x + ^-€))n°(x + -ι
-
2
£ )d£
2
a result e n a b l i n g one to calculate the pore dlffuslvlty f r o m the e q u i l i b r i u m density distribution function. E q u a t i o n 28 Is s i m i l a r to the L A D M f o r m u l a for pore dlffuslvlty, e x c e p t that In L A D M D ^ x ) Is replaced b y (k T/7rm)
1 / 2
B
D°(n(x)) =
, , (30)
2
(8σ /3) °(σ,η"(χ))η-(χ) δ
T h e C h a p m a n - E n s k o g t h e o r y o f flow In a o n e - c o m p o n e n t fluid yields the f o l l o w i n g a p p r o x i m a t i o n to the m o m e n t u m balance e q u a t i o n ( J H ) .
e
nd v + n v v v + — v u m t
-
V ' P =-
M,:vv + M^v V v
(31)
w h e r e Ρ Is the local pressure tensor a n d M a n d are t h i r d a n d f o u r t h r a n k tensors a c c o u n t i n g for v i s c o u s dissipation. I n Isotropic fluid Ρ = PI, I the u n i t tensor, M = 0 a n d M 2 Is a f o u r t h rank Isotropic tensor. T h e s y m m e t r i e s o f P, M a n d M d e p e n d o n the s y m m e t r y o f the l n h o m o geneous fluid. T h e general C h a p m a n - E n s k o g f o r m u l a s f o r M j a n d M are v e r y c o m p l i c a t e d a n d w i l l n o t be r e c o r d e d h e r e . H o w e v e r , If the d e v i a t i o n o f the v e l o c i t y d i s t r i b u t i o n f u n c t i o n f r o m Its local M a x w e l l l a n f o r m (φ = ( π ι / 2 π ^ Τ ) e x p [ - m ( v - v ( r ) ) / k T ] ) Is neglected, the f o l l o w i n g r e l a t i v e l y s i m p l e f o r m u l a s are o b t a i n e d (10) x
x
l
2
2
3 / 2
2
B
In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
15.
DAVIS ET A L .
( Downloaded by UNIV OF CALIFORNIA SANTA CRUZ on October 14, 2014 | http://pubs.acs.org Publication Date: October 22, 1987 | doi: 10.1021/bk-1987-0353.ch015
M2 =
Structure,
265
Transport, and Flow of Fluid
7
mk T V * —^—\ * n ( r ) / n ( r +
(
5 1
)
Δ\ι
w h e r e ν ls the flow v e l o c i t y at the l o c a t i o n o f particle 1 a n d D
z
p
o
r
e
=
l l m ^ Σ ^ - < [Φ) oo IN 2,1
W
~
z
i(°)l
2
>
(52)
i = = 1
since there ls n o flow In the ζ d i r e c t i o n . T h e shear stress ls u n i f o r m t h r o u g h o u t the m a i n l i q u i d slab for C o u e t t e flow (£.). T h e r e f o r e , two Independent m e t h o d s f o r the calcula t i o n o f the shear stress are available; It can be calculated either f r o m the y c o m p o n e n t o f the force e x e r t e d b y the particles o f the l i q u i d slab u p o n each r e s e r v o i r o r f r o m the v o l u m e average o f the shear stress d e v e l o p e d Inside the l i q u i d slab f r o m the I r v l n g - K l r k w o o d f o r m u l a (lu). F o r rea sons e x p l a i n e d In R e f e r e n c e (£L) the s i m p l e r v e r s i o n o f this f o r m u l a can be u s e d In b o t h o u r systems a l t h o u g h this v e r s i o n does n o t apply In gen eral to s t r u c t u r e d systems. T h e I r v l n g - K l r k w o o d e x p r e s s i o n f o r the x y c o m p o n e n t o f the stress tensor u s e d In o u r s i m u l a t i o n ls
In Supercomputer Research in Chemistry and Chemical Engineering; Jensen, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.
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SUPERCOMPUTER RESEARCH _ r
Ν, V
xy pore=
