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Mitra’s short time expansion
Outline
-Mitra, who’s he?
-The model, a dimensional argument
-Evaluating the leading order correction term to the restricted diffusion at short observation times
-Second order corrections, and their effect on the restricted diffusion
-Example of application: diffusion amongst compact monosized spheres
Short Bio
Partha Mitra received his PhD in theoretical physics from Harvard in 1993. He worked in quantitative neuroscience and theoretical engineering at Bell Laboratories from 1993-2003 and as an Assistant Professor in Theoretical Physics at Caltech in 1996 before moving to Cold Spring Harbor Laboratory in 2003, where he is currently Crick-Clay Professor of Biomathematics. Dr. Mitra’s research interests span multiple models and scales, combining experimental, theoretical and informatic approaches toward achieving an integrative understanding of complex biological systems, and of neural systems in particular.
Short-time behaviour of the diffusion coefficient as a geometrical probe of porous media
(Physical Review B, Volume 47, Number 14, 8565-8574
Physical argument:
-In the bulk phase the mean squared displacement is given by the Einstein relation (6 D0 t)1/2
-When there are restrictions, the early time departure from unrestricted diffusion must be proportional to the number ( or volume fraction) of molecules sensing the restriction. This volume fraction is given by ((D0 t)1/2 S)/V
-Thus the diffusion coeffcient at short observation times is reduced from the bulk value as
))(/)(1()( 02/1
00 tDVStDDtD
A two-dimensional slice of a porous system. The black area corresponds to the cavities that can be filled with brine while the gray areas correspond to the sold matrix. The interface between the black and grey area is the surface S while r0 and r correspond to the initial position of a water molecule and the position after a time t.
where the diffusion propagator ),,( 0 tGG rr , is the conditional probability, defined
as
),,()0,( 00 tPpG rrr
where )0,( 0rp is the probability of finding the polarized particle at position 0r at time
t = 0, and ),,( 0 tP rr is the probability of finding this particle at position r at at a later
time t. When including the effect from relaxation at the pore walls, the boundary condition can be stated as
0 0r S r SD G G n
Here n is the outward normal vector on the pore surface S and ρ is the surface relaxivity.
The equation of motion for the diffusing molecules within the cavities may be described by the standard diffusion equation:
GDt
G 20
A solution to the diffusion equation without any restricting geometries, is given by
Dt
DttG 4/)(exp
4
1),,( 2
02/300 rrrr
Later on we will make this solution as the initial solution and thus a starting point in a petrubation expansion for a solution in the presence of restricted diffusion.
Pertubative expansion for the propagator
),,(),,( '2
'
tGDt
tGrr
rr
One may remove the partial time derivative by applying the Laplace transform
),,(~
)(),,(~ '2'' sGDrrsGs rrrr
Consider the diffusion equation on the form
(1)
(2)
Laplace transform on the border conditions gives:
0|),,(~
),,(~
ˆ ''0 rsGsGnD rrrr
Now, let be any other function that satisfies the diffusion equation in the cavities of the porous medium:
),,(~ '''
0 sG rr
),,(~
)(),,(~ '''
02'''''''
0 sGDrrsGs rrrr
(3)
(4)
Multiplying (2) by , (4) by , integrating over r’, gives us the two equations
),,(~ '''
0 sG rr ),,(~ ' sG rr
''2''''00
'''''0
''''0
'
r),,(~
),,(~
r)(),,(~
r),,(~
),,(~
dsGsGD
drrsGdsGsG
rrrr
rrrrrr
''''0
2''0
''''''''''0
r),,(~
),,(~
r)(),,(~
r),,(~
),,(~
dsGsGD
drrsGdsGsG
rrrr
rrrrrr
Subtraction of those two equations then yields
''''0
2''0
''2''''00
''''''''''0
'''''0
''''0
'
r),,(~
),,(~
r),,(~
),,(~
r)(),,(~
r),,(~
),,(~
r)(),,(~
r),,(~
),,(~
dsGsGD
dsGsGD
drrsGdsGsG
drrsGdsGsG
rrrr
rrrr
rrrrrr
rrrrrr
''''0
2''
''2''''0
0
''0
''
r),,(~
),,(~
r),,(~
),,(~
),,(~
),,(~
dsGsG
dsGsGD
sGsG
rrrr
rrrr
rrrr
Green’s theorem
0GGu
02
0 GGGGu
dVGGGGdSudVu 02
0
dSGGdVGGGG 002
0
Green’s theorem
GGv 0
GGGGv 200
dVGGGGdSvdVv 200
dSGGdVGGGG 02
00
dSGGGGdVGGGG )()( 002
002
dVGGGGGGGG
dVvu
)(
)(
2000
20
Insertion and use of the border conditions in (2) then gives us the first two terms in a series expansion when G has been substituted with G0 on the right hand side:
......),,(~
]ˆ[),,(~
),,(~
),,(~
'''0
0
''00
''0
''
dSsGD
nsGD
sGsG
rrrr
rrrr
(5)
SHORT-TIME EXPANSION Reflecting boundary conditions
),',()'(')/1()( 22 trrGrrdrdrVtR
From the time derivative of the equation above, one gets:
),',()'(')/(
),',()'(')/1()(
220
22
trrGrrdrdrVD
trrGt
rrdrdrVtRt
The mean squared displacement may be written as
),',()'(')/(),',()'(')/1()( 220
22 trrGrrdrdrVDtrrGt
rrdrdrVtRt
Working with the laplace transform, one then has
),',(~
)'(')/()(~ 22
02 srrGrrdrdrVDsRs
Using a 2-step partial integration, and remembering that vanishes at the surface (remember reflecting boundaries!)
),',(~
srrG
),',(~
'2
),',(~
)'(ˆ'2
)(~
0
02
srrGdrdrV
D
srrGrrnddrV
DsRs
u v’
drvudvudrvudrvuvdru ')(')('
(6)
The last term in (6) gives us the Einstein relation, as it is an integral over a normalized density distribution function. Conducting the inverse laplace transform one then gets the Einstein relation in three dimensions:
tDR 02 6
(6) is then written
),',(~
)'(ˆ'26
)(~ 002 srrGrrnddr
V
D
s
DsRs
0
00
00 66),',('
2),',(
~'
2
s
DdteDetrrdrdtGdr
V
DsrrGdrdr
V
D stst
(7)
The second term of (7) consists of a surface integral over the poinr r and a volume integral over r’. Now we do the approximation that disregards curvature of the surface: At the shortest observation times, the surface may be approximated by a plane transverse to z, i.e the tangent plane at r.
2
r)(s/D
1
r)(s/D
0
''0
21/2
011/2
0
41
),,'(~r
er
eD
sG
rr
2221 )'''()'''()'''( zzyyxxr
2222 )'''()'''()'''( zzyyxxr
2
2
1
2
2
1
R
y
R
xz
Then one must make use of the pertubation expansion for the propagator (5) and put this into (6). The inital propagator is the Gaussian diffusion propagator with reflecting boundary conditions at a flat surface.
''),,(~
ˆ),,(~
),,(~
)'('26
)(~
'''0
''''''00
'0
002
dsGnsGDsG
rrddrV
D
s
DsRs
rrrrrr
Before evaluating the integrals above, it is convenient to scale the r-variable with aim to simplify the expression. By choosing , the exponent will contain only the dimensionless variable
s
D0
r
r 0
(8)
The first part of the second term in (8)
),,(~
)'('2 '
00 sGrrddr
V
Drr
By placing the coordinate system as shown in the figure below with r in origo and assuming a piecewise flat surface (i.e n=[0,0,1] and z0 = 0 ) the diffusion propagator is written
'
0
'
0
0000
/
2),,(
~ 0
r
sD
rrwhere
rD
esG
r
rr
),,(~)'(ˆ'2 '
00 sGrrnddr
VD
rr
0
0
0
20
20
200
0
0 )(
r
ze
dz
ed
zyxr
rr
000000
0
2
2 0
rDe
zddzdydxV
D r
20
20
0
000
3
)0(
00
3yx
z
r edydxdV
dedydxdV
areasurfacethedenotesSwhereVS
dV
33 22
0
R3
0R
3
0
R3
R2R2RR2 dedV
edV
dedV
)0( 0
0000
3
0
0
z
r
re
zdzdydxdV
When performing an inverse Laplace transform of the two first parts, and denoting the mean squared displacement as 6D(t), one finds
''),,(~
ˆ),,(~226
)(~ '''
0''''''
0
20
302
dsGnsGV
D
V
S
s
DsRs rrrr
The mean squared displacement is now written
4
3)
2
1(
2
1
2
3
)2
3(
2
3)1
2
3()
2
5(
43
26
)2/5(2
62/32/3
00
2/32/30
0
tVSD
tDt
VSD
tD
)1
(2
626
)(6 2/51
2/30
02/5
2/30
201
sVSD
tDVsSD
sD
ttD
LL
tD
VS
tD 00 9
416
By assuming piecewise smooth and flat surfaces and that only a small fraction of the particles are sensing the restricting geometries, the restricted diffusion coefficient can be written as
),,(
9
41
)(0
0
tRV
StD
D
tD
where D(t) is the time dependent diffusion coefficient, D0 is the unrestricted diffusion coefficient, in bulk fluid, and t is the observation time. The higher order terms in t,
),,( tR holds the deviation due to finite surface relaxivity and curvature (R) of the surfaces. At the shortest observation times these terms may be neglected such that the deviation from bulk diffusion depends on the surface to volume ratio alone.
Conclusion
Second order corrections
2
2
1
2
R
y
R
x
2
1z
......),,(~
]ˆ[),,(~
),,(~
),,(~
'''0
0
''00
''0
''
dSsGD
nsGD
sGsG
rrrr
rrrr
Surface relaxivity introduces sinks at the boundaries
Curvature depenency on z introduces curved surfaces
tDRV
St
V
S
V
StD
D
tD00
0
1
669
41
)(
The final expression for restricted diffusion at short observation times, taking into account curvature and surface relaxation, is to the first order
Restricted diffusion
0,65
0,7
0,75
0,8
0,85
0,9
0,95
1
1,05
0 5 10 15 20 25
square root of observation time
D/D
o
No curvature, reflecting boundaries No curvature, non zero surface relaxivity
Negative curvature, reflecting boundaries Positive curvature, non zero surface relaxivity
Negative curvature, non zero curvature
Diffusion amongst compact monosized spheres
Restricted diffusion at short observation times
y = -0,0801x + 2,3402
R2 = 0,998
1,8
1,9
2
2,1
2,2
2,3
2,4
0 1 2 3 4 5 6 7
square root of observation time
dif
fusi
on
co
effi
cien
t
VS
d)1
1(6
As we are measuring the S/V ratio of the water phase, we need to quantify the volume of the water before beeing able to solve out the diameter of the spheres. This is done by measuring the NMR signal of the water and calibrating this signal against a signal of known volume ( as a 100% water sample). Then we find the porosity, , of the sample, which is used to find the diameter of the spheres
Then we find a mean diameter of 100,6 µm while the certified sphere diameter was 98,7 µm
( uncertainty for both numbers are approximately ± 4 µm )
