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 CrickClay 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
D(t )  D0 (1   (D0t )1/ 2 S / V  (D0t ))
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.
The equation of motion for the diffusing molecules within the cavities may be
described by the standard diffusion equation:
G
 D0  2 G
t
where the diffusion propagator G  G(r0 , r, t ) , is the conditional probability, defined
as
G  p(r0 ,0)  P(r0 , r, t )
where p(r0 ,0) is the probability of finding the polarized particle at position r0 at time
t = 0, and P(r0 , r, t ) 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
D0 n GrS   GrS  0
Here n is the outward normal vector on the pore surface S and ρ is the surface
relaxivity.
A solution to the diffusion equation without any restricting
geometries, is given by

1
2
G0 (r, r0 , t ) 
exp

(
r

r
)
/ 4 Dt
0
3/ 2
4Dt 

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
Consider the diffusion equation on the form
G(r, r , t )
 D 2G(r, r ' , t )
t
'
(1)
One may remove the partial time derivative by applying the
Laplace transform
~
'
'
2~
sG(r, r , s)   (r  r )  D G(r, r' , s)
(2)
Laplace transform on the border conditions gives:
~
~
'
D0nˆG(r, r , s)   G(r, r' , s) |r  0
(3)
~ ' ''
G0 (r , r , s) be any other function that satisfies the
Now, let
diffusion equation in the cavities of the porous medium:
~ ' ''
'
''
'2 ~
sG0 (r , r , s)   (r  r )  D G0 (r' , r'' , s)
(4)
~
~
Multiplying (2) by G0 (r' , r '' , s) , (4) by G(r, r ' , s) , integrating
over r’, gives us the two equations
~
~ ' ''
~ ' ''
'
'
'
'
G
(
r
,
r
,
s
)
G
(
r
,
r
,
s
)
d
r

G
(
r
,
r
,
s
)

(
r

r
)
d
r

0

 0
~ ' ''
'2 ~
'
'
  D0G0 (r , r , s) G(r, r , s)d r
~ ' '' ~
~
'
'
'
'
''
'
G
(
r
,
r
,
s
)
G
(
r
,
r
,
s
)
d
r

G
(
r
,
r
,
s
)

(
r

r
)
d
r
 0

~
'
'2 ~
  D0G(r, r , s) G0 (r ' , r '' , s)d r '
Subtraction of those two equations then yields
~
~ ' ''
~ ' ''
'
'
'
'
G
(
r
,
r
,
s
)
G
(
r
,
r
,
s
)
d
r

G
(
r
,
r
,
s
)

(
r

r
)
d
r

0

 0
~ ' '' ~
~
'
'
'
'
''
'
G
(
r
,
r
,
s
)
G
(
r
,
r
,
s
)
d
r

G
(
r
,
r
,
s
)

(
r

r
)
d
r
 0

~ ' ''
'2 ~
'
'
  D0G0 (r , r , s) G (r, r , s )d r
~
'
'2 ~
  D0G (r, r , s ) G0 (r ' , r '' , s)d r '
~
~
''
''
G (r, r , s)  G0 (r, r , s) 
~ ' ''
'2 ~
'
' 
 G
0 (r , r , s ) G (r, r , s )d r


D0  ~
~ ' ''
'
 G (r, r ' , s) '2G
(
r
,
r
,
s
)
d
r
0
 

Green’s theorem
u  GG0
u  GG0  G2G0
2

u
dV

u
dS


G

G

G

G0 dV
0



 GG
0
 G G0 dV   GG0 dS
2
Green’s theorem
v  G0G
v  G0G  G02G
2

v
dV

v
dS


G

G

G

G dV
0


 0
 G G  G  G dV   G GdS
2
0
0
0
 (u  v) dV 
 (GG  G G
2
0
0
 G0G  G0 G) dV
2
2
2
(
G

G

G

G )dV   (GG0  G0G )dS
0
0

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:
~
~
''
''
G(r, r , s)  G0 (r, r , s) 
 ~ ' ''
~
'
'
D0  G0 (r, r , s) [nˆ  ] G0 (r , r , s)dS  ...... (5)
D0
SHORT-TIME EXPANSION
Reflecting boundary conditions
The mean squared displacement may be written as
R 2 (t )  (1 / V )   drdr ' (r  r ' ) 2 G (r , r ' , t )
From the time derivative of the equation above, one gets:
 2
2 
R (t )  (1 / V )   drdr' (r  r ' )
G (r , r ' , t )
t
t
 ( D0 / V )   drdr' (r  r ' ) 2  2G (r , r ' , t )
 2

R (t )  (1 / V )   drdr ' (r  r ' ) 2 G (r , r ' , t )  ( D0 / V )   drdr ' (r  r ' ) 2  2G (r , r ' , t )
t
t
Working with the laplace transform, one then has
u
v’
~2
2
2~
sR ( s )  ( D0 / V )   drdr ' (r  r ' )  G (r , r ' , s )
~
Using a 2-step partial integration, and remembering that G(r, r ' , s)
vanishes at the surface (remember reflecting boundaries!)
 u'vdr   (u  v)dr   u  v' dr   (u  v)d   u  v' dr
2 D0
~
~2
sR ( s )  
dr' d nˆ  (r  r ' )G (r , r ' , s )


V
(6)
2 D0
~

dr' drG (r , r ' , s )


V
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:
R  6D0t
2
2 D0
V
2 D0
~
dr
'
dr
G
(
r
,
r
'
,
s
)


V

 st
 st
 dr' drdtG(r, r ' , t )e  6D0  e dt 
0
6 D0
s
(6) is then written
6 D0 2 D0
~2
sR ( s ) 

s
V
~
  dr ' d nˆ  (r  r ' )G (r , r ' , s)
(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.
 (s/D 0 )

1
e
~
G0 (r ' , r '' , s) 

4 D0  r1
1/2
r1

e
 (s/D 0 )1/2 r2
r2
r1  ( x' x' ' )2  ( y' y' ' )2  ( z' z' ' )2
r2  ( x' x' ' ) 2  ( y' y' ' ) 2  ( z' z' ' )2
1  x2 y2 
z   
2  R1 R2 



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.
6 D0 2 D0
~2
sR ( s ) 

dr' d (r  r ' )


s
V
~
~
'
''
'' '' ~
G0 (r, r , s )  D0  G0 (r, r , s ) nˆ  G0 (r '' , r ' , s )d ' '


(8)
Before evaluating the integrals above, it is convenient to scale
the r-variable with aim to simplify the expression. By
choosing   Ds , the exponent will contain only the
dimensionless variable r0  r
0

The first part of the second term in (8)
2 D0
V
~
'
dr
'
d

(
r

r
'
)
G
(
r
,
r
, s)
0

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
~
G0 (r, r0 , s ) 
where r0 
e  r0
2 D0 r0
r'
r'

D0 / s 
2 D0
~ (r, r ' , s )

dr
'
d

n

(
r

r
'
)
G
ˆ
0


V
2D0
e  r0

dx0dy0 dz0  d z0


V
2 D0 r0
3

V
3

V
3

V

r0  x02  y02  z02
z
d (e r0 )

 e  r0 0
dz0
r0

e  r0
 d dx0 dy0 z (0) dz 0 z0 r0
0


 d  dx dy  de
0


0
z0 ( 0 )

 r0
3

V

 d  dx dy e

0
 x02  y02
0


3
3
R
R 
R


d

2

R
e
d
R


d


2

R
e

d

2

e
dR
0
 0



V 
V  0
2 3
2 3 S

d  

V 
V
where S denotesthe surface area 
The mean squared displacement is now written
6D0 2 3 S 2D02
~2
sR (s) 


s
V
V
 G~ (r, r , s) nˆ  G~ (r , r , s)d ' '
''
''
''
0
''
'
0
When performing an inverse Laplace transform of the two first parts,
and denoting the mean squared displacement as 6D(t), one finds
3/ 2
3/ 2


6
D
2
D
S
2
D
S
1
6 D(t )t  L1  2 0  5 0/ 2   6 D0t  0 L1 ( 5 / 2 )
s V 
V
s
 s
3/ 2
0
3/ 2
3/ 2
0
3/ 2
2D S t
2D S t
 6 D0t 
3
V
(5 / 2)
V

4
4 S


 6 D0t 1 
D0t 
 9  V

 6 D0t 
5
3
3 3
( )  (  1)  ( )
2
2
2 2
31 1
3

( ) 

22 2
4
Conclusion
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
D(t )
4
 1
D0
9 
S
D0 t   (  , R, t )
V
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,
 (  , R, t ) 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.
Second order corrections
Surface relaxivity introduces sinks at the boundaries
~
~
''
G(r, r , s)  G0 (r, r '' , s) 
 ~ ' ''
~
'
'
D0  G0 (r, r , s) [nˆ  ] G0 (r , r , s)dS  ......
D0
Curvature depenency on z introduces curved surfaces
1  x 2 y2 
z   
2  R1 R 2 
The final expression for restricted diffusion at short
observation times, taking into account curvature and
surface relaxation, is to the first order
D(t )
4
 1
D0
9 
S  S
S 1
D0t 
t
D0t
V 6V
6V R
Restricted diffusion
No curvature, reflecting boundaries
Negative curvature, reflecting boundaries
Negative curvature, non zero curvature
No curvature, non zero surface relaxivity
Positive curvature, non zero surface relaxivity
1,05
1
0,95
D/Do
0,9
0,85
0,8
0,75
0,7
0,65
0
5
10
15
square root of observation time
20
25
Diffusion amongst compact monosized spheres
Restricted diffusion at short observation times
2,4
diffusion coefficient
2,3
2,2
y = -0,0801x + 2,3402
2
R = 0,998
2,1
2
1,9
1,8
0
1
2
3
4
square root of observation time
5
6
7
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
6(
d 
1

 1)
S
V
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 )