Mitra

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Mitras short time expansion

• Outline
• Mitra, whos 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

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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. Mitras 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.
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Short-time behaviour of the diffusion coefficient
as a geometrical probe of porous
media (Physical Review B, Volume 47, Number 14,
8565-8574
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• 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

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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.
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A solution to the diffusion equation without any
restricting geometries, is given by
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.
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Pertubative expansion for the propagator
Consider the diffusion equation on the form
(1)
One may remove the partial time derivative by
applying the Laplace transform
(2)
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Laplace transform on the border conditions gives
(3)
Now, let be any other
function that satisfies the diffusion equation in
the cavities of the porous medium
(4)
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Multiplying (2) by , (4) by
, integrating over r, gives us
the two equations
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Subtraction of those two equations then yields
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Greens theorem
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Greens theorem
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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
(5)
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SHORT-TIME EXPANSION
Reflecting boundary conditions
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The mean squared displacement may be written as
From the time derivative of the equation above,
one gets
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Working with the laplace transform, one then has
u v
Using a 2-step partial integration, and
remembering that vanishes at the surface
(remember reflecting boundaries!)
(6)
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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
(6) is then written
(7)
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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.
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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.
(8)
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
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The first part of the second term in (8)
By placing the coordinate system as shown in the
figure below with r in origo and assuming a
piecewise flat surface (i.e n0,0,1 and z0 0
) the diffusion propagator is written
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The mean squared displacement is now written
When performing an inverse Laplace transform of
the two first parts, and denoting the mean
squared displacement as 6D(t), one finds
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Conclusion
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Second order corrections
Surface relaxivity introduces sinks at the
boundaries
Curvature depenency on z introduces curved
surfaces
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The final expression for restricted diffusion at
short observation times, taking into account
curvature and surface relaxation, is to the first
order
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Diffusion amongst compact monosized spheres
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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 )