1. 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

2. 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.

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

4. Physical argument: -In the bulk phase the mean squared displacement is given by the Einstein relation (6 D 0 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 ((D 0 t) 1/2 S)/V -Thus the diffusion coeffcient at short observation times is reduced from the bulk value as

5. 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 r 0 and r correspond to the initial position of a water molecule and the position after a time t.

7. 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.

8. Pertubative expansion for the propagator One may remove the partial time derivative by applying the Laplace transform Consider the diffusion equation on the form (1) (2)

9. Laplace transform on the border conditions gives: Now, let be any other function that satisfies the diffusion equation in the cavities of the porous medium: (3) (4)

10. Multiplying (2) by, (4) by, integrating over r’, gives us the two equations

11. Subtraction of those two equations then yields

13. Green’s theorem

16. 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 G 0 on the right hand side: (5)

17. SHORT-TIME EXPANSION Reflecting boundary conditions

18. From the time derivative of the equation above, one gets: The mean squared displacement may be written as

19. Working with the laplace transform, one then has Using a 2-step partial integration, and remembering that vanishes at the surface (remember reflecting boundaries!) u v’ (6)

20. 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)

21. 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.

22. 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. 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 (8)

23. 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 n=[0,0,1] and z 0 = 0 ) the diffusion propagator is written

25. When performing an inverse Laplace transform of the two first parts, and denoting the mean squared displacement as 6D(t), one finds The mean squared displacement is now written

26. Conclusion

27. Second order corrections Surface relaxivity introduces sinks at the boundaries Curvature depenency on z introduces curved surfaces

28. The final expression for restricted diffusion at short observation times, taking into account curvature and surface relaxation, is to the first order

31. Diffusion amongst compact monosized spheres

32. 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 )