Diffusion From a Micropipette

In Chapter 4 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I solve the diffusion equation. We consider the classic one-dimensional example of particles released at a point (x = 0) and at one instant (t = 0). The particles diffuse, and the concentration C(x,t) has a Gaussian distribution

where D is the diffusion constant and N is the number of particles per unit area (assuming diffusion along a tube of fixed cross-sectional area).

Often this concentration distribution is drawn as a function of x at a fixed time t (a snapshot). Russ and I include such an illustration in IPMB’s Figure 4.13. Below is a modified version of that figure, showing the Gaussian distribution at three times.

The concentration C(x,t) as a function of x at three times t, 2t, and 3t.

Alternatively, we could plot the concentration as a function of t, for a particular location x. Such a plot illustrates how a wave of particles diffuses outward, so at any point x the concentration starts at zero, rises quickly to a peak, and then slowly decays.

The concentration C(x,t) as a function of t at three locations x, 2x, and 3x.

We can calculate the time when the concentration reaches its peak, t_peak, by setting the time derivative of C(x,t) equal to zero and solving for t. The result is t_peak = x²/2D. To find the maximum value of the concentration, C_max, at any location we plug t_peak into the expression for C(x,t) and find C_max = 0.242N/x.

Random Walks in Biology,
by Howard Berg.

I was motivated to draw the concentration as a function of time by the discussion of diffusion in Howard Berg’s book Random Walks in Biology. He also analyzes the three-dimensional version of this problem. 

Diffusion from a micropipette: A micropipette filled with an aqueous solution of a green fluorescent dye is inserted into a large body of water. At time t = 0, particles of the dye are injected into the water… The total number of particles injected is N… [The diffusion equation] has the solution

This is a three-dimensional Gaussian distribution… Looking through a microscope, one sees the sudden appearance of a green spot that spreads rapidly outward and fades away. The concentration remains highest at the tip of the pipette, but it decreases there as the three-halves power of time.

I’ll let the reader analyze this case by writing a new homework problem. Enjoy!

Section 4.8

Problem 16 ½. When N particles released at time t = 0 and location r = 0 diffuse, the concentration C(r,t) is governed by

(a) Show that this expression for C(r,t) obeys the diffusion equation written in spherical coordinates

(b) Integrate C(r,t) over all space and show that the number of particles is always N.

(c) Calculate the variance (the mean value of r²) and show that σ² = 6Dt, as found in Problem 16. You may need an integral from Appendix K.

(d) Calculate the time t_peak when the concentration at a distance r is maximum.

(e) Calculate the maximum concentration, C_max, at distance r.

(f) Sketch a plot of C(r,t) as a function of r for three times, and then plot C(r,t) as a function of t for three locations.