# Diffusion through a Membrane

Diffusion through a Membrane

This Demonstration shows the effect of time delay on the dynamics of a simple diffusion model.

Consider a system of two compartments with lengths and containing the same species separated by a permeable membrane of unit area. Each compartment is assumed to be well stirred and homogeneous, so each can be represented by a single concentration variable, and .

L

1

L

2

c

c

1

c

2

Fick's first law of diffusion gives

d

c

i

dt

q

i

c

j

c

i

q

i

D

L

L

i

where represents time, is the diffusion coefficient of , and is the thickness of the membrane. Implicit in this equation is the assumption that the time required for diffusion through the membrane is zero; in reality, the time is not zero, in fact there is a distribution of times for molecules to cross the membrane. If we consider a model [1] in which all molecules take the same time to transverse the membrane, then Fick's law becomes

t

D

c

L

τ

d

c

i

dt

q

i

c

j

c

i

with initial history functions

c

i

c

i

q

i

This system of delayed differential equations is solved with =1. The solution shows remarkable features of the model. If the compartments have equal volumes and different initial concentrations, the system approaches equilibrium with damped oscillations. Further, if the volumes of the compartments are different, we can start with equal concentrations and the system will approach equilibrium in a damped oscillatory fashion. Even more surprising, we can start with equal concentrations and equal volumes and equilibrium will be approached with oscillations.

q

1