ECE532 Biomedical Optics|
Steven L. Jacques, Scott A. Prahl
Oregon Graduate Institute
Fick's 1st law of diffusion
Diffusion occurs in response to a concentration gradient expressed as the change in concentration due to a change in position, . The local rule for movement or flux J is given by Fick's 1st law of diffusion:
in which the flux J [cm-2 s-1] is proportional to the diffusivity [cm2/s] and the negative gradient of concentration, [cm-3 cm-1] or [cm-4]. The negative sign indicates that J is positive when movement is down the gradient, i.e., the negative sign cancels the negative gradient along the direction of positive flux.
Consider a local concentration of 106 units per cm3 which drops by 10% over a distance of 1 cm. Then the gradient is -105 [units cm-4]. Assume the diffusivity is 103 [cm2/s]. Then the flux J equals:
For light, the diffusivity is proportional to the diffusion length D [cm] and the speed of light c:
where D = 1/(3 µs(1-g)). The units of velocity [cm/s] times the units of length [cm] yield the units of diffusivity [cm2/s]. The following example describes the local diffusion of red light in milk.
Consider a local fluence rate F of 1 W/cm2 in milk (n = 1.33, µs(1-g) = 10 cm-1).
For optical diffusion, Fick's 1st law is expressed as the energy flux J [W cm-2] proportional to the diffusion constant D [cm] and the negative fluence gradient dF/dx:
which was obtained by substituting cD for and substituting F/c for C. The factors c and 1/c cancel to yield the above equation.