.
Solutions of Fick's diffusion equation for planar geometry.
For a constant D, independent of composition, position, direction and time, the fundamental differential equation for diffusion in a planar, one dimensional geometry, is:
.
This can be derived from Fick's Law (
), and
it is sometimes referred to as Fick's Second Law. Its solutions are profiles of concentration of diffusant as a function of time and position in the diffusion medium.
The planar geometry case covers an initially empty barrier sheet of thickness L exposed on one side to a diffusant, as well as an initially empty planar sheet of thickness 2L immersed in diffusant. In the diagram below, concentration profiles are plotted for several dimensionless times, Dt/L2. At dimensionless time 0.1, diffusant is starting to arrive at the rear
of the barrier or the center of the sheet in abundance. We will refer to this as the Breakthrough
Time, and use it to characterize barriers. At dimensionless time 1.0, the diffusant is approaching
saturation throughout the barrier. For more on the subject see Crank.
Hubble et. al. report a diffusivity D for water in Parylene C of 2.60 x 10-9 cm2/s at 30°C. Similar values were reported for other polymers, so it is anticipated that other Parylenes should not much differ.
Setting the Breakthrough Criterion at 0.1, D, t and L are linked together: once two are selected, the third is fixed. The plot below summarizes how long a barrier will last over a range of times and thicknesses of general interest. Note the particularly strong influence of barrier thickness, which enters the relation as the square. If we use Parylene C as a printed wiring board coating at 0.5 mil, the time from exposure of the coated board to water to water being available at the underlying circuits is about 1 minute. Any long term benefits to be achieved with Parylene C a conformal circuit coating must be explained on the basis of something other than simply keeping water away.
For very thin films, the situation is serious. A 30 Angstrom Parylene film, depending somewhat on which permeant is being considered, should serve as a barrier for under a millisecond. A diffusivity of less than 10-25 cm2/s would be required to achieve a useful barrier life.