Heat transfer and diffusion of particles
Fondamental : Fick's law and equation of particle scattering
Fick's three-dimensional law :
with the coefficient of scattering and the particle density.
This law shows that scattering can only take place in a medium where the particle density is not uniform.
Moreover, the gradient being oriented towards the increasing , the - symbol in Fick's law indicates that particles spontaneously scatter from the more concentrated places to the less concentrated ones.
Let's determine the one-dimensional equation of scattering without internal source of creation of particles.
A reasoning similar to the one of thermal transfers (we write the conservation of matter considering a cube, which has a volume ) :
It results in the equation of conservation of matter :
Using Fick's law, we obtain the one-dimensional equation of scattering :
In three dimensions, we have (see the lesson about "Vector calculus") :
And the equation of scattering (without source) :
Attention : Fick's law and equation of particle scattering
Fick's law :
Equation of scattering (without source) :