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Diffusion, concentration profile

  1. Jul 24, 2015 #1
    I'm right now working with modelling a system, where atoms diffuse along a coordinate z until they reach a certain point z=z0 at which they are taken out of the system. The diffusion equation is:

    ∂n/∂t = D∂2n/∂z2

    The question is now what boundary conditions to use for the concentration profile n(z) of the system. I want to take n(z0)=0 since for me this would reflect the fact that atoms are taken instantaneously out of the system, when they diffuse to this point. But my intuition also tells me that there can be no flux of atoms out of the system at z=z0 if the concentration at this point is zero.
    Does anyone have anything that can enlighten me on this problem?
     
  2. jcsd
  3. Jul 24, 2015 #2

    Orodruin

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    This is a common misconception among students who start encountering the diffusion (or heat) equation. The current is not proportional to the density, it is proportional to the density gradient. Therefore, there can be a current even if the density is zero.

    As a more realistic model than the substance being taken out directly, you can consider the case when the current is proportional to the density and very large as soon as the density is non-zero, i.e.,
    $$
    j/D = -\partial_z n = \alpha n,
    $$
    where ##\alpha## is a large number. Dividing by ##\alpha## gives ##n = - \alpha^{-1} \partial_z n##, for which you recover the limit ##n = 0## when ##\alpha \to \infty##, i.e., as you get really really fast at removing the substance.
     
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