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20.2: Fick's Second Law of Diffusion

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    7914
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    Fick’s second law is concerned with concentration gradient changes with time.

    Fick2aa.gif

    By considering Fick’s 1st law and the flux through two arbitrary points in the material it is possible to derive Fick’s 2nd law.

    \[\dfrac{\partial C}{\partial t} = D\left( \dfrac{\partial ^2C}{\partial x^2} \right)\]

    This equation can be solved for certain boundary conditions:

    Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.

    1. “Thin source”

    Consider a semi-infinite bar with a small, fixed amount of solute material diffusing in from one end.

    altfick2b.jpg

    The amount of solute in the system must remain constant, therefore

    \[\int \limits_0^\infty C\left( {x,t} \right ) \,dx = B\]

    where \(B\) is a constant.

    The initial concentration of solute in the bar is zero, therefore

    \[C\left( x,t = 0 \right ) = C_0\]

    These boundary conditions give the following solution:

    \[C\left( x,t \right ) = \frac{B}{\sqrt {\pi Dt}}\exp \left( \frac{ - x^2}{4Dt} \right ) \]

    2. “Infinite source”

    A semi-infinite bar with a constant source (i.e. constant concentration) of solute material diffusing in from one end.

    bar.gif

    In this case, the solution is obtained by stacking a series of “thin sources” at one end of the bar, and summing the effects of all of the sources over the whole bar.

    The initial concentration of solute in the bar is \(C_0\), therefore

    \[C\left( {x,t = 0} \right ) = {C_0}\]

    The concentration of solute at the end of the bar is a constant, \(C_s\), therefore

    \[C\left( {x = 0,t} \right ) = {C_s}\]

    These boundary conditions give the following solution:

    \[C\left( x,t \right ) = C_s - (C_s - C_0) erf \left( \frac{x}{2\sqrt {Dt}} \right ) \]

    erf{x} is known as the error function and results from the summation of the thin sources at the end of the bar. It is defined as

    \[erf\left( x \right ) = \frac{2}{\sqrt \pi } \int \limits_0^x \exp ( - u^2 ) du\]

    The integral can only be solved numerically with a computer, so erf tables are used to solve the diffusion equation where necessary.

    This animation shows the applications of Fick’s 2nd law and its solutions.

    3. Non-Analytical Solutions

    For more complicated situations we cannot obtain an analytical solution for Fick’s 2nd law. In these cases numerical analysis is used. Solutions obtained in this way are approximations, however, they can be made as precise as needed.

    The following demonstration shows how numerical analysis can be used to approximate solutions for various conditions.

    Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.


    This page titled 20.2: Fick's Second Law of Diffusion is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Dissemination of IT for the Promotion of Materials Science (DoITPoMS) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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