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Steady State Solidification

We can also consider another situation, where there is still no diffusion in the solid phase, but in the liquid phase there is now limited diffusion, rather than complete mixing.

From Fick’s second law, we have:

\[\frac{{{\rm{d}}C}}{{{\rm{d}}t}} = {D_{\rm{L}}}\frac{{{{\rm{d}}^2}C}}{{{\rm{d}}{x^2}}}\]

also:

\[\frac{{{\rm{d}}C}}{{{\rm{d}}t}} = \frac{{{\rm{d}}C}}{{{\rm{d}}x}}\frac{{{\rm{d}}x}}{{{\rm{d}}t}} = v\left( {\frac{{{\rm{d}}C}}{{{\rm{d}}x}}} \right)\]

These combine to give:

\[{D_{\rm{L}}}\left( {\frac{{{{\rm{d}}^2}C}}{{{\rm{d}}{x^2}}}} \right) + v\left( {\frac{{{\rm{d}}C}}{{{\rm{d}}x}}} \right) = 0\]

which can be evaluated with the appropriate boundary conditions to give us:

\[C = {C_0} + \frac{{{C_0}\left( {1 - k} \right)}}{k}\exp \left( {\frac{x}{{{D_{\rm{L}}}/v}}} \right)\]

This equation describes the solute profile in the liquid ahead of the interface, in the steady state. For a full derivation, click here. In the following pages, x should, strictly speaking, be written everywhere as x’ (distance ahead of the moving front)

This model describes the solute profile in the liquid ahead of the interface once the situation has reached a steady state. However, when solidifying a bar, for example; it will take a period of time for the solute ‘bow wave’ to build up ahead of the interface. During this initial transient solid will be forming with a concentration less than C0, (initially it will be k C0) until a steady state is achieved. The solid then advances along the length of the bar with the solute bow wave ahead of the interface. When the bow wave, with a characteristic length, DL / v, begins to run into the end of the bar, the solute ‘piles up’ giving a rapid increase in the solute concentration of the liquid. This final transient sees an increase in the concentration of the solid, and may result in the formation of some non-equilibrium eutectic at the end of the bar.

The simulation below uses a numerical model to predict the solute profiles for the solidification of a 10 mm bar, with a bulk concentration of 10 %. The partition coefficient, k, the size of the solute bow wave; which is dictated by DL/v, and the eutectic composition for the system can all be changed using the scroll bars.

Adjust the variables to see how they each affect the solute profile, and consider why the simulation responds as it does.

Click ‘Run’ to see how the solute profile develops during this schematic movie of steady state solidification.

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


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