I am trying to simulate 1D simple diffusion problems with diffusion coefficient/mobility values a function of concentration. For example, for homogenization of A-B system, I am going to use mobility values of B, corresponding to the concentration of B at that point in A-B system. I presume this must be the case for CALPHAD diffusional simulations. However, I observed that mass conservation is not maintained after simulated diffusion, as a result of varying mobility parameter across the simulation distance. I mean, the total B content after simulation is different from the initial value. Should I consider any boundary condition to fix this?
Thanks in advance.
If the diffusion coefficient is a function of concentration we write D(c) for the diffusion coefficient. J is the flux, t is time, x is distance and c is concentration.
hence
dc/dt=-dJ/dx
where J=-D(c)*(dc/dx)
hence
dc/dt=d/dx(D(c)dc/dx)
dc/dt=D(c)d2c/dx2+(dc/dx)*(dD(c)/dx)
dc/dt=D(c)d2c/dx2+(dc/dx)*(dD(c)/dc)*(dc/dx)
dc/dt=D(c)d2c/dx2+(dD(c)/dc)*(dc/dx)^2
hence an extra term appears.
dc/dt is the derivative of c with respect to time
d2c/dx2 is the second derivative of c with respect to x
(dc/dx) is the derivative of of c with respect to x
(dc/dx)^2 is the above derivative squared
dD(c)/dc is the derivative of the diffusion coefficient with respect to c
This is the equation you need when the diffusion constant D depends on c
To what extent is mass conservation not observed? Is it small and potentially attributed to rounding errors?
Dear Eric Teitelbaum
The difference is significant, depending on the mobility parameter and evolved time steps. The regions with higher mobility parameter evolves faster. I have attached a simplified homogenization MATLAB code, in which one of diffusivity value is given as 1, while other is 2*concentration. You can check how the second case fails to conserve mass. I am wondering whether I should add any boundary conditions to this.
Thank you.
really interesting questions. Actually, I have performed a diffusion simulation employing my own code and I am getting the same issue with the mass conservation.
Dear Viswanathan Ramamoorthy
You should not be experiencing a conservation problem. This process is described by the general diffusion equation (https://en.wikipedia.org/wiki/Diffusion_equation), which is a conservation equation. Thermal conduction with temperature-dependent thermal conductivity is an equivalent example of what you are doing.
You do need to impose the appropriate boundary conditions for your problem. If it is simply homogenisation based on an initial state, you can use zero rate boundary conditions (dC/dx = 0).
All the best
Johan
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