Diffusion and Mass Transfer


 
 

Within a microstructure, there is generally mass transfer whenever there are gradients in chemical potential. There are three processes that are important to model to have mass transfer within the microstructure:

1) Lattice Diffusion - mass transfer within a grain

2) Grain Boundary Diffusion - mass transfer along a grain boundary

3) Precipitation/Dissolution - mass transfer between a grain and grain boundary

Currently we have modelled latice diffusion and grain-boundary diffusion, but precipitation/dissolution is only in the conceptual stage. To calculate diffusion, we use a discrete approach that divides the microstructure into discrete triangles (latice diffusion) or line segments (grain-boundary diffusion) and calculates the gradients in a step-wise fashion across these elements. While this approach may not be as accurate as a finite element approach, it is easy to implement, can be infinitely extensible to smaller and smaller elements, and provides suitable accuracy if the elements are small enough.

Diffusion Equation:

     (1D)       or             (3D)
Mass Flux
        (1D)        or               (3D)
where C is chemical concentration or potential, D is diffusion constant (d is also, but may have different units), and q is mass flux.

1) Lattice Diffusion

We first discretize the elle data structure into a series of triangles. We then assign the attributes of each flynn to the triangles contained in it and then run the diffusion code.
 

where C0 is the concentration in Triangle-0, Q1 is the flux from Triangle-1 into Triangle-0, A0 is the area of Triangle-0, and L1 is the length of the boundary between Triangle-1 and Triangle-0.
Flux into Triangle 0 from Triangle 1: q1 = k (C1-C0)L1/A0

Flux into Triangle 1 from Triangle 0: q1' = k (C0-C1)L1/A1

For Mass Balance: q1 should = q1' therefore define balance flux Q1 = 1/2(q1- q1') and Q1'=1/2(q1'- q1) Total influx of mass into Triangle-0 = K(Q1 L1 + Q2 L2 + Q3 L3)
Total change in concentration in Triangle-0 per time = total influx/volume: 2) Grain Boundary Diffusion

The concentration or chemical potential on each grain boundary segment is determined by the values at each of the end-point nodes. Each node carries attributes, which are allowed to diffuse as the grain boundary diffusion code is run.

where C0 is the concentration at Node-0, q1 is the flux from Node-1 into Node-0, x1 is the length of the boundary segment between Node-1 and Node-0, and w1 is the width of the boundary segment between Node-1 and Node-0.
Flux into Node-0 from Node-1: q1 = k (C1-C0)/x1
Total influx of mass into Node-0 = A(q1 w1 + q2 w2 + q3 w3)
Total change in concentration at Node-0 per time = total influx/volume:

 
 
 

last updated 17 November 1999