Here u represents the concentration of solute atoms,
takes the values +1 and -1 in the two perfect
crystal grains and intermediate values in the boundary between them,
, 
and 
are constants characterizing the material,
is an interaction energy density,
and the diffusivity 
is large in the grain boundary
but zero in the grains 
.
The model is thermodynamically consistent,
being derivable from a free energy functional.
The aim of the work is to understand what interactions
can or cannot account for the observed results.
For small the speed of travelling wave solutions can be
calculated approximately using a successive approximations scheme.
The results indicate that the simple interaction,
,
corresponding to differing solubility in the grain boundary
and in the bulk crystal, cannot explain all the observed data.
An interaction modelling the elastic coherency strain energy is also
considered, and its consequences are consistent with the
observed features of DIGM in nearly all cases.
The same interaction also provides possible mechanisms for
the initiation of DIGM and for liquid film migration.
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