The equations that model the mixing process originate in references[1-3]
and they are summarised in this section.
Let the material system consist of n pure material species.
The governing equations for the concentration of each
i-material is as follows:
¶qi(x,f)
¶f
= (q(x,f)+u(f))
¶qi(x,f)
¶x
+
¶q(x,f)
¶x
qi(x,f) -
¶qi(x,f)
¶x
(1)
for i=1,2,..,n, 0 £ x < ¥, 0 £ f < ¥,
where x is the depth below the receding surface and f is the
dose.
At any dose f the material distribution is determined by the
functions qi(x,f) for i=1,2, ..,n,
the fraction by volume of i-atoms at
depth x. The functions qi satisfy the constraints
Si=1n qi(x,f) = 1, qi(x,f) ³ 0 for all x, f.
The surface recession speed is denoted by u(f). If X0(f)
denotes the depth of erosion at dose f then X0(f) = ò0f u(f¢) d f¢.
The relationship between the fixed and moving coordinate system is such that
X=X0(f) + x.
At this stage it is satisfactory to
assume that the material structure model has infinite depth.
The scaled current of i-atoms crossing
the point at depth x below the surface in the direction of
increasing depth is denoted qi(x,f) and the collective current
by q(x,f) where
q(x,f)=Si=1n qi(x,f).
The qi are given by the following equation
qi(x,f) = -
¶
¶x
{ Di(x,f) qi(x,f) } ,
(2)
for i=2,3,..,n and
q1(x,f) = -
¶
¶x
{D1(x,f) q1(x,f) }+ t1
ó õ
¥
x
fR(x¢,f) dx¢
(3)
in the domain (x,f) Î [0,¥) ×[0,¥), where fR(x,f) is the probability density functions (with respect to
depth x) for the range of the individual primary particle in the material
structure at dose f, Di(x,f) is the diffusivity of the
i-atoms at depth x and dose f in the material structure and
t1 is the effective volume of a primary particle.
Conditions are applied to the above model to define the solution.
The material distribution is given at f = 0 so that
qi(x,0) = Gi(x) is known for x Î [0,¥).
A boundary conditon is given at x=0 by applying a
conservation of material condition in the surface layer.
In practice, the initial material concentration functions
Gi(x) can be assumed to belong to a simple space. In the
most trivial case,
the Gi(x) functions could be assumed to take constant values between
each pair of interfaces. In practical circumstances, the interfaces
may not be sharp, hence the space of functions will need to be extended
to either ramped functions where there is a linear transition from
one value to another at each interface and/or to an error function
transition which represents a sharp interface subjected to a quantity
of linear diffusion. In IMPETUS II, the initial material concentration
take constant values between points (interfaces) or is in linear
progression from point to point. The interfaces between layers of
different concentrations are assumed to be sharp. A typical original
material distribution is given in figure 1.