ME574 Team 1
Raymond Jonathan
Sean Bong
Jared Slaybaugh
Daniel Kim
Free Energy
One can assign free energy densities to the volume of bulk
solid phases, the area of interfaces, and the length of triple junctions.
Let g be the free energy per unit volume of atoms
in a bulk solid
phase. Here we assume the long range field
and volume change associated with the phase change are
neglected. Also, we assume
that g is uniform in each solid
phase, but may differ in different
phases. All solid phases are
taken to have the same chemical composition. Let Y be the interfacial tension, which depends on
the crystalline orientation
of the interface. The free energy per unit length of a triple
junction, Г, may also affect the structural change
when the
grain size is extremely small. The total free energy of the
system, G, has the three types of contributions:
However, we neglected the triple junction term in our project
which
can be further reduced to:
The system would always wants to minimize its free energy
and interfaces change their shape in such a way to reduce G.
Kinetic Law
Let vn be the actual velocity of the interface in the direction
normal to the interface. The actual velocity is taken be a
function of the driving pressure, P and kinetic coefficient, L.
The above equation is only valid when ΩP<<kT where Ω is
the atomic volume, k is the Boltzmann’s constant, and T the
absolute temperature.
Weak Statement
The weak statement can be describe by the driving force
which is given by:
We can eliminate the driving pressure P by using the kinetic
law. Thus, the
equation becomes:
Hence, this equation is defined as the weak statement of
interface migration problems.