60
Point Defect Thermodymanics
Equilibrium Concentration of Solute Atoms
We consider a dilute substitutional solution of B atoms in an A matrix.
We let
N
be the number of A atoms in the system (which is fixed) and
xN
be the number
of B atoms dissolved in the lattice.
Thus
x
is the fraction of atomic sites where
the wrong kind of atom is located.
We follow the regular solution/quasi
chemical approach in which the formation energy of the point defect is
dominated by the energies of the chemical bonds associated with the impurity
defect (quasichemical) and where the mixing entropy is that for an ideal
solution (regular solution).
The system under consideration is one with an
infinite supply of B atoms.
Each time a B atom is dissolved in the lattice the A
atom it replaces takes up a site at the A/B interface and extends the A lattice by
one atomic volume.
Unit process of dissolving one B atom into an A lattice
We wish to describe the equilibrium state of the system.
Will any B atoms
dissolve in A at equilibrium?
Adding B atomic defects to the A lattice increases
the internal energy of the system and that discourages the introduction of
foreign, defect atoms.
But the entropy of the system also increases when B
atoms are introduced and that causes some B atoms to be present at equilibrium,
in spite of the increased internal energy of the system.
We know this result in
another context.
Everyone is familiar with the fact that solutes usually dissolve
spontaneously in pure components and the equilibrium concentration increases
with temperature.
This is shown as the solvus line on a phase diagram.
At a
particular temperature, the equilibrium concentration of solute is given by the
intersection of the isotherm with the solvus line.
Solvus line in a simple phase diagram
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61
If the pressure and temperature are held constant, then the equilibrium state of
the system is determined by the lowest possible Gibbs free energy.
So we
consider the change in Gibbs free energy that occurs when
N
B
=
xN
B atoms are
introduced into the A lattice.
This may be written as
±
G
=
±
H
²
T
±
S
,
where
±
H
and
±
S
are the change in enthalpy and entropy, respectively, when
the defects are introduced.
We first consider the factors that make up
±
H
.
Enthalpy
The change in enthalpy can be expressed as
±
H
=
±
E
+
p
ext
±
V
,
where
±
E
and
±
V
are the change in internal energy and volume associated with
creation of the defects and
p
ext
is the external hydrostatic pressure.
The internal
energy change can be expressed as
±
E
=
xN
±
e
f
,
where
±
e
f
is the formation (internal) energy of each defect, usually just called
the formation energy of the defect.
It, in turn, can be composed of two parts,
±
e
f
=
±
e
f
bond
+
±
e
f
strain
,
where
±
e
f
bond
is the internal energy change associated with making the wrong
kinds of chemical bonds at the defect and
±
e
f
strain
is the elastic strain energy
associated with creating the defect.
We may estimate
±
e
f
bond
if the energies of the
near neighbor chemical bonds are known or can be estimated.
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 Spring '08
 nix
 Thermodynamics, Energy, Entropy

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