12.480 Handout #7: Effects of Ordering Species on Sites in a
Crystalline Solution
Equilibrium in a “closed” crystal means one that does not vary in composition.
There are independently variable “reactions” that result in distributions of species on sites. We
want to look at how the free energy of crystals varies as these reactions occur.
At equilibrium the Gibbs function will be a minimum.
The free energy of the crystal consists of two parts. The first is the ideal configurational entropy,
approprialtely readjusted for species and sites.
��
Sideal = −nK
qNsq ln Nsq ,
s
q
�
�
where −nK = R, s = sum over all species, q = sum over all sites, q = multiplicity of site, and
Nsq = atoms of types on site q.
Remeber the equivalent derivation we carried out earlier for ways of distributing species over
sites...
W =
(nN )!
(XnN )!((1 − X)nN )!
Now — define thermodynamic properties like
S̄ ∗ ≡ S̄soln − S̄ideal ← excess entropy for sites
Ḡ∗ ≡ Ḡsoln + T Sideal = H̄soln − T S ∗ ← free energies for sites
then
Gsoln = Ḡ∗ −
�
T S̄i = Ḡ∗ + RT
i
��
s
qNsq ln Nsq
q
This G∗ is defined as a deviation from ideal solution behavior. It arises from:
1. long-range ordering on sites
2. internal reactions (oxidation-reduction) among species and vacancies
3. excess energy and volume terms for sites
Application to Pyroxene
(F e, M g)M 2 (F e, M g)M 1 Si2 O6
XFs =
XFxtal
XFMe1 + XFMe
2
e
← this is a familiar descriptive parameter
=
xtal
2
XFxtal
e + XM g
We’ll use different compositional parameters rads that reflect distribution of species on sites.
r ≡ XFMe2 + XFMe1 − 1 = 2XF s − 1
s ≡ XFMe2 − XFMe1
Physically accessible composition space is then defined in terms of r and s and the possibility of
ordering is included.
1
s
Fe M²Mg M¹Si₂O₆
Fe₂Si₂O₆
Mg₂Si₂O₆
r
Mg M²Fe M¹Si₂O₆
Now - composition space can be recast into r and s ordering parameters and G∗ expanded in a
power series about a reference state.
¯∗
G
= g0 + gr r + gs s + grr r2 + grs rs + gss s2
RT
Here is the key to the model. We assume that G∗ can be accounted for by a power series expansion.
Conditions of equilibrium are used to determine G∗ .
In this example we are dealing with a closed system (crystal). The Gibbs function must be minimized
with respect to all variations in independent parameters.
�
�
∂G
=0
∂Xs P,T,Xi
If this is not the case, some variation of Xi ’s could lower the free energy.
G₁
G
G₁
G eq  slope = 0
s = ordering parameter
We could also expand our expression in terms of terms which look a bit more familiar....
¯ = 2RT [X11 ln X11 + X21 ln X21 + X12 ln X12 + X22 ln X22 ] ← ideal entropy of mixing
G
¯0
¯ 0 X21 X22 + G
¯ 0 X11 X22 ← ordering energy
+Ḡ010
¯ X11 X12 + G10 X21 X22 + G0¯
01
1
+WM 1 X11 X21 + WM 2 X12 X22 ← deviation from ideality
2
using JBT’s ordering parameters....
⎡ � ¯ �⎤
G
M1
∂ RT
XFMe2 XM
g
⎣
⎦
at equilibrium
= 0 = ln M 2 M 1 + gs + grs r + 2gss s
∂s
XM g X F e
P,T,r
intra
ln KD
= −gs − grs r − 2gss s
�
1 � 0
ḠF eM g Ḡ0M gF e + Ḡ0F eF e + Ḡ0M gM g + WM 1 + WM 2
4RT
�
1 � 0
gs =
ḠF eM gSi2 Ō6 Ḡ0M gF eSi2 O6
2RT
g0 =
grs =
1
(WM 1 − WM 2 )
2RT
and gss ... contains all of these terms.
If we plug in and assume GF eF e = GM gM g , GM gF e = GF eM g and WM =
intra
ln KD
=
Ḡ0F eM gSi2 O6 − Ḡ0M gF eSi2 O6
WM
−
2RT
RT
3
WM 1 +WM 2
.
2