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Studiengang Geowissenschaften B.Sc.Wintersemester 2004/5
Mineralogie (Wahlpflicht)
Andrew Putnis & Thorsten Geisler-Wierwille
Topic 2 : Solid Solutions and Binary Phase Diagrams
A solid solution - a single crystalline phase which exists over a range of chemical compositions.
Some minerals have a wide and varied chemistry e.g. garnets, while others permit very little chemical deviation from their ideal chemical formulae e.g. quartz.
The extent of solid solution is a strong function of the crystalstructure and also of temperature –solid solution is favoured at high temperatures and unmixing and/or cation orderingfavoured at low temperatures.
Binary solid solutions - the chemical composition of the mineral can be specified in terms of just two chemical components.
Olivine crystal chemistry
solid solution
solid solution
monticelliteCaMgSiO4
kirschsteiniteCaFe2+SiO4
forsteriteMg2SiO4
fayaliteFe22+SiO4
Miscibility gap
In the (Ca,Mg,Fe) olivines Ca is in M2 while Fe,Mg are randomly mixed over the M1 sites.
In the (Mg,Fe) olivines Fe,Mg are essentially random over all M1 and M2 sites, with a slight preference for Fe to occupy M1 sites.
Fayalite Fe2SiO4
Forsterite Mg2SiO4
The equilibrium behaviour of the system as a function of temperature and composition is specified by a binary phase diagram.
To illustrate solid solutions and phase diagrams we will refer to the important igneous mineral olivine.
The crystal structure of olivine, M2SiO4
The tetrahedra and cations in darker print are at level 0, the rest are at level 1/2.
The lower half of the unit cell showing the connections between tetrahedra and octahedra
M1 and M2 octahedra all share edges.
Olivine can also be considered as an approximately hexagonal close-packing of O atoms, with one half of the octahedral sites occupied by M cations and one eighth of the tetrahedral sites occupied by Si. The close packed layers are parallel to (100) planes.
M2SiO4 : Note, in a close-packed structure there are the same number of octahedral sites and twice the number of tetrahedral sites as close-packed atoms.
Olivine crystal chemistry
solid solution
solid solution
monticelliteCaMgSiO4
kirschsteiniteCaFe2+SiO4
forsteriteMg2SiO4
fayaliteFe22+SiO4
Miscibility gap
In the (Ca,Mg,Fe) olivines Ca is in M2 while Fe,Mg are randomly mixed over the M1 sites.
In the (Mg,Fe) olivines Fe,Mg are essentially random over all M1 and M2 sites, with a slight preference for Fe to occupy M1 sites.
Substitutional solid solution
Olivine: e.g. (Mg0.4Fe0.6)2SiO4 40% Forsterite; 60% Fayalite
or Fo40Fa60
If the Mg,Fe are randomly distributed over the M sites, the probability of an M site being occupied by Mg is 40%
Other types of solid solution
Coupled substitution
e.g. Al3+ + Ca2+ ⇔ Si4+ + Na+
e.g. Mg2+ + Si4+ ⇔ 2 Al3+ (Tschermaks substitution - 1 Al substitutes on an octahedral site, 1 Al substitutes on a tetrahedral site)
Other types of solid solution
Omission solid solution
Omitting cations from normally occupied sites
e.g. pyrrhotite FeS - Fe7S8
e.g. magnetite-maghemite Fe3O4 - Fe8/3O4 (=Fe2O3)
Other types of solid solution
Interstitial solid solution
Adding cations to sites not normally occupied in the structure
e.g. the composition of tridymite (siO2) can be varied towards nepheline (NaAlSiO4) by stuffing Na into the larger channel sites (empty in tridymite) and substituting Al3+ for Si4+ in the tetrahedral framework.
Factors controlling the extent of solid solution
1. Cation size
size difference should be <15%
e.g. Mg2+ has ionic radius 0.86Å
Fe2+ has ionic radius 0.92Å
Ca2+ has ionic radius 1.14Å
Factors controlling the extent of solid solution
2. Temperature
• higher temperatures favor the formation of solid solutions, so that end members which are immiscible at low temperatures may form more extensive or complete solid solutions at high temperatures
Factors controlling the extent of solid solution
3. Structural flexibility
In forming solid solutions much depends on the ability of the structure to bend bonds to accommodate different cations.
e.g. Mg - Ca mixing is very limited in olivine at all temperatures, while there is extensive solid solution at high temperatures in MgCO3-CaCO3, and complete solid solution at high temperatures in grossular - pyrope (Ca3Al2Si3O12 - Mg3Al2Si3O12) garnets
Factors controlling the extent of solid solution
4. Cation charge
• heterovalent substitutions (those involving cations with different charge) rarely lead to complete solid solution at low temperatures
Thermodynamics of solid solutions
Entropy as a function of composition
Enthalpy (internal energy) as a function of composition
Free energy as a function of composition
Entropy
Vibrational entropy of the end members
Configurational entropy of mixing
S = k ln w
w =N!
NA
!NB
!=
N!(x
AN)!(x
BN )!
ln N! = N ln N - N
S = -Nk (xA ln xA + xB ln xB)
S = -R (xA ln xA + xB ln xB) per mole of sites
Entropy
S = -R (xA ln xA + xB ln xB) per mole of sites
e.g. in olivine there are 2 M sites per formula unit (M2SiO4)
Therefore if Mg and Fe mix randomly over both sites, the configurational entropy of mixing is
-2R(xMg ln xMg + xFe ln xFe))
Enthalpy (or internal energy)
In addition to the energy of the bonds in the end members, the enthalpy of mixing depends on whether the mixing of A and B cations introduces any extra energy into the system.
This energy comes from the differences in the interaction energy of A-A, B-B and A-B bonds
Define: WAA , WBB , and WAB interaction energies
Enthalpy (or internal energy)
Simple nearest neighbour mixing modelAssume: A and B atoms mixing on sites which have coordination number of z (number of bonds between A and B atoms)
Then: if total number of sites is N, the total number of nearest neighbour bonds is 1/2Nz
The probability of A-A, B-B and A-B bonds is xA2, xB
2 and 2xAxB respectively
The total enthalpy of the solid solution is then
H = 1/2 Nz(xA2WAA + xB
2WBB + 2xAxB WAB)
H = 1/2Nz(xAWAA + xBWBB) + 1/2 Nz xAxB[2WAB - WAA - WBB]
mechanical mixture of end members enthalpy of mixing
Enthalpy (or internal energy)
Simple nearest neighbour mixing model
The total enthalpy of the solid solution is
H = 1/2 Nz(xAWAA + xBWBB) + 1/2 Nz xAxB[2WAB - WAA - WBB]mechanical mixture of end members enthalpy of mixing
∆Hmix = 1 /2 Nz xAxB[2WAB - WAA - WBB] = 1 /2 Nz xAxB W
W is the regular solution interaction parameter
Enthalpy
The sign of W determines the sign of the enthalpy of mixing
Enthalpy
The sign of W determines the sign of the enthalpy of mixing
When W > 0 the solid solution will attempt to maximise the number of A-A and B-B nearest neighbours (unmixing or exsolution)
When W < 0 the solid solution will attempt tomaximise the number of A-B nearest neighbours(ordering)
Free energy of a solid solution
∆Gmix = ∆Hmix - T ∆Smix
3 cases : ∆Hmix = 0 ∆Hmix > 0 ∆Hmix < 0
3 cases : ∆Hmix = 0 ∆Hmix > 0 ∆Hmix < 0
Case 1: ∆Hmix = 0 Ideal solid solution
∆Gmix = - T ∆Smix
As ∆Smix is always positive
∆Gmix is always negative
Case 2: ∆Hmix > 0 Non - ideal solid solution
∆Gmix = ∆Hmix - T∆Smix
Case 2: ∆Hmix > 0 Non - ideal solid solution
Lever rule:
Amount of Q. (C0 - C1) = Amount of R. (C2 - C0)
Exsolution lamellae in clinopyroxenes
Exsolution II
The scale of the exsolution microstructure depends on the diffusion rate of the ions in the structure, and on the cooling rate.
Fast cooling ⇒ fine scale intergrowth of the two phases
Slow cooling ⇒ larger scale intergrowth of the two phases
Fine-scale intergrowths can only be seen by electron microscopy
10µm
Case 3: ∆Hmix < 0 Non - ideal solid solution
Crystallisation of an ideal solid solution from a melt
Crystallisation of an ideal solid solution from a melt
Crystallisation of an ideal solid solution from a melt
Crystallisation where two end members have limited solid solution
Crystallisation where two end members have no solid solution
Solid solutions where the end members undergo reconstructive phase transitions.
More complex phase diagrams
Congruent melting
More complex phase diagrams
Incongruent melting
More complex phase diagrams : forsterite - silica
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