Solid Solutions

A solid solution is a single phase which exists over a range in chemical compositions.

Almost all minerals are able to tolerate variations in their chemistry (some more than others).

Chemical variation greatly affects the stability and the behaviour of the mineral.

Therefore it is crucial to understand:

the factors controlling the extent of solid solution tolerated by a mineral

the variation in enthalpy and entropy as a function of chemical composition

different types of phase transition that can occur in solid solutions

M1 and M2 octahedral sites in olivineM1 forms ribbons of edge-sharing octahedra parallel to z

M2 sites share edges and corners with M1

M1 and M2 share edges and corners with tetrahedra

Tetrahedra remain unchanged with changing T and P, but M1 and M2 can expand and contract, and their relative sizes can change

M2 is larger and more disorted than M1

Chemistry of the olivinesThe two most important forms of olivine are:

Forsterite

Mg2SiO4

Fayalite

Fe2SiO4

and

Both forms have identical structures and identical symmetry. They differ only in the type of cation occupying the M sites.

Most natural olivines contain a mixture of Mg and Fe on the M sites: (Mg, Fe)2SiO4. This is an example of substitutional solid solution.

The composition of the solid solution is specified in terms of the mole fraction of the two endmembers forsterite and fayalite.

e.g. (Mg0.4Fe2+0.6)2SiO4 is said to contain 40% forsterite and 60% fayalite

(often abbreviated to Fo40Fa60)

Substitutional solid solutionsSpecific sites in the structure (e.g. M-sites in olivine) are occupied by either Mg or Fe.

In the ideal case, Mg and Fe are randomly distributed.

The probability of any one site being Mg is equal to the mole fraction of Mg in the system.

e.g. in Fo40Fa60, each site has a 40% chance of being occupied by Mg and a 60% change of being occupied by Fe.

Other types of solid solution

Coupled substitution : Cations of different charge are substituted for each other. Requires two coupled substitutions to maintain charge balance. e.g. Al3+ + Ca2+ = Si4+ + Na+ in plagioclase feldspars.

Omission solid solution : Chemical variation achieved by omitting cations from sites that are normally occupied. e.g. pyrhottite solid solution between FeS and Fe7S8. Charge balance is achieved by changing the valance of transition metal cations (e.g. Fe2+ is converted to Fe3+)

Interstitial solid solution : Cations are inserted into sites not normally occupied in the structure. e.g. solid solution between tridymite (SiO2) and nephaline (NaAlSiO4) achieved by stuffing Na into channel sites and substituting Al for Si in framework.

Factors controlling the extent of solid solution

Cation SizeIf cation sizes are very similar (i.e. ionic radii differ by less than 15%) then extensive or complete solid solution is often observed.

Mg

0.86 Å

Fe2+

0.92 Å

Ca

1.14 Å

7%32%

Factors controlling the extent of solid solution

Temperature

Cation disorder in a solid solution increases the

configurational entropy: solid solution is stabilised at high

temperature

G = H – TS

Cation-size mismatch increases the enthalpy

(structure must strain to accommadate cations of

different size): solid solution is destabilised at low

temperatures

Extent of solid solution tolerated is greater at higher temperatures

Factors controlling the extent of solid solution

Structural flexibilityCation size alone is not enough to determine the extent of solid solution, it also depends on the ability of the structural framework to flex and accommodate differently-sized cations

e.g. there is extensive solid solution between MgCO3 and CaCO3 at high temperature

Cation charge

Complete solid solution is usually only possible if the substituting cations differ by a maximum of ± 1. Heterovalent (coupled) substitutions often lead to complex behaviour at low temperatures due to the need to maintain local charge balance.

Thermodynamics of solid solutions

Enthalpy

Hmix = 1/2 Nz xAxB [2WAB - WAA - WBB] = 1/2 Nz xAxB W where W is referred to as the regular solution interaction parameter.

H = WAA NAA + WBB NBB + WAB NAB

H = 1/2 Nz (xA2 WAA + xB

2 WBB + 2xAxB WAB)

H = 1/2 Nz (xA WAA + xB WBB) + 1/2 Nz xAxB [2WAB - WAA - WBB] 

The fist term is the enthalpy of the mechanical mixture. The second term is the enthalpy of mixing.

 

Thermodynamics of solid solutionsEnthalpy

Hmix = 1/2 Nz xAxB [2WAB - WAA - WBB]

= 1/2 Nz xAxB W

If W > 0 then like neighbours (AA and BB) are favoured over unlike neighbours (AB)

If W < 0 then unlike neighbours (AB) are favoured over like neighbours (AA and BB)

Thermodynamics of solid solutions

Exsolution

Cation ordering

Thermodynamics of solid solutions

EntropyS = k ln w

w N!

NA!NB!

N!

(xAN)!(xBN)!

w = no. of degenerate configurations of the system

For a collection of N atoms, consisting of NA A atoms and NB B atoms, the no. of configurations is:

Simplifies to:

S = - R (xA ln xA + xB ln xB)

per mole of sites.

Free energy of mixing

Hmix = 1/2 Nz xAxB W Smix = - R (xA ln xA + xB ln xB)

Gmix = Hmix - TSmix

Ideal solid solution (Hmix = 0)

In an ideal solid solution, the two cations substituting for each other are very similar (same charge and similar size).

Gmix = - TSmix

Solid solution is stable at all compositions and all temperatures.

The forsterite-fayalite solid solution is very close to ideal!

Non-ideal solid solution (Hmix > 0)

Free energy develops minima at low temperature due to competition between positive Hmix term and negative -TSmix term.

Solid solution is unstable at intermediate compositions. Equilibrium behaviour is given by the common tangent construction.

Non-ideal solid solution (Hmix > 0)

Lever Rule:

Phase with bulk composition C0 splits into two phases Q and R with compositions C1 and C2.

Proportion of Q = (C2-C0)/(C2-C1)

Proportion of R = (C0-C1)/(C2-C1)

Non-ideal solid solution (Hmix < 0)

Cation ordering phase transition occurs below a particular transition temperature Tc

Tc varies with composition - highest in the centre when A:B ratio is 1:1