Thermodynamics of solid solutions

Entropy

The entropy of the two endmembers, A and B, of a solid solution are S_A and S_B, and are mainly vibrational in origin (i.e. related to the structural disorder caused by thermal vibrations of the atoms at finite temperature). The entropy of the solid solution will always be greater than the entropy of the mechanical mixture.

The entropy of the mechanical mixture is given by:

S = x_AS_A + x_BS_B

The excess entropy is called the entropy of mixing (ΔS_mix), and is mainly configurational in origin (i.e. it is associated with the large number of energetically-equivalent ways of arranging atoms/ions on the available lattice sites).

The configurational entropy is defined as:

S = k lnW

where k is Boltzmann’s constant (1.38 x 10^-23 JK^-1), and W is the number of possible configurations.

If it is assumed that the entropy of mixing is equal to the configurational entropy,

ΔS_mix = k lnW

If we consider mixing N_A A atoms and N_B B atoms on N lattice sites at random, then the number of different configurations of A and B cations is given by:

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where x_A and x_B are the mole fractions of A and B respectively.

Hence

Δ�mix=�ln⁡�!(���)!(���)!=�[ln⁡�!−ln⁡(���)!−ln⁡(���)!]

Stirling’s approximation states that, for large N:

lnN! ≈ N lnN − N

Hence

ΔS_mix = k(N lnN – N) – k(x_AN lnx_AN – x_AN) – k(x_BN lnx_BN – x_BN)

ΔS_mix = –Nk(–lnN + 1 + x_Alnx_A + x_AlnN – x_A + x_Blnx_B + x_BlnN – x_B)

ΔS_mix = –Nk(x_Alnx_A + x_Blnx_B + (x_A + x_B)lnN – (x_A + x_B) – lnN + 1)

Since x_A + x_B = 1,

ΔS_mix = –Nk(x_Alnx_A + x_Blnx_B)

If N is taken to be equal to Avogadro’s number, then per mole of sites:

ΔS_mix = –R(x_Alnx_A + x_Blnx_B)

Enthalpy

The enthalpy of the two endmembers of the solid solution, A and B, are equal to H_A and H_B respectively. For a mechanical mixture of these two endmembers, the enthalpy is given by:

H = x_AH_A + x_BH_B

The excess enthalpy relative to the mechanical mixture is known as the enthalpy of mixing, ΔH_mix. This can either be positive or negative, or zero.

If ΔH_mix = 0, the solution is said to be ideal, and for ΔH_mix ≠ 0, the solid solution is said to be non-ideal.

A simple expression for the enthalpy of mixing can be derived by assuming that the energy of the solid solution arises only from the interaction between nearest-neighbour pairs.

Let z be the coordination number of the lattice sites on which mixing occurs. If the total number of sites is N, then the total number of nearest-neighbour bonds is 0.5Nz. (The factor of 0.5 arises since there are two atoms/ions per bond).

Let the energy associated with A-A, B-B and A-B nearest-neighbour pairs be W_AA, W_BB and W_AB respectively. If the cations are mixed randomly, then the probability of A-A, B-B and A-B neighbours is x_A², x_B² and 2x_Ax_B respectively.

Hence the total enthalpy of the solid solution is given by:

H = 0.5 Nz(x_A²W_AA + x_B²W_BB + 2x_Ax_BW_AB)

This can be rearranged to:

H = 0.5 Nz(x_AW_AA + x_BW_BB) + 0.5 Nzx_Ax_B(2W_AB –W_AA –W_BB)

The first term in this expression is equal to the enthalpy of the mechanical mixture. Hence:

ΔH = 0.5 Nzx_Ax_B(2W_AB –W_AA –W_BB) = 0.5 Nzx_Ax_BW

where W (= 2W_AB –W_AA –W_BB) is known as the regular solution interaction parameter, and its sign determines the sign of ΔH_mix.

A positive value of W indicates that it is energetically more favourable to have A-A and B-B neighbours, rather than A-B neighbours. In order to maximise the number of A-A and B-B neighbours, the solid solution unmixes into A-rich and B-rich regions. This process is called exsolution.

A negative value of W indicates that it is energetically more favourable to have A-B neighbours, rather than A-A or B-B neighbours. To maximise the number of A-B neighbours, the solid solution forms an ordered compound.