Ideal Solution Mixing: A-B Lattice

Here we will review the ideal solution mixing model for simple A-B lattice random mixing alloy system. The first step is to recall that for any system (e.g. state or phase) we can write the Gibbs free energy as:

$$ G = H-TS $$

where $H$ is the enthalpy, $T$ the temperature, and $S$ the entropy. We now propose that we have two isolated systems, lattice A and lattice B, and we want to find the change in Gibbs free energy when the two are combined to form a lattice with both A and B sites (randomly). An illustrative example would look something like below.

The next step is to write the change in Gibbs free energy as:

\begin{align}
\Delta G^{mix} & = G_{initial} - G_{final} \\
& = \Delta H^{mix} - T \Delta S^{mix} \\
\end{align}
Notice that we are using the label $mix$ to indicate that the change in Gibbs free energy is due to the mixing of the two lattices into one (i.e. Gibbs free energy of mixing).

In the ideal solution mixing model, we first approximate that $\Delta H^{mix}$  is negligible and taken to be zero. We can think of this as meaning that we assume no change in internal energy due to the chemical interactions between A and B. The next assumption is that the change in entropy is strictly due to configurational arrangement of A and B points on the combined lattice. This means that entropic effects due to lattice vibrations or magnetic ordering are not accounted for. Thus the Gibbs free energy has a simple relation to entropy:

$$ \Delta G^{mix} = - T \Delta S^{c} $$

The next step is to define the representation for the configurational entropy. To do this we will use to facts that each microstate is probabilistic and given by the combinatorics (i.e. possible configurations ). This is compactly represented by the famous equation:

$$ S^{c} = k_b \ln \omega^{c} $$

with $k_b$ being the Boltzmann constant and $\omega^{c}$ the configuration combinatorics. For the lattice AB this is going to be given by:

$$ \omega^{c} = \frac{N!}{N_{A}!N_{B}!}$$

where $N=N_{A}+N_{B}$ and $N_{A}$ and $N_{B}$ are the number of sites of a given type. At first glance calculating the configurational entropy may not seem daunting, however, logarithms of factorials can become demanding to calculate very quickly. Fortunately enough there is an approximation provided by mathematician James Stirling that allows one to approximate logarithms of factorials and is given by:

$$ \ln N! \approx N \ln N - N $$
Using this approximation we can determine $\omega^{c}$ and distill the expression of $S^{c}$ into something that is relative compact and meaningful. Apply the approximation we get:
\begin{align}
 \ln \omega^{c} &= N \ln N - N - \left[\ln\left(N_{A}!N_{B}!\right)\right] \\
&= N \ln N - N - \left[ N_{A} \ln N_{A} - N_{A} + N_{B} \ln N_{B} - N_{B}\right] \\
&= N \ln N - N_{A} \ln N_{A} - N_{B} \ln N_{B} - N + N_A + N_B \\
\end{align}
The last three terms cancel out, e.g., $N_A + N_B = N$ and we then rewrite the first term as:
\begin{align}
\ln \omega^{c} &= \left(N_A + N_B\right) \ln N  - N_{A}\ln N_A - N_{B}\ln N_B \\
&=-\left[N_A \ln \left(\frac{N_A}{N}\right) + N_{B}\ln\left( \frac{N_B}{N}\right) \right]
\end{align}
the ratio of $X_A = \frac{N_A}{N}$ or $X_B = \frac{N_B}{N}$  are the fraction of sites on the mixed lattice with A and B sites, respectively. Let us take one further step by multiplying the equation above by $\frac{N}{N}$ to get
$$ \ln \omega^{c} = -N \left[ X_A \ln X_A + X_B \ln X_B \right] $$
Now we can write $\Delta S^{mix}$ as,
$$ \Delta S^{mix} = -k_{b} N \left[ X_A \ln X_A + X_B \ln X_B \right] $$
if we assume that the total number of N sites on the alloy lattice is comparable to the number of particles in 1 mole, i.e., Avogadro's number $N_a = \text{6.022}\times \text{10}^{\text{23}}$, then we can write the Gibbs free energy of mixing in most familiar form as:
\begin{align}
 \Delta G^{mix} &= -T \Delta S^{mix} \\
&= -T \cdot -k_{b} N_{a} \left[ X_A \ln X_A + X_B \ln X_B \right]  \\
&= \boxed{RT \left[ X_A \ln X_A + X_B \ln X_B \right]}
\end{align}
where $R$ is the gas constant given by  $k_b N_a$. We can get a sense for how the Gibbs free energy of mixing changes with temperature as shown in the graph below,

From the graph we observe two features, 1.) the Gibbs free energy of mixing for an ideal solution is a symmetric function, 2.) as the temperature is increased $\Delta G^{mix} is decreases. Not that in the graph the line(s) do not extend to zero and one, this is because these would be given by the Gibbs free energy of the reference states of lattice A and B.

Ideal solution mixing is typically not suitable for real material alloy systems and thus other approximations such as the regular solution model are used. In the regular solution model we use the same $\Delta S^{mix}$ and include a non-zero expression for $\Delta H^{mix}$. The most accurate approach for calculating Gibbs free energy of mixing for real materials is to use CALPHAD methodologies.

For this blog postings quote we will get two quotes:

"Nothing in life is certain except death, taxes and the second law of thermodynamics."
-Seth Lloyd, MIT Professor 

"In this house, we obey the laws of thermodynamics!"
-Homer Simpson, response to Lisa's perpetual motion machine

References & Additional Reading

[1] Thermodynamics in Materials Science, Robert DeHoff, 2nd ed.
[2] Introduction to the Thermodynamics of Materials, D.R. Gaskell, 5th ed.

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