Theory
In this post we report our findings on our study to determine whether pure Li or LiH is more energetically favorable for different hydrogen pressures P\(_{H_2}\) and temperatures T. For this we need to determine which material minimizes the free energy of the a system containing gaseous H\(_2\) and Li at a given pressure P\(_{H_2}\) and temperature T. We use the grand potential of a given system as a proxy for the free energy since the grand potential of a system is lowest for a system with lowest free energy.
The grand potential for a system containing N\(_{Li}\) Li atoms and N\(_H\) hydrogen atoms is defined by,
\begin{equation}\Omega\left(T,\mu_{Li},\mu_H,N_{Li},N_H\right)=E\left(N_{Li},N_H\right)-TS-\mu_H N_H- \mu_{Li}N_{Li}.\label{gp_def}\end{equation}
Where \(E\left(N_{Li},N_H\right)\) is the internal energy of the material and is determined using a DFT calculation, \(S\) is the material entropy and respectively \(\mu_{species}\) and \(N_{species}\) define the chemical potential and number of atoms of the given species of atoms in the material.
Now we are interested in comparing the grand potentials for Li and LiH. Before proceeding with our calculations we make a few adjustments to equation \eqref{gp_def} to simplify our calculations. First the difference in internal energy between Li and LiH is much larger than the entropic differences between, hence the second term in the right hand side of equation \eqref{gp_def} can be neglected. Next we can eliminate the last term in equation \eqref{gp_def} by choosing to look at a pure Li unit cell and a LiH unit cell with equal number of Li atoms. Since \(\mu_{Li}\) is independent of P\(_{H_2}\) and we set \(\left(N_{Li}\right)_{pure\;Li}= \left(N_{Li}\right)_{LiH}\), the term \(\mu_{Li}N_{Li}\) is the same for both systems and will cancel out when considering the difference in grand potential between the two systems.
With above mentioned simplifications equation \eqref{gp_def} now reads as,
\begin{equation}\Omega\left(T,\mu_H,N_H\right)=E\left(N_H\right)-\mu_H N_H.\label{gp_red}\end{equation}
In equilibrium condition the chemical potential of molecular hydrogen and atomic hydrogen are related by,
\begin{equation}\mu_{H}=\frac{1}{2}\mu_{H_2}.\label{muH2_H}\end{equation}.
Treating H\(_2\) as an ideal gas the definition of it’s chemical potential is,
\begin{equation}\mu_{H_2}=E_{H_2}^{total}+\tilde{\mu}_{H_2}\left(T,p^0\right)+k_BTln\left(\frac{p}{p^0}\right).\label{H2_ideal_chem_pot}\end{equation}
Here \(k_B\) is the Boltzmann constant, \(E_{H_2}^{total}\) is the total energy of an isolated H\(_2\) molecule and can be obtained from a DFT calculation for an isolated H\(_2\) molecule. The second term in equation \eqref{H2_ideal_chem_pot}, \(\tilde{\mu}_{H_2}\left(T,p^0\right)\) is the difference between the \(T=0\;K\) and the temperature of interest chemical potential of molecular H\(_2\). \(\tilde{\mu}_{H_2}\left(T,p^0\right)\) can be calculated using data tabulated in NIST-JANAF Thermochemical Tables for molecular hydrogen. In terms of quantities defined in the tables referred to; by noting the chemical potential of an ideal gas is equal to the free energy \(G=H-TS\) of an ideal gas,
\begin{equation}\tilde{\mu}_{H_2}\left(T,p^0\right)=\left[H^0\left(T\right)-H^0\left(T_r\right)\right]-TS\left(T\right)-\left[H^0\left(0\right)-H^0\left(T_r\right)\right].\label{mu_tilde}\end{equation}
Calculations
Now we have the recipe to obtain the Li-LiH phase diagram. From DFT we need the internal energies of a pure Li unit cell, LiH unit cell and a Hydrogen molecule. All energy calculations were performed using DFT calculations using the DMol3 software package as implemented by Materials Studio. DMol3 employs numerical radial functions centered at atomic positions as basis sets. The GGA base PBE functional was used to treat exchange correlational effects and the double numerical precision – DNP 3.5 basis set was used. All energy calculations were deemed to be converged once an SCF convergence of \(10^{-6}\) was achieved. Additionally for the periodic systems a k-point mesh with \(0.03\;\mathring{A}^{-1}\) maximum spacing between mesh points to have the internal energies calculated to an accuracy of \(1\;m\,eV\).
The systems for which we require the internal energies to perform our calculations are a pure Li primitive unit cell, a LiH primitive unit cell and the hydrogen molecule. First the primitive unit cells of Li and LiH and also the hydrogen molecule were geometry optimized to obtain the optimal structures. The optimal structures of the Li unit cell (primitive unit cell of a BCC lattice with lattice constant = 3.0155 Å), LiH unit cell (primitive unit cell of a FCC lattice with lattice constant = 2.9409 Å) and the hydrogen molecule (H-H bond length = 0.748444 Å) are shown in figures 1, 2 and 3 respectively. The internal energies of the systems of interest are shown in table 01.
Figure 1 : Primitive unit cell of Li. This is a primitive unit cell of a BCC lattice with lattice constant = 3.0155 Å.
Figure 2 : Primitive unit cell of LiH. This is a primitive unit cell of a FCC lattice with lattice constant = 2.9409 Å.
Figure 3 : Hydrogen molecule. Hydrogen bond length = 0.748444 Å.
Table 01 : Internal Energies of systems of interest.
| System | Internal Energy \(\left(e\,V\right)\) |
|---|---|
| Li | -204.660 |
| LiH | -221.233 |
| H\(_2\) | -31.678 |
Now using the primitive unit cells of Li and LiH and comparing their respective grand potentials, we see that \(\Omega_{Li}=\Omega_{LiH}\) or Li and LiH are thermodynamic equilibrium when,
\begin{equation}\mu_H^{eq}=E_{LiH}-E_{Li}=-16.573\;eV;\label{req_mu_H}\end{equation}
Or when, (using equation \eqref{muH2_H})
\begin{equation}\mu_{H_2}^{eq}=-33.146\;eV.\label{req_mu_H2}\end{equation}
Now in order to determine the phase diagram we need to determine the pressure at which \(\mu_{H_2}=-33.146\;eV\) for a given temperature. To do this we look at equation \eqref{H2_ideal_chem_pot}. From the NIST-JANAF Thermochemical Tables for molecular hydrogen we have obtained the plot , shown in figure 4, of \(\tilde{\mu}_{H_2}\left(T,p^0\right)\) vs \(T\) for a reference pressure \(p^0=0.1\;MPa\).
Figure 4 : Plot of \(\tilde{\mu}_{H_2}\left(T,p^0\right)\) vs \(T\) for a reference pressure \(p^0=0.1\;MPa\).
Now for a given temperature we can solve for the pressure \(p\) that makes \(\mu_{H_2}=\mu_{H_2}^{eq}\), by rewriting equation \eqref{H2_ideal_chem_pot} as,
\begin{equation}\ln\left(\frac{p^{eq}}{p^0}\right)=\frac{\mu_{H_2}^{eq}-E_{H_2}^{total}-\tilde{\mu}_{H_2}\left(T,p^0\right)}{k_BT}.\label{solve_for_p}\end{equation}
The curve \(\ln\left(\frac{p^{eq}}{p^0}\right)\) vs \(T\) which is the phase boundary between Li and LiH is shown in figure 5. For any point in phase space, corresponding to a point above and/or to the left of the \(\ln\left(\frac{p^{eq}}{p^0}\right)\) vs \(T\) curve, LiH is the thermodynamically preferred structure. Conversely, any point in phase space, corresponding to a point below and/or to the right of the \(\ln\left(\frac{p^{eq}}{p^0}\right)\) vs \(T\) curve, pure Li is the thermodynamically preferred structure.
Figure 5 : Plot of \(\ln\left(\frac{p^{eq}}{p^0}\right)\) vs \(T\).
