First-principles estimation of partition functions representing disordered lattices such as the cubic phases of ${\mathrm{Li}}_2\mathrm{OHCl}$ and ${\mathrm{Li}}_2\mathrm{OHBr}$

In order to develop computational methods that can simulate thermodynamic properties of disordered materials at a first-principles level, we investigate the use of a random set of configurations to evaluate the canonical partition function of lattice-based disordered systems. Testing the sampling method on the one- and two-dimensional Ising models indicates that for the ordered system at low temperature, convergence is achieved when the number of samples $\mathcal{S}$ is comparable to or larger than the number of configurations $\mathrm{\Omega }$, while for the partially disordered system at high temperature, convergence is achieved for smaller sample sizes as low as $\mathcal{S}\approx \mathrm{\Omega }/100$ or $\mathcal{S}\approx \mathrm{\Omega }/1000$. The sampling method is combined with first-principles calculations to examine the ordered $\leftrightarrow$ disordered phase transition for the Li ion electrolyte materials ${\mathrm{Li}}_2\mathrm{OHCl}$ and ${\mathrm{Li}}_2\mathrm{OHBr}$. Static-lattice internal energies and harmonic-phonon free energies were incorporated into the evaluation of the partition function. The evaluation of the partition function depends on the value of $\mathrm{\Omega }$ corresponding to the number of metastable states of the system. Accordingly, we developed a method of approximating $\mathrm{\Omega }$ using the properties of the sampled configurations. The results of the calculations are consistent with the experimental observation that the transition temperature for the orthorhombic $\leftrightarrow$ cubic phase transition is higher for ${\mathrm{Li}}_2\mathrm{OHCl}$ than for ${\mathrm{Li}}_2\mathrm{OHBr}$. We expect the sampling method to be generally useful for investigating the thermodynamic properties of other disordered-lattice systems. We also investigate a “disordered-subspace function” which is shown to satisfy inequality relationships with respect to the Helmholtz free energy.

Figure 1

Helmholtz free energy per lattice site as a function of the scaled temperature for the one-dimensional Ising model for $n=16$, comparing the exact result (“Exact”, black line) given by Eq. (7) with the sampling density of states (“SDOS”, red lines) calculated from the sampled partition function Eq. (5). The sample sizes are $\mathcal{S}=\mathrm{\Omega }/100,\mathrm{\Omega }/10$, and $\mathrm{\Omega }$ as indicated. Each calculation was repeated five times to show the variability.

Figure 2

Helmholtz free energy per lattice site as a function of the scaled temperature for the two-dimensional Ising model for various numbers of sites $n$ in each of the two directions. The $n\to \infty$ result was evaluated from Eq. (9).

Figure 3

Convergence of the free energy of the two-dimensional square Ising model for $n=5$ with respect to number of samples $\mathcal{S}$, comparing sampling density of states (“SDOS”, red lines) results from Eq. (5) with the exact result (black line). The sample sizes are $\mathcal{S}=\mathrm{\Omega }/10000,\mathrm{\Omega }/1000,\mathrm{\Omega }/100,\mathrm{\Omega }/10$, and $\mathrm{\Omega }$ as indicated. Each calculation was repeated five times to show the variability.

Figure 4

Ball and stick diagram of O1 orthorhombic structure of ${\mathrm{Li}}_2\mathrm{OHCl}$ having the space group $Pmc{2}_1$. The ball colors gray, red, blue, and green indicate Li, O, H, and Cl sites, respectively.

Figure 5

Ball and stick diagram of O2 orthorhombic structure of ${\mathrm{Li}}_2\mathrm{OHCl}$ having the space group $Cmcm$. The ball colors gray, red, blue, and green indicate Li, O, H, and Cl sites, respectively.

Figure 6

Comparison of simulated and experimental x-ray diffraction patterns at wavelength $\lambda =1.54056\AA$ for the orthorhombic structure of ${\mathrm{Li}}_2\mathrm{OHCl}$. The experimental result was measured at room temperature by Zachary Hood. The simulated results were based on the the optimized O1 and O2 structures in the static-lattice approximation and generated using the Mercury [24] software package. The “Sim: O1 model” and “Exp: Hood” results were previously presented in Ref. [15].

Figure 7

Ball and stick model of a unit cell of the disordered cubic structure of ${\mathrm{Li}}_2\mathrm{OHCl}$ and ${\mathrm{Li}}_2\mathrm{OHBr}$. Ball colors gray, red, blue, and green indicate Li, O, H, and Cl or Br sites, respectively. The shading of the Li sites indicates their fractional (two-thirds) occupancy.

Figure 8

Histogram plots of static-lattice internal energies ${\left\{u_s^{\mathrm{SL}}\right\}}_{{\mathcal{S}}^c}$ for the cubic phases of ${\mathrm{Li}}_2\mathrm{OHCl}$ (upper panel) and ${\mathrm{Li}}_2\mathrm{OHBr}$ (lower panel). The static-lattice energies are given relative to their lowest energy values and have units of eV/supercell. The histograms are normalized so that their integral is equal to ${\mathcal{S}}^c=10000$. This sample set was constructed from randomly occupying the Li sublattice and randomly choosing the OH bond orientations in the $2\times 2\times 2$ simulation cell as described in the text.

Figure 9

Detailed plot of histograms shown in Fig. 8 for the sampled energies of the cubic phases of ${\mathrm{Li}}_2\mathrm{OHCl}$ and ${\mathrm{Li}}_2\mathrm{OHBr}$. The lower blue panels show the 20 lowest static-lattice energies $\left\{u_s^{\mathrm{SL}}\right\}$, while the upper red panels show the corresponding full energies $\{e_s\left(T\right)\equiv u_s^{\mathrm{SL}}+f_s^{\mathrm{vib}}\left(T\right)\}$ evaluated at $T=300$ K. In each plot the energy scale is adjusted relative to $u_1^{\mathrm{SL}}$ or $e_1\left(T\right)$, as appropriate.

Figure 10

Helmholtz free energy differences [Eq. (18)] in units of eV per formula unit of the disordered cubic and ordered orthorhombic phases of ${\mathrm{Li}}_2\mathrm{OHCl}$ and ${\mathrm{Li}}_2\mathrm{OHBr}$. The upper and lower panels show $\mathrm{\Delta }F$ based on the O1 and O2 orthorhombic structures, respectively. The full lines represent the values obtained using the ${\mathrm{\Omega }}^c$ values given in Table 2 while the dashed lines indicate the range of values due to the estimated standard errors in the estimate of ${\mathrm{\Omega }}^c$.

Figure 11

Components of the free energy difference $\mathrm{\Delta }F\left(T\right)$ for ${\mathrm{Li}}_2\mathrm{OHCl}$ (upper panel) and ${\mathrm{Li}}_2\mathrm{OHBr}$ (lower panel) given per formula unit, referenced to O2 orthorhombic structures of both materials, evaluated from Eq. (20). The full green lines were evaluated using the estimated values of ${\mathrm{\Omega }}^c$ listed in Table 2 while the dashed green lines indicated the range of the estimated error in the estimate of ${\mathrm{\Omega }}^c$.

Figure 12

Comparison of the number of microcanonical configurations $\mathcal{M}$, calculated from the analytic equations (black), and calculated from the disordered-subspace optimized value determined from the right-hand side of Eq. (32) (red), evaluated for the one-dimensional Ising model with $n=10$, and sample size $\mathcal{S}=10\mathrm{\Omega }$.

Figure 13

Helmholtz free energy differences [Eq. (18)] in units of eV per formula unit of the disordered cubic and ordered orthorhombic phases of ${\mathrm{Li}}_2\mathrm{OHCl}$ and ${\mathrm{Li}}_2\mathrm{OHBr}$ analogous to results presented in Fig. 10 but evaluated within the disordered-subspace formalism.

Figure 14

Components of the free energy difference $\mathrm{\Delta }F\left(T\right)$ for ${\mathrm{Li}}_2\mathrm{OHCl}$ (upper panel) and ${\mathrm{Li}}_2\mathrm{OHBr}$ (lower panel) given per formula unit, referenced to O2 orthorhombic structures of both materials, analogous to the results presented in Fig. 11 but evaluated within the disordered-subspace formalism.

Figure 15

Histogram of duplicate pair values (doubly counted) $d$ for 512 000 sets of configurations ${\left\{{\mathbf{V}}_s\right\}}_{\mathcal{S}}$ for sample size $\mathcal{S}=10000$ representing the ideal Li sublattice of the ${\mathrm{Li}}_2\mathrm{OHCl}/\mathrm{Br}$ system for $n=2$ supercells.