# Analytic Free-Energy Expression for the 2D-Ising Model and Perspectives for Battery Modeling

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory: Statistical Thermodynamics of the Ising Model

## 3. Materials and Methods

#### 3.1. Free Energy of Independent Ising Chains

#### 3.2. Inclusion of Inter-Chain Interactions

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DOS | Density of state |

FBC | Free boundary conditions |

PBC | Periodic boundary conditions |

MC | Monte Carlo |

LT | Low-temperature |

HT | High-temperature |

SMI | Shannon measure of information |

## Appendix A

#### Appendix A.1

**Figure A1.**Explicit examples for simple 1D-Ising models (

**a**) and 2D-Ising models (

**b**) with free boundary conditions (FBC) and periodic boundary conditions (PBC). Color scheme: black spheres correspond to spins in the up-state (${s}_{i}=+1$), white spheres to spins in the down-state (${s}_{i}=-1$). For 1D and 2D, the given values of the Hamiltonian H associated with the presented configurations were calculated according to Equations (A1a), (A1b) and Equations (A2a)–(A2c), respectively.

#### Appendix A.2

**Figure A2.**Reduced entropy $S/{k}_{\mathrm{B}}$ as function of temperature for a couple of quadratic ($N\times N$) 2D-Ising systems with free boundary conditions (FBC): $2\times 2$ (blue), $3\times 3$ (green), $4\times 4$ (orange), $5\times 5$ (red). The low- and high-temperature limiting values are given by $S\left(0\right)/{k}_{\mathrm{B}}=ln2$ and $S(\infty )/{k}_{\mathrm{B}}={N}_{\mathrm{spins}}\xb7ln2$, respectively. Full colored lines refer to the calculation based on the Gibbs-Helmholtz equation (Equation (A3c)), while black dashed lines refer to the calculation based on the SMI (Equation (A10)).

#### Appendix A.3

**Figure A3.**All ${\Omega}_{\mathrm{tot}}={2}^{{N}_{\mathrm{spins}}}=16$ possible configurations of the minimal $2\times 2$-Ising model (FBC). Color scheme: black spheres correspond to spins in the up-state (${s}_{i}=+1$), white spheres to spins in the down-state (${s}_{i}=-1$). The numbering scheme is shown in the upper left corner. The configurations are classified according to the order parameter ${n}_{\mathrm{up}}$, denoting the number of spins in the up-state with ${n}_{\mathrm{up}}=\left(\right)open="\{"\; close="\}">0,1,\dots ,{N}_{\mathrm{spins}}$. The value of the Hamiltonian H associated with a particular configuration (given in the center of configuration) was calculated according to Equation (A15a).

#### Appendix A.4

#### Appendix A.5

**Figure A4.**Influence of treating the offset-parameter d of $\Delta {A}_{\mathrm{bond}}^{\mathrm{fit}}$ as free adjustable parameter. (

**a**) reduced heat capacity per spin as function of inverse temperature for selected $N\times N$-systems: $2\times 2$ (blue), $40\times 40$ (red), infinite lattice size (black). (

**b**) corresponding model parameters a (blue), b (green), c (orange), d (red) as function of inverse temperature. For the fitting procedure, 200 equidistant points within the interval $\beta J=[0.01,\phantom{\rule{0.166667em}{0ex}}2]$ were applied for system sizes $2\le N\le 20$ without constraining d to the limiting exact Onsager solution. The black dashed-dotted line is shown as guide to the eye and corresponds to the inverse critical temperature in the thermodynamic limit (${\beta}_{c}J={({k}_{\mathrm{B}}{T}_{c}/J)}^{-1}\approx 0.44$) as given by Equation (8).

**Figure A5.**Influence of increased temperature-resolution in the parameter optimization step: reduced heat capacity per spin as function of inverse temperature for selected $N\times N$-systems: $2\times 2$ (blue), $5\times 5$ (cyan), $10\times 10$ (green), $20\times 20$ (orange), $40\times 40$ (red). The model parameters were parametrized based on exact results of systems with $2\le N\le 20$, using 400 (

**a**) and 800 (

**b**) equidistant points in the temperature-range $\beta J=[0.01,\phantom{\rule{0.166667em}{0ex}}2]$. The parameter d was constrained to the exact limiting value of $\Delta {A}_{\mathrm{bond}}/J$ for $N\to \infty $ according to Equation (18). Colored and black lines refer to exact and modelled results, respectively. The black dashed-dotted line is shown as guide to the eye and corresponds to the inverse critical temperature in the thermodynamic limit (${\beta}_{c}J={({k}_{\mathrm{B}}{T}_{c}/J)}^{-1}\approx 0.44$) as given by Equation (8).

**Figure A6.**Influence of including different numbers of exact reference solutions into the parameter optimization step of the modeling approach: reduced heat capacity per spin as function of inverse temperature for a 2 × 2-FBC-system (

**a**) and 40 × 40-FBC-system (

**b**). Exact reference solutions are shown in black. Colored lines refer to modeling results for which an increasing number of exact reference solutions of N × N-systems were considered in the fitting step: 2 × 2 − 10 × 10 (blue), 2 × 2 − 20 × 20 (green), 2 × 2 − 30 × 30 (orange), 2 × 2 − 40 × 40 (red). This means, for example, the underlying model parameters (a, b, c, d according to Equation (21a)) for the blue curve in the left graph which represents the prediction for c* for a 2 × 2-FBC-system were optimized from fitting to exact reference solutions of systems 2 × 2, 3 × 3, …, 10 × 10. All variants were fitted to 200 equidistant temperature reference points within the interval βJ = [0.01, 2]. The parameter d was constrained to the exact limiting value of ΔA

_{bond}/J for N → ∞ according to Equation (18). The black dasheddotted line is shown as guide to the eye and corresponds to the inverse critical temperature in the thermodynamic limit (β

_{c}J = (k

_{B}T

_{c}/J)

^{−1}≈ 0.44) as given by Equation (8).

## References

- Lenz, W. Beitrag zum Verständnis der magnetischen Erscheinungen in festen Körpern. Phys. Z.
**1920**, 21, 613–615. [Google Scholar] - Ising, E. Beitrag zur Theorie des Ferromagnetismus. Z. Phys.
**1925**, 31, 253–258. [Google Scholar] [CrossRef] - Lipowski, A. Ising model: Recent developments and exotic applications. Entropy
**2022**, 24, 1834. [Google Scholar] [CrossRef] - Pan, A.C.; Chandler, D. Dynamics of nucleation in the Ising model. J. Phys. Chem. B
**2004**, 108, 19681–19686. [Google Scholar] [CrossRef] - De Oliveira, M.; Griffiths, R.B. Lattice-gas model of multiple layer adsorption. Surf. Sci.
**1978**, 71, 687–694. [Google Scholar] [CrossRef] - Stauffer, D. Social applications of two-dimensional Ising models. Am. J. Phys.
**2008**, 76, 470–473. [Google Scholar] [CrossRef] - Tarascon, J.M.; Armand, M. Issues and challenges facing rechargeable lithium batteries. Nature
**2001**, 414, 359–367. [Google Scholar] [CrossRef] - Viswanathan, G.M.; Portillo, M.A.G.; Raposo, E.P.; da Luz, M.G. What does it take to solve the 3D Ising model? Minimal necessary conditions for a valid solution. Entropy
**2022**, 24, 1665. [Google Scholar] [CrossRef] [PubMed] - Fisher, M.E.; Selke, W. Infinitely many commensurate phases in a simple Ising model. Phys. Rev. Lett.
**1980**, 44, 1502. [Google Scholar] [CrossRef] - Potts, R.B. Some generalized order-disorder transformations. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 1952; Volume 48, pp. 106–109. [Google Scholar]
- Scheck, F. Theoretische Physik 5: Statistische Theorie der Wärme; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Newell, G.F.; Montroll, E.W. On the theory of the Ising model of ferromagnetism. Rev. Mod. Phys.
**1953**, 25, 353. [Google Scholar] [CrossRef] - Brush, S.G. History of the Lenz-Ising model. Rev. Mod. Phys.
**1967**, 39, 883. [Google Scholar] [CrossRef] - Landau, D.P.; Wang, F. A new approach to Monte Carlo simulations in statistical physics. Braz. J. Phys.
**2004**, 34, 354–362. [Google Scholar] [CrossRef] - Janke, W. Monte Carlo simulations in statistical physics—From basic principles to advanced applications. In Order, Disorder and Criticality: Advanced Problems of Phase Transition Theory Volume 3; World Scientific: Singapore, 2013; pp. 93–166. [Google Scholar]
- Onsager, L. Crystal statistics. I. A two-dimensional model with an order-disorder transition. Phys. Rev.
**1944**, 65, 117–149. [Google Scholar] [CrossRef] - Kaufman, B. Crystal statistics. II. Partition function evaluated by spinor analysis. Phys. Rev.
**1949**, 76, 1232. [Google Scholar] [CrossRef] - Ferdinand, A.E.; Fisher, M.E. Bounded and inhomogeneous Ising models. I. Specific-heat anomaly of a finite lattice. Phys. Rev.
**1969**, 185, 832. [Google Scholar] [CrossRef] - Beale, P.D. Exact distribution of energies in the two-dimensional Ising model. Phys. Rev. Lett.
**1996**, 76, 78. [Google Scholar] [CrossRef] - Binder, K. Statistical mechanics of finite three-dimensional Ising models. Physica
**1972**, 62, 508–526. [Google Scholar] [CrossRef] - Kim, S.Y. Ising model on L× L square lattice with free boundary conditions up to L= 19. J. Phys. Conf. Ser.
**2013**, 410, 012050. [Google Scholar] [CrossRef] - Carra, J.H.; Murphy, E.C.; Privalov, P.L. Thermodynamic effects of mutations on the denaturation of T4 lysozyme. Biophys. J.
**1996**, 71, 1994–2001. [Google Scholar] [CrossRef] - Karandashev, I.M.; Kryzhanovsky, B.V.; Malsagov, M.Y. Analytical expressions for a finite-size 2D Ising model. Opt. Mem. Neural Netw.
**2017**, 26, 165–171. [Google Scholar] [CrossRef] - Kramers, H.A. Statistics of the two-dimensional ferromagnet. Part I. Phys. Rev.
**1941**, 60, 252–262. [Google Scholar] [CrossRef] - Landau, D.; Binder, K. A Guide to Monte Carlo Simulations in Statistical Physics; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Rehner, P.; Bauer, G. Application of generalized (hyper-) dual numbers in equation of state modeling. Front. Chem. Eng.
**2021**, 3, 758090. [Google Scholar] [CrossRef] - Karandashev, Y.M.; Malsagov, M.Y. Polynomial algorithm for exact calculation of partition function for binary spin model on planar graphs. Opt. Mem. Neural Netw.
**2017**, 26, 87–95. [Google Scholar] [CrossRef] - Implementation 2D-Partition-Function Code. Available online: https://github.com/Thrawn1985/2D-Partition-Function (accessed on 21 April 2023).
- Van Rossum, G. The Python Library Reference, Release 3.8.2; Python Version 3.9.16; Python Software Foundation: Wilmington, DE, USA, 2020. [Google Scholar]
- Virtanen, P.; Gommers, R.; Oliphant, T.E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; et al. SciPy 1.0: Fundamental algorithms for scientific computing in Python. Nat. Methods
**2020**, 17, 261–272. [Google Scholar] [CrossRef] [PubMed] - Ceder, G. Opportunities and challenges for first-principles materials design and applications to Li battery materials. MRS Bull.
**2010**, 35, 693–701. [Google Scholar] [CrossRef] - Chen, G.; Song, X.; Richardson, T.J. Electron microscopy study of the LiFePO4 to FePO4 phase transition. Electrochem.-Solid-State Lett.
**2006**, 9, A295. [Google Scholar] [CrossRef] - Martínez-Herrera, J.; Rodríguez-López, O.A.; Solís, M. Critical temperature of one-dimensional Ising model with long-range interaction revisited. Phys. A Stat. Mech. Appl.
**2022**, 596, 127136. [Google Scholar] [CrossRef] - Persson, K.; Sethuraman, V.A.; Hardwick, L.J.; Hinuma, Y.; Meng, Y.S.; Van Der Ven, A.; Srinivasan, V.; Kostecki, R.; Ceder, G. Lithium diffusion in graphitic carbon. J. Phys. Chem. Lett.
**2010**, 1, 1176–1180. [Google Scholar] [CrossRef] - Xu, B.; Meng, S. Factors affecting Li mobility in spinel LiMn
_{2}O_{4}—A first-principles study by GGA and GGA+ U methods. J. Power Sources**2010**, 195, 4971–4976. [Google Scholar] [CrossRef] - Dyson, F.J. Existence of a phase-transition in a one-dimensional Ising ferromagnet. Commun. Math. Phys.
**1969**, 12, 91–107. [Google Scholar] [CrossRef] - Fröhlich, J.; Spencer, T. The phase transition in the one-dimensional Ising model with 1/r
^{2}interaction energy. Commun. Math. Phys.**1982**, 84, 87–101. [Google Scholar] [CrossRef] - Göpel, W.; Wiemhöfer, H.D. Statistische Thermodynamik; Springer: Berlin/Heidelberg, Germany, 2000. [Google Scholar]
- Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Ben-Naim, A. Information, entropy, life, and the universe. Entropy
**2022**, 24, 1636. [Google Scholar] [CrossRef] [PubMed] - Mueller, M.; Johnston, D.A.; Janke, W. Exact solutions to plaquette Ising models with free and periodic boundaries. Nucl. Phys. B
**2017**, 914, 388–404. [Google Scholar] [CrossRef]

**Figure 1.**Calculated densities of states (DOS) for a couple of quadratic ($N\times N$) 2D-Ising systems: $2\times 2$ (blue), $3\times 3$ (green), $4\times 4$ (orange), $5\times 5$ (red). (

**a**) Free boundary conditions (FBC), (

**b**) periodic boundary conditions (PBC).

**Figure 2.**Thermodynamics of $2\times 2$-Ising model with FBC: (

**a**) fractions (=probabilities) ${p}_{0}$ (black), ${p}_{1}$ (blue) and ${p}_{2}$ (red) of finding the system in states with system energy $-4J$, 0, and $4J$, respectively, as function of temperature according to Equations (6b)–(6d). High-temperature asymptotic values for $T\to \infty $ derived from Equations (6b)–(6d) are given by: ${p}_{0}(\infty )={p}_{2}(\infty )=0.125,{p}_{1}(\infty )=0.75$. (

**b**) Temperature-dependence of free energy A (black) and decomposition into internal energy U (blue) and entropic part $-TS$ (red) according to the Gibbs-Helmholtz equation: $A=U-TS$. All quantities were calculated from the corresponding partition function Equation (4) using Equations (A7a), (A7b) and (A3c). The black dashed line which is shown as guide to the eye corresponds to the high-temperature asymptotic given by $-{k}_{\mathrm{B}}Tln{\Omega}_{\mathrm{tot}}$ with ${\Omega}_{\mathrm{tot}}={2}^{4}=16$.

**Figure 3.**Thermodynamics of $5\times 5$-Ising model with FBC: (

**a**) fractions (=probabilities) ${p}_{i}$ of finding the system in one of the 39 distinct energy states (cf. Figure 1a) as function of temperature according to Equation (6a). The 39 curves are colored according to ascending energy levels from ${E}_{0}=-40J$ (dark blue) to ${E}_{38}=40J$ (dark red). The black dashed lines corresponds to state 20 at the maximum of the DOS distribution in Figure 1a with corresponding energy level ${E}_{19}=0$, showing the highest high-temperature asymptotic value (${p}_{19}(\infty )\approx 0.130$) among all states. (

**b**) Temperature-dependence of free energy A (black) and decomposition into internal energy U (blue) and entropic part $-TS$ (red) according to the Gibbs-Helmholtz equation: $A=U-TS$. All quantities were calculated from the corresponding partition function using Equations (A7a), (A7b) and (A3c). The black dashed line which is shown as guide to the eye corresponds to the high-temperature asymptotic given by $-{k}_{\mathrm{B}}Tln{\Omega}_{\mathrm{tot}}$ with ${\Omega}_{\mathrm{tot}}={2}^{25}\approx 3.36\times {10}^{7}$.

**Figure 4.**Reduced free energy per spin ${a}^{*}=\beta A/{N}_{\mathrm{spins}}$ as function of inverse temperature $\beta J=J/{k}_{\mathrm{B}}T$ for a couple of quadratic ($N\times N$) 2D-Ising systems: $2\times 2$ (blue), $3\times 3$ (green), $4\times 4$ (orange), $5\times 5$ (red). (

**a**) Free boundary conditions (FBC), (

**b**) periodic boundary conditions (PBC). For completeness, the Onsager solution for the case of an infinitely sized system according to Equation (7) is also shown (black curve). Dashed lines in (

**a**) correspond to the low-temperature (LT) approximation as applicable for FBC given by ${a}_{N}^{*\mathrm{LT}}\left({\beta}^{*}\right)=-2{\beta}^{*}\phantom{\rule{0.166667em}{0ex}}\left(\right)open="("\; close=")">1-1/N$ with ${\beta}^{*}=\beta \phantom{\rule{0.166667em}{0ex}}J$ [23].

**Figure 5.**Reduced heat capacity per spin ${c}^{*}=C/{k}_{\mathrm{B}}{N}_{\mathrm{spins}}$ as function of temperature for a couple of quadratic ($N\times N$) 2D-Ising systems: $2\times 2$ (blue), $3\times 3$ (green), $4\times 4$ (orange), $5\times 5$ (red). (

**a**) Free boundary conditions (FBC), (

**b**) periodic boundary conditions (PBC). The corresponding heat capacity for the thermodynamic limit was derived from the Onsager expression for ${a}_{\infty}^{*}$ in Equation (7) (black curve). The black dashed line corresponds to the critical temperature in the thermodynamic limit (${k}_{\mathrm{B}}{T}_{c}/J\approx 2.269$) as given by Equation (8). The left graph also shows the limiting heat capacity for the 1D-Ising model (fine purple dashed-dotted line).

**Figure 6.**Schematic representation of the approach: starting from a system of non-interacting spins for the 2D-topology of interest (=state 0), the spins are first summarized to a set of non-interacting 1D-systems, i.e., Ising chains (=state 1). In the following steps, successive layers of inter-chain interactions (=bonds) between neighboring chains are added sequentially, resulting in a fully connected graph at the final state (here a $5\times 5$-FBC-system).

**Figure 7.**Free energy contribution of a single inter-chain bond $\Delta {A}_{\mathrm{bond}}/J$ as function of lattice size (left) and temperature (right): (

**a**) circles (closed and open) represent exact solutions for $\Delta {A}_{\mathrm{bond}}/J$ calculated according to Equation (17). Dashed lines correspond to fitting results according to Equation (21a) at constant temperature within $\beta J=[0.01,\phantom{\rule{0.166667em}{0ex}}2]$ using 200 equidistant points. Closed (open) circles represent points that were (not) incorporated into the fitting process. The coloring scheme of the lines refers to ascending order in temperature, with blue curves at the bottom corresponding to low temperature (high $\beta $) and red curves at the top to high temperature (low $\beta $). (

**b**) $\Delta {A}_{\mathrm{bond}}/J$ as function of reduced temperature ${k}_{\mathrm{B}}T/J$ for selected $N\times N$-systems: $2\times 2$ (blue), $5\times 5$ (cyan), $10\times 10$ (green), $20\times 20$ (orange), $40\times 40$ (red). The limiting Onsager solution is shown in magenta. Colored curves correspond to exact results according to Equation (17) whereas superimposed black dashed lines refer to modeling results from (

**a**). The shown $40\times 40$-system (red) was not taken explicitly into account into the fitting process but is a prediction.

**Figure 8.**Model parameters a (blue), b (green), c (orange) and d (red) of the generalized hyperbola function for $\Delta {A}_{\mathrm{bond}}^{\mathrm{fit}}/J$ (see Equation (21a)), corresponding to Figure 7a as function of inverse temperature. For the fitting procedure, the c-parameter was constrained to positive values and the parameter d was not taken as free parameter but constrained to the exact limiting value of $\Delta {A}_{\mathrm{bond}}/J$ for $N\to \infty $ according to Equation (18). The black dashed-dotted line is shown as guide to the eye and corresponds to the inverse critical temperature in the thermodynamic limit (${\beta}_{c}J={({k}_{\mathrm{B}}{T}_{c}/J)}^{-1}\approx 0.44$) as given by Equation (8).

**Figure 9.**Reduced free energy (

**a**) and reduced heat capacity per spin (

**b**) as a function of inverse temperature for selected $N\times N$-systems: $2\times 2$ (blue), $5\times 5$ (cyan), $10\times 10$ (green), $20\times 20$ (orange), $40\times 40$ (red). The left graph also includes the limiting Onsager solution (magenta). For the fitting procedure, 200 equidistant points within the interval $\beta J=[0.01,\phantom{\rule{0.166667em}{0ex}}2]$ were applied for system sizes $2\le N\le 20$. Systems with dimensions $21<N<\infty $ represent predictions. Colored curves correspond to exact results while black (dashed) lines represent modeling results based on Equations (16b) and (19b), respectively. The black dashed-dotted line in the right graph is shown as guide to the eye and corresponds to the inverse critical temperature in the thermodynamic limit (${\beta}_{c}J={({k}_{\mathrm{B}}{T}_{c}/J)}^{-1}\approx 0.44$) as given by Equation (8).

**Figure 10.**Relative deviation (in percent) between exact and approximated reduced free energy per spin $({a}_{N\times N}^{*\mathrm{app}.}-{a}_{N\times N}^{*})/\left|{a}_{N\times N}^{*}\right|\times 100\%$ as function of inverse temperature for selected $N\times N$-systems (

**a**) and as function of lattice size for specified inverse temperatures (

**b**). ${a}_{N\times N}^{*\mathrm{app}.}$ denotes the modeled reduced free energy per spin according Equation (16b) while ${a}_{N\times N}^{*}$ corresponds to the exact reference solution, evaluated via the code of Karandashev et al. [27]. For the fitting procedure, 200 equidistant points within the interval $\beta J=[0.01,\phantom{\rule{0.166667em}{0ex}}2]$ were applied for system sizes $2\le N\le 20$. Systems with dimensions $21<N<\infty $ represent predictions. Shown system sizes in (

**a**): $2\times 2$ (blue), $5\times 5$ (cyan), $10\times 10$ (green), $20\times 20$ (orange), $30\times 30$ (dark red), $40\times 40$ (red). The black dashed-dotted line corresponds to the inverse critical temperature in the thermodynamic limit (${\beta}_{c}J={({k}_{\mathrm{B}}{T}_{c}/J)}^{-1}\approx 0.44$) as given by Equation (8). Shown isothermes in (

**b**): $\beta J=0.01$ (blue), $\beta J=0.44$ (cyan), $\beta J=0.50$ (green), $\beta J=0.70$ (orange), $\beta J=1.00$ (red), $\beta J=2.00$ (magenta).

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Markthaler, D.; Birke, K.P.
Analytic Free-Energy Expression for the 2D-Ising Model and Perspectives for Battery Modeling. *Batteries* **2023**, *9*, 489.
https://doi.org/10.3390/batteries9100489

**AMA Style**

Markthaler D, Birke KP.
Analytic Free-Energy Expression for the 2D-Ising Model and Perspectives for Battery Modeling. *Batteries*. 2023; 9(10):489.
https://doi.org/10.3390/batteries9100489

**Chicago/Turabian Style**

Markthaler, Daniel, and Kai Peter Birke.
2023. "Analytic Free-Energy Expression for the 2D-Ising Model and Perspectives for Battery Modeling" *Batteries* 9, no. 10: 489.
https://doi.org/10.3390/batteries9100489