# 2D Ising Model for Enantiomer Adsorption on Achiral Surfaces: L- and D-Aspartic Acid on Cu(111)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Langmuir Adsorption Isotherms

## 3. Enantiomer Co-Adsorption: D- and L-Asp on Cu(111)

^{13}CO

_{2}from 1,4-

^{13}C

_{2}-L-Asp and

^{12}CO

_{2}from D-Asp.

^{R&S}[8], Asp/Cu(653)

^{R&S}[11], and Pro/Cu(643)

^{R&S}[9].

## 4. Implications of the 2D Ising Model for Enantiomer and Prochiral Adsorption

_{g}and ꟼ

_{g}are neither chiral nor enantiomers of one another in the gas phase. This renders the model for equilibrium adsorption of prochiral molecules onto achiral surfaces exactly equivalent to the case of the 2D Ising model for which Onsager’s solution applies. In fact, one need not consider adsorption at all, because the interconversion between P and ꟼ can be achieved simply by literally flipping the molecule over to change configurations on the surface. For prochiral adsorbates with $\Delta \Delta {E}_{exch}^{\mathrm{P-\ua7fc}}>0$ and conditions $T<{T}_{c}$, the prochiral gas phase species will form an adsorbed monolayer with arbitrarily high $e{e}_{s}$.

## 5. Implications of the 2D Ising Model for the Origins of Homochirality in Life

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**A**) Illustration of the 2D Ising model for spins interacting through nearest neighbor exchange interaction, $J$, in the presence of an applied magnetic field, $\stackrel{\rightharpoonup}{\mathit{H}}$. (

**B**) Equivalent illustration for adsorbate, $A$, with adsorption energy, $\Delta {E}_{ads}^{A}$, from a gas phase with chemical potential, ${\mu}_{g}^{A}$, onto a square lattice of adsorption sites. The interaction $\Delta \Delta {E}_{exch}^{\mathit{\text{A-A}}}$ occurs between $A$ ’s adsorbed on adjacent nearest neighbor sites. (

**C**) Illustration of enantiomer adsorption at saturation coverage. The difference in adsorption energies is $\Delta \Delta {E}_{ads}^{\mathit{\text{D-L}}}=0$ on an achiral surface. The adsorbate–adsorbate interaction is quantified by the difference in energy between homochiral pairs and heterochiral pairs of adjacent adsorbates. Figure 1A,C are reprinted/adapted with permission from [5]. Copyright 2020, John Wiley and Sons.

**Figure 2.**(

**A**) Plot of $e{e}_{s}$ versus $e{e}_{g}$ for gas phase mixtures of D- and L-Asp in equilibrium with Asp adsorbed on the Cu(111) surface at 460 K: experimental measurements (solid black squares), predictions of Monte Carlo simulations using the 2D Ising model and a 100 × 100 square lattice (open red circles). (

**B**) Plot of the residual, ${\chi}^{2}$, arising from fitting the results of the 100 × 100 Monte Carlo simulation obtained using values of $\Delta \Delta {E}_{exch}^{\mathit{\text{D-L}}}$ spanning the range 2.1 to 2.7 kJ/mole. The red line is a fit of a cubic polynomial to the values of ${\chi}^{2}$, showing the minimum at $\Delta \Delta {E}_{exch}^{\mathit{\text{D-L}}}=2.31$ kJ/mole. Data in Figure 2A are reproduced from [7].

**Figure 3.**Illustrations of the enantiomer distributions on the 100 × 100 square lattice used for 2D Ising model simulation of competitive enantiomer adsorption. Blue sites are occupied by D-enantiomers and orange sites are occupied by L-enantiomers. Simulations were conducted using T = 460 K, $\Delta \Delta {E}_{exch}^{\mathit{\text{D-L}}}=2.31\mathrm{kJ}/\mathrm{mol}$, and values of $e{e}_{g}$ in the range of 0 to 0.63. The values of $e{e}_{g}$ and resulting $e{e}_{s}$ are shown for each MC simulation.

**Figure 4.**Illustration of the 2D Ising model predictions for equilibrium adsorption of prochiral molecules onto an achiral surface at $T>{T}_{c}$ and $T<{T}_{c}$. Note that P and ꟼ are achiral and truly identical in the gas phase because they are free to rotate out of the plane of the page. That degree of freedom is frozen out in the adsorbed state, although the molecules can ‘flip’ between states, but not freely, i.e., there is a barrier to flipping. The 2D Ising model predicts that for $T<{T}_{c}$, the equilibrium adsorbed state can be arbitrarily close to homochiral. Figure 4 reprinted/adapted with permission from [5]. Copyright 2020, John Wiley and Sons.

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**MDPI and ACS Style**

Dutta, S.; Gellman, A.J.
2D Ising Model for Enantiomer Adsorption on Achiral Surfaces: L- and D-Aspartic Acid on Cu(111). *Entropy* **2022**, *24*, 565.
https://doi.org/10.3390/e24040565

**AMA Style**

Dutta S, Gellman AJ.
2D Ising Model for Enantiomer Adsorption on Achiral Surfaces: L- and D-Aspartic Acid on Cu(111). *Entropy*. 2022; 24(4):565.
https://doi.org/10.3390/e24040565

**Chicago/Turabian Style**

Dutta, Soham, and Andrew J. Gellman.
2022. "2D Ising Model for Enantiomer Adsorption on Achiral Surfaces: L- and D-Aspartic Acid on Cu(111)" *Entropy* 24, no. 4: 565.
https://doi.org/10.3390/e24040565