The equilibrium shape of a crystal is a fundamental property influencing its various behaviors, including catalytic, light-emitting, sensing, magnetic, and plasmonic properties. For crystals with known surface/edge energies for different directions, the Wulff construction provides a method to predict the shape. However, for crystals lacking sufficient symmetry, the edge energy becomes undefinable, posing a significant theoretical hurdle. This paper addresses this challenge by introducing a novel method that utilizes auxiliary edge energies to predict the equilibrium shape of 2D crystals, even those with very low symmetry. The study’s significance lies in its ability to predict the shape of crystals previously inaccessible to theoretical prediction, offering a significant advancement in computational materials science. The approach moves beyond the limitations of traditional methods based on surface energy minimization, which often fail for low-symmetry materials. By using a combination of well-planned computations and auxiliary parameters, the authors demonstrate a way to overcome the inherent indeterminacy associated with undefined edge energies. This research opens avenues for understanding and predicting the shapes of a vast array of materials not previously amenable to theoretical analysis, with potential implications across numerous fields.

The Wulff construction, a cornerstone of crystal physics, allows for the prediction of crystal shape based on angle-dependent surface energy. However, this method fails when the crystal lacks sufficient symmetry, leading to undefinable edge energies. Previous studies by Cahn and colleagues highlighted the gauge invariance of surface energy, demonstrating that certain changes to angle-dependent surface energy can yield the same Wulff shape. This signifies that for low-symmetry crystals, the determination of surface energy is fundamentally impossible. While this paradox highlighted the limitations of existing theoretical approaches, it did not offer a solution for predicting the shape of such crystals. Existing methods for defining edge energy, such as using ribbons or symmetric polygons, only work for crystals with sufficient symmetry to create samples with identical edges. The lack of a generalized method for low-symmetry crystals leaves a significant gap in theoretical understanding of crystal morphology. This paper aims to fill this gap by proposing a new methodology that addresses the issue of undefinable edge energies.

The authors propose a methodology that uses auxiliary edge energies to predict the equilibrium shapes of 2D crystals, even in the absence of sufficient symmetry to define edge energies directly. This approach centers on defining basic edges along Bravais lattice vectors and diagonals. The total energies of polygons constructed using these basic edges are calculated (e.g., using density functional theory, DFT). These energies are then used to set up a system of linear algebraic equations relating the basic edge energies. Since the number of unknowns (edge energies) exceeds the number of independent equations, the system is underdetermined. To address this, the authors introduce auxiliary parameters (closure equations) that constrain the system and permit its solution. Crucially, they demonstrate that the predicted shape remains independent of the choice of auxiliary parameters, thus revealing the true equilibrium shape. The process involves several key steps: 1. **Identifying Basic Edges:** Determining non-equivalent basic edges from the crystal structure, forming the basis for constructing polygons. 2. **Energy Computations:** Computing the total energies of chosen polygons (ribbons and triangles) using DFT or similar methods. This yields a system of equations relating the basic edge energies. 3. **Introducing Closure Equations:** Augmenting the underdetermined system of equations with one or more closure equations to introduce additional constraints and enable solving for edge energies. 4. **Interpolation:** Employing an interpolation ansatz to obtain a continuous function representing the edge energy as a function of angle. 5. **Wulff Construction:** Using the calculated edge energy function in the Wulff construction to predict the equilibrium crystal shape. The authors apply this methodology to both hypothetical (Y- and γ-crystals) and real materials (SnSe and AgNO2). For binary and ternary compositions, they account for chemical potentials, which influence the RHS of the system of equations. The interpolation ansatz used for the edge energy is ε(α) = ε₀cos(α + C), where ε₀ is the amplitude and C is the phase, determined by lattice geometry and basic edge energies. The robustness of the method is tested against various choices of auxiliary parameters, showcasing the invariance of the predicted shape. The method was also validated by comparing the predicted shapes with experimental data.

The research demonstrates that the equilibrium shape of 2D crystals, even those with low symmetry and undefinable edge energies, can be accurately predicted using a combination of DFT calculations, a system of linear algebraic equations, and closure equations involving auxiliary parameters. This approach was successfully validated for the following systems: * **SnSe (C2v symmetry):** The method correctly predicted the SnSe shape as a function of chemical potential, mirroring experimental observations of rhombus and truncated rectangle shapes. The variation of the shape with the chemical potential was accurately captured. * **AgNO2 (C1 symmetry, no symmetry):** The highly unusual, needle-like shape of AgNO2 was predicted and confirmed by existing experimental evidence. This confirmed the robustness of the proposed method, even for completely asymmetric systems. The study highlights the following key aspects: * **Indeterminacy and Closure:** The underdetermined nature of the system of equations representing basic edge energies is addressed effectively using auxiliary parameters to provide sufficient constraints. This demonstrates that the equilibrium shape is independent of the auxiliary parameters themselves, proving that the method is robust. * **Gauge Invariance:** The method correctly accounts for gauge invariance, where changes in the edge energy do not affect the Wulff construction shape. This demonstrates the theoretical soundness of the approach. * **Chemical Potential:** The influence of chemical potential on the equilibrium shape was incorporated into the calculations for binary and ternary systems (SnSe, AgNO2). This demonstrated the adaptability of the method to various material conditions. * **Interpolation Ansatz:** The interpolation ansatz used for edge energy proved effective and robust, offering a practical way to obtain the continuous edge energy function. The authors also provided a classification system to rank materials based on the definability of edge energies: 1. Trivial-definable (inversion symmetry allows for direct calculation). 2. Non-trivial-definable (regular polygons can be used to define edge energies). 3. Partially undefinable (only a pair of opposite edges directly definable). 4. Fully undefinable (no symmetry for defining edge energies). The auxiliary edge energy approach is successfully applicable to materials in categories 3 and 4.

This work provides a significant advancement in the theoretical prediction of crystal shapes, particularly for materials with low symmetry and undefinable edge energies. The successful application of the auxiliary edge energy approach to both hypothetical and real materials, including the highly asymmetric AgNO2, validates its general applicability. The robustness of the method is demonstrated by its independence from the choice of auxiliary parameters, highlighting the inherent uniqueness of the predicted shapes. The inclusion of chemical potentials adds to the practical relevance of the approach, allowing for the prediction of shape variations depending on material conditions. This opens new avenues for understanding the morphology of a wide range of materials with low symmetry, which were previously inaccessible to theoretical methods. The findings have significant implications for various fields, including catalysis, light-emission, electronics, sensing, and plasmonics, where crystal shape plays a crucial role.

This paper presents a novel and effective methodology for predicting the equilibrium shapes of 2D crystals, even those with low symmetry and undefinable edge energies. The approach, which involves calculating total energies of polygons, introducing closure equations, and employing an interpolation ansatz, has been successfully validated against experimental data. Future research could extend this method to 3D crystals and explore the influence of various factors such as temperature, substrates, and solvents on crystal morphology. The ability to predict shapes for a wider range of materials promises to be invaluable across diverse scientific and engineering applications.

While the method successfully predicts the equilibrium shapes of various 2D crystals, certain limitations exist. The accuracy of the predictions is inherently linked to the accuracy of the underlying DFT calculations or other computational methods used to determine the total energies of the polygons. The choice of the interpolation ansatz, while effective in the examples presented, might need adjustments for specific materials with complex edge structures. The computational cost of DFT calculations can be significant, particularly for larger systems. Furthermore, the method currently does not explicitly account for kinetic effects during crystal growth which may deviate from pure thermodynamic equilibrium shapes. Future work could improve this by incorporating kinetic models.