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Unraveling the Mystery of Aperiodic Patterns: A Chemical Solution to the Einstein Problem
Can a single shape be used to tile a surface in a manner that the pattern never repeats? This intriguing question, known as the “Einstein problem,” received a mathematical solution in 2022. Recently, researchers from Empa have explored a chemical answer, revealing a molecule that organizes itself into complex, non-repeating patterns on a surface, potentially leading to unique physical properties.
The Einstein problem is a fascinating intersection of mathematics and practical tiling. Despite its name, it does not relate to the physicist Albert Einstein but rather poses the question: Is it possible to cover an infinite surface seamlessly using just one shape (referred to as an “Einstein”) without ever creating a repeating pattern? This concept was first illustrated mathematically by amateur mathematician David Smith, who identified a “proto-tile” that accomplishes this feat.
Within this context, Empa researcher Karl-Heinz Ernst, primarily focused on chemical processes rather than mathematics, found his attention drawn to the Einstein problem when his doctoral student, Jan Voigt, presented unexpected experimental results involving crystal formations. Instead of forming expected regular structures on a silver surface, certain molecules created irregular and aperiodic patterns, defying prior expectations. Furthermore, when the experiments were repeated, different aperiodic arrangements emerged each time.
Initially, Ernst and Voigt believed they might have encountered an experimental anomaly. However, subsequent investigations confirmed the accuracy of their observations. Their findings were recently published in the journal Nature Communications.
Exploring Chirality and Its Implications
Central to their research is the concept of chirality – a property exhibited by many organic molecules characterized by “handedness.” Chiral molecules, while chemically identical, are distinguishable and cannot be converted into one another, analogous to left and right hands. This property holds significant implications in pharmaceuticals; more than half of all modern drugs are chiral, necessitating precise control over their structure to ensure effectiveness and safety.
Thus, understanding how to manage chirality during the synthesis of organic molecules is a high priority in the field of chemistry. One approach involves the crystallization of chiral molecules, a method that is cost-efficient and widely practiced, though still not fully understood. The researchers aimed to delve deeper into this topic using a specific molecule known for its ability to change handedness at room temperature, a rarity among chiral substances.
“We anticipated that the molecules would organize themselves in a crystal structure according to their handedness,” stated Karl-Heinz Ernst, “alternating or grouping by their handedness.” Contrary to expectations, the molecules formed triangles of varying sizes, resulting in spirals across the surface—an irregular and aperiodic structure that initially appeared erroneous.
Cracking the Code of Molecular Patterns
After considerable investigation, Voigt and Ernst pieced together the molecular organization, combining insights from physics and mathematics with hands-on experimentation using puzzle pieces that mimicked molecular arrangements. The molecules did not align randomly; instead, they periodically formed triangles ranging from two to fifteen units per side, with a dominant triangle size present in each experiment, accompanied by slightly larger and smaller triangles.
According to Ernst, the molecules aim to occupy the silver surface as densely as possible under the experimental conditions, which is energetically favorable. Yet, because of their handedness, the triangles cannot align perfectly at the edges and need to offset slightly. This leads to the necessity of additional triangles to achieve maximum surface coverage, creating defects that can initiate spiral formations.
The Role of Entropy in Pattern Formation
Surprisingly, while defects are typically seen as energetically unfavorable, they can facilitate a denser arrangement of triangles, compensating for any energy costs incurred. This imbalance explains why identical patterns were never replicated during experiments; the concept of entropy plays a pivotal role here, determining the variability of patterns as energy levels remain constant.
Implications for Future Research
With the mystery of the “molecular Einstein” unraveled, the question remains: what are the broader implications of these findings? Ernst notes that surfaces exhibiting atomic or molecular-level defects can yield unique and significant properties. In the case of their aperiodic surface, it is hypothesized that electron behavior might differ, potentially opening new avenues in physics. However, further investigation will be needed, especially under various magnetic field influences on different surfaces. As Ernst, now retired, reflects on the future work, he acknowledges a healthy respect for the complexities of physics.
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