The molecular einstein

Standing at the intersection between mathematics and the tiler’s trade is the so-called einstein problem. Despite its name, this mathematical question has nothing to do with the Nobel Prize winner Albert Einstein. It asks: Can you seamlessly tile an endless surface with a single shape (an “einstein”) in such a way that the resulting pattern is never repeated? Such a “proto-tile” was first discovered in 2022 by the English amateur mathematician David Smith.

Empa researcher Karl-Heinz Ernst is neither a mathematician nor a tiler. As a chemist, he researches the crystallization of molecules on metal surfaces. He never expected to deal with the einstein problem in his professional life – until his doctoral student Jan Voigt approached him with the unusual results of an experiment. When a certain molecule crystallized on a silver surface, instead of the expected regular structure, irregular patterns were formed that never seemed to repeat themselves. Even more surprising: Each time he repeated the experiment, different aperiodic patterns emerged.

Like all good researchers, Ernst and Voigt initially suspected an experimental error. But it soon became clear that the strange finding was real. The next step was to find out why the molecules behaved in such a unique way. The researchers recently published the answer to this question in the journal Nature Communications.

Unexpected effects

Ernst and Voigt are interested in so-called chirality, the “handedness” that characterizes many organic molecules. Although chiral structures are chemically identical, they cannot be rotated into one another – similar to our right and left hands. This property is particularly important in the pharmaceutical industry. More than half of all modern medicines are chiral. Since biomolecules such as amino acids, sugars and proteins in our body all have the same handedness, active pharmaceutical ingredients must also be chiral. A drug with the wrong handedness is ineffective at best and at worst even harmful.

Controlling handedness during the synthesis of organic molecules is therefore of enormous interest in chemistry. One of the possibilities is the crystallization of chiral molecules. It is cheap, effective and widely used – and yet not fully understood. The Empa researchers originally wanted to further this understanding with their experiment. To do this, they took a very special molecule, one that easily changes its handedness at room temperature – something that most chiral molecules practically never do.

“We expected the molecules to arrange themselves in the crystal according to their handedness,” explains Karl-Heinz Ernst, ”that is, either alternating or in groups with the same handedness.” Instead, the molecules seemingly randomly arranged themselves into triangles of different sizes, which in turn formed irregular spirals on the surface – the non-repeating or aperiodic structure that the researchers initially thought was a mistake.

From puzzle pieces to physics

After a lot of puzzling, Voigt and Ernst finally managed to decipher the molecular patterns – not only through physics and mathematics, but also by trying them out with actual puzzle pieces on the computer or even at home at the kitchen table. The arrangement of the molecules is not completely random. They form triangles that measure between two and 15 molecules per side. In each experiment, one triangle size dominated. What’s more, triangles one size larger and one size smaller were also represented – but no others.

“Under our experimental conditions, the molecules ‘want’ to cover the silver surface as densely as possible because this is the most energetically favorable outcome,” explains Ernst. “However, due to their chirality, the triangles they form do not fit together exactly at the edges and have to be slightly offset.” The smaller and larger triangles are needed to fill the surface as efficiently as possible. This arrangement also creates defects in some places – small inconsistencies or holes that can become the center of a spiral.

Entropy decides

“Defects are actually unfavorable in terms of energy,” Ernst continues. “In this case, however, they enable a denser arrangement of the triangles, which compensates for the ‘lost’ energy.” This balance also explains why the researchers never found the same pattern twice: If all patterns are the same in terms of their energy cost, entropy decides.

The mystery of the “molecular einstein” has been solved – but how does this insight benefit us? “Surfaces with defects at the atomic or molecular level can have unique properties,” explains Ernst. “For an aperiodic surface like ours in particular, it has been predicted that the electrons in it would behave differently and that this could give rise to a new kind of physics.” To investigate this, however, the aperiodic molecule would have to be studied under the influence of magnetic fields on a different surface. Karl-Heinz Ernst, who has recently retired, is leaving this task to others. “I have a little too much respect for physics,” smiles the chemist.