University of Houston Cullen College of Engineering


Cullen College Grad Students Track Down the Elusive Gaussian Modulus

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Fatemeh Ahmadpoor
Fatemeh Ahmadpoor
Matthew Zelisko (left) and Pradeep Sharma
Matthew Zelisko (left) and Pradeep Sharma

Two Cullen College mechanical engineering doctoral students have found a way to determine and understand the often-elusive Gaussian curvature modulus, a critically important property of two-dimensional materials like graphene and biological membranes. Fatemeh Ahmadpoor and Matthew Zelisko, whose work is now published in the journal Physical Review Letters, established a detailed mathematical design that works in conjunction with atomistic simulations to precisely assess this hard-to-find property.

“Our work provides perhaps one of the clearest benchmark results for the Gaussian modulus, especially at finite temperature,” said Ahmadpoor. She and Zelisko performed the research under the supervision of Pradeep Sharma, professor and chair of the mechanical engineering department. Professor Huajian Gao from Brown University collaborated with the UH team on this project. 

It’s significant

If you’ve ever used a pencil (and let’s face it, who hasn’t?) you’ve utilized (in a manner of speaking) one of the most coveted two-dimensional materials, graphene. One of the marvels of this 2D material made of carbon is that it is the lightest, thinnest and strongest conductor of heat and electricity.

Two-dimensional materials are essentially “thin” sheets; so thin, in fact, that their thickness is at the atomistic scale. Not surprisingly, they are highly flexible and bend quite easily. This mechanical characteristic, in addition to other physical and chemical attributes, has opened up an entirely new field of research in the sciences with tantalizing applications that range from next generation electronics to drug delivery, energy harvesters and structural composites. The laboratory isolation of graphene even resulted in the 2010 Nobel Prize in Physics.

Nature also has its own version of 2D materials – the thin “skin” of the cell, known as the biological membrane, is one example. It controls when and what enters (and exits) the cell. In short, biological membranes play a critical role in various biological functions.

Whether it is graphene or squishy biological membranes, the mechanical behavior of these thin sheets is critical for understanding their application as well as various physical and biological phenomena. To fully understand the mechanical behavior of 2D materials, scientists measure the spontaneous curvatures and membrane elasticity, which are determined by mathematical equations called the bending modulus and the Gaussian modulus.

The elusiveness of the Gaussian modulus stems from the difficulty of measuring it. To estimate it, you  must measure the energy required for events that are often not easily controlled in simulations or experiments, such as cellular fission or fusion – the very events which are necessary for the creation of life. Only a handful of works exist that have provided reasonable estimates.

“While we found nothing surprising for biological membranes and simply confirmed past estimates, we find some paradoxical results for graphene. We show that if existing Gaussian modulus estimates of graphene are used, unstable behavior for the material will be predicted, which is not correct,” said Zelisko.

Even more surprisingly, the new estimates found the Gaussian modulus for graphene appears to contradict the well-established notion that this property must be negative.
“The fact that we find a positive Gaussian modulus is truly paradoxical and we are now carrying out follow-up work to explain this,” said Ahmadpoor, who is currently a postdoctoral fellow at Brown University. Zelisko is currently a lecturer at the Cullen College of Engineering.

Their research was partially funded by Sharma’s M.D Anderson Professorship and grants from the National Science Foundation (NSF).

To view the full paper, please visit



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