May 05, 2014  
A new mathematical framework to characterize shape of graphene 

(Nanowerk News) Scientists studying graphene’s properties are using a new mathematical framework to make extremely accurate characterizations of the twodimensional material’s shape.  
“The properties of twodimensional materials depend on shape,” said Salvador BarrazaLopez, an assistant professor of physics at the University of Arkansas. “And this mathematical framework allows you to make extremely accurate characterizations of shape. This framework is a novel tool to understand shape in materials that behave as atomthin membranes.”  
Salvador BarrazaLopez, University of Arkansas.  
The mathematical framework being used is known as discrete differential geometry, which is the geometry of twodimensional interlaced structures called meshes. When the nodes of the structure, or mesh points, correspond with atomic positions, discrete differential geometry provides direct information on the potential chemistry and on the electronic properties of twodimensional materials, BarrazaLopez said.  
The application of discrete differential geometry to understand twodimensional materials is an original interdisciplinary development, he said.  
An international research group, led by BarrazaLopez, published its findings on Jan. 8 in the journal ACS Nano ("Quantitative Chemistry and the Discrete Geometry of Conformal AtomThin Crystals").  
A second article describing the research was published March 6 as a rapid communication in the journal Physical Review B ("Graphene's morphology and electronic properties from discrete differential geometry ").  
Graphene was once thought of as existing on a continuum — think of a smooth, continuous “blanket” — but the new mathematical framework allows the consideration of the blanket’s “fibers,” which provides an accurate understanding of the blanket’s properties that complements the continuum perspective.  
“Since twodimensional materials can be easily visualized as meshes, we asked ourselves how these theories would look if you express them directly in terms of the positions of the atoms, bypassing entirely the common continuum approximation,” BarrazaLopez said. “These two papers provide our latest strides towards that direction.”  
The results for the study published in ACS Nano on Jan 8 were obtained through a collaborative effort with Alejandro A. Pacheco Sanjuan at Universidad del Norte, Barranquilla, Colombia; Edmund O. Harriss, a clinical assistant professor of mathematics at the University of Arkansas; Mehrshad Mehboudi, a master’s student in microelectronicsphotonics at the University of Arkansas and Humberto Terrones, then at Pennsylvania State University, now at Rensselaer Polytechnic Institute.  
Collaborators on the rapid communication published in Physical Review B on March 6 were Pacheco Sanjuan; Hamed Pour Imani, a physics doctoral student at the University of Arkansas; Zhengfei Wang at the University of Utah; and Mihajlo Vanevic at the University of Belgrade, Serbia. 
Source: University of Arkansas  
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