Temperature dependence bending rigidity of 2D membranes: Graphene as an example

Bending rigidity plays an important role in graphene from mechanical behavior to magnetic and electrical properties. However, it is still in a theoretical debate whether the bending rigidity of graphene increase or decrease with increasing temperature. The liquid membranes renormalization theory is always used to calculate the bending modulus of 2D membrane (graphene) at different temperatures. Although this theory has been successfully used to describe the mechanical behavior of liquid membranes like cell membrane, we point out some possible unsuitable places when it is used to evaluate the temperature effect on the bending rigidity of graphene. The energy difference between the notional planar and pure bending graphene is related to the definition of the bending rigidity directly. Based on this energy variation analysis, we demonstrate that the bending rigidity of graphene increases with increasing temperature. Moreover, we reveal the mechanism is that the configurational entropy plays a crucial role in the variation of the free energy of graphene with increasing temperature. Our approach also paves a way to investigate the temperature effect on the bending rigidity of other 2D materials.