Higher order bending

Klosgen, Beate, Membrane Roughness and Dispersive Phase as Effects of Higher-order Bending in Fluid Membranes, 6, 243 see also Thimmel, Johannes, 6, 253. 

Fluid membranes are molecularly thin two-dimensional objects in three-dimensional space. Their extreme deformability distinguishes them from liquid crystals and other three-dimensional systems. Moreover, their fluidity and complex internal structure sets them apart from polymers. The intervention of higher order bending elasticity and its consequences for membrane shape and fluctuations seem to be related to these characteristic properties. 

Membrane Roughness and Dispersive Phase as Effects of Higher-order Bending in Fluid Membranes... 

Whitney and Pagano [6-32] extended Yang, Norris, and Stavsky s work [6-33] to the treatment of coupling between bending and extension. Whitney uses a higher order stress theory to obtain improved predictions of a, and and displacements at low width-to-thickness ratios [6-34], Meissner used his variational theorem to derive a consistent set of equations for inclusion of transverse shearing deformation effects in symmetrically laminated plates [6-35]. Finally, Ambartsumyan extended his treatment of transverse shearing deformation effects from plates to shells [6-36]. 

Thus, the metric elements Mtj that underlie the geometry of Ms themselves become geometrical vectors My) of Ms, if the higher-order derivative vectors mLj (or conjugate m-) are known. This testifies to the rather mind-bending mathematical richness of thermodynamic geometry. 

In Fig. 3 c the schematic volume-temperature curve of a non crystallizing polymer is shown. The bend in the V(T) curve at the glass transition indicates, that the extensive thermodynamic functions, like volume V, enthalpy H and entropy S show (in an idealized representation) a break. Consequently the first derivatives of these functions, i.e. the isobaric specific volume expansion coefficient a, the isothermal specific compressibility X, and the specific heat at constant pressure c, have a jump at this point, if the curves are drawn in an idealized form. This observation of breaks for the thermodynamic functions V, H and S in past led to the conclusion that there must be an internal phase transition, which could be a true thermodynamic transformation of the second or higher order. In contrast to this statement, most authors... 

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