What determines the shape of the temperature-entropy graph?

[iScience]

in my thermo class when we were formalizing the definition of temperature (\frac{1}{T}=\frac{∂S}{∂U}), we drew out all the combinations of various slopes and concavities of the ∂S/∂U graphs.

http://imgur.com/cR4V8K8

The shape of this graph i figure should be a reflection of the inherent nature of the material. so, my question is, what properties of the material determines the shape of this graph?

 

[itaischles]

I think the graph should be monotonically increasing since the temperature is the slope. Furthermore, the second derivative should also be considered since it relates to the heat capacity.