I'm reading an article about bi layered membranes which state that for the free energy function
$f(\theta) = \theta \ln \theta + (1-\theta)\ln(1-\theta) + \chi \theta (1-\theta)$
Where $\phi_i$ is the mole fraction of certain types of lipids in layer $i$ and $\chi$ is the energy of interaction between two lipids molecules composing the layer. The article states that a critical point exist at $\chi=2$ and $\theta=0.5$. Reading in wikipedia, I concluded that removing the layers interaction part, I simply need to take the second and third derivative of the free energy of the membrane. How do I do this with the interactions? Where can I read about this?