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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?

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Which article are you reading? –  Qmechanic Oct 6 '13 at 9:11

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removing the layers interaction part, I simply need to take the second and third derivative of the free energy of the membrane

What do you mean by “removing the interactions”? The critical point is the point where the second and third derivatives of $f(\theta)$ are equal to zero. This gives directly the conditions you cited:

              enter image description here

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