line tension $\tau$ of the interface as calculated in Sect. VII) is given by $$\gamma = \int_{-\infty}^{\infty} {\cal I} \d x$$ where the free energy density ${\cal I}$ includes contributions associated with spatial variations of the concentrations. Defining the “stiffness coefficients” $g_\phi$ and $g_\eta$ for the protein and lipid concentration profiles, respectively, the free energy density ${\cal I}$ is given by $${\cal I}= {\cal G}/{T} +\frac{1}{2} g_{\phi}
\left( \frac{\d \phi}{\d x} \right)^2 +\frac{1}{2} g_{\eta}
\left( \frac{\d \eta}{\d x} \right)^2$$ In the next section we study the bulk phase diagram based on the thermodynamic potential (8). In the subsequent sections we use the simplified expression (13) and determine the concentration profiles $\phi(x)$ and $\eta(x)$ by applying a variational principle to the free energy functional $\gamma$.

The phase diagram
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The phase diagram as a function of the chemical potentials $\mu_{\eta}$ and $\mu$ can be obtained from the thermodynamic potential (8) by minimizing ${\cal G}$ with respect to the order parameters $\eta$ and $\phi$ in the two-phase coexistence region[@note2]. The coexisting solutions, denoted by $(\eta_1,\phi_1)$ and $(\eta_2, \phi_2)$, are determined from the equations $$\mu_{\eta}= 
\left. \frac{\partial {\cal F}}{\partial \eta} \right|_{\eta_1,\phi_1}=
\left. \frac{\partial {\cal F}}{\partial \eta} \right|_{\eta_2,\phi_2}=
\frac{{\cal F}(\eta_1,\phi_1)-{\cal F}(\eta_2,\phi_2)}{\eta_1-\eta_2}$$ $$\mu+\log 2=
\left. 