5. Thermodynamics of mixed monolayers at the water-air interface

The monolayer deposited can be made of two or more surface active molecules at varying composition. In this chapter, we are concerned with mixed monolayers of glycolipids and other membrane components including membrane phospholipids such as dipalmitoylphosphatidylcholine (DPPC), dimyristoylphosphatidycholine (DMPC), distearoylphosphatidylcholine (DSPC), dioleoylphosphatidylcholine (DOPC), dipalmitoylphosphatidylethanolamine (DPPE) and others. Three main types of interactions exist between such molecules in the mixed monolayer, van der Waal attractions between the hydrocarbon chains, and hydrogen-bonding and ionic interaction between the head-groups. These interactions, especially the ionic interactions, determine the stability of the mixed monolayer and are also responsible for the deviations from ideality in mixing. A quantitative study of such forces can be done by applying the concepts of thermodynamics.

The excess Gibbs free energy of mixing, ΔGexc is one such measure which helps determine the stability of the mixed monolayer. ΔGexc can be determined by using the following relation.

$$
\Delta \mathbf{G}^{\rm exc} = \int\_0^\pi (A\_{1,2} - \mathbf{x}\_1 A\_1 - \mathbf{x}\_1 A\_2) d\pi. \tag{3}
$$

A1,2 is mean molecular area of the mixed monolayer, A<sup>1</sup> and A<sup>2</sup> are the molecular areas of individual monolayer components, and X<sup>1</sup> and X<sup>2</sup> are their mole fractions, respectively. The term inside parenthesis is the excess surface area. Negative values of ΔGexc indicate favorable interactions resulting in a more stable monolayer and positive ΔGexc indicates unfavorable interactions and possible phase separation. Furthermore, one could calculate a free energy of mixing, ΔGmix using the following equations:

$$
\Delta G^{\rm mix} = \Delta G^{\rm id} + \Delta G^{\rm exc} \tag{4}
$$

ΔGid is the ideal value that can be calculated using

$$
\Delta G^{id} = RT(\mathbf{x}\_1 \ln \mathbf{x}\_1 + \mathbf{x}\_2 \ln \mathbf{x}\_2) \tag{5}
$$

where R and T are gas constant and temperature respectively. Phase separation may also occur upon compression in mixed monolayers, resulting in coexisting phases of different composition. Occurrence of critical points and azeotropes in two-dimensions is also possible.
