Scaling Relations for Fluid Membrane Phases

It has proven extremely difficult to make an exhaustive assessment of all configurations available to the disordered L3 structure. For this reason, the explanation of phase behaviour in terms of curvature moduli K and K remains rather qualitative. There is nevertheless one aspect which is accessible to quantitative prediction, namely the way structural properties vary with dilution. 

These scaling relations are an immediate consequence of the scale invariance noted for the curvature elasticity (5.9) of fluid membranes. An isotropic dilation of factor A transforms Ci and to Ci/A and C2/A, respectively, whilst dA transforms to A dA. The whole thing leaves d ei unchanged. 

Furthermore, as long as the dual configurations of the small and large systems have the same statistical weighting, they contribute equally to their respective free energies F. In other words, the free energy F of fluid membrane phases is also scale invariant. Since F is an extensive quantity, it must take the form 

The free energy per unit volume F/V is then proportional to 

In Sect. 5.5.1, we saw that the free energy density per unit membrane area 

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