Homogeneous IPMS

The pressure drop over a given length of pipe must be determined by a stepwise procedure, as described for homogeneous flow. The major additional complication in this case is evaluation of the holdup (ipm) or the equivalent slip ratio (S) using one of the above correlations. 

Cut the tissue into small pieces and homogenize in a Potter homogenize at 1000 ipm for 2 min. 

Table 4.1 Geometric properties of some periodic minimal surfaces. The "genus" of each three-periodic minimal surface (IPMS) is the genus of a unit cell of the IPMS (with symmetrically distinct sides). The "symmetry" refers to the crystallographic space group for the surface (assuming equivalent sidesl.liie surfaces are tabulated in order of deviation of the homogeneity index from the "ideal" value of 3/4. ...

Notice that the homogeneity indices of these IPMS are close to those of perfectly homogeneous minimal surfaces. Clearly then, three-periodic minimal surfaces are quasi-homogenous hyperbolic surfaces, in contrast to... 

Figure 4.8 Plot of the relations between the surfactant-water conrposition (characterised by 4>int) and the surfactant parameter for nomud (v/al>l) and reversed (v/al

So far, we have assumed the ideal value of (derived above) for the homogeneity index, H. In reality, this index varies according to the S)mimetry and topology of the IPMS (Table 4.1). If the true values of H are inserted into eq. 4.4, more accurate estimates of the local/global relations can be made for various IPMS. These estimates are plotted in Fig. 4.9 below. 

The relative stability of mesh and IPMS structures is still unclear. For example, the Ri mesophase (of rhombohedral symmetry) in the SDS-water system transforms continuously into the neighbouring bicontinuous cubic phase (Fig. 4.14) [20]. This suggests that this mesophase is a hyperbolic (reversed) bilayer Ijring on a rhombohedral IPMS. Indeed, the rhombohedral rPD surface is only marginally less homogeneous than its cubic counterparts, the P- and D-svu-faces. 

It is surely no coincidence then that the symmetries of the lower temperature blue phases ("BPI" and "BPII") are precisely those of the D-surface and the gyroid - Pn3.m and laid respectively. These IPMS, the D-surface and the gyroid, are the most homogeneous "leaves" upon which the foliation of space is built [60]. 

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