van der Waals energy of lecithins in the ripple phase

The van der Waals energy of a two-dimensional array of tilted alkyl chains is evaluated numerically by summing the contributions of pairs of ${\mathrm{CH}}_2$ groups and water molecules. The energy is found to be lower for a system with a spatially oscillating tilt angle theta than for a flat layer. The cost in free energy of these oscillations is estimated by assuming that where the magnitude of theta is reduced, there is an increase in chain area and hence in chain fluidity. It is found that the free-energy reduction due to the van der Waals interaction is sufficient to compensate for increased chain fluidity at temperatures a few degrees below the chain-melting temperature. Thus the van der Waals energy provides a plausible physical mechanism for the occurrence of the ripple phase. The surface profile predicted is in good agreement with that observed in the $P_{\beta }$ or ripple phase with electron microscopy and characterized by x-ray diffraction. An analytic expression for a negative term in the free energy of the form z’z’ ’ ’, where z’=tantheta is derived on the basis of this physical mechanism.