Length-scale cutoff range

Here, the actual thickness of adsorbed molecule layers /ad is obtained by multiplying n by adsorbed molecule diameter a0. The actual thickness range in which the fractal geometry is satisfied represents the length-scale cutoff range of fractality.116,118 Here, the value of the lower (inner) cutoff length tadjmfa is equal to... 

Therefore, we can define a length scale range where the rough surface shows the fractal behavior by determining the inner and outer cutoffs using Eq. (28).121 In other words the fractal dimensions determined from the power relation between 7peak and... 

As an essential feature of these theories, crossover behavior is governed by two physical parameters [317] (1) a scaled coupling constant u which reflects the strength and range of the intermolecular forces as represented by 0 and (2) a cutoff wave number A which is assumed to be inversely proportional to a structural length When = 0, one has only one length scale, and one recovers the Ginzburg number with Afa oc (wA)2. 

If the lateral tension X vanishes, the second bracket becomes unity. This form corresponds to the famous Helfrich potential [53], The numerical prefactor 6fe2 0.1 can be estimated by Monte-Carlo simulations [54]. In the presence of a lateral tension Z, which pulls at the membrane, the power law decay of the first bracket is cutoff by the second factor at a length scale l = (4br/Z) [55,52]. The best choice for b in this case is 6 = 1 jin. The form of eq. (7) covering the full range of tension and separation values has been derived using a simple self-consistent picture [50]. 

Where a is a microscopic lower distance cutoff = 5k, Thus, one expects that if kc/kfiT 1, which is what is found in most single lipid or surfactant membranes, (e.g. ke/kfiT ranges between 20 and 50), the membrane is flat on macroscopic length scales and fluctuations are not important. These are the class of rigid membranes. On the other hand, when kc/kgT = 1 we expect that the membranes will be flexible and layer undulations will be important. We now consider the effect that, thermal fluctuations have on the interlayer interactions for multilayers consisting of flexible membranes. We shall return to the role of the persistence length in understanding the ultimate stability of extremely dilute phases in the next section. 

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