Lamellar structure correlation function

Lattice models for bulk mixtures have mostly been designed to describe features which are characteristic of systems with low amphiphile content. In particular, models for ternary oil/water/amphiphile systems are challenged to reproduce the reduction of the interfacial tension between water and oil in the presence of amphiphiles, and the existence of a structured disordered phase (a microemulsion) which coexists with an oil-rich and a water-rich phase. We recall that a structured phase is one in which correlation functions show oscillating behavior. Ordered lamellar phases have also been studied, but they are much more influenced by lattice artefacts here than in the case of the chain models. 

Figure 5.23 The correlation function y x) calculated for the ideal two-phase lamellar structure given in Figure 5.20.

Figure 5.24 The correlation functions calculated for lamellar structures in which the thickness of the lamellae varies according to a Gaussian distribution function. The solid curve is based on the model that gave the solid intensity curve in Figure 5.22, and the broken curve here matches the broken curve in Figure 5.22.

Form and Properties of the One-Dimensional Correlation Function A periodic structure with a given electron density function p r) can be described by means of the long period L, the erystalline thickness and the difference of electronic densities Aq = p. - p. The effect of different lamellar systems on the form and properties of the one-dimensional correlation function was discussed by Strobl and Schneider [36]. Next, systems of two phases and their impact are presented, giving the form and characteristics of the correlation function Ki(r), in accordance with these authors. 

These conclusions have been strengthened by an analysis of suitable correlation functions and structure factors [99]. These results show (Fig. 31) that a cylindrical bottle brush is a quasi-lD object and, as expected for any kind of ID system, from basic principles of statistical thermodynamics, statistical fluctuations destroy any kind of long-range order in one dimension [108]. Thus, for instance, in the lamellar structure there cannot be a strict periodicity of local composition along the z-axis, rather there are fluctuations in the size of the A-rich and B-rich domains as one proceeds along the z-axis, these fluctuations are expected to add up in a random fashion. However, in the molecular dynamics simulations of Erukhimovich et al. [99] no attempt could be made to study such effects quantitatively because the backbone contour length L was not very large in comparison with the domain size of an A-rich (or B-rich, respectively) domain. 

SAXS is a widely used method for the investigation of lamellar structure in the two phase systems. To obtain structural parameters, such as the average crystal separation and crystal thickness, the one dimensional elecbon density correlation function is often used. In syndiotactic poly(propylene) the one dimensional model calculation [24] can be applied since the amorphous phase and crystalline phase form one-dimensional stacks of crystalline lamellae. But in the case of monoclinic iPP, electron microscopy reveals the existence of unique cross-hatched lamellar stracture [25-30], and the applicability of the one dimensional model has been questioned by Albrecht and Strobl [31]. These researchers used SAXS and dilatometry to study structure development in PP, and presented a scheme to check for the feilure of the... 

A quantitative analysis of the lamellar structure can be made by using the correlation function analysis [7,8]. Performing the Fourier transformation on the observed SAXS data in Figure 11.2, the one-dimensional electron density correlation function y(r) can be calculated [7],... 

Figure 9.5 Correlation function K(z) for an ideal lamellar stack and the first and second derivatives K z) and K"(z). Deviations from the ideal lamellar structure lead to a broadening (dotted line) of the 5-Peaks (arrows).

The above theoretical framework was developed in real space. However, computing the correlation functions in real space can be carried out only for simple geometries, such as the lamellar phase [31]. In order to apply the theory to more complex structures, efficient methods other than the direct real-space computation have to be developed. One particularly useful method is the reciprocal-space technique, which utilizes the symmetries of the ordered phases. The key observation is that the mean-field solution w (r) = is a periodic... 

Hansen A, Rolen SH, Anderson K, Morita Y, Caprio J, Finger TE (2003) Correlation between olfactory receptor cell type and function in the channel catfish. J Neurosci 23 347-359 Hansen A, Anderson KT, Finger TE (2004) Differential distribution of olfactory receptor neurons in goldfish structural and molecular correlates. J Comp Neurol 477 347-359 Hansen A, Zielinski BS (2005) Diversity in the olfactory epithelium of bony fishes development, lamellar arrangement, sensory neuron cell types and transduction components. J Neurocytol 34 183-208... 

To correlate the macroscopic properties and the structural changes occurring dtuing deformation the orientation functions evaluated from Eqs. (20 a—c) have been plotted in dependence of strain in Fig. 14. Onc and Asada have discussed in detail the expected changes of infrared dichroism and orientation functions due to the different molecular processes of lamellar orientation and chain unfolding. 

---