Lamellar peaks

A longitudinal slice (i.e., parallel to the fiber axis z-axis) through the lamellar reflections showing the lamellar peaks. 

There are two widths to the lamellar peak, the axial width in scans parallel to the fiber-axis and the transverse width in scans perpendicular to the fiber-axis. The axial width of the peak in longitudinal scans (Figure 2) is used to evaluate the coherence length or height of the lamellar stack (IJ using the Scherrer equation"... 

The transverse width of the lamellar peak in azimuthal scans (Figure 4) is used to calculate the size of die lamellae in the equatorial plane using the above Scherrer equation. Alternatively, the intensity distribution in Figure 4 can be plotted as a Guinier plot (Figure S), and the slope of this curve is used to evaluate the diameter of the lamellae according to the equation... 

Variations in the Axial Widths of the Lamellar Peak with x... 

If all the lamellar stacks are p fectly oriented parallel to the fiber-axis, then the lamellar peak would be streak or a layer line of constant axial widdL But in a typical SAS pattern, the width of the layer line increases with the distance from the meridian, as shown In Figure 6. Whole body rotation of the lamellar stack (misorientation) causes the width Azj of the layer line to increase with the distance fit)m the meridian. The rate of such increase in the width is determined by the average angle that the lamellar stack makes with the fiber-axis. This orientation angle O) of the lamellar stacks is calculated using the expression... 

The lamellar reflections are not flat, but are curved i.e., there is a continuous shift in the z-position of the maxima (z ) in the lamellar peaks as a function of x (Figure 1) Because of this curvature, the two-dimensional (2-D) data could not be fitted in Cartesian coordinates. But they are not curved enough to be a circle, hence the polar coordinates ordinarily used in analyzing the wide-angle x-ray diffraction patterns cannot be used either. It appears that the description in elliptical coordinates provides the best fit to the data. This feature of the scattering curve will be analyzed in detail in this paper. 

Two new parametets for describing the SAS data from semictystalline polymers are introduced. These are the ellipticity of the trace of the lamellar peak-maxima, and the orientation parameter determined from the increase in the longitudinal width of the lamellar peak with the distance from the meridional axis. These two parameters along with Ae lamellar spacing, tilt-angle of the lamellar plane, the diameter and the coherence length of the lamellar stack, and the lamellar intensity completely describe the SAS data from oriented semiciystalline polymers. These parameters can be obtained by fitting the 2-D SAS from uniaxially oriented semicrystalline polymers in elliptical coordinates. 

The behavior of the internal energy, heat capacity, Euler characteristic, and its variance ( x ) x) ) the microemulsion-lamellar transition is shown in Fig. 12. Both U and (x) jump at the transition, and the heat capacity, and (x ) - (x) have a peak at the transition. The relative jump in the Euler characteristic is larger than the one in the internal energy. Also, the relative height of the peak in x ) - x) is bigger than in the heat capacity. Conclude both quantities x) and x ) - can be used to locate the phase transition in systems with internal surfaces. 

In this section we will discuss in some detail the application of X-ray diffraction and IR dichroism for the structure determination and identification of diverse LC phases. The general feature, revealed by X-ray diffraction (XRD), of all smectic phases is the set of sharp (OOn) Bragg peaks due to the periodicity of the layers [43]. The in-plane order is determined from the half-width of the inplane (hkO) peaks and varies from 2 to 3 intermolecular distances in smectics A and C to 6-30 intermolecular distances in the hexatic phase, which is characterized by six-fold symmetry in location of the in-plane diffuse maxima. The lamellar crystalline phases (smectics B, E, G, I) possess sharp in-plane diffraction peaks, indicating long-range periodicity within the layers. 

Below the ODT such a label highlights the polymer-polymer interface. A main peak around Q" =0.02 A" corresponding to a lamellar periodocity 2 n/diain with di j =3l5 A is observed. Its visibility results from the asymmetric nature of the diblock. We note the existence of a second order peak, which is well visible at Todt=433 K. At large Q>Q the scattering is dominated by the form factor of the PEP-label in the environment of the deuterated monomers at the interface. This form factor may be described by a Debye function A)ebye( ) (Eq. 3.23). The absolute cross-section for these labels is given by ... 

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