The integrals determining orientational order in liquid crystals by x-ray diffraction revisited

ABSTRACT

The orientational distribution function f(β) relative to the average orientation direction or director of liquid crystals and related materials can be obtained from the angular variation of the diffracted x-ray intensity distribution I(θ) of the wide-angle or equatorial arcs. The two quantities f(β) and I(θ) are related by an integral equation. Two different integral equations, one by Kratky and the other by Leadbetter and Norris describing the same phenomena, are the basis of the analyses. There has been discussion of which of the equations is correct; however, the form first derived by Kratky is correct. In this paper, solutions to the Kratky form of the equation are presented. Analytical, closed-form expressions for for n = 0, 1, 2 and 3 are presented. These allow calculation of the first three non-zero orientational order parameters. Experimental data are analysed using solutions to the Kratky kernel and the often used Leadbetter–Norris kernel as a series and in integral form, and the method of Lovell and Mitchell, based on symmetry considerations. The results show that the five approaches obtain indistinguishable values for the second- and fourth-order parameters in the range commonly encountered in liquid crystal studies.

Graphical Abstract

KEYWORDS:

• X-ray diffraction
• orientational order
• order parameter
• Kratky integral equation

Acknowledgments

The authors wish to thank the referee for invaluable comments and suggestions that greatly improved this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

We acknowledge the US–Ireland R&D partnership programme of the National Science Foundation award No. DMR-1410649 for it support of this work. This work was also supported by the Basic Energy Science program of the Office of Science, Department of Energy Award No. SC-0001412.