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Extending the capillary wave model to include the effect of bending rigidity: X-ray reflectivity and diffuse scattering

Chen Shen (沈辰)^1,*, Honghu Zhang (张洪湖)², Beate Klösgen ^1,3, and Benjamin M. Ocko ^2,†

¹Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany
²National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA
³Department of Physics, Chemistry and Pharmacy & PhyLife, University of Southern Denmark, Campusvej 55, 5230 Odense, Denmark

^*Contact author: chen.shen@desy.de
^†Contact author: ocko@bnl.gov

Phys. Rev. Research 7, 043016 – Published 3 October, 2025

DOI: https://doi.org/10.1103/znt1-fmx6

Abstract

The surface roughness of a thin film at a liquid interface exhibits contributions of thermally excited fluctuations. This thermal roughness depends on temperature ( $T$ ), surface tension ( $γ$ ), and elastic material properties, specifically the bending modulus ( $κ$ ) of the film. A nonzero $κ$ suppresses the thermal roughness at small length scales compared to an interface with zero $κ$ , as expressed by the power spectral density of the thermal roughness. The description of the x-ray scattering of the standard capillary wave model (CWM), which is valid for zero $κ$ , is extended to include the effect of $κ$ . The extended CWM (eCWM) provides a single analytical form for both the specular x-ray reflectivity (XRR) and the diffuse scattering around the specular reflection, and recovers the expression of the CWM at its zero $κ$ limit. This theoretical approach enables the use of single-shot grazing incidence x-ray off-specular scattering (GIXOS) measurements for characterizing the structure of thin films on a liquid surface. The eCWM analysis approach decouples the thermal roughness factor from the surface scattering signal, providing direct access to the intrinsic surface-normal structure of the film and its bending modulus. Moreover, the eCWM facilitates the calculation of reflectivity at any desired resolution (pseudo-XRR approach). The transformation into pseudo-XRR provides the benefit of using widely available XRR software to perform GIXOS analysis. The extended range of the vertical scattering vector ( $Q_{z}$ ) available with the GIXOS pseudo-XRR approach allows for a higher spatial resolution than with conventional XRR. Experimental results are presented for various lipid systems, showing strong agreement between conventional specular XRR and pseudo-XRR methods. This agreement validates the proposed approach and highlights its utility for analyzing soft, thin films.

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References (70)

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Phys. Rev. Research 7, 043016– Published 3 October, 2025

Vol. 7, Iss. 4 — October - December 2025

Received 6 June 2025
Accepted 1 September 2025

DOI: https://doi.org/10.1103/znt1-fmx6

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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Extending the capillary wave model to include the effect of bending rigidity: X-ray reflectivity and diffuse scattering

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