^{1,a)}, Wei Bu

^{1}and Alex Travesset

^{1}

### Abstract

We show that the structure factor of water can be obtained from x-ray synchrotron experiments at grazing angle of incidence (in reflection mode) by using a liquid surfacediffractometer. The corrections used to obtain self-consistently are described. Applying these corrections to scans at different incident beam angles (above the critical angle) collapses the measured intensities into a single master curve, without fitting parameters, which within a scale factor yields . Performing the measurements below the critical angle for total reflectivity yields the structure factor of the top most layers of the water/vapor interface. Our results indicate water restructuring at the vapor/water interface. We also introduce a new approach to extract , the pair distribution function (PDF), by expressing the PDF as a linear sum of error functions whose parameters are refined by applying a nonlinear least square fit method. This approach enables a straightforward determination of the inherent uncertainties in the PDF. Implications of our results to previously measured and theoretical predictions of the PDF are also discussed.

We thank C. Lorentz for helpful insights on SPC/E and TIP3P models. The MUCAT sector at the APS is supported by the U.S. DOE Basic Energy Sciences, Office of Science, through Ames Laboratory under Contract No. W-7405-Eng-82. Use of the Advanced Photon Source is supported by the U.S. DOE, Basic Energy Sciences, Office of Science, under Contract No. W-31-109-Eng-38. A.T. is supported by NSF Grant No. DMR-0426597 and partially by DOE-BES through the Ames Lab under Contract no. DE-AC02-07CH11358.

I. INTRODUCTION

II. DETERMINATION OF IN REFLECTION MODE

A. Reflection mode setup

III. A NEW METHOD OF EXTRACTING FROM FOR BULK WATER

IV. EXPERIMENTAL DETAILS

V. RESULTS AND ANALYSIS

A. Bulk water

B. Restructured water surface

VI. DISCUSSION AND SUMMARY

A. Comparison with previous determinations of the water PDF

B. Implications for theoretical models of water

C. Theoretical predictions of the properties of interfacial water

D. Summary

### Key Topics

- X-ray scattering
- 25.0
- Interface structure
- 16.0
- Liquid surfaces
- 13.0
- X-ray diffraction
- 11.0
- Gas liquid interfaces
- 10.0

## Figures

Setup for measuring bulk structure factor of a liquid in reflection mode. The incident beam with a wave vector hits the surface at a fixed angle with respect to the liquid surface. For bulk measurements, is kept above the critical angle for total external reflection. The scattered beam is collected at an angle with respect to the surface and at an angle with respect to the axis in the plane.

Setup for measuring bulk structure factor of a liquid in reflection mode. The incident beam with a wave vector hits the surface at a fixed angle with respect to the liquid surface. For bulk measurements, is kept above the critical angle for total external reflection. The scattered beam is collected at an angle with respect to the surface and at an angle with respect to the axis in the plane.

The coherent form factor (dashed line) and the total incoherent Compton scattering (dotted line) as given in Ref. 23. The solid line shows the sum of the incoherent and coherent scattering .

The coherent form factor (dashed line) and the total incoherent Compton scattering (dotted line) as given in Ref. 23. The solid line shows the sum of the incoherent and coherent scattering .

(a) Scattered intensities vs momentum transfer for various incident-beam angles at . The background at all values is at the level, in the same units as shown in the figure. (b) Same data after normalization by the effective volume of scattering . All the data at different incident angles collapse to a single master curve without any fitting parameters. The data are also corrected by the polarization factor. This is up to a scale factor.

(a) Scattered intensities vs momentum transfer for various incident-beam angles at . The background at all values is at the level, in the same units as shown in the figure. (b) Same data after normalization by the effective volume of scattering . All the data at different incident angles collapse to a single master curve without any fitting parameters. The data are also corrected by the polarization factor. This is up to a scale factor.

(a) (at ) obtained after scaling the data shown in Fig. 3 (circles), the best fit (solid line) and a second fit (dashed line) with different PDF but within the uncertainty range. (b) Two PDFs used to calculate the best fit for the shown in (a). (c) calculations extended to large values using the two PDFs shown in (b) showing the high values of are almost identical.

(a) (at ) obtained after scaling the data shown in Fig. 3 (circles), the best fit (solid line) and a second fit (dashed line) with different PDF but within the uncertainty range. (b) Two PDFs used to calculate the best fit for the shown in (a). (c) calculations extended to large values using the two PDFs shown in (b) showing the high values of are almost identical.

Raw GIXD data after background subtraction above and below the critical incident angle for total reflectivity as indicated. Solid lines are the best fit as discussed in the text. Vertical dashed-dotted lines indicate main peak positions of the bulk and surface structure factor.

Raw GIXD data after background subtraction above and below the critical incident angle for total reflectivity as indicated. Solid lines are the best fit as discussed in the text. Vertical dashed-dotted lines indicate main peak positions of the bulk and surface structure factor.

A comparison of the PDFs from our study and previous studies as indicated. The two PDFs of our study are the same as those shown in Fig. 4(b).

A comparison of the PDFs from our study and previous studies as indicated. The two PDFs of our study are the same as those shown in Fig. 4(b).

Illustration of a side view of the beam footprint (upper panel). As the beam penetrates the bulk, the center of the footprint is shifted along . The top view of the beam footprint (, middle panel) shows the cross section with the footprint of an outgoing beam at angle . The lower panel shows a footprint of the incident beam at a finite value.

Illustration of a side view of the beam footprint (upper panel). As the beam penetrates the bulk, the center of the footprint is shifted along . The top view of the beam footprint (, middle panel) shows the cross section with the footprint of an outgoing beam at angle . The lower panel shows a footprint of the incident beam at a finite value.

Attenuation length for an external incident beam (solid line) and for an internal incident beam (dashed line) at the vapor/water interface, calculated by Eq. (A3) for a x-ray beam. The two curves converge at angles larger than the critical angle for total reflection. (Arrow indicates the location of the critical angle.)

Attenuation length for an external incident beam (solid line) and for an internal incident beam (dashed line) at the vapor/water interface, calculated by Eq. (A3) for a x-ray beam. The two curves converge at angles larger than the critical angle for total reflection. (Arrow indicates the location of the critical angle.)

Effective volume of scattering using Eq. (A2) at various angles of incident beam and . The calculation is for a beam and slit parameters , , .

Effective volume of scattering using Eq. (A2) at various angles of incident beam and . The calculation is for a beam and slit parameters , , .

Illustrations for attenuation factors and the integration range of Eq. (B3).

Illustrations for attenuation factors and the integration range of Eq. (B3).

(a) data from Ref. 14 (circles). Solid and dashed lines are the best fits using the method described in Sec. III. (b) Two extracted PDFs from (a) that fit the data equally well. Despite the higher range of the measured values, possible uncertainty in height, width, and position of the first peak in is evident.

(a) data from Ref. 14 (circles). Solid and dashed lines are the best fits using the method described in Sec. III. (b) Two extracted PDFs from (a) that fit the data equally well. Despite the higher range of the measured values, possible uncertainty in height, width, and position of the first peak in is evident.

## Tables

Parameters that generate the best-fit calculated structure factor using Eq. (7).

Parameters that generate the best-fit calculated structure factor using Eq. (7).

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