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X-ray reflectivity theory for determining the density profile of a liquid under nanometre confinement.

Perret E, Nygård K, Satapathy DK, Balmer TE, Bunk O, Heuberger M, van der Veen JF - J Synchrotron Radiat (2010)

Bottom Line: The confinement geometry acts like an X-ray interferometer, which consists of two opposing atomically flat single-crystal mica membranes with an intervening thin liquid film of variable thickness.An expression for the reflected intensity as a function of momentum transfer is given.The total structure factor intensity for the liquid-filled confinement device is derived as a sum of contributions from the inner and outer crystal terminations.

View Article: PubMed Central - HTML - PubMed

Affiliation: Paul Scherrer Institut, Villigen PSI, Switzerland.

ABSTRACT
An X-ray reflectivity theory on the determination of the density profile of a molecular liquid under nanometre confinement is presented. The confinement geometry acts like an X-ray interferometer, which consists of two opposing atomically flat single-crystal mica membranes with an intervening thin liquid film of variable thickness. The X-rays reflected from the parallel crystal planes (of known structure) and the layered liquid in between them (of unknown structure) interfere with one another, making X-ray reflectivity highly sensitive to the liquid's density profile along the confinement direction. An expression for the reflected intensity as a function of momentum transfer is given. The total structure factor intensity for the liquid-filled confinement device is derived as a sum of contributions from the inner and outer crystal terminations. The method presented readily distinguishes the confined liquid from the liquid adsorbed on the outer mica surfaces. It is illustrated for the molecular liquid tetrakis(trimethyl)siloxysilane, confined by two mica surfaces at a distance of 8.6 nm.

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Measured and calculated structure factor amplitudes with corresponding electron density profile for TTMSS confined by mica membranes at a distance of 8.6 nm. (a) Measured amplitudes are indicated by the grey dots, amplitudes for the best-fit model by the red solid curve and amplitudes for a deviating model by the black dashed curve. (b) Corresponding best-fit and deviating electron density profiles are indicated by solid red and black dashed curves, respectively.
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fig6: Measured and calculated structure factor amplitudes with corresponding electron density profile for TTMSS confined by mica membranes at a distance of 8.6 nm. (a) Measured amplitudes are indicated by the grey dots, amplitudes for the best-fit model by the red solid curve and amplitudes for a deviating model by the black dashed curve. (b) Corresponding best-fit and deviating electron density profiles are indicated by solid red and black dashed curves, respectively.

Mentions: We now apply the reflectivity theory to the XRR data obtained for confined TTMSS. Specularly reflected intensities I int(q ⊥) were integrated for momentum transfers q ⊥ up to 1.4 Å−1. Using equation (3), with f initially set to 1, values for the corresponding structure factor amplitudes /F exp(q ⊥)/ were derived. Sets of calculated values /F calc(q ⊥)/ were generated for a variety of structure models for the confined and adsorbed liquids as discussed in §3. The measured and calculated sets of values were compared using the logarithmic residual (Hirano et al., 1998 ▶) The model structural parameters, including the number of layers and f, were varied so as to minimize the residual E and thus to find the structure providing the best fit. In order to reduce the number of fitting parameters, the confinement arrangement was taken to be symmetric, the TTMSS layers in the gap were assumed to have equal electron density and width. The following additional constraints were applied: the liquid was not allowed to penetrate the mica, areal densities of the liquid layers were not to exceed the electron density for triangular closest packing of TTMSS molecules (calculated for a molecule diameter of 9.0 Å), and the width of the layers was kept to a lower limit of σ = 2 Å. In total, 23 fitting parameters were used: 12 symmetrical confined Gaussian peaks were fitted, each having a position (six parameters), a width and a height. The widths and heights of the inner eight density peaks were assumed to be equal, which results in two parameters plus four parameters from the boundary layers. The liquid on the outer mica surfaces were fitted with three layers, each having a position, a width and a height (nine parameters). Furthermore, the gap width and the correction factor f were two additional fitting parameters. We note that a number of fitting parameters are correlated, for example the width and the height of the Gaussian peaks. Fig. 6 ▶ shows the best-fit structure factor amplitudes in comparison with the measured values and the corresponding best-fit electron density profiles. All density profiles have been broadened with the experimental resolution (π/q ⊥,max = 2.2 Å) (Fenter, 2002 ▶). The best fit has been achieved for E = 0.30 and f ≃ 0.6.


X-ray reflectivity theory for determining the density profile of a liquid under nanometre confinement.

Perret E, Nygård K, Satapathy DK, Balmer TE, Bunk O, Heuberger M, van der Veen JF - J Synchrotron Radiat (2010)

Measured and calculated structure factor amplitudes with corresponding electron density profile for TTMSS confined by mica membranes at a distance of 8.6 nm. (a) Measured amplitudes are indicated by the grey dots, amplitudes for the best-fit model by the red solid curve and amplitudes for a deviating model by the black dashed curve. (b) Corresponding best-fit and deviating electron density profiles are indicated by solid red and black dashed curves, respectively.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2927908&req=5

fig6: Measured and calculated structure factor amplitudes with corresponding electron density profile for TTMSS confined by mica membranes at a distance of 8.6 nm. (a) Measured amplitudes are indicated by the grey dots, amplitudes for the best-fit model by the red solid curve and amplitudes for a deviating model by the black dashed curve. (b) Corresponding best-fit and deviating electron density profiles are indicated by solid red and black dashed curves, respectively.
Mentions: We now apply the reflectivity theory to the XRR data obtained for confined TTMSS. Specularly reflected intensities I int(q ⊥) were integrated for momentum transfers q ⊥ up to 1.4 Å−1. Using equation (3), with f initially set to 1, values for the corresponding structure factor amplitudes /F exp(q ⊥)/ were derived. Sets of calculated values /F calc(q ⊥)/ were generated for a variety of structure models for the confined and adsorbed liquids as discussed in §3. The measured and calculated sets of values were compared using the logarithmic residual (Hirano et al., 1998 ▶) The model structural parameters, including the number of layers and f, were varied so as to minimize the residual E and thus to find the structure providing the best fit. In order to reduce the number of fitting parameters, the confinement arrangement was taken to be symmetric, the TTMSS layers in the gap were assumed to have equal electron density and width. The following additional constraints were applied: the liquid was not allowed to penetrate the mica, areal densities of the liquid layers were not to exceed the electron density for triangular closest packing of TTMSS molecules (calculated for a molecule diameter of 9.0 Å), and the width of the layers was kept to a lower limit of σ = 2 Å. In total, 23 fitting parameters were used: 12 symmetrical confined Gaussian peaks were fitted, each having a position (six parameters), a width and a height. The widths and heights of the inner eight density peaks were assumed to be equal, which results in two parameters plus four parameters from the boundary layers. The liquid on the outer mica surfaces were fitted with three layers, each having a position, a width and a height (nine parameters). Furthermore, the gap width and the correction factor f were two additional fitting parameters. We note that a number of fitting parameters are correlated, for example the width and the height of the Gaussian peaks. Fig. 6 ▶ shows the best-fit structure factor amplitudes in comparison with the measured values and the corresponding best-fit electron density profiles. All density profiles have been broadened with the experimental resolution (π/q ⊥,max = 2.2 Å) (Fenter, 2002 ▶). The best fit has been achieved for E = 0.30 and f ≃ 0.6.

Bottom Line: The confinement geometry acts like an X-ray interferometer, which consists of two opposing atomically flat single-crystal mica membranes with an intervening thin liquid film of variable thickness.An expression for the reflected intensity as a function of momentum transfer is given.The total structure factor intensity for the liquid-filled confinement device is derived as a sum of contributions from the inner and outer crystal terminations.

View Article: PubMed Central - HTML - PubMed

Affiliation: Paul Scherrer Institut, Villigen PSI, Switzerland.

ABSTRACT
An X-ray reflectivity theory on the determination of the density profile of a molecular liquid under nanometre confinement is presented. The confinement geometry acts like an X-ray interferometer, which consists of two opposing atomically flat single-crystal mica membranes with an intervening thin liquid film of variable thickness. The X-rays reflected from the parallel crystal planes (of known structure) and the layered liquid in between them (of unknown structure) interfere with one another, making X-ray reflectivity highly sensitive to the liquid's density profile along the confinement direction. An expression for the reflected intensity as a function of momentum transfer is given. The total structure factor intensity for the liquid-filled confinement device is derived as a sum of contributions from the inner and outer crystal terminations. The method presented readily distinguishes the confined liquid from the liquid adsorbed on the outer mica surfaces. It is illustrated for the molecular liquid tetrakis(trimethyl)siloxysilane, confined by two mica surfaces at a distance of 8.6 nm.

No MeSH data available.


Related in: MedlinePlus