Density Fluctuations of Hard-Sphere Fluids in Narrow Confinement

Kim Nygård, Sten Sarman, Kristin Hyltegren, Shirish Chodankar, Edith Perret, Johan Buitenhuis, J. Friso van der Veen, and Roland Kjellander Department of Chemistry and Molecular Biology, University of Gothenburg, SE-41296 Gothenburg, Sweden Department of Materials and Environmental Chemistry, Stockholm University, SE-10691 Stockholm, Sweden Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland Forschungszentrum Jülich, ICS-3, D-52425 Jülich, Germany ETH Zürich, CH-8093 Zürich, Switzerland (Received 11 June 2015; revised manuscript received 17 December 2015; published 16 February 2016)


I. INTRODUCTION
Nanoscopic confinement of simple fluids, which is confinement of such fluids between solid surfaces at separations of a few particle diameters, is known to strongly modify the fluids' dynamic and thermodynamic properties [1,2].The rationale for these "confinement phenomena" is as follows.Spatial confinement induces microscopic ordering of the fluid constituents because of competing packing constraints imposed by both the confining surfaces and the other particles in the system.This reasoning holds for atoms, molecules, and colloidal particles alike, and is routinely observed as a layered or stratified density profile across the confining slit, irrespective of whether the study is theoretical [3], experimental [4,5], or based on simulations [6].This microscopic layering, in turn, shows up in emergent properties of the confined fluid, such as slit-width and position-dependent diffusivity [7].However, the density profile of a confined fluid, or more generally an inhomogeneous fluid, does not always provide the most relevant measure of the fluid's microscopic structure, as illustrated by water near hydrophobic interfaces; although much effort has been devoted to determining small density depletions near hydrophobic interfaces [8][9][10][11], it has recently been shown that water's density fluctuations or isothermal compressibility provides superior measures of the link between microscopic structure and contact angle [12].
The next level of sophistication is provided by the pair density, which describes the density around a central particle at a given position in the slit.While the theoretical framework for studying confined fluids at this level was developed a long time ago [13][14][15][16][17], it is only recently that theoretical studies of such sophistication have started to gain broader interest [18][19][20][21][22][23].In terms of experiments, in turn, we have only very recently developed the first method for probing pair distributions of confined fluids, based on x-ray scattering from colloid-filled nanofluidic channel arrays [24,25].We emphasize that the pair density is a fundamental quantity of the confined fluid for several reasons: First, because of the hierarchy of distribution functions [3], i.e., the dependence of distribution functions on higher-order distributions, a mechanistic analysis of density profiles requires pair densities as input [23].Second, pair densities provide a straightforward approximation for the fluid's diffusive properties [26,27], which are otherwise computationally demanding to determine for spatially confined fluids [7].Third, pair densities can be directly related to thermodynamic properties of the system, most notably to the above-mentioned density fluctuations and isothermal compressibility [28], thereby providing a formal connection between microscopic many-body interactions of the confined fluid and its macroscopic, thermodynamic properties.This latter point, which is largely overlooked in the literature, is the main focus of the present study.
In this article, we address the microscopic structure of confined fluids explicitly, both in terms of density profiles and structure factors, by combining state-of-the-art x-ray scattering from colloid-filled nanofluidic channel arrays and integral-equation-based statistical mechanics.As a model system, we use the extensively studied hard-sphere fluid [29], which we confine between planar surfaces at short separation.Most importantly, we demonstrate that density fluctuations and isothermal compressibilities in confined fluids can be determined directly by analyzing the long-wavelength limit of the structure factor.To the best of our knowledge, we report the first direct experimental determination of isothermal compressibilities in spatially confined fluids.Finally, we comment on the potential for directly observing density fluctuations in confined molecular liquids and water near hydrophobic interfaces using x-ray scattering.

II. RESULTS AND DISCUSSION
In this study, we consider a hard-sphere fluid confined between planar surfaces at close separation of a few times the (average) particle diameter σ (see Fig. 1 for a schematic of the confinement geometry).We keep the fluid in equilibrium with a bulk reservoir of either number density n b ¼ 0.38σ −3 or 0.75σ −3 , corresponding to volume fractions of ϕ b ≈ 0.20 or 0.39, respectively.These two cases will be denoted "moderately dense" and "dense" fluids, respectively.We denote the slit width by H, as measured between the surfaces of the confining walls, and orient the z axis perpendicular to the surfaces with the origin, z ¼ 0, placed at the middle of the slit.
From a methodological point of view, our approach is twofold.First, we carry out small-angle x-ray scattering experiments on colloid-filled nanofluidic channel arrays.As a model for a hard-sphere fluid, we use octadecyl-coated spherical silica particles suspended in toluene [30], which we confine in nanofluidic channel arrays.From the observed diffraction pattern, we reconstruct the fluid's density profile across the confining slit in a model-independent manner using phase-retrieval schemes [31,32].Furthermore, and more importantly for the present study, we probe the fluid's pair densities by determining anisotropic structure factors from the observed diffuse scattering patterns [24,25], thereby allowing us to directly measure density fluctuations and isothermal compressibilities in the system.Second, we complement the experimental results by carrying out theoretical calculations using integral-equation-based statistical mechanics at the level of pair distributions for the inhomogeneous fluid, within the anisotropic Percus-Yevick (APY) approximation [14,15].

A. Density profiles
Let us begin by considering density profiles of the fluid across the confining slit.In Fig. 2(a), we present experimental density profiles of the dense hard-sphere fluid across slits of selected widths H, obtained from Ref. [25].The data are shown as the local volume fraction profile CðzÞ [31], which is formally given by CðzÞ ¼ nðzÞ ⊗ PðzÞ, with n denoting the number-density profile of the fluid and ⊗ the convolution operator, while P accounts for the finite size of the particles (see Sec. IV G below).For comparison, we also present the corresponding theoretical CðzÞ profiles, as obtained within the APY theory assuming monodisperse hard spheres and hard confining surfaces.The theoretical number-density profiles nðzÞ are presented in Fig. 3(a) for completeness.As noted already in Ref. [25], the experimental and theoretical CðzÞ profiles are in very good agreement; the minor shift of the experimental peak positions is attributed to effects not taken into account in the theoretical calculations, namely, polydispersity of the particles (Δσ=σ ≈ 12%), a minor taper of the confining surfaces (≈0.1°), and possible weak particle-wall attractions (see discussion below).
For comparison, we present CðzÞ profiles in Fig. 2(b) for the moderately dense fluid in slits of selected widths.The ordering is less pronounced in this system compared to the dense fluid, as expected because of weaker particle correlations.In contrast to the dense fluid, however, the experimental data are not accurately described by 1. Schematic of the confinement geometry.The surface separation is denoted by H, the z axis is aligned perpendicularly to the confining surfaces, and the center of the slit is positioned at z ¼ 0. The scattering vector components parallel (q ∥ ; in plane) and perpendicular (q ⊥ ; out of plane) to the confining surfaces are also shown.
theoretical calculations assuming hard particle-wall interactions [colored dashed lines in Fig. 2(b)].A possible explanation could be that the single octadecyl layers grafted on the colloidal particles are not thick enough to fully cancel out van der Waals attractions between particles and confining surfaces, thereby leading to a minor increase of the particle concentration in the slit.To mimic such a remnant attraction, we have also carried out theoretical calculations for the moderately dense fluid, including shortranged and weakly attractive square-well particle-wall interactions.The black dashed lines in Fig. 2(b), which we have obtained using a potential well depth βϵ ¼ 1.1 and width δ ¼ 0.05σ [defined in Eq. ( 10) below], where β ¼ ðk B TÞ −1 with k B Boltzmann's constant and T the temperature, are in good agreement with experimental data.We denote the latter case as "slightly sticky hard surfaces" throughout this study.For the corresponding theoretical number-density profiles nðzÞ, we refer to Fig. 3(b).

B. Structure factors
Next, we turn to the anisotropic structure factors SðqÞ of the confined fluids.Here, we recall that SðqÞ is formally related to the total pair correlation function hðr 1 ; r 2 Þ ¼ gðr 1 ; r 2 Þ − 1 of the confined fluid by [25] where g is the pair distribution function, ng the pair density, N the number of particles in the slit, and r i a position vector.The integrations are performed over the whole space between the surfaces.Throughout this study, we decompose the scattering vector q into its in-plane (q ∥ ) and out-of-plane (q ⊥ ) components (see Sec. IV D below) and give it in units of 2πσ −1 .In essence, SðqÞ probes pair-density correlations of confined fluids, averaged over all particles in the slit [21].
We have shown in Ref. [25] that the experimental structure factors of the dense hard-sphere fluid between planar surfaces are in good agreement with theoretical APY data, assuming hard surfaces.For completeness, we exemplify this good agreement in Fig. 4 for the same slit widths H ¼ 2.45σ and 3.10σ as discussed above.We further note that the agreement is not systematically improved by including a weak particle-wall square-well attraction, unlike for the moderately dense fluid to be discussed below.
In Fig. 5, we present structure factors SðqÞ for the moderately dense hard-sphere fluid between planar surfaces  with the same slit widths as in Fig. 2(b): H ¼ 1.7σ and 2.0σ.Because of the statistical inaccuracy at large scattering vectors (see Sec. IV C below), we present the data for the moderately dense fluid in a smaller q range compared to the dense fluid.In agreement with previous studies [24,25], spatial confinement is found to induce anisotropy in SðqÞ.For this particular system, the features in SðqÞ are rather weak-the magnitude of the first peak in SðqÞ is only ≈1.3-reflecting the modest particle-particle correlations in the moderately dense hard-sphere fluid.The experimental data are therefore noisier than for the dense fluid.Nevertheless, systematic changes are observed in SðqÞ as a function of slit width H, most notably the "diamondshaped" small-q region and the maxima at nonzero out-ofplane scattering vectors, ðq ⊥ ; q ∥ Þ ≈ ðAE1=2; AE1Þ, occurring for H ¼ 2.0σ.We note that the out-of-plane maxima are reminiscent of those observed earlier for both charged [24] and hard-sphere fluids [25] under spatial confinement at high densities, and they can be attributed to particle packing at the level of pair distributions [21].Most of these H-dependent features in SðqÞ are well described by the APY theory, assuming hard confining surfaces, as shown in the middle panels of Fig. 5.However, the out-of-plane maxima of Fig. 5(d) cannot be reproduced theoretically for hard surfaces in this case, but they can when we assume that the surfaces are slightly sticky [see Fig. 5(f)].
In order to quantify the anisotropy, we present in Fig. 6 the value of SðqÞ at a fixed scattering vector magnitude q max of the primary peak (red arrows in Figs. 4 and 5), as a function of the azimuthal angle α ¼ arctanðq ⊥ =q ∥ Þ, i.e., on a circle of radius q max in Figs. 4 and 5, with angle α counted from the positive q ∥ axis.A similar approach has recently been used to quantify orientational order of confined anisotropic nanoparticles [33].The data exhibit three noteworthy features.First, all the confined fluids show anisotropy, in stark contrast to isotropic bulk fluids.Second, the moderately dense fluid in a slit of width H ¼ 2.0σ (magenta curve) has maxima at q ⊥ ≠ 0 (i.e., α ≠ 0), as has already been inferred from Fig. 5(d).Note that the data ideally should be symmetric around α ¼ 0 and that more detailed features are difficult to assess because of the scattering of the experimental data.Third, the azimuthal variation of SðqÞ is overall well described by the APY theory, in particular, for the dense fluid.The deviations FIG. 4. Experimental (top panels) and theoretical (bottom panels) anisotropic structure factors SðqÞ for a dense hard-sphere fluid confined between planar surfaces.Data are presented as a function of in-plane (q ∥ ) and out-of-plane (q ⊥ ) scattering vector components.The surface separations are (a,b) H ¼ 2.45σ and (c,d) 3.10σ, and the confined fluid is kept in equilibrium with a bulk fluid of number density n b ¼ 0.75σ −3 .The theoretical data have been obtained assuming hard confining walls (βϵ ¼ 0).The dark red feature at q ∥ ¼ 0 in the experimental data, which is due to diffraction from the colloid-filled nanofluidic channel array, should be neglected in the comparison.The solid and dashed white lines illustrate in-plane cuts of SðqÞ for q ⊥ ¼ 0 and 1=2, respectively, as displayed in Fig. 7, while the red arrows depict the scattering vector q max of the primary peaks in the structure factor, as utilized in Fig. 6.
As in Fig. 4, except for a moderately dense hard-sphere fluid (n b ¼ 0.38σ −3 ).The surface separations are (a)-(c) H ¼ 1.7σ and (d)-(f) 2.0σ, and the theoretical data in the middle and bottom panels have been obtained using hard (βϵ ¼ 0) and slightly sticky hard surfaces (βϵ ¼ 1.1), respectively.Note that the q ∥ and q ⊥ scales and the color coding differ from those in Fig. 4.
between the experimental and theoretical data are most likely due to imperfections of the experiments, as mentioned above (see also below in connection to Fig. 7).The experimental system does not consist of monodisperse hard spheres between perfectly parallel walls, as assumed in the theoretical model.It is unlikely that the APY approximation is the cause of the deviations [34], implying that the theoretical data show the actual SðqÞ for such an idealized hard sphere system.For the moderately dense fluid in a slit of width H ¼ 1.7σ (the bottom case in Fig. 6), the fairly pronounced maximum in the dashed theoretical curve is not found in the experimental results.The structure factor around α ¼ 0 is better represented in a theoretical model with hard walls in this case, as apparent also from Figs. 5(a)-5(c).The scattering of the experimental data makes it difficult to discern whether the red curve shows an extremum or is fairly level in the middle.Effects of a possible residual attraction of the particles to the walls are further discussed below in connection to Fig. 7.
Nevertheless, the overall good agreement between experimental and theoretical data, even for weakly correlated inhomogeneous fluids, is a testimony to the sensitivity of the experimental approach and the accuracy of the theoretical scheme.The results of Figs.4-6 constitute a considerable achievement, given that the parameters used in the theoretical calculations-the average particle size σ, the bulk density n b , and the surface separation H-have been obtained from independent experiments.

C. Density fluctuations and isothermal compressibility
Having quantified the microscopic structure of the confined hard-sphere fluids both in terms of density profiles nðzÞ and structure factors SðqÞ, we are now in the position to address the main objective of the present paper-density fluctuations and the isothermal compressibility in confined hard-sphere fluids.The formal connection between a fluid's microscopic structure, on the one hand, and its density fluctuations and isothermal compressibility, on the other hand, is provided by the long-wavelength limit of the structure factor.For bulk fluids, this well-known relation is given by lim q→0 S b ðqÞ ¼ ðhN with S b ðqÞ denoting the isotropic bulk structure factor, hNi the mean number of particles in the open system and hN 2 i − hNi 2 ¼ hðN − hNiÞ 2 i its variance, μ the chemical potential, and χ T the isothermal compressibility [28].For spatially confined fluids, we must instead consider a slit-width-dependent isothermal compressibility χ H;T , which is related to density fluctuations in the presence of confining surfaces [35].The anisotropic structure factor of Eq. ( 1) is thus related to χ H;T via where n H ¼ H −1 R H=2 −H=2 nðzÞdz is the average number density of particles in the slit.Spatial confinement imposes slit-width-dependent packing constraints on the fluid, thereby modifying the fluctuations in number of particles ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi hN 2 i − hNi 2 p and resulting in an H-dependent compressibility χ H;T .In the rest of this study, we focus on the experimentally and theoretically accessible isothermal compressibility as obtained from the low-q limit of SðqÞ.
In order to exemplify the long-wavelength limit of the structure factor, we present in Fig. 7 experimental and corresponding theoretical in-plane values of SðqÞ as a function of q ∥ for all the systems considered so far.The data are shown for both q ⊥ ¼ 0 [denoted by Sð0; q ∥ Þ] and 1=2 [denoted by Sð1=2; q ∥ Þ], as illustrated by the solid and dashed white lines in Figs. 4 and 5.In all cases, we have an agreement between experimental and state-of-the-art theoretical data that varies between very good to reasonable.Quantitative differences remain around the first peak in the 3.10σ 2.45σ 2.0σ 1.7σ FIG. 6. Azimuthal structure factor Sðq max ; αÞ for the dense (n b ¼ 0.75σ −3 ; green and blue lines) and moderately dense (n b ¼ 0.38σ −3 ; magenta and red lines) confined hard-sphere fluid.The data are presented at a fixed scattering vector magnitude q max of the primary peak (red arrows in Figs. 4  and 5), as a function of the azimuthal angle α ¼ arctanðq ⊥ =q ∥ Þ on a circle of radius q max in Figs. 4 and 5.For the top two cases, q max ¼ 1.03, and for the bottom two, 0.97.The solid and dashed black lines depict corresponding APY data assuming hard (βϵ ¼ 0) and slightly sticky hard walls (βϵ ¼ 1.1.),respectively.The data are vertically offset by multiples of 1.0 for clarity in the top three cases.structure factor (q ∥ ≈ 1) related to particle packing [25].We attribute these deviations to imperfections of the experimental system, such as particle-size polydispersity, a minor taper of the confining surfaces, and remnant particle-wall attractions.In the low-q limit, the agreement between experiment and theory is good; the long-wavelength limit of Sð0; q ∥ Þ is well described for the dense fluid by first-principles calculations assuming hard walls (βϵ ¼ 0), while for the moderately dense fluid, the agreement becomes better by assuming slightly sticky hard walls (βϵ ¼ 1.1).
Focusing on the moderately dense fluid in the slit of width H ¼ 2.0σ, Fig. 7(b), we notice that the theoretical model with slightly sticky hard walls (βϵ ¼ 1.1) overestimates Sð0Þ and hence the compressibility.This deficit can be overcome by deepening the potential well to βϵ ¼ 1.7 (dashed-dotted line), although at the expense of the good agreement between experiment and theory in the full SðqÞ of Figs.5(d) and 5(f).For βϵ ¼ 1.7, too many particles enter into the slit, and the system becomes a bit too structured compared to the experiments.The value βϵ ¼ 1.1 gives the best overall agreement with experiment for both the density profile and the structure factor.A similar reasoning holds for the dense fluid in a slit of width H ¼ 3.10σ as shown in Fig. 7(b), where good overall agreement is found assuming hard walls (βϵ ¼ 0), but the low-q regime is best described by including a shallow potential well of βϵ ¼ 0.3.Clearly, the full SðqÞ is sensitive to the strength and detailed form of interaction potentials in the system, and a quantitative agreement between experimental and theoretical SðqÞ for all slit widths H and bulk densities n b would require a more accurate theoretical description of the particle-wall interactions as well as inclusion of the aforementioned size polydispersity and tapering of confining walls.Since the particle-wall interaction potential of the experimental system is unknown, we have not pursued this question here.
The structure factor for q ⊥ ¼ 0 behaves as Sð0; q ∥ Þ ¼ a þ bq 2 ∥ þ higher-order corrections when q ∥ → 0, where a and b are constants (see Sec. IV D below).In order to highlight the long-wavelength limit of SðqÞ, we therefore present the experimental and selected theoretical Sð0; q ∥ Þ as a function of q 2 ∥ in Figs.8(a) and 8(b).The good agreement between theoretical and experimental structure factors SðqÞ for long wavelengths shows that the latter data can easily be extrapolated to q ¼ 0, thus providing the experimental n H k B Tχ H;T according to Eq. (2).To the best FIG. 7. In-plane structure factor for the confined hard-sphere fluid, presented as a function of parallel scattering vector component q ∥ for either q ⊥ ¼ 0 (left panels) or 1=2 (right panels).Experimental data are shown for the dense fluid in slits of width H ¼ 2.45σ (blue diamonds) and 3.10σ (green squares) as well as for the moderately dense fluid in slits of width H ¼ 1.7σ (red circles) and 2.0σ (magenta triangles).The lines depict corresponding theoretical data obtained within the APY theory for either hard or slightly sticky hard surfaces.Plots (a) and (b) (to the left in the figure) are relevant for the extrapolation to q ¼ 0.
of our knowledge, this is the first demonstration that density fluctuations and isothermal compressibilities in confined fluids can be probed directly by x-ray scattering experiments.
Note that the experimental data in Figs.8(a) and 8(b) can, in some cases, be linearly extrapolated to zero when plotted as a function of q 2 ∥ .One can alternatively make a curve fitting to a polynomial a þ bq 2 ∥ þ dq 4 ∥ for small q 2 ∥ (see Sec. IV D), but then it is important not to include points at too large q 2 ∥ values in the fitting (terms beyond dq 4 ∥ in the power expansion must remain small).Such a fitting can therefore easily lead astray when it is extrapolated to zero if there is no theoretical curve to compare with.Our theoretical curves constitute at most one-parameter fits (the value of ϵ), compared to three-parameter fits for the polynomial curve or two-parameter fits for the linear extrapolations.
In Fig. 8(c), we present the experimental n H k B Tχ H;T , as obtained directly from the results in Figs.8(a) and 8(b) by extrapolating the fitted theoretical curves to q ∥ ¼ 0. For completeness, we have also included results for the dense fluid in slits of width H ¼ 2.65σ and 2.90σ, as obtained in the same manner from the structure factors in Ref. [25] for these two cases.The corresponding experimental bulk data are shown for comparison.In Fig. 8(d), we show the experimental average number density n H in the slit, obtained from Fig. 2 by integration.For comparison, we also present the bulk density n b obtained from the composition of the suspensions and verified by SAXS.In Fig. 8(e), the experimental isothermal compressibility is presented as k B Tχ H;T .It has been obtained by dividing the data in panel (c) with those in panel (d).The temperature is T ¼ 297 K.
The average density n H varies relatively little with the slit width, which implies that n H k B Tχ H;T in panel (c) and k B Tχ H;T in panel (e) of Fig. 8 behave in a very similar manner.Previous theoretical studies on the hard-sphere fluid between hard planar walls have shown that n H is slightly smaller than n b because of excluded volume near the walls, and that it varies rather weakly with H [21].These features can be seen in our experimental data, but the residual particle-wall attraction affects the difference between n H and n b and makes it smaller in some cases.According to Eqs. ( 1) and ( 2), n H k B Tχ H;T is intimately related to (i) the number of particles in the slit and (ii) particle packing at the level of pair densities.The former effect is most clearly manifested in the different compressibilities of the dense and moderately dense fluids in Figs.8(c) and 8(e); the compressibility is larger in the moderately dense case because of the lower particle density between the surfaces.Since the confined fluid alternates between being more and less structured depending on the surface separation [21], the compressibility can be both above and below the bulk value, and in the limit of large surface separation, the difference between the compressibility of the confined fluid and the bulk will tend to zero.Because of the imperfections of the experimental system, like the polydispersity and the residual attraction between the particles and the walls, the situation is, however, not simple.
A notable feature of Figs.8(c) and 8(e) is the smaller relative difference in compressibility between the bulk and confined fluid for the dense case compared to the moderately dense case.In qualitative terms, density fluctuations in the former case are already strongly suppressed by the fact that the particles are densely packed.Hence, the additional effect of confinement is relatively small.Furthermore, in the experiments, the surface separations are larger in the dense case, so the differences from the bulk are smaller.The smaller slit widths for the moderately dense fluid mean that the compressibility is affected more by the remnant particle-wall attractions.These experimental observations are also seen in the theoretical results of Figs.7(a) and 7(b), in which we can note a distinct effect of particle-wall attractions; n H k B Tχ H;T is smaller for the fluid between sticky hard surfaces compared to hard surfaces, implying a suppression of density fluctuations in the confined hard-sphere fluid due to particle-wall attractions.However, the quantitative effect of the particle-wall attraction on the compressibility depends in a nontrivial manner on the slit width H, and hence on detailed particle packing, as can be seen by comparing theoretical Sð0; q ∥ Þ for hard slits and sticky hard slits of widths H ¼ 1.7σ and 2.0σ.

D. Scattering intensity
We have so far discussed experimental structure factors obtained from the scattering pattern of polydisperse colloidal suspensions (see Sec. IV C below for details) and compared these with theoretical data for monodisperse particles.However, we emphasize that the main observations of the present study are independent of these approximations and can already be inferred from the scattering intensities.In order to demonstrate this fact, we present in Fig. 9 experimental scattering intensities obtained from bulk and one selected slit width H for both the dense and moderately dense fluids.For comparison, we also present the theoretical data for the monodisperse system as Ið0; q ∥ Þ ¼ jFðq ∥ Þj 2 Sð0; q ∥ Þ, with jFj 2 denoting the particle's form factor.The largest deviations between experimental and theoretical data, namely, the deep minima at q ∥ ≈ 1.5 and 2.5, are due to experimental imperfections such as particle-size polydispersity, multiple scattering, and experimental resolution, and should be disregarded in the comparison.Most importantly, all the findings discussed in the context of the Sð0; q ∥ Þ of Fig. 7-the effects of bulk number density and particle-wall interactions on the density fluctuations as well as the small shift of the position and height of the first peak in the experimental Sð0; q ∥ Þ 0. In both cases we present experimental data obtained from both bulk (solid black line) and confinement (red circles), as well as corresponding theoretical data for bulk (blue solid line), hard confinement (dashed red line), and a single sphere (i.e., the form factor; green dashed line).For the moderately dense fluid we also present theoretical data for slightly sticky hard confinement (dasheddotted black line).See text for details on the theoretical calculations.The experimental data are offset vertically by a factor of ten for clarity, and we present one of the theoretical curves (gray curve) also vertically offset to facilitate comparison.The deviation between the theoretical and experimental data around the deep minima should be disregarded in the comparison, see text.compared to bulk data-are clearly visible in the scattering intensity also.
The fact that the intensities Ið0; q ∥ Þ for the confined fluid and IðqÞ for the bulk fluid in Fig. 9(b) are very similar is coincidental.For other surface separations, they differ more.Note that IðqÞ for q ⊥ ≠ 0 differs a great deal from the corresponding scattering intensity for the bulk fluid, which, for instance, can be inferred from substantial anisotropy of the structure factors SðqÞ in Fig. 4.However, as discussed in detail in our previous work, Ref. [21], the anisotropy is less for surface separations about halfway between integer multiples of the sphere diameter [as in Fig. 9(b)] compared to that for integer multiple separations.So SðqÞ is more "bulklike" in the former case.

III. CONCLUSIONS
The detailed structural information for confined fluids that is obtained by our combined experimental and theoretical approach provides, as we have seen, in-depth information about the properties of the fluid and its interactions with the confining surfaces.The density profile data give important insights, which in the present case is most clearly manifested by the number of particles in the confining slit and the formation of dense particle layers between the surfaces.The same effects are visible in the structure factor and pair-correlation data.However, the latter data, which are unique to the kind of approach used here, also provide new possibilities for analysis of confined fluids that have a great potential; see, for example, Ref. [23], where a mechanistic analysis of the structure is done.
The foremost novelty of the present study is the introduction of a slit-width-dependent isothermal compressibility for confined fluids, χ H;T , which is accessible both experimentally and theoretically from the longwavelength limit of the anisotropic structure factor.Here, we have focused on a model system, the hard-sphere fluid in narrow confinement, and provided the first experimental demonstration that isothermal compressibilities (and density fluctuations) in confined fluids can be determined in a direct manner using x-ray scattering.Our approach thus provides direct means to relate the microscopic structure of spatially confined fluids with thermodynamics, at the level of pair densities.
We emphasize that the applicability of the isothermal compressibility considered here goes beyond the model colloidal systems of the study.Recently, Lippmann et al. have reported an x-ray scattering setup, which allows the determination of in-plane structure factors of molecular liquids in nanoslits [36].We note that their confining geometry is equivalent to the one of the present study.Once the experimental scheme of Ref. [36] is fully operational, it will extend the experimental analysis of isothermal compressibilities, as investigated here, to confined molecular liquids.
Finally, we comment on the application to one of the most important examples of density fluctuations in inhomogeneous liquids-namely, that of water at hydrophobic interfaces.Extensive theoretical work and simulation studies have shown that water exhibits enhanced density fluctuations at hydrophobic interfaces, but these fluctuations are yet to be directly observed in experiments (see, e.g., Ref. [37] for a brief introduction).In the course of analyzing these enhanced density fluctuations of water near hydrophobic interfaces, or more generally, of liquids at solvophobic interfaces, many different compressibilities have been devised [38][39][40].Of particular interest are the recent studies by Evans et al. [12,40], who demonstrate the sensitivity of a properly defined local compressibility χðzÞ ≡ ½∂nðzÞ=∂μ H;T to diverging density fluctuations of water (or other liquids) near hydrophobic (or solvophobic) interfaces.The isothermal compressibility n H k B Tχ H;T investigated in the present work is an integral of the local quantity χðzÞ over the slit width, and for narrow confinement (as considered here), it is sensitive to the liquid's density fluctuations near the interfaces.Since our isothermal compressibility is experimentally accessible for liquidlike colloids, and even on the molecular scale in the near future [36], it provides unique means for directly verifying the enhanced density fluctuations of water and other liquids in contact with solvophobic interfaces.

A. Colloidal suspensions
Following Ref. [25], we used octadecyl-grafted spherical silica particles suspended in toluene as a model fluid.At room temperature, this particular colloidal suspension acts essentially like a hard-sphere fluid.The particles had an average diameter of σ ¼ 69.2 nm, including the 2.3-nmthick octadecyl layer grafted on the particles, and a polydispersity of Δσ=σ ¼ 0.124.The bulk density in the present study was either n b ¼ 0.38σ −3 or 0.75σ −3 , corresponding to bulk volume fractions ϕ b ≈ 0.20 and ≈0.39, respectively.The synthesis and characterization of the colloidal suspension has been reported elsewhere [30].

B. Nanofluidic channel arrays
The confining containers consisted of silicon-based nanofluidic channel arrays with a period of 400 nm, a depth of 1.4−1.8μm, and a total area of 0.5 × 0.5 mm 2 per channel array, as obtained by electron-beam lithography and KOH etching following Ref.[32].The resulting scattering volume is thus 3 orders of magnitude smaller than in a standard small-angle x-ray scattering experiment on bulk colloids.The confining walls were smooth and nearly parallel (with tapering angle of about 0.1°), which greatly facilitates studies of confinement-induced ordering of the fluid.The surface separation was systematically varied in the range H ¼ 118 − 215 nm, corresponding to H ≈ 1.7σ − 3.10σ in units of average particle diameters.

C. X-ray scattering experiment
The small-angle x-ray scattering experiment was carried out at the cSAXS beamline of the Swiss Light Source (Paul Scherrer Institut, Switzerland), using an x-ray wavelength of λ ¼ 0.10 nm.The incident x rays impinged parallel to the confining surfaces, and we collected diffracted and diffusely scattered x rays simultaneously 7 m behind the sample using the PILATUS 2M detector [41].The incident x-ray beam had a size of 0.2 × 0.2 mm 2 at the sample position, and it was focused onto the detector plane in order to maximize the angular resolution.Finally, we inserted an evacuated flight tube between the sample and the detector in order to minimize parasitic scattering.The experiment was carried out at a temperature of T ¼ 297 K.
Small-angle x-ray scattering data from colloid-filled nanofluidic channel arrays are twofold: First, we observe a diffraction pattern, which can be inverted to yield the average density profile of the fluid across the slit.In the present experiment, we determined diffraction efficiencies up to 50 diffraction orders, allowing a reconstruction of the confined fluid's density profile with a real-space sampling interval of ≈4.0 nm.We have carried out model-independent density reconstructions using both iterative [31] and noniterative [32] phase-retrieval schemes.Second, and more importantly for the present study, we observe diffuse scattering, which probes the confined fluid's pair distributions in terms of the anisotropic structure factor of Eq. (1) [24,25].We obtained the scattering intensity of Fig. 9 from the raw scattering data by carefully subtracting both parasitic scattering and the diffuse scattering contribution from the fluid reservoir on top of the channel array.In order to facilitate the comparison in Figs.4-8 with theoretical structure factors for monodisperse spheres, we finally determined (effective) experimental structure factors by dividing out the form factor for polydisperse spheres within the monodisperse approximation.
The data in Figs. 4 and 5 are noisy at large scattering vectors q, especially for the moderately dense fluid, and as a consequence, we plot the data in a limited q range.This effect is readily understood as follows.The experimental structure factors are obtained by dividing the scattering intensity with the particles' form factor.Since the value of the latter decreases strongly with increasing q (see, e.g., the green dashed curves in Fig. 9), the statistical inaccuracy in SðqÞ is magnified with increasing q.For a bulk fluid exhibiting an isotropic SðqÞ, this problem could be partly overcome by radial averaging of the data.Since the confined fluid exhibits an anisotropic SðqÞ, one cannot make use of radial averaging, resulting in an enhanced statistical inaccuracy with increasing q.In order to partially compensate for this problem, the data of Figs. 6 and 7, as well as Figs.8(a) and (b), have been obtained by averaging over ≈10 detector pixels.

D. Anisotropic structure factor
For a fluid confined by planar smooth surfaces, the structure factor in Eq. ( 1) can be written as where we have introduced the in-plane and out-of-plane components of q as q ¼ ðq ∥ ; q ⊥ Þ and where A is the surface area, R 21 ¼ ðx 1 − x 2 ; y 1 − y 2 Þ, R 21 ¼ jR 21 j, and the integrations (denoted by a single integration symbol) are performed over the whole space between the surfaces.The pair correlation function hðr 1 ; r 2 Þ depends only on the three independent coordinates R 21 , z 1 , and z 2 for this geometry.Because of the radial symmetry in R 21 for fixed z 1 and z 2 , the structure factor depends only on q ∥ ¼ jq ∥ j, and we have where J 0 ðxÞ is the ordinary Bessel function of order 0. By expanding the Bessel function in a power series, J 0 ðxÞ ¼ , we obtain from Eq. ( 4) for q ⊥ ¼ 0 and q ∥ ≈ 0, and d is given by a similar expression.The integral in Eq. ( 6) converges since hðR 21 ; z 1 ; z 2 Þ decays quickly with R 21 and the z 1 and z 2 ranges are finite.

E. Anisotropic Percus-Yevick theory
For the theoretical analysis, we use statistical mechanics of inhomogeneous fluids at the pair distribution level.Following Refs.[14,15], we determine the singlet density profiles nðzÞ and total pair correlation functions hðr 1 ; r 2 Þ of the hard-sphere fluid in a planar slit by self-consistently solving two exact integral equations: the inhomogeneous Ornstein-Zernike equation and the Lovett-Mou-Buff-Wertheim equation, where c denotes the direct pair correlation function, v the particle-wall interaction potential, and the other quantities have been defined earlier.As the only approximation, we adopt the Percus-Yevick closure for anisotropic pair correlations, c ¼ g − y, where yðr 1 ; r 2 Þ is the cavity function that satisfies g ¼ y expð−βuÞ, and the hard particle-particle interaction potential is given by Together with a special integration routine that ensures equilibrium with a bulk fluid reservoir of given density, this constitutes the anisotropic Percus-Yevick theory.For a brief review of the theory and computational details, see Ref. [21].

F. Particle-wall interaction potential
Throughout this study, we assume the particle-wall interaction potential v of Eq. ( 8) to have the simple form where ϵθðjzjÞ, with is a square-well potential of depth ϵ and width δ next to each wall (we have ϵ ≥ 0).We take the attraction to be short-ranged, δ ¼ 0.05σ, and we have used values βϵ ¼ 0 and βϵ ¼ 1.1 (in the fitting βϵ ≤ 1.7) to describe hard and slightly sticky hard surfaces, respectively.
In practice, we include the attractive particle-wall interaction in the computations as follows.Let us consider an infinitesimal change δvðrÞ ¼ dϵθðjzjÞ in the particle-wall potential when ϵ is changed by dϵ.The corresponding changes in the density profile, δn, and potential of mean force, δw ¼ −δ ln n=β, are then given by the first Yvon equation [28] which describes the linear response upon a change δv in the external potential.Starting with the density profile and pair functions for hard walls (ϵ ¼ 0), we can thereby gradually deepen the potential well of the system using Eq. ( 12) and determine new values for the profile and pair functions for each ϵ.By integrating Eq. ( 12) with Eq. ( 10) inserted, we obtain the exact relationship Δwðr 1 ; ϵÞ ¼ ϵθðjz 1 jÞ þ Z ϵ 0 dϵ 0 Z dr 2 hðr 1 ; r 2 ; ϵ 0 Þnðr 2 ; ϵ 0 Þθðjz 2 jÞ; ð13Þ where Δwðr 1 ; ϵÞ ¼ wðr 1 ; ϵÞ − wðr 1 ; 0Þ.Here, we have explicitly stated the ϵ dependence of w, h, and n, but such a dependence is implicitly assumed also in Eqs. ( 7), (8), and (12).This integration process ensures that the chemical potential of the confined fluid remains the same for all values of ϵ.The profile in the presence of the potential wells is given by nðr 1 ; ϵÞ ¼ nðr 1 ; 0Þe −βΔwðr 1 ;ϵÞ : ð14Þ The first term on the right-hand side of Eq. ( 13) gives the direct influence from the deepening square wells, while the second term gives the indirect influence (via correlations) in the whole slit (all r 1 ) from the changes in the well (i.e., from particles with coordinate r 2 inside the wells).These different changes of the profile can clearly be seen in Fig. 3(b).

G. Local volume fraction
The density profile n describes the distribution of particle centers.However, in the experiments, we probe electron densities, and therefore the experimental density profiles include the finite size of the colloidal particles.Formally, the experimental density profiles are given as a local volume fraction CðzÞ ¼ nðzÞ ⊗ PðzÞ, where ⊗ is the convolution operator and P a projection of the particles' finite size [31].Moreover, because of the negligible contrast between toluene and octadecyl in the x-ray regime, we are only sensitive to the particles' silica cores in the experiment.Assuming monodisperse particles, we thus obtain where σ c is the diameter of the particles' cores and ðσ=σ c Þ 3 a scaling factor to obtain the particles' (i.e., both cores and shells) local volume fraction.For the quantitative comparison between experimental and theoretical density profiles in Fig. 2, we have convoluted the theoretical n with the factor P of Eq. ( 15).

3 FIG. 3 .
FIG.3.Theoretical number-density profile nðzÞ corresponding to the local volume fraction profiles CðzÞ of Fig.2, obtained within the APY theory.Data are presented for (a) the dense hardsphere fluid (n b ¼ 0.75σ −3 ) between hard planar walls as well as for (b) the moderately dense hard-sphere fluid (n b ¼ 0.38σ −3 ) between either hard (solid lines; βϵ ¼ 0) or slightly sticky hard planar walls (dashed lines; βϵ ¼ 1.1).Data for different slit widths H are vertically offset by 1.0 units for clarity.

FIG. 8 .
FIG. 8. (a,b) Long-wavelength limit of experimental and selected theoretical structure factors of Figs.7(a) and 7(b), presented as functions of q 2 ∥ (see text for details and Fig. 7 for the color and curve symbol codings).(c) The isothermal compressibility in the form n H k B Tχ H;T , obtained from panels (a) and (b) by extrapolation of the experimental structure factor to q ∥ ¼ 0. The error bars depict the inaccuracy of the extrapolation.The inset shows a magnification of the data for the dense fluid.(d) Experimental average number densities n H , as obtained from the density profiles of Fig. 2. The error bars are smaller than the symbol size.(e) Same as in panel (c), but showing the experimental isothermal compressibility as k B Tχ H;T .The dashed and dashed-dotted lines in panels (c)-(e) represent experimental bulk values for the dense and the moderately dense fluid, respectively.The data depicted by black stars in panels (c)-(e) have been determined from density profiles and structure factors for the dense fluid in slits of width H ¼ 2.65σ and 2.90σ, obtained from Ref. [25].

FIG. 9 .
FIG. 9. Mutually normalized x-ray scattering intensities obtained from (a) the moderately dense (n b ¼ 0.38σ −3 ) and (b) dense (n b ¼ 0.75σ −3 ) hard-sphere fluid.In both cases we present experimental data obtained from both bulk (solid black line) and confinement (red circles), as well as corresponding theoretical data for bulk (blue solid line), hard confinement (dashed red line), and a single sphere (i.e., the form factor; green dashed line).For the moderately dense fluid we also present theoretical data for slightly sticky hard confinement (dasheddotted black line).See text for details on the theoretical calculations.The experimental data are offset vertically by a factor of ten for clarity, and we present one of the theoretical curves (gray curve) also vertically offset to facilitate comparison.The deviation between the theoretical and experimental data around the deep minima should be disregarded in the comparison, see text.