^{1,a)}, Sten Sarman

^{2,b)}and Roland Kjellander

^{1,c)}

### Abstract

Pair distributions of fluids confined between two surfaces at close distance are of fundamental importance for a variety of physical, chemical, and biological phenomena, such as interactions between macromolecules in solution, surface forces, and diffusion in narrow pores. However, in contrast to bulk fluids, properties of inhomogeneous fluids are seldom studied at the pair-distribution level. Motivated by recent experimental advances in determining anisotropic structure factors of confined fluids, we analyze theoretically the underlying anisotropic pair distributions of the archetypical hard-sphere fluid confined between two parallel hard surfaces using first-principles statistical mechanics of inhomogeneous fluids. For this purpose, we introduce an experimentally accessible ensemble-averaged local density correlation function and study its behavior as a function of confining slit width. Upon increasing the distance between the confining surfaces, we observe an alternating sequence of strongly anisotropic versus more isotropic local order. The latter is due to packing frustration of the spherical particles. This observation highlights the importance of studying inhomogeneous fluids at the pair-distribution level.

R.K. and K.N. acknowledge support from the Swedish Research Council (Grant Nos. 621-2009-2908 and 621-2012-3897, respectively). The computations were supported by the Swedish National Infrastructure for Computing (SNIC 001-09-152) via PDC.

I. INTRODUCTION

II. METHODS

A. Inhomogeneous integral-equation theory

B. Boundary conditions

C. Computational details

III. RESULTS AND DISCUSSION

A. Anisotropic structure factor

B. Anisotropic local order

C. Anisotropic local density

IV. CONCLUSION

### Key Topics

- Anisotropy
- 24.0
- Correlation functions
- 12.0
- Photon density
- 6.0
- Density functional theory
- 5.0
- Statistical properties
- 5.0

## Figures

Anisotropic structure factor for a hard-sphere fluid confined between hard planar surfaces. (a) Theoretical and (b) experimental S(q ⊥, q ‖) are shown for a reduced slit width of L = 2.10σ and bulk number density n 0 = 0.75σ^{−3}. The dark red feature at q ‖ = 0 in the experimental data is diffraction from the confining channel array, which should be neglected in the comparison. The experimental data are taken from Ref. 29 . (c) The corresponding isotropic bulk S(q) for n 0 = 0.75σ^{−3}.

Anisotropic structure factor for a hard-sphere fluid confined between hard planar surfaces. (a) Theoretical and (b) experimental S(q ⊥, q ‖) are shown for a reduced slit width of L = 2.10σ and bulk number density n 0 = 0.75σ^{−3}. The dark red feature at q ‖ = 0 in the experimental data is diffraction from the confining channel array, which should be neglected in the comparison. The experimental data are taken from Ref. 29 . (c) The corresponding isotropic bulk S(q) for n 0 = 0.75σ^{−3}.

Theoretical anisotropic structure factor as in Fig. 1 , but for different slit widths. The reduced slit widths are (a) L = 1.05σ, (b) 1.60σ, (c) 2.05σ, (d) 2.55σ, (e) 3.00σ, and (f) 3.50σ.

Theoretical anisotropic structure factor as in Fig. 1 , but for different slit widths. The reduced slit widths are (a) L = 1.05σ, (b) 1.60σ, (c) 2.05σ, (d) 2.55σ, (e) 3.00σ, and (f) 3.50σ.

Number density profiles n(z) for the hard-sphere fluid confined between hard planar surfaces. The reduced slit widths range from L = 1.05σ to 4.00σ.

Number density profiles n(z) for the hard-sphere fluid confined between hard planar surfaces. The reduced slit widths range from L = 1.05σ to 4.00σ.

Excess adsorption Γ and average volume fraction ϕ av of hard spheres in the slit between two surfaces as functions of surface separation. The dashed curve shows the volume fraction in bulk.

Excess adsorption Γ and average volume fraction ϕ av of hard spheres in the slit between two surfaces as functions of surface separation. The dashed curve shows the volume fraction in bulk.

Ensemble-averaged local density correlation function ⟨n(z)h(z, R, 0)⟩ for the reduced slit width L = 1.05σ. In the bottom part of the figure a contour plot of the function is shown and in the top part the same plot is illustrated in a 3D manner with peak heights proportional to the function value. The gray color denotes a narrow interval around the value zero.

Ensemble-averaged local density correlation function ⟨n(z)h(z, R, 0)⟩ for the reduced slit width L = 1.05σ. In the bottom part of the figure a contour plot of the function is shown and in the top part the same plot is illustrated in a 3D manner with peak heights proportional to the function value. The gray color denotes a narrow interval around the value zero.

Ensemble-averaged local density correlation function ⟨n(z)h(z, R, 0)⟩ corresponding to the anisotropic structure factors of Fig. 2 . The reduced slit widths are (a) L = 1.05σ, (b) 1.60σ, (c) 2.05σ, (d) 2.55σ, (e) 3.00σ, and (f) 3.50σ.

Ensemble-averaged local density correlation function ⟨n(z)h(z, R, 0)⟩ corresponding to the anisotropic structure factors of Fig. 2 . The reduced slit widths are (a) L = 1.05σ, (b) 1.60σ, (c) 2.05σ, (d) 2.55σ, (e) 3.00σ, and (f) 3.50σ.

Local density n(z 1)g(z 1, z 2, R 12) at coordinate (R 12, z 1) around a particle in the slit between two hard surfaces, when the particle is located on the z axis at coordinate z 2. Data are shown for the reduced slit width L = 2.05σ and three different particle positions: (a) in contact with one surface, (b) at the density minimum, and (c) in the slit center. The gray region depicts the excluded volume around the particle.

Local density n(z 1)g(z 1, z 2, R 12) at coordinate (R 12, z 1) around a particle in the slit between two hard surfaces, when the particle is located on the z axis at coordinate z 2. Data are shown for the reduced slit width L = 2.05σ and three different particle positions: (a) in contact with one surface, (b) at the density minimum, and (c) in the slit center. The gray region depicts the excluded volume around the particle.

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