Structural properties of liquids in extreme confinement

We simulate a hard-sphere liquid in confined geometry where the separation of the two parallel, hard walls is smaller than two particle diameters. By systematically reducing the wall separation we analyze the behavior of structural and thermodynamic properties, such as inhomogeneous density profiles, structure factors, and compressibilities when approaching the two-dimensional limit. In agreement with asymptotic predictions, we find for quasi-two-dimensional fluids that the density profile becomes parabolic and the structure factor converges toward its two-dimensional counterpart. To extract the compressibility in polydisperse samples a perturbative expression is used which qualitatively influences the observed nonmonotonic dependence of the compressibility with wall separation. We also present theoretical calculations based on fundamental-measure theory and integral-equation theory, which are in very good agreement with the simulation results.

Figure 1

Schematic of the slit geometry highlighting the important length scales: the wall separation $H$, the accessible slit width $L$, and the average particle diameter $\sigma .$

Figure 2

Transverse form factors $A_{\mu }\left(q_{\perp }\right)$ defining the transformation from the anisotropic to the generalized structure factor as described by Eq. (12).

Figure 3

Inhomogeneous density profile $n\left(z\right)$ for polydisperse ($\delta =0.15$) hard spheres at constant packing fraction $\phi =0.4$ for accessible slit widths $L/\sigma =1.0,0.9,0.75,$ and 0.6. Shown are simulations (solid lines) and FMT results (dashed lines) which strongly overlap. Profiles are shifted along the $y$ axis for the sake of visibility. The black dotted line represents a parabola. (a) and (b) show the same data in different representation.

Figure 4

Inhomogeneous density profile $n\left(z\right)$ for monodisperse hard spheres at constant area density $n_0=0.573{\sigma }^{-2}$ for accessible slit widths $L/\sigma =0.75,0.5,0.3,0.2,0.15$ and 0.1 (bottom to top). (a) Shown are simulations (solid lines), FMT (dashed lines) and quasi-2D theory (14) (dotted lines) which strongly overlap. Profiles are shifted along the $y$ axis for the sake of visibility. The insets show a zoom for $L=0.1\sigma$ (left) and $L=0.3\sigma$ (right). For $L\le 0.15\sigma$ no convergence was achieved for the FMT calculations. (b) Similar simulation data but rescaled according to Eq. (14). The dashed line shows a parabolic law fitted to $L=0.75\sigma$ between $-0.3\le z/L\le 0.3.$

Figure 5

Prefactor $C$ as defined in Eq. (14) determined by fitting the density profiles for $n_0{\sigma }^2=0.573$ shown in Fig. 4. The arrow marks the contact value of hard disks as determined from simulation results of the radial distribution function $g\left({\sigma }_{+}\right)$.

Figure 6

GSF $S_{00}\left(q\right)$ (top) and $S_{01}\left(q\right)$ (bottom) for polydisperse ($\delta =0.15$) hard spheres at constant packing fraction $\phi =0.4$.

Figure 7

Snapshot of the slit geometry for two accessible slit widths $L.$ The color code indicates the diameter of the particles [from diameter $<0.9\sigma$ (blue) to diameter $>1.1\sigma$ (red) and interpolation between]. The snapshot has been created with the visualization tool of DynamO [40]. For $L=0.6\sigma$ many pairs of small spheres with diameter smaller than $\sigma$ can be found (blue), while at $L=0.9\sigma$ several pairs of small and large spheres with diameter larger than $\sigma$ emerge (red and blue).

Figure 8

Compressibility as calculated from the small-wave-number limit of the in-plane GSF Eq. (16). Data are shown for polydisperse samples ($\delta =0.15$) at $\phi =0.4$, as characterized in Fig. 6, and for monodisperse samples at $\phi =0.3$, as shown in Fig. 15. The arrows show the compressibility of the corresponding two-dimensional fluids at density $n_{\text{2D}}=6\phi /\pi {\sigma }^2$.

Figure 9

In-plane GSF $S_{00}\left(q\right)=k_{B}T{n}_{\text{3D}}{\chi }_{\text{poly}}^{\left(0\right)}\left(q\right)$ and higher order correction $k_{B}T{n}_{\text{3D}}{\chi }_{\text{poly}}^{\left(2\right)}\left(q\right)$ to account for the polydispersity of the system [35]. Data are shown for polydisperse samples ($\delta =0.15$) at $\phi =0.4$.

Figure 10

Color plots of the anisotropic structure factor $S(q,q_{\perp })$ for monodisperse hard spheres as calculated from Eq. (8), visualized as a function of the wave-vector components in lateral ($q$) and transverse ($q_{\perp }$) direction to the confining walls. Shown are theory results ($\text{FMT}+\text{OZ}+\text{PY}$) and simulations (EDMD) for wall separations $L/\sigma =0.2,0.3,0.5$, and 0.75. The black dashed and dotted lines highlight the positions of $S_{00}\left(q\right)$ and $S_{11}\left(q\right)$, respectively.

Figure 11

GSF $S_{\mu \nu }\left(q\right)$ for monodisperse hard spheres at constant area density $n_0=0.573$ for modes $\nu =\nu =0$ (top panel), $\mu =\nu =1$ (middle panel), and $\mu =0,\nu =1$ (bottom panel). Results are shown for computer simulations (full line) and $\text{FMT}+\text{OZ}+\text{PY}$ (dashed line). For $S_{00}\left(q\right)$ the results of theory and simulations overlap.

Figure 12

Difference between the generalized structure factor $S_{\mu \nu }\left(q\right)$ for monodisperse hard spheres at constant area density $n_0=0.573$ and the 2D limit. The difference are scaled by $L^{-2}$.

Figure 13

Differences between the generalized structure factor $S_{00}\left(q\right)$ for monodisperse hard spheres at constant packing fraction $\phi =0.3$ and the corresponding 2D limit. The differences are scaled by $L^{-1}$.

Figure 14

Inhomogeneous density profile $n\left(z\right)$ for monodisperse hard spheres at constant packing fraction $\phi =0.3$ for accessible slit widths $L/\sigma =0.6,0.5,0.4,0.2$, and 0.1 (bottom to top). Shown are simulations (solid lines), FMT (dashed lines) and quasi-2D theory (14) (dotted lines). Profiles are shifted along the $y$ axis for the sake of visibility. The insets show a zoom of the $L/\sigma =0.1$ (left) and $L/\sigma =0.2$ (right). For $L/\sigma =0.1$ the FMT result did not converge, while for $L/\sigma =0.2$ the FMT and quasi-2D results perfectly overlap (even in the inset).

Figure 15

GSF $S_{\mu \nu }\left(q\right)$ for monodisperse hard spheres at $\phi =0.3$. Results are shown for computer simulations (full line) and $\text{FMT}+\text{OZ}+\text{PY}$ (dashed line).

Figure 16

Differences between the GSF $S_{\mu \nu }\left(q\right)$ for monodisperse hard spheres at $\phi =0.3$ and the corresponding 2D limit. The differences are scaled by $L^{-2}$.

Figure 17

Radial distribution function $g\left(r\right)$ for polydisperse ($\delta =0.15$) hard spheres at constant packing fraction $\phi =0.4$, calculated for different pairs of species: small-small (ss), small-large (sl), and large-large (ll). All particles with diameter $d<0.9\sigma$ ($d>1.1\sigma$) are accounted to the small (large) species, respectively.

Figure 18

GSF $S_{\mu \nu }\left(q\right)$ for polydisperse ($\delta =0.15$) hard spheres at constant packing fraction $\phi =0.4$ for small wave numbers $q\sigma <5$. For $L=0.6\sigma$, the figure also shows data for polydispersity $\delta =0.10$ (green dashed) and $\delta =0.05$ (blue dotted), respectively.