Areal pressures

The foregoing considerations serve to confirm that the same colloidal silica spheres experience qualitatively different equilibrium pair interactions when confined between parallel glass walls separated by $H = 195~\ensuremath{\unit{\mu m}}\xspace$ and $H = 9~\ensuremath{\unit{\mu m}}\xspace$, with the more strongly confined dispersion exhibiting anomalous long-ranged attractions. Such a qualitative difference in the system's pair potential also should manifest itself in the system's other thermodynamic properties, such as its pressure.

Figure 12: Finite-size scaling of the normalized areal pressure for systems with repulsive ( $H = 195~\ensuremath{\unit{\mu m}}\xspace$) and attractive ( $H = 9~\ensuremath{\unit{\mu m}}\xspace$) potentials. Squares and circles were computed for components of the force in the $x$ and $y$ directions respectively.

Given the particles' locations and an estimate for their pair interactions, we can calculate the monolayers' pressure as $P = n k_B T + p^{pot}$, where $p^{pot}$ is a departure from ideal gas behavior due to the particles' interactions. This so-called ``virial pressure'' is given by

$$Vp^{pot} = \frac{1}{d} \, \left< \sum_i \vecr_i \cdot {\mathbf F}_i \right> = \frac{1}{d} \, \sum_{i<j} \vecr_{ij} \cdot {\mathbf F}_{ij}$$ (35)

where $d$ is the dimension of the space, $\vecr_{ij} = \vecr_j - \vecr_i$ and ${\mathbf F}_{ij} = - \nabla_i u(\vecr_{ij})$. The former definition based on absolute coordinates works best for systems with open boundaries, which are described in the grand canonical ensemble, but fails in systems with periodic boundary conditions, which often are used in computer simulations. The latter definition based on relative coordinate works in both. Either should apply to our experimental data.

Comparing the different systems' interaction-driven departures from ideal gas behavior is facilitated by defining

$$B = \frac{\beta P}{n} = 1 + \frac{\beta p^{pot}}{n},$$ (36)

whose dependence on system size is plotted in Fig. 12. These data were obtained with the definition based on absolute coordinates measured from the center of the field of view. Equivalent results obtained with relative coordinates differ by less than 1.5%. As for the configurational temperature definitions, Eq. (35) applies in the thermodynamic limit, and $B$ is obtained by extrapolating to large system size. In this case, we see that the system at $H = 195~\ensuremath{\unit{\mu m}}\xspace$ with long-ranged repulsive interactions has a substantially higher pressure than would be expected for an ideal gas at the same areal density. The increase is much smaller in the more strongly confined system, presumably because of the potential's long-ranged attractive tail.

Like the configurational temperature, the pressure also depends sensitively on the input potential and length scale calibration. Without appropriate rescaling, $P_x$ is 10% higher than $P_y$ in the $H = 195~\ensuremath{\unit{\mu m}}\xspace$ data set. After rescaling using the factors obtained by requiring isotropy in the configurational temperatures, the two components agree well, as can be seen in Fig. 12. This further confirms that the observed anisotropy is due to artificial image distortion that can be corrected with a single rescaling factor.