Accounting for sample imperfections in $T_h^{(s)}$

Colloidal samples quite often include small populations of aggregated pairs of spheres, also known as dimers. These can have a striking effect on the configurational temperature, as the data in Fig. 8 demonstrate. This sample also consists of $\sigma = 1.58~\ensuremath{\unit{\mu m}}\xspace$ diameter silica spheres, but sedimented to the bottom wall of a slit pore $H = 195~\ensuremath{\unit{\mu m}}\xspace$ thick at an areal density $n\sigma^2 = 0.080$. Unlike samples in relatively thin ( $H = 9~\ensuremath{\unit{\mu m}}\xspace$) slit pores considered so far, such weakly confined silica monolayers are found to have purely repulsive screened-Coulomb interactions (19,10) in at least qualitative agreement with mean-field theory. The presence of a few dimers, however, contribute a small peak to the radial distribution function at $r = 0.8~\sigma$ whose tail extends to $1.1~\sigma$. These excess correlations are dramatically magnified in $T(r)/T$ because any two nominally repulsive particles ought to exert exceptionally large forces on each other near contact. The overall result is extremely large configurational temperatures.

At first glance, such defects in the sample would appear to render meaningful measurements of the configurational temperature impossible. However, the function $T(r)/T$ can be used to cut the spurious data while retaining enough useful information for an accurate assessment. In particular, the peak in $T(r)/T$ at small $r$ is well separated from the principal peak at $r = 1.8~\sigma$. This suggests that the former can be ascribed entirely to dimers and the latter to genuine long-ranged interactions, with a clean division at about $r = 1.3~\sigma$. It seems reasonable, therefore, to eliminate dimers' contribution to the configurational temperatures by excluding any pair separations smaller than $1.3~\sigma$.

Restricting the range of $u(r)$ too much would exclude relevant interactions, causing the net force on many particles to appear unbalanced and the configurational temperature to increase accordingly. For the present data set, we find that increasing the lower cutoff to $r = 1.45~\sigma$ only increases the apparent temperature by a few percent. This indicates that three-body correlations are weak at this concentration range and that the proposed cutoff at $r = 1.3~\sigma$ will not distort the results. Similarly, ignoring contributions from pairs separated by more than $r = 2~\sigma$ has little influence on the estimated temperature and establishes the interaction range for accurate temperature estimates.

Figure 8: Contribution to the configurational temperature at different pair separations assuming interactions described by Eq. (23) with $\kappa^{-1} = 180~\unit{nm}$, $Z = 7563$ for a silica monolayer at $H = 195~\ensuremath{\unit{\mu m}}\xspace$. The peak at $1 < r/\sigma < 1.2$ is caused by dimers. The tiny peak at $r = 1.5~\sigma$ is due to one particle stuck to the bottom surface. Right Inset: $g(r)$ with a dimer peak around $r = 0.8~\sigma$. Left Inset: a more detailed view of $g(r)$ near $r = \sigma$.

Figure 9 shows typical finite-size scaling results for a trial fourth-order polynomial fit to the experimentally obtained pair potential over the range $1.435 < r/\sigma < 1.89$. The fit potential is plotted as a dashed curve in Fig. 10(a). Replacing this with a fit to the predicted mean field potential described by Eq. (23) over the range $1.2 < r/\sigma < 1.95$ yields comparably good convergence, and increasing the range to $1.2 < r /\sigma < 10$ decreases the extrapolated configurational temperature by just 4 %. All of these trial potentials fall within error estimates for the measured potential. On this basis, and because the various definitions of the configurational temperature all extrapolate to unity, we conclude that the measured potential once again accurately describes the system's equilibrium pair interactions. In this case, however, the potential appears to be purely repulsive.

Figure 9: Finite size scaling of three configurational temperature definitions computed for a purely repulsive monolayer of $\sigma = 1.58~\ensuremath{\unit{\mu m}}\xspace$ silica spheres in a slit pore of height $H = 195~\ensuremath{\unit{\mu m}}\xspace$. Solid curves are weighted fits to second-order polynomials in $1/N$ used to extrapolate to the thermodynamic limit. Data at $1/N < 0.005$ are influenced by the stuck particle, as discussed in the text.

The sample used to compile these data also included one particle that had deposited irreversibly onto the lower glass surface near the edge of the field of view. A single immobilized sphere might not be expected to influence the free monolayer's structure and dynamics much. The influence on the radial distribution function is indeed subtle, with the slight peak at $r = 1.5~\sigma$ in $T(r)/T$ (Fig. 8) disappearing when the region containing the stuck particle is excluded from the calculation. The effect on the candidate pair potential is somewhat more pronounced, particularly for $r < 1.5~\sigma$, as can be seen in Fig. 10(a). But does the potential from the restricted data set better reflect the system's interactions? After all, the configurational temperatures calculated with the unrestricted data in Fig. 9 all extrapolate reasonably well to the thermodynamic value. Figures 10(b) and 10(c) show that the hierarchy of hyperconfigurational temperatures collapses to unity only when the region containing the stuck particle is excluded. The restricted data set, therefore, offers a more accurate picture of the pair potential. This is consistent with our earlier observation that small uncertainties in $g(r)$ at small separations can be dramatically magnified in $u(r)$.

Figure 10: Influence of an immobilized sphere on the effective pair potential and hyperconfigurational temperatures. (a) Measured pair potentials together with fourth-order polynomial fits yielding optimal collapse of the hyperconfigurational temperatures. Squares: results including all data. Circles: results obtained by excluding the region around the stuck sphere. (b) Hyperconfigurational temperatures obtained for the entire field of view. (c) Hyperconfigurational temperatures obtained for the restricted data set excluding the region around the stuck particle.