Sensitivity to input potential

The apparently good extrapolation of the configurational temperatures to the thermodynamic temperature would offer little insight into the nature of the confined colloids' interactions if these results were insensitive to relevant features in the pair potential. For example, considerable attention has been paid in the literature to the possibility that anomalous confinement-induced like-charge attractions such as the example in Fig. 1 might be artifactual. However, if we truncate the negative region of $u(r)$ to create a purely repulsive potential and recalculate the configurational temperatures, $T_{conF}$, $T_{con1}$ and $T_{con2}$ all extrapolate to 1.5, an error of $150^\circ\unit{C}$. The large deviation resulting from this admittedly crude test suggests that the observed attraction is indeed an integral part of the charged particles' interaction.

More sensitive tests for thermodynamic consistency are provided by the hyperconfigurational temperatures defined in Eq. (9). Figure 7 demonstrates how small variations in $u(r)$ can cause the hyperconfigurational temperatures to deviate with respect to each other and also with respect to the thermodynamic temperature. Two smoothed versions of the potential are plotted in Fig. 7(a), one fit over the range $0.93~\sigma \le r \le R$ and the other over the more restricted $\sigma \le r \le R$. The former collapses the entire hierarchy of hyperconfigurational temperatures plotted in Fig. 7(b) to $\left< T_h^{(s)} \right>_s \approx 1$ in the extrapolated thermodynamic limit, with $T_h^{(1)} = 1.012$, which compares favorably to $T_{conF} = 0.965$ in Fig. 6. The latter yields the far less satisfactory results in Fig. 7(c). Rather than collapsing onto the thermodynamic temperature, $T_h^{(s)}$ deviates systematically to lower values with larger index $s$.

This qualitative difference is due to substantial contributions from pairs of particles with $r < \sigma$. Such pairs should not be present in a monodisperse sample of impenetrable spheres, but appear in practice because of the sample's 3% polydispersity in radius and because of projection errors due to the particles' out-of-plane fluctuations. These two effects are responsible for the observed correlations at $r < \sigma$ in Fig. 1, and for the unreasonably small values of $u(r)$ in the unphysical range $0.5~\sigma < r < \sigma$. The successful collapse of the configurational and hyperconfigurational temperatures under these conditions demonstrates that the effective potential accounts for the apparent particle distribution $\rho(\ensuremath{{\mathbf r}}\xspace )$ and may differ subtly from the true pair potential.

The results in Fig. 7(b) and 7(c) reflect the general trend that higher order hyperconfigurational temperatures are more sensitive to details of the input potential. Even so, we can adjust the input potential within the experimental error bounds so that all of the hyperconfigurational temperatures converge to unity. In this sense, the hyperconfigurational temperatures not only strengthen our conclusions regarding the nature of anomalous like-charge attractions, but also enable us to improve our estimates for $u(r)$ by adjusting for improved thermodynamic self-consistency.

The data in Fig. 7(c) also highlight another general feature of the configurational temperatures. Even though the more restricted trial potential does not successfully collapse the data, it does yield consistent results for configurational temperatures factored along orthogonal directions. This is a good indication that, indeed, the system is isotropic.

Figure 7: (a) Measured pair potential $u(r)$ together with least-squares fits to fifth-order polynomials over the range $0.93 < r/\sigma < 2$ and $1 < r/\sigma < 2$. (b) Hyperconfigurational temperatures including data from $r < \sigma$. $T_h^{(1)} = 1.012$. (c) Hyperconfigurational temperatures over the more restricted range, factored into Cartesian components.