The range of interactions

These thermodynamic tests turn out to be exceedingly sensitive to imperfections in experimental data, and care is required to apply them meaningfully. For example, video microscopy data necessarily is restricted to a limited field of view, even if the sample itself is substantially larger. Particles near the edge of the field of view may have strongly interacting neighbors just out of sight whose contributions to their net force would be overlooked. The large apparently unbalanced forces due to this pernicious edge effect would grossly distort estimates of the configurational temperature were they included in averages such as Eq. (6).

To avoid this, we calculate force distributions only for particles whose relevant neighbors all lie within the field of view. Such particles lie no closer than the interaction's range $R$ to the edge of the field of view. We estimate $R$ from $u(r)$ and $g(r)$ by computing

$$\frac{T(r)}{T} = 2\pi \frac{r}{\sigma} g(r) \, \frac{\vert \nabla \beta u(r) \vert ^2}{\nabla^2 \beta u(r)},$$ (33)

which heuristically describes an effective contribution to the configurational temperature of $2 \pi r g(r)$ pairs of spheres interacting with potential $u(r)$ at range $r$ (5). A typical example appears in Fig. 3. This should be considered no more than a heuristic guide because it neglects three-particle correlations, which are known (21) to be important for estimating the temperature.

Figure 3: Contributions to the configurational temperature due to pairs of colloidal spheres separated by distance $r$, calculated from the fifth-order polynomial fit to the data in Fig. 1. For this system, spheres separated by more than $R = 2 \sigma$ contribute negligibly to the configurational temperature.

Pairs at large enough separation that their interactions are vanishingly weak contribute negligibly to $T(r)$. The interaction range $R$ therefore can be estimated from the trailing edge of $T(r)$ in Fig. 3. By considering particles no closer than $R$ to the edge of the field of view, we can ensure that no relevant pair interactions will be missed in calculating the full configurational temperature. The restricted field of view is indicated by the small rectangle overlaid on the photograph in Fig. 2. Given this cut on the data, we can proceed to calculate the full configurational temperature.

The data in Fig. 1 display two features whose validity we can assess using the configurational temperature formalism. The first is the anomalous minimum that can be interpreted as evidence for long-ranged attractions between like-charged particles. The second is the surprisingly weak contact repulsion. This low barrier to aggregation probably is not real, otherwise particles would aggregate rapidly by van der Waals attraction. Could the minimum similarly be an artifact?

Closer inspection of the data in Fig. 1 reveals that $g(r)$ is finite even at $r < \sigma$, which should be impossible. Two sources of experimental error, projection error due to particles' small out-of-plane motions and polydispersity in the spheres' diameters, artificially increase $g(r)$ near contact and thus dramatically reduce the apparent interaction energy near $r = \sigma$. What Fig. 1 shows is the effective potential consistent with the raw data for the particles' positions $\rho(\vec{r},t)$, including both of these contributions. The configurational temperature calculation also is based on the same raw position data and so requires the associated effective potential as an input. The consistency condition $T_{config} = 1$ thus tests the accuracy and thermodynamic self-consistency of $u(r)$ for the measured set of $\rho(\ensuremath{{\mathbf r}}\xspace ,t)$ data.

Given $\rho(\ensuremath{{\mathbf r}}\xspace ,t)$ and $u(r)$, the net force ${\mathbf F}_j(t)$ on the $j$-th particle at time $t$ can be estimated using Eq. (5), with the sum over neighboring particles being restricted to those with $\vert \vecr_i - \vecr_j \vert \leq R$. The set of single-particle forces then can be compiled into estimates for the configurational temperature for that particular snapshot, and a sequence of snapshots averaged to obtain a final result. Averaging is not necessary if each snapshot captures a large enough number of particles. The dilute samples in our study, however, typically yield $N = {\cal O}\left( 100 \right)$ so that several thousand frames are required for adequate statistics.