# § VI. Conclusion and discussion

The recently introduced notion of configurational temperature provides a valuable new tool for assessing experimental systems' thermodynamic state from static snapshots. We have introduced a hierarchy of hyperconfigurational temperatures that emerge naturally from the generalized definition of temperature, and have shown that both these and the generalized definition of the temperature can be derived from the classical hypervirial theorem. Colloidal monolayers provide an ideal experimental test bed for these new concepts in statistical mechanics.

We have tested the configurational and hyperconfigurational temperature to within 1% accuracy using particles' distribution and pair potentials measured in Ref. (19). The effect of finite system size is clearly observed and accounted for in these measurements. Since the configurational temperatures' derivation requires pairwise additivity and their computation depends sensitively on the input potential, they can be used as thermodynamic self-consistency checks for measured pair potentials. The configurational temperatures calculated for our experimental data on confined colloidal silica monolayers confirm that our measured potentials are accurate and that they are consistent with the assumption of pairwise additivity.

We have found that higher-order hyperconfigurational temperatures are increasingly sensitive to errors in the potential because of their dependence on higher moments of the force distribution. Even so, we have found that we can adjust the pair potentials within the measured uncertainties to converge the entire hierarchy of hyperconfigurational temperatures to unity for all of our data sets. This provides substantial independent evidence that our directly measured potentials reliably reflect equilibrium pair potentials for our systems.

A sum rule introduced in Sec. V.4 complements the information provided by the configurational and hyperconfigurational temperatures by providing additional insights into the system's degree of equilibration. Very small errors in the pair potential, moreover, are dramatically emphasized by the derivatives in the sum rule's definition. We find, nevertheless, that the sum rule can be satisfied for the colloidal samples in our study by adjusting the potential within the range of their uncertainties.

Applying these analytical tools to our system of sedimented colloidal silica spheres allows us to draw new conclusions regarding the nature of electrostatic interactions in this deceptively simple system. Rather than being purely repulsive, as mean-field theory predicts, confined colloids' interactions can be characterized by a strong and long-ranged attraction. This result echos those reported more than a decade ago in the first generation of colloidal interaction measurements (14); (23); (24); (11); (16). Now, however, we can assert with confidence that the observed anomalous attractions constitute pairwise-additive contributions to the systems' equilibrium free energies, and not from any of the myriad of possible experimental artifacts that have been proposed. This interpretation is further bolstered by measurements of the areal pressure in these monolayers, which show clear signatures of their differing interactions. Resolving the anomaly of confinement-induced like-charge attractions therefore requires a fresh assessment of the nature of colloidal electrostatic interactions in simple electrolytes.

Hyperconfigurational temperatures also can be used as a set of constraints to determine the free parameters in a model for a system's pair interactions. We have demonstrated this by determining the two free parameters in a screened-Coulomb model for charge-stabilized colloids' interactions in the repulsive regime. In principle, the unbounded hierarchy of hyperconfigurational temperatures can be used in this way to determine any pairwise additive potential given the position, particularly if that potential can be modeled as a polynomial or comparably simple function.

One advantage of this method is that it circumvents the uncontrolled approximations that have dogged other approaches to measuring macroionic interactions in equilibrium. This method in more general in that it can be applied at arbitrary particle densities. Unlike measurement techniques based on the radial distribution function, , furthermore, configurational temperature measurements can be applied to inhomogeneous systems. This new method therefore should be useful in phase separated systems at equilibrium.

We are grateful to Owen Jepps, Sven Behrens and Brian Koss for helpful discussions. This work was supported by the Donors of the Petroleum Research Fund of the American Chemical Society.