Static Structure and the Bulk Modulus

  

Figure 7: Detailed views of the structure factors in Fig. 4 at long wavelengths. The shaded regions indicate the ranges of values expected for $\lim_{q \rightarrow 0} S(q)$ based on the measured pair interactions and the resolution limit for the field of view. The data points (circles) are averaged over 100 video frames (roughly 3 seconds) to reduce scatter. The circles themselves are comparable in size to the resulting error bars. The dense crystal's structure factor is inconsistent with the anticipated behavior while the dilute crystal's structure is no more than marginally consistent.

In addition to gauging long-ranged order, the static structure factor S(q) also provides a measure of the sphere ensemble's bulk modulus B through its limiting behavior at small wavenumbers [33,34]:

$$B = \lim_{q \rightarrow 0} \frac{n k_B T}{S(q)},$$ (32)

where $n = \sqrt{2}/a^3$ is the mean number density of particles at temperature T. Fig. 7 shows S(q)near q = 0 for the crystals in this study.

Errors in S(q) due to the limited extent of the field of view were minimized by applying the Hann windowing function to the structural data[35] before calculating S(q). We estimated the resolution limit for the measured structure factors by calculating S(q) for simulated distributions of points using the same lattice constants and vacancy concentrations as the actual crystals and including normally distributed random displacements large enough to account for the crystals' Lindemann factors. These simulated defects constitute small departures from an otherwise perfect and thus ideally rigid lattice. The crystals' long-wavelength structure factors should exceed the simulated results because of order-disrupting elastic and plastic deformations. We also expect the experimental structure factors to verge below estimates based on the bulk moduli calculated in Section 5. The anticipated ranges of values for $\lim_{q \rightarrow 0} S(q)$appear as shaded regions in Fig. 7.

As can be seen in Fig. 7(a), data for the dilute, weakly interacting crystal fall quite close to the anticipated range. The small discrepancy might be accounted for by the $17 \pm 5$ vacancies in the field of view. Given the $N = 722 \pm 5$ spheres in the field of view, these would tend to depress the bulk modulus by a bit more than two percent both because of the reduction in density and because of the loss of supporting interactions. With these caveats, the linear interaction theory appears to be consistent with both the dynamics and the elasticity of the dilute crystal.

The long-wavelength behavior of S(q) for the dense, comparatively rigid crystal, on the other hand, falls significantly outside the anticipated range. In fact, the bulk modulus corresponding to $\lim_{q \rightarrow 0} S(q)$is only $B = 0.16 \pm 0.05~\mathrm{dynes/cm}^2$, less than a third of the expected value. In other words, the crystal is much softer than expected. At least part of this difference can be attributed to structural defects such as vacancies. This cannot be the whole story, however. The rigid crystal has only $5 \pm 1$ vacancies out of $N = 1419\pm 13$ spheres. Disorder's effect on the dense crystal's bulk modulus should be proportionately smaller than it was in the dilute crystal. And yet the discrepancy is much larger.

The discrepancy for the dense crystal also cannot be ascribed to inadequacies in the charge renormalization formula in Eq. (25). If this were the case, then alternative values of Z and $\kappa$could be found which would be consistent with both the dynamics in the potential of mean force and the elasticity of the crystal. No such values can be found. Combining Eq. (24) for the potential of mean force with Eq. (30) for the bulk modulus yields physically meaningful values for $\kappa$ only if

$$B > \frac{\sqrt{2}}{3} \, \frac{k}{a}.$$ (33)

Given the measured values of $\frac{1}{2} \beta k a^2$, the smallest possible bulk modulus for the rigid crystal is B > 0.48 dynes/cm². This is still a factor of three larger than the measured value. Therefore, no values of Z and $\kappa$ describe both the single-sphere dynamics and the crystal's collective behavior.

Finally, we cannot simply dismiss the DLVO theory and seek consistency in other models for the pair potential. The DLVO theory successfully describes the electrostatic interactions between isolated pairs of the spheres used in the present study [12]. It very nearly accounts for the elastic properties of the more dilute crystal. When viewed in this light, its inability to describe the dense crystal's properties indicates not just a failure of the DLVO theory, but rather a failure of linear superposition. This is more serious since the linear superposition approximation underlies all linearized theories for colloidal electrostatic interactions. Not surprisingly, it seems to fail when used to describe strongly interacting many-body systems. Under these conditions, the nonlinearity ignored in the derivation of Eq. (21) apparently results in non-additive electrostatic interactions which significantly affect the crystals' elastic properties. This interpretation is consistent with recent numerical [36] and analytical [29] studies of nonlinear coupling in charge-stabilized suspensions. Such many-body effects should be less important in more weakly coupled systems. Thus, the dilute crystal's properties are adequately described by the linear theory while the dense crystal's are not. Perhaps the most troubling point is that there seems to be no simple way to anticipate when the linearized description will suffice and when it will not.