Failure of Linear Superposition in Equilibrium Crystals

Despite its shortcomings, the DLVO theory is appealing in its simplicity. Even if it does not accurately describe pair interactions in strongly interacting suspensions, it might adequately describe suspensions' bulk properties, perhaps with appropriately renormalized interaction parameters. This approach has been invoked to reconcile experimentally observed fluid-FCC-BCC phase transitions with those seen in simulations on particles interacting with screened-Coulomb pair repulsions [31,32]. Superposition of effective pairwise interactions similarly has been used to account for colloidal crystals' elastic properties [4,29,33,34]. Unfortunately, microscopic analysis of colloidal crystals' microscopic structure and dynamics reveals that such a parameterization fails to describe their bulk properties consistently [35] In particular, no combination of effective interaction parameters simultaneously parameterizes the potential of mean force and the bulk modulus.

Figure 3: (a) Potential of mean force along an arbitrary direction for spheres in a colloidal crystal at $\phi = 0.026$. Distances are measured in units of the crystal's equilibrium lattice constant, $a$. The solid line is a fit to the form $W(x) = Ax^2$. (b) Structure factor calculated from spheres' measured locations together with an extrapolation to long wavelengths, $\lim_{q \rightarrow 0} S(q)$. (c) Possible values for $Z$ and $\kappa ^{-1}$ derived from the data in (a) and (b). Dashed horizontal lines indicate sphere charges measured by conductimetric titration and by fitting the pair potential in figure 1(a).

Figure 3(a) shows a cross-section through the potential of mean force for colloidal spheres in a FCC colloidal crystal at volume fraction $\phi = 0.026$. This crystal was grown from a suspension of polystyrene sulfate spheres of radius $\sigma = 0.327~\mu\mathrm{m}$ dispersed in deionized water in diffusive contact with mixed bed ion-exchange resin. Its volume fraction was computed from a measured nearest-neighbor spacing of $a = 2.50 \pm 0.10~\mathrm{\mu m}$. The potential of mean force, $W(\vec r)$, describes a sphere's potential energy due to interactions with its neighbors and can be measured through the probability $P (\vec r) = \exp [ - \beta W(\vec r) ]$ for finding spheres displaced by $\vec r$ from their equilibrium positions.

If spheres in the crystals repel each other according to (9), then the potential of mean force can be built up by superposition of nearest-neighbor interactions [35]:

$$W(r) \approx 12 \, U(a) \left( \frac{\sinh \kappa r}{\kappa r} - 1 \right) \approx 2 \, U(a) (\kappa r)^2,$$ (13)

for a 12-fold coordinated FCC crystal. The quality of the quadratic fit to the data in figure 3(a) suggests that this indeed might be an adequate description of colloidal crystals' microscopic dynamics. Comparably good results are obtained for other crystals.

Measuring a single property leaves us with an entire family of possible values for $Z$ and $\kappa$. Identifying a single $(Z,\kappa)$ pair describing the crystal's collective properties requires a simultaneous measurement of an independent property.

Fourier transforming the crystal's pair correlation function $g(r)$ yields its angle-averaged static structure factor:

$$S(q) = 1 + 2 \pi \int_0^\infty \left[ g(r) - 1 \right] \, J_0(qr) r \, dr.$$ (14)

A crystal's structure at small wavenumbers $q$ is inversely proportional to its bulk modulus,

$$B = \lim_{q \rightarrow 0} \, \frac{\bar \rho \, k_B T}{S(q)},$$ (15)

which also can be related to an effective pair interaction [35],

$$B = \frac{2 \sqrt{2}}{3} \: \frac{U(a)}{a^3} \: \left[ 2 (\kappa a)^2 + \kappa a + 1 \right],$$ (16)

to yield a family of $(Z,\kappa)$ values. The crystal's structure factor at small wavenumbers is shown in figure 3(b).

Rather than intersecting at a particular set of effective interaction parameters for the crystal, the families of values obtained by these two methods are disjoint, as can be seen in figure 3(c). Thus, this colloidal crystal's collective properties are not parameterized by (9) even though this effective potential nicely describes interactions between isolated pairs of its spheres. The breakdown of pairwise additivity is manifest not only in biphasic or strongly confined suspensions, but even in colloidal crystals' bulk properties, and even for quite weakly interacting crystals.