Free energy of singular sticky-sphere clusters

Networks of particles connected by springs model many condensed-matter systems, from colloids interacting with a short-range potential and complex fluids near jamming, to self-assembled lattices and various metamaterials. Under small thermal fluctuations the vibrational entropy of a ground state is given by the harmonic approximation if it has no zero-frequency vibrational modes, yet such singular modes are at the epicenter of many interesting behaviors in the systems above. We consider a system of $N$ spherical particles, and directly account for the singularities that arise in the sticky limit where the pairwise interaction is strong and short ranged. Although the contribution to the partition function from singular clusters diverges in the limit, its asymptotic value can be calculated and depends on only two parameters, characterizing the depth and range of the potential. The result holds for systems that are second-order rigid, a geometric characterization that describes all known ground-state (rigid) sticky clusters. To illustrate the applications of our theory we address the question of emergence: how does crystalline order arise in large systems when it is strongly disfavored in small ones? We calculate the partition functions of all known rigid clusters up to $N\le 21$ and show the cluster landscape is dominated by hyperstatic clusters (those with more than $3N-6$ contacts); singular and isostatic clusters are far less frequent, despite their extra vibrational and configurational entropies. Since the most hyperstatic clusters are close to fragments of a close-packed lattice, this underlies the emergence of order in sticky-sphere systems, even those as small as $N=10$.

Figure 1

Left: the smallest singular cluster of sticky spheres, consisting of nine spheres. Right: a framework representation of the cluster with the singular mode indicated by red arrows. The singular mode is a vibrational mode with zero frequency, which cannot be extended to any finite deformation without incurring an energy cost, or equivalently without breaking a contact.

Figure 2

The sticky sphere limit of a Morse pair potential, $V\left(r\right)=\mathcal{E}{e}^{-\rho (r-1)}[e^{-\rho (r-1)}-2]$, where the depth $\mathcal{E}\to \infty$ and the range $\epsilon =1/\rho \to 0$.

Figure 3

Simulation results for $N=9$ sticky spheres with $\kappa \approx 220$ and $\rho =30,40,50,60,70,80,100,120,$ and 140 interpreted using different bond cutoffs. The dashed line shows the theoretically predicted frequency of the singular cluster, Eq. (15). Error bars are standard deviation estimates.

Figure 4

Top: phase diagram for $N=10$ spheres. Colors (shading) indicate which kind of cluster has the largest partition function for each value of $\gamma$, $\alpha$, where the clusters are grouped by the number of extra contacts $\mathrm{\Delta }B$ and the number of singular modes $d_X$. Dashed lines are $\kappa =10$ (top), $\kappa =1000$ (bottom). Bottom: Relative frequency of the types of rigid clusters of 10 spheres as a function of $\alpha$ (roughly the inverse square root width of the potential), at $\kappa =40$.

Figure 5

Phase diagrams for rigid clusters of $N=11$–14 spheres. Labeling is the same as described in the caption of Fig. 4.

Figure 6

Phase diagrams for rigid clusters of 15–21 spheres. Labeling is the same as described in the caption of Fig. 4.

Figure 7

Top row: the total geometric partition function of rigid clusters with $\mathrm{\Delta }B$ bonds above the isostatic number of bonds as a function of $\mathrm{\Delta }B$. The dashed line is an exponential fit, $log{z}_{\mathrm{\Delta }B}=c_1+c_2\mathrm{\Delta }B$, with the best fit value of $c_2$ indicated. The bottom row shows different contributions to the value of $z_{\mathrm{\Delta }B}$: $n_{\mathrm{\Delta }B}$ is the number of distinct rigid clusters, ${\overline{z}}_{\mathrm{\Delta }B}$ is the mean geometric partition function of such a cluster, and ${\overline{v}}_{\mathrm{\Delta }B}$ is the mean vibrational contribution to the latter.

Figure 8

Mean value of the bond-orientational order parameter $Q_6$ as a function of the number of extra bonds $\mathrm{\Delta }B$. The mean includes all clusters, regular and singular, weighted uniformly.

Figure 9

Simulation results for $N=9$ sticky spheres with $\kappa \approx 220$ and $\rho =30,40,50,60,70,80,100,120,$ and 140 interpreted using different bond cutoffs. We show the observed frequencies of four of the 51 nonsingular clusters. The dashed line shows the theoretically predicted frequency. Compare to Fig. 3, which plots the same data for the singular cluster.

Figure 10

Predicted vs. observed frequencies for the 52 different rigid cluster geometries for $N=9$ sticky spheres with $\kappa \approx 220$ and $\rho =120$, using a cutoff of $1+1/\rho$.

Figure 11

Scaling laws for $N=10–14$. Same labeling as described in the caption of Fig. 7.

Figure 12

Mean symmetry number $1/\sigma$ and mean moment of inertia factor ${\left|I\right|}^{1/2}$ for regular clusters as a function of $\mathrm{\Delta }B$, for each $N=10–21$. The dashed line is the best-fit line with best-fit slope shown beside it. Vertical axis is a logarithmic scale. This shows that neither factor contributes strongly to the exponential scaling of $z_{\mathrm{\Delta }B}$.