The Free Energy of Sticky Sphere Clusters

Yoav Kallus and Miranda Holmes-Cerfon

APS March Meeting
New Orleans, March 13, 2017

Colloidal particles with short range interactions

Meng, G., Arkus, N., Brenner, M. P., & Manoharan, V. N., Science (2010) Angioletti-Uberti, S., Mognetti, B. M. & Frenkel, D., Nature Materials (2012)

The sticky sphere limit:
E➞∞, ϵ➞0

Energy landscape of a system of sticky spheres

Continuous potential Kasha-Katuwe Tent Rocks National Monument

$U = -(\text{# of bonds})E$ Rio Grande del Norte National Monument

Rigid Clusters and Energy Minima

Energy minima ≈ rigid bar frameworks

Can have infinitesimal flex and still be rigid

Cluster Chemistry

Meng, G., Arkus, N., Brenner, M. P., & Manoharan, V. N., Science (2010)

Cluster Chemistry

$U_1 = -12E$

$U_2 = -12E$

$\displaystyle{Z_i = \int_{\mathcal{N}_i\subseteq \mathbb{R}^{3N}} e^{-\beta U(\mathbf{r})}\mathrm{d}^{3N}\mathbf{r}\qquad\qquad Z_2 \approx 20 Z_1}$

Entropic contributions

$Z_\mathrm{rot} = e^{S_\mathrm{rot}} = (\det I)^{1/2}/\sigma$

$Z_\mathrm{vib} = e^{S_\mathrm{vib}} = \prod_{a=1}^{3N-6} (\beta k_a)^{-1/2}$

$Z_\text{vib} \to \infty$

Asymptotically leading term

$Z_i ~\propto~ \alpha^{f} e^{\beta E B} z_i + O(\cdots)$
$\alpha = \left[d^2 \beta V''(d) \right]^{1/4}$, $f=$ # of flexes, $B=$ # of bonds, $z_i =$ geometric factor, independent of the potential

N=9 comparison with simulations

$P_1 = Z_1/\sum_{i=1}^{52}Z_i \approx \frac{\alpha}{235+\alpha}$

exact only for $\alpha\to 0$ (where $P_1\to 1$), but accurate even when $P_1=.08$.

Baxter's sticky sphere fluid-fluid transition

When $0 < \lim_{1/\epsilon,E\to\infty}\epsilon e^{\beta E} < \infty$,
$e^{-\beta U(\mathbf{r})} \to \prod_{i,j=1}^N (1 + \kappa\delta(r_{ij}-d)) \theta(r_{ij}-d)$

Based on the second order virial coefficient, Baxter (1968) calculated $\kappa_c$ for a gas-liquid transition

Sticky spheres and polydispersity

$\lim_{\epsilon\to 0}\int \delta(y+\epsilon \xi_1)\delta(y-x+\epsilon \xi_2)$
$\quad\delta(y+x+\epsilon \xi_3)\,\mathrm{d}x\mathrm{d}y < \infty$

$\lim_{\epsilon\to 0}\int \delta(y+x^2+\epsilon \xi_1)$
$\quad\delta(y-x^2+\epsilon \xi_2)\,\mathrm{d}x\mathrm{d}y = \infty$