The Free Energy Landscape of Hard Sphere Clusters

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Entry by Leon Furchtgott, APP 225 Fall 2010.

The Free-Energy Landscape of Clusters of Attractive Hard Spheres. Meng, G., Arkus, N., Brenner, M. P., & Manoharan, V. N. (2010). Science, 327, 560-563


The paper is interested in the behavior of small (10 or fewer colloidal particles) clusters and their relation to bulk behavior. In particular, the paper discusses the thermodynamics of small clusters: what structures are favored by entropy or by the potential energy, how this competition changes as N grows larger. Through careful experimentation the authors succeed in measuring structures and free energies of small equilibrium clusters. They compare these structures to theoretical predictions and draw conclusions regarding highly favored configurations.

Experimental Setup

Small numbers of polystyrene (PS) microspheres were placed in cylindrical microwells filled with polyNIPAM nanoparticles (Fig 1A, 1D). The microwells have depth and diameter 30 <math>\mu</math>m, and they are chemically functionalized so that the particles cannot stick to the surfaces. The microspheres have diameter 1.0 <math>\mu</math>m. The nanoparticles have diameter 80 nm and induce a depletion attraction between 2 microspheres (sticky spheres, see Fig 1B). This depletion attraction is very short-ranged (< 1/10 PS sphere diameter) which means that the interactions are pairwise additive (see Fig 1C). Therefore the total potential energy U of a given structure is well approximated by <math>U = CU_m</math>, where <math>C</math> is the number of contacts or depletion bonds and <math>U_m</math> is the depth of the pair potential.

The authors do this for thousands of clusters which they then image using optical microscopy. For each value of N <math>\leq</math> 10 they determine different cluster configurations and their probabilities <math>P_i</math> and thus the free energies <math> F_i = -k_B T ln P_i </math>.

Fig. 1. A. Experimental system: microwells filled with polyNIPAM microgel particles and PS microbeads. B. Close-up view of two sticky microspheres and surrounding nanoparticles. C. Pair potential. Note that the depletion attraction is very short-range, so the interaction is strictly pairwise additive. D. Micrograph of several microwells, each with different individual clusters. E. Optical micrographs of colloidal clusters in microwells with N = 2, 3, 4, 5 particles.


The cluster classifications can then be compared to experimental predictions, which are found in a previous theoretical PRL paper (Arkus, Manoharan, Brenner. Phys. Rev. Lett. 103, 118303 (2009)). Observed structures agree with experimental predictions. For N = 2, 3, 4, 5, one unique structure (dimer, trimer, tetrahedron, triangular dipyramid), as shown in Fig 1E.

The first interesting case is N = 6. Two structures are observed, an octogon and a polytetrahedron. Both have C = 12 contacts and thus have the same potential energy. What explains the 20-fold preference for the polytetrahedron (3 kT difference) is an entropic difference. The entropy can be divided into two factors: a rotational entropy and a vibrational entropy. The rotational entropy makes the largest contribution to the free-energy difference and is proportional to the moment of inertia or equivalently the permutational degeneracy. This contributes a factor of 12, the remaining factor of 2 coming from the vibrational entropy. In the case of N = 6 as for N = 7 and 8, highly symmetric structures are extremely unfavorable among clusters with the same potential energy.

Cases of N = 7 and N = 8 are similar to N = 6, although the number of structures increases to 6 and 16. For all cases, all the structures have the same number of contacts and entropic effects dominate, with highly symmetric structures extremely unfavored. These are all shown in Fig 2.

Fig. 2. Comparison of experimental and theoretical cluster probabilities P at N = 6, 7, and 8. Structures that are difficult to differentiate experimentally have been binned together at N = 7 and 8 to compare to theory. The calculated probabilities for the individual states are shown as light gray bars, and binned probabilities are dark gray. Orange dots indicate the experimental measurements, with 95% confidence intervals given by the error bars.

The landscape undergoes a qualitative change when N reaches 9. The number of structures predicted theoretically reaches 77 for N = 9 and 393 for N = 10, too many to catalog experimentally. The authors concentrate on measuring the probability of having structures of two types:

1. nonrigid structures, in which one of the vibrational modes is a large-amplitude, anharmonic shear mode. These have high vibrational entropy. (Fig 3A, 3B) 2. structures with more than 3N - 6 bonds. These often have high symmetry, but they have extra bonds. The potential-energy gain is therefor large enough to overcome the deficiency in rotational entropy. (Fig 3C).

Fig. 3. A. Optical micrographs and renderings of nonrigid structures at N = 9 and (B) N = 10 (C) Structures of 3N – 5 = 25 bond packings at N = 10. The anharmonic vibrational modes of the nonrigid structures are shown by red arrows. Experimentally measured probabilities are listed at top. Annotations in micrographs indicate clusters corresponding to subsets of HCP or FCC lattices. Scale bars, 1 µm.
Fig. 4. Calculated minima of the free-energy landscape for 6 <math>\leq</math> N <math>\leq</math> 10. The x axis is in units of the rotational partition function, where I is the moment of inertia (calculated for a particle mass equal to 1) and <math>\sigma</math> is the rotational symmetry number. Each black symbol represents the free energy of an individual cluster. The number of spokes in each symbol indicates the symmetry number (dot = 1, line segment = 2, and so on). Orange symbols are nonrigid structures, which first appear at N = 9, and violet symbols have extra bonds, first appearing at N = 10. Vertical gray lines indicate the contribution to the free energy due to rotational and vibrational entropy. The reference states are chosen to be the highest free-energy states at each value of N.