Hard spheres crystalization

The fonnation of colloidal crystals requires particles tliat are fairly monodisperse—experimentally, hard sphere crystals are only observed to fonn in samples witli a polydispersity below about 0.08 [69]. Using computer... 

An intrinsic surface is built up between both phases in coexistence at a first-order phase transition. For the hard sphere crystal-melt interface [51] density, pressure and stress profiles were calculated, showing that the transition from crystal to fluid occurs over a narrow range of only two to three crystal layers. Crystal growth rate constants of a Lennard-Jones (100) surface [52] were calculated from the fluctuations of interfaces. There is evidence for bcc ordering at the surface of a critical fee nucleus [53]. 

The structure of the hard sphere crystal has been shown to be fee rather than hep by Bruce et al. [49], Their lattice-switch Monte Carlo method allows the crystal to change its lattice during the course of the simulation. 

Fig. 4.8 Micelle volume fraction () versus polymer concentration at different temperatures for solutions of PEO26PPO39PEO26 in D20 (Mortensen 1993a). 4> was obtained from fits of the hard sphere Percus-Yevick model to neutron scattering profiles (see Fig. 3.9). At high concentration the asymptote = for hard sphere crystallization is reached.

ESPS remains a computationally intensive strategy, though not prohibitively so on the scale of its competitors One explicit comparison (in the case hard-sphere crystals) indicates that ESPS and NIRM deliver similar precision for similar compute resource [34]. 

Hoogenboom JP, van Langen-Suurling AK, Romijn J, van Blaaderen A (2003) Hard-sphere crystals with hep and non-close-packed structure grown by colloidal epitaxy. Phys Rev Lett 90 1-4... 

Pronk, S., and Frenkel, D., Can stacking faults in hard-sphere crystals anneal out spontaneously J. Chem. Phys., 110, 4589, 1999. 

Perhaps more than any other tool, molecular simulation has played an indispensable role in the development of the insights into molecular behavior we have reviewed in this Chapter. Before simulation was possible, even the existence of a stable hard-sphere crystal was in doubt, despite many years of attention to the question. Much of the progress reviewed here has occurred in the past decade, coinciding with the widespread availability of very powerful, inexpensive computers equally important have been advances in molecular simulation methodology as applied to solid phases. 

Tarazona, P. Density functional for hard sphere crystals A fundamental measure approach. 2000. Phys. Rev. Lett. 84 694. 

For the cubic phase the SANS experiment shows stUl the same type of aggregates hut more concentrated. However, here the packing exceeds the critical volume fraction of 53 vol%, which is typical for a hard sphere crystallization [63, 64]. In the cubic phase the spherical aggregates are packed similar to metal atoms in a cubic lattice. From SANS investigations of samples in the cubic phase it seems that the packing is not primitive cubic but either face-centred cubic (fee) or body-centred cubic (bcc). Between these two possibilities an experimental distinction was not feasible [65]. 

The lattice parameter of a colloidal hard-sphere crystal formed by sedimentation on a template depends on the size of the particles and the thickness of the crystal [27], For example, silica particles with a diameter of 1.55 pm and a size spread ( polydispersity ) of less than 3.5% form a perfect crystal on a (100) template with nearest neighbor distance do = 163 pm (which corresponds to an fee lattice parameter a — 2.31 pm). The lattice spacing decreases slightly with increasing thickness due to the increasing pressure head. 

Fig. 13 shows confocal images of the dislocations on the particle scale. As expected in a hard-sphere crystal, the defects that nucleate are stacking faults on 111 planes, each bounded by a Shockley partial dislocation. The fluctuations that lead up to the nucleation of the defect can be observed directly the defect in Figs 13(b) and 13(e) disappears after about 5 min, and four more of these fluctuations are observed before the nucleation is successful and grows into a large dislocation loop [Figs 13(c and d) and 13(f and g)]. 

In two recent and independent studies, Kyriidis and Brown and Mori, Manabe and Nishioka performed simulations of the hard sphere crystal-fluid interface using MC and MD techniques, respectively. (The study by Kyriidis and Brown also compared their results with a variety of DFTs, a more detailed discussion of which will be included in Section 6.) Both simulations used the phase coexistence data calculated for this system by Hoover and Ree, namely that a fluid of density = 0.943 coexists with an fee crystal with density = 1.041. Three crystal faces were studied [100], [110] and [111]. Note that the details of setting up a stable equilibrium interface for a hard sphere system are complicated somewhat by the discontinuous nature of the potential compared with that for continuous potentials described in Section 3.2. 

Fig. 12. Snapshot of a cross-section of a critical nucleus of a hard-sphere crystal at a liquid volume fraction (j) = 0.5207. The figure shows a three-layer thick slice through the center of the crystallite. Solid-like particles are shown in yellow and liquid-like particles in blue. The layers shown in the figure are close-packed hexagonal crystal planes. The stacking shown in this figure happens to be fcc-like, i.e. ABC-stacking — however, analysis of many such snapshots showed that fee and hep stackings were equally likely...

Fig. 13. Structure analysis of (pre) critical crystal nuclei. The figure shows the relative weight of the structural signatures for rhep, bcc, icosahedral and liquidlike ordering in hard-sphere crystal nuclei of size n...

Cheng Z (1998) Colloidal Hard Sphere Crystallization and Glass Transition, Ph.D. thesis, Pinceton University, Princeton, US 163,167... 

Speedy RJ. 1998. Pressure and entropy of hard-sphere crystals. Journal of Physics Condensed Matter 10 4387-4391. 

Davidchack RL and Laird BB. 1998. Simulation of the hard-sphere crystal-melt interface. Journal of... 

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