Prediction of absolute crystal-nucleation rate in hard-sphere colloids

Abstract

Crystal nucleation is a much-studied phenomenon, yet the rate at which it occurs remains dif®cult to predict. Small crystal nuclei form spontaneously in supersaturated solutions, but unless their size exceeds a critical valueÐthe so-called critical nucleusÐthey will re-dissolve rather than grow. It is this rate-limiting step that has proved dif®cult to probe experimentally. The crystal nucleation rate depends on P crit , the (very small) probability that a critical nucleus forms spontaneously, and on a kinetic factor (k) that measures the rate at which critical nuclei subsequently grow. Given the absence of a priori knowledge of either quantity, classical nucleation theory 1 is commonly used to analyse crystal nucleation experiments, with the unconstrained parameters adjusted to ®t the observations. This approach yields no`®rst principles' prediction of absolute nucleation rates. Here we approach the problem from a different angle, simulating the nucleation process in a suspension of hard colloidal spheres, to obtain quantitative numerical predictions of the crystal nucleation rate. We ®nd large discrepancies between the computed nucleation rates and those deduced from experiments 2±4 : the best experimental estimates of P crit seem to be too large by several orders of magnitude. The probability (per particle) that a spontaneous¯uctuation will result in the formation of a critical nucleus depends exponentially on the free energy DG crit that is required to form such a nucleus:where T is the absolute temperature and k B is Boltzmann's constant. According to classical nucleation theory (CNT), the total free energy of a crystallite that forms in a supersaturated solution contains two terms: the ®rst is a`bulk' term that expresses the fact that the solid is more stable than the supersaturated¯uidÐthis term is negative and proportional to the volume of the crystallite. The second is à surface' term that takes into account the free-energy cost of creating a solid±liquid interface. This term is positive and proportional to the surface area of the crystallite. According to CNT, the total (Gibbs) free-energy cost to form a spherical crystallite with radius R is DG 4 3 pR 3 r S Dm 4pR 2 g 2 where r S is the number-density of the solid, Dm (,0) is the difference in chemical potential of the solid and the liquid, and g is the solid±liquid interfacial free energy density. The function DG goes through a maximum at R 2g= r S jDmj and the height of the nucleation barrier is:The crystal-nucleation rate per unit volume, I, is the product of P crit and the kinetic prefactor k:The CNT expression for the nucleation rate then becomes: (6). A prerequisite for the calculation of the nucleation barrier is the choice of à reaction coordinate' that measures the progress from liquid to solid. As our reaction coordinate we use n, the number of particles that constitute the largest solid-like cluster in the system. A criterion based on that in ref. 8 was used to identify which particles are solid-like. If two solid-like particles are less t...

Crystallization in suspensions of hard spheres: a Monte Carlo and molecular dynamics simulation study

“…Our data seem to lie below the simulation data from [3], but this effect might again still be within the error bars. It is not possible to extract an accurate statement regarding the slope of our data, as the interval of packing fractions we covered is relatively small.…”

“…The simulation data of Auer and Frenkel [3] for samples with 5% polydispersity have been scaled to the freezing point of monodisperse spheres, allowing a direct comparison with the older experiments. Our data seem to lie below the simulation data from [3], but this effect might again still be within the error bars.…”

“…However, in the meantime computers have become fast enough to sample crystal nucleation by 'brute force' simulation in simple model systems, such as hard spheres. Hence hard spheres have recently been used to test the predictions of rare event sampling techniques (such as results obtained by Umbrella Sampling [3]) and to compare nucleation pathways directly to experiment [6,10].…”

Monte Carlo simulations of densely-packed athermal polymers in the bulk and under confinement

“…It is now wellestablished that the face-centered cubic (fee) is marginally thermodynamically more stable than the hexagonal close packed (hep) crystal structure (Bolhuis et al, 1997;Woodcock, 1997). In spite of this, different ordered morphologies can also be observed in experiments and simulations like the random hexagonal close packed (rhep) layered structure or close packed crystallites, randomly oriented with defects being strongly correlated with twinning planes Auer and Frenkel, 2001;Bagley, 1970;Bolhuis et al, 1997;Cheng et al, 2002;Frenkel, 1999;Harland and van Megen, 1997;He et al, 1997;Henderson and van Megen, 1998;Karayiannis et al, 2011Karayiannis et al, , 2012Kawasaki and Tanaka, 2010;Leocmach and Tanaka, 2012;O'Malley and Snook, 2003;Pusey and Vanmegen, 1986;Pusey et al, 1989Pusey et al, , 2009Rintoul and Torquato, 1996;Russo and Tanaka, 2012;Schilling et al, 2010;Zaccarelli et al, 2009). These later crystal structures can be viewed, according to Ostwald's rule (Ostwald, 1897), as intermediate (metastable) thermodynamic stages between the amorphous (random) state and the fee crystal.…”

Phase field approach to heterogeneous crystal nucleation in alloys

“…However, the liquid will exist in a metastable state until a nucleation event occurs. In the study of nucleation, a distinction is made between homogeneous and heterogeneous nucleation [1,2]. Homogeneous nucleation occurs in an idealized pure material, where the only source of nucleation in an undercooled melt is due to fluctuation phenomena [1,2].…”