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...
density increases. This is reasonable, since the cooling timescale of the X-ray gas is long compared with the dynamical timescale of the system (see above). Thus, from the X-ray derivation, the average density in the region producing the bulk of the optical emission lines should be n o n X T=T o < 30±100 n X or about 1,700±5,000 cm -3 . These values are consistent with the optical model. The difference between the shock velocities in the optical model and that derived for the X-rays could be real or could indicate a somewhat lower X-ray temperature, T < 0:6 3 10 6 K. A consistent theory of the emission from HH2H from optical through X-rays appears within reach. However, it is also possible that while the X-ray emission arises from shock-heated post-shock¯uid, the optical emission is dominated by a reverse shock propagating back into the fast material entering the HH2H region from the source (upstream) direction. On a much larger and faster scale, reverse shocks 23 are successful in explaining emission from supernova remnants. If the fast-moving upstream¯uid is much denser than the downstream¯uid before entering the reverse shock, the reverse shock speed will be much lower than the forward shock speed, and the post-shock material will predominantly radiate in lower excitation ultraviolet, optical and infrared lines. Deeper X-ray observations and nearly simultaneous optical images are needed to determine if the hot and relatively tenuous X-ray plasma does indeed lie at the leading (downstream) edge of the HH2H shock.
Special computational techniques are required to compute absolute crystal nucleation rates of colloidal suspensions. Using crystal nucleation of hard-sphere colloids as an example, we describe in some detail the novel computational tools that are needed to perform such calculations. In particular, we focus on the definition of appropriate order parameters that distinguish liquid from crystal, and on techniques to compute the kinetic prefactor that enters in the expression for the nucleation rate. In addition, we discuss the relation between simulation results and theoretical predictions based on classical nucleation theory.
Aggregation of proteins and peptides is a widespread and muchstudied problem, with serious implications in contexts ranging from biotechnology to human disease. An understanding of the proliferation of such aggregates under specific conditions requires a quantitative knowledge of the kinetics and thermodynamics of their formation; measurements that to date have remained elusive. Here, we show that precise determination of the growth rates of ordered protein aggregates such as amyloid fibrils can be achieved through real-time monitoring, using a quartz crystal oscillator, of the changes in the numbers of molecules in the fibrils from variations in their masses. We show further that this approach allows the effect of other molecular species on fibril growth to be characterized quantitatively. This method is widely applicable, and we illustrate its power by exploring the free-energy landscape associated with the conversion of the protein insulin to its amyloid form and elucidate the role of a chemical chaperone and a small heat shock protein in inhibiting the aggregation reaction.protein aggregation ͉ quartz crystal microbalance ͉ biosensors
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