31. The MWNT E values were calculated with the assumption that the nanotubes are solid cylinders. Lett. 79, 1297Lett. 79, (1997. 37. The discontinuity in F-d is not due to a discontinuity in the topography. First, high-resolution images demonstrate that the MoS 2 substrate is atomically flat in the region where the force discontinuity is observed. Furthermore, the topographic signal, which was recorded at the same time as F, is constant across the region of force discontinuity. Enhancement of Protein Crystal Nucleation by Critical Density FluctuationsPieter Rein ten Wolde and Daan Frenkel* Numerical simulations of homogeneous crystal nucleation with a model for globular proteins with short-range attractive interactions showed that the presence of a metastable fluid-fluid critical point drastically changes the pathway for the formation of a crystal nucleus. Close to this critical point, the free-energy barrier for crystal nucleation is strongly reduced and hence, the crystal nucleation rate increases by many orders of magnitude. Because the location of the metastable critical point can be controlled by changing the composition of the solvent, the present work suggests a systematic approach to promote protein crystallization.
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...
We have mapped out the complete phase diagram of hard spherocylinders as a function of the shape anisotropy L/D. Special computational techniques were required to locate phase transitions in the limit L/D→ϱ and in the close-packing limit for L/D→0. The phase boundaries of five different phases were established: the isotropic fluid, the liquid crystalline smectic A and nematic phases, the orientationally ordered solids-in AAA and ABC stacking-and the plastic or rotator solid. The rotator phase is unstable for L/Dу0.35 and the AAA crystal becomes unstable for lengths smaller than L/DϷ7. The triple points isotropic-smectic-A-solid and isotropic-nematic-smectic-A are estimated to occur at L/D ϭ 3.1 and L/D ϭ 3.7, respectively. For the low L/D region, a modified version of the Gibbs-Duhem integration method was used to calculate the isotropic-solid coexistence curves. This method was also applied to the I-N transition for L/DϾ10. For large L/D the simulation results approach the predictions of the Onsager theory. In the limit L/D→ϱ simulations were performed by application of a scaling technique. The nematic-smectic-A transition for L/D→ϱ appears to be continuous. As the nematic-smectic-A transition is certainly of first order nature for L/Dр5, the tri-critical point is presumably located between L/Dϭ5 and L/Dϭϱ. In the small L/D region, the plastic solid to aligned solid transition is first order. Using a mapping of the dense spherocylinder system on a lattice model, the initial slope of the coexistence curve could even be computed in the close-packing limit.
We present a new method to compute the absolute free energy of arbitrary solid phases by Monte Carlo simulation. The method is based on the construction of a reversible path from the solid phase under consideration to an Einstein crystal with the same crystallographic structure. As an application of the method we have recomputed the free energy of the fcc hard-sphere solid at melting. Our results agree well with the single occupancy cell results of Hoover and Ree. The major source of error is the nature of the extrapolation procedure to the thermodynamic limit. We have also computed the free energy difference between hcp and fcc hard-sphere solids at densities close to melting. We find that this free energy difference is not significantly different from zero: −0.001<ΔF<0.002.
We propose a novel approach that allows efficient numerical simulation of systems consisting of flexible chain molecules. The method is especially suitable for the numerical simulation of dense chain systems and monolayers. A new type of Monte Carlo move is introduced that makes it possible to carry out large scale conformational changes of the chain molecule in a single trial move. Our scheme is based on the selfavoiding random walk algorithm of Rosenbluth and Rosenbluth. As an illustration, we compare the results of a calculation of mean-square end to end lengths for single chains on a two-dimensional square lattice with corresponding data gained from other simulations.
We present a systematic numerical study of the phase behavior of square-well fluids with a ''patchy'' short-ranged attraction. In particular, we study the effect of the size and number of attractive patches on the fluid-fluid coexistence. The model that we use is a generalization of the hard sphere square well model. The systems that we study have a stronger tendency to form gels than the isotropic square-well system. For this reason, we had to combine Gibbs ensemble simulations of the fluidfluid coexistence with a parallel tempering scheme. For moderate directionality, changes of the critical density and the width of coexistence curves are small. For strong directionality, however, we find clear deviations from the extended law of corresponding states: in contrast to isotropic attractions, the critical point is not characterized by a universal value of the reduced second virial coefficient. Furthermore, as the directionality increases, multiparticle bonding affects the critical temperature. We discuss implications for the phase behavior, and possibly crystallization, of globular proteins.
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