Peter G. Bolhuis

Tracing the phase boundaries of hard spherocylinders

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→∞ 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/D⩾0.35 and the AAA crystal becomes unstable for lengths smaller than L/D≈7. The triple points isotropic-smectic-A-solid and isotropic-nematic-smectic-A are estimated to occur at L/D=3.1 and L/D=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/D>10. For large L/D the simulation results approach the predictions of the Onsager theory. ...

Accurate Effective Pair Potentials for Polymer Solutions

Dilute or semidilute solutions of nonintersecting self-avoiding walk (SAW) polymer chains are mapped onto a fluid of “soft” particles interacting via an effective pair potential between their centers of mass. This mapping is achieved by inverting the pair distribution function of the centers of mass of the original polymer chains, using integral equation techniques from the theory of simple fluids. The resulting effective pair potential is finite at all distances, has a range of the order of the radius of gyration, and turns out to be only moderately concentration-dependent. The dependence of the effective potential on polymer length is analyzed in an effort to extract the scaling limit. The effective potential is used to derive the osmotic equation of state, which is compared to simulation data for the full SAW segment model, and to the predictions of renormalization group calculations. A similar inversion procedure is used to derive an effective wall–polymer potential from the center of mass density pro...

Entropy difference between crystal phases

In a recent Letter, Woodcock reported the results of a molecular dynamics study in which he claims to have finally determined the free-energy difference between the hexagonal close-packed (h.c.p.) and face-centred cubic (f.c.c.) phases of a crystal of (classical) hard spheres. Woodcock reports a small positive difference in the reduced Gibbs free-energy, which is equivalent to a difference in the reduced Helmholtz free-energy of ΔF*≡(Fhcp −Ffcc)/RT=0.005(1) at the melting density (R is the gas constant, T is the absolute temperature, and the number in parentheses is the estimated error in the last digit). As Woodcock correctly points out, the calculation of the relative stability of the f.c.c. and h.c.p. phases of hard spheres is a long-standing problem in statistical physics. Attempts to resolve it date back to the work of Alder, Hoover and colleagues, and most recently, a direct simulation by Frenkel and Ladd, obtaining the bounds of Helmholtz free-energy of −0.001⩽ΔF*⩽0.002. Woodcocks estimate is incompatible with this latter result.

Elaborating transition interface sampling methods

We review two recently developed efficient methods for calculating rate constants of processes dominated by rare events in high-dimensional complex systems. The first is transition interface sampling (TIS), based on the measurement of effective fluxes through hypersurfaces in phase space. TIS improves efficiency with respect to standard transition path sampling (TPS) rate constant techniques, because it allows a variable path length and is less sensitive to recrossings. The second method is the partial path version of TIS. Developed for diffusive processes, it exploits the loss of long time correlation. We discuss the relation between the new techniques and the standard reactive flux methods in detail. Path sampling algorithms can suffer from ergodicity problems, and we introduce several new techniques to alleviate these problems, notably path swapping, stochastic configurational bias Monte Carlo shooting moves and order-parameter free path sampling. In addition, we give algorithms to calculate other interesting properties from path ensembles besides rate constants, such as activation energies and reaction mechanisms.

Transition Path Sampling

Often, the dynamics of complex condensed materials is characterized by the presence of a wide range of different time scales, complicating the study of such processes with computer simulations. Consider, for instance, dynamical processes occurring in liquid water. Here, the fastest molecular processes are intramolecular vibrations with periods in the 10–20 fs range. The translational and rotational motions of water molecules occur on a significantly longer time scale. Typically, the direction of translational motion of a molecule persist for about 500 fs, corresponding to 50 vibrational periods. Hydrogen bonds, responsible for many of the unique properties of liquid water, have an average lifetime of about 1 ps and the rotational motion of water molecules stays correlated for about 10 ps. Much longer time scales are typically involved if covalent bonds are broken and formed. For instance, the average lifetime of a water molecule in liquid water before it dissociates and forms hydroxide and hydronium ions is on the order of 10 h. This enormous range of time scales, spanning nearly 20 orders of magnitude, is a challenge for the computer simulator who wants to study such processes.

Influence of Polymer-Excluded Volume on the Phase-Behavior of Colloid-Polymer Mixtures

We determine the depletion-induced phase-behavior of hard-sphere colloids and interacting polymers by large-scale Monte Carlo simulations using very accurate coarse-graining techniques. A comparison with standard Asakura-Oosawa model theories and simulations shows that including excluded-volume interactions between polymers leads to qualitative differences in the phase diagrams. These effects become increasingly important for larger relative polymer size. Our simulation results agree quantitatively with recent experiments.

Rate Constant and Reaction Coordinate of Trp-Cage Folding in Explicit Water

We report rate constant calculations and a reaction coordinate analysis of the rate-limiting folding and unfolding process of the Trp-cage mini-protein in explicit solvent using transition interface sampling. Previous transition path sampling simulations revealed that in this (un)folding process the protein maintains its compact configuration, while a (de)increase of secondary structure is observed. The calculated folding rate agrees reasonably with experiment, while the unfolding rate is 10 times higher. We discuss possible origins for this mismatch. We recomputed the rates with the forward flux sampling method, and found a discrepancy of four orders of magnitude, probably caused by the methods higher sensitivity to the choice of order parameter with respect to transition interface sampling. Finally, we used the previously computed transition path-sampling ensemble to screen combinations of many order parameters for the best model of the reaction coordinate by employing likelihood maximization. We found that a combination of the root mean-square deviation of the helix and of the entire protein was, of the set of tried order parameters, the one that best describes the reaction coordination.

Numerical study of the phase behavior of rodlike colloids with attractive interactions

We examine the influence of attractive interactions on the phase behavior of rodlike colloids. We model the rodlike particles by spherocylinders, for which the phase diagram, in the absence of attraction, is known for arbitrary aspect ratio. We consider the case that the attraction is due to depletion forces caused by the addition of nonadsorbing polymer. The range of this attraction is determined by the size of the polymer. If the radius of gyration of the polymer is small compared to the diameter of the rods, we can model the polymer-induced attraction by a suitable generalization of the square-well model for spherical particles. However, for longer ranged attractions, pairwise additive attractions lead to phase behavior that is very different from what is found when the nonadditivity of depletion forces is taken into account. In our simulations, we find evidence for demixing transitions in the isotropic, nematic and solid phases. We compare our simulation results with predictions based on the perturbation theory of Lekkerkerker and Stroobants [Nuovo Cimento D 16, 949 (1994)]. A crucial input in this theory is the so-called free-volume fraction of the hard spherocylinder reference system. In the work of Lekkerkerker and Stroobants, this quantity is estimated using scaled-particle theory. We test the validity of this approach by comparing it to numerical results for the free-volume fraction.

Transition path sampling and other advanced simulation techniques for rare events

Computer simulations of molecular processes such as nucleation in first-order phase transitions or the folding of a protein are often complicated by widely disparate time scales related to important but rare events. Here, we will review sev eral recently developed computational methods designed to address the rare-events problem. In doing so, we will focus on the transition path sampling methodology.

Polymer induced depletion potentials in polymer-colloid mixtures

The depletion interactions between two colloidal plates or between two colloidal spheres, induced by interacting polymers in a good solvent, are calculated theoretically and by computer simulations. A simple analytical theory is shown to be quantitatively accurate for the case of two plates. A related depletion potential is derived for two spheres; it also agrees very well with direct computer simulations. Theories based on ideal polymers show important deviations with increasing polymer concentration: They overestimate the range of the depletion potential between two plates or two spheres at all densities, with the largest relative change occurring in the dilute regime. They underestimate the well depth at contact for the case of two plates, but overestimate it for two spheres. Depletion potentials are also calculated using a coarse graining approach which represents the polymers as “soft colloids;” good agreement is found in the dilute regime. Finally, the effect of the polymers on colloid–colloid osmot...