In recent years attempts have been made by various groups to study lipid aggregates (bilayers, vesicles) in water by means of coarse-grained simulation models (coarse-grained molecular dynamics simulations and dissipative particle dynamics (DPD) simulations in particular) in order to make phenomena occurring on mesoscopic time and length scales (e.g., lateral shape fluctuations) accessible to simulation studies. In this talk I will shortly review some of these models, and then focus on two questions related to the problem of system equilibration: (1) In the case of lipid bilayers, to what extent is the lateral pressure profile useful for that purpose? (2) Is there kind of a "finger-print" method which can easily be employed even in the case of more complex lipid aggregates to monitor proper equlibration?
In this presentation I will give a brief overview of transition path sampling
(TPS), a method to study activated processes in complex environment. The TPS
technique harvests a collection of transition paths that connect the reactant
with the product states. This ensemble of true dynamical paths allows detailed
understanding of the kinetics and mechanism of the reaction. In addition, rate
constants can be computed. The main advantage is that the method does in
principle not need prior knowledge of the reaction coordinate. Processes as
diverse as cluster isomerization, auto dissociation of water, ion pair
dissociation, the folding of a polypeptide and reactions in aqueous solution
have been studied with TPS.
The rate constant calculation is rather time consuming and improvements in
efficiency were achieved by the introduction of the Transition Interface
Sampling approach. This approach is based on the sampling of paths that cross
hypersurfaces in phase spaces defined by an order parameter that does not have
to be equal to the reaction coordinate. TIS allows for a variable path length,
thus greatly enhancing efficiency. Recently, we improved the efficiency even
more for the calculation of rate constants for the case diffusive processes by
sampling only partial paths.
Having introduced the methods I will show their applicability by giving a few
examples.
There are also disadvantages to TPS and related techniques. For instance,
qualitatively different reactive pathways are hard to sample. Another problem
is the existence of unknown local traps (metastable states) in trajectory space
which can hamper the sampling. I will discuss the problems occurring in the
sampling of paths and their possible solutions.
Markov-Chain Monte Carlo methods have been used over the last decades with great success, in a variety of disciplines from physics to economics. The clear advantage of these methods has been demonstrated in cases where the dimensionality of the sampled phase space is large. In this presentation we will provide a general derivation of a novel method that improves the sampling efficiency of ensemble averages, computed via Monte Carlo. The former is achieved by collecting information not only for the microstates sampled via the importance sampling, but also for all points in phase space that can be linked to the sampled point via an "independent" Markov transition matrix between microstates. As a result, the efficiency of Monte Carlo simulations can be enhanced, sometimes dramatically, by properly sampling configurations that are normally rejected. The method we propose has been developed for the case of transition matrices between microstates and does not require any discretization of macro-states, in contrast with recently proposed "transition matrix" methods which, through appropriate bookkeeping, are able to reconstruct a transition matrix between discrete macro-states. Furthermore, the new method allows the direct evaluation of ensemble averages. The derivation of the method is general; application to specific problems may lead to the development of new methods to improve sampling efficiency especially in the case where out of a large set of trial configurations, only one is accepted, and in the case of parallel Monte Carlo applications.
In the past few years, the mesoscopic modelling of flow in nematic liquid
crystals has been successfully undertaken using lattice Boltzmann (LB)
methods. The methods have been designed to recover a number of different
macroscopic descriptions of the flow, and order, fields. A brief review will be
given of the motivation for using a mesoscopic approach and of the techniques
which have been developed. The problems associated with modelling the slow and
fast degrees of freedom within the same simulation framework will be discussed.
A number of different physical problems will then be presented to illustrate
the way in which the LB method provides a route to achieving the modelling of
the physical system through the introduction of additional physics. The
problems considered will include (i) solid colloidal particles embedded in a
nematic liquid crystal (ii) a droplet of an isotropic fluid embedded in a
nematic (iii) colloidal suspensions of multiple deformable droplets (iv) the
influence of flexo-electricity, and surface structure or patterning, on liquid
crystal device switching.
Despite the recent advent of multi-teraflop computing platforms, long time-scale events such as phase transition and protein folding remain out of reach for conventional molecular simulation. The sampling difficulties for many rare events are caused by large free energy barriers and the inherent micro-heterogeneity of the phase space. Although the former problem can now be surmounted by a host of free-energy based methods (including umbrella sampling), a separate approach is still required to deal with the latter problem. For example, for vapor-liquid nucleation the micro-heterogeneity arises from the presence of a spectrum of microphase regions (e.g., monomers and clusters). These microphase regions differ to a great extent on both energetic and entropic factors. In contrast, the random displacements used in the conventional Metropolis Monte Carlo scheme and the force-driven diffusion employed by molecular dynamics lack the balance of these two factors. This leads to slow transfer of particles between the micro-phase regions. A novel technique, called aggregation-volume-bias Monte Carlo (AVBMC), can overcome this problem by explicitly dividing the space surrounding a molecule into the associating and non-associating regions. This allows for direct transfer between microphase regions, thereby bypassing the time and spatial constraints imposed on molecular dynamics and Metropolis Monte Carlo techniques. AVBMC can be combined with configurational-bias Monte Carlo, umbrella sampling, histogram-reweighting, and density of states methods to study nucleation phenomena for complex systems. Applications to nucleation of neat water using polarizable force fields and mixtures containing water, alcohols, and alkanes will be presented.
Jarzynski's recent theorem relates equilibrium free energies to the statistics of work accumulated in irreversible transformations between ensembles. Fast switching simulations based on this theorem can be carried out with arbitrary switching rate promising inexpensive free energy estimation algorithms. Straightforward application of the fast switching method, however, is efficient only if the system is driven away from equilibrium only mildly thus requiring the generation of long non-equilibrium trajectories. The reason for this limitation lies in the distribution of work at fast switching rates: the values of work contributing most to the free energy are rarely sampled causing large statistical errors. By combining transition path sampling methods with biased sampling procedures one can focus on the trajectories carrying the largest contribution to the non-equilibrium averages. Here we discuss whether such biased fast switching methods can be more efficient than conventional methods such as thermodynamic integration, umbrella sampling, or Wang-Landau sampling. Numerical results are presented for a simple toy model.
An efficient description of soft matter almost invariably involves some degree of coarse-graining, whereby a large fraction of the initial microscopic degrees of freedom are traced out, leaving a much reduced space of variables associated with the composite (often mesoscopic) particles. The effective potentials arising from these procedures are usually of a many-body character, but, to make them tractable, are usually truncated at the pair level. Classical examples of such effective pair interactions include the DLVO potential between charged colloids (which results from integrating out the microscopic salt ions), the Asakura-Oosawa depletion potential (which results from integrating out the polymers in a colloid-polymer mixture), and the depletion potential in binary hard sphere mixtures obtained from density functional theory calculations. Employing the effective depletion pair potentials in simulations, we show that the structure and the phase behaviour of binary hard sphere mixtures is well-predicted by the effective pair potential description. Over the past few years evidence has been accumulated that suggests a qualitative breakdown of the pair-only picture: the many-body effects are not necessarily small as in simple fluids but may completely dominate and drive the physical phenomena in soft matter. This is, for instance, the case for charged colloids at extremely low salt concentrations (such that the Debye length is of the order of the colloidal diameter), where experimental observations show like-charge attractions. The source of these attractions has been suggested to be of many-body nature, either through volume terms or through triplet attractions. We determine the effect of three-body attractions on the phase behaviour of charged colloids. Another example of many-body interactions is encountered in mixtures of similarly sized colloids and polymer coils, where the repulsive triplet interactions between the colloids compete with the pairwise Asakura-Oosawa attractions. We determined a new simulation technique to simulate the effective one-component system with the full effective Hamiltonian that includes all the many-body interactions. For ideal polymers this was recently shown to have pronounced effects on the bulk phase behaviour and wetting phenomena.
We derive the method recently proposed by Maggs et al. for obtaining Coulomb interactions as the potential of mean force between charges which are dynamically coupled to a local electromagnetic field. We focus on the Molecular Dynamics version of the method and show that it is intimately related to the Car-Parrinello approach, while being equivalent to solving Maxwell's equations with freely adjustable speed of light. Unphysical self-energies arise as a result of the lattice interpolation of charges, and are corrected by a subtraction scheme based on the exact lattice Green's function. The method can be straightforwardly parallelized using standard domain decomposition. Some preliminary benchmark results are presented.
In this work we discuss methods for effectively extracting density of states (Omega) and derived functions (like free-energy landscapes) over large domains of phase space via (i) Monte Carlo simulations and (ii) equations of state (EoS). The simulation approach (ii) relies on the use of replica-exchange and Bennett's method in expanded-ensembles and multicanonical-type calculations to map out free energy landscapes in domains that bridge across multiphase fluid regions. Various implementations corresponding to the isothermal-isobaric, grand canonical, and semi-grand ensembles are considered in order to speed up convergence of simulation with single component and multicomponent systems. Central to the implementation of the EoS approach (ii) is the fact that the configurational Omega is related to thermodynamic properties accessible by an EoS via Bolztmann's equation. Sample calculations are presented for the EoS Omega relevant to isothermal-isobaric and grand canonical ensemble simulations using the several model fluids and model EoS. We further demonstrate how an EoS Omega can be used to aid the multicanonical-type simulations.
I will first give a short overview over "PERM", our own depth-first implementation of the sequential Monte Carlo method with biasing and resampling. In the following, I will deal in some depth with two recent applications. The first is the simulation of star polymers, including the estimation of the effective potential between two star polymers of equal size and structure. In this problem we find clear deviations from the cone approximations, and we find potentials which are cut off much faster than preposed by Likos et al. The second problem is that of lattice animals, which are a standard model for randomly branched flexible plymers. The strategy there is to use a breadth first algorithm for simulating percolation clusters, to re-weigh the clusters according to the animal ensemble, and to re-sample (prune/clone) the clusters according to a heuristic fitness function. We obtain precise estimates of the radii of gyration and of the free energy for large animals in 2 to 8 dimensions, which allows the first precise test of the Parisi-Sourlas conjecture. Finally, we consider two collapse thermal phase transitions of lattice animals: the adsorption to a plane wall, and the collapse due to a bad solvent. In the first we verify the superuniversality of the crossover exponent predicted by Janssen & Lyssy, in the second we obtain a detailed description of the transition in d=3.
Modelling mixtures of colloids and polymers poses a classic multiple length scale problem. Whereas the colloids can easily be viewed as single "giant atoms", the polymers are typically modelled by hundreds or even thousands of small constituents, all strung together. In a simulation most of the time is then spent moving polymer units. This talk will describe how to "coarse-grain" polymers to single objects, on the same footing as the colloids, leading to several orders of magnitude speedup in simulation times. Moreover, this approach clearly highlights some often overlooked subtleties in describing condensed matter with pair potentials.
Because polymers are very large molecules their dynamics are, by atomic
standards, very slow. To access time-scales long enough to calculate the real
(long-time, or low frequency) transport coefficients we need to be able to
simulate polymeric systems on times approaching seconds. We also need to model
hydrodynamic interactions, which significantly influence polymer dynamics, and
the thermal fluctuations that cause the instantaneous shape to change. To do
so we use a simple off-lattice particle model solvent that I will show does a
surprisingly good job of this. Reaching the time-scales we are interested in
is only possible, however, if we represent real long polymers with polymers
modelled using a vastly smaller number of repeating units. This raises the
issue, can we show that in any meaningful way we model the dynamic behaviour of
a real long polymer with such a short model one? After all, the dynamics of
short and long polymers normally differ significantly. Putting things together
I will argue that we can indeed claim to study long time dynamics of long ideal
polymers with simulations of chains composed of only a few beads. We have used
these short model chains to calculate the long-chain, low density,
visco-elastic response and polymer centre of mass motion . The simulations
shed light on how hydrodynamic interactions propagate, the role of intrapolymer
hydrodynamics and the limitations of theories that neglect the time dependence
of these interactions.
To now move beyond the ideal chain we need to be able to model chains with
interacting monomers (excluded volume chains). Before we can try to look at
their dynamics, we need to understand how to renormalize the non-trivial static
properties of these chains. As this is a workshop I will admit I can't do
it. However, I can illustrate one method that does not work and another method
that might.
One of the central aspects of the Monte Carlo method is the possibility to introduce non-physical dynamics, permitting the study of systems that evolve over otherwise prohibitively large time scales. A well-known example is the cluster algorithm for lattice spin models introduced by Swendsen and Wang, which suppresses dynamic slowing down near a critical point. Since the conception of this method, its generalization to off-lattice fluids of interacting particles has been an elusive goal, the main bottleneck being the absence of particle--hole symmetry. Also away from the critical point the existence of several different time and length scales constitutes a major obstacle in the simulation of complex fluids. This situation commonly arises in multi-component systems, such as binary mixtures, colloidal suspensions and colloid--polymer mixtures, and has essentially precluded the computational study of many such systems. We present a novel, rejection-free cluster Monte Carlo method of considerable generality that alleviates this problem. Geometric transformations are used to identify clusters of particles in such a manner that every cluster move is accepted, irrespective of the nature of the pair interactions. The rejection-free and non-local nature of the algorithm make it particularly suitable for the efficient simulation of complex fluids with components of widely varying size, such as colloidal mixtures. Compared to conventional simulation algorithms, typical efficiency improvements amount to several orders of magnitude.
We investigate how the single chain dynamics influences the kinetics of collective concentration fluctuations during the early stages of phase separation. Comparing Monte Carlo simulations of the bond fluctuation model at intermediate chain length, N=64, with dynamic self-consistent field calculations we show that a quantitative description of the growth of composition fluctuations has to account for the Rouse-like dynamics of the molecules. The information about the single chain dynamics enters the dynamic self-consistent field theory. In the latter, either the composition (DSCFT) or the effective external fields (External Potential Dynamics) can be regarded as dynamic variables. The latter scheme is computationally advantageous and a local Onsager coefficient in the EPD calculations corresponds to Rouse-like dynamics.
A Fourier accelerated simulation method is presented that allows one to simulate (soft) elastic bodies in external potentials eliminating hydrodynamic slow down. The method will be discussed in the context of Monte Carlo and molecular dynamics simulations. While the applications are mainly related to driven (quantum mechanical) charge-density waves, the method itself is applicable to many different problems in condensed matter physics. If time allows, I will also discuss how to calculate electro-mechanical properties, such as piezo-electrical coefficients, from simulations that are based on classical force fields. The calculation of such properties is less trivial than that of thermo-mechanical coefficients. The reason is that polarization is ill-defined in the NVT ensemble and while it is yet possible to cleanly define a a polarization difference in NVT, this is not as easily done if the stress is kept constant and the simulation cell is allowed to fluctuate.
Dissipative particle dynamics (DPD) is a coarse-grained simulation technique that employs soft interaction potentials and a momentum conserving thermostat to model complex fluids while persevering hydrodynamic modes. The details of the method and its limitations for the treatment of real thermodynamics systems will be discussed. The recent extension of the basic DPD method to allow the specification of an arbitrary equation of state, the so-called many-body DPD, will be described. The differing physical interpretations of many-body DPD will be outlined, with particular reference to the description of the potential energy function as a sum of local density-dependent self-energies. Brief mention of the applications of dissipative particle dynamics to various problems will be reviewed including future directions.
I will discuss different strategies to model the dynamics of complex fluids at mesoscopic scales. In particular, I will concentrate on the proper description of the hydrodynamic coupling in these systems by analyzing different lattice Boltzmann models. I will discuss the kinetics of binary mixtures and the hydrodynamics of electrolyte and charged colloidal suspensions.
This presentation summarizes recent work in the author's group on modeling phase transitions in model polymer and surfactant systems. A key characteristic of these fluids is the close interplay between microstructure and macroscopic properties and the existence of interactions on multiple length and time scales. Two complementary approaches are used to render the computational problem tractable, namely drastic simplification of the model studied to retain only essential physical characteristics and development of appropriate sampling methodologies to avoid getting trapped in local free energy minima. We have developed a methodology based on grand canonical Monte Carlo combined with histogram reweighting to distinguish between phase separation on one hand and micellization to finite-size aggregates on the other. The effects of chain flexibility on polymer and surfactant phase and aggregation behavior have been examined in detail. Ongoing investigations of surfactant systems under shear will also be described, in connection to the development of long-range order observed experimentally in PS-PEP thin films.
This talk outlines how grand canonical Monte Carlo simulation can be applied to the study of phase behaviour in polydisperse fluids. Attention is focused on the case of fixed polydispersity in which the form of the `parent' density distribution of the polydisperse attribute is prescribed by the chemical synthesis of the fluid. A recently proposed method--non-equilibrium potential refinement can be used to determine the chemical potential distribution conjugate to this density distribution. By additionally incorporating extended sampling techniques within this approach, the compositions of coexisting (`daughter') phases can be obtained and fractionation effects quantified. These techniques are applied to study the liquid-vapor phase equilibria of a size-disperse Lennard-Jones fluid exhibiting a large (40%) degree of polydispersity. Cloud and shadow curves are obtained, the latter of which exhibit considerable fractionation with respect to the parent. Additionally, a surprisingly large degree of broadening of the coexistence region is observed relative to the monodisperse limit.