Parallel Wang–Landau study of micelle formation ∗

Thomas Vogel and David P. Landau

Center for Simulational Physics, Department of Physics and Astronomy, University of Georgia, USA

1 Introduction

The self-assembly process in amphiphilic solutions is a phenomenon ofbroad interest and vast computational studies of these processes are chal-lenging. We present a framework for highly parallel Monte Carlo (MC)computer simulations using the Wang–Landau method, which we applyto a generic coarse-grained model for amphiphilic molecules.

2 Generic model for amphiphilic molecules

The generic model we use is based on models previously employed to studythe self-assembly of molecules, see [1] and references therein.

Illustration of particles and interaction parameters in the generic

coarse-grained model. P: polar head monomer, H: hydrophobic tail

monomer, W: solution particle (water molecule).

We use three different types of coarse-grained particles, representingpolar (P) and hydrophobic (H) parts of a larger chain molecule and thesurrounding water (W) molecules. The amphiphilic molecules are builtup of a polar head and, in this study, two hydrophobic tail monomers(P–H–H). The interaction between solution particles and tail monomersas well as between head monomers and tail monomers is purely repul-sive and modeled by a repulsive soft-core potential, all other non-bondedinteractions are of Lennard-Jones type. Bonds are finitely extendible,non-elastic (FENE) in nature.

Conformations of a system containing m = 25 amphiphilic molecules and a total of n = 1000 particles.

a) Random configuration (initial configuration in simulation), b) molecules assemble and form clusters,

and c) low-energy single-cluster (micelle) configuration. Upper row: Only the amphiphilic molecules are

shown, colors mark different clusters. Periodic copies of the simulated system are shown. Bottom row:

Simulation boxes including all particles are shown. Periodic boundary conditions apply.

3 Parallelization of the Wang–Landau sampling

In a single-processor Wang–Landau flat-histogram MC simulation [2],a “walker” samples the conformational space and estimates the densityof states g(E) in a certain energy range between Emin and Emax. Thisis done in an iterative way, where the histogram of visited energies H(E)has to become “flat” in order to proceed to the next iteration.For large-size systems, this very efficient generalized-ensemble samplingcan be enhanced by making use of multiple processors working in parallel.This can be done, for example, by introducing multiple walkers samplingan identical energy range and contributing to the same histogram, or bysplitting up the energy range and estimating the density of states for therespective energy intervals by independent walkers [3].

∗ This work is supported by the National Science Foundation.

Examples of possible partitions of the energy range of interest into nine intervals. a) overlap 100%,

b) overlap 90%, c) overlap 50%, and d) overlap 75%. Several walkers might work in each interval.

In an attempt to improve the latter approach, we now also allow for con-formation exchanges between independent walkers.

4 Performance and Results

In order to get reliable results, we want each sample to perform walksthrough the whole energy range, i.e., it should walk back and forth throughall single energy intervals. For the given problem we find that a splittinginto nine energy intervals with an overlap of 75% is a good choice. Hence,the width of each single energy interval is 1/3 of the whole energy range.In the following example, we have a total of 81 walkers, with nine walkersin each energy interval. For conformation exchanges, every walker choosesa walker from a neighboring interval at random.

Top: Path of a specific sample through energy (left)

and walkers (right). It starts at high energies and

performs a complete “round-trip” within the first 107

sweeps. A conformational exchange between walkers

is proposed every 103 sweeps, with acceptance rates

between 30 and 40%. Left: Sketch of the splitting of

the energy range for this example.

We test the overall performance on a smaller energy range. The singlewalker needed ≈ 1.3 × 107 sweeps to complete the Wang–Landau itera-tions and estimate the density of states (ln g(E) is shown in the figure,dashed green line). The red symbols in the figure represent the samefunction composed of the results of seven parallel walkers. The walker inthe lowest energy interval completed the iteration after ≈ 7× 106 sweeps.Due to the overlap of the energy intervals and the multiple walkers perinterval, the statistical precision is also much higher.

Estimate of the logarithm of density of states

for a system containing m = 75 amphiphilic

molecules and a total of n = 1000 particles.

Comparison between result from single run with

Emin = −4500 and the respective walkers from

a parallel run. Setup as above, no error bars

shown.

[1] R. Goetz and R. Lipowsky, J. Chem. Phys. 108, 7397 (1998)[2] F. Wang, D. P. Landau, Phys. Rev. Lett. 86, 2050 (2001)[3] D. P. Landau et al., Am. J. Phys. 72, 1294 (2004)