W. K. den Otter

Free energy of a trans-membrane pore calculated from atomistic molecular dynamics simulations

Atomistic molecular dynamics simulations of a lipid bilayer were performed to calculate the free energy of a trans-membrane pore as a function of its radius. The free energy was calculated as a function of a reaction coordinate using a potential of mean constraint force. The pore radius was then calculated from the reaction coordinate using Monte Carlo particle insertions. The main characteristics of the free energy that comes out of the simulations are a quadratic shape for a radius less than about 0.3 nm, a linear shape for larger radii than this, and a rather abrupt change without local minima or maxima between the two regions. In the outer region, a line tension can be calculated, which is consistent with the experimentally measured values. Further, this line tension can be rationalized and understood in terms of the energetic cost for deforming a part of the lipid bilayer into a hydrophilic pore. The region with small radii can be described and understood in terms of statistical mechanics of density fluctuations. In the region of crossover between a quadratic and linear free energy there was some hysteresis associated with filling and evacuation of the pore with water. The metastable prepore state hypothesized to interpret the experiments was not observed in this region.

Liquid-crystalline ordering in rod-coil diblock copolymers studied by mesoscale simulations

Using mesoscale dissipative particle dynamics (DPD) simulations, which ignore all atomistic details, we show the formation of lamella mesophases by cooling a fully disordered system composed of symmetric (A7B7) rod–coil diblock copolymers. Equilibration is achieved very rapidly using DPD, and isotropic, smectic A and crystalline phases of the rod–like blocks can be observed either by heating or cooling. An interesting pseudo–smectic phase can be characterized when the order–disorder transition temperature is above the clearing temperature. This phase gradually fades into a normal microphase–separated structure as the system is heated through the clearing temperature. Simulations of pure rods, however, show the formation of isotropic, nematic, smectic A and crystalline phases.

Simulations of stable pores in membranes: System size dependence and line tension

Amphiphilic bilayers with a pore were simulated using a coarse grained model. By stretching the bilayer to 70% beyond its equilibrium surface area, we established the phase diagram of pores, identifying regions where pores are stable, metastable, or unstable. A simple theoretical model is proposed to explain the phase diagram, and to calculate the critical and equilibrium relative stretches. Interestingly, these are found to scale with the inverse cubic root of the number of amphiphiles in the bilayer, thus explaining the order of magnitude difference between the simulated and the measured values. Three different methods are used to calculate a line tension coefficient of (3.5-4.0) x 10(-11) J/m, in good agreement with experimental data.

The bending rigidity of an amphiphilic bilayer from equilibrium and nonequilibrium molecular dynamics

Helfrichs theory predicts that the bending free energy of a tensionless amphiphilic bilayer is proportional to the square of the Fourier coefficients of the undulation modes. Equilibrium molecular dynamics simulations with coarse-grained amphiphiles confirm the correctness of this prediction for thermally excited undulations. The proportionality constant then provides the bending rigidity of the layer. Non-equilibrium methods, in particular umbrella sampling, potential of mean constraint force, and thermodynamic integration in Cartesian coordinates, have been used to extend the range of sampled amplitudes. For small amplitudes there is a good agreement with the equilibrium simulations, while beyond the thermally accessible amplitudes a clear deviation from theory is observed. Calculations of the elastic modulus showed a pronounced system size dependence.

Microphase separation and liquid-crystalline ordering of rod-coil copolymers

Microphase separation and liquid-crystalline ordering in diblock and triblock rod-coil copolymers (with rod-to-coil fraction f=0.5) were investigated using the dissipative particle dynamics method. When the isotropic disordered phases of these systems were cooled down below their order-disorder transition temperatures T(ODT), lamellar structures were observed. For rod-coil diblock copolymers, the lamellar layers were obtained below T=2.0. This temperature was found to be higher than the T(ODT) for normal coil-coil diblock copolymers. Significant ordering of the rods was observed only below T=0.9 which is the isotropic-nematic transition temperature for rodlike fluids. For the triblock rod-coil copolymers, both microphase separation and rod ordering occurred at T=0.9. Normal coil-coil triblock copolymers were found to undergo microphase separation at T=0.8, which is about half the T(ODT) of the normal diblock copolymers. Investigations of the mean square displacement and the parallel and the perpendicular components of the spatial distribution function revealed that at low temperatures, the rod-coil diblock copolymers exhibit smectic-A and crystalline phases, while the triblock copolymers show smectic-C and crystalline phases. No nematic phases were observed at the density and interaction parameters used in this study.

Area compressibility and buckling of amphiphilic bilayers in molecular dynamics simulations.

The elastic modulus or area compressibility of a membrane is routinely calculated in molecular dynamics simulations as the proportionality constant relating surface tension and projected surface area. Recent studies, however, have revealed a marked system size dependence of these moduli, which we attribute to the neglect of thermal undulations in the area calculation. We discuss several methods, based on the Helfrich model and on numerical triangulation, to remedy this situation, and find a satisfying agreement between them. The Helfrich model also quantitatively describes a buckling transition observed for compressed bilayers.

Brownian dynamics simulations of the self- and collective rotational diffusion coefficients of rigid long thin rods

Recently a microscopic theory for the dynamics of suspensions of long thin rigid rods was presented, confirming and expanding the well-known theory by Doi and Edwards [The Theory of Polymer Dynamics (Clarendon, Oxford, 1986)] and Kuzuu [J. Phys. Soc. Jpn. 52, 3486 (1983)]. Here this theory is put to the test by comparing it against computer simulations. A Brownian dynamics simulation program was developed to follow the dynamics of the rods, with a length over a diameter ratio of 60, on the Smoluchowski time scale. The model accounts for excluded volume interactions between rods, but neglects hydrodynamic interactions. The self-rotational diffusion coefficients D(r)(phi) of the rods were calculated by standard methods and by a new, more efficient method based on calculating average restoring torques. Collective decay of orientational order was calculated by means of equilibrium and nonequilibrium simulations. Our results show that, for the currently accessible volume fractions, the decay times in both cases are virtually identical. Moreover, the observed decay of diffusion coefficients with volume fraction is much quicker than predicted by the theory, which is attributed to an oversimplification of dynamic correlations in the theory.

Alignment of particles in sheared viscoelastic fluids

We investigate the shear-induced structure formation of colloidal particles dissolved in non-Newtonian fluids by means of computer simulations. The two investigated visco-elastic fluids are a semi-dilute polymer solution and a worm-like micellar solution. Both shear-thinning fluids contain long flexible chains whose entanglements appear and disappear continually as a result of Brownian motion and the applied shear flow. To reach sufficiently large time and length scales in three-dimensional simulations with up to 96 spherical colloids, we employ the responsive particle dynamics simulation method of modeling each chain as a single soft Brownian particle with slowly evolving inter-particle degrees of freedom accounting for the entanglements. Parameters in the model are chosen such that the simulated rheological properties of the fluids, i.e., the storage and loss moduli and the shear viscosities, are in reasonable agreement with experimental values. Spherical colloids dispersed in both quiescent fluids mix homogeneously. Under shear flow, however, the colloids in the micellar solution align to form strings in the flow direction, whereas the colloids in the polymer solution remain randomly distributed. These observations agree with recent experimental studies of colloids in the bulk of these two liquids.

Coarse graining of slow variables in dynamic simulations of soft matter

A new Brownian dynamics model is presented to describe the coarse grain dynamics of particles with long-lived memory. Instead of solving a set of generalized Langevin equations we introduce a set of variables describing the slowly fluctuating thermodynamic state of the ignored degrees of freedom. These variables give rise to additional transient forces on the simulated particles, whose interpretation provides a new way of thinking about memory effects in soft-matter physics. We illustrate the proposed method by simulating shear thinning of synthetic resins.

Isotropic-nematic spinodals of rigid long thin rodlike colloids by event-driven Brownian dynamics simulations.

The isotropic-nematic spinodals of solutions of rigid spherocylindrical colloids with various shape anisotropies L/D in a wide range from 10 to 60 are investigated by means of Brownian dynamics simulations. To make these simulations feasible, we developed a new event-driven algorithm that takes the excluded volume interactions between particles into account as instantaneous collisions, but neglects the hydrodynamic interactions. This algorithm is applied to dense systems of highly elongated rods and proves to be efficient. The calculated isotropic-nematic spinodals lie between the previously established binodals in the phase diagram and extrapolate for infinitely long rods to Onsagers [Ann. N. Y. Acad. Sci. 51, 627 (1949)] theoretical predictions. Moreover, we investigate the shear induced shifts of the spinodals, qualitatively confirming the theoretical prediction of the critical shear rate at which the two spinodals merge and the isotropic-nematic phase transition ceases to exist.