Universality of the collapse transition of sticky polymers.

The universality of the swelling of the radius of gyration of a homopolymer relative to its value in the θ state, independent of polymer-solvent chemistry, in the crossover regime between θ and athermal solvent conditions, is well known. Here we study, by Brownian dynamics, a polymer model where a subset of monomers is labelled as "stickers". The mutual interaction of the stickers is more attractive than those of the other ("backbone") monomers, and has an additional important characteristic of "functionality" φ, i.e., the maximum number of stickers that can locally bind to a given sticker. A saturated bond formed in this manner remains bound until it breaks due to thermal fluctuations, a requirement which can be viewed as an additional Boolean degree of freedom that describes the bonding. This, in turn, makes the question of the order of the collapse transition a non-trivial one. Nevertheless, for the parameters that we have studied (in particular, φ = 1), we find a standard second-order θ collapse, using a renormalised solvent quality parameter that takes into account the increased average attraction due to the presence of stickers. We examine the swelling of the radius of gyration of such a sticky polymer relative to its value in the altered θ state, using a novel potential to model the various excluded volume interactions that occur between the monomers on the chain. We find that the swelling of such sticky polymers is identical to the universal swelling of homopolymers in the thermal crossover regime. Additionally, for our model, the Kuhn segment length under θ conditions is found to be the same for chains with and without stickers.

Equilibrium binding energies from fluctuation theorems and force spectroscopy simulations.

Brownian dynamics simulations are used to study the detachment of a particle from a substrate. Although the model is simple and generic, we attempt to map its energy, length and time scales onto a specific experimental system, namely a bead that is weakly bound to a cell and then removed by an optical tweezer. The external driving force arises from the combined optical tweezer and substrate potentials, and thermal fluctuations are taken into account by a Brownian force. The Jarzynski equality and Crooks fluctuation theorem are applied to obtain the equilibrium free energy difference between the final and initial states. To this end, we sample non-equilibrium work trajectories for various tweezer pulling rates. We argue that this methodology should also be feasible experimentally for the envisioned system. Furthermore, we outline how the measurement of a whole free energy profile would allow the experimentalist to retrieve the unknown substrate potential by means of a suitable deconvolution. The influence of the pulling rate on the accuracy of the results is investigated, and umbrella sampling is used to obtain the equilibrium probability of particle escape for a variety of trap potentials.

Size, shape, and diffusivity of a single Debye-Hückel polyelectrolyte chain in solution.

Brownian dynamics simulations of a coarse-grained bead-spring chain model, with Debye-Hückel electrostatic interactions between the beads, are used to determine the root-mean-square end-to-end vector, the radius of gyration, and various shape functions (defined in terms of eigenvalues of the radius of gyration tensor) of a weakly charged polyelectrolyte chain in solution, in the limit of low polymer concentration. The long-time diffusivity is calculated from the mean square displacement of the centre of mass of the chain, with hydrodynamic interactions taken into account through the incorporation of the Rotne-Prager-Yamakawa tensor. Simulation results are interpreted in the light of the Odjik, Skolnick, Fixman, Khokhlov, and Khachaturian blob scaling theory (Everaers et al., Eur. Phys. J. E 8, 3 (2002)) which predicts that all solution properties are determined by just two scaling variables-the number of electrostatic blobs X and the reduced Debye screening length, Y. We identify three broad regimes, the ideal chain regime at small values of Y, the blob-pole regime at large values of Y, and the crossover regime at intermediate values of Y, within which the mean size, shape, and diffusivity exhibit characteristic behaviours. In particular, when simulation results are recast in terms of blob scaling variables, universal behaviour independent of the choice of bead-spring chain parameters, and the number of blobs X, is observed in the ideal chain regime and in much of the crossover regime, while the existence of logarithmic corrections to scaling in the blob-pole regime leads to non-universal behaviour.

Numerical electrokinetics.

A new lattice method is presented in order to efficiently solve the electrokinetic equations, which describe the structure and dynamics of the charge cloud and the flow field surrounding a single charged colloidal sphere, or a fixed array of such objects. We focus on calculating the electrophoretic mobility in the limit of small driving field, and systematically linearize the equations with respect to the latter. This gives rise to several subproblems, each of which is solved by a specialized numerical algorithm. For the total problem we combine these solvers in an iterative procedure. Applying this method, we study the effect of the screening mechanism (salt screening versus counterion screening) on the electrophoretic mobility, and find a weak non-trivial dependence, as expected from scaling theory. Furthermore, we find that the orientation of the charge cloud (i.e. its dipole moment) depends on the value of the colloid charge, as a result of a competition between electrostatic and hydrodynamic effects.

Optimization of a Brownian-dynamics algorithm for semidilute polymer solutions.

Simulating the static and dynamic properties of semidilute polymer solutions with Brownian dynamics (BD) requires the computation of a large system of polymer chains coupled to one another through excluded-volume and hydrodynamic interactions. In the presence of periodic boundary conditions, long-ranged hydrodynamic interactions are frequently summed with the Ewald summation technique. By performing detailed simulations that shed light on the influence of several tuning parameters involved both in the Ewald summation method, and in the efficient treatment of Brownian forces, we develop a BD algorithm in which the computational cost scales as O(N(1.8)), where N is the number of monomers in the simulation box. We show that Beenakker's original implementation of the Ewald sum, which is only valid for systems without bead overlap, can be modified so that θ solutions can be simulated by switching off excluded-volume interactions. A comparison of the predictions of the radius of gyration, the end-to-end vector, and the self-diffusion coefficient by BD, at a range of concentrations, with the hybrid lattice Boltzmann-molecular dynamics (LB-MD) method shows excellent agreement between the two methods. In contrast to the situation for dilute solutions, the LB-MD method is shown to be significantly more computationally efficient than the current implementation of BD for simulating semidilute solutions. We argue, however, that further optimizations should be possible.

Dynamic crossover scaling in polymer solutions.

The crossover region in the phase diagram of polymer solutions, in the regime above the overlap concentration, is explored by Brownian dynamics simulations to map out the universal crossover scaling functions for the gyration radius and the single-chain diffusion constant. Scaling considerations, our simulation results, and recently reported data on the polymer contribution to the viscosity obtained from rheological measurements on DNA systems support the assumption that there are simple relations between these functions, such that they can be inferred from one another.

Studying flow close to an interface by total internal reflection fluorescence cross-correlation spectroscopy: quantitative data analysis.

Total internal reflection fluorescence cross-correlation spectroscopy (TIR-FCCS) has recently [S. Yordanov et al., Optics Express 17, 21149 (2009)] been established as an experimental method to probe hydrodynamic flows near surfaces, on length scales of tens of nanometers. Its main advantage is that fluorescence occurs only for tracer particles close to the surface, thus resulting in high sensitivity. However, the measured correlation functions provide only rather indirect information about the flow parameters of interest, such as the shear rate and the slip length. In the present paper, we show how to combine detailed and fairly realistic theoretical modeling of the phenomena by Brownian dynamics simulations with accurate measurements of the correlation functions, in order to establish a quantitative method to retrieve the flow properties from the experiments. First, Brownian dynamics is used to sample highly accurate correlation functions for a fixed set of model parameters. Second, these parameters are varied systematically by means of an importance-sampling Monte Carlo procedure in order to fit the experiments. This provides the optimum parameter values together with their statistical error bars. The approach is well suited for massively parallel computers, which allows us to do the data analysis within moderate computing times. The method is applied to flow near a hydrophilic surface, where the slip length is observed to be smaller than 10nm, and, within the limitations of the experiments and the model, indistinguishable from zero.

Simple and robust solver for the Poisson-Boltzmann equation.

A variational approach is used to develop a robust numerical procedure for solving the nonlinear Poisson-Boltzmann equation. Following Maggs and Rossetto [Phys. Rev. Lett. 88, 196402 (2002)], we construct an appropriate constrained free energy functional such that its Euler-Lagrange equations are equivalent to the Poisson-Boltzmann equation. This is a formulation that searches for a true minimum in function space, in contrast to previous variational approaches that rather searched for a saddle point. We then develop, implement, and test an algorithm for its numerical minimization, which is quite simple and unconditionally stable. The analytic solution for planar geometry is used for validation. Some results are presented for a charged colloidal sphere surrounded by counterions and optimizations based upon fast Fourier transforms and hierarchical preconditioning are briefly discussed.

Suppression of capillary wave broadening of interfaces in binary alloys due to elastic interactions.

By Monte Carlo simulations in the constant-temperature--constant-pressure ensemble a planar interface between unmixed A-rich and B-rich phases of a binary (A, B) alloy on a compressible diamond lattice is studied. No significant capillary wave broadening of the concentration profile across the interface is observed, unlike lattice models of incompressible mixtures and fluids. The distortion of the lattice structure across the interface is studied.

Shear-induced undulation of smectic-A: Molecular Dynamics simulations vs. analytical theory.

Experiments on a variety of systems have shown that layered liquids are unstable under shear even if the liquid layers are planes of constant velocity. We investigate the stability of smectic- A like liquids under shear using Molecular Dynamics simulations and a macroscopic hydrodynamic theory (including the layer normal and the director as independent variables). Both methods show an instability of the layers, which sets in above a critical shear rate. We find a remarkable qualitative and reasonable quantitative agreement between both methods for the spatial homogeneous state and the onset of the instability.

Screening of hydrodynamic interactions in semidilute polymer solutions: a computer simulation study.

We study single-chain motion in semidilute solutions of polymers of length N=1000 with excluded-volume and hydrodynamic interactions by a novel algorithm. The crossover length of the transition from Zimm (short lengths and times) to Rouse dynamics (larger scales) is proportional to the static screening length. The crossover time is the corresponding Zimm time. Our data indicate Zimm behavior at large lengths but short times. There is no hydrodynamic screening until the chains feel constraints, after which they resist the flow: "Incomplete screening" occurs in the time domain.

Parallel excluded volume tempering for polymer melts.

We have developed a technique to accelerate the acquisition of effectively uncorrelated configurations for off-lattice models of dense polymer melts that makes use of both parallel tempering and large-scale Monte Carlo moves. The method is based upon simulating a set of systems in parallel, each of which has a slightly different repulsive core potential, such that a thermodynamic path from full excluded volume to an ideal gas of random walks is generated. While each system is run with standard stochastic dynamics, resulting in an NVT ensemble, we implement the parallel tempering through stochastic swaps between the configurations of adjacent potentials, and the large-scale Monte Carlo moves through attempted pivot and translation moves that reach a realistic acceptance probability as the limit of the ideal gas of random walks is approached. Compared to pure stochastic dynamics, this results in an increased efficiency even for a system of chains as short as N=60 monomers, however at this chain length the large-scale Monte Carlo moves were ineffective. For even longer chains, the speedup becomes substantial, as observed from preliminary data for N=200. We also compare our scheme to the end bridging algorithm of Theodorou et al. For N=60, end bridging must allow a polydispersity of more than 10% in order to relax the end-to-end vector more quickly than our method. The comparison is, however, hampered by the fact that the end-to-end vector becomes a somewhat artificial quantity when one implements end bridging, and is perhaps no longer the slowest dynamic variable.

Conformations of random polyampholytes

We study the size R(g) of random polyampholytes (i.e., polymers with randomly charged monomers) as a function of their length N. All results of our extensive Monte Carlo simulations can be rationalized in terms of the scaling theory we develop for the Kantor-Kardar necklace model, although this theory neglects the quenched disorder in the charge sequence along the chain. We find approximately N1/2. The elongated globule model, the initial predictions of both Higgs and Joanny ( approximately N1/3) and Kantor and Kardar ( approximately N), and previous numerical estimates are ruled out.