Christer Elvingson

Monte Carlo simulation of polymer brushes attached to a spherical surface

We present results of Monte Carlo simulations of polymer brushes attached to a spherical surface. We have varied the chain length, surface radius, and the grafting density, and the analysis has included the size, shape, and orientation as well as segment density profiles. We also compared different methods of generating the starting configurations to establish the results being independent of the initial structure. For flexible molecules, the segment density profiles agree with earlier results from self-consistent field calculations, and there is no depletion zone for the chain ends close to the surface, except for the largest molecules. Increasing the persistence length, the orientation of the molecules increases and the chain density close to the surface decreases, but there is still a finite probability of finding a chain end close to the particle surface. We have also looked at the effect of restricting the available space for the chains by an outer bounding surface and the change in molecular shape and size is then seen much earlier compared to the effect on the subunit density profiles.We present results of Monte Carlo simulations of polymer brushes attached to a spherical surface. We have varied the chain length, surface radius, and the grafting density, and the analysis has included the size, shape, and orientation as well as segment density profiles. We also compared different methods of generating the starting configurations to establish the results being independent of the initial structure. For flexible molecules, the segment density profiles agree with earlier results from self-consistent field calculations, and there is no depletion zone for the chain ends close to the surface, except for the largest molecules. Increasing the persistence length, the orientation of the molecules increases and the chain density close to the surface decreases, but there is still a finite probability of finding a chain end close to the particle surface. We have also looked at the effect of restricting the available space for the chains by an outer bounding surface and the change in molecular shape a...

Algorithm for generating a Brownian motion on a sphere

We present a new algorithm for generation of a random walk on a two-dimensional sphere. The algorithm is obtained by viewing the 2-sphere as the equator in the 3-sphere surrounded by an infinitesimally thin band with boundary which reflects Brownian particles and then applying known effective methods for generating Brownian motion on the 3-sphere. To test the method, the diffusion coefficient was calculated in computer simulations using the new algorithm and, for comparison, also using a commonly used method in which the particle takes a Brownian step in the tangent plane to the 2-sphere and is then projected back to the spherical surface. The two methods are in good agreement for short time steps, while the method presented in this paper continues to give good results also for larger time steps, when the alternative method becomes unstable.

Brownian dynamics simulations on a hypersphere in 4-space

We describe an algorithm for performing Brownian dynamics simulations of particles diffusing on S3, a hypersphere in four dimensions. The system is chosen due to recent interest in doing computer simulations in a closed space where periodic boundary conditions can be avoided. We specifically address the question how to generate a random walk on the 3-sphere, starting from the solution of the corresponding diffusion equation, and we also discuss an efficient implementation based on controlled approximations. Since S3 is a closed manifold (space), the average square displacement during a random walk is no longer proportional to the elapsed time, as in R3. Instead, its time rate of change is continuously decreasing, and approaches zero as time becomes large. We show, however, that the effective diffusion coefficient can still be obtained from the time dependence of the square displacement.

Effect of compression on the molecular shape of polymer mushrooms with variable stiffness

Under confinement, the average shape of a polymer chain is modified in interesting ways. In this work, we discuss how confinement affects the mean geometrical properties of wormlike polymers with variable flexibility and monomer–monomer interaction. Here, we consider a polymer mushroom, i.e., a single chain that is permanently anchored to a flat surface by an end point. Compression is introduced by confining the chains inside an infinite slab with parallel hard walls. Regarding polymer shape, we focus on two large-scale geometrical properties that are not correlated a priori: the chain’s size and its entanglement complexity. Using Monte Carlo simulations, we have analyzed the behavior of these two properties under confinement for a range of potential energy functions. A recurrent pattern of shape transitions emerges, as indicated by changes in the correlation between mean size and entanglements. Our results show that, whereas a flexible polymer with strong self-attraction sustains high compression without...

Cluster identification and percolation analysis using a recursive algorithm

Abstract A recursive algorithm for sampling properties of physical clusters such as size distribution and percolation is presented. The approach can be applied to any system with periodic boundary conditions, given a spatial definition of a cluster. We also introduce some modifications in the algorithm that increases the efficiency considerably if one is only interested in percolation analysis. The algorithm is implemented in Fortran 90 and is compared with a number of iterative algorithms. The recursive cluster identification algorithm is somewhat slower than the iterative methods at low volume fraction but is at least as fast at high densities. The percolation analysis, however, is considerably faster using recursion, for all systems studied. We also note that the CPU time using recursion is independent on the static allocation of arrays, whereas the iterative method strongly depends on the size of the initially allocated arrays.

Computer simulations of polymer chain structure and dynamics on a hypersphere in four-space

There is a rapidly growing interest in performing computer simulations in a closed space, avoiding periodic boundary conditions. To extend the range of potential systems to include also macromolecules, we describe an algorithm for computer simulations of polymer chain molecules on S3, a hypersphere in four dimensions. In particular, we show how to generate initial conformations with a bond angle distribution given by the persistence length of the chain and how to calculate the bending forces for a molecule moving on S3. Furthermore, we discuss how to describe the shape of a macromolecule on S3, by deriving the radius of gyration tensor in this non-Euclidean space. The results from both Monte Carlo and Brownian dynamics simulations in the infinite dilution limit show that the results on S3 and in R3 coincide, both with respect to the size and shape as well as for the diffusion coefficient. All data on S3 can also be described by master curves by suitable scaling by the corresponding values in R3. We thus show how to extend the use of spherical boundary conditions, which are most effective for calculating electrostatic forces, to polymer chain molecules, making it possible to perform simulations on S3 also for polyelectrolyte systems.

Tracer diffusion in a polymer gel: simulations of static and dynamic 3D networks using spherical boundary conditions

We have investigated an alternative to the standard periodic boundary conditions for simulating the diffusion of tracer particles in a polymer gel by performing Brownian dynamics simulations using spherical boundary conditions. The gel network is constructed by randomly distributing tetravalent cross-linking nodes and connecting nearest pairs. The final gel structure is characterised by the radial distribution functions, chain lengths and end-to-end distances, and the pore size distribution. We have looked at the diffusion of tracer particles with a wide range of sizes, diffusing in both static and dynamic networks of two different volume fractions. It is quantitatively shown that the dynamical effect of the network becomes more important in facilitating the diffusional transport for larger particle sizes, and that one obtains a finite diffusion also for particle sizes well above the maximum in the pore size distribution.

Construction of a closed polymer network for computer simulations

Computer simulations are an important tool for linking the behaviour of polymer materials to the properties of the constituent polymer chains. In simulations, one normally uses periodic boundary conditions to mimic a macroscopic system. For a cross-linked polymer network, this will impose restrictions on the motion of the polymer chains at the borders of the simulation cell. We present a new method for constructing a three-dimensional closed network without periodic boundaries by embedding the system onto the surface of a sphere in four dimensions. This method can also be used to construct finite-sized gel particles for simulating the swelling of particles in a surrounding solvent. The method is described in algorithmic detail to allow the incorporation of the method into different types of simulation programs. We also present the results of Brownian dynamics simulations, analyzing the end-to-end distribution, radial distribution function, and the pore size distribution for different volume fractions and for chains with varying stiffness.

Role of non-uniform confinement in shape transitions of semi-stiff polymers

New types of nanostructures are constantly being developed synthetically but are also found in biological systems. Specific examples include the production of carbon nanocones as well as the conical core in some viruses. Such conical structures can be used to investigate the role of non-uniform confinement on the stability of e.g. toroidal structures formed by semi-stiff circular polymers, such as DNA. In this communication we are interested in the principal features of the compaction process. Using an external field and a conical confinement we observe several distinct shape transitions from a circle-like shape to several toroidal-like loops for both a two-dimensional and a three-dimensional system. The thermodynamic stability of these toroidal-like structures was investigated by evaluating their relative free energies using Monte Carlo simulations in the Extended Ensemble in the case of a two-dimensional system and by observing a hysteresis of the compaction-extension curve for the three-dimensional case.

Determination of the equilibrium charge distribution for polyampholytes of different compactness in a single computer experiment

The conformational properties of charge-balanced polyampholytes described by the end-to-end distance or radius of gyration depend on parameters such as the temperature and pH as well as on the detailed charge distribution along the backbone. In this work we present a method to determine the charge distribution along a semi-stiff polyampholyte backbone which will result in a thermodynamically stable structure for the compactness of interest, from several loops to an uncoiled structure, performed in a single computer experiment.