Peter Biller

A numerical stochastic approach to network theories of polymeric fluids

A new numerical approach is presented to exactly solve the convection equation arising in network theories. The method is based on a direct stochastic interpretation of the convection equation. We show that with this approach models can be studied extensively which are not solvable analytically. It turns out that a conceptually simple approach to network theories predicts a qualitatively satisfying rheological behavior.

Continuous time simulation of transient polymer network models

A continuous time simulation algorithm for polymer melts is presented. The method is introduced via an explicit application to transient polymer network models, but it may be applied to a much larger class of models. The central quantity of the simulation is the lifetime of a strand. This can be calculated according to a proper distribution, which may depend on the orientation and length of the generated new strand. As the equation of motion of the strand can be integrated analytically during the lifetime of the strand, the procedure works much more efficiently than a previous simulation algorithm for transient network models. The material functions can be calculated as ensemble averages. The new method is shown to work for time‐dependent flows. A simplified algorithm for steady flows is also given. Some predictions are made for a specific configuration‐dependent transient network model.

Consistently averaged hydrodynamic interaction for dumbbell models in elongational flow

We apply the recently proposed consistent averaging approximation for the hydrodynamic interaction to elongational flow. This flow situation can be treated to a great extent analytically for Rouse and FENE‐P dumbbells. In both cases expansions for the elongational viscosity are presented for low and high elongational rates.

A New Model for Polymer Melts and Concentrated Solutions

A new mesoscopic model is presented for polymer melts and concentrated solutions. It is a single Kramers chain model in which elementary motions of the Orwoll–Stockmayer type are allowed. However, for this model, the bead jumps are no longer given by a Markovian probability, but rather are described by ‘‘a waiting time distribution function.’’ Such a distribution is supposed to occur when the chain is ‘‘frozen’’ in space until a ‘‘gap’’ in the solution or melt meets with the bead or chain segment. The time a bead must wait to jump is given by a distribution function with a single adjustable parameter β, which describes the long‐time behavior of the distribution: ∼1/t1+β . We find that the model predicts non‐Fickian diffusion in agreement with experimental data and Fickian diffusion for longer times which scales with chain length as 1/N2/α−1, where α is a function of β. For β=1.3, D∼1/N2.28. The autocorrelation of the end‐to‐end vector of the chain is a stretched‐exponential form with a time constant which...

Continuous time simulation of the Doi-Edwards model

A continuous time simulation algorithm for polymer melts is applied to the Doi–Edwards model. In this model a certain tube segment disappears when it is reached by either end of the primitive chain. The probability that this segment remains at a certain time is well‐known analytically. In the underlying stochastic process a unit vector is chosen with new random orientation every time the diffusion process reaches one of its boundaries. To simulate this stochastic process an algorithm is used which is very similar to one introduced for transient network polymer models. In the special case of the Doi–Edwards model one can draw random birth times for the unit vectors from the appropriate probability distribution. The deterministic equations of motion are integrated analytically. This leads to a very efficient method for the simulation of the stochastic process.We compare the simulation results with results which were already obtained by other methods.

Nonlinear dumbbell model for flexible polymers: dynamical phenomena

Abstract We study the stress growth upon the inception of steady shear and steady elongational flow for a nonlinear dumbbell model for flexible macromolecules suspended in a Newtonian fluid. The internal force law is made nonlinear by adding a cubic term to the Hookean force law. The generalization to a nonlinear chainlike bead—spring model is outlined. In order to consider higher concentration effects also the configuration-dependent tensorial mobility is included. We use the mean-field approximation and test this approximation with a Brownian dynamics simulation. The mean-field results for the material functions are compared with experimental results for both flow situations.

Chaotic inflation: A numerical approach

Abstract A numerical study of chaotic inflation is presented. Following a semiclassical treatment of quantum effects, the dynamics is described as a random process. The relevant Langevin equation is then integrated numerically for a large number of realizations and results are evaluated as ensemble averages. For the understanding of the global structure of the universe the fact that different domains of the universe have different growth rates is important. This is handled by a new modified algorithm. The simulation results for the probability distribution functions at constant and proper volume are given for two typical initial conditions. We compare them to the approximate results of an already existing analytical approach. The picture of an eternally existing self-reproducing universe is confirmed.

Algorithms for the simulation of network models of the Yamamoto type

Abstract Concentrated polymer solutions and melts are usually simulated by molecular dynamics, Brownian dynamics or Monte Carlo methods. In this paper we describe another simulation approach. It is based on transient polymer network theories which can correctly describe the rheological behaviour. Due to a stochastic interpretation of the fundamental equations of these theories it is possible to formulate algorithms to simulate the dynamics of the network. Network models that cannot be treated analytically can then be studied. One of the algorithms presented here works with small discrete time steps. It can be applied to the modt general models. For some simpler models it is possible to formulate a continuous-time algorithm which is generally more efficient.