# Simulation of the Coiling of a Polymer Strand

Tsvetoslav Pavlov Department of Materials, Imperial College London, United Kingdom

27th Feb 2012

1. Abstract

This study will examine the temperature dependence of internal energy, heat capacity and R2 value (representative of the end to end distance) using a Langevin Dynamics simulation. It will also consider the dependence of the internal energy, heat capacity and R2 value with increasing polymer chain length. Internal energy has been found to be is negative at low temperatures, increases linearly with temperature and becomes positive at high temperatures. Heat capacity seems to increase linearly with a higher number of atoms in the chain due to additional atoms storing additional energy. The internal energy increases initially with a higher number of atoms. with increasing atomic separation. It then reaches a peak and decreases rapidly with further increases in chain length. This has been explained to be due to the negative eect of non- bonding interactions

2. Introduction

For many applications the interaction of a polymer with a solvent is vital. Polymer interactions can be examined using computational methods such as a Metropolis Monte Carlo method, Molecular Dynamics or Langevin Dynamics. It is important to know when for a certain system a solvent is good, bad or in-between (theta solvent). A good solvent would maximise the polymer-solvent interaction by unfolding while a bad solvent will minimize these interactions by keeping the polymer folded. It is also important to understand how the polymer behaves at dierent temperatures and chain lengths. Computer simulations allow for a polymer to be examined in great detail whereby very accurate values can be obtained, the accuracy of which depends on the accuracy of the applied model. Using computer simulations removes the costs associated with numerous lab experiments.

3. Method

Langevin Dynamics has been used to set up the simulation. It is identical to molecular dynamics in the respect that it uses Newton's second law. However, Langevin Dynamics takes into account two additional forces (drag and random impulse forces) which are due to the polymer being surrounded by a uid. The drag force as the name suggests is the stopping force exerted due to relative motion between the polymer and the uid. The random impulse force arises from uid particles hitting the polymer. Molecular Dynamics works on the basis of moving atoms a step at a time under an applied force. These atoms obey Newton's second law. The displacement for example can be expressed as a Taylor series with a time step of

δt.

Only the rst three terms of the Taylor expansion are considered for most applications (these

involve displacement, velocity and acceleration) as the next terms decay exponentially for a relatively small time step. The Taylor series for the velocity would consist only of two terms respectively (velocity and acceleration). By substituting the velocity series into the displacement series it is possible to obtain an expression for the acceleration and thus for the force exerted on a xed mass. However, a better method for approximating these values is the Verlet algorithm which considers the Taylor series of displacements at times (t-δt) and (t+δt). Combining the two Taylor series makes it possible to take out the velocity dependence in the equation. This approach greatly simplies the expression for the force. Having obtained an expression for the force exerted on a particle/atom based on positions, velocities and accelerations it is possible to compute quantities which arise due to motion. These are internal energy, stress and temperature.

1

4. Results

Table 1: The average mean and standard deviation values for the dierent time steps. The avrages have been taken over 10 runs. Time step 0.003 0.005

Average Standard Deviation 1.6 x 10

-5

Average Mean 0.218

0.234

Mean/Standard Deviation 7.34 x 10...

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