Elsevier

Chemical Physics Letters

Volume 316, Issues 3–4, 14 January 2000, Pages 285-296
Chemical Physics Letters

Time-averaged normal coordinate analysis of polymer particles and crystals

https://doi.org/10.1016/S0009-2614(99)01152-5Get rights and content

Abstract

A common problem in the application of normal coordinate analysis to study low-frequency modes of large molecular systems is the occurrence of a large number of negative eigenvalues (unstable modes). By averaging the terms of the Hessian matrix over a short classical trajectory, the unstable modes were found to be completely eliminated for 6000 atom model polymer particles and crystals. The time-averaged matrices were made possible by an efficient analytical formulation of the Cartesian second derivatives and diagonalization was achieved using a sparse matrix solver (ARPACK).

Introduction

Normal coordinate analysis (NCA) is an important tool to study spectral and thermal details of polymer processes at the atomic or molecular level (see, e.g., Refs. 1, 2). The reason for this is that once the spectral information has been obtained, it is very simple to link these observations to experimentally accessible macroscopic properties of polymeric materials. Although in the past normal mode calculations have been limited to relatively small systems (i.e. ∼1000 atoms), new methods for diagonalizing large sparse matrices and the rapid increase in the available computer memory allow routine calculations of much larger systems. For example, systems with 1 000 000 normal modes (∼333 333 atoms) are now very feasible using new programs such as ARPACK[3] for the calculation of eigenvalues for large sparse matrices. Unfortunately, we have found that the traditional methods for the calculation of the Hessian matrix results in large numbers of negative eigenvalues, indicating many unstable degrees of freedom. Because of this problem, reliable calculation of the low-frequency modes of large polymer systems is very difficult. However, these particular modes are extremely important for studying the structural stability of nano-devices and various biological processes in enzymes and proteins.

Recently an experimental technique was developed in our laboratory for creating very fine polymer particles of arbitrary composition and size 4, 5, 6. In the experiment, we used previously developed instrumentation for generation and characterization of droplet streams with small (⩽1–2 μm) average diameter and monodispersity for probing single molecules in solution 7, 8. This technique makes the initial volume of dilute solution sufficiently small so that the solvent evaporates on a short timescale. The particles in nanometer and micrometer size range provide many unique physical properties due to size reduction to the point where critical length scales of physical phenomena become comparable to or larger than the size of the structure. Applications of such particles take advantage of high surface area and confinement effects, which leads to nanostructures with different properties than conventional materials. Clearly such changes offer extraordinary potential for development of new materials in the form of bulk composites and blends which can be used for nano-manufacturing and bioengineering [9].

A computational algorithm for generating and modeling polymer particles for our experiments was developed to construct particles that are as similar as possible to those experimentally generated. We have examined a variety of polyethylene (PE) nano-scale particles, allowing the systematic study of size-dependent physical properties of these particles [10]. The model parameters have been well tested and shown to provide realistic representation of the structure and vibrational spectroscopy of a number of polymer systems: harmonic/Morse potentials for the bond stretches, harmonic potential for bending between two bonds, a truncated Fourier series for the torsional potential, and Lennard-Jones 6-12 potentials for the non-bonded interactions (both chain–chain and intra-chain) 11, 12, 13, 14, 15.

In this Letter, we will for the first time use NCA to characterize the large-amplitude low-frequency vibrational modes of polymer nanoparticles and contrast them with the transverse modes of polymer crystals of similar size. Because of the large number of negative eigenvalues found for a polymer particle, the standard application of the NCA method to determine the low-frequency modes was found to be inadequate. In the evaluation of the Hessian matrix elements for a set of closely related polymer particle structures, large variations were found. Using molecular dynamics (MD) to study the time dependence of these matrix elements, we found that while variations in the individual terms were large, they oscillated about an average value. Using the time-averaged values for the Hessian matrix, the negative eigenvalues can be eliminated. In Section 2, we present the models used with the results of the calculations given in Section 3. Our conclusions of this study are presented in Section 4.

Section snippets

Model development

Normal mode analysis and MD method are well-known methods based on classical mechanics and have been reviewed in several books and papers 1, 2. The standard normal mode analysis method involves solving the secular equation|F−λI|=0,where λ are the eigenvalues and F is the force constant matrix in mass weighted Cartesian coordinates. The force matrix is obtained from the second derivatives of potential function V,F=M−1/2(∇2V)M−1/2.and M is the mass matrix.

Molecular dynamics, molecular mechanics,

Calculations

Using the methods described above, the time dependence for typical diagonal, off-diagonal (but large) and non-bonded matrix elements for the polymer particle and crystal are shown in Fig. 2a,b. As can be easily seen, the resulting plots of the fluctuations of a randomly chosen matrix element are largest for the diagonal crystal terms (Fig. 2a) with a variation of ∼100%. The corresponding randomly chosen matrix element variations for the particle are also considerable: on the order of ±10%. The

Conclusions

In this study we have found that averaging the terms of the Hessian matrix (second derivatives of the energy function) over a short trajectory reduces the number of unstable modes for 6000 atom model polymer particles and crystals used in this study and improves the resulting eigenvectors so that they can easily be interpreted. Recently other methods by Karplus et al. 31, 32 have been proposed for eliminating the negative eigenvalues for bio-polymers. One of proposed methods involved shifting

Acknowledgements

C.Y. was supported in part by a Householder fellowship at Oak Ridge National Laboratory (ORNL) and K.F. is supported by the Postdoctoral Research Associates Program administered jointly by ORNL and Oak Ridge Institute for Science and Education. This research was sponsored by the Division of Materials Sciences, Office of Basic Energy Sciences, US Department of Energy under Contract DE-AC05-96OR22464 with Lockheed–Martin Energy Research. We thank NEC for assistance in using the NEC SX-4

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