Robert E. Tuzun

Continuum methods of mechanics as a simplified approach to structural engineering of nanostructures

Recent studies of potential components for nanomachines reveal that for a wide variety of structures, the rigidity of the structure is a key element in its proper performance. Vibrational analysis is an ideal way to study structural rigidity, but standard methods of molecular vibrational analysis are computationally prohibitive for nanostructures with large numbers of atoms. Herein, the vibration of nanotubes is used to demonstrate that continuum methods of vibrational analysis have potential utility in the engineering of nanostructures.

The dynamics of molecular bearings

Various types of molecular bearings have recently been proposed in the growing nanotechnology literature. Using novel molecular dynamics methods, we have simulated several model graphite bearings. The bearings varied in size from inner shafts of between 4 and 16 A in diameter, up to 120 A in length, and outer cylinders of between 10 and 23 A in diameter, up to 40 A in length. The turning shaft was either instantaneously started or torqued up to the desired rotational speeds. Frictional properties were size-, temperature- and velocity-dependent. The presence of more than one bearing vibrational mode in some simulations created beats that could possibly adversely affect bearing performance; placing a stretching tension on the bearing suppressed one of the modes and therefore the beats. These and future studies will help evaluate the performance, wear and load-bearing properties of fundamental components of nanomachines such as bearings.

Dynamics of flow inside carbon nanotubes

Future nanotechnology applications are likely to involve reactive or non-reactive species carried along a fluid stream. We have performed several molecular dynamics simulations of a buckyball, , cage or idealized atom in a helium fluid flowing axially inside a carbon nanotube. The fluid was started at some initial velocity and both the fluid and buckyball allowed to recycle axially via minimum image boundary conditions. A buckyball introduced into the feedstream (started at zero velocity) usually reached fluid velocity within 5 ps. Leakage rates of helium past the depended on the nanotube diameter and fluid velocity. These leakage rates and other important features of the dynamics changed significantly when was modelled as an idealized atom or when the nanotube was held rigid, suggesting that simulations of fluid dynamics inside nanomachines should be fully dynamic and atomistic.

On the importance of quantum mechanics for nanotechnology

In this article it is argued that classical molecular dynamics studies of nanomachines may not give an accurate representation of their performance. Fortunately a new method, internal coordinate quantum Monte Carlo, an improved technique for computing quantum mechanical ground-state energies and wavefunctions, has the potential capability to model these systems. Some relevant examples demonstrate that the quantum ground state for many-body systems similar to those of interest in nanotechnology has a qualitatively different structure than that obtained from a molecular dynamics calculation which exhibited chaos and gross instabilities at energies of only a fraction of the ground-state energy. This result casts uncertainty on the reliability of using the molecular dynamics method to calculate the structure or any other dynamical quantity relevant to nanotechnology.

Docking envelopes for the assembly of molecular bearings

To design a process for the assembly or self-assembly of a nanomachine requires knowing the degree of spatial control needed to put the components together. A useful starting point in this area of study is the concept of a docking envelope, which is a continuous region of initial condition parameter space for which two structures will dock. In this paper docking envelopes, determined from molecular dynamics simulation, are presented for the assembly of a molecular bearing consisting of two concentric carbon nanotubes. In the beginning of each simulation the outer nanotube (sleeve) is held in place and the inner nanotube (shaft) starts far away from, but is given an initial velocity toward, the sleeve. The docking envelope in this case is delineated by the initial offset from a coaxial geometry. In order to address recent concerns about the effects of zero-point energy leakage and chaos in classical simulations of nanomachine components, docking envelopes from two types of simulations are presented: fully atomistic (all degrees of freedom included) and rigid body (each nanotube rigid but shaft allowed to rotate and translate).

Large-Scale Normal Coordinate Analysis for Molecular Structures

We apply truncated RQ-iteration (TRQ) and the Jacobi--Davidson (JD) method to perform vibrational (eigenvalue) analysis for large-scale molecular systems. Both algorithms employ a preconditioned iterative solver to construct a low-dimensional subspace that contains desired vibrational modes. We discuss several strategies for speeding up the eigenvalue calculation. In particular, we illustrate how to construct effective preconditioners and analyze the quality of these preconditioners. We show that convergence can be improved by choosing appropriate shifts and deflating the translational and rotational modes. Numerical examples are provided to demonstrate the efficiency of our computation.

Time-averaged normal coordinate analysis of polymer particles and crystals

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 ).

Efficient treatment of out-of-plane bend and improper torsion interactions in MM2, MM3, and MM4 molecular mechanics calculations

Simple and very efficient formulas are presented for four‐body out‐of‐plane bend (used in MM2 and MM3 force fields) and improper torsion (used in the MM4 force field) internal coordinates and their first and second derivatives. The use of a small set of bend and stretch intermediates allows for order of magnitude decreases in calculation time for potential energies and their first and second derivatives, which are required in molecular mechanics calculations. The formulas are eminently suitable for use in molecular simulations of systems with complicated bond networks. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1804–1811, 1997

Accurate computation of individual and tables of 3-j and 6-j symbols

By rewriting the formulas for 3-j and 6-j symbols in terms of several possible alternating binomial sums, it is possible to calculate these quantities quickly and accurately, often exactly, using floating point operations. The binomial sums can be calculated by direct summation or by recursion. A simple method for uniquely parameterizing the well-known Regge symmetries of the 3-j and 6-j symbols makes it possible to systematize the choice of the smallest magnitude binomial sum (which enhances the accuracy of floating point calculations and speeds up exact calculations using large integer routines). Formulas for special cases of the 3-j symbols enable the construction of recursion sequences which are often substantially faster than direct summation, especially for very large angular momentum arguments. For both 3-j and 6-j symbols, recursion offers several advantages over direct summation in exact calculations and for calculating tables.

An internal coordinate quantum Monte Carlo method for calculating vibrational ground state energies and wave functions of large molecules: A quantum geometric statement function approach

An internal coordinate quantum Monte Carlo (ICQMC) method using importance sampling is illustrated for a 100 atom model polyethylene chain. Importance sampling with an internal coordinate guiding wave function yields smoother, more physically reasonable wave functions and lower ground state energies than Cartesian importance sampling, in good agreement with normal coordinate analysis results. A novel geometric statement function (GSF) method for economizing expressions involving first and second derivatives of stretch, bend, and torsion internal coordinates by up to 2 orders of magnitude allows QMC calculations to be performed even for large molecules in reasonable times on standard workstations. The ICQMC method with quantum GSF is eminently suitable for large molecules with complicated, strongly coupled potential energy surfaces such as polymer chains.