 ZENO Efficient Method for Characterizing Object Shape and for Calculating Transport                                            Properties of Nanoparticles and Synthetic and Biological Macromolecule Movie (will be available soon)   Computational Elements: Home Computation References Slide Show ZENO computation: Downloads new Principle Behind Calculation Analytic Approximation Summary of Important Properties Dr. Marc L. Mansfield

(Stevens Institute of Technology) Dr. Jack F. Douglas

(Polymer Division at NIST)

A Monte Carlo numerical path integration that generates a large number of random walks in the space outside the body.  Because the Laplacian operator governs the statistics of these walks, sums taken over these random walks yield:

electrostatic capacity, C

polarizability tensor, a

intrinsic conductivity, [s]

translational diffusion coefficient, D

translational friction coefficient, f

intrinsic viscosity, [h]

hydrodynamic volume, Vh INTERIOR computation:

Protein Properties/Zeno:

PDB ID code (will be available soon)

contact: ekang1@stevens.edu

A Monte Carlo integration that generates a large number of points distributed randomly

throughout the interior of the body.  Sums taken over these points yield:

particle volumes, V

radius of gyration (evaluated for the interior of the particle) Rgi

particle scattering function, P(q) SURFACE computation:

A Monte Carlo integration that generates a large number of points distributed randomly

over the surface of the body.  Sums takes over these points yield:

surface area, S

radius of gyration (evaluated for the surface of the particles) Rgs

translational diffusion coefficient, D, by the Kirkwood double-sum formula

Acknowledgements:

We thank CTCMS at NIST for funding support of this development

Figure 1:   Collection of spheres (green) represents a model soot particles (cluster-cluster aggregate)  and the path (yellow) represents a probing random walk trajectory.    C  is determined by fraction trajectories that hit the sphere.

f (translational) » 6p h C

Purpose of Zeno:

Algorithm (Zeno) for calculating the Stokes friction coefficient, electrostatic

capacity, intrinsic viscosity, intrinsic conductivity and electrical polarizability

of essentially arbitrarily-shaped objects to unprecedented accuracy.

Idea Behind Calculation:

There is a fundamental relation between the Laplacian operator and random

paths whose step size has a finite variance.   This correspondence allows

for the solution of the equations of mathematical physics to be formally expressed

as averages over random walk trajectories.  There are many famous mathematicians

that have contributed to this formal computational scheme.

Kakutani, Wiener, Kac, Ito and MacKean and many others

The advantage of this method is that it allows for the calculation of

transport properties for objects having essentially arbitrary shape.  This method

becomes a practical and highly accurate method for performing transport

property calculations when random walks are generated by computer.

Computational Method:

The Zeno computational method involves enclosing an arbitrary-shaped probed object

within a sphere and launching random walks from this sphere.

The probing trajectories either hit or return to the launch surface ('loss')

as shown in the fig.1 for a model soot particles aggregate, whereupon

the trajectory is either terminated or reinitiated.

Electrostatic-Hydrodynamic Analogy:

Hydrodynamic and electrostatic  properties are determined, respectively, by the Navier-

Stokes and Laplacian equation.  However, a specific orientational averaging of the Navier-

Stokes equations brings them into the form of Laplace's equation.  This means that an

approximate analogy exists between certain hydrodynamic and electrostatic properties.

Particularly, the hydrodynamic radius and the intrinsic viscosity of a macromolecule are

proportional, respectively, to the capacitance and polarizability of a perfect conductor

having the same shape as the macromolecule.  These proportionalities have been

extensively tested on diverse shapes, and are always found to be accurate to 1% for the

hydrodynamic radius and 5% for the intrinsic viscosity.  Zeno determines the electrostatic

properties directly by Monte Carlo path integration, and then infers hydrodynamic

properties from these proportionalities.  See References ,, and .