strategies for multiscale modeling and simulation of...

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Strategies for multiscale modeling and simulation of organic materials:polymers and biopolymers

William A. Goddard III*, Tahir Cagin, Mario Blanco, Nagarajan Vaidehi, Siddharth Dasgupta,Wely Floriano, Michael Belmares, Jeremy Kua, Georgios Zamanakos, Seichi Kashihara,

Mihail Iotov, Guanghua Gao

Division of Chemistry and Chemical Engineering, California Institute of Technology, Materials and Process Simulation Center, 139-74,

Pasadena, CA 91125, USA

Received 21 September 2000; revised 7 February 2001; accepted 9 February 2001

Abstract

Advances in theory and methods are making it practical to consider fully ®rst principles (de novo) predictions of structures, properties and

processes for organic materials. However, despite the progress there remains an enormous challenge in bridging the vast range of distances

and time scales between de novo atomistic simulations and the quantitative continuum models for the macroscopic systems essential in

industrial design and operations. Recent advances relevant to such developments include: quantum chemistry including continuum solvation

and force ®eld embedding, de novo force ®elds to describe phase transitions, molecular dynamics (MD) including continuum solvent, non

equilibrium MD for rheology and thermal conductivity and mesoscale simulations. To provide some ¯avor for the opportunities we will

illustrate some of the progress and challenges by summarizing some recent developments in methods and their applications to polymers and

biopolymers. Four different topics will be covered: (1) hierarchical modeling approach applied to modeling olfactory receptors, (2)

stabilization of leucine zipper coils by introduction of tri¯uoroleucine, (3) modeling response of polymers sensors for electronic nose,

and (4) diffusion of gases in amorphous polymers. q 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Quantum mechanics; Force ®eld; Molecular dynamics

1. Introduction

In order to develop new materials and compositions with

designed new properties, it is essential that these properties

be predicted before preparation, processing, and experimen-

tal characterization. Despite the tremendous advances made

in the modeling of the structural, thermal, mechanical and

transport properties of materials at the macroscopic level

(®nite element analysis of complicated structures) there

remains tremendous uncertainty about how to predict

many critical properties related to performance. The funda-

mental problem here is that these properties depend on the

atomic level interactions and chemistry (e.g. making and

breaking of bonds) dealing with the electronic and atomic

level description at the level of nanometers and pico-

seconds. The materials designer needs answers from macro-

scopic modeling (®nite element paradigm) of components

having scales of centimeters and milliseconds or larger. To

dramatically advance the ability to design useful high

performance materials, it is essential that we insert the

chemistry into the mesoscopic and macroscopic (®nite

element) modeling.

The dif®culties in doing this are shown in Fig. 1, where

we see that vast length and time scales separate the quantum

mechanics (QM) from the macroscopic world of engineer-

ing design. Tremendous advances have been made recently

in ®rst principles QM predictions of chemical reactions, but

the state of the art can handle accurately reactions with only

,50 atoms. There is no practical approach to carrying out a

QM calculation on the initiation and propagation of a crack

through a stabilized zirconia ceramic. Despite this dif®culty,

the computations MUST be based on accurate ®rst-principles

QM if we are to predict the properties of new materials.

Our strategy for accomplishing this objective is to

develop an overlapping array of successively coarser

modeling techniques. At each plateau (a range of length

and time scales), the parameters of the coarse description

are based on the parameters of the immediately ®ner

description, as shown in Fig. 1. Thus based on accurate

QM calculations we ®nd a force ®eld (FF) including

charges, force constants, polarization, van der Waals

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Computational and Theoretical Polymer Science 00 (2001) 000±000

CTPS221

1089-3156/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.

PII: S1089-3156(01)00025-3

www.elsevier.nl/locate/ctps

* Corresponding author. Tel.: 11-626-395-2731; fax: 11-626-585-0918.

E-mail address: [email protected] (W.A. Goddard III).

Computational and Theoretical Polymer Science ± Model 5 ± Ref style 3 ± AUTOPAGINATION 2 14-06-2001 11:48 all ABAlden

UNCORRECTED PROOF

interactions etc that accurately reproduces the QM. With the

FF, the dynamics is described with Newton's equations

[molecular dynamics (MD)], instead of the Schrodinger

Equation.

The MD level allows one to predict the structures and

properties for systems ,105 times larger than for QM,

allowing direct simulations for the properties of many inter-

esting systems. This leads to many results relevant and

useful in materials design, however, many critical problems

in materials design require time and length scales for too

large for practical MD.

Thus we need to develop methods treating the mesoscale

in between the atomic length and time scales of MD and the

macroscopic length and time scales (microns to mm and ms

to s) of ®nite element analysis (FEA). This linking through

the mesoscale in which we can describe microstructure is

probably the greatest challenge to developing reliable

®rst principles methods for practical materials' design

applications.

Only by establishing this connection from microscale to

mesoscale it is possible to build ®rst principles methods for

describing the properties of new materials and composites.

Our aim is to reach the domain of materials science and

engineering by building from fundamental principles of

physics and chemistry. Thus, for fundamental predictions

to play a direct role in materials innovation and design, it

is essential to bridge the micro±meso gap. The problem here

is that the methods of coarsening the description from

atomistic to mesoscale or mesoscale to continuum is not

so obvious as it was in going from electrons to atoms. For

example, the strategy for polymers seems quite different

than for metals, which seem different from ceramics or

semiconductors.

Given the concepts, it is necessary to carry out calcula-

tions for realistic time scales fast enough to be useful in

design. This requires developing software tools useful by

design engineers, by incorporating the methods and results

of the QM to MD to mesoscale simulations. To accomplish

the goals of developing methods for accurate calculations of

materials and properties, we focus on: (i) implementations

that make full use of modern highly parallel computers, and

(ii) building in knowledge based heuristic methods of acces-

sing this information automatically so that designers can

focus on the macroscopic issues without concern for the

details of theory and simulation. At this point, we expect a

revolution in materials design and innovations where the

®rst-principles multiscale modeling and simulations play

increasing role in the design stage and complementing the

experiments.

2. Progress in methods developments

Our strategy is to transcend from the most fundamental

theory (QM) to practical engineering designs in a sequence

of several levels as indicated in Fig. 1.

2.1. Quantum mechanics

2.1.1. Ab initio quantum chemistry applications

It is important to use QM to describe systems in which

bonds are being broken and formed. Only then can we be

sure to obtain accurate barrier heights and bond energies.

The modern methods of QM (generalized valence bond

(GVB) [1], pseudo spectral generalized valence bond (PS-

GVB) [2], multi-reference con®guration integral (MR-CI)

[3], and (gaussian dual space density functional theory

(GDS-DFT) [4]) can give accurate barriers for reactions

useful in studying the properties of nanoscale materials.

However, despite the progress in ®rst principles electronic

structure theory, the calculations are often far too slow for

studying the dynamics of nanotechnology applications. This

is necessary to have general approaches for averaging the

electrons from QM to obtain an FF in terms of atomic posi-

tions. For ®nite molecules; A new methodology (PS-GVB)

combining pseudo-spectral (PS) multi-grid and de-aliasing

strategies with the sophisticated many-body wave functions

[generalized valence bond (GVB)] were implemented and

applied to large scale problems. PS-GVB led to consider-

ably better scaling with size (N2 rather than the N3, N4,

N5, N6 characteristic of alternative methods) and simpler

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Fig. 1. Multiscale modeling hierarchy.


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parallelization. PS-GVB has been extended to treat all

atoms of the periodic table (using core effective potentials),

handle new sophisticated wave functions (GVB-RCI, MP2),

and describe important properties (solvation energies,

reaction rates, activation/reaction barriers). We will opti-

mize these methods for parallel implementations, and

extend the methodology to include GVB-RCI-MP2 and

self-consistent GVB-RCI.

2.1.2. Density functional theory applications

Most practical materials properties require a description

of in®nite systems using periodic boundary conditions

(PBC). This is three-dimensional (3D) for bulk properties

or two-dimensional (2D) for surface growth and interfaces.

For this purpose, we have developed a new method, Gaus-

sian dual space density functional theory (GDS±DFT), in

which most parts scale linearly with N. In implementing

this, we have developed a new separable pseudo-potential

that can be applied to all atoms of the periodic table. We

reformulated the theories for electronic structure calcula-

tions of periodic systems in a way suitable for large-scale

calculations using Gaussian basis functions. An accurate

grid is introduced for ef®cient calculation of matrix

elements. A dual-space approach is used to calculate the

Coulomb potential with computational cost that scales line-

arly with the size of basis set. Preconditioned generalized

conjugate gradients approach is introduced for rapidly

converging wave functions expressed in terms of Gaussian

basis functions. This method is applied to a variety of

systems with excellent results.

2.2. Force ®elds (FF)

Using quantum mechanical results we develop FF

descriptions to provide the energetics needed for the simu-

lations of the nano-phase materials and their properties. The

FF must even be accurate enough to obtain the proper

energy differences for representing phase behavior of the

materials and transferable so that one can apply it to

phase transformations and interface phenomena. Standard

FF generally uses simple springs to represent bonds and

angles in describing structures and vibrations of molecules.

Generally this is not suf®ciently accurate to obtain a FF that

accurately describes the properties of a speci®c class of

molecules or polymers. For better FF, we ®t to the QM

using the Hessian-Biased FF (HBFF) [5] approach, which

combines normal mode information from HF theory with

the frequency information from theory or experiment. This

HBFF approach has been used to develop accurate FF for

polymers (e.g. PE, PVDF, nylon, POM, SiH) [6±10], cera-

mics (e.g. Si3N4, C3N4), [11,12] semiconductors [13] and

metals [14].

On the other hand, for fast qualitatively considerations of

new systems, we ®nd that generic FF suitable for general

classes of systems are most useful. DREIDING FF [15] (for

the main group elements) and the Universal force ®eld

(UFF) [16] (all elements, including inorganics and organo-

metallics) are such FF.

In recent years we made critical advances made in devel-

oping ab QM based FF for describing

(a) metals where many-body interactions play critical role

on their physical properties; [17±21]

(b) oxides, ceramics, and zeolites where competition

between ionic and covalent bonding is often very impor-

tant, especially in describing polymorphic phase transi-

tions, reactions, surface and interface properties; [22±24]

(c) covalent bonded systems such as carbon, hydrocar-

bons, silicon, germanium and their behavior far from

equilibrium where the description of bond breaking and

forming must be a part of an accurate classical description

[24,25].

2.3. Molecular mechanics and molecular dynamics: MPSim

Using these FFs in large-scale MD simulations allows

practical calculations on up to several millions of atoms.

Our objective is to develop new strategies and algorithms

in addition to taking advantage evolving hardware and soft-

ware technologies to extend the time and distance scale to

100s of ns and close to microns. This will involve using fast

multi-pole techniques, multiple time step approaches,

NEIMO (Newton Euler inverse mass operator) method,

CFA (constrained force algorithms) method, hyper MD

approaches where they are suitable.

The focus here is on extending the methods of MD to

physical systems of molecules, polymers, liquids, and inor-

ganic materials with up to 100 million atoms while accu-

rately treating long-range interactions using the cell multi-

pole method (CMM). For fast internal coordinate dynamics

on a million atoms, we have developed the NEIMO method.

This methodology handles periodic systems and will be

extended (Gibbs-Ensemble MD. The applications for this

method will enable us to investigate the long time dynamics

of liquid polymer and solid interfaces; which has tremen-

dous impact on broad range of technological applications;

such as wetting, adhesion, phase separation, coatings.

In MD simulations, the FF is used to predict the equations

of motion. This leads to trajectories �xi�t�; vi�t�;¼ �1;¼; 3N� that can be analyzed (using statistical mechanics

and thermodynamics [26] principles) to obtain macroscopic

properties. MD simulations of heterogeneous nano-phase

materials may require millions of atoms to be considered

explicitly (a 25 nm cube of polyethylene has 1 million

atoms). The most time-consuming aspect of the MD simu-

lations of large systems is the accurate evaluation of the

long-range interactions (electrostatic and dispersion),

which decrease slowly with distance. Without cutoffs, this

cost scales as order (N2) for N particles (a system of 10

million atom leads to 1014 terms to be evaluated each

step). Using cutoffs may reduce the computational cost.

However, the cutoffs can lead to excessive errors. For a

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UNCORRECTED PROOF

periodic system, the Ewald procedure leads to accurate

summations for these interactions, but the problem scales

as N1.5, totally impractical for systems with million atoms

[27,28]. In order to simulate systems with millions of atoms,

we developed methods and optimized parallel computer

programs ef®cient for high capacity MD with the following

advanced features:

(i) Cell multipole method [29] (CMM) which dramati-

cally reduces the cost of long-range Coulomb and van

der Waals interactions while retaining high accuracy.

The cost scales linearly with size, allowing atomic-level

simulations for million atom systems [30±33].

(ii) Reduced cell multipole method [34] (RCCM) which

handles the special dif®culties with long-range Coulomb

interactions for crystals by combining a reduced unit cell

plus CMM for interactions between the unit cell with its

adjacent cells. The cost scales linearly with size while

retaining high accuracy, allowing simulation of crystals

having a million atoms per unit cell (the major use is for

models of amorphous and semi-crystalline materials).

(iii) Newton Euler inverse mass operator method

(NEIMO) [35±37] for internal coordinate dynamics

(e.g. treats torsions only). This allows the solution of

the dynamical equations for internal coordinates without

inverting the mass tensor (moment of inertia tensor). The

cost of NEIMO is linear in the number of degrees of

freedom and small compared to other costs for million

atom systems. More recently, we also developed a new

constrained force algorithm (CFA) for massively parallel

MD simulation of polymers and dendrimer [38,39].

(iv) Advanced MD algorithms to simulate systems under

constant temperature and constant pressure conditions

[40,41].

(v) Nonequilibrium MD We have implemented synthetic

equations of motions to simulate various nonequilibrium

conditions to predict transport properties such as viscos-

ity, thermal conductivity of materials [42±45]. Using

these methods, we have studied the effect of molecular

topology of liquid alkanes on their measured viscosity

indices [46,47].

(vi) Steady state MD Methods are used to simulate non-

equilibrium processes such as friction and wear in nano-

scopically con®ned lubricants and diamond surfaces.

Here, the external work is dissipated through the material

and coupled to a thermal bath using the Langevin

equation [48±52].

2.4. Continuum solvent approaches

Solvents can have a major effect on the structure and

properties of polymers. Consequently, we must include

the solvent in the MD simulations. The costs are signi®cant.

For instance, in studying the Frechet Stimuli-responsive

dendrimer our simulations included ,1609 atoms of the

dendrimer plus ,80,000 atoms of solvent. Such explicit

solvent calculations are compute-intensive and we have

searched for accurate ways to obtain an implicit description

of these solvent effects. In the biological literature, it is

common to ignore the solvent and to modify the coulomb

interactions between the atoms of the proteins by using a

distance dependent dielectric constant. This is too crude for

our purposes. Instead, we replace the solvent with a conti-

nuum solvent, the Poisson±Boltzmann approximation [53].

This approach provides a solution for large-scale solvation

problems. It includes local solvent reorganization using

explicit ®rst solvation layer. This PB continuum solvation

description serves as an intermediate step between implicit

and explicit solvent simulations. We have used this the

continuum solvent approach to study various polymers

and biopolymers. Although the PB approximation greatly

decreases the cost of accurate solvation, the PB calculation

is still the expensive component of force evaluation in each

MD step. Consequently, we are currently using another

promising approach generalized Born (GB) method, in our

MD simulations [54].

3. Applications

3.1. Multiscale modeling of biopolymers

Multi-scale modeling techniques are very vital to

research problems in biology. From the ®ne level calcula-

tion of accurate binding energies for drug molecules using

QM to coarse level structure prediction for proteins (both

globular and membrane) and understanding viral protein

coat assembly are some of the typical cases that require

multi-scale modeling for biological systems. Such modeling

schemes are also critical in bioengineering problems that

offer excellent control in growth of a self-assembly at

nanoscale. We have developed methods using CCBB MC

[55,56] to predict the structure of globular proteins [57±59]

and transmembrane proteins (G-protein coupled receptors).

We have used this method to predict the structures of odor

receptors in mammalian olfaction system. To understand the

molecular basis of odor recognition we need to have an

atomic level model of these odor receptors.

3.1.1. Multi-scale modeling techniques for deriving the

atomic level structure of odor receptors

Odor receptors are seven helical domain membrane

proteins that belong to the family of G-protein coupled

receptors. We have used a combination of hydrophobicity

pro®le prediction methods [60,61] and large-scale coarse

grain MD methods with proper description of differential

solvent environment to derive the atomic model for odor

receptor OR S25. The protocol used for the multiscale

modeling is shown in Fig. 2

Prediction of helical regions using hydrophobicity

pro®les and optimization: The trans-membrane helices


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UNCORRECTED PROOF

were identi®ed on the basis of hydrophobicity by the multi-

sequence pro®le method of Donelly et al. [60], as imple-

mented in PERSCAN. The sequence used for ORS25 was

used from the data by Malnic et al. [62]. For validation, the

analysis was done on 21 rat olfactory receptors reported by

Singer et al. [61]. Sequences were aligned by the iterative

pro®le alignment utility of whatif [63]. The sequence from

Ref. [62] was used for the odor receptor S25 to build cano-

nical right-handed a-helices with Builder and Homology

software (MSI). The structure of the helices thus built

were optimized using NEIMO torsional MD method

[35,36]. NEIMO algorithm is a fast torsional dynamics

method that scales linearly with the number of torsional

degrees of freedom.

Helix assembly: Helical rotations and the orientations of

the helical axes were built using the bovine rhodopsin 7.5 AÊ

electron density map [64]. We found that the proper descrip-

tion of the membrane bilayer is critical to packing the

helices using rigid body MD. Dreiding FF [15] was used

for the hydrophobic tail of the lipids. The charges on the

polar heads of the lipids were assigned using the charge

equilibration scheme. Crystal simulations of the lipid

bilayer at constant pressure and temperature were

performed using the PBC. These simulations reproduced

the experimental crystal cell dimensions and density accu-

rately. Following the assembly of the helices in the trans-

membrane domain, we performed 200 ps of rigid body

dynamics of the helices with a barrel of lipid surrounding

the helical barrel. For the protein Dreiding FF with charges

from CHARMM [65] was used.

Optimization of the full atomic model: Following the rigid

body dynamics described in Section 3.1 we further added

the loops to the helices according to the sequence given in

Malnic et al. [62] Loops were added using whatif [63]

software. Following the addition of loops we performed a


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Fig. 2. Modeling strategy for odor receptors.

Table 1

Carbon alpha root mean square (CaCRMS) deviation for the modeled bacteriorhodopsin structure after MD compared to the crystallographic structure

BRDP structure CaCRMS (AÊ )

Helicesa Loopsb Completec

Xtal Crystallographic structure (xtal)d 0.0 0.0 0.0

Modeled Modeled helix bundle (raw) e 3.16 3.15 (163 aa)

Helix Modeled helix bundle minimizedf 3.29 3.28 (163 aa)

Bundle Modeled helix bundle after 180 ps

RBMD at 300 Kg

3.32 3.31 (163 aa)

Complete model (raw) 3.32 7.22 4.62 (221 aa)

Complete Complete model (minimized DPG)h 3.30 7.41 4.69 (221 aa)

Model Complete model after 50 ps of

NEIMO/SGB TVNi

3.29 8.57 5.98 (221 aa)

a Helices refers to residues (aa) in the initial helix bundle (residues predicted as helical) compared to the corresponding region of the crystallographic

structure (see text for the predicted transmembrane regions) (162 residues).b Loops refers to inter helical regions that were not predicted as helical compared to the crystallographic structure (55 residues).c Complete compares the corresponding complete structure to the crystallographic one.d The crystallographic (xtal) structure corresponds to pdb code 2brd (resolution 3.5 AÊ ), without residue Ala228 (which had missing atoms).e Raw refers to model before any molecular mechanics (MM) or dynamics (MD) calculation.f This is a raw model after conjugate gradient minimization in vacuum, in presence of Sodium Diphosphatidylglycerophosphate (DPG) bilayers.g RBMD is Rigid Body MD with DPG bilayers.h The complete model was minimized in the presence of DPG bilayers.i Newton±Euler Inverse Mass Operator (NEIMO) method with H-NEIMO for protein. SGB (Surface Generalized Born) approach for solvation with a

dielectric constant of 60. This was done with helices treated using H-NEIMO and loops with NEIMO, the counterions as Cartesian atoms. The membrane was

simulated by a ring of DPG bilayers that were kept as rigid bodies.


UNCORRECTED PROOF

full minimization of all the atoms with a barrel of lipid

surrounding the protein. Counterions Na1 and Cl21 were

added to neutralize acidic and basic residue side chains.

The outside of the lipid layer was simulated using a conti-

nuum solvent description model (the surface generalized

Born model [66]). A dielectric of 60.0 was used to simulate

the low dielectric region surrounding the membrane. To

optimize the solution structure further we then performed

a multiscale mixed mode dynamics. The helices and loops

in the protein were modeled using the NEIMO torsional

MD, the lipids were treated as rigid bodies and the counter-

ions as free Cartesian atoms. Constant temperature mixed

mode dynamics yielded an optimum model for the atomic

structure.

Control simulations: Our protocol described above was

®rst tested on bacteriorhodopsin, a membrane protein for

which the crystal structure has been ®tted with fair accuracy

in the transmembrane region of the protein. We started from

the sequence of bacteriorhodopsin, used no knowledge of

the crystal structure, and built the complete model using the

above protocol. The overall RMS deviation in coordinates

of Ca atoms for the ®nal model is 5.98 AÊ for all the 221

amino acids. The CRMS for the residues in the membrane

barrel is 3.29 AÊ while that for the loops is 8.57 AÊ . It is seen

from Table 1 that the multiscale modeling improved the

model by making the loops more ¯exible and de®ning the

helical regions consistently. Thus, this modeling procedure

gives a very reasonable model for a known membrane

protein. Having evaluated the control protein, we further

used the same protocol for modeling olfactory receptors.

Modeling of S25 olfactory receptor: The odor receptor

S25 from Malnic et al. [64] has odor detection responses

only to alcohols and not the other acids tested. We built the

model using the sequence for S25 from Malnic et al. [64]

and the procedures described above. The membrane is simu-

lated using explicit lipid bilayers of Dilauroylphosphatidyl

choline (DPC). The choice of lipid in the OR case is

supported by experimental indications that the membrane

surrounding the ORs in vivo can be satisfactory simulated

using a single component lipid system of DPC. The ®nal

atomic level model used in further docking studies is

showed in Fig. 3. Using the HEIR-Dock protocol [67±69]

we have predicted the binding site for alcohol and acid

odorants in ORS25. This dynamic receptor structure not

only points to a likely odor-binding site but also indepen-

dently predicts the two compounds that experimentally best

activate ORS25. The results provide a mechanistic model

for olfactory transduction at the molecular level and show

how the basic GPCR template is adapted for encoding the

enormous odor space.

3.1.2. Stabilization of leucine zipper coiled coils by

introduction of tri¯uoroleucine [70]

Engineering of stable enzymes and robust therapeutic

proteins is of central importance to the biotechnology and

pharmaceutical industries. Although protein engineering

provides powerful tools for the enhancement of enzymatic

activity and protein stability, the scope of in vivo engineer-

ing methods is limited by the availability of just 20 naturally

occurring proteinogenic amino acids. Increasing success in

the incorporation of non-canonical amino acids into recom-

binant proteins in vivo has allowed the introduction of novel

side-chain functionality into engineered proteins. This

raises prospects of new approaches to the design of peptides

and proteins of enhanced activity and/or stability. The effect

of ¯ourination of the side chains of substitution of leucine

residues by 5,5,5-tri¯uoroleucine at the d-positions of the

leucine zipper peptide GCN4-p1 increases the thermal stabi-

lity of the coiled-coil structure [70]. To determine the

origins of the stabilizing effect of side-chain ¯uorination,

we carried out MD calculations using a continuum descrip-

tion of the solvent (Ref. [20]). A good description of solvent

is critical to the simulation of biological systems. While

using explicit solvent molecules in the simulation is compu-

tationally tedious Ghosh et al. [66] and Tannor et al. [2]

have recently derived continuum solvent models that

describe the solvent reaction ®eld fairly accurate. In this

study, we have clearly shown that the effect of solvation

is well described both quantitatively on the binding energies

and the structure prediction for ¯uorinated leucine zippers.

The mpsim md program and the dreiding ff were used for

all calculations. The starting structure for the Leu-GCN4-p1

dimer was taken from the RCSB Protein Data Bank. The

¯uorinated dimers were derived from the native dimer struc-

ture by replacement of the appropriate methyl hydrogens

with ¯uorines, followed by re-optimization of the structure.

Because the g-carbon of tri¯uoroleucine (TFL) is asym-

metric, multiple arrangements of adjacent diastereotopic


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Fig. 3. The tri¯uoroleucine subsituted gcn4-pl leucine zipper is shown in

the center. Possible con®gurations of the leucine/tri¯uoroleucine stereo-

center at the g-carbon on tri¯uoroleucine. Shown here are the various

packing possibilities considered in the MD simulations. (A) Leu/Leu; (B)

4S/4S, `close'; (C) 4R/4R, `far'; (D) 4R/4S; (E) 4S/4R; and (F) H¯/H¯,

where both leucines are substituted with hexa¯uoroleucine.


UNCORRECTED PROOF

tri¯uoro-methyl groups must be considered (Fig. 3). When

both T¯ residues at a given d-position are of the (2S,4S)

con®guration, the two tri¯uoromethyl groups are relatively

close to one another; the ¯uorinated carbon centers are sepa-

rated by ca. 6 AÊ . On the other hand, when two (2S,4R)

isomers are juxtaposed, the corresponding carbon±carbon

distance increases to about 8 AÊ . The tri¯uoromethyl groups

are separated by intermediate distances, in the remaining

con®gurations (where the two strands carry different

isomers). We performed simulations on all con®gurations,

to determine how side-chain stereochemistry affects dimer

stability. For each dimer we carried out 1 ns of constant

temperature (300 K) Nose±Hoover MD with the SGB

description of the water solvent.

From the 1 ns trajectory, we calculated the average prop-

erties over 800 ps after equilibration. The binding energies

were calculated as the difference in potential energies of the

solvated dimer and the corresponding solvated monomers.

Table 2 reports the average values of the binding energy

(per monomer) for the native and ¯uorinated forms.

The T¯-GCN4-p1 dimers are predicted to exhibit binding

energies (Table 2) ca. 55% larger than that of the leucine

form (calculated relative to the respective random coil

monomers). The various stereochemical arrangements lead

to increases ranging from 44 to 71%, indicating that side-

chain con®guration may have some differential effect on

dimer stability. This is in excellent agreement with the

experimentally measured free energy difference between

T¯-GCN4-p1 and GCN4-pl which is 57%. Similar analysis

of a hexa¯uoroleucine (H¯) dimer leads to the prediction

that such dimers (which were not prepared experimentally

in this work) would be less stable than the T¯ dimers but

marginally more stable (19%) than the wild type.

To investigate the source of stability of the ¯uorinated

dimers we analyzed the components of the binding energy

for each peptide. The primary driving forces for stabilizing

the T¯-GCN4-p1 dimers are predicted to arise from van der

Waals (vdW) and hydrogen bonding interactions. Consid-

eration of electrostatic (intra- and inter-peptide coulomb

forces) and solvation interactions suggests a hydrophobic

effect that favors burial of CF3 rather than CH3 in the

dimer. If just coulomb and solvation interactions are consid-

ered, the driving force for dimerization is predicted to

decrease in the order Hfl . Tfl . Leu: It is the balance of

desolvation, electrostatics, H-bonding and vdW forces that

leads to the prediction that the T¯ dimers are more stable

than the H¯ dimer which is more stable than the native

leucine dimer. The average helicities of dimers are predicted

to be 90.8% for Leu-GCN4-p1, 83.5% for T¯, and 78.5% for

H¯.

Our results demonstrate that the subtle change from four

leucine methyl groups to four tri¯uoromethyl groups results

in a large gain in stability of the folded structure. It is

remarkable that the inclusion of the solvation effects predict

the structures and binding energies between the two

peptides with good accuracy.

3.2. Modeling polymer sensors for the electronic nose

The Lewis group at Caltech [71±74] has experimentally

constructed a particular design of an electronic nose.

Sensors are built with conducting leads connected through

thin ®lm polymers loaded with carbon black. Odorant detec-

tion relies on a change in electric resistivity of the polymer

®lm as function of the amount of swelling caused by the

odorant vapor. The amount of swelling depends upon the

chemical composition of the polymer and the odorant mole-

cule. An array of 20 carbon black loaded polymers give rise

to a speci®c change in resistivity patterns upon exposure to a

given molecular species. The pattern is unique and unam-

biguously identi®es the compound [73,74].

It is of great value to be able to correlate from ®rst prin-

ciples the change of resistivity of a given polymer sensor

with the chemical nature of the solvent. Predictions of this

type may be of practical use for increasing the electronic

nose sensitivity for speci®c compounds such as those found

in the wine, cheese or perfume industries, or for detecting

nerve gases and air bound compounds emanating from

explosives. Here we report a molecular modeling method

that provides a strong correlation between the calculated

properties of a series of polymers and solvents and the

experimental relative change in the resistivity of the sensors

upon exposure to these solvents.

3.2.1. Method

The permeability of a solvent in a polymer is related to

the heat of sorption as follow [75]

P � P0exp�2DEs=RT� exp�2DHs=RT� �1�P0 is the exponential pre-factor related to entropy, DHs is


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Table 2

Binding energies (BE, kcal/mol) of Leu-GCN4-p1 and ¯uorinated dimers.

BE is the difference in the potential energy (averaged over 800 ps of MD

after equilibration) of solvated monomers and the solvated dimer each from

separate SGB MD calculations (®nal solvation energies with PBF). BE is

quoted per mole of the monomer. % increase is the increase in BE

compared to the Leu-GCN4-p1 structure. Also shown is the % helicity of

each

Structure BE Increase (%) Helicity (%)a

Leu-GCN4-p1 65.08 0 90.8

Close (4S,4S)b 93.75 44 84.3

Far (4R,4R) 98.14 51 79.4

Mixed (4S,4R) 99.20 52 81.1

Mixed (4R,4S) 111.15 71 89.3

T¯-average 100.56 55 83.5

H¯-GCN4-1p 77.21 19 78.5

a Helicity quoted here has been calculated as the ratio of the residues with

torsion angles f and c in the helical region of the Ramachandran plot to the

total number of residues in the protein.b Close, Far, Mixed: con®guration of the pair of tri¯uoromethyl groups as

shown in Fig. 3. T¯-average: the averaged binding energy of the four

con®gurations.


UNCORRECTED PROOF

the heat of sorption of the solvent in the polymer, and Es is

the activation energy for diffusion of the molecule in the

polymer. Because the amount of swelling at a ®xed time is

proportional to the permeability of the polymer towards the

solvent, the change in resistivity, DR/R, could be modeled

with an expression similar to Eq. (1). A computational

method for the estimation of the heat of sorption, or other

strongly correlated property, would be valuable.

Hansen [76,77] proposed an extension of the Hildebrand

parameter method to estimate the relative miscibility of

polar and hydrogen bonding systems

d2 � d2d 1 d2

p 1 d2h �2�

Where d corresponds to the Hildebrand solubility para-

meter,

d2 � DHv=Vm �3�The two quantities are not expected to be identical any

more than the Hildebrand parameters of liquids with speci®c

interactions are identical when determined by different

methods. DHv is the heat of vaporization and Vm the molar

volume. The Hansen solubility parameters in Eq. (2) are

determined empirically based on many experimental obser-

vations. Here we introduced a new modeling method to

estimate Hansen solubilities. Furthermore, we use the

predicted Hansen enthalpies Hi � Vmd2i of the polymer

and solvent molecules to construct a model for the change

in relative resistivity

DR=R � R0exp�2gVk=RT� expX

bi�H�k�i 2 H�n�i ��

�4�Where DR/R is the change in resistivity upon exposing

polymer sensor n to odorant k, R0 is sensor's resistivity in

air. Hi(k) and H�n�i �i � 1; 2; 3� are the three components (elec-

trostatic, dispersion, and hydrogen bond) of the solvent and

polymer Hansen enthalpies, respectively. These compo-

nents are calculated using ®rst principles MD in a suitable

FF [15,16]. DEk � gVk is a diffusion barrier directly propor-

tional to the molecular volume of the solvent. This relation

follows from the fact that the diffusion coef®cient is linearly

related to the molar volume of the penetrant, particularly

when the temperature is greater than the glass transition (Tg)

of the polymer [78].

3.2.2. Hansen parameters from MD simulations

A precise method was developed to estimate the Hansen

solubilities of solvents and polymers from MD calculations

under PBC. The procedure has been coded as a single appli-

cation under the Software Developer's Kit (SDK) distribu-

ted by MSI [79]. A unit cell of twelve solvent molecules (or

four polymer chains) is built at 50% of the target density.

This step is accomplished with the amorphous builder in the

Cerius2 software package [80] with a Van der Waals radius-

scaling factor of 0.30. If available the experimental density

of the solvent or polymer are used as a target value, although

the ®nal density may differ slightly depending on the FF

employed. The charges of the isolated solvent or polymer

molecules were based on the charge equilibration method

[81] and the rest of the parameters were taken from the

Dreidiing force®eld [15]. The potential energy of the bulk

system is minimized for 5000 steps or until the atom rms

force converges to 0.10 kcal/mol AÊ . 750 steps of MD (1 fs/

step) at a temperature of 700 K using canonical ®xed

volume dynamics (TVN) is carried out to anneal the sample.

The cell is then minimized with the previous procedure. The

reduced cell coordinates are shrunk such that the density is

64% of the target density. The atoms' coordinates are mini-

mized and dynamics is done on the system with the

previously described procedure holding the cell ®xed. A

total of ®ve compression, dynamics, and minimization

cycles are performed until the density reaches 120% of

the target density. Then the cell parameters are increased

in ®ve reverse cycles of expansion, dynamics, and minimi-

zation until the target density is reached. Finally, the sample

is allowed to relax in a minimization involving the cell and

the atoms' coordinates. The Hansen enthalpy components

are calculated by subtracting the potential energy of the bulk

system from the sum of the potential energies of the indivi-

dual molecules separated at an in®nite distance. This

process is repeated ten times with different initial random

packing. Hansen solubility parameters and molar volumes

were computed as well as the standard deviations. In poly-

mer calculations, the number of monomers in each chain

was determined such that the total volume of the four chains

was approximately 5900 AÊ 3. The initial polymer amorphous

structures were constructed using a one-dimensional rota-

tional isomeric states (RIS) approach to achieve a correct

distribution of conformational states.

3.2.3. Results and discussion

The experimentally determined relative resistivity of

seven polymer sensors upon exposure to twenty-four

solvent vapors was correlated with the calculated Hansen

enthalpy components and molar volumes via Eq. (4). The

correlation is shown in Fig. 4 for seven electronic nose

polymer sensors and 24 solvents.

Strong linear correlations (Table 3) between the experi-

mentally determined [82] change in resistivity and the

permeability-related expression Eq. (4) were found for

seven different polymer sensors (polymethylmethacrylate


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Table 3

Pearson R correlation values and slopes of predicted versus experimental

DR/R for each of seven polymer odorant detectors

Polymer sensor Slope R

Polycaprolactone 0.858 0.925

Polysulfone 0.932 0.962

PMMA 0.678 0.827

PEVA 0.888 0.936

Polyethylene 0.870 0.933

Polyethyleneoxide 0.746 0.874

4-HydroxyPS 1.018 0.991


UNCORRECTED PROOF(PMMA), 4-hydroxypolystyrene (4HPS), polyethyleneoxide

(PEO), polyethylene (PE), polyethylenevinylacetate (PEVA),

polysulfone, and caprolactone). Eq. (4) ®tting parameters

for the various polymer sensors are given in Table 4.

Calculated densities and Hansen solubilities for the seven

polymers and 24 solvents are summarized in Table 5.

The slopes and the Pearson R values for the correlation

plots are listed in Table 4. The correlation was particularly

good for polysulfone, 4-hydroxypolystyrene and PEVA

(polyethylene-co-vinyl acetate) and especially poor for

polymethylmethacrylate based on both correlation slope

and the Pearson R values for the linear ®t. Polysulfone

appears to discriminate between solvents of different sizes

since the free volume fraction is small and the free volume

distribution may be narrow, resulting in a `molecular' sieve

effect. Additionally, the experimental relative change in

resistivity in polysulfone ranges from zero to 1.0, which

makes it a particularly good high-resolution sensor.

The polyethylene-co-vinyl acetate detector also corre-

lates reasonably well with the theoretical relative change

in resistivity. However, the relative change in resistivity

range is smaller compared to polysulfone indicating that it

is less discriminating towards ester and alcohol solvents. A

possible explanation that accounts for this observation is

that PEVA contains polar ester functional groups due to

the vinyl content (18%), as well as non-polar components

due to the polyethylene content (82%). PEVA has a glass

transition below room temperature and as a result contains a

large free volume fraction. This decreases the sensitivity

towards molecules of different sizes compared to high Tg

polymers such as polysulfone. The third particularly good

detector in terms of signal correlation with theoretical

prediction is 4-hydroxy-polystyrene. This detector is parti-

cularly sensitive to molecules functionalized with highly

polar groups such as alcohol, obviously due to the hydroxyl

functional group. However, the sensitivity of this sensor to

moderately polar or non-polar solvents such as esters is

particularly low.

3.3. Diffusion of gases in amorphous polymers

The diffusion of molecules in polymer matrices involves

mesoscale times [microseconds (ms) to seconds (ms)], far

too long for routine MD simulations [83±86]. Consequently

we formulated the Monte Carlo Void Diffusion (MCVD)

method to calculate diffusion of gases in polymer matrices

for very long time scales based on coarse-grained informa-

tion extracted from short [sub nanoseconds (ns)] MD simu-

lations. The MD (over ,200 ps) is used to de®ne a

probability of having a void at each point of a grid. The

MCVD considers random motion of a penetrant molecule

on this grid. We show that the MC and MD both give rise to

an anomalous �R2 / t1=2� behavior at short time and

Einstein diffusion behavior �R2 / t� at long time. Compar-

ing the MD and MC at short time (,200 ps) provides the

corresponding time for the MCVD step. It is practical to

carry out MCVD for mesoscale times (ms to s). This repre-

sents an example of how to connect dynamics from the

atomistic scale to the meso scale. The results reported are

on studies of the diffusion of He in an amorphous poly-

ethylene (PE) matrix.

The permeability (P) of a gas through a membrane can be

written as [87]

P � DS �5�where D is the diffusion coef®cient and S is the solubility.

Generally, only P is available experimentally, but D and S

often depend differently on the various design parameters.

The total distance (R) travelled in a time (t) is given by the

Einstein relation

R2D E

� 6Dt as t ! 1 �6�where k l designates averaging over the ensemble of starting


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0.00 0.20 0.40 0.60 0.80 1.00

Experimental ∆R/Ro

Equ

atio

n(4

)

Fig. 4. Correlation between modeled and experimental changes in resistiv-

ity of seven polymer sensors exposed to 24 solvents.

Table 4

Model parameters (Eq. (4)) for seven electronic nose polymer sensors

Model parameter PMMA 4HPS PEO PE PEVA Poly sulfone Capro-lactone

Pre-exponential LnR0 0.676 0.233 20.761 2.296 0.770 1.092 20.813

Electrostatic b 1 20.001 20.028 0.014 0.040 0.018 20.001 0.019

Dispersion b 2 0.007 20.007 0.012 0.047 0.020 20.007 0.031

Hydrogen bond b 3 0.026 0.042 0.016 0.060 0.030 0.029 0.040

Diffusion g 0.009 0.033 0.003 0.011 0.007 0.012 0.006


UNCORRECTED PROOF

and ending points for the given time interval t. To predict D

for a He in PE, we perform a number of MD calculations,

each starting with He in various sites, for times suf®ciently

long that Eq. (6) is obeyed.

An example is given in Fig. 5a below on the left of He

diffusion in PE. Here we see that 1.5 ns of MD leads to the

behavior in Eq. (6). Hence, the MD can be used to determine

D. The Fig. 5b on the right shows that for O2 in a PVC/

PVDC copolymer, 1.5 ns of MD is not suf®cient to display

Einsteinian diffusion, and we estimate that this latter case

requires a ms time scale.

Fig. 6 shows the trajectories for ®ve He atoms diffusing

for 1.5 ns in PE. There are three-dimensional regions (we

call them felicitons) in which the gases spend signi®cant

times [ < 20±70 picoseconds (ps)] separated by pseudo

one-dimensional channels where they spend shorter times

( < 5±20 ps). Calculations on various gas molecules (H2,

CO2, O2, He, Ar) diffusing in various amorphous polymers

give similar results, suggesting that this feliciton-channel

network is generally observed in MD.

Fig. 7 shows the dynamical void distributions in the

polymer. Here we partitioned the unit cell into one million


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1120

Table 5

Estimated densities and Hansen solubilities

Density (g/cc) d 2/Vm (Kcal/mol) H1 H2 H3Solvents

Electrostatic Dispersion H-bonding

2-Pentanol 0.89 2151.42 253.32 276.48 221.62

3-Pentanol 0.88 2142.40 247.89 276.87 217.64

Amylacetate 0.95 2127.31 240.19 287.13 0.00

Butylacetate 0.97 2132.03 241.75 290.28 0.00

Decylacetate 0.91 2104.70 221.02 283.68 0.00

Ethanol 0.88 2257.64 2146.00 251.35 260.29

Ethylacetate 1.01 2159.31 268.99 290.33 0.00

Hexylacetate 0.95 2122.55 234.83 287.72 0.00

Iso-amylalcohol 0.89 2159.46 259.82 273.87 225.77

Isoamylacetate 0.96 2125.90 238.67 287.24 0.00

Isoamylbenzoate 1.03 2119.56 223.04 296.52 0.00

Isoamylbutyrate 0.94 2111.52 225.34 286.17 0.00

Isoamylcaproate 0.92 2104.57 220.83 283.74 0.00

Isoamylpropionate 0.93 2113.17 230.36 282.81 0.00

Isobutylacetate 0.96 2130.92 245.05 285.87 0.00

Isopropylacetate 0.98 2143.46 257.20 286.26 0.00

n-amylalcohol 0.88 2159.42 259.53 275.46 224.44

n-Heptanol 0.87 2130.23 237.63 276.59 216.01

n-Hexanol 0.88 2141.38 246.42 277.97 216.99

n-Propanol 0.86 2193.82 294.68 260.77 238.37

Octanol 0.88 2127.59 233.80 279.91 213.88

Octylacetate 0.92 2112.37 226.42 285.95 0.00

Propylacetate 0.98 2142.96 254.90 288.06 0.00

n-Butanol 0.84 2152.72 264.31 261.58 226.82

Polymers

PMMA 1.11 290.51 231.19 259.32 0.00

4HPS 1.09 2106.66 228.66 264.48 213.51

PEO 1.13 2168.10 268.36 295.90 23.84

PE 0.88 285.45 21.00 284.46 0.00

PEVA 0.96 285.02 210.82 274.20 0.00

Polysulfone 1.30 2138.74 229.76 2108.98 0.00

Caprolactone 1.09 2122.66 235.31 287.34 0.00

Table 6

The MD to MCVD time constants

Temperature D (AÊ 2/ps) Time conversion (fs/MCVD step) Ratio

Crossover Anomalous Fickian

tanomalous/tFickian

400 1.77 4.23 8.03 8.47 0.95

350 1.30 10.00 10.00 9.23 1.08

300 0.60 8.91 8.91 5.83 1.53

250 0.20 9.97 9.97 6.00 1.66

200 0.12 7.84 7.84 8.33 0.94


UNCORRECTED PROOF

cells �100 £ 100 £ 100� and examined the voids over a

period of 200 ps. Every 5 ps we examined whether a

probe radius 0.7 AÊ would contact any part of the polymer.

Here we see that the void analysis leads to an excellent

match with the felicitons and channels de®ned by the gas

particle in MD.

To calculate the mean square displacement (MSD) with

time, we considered each ps step of the 10 ns trajectory to be


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Fig. 5. (a) He diffusion in PE. Here we see that 1.5 ns of MD leads to the behavior in (2). Hence the MD can be used to determine D. (b) O2 diffusion in a PVC/

PVDC copolymer, 1.5 ns of MD is not suf®cient to display Einsteinian diffusion, and we estimate that this latter case requires a ms time scale.


UNCORRECTED PROOF

a possible starting point

R2�t� �XT 2 t

t0�0

uR�t 1 t�2 R�t0�u2=XT 2 t

t0�0

1

where, T is the total time. To determine whether the system

is in the diffusion region, where the Einstein relation Eq. (2)

holds, we plot logkR2l vs. logt: This should have a slope of

unity with the intercept providing the value for 6D. Indeed,

Fig. 8 shows diffusive behavior for t $ 0:1 ns averaged

over all 5 He atoms. The intercept, (6D), leads to D �1:77 A2

=ps � 29:5 £ 1026 cm2=s �r � 0:854 g=cc�:=

The experimental diffusion coef®cient [88] of He in

LDPE �r � 0:914 g=cc� at 258C is D � 6:8 £ 1026 cm2=s

and in HDPE �r � 0:964 g=cc� at 258C is D � 3:07 £1026 cm2

=s: The lower density and lower MW in the calcu-

lations is consistent with the higher D.

Both from the MD and MCVD, we observe three regions:

² Ballistic: Up to ,0.03 ps, the distance increases linearly

with time �kR2l / t2�: This is the ballistic regime, before

the particle hits a wall of the feliciton.

² Anomalous: From ,0.25 ps to ,4 ps, the trajectory ®ts

the kR2l / t1=2 line quite well. This is the anomalous

diffusion region, corresponding to intra-feliciton motion.

During this time the motion is primarily within felicitons.

We have found such anomalous diffusion regions in all

studies of amorphous polymers.

² Fickian: For times longer than 0.1 ns we ®nd true Einstei-

nian diffusion conditions.

Given the void probability grid from MD we design a

random walk MC algorithm in which a particle on a grid

point moves to an adjacent grid point. We assume the prob-

ability to be proportional to the void weight (Wg � 1 if

always a void to Wg � 0 if never a void). In order for a

random walk process to mimic this motion, we include a

momentum bias in the jump probability with an angular

dependence based on the last previous step direction. We

®nd that a simple cosine term, of the angle between the last

step and the next possible jump direction, leads to a MCVD

trajectory within a feliciton that mimics the MD trajectory.

A second issue concerns which neighboring points are

allowed for the jump. Simple choices are:

(a) only the six nearest neighbor points (forward, back-

ward, up, down, right, left),

(b) 18 points also including next nearest neighbors, or

(c) 26 points also including third nearest neighbors.

We chose case (c) since it ef®ciently spans the choices of

solid angle, allowing channels in diagonal direction to be

found.

The remaining issue concerns converting the MCVD step

to physical time. We have considered three ways to do this:

(1) Fickian match: If MD for suf®ciently long time is

available (200 ps for He/PE and several ns for other

combinations such as O2/PVC-PVDC), then one can

obtain the diffusion constant from the Fickian regime of

the MD and use this to get the conversion from MC step


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Fig. 6. The trajectories for ®ve He atoms diffusing for 1.5 ns in PE.


UNCORRECTED PROOF

to actual time. This corresponds to matching the log R2 2log t intercepts for the Fickian regimes.

(2) Crossover match: If MD for less time is available

(50 ps for He/PE), then one can match the crossover

time from the anomalous to the Fickian regime and obtain

the conversion from MC step to actual time.

(3) Anomalous match: For shorter time scales, or poly-

mers where the crossover is not observed in MD, one can

match the log R2 2 log t intercepts of the anomalous

regime from MD and MCVD.

The anomalous match is preferred since it requires

the least time for the MD simulations. Most polymer

systems show a well-de®ned anomalous behavior within

ps to ns MD time. However, the question is whether

this corresponds to the conversion from the longer time

MD. For He/PE the MD was suf®ciently long to

consider all three methods. Table 6 shows the MD to

MCVD time constant for all 3 cases. We see that while

the diffusion constant changes by a factor of 15 between

the temperature extremes �1:77@400 K! 0:12@200 K�


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1456Fig. 8. The diffusive behavior for t $ 0:1 ns averaged over all 5 He atoms. The intercept (6D) leads to D � 1:77 A2=ps � 29:5 £ 1026 cm2

=s �r � 0:854 g=cc�.

Fig. 7. The dynamical void distributions in the polymer.


UNCORRECTED PROOF

the time factor from the anomalous diffusion match differs

only by 20%.

The research projects reported in this paper are supported

by grants from DOE-ASCI, NASA/Ames, NASA-JPL,

Owens Corning, Chevron Research Technology Co. Facil-

ities of MSC is also supported by funds from NSF (CHE 95-

22179), ARO/DURIP, ARO-MURI, ONR; Asahi Chemical,

Avery Dennison, BP Chemical, Beckman Institute, Dow

Chemical, Chevron Petroleum Technology Co., Chevron

Chemical Co., Exxon, Kellogg, Seiko-Epson

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