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Molecular dynamics and kinetic study of carbon coagulation in the releaseMolecular dynamics and kinetic study of carbon coagulation in the release

wave of detonation productswave of detonation products

Guillaume Chevrot, Arnaud Sollier, and Nicolas Pineau

Citation: J. Chem. Phys. 136 136, 084506 (2012); doi: 10.1063/1.3686750

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THE JOURNAL OF CHEMICAL PHYSICS 136, 084506 (2012)

Molecular dynamics and kinetic study of carbon coagulation in the release

wave of detonation products

Guillaume Chevrot, Arnaud Sollier, and Nicolas Pineaua)

CEA/DAM/DIF, F-91297 Arpajon, France

(Received 10 October 2011; accepted 1 February 2012; published online 29 February 2012)

We present a combined molecular dynamics and kinetic study of a carbon cluster aggregation pro-

cess in thermodynamic conditions relevant for the detonation products of oxygen deficient explo-

sives. Molecular dynamics simulations with the LCBOPII potential under gigapascal pressure and

high temperatures indicate that (i) the cluster motion in the detonation gas is compatible with Brow-

nian diffusion and (ii) the coalescence probability is 100% for two clusters entering the interac-

tion cutoff distance. We used these results for a subsequent kinetic study with the Smoluchowski

model, with realistic models applied for the physical parameters such as viscosity and cluster size.

We found that purely aggregational kinetics yield too fast clustering, with moderate influence of the

model parameters. In agreement with previous studies, the introduction of surface reactivity through

a simple kinetic model is necessary to approach the clustering time scales suggested by experiments

(1000 atoms after 100 ns, 10 000 atoms after 1 μs). However, these models fail to reach

all experimental criteria simultaneously and more complex modelling of the surface process

seems desirable to go beyond these current limitations. © 2012 American Institute of Physics.

[http://dx.doi.org/10.1063/1.3686750]

I. INTRODUCTION

The detonation of CHON explosives with a negative

oxygen balance (e.g., RDX or TATB) produces soots which

contain a large variety of carbon particles (graphitic ribbons,

nanodiamonds, nano-onions, etc.) and modify significantly

the way the detonation energy is released. Two phenomenons

contribute to slow down the energy release: the kinetics of

carbon coagulation and the gradual release of heteroatoms en-

closed in the core of carbon clusters. Besides, the carbon clus-

ters interact with the gaseous detonation products and modify

their chemical and thermodynamic equilibrium properties.1

Therefore the description of the kinetics of carbon cluster co-

agulation is crucial to describe the time-dependent properties

of the detonation products.

Experimental data on the time evolution and final state

of the solid carbon phase along the full detonation/expansion

process are scarce in terms of structure, size, and ki-

netic/thermodynamic history. Besides numerous postmortem

analysis of the carbon soots which usually contain up to a

few ten mass-percent of nanodiamonds with 5 nm diameter

(corresponding to about 10 000 atoms), one can mention a

SAXS experiment2on TNT and mixed TNT/RDX which sug-

gests that carbon clusters with diameters larger than 2.8 nm

(i.e., about 1000 atoms) may form in the reaction zone (t ∼

100 ns), while further aggregation may proceed in the expan-

sion wave for a few microseconds, in agreement with various

references.3,4

Several previous studies dealt with the kinetics of car-

bon cluster coagulation,4–8based on the Smoluchowski the-

a)Author to whom correspondence should be addressed. Electronic mail:

nicolas.pineau@cea.fr.

ory developed in 1916 to describe coagulation kinetics in vis-

cous media.9,10In 1987, Shaw and Johnson5mentioned such

a study applied to the coalescence of carbon in a detonation

wave, using a dilute atomic carbon gas in a constant viscos-

ity medium as a starting point. The output data were used

to estimate the energy release during the aggregation pro-

cess. The thermodynamic properties (pressure, temperature,

anddensity)weresettotheChapman-Jouguetvaluesthrough-

out the calculations, neglecting their time evolution during the

full expansion process, reaction zone + release of gaseous

products on the order of a few microseconds, having a non-

negligible impact on viscosity thus on aggregation kinetics.

Besides, more recent studies showed that detonation waves

lead to the initiation of carbon cluster formation inside the re-

action zone,11,12contrary to the basic assumption of a dilute

gas of carbon atoms. In 1994, Mal’kov presented a similar

study using empirical values for the mean volume of clusters

and viscosity as a function of time, yielding reasonable or-

ders of magnitude for carbon cluster formation.6Finally be-

tween 1998 and 2002, Ree, Viecelli, and Glosli4,7,8published

several studies on the kinetics of carbon coalescence, using

the same approximations as Shaw and Johnson.5Starting with

similar initial conditions, they found that pure aggregation ki-

netics yield mean cluster sizes of 10 000 atoms after 100 ns,

when experiment suggests values close to 1000 atoms. There-

fore, they improved the Smoluchowski model with a fragmen-

tation term designed to slow down the kinetics, which they

interpreted as a process of surface evaporation of atomic or

dimer fragments.

These experimental and theoretical observations suggest

that the previous Smoluchowski models may be improved

in terms of properties of the viscous fluid with respect to

real detonation gas expansion: indeed the thermodynamic

0021-9606/2012/136(8)/084506/10/$30.00© 2012 American Institute of Physics

136, 084506-1

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Page 3

084506-2 Chevrot, Sollier, and PineauJ. Chem. Phys. 136, 084506 (2012)

properties of the detonation gas vary substantially during the

release process which has some impact on viscosity. Be-

sides, the size of the clusters appears explicitly in the model

and should therefore be modeled consistently, and their ini-

tial structure is far from mono-atomic for many carbon-rich

explosives such as TATB: Zhang et al. used molecular dy-

namics simulations with the ReaxFF potential11to show that

clusters formed right after the shock front can contain as

many as 145 atoms, including many H-N-O heteroatoms as

impurities. Finally the evaporation mechanism suggested by

Glosli, Viecelli, and Ree was not observed in recent molecu-

lar dynamics simulations of carbon clusters under detonation

conditions,13,14and therefore alternate mechanisms need to

be proposed when too fast kinetics is obtained for pure aggre-

gation processes.

In this paper, we present a combined molecular dynamic

(MD) and kinetic study of the carbon clustering process un-

der time-dependent thermodynamic conditions relevant for

the expansion of the gaseous products of carbon rich explo-

sives. First we use MD simulations in the NVT ensemble in

order to assess (i) the Brownian motion behavior of carbon

clusters in a high temperature, high pressure viscous media

and (ii) the usual approximation that two encountering clus-

ters undergo coalescence with a 100% probability under those

conditions. We describe the coalescence mechanism of nano-

onion clusters, with particular emphasis on the influence of

pressure, temperature, mass, and mass-mismatch effects on fi-

nal topologies. Then we use the Smoluchowski model to eval-

uate the kinetics of formation and the impact of the evolution

of the thermodynamic properties in the detonation gas expan-

sion. We discuss on the relevance of various surface processes

(evaporation, reactivity, etc.) to balance the too fast aggrega-

tion kinetics yielded by the original Smoluchowski model.

II. MOLECULAR DYNAMICS SIMULATIONS

A. Simulation details

We used the molecular dynamics code STAMP de-

veloped at CEA and the LCBOPII empirical potential for

carbon.15Previous studies showed that LCBOPII provides

good descriptions for a large variety of carbon systems under

high temperature and pressure, typically up to 10.000 K and

100 GPa.13,14,16,17All the simulations were run in the NVT

ensemble using a Langevin thermostat with a friction param-

eter of 1014s−1and periodic boundary conditions whenever

external pressure was applied. The equations of motion were

integrated with a Velocity-Verlet scheme and a time step of

0.5 fs, yielding good conservation of the total energy over

long simulation times in the NVE ensemble (several hundreds

of picoseconds). All the simulations were stopped when the

energy of the system was converged, reaching times of the

order of a nanosecond.

We studied the coalescence processes under thermody-

namic conditions close to the Chapman-Jouguet (CJ) point

of TATB (P ∼ 30 GPa and T ∼ 3000 K), a high ex-

plosive that produces substantial amounts of solid carbon

residues. Temperatures up to 6000 K were used to account for

local thermodynamic inhomogeneity of the surrounding det-

onation gas,18and some simulations were run in vacuum to

evidence the pressure effects. Although the structure of

nanocarbons has never been observed during the explosive

process itself, the postmortem analyses of detonation carbon

soot show 3 main types of nanocarbons, nano-onions, nano-

diamonds, and nano-graphites, which proportions vary with

the explosive and experimental conditions.19–22However, a

previous study14showed that nano-onions, consisting of con-

centric piles of nearly spherical fullerenes with increasing

diameter, are likely to be the most stable structures for car-

bon clusters containing up to a few thousand atoms, making

them good candidates for a model of detonation soot. There-

fore, we designed a set of samples using an initial 1000-atom

nano-onion thatwasprogressivelypeeledofftoobtainsmaller

clusters of 44/127/226/426/634 atoms, respectively. Each

cluster was equilibrated at 3000 K, the lowest temperature

used in this study, resulting in a decrease of the number of

internal layers for clusters of 226 atoms and above. For each

cluster pair, three constant temperature simulations were run

(3000 K, 4000 K, and 6000 K). The final state of the 3000

K cases was used as the starting point for the 4000 K cases,

while the 6000 K cases were run independently starting from

liquidclusterpairsobtainedbyindividual equilibrations.Sim-

ulations with pairs of clusters of equal and unequal masses

were run: in the following parts of this paper they are referred

to as “symmetric” and “asymmetric,” respectively.

The initial configuration of each system was set such

that the surface-to-surface distance between two carbon clus-

ters was inferior to the LCBOPII long-range cutoff (6.0 Å),

while preserving an overall attractive interaction. We found

that surface-to-surface distances ranging from 2.5 to 4.0 Å

were reasonable, yielding cluster separations short enough to

avoid insertion of the LJ fluid, while large enough to avoid the

repulsive part of the LCBOPII potential. In order to mimic the

high pressures achieved in the detonation products of high ex-

plosives, the carbon clusters were immersed in a cubic simu-

lation cell containing a Lennard-Jones (LJ) fluid of argon: the

thickness of the rare gas layer around the clusters was at least

1 nm in all directions, resulting in simulation cell sizes rang-

ing from 30 Å to 60 Å. The LJ parameters provided in Table I

are the same as those used in previous simulations of the nu-

cleation and growth of carbon clusters under thermodynamic

conditions close to the CJ point of nitromethane.13Using a

rare-gas pressure bath neglects the influence of the reactivity

of the clusters with the hot mixture of detonation gas. Multi-

species potentials such as ReaxFF (Ref. 23) have the capabil-

ity to model this type of reactive medium; however, we pref-

ered to use the LCBOPII potential with a LJ bath because (i)

TABLE I. Interaction parameters Ar-Ar and C-Ar. The Ar-Ar parameters

are taken from Ref. 24. We combined these parameters and the C-C param-

eters used in Ref. 13 to calculate the C-Ar parameters, using usual Lorentz-

Berthelot rule.

?σ

Cutoff radius

(Å)(J) (Å)

Ar-Ar

C-Ar

1.654 × 10−21

1.0 × 10−21

3.405

3.800

10.0

10.0

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084506-3 MD and kinetic study of carbon coagulationJ. Chem. Phys. 136, 084506 (2012)

experiments suggest that coagulation occurs mainly beyond

the reaction zone of the detonation wave,2(ii) we focus pri-

marily on the carbon/carbon interactions for which LCBOPII

is one of the best known potentials in this thermodynamic

regime, and (iii) the computational cost of simulating large

systems with a fully reactive potential would have been pro-

hibitive. Although the simulations were not run in the NPT

ensemble, we checked carefully that the pressure remained

reasonably constant in the course of each NVT trajectory.

B. Results

1. Brownian motion assessment

In order to assess the possibility of non-Brownian behav-

ior of the carbon clusters in a high viscosity medium, we ran

a first set of simulations imposing various orders of magni-

tude of collisional velocities to the clusters: we found these

initial velocities vanished before the formation of any inter-

cluster bond due to the interaction with the surrounding argon

bath, implying that the carbon clusters formed in the detona-

tion gas undergo Brownian motion exclusively. This obser-

vation is consistent with the use of the Smoluchowski model

in the forthcoming kinetic study. Accordingly, no collisional

velocity was applied in the subsequent simulations.

2. Pressure effect

A set of symmetric and asymmetric simulations were run

at 3000 K both in vacuum and under 30 GPa pressure. Some

of the final structures are presented in Fig. 1 and show that

FIG. 1. Final configurations of a selection of coalescence simulations with

or without applied pressure (30 GPa). Pressure improves the coalescence sub-

stantially, yielding higher merge of the clusters. Simulations in vacuum were

not 100% efficient with a certain probability of rebound of the interacting

clusters. Simulations under pressure yielded 100% coalescence.

pressure enhances the coalescence yield, with more spheri-

cal final structures due to a more efficient merge of the two

clusters. We did not obtain sphericity for large clusters on

the time scale of the simulations, showing such processes

are beyond the capability of conventional molecular dynamics

methods. However, all the simulations under pressure yielded

coalescence, implying a 100% efficiency: this result is consis-

tent with the Smoluchowski model and rules out the need for

an estimate of reaction probabilities with respect to cluster

size or mass mismatch. In comparison, we obtained a non-

negligible probability of rebound for the simulations in vac-

uum, using a statistical estimate of the coalescence efficiency

averaged over relative cluster orientation. For example the 44

+ 44 and 44 + 226 cases in vacuum yielded coalescence

probabilities of 41% and 29%, respectively: this decrease is

probably related to the lower curvature of the surface of large

clusters, resulting in lower surface energy and reactivity.

3. Temperature effect

The full set of symmetric and asymmetric cases were

simulated under 30 GPa pressure and temperatures ranging

from 3000 K to 6000 K, a range of thermodynamic condi-

tions that is compatible with local fluctuations at the CJ point

for carbon-rich explosives such as TATB.

As an example, the coalescence mechanism of the 127

+ 127 case is detailed in Fig. 2. At 3000 K, the coalescence

process consists in the merge of the equivalent layers of the

FIG. 2. Slices of 6 Å of the coalescence process of two 226-atom clus-

ters from molecular dynamics simulations under 30 GPa pressure and

3000 K, 4000 K, and 6000 K temperatures. A first simulation was run for

1 ns at 3000 K, starting from nano-onion clusters, then the temperature was

increased to 4000 K for another 1 ns. A second simulation was run indepen-

dently at 6000 K starting from liquid nanocarbons. The color code is a guide

to distinguish the carbon atoms from each initial cluster.

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Page 5

084506-4 Chevrot, Sollier, and PineauJ. Chem. Phys. 136, 084506 (2012)

FIG. 3. Initial and final internal structures for the full set of coalescence

simulations at 30 GPa and with increasing temperature, for initial clusters

of the same size (symmetric case). No simulation was run for the 426 + 426

case at 4000 K.

two clusters, that is first the external layers then the internal

ones merge, resulting in a peanut-like structure. Mixing re-

mains incomplete after up to 1 ns of simulation with a clear

spatial separation evidenced by the distinct colors used for the

initial leftand right clusters.The final cluster remains solid, as

shown by the absence of sphericity and the preservation of the

layered structure throughout the coalescence process. Heating

up to 4000 K leads to a spherical cluster consisting in a solid

(fullerene-like) external layer and a liquid core (the internal

atoms of both initial clusters are mixed, which is character-

istic of a diffusive, i.e., liquid-like, behavior). The existence

of this “quasi-liquid” cluster was predicted previously14and

interpreted in terms of location of the structural defects (the

highly curved inner layers of the nano-onion clusters), where

melting is more likely to nucleate. At 6000 K the coalescence

process results in a perfectly spherical liquid cluster within a

few ten picoseconds.

All the final structures are presented in Figs. 3 and 4

(symmetric and asymmetric case, respectively). The trends

correspond to those observed for the 127 + 127 case except-

ing for the mono-layer 44 + 44 case which yields a liquid

FIG. 4. Same as Fig. 3 for clusters of different sizes (asymmetric case).

Smaller system snapshots are magnified for better visualization, which re-

sults in unrespected spatial scale.

FIG. 5. Energy and pressure of the minimized structures along the 226

+ 226 coalescence simulations at 3000 K, 4000 K, and 6000 K (black-red-

green curves, respectively). The total (full), LJ (dotted), LCBOPII (dashed),

and LCBOPII + surface (dotted-dashed) energy contributions are presented

(LCBOPII + surface contains the LCBOPII and interfacial C/Ar LJ energies).

Note that the LCBOPII contributions are given “per carbon atom” contrary

to the total and LJ energies which are given “per atom”. The lower plots give

absolute values, while the upper plots give relative evolutions with respect to

the beginning of each simulation. The snapshots represent slices of the final

0 K-structures and show little structural rearrangements occurred along the

minimization process.

cluster as soon as 4000 K due to the high surface curvature

of the unique shell. For the asymmetric cases with high mass

mismatch (Fig. 4) low temperatures lead to merging of the

small cluster with the external layer of the large cluster, while

at high temperatures diffusion of the small cluster atoms is

observed throughout the final cluster.

For the 226 + 226 case, we performed an energy mini-

mization of the full system every 50 ps using a steepest de-

scent algorithm, yielding 20 optimized structures for each

temperature.

The time evolution of the 0 K energy and pressure are

presented in Fig. 5. They show that (i) at 3000 K and 4000 K

the coalescence process results in a decrease of all the compo-

nents of the potential energy and (ii) increasing the tempera-

ture from 3000 K to 4000 K leads to higher energy structures

as shown by the increase in total/LCBOPII/LCBOPII + sur-

face energies (although the energy/pressure contributions are

not fully converged after 1 ns, we believe our final structures

are reasonably close to fully equilibrated ones). This latter

result is unexpected and suggests the existence of an energy

barrier along the coalescence reaction coordinate, resulting in

a temperature-dependent coalescence rate. Therefore, we sug-

gest that fully coalesced spherical clusters cannot be obtained

below a threshold temperature, which cannot be evaluated di-

rectly from our simulations.

In the mean time, the 0 K pressure tends to decrease (i) in

the course of individual simulations (fixed T, except at 6000 K

for which the equilibrium is reached very fast, resulting in

fluctuations around a constant 8 GPa value) and (ii) with in-

creasing temperature. This can be explained by the decrease

in volume of the carbon cluster during coalescence, resulting

in the expansion of the argon bath (the total volume is con-

stant). For NPT simulations we would expect a similar effect

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084506-5 MD and kinetic study of carbon coagulationJ. Chem. Phys. 136, 084506 (2012)

with presumably enhanced coalescence due to the extra-work

of non-decreasing pressure forces.

We conclude that the driving force for carbon cluster coa-

lescence is the external pressure imposed by the LJ bath, with

possibly some additional entropic contribution due to the in-

crease of the LJ volume (although counter-balanced by the

assumed entropy decrease associated to cluster coalescence).

Note that the pressure jump observed between 4000 K and

6000 K may be mainly attributed to the lower argon density

used at 6000 K to obtain a 30 GPa pressure, while minor con-

tributions may come from the decrease in cluster volume due

to a higher coalescence rate combined with the solid/liquid

transition.

C. Discussion

TheseresultsvalidatetheuseoftheSmoluchowskimodel

for the subsequent kinetic studies of the full-scale cluster-

ing process since (i) they illustrate the Brownian behavior of

the carbon clusters (viscosity dissipates any initial cluster ve-

locity on very short time scales) and (ii) they show coales-

cence occurs as soon as the two carbon clusters come within

interaction distance, provided sufficient external pressure is

applied.

Besides, these simulations give some interesting details

on the clustering mechanism and driving forces at play dur-

ing the detonation of carbon rich explosives, showing that de-

pending on the local thermodynamic properties of the det-

onation gas, a variety of final structures may be formed,

over time scales that are not always compatible with the MD

capabilities.

Weshouldstressthatduringthecourseofoursimulations

(up to 1 ns), we did not observe the evaporation of surface

atoms or groups of atoms, which confirms some observations

made, but not reported, in previous MD simulations of the

nucleation/growth of carbon clusters in detonation products

conditions.13This comment is important in view to determine

the surface processes relevant to extend the Smoluchowski

model.

III. KINETIC STUDY

A. The Smoluchowski model

We used the Smoluchowski model9,10in order to sim-

ulate the clustering kinetics of carbon clusters in thermody-

namic conditions relevant for detonation products. The basic

model is based on the following assumptions: (i) the solvent

particles are all similar with same spherical size, (ii) the so-

lute particle concentration is uniform throughout the solution,

and (iii) the solute particles undergo Brownian motion so long

as no other solute particle comes within their sphere of influ-

ence. In our case, the carbon clusters and the detonation gas

molecules are defined as the solute and solvent particles, re-

spectively. The sphere of influence of a solute particle i is de-

fined as the distance Ribelow which this particle coalesces

with another solute particle. Then the new “double” particle

undergoes Brownian motion at a reduced rate due to its in-

creased size. The model is thus defined by a set of differential

equations giving the production rate dCk/dt of solute particles

of size k

?

j > i

dCk

dt

= 4π

i + j = k

CiCjDijRij

?

1 +

Rij

(πDijt)1/2

?

−4π(1 + δj,k)Ck

∞

?

j=1

CjDkjRkj

?

1 +

Rkj

(πDkjt)1/2

?

(1)

,

with

δj,k=

?

0

1

ifj ?= k,

ifj = k,

where the Kronecker delta is added to correct for the single

occurrence of the kk consumption case when all other kj cases

(with j ?= k) occur twice in the algorithm (kj and jk). The first

sum in Eq. (1) corresponds to the formation of clusters of size

k by coalescence of i + j and the second sum corresponds to

the consumption of k due to coalescence with clusters j. Ci

is the concentration of clusters i, Dijis the relative diffusion

constant of i with respect to j, and Rijis the coalescence dis-

tance for an ij pair, with Dij= Di+ Djand Rij= (Ri+ Rj)/2.

The diffusion constant is obtained with the Stokes-Einstein

relation

Di=

kBT

6πRiη,

(2)

where η is the viscosity and can be calculated for a dense fluid

of hard spheres using the Enskog theory

?1

with

?√MRT

η = η0b0ρ

Y+ 0.800 + 0.761Y

?

,

(3)

η0=

5

16√π

Naσ2

?

,

(4)

b0=2

3πσ3,

(5)

Y =

P

ρkBT− 1,

(6)

where σ is the diameter of the hard sphere, M the mo-

lar mass, R, kB, and Na the Rydberg, Boltzmann, and

Avogadro constants, respectively, and (ρ, P, T) the thermody-

namic properties of the system. Applied to the CJ properties

of TATB (ρ = 2.41 g/cm3, xH2O= 0.3959, xCO2= 0.2027,

xN2= 0.4014), with the excluded molecular volumes σH2O

= 2.4 Å, σN2= 2.9 Å, σCO2= 3.2 Å, the Enskog theory

yields a viscosity value of 2.447 g m−1s−1in qualitative

agreement with Shaw5(η = 1 g m−1s−1) but smaller by

two orders of magnitude than the value proposed by Mal’kov

(η = 100 g m−1s−1).6

In previous studies, the Smoluchowski equation was

used in forms that neglect the influence of cluster size

on the diffusion/coagulation rate constants,4,5based on the

neglect of the first-order correction Rij/(πDijt)1/2in the

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Page 7

084506-6 Chevrot, Sollier, and Pineau J. Chem. Phys. 136, 084506 (2012)

original Smoluchowski rate constant kijdt = 4πCiCjDijRij

(1 + Rij/(πDijt)1/2)dt. However, neglecting this first-order cor-

rection may be unsafe since the radii of detonation carbon

clusters range from fractions of up to several nanometers, the

diffusion constants are of the order of 10−10m2s−1, and the

coalescence times range up to 10−6s, yielding first-order cor-

rections between 0.05 and 104, that is potentially ?1. Be-

sides a constant viscosity was used for the detonation gas

which does not account for the evolution of the thermody-

namic properties in the release wave beyond the CJ point (the

time scale of the coalescence process is not negligible with

respect to the full release to ambient conditions).

Therefore we decided to use a complete formulation

of the Smoluchowski equation, including first-order correc-

tions to the diffusion rates and introducing time-dependent

thermodynamic properties (ρ, P, T) consistent with a high

explosive release wave. The resulting dependence of the dif-

fusion rates on Riand Rjcalled for a realistic model for cluster

size. We did not include any correction due to clustering ef-

ficiency since MD simulations under pressure yielded 100%

coalescence.

1. Size model for carbon clusters

The cluster size is defined by the number of atoms com-

posingthecluster,althoughtheSmoluchowskimodelrequires

the knowledge of the cluster radius. Previous direct molecular

dynamics simulations of the nucleation/growth mechanism of

carbon clusters in nitromethane detonation products yielded

mono-layer fullerene structures of up to 140 atoms,13al-

though typical cluster sizes involved in the full process range

up to a few ten thousands (experiments give indications on

the formation times of 1000 and 10 000 atom clusters approx-

imately). Therefore direct simulations do not cover the full

range of sizes that are desirable for the full aggregation pro-

cess and an approximate model is necessary for N > 140. For

N < 140, we used the MD simulations of Ref. 13 to get a

reasonable statistical distribution of cluster shapes and we de-

rived the R = f(N) relationship based on an estimated effective

radius Refffor each cluster

R(N) = RvdW+1

n

n

?

i=1

Reff(N),

with

Reff(N) =

?

1

N

N

?

i=1

r2

i

?1/2

,

(7)

with ri the distance between the atom i and the center of

mass of the cluster, N the number of atoms in the cluster,

and n the number of clusters of size N. RvdWis defined as

the van der Waals exclusion radius of a single carbon atom

(set to 1.75 Å in Ref. 25). Including all the encountered clus-

ter types allowed to account for the existence of non-fullerene

shapes such as chains, rings, or flakes. This approach makes

the rather harmless assumption that shape and size distribu-

tions only weakly depend on the explosive excess in carbon.

For N > 140 we used the thermodynamic equilibrium re-

sults from Ref. 14 which showed that clusters of up to a few

0

500

1000

1500

2000

2500

3000

3500

4000

Number of atoms

0

5

10

15

20

25

Cluster radius (angtsrom)

TATB

Nitromethane

FIG. 6. Average cluster radius as a function of the number of atoms. For

N < 140, the radius is obtained from a statistical average over structures

extracted from previous molecular dynamics simulations.13For N > 140, a

phenomenological model is used:26the changes of slope are caused by the

discrete increase of the number of nano-onion shells when N reaches critical

values. Size models for the detonation products of TATB and nitromethane

are plotted.

thousand atoms are more likely to adopt a nano-onion struc-

ture. We used a size model designed for these nano-onions

based on their russian-doll structure26yielding a typical layer

by layer increase of the cluster radius with increasing N

R0= Ri+ (ns− 1) × ?r + RvdW,

with Rithe radius of the inner shell, nsthe number of shells,

?r the inter-shell distance, and RvdWthe previously defined

van der Waals radius. Pressure and temperature were included

in the model through a thermal expansion and compressibility

coefficient.27The subsequent R = f(N) relationship is plotted

in Fig. 6 for nitromethane and TATB, showing larger cluster

sizes for nitromethane due to lower pressure and larger tem-

perature at the CJ point.28

(8)

2. Time dependence of the thermodynamic properties

in the release wave

In order to evaluate the time evolution of the thermody-

namic properties in a high explosive release wave, we per-

formed Direct Numerical Simulations of a detonation wave

propagating in TATB using the HESIONE hydrocode devel-

oped at CEA. We first performed 1D Lagrangian calculations

in plane geometry, corresponding to the assumptions of the

ZND model for a 1D detonation wave. For the sake of com-

parison, we have also performed 2D Lagrangian and Eule-

rian calculations in both plane and axisymetric geometries,

in order to study the influence of dimensions and boundary

conditions on the late time dependence of the hydrodynam-

ics. In all calculations, the dimensions of the explosive sam-

ple were chosen large enough to ensure the formation of a

stationary sustained detonation wave. Moreover, we used a

constant mesh size of 5 μm ensuring convergence in all our

configurations. For all these calculations, we used the classi-

cal Empirical Hot Spot (or JTF) reactive burn model29,30cou-

pled with a Cochran-Chan EOS (Ref. 31) for the unreacted

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Page 8

084506-7 MD and kinetic study of carbon coagulationJ. Chem. Phys. 136, 084506 (2012)

1

1.5

2

2.5

3

2500

3000

3500

4000

024

6

8

0

10

20

30

40

024

6

80

1

2

3

Time (μs)

Mass density (g.cm-3) Temperature (K)

Viscosity (g.m-1.s-1) Pressure (GPa)

FIG. 7. Time evolution of the density, temperature, pressure, and resulting

viscosity of the detonation products of TATB calculated from the hydrody-

namic simulation of the propagation of a planar detonation wave. The viscos-

ity is calculated with the Enskog theory (Eqs. (3)and(6)) using the computed

thermodynamic properties. t = 0 corresponds to the ZND peak.

TATB and a JWL EOS (Ref. 32) for the reaction products.

We stored the evolution of density, temperature, and pressure

of the detonation products at the rear face of the sample as a

function of time, starting from the ZND peak. Using the En-

skog theory (Eqs. (3)and(6)) we could then estimate the re-

sulting time dependence of the viscosity of the detonation gas.

The only difference between our 1D and 2D calculations lies

in the fact that in 2D geometry release waves coming from

the different boundaries are going to induce a faster release

of the detonation wave compared to 1D, resulting in discrep-

ancies for late times only (typically several microseconds for

reasonably large samples). However, the aggregation process

is assumed to be completed at these times, therefore these dif-

ferences can be neglected in the kinetic study. The results for

the 1D plane simulations, corresponding to the ideal 1D deto-

nation wave of the ZND theory, are plotted in Fig 7. Besides,

we use the time dependent mass density to compute the vol-

ume expansion coefficient (ρ(t + ?t)/ρ(t)) of the system and ac-

count for a realistic decrease of the cluster concentrations by

Ci= Ci× ρ(t+?t)/ρ(t).

B. Results

We computed the time evolution of the size distribution

of carbon clusters by solving the system of differential equa-

tions (1) with a time step of 1 fs, allowing good balance be-

tween computational efficiency and numerical stability. The

maximal cluster size was set to 20.000 atoms, about twice the

size of the nanodiamonds observed in experiments. The ini-

tial excess carbon concentration C0can be obtained using the

dissociation reaction of TATB

2C6H6N6O6(s)→ 6H2O(g)+ 6N2(g)+ 3CO2(g)+ 9C(s)

which shows that C0= (9/2)CTATB. At the ZND peak, the

density of TATB is 2.835 g/cm3corresponding to CTATB

= 0.011 mol/cm3(the molar mass of TATB is 258 g/cm3),

0

0.05

0.1

0.15

0.2

Time (μs)

0

5000

10000

15000

20000

Average cluster size (number of atoms)

C1

C6

C12

C24

FIG. 8. Time evolution of the average size of carbon clusters depending on

the initial size distribution. Initial sizes of 1, 6, 12, and 24 atoms are shown.

and therefore to a total excess carbon concentration

C0= 0.0495 mol/cm3.

1. Influence of the initial distribution

In a first set of simulations we investigated the influ-

ence of the initial size distribution of the carbon clusters. In

four separate calculations, we successively assigned the ex-

cess carbon concentration C0to the cluster types C1, C6, C12,

and C24. We preferentially used C6*nspecies assuming a coa-

lescence process based on the six-membered ring of TATB, as

suggested by the dimerization/polymerization processes ob-

served in Ref. 11. The time evolution of the mean size of

the carbon clusters plotted in Fig. 8 shows very fast cluster-

ing kinetics with 1000 atom average sizes reached over a few

nanoseconds, to be compared to the experimental estimate of

100 ns, that is nearly two orders of magnitude faster. Accord-

ingly, 10 000 atom sizes are reached over ∼25 ns,33against

an experimental order of magnitude of 1 μs.

The initial distribution does not seem to influence the ki-

netics since all the curves are superimposed and barely dis-

tinguishable. Two practical consequences of this result are

that (i) we can leave the initial distribution issue which is ir-

relevant here and (ii) all subsequent calculations will be run

starting from C24distributions for the sake of computational

efficiency.

2. Influence of the parameters

Weanalyzed theinfluence ofthemainparameters andap-

proximations applied to the Smoluchowski equation, namely

the cluster size model, the total excess carbon concentration

C0, the time-dependent thermodynamic properties, and the

viscosity model. A new set of simulations were run, each of

which had one parameter modified with respect to the ref-

erence simulation set: respectively, we increased all cluster

radii arbitrarily by 25%, we changed the total carbon density

to match that of nitromethane (an explosive less carbon-rich

than TATB with C0= 0.0072 mol cm−3), we disconnected the

time-dependent thermodynamic properties, and we used the

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Page 9

084506-8 Chevrot, Sollier, and PineauJ. Chem. Phys. 136, 084506 (2012)

0 0.10.20.3 0.4

0.5

Time (μs)

0

5000

10000

15000

20000

Average cluster size (number of atoms)

Reference data

Increased size model

Constant (ρ,P,T)

Nitromethane carbon excess

Mal’kov

FIG. 9. Influence of the model parameters and approximations on the time

dependence of the average cluster size for initial C24cluster distributions.

The reference curve of Fig. 8 is in black. Increased cluster size model in

red, disconnected hydrodynamic parameters in orange, nitromethane carbon

concentration in green, and Mal’kov viscosity in brown.

constant viscosity parameter of Mal’kov.6Accordingly with

the previous observation, we used the C24initial distribution

to speed up the calculations. The resulting time dependence

of the average cluster size is presented in Fig. 9, with compar-

ison to the reference data of Fig. 8.

We found that most modifications yielded negligible dif-

ferences with the reference data, except for the total excess

carbon concentration C0and the viscosity model. When using

the carbon excess concentration of nitromethane the kinetics

is decreased by a factor of five approximately, yielding cluster

sizes of about 6.400 atoms after 100 ns (∼15.000 for TATB),

and converging towards 15.000 atoms after half a microsec-

ond, about 15% lower than TATB.

Although these values are still too large with respect to

experiment, the initial concentration seems to play an im-

portant role on kinetics, yielding clustering rate differences

that may be evidenced in future experiments. However, C0

is an intrinsic property that depends on the stoichiometry of

the CwHxOyNzexplosive and on the efficiency of the burning

process, which in turn depends on the experimental setup (ge-

ometry and dimensions of the explosive, external constraints,

etc.). Therefore, it cannot be considered a parameter of the

model contrary to the viscosity model which appears to have

a strong influence: when applying the constant value pro-

posed by Mal’kov, the clustering kinetics is dramatically re-

duced and yields a profile consistent with the experimental es-

timates. However, the viscosity of Mal’kov is 50 times larger

than the one obtained through the Enskog theory, which is

consistent with the value proposed by Shaw and Johnson.5In

the absence of clear justification for this choice of viscosity,

we can only assume that it has actually been adjusted to match

the experimental data. The absence of influence of parame-

ters such as the cluster size model, or the time-dependence of

the thermodynamic properties show that further investigations

or improvements in those directions are clearly a dead-end.

Although the thermodynamic properties are strongly varying

in the release wave of the detonation product, we found that

the decrease of the corresponding viscosity is compensated

by the concomitant temperature decrease in the calculation of

the diffusion constant Di(Eq. (2)).

Last, we did not test numerically the assumption that

all cluster collisions yield coalescence, implying a reaction

probability of one which may be questioned for large clusters

whose surface reactivity should be lower due to their high cur-

vature radius. Moreover, we showed that the efficiency of the

clustering mechanism is sensitive to external pressure, which

decreases along the release process. However, after one mi-

crosecond of release, the external pressure computed from

hydrodynamic simulations is still superior to 7 GPa (Fig. 7)

and MD simulations show that clustering is still 100% effi-

cient for clusters up to more than 400 atoms. Therefore, we

assume this approximation is reasonably safe for all cluster

sizes along the relevant clustering time scale.

C. Surface reactivity effects

Carbon clusters under high temperatures in a chemically

complex environment may be subjected to many surface pro-

cesses such as atom/cluster evaporation, surface reactivity,

andsoon.Forexample,endothermicsurfacereactionssuchas

C(s)+ CO2(g)? 2CO(g)occur spontaneously at high temper-

atures and may compete with the clustering process to yield

an equilibrium cluster size.

Viecelli et al.4suggested fragmentation mechanisms

could provide such a counter-balancing effect and added an

“evaporation” term to the Smoluchowski model. Although we

never observed any evaporation in our MD simulations under

detonation product conditions, we applied a similar approach

in view of surface reactivity. We expect surface reactivity to

be a very complex process involving a large number of chem-

ical reactions with variable rate constants. However, a global

approach including a limited number of simple contributions

may be sufficient to correct the behavior of the clustering ki-

netics, such as

dCk

dt

= 4π

?

j > i

i + j = k

CiCjDijRij

?

1 +

Rij

(πDijt)1/2

?

−4π(1 + δj,k)Ck

∞

?

j=1

CjDkjRkj

?

1 +

Rij

(πDijt)1/2

?

+

?

1≤n<N

λn(Ck+n− Ck) + λnδk,n

∞

?

l=n+1

Cl,

(9)

where the third and fourth terms include surface reactions in-

volving up to N carbon atoms with λnan average rate constant

for n-atom processes. For each cluster size we expect the re-

action rates to be proportional to the surface area, so the rate

constants are defined as λn= λ0

stant and Natis the number of atoms of the clusters.34

We computed the time evolution of the size distribution

of carbon clusters by solving the system of differential equa-

tions (9). For viscosity we kept the physically based Enskog

model, rather than the estimate of Mal’kov, assuming that the

discrepancy between our previous calculations and the exper-

imental data is due to the absence of any surface reaction

nN2/3

ats−1, where λ0

nis a con-

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Page 10

084506-9 MD and kinetic study of carbon coagulationJ. Chem. Phys. 136, 084506 (2012)

0

0.0250.05 0.075

0.1

Times (μs)

0

300

600

900

1200

Average cluster size (number of atoms)

C24

C24+C48

C24+C48+C72

C24+C48+C72+C96

FIG. 10. Dependence of the average cluster size on the model of surface

reaction processes. Surface reactions involving 24/48/72/96 atom clusters are

considered. The rate constant λ0is adjusted to 5 × 106s−1to approach the

first experimental estimate of 1000 atoms at 100 ns.

process. As for the aggregation process, the basic block for

surface reactivity is a 24-atom cluster. We optimized the λ0

constant to approach the first experimental criteria of 1000

atom clusters after ∼100 ns, yielding a value of 5 × 106s−1.

Figure 10 summarizes the results obtained for four cases con-

sisting in surface reactions involving 24, 24 → 48, 24 → 72,

and 24 → 96 atom clusters. The results show consistent con-

vergence of the average cluster size after 100 ns for all the

calculated cases, with a clear but limited decrease of cluster-

ing kinetics when surface reactivity involves particles of in-

creasing size. However, with this optimized value of λ0, the

time required to reach 10 000 atom sizes tends to infinity sug-

gesting the second experimental estimate (10 000 atoms after

1 μs) cannot be met simultaneously. The subsequent yield of

the surface reactions was estimated to a few tens of atoms per

nanosecond, a time scale compatible with our MD simula-

tions: the absence of surface evaporation in these simulations

suggests that surface reactivity with the external detonation

gas is a more appropriate counter-balancing process.

We conclude that a simple surface reaction model slows

down substantially the clustering kinetics yielded by the

purely aggregational Smoluchowski model, although we were

unsuccessful with its adjustement to all the available experi-

mental estimates, i.e., 1000 atoms at 100 ns and 10 000 at

1 μs. This empirical fit is not satisfactory; however, the ex-

tensive treatment of the surface reactions involved, and the

estimate of the corresponding rate constants, is clearly a very

difficult task.

IV. CONCLUSION

In this study, we performed a combined molecular dy-

namics and kinetic study of the carbon clustering process in

thermodynamic conditions relevant for the detonation prod-

ucts of carbon-rich explosives.

The molecular dynamics simulations confirm that under

gigapascal pressure the clusters undergo Brownian motion

and that cluster-cluster encounters yield 100% coalescence

probabilities, which partially validates the use of a Smolu-

chowski model for the subsequent kinetic studies. Besides,

we found that the coalescence process under high tempera-

ture and pressure is spontaneous and temperature-dependent,

yielding clusters that can be (i) solid at 3000 K, adopt-

ing peanut-like structure, (ii) “quasi-liquid” and spherical at

4000 K (consisting of a liquid core surrounded by a solid sur-

face), or (iii) liquid and spherical at 6000 K.

The subsequent kinetic study was based on an improved

Smoluchowski model with realistic physical properties spe-

cific of the detonation gas conditions. We found purely aggre-

gational Smoluchowski models yield too fast kinetics, with

10 000 atom clusters formed within 10 ns when the exper-

imental estimates point towards an order of magnitude of a

microsecond. Including adjustable surface reactivity allows to

slow down the kinetics so that one experimental criterion can

be met at a time (1000 atoms in 0.1 μs or 10 000 atoms in

1μs),butnotboth.Apartfromthesurfacereactivityforwhich

we expect the modeling to be a tedious task, further improve-

ments to the kinetic model may come from a more accurate

description of the nature of the reactive systems and physical

and chemical processes at play, such as including the influ-

ence of the non-negligible fraction of heteroatoms in the co-

alescing clusters, or using a two-step kinetics to model more

accurately the first part of the reactive process, when the det-

onation mixture is essentially composed of small fractions of

gaseous molecules embedded in a polymeric CHNO matrix

(therefore very far from the Smoluchowski approximations).

We expect the latter modification will strongly decrease the

initial rates of formation of clusters, thus hopefully allowing

for a better match with the 1000 atom/0.1 μs criterion.

ACKNOWLEDGMENTS

Nicolas Pineau wishes to thank Laurent Soulard and

Emeric Bourasseau for fruitful discussions.

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27Based on our MD simulations of nano-onion aggregates,14we ob-

served that the cluster volume increases linearly with temperature.

We were then able to determine a thermal expansion coefficient (αV

= 1.58 × 10−4Å3/at/K) yielding the cluster volume as a function of T and

N: V = V0+ αV× N × T with N the number of atoms and T the temper-

ature. V0, the volume at 0 K, depends on the potential pressure P0Kand

its value can be obtained by solving numerically the pressure relationship

derived from the universal binding energy35:

?

P0K= exp

−α

??V0

V00

?1/3

− 1

??

×α2V−2/3

3V−1/3

0

00

E0

??V0

V00

?1/3

− 1

?

where V00the zero-pressure volume, E0the potential energy and α are de-

rived from the MD simulations.

28Estimated by thermochemical calculations around 12 GPa vs. 25 GPa, and

3500 K vs. 2700 K for nitromethane and TATB, respectively.

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33These 25 ns may be over-estimated due to the maximal cluster size used

in the calculation which was set to 20 000, thus limiting the “reactivity”

of the larger clusters. However, this does not impact on the conclusion that

pure Smoluchowski kinetics are too fast with respect to experimental esti-

mates since increasing the maximal cluster size can only yield even faster

kinetics.

34For the sake of simplicity we assumed all clusters are perfectly spherical

and did not account for the temperature-dependent topologies observed in

the MD simulations. More realistic accounting of the cluster topologies

would only be useful if combined with a surface reactivity model based on

accurate estimates of the rate constants.

35A. Banerjea and J. R. Smith, Phys. Rev. B 37, 6632 (1988).

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