Download - Dmitry Bedrov et al- Temperature-dependent shear viscosity coefficient of octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX): A molecular dynamics simulation study

8/3/2019Dmitry Bedrov et al- Temperature-dependent shear viscosity coefficient of octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX): A molecular dynamics simulatio

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FORCE FIELD AND SIMULATION METHODOLOGY

In previous work4 we developed a classical, explicit-

atom force field for flexible HMX molecules using results ofhigh-level quantum chemistry calculations for conforma-tional energies and geometries of gas-phase HMX one ofthe optimal geometries is illustrated in Fig. 1 and other nit-ramine compounds 1,3-dimethyl-1,3-dinitro methyldiamineDDMD4 and dimethylnitramine DMNA7. The lack ofexperimental data for liquid HMX precludes direct validationof the developed force field. However, good agreement be-tween experimental results and simulation predictions wasobtained for thermophysical properties of gaseous and liquidDMNA.7

Molecular dynamics MD simulations were carried outat six different temperatures 550800 K, at 50 K intervals

and atmospheric pressure. Isothermal-isobaric NpT simula-tions were performed for 4 ns in order to establish the equi-librium density, using the final configuration from a higher-temperature equilibration as the starting point for the nextlower temperature. Isothermal-isochoric NVT productionruns of 1030 ns duration depending upon temperaturewere performed using the NoseHoover thermostat8 with anintegration step size of 1.0 fs. Periodic boundary conditionswere employed. The standard Shake algorithm9 was used toconstrain bond lengths. The Ewald summation method10 wasemployed to evaluate long-range electrostatic interactions.All simulation cells contained 50 HMX molecules.

VISCOSITY CALCULATION

The shear viscosity can be calculated using equilib-rium fluctuations of the off-diagonal components () ofthe stress tensor.10 It was shown by Daivis and Evans11 that,for an isotropic system, the convergence of viscosity calcu-lations can be improved by including equilibrium fluctua-tions of diagonal components of the stress tensor. In this casethe generalized GreenKubo formula is applied to the sym-metrized traceless portion ( P) of the stress tensor withappropriate weight factors for diagonal and off-diagonal el-ements:

V

10kBT

0

qP tP0dt, 1

where V and Tare volume and temperature of the system, kBis the Boltzmann constant, q is a weight factor (q1 if, q

43 if, and P is defined as

P/2

3

, 2where is the Kronecker delta.

Einstein relations are often used in place of the GreenKubo expression for the calculation of transport coefficients.Haile12 has shown that in a system with periodic boundaryconditions the viscosity cannot be calculated using the con-ventional Einstein formula,10 which involves atomic coordi-nates and velocities. However, it can be employed afterslight modifications,12,13 yielding

limt

V

20kBTt

qA tA02

lim

t

V

20kBTt

qA t2 , 3

where

A t0

t

P tdt. 4

It was shown by Mondello and Grest13 that Eqs. 3 and 4give the same results as the GreenKubo formulation forMD simulations of short-chain alkanes. We considered bothforms Eqs. 1 and 3 for the shear viscosity of HMX andfound the results to differ by two percent or less. Thus, allresults reported below were obtained using the Einstein rela-tions defined in Eqs. 3 and 4. In addition, the self-diffusion coefficient D was computed using the standard Ein-stein relationship10

D limt

Rcm tRcm0 2

6t, 5

where Rcm( t)Rcm(0) is the time-dependent center-of-massdisplacement of a given molecule.

The choices of integration time step and sampling fre-quency are important for obtaining accurate results from MDsimulations. Previous experience involving simulations ofexplicit-atom systems e.g., DMNA,7 ethers,14 andpolystyrene15 with bond constraints has shown that a 1 fsintegration time step is sufficiently short to yield accurateintegration of the equations of motion. In order to determinehow often the stress tensor should be sampled during pro-duction runs to yield reliable calculations of the shear vis-cosity we performed a short 2 ns simulation at high tem-perature 750 K, recording the stress tensor at every timestep 1 fs. The time history of one of the elements of thestress tensor is shown in Fig. 2, where it is plotted at inter-

FIG. 1. Conformation of the HMX molecule in the form. Solid atoms arecarbons, cross-hatched are nitrogens, patterned are oxygens.

7204 J. Chem. Phys., Vol. 112, No. 16, 22 April 2000 Bedrov, Smith, and Sewell

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vals of 1, 10, and 20 fs. Note that a time integration step of1.0 fs is used in each case; only the frequency of stress tensor

output is varied. As would be expected for an atomicrepresentation,13 the stress tensor exhibits strong oscillatorybehavior. It is clear that a sampling frequency of 20 fs resultsin loss of information, whereas a 10 fs time step captures allbut the fastest oscillations. In order to further ensure that a 10fs sampling interval does not influence the viscosity calcula-tions, we computed the apparent shear viscosity versus timeEq. 3,16 sampling the stress tensor at intervals of 1, 10,20, and 40 fs from the 2 ns run at 750 K. The results, whichare shown in Fig. 3, indicate a nearly imperceptible differ-ence between a sampling interval of 10 fs versus 1 fs. It isinteresting to note that, although sampling the stress tensor at20 fs intervals led to significant differences in the apparent

time history Fig. 2, the resulting shear viscosity is onlyweakly affected. We tentatively attribute this to cancellationof errors during integration in Eq. 4. Based on the preced-ing considerations, the stress tensor was sampled at an inter-val of 10 fs in all simulations. The center-of-mass positionsof the molecules were recorded at intervals of 1 ps for use incalculating the self-diffusion coefficients.

The length of the trajectory ( tsim) required for the calcu-lation of a given property is another important, but difficult

to estimate, parameter in MD simulations. In order to obtainaccurate transport coefficients from equilibrium MD meth-ods, the length of the trajectory should be much longer thanany relevant characteristic relaxation times of the system.Mondello and Grest13 showed that for n-alkanes the rota-tional diffusion time (R) is the longest relevant relaxationtime for viscosity calculations. Dysthe et al.17 pointed outthat other relaxation times could be important for small mol-ecules, e.g., the mean time for a molecule to move one mo-lecular diameter (D). HMX does not resemble a long poly-meric chain, but neither can it be considered a simplemolecule. Thus, it is difficult to predict which of the charac-teristic times will most closely correlate with the shear vis-

cosity for liquid HMX; and hence which will determine thelength of the simulation required to obtain accurate values ofthe shear viscosity. However, one can define a viscosity re-laxation time as

17

1e1 tsim. 6

For very long trajectories minimal statistical error, we ex-pect the rhs. of Eq. 3 to be time dependent for times t and independent of time for t . Hence, is acharacteristic time for the approach of the rhs. of Eq. 3 to aconstant value, and simulations must be much longer than in order to obtain reliable values for the shear viscosity.

While the criteria tsim , tsimD , and tsimR pro-

vide guidance in determining the length of the simulationtrajectories tsim required to obtain accurate values of theshear viscosity, in practice, we used the following ultimatecriterion in determining when to discontinue the simulationfor a particular temperature. Using Eq. 3, we calculate theviscosity by averaging values of the rhs of Eq. 3 for 10ttsim/2 as a function of the length of the simulation, tsim .In Fig. 4 we show the dependence of the average viscosity asfunction of trajectory length for 650 and 700 K. It can beseen that, after some tmin additional simulation time results inminor fluctuations less than 10% from some mean value ofthe average viscosity. We define tmin as the trajectory lengthsuch that 1 the time-dependent average viscos-

FIG. 2. Representation of the stress tensor evolution using different outputfrequencies for one of the off-diagonal components at 750 K.

FIG. 3. Influence of the stress tensor output frequency on the rhs of Eq. 3as a function of time at 750 K.

FIG. 4. Average viscosity see the text for a definition obtained from MDsimulation as a function of trajectory length for 650 and 700 K.

7205J. Chem. Phys., Vol. 112, No. 16, 22 April 2000 Shear viscosity of HMX



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ity Fig. 4 has at least two maxima in the interval 10t tmin and 2 after tmin further simulation time does notchange the average viscosity by more than 10%. Therefore,for all temperatures except 550 K, our conservative estimateof the uncertainty in the shear viscosity is 10% or less. At550 K, the simulation was too short to allow estimation ofuncertainty in the viscosity.

RESULTS AND DISCUSSION

Characteristics relaxation times

In Table I we report, for each temperature, tmin , tsim ,and , as defined in Eq. 6; the rotational diffusion corre-lation time, obtained by fitting the end-to-end vector au-tocorrelation function T1cos (t) for nitro group nitro-gen atoms on opposite sides of the molecule to a simpleexponential decay law,

T1exp t2R ; 7and D calculated as

DR

g

2

6D , 8

where D is the HMX center-of-mass self-diffusion coeffi-cient determined from Eq. 5 and R g

2 is the averagesquared radius of gyration 7.3 2. With the exception of550 K, each trajectory was at least 45 times longer than thelargest relaxation time at a given temperature. The system at550 K would have required upward of 70 ns by these criteria,which is beyond the practical limits imposed by our compu-tational resources. Therefore we stopped the simulations atthis temperature after 20 ns, which is sufficient to accuratelycalculate the self-diffusion coefficient.

Viscosity and self-diffusion

The atmospheric pressure, temperature-dependent shearviscosity and self-diffusion coefficients for liquid HMX inthe temperature domain 550 KT800 K are summarizedin Table I. We also include the equilibrium density for eachtemperature. The shear viscosity is predicted to range from0.0055 Pa *s at 800 K up to 0.45 Pa *s at 550 K.

The temperature dependence of the shear viscosity andself-diffusion coefficients are well described by Arrheniusexpressions, as shown in Fig. 5. There is no evidence for theonset of nonArrhenius behavior over the temperature domainconsidered, which extends down to near the melting point of

HMX. The calculated apparent activation energies are 14.4and 14.5 kcal/mol for self-diffusion and shear viscosity, re-spectively. The Eyring rate expression for dense fluids18 in-dicates that the viscosity activation energy is proportional tothe energy of vaporization. For more than 100 substances,including associated liquids, the simple relation (EvapnEvis , where 2n5 holds.

18 Assuming that the en-ergy of vaporization is approximately equal to the cohesiveenergy, it is possible to compare Evap and Evis determineddirectly from MD simulations. We obtain Evap36.3 kcal/mol from simulation 800 K, yielding a ratioEvap/Evis2.5. The large value for the viscosity activa-tion energy is consistent with the high energy of vaporizationof HMX.

Correlation of viscosity with diffusion and rotationof HMX

In this section, based on the MD simulation results, weexamine correlations between the temperature dependence ofviscosity with that of other dynamic properties, in particularself-diffusion and rotational diffusion. There are many em-

pirical, semiempirical, and theoretically based relations todescribe the correlation between these thermophysical prop-erties. Most of these correlations can be represented as

DKT

D9

TABLE I. Relaxation times and transport coefficients of liquid HMX obtained from MD simulations.

T K tmin ns tsim ns ps R ps D ps kg m3

D109

(m2 s1) Pa s

550 20 1250 310.0 1664 1650.9 0.006 0.450600 25.0 30 520 70.0 675 1614.4 0.018 0.120650 17.5 20 270 33.6 304 1586.9 0.040 0.040700 8.0 15 120 20.7 129 1554.5 0.094 0.022750 6.0 15 40 12.9 54 1520.1 0.225 0.010

800 4.0 10 20 9.5 37 1488.2 0.325 0.0055

FIG. 5. Arrhenius fit solid lines of viscosity in the temperature domain600800 K and self-diffusion in the temperature domain 550800 K ob-tained from MD simulations symbols and extrapolation of viscosity fit to550 K dashed line.




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ergy under the auspices of the Los Alamos ASCI High Ex-plosives Project. TDS wishes to thank John K. Dienes andRalph S. Menikoff for useful discussions concerning meso-mechanics modeling. Computational resources for some ofthe calculations were provided by the Los AlamosT-division/CNLS Avalon Beowulf cluster

1 E. Kober and R. Menikoff, Los Alamos Report, LA-13546 MS, Com-

paction waves in granular HMX, January 1999.2 For example, T. D. Sewell and C. M. Bennett, Monte Carlo calculationsof the anisotropic engineering moduli for RDX, submitted, J. Appl.Phys.

3LASL Explosive Property Data, edited by T. R. Gibbs and A. PopolatoUniversity of California, Berkeley, 1980.

4 G. D. Smith and R. K. Bharadwaj, J. Phys. Chem. B 103, 3570 1999.5 H. H. Cady, A. C. Larson, and D. T. Cromer, Acta Crystallogr. 16, 6171963.

6 J. K. Dienes, P. Conley, and R. Menikoffprivate communication.7 G. D. Smith, R. K. Bharadwaj, D. Bedrov, and C. Ayyagari, J. Phys.

Chem. B 103, 705 1999.

8 S. Nose, J. Chem. Phys. 81, 511 1984.9 J. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comput. Phys. 23, 3271977.

10 M. P. Allen and D. T. Tildesley, Computer Simulation of Liquids Oxford,New York, 1987.

11 P. J. Daivis and D. J. Evans, J. Chem. Phys. 103, 4261 1996.12 J. M. Haile, Molecular Dynamics Simulation Wiley, New York, 1992.13 M. Mondello and G. S. Grest, J. Chem. Phys. 106, 9327 1997.14 D. Bedrov, O. Borodin, and G. D. Smith, J. Phys. Chem. B 102, 56831998; D. Bedrov, O. Borodin, and G. D. Smith, ibid. 102, 9565 1998;

D. Bedrov and G. D. Smith, J. Chem. Phys. 109, 8118 1998; G. D.Smith, O. Borodin, and D. Bedrov, J. Phys. Chem. A 102, 10 318 1998.

15 G. D. Smith, C. Ayyagari, R. L. Jaffe, M. Pekny, and A. Bernarbo, J.Phys. Chem. A 102, 4694 1998.

16 We refer to the rhs of Eq. 3 as the time-dependent apparent shear vis-cosity. The shear viscosity itself is defined as the long-time limit of the rhsEq. 3 and is independent of time.

17 D. K. Dysthe, A. H. Fuchs, and B. Rousseau, J. Chem. Phys. 110, 40471999.

18 S. Glasstone, K. J. Laidler, and H. Eyring, The Theory of Rate ProcessesMcGraw-Hill, New York, 1941.


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Download - Dmitry Bedrov et al- Temperature-dependent shear viscosity coefficient of octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX): A molecular dynamics simulation study

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