We study the rotational relaxation process in nitrogen using all-atom molecular dynamics (MD) simulations and direct simulation Monte Carlo (DSMC). The intermolecular model used in the MD simulations is shown to (i) reproduce very well the shear viscosity of nitrogen over a wide range of temperatures, (ii) predict the near-equilibrium rotational collision number in good agreement with published trajectory calculations done on ab initio potential energy surfaces, and (iii) produce shock wave profiles in excellent accordance with the experimental measurements. We find that the rotational relaxation process is dependent not only on the near-equilibrium temperature (i.e., when systems relax to equilibrium after a small perturbation), but more importantly on both the magnitude and direction of the initial deviation from the equilibrium state. The comparison between MD and DSMC, based on the Borgnakke-Larsen model, for shock waves (both at low and high temperatures) and one-dimensional expansions shows that a judicious choice of a constant Zrot can produce DSMC results which are in relatively good agreement with MD. However, the selection of the rotational collision number is case-specific, depending not only on the temperature range, but more importantly on the type of flow (compression or expansion), with significant limitations for more complex simulations characterized both by expansion and compression zones. Parker's model, parametrized for nitrogen, overpredicts Zrot for temperatures above about 300 K. It is also unable to describe the dependence of the relaxation process on the direction to equilibrium. Finally, we present a demonstrative cell-based formulation of a rotational relaxation model to illustrate how, by including the key physics obtained from the MD data (dependence of the relaxation process on both the rotational and the translational state of the gas), the agreement between MD and DSMC solutions is drastically improved.
REFERENCES
1.
2.
H.
Rabitz
and S.-H.
Lam
, “Rotational energy relaxation in molecular hydrogen
,” J. Chem. Phys.
63
(8
), 3532
–3542
(1975
).3.
B. K.
Annis
and A. P.
Malinauskas
, “Temperature dependence of rotational collision numbers from thermal transpiration
,” J. Chem. Phys.
54
(11
), 4763
–4768
(1971
).4.
R. N.
Healy
and T. S.
Storvick
, “Rotational collision number and Eucken factors from thermal transpiration measurements
,” J. Chem. Phys.
50
(3
), 1419
–1427
(1969
).5.
A.
Tip
, J.
Los
, and A. E.
De Vries
, “Rotational relaxation numbers from thermal transpiration measurements
,” Physica
35
(4
), 489
–498
(1967
).6.
G. T.
McConville
, W. L.
Taylor
, and R. A.
Watkins
, “Analysis of the determination of rotational relaxation numbers from thermal transpiration
,” J. Chem. Phys.
53
(3
), 912
–919
(1970
).7.
G.
Ganzi
and S. I.
Sandler
, “Determination of thermal transport properties from thermal transpiration measurements
,” J. Chem. Phys.
55
(1
), 132
–140
(1971
).8.
E. H.
Carnevale
, C.
Carey
, and G.
Larson
, “Ultrasonic determination of rotational collision numbers and vibrational relaxation times of polyatomic gases at high temperatures
,” J. Chem. Phys.
47
(8
), 2829
–2835
(1967
).9.
T. G.
Winter
and G. L.
Hill
, “High-temperature ultrasonic measurements of rotational relaxation in hydrogen, deuterium, nitrogen, and oxygen
,” J. Acoust. Soc. Am.
42
(4
), 848
–858
(1967
).10.
P. G.
Kistemaker
, A.
Tom
, and A. E.
De Vries
, “Rotational relaxation numbers for the isotopic molecules of N2 and CO
,” Physica
48
(3
), 414
–424
(1970
).11.
J. G.
Parker
, “Rotational and vibrational relaxation in diatomic gases
,” Phys. Fluids
2
(4
), 449
–462
(1959
).12.
C.
Nyeland
, “Rotational relaxation of homonuclear diatomic molecules
,” J. Chem. Phys.
46
(1
), 63
–67
(1967
).13.
J. A.
Lordi
and R. E.
Mates
, “Rotational relaxation in nonpolar diatomic gases
,” Phys. Fluids
13
(2
), 291
–308
(1970
).14.
C. A.
Brau
and R. M.
Jonkman
, “Classical theory of rotational relaxation in diatomic gases
,” J. Chem. Phys.
52
(2
), 477
–484
(1970
).15.
A.
van der Avoird
, P. E. S.
Wormer
, and A. P. J.
Jansen
, “An improved intermolecular potential for nitrogen
,” J. Chem. Phys.
84
(3
), 1629
–1635
(1986
).16.
D.
Cappelletti
, F.
Vecchiocattivi
, F.
Pirani
, E. L.
Heck
, and A. S.
Dickinson
, “An intermolecular potential for nitrogen from a multi-property analysis
,” Mol. Phys.
93
(3
), 485
–499
(1998
).17.
M. S. H.
Ling
and M.
Rigby
, “Towards an intermolecular potential for nitrogen
,” Mol. Phys.
51
(4
), 855
–882
(1984
).18.
L.
Gomez
, B.
Bussery-Honvault
, T.
Cauchy
, M.
Bartolomei
, D.
Cappelletti
, and F.
Pirani
, “Global fits of new intermolecular ground state potential energy surfaces for N2H2 and N2N2 van der waals dimers
,” Chem. Phys. Lett.
445
(46
), 99
–107
(2007
).19.
R. M.
Berns
and A.
van der Avoird
, “N2-N2 interaction potential from ab initio calculations, with application to the structure of (N2)2
,” J. Chem. Phys.
72
(11
), 6107
–6116
(1980
).20.
V.
Aquilanti
, M.
Bartolomei
, D.
Cappelletti
, E.
Carmona-Novillo
, and F.
Pirani
, “The N2-N2 system: An experimental potential energy surface and calculated rotovibrational levels of the molecular nitrogen dimer
,” J. Chem. Phys.
117
(2
), 615
–627
(2002
).21.
C.
Nyeland
and G. D.
Billing
, “Transport coefficients of diatomic gases: Internal-state analysis for rotational and vibrational degrees of freedom
,” J. Phys. Chem.
92
, 1752
–1755
(1988
).22.
G. D.
Billing
and L.
Wang
, “Semiclassical calculations of transport coefficients and rotational relaxation of nitrogen at high temperatures
,” J. Phys. Chem.
96
, 2572
–2575
(1992
).23.
J. I.
Steinfeld
, P.
Ruttenberg
, G.
Millot
, G.
Fanjoux
, and B.
Lavorel
, “Scaling laws for inelastic collision processes in diatomic molecules
,” J. Phys. Chem.
95
, 9638
–9647
(1991
).24.
I. J.
Wysong
and D. C.
Wadsworth
, “Assessment of direct simulation Monte Carlo phenomenological rotational relaxation models
,” Phys. Fluids
10
(11
), 2983
–2994
(1998
).25.
G. A.
Bird
, Molecular Gas Dynamics and the Direct Simulation of Gas Flows
(Clarendon
, Oxford
, 1994
).26.
C.
Borgnakke
and P. S.
Larsen
, “Statistical collision model for monte carlo simulation of polyatomic gas mixture
,” J. Comp. Phys.
18
(4
), 405
–420
(1975
).27.
B. L.
Haas
, D. B.
Hash
, G. A.
Bird
, F. E.
Lumpkin
III, and H. A.
Hassan
, “Rates of thermal relaxation in direct simulation Monte Carlo methods
,” Phys. Fluids
6
(6
), 2191
–2201
(1994
).28.
I. D.
Boyd
, “Rotational-translational energy transfer in rarefied nonequilibrium flows
,” Phys. Fluids A
2
(3
), 447
–452
(1990
).29.
I. D.
Boyd
, “Temperature dependence of rotational relaxation in shock waves of nitrogen
,” J. Fluid Mech.
246
, 343
–360
(1993
).30.
I. D.
Boyd
, “Relaxation of discrete rotational energy distributions using a Monte Carlo method
,” Phys. Fluids A
5
(11
), 2278
(1993
).31.
K.
Koura
, “Monte Carlo direct simulation of rotational relaxation of diatomic molecules using classical trajectory calculations: Nitrogen shock wave
,” Phys. Fluids
9
(11
), 3543
–3549
(1997
).32.
K.
Koura
, “Monte Carlo direct simulation of rotational relaxation of nitrogen through high total temperature shock waves using classical trajectory calculations
,” Phys. Fluids
10
(10
), 2689
–2691
(1998
).33.
F.
Robben
and L.
Talbot
, “Experimental study of the rotational distribution function of nitrogen in a shock wave
,” Phys. Fluids
9
(4
), 653
–662
(1966
).34.
R. B.
Smith
, “Electron-beam investigation of a hypersonic shock wave in nitrogen
,” Phys. Fluids
15
(6
), 1010
–1017
(1972
).35.
T.
Tokumasu
and Y.
Matsumoto
, “Dynamic molecular collision (DMC) model for rarefied gas flow simulations by the DSMC method
,” Phys. Fluids
11
(7
), 1907
–1920
(1999
).36.
M. P.
Allen
and D. J.
Tildesley
, Computer Simulation of Liquids
(Clarendon
, Oxford
, 1987
).37.
D.
Frenkel
and B.
Smit
, Understanding Molecular Simulation
(Academic
, San Diego
, 2002
).38.
S.
Schlamp
and B. C.
Hathorn
, “Higher moments of the velocity distribution function in dense-gas shocks
,” J. Comp. Phys.
223
, 305
–315
(2007
).39.
J. P.
Ryckaert
, G.
Ciccotti
, and H. J. C.
Berendsen
, “Numerical integration of the Cartesian equations of motion of a system with constraints: Molecular dynamics of n-alkanes
,” J. Comp. Phys.
23
(3
), 327
–341
(1977
).40.
D. A.
McQuarrie
and J. D.
Simon
, Physical Chemistry: A Molecular Approach
(University Science Books
, Sausalito, CA
, 1997
).41.
J. O.
Hirschfelder
, C. F.
Curtiss
, and R. B.
Bird
, Molecular Theory of Gases and Liquids
(Wiley
, 1954
).42.
C. S.
Wang-Chang
and G. E.
Uhlenbeck
, “Transport phenomena in polyatomic gases
,” University of Michigan Engineering Research Technical Report No. CM-681, 1951
.43.
H. J. M.
Hanley
and J. F.
Ely
, “The viscosity and thermal conductivity coefficients of dilute nitrogen and oxygen
,” J. Phys. Chem. Ref. Data
2
(4
), 735
–755
(1973
).44.
E.
Vogel
, “Przisionsmessungen des viskosittskoeffizienten von stickstoff und den edelgasen zwischen raumtemperatur und 650 K
,” Ber. Bunsenges. Phys. Chem.
88
(10
), 997
–1002
(1984
).45.
E.
Salomons
and M.
Mareschal
, “Usefulness of the Burnett description of strong shock waves
,” Phys. Rev. Lett.
69
(2
), 269
–272
(1992
).46.
B. L.
Holian
, “Modeling shock-wave deformation via molecular dynamics
,” Phys. Rev. A
37
(7
), 2562
–2568
(1988
).47.
P.
Valentini
and T. E.
Schwartzentruber
, “Large-scale molecular dynamics simulations of normal shock waves in dilute argon
,” Phys. Fluids
21
(6
), 066101
–1
(2009
).48.
P.
Valentini
and T. E.
Schwartzentruber
, “A combined event-driven/time-driven molecular dynamics algorithm for the simulation of shock waves in rarefied gases
,” J. Comp. Phys.
228
(23
), 8766
–8778
(2009
).49.
P.
Kowalczyk
, A.
Palczewski
, G.
Russo
, and Z.
Walenta
, “Numerical solutions of the Boltzmann equation: Comparison of different algorithms
,” Eur. J. Mech. B/Fluid
27
, 62
–74
(2008
).50.
S. J.
Plimpton
, “Fast parallel algorithms for short-range molecular dynamics
,” J. Comp. Phys.
117
, 1
–19
(1995
).51.
LAMMPS molecular dynamics simulator, open source code, primary developers S. Plimpton, A. Thompson, and P. Crozier, Sandia Laboratories, Albuquerque, NM (as of September 2012, available at http://lammps.sandia.gov/index.html).
52.
D.
Gao
, C.
Zhang
, and T. E.
Schwartzentruber
, “Particle simulations of planetary probe flows employing automated mesh refinement
,” J. Spacecr. Rockets
48
(3
), 397
–405
(2011
).© 2012 American Institute of Physics.
2012
American Institute of Physics
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