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Computer simulation of the system N2 in fluid argon-correlation functions and relaxation times
Volume 8, number 1
COMPUTER
CHEiCIICALPHYSICSLETTERS
SIMULATION
CORRELATION
OF
THE
FUNCTIONS
SYSTEM AND
1 January 19il
N2 IN
FLUID
RELAXATION
ARGON-
TIMES*
E. F. O’BRIEN and G. WILSE ROBINSON Arthur Amos Noyes Laboratory of Chemical Physics **, California Institde of Technology. Pasadena, Catifotnia 91109, USA
Received 20 October 1970 The correlation functions of the dipole moment, P [u(O) u(t)]. angular and linear velocity, and bond forces have been calculated from computer aim f ated data for four different density-temperature states of N2 in fluid argon, From these functions infrared and Raman line shapes, NMR relaxation times, and rotational and classical vibrational relaxation times have been computed. l
In an effort to understand more about the microscopic processes responsible for relaxation and line broadening in fluids, we have carried out a computer simulation of the three-dimensional system N2 in argon using classical equations of motion. The wholly classical computations are expected to be applicable to NMR and pure rotational spectra as well as to rotational motions within a given vibrational state but not to “quantum mechanical processes”, such as vibrational relaxation, at the temperatures considered. Ar-Ar interactions were taken to be of the Lennard-Jones (6-12) type, while the N2-Ar potential used was constructed by assuming a rigid N2 bond and a Lennard-Jones 3-center atom-atom interaction model between the two centers on the diatomic and each Ar atom$. Combining laws for the force constants E and u were used $.$, and all interactions at distances greater than 2.5 o(Ar) were set equal to zero. This potential is qualitatively different from the Stockmeyer potentials used by Berne and Hays [Z]. The algorithm is similar to that used previously in this laborzitory for two-dimensional computations of pure liquid argon[3] $$$. The systems calculated consist of 63 or 156 Ar atoms (depending on density) and one N2 molecule, periodic boundary conditions being imposed. l
This work was supported in part by a grant;(No.GP-
12381) from the National Science *f Contribution No.4151. -
Foundation.
$ This potential is similar to the one formulated by
Sweet and SteeIe.[7] for two interacting N2_ $$ The combining laws Used were EN2_A= (EN,E&~!?
’ @N -A=& Na+oA)F&der% a! goritlnn [3] was based on one described earlier by Rahman 141. .-
$$$
While dipole transitions between rotationaL or rovibrational levels in N2 are forbidden, the computational model used here should closely approach the physical situation exemplified by CO in fluid solution. The small dipole moment in CO is not expected to affect substantially the correlation functions t . The following correlation functions and associated physical quantities were considered. 1) (u(O) u(t)) (infrared rotational and rovibrational line shapes [5]); 2) (P2[u(O) u(t)]) (Raman rotational and rovibrational line shapes and NMR quadrupole and magnetic dipole-dipole relaxation times [5]); 3) {o(O) - o(C)) (NMR spin-rotation relaxation times [6]); 4) {v(O) - v(t)) (diffusion constant {7]]; 5) cf(O)f(t)> (classical vibrational relaxation [8]). In the above, u is the unit vector along the N2 bond, P2 is the second Legendre polynomial, w is the angular frequency of N2, v labels velocity vectors of the particles, and $is the component of force along the N2 bond. Four systems have thus far beeu studied, three with a density of 1.227 g/cm3 at the temperatures 25’7.3OK (system I), 142.6oK (system 2) and 112.6OK (system 31, and the four& having a temperature of 289.3OK and a density of 0.506 g/cm3 (system 4). AH the above correlation functions were calculated for each of these systems. Since statistics were poor in the molecular dynamics data for Iong times, the tails of the correktionfunCtiOnSare very pgorly represented. l
l
t The addition of the dipole term in the potential fun+ tion changes the.potential by less than 0.1% of the. L-J E thrOUghOUt tie range Of the potential considered , . here. :,
.-
_. .’
;
79
Volume 8, number 1
CHEMICAL PHYSICS LETTERS T&e
Ccrr&ation function a) (Diatomic) (u(O) - u(t)) &IWO) - W)l) I WO). W,>
(fWf(l!)
(V,(O) . V,(0)
Correlation 1 1.71*0.09 0.91* 0.07 1.18* 0.09 0.60* 0.07 1.68*0.09
1 January 1971.
1 times (lo-l3 set) at half-width System 2. 3
2.58a0.14 1.32*0.10 1.12*0.09 0.6la0.07 1.80*0.10
3.60*0.20 1.64*0.12 1.12*0.09 0.57*0.07 1.95~0.11
a) The error limits were determined by the method of Zwanzig and Ailwadi 191.
4 1.30*0.07 O-76*0.05 3.24*0.10 6.70* 0.30 6.45+0.13
. Thus the correlation times are reported at halfcorrespondence between the calculated band width, i.e., the width at half the maximum height. shapes and the experimental ones used by Cordon. See table 1. 14N2 and 15N2 NMR relaxation times were calThe (u(0) - u(t)> and &[u(O) - u(t)]) correculated using the formalisms that relate relaxlation functions approach zero in a nearly expoation rates to the appropriate correlation funcnential manner for systems 1, 2, 3. This result tions [ (fi[U(O)*U(t)]) or (~(0) - o(t))] for the contrasts that obtained from experimental data following relaxation mechanisms: quadrupolar, by Cordon where a prominent shoulder occurs in spin-rotational and magnetic dipole-dipole. the tail of the correlation function. System 4 These calculations were performed for systems yields a (u(0) - u(t)> correlation function that 2 and 3. They indicate, as expected, that the dips below zero. This functional form is similar quadrupolar relaxation mechanism is dominant to the one calculated by Cordon at about this for 14N2 givin times of the order of miIliseconds.’ For 1gN2, the spin-rotation mechanism density. The above two correlation functions also show dominates giving relaxation times for systems 2 a strong temperature dependence, the correlation and 3, respectively, of 746 set and 964 sec. The magnetic dipole-dipole relaxation mechanism half-width times at 112.6oK being approximately 1.5 times those for 142.6oK and double those for yields times two orders of magnitude longer. 257.3OK at the same density. This effect is at These times show the correct temperature deleast partly caused by the fact that the angular pendence: for spin-rotation coupling they dedependence of the N2-Ar interactions is itself crease as the temperature is increased, while dependent on temperature in the sense that diffor the other couplings the relaxation times inferent regions of the potential function become crease. The force on the bond f(t) can be separated important at different temperatures. . Naturally then the exact temperature dependence of these into two parts, the force due to solvent interactions, and the centrifugal force. The correlation correlation times is expected to be a sensitive functions v(tjf(O)> for the two contributions do function of the form chosen for the N2-Ar potennot tend to zero in the limit t - 00. The nonzero tial, with the short-ranged, more strongly average value of the force on the bond due to solangular-dependent repulsive part dominating the vent interactions is analogous to the environmeneffect., Conversely, an experimental study of the tal forces that give rise to frequency shifts and correlation functions {U(O) - u( 0) and vibrational exciton splittings in the crystalline W2[U(O) u(t)]>- could provide insight into the state and of course are not expected to average nature of the actual potential. to zero in the fluid phase. Neither does the cenThese same correlation functions can be trifugal force along the bond due to the rotational Fourier transformed to_yield infrared and Raman motions of the diatomic molecule average to zero rotational and rovfbrational band and line shapes. at finite temperatures. These two types of bond Since. the. calcuktions are classical, the correforce contributions give o@posite cantributions to I lation-functions-d(t) exhibit a time-reversal pro- .. he total force aldng:the bond,- the averaged ten: pe&y-c(f) 5 c(-t), givingrise t’o bands a&lines trifug4 contribution tending to.spread the diatom.that are-symmetrical. :FourJer transforms of ic aptit and the averaged. solvent j&era&ions, -. ‘k(O) ;.rr(&.and (3[ti(O) i-h(t)]) for t&e three -. tending to compress. it; While tim+ver&?d hikhee deli&y systems Fe- singie-peeed an& / I : do.-&& &5ntri~ke .t&elax&i6n phenomena, ._I : -,f&es -,symmet&: _For.s&em 4i‘the trayfoFm_of key _d$‘@e~$q63,0s k__&i&‘df :the:tot&.?br&cnal I (u(O)-{ :e(tj} -is a dquble:pe&ed .batid~f-Except for . ._:banii through,tlieir dqendence on~diatqmic inter- : ” the band shape ,frdrh sy&erq, 4,‘ ‘there exists l$tt$+ :; ._ . . : _; ,: : .:. .,:: - :, ::, :.:- __ 1. .I, ,-.-.:. .. l
Vo!ume 8, number 1 separation. shifts was
nuclear sulting work.
CHEMICAL PHYSICS LETTERS
No calcuIation of these reattempted in this preliminary
REFERENCES
[l] J. R. Sweet and W. A. Steele, J. Chem. Phys. 47 (1967) 3022, 3029.
1 January 1971
: [Z] B. J. Berne and G. D. Hays. Advan. Chem. Phys:17
(1970) 63. p] P. L. Fehder. J. Chem. Phys,SO (L969} 2617. [4] A. Rahman, Phys. Rev. 136 (196-Q A405. [5] R. G. Gordon. in: Advances in magnetic resonance 1 Vol. 3, ed. J. S. Waugh (Academic Press. Kew York, 1968) p. 1. [G] R. G. Gordon, J. Chem. Phys. 44 (19GG) 1830. p] R. Zwanzig. Ann. Rev. Phys.Chem. 16 (1965)67. [BJ13.J. Berne, R. G. Gordon and V. F. Sears. J. Chem. Phys. 49 (1968) 475. [9] R. Zwanzig and Ailwadi, Phys.Rev. I82 (1969) 280.
.. --
.,
_
-_ ._ --
-:
_’
.
’
,
-. _‘
,-,
_’ ._-
:
__--.
:
,-
:
.‘ ,.-
_,.
’ ._
.-
.,
._
;.:
_
81
COMPUTER
CHEiCIICALPHYSICSLETTERS
SIMULATION
CORRELATION
OF
THE
FUNCTIONS
SYSTEM AND
1 January 19il
N2 IN
FLUID
RELAXATION
ARGON-
TIMES*
E. F. O’BRIEN and G. WILSE ROBINSON Arthur Amos Noyes Laboratory of Chemical Physics **, California Institde of Technology. Pasadena, Catifotnia 91109, USA
Received 20 October 1970 The correlation functions of the dipole moment, P [u(O) u(t)]. angular and linear velocity, and bond forces have been calculated from computer aim f ated data for four different density-temperature states of N2 in fluid argon, From these functions infrared and Raman line shapes, NMR relaxation times, and rotational and classical vibrational relaxation times have been computed. l
In an effort to understand more about the microscopic processes responsible for relaxation and line broadening in fluids, we have carried out a computer simulation of the three-dimensional system N2 in argon using classical equations of motion. The wholly classical computations are expected to be applicable to NMR and pure rotational spectra as well as to rotational motions within a given vibrational state but not to “quantum mechanical processes”, such as vibrational relaxation, at the temperatures considered. Ar-Ar interactions were taken to be of the Lennard-Jones (6-12) type, while the N2-Ar potential used was constructed by assuming a rigid N2 bond and a Lennard-Jones 3-center atom-atom interaction model between the two centers on the diatomic and each Ar atom$. Combining laws for the force constants E and u were used $.$, and all interactions at distances greater than 2.5 o(Ar) were set equal to zero. This potential is qualitatively different from the Stockmeyer potentials used by Berne and Hays [Z]. The algorithm is similar to that used previously in this laborzitory for two-dimensional computations of pure liquid argon[3] $$$. The systems calculated consist of 63 or 156 Ar atoms (depending on density) and one N2 molecule, periodic boundary conditions being imposed. l
This work was supported in part by a grant;(No.GP-
12381) from the National Science *f Contribution No.4151. -
Foundation.
$ This potential is similar to the one formulated by
Sweet and SteeIe.[7] for two interacting N2_ $$ The combining laws Used were EN2_A= (EN,E&~!?
’ @N -A=& Na+oA)F&der% a! goritlnn [3] was based on one described earlier by Rahman 141. .-
$$$
While dipole transitions between rotationaL or rovibrational levels in N2 are forbidden, the computational model used here should closely approach the physical situation exemplified by CO in fluid solution. The small dipole moment in CO is not expected to affect substantially the correlation functions t . The following correlation functions and associated physical quantities were considered. 1) (u(O) u(t)) (infrared rotational and rovibrational line shapes [5]); 2) (P2[u(O) u(t)]) (Raman rotational and rovibrational line shapes and NMR quadrupole and magnetic dipole-dipole relaxation times [5]); 3) {o(O) - o(C)) (NMR spin-rotation relaxation times [6]); 4) {v(O) - v(t)) (diffusion constant {7]]; 5) cf(O)f(t)> (classical vibrational relaxation [8]). In the above, u is the unit vector along the N2 bond, P2 is the second Legendre polynomial, w is the angular frequency of N2, v labels velocity vectors of the particles, and $is the component of force along the N2 bond. Four systems have thus far beeu studied, three with a density of 1.227 g/cm3 at the temperatures 25’7.3OK (system I), 142.6oK (system 2) and 112.6OK (system 31, and the four& having a temperature of 289.3OK and a density of 0.506 g/cm3 (system 4). AH the above correlation functions were calculated for each of these systems. Since statistics were poor in the molecular dynamics data for Iong times, the tails of the correktionfunCtiOnSare very pgorly represented. l
l
t The addition of the dipole term in the potential fun+ tion changes the.potential by less than 0.1% of the. L-J E thrOUghOUt tie range Of the potential considered , . here. :,
.-
_. .’
;
79
Volume 8, number 1
CHEMICAL PHYSICS LETTERS T&e
Ccrr&ation function a) (Diatomic) (u(O) - u(t)) &IWO) - W)l) I WO). W,>
(fWf(l!)
(V,(O) . V,(0)
Correlation 1 1.71*0.09 0.91* 0.07 1.18* 0.09 0.60* 0.07 1.68*0.09
1 January 1971.
1 times (lo-l3 set) at half-width System 2. 3
2.58a0.14 1.32*0.10 1.12*0.09 0.6la0.07 1.80*0.10
3.60*0.20 1.64*0.12 1.12*0.09 0.57*0.07 1.95~0.11
a) The error limits were determined by the method of Zwanzig and Ailwadi 191.
4 1.30*0.07 O-76*0.05 3.24*0.10 6.70* 0.30 6.45+0.13
. Thus the correlation times are reported at halfcorrespondence between the calculated band width, i.e., the width at half the maximum height. shapes and the experimental ones used by Cordon. See table 1. 14N2 and 15N2 NMR relaxation times were calThe (u(0) - u(t)> and &[u(O) - u(t)]) correculated using the formalisms that relate relaxlation functions approach zero in a nearly expoation rates to the appropriate correlation funcnential manner for systems 1, 2, 3. This result tions [ (fi[U(O)*U(t)]) or (~(0) - o(t))] for the contrasts that obtained from experimental data following relaxation mechanisms: quadrupolar, by Cordon where a prominent shoulder occurs in spin-rotational and magnetic dipole-dipole. the tail of the correlation function. System 4 These calculations were performed for systems yields a (u(0) - u(t)> correlation function that 2 and 3. They indicate, as expected, that the dips below zero. This functional form is similar quadrupolar relaxation mechanism is dominant to the one calculated by Cordon at about this for 14N2 givin times of the order of miIliseconds.’ For 1gN2, the spin-rotation mechanism density. The above two correlation functions also show dominates giving relaxation times for systems 2 a strong temperature dependence, the correlation and 3, respectively, of 746 set and 964 sec. The magnetic dipole-dipole relaxation mechanism half-width times at 112.6oK being approximately 1.5 times those for 142.6oK and double those for yields times two orders of magnitude longer. 257.3OK at the same density. This effect is at These times show the correct temperature deleast partly caused by the fact that the angular pendence: for spin-rotation coupling they dedependence of the N2-Ar interactions is itself crease as the temperature is increased, while dependent on temperature in the sense that diffor the other couplings the relaxation times inferent regions of the potential function become crease. The force on the bond f(t) can be separated important at different temperatures. . Naturally then the exact temperature dependence of these into two parts, the force due to solvent interactions, and the centrifugal force. The correlation correlation times is expected to be a sensitive functions v(tjf(O)> for the two contributions do function of the form chosen for the N2-Ar potennot tend to zero in the limit t - 00. The nonzero tial, with the short-ranged, more strongly average value of the force on the bond due to solangular-dependent repulsive part dominating the vent interactions is analogous to the environmeneffect., Conversely, an experimental study of the tal forces that give rise to frequency shifts and correlation functions {U(O) - u( 0) and vibrational exciton splittings in the crystalline W2[U(O) u(t)]>- could provide insight into the state and of course are not expected to average nature of the actual potential. to zero in the fluid phase. Neither does the cenThese same correlation functions can be trifugal force along the bond due to the rotational Fourier transformed to_yield infrared and Raman motions of the diatomic molecule average to zero rotational and rovfbrational band and line shapes. at finite temperatures. These two types of bond Since. the. calcuktions are classical, the correforce contributions give o@posite cantributions to I lation-functions-d(t) exhibit a time-reversal pro- .. he total force aldng:the bond,- the averaged ten: pe&y-c(f) 5 c(-t), givingrise t’o bands a&lines trifug4 contribution tending to.spread the diatom.that are-symmetrical. :FourJer transforms of ic aptit and the averaged. solvent j&era&ions, -. ‘k(O) ;.rr(&.and (3[ti(O) i-h(t)]) for t&e three -. tending to compress. it; While tim+ver&?d hikhee deli&y systems Fe- singie-peeed an& / I : do.-&& &5ntri~ke .t&elax&i6n phenomena, ._I : -,f&es -,symmet&: _For.s&em 4i‘the trayfoFm_of key _d$‘@e~$q63,0s k__&i&‘df :the:tot&.?br&cnal I (u(O)-{ :e(tj} -is a dquble:pe&ed .batid~f-Except for . ._:banii through,tlieir dqendence on~diatqmic inter- : ” the band shape ,frdrh sy&erq, 4,‘ ‘there exists l$tt$+ :; ._ . . : _; ,: : .:. .,:: - :, ::, :.:- __ 1. .I, ,-.-.:. .. l
Vo!ume 8, number 1 separation. shifts was
nuclear sulting work.
CHEMICAL PHYSICS LETTERS
No calcuIation of these reattempted in this preliminary
REFERENCES
[l] J. R. Sweet and W. A. Steele, J. Chem. Phys. 47 (1967) 3022, 3029.
1 January 1971
: [Z] B. J. Berne and G. D. Hays. Advan. Chem. Phys:17
(1970) 63. p] P. L. Fehder. J. Chem. Phys,SO (L969} 2617. [4] A. Rahman, Phys. Rev. 136 (196-Q A405. [5] R. G. Gordon. in: Advances in magnetic resonance 1 Vol. 3, ed. J. S. Waugh (Academic Press. Kew York, 1968) p. 1. [G] R. G. Gordon, J. Chem. Phys. 44 (19GG) 1830. p] R. Zwanzig. Ann. Rev. Phys.Chem. 16 (1965)67. [BJ13.J. Berne, R. G. Gordon and V. F. Sears. J. Chem. Phys. 49 (1968) 475. [9] R. Zwanzig and Ailwadi, Phys.Rev. I82 (1969) 280.
.. --
.,
_
-_ ._ --
-:
_’
.
’
,
-. _‘
,-,
_’ ._-
:
__--.
:
,-
:
.‘ ,.-
_,.
’ ._
.-
.,
._
;.:
_
81