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The planar itinerant oscillator model. A discussion of its use in reproducing experimental data from three separate sources
Vohunc 50, number 1
15 August1977
CHEMICAL PHYSICS Ll3TERS
THE PLANAR iTlNERANT
OSClLLATOR MODEL. A DISCUSSION OF ITS USE IN
REPRODUCING EXPERIMENTAL
DATA FROM THREE SEPARATE SOURCES
G.J. EVANS, M.W. EVANS Chctttt~fry
Dcparttttcttt.
Uttivcrriiy
Colkge
of IVales. Aherystwyth,
Dyf~vY S Y23
ZNE,
UK
J.11. CALDERWOOD Dqmrttttetzt
of I:‘lcctricol
Ettgittcerittg.
Uttivcrsity
of Salford,
M_54 WT, UK
Mattclte~ter
W.T. COFFEY Sciiool
of Ehgitteerittg,
Ttlttity
Cltll~~ge, Lhfthn
4. Em
and
WEGDAM
G.H.
Robot-atory for Physiral
Chnistry.
University
of Atnstcrdutn,
Amstcrdatn.
The iVcthcrlortd~
Kece~vcd 17 hlny 1977
Autocorrclation functions c.~lcul~tcd I’rom the itirwrant osalldlor rnodcl arc c0mpJrcd wltil eupcrimental data from far infrared absorption mcxsurcmcnts, dcpolar~scd I?aylcigb scattermg mcrlsurcmcnt\ and molecuLlr dynamic5 comput;ltlons. It 15 conhded that the model reprc5cnts the reoricntatiorull motion of a molecule in the flutd (at short times) s:lti5factorily The predicted angular vcloclty xf is not :I pure cxponenthl but r:lthcr nn oscillatory function of tims. ‘I%Is seem5 to be the cdSc ::encr.llly
1. Introduction
and theory
compare autocorrelation functions (a&s) calculated from the itinerant oscilmotion developed lcccntly by Coffey et ad. [l-3] with experimental results from three separate fields. Thcsc arc (a) depolarised light scattering 141 from liquids subject to kilobars of external pressure, (b) molecular dynamics results using the atom- atompotential of Tildesley and Streett [SJ, (c) far infrared absorption of 2 highly dipolnr species in dilute solution. Several intcrrelatcd autocorrelation functions are obtained by using these three techniques simultaneously, thus providing a stern test for the model.For details of this planar rncchanism see refs. [ l- -31. IIerc WCstate merely that the librations of a molecule within a cage of its neighbours are represented by the angular motion of a disk harmonically bound within an un1~u1us or ring which is ltsclf undergoing rotational brownian motion. The equations of motion governing this system lead to an angular velocity acf, C,(t), of the form In
lator
this
for snidll rnoledcs.
letter
we shall
a~tcmpt
to
model of rcorientntional molecular
C,(r)
= i(r)~(o))~/(d2(o):~~
(t > 0)
= (1 + 1’): ’ { [Cos wit + (ol + l%2)wi1
sin olt]
c-urf
f re-a2f},
(1)
where 0 is a coordinate which specifics the angular position of a dipole I, lying along the axis of the disk at any time t > 0. For convenience, O(O) is taken to be coincident with the direction of a steady electric field E, and it is rssumed that this field is suddenly removed at the time E = 0. Further, hI, hz and h, are the roots of the equation: 142
Volume
50, number
CHEMICAL WYSICS LETIERS
1
,3 -I- ps2-I- (0; + qj)s + i.e. A, =ol
+iw,,h2=u1
r = -2&
15 August 1977
pw; =0,
--iwl,
(2)
A3 = 02~ The factor l? is given by:
+ &o2(3o~
- “22 - 0:).
(3)
If y?(r) is the angle which a point on the rim of the annulus makes with the reference direction at time f and if II is the moment of inertia of the annulus, then i#$ is the frictional couple acting on the annulus due to the surroundings. o. is the angular frequency of oscillation of the disk when the annulus is held in a fixed position and $ = (12/I,)o& w here f2 is the moment of inertia of the disk. Uy means of a theorem about characteristic functions of gaussian distributions [l-3], it may be shown from cq. (1) that the orientational acfs appropriate to dielectric relaxation and dcpolarisecl Rayleigh scattering of light are given by: p1 (t) = (cos 0(r) cos 0(0)>/ = exp [-y(f)], p2(f) = &OS 20(r) cos 20(0)>/
,
(4)
where v(r) is defined as follows. -
3u: +2u1u2r-oJ~ +-.
_--
__
1
_L---____-1 cosw~T+c: - 30,o;
II
(UT + w:y
.---
w#of
+ u* r(u?
I--0;)
sin w,t
+2a&kdf)
r -qt
e--c’lr -ky
c
.
(5)
02
One may readily calculate
acf, and the quantities
from cqs. (1) and (3) the quantity Cdit)~(O)>/<~jO)ci~O)>, which we shall call the torque -ijl(t) and -j2(t) needed [6] to fit the raw experimental data.
2. Experimental The original data for this latter consists of the far infrared spectrum of the highly dipolar species CfI$l, in dilute Ccl4 solution. This spectrum was chosen to eliminate: (i) dipole--dipole coupling [7] ; (ii) collision induced effects 181. The data were obtained wit!1 a Mark 3 Grubb-Parsons/N.P.L. Fourier transform intcrferometcr using phase modulation of the radiation arriving at a solid-state Golay iK50 detector. In comparison with the pure liquid CH2C12 the solution absorption band peaks 30 cm--l lower in frequency. Solute and solvent were Analat spectroscopic grade-
3. Results and discussion Least mean squares fits (N-A-G. system E04FAA) to some depolariscd light scattering data of Iitovitz et al. [4], to our own molecular dynamics data and far infrared spectrum are shown in figs. 1-3. The optimum values of o& !$ and /3 (regarded as variables), are tabulated in table 1. 3.1. Ray&h
scattering
The function $2(t) was used to fit the “angular velocity acf’ of Litovitz et al. [4] _ It is seen that 06 (which is proportional [Q] to the mean square torque,
Volume SO. number
-OS-
C’!CAIICAL
1
___--__
---
.-.-
----_
PHYSICS
1
LETTERS
_05L-__--__.
15 August
--
-
___-_
_-
1977
______I
Fig. 2. (1) Angular vcloaty acf comuutcd with the iitom -atom potential for ;1 rcduccd mtcratomic separation, d* = 0.5, at o reduced number density of 0.643 and reduced temperaturc of 2.3. -(2) Torque acf likcwrsc computed. - -- (1) Hcsttlt oft8(t)ci(o))/t~(O)ci(o), to (1). --(2) Prcdictcd itinera7t os~lll,ltor torque acf from the tittmg with the paramctcrs tabulated 111the tcut.
-0s
L -__
__-
-__
-___ -_-----I - ___-.-
--_-__-
--
-.--
--
---
-_
(1) I~xpcrllnental -d*pz(r)/dtz for 3-D rcr1g. 1. (a)-onentations. (2) Eapcrimcntal p*(t); - - - (1) ([d co\ 20(f)ldf] (cl co\ 20(O)ldr]) fittcd to (11, (2) (co% 20(t) cob 20(O)) zlnd (3) (o’(f)J(O)) c~lculatcd from the fittmg for liquid methyl mdidc at 1 bar, 296 K. Dat,i of htovltz ct al. [4]. (b) As for (d), 1 kbar. (c) As for (a), 2.5 kbar.
IQ 3. (1) t.uncd [ 6 J absorption dt]) fitted (8(O)ri(ON,
Expcrimcntdl (i(t) . I;(O))/ti(O) . I;(O)) obfrom Fourier transrorming the far infrared power cocfficicnt. ---(I) ([d cos o(t)/dt) [d COPO(O)/ to (I), (2) tcos 00) cos s(o),, (3) &)0’(o),/ (4) ~B(t)Zi(o),/cii(o)ii(o),.
is increased a thousandfold by application of extefnal hydraulic pressure. The values of p and St: vary much more rapidly, and the latter (which is related to (7$/C$ -- <7$) increases a thousandfold in a few instances as the external pressure is increased over the same range. It seems that the interpretation of S26 as (12/II)wg is not meaningful experirncntaily, but in general, a very high value of S2$08 means that (Ti) changes very rapidly with time at certain points in its domain of definition. The p2(t) function predicted by this fitting decays consistently more slowly than the experimental response function (corrected by Litovitz et al. for collision induced polarisabilities). This is due in some measure to our USCof a planar model. 144
Volume
50, number
CHFMICAL. PHYSICS LiTI,-!.X
1
15 August 1977
Table f Vaiucs of ~‘0, ~2%dnd (3 for best tit _ ___-__---Se
I’xperlrnentJI tunctton
LlCpld
____-_
-__-
meti1yl
_
q---p
* Reduced
Pretsun: (bar)
-_
7-m-_-_--
--
‘kmpcr~~lurc -
-__.-...-w-s_
43.4
296
97.7
296
115.2
molecular dynamics, torque dcf
p* = 0.643
T* = 2.3
80.9 *
far infrdrcd
1.97 x 102’ n~olecules cm 3 --
298
16.4
units corresponding
_--
---
to 200 @T/l)‘/’
--
--_
-
P-m-(f/k7.)“*@
kT
296
___
_ ,azg
g
WI
1000 2500
sc.ittcrin::
CfI*C12/CC1~
- --
1
Ksylc1gl,h
iodide
-
. -,_
_...--
102.9
16.1
30187
3254
3645 1
4250
252.8 *
25.2 *
80.7
6.2
-_.---
_-
time steps per time unfit.
3.2. iVolecrtlar dytwnics
In this case angular velocity and torque acfs were computed for a d~irnbbel~ potential using an atom-atom algorithm (described fully by Evans et al. [lOI and by Tildesiey and Streett [S] ) under the conditions dcscribcd
in the caption to fig. 2. For an interatomic separatiorl of d * = 0.5 in reduced units [5] the computed angular ncf is fitted accurately by the analytical model and the torque acf realistically predicted. Thus work wilt be expanded in a fort~~~otning publication in order to evaluate the undcr~ying dependence of ($> on interatomic separation d*. This result indicates that a planar molecule is capable of reproducing the molecular dynamics results, which arc derived from 3-D space reorientations. It might be interesting to use an algorithm based on an enscmbie of rough disks, but previous computations have shown that in this case the angular velocity acf is always exponential. This indicates that an atom-atom potential is the more realistic, since an ~~ci~~at~~~angular velocity acf is needed to fit both Rayleigh scattering and far infrared data. velocity
3.3. Far ittfrared da Ta fn this case, data for a ili~~t~y dipolar species, CH+$I,, in dilute solution function: Ft =
l
;(o)YG(o)
were used to evaluate the cxpcritnenta~
- G(O)),
where fc is the unit dipole vector. In a plane, this wouid be --iI (t). This is true regardless of mol&~~lar symmetry. The match is less satisfactory than for the results of the other two techniques considered hcrc but stili within about 210% [6] over most of the time interval up to 4 reduced units. At short times the fit is far closer than in the case of the Wyllie/Larkirl version [S,l 11 of the bamc model. From fig. 3 the lich variety of dynamical infortnation obtainable is clearly apparent. In particular the angular velocity acf is oscillatory. This seems to be tl-Jc generally [l I], since F1 cxpe~inen~~ly becomes negative at ix~terlnediate times. 4. Cunclusions (i) The 2-D itiner~t oscillator modeli reproduces closely the molccuiar dynaluics of molecules in fluids as manifested by the three different probes used here. (ii) The predicted angular velocity acf is not a pure exponential but rather an oscillatory function of time. This seems to be the cast gcncrally for small molcculcs. 145
V01u111c
50,
llumbcr
1
CIIEMICAL PIIYSICS LETTERS
15 August 1977
Acknowledgement The Rarnsay Memorial Trust is thanked for a fellowship to M.W.E., the S.R.C. for an equipment studentship to G.J.B., and the University of Wales for a sabbatical year to G.W.
grant
and a
References [l] W-T. Coffey. T. Ambrose and J.11. Calderwood, J. Phys. D9 (1976) L115. [2] J.11. Calderwood and W.T. Coffey, Proc. Roy. Sec. A (1977), to bc published. [ 31 W.T. Coffey, J.G. F%XWX ZNI~ .l.II. Calderwood, J. I’hys. D, to bc pubhshc& [4] J.F. Dill. T.A. Litovitz and J.A. Bucaro, J. Chern. Phys. 62 (1975) 3839. [S] W-H. Strcctt and D.J. TildesIcy. Proc. Roy. Sot. 348A (1976) 485. [6] C. Brot, in* Dielectric and related molecular proccsccs, Vol. 2 (1 he Chcrnical Society. London, 1975) p. 1. [ 7] B.K.P. Scalfe, Complex penuittivity (tnglish Univ. Press, London, 1971). [S] M. Evans, AC~V.UI. Mol. Rcl. Int. Proc. 10 (1977) 203. [9] M. Evans. Chern. Phys. Letters 39 (1976) 601. [IO] G-J. Evans, G.H. Wcgdarn and hf. Evms, Mol. Phy5. (1977). to be publ~shcd: Advan. hIo1. Rei. Int. Proc. (1977). to be pubkhed. [I I] hl. Evans. J. Chern. Sot. 1.araday II 71 (1975) 2051.
146
15 August1977
CHEMICAL PHYSICS Ll3TERS
THE PLANAR iTlNERANT
OSClLLATOR MODEL. A DISCUSSION OF ITS USE IN
REPRODUCING EXPERIMENTAL
DATA FROM THREE SEPARATE SOURCES
G.J. EVANS, M.W. EVANS Chctttt~fry
Dcparttttcttt.
Uttivcrriiy
Colkge
of IVales. Aherystwyth,
Dyf~vY S Y23
ZNE,
UK
J.11. CALDERWOOD Dqmrttttetzt
of I:‘lcctricol
Ettgittcerittg.
Uttivcrsity
of Salford,
M_54 WT, UK
Mattclte~ter
W.T. COFFEY Sciiool
of Ehgitteerittg,
Ttlttity
Cltll~~ge, Lhfthn
4. Em
and
WEGDAM
G.H.
Robot-atory for Physiral
Chnistry.
University
of Atnstcrdutn,
Amstcrdatn.
The iVcthcrlortd~
Kece~vcd 17 hlny 1977
Autocorrclation functions c.~lcul~tcd I’rom the itirwrant osalldlor rnodcl arc c0mpJrcd wltil eupcrimental data from far infrared absorption mcxsurcmcnts, dcpolar~scd I?aylcigb scattermg mcrlsurcmcnt\ and molecuLlr dynamic5 comput;ltlons. It 15 conhded that the model reprc5cnts the reoricntatiorull motion of a molecule in the flutd (at short times) s:lti5factorily The predicted angular vcloclty xf is not :I pure cxponenthl but r:lthcr nn oscillatory function of tims. ‘I%Is seem5 to be the cdSc ::encr.llly
1. Introduction
and theory
compare autocorrelation functions (a&s) calculated from the itinerant oscilmotion developed lcccntly by Coffey et ad. [l-3] with experimental results from three separate fields. Thcsc arc (a) depolarised light scattering 141 from liquids subject to kilobars of external pressure, (b) molecular dynamics results using the atom- atompotential of Tildesley and Streett [SJ, (c) far infrared absorption of 2 highly dipolnr species in dilute solution. Several intcrrelatcd autocorrelation functions are obtained by using these three techniques simultaneously, thus providing a stern test for the model.For details of this planar rncchanism see refs. [ l- -31. IIerc WCstate merely that the librations of a molecule within a cage of its neighbours are represented by the angular motion of a disk harmonically bound within an un1~u1us or ring which is ltsclf undergoing rotational brownian motion. The equations of motion governing this system lead to an angular velocity acf, C,(t), of the form In
lator
this
for snidll rnoledcs.
letter
we shall
a~tcmpt
to
model of rcorientntional molecular
C,(r)
= i(r)~(o))~/(d2(o):~~
(t > 0)
= (1 + 1’): ’ { [Cos wit + (ol + l%2)wi1
sin olt]
c-urf
f re-a2f},
(1)
where 0 is a coordinate which specifics the angular position of a dipole I, lying along the axis of the disk at any time t > 0. For convenience, O(O) is taken to be coincident with the direction of a steady electric field E, and it is rssumed that this field is suddenly removed at the time E = 0. Further, hI, hz and h, are the roots of the equation: 142
Volume
50, number
CHEMICAL WYSICS LETIERS
1
,3 -I- ps2-I- (0; + qj)s + i.e. A, =ol
+iw,,h2=u1
r = -2&
15 August 1977
pw; =0,
--iwl,
(2)
A3 = 02~ The factor l? is given by:
+ &o2(3o~
- “22 - 0:).
(3)
If y?(r) is the angle which a point on the rim of the annulus makes with the reference direction at time f and if II is the moment of inertia of the annulus, then i#$ is the frictional couple acting on the annulus due to the surroundings. o. is the angular frequency of oscillation of the disk when the annulus is held in a fixed position and $ = (12/I,)o& w here f2 is the moment of inertia of the disk. Uy means of a theorem about characteristic functions of gaussian distributions [l-3], it may be shown from cq. (1) that the orientational acfs appropriate to dielectric relaxation and dcpolarisecl Rayleigh scattering of light are given by: p1 (t) = (cos 0(r) cos 0(0)>/
,
(4)
where v(r) is defined as follows. -
3u: +2u1u2r-oJ~ +-.
_--
__
1
_L---____-1 cosw~T+c: - 30,o;
II
(UT + w:y
.---
w#of
+ u* r(u?
I--0;)
sin w,t
+2a&kdf)
r -qt
e--c’lr -ky
c
.
(5)
02
One may readily calculate
acf, and the quantities
from cqs. (1) and (3) the quantity Cdit)~(O)>/<~jO)ci~O)>, which we shall call the torque -ijl(t) and -j2(t) needed [6] to fit the raw experimental data.
2. Experimental The original data for this latter consists of the far infrared spectrum of the highly dipolar species CfI$l, in dilute Ccl4 solution. This spectrum was chosen to eliminate: (i) dipole--dipole coupling [7] ; (ii) collision induced effects 181. The data were obtained wit!1 a Mark 3 Grubb-Parsons/N.P.L. Fourier transform intcrferometcr using phase modulation of the radiation arriving at a solid-state Golay iK50 detector. In comparison with the pure liquid CH2C12 the solution absorption band peaks 30 cm--l lower in frequency. Solute and solvent were Analat spectroscopic grade-
3. Results and discussion Least mean squares fits (N-A-G. system E04FAA) to some depolariscd light scattering data of Iitovitz et al. [4], to our own molecular dynamics data and far infrared spectrum are shown in figs. 1-3. The optimum values of o& !$ and /3 (regarded as variables), are tabulated in table 1. 3.1. Ray&h
scattering
The function $2(t) was used to fit the “angular velocity acf’ of Litovitz et al. [4] _ It is seen that 06 (which is proportional [Q] to the mean square torque,
Volume SO. number
-OS-
C’!CAIICAL
1
___--__
---
.-.-
----_
PHYSICS
1
LETTERS
_05L-__--__.
15 August
--
-
___-_
_-
1977
______I
Fig. 2. (1) Angular vcloaty acf comuutcd with the iitom -atom potential for ;1 rcduccd mtcratomic separation, d* = 0.5, at o reduced number density of 0.643 and reduced temperaturc of 2.3. -(2) Torque acf likcwrsc computed. - -- (1) Hcsttlt oft8(t)ci(o))/t~(O)ci(o), to (1). --(2) Prcdictcd itinera7t os~lll,ltor torque acf from the tittmg with the paramctcrs tabulated 111the tcut.
-0s
L -__
__-
-__
-___ -_-----I - ___-.-
--_-__-
--
-.--
--
---
-_
(1) I~xpcrllnental -d*pz(r)/dtz for 3-D rcr1g. 1. (a)-onentations. (2) Eapcrimcntal p*(t); - - - (1) ([d co\ 20(f)ldf] (cl co\ 20(O)ldr]) fittcd to (11, (2) (co% 20(t) cob 20(O)) zlnd (3) (o’(f)J(O)) c~lculatcd from the fittmg for liquid methyl mdidc at 1 bar, 296 K. Dat,i of htovltz ct al. [4]. (b) As for (d), 1 kbar. (c) As for (a), 2.5 kbar.
IQ 3. (1) t.uncd [ 6 J absorption dt]) fitted (8(O)ri(ON,
Expcrimcntdl (i(t) . I;(O))/ti(O) . I;(O)) obfrom Fourier transrorming the far infrared power cocfficicnt. ---(I) ([d cos o(t)/dt) [d COPO(O)/ to (I), (2) tcos 00) cos s(o),, (3) &)0’(o),/ (4) ~B(t)Zi(o),/cii(o)ii(o),.
is increased a thousandfold by application of extefnal hydraulic pressure. The values of p and St: vary much more rapidly, and the latter (which is related to (7$/C$ -- <7$) increases a thousandfold in a few instances as the external pressure is increased over the same range. It seems that the interpretation of S26 as (12/II)wg is not meaningful experirncntaily, but in general, a very high value of S2$08 means that (Ti) changes very rapidly with time at certain points in its domain of definition. The p2(t) function predicted by this fitting decays consistently more slowly than the experimental response function (corrected by Litovitz et al. for collision induced polarisabilities). This is due in some measure to our USCof a planar model. 144
Volume
50, number
CHFMICAL. PHYSICS LiTI,-!.X
1
15 August 1977
Table f Vaiucs of ~‘0, ~2%dnd (3 for best tit _ ___-__---Se
I’xperlrnentJI tunctton
LlCpld
____-_
-__-
meti1yl
_
q---p
* Reduced
Pretsun: (bar)
-_
7-m-_-_--
--
‘kmpcr~~lurc -
-__.-...-w-s_
43.4
296
97.7
296
115.2
molecular dynamics, torque dcf
p* = 0.643
T* = 2.3
80.9 *
far infrdrcd
1.97 x 102’ n~olecules cm 3 --
298
16.4
units corresponding
_--
---
to 200 @T/l)‘/’
--
--_
-
P-m-(f/k7.)“*@
kT
296
___
_ ,azg
g
WI
1000 2500
sc.ittcrin::
CfI*C12/CC1~
- --
1
Ksylc1gl,h
iodide
-
. -,_
_...--
102.9
16.1
30187
3254
3645 1
4250
252.8 *
25.2 *
80.7
6.2
-_.---
_-
time steps per time unfit.
3.2. iVolecrtlar dytwnics
In this case angular velocity and torque acfs were computed for a d~irnbbel~ potential using an atom-atom algorithm (described fully by Evans et al. [lOI and by Tildesiey and Streett [S] ) under the conditions dcscribcd
in the caption to fig. 2. For an interatomic separatiorl of d * = 0.5 in reduced units [5] the computed angular ncf is fitted accurately by the analytical model and the torque acf realistically predicted. Thus work wilt be expanded in a fort~~~otning publication in order to evaluate the undcr~ying dependence of ($> on interatomic separation d*. This result indicates that a planar molecule is capable of reproducing the molecular dynamics results, which arc derived from 3-D space reorientations. It might be interesting to use an algorithm based on an enscmbie of rough disks, but previous computations have shown that in this case the angular velocity acf is always exponential. This indicates that an atom-atom potential is the more realistic, since an ~~ci~~at~~~angular velocity acf is needed to fit both Rayleigh scattering and far infrared data. velocity
3.3. Far ittfrared da Ta fn this case, data for a ili~~t~y dipolar species, CH+$I,, in dilute solution function: Ft =
l
;(o)YG(o)
were used to evaluate the cxpcritnenta~
- G(O)),
where fc is the unit dipole vector. In a plane, this wouid be --iI (t). This is true regardless of mol&~~lar symmetry. The match is less satisfactory than for the results of the other two techniques considered hcrc but stili within about 210% [6] over most of the time interval up to 4 reduced units. At short times the fit is far closer than in the case of the Wyllie/Larkirl version [S,l 11 of the bamc model. From fig. 3 the lich variety of dynamical infortnation obtainable is clearly apparent. In particular the angular velocity acf is oscillatory. This seems to be tl-Jc generally [l I], since F1 cxpe~inen~~ly becomes negative at ix~terlnediate times. 4. Cunclusions (i) The 2-D itiner~t oscillator modeli reproduces closely the molccuiar dynaluics of molecules in fluids as manifested by the three different probes used here. (ii) The predicted angular velocity acf is not a pure exponential but rather an oscillatory function of time. This seems to be the cast gcncrally for small molcculcs. 145
V01u111c
50,
llumbcr
1
CIIEMICAL PIIYSICS LETTERS
15 August 1977
Acknowledgement The Rarnsay Memorial Trust is thanked for a fellowship to M.W.E., the S.R.C. for an equipment studentship to G.J.B., and the University of Wales for a sabbatical year to G.W.
grant
and a
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