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Inertial effects in the dielectric relaxation of an assembly of molecules containing rotating polar groups
Volumc9Z,numbcr3
INERTIAL
PHYSICS
LETTERS
EFFECTS IN THE DIELECTRIC RELAXATION
CONTAINING W.T.
CHIIBIICAL
OF
31 Ocrobcr 1982
ANASSEMBLY
OF MOLECULES
ROTATING POLAR GROUPS
COFFEY
and C.
RYBARSCH
and W. SCHROER
Fmlrbercielr Clreme. Unrrersu~c Brmetr. D-ZOO Brcnwn, Ir’esr Gernmy Recewd
76 June 1982
It IS shos n how the problem oi includmg incrtul ciiws in rlw dwAcctrlc rrlx~uon rorarlng polx groups ma)’ be reduced to the solurron oi NO srmplcr problems
IUUII~
ireely romrmg drpolc. (u) die brownian mowmcnl
oi an xsrcmbly oi molcculcs con(I) the browman movement oi 3
oia drpolr m a cosme porcnrial
I. Introduction In the ongmal theory of dielectric
molecules due to Debye [I] and the molecules are electrxally inert. Thus this theory cannot explam the Poley or far-mfrared absorption [3]. An mteresting variant of this theory which can be treated analyrically IS the problem of the dielecrric relaxation of an assembly of molecules containing rotating polar groups. Each molecule does not interact electncally w1t.hits neighbours, only the groups mside an mdividual molecule do so. If this sirnphfication is not made, then we are immediately led into many-body considerations. The rotatinggroup problem was fust treated by Bud6 [4] who gave a full solution of it (subject to the constramt mentioned above) in the low-frequency limit where inertial effects are not operative. The problem was reconsidered by Coffey [S ,6], who showed how mertial effects could be included subject to the constraint that frictional effects rather than the strength of the mteraction potential dominated rhe motion. Thus his solutioncould not be expectedto providean eaplawtion of the high-frequency absorprlonbehmour. AU I[ protidcs is a fh=.r-order kerual correction UI the ~XU-UI~T of the Roard equation [2] for the Freely rotating dipole. Later a scheme was proposed [7] whereby an exact solution of the problem could be obtained numerically. In this letter we show how the set of equations obtained by Coffey may be factorised to obtain part of the solution in closed form. Some numerical results for the relaxation behaviour of the system are also given. inerna-connected
version
of it [2],
relaxation
of an assembly
one always assumes
that
of dlpolar
the
2. Dielectric relaxation of an assembly of molecules containing rotating polar groups In the theory studied here, an electric field E is assumed to be apphed in the same plane as the plane of rota-
tion of two dipoles pl and ~2 of a group. These dipoles are compelled to rotate about an axis through their common centre normal to the plane containing E. The equations of motion of the dipoles are
I~~(t)+T~,(f)+v’(Q1-91)+~,Esin[~,(f)l=gl(~), 0 009-2614/82/0000-0000/S
02.75 0 1982 North-Holland
(1)
CHEMICAL PHYSICS LETTERS
Volume 92. number 3 @2(t)
+ S&(r)
-
V’(dl
- #2) + I2_rE sin
22 October 1982
0)
I~&)1 =&T&j*
(PlO),f#4f) are the ad= pl and pT make with Eat any time. I is the moment of inerua of a dipole about a cento Its own plane. t h the friction coefficient arising from the brownian movement of the surroundare the white noise driving torques acting on p, and ~2. V($t - #Q) is the potential arising from the dipole-dzpole coupling. It is assumed that the field E which had been applied at t = -m is switched of at t = 0. This allows the system to relax from the ordered polarised state to the onginal chaotic state. Thus for f > 0 the equations of motion of the system are tral aknorm~
inp5;gl(f),g2(t)
,
(31
+ S&Q) - VI@, - 02) = Q(f) .
(4)
f&t@) +GtW G$)
-f W,
-&I
=gtM
These Langevin equations are not tractable as they stand except m the linear case (harmonic potential) where V = 5 ~(91 - #I)‘. Thus in general we must appeal ro the underlying probability denaty d~fu~oo equation - r.@ Framers equafion IS]. We note that eqs. (3) and (4) are marked& simplified if we introduce the sum and difference variables +@?I,
(5)
71=&75*-@2),
(6)
x=!lrpt
whence
(7)
~~~~~=~~t(r}~g~(r)l,
I;i+~~++Ir’(‘Ls)=12~~(t)-g2(1)1. Eqs.(7)and (8) are decoupled, x 1sessentially the angle turned
09
dour by the dipoles rotating as a unit. The vanabon ofx is like that of a dipole free to rotate without any electrical interaction [g]. “Ike variation of T)is iike that Of a rotator moving in a potential V(29) [9]. The foregoing suggests that we should also use the x, q variables in constructing Kramers equation rather than Qt and q$_ Kramers equation is_[5,6]
I%‘&,g, T?,;, t) is the distribution function in configuration-angular velocity space. Eq. (9) must be solved subject “Je initial condition, which in the linear approximation @E 6 KQ, is
i0
h’ = A exp {- [l@ f ri2) + V(?&+]/Wj [ 1 + (2&T/k?+)cos x cos q] .
00)
In wntmg down “q. (10) we have assumed that pl = pz_ This affords much mathematical simplification without any significant loss of generality.
3. Solution of eq- (9) We dewmnine Iv=A
W subject
exp [-Iti
to the titial
+ i)/kTJ
C
C
condruon
mn p.q
(10) as follows.
H,,n((21/kT)lE);,
We ass-e
(2f/kT)1/2jl)A$‘(f)
that for I > 0 W may be written
exp[i@x +p71)] ,
0 1)
where the HM n are the two~~~n~on~ Semite polynom~s defmed by Appeli and Kampt de F&&t [IO]. One then fmds thai the .4z;n(r) satisfy the set of differential-difference equations (we take ~(~11)= -vi, cos 2~)
2, Ocfober 1982
CHEMICAL PHYSICS LETTERS
Volume 92. number 3
+i(kT/X)t12[q(m
t l)A~r’*“(r)+P(rr
+ l)Azbn+‘(f)
= 0.
+qAEP1*n(f) +pA,7;:-‘(f)]
Our initial condition becomes rn terms of the A d$O)
= j: i” --II 0
[ 1 + (2/.IE/R7-)cos x cos ‘11exp[(Vu/kT) cos 7~71exp[i(qX tpr))]dx
=0 ,
d:;‘(O)
dn ,
m,n>O.
The mean dipole moment (m*e)may be determined from
Z/.X .ZOS x cos p k’di dx d; dq
1”_,J8”/:W1-“.. (We)=(/rcos$J,
t~cos~~~=~~!II(cos~cos~~= ,!!,J&f%og,_“,
wd;r
dx d; drl
Thrs reduces with the aid of the orthogonahty property of the circular fimctrons and the H,,, On-e)= 2~[.4!y,_~(r) + dt!t(r)
+4?.,,(r)
to
,
+.4~~(r)]/k4$$)
d%!(r) = constant _ The complex polarisabrhty IS
o(s) = 1 - sL?lm*e) wheregdenotes
,
the Laplace transform.
4. Factorization of the A Eq. (12) is extremely complicated, betng essentrally a double matrix differentral equation. A great srmplificatton can be made rf we note that A;;‘(t)
=~$‘(04#)
(17)
,
where A,“(t) satisfies the equation dA~(f)/dr+(E/l)nzrl~(r)+i(kT/~1~7[q(nrt
l)A~+‘(t)+qA~-l(f)]
=O
(18)
and ,4,“(r) satisfies dA#)/dt
t (T/r)nAi(r)
t i(kW)*72[p(tz
+ iVo(2Lk7)-1~2 [cl,“: i(r) -A,“;:(r)]
+ I)Ai”(r)
t&j-l(r)]
=0 .
(19)
Eq, (18) is exactly the same as that which one obtains for the brownian movement of a free rotator having the equation of motion Ijit
(20)
The solution of the set (18) has been described by Coffey rn ref. [S]. One fmds that for m = 0 say that ~~O)M~(O) = exp (-W/X*)
4’ Wr) r - 1+ exp HWI
fl II .
(21) 247
CHEMICAL
Volume 91. number 3
PHYSICS LETTERS
22 October
1982
b(t)
b(t) I
t
a -
.s
8 2
4
6
B
t
-.S
-.5
-I
-I
Fig 1. Tie behaviour of the dtpotr relaxation fonclion b(r) = m-e> for values oi the paramerers grven bq VofkT= 1, r/f
=ol,~~~/ZT=l.ume~~~sirmurulsoi(kT/~l”.Cu~e(~) ISA;(~) for n = 0.p = 1 (that IS the paxi~cb UI a cos 2n
poren-
lull tune fibi . lss!?lor.7 = I.m = 0 (1.Ces. (11) with q = i, in othci t&j the II& brownbn rotatorf. Curve (c) 1s the product
Of cuwzs
(a) and (b).
tbh3t IS tie
dtpole
of b(f) for negame ~31~s Rg. 3. As fg 1 but tone behmour l,~~f21~-t,time~~sisinuruts of Yo. Folk7 G-l,r@=O of (kT/#n. Cum (a) isApR(0.n = 0.p = I. carve(b) fiiIT(t), q = 1, m = 0. CUFW (c) ISthe product of c”~cs (a) and (b), I c the dipole rela~tion iunction).
ielrr~s-
tlon iunction.
b(t) 1
-I -I
a iunction oi the damping for values of the pmmeters given by Y&%7= 1, @-(J/x= 1. Cune (a) $fI = 0.1. cune cuxve fcf r/f = 5, time stale m UN~S of (kT/J)ln
248
@)
rrr =
1.
CHEMICAL PHYSICS LEnERS
Volume 92, number 3
22 October 1987
The set of equations (19) ISessentially that obtained by considering the browman movement of a rotator havmg
the equation of motion
Also, one set of differentral difference equations (19) is very similar to the problem of a partrcle moving in a cosine potential considered earlier by us [2]. The only difference is that one has a cos 2~ potential in this (=~se rather than a cos 5]. Thus the problem of the dielectric relaxation of polar groups inside a molecule may be reduced to the solution of two problems which have already been solved. These are respectively the free rotator and the pendulum. By way of illustration of our results we show the behaviour of (m-e) in figs. l-4 for typical values of the parameters I, 5, Vo_ The factorization property of the A simplifies the calculation of a(w) (essentially 9 {(m-e)) where FFdenotes Founer transformation), since we may now utihse the convolution theorem for Fourrer transforms. According to the theorem if
(23) then
F(w) = (11%)
7 G(h) H(w - 1) dk ,
e-0
-m
F, C, H are the Fourier transforms off,g, and 11,respectively. The advantage of domg this is that the Founer transform of eq. (21) can always be wntten down KI a simple form.
where
Acknowledgement We thank Professor BKP. Scaife for drawing our attention to this problem. One of us (WS) would hke to express hrs thanks to the Deutsche Forschungsgemeinschaft and the Fonds der Chermsche Industrie for financial support for this project. WIT thanks the University of Bremen for a Visiting Fellowship. Thus work forms part of the coordinated research programme of the European Molecular hquids Group (EMLG).
[I] P. Dcbye, Polar molecules (Cbemlcal Catalog Co, New York. 1929). [2] J.R. McConnell, Rotational brouruan monon and rhelectnc theory (Academic Press. New York, 1980). [3] J.P. Poley,J. Appl. Sci. Res. 83 (1955) 337. 141 A. Bud& J. Chem. Phys. 17 (1949) 686. [S] W.T. Coffey, hlol. Phys. 37 (1979) 373. [6] W.T. Coffey, Mol. Phys 39 (1980) 227. [7] W.T. Coffey, MoL Phys. 41 (1980) 229. 181 . . h1.W. Evans, C I. Evans, W-T. Coffey and P. Gngohm. hlotecular dynamlcs and theory oi broad band spectroscopy (Wiley. New York,.1982). [9] W.T. Coifey, C. Rybarsch and W. Schriicr. Phys. Letters 88A (1982) 331. [lo] P. AppeUand J. Kamp6 de F&et. Fonctions hypergkomkiques et hypcrsphkiqucs polynomes d’Hermite (Gautiucr-Ifus. Paris. 1926).
249
INERTIAL
PHYSICS
LETTERS
EFFECTS IN THE DIELECTRIC RELAXATION
CONTAINING W.T.
CHIIBIICAL
OF
31 Ocrobcr 1982
ANASSEMBLY
OF MOLECULES
ROTATING POLAR GROUPS
COFFEY
and C.
RYBARSCH
and W. SCHROER
Fmlrbercielr Clreme. Unrrersu~c Brmetr. D-ZOO Brcnwn, Ir’esr Gernmy Recewd
76 June 1982
It IS shos n how the problem oi includmg incrtul ciiws in rlw dwAcctrlc rrlx~uon rorarlng polx groups ma)’ be reduced to the solurron oi NO srmplcr problems
IUUII~
ireely romrmg drpolc. (u) die brownian mowmcnl
oi an xsrcmbly oi molcculcs con(I) the browman movement oi 3
oia drpolr m a cosme porcnrial
I. Introduction In the ongmal theory of dielectric
molecules due to Debye [I] and the molecules are electrxally inert. Thus this theory cannot explam the Poley or far-mfrared absorption [3]. An mteresting variant of this theory which can be treated analyrically IS the problem of the dielecrric relaxation of an assembly of molecules containing rotating polar groups. Each molecule does not interact electncally w1t.hits neighbours, only the groups mside an mdividual molecule do so. If this sirnphfication is not made, then we are immediately led into many-body considerations. The rotatinggroup problem was fust treated by Bud6 [4] who gave a full solution of it (subject to the constramt mentioned above) in the low-frequency limit where inertial effects are not operative. The problem was reconsidered by Coffey [S ,6], who showed how mertial effects could be included subject to the constraint that frictional effects rather than the strength of the mteraction potential dominated rhe motion. Thus his solutioncould not be expectedto providean eaplawtion of the high-frequency absorprlonbehmour. AU I[ protidcs is a fh=.r-order kerual correction UI the ~XU-UI~T of the Roard equation [2] for the Freely rotating dipole. Later a scheme was proposed [7] whereby an exact solution of the problem could be obtained numerically. In this letter we show how the set of equations obtained by Coffey may be factorised to obtain part of the solution in closed form. Some numerical results for the relaxation behaviour of the system are also given. inerna-connected
version
of it [2],
relaxation
of an assembly
one always assumes
that
of dlpolar
the
2. Dielectric relaxation of an assembly of molecules containing rotating polar groups In the theory studied here, an electric field E is assumed to be apphed in the same plane as the plane of rota-
tion of two dipoles pl and ~2 of a group. These dipoles are compelled to rotate about an axis through their common centre normal to the plane containing E. The equations of motion of the dipoles are
I~~(t)+T~,(f)+v’(Q1-91)+~,Esin[~,(f)l=gl(~), 0 009-2614/82/0000-0000/S
02.75 0 1982 North-Holland
(1)
CHEMICAL PHYSICS LETTERS
Volume 92. number 3 @2(t)
+ S&(r)
-
V’(dl
- #2) + I2_rE sin
22 October 1982
0)
I~&)1 =&T&j*
(PlO),f#4f) are the ad= pl and pT make with Eat any time. I is the moment of inerua of a dipole about a cento Its own plane. t h the friction coefficient arising from the brownian movement of the surroundare the white noise driving torques acting on p, and ~2. V($t - #Q) is the potential arising from the dipole-dzpole coupling. It is assumed that the field E which had been applied at t = -m is switched of at t = 0. This allows the system to relax from the ordered polarised state to the onginal chaotic state. Thus for f > 0 the equations of motion of the system are tral aknorm~
inp5;gl(f),g2(t)
,
(31
+ S&Q) - VI@, - 02) = Q(f) .
(4)
f&t@) +GtW G$)
-f W,
-&I
=gtM
These Langevin equations are not tractable as they stand except m the linear case (harmonic potential) where V = 5 ~(91 - #I)‘. Thus in general we must appeal ro the underlying probability denaty d~fu~oo equation - r.@ Framers equafion IS]. We note that eqs. (3) and (4) are marked& simplified if we introduce the sum and difference variables +@?I,
(5)
71=&75*-@2),
(6)
x=!lrpt
whence
(7)
~~~~~=~~t(r}~g~(r)l,
I;i+~~++Ir’(‘Ls)=12~~(t)-g2(1)1. Eqs.(7)and (8) are decoupled, x 1sessentially the angle turned
09
dour by the dipoles rotating as a unit. The vanabon ofx is like that of a dipole free to rotate without any electrical interaction [g]. “Ike variation of T)is iike that Of a rotator moving in a potential V(29) [9]. The foregoing suggests that we should also use the x, q variables in constructing Kramers equation rather than Qt and q$_ Kramers equation is_[5,6]
I%‘&,g, T?,;, t) is the distribution function in configuration-angular velocity space. Eq. (9) must be solved subject “Je initial condition, which in the linear approximation @E 6 KQ, is
i0
h’ = A exp {- [l@ f ri2) + V(?&+]/Wj [ 1 + (2&T/k?+)cos x cos q] .
00)
In wntmg down “q. (10) we have assumed that pl = pz_ This affords much mathematical simplification without any significant loss of generality.
3. Solution of eq- (9) We dewmnine Iv=A
W subject
exp [-Iti
to the titial
+ i)/kTJ
C
C
condruon
mn p.q
(10) as follows.
H,,n((21/kT)lE);,
We ass-e
(2f/kT)1/2jl)A$‘(f)
that for I > 0 W may be written
exp[i@x +p71)] ,
0 1)
where the HM n are the two~~~n~on~ Semite polynom~s defmed by Appeli and Kampt de F&&t [IO]. One then fmds thai the .4z;n(r) satisfy the set of differential-difference equations (we take ~(~11)= -vi, cos 2~)
2, Ocfober 1982
CHEMICAL PHYSICS LETTERS
Volume 92. number 3
+i(kT/X)t12[q(m
t l)A~r’*“(r)+P(rr
+ l)Azbn+‘(f)
= 0.
+qAEP1*n(f) +pA,7;:-‘(f)]
Our initial condition becomes rn terms of the A d$O)
= j: i” --II 0
[ 1 + (2/.IE/R7-)cos x cos ‘11exp[(Vu/kT) cos 7~71exp[i(qX tpr))]dx
=0 ,
d:;‘(O)
dn ,
m,n>O.
The mean dipole moment (m*e)may be determined from
Z/.X .ZOS x cos p k’di dx d; dq
1”_,J8”/:W1-“.. (We)=(/rcos$J,
t~cos~~~=~~!II(cos~cos~~= ,!!,J&f%og,_“,
wd;r
dx d; drl
Thrs reduces with the aid of the orthogonahty property of the circular fimctrons and the H,,, On-e)= 2~[.4!y,_~(r) + dt!t(r)
+4?.,,(r)
to
,
+.4~~(r)]/k4$$)
d%!(r) = constant _ The complex polarisabrhty IS
o(s) = 1 - sL?lm*e) wheregdenotes
,
the Laplace transform.
4. Factorization of the A Eq. (12) is extremely complicated, betng essentrally a double matrix differentral equation. A great srmplificatton can be made rf we note that A;;‘(t)
=~$‘(04#)
(17)
,
where A,“(t) satisfies the equation dA~(f)/dr+(E/l)nzrl~(r)+i(kT/~1~7[q(nrt
l)A~+‘(t)+qA~-l(f)]
=O
(18)
and ,4,“(r) satisfies dA#)/dt
t (T/r)nAi(r)
t i(kW)*72[p(tz
+ iVo(2Lk7)-1~2 [cl,“: i(r) -A,“;:(r)]
+ I)Ai”(r)
t&j-l(r)]
=0 .
(19)
Eq, (18) is exactly the same as that which one obtains for the brownian movement of a free rotator having the equation of motion Ijit
(20)
The solution of the set (18) has been described by Coffey rn ref. [S]. One fmds that for m = 0 say that ~~O)M~(O) = exp (-W/X*)
4’ Wr) r - 1+ exp HWI
fl II .
(21) 247
CHEMICAL
Volume 91. number 3
PHYSICS LETTERS
22 October
1982
b(t)
b(t) I
t
a -
.s
8 2
4
6
B
t
-.S
-.5
-I
-I
Fig 1. Tie behaviour of the dtpotr relaxation fonclion b(r) = m-e> for values oi the paramerers grven bq VofkT= 1, r/f
=ol,~~~/ZT=l.ume~~~sirmurulsoi(kT/~l”.Cu~e(~) ISA;(~) for n = 0.p = 1 (that IS the paxi~cb UI a cos 2n
poren-
lull tune fibi . lss!?
Of cuwzs
(a) and (b).
tbh3t IS tie
dtpole
of b(f) for negame ~31~s Rg. 3. As fg 1 but tone behmour l,~~f21~-t,time~~sisinuruts of Yo. Folk7 G-l,r@=O of (kT/#n. Cum (a) isApR(0.n = 0.p = I. carve(b) fiiIT(t), q = 1, m = 0. CUFW (c) ISthe product of c”~cs (a) and (b), I c the dipole rela~tion iunction).
ielrr~s-
tlon iunction.
b(t) 1
-I -I
a iunction oi the damping for values of the pmmeters given by Y&%7= 1, @-(J/x= 1. Cune (a) $fI = 0.1. cune cuxve fcf r/f = 5, time stale m UN~S of (kT/J)ln
248
@)
rrr =
1.
CHEMICAL PHYSICS LEnERS
Volume 92, number 3
22 October 1987
The set of equations (19) ISessentially that obtained by considering the browman movement of a rotator havmg
the equation of motion
Also, one set of differentral difference equations (19) is very similar to the problem of a partrcle moving in a cosine potential considered earlier by us [2]. The only difference is that one has a cos 2~ potential in this (=~se rather than a cos 5]. Thus the problem of the dielectric relaxation of polar groups inside a molecule may be reduced to the solution of two problems which have already been solved. These are respectively the free rotator and the pendulum. By way of illustration of our results we show the behaviour of (m-e) in figs. l-4 for typical values of the parameters I, 5, Vo_ The factorization property of the A simplifies the calculation of a(w) (essentially 9 {(m-e)) where FFdenotes Founer transformation), since we may now utihse the convolution theorem for Fourrer transforms. According to the theorem if
(23) then
F(w) = (11%)
7 G(h) H(w - 1) dk ,
e-0
-m
F, C, H are the Fourier transforms off,g, and 11,respectively. The advantage of domg this is that the Founer transform of eq. (21) can always be wntten down KI a simple form.
where
Acknowledgement We thank Professor BKP. Scaife for drawing our attention to this problem. One of us (WS) would hke to express hrs thanks to the Deutsche Forschungsgemeinschaft and the Fonds der Chermsche Industrie for financial support for this project. WIT thanks the University of Bremen for a Visiting Fellowship. Thus work forms part of the coordinated research programme of the European Molecular hquids Group (EMLG).
[I] P. Dcbye, Polar molecules (Cbemlcal Catalog Co, New York. 1929). [2] J.R. McConnell, Rotational brouruan monon and rhelectnc theory (Academic Press. New York, 1980). [3] J.P. Poley,J. Appl. Sci. Res. 83 (1955) 337. 141 A. Bud& J. Chem. Phys. 17 (1949) 686. [S] W.T. Coffey, hlol. Phys. 37 (1979) 373. [6] W.T. Coffey, Mol. Phys 39 (1980) 227. [7] W.T. Coffey, MoL Phys. 41 (1980) 229. 181 . . h1.W. Evans, C I. Evans, W-T. Coffey and P. Gngohm. hlotecular dynamlcs and theory oi broad band spectroscopy (Wiley. New York,.1982). [9] W.T. Coifey, C. Rybarsch and W. Schriicr. Phys. Letters 88A (1982) 331. [lo] P. AppeUand J. Kamp6 de F&et. Fonctions hypergkomkiques et hypcrsphkiqucs polynomes d’Hermite (Gautiucr-Ifus. Paris. 1926).
249