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Extended rotational diffusion and dielectric relaxation in liquid solutions
Physica A 193 (1993) 394-412 North-Holland
Extended rotational diffusion and dielectric relaxation in liquid solutions Yuri P. Kalmykov Institute of Radio Engineering & Electronics of the Russian Academy of Sciences, Vvedenskii Sq. 1, Fryazino, Moscow Region 141120, Russian Federation
James
McConnell
School of Theoretical Physics, Dublin Institute for Advanced
Studies, Dublin 4, Ireland
Received 2 April 1992 The complex dielectric permittivity for a dilute solution of polar molecules dissolved in a nonpolar solvent is calculated in the context of a generalized J-diffusion model. This model takes into account both inertial effects and finite duration of collisions. The theory is compared with experimental far-infrared absorption spectra of HCl and DC1 in nonpolar solvents.
1. Introduction
Nuclear magnetic relaxation, Raman, Rayleigh, infrared, far infrared and dielectric spectroscopy constitute an important source of information about molecular motion in both gas mixtures and liquid solutions. To give a theoretical description of experimental spectroscopic results use is frequently made of the J-diffusion model first introduced by Gross [l] and Sack [2] and independently by Gordon [3]. The majority of applications of the J-diffusion model have been in nuclear magnetic relaxation, infrared and Raman band shapes (see review article [4] and references cited therein). Attempts to apply the model to far infrared spectra of polar liquids and solutions have been unsuccessful (see e.g., refs. [5-71). The fundamental reason for this is that the J-diffusion model is based on the assumptions 7c 4 T and T, -+ ~IT/W [l], where 7c is the duration of molecular collisions, 7 is the mean time between collisions, which is equal to angular momentum correlation time, and ~IT/Wis the period of an applied electric field E,, cos(wt). The second assumption is not valid in the far infrared region even for the gas phase, since rc is of the order d/u -lo-‘* - lo-l3 s, where d is a molecular diameter and u is the mean translation velocity. 037%4371/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved
Y.P. Kalmykov, .I. McConnell I Rotational diffusion and dielectric relaxation
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In the recent investigations [8] the J-diffusion model has been generalized to a three-parameter theory based on the collision time T,, the angular momentum correlation time r and the Debye relaxation time in, and the experimental implications of this theory have been examined. The generalization admits the possibility of a finite time for molecular collision. When the collision time TV becomes of order lo-l3 s, the model allows us to explain the Poley absorption in the far-infrared spectra of polar liquids. Unfortunately the approach of ref. [8] has several limitations. In particular, (i) in the limit ~~-0, the results of the Burshtein-McConnell (BM) model do not reduce to those of the J-diffusion one (due to the fact that crude approximations have been used in ref. [S]), (ii) only the linear molecule model has been considered, and (iii) the conditions for the applicability of the model have not been established in ref. [8]. The main aim of the present article is to propose an exact generalization of the J-diffusion model in order to take into account a finite duration of collisions for molecules of arbitrary symmetry and to find conditions of applicability of the BM model [8]. Our approach is based on the ZwanzigMori memory function formalism [9-121. In section 2 general formulae for the complex dielectric permittivity E(W) for the J-diffusion and BM models are presented. Application of the ZwanzigMori memory function formalism to dielectric relaxation in polar fluids is described in section 3. Generalization of the J-diffusion model in the framework of memory function formalism is given in section 4. Then section 5 contains a discussion of results obtained and a comparison of the theory with experimental far-infrared spectra of HCl and DC1 in nonpolar solvents.
2. Dielectric permittivity E(O) and molecular dipole autocorrelation function for the J-diffusion model We consider a system of molecules having permanent electric dipoles, the molecules being dissolved in a nonpolar liquid. We express the complex permittivity E(O) by (ref. [13], section 2.3) E(O) = E’(O) - i&“(w),
(2.1)
where o is the angular frequency of a periodic electric field and E’(W), E”(O) are both real. Since c”(O) vanishes, E(O) is real and we write it E,. It may also be proved that i& E”(W) vanishes and hence that
pm E(O) =
liz_ E’(O)
)
396
Y.P. Kalmykov,
J. McConnell
I Rotational
dif@sion
and dielectric
relaxation
which we write as E,. For a dilute solution of polar molecules in nonpolar solvent
E,
-
E,
=
U2N q,kBT ’
(2.2)
2--_--
where p is the dipole moment of the molecule, N is the concentration of polar molecules, a,, is the permittivity of free space, k, is the Boltzmann constant, and T is the temperature. The absorption coefficient (Y(O) is expressed by O&“(0) (.y(w) = ___ en(w) ’
(2.3)
where n(w) is the refractive index and c is the velocity of light. According to linear response theory [13,14], in the zero wave vector limit, the complex dielectric permittivity E(W) of a system of noninteracting polar linear molecules dissolved in a nonpolar solvent is given by CZ e(w) - ace = 1 - io E,
-
63
I 0
C(t) e-‘“’ dt
where the dipole autocorrelation C(t) = (u(O) - 44 > 3
,
(2.4)
function C(t) is defined by (2.5)
u(t) is the unit vector along the dipole vector p and the brackets ( ) indicate equilibrium ensemble average. The electric field induced moment caused by time dependent collisional motions is assumed to be much smaller than the permanent moment. We apply the foregoing general results to the J-diffusion model. The J-diffusion model was discussed in detail elsewhere [4]. Here only a brief description of the model is given. It is assumed that a classical rigid rotator, whose permanent dipole moment is p, rotates freely in space. The rotation of the molecule is interrupted by “instantaneous” collisions, that is to say, the duration of collisions is assumed to be much smaller than the average time between collisions 7. The collisions take place at random times governed by a Poisson law and they randomize both the magnitude and direction of the angular momentum vector J, the random values of J being governed by a Boltzmann law. It is also assumed that collisions do not change the orientation of the rotator in space. It should be emphasized that the term “collision”, in the context of the J-diffusion model, does not signify a bimolecular collision as
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is presumed to occur in dilute gases. A “collision” is an idealized event wherein the molecule experiences an impulsive torque [4]. The dipole autocorrelation function C(t) for the J-diffusion model can be evaluated in closed form for a molecule of arbitrary symmetry and is given by
PI C(t)
=
7-l
e-“’ C”“(t) +
I
e-(r-‘l)‘T CFR(t - cl) C(t,) dt,
0
=
5 .-(“-‘)
e-“’
n=l
i 0
or in the frequency
dt,_,
‘[’ dt,_, * * * f dt, fi [ CFR(ti - ti_r)] 0
0
domain
EFR(w- i/r)
C(w)=
(2.6)
l-CFR(W-i/7)/7’
where C”“(t) model [4],
i=l
is the dipole autocorrelation
function
for the free rotational
C(w) = f C(t) e-‘“I dt 0
and m
e;“R(@ _
i/7)
=
I
c”“(t)
e-i(w-i/T)f
dt.
0
Hence from (2.4) and (2.6)
E(O)- cc = E, - E,
1 - i(w - i/r)CFR(W - i/r)
l-
Z;““(w -i/r)/7
’
(2.7)
It should be noted that eqs. (2.6) and (2.7) are exact and are valid for a molecule of arbitrary symmetry (linear, spherical, symmetric and asymmetric top molecules). Equations for CFR(z) for all types of molecules are to be found in table 8.2 of ref. [7]. The assumption, which limited the range of applicability of the J-diffusion model, is that of instantaneous collisions, Burshtein and McConnell [8] have tried to take into account a finite duration of collisions in the framework of the
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/ Rotational diffusion and dielectric relaxation
J-diffusion model for a linear molecule by the following procedure. They started from the approximate equation for the autocorrelation function C(t) obtained with the help of a perturbation theory (ref. [15], p. 38),
dc(t) = -I-’ j- C,(t 27
t’) C(t’) dt
,
P-8)
0
where CJ(t), the autocorrelation molecule, is given by C,(t) = (J(0)
*J(t))
function of the angular momentum J(t) of the
= 2k,TZ eet” ,
Z being the moment of inertia of the linear molecule. It should be noted that eq. (2.8) is an approximate equation obtained by the application of perturbation theory. Then they employ the memory function formalism as presented by Berne, Boon and Rice [ll]. In this formalism C,(t) obeys the equation
$ C,(t)
= - j R(t - t’) C,(t’) dt’ ,
(2.9)
0
where the memory function specified by (2.10)
R(t) = 26(t) /r
is related to the mechanism of instantaneous collisions involving the occurrence of infinitely large torques and s(t) is the Dirac S-function. In order to take into account the finite duration of collisions TVBurshtein and McConnell have made the simple assumption R(t) =
$
e-“’
.
(2.11)
c
In the limit T~+O, eq. (2.11) reduces to eq. (2.10). On using the Fourier-Laplace transform the system of eqs. (2.4), (2.8), (2.9) and (2.11) can be easily solved to yield the results: iw(w - i/r,) - i/77, c(w) = (2k,TIZ E(W) - &cc=
E, - G,
(2.12)
- w’)(w - i/r,) + GJ/~T, ’ (2k,TlZ)(w
(2k,TIZ
- o*)(o
-i/r,) - i/r,) + wlrr,
’
(2.13)
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Eqs. (2.12) and (2.13) are the main results of ref. [8]. This model may be considered as a generalization of the J-diffusion model. As already mentioned, the approach of ref. [8] has several shortcomings: (i) only the linear molecule model was considered, (ii) in the limit TV-+0, eq. (2.13) does not reduce to the exact formula (2.7) since eq. (2.8) is an approximate one, and (iii) the conditions for the applicability of the model were not established in ref. [S]. In the next sections we propose a precise extension of the J-diffusion model for molecules of arbitrary symmetry and we obtain conditions for the applicability of the BM model.
3. The memory function formalism At this stage it is opportune to introduce some notation which will be useful for the application of the memory function formalism [9-121. The fundamental equation is
-$ A(t) = &A(t)
-
I 0
@*(t - t’) A(t’) dt’ +fi(t)
.
(3.1)
Here A(t) is a dynamic variable, w. defined by (3.2) is a frequency
which determines
A(O)= $ A(t)I,
the collective oscillation of A(t),
,
the function @r(t) given by
@,l(t)= (f,(O)tf,(t))I(A(O)IA(O)) is called the memory function of A(t),
f&k i(l-pO)ir
i(l
_
and the function fi(t) defined by
~o),f+lA(()))
is the random force acting on the variable A(t) and is orthogonal
(A(O)I
= 0.
(3.3)
to A(O), i.e.,
Y. P. Kalmykov, J. McConnell
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I Rotational diffusion and dielectric relaxation
i is the Liouville operator, IA(t)) = eii’IA(0))
i), is the projection rule
,
(3.4)
operator which acts on variables R and S according to the
(Rl$olS> = (RlA(O))(A(O)IA(O))-‘(A(O)IS) 7
(3.5)
(R(S) denotes the scalar product of the two vectors (RIand IS) in the Hilbert space of dynamic variables and ( AJB ) = (AB ) by definition [12]. It follows from (3.1) that the correlation function C,(t) defined by
C,(f) = (A(O)kW))&W)kWV)
(3.6)
satisfies I
$ C,(t) = iw,,C,(t)
-
I
G1(t - t’) C,(t’) dt’ .
(3.7)
0
The one-sided Fourier transformation C,(o)
=
of eq. (3.7) yields
1 (3.8)
- iw, + iw + &((w) *
Since G$(t) is the time correlation function of fi(t), it is immediately seen that Q,(t), which in (3.7) is a memory function, will also satisfy equations similar to (3.7) and (3.8) in the time domain and hence f
$ @1(t)= iw,G1(t)
-
I
@*(t - t’) Q1(t’) dt’ ,
(3.9)
0
$ c&(t) = iWZGz(t) -
1 Q3(t - t’) $(t’)
dt’
,
(3.10)
0
$ cDn((t)= iW,Qn((t) -
1 @n+l(t - t’) Qn(t’) dt’ .
(3.11)
Y. P. Kalmykov, .I. McConnell
In the frequency
I Rotational diffusion and dielectric relaxation
domain eqs. (3.8)-(3.11)
are similarly extended
to
1
C,(w) =
,
A1
iw-io,+
401
(3.12)
A2
iw-iw,+
...
iw - iw, +
4
iw - iw, + Gn+,(w)
where % =
~~(o)lf,(o))~(f,-l(o)lf,-l(o)) 7
(3.14)
(f,(o)lf,(t))~(f,-l(o)lf,-l(o)) 7
(3.15)
A, = (.L(O)l.L(O))KL1(O)lf,-1(O)) @n=
(3.13)
9
If,(t))
= i exp[i(l
- i), - P, -a..
- Pn_,)it]
x (1 - i), - P, - * * *Pn_,)ilfn_l(o))
)
Wnl~> = (Rlf,(O))(f,(O)lf”(O))-‘(f,(O)lS). It can be shown [16] that the quantities moments M,, defined by
A, are expressed
(3.16) (3.17) in terms of the
(3.18) and which appear in the expansion
C,(t) =
c
&
(-l)“M,,
n=l
(3.19)
as follows:
4=M2, A2 = (M, - M;) lM,
(3.20)
,
A3 = (M6M2 - M:)I(M,M, and so on. Eqs. (3.20)-(3.22)
(3.21) - M;)
,
(3.22)
allow us to evaluate A, since the quantities Mzn
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Y.P. Kalmykov,
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/ Rotational
diffusion
can in principle be both evaluated theoretically For dielectric relaxation with
and dielectric
relaxation
and measured experimentally.
A(t) = u(t)
(3.23)
and C,(t) = C(t) = ( u(0) * 40)
(3.24)
7
where u(t) = p/p is the unit vector along the dipole vector ~1, wn = 0 for all 12 [16]. Hence, eq. (3.12) becomes 1
e(0) =
(3.25)
A1
io +
A2
iw +
..
iw +
-4
iw + Gn+,(w)
As already mentioned, all quantities A, in eq. (3.25) can in principle be evaluated theoretically. For example, A, can be calculated by the formula (ref. [I719 P. 184) (3.26) where I,, Z, and I, are the principal moments of inertia of a molecule, ux, uy and uz are the projections of the vector u onto the principal axes of inertia XYZ. It is of importance that A, is independent of intermolecular interactions. Moreover, the quantities A,, can be determined experimentally from eqs. (3.20)-(3.22) and the following sum rules [17]: (3.27)
However, only M, and M, can actually accuracy [ 171.
4. Memory
function
approach
be measured
to the J-diffusion
with a satisfactory
model
There are at least three equivalent approaches to the mathematical formulation of the Z-diffusion model, viz. the kinetic equation [1,2,7], analytical
Y.P. Kalmykov, J. McConnell
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dynamics [4] and memory function [4] approaches. The kinetic equation and analytical dynamics approaches are straightforward but rather tedious. The much more compact exposition is afforded by the use of the memory function formalism [4]. In the framework of the memory function approach the J-diffusion model may be formulated as follows [4]. The memory function Q+((t) from eq. (3.3) can be considered to be the correlation function for the random torques which act on the molecule. In the extended diffusion picture where the molecules are assumed to rotate freely between collisions which change the angular momenta, the memory function for a molecule which has experienced no collisions is to be the free rotor memory function @r”(t) which corresponds to the dipole autocorrelation function C”“(t) and the memory function for molecules which have experienced collisions is to be zero. Then the memory function Q1(t) for the ensemble is given by Q1(t) = @y(t)
e-“’ ,
(4.1)
since the times between collisions are assumed to follow a Poisson distribution [4]. It is readily deduced from eq. (4.1) that &i(w) = @“(o The function
p+))
-i/7).
&p(z)
(4.2)
can be evaluated
by employing (3.25) to write down
l
=
io + S;“(w)
(4.3)
’
from which it is deduced that
CiqyoJ)=
1 - ifOEFR(W)
CFR(0)
(4.4)
.
We deduce from (4.2) and (4.4) that ~)l(w)
= ~~“
_
i/7)
=
(w ’ - i(w- i’T)cwFR 2;““(,
- i/7)
-i/7) ’
(4.5)
Hence C(w) =
l, iw + al(o)
which is just eq. (2.6).
eFR(w-i/r) = 1 - E:““(w -i/7)/7
(4.6)
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Y. P. Kalmykov, J. McConnell
1 Rotational diffusion and dielectric relaxation
We may introduce a set of memory functions &FR(t), $yR(t), . . . associated with C”“(t) and defined by
(4.7) (4.8)
(4.9) where A? 1s . the value at time t = 0 of G’,‘“(t). It should be noted that AyR = A,, for it = 1 only, since only A, is independent of intermolecular interactions. These equations will provide equations for the memory functions of J-diffusion at any level. Thus for the second level we obtain from (4.5) and (4.8)
&(w) = @“(LO-
(4.10)
i/r) + 7-l ,
and this leads to C&(t) = e -t’T@;R(t)
+
2S(t)/r
.
(4.11)
Here we have taken into account that AyR = A,. One can show that both (4.10) and (4.11) lead to eqs. (4.6). The appearance of the S-function in eq. (4.11) is the consequence of the assumption of instantaneous collisions which implies the occurrence of infinitely strong torques during the collisions. The finite duration of collisions can be taken into account by the same manner as was done in ref. [LX];that is to say, we introduce a model memory function -lhc
C&(t) = e -“‘~;“p)
+ %
77, ’
(4.12)
where T, is the collision time and lim L e-“‘c = 26(t) .
r,-0 Tc
(4.13)
It should be noted that the semi-classical version of a similar model, in which
Y.P. Kalmykov,
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I Rotational diffusion and dielectric relaxation
405
the molecular rotation during the diffusive steps is described quantummechanically, has been proposed in ref. [18] to explain the rotational spectrum of atmospheric water vapour. From eq. (3.25) iw + &(w)
1
z;(o)= io +
= A1 - w2 + k&(w) A* iw + C&(o)
(4.14) ’
where according to eq. (4.12) (4.15) We substitute eq. (4.15) into the numerator of eq. (4.14). Thus iw(w -i/r,)
and denominator
of the last term
- ilrr, + (6~ - i/7,)&)2FR(6J-i/r)
c(w) = (A1 - w’)(w - i/r,) + w/rr, + iw(o - i/r,)&F(o
- i/T) ’ (4.16)
where the quantity AI is given by eq. (3.26). Substituting eq. (4.16) into eq. (2.4) we obtain
E(W)-
Al
EC?2
E, - E,
= (A, - o*) + iw[T(l
+ OK-,)]-’ + io&yR(o
- i/T) ’
(4.17)
Eqs. (4.16) and (4.17) are quite general and are valid for any model of a polar molecule, viz. planar, linear, spherical, symmetric and asymmetric top molecules. We need only to evaluate the Fourier-Laplace transforms &F(w - i/T). According to eqs. (4.7) and (4.8) the function $,,““(oJ -ir) in (4.17) can be expressed in terms of ?‘“( w - i/T) by the following equation:
E:‘“(aJ
-
i/T)
1
=
io +
7-l
(4.18)
AI
+ iW
+
7-l
+
@“(w
-i/T)
Hence GFR(o _ i,T) = A,cFR(o 2
-i/T)
- i(w - i/r)[l
- i(w - i/T)eFR(o
- i/T)]
1 - i(w - i/7)CFR(W - i/r)
(4.19)
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I Rotational diffusion and dielectric relaxation
According to eq. (4.17) the complex permittivity depends on the two model parameters, viz. angular momentum relaxation time r and the mean duration of collisions rc. The time r can be measured in NMR relaxation and/or infrared experiments (for details see ref. [8]). The other model parameter TVcan be evaluated as follows. We have from eq. (4.12) CD*(O)=
@y(O) + l/77,
)
(4.20)
or AZ = ApR + ~/TT, .
(4.21)
On the other hand A2 is given by eq. (3.21), where for molecules of the C,, or C,, symmetry [16] M, =
kBT(z;’ + zyl)
(4.22)
and
(4.23)
where I, and I, are the principal moments of inertia about the axes perpendicular to the direction of the dipole vector cr_,I, is the third principal moment of inertia, N, and NY are the components of the torque N acting on the molecule. Thus (N;) (N9 kBTZX(l + IX/Z,) + k,TZ,(l -t Z,lZ,)
(rrJ’=A2-Ay=
*
(4.24)
Hence @J-l
=
7
k, T
(
(Ni)
0%)
Z,(l -t- IX/Z,) + Z,( 1 + Z,lZ,) > ’
(4.25)
Since both r and A, - AFR can be measured experimentally [8,16], eq. (4.25) allows us to evaluate TV. We shall now consider only linear molecules. In this case Z, = I, = Z and Z, = 0, we have from eqs. (4.22) and (4.23)
Y. P. Kalmykov,
.I. McConnell
407
I Rotational diffusion and dielectric relaxation
44, = 2k, TII
(4.26)
and (4.27)
(N*=N;+N;).
Hence taking into account eqs. (3.20) and (3.21) we obtain (4.28)
A, = 2k,TII, A2=I
2kaT + (N*) 2k,TI
(4.29)
’
Thus (g’
T(N2)
=
2k,TI
(4.30)
’
in accordance with the result of eq. (55) of ref. [8]. Furthermore the FourierLaplace transform of the autocorrelation function C’“(t) for linear molecules is given by [4]
C”“(o -
i/7) = iv2 e-Z2E,(-Z2)
where Z = ~(0 -i/T),
E,(x) =
exp(-0 Im
(4.31)
,
v = (I/2k,T)“’
,
(4.32)
dt
7
,
x
the exponential
E(O)-
50
E, - &Xx
integral function [19]. Whence 1
=
(1 - v2w2) + ion2[7(1 + iwr,)]-’
+ if_07j2SyR(W-i/r)
’
(4.33) where “FR
G2 (w - i/7) = iT_l
Z e-”
E,(-Z2)
- Z[l
+ Z2e-L2 E,(-Z’)]
1 + 2’ ewz2 E,(-
Z’)
(4.34)
408
Y. P. Kalmykov, J. McConnell
I Rotational diffusion and dielectric relaxation
5. Discussion of results and comparison with experimental
data
On comparing eqs. (2.13) and (4.33) it is easy to show in the case of &F”(w - i/T) = 0 th a t eq. (4.33) coincides precisely with eq. (2.13) while in the limit r,+O eq. (4.33) reduces to the well-known equation (2.6). Moreover, eq. (4.12) allows us to obtain the condition of applicability of the BM model [8], viz.
(5.1) i.e. the BM model is valid only in the case when we can ignore the term exp(--t/T) @FR(t) in eq. (4.12). Figs. 1 and 2 illustrate a comparison of the absorption spectra a(~) VW&“(W)predicted by the both models for different values of the model parameters 7 and TV.One may see by inspection of figs. 1 and 2 that when eq. (5.1) is valid both models lead practically to the same result (curves 3 and 3’). However when eq. (5.1) is not satisfied, these models predict an essential difference in the calculated spectra. To test our model we consider dilute solutions of small polar molecules HCl and DC1 in nonpolar solvents. Datta and Barrow [20], Birnbaum and Ho [21], Morita et al. [22] and others found a similarity between the far infrared spectra of small molecules like HCl, DCl, HF, NH, and H,O dissolved in nonpolar solvents and those in the gas phase. This highlights the importance of inertial effects for such systems.
Fig. 1. Comparison of r)w”(w)/( E, - cl) as a function of 07 for the BM and present models at different values of the model parameters: T/T, = 1.0 and V/T = 0.5 (curves 1 and l’), V/T = 2.0 (curves 2 and 2’), q/r = 10.0 (curves 3 and 3’). The curves numbered 1, 2, 3 refer to the BM model.
Y. P, Kalmykov, J. McConnell
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Fig. 2. The same as in fig. 1 for T/T = 2.0 and q/rC = 0.5 (curves 1 and l’), V/T, = 1.0 (curves 2 and 2’), qhC = 10.0 (curves 3 and 3’).
Morita et al. [22] have applied the free rotational model, which in fact is a particular case of our model when 7 + 00, to calculate the far infrared spectra of small molecules in nonpolar solvents. It was found that for linear molecules reasonable values for frequencies w,,, at which the maximum absorption occur can be calculated as predicted by the free rotational model [22] to be 0 max
= (3k,T/z)“*.
(5.2)
For HCl in several nonpolar solvents this was also supported by the linear relationship between w,,,,~ and T”*, which is given by eq. (5.2). Furthermore, shapes of theoretical spectra were in good agreement with those observed [22]. This agreement of the experimental and theoretical spectra shows clearly the importance of inertial effects. However the free rotational model could not explain the distinctive feature of the far infrared spectra of light molecules, namely a high frequency shift A%,, of the maximum absorption frequency w,,, predicted by eq. (5.2). For example, eq. (5.2) gives v,,, = 114cm-’ (v,,, = w,,,,,/~ITc) for HCl in cyclohexane at 298 K, while the observed value of v,,,,, is 125 cm-’ [22]. This is due to the fact that the free rotational model ignores intermolecular interactions completely. We apply our model in order to explain the above features of the far infrared spectra of light molecules. It should be noted that both the J-diffusion model and BM model have failed to explain far infrared spectra of such systems. The J-diffusion model predicts a negative frequency shift AU,,,,, while the BM model cannot describe the shape of the spectra (see fig. 1).
410
Y. t? Kalmykov, J. McConnell
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The absorption coefficient a(w) is given by eq. (2.3). For the dilute solutions of polar liquids in nonpolar solvents which we shall consider, the variation of the refractive index n(w) for the range of o with which we are concerned is small and so n(u) may be taken to be constant. Then we can evaluate CZ(W) using only values of the temperature T and moments of inertia and fitting the parameters r and TV,provided that E, - E, is known. In principle the calculations are capable of predicting the magnitude as well as the shape of the absorption band, but this requires that both E, and E, be known with a high degree of accuracy, since their difference is the important factor [22]. However, such data were not available. We shall take experimental data concerning HCl in cyclohexane at 298 K [22], HCl in Ccl, at 293 K [23] and DC1 in SF, at 273 K [21]. The values of the moments of inertia of HCl and DC1 are respectively 2.68 x 10P4’ and 5.19 X 10e4’ g cm2 [22]. For these solutions the theoretical absorption coefficient a( V) has been calculated using eqs. (2.3), (4.31)-(4.34). The values of the model parameters r and 7c have been chosen from the best fit of the theoretical and experimental profiles provided that the observed and calculated maximum absorption frequencies vmaXshould coincide. In order to compare the theoretical spectra with the observed profiles, the absorption maxima of the theoretical curves are set equal to those of the experimental ones. The theoretical and experimental curves are compared in figs. 3-5, where the spectra predicted by the free rotational model are also shown in comparison. In the foregoing sections a model of molecular rotation in liquids has been
Fig. 3. Comparison of the theoretical absorption coefficient CX(V)for HCl in cyclohexane (curve 1) with the observed profile (filled circles) at 298 K from Morita et al. [22] and with that for the free rotational model (curve 2). The best fit is at T = 0.115 x lo-” s and T, = 0.357 x lo-” s.
Y. P. Kalmykov, J. McConnell I Rotational diffusion and dielectric relaxation
411
Fig. 4. Comparison of the theoretical absorption cross section cr(v)IN for HCl in CCI, (curve 1) with the observed profile (filled circles) at 293 K from Ohkubo et al. [23] and with that for the free rotational model (curve 2). The best fit is at T = 0.689 X lo-l3 s and r, = 0.443 X lo-” s.
0 frequency
~r/cm)
a
Fig. 5. Comparison of the theoretical absorption coefficient a(v) for DCI in SF, (curve 1) with the observed profile (stars) at 273 K from Bimbaum and Ho 1211and with that for the free rotational model (curve 2). The best fit is at T = 0.224 X lo-‘* s and rc = 0.395 x lo-l3 s.
considered. The model is a generalization of the Gordon J-diffusion and BM models. It is applicable to molecules of an arbitrary symmetry. This generalization allows account to be taken of (i) a finite time of molecule collisions and (ii) the inertial effects. This permits us to apply the model to far infrared absorption of small polar molecules in nonpolar solvents where the both factors are of importance. The values of the model parameters r and TVobtained are reasonable and are in agreement with the previous reports [7,8,16].
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Acknowledgements Support of this work by the Dublin Institute for Advanced Studies and the Institute of Radio Engineering and Electronics of the Russian Academy of Sciences is gratefully acknowledged.
References [l] [2] [3] [4] [S]
E.P. Gross, J. Chem. Phys. 23 (1955) 1415. R.A. Sack, Proc. Phys. Sot. London B 70 (1957) 402, 414. R.G. Gordon, J. Chem. Phys. 44 (1966) 1830. R.E.D. McClung, Adv. Mol. Rel. Int. Proc. 10 (1977) 83. C.J. Reid, R.A. Yadav, G.J. Evans, M.W. Evans and G.J. Davies, J. Chem. Sot. Faraday Trans. 2 74 (1978) 2143. [6] A. Gerschel, C. Brot, I. Dimicoli and A. Riou, Mol. Phys. 33 (1977) 527. [7] V.I. Gaiduk and Yu.P. Kalmykov, J. Mol. Liquids 34 (1987) 1. [8] AI. Burshtein and J. McConnell, Physica A 157 (1989) 933, to be quoted as BM. [9] H. Mori, Progr. Theor. Phys. 33 (1965) 424. [lo] H. Mori, Progr. Theor. Phys. 34 (1965) 399. [ll] B.J. Berne, J.P. Boon and S.A. Rice, J. Chem. Phys. 45 (1966) 1086. [12] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Benjamin, New York, 1975). [13] J. McConnell, Rotational Brownian Motion and Dielectric Theory (Academic Press, New York, 1980). [14] M.W. Evans, G.J. Evans, W.T. Coffey and P. Grigolini, Molecular Dynamics and Theory of Broad Band Spectroscopy (Wiley, New York, 1982). [15] A.I. Burshtein and S.I. Temkin, Spectroscopy of Molecular Rotation in Gases and Liquids (Nauka, Novosibirsk, 1982) [in Russian]. [16] S. Ikawa, K. Sato and M. Kimura, Chem. Phys. 47 (1980) 65. 1171 C.J.F. Bottcher and P. Bordewijk, Theory of Electric Polarization, vol. 2, Dielectrics in Time-Dependent Fields (Elsevier, Amsterdam, 1978). [18] Yu.P. Kalmykov and S.V. Titov, Radiotekh. Elektron. 34 (1989) 13. [19] M. Abramoviz and I.A. Stegun, eds., Handbook of Mathematical Functions, NBS Series, vol. 55 (National Bureau of Standards, New York, 1964). [20] P. Datta and G.M. Barrow, J. Chem. Phys. 43 (1965) 2137. [21] G. Birnbaum and W. Ho, Chem. Phys. Lett. 5 (1970) 334. [22] A. Morita, S. Walker and J.H. Calderwood, J. Phys. D 9 (1976) 2485. [23] Y. Ohkubo, S. Ikawa and M. Kimura, Chem. Phys. Lett. 43 (1976) 138.
Extended rotational diffusion and dielectric relaxation in liquid solutions Yuri P. Kalmykov Institute of Radio Engineering & Electronics of the Russian Academy of Sciences, Vvedenskii Sq. 1, Fryazino, Moscow Region 141120, Russian Federation
James
McConnell
School of Theoretical Physics, Dublin Institute for Advanced
Studies, Dublin 4, Ireland
Received 2 April 1992 The complex dielectric permittivity for a dilute solution of polar molecules dissolved in a nonpolar solvent is calculated in the context of a generalized J-diffusion model. This model takes into account both inertial effects and finite duration of collisions. The theory is compared with experimental far-infrared absorption spectra of HCl and DC1 in nonpolar solvents.
1. Introduction
Nuclear magnetic relaxation, Raman, Rayleigh, infrared, far infrared and dielectric spectroscopy constitute an important source of information about molecular motion in both gas mixtures and liquid solutions. To give a theoretical description of experimental spectroscopic results use is frequently made of the J-diffusion model first introduced by Gross [l] and Sack [2] and independently by Gordon [3]. The majority of applications of the J-diffusion model have been in nuclear magnetic relaxation, infrared and Raman band shapes (see review article [4] and references cited therein). Attempts to apply the model to far infrared spectra of polar liquids and solutions have been unsuccessful (see e.g., refs. [5-71). The fundamental reason for this is that the J-diffusion model is based on the assumptions 7c 4 T and T, -+ ~IT/W [l], where 7c is the duration of molecular collisions, 7 is the mean time between collisions, which is equal to angular momentum correlation time, and ~IT/Wis the period of an applied electric field E,, cos(wt). The second assumption is not valid in the far infrared region even for the gas phase, since rc is of the order d/u -lo-‘* - lo-l3 s, where d is a molecular diameter and u is the mean translation velocity. 037%4371/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved
Y.P. Kalmykov, .I. McConnell I Rotational diffusion and dielectric relaxation
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In the recent investigations [8] the J-diffusion model has been generalized to a three-parameter theory based on the collision time T,, the angular momentum correlation time r and the Debye relaxation time in, and the experimental implications of this theory have been examined. The generalization admits the possibility of a finite time for molecular collision. When the collision time TV becomes of order lo-l3 s, the model allows us to explain the Poley absorption in the far-infrared spectra of polar liquids. Unfortunately the approach of ref. [8] has several limitations. In particular, (i) in the limit ~~-0, the results of the Burshtein-McConnell (BM) model do not reduce to those of the J-diffusion one (due to the fact that crude approximations have been used in ref. [S]), (ii) only the linear molecule model has been considered, and (iii) the conditions for the applicability of the model have not been established in ref. [8]. The main aim of the present article is to propose an exact generalization of the J-diffusion model in order to take into account a finite duration of collisions for molecules of arbitrary symmetry and to find conditions of applicability of the BM model [8]. Our approach is based on the ZwanzigMori memory function formalism [9-121. In section 2 general formulae for the complex dielectric permittivity E(W) for the J-diffusion and BM models are presented. Application of the ZwanzigMori memory function formalism to dielectric relaxation in polar fluids is described in section 3. Generalization of the J-diffusion model in the framework of memory function formalism is given in section 4. Then section 5 contains a discussion of results obtained and a comparison of the theory with experimental far-infrared spectra of HCl and DC1 in nonpolar solvents.
2. Dielectric permittivity E(O) and molecular dipole autocorrelation function for the J-diffusion model We consider a system of molecules having permanent electric dipoles, the molecules being dissolved in a nonpolar liquid. We express the complex permittivity E(O) by (ref. [13], section 2.3) E(O) = E’(O) - i&“(w),
(2.1)
where o is the angular frequency of a periodic electric field and E’(W), E”(O) are both real. Since c”(O) vanishes, E(O) is real and we write it E,. It may also be proved that i& E”(W) vanishes and hence that
pm E(O) =
liz_ E’(O)
)
396
Y.P. Kalmykov,
J. McConnell
I Rotational
dif@sion
and dielectric
relaxation
which we write as E,. For a dilute solution of polar molecules in nonpolar solvent
E,
-
E,
=
U2N q,kBT ’
(2.2)
2--_--
where p is the dipole moment of the molecule, N is the concentration of polar molecules, a,, is the permittivity of free space, k, is the Boltzmann constant, and T is the temperature. The absorption coefficient (Y(O) is expressed by O&“(0) (.y(w) = ___ en(w) ’
(2.3)
where n(w) is the refractive index and c is the velocity of light. According to linear response theory [13,14], in the zero wave vector limit, the complex dielectric permittivity E(W) of a system of noninteracting polar linear molecules dissolved in a nonpolar solvent is given by CZ e(w) - ace = 1 - io E,
-
63
I 0
C(t) e-‘“’ dt
where the dipole autocorrelation C(t) = (u(O) - 44 > 3
,
(2.4)
function C(t) is defined by (2.5)
u(t) is the unit vector along the dipole vector p and the brackets ( ) indicate equilibrium ensemble average. The electric field induced moment caused by time dependent collisional motions is assumed to be much smaller than the permanent moment. We apply the foregoing general results to the J-diffusion model. The J-diffusion model was discussed in detail elsewhere [4]. Here only a brief description of the model is given. It is assumed that a classical rigid rotator, whose permanent dipole moment is p, rotates freely in space. The rotation of the molecule is interrupted by “instantaneous” collisions, that is to say, the duration of collisions is assumed to be much smaller than the average time between collisions 7. The collisions take place at random times governed by a Poisson law and they randomize both the magnitude and direction of the angular momentum vector J, the random values of J being governed by a Boltzmann law. It is also assumed that collisions do not change the orientation of the rotator in space. It should be emphasized that the term “collision”, in the context of the J-diffusion model, does not signify a bimolecular collision as
Y. P. Kalmykov,
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I Rotational diffusion and dielectric relaxation
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is presumed to occur in dilute gases. A “collision” is an idealized event wherein the molecule experiences an impulsive torque [4]. The dipole autocorrelation function C(t) for the J-diffusion model can be evaluated in closed form for a molecule of arbitrary symmetry and is given by
PI C(t)
=
7-l
e-“’ C”“(t) +
I
e-(r-‘l)‘T CFR(t - cl) C(t,) dt,
0
=
5 .-(“-‘)
e-“’
n=l
i 0
or in the frequency
dt,_,
‘[’ dt,_, * * * f dt, fi [ CFR(ti - ti_r)] 0
0
domain
EFR(w- i/r)
C(w)=
(2.6)
l-CFR(W-i/7)/7’
where C”“(t) model [4],
i=l
is the dipole autocorrelation
function
for the free rotational
C(w) = f C(t) e-‘“I dt 0
and m
e;“R(@ _
i/7)
=
I
c”“(t)
e-i(w-i/T)f
dt.
0
Hence from (2.4) and (2.6)
E(O)- cc = E, - E,
1 - i(w - i/r)CFR(W - i/r)
l-
Z;““(w -i/r)/7
’
(2.7)
It should be noted that eqs. (2.6) and (2.7) are exact and are valid for a molecule of arbitrary symmetry (linear, spherical, symmetric and asymmetric top molecules). Equations for CFR(z) for all types of molecules are to be found in table 8.2 of ref. [7]. The assumption, which limited the range of applicability of the J-diffusion model, is that of instantaneous collisions, Burshtein and McConnell [8] have tried to take into account a finite duration of collisions in the framework of the
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Y. P. Kalmykov, J. McConnell
/ Rotational diffusion and dielectric relaxation
J-diffusion model for a linear molecule by the following procedure. They started from the approximate equation for the autocorrelation function C(t) obtained with the help of a perturbation theory (ref. [15], p. 38),
dc(t) = -I-’ j- C,(t 27
t’) C(t’) dt
,
P-8)
0
where CJ(t), the autocorrelation molecule, is given by C,(t) = (J(0)
*J(t))
function of the angular momentum J(t) of the
= 2k,TZ eet” ,
Z being the moment of inertia of the linear molecule. It should be noted that eq. (2.8) is an approximate equation obtained by the application of perturbation theory. Then they employ the memory function formalism as presented by Berne, Boon and Rice [ll]. In this formalism C,(t) obeys the equation
$ C,(t)
= - j R(t - t’) C,(t’) dt’ ,
(2.9)
0
where the memory function specified by (2.10)
R(t) = 26(t) /r
is related to the mechanism of instantaneous collisions involving the occurrence of infinitely large torques and s(t) is the Dirac S-function. In order to take into account the finite duration of collisions TVBurshtein and McConnell have made the simple assumption R(t) =
$
e-“’
.
(2.11)
c
In the limit T~+O, eq. (2.11) reduces to eq. (2.10). On using the Fourier-Laplace transform the system of eqs. (2.4), (2.8), (2.9) and (2.11) can be easily solved to yield the results: iw(w - i/r,) - i/77, c(w) = (2k,TIZ E(W) - &cc=
E, - G,
(2.12)
- w’)(w - i/r,) + GJ/~T, ’ (2k,TlZ)(w
(2k,TIZ
- o*)(o
-i/r,) - i/r,) + wlrr,
’
(2.13)
Y. P. Kalmykov,
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Eqs. (2.12) and (2.13) are the main results of ref. [8]. This model may be considered as a generalization of the J-diffusion model. As already mentioned, the approach of ref. [8] has several shortcomings: (i) only the linear molecule model was considered, (ii) in the limit TV-+0, eq. (2.13) does not reduce to the exact formula (2.7) since eq. (2.8) is an approximate one, and (iii) the conditions for the applicability of the model were not established in ref. [S]. In the next sections we propose a precise extension of the J-diffusion model for molecules of arbitrary symmetry and we obtain conditions for the applicability of the BM model.
3. The memory function formalism At this stage it is opportune to introduce some notation which will be useful for the application of the memory function formalism [9-121. The fundamental equation is
-$ A(t) = &A(t)
-
I 0
@*(t - t’) A(t’) dt’ +fi(t)
.
(3.1)
Here A(t) is a dynamic variable, w. defined by (3.2) is a frequency
which determines
A(O)= $ A(t)I,
the collective oscillation of A(t),
,
the function @r(t) given by
@,l(t)= (f,(O)tf,(t))I(A(O)IA(O)) is called the memory function of A(t),
f&k i(l-pO)ir
i(l
_
and the function fi(t) defined by
~o),f+lA(()))
is the random force acting on the variable A(t) and is orthogonal
(A(O)I
= 0.
(3.3)
to A(O), i.e.,
Y. P. Kalmykov, J. McConnell
400
I Rotational diffusion and dielectric relaxation
i is the Liouville operator, IA(t)) = eii’IA(0))
i), is the projection rule
,
(3.4)
operator which acts on variables R and S according to the
(Rl$olS> = (RlA(O))(A(O)IA(O))-‘(A(O)IS) 7
(3.5)
(R(S) denotes the scalar product of the two vectors (RIand IS) in the Hilbert space of dynamic variables and ( AJB ) = (AB ) by definition [12]. It follows from (3.1) that the correlation function C,(t) defined by
C,(f) = (A(O)kW))&W)kWV)
(3.6)
satisfies I
$ C,(t) = iw,,C,(t)
-
I
G1(t - t’) C,(t’) dt’ .
(3.7)
0
The one-sided Fourier transformation C,(o)
=
of eq. (3.7) yields
1 (3.8)
- iw, + iw + &((w) *
Since G$(t) is the time correlation function of fi(t), it is immediately seen that Q,(t), which in (3.7) is a memory function, will also satisfy equations similar to (3.7) and (3.8) in the time domain and hence f
$ @1(t)= iw,G1(t)
-
I
@*(t - t’) Q1(t’) dt’ ,
(3.9)
0
$ c&(t) = iWZGz(t) -
1 Q3(t - t’) $(t’)
dt’
,
(3.10)
0
$ cDn((t)= iW,Qn((t) -
1 @n+l(t - t’) Qn(t’) dt’ .
(3.11)
Y. P. Kalmykov, .I. McConnell
In the frequency
I Rotational diffusion and dielectric relaxation
domain eqs. (3.8)-(3.11)
are similarly extended
to
1
C,(w) =
,
A1
iw-io,+
401
(3.12)
A2
iw-iw,+
...
iw - iw, +
4
iw - iw, + Gn+,(w)
where % =
~~(o)lf,(o))~(f,-l(o)lf,-l(o)) 7
(3.14)
(f,(o)lf,(t))~(f,-l(o)lf,-l(o)) 7
(3.15)
A, = (.L(O)l.L(O))KL1(O)lf,-1(O)) @n=
(3.13)
9
If,(t))
= i exp[i(l
- i), - P, -a..
- Pn_,)it]
x (1 - i), - P, - * * *Pn_,)ilfn_l(o))
)
Wnl~> = (Rlf,(O))(f,(O)lf”(O))-‘(f,(O)lS). It can be shown [16] that the quantities moments M,, defined by
A, are expressed
(3.16) (3.17) in terms of the
(3.18) and which appear in the expansion
C,(t) =
c
&
(-l)“M,,
n=l
(3.19)
as follows:
4=M2, A2 = (M, - M;) lM,
(3.20)
,
A3 = (M6M2 - M:)I(M,M, and so on. Eqs. (3.20)-(3.22)
(3.21) - M;)
,
(3.22)
allow us to evaluate A, since the quantities Mzn
402
Y.P. Kalmykov,
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/ Rotational
diffusion
can in principle be both evaluated theoretically For dielectric relaxation with
and dielectric
relaxation
and measured experimentally.
A(t) = u(t)
(3.23)
and C,(t) = C(t) = ( u(0) * 40)
(3.24)
7
where u(t) = p/p is the unit vector along the dipole vector ~1, wn = 0 for all 12 [16]. Hence, eq. (3.12) becomes 1
e(0) =
(3.25)
A1
io +
A2
iw +
..
iw +
-4
iw + Gn+,(w)
As already mentioned, all quantities A, in eq. (3.25) can in principle be evaluated theoretically. For example, A, can be calculated by the formula (ref. [I719 P. 184) (3.26) where I,, Z, and I, are the principal moments of inertia of a molecule, ux, uy and uz are the projections of the vector u onto the principal axes of inertia XYZ. It is of importance that A, is independent of intermolecular interactions. Moreover, the quantities A,, can be determined experimentally from eqs. (3.20)-(3.22) and the following sum rules [17]: (3.27)
However, only M, and M, can actually accuracy [ 171.
4. Memory
function
approach
be measured
to the J-diffusion
with a satisfactory
model
There are at least three equivalent approaches to the mathematical formulation of the Z-diffusion model, viz. the kinetic equation [1,2,7], analytical
Y.P. Kalmykov, J. McConnell
I Rotational diffusion and dielectric relaxation
403
dynamics [4] and memory function [4] approaches. The kinetic equation and analytical dynamics approaches are straightforward but rather tedious. The much more compact exposition is afforded by the use of the memory function formalism [4]. In the framework of the memory function approach the J-diffusion model may be formulated as follows [4]. The memory function Q+((t) from eq. (3.3) can be considered to be the correlation function for the random torques which act on the molecule. In the extended diffusion picture where the molecules are assumed to rotate freely between collisions which change the angular momenta, the memory function for a molecule which has experienced no collisions is to be the free rotor memory function @r”(t) which corresponds to the dipole autocorrelation function C”“(t) and the memory function for molecules which have experienced collisions is to be zero. Then the memory function Q1(t) for the ensemble is given by Q1(t) = @y(t)
e-“’ ,
(4.1)
since the times between collisions are assumed to follow a Poisson distribution [4]. It is readily deduced from eq. (4.1) that &i(w) = @“(o The function
p+))
-i/7).
&p(z)
(4.2)
can be evaluated
by employing (3.25) to write down
l
=
io + S;“(w)
(4.3)
’
from which it is deduced that
CiqyoJ)=
1 - ifOEFR(W)
CFR(0)
(4.4)
.
We deduce from (4.2) and (4.4) that ~)l(w)
= ~~“
_
i/7)
=
(w ’ - i(w- i’T)cwFR 2;““(,
- i/7)
-i/7) ’
(4.5)
Hence C(w) =
l, iw + al(o)
which is just eq. (2.6).
eFR(w-i/r) = 1 - E:““(w -i/7)/7
(4.6)
404
Y. P. Kalmykov, J. McConnell
1 Rotational diffusion and dielectric relaxation
We may introduce a set of memory functions &FR(t), $yR(t), . . . associated with C”“(t) and defined by
(4.7) (4.8)
(4.9) where A? 1s . the value at time t = 0 of G’,‘“(t). It should be noted that AyR = A,, for it = 1 only, since only A, is independent of intermolecular interactions. These equations will provide equations for the memory functions of J-diffusion at any level. Thus for the second level we obtain from (4.5) and (4.8)
&(w) = @“(LO-
(4.10)
i/r) + 7-l ,
and this leads to C&(t) = e -t’T@;R(t)
+
2S(t)/r
.
(4.11)
Here we have taken into account that AyR = A,. One can show that both (4.10) and (4.11) lead to eqs. (4.6). The appearance of the S-function in eq. (4.11) is the consequence of the assumption of instantaneous collisions which implies the occurrence of infinitely strong torques during the collisions. The finite duration of collisions can be taken into account by the same manner as was done in ref. [LX];that is to say, we introduce a model memory function -lhc
C&(t) = e -“‘~;“p)
+ %
77, ’
(4.12)
where T, is the collision time and lim L e-“‘c = 26(t) .
r,-0 Tc
(4.13)
It should be noted that the semi-classical version of a similar model, in which
Y.P. Kalmykov,
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I Rotational diffusion and dielectric relaxation
405
the molecular rotation during the diffusive steps is described quantummechanically, has been proposed in ref. [18] to explain the rotational spectrum of atmospheric water vapour. From eq. (3.25) iw + &(w)
1
z;(o)= io +
= A1 - w2 + k&(w) A* iw + C&(o)
(4.14) ’
where according to eq. (4.12) (4.15) We substitute eq. (4.15) into the numerator of eq. (4.14). Thus iw(w -i/r,)
and denominator
of the last term
- ilrr, + (6~ - i/7,)&)2FR(6J-i/r)
c(w) = (A1 - w’)(w - i/r,) + w/rr, + iw(o - i/r,)&F(o
- i/T) ’ (4.16)
where the quantity AI is given by eq. (3.26). Substituting eq. (4.16) into eq. (2.4) we obtain
E(W)-
Al
EC?2
E, - E,
= (A, - o*) + iw[T(l
+ OK-,)]-’ + io&yR(o
- i/T) ’
(4.17)
Eqs. (4.16) and (4.17) are quite general and are valid for any model of a polar molecule, viz. planar, linear, spherical, symmetric and asymmetric top molecules. We need only to evaluate the Fourier-Laplace transforms &F(w - i/T). According to eqs. (4.7) and (4.8) the function $,,““(oJ -ir) in (4.17) can be expressed in terms of ?‘“( w - i/T) by the following equation:
E:‘“(aJ
-
i/T)
1
=
io +
7-l
(4.18)
AI
+ iW
+
7-l
+
@“(w
-i/T)
Hence GFR(o _ i,T) = A,cFR(o 2
-i/T)
- i(w - i/r)[l
- i(w - i/T)eFR(o
- i/T)]
1 - i(w - i/7)CFR(W - i/r)
(4.19)
Y.P. Kalmykov, J. McConnell
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I Rotational diffusion and dielectric relaxation
According to eq. (4.17) the complex permittivity depends on the two model parameters, viz. angular momentum relaxation time r and the mean duration of collisions rc. The time r can be measured in NMR relaxation and/or infrared experiments (for details see ref. [8]). The other model parameter TVcan be evaluated as follows. We have from eq. (4.12) CD*(O)=
@y(O) + l/77,
)
(4.20)
or AZ = ApR + ~/TT, .
(4.21)
On the other hand A2 is given by eq. (3.21), where for molecules of the C,, or C,, symmetry [16] M, =
kBT(z;’ + zyl)
(4.22)
and
(4.23)
where I, and I, are the principal moments of inertia about the axes perpendicular to the direction of the dipole vector cr_,I, is the third principal moment of inertia, N, and NY are the components of the torque N acting on the molecule. Thus (N;) (N9 kBTZX(l + IX/Z,) + k,TZ,(l -t Z,lZ,)
(rrJ’=A2-Ay=
*
(4.24)
Hence @J-l
=
7
k, T
(
(Ni)
0%)
Z,(l -t- IX/Z,) + Z,( 1 + Z,lZ,) > ’
(4.25)
Since both r and A, - AFR can be measured experimentally [8,16], eq. (4.25) allows us to evaluate TV. We shall now consider only linear molecules. In this case Z, = I, = Z and Z, = 0, we have from eqs. (4.22) and (4.23)
Y. P. Kalmykov,
.I. McConnell
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I Rotational diffusion and dielectric relaxation
44, = 2k, TII
(4.26)
and (4.27)
(N*=N;+N;).
Hence taking into account eqs. (3.20) and (3.21) we obtain (4.28)
A, = 2k,TII, A2=I
2kaT + (N*) 2k,TI
(4.29)
’
Thus (g’
T(N2)
=
2k,TI
(4.30)
’
in accordance with the result of eq. (55) of ref. [8]. Furthermore the FourierLaplace transform of the autocorrelation function C’“(t) for linear molecules is given by [4]
C”“(o -
i/7) = iv2 e-Z2E,(-Z2)
where Z = ~(0 -i/T),
E,(x) =
exp(-0 Im
(4.31)
,
v = (I/2k,T)“’
,
(4.32)
dt
7
,
x
the exponential
E(O)-
50
E, - &Xx
integral function [19]. Whence 1
=
(1 - v2w2) + ion2[7(1 + iwr,)]-’
+ if_07j2SyR(W-i/r)
’
(4.33) where “FR
G2 (w - i/7) = iT_l
Z e-”
E,(-Z2)
- Z[l
+ Z2e-L2 E,(-Z’)]
1 + 2’ ewz2 E,(-
Z’)
(4.34)
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5. Discussion of results and comparison with experimental
data
On comparing eqs. (2.13) and (4.33) it is easy to show in the case of &F”(w - i/T) = 0 th a t eq. (4.33) coincides precisely with eq. (2.13) while in the limit r,+O eq. (4.33) reduces to the well-known equation (2.6). Moreover, eq. (4.12) allows us to obtain the condition of applicability of the BM model [8], viz.
(5.1) i.e. the BM model is valid only in the case when we can ignore the term exp(--t/T) @FR(t) in eq. (4.12). Figs. 1 and 2 illustrate a comparison of the absorption spectra a(~) VW&“(W)predicted by the both models for different values of the model parameters 7 and TV.One may see by inspection of figs. 1 and 2 that when eq. (5.1) is valid both models lead practically to the same result (curves 3 and 3’). However when eq. (5.1) is not satisfied, these models predict an essential difference in the calculated spectra. To test our model we consider dilute solutions of small polar molecules HCl and DC1 in nonpolar solvents. Datta and Barrow [20], Birnbaum and Ho [21], Morita et al. [22] and others found a similarity between the far infrared spectra of small molecules like HCl, DCl, HF, NH, and H,O dissolved in nonpolar solvents and those in the gas phase. This highlights the importance of inertial effects for such systems.
Fig. 1. Comparison of r)w”(w)/( E, - cl) as a function of 07 for the BM and present models at different values of the model parameters: T/T, = 1.0 and V/T = 0.5 (curves 1 and l’), V/T = 2.0 (curves 2 and 2’), q/r = 10.0 (curves 3 and 3’). The curves numbered 1, 2, 3 refer to the BM model.
Y. P, Kalmykov, J. McConnell
I Rotational diffusion and dielectric relaxation
409
Fig. 2. The same as in fig. 1 for T/T = 2.0 and q/rC = 0.5 (curves 1 and l’), V/T, = 1.0 (curves 2 and 2’), qhC = 10.0 (curves 3 and 3’).
Morita et al. [22] have applied the free rotational model, which in fact is a particular case of our model when 7 + 00, to calculate the far infrared spectra of small molecules in nonpolar solvents. It was found that for linear molecules reasonable values for frequencies w,,, at which the maximum absorption occur can be calculated as predicted by the free rotational model [22] to be 0 max
= (3k,T/z)“*.
(5.2)
For HCl in several nonpolar solvents this was also supported by the linear relationship between w,,,,~ and T”*, which is given by eq. (5.2). Furthermore, shapes of theoretical spectra were in good agreement with those observed [22]. This agreement of the experimental and theoretical spectra shows clearly the importance of inertial effects. However the free rotational model could not explain the distinctive feature of the far infrared spectra of light molecules, namely a high frequency shift A%,, of the maximum absorption frequency w,,, predicted by eq. (5.2). For example, eq. (5.2) gives v,,, = 114cm-’ (v,,, = w,,,,,/~ITc) for HCl in cyclohexane at 298 K, while the observed value of v,,,,, is 125 cm-’ [22]. This is due to the fact that the free rotational model ignores intermolecular interactions completely. We apply our model in order to explain the above features of the far infrared spectra of light molecules. It should be noted that both the J-diffusion model and BM model have failed to explain far infrared spectra of such systems. The J-diffusion model predicts a negative frequency shift AU,,,,, while the BM model cannot describe the shape of the spectra (see fig. 1).
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I Rotational diffusion and dielectric relaxation
The absorption coefficient a(w) is given by eq. (2.3). For the dilute solutions of polar liquids in nonpolar solvents which we shall consider, the variation of the refractive index n(w) for the range of o with which we are concerned is small and so n(u) may be taken to be constant. Then we can evaluate CZ(W) using only values of the temperature T and moments of inertia and fitting the parameters r and TV,provided that E, - E, is known. In principle the calculations are capable of predicting the magnitude as well as the shape of the absorption band, but this requires that both E, and E, be known with a high degree of accuracy, since their difference is the important factor [22]. However, such data were not available. We shall take experimental data concerning HCl in cyclohexane at 298 K [22], HCl in Ccl, at 293 K [23] and DC1 in SF, at 273 K [21]. The values of the moments of inertia of HCl and DC1 are respectively 2.68 x 10P4’ and 5.19 X 10e4’ g cm2 [22]. For these solutions the theoretical absorption coefficient a( V) has been calculated using eqs. (2.3), (4.31)-(4.34). The values of the model parameters r and 7c have been chosen from the best fit of the theoretical and experimental profiles provided that the observed and calculated maximum absorption frequencies vmaXshould coincide. In order to compare the theoretical spectra with the observed profiles, the absorption maxima of the theoretical curves are set equal to those of the experimental ones. The theoretical and experimental curves are compared in figs. 3-5, where the spectra predicted by the free rotational model are also shown in comparison. In the foregoing sections a model of molecular rotation in liquids has been
Fig. 3. Comparison of the theoretical absorption coefficient CX(V)for HCl in cyclohexane (curve 1) with the observed profile (filled circles) at 298 K from Morita et al. [22] and with that for the free rotational model (curve 2). The best fit is at T = 0.115 x lo-” s and T, = 0.357 x lo-” s.
Y. P. Kalmykov, J. McConnell I Rotational diffusion and dielectric relaxation
411
Fig. 4. Comparison of the theoretical absorption cross section cr(v)IN for HCl in CCI, (curve 1) with the observed profile (filled circles) at 293 K from Ohkubo et al. [23] and with that for the free rotational model (curve 2). The best fit is at T = 0.689 X lo-l3 s and r, = 0.443 X lo-” s.
0 frequency
~r/cm)
a
Fig. 5. Comparison of the theoretical absorption coefficient a(v) for DCI in SF, (curve 1) with the observed profile (stars) at 273 K from Bimbaum and Ho 1211and with that for the free rotational model (curve 2). The best fit is at T = 0.224 X lo-‘* s and rc = 0.395 x lo-l3 s.
considered. The model is a generalization of the Gordon J-diffusion and BM models. It is applicable to molecules of an arbitrary symmetry. This generalization allows account to be taken of (i) a finite time of molecule collisions and (ii) the inertial effects. This permits us to apply the model to far infrared absorption of small polar molecules in nonpolar solvents where the both factors are of importance. The values of the model parameters r and TVobtained are reasonable and are in agreement with the previous reports [7,8,16].
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Acknowledgements Support of this work by the Dublin Institute for Advanced Studies and the Institute of Radio Engineering and Electronics of the Russian Academy of Sciences is gratefully acknowledged.
References [l] [2] [3] [4] [S]
E.P. Gross, J. Chem. Phys. 23 (1955) 1415. R.A. Sack, Proc. Phys. Sot. London B 70 (1957) 402, 414. R.G. Gordon, J. Chem. Phys. 44 (1966) 1830. R.E.D. McClung, Adv. Mol. Rel. Int. Proc. 10 (1977) 83. C.J. Reid, R.A. Yadav, G.J. Evans, M.W. Evans and G.J. Davies, J. Chem. Sot. Faraday Trans. 2 74 (1978) 2143. [6] A. Gerschel, C. Brot, I. Dimicoli and A. Riou, Mol. Phys. 33 (1977) 527. [7] V.I. Gaiduk and Yu.P. Kalmykov, J. Mol. Liquids 34 (1987) 1. [8] AI. Burshtein and J. McConnell, Physica A 157 (1989) 933, to be quoted as BM. [9] H. Mori, Progr. Theor. Phys. 33 (1965) 424. [lo] H. Mori, Progr. Theor. Phys. 34 (1965) 399. [ll] B.J. Berne, J.P. Boon and S.A. Rice, J. Chem. Phys. 45 (1966) 1086. [12] D. Forster, Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions (Benjamin, New York, 1975). [13] J. McConnell, Rotational Brownian Motion and Dielectric Theory (Academic Press, New York, 1980). [14] M.W. Evans, G.J. Evans, W.T. Coffey and P. Grigolini, Molecular Dynamics and Theory of Broad Band Spectroscopy (Wiley, New York, 1982). [15] A.I. Burshtein and S.I. Temkin, Spectroscopy of Molecular Rotation in Gases and Liquids (Nauka, Novosibirsk, 1982) [in Russian]. [16] S. Ikawa, K. Sato and M. Kimura, Chem. Phys. 47 (1980) 65. 1171 C.J.F. Bottcher and P. Bordewijk, Theory of Electric Polarization, vol. 2, Dielectrics in Time-Dependent Fields (Elsevier, Amsterdam, 1978). [18] Yu.P. Kalmykov and S.V. Titov, Radiotekh. Elektron. 34 (1989) 13. [19] M. Abramoviz and I.A. Stegun, eds., Handbook of Mathematical Functions, NBS Series, vol. 55 (National Bureau of Standards, New York, 1964). [20] P. Datta and G.M. Barrow, J. Chem. Phys. 43 (1965) 2137. [21] G. Birnbaum and W. Ho, Chem. Phys. Lett. 5 (1970) 334. [22] A. Morita, S. Walker and J.H. Calderwood, J. Phys. D 9 (1976) 2485. [23] Y. Ohkubo, S. Ikawa and M. Kimura, Chem. Phys. Lett. 43 (1976) 138.