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On the separation of long-range and short-range dipole correlations in polar fluids
Chemical Physics 33 (1978) 451464 0 Xorth-Holland Publishing Company
ON THE SEPARATION OF LONG-RANGE AND SHORT-RANGE DIPOLE CORRELATIONS IN POLAR FLUIDS
P. BORDEWIJK Gonku~ Laboratories of the Universi~, Department GfPhysical Chemistry, Leyden. The Net’Ierlands Received 30 December 1977
Revised manuscript received 7 June 1978
A new method for the separation of long-range and short-range dipole correlations in polar fleids with pokimble molecules is applied to the correlation functions connected with the static dielectic permittivity, the frequency-dependent Permittivity, and the electrically induced alignment. The method avoids the division of the sample by a distinct boundary into a region considered on a microscopic basis and a surrounding dielectric contikm, and yields rigorous resnhs for a system with identical isotropic polarisabilities on a cubic lattice, among which the E&wood-Friihhch equation for the static permittivity. in other cases it makes sense to replace the dipole moment of the isolated molecule occurring in this equation by an effective value, for which expressions applicable under different conditions are suggested
1. Introduction Information about local order in polar fluids can be obtained f;om the response to a static electric field. This was fast observed by Kirkwood [l], who related the dielectric permittivity to the averageofor an unperturbed spherical sample, M denoting the instantaneous moment of the sphere and M the magnitude of this quarttity- It was pointed out that to obtain information about the local order, this average should be split in terms due POlong-range and short-range dipole correlation. The same holds for the time-dependent function (M(O) -.M(f))o, which is related to the frequency-dependent permittivity G?(W)[2], the function (3(e, *IV)* -M2jo, which plays a role in the theory of the Kerr effect,and of the electrically induced NMR line splitting in polar liquids [3/t] (er denotes the unit vector along the permanent dipole of molecule l), and the function 3(M4jo - 5(M2)& which plays-a role in the dielectric saturation [5]. In performing the separation between long-range and short-range dipole correlations, two problems may arise. One of these concerns the contribution to the averages mentioned by the induced moments in the molecules, which are fluctuating quantities that depend on the positions and orientations of all other molecules. Apart from restricting oneself to a hypothetical system ofnon~polarizablemolecules [6,7];achievements have been made to account for the molecular polarizabilities on a continuum basis, and by using for the induced moment the value from Onsager’s model [8], iu which the surroundings of a single molecule are represented by a dielectric continuum, thus ignoring the fluctuations of these moments. The fast method was introduced by Friihlich [9] to account for the polarizabilities in the calculation of the static permittivity, and was also applied with respect to the dielectric saturation [lo-121 ; this method is very attractive from an intuitive point of view, but the approximations involved stay somewhat obscure. The second method has raised much controversy with respect to the stage of the derivation where the approximation is allowed, yielding conflicting expressions both for the static permittivity [1,3,13-181 and for the frequency-dependent permittivity [19-211. The other problem concerns the introduction in the dielectric of a distinct boundary between a small sphere considered on a microscopic basis and a surrounding dielectric continuum with the macroscopic properties of the sampIe. Such a boundary has been introduced in calculations on the static electric permittivity [9,22], the fre-
452
P. BordewijklSeparation
of dipole correlations in polarfluids
quency-dependent permittivity [2,6,7,23-251, the alignment [3], and the dielectric saturation [lo-121. In the case of the frequency-dependent permittivity, the time-dependent behaviour of the virtual surface charge at the boundary has been the object of much discussion. Moreover, the method ignores short-range correlations between molecules atboth sides of the boundary, which is not a priori justified, as pointed out by Mandel [26]. In this article the averages uM2)u, (M(0) . M(r))u, and (3(el •LU)~ -M2) will be calculated by a method which avoids either of the assumptions mentioned concerning the induced polarization, and also avoids the division of the sample into a small sphere considered on a microscopic basis and a surrounding continuous dielectric. It appears that for a system with isotropic polarizabihties on a cubic lattice the Kirkwood-FrChlich equation, the expression for the complex permittivity derived by Klug et al. [ 191 and by Rivail [20], and the expression for the alignment derived by Rarnshaw et al. [3] are rigorous results, whereas in other cases a good approximation can be obtained by replacing the dipole moment of the isolated molecule in these expressions by an effective value.
2. The relationship between the instantaneous moment and the polarizabilities As a fmt step, we will derive an expression for the instantaneous morrientM of a spherical system of n polarizable dipoles. We assume that the dipole moment of each molecule is completely determined by its orientation and the electric field working on it, i.e., we ignore fhrctuations of the atomic and electronic polarization. These dipole moments can then be written as: mi=pi+ai-
where mi, pi, andai are the total moment, the permanent dipole moment, and the polarizability tensor of molecule i, E. denotes the external field, and Tji is the dipole-dipole interaction tensor [27] _ Eq. (1) can be written in Sn-dimensional notation [Z&27, sec. 361 m=R+a-(e&-T-m),
(2)
where m, p, and e are 3ndimensional vectors whose projections on the three-dimensional subspace i are equal to mi,ti, and the unit vector e ilong the external field E,,, respectively, and czand T are 3ndimensional tensors with projections on the product of subspaces i andi equal to ar Sij and Tii(l - Q), Sii being the Kronecker delta. In contrast to previous articles [22,29,30] in this article we consider a sphere in vacua, so that no contribution due to the reaction field should be incorporated. From eq. (2) it follows: m=(1+a-T)-~-(~+a~eEg)=(l+a-T)-1~~+a~(1+T~a)-l~eEO,
(3)
where the last equality can be proved by developing (1 + a - T)-1 and (1 + T - a)-’ into a series. In eq. (3), 1 denotes the 3n-dimensional unit tensor. With A = (1 + T * a)-l, eq. (2) yields iu three-dimensional notation:
where it is used that (1 + u . T)-l is the transpose of (1 + T - a)-l, and Aii denotes the projection ofA on the product of subspaces i andj. From eq. (4) it follows that Zi Aii is a three-dimensional tensor transforming the external field to the part of the local field at molecule i that depends on the external field E,. This tensor is independent of the permanent dipoles; we wilI cah ii the local-field tensor. In the absence of an external field, eq. (4) yields for the instantaneous moment M:
P. BordewijklSeparation of dipole correlations in polarflrtids
453
where we call
the effective moment of molecule i. This effective moment consists of the permanent dipole moment of molecule iand the moments induced directly or indirectly by this moment in all molecules. In principle, the effective moment, like the moment mi, is a fluctuating quantity that depends on the positions and orientations of all other molecules. In contrast to mi, however, for isotropic polarlzabllities the effective moment does not depend on the orientations of the other molecules. This makes the neglect of fluctuations of the effective moment for given orientation of the molecule, which will be made in the following, a less crude approximation than the neglect of fluctuations OfITZr,which is made in other derivations of the Kirkwood-Friihlich equation. The approximation made here is equivalent with the neglect of fluctuations in the internal field for given orientation of the molecule, when studying the dielectric constant of non-polar systems or the dielectric constant at optical frequencies. Fluctuations of the effective moment still occur due to the translational freedom in liquids, however, and for molecules with non-isotropic polarizabilities due to the orientational freedom; these fluctuations contribute to the SOcalled induced-dipole absorption in the far-infrared frequency range [31]. Eqa.(5) and(6) are valid both for systems of identical particles and for mixtures. In the former case, the average local field tensor will be the same for all molecules, In the second it will be different for different components in the mixture. Since the equations also keep their validity if some molecules do not have a permanent dipole, they can also be applied to solutions of polar compounds in non-polar solvents. For identical molecules where the anisotropy of the polarizability and of the shape can be ignored, the dipoles induced directly by the external field are equal, and their contributions to the local field at a certain molecule cancel after averaging, so that when translational fluctuations are ignored, or the molecules are situated on a cubic lattice, the tensor Xj Aii reduces to the unit tensor. This does not hold for molecules which are near to the boundary of the sample, but for a macroscopic sphere this regards only a negligible fraction of the molecules. It follows that for identical molecules where the anisotropy of the polarizability and of the shape can be ignored, the effective moments are equal to the permanent dipole moments of the isolated moIecules. This holds rigorously if the molecules are situated on a cubic lattice. In other cases, more drastic approximations should be made. If the molecules are not identical, the neglect of translational fluctuations is an approximation even if the molecules are situated on a cubic lattice, and if the anisotropy of the molecular shape and of the polarizabillties cannot be ignored, the neglect of fluctuations of the effective moment involves not only a neglect of translational fluctuations, but also ofcertain orientational fluctuations. Making this approximation, the tensor Xi Aij can be estimated from the model of a molecule in a cavity in a dielectric continuum with permittivity equa1 to the dielectric constant of induced polarization E, of the system under Investigation. It is known that such a model may yield results different from a corresponding model with point polarizablhties [32], but this does not necessarily invalidate the continuum model, since, in a way, it accounts for the extensiveness of the polarizabilities, which is ignored in the model of point polar&abilities. In fact, for solids of elongated molecules the point-polarizability approach was shown to be inapplicable [33]. Assuming an ellipsoidal shape of the cavity, with the dipole along one of the axes, and the induced dipole distributed homogeneously over.the cavity, one obtains the Beld E, working on the molecule, from the cavity field&:
(7) where eqs. (2.84), (4.100) and (2.71) from ref. [27] are used. In eq. (7) cr3 denotes the component of the polarizability along the dipole, Q, b, and c are the half axes of the
’
454
P. BordewdkjSeparation of dipole correlations in polar jhids
~chfpsoidal cavity, and A is a shape factor. From eq. (7) it follows: ‘*=‘(e_,
3% +2){e,__+(1-&4[1+3~~(1-&~~c]}’
(8)
If the dipole is not along one of the axes of the ellipsoid, the components of the effective moment along the different axes should be calculated separateIy with eq. (S), using the appropriate value of the polar&ability and the shape factor for each axis. Although in that case the effective moment has not the same direction as the moment of the isolated molecule, it is still fiied in molecular coordinates. Since in eq. (8) the polarizabihty is thought to be distributed over the ellipsoid, it can be related to an internal dielectric constant Ei,3 by: ~~ ‘U~C(Ei,3 - 1)/3 [I + A(Ei,J - l)] ,
(9)
yielding P* = 3em ]l +A(ei,3 - l)ll(eca + 2)]eca +A(ei,3 - eta)] * ,
(10)
For dilute solutions, E, may be taken equal to e, and eq. (10) changes into an expression for the moment in solution derived by Ross and Sack [34], if the internal dielectric constant Ei,3 is taken to be isotropic and equal to the square of the refractive index of the pure solute. The assumption of an isotropic internal dielectric constant seems to underestimate the anisotropy of the polarizability, however. For, if in the crystal the moIecule can be represented by an ellipsoid with the same shape as io the liquid, filling up a cavity with volume equal to the volume available to each molecule, the assumption of an isotropic internal dielectric constant yields an isotropic dielectric constant of the crystal with the same magnitude, from which it follows that the anisotropy of the dielectric constant of the crystal is ignored in this way. Therefore, for compounds crystallizing with all molecular axes parallel; it should be preferred to take the internal dielectric constant equal to the dielectric constant of the crystal in the corresponding direction. If apart from the anisotropy of the dielectric constant, also the shape anisotropy cm be ignored, one has A = :, and eq. (10) changes into cr* = [3E,(fi+ 2)I(E, c 2)(2~~ f Ei)] 12.
(11)
For dilute solutions, this corresponds with the ratio between the apparent moment and the moment of the isolated molecule given by Onsager [a]. For the pure compound, E, and Ei should be taken equal, and the effective moment is equal to the moment of the isolated molecule, as for isotropic point polarizabilities on a cubic lattice. A different relationship between the internal field and the Maxwell field has been shown to be applicable for compounds forming a liquid crystalline phase [35] : Ei=K-E,
.W)
with KgJ
=
(1 - 47rN&)-’
,
(13)
where Qand t refer to the principal values along and transverse to the long axis of the molecule. In this way one obtains for the components of the effective moment. P;,t = 3Pp,&o
+ 2)(1 - 47n%,,~,,t)
-
(14)
The effective moment p* should be distinguished from the moment pd used in ref. [30]. With eq. (9), eq. (14) can be rewritten to: P& = ]3Pa,tI(e~ + 211[I -AQ,,+A&~(EiA?,J -
Thisis identical
to eq. (10) for fi = E,, but not for other cases.
(1%
P. BoniewijkSeparationof dipolecon-elations in polar fluids 3.
The Kirkwood-Ftihlich
455
equation
The average value of the total moment of a sphere with radius a in an external field can be written in terms of the dielectric permittivity, but can also be obtained from eq. (4):
where only effects linear in the applied electric field are considered. If in the last member of eq. (16) the permanent dipoles are ignored, the remaining term is related in an analogous way to the dielectric constant of induced polarization
3@ - CJ @ + 2k
Q3&-, = c
+ 2)
i
($
lE
(18)
,
or [28;27, sec. 361
where U= U. -Mm E. f O(E$ is the energy of the system. Eq. (19) shows the relationship between the permittivity and the average Gl@>o . For a further evaluation, we consider the average value of the moment M for. fmed orientation of mplecule 1, which can be written, if only one polar compound is present, as:
where el denotes the unit vector along the effective moment of molecule 1, r12 the angle between the effective moments of molecules 1 and 2, and n the number of polar molecules. To obtain from the average (WeI) an expression accounting for local order only, it should be reduced to: (21) vIR ‘,m
ri~R
where ri is the distance from molecule i to molecule 1, and the super index m in the last member indicates that tire average is taken over a macroscopic sphere (with radius R) surrounded by an infinite sample of the same composition. The symbol g stands for the Kirkwood correlation factor defmed by:
(22) To perform the reduction, we start from the average potential outside the sphere, for fmed orientation and position of molecule 1, using polar coordinates with the origin at the centre of the sphere and the z axis parallel with el :
n (~llel>o
= U.flel>O
cos e/r2 + n12
rP_l
C
m=-n
B~m)P~m)(cos
f3) eim@ ,
(23)
456
P. BordewijkjSepararion
of dipole >orrelations in polar fluids
where the terms in Bfm) account for the higher electric moments, PLm)(cos 0) denoting the associated Legendre polp~tials. In eq_ ($3) it is assumed that the average dipole moment of the sphere is independent of the position of the molecule, which seems appropriate as long as molecule 1 is not near the boundary. The constraint imposed at the averaging, i.e., the fixed orientation and position of molecule 1, does not affect the behaviour of the sample at the boundary as a dielectric continuum, except if molecule 1 is kept near the boundary, which case yields a negligible contribution to ( p=iM), however. Therefore, at the boundary the electric potential and the normal component of the dielectric displacement should be continuous_ From this it follows for the average potential inside the cavity [21]
(24) The terms in negative power of r are related to a field with its sources inside the region of the sphere that may not be considered as a dielectric continuum due to the constraint_ This field vanishes after averaging outside a small sphere around moIecuIe 1 which contains the region where it has its sources. The term in r accounts for a homogeneous field -2(~ - 1)(Mlet)/3s03, the field corresponding with the terms in higher powers ofr vanish after averaging over the sphere, due to the orthogonality of the associated Legendre polynomials. The average vaIue of p* for given value of Ecan be inferred from eq. (18):
(25) One thus obtains from the homogeneous field corresponding with the second term of eq. (24), for molecules far from molecule 1, i.e., in the region that can be considered as a dielectric continuum: lim C$ leljo = -[2(15 - I)(E - e,)/3e(e,
rp-
+ 2)nl LWel)o .
(26)
Subtracting these contributions from (Mlet),,, one obtains: g$
= Wei)
+
(2c + E,)(E + 2) 2(E - i)(e - E,) 3E(E, + 2) (Mlel)o = 3E(E, + 2) (“lel+l
’
(27)
or, for an isotropic system:
= [3e(e- + 2)/(2E + E,)(E + 2)]g(rz*)2 -
(28)
Substituting this into the penultimate member of eq. (19), one obtains a generalization of the Kirkwood-Friihlich equation: (E - EJ(2E + E,)/E(E, +.2)2 = (4rrN9kT) g(JP
,
(29)
where the moment of the isolated molecule p has been replaced by the effective moment 0. In eq. (29) N= 3n/4ruz3 denotes the number density. For identical molecules with isotropic shape and polarizability the neglect of translational fluctuations implies that the effective moment is equal to the moment of the isolated molecule, and eq. (29) reduces to the prigind Kirkwood-FriWch equation: (e - fJ(2E
+ eJle(e,
+ 2)2 = (47rN/9kT) 9112.
For other cases, however, eq. (10) or eq,(ll)
may be preferred, yieldivg respectively:
(30)
P. BordewijklSparation of dipole correhtions in polar
E~)(~Ef E,) [em +A(Q
(E -
fluids
- cm)] 2/ee$ [l f d(ei,s - l)] 2 = (4dV/kT)gp2 9
(~ - f,)(2E + E,)(2~~ + ~i)2/~P(~i+ 2)2 = (4~~/~~)g~2 .
451
(31) (32)
For compounds forming a liquid-crystalline phase, and where the dipole is along the long axis, like for the p-alkylcyanobiphenyls, eq. (14) yields: (e - E,)(2E + e&(1
- 47rN+4,)2
= (47rN/kT) g$
)
(33)
as is also obtained from eq. (22) in ref. [30] by taking S = 0.
4. The frequency dependent permittivity The moment induced by an external field with harmonic time dependence, written in complex notation as: = e El exp(iwf) ,
&f)
(34)
is given by: (35) It follows: (3 [+I)
- e,]&(o)
+
2](c_ + 2)la3io(t) = n@ 1&j .
(36)
The right-hand side can be rewritten with statistical mechanics of linear dissipative systems [36] :
(I$&*> =(&J =- 1
t
3kT s
-05
+F
1
t dt’&(t’)((lr;(f) $ -ca
dT)&,(t))(&
- e)(h(r’) * e)$,
- ni(t’)>, =&&,(&J~f(0)
l
M(t)$ ,
(37)
where ~iiw denotes the Laplace transform. It follows (38) This is equivalent with the starting-point of the calculation by Glarum [2]. For w = 0, eq. (38) reduces to eq. (19). TO split up the average in eq. (38) into contributions due to short-range and long-range dipole correlation, we have to calculate the potential (Gz(r, t)]el(0)&. To this end, we should know the time-dependent equivalent of the relation U)(r)]el),, = e(E(~)]et$, used in the derivation of eq. (24). This relation yields fort = 0: (D(r, O)lel(0)J = dE(r, O)lq(O)& .
(39)
This implies that in the region that can be considered as a dielectric continuum, the distribution of orientations at 1= 0 is in equilibrium with the field E(r, 0). Therefore, in the case that the time dependence of the field E were such that this field would disappear immediately after t = 0, the dielectric displacement would be given by: W,
r)let(Wo
= eW,
O)lel(O)WW
,
(40)
where, by defmition, F(t) is the step-response function of the dielectric displacement. In reality, however, the field E does not disappear instantaneously, and the behaviour of E(t’) between t’ = 0 and r’ = t also contributes
458
P. BordewijklSepamtion
of dipole correlations in polar fluids
$0 D(t), yielding:
(D(r, t)lel(0)lo = dE(r, O)le,(0)>oF(t)
+j
dr’L??(r, t)lel(0)>,,f(r
- t’),
t>o,
(41)
0
where the superposition principle is used, from which it also follows that the pulse-response function, f(t) =-i(t), is related to the complex permittivity by: p(o) =ePi,(f)
>
=E~~(.f)
(42)
FF,standing for the bilateral Fourier transform: Yw(f) = 7 dff(f) exp(-iwr) . _m
(43)
In eq. (42) the bilateral Fourier transform may be used since f(t) is a causal function [f(t) = 0 for t < 01. We now introduce a time-dependent field B’(f) by: fd0
E’(t) = 6tio (E(r, O)lel(0)>oe6f, -f =W~,t)lel&9)o,
;
t>O.
WI
tz=o.
(45)
Eq. (41) can then be rewritten to: (D(r, t)lel(0)>o = E 7 dt’E’(r, t’)f(f - t’), -a
It appears from eq.445) that the dielectric displacement can be written as a convolution of the field E’ defmed in eq. (44) with the pulse-response function, and not of the field (Efr, f)lel(0)>o. This is due to the fact that for t = 0 one should.use the equilibrium distribution function, as pointed out already by Kirkwood [ 11. Similar problems arise with respect to the reaction field [37,38], where also the average value at t = 0 should be calculated from the equilibrium distriiution function [39]. Eq. (45) can be extended to values t< 0 by replacing the left-hand side by a field D’(t) defined in analogy with E’(& Taking the Fourier transform of the resulting equation, and using the convolution property, one obtains: g:, CD’@,t)) = C(w) FU (E’(r, t)} .
(46)
It follows that a relation analogous with eq. (24) can be derived for each frequency component of the timedependent potential&(t) connected with the fieldE’(r, r) in the sphere. The resulting contribution to lin+_ $#)lel(0)> is then obtained by combining eqs. (37) and (38), and one obtains by subtracting the long-range contribution to (M(t)lel(0)): (47) where M’(f) andg’(?) stand for: M’(f) = she (M(0)lel(O))OeSr,
t
= Uf(~le~(0)~o,
t>O,
g’(t) = she ge6*, -t
t
= 1 + (n - l)kos(el(0)
l
el(t))$,
;
t> 0 .
(48)
(49)
i! Bordew~k/~eparation of dipole correhtions in pohu jluids
459
From eq. (47) a similar relation follows between k’(t) andg’(T), and since these derivatives are zero for t < 0, the double-sided Fourier transforms then reduce to Laplace transforms: 3(e, + 2)h(w) -@icdMt)
lel(‘)‘O
7 p(o)
+ E, J p(u)
f 21 -@iu
(50)
-
( ~Pj(41~1(0)),
With eq. (38) it then follows:
or, with eq. (29):
(52) For non-polarizable molecules (e_, = 1) this is identical to the result by Fatuzzo and Mason and by Titulaer and De&h; when ignoring the cross-correlation terms eq. (52) reduces to the result by Klug et al. 1191 and by RivaiI [20], usingg= 1.
5. The alignment
From eq. (5) it directly follows:
(3(el - M)* -
(p’)*$j & [3(q -
M*)o=
ei)(el - ek) - ei -
ek]>,=2(p*)*R~,
I
where R, ,is a macroscopic correlation factor, introduced by Kielich [lo] as the reduction factor for the Kerr effect, and defined by:
= 1 +(n - 1)(P2(COS’y12))o iFyV
+ $(n - l)(tZ -2)(COS~@OS~&
-:(n
- l)(n
- ~)(cos~~~)~
denotes the angle between the effective moments JI: and pf _R, canalsobe kitten RK=(3el~M+3(~-l)cos712eI~M--neI
-M)0/2~*=&~-l)(cos7~2e~
-Ml&*
,
(54)
as: -:(n-3)(el-M)/p*. (5%
To rewrite the first term we write the average moment of the sphere for fxed positions and orientations of molecules 1 and 2 as a sum of the average moments obtained if molecules 1 and 2 kept fured separately, and a correction term: (Mlel,e2, rl, r2jo = (Mlel)o + (Mle2)o + AM.
(56)
Substituting this into eq. (55) one obtains R, = $z - l)(cosr12el*
AM>,/r.c* +gB[&B
- 1) +gz] ,
(57)
where B stands for [cf. eq. (28)] : B = lim (elmM)/gp* = ~E(E, + 2)/(2~ + E,)(E + 2) ,
n*m
(5%
460 adg2
P. Bordewi~k&mmtion of dipole correlationsin polarfluids stands for the analogue of the Kirkwood correlation factor for the second Legendre polynomial: g2 =
f
c P,(cos7li) )o = 1+(n-l)w~(cos7& j
--
.
This correlation parameter contains no long-range contriiution. For E = E, one has 3 = 1; in that case the longrange dipole correlation also vanishes. The moment AM depends on the orientations and positions of molecules 1 and 2. We will make a distinction between the cases that the distance between molecules 1 and 2 is so large, that when molecules 1 and 2 are kept f=ed, there is nevertheless a region in between them that can be regarded as a dielectric continuum, and the case that the distance is too small for this. For the result it is irrelevant at which distance the distinction should be made, as long as it can be made at a distance that is small with respect to the size of the macroscopic sphere. In the second case it makes sense to defme a moment Miq by:
which depends on the positions and orientations of molecules 1 and 2, and decreases with r12. The moment of the macroscopic sphere is then given by: ~lel,e2,~l,~~~0
(61)
=g@*@l +++BM;z,
from which it foUows that AM for this case can be taken equal to BMiz. In spite of the decrease ofMi2 for large r12, it is not allowed to take AMequal to zero for the case that the separation between molecules 1 and 2 is so large that the region in between them can be considered as a dielectric continuum. T-his is due to the fact that for futed orientation of molecule 2, the long-range polarization connected with molecule 1 is slightly affected. Although this effect is only of order l/n, its contribution to R, is nevertheless of importance due to the factor (n - 1) in the first term of the second member of eq. (58). To calculate this effect, we consider the moment induced in the sphere by an external field, if one molecule is kept fiied: SdXn-lMexp(-UolkT+
M - EolkT)
IC
$dXnmlexp(-U,$kT
+M * EolkT)
-1 (62) I ,
where dX”-l points to an integration over all positions and orientations except one, and U. stands for the interaction energy of the molecules iu the absence of the field. In eq. (62) terms in the energy of the system in higher powers of E have been neglected, since they do not play a role in the following. From eq. (62) it fo!.lows:
(auflel)E/aEo)E=o
= (ahzjaml~o
+ {[(IWMI ello - (Mlel~o(Mlel~o] mel/kT
= taM/aElel>of {[WMl;l)o
-g2B2(p*)2elel]
l
e)/kT .
(63)
Neglecting the anisotropy of(aM/aEle,>,, it follows that the polarizabiity of the sphere under the constraint mentioned differs from the one without this constraint by: Aal = [(MMlel)o -g2B2(u*)2e1e1
- (MM),-,]/kT.
(64)
Now one has: (M-Mle,)o=(M~M)o
_
(65)
of dipole correlations in polar fluids
P. BordewijklSeparation
461
This makes (MMlel)o -0, e traceless tensor. Furthermore, this tensor has axial symmetry about el. Therefore, the tensor Aa is known, if the principal value of (MMlel)o - (MM), along e, is known. This value is given by: *MJJ = &.L*)2RK . el - ((MMleI)o - (MM)o) . e1 = &I* MM el>o - ‘W j
056)
l
It follows, if we denote the three-dimensional unit tensor by I Aa = (P*)~ [(RK - g2B2)e 11 e -;RKl]/kT.
(67)
The homogeneous field over the sphere for ftved orientation of molecule 1 (eq. (24)), corresponds with an external field given by: 2(E - l)(e_ + 2)
3kT(B - I)
033) e1 ’ tip* where the generalized Kirkwood-Froblich equation, eq. (29), was applied. The field given in eq. (68) polarizes the sphere. If molecule 2 is kept fixed, the moment induced by the field changes by an amount (E& - Aua. Analogously, the moment induced by the field corresponding with molecule 2 will change by an amount (E& . Aa,, if molecule 1 is kept fuced. One thus obtains for AM: 3a3(2e + e,)
AM=BM;2+(E&a
Aa2+(E&-
=BM& + [3(B - l)~*/nB](el
gfi**el =
Aal
+eZ)[(R~ -g2B2)cos~12
-$&I
.
(6%
Substituting this into eq. (58) ignoring terms in l/n, and taking (cos yL2) =O, (cos2yr2) =i, and (~0s~‘)‘~)
=
0, which again impties the ignorance of terms in l/n, one obtains: R, =gB[i(g - 1) + g2] + [(B - l)jB]RK + $B(n - l)(cos r12el - h&)/p*
,
(70)
or: R, =gB2&g-l)+g2]
+zB*(n -I)(cos~~~~~*M~~)/,!I*.
(71)
From this expression one obtains the local correlation parameter, to be denoted by GK, by takingB = 1: GK =g[:(g
- 1) fgz]
f $I - ~)(COS-r12el M;z)/p* l
(72)
Comparing this with eq. (71) one obtains: RK =B2G, .
(73)
6. Discussion
The present article shows that the instantaueous’moment of a dielectric sphere in the absence of an external electric tield can be written as a sum of effective moments of the MOlecUleS.These effective moments are related to the permanent dipole moments of the molecules by a tensor that depends on the polarizabilities of the molecules, but not on their permanent moments. This tensor is the same as the tensor transforming the external field into the field dependent part of the local field. Ignoring fluctuations of this tensor for given orientation of the
molecule, it can be calculated with a suitable model for the internal field at optical frequencies. For a system of identical, sphkrical molecules with isotropic polar&ability, the tensor then reduces to the unit tensor, and the effective moment is equal to the permanent dipole moment of the isolated molecule. It was further shown that the elimination of the long-range dipole correlation can be performed without mtroduc% a distinct boundary between a small sphere considered on a microscopic basis and a surrounding dielectric to be considered as a continuum. This Is of interest since in such a procedure the short-range interactions between mole-
462
P. BordewijkSeparation
of dipole com&tions in polarfluids
atboth sides of the boundary between the two regions are ignored, and it is not a priori clear if this is justified In the present derivation, however, the only assumption made is that far from a molecule whose position and orientation is kept fmed in the averaging, the sample behaves as a continuum, which seems an essential property of liquids- The result a posteriori justifies the neglect of short-range interactions between molecules at both s&es of the boundary in the calculation of (p,- &, (~~(0) - M(t)&,, and (3(el- &Q2 - M2 J-,.In avoiding the criticized introduction of a distinct boundary, our derivation agrees with the one by Kirkwood but it seems new as concerns the application to polarizable molecules. It should be pointed out that although the treatment eliminates long-range interactions that are essentially dipolar in origin, this does not imply that the remaining short-range correlations are completely non&polar in origin. On the contrary, it has been shown that also systems with only dipolar interactions between the molecules yield deviations from Onsager’s equation, corresponding with Kirkwood factors deviating from unity [40,41]. Apart from accounting for the anisotropy of the polarizability and of the molecular shape, our derivation also makes it possible to generalize the Kirkwood-Frijhlich equation to solutions, as shown by eqs. (3 1) and (32). in contrast to a previous derivation [27, sec. 401, the validity is not restricted to the case that the molecules form associates of a Iimited size. It seems worthwhile to compare the present result for spherical molecules with isotropic polarizability, eq. (32), with the expression derived previously:
c&s
@l(Ei - 1) I O Q(2e + eg) -3x1’o(2~t~i)
3x @o(eo- 1)
t3x
1’
@i(Eu-ei)+3A$eO-1 l#J 2c+l ’ ~(2E+Ei)
(74)
where $I=+,$Q +x~@~ t A@stands for the molar volume, and the indices 0 and 1 refer to tire non-polar solvent and the polar soIute, respectively. If in the expression within the brackets in the last member of eq. (74) E is replaced by e-, it should yield zero. Subtracting the resulting expression, we have: 9(2E t
Ei)*(E- E,)
(Ei f 2)2(2E + Eg)
EO
&CEO- El1
- -6x ’ ~(2E + Ei)(2E, + ei) ‘Em
-hAQ 9
E(j
-
1
1
(26 + 1)(2f, + 1) -
(75)
For dilute solutions, one may take, except in the factor E - E,, the permittivities B and oooequal to or-,,and take x1 = 0, A$ = 0. Eq. (75) then yields: (4~~/kT)g~2
= 3(260 + Ei)2(E - E,)/(Ei f 2)2 Ed ,
(76)
which is the same result as obtained from eq. (32). For the pure solute, both eq. (32) and eq. (75) reduce to the original Rirkwood-FrBhlich equation- For the intermediate case, however, expressions (32) and (75) may yield slightly different values of g. It then seems preferable to use eq. (32) since eq. (75) accounts for the correlations between the orientation of a given molecule and the induced dipoles less thoroughly than eq. (32). In practice, the differences will be small, however_ : For dilute solutions expressions (10) and (11) for the effective moment are equal to those obtained by Ross and Sack and by Onsager for the moment in solution; This was to be expected, since the moment in solution is the moment determined according to the Debye equation_ This equation differs from the Kirkwood-FrGhlich equation in that the long-range dipole correlation is ignored_ For dilute solutions this is justified (cf. eq. (22) for E - e, = 0), and both approaches are equivalent. The generalization to time-d.ependent correlation functions shows that the result by Fatuzzo and Mason and by Titulaer and Deutch, obtained for a system of non-polarizable molecules, can also be derived without splitting up the sample into a small sphere considered on a microscopic basis and a surrounding dielectric continuum. As concerns the generalization to polarizable molecules, our derivation shows that the result obtained by Klug et al. [19] and by Rivail [20] is correct, and not the one by J5ill to which Titulaer and Deutch refer. The same result was ob-
P. BardewijklSeparation of dipoiecorrelationsin polar fluids
463
tained by Fulton [42,43] but in the opinion of the author the derivation given here is preferable for its simplicity. An account of the consequences for the relationship between macroscopic and molecular relaxation behaviour has been given before 1441. AS concerns the alignment, it follows from the above that the reduction from the macroscopic correlation parameter RK to the local parameter GK can be performed by a simple division by &_ This is consistent with the result by Ramshaw et al. 133. Our derivation shows that their reserve with respect to systems of polarizable molecules is not necessary for systems of isotropic point polarizabilities on a cubic lattice. Although the result by Ramshaw et al. is correct with respect td the aligriment, care should be taken in using their expression for the effective moment, since this applies to the average value of the moment mC whose fluctuations are not independent oftbe moments of the surrounding molecules. This invalidates their expression (5) for the static permittivity [lS]. The resulting expression for the alignment is: P,(cos 0)1&C = & (&f*/rcT)2[S(E,
+ 2)2/(2e + E,)2]C#
,
(77)
which is obtained by substituting eq. (73) into eq. (53), substituting the result into eq. (32) of ref. [3] with ignorance of the poiarizability terms, and using eq. (56). In the absence of local ordering (G = l), eq. (77) reduces to the expression obtained if the molecules are supposed to orient themselves independently in a directing field ln agreement with the Onsager theory [4,8]. Biemond et al. [45-471 used the symbol G to account simply for deviations of the alignment from the value expected according to this theory, and gave values for a number of nitrocompounds. The above treatment demonstrates that these values may be identified with those of the local correlation parameter defined in eq. (72). Finally, one may ask about the applicability of the method to the separation of long-range and short-range correlations in the dielectric saturation. In analogy to the calculation leading to eq. (58), one obtains for the correlation factor for an isolated macroscopic sphere, which we will denote by R, (where we stress the analogy withRK, in contrast to Kielich, who uses R, for the corresponding local correlation factor): R,=[~(~~*M)(M.M)-~(c.MM-M)]/(c~*)~ =gB(9gB f 2g2 - 6) - 2RK(gB f 1) - $z - I)@ - 2)(cos ‘y12e3 - AM)/p* , when AM now stands for: AM=(Mlel,e2,e3,rl,r2,r3)0
(78)
3
-i=l
c (Mlej+, .
(7%
It follows that in the case of the dielectric saturation terms in AMiz-* should be accounted for, making the evaluation much more complicated, e.g. since non-linear terms in the polarization induced by the homogeneous field should be incorporated. We are now attempting such a calculation.
References [1J J.G. Kirkwood,J. Chem.Phys.7 (1939) 911. [2] S.H. Glarum, J. Chem.F’hys.33 (1960) 1371. (31 J.D. Rarnshaw, D-W_Schaefer. J.S. Waughand J.M.Deutch, I. Chem. Phys. 54 (1971) 1239. [4] C.W. Hitbers and C. MacLean, in: NMR -basic principles and progress, (Springer, Berlin, 1972). [S] A.D. Buckiq$am, J. Chem. Phys. 25 (1956) 428. [6] E. Fatuzzo andP.R. Mason, Proc. Phys. Sot. 90 (1967) 741. [7] U.M. Titulaer and J.M. Deutch, J. Chem. Phys. 60 (1974) 1502. [8] L. Onsager, J. Am. Chem. Sot. 58 (1936) 1486. [9] H. Fr&lich, Theory of dielectrics(Oxford Univ. Press, London, 1949). [lOI S. Kielich, Acta Phys. Poion. 18 (1958) 239
Vol. 7, eds. P. Diehl, E. Fluck and R. Kosfeld
464
P. BordewijkfSeparation
of d@ole comzlations in polarfluids
1111 P.‘Bordewijk,J. Chem.Phys. 58 (1973) 1220. [l2] .W.T. Coffey and B.K.P. Sctie, Proc. Roy. Irish Acid. 76 (1976) 195. [13] FE. Harris and B.J. Alder, 1. Chm. Phys. 21(1953) 1031. [14] H. Frchlich, J. Chem. Phys. 22 (1954) 1804. [15] F-E. Harris and B.J. Alder, J. (Ihem. Phys. 22 (1954) 1806. 1161 AD. Buckingham, Proc. Roy. SOL 238A (1957) 235. [17] R.H. Cole, J. Chem. Phys. 27 (1957) 33, [18] J.D. Ramshaw, J. Chem. Phys. 61(1974) 889. 1191 D.D. Klug, D.E. Kranbuchland W.E. Vaughan, J. Chem. Phys. 50 (1969) 3904. [20] J.L. Rivail, I. Chim. Phys. 66 (1969) 981. [21] N.E. Hill, J. Phys. C5 (1972) 415. [22] P. Bordewijk, Physica 69 (1973) 422. 1231 S.H. Glarum, Mol. Phys 24 (1972) 1327. (241 R.H. CoIe, MoI. Phys. 26 (1973) 969. [25] N.E. Hill, hfol. Phys. 27 (1974) 185. [26] M. Mandel, Physica 57 (1972) 141. [27] C.J.F. Biittcher, Theory of electricpolarization,Vol. 1,2nd Ed. (Elsevier, Amsterdam, 1973) p. 18. [28] M. Mandel and P. Maaur,Physica 24 (1958) 116. 1291.P. Bordewijk, Physica 75 (1974) 146. [30] P. Bordewijkand W.H. de Jeu, J. Chem. Phys. 68 (1978) 116. [31] A. Gerschel, L Darmon and C. Brot, Mol. Phys. 23 (1972) 317. [32] J.H. van Vleck, Mol. Phys. 24 (1972) 341. 1331 P.J. Boundsand R.W. Munn, Chem. Phys. 24 (1977) 343. 1341 J.G. Ross and R.A. Sack, Proc. Phys. Sot. (London) B63 (1950) 893: B64 (1951) 620. [35] W.H. de Jeu and?_ Bordewijk, 3. Chem. Phys. 68 (1978) 109. 1361 R. Kubo, J. Phys. Sot. Japan 6 (1957) 570. 1371 C.J.F. Btittcher, Theory of electric pohuisation, 1st Ed. (Elsevier, Amsterdam, 1952) p. 175. [38] T.S. Nee and R. Zwanzig, I. Chem. Phys. 52 (1970) 6353. [39] T.H. Tjia, P. Bordetijk and C.J.F. Bottcher, Advan. Mol. Relaxation Processes6 (1974) 19. [40] M.S. Wertheim, J. Chem. Phys; 55 (1971) 4291. [41] G.S. Rushbrooke, G. Stell and J.S. Hdye, Mol. Phys. 26 (1973) 1199. [42] R-L. Fulton, Mol. Phys. 29 (1975) 405. [43] RL. Fulton, J. Chem. Phys. 62 (1975) 4355. [44] P. Bordewijk, Advan. Mol. Relaxation Processes 5 (1973) 285. [45] J. Biemond, Ph.D. Thesis, Free Universityof Amsterdam, The Netherlands. 1975. [46] J. Biemond, B.J.M. Nejjzen and C. MacLean, Chem. whys. l(1973) 335. [47] J. Biemond, S. van der Gpot and C. MacLean, Chem. Phys. Letters 32 (1975) 390.
ON THE SEPARATION OF LONG-RANGE AND SHORT-RANGE DIPOLE CORRELATIONS IN POLAR FLUIDS
P. BORDEWIJK Gonku~ Laboratories of the Universi~, Department GfPhysical Chemistry, Leyden. The Net’Ierlands Received 30 December 1977
Revised manuscript received 7 June 1978
A new method for the separation of long-range and short-range dipole correlations in polar fleids with pokimble molecules is applied to the correlation functions connected with the static dielectic permittivity, the frequency-dependent Permittivity, and the electrically induced alignment. The method avoids the division of the sample by a distinct boundary into a region considered on a microscopic basis and a surrounding dielectric contikm, and yields rigorous resnhs for a system with identical isotropic polarisabilities on a cubic lattice, among which the E&wood-Friihhch equation for the static permittivity. in other cases it makes sense to replace the dipole moment of the isolated molecule occurring in this equation by an effective value, for which expressions applicable under different conditions are suggested
1. Introduction Information about local order in polar fluids can be obtained f;om the response to a static electric field. This was fast observed by Kirkwood [l], who related the dielectric permittivity to the average
452
P. BordewijklSeparation
of dipole correlations in polarfluids
quency-dependent permittivity [2,6,7,23-251, the alignment [3], and the dielectric saturation [lo-121. In the case of the frequency-dependent permittivity, the time-dependent behaviour of the virtual surface charge at the boundary has been the object of much discussion. Moreover, the method ignores short-range correlations between molecules atboth sides of the boundary, which is not a priori justified, as pointed out by Mandel [26]. In this article the averages uM2)u, (M(0) . M(r))u, and (3(el •LU)~ -M2) will be calculated by a method which avoids either of the assumptions mentioned concerning the induced polarization, and also avoids the division of the sample into a small sphere considered on a microscopic basis and a surrounding continuous dielectric. It appears that for a system with isotropic polarizabihties on a cubic lattice the Kirkwood-FrChlich equation, the expression for the complex permittivity derived by Klug et al. [ 191 and by Rivail [20], and the expression for the alignment derived by Rarnshaw et al. [3] are rigorous results, whereas in other cases a good approximation can be obtained by replacing the dipole moment of the isolated molecule in these expressions by an effective value.
2. The relationship between the instantaneous moment and the polarizabilities As a fmt step, we will derive an expression for the instantaneous morrientM of a spherical system of n polarizable dipoles. We assume that the dipole moment of each molecule is completely determined by its orientation and the electric field working on it, i.e., we ignore fhrctuations of the atomic and electronic polarization. These dipole moments can then be written as: mi=pi+ai-
where mi, pi, andai are the total moment, the permanent dipole moment, and the polarizability tensor of molecule i, E. denotes the external field, and Tji is the dipole-dipole interaction tensor [27] _ Eq. (1) can be written in Sn-dimensional notation [Z&27, sec. 361 m=R+a-(e&-T-m),
(2)
where m, p, and e are 3ndimensional vectors whose projections on the three-dimensional subspace i are equal to mi,ti, and the unit vector e ilong the external field E,,, respectively, and czand T are 3ndimensional tensors with projections on the product of subspaces i andi equal to ar Sij and Tii(l - Q), Sii being the Kronecker delta. In contrast to previous articles [22,29,30] in this article we consider a sphere in vacua, so that no contribution due to the reaction field should be incorporated. From eq. (2) it follows: m=(1+a-T)-~-(~+a~eEg)=(l+a-T)-1~~+a~(1+T~a)-l~eEO,
(3)
where the last equality can be proved by developing (1 + a - T)-1 and (1 + T - a)-’ into a series. In eq. (3), 1 denotes the 3n-dimensional unit tensor. With A = (1 + T * a)-l, eq. (2) yields iu three-dimensional notation:
where it is used that (1 + u . T)-l is the transpose of (1 + T - a)-l, and Aii denotes the projection ofA on the product of subspaces i andj. From eq. (4) it follows that Zi Aii is a three-dimensional tensor transforming the external field to the part of the local field at molecule i that depends on the external field E,. This tensor is independent of the permanent dipoles; we wilI cah ii the local-field tensor. In the absence of an external field, eq. (4) yields for the instantaneous moment M:
P. BordewijklSeparation of dipole correlations in polarflrtids
453
where we call
the effective moment of molecule i. This effective moment consists of the permanent dipole moment of molecule iand the moments induced directly or indirectly by this moment in all molecules. In principle, the effective moment, like the moment mi, is a fluctuating quantity that depends on the positions and orientations of all other molecules. In contrast to mi, however, for isotropic polarlzabllities the effective moment does not depend on the orientations of the other molecules. This makes the neglect of fluctuations of the effective moment for given orientation of the molecule, which will be made in the following, a less crude approximation than the neglect of fluctuations OfITZr,which is made in other derivations of the Kirkwood-Friihlich equation. The approximation made here is equivalent with the neglect of fluctuations in the internal field for given orientation of the molecule, when studying the dielectric constant of non-polar systems or the dielectric constant at optical frequencies. Fluctuations of the effective moment still occur due to the translational freedom in liquids, however, and for molecules with non-isotropic polarizabilities due to the orientational freedom; these fluctuations contribute to the SOcalled induced-dipole absorption in the far-infrared frequency range [31]. Eqa.(5) and(6) are valid both for systems of identical particles and for mixtures. In the former case, the average local field tensor will be the same for all molecules, In the second it will be different for different components in the mixture. Since the equations also keep their validity if some molecules do not have a permanent dipole, they can also be applied to solutions of polar compounds in non-polar solvents. For identical molecules where the anisotropy of the polarizability and of the shape can be ignored, the dipoles induced directly by the external field are equal, and their contributions to the local field at a certain molecule cancel after averaging, so that when translational fluctuations are ignored, or the molecules are situated on a cubic lattice, the tensor Xj Aii reduces to the unit tensor. This does not hold for molecules which are near to the boundary of the sample, but for a macroscopic sphere this regards only a negligible fraction of the molecules. It follows that for identical molecules where the anisotropy of the polarizability and of the shape can be ignored, the effective moments are equal to the permanent dipole moments of the isolated moIecules. This holds rigorously if the molecules are situated on a cubic lattice. In other cases, more drastic approximations should be made. If the molecules are not identical, the neglect of translational fluctuations is an approximation even if the molecules are situated on a cubic lattice, and if the anisotropy of the molecular shape and of the polarizabillties cannot be ignored, the neglect of fluctuations of the effective moment involves not only a neglect of translational fluctuations, but also ofcertain orientational fluctuations. Making this approximation, the tensor Xi Aij can be estimated from the model of a molecule in a cavity in a dielectric continuum with permittivity equa1 to the dielectric constant of induced polarization E, of the system under Investigation. It is known that such a model may yield results different from a corresponding model with point polarizablhties [32], but this does not necessarily invalidate the continuum model, since, in a way, it accounts for the extensiveness of the polarizabilities, which is ignored in the model of point polar&abilities. In fact, for solids of elongated molecules the point-polarizability approach was shown to be inapplicable [33]. Assuming an ellipsoidal shape of the cavity, with the dipole along one of the axes, and the induced dipole distributed homogeneously over.the cavity, one obtains the Beld E, working on the molecule, from the cavity field&:
(7) where eqs. (2.84), (4.100) and (2.71) from ref. [27] are used. In eq. (7) cr3 denotes the component of the polarizability along the dipole, Q, b, and c are the half axes of the
’
454
P. BordewdkjSeparation of dipole correlations in polar jhids
~chfpsoidal cavity, and A is a shape factor. From eq. (7) it follows: ‘*=‘(e_,
3% +2){e,__+(1-&4[1+3~~(1-&~~c]}’
(8)
If the dipole is not along one of the axes of the ellipsoid, the components of the effective moment along the different axes should be calculated separateIy with eq. (S), using the appropriate value of the polar&ability and the shape factor for each axis. Although in that case the effective moment has not the same direction as the moment of the isolated molecule, it is still fiied in molecular coordinates. Since in eq. (8) the polarizabihty is thought to be distributed over the ellipsoid, it can be related to an internal dielectric constant Ei,3 by: ~~ ‘U~C(Ei,3 - 1)/3 [I + A(Ei,J - l)] ,
(9)
yielding P* = 3em ]l +A(ei,3 - l)ll(eca + 2)]eca +A(ei,3 - eta)] * ,
(10)
For dilute solutions, E, may be taken equal to e, and eq. (10) changes into an expression for the moment in solution derived by Ross and Sack [34], if the internal dielectric constant Ei,3 is taken to be isotropic and equal to the square of the refractive index of the pure solute. The assumption of an isotropic internal dielectric constant seems to underestimate the anisotropy of the polarizability, however. For, if in the crystal the moIecule can be represented by an ellipsoid with the same shape as io the liquid, filling up a cavity with volume equal to the volume available to each molecule, the assumption of an isotropic internal dielectric constant yields an isotropic dielectric constant of the crystal with the same magnitude, from which it follows that the anisotropy of the dielectric constant of the crystal is ignored in this way. Therefore, for compounds crystallizing with all molecular axes parallel; it should be preferred to take the internal dielectric constant equal to the dielectric constant of the crystal in the corresponding direction. If apart from the anisotropy of the dielectric constant, also the shape anisotropy cm be ignored, one has A = :, and eq. (10) changes into cr* = [3E,(fi+ 2)I(E, c 2)(2~~ f Ei)] 12.
(11)
For dilute solutions, this corresponds with the ratio between the apparent moment and the moment of the isolated molecule given by Onsager [a]. For the pure compound, E, and Ei should be taken equal, and the effective moment is equal to the moment of the isolated molecule, as for isotropic point polarizabilities on a cubic lattice. A different relationship between the internal field and the Maxwell field has been shown to be applicable for compounds forming a liquid crystalline phase [35] : Ei=K-E,
.W)
with KgJ
=
(1 - 47rN&)-’
,
(13)
where Qand t refer to the principal values along and transverse to the long axis of the molecule. In this way one obtains for the components of the effective moment. P;,t = 3Pp,&o
+ 2)(1 - 47n%,,~,,t)
-
(14)
The effective moment p* should be distinguished from the moment pd used in ref. [30]. With eq. (9), eq. (14) can be rewritten to: P& = ]3Pa,tI(e~ + 211[I -AQ,,+A&~(EiA?,J -
Thisis identical
to eq. (10) for fi = E,, but not for other cases.
(1%
P. BoniewijkSeparationof dipolecon-elations in polar fluids 3.
The Kirkwood-Ftihlich
455
equation
The average value of the total moment of a sphere with radius a in an external field can be written in terms of the dielectric permittivity, but can also be obtained from eq. (4):
where only effects linear in the applied electric field are considered. If in the last member of eq. (16) the permanent dipoles are ignored, the remaining term is related in an analogous way to the dielectric constant of induced polarization
3@ - CJ @ + 2k
Q3&-, = c
+ 2)
i
($
lE
(18)
,
or [28;27, sec. 361
where U= U. -Mm E. f O(E$ is the energy of the system. Eq. (19) shows the relationship between the permittivity and the average Gl@>o . For a further evaluation, we consider the average value of the moment M for. fmed orientation of mplecule 1, which can be written, if only one polar compound is present, as:
where el denotes the unit vector along the effective moment of molecule 1, r12 the angle between the effective moments of molecules 1 and 2, and n the number of polar molecules. To obtain from the average (WeI) an expression accounting for local order only, it should be reduced to: (21) vIR ‘,m
ri~R
where ri is the distance from molecule i to molecule 1, and the super index m in the last member indicates that tire average is taken over a macroscopic sphere (with radius R) surrounded by an infinite sample of the same composition. The symbol g stands for the Kirkwood correlation factor defmed by:
(22) To perform the reduction, we start from the average potential outside the sphere, for fmed orientation and position of molecule 1, using polar coordinates with the origin at the centre of the sphere and the z axis parallel with el :
n (~llel>o
= U.flel>O
cos e/r2 + n12
rP_l
C
m=-n
B~m)P~m)(cos
f3) eim@ ,
(23)
456
P. BordewijkjSepararion
of dipole >orrelations in polar fluids
where the terms in Bfm) account for the higher electric moments, PLm)(cos 0) denoting the associated Legendre polp~tials. In eq_ ($3) it is assumed that the average dipole moment of the sphere is independent of the position of the molecule, which seems appropriate as long as molecule 1 is not near the boundary. The constraint imposed at the averaging, i.e., the fixed orientation and position of molecule 1, does not affect the behaviour of the sample at the boundary as a dielectric continuum, except if molecule 1 is kept near the boundary, which case yields a negligible contribution to ( p=iM), however. Therefore, at the boundary the electric potential and the normal component of the dielectric displacement should be continuous_ From this it follows for the average potential inside the cavity [21]
(24) The terms in negative power of r are related to a field with its sources inside the region of the sphere that may not be considered as a dielectric continuum due to the constraint_ This field vanishes after averaging outside a small sphere around moIecuIe 1 which contains the region where it has its sources. The term in r accounts for a homogeneous field -2(~ - 1)(Mlet)/3s03, the field corresponding with the terms in higher powers ofr vanish after averaging over the sphere, due to the orthogonality of the associated Legendre polynomials. The average vaIue of p* for given value of Ecan be inferred from eq. (18):
(25) One thus obtains from the homogeneous field corresponding with the second term of eq. (24), for molecules far from molecule 1, i.e., in the region that can be considered as a dielectric continuum: lim C$ leljo = -[2(15 - I)(E - e,)/3e(e,
rp-
+ 2)nl LWel)o .
(26)
Subtracting these contributions from (Mlet),,, one obtains: g$
= Wei)
+
(2c + E,)(E + 2) 2(E - i)(e - E,) 3E(E, + 2) (Mlel)o = 3E(E, + 2) (“lel+l
’
(27)
or, for an isotropic system:
= [3e(e- + 2)/(2E + E,)(E + 2)]g(rz*)2 -
(28)
Substituting this into the penultimate member of eq. (19), one obtains a generalization of the Kirkwood-Friihlich equation: (E - EJ(2E + E,)/E(E, +.2)2 = (4rrN9kT) g(JP
,
(29)
where the moment of the isolated molecule p has been replaced by the effective moment 0. In eq. (29) N= 3n/4ruz3 denotes the number density. For identical molecules with isotropic shape and polarizability the neglect of translational fluctuations implies that the effective moment is equal to the moment of the isolated molecule, and eq. (29) reduces to the prigind Kirkwood-FriWch equation: (e - fJ(2E
+ eJle(e,
+ 2)2 = (47rN/9kT) 9112.
For other cases, however, eq. (10) or eq,(ll)
may be preferred, yieldivg respectively:
(30)
P. BordewijklSparation of dipole correhtions in polar
E~)(~Ef E,) [em +A(Q
(E -
fluids
- cm)] 2/ee$ [l f d(ei,s - l)] 2 = (4dV/kT)gp2 9
(~ - f,)(2E + E,)(2~~ + ~i)2/~P(~i+ 2)2 = (4~~/~~)g~2 .
451
(31) (32)
For compounds forming a liquid-crystalline phase, and where the dipole is along the long axis, like for the p-alkylcyanobiphenyls, eq. (14) yields: (e - E,)(2E + e&(1
- 47rN+4,)2
= (47rN/kT) g$
)
(33)
as is also obtained from eq. (22) in ref. [30] by taking S = 0.
4. The frequency dependent permittivity The moment induced by an external field with harmonic time dependence, written in complex notation as: = e El exp(iwf) ,
&f)
(34)
is given by: (35) It follows: (3 [+I)
- e,]&(o)
+
2](c_ + 2)la3io(t) = n@ 1&j .
(36)
The right-hand side can be rewritten with statistical mechanics of linear dissipative systems [36] :
(I$&*> =(&J =- 1
t
3kT s
-05
+F
1
t dt’&(t’)((lr;(f) $ -ca
dT)&,(t))(&
- e)(h(r’) * e)$,
- ni(t’)>, =&&,(&J~f(0)
l
M(t)$ ,
(37)
where ~iiw denotes the Laplace transform. It follows (38) This is equivalent with the starting-point of the calculation by Glarum [2]. For w = 0, eq. (38) reduces to eq. (19). TO split up the average in eq. (38) into contributions due to short-range and long-range dipole correlation, we have to calculate the potential (Gz(r, t)]el(0)&. To this end, we should know the time-dependent equivalent of the relation U)(r)]el),, = e(E(~)]et$, used in the derivation of eq. (24). This relation yields fort = 0: (D(r, O)lel(0)J = dE(r, O)lq(O)& .
(39)
This implies that in the region that can be considered as a dielectric continuum, the distribution of orientations at 1= 0 is in equilibrium with the field E(r, 0). Therefore, in the case that the time dependence of the field E were such that this field would disappear immediately after t = 0, the dielectric displacement would be given by: W,
r)let(Wo
= eW,
O)lel(O)WW
,
(40)
where, by defmition, F(t) is the step-response function of the dielectric displacement. In reality, however, the field E does not disappear instantaneously, and the behaviour of E(t’) between t’ = 0 and r’ = t also contributes
458
P. BordewijklSepamtion
of dipole correlations in polar fluids
$0 D(t), yielding:
(D(r, t)lel(0)lo = dE(r, O)le,(0)>oF(t)
+j
dr’L??(r, t)lel(0)>,,f(r
- t’),
t>o,
(41)
0
where the superposition principle is used, from which it also follows that the pulse-response function, f(t) =-i(t), is related to the complex permittivity by: p(o) =ePi,(f)
>
=E~~(.f)
(42)
FF,standing for the bilateral Fourier transform: Yw(f) = 7 dff(f) exp(-iwr) . _m
(43)
In eq. (42) the bilateral Fourier transform may be used since f(t) is a causal function [f(t) = 0 for t < 01. We now introduce a time-dependent field B’(f) by: fd0
E’(t) = 6tio (E(r, O)lel(0)>oe6f, -f =W~,t)lel&9)o,
;
t>O.
WI
tz=o.
(45)
Eq. (41) can then be rewritten to: (D(r, t)lel(0)>o = E 7 dt’E’(r, t’)f(f - t’), -a
It appears from eq.445) that the dielectric displacement can be written as a convolution of the field E’ defmed in eq. (44) with the pulse-response function, and not of the field (Efr, f)lel(0)>o. This is due to the fact that for t = 0 one should.use the equilibrium distribution function, as pointed out already by Kirkwood [ 11. Similar problems arise with respect to the reaction field [37,38], where also the average value at t = 0 should be calculated from the equilibrium distriiution function [39]. Eq. (45) can be extended to values t< 0 by replacing the left-hand side by a field D’(t) defined in analogy with E’(& Taking the Fourier transform of the resulting equation, and using the convolution property, one obtains: g:, CD’@,t)) = C(w) FU (E’(r, t)} .
(46)
It follows that a relation analogous with eq. (24) can be derived for each frequency component of the timedependent potential&(t) connected with the fieldE’(r, r) in the sphere. The resulting contribution to lin+_ $#)lel(0)> is then obtained by combining eqs. (37) and (38), and one obtains by subtracting the long-range contribution to (M(t)lel(0)): (47) where M’(f) andg’(?) stand for: M’(f) = she (M(0)lel(O))OeSr,
t
= Uf(~le~(0)~o,
t>O,
g’(t) = she ge6*, -t
t
= 1 + (n - l)kos(el(0)
l
el(t))$,
;
t> 0 .
(48)
(49)
i! Bordew~k/~eparation of dipole correhtions in pohu jluids
459
From eq. (47) a similar relation follows between k’(t) andg’(T), and since these derivatives are zero for t < 0, the double-sided Fourier transforms then reduce to Laplace transforms: 3(e, + 2)h(w) -@icdMt)
lel(‘)‘O
7 p(o)
+ E, J p(u)
f 21 -@iu
(50)
-
( ~Pj(41~1(0)),
With eq. (38) it then follows:
or, with eq. (29):
(52) For non-polarizable molecules (e_, = 1) this is identical to the result by Fatuzzo and Mason and by Titulaer and De&h; when ignoring the cross-correlation terms eq. (52) reduces to the result by Klug et al. 1191 and by RivaiI [20], usingg= 1.
5. The alignment
From eq. (5) it directly follows:
(3(el - M)* -
(p’)*$j & [3(q -
M*)o=
ei)(el - ek) - ei -
ek]>,=2(p*)*R~,
I
where R, ,is a macroscopic correlation factor, introduced by Kielich [lo] as the reduction factor for the Kerr effect, and defined by:
= 1 +(n - 1)(P2(COS’y12))o iFyV
+ $(n - l)(tZ -2)(COS~@OS~&
-:(n
- l)(n
- ~)(cos~~~)~
denotes the angle between the effective moments JI: and pf _R, canalsobe kitten RK=(3el~M+3(~-l)cos712eI~M--neI
-M)0/2~*=&~-l)(cos7~2e~
-Ml&*
,
(54)
as: -:(n-3)(el-M)/p*. (5%
To rewrite the first term we write the average moment of the sphere for fxed positions and orientations of molecules 1 and 2 as a sum of the average moments obtained if molecules 1 and 2 kept fured separately, and a correction term: (Mlel,e2, rl, r2jo = (Mlel)o + (Mle2)o + AM.
(56)
Substituting this into eq. (55) one obtains R, = $z - l)(cosr12el*
AM>,/r.c* +gB[&B
- 1) +gz] ,
(57)
where B stands for [cf. eq. (28)] : B = lim (elmM)/gp* = ~E(E, + 2)/(2~ + E,)(E + 2) ,
n*m
(5%
460 adg2
P. Bordewi~k&mmtion of dipole correlationsin polarfluids stands for the analogue of the Kirkwood correlation factor for the second Legendre polynomial: g2 =
f
c P,(cos7li) )o = 1+(n-l)w~(cos7& j
--
.
This correlation parameter contains no long-range contriiution. For E = E, one has 3 = 1; in that case the longrange dipole correlation also vanishes. The moment AM depends on the orientations and positions of molecules 1 and 2. We will make a distinction between the cases that the distance between molecules 1 and 2 is so large, that when molecules 1 and 2 are kept f=ed, there is nevertheless a region in between them that can be regarded as a dielectric continuum, and the case that the distance is too small for this. For the result it is irrelevant at which distance the distinction should be made, as long as it can be made at a distance that is small with respect to the size of the macroscopic sphere. In the second case it makes sense to defme a moment Miq by:
which depends on the positions and orientations of molecules 1 and 2, and decreases with r12. The moment of the macroscopic sphere is then given by: ~lel,e2,~l,~~~0
(61)
=g@*@l +++BM;z,
from which it foUows that AM for this case can be taken equal to BMiz. In spite of the decrease ofMi2 for large r12, it is not allowed to take AMequal to zero for the case that the separation between molecules 1 and 2 is so large that the region in between them can be considered as a dielectric continuum. T-his is due to the fact that for futed orientation of molecule 2, the long-range polarization connected with molecule 1 is slightly affected. Although this effect is only of order l/n, its contribution to R, is nevertheless of importance due to the factor (n - 1) in the first term of the second member of eq. (58). To calculate this effect, we consider the moment induced in the sphere by an external field, if one molecule is kept fiied: SdXn-lMexp(-UolkT+
M - EolkT)
IC
$dXnmlexp(-U,$kT
+M * EolkT)
-1 (62) I ,
where dX”-l points to an integration over all positions and orientations except one, and U. stands for the interaction energy of the molecules iu the absence of the field. In eq. (62) terms in the energy of the system in higher powers of E have been neglected, since they do not play a role in the following. From eq. (62) it fo!.lows:
(auflel)E/aEo)E=o
= (ahzjaml~o
+ {[(IWMI ello - (Mlel~o(Mlel~o] mel/kT
= taM/aElel>of {[WMl;l)o
-g2B2(p*)2elel]
l
e)/kT .
(63)
Neglecting the anisotropy of(aM/aEle,>,, it follows that the polarizabiity of the sphere under the constraint mentioned differs from the one without this constraint by: Aal = [(MMlel)o -g2B2(u*)2e1e1
- (MM),-,]/kT.
(64)
Now one has: (M-Mle,)o=(M~M)o
_
(65)
of dipole correlations in polar fluids
P. BordewijklSeparation
461
This makes (MMlel)o -0, e traceless tensor. Furthermore, this tensor has axial symmetry about el. Therefore, the tensor Aa is known, if the principal value of (MMlel)o - (MM), along e, is known. This value is given by: *MJJ = &.L*)2RK . el - ((MMleI)o - (MM)o) . e1 = &I* MM el>o - ‘W j
056)
l
It follows, if we denote the three-dimensional unit tensor by I Aa = (P*)~ [(RK - g2B2)e 11 e -;RKl]/kT.
(67)
The homogeneous field over the sphere for ftved orientation of molecule 1 (eq. (24)), corresponds with an external field given by: 2(E - l)(e_ + 2)
3kT(B - I)
033) e1 ’ tip* where the generalized Kirkwood-Froblich equation, eq. (29), was applied. The field given in eq. (68) polarizes the sphere. If molecule 2 is kept fixed, the moment induced by the field changes by an amount (E& - Aua. Analogously, the moment induced by the field corresponding with molecule 2 will change by an amount (E& . Aa,, if molecule 1 is kept fuced. One thus obtains for AM: 3a3(2e + e,)
AM=BM;2+(E&a
Aa2+(E&-
=BM& + [3(B - l)~*/nB](el
gfi**el =
Aal
+eZ)[(R~ -g2B2)cos~12
-$&I
.
(6%
Substituting this into eq. (58) ignoring terms in l/n, and taking (cos yL2) =O, (cos2yr2) =i, and (~0s~‘)‘~)
=
0, which again impties the ignorance of terms in l/n, one obtains: R, =gB[i(g - 1) + g2] + [(B - l)jB]RK + $B(n - l)(cos r12el - h&)/p*
,
(70)
or: R, =gB2&g-l)+g2]
+zB*(n -I)(cos~~~~~*M~~)/,!I*.
(71)
From this expression one obtains the local correlation parameter, to be denoted by GK, by takingB = 1: GK =g[:(g
- 1) fgz]
f $I - ~)(COS-r12el M;z)/p* l
(72)
Comparing this with eq. (71) one obtains: RK =B2G, .
(73)
6. Discussion
The present article shows that the instantaueous’moment of a dielectric sphere in the absence of an external electric tield can be written as a sum of effective moments of the MOlecUleS.These effective moments are related to the permanent dipole moments of the molecules by a tensor that depends on the polarizabilities of the molecules, but not on their permanent moments. This tensor is the same as the tensor transforming the external field into the field dependent part of the local field. Ignoring fluctuations of this tensor for given orientation of the
molecule, it can be calculated with a suitable model for the internal field at optical frequencies. For a system of identical, sphkrical molecules with isotropic polar&ability, the tensor then reduces to the unit tensor, and the effective moment is equal to the permanent dipole moment of the isolated molecule. It was further shown that the elimination of the long-range dipole correlation can be performed without mtroduc% a distinct boundary between a small sphere considered on a microscopic basis and a surrounding dielectric to be considered as a continuum. This Is of interest since in such a procedure the short-range interactions between mole-
462
P. BordewijkSeparation
of dipole com&tions in polarfluids
atboth sides of the boundary between the two regions are ignored, and it is not a priori clear if this is justified In the present derivation, however, the only assumption made is that far from a molecule whose position and orientation is kept fmed in the averaging, the sample behaves as a continuum, which seems an essential property of liquids- The result a posteriori justifies the neglect of short-range interactions between molecules at both s&es of the boundary in the calculation of (p,- &, (~~(0) - M(t)&,, and (3(el- &Q2 - M2 J-,.In avoiding the criticized introduction of a distinct boundary, our derivation agrees with the one by Kirkwood but it seems new as concerns the application to polarizable molecules. It should be pointed out that although the treatment eliminates long-range interactions that are essentially dipolar in origin, this does not imply that the remaining short-range correlations are completely non&polar in origin. On the contrary, it has been shown that also systems with only dipolar interactions between the molecules yield deviations from Onsager’s equation, corresponding with Kirkwood factors deviating from unity [40,41]. Apart from accounting for the anisotropy of the polarizability and of the molecular shape, our derivation also makes it possible to generalize the Kirkwood-Frijhlich equation to solutions, as shown by eqs. (3 1) and (32). in contrast to a previous derivation [27, sec. 401, the validity is not restricted to the case that the molecules form associates of a Iimited size. It seems worthwhile to compare the present result for spherical molecules with isotropic polarizability, eq. (32), with the expression derived previously:
c&s
@l(Ei - 1) I O Q(2e + eg) -3x1’o(2~t~i)
3x @o(eo- 1)
t3x
1’
@i(Eu-ei)+3A$eO-1 l#J 2c+l ’ ~(2E+Ei)
(74)
where $I=+,$Q +x~@~ t A@stands for the molar volume, and the indices 0 and 1 refer to tire non-polar solvent and the polar soIute, respectively. If in the expression within the brackets in the last member of eq. (74) E is replaced by e-, it should yield zero. Subtracting the resulting expression, we have: 9(2E t
Ei)*(E- E,)
(Ei f 2)2(2E + Eg)
EO
&CEO- El1
- -6x ’ ~(2E + Ei)(2E, + ei) ‘Em
-hAQ 9
E(j
-
1
1
(26 + 1)(2f, + 1) -
(75)
For dilute solutions, one may take, except in the factor E - E,, the permittivities B and oooequal to or-,,and take x1 = 0, A$ = 0. Eq. (75) then yields: (4~~/kT)g~2
= 3(260 + Ei)2(E - E,)/(Ei f 2)2 Ed ,
(76)
which is the same result as obtained from eq. (32). For the pure solute, both eq. (32) and eq. (75) reduce to the original Rirkwood-FrBhlich equation- For the intermediate case, however, expressions (32) and (75) may yield slightly different values of g. It then seems preferable to use eq. (32) since eq. (75) accounts for the correlations between the orientation of a given molecule and the induced dipoles less thoroughly than eq. (32). In practice, the differences will be small, however_ : For dilute solutions expressions (10) and (11) for the effective moment are equal to those obtained by Ross and Sack and by Onsager for the moment in solution; This was to be expected, since the moment in solution is the moment determined according to the Debye equation_ This equation differs from the Kirkwood-FrGhlich equation in that the long-range dipole correlation is ignored_ For dilute solutions this is justified (cf. eq. (22) for E - e, = 0), and both approaches are equivalent. The generalization to time-d.ependent correlation functions shows that the result by Fatuzzo and Mason and by Titulaer and Deutch, obtained for a system of non-polarizable molecules, can also be derived without splitting up the sample into a small sphere considered on a microscopic basis and a surrounding dielectric continuum. As concerns the generalization to polarizable molecules, our derivation shows that the result obtained by Klug et al. [19] and by Rivail [20] is correct, and not the one by J5ill to which Titulaer and Deutch refer. The same result was ob-
P. BardewijklSeparation of dipoiecorrelationsin polar fluids
463
tained by Fulton [42,43] but in the opinion of the author the derivation given here is preferable for its simplicity. An account of the consequences for the relationship between macroscopic and molecular relaxation behaviour has been given before 1441. AS concerns the alignment, it follows from the above that the reduction from the macroscopic correlation parameter RK to the local parameter GK can be performed by a simple division by &_ This is consistent with the result by Ramshaw et al. 133. Our derivation shows that their reserve with respect to systems of polarizable molecules is not necessary for systems of isotropic point polarizabilities on a cubic lattice. Although the result by Ramshaw et al. is correct with respect td the aligriment, care should be taken in using their expression for the effective moment, since this applies to the average value of the moment mC whose fluctuations are not independent oftbe moments of the surrounding molecules. This invalidates their expression (5) for the static permittivity [lS]. The resulting expression for the alignment is: P,(cos 0)1&C = & (&f*/rcT)2[S(E,
+ 2)2/(2e + E,)2]C#
,
(77)
which is obtained by substituting eq. (73) into eq. (53), substituting the result into eq. (32) of ref. [3] with ignorance of the poiarizability terms, and using eq. (56). In the absence of local ordering (G = l), eq. (77) reduces to the expression obtained if the molecules are supposed to orient themselves independently in a directing field ln agreement with the Onsager theory [4,8]. Biemond et al. [45-471 used the symbol G to account simply for deviations of the alignment from the value expected according to this theory, and gave values for a number of nitrocompounds. The above treatment demonstrates that these values may be identified with those of the local correlation parameter defined in eq. (72). Finally, one may ask about the applicability of the method to the separation of long-range and short-range correlations in the dielectric saturation. In analogy to the calculation leading to eq. (58), one obtains for the correlation factor for an isolated macroscopic sphere, which we will denote by R, (where we stress the analogy withRK, in contrast to Kielich, who uses R, for the corresponding local correlation factor): R,=[~(~~*M)(M.M)-~(c.MM-M)]/(c~*)~ =gB(9gB f 2g2 - 6) - 2RK(gB f 1) - $z - I)@ - 2)(cos ‘y12e3 - AM)/p* , when AM now stands for: AM=(Mlel,e2,e3,rl,r2,r3)0
(78)
3
-i=l
c (Mlej+, .
(7%
It follows that in the case of the dielectric saturation terms in AMiz-* should be accounted for, making the evaluation much more complicated, e.g. since non-linear terms in the polarization induced by the homogeneous field should be incorporated. We are now attempting such a calculation.
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464
P. BordewijkfSeparation
of d@ole comzlations in polarfluids
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