J. Phys. Chem. 1983, 87, 4823-4826
4823
Theory of Dielectric Relaxation for a Liquid Consisting of Two Nonpolarizabie Dipolar Species Danlel Klvelson Depertment of Chemistry, University of Caiifofnla, Los Angeies, California 90024
and Paul A. Madden Theoretical Physics Section, Royal Slgnals and Radar Establishment, &eat Malvern, WORCS WR 14 3PS. England (Received: April 28, 1983)
The theory of dielectric relaxation for a fluid consisting of two nonpolarizable dipolar species is worked out in the rotational diffusion limit. [ ~ ( w -) 1][40)- 11-l can be expressed as the sum of two Lorentzians which can be analyzed to yield three "adjustable parameters", whereas the corresponding molecular theory depends upon seven hard-to-calculatetransport and equilibrium quantities. The three so-calleddynamic cross-correlation parameters are often neglected, and the two single molecule rotational correlation frequencies can often be determined from NMR or Raman experiments; we are then left with two molecular equilibrium parameters which can be obtained from the three "adjustable" parameters or possibly calculated by means of a Monte Carlo calculation. Only under conditions where the molecules of the two species have identical properties in solution can the dielectric spectrum be described by one Lorentzian. For two-component solutions of certain alcohols, only one Lorentzian is observed, and its width is between that observed for the neat liquids of the constituents. A rationalization of these results is given in terms of the aforementioned theory; this rationalization requires that all static dipolar properties, as well as the molecular volumes of the two species be nearly the same, that the viscosities (and hence the rotational correlation times) be different, and that in solution the rotational correlation times of both species be the same because both species are immersed in the same environment with a common viscosity.
I. Introduction In this article we study the dielectric relaxation of a fluid composed of two components, each consisting of molecules with rigid (nonpolarizable) dipoles. The molecules are assumed to rotate diffusionally, but the molecules of each species have both different dipole moments and different rotational diffusion constants; consequently, the dielectric permittivity t(w) can be described as the sum of two Lorentzian functions but the relaxation times of the Lorentzians are not those of the individual species. We obtain a molecular expression for ~ ( w )under these conditions; in spirit this treatment is similar to that described by Tsay and Kivelsonl for the depolarized light scattering from two-component fluids. We then investigate those special cases where two alcohols, each with a different ) dielectric relaxation time, form a solution in which ~ ( w can be described by a single Lorentzian with a relaxation time roughly equal to an average of the two for the pure alcohols;2this is a phenomenon which has not previously been explained. 11. Macroscopic Theory The dielectric properties of liquids composed of molecules with permanent dipoles are dependent upon the interactions of these dipoles. If in a two-component system the dipoles of the component molecules are g i and gL for species A and B, respectively, we expect the dielectric properties to depend upon the collective moment M? M = MA + Mg (1)
(1) S. Tsay and D. Kivelson, Mol. Phys., 29, 29 (1975). (2) A summary discussion of experiments is given by M. Davies in 'Dielectric Properties and Molecular Behavior",N. Hill et al., Ed., Van Nostrand-Reinhold, London, 1969, pp 340-4. See ref 13-21. 0022-3654/83/2087-4823$01.50/0
where I-$, is the position of the jth molecule of species A, and an analogous relation holds for Mg. In particular, we know that the dielectric properties depend upon the correlation function ( [M(k,t)],[M(-k,O)],),where ( ) indicates an equilibrium ensemble average, [MI, is the component of M along k, and [MI, is the component perpendicular to k. Because of the long-range character of the dipolar interactions the transverse component, ( [A4lx[A4I,), differs ), the from the longitudinal component ( [ ~ z [ M I zalthough dielectric permittivity c(w), which is the quantity actually determined in the laboratory, is i s ~ t r o p i c .We ~ shall be interested in very small lz, i.e., 12 0, but even in this limit the longitudinal and transverse components differ from each other.3 For an infinite sample, it can readily be shown that in the k 0 limit,3 kBTVto t(0) - 1 = (l[MI,I2) (3) N where ( N / V) is the molecular number density, kB is the Boltzmann constant, T i s the temperature, to is the permittivity of free space, and c(0) is the dielectric constant or zero frequency permittivity. At finite frequency, from linear response theory3n4one can show that for an infinite sample
-
-
-1 - 1 + iwaJ,(w) 40) - 1
--
€(W)
(4)
where is the transform of the normalized correlation function of [Ma:
~~~
(3) P. Madden and D. Kivelson, Adu. Chern. Phys., to be published. (4) R. Kubo, J . Phys. SOC.Jpn., 12,570 (1957).
0 1983 American Chemical Society
Kivelson and Madden
The Journal of Physical Chemistry, Vol. 87, No. 24, 1983
4024
Thus not only must the static correlation function (I[M(0)]a12)be anisotropic, but so too must be the dynamic one, ([M(t)],[M(O)],*). If [MI, relaxes with a single decay time, then [t(o) - 11 is a simple Lorentzian function of o. This is the Debye or diffusional model? But in a two-component system one might expect the molecular dipoles of each species to rotate with quite different reorientation times so that the moment M A for species A should relax at a considerably different rate than the moment M B for species B. We might, therefore, expect [MI, as specified in eq 1, to relax with at least two relaxation times, and the description of [ ~ ( w ) - 11 to require at least two Lorentzians. In some liquid solutions one finds that two Lorentzians are indeed needed while in others one suffices.2 This is one of the problems that we wish to consider: Under what conditions are two and under what conditions is one Lorentzian needed? If one relaxation time suffices6
7D
=
(ar)-l
111. Molecular Theory
In the last section we discussed the macroscopic equations pertaining to dielectric relaxation; here we obtain molecular expressions. To do this we employ the Mori formalism.6 It can readily be seen that the components of the transport matrix r appearing in eq 8 can be written as rAA =
- xAxB~ABfAB(?'A?'B)'/21
[gAAk?BB - gAB21-'[Y&AAk!BB
(19) =
IlAB
[gAAk?BB
- g A B 2 1 - 1 [ ( ~ A 7 B ) 1 ' 2 f A B - EABYA&?AA/gAAI (20)
where if C and D represent A and B gCD
=
~
~
~
C
~
x
~
~
~
bcD
where TD is called the Debye relaxation time. The corresponding diffusional equation of motion is
a
-( at
[M(t)l,[~(o)l,*)= -7n-Y [M(t)l,[M(O)I,*) (7)
At low k we assume that there is no k dependence. It is a simple matter to show that eq 8 , 4 , and 5 lead to 4w) -1 1 -iau --= 40) - 1
~
x
*
=~
+ {cDXc'12XD1/2
/
~
~
(21)
XC and
X D are the mole fractions of species C and D, respectively, p e ) and pg) are the corresponding gas-phase molecular dipoles for molecule (1) of species C and molecule (2) of species D.
gcc = 1
If M A and M B relax at quite different rates, then we expect the following coupled diffusional equations of motion to hold:
(18)
EcD
= lim k-0
+ iccxc
(22)
N ( [~fi)l,*[~g)l, exp[ik.(r@)-
)
(23)
[ ( I rPPlxl2) ( l[PPlXl2)IllZ
EDC
1 - ipu - u2
A ( l - iu/X+)-' + (1 - A)(1 - ~ U / L )(9) -'
where
u =w/r
P is the Mori-Zwanzig projection operator which we shall discuss below, L is the Liouville operator, rv) is the position of the j t h molecule, and N is the total number of molecules; r B B and r B A can be obtained from the analogous ex ressions above by reversing A and B. If C = D, &) and rg become p 8 ) and rg), respectively, and we have
P
r2
= [gAAk?BB - x A x B E A B 2 1 - ' [YAdAAYEdBB - X A X B Y A Y B f A B 2 1
(26)
and an analogous relation holds for
qB
rfl
=
k d B B - XAXB6AB21-1[YAkAAk?BB
+YdBdAA (27)
2XAXBfAB~AB(YAYB)'izl
A =
p - [p2 - 4]'12 - 2a 2[p2 - 4]'12
The first equality in eq 9 is a compact expression, but the second one displays explicitly the two-Lorentzian character of the results. The widths Xi of the Lorentzians are not characteristic of the individual species since they are affected by the interactions between species. The two-Lorentzian solution reduces to a single Lorentzian if
p = a + a-1
(17) If this equality holds, then eq 9 reduces to eq 6 with (5) P. Debye, "Polar Molecules", Dover Publishers, New York, 1945.
=
[ ( ~ [ ~ X ~ ~ ) I - ' N [ Y A ~ A A+PYAd ~B B~ yAB 2 x B
+
2PAPBYAYBtABXAXBI
(28)
and ~ A Ag ,B B , ~ A A and , %BB can be related to EAA, EBB, ~ A A , and EBB by means of eq 21 and 22. E l ~ e w h e r ewe ~ , ~have shown that useful comparisons can be made between the hydrodynamic results obtained with collective variables, such as M A and M B , and the corresponding results obtained with linear transport equations for the molecular variables PI)exp(ik.rk)) and pg) exp (ik-rg)). The conclusion, called the corresponding macro-micro correlation theorem (CMMC), indicates that the collective projection operator in eq 26-28, an operator (6) H. Mori, Prog. Theor. Phys., 33, 423 (1965). (7)D. Kivelson and P. Madden, Mol. Phys., 30, 1749 (1975).
~
~
f
Dielectric Relaxation in a Two-Component Fluid
The Journal of Physical Chemistry, Voi. 87,
which projects onto both M A and MB, can be replaced by the operator, which pro’ects onto each of the molecular quantities pa) exp(ik.ra ) and pi) exp(ik-rg))for all j and 1. This replacement enables us to interpret YA, which is given in eq 26, as the molecular correlation frequency for a single dipole of species A; this should be a short-ranged term, independent of k. The formulation above becomes particularly useful if t u , EBB, SAB, L A , kBB, YA, YB, and L B are a t most weakly dependent upon concentration; at constant viscosity this may often be the case. If we now rewrite eq 3 in terms of the quantities specified above we find that [t(o) -
pA2gAAxA
+ p B 2 g B B x B + 2 p A p B E A B X A X B (29)
This is a condition imposed upon the parameters. We can rewrite eq 28, making us of eq 3, as
2pApBYA1’2YB1’2fABXAXB]
(30)
IV. Approximations If two Lorentzians are observed for [t(w) - 11, then according t~ eq 9 and 10, we can determine three parameters experimentally, either the set [a,r, p] or [ A , X+, A_]. In the molecular theory there are the ten parameters: YA, Y ~ , &?AA,%BB, AB, g B B , AB, PA, and PB 88 well 88 the readily determined quantities T, V, X A , and ~ ( 0 ) .The dipoles pA and p B can be determined from gas-phase experiments, and eq 29 is an interrelationship among the parameters. Y~ and Y B can in principle be independently determined since 3 Y A and 3 Y B can be associated with the molecular rotational correlation times for A and B molecules, respectively, obtained by NMR or Raman techniques. This means we have to determine six parameters gut &B, SAB, g,, g B B , and (AB reduced by the one interrelationship of eq 29. This is still too many “adjustable” parameters, but the problem is simplified by the assumption that the “dynamic crosscorrelation parameters” vanish:*n9
& = iAA = &BB = o i.e., that,g becomes
=
(31)
= 1. Under these conditions, eq 30
&B
rff= ( Y A / 1 A 2 x A + Y B / l B 2 x B ) / ( p A 2 x A [ 1 + EAAX.41 + pB2xB[1
+ SBBXBI + 2kApBEABXAXB)
(32)
Equation 26 becomes r2
= YAYB[l
+ SAAXA + 6BBXB + k 4 A l B B - ~ A B f . B A ~ X A X B I - l (33)
Equation 27 becomes
rp =
YA
+ hAXA
+ YB + YASBBXB + YBEAAXA + S B B X B + [EAAtBB - E A B E B A l X A X B (34)
and eq 29 becomes
~ A ~ X+ A [AAXA] [ ~ + + S B B X B I + 2 p A k B f A B X A X B (35) We now have only two adjustable parameters: tu, EBB, and f m reduced by the condition in eq 35. Actually since (kBTtoV/N)[e(O)
we have molecular expressions for the equilibrium cross particle correlation parameters t u , EBB and tm, they can, in principle, be obtained by computer calculations. V. Conditions for a Single Lorentzian We now return to the question of “when is [t(w) - 11 described by a single Lorentzian?” The condition for a single Lorentzian is given in eq 17; this equation is satisfied under any of the following sets of conditions: 6) X A = 1 and X B = O (ii) (iii)
l][k~TtoV/N]-’ =
- 11 =
pB2xB[1
(8) T. Keyes and D. Kivelson, J. Chem. Phys., 56, 1057 (1972). (9) T. D. Gierke, J. Chem. Phys., 65, 3873 (1976).
No. 24, 1983 4825
pg
pA
=
pB
= 0 and
and
YA
If 5, = SAB = S B B and becomes (YA - Y B ) 2 x A x B
=
pA
EBB YB
= [AB = 0
and SAA = SAB = SBB
= p B , but
YA # Y B
then eq 17
+ 6AB*[(YAXA + Y B x B ) 2 - Y A Y B l
=0 (36)
which, except at a few special points, can only be satisfied = EBB = Em and YA = YB, but if YA = YB. Similarly, if p A # pg, then eq 17 becomes ( P A - PB)’
=0
(37)
which can be satisfied only if p~ = p B . It would appear, therefore, that if both components are dipolar, eq 17 cannot be satisfied at arbitrary concentration unless EAA = SBB = SAB, YA = YB, and p A = pB. In other words, unless these equalities are satisfied, i.e., the conditions of (iii) above, two Lorentzians are needed to describe [t(w) - 11 for solutions composed of two dipolar species. Note that in contrast to pA and p B , which are properties of the isolated gas molecules, YA and YB are very much affected by intermolecular interactions even though they are single particle relaxation frequencies. One of our aims has been an explanation of some experimental results on solutions of alcohols.2 What has been observed is that for a solution of two alcohols, A and B, the dielectric relaxation can be represented by a single feature which we shall approximate by a single Lorentzian with a width XAB which is between the width XA of the single feature observed for pure A and the width AB of the single feature observed for pure B. For this to be understood on the basis of the diffusional theory which we have developed, conditions (iii) given above must be satisfied; otherwise, the solution should give rise to two “distinct” features. The equality of the dipoles, pA = pB = p, can be understood by the fact that the dielectric properties of alcohols are in large part due to the dipoles of the -OH groups, and these are quite similar in most alcohols. The correlations of these dipoles should be similar in many alcohol environments so it is reasonable to expect EM = EBB = tm = 5. For alcohols the reorientational motion is described quite well by a diffusional model with stick boundary conditions;1°thus for dipolar reorientation”
Y = k~T/3umf
(38)
where k B is the Boltzmann constant, T is the temperature, u is the molecular volume, qBolis the coefficient of shear viscosity of the solution, and f is a shape factor.locSd (For alcohols the microviscosity ~v~~~equals the solution viscosity ?.lob) For two alcohols of comparable size (u), the frequencies y thus differ only if the viscosities differ, and (10) (a) A. Gierer and K. Wirtz, 2.Naturforsch. A , 8 , 532 (1953); (b) D. Hoe1 and D. Kivelson, J. Chem. Phys., 62, 4535 (1975); ( c ) F.Perrin, J. Phys. Radium, 5, 497 (1934); (d) J. L. Dote, D. Kivelson, and R. N. Schwartz, J. Phys. Chem., 85, 2169 (1981). (11) B. Kowert and D. Kivelson, J. Chem. Phys., 64, 5206 (1976).
4026
The Journal of Physical Chemistry, Vol. 87, No. 24, 1983
we expect the viscosity of a solution A and B to be intermediate between those of the (qA and TB) neat liquids.12 It follows that in solution, yA = yB = y, but y varies with concentration since qaol varies with concentration. In conclusion, if the molecular volumes of the two alcohols are equal, the dielectric relaxation should be describable by a single “feature” at all concentrations; the width of this feature, approximated by a Lorentzian, is XAB = ( k B T / 3 W +
(39)
and since qaol lies between those of the neat liquids, so should Am. The only other model that we can identify that gives a single Lorentzian for the relaxation of a solution of dipolar liquids is one in which a strong complex of the two species is formed so that the rotating dipole in solution is the complex. In this case the rotation of the complex would be slower than that of either molecule alone in the neat liquids, and we would not, therefore, expect the Lorentzian width of the solution to lie between the widths observed for the neat liquids. S~hallamach,’~ in his model, envisaged a relaxing entity which included enough molecules so that the concentrations of species within it approximated the concentrations in the bulk. It is not clear just what is meant by these relaxing volume elements; if they relax as a unit we would expect very slow rotational relaxation, but if they merely provide the local environment for individual molecular relaxation, then we might expect molecular relaxation dependent upon gaol,as in our model. Schallamach13was the first to observe the fusing of the relaxation channels (Lorentzians) associated with the two component alcohols into one; he also reported similar fusing of relaxation channels for other liquid mixtures, but concern has been expressed because his measurements were made at one frequency as a function of temperature.2 B0s,14however, found that solutions of bromobenzene in benzyl alcohol and of bromobenzene in cyclohexanol exhibited two distinct relaxation channels whereas a solution of benzyl alcohol and cyclohexanol appeared to relax via a single channel with a relaxation time intermediate between those of the two alcohols. The two alcohols have roughly the same molecular volumes but very different viscosities, and so these results are readily understood on the basis of our theory. Denney and Colel5found a single relaxation channel for mixtures of methanol and 1propanol, and Denney16 found two relaxation channels for mixtures of alcohols and allcyl halides; however, Denney16 found only a single channel for a mixture of isobutyl bromide and isobutyl chloride. Similar results for alcohol mixtures have been obtained by Sixou, Daumezor, and Dansas,2O by McDuffie, LaMacchia, and Conord,14and by Gary and Kadoba.l8 For the most part people have found a single channel for mixtures of alcohols and two channels in other case^.^^-^ Actually a Cole-Davidson curve seems to describe associated liquids better than does a Lorentzian, so in the discussion above when we talk about one (12) (1955). (13) (14) 2. (15) (16) (17) (18) (19)
N. E. Hill, Proc. R SOC. London Ser. B , 67, 149 (1954); 68, 209 A. Schallamach, Trans. Faraday SOC.,42, 180 (1946). F. F. Bos, Thesis, University of Leiden, 1958. As reported in ref D. J. Denney and R. H. Cole, J. Chem. Phys., 23, 1767 (1955). D. J. Denney, J. Chem. Phys., 30, 1019 (1959). P. K. Kadoba, J. Phys. Chem., 62, 887 (1958). S. K. Gary and P. K. Kadoba, J. Phys. Chem., 69, 674 (1965). G . E. McDuffie, Jr., J. T. LaMacchia, and A. E. Conord, J. Chem.
Phys., 39, 1878 (1963). (20) P. Sixou, P. Daumezon, and P. Dansas, J . Chem. Phys., 64, 824 (1967).
Klvelson and Madden
or two Lorentzians we really mean one or two “distinct” relaxation features. Even for pure alcohols several relaxation “features” have been identified, with different mean relaxation times,22but they are not “distinct” since they are not clearly resolved; thus to a first approximation we have assumed that every “distinct feature” can be described by a single Lorentzian or a single relaxation channel. Whether or not we have identified the correct mechanism associated with the single-channel dielectric relaxation in solutions of dipolar liquids must be determined by future experiments. In the Appendix we analyze dielectric relaxation for two-component systems in somewhat more detail.
Acknowledgment. The authors thank Professor R. H. Cole for introducing them to this problem, and Dr. Richard MacPhail for his many useful comments. This work was supported in part by the National Science Foundation (Grant CHE81-09068). Appendix Above, we discussed the behavior of the transverse components of the dipole, i.e., the x component. For completeness we here discuss the longitudinal or z comp~nent.~ In analogy to eq 6 we have, for the longitudinal relaxation5
where TD’
=
(-4-2)
TD/E(O)
We shall place primes on longitudinal quantities. Equation A-1 is a consequence of the fact that 4 w ) is isotropic and both eq 6 and A-1 must hold. All the equations in sections I1 and I11 hold for the longitudinal components as well as for the transverse components, except that for longitudinal modes everything but c(o) is primed, all x ’ s are replaced by z’s, and the left side of eq 15 is replaced by the left side of eq A-1 above. In addition we have
r’ = E(o)1/2r
(A-3)
a’ = t(0)’/2a
(A-4) (A-5)
(8 - a) = €(O)’/Z(P,- a’)
these conditions arise for reasons similar to those for eq A-2. The frequencies yAand yB are, as discussed, single particle frequencies and so yA = YA, and YB = Y B ~ . Equation 29 is specific to the transverse component. If eq 28, 29, A-3, and A-4 and the condition5 (A-6) are combined, then the numerator on the right side of eq 28 is identical for x or z directions; we thus expect gAA, gBB, and & to be independent of direction, i.e., we expect them to be short ranged. I t then follows from eq 26, its longitudinal analogue, and eq A-3 that k&BB
- ( s A B ) 2 x A x B I = t(o)[g,’gBB’
- (EAB’)2XAXBI
(-4-7) In section IV, the discussion remains valid for longitudinal FBB, and .$AB are replaced by primed behavior if r, a,p, quantities, and the left side of eq 35 is multiplied by c(O)-l. (21) E. Forest and C. P. Smyth, J. Phys. Chem., 69, 1302 (1965). (22) C. P. Smyth and S. K. Gary, J. Phys. Chem., 69, 1294 (1965).
