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Nonlinear dielectric relaxation in non-interacting dipolar systems
19 June 1998
Chemical Physics Letters 289 Ž1998. 541–545
Nonlinear dielectric relaxation in non-interacting dipolar systems P. Ke¸dziora a , J. Jadzyn ˙ a b
a,)
, K. De Smet b, L. Hellemans
b
Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan, ´ Poland Department of Chemistry, Katholieke UniÕersiteit LeuÕen, Celestijnenlaan 200 D, B-3001 LeuÕen, Belgium Received 19 February 1998; in final form 21 April 1998
Abstract The frequency dependence of the nonlinear dielectric effect caused by the Langevin dipolar saturation has been studied for dilute benzene solutions of 4,4X-n-hexylcyanobiphenyl and 4-Žtrans-4X-n-hexylcyclohexyl.isothiocyanatobenzene. The nonlinear dielectric increment induced by the coupling between the static electric field of high strength Ž10 7 Vrm. and a field of small intensity and changeable frequency Ž1 MHz–3 GHz. shows a relaxation which is interpreted by Coffey theory. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Electric fields of high intensity ŽG 10 6 Vrm. applied to polar liquids allows one to observe a nonlinearity of the dielectric polarization Ž P . on the field strength Ž E . dependence. There are two main molecular phenomena which may cause this nonlinearity. The first one concerns the systems in which the intermolecular interactions Že.g., dipole–dipole or hydrogen bonding. lead to the formation of aggregates with a compensated dipole moment. In the liquid state the entities of low polarity are in equilibrium with the other entities Žof different polarity. andror with the non-associated molecules Žmonomers.. In a strong electric field such an equilibrium undergoes a shift in favor of more polar species Žor monomers.. This ‘chemical effect’ makes the dielectric polarization of the liquid increase more rapidly with an increase of field intensity than in the linear case ŽFig. 1.. The nonlinear ‘chemical effect’ mea)
Corresponding author. Fax: q48-61-868-4524.
sured with a weak alternating field EŽ v . shows a relaxation in a frequency region dependent on the rate of the molecular aggregation process. Nonlinear dielectric spectroscopy has been applied to the study of the kinetics of the cis-lactams dimerization w1,2x and cholesterol aggregation w3,4x. Here we will discuss the second main phenomenon causing a nonlinearity of the P Ž E . dependence, i.e. the Langevin saturation of the dipoles’ orientation in strong electric fields. This phenomenon leads to a decrease in permittivity of liquids. The nonlinear dielectric increment, defined as the difference between the permittivity value measured when a strong electric field E0 is applied to the dielectric Ž ´ E 0 . and when E0 is not applied Ž ´ .: D ´ L s ´ E 0 y ´ , is negative. For most liquids the D ´ L is proportional to the square of the biasing field strength E0 w5x. The increment measured with an alternating field EŽ v . of small amplitude shows a frequency dependence and can be presented in the complex form D ´ L) Ž v . s D ´ XL Ž v . y iD ´ YL Ž v . ,
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 5 7 - 6
Ž 1.
542
P. Ke¸dziora et al.r Chemical Physics Letters 289 (1998) 541–545
3. Results and discussion
Fig. 1. Schematic presentation of the nonlinear dielectric behavior of dipolar liquids caused by the chemical effect Žpositive contribution. and the Langevin dipolar saturation effect Žnegative contribution..
The Langevin saturation phenomenon can be investigated only for non-interacting dipolar systems. As we have mentioned, the dipolar interactions lead to molecular aggregates, which in presence of strong electric field can give a positive contribution to the nonlinear dielectric increment. The experimental results presented in Figs. 2 and 3 have been obtained for diluted solutions of 6-CB and 6-CHBT in benzene. Dilution of a dipolar substance in a nonpolar medium is an efficient way for the reduction of dipole–dipole interactions. Unfortunately, a decrease of the dipole number in the unit volume leads simultaneously to a decrease of the measured signal amplitude. The concentration of dipoles in our experiments Ž1% in molar fraction. was the compromise between the effectiveness of the
where D ´ XL and D ´ YL are the real and imaginary parts of the nonlinear dielectric increment, respectively. In this Letter we present the experimental results of the relaxational behavior of D ´ L) Ž v . for two systems of molecular rotators of different dipole moment.
2. Experimental 4,4X-n-Hexylcyanobiphenyl ŽC 6 H 13 -f-f-C[N, 6-CB. and 4-Žtrans-4X-n-hexylcyclohexyl. isothiocyanatobenzene ŽC 6 H 13 -CyHx-f-N5C5S, 6CHBT. of high purity were kindly supplied by R. Da¸browski, Institute of Chemistry, Technical Military Academy, Warsaw. The 6-CB and 6-CHBT molecules are, to a good approximation axially symmetric and the direction of the dipole moment vectors is close to the axis of symmetry. The nonlinear dielectric increment measurements were carried out with the stationary relaxation method in which the quasi-static electric field Ž; 85 Hz. of high amplitude Ž1.1 = 10 7 Vrm. is superimposed on the field EŽ v . of weak amplitude and changeable frequency: from 1 MHz to 3 GHz. The details concerning the apparatus and measuring procedures are described previously w1–4,6x.
Fig. 2. Nonlinear dipolar relaxation spectrum Ža. and nonlinear Cole–Cole plot Žb. for 6-CB solution in benzene. Molar fraction of 6-CB: x s 0.0107, E0 s1.1=10 7 Vrm, T s 258C. Solid lines represent the Coffey equations Ž2. and Ž3. for Lsy2.01=10y4 and t s 0.205 ns.
P. Ke¸dziora et al.r Chemical Physics Letters 289 (1998) 541–545
543
of high amplitude is applied to the dielectrics w8,9,12x and Ž2. a weak alternating field is superimposed on a strong direct field w7,11x. Despite of the different basis of the Debye model extension to the nonlinear case, all theories give quantitatively similar predictions for the frequency dependence of the increment D ´ L) . In particular, the most general theory, proposed by Kielich et al. w8,9x, covering a great variety of nonlinear effects, predicts exactly the same dependence D ´ L) Ž v . as the theory of Coffey and Paranjape w7x. Here we will use the following Coffey equations w7x, which fit well to our experimental arrangement: 1
D ´ XL s L D ´ YL s L
1 q v 2t 2 vt 1 q v 2t
F1 Ž v . ,
Ž 2.
F v. , 2 2Ž
where t is the rotational relaxation time, F1Ž v . and F2 Ž v . refer to the nonlinearity of the dipolar relaxation process and are given by F1 Ž v . s Fig. 3. Nonlinear dipolar relaxation spectrum Ža. and nonlinear Cole–Cole plot Žb. for 6-CHBT solution in benzene. Molar fraction of 6-CHBT: x s 0.0108, E0 s1.1=10 7 Vrm, T s 258C. Solid lines represent the Coffey equations Ž2. and Ž3. for Ls y0.40=10y4 and t s 0.254 ns.
dipole–dipole interactions reduction and the limit of the measuring apparatus sensitivity. A theoretical description of the frequency dependence of the nonlinear dielectric phenomenon caused by the saturation of the dipoles orientation in strong electric fields has been proposed by Coffey and Paranjape w7x, Kasprowicz-Kielich and co-workers w8–11x, and Kimura and Hayakawa w12x. The theories concern the isotropic dielectrics containing N mutually non-interacting dipolar, axially symmetric molecules, with the dipole moment vector parallel to the axis of symmetry. The rotational relaxation of such an assembly, placed in a strong electric field, is described by the extended Debye rotational diffusion model w13,14x. Two particular cases of the permittivity variation with the frequency and field strength, corresponding to two possible experimental options, have been considered: Ž1. only one alternating field
F2 Ž v . s
27 q v 2t 2 y 2 v 4t 4 3 Ž 1 q v 2t 2 . Ž 9 q v 2t 2 . 42 q 19v 2t 2 q v 4t 4 3 Ž 1 q v 2t 2 . Ž 9 q v 2t 2 .
,
Ž 3. .
The quantity L in Eqs. Ž2. denotes the strength of the nonlinear dielectric increment and is equal to the static value of the increment Ž D ´ Ls . Žthe highfrequency value of the increment Ž D ´ L` . is equal to zero.. Studies of the static nonlinear dielectric effect have a long history and were carried out mainly by Piekara 1 and co-workers w15,16x, Chełkowski w17x, Małecki w18x, Thiebaut and co-workers w19–21x and Parry Jones w22x. The strength L is related to the molecular quantities in the following way w6x: L ' D ´ Ls s y
Nm4 45´ 0 k 3 T 3
E02 F Ž ´s , ´` . ,
Ž 4.
where N is the number of dipoles per unit volume,
1 In 1936 A. Piekara, during his studies of the nitrobenzene solutions, discovered the positive nonlinear dielectric effect Ž‘chemical effect’. w15x.
P. Ke¸dziora et al.r Chemical Physics Letters 289 (1998) 541–545
544
m is the molecular dipole moment, k is the Boltzmann constant and T is the absolute temperature. The value of L is negative and depends on the square of the biasing electric field strength E0 and the fourth power of the molecular dipole moment. The function F Ž ´s , ´` . in Eq. Ž4. expresses the fact that the dipoles are immerged in the dielectric medium of the static permittivity ´s and the highfrequency permittivity ´` . The form of the function depends on the model of the local field in the dielectrics and, as has been shown by Thiebaut w19x for the Onsager model w23x, is given by F Ž ´s , ´` . s
´s4 Ž ´` q 2 .
4
2 Ž 2 ´s q ´` . Ž 2 ´s q ´`2 .
.
Ž 5.
The solid lines in Figs. 2 and 3 present the best fitting of the theoretical predictions given by Eqs. Ž2. and Ž3. to the experimental results Žpoints.. The value of the fitting parameters Ž L and t . are gathered in Table 1. The data presented in Figs. 2 and 3 allow us to conclude that the relaxational behavior of the Langevin nonlinear dielectric increment, including the sign change of its real part in the highfrequency region, is perfectly reproduced by the Coffey equations. From the static value of the nonlinear increment L one can evaluate the dipole moment of rotating molecules. The data presented in Table 1 indicate that, for both 6-CB and 6-CHBT, the dipole moment values Ž mcalc . calculated from Eqs. Ž4. and Ž5. are not too far from the values resulting from the linear polarization measurements in diluted solutions. The small difference of mexp y mcalc f 0.6 D shows that for the concentration of ; 1% Žin molar fraction. the
Table 1 Values of linear a and nonlinear dielectric data for the diluted solutions of 6-CB and 6-CHBT in benzene at 258C Ž x denotes the molar fraction of dipolar compound.
t Žns. L mcalc ŽD. mexp ŽD. ´s a
6-CB x s 0.0107
6-CHBT x s 0.0108
0.205 y2.01=10y4 4.2 4.8–4.9 w24x 2.66
0.254 y0.40=10y4 2.9 3.48 w25x 2.48
´` s 2.278, density r s 0.881 grcm3 Žfor both solutions..
Fig. 4. Comparison of the nonlinear Cole–Cole plots ŽFigs. 2b and 3b. for two molecular rotators with different values of the dipole moment: m6-CB s 4.2 D, m6-CHBT s 2.9 D.
dipole–dipole association is not completely eliminated. This conclusion agrees with the results obtained by Dunmur et al. w24x. The strong dependence of the Langevin nonlinear effect on the dipole moment of molecular rotators Ž L ; m4 . is illustrated in Fig. 4, where two nonlinear Cole–Cole plots corresponding to the 6-CB and 6CHBT molecules are compared. The difference in the dipole moment values of these two molecules is relatively small and is equal to ; 1.3 D. Since a strong electric field applied to a dipolar liquid causes a partial ordering of dipoles and hampers the thermal rotation, one can expect an influence of the field intensity on the relaxation time. It has been shown by Coffey et al. w26x that the difference between the relaxation time obtained from nonlinear Žt . and linear Žt D . dipolar relaxation spectroscopies depends on the ratio j of the electric energy of dipoles Ž m E0 . and the thermal energy kT. The theory predicts that for a small j value Ž j < 1., t and t D should be equal to each other w27x. In our experiment the j value is ; 4 = 10y2 , thus one can expect that t and t D are close to each other; unfortunately, the lack of the linear relaxation data for 6-CB and 6-CHBT diluted solutions makes one unable to verify this expectation.
Acknowledgements This work was partially supported by the Polish Research Project No. 2P03B 160 09 coordinated by
P. Ke¸dziora et al.r Chemical Physics Letters 289 (1998) 541–545
the Committee for Scientific Research ŽKBN.. PK is grateful to the K.U. Leuven for the award of fellowships and KDS obtained a research grant from the Flemish Executive ŽI.W.T...
References w1x L. Hellemans, L. De Maeyer, J. Chem. Phys. 63 Ž1975. 3490. w2x K. De Smet, P. Ke¸dziora, J. Jadzyn, ˙ L. Hellemans, J. Phys. Chem. 100 Ž1996. 7662. w3x J. Jadzyn, ˙ L. Hellemans, Acta Phys. Pol. A 67 Ž1985. 1093. w4x J. Jadzyn, ˙ L. Hellemans, Ber. Bunsenges. Phys. Chem. 97 Ž1993. 205. w5x C.J.F. Bottcher, Theory of Electric Polarization, vol. 1, Else¨ vier, Amsterdam, 1973. w6x M. Vints, L. Hellemans, I.E.E. Conf. Publ. 289 Ž1988. 340. w7x W.T. Coffey, B.V. Paranjape, Proc. R. Ir. Acad. 78 Ž1978. 17. w8x B. Kasprowicz-Kielich, S. Kielich, J.R. Lalanne, in: J. Lascombe ŽEd.., Molecular Motions in Liquids, D. Reidel, Dordrecht, 1974, pp. 563–573. w9x B. Kasprowicz-Kielich, S. Kielich, Adv. Mol. Relaxation Proc. 7 Ž1975. 275. w10x W. Alexiewicz, B. Kasprowicz-Kielich, in: M. Evans, S. Kielich ŽEds.., Modern Nonlinear Optics, part 1, Wiley, New York, 1993, p.1.
545
w11x W. Alexiewicz, Mol. Phys. 83 Ž1994. 245. w12x Y. Kimura, R. Hayakawa, Jpn. J. Appl. Phys. 31 Ž1992. 3387. w13x W.T. Coffey, in: M.W. Evans ŽEd.., Dynamical Processes in Condensed Matter, Advances in Chemical Physics, vol. 63, Wiley, New York, 1985, pp. 69–252. w14x B.K.P. Scaife, Principles of Dielectrics, Oxford University Press, Oxford, 1989. w15x A. Piekara, B. Piekara, C.R. Acad. Sci. ŽParis. 203 Ž1936. 852. w16x A. Piekara, S. Kielich, J. Phys. Radium 18 Ž1957. 490. w17x A. Chełkowski, J. Chem. Phys. 28 Ž1958. 1249. w18x J. Małecki, Acta Phys. Pol. 21 Ž1962. 13. w19x J.M. Thiebaut, Thesis, University of Nancy I, Nancy, 1968. w20x J.M. Thiebaut, A. Weisbecker, C. Ginet, C.R. Acad. Sci. ŽParis. 267 Ž1968. 661. w21x J.M. Thiebaut, J.L. Rivail, J. Barriol, J. Chem. Soc., Faraday Trans. II 68 Ž1972. 1253. w22x G. Parry Jones, in: M. Davies ŽEd.., Dielectric and Related Molecular Processes, vol. 2, The Chemical Society, London, 1975, p.198. w23x L. Onsager, J. Am. Chem. Soc. 58 Ž1936. 1486. w24x D.A. Dunmur, M.R. Manterfield, W.H. Miller, J.K. Dunleavy, Mol. Cryst. Liq. Cryst. 45 Ž1978. 127. w25x P. Ke¸dziora, J. Jadzyn, ˙ Mol. Cryst. Liq. Cryst. 192 Ž1990. 31. w26x W.T. Coffey, Y.P. Kalmykov, K.P. Quinn, J. Chem. Phys. 96 Ž1992. 5471. w27x K. De Smet, L. Hellemans, J.F. Rouleau, R. Courteau, T.K. Bose, Phys Rev. E 57 Ž1998. 1384.
Chemical Physics Letters 289 Ž1998. 541–545
Nonlinear dielectric relaxation in non-interacting dipolar systems P. Ke¸dziora a , J. Jadzyn ˙ a b
a,)
, K. De Smet b, L. Hellemans
b
Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Poznan, ´ Poland Department of Chemistry, Katholieke UniÕersiteit LeuÕen, Celestijnenlaan 200 D, B-3001 LeuÕen, Belgium Received 19 February 1998; in final form 21 April 1998
Abstract The frequency dependence of the nonlinear dielectric effect caused by the Langevin dipolar saturation has been studied for dilute benzene solutions of 4,4X-n-hexylcyanobiphenyl and 4-Žtrans-4X-n-hexylcyclohexyl.isothiocyanatobenzene. The nonlinear dielectric increment induced by the coupling between the static electric field of high strength Ž10 7 Vrm. and a field of small intensity and changeable frequency Ž1 MHz–3 GHz. shows a relaxation which is interpreted by Coffey theory. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Electric fields of high intensity ŽG 10 6 Vrm. applied to polar liquids allows one to observe a nonlinearity of the dielectric polarization Ž P . on the field strength Ž E . dependence. There are two main molecular phenomena which may cause this nonlinearity. The first one concerns the systems in which the intermolecular interactions Že.g., dipole–dipole or hydrogen bonding. lead to the formation of aggregates with a compensated dipole moment. In the liquid state the entities of low polarity are in equilibrium with the other entities Žof different polarity. andror with the non-associated molecules Žmonomers.. In a strong electric field such an equilibrium undergoes a shift in favor of more polar species Žor monomers.. This ‘chemical effect’ makes the dielectric polarization of the liquid increase more rapidly with an increase of field intensity than in the linear case ŽFig. 1.. The nonlinear ‘chemical effect’ mea)
Corresponding author. Fax: q48-61-868-4524.
sured with a weak alternating field EŽ v . shows a relaxation in a frequency region dependent on the rate of the molecular aggregation process. Nonlinear dielectric spectroscopy has been applied to the study of the kinetics of the cis-lactams dimerization w1,2x and cholesterol aggregation w3,4x. Here we will discuss the second main phenomenon causing a nonlinearity of the P Ž E . dependence, i.e. the Langevin saturation of the dipoles’ orientation in strong electric fields. This phenomenon leads to a decrease in permittivity of liquids. The nonlinear dielectric increment, defined as the difference between the permittivity value measured when a strong electric field E0 is applied to the dielectric Ž ´ E 0 . and when E0 is not applied Ž ´ .: D ´ L s ´ E 0 y ´ , is negative. For most liquids the D ´ L is proportional to the square of the biasing field strength E0 w5x. The increment measured with an alternating field EŽ v . of small amplitude shows a frequency dependence and can be presented in the complex form D ´ L) Ž v . s D ´ XL Ž v . y iD ´ YL Ž v . ,
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 4 5 7 - 6
Ž 1.
542
P. Ke¸dziora et al.r Chemical Physics Letters 289 (1998) 541–545
3. Results and discussion
Fig. 1. Schematic presentation of the nonlinear dielectric behavior of dipolar liquids caused by the chemical effect Žpositive contribution. and the Langevin dipolar saturation effect Žnegative contribution..
The Langevin saturation phenomenon can be investigated only for non-interacting dipolar systems. As we have mentioned, the dipolar interactions lead to molecular aggregates, which in presence of strong electric field can give a positive contribution to the nonlinear dielectric increment. The experimental results presented in Figs. 2 and 3 have been obtained for diluted solutions of 6-CB and 6-CHBT in benzene. Dilution of a dipolar substance in a nonpolar medium is an efficient way for the reduction of dipole–dipole interactions. Unfortunately, a decrease of the dipole number in the unit volume leads simultaneously to a decrease of the measured signal amplitude. The concentration of dipoles in our experiments Ž1% in molar fraction. was the compromise between the effectiveness of the
where D ´ XL and D ´ YL are the real and imaginary parts of the nonlinear dielectric increment, respectively. In this Letter we present the experimental results of the relaxational behavior of D ´ L) Ž v . for two systems of molecular rotators of different dipole moment.
2. Experimental 4,4X-n-Hexylcyanobiphenyl ŽC 6 H 13 -f-f-C[N, 6-CB. and 4-Žtrans-4X-n-hexylcyclohexyl. isothiocyanatobenzene ŽC 6 H 13 -CyHx-f-N5C5S, 6CHBT. of high purity were kindly supplied by R. Da¸browski, Institute of Chemistry, Technical Military Academy, Warsaw. The 6-CB and 6-CHBT molecules are, to a good approximation axially symmetric and the direction of the dipole moment vectors is close to the axis of symmetry. The nonlinear dielectric increment measurements were carried out with the stationary relaxation method in which the quasi-static electric field Ž; 85 Hz. of high amplitude Ž1.1 = 10 7 Vrm. is superimposed on the field EŽ v . of weak amplitude and changeable frequency: from 1 MHz to 3 GHz. The details concerning the apparatus and measuring procedures are described previously w1–4,6x.
Fig. 2. Nonlinear dipolar relaxation spectrum Ža. and nonlinear Cole–Cole plot Žb. for 6-CB solution in benzene. Molar fraction of 6-CB: x s 0.0107, E0 s1.1=10 7 Vrm, T s 258C. Solid lines represent the Coffey equations Ž2. and Ž3. for Lsy2.01=10y4 and t s 0.205 ns.
P. Ke¸dziora et al.r Chemical Physics Letters 289 (1998) 541–545
543
of high amplitude is applied to the dielectrics w8,9,12x and Ž2. a weak alternating field is superimposed on a strong direct field w7,11x. Despite of the different basis of the Debye model extension to the nonlinear case, all theories give quantitatively similar predictions for the frequency dependence of the increment D ´ L) . In particular, the most general theory, proposed by Kielich et al. w8,9x, covering a great variety of nonlinear effects, predicts exactly the same dependence D ´ L) Ž v . as the theory of Coffey and Paranjape w7x. Here we will use the following Coffey equations w7x, which fit well to our experimental arrangement: 1
D ´ XL s L D ´ YL s L
1 q v 2t 2 vt 1 q v 2t
F1 Ž v . ,
Ž 2.
F v. , 2 2Ž
where t is the rotational relaxation time, F1Ž v . and F2 Ž v . refer to the nonlinearity of the dipolar relaxation process and are given by F1 Ž v . s Fig. 3. Nonlinear dipolar relaxation spectrum Ža. and nonlinear Cole–Cole plot Žb. for 6-CHBT solution in benzene. Molar fraction of 6-CHBT: x s 0.0108, E0 s1.1=10 7 Vrm, T s 258C. Solid lines represent the Coffey equations Ž2. and Ž3. for Ls y0.40=10y4 and t s 0.254 ns.
dipole–dipole interactions reduction and the limit of the measuring apparatus sensitivity. A theoretical description of the frequency dependence of the nonlinear dielectric phenomenon caused by the saturation of the dipoles orientation in strong electric fields has been proposed by Coffey and Paranjape w7x, Kasprowicz-Kielich and co-workers w8–11x, and Kimura and Hayakawa w12x. The theories concern the isotropic dielectrics containing N mutually non-interacting dipolar, axially symmetric molecules, with the dipole moment vector parallel to the axis of symmetry. The rotational relaxation of such an assembly, placed in a strong electric field, is described by the extended Debye rotational diffusion model w13,14x. Two particular cases of the permittivity variation with the frequency and field strength, corresponding to two possible experimental options, have been considered: Ž1. only one alternating field
F2 Ž v . s
27 q v 2t 2 y 2 v 4t 4 3 Ž 1 q v 2t 2 . Ž 9 q v 2t 2 . 42 q 19v 2t 2 q v 4t 4 3 Ž 1 q v 2t 2 . Ž 9 q v 2t 2 .
,
Ž 3. .
The quantity L in Eqs. Ž2. denotes the strength of the nonlinear dielectric increment and is equal to the static value of the increment Ž D ´ Ls . Žthe highfrequency value of the increment Ž D ´ L` . is equal to zero.. Studies of the static nonlinear dielectric effect have a long history and were carried out mainly by Piekara 1 and co-workers w15,16x, Chełkowski w17x, Małecki w18x, Thiebaut and co-workers w19–21x and Parry Jones w22x. The strength L is related to the molecular quantities in the following way w6x: L ' D ´ Ls s y
Nm4 45´ 0 k 3 T 3
E02 F Ž ´s , ´` . ,
Ž 4.
where N is the number of dipoles per unit volume,
1 In 1936 A. Piekara, during his studies of the nitrobenzene solutions, discovered the positive nonlinear dielectric effect Ž‘chemical effect’. w15x.
P. Ke¸dziora et al.r Chemical Physics Letters 289 (1998) 541–545
544
m is the molecular dipole moment, k is the Boltzmann constant and T is the absolute temperature. The value of L is negative and depends on the square of the biasing electric field strength E0 and the fourth power of the molecular dipole moment. The function F Ž ´s , ´` . in Eq. Ž4. expresses the fact that the dipoles are immerged in the dielectric medium of the static permittivity ´s and the highfrequency permittivity ´` . The form of the function depends on the model of the local field in the dielectrics and, as has been shown by Thiebaut w19x for the Onsager model w23x, is given by F Ž ´s , ´` . s
´s4 Ž ´` q 2 .
4
2 Ž 2 ´s q ´` . Ž 2 ´s q ´`2 .
.
Ž 5.
The solid lines in Figs. 2 and 3 present the best fitting of the theoretical predictions given by Eqs. Ž2. and Ž3. to the experimental results Žpoints.. The value of the fitting parameters Ž L and t . are gathered in Table 1. The data presented in Figs. 2 and 3 allow us to conclude that the relaxational behavior of the Langevin nonlinear dielectric increment, including the sign change of its real part in the highfrequency region, is perfectly reproduced by the Coffey equations. From the static value of the nonlinear increment L one can evaluate the dipole moment of rotating molecules. The data presented in Table 1 indicate that, for both 6-CB and 6-CHBT, the dipole moment values Ž mcalc . calculated from Eqs. Ž4. and Ž5. are not too far from the values resulting from the linear polarization measurements in diluted solutions. The small difference of mexp y mcalc f 0.6 D shows that for the concentration of ; 1% Žin molar fraction. the
Table 1 Values of linear a and nonlinear dielectric data for the diluted solutions of 6-CB and 6-CHBT in benzene at 258C Ž x denotes the molar fraction of dipolar compound.
t Žns. L mcalc ŽD. mexp ŽD. ´s a
6-CB x s 0.0107
6-CHBT x s 0.0108
0.205 y2.01=10y4 4.2 4.8–4.9 w24x 2.66
0.254 y0.40=10y4 2.9 3.48 w25x 2.48
´` s 2.278, density r s 0.881 grcm3 Žfor both solutions..
Fig. 4. Comparison of the nonlinear Cole–Cole plots ŽFigs. 2b and 3b. for two molecular rotators with different values of the dipole moment: m6-CB s 4.2 D, m6-CHBT s 2.9 D.
dipole–dipole association is not completely eliminated. This conclusion agrees with the results obtained by Dunmur et al. w24x. The strong dependence of the Langevin nonlinear effect on the dipole moment of molecular rotators Ž L ; m4 . is illustrated in Fig. 4, where two nonlinear Cole–Cole plots corresponding to the 6-CB and 6CHBT molecules are compared. The difference in the dipole moment values of these two molecules is relatively small and is equal to ; 1.3 D. Since a strong electric field applied to a dipolar liquid causes a partial ordering of dipoles and hampers the thermal rotation, one can expect an influence of the field intensity on the relaxation time. It has been shown by Coffey et al. w26x that the difference between the relaxation time obtained from nonlinear Žt . and linear Žt D . dipolar relaxation spectroscopies depends on the ratio j of the electric energy of dipoles Ž m E0 . and the thermal energy kT. The theory predicts that for a small j value Ž j < 1., t and t D should be equal to each other w27x. In our experiment the j value is ; 4 = 10y2 , thus one can expect that t and t D are close to each other; unfortunately, the lack of the linear relaxation data for 6-CB and 6-CHBT diluted solutions makes one unable to verify this expectation.
Acknowledgements This work was partially supported by the Polish Research Project No. 2P03B 160 09 coordinated by
P. Ke¸dziora et al.r Chemical Physics Letters 289 (1998) 541–545
the Committee for Scientific Research ŽKBN.. PK is grateful to the K.U. Leuven for the award of fellowships and KDS obtained a research grant from the Flemish Executive ŽI.W.T...
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