Electric Relaxational Effects Induced by Displacement Current in

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Electric Relaxational Effects Induced by Displacement Current in Dielectric Materials Jan Jad_zyn and Jolanta Swiergiel* Institute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 17, 60-179 Poznan, Poland ABSTRACT: An investigation was carried out on frequency behavior of the complex electric modulus, the impedance and the conductance, related to the displacement current resulting from the normal Brownian dynamics of relaxing molecular dipoles in dielectric materials. The experiment was performed for the liquid crystalline smectic A phase of n-octylcyanobiphenyl (8CB) in the frequency region of 200 kHz50 MHz, where the single dielectric relaxation of the Debye-type occurs and, at the same time, the ionic conductivity in the compound is negligibly low (σDC ≈ 0).

1. INTRODUCTION In recent paper1 we presented the frequency behavior of the complex impedance, Z*(ω) = Z0 (ω) + jZ00 (ω), the dielectric permittivity, ε*(ω) = ε0 (ω) jε00 (ω), the electric modulus, M*(ω) = M0 (ω) + jM00 (ω), and the conductance, σ*(ω) = σ0 (ω) + jσ00 (ω), for polar molecular liquids containing some thermally activated ionic admixtures. All these complex quantities are alternative representation of the same macroscopic relaxation data and can be transformed to each other according to the scheme: 1 jωε0 ¼ M ðωÞ ¼ jωC0 ZðωÞ ¼ ð1Þ ε ðωÞ σ ðωÞ

with the real (ε0 ) and imaginary (ε00 ) parts: σ0ion ε0 ¼ εs , ε00 ðωÞ ¼ ωε0

where C0 = kε0 is the electric capacity of empty measuring cell (the value C0 = 10 pF was taken in all theoretical calculations), k = S/l, S and l are the electrode surface and the distance between the electrodes, respectively, ω = 2πf is the angular frequency of the electric stimulus, f is the linear frequency, ε0= 8.85 pF/m is the permittivity of free space, and j = (1)1/2. According to eq 1, the general relations between the real (ε0 ) and imaginary (ε00 ) parts of the dielectric permittivity and those of the electric modulus, the impedance, and the electric conductivity are, respectively, the following:

and

M 0 ðωÞ ¼

0

ε , ε02 þ ε002

1 ε00 , Z0 ðωÞ ¼ 02 ωC0 ε þ ε002

00

ε ε02 þ ε002

ð2Þ

1 ε0 Z00 ðωÞ ¼ 02 ωC0 ε þ ε002

ð3Þ

M 00 ðωÞ ¼

The simplicity of eq 6 allows one to obtain, from general eqs 2 and 3, the following exact analytical expressions for frequency dependence of the electric modulus and the impedance of the molecular material in its static dielectric regime, where ionic carriers are moving due to an applying of the electric stimulus of angular frequency ω: 0

Mion ðωÞ ¼ εs 1

0

Zion ðωÞ ¼

σ00 ðωÞ ¼ ωε0 ε0

R0 , 1 þ ω2 τion 2

Mion ðωÞ ¼ εs 1

00

Mion ðωÞ ¼

00

Zion ðωÞ ¼

εs 1 ωτion 1 þ ω2 τion 2

R0 ωτion 1 þ ω2 τion 2

ð7Þ

ð8Þ

εs 1 1 þ jωτion

ð9Þ

and

Zion ðωÞ ¼ ðkσ0ion Þ1

1 1 þ jωτion

ð10Þ

where

ð4Þ

τion ¼ ε0

The paper1 concerns the ionically conducting molecular material being in a static dielectric regime. In such a limiting case, the material is fully described by its static dielectric permittivity (εs) and the dc ionic conductivity (σ0ion) and the complex permittivity is given by the following relation: σ 0ion ð5Þ εðωÞ ¼ εs þ jωε0 r 2011 American Chemical Society

εs 1 , 1 þ ω2 τion 2

R0 is the resistance of the sample recorded for ω f 0 which, in general, is presented as the dc conductivity (σ0ion) of studied material, R0 = 1/kσ0ion. In the complex notation, eqs 7 and 8 take, respectively, the following form:

and σ0 ðωÞ ¼ ωε0 ε00 ,

ð6Þ

εs σ0ion

ð11Þ

is known as the (ionic) conductivity relaxation time.2 Received: September 30, 2011 Accepted: December 9, 2011 Revised: November 23, 2011 Published: December 09, 2011 807

dx.doi.org/10.1021/ie202243x | Ind. Eng. Chem. Res. 2012, 51, 807–813

Industrial & Engineering Chemistry Research

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The above relations show that in the static dielectric regime of conducting dielectric material, both the impedance and the electric modulus exhibit the relaxational behavior of the Debye-type and their spectral characteristics are determined by the static dielectric permittivity (εs) and the dc ionic conductivity (σ0ion). The relaxation process of both Z/ion(ω) and M/ion(ω) is characterized by the same relaxation time, τion, which is proportional to the ratio of the static permittivity to the dc conductivity (eq 11). Finally, in the static dielectric state, eq 4 describing the real (σ0 ion) and imaginary (σ00 ion) parts of the complex ionic conductivity, reduce themselves to a simple form: 0

00

σion ¼ σ0ion ,

σion ðωÞ ¼ ωε0 εs

ð12Þ

The frequency dependence of the imaginary part of the conductivity, according to eq 12, has a form of straight line of the slope + 1 (in loglog scale). In the present paper, another important limiting case will be considered. Here we will discuss the relaxational behavior of the electric modulus, the impedance, and the electric conductivity (all as the complex quantities), for molecular material where the ionic conductivity is negligibly low (σ0ion ≈ 0) but the displacement current due to the molecular dipoles relaxation occurs.3 So, the molecular system will be characterized by the following three quantities: the low (εs) and the high (ε∞) frequency limits of the dielectric permittivity and the dielectric (dipolar) relaxation time (τD). In the simplest case when in the studied system one is dealing with one type of the molecular relaxor and its relaxation process proceeds throughout the normal Brownian rotational diffusion, then, as it was shown by P. Debye,4 these three quantities are linked by the following relaxational equation: εðωÞ ¼ ε∞ þ

εs ε∞ 1 þ jωτD

Figure 1. Sketch presenting the dielectric relaxation spectra (a) [the real (ε0 ) and imaginary (ε00 ) parts] calculated with eqs 14 for εs = 50, ε∞ = 3, and three different values of the dipoles relaxation time: τD1 = 106 s, τD2 = 107 s, and τD3 = 3 107 s. In panel b the spectra are presented in the complex plane.

ð13Þ

with the real (ε0 ) and imaginary (ε00 ) parts of the complex permittivity as follows: ε0 ðωÞ ¼ ε∞ þ

εs ε∞ , 1 þ ω2 τ D 2

ε00 ðωÞ ¼

ðεs ε∞ ÞωτD 1 þ ω2 τ D 2

ð14Þ

As the terms of eq 14 are more complicated than those of eq 6, their substitution to the general eqs 24 does not lead to any comprehensive analytical form of the expressions representing the electric relaxational effects due to an existence of the displacement current in dielectric materials. Hence, in the paper, the numerical dependencies representing eqs 24 will be considered, calculated for given sets of the parameters of eq 14. Next, some empirical equations describing the dependencies obtained will be formulated. Finally, the experimental data will verify the theoretical consideration and the formulated equations.

2. THEORETICAL CONSIDERATIONS In Figure 1a are sketched the dielectric relaxation spectra of the Debye-type [real, ε0 (ω), and imaginary, ε00 (ω)], which correspond to the following values of the parameters of eq 14: εs = 50, ε∞ = 3 and three different dielectric relaxation times: τD1 = 106 s, τD2 = 107 s and τD3 = 3 107 s. In Figure 1b the spectra are presented in the complex plane: one semicircle represents all the spectra (the same εs and ε∞) and in that representation only the different frequency at the maximum of the circle correspond to the different dipolar relaxation time: = (2πτDi)1, (i = 1, 2, 3). fmax i

Figure 2. The spectra of the electric modulus corresponding to the dielectric spectra from Figure 1: (a) the real M0displ and imaginary M00displ parts as a function of the frequency [calculated with eqs 2], (b) the modulus spectra in the complex plane.

The dielectric permittivity data from Figure 1, transformed into the electric modulus spectra (according to eq 2), are depicted in Figure 2 as the frequency dependence (a) and in the complex plane (b). It can be easy verified that the obtained electric 808

dx.doi.org/10.1021/ie202243x |Ind. Eng. Chem. Res. 2012, 51, 807–813


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Figure 4. The impedance spectra in the complex plane, ΔZ00displ[ Z00displ + (ωεsC0)1] vs Z0displ, for the system where the displacement current occurs.

figure, the real part of the impedance, Z0displ(ω), exhibits a typical relaxational behavior, while the relaxational effects in the imaginary part, Z00displ(ω), are strongly masked by the capacity effect related to the static dielectric permittivity, (ωεsC0)1. After its removal, one obtains an increment of the imaginary part of impedance, ΔZ00displ(ω) = Z00displ(ω) + (ωεsC0)1, which has a form of the Debye-type absorption band, shown in Figure 3b as the dashed lines. As shown in Figure 4, in the complex plane the increment (ΔZ00displ) and the real part (Z0displ) of the impedance form the semicircles with the centers placed on the real axis of the impedance. Empirical equations describing the frequency dependence of the real, Z0displ(ω), and imaginary, Z00displ(ω) = ΔZ00displ(ω) + (ωεsC0)1, parts of the impedance of the system where the displacement current occurs (Figure 3a,b, respectively), can be formulated as following:

Figure 3. The impedance real (a) and imaginary (b) spectra (calculated with eqs 3) corresponding to the dielectric relaxation spectra from Figure 1.

modulus spectra are of the Debye-type and the spectra can be described with the following empirical equation: 0

Mdispl ðωÞ ¼ ε∞ 1

ε∞ 1 εs 1 , 1 þ ω2 τdispl 2

00

Mdispl ¼

0

Zdispl ðωÞ¼

Z0displ ωτdispl 1 þ ωεs C0 1 þ ω2 τdispl 2

or in the complex notation

Zdispl ðωÞ ¼

or in the complex notation: ε∞ 1 εs 1 1 þ jωτdispl

00

Zdispl ðωÞ ¼

ð18Þ

ðε∞ 1 εs 1 Þωτdispl 1 þ ω2 τdispl 2

ð15Þ

Mdispl ðωÞ ¼ ε∞ 1

Z0displ , 1 þ ω2 τdispl 2

ð16Þ

Z0displ 1 þ jωεs C0 1 þ jωτdispl

ð19Þ

with the relaxation time identical to that corresponding to the electric modulus (eq 17). Finally, the conductivity of a system, in which the displacement current occurs, is presented in Figure 5 in form of the conductivity spectra. The spectra were calculated for the dielectric relaxation data from Figure 1 with the use of eq 4. The real part of the conductivity shows the frequency dependence formally analogous to that observed for the real part of the electric modulus (Figure 2). However, the relaxation time of the displacement conductivity is identical to the dipolar relaxation time (τD). The high frequency limit of the conductivity (σ∞ displ, Figure 5a) can be obtained from the following equation:

It can be also found that the relation between the relaxation time of the displacement current, τdispl, and the relaxation time of the molecular dipoles, τD, contains the ratio between the two limiting dielectric permittivities of the material studied: ε∞ τdispl ¼ τD ð17Þ εs According to eq 17, the relaxation time τdispl is always shorter than τD and, for example, for the data from Figure 1, τdispl = 6 102τD. Besides, as ε∞ only weakly depends on temperature and both the temperature dependences εs(T) and τD(T) are, in general, of the same type (with increasing temperature both the quantities are decreasing); it results from eq 17 that in a typical liquid behavior, τdispl is expected to be rather weakly dependent on the temperature. In the complex plane, similarly to the dielectric relaxation spectra (Figure 1b), the electric modulus spectra are also reand ε1 presented by one semicircle, determined by ε1 s ∞ (Figure 2b). The dielectric spectra transformed into the impedance spectra with the use of eq 3 are presented in Figure 3. As seen in the

0

σ displ ðωÞ ωε0 ε0 ¼ ε0 ðεs ε∞ Þ which for ω f ∞ leads to expression εs ε∞ 0 σ displ f σ∞ displ ¼ ε0 τD

ω2 τ D 1 þ ω2 τ D 2

ð20Þ

ð21Þ

As seen in Figure 5b, the frequency dependence of the imaginary part of the conductivity is strongly masked by the high-frequency 809



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Figure 6. The conductivity spectra in the complex plane, Δσ00displ[σ00displ ωε0ε∞] vs σ0displ, for the system where the displacement current occurs.

Figure 5. The conductivity real (a) and imaginary (b) spectra [calculated with eqs 4] corresponding to the dielectric relaxation spectra from Figure 1.

dielectric permittivity effect. After removal of ωε0ε∞ part from the total imaginary conductivity, one obtains the increment Δσ00displ(ω) = σ00displ(ω) ωε0ε∞, which has a form of the Debye-type absorption band (dashed lines in Figure 5b). As presented in Figure 6, the increment Δσ00displ versus the real part of the conductivity σ0displ, form the semicircles which intercept the real conductivity axis at zero, in the low frequencies side and at σ∞ displ, in the high frequencies side. The maximum of the circle corresponds to the dipolar relaxation time of the system investigated. Empirical equations describing the frequency dependence of the real, σ0displ(ω), and imaginary, σ00displ(ω) = Δσ00displ(ω) + ωε0ε∞, parts of the conductivity of the system (Figure 4 panels a and b, respectively) where the displacement current occurs, can be formulated as follows: 1 þ

ω2 τ D 2

,

00

σ displ ðωÞ ¼

σ∞ displ ωτ D 1 þ ω2 τ D 2

þ ωε0 ε∞

)

σ∞ displ

0

σ displ ðωÞ¼ σ ∞ displ

Figure 7. The experimental dielectric relaxation spectra, real (a) and imaginary (b), recorded in the liquid crystalline smectic A phase of n-octylcyanobiphenyl (8CB) when the long axes of the mesogenic molecules are oriented parallel to the probing electric field (ε ). The solid lines represent the best fit of eq 14 to the experimental data (points).

ð22Þ or in the complex notation:

σdispl ðωÞ ¼ σ∞ displ

σ∞ displ 1 þ jωτD

þ jωε0 ε∞

(the Debye-type spectrum), should be recorded. As a matter of fact, it is not very easy to find out the soft material which fulfills these seemingly simple requirements. It was found that those requirements are quite well fulfilled by the liquid crystalline (high viscous) smectic A phase of the compound belonging to the well-known cyanobiphenyls family, so the system was taken as a subject of our studies.

ð23Þ

For experimental illustration of the electric relaxational effects due to the displacement current occurring in dielectric materials, the following two requirements related to the studied material should be fulfilled in the frequency range used: (i) An ionic conductivity should be negligible low (σ0ion ≈ 0), and (ii) a single dielectric spectrum, representing the relaxation of dipolar molecules throughout the normal Brownian rotational diffusion

3. EXPERMENTAL PROCEDURE The studied mesogenic compound, n-octylcyanobiphenyl (C 8H17PhPhCtN, 8CB), was synthesized and purified at the 810



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Figure 8. Temperature dependences of the static (εs) and the high frequency (ε∞) permittivities of 8CB in the smectic A phase.

Figure 10. The electric modulus spectra of 8CB in the smectic A phase resulting from transformation of the dielectric relaxation spectra with equation M* = 1/ε*. Figure 9. Arrhenius plots for the dielectric (τD) and displacement (τdispl) relaxation times of 8CB in the smectic A phase.

4. RESULTS AND DISCUSSION Figure 7 presents the experimental dielectric spectra (the real (a) and the imaginary (b) parts) recorded in the temperature range of existence of the liquid crystalline smectic A phase of 8CB, where the long molecular axis of the mesogenic molecules is oriented perpendicular to the surface of the measuring cell electrodes. One absorption band, corresponding to the molecular rotation around the short axis, is strongly dominating in the spectra. The experimental data (points) can be perfectly described with the Debye’s eq 14, represented in Figure 7a,b by the solid lines. However, a slight deviation from the Debye’s behavior appears for the frequencies higher than 107 Hz (seen in Figure 7b) that reflects the beginning of the next dielectric absorption band, related to the molecular rotations around the long axis. The maximum of that second band appears in the gigahertz region.68 From the best fit of the Debye eq 14 to the experimental spectra one obtains the values of three quantities: the dielectric permittivities εs and ε∞, and the relaxation time of dipolar reorientations τD. The temperature dependences of these parameters are presented in Figure 8 and Figure 9, respectively. The dielectric relaxation spectra of 8CB, transformed with the use of eq 2 into the electric modulus spectra, are presented in Figure 10. The solid lines in the picture represent the best fit of eq 15 to the modulus data (points). As it is clearly seen, the electric modulus spectra are perfectly Debye shaped. As expected, the dielectric permittivities resulting from the best fit of eq 15 to the spectra are identical to those depicted in Figure 8. The relaxation time of the electric modulus (τdispl), together

)

Institute of Chemistry, Military University of Technology, Warsaw, Poland. The temperatures of the phase transitions of the compound are the following: crystal (22 °C), smectic A (32.5 °C), nematic (43 °C), isotropic liquid. The dielectric spectra of 8CB were recorded with the use of an HP 4194A impedance/gain phase analyzer in the frequency range from 200 kHz to 50 MHz and in the temperature range from 22 to 32.4 °C. A measuring capacitor consisted of three plane electrodes (with a surface of about 0.8 cm2): one central and two grounded on each side, with a distance between them of about 0.1 mm. The shape of the capacitor electrodes is rectangular, and they are made with a gold-plated copper. The threeelectrodes capacitor construction was taken due to its good frequency characteristics up to about 100 MHz.5 The capacity of the empty cell used (C0) was equal about 15 pF. The probing electric field intensity E was equal to about 1 V/mm. An additional biasing dc electric field of intensity about 5 V/mm was applied to the sample in the nematic phase to orient the molecules perpendicular to the electrode surface, and next, conserving that orientation, the temperature was decreased to reach the smectic phase. So, in our experiment the parallel component (ε ) of the permittivity tensor was measured. The electrical heating of high performance with the use of a “Scientific Instruments” temperature controller, model 9700, assured very good temperature stabilization (at millikelvin level). Such equipment allows one to determine the impedance with an uncertainty less than 1%. 811



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Figure 11. Experimental verification of eq 17.

Figure 13. The conductivity spectra, real (a) and imaginary (b), of 8CB in the smectic A phase resulting from transformation of the dielectric relaxation spectra with equation σ* = jωε0ε*.

dielectric absorption band reflecting the molecular rotations around their long axis. A fundamental problem arises in presentation of the dielectric relaxation spectra (Figure 7) in the form of the impedance (Figure 12) and the conductance (Figure 13) spectra. In the normal physical mining of the term of “electric conductivity”, an apparent inconsistency between these two presentations manifests itself when one compares the frequency dependences of the real parts of the impedance and the conductance (Figures 12a and 13a). In the low frequency range, the “static” impedance Z0displ value of about 200 Ω, corresponds to the “dc” conductivity σ0displ equal to zero. In the high frequency range, an inverse phenomenon is observed: a decreasing of Z0displ to zero is followed by an increasing of σ0displ to the final, quite important value 3 S 3 m1. The comprehensive physical basis of σ∞ displ ≈ 2 10 the effect was given by J. R. Macdonald.9

Figure 12. The impedance spectra, real (a) and imaginary (b), of 8CB in the smectic A phase resulting from transformation of the dielectric relaxation spectra with equation Z* = (jωC0ε*)1.

with the dipolar relaxation time (τD), is shown in Figure 9 in a form of an Arrhenius temperature dependence. As can be noticed, the activation energy of the both relaxation times are very close to each other. As shown in Figure 11, the relation between the displacement relaxation times, τdispl, and the dipolar relaxation time, τD, corresponds perfectly to eq 17. The solid lines in Figures 12 and 13 represent the best fit of eqs 18 and 22 to the impedance and the conductance spectra, respectively, resulting from the transformation (with the use of eqs 3 and 4, respectively) of the dielectric relaxation spectra recorded for the smectic A phase of 8CB. A small increase of the experimental impedance, observed in the low frequency range in Figure 12a, points out the beginning of the ionic conductance area of the compound studied. However, an analogous effect, observed in the high frequencies of the conductance spectra (Figure 13a), is a consequence of the beginning of the second

5. CONCLUSIONS Indeed, the displacement current, reflecting the changes in the electrons density on the surfaces of electrodes of the measuring cell, that is, the charge transporting in the external electric circuit, cannot be considered in the same way as the ionic conduction current, that is, the transporting charge throughout the dielectric material. That circumstance may lead to serious difficulties in analysis and interpretation of the electric spectra in systems where two effects, the ionic current and the displacement current (the dielectric relaxation), are occurring simultaneously in the frequency range used.10 An analysis of the papers published up to 812



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now in that field may show that the studied systems where the two discussed currents are well separated in the frequency range used are probably rather rare. It is worth mentioning here that, in principle, the Maxwell’s equations preclude distinguishing between conduction and displacement currents by means of external electrical measurements, as discussed by J. R. Macdonald.11 Therefore, in future work we plan to compare dielectric-model fits of the present data with a conductive-system ionic diffusional model, such as that discussed in ref 10.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ REFERENCES

(1) Swiergiel, J.; Jad_zyn, J. Electric relaxational effects induced by ionic conductivity in dielectric materials. Ind. Eng. Chem. Res. 2011, 50, 11935–11941. (2) Barsoukov, E.; Macdonald, J. R. Impedance Spectroscopy: Theory Experiment & Applications, 2nd ed.; John Wiley & Sons: London, 2005. (3) Heras, Jose A. A formal interpretation of the displacement current and the instantaneous formulation of Maxwell’s equations. Am. J. Phys. 2011, 79, 409–416. (4) B€ottcher, C. J. F.; Bordewijk, P. Theory of Electric Polarization: Dielectric in Time-Dependent Fields; Elsevier: Amsterdam: 1992; Vol 2. (5) Legrand, C.; Parneix, J. P.; Chapoton, A.; Tinh, N. H.; Destrade, C. Study of reentrant nematic and smectic phases using dielectric relaxation. Mol. Cryst. Liq. Cryst. 1985, 124, 277–285. (6) Jad_zyn, J.; Czechowski, G.; Dejardin, J.-L.; Ginovska, M. Anomalous rotational diffusion in the vicinity of the isotropic to nematic phase transition. J. Phys.: Cond. Matter 2005, 17, 813–819. (7) Bose, T. K.; Campbell, B.; Yagihara, S.; Thoen, J. Dielectricrelaxation study of alkylcyanobiphenyl liquid crystals using time-domain spectroscopy. Phys. Rev. A 1987, 36, 5767–5773. (8) Markwick, P.; Urban, S.; Wurflinger, A. High pressure dielectric studies of 8CB in the isotropic, nematic, and smectic A phases. Z. Naturforsch. A 1999, 54, 275–280. (9) Macdonald, J. R. Utility of continuum diffusion model for analyzing mobile-ion immittance data: Electrode polarization, bulk, and generation-recombination effects. J. Phys.: Cond. Matter 2010, 22, 495101 (1–15). (10) Macdonald, J. R. Comments on the electric modulus formalism model and superior alternatives to it for the analysis of the frequency response of ionic conductors. J. Phys. Chem. Solids 2009, 70, 546–554. (11) Macdonald, J. R. Dispersed electrical-relaxation response: Discrimination between conductive and dielectric relaxation processes. Braz. J. Phys. 1999, 29, 332–346.

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Electric Relaxational Effects Induced by Displacement Current in

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