multiple representation of linear dielectric response

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Note

Nihon Reoroji Gakkaishi Vol.38, No.3, 149~155(Journal of the Society of Rheology, Japan)©2010 The Society of Rheology, Japan

1. INTRODUCTION

Dielectric responses of materials under weak electric field E and/or small dielectric displacement D (corresponding to an effective charge density) satisfy the linearity and are described by the Boltzmann superposition formally identical to that for the linear viscoelastic responses. Thus, the complex dielectric constant e*(w) and the complex dielectric modulus M*(w), the former representing D* induced by sinusoidally oscillating E* while the latter describing E* induced by D*, are completely equivalent to each other in a phenomenological sense and satisfy a simple relationship,1) e*(w)M*(w) = 1. In this sense, there is no preference of using e* or M* for representing the dielectric relaxation behavior of materials. This situation of dual representation is similar to that for the representation of linear viscoelastic relaxation behavior in terms of the complex modulus and compliance, G*(w) and J*(w).

However, in many cases, we naturally prefer e* rather than M* when we attempt to analyze a molecular process(es) underlying the dielectric relaxation and extract a characteristic quantity related to the molecular process, for example, a re-orientational (rotational) time of molecules having electric dipoles. This situation is again similar to the situation for the viscoelastic response, the preference of G*(w) for materials

obeying the stress-optical rule.2-5)

In this article, we first start from the most fundamental linear stimuli-response framework to re-visit the above duality in the representation of the dielectric response. Then, we focus on the relaxation functions involved in this framework and examine, for some model cases, molecular expressions of these functions that naturally results in the preference of e*. Readers familiar with the linear stimuli-response framework can skip the next section and directly proceed to the Discussion section where the preference of e* is examined.

2. BASIC PHENOMENOLOGICAL FRAMEWORK

For a material charged in a dielectric cell, we consider a polarization P under a given electric field E. If the field is zero at time t < 0 and set at a constant E at t > 0, the growth of the polarization is described by6)

(1)

Here, P~

a/e and P~

m/i, respectively, correspond to the fully-grown polarization due to fast atomic/electronic displacement (being regarded to be instantaneous in eq (1)) and slow motion of molecules/ions, and F(t) is a normalized dielectric relaxation function satisfying F (0) = 1 and F (∞) = 0. (The function {1-F(t)} is the normalized dielectric retardation function.) In the linear response regime under a small field, P(t) under an

Multiple Representation of Linear Dielectric Response

Ayoung Lee*,**, Yumi Matsumiya*, Hiroshi Watanabe*,†, Kyung Hyun Ahn**, and Seung Jong Lee**

*Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011 Japan**School of Chemical and Biological Engineering, Seoul National University, Seoul 151744, South Korea

(Received: February 5, 2010)

Within the linear stimulus-response framework, dielectric responses of materials can be described in a dual way in terms of the complex dielectric constant e*(w ) and complex dielectric modulus M*(w ) (= 1/e*(w )), both being defined as functions of angular frequency w . In a phenomenological sense, e*(w ) and M*(w ) are completely equivalent to each other. In this sense, a characteristic dielectric relaxation time t can be defined for either quantity. However, in many cases, t directly reflecting a slow molecular process(es) underlying the dielectric relaxation is defined for e*(w ) but not for M*(w ), which is similar to the situation for the viscoelastic modulus and compliance, G*(w ) and J*(w ), the former often serving as the fundamental quantity detecting the slow molecular process. Focusing on some examples, this article discusses the duality in the representation of the dielectric responses and the superiority of e*(w ) compared to M*(w ).Key Words: Dielectric response / Duality / Complex dielectric constant / Complex dielectric modulus

† to whom correspondence should be addressed. E-mail: [email protected], Tel: -81 774-38-3135, Fax: -81 774-38-3139

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Nihon Reoroji Gakkaishi Vol.38 2010

arbitrary, time (t)-dependent electric field E(t) is given by the Boltzmann superposition of the response shown in eq (1) and expressed as6)

(2)

The dielectric displacement D(t) is defined as a sum of P(t) of the material and a charge density evacE(t) for the cell itself, with evac being the absolute dielectric permittivity of vacuum. For a sinusoidal electric field E*(t) = Eexp(iw t), D*(t) obtained from eq (2) (after integral by parts) is expressed as

(3)

Thus, the complex dielectric constant, e*(w) = e '(w) - ie '(w) appearing in eq (3), is related to F(t) as6)

(4a)

(4b)

with

(high-frequency dielectric constant)(4c)

and

(dielectric relaxation intensity) (4c)

Namely, the dynamic dielectric constant e '(w) and dielectric loss e ''(w ) are directly related to the dielectric relaxation function F(t) of the material.

For the same material, we here consider the electric field E induced by the dielectric displacement D. If the dielectric displacement is zero at time t < 0 and set at a constant D at t > 0, the time evolution of the electric field (in the material) is described in terms of e vac, P

~a/e, P

~m/i, and a normalized

relaxation function Y(t) (with Y(0) = 1 and Y(∞) = 0) defined for the electric field as

(5)

Note that Y(t) is totally different from F(t), the latter being defined for the polarization under a constant electric field. The functional form of eq (5) can be easily understood in the following way. The time evolution of E(t) reflects growth

of the polarization of the material under the constraint of constant D. Clearly, E(t) agrees with D/(e vac+ P

~a/e) at t = 0

where the material just exhibits the instantaneous response and with D/(evac+ P

~a/e + P

~m/i) at t = ∞ where the fully-grown

polarization P~

m/iE(∞) (due to slow motion of molecules/ions; cf. eq (1)) contributes to the constant D. Thus, as shown in eq (5), the time evolution of E(t) is written in terms of Y(t) that decays from 1 to 0 with time.

In the linear response regime, the time evolution of E(t) under arbitrary time-dependent D(t) is given by the Boltzmann superposition of the response shown in eq (5). Thus, E*(t) for sinusoidal D*(t) = Dexp(iwt ) is given by an integral formally similar to that in eq (3), and the complex dielectric modulus, M*(w) = M'(w) + i M''(w) = evac E*(t)/D*(t), can be expressed in terms of Y(t) as

(6a)

(6b)

with De = P~

m/i /e vac and e∞ = 1 + P~

a/e (cf. eqs (4c) and (4d)). Namely, the real and imaginary parts of the complex dielectric modulus, M'(w) and M''(w), are directly related to Y(t) of the material.

As can be noted from eqs (1)-(6), the relaxation functions F(t) and Y(t) are related to each other through a convolution equation to give the equivalent quantities, e *(w ) and M*(w ) = 1/e*(w ). This situation is similar to that for the relaxation modulus G(t) and creep compliance J(t) (giving G*(w) and J*(w) = 1/G*(w), respectively).

3. Discussion

The complex dielectric constant e*(w ) and the complex dielectric modulus M*(w ) are equivalent to each other in a phenomenological sense, as explained in the previous section. In this sense, we may utilize either e*(w) or M*(w) for representing the dielectric behavior of materials. However, we naturally prefer one of these quantities when we focus on a molecular/structural expression of the underlying relaxation functions, F(t) and Y(t). This preference is examined below for some simple cases.

3.1 MolecularMotion

Most of organic molecules have permanent dipoles attached to their functional groups. Thus, their thermal, re-orientational (rotational) motion results in the dielectric relaxation. At

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A. LEE • MATSUMIYA • WATANABE • AHN • S. J. LEE : Multiple Representation of Linear Dielectric Response

equilibrium, the dielectric relaxation function defined for the polarization, F(t) introduced in eq (1), is unequivocally related to the orientational memory of the molecular dipoles pj ( j = index for molecules) and expressed as6,7)

(7)

with <…> being the ensemble average taken for all dipoles at equilibrium. For the molecules having weak dipoles, the dipole-dipole interaction energy is negligible compared to the thermal energy kBT (kB = Boltzmann constant, T = absolute temperature). For this case, pj of a given molecule j has no correlation with pq of the other molecule q in a sense that the opposite dipole orientations pq and –pq of the molecule q occur with the same probability for a given orientation pj of the molecule j. Then, F(t) detects the orientational auto-correlation of individual dipoles (i.e., no cross-correlation), as shown in eq (7). This dipolar auto-correlation can be replaced by the auto-correlation of the molecular orientation for some cases, as explained below.

For rigid, rod-like molecules having the dipole p parallel along the molecular end-to-end vector u, F(t) is given by F(t) = <u(t)•u(0)>/<u2> = exp(-t/t r) with t r being the rotational relaxation time of the molecule. The corresponding dynamic dielectric constant and dielectric loss exhibit a Debye-type single relaxation (cf. eq (4)),

(8)

It should be noted that the angular frequency for the dielectric loss peak, 1/t r, unequivocally represents the characteristic time for the molecular motion (rotation). In addition, the dielectric relaxation intensity for the rod-like molecules, De ∝ np2, gives the information for the number density of the molecules, n .

From eq (8), the real and imaginary parts of the complex dielectric modulus of those rod-like molecules are simply obtained as

(9a)

(9b)

with

(9c)

The corresponding Y(t) is given by Y(t) = exp(-t/t r' ). For

analysis of the molecular dynamics, this single Maxwellian relaxation behavior of M* associated with a non-zero equilibrium plateau of M' may appear to be as useful as the Debye behavior of e* (eq (8)). However, we should note that the Maxwellian relaxation time t r' obtained for M* is affected by the dielectric relaxation intensity De (∝ np2) as well as the high-frequency dielectric constant e ∞, meaning that molecules exhibiting the same dynamics (having the same t r) but different magnitudes of the dipole |p| exhibit different t r' for M*; see eq (9c). This fact demonstrates that M* does not straightforwardly reflect the dynamics of the rod-like molecules. For this reason, the study of the dynamics of those molecules is preferably made for the complex dielectric constant, e*(w).

As an example of the dynamic behavior of rod-like molecules, Figure 1 shows the literature data for the linear viscoelastic storage and loss moduli, G' and G'', and the dynamic dielectric loss, e '', measured for N,N’,N”-tris(3,7-dimethyloctyl)benzene-1,3,5-tricarboxamide (DO3B) in n-decane at 25ºC (DO3B concentration = 30 mM).8) The imaginary (loss) part of the complex dielectric modulus, M'', evaluated from the e ' and e '' data (with the e ' data being not reported in ref.8 but kindly supplied by Prof. T. Shikata at Osaka University), is also shown. In n-decane, DO3B molecules form helical columnar aggregates through hydrogen bonding to form supramolecular chains that can be regarded as the rods up to a considerably large persistent length.8,9) These supramolecular chains have huge dipoles (so-called macro-dipoles) parallel along the chain backbone due to the helical columnar stacking of the DO3B molecules.8,9) The viscoelastic relaxation observed for the G' and G'' data

Fig. 1. Viscoelastic and dielectric behavior of DO3B supramolecular chains formed in n-decane at 25 °C (DO3B concentration = 30 mM).

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results from the phantom crossing of the supramolecular chains through the exchange of the hydrogen bonds,8,9) and their mode distribution is close to the single Maxwellian distribution,

(10)

In Figure 1, the thin solid curves well describing the G' and G'' data show the modulus given by eq (10) with the viscoelastic relaxation intensity G = 250 Pa and the viscoelastic relaxation time t = 0.82 s. For the phantom crossing mechanism, the DO3B supramolecular chain would behave transiently as a rod-like fragment that rotates and then recombines itself with the other fragments. For such rotational motion of the rods, the dielectric relaxation time t r appearing in eq (8) is expected to be three times of the viscoelastic t .10) In Figure 1, the thick solid curve indicates e '' given by eq (8) with t r = 3t = 2.46 s. This curve well describes the e '' data, suggesting that the phantom passing process of the rod-like fragments is consistently detected with the G* and e '' data. On the other hand, for the M'' data, the relaxation time (= 0.25 s) evaluated from the peak frequency is much smaller than t r (as explained for eq (9)), and no straightforward correlation can be easily noted between the G* and M'' data. This result demonstrates the preference of using the e* data for the study of the slow dynamics of the rod-like molecules.

Another example demonstrating the preference of using e * is found for so-called type-A polymers such as cis-polyisoprene that have segmental, type-A dipoles parallel to the end-to-end vector un of the segments.11,12) This type-A dipole is somewhat similar to the macro-dipole of the DO3B supramolecular chain, but the covalent bonds along the backbone of the type-A chain never allows this chain to relax through the phantom crossing mechanism. Since the total dipole of the type-A chain is proportional to the end-to-end vector of the chain as a whole, R = Snun, the dielectric relaxation function F(t) of the chain in long time scales (where the local segmental relaxation has completed) detects the end-to-end vector fluctuation due to the thermal, global motion of the chain; namely, F (t) = <R(t)•R(0)>/<R2>. Thus, the longest relaxation time detected for e*(w) coincides with the characteristic time for the slowest mode of this fluctuation, meaning that the e *(w ) data give the straightforward information for the chain dynamics. This information is smeared in the M*(w) data, demonstrating the preference of using the e*(w) data for the study of the slow chain dynamics.

Furthermore, the slow viscoelastic relaxation of flexible

chains detects a decay of the isochronal orientational anisotropy, Sn<un(t)un(t)> with un(t)un(t) being the dyadic of un(t) at time t, as evidenced from the stress-optical rule.11) Since F(t) detects fluctuation of R(t) = Sn un(t), the chain dynamics (time evolution of un(t)) activating the relaxation of F(t) is identical to that for the relaxation of the relaxation modulus G(t). However, the averaging moment of un(t) at time t is different for F(t) and G(t); the first and second-order moments for F (t) and G(t), respectively. This difference enables us to examine detailed aspects of chain dynamics (such as the motional correlation of entanglement segments along the chain backbone11,13,14)) through comparison between the F(t) and G(t) data, or equivalently, between the e*(w) and G*(w) data. Similar comparison cannot be straightforwardly made for M*. This fact also demonstrates the preference of using e*(w) in the study of slow chain dynamics.

3.2 SlowIonicConduction

In the idealized case of direct current (dc) conduction due to ions in a matrix exhibiting no dielectric relaxation/loss, the dynamic dielectric constant and dielectric loss are simply written as

(11)

where em is the (static) dielectric constant of the matrix and s 0 is the ionic conductance. The corresponding complex dielectric modulus is given by

(12a)

with

(12b)

The simplest dc conduction considered here corresponds to steady drift of ions so that it is associated with no characteristic time. Nevertheless, M* exhibit the single-Maxwellian relaxation with the characteristic time t'' that merely corresponds to a matching frequency defined for e '(w) and e ''(w) given by eq (11) as e '(1/t) = e ''(1/t '') ; also see Figure 2. This t '' is not related to the ion dynamics at all, meaning that the dynamics underlying the dc conduction is not well represented by M*. This fact has been pointed out in several studies15-18) for ionic conduction.

In the actual ionic conduction process observed for mixtures of salt in organic matrices (such as lithium salt dissolved in polar polymers), the ions cannot discharge at the electrodes. Then, the observed conduction is not the simple/idealized dc conduction but is attributed to the electrode polarization

153


(EP).19-21) When the matrix exhibits no dielectric relaxation/loss, the complex dielectric constant for the EP process at low w is satisfactorily approximated as that for the Debye relaxation as20,21)

(13)

(Calculation of e* based on the Macdonald theory19) is briefly described in Appendix A.) The corresponding complex dielectric modulus is given by

(14a)

(14b)

with

(14c)

Reflecting the interaction between the ions, the characteristic time tEP appearing in eqs (13) and (14c) depends on the mobility m and number density n of the ions while the dielectric intensity DeEP depends on n .20,21) In addition, tEP and DeEP depend on the gap L between the electrodes because the time required for the ion motion between electrodes and the electrostatic energy stored by ions concentrated near the electrodes increase with L; actually, we find tEP ∝ L and DeEP ∝ L (cf. Appendix A). The e* data straightforwardly give tEP and DeEP exhibiting these

characteristic EP features, while the M* data do not, as noted from comparison of eqs (13) and (14). In particular, the M* data give t 'EP being insensitive to L and much smaller than the real tEP for the case of em << DeEP (which is the case for most of ion conductors). Thus, the slow EP phenomenon is better analyzed for e* data than for the M* data.

3.3 FastConductingProcess

In the slow conducting process described in the previous section, the ions are regarded as an object having a constant friction coefficient and moving over a long distance in an averaged medium. In a fast conducting process at short times, the ion moves over just a short distance and thus its dynamic behavior detects local (and heterogeneous) structure of the medium. At such short times and/or high angular frequencies w , the simple dc conduction vanishes and the characteristic behavior such as the hopping conduction16,17,22,23) is observed. This behavior is conveniently described for the real part of the complex conductance, s '(w), as

(15)

where s 0 is the static conductivity, w c is the angular frequency characterizing the hopping, and s (≤ 1) is a weakly w-dependent exponent reflecting the hopping barrier height; in the simplest case, s = 1-2/ln(w/w*) with w* being the attempt frequency for hopping the highest barrier that determines the dc conductivity.23)

Apart from those details of the hopping mechanism, we should emphasize that the quantity s '(w) straightforwardly representing the high-w conduction is uniquely related to e ''(w ) (s '(w ) = iwe ''(w )) and simply expressed in terms of the relaxation function F(t) introduced in eq (1). Thus, the complex dielectric constant e*(w ) described by this F (t) (cf. eq (4)), not the complex dielectric modulus M*(w ), serves as the fundamental quantity for representation of the hopping conduction equally useful compared to the complex conductivity s*(w) .

3.4 DielectricRelaxationofMulti-phaseSystems

The dielectric response of multiple-phase systems can be expressed as a linear combination of the responses of all phases (unless the surface polarization is important). Specifically, for the system composed of the phases of two components stacked in series between the electrodes, the response of the system is conveniently represented as

(16)Fig. 2. Comparison of complex dielectric constant e * and complex dielectric modulus M* for dc-conducting materials.

154


where f j and M j* represent the volume fraction and complex dielectric modulus of the component j (= 1 and 2). Thus, the response of the series-phase system is more conveniently represented with M* than with the complex dielectric constant e*. (However, we should also note that e* is more convenient for systems containing side-by-side phases each occupying the whole gap between the electrodes and that the response of the molecules/ions in each phase is more easily analyzed for e* than for M*.)

4. Concluding Remarks

We have summarized the phenomenological framework common for the complex dielectric constant e*(w) and complex dielectric modulus M*(w) and further demonstrated, from a molecular consideration, that e*(w) is superior to the complex dielectric modulus M*(w ) in most of the studies for slow dielectric relaxation processes due to motion of molecules/ions. At the same time, we should emphasize that e*(w) and M*(w) are equivalent in the phenomenological sense, and this superiority of e*(w) emerges only when the molecular analysis of the dielectric relaxation processes is conducted.

Acknowledgment

This work was partly supported by the National Research Foundation of Korea (NRF) grant (No. 0458-20090039) funded by the Korea government(MEST), by Grant-in-Aid for Scientific Research on Priority Area “Soft Matter Physics” and also by Grant-in-Aid for Young Scientists (B)

from the Ministry of Education, Culture, Sports, Science and Technology, Japan (No. 18068009 and 22750204). The authors cordially thank Prof. T. Shikata at Osaka University for kindly supplying the dynamic data of the DO3B/n-decane system examined in Figure 1.

Appendix A. Macdonald theory for electrode polarization.

Macdonald19) formulated a time-evolution equation for the concentration profile of ions in a medium containing a dissociated monovalent salt. The dissociation of salt is not affected by the applied electric field if the field is sufficiently weak. For this case, the dynamic dielectric constant e '(w) and dielectric loss e ''(w) can be explicitly calculated as functions of the angular frequency w, the mobility m and number density n of the ions, the dielectric constant of the medium em, and the gap between the electrodes L. Although this calculation is based on a rather complicated set of equations summarized in ref.20, the resulting e '(w) and e ''(w) are satisfactorily approximated as the simple Debye function,20,21) eq (13). As an example, Figure 3a shows e '(w) and e ''(w) calculated for a set of parameters mimicking those in a LiClO4/poly(ethylene oxide) mixture,20) n = 1 × 1024 m-3, m = 1 × 10-9 m2 V-1 s-1, em = 10, and L = 0.02-2.5 mm. e '(w) and e ''(w) clearly change with L, which is the characteristic to the electrode polarization (EP).

In Figure 3a, we note that e '(w) and e ''(w) for each L value are well described by the simple Debye function, eq (13). Thus, the low-w parts of the e '(w ) and e ''(w ) curves for different L values are excellently superposed by vertical and

Fig. 3. (a): Complex dielectric constant e* calculated from Macdonald theory with n = 1×1024 m-3, m = 1×10-9 m2 V-1 s-1, e m = 10. The gap between the electrodes L is varied.

(b): Superposition of the e* curves in panel (a) after diagonal shift in -45˚ direction in the double logarithmic scale. Thick solid curves indicate the Debye relaxation due to the electrode polarization. (The high-frequency plateau of e ' is not included in this curve.)

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horizontal shifts of the curves in the double-logarithmic scale, as shown in Figure 3b. The vertical shift factor, Q, represents a change of the dielectric EP intensity De EP with L, while the horizontal shift factor tEP bringing the e '' peak to wtEP = 1 coincides with the EP time appearing in eq (13). Figure 4 shows changes of Q and tEP with L. Clearly, both Q and tEP are proportional to L, which naturally reflects a fact that both of the time for the ion motion between the electrodes and the electrostatic energy stored between the electrodes increase with L. Although not shown here, Q is independent of m but increases with increasing n , while tEP decreases with increasing m and n .20) These results can be cast in an empirical equation,20)

(A1)

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Fig. 4. Shift factors utilized in the superposition in Figure 3.

multiple representation of linear dielectric response

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