About dielectric relaxation in highly cross-linked poly(ethylene oxide)

Abstract

The effect of lithium salt on dynamics in highly cross-linked poly(ethylene oxide) (PEO) has been investigated. The elaboration uses results of dielectric relaxation studies. It turns out that competition of electric and structural relaxation coins conductivity mechanism. Neat cross-linked PEO with low mesh size can be transferred in super-cooled liquid state. Then, cross-linked PEO behaves like a hydrogen-bonded liquid since crystallization is strongly suppressed. As a result, one observes slow Debye-like relaxation at low temperature. It disappears after addition of salt since interaction of salt with polymer chains is stronger than the hydrogen-bonded network in the neat polymer. Analysis of tangent-loss spectra shows: Particle density governing dc conductivity does not depend on temperature at low concentration of added salt. It increases with temperature for neat cross-linked PEO and PEO loaded with sufficiently high concentration of salt. Prevention of crystallization requires a tight network of cross-links. Scaled representations of relevant impedance data for neat cross-linked PEO over extended ranges of frequency and temperature reveal that electric and structural relaxations are independent of temperature to good approximation at low and high frequency. There is a range of damped network oscillations, sandwiched between these limits, where relaxation becomes dependent on temperature. This range lessens with temperature. It does not occur at all in salt-comprising cross-linked PEO.

Appendix. Impedance spectra—phenomenological background

The summarizing view is appended for illustration of impedance spectra. In the following, we are chiefly focusing on impedance spectra of polar polymers. One observes frequently in imaginary part Z″(ω) of those spectra two characteristic frequencies, indicating extreme values of imaginary part Z″

$$ {\omega}_{\mathrm{min}}^{Z"}<\omega <{\omega}_{\mathrm{max}}^{Z"} $$

(24)

The main dipolar relaxation peak appears at frequency ω^Z"_max. In frequency range (24), polarization develops or electric relaxation that is local and non-local motions of charged entities appears, respectively. It depends critically on crossing of Z′ and Z″ at ${\omega}_{\mathrm{max}}^{Z"}$ or beyond it. Long-range electric relaxation can be observed only if

$$ {\omega}_{\mathrm{max}}^{Z"}={\omega}_{cross}^{Z\hbox{'}-Z"} $$

(25)

One observes only local relaxation if

$$ {\omega}_{\mathrm{max}}^{Z"}<{\omega}_{cross}^{Z\hbox{'}-Z"} $$

(26)

Imaginary part of permittivity ε″, closely related to conductivity σ′, varies with frequency according to power law ε" ∝ ω^-n with n ≤ 1

$$ {\varepsilon}^{"}\propto {\omega}^{-n}\kern0.6em with\;n\le 1 $$

(27)

in range (24). Power law (27) characterizes polarization relaxation in low-frequency range. Exponent n = 1 gives Debye relaxation. We call relaxation Debye-like when n < 1 but close to unity.

Real part of impedance Z′ does not exhibit extreme values. It approaches Ohm’s resistance of the bulk, R_b, in the limit ω → 0. Generally, we may say Z′ reflects stored dipoles in the system whereas imaginary part focuses on mobile dipoles. In the same way, one may see components of permittivity. dc conductivity constitutes an example, σ_dc ∝ ε″.

The ratio of the two impedance components represents the tangent-loss spectrum of dipolar medium

$$ \left(\tan \delta \right)\left(\omega \right)=\frac{Z^{\hbox{'}}\left(\omega \right)}{Z^{"}\left(\omega \right)} $$

(28)

One immediately recognizes that (tanδ)(ω) will have a maximum at frequency where Z″ displays its minimum. Thus, we observe in Debye-like relaxation (tanδ)_max at

$$ {\omega}_{\mathrm{max}}^{\tan \delta }={\omega}_{\mathrm{min}}^{Z"} $$

(29)

Characteristic frequency ${\omega}_{\mathrm{max}}^{\tan \delta }$ shifts to lower frequency when polarization relaxation is sufficiently far from Debye-like relaxation. Analogously to inequality (26), we have

$$ {\omega}_{\mathrm{max}}^{\tan \delta }<{\omega}_{\mathrm{min}}^{Z"} $$

(30)

We note frequency ω_max^tan δ represents the longest relaxation time 1/ω_max^tan δ.

Electric modulus M* is defined by inverse permittivity, M* = (ε*)⁻¹. One may formulate the imaginary part of modulus by M” = C_oωZ’ (ω)with capacity C_o of the empty cell. It exhibits a maximum at

$$ {\omega}_{\mathrm{max}}^{M"}={\omega}_{\mathrm{max}}^{Z"} $$

(A9)

for Debye-like relaxation. Comparison of equalities (25) and (32) shows that electric modulus reflects long-range electric relaxation. In distance to Debye-like relaxation, frequency${\omega}_{\mathrm{max}}^{M"}$ shifts to higher frequency

$$ {\omega}_{\mathrm{max}}^{M"}>{\omega}_{\mathrm{max}}^{Z"} $$

(33)

One easily confirms Eq. (32) for Debye-like relaxation. It is M″_max at ω = ${\omega}_{\mathrm{max}}^{M"}$. Moreover, components of Z* cross at ${\omega}_{\mathrm{max}}^{Z"}$. Hence, it follows

$$ M{"}_{\mathrm{max}}={C}_o{\omega}_{\mathrm{max}}^{Z"}Z{"}_{\mathrm{max}} $$

(34)

when Eq. (32) is obeyed. In conclusion, one may state tangent-loss spectra acquire dielectric relaxation and electric-modulus spectra do so for electric relaxation.

The sample in impedance spectroscopy has resistance R_b and capacitance C_b = ε_∞C_o. Additional capacities may occur when polarization appears in the system. Polarization causes double-layer capacities in in-homogeneities regions of the system, preferably in the interfacial region electrolyte-electrode.

Structural in-homogeneities in sample generate double-layer capacity, C_dl, responsible for polarization effect. Taking this into account, besides bulk capacity C_b, leads to a slightly generalized Debye approximation with more than one relaxation time. Main relaxation frequency ${\omega}_{\mathrm{max}}^{Z"}$ is seen as average over distribution of relaxation times. Accordingly, response of the dielectric sample to action of an alternating electric field is coined by relaxation time τ_max and capacity ratio A_C

$$ {\tau}_{\mathrm{max}}^{Z"}={R}_b{C}_b\kern0.48em {A}_C\equiv \frac{2{C}_b}{C_{dl}}=\frac{\lambda }{d} $$

(35)

where λ and 2d represent double-layer and sample thickness, respectively. Capacities are given by C_b = ε_∞ε_oA/2d and C_dl = ε_∞ε_oA/λ. We may say characteristic quantities (35) provide an acceptable means for elucidating impedance spectra under condition

$$ {A}_C<<1\ obeyed\ together\ with\ equations\ (29)\ and\ (32) $$

(36)

Under action of periodic electric field, controversial effects occur in the low-frequency regime (24): One observes polarization caused by boundaries or interfacial areas between different phases and electric conductance; actually fluctuation-dissipation rules flowing effects. In other words, depending on structural homogeneity of the system, short-range and long-range relaxations are observed in low-frequency range (24). For frequencies beyond ${\omega}_{\mathrm{max}}^{Z"}$, relaxations are trap-controlled; only short-range relaxations appear. These effects are in our model related to capacity ratio A_C and to time constant R_bC_b, respectively. Therefore, we may express these quantities adequately by characteristic quantities of impedance spectra. In the low-frequency range, we may unify the two components of impedance as follows

$$ \left(\frac{Z^{"}}{R_b}\right)\left(\omega \right)=\omega \tau \left(\frac{Z^{\hbox{'}}}{R_b}\right)\left(\omega \right)+\frac{A_C}{\omega \tau} $$

(37)

Accordingly, one verifies (Z″/R_b)_min at $\left({\omega}_{\mathrm{min}}^{Z"}\tau \right)=\sqrt{A_C}$ with (Z′/R_b) = 1 in the low-frequency region. It follows

$$ \left(\frac{Z^{"}}{R_b}\right)\left({\omega}_{\mathrm{min}}^{Z"}\right)=2\sqrt{A_C}=\left(\frac{Z^{"}}{Z^{\hbox{'}}}\right)\left({\omega}_{\mathrm{max}}^{\tan \delta}\right) $$

(38)

Capacity ratio A_C is closely related to tangent-loss spectra

$$ {A}_C=\frac{1}{{\left(2\tan \delta \right)}_{\mathrm{max}}^2} $$

(39)

This result leads straightforwardly to R_b

$$ 2{R}_b=Z{{\prime\prime}_{min}}^{.}{\left( tan\delta \right)}_{max} $$

(40)

We emphasize again, relationships (39) and (40) are good approximations when proposition (36) is obeyed.