- Email: [email protected]
LiClO4 complex1
Journal of Electrostatics 43 (1998) 203—213
Linear and nonlinear conductivity spectra of tetraethylene oxide-organosiloxane copolymer/LiClO4 complex1 Y. Tajitsu* Department of Materials Science and Engineering, Faculty of Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa, Yamagata 992-8510, Japan Received 6 September 1997; received in revised form 22 November 1997; accepted 20 December 1997
Abstract The temperature dependence of the frequency spectra of linear to third-order nonlinear complex conductivities in tetraethylene oxide-organosiloxane copolymer/LiClO complexes 4 was measured. At each temperature, a characteristic conduction relaxation phenomenon was observed in the spectra, which suggested the existence of different ion conduction mechanisms between the low- and high-frequency regions. We found that the ratio of linear to nonlinear conductivities was closely related to the elementary process of ionic transport in the samples. Without the need for any additional assumptions, we also found that the ratio obtained from nonlinear measurements allowed an estimate of the important parameters which characterize ionic transport in the samples, such as the size of a connected cluster of the site capable of ion hopping or the hopping distance of ion. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear conductivity spectra; Ion conduction; Tetraethylene oxide-organosiloxane copolymer
1. Introduction The problem of ionic transport in polymers has been considered many times over the past years. Recently, the study of ionic transport in polymers has developed rapidly and the findings have become a centre of attention in basic and applied research [1,2]. In fact, the frequency spectra of the complex conductivities have been shown to provide useful information concerning the ion transport processes in various polymers [1—8]. However, most conductivity measurements have been made in * Corresponding author. Tel.:#81 238 26 3410; fax: #81 238 26 3410; e-mail: [email protected]. 1 This paper first appeared, in Japanese, in the Proceedings of the Institute of Electrostatics, Japan, Vol. 21, No. 5, 1997, pp. 220—224. 0304-3886/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 8 8 6 ( 9 8 ) 0 0 0 0 5 - 9
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a linear regime where the current response is proportional to the applied electric field [3—8]. In any event, we have taken note of the possibility that studies on nonlinear spectra of conductivity may provide more detailed information regarding the elementary processes of ionic transport and the formation of potential energy, which is difficult to obtain from measurements in the linear region. Using the nonlinear conductivity measurements, we have already reported the evaluation of the basic parameters, such as the size of a connected cluster of the effective sites capable of ion hopping on the basis of the dynamic percolation model, which characterize the behavior of ion transport in ion-conducting polymers [9]. On the other hand, we have synthesized a new polymer, tetraethylene oxide-organosiloxane copolymer (TEOS) in order to obtain an amorphous polymer in which ionic transport is stable [10]. In this work, we extend the measurements of the conductivity spectra to a nonlinear regime in order to gain additional knowledge about the conduction mechanisms in an amorphous polymer in which ionic transport is stable.
2. Experiments 2.1. Measurements When the current I for a nonlinear system is expanded in odd powers of the electric field E, I"p E#p E3#p E5#2, (1) 1 3 5 we can express the linear and nonlinear conductivities in terms of coefficients of exponents. Here, p is the linear conductivity. The coefficient p for n*3 defines the 1 n nonlinear conductivity. However, when excitation is not a static field but a sinusoidal electric field, it is complicated in that complex nonlinear conductivity is defined. The details of the principles of complex nonlinear conductivity measurements have been described previously [9]. Here, we touch briefly upon the principal points. The phenomenological theory of stationary linear response is well established on the basis of Boltzmann’s superposition principle. On the other hand, extending this theory of stationary linear response to a nonlinear system, Nakada developed a phenomenological theory of the nonlinear relaxation response [11]. According to his theory, the resulting response can be expressed by a sum of convolution integrals of the applied electric field at multiple time points and the nonlinear aftereffect functions. For the measurements of the linear and nonlinear conductivities, we applied a sinusoidal electric field with amplitude E and angular frequency u, 0 E(t)"E cos ut, (2) 0 and detected the in-phase and 90° out-of-phase components of the electric currents with frequency nu (n"1, 3, 5,2). = I(t)" + (I@ cos nut#IA sin nut). n n n/1
(3)
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205
When the excitation is given by Eq. (2), the resulting response I(t) is calculated as follows: I(t)"(p@ (u) cos ut#pA (u) sin ut)E #(p@ (3u) cos 3ut#pA (3u) sin 3ut 3 3 1 0 1 #B@ (u) cos ut#BA (u) sin ut)E3#2 . 0 3 3
(4)
Here p@ (nu) and pA(nu) are the real and imaginary parts of linear (n"1) and nonlinn n ear (n'1) complex conductivities p*(nu)"p@ (nu)#jpA(nu), respectively. The other n n n harmonic coefficients B@ (u) and BA(u) are represented. We must point out that p@ (nu) n n n and B@ (u) are entirely different in their coefficients of the higher harmonic wave n although the outer shape is similar. Comparing Eq. (4) with Eq. (3), we find the following relationships between the amplitude of the in-phase component I@ and the n amplitude of the applied field E : 0 I@ "p@ (u)E # 3B@ (u)E3#2 , 4 3 1 0 1 0
(5)
I@ " 1p@ (3u)E3# 5 B@ (u)E5#2 . 4 3 3 16 5 0 0
(6)
Furthermore, the relationships between 90° out-of phase components IA (n"1, 3, n 5,2) and E are of the same forms. Due to their complexity, the terms p*(nu), p@ (nu) n n 0 and pA(nu) will hereinafter be abbreviated to Ap*, p@ and p A, respectively. As a result, n n n n we obtained the real and imaginary components of complex conductivities p*"p@ #jp@ (n"1, 3, 5,2), using n n n 2n~1I@ n, p@ " n En 0
2n~1IA n. pA" n En 0
(7)
Based on the above principle, we developed a new experimental system which enables measurements of linear as well as nonlinear complex conductivities in a 25 mHz— 1 MHz frequency range. Fig. 1 shows a schematic diagram of the experimental system [9]. The excitation wave propagates to the sample. The induced current of the electrode is detected using a current amplifier. The voltage and current signals are simultaneously stored in a wave memory. The detected data are used in calculations by the microcomputer. 2.2. Materials We have synthesized tetraethylene oxide-organosiloxane copolymer (TEOS) using the same method as in a previous work [10]. Tetrahydrofuran (THF) was purchased as the source material of methyltrichlorosilane and tetraethylene oxide, and was distilled on sodium. LiClO was dissolved in dry THF with tetraethylene oxide. The 4 solution was spread on a teflon sheet and the reaction occurred in dry Ar at room temperature. We obtained TEOS after one week. Then the films were annealed at 140°C in vacuo for 1 day. The samples thus prepared were placed in a cryostat. Circular lithium electrodes (2 mm/) were used for nonlinear measurements. Fig. 2
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Fig. 1. Schematic diagram of the experimental system for nonlinear complex conductivity measurements.
Fig. 2. Structural formula of tetraethylene oxide-organosiloxane (TEOS) copolymer.
shows the structural formula of this copolymer. We believe that TEOS copolymer is noncrystalline on the basis of its diffuse X-ray diffraction pattern. The elastic constant, c*"c@#jcA, was determined by increasing the temperature at a frequency of 10 Hz. The results for TEOS are shown in Fig. 3. The relaxation of c* is observed at around !60°C, which is related to the glass—rubber transition in the noncrystalline phase.
3. Results Typical experimental results of the electric field dependence of the in-phase component I@ (n"1, 3) in the electric current are shown in Fig. 4. In the top figure, I@ is n 1 plotted against the amplitude E of the electric field. Using Eq. (7), we obtain p@ from 0 1 the gradient of the linear relationship. When I@ is plotted against the third power of 3 E , we obtain p@ from the slope, referring to Eq. (7). Using the same method, we can 0 3 obtain the imaginary part pA of the complex linear and nonlinear conductivities from n the gradient of the linear relationship between IA and En (n"1, 3). Fig. 5 shows the n 0
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207
Fig. 3. The temperature dependence of the elastic constant, c*"c@#jcA, of TEOS.
frequency spectra of the linear conductivity p*"p@ #jpA . At 0°C, p@ remains nearly 1 1 1 1 constant in the low-frequency range. Both p@ and pA increase with increasing fre1 1 quency. With decreasing temperature, the spectra shift to the lower frequency region. Results for the third-order nonlinear conductivity p*"p@ #jpA are also shown as 3 3 3 circles in Fig. 6. p* shows the characteristic frequency spectra consisting of a negative 3 peak of p@ , and positive and negative peaks of pA . As the temperature is decreased, the 3 3 spectra shift to the lower frequency region with a considerable reduction of peak height. We attempted to reproduce the observed linear and nonlinear conductivity spectra using equations of the form juq n p* "*p n3 n (1#(juq )bn)nan n
(8)
with n"1 for linear conductivity spectra and n*3 for nth-order nonlinear conductivity spectra. Here, *p is the linear/nonlinear relaxation strength, q is the relaxation / n time, and a and b are parameters expressing the distribution of relaxation times n n (n"1, 3,2). We actually use p*"p #p* #jue 1 1$# 13 */
(9)
for the linear conductivity spectra. Here p is linear dc (low-frequency) conductivity 1$# and e is the linear permittivity which is assumed to be independent of frequency. As */ shown in Fig. 7, solid curves, calculated using Eq. (9) with Eq. (8) (n"1) reproduce the observed spectra reasonably well. We also use p*"p #p* 3 3$# 33
(10)
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Fig. 4. Dependence of the in-phase components, I@ and I@ , of the linear and third-order harmonic electric 1 3 currents as a function of the amplitude of applied field E . 0
for the third-order nonlinear conductivity spectra. Here p is the third-order 3$# nonlinear dc (low frequency) conductivity. Solid curves, calculated using Eq. (10) with Eq. (8) (n"3), also reproduce the observed spectra reasonably well, as shown in Fig. 8. 4. Discussion We described, in the preceding section, that conductive relaxation is indicated in the frequency spectra of TEOS. Conductive relaxation results in a decrease in
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209
Fig. 5. Frequency spectra of the real and imaginary components of the linear conductivity.
conductivity from p #*p to p with decreasing frequency. The relaxation term 1$# 1 1$# in Eq. (9) is essentially the same as the linear dielectric relaxation function. The corresponding dielectric relaxation strength *e "q *p /e (e the permittivity of free 1 1 1 0 0 space) reaches 100, which suggests that ionic motions accompany a large dielectric polarization. On the basis of these results, we speculate that there are two separate conduction mechanisms in polymers. The high-frequency conduction can be attributed to free motions of ions due to their hopping to adjacent sites to yield a conductivity of p #*p . Such motions seem to occur in limited domains. At low 1$# 1 frequencies, ions reach the edge of these domains, generate polarization and yield a low conductivity (p ) governed by their hopping to adjacent domains. The 1$# relaxation term in Eq. (8) is also the same as the nonlinear dielectric relaxation function. However, from the calculated results following Eq. (8), we found, this relaxation does not affect the nonlinear conductivities at low or high frequencies. Upon further development of these speculations, we reach a similar conclusion with regard to ion transport, based on the dynamic percolation theory [1,9,12,13]. The dynamic percolation model characterizes ionic motion in terms of hopping between neighboring positions in polymers [1,12]. This model developed by Ratner and
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Fig. 6. Frequency spectra of the third-order nonlinear conductivity.
coworkers [12], takes into account the dependence of ionic motion rates on the fluidity, or rate of segmental motion, of the polymer host. A characteristic rate of renewal is defined, which characterizes the rate at which a motion pathway from one site to another becomes available for ion motion. That is to say, the motion of ions is controlled by the dynamics of the polymer host and cannot occur unless it is promoted by the segmental motion of the polymer host [1,12]. On the other hand, we found that the ratio of linear to nonlinear conductivities was closely related to the elementary process of ionic transport [9]. Furthermore, this ratio obtained from nonlinear measurements allowed an estimate the important parameters which characterized ionic transport in ion-conducting polymers, such as the hopping distance of ion or the size of a connected cluster of the site capable of ion hopping [9]. Actually, on the basis of dynamic percolation theory, we could obtain, from nonlinear measurements, that in a polyethylene oxide/salt complex, the typical size of a connected cluster of effective sites capable of ion hopping was approximately 4 nm, without the need for any additional assumptions [9]. However, we need more detailed information on the structure of the TEOS in order to analyze the ion motion in the TEOS on the basis of the dynamic percolation theory. At present, we do not have information on the structure of TEOS because TEOS is almost an amorphous polymer, as described above. Therefore, we evaluate the ion motion on the basis of the hopping model [14] which is an often used technique when information concerning the structure of amorphous polymers is lacking [1].
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211
Fig. 7. Comparison of observed (circles) and fitted (solid lines) frequency spectra of the linear conductivity at !20°C.
If the ionic motion is of the hopping type, the resulting electric current I [14] is given as
A
B A B
eaE *º I"2eNaP exp ! sinh . 0 2k¹ k¹
(11)
Here a is the hopping distance, E is the electric field, ¹ is the absolute temperature, P is the hopping probability at the high-temperature limit, e is the elementary electric 0 charge, N is the carrier density, k is the Boltzmann constant and *º is the activation energy. When Eq. (11) for a nonlinear system is expanded with odd powers of the electric field E, we have I"p E#p E3#2 1 3
(12)
with
A
B
*º 1 p " e2a2P N exp ! 0 1 k¹ k¹
(13)
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Y. Tajitsu/Journal of Electrostatics 43 (1998) 203–213
Fig. 8. Comparison of observed (circles) and fitted (solid lines) frequency spectra of the third-order nonlinear conductivity at !20°C.
and
A
B
1 *º p " e4a4P N exp ! . 3 24k3¹3 0 k¹
(14)
The expressions obtained for p and p contain many physical quantities which 1 3 cannot be obtained experimentally. However, the ratio of p and p , 1 3 e2a2 p 3" (15) 24K2¹2 p 1 yields the hopping distance a without the need for any additional assumptions. Using the values for p and p at high and low frequencies, we obtain a"5nm and 80 nm for 1 3 the respective processes. The former corresponds to the distance between adjacent hopping sites and the latter to the average size of domains where ions can move freely. TEOS is an extremely large molecular aggregate of chains whose individual units are covalently bonded to each other. Also, the TEOS examined here is completely amorphous, judging from its very diffuse X-ray pattern. TEOS does not have a complex highly ordered structure with intermingled crystalline and amorphous regions. However, the nonlinear conductivity investigation presented here has revealed that a microscopically inhomogeneous structure that affects ionic transport processes in this polymer exists. Thus, nonlinear measurements are extremely useful in determining the mechanism of ion transport in polymers.
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213
5. Summary We developed a new experimental system which enables measurements of linear as well as nonlinear complex conductivities. Using this system, we measured the frequency dependence of linear to third-order nonlinear complex conductivities in tetraethylene oxide-organosiloxane copolymers. We proposed a new empirical function to reproduce the observed linear and nonlinear conductivity spectra which show a complicated characteristic frequency dependence together with a relaxation phenomenon. Furthermore, we found that the validity of the basic parameters can be examined without any additional assumptions by using the values of nonlinear conductivities obtained from experiments. We then evaluated the basic parameters, such as the hopping distance in the two-site hopping model, without making any additional assumptions. Thus, the nonlinear technique developed here appears to be useful for the empirical confirmation of the validity of the ionic transport theory in polymers. Acknowledgements We thank Dr. Y. Ueno of Waseda University for providing us with various polymers. We are also grateful to Professor H. Ohigashi of Yamagata University and Dr. M. Date of the Institute of Physical and Chemical Research for their support. This work was supported in part by the Casio Foundation for promotion of Science and a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan. References [1] J.R. MacCallum, in: C.A. Vincent (Ed.), Polymer Electrolyte Reviews 1, Elsevier Appl. Sci., London, 1987. [2] J.R. MacCallum, in: C.A. Vincent (Ed.), Polymer Electrolyte Reviews 2, Elsevier Appl. Sci., London, 1989. [3] C.A. Angell, Solid State Ionics 9/10 (1983) 3—12. [4] J.J. Fontanella, M.C. Wintersgill, M.K. Smith, J. Semancik, C.G. Andeen, J. Appl. Phys. 60 (1986) 2665—2671. [5] T. Wong, M. Brodwin, J.I. McOmber, D.F. Shriver, Solid State Commun. 35 (1980) 591—595. [6] R. Vatikus, A. Kezionics, A. Saumlionis, A. Orliukas, V. Skritskij, Solid State Ionics 40/41 (1990) 922—927. [7] K. Funk, Solid State Ionics 18/19 (1986) 183—188. [8] A.I. Tipton, M.C. Lonergan, M.A. Ratner, D.F. Shriver, T.Y. Wong, K. Han, J. Phys. Chem. 98 (1994) 4148—4154. [9] Y. Tajitsu, J. Mater. Sci. 31 (1996) 2081—2089. [10] Y. Ueno, Y. Tajitsu, T. Furukawa, Polymer 33 (1992) 665—667. [11] O. Nakada, J. Phys. Soc. Japan 15 (1960) 2280—2288. [12] S.D. Druger, M.A. Ratner, A. Nitzan, Phys. Rev. B 31 (1985) 3939—3947. [13] A.K. Harison, R. Zwanzig, Phys. Rev. A 32 (1985) 1072—1075. [14] N.F. Mott, E.A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd Ed., Clarendon, London, 1979.
Linear and nonlinear conductivity spectra of tetraethylene oxide-organosiloxane copolymer/LiClO4 complex1 Y. Tajitsu* Department of Materials Science and Engineering, Faculty of Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa, Yamagata 992-8510, Japan Received 6 September 1997; received in revised form 22 November 1997; accepted 20 December 1997
Abstract The temperature dependence of the frequency spectra of linear to third-order nonlinear complex conductivities in tetraethylene oxide-organosiloxane copolymer/LiClO complexes 4 was measured. At each temperature, a characteristic conduction relaxation phenomenon was observed in the spectra, which suggested the existence of different ion conduction mechanisms between the low- and high-frequency regions. We found that the ratio of linear to nonlinear conductivities was closely related to the elementary process of ionic transport in the samples. Without the need for any additional assumptions, we also found that the ratio obtained from nonlinear measurements allowed an estimate of the important parameters which characterize ionic transport in the samples, such as the size of a connected cluster of the site capable of ion hopping or the hopping distance of ion. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear conductivity spectra; Ion conduction; Tetraethylene oxide-organosiloxane copolymer
1. Introduction The problem of ionic transport in polymers has been considered many times over the past years. Recently, the study of ionic transport in polymers has developed rapidly and the findings have become a centre of attention in basic and applied research [1,2]. In fact, the frequency spectra of the complex conductivities have been shown to provide useful information concerning the ion transport processes in various polymers [1—8]. However, most conductivity measurements have been made in * Corresponding author. Tel.:#81 238 26 3410; fax: #81 238 26 3410; e-mail: [email protected]. 1 This paper first appeared, in Japanese, in the Proceedings of the Institute of Electrostatics, Japan, Vol. 21, No. 5, 1997, pp. 220—224. 0304-3886/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 8 8 6 ( 9 8 ) 0 0 0 0 5 - 9
204
Y. Tajitsu/Journal of Electrostatics 43 (1998) 203–213
a linear regime where the current response is proportional to the applied electric field [3—8]. In any event, we have taken note of the possibility that studies on nonlinear spectra of conductivity may provide more detailed information regarding the elementary processes of ionic transport and the formation of potential energy, which is difficult to obtain from measurements in the linear region. Using the nonlinear conductivity measurements, we have already reported the evaluation of the basic parameters, such as the size of a connected cluster of the effective sites capable of ion hopping on the basis of the dynamic percolation model, which characterize the behavior of ion transport in ion-conducting polymers [9]. On the other hand, we have synthesized a new polymer, tetraethylene oxide-organosiloxane copolymer (TEOS) in order to obtain an amorphous polymer in which ionic transport is stable [10]. In this work, we extend the measurements of the conductivity spectra to a nonlinear regime in order to gain additional knowledge about the conduction mechanisms in an amorphous polymer in which ionic transport is stable.
2. Experiments 2.1. Measurements When the current I for a nonlinear system is expanded in odd powers of the electric field E, I"p E#p E3#p E5#2, (1) 1 3 5 we can express the linear and nonlinear conductivities in terms of coefficients of exponents. Here, p is the linear conductivity. The coefficient p for n*3 defines the 1 n nonlinear conductivity. However, when excitation is not a static field but a sinusoidal electric field, it is complicated in that complex nonlinear conductivity is defined. The details of the principles of complex nonlinear conductivity measurements have been described previously [9]. Here, we touch briefly upon the principal points. The phenomenological theory of stationary linear response is well established on the basis of Boltzmann’s superposition principle. On the other hand, extending this theory of stationary linear response to a nonlinear system, Nakada developed a phenomenological theory of the nonlinear relaxation response [11]. According to his theory, the resulting response can be expressed by a sum of convolution integrals of the applied electric field at multiple time points and the nonlinear aftereffect functions. For the measurements of the linear and nonlinear conductivities, we applied a sinusoidal electric field with amplitude E and angular frequency u, 0 E(t)"E cos ut, (2) 0 and detected the in-phase and 90° out-of-phase components of the electric currents with frequency nu (n"1, 3, 5,2). = I(t)" + (I@ cos nut#IA sin nut). n n n/1
(3)
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205
When the excitation is given by Eq. (2), the resulting response I(t) is calculated as follows: I(t)"(p@ (u) cos ut#pA (u) sin ut)E #(p@ (3u) cos 3ut#pA (3u) sin 3ut 3 3 1 0 1 #B@ (u) cos ut#BA (u) sin ut)E3#2 . 0 3 3
(4)
Here p@ (nu) and pA(nu) are the real and imaginary parts of linear (n"1) and nonlinn n ear (n'1) complex conductivities p*(nu)"p@ (nu)#jpA(nu), respectively. The other n n n harmonic coefficients B@ (u) and BA(u) are represented. We must point out that p@ (nu) n n n and B@ (u) are entirely different in their coefficients of the higher harmonic wave n although the outer shape is similar. Comparing Eq. (4) with Eq. (3), we find the following relationships between the amplitude of the in-phase component I@ and the n amplitude of the applied field E : 0 I@ "p@ (u)E # 3B@ (u)E3#2 , 4 3 1 0 1 0
(5)
I@ " 1p@ (3u)E3# 5 B@ (u)E5#2 . 4 3 3 16 5 0 0
(6)
Furthermore, the relationships between 90° out-of phase components IA (n"1, 3, n 5,2) and E are of the same forms. Due to their complexity, the terms p*(nu), p@ (nu) n n 0 and pA(nu) will hereinafter be abbreviated to Ap*, p@ and p A, respectively. As a result, n n n n we obtained the real and imaginary components of complex conductivities p*"p@ #jp@ (n"1, 3, 5,2), using n n n 2n~1I@ n, p@ " n En 0
2n~1IA n. pA" n En 0
(7)
Based on the above principle, we developed a new experimental system which enables measurements of linear as well as nonlinear complex conductivities in a 25 mHz— 1 MHz frequency range. Fig. 1 shows a schematic diagram of the experimental system [9]. The excitation wave propagates to the sample. The induced current of the electrode is detected using a current amplifier. The voltage and current signals are simultaneously stored in a wave memory. The detected data are used in calculations by the microcomputer. 2.2. Materials We have synthesized tetraethylene oxide-organosiloxane copolymer (TEOS) using the same method as in a previous work [10]. Tetrahydrofuran (THF) was purchased as the source material of methyltrichlorosilane and tetraethylene oxide, and was distilled on sodium. LiClO was dissolved in dry THF with tetraethylene oxide. The 4 solution was spread on a teflon sheet and the reaction occurred in dry Ar at room temperature. We obtained TEOS after one week. Then the films were annealed at 140°C in vacuo for 1 day. The samples thus prepared were placed in a cryostat. Circular lithium electrodes (2 mm/) were used for nonlinear measurements. Fig. 2
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Fig. 1. Schematic diagram of the experimental system for nonlinear complex conductivity measurements.
Fig. 2. Structural formula of tetraethylene oxide-organosiloxane (TEOS) copolymer.
shows the structural formula of this copolymer. We believe that TEOS copolymer is noncrystalline on the basis of its diffuse X-ray diffraction pattern. The elastic constant, c*"c@#jcA, was determined by increasing the temperature at a frequency of 10 Hz. The results for TEOS are shown in Fig. 3. The relaxation of c* is observed at around !60°C, which is related to the glass—rubber transition in the noncrystalline phase.
3. Results Typical experimental results of the electric field dependence of the in-phase component I@ (n"1, 3) in the electric current are shown in Fig. 4. In the top figure, I@ is n 1 plotted against the amplitude E of the electric field. Using Eq. (7), we obtain p@ from 0 1 the gradient of the linear relationship. When I@ is plotted against the third power of 3 E , we obtain p@ from the slope, referring to Eq. (7). Using the same method, we can 0 3 obtain the imaginary part pA of the complex linear and nonlinear conductivities from n the gradient of the linear relationship between IA and En (n"1, 3). Fig. 5 shows the n 0
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207
Fig. 3. The temperature dependence of the elastic constant, c*"c@#jcA, of TEOS.
frequency spectra of the linear conductivity p*"p@ #jpA . At 0°C, p@ remains nearly 1 1 1 1 constant in the low-frequency range. Both p@ and pA increase with increasing fre1 1 quency. With decreasing temperature, the spectra shift to the lower frequency region. Results for the third-order nonlinear conductivity p*"p@ #jpA are also shown as 3 3 3 circles in Fig. 6. p* shows the characteristic frequency spectra consisting of a negative 3 peak of p@ , and positive and negative peaks of pA . As the temperature is decreased, the 3 3 spectra shift to the lower frequency region with a considerable reduction of peak height. We attempted to reproduce the observed linear and nonlinear conductivity spectra using equations of the form juq n p* "*p n3 n (1#(juq )bn)nan n
(8)
with n"1 for linear conductivity spectra and n*3 for nth-order nonlinear conductivity spectra. Here, *p is the linear/nonlinear relaxation strength, q is the relaxation / n time, and a and b are parameters expressing the distribution of relaxation times n n (n"1, 3,2). We actually use p*"p #p* #jue 1 1$# 13 */
(9)
for the linear conductivity spectra. Here p is linear dc (low-frequency) conductivity 1$# and e is the linear permittivity which is assumed to be independent of frequency. As */ shown in Fig. 7, solid curves, calculated using Eq. (9) with Eq. (8) (n"1) reproduce the observed spectra reasonably well. We also use p*"p #p* 3 3$# 33
(10)
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Fig. 4. Dependence of the in-phase components, I@ and I@ , of the linear and third-order harmonic electric 1 3 currents as a function of the amplitude of applied field E . 0
for the third-order nonlinear conductivity spectra. Here p is the third-order 3$# nonlinear dc (low frequency) conductivity. Solid curves, calculated using Eq. (10) with Eq. (8) (n"3), also reproduce the observed spectra reasonably well, as shown in Fig. 8. 4. Discussion We described, in the preceding section, that conductive relaxation is indicated in the frequency spectra of TEOS. Conductive relaxation results in a decrease in
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209
Fig. 5. Frequency spectra of the real and imaginary components of the linear conductivity.
conductivity from p #*p to p with decreasing frequency. The relaxation term 1$# 1 1$# in Eq. (9) is essentially the same as the linear dielectric relaxation function. The corresponding dielectric relaxation strength *e "q *p /e (e the permittivity of free 1 1 1 0 0 space) reaches 100, which suggests that ionic motions accompany a large dielectric polarization. On the basis of these results, we speculate that there are two separate conduction mechanisms in polymers. The high-frequency conduction can be attributed to free motions of ions due to their hopping to adjacent sites to yield a conductivity of p #*p . Such motions seem to occur in limited domains. At low 1$# 1 frequencies, ions reach the edge of these domains, generate polarization and yield a low conductivity (p ) governed by their hopping to adjacent domains. The 1$# relaxation term in Eq. (8) is also the same as the nonlinear dielectric relaxation function. However, from the calculated results following Eq. (8), we found, this relaxation does not affect the nonlinear conductivities at low or high frequencies. Upon further development of these speculations, we reach a similar conclusion with regard to ion transport, based on the dynamic percolation theory [1,9,12,13]. The dynamic percolation model characterizes ionic motion in terms of hopping between neighboring positions in polymers [1,12]. This model developed by Ratner and
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Fig. 6. Frequency spectra of the third-order nonlinear conductivity.
coworkers [12], takes into account the dependence of ionic motion rates on the fluidity, or rate of segmental motion, of the polymer host. A characteristic rate of renewal is defined, which characterizes the rate at which a motion pathway from one site to another becomes available for ion motion. That is to say, the motion of ions is controlled by the dynamics of the polymer host and cannot occur unless it is promoted by the segmental motion of the polymer host [1,12]. On the other hand, we found that the ratio of linear to nonlinear conductivities was closely related to the elementary process of ionic transport [9]. Furthermore, this ratio obtained from nonlinear measurements allowed an estimate the important parameters which characterized ionic transport in ion-conducting polymers, such as the hopping distance of ion or the size of a connected cluster of the site capable of ion hopping [9]. Actually, on the basis of dynamic percolation theory, we could obtain, from nonlinear measurements, that in a polyethylene oxide/salt complex, the typical size of a connected cluster of effective sites capable of ion hopping was approximately 4 nm, without the need for any additional assumptions [9]. However, we need more detailed information on the structure of the TEOS in order to analyze the ion motion in the TEOS on the basis of the dynamic percolation theory. At present, we do not have information on the structure of TEOS because TEOS is almost an amorphous polymer, as described above. Therefore, we evaluate the ion motion on the basis of the hopping model [14] which is an often used technique when information concerning the structure of amorphous polymers is lacking [1].
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211
Fig. 7. Comparison of observed (circles) and fitted (solid lines) frequency spectra of the linear conductivity at !20°C.
If the ionic motion is of the hopping type, the resulting electric current I [14] is given as
A
B A B
eaE *º I"2eNaP exp ! sinh . 0 2k¹ k¹
(11)
Here a is the hopping distance, E is the electric field, ¹ is the absolute temperature, P is the hopping probability at the high-temperature limit, e is the elementary electric 0 charge, N is the carrier density, k is the Boltzmann constant and *º is the activation energy. When Eq. (11) for a nonlinear system is expanded with odd powers of the electric field E, we have I"p E#p E3#2 1 3
(12)
with
A
B
*º 1 p " e2a2P N exp ! 0 1 k¹ k¹
(13)
212
Y. Tajitsu/Journal of Electrostatics 43 (1998) 203–213
Fig. 8. Comparison of observed (circles) and fitted (solid lines) frequency spectra of the third-order nonlinear conductivity at !20°C.
and
A
B
1 *º p " e4a4P N exp ! . 3 24k3¹3 0 k¹
(14)
The expressions obtained for p and p contain many physical quantities which 1 3 cannot be obtained experimentally. However, the ratio of p and p , 1 3 e2a2 p 3" (15) 24K2¹2 p 1 yields the hopping distance a without the need for any additional assumptions. Using the values for p and p at high and low frequencies, we obtain a"5nm and 80 nm for 1 3 the respective processes. The former corresponds to the distance between adjacent hopping sites and the latter to the average size of domains where ions can move freely. TEOS is an extremely large molecular aggregate of chains whose individual units are covalently bonded to each other. Also, the TEOS examined here is completely amorphous, judging from its very diffuse X-ray pattern. TEOS does not have a complex highly ordered structure with intermingled crystalline and amorphous regions. However, the nonlinear conductivity investigation presented here has revealed that a microscopically inhomogeneous structure that affects ionic transport processes in this polymer exists. Thus, nonlinear measurements are extremely useful in determining the mechanism of ion transport in polymers.
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5. Summary We developed a new experimental system which enables measurements of linear as well as nonlinear complex conductivities. Using this system, we measured the frequency dependence of linear to third-order nonlinear complex conductivities in tetraethylene oxide-organosiloxane copolymers. We proposed a new empirical function to reproduce the observed linear and nonlinear conductivity spectra which show a complicated characteristic frequency dependence together with a relaxation phenomenon. Furthermore, we found that the validity of the basic parameters can be examined without any additional assumptions by using the values of nonlinear conductivities obtained from experiments. We then evaluated the basic parameters, such as the hopping distance in the two-site hopping model, without making any additional assumptions. Thus, the nonlinear technique developed here appears to be useful for the empirical confirmation of the validity of the ionic transport theory in polymers. Acknowledgements We thank Dr. Y. Ueno of Waseda University for providing us with various polymers. We are also grateful to Professor H. Ohigashi of Yamagata University and Dr. M. Date of the Institute of Physical and Chemical Research for their support. This work was supported in part by the Casio Foundation for promotion of Science and a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan. References [1] J.R. MacCallum, in: C.A. Vincent (Ed.), Polymer Electrolyte Reviews 1, Elsevier Appl. Sci., London, 1987. [2] J.R. MacCallum, in: C.A. Vincent (Ed.), Polymer Electrolyte Reviews 2, Elsevier Appl. Sci., London, 1989. [3] C.A. Angell, Solid State Ionics 9/10 (1983) 3—12. [4] J.J. Fontanella, M.C. Wintersgill, M.K. Smith, J. Semancik, C.G. Andeen, J. Appl. Phys. 60 (1986) 2665—2671. [5] T. Wong, M. Brodwin, J.I. McOmber, D.F. Shriver, Solid State Commun. 35 (1980) 591—595. [6] R. Vatikus, A. Kezionics, A. Saumlionis, A. Orliukas, V. Skritskij, Solid State Ionics 40/41 (1990) 922—927. [7] K. Funk, Solid State Ionics 18/19 (1986) 183—188. [8] A.I. Tipton, M.C. Lonergan, M.A. Ratner, D.F. Shriver, T.Y. Wong, K. Han, J. Phys. Chem. 98 (1994) 4148—4154. [9] Y. Tajitsu, J. Mater. Sci. 31 (1996) 2081—2089. [10] Y. Ueno, Y. Tajitsu, T. Furukawa, Polymer 33 (1992) 665—667. [11] O. Nakada, J. Phys. Soc. Japan 15 (1960) 2280—2288. [12] S.D. Druger, M.A. Ratner, A. Nitzan, Phys. Rev. B 31 (1985) 3939—3947. [13] A.K. Harison, R. Zwanzig, Phys. Rev. A 32 (1985) 1072—1075. [14] N.F. Mott, E.A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd Ed., Clarendon, London, 1979.