- Email: [email protected]
Nonlinear ionic-conductivity spectra of antistatic polymer films
Journal of Electrostatics 55 (2002) 299–310
Nonlinear ionic-conductivity spectra of antistatic polymer films Yoshiro Tajitsu* Faculty of Engineering, Department of Polymer Science and Engineering, Yamagata University, Yonezawa, Yamagata 992-8510, Japan Received 8 December 2000; received in revised form 3 September 2001
Abstract We measured the frequency dependence of linear to third-order nonlinear complex conductivities and permittivities of a new antistatic polymer film (NASPF), that is, a composite film which consists of highly conductive butadiene rubber (BUT) particles containing Li+ and poly(methyl methacrylate), at room temperature. The nonlinear conductivities obtained allowed an estimate of the important parameters which characterize ionic transport in the composite film, such as the hopping distance of ions or the size of a cluster connected to the site capable of ion hopping, without any additional assumptions. We obtained two hopping distances, 68 and 380 nm. The former may correspond to the distance between adjacent hopping sites in one BUT particle. The latter corresponds to the average size of the clusters formed by some BUT particles, in which ions can move freely. The results are consistent with transmission electron microscopic image data concerning BUT particle dispersion in the NASPF. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear conductivity; Antistatic polymer; Ionic conduction; Frequency spectra
1. Introduction Recently, the study of the antistatic mechanism due to the ionic transport in antistatic polymers (ASPs) has advanced rapidly and the resulting developments have become the focus of attention [1–3]. However, for high-grade practical application, for example, IC packaging, some critical requirements must be satisfied, which is not possible with the present level of technology, and many points must be *Tel.: +81-238-26-3410; fax: +81-238-26-3410. E-mail address: [email protected] (Y. Tajitsu). 0304-3886/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 8 6 ( 0 1 ) 0 0 2 1 2 - 1
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
300
improved. Therefore, the antistatic mechanism of the ionic transport process in ASPs must be studied. On the other hand, the mechanism of ionic transport in polymers has been studied for many years [4]. Detailed information regarding the correlation between ionic transport and dynamic behavior of the polymer chain are expected to be obtained through the investigation of the frequency dependence of conductivity [4]. In those studies, the frequency spectra of the complex conductivities have been shown to provide useful information concerning the ionic transport processes. However, most measurements have been carried out in a linear regime where the current response is proportional to the applied electric field. On the other hand, we have taken note of the possibility that studies on nonlinear spectra of conductivity may provide 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 [5–8]. In this paper, we extend the measurements of the conductivity spectra to a nonlinear regime in order to gain additional knowledge about the antistatic mechanism of the ionic transport process in ASPs.
2. Experiments 2.1. Materials The sample used in this study was a new ASP film (NASPF) which consists of highly conductive butadiene rubber (BUT) particles (80 nm diameter) containing Li+ (½Liþ =½O ¼ 0:05) dispersed in poly(methyl methacrylate) (PMMA) (reforming BAYONs: Kureha Chemical Industry, Co., Ltd.) [3]. The BUT particle volume fractions f of 7.5, 15.0 and 30.0 wt% in the NASPFs were obtained. The thickness of NASPFs obtained here is 50 mm. Circular lithium electrodes (10 mm diameter) were used for nonlinear measurements. 2.2. Measurements We have developed a new experimental system which enables measurements of linear as well as nonlinear complex conductivities sn ¼ s0n þ js00n (n ¼ 1 5) in a 10 mHz to 1 MHz frequency range [5–9]. Details of the experiments have been described previously [5–9]. Here, we touch briefly upon the principal points. The schematic diagram of the measuring system is shown in Fig. 1. The excitation wave is applied to the sample. The induced current at the electrode is detected by a current amplifier. The voltage and current signals are simultaneously stored in a wave memory. The detected data are used in calculations with the microcomputer. Actually, for the measurements of the complex linear and nonlinear conductivities, we applied a sinusoidal electric field with amplitude E0 and angular frequency o; EðtÞ ¼ E0 cos ot;
ð1Þ
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
SYNTHESIZER
fc
301
MICROCOMPUTER
f c'
VOLTAGE AMPLIFIER
WAVE MEMORY V (t)
ATTENUATOR
I (t)
V = V0 cos t SAMPLE
CURRENT AMPLIFIER I
Fig. 1. The schematic diagram of measuring system.
and detected the in-phase and 901-out-of-phase components of the electric currents with frequency no (n=1, 2, 3,y). N X IðtÞ ¼ ðIn0 cos not þ In00 sin notÞ: ð2Þ n¼1
As a result, we obtained the real and imaginary components of complex conductivities sn ¼ s0n þ js00n and nonlinear permittivity en ¼ e0n je00n (n=1, 2, 3,y) with 0
0
sn ¼
2n1 In ; E0n
0
sn ¼
2n1 In00 : E0n
sn ¼ jnoen :
ð3Þ ð4Þ
Using this system, we have measured sn (n ¼ 1;2,3) and en ¼ e0n je00n (n ¼ 1;2,3) of the NASPF. To measure the nonlinear permittivity spectrum of the NASPF [9], we applied an electric field E perpendicular to the film surface. Therefore, we can formally express the electric displacement D as D ¼ e1 E þ e2 E 2 þ e3 E 3 þ ? :
ð5Þ
Here, e1 is the linear permittivity. The coefficient en for n > 1 defines the nonlinear permittivity. In the same way, when the equation for current I for a nonlinear system is expanded in powers of the electric field E; I ¼ s1 E þ s3 E 2 þ s3 E 3 þ ? ;
ð6Þ
we can express the linear and nonlinear conductivities in terms of coefficients of the exponents. Here, s1 is the linear conductivity. The coefficient sn for n > 1 defines the nonlinear conductivity.
3. Results We observed the BUT particle dispersion in the NASPF using a transmission electron microscope (TEM). Typical TEM images are shown in Fig. 2. In these
302
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
Fig. 2. TEM images of a new ASP film. (a) f ¼ 30:0 wt%, (b) f ¼ 15:0 wt%, (c) f ¼ 7:5 wt%.
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
303
Fig. 3. TEM image data processing.
figures, two phases are seen to exist in the NASPF: one is made up of BUT particles with diameters of 70–100 nm which form clusters of about 400–500 nm in size, and the other contains PMMA. The BUT particle dispersion in the NASPF is determined by TEM graphic image data processing. Fig. 3 shows an example of typical data processing of a TEM image. The number of clusters, the number of BUT particles in a cluster, the area of a cluster, the area of BUT particles, the length of the major axis of the clusters, the length of the minor axis of the clusters, and the aspect ratio (=the minor axis length to major axis length of a cluster) were obtained. Table 1 lists the averages of the above data of clusters and BUT particles in NASPFS. Fig. 4 shows the frequency spectra of e1 ¼ e01 je001 obtained in the 0.025 HzB5 MHz frequency range plotted as circles, for a NASPF with f ¼ 0:15; at 201C. The spectra show a large relaxation and an increase with decreasing frequency. The first increase in the high-frequency range is associated with an inconspicuous loss peak, suggesting the existence of a relaxation process. The relaxation strength is >50. Furthermore, we found a second rapid increase in the low-frequency range. The real part s01 of complex linear conductivity s1 ¼ s01 þ js001 of a NASPF is then plotted in Fig. 5 as circles. s01 remains nearly constant in the low-frequency range and increases with
304
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
Table 1 Average data obtained by TEM image data processing BUT volume fraction f 30 wt%
15 wt%
7.5 wt%
Clusters Area (102 nm2) Major axis (nm) Minor axis (nm) Aspect ratio Total number of clusters Total number of BUT particles in a cluster BUT particles forming a cluster (%)
368 337 165 2.04 32 5.03 85.2
320 282 152 1.85 22 3.52 67.5
220 272 115 2.36 10 2.5 27.1
Butadiene rubber (BUT) particles Area (102 nm2) Major axis (nm) Minor axis (nm) Diameter (nm) Aspect ratio
55 114 63 82 1.8
81 138 104 120 1.32
89 145 98 112 1.47
increasing frequency. Figs. 6 and 7 show the plots of complex third-order nonlinear permittivity e3 ¼ e03 je003 and conductivity s3 ¼ s3 0 þ js003 versus log f (circles), respectively. It is found that e03 has a characteristic frequency dependence wherein e03 with a positive sign shows a positive peak and then decreases to a negative constant with decreasing frequency. It is also found that s03 has a characteristic frequency dependence wherein s03 with a positive sign shows a negative peak and then increases to a positive constant with decreasing frequency. Furthermore, we emphasize that the complex second-order nonlinear permittivity e2 and conductivity s2 do not exist in all NASPFs. We then attempted to reproduce the observed spectra using a phenomenological nonlinear conductive relaxation [5–11]: s1 ¼ s1dc þ joein þ Ds1
jot1 ð1 þ ðjot1 Þb1 Þa1
ð7Þ
and s3 ¼ s3dc þ Ds3
jot3 ð1 þ ðjot3 Þb3 Þ3a3
:
ð8Þ
Here, sndc represents dc (low frequency) conductivity, Dsn the relaxation strength, tn the relaxation time, and an and bn the parameters expressing the distribution of relaxation times (n ¼ 1 and 3). The quantity ein in Eq. (7) is the linear permittivity which is assumed to be independent of frequency. As shown in Figs. 4–7, solid curves calculated from Eqs. (4), (7) and (8) reproduce the observed spectra reasonably well.
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
305
6
log ε1'/ε0
4
2 εin 0
-24
log ε1"/ε0
2
0
-2
-4 -2
0
2
4
6
log f (f : Hz) Fig. 4. Linear permittivity e1 ¼ e01 je001 spectra for a new ASP film.
4. Discussion If the ionic motion is of the hopping type, the resulting electric current I is given by [12] DU eaE I ¼ 2eNaP0 exp sin h : ð9Þ kT 2kT Here, a is the hopping distance, E the electric field, T the absolute temperature, P0 the hopping probability at the high-temperature limit, e the elementary electric
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
306
-4
log σ1' ( S/m )
-6
-8
σ1dc
-10
-12 -4
log σ1" ( S/m )
-6
-8
-10
-12 -2
0
2
4
6
log f (f : Hz) Fig. 5. Linear conductivity s1 ¼ s01 þ js001 spectra for a new ASP film.
charge, N the carrier density, k the Boltzmann constant and DU the activation energy. Expanding the hyperbolic function in powers of E; we have 1 2 2 DU e a P0 N exp s1 ¼ ; ð10Þ kT kT s3 ¼
1 DU 4 4 e a P N exp : 0 24k3 T 3 kT
ð11Þ
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
307
1
ε3'/ε0 (10-12m2V-2)
0
-1 ∆ε3 -2
-3
-4 -4 1
ε3"/ε0 (10-12m2V-2)
0
-1
-2
-3
-4 -1
0
1
2
3
4
log f (f :Hz) Fig. 6. Third-order nonlinear permittivity e3 ¼ e03 je003 spectra for a new ASP film.
If we take the ratio of s1 to s3 ; we obtain [5–8] s3 e 2 a2 ¼ ; s1 24k2 T 2
ð12Þ
yielding a hopping distance without the need for any additional assumptions. Using the values for s1 and s3 at high and low frequencies in the case of the NANSP with f ¼ 0:30; we obtain a ¼ 68 and 380 nm for the respective processes. Table 2 lists the average lengths of the major axis of clusters and BUT particles obtained by TEM data processing and from the nonlinear measurements of the NASPFS. We found
308
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
1.0
σ3' (10-20SmV-2)
0.5
σ3dc
0.0
-0.5
1.0 -1.0
σ3" (10-20SmV-2)
0.5
0.0
-0.5
-1.0 -2
-1
0
1
2
3
4
log f (f :Hz) Fig. 7. Third-order nonlinear conductivity s3 ¼ s03 þ js003 spectra for a new ASP film. Table 2 Data obtained from nonlinear measurements and TEM image BUT volume fraction f 30 wt% 15 wt% 7.5 wt% Long-range hopping distance obtained from nonlinear measurements (nm) 380 Major axis of clusters obtained from TEM image data (nm) 337 Short-range hopping distance obtained from nonlinear measurements (nm) 68 Diameter of BUT particles obtained from TEM image data (nm) 82
320 282 72 120
300 272 70 112
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
309
that long hopping distances, 380, 320 and 300 nm, were in good agreement with the average lengths of the major axis of clusters, 337, 282 and 272 nm, respectively, and that short hopping distances, 68, 72 and 70 nm, were in good agreement with the average lengths of BUT particles. On the basis of these results, we speculate that there are two separate conduction mechanisms in the NASPF. The high-frequency conduction could be attributed to free motions of ions due to their hopping to adjacent sites. Such motions seem to occur in limited domains. At low frequencies, ions reach the edge of these domains and generate polarization and yield a low conductivity governed by their hopping to adjacent domains. The former could be associated with the distance between adjacent hopping sites and the latter with the average size of domains in which ions can move rather freely. The nonlinear conductivity investigation presented here has revealed that there exists a microscopically inhomogeneous structure affecting the ionic transport processes in the NASPF.
5. Summary We have 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 the NASPF. We proposed a new empirical function to reproduce the observed linear and nonlinear conductive spectra. These spectra 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 making any additional assumptions, by using the values of nonlinear conductivities obtained experimentally. We then evaluated the hopping distance without making any additional assumptions. The short-range hopping distance obtained here may correspond to the distance between adjacent hopping sites in one rubber particle. The long-range hopping distance obtained here corresponds to the average size of the clusters formed by some rubber particles, in which ions can move freely. Namely, at low frequencies, ions reach the edge of such domains and give rise to polarization which yields the dc conductivity governed by the hopping distance between adjacent domains.
Acknowledgements We would like to thank Kureha Chemical Industry, Co., Ltd. for providing us with the antistatic polymer, BAYONs. This work was supported in part by a Grantin-Aid (Nos. 10875194 and 13650944) for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan. We also thank M. Yonezawa of Yamagata University, for his technical assistance.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
The Institute of Electrostatics Japan, (Ed.), Handbook of Electrostatics, Ohm, Tokyo, 1998. B.L. Lee, Polym. Eng. Sci. 32 (1992) 36–42. M. Suzuki, T. Fukumoto, Y. Taka, Kogyozairyo 42 (1994) 93–99 (in Japanese). J.R. Maccallum, C.A. Vincent (Eds.), Polymer Electrolyte Reviews 1& 2, Elsevier Appl. Sci., London and New York, 1989. Y. Tajitsu, J. Mater. Sci. 31 (1996) 2081–2089. Y. Tajitsu, J. Mater. Sci. 16 (1997) 1741–1744. Y. Tajitsu, J. Electrostatics 43 (1998) 203–213. Y. Tajitsu, J. Mater. Sci. Lett. 17 (1998) 989–992. Y. Tajitsu, J. Mater. Sci. Lett. 17 (1999) 1287–1290. S. Oka, O. Nakada, Theory of Solid State Dielectrics, Iwanami, Tokyo, 1964, pp. 31–51. O. Nakada, J. Phys. Soc. Jpn. 15 (1960) 2280–2288. N.F. Motto, E.A. Davis, Electronic Processes in Non-Crystalline Materials, 2nd Edition, Clarendon, London, 1979, pp. 125–135.
Nonlinear ionic-conductivity spectra of antistatic polymer films Yoshiro Tajitsu* Faculty of Engineering, Department of Polymer Science and Engineering, Yamagata University, Yonezawa, Yamagata 992-8510, Japan Received 8 December 2000; received in revised form 3 September 2001
Abstract We measured the frequency dependence of linear to third-order nonlinear complex conductivities and permittivities of a new antistatic polymer film (NASPF), that is, a composite film which consists of highly conductive butadiene rubber (BUT) particles containing Li+ and poly(methyl methacrylate), at room temperature. The nonlinear conductivities obtained allowed an estimate of the important parameters which characterize ionic transport in the composite film, such as the hopping distance of ions or the size of a cluster connected to the site capable of ion hopping, without any additional assumptions. We obtained two hopping distances, 68 and 380 nm. The former may correspond to the distance between adjacent hopping sites in one BUT particle. The latter corresponds to the average size of the clusters formed by some BUT particles, in which ions can move freely. The results are consistent with transmission electron microscopic image data concerning BUT particle dispersion in the NASPF. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Nonlinear conductivity; Antistatic polymer; Ionic conduction; Frequency spectra
1. Introduction Recently, the study of the antistatic mechanism due to the ionic transport in antistatic polymers (ASPs) has advanced rapidly and the resulting developments have become the focus of attention [1–3]. However, for high-grade practical application, for example, IC packaging, some critical requirements must be satisfied, which is not possible with the present level of technology, and many points must be *Tel.: +81-238-26-3410; fax: +81-238-26-3410. E-mail address: [email protected] (Y. Tajitsu). 0304-3886/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 8 6 ( 0 1 ) 0 0 2 1 2 - 1
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
300
improved. Therefore, the antistatic mechanism of the ionic transport process in ASPs must be studied. On the other hand, the mechanism of ionic transport in polymers has been studied for many years [4]. Detailed information regarding the correlation between ionic transport and dynamic behavior of the polymer chain are expected to be obtained through the investigation of the frequency dependence of conductivity [4]. In those studies, the frequency spectra of the complex conductivities have been shown to provide useful information concerning the ionic transport processes. However, most measurements have been carried out in a linear regime where the current response is proportional to the applied electric field. On the other hand, we have taken note of the possibility that studies on nonlinear spectra of conductivity may provide 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 [5–8]. In this paper, we extend the measurements of the conductivity spectra to a nonlinear regime in order to gain additional knowledge about the antistatic mechanism of the ionic transport process in ASPs.
2. Experiments 2.1. Materials The sample used in this study was a new ASP film (NASPF) which consists of highly conductive butadiene rubber (BUT) particles (80 nm diameter) containing Li+ (½Liþ =½O ¼ 0:05) dispersed in poly(methyl methacrylate) (PMMA) (reforming BAYONs: Kureha Chemical Industry, Co., Ltd.) [3]. The BUT particle volume fractions f of 7.5, 15.0 and 30.0 wt% in the NASPFs were obtained. The thickness of NASPFs obtained here is 50 mm. Circular lithium electrodes (10 mm diameter) were used for nonlinear measurements. 2.2. Measurements We have developed a new experimental system which enables measurements of linear as well as nonlinear complex conductivities sn ¼ s0n þ js00n (n ¼ 1 5) in a 10 mHz to 1 MHz frequency range [5–9]. Details of the experiments have been described previously [5–9]. Here, we touch briefly upon the principal points. The schematic diagram of the measuring system is shown in Fig. 1. The excitation wave is applied to the sample. The induced current at the electrode is detected by a current amplifier. The voltage and current signals are simultaneously stored in a wave memory. The detected data are used in calculations with the microcomputer. Actually, for the measurements of the complex linear and nonlinear conductivities, we applied a sinusoidal electric field with amplitude E0 and angular frequency o; EðtÞ ¼ E0 cos ot;
ð1Þ
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
SYNTHESIZER
fc
301
MICROCOMPUTER
f c'
VOLTAGE AMPLIFIER
WAVE MEMORY V (t)
ATTENUATOR
I (t)
V = V0 cos t SAMPLE
CURRENT AMPLIFIER I
Fig. 1. The schematic diagram of measuring system.
and detected the in-phase and 901-out-of-phase components of the electric currents with frequency no (n=1, 2, 3,y). N X IðtÞ ¼ ðIn0 cos not þ In00 sin notÞ: ð2Þ n¼1
As a result, we obtained the real and imaginary components of complex conductivities sn ¼ s0n þ js00n and nonlinear permittivity en ¼ e0n je00n (n=1, 2, 3,y) with 0
0
sn ¼
2n1 In ; E0n
0
sn ¼
2n1 In00 : E0n
sn ¼ jnoen :
ð3Þ ð4Þ
Using this system, we have measured sn (n ¼ 1;2,3) and en ¼ e0n je00n (n ¼ 1;2,3) of the NASPF. To measure the nonlinear permittivity spectrum of the NASPF [9], we applied an electric field E perpendicular to the film surface. Therefore, we can formally express the electric displacement D as D ¼ e1 E þ e2 E 2 þ e3 E 3 þ ? :
ð5Þ
Here, e1 is the linear permittivity. The coefficient en for n > 1 defines the nonlinear permittivity. In the same way, when the equation for current I for a nonlinear system is expanded in powers of the electric field E; I ¼ s1 E þ s3 E 2 þ s3 E 3 þ ? ;
ð6Þ
we can express the linear and nonlinear conductivities in terms of coefficients of the exponents. Here, s1 is the linear conductivity. The coefficient sn for n > 1 defines the nonlinear conductivity.
3. Results We observed the BUT particle dispersion in the NASPF using a transmission electron microscope (TEM). Typical TEM images are shown in Fig. 2. In these
302
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
Fig. 2. TEM images of a new ASP film. (a) f ¼ 30:0 wt%, (b) f ¼ 15:0 wt%, (c) f ¼ 7:5 wt%.
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
303
Fig. 3. TEM image data processing.
figures, two phases are seen to exist in the NASPF: one is made up of BUT particles with diameters of 70–100 nm which form clusters of about 400–500 nm in size, and the other contains PMMA. The BUT particle dispersion in the NASPF is determined by TEM graphic image data processing. Fig. 3 shows an example of typical data processing of a TEM image. The number of clusters, the number of BUT particles in a cluster, the area of a cluster, the area of BUT particles, the length of the major axis of the clusters, the length of the minor axis of the clusters, and the aspect ratio (=the minor axis length to major axis length of a cluster) were obtained. Table 1 lists the averages of the above data of clusters and BUT particles in NASPFS. Fig. 4 shows the frequency spectra of e1 ¼ e01 je001 obtained in the 0.025 HzB5 MHz frequency range plotted as circles, for a NASPF with f ¼ 0:15; at 201C. The spectra show a large relaxation and an increase with decreasing frequency. The first increase in the high-frequency range is associated with an inconspicuous loss peak, suggesting the existence of a relaxation process. The relaxation strength is >50. Furthermore, we found a second rapid increase in the low-frequency range. The real part s01 of complex linear conductivity s1 ¼ s01 þ js001 of a NASPF is then plotted in Fig. 5 as circles. s01 remains nearly constant in the low-frequency range and increases with
304
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
Table 1 Average data obtained by TEM image data processing BUT volume fraction f 30 wt%
15 wt%
7.5 wt%
Clusters Area (102 nm2) Major axis (nm) Minor axis (nm) Aspect ratio Total number of clusters Total number of BUT particles in a cluster BUT particles forming a cluster (%)
368 337 165 2.04 32 5.03 85.2
320 282 152 1.85 22 3.52 67.5
220 272 115 2.36 10 2.5 27.1
Butadiene rubber (BUT) particles Area (102 nm2) Major axis (nm) Minor axis (nm) Diameter (nm) Aspect ratio
55 114 63 82 1.8
81 138 104 120 1.32
89 145 98 112 1.47
increasing frequency. Figs. 6 and 7 show the plots of complex third-order nonlinear permittivity e3 ¼ e03 je003 and conductivity s3 ¼ s3 0 þ js003 versus log f (circles), respectively. It is found that e03 has a characteristic frequency dependence wherein e03 with a positive sign shows a positive peak and then decreases to a negative constant with decreasing frequency. It is also found that s03 has a characteristic frequency dependence wherein s03 with a positive sign shows a negative peak and then increases to a positive constant with decreasing frequency. Furthermore, we emphasize that the complex second-order nonlinear permittivity e2 and conductivity s2 do not exist in all NASPFs. We then attempted to reproduce the observed spectra using a phenomenological nonlinear conductive relaxation [5–11]: s1 ¼ s1dc þ joein þ Ds1
jot1 ð1 þ ðjot1 Þb1 Þa1
ð7Þ
and s3 ¼ s3dc þ Ds3
jot3 ð1 þ ðjot3 Þb3 Þ3a3
:
ð8Þ
Here, sndc represents dc (low frequency) conductivity, Dsn the relaxation strength, tn the relaxation time, and an and bn the parameters expressing the distribution of relaxation times (n ¼ 1 and 3). The quantity ein in Eq. (7) is the linear permittivity which is assumed to be independent of frequency. As shown in Figs. 4–7, solid curves calculated from Eqs. (4), (7) and (8) reproduce the observed spectra reasonably well.
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
305
6
log ε1'/ε0
4
2 εin 0
-24
log ε1"/ε0
2
0
-2
-4 -2
0
2
4
6
log f (f : Hz) Fig. 4. Linear permittivity e1 ¼ e01 je001 spectra for a new ASP film.
4. Discussion If the ionic motion is of the hopping type, the resulting electric current I is given by [12] DU eaE I ¼ 2eNaP0 exp sin h : ð9Þ kT 2kT Here, a is the hopping distance, E the electric field, T the absolute temperature, P0 the hopping probability at the high-temperature limit, e the elementary electric
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
306
-4
log σ1' ( S/m )
-6
-8
σ1dc
-10
-12 -4
log σ1" ( S/m )
-6
-8
-10
-12 -2
0
2
4
6
log f (f : Hz) Fig. 5. Linear conductivity s1 ¼ s01 þ js001 spectra for a new ASP film.
charge, N the carrier density, k the Boltzmann constant and DU the activation energy. Expanding the hyperbolic function in powers of E; we have 1 2 2 DU e a P0 N exp s1 ¼ ; ð10Þ kT kT s3 ¼
1 DU 4 4 e a P N exp : 0 24k3 T 3 kT
ð11Þ
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
307
1
ε3'/ε0 (10-12m2V-2)
0
-1 ∆ε3 -2
-3
-4 -4 1
ε3"/ε0 (10-12m2V-2)
0
-1
-2
-3
-4 -1
0
1
2
3
4
log f (f :Hz) Fig. 6. Third-order nonlinear permittivity e3 ¼ e03 je003 spectra for a new ASP film.
If we take the ratio of s1 to s3 ; we obtain [5–8] s3 e 2 a2 ¼ ; s1 24k2 T 2
ð12Þ
yielding a hopping distance without the need for any additional assumptions. Using the values for s1 and s3 at high and low frequencies in the case of the NANSP with f ¼ 0:30; we obtain a ¼ 68 and 380 nm for the respective processes. Table 2 lists the average lengths of the major axis of clusters and BUT particles obtained by TEM data processing and from the nonlinear measurements of the NASPFS. We found
308
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
1.0
σ3' (10-20SmV-2)
0.5
σ3dc
0.0
-0.5
1.0 -1.0
σ3" (10-20SmV-2)
0.5
0.0
-0.5
-1.0 -2
-1
0
1
2
3
4
log f (f :Hz) Fig. 7. Third-order nonlinear conductivity s3 ¼ s03 þ js003 spectra for a new ASP film. Table 2 Data obtained from nonlinear measurements and TEM image BUT volume fraction f 30 wt% 15 wt% 7.5 wt% Long-range hopping distance obtained from nonlinear measurements (nm) 380 Major axis of clusters obtained from TEM image data (nm) 337 Short-range hopping distance obtained from nonlinear measurements (nm) 68 Diameter of BUT particles obtained from TEM image data (nm) 82
320 282 72 120
300 272 70 112
Y. Tajitsu / Journal of Electrostatics 55 (2002) 299–310
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that long hopping distances, 380, 320 and 300 nm, were in good agreement with the average lengths of the major axis of clusters, 337, 282 and 272 nm, respectively, and that short hopping distances, 68, 72 and 70 nm, were in good agreement with the average lengths of BUT particles. On the basis of these results, we speculate that there are two separate conduction mechanisms in the NASPF. The high-frequency conduction could be attributed to free motions of ions due to their hopping to adjacent sites. Such motions seem to occur in limited domains. At low frequencies, ions reach the edge of these domains and generate polarization and yield a low conductivity governed by their hopping to adjacent domains. The former could be associated with the distance between adjacent hopping sites and the latter with the average size of domains in which ions can move rather freely. The nonlinear conductivity investigation presented here has revealed that there exists a microscopically inhomogeneous structure affecting the ionic transport processes in the NASPF.
5. Summary We have 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 the NASPF. We proposed a new empirical function to reproduce the observed linear and nonlinear conductive spectra. These spectra 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 making any additional assumptions, by using the values of nonlinear conductivities obtained experimentally. We then evaluated the hopping distance without making any additional assumptions. The short-range hopping distance obtained here may correspond to the distance between adjacent hopping sites in one rubber particle. The long-range hopping distance obtained here corresponds to the average size of the clusters formed by some rubber particles, in which ions can move freely. Namely, at low frequencies, ions reach the edge of such domains and give rise to polarization which yields the dc conductivity governed by the hopping distance between adjacent domains.
Acknowledgements We would like to thank Kureha Chemical Industry, Co., Ltd. for providing us with the antistatic polymer, BAYONs. This work was supported in part by a Grantin-Aid (Nos. 10875194 and 13650944) for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan. We also thank M. Yonezawa of Yamagata University, for his technical assistance.
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