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Dielectric relaxation of porous glasses
Journal of Non-Crystalline Solids 235±237 (1998) 302±307
Dielectric relaxation of porous glasses Anna Gutina a, Ekaterina Axelrod a, Alexander Puzenko a, Ewa Rysiakiewicz-Pasek b, Nick Kozlovich a, Yuri Feldman a,* a
Department of Applied Physics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel b Institute of Physics, Technical University of Wroclaw, 50-370 Wrocøaw, Poland
Abstract The dielectric properties of porous glasses, obtained from sodium borosilicate glass, were investigated in the frequency range 20 Hz to 1 MHz and temperature range )100°C to +300°C for the purpose of inferring the geometric properties of porous materials. The features of the dielectric properties due to the geometrical disorder were analysed by using models describing the non-Debye slow decay dynamics. The dielectric response is aected by the geometrical micro- and mesostructural properties of the porous matrix and the properties of the material ®lling the pores. It provides information on the hindered dynamics of water molecules, located within the pores and aected by the surfaces. An analysis of the dielectric parameters enables us to describe the porosity of the materials. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction A long-standing problem of considerable scienti®c and technological importance is the improvement of the understanding of the correlation between the morphology of porous materials and their electrical and dielectric properties [1±5]. One of the most important examples of porous systems is porous glasses [2,3,6,7] which could by itself be the subject of the scienti®c investigation. The glasses demonstrate interesting properties that could be used for various applications in optics, optoelectronics, medicine and biotechnology [8± 10]. They also can be considered as model media, each with a huge inner surface, high porosity, and a developed fractal structure [5,7]. The matrix represents a rigid sponge-like SiO2 framework that
* Corresponding author. Tel.: +972 2 6586187; fax: +972 2 5663878; e-mail: [email protected].
0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 5 6 2 - 6
can be ®lled with dierent materials [11,12]. Water can easily be adsorbed and desorbed from silica surfaces. The dielectric response is found to be aected by the geometrical nano- and mesostructural features of the porous matrix and dierent materials ®lling the pores. Dielectric properties of ®llers con®ned in small volumes of porous media have recently been studied using a variety of theoretical approaches and dielectric experimental techniques [2,3,11,12]. The main purpose of these studies is to understand the eect of pore size on structure and dynamics of ®llers. Another approach in a dielectric investigation of porous systems is to study the aects of the pore-space geometry and insulating porous material on the high- and low-frequency dielectric response and electric conductivity of the system when the pore space is ®lled with a conducting or non-conducting dielectric matter [13]. In this paper we shall study porous glasses with dierent controlled parameters of the pores ®lled
A. Gutina et al. / Journal of Non-Crystalline Solids 235±237 (1998) 302±307
with water molecules by dielectric spectroscopy in a wide frequency and temperature ranges. The main purpose of this investigation is to infer information about the dierent structures of the water adsorbed on the pore surface and the matrix morphology.
2. Experimental section Sample preparation: We studied three dierent porous glass samples A, B and C, obtained from one sodium borosilicate glass. Sample A was heated at 490°C for 165 h and sample C was heated at 650°C for 100 h. All the samples were then etched in hydrochloric acid and rinsed in deionized water. Sample B was obtained from sample A by additional immersion in KOH. This sample was also rinsed in deionized water. The samples had dierent structures. We can assume that sample A contains silica gel in the pore volumes albeit the silica gel is almost not present in samples B and C. Porosity of the samples was determined by relative mass decrement measurement after sodium borate extraction from samples A and C and after treatment in KOH for sample B. The porosity of samples A and B were found to be 38% and 48%, respectively and 38% was obtained for sample C and it did not change after the additional treatment in KOH. The dimensions of pores were calculated from absorption/desorption isotherms and from microscope photographs of porous glass samples [14]. The water content, h, de®ned as the ratio of the mass of sorbed water to the mass of the dry sample, was determined by weighing the samples prior to and after the dielectric measurements. The number of monolayers (samples B and C) were calculated according to the following equation:
N
h 100 ÿ upd 2 qD ; 12uM
303
1
where u Vp =V is the porosity, D is the average diameter of the pores, q is the density of the dry glass (q 1.5 g/cm3 ), d and M the diameter and the molecular mass of a water molecule, respectively, Vp the volume of pores and V the entire volume of a sample. The dimensions of pores, porosity of the samples, humidity and the number of water monolayers on the pore surfaces for samples B and C are presented in Table 1. The signi®cant amount of silica gel inside the pores of sample A did not make it possible to determine correctly the number of monolayers in this sample. The size and the thickness of the square plate samples, used for dielectric measurements, were 38 and 0.31 0.01 mm, respectively. Experimental technique: Dielectric measurements in the frequency range: 20 Hz to 1 MHz were performed by using a broad band dielectric spectrometer (BDS 4284, Novocontrol) with automatic temperature control (Quatro Cryosystem). The accuracy of the measured dielectric permittivity and losses is estimated to be <3%. The measurements were carried out in the following way: each of the samples was placed into the sample cell at room temperature, the measurements were then performed by cooling the samples from 20°C to )100°C. In the second stage the samples were measured by heating them to 300°C (a step of 5°C was used in both cases). The samples were then held at 300°C for an hour, after which they were measured with the temperature reduced to 25°C in steps of 15°C. 3. Results The typical spectra of the dielectric permittivity and losses for the porous samples as a function of
Table 1 Structural parameters and water content of A, B and C samples Sample
Dimension of the pores (nm)
Porosity / (%)
Humidity h (%)
Number of water monolayers N
A B C
50±70 50±70 280±400
38 1 48 1 38 1
1.19 0.05 1.21 0.05 3.20 0.05
not-determined 1.1 0.2 18 2
304
A. Gutina et al. / Journal of Non-Crystalline Solids 235±237 (1998) 302±307
frequency and temperature are displayed in Fig. 1. One can see that the complex dielectric behavior can be described in terms of the four distributed relaxation processes. The e00 f ; T cuts at a constant low-frequency plane (20 Hz) for samples A, B and C represent the temperature dependence of the dielectric losses versus temperature (see Fig. 2).
Fig. 1. Three dimension plot of the frequency and temperature dependence of the dielectric permittivity e0 (a) and losses e00 (b) for the porous glass, sample C.
The ®rst relaxation process is observed in the low temperature region )100°C to +10°C in samples A and C (see Fig. 1(b)). In sample B there are only the traces of this process, which most likely shifts to the high frequency limit that is out of the frequency range. The second relaxation process is well marked for all the samples and has a speci®c behavior in the temperature range: )50°C to +150°C. The kink in the spectrum of dielectric losses is observed in the temperature interval: 60±70°C. The magnitude of dielectric strength of this process does not change beyond this temperature region (see Fig. 2). The third process is located in the low-frequency region and the temperature interval: 20±80°C. The amplitude of this process essentially decreases when the frequency increases (see Fig. 1(a)) and the maximum of dielectric losses (see Fig. 1(b)) has almost no temperature dependence. The largest amplitude is observed for sample B (see Fig. 2). Sample C demonstrates a decrease of the amplitude and in sample A the process is hardly detected (see Fig. 2). In the high temperature region, above 150°C, all the materials become electrically conductive and show an increase of the dielectric losses in the low frequency limit. To analyze the fourth relaxation process subtraction of the DC-conductivity is required. After additional one-hour heating at 300°C the samples were cooled to 25°C and no relaxation process was detected.
Fig. 2. Temperature dependence of the low (20 Hz) frequency behavior of the dielectric losses e00 of samples A, B and C.
A. Gutina et al. / Journal of Non-Crystalline Solids 235±237 (1998) 302±307
4. Discussion
305
ior in silica±water systems at low water content has been observed recently [2,3] and ascribed to reorientation of water molecules in ice-like structures. The formation of the structure of this ice-like water is dependent on the number of water monolayers covering the pore surfaces. That is why in sample B (with N 1.1), the ®rst process is negligibly small. At the same time a similar amount of water in sample A gives a process that can be explained by the large amount of the nanometer sized pores in silica-gel inside the pores. The second relaxation process in the temperature interval below the kink region (60°C) displays a non-Arrhenius behavior for all the three samples (see Fig. 4). The temperature dependencies of the relaxation times can be ®tted to Vogel±Fulcher±Tammann (VFT) equations [18] that describe the relaxations of glass-forming liquids:
For a quantitative analysis of the dielectric spectra for the ®rst, second and third processes a superposition of Havriliak±Negami (HN) formula b Dej = 1 ixsj aj j [15] and Jonscher's empirical nÿ1 [16] has been ®tted to the isothermal term ix complex dielectric permittivity data. Here Dej is the dielectric strength and sj the mean relaxation time. The index j refers to the dierent processes which contribute to the dielectric response. The parameters aj and bj describe the symmetric and asymmetric broadening of the relaxation process; n is a Jonscher parameter for the high-frequency part of the relaxation process. In the case of the fourth process the superposition of the HN function and a conductivity contribution was used. The temperature dependencies of the relaxation times of the ®rst process for samples A and C (see Fig. 3) can be ®t with an Arrhenius equation with energies of activation, 42 3 kJ/mol and 80 3 kJ/ mol, respectively, and correlate with the activation energy of pure ice [17]. However, unlike the Debye relaxation of pure ice, the relaxation in samples A and C demonstrate a non-Debye dependence. The ®tting parameters a and b show dierent temperature dependencies for sample A and C, that can be connected to variance of the morphology of pores and the interaction of water molecules with the pores surface. A similar dielectric relaxation behav-
where s1 is the time characteristic of a molecular vibration, T0 the so-called Vogel or ideal glass transition temperature and A a constant. At higher temperatures, the deviation of the data from the VFT description that can be due to the evaporation of water molecules. We assume that the third relaxation process is related to the percolation of charge carriers within the fractal structure of connected pores. Analysis of dielectric relaxation parameters of this process
Fig. 3. Temperature dependence of the characteristic relaxation time of the ®rst process for samples A and C. The solid lines are a ®t of the Arrhenius equation to the data. The correlation coeecients of the ®ts are RA 0:96 and RC 0:94.
Fig. 4. Relaxation plot of the second process for the porous glasses A, B and C. The solid lines correspond to a ®t of the VFT equation. The correlation coecients are RA 0:97, RB 0:97, and RC 0:97.
log s log s1 A= T ÿ T0 ;
2
306
A. Gutina et al. / Journal of Non-Crystalline Solids 235±237 (1998) 302±307
allowed us to determine the geometrical properties of the porous medium. The dielectric response for this process in time domain can be described by a Kohlrausch±Williams±Watts (KWW) expression, W t=s exp ÿ t=sm , where W is the dipole correlation function, s the average relaxation time and m the stretched parameter, 0 < m 6 1. It was shown in Ref. [19] that in complex fractal systems, the stretched parameter m can be related to the fractal dimension Df of the path, m Df =3 for the relaxation related to the charge transfer along the rami®ed path. To describe the porous medium, the relationship between the fractal dimension and porosity of the material was obtained by using the model of a random fractal [20]. In the simplest case of the model, the mean deviations of the scaling parameters of the random fractal from the scaling parameters of the regular dominating fractal are zero. This makes it possible to obtain the approximate relationship connection between the porosity, /, and the fractal dimension, Df , of the porous structure. /
1 1 ÿ l4ÿDf 1 ; 4 ÿ Df 1 ÿ l 4 ÿ Df
3
where l k=K; k is the minimal scale (pore size), and K the maximal scale (sample size). The experimental macroscopic dipole correlation functions for samples B and C at the temperature of the kink were obtained by inverse Fourier transform (see Fig. 5). The correlation functions have been ®tted by the sum of two KWW functions. The parameter, m, for the long relaxation process was used for the calculation of fractal dimension, Df , for samples B and C. The results of the calculation are presented in Table 2. In this table we also show the porosities determined from the dipole correlation function analysis, which are compared and show agreement with the poros-
Fig. 5. Macroscopic dipole correlation functions of samples B and C at the corresponding kink point.
ities determined from relative mass decrement measurements. The fractal dimensions (Df < 2) indicate that the diusion of the charge carriers is on the fractal surface of the connective pores rather than in the bulk. The loss maximum that belongs to the fourth polarization process is hardly visible for all the samples because of the conduction losses. A subtraction of the conductivity from the relaxation spectrum in samples B and C reveals a peak without temperature dependent relaxation times (see Fig. 6) due to the interfacial polarization [21]. On the other hand, sample A has an activation energy (170 kJ/mol) due to the silica gel ®lling the pores. This relaxation process can be related to, Maxwell±Wagner±Sillars (MWS), polarization process as a result of trapping of free charge carriers at the interface, thus causing build-up of macroscopic charge separation, or space charge with a relatively long relaxation time [22]. Samples B and C also demonstrate correlation between the relaxation times of the fourth process and the pores sizes (see Fig. 6).
Table 2 Values of fractal dimension and porosity for samples B and C Sample
Stretched parameter m
Fractal dimension Df
Porosity / (%) (Obtained from Porosity / (%) (Obtained from sample preparation) dielectric measurements)
B C
0.64 0.02 0.46 0.02
1.92 0.06 1.38 0.06
48 1 38 1
48 1 38 1
A. Gutina et al. / Journal of Non-Crystalline Solids 235±237 (1998) 302±307
307
calculations. One of the authors (A.G.) acknowledges the support from the Israel Ministry of Science and Technology. References
Fig. 6. Temperature dependence of the characteristic relaxation time of the fourth process for samples A, B, and C. The solid lines correspond to a ®t of the Arrhenius equation. The correlation coecients are RA 0:96, RB 0:96, and RC 0:94.
5. Conclusions The dielectric relaxation of our porous glasses has a complex non-exponential behavior with some common features for all the samples. The dielectric relaxation is associated with the complex dynamics of the water molecules and is due to their interactions with pore surfaces. Analysis of the dielectric spectra of the porous samples enables us to calculate their fractal dimensions and porosities. The magnitudes agree with the data determined from measurements of the relative mass decrements. Acknowledgements The authors would like to express their appreciation to Professor Ya.O. Roizin for reading the manuscript and for helpful discussion, to Professor O. Lev and to Dr J. Gun for inspiration in writing this paper, to Mr Yu. Alexandrov for assistance in
[1] S. Kurosaki, J. Chem. Phys. 58 (1954) 320. [2] P. Pissis, J. Laudat, D. Daoukaki, A. Kyritsis, J. NonCryst. Solids 171 (1994) 201. [3] P. Pissis, A. Anagnostopoulou-Konsta, L. Apekis, D. Daoukaki-Diamanti, C. Christodoulides, J. Non-Cryst. Solids 131 (1991) 1174. [4] T. Sakamoto, H. Nakamura, A. Wada, J. Phys. Chem. 93 (1989) 357. [5] R. Hilfer, Adv. Chem. Phys. 42 (1996) 299. [6] G. Allen, P.C.L. Stephenson, J.H. Strange, J. Chem. Phys. 106 (1997) 7802. [7] J.-C. Li, D.K. Ross, J. Phys.: Condens. Matter 6 (1994) 351. [8] A.L. Huston, B.L. Justus, Opt. Lett. 20 (1995) 952. [9] D. Avnir, R. Gutfraind, D. Farin, in: A. Bunde, S. Halvin (Eds.), Fractals in Science, Springer, Berlin, 1995. [10] A.B. Shelekhin, S. Pien, Y.H. Ma, J. Membrane Sci. 103 (1995) 39. [11] J. Sch uller, R. Richert, E.W. Fisher, Phys. Rev. B 52 (1995) 15232. [12] M. Arndt, R. Stannarius, W. Gorbatschow, F. Kremer, Phys. Rev. E 54 (1996) 5377. [13] R. Hilfer, W.E. Kenyon, J. Appl. Phys. 55 (1983) 3153. [14] A.V. Alexeev-Porov, Ya.O. Roizin, E. Rysiakiewicz-Pasek, K. Marczuk, Opt. Mater. 2 (1993) 249. [15] S. Havriliak, S. Negami, J. Polym. Sci. C 14 (1966) 99. [16] A.K. Jonscher, Universal Relaxation Law, Chelsea Dielectrics, London, 1996. [17] R.K. Chan, D.W. Davidson, E. Whalley, J. Chem. Phys. 43 (1965) 2376. [18] A. Sch onhals, F. Kremer, J. Non-Cryst. Solids 172 (1994) 336. [19] Yu. Feldman, N. Kozlovich, Yu. Alexandrov, R. Nigmatullin and Ya. Ryabov, Phys. Rev. E 54 (1996) 5420. [20] R. Nigmatullin, Phys. Stat. Sol. B 153 (1989) 49. [21] S. Takashima, Electric Properties of Biopolymers and Membranes, Adam Hilger, Bristol, UK, 1989. [22] G. Perrier, A. Bergeret, J. Appl. Phys. 77 (1995) 2651.
Dielectric relaxation of porous glasses Anna Gutina a, Ekaterina Axelrod a, Alexander Puzenko a, Ewa Rysiakiewicz-Pasek b, Nick Kozlovich a, Yuri Feldman a,* a
Department of Applied Physics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel b Institute of Physics, Technical University of Wroclaw, 50-370 Wrocøaw, Poland
Abstract The dielectric properties of porous glasses, obtained from sodium borosilicate glass, were investigated in the frequency range 20 Hz to 1 MHz and temperature range )100°C to +300°C for the purpose of inferring the geometric properties of porous materials. The features of the dielectric properties due to the geometrical disorder were analysed by using models describing the non-Debye slow decay dynamics. The dielectric response is aected by the geometrical micro- and mesostructural properties of the porous matrix and the properties of the material ®lling the pores. It provides information on the hindered dynamics of water molecules, located within the pores and aected by the surfaces. An analysis of the dielectric parameters enables us to describe the porosity of the materials. Ó 1998 Elsevier Science B.V. All rights reserved.
1. Introduction A long-standing problem of considerable scienti®c and technological importance is the improvement of the understanding of the correlation between the morphology of porous materials and their electrical and dielectric properties [1±5]. One of the most important examples of porous systems is porous glasses [2,3,6,7] which could by itself be the subject of the scienti®c investigation. The glasses demonstrate interesting properties that could be used for various applications in optics, optoelectronics, medicine and biotechnology [8± 10]. They also can be considered as model media, each with a huge inner surface, high porosity, and a developed fractal structure [5,7]. The matrix represents a rigid sponge-like SiO2 framework that
* Corresponding author. Tel.: +972 2 6586187; fax: +972 2 5663878; e-mail: [email protected].
0022-3093/98/$19.00 Ó 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 8 ) 0 0 5 6 2 - 6
can be ®lled with dierent materials [11,12]. Water can easily be adsorbed and desorbed from silica surfaces. The dielectric response is found to be aected by the geometrical nano- and mesostructural features of the porous matrix and dierent materials ®lling the pores. Dielectric properties of ®llers con®ned in small volumes of porous media have recently been studied using a variety of theoretical approaches and dielectric experimental techniques [2,3,11,12]. The main purpose of these studies is to understand the eect of pore size on structure and dynamics of ®llers. Another approach in a dielectric investigation of porous systems is to study the aects of the pore-space geometry and insulating porous material on the high- and low-frequency dielectric response and electric conductivity of the system when the pore space is ®lled with a conducting or non-conducting dielectric matter [13]. In this paper we shall study porous glasses with dierent controlled parameters of the pores ®lled
A. Gutina et al. / Journal of Non-Crystalline Solids 235±237 (1998) 302±307
with water molecules by dielectric spectroscopy in a wide frequency and temperature ranges. The main purpose of this investigation is to infer information about the dierent structures of the water adsorbed on the pore surface and the matrix morphology.
2. Experimental section Sample preparation: We studied three dierent porous glass samples A, B and C, obtained from one sodium borosilicate glass. Sample A was heated at 490°C for 165 h and sample C was heated at 650°C for 100 h. All the samples were then etched in hydrochloric acid and rinsed in deionized water. Sample B was obtained from sample A by additional immersion in KOH. This sample was also rinsed in deionized water. The samples had dierent structures. We can assume that sample A contains silica gel in the pore volumes albeit the silica gel is almost not present in samples B and C. Porosity of the samples was determined by relative mass decrement measurement after sodium borate extraction from samples A and C and after treatment in KOH for sample B. The porosity of samples A and B were found to be 38% and 48%, respectively and 38% was obtained for sample C and it did not change after the additional treatment in KOH. The dimensions of pores were calculated from absorption/desorption isotherms and from microscope photographs of porous glass samples [14]. The water content, h, de®ned as the ratio of the mass of sorbed water to the mass of the dry sample, was determined by weighing the samples prior to and after the dielectric measurements. The number of monolayers (samples B and C) were calculated according to the following equation:
N
h 100 ÿ upd 2 qD ; 12uM
303
1
where u Vp =V is the porosity, D is the average diameter of the pores, q is the density of the dry glass (q 1.5 g/cm3 ), d and M the diameter and the molecular mass of a water molecule, respectively, Vp the volume of pores and V the entire volume of a sample. The dimensions of pores, porosity of the samples, humidity and the number of water monolayers on the pore surfaces for samples B and C are presented in Table 1. The signi®cant amount of silica gel inside the pores of sample A did not make it possible to determine correctly the number of monolayers in this sample. The size and the thickness of the square plate samples, used for dielectric measurements, were 38 and 0.31 0.01 mm, respectively. Experimental technique: Dielectric measurements in the frequency range: 20 Hz to 1 MHz were performed by using a broad band dielectric spectrometer (BDS 4284, Novocontrol) with automatic temperature control (Quatro Cryosystem). The accuracy of the measured dielectric permittivity and losses is estimated to be <3%. The measurements were carried out in the following way: each of the samples was placed into the sample cell at room temperature, the measurements were then performed by cooling the samples from 20°C to )100°C. In the second stage the samples were measured by heating them to 300°C (a step of 5°C was used in both cases). The samples were then held at 300°C for an hour, after which they were measured with the temperature reduced to 25°C in steps of 15°C. 3. Results The typical spectra of the dielectric permittivity and losses for the porous samples as a function of
Table 1 Structural parameters and water content of A, B and C samples Sample
Dimension of the pores (nm)
Porosity / (%)
Humidity h (%)
Number of water monolayers N
A B C
50±70 50±70 280±400
38 1 48 1 38 1
1.19 0.05 1.21 0.05 3.20 0.05
not-determined 1.1 0.2 18 2
304
A. Gutina et al. / Journal of Non-Crystalline Solids 235±237 (1998) 302±307
frequency and temperature are displayed in Fig. 1. One can see that the complex dielectric behavior can be described in terms of the four distributed relaxation processes. The e00 f ; T cuts at a constant low-frequency plane (20 Hz) for samples A, B and C represent the temperature dependence of the dielectric losses versus temperature (see Fig. 2).
Fig. 1. Three dimension plot of the frequency and temperature dependence of the dielectric permittivity e0 (a) and losses e00 (b) for the porous glass, sample C.
The ®rst relaxation process is observed in the low temperature region )100°C to +10°C in samples A and C (see Fig. 1(b)). In sample B there are only the traces of this process, which most likely shifts to the high frequency limit that is out of the frequency range. The second relaxation process is well marked for all the samples and has a speci®c behavior in the temperature range: )50°C to +150°C. The kink in the spectrum of dielectric losses is observed in the temperature interval: 60±70°C. The magnitude of dielectric strength of this process does not change beyond this temperature region (see Fig. 2). The third process is located in the low-frequency region and the temperature interval: 20±80°C. The amplitude of this process essentially decreases when the frequency increases (see Fig. 1(a)) and the maximum of dielectric losses (see Fig. 1(b)) has almost no temperature dependence. The largest amplitude is observed for sample B (see Fig. 2). Sample C demonstrates a decrease of the amplitude and in sample A the process is hardly detected (see Fig. 2). In the high temperature region, above 150°C, all the materials become electrically conductive and show an increase of the dielectric losses in the low frequency limit. To analyze the fourth relaxation process subtraction of the DC-conductivity is required. After additional one-hour heating at 300°C the samples were cooled to 25°C and no relaxation process was detected.
Fig. 2. Temperature dependence of the low (20 Hz) frequency behavior of the dielectric losses e00 of samples A, B and C.
A. Gutina et al. / Journal of Non-Crystalline Solids 235±237 (1998) 302±307
4. Discussion
305
ior in silica±water systems at low water content has been observed recently [2,3] and ascribed to reorientation of water molecules in ice-like structures. The formation of the structure of this ice-like water is dependent on the number of water monolayers covering the pore surfaces. That is why in sample B (with N 1.1), the ®rst process is negligibly small. At the same time a similar amount of water in sample A gives a process that can be explained by the large amount of the nanometer sized pores in silica-gel inside the pores. The second relaxation process in the temperature interval below the kink region (60°C) displays a non-Arrhenius behavior for all the three samples (see Fig. 4). The temperature dependencies of the relaxation times can be ®tted to Vogel±Fulcher±Tammann (VFT) equations [18] that describe the relaxations of glass-forming liquids:
For a quantitative analysis of the dielectric spectra for the ®rst, second and third processes a superposition of Havriliak±Negami (HN) formula b Dej = 1 ixsj aj j [15] and Jonscher's empirical nÿ1 [16] has been ®tted to the isothermal term ix complex dielectric permittivity data. Here Dej is the dielectric strength and sj the mean relaxation time. The index j refers to the dierent processes which contribute to the dielectric response. The parameters aj and bj describe the symmetric and asymmetric broadening of the relaxation process; n is a Jonscher parameter for the high-frequency part of the relaxation process. In the case of the fourth process the superposition of the HN function and a conductivity contribution was used. The temperature dependencies of the relaxation times of the ®rst process for samples A and C (see Fig. 3) can be ®t with an Arrhenius equation with energies of activation, 42 3 kJ/mol and 80 3 kJ/ mol, respectively, and correlate with the activation energy of pure ice [17]. However, unlike the Debye relaxation of pure ice, the relaxation in samples A and C demonstrate a non-Debye dependence. The ®tting parameters a and b show dierent temperature dependencies for sample A and C, that can be connected to variance of the morphology of pores and the interaction of water molecules with the pores surface. A similar dielectric relaxation behav-
where s1 is the time characteristic of a molecular vibration, T0 the so-called Vogel or ideal glass transition temperature and A a constant. At higher temperatures, the deviation of the data from the VFT description that can be due to the evaporation of water molecules. We assume that the third relaxation process is related to the percolation of charge carriers within the fractal structure of connected pores. Analysis of dielectric relaxation parameters of this process
Fig. 3. Temperature dependence of the characteristic relaxation time of the ®rst process for samples A and C. The solid lines are a ®t of the Arrhenius equation to the data. The correlation coeecients of the ®ts are RA 0:96 and RC 0:94.
Fig. 4. Relaxation plot of the second process for the porous glasses A, B and C. The solid lines correspond to a ®t of the VFT equation. The correlation coecients are RA 0:97, RB 0:97, and RC 0:97.
log s log s1 A= T ÿ T0 ;
2
306
A. Gutina et al. / Journal of Non-Crystalline Solids 235±237 (1998) 302±307
allowed us to determine the geometrical properties of the porous medium. The dielectric response for this process in time domain can be described by a Kohlrausch±Williams±Watts (KWW) expression, W t=s exp ÿ t=sm , where W is the dipole correlation function, s the average relaxation time and m the stretched parameter, 0 < m 6 1. It was shown in Ref. [19] that in complex fractal systems, the stretched parameter m can be related to the fractal dimension Df of the path, m Df =3 for the relaxation related to the charge transfer along the rami®ed path. To describe the porous medium, the relationship between the fractal dimension and porosity of the material was obtained by using the model of a random fractal [20]. In the simplest case of the model, the mean deviations of the scaling parameters of the random fractal from the scaling parameters of the regular dominating fractal are zero. This makes it possible to obtain the approximate relationship connection between the porosity, /, and the fractal dimension, Df , of the porous structure. /
1 1 ÿ l4ÿDf 1 ; 4 ÿ Df 1 ÿ l 4 ÿ Df
3
where l k=K; k is the minimal scale (pore size), and K the maximal scale (sample size). The experimental macroscopic dipole correlation functions for samples B and C at the temperature of the kink were obtained by inverse Fourier transform (see Fig. 5). The correlation functions have been ®tted by the sum of two KWW functions. The parameter, m, for the long relaxation process was used for the calculation of fractal dimension, Df , for samples B and C. The results of the calculation are presented in Table 2. In this table we also show the porosities determined from the dipole correlation function analysis, which are compared and show agreement with the poros-
Fig. 5. Macroscopic dipole correlation functions of samples B and C at the corresponding kink point.
ities determined from relative mass decrement measurements. The fractal dimensions (Df < 2) indicate that the diusion of the charge carriers is on the fractal surface of the connective pores rather than in the bulk. The loss maximum that belongs to the fourth polarization process is hardly visible for all the samples because of the conduction losses. A subtraction of the conductivity from the relaxation spectrum in samples B and C reveals a peak without temperature dependent relaxation times (see Fig. 6) due to the interfacial polarization [21]. On the other hand, sample A has an activation energy (170 kJ/mol) due to the silica gel ®lling the pores. This relaxation process can be related to, Maxwell±Wagner±Sillars (MWS), polarization process as a result of trapping of free charge carriers at the interface, thus causing build-up of macroscopic charge separation, or space charge with a relatively long relaxation time [22]. Samples B and C also demonstrate correlation between the relaxation times of the fourth process and the pores sizes (see Fig. 6).
Table 2 Values of fractal dimension and porosity for samples B and C Sample
Stretched parameter m
Fractal dimension Df
Porosity / (%) (Obtained from Porosity / (%) (Obtained from sample preparation) dielectric measurements)
B C
0.64 0.02 0.46 0.02
1.92 0.06 1.38 0.06
48 1 38 1
48 1 38 1
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calculations. One of the authors (A.G.) acknowledges the support from the Israel Ministry of Science and Technology. References
Fig. 6. Temperature dependence of the characteristic relaxation time of the fourth process for samples A, B, and C. The solid lines correspond to a ®t of the Arrhenius equation. The correlation coecients are RA 0:96, RB 0:96, and RC 0:94.
5. Conclusions The dielectric relaxation of our porous glasses has a complex non-exponential behavior with some common features for all the samples. The dielectric relaxation is associated with the complex dynamics of the water molecules and is due to their interactions with pore surfaces. Analysis of the dielectric spectra of the porous samples enables us to calculate their fractal dimensions and porosities. The magnitudes agree with the data determined from measurements of the relative mass decrements. Acknowledgements The authors would like to express their appreciation to Professor Ya.O. Roizin for reading the manuscript and for helpful discussion, to Professor O. Lev and to Dr J. Gun for inspiration in writing this paper, to Mr Yu. Alexandrov for assistance in
[1] S. Kurosaki, J. Chem. Phys. 58 (1954) 320. [2] P. Pissis, J. Laudat, D. Daoukaki, A. Kyritsis, J. NonCryst. Solids 171 (1994) 201. [3] P. Pissis, A. Anagnostopoulou-Konsta, L. Apekis, D. Daoukaki-Diamanti, C. Christodoulides, J. Non-Cryst. Solids 131 (1991) 1174. [4] T. Sakamoto, H. Nakamura, A. Wada, J. Phys. Chem. 93 (1989) 357. [5] R. Hilfer, Adv. Chem. Phys. 42 (1996) 299. [6] G. Allen, P.C.L. Stephenson, J.H. Strange, J. Chem. Phys. 106 (1997) 7802. [7] J.-C. Li, D.K. Ross, J. Phys.: Condens. Matter 6 (1994) 351. [8] A.L. Huston, B.L. Justus, Opt. Lett. 20 (1995) 952. [9] D. Avnir, R. Gutfraind, D. Farin, in: A. Bunde, S. Halvin (Eds.), Fractals in Science, Springer, Berlin, 1995. [10] A.B. Shelekhin, S. Pien, Y.H. Ma, J. Membrane Sci. 103 (1995) 39. [11] J. Sch uller, R. Richert, E.W. Fisher, Phys. Rev. B 52 (1995) 15232. [12] M. Arndt, R. Stannarius, W. Gorbatschow, F. Kremer, Phys. Rev. E 54 (1996) 5377. [13] R. Hilfer, W.E. Kenyon, J. Appl. Phys. 55 (1983) 3153. [14] A.V. Alexeev-Porov, Ya.O. Roizin, E. Rysiakiewicz-Pasek, K. Marczuk, Opt. Mater. 2 (1993) 249. [15] S. Havriliak, S. Negami, J. Polym. Sci. C 14 (1966) 99. [16] A.K. Jonscher, Universal Relaxation Law, Chelsea Dielectrics, London, 1996. [17] R.K. Chan, D.W. Davidson, E. Whalley, J. Chem. Phys. 43 (1965) 2376. [18] A. Sch onhals, F. Kremer, J. Non-Cryst. Solids 172 (1994) 336. [19] Yu. Feldman, N. Kozlovich, Yu. Alexandrov, R. Nigmatullin and Ya. Ryabov, Phys. Rev. E 54 (1996) 5420. [20] R. Nigmatullin, Phys. Stat. Sol. B 153 (1989) 49. [21] S. Takashima, Electric Properties of Biopolymers and Membranes, Adam Hilger, Bristol, UK, 1989. [22] G. Perrier, A. Bergeret, J. Appl. Phys. 77 (1995) 2651.