ARTICLE pubs.acs.org/JPCC
Silver Ion Dynamics in Ag2S-Doped Silver MolybdateGlass Nanocomposites: Correlation of Conductivity and Scaling with Structure B. Deb and A. Ghosh* Department of Solid State Physics, Indian Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India ABSTRACT: This work reports the study of silver ion dynamics in Ag2S-doped silver molybdateglass nanocomposites of compositions xAg2S(1 x)(yAg2O(1 y)MoO3). The volume fraction of crystalline phases in these glass nanocomposites increases with the increase of Ag2S content and considerably influences the dc conductivity. It is observed that a significant amount of volume fraction of crystalline phases for x g 0.15 for the y = 0.20 series and for x = 0.20 for the y = 0.30 series causes the conductivity to decrease. The power law exponent has been obtained from ac conductivity spectra using the power law model and is observed to be almost constant at lower volume fraction of the nanocrystalline phases embedded in the glass matrix, but it decreases for samples with high crystalline volume fraction. The electric modulus data have been analyzed on the basis of the HavriliakNegami (HN) equation. The KohlrauschWilliamsWatts (KWW) stretched exponent β, obtained from modulus data, indicates a strong nonexponential relaxation. The scaling of the conductivity spectra reveals that the relaxation dynamics is independent of temperature but is dependent on compositions affected by the increased volume fraction of crystalline phases of the glass nanocomposites.
1. INTRODUCTION Salt-doped ion-conducting glasses and glass nanocomposites are important from a scientific point of view to understand the ion dynamics in complex disorder materials. Several theoretical and experimental studies14 have been devoted to such ionconducting glasses to account for the variation of the conductivity with composition and to correlate the conductivity with their structures. It is assumed in a few studies that all the cations contribute to the conductivity and the mobility of the cations mainly increases with the increase in temperature.3 On the other hand, in some models4 it has been observed that the mobility is constant and the change in the concentration of mobile ions is correlated to the change in the conductivity. Recent reports based on extended X-ray absorption fine structure (EXAFS)5 and reverse Monte Carlo (RMC) modeling6 reveal that the expansion of the glassy network caused by an addition of dopant salts increases the volume of the conduction pathways and hence the mobility of the cations leading to the enhancement of the conductivity. Thus, there exists a great diversity regarding the assumptions in various theoretical models considered and any general unification of structuretransport correlation is still lacking.7 Impedance spectroscopy has been used to probe the relaxation phenomena occurring in various ion-conducting disorder materials,8 and the different dynamical processes involved in the conductivity relaxation mechanism have been predicted. The conductivity spectra have been analyzed by invoking a characteristic length scale denoting the average displacement of ions per activated jump during hoping motion of the ions, which is correlated r 2011 American Chemical Society
with the structure of the glassy network.9 The frequency-dependent ac conductivity of different ionic conductors shows nearly universal behavior of the ionic relaxation process, which is conveniently described by a power law.10,11 The physical interpretation of this universal behavior, however, remains contradictory, as it is sometimes considered to be a manifestation of the same underlying microscopic phenomenon,12 whereas in some cases it is just considered as a macroscopic phenomenon.13 Thus, the deviation from universality has to be mooted carefully to understand the microscopic parameters influencing the relaxation dynamics. Alternatively, the electric modulus formalism14,15 has been used in the time domain to study the ionic relaxation. However, the issue of which formalism describes best the relaxation in ionconducting glasses is controversially discussed in the literature.1618 Nevertheless, the modulus formalism is useful in various cases, and even severe effects due to polarization of electrodes and other interfaces have negligible contribution in this formalism.14,15 The complex electric modulus, M* is given by the inverse of complex dielectric permittivity, ε*. The relaxation phenomena occurring in various ion-conducting glasses are generally observed to be non-Debye type characterized by a nonexponential decay of the relaxation function or the corresponding correlation function. This nonexponential behavior arises from the distribution of hopping sites of the mobile ions having different environments Received: May 13, 2011 Revised: June 18, 2011 Published: June 20, 2011 14141
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resulting from the disordered topology of the glass structure.19 When the interaction among the charge carriers becomes prominent, the effect of ionion interaction and correlation among the ions can also influence the relaxation dynamics.20 The formation and growth of nanocrystallites within the glassy matrix can considerably influence the ion transport behavior.21 In a few silver molybdateglass nanocomposites, an increase of the conductivity has been observed in comparison to the parent glass matrix and is attributed to the decrease of particle size embedded in the glass matrix.22 In contrary, in a few lithium ion conducting glass ceramics, an increase of the conductivity with the increase of particle size and crystallinity has been observed.23 Thus, the microstructure of the glass nanocomposites plays a crucial role in determining the variation of the conductivity and relaxation dynamics with composition. Although a lot of studies have been devoted to AgI-doped glasses2427 and glass nanocomposites,23 the study of the ionic conductivity of the Ag2S-doped glasses and glass nanocomposite is rare.28 In this work we focus our attention to study the relaxation dynamics in the Ag2S-doped silver molybdateglass nanocomposites with an attempt to correlate the ion dynamics to the structure. The relaxation behavior is observed to be highly nonDebye. It is further revealed that the scaling property of the relaxation dynamics depends strongly on their microstructure.
2. EXPERIMENTAL PROCEDURE Two series of compositions xAg2S(1 x)(yAg2O(1 y)MoO3), where x = 0.05, 0.10, 0.15, and 0.20 and y = 0.20 and 0.30, were prepared by the melt quenching technique. The details of preparation have been reported elsewhere.29 Appropriate amounts of Ag2S, AgNO3, and MoO3 were mixed and heated initially in alumina crucibles in an electric furnace at 450 °C for 4 h for denitrogenation of AgNO3. Then the mixtures were melted in the temperature range of 850950 °C depending on compositions and equilibrated for 2 h and then quenched to room temperature by pressing them between two aluminum plates. The formation of glass nanocomposites was ascertained using X-ray diffraction (XRD) and transmission electron microscopic (TEM) studies.29 The density of the glass samples was measured using Archimedes principle with acetone as immersion liquid. The molar volumes for the Ag2S-doped samples were calculated from the sample density and composition. The conductivity measurements were accomplished using an LCR meter (QuadTech, model 7600) in the frequency range of 10 Hz to 2 MHz and in a wide temperature range, using silver paste as electrodes. The measurements were made in a cryostat with a stability of temperature ∼( 0.1 K. 3. RESULTS AND DISCUSSION The dc conductivity (σdc) for all the glass compositions has been calculated from the complex impedance plots. Figure 1a shows the reciprocal temperature dependence of the dc conductivity for all the samples. It is noted that the dc conductivity follows the Arrhenius relation of the form σ dc ¼ σ0 expð Eσ =kB TÞ
ð1Þ
where σ0 is the pre-exponential factor, T is the absolute temperature, Eσ is the activation energy, and kB is the Boltzmann constant. The values of activation energy Eσ have been calculated from the least-squares straight line fits to the data and are listed in
Figure 1. (a) Variation of the dc conductivity (σdc) as a function of reciprocal temperature for the glass nanocomposites of compositions xAg2S(1 x)(yAg2O(1 y)MoO3). The solid lines are the leastsquares linear fit to the data. (b) Composition dependence of the dc conductivity (open symbols) at 303 K and the crystalline volume fraction, Vf (solid symbols) (ref 29), for the same compositions as in panel a. The solid lines are the guide to the eye.
Table 1. The variation of σdc with Ag2S content (x) at a fixed temperature (303 K) is shown in Figure 1b. It is observed that σdc for the y = 0.20 series increases with increase of Ag2S content up to x = 0.10 and then decreases at Ag2S contents x g 0.15, whereas the conductivity for the y = 0.30 series increases up to x = 0.15 and then decreases for x = 0.20. It is thus noted that the conductivity depends strongly on the composition. The compositional variation of the dc conductivity can be explained on the basis of physical and microstructural details of the glass nanocomposites. The volume expansion (ΔV) of the samples was determined using the relation ΔV = (Vd Vm)/Vm, where Vd and Vm are the molar volumes of Ag2S-doped and undoped samples, respectively. The value of Vm for undoped binary glasses was taken from ref 30. The percentages of the volume expansion of the glass structure for all the compositions are listed in Table 1. It is clearly noted in Table 1 that the volume expansion for the y = 0.20 series increases with increase of x up to 0.10 and then decreases as x increases further, whereas for the y = 0.30 series the volume expansion increases up to x = 0.15 and then decreases for x = 0.20. Thus, the composition dependence of the volume expansion is similar to that of the dc conductivity. The increase of the volume expansion favors the creation of a more open network structure for ion migration so that the mobility of ions increases and thereby the conductivity increases.5,6 Similarly, the decrease of the volume expansion decreases the pathway volume for ion 14142
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Table 1. Volume Expansion (%ΔV), Crystalline Volume Fraction Vf (%), dc Conductivity (σdc), dc Activation Energy (Eσ), Crossover Frequency (ωc), ac Activation Energy (Ec), and the Power Law Exponent (n) for the Compositions xAg2S(1 x)(yAg2O(1 y)MoO3) log10[σdc (Ω1 cm1)] at
log10[ωc (rad s1)] at
composition
ΔV (%)
Vf (%)
303 K ((0.05)
x = 0.05
1.13
0.5
7.53
0.69
4.29
0.70
0.64
x = 0.10
2.05
1.5
7.07
0.64
5.05
0.63
0.66
x = 0.15 x = 0.20
1.65 0.92
7.34 7.89
0.65 0.66
4.58 3.23
0.64 0.66
0.63 0.48
x = 0.05
0.97
0.3
6.60
0.58
5.64
0.59
0.65
x = 0.10
2.47
1
6.23
0.56
5.91
0.57
0.63
Eσ (eV) ((0.01)
303 K ((0.05)
Ec (eV) ((0.01)
n ((0.02)
y = 0.20
8 14
y = 0.30
x = 0.15
2.84
x = 0.20
1.95
2.3 17
5.99
0.55
6.05
0.54
0.64
6.42
0.59
5.36
0.59
0.51
Figure 2. TEM images for two compositions of xAg2S(1 x)(yAg2O(1 y)MoO3): (a) x = 0.10, y = 0.20 and (b) x = 0.15, y = 0.30. The inset in panel a is the selected area electron diffraction pattern.
migration and hence the conductivity. The microstructural studies of these glass nanocomposites were reported earlier.29 Figure 2 shows the TEM images from some selected compositions. It is observed that the samples contain nanocrystallites dispersed within the glass matrix. It was observed that in these samples different nanocrystalline phases corresponding to R-Ag2MoO4, Ag2Mo2O7, and γ-Ag2S were present.29 The average size of the crystallites was found to increase slightly with the increase in Ag2S content for both the series y = 0.20 and y = 0.30. The crystalline volume fraction of these glass nanocomposites was determined earlier29 from the XRD data in order to assess the amount of crystallinity present in the glass compositions. The values of crystalline volume fraction for all the compositions are listed in Table 1. In Figure 1b the compositional variation of crystalline volume fraction (Vf) is plotted along with the dc conductivity. It is noted that for x e 0.10 for the y = 0.20 series the crystallinity is very low (10%) for x g 0.15. For the y = 0.30 series the crystallinity is very less (16%) for x = 0.20. Thus, an initial increase of the conductivity for x e 0.10 for the y = 0.20 series and for x e 0.15 for the y = 0.30 series is attributed to the increase of the free volume or the volume expansion of the glass structure as discussed in the preceding paragraph. However, for Ag2S content x g 0.15 for the y = 0.20 series and for x = 0.20 for the y = 0.30
Figure 3. (a) Conductivity spectra for the composition 0.05Ag2S 0.95(0.20Ag2O0.80MoO3) at several temperatures shown. The solid lines are the fits to eq 2. (b) The temperature dependence of the crossover frequency ωc for different compositions of xAg2S(1 x)(yAg2O (1 y)MoO3). The solid lines are the least-squares linear fit to the data. The inset shows the variation of the power law exponent n with composition.
series, the crystalline volume fraction is considerably high so that the ion dynamics is influenced by these nanocrystallites embedded within the glass matrix. The increase of the nanocrystallite volume fraction at higher Ag2S content affects the mobility of ion and hence the conductivity. The ac conductivity (σac) spectra at several temperatures for a composition are shown in Figure 3a. At lower frequencies the ac 14143
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conductivity is almost independent of frequency corresponding to the dc conductivity. The ac conductivity shows dispersive behavior with the increase of frequency. The frequency dependence of the ac conductivity spectra can be well-described by a power law model given by11 σðωÞ ¼ σ dc ½1 þ ðω=ωc Þn
ð2Þ
where n is power law exponent having a value of 0 < n e 1 and ωc is the crossover frequency from the dc to dispersive region. The temperature dependence of ωc is shown in Figure 3b, which indicates that the crossover frequency follows the Arrhenius relation with activation energy Ec (Table 1). The values of ωc for different compositions shown in Table 1 indicate the composition dependence of ωc similar to that of the dc conductivity. It may be noted that the values of Eσ and Ec (Table 1) are very close, indicating a common conduction and relaxation mechanism.7 The variation of the power law exponent n (Table 1) with Ag2S content is shown in the inset of Figure 2b. It is clearly observed that the value of n is almost constant (∼0.65) for x e 0.15 for both series y = 0.20 and y = 0.30, whereas for x = 0.20 the power law exponent decreases to ∼0.50. The power law exponent n is related to the dimensionality of conduction pathway.31 The value of n ∼ 0.65 corresponds to a three-dimensional conduction, whereas n ∼ 0.50 corresponds to a two-dimensional conduction.31 The conduction pathways available for ion transport decrease with the decrease of the dimensionality, which decreases the mobility and the conductivity. Thus, the change in the dimensionality along with change in the crystallinity influences the ion dynamics. This result is consistent with that observed for change in the volume expansion of the glass structure. The study of ion dynamics in the framework of complex modulus formalism has shown growing interest.3234 In this formalism the electric modulus M* is given by the inverse of the complex dielectric permittivity (ε*(ω)) and can be expressed in the form as M ðωÞ ¼ 1=εðωÞ ¼ M 0 ðωÞ þ iM 00 ðωÞ
ð3Þ
where M0 and M00 are, respectively, the real and imaginary parts of the complex modulus. Figure 4 shows the frequency dependence of M0 (ω) and M00 (ω) for a composition at several temperatures. It is observed that M0 (ω) gradually increases with the increase of frequency, and at sufficiently high frequency a plateau-like behavior is observed corresponding to the high-frequency limiting value of M0 (ω) denoted by M∞. In the M00 (ω) spectra, distinct peaks are observed for all temperatures. It is noted that the peaks shift toward higher frequency as the temperature increases indicating the relaxation behavior. The region in the lower frequency side of these peaks determines the long-range motion of mobile ions, whereas the region in the higher frequency side indicates the localized motion of mobile ions. It is also noted that the M00 (ω) plots are asymmetric in nature and are skewed toward the high-frequency side. Also it is observed that the full width at half-maximum (fwhm) of the M00 (ω) curves is considerably broader than that of the ideal Debye case (1.14 decades). These facts clearly indicate the occurrence of non-Debye relaxation in the present samples. The non-Debye relaxation observed in different ion-conducting systems has been described using the empirical Havriliak Negami (HN) function33,35 given by " # 1 M ðωÞ ¼ M ∞ þ ðM s M ∞ Þ ð4Þ ½1 þ ðiωτhn ÞRhn γhn
Figure 4. Frequency dependence of (a) the real part (M0 ) and (b) imaginary part (M00 ) of complex electric modulus M* for the composition 0.05Ag2S0.95(0.20Ag2O0.80MoO3). The solid lines are the best fits to eq 4.
where M∞ and Ms are, respectively, the high- and low-frequency limiting values of the electric modulus, τhn is the relaxation time, and Rhn and γhn are the shape parameters which can assume values such that 0 < Rhn e 1 and 0 < Rhn γhn e 1. The parameters Rhn and γhn characterize the symmetric and asymmetric broadening of the modulus peak. The Debye relaxation is recovered when both Rhn and γhn are unity. We have fitted the present experimental data to the HN relaxation function [eq 4]. The best fits are shown as solid lines in Figure 4. The parameters obtained from fits are shown in Table 2. Figure 5a shows the temperature dependence of relaxation time, τhn, for different compositions. It is clearly observed that the relaxation time follows an Arrhenius relation with activation energy Ehn similar to the dc conductivity and the values of Ehn (Table 2) and Eσ (Table 1) are very close. Figure 5b shows the temperature variation of the parameters Rhn and γhn for a selected composition. It is noted that Rhn is almost independent of temperature varying slightly at lower temperatures. The γhn is observed to increase initially at low temperatures and is almost constant at high temperatures. These results indicate a slight initial increase in asymmetry of the peaks with increase of temperature. Similar types of behavior have been also observed for others compositions. The compositional variation of these two exponents is shown in the inset of Figure 5b. It is observed that the values of Rhn and γhn are almost independent of composition up to x = 0.15 for both the series y = 0.20 and y = 0.30, but for x = 0.20 these two parameters change rapidly and show opposite behavior. This may be related to the microstructure of the glass nanocomposites, where increased crystalline volume fraction and change in the 14144
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Table 2. Relaxation Time τhn, Activation Energy Ehn, HN Parameters Rhn and γhn Obtained Using the HN Equation, τKWW and Stretched Exponent β Obtained from Electric Modulus Data for Different Compositions xAg2S(1 x)(yAg2O(1 y)MoO3) log10[τhn (s)] ((0.05)
Ehn (eV)
Rhn ((0.02)
γhn ((0.02)
log10[τKWW (s)] ((0.05)
composition
(303 K)
((0.01)
(303 K)
(303 K)
(303 K)
β ((0.02)
x = 0.05 x = 0.10
3.67 4.22
0.68 0.64
0.87 0.92
0.37 0.41
4.39 4.69
0.41 0.47
y = 0.20
x = 0.15
3.76
0.65
0.88
0.44
4.30
0.47
x = 0.20
4.04
0.67
0.69
0.66
4.29
0.46
y = 0.30 x = 0.05
4.69
0.59
0.94
0.43
5.24
0.49
x = 0.10
5.19
0.56
0.92
0.44
5.72
0.50
x = 0.15
5.46
0.54
0.93
0.46
5.94
0.51
x = 0.20
5.45
0.60
0.83
0.52
5.80
0.49
where the function j(t) describes the decay of electric field within ionic conductors in the time domain. In the case of non-Debye behavior, j(t) can be well-approximated by the Kohlrausch WilliamsWatts (KWW) decay function given by36,37 " β # t jðtÞ ≈ exp ð6Þ τKWW where τKWW is the characteristic relaxation time and β is the stretched exponent (0 < β e 1), which indicates the deviation from ideal Debye relaxation. A quantitative analysis of the modulus data in the time domain can be obtained by calculating the function j(t) using the inverse transform of eq 5 such as Z 2 ∞ M 00 cosðωtÞ dω ð7Þ jðtÞ ¼ π 0 ωM ∞
Figure 5. (a) Reciprocal temperature dependence of relaxation time, τhn, for different compositions xAg2S(1 x)(yAg2O(1 y)MoO3). (b) Temperature dependence of HN parameters Rhn and γhn for the composition 0.05Ag2S0.95(0.20Ag2O0.80MoO3). The solid lines are the guide to the eye. The inset shows the composition dependence of Rhn and γhn at 303 K for compositions xAg2S(1 x)(yAg2O(1 y)MoO3). The solid lines are the guide to the eye.
dimensionality of conduction pathways for x = 0.20 influence the relaxation dynamics to a large extent. The complex modulus M* can be expressed in terms of the Fourier transform of a decay function j(t) given as34 Z M ¼ M∞ 1 0
∞
dj expð iωtÞ dt
dt
ð5Þ
The experimental decay curves, obtained from such analysis, are plotted in Figure 6a for different compositions. The decay function is found to be asymmetric with respect to time. The experimental j(t) curves resulting from such transformation have been fitted to the KWW function [eq 6], represented by solid lines in Figure 6a. The values of τKWW and stretched exponent β, obtained from such fits, are shown for different compositions in Table 2. It has been observed that the temperature dependence of τKWW follows an Arrhenius relation indicating a thermally activated behavior with activation energy Eτ. Figure 6b shows the composition dependence of β for both the series y = 0.20 and y = 0.30. The values of β obtained for the present samples are much lower than unity signifying that the relaxation is highly nonexponential.38,39 The nonexponential nature of the relaxation may result from the distribution of relaxation processes due to the distribution of energy barriers or sites.40 On the other hand, the correlated motion between successive relaxation events resulting from a cooperative interaction between relaxing ions may also lead to nonexponential relaxation. As the interaction between the mobile ions becomes strong, the relaxation shows a stretched exponential behavior, and consequently, the value of β deviates further from ideal Debye behavior. According to Ngai’s coupling model39 the stretched exponent β is identified as β = (1 n0 ), where n0 (0 < n0 < 1), called a coupling parameter, is a measure of coupling or interaction between the mobile ions during conduction, or in other words, the degree of cooperativity. This exponent n0 is sometimes compared to the 14145
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Figure 6. (a) Temperature and time dependence of the KWW decay function, j(t), for the composition 0.05Ag2S0.95(0.30Ag2O 0.70MoO3). Solid lines are the best fit to eq 6. (b) Composition dependence of β for the compositions xAg2S(1 x)(yAg2O(1 y)MoO3). Solid lines are the guide to the eye.
power law exponent n obtained from the ac conductivity data.41 It is noted in Tables 1 and 2 that the values of n do not exactly match with those of n0 (= 1 β) for the present samples. The difference might arise from different frequency windows employed for determining the stretched exponent β and power law exponent n. The other reason might be the fact that the shape of the modulus spectra is influenced by the values of the high-frequency dielectric constant.42 Nevertheless, these values for x e 0.15 for both the series y = 0.20 and y = 0.30 are close and show similar compositional dependence, but a difference is indeed observed for x = 0.20. Similar behavior is also observed for the composition dependence of HN parameters Rhn and γhn as discussed in the preceding paragraph. The formation and growth of different nanocrystallites affect the ionion interaction and hence the coupling parameter when the crystalline volume fraction increases by a significant amount. Using the approximate correspondence of Fourier transformation of dj/dt and the HN function, it has been shown43 that the relation between KWW exponent β and HN parameters Rhn and γhn can be written as β ¼ ðRhn γhn Þ1=1:23
ð8Þ
It is noted in Table 2 that this relation holds good for the present samples, further suggesting that either of the representation of the HN function or KWW function is valid and has the same physical significance. To understand the relaxation dynamics in these glass nanocomposites, we have tried to scale the ac conductivity spectra following the procedure described in ref 44. Here the conductivity axis is scaled by the dc conductivity for a particular temperature, whereas the frequency axis is scaled by the corresponding crossover frequency. Figure 7a shows the scaled master curve for a particular composition at several temperatures, and Figure 7b
Figure 7. (a) Scaling of the conductivity spectra shown at several temperatures for the composition 0.05Ag2S0.95(0.20Ag2O0.80MoO3). (b) Scaling of the conductivity spectra at 293 K for several compositions of xAg2S(1 x)(0.20Ag2O0.80MoO3). The inset shows the scaled spectra at 293 K for xAg2S(1 x)(0.30Ag2O0.70MoO3).
shows the same for different compositions at a fixed temperature. It is observed that a master curve is indeed obtained for different temperatures but not for different compositions implying that the relaxation dynamics is independent of temperature but depends strongly on compositions. It is already pointed out in Figure 3b that the value of the power law exponent n changes considerably for x = 0.20 for both the series y = 0.20 and y = 0.30. Thus, the nonsuperposition of different spectra to a master curve in Figure 7b can intimately be related to the change in dimensionality of conduction pathways making the macroscopic ion transport property to depend on the microstructure of the glasses.
4. CONCLUSIONS The Ag+ ion dynamics in Ag2SAg2OMoO3 glass nanocomposites have been studied in a wide frequency and temperature range. The dc conductivity initially increases with the increase of Ag2S content, which is attributed to the increase in volume expansion of the glass network structure. However, a decrease of the conductivity at higher Ag2S content is observed due to significant increase in the crystalline volume fraction. The power law exponent obtained from ac conductivity spectra is almost independent of composition with a value ∼0.65 for samples with lower crystalline volume fraction (
