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DIVISION S-2—SOIL CHEMISTRY

Low Frequency Impedance Behavior of Montmorillonite Suspensions

Polarization Mechanisms in the Low Frequency Domain

^a Dep. of Plants, Soils and Biometeorology, Utah State University, Logan, UT 84322
^b Northeast Foundation Utilities Professor, Dep. Civil and Environmental Engineering, Univ. of Connecticut, 261 Glenbrook Road, Unit 2037, Storrs, CT 06269-2037
^c Chemistry and Biochemistry Dep., Utah State University, Logan, UT 84322
^d Physics Dep., Utah State University, Logan, UT 84322

^* Corresponding author (LDUD{at}MENDEL.USU.EDU)

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Large changes in permittivity have been observed as the frequency of an electromagnetic (EM) field applied to systems containing phases of contrasting permittivity is changed. Two mechanisms, polarization of a diffuse double layer (DDL) and polarization of the charge imbalance created by contact of two phases of different permittivity (the Maxwell-Wagner [MW] effect), are responsible for the frequency dependence of dielectric properties. To use the frequency dependence of dielectric properties to determine soil geometrical and electrochemical properties, the two mechanisms must be quantified. Three models of the frequency dependent dielectric properties, based on terms representing polarization of the electrical double layer that develops at the electrode surface, polarization of the DDL and the MW effect, were used to investigate the dielectric spectrum of montmorillonite suspensions. Dielectric spectra of suspensions of three particle-size separates (r > 1.0 µm, 1.0 µm > r > 0.2 µm, 0.2 µm > r) of homoionic (Na⁺ or Ca²⁺) were measured at a suspension density of 5.0 g of clay in 50 mL of water. Impedance plane plots suggested the contribution of three relaxation processes to the spectra. While all three models reproduced the data, they gave different interpretations of the data. Two models attributed relaxation in the kHz range to electrode polarization, relaxation at approximately 10 kHz to DDL polarization and relaxation at 1 MHz to MW polarization. The third model assigned MW polarization to the relaxation at 10 kHz and DDL polarization to the relaxation at 1 MHz.

Abbreviations: dc, direct current • DDL, diffuse double layer • DM, Debye's relaxation model • ECM, equivalent-circuit model • EM, electromagnetic • MM, mixing model • MW, Maxwell-Wagner • TDR, time domain reflectometry

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
THE SUCCESS OF time domain reflectometry (TDR) as a valuable technique for soil water content and bulk electrical conductivity measurements (Topp et al., 1980; Dalton et al., 1984) prompted considerable interest in the usefulness of other methods based on interactions between EM fields and soil constituents. In the past few years numerous new EM-based sensors have been introduced primarily for measurement of soil water content. A common feature for many EM methods is the induction of a perturbation in an EM field affecting a soil sample and inference of its electrical properties from the measured response. Electromagnetic interactions of induced electric fields with matter are not simply a function of the bulk electric properties, but with proper interpretation, may reveal spatial arrangement and interactions among various components in heterogeneous systems such as soils.

Systems containing material of contrasting permittivity (dielectric constant) such as soils, rocks, and solutions containing macromolecules or alumino-silicate colloids exhibit extremely large changes in electrical properties such as permittivity and impedance as the frequency of the applied EM field changes (Mudgett et al., 1977; Hasted, 1973). These large changes in impedance or permittivity have been used in geophysics to study electrochemical properties of porous media (Olhoeft, 1985; Knight and Nur, 1987; Garrouch and Sharma, 1994), and could be used for determination of electrochemical and geometric properties of soils. Noninvasive EM techniques have the potential to provide powerful tools for identification and monitoring of soil geometrical and electrochemical properties with minimal disturbance and possibly in situ.

The permittivity of a material is a measure of the extent of distortion of its electric charge distribution in response to an electric field. The main mechanism for dielectric behavior at microwave frequencies (MHz–GHz) is orientation of polar molecules in the presence of an applied alternating electric field (von Hippel, 1993). At low frequencies (Hz–MHz), the primary mechanism for a permittivity or an impedance response is the perturbation of charges at the solution-solid interface. Very high permittivity values (Hasted, 1973) result from two mechanisms: (i) polarization of the counter ions in the DDL, and (ii) the MW that arises from polarization of the charge created by contact of two phases with different permittivity.

The application of an electric field causes counter ions in the DDL of charged colloidal particles (clay or organic matter [OM]) to move along the surface in response to the electric gradient. The motion is mostly tangential along the surface because of the existence of a much larger energy barrier normal to the surface (de Lima and Sharma, 1992). For impedance spectroscopy, the applied field (50 mV) is about an order of magnitude less than the zeta potential (Sposito, 1981). Consequently, electric charges accumulate at particle interfaces such as particle edges and water-air interfaces. When the field is removed, the charges relax back to their original distribution by diffusion. A second mechanism operates similarly to polarization of the DDL, but results from polarization of the charge created by contact of two phases with different permittivity. Herein, we refer the two mechanisms as DDL polarization and the MW, respectively.

Often these two mechanisms are not differentiated (see e.g., Knight and Nur, 1987) leading to ambiguities in the literature, however for particles >1µm, two relaxations in the low frequency spectrum were observed by Blum et al. (1995). Differentiating the two mechanisms may be important in interpreting impedance spectra because relaxation of the DDL is a function of geometry and surface charge density and the MW relaxation is only a function of geometry. Our objective in this study was to identify the mechanism associated with the relaxations observed in the low frequency (Hz–MHz) impedance spectra of Na and Ca saturated montmorillonite as a first step in formulating a conceptual model of the impedance behavior of soils.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
A review of the theory of impedance and dielectric properties is found in many text books, for example, Hasted (1973); Macdonald and Johnson (1987); von Hippel (1993). Herein, we use impedance, permittivity, and conductivity to describe the frequency dependant electrical properties of clay suspensions. The three quantities are interrelated as follows. The complex impedance Z* (Z* = Z' - iZ," where Z' is the real resistance R, Z" is the imaginary reactance, and i = [-1]^1/2) is related through the complex resistivity * (impedance times sample area divided by sample thickness) to the complex conductivity * ([*]^-1 = * = ' + i"). Von Hippel (1993) discusses the relationships between the complex conductivity and the complex dielectric permittivity (* = ' - i"). Both quantities (* and *) define the total current density in a sample subjected to an alternating electric field. These two complex scalars describe the mode of current flow (* = conduction through charge carriers, or * = polarization currents). They are interrelated by:

[1]

where = 2f is the angular frequency of the applied field. Complex relative permittivity, E*, is obtained by scaling the permittivity, *, by ₀, the permittivity of free space (8.854 x 10^-12 F m^-1), E* = */₀. The dissipation factor , or the loss tangent are often used to describe energy dissipation in the sample, and to relate the real and imaginary parts of these two quantities as:

[2]

and relative permittivity that is related to the complex impedance, Z*, by:

[3]

where C₀ is the capacitance of empty measurement cell.

Models are often used to interpret impedance, permittivity, or conductivity data, and a number of models have been proposed for colloidal suspensions, water-saturated rocks, and biological membranes. Models applicable to a suspension of charged clay particles are based on effective medium theory (see e.g., Bonanos et al., 1987) in which the suspension is treated as two equivalent volume elements: bulk solution and a wet, charged sphere aggregate of the clay particles. Equivalent volume models may be derived from empirical equivalent-circuit models (ECMs), empirical Debye-relaxation models (DMs) or more physically based mixing models (MMs). The ECM (Bonanos et al., 1987) shown in Fig. 1 has been used for analyzing the impedance behavior of suspensions of charged latex spheres and thus, was used herein to model clay suspensions. A second model (Kuang and Nelson, 1997) used to analyze the data was based on a Debye-type (Debye, 1929) representation of the dielectric dispersion. Nettelblad and Niklasson (1996) compared MMs and found that the models provided comparable predictions the total conductance of colloidal suspensions, as long as fundamental assumptions of the models were not violated by the test case. A variation of the model developed by de Lima and Sharma (1992), appropriate for computing the complex conductivity and permittivity of clay suspensions, was selected for use in this study.

View larger version (17K): [in this window] [in a new window]   Fig. 1. (a) A schematic representation (brick model) of the assembly of the clay and solution effective media and (b) the corresponding equivalent circuit (Bonanos et al., 1981). Re is the electrode impedance, gs is the conductivity of the solution element, cs is the capacitance of the solution element, gc is the conductivity of the moist clay element, and cc is the capacitance of the moist clay element.

[4]

where

[5]

and where the subscript s denotes the electrolyte solution and c the clay, 0 denotes the low frequency limit and denotes the high frequency limit, _s = 6.98 x 10^-10 F m^-1 at 25°C and _c 6.73 x 10^-11 F m^-1 (Shugg, 1995), _s0 was the direct current (dc) conductivity of the bulk and the estimation of _c0 is described in the following section. The impedance of the circuit is given by

[6]

where k is a cell constant (=length/area), R_e0 is the low-frequency resistance ( 0) of the electrode impedance element, represented as a Warburg impedance (MacDonald and Johnson, 1987), and t_c = c_c/g_c, t_s = c_s/g_s.

Estimation of _c0
The clay particles were assumed to have lengths and widths ten times greater than their thickness. When Na⁺ was the counter ion, expansion of the DDL might permit both internal and external charge to participate in propagating the applied field and thus, the total particle charge was used to compute the effective clay conductivity. From the measured radius (described in the Materials and Methods section below), a particle volume was computed and that value multiplied by a particle density of 2.63 kg m^-3 gave a particle mass of 8.71 x 10^-20 kg. Dividing by the molecular weight of SWy-1 montmorillonite and multiplying by the charge deficit of 0.67 mol_c mol^-1 (Sposito, 1981) produced a total particle charge of 7.85 x 10^-20 mol_c. The equivalent conductivity for Na in the DDL (⁰_Na) was computed from the surface diffusion coefficient for Na (D_Na = 3.0 x 10^-11 m² s^-1) (Jurinak et al., 1987) using the Nernst-Einstein relationship (Bockris and Reddy, 1973)

[7]

where D_i is the surface diffusion coefficient for the ith cation, F is the Faraday constant (9.6487 x 10⁴ C mol^-1), R is the universal gas constant (8.3143 J K^-1 mol^-1), and T is the temperature (K). Given the uncertainty in computing the conductivity of the clay, the small correction to the equivalent conductivity for background electrolyte concentration was neglected. A value of ^Na_c0 = 2.65 S m^-1 was computed from the equivalent conductivity and the total surface charge.

When Ca²⁺ was the counter ion, the DDL should be collapsed (Bolt, 1982) and only cations associated with external exchange sites of the tactoid might participate in field propagation. The total area of two faces of the tactoid was computed from the measured particle radius and the geometry described above. The surface charge per particle was obtained using a specific surface charge of 1.0 x 10⁶ mol_c m^-2 (Sposito, 1981). The equivalent conductivity was computed from Eq. [7] with a value for D_Ca of 1.3 x 10^-11 m² s^-1 (Jurinak et al., 1987) and the values of the particle surface charge and equivalent conductivity gave a value for ^Ca_c0 of 0.162 S m^-1.

Debye-Type Relaxation Model
Kuang and Nelson (1997)(1998) expressed the real and imaginary components of the dielectric dispersion of a biological membrane by the equations:

[8]

where _i = _is - _i, _i is the time constant for the i^th dielectric relaxation, _s0 is the dc conductivity of the solution and _s is the permittivity of water. Over the frequency range of hertz to a few megahertz, three possible dielectric relaxation mechanisms are: (i) a relaxation arising from polarization at the electrode-electrolyte interface, denoted e, (ii) polarization of the DDL (often referred to as dispersion), and (iii) MW polarization (referred to as ß dispersion). Thus,

[9]

[10]

Multiple relaxation constants in the dispersion may result, at least in part, from variation in the path length for movement of the ion swarm, as shown by the equation (Schwartz, 1962)

[11]

where r is the particle radius, ^-1 is the Debye screening length, and D is the surface diffusion coefficient. The inverse Debye length is given by (Sposito, 1981)

[12]

where m_c is the background electrolyte concentration (mol_c L^-1) and E_s is the static relative permittivity of water. While _a was computed from the mean particle radius, Eq. [11] suggests that variation in particle size produces a corresponding variation in relaxation time constants. Thus, a distributed relaxation model was selected to represent dispersion.

The Cole-Cole model (Cole and Cole, 1941) has been used to describe dielectric relaxation in a system where the relaxation process is distributed and thus is more appropriate for the term in Eq. [12] and is given by

[13]

where ( = 0.4 in this study) is a distribution parameter. Substituting the rational forms of Eq. [13] for the terms in Eq. [9] and [10] yields

[14]

[15]

Grosse and Foster (1987) derived the equation for the permittivity increment for dispersion of a charged colloidal particle of radius r

[16]

The permittivity increment for ß dispersion was obtained from the increase in conductivity, _ß, observed at frequencies near 1 MHz as follows

[17]

Mixing Model
A third approach to modeling the EM field interactions with the clay suspensions was derived by adding terms accounting for electrode polarization and the MW effect to the de Lima and Sharma model (de Lima and Sharma, 1992) of the complex conductivity of the suspension giving

[18]

and

[19]

where the static conductivity, ₀, high frequency conductivity, , high frequency permittivity, , and mean relaxation time, _m, of the suspension are given by

[20]

and where

[21]

and where all values are suspension averages except those denoted ef (effective), 0 is the static value and the high frequency value, G is the surface charge density (1.0 mmol_c m^-2 Sposito, 1981), r is the particle radius and m_c (m_c = 8.433₀^1.037, Marion and Babcock, 1976) is the background electrolyte concentration. Equations [18] through [21] were solved in a Mathcad 8 worksheet (Mathsoft, 1998).

	MATERIALS AND METHODS

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Homoionic SWy-1 montmorillonite (obtained from Ward's Natural Science Establishment, Inc., Rochester, NY) was prepared by washing 5 g of clay with 30 to 40 mL of a 1.0 M solution of either CaCl₂ or NaCl three times. Clay and electrolyte were placed on a reciprocal shaker for 15 min and the clay and electrolyte solution was subsequently separated by centrifugation. After the third washing, the clay was placed into diffusion tubing for removal of excess salt. The diffusion tubing was placed in a container filled with distilled-deionized water and the water was replaced daily until its conductivity was <0.01 S m^-1 at the end of a 240-h period. The clay was then suspended in double-deionized water and separated into three particle size ranges by centrifugation. Centrifugation times for target particle size ranges of r > 1.0 µm, 1.0 µm > r > 0.2 µm, and 0.2 µm > r were computed from Stoke's law assuming a flat-disk particle geometry. The resulting size fractions were each then suspended in double-deionized water at a ratio of 5.0 g of clay in 0.050 L of water. The particle-size distribution was measured by laser diffraction using Coulter LS Particle Size Analyzer (Beckman and Coulter, Fullerton, CA). The means and standard deviations of the particle radii for the sizes separates were: large—Na saturated 11.12 ± 13.24 µm, medium—Na saturated 1.99 ± 1.25 µm, small—Na saturated 0.525 ± 0.474 µm, and small—Ca saturated 0.225 ± 0.139 µm.

The various models used in this study required estimates of the electrochemical properties of the clay and suspension such as surface charge density, electrolyte and clay conductivity, and the Debye screening length. The conductivity of the suspension increased during storage and clay and electrolyte suspensions were separated by centrifugation for measurement of the _s0 at the time of impedance measurements. These dc conductivity measurements were corrected to the appropriate temperature when used to model the impedance measurements.

Impedance measurements were taken with a Hewlett-Packard 4194A Impedance/Gain-Phase Analyzer, with a Hewlett-Packard 16452A Liquid Test Cell (Agilent, Tempe, AZ). The 4194A with the liquid test cell has a measurement range of 100 Hz to about 14 MHz. The cell has a two-electrode configuration with stainless steel electrodes that were cleaned and polished as needed. The electrode spacing was 0.3 mm and an electrical field of 160 V m^-1 was applied across the sample. The cell and samples were placed in a temperature bath and impedance spectra of the suspensions were measured at 5, 25, and 40°C. The number of data points that could be collected was limited. Thus, to have sufficient data density the low (0.1–300 kHz) and high frequency (1.0 kHz to 10 MHz) portions of the spectra were collected in separate measurements. Overlap of the spectra created a check on instrument calibration and function.

The distribution of relaxation times was determined directly from the data using and expectation-maximization algorithm. The expectation-maximization algorithm has been shown to be a stable method for solving inverse problems, producing only physically realistic distributions (Bialkowski, 1991; 1998). Relaxation time distributions were obtained by fitting the imaginary component of the immittance to the Lorentzian objective function (Macdonald and Johnson, 1987)

[22]

where () is the fraction of relaxations times occurring at .

	RESULTS AND DISCUSSION

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Data Analysis
Similar trends were observed in the impedance plane plots spectra for all of the particle sizes and temperatures and the impedance plane plot for small particle size, Ca-saturated clay is shown in Fig. 2 as an example of the features. Impedance plane plots reverse the order of the data with respect to frequency so that the low frequency data are at the right of the figure and frequency increases to the left. To the right of the figure at about 45 ohms, a slight offset in the two sets of data is apparent. The offset was attributed to slightly different instrument conditions in the collection of the data. The figure shows a worst case for offset and generally agreement between data collected over the two frequency ranges was better than shown. Two features were observed in the impedance plane plots: a linear element occurring at about 0.1 to 2.0 kHz with slope approximately equal to ^-1/2 and a suppressed arc centered at 200 kHz. The linear section results from diffusion-controlled impedance, referred to as Warburg impedance (Macdonald and Johnson, 1987) and the suppressed arc is characteristic of some dielectric mechanism with distributed relaxation times (Cole and Cole, 1941). A DDL not only exists at the clay-water interface, but also at the electrode interface and motion of ions associated with either DDL could result in Warburg impedance. Distributed relaxation times result in a suppressed arc in an impedance plane plot and such a shape is often attributed to physical cause such as variation in particle size (Garrouch and Sharma, 1994) or edge-to-edge and edge-to-face particle arrangement of clay particles (Lockhart, 1980; Ishida et al., 2000). An impedance plane plot of the modulus (Fig. 3) suggested a third relaxation near the high frequency limit of the measurements evidenced by the beginning of an upturned curve at 7.0 MHz in Fig. 3. Figure 3 also shows a suppressed arc centered at about 0.4 MHz, which is approximately the same frequency (within the resolution of the data) as the center of the suppressed arc in Fig. 2. Impedance plane plots of the modulus emphasize high frequency relaxations in contrast to the impedance, which emphasizes lower frequency processes (Macdonald and Johnson, 1987).

View larger version (21K): [in this window] [in a new window]   Fig. 2. An impedance plane plot of the real (') and imaginary (") components of the impedance, Z, and modulus, M (-1), for the small particle-size separate of Ca-saturated clay (data collected at 55°C).

View larger version (18K): [in this window] [in a new window]   Fig. 3. Real and imaginary components of the immittance (Z/R0) spectrum for the small particle size separate of Ca-saturated clay (at 55°C). The fit curve is reproduced from the relaxation time distributions estimated from an expectation maximization algorithm.

[23]

The set of relaxation times estimated from I" reproduced the I' data (an example is shown in Fig. 4) validating the expectation-maximization approach to data analysis. The advantage to this approach is that no a priori assumption of the distribution of relaxation times is required. This is in contrast to other approaches that require some assumption such as fitting to Cole-Cole or other distributed relaxation-time models.

View larger version (14K): [in this window] [in a new window]   Fig. 4. Relaxation time distribution (a) extracted from the imaginary component of the immittance (Z"/R0) spectrum for the small particle size separate, Ca-saturated clay suspension. The temperature of the suspension was 55°C.

View larger version (21K): [in this window] [in a new window]   Fig. 5. Measured and equivalent-circuit model predicted impedance spectra for the small particle-size separate of the Ca-saturated (a) and Na-saturated (b) clay at 25°C.

View larger version (17K): [in this window] [in a new window]   Fig. 6. Measured and equivalent circuit model predicted impedance spectra for the Ca-saturated clay using the clay element conductivity computed for the Na-saturated clay.

The real component of the permittivity computed by DM closely approximated the Ca data and gave a reasonable representation of the Na-clay suspensions (Fig. 7 and 8) . Order of magnitude agreement was obtained for the conductivity (Fig. 7 and 8). For Fig. 7 and 8, _e was assigned a value of 5 ms and thus, the model predicted that DDL polarization occurred in the same frequency range as electrode polarization. The computed value of ^Ca_a was 4.0 ms and ^Na_a was 0.75 ms. The values for ^Ca_b = 137 ns and ^Na_b = 28.2 ns used in the computations were taken from the position of the well-defined peak in the expectation-maximization estimated relaxation time distributions. The relaxation time distribution predicted by the DM was consistent with the observed distribution. Electrode and DDL polarization occurred in the same region of the spectrum of the broad band of relaxation times and MW polarization occurred at a much higher frequency.

View larger version (19K): [in this window] [in a new window]   Fig. 7. The real components of the permittivity (a) and conductivity (b) computed by the Debye-relaxation model and measured for the Ca-clay suspension.

View larger version (20K): [in this window] [in a new window]   Fig. 8. The real components of the permittivity (a) and conductivity (b) computed by the Debye-relaxation model and measured for the Na-clay suspension.

View larger version (18K): [in this window] [in a new window]   Fig. 9. The real components of the permittivity (a) and conductivity (b) computed by the mixing model and measured for the Ca-clay suspension.

View larger version (18K): [in this window] [in a new window]   Fig. 10. The real components of the permittivity (a) and conductivity (b) computed by the mixing model and measured for the Na-clay suspension.

	CONCLUSIONS

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Three different models were used in an attempt to identify the physical mechanism for the three relaxations. The ECM gave a different interpretation of the data than the other two models. The ECM predicted that DDL polarization occurred at higher frequency than MW and that both relaxations occurred in the megahertz range (time constants in the nanosecond range). The other two models predicted that DDL polarization occurred at much lower frequency (1 to 10 kHz = millisecond time constants) and that MW occurred at frequencies >1 MHz. The ECM and the MM were in good agreement with the data and so interpretation of the data remains ambiguous.

The agreement between computed and measured real permittivity suggests that the MM provides the correct interpretation of the relaxation mechanisms, at least in a general sense. According to the MM, the relaxations observed for the clay suspensions resulted from three mechanisms: electrode impedance with a relaxation time in the ms range, DDL polarization with a distributed relaxation time that had a mean value in the millisecond range, and a MW relaxation with a relaxation time in nanoseconds range. However, a degree of uncertainty remains for this interpretation because the relaxation time distribution predicted by the model did not correspond particularly well to the observed distribution. The relaxation frequency distribution for the MM computed permittivity spectrum (not shown) would have narrow peaks at 5 ms and 173 ns superimposed on a broad Gaussian peak (Hasted, 1973) centered at 0.4 ms, features not evident in Fig. 5.

While existing models may be adequate tools for providing the general information on a system necessary to detect oil deposits and material defects, or monitoring the progress of oxidation-reduction reactions (Olhoeft, 1985), they may not fully reflect the underlying physicochemical processes. Nettelblad and Niklasson, (1996) compared a number of dielectric relaxation models and found that, even though physical mechanisms on which the models were based differed, the models adequately fit the data. Our results are similar in that the three models reproduced the data, but were inconsistent in assigning relaxation mechanisms in the spectra. The ability of all three models to fit the data may be attributed, at least in part, to the need to fit parameters for one relaxation process per model. For impedance spectroscopy to provide a means of determining soil geometrical and electrochemical properties significant advances in theory will be necessary.

	ACKNOWLEDGMENTS

 
Funding for this research was provided by a grant from the NRI Competitive Grants Program/USDA (award number 99-35107-7829) and the Utah Agricultural Experiment Station.

Received for publication January 25, 2002.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

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