Electroacoustic and Dielectric Dispersion of Concentrated Colloidal Suspensions

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1 IEEE Transactions on Dielectrics an Electrical Insulation Vol. 13, No. 3; June Electroacoustic an Dielectric Dispersion of Concentrate Colloial Suspensions S. hualli, M. L. Jiménez,. V. Delgao Department of pplie Physics University of Granaa, Faculty of Science 1871 Granaa, Spain F. J. rroyo Department of Physics University of Jaén, Faculty of Experimental Sciences 2371 Jaén, Spain an F. Carrique Department of pplie Physics II University of Málaga, Faculty of Science 2971 Málaga, Spain STRCT The etermination of the permittivity of colloial ispersions has been emonstrate to be a very useful technique to characterize the electrical state of the soli/liqui interface. This is so because of its sensitivity to such features as particle size an shape, state of aggregation, surface charge, an so on. The ielectric ispersion ata are particularly suite to the evaluation of the surface characteristics when the suspension contains a high solis concentration, as in such conitions the usual electrokinetic techniques linke to optical methos are not applicable. The same avantage is share by electroacoustic techniques, where it is a collective response of the system that matters. In this case, the experimental quantity is the ynamic (or ac) electrophoretic mobility. In this work, both methos will be use for the investigation of the electrical surface characteristics of concentrate alumina suspensions with volume fractions of solis ranging between 1 an 2%. The ata will be analyze in the frame of the so-calle cell moel, often use to analyze the hyroynamic an electrical interactions between particles in concentrate suspensions. We will show that the moel is capable of properly escribing both their ielectric ispersion an ynamic mobility, an that the surface (zeta) electric potential calculate from electroacoustic ata can be use as an input parameter to reprouce the permittivity. This will emonstrate the internal coherence an accuracy of the cell moel an give clues to improve our unerstaning of the electrokinetic behavior of concentrate slurries. Inex Terms coustoelectric effects, concentrate colloial ispersions, electrophoresis, permittivity measurement. 1 INTRODUCTION THE ielectric spectroscopy of colloial ispersions is a powerful tool for the precise characterization of the electrical properties of interfaces. Its etermination can be carrie out by performing measurements of the impeance of a conuctivity cell immerse in the suspension. One ifficulty involve in these experiments is that the frequency interval where ielectric relaxation takes place (khz) can inclue the region where electroe polarization interferes with the results. Manuscript receive on 27 June 25, in final form 15 March 26. Since impeance measurements inform about a collective response of the system, they woul be an excellent tool for the analysis of electrokinetics of concentrate suspensions. oth the frequency epenence of the ielectric constant an the amplitue of its relaxations are very sensitive to such properties of particles as the surface charge, size an shape. On the other han, electroacoustic techniques are recently gaining interest from the colloi science community as powerful tools to obtain information on the electrical state of many interfaces, particularly in a number of situations where traitional electrokinetic methos are of limite use. One of such situations is that of concentrate suspensions an slurries, /6/$2. 26 IEEE

2 658 in which we can get no information on the behavior of iniviual particles by performing, for instance, microelectrophoresis experiments. There are two such techniques. One involves the generation of a pressure wave when an ac electric fiel is applie to the suspension: we speak of the ES effect, use in this work. The secon metho, reciprocal of ES, is base on the etermination of the electric potential inuce by the passage of a soun wave through the system. It is calle colloi vibration potential (CVP) or colloi vibration current (CVI), epening on the quantity measure. Current electroacoustic technology allows to obtain the ynamic mobility, u, a complex quantity that is in fact proportional to the ES signal, an relate it to the zeta potential, the particle shape an imensions, the ionic composition of the meium, an, very important, the volume fraction of solis, φ. It can be sai that ynamic mobility is the ac (alternating current) counterpart of classical c (irect current) electrophoresis. Two inepenent approaches have been followe to solve specifically the problem of the volume fraction epenence of u in the case of concentrate ispersions. O rien et al. use a first-principles approach, consiering a particle interacting with its nearest neighbors [1]. nother evaluation, first carrie out by Dukhin et al. [2,3], is base on a so-calle cell moel, that has also been successfully applie to the analysis of a variety of electrokinetic phenomena in concentrate suspensions, incluing electrophoresis, seimentation potential or ielectric ispersion [4-8]. In this sort of calculation, the hyroynamic an electrical interactions between particles are accounte for in an average form by solving the electrokinetic equations for a single spherical particle of raius a, enclose by a spherical shell of solution of raius b, concentric with the particle. The raius of the sphere is chosen in such a way that the volume fraction of solis is the same in the cell an in the suspension: 3 a φ= (1) b In this paper, our aim is to report experimental ynamic mobility an electric permittivity ata on suspensions of alumina of ifferent volume fractions, checking these results against the preictions of cell moels. In a previous work [9], we have carrie out a similar stuy on the ynamic mobility as a function of electrolyte concentration. 2 EXPERIMENTL S. Hawalli et al.: Electroacoustic an Dielectric Dispersion of Concentrate Colloial Suspensions polyisperse. The average hyroynamic raius of lumina was 165 ± 3 nm an for lumina it was 288 ± 5 nm, as measure by photon correlation spectroscopy in a Malvern 47c PCS evice (Malvern Instruments, U. K.). The original suspensions were repeately washe with mol/l KCl solution. The process was carrie out by centrifugation at 85 g in a Kontron T-124 centrifuge, an reispersion in the KCl solution. The volume fraction of solis, φ, in the suspensions prepare was etermine from the ry weight of known volumes using p = 2.4 g/cm 3 as the ensity of alumina. Mobility moulus (1-8 m 2 V -1 s -1 ) Mobility moulus (1-8 m 2 V -1 s -1 ) 5, 4,5 4, 3, 5, 4,5 4, 3, 2, 1,5 ζ=9 mv ζ=9 mv u e ζ=95 mv ζ=1 mv u e ζ=9 mv ζ=91 mv ζ=83 mv ζ=85 mv ζ=87 mv 2.5% 5% 1% 2% 1 Freq (MHz) Freq (MHz) From 3% to 29% Figure 1. Comparison between the moulus of the ynamic mobility of alumina suspensions in.2 mm KCl (symbols) an the corresponing theoretical preictions obtaine from cell moel (soli lines), for the zeta potentials inicate. The c mobility measure in ilute suspensions is shown as a short otte line labelle u e. an correspon to alumina samples an, respectively. 2.1 MTERILS Two ifferent samples of hyrate alumina were use in this investigation, both kinly provie by Dispersion Technology Inc. (US). Sample was given to us in the form of a concentrate suspension in an unknown ispersion meium, while sample was a ry power. SEM micrographs showe that in both cases the particles were crystalline an moerately 2.2 METHODS The classical or c electrophoretic mobility of the particles, u e, was measure in ilute suspensions (φ 1-4 ), using a Malvern Zetasizer 2 (Malvern Instruments, U.K.). These measurements provie an alternative or complementary way to evaluate the electrokinetic or zeta potential of the particles.

3 IEEE Transactions on Dielectrics an Electrical Insulation Vol. 13, No. 3; June The ynamic or ac mobility of the particles, u, was measure for a frequency range between 1 an 18 MHz in a coustosizer II (Colloial Dynamics, US). For the etermination of the ielectic properties of the samples we use a HP4284 LCZ meter operating between 1 Hz an 1 MHz, connecte to a conuctivity cell with variable electroe istance. ecause of the high solis concentrations typically use, we performe the measurements keeping the systems flowing through the conuctivity (an, eventually, also through the electroacoustic) cell, by pumping them from a stirre an thermostatte beaker. The temperature was kept constant at 25.±.1 C using a Haake F3K water circulator. For the frequencies of interest in the eterminations of permittivity in colloial suspensions, an important source of perturbation is the polarization of the electroe/solution interface. We use some proceures to minimize this unwante effect: i) measure at ifferent separations between the electroes; ii) use calibration solutions an perform relative measurements (quarupole metho, see Ref. [1]); iii) work with the logarithmic erivative of the real permittivity, so as to separate the frequency epenences of the electroe an sample contributions to the measure permittivity [11]. Specifically, the results presente in this work were obtaine using methos (ii) for getting a general picture of the permittivity-frequency trens, an (iii) to obtain the relaxation frequency an the low-frequency permittivity. 3 RESULTS 3.1 DYNMIC MOILITY The moulus an phase of the complex ynamic mobility of an alumina particles with varying volume fractions (φ %) as obtaine from the ES instrument are shown in Figures 1 an 2. Note that, as Figure 1 shows, increasing the volume fraction provokes a reuction in the mobility. This was to be expecte, as it is preicte by all theoretical moels [1-8, 12] an justifie because of the hyroynamic an electrostatic interactions of a given particle with its neighbors. In aition, increasing the frequency f can bring about two istinct behaviors in the mobility: for frequencies on the orer of 2-3 MHz, u goes through a maximum, that occurs at higher frequencies the more concentrate the suspension. If we increase the frequency even further, the moulus of the mobility monotonously ecreases. On the other han, phase angle ata in Figure 2 show consistently negative values of that angle, i.e., the velocity lags behin the applie ac fiel. This must be ue to two phenomena that, for certain frequency ranges, affect the electrophoretic motion. One is the particle an flui inertia, that become increasingly important for frequencies above 2 f η/ ρ a ( 5 MHz for our particles), where is the I p Moblity phase (eg) Moblity phase (eg) Freq (MHz) 1 Freq (MHz) 1 2% 1% 2.5% 24 % 16 % Figure 2. Comparison between the phase angle of the ynamic mobility (symbols) an the corresponing theoretical preictions obtaine from cell moel (soli lines) for the same suspensions as in Figure 1. an correspon to alumina samples an, respectively. viscosity of the liqui meium. ecause of this inertia, the phase lag of the mobility will become increasingly negative with frequency for f > f I. This inertial effect also explains the ecrease observe in u beyon the maxima in Figure 1. The other phenomenon that contributes to the value of the mobility is the inuction of an electrical ipole at the soli/liqui interface, that epens on the frequency an the zeta potential. Roughly speaking [12-14], the ynamic mobility can be expresse as: εm u = ζ(1 C ) Ginertia( ω) (2) η where C is the inuce ipole coefficient, an G ( ) is the inertia complex function that accounts for the particle an flui inertia above mentione, for an applie fiel of frequency. The inuce ipole moment has a somewhat complex frequency epenence, that can be summarize as shown in Figure 3, where we have plotte 1 C : at relatively low frequencies (in the khz region), a ecrease is observe, associate to the 7 %

4 66 so-calle -relaxation. This curve has been obtaine for a single spherical particle, using the DeLacey an White s theory [15]. S. Hawalli et al.: Electroacoustic an Dielectric Dispersion of Concentrate Colloial Suspensions a lower curvature inicates that the electrocaoustic measurements are very sensitive to the larger polyispersity of sample. The cell moel, elaborate for monoisperse particles, can harly reprouce such flat epenences. Mo (1-C ) 1,3 1,2 1,1 1,,9,8,7 α-relaxation MW-relaxation, Freq (Hz) Figure 3. Frequency epenence of the moulus of (1- C ), where C is the inuce ipole coefficient, for colloial particles (15 nm in raius) with a zeta potential of 1 mv. The approximate positions of the - an Maxwell- Wagner (MW) relaxations are inicate. t this frequency, the iffusion processes in the ionic atmosphere surrouning the particles (the electrical ouble layer) cannot follow the fiel changes an the inuce ipole moment increases. t still higher frequencies, the Maxwell- Wagner (MW) relaxation gives rise to a ecrease in the ipole moment, hence the increase in 1 C an in the moulus of the mobility, observe in Figure 1. This relaxation may also provoke a phase lea, observable in some cases in Figure 2. Let us now consier how the cell moel can escribe the effects of volume fraction on the ynamic mobility. In Figures 1 an 2, we have superimpose on the experimental points the best fit relationships between the moulus (or phase) of the mobility an the frequency of the fiel, obtaine from the moel, using the zeta potential ( ) of the interface as the only parameter. The best-fit values are inclue in the Figures. It must be mentione that, accoring to Shilov et al [16], the zeta potential is a suitable parameter escribing the electrical state of the interface as long as the ouble layer thickness an/or the particle concentration are such that the interparticle istance is larger than the ouble layer thickness. If this conition is not fulfille, it is the surface charge an not the potential that remains physically soun. Throughout this paper, we assume that we are far from the strong-overlap situation. Note that in general the cell moel is capable of properly fitting the ata, although sometimes it fails to reach the highfrequency sie because of convergence limitations of our computer program. In the case of sample, the moel can reprouce the maxima in the moulus of the mobility at frequencies close to the experimental ones (Figure 1). However, the fitting is not as goo for sample. The fact that u -frequency (an, mostly, phase-frequency) curves isplay 3.2 ELECTRIC PERMITTIVITY n equally promising an useful proceure is the comparison between permittivity an ynamic mobility ata obtaine on exactly the same suspensions. Thus, Figure 4 shows the real part of the relative permittivity, ε (ω)/ε, of an lumina suspensions as a function of the frequency of the applie electric fiel for the volume fractions inicate. Note the rough fall corresponing to the low frequency or α-relaxation in the range Hz. The Maxwell-Wagner (MW) relaxation cannot be observe because its amplitue is much smaller than the low-frequency one. Figure 2 shows that MW relaxation is clearly noticeable in ynamic mobility ata, so that both techniques probe the frequency epenence of the electrokinetic quantities in a complementary fashion. The effect of volume fraction on the ielectric relaxation is appreciate in Figure 4: both the very low frequency (ω ) electric permittivity, ε (), an the alpha relaxation frequency, increase with the volume fraction φ. 3.3 COMPRISON ETWEEN OTH TECHNIQUES We will now check if these experimental ata are well reprouce by the theoretical preictions of the cell moel, alreay verifie for the ac mobility results. This requires a previous knowlege of the zeta potential of the particles, a task often performe by aitionally using c electrophoresis measurements in ilute suspensions of the same particles an the same ionic meium. The present work explores the interesting possibility of using the same concentrate suspensions in both electrophoresis an ielectric ispersion experiments: the former will be use to estimate the zeta potential, which in turn will be use to calculate the permittivity using the same cell moel. This offers the possibility of testing both the theory itself an the joint use of the two electrokinetic techniques. Figures 5 an 6 show, respectively, the values of the low-frequency relative permittivity ε ()/ε, an of the -relaxation frequency as a function of the volume fraction of suspensions. Using the zeta potentials obtaine from ynamic mobility, the cell moel allows to explain the overall features of the ielectric relaxation in the suspensions, an to preict the variation of the relaxation amplitue an the increase of the relaxation frequency with volume fraction. However, similar to previous finings regaring the comparison of permittivity an c electrophoresis, it is clear that the electrokinetic moel tens to unerestimate ε () an to overestimate ω α, the ifferences amounting to about one orer of magnitue. These ifferences have been explaine consiering that the classical or stanar electrokinetic moel has funamental limitations:

5 IEEE Transactions on Dielectrics an Electrical Insulation Vol. 13, No. 3; June Relative permitivity % 1 % 5 % 2.5 % 1 % Relative permittivity (ω ) experimental ata Zeta potential obtaine from electroacoustic ata Zeta potential obtaine from 11 electrophoresis ata Volume fraction (%) Relative permittivity (ω ) theoretical ata Freq (Hz) Relative permitivity % 19 % 1 % 16 % 7% 4 % 3 % 29 % Freq (Hz) Relative permitivity (ω ) experimental ata Experimental ata Zeta potential obtaine from electrophoresis ata Zeta potential obtaine from electroacoustic ata Volume fraction (%) Figure 5. Experimental (full squares) an theoretical (open symbols) ata of the low-frequency relative permittivity of alumina suspensions. The latter were calculate using the zeta potential obtaine from electroacoustic measurements (circles) an from electrophoresis (triangles). an as in Figure Relative permitivity (ω ) theoretical ata Figure 4. Real part of the relative permittivity, ε (ω)/ε, as a function of the angular frequency for ifferent volume fractions of alumina particles in KCl (2 1-4 mol/). an correspon to alumina samples an, respectively. the hypothesis that the stagnant layer (extening between the soli surface an the shear plane) is non-conucting has been revise an so-calle ynamic-stern-layer moels have been elaborate that emonstrate a better quantitative agreement between ifferent techniques [6-8]. The behavior of ε ()/ε (Figure 5) is similar to the preictions of our previous work [17]: although the permittivity of the particles is much smaller than that of the meium, only for φ larger than.14 is the permittivity of the suspension lower than that of the ispersion meium; for lower concentrations of particles, the ouble layer polarization more than compensates this, an very high values of the relative permittivity are foun at low frequencies. In the case of sample (Figure 5), the theoretical preictions are very similar to the experimental ata; however, the maximum reache by ε () occurs at ifferent volume fractions. This iffrence may be ue to the fact that the zeta potential obtaine from u has some egree of uncertainty because of the polyispersity of the sample, that we mentione above. It must be taken into account that the maximum in the permittivity-volume fraction epenence is very sensitive to the value of [17]. Let us finally mention that both the calculations an the eexperimental ata emonstrate that the relaxation frequency increases with the volume fraction of particles (Figure 6). In orer to justify this, we must recall that the high values of ε () are ue to the iffusive currents of counterions in the electical ouble layer; the shorter the characteristic iffusion length, the higher the relaxation frequency. It is reasonable to amit that the length will ecrease (an the frequency will increase) by the presence of closer neighbor particles as we increase the volume fraction, as in fact observe. CONCLUSIONS We can conclue that using the zeta potential obtaine from electroacoustics is a reasonable approximation to the true electrical potential at the soli/liqui interface, better, in any case, than that euce from c electrophoresis in ilute

6 662 S. Hawalli et al.: Electroacoustic an Dielectric Dispersion of Concentrate Colloial Suspensions experiemntal ata (khz) experimental ata (khz) 4, 3, 2, 1,5 1, 5, 4,5 4, 3, 2, Volume 1 fraction 12 (%) Volume fraction (%) theoretical ata (khz) theoretical ata (khz) Figure 6. Experimental (full squares) an theoretical (open symbols) ata of the -relaxation frequency of alumina suspensions. The latter were calculate using the zeta potential obtaine from electroacoustic measurements (circles) an from electrophoresis (triangles). an as in Figure 4. Note the ifferent scales of the left an right orinate axes. suspensions. The cell moel use appears to properly escribe the main features of the volume fractions epenence of electrokinetic quantities. The fact that it unerestimates the permittivity an overestimates the relaxation frequency is a point open for further investigation, perhaps in relation with bounary conitions for ions an liqui in the vicinity of the soli surface. CKNOWLEDGMENT Financial support from the Spanish Ministry of Science an Technology an Feer Funs (EU) (FIS C2-1/2) is acknowlege. S.. woul like to thank the Spanish Ministry of Eucation for a grant covering this research. REFERENCES [1] R.W. O rien,. Jones an W. N. Rowlans, new formula for the ynamic mobility in a concentrate colloi, Collois Surf., Vol. 218, pp , 23. [2].S. Dukhin, H. Ohshima, V.N. Shilov an P.J. Goetz, Dynamic electrophoretic mobility in concentrate isperse systems. Cell moel, Langmuir, Vol. 15, pp , [3].S. Dukhin, V. N. Shilov, H. Oshima an P. J. Goetz, Electroacoustic phenomena in concentrate ispersions. New theory an CVI experiments, Langmuir, Vol. 15, pp , [4] S. Kuwabara, The forces experience by ranomly istribute parallel circular cyliners or spheres in a viscous flow at small Reynols numbers, J. Phys. Soc. Jpn., Vol. 14, pp , [5] H. Ohshima, Dynamic electrophoretic mobility of spherical colloial particles in concentrate suspensions: approximation of nonoverlapping ouble layers, Collois Surf., Vol. 159, pp , [6] F. Carrique, F. J. rroyo an. V. Delgao, Seimentation velocity an potential in a concentrate colloial suspension. Effect of a ynamic Stern layer, Collois Surf., Vol. 195, pp , 21. [7] F. Carrique, F. J. rroyo an. V. Delgao, Electrokinetics of concentrate suspensions of spherical colloial particles: effect of a ynamic Stern layer on electrophoresis an DC conuctivity, J. Colloi Interface Sci., Vol. 243, pp , 21. [8] F. Carrique, F. J. rroyo an. V. Delgao, Electrokinetics of concentrate suspensions of spherical colloial particles with surface conuctance, arbitrary zeta potential, an ouble layer thickness in static electric fiels, J. Colloi Interface Sci., Vol. 252, pp , 22. [9]. V. Delgao, S. hualli, F. J. rroyo an F. Carrique, Dynamic electrophoretic mobility of concentrate suspensions. Comparison between experimental ata an theoretical preictions, Collois Surf., Vol. 267, pp , 25. [1] C. Grosse an M. Tirao, Measurement of the ielectric properties of polystyrene particles in electrolyte solution, Mater. Res. Soc. Symp. Proc., Vol. 43, pp , [11] M.L. Jiménez, F.J. rroyo, J. van Turnhout,.V. Delgao, nalysis of the ielectric permittivity of suspensions by means of the logarithmic erivative of its real part, J. Colloi Interface Sci., Vol. 249, , 22. [12] F. J. rroyo, F. Carrique, S. hualli an. V. Delgao, Dynamic mobility of concentrate suspensions. Comparison between ifferent calculations, Phys. Chem. Chem. Phys., Vol. 6, pp , 24. [13] R. W. O rien, Electro-acoustic effects in a ilute suspension of spherical particles, J. Flui Mech., Vol. 19, pp , [14]. S. Dukhin, V. N. Shilov, H. Ohshima an P. J. Goetz in. V. Delgao (E.), Interfacial Electrokinetics an Electrophoresis, Surfactant Science Series, Marcel Dekker, Inc., New York, 22, Chapter 17. [15] E. H.. DeLacey an L. R. White, Dielectric response an conuctivity of ilute suspensions of colloial particles, J. Chem. Soc., Faraay Trans. II, Vol. 77, pp , [16] V. N. Shilov, Y.. orkovskaja an. S. Dukhin, Electroacoustic theory for concentrate collois with overlappe DLs at arbitrary a. 1 pplication to nanocollois an nonaqueous collois, J. Colloi Interface Sci., Vol. 277, pp , 24. [17] F. Carrique, F. J. rroyo, M. L. Jiménez an. V. Delgao, Dielectric response of concentrate colloial suspensions, J. Chem. Phys., Vol. 118, pp , 23. PHOTO 24mm X 3mm S. hualli was born in Tucumán, rgentina in She receive the.s. (Physics) egree from the University of Tucumán, rgentina in 21, an the M.S.C. (Electroacustic phenomena) egree from the University of Granaa, Granaa, Spain in 24. Currently, she is pursuing a Ph.D. egree in the Department of pplie Physics of the Unversity of Granaa. Her areas of interest inclue electrokinetic an interface phenomena, particularly, electroacoustic an ielectric properties of concentrate suspensions. of Milano, Italy. M.L. Jiménez was born in Seville, Spain in She receive the.s. (Physics) an the M.S.C. (Permittivity of suspensions) egrees from the University of Granaa, Spain in 1998 an 2, respectively She efene her Ph.D. issertation in 23, ealing with the permittivity of suspensions of nonspherical particles. Her areas of interest inclue electrokinetic phenomena, ielectric properties of suspensions, an Kerr effects. She is presently oing her post-oc research in the University

7 IEEE Transactions on Dielectrics an Electrical Insulation Vol. 13, No. 3; June V. Delgao was born in aajoz, Spain in He receive the.s. (Physics) an Ph.D. (Electrokinetics) egrees from the University of Granaa, Spain in 1978 an 1984, respectively. He worke as an associate professor between 1986 an 1988 an since 1998 he is Professor at the Department of pplie Physics of the same University. His research is evote to the electrical properties of isperse systems. former member of the Eitorial oar of the Journal on Colloi an Interface Science, he has been the Eitor of a multiauthor book on Electrokinectic Phenomena, an coauthor of a number of publications in the fiel. F. Carrique was born in lmería, Spain, in He receive the.s. (Physics) an Ph.D. (Electrokinetics) egrees from the University of Granaa, Spain in 1988 an 1993, respectively. He worke as assistant professor between 1989 an 1996 an since 1996 he is an ssociate Professor at the Department of pplie Physics of the University of Málaga. His research is evote to the electrical properties of isperse systems. He has coauthore a number of papers on the calculation of the ielectric properties of isperse systems. F.J. rroyo was born in Granaa, Spain, in He receive the.s. (Physics) an Ph.D. (Electrokinetics) egrees from the University of Granaa, Spain in 1994 an 1998, respectively. He worke as assistant professor between 2 an 24 in the Universities of Jaén an Granaa, Spain. Since 24 he is an ssociate Professor at the Department of Physics of the University of Jaén. His research is evote to the ielectric properties of isperse systems.

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