Optical Measurement of Thermal Conductivity and Absorption Cross

Mar 23, 2012 - ... HuVignesh SundaresanAaron J. McLeodDavid J. MasielloKatherine A. Willets ... Susil Baral , Ali Rafiei Miandashti , Hugh H. Richards...
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Optical Measurement of Thermal Conductivity and Absorption Cross-Section of Gold Nanowires Michael T. Carlson, Andrew J. Green, Aurangzeb Khan,† and Hugh H. Richardson* Department of Chemistry and Biochemistry, Ohio University, Athens, Ohio 45701, United States S Supporting Information *

ABSTRACT: A thin film of Al0.94Ga0.06N doped with Er3+ is used to measure the heat dissipation from lithographically prepared gold nanowires (330 nm × 20 nm × 1.6 −13.2 μm) excited at 532 nm with a Nd:YAG continuous wave laser. A temperature image of the optically excited gold nanowire is processed from the relative peak intensities from Er3+ photoluminescence. The thermal profile along the nanowire is fit using an elementary heat transfer model with the thermal conductivity as an adjustable parameter. We observe a reduction in the temperature dependent thermal conductivity of the nanowires compared to the bulk value that is attributed to an increase in scattering from grain boundaries having an average diameter of 8 nm. Energy balance yields the absorption cross-section at 532 nm, when the laser polarization is aligned with the nanowire, as 1.4 ± 0.5 × 10−13 m2.

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INTRODUCTION Optical absorption of nanostructures impacts many important areas of research ranging from noninvasive cancer treatments to solar technologies.1,2 Nanorods, compared to spherical nanoparticles, show promise in solar energy conversion due to the significant increase in absorption cross-section in the solar spectrum.3 Currently, most measurements of absorption crosssections encompass either extinction measurements or use ensemble techniques.4,5 In contrast, there are few techniques that can measure thermal conductivity of nanostructures or nanowires without electrical contacts.6−9 Thermal conductivity and electrical conductivity are related through the Wiedemann−Franz law,10 but exceptions have been reported.11,12 We have pioneered a method of determining the absorption cross-section of individual nanostructures by measuring the heat generation from optically excited nanostructures under steady-state conditions. At steady-state, the rate of heat generation and dissipation are equal. The absorption crosssection of the nanostructure at the excitation wavelength is elucidated by solving the heat dissipation problem and then equating the amount of heat dissipated to the amount of heat generated. Previously, we examined the optical absorption and heat transfer mechanisms of individual spherical gold nanoparticles (40 nm diameter) and gold nanodots (effective radius of 58 nm) excited by a 532 nm Nd:YAG continuous wave laser.13 These nanostructures absorbed the incident light via excitation of localized surface plasmon modes. The energy absorbed by the nanostructure is converted into heat that is dissipated into the underlying temperature sensing thin film of Al0.94Ga0.06N doped with Er3+ ions. The relative intensity of the Er3+ photoluminescence peaks are temperature dependent.14,15 © 2012 American Chemical Society

We use this property to measure the local temperature in the thin film during optical excitation of the nanostructure. In this article, we show that this technique can be applied to measure the thermal conductivity and absorption cross-section of lithographically prepared gold nanowires of different lengths. Under steady-state conditions, elementary heat transfer equations can be used to model the temperature profile of the nanowire. The model is fit to the experimental data using only the thermal conductivity of the gold nanowire as an adjustable parameter. Our fit yields the temperature-dependent thermal conductivity. The reduced thermal conductivity is consistent with scattering from small grain boundaries (average diameter of 8 nm). This measurement is in good agreement with thermal conductivity measurements of similar sized gold nanofilms.12,16,17 Energy balance provides a method for determining the absorption cross-section for the nanowires.

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MATERIALS AND METHODS Optical Measurements of Temperature. The optical measurements and characterization of the thermal sensor film has been described in detail previously.13 We use a thin film (∼270 nm thick) of Al0.94Ga0.06N embedded with Er3+ ions on a silicon substrate as a thermal sensor film by measuring the relative photoluminescent intensities of the 2H11/2 → 4I15/2 and the 4S3/2 → 4I15/2 energy transitions of the Er3+ ions. These intensities have been shown to be temperature dependent14,15 and are related by a Boltzmann factor (exp(−ΔE/kT) where ΔE is the energy difference between the two levels, k is the Received: December 29, 2011 Revised: March 23, 2012 Published: March 23, 2012 8798

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composite Er3+ emission over the collected volume and converting this composite photoluminescence spectrum into a temperature. This procedure results in a composite temperature within the collected volume that is the convolution of the true thermal image with the diffraction limited optical transfer function of the microscope. The parameter relating the measured composite temperature to the temperature at the nanostructure (local temperature) is TTP. The measured composite temperature can be scaled to give the local temperature by multiplying the measured temperature by a factor of 12.8.13 This factor relates the projection of the temperature image from the substrate to the focal plane of the microscope. Interestingly, to a first approximation, TTP is a scalar and does not depend upon laser intensity.13 The procedure for determining TTP has been published.13 A brief synopsis of how TTP is determined is given in the Supporting Information. The validity of using this procedure to relate the measured composite temperature to the local temperature has been checked by measuring phase transitions that are intrinsic to the heated nanostructure. We have shown that gold nanodots heating in air shows a linear trend until the melting temperature is reached where an irreversible positive deviation is observed. The deviation from linearity is because of changes in the optical properties of the nanodot. Melting of the nanodot is confirmed with SEM by imaging the nanodot before and after laser excitation.13 Recently, we have shown that heating nanodots in water produces a linear trend until a temperature of 594 K ± 17 K is reached.18 At this temperature an increase in laser intensity causes temperature instabilities that are observed as large temperature variation between nanodots at the same laser intensity. The reason for the temperature instability is that 594 K is the spinodal decomposition temperature of water, and at this temperature, the water spinodal grows until its correlation length is on the same magnitude as the wavelength of the excitation laser (532 nm). This causes scattering of the excitation laser (critical opalescence) that decreases the amount of light absorbed by the nanodot. The onset of critical opalescence is confirmed by plotting the normalized backscattered laser intensity relative to the background scattering as a function of laser intensity. Below the spinodal decomposition temperature, the normalized back scattering is a positive constant, while above the spinodal decomposition temperature, the back scattering is a negative constant. These results substantiate that the procedure of converting the measured temperature to local temperature is valid.

Boltzmann constant, and T is the absolute temperature). The optical temperature measurements using the thermal sensor with isolated gold nanostructures were made with a WITec αSNOM300s upgraded with Raman imaging, dark-field, and AFM capabilities. A CW 532 nm wavelength Nd:YAG laser with adjustable power is focused with a 50× objective (Zeiss EC-Epiplan-NEOFLUAR 50X/0.8 HD) onto the thin film thermal sensor. The emitted light from the thermal sensor is collected with the same objective and sent to the CCD spectrograph with a collection fiber. The highest spatial resolution is achieved with a 25 μm collection fiber, and the data in this article has been collected with this fiber. The 50× objective used with the 25 μm collection fiber produced an Airy diffraction pattern that has a fwhm of ∼500 nm. This pattern was obtained by imaging in dark-field a small scattering object (single array dot). We used dark-field imaging and scattering to determine the location of the gold nanostructure on the thermal sensor film. Once the position of the nanostructure is located, the thermal sensor film is translated, with nanometer control, under the objective during laser excitation. An image is collected by storing the full photoluminescence spectrum at pixel locations within the image (100 × 100 pixels). A temperature image is constructed from the photoluminescence spectra at each pixel location. The temperature images are obtained by summing and baseline correcting a wavelength range corresponding to the 2H11/2 → 4I15/2 and the 4S3/2 → 4 I15/2 energy transitions. A ratio of the peak areas is then placed in the Boltzmann equation to produce a temperature that is dependent upon the value of the intercept. The intercept is chosen so that the temperature in the limit of no laser intensity is room temperature. We have confirmed that the relative peak intensities are related by a Boltzmann factor.13 Fabrication of Gold Nanowires. The gold nanowires are fabricated using conventional e-beam lithography with lift-off. The height of the nanowires is 20 with a roughness of 4 nm. The roughness is measured with a Nanoscope IIIA Multimode AFM. We also fabricated periodic arrays of Au nanodots that are elliptical in shape having short and long axis diameters of 380 and 550 nm using electron beam lithography. The height of the nanodots is 30 ± 1.5 nm. Single layers of around 100 nm of poly(methyl methacrylate) (PMMA) (3% 495 K PMMA in Anisole) positive resist were spin coated onto the Al0.94Ga0.06N:Er thin film on Si. The PMMA coated samples were baked in an oven for 30 min at 180 °C . After baking, electron beam lithography (∼30 KV in a JEOL 6400) was performed to draw the desired pattern/dot arrays with a resolution of ∼100 nm. The samples were then developed with MIBK/IPA= 1:3 (volume) for 30 s and rinsed/cleaned with high purity water and then dried. A thin adhesion layer of Ti was first deposited, then followed by a thicker (50−70 nm) layer of Au using DC sputtering. A lift-off procedure with laboratory grade acetone (heated at 60 °C) for 30 min with sonication was carried out in order to remove the Au/PMMA residual layer. The thermal sensor film with nanostructures is cleaned by sonication in chloroform. Over time, with repeated cleanings, the height of the gold nanostructures gradually decreases. Thermal Transfer Parameter (TTP) Relating Measured Composite Temperature to the Local Temperature. The thermal transfer parameter (TTP) results because the true thermal image within the thermal sensor film is much smaller that the collected volume that is sampled in the photoluminescence measurement. Also, we are collecting the

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RESULTS Heat Generation and Temperature Change from a Single Gold Nanowire. Figure 1 shows a SEM image of the gold nanowire. The width of the nanowire is 330 nm with a height of 20 nm and a length of 6600 nm. Gold nanowires of lengths ranging from 1650 nm to 13 200 nm were fabricated with e-beam lithography. Figure 2 shows a temperature image and temperature profile for a 9000 nm long gold nanowire. The wire is excited with 2.2 × 1010 W/m2 laser light and the polarization direction is along the wire. The laser spot has a two-dimensional Gaussian profile with a fwhm of ∼800 nm. The collection optics samples an area that can be represented by a two-dimensional Gaussian with a fwhm of ∼500 nm. Figure 3 shows the temperature profile along the wire during excitation with different polarizations. This nanowire is excited with 1.3 × 1010 W/m2 laser light. Changing the direction of polarization did not change the profile appearance/shape; only 8799

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thermal profile at the right side of the nanowire is due to an imperfection in the structure of the nanowire. Generally, the thermal profiles are symmetrical. Figure 4 compares the

Figure 1. SEM Image of a 6.6 μm long gold nanowire. The width of the nanowire is 330 nm with a height of 20 nm.

Figure 4. Thermal profiles from nanowires of different lengths excited with a laser excitation of 1.3 × 1010 W/m2 and parallel polarization. The red line represents the fit to the experimental data from the heat transfer model.

temperature profile of different nanowires to a nanodot. The nanowires and nanodot are excited with 1.3 × 1010 W/m2 laser light and the polarization direction is along the wire.

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DISCUSSION Model of Heat Transfer. We believe that the increase in temperature at the ends of the gold wire is due to diminished heat dissipation and not due to an increase in light absorption. The excitation of surface plasmon polariton (spp) modes previously reported for silver nanowires19 shows a strong dependence upon laser polarization. The polarization direction along the wire excites ssp modes, while the perpendicular polarization does not. Excitation of ssp modes causes an increase in absorption at the end of the wire compared to exciting the middle of the wire. We observe no change in the appearance of the temperature profile when the laser polarization is changed (see Figure 3) and observe only a change in the relative temperature of the entire profile with polarization. Because there is no polarization effect, we rule out assigning this temperature increase due to an increase in absorption because of excitation of spp modes. The temperature profile along the wire can be modeled using eq 1 where x is the direction along the wire, kwire is the thermal conductivity of the nanowire, A is the cross-sectional area of the wire, g is the thermal transfer coefficient for heat dissipation into the substrate, and P is the perimeter at x under the wire in contact with the substrate. In this model, the heat only dissipates through the substrate from the bottom of the gold wire and not through air. The thermal transfer coefficient (g) is just the thermal conductivity of the sensor film (1.5 W/m·K) divided by the thickness of the film (270 nm). This gives 5.55 × 106 W/m2·K for the thermal transfer coefficient.

Figure 2. Top image is the thermal profile along the nanowire taken from the thermal image shown in the bottom panel. The black squares represent data points from the thermal image. The solid red line shows the calculated theoretical temperature profile as dictated by the heat transfer equations. The red dotted line is the laser excitation profile, while the black dotted line is the temperature profile at the point of excitation. The nanowire is excited with a laser intensity of 2.2 × 1010 W/m2.

Figure 3. Polarization dependence of thermal profiles from a 9 μm long nanowire. The wire was excited with a laser intensity of 1.3 × 1010 W/m2, and the excitation polarization was changed from along the wire (0 degrees) to orthogonal to the nanowire (90 degrees) to along the nanowire but in the opposite direction (180 degrees).

dT + qA ̇ dx + gP(ΔT )dx dx ⎛ d2T ⎞ dT = −k wireA − k wireA⎜⎜ 2 ⎟⎟dx dx ⎝ dx ⎠

−k wireA

a change in the relative temperature is observed. The larger temperature changed is observed with a polarization perpendicular to the gold nanowire long axis. The change in the 8800

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Figure 5. (A) Root-mean-squared deviation (rmsd) of the fit from the heat transfer model to the experimental data using m (m = (gP)/(kwireA)1/2) as an adjustable parameter. (B) Root-mean-squared deviation (rmsd) of the fit from the heat transfer model to the experimental data using kwire (thermal conductivity of the nanowire) as an adjustable parameter. The minimum rmsd of the fit occurs when kwire is 108 W/m·K.

θ/θo = cosh(m(L − x))/cosh(mL) where θo is θ at x = 0 and L is the length of the nanowire. Thermal Profile Fitting with Model. Figure 2 shows the results of fitting the parameter m to a single gold nanowire 9000 nm long with the solid red line as the fit to the experimental profile. The thermal image of the wire is shown in the lower part of the figure with the temperature profile shown in the top of the figure. The black dashed line is the temperature profile in the wire, while the red dashed line is the laser excitation profile. When the middle of the wire is excited, the heat dissipation is larger than when excited at the end of the nanowire because the thermal profile stretches in both directions toward the end of the wire. The larger heat dissipation causes a decrease in the temperature in the middle of the wire relative to the end of the wire. In contrast, when the wire is excited at either end, heat only propagates in one direction with a reduction in heat dissipating. The decrease in heat dissipation produces an increase in temperature. The change in the temperature profile when moving from excitation at the end of the wire to excitation toward the middle of the wire is sensitive to the parameter m2, which is equal to (gP)/(kwireA) or inversely proportional to kwire. Fitting the temperature decay from excitation at the end of the wire toward excitation in the middle of the wire yields a value for the thermal conductivity of the wire. Thermal Conductivity from Nanowires of Different Lengths. Figure 4 shows the thermal profile along gold wires of different lengths compared to the thermal profile of a gold nanodot. The solid red lines are the model fit using the same optimized m parameter. Figure 5A shows the root-meansquared deviation (rmsd) of the model fit to the experimental data as a function of the parameter m for a nanowire of 6600 nm length. Each value for m can be solved for the thermal conductivity using kwire = (gP)/(m2A). The rmsd of the model fit to thermal conductivity is shown in Figure 5B. The best fit for the thermal conductivity of the 6600 nm nanowire is 108 ± 5 W/m·K. The temperature of the nanowire is given by Twire = 300 K + (12.8ΔTmiddle) and is equal to 773 K for the nanowire with a length of 6600 nm. The parameter ΔTmiddle is the measured temperature change when the nanowire is excited and measured in the middle of the nanowire, and the factor, 12.8, is TTP for the optical configuration used in the measurements (50× objective lens and 25 μm fiber). The local

When the wire is excited in the middle and the temperature measured at that point, heat can propagate in both directions to either end of the wire. The magnitude of heat dissipation increases with the temperature gradient into the thermosensing substrate. Unlike the optically excited nanodot or nanoparticle where the temperature of the nanostructure is nearly uniform across the nanostructure,13 the temperature in an optically excited nanowire is different along the wire. This difference increases the complexity of the dissipation part of the problem because the rate of dissipation changes along the wire. Equation 1 takes this effect into account with the term gP(ΔT)dx, which models the heat dissipation into the substrate. At steady-state, the heat generation term, ∫ q̇Adx′ = ∫ ((CabsI(x′))/(Vexc))Adx′ = CabsIavg, and the heat dissipation term, 12.8∫ gP(ΔT)dx are equal. In the heat generation term, the laser intensity is not constant over the gold wire but varies as a Gaussian profile along the wire. The variable x′ refers to the distance along the wire that is excited with the laser light, and Iavg is the average constant laser intensity illuminating the nanostructure equivalent to the spatial size where half of the laser intensity excites the nanowire. The average laser intensity is determined by integrating the laser intensity over the dimensions of the nanowire and then restricting the dimension along the nanowire until half of the integrated intensity is obtained. The laser intensity at this restricted dimension is Iavg. The absorption cross-section is now Cabs = (12.8gP∫ (ΔT(x)) dx)/Iavg. The factor 12.8 in the heat dissipation term relates the local temperature of the nanowire to the measured temperature.13 This factor takes into account that our optical measurement of temperature is resolution limited and needs to be convoluted with the collection volume of our microscope and the true thermal image in the substrate. Equation 1 can be recast into eq 2 and solved using the appropriate boundary conditions of the nanowire.

d2θ d

x2

+

gP k wireA

θ=0

(2)

In this equation, temperature is transformed into θ where θ = T − T∞ + q̇/kwire. The solution of this differential equation is θ = C1 exp(−mx) + C2 exp(mx) where m2 = (gP)/(kwireA). After applying appropriate boundary conditions, the solution is 8801

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thermal conductivity at 300 K assuming that the Wiedemann− Franz law holds.

temperature change depends upon the length of the nanowire. Figure 6 plots the thermal conductivity from different length

kf ⎛ α + 1⎞ 3α ⎟ =1− + 3α2 − 3α3 ln⎜ ⎝ α ⎠ ko 2

(4)

In this equation kf and ko are the film and bulk (317 W/m·K) thermal conductivity at 300 K, respectively. The width of the nanowire is 330 nm, which suggests that a two-dimensional film with the thickness of the nanowire should have generally similar thermal conductivity as our nanowire. The modeled results using eq 4 for the thermal conductivity at 300 K and assuming a similar temperature dependence as the bulk is shown as the green and blue dotted lines in Figure 6. In the green dotted line fit, the average grain boundary size is taken as equal to the nanowire thickness, suggested by De Vries,16 while in the blue dotted line fit, the average grain boundary size is adjusted to fit the experimental data. The best fit for the average grain boundary size is 8 nm. Absorption Cross-Section for Nanodot and Nanowires. The absorption cross-section is calculated using energy balance after fitting the temperature profile for the nanowire thermal conductivity. The temperature in the middle of the nanostructure is used to determine the absorption cross-section using C abs = (12.8gP∫ (ΔT(x))dx)/I avg . The integral ∫ (ΔT(x))dx is solved after the parameter m2 is known from the fitting. The black dashed curve shown in Figure 2 is equal to the temperature change as a function of x along the nanowire with the area under the curve equal to ∫ (ΔT(x))dx. Iavg is the average laser intensity illuminating the nanostructure. The laser intensity is not constant over the gold wire but varies as a Gaussian profile along the wire. Iavg is determined by integrating the laser intensity over the dimensions of the nanowire and then restricting the dimension along the nanowire until half of the integrated intensity is obtained. The length of the nanowire that gives Iavg is 480 nm. The absorption cross-section at 532 nm with the polarization of the exciting laser pointed parallel to the wire is 1.4 ± 0.5 × 10−13 m2.

Figure 6. Temperature-dependent nanowire thermal conductivity compared to the bulk thermal conductivity. The black squares with red dotted line data is the temperature-dependent thermal conductivity of bulk gold. The experimental temperature-dependent thermal conductivity for the 20 nm thick gold nanowire is shown in black with error bars determined from the uncertainty in the fit of the heat transfer model to the experimental thermal profile. The green dotted line is the temperature-dependent extrapolation using the temperature trend from bulk gold but reduced thermal conductivity because of grain boundary scattering. The average grain boundary size is the same as the nanowire thickness. The blue dotted line is the same as the green dotted line except the average grain boundary size is reduced to 8 nm.

gold nanowires compared to the bulk value20 as a function of temperature. The electrical conductivity, and conversely thermal conductivity, has been solved by Mayadas and Shatzkes for a polycrystalline metallic film.21 Their result is given by eq 3 where α = (loR)/(D(1−R)), B = (6(1−p))/(πko), ko = d/lo, H(t,θ) = 1+d/(cos(θ)(1 − t−2)1/2); σf and σo are the electrical conductivities for the film and bulk, respectively, lo is the mean free path of an electron within a grain (41 nm),10 d is the film thickness, D is the average grain size, p is the specular reflection parameter of electrons at the film surface, and R is the reflection coefficient of electrons at grain boundaries. The reflection coefficient for gold grain boundaries is 0.17.16

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CONCLUSIONS

In summary, gold nanowires have been fabricated and deposited onto a Al0.94Ga0.06N:Er3+ thin film on Si. The temperature of the film is determined optically by measuring the relative intensities of the Er3+ photoluminescence. The heat produced by irradiating gold nanowires at 532 nm is determined by measuring the temperature of the Al0.94Ga0.06N:Er3+ thin film. General heat transfer equations were used to model the heat transfer along the wire and dissipation into the underlying film. We were able to fit the heat transfer model using the thermal conductivity of the gold nanowire as an adjustable parameter. The temperature-dependent thermal conductivity from different length gold nanowires is reduced compared to the bulk. The reduction in thermal conductivity is in agreement with theory16,21 if a grain size of 8 nm is used. This experimental result is in reasonable agreement with previously published thermal conductivity from similar sized gold nanofilms12 where the reduced thermal conductivity is attributed to an increase in grain boundary scattering. The heat generation from the optically excited nanowire is directly related to the absorption cross-section. Energy balance, at steady-state, allows us to determine an absorption cross-section of 1.4 ± 0.5 × 10−13 m2 when the laser polarization is parallel with the nanowire axis.

π /2 σf ⎛ α + 1⎞ 3α ⎟ − B =1− + 3α2 − 3α3 ln⎜ dθ ⎝ α ⎠ σo 2 0 ⎛ ⎞ ∞ cos2(θ) −3 1 − e−koH(t , θ)t ⎟ dt 2 (t − t −5)⎜⎜ −k H(t , θ)t ⎟ 1 H (t , θ) ⎝ 1 − pe o ⎠

∫

∫

(3)

Equation 3 can be approximated as σf/σo = 1 − (3α)/2 + 3α2 − 3α3 ln((α+1)/α) because grain boundary scattering makes the largest contribution to the resistivity of thin polycrystalline films.16 Equation 4 gives the ratio of film to bulk 8802

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ASSOCIATED CONTENT

S Supporting Information *

Determination of the thermal transfer parameter. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Present Address †

Department of Physics, Faculty of Science, University of Tabuk, Tabuk, Saudi Arabia. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Savas Kaya and Martin Kordesch for their help in fabrication of the nanowires. The work was supported by a grant from the Biomimetic Nanoscience and Nanotechnology Initiative at Ohio University.

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REFERENCES

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