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Sep 21, 2008 - Abstract We present new data for the thermal conductivity enhancement in seven nanofluids con- taining 8–282 nm diameter alumina ...
J Nanopart Res (2009) 11:1129–1136 DOI 10.1007/s11051-008-9500-2

RESEARCH PAPER

The effect of particle size on the thermal conductivity of alumina nanofluids Michael P. Beck Æ Yanhui Yuan Æ Pramod Warrier Æ Amyn S. Teja

Received: 5 June 2008 / Accepted: 29 August 2008 / Published online: 21 September 2008 Ó Springer Science+Business Media B.V. 2008

Abstract We present new data for the thermal conductivity enhancement in seven nanofluids containing 8–282 nm diameter alumina nanoparticles in water or ethylene glycol. Our results show that the thermal conductivity enhancement in these nanofluids decreases as the particle size decreases below about 50 nm. This finding is consistent with a decrease in the thermal conductivity of alumina nanoparticles with decreasing particle size, which can be attributed to phonon scattering at the solid–liquid interface. The limiting value of the enhancement for nanofluids containing large particles is greater than that predicted by the Maxwell equation, but is predicted well by the volume fraction weighted geometric mean of the bulk thermal conductivities of the solid and liquid. This observation was used to develop a simple relationship for the thermal conductivity of alumina nanofluids in both water and ethylene glycol.

Nomenclature C Exponent of Euler’s constant k Thermal conductivity (W m-1 K-1) q Heat dissipated per length of wire (W m-1) rw Radius of wire (m) T Temperature of wire (K) t Time (s)

Keywords Nanofluids  Thermal conductivity  Transient hot wire method  Phonon scattering  Nanoparticles  Colloids

Nanofluids (which are dispersions of solid or liquid nanoparticles in a liquid) have attracted considerable attention recently because of their potential as high performance heat transfer fluids in automotive and electronic cooling (Maiga et al. 2006; Wen and Ding 2006) and in microchannel heat sinks (Jang and Choi 2006). This interest stems from the work of Choi, Eastman, and others (Eastman et al. 1997; Lee et al. 1999; Wang et al. 1999) who reported large enhancements in the effective thermal conductivity of common heat transfer fluids such as water, ethylene glycol, and oil, upon the addition of small amounts of

M. P. Beck  Y. Yuan  P. Warrier  A. S. Teja (&) School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, USA e-mail: [email protected]

Greek symbols a Thermal diffusivity of liquid (m2 s-1) / Volume fraction n Thermal conductivity enhancement Subscripts P Particle l Liquid

Introduction

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solid nanoparticles. For example, Eastman et al. (2001) reported a 40% enhancement in the effective thermal conductivity of ethylene glycol when 0.3% (v/v) copper nanoparticles were dispersed in the liquid, and Choi et al. (2001) reported a 150% enhancement in the effective thermal conductivity of synthetic oil containing 1% (v/v) carbon nanotubes. These increases were attributed to the high surface area of the nanoparticles and their high thermal conductivity, as well as to other factors including Brownian motion of particles, the ordering of liquid molecules near the surface of particles, and interfacial resistance at the fluid-particle interface. The volume fraction dependence of the thermal conductivity of nanofluids has been studied extensively, but there are few systematic studies related to the particle size dependence of the thermal conductivity. Indeed, reported trends in thermal conductivity enhancement (or the ratio of effective thermal conductivity of the nanofluid to that of the base liquid) with particle size range from positive (Kim et al. 2007; Li and Peterson 2007) to negative (Yu et al. 2007). As noted above, few data are available to determine the effect of size-dependent phenomena on thermal transport in nanofluids. Most experimental studies have reported data on nanofluids containing one or two sizes of particles, so that general trends with particle size have been difficult to discern. The work of Xie et al. (2002) does report data for nanofluids containing five different sizes of alumina particles. However, they observed an increase followed by a decrease in the thermal conductivity with particle size for alumina nanoparticles in ethylene glycol, as well as in pump oil. In contrast, other studies have reported monotonic increases (Kim et al. 2007; Li and Peterson 2007) or decreases (Yu et al. 2007) in the thermal conductivity with decreasing particle size. In this work, we report measurements of the thermal conductivity of nanofluids containing seven different sizes of nanoparticles in order to ascertain the effect of particle size on thermal conductivity enhancement. We also report several estimates of particle size using different measurement techniques so that the appropriate particle characteristics can be used to correlate data. Our work should therefore facilitate the development of models for the thermal conductivity of nanofluids, and also aid in identifying mechanisms of thermal transport in dispersions of nanoparticles.

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Experimental Particles were purchased from Nanostructured and Amorphous Materials, Inc., Nanophase Technologies Corporation, and Electron Microscopy Sciences. The average particle sizes reported by these vendors based on their surface area measurements are listed in Table 1. We also determined average particle sizes using BET (Brunauer–Emmett–Teller) surface area measurements using nitrogen adsorption and have listed these values in Table 1 for comparison. In addition, we determined the shape of the particles and their polydispersity via transmission electron microscopy. We found that all the alumina particles were spherical or ellipsoidal (sphericity [ 0.9). However, their polydispersity varied significantly, and we have therefore listed standard deviations for particle sizes obtained from TEM micrographs in Table 1. Since nanoparticles may aggregate in dispersions, we obtained estimates of the aggregate size from measurements of the average hydrodynamic radius using dynamic light scattering (after diluting the dispersions). Values of hydrodynamic radii are also listed in Table 1, as are the crystalline phases of the particles obtained from the vendors. It should be added that the crystalline phases of particles obtained from different vendors were different. However, the bulk thermal conductivity of the three phases of alumina is about the same. Furthermore, as noted by Xie et al. (2002), the crystalline phase of the solid particles apparently has no effect on the thermal conductivity of nanofluids. Nanofluids were prepared by dispersing preweighed quantities of alumina nanoparticles into either deionized water or ethylene glycol. The samples were then subjected to ultrasonic processing to obtain uniform dispersions. In addition, the pH of each dispersion was adjusted to a value of 4 by adding HCl. The isoelectric point of alumina lies between a pH of 7 and 9, so that the acidic pH of the nanofluids ensured electrostatic stabilization of the suspension from surface charges. The thermal conductivity of each nanofluid was measured by a liquid metal transient hot wire device. This method of thermal conductivity measurement was chosen because it is rapid, robust, and has been used successfully to measure the thermal conductivity of electrically conducting liquids at temperatures as high as 465 K in our laboratory (Bleazard and Teja

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Table 1 Source, crystal phase, and average diameter of particles used in this study Manufacturer (product name)

Crystal phase & material

Average particle diameter (nm) Reported by manufacturer

From BET surface area measurements

Standard deviation From TEM measurements

Hydrodyn. radius (nm) from light scattering

Electron Microscopy Sciences

c-Al2O3

50

16

5

520

Electron Microscopy Sciences

a-Al2O3

300

71

95

492

Electron Microscopy Sciences

a-Al2O3

1000

282

51

522

Nanophase (NanoTek)

d/c-Al2O3

47

46

110

205

Nanostructured and Amorphous Materials Nanostructured and Amorphous Materials

c-Al2O3

11

8

2

320

c-Al2O3

20

12

2

265

Nanostructured and Amorphous Materials

a-Al2O3

150

245

38

790

1995; Sun and Teja 2004a, b). We have also verified (Beck et al. 2007) that the method can be used to measure the thermal conductivity of alumina nanofluids over a broad range of temperatures. Thermal conductivities were measured using a transient hot-wire technique with a mercury-filled glass capillary acting as the ‘‘wire.’’ The glass capillary was suspended vertically in the test liquid or dispersion and served to insulate the mercury ‘‘wire’’ from the electrically conducting liquid or dispersion. The mercury wire formed one resistor in a Wheatstone bridge and was resistively heated by application of a constant potential to the bridge. The temperature change of the wire was computed from the change in resistance of the mercury wire over time, determined by measuring the voltage across the initially balanced Wheatstone bridge. Subsequently, the temperature rise versus time data were used to calculate the thermal conductivity from the solution to Fourier’s equation for a linear heat source of infinite length in an infinite medium:   q 4at DT ¼ ln 2 ð1Þ 4pk rw C where DT is the temperature change of the wire, q the dissipated heat per unit length, k the effective thermal conductivity of the fluid, t the time from the start of heating, a the thermal diffusivity of the fluid, rw the radius of the wire, and C the exponent of Euler’s constant. The existence of a linear relationship

between the temperature change of the wire and the natural log of time was used to confirm that the primary mode of heat transfer during the measurement is conduction. Corrections to DT are applied to account for the insulating layer around the wire, the finite dimensions of the wire, the finite volume of the fluid, and heat loss due to radiation. In order to account for the nonuniform thickness of the capillary and end effects, an effective length of the wire was obtained by calibration with a reference fluid. In the present study, water (Meyer 1993) and dimethyl phthalate (Marsh 1987) were used as the reference fluids. Each thermal conductivity measurement was repeated 5 times and the values were averaged, with a reproducibility of ±1% and an estimated error of ±2%. Additional details of the technique are available elsewhere (Bleazard and Teja 1995).

Results Table 2 gives our measured values of the thermal conductivity enhancement for seven alumina nanofluids in water. As noted previously, each data point represents the average of five measurements at a specific concentration and room temperature. We were able to correlate the data with the primary particle size obtained from BET measurements, but not with aggregate size estimated from the hydrodynamic radius. The data and correlations are plotted in

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Table 2 Thermal conductivity of alumina nanofluids in water

Average particle diameter (nm)

T (K)

Volume fraction of Al2O3 (%)

Mean k W m-1 K-1

Standard deviation

8

299.1

1.93

0.620

0.017

8

297.4

2.99

0.621

0.008

8 12

297.7 297.4

3.99 2.00

0.623 0.622

0.016 0.006

12

297.4

3.00

0.630

0.009

12

297.8

4.00

0.640

0.005

16

296.8

2.00

0.635

0.003

16

298.0

3.00

0.642

0.015

16

298.3

3.98

0.662

0.015

46

298.0

2.00

0.637

0.017

46

298.3

2.99

0.644

0.018

46

297.8

3.99

0.660

0.011

71

296.8

1.86

0.657

0.010

71

297.7

3.00

0.696

0.017

71

296.6

3.99

0.712

0.016

245

300.5

1.86

0.656

0.013

245

298.5

3.00

0.678

0.012

245

299.1

4.00

0.709

0.013

282 282

297.4 298.0

2.00 3.00

0.669 0.693

0.011 0.008

282

298.7

4.00

0.719

0.020

Fig. 1 and show that the thermal conductivity of alumina nanofluids decreases as the particle size decreases below about 50 nm. The correlation of the thermal conductivity enhancement n = (k - kl)/kl with particle size can be expressed as follows: n¼

k  kl ¼ nmax ð1  e0:025d Þ kl

ð2Þ

with nmax ¼ 4:4134/

ð3Þ

where k is the thermal conductivity of the nanofluid, kl is the thermal conductivity of the base fluid, d is the diameter of the particles in nm, and / is the volume fraction of the particles. Note that the thermal conductivity attains a limiting value nmax at each volume fraction as the particle size increases. Agreement between calculated and experimental values is satisfactory (R2 values between 0.90 and 0.93) as shown in Fig. 1, although the data for 46 and 71 nm nanoparticles exhibit larger than average deviations because of the polydispersity of the particles. We determined the polydispersity from

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TEM images, as shown in Fig. 2 for 46 nm particles. Estimates of the polydispersity from Fig. 2 and other TEM images are listed in Table 1 in terms of the standard deviation of the particle size. Polydispersity is difficult to control in most nanofluid studies because of the method of preparation of the starting materials. Nevertheless, the fit of the data by Eq. 2 is quite satisfactory. Similar behavior was also exhibited by alumina nanofluids in ethylene glycol, as shown in Fig. 3 and Table 3. Equation 2 can also be applied to these nanofluids, with: nmax ¼ 5:527/

ð4Þ

We can therefore conclude that the thermal conductivity enhancement in nanofluids decreases with particle size from its limiting (maximum) value for nanofluids containing large particles.

Discussion One of the first thermal conductivity relations for a dilute suspension of randomly dispersed spherical

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20

18 16

(k - k ) / k (%)

14

l

l

(k - k ) / k (%)

15

l

l

10

10 8

5 2% 3% 4%

0

12

6 4

0

50

100

150

200

250

300

0

50

100

150

200

250

300

Average Particle Diameter (nm)

Average Particle Diameter (nm) Fig. 1 Thermal conductivity enhancement for alumina dispersions in water at 298 K. Error bars are shown only for 4% (v/v) dispersions. Lines represent calculated values using Eqs. 2 and 3

Fig. 3 Thermal conductivity enhancement for 2% (v/v) (d) and 3% (v/v) (j) alumina particles dispersed in ethylene glycol at 298 K. Lines represent calculated values using Eqs. 2 and 4

the solid particles, and kl is the thermal conductivity of the fluid. As discussed in detail by Turian et al. (1991), the Maxwell equation gives good predictions of thermal conductivity enhancements in heterogeneous systems when the ratio kP/kl * 1, and increasingly poor predictions as this ratio increases. Based on limiting cases of particle arrangements in the dispersion, they proposed a volume fraction averaged geometric mean to obtain the effective thermal conductivity of dispersions as follows: k=kl ¼ ðkP =kl Þ/

ð6Þ

or ngm ¼ ðkP =kl Þ/  1

Fig. 2 TEM images of alumina particles with a nominal size of 47 nm from Nanophase Technologies. Magnification = 100,000

particles in a matrix was developed by Maxwell (1892) and is given by: nM ¼

3ðkP  kl Þ/ kP þ 2kl  ðkP  kl Þ/

ð5Þ

where nL is the thermal conductivity enhancement (the subscript M denotes the value obtained using the Maxwell equation), kP is the thermal conductivity of

ð7Þ

Turian et al. found that Eq. 6 worked well for solid– liquid dispersions when kP/kl [ 4. Note, however, that neither the Maxwell equation nor Eq. 6 incorporate any dependence on particle size. Since Eqs. 5–7 were developed for dispersions of large particles, they provide estimates of the maximum thermal conductivity enhancement when the thermal conductivity of the bulk solid is used to obtain kP. Maximum thermal conductivity enhancements calculated from these equations are listed in Table 4. As can be seen from this table, experimental values (calculated from the mean of the enhancements at the two largest particle sizes) are in agreement with values calculated using the geometric

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Table 3 Thermal conductivity of alumina nanofluids in ethylene glycol

Average particle diameter (nm)

T (K)

Volume fraction of Al2O3 (%)

Mean k (W m-1 K-1)

Standard deviation

12

298.6

2.00

0.272

0.001

12

298.0

3.00

0.276

0.002

16 16

298.6 297.9

2.00 3.00

0.273 0.281

0.001 0.002

245

298.6

2.00

0.280

0.002

245

298.3

2.99

0.294

0.003

282

297.9

2.00

0.283

0.002

282

298.0

3.01

0.299

0.002

Table 4 Calculated values of the maximum thermal conductivity enhancement in alumina nanofluids in water at 298 K Volume fraction /

nexpt

nM Eq. 5

ncorr Eq. 3

ngm Eq. 6

0.02

0.0918

0.0585

0.0902

0.0873

0.03

0.1298

0.0886

0.1338

0.1338

0.04

0.1767

0.1193

0.1781

0.1822

nexpt obtained by averaging values for the two largest particle sizes

mean (Eq. 6). The Maxwell equation significantly underpredicts these enhancements. These results suggest that models for nanofluid thermal conductivity should emphasize the thermal conductivity of the individual phases and, in particular, the size dependence of the thermal conductivity of the solid nanoparticles. The relationship between the thermal conductivity of a solid and its characteristic dimensions has been explored experimentally (Li et al. 2003; Liu and Asheghi 2004) as well as computationally (Fang et al. 2006; Ziambaras and Hyldgaard 2006). In two experimental studies, the thermal conductivity of ultrathin silicon films (Liu and Asheghi 2004) and silicon nanowires (Li et al. 2003) was shown to be an order of magnitude less than the bulk thermal conductivity of bulk silicon. In a molecular dynamics (MD) study, Fang et al. (2006) showed that the thermal conductivity of spherical silicon nanoparticles decreased by two orders of magnitude as their size decreased below about 10 nm. They also found a linear relationship between thermal conductivity and particle size for very small (\10 nm diameter) nanoparticles. Linear relationships with characteristic dimensions have also been found for nanowires and ultrathin films (Ziambaras and Hyldgaard 2006) using

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MD simulations, with a gradual change to the bulk value as the radius or thickness becomes larger. This decrease in the thermal conductivity of the solid as its characteristic dimensions become of the same order as the phonon mean free path (Cahill et al. 2003) must therefore be accounted for when determining the effective thermal conductivity of nanofluids. Thermal resistance at the solid–liquid interface could also play a role in thermal transport in nanofluids. Such a resistance was first detected by Kapitza (Cahill et al. 2003) and attributed to different rates of phonon transport in the two materials. Unfortunately, the complex interactions between phonons and interfaces are poorly understood. Nevertheless, Nan et al. (1997) have proposed a model that shows a decrease in the effective thermal conductivity of solid composites due to the Kapitza resistance, as well as factors such as poor contact between the materials (Ge et al. 2006) and phonon dynamics. Our data show that the thermal conductivity enhancement in alumina nanofluids decreases as the particle size decreases below about 50 nm. The trend is similar to that predicted by Nan et al. (1997) in solid composites. However, their model reduces to the Maxwell equation when the particles are large, as shown in Fig. 4. This figure also shows that the limiting value exhibited by our data is greater than that predicted by the Maxwell equation. As noted above, our limiting value is predicted very well by the volume fraction weighted geometric mean given in Eq. 5. Prasher et al. (2006) have suggested that particle aggregation may lead to thermal conductivity enhancements that are greater than those predicted by the Maxwell equation. They suggest that the Maxwell equation therefore provides a lower bound for the enhancement in nanofluids, since it is valid for a welldispersed system of large particles. However, it is not

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greater than that predicted by the Maxwell equation, but is predicted well by the volume fraction weighted geometric mean of the bulk thermal conductivities of the two phases. This observation can be used to develop simple correlations for the thermal conductivity of nanofluids.

20

Geometric Mean

Maxwell

l

(k - k ) / k (%)

15

l

10

References 5

0

0

50

100

150

200

250

300

Mean Particle Diameter (nm) Fig. 4 Thermal conductivity enhancement for 4% (v/v) alumina in water at 298 K. The horizontal lines represent the Maxwell Eq. 5 and the geometric mean (Eq. 6). The solid curve represents the empirical fit using Eq. 2 and the dashed curve represents predictions with the model by Nan et al. (1997)

clear why the geometric mean would more accurately reflect particle arrangement in suspensions. As discussed by Turian et al. (1991), the effective thermal conductivity of a suspension of solid particles depends on the volume fraction of the particles / and the detailed microstructure of the dispersion, as well as on any thermal resistance at the fluid-particle interface. We note that the systems studied in our work were hydrophilic and were chosen to minimize the effect of poor contact on the Kapitza resistance. For these systems, the primary source of interfacial thermal resistance should thus be phonon scattering at the solid–liquid interface.

Conclusions We have measured the thermal conductivity of nanofluids containing seven sizes of alumina nanoparticles ranging from 8 to 282 nm in diameter. To date this is the largest number of particle sizes investigated in one study. Our results indicate that the thermal conductivity enhancement decreases as the particle size decreases below about 50 nm. We attribute this decrease in enhancement to a decrease in the thermal conductivity of the nanoparticles themselves as the particle size becomes small enough to be affected by increased phonon scattering. The limiting value of the enhancement for nanofluids containing large particles is

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