Chapter 8Thermal Conductivity of Nanofluids

Pawel Keblinski

Nanofluids (colloidal suspensions of solid nanoparticles) sparked excitement as wellas controversy. In particular, a number of researches reported dramatic increases ofthermal conductivity with small nanoparticle loading, while others showed moder-ate increases consistent with the effective medium theories on well-dispersed con-ductive spheres. Here we discuss potential mechanisms that were put forward inorder to understand nanoflouid thermal conductivity and demonstrate that particleaggregation is currently the only physically reasonable mechanism that can explainthe majority of the experimental data.

8.1 Introduction

Cooling is one of the most important technical challenges facing numerous diverseindustries, including microelectronics, transportation, solid-state lighting, and man-ufacturing. Developments driving the increased thermal loads that require advancesin cooling include faster speeds (in the multi-GHz range) and smaller features (to< 100 nm) for microelectronic devices, higher power engines, and brighter opti-cal devices. The conventional method for increasing heat dissipation is to increasethe area available for exchanging heat with a heat transfer fluid, but this approachrequires an undesirable increase in the size of thermal management system. Thereis therefore an urgent need for new and innovative coolants with improved perfor-mance. About a decade ago a novel concept of nanofluids, i.e., heat transfer fluidscontaining suspensions of nanoparticles, was proposed as a means of meeting thesechallenges [1].

Nanofluids are solid–liquid composite materials consisting of solid nanoparticlesor nanofibers, with sizes typically on the order of 1–100 nm, suspended in a liquid.Nanofluids have attracted great interest recently due to reports of greatly enhancedthermal properties at low volume fractions. For example, a small amount (lessthan 1% volume fraction) of copper nanoparticles and carbon nanotubes dispersed

S. Volz (ed.), Thermal Nanosystems and Nanomaterials, Topics in Applied Physics, 118c

213DOI 10.1007/978-3-642-04258-4 8, © Springer-Verlag Berlin Heidelberg 2009

214 Pawel Keblinski

in ethylene glycol and oil, respectively, was reported to increase the inherently poorthermal conductivity of the liquid by 40% and 150%, respectively [2–5]. Conven-tional suspensions of well-dispersed particles require high concentrations (> 10%)of particles to achieve such enhancement; problems of rheology and stability, whichare amplified at high concentrations, preclude the widespread use of conventionalslurries as heat transfer fluids. In several cases, the observed enhancement in ther-mal conductivity of nanofluids is orders-of-magnitude larger than predicted by well-established theories of dispersed particles [6–12].

8.2 Excitement, Controversy, and New Physics

The large thermal conductivity enhancements reported by experiment led to excite-ment but also to controversy. The origin of the excitement was that the measuredthermal conductivity was often much larger than that predicted by well-establishedeffective medium theories under the assumption of well-dispersed particles. For asystem of well dispersed particles, in the limit of low particle volume fraction φ , andwith much higher particle conductivityΛNP than fluid conductivityΛfluid, the effec-tive medium (Maxwell) theory for vanishing interfacial thermal resistance leads tothe relation [13, 14]

ΛNF

Λfluid= 1 + 3φ . (8.1)

Equation (8.1) predicts only moderate thermal transport increase, independently ofthe conductivity of the filler. As already discussed, many experiments reported muchlarger increases.

This mismatch between predicted and observed values led not only to excitementbut also to controversy, since in some experiments, the measured thermal conduc-tivity was not very large and consistent with the prediction of (8.1) [15–19]. Thisinitially led to the belief that some experiments had to be wrong, particularly con-sidering that different research groups often obtained different results for what werepresumably the same nanofluids.

Further excitement and controversy were associated with potential mechanismsput forward to explain the ‘anomalous’ experimental observations. This discussionwas initiated by Keblinski et al. [20]. In the following we will review the key mech-anism analyzed and discussed in the literature.

8.2.1 Brownian Motion

Motivated by the ‘unusual’ thermal conductivity enhancements, a number of resear-chers promoted the concept of Brownian motion induced micro- or nanoconvec-tion [21–23]. They argued in different ways that each Brownian particle generates along-range velocity field in the surrounding fluid, akin to that present around a par-

8 Thermal Conductivity of Nanofluids 215

ticle moving with a constant velocity, which decays approximately as the inverse ofthe distance from the particle center. The ability of large volumes of fluid draggedby the nanoparticles to carry substantial amounts of heat was credited for the largethermal conductivity increases in nanofluids.

A key weakness of this argument is that the thermal diffusivity DT of the basefluid, which measures the rate of the heat flow via thermal conduction, is several or-ders of magnitude larger than nanoparticle diffusivity DNP, which measures the rateof mass motion due to nanoparticle diffusion, whence the magnitude of possiblenanoconvection effects is negligible [24]. Furthermore, the velocity field around aBrownian particle is much shorter-ranged than that around a particle moving with aconstant velocity [25]. The low estimates for the contribution to thermal transport ofBrownian motion were supported by the results of molecular dynamics simulations[24,26], and by the results of experimental measurements on well-dispersed spheri-cal nanoparticle fluids [16–19,27], all showing thermal conductivity enhancements(positive and negative) that agree with (8.1). In a direct experimental investigation,the density effects associated with the postulated nanoconvection [23] were ex-perimentally tested with lighter silica and Teflon particles, and were shown to beincorrect [28]. The nanoconvection velocities were further shown to be of the or-der of thermophoretic velocities, which for most nanofluids were insignificant (aslow as 10−9 m/s). Even for sub-nanometer clusters, as evidenced from moleculardynamics simulations, the thermophoretic velocities are exceedingly small, and thiseffectively precludes a discernible contribution to the nanofluid thermal conductiv-ity from any conceivable nanoconvection mechanism [29].

8.2.2 Interfacial Liquid Layer

Considering that the molecular structure of liquid at the solid interface is more or-dered, possibilities of larger thermal conductivity of this ordered liquid layer and‘tunneling’ of heat carrying phonons from one particle to another were put forward[20]. The subsequent molecular dynamics simulation work concluded that thosemechanisms do not contribute significantly to heat transfer [30]. For strong solid–liquid interactions, typical of those in nanofluids with metallic nanoparticles, a per-colating network of amorphous-like fluid structures can emerge which can facilitateadditional thermal conduction paths [31, 32]. However, a discernible increase inthermal conductivity is possible only for exceedingly small colloidal particles (lim-ited to few tens of atoms). Experiments have shown that the structured interfacialfluid layers are limited to a few molecular spacings from a solid surface [33], whichmakes them inadequate to influence the thermal transport in common nanofluids.Furthermore, interpretation of the cooling rates of Au nanoparticles suspended inwater and organic solvents does not appear to require any unusual thermophysicalproperties of the surrounding liquid to explain the experimental results [34].

216 Pawel Keblinski

8.2.3 Interfacial Thermal Resistance

As described above, the liquid layering does not generate thermal conductivity in-creases. There is, however, a well-known interfacial effect degrading thermal con-ductivity: the thermal interfacial resistance Rk, most commonly defined via its in-verse, the interfacial conductance G = 1/Rk. G is related to the heat flux JQ and thetemperature drop ΔT at the interface via

JQ = GΔT . (8.2)

A simple measure of the relative importance of the interfacial resistance in the over-all heat flow in a composite can be obtained from the equivalent thickness h, definedas the distance over which the temperature drop is the same as at the interface. Thisthickness is given by the ratio of the fluid thermal conductivity Λfluid to the interfa-cial conductance, i.e., h =Λfluid/G.

The interfacial thermal resistance effect on well-dispersed spherical particle com-posites can be estimated by effective medium theory to be [15]

ΛNF

Λfluid−1 = 3φ

γ−1γ+ 2

, (8.3)

where γ is the ratio of the particle radius to the equivalent matrix thickness h. Ac-cording to (8.3), when the particle radius becomes equal to the equivalent matrixthickness (γ = 1), there is no enhancement at all, while for larger interfacial resis-tance (γ < 1), the addition of particles decreases the thermal conductivity of thecomposite. This effect can explain the decrease of thermal conductivity of nanoflu-ids below the base fluid value [18, 19] and much lower than expected thermal con-ductivity increases of carbon nanotube nanofluids and carbon nanotube polymercomposites [35, 36].

8.2.4 Near Field Radiation

Somewhat less popular was the idea of anomalously high radiative energy transfersbetween nanoparticles [37]. Since the classical radiation theory does not predictradiative transfer of significance for nanofluids, molecular dynamics simulations ofthermal energy exchange between two silica nanoparticles were carried out, result-ing in the prediction of enormous thermal conductance due to near-field interactionswhen particles were closely separated [37]. However, the thermal conductance ob-tained from molecular dynamics was significantly larger than that obtained underthe assumption that all the thermal energy is exchanged between two nanoparticleswithin a single atomic vibration period. Further analysis demonstrated that near-field interactions do not affect the nanofluid thermal conductivity [38].

8 Thermal Conductivity of Nanofluids 217

8.2.5 Particle Clustering

Maxwell’s expression, the limiting case of which is given by (8.1), corresponds tothe situation of well-dispersed nanoparticles. Such structure is the least efficientfrom the point of view of thermal transport enhancement since conductive particlesare separated from each other by low conductivity liquid. Aggregation of particlesinto sparse clusters, or ideally into linear chains, leads to extended and highly con-ductive paths for the heat flow. In fact, there is a well established understanding ofthe impact of morphology on thermal conductivity of a composite. Depending onthe morphology, the conductivity can vary greatly, even at the same volume fractionof the components. It is generally accepted that the conductivity has to fall betweenso-called Hashin–Shtrikman (HS) bounds, obtained under an effective medium anal-ysis [39] without any restriction on the volume fraction. The HS bounds are givenby

Λfluid

[1+

3φ(ΛNP−Λfluid)3Λfluid +(1−φ)(ΛNP−Λfluid)

]≤ΛNF ≤

[1− 3(1−φ)(ΛNP−Λfluid)

3ΛNP−φ(ΛNP−Λfluid)

]ΛNP .

(8.4)In the inequality given by (8.4), the lower (Maxwell) bound corresponds physicallyto a set of well-dispersed nanoparticles in a fluid matrix, while the upper limit cor-responds to large pockets of fluid separated by linked, chain-forming or clusterednanoparticles. From the point of view of circuit analysis, the lower HS limit liescloser to conductors connected in series, while the upper limit lies closer to con-ductors in a parallel mode. The HS bounds do not give a precise mechanism ofthermal conductance, but set the most restrictive limits based on knowledge of thevolume fraction alone. It is relatively well-known that a large number of experi-mental data on solid composites, and to a lesser extent the data on liquid mixtures,fall between the HS bounds [29]. An unbiased estimate (favoring neither series norparallel modes) predicts thermal conductivity values that lie between the upper andlower HS bounds [40].

The evidence from scanning electron microscopy (SEM) points to the existenceof partial clustering and chain-like linear aggregation [10–12, 41]. Viscosity dataon nanofluids has also shown an anomalous increase as compared to the Einsteinmodel for well dispersed particles [42]. A large increase in the viscosity is anotherindication of aggregation in the nanofluids. Interestingly, for a carbon nanotube sus-pension, an effective medium theory accounting for the very high aspect ratio of thenanotubes [14], predicts thermal conductivity enhancements that are in fact wellabove the reported values. This was attributed to a significant interfacial resistanceto heat flow between the carbon nanotubes and the fluid [35].

218 Pawel Keblinski

8.3 Discussion

Clear experimental evidence for extensive clustering observed in nanofluids indi-cates that the expected thermal conductivity can fall within a wide range boundedby (8.4) and strongly depends on the actual aggregation. Recently, using a multi-level effective medium theory, it was demonstrated that the thermal conductivity ofnanofluids can be significantly enhanced by the aggregation of nanoparticles intochain-like clusters, and this enhancement can be quite dramatic for large, but sparseclusters [43]. Predictions of this aggregation model were in excellent agreementwith detailed numerical calculations on model nanofluids involving fractal clus-ters [43]. Thus, allowing for clustering effects dramatically broadens the thermalconductivity range that is consistent with the effective medium theory [29]. Quitestrikingly, it was demonstrated that the vast majority of experimental results fallwithin these bounds, supporting the classical nature of thermal conduction behaviorin nanofluids [44].

The increase in the relative thermal conductivity of nanofluids with tempera-ture [4] is another example of anomalous behavior that cannot be explained on theassumption of well-dispersed particles. Similarly, the increase in the thermal con-ductivity with decreasing particle size cannot be explained if the particles are welldispersed. The probability of aggregation increases with increasing temperature anddecreasing particle size [45]. Therefore the apparent contradictions between exper-iment and theory, such as particle size effects, can be resolved by weighing in theability of nanoparticles to form linear clusters. Furthermore, the temperature depen-dence is not as strong as it was previously believed to be, with recent experimentsshowing a similar variation for both nanofluids and the base fluid [19, 29]. Thisimplies that the mechanism for increase in the thermal conductivity of water (pre-sumably from the hydrogen bonded structures) is partly responsible for the thermalconductivity increase in nanofluids as well. Conversely, it is reasonable to expect adecrease in the nanofluid thermal conductivity for a base fluid that has a negativechange in conductivity with increasing temperature.

It remains a challenge to accurately identify and manipulate the cluster config-uration to modify the thermal transport properties of a nanofluid. The two charac-terization techniques, DLS and SEM, have limitations in assessing the structure ofnanoparticles. DLS measurements are limited to dilute suspensions (φ < 1%) formost nanofluids, while SEM imaging can be performed only after drying the basefluid. While the science of making well-dispersed colloids has reached a fair levelof maturity, attempts to generate targeted nanoparticle configurations are still in anevolving phase.

The fact that significant aggregation is required to obtain substantial increases inthermal transport has an important consequence for the potential application of suchfluids in flow-based cooling, which is the most important benefit from the techno-logical point of view. It is well known that aggregation into sparse but large clustersincreases fluid viscosity. Such increases become very dramatic when the aggregatesstart to touch each other, which can occur at very low volume fractions, as low as0.2% [46]. Therefore, the same aggregate structures that are most beneficial to the

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thermal transport are also the most detrimental to the fluid flow characteristics. Fu-ture research should therefore address the issue of optimizing nanofluid structurewith the best combination of thermal conductivity and viscosity.

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