Toward nanofluids of ultrahigh thermal conductivity
 Liqiu Wang†^{1}Email author and
 Jing Fan†^{1}
DOI: 10.1186/1556276X6153
© Wang and Fan; licensee Springer. 2011
Received: 6 December 2010
Accepted: 18 February 2011
Published: 18 February 2011
Abstract
The assessment of proposed origins for thermal conductivity enhancement in nanofluids signifies the importance of particle morphology and coupled transport in determining nanofluid heat conduction and thermal conductivity. The success of developing nanofluids of superior conductivity depends thus very much on our understanding and manipulation of the morphology and the coupled transport. Nanofluids with conductivity of upper HashinShtrikman (HS) bound can be obtained by manipulating particles into an interconnected configuration that disperses the base fluid and thus significantly enhancing the particlefluid interfacial energy transport. Nanofluids with conductivity higher than the upper HS bound could also be developed by manipulating the coupled transport among various transport processes, and thus the nature of heat conduction in nanofluids. While the direct contributions of ordered liquid layer and particle Brownian motion to the nanofluid conductivity are negligible, their indirect effects can be significant via their influence on the particle morphology and/or the coupled transport.
Introduction
Nanofluids are a new class of fluids engineered by dispersing nanometersize structures (particles, fibers, tubes, droplets, etc.) in base fluids. The very essence of nanofluids research and development is to enhance fluid macroscopic and systemscale properties through manipulating microscopic physics (structures, properties, and activities) [1, 2]. One of such properties is the thermal conductivity that characterizes the strength of heat conduction and has become a research focus of nanofluid society in the last decade [1–9].
The importance of highconductivity nanofluids cannot be overemphasized. The success of effectively developing such nanofluids depends very much on our understanding of mechanism responsible for the significant enhancement of thermal conductivity. Both static and dynamic reasons have been proposed for experimental finding of significant conductivity enhancement [1–9]. The former includes the nanoparticle morphology [10, 11] and the liquid layering at the liquidparticle interface [12–17]. The latter contains the coupled (cross) transport [18–20] and the nanoparticle Brownian motion [21–26]. Here, the effect of particle morphology contains those from the particle shape, connectivity among particles (including and generalizing the nanoparticle clustering/aggregating in the literature [10, 11]), and particle distribution in nanofluids. This short review aims for a concise assessment of these contributions, thus identifying the future research needs toward nanofluids of high thermal conductivity. The readers are referred to, for example, [1–9] for stateoftheart expositions of major advances on the synthesis, characterization, and application of nanofluids.
Static mechanisms
Morphology
The nanoparticle morphology in nanofluids can vary from a welldispersed configuration in base fluids to a continuous phase of interconnected configuration. Such a morphology variation will change nanofluid's effective thermal conductivity significantly [27–32], a phenomenon credited to the particle clustering/aggregating in the literature [1–9]. This appears obvious because the nanofluid's effective conductivity stems mainly from the contribution of continuous phase that constitutes the continuous path for thermal flow [27, 28]. Although particle clustering/aggregating offers a way of changing particle morphology, it is not necessarily an effective means. The research should thus focus not only on the clustering/aggregating, but also on the general ways of varying morphology.
Here k_{p}, k_{f}, and k_{e} are the conductivities of particle, base fluid, and nanofluid, respectively, and φ is the particle volume fraction. For the case of k_{p}/k_{f} ≥1, Equations (1) and (2) give the lower and the upper bounds for nanofluid effective thermal conductivity, corresponding to the two limiting morphologies where the liquid serves as the continuous phase for the lower bound and the particle disperses the liquid for the upper bound, respectively. When k_{p}/k_{f} ≤1, their roles are interchanged, so that Equations (1) and (2) provide the upper and the lower bounds, respectively. Therefore, the upper bound always takes a configuration (morphology) where the continuous phase is made of the higherconductivity material.
The morphology dependence of nanofluid's conductivity has been recently examined in detail by either of the two approaches: the constructal approach [1, 2, 29–32] and the scalingup by the volume average [1, 2, 27, 28]. Such studies not only confirm the features captured in the HS bounds but also uncover the microscopic mechanism responsible for the morphology dependence of nanofluid's conductivity. As higherconductivity particles interconnect each other and disperse the lowerconductivity base fluid into a dispersed phase, the interfacial energy transport between particle and base fluid becomes enhanced significantly such that the nanofluid's conductivity takes its value of upper HS bound (Fan J and Wang LQ: Heat conduction in nanofluids: structureproperty correlation, submitted).
As k_{p}/k_{f} moves away from the unity along both directions, the separation between the upper and lower HS bounds becomes pronounced (Figures 1 and 2) so that the room for manipulating nanofluid conductivity via changing the particle morphology becomes more spacious. The HS bounds are respected by some nanofluids for which their thermal conductivity is strongly dependent on particle morphology, such as whether nanoparticles stay welldispersed in the base fluid, form aggregates, or assume a configuration of continuous phase that disperses the fluid into a dispersed phase (Figure 1). There are thermal conductivity data that fall outside the HS bounds (Figures 1 and 2).
Ordered liquid layer
Both experimental and theoretical evidences have been reported of the presence of ordered liquid layer near a solid surface by which the atomic structure of the liquid layer is significantly more ordered than that of bulk liquid [64–67]. For example, two layers of icelike structures are experimentally observed to be strongly bounded to the crystal surface on a crystalwater interface, followed by two diffusive layers with less significant ordering [65]. Three ordered water layers have also been observed numerically on the Pt (111) surface [64].
where N_{a} is the Avogadro's number, and ρ_{f} and M_{f} are the density and the molecular weight of base fluids, respectively. The liquid layer thickness is thus 0.28 nm for waterbased nanofluids, which agrees with that from experiments and molecular dynamic simulation on the order of magnitude.
Dynamic mechanisms
Coupled transport
In a nanofluid system, normally, there are two or more transport processes that occur simultaneously. Examples are the heat conduction in dispersed phase, heat conduction in continuous phase, mass transport, and chemical reactions either among the nanoparticles or between the nanoparticles and the base fluid. These processes may couple (interfere) and cause new induced effects of flows occurring without or against its primary thermodynamic driving force, which may be a gradient of temperature, or chemical potential, or reaction affinity. Two classical examples of coupled transport are the Soret effect (also known as thermodiffusion or thermophoresis) in which directed motion of particles or macromolecules is driven by thermal gradient and the Dufour effect that is an induced heat flow caused by the concentration gradient.
where T is the temperature; subscripts β and σ refer to the β and σ phases, respectively. γ_{ β }= (1  φ)(ρc)β and γ_{ σ }= φ(ρc)_{ σ }are the effective thermal capacities of β and σ phases, respectively, with ρ and c as the density and the specific heat. φ is the volume fraction of the σ phase. h and a_{ υ }come from modeling of the interfacial flux and are the film heat transfer coefficient and the interfacial area per unit volume, respectively. k_{ ββ }and k_{ σσ }are the effective thermal conductivities of the β and σ phases, respectively; k_{ βσ }and k_{ σβ }are the coupling (cross) effective thermal conductivities between the two phases.
Equation (12) is not a classical heatconduction equation, but can be regarded as a dualphaselagging (DPL) heatconduction equation with ((k_{ βσ }k_{ σβ } k_{ ββ }k_{ σσ })/(ha_{ υ }))Δ^{2}T_{ i }as the DPL sourcerelated term $F(\text{r},t)+{\tau}_{q}\frac{\partial F(\text{r},t)}{\partial t}$ and with τ_{ q }and τ_{ T }as the phase lags of the heat flux and the temperature gradient, respectively [2, 18, 69]. Here, F(r, t) is the volumetric heat source. k, ρc, and α are the effective thermal conductivity, capacity and diffusivity of nanofluids, respectively.
Therefore, by the condition for the existence of thermal waves that requires τ_{ T }/τ_{ q }< 1 [18, 70], thermal waves may be present in nanofluid heat conduction.
Note also that, for heat conduction in nanofluids, there is a timedependent source term F(r, t) in the DPL heat conduction (Equations (12) and (13)). Therefore, the resonance can also occur. When k_{ βσ }= k_{ σβ }= 0 so that τ_{ T }/τ_{ q }is always larger than 1, thermal waves and resonance would not appear. Therefore, the coupled transport could change the nature of heat conduction in nanofluids from a diffusion process to a wave process, thus having a significant effect on nanofluid heat conduction.
Therefore, the cross coupling between the heat conduction in the fluid and particle manifests itself as thermal waves at the macroscale. Depending on factors such as material properties of nanoparticles and base fluids, nanoparticles' geometrical structure and their distribution in the base fluids, and interfacial properties and dynamic processes on particlefluid interfaces, the crosscouplinginduced thermal waves may either enhance or counteract with the moleculardynamicsdriven heat diffusion. Consequently, the heat conduction may be enhanced or weakened by the presence of nanoparticles. This explains the thermal conductivity data that fall outside the HS bounds (Figures 1 and 2).
This can be regarded as a DPL heatconduction equation regarding Δu with τ_{ q }, τ_{ T }, and $F(\text{r},t)+{\tau}_{q}\frac{\partial F(\text{r},t)}{\partial t}$ as the phase lags of the heat flux and the temperature gradient, and the sourcerelated term, respectively. Therefore, the coupled heat and mass transport is capable of varying not only thermal conductivity from that in Equation (13) to the one in Equation (21) but also the nature of heat conduction from that in Equation (12) to the one in Equation (19). As practical nanofluid system always involves many transport processes simultaneously, the coupled transport could play a significant role. For assessing its effect and understanding heat conduction in nanofluids, future research is in great demand on coupling (cross) transport coefficients that are derivable by approaches like the upscaling with closures [2, 27, 28], the kinetic theory [71, 72], the timecorrelation functions [73, 74], and the experiments based on phenomenological flux relations [68]. While the uncoupled form of conservation equations, such as Equations (12) and (19), is very useful for examining nature of heat transport, its coupled form, such as Equations (8), (9), (16)(18), is normally more readily to be resolved for the temperature or concentration fields after all the transport coefficients are available.
Brownian motion
where subscript BC refers to the Brownianmotioninduced convection, and α_{f} is the thermal diffusivity of the base fluid.
Consider a 1% volume fraction of d_{p} = 10 nm copper nanoparticle in water suspension at T = 300 K. (ρc)_{P} = 8900 kg/m^{3} × 0.386 kJ/(kg K) = 3435.4 kJ/(m^{3} K), μ = 0.798 × 10^{3}kg/(ms), k_{f} = 0.615 W/(mK), and α_{f} = 1.478 × 10^{7} m^{2}/s. These yield k_{BD}/k_{f} = 3.076 × 10^{6} and k_{BC}/k_{f} = 3.726 × 10^{4}. Therefore, both contributions are negligibly small.
Although the direct contribution of particle Brownian motion to the nanofluid conductivity is negligible, its indirect effect could be significant because it plays an important role in processes of particle aggregating and coupled transport.
Concluding remarks
Under the specified volume fractions and thermal conductivities of the two phases in the colloidal state, the interfacial energy transport between the two phases favors a configuration in which the higherconductivity phase forms a continuous path for thermal flow and disperses the lowerconductivity phase. The effective thermal conductivity is thus bounded by those corresponding to the two limiting morphologies: the welldispersed configuration of the higherconductivity phase in the lowerconductivity phase and the welldispersed configuration of the lowerconductivity phase in the higherconductivity phase, corresponding to the lower and the upper bounds of thermal conductivity, respectively. Without considering the effect of interfacial resistance and cross coupling among various transport processes, the classical effectivemedium theory gives these bounds known as the HS bounds. A wide separation of these two bounds offers spacious room of manipulating nanofluid thermal conductivity via the morphology effect.
In a nanofluid system, there are normally two or more transport processes that occur simultaneously. The cross coupling among these processes causes new induced effects of flows occurring without or against its primary thermodynamic driving force and is capable of changing the nature of heat conduction via inducing thermal waves and resonance. Depending on the microscale physics (factors like material properties of nanoparticles and base fluids, nanoparticles' morphology in the base fluids, and interfacial properties and dynamic processes on particlefluid interfaces), the heat diffusion and thermal waves may either enhance or counteract each other. Consequently, the heat conduction may be enhanced or weakened by the presence of nanoparticles.
The direct contributions of ordered liquid layer and particle Brownian motion to the nanofluid conductivity are negligible. Their influence on the particle morphology and/or the coupled transport could, however, offer a strong indirect effect to the nanofluid conductivity.
Therefore, nanofluids with conductivity of upper HS bound can be obtained by manipulating particles into an interconnected configuration that disperses the base fluid, and thus significantly enhancing the particlefluid interfacial energy transport. Nanofluids with conductivity higher than the upper HS bound could also be developed by manipulating the cross coupling among various transport processes and thus the nature of heat conduction in nanofluids.
Notes
Abbreviations
 DPL:

dualphaselagging
 HS:

HashinShtrikman.
Declarations
Acknowledgements
The financial support from the Research Grants Council of Hong Kong (GRF718009 and GRF717508) is gratefully acknowledged.
Authors’ Affiliations
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