Enhancing surface heat transfer by carbon nanofins: towards an alternative to nanofluids?
 Eliodoro Chiavazzo^{1} and
 Pietro Asinari^{1}Email author
DOI: 10.1186/1556276X6249
© Chiavazzo and Asinari; licensee Springer. 2011
Received: 23 November 2010
Accepted: 22 March 2011
Published: 22 March 2011
Abstract
Background
Nanofluids are suspensions of nanoparticles and fibers which have recently attracted much attention because of their superior thermal properties. Nevertheless, it was proven that, due to modest dispersion of nanoparticles, such high expectations often remain unmet. In this article, by introducing the notion of nanofin, a possible solution is envisioned, where nanostructures with high aspectratio are sparsely attached to a solid surface (to avoid a significant disturbance on the fluid dynamic structures), and act as efficient thermal bridges within the boundary layer. As a result, particles are only needed in a small region of the fluid, while dispersion can be controlled in advance through design and manufacturing processes.
Results
Toward the end of implementing the above idea, we focus on single carbon nanotubes to enhance heat transfer between a surface and a fluid in contact with it. First, we investigate the thermal conductivity of the latter nanostructures by means of classical nonequilibrium molecular dynamics simulations. Next, thermal conductance at the interface between a single wall carbon nanotube (nanofin) and water molecules is assessed by means of both steadystate and transient numerical experiments.
Conclusions
Numerical evidences suggest a pretty favorable thermal boundary conductance (order of 10^{7} W·m^{2}·K^{1}) which makes carbon nanotubes potential candidates for constructing nanofinned surfaces.
Background and motivations
Nanofluids are suspensions of solid particles and/or fibers, which have recently become a subject of growing scientific interest because of reports of greatly enhanced thermal properties [1, 2]. Filler dispersed in a nanofluid is typically of nanometer size, and it has been shown that such nanoparticles are able to endow a base fluid with a much higher effective thermal conductivity than fluid alone [3, 4]: significantly higher than those of commercial coolants such as water and ethylene glycol. In addition, nanofluids show an enhanced thermal conductivity compared to theoretical predictions based on the Maxwell equation for a welldispersed particulate composite model. These features are highly favorable for applications, and nanofluids may be a strong candidate for new generation of coolants [2]. A review about experimental and theoretical results on the mechanism of heat transfer in nanofluids can be found in Ref. [5], where those authors discuss issues related to the technology of nanofluid production, experimental equipment, and features of measurement methods. A large degree of randomness and scatter has been observed in the experimental data published in the open literature. Given the inconsistency in these data, we are unable to develop a comprehensive physicalbased model that can predict all the experimental evidences. This also points out the need for a systematic approach in both experimental and theoretical studies [6].
In particular, carbon nanotubes (CNTs) have attracted great interest for nanofluid applications, because of the claims about their exceptionally high thermal conductivity [7]. However, recent experimental findings on CNTs report an anomalously wide range of enhancement values that continue to perplex the research community and remain unexplained [8]. For example, some experimental studies showed that there is a modest improvement in thermal conductivity of water at a high loading of multiwalled carbon nanotubes (MWCNTs), approximately of 35% increase for a 1 wt% MWNT nanofluid [9]. Those authors attribute the increase to the formation of a nanotube network with a higher thermal conductivity. On the contrary, at low nanotube content, <0.03 wt%, they observed a decrease in thermal conductivity with an increase of nanotube concentration. On the other hand, more recent experimental investigations showed that the enhancement of thermal conductivity as compared with water varied linearly when MWCNT weight content was increased from 0.01 to 3 wt%. For a MWNT weight content of 3 wt%, the enhancement of thermal conductivity reaches 64% of that of the base fluid (e.g., water). The average length of the nanotubes appears to be a very sensitive parameter. The enhancement of thermal conductivity compared with water alone is enhanced when nanotube average length is increased in the 0.55 μ m range [10].
This article investigates a possible implementation of the above idea using CNTs, because of their unique geometric features (slimness) and thermophysical properties (high thermal conductivity). CNTs have attracted the attention of scientific community, since their mechanical and transport (both electrical and thermal) properties were proven to be superior compared with traditional materials. This observation has motivated intensive theoretical and experimental efforts during the last decade, toward the full understanding/exploitation of these properties [12–16]. Despite these expectations, however, it is reasonable to say that these efforts are far from setting out a comprehensive theoretical framework that can clearly describe these phenomena. First of all, the vast majority of CNTs (mainly multiwalled) exhibits a metallic behavior, but the phonon mechanism (lattice vibrations) of heat transfer is considered the prevalent one close to room temperature [17, 18]. The phonon mean free path, however, is strongly affected by the existence of lattice defects, which is actually a very common phenomenon in nanotubes and closely linked to manufacturing methods. Second, there is the important issue of quantifying the interface thermal resistance between a nanostructure and the surrounding fluid, which affects the heat transfer and the maximum efficiency. It is noted that, according to the classical theory, there is an extremely low thermal resistance when one reduces the characteristic size of the thermal "antenna" promoting heat transfer [19], as confirmed by numerical investigations for CNTs [20–22].
This article investigates, by molecular mechanics based on force fields (MMFF), the thermal performance of nanofins made of single wall CNTs (SWCNTs). The SWCNTs were selected mainly because of time constraints of our parallel computational facilities. The following analysis can be split into two parts. First of all, the heat conductivity of SWCNTs is estimated numerically by both simplified model (section "Heat conductivity of singlewall carbon nanotubes: a simplified model", where this approach is proved to be inadequate) and a detailed threedimensional model (section "Heat conductivity of singlewall carbon nanotubes: detailed three dimensional models"). This allows one to appreciate the role of model dimensionality (and harmonicity/anharmonicity of interaction potentials) in recovering standard heat conduction (i.e., Fourier's law). This first step is used for validation purposes in a vacuum and for comparison with results from literature. Next, the thermal boundary conductance between SWCNT and water (for the sake of simplicity) is computed by two methods: the steadystate method (section "Steadystate simulations"), mimicking ideal cooling by a strong forced convection (thermostatted surrounding fluid), and the transient method (section "Transient simulations"), taking into account only atomistic interactions with the local fluid (defined by the simulation box). This strategy allows one to estimate a reasonable range for the thermal boundary conductance.
Heat conductivity of SWCNTs: a simplified model
with k_{b}, T_{0}, N_{f}, and τ_{T} denoting the Boltzmann constant, the thermostat temperature, number of degrees of freedom, and relaxation time, respectively, while the auxiliary variable ξ is typically referred to as friction coefficient[31]. NoséHoover thermostatting is preferred since it is deterministic and it typically preserves canonical ensemble. However, we notice that (5) represent the equations of motions with a single thermostat. In this case, it is known that the latter scheme may run into ergodicity problems and thus fail to generate a canonical distribution. Although stochastic thermostats (see, e.g., Andersen [32]) are purposely devised to generate a canonical distribution, they are characterized by a less realistic dynamics. Hence, to the end of overcoming the above issues, using deterministic approaches, Martyna et al. have introduced the idea of NoséHoover chain [33] (see also [34, 35] for the equation of motion of NoséHoover chains and further details on thermostats in molecular dynamics simulations). Simulations presented below were carried out using both a single thermostat and a NoséHoover chain (with two thermostats), and no differences were noticed.
The above simplified model has been tested in a range of low temperature (300 K < T < 1000 K), where we notice that it is not suitable to predict normal heat conduction (Fourier's law). In other words, at steady state (i.e., when heat flux is uniform along the chain and constant in time) is observed a finite heat flux although no meaningful temperature gradient could be established along the chain (see Figure 2). Thus, the above results predict a divergent heat conductivity. In this context, it is worth stressing that onedimensional lattices with harmonic potentials are known to violate Fourier's law, and they exhibit a flat temperature profile (and divergent heat conductivity). On the one hand, the results of the simplified model in Figure 2 are likely due to a nonsufficiently strong anharmonicity. Indeed, as reported on the righthand side of Figure 3, the Morse function (1) can be safely approximated by an harmonic potential in the range of maximal deviation x observed at low temperature (T < 1000 K), namely, V_{b} (x) ≈ V_{0} (x^{2}/a^{2}  1).
On the other hand, it is worth stressing that it has been demonstrated that anharmonicity alone is insufficient to ensure normal heat conduction [23], in onedimensional lattice chains.
Heat conductivity of SWCNTs: detailed threedimensional models
Carboncarbon interactions  

 47890 kJ·mol^{1}·nm^{2} 
 562.2 kJ·mol^{1} 
 25.12 kJ·mol^{1} 
ϵ_{CC}  0.4396 kJ·mol^{1} 
σ _{CC}  3.851 Å 
Carbonoxygen interactions  
ϵ_{CO}  0.3126 kJ·mol^{1} 
σ _{CO}  3.19 Å 
Oxygenoxygen interactions  
ϵ_{OO}  0.6502 kJ·mol^{1} 
σ _{OO}  3.166 Å 
Oxygenhydrogen interactions  
q _{O}  0.82e 
q _{H}  0.41 e 
where the cross section S_{A} is defined as S_{A} = 2πrb, with b = 0.34 nm denoting the van der Waals thickness (see also [43]). In this case, the use of formula (12) is particularly convenient since the quantity 〈ξ〉 can be readily extracted from the output files in GROMACS.
Summary of the results of MD simulations in this study.
Chirality, case  Box  L _{NH}  L  λ  α _{st}  α _{tr}  τ _{d}  mL/2 

(nm^{3})  (nm)  (nm)  W·m^{2}·K^{1}  W·m^{2}·K^{1}  W·m^{2}·K^{1}  (ps)  
(5, 5), BADLJ (vac)  12 × 12 × 12  1.5  5.5  67         
(5, 5), BwADLJ (vac)  12 × 12 × 12  1.5  5.5  64         
(5, 5), BAD (vac)  12 × 12 × 12  1.5  5.5  65         
(5, 5), BA (vac)  12 × 12 × 12  1.5  5.5  49         
(5, 5), BwA (vac)  12 × 12 × 12  1.5  5.5  48.9         
(5, 5), BADLJ (vac)  20 × 20 × 20  2  10  96.9         
(5, 5), BADLJ (vac)  105 × 105 × 105  25  25  216.1         
(5, 5), BADLJ (sol)  2.5 × 2.5 × 14  2  10    5.18 × 10^{7}      0.28 
(5, 5), BADLJ (sol)  4 × 4 × 14  2  10    5.18 × 10^{7}      0.28 
(5, 5), BADLJ (sol)  4 × 4 × 14  0  14      1.70 × 10^{7}  33   
(5, 5), BADLJ (sol)  5 × 5 × 5  0  3.7      1.37 × 10^{7}  41   
(15, 0), BADLJ (sol)  5 × 5 × 5  0  4.7      1.60 × 10^{7}  35   
(15, 0), BADLJ (sol)  5 × 5 × 5  0  3.8      1.43 × 10^{7}  39   
(3, 3), BADLJ (sol)  5 × 5 × 5  0  3.7      8.90 × 10^{6}  63   
Thermal boundary conductance of a carbon nanofin in water
Steadystate simulations
where ε_{0} is the permittivity in a vacuum, while q_{ i } and q_{ j } are the partial charges with q_{O} = 0.82 e and q_{H} = 0.41 e (see also [41]).
Transient simulations
In our computations, based on [20], we considered the heat capacity per unit area of an atomic layer of graphite c_{T} = 5.6 × 10^{4} (J·m^{2}·K^{1}).
and λ_{w} is the thermal conductivity of the medium, while the square box has dimensions w × w × L. Let us consider the following example, corresponding to the row '(5, 5), BADLJ (sol)' in Table 2. Assuming λ_{w} = 0.58 (W·m^{1}·K^{1}), D = 0.68 nm, w = 4 nm, it yields α_{csf} = 9.2 × 10^{8} W m^{2} K^{1}. The analytic results are even larger than those obtained by the steadystate simulation (usually larger than those obtained by the transient method). Moreover, the continuumbased formula prescribes that thermal conductance (weakly) diverges by reducing the cylinder diameter. On the contrary, MD simulations is in line with the expectation of a bounded thermal boundary conductance. In fact, in agreement with others [45], we even observe a slight decrease with the tube diameter.
We point out that neither the steadystate method nor the transient method fully reproduce the setup described by the analytic formula (23). In fact, in the steadystate method, the entire water bath is thermostatted (while in the analytic formula, only the water boundaries are thermostatted) and, in the transient method, the water temperature changes in time (while the analytic formula is derived under steadystate condition). Nevertheless, from the technological point of view, the above results are in line with the basic idea that high aspectratio nanostructures (such as CNTs) are suitable candidates for implementing the above idea of nanofin, and thus can be utilized for exploiting advantageous heat boundary conductances.
Conclusions
In this study, we first investigated the thermal conductivity of SWCNTs by means of classical nonequilibrium MD using both simplified onedimensional and fully threedimensional models. Next, based on the latter results, we have focused on the boundary conductance and thermal efficiency of SWCNTs used as nanofins within water. More specifically, toward the end of computing the boundary conductance α, two different approaches have been implemented. First, α = α_{st} was estimated through a fitting procedure of results by steadystate MD simulations and a simple onedimensional continuous model. Second, cooling of SWNT (at T_{CNT} ) within water (at T_{w}) was accomplished by NVE simulations. In the latter case, the time constant τ_{d} of the temperature difference (T_{CNT}  T_{w}) dynamics enables us to compute α = α_{tr}. Numerical computations do predict pretty high thermal conductance at the interface (order of 10^{7} W·m^{2}·K^{1}), which indeed makes CNTs ideal candidates for constructing nanofins. We should stress that, consistently with our results α_{st}> α_{tr}, it is reasonable to expect that α_{st} represents the upper limit for the thermal boundary conductance, because (in steadystate simulations) water is forced by the thermostat to the lowest temperature at any time and any position in the computational box. Finally, it is worthwhile stressing that, following the suggestion in [46], all the results of this study can be generalized to different fluids using standard nondimensionalization techniques, upon a substitution of the parameterization (ϵ_{CO}, σ_{CO}) representing a different LennardJones interaction between SWNT and fluid molecules.
Methods
The CNTs geometries simulated in this article were generated using the program Tubegen [39], while water molecules were introduced using the SPC/E model implemented by the genbox package available in GROMACS [38]. Numerical results in this study are based on nonequilibrium MD where the allatom forcefields OPLSAA is adopted for modeling atom interactions. Visualization of simulation trajectories is accomplished using VEGA ZZ [47].
Abbreviations
 CNTs:

carbon nanotubes
 GROMACS:

GROningen MAchine for Chemical Simulations
 MD:

molecular dynamics
 MMFF:

molecular mechanics based on force fields
 MWCNTs:

multiwalled carbon nanotubes
 NEMD:

nonequilibrium molecular dynamics
 SWCNTs:

single wall CNTs
 SWNT:

single wall nanotube.
Declarations
Acknowledgements
The above research has received funding from the European Community Seventh Framework Program (FP7 20072013) under grant agreement N. 227407Thermonano. The authors owe their appreciation to Mr. Marco Giardino for his kind assistance whenever the authors had difficulites with computational facilities. The authors also thank Dr. Andrea Minoia and Dr. Thomas Moore for the fruitful discussions with them on the usage of GROMACS in simulating carbon nanotubes. The authors acknowledge also the inspiring discussions with Dr. JeanAntoine Gruss (CEA DTS/LETH, France) about CNTbased nanofluids.
Authors’ Affiliations
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