# A Novel Method to Determine the Thermal Conductivity of Interfacial Layers Surrounding the Nanoparticles of a Nanofluid

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Viscosity of Nanofluids

_{r}is the relative viscosity of a suspension defined as the ratio of suspension viscosity to the base fluid viscosity and φ is the volume fraction of particles. The Einstein equation is accurate only for small values of φ, on the order of 2% or so [17]. However, even very dilute nanofluids with φ ≤ 0.02 do not obey the Einstein equation. The Einstein equation severely underpredicts the nanofluid viscosity. It has been shown by Pal [14] that the viscosity of nanofluids, in general, is strongly influenced by the “solvation” and “clustering” of nanoparticles. The thermal conductivity of nanofluids is also significantly affected by the “solvation” and “clustering” of nanoparticles [18]. In very dilute nanofluids, however, only the “solvation effect” is present with negligible or no clustering of nanoparticles. Due to solvation of particles, the effective volume fraction of the dispersed-phase (φ

_{eff}) is significantly larger than the actual volume fraction (φ) of nanoparticles (un-solvated). The relationship between φ

_{eff}and φ is given as:

_{o}is the radius of the dry (un-solvated) nanoparticle; δ is the thickness of the interfacial solvation layer; and k

_{s}is the solvation coefficient. Note that k

_{s}= (1 + δ/R

_{o})

^{3}.

_{r}versus φ plot at φ → 0 (φ ≤ 0.02). Alternatively, [η] can be determined from the slope of 1/ η

_{r}versus φ plot at φ → 0. Note that the Einstein equation could be re-written as:

_{r}versus φ data in the limit of φ → 0 is −[η]. Once the intrinsic viscosity is known, the solvation coefficient k

_{s}of a nanofluid can be determined from Equation (6).

#### 2.2. Thermal Conductivity of Nanofluids

_{m}is the thermal conductivity of the matrix; K

_{d}is the thermal conductivity of particles and φ is the volume fraction of particles. This equation could be re-written as:

_{r}is the relative thermal conductivity defined as K / K

_{m}; and λ is the thermal conductivity ratio defined as K

_{d}/ K

_{m}. Equation (9) follows from the following Maxwell–Eucken equation for the thermal conductivity of suspensions [20]:

_{3}, a shell (solvated layer) of thermal conductivity K

_{2}and a matrix fluid of thermal conductivit K

_{1}. The exact expression for the relative thermal conductivity of an infinitely dilute suspension of core-shell particles is given as [25]:

_{s}is the solvation coefficient, defined earlier as k

_{s}= (1 + δ/R

_{o})

^{3}, where R

_{o}is the un-solvated particle radius and δ is the interfacial nanolayer thickness. Equation (11) is accurate for small values of φ, on the order of 5% or so [25], and it could be expressed in terms of φ

_{eff}(= k

_{s}φ) as:

_{31}= K

_{3}/ K

_{1}, λ

_{21}= K

_{2}/ K

_{1}and λ

_{32}= K

_{3}/ K

_{2}. Upon further rearrangement, Equation (13) could be expressed as:

_{s}= 1, B = 3(K

_{3}/ K

_{1}), A = 3, φ

_{eff}= φ, and consequently, Equation (14) reduces to Equation (8). Note that K

_{d}= K

_{3}and K

_{m}= K

_{1}.

_{d,core-shell}) to the matrix thermal conductivity, that is:

_{r}versus φ plot at φ → 0 (φ < 0.05). Alternatively, [K] can be determined from the slope of 1/ K

_{r}versus φ plot at φ → 0. Note that Equation (19) could be re-written as:

_{r}versus φ data in the limit of φ → 0 is −[K].

## 3. Estimation of Interfacial Layer Thermal Conductivity

_{eff}/ φ is the solvation coefficient k

_{s}, Equation (21) becomes:

_{2}) from Equation (22). Equation (22) could be re-cast as follows:

_{2}, as all other quantities ([K], k

_{s}, K

_{1}and K

_{3}) are known. Note that the solvation coefficient k

_{s}is known from the intrinsic viscosity through Equation (6). In the absence of any solvation of particles, k

_{s}= 1, and the intrinsic thermal conductivity [K] reduces to:

_{2}can be calculated from the following expression:

## 4. Simulation Results

_{1}= 0.6 W / m ⋅K, K

_{3}= 12 W / m ⋅K, [η] = 5 and [K] = 4. In this case, the enhancement of the thermal conductivity of the nanofluid is more than that predicted from the Maxwell equation (Equation (24)), which gives [K] of 2.59. With the given values of K

_{1}, K

_{3}, [η], and [K], we obtain the following results: k

_{s}= 2, a = −6.66667, b = −90, c = 168, K

_{2}= 1.662 W / m ⋅K and K

_{2}/ K

_{1}= 2.77. Thus, the thermal conductivity of the interfacial layer of nanoparticles is 2.77-times the thermal conductivity of the base (matrix) fluid.

_{1}= 0.258 W / m ⋅K, K

_{3}= 400 W / m ⋅K, [η] = 10 and [K] = 8. In this case, the enhancement of thermal conductivity of nanofluid is much more than that predicted from the Maxwell equation (Equation (24)), which gives a [K] of 2.994. With the given values of K

_{1}, K

_{3}, [η], and [K], we obtain the following results: k

_{s}= 4, a = −93.023, b = −36,957.3, c = 33,600, K

_{2}= 0.907254 W / m ⋅K and K

_{2}/ K

_{1}= 3.52. Thus, the thermal conductivity of the interfacial layer of nanoparticles is 3.52-times the thermal conductivity of the base (matrix) fluid.

## 5. Experimental Validation

_{2}of the interfacial layer of nanoparticles: K

_{1}, K

_{3}, [η], and [K]. For a given nanofluid, K

_{1}(base fluid) and K

_{3}(unsolvated nanoparticles) are known. The determination of intrinsic viscosity [η] requires experimental viscosity data on a dilute nanofluid at low values of nanoparticle concentration. Likewise the determination of intrinsic thermal conductivity [K] requires experimental thermal conductivity data on a dilute nanofluid at low values of nanoparticle concentration.

_{1}and K

_{3}are 0.258 and 400 W / m ⋅K, respectively. The data are plotted as 1/ η

_{r}versus φ and as 1/ K

_{r}versus φ. The plots are linear over the φ range of 0 to 0.02. The slope of 1/ η

_{r}versus φ plot is −9 and the slope of 1/ K

_{r}versus φ plot is −5.75. It should be noted that these nanofluids are Newtonian in nature with the negligible dependence of viscosity on the shear rate. The shear rate range covered in the experiments is 3–3,000 s

^{−1}. The linear dependence of η

_{r}on φ and the lack of shear dependence of viscosity (Newtonian behavior) suggest that clustering of nanoparticles is negligible in these nanofluids over the φ range of 0 to 2%. The clustering of nanoparticles is expected to impart non-Newtonian shear-thinning behavior to nanofluids [14]. The dependence of η

_{r}on φ is also expected to be non-linear [14].

_{1}, K

_{3}, [η], and [K], we obtain the following results: k

_{s}= 3.6, a = −101.783, b = −43,662.1, c = 23,192, K

_{2}= 0.5305 W / m ⋅K and K

_{2}/ K

_{1}= 2.056. Thus, the thermal conductivity of the interfacial layer of copper nanoparticles in Cu-EG nanofluid is 2.056-times that of the thermal conductivity of the base (matrix) fluid.

_{2}-EG nanofluid), based on the experimental data of Chen et al. [11]. The thermal conductivity is measured at 20 °C, and the viscosity is measured over a temperature range of 20–60 °C. The relative viscosity is observed to be independent of the temperature. The un-solvated TiO

_{2}nanoparticles are 25 nm in diameter, and the values of K

_{1}and K

_{3}are 0.256 and 8.5 W / m ⋅K, respectively. The data are plotted as 1/ η

_{r}versus φ and as 1/ K

_{r}versus φ. The plots are linear over the φ range of 0 to 0.018. The slope of 1/ η

_{r}versus φ plot is −10.5, and the slope of 1/ K

_{r}versus φ plot is −6.7. Thus, the values of [η] and [K] for TiO

_{2}-EG nanofluid are 10.5 and 6.7, respectively. Equation (24) gives a [K] of 2.74. Obviously, the actual enhancement of thermal conductivity of the nanofluid ([K]) is substantially larger than that predicted from the Maxwell equation (Equation (24)). It should be noted that these nanofluids are Newtonian in nature with the negligible dependence of viscosity on the shear rate. The shear rate range covered in the experiments is 0.5–10,000 s

^{−1}. The linear dependence of η

_{r}on φ and the lack of shear dependence of viscosity (Newtonian behavior) suggest that the clustering of nanoparticles is negligible in these nanofluids over the φ range of 0 to 1.8%. For the given values of K

_{1}, K

_{3}, [η], and [K], we obtain the following results: k

_{s}= 4.2, a = −147.5, b = −964.81, c = 707.2, K

_{2}= 0.6653 W / m ⋅K and K

_{2}/ K

_{1}= 2.6. Thus, the thermal conductivity of the interfacial layer of copper nanoparticles in TiO

_{2}-EG nanofluid is 2.6-times that of the thermal conductivity of the base (matrix) fluid.

_{matrix}to 2.5 K

_{matrix}.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Das, S.K.; Choi, S.U.S.; Yu, W.; Pradeep, T. Nanofluids; Wiley: New York, NY, USA, 2008; pp. 1–37. [Google Scholar]
- Wang, Y.; Zheng, Y.; Zhang, L.; Wang, Q.; Zhang, D. Stability of nanosuspensions in drug delivery. J. Control. Release
**2013**, 172, 1126–1141. [Google Scholar] [CrossRef] [PubMed] - Mahbubul, I.M.; Saidur, R.; Amalina, M.A. Latest developments on the viscosity of nanofluids. Int. J. Heat Mass Transf.
**2012**, 55, 874–885. [Google Scholar] [CrossRef] - Duan, F.; Kwek, D.; Crivoi, A. Viscosity affected by nanoparticle aggregation in Al
_{2}O_{3}-water nanofluids. Nanoscale Res. Lett.**2011**, 6, 248–252. [Google Scholar] [CrossRef] [PubMed] - Xu, J.; Yang, B.; Hammouda, B. Thermal conductivity and viscosity of self-assembled alcohol/polyalphaolefin nanoemulsion fluids. Nanoscale Res. Lett.
**2011**, 6, 274–279. [Google Scholar] [CrossRef] [PubMed] - Lee, S.W.; Park, S.D.; Kang, S.; Bang, I.C.; Kim, J.H. Investigation of viscosity and thermal conductivity of SiC nanofluids for heat transfer applications. Int. J. Heat Mass Transf.
**2011**, 54, 433–438. [Google Scholar] [CrossRef] - Namburu, P.K.; Kulkarni, D.P.; Misra, D.; Das, D.K. Viscosity of copper oxide nanoparticles dispersed in ethylene glycol and water mixture. Exp. Therm. Fluid Sci.
**2007**, 32, 397–402. [Google Scholar] [CrossRef] - Nguyen, C.T.; Desgranges, F.; Galanis, N.; Roy, G.; Mare, T.; Boucher, S. Viscosity data for Al
_{2}O_{3}-water nanofluid-hysteresis: Is heat transfer enhancement using nanofluids reliable? Int. J. Therm. Sci.**2008**, 47, 103–111. [Google Scholar] [CrossRef] - Wang, X.Q.; Mujumdar, A.S. Heat transfer characteristics of nanofluids: A review. Int. J. Therm. Sci.
**2007**, 46, 1–19. [Google Scholar] [CrossRef] - Wang, X.Q.; Mujumdar, A.S. A review on nanofluids. Braz. J. Chem. Eng.
**2008**, 25, 613–630. [Google Scholar] - Chen, H.; Ding, Y.; He, T.; Tan, C. Rheological behavior of ethylene glycol based titania nanofluids. Chem. Phys. Lett.
**2007**, 444, 333–337. [Google Scholar] [CrossRef] - Nguyen, C.T.; Desgranges, F.; Roy, G.; Galanis, N.; Mare, T.; Boucher, S.; Angue Mintsa, H. Temperature and particle-size dependent viscosity data for water-based nanofluids—Hysteresis phenomenon. Int. J. Heat Fluid Flow
**2007**, 28, 1492–1506. [Google Scholar] [CrossRef] - Yu, W.; Choi, S.U.S. The role of interfacial layers in the enhanced thermal conductivity of nanofluids: A renovated Maxwell model. J. Nanoparticle Res.
**2003**, 5, 167–171. [Google Scholar] [CrossRef] - Pal, R. New models for the viscosity of nanofluids. J. Nanofluids
**2014**, 3, 260–266. [Google Scholar] [CrossRef] - Einstein, A. Eine neue Bestimmung der Molekuldimension. Ann. Phys. (Leipzig)
**1906**, 19, 289–306. (in German). [Google Scholar] [CrossRef] - Einstein, A. Berichtigung zu meiner Arbeit: Eine neue Bestimmung der Molekuldimension. Ann. Phys. (Leipzig)
**1911**, 34, 591–592. (in German). [Google Scholar] [CrossRef] - Batchelor, G.K.; Green, J.T. The determination of the bulk stress in a suspension of spherical particles to order c
^{2}. J. Fluid Mech.**1972**, 56, 401–427. [Google Scholar] [CrossRef] - Okeke, G.; Witharana, S.; Antony, S.J.; Ding, Y. Computational analysis of factors influencing thermal conductivity of nanofluids. J. Nanoparticle Res.
**2011**, 13, 6365–6375. [Google Scholar] [CrossRef] - Pal, R. New models for thermal conductivity of particulate composites. J. Reinf. Plast. Compos.
**2007**, 26, 643–651. [Google Scholar] [CrossRef] - Pal, R. Electromagnetic, Mechanical, and Transport Properties of Composite Materials; CRC Press: Boca Raton, FL, USA, 2014; pp. 261–264. [Google Scholar]
- Yu, W.; Choi, S.U.S. The role of interfacial layers in the enhanced thermal conductivity of nanofluids: A renovated Hamilton-Crosser model. J. Nanoparticle Res.
**2004**, 6, 355–361. [Google Scholar] [CrossRef] - Liang, Z.; Tsai, H.L. Thermal conductivity of interfacial layers in nanofluids. Phys. Rev.
**2011**, 83, 0416021–0416028. [Google Scholar] - Yu, C.J.; Richter, A.G.; Datta, A.; Durbin, M.K.; Dutta, P. Molecular layering in a liquid on a solid substrate. Phys. B
**2000**, 283, 27–31. [Google Scholar] [CrossRef] - Okeke, G.; Hammond, R.B.; Joseph Antony, S. Influence of size and temperature on the phase stability and thermophysical properties of anatase TiO
_{2}nanoparticles: Molecular dynamics simulation. J. Nanoparticle Res.**2013**, 15, 1584–1592. [Google Scholar] [CrossRef] - Garboczi, E.J.; Schwartz, L.M.; Bentz, D.P. Modeling the influence of the interfacial zone the dc electrical conductivity of mortar. Adv. Cem. Based Mater.
**1995**, 2, 169–181. [Google Scholar] [CrossRef] - Garg, J.; Poudel, B.; Chiesa, M.; Gordon, J.B.; Ma, J.J.; Wang, J.B.; Ren, Z.F.; Kang, Y.T.; Ohtani, H.; Nanda, J.; et al. Enhanced thermal conductivity and viscosity of copper nanoparticles in ethylene glycol nanofluid. J. Appl. Phys.
**2008**, 103, 074301–074306. [Google Scholar] [CrossRef]

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**MDPI and ACS Style**

Pal, R. A Novel Method to Determine the Thermal Conductivity of Interfacial Layers Surrounding the Nanoparticles of a Nanofluid. *Nanomaterials* **2014**, *4*, 844-855.
https://doi.org/10.3390/nano4040844

**AMA Style**

Pal R. A Novel Method to Determine the Thermal Conductivity of Interfacial Layers Surrounding the Nanoparticles of a Nanofluid. *Nanomaterials*. 2014; 4(4):844-855.
https://doi.org/10.3390/nano4040844

**Chicago/Turabian Style**

Pal, Rajinder. 2014. "A Novel Method to Determine the Thermal Conductivity of Interfacial Layers Surrounding the Nanoparticles of a Nanofluid" *Nanomaterials* 4, no. 4: 844-855.
https://doi.org/10.3390/nano4040844