Effect of interfacial thermal resistance and nanolayer on estimates of effective thermal conductivity of nanofluids interfacial thermal resistance and nanolayer on estimates of effective thermal conductivity of nanofluids, Case Studies in Thermal Engineering,

Colloidal suspensions of nanoparticles (nanoﬂuids) are materials of interest for thermal engineering, because their heat transfer properties are typically enhanced as compared to the base ﬂuid one. Eﬀective medium theory provides popular models for estimating the overall thermal conductivity of nanoﬂuids based on their composition. In this article, the accuracy of models based on the Bruggeman approximation is assessed. The sensitivity of these models to nanoscale interfacial phenomena, such as interfacial thermal resistance (Kapitza resistance) and ﬂuid ordering around nanoparticles (nanolayer), is considered for a case study consisting of alumina nanoparticles suspended in water. While no signiﬁcant diﬀerences are noticed for various thermal conductivity proﬁles in the nanolayer, a good agreement with experiments is observed with Kapitza resistance ≈ 10 − 9 m 2 K/W and sub-nanometer nanolayer thickness. These results conﬁrm the classical nature of thermal conduction in nanoﬂuids and highlight that future studies should rather focus on a better quantiﬁcation of Kapitza resistance at nanoparticle-ﬂuid interfaces, in order to allow bottom up estimates of their eﬀective thermal conductivity.


Introduction
Since the first report on their peculiar thermal conductivity in 1995, thermophysical properties of colloidal suspensions of nanoparticles (nanofluids) have been widely investigated in the biomedical and engineering fields [1].
In particular, suspending thermally conductive nanoparticles in conventional fluids with the aim of improving their heat transfer properties has been among the most investigated and controversial research areas [18][19][20]. Researchers have studied experimental, semi-empirical and theoretical models for the thermal conductivity of nanofluids, which is typically enhanced respect to the base fluid one [21][22][23][24][25]. While the classical nature of thermal conduction in nanofluids forces their thermal conductivities to fall between lower and upper Maxwell bounds for homogeneous systems [26], a general model accommodating the numerous experimental evidences has been under debate for more than two decades. In particular, classic Effective Medium Theories (EMTs), such as Maxwell-Garnett (MG) [27] or Bruggeman (BR) [28] approximations, have been progressively amended to include the nanoscale effects at nanoparticle-fluid interface, as well as the nanoparticle size, shape and aggregation [13].
In this work, we analyze two nanoscale phenomena involved in the effective thermal conductivity of nanofluids, namely interfacial thermal resistance (Kapitza resistance) and fluid ordering around nanoparticles (nanolayer). The sensitivity of BR approximation to these interfacial effects is systematically evaluated. Results show that Kapitza resistance plays a significant role in determin-ing the effective thermal conductivity of nanofluids; whereas, the approximation employed for the thermal conductivity profile within the nanolayer has not a sensible effect. Moreover, the influence of nanolayer on the effective thermal conductivity appears as negligible if realistic values of nanolayer thickness are considered, namely less than 1 nm. This analysis suggests that future studies on thermal properties of nanofluids should focus on a better quantification of Kapitza resistance at the nanoparticle-fluid interface.

Methods
Several models based on EMT have been proposed to predict the thermal conductivity of nanofluids. In particular, Bruggeman approximation predicts the effective thermal conductivity of homogeneous suspensions as: where φ is the particle volume fraction, while λ p , λ f and λ ef f are the particle, base fluid and effective thermal conductivity of nanofluid, respectively. The Bruggeman approximation is particularly suitable for nanosuspensions with unbiased configuration, namely a mix of linearly aggregated and well-dispersed nanoparticles [26]. <20% mismatch) in the range φ = 0.001-3% [29].

Nanolayer
The nanolayer is a structured layer of fluid molecules at the interface with nanoparticle surface, and it generally shows properties different from the bulk fluid ones [30][31][32]. In particular, in case of hydrophilic nanoparticles immersed in aqueous media, the average thermal conductivity of nanolayer (λ l ) has reportedly higher values respect to the base fluid one (see Fig. 1) [33]. The nanolayer nanoparticle (λ f = 0.60 W/m·K, λp = 35 W/m·K [29], rp = 20 nm and t = 0.30 nm [26,30]).
The thermal conductivity profiles in the nanolayer predicted by Eqs. 6-9 are compared.
typically shows sub-nanometer thickness, namely a few layers of water molecules in the proximity of nanoparticle surface [34,35].
Several models have been proposed to capture the effect of nanolayer on the thermal conductivity of nanofluids. For example, the EMT model modified by Yu and Choi accounts for the effect of liquid layering on the thermal conductivity of nanofluids [36]. In this model, nanoparticle (radius: r p ) and the surrounding nanolayer (thickness: t) are treated as a single particle with an equivalent radius equal to r p + t. The resulting equivalent volume concentration (φ e ) is thus evaluated as where δ = t rp is the ratio between nanolayer thickness and particle radius. The thermal conductivity of equivalent particles (λ pe ) is subsequently derived from effective medium theory as being γ =λ l λp . The BR model in Eq. 1 can be then modified as Since the average thermal conductivity of nanolayer should present values higher than base fluid one and possibly lower than that of the particle (λ p ≥ λ l ≥ λ f ), a continuous thermal conductivity profile (λ l (r)) has been typically hypothesized within the nanolayer (r p ≤ r ≤ r p + t) [37], andλ l computed as Different thermal conductivity profiles in the nanolayer have been proposed in the literature. For instance, Xie et al. [37] investigated the effect of a linear λ l (r) profile, namely Jiang et al. [38], instead, introduced a cubic polynomial model for the nanolayer thermal conductivity: whereas, Kotia et al. [39] a logarithmic one Finally, Pasrija et al. [40] proposed a exponential profile for λ l (r), that is where m is a real positive value (m = 2 [40]). Considering Eqs. 6-9, Fig. 1 compares the different thermal conductivity profiles in a representative case study, which is made of an alumina nanoparticle immersed in water and surrounded by water nanolayer.

Kapitza resistance
The Kapitza resistance at nanoparticle-fluid interface also influences the effective thermal conductivity of nanofluids [41,42]. Such interfacial thermal resistance arises from the phonon scattering due to acoustic mismatch at the interface of dissimilar materials (e.g. solid-liquid phases). Kapitza resistance can be expressed as where ∆T is the temperature jump at the interface generated by a specific heat flux q.
To take into account this additional resistance to heat transfer in nanoparticle suspensions, the thermal conductivity of nanoparticles can be modified where R k refers to the nanoparticle-fluid interface [43]. The thermal conductivity of equivalent particles (λ pe , Eq. 3) can be then computed using λ * p instead of λ p ; finally, BR model (Eq. 4) can be adopted to estimate λ ef f .

Sensitivity of thermal conductivity to nanolayer and Kapitza resistance
The sensitivity of EMT-based thermal conductivity models to nanolayer thermal conductivity and Kapitza resistance has been then assessed (see Tab. 1 for a detailed list of tested models). A water-alumina nanofluid has been considered as a case study (r p = 20 nm, λ p = 35 W/m·K, λ f = 0.60 W/m·K [29]). On the one hand, the influence of different λ(r) profiles on λ ef f is shown in Fig. 2a. As generally predicted by EMT approximations, effective thermal conductivity increases with nanoparticle volume fraction. In accordance with experimental and numerical studies in the literature [26,30], the nanolayer ≤ r p ≤ 30 nm, namely t = 2.55 nm [44]. Under such assumption, λ ef f appears to be significantly enhanced by nanolayer effect (up to 4.1% increase), while no relevant discrepancies between the different λ l (r) profiles are still observable (less than 0.6% differences). However, nanolayer thicknesses larger than 1 nm are at variance with consolidated experimental and numerical evidences [26].
On the other hand, Fig. 2b

Experimental validation of EMT models
Experimental data from the literature are then used to assess the accuracy of effective thermal conductivity predictions by EMT-based models [45,[47][48][49][50][51][52][53][54][55]. The detailed list of considered experimental data is given in Tab. 2. As a first approximation, t = 0.30 nm [26,30] and R k = 0.2 × 10 −8 m 2 K/W (fitting value adopted in reference [45]) are taken as nanolayer thickness and Kapitza resistance at the alumina-water interface, respectively. Nanofluids with nanoparticles characterized by a diameter approximately equal to 39 nm are initially considered. In Fig. 3

Optimal nanolayer thickness and Kapitza resistance
Since both interfacial thermal resistance [57][58][59] and nanolayer [30,60,61] at the nanoscale interface between solid and liquid phase have been widely observed by experiments and simulations, the BR-LIN-RK model should -in principlebest represent the heat transfer mechanisms determining λ ef f . Therefore, the significant discrepancy between BR-LIN-RK model and experiments in Fig. 3 may be due to sub-optimal estimations of R k , t or both. The sensitivity analysis reported in Fig. 4 indeed shows that, with proper combinations of R k and t values, the BR-LIN-RK model could potentially achieve an accurate match (up to R 2 =0.87) of the experimental results in Fig. 3. For example, a large coefficient of determination (R 2 =0.86) is observed with R k = 5.0×10 −9 m 2 K/W and t = 0.35 nm, namely typical values of Kapitza resistance [59,62] and nanolayer thickness [30] observed in case of other metal oxide-water interfaces under fully hydrated conditions.
The latter values are then employed to evaluate the applicability of these EMT-based models with other nanoparticles sizes. Results in Fig. 5 illustrate that a better match between experiments and models is generally observed with larger particles (Figs. 5b and c), while an higher variability can be noticed with the smaller ones (Fig. 5a). In the latter case, the nanolayer extension becomes comparable with nanoparticle diameter, therefore determining an increased influence of interfacial phenomena on λ ef f ; hence, small variations in the value of nanolayer thickness or Kapitza resistance may cause a large scattering of results.
Nonetheless, the linear fitting (i.e. the average value) of experimental results in Fig. 5a presents a difference less than 0.8% with respect to the BR-LIN-RK model.
Hence, models based on effective medium theory that account for nanoscale thermal transport phenomena (interfacial thermal resistance and nanolayer) have the potential to provide good approximations of the effective thermal conductivity of suspensions of nanoparticles with a broad range of diameters.
However, further researches should be devoted to measure, both numerically

Conclusions
Enhanced physical properties of nanofluids have led to their widespread exploitation in different fields, for instance the energy, mechanical, automotive, and biomedical ones. In particular, the thermal conductivity of such nanoparticle suspensions is generally improved respect to the base fluid one. However, nanoscale characteristics and the resulting macroscopic properties have to be better understood to achieve a more rational design of nanosuspensions.
In this study, different models based on Bruggeman approximation have been compared with experimental data from the literature, in order to assess the sen-