1 Introduction

Anomalous transport properties of nanofluids (suspensions of nanoparticles in liquids) have become a subject of intense research recently [1–4]. Most efforts have been focused on so-called anomalous heat transfer enhancement, which means an increase in thermal conductivity beyond the predictions of the classical effective medium theory [5]. Other transport properties, such as tracer diffusion coefficient [6], have also been reported as anomalously high, and a connection between the heat- and mass-transport enhancement has been proposed in microconvection-based theories [1, 7–9]. A large body of literature also reports that addition of nanoparticles to liquids can cause interfacial mass-transport rates, such as the rate of gas sorption into liquid under convection [10–14], to increase several times. As this phenomenon is more complex than transport in the bulk of stagnant liquid, and may involve contributions from interfacial area, hydrodynamic boundary layer thickness, adsorption of the gas on the nanoparticles, etc., it seems appropriate to investigate first the intrinsic mass-transport in nanofluids in absence of convective, interfacial and other complicating effects.

The goals of the present study were to determine the thermal conductivity and the solvent self-diffusion coefficient (SDC) of silica nanofluids studied previously in the context of thermal conductivity enhancement [15, 16]. The thermal conductivity was measured using the transient hot wire method employed in our earlier work on alumina nanofluids [4]. For the measurement of the proton SDC, we used pulsed field gradient nuclear magnetic resonance (PFG NMR) method. This is a well-established technique for diffusion studies in liquids in the time scale from a few milliseconds to a few seconds [17, 18], and it has been applied successfully to study both the solvent self-diffusion [19–21] as well as the nanoparticle diffusion [22–24] in colloidal suspensions.

2 Experimental

The samples used in our study were Ludox brand aqueous suspension of colloidal silica (Grace Davison). According to the manufacturer and previous studies [16], they contain quasi-monodisperse non-agglomerated spherical nanoparticles suspended in a dilute (mM) aqueous carbonate buffer at pH 9. Additional data on the structure of the nanofluids are provided in our forthcoming publication [25]. The properties of the stock solutions are summarized in Table 1. We would like to note here that we calculated the volume fraction of nanoparticles on the basis of the experimentally determined mass fraction of nanoparticles and density of nanofluid rather than using assumptions about the density of nanoparticles. Although the nanoparticle densities calculated this way are noticeably higher (2.51 g/cm3 in for SM, 2.36 for HM and TM, and 2.33 for INPBE) than those assumed in earlier studies (2.2 g/cm3 in Ref. [16]), the resulting differences in volume fractions are insignificant (≤1%). The stock solutions were diluted, if needed, using ultrapure water produced with Nanopure Diamond apparatus (Barnstead) with the resistivity >18.2 M Ω cm−1 and <2 ppb of total organic carbon.

Table 1 Properties of Ludox brand colloidal silica suspensions
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For the thermal conductivity measurements we used a transient-hot wire thermal property analyzer (KD2pro from Decagon Devices Inc.). The details of our experimental set up can be found in Ref. [4] The values we obtained for thermal conductivity of pure water [0.608 ± 0.002 W/(m K) at 20 ± 0.5°C] are slightly higher that the reference value [0.597 ± 0.001 W/(m K)] [26]. Based on our extensive earlier studies we attribute this discrepancy to a convective effect in water. Such effects are expected to be smaller in nanofluids which have higher viscosities. At any rate, the magnitude of the discrepancy even in water is within 1.5% and it is smaller than the effects on which this work is focused.

The NMR measurements have been performed at 20±0.5°C (unless stated otherwise) on water protons using Varian Unity/Inova 500 MHz spectrometer equipped with narrow-bore Varian XYZ gradient probe with a maximum gradient strength 28 G/cm in X or Y direction. To minimize convection in the liquid samples, 1.6 mm inner diameter NMR tubes were used. The height of the liquid samples was 15 mm which is within the region of a uniform magnetic field gradient of the probe.

The SDC was measured by means of Stejskal–Tanner sequence (90°–τ–180°–τ echo) with PFGs [27]. The main magnetic field was in the Z direction (vertically). Pulsed gradients of duration δ = 2 ms were applied perpendicular to the main magnetic field after τ = 50 ms the first 90° (4.1 μs) and second 180° (8.2 μs) pulses. Thus, the peaks of the gradient pulses are separated by the diffusion time Δ = 100 ms. The gradients, G, were varied in five to ten steps up to the gradient of 17.5 G/cm. We did not use the maximum gradient to avoid the occurrence of temperature gradients and a mechanical “ringing” of coils. A delay of 10 s was applied before every pulse sequence to allow for a full equilibration of the protons polarization (in our samples T 1 = T 2 ∈ [0.5 s; 2.0 s]). In the experiments dealing with the time dependence of the diffusion coefficient, Δ was varied between 10 and 750 ms.

The NMR signal, A, was obtained by integrating the Fourier transform of the echo signal. Signal attenuation due to unrestricted random walk diffusion is given by the Stejskal–Tanner formula [27]

$$ \ln \left( {{\frac{A}{{A_{0} }}}} \right) = - \gamma^{2} G^{2} D\delta^{2} \left( {\Updelta - {\frac{\delta }{3}}} \right), $$
(1)

where D and γ are the SDC and the gyromagnetic ratio of the protons, A 0 is A at G = 0. In order to obtain an adequate signal to noise ratio in the measurements of D, we averaged A values of 16 scans cycling the phases of 90° and 180° pulses. A versus G data were repeated for each sample at least five times on different days and the standard deviations of D shown in this work reflect this repeatability. Additional experiments, where G was varied keeping Gδ constant, proved the heating of the gradient coils was not the limiting factor in the precision of these measurements. In our opinion, the precision (and accuracy) of D is determined by the long-term stability of the temperature control, which in turn influences the convection in the samples. All error bars and uncertainties for experimental values in this paper refer to the standard deviation estimated from a series of consecutive measurements rather than to a standard error of the mean.

3 Results and discussion

3.1 Thermal conductivity of nanofluids

Figure 1 displays the results of our thermal conductivity measurement of different colloidal silica suspensions. The dependence of normalized thermal conductivity k nf /k 0, where k nf and k 0 are the thermal conductivity of the base fluid (water) and of the nanofluid, respectively, on the volume fraction of colloidal particles, ϕ, is given in the effective medium theory by Eq. 2 [1, 4]:

Fig. 1
figure 1

Normalized thermal conductivity of different colloidal silica suspensions at 20°C plotted versus the volume fraction of the nanoparticles (squares). The dashed line is the best linear fit with a fixed intercept of 1. The solid line is calculated based on Eq. 2. The standard deviation in k nf /k 0 is approximately the same as the height of the symbols used

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$$ {\frac{{k_{nf} }}{{k_{0} }}} = 1 + 3{\frac{{k_{1} /k_{0} - 1}}{{k_{1} /k_{0} + 2}}}\phi . $$
(2)

In the latter formula, k 1 is the thermal conductivity of the material of the colloidal particles, and this equation is strictly applicable only to spherical particles. The slope for the dependence shown in Fig. 1 predicted on the basis on Eq. 3 is 0.92 if the values of k 0 = 0.605 and k 1 = 1.4 W/(m K) are used as in Ref. [16] The experimental results show a smaller slope, i.e. 0.53 ± 0.02. The most likely reason for the thermal conductivity enhancement being below the prediction of Eq. 2 is the presence of Kapitza (heat-transfer) resistance on the interface between silica particles and water, as was noticed in our earlier paper dealing with alumina nanofluids [4]. In this case, the appropriate formula is [1, 4]

$$ {\frac{{k_{nf} }}{{k_{0} }}} = 1 + 3{\frac{{k_{1} /k_{0} - 1}}{{k_{1} /k_{0} + 2 + 2k_{1} /(R\beta )}}}\phi , $$
(3)

where R is the radius on the suspended particles and β is the interfacial thermal conductance. The introduction of the Kapitza resistance into consideration may explain the increase in the normalized conductivity with increase in the silica particle size reported in Ref. [16]. We believe that data in Fig. 1 are not sufficiently accurate to draw reliable conclusions on the particle size effect. Instead, we estimate the upper and lower limits of β assuming 5 and 15 nm particle radii, respectively. Such analysis leads to β = (0.7–2.2) × 108 K/W, which has the same order of magnitude as the value we found earlier for the alumina/water interface (5 × 108 K/W) [4].

3.2 Solvent self-diffusion coefficient in nanofluids

Pulsed field gradient nuclear magnetic resonance diffusion measurements with pure water at 20 ± 0.5°C yielded the values for SDC of 1H as 2.00 ± 0.08 × 10−9 m2 s−1 which is close to the literature values obtained with the same technique, i.e. 2.025 × 10−9 m2 s−1 [28]. In following studies we focused on 1H SDC measurements with Ludox TM samples at different dilutions.

For all studied samples and diffusion times the dependence of the echo amplitude, A, on the square of the magnetic field gradient, G 2, followed a single exponential decay predicted by (1) in the whole range of the gradients studied (1.5 decades in A, see Fig. 2). No time dependence of D was observed for any of the samples in the range of 10 ms ≤ Δ ≤ 750 ms (see Fig. 3) within the precision of our experiment (relative standard deviation of 4%).

Fig. 2
figure 2

Stejskal–Tanner plots for Ludox TM colloidal silica at different volume fraction of the nanoparticles (filled squares 0%, open squares 3.8%, filled circles 11.5%, open circles 15.3%, filled triangles 23%). The standard deviation in ln(A/A 0) is smaller than the height of the symbols used

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Fig. 3
figure 3

Time dependence of the 1H SDC of selected samples (filled squares ϕ = 0%, filled circles ϕ = 11.5%, filled triangles ϕ = 23%) as determined by PFG NMR. These data were obtained at 25°C and they show a bigger scatter that we observed at 20°C in other experiments because of the convection induced by temperature control. The solid lines and the numbers is boxes show the average value of D for given ϕ

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At short times Δ, normal diffusion, with the solvent SDC value the same as for pure solvent, can be expected. At longer times, when the distance that the solvent molecules travel within the duration of the experiment, Δ, becomes comparable to the distance between the obstructive particles, d, a third regime, restricted diffusion, with the solvent SDC decreasing with time, is expected to show up. For particles with R = 10 nm, the values of d are 18 and 66 nm for ϕ = 0.22 and ϕ = 0.04, respectively, if the hexagonal close packing is assumed for calculating d, i.e. \( d = \left\{ {\sqrt[3]{{{\frac{8\pi }{3\sqrt 3 \phi }}}} - 2} \right\}R. \) Therefore, the transition to restricted diffusion regime should start around ΔRD = 27 ns for ϕ = 0.22 and 363 ns for ϕ = 0.04, respectively. Finally, at even longer times Δ, when the solvent molecules travel distances significantly larger than the distance between the suspended nanoparticles, the volume-averaged solvent SCD reaches a constant lower value determined by the excluded volume of the obstructive phase [29]. Since PFG NMR diffusivity measurements are limited at present to the experiments with Δ orders of magnitude larger than ΔRD calculated above, we expect to work in the excluded volume diffusion regime. It should be noted here that the mean square displacement of the solvent molecule changes linearly with time only under the normal and excluded volume diffusion regimes, and that under the intermediate restricted diffusion regime Eq. 1 is not valid.

Under the excluded volume diffusion regime, one can expect a reduction in the value of D with the increase of the volume fraction of the obstructive phase (nanoparticles), i.e. the behavior we found experimentally (see Fig. 4). The simplest theory (spherical particles in spherical cells) gives the following result for the obstruction (excluded volume) effect [29]:

$$ {\frac{D}{{D_{0} }}} = {\frac{1}{1 + 0.5\phi }} $$
(4)

where D 0 is the solvent SDC in pure solvent, D is the macroscopic (volume averaged) solvent SDC in the suspension, and ϕ is the volume fraction of the particles. Corrections to this formula within a lattice model have been developed in Ref. [20] but they are important only for ϕ > 0.5 and can be neglected here. The data in Fig. 4 show that the reduction of the D upon increase in ϕ is larger than predicted by Eq. 4. Particularly interesting is the fact that (4) predicts the limit of D/D 0 = 2/3 at ϕ = 1, but experimentally this value of D/D 0 is reached at ϕ = 0.25. Thus, we turned our attention to other factors which may contribute to a larger-than-expected decrease of the macroscopic solvent SDC in the nanofluids.

Fig. 4
figure 4

Dependence of the 1H SDC (normalized to SDC in pure water, D 0) in Ludox TM nanofluids on the volume fraction of nanoparticles (filled squares) and the best fits from different models (see text): line 1 (solid) Y = 1/(1 + 0.5X); line 2 (dash) Y = exp{−7.5X 2}; line 3 (dot) Y = (1 − 1.5X)/{(1 − X)(1 + 0.5 × 1.5X)}

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Because Ludox TM colloids are known to consist of fairly monodisperse spherical particles and because the shape of D versus ϕ for spheroidal particles is predicted to be convex [29] rather than concave as found in this work (Fig. 4) we did not consider the particle shape effects. This exclusion is further validated by the fact that if a fitting using Eq. (1.3) from Ref. [20] is attempted, an unreasonable value of the form factor (0.75) is obtained. Also, the polydispersity and disorder of the position of nanoparticles is expected to increase rather than decrease D [29].

On the other hand, silica is well known for its hygroscopic properties; therefore, the adsorption of water onto (and even absorption into) the nanoparticles resulting in the partition of the solvent between fast “free” water and slow “bound” fractions should be considered.

The obstructive effects of dissolved polymers on solvent SDC have been modeled using (5) [30]. This model also worked well for other colloidal suspensions with solvent penetrating the particles [21]. This formula gives a better fit to our data (with α = 7.5 and ν = 2) but the physical meaning of the adjustable parameters remains elusive [21].

$$ {\frac{D}{{D_{0} }}} = \exp \left( { - \alpha \phi^{\nu } } \right) $$
(5)

A better fit can also be obtain using Eq. 6 derived under assumption of solvent layering around the particle [19]:

$$ {\frac{D}{{D_{0} }}} = {\frac{{1 - k^{3} \phi }}{{\left( {1 - \phi } \right)\left( {1 + 0.5k^{3} \phi } \right)}}}, $$
(6)

where k is the ratio of hydrodynamic particle radius to the radius of the solid particle core. In our case, the fitting gives k = 1.51/3 = 1.145. This corresponds to a water layer of 1.67 nm, or six water molecules (assuming 0.275 nm as water molecule radius) [31]. Other formulas derived for layering type of corrections (Eqs. 28 and 32 in Ref. [29]) did not produce a better fit or required unphysical values of adjustable parameters.

Our observation of the reduction of the solvent diffusion coefficient with the increase in the nanoparticle volume fraction, and, more specifically, a reduction larger can predicted by the simple excluded volume model, agrees with recent data obtained with aqueous suspension of alumina particles of somewhat larger size (50 nm) [32].

4 Conclusion

We measured thermal conductivity and solvent SDC in well-characterized quasi-monodisperse suspensions of silica nanoparticles in water. We did not find anomalous enhancement in either property. In fact, the observed values the thermal conductivities and solvent SDC as a function of the volume fraction of nanoparticles are below the predictions of the simple effective medium theory; and the deviations can be explained by interaction of the water with the silica particles. The results of our combined heat- and mass-transport study agree with recent experimental [4, 32, 33] and theoretical [4, 34] findings that disprove the existence of anomalous enhancement in the transport properties of nanofluids.