Characterization of nanofluids by multifractal analysis of a liquid droplet trace


The base material was a commercially available hydrophilic amorphous colloidal silica with a specific surface area (BET) of 200 ± 25 m2/g, pH in dispersion at 4% in the range 3.7–4.5 and packed density around 50 g/l. To prepare nanofluids, SiO2 the nanoparticles (~0.01 g) were suspended in 100 ml of deionized water (HYDROLAB HLP 5sp, HYDROLAB, Poland) and in 100 ml of 99.5% isopropanol, labeled sample A and sample B, respectively. To taste A was sonicated in common mode, while the sample B in degassing mode using an ultrasonic cleaner (ZX-615FTS, Shanghai ZX Trading Co, LTD, Shanghai) for 60 min according to26. Both samples were sonicated without heating, at 95% device power (342 W ultrasonic power).

Experimental techniques

Classic approach

Transmission electron microscopy (TEM) was used to perform visualization of the silica nanopowder using a scanning TEM (Carl Zeiss Libra 200 MC Cs, Carl Zeiss AG, Oberkochen, Germany), operating at a acceleration voltage of 200 kV (see Fig. S10).

The two nanofluids were analyzed by conventional characterization methods: determination of the contact angle, determination of the zeta potential, of the pH and examination with a granulometer. In summary, this work mainly focuses on two of the five parameters characterizing nanofluids according to27: particulate and colloidal properties.

The contact angle was measured by an Ossila (UK) contact angle goniometer. The contact angle analyzes were carried out by the sessile drop technique at room temperature and atmospheric pressure. Ten independent measurements for nanofluids (water and nanosilica, isopropanol and nanosilica) and five independent measurements for pure water and isopropanol were performed for each sample, each with a 15 µL drop of water, and the results obtained were averaged to reduce the impact of surface non-uniformity.

The silica particle size after the sonication process in an ultrasonic cleaner was measured by the SALD-7500nano Nano Particle Size Analyzer for the water-silica solution in two steps: (1) immediately after the liquid preparation and (2) 24 h after liquid preparation see Figs. 5 and 6.

Measurements of zeta potential and particle size were performed by the technique of multi-angle dynamic light scattering (MADLS) and dynamic light scattering (DLS). Zeta, MADLS and DLS potential with horizontal or vertical polarizer were performed using a Zetasizer Ultra (Malvern Panalytical Ltd., Malvern, UK) equipped with a 10 mW helium/neon laser (λ = 633 nm ) at 25°C. Instrument parameters were automatically optimized using ZS XPLORER software (Malvern Panalytical Ltd., Malvern, UK). MADLS measurements were performed after 24 h, one week and one month after the suspension. The particle sizes are expressed as the average hydrodynamic diameter of 5 measurements.

For SEM imaging, droplets of Aerosil A200 + water (sample A) and Aerosil A200 + isopropanol (sample B) solution were placed on an aluminum pin disk covered with a double-sided adhesive carbon tape on the outside. using an Eppendorf Research automatic pipette (volume of the drop: 2 μl). The evaporation process took place at room temperature for 24 h. For the highest resolution scanning electron microscopy (FE-SEM) imaging, samples were coated with 10 nm Au using a high vacuum spray coater (Compact Coating Unit CCU -010, Safematic, Switzerland). Imaging was performed with a dual-beam FIB-SEM tool (Scios 2, ThermoFisher, USA) using secondary electron detectors: ETD and top detector in the objective – T2, accelerating voltage of 10 kV for electrons, 3 mm and Working distance of 7 mm. Figures 2b and 3b present photos of example SEM pin stubs showing the effect of evaporation (which occurred without additional evaporation conditions without heating) on ​​the shape of the dried nanofluid. Edge magnification is shown via SEM images in Figs. 2d and 3d, respectively (~100×).

A new approach with the use of multifractals

The multifractal approach opens up possibilities for viewing materials as a heterogeneous system with all aspects as subsets of the fundamental elements. The result of this analysis presents a multifractal spectrum that describes the dimension of a fractal subset of function points. To obtain an adaptive form of division in both time and space, the result of which is the spectrum of singularities, the Legendre transformation (τ(q)) is used. This transformation defines the relationship between itself and the global spectrum of singularities D(h)28:

$$ D(h) = qh(q) – tau(q), $$


$$ h(q) = frac{dtau (q)}{{dq}}, $$


where h(q) is the Hölder exponent at the moment q. Current negative values q refer to weak exponents, when positive – analogous to strong exponents. The spectrum itself can be written:

$$ Dleft( h right) = lim_{l to 0} frac{{sumnolimits_{i = 1}^{Nleft( l right)} {mu_{i} left( {q,l} right)ln left[ {mu_{i} left( {q,l} right)} right]} }}{ln l}, $$


where D(h) is the moment function q, µI(q, I) is a normalized measure such that qe mass probability moment PI(I) where to estimate multifractal properties over a small interval of scales a constant range of I is taken advantage of29:

$$ mu_{i} left( {q,l} right) = frac{{P_{i}^{q} left( l right)}}{{sumnolimits_{i = 1 }^{Nleft( l right)} {P_{i}^{q} left( l right)} }}. $$


The exemplary result of the algorithm for multifractal analysis of SEM images in the form of a singularity spectrum for the sample A is shown in Figure S9.

The polynomial approximation method makes it possible to determine three characteristic points on the graph (see Fig. S11).

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