Prediction of Contact Angle of Nanofluids by Single-Phase Approaches

Wettability is the ability of the liquid to contact with the solid surface at the surrounding fluid and its degree is defined by contact angle (CA), which is calculated with balance between adhesive and cohesive forces on droplet surface. Thermophysical properties of the droplet, the forces acting on the droplet, atmosphere surrounding the droplet and the substrate surface are the main parameters affecting on CA. With nanofluids (NF), nanoparticle concentration and size and shape can modify the contact angle and thus wettability. This study investigates the validity of single-phase CA correlations for several nanofluids with different types of nanoparticles dispersed in water. Geometrical parameters of sessile droplet (height of the droplet, wetting radius and radius of curvature at the apex) are used in the tested correlations, which are based on force balance acting on the droplet surface, energy balance, spherical dome approach and empirical expression, respectively. It is shown that single-phase models can be expressed in terms of Bond number, the non-dimensional droplet volume and two geometrical similarity simplexes. It is demonstrated that they can be used successfully to predict CA of dilute nanofluids’ at ambient conditions. Besides evaluation of CA, droplet shape is also well predicted for all nanofluid samples with ±5% error.


Introduction
Wettability is the property of a solid surface contacted with a liquid within a surrounding fluid (liquid or gas) and its quantity is defined by the contact angle (CA). Wettability is important for the industrial applications such as phase change heat transfer, oil recovery and liquid coating. Contact angle (CA) is the angle between liquid-gas (or liquid-liquid) interface and liquid-solid interface for a droplet on a solid surface. The CA strongly depends on solid, liquid and surrounding fluid properties. If CA is smaller than 90 • , it means that the liquid wets the solid surface. For CA greater than 90 • , the liquid has lower wettability. CA is also dependent on liquid type and surface properties. Surface material including chemical composition and morphology, roughness and contamination make the CA and thus wetting properties. Since CA is predicted by the balance of interfacial/surface

Nanofluids
To assess the relevance of different single-phase models for the prediction of nanofluid contact angle, experimental data for distilled water (DIW) as a single-phase reference fluid and base fluid of nanofluids with GO, Au, SiO 2 and Al 2 O 3 nanoparticles were utilized. Au, Al 2 O 3 and GO NFs here employed were fully characterized in [19] where preparation and thermophysical properties evaluation including density are also reported.
For preparation of SiO 2 NF, silica raw material with density of 2000 kg/m 3 at 20 • C (SIPERNAT ® 22S, Evonik Industries AG, Germany) was dispersed unfractionated. NF had pH-value of 10.5 with stabilization by KOH. Mean agglomerate size of particles was measured as 177 nm by using Dynamic Light Scattering analysis (Zetasizer Nano ZS, Malvern Instruments GmbH, Germany). Density of SiO 2 Energies 2019, 12, 4558 3 of 16 NF was calculated with the mixture rule (Equation (1)) [25]. Thermal conductivity of SiO 2 nanofluid was measured by using 3ω method [26]. Measurements were conducted at 24 • C with three repetitions. Main information for all nanofluids are compiled in Table 1 showing the difference in concentration, size, shape and density of nanoparticles, and density and thermal conductivity of NF. (1)

Experiments on Contact Angle Measurements and Determination of Geometrical Parameters
Experiments were performed in different institutes. The CA data of DIW, Au, Al 2 O 3 and GO NFs were collected atİzmir Kâtip Çelebi University (İKÇÜ), Universitat Jaume I Castelló (UJI) and Université Rennes 1 (UR1) and were taken from [19]. CA of SiO 2 nanofluid was measured at ILK-Dresden (ILK). Sessile drop method was used in all institutes. Devices used in institutes and experimental conditions such as temperature T, droplet volume V and relative humidity RH used by each institute for the different liquids are described in Table 2.
Main steps of the CA measurements performed by each institute are described in the following paragraph. The reader is also referred to [19] for more details. Attention Theta Goniometer (Biolin Scientific, (Sweden/Finland)) was used in the experiments carried out byİKÇÜ. After calibration of the device, surface tension of liquids was measured and then introduced to device software as property to find CA. In 10 s, 125-140 images of droplet were captured. CA was evaluated by fitting the droplet shape with the Young-Laplace equation.
UR1 calibrated the commercial device (DSA-30 Drop Shape Analyzer (KRÜSS GmbH, Germany)) from gauges with known CAs of 30 • , 60 • and 120 • , and a maximum relative deviation was obtained as 0.52%. Droplet was positioned at the surface by using 15-gauge needle with an outer diameter of 1.835 mm. After positioning, CA values were measured in a few seconds. Further information on setup, experimental procedure and image analysis was given in [19].
Lab-made device developed by UJI was combination of camera, syringe, indirect LED light source and micrometer carrier. The droplet regulator, which holds the syringe, was controlled by a micrometer carrier. Syringe was used to position the droplet on the solid surface at a controlled volume. LED panel light was used to obtain high contrast by background illumination. This device allows the prediction of CA and volume of the droplet by image processing.
CA was measured at ILK by using the lab-made device [19]. In the setup [19], the droplet was positioned by a micropipette (VWR Pipettor 2-20 µL) on the solid substrate. Digital reflex camera (Canon EOS 40D) equipped with close-up lens (Tamron, SP 90 mm F/2.8, Di MACRO 1:1, VC USD; Japan) was used to capture the droplet images. In order to obtain accurate CA, contrast of images improved by using light disperser and light source with 129 LEDs. Images of droplet were stored at a laptop and contact angle of each droplet was predicted by using ImageJ [27] software with extension "drop analysis" [28]. Contact angle of the droplet was determined for both sides and mean contact angle was used. Experiments of CA measurement were conducted in ILK for constant volume at 10 µL.
CA of DIW and NFs was measured on the solid stainless steel substrate with thickness of 5 mm and an averaged roughness of 1.4 µm. For each institution, solid substrates were manufactured from single round stock with same lathe to provide same material and surface quality for all substrates. Properties of the solid substrate were presented in [19] with details.
In addition to CA, surface tension (σ lg ) of the working fluids was evaluated experimentally at UR1 (DIW, Au NF, Al 2 O 3 and GO NF) andİKÇÜ (SiO 2 NF; Table 3). In both institutes, the pendant drop method was employed to define the σ lg . DSA-30 Drop Shape Analyzer (KRÜSS GmbH, Germany) and Biolin Scientific, (Sweden/Finland) were used at UR1 andİKÇÜ, respectively. InİKÇÜ, surface tension of DIW was measured before SiO 2 NF. For both working fluids, the volume was kept constant at 10 µL and three measurements were carried out. There is no significant deviation between replicates and between experimental values of DIW with NIST data [29] at a given temperature. The comprehensive description of the experimental procedure used for the surface tension measurement at UR1 can be found in [30]. Finally, the geometrical parameters (δ, r d and R 0 ) used in the models that are described in the next section were obtained by CA image analysis. Such an analysis has been performed with an image processing program called Fiji [31] as shown in Figure 1. Pixel-to-mm ratio with respect to a reference dimension for the images from the institutes is defined to spatial calibration of each data set.

Single-Phase Models for Prediction of Contact Angle
In this section, the models developed for the CA evaluation of single-phase liquids are presented. The first model considered was developed by Vafaei and Podowski [32], which is based on the force balance of gravity and surface forces on the droplet surface. Reason behind the gravity force is to the hydrostatic force of the variable droplet height and the surface forces are result of external pressure and surface tension. Vafaei and Podowski [32] proposed a model for the prediction of droplet volume that writes as follow: Here ρ denotes density and V are droplet baseline radius and volume. As geometrical parameters r d , δ and R 0 are the droplet baseline radius, apex height and curvature, respectively. θ d is the contact angle of the droplet. Droplet volume equation (2) was reformulated by using non-dimensional numbers: Bond number (Bo), which is the ratio of gravitational and surface forces, two geometrical similarity simplexes-G 1 and G 2 -which describe the droplet geometry and V* as the non-dimensional droplet volume.
where Bo is In the definition of Bo number characteristic length equals to r d , which is defined by Kuiken [33]. Moreover, Stacy [34] reported that characteristic length equals to radius at the droplet equator and wetting radius for non-wetting droplets and for wetting droplets, respectively.
The second model we have considered was proposed by Yonemoto and Kunugi [35] (YK model), which is based on the energy balance for various-sized droplets on a solid surface. This correlation writes A non-dimensional form is obtained by dividing (5) by the gravitational potential ρgδV/2.
YK model is used for only spherical droplets. Therefore, employing the correlation for θ s = arcsin r d sin θ d r ds given in [32] Equation (5) is rewritten. The contact angle for spherical droplets is denoted by θ s and its baseline radius by r ds .
Moreover, the spherical dome model (SD model), which is based on the tangent of only dimensions of the droplet (δ and r d ), analyzed in [24] was rewritten in terms of geometrical similarity simplex, G 1 , and validity was studied for nanofluids.
Finally, Wong et al. [24] (W model) developed an empirical formulation, expressed in Equation (9), to predict contact angle of liquids, which was reported as being valid for any substrate and droplet materials. This empirical model uses the contact angle, droplet volume, height and width (W = 2r d ) and allows us to find two of them by using known other two parameters.

Prediction of Droplet Shape
In addition to CA prediction from single phase models, the sessile droplet shape of NFs is compared to the following approach. Thus, Vafaei and Podowski [32] reported that droplet shape could be predicted by force balance on the droplet surface as well as CA as following: Coefficients (c 1 , c 2 , c 3 , c 4 ) were determined by applying boundary conditions: Applying one more boundary condition z(−r d ) = 0 and power series assumption (c 3 = 0), Equation (10) was re-arranged by using geometrical similarity simplexes. Considering the origin is at the center of the wetting circle, r is the horizontal axis and z is the vertical axes of the droplet (Figure 1).

Results and Discussion
In this study, validity of contact angle models, which were developed for single-phase fluids, was investigated. In the figures, filled symbols (•) were for models and empty symbols (•) were used for experimental results. Due to the measurement uncertainties and in order to evaluate their influence, numerical results from the models were corrected with increasing R 0 by 2% and it was presented as a red cross in the figures ().
Error between experimental and predicted CA was determined by using mean absolute percentage error (MAPE) and was denoted as Ω in the study. It is found as:

Contact Angle Prediction with Single-Phase Models
Figures 2 and 3 show the validity of VP model for DIW and the different nanofluids. DIW as expected for a single-phase liquid has good agreement with the model (for various volume Ω = 2.32% and for constant volume Ω = 6.34%). Due to the low concentration and the small particle size, the Au and Al 2 O 3 NF shows also reasonable good agreement with Ω = 2.07% and Ω = 4.07%, respectively. However, the VP model seems to be not valid for GO (Ω = 8.52%) and SiO 2 NFs (Ω = 30.46%). Large graphene flakes in GO NF and higher concentration (3.935 wt.%) of SiO 2 NF may be the reason of preventing the application of the VP model. vertical axes of the droplet (Figure 1).

Results and Discussion
In this study, validity of contact angle models, which were developed for single-phase fluids, was investigated. In the figures, filled symbols ( ) were for models and empty symbols ( ) were used for experimental results. Due to the measurement uncertainties and in order to evaluate their influence, numerical results from the models were corrected with increasing R0 by 2% and it was presented as a red cross in the figures (✕).
Error between experimental and predicted CA was determined by using mean absolute percentage error (MAPE) and was denoted as Ω in the study. It is found as: Figures 2 and 3 show the validity of VP model for DIW and the different nanofluids. DIW as expected for a single-phase liquid has good agreement with the model (for various volume Ω = 2.32% and for constant volume Ω = 6.34%). Due to the low concentration and the small particle size, the Au and Al2O3 NF shows also reasonable good agreement with Ω = 2.07% and Ω = 4.07%, respectively. However, the VP model seems to be not valid for GO (Ω = 8.52%) and SiO2 NFs (Ω = 30.46%). Large graphene flakes in GO NF and higher concentration (3.935 wt.%) of SiO2 NF may be the reason of preventing the application of the VP model.  It was found, that the VP model was very sensitive against tiny experimental errors following the determination of R 0 . To illustrate this, R 0 was increased artificially by just 2% and additional results presented in Figure 2 for GO NF, and Figure 3 for DIW and SiO 2 . CA data of GO NF fit much better with the VP model with this correction. Ω decreased from 8.52% to 4.19% for GO NF. Unlike GO NF, CA of SiO 2 NF was not affected from the increase of R 0 (Ω = 27.79%). This shows the influence of nanoparticle content on the applicability of V-P model with nanofluids.  It was found, that the VP model was very sensitive against tiny experimental errors following the determination of R0. To illustrate this, R0 was increased artificially by just 2% and additional results presented in Figure 2 for GO NF, and Figure 3 for DIW and SiO2. CA data of GO NF fit much better with the VP model with this correction. Ω decreased from 8.52% to 4.19% for GO NF. Unlike GO NF, CA of SiO2 NF was not affected from the increase of R0 (Ω = 27.79%). This shows the influence of nanoparticle content on the applicability of V-P model with nanofluids. The modified YK model was in good agreement with experimental results for DIW, Au, Al2O3 and GO nanofluids. As shown in Figure 4, all the data was in the range of the ±10% error band. However, the results for the SiO2 NF ( Figure 5) show again that a single-phase model was not appropriate for a highly concentrated nanofluid.  The modified YK model was in good agreement with experimental results for DIW, Au, Al 2 O 3 and GO nanofluids. As shown in Figure 4, all the data was in the range of the ±10% error band. However, the results for the SiO 2 NF ( Figure 5) show again that a single-phase model was not appropriate for a highly concentrated nanofluid.  It was found, that the VP model was very sensitive against tiny experimental errors following the determination of R0. To illustrate this, R0 was increased artificially by just 2% and additional results presented in Figure 2 for GO NF, and Figure 3 for DIW and SiO2. CA data of GO NF fit much better with the VP model with this correction. Ω decreased from 8.52% to 4.19% for GO NF. Unlike GO NF, CA of SiO2 NF was not affected from the increase of R0 (Ω = 27.79%). This shows the influence of nanoparticle content on the applicability of V-P model with nanofluids. The modified YK model was in good agreement with experimental results for DIW, Au, Al2O3 and GO nanofluids. As shown in Figure 4, all the data was in the range of the ±10% error band. However, the results for the SiO2 NF ( Figure 5) show again that a single-phase model was not appropriate for a highly concentrated nanofluid.    Figures 6 and 7 show that the SD model was valid for DIW (Ω = 4.71%), Al2O3 NF (Ω = 3.78%) and Au NF (Ω = 7.18%). Vafaei and Podowski [32] mentioned that as volume decreases the shape of the droplet is more spherical. Although the volume range was narrow for this study, change of the droplet shape with volume was obviously clear. For GO nanofluid at the highest volume, CA was underpredicted compared to experimental results with Ω = 7.35%. The reason could be the shape and size of the graphene flakes. Different from the other theoretical models, predicted SiO2 data with the SD model was much collapsed with the experimental data with Ω = 10.71% where DIW had Ω = 3.90%.   [32] mentioned that as volume decreases the shape of the droplet is more spherical. Although the volume range was narrow for this study, change of the droplet shape with volume was obviously clear. For GO nanofluid at the highest volume, CA was underpredicted compared to experimental results with Ω = 7.35%. The reason could be the shape and size of the graphene flakes. Different from the other theoretical models, predicted SiO 2 data with the SD model was much collapsed with the experimental data with Ω = 10.71% where DIW had Ω = 3.90%.  Figures 6 and 7 show that the SD model was valid for DIW (Ω = 4.71%), Al2O3 NF (Ω = 3.78%) and Au NF (Ω = 7.18%). Vafaei and Podowski [32] mentioned that as volume decreases the shape of the droplet is more spherical. Although the volume range was narrow for this study, change of the droplet shape with volume was obviously clear. For GO nanofluid at the highest volume, CA was underpredicted compared to experimental results with Ω = 7.35%. The reason could be the shape and size of the graphene flakes. Different from the other theoretical models, predicted SiO2 data with the SD model was much collapsed with the experimental data with Ω = 10.71% where DIW had Ω = 3.90%.  In addition to these theoretical models, the empirical model developed by Wong et al. [24] allowed us to predict contact angle of nanofluids at lower concentrations ( Figure 8). Ω value was calculated for DIW for various volumes, DIW at constant volume, Au NF and GO NF were 5.77%, 3.53%, 4.70% and 5.26%, respectively. CA data of Al2O3 NF (Ω = 9.78%) predicted by the W model is different for smaller volumes. However, increase in volume results in more accurate results with the W model. The reason could be that the concentration highly affects on CA at lower volumes for this model. Similar behavior was observed for also SiO2 NF. The W model was not suitable for highly concentrated nanofluids as shown in SiO2 NF (Ω = 144.46%) results in Figure 9.  In addition to these theoretical models, the empirical model developed by Wong et al. [24] allowed us to predict contact angle of nanofluids at lower concentrations ( Figure 8). Ω value was calculated for DIW for various volumes, DIW at constant volume, Au NF and GO NF were 5.77%, 3.53%, 4.70% and 5.26%, respectively. CA data of Al 2 O 3 NF (Ω = 9.78%) predicted by the W model is different for smaller volumes. However, increase in volume results in more accurate results with the W model. The reason could be that the concentration highly affects on CA at lower volumes for this model. Similar behavior was observed for also SiO 2 NF. The W model was not suitable for highly concentrated nanofluids as shown in SiO 2 NF (Ω = 144.46%) results in Figure 9. In addition to these theoretical models, the empirical model developed by Wong et al. [24] allowed us to predict contact angle of nanofluids at lower concentrations ( Figure 8). Ω value was calculated for DIW for various volumes, DIW at constant volume, Au NF and GO NF were 5.77%, 3.53%, 4.70% and 5.26%, respectively. CA data of Al2O3 NF (Ω = 9.78%) predicted by the W model is different for smaller volumes. However, increase in volume results in more accurate results with the W model. The reason could be that the concentration highly affects on CA at lower volumes for this model. Similar behavior was observed for also SiO2 NF. The W model was not suitable for highly concentrated nanofluids as shown in SiO2 NF (Ω = 144.46%) results in Figure 9.

Droplet Shape Prediction
Vafaei and Podowski [32] also developed a model to predict droplet shape based on force balance (Equation 15). This subsection presents the validity of this model for nanofluids considered in this study. For volume dependent analysis, droplets had a maximum 21 μL of volume ( Figure 10). For constant volume analysis of DIW and SiO2 NF, droplets that had maximum, minimum and average contact angles were chosen ( Figure 11).
Droplet shape was predicted with ±5% error for all cases without influence of shape and concentration of nanoparticles, and volume of the droplet. This ±5% error band could include measurement errors in geometrical parameters and experimental errors.

Droplet Shape Prediction
Vafaei and Podowski [32] also developed a model to predict droplet shape based on force balance (Equation (15)). This subsection presents the validity of this model for nanofluids considered in this study. For volume dependent analysis, droplets had a maximum 21 µL of volume ( Figure 10). For constant volume analysis of DIW and SiO 2 NF, droplets that had maximum, minimum and average contact angles were chosen ( Figure 11).
Droplet shape was predicted with ±5% error for all cases without influence of shape and concentration of nanoparticles, and volume of the droplet. This ±5% error band could include measurement errors in geometrical parameters and experimental errors.

Conclusions
The aim of this study was to evaluate the reliability of three theoretical models and one empirical model developed for single-phase liquids for describing the correlation between contact angle, geometrical parameters and forces/energy balance of nanofluid sessile droplets. Four types of waterbased nanofluids were considered varying the size, shape and nature of nanoparticles as well as the concentration.
Single-phase models were expressed in terms of Bo, which depends on surface tension and density, non-dimensional volume V* and geometrical simplexes and they were used for prediction of dilute NFs' CA. Although the measurement of the required geometrical parameters for prediction of CA demands high precision to obtain accurate results, it is concluded that concentration is the main factor to affect applicability of models for NFs. Although the force balance-based VP model was too sensitive to measurement errors for GO NF and CA of dilute NFs (Au, GO and Al2O3 NF) could be predicted by both the VP model and energy balance-based YK model. Moreover, the empirical W model was valid for dilute NFs (Au and GO NF) and higher volumes of Al2O3 NF. However, significant differences were obtained for highly concentrated SiO2 NF in these theoretical models and empirical model. Additional effects like disjoining pressure, convective flows inside the droplet, etc. could be the reasons of differences in results of different models. Beside them, the SD model is suitable for almost all samples due to the spherical shape of droplets as a result of smaller volumes. However, higher error band of SD model is the limitation for accurate prediction of CA. It was also shown that the droplet shape of all NFs could be well predicted from a model based on force balance.
To conclude, further research is now needed to resolve and improve the limitation of the measurement of dimensions and for the development of a useful and generic CA model by investigation of additional possible effects such as temperature, a wider range of nanoparticle size and shape varying the nanoparticle concentration.   (14) and-shows the ±5% errors).

Conclusions
The aim of this study was to evaluate the reliability of three theoretical models and one empirical model developed for single-phase liquids for describing the correlation between contact angle, geometrical parameters and forces/energy balance of nanofluid sessile droplets. Four types of water-based nanofluids were considered varying the size, shape and nature of nanoparticles as well as the concentration.
Single-phase models were expressed in terms of Bo, which depends on surface tension and density, non-dimensional volume V* and geometrical simplexes and they were used for prediction of dilute NFs' CA. Although the measurement of the required geometrical parameters for prediction of CA demands high precision to obtain accurate results, it is concluded that concentration is the main factor to affect applicability of models for NFs. Although the force balance-based VP model was too sensitive to measurement errors for GO NF and CA of dilute NFs (Au, GO and Al 2 O 3 NF) could be predicted by both the VP model and energy balance-based YK model. Moreover, the empirical W model was valid for dilute NFs (Au and GO NF) and higher volumes of Al 2 O 3 NF. However, significant differences were obtained for highly concentrated SiO 2 NF in these theoretical models and empirical model. Additional effects like disjoining pressure, convective flows inside the droplet, etc. could be the reasons of differences in results of different models. Beside them, the SD model is suitable for almost all samples due to the spherical shape of droplets as a result of smaller volumes. However, higher error band of SD model is the limitation for accurate prediction of CA. It was also shown that the droplet shape of all NFs could be well predicted from a model based on force balance.
To conclude, further research is now needed to resolve and improve the limitation of the measurement of dimensions and for the development of a useful and generic CA model by investigation of additional possible effects such as temperature, a wider range of nanoparticle size and shape varying the nanoparticle concentration.