Dynamic contact angle of water-based titanium oxide nanofluid
© Radiom et al.; licensee Springer. 2013
Received: 27 February 2013
Accepted: 15 April 2013
Published: 11 June 2013
This paper presents an investigation into spreading dynamics and dynamic contact angle of TiO2-deionized water nanofluids. Two mechanisms of energy dissipation, (1) contact line friction and (2) wedge film viscosity, govern the dynamics of contact line motion. The primary stage of spreading has the contact line friction as the dominant dissipative mechanism. At the secondary stage of spreading, the wedge film viscosity is the dominant dissipative mechanism. A theoretical model based on combination of molecular kinetic theory and hydrodynamic theory which incorporates non-Newtonian viscosity of solutions is used. The model agreement with experimental data is reasonable. Complex interparticle interactions, local pinning of the contact line, and variations in solid–liquid interfacial tension are attributed to errors.
Industrial operations such as spin coating, painting, and lubrication are based on spreading of fluids over solid surfaces. The fluid may be simple [1–3] or particulate such as paint, ink, or dye . For many years, capillary flow of simple fluids has received considerable attention, and physics of capillary action is known for a long time [5–9]. In addition, capillary flow of micellar surfactant solutions which contain monodisperse and naturally stabilized nanoparticles has been studied [10–14]. However, the same study on liquids laden with metallic and oxide nanoparticles such as silver, copper, zinc oxide, and titanium oxide is scarce. These fluid suspensions are termed as nanofluids after the seminal work by Choi and Eastman . The application of nanofluids is coined with enhanced heat transfer performance compared with their base fluids. They are proposed for applications in cooling of electronic devices, ventilation and air conditioning, and biomedical applications [14, 16–24].
It is known that out-of-equilibrium interfacial energy (σ(cos θ0 − cos θ)) provides free energy of capillary flow where σ is the liquid-air surface tension and θ0 and θ are the equilibrium and dynamic contact angles, respectively. During capillary flow, the free energy is dissipated by two mechanisms : (1) contact line friction (T ∑ l ) which occurs in proximity of three-phase contact line (solid–liquid–air). The friction at the three-phase contact line is due to intermolecular interactions between solid molecules and liquid molecules. (2) Wedge film viscosity (TΣ W ) which occurs in the wedge film region behind the three-phase contact line. Lubricating and rolling flow patterns in the wedge film region result in the dissipation of the free energy. For each mechanism of energy dissipation, a theory is developed: (1) molecular kinetic theory (MKT) [25, 26] models the contact line friction, and (2) hydrodynamic theory (HDT) [27, 28] models the wedge film viscosity. For partial wetting systems (θ0 > 10°), it is assumed that both dissipative mechanisms coexist and models that combine MKT and HDT are developed by Petrov  and De Ruijter . In Petrov's model, it is assumed that the equilibrium contact angle θ0 is not constant and its change is described by MKT. In De Ruijter's model, it is assumed that θ0 is constant and the dissipation functions are added to form the total dissipation function, TΣtot = T ∑ l + TΣ W . These models are developed for Newtonian fluids and show generally good agreement with experimental data .
This paper presents an investigation into spreading dynamics and dynamic contact angle of TiO2-deionized (DI) water nanofluids. Metal oxide TiO2 nanoparticle was chosen for its ease of access and popularity in enhanced heat removal applications. Various nanoparticle volume concentrations ranging from 0.05% to 2% were used. The denser solutions exhibit non-Newtonian viscosity at shear rate ranges that are common to capillary flow. To model experimental data a theoretical model based on combination of MKT and HDT similar to De Ruijter's model is used. The non-Newtonian viscosity of the solutions is incorporated in the model.
Preparation of nanofluids
Measurement of viscosity
Viscosity of the solutions was measured using a controllable low shear rate concentric cylinders rheometer (Contraves, Low Shear 40, Zurich, Switzerland). The viscosity was measured at shear rates ranging from 0 to 50 s−1. This range corresponds to the shear rates that are common to capillary flow.
Measurement of surface tension
Surface tension of the solutions was measured by pendant droplet method using FTA200 system (First Ten Angstroms, Inc., Portsmouth, VA, USA). To form the pendant droplets, the solutions were pumped out of a syringe system at a very low rate, namely 1 μl/s, to minimize inertia effects. To minimize errors due to evaporation, surface tension was measured right after the pendant droplet reached its maximum volume, namely 10 μl for the dense solutions.
Measurement of dynamic contact angle
Empirical analysis of viscosity
Power-law viscosity, surface tension, and equilibrium contact angle of TiO 2 -DI water solutions
TiO2volume concentration (ϕ)
Power-law index (n)
Proportionality factor (K)
Surface tension (σ[N/m])
Equilibrium contact angle (θ0)
Molecular kinetic theory
In the next section, the wedge film viscous dissipation is calculated and added to Equation 8 to form the total dissipation function from which the total drag force is calculated. The total drag force is then equated to the LHS of Equation 7 to form the complete equation of the three-phase contact line motion.
Dynamic contact angle
It is noted that for n = 1 (Newtonian fluid), the integral of Equation 12 results in logarithm ln(r/x m ). In this case the final form of Equation 16 is similar to De Ruijter's model  (σ(cos θ0 − cos θ) = ζU + 6ηΦ(θ)U ln(r/a)) where Φ = sin 3θ/2 − 3 cos θ + cos 3θ and a is the cutoff length in De Ruijter's model).
in which the dynamic contact angle θ = π − α. To compare with experimental data θ is used. Equation 19 is an implicit ordinary differential equation, which cannot be solved analytically, and thus numerical solutions to this equation will be sought.
Results and discussion
The effective diameter of nanoparticles was equal to 260 nm at the lowest solution concentration of 0.05 vol.%. At higher particle concentrations, the increased interparticle interactions result in larger clusters. This increases the possibility of clusters to deposit on the surface of solid and form a new hydrophilic surface. Due to their larger size, these clusters are less possible to deposit on the three-phase contact line, and thus a heterogeneous surface will form: within the wedge film and away from the three-phase contact line, deposition of TiO2 clusters results in a hydrophilic surface with higher surface energy (approximately 2.2 J/m2) than the three-phase contact line where the bare borosilicate glass is present (approximately 0.11 J/m2). The higher surface energy inside the droplet shrinks the wetted area by increasing the equilibrium contact angle (denser solutions are more hydrophilic inside than outside). As a result, solid–liquid interfacial tension increases which on the other hand enhances the equilibrium contact angle . Surface tension of these solutions decreases with particle concentration that is in accordance with Gibb's adsorption isotherm. The shear thinning viscosity of the solutions is due to strong interparticle interaction of the nanoparticle clusters [19, 23, 36]. Other nanofluids such as ethylene glycol-based ZnO nanofluid  and CuO nanofluid  also exhibited shear thinning viscosity at low shear rates.
Figure 6 shows the dynamic contact angle of TiO2-DI water nanofluids at various nanoparticle volume concentrations ranging from 0.05% to 2%. Due to limitation in camera frame per second speed (30 fps), the onset of pendant droplet touching the surface of solid cannot be determined accurately. Hence, the time axis in Figure 6 was shifted to where all of the captured images were readable to the FTA200 software. From Figure 6, it is obvious that for higher nanoparticle concentrations, the contact angles are higher. Figure 6 also shows that the spreading of these nanofluids starts from a primary region where the contact angle changes rapidly followed by a region where the contact angle changes more gradually (note that in a very short period of time (less than 300 ms), the contact angle evolves from 180° at point of contact to angles that are readable to our software and are plotted in Figure 6 at the shifted zero time). In the primary region, the contact line friction dissipation predominates the wedge film viscous dissipation causing fast reduction in the contact angle; then the wedge film viscous dissipation controls the droplet spreading .
Coefficient of contact line friction ζ , theoretical equilibrium contact angle, and error of comparison between theory and experiment
Table 2 shows values of ζ for various nanoparticle volume concentrations. From solution concentration of 0.05% to 0.5% ζ only changes by 5%; however, it drops rapidly for denser solutions. It is possible that the relative higher hydrophobicity at the three-phase contact line for denser solutions lowers the affinity of surface molecules to water molecules, thereby lowering the friction. At dense concentrations, the presence of large amount of nanoparticles in the wedge film varies the flow field structure. Without nanoparticles, it has been stated that there are two flow patterns in the wedge film: rolling and lubricating patterns . Nanoparticles in the wedge film can change these flow patterns and result in more complex flow structures. As a result of these interparticle interactions, dissipation is more pronounced in the wedge film. Equation 19 gives better results at lower nanoparticle concentrations since complex interparticle interactions are less frequent in dilute solutions (see Table 2). Other sources of disagreement between experiment and theory can be local variations in the concentration of the nanoparticles in the nanofluid , pinning of the contact line, and variations in solid–liquid interfacial tension (σsl) [18, 21]. It is not possible to model all these effects in theory, and only simple models which can accommodate some of these effects can be developed. Also shown in Table 2 are the theoretical equilibrium contact angles, , which are in reasonable agreement with the experimental equilibrium contact angles, (see Table 1).
Due to a wide range of industrial applications, studying capillary flow of liquids laden with metallic and metal oxide nanoparticles is important. Metal oxide TiO2 nanoparticles are especially interesting in enhanced heat removal applications. Agglomeration of nanoparticles results in clusters that have larger effective diameter than the actual particle size. These clusters can deposit on the surface of solid substrates and form a heterogeneous surface condition inside the droplet away from the three-phase contact line that increases the equilibrium contact angle. Dynamic contact angle of metal oxide TiO2 nanoparticles dispersed in DI water revealed two stages of spreading: rapid reduction in contact angle coincides with contact line friction dissipation governed by MKT while gradual reduction in contact angle coincide with wedge film viscous dissipation governed by HDT. Non-Newtonian viscosity of the solution is incorporated in HDT model to give reasonable comparison with experimental data. Nanoparticles in the wedge film change lubricating and rolling flow patterns and result in complex flow field structures. Including all physical aspects of such complex flow in theory is not feasible at the current stage. Simple theoretical equations can only give reasonable comparisons with experiment.
The authors gratefully acknowledge the financial support of the research grant (MOE2009-T2-2-102) from the Ministry of Education of Singapore to CY and the Singapore A*STAR scholarship to MR.
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