Department of Chemical Engineering, Virginia Tech, Blacksburg, VA 24060, USA

School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore

Abstract

This paper presents an investigation into spreading dynamics and dynamic contact angle of TiO_{2}-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.

Background

Industrial operations such as spin coating, painting, and lubrication are based on spreading of fluids over solid surfaces. The fluid may be simple

It is known that out-of-equilibrium interfacial energy (^{0} − cos ^{0} and _{
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 (_{
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) ^{0} > 10^{0} is not constant and its change is described by MKT. In De Ruijter's model, it is assumed that ^{0} is constant and the dissipation functions are added to form the total dissipation function, _{tot} = _{
l
} + _{
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 TiO_{2}-deionized (DI) water nanofluids. Metal oxide TiO_{2} 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.

Methods

Preparation of nanofluids

The solutions were prepared by dispersing 15 nm TiO_{2} nanoparticles (anatase, 99%, Nanostructured and Amorphous Materials Inc., Houston, TX, USA) in DI water. Oleic acid is reported to stabilize TiO_{2} nanoparticles in DI water _{2} nanoparticles to examine these factors. An aliquot of dilute solution was dropped and dried on a carbon-coated copper grid. TEM images were then taken immediately. Figure _{2} particles and dissociation of proton from carboxylic acid head groups result in net negative charges on the surface of particles and thus formation of electric double layer around them. Thick electric double layers cause the deviation of particle-particle interactions from hard-sphere interactions. The (Debye) length in nanometer of an electric double layer of 1:1 electrolyte in water at 25°C can be approximated by ^{-4} molar), the Debye length is estimated to be about 16.9 nm. Such a small increase in the effective diameter of particles allows for an assumption of hard-sphere interactions between particles in the solution which is an important assumption in using Krieger's formula

TEM nanographs of 15 nm TiO_{2} nanoparticles

**TEM nanographs of 15 nm TiO**
_{
2
}
**nanoparticles.**

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

Dynamic contact angle of the solutions was measured using the FTA200 system. A droplet of solution was generated at a very low rate (1 μl/s) and detached from the syringe needle tip as soon as it touched the borosilicate glass slide. A video was captured while the droplet was spreading over the glass slide from initial contact to equilibrium position (see Figure

Consecutive photographs of spreading droplet detached from syringe needle tip

**Consecutive photographs of spreading droplet detached from syringe needle tip.**

Theory

Empirical analysis of viscosity

From Figure ^{−1}. At higher shear rates, Newtonian behavior was observed for all solutions. For dilute solutions, 0.1 vol.% and 0.05 vol.%, a weak shear thinning behavior was also observed at very low shear rates

Viscosity of TiO_{2}-DI water solutions

**Viscosity of TiO**
_{
2
}
**-DI water solutions.**

A power-law equation is used to model the shear rate and nanoparticle concentration dependent viscosity:

where _{
b
} is the viscosity of DI water equal to 0.927 mPa s,

where _{max} is the fluidity limit that is empirically equal to 0.68 for hard spherical particles. In Equation 1,

**TiO**
_{
2
}**volume concentration (**
**)**

**Power-law index (****
n
**

**Proportionality factor (****
K
**

**Surface tension (****
σ
**

**Equilibrium contact angle (****
θ
**

2%

0.04

2,932

0.0543

51.7

1%

0.18

432

0.0606

47.5

0.5%

0.76

5

0.0612

46.7

0.1%

0.89

2

0.0623

45.7

0.05%

0.92

1

0.0632

44.5

Molecular kinetic theory

Schematic of a spreading droplet of radius _{
W
} (_{
W
} = ^{+} − ^{−}, where ^{+} is the frequency of forward motion and ^{−} is the frequency of backward motion), multiplied by average distance between the adsorption sites,

Schematic of a spreading droplet

**Schematic of a spreading droplet.**

The equilibrium frequency of the three-phase contact line motion (

where _{sl}), solid-vapor (_{sa}), and liquid–vapor (^{0} − cos

where

For small arguments of sinh, Equations 3 and 6 result in linear MKT

where ^{0}) = _{sa} − _{sl} (Young's equation). Left hand side (LHS) of Equation 7 is the out-of-equilibrium interfacial energy which is the driving force of capillary flow. Right hand side (RHS) of Equation 7 only includes dissipation of the free energy due to the contact line friction. De Ruijter et al. _{
l
}) is:

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.

Hydrodynamic theory

To calculate the wedge film viscous dissipation (_{
W
}), Navier–Stokes equation of motion is solved in the wedge film region. From Figure

where

where _{
n
} is replaced by its expression in Equation 1. The average cross-sectional fluid velocity in the wedge film (

The viscous dissipation in the wedge film can be obtained as follows

where _{
n
} ∂ _{
m
} is the cutoff length similar to slip length in HDT _{
m
}, dissipation of energy at the wedge film grows infinitely close to the three-phase contact line. For a thin wedge film, solution to Equation 12 gives the wedge film viscous dissipation function:

Dynamic contact angle

Combining Equations 8 and 13 gives the total dissipation function

Taking derivative of the total dissipation function with respect to contact line velocity (∂ [_{
l
} + _{
W
}]/∂

Finally, equating Equation 15 with LHS of Equation 7 gives:

It is noted that for _{
m
}). In this case the final form of Equation 16 is similar to De Ruijter's model ^{0} − cos ^{3}
^{3}

In Equation 16, the base radius (_{m}) is in nanometer length scale. Thus, _{
m
}, and consequently ^{
1−n
} ≫ _{
m
}
^{
1−n
} for

where _{m}
^{1 − n
} and substituting

Equation 18 shows the dynamic contact angle (

in which the dynamic contact angle

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 TiO_{2} clusters results in a hydrophilic surface with higher surface energy (approximately 2.2 J/m^{2}
^{2}

Equation 19 suggests that the contact line friction dissipation (first term on the RHS of Equation 19) and the wedge film viscous dissipation (second term on the RHS of Equation 19) can occur at different time scales

Experimental three-phase contact line velocity (

**Experimental three-phase contact line velocity (**
**
U
**

Dynamic contact angle of TiO_{2}-DI water solutions

**Dynamic contact angle of TiO**
_{
2
}
**-DI water solutions.**

Figure _{2}-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

Using Equation 19,

where

Dynamic contact angle of TiO_{2}-DI water nanofluid, comparison of experiment and theory

**Dynamic contact angle of TiO**
_{
2
}
**-DI water nanofluid, comparison of experiment and theory.**

**Nanoparticle concentration**

**
ζ
**

**Error**

2%

32

52.1

1.1

1%

99

48.2

1

0.5%

464

46.4

0.65

0.1%

483

45.3

0.54

0.05%

486

44.8

0.34

Table _{sl})

Conclusions

Due to a wide range of industrial applications, studying capillary flow of liquids laden with metallic and metal oxide nanoparticles is important. Metal oxide TiO_{2} 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 TiO_{2} 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.

Competing interests

The authors declare that they have no competing interests.

Acknowledgments

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.