# Modeling the Viscosity of Concentrated Nanoemulsions and Nanosuspensions

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Dilute Dispersions

#### 2.2. Non-Dilute Dispersions

## 3. New Viscosity Model for Concentrated Nanoemulsions and Nanosuspensions

_{2}O

_{3}nanosuspensions [14]. The reason for the poor predictability of the nanofluid viscosity by the Oldroyd model is that this model does not consider solvation and aggregation of nanoparticles/nanodroplets. Due to the small size of nanoparticles or nanodroplets, the influence of solvation films present on the surfaces of the particles could not be neglected. It is a well-known fact that lyophilic surfaces are solvated [45], that is, they are coated with thin films of the matrix fluid. Due to the strong attractive interactions between the dispersion medium (matrix) molecules and the particle surfaces, the solvation films become a part of the particles and the particles behave as swollen particles.

#### Estimation of the Solvation and Aggregation Coefficients

## 4. Comparison of Model Predictions with Experimental Data and Discussion

#### 4.1. Scaling of Relative Viscosity of Nanoemulsions and Nanosuspensions

#### 4.2. Influence of Viscosity Ratio on the Relative Viscosity of Nanoemulsions

## 5. Conclusions

- •
- The relative viscosity of a nanofluid is strongly affected by factors such as solvation and aggregation of nanoparticles/nanodroplets. In the case of nanoemulsions, the additional factor affecting the viscosity is the viscosity ratio (ratio of nanodroplet viscosity to base fluid viscosity).
- •
- The relative viscosity data for different nanofluids can be collapsed together on to a single unique curve if the data are plotted as relative viscosity versus volume fraction of solvated nanoparticles/nanodroplets. This scaling approach is valid for both nanosuspensions and nanoemulsions.
- •
- A new modified version of the Oldroyd model describes the relative viscosity versus particulate concentration behavior of nanoemulsions and nanosuspensions reasonably well. The model takes into consideration the influences of the viscosity ratio, solvation and aggregation of nanoparticles/nanodroplets.
- •
- The influence of the viscosity ratio on the relative viscosity of nanoemulsions is important. The relative viscosity of a nanoemulsion increases substantially with the increase in the viscosity ratio.
- •
- Systematic experimental studies on the effect of viscosity ratio on viscous behavior of nanoemulsions are lacking. More work needs to be done in this area.

## Acknowledgements

## Conflicts of Interest

## References

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**Figure 1.**Comparison of experimental viscosity data for nanoemulsions [44] with the predictions of the Oldroyd model.

**Figure 2.**Comparison of experimental viscosity data for nanosuspensions [14] with the predictions of the Oldroyd model.

**Figure 3.**Effective volume fraction ${\mathsf{\phi}}_{eff}$ versus volume fraction of un-aggregated solvated particles ${\mathsf{\phi}}_{s}$ generated from Equations (34) to (38).

**Figure 4.**Experimental data for the viscosity of oil-in-water nanoemulsions of different droplet sizes.

**Figure 6.**Estimation of intrinsic viscosity of nanoemulsion (set 4, see Table 2).

**Figure 7.**Estimation of intrinsic viscosity of nanosuspension (set 6, see Table 2).

Set No | Type of Nanofluid | Type and Diameter of un-Solvated Nanoparticles (nm) | Temperature (°C) | Reference |
---|---|---|---|---|

1 | nanoemulsion | oil nanodroplets; 205 nm | 20 | Van der Waarden [44] |

2 | nanoemulsion | oil nanodroplets; 102 nm | 20 | Van der Waarden [44] |

3 | nanoemulsion | oil nanodroplets; 58.5 nm | 20 | Van der Waarden [44] |

4 | nanoemulsion | oil nanodroplets; 27.5 nm | 20 | Van der Waarden [44] |

5 | nanosuspension | Al_{2}O_{3} ; 36 nm | 22–25 | Nguyen et al. [14] |

6 | nanosuspension | Al_{2}O_{3} ; 47 nm | 22–25 | Nguyen et al. [14] |

7 | nanosuspension | CuO ; 29 nm | 22–25 | Nguyen et al. [18] |

8 | nanosuspension | Poly(styrene) latex; 146 nm | 20 | Weiss et al. [53] |

9 | nanosuspension | Polymer; 56 nm | 20 | Jones et al. [54] |

10 | nanosuspension | Silica; 50 nm | 20 | Jones et al. [55] |

**Table 2.**Intrinsic viscosity, solvation coefficient, thickness of solvation layer, and ${\mathsf{\phi}}_{SL}$ of the nanofluids.

Set No | Type of Nanofluid and Diameter (nm) | Intrinsic Viscosity, [η] | Solvation Coefficient, k_{s} | Thickness of Solvation Nanolayer, $\mathsf{\delta}$ (nm) | Volume Fraction of Solvated Layer in the Solvated Droplet, ${\mathsf{\phi}}_{\mathit{S}\mathit{L}}$ |
---|---|---|---|---|---|

1 | Nanoemulsion (205) | 2.65 | 1.077 | 2.58 | 0.072 |

2 | Nanoemulsion (102) | 3.05 | 1.243 | 3.83 | 0.195 |

3 | Nanoemulsion (58.5) | 3.8 | 1.555 | 4.64 | 0.357 |

4 | Nanoemulsion (27.5) | 4.9 | 2.018 | 3.63 | 0.504 |

5 | Nanosuspension (36) | 5.65 | 2.26 | 5.62 | 0.557 |

6 | Nanosuspension (47) | 10 | 4.0 | 13.80 | 0.75 |

7 | Nanosuspension (29) | 11 | 4.4 | 9.26 | 0.773 |

8 | Nanosuspension (146) | 3.9 | 1.56 | 11.66 | 0.359 |

9 | Nanosuspension (56) | 3.39 | 1.36 | 3.02 | 0.265 |

10 | Nanosuspension (50) | 2.5 | 1.0 | $\approx $0 | $\approx $0 |

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Pal, R.
Modeling the Viscosity of Concentrated Nanoemulsions and Nanosuspensions. *Fluids* **2016**, *1*, 11.
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Pal R.
Modeling the Viscosity of Concentrated Nanoemulsions and Nanosuspensions. *Fluids*. 2016; 1(2):11.
https://doi.org/10.3390/fluids1020011

**Chicago/Turabian Style**

Pal, Rajinder.
2016. "Modeling the Viscosity of Concentrated Nanoemulsions and Nanosuspensions" *Fluids* 1, no. 2: 11.
https://doi.org/10.3390/fluids1020011