# Size-Exclusion Particle Separation Driven by Micro-Flows in a Quasi-Spherical Droplet: Modelling and Experimental Results

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## Abstract

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## 1. Introduction

## 2. Experimental Design and Model

#### 2.1. Fabrication of Nano-Patterned Super-Hydrophobic Surfaces

#### 2.2. Particle Image Velocimetry (PIV) Analysis of Fluid Flows in an Evaporating Microliter Drop

#### 2.3. Finite Elements Method (FEM) Simulations of Recirculating Marangoni Flows in a drying droplet upon a Superhydrophobic Surface

#### 2.3.1. Mathematical Background

_{r}is the mean velocity of the flux along the direction parallel to the substrate. Based on the values in Table 2, we estimate a low value of $\mathrm{Re}~0.003,$ implying that the contribution of the inertial forces can be neglected and indicating the presence of lamellar flow. The heat transfer is expressed as:

#### 2.3.2. Boundary Conditions

#### 2.3.3. Numerical Model Implementation

#### 2.4. Numerical Solution of the Langevin Equation and Solute Distribution in a Drop

## 3. Results

#### 3.1. Experimental Analysis of Fluid Flow Fields in the Drop

_{0}= 0.05 mm, y

_{0}= 0.89 mm, marked with a white arrow, and the final position at the bottom is reached after about 20 s. The trajectory of the particle was calculated from subsequent PIV images during the evaporation of the droplet using a method written in MATLAB (R2017b, MathWorks) by G.M., which includes the applications of image processing algorithms and a particle tracking method (for more details, contact G.M.). Particle tracking was limited to larger particles. However, as shown in the video of Supplementary Material, smaller particles continue circulating after 2 min. The recirculation goes on for a longer time until the volume of the drying droplet is further shrunk.

#### 3.2. FEM Simulations of Fluid Flow Fields in the Drop

#### 3.3. Solute Transport in the Droplet and Size-Dependent Particle Separation

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Scanning Eelectron Micrograph image of the pillared surface and zoom in an onset (

**A**). Top view optical image of the drying droplet on the Superhydrophobic SurfaceS (

**B**). Residue of polystyrene particles after evaporation (

**C**). Particle image velocimetry (PIV) setup and speckle imaging. The $\mathrm{x}/\mathrm{y}/\mathrm{z}$ stage allows positioning the droplet relative to the sheet-like $0.2\times 5.6{\mathrm{mm}}^{2}$ laser beam (

**D**).

**Figure 2.**Particle image velocimetry (PIV) analysis of the water droplet loaded with polystyrene (PS) particles evaporating on an superhydrophobic surface (SHS). (

**a**,

**c**) Vector flow field of droplet at $\mathrm{t}=5$ and $20\mathrm{min}$ after the start of evaporation. The limit of the droplet is schematically indicated by a dashed yellow circle. The local magnitude of the vortex-like flow is proportional to the length of the vectors. (

**b**,

**d**) Heat maps of velocity magnitude distribution (in $\mathrm{mm}/\mathrm{s}$) overlaying the vector distribution at $\mathrm{t}=5$ and $20\mathrm{min}$. Note the change in vector orientation for the two evaporation times.

**Figure 5.**Finite element method (FEM) analysis of the evaporating droplet and its environment for $21\xb0\mathrm{C}$ and $40\%$ humidity. (

**a**) Heat map of vapor concentration based on parula color map array (also

**b**,

**d**. (

**b**) Heat map of the evaporation flux. (

**c**) Evaporation flux along the rim of the droplet. (

**d**) Heat map of temperature distribution. Overlay of temperature gradient curves at the rim ($\mathsf{\Delta}\mathrm{T}$) for central recirculating flow (solid red curve) and recirculating flows in two hemispheres (dotted yellow curve). Vertical axis: $\mathsf{\Delta}\mathrm{T}$ ($\xb0\mathrm{C}$), horizontal axis: ($\mathrm{mm}$). The same as for the droplet.

**Figure 6.**Recirculating flows in the two hemispheres of the droplet revealed by flow vectors. Evaporation flux vectors normal to rim. Tangential stress vectors in pink (

**A**). Central recirculating flow (

**B**). Recirculating flows in one hemisphere and central recirculating flow shown with a refined mesh (

**C**,

**D**). Heat map of magnitude of flow vectors for recirculating flow in two hemispheres (

**E**). Heat map of magnitude of flow vectors central recirculating flows (

**F**).

**Figure 7.**The substrate used in the study uses super-hydrophobic surfaces for maintaining solutions in a quasi-spherical shape (

**a**). Because of its curvature and a temperature gradient, in the drop, develop convective flows with characteristic streamlines reported in (

**b**) and values of velocity along the horizontal ($x$) (

**c**) and vertical ($y$) (

**d**) direction that vary between $0$ and a maximum of $~350\mathsf{\mu}\mathrm{m}/\mathrm{s}$ for the $x$ coordinate and of $~150\mathsf{\mu}\mathrm{m}/\mathrm{s}$ for the $y$ coordinate. The flow fields determined experimentally were used to estimate the transport of large ($36\mathsf{\mu}\mathrm{m}$–(

**e**)) and small ($3\mathsf{\mu}\mathrm{m}$–(

**f**)) particulates within the droplet. The particle density plot against the position in the drop reported for different particle sizes (

**g**) illustrates that larger particles are mostly transported toward the center of the substrate.

Equation Name | Equation | Approximation | Equation after Approximation |
---|---|---|---|

Diffusion | $\mathrm{D}\mathsf{\Delta}\mathrm{C}=\frac{\partial \mathrm{C}}{\partial \mathrm{t}}$ | $\mathsf{\rho}>>\mathrm{C}$ | $\mathsf{\Delta}\mathrm{C}=0$ |

Navier-Stokes | $\mathsf{\mu}\mathsf{\Delta}\overrightarrow{\mathrm{v}}-\nabla \mathrm{p}-\mathsf{\rho}\overrightarrow{\mathrm{v}}\nabla \overrightarrow{\mathrm{v}}-\mathsf{\rho}\frac{\partial \overrightarrow{\mathrm{v}}}{\partial \mathrm{t}}=0$ | $\mathrm{Re}=\frac{\mathsf{\rho}{\mathrm{u}}_{\mathrm{r}}\mathrm{R}}{\mathsf{\mu}}\approx 0$ | $\mathsf{\Delta}\overrightarrow{\mathrm{v}}=0$ |

Heat transfer | $\frac{\partial \mathrm{T}}{\partial \mathrm{t}}+\overrightarrow{\mathrm{v}}\nabla \mathrm{T}=\mathrm{k}\mathsf{\Delta}\mathrm{T}$ | $\mathrm{S}{\mathrm{t}}^{-1}<<0$ | $\mathsf{\Delta}\mathrm{T}=0$ |

Variable | Significance | Value |
---|---|---|

$\mathsf{\rho}\left(\mathrm{g}/{\mathrm{cm}}^{3}\right)$ | density of liquid phase | $1$ |

$\mathsf{\mu}\left(\mathrm{Pa}\mathrm{s}\right)$ | viscosity | ${10}^{-3}$ |

${\overline{\mathrm{v}}}_{\mathrm{r}}\left(\mathrm{m}/\mathrm{s}\right)$ | mean velocity of the flux along the direction parallel to the substrate | $~{10}^{-6}$ |

$\mathrm{R}\left(\mathrm{m}\right)$ | radius of contact area of the droplet with the substrate | $~{10}^{-3}$ |

$\mathrm{Re}$ | Reynolds number | $~{10}^{-3}$ |

$\mathsf{\vartheta}(\xb0)$ | droplet contact angle | $\ge 150\xb0$ on SHS |

$\mathrm{D}\left({\mathrm{mm}}^{2}/\mathrm{s}\right)$ | diffusion coefficient in air | $26.1$ |

$\mathrm{C}\left(\mathrm{mol}/{\mathrm{m}}^{3}\right)$ | vapor molar concentration | |

$\mathrm{Q}\left(\mathrm{kJ}/\mathrm{kg}\right)$ | latent heat of water evaporation | $2264$ |

$\mathrm{k}\left(\mathrm{W}/\mathrm{m}\mathrm{K}\right)$ | thermal conductivity |

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**MDPI and ACS Style**

Marinaro, G.; Riekel, C.; Gentile, F. Size-Exclusion Particle Separation Driven by Micro-Flows in a Quasi-Spherical Droplet: Modelling and Experimental Results. *Micromachines* **2021**, *12*, 185.
https://doi.org/10.3390/mi12020185

**AMA Style**

Marinaro G, Riekel C, Gentile F. Size-Exclusion Particle Separation Driven by Micro-Flows in a Quasi-Spherical Droplet: Modelling and Experimental Results. *Micromachines*. 2021; 12(2):185.
https://doi.org/10.3390/mi12020185

**Chicago/Turabian Style**

Marinaro, Giovanni, Christian Riekel, and Francesco Gentile. 2021. "Size-Exclusion Particle Separation Driven by Micro-Flows in a Quasi-Spherical Droplet: Modelling and Experimental Results" *Micromachines* 12, no. 2: 185.
https://doi.org/10.3390/mi12020185