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

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 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

- Marinaro, G. Contributions to Modeling and Applications of Superhydrophobic Surfaces for Self-Assembly of Biological Materials. Ph.D. Thesis, Università degli studi di Genova, Genova, Italy, 2015. Available online: https://www.tesionline.it/tesi/preview/contributions-to-modeling-and-applications-of-superhydrophobic-surfaces-for-self-assembly-of-biological-materials/55181/1 (accessed on 23 May 2015).
- Marinaro, G.; Accardo, A.; Benseny-Cases, N.; Burghammer, M.; Castillo-Michel, H.; Cotte, M.; Dante, S.; De Angelis, F.; Di Cola, E.; Di Fabrizio, E.; et al. Probing droplets with biological colloidal suspensions on smart surfaces by synchrotron radiation micro- and nano-beams. Opt. Lasers Eng.
**2014**, 76, 57–63. [Google Scholar] [CrossRef] - Marinaro, G.; La Rocca, R.; Toma, A.; Barberio, M.; Cancedda, L.; Di Fabrizio, E.; Decuzzi, P.; Gentile, F. Networks of neuroblastoma cells on porous silicon substrates reveal a small world topology. Integr. Biol.
**2014**, 7, 184–197. [Google Scholar] [CrossRef] [PubMed] - Gentile, F.; Coluccio, M.L.; Coppedè, N.; Mecarini, F.; Das, G.; Liberale, C.; Tirinato, L.; Leoncini, M.; Perozziello, G.; Candeloro, P.; et al. Superhydrophobic Surfaces as Smart Platforms for the Analysis of Diluted Biological Solutions. ACS Appl. Mater. Interfaces
**2012**, 4, 3213–3224. [Google Scholar] [CrossRef] - Sperling, M.; Gradzielski, M. Droplets, Evaporation and a Superhydrophobic Surface: Simple Tools for Guiding Colloidal Particles into Complex Materials. Gels
**2017**, 3, 15. [Google Scholar] [CrossRef][Green Version] - Gentile, F.; Coluccio, M.L.; Accardo, A.; Marinaro, G.; Rondanina, E.; Santoriello, S.; Marras, S.; Daş, G.; Tirinato, L.; Perozziello, G.; et al. Tailored Ag nanoparticles/nanoporous superhydrophobic surfaces hybrid devices for the detection of single molecule. Microelectron. Eng.
**2012**, 97, 349–352. [Google Scholar] [CrossRef][Green Version] - Marinaro, G.; Das, G.; Giugni, A.; Allione, M.; Torre, B.; Candeloro, P.; Kosel, J.; Di Fabrizio, E. Plasmonic Nanowires for Wide Wavelength Range Molecular Sensing. Materials
**2018**, 11, 827. [Google Scholar] [CrossRef][Green Version] - Accardo, A.; Di Fabrizio, E.; Limongi, T.; Marinaro, G.; Riekel, C. Probing droplets on superhydrophobic surfaces by synchrotron radiation scattering techniques. J. Synchrotron Radiat.
**2014**, 21, 643–653. [Google Scholar] [CrossRef][Green Version] - Yang, H.-Y.; Moerner, W.E. Resolving Mixtures in Solution by Single-Molecule Rotational Diffusivity. Nano Lett.
**2018**, 18, 5279–5287. [Google Scholar] [CrossRef][Green Version] - Huang, L.R.; Cox, E.C.; Austin, R.H.; Sturm, J.C. Continuous Particle Separation Through Deterministic Lateral Displacement. Science
**2004**, 304, 987–990. [Google Scholar] [CrossRef] [PubMed] - Gentile, F.; La Rocca, R.; Marinaro, G.; Nicastri, A.; Toma, A.; Paonessa, F.; Cojoc, G.; Liberale, C.; Benfenati, F.; Di Fabrizio, E.; et al. Differential Cell Adhesion on Mesoporous Silicon Substrates. ACS Appl. Mater. Interfaces
**2012**, 4, 2903–2911. [Google Scholar] [CrossRef] [PubMed] - Coppedè, N.; Villani, M.; Gentile, F. Diffusion Driven Selectivity in Organic Electrochemical Transistors. Sci. Rep.
**2015**, 4, 4297. [Google Scholar] [CrossRef][Green Version] - Gentile, F.; Ferrara, L.; Villani, M.; Bettelli, M.; Iannotta, S.; Zappettini, A.; Cesarelli, M.; Di Fabrizio, E.; Coppedé, N. Geometrical Patterning of Super-Hydrophobic Biosensing Transistors Enables Space and Time Resolved Analysis of Biological Mixtures. Sci. Rep.
**2016**, 6, srep18992. [Google Scholar] [CrossRef][Green Version] - Pradhan, T.K.; Panigrahi, P.K. Evaporation induced natural convection inside a droplet of aqueous solution placed on a superhydrophobic surface. Colloids Surfaces A: Physicochem. Eng. Asp.
**2017**, 530, 1–12. [Google Scholar] [CrossRef] - Kang, K.H.; Lim, H.C.; Lee, H.W.; Lee, S.J. Evaporation-induced saline Rayleigh convection inside a colloidal droplet. Phys. Fluids
**2013**, 25, 042001. [Google Scholar] [CrossRef][Green Version] - Pan, Z.; Dash, S.; Weibel, J.A.; Garimella, S.V. Assessment of Water Droplet Evaporation Mechanisms on Hydrophobic and Superhydrophobic Substrates. Langmuir
**2013**, 29, 15831–15841. [Google Scholar] [CrossRef] - Adrian, R.L.; Adrian, J.; Westerweel, J. Particle Image Velocimetry; Cambridge University Press: Cambridge, UK, 2011. [Google Scholar]
- Marinaro, G.; Accardo, A.; De Angelis, F.; Dane, T.; Weinhausen, B.; Burghammer, M.; Riekel, C. A superhydrophobic chip based on SU-8 photoresist pillars suspended on a silicon nitride membrane. Lab Chip
**2014**, 14, 3705–3709. [Google Scholar] [CrossRef][Green Version] - Marinaro, G.; Burghammer, M.; Costa, L.; Dane, T.; De Angelis, F.; Di Fabrizio, E.; Riekel, C. Directed Growth of Virus Nanofilaments on a Superhydrophobic Surface. ACS Appl. Mater. Interfaces
**2015**, 7, 12373–12379. [Google Scholar] [CrossRef] [PubMed] - Thielicke, W.; Stamhuis, E.J. PIVlab—Towards User-friendly, Affordable and Accurate Digital Particle Image Velocimetry in MATLAB. J. Open Res. Softw.
**2014**, 2, e30. [Google Scholar] [CrossRef][Green Version] - Morse, P.M.C.; Feshbach, H. Methods of Theoretical Physics; McGraw-Hill: New York, NY, USA, 1953. [Google Scholar]
- Geuzaine, C.; Remacle, J.F. Gmsh: A Three-Dimensional Finite Element Mesh Generator with Built-In Pre- and Post-processing Facilities. Int. J. Numer. Methods Eng.
**2009**, 79, 1309–1331. [Google Scholar] [CrossRef] - Galerkin, B.G. Rods and Plates: Series in Some Questions of Elastic Equilibrium of Rods and Plates; National Tech-nical Information Service: Alexandria, VA, USA, 1968. [Google Scholar]
- Astier, Y.; Datas, L.; Carney, R.; Stellacci, F.; Gentile, F.; DiFabrizio, E. Artificial Surface-Modified Si3N4 Nanopores for Single Surface-Modified Gold Nanoparticle Scanning. Small
**2010**, 7, 455–459. [Google Scholar] [CrossRef] [PubMed] - Kim, M.-M. Effect of electrostatic, hydrodynamic, and Brownian forces on particle trajectories and sieving in normal flow filtration. J. Colloid Interface Sci.
**2004**, 269, 425–431. [Google Scholar] [CrossRef] [PubMed] - Gentile, F.; Coluccio, M.L.; Zaccaria, R.P.; Francardi, M.; Cojoc, G.; Perozziello, G.; Raimondo, R.; Candeloro, P.; Di Fabrizio, E. Selective on site separation and detection of molecules in diluted solutions with super-hydrophobic clusters of plasmonic nanoparticles. Nanoscale
**2014**, 6, 8208–8225. [Google Scholar] [CrossRef] - Dubin, D.; Diego, S. Numerical and Analytical Methods for Scientists and Engineers, Using Mathematica, USA; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2003. [Google Scholar]
- Pearson, J.E. Complex Patterns in a Simple System. Science
**1993**, 261, 189–192. [Google Scholar] [CrossRef][Green Version] - Hu, H.; Larson, R.G. Evaporation of a Sessile Droplet on a Substrate. J. Phys. Chem. B
**2002**, 106, 1334–1344. [Google Scholar] [CrossRef] - Hu, H.; Larson, R.G. Analysis of the Microfluid Flow in an Evaporating Sessile Droplet. Langmuir
**2005**, 21, 3963–3971. [Google Scholar] [CrossRef] - Hu, H.; Larson, R.G. Analysis of the Effects of Marangoni Stresses on the Microflow in an Evaporating Sessile Droplet. Langmuir
**2005**, 21, 3972–3980. [Google Scholar] [CrossRef] - Gentile, F.; Ferrari, M.; Decuzzi, P. The Transport of Nanoparticles in Blood Vessels: The Effect of Vessel Permeability and Blood Rheology. Ann. Biomed. Eng.
**2008**, 36, 254–261. [Google Scholar] [CrossRef] [PubMed][Green Version] - Wong, T.-S.; Chen, T.-H.; Shen, X.; Ho, C.-M. Nanochromatography Driven by the Coffee Ring Effect. Anal. Chem.
**2011**, 83, 1871–1873. [Google Scholar] [CrossRef]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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