# Colloidal Shear-Thickening Fluids Using Variable Functional Star-Shaped Particles: A Molecular Dynamics Study

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Shear Thickening Due to Functional Particle Infusion

#### 3.2. Pressure, Viscosity, Diffusivity and the Onset of Jamming

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

MR | Magnetorheological |

ER | Electrorheological |

SRD | Stochastic rotation dynamics |

DPD | Dissipative particle dynamics |

LAMMPS | Large-scale Atomic/Molecular Massively Parallel Simulator |

GA | Geometric asperity |

## References

- De Vicente, J.; Klingenberg, D.J.; Hidalgo-Alvarez, R. Magnetorheological fluids: A review. Soft Matter
**2011**, 7, 3701–3710. [Google Scholar] [CrossRef] - Jolly, M.R.; Carlson, J.D.; Munoz, B.C. A model of the behaviour of magnetorheological materials. Smart Mater. Struct.
**1996**, 5, 607. [Google Scholar] [CrossRef] - Ruzicka, M. Electrorheological Fluids: Modeling and Mathematical Theory; Springer Science & Business Media: New York, NY, USA, 2000. [Google Scholar]
- Jordan, T.C.; Shaw, M.T. Electrorheology. IEEE Trans. Electr. Insul.
**1989**, 24, 849–878. [Google Scholar] [CrossRef] - Coulter, J.P.; Weiss, K.D.; Carlson, J.D. Engineering applications of electrorheological materials. J. Intell. Mater. Syst. Struct.
**1993**, 4, 248–259. [Google Scholar] [CrossRef] - Hong, S.; Choi, S.B.; Jung, W.; Jeong, W. Vibration isolation of structural systems using squeeze mode ER mounts. J. Intell. Mater. Syst. Struct.
**2002**, 13, 421–424. [Google Scholar] [CrossRef] - Tan, K.; Stanway, R.; Bullough, W. Shear mode ER transfer function for robotic applications. J. Phys. D Appl. Phys.
**2005**, 38, 1838. [Google Scholar] [CrossRef] - Atkin, R.; Ellam, D.; Bullough, W. Electro-structured fluid seals. J. Intell. Mater. Syst. Struct.
**2002**, 13, 459–464. [Google Scholar] [CrossRef] - Ouellette, J. Smart fluids move into the marketplace. Ind. Phys.
**2004**, 9, 14–17. [Google Scholar] - Janocha, H.; Clephas, B.; Claeyssen, F.; Hesselbach, J.; Bullough, W.; Carlson, J.D.; zur Megede, D.; Wurmus, H.; Kallenbach, M. Actuators in adaptronics. In Adaptronics and Smart Structures; Springer: Berlin/Heidelberg, Germany, 1999; pp. 99–240. [Google Scholar]
- Stanway, R. Smart fluids: Current and future developments. Mater. Sci. Technol.
**2004**, 20, 931–939. [Google Scholar] [CrossRef] - Fox, R.W. AT McDonald. Introduction to Fluid Mechanics; Wiley: New York, NY, USA, 2011. [Google Scholar]
- Viswanath, D.S.; Ghosh, T.K.; Prasad, D.H.; Dutt, N.V.; Rani, K.Y. Theories of Viscosity. In Viscosity of Liquids; Springer: Berlin/Heidelberg, Germany, 2007; pp. 109–133. [Google Scholar]
- Müller-Plathe, F. Reversing the perturbation in nonequilibrium molecular dynamics: An easy way to calculate the shear viscosity of fluids. Phys. Rev. E
**1999**, 59, 4894. [Google Scholar] [CrossRef] [Green Version] - Kirova, E.; Norman, G. Viscosity calculations at molecular dynamics simulations. J. Physics: Conf. Ser.
**2015**, 653, 012106. [Google Scholar] [CrossRef] [Green Version] - Lee, S.H.; Cummings, P.T. Shear viscosity of model mixtures by nonequilibrium molecular dynamics. I. Argon–krypton mixtures. J. Chem. Phys.
**1993**, 99, 3919–3925. [Google Scholar] [CrossRef] [Green Version] - Olsson, P.; Teitel, S. Critical scaling of shear viscosity at the jamming transition. Phys. Rev. Lett.
**2007**, 99, 178001. [Google Scholar] [CrossRef] [Green Version] - Hecht, M.; Harting, J.; Bier, M.; Reinshagen, J.; Herrmann, H.J. Shear viscosity of claylike colloids in computer simulations and experiments. Phys. Rev. E
**2006**, 74, 021403. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rao, M.A. Flow and functional models for rheological properties of fluid foods. In Rheology of Fluid, Semisolid, and Solid Foods; Springer: Berlin/Heidelberg, Germany, 2014; pp. 27–61. [Google Scholar]
- Gravish, N.; Franklin, S.V.; Hu, D.L.; Goldman, D.I. Entangled granular media. Phys. Rev. Lett.
**2012**, 108, 208001. [Google Scholar] [CrossRef] [Green Version] - Heine, D.R.; Petersen, M.K.; Grest, G.S. Effect of particle shape and charge on bulk rheology of nanoparticle suspensions. J. Chem. Phys.
**2010**, 132, 184509. [Google Scholar] [CrossRef] [Green Version] - Papanikolaou, S.; O’Hern, C.S.; Shattuck, M.D. Isostaticity at frictional jamming. Phys. Rev. Lett.
**2013**, 110, 198002. [Google Scholar] [CrossRef] [Green Version] - Liu, A.; Nagel, S. (Eds.) Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales; Taylor&Francis: London, UK, 2001. [Google Scholar]
- Liu, A.J.; Nagel, S.R. Jamming is not just cool any more. Nature
**1998**, 396, 21–22. [Google Scholar] [CrossRef] - Marschall, T.A.; Franklin, S.V.; Teitel, S. Compression-and shear-driven jamming of U-shaped particles in two dimensions. Granul. Matter
**2015**, 17, 121–133. [Google Scholar] [CrossRef] [Green Version] - Bi, D.; Zhang, J.; Chakraborty, B.; Behringer, R.P. Jamming by shear. Nature
**2011**, 480, 355–358. [Google Scholar] [CrossRef] - van Hecke, M. Jamming of soft particles: Geometry, mechanics, scaling and isostaticity. J. Phys. Condens. Matter
**2009**, 22, 033101. [Google Scholar] [CrossRef] - Torquato, S.; Stillinger, F.H. Jammed hard-particle packings: From Kepler to Bernal and beyond. Rev. Mod. Phys.
**2010**, 82, 2633. [Google Scholar] [CrossRef] [Green Version] - Shen, T.; Papanikolaou, S.; O’Hern, C.S.; Shattuck, M.D. Statistics of frictional families. Phys. Rev. Lett.
**2014**, 113, 128302. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zhang, K.; Wang, M.; Papanikolaou, S.; Liu, Y.; Schroers, J.; Shattuck, M.D.; O’Hern, C.S. Computational studies of the glass-forming ability of model bulk metallic glasses. J. Chem. Phys.
**2013**, 139, 124503. [Google Scholar] - Sims, N.D.; Stanway, R.; Johnson, A.R.; Peel, D.J.; Bullough, W.A. Smart fluid damping: Shaping the force/velocity response through feedback control. J. Intell. Mater. Syst. Struct.
**2000**, 11, 945–958. [Google Scholar] [CrossRef] - Cho, S.W.; Jung, H.J.; Lee, I.W. Smart passive system based on magnetorheological damper. Smart Mater. Struct.
**2005**, 14, 707. [Google Scholar] [CrossRef] - Jung, H.J.; Kim, I.H.; Koo, J.H. A multi-functional cable-damper system for vibration mitigation, tension estimation and energy harvesting. Smart Struct. Syst.
**2011**, 7, 379–392. [Google Scholar] [CrossRef] - Shäfer, J.; Dippel, S.; Wolf, D. Force schemes in simulations of granular materials. J. Phys. I
**1996**, 6, 5–20. [Google Scholar] [CrossRef] - Espanol, P. Fluid particle model. Phys. Rev. E
**1998**, 57, 2930. [Google Scholar] [CrossRef] [Green Version] - Espanol, P.; Warren, P. Statistical mechanics of dissipative particle dynamics. EPL Europhys. Lett.
**1995**, 30, 191. [Google Scholar] [CrossRef] [Green Version] - Hütter, M. Local structure evolution in particle network formation studied by brownian dynamics simulation. J. Colloid Interface Sci.
**2000**, 231, 337–350. [Google Scholar] [CrossRef] [PubMed] - Ladd, A.; Verberg, R. Lattice-Boltzmann simulations of particle-fluid suspensions. J. Stat. Phys.
**2001**, 104, 1191–1251. [Google Scholar] [CrossRef] - Komnik, A.; Harting, J.; Herrmann, H. Transport phenomena and structuring in shear flow of suspensions near solid walls. J. Stat. Mech. Theory Exp.
**2004**, 2004, P12003. [Google Scholar] [CrossRef] [Green Version] - Brady, J.F.; Bossis, G. Stokesian dynamics. Annu. Rev. Fluid Mech.
**1988**, 20, 111–157. [Google Scholar] [CrossRef] - Phung, T.N.; Brady, J.F.; Bossis, G. Stokesian dynamics simulation of Brownian suspensions. J. Fluid Mech.
**1996**, 313, 181–207. [Google Scholar] [CrossRef] [Green Version] - Brady, J.F. The rheological behavior of concentrated colloidal dispersions. J. Chem. Phys.
**1993**, 99, 567–581. [Google Scholar] [CrossRef] [Green Version] - Inoue, Y.; Chen, Y.; Ohashi, H. Development of a simulation model for solid objects suspended in a fluctuating fluid. J. Stat. Phys.
**2002**, 107, 85–100. [Google Scholar] [CrossRef] - Gompper, G.; Ihle, T.; Kroll, D.; Winkler, R. Multi-particle collision dynamics: A particle-based mesoscale simulation approach to the hydrodynamics of complex fluids. In Advanced Computer Simulation Approaches for Soft Matter Sciences III; Holm, C., Kremer, K., Eds.; Springer: Berlin/Heidelberg, Germany, 2009; pp. 1–87. [Google Scholar]
- Plimpton, S. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] [Green Version] - Lykov, K.; Li, X.; Lei, H.; Pivkin, I.V.; Karniadakis, G.E. Inflow/outflow boundary conditions for particle-based blood flow simulations: Application to arterial bifurcations and trees. PLoS Comput. Biol.
**2015**, 11, e1004410. [Google Scholar] [CrossRef] [PubMed] - Miskin, M.Z.; Jaeger, H.M. Adapting granular materials through artificial evolution. Nat. Mater.
**2013**, 12, 326–331. [Google Scholar] [CrossRef] - Li, Y.; Li, X.; Li, Z.; Gao, H. Surface-structure-regulated penetration of nanoparticles across a cell membrane. Nanoscale
**2012**, 4, 3768–3775. [Google Scholar] [CrossRef] - Bogart, L.K.; Pourroy, G.; Murphy, C.J.; Puntes, V.; Pellegrino, T.; Rosenblum, D.; Peer, D.; Lévy, R. Nanoparticles for imaging, sensing, and therapeutic intervention. ACS Nano
**2014**, 8, 3107–3122. [Google Scholar] [CrossRef] - Van Lehn, R.C.; Ricci, M.; Silva, P.H.; Andreozzi, P.; Reguera, J.; Voïtchovsky, K.; Stellacci, F.; Alexander-Katz, A. Lipid tail protrusions mediate the insertion of nanoparticles into model cell membranes. Nat. Commun.
**2014**, 5, 1–11. [Google Scholar] [CrossRef] [PubMed] - O’hern, C.S.; Silbert, L.E.; Liu, A.J.; Nagel, S.R. Jamming at zero temperature and zero applied stress: The epitome of disorder. Phys. Rev. E
**2003**, 68, 011306. [Google Scholar] [CrossRef] [Green Version] - Donev, A.; Torquato, S.; Stillinger, F.H.; Connelly, R. Jamming in hard sphere and disk packings. J. Appl. Phys.
**2004**, 95, 989–999. [Google Scholar] [CrossRef] [Green Version] - Donev, A.; Connelly, R.; Stillinger, F.H.; Torquato, S. Underconstrained jammed packings of nonspherical hard particles: Ellipses and ellipsoids. Phys. Rev. E
**2007**, 75, 051304. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zeravcic, Z.; Xu, N.; Liu, A.; Nagel, S.; van Saarloos, W. Excitations of ellipsoid packings near jamming. EPL Europhys. Lett.
**2009**, 87, 26001. [Google Scholar] [CrossRef] [Green Version] - Mailman, M.; Schreck, C.F.; O’Hern, C.S.; Chakraborty, B. Jamming in systems composed of frictionless ellipse-shaped particles. Phys. Rev. Lett.
**2009**, 102, 255501. [Google Scholar] [CrossRef] [Green Version] - Hidalgo, R.C.; Zuriguel, I.; Maza, D.; Pagonabarraga, I. Role of particle shape on the stress propagation in granular packings. Phys. Rev. Lett.
**2009**, 103, 118001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Desmond, K.; Franklin, S.V. Jamming of three-dimensional prolate granular materials. Phys. Rev. E
**2006**, 73, 031306. [Google Scholar] [CrossRef] [Green Version] - Azéma, E.; Radjaï, F. Stress-strain behavior and geometrical properties of packings of elongated particles. Phys. Rev. E
**2010**, 81, 051304. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jiao, Y.; Torquato, S. Maximally random jammed packings of Platonic solids: Hyperuniform long-range correlations and isostaticity. Phys. Rev. E
**2011**, 84, 041309. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications; Elsevier: Amsterdam, The Netherlands, 2001; Volume 1. [Google Scholar]
- Malevanets, A.; Kapral, R. Mesoscopic model for solvent dynamics. J. Chem. Phys.
**1999**, 110, 8605–8613. [Google Scholar] [CrossRef] - Malevanets, A.; Kapral, R. Solute molecular dynamics in a mesoscale solvent. J. Chem. Phys.
**2000**, 112, 7260–7269. [Google Scholar] [CrossRef] [Green Version] - Hecht, M.; Harting, J.; Ihle, T.; Herrmann, H.J. Simulation of claylike colloids. Phys. Rev. E
**2005**, 72, 011408. [Google Scholar] [CrossRef] [Green Version] - Petersen, M.K.; Lechman, J.B.; Plimpton, S.J.; Grest, G.S.; in’t Veld, P.J.; Schunk, P. Mesoscale hydrodynamics via stochastic rotation dynamics: Comparison with Lennard-Jones fluid. J. Chem. Phys.
**2010**, 132, 174106. [Google Scholar] [CrossRef] - Bair, S.; Winer, W. The high pressure high shear stress rheology of liquid lubricants. J. Tribol.
**1992**, 114, 1–9. [Google Scholar] [CrossRef] - Moore, J.D.; Parsons, B. Scaling of viscous shear zones with depth-dependent viscosity and power-law stress–strain-rate dependence. Geophys. J. Int.
**2015**, 202, 242–260. [Google Scholar] [CrossRef] [Green Version] - Kubečka, J.; Uhlík, F.; Košovan, P. Mean squared displacement from fluorescence correlation spectroscopy. Soft Matter
**2016**, 12, 3760–3769. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Olsson, P.; Teitel, S. Herschel-Bulkley shearing rheology near the athermal jamming transition. Phys. Rev. Lett.
**2012**, 109, 108001. [Google Scholar] [CrossRef] [PubMed] - Stalter, S.; Yelash, L.; Emamy, N.; Statt, A.; Hanke, M.; Lukáčová-Medvid’ová, M.; Virnau, P. Molecular dynamics simulations in hybrid particle-continuum schemes: Pitfalls and caveats. Comput. Phys. Commun.
**2018**, 224, 198–208. [Google Scholar] [CrossRef] [Green Version] - Grady, D. Strain-rate dependence of the effective viscosity under steady-wave shock compression. Appl. Phys. Lett.
**1981**, 38, 825–826. [Google Scholar] [CrossRef] - Webb, S.L.; Dingwell, D.B. Non-Newtonian rheology of igneous melts at high stresses and strain rates: Experimental results for rhyolite, andesite, basalt, and nephelinite. J. Geophys. Res. Solid Earth
**1990**, 95, 15695–15701. [Google Scholar] [CrossRef] [Green Version] - Magnin, A.; Piau, J. Cone-and-plate rheometry of yield stress fluids. Study of an aqueous gel. J. Non-Newton. Fluid Mech.
**1990**, 36, 85–108. [Google Scholar] [CrossRef] - Olsson, P.; Teitel, S. Critical scaling of shearing rheology at the jamming transition of soft-core frictionless disks. Phys. Rev. E
**2011**, 83, 030302. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Otsuki, M.; Hayakawa, H. Critical scaling near jamming transition for frictional granular particles. Phys. Rev. E
**2011**, 83, 051301. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Mewis, J.; Wagner, N.J. Colloidal Suspension Rheology; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]

**Figure 1.**Functional particle configuration construction, for single (

**a**) three, (

**b**) five and (

**c**) seven legs in a single star particle with core radius of 0.65 σ and leg length of 2.5 σ. Randomly dispersed configurations of 30 star particles in each system for (

**d**) three, (

**e**) five and (

**f**) seven legs on each functional particle. Snapshot of MD simulation movies, before the pure shear applied on the system of star particles, with (

**g**) three, (

**h**) five and (

**i**) seven legs infused in square-shaped (30 σ × 30 σ) periodic boxes filled with solvent SRD molecules and after the pure shear applied on the same system of star particles with (

**j**) three, (

**k**) five and (

**l**) seven legs.

**Figure 2.**Fitting plots of shear stress ($\u03f5/{\sigma}^{3}$) vs. strain rate (1/$\tau $) for: (

**a**) 3-legged star particles with packing fraction, $\varphi $, of 0.18, 0.23, 0.28, 0.34, 0.44, 0.49, 0.54, 0.59, 0.69, 0.74, 0.79, 0.84 and 0.94; (

**b**) 5-legged star particles with packing fraction, $\varphi $, of 0.22, 0.30, 0.40, 0.42, 0.45, 0.47, 0.54, 0.62, 0.72 and 0.80; (

**c**) 7-legged star particles with packing fraction, $\varphi $, of 0.25, 0.37, 0.50, 0.59, 0.60, 0.70, 0.72, 0.74, 0.84 and 0.96. The log–log plot of shear stresses normalized by removing the respective yield stresses with respect to the strain rates for (

**d**) 3-legged, (

**e**) 5-legged and (

**f**) 7-legged star particles.

**Figure 3.**Viscosity ($\u03f5\tau /{\sigma}^{3}$) vs. strain rate (1/$\tau $) with leg length L ($\sigma $) of: (

**a**) 0.0 $\sigma $, 0.5 $\sigma $, 1.0 $\sigma $, 1.5 $\sigma $, 2.0 $\sigma $ and 2.5 $\sigma $ for 3-legged star particles; (

**b**) 0.0 $\sigma $, 0.5 $\sigma $, 1.0 $\sigma $, 1.2 $\sigma $, 1.3 $\sigma $ and 1.5 $\sigma $ for 5-legged star particles; (

**c**) 0.0 $\sigma $, 0.5 $\sigma $, 1.0 $\sigma $, 1.3 $\sigma $ and 1.5 $\sigma $ for 7-legged star particles.

**Figure 4.**Total Pressure and Packing Fraction. (

**a**) Total pressure ($\u03f5/{\sigma}^{3}$) vs. packing fraction ($\varphi $) for strain rates equal to 0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08 and 0.09 for 3-legged star particles, and with $\varphi $ equal 0.18, 0.23, 0.28, 0.34, 0.44, 0.49, 0.54, 0.59, 0.69, 0.74, 0.79, 0.84 and 0.94. (

**b**) $\u03f5/{\sigma}^{3}$ vs. $\varphi $ for same strain-rates as (

**a**), for 5-legged star particles with $\varphi $ equals 0.22, 0.30, 0.40, 0.42, 0.45, 0.47, 0.54, 0.62, 0.72 and 0.80. (

**c**) $\u03f5/{\sigma}^{3}$ vs. $\varphi $ for same strain-rates as (

**a**), for 7-legged star particles with $\varphi $ equals 0.25, 0.37, 0.50, 0.59, 0.60, 0.84 and 0.96. (

**d**–

**f**) The total pressure is shown as function of $\varphi -{\varphi}_{c}$ in log-scale, in correspondence to (

**a**–

**c**).

**Figure 5.**Pressure to strain rate ($\u03f5\tau /{\sigma}^{3}$) vs. strain rate (1/$\tau $) plot for system infused with (

**a**) 3-legged, (

**b**) 5-legged and (

**c**) 7-legged star particles.

**Figure 6.**Average total pressure ($\u03f5/{\sigma}^{3}$) and shear stress ($\u03f5/{\sigma}^{3}$) vs. strain for constant strain rate (1/$\tau $) of 0.001 for: (

**a**,

**d**) 3-legged star particles with packing fraction $\varphi $ of 0.18, 0.23, 0.28, 0.34, 0.44, 0.49, 0.54, 0.59, 0.69, 0.74, 0.84 and 0.94; (

**b**,

**e**) 5-legged star particles with packing fraction $\varphi $ of 0.22, 0.30, 0.40, 0.47, 0.62, 0.72 and 0.80; (

**c**,

**f**) 7-legged star particles with packing fraction $\varphi $ of 0.25, 0.37, 0.50, 0.60, 0.84 and 0.96, with standard error of the mean error bars. Each point of the plots represents the (

**a**–

**c**) average of pressure and (

**d**–

**f**) average of shear stress over a strain window $\Delta t$ = 1000. The non-monotonic behavior seen at very high packing fractions is a direct evidence of yielding behavior from the jammed state of the hybrid fluid/functional particles system.

**Figure 7.**Diffusivity D (${\sigma}^{2}/\tau $) of small SRD fluid and big star particles vs. leg length L ($\sigma $) of (

**a**) 3-legged, (

**b**) 5-legged and (

**c**) 7-legged star particles system; radial distribution function g(r) of big star particles center of mass for radial increment of 0.2 $\sigma $ and maximum radial distance 10 $\sigma $ in: (

**d**) average of all 30 configurations for 3 legs with maximum leg length L = 1.5 $\sigma $, L = 5.0 $\sigma $ and L = 6.0 $\sigma $; (

**e**) average of all 30 configurations for 5 legs with maximum leg length L = 1.0 $\sigma $, L = 2.5 $\sigma $ and L = 3.0 $\sigma $; (

**f**) average of all 30 configurations for 7 legs with maximum leg length L = 1.0 $\sigma $, L = 2.0 $\sigma $ and L = 2.5 $\sigma $.

**Figure 8.**Viscosity ($\u03f5\tau /{\sigma}^{3}$) of the SRD fluids with respect to the (

**a**) leg length L ($\sigma $) and (

**b**) packing fraction $\varphi $ of infused star particles. The number of star particles N = 30 and number of legs ${N}_{leg}=$ 3, 5 and 7 for star particles is indicated with red, green and blue color, respectively. It is worth noting that the viscosity can increase by multiple orders of magnitude, turning water-like fluids into gel-like texture by altering the functional particle shapes. Furthermore, for a given functional particle shape, the viscosity displays a divergence that clearly violates typical colloidal suspension rules, such as the Einstein formula for effective viscosity of a solvent with solutes ${\mu}_{eff}={\mu}_{0}(1+B\varphi )$, where ${\mu}_{0}$ is the solvent viscosity, $\varphi $ the solute packing fraction and B a finite geometrical factor [75].

**Figure 9.**Five-legged star particles with core radius of 0.65 cm and leg lengths of (

**a**) 1.5 cm and (

**b**) 3.0 cm, created by laser cutting on Plexiglas.

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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Salehin, R.; Xu, R.-G.; Papanikolaou, S.
Colloidal Shear-Thickening Fluids Using Variable Functional Star-Shaped Particles: A Molecular Dynamics Study. *Materials* **2021**, *14*, 6867.
https://doi.org/10.3390/ma14226867

**AMA Style**

Salehin R, Xu R-G, Papanikolaou S.
Colloidal Shear-Thickening Fluids Using Variable Functional Star-Shaped Particles: A Molecular Dynamics Study. *Materials*. 2021; 14(22):6867.
https://doi.org/10.3390/ma14226867

**Chicago/Turabian Style**

Salehin, Rofiques, Rong-Guang Xu, and Stefanos Papanikolaou.
2021. "Colloidal Shear-Thickening Fluids Using Variable Functional Star-Shaped Particles: A Molecular Dynamics Study" *Materials* 14, no. 22: 6867.
https://doi.org/10.3390/ma14226867