# Optimized Simulation and Validation of Particle Advection in Asymmetric Staggered Herringbone Type Micromixers

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Mixing at the Microscale

#### 1.2. Chaotic Advection

## 2. Experimental Section

#### 2.1. Applied Geometries

**Figure 1.**Parameters of the Staggered Herringbone Mixer (SHM): (

**a**) width (W) and height (H) of the channel, depth (h) and width of the herringbones (a), and the schematic structure of the overall microfluidic system (

**b**) including inlets and outlet and one mixing unit.

**Table 1.**Structural parameters of the applied micromixers. Width of the herringbone grooves, number of herringbones per half unit, and the number of mixing units were considered in our study.

Herringbone groove width | Number of herringbones per half unit | Number of mixing units | Name |
---|---|---|---|

30 μm | 4 | 6 | 30/4/6 |

30 μm | 6 | 4 | 30/6/4 |

35 μm | 4 | 5 | 35/4/5 |

35 μm | 5 | 4 | 35/5/4 |

40 μm | 4 | 5 | 40/4/5 |

40 μm | 5 | 4 | 40/5/4 |

#### 2.2. Modeling

^{3}, kinematic viscosity: 10

^{−6}m

^{2}/s) were applied as material parameters. The maximal 0.0112 Reynolds number was calculated by the simulator solver considering 0.002 m/s average inlet velocity. The average Reynolds number was estimated to be 0.00244.

Boundary | Model | Boundary condition | Value |
---|---|---|---|

Inlet | CFD | Laminar inflow with average flow velocity | 0.002 m/s |

Inlet | Trajectory | Particle inlet | 6000 particles with uniform density |

Inflow_{1} | Concentration | Concentration | 100 mol/m^{3} |

Inflow_{2} | Concentration | Concentration | 0 mol/m^{3} |

Channel wall | CFD | No slip | - |

Channel wall | Trajectory | Bounce (for the trajectory model) | - |

Channel wall | Concentration | No flux | - |

^{3}on the left side and zero on the right side of the inlet section. At the channel walls and the outlet, no flux and outflow boundary conditions were defined, respectively. Diffusion coefficient of the model molecule was set to 10

^{−10}m

^{2}/s. The mesh resolution was refined to avoid numerical diffusion effects [13], which could be a key issue in modeling molecular diffusion processes and typically induces false mixing effects in the channel at low mesh resolutions (see Table 3 for more details).

^{3}, particle diameter: 6 μm [14,15]). Spherical particle geometry was used as an approximation of the cell geometry and this approach was in accordance with the experimental methods, as well. Particle trajectories follow the fluid streamlines established by prior hydrodynamic simulations, thus reducing the need for higher mesh resolution and higher computational demand as compared to the concentration-based model.

Mesh name | Number of elements | Average element size (μm) | Standard deviation of element size (μm) |
---|---|---|---|

Fine | 11,828,167 | 2.3752 | 1.8527 |

Medium | 4,922,845 | 3.7606 | 2.4241 |

Low | 1,788,920 | 5.4645 | 3.2021 |

Coarse | 121,404 | 13.8776 | 8.5472 |

#### 2.3. Fabrication

_{3}C)

_{3}SiO[Si(CH

_{3})

_{2}O]

_{n}Si(CH

_{3})

_{3}which has become a versatile material to realize microfluidic test structures due to its transparency, flexibility, reliable geometry transfer, price, and biocompatibility. This material provides an excellent solution for fast prototyping of simple microfluidic test structures, and can also be convenient for more complex demonstrator applications, although its applicability for large scale production is not yet proven considering its long term chemical and biological stability.

**Figure 2.**(

**a**) SU-8 multilayer lithography steps and (

**b**) the 3D SU-8 molding replica imaged by Scanning Electron Microscopy (SEM).

#### 2.4. Measurement

^{−11}m

^{3}/s inlet flow rate).

^{12}L

^{−1}(woman) and the applied yeast cell concentration was 7.12 × 10

^{10}L

^{−1}to avoid clogging and multiple scattering during the measurements. The hematology analyzer uses 25 μL sample volume for one test. The spherical shape used in the model calculations is a good approximation for the yeast cells. To enhance and visualize cell trajectories and the resulting mixing states in the microchannel, dark field imaging was used. This imaging method facilitates the recording of the light scattered from the cells crossing the light beam, and we estimated the local cell concentration from the lateral distribution of the scattered light intensities, i.e. from the local brightness levels of the image. A relative light intensity was calculated by transforming the recorded intensities to follow the following conservation law:

_{0}and y represents the local coordinates in the channel at a given cross-section and w denotes the width of the channel. This calibration ensures that the integral of the relative scattered light intensity is constant and equal to the initial value (0.5 at the inlet).

**Figure 3.**Size distribution of yeast cells and red blood cells. The two cell types have a similar range of cell diameters making yeast cells a well suited model of human cells in our experiments. The significant peaks in the volume distribution of the RBCs and the yeast cells are around 100 fL, and the spherical equivalent diameter belonging to this value is 5.76 μm. In the size distribution of the yeast cell, there is another peak at around 30 fL with an equivalent diameter of 4.16 μm. The total number of counted yeast and human red blood cells was 1.78 × 10

^{6}and 1.25 × 10

^{8}, respectively.

## 3. Results and Discussion

#### 3.1. Modeling Results

**Figure 4.**Velocity field in the cross section of the channel and grooves. Coloring and arrows denote the amplitude (blue: left direction, yellow to red: right direction) and the direction of the y component of the local flow velocity vector, respectively. Rotating effect of the herringbone shaped grooves is clearly observable due to the flow direction modification effect of the grooves.

**Figure 5.**Effect of mesh element size on the quality and reliability of the numerical solution in case of the concentration based model. Concentration fields after the first mixing unit with fine (

**a**), medium (

**b**), low (

**c**) and coarse (

**d**) mesh resolutions. The numerical diffusion caused by the poor mesh resolution is well observable on the more extensive green colored areas on (

**b**) and (

**d**) compared to the respective areas on (

**a**) and (

**c**).

**Figure 6.**Mesh convergence study for the trajectory model, by calculating the ratio of the number of transferred and remained particles. The difference between values is calculated from to the actual and the previous mesh resolution results (depicted with grey bars). Poincaré maps at the outlet for the extremely coarse and the fine mesh show minor differences in the particle trajectories compared to the diffusion model.

**Figure 7.**Mixing evolution of two different solutions modeled by concentration based method (

**a**–

**d**) and particles modeled by trajectory based method (

**e**–

**h**) along the microchannel. The mixing distributions are qualitatively similar which is observable on the similar distributions of shapes on (

**a**–

**d**) compared to (

**e**–

**h**), respectively.

**Figure 8.**Outlines of the mixed particles at the outlet plane of the mixer channel. The color of the lines denote the width of the grooves, line type indicates the number of grooves. Shrinkage of the area with large number of particles at their initial side (left) and expansion of the area rich in particles on the right side is well observable, which results in the narrowing of the unmixed region in the middle of the channel.

Geometry | 40/5/4 | 40/4/5 | 35/5/4 | 35/4/5 | 30/4/6 | 30/6/4 |
---|---|---|---|---|---|---|

Efficiency | 0.4526 | 0.4513 | 0.3655 | 0.2998 | 0.2105 | 0.1984 |

#### 3.2. Experimental Validation

**Figure 9.**Modeled and measured concentration fields in case of diffusion based approach. (

**a**) Result of finite element concentration modeling. (

**b**) Mixing of food dies in the microchannel monitored by bright field microscopy imaging. (

**c**) The recorded histograms represent the calculated and measured relative concentrations along a line section perpendicular to the channel direction.

**Figure 10.**Relative scattered light intensity distribution corresponding to the local yeast cell concentrations recorded by dark field microscopy. (

**a**) The cell-depleted region is clearly observable and coinciding with the food color dye concentration field presented in Figure 9. (

**b**) The histogram extracted from the intensity field along a line section perpendicular to the channel direction is in good agreement with the modeled Poincaré map evolved at the corresponding plane.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Tóth, E.L.; Holczer, E.G.; Iván, K.; Fürjes, P.
Optimized Simulation and Validation of Particle Advection in Asymmetric Staggered Herringbone Type Micromixers. *Micromachines* **2015**, *6*, 136-150.
https://doi.org/10.3390/mi6010136

**AMA Style**

Tóth EL, Holczer EG, Iván K, Fürjes P.
Optimized Simulation and Validation of Particle Advection in Asymmetric Staggered Herringbone Type Micromixers. *Micromachines*. 2015; 6(1):136-150.
https://doi.org/10.3390/mi6010136

**Chicago/Turabian Style**

Tóth, Eszter L., Eszter G. Holczer, Kristóf Iván, and Péter Fürjes.
2015. "Optimized Simulation and Validation of Particle Advection in Asymmetric Staggered Herringbone Type Micromixers" *Micromachines* 6, no. 1: 136-150.
https://doi.org/10.3390/mi6010136