# Optimization and parametric study of AC electroosmotic micropumping by response surface method

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## Abstract

Alternative current electroosmotic micropumps (ACEO) have gained a great interest in microfluidics research due to their low operating potential. In this paper, a parametric study is performed on various geometrical parameters of an ACEO micropump design. The problem is modeled and solved numerically using COMSOL Multiphysics. This micropump relies on the grouping of a periodic electrode array in terms of the applied voltage to create asymmetries in the electrode configuration. The pumping velocity is obtained and investigated for various geometry designs. In addition, the optimal parameters are obtained by employing the response surface methodology (RSM). The RSM could predict the effects of different factors affecting the performance of the system and their interactions. The most important parameters affecting the pumping velocity are identified. It is shown that the pumping velocity is controllable. Three various micropump designs with different cost to benefit ratios are presented. These designs are different due to their fabrication cost and the pumping velocity.

## Keywords

AC electroosmotic flow Lab-on-chip Micropump Optimization Response surface methodology (RSM)## List of symbols

- \(W\)
Electrode width (μm)

- \(G\)
Inter-electrode gap (μm)

*H*Microchannel height (μm)

*ϕ*Electrical potential (V)

*Z*Area-specific impedance (Ω m

^{2})*C*Capacitance per unit area (F m

^{−2})- \(V\)
Electrical potential at electrode (V)

- \(P\)
Pressure (Pa)

- \(\varvec{u}\)
Velocity field (m s

^{−1})*D*Diffusion constant (m

^{2}s^{−1})- \(x\)
Design parameter

## Greeks

- \(\lambda\)
Debye length (m)

- \(\varLambda\)
Ratio of the diffuse layer of EDL potential drop to the total double layer potential drop

- \(\mu\)
Fluid viscosity (Pa s)

*ϵ*Permittivity (F m

^{−1})- \(\omega\)
Angular frequency (rad s

^{−1})- \(\sigma\)
Electrolyte conductivity (S m

^{−1})- \(\beta\)
Polynomial coefficient

## Subscripts

- ep
Outer edge of the EDL

- DL
Double layer

- app
Applied voltage to the electrode

- non
Non-dimensional

## 1 Introduction

ACEO is an effective technique for the manipulation of fluids and particles in microscale systems [1, 2, 3, 4, 5]. ACEO is defined as the flow generation in a microchannel at the surface of electrodes where low AC voltage is applied. It has proved to be a great potential to be employed for the fabrication of micropumps [6, 7]. Such micropumps benefit from low operating potential, use of no moving parts, minimum electrolysis, and consequent elimination of bubble formation and generation of new species over the electrodes [8].

ACEO was firstly introduced by Ramos et al. [9, 10]. Taking advantage of ACEO principles, Ajdari [11] developed an array of asymmetric electrodes for liquid pumping. Various designs of ACEO micropumps have been developed mainly employing asymmetric electrodes or asymmetric applied voltage [12, 13], transverse traveling waveforms [14, 15] and AC voltage with a DC bias [16, 17]. A detailed review of different designs of micropumps and their applications was presented in Ref. [18].

Loucaides et al. [13] proposed a novel ACEO micropump that works by the grouping of electrodes. The asymmetry generated in terms of applied voltage to produce a net flow in the microchannel. The method is useful or pumping using low voltages. They showed that by switching voltages the degree of asymmetry changes; therefore, a bidirectional flow is achieved. In another study, Loucaides et al. [19] investigated a configurable ACEO-based microdevice that showed dual functions of simultaneous pumping and mixing in microcapillaries. They examined various electrode excitation modes that can be applied to a single device. They achieved significant mixing efficiency; however, there is a trade-off between mixing performance and the pumping performance. Yoshida et al. [20] proposed a new ACEO micropump using a square pole—slit electrode array. They showed that the proposed method could generate a total flow by the slip velocity mainly on the slit electrodes. The COMSOL Multiphysics software was used to simulate the micropump and obtain the optimum parameters for maximizing the effective slip velocity. They also fabricated and experimentally studied the micopump. A maximum slip velocity of 1.6 mm s^{−1} was observed in their study.

Commonly, the goal in a pump design is to maximize the pumping rate [21] with minimized power consumption. To this end, few investigations on the optimization of ACEO-based have been performed. Ramos et al. [22] performed a theoretical and numerical analysis on ACEO micropumps with asymmetric pairs of microelectrodes. They obtained the pumping velocity by changing a geometrical parameter while keeping other parameters constant. Olesen et al. [23] used an unconstrained optimization method to optimize the geometry of a ACEO micropump with two asymmetric electrodes. However, they did not use an optimization method to reveal the interaction of various parameters. Surveying the literature suggests that a comprehensive optimization and parametric study on the ACEO micropumps to show the interaction of different parameters in the design of the micropump is rare. This type of ACEO micropump consists of planner electrodes, which has the advantage of simple and straightforward fabrication.

The aim of this study is to investigate the effects of different geometrical parameters on an ACEO micropump and to provide statistical analysis for the performance optimization of the electrode array in an ACEO micropump. A three-electrode micropump is considered in this study because this type of micropump benefits from enabling a bidirectional flow. Additionally, the micropump has the advantage of simple fabrication, configurable geometry, and versatility in terms of applied voltage. The geometrical parameters are optimized to achieve higher pumping velocities; therefore, lower voltages could be applied, and the design would be more energy-efficient and eliminates bubble formation and faradic reactions. In addition, in this study, the different electrode configurations of micropump are investigated and the response surface methodology (RSM) is used to obtain the optimum results. Three various optimum designs are suggested based on their fabrication cost and pumping velocities. The RSM is a collection of mathematical and statistical techniques used to explore the influence and interaction of several independent factors (i.e., the design parameters) on the design responses [24]. Therefore, the identified relation (the response surface) shows the effects of the design parameters using a limited number of controlled experiments/simulations and design parameters can be optimized to produce the most desirable design response [25, 26].

## 2 Theory and mathematical modeling

Since the channel depth is considered much larger than its height, a two-dimensional model is employed. The model is established based on the Debye–Huckel double layer theory [29]. The mathematical model presented by the references [19, 22] is used here. Experimental studies show that for RMS voltage value higher than 1.5 Faradic reactions take place [15, 28, 30]. Therefore, the effects of Faradaic currents are not taken into consideration.

### 2.1 Electrical field

The inlet and outlet of the domain are set to periodic boundary condition. The inter-electrode walls have Neumann boundary condition of \(\varvec{n} \cdot \nabla \cdot \phi = 0\), where \(\varvec{n}\) is the normal to plane vector.

### 2.2 Fluid flow

The properties of electrolyte material used in the numerical study [19]

Property | Value |
---|---|

Dynamic viscosity ( | \(1 \times 10^{ - 3} \,{\text{Pa}}\,{\text{s}}\) |

Relative permittivity ( | 80 |

Vacuum permittivity ( | \(8.8542 \times 10^{ - 12} \,{\text{F}}\,{\text{m}}^{ - 1}\) |

Debye length \((\lambda_{\text{Debye}} )\) | \(3 \times 10^{ - 8} \,{\text{m}}\) |

Conductivity | \(1.23 \times 10^{ - 3} \,{\text{S}}\,{\text{m}}^{ - 1}\) |

Diffusion constant ( | \(10^{ - 10} \,{\text{m}}^{2} \,{\text{s}}^{ - 1}\) |

Microchannel height | \(2 0 0\,\upmu{\text{m}}\) |

The numerical model is established in COMSOL Multiphysics for simultaneous solving of the electrical field and fluid flow equations.

Non-dimensional frequency and slip velocity are employed to facilitate the explanation of the obtained results. Non-dimensional frequency is defined as \(\varOmega = {\raise0.7ex\hbox{${\omega \epsilon L}$} \!\mathord{\left/ {\vphantom {{\omega \epsilon L} {\sigma \lambda_{\text{Debye}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\sigma \lambda_{\text{Debye}} }$}}\) where \(\omega\) is the angular frequency of AC voltage, \(\lambda_{\text{Debye}}\) is the Debye length, \(\sigma\) is the electrolyte conductivity, and \(L\) is the characteristic length of the micropump. Height of the channel is used as the characteristic length (200 μm). The non-dimensional slip velocity on the periodical length of the microelectrode array that indicates the pumping capacity of the micropump is defined as \(U_{\text{non}} = {\raise0.7ex\hbox{${4\mu Lu_{\text{slip}} }$} \!\mathord{\left/ {\vphantom {{4\mu Lu_{\text{slip}} } {\epsilon \left( {\Delta V} \right)^{2} \varLambda }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\epsilon \left( {\Delta V} \right)^{2} \varLambda }$}}\) where \(\mu\) is the fluid viscosity, \(u_{\text{slip}}\) is the fluid velocity at the electrode surface, and \(\Delta V\) is the potential difference between the electrodes (\(\Delta V = 2V_{0}\)). This slip velocity is referred to as the pumping velocity in the present text as it is an indication of pumping capacity.

## 3 Results and discussion

### 3.1 Mesh study and validation

Value of the electrodes parameters used for the baseline case [13]

Parameters | Value |
---|---|

Electrodes widths \(\left( {W_{1} = W_{2} = W_{3} } \right)\) | \(1 0 0\, (\upmu{\text{m)}}\) |

Inter-electrodes gaps \(\left( {G_{1} = G_{2} } \right)\) | \(1 0\, (\upmu{\text{m)}}\) |

Third inter-electrodes gap \(\left( {G_{3} } \right)\) | \(2 8 0\, (\upmu{\text{m)}}\) |

Voltage \((V_{0} )\) | \(0. 5\, ( {\text{V)}}\) |

Frequency \(\left( f \right)\) | \(2 5 0\, ( {\text{Hz)}}\) |

### 3.2 Parametric study

Maximum pumping velocity increases once \(W_{2}\) and \(W_{3}\) are increased; however, by increasing the \(G_{1}\), \(G_{2}\), and \(G_{3}\) the maximum velocity decreases. Moreover, the increase of \(W_{1}\) value to 0.2 results in the initial increase of maximum velocity followed by its decline. For most cases, the maximum pumping velocity obtains at the non-dimensional frequency of 0.5. Figure 5a reveals that the maximum pumping velocity occurs at the non-dimensional frequency of 0.55. Furthermore, Figs. 6 and 7 indicate that the second and third gap (\(G_{2}\) and \(G_{3}\)) have a small effect on the pumping velocity.

Figure 8 shows that using \(W_{1} = 0.2\) results in the highest pumping velocity. It can be seen from Fig. 8 that there is a highly nonlinear relation between the parameter \(W_{1}\) and pumping velocity, making \(W_{1}\) an important parameter in the design of an ACEO micropump. Furthermore, Fig. 9 (a) shows that using \(W_{2} = 0.1\) would cause flow reversal at higher AC frequencies. Figure 10 also shows that \(W_{3}\) affects the pumping velocity significantly.

### 3.3 Statistical analysis

#### 3.3.1 Overview of the response surface model

*y*is the response value, \(\beta_{0} , \beta_{p} , \beta_{pp} ,\) and \(\beta_{pq}\) are the coefficients, \(x_{p}\) and \(x_{q}\) are the

*p*th and

*q*th design parameter, \(k\) is total number of parameters, respectively.

*X*

_{1},

*X*

_{2}, and

*X*

_{3}as an example. The design points are chosen at the edges and the central point, whereas the corners points are avoided to avoid extreme conditions [25]. The central point is repeated in order to comprise the errors produced by simulations.

#### 3.3.2 The response surface method implementation

*C*), \(G_{1}\) (D), \(G_{2}\) (E), and \(G_{3}\) (F) are the design factors, Table 3. The response value is the peak of pumping velocity. The number of runs for RSM in this study is 54.

Variables and their corresponding levels used for BBD

Parameters | Factor symbol | Level | ||
---|---|---|---|---|

− 1 | 0 | 1 | ||

\(W_{1}\) | A | 20 | 110 | 200 |

\(W_{2}\) | B | 20 | 110 | 200 |

\(W_{3}\) | C | 20 | 110 | 200 |

\(G_{1}\) | D | 1 | 10.5 | 20 |

\(G_{2}\) | E | 1 | 10.5 | 20 |

\(G_{3}\) | F | 100 | 200 | 300 |

The statistical analysis is performed in terms of analysis of variance (ANOVA) in order to find significant model terms. The values of the partial probability (*p* value) less than 0.05 and high *F* value imply that the model factors are significant, whereas the *p* values higher than 0.1 indicate the statistically insignificant terms [31]. From Online Resource Table S1, it can be seen that the quadratic model used here has the *p* value of < 0.0001 indicating that the model response model is significant and there is only < 0.01% chance that the *F* value of the model is because of the random error. The model terms A, B, D, AD, A^{2}, C^{2} (\(W_{1}\), \(W_{2}\), \(G_{1}\), \(W_{1} G_{1}\), \(W_{1}^{2}\), and \(W_{3}^{2}\)) are significant model terms. Therefore, the terms with the *p* value higher than 0.05 are removed from the model. The term *A* (\(W_{1}\)) with the smallest *p* value of less than 0.0001 and the largest *F* value of 54.48 verifies that the \(W_{1}\) has the most effect on the pumping capacity. The effects of six geometrical parameters and their interactions on the maximum of non-dimensional pumping velocity (*V*_{pumping}) are shown in Online Resource Figure S1. The 3D response surfaces are obtained when two variables are changed in the range of experiments and other variables are kept at their central level. It can be seen in Online Resource Figure S1 that the model variables are interdependent. Moreover, the results show that the maximum of non-dimensional pumping velocity is varied from 0 to 0.095 using different design parameters. The interactions between \(W_{1}\) and \(G_{1}\) affected the pumping velocity significantly (*p* < 0.05) (Online Resource Figure S1(b)).

Various optimum designs to maximize the pumping velocity in the micropump [geometrical parameters are non-dimensionalized with respect to the height of micropump in this case \(H = 200\, (\upmu{\text{m)}}\)]

Design | \(W_{1}\) | \(W_{2}\) | \(W_{3}\) | \(G_{1}\) | \(G_{2}\) | \(G_{3}\) | Frequency \(\left( f \right)\) | Maximum pumping velocity |
---|---|---|---|---|---|---|---|---|

1 | 0.1 | 1 | 0.2 | 0.005 | 0.1 | 0.5 | 11.2 | 0.1 |

2 | 0.1 | 1 | 0.25 | 0.05 | 0.09 | 0.5 | 10 | 0.044 |

3 | 1 | 1 | 0.35 | 0.1 | 0.1 | 0.5 | 3 | 0.011 |

### 3.4 Applications of the studied micropump

In this section, the applications and the fluid types and the range of pumping capacity of the presented micropump are discussed. In general, the ACEO micropumps work with fluids with a conductivity below 85 ms m^{−1} [7]. The velocity as shown in various cases in this study has a bell-shaped relation with frequency and there is an optimal frequency. The velocity also depends on the amplitude of the AC electric field and the electrolyte concentration [33]. The ACEO occurs in frequencies lower than 100 kHz [7, 34]. The concentration of the biological electrolyte is usually diluted to 10–100 mM buffers for ACEO micropumping [33]. Using the proposed micropump, it is possible to achieve the velocities up to 1 mm s^{−1}. The studied ACEO micropump is used in low-voltage pumping in microfluidics. Furthermore, it can be used in flow control in microfluidic networks. Therefore, the direction of flow is controlled locally. The secondary flow generated is also used as a mixing enhancer.

## 4 Conclusion

This paper presents the parametric study of an ACEO micropump using finite element method. The proposed micropump works with the asymmetries in electrodes dimensions and their applied voltage. The electrode widths and inter-electrode gaps are varied in reasonable ranges, and their effects on the pumping velocity are studied. In addition, the response surface methodology (RSM) is employed to show the effects of the independent variables and their interactions on the output variable (non-dimensional pumping velocity). Moreover, the most important parameters affecting the pumping velocity are found. The micropump is optimized using RSM to enhance the pumping velocity, and the optimized design parameters have been identified. Three optimized designs are suggested based on the fabrication cost and pumping output. The design with the highest output and highest fabrication cost has a maximal non-dimensional pumping velocity of 0.1. Based on the finding of this study, it can be concluded that an optimized geometry of micropump is required to achieve fast micro pumping.

## Notes

### Acknowledgements

We kindly acknowledge that some parts of the numerical study were performed on the High Performance Computing Center of Ferdowsi University of Mashhad.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

## Supplementary material

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