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To achieve fast and accurate cell manipulation in a microfluidic channel, it is essential to know the true nature of its input-output relationship. This paper aims to reveal the transfer function of such a micro manipulation controlled by a macro actuator. Both a theoretical model and experimental results for the manipulation are presented. A second-order transfer function is derived based on the proposed model, where the polydimethylsiloxane (PDMS) deformation plays an important role in the manipulation. Experiments are conducted with input frequencies up to 300 Hz. An interesting observation from the experimental results is that the frequency responses of the transfer function behave just like a first-order integration operator in the system. The role of PDMS deformation for the transfer function is discussed based on the experimentally-determined parameters and the proposed model.

There are various situations where cell manipulation is required in microfluidic applications [

However, the manipulation using a macro actuator is challenging because even a slight motion from the actuator may result in a very large displacement for the target object. This is due to the large ratio between the micro and macro cross-sectional areas, and a reduction mechanism is necessary for making this possible. Fortunately, polydimethylsiloxane (PDMS), one of the most common materials for making a microfluidic system, is embedded with a natural reduction mechanism from its deformable characteristic [

The transfer function and its frequency responses are experimentally investigated in this work. An open-loop control system is employed for determining the transfer function of the system. Different frequencies of sinusoidal inputs are applied to the PZT actuator as the inputs of the transfer function, while the motions of micro-objects are tracked as the outputs. The maximum frequency of the input signal is up to 300 Hz SHM, which is more than double that of previous works [

In summary, we focus on directly identifying the transfer function from the experimental inputs and outputs. A mechanical model of the system is proposed, and the experimental results are discussed with the model. The rest of this paper is organized as follows. After briefly reviewing the related works in

Various approaches have been developed for cell manipulation, for example, using flow control in a microfluidic channel [

The 2 ms may be negligible at lower frequencies but is significant when the frequency goes over 100 Hz. For example, when the input frequency is 300 Hz, the 2 ms becomes

The system can be simplified by considering the similarity to an integrator as stated at the end of

While Equation (5) shows a visually-identified transfer function based on the experimental results in

The coefficient of determination (

An interesting insight of the PDMS system can be found by comparing Equations (4) and (5). The general transfer function in Equation (4) is a second-order system with two zeros and poles while the Equation (5) is a very simple first-order system with a pole at zero frequency. That means the dominant terms that govern the transfer function are only the

This paper reveals the frequency characteristics between a macro actuator and a micro object in a PDMS microfluidic channel and aims at improving cell manipulation systems at high speed. Both the theoretical model and experimental validations are presented. The gain and phase of the transfer function are obtained. The theoretical model can fit well with the experimental results, and the system parameters are identified. According to the experimental results, the PDMS microfluidic device works like an integrator,

This work was supported by JSPS KAKENHI Grant Nos. JP15H05761, JP16K14197, and JP16K18051.

All authors conceived and designed the experiments; Kaoru Teramura and Chia-Hung Dylan Tsai performed the experiments; Koji Mizoue, Kaoru Teramura, and Chia-Hung Dylan Tsai analyzed the data; Koji Mizoue contributed materials/analysis tools; Chia-Hung Dylan Tsai and Makoto Kaneko wrote the paper.

The authors declare no conflict of interest.

The derivations and general discussion on the theoretical modeling are presented in this appendix. Equations (1)–(3) can be converted from time domain to frequency domain by Laplace transform as:

When

If we assume the outlet of the channel is opened to the atmosphere, which gives

When the input frequency is infinity

In typical microfluidic modeling, the inertia of the fluid in the microchannel is neglected as

Based on Equation (4), the zeros and poles of the transfer function are defined as:

When the manipulation frequency hits the zeros, no motion of the micro object will be detected since the gain is zero as the numerator in Equation (4) equals zero. On the other hand, the gain will be out of control if the manipulation frequency hits the poles, which makes the denominator of Equation (4) zero. Equation (4) will later be used to perform the numerical fitting for determining the parameters from experimental results.

The response of a sinusoidal input can be calculated by letting

The gain can then be derived as:

In order to obtain the frequency responses of the system, such as phase and gain, the experimental results, as the input and output signals are plugged into FFT for performing spectrum analysis. A series of complex values with respect to frequencies can be obtained after FFT. The gain and phase of them can be calculated as their distance and angle from the origin on a complex coordinate. This appendix explains the step-by-step procedure of how the gain and phase is determined using an example of the results at 10 Hz.

An example of how the gain and phase are determined using fast Fourier transform (FFT) and its results.

An illustrative diagram demonstrates how cell manipulation is controlled by a macro actuator outside the microfluidic chip.

The proposed mechanical model for describing the relation between the movement between a macro actuator (

An overview of the experimental system. (

The experimental results. (

The signal distortion due to the limit of temporal resolution in C program. (

The measurement of time delay on the computer. (

The Bode plots from experimental results. The upper and lower plots are the gain and phase response, respectively. The delay of 2 ms in the recording system is essential for the phase. Both before and after the compensation plots are shown.

The Bode plots based on the proposed model in Equation (6) and the curve fitting on the gain response. Both the gain and the phase match well to the experimental analysis in