A stretchable micrometer with enhanced performance for intermediate Reynolds numbers

Chemical reactions in microscale require good mixing at a relatively low owrate. However, mixing in microscale faces the major challenge of stable laminar ow associated with the low Reynolds number, the relative ratio between inertial force and viscous force. For low Reynolds numbers of less than unity, mixing occurs due to molecular diffusion. For high Reynolds number of more than several tens, chaotic advection enhances mixing. However, in the intermediate regime, mixing is not ecient. This paper reports a stretchable micromixer with dynamically tuneable channel dimensions. Periodically stretching the device changes the channel geometry and the curvature induced secondary Dean ows. The dynamically evolving secondary and main ows in the mixing channel result in chaotic advection and enhance mixing. The concept was demonstrated in a stretchable micromixer with a serpentine channel. We evaluated the performance of this stretchable micromixer both experimentally and numerically. At the intermediate range of Reynolds numbers from 4 to 17, the periodically stretched micromixer showed a better mixing eciency than the non-stretched counterpart. Therefore, our stretchable micromixer is a potential candidate for applications where precious reagents need to be mixed at relatively low ow rate conditions.


Introduction
Micro uidics is the science and engineering of handling and analysing a tiny amount of liquid for a broad range of applications in chemistry, food science, and medicine [1]. The small size of micro uidic devices brings advantages such as controllable reaction dynamics, high reaction e ciency and low cost [2]. Many micro uidic components and devices have been developed, examples are micromixers [3], microdispensers [4], micropumps [5], microvalves [6], and micro uidic sorters [7].
Micromixer is an indispensable component in miniaturised platforms for chemical, biochemical and biomedical applications such as drug delivery [8], detection and synthesis of DNA and RNA [9], enzyme assay [10], disease diagnosis [11,12], bioreactors [13], and microreactors [14,15]. These applications require thorough mixing of the reagents. Thus, a reliable micromixer is essential. However, since viscosity dominates the ow characteristics in microchannels, uid ow is in the laminar regime of low Reynolds numbers (Re < 1) [16][17][18]. In a laminar ow regime, the uids move alongside each other in parallel streams. The molecules diffuse from one uid stream into the other by Brownian motion from the higher concentration to the lower concentration regions [19]. The diffusivity depends on the uid viscosity, molecular size, and temperature. One of the solutions to enhance mixing is enlarging the interfacial area.
Both passive and active strategies have been employed to increase the interface between the two uids [20]. Active mixing techniques require external forces to disturb the uid ows through mechanical, magnetic, electrical, thermal, or ultrasonic means [20]. Passive micromixers, on the other hand, rely on intrinsic uid dynamics to stretch, fold, split, and recombine uids to increase the interface between uids [21,22]. The performance of passive mixing is determined by the dimension, con guration, and geometry of the mixing channels. Passive micromixers with both two-dimensional planar structures [23][24][25][26] and three-dimensional structures [27][28][29][30][31] have been studied extensively.
Curved microchannels have been used for developing passive micromixers [26][27]. Introducing curvature in the ow direction induces centrifugal force, leading to the formation of a pair of circulating vortices perpendicular to the mainstream direction [32,33]. These vortices, also called Dean vortices, result in subsequent chaotic advection, improving the mixing e ciency by folding the uids [34,35]. Following this mechanism, uid mixing has been investigated in serpentine [36], spiral [37], square wave [38], zigzag [39], and convergent-divergent [40] channels. However, in all these past works, micromixers are rigid, and their geometries and dimensions cannot be tailored or adjusted on-demand and in real time. Thus, mixing e ciencies were evaluated only for stationary conditions. The uid ow was steady, and the channel shape and dimensions were xed. Currently, no existing works in the literature reported on the effects of dynamic and real-time modi cation of channel curvature and dimensions for enhanced mixing.
We recently proposed the concept for exible and stretchable micro uidics, which enables tunning the channel dimensions after their fabrication [41][42][43]. Stretchability allows for elongating the micro uidic channel in a desired manner and accordingly changing its dimensions. In the present paper, we expand the concept of exible micro uidics further and propose a stretchable micromixer with dynamically tuneable dimensions and evaluate its mixing performance. The geometry of the stretchable micromixer can be dynamically tuned by adjusting the stretching length and the frequency. Periodic deformation alters the dimensions of a stretchable serpentine channel. The coupled effects of the dynamic geometry changes lead to periodic and rapid transformation of the ow eld, particularly of the Dean vortices, resulting in improved mixing. Figure 1 illustrates the concept of stretchable mixing. We rst experimentally investigated the mixing performance of the stretchable micromixer under periodic elongations. Next, we performed numerical simulation to gain insight into the dynamic changes in the ow eld and the induced Dean vortices. The numerical results agreed well with the experimental data.
Our stretchable micromixer provides an innovative way to modify uid mixing in microchannels in the intermediate regime, where both molecular diffusion and chaotic advection do not work.

Device fabrication and design
Standard photo lithography and soft lithography were used to fabricate the silicon mould and poly(dimethylsiloxane) (PDMS) micro uidic devices, respectively [44]. To start with, a 4-inch silicon wafer was spin-coated with SU-8 3050 (MicroChem Corp.) photoresist which was followed by soft baking. Next, the photoresist was selectively exposed to UV light through a photomask. Subsequently, the unexposed photoresist was dissolved by the developer, leaving behind the mould with the channel patterns.
Following the fabrication of the mould, soft lithography was employed to make the thin stretchable channel layers. Liquid PDMS (Sylgard 184, Dow Corning), prepared by mixing the base and curing agent at a 10: 1 ratio was spread on the mould to replicate the channels. The PDMS layer was subsequently cured for 2 hrs at 75°C. Next, two thick PDMS slabs were bonded to the inlet and outlet regions of the thin channel layer after plasma treatment (PDC-32G-2, Harrick Plasma). These slabs were used to hold the tubing connected to the chip. The inlet and outlet holes were then punched manually. Finally, the PDMS layer comprising the channel patterns with the thick slabs on it was plasma bonded to a plain PDMS layer.
Using the above steps, we designed and fabricated a stretchable micromixer with a curved serpentine mixing channel. The serpentine channel with 20 mixing units has a uniform width of 250 µm, and a height of 113 µm. The radius of curvatures is 312 µm. The total thickness of the stretchable microdevice is 1 mm in the mixing channel region and 5 mm at the inlet/outlet region. Fig. 2a illustrates the entire experimental setup for the stretchable micromixer. A custom-made stretching platform was designed and manufactured for automatic, unidirectional, periodic elongation of the micromixer, Fig. 2b. This platform is assembled on top of a poly (methyl methacrylate) (PMMA) sheet that can be tted onto the stage of an inverted microscope for observation. A translation stage, driven by a hybrid stepper motor (JKM NEMA17 Two Phase 42 mm), is mounted on the platform for applying automated single-axis motion to the micromixer, Fig. 2b. This motor is controlled by a microcontroller board (Arduino), and can apply a periodic translational motion to dynamically stretch the micromixer along its length. The inlet and outlet areas of the stretchable device are rmly clamped by the PMMA holders so that no slip will happen during the stretching process, and the mixing unit (the channel section) can be stretched freely, Fig. 2c. The platform can stretch the micromixer along its length up to 4 mm with different frequencies at constant stretching velocity (triangle-wave displacement).

Experimental setup
The stretchable micromixer has two inlets for the introduction of the two uids to be mixed. The rst uid is deionised (DI) water (Milli-Q® Direct 8 water puri cation system). The second is a 0.1% (w/v) solution of uorescein sodium salt (Sigma-Aldrich) in DI water. The uorescent solution and DI water were loaded in glass syringes and delivered into the microdevice using a ow-rate-controlled syringe pump (neMESYS, Centoni GmbH). We used a uorescence inverted microscope (Nikon Eclipse Ti2) coupled with an LED illumination source (pE-4000, CoolLED) to visualise of the mixing behaviour of the uorescent stream in the stretchable micromixer. A digital camera (Nikon, DS-Qi2) was mounted on the microscope for recording uorescent images of mixing channels. The mixing behaviour was evaluated at the outlet region where the device oscillation does not impact the focusing plane of the camera.

Data analysis
The recorded images from both the experimental setup and the COMSOL simulations were analysed using MATLAB to evaluate the mixing performance quantitively. For this purpose, the mixing index (MI) of the uorescent solution in water was calculated as [45]: Where N is the total number of discrete samples taken (total number of pixels); is the average of the normalised concentration intensity eld; and is the normalised concentration intensity for the ith pixel. The value of mixing index varies from 0 for the unmixed condition (binary mixture) to 1 for complete uniform mixing. This formula was employed over all pixels of the selected frame. The reported mixing indexes in this paper have been measured based on one period of elongation.

Numerical modelling
We further studied the effect of periodic elongation on the uid ow and mixing performance of the stretchable micromixer using numerical simulation. The three-dimensional (3D) numerical model of the stretchable serpentine micro uidic chip was formulated in COMSOL Multiphysics 5.6 software. The governing equations for the physics of solid deformation, Navier-Stokes equations for the incompressible uid with no external forces, and the diffusion equation for mass transport were solved numerically.
The whole process of periodic elongation, uid ow, and mixing were modelled with three different physics provided in COMSOL: Solid Mechanics, Single-Phase Laminar Flow, and Transport of Diluted Species. Moving Mesh was also applied to the model for the time-dependent simulation. Water, as the input ow, was considered as a Newtonian incompressible ow with the following properties: ρ = 1,000 kgm -3 , μ = 10 −3 kgs -1 m -1 . The concentration of the dye was set at 0 mol/m 3 for the two side inlets and at 1 mol/m 3 for the middle inlet. The diffusion coe cient of the uorescent dye was set at 10 -9 m 2 s −1 . The boundary conditions were no slip and no ux for the walls and zero pressure at the outlet.

Fluid mixing in the stretchable micromixer with serpentine channel
As mentioned previously, most micromixers perform well only at either very low or high Re. However, several micromixing applications are slow processes associated with Re ranging from 1 to several tens [46]. The dominant mixing mechanism at low Re is molecular diffusion, which requires a long residence time and correspondingly a long channel. By introducing curves, chaotic advection can be generated and dominate the molecular diffusion in the high Re range, leading to better mixing. As such, various micromixer designs have been proposed with serpentine channels with zigzag, sinusoidal, and square wave forms. These micromixers work well at relatively high Reynolds numbers (Re > > 10) but are not very e cient at the intermediate Reynolds numbers (approximately 1 < Re < 10) [13,14,[46][47][48]. Here, we demonstrate that dynamic elongation can improve the mixing e ciency of a stretchable micromixer with a serpentine channel operated at this critical intermediate range of Reynolds numbers.
We fabricated a simple curved serpentine micromixer with a uniform width and two ow-focusing inlets, Fig. 1. The micromixer has in total 20 curved mixing units. To evaluate the mixing performance of the stretchable serpentine micromixer, the uorescent solution was delivered through the centre inlet, with DI water introduced through the two side inlets. The owrates of the uorescent solution and DI-water were adjusted with a constant ratio of 1:6 to vary the Reynolds number from 0.09 up to 36. This ow rate ratio remains xed across all Reynolds numbers under investigation. We selected a non-equal owrate ratio because it is more representative of real chemical mixing applications. Figure 3 shows the mixing index as a function of Reynolds number Re without oscillation and at three representative oscillation frequencies. As expected, the micromixer exhibited good mixing performance. For Reynolds number of less than ~ 4, good mixing was achieved due to the long residence time of the uids, allowing for molecular diffusion, while for Reynolds number greater than ~ 17 the good mixing can be attributed to chaotic advection. However, the performance in the intermediate range 4 < Re < 17 is poor, since neither mixing mechanism has a signi cant effect. We demonstrated that periodic elongation improved the mixing index for this low Re range. The best improvement was observed for 1-Hz stretching, which increased the mixing index by as much as 0.2 as compared to without stretching. Increasing the frequency beyond 1 Hz slightly decreases the performance, but it is still much better than without oscillation. We hypothesize that this improvement in mixing performance is attributed to the disturbance caused by the oscillation, as well as the dynamic changes of the microchannel dimensions. Under elongation, not only does the channel length increase, but the cross section also shrinks [41]. These rapid changes in the geometry of the stretchable micromixer induce disturbance to the ow, resulting in an improved mixing. The dynamic changes affect the ow in different ways, including generating variations in the ow velocity at different locations of the microchannel. This could potentially lead to the continuous transformation of Dean vortices and create more chaotic advection in the ow eld.
To test the above hypothesis, we investigated the changes in channel dimensions including the height, width, and the radius of the curvature both experimentally and numerically. Figure 4(a-b) show respectively the actual and simulated deformation of the mixing channel for different stretching lengths. Stretching the entire micromixer device along its length alters the width of the channel non-uniformly. The width parallel to the stretching direction, the horizontal width W h , increases, Fig. 4c. In contrast, the width perpendicular to the stretching direction, the vertical width W v , decreases, Fig. 4c. Figure 4c compares the simulated channel deformation to the experimental data, and shows a good agreement. The simulation results also showed that the height of the channel reduced slightly, Fig. 4d. An additional change to the geometry was that the radius of curvature altered signi cantly under stretching, Fig. 4e. Therefore, under oscillatory elongation, the channel dimensions and the cross-sectional area of the channel vary periodically. At a constant total ow rate, periodic change of the cross section disturbs the ow eld, leading to chaotic advection and promoting mixing.

Numerical simulation of mixing in the stretchable micromixer with serpentine channel
In the above section, we demonstrated experimentally and numerically that stretching leads to change of cross section and curvature. Here, we hypothesise that the geometry change will affect the ow eld as well as the Dean vortices at different locations of the mixing channel. To test this hypothesis, we carried out a numerical simulation to model the complex mixing process with a time-dependent 3D model. Figure 5 shows the velocity magnitude in the microchannel at the zero-stretch condition (t = 1s, Fig. 5a) and maximal stretch condition (t = 1.5s, Fig. 5b). Figures 5(c-h) show the close-up images of the channel curvatures at different times from zero-stretch (t = 1s) to maximum stretch (t = 1.5s). The results indicate that the velocity of the uid increases during the stretching process. Thus, the axial velocity has varying values along the mixing channel, with the highest value while passing through W v and the lowest value while passing through W h . As the uid passes through the device, it encounters continuous changes in ow velocity, which disrupts the stable ow and hence promotes mixing.
Another critical factor affecting mixing is the changes in Dean vortices generated in the curvatures. Dean vortices are secondary ows that form in a curved channel due to inertial forces. Dean vortices promote mixing by inducing transversal transport and chaotic advection. We studied the variation of Dean vortices numerically at different times during the periodic elongation. Figure 5c shows the simulation results. Dean vortices altered in shape and magnitude from the initial condition (zero-stretch, t = 1 s) to maximum stretch condition (t = 1.5 s). Under the periodic elongation, the deformation of the channel cross-section and the corresponding dynamic changes in the axial ow led to changes of the Dean vortices.
Finally, we compared the numerical mixing performance with experimental data. The ow rates were 10 µL/min for the dye stream in the centre and 60 µL/min for the side inlets, which correspond to a Re of 6.43. Figure 6a shows the experimental images of the outlet section of the serpentine channel. Figure 6b shows the simulated results at the same channel locations and ow conditions as the experimental results. We can see that the numerical results are consistent with the experimental data. In addition, the quantitative mixing index from both the experimental and numerical studies agree relatively well, Fig. 6c. These results demonstrate that the uid mixing can be enhanced by oscillatory stretching.

Conclusion
This paper reports the concept of a stretchable micromixer with a serpentine channel design. By periodically stretching the device along its length, we demonstrated an improvement in the mixing performance at intermediate Reynolds number range. The improvement in mixing performance can be attributed to periodic changes in the geometry and cross section of the micromixer, affecting the main ow eld and the secondary Dean vortices. These coupled effects lead to disruption of the stable uid ow and enhance mixing. We veri ed that this is the mechanism of mixing improvement via both experiment and numerical studies. This proof-of-concept investigation demonstrates that tuneable micromixers can be achieved. The results show a way of improving the e ciency of existing micromixers operating at intermediate regime, between diffusive mixing and chaotic advection, which is important for many practical applications.

Declarations Declaration of competing interest
The authors declare no competing nancial interests.  (a) Experimental setup including the inverted microscope, syringe pump, and Arduino board; (b) Custommade automated stretching platform with a translation stage driven by a hybrid stepper motor moving the PMMA holder. (c) The stretchable micromixer with its inlet and outlet areas rmly clamped by PMMA holder.

Figure 3
Mixing index as a function of Re for the stretchable micromixer at different frequency conditions. The inset shows an elongated micromixer with the window for evaluation of mixing index.   Numerical simulation of the stretchable micromixer: (a) Velocity magnitude of the 3D dynamic simulation of the stretchable serpentine micromixer at (i) the initial condition (zero-stretch), t=1s and (ii) maximum stretch condition, t=1.5s; (b) Close-ups of the curved channel sections at different times during the oscillation. (c) Cross-sectional streamlines of the secondary ow during the stretching process.