**4. Mass‐producible rapid mixer based on Baker's transformation**

We developed a novel methodology to fabricate three‐dimensional passive‐type mixer based on the baker's transformation (BT). BT is the best transformation for mixing fluids of laminar flow. We newly designed the BT structure with isovolumetric change without any separation/ joining process of two channels. It is a suitable solution for mass‐producing BT mold structures by utilizing precision cutting techniques. Two scales of BT mixers with similar structures are introduced herein. The one is for microfluidic analytical systems to accomplish well‐mixed solutions in a short channel length, and the other one in miniature scale aims at high perform‐ ance mixing of high viscosity fluids in food processing or resin blending. An ultraprecision five‐axis planing machine and diamond cutting tools were used for a microfluidic BT mixer mold on a oxygen‐free copper block, in which the flow passage area was 3.2E‐9 m2 . For a miniature BT mixer mold on an aluminium block, a precision machining center and an end mill with a 1 mm radius were used. The flow passage area was 3.2E‐5 m2 . We studied their mixing performances by numerical analyses and obtained the BT mixing results showing good similarities with that of numerical analyses. Moreover, the mixing performance of the micro‐ BT mixer was quantitatively examined to accomplish complete mixing over a wide range of flow rates.

#### **4.1. Background of micromixers for bio‐informatics**

In the past decades microfluidic systems have been widely used in chemistry, biology, and nanobiotechnology, including DNA [18] or protein analysis [19], cell sorting [20], and chemical

reactions [21]. Mixing inside microchannels plays an important role in those microfluidic analysis systems, and many researchers have made efforts to develop innovative mixing techniques inside microchannels [22]. In particular, mixing of solutes with a low diffusion coefficient inside microchannels is important and useful for a variety of applications including immunoassay [23], and it is desirable to get the most efficient and rapid mixing possible inside them.

tions of the numerical simulation were: 20 mm/s flow rate for the inlets; 0 Pa static pressure for the outlets; and 0 mm/s flow rate for the channel wall (the wall was regarded as having no roughness and skidding did not occur). The numerical simulation results in Figure 16b.

Precision Micro Machining Methods and Mechanical Devices 65

We have fabricated baker's transformation structures with isovolumetric change (without dead volume) on an oxygen‐free copper block. We selected a planing process by using an ultraprecision planing machine, NIC‐300 (Nagase Integrex Co., Ltd.), which consists of three feed tables with double hydrostatic oil guide ways on the XYZ axes, two rotary index tables

A custom‐ordered ultraprecision diamond cutting tool (A.L.M.T. Diamond Corp.), which had a 1 mm nose radius, was used to finish the top surface ofthe mold. The proposed microchannels were then machined with another ultraprecision diamond tool (A.L.M.T. Diamond Corp.). A schematic of the mold is shown in Figure 17a. On an oxygen‐free copper block (30×60 mm), the BT device is designed to have two inlets, injection microchannels with 10 mm length and 20 μm height, 10.4 mm length BT structures, and outlet channel with 33 mm length and 20 μm

PDMS type microfluidic BT mixer was then developed. PDMS (Dow Corning Inc.) and curing agent (Dow Corning Inc.) were mixed at a ratio of 10 to 1, and then the mixture was poured onto the mold, cured at 65 ºC for 2 h, peeled from the mold, and baked at 120 ºC for 30 min. Because the baked PDMS was sequentially used for the second mold, it was soaked in a commercial detergent solution (5%) for 5 min. Then, it was rinsed with double distilled water and dried in a vacuum chamber. After drying, the PDMS mixture was poured onto the PDMS mold, cured, and peeled from the mold as above. Before bonding of the second replication PDMS and a commercial slide glass, access holes were punched into the PDMS and then the

The 3D configuration of the structures is shown in Figure 17b. These structures were designed to have transverse (x‐y) and longitudinal (x‐z) movement of solutions at the same time to realize the BT like that of Figure 16. Vertical cross‐sections of the BT device along the y axis are illustrated in Figure 17c. The dotted lines indicate the replication point in Figure 17b. One cycle of the BT device is 1040 μm and there are 10 cycles in all. SEM image of the BT device is

PDMS and slide glass were both treated under oxygen plasma.

given as a cross‐sectional view (Figure 17d). The scale bar is 100 μm.

*4.2.3. Fabrication method*

on the BC axes and a five‐axis control system.

**Figure 15.** Schema of Baker's transformation (BT)

height.

Most passive‐type mixers [24‐27] depend on simple mixing techniques without any external power sources, unlike active‐type mixers, which use such sources as ultrasonic [28], magnetic stirring [29], and bubble induced acoustic actuation [30]. In the passive‐type mixers, only the structural design induces the mixing of fluids affected by the convection flow and large interface of fluids.

Comparison of active type micromixers with the passive type shows that the former can realize excellent mixing performance, but with some disadvantages: 1) difficulty in integrating other microfluidic components; 2) high cost from the standpoint of being a disposable device; 3) complex control units or an external power source. On the other hand, passive (static) type micromixers have three advantages: 1) easy integration with other microfluidic components; 2) low cost; 3) no external power source is needed. These passive structures, however, did not have all the desired features of providing rapid mixing, high mixing efficiency, a wide range of flow rates, and having no dead volume.

## **4.2. Microfluidic mixer utilizing Baker's transformation**

In this paper, we propose a new design for the 3D micromixer, which is based on the baker's transformation (BT in short), which provides the highest mixing performance as demonstrated in chaotic theories [31].

### *4.2.1. Baker's transformation*

The schematic illustration of the BT process (stretch, cut and fuse) is shown in Figure 15. The BT process transform fluid layers from one into two, and therefore the transformation of *n* times produces *2n* fluid layers. Consequently, the baker's transformation can exponentially shorten the diffusion length.

Figure 16a illustrates the BT mixing process. Two types of fluids are indicated in blue and yellow colors. The successive 3D configuration changes fold the blue fluid onto the yellow fluid gradually. After completing the folding, the combined fluids are stretched and then half of the fluids are bent 90º, which these steps make a great difference between baker's transfor‐ mation and parallel lamination. The bent part of the fluids are moved to the opposite side. Then the moved part is gradually folded onto the original fluids again.

### *4.2.2. Numerical CFD analysis*

The microfluidic distribution of vertical cross‐sectional views was investigated numerically, by using CFD (computational fluid dynamics) software (ANSYS CFX). The boundary condi‐ tions of the numerical simulation were: 20 mm/s flow rate for the inlets; 0 Pa static pressure for the outlets; and 0 mm/s flow rate for the channel wall (the wall was regarded as having no roughness and skidding did not occur). The numerical simulation results in Figure 16b.

### *4.2.3. Fabrication method*

reactions [21]. Mixing inside microchannels plays an important role in those microfluidic analysis systems, and many researchers have made efforts to develop innovative mixing techniques inside microchannels [22]. In particular, mixing of solutes with a low diffusion coefficient inside microchannels is important and useful for a variety of applications including immunoassay [23], and it is desirable to get the most efficient and rapid mixing possible inside

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

Most passive‐type mixers [24‐27] depend on simple mixing techniques without any external power sources, unlike active‐type mixers, which use such sources as ultrasonic [28], magnetic stirring [29], and bubble induced acoustic actuation [30]. In the passive‐type mixers, only the structural design induces the mixing of fluids affected by the convection flow and large

Comparison of active type micromixers with the passive type shows that the former can realize excellent mixing performance, but with some disadvantages: 1) difficulty in integrating other microfluidic components; 2) high cost from the standpoint of being a disposable device; 3) complex control units or an external power source. On the other hand, passive (static) type micromixers have three advantages: 1) easy integration with other microfluidic components; 2) low cost; 3) no external power source is needed. These passive structures, however, did not have all the desired features of providing rapid mixing, high mixing efficiency, a wide range

In this paper, we propose a new design for the 3D micromixer, which is based on the baker's transformation (BT in short), which provides the highest mixing performance as demonstrated

The schematic illustration of the BT process (stretch, cut and fuse) is shown in Figure 15. The BT process transform fluid layers from one into two, and therefore the transformation of *n* times produces *2n* fluid layers. Consequently, the baker's transformation can exponentially

Figure 16a illustrates the BT mixing process. Two types of fluids are indicated in blue and yellow colors. The successive 3D configuration changes fold the blue fluid onto the yellow fluid gradually. After completing the folding, the combined fluids are stretched and then half of the fluids are bent 90º, which these steps make a great difference between baker's transfor‐ mation and parallel lamination. The bent part of the fluids are moved to the opposite side.

The microfluidic distribution of vertical cross‐sectional views was investigated numerically, by using CFD (computational fluid dynamics) software (ANSYS CFX). The boundary condi‐

Then the moved part is gradually folded onto the original fluids again.

them.

64

interface of fluids.

Biomedical Engineering

in chaotic theories [31].

*4.2.1. Baker's transformation*

shorten the diffusion length.

*4.2.2. Numerical CFD analysis*

of flow rates, and having no dead volume.

**4.2. Microfluidic mixer utilizing Baker's transformation**

We have fabricated baker's transformation structures with isovolumetric change (without dead volume) on an oxygen‐free copper block. We selected a planing process by using an ultraprecision planing machine, NIC‐300 (Nagase Integrex Co., Ltd.), which consists of three feed tables with double hydrostatic oil guide ways on the XYZ axes, two rotary index tables on the BC axes and a five‐axis control system.

A custom‐ordered ultraprecision diamond cutting tool (A.L.M.T. Diamond Corp.), which had a 1 mm nose radius, was used to finish the top surface ofthe mold. The proposed microchannels were then machined with another ultraprecision diamond tool (A.L.M.T. Diamond Corp.). A schematic of the mold is shown in Figure 17a. On an oxygen‐free copper block (30×60 mm), the BT device is designed to have two inlets, injection microchannels with 10 mm length and 20 μm height, 10.4 mm length BT structures, and outlet channel with 33 mm length and 20 μm height.

PDMS type microfluidic BT mixer was then developed. PDMS (Dow Corning Inc.) and curing agent (Dow Corning Inc.) were mixed at a ratio of 10 to 1, and then the mixture was poured onto the mold, cured at 65 ºC for 2 h, peeled from the mold, and baked at 120 ºC for 30 min. Because the baked PDMS was sequentially used for the second mold, it was soaked in a commercial detergent solution (5%) for 5 min. Then, it was rinsed with double distilled water and dried in a vacuum chamber. After drying, the PDMS mixture was poured onto the PDMS mold, cured, and peeled from the mold as above. Before bonding of the second replication PDMS and a commercial slide glass, access holes were punched into the PDMS and then the PDMS and slide glass were both treated under oxygen plasma.

The 3D configuration of the structures is shown in Figure 17b. These structures were designed to have transverse (x‐y) and longitudinal (x‐z) movement of solutions at the same time to realize the BT like that of Figure 16. Vertical cross‐sections of the BT device along the y axis are illustrated in Figure 17c. The dotted lines indicate the replication point in Figure 17b. One cycle of the BT device is 1040 μm and there are 10 cycles in all. SEM image of the BT device is given as a cross‐sectional view (Figure 17d). The scale bar is 100 μm.

**Figure 17.** (a) Schematic of the BT device. (b) Schematic 3D diagram of the mold for the BT device. (c) Schematic illus‐ trations of vertical cross-sections of the BT device. (d) SEM image of the BT device. (e)&(f) Series of confocal micro‐

We quantitatively studied its mixing performance to attain complete mixing over a wide range of flow rates. The mixing performance of the BT device is evaluated using FITC (fluorescein

s‐1

A confocal microscope (FV1000, Olympus) was used to observe the mixing behaviourthrough a focus lens 40x/0.90 (UPLSAPO, Olympus). Excitation laser was 488 nm. Confocal images were captured at a data acquisition rate of 0.90 frames per second. Stacks of each confocal x‐y scan of 512×512 pixels were collected with a step of 0.5 μm in the z direction. Scan speed was 200 μs/pixel. Z‐series images were merged into vertical cross‐sectional images and analyzed. The flames in Figure 17e and Figure 17f show confocal micrographs of the vertical cross‐ sections of a channel similar to those in Figure 17c for one cycle; the mixing of FITC solution and PBS (Figure 17e) and the mixing of IgG solution and PBS (Figure 17f). Flow rate was 100 mm/s. The scale baris 100 μm and the white dotted lines indicate channel outlines. The mixing behaviour showed a good agreement with the fluid distributions of numerical simulations.

To evaluate the mixing performance, the mixing ratio was calculated using the following

s‐1

) [32] and goat anti‐mouse IgG (immuno‐

Precision Micro Machining Methods and Mechanical Devices 67

) [33] dissolved in phosphate buffered saline

*4.2.4. Analytical evaluation of BT mixing performance[Yasui et al., 2011]*

isothiocyanate; diffusion coefficient: 2.6×10‐<sup>10</sup> m2

globulin G; diffusion coefficient: 4.6×10‐<sup>11</sup> m2

graphs for one cycle in a microchannel.

*4.2.4.1. Confocal microscopy*

(PBS).

formula [34];

**Figure 16.** Schematics of (a)mixing process of BT device and (b)microfluidic distributions derived from numerical simulations.

**Figure 17.** (a) Schematic of the BT device. (b) Schematic 3D diagram of the mold for the BT device. (c) Schematic illus‐ trations of vertical cross-sections of the BT device. (d) SEM image of the BT device. (e)&(f) Series of confocal micro‐ graphs for one cycle in a microchannel.

#### *4.2.4. Analytical evaluation of BT mixing performance[Yasui et al., 2011]*

We quantitatively studied its mixing performance to attain complete mixing over a wide range of flow rates. The mixing performance of the BT device is evaluated using FITC (fluorescein isothiocyanate; diffusion coefficient: 2.6×10‐<sup>10</sup> m2 s‐1 ) [32] and goat anti‐mouse IgG (immuno‐ globulin G; diffusion coefficient: 4.6×10‐<sup>11</sup> m2 s‐1 ) [33] dissolved in phosphate buffered saline (PBS).

#### *4.2.4.1. Confocal microscopy*

**Figure 16.** Schematics of (a)mixing process of BT device and (b)microfluidic distributions derived from numerical

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

simulations.

Biomedical Engineering

66

A confocal microscope (FV1000, Olympus) was used to observe the mixing behaviourthrough a focus lens 40x/0.90 (UPLSAPO, Olympus). Excitation laser was 488 nm. Confocal images were captured at a data acquisition rate of 0.90 frames per second. Stacks of each confocal x‐y scan of 512×512 pixels were collected with a step of 0.5 μm in the z direction. Scan speed was 200 μs/pixel. Z‐series images were merged into vertical cross‐sectional images and analyzed. The flames in Figure 17e and Figure 17f show confocal micrographs of the vertical cross‐ sections of a channel similar to those in Figure 17c for one cycle; the mixing of FITC solution and PBS (Figure 17e) and the mixing of IgG solution and PBS (Figure 17f). Flow rate was 100 mm/s. The scale baris 100 μm and the white dotted lines indicate channel outlines. The mixing behaviour showed a good agreement with the fluid distributions of numerical simulations.

To evaluate the mixing performance, the mixing ratio was calculated using the following formula [34];

$$\left\{1 - \sqrt{\frac{1}{N} \sum\_{i=1}^{N} \left(I\_i - I\_i^{\text{perf\\_mix}}\right)^2 \Big| \frac{1}{N} \sum\_{i=1}^{N} \left(I\_i^0 - I\_i^{\text{perf\\_mix}}\right)^2}\right\} \times 100\tag{1}$$

Figure 18d shows confocal micrographs of the vertical cross‐sections of the most stretched channel (160 μm × 20 μm, width and height, respectively) for different cycles as indicated in

Precision Micro Machining Methods and Mechanical Devices 69

In these micrographs, FITC solution and PBS in the upper part of Figure 18d or IgG solution and PBS in the lower part of Figure 18d were introduced to observe the mixing behaviour inside the BT device; a syringe pump operated at constant flow velocity of 100 mm/s was used. As the simulation results indicated, the right edge of the channel was not fully mixed for both of FITC and IgG after 5 cycles. At this flow velocity, FITC solution and PBS were completely mixed after 10 cycles, but IgG solution and PBS were not. This insufficient mixing comes from

To determine residence time to attain complete mixing, we calculated the time to achieve complete mixing from Figure 18a andFigure 18b bydividing the length by flow velocity. Figure 19 shows the mixing ratio vs.residence time after 10 cycles with the BT device. Filled and open symbols show FITC and IgG mixing, respectively. The dotted line indicates 90% mixing ratio. The logarithmic fitting curves are expressed as Y=98.35+6.49logX (red filled circles), Y=94.50+8.74logX (red open circles), Y=52.59+62.71logX (black filled triangles), and Y=6.12+34.12logX (black open triangles); where Y is percentage of mixing and X is residence

The residence time for FITC in the BT device was 51 ms and for IgG 306 ms. Considering the residence times in the microchannel without BT structures of 4.0 s for FITC and 297 s for IgG (that could not be attained in this mixing length but was calculated from the fitting curve), our BT device showed significant potential for mixing FITC solution and IgG solution more efficiently and rapidly in a 10.4 mm mixing length microchannel than in a microchannel without BT structures, and mixing rate was more than 70‐fold fasterfor FITC solution and 900‐

**Figure 19.** The mixing ratio vs. residence time in the BT device (circles) and microchannel without BT structures

the 10‐fold smaller diffusion coefficient of IgG compared to FITC.

Figure 18c. The scale bar is 100 μm.

time.

(triangles).

fold faster for IgG solution.

where *N*, *Ii* , *Ii 0* , and *Ii perf.mix* are the total number of pixels, the fluorescence intensity at pixel *i*, the fluorescence intensity at pixel *i* without mixing or diffusion, and the fluorescence intensity of the completely mixed solution at pixel *i*, respectively. Generally, the 90% mixing ratio was regarded as complete mixing.

#### *4.2.4.2. Analysis of mixing ratio*

The mixing ratios of FITC solution vs. PBS and IgG solution vs. PBS are shown in Figures 18a and 18b, respectively. The mixing ratio was calculated using formula (1). Circles, squares, and triangles show ∆y=1.04, 5.02, and 10.4 mm (i.e. after 1, 5, and 10 cycles of mixing), respectively. The dotted line indicates 90% mixing ratio. The BT device provided complete mixing of FITC solution and PBS at flow velocities up to 400 mm/s for 10 cycles, and complete mixing of IgG solution and PBS was attained at velocities up to 50 mm/s for 10 cycles; these were character‐ ized by the values of low Reynolds number (*Re = Ul/ν* < 100, where *U* is the average flow velocity, *l* is the typical cross‐sectional dimension, and *ν* is the kinematic viscosity of the fluid). The difference in maximum flow rate to attain complete mixing between FITC (400 mm/s) and IgG solution (50 mm/s) was attributed to the different diffusion coefficients. From these figures, we saw the mixing performance increased as the number of cycles increased, e.g. when mixing IgG solution and PBS at 4 mm/s flow velocity, the mixing ratios were 44, 72, and 97% after 1, 5 and 10 cycles, respectively.

**Figure 18.** (a) Mixing ratio of FITC solution and PBS vs. flow rate. (b) Mixing ratio of IgG solution and PBS vs. flow rate. (c) Schematic diagrams of the BT device. (d) Confocal micrographs of vertical cross-sections of a microchannel.

Figure 18d shows confocal micrographs of the vertical cross‐sections of the most stretched channel (160 μm × 20 μm, width and height, respectively) for different cycles as indicated in Figure 18c. The scale bar is 100 μm.

(1 ‐ <sup>1</sup> *<sup>N</sup>* ∑ *i*=1 *N* (*Ii* ‐ *Ii*

where *N*, *Ii*

68

, *Ii 0* , and *Ii*

Biomedical Engineering

regarded as complete mixing.

*4.2.4.2. Analysis of mixing ratio*

5 and 10 cycles, respectively.

*perf* .*mix*)<sup>2</sup>

/ 1 *<sup>N</sup>* ∑ *i*=1 *N* (*Ii* <sup>0</sup> ‐ *Ii*

the fluorescence intensity at pixel *i* without mixing or diffusion, and the fluorescence intensity of the completely mixed solution at pixel *i*, respectively. Generally, the 90% mixing ratio was

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

The mixing ratios of FITC solution vs. PBS and IgG solution vs. PBS are shown in Figures 18a and 18b, respectively. The mixing ratio was calculated using formula (1). Circles, squares, and triangles show ∆y=1.04, 5.02, and 10.4 mm (i.e. after 1, 5, and 10 cycles of mixing), respectively. The dotted line indicates 90% mixing ratio. The BT device provided complete mixing of FITC solution and PBS at flow velocities up to 400 mm/s for 10 cycles, and complete mixing of IgG solution and PBS was attained at velocities up to 50 mm/s for 10 cycles; these were character‐ ized by the values of low Reynolds number (*Re = Ul/ν* < 100, where *U* is the average flow velocity, *l* is the typical cross‐sectional dimension, and *ν* is the kinematic viscosity of the fluid). The difference in maximum flow rate to attain complete mixing between FITC (400 mm/s) and IgG solution (50 mm/s) was attributed to the different diffusion coefficients. From these figures, we saw the mixing performance increased as the number of cycles increased, e.g. when mixing IgG solution and PBS at 4 mm/s flow velocity, the mixing ratios were 44, 72, and 97% after 1,

**Figure 18.** (a) Mixing ratio of FITC solution and PBS vs. flow rate. (b) Mixing ratio of IgG solution and PBS vs. flow rate. (c) Schematic diagrams of the BT device. (d) Confocal micrographs of vertical cross-sections of a microchannel.

*perf* .*mix*)<sup>2</sup>

*perf.mix* are the total number of pixels, the fluorescence intensity at pixel *i*,

) × 100 (1)

In these micrographs, FITC solution and PBS in the upper part of Figure 18d or IgG solution and PBS in the lower part of Figure 18d were introduced to observe the mixing behaviour inside the BT device; a syringe pump operated at constant flow velocity of 100 mm/s was used. As the simulation results indicated, the right edge of the channel was not fully mixed for both of FITC and IgG after 5 cycles. At this flow velocity, FITC solution and PBS were completely mixed after 10 cycles, but IgG solution and PBS were not. This insufficient mixing comes from the 10‐fold smaller diffusion coefficient of IgG compared to FITC.

To determine residence time to attain complete mixing, we calculated the time to achieve complete mixing from Figure 18a andFigure 18b bydividing the length by flow velocity. Figure 19 shows the mixing ratio vs.residence time after 10 cycles with the BT device. Filled and open symbols show FITC and IgG mixing, respectively. The dotted line indicates 90% mixing ratio. The logarithmic fitting curves are expressed as Y=98.35+6.49logX (red filled circles), Y=94.50+8.74logX (red open circles), Y=52.59+62.71logX (black filled triangles), and Y=6.12+34.12logX (black open triangles); where Y is percentage of mixing and X is residence time.

The residence time for FITC in the BT device was 51 ms and for IgG 306 ms. Considering the residence times in the microchannel without BT structures of 4.0 s for FITC and 297 s for IgG (that could not be attained in this mixing length but was calculated from the fitting curve), our BT device showed significant potential for mixing FITC solution and IgG solution more efficiently and rapidly in a 10.4 mm mixing length microchannel than in a microchannel without BT structures, and mixing rate was more than 70‐fold fasterfor FITC solution and 900‐ fold faster for IgG solution.

**Figure 19.** The mixing ratio vs. residence time in the BT device (circles) and microchannel without BT structures (triangles).

### **4.3. Miniature BT device for high viscosity fluid mixing**

We have also developed the miniature BT mixer, which was a scale‐up model of the mass‐ producible micro BT mixer, to make it possible that the mixing methodology we proposed was available for multi‐scale structures [35].With the mass‐producible mold design, its shortlength and high mixing ratio characteristics, the miniature BT mixer would feasible for commercial use of treating high viscosity solutions in mixing/heating/chemistry. Herein, we developed the miniature fluid channel structure along with the basic guideline just like as the development of micro BT mixer.

High viscosity solution forms the laminar flow when flowing inside the miniature fluid channel with tens of mm2 passage area. In the food industry, oils or flavouring materials need to be mixed smoothly and homogeneously. The same need is for material industries like mixing resins or other chemicals. The viscosity of foods has the distribution of 0.1 – 100 Pa∙s, which is 100 – 100,000 times bigger than that of water. Just like the water solutions in a simple micro channel, a simple macro channel cannot mix food solutions, therefore baker's transformation (BT) is feasible.

**Figure 21.** Prototype BT mixer made of hard aluminium alloy

*4.3.2. Preliminary experiment*

distinction.

*4.3.3. Results*

In Figure 20, the side, top, and cross‐section view are shown in upper, middle, and lowerlevel, respectively. The dotted lines indicate the replication points in the flow channel. The far left ofthe channel illustration is connected to two separate inlets. Two colors representthe different fluids mixed by BT ideally: (1) 1st BT forming 2 folds; (2) 2nd BT forming 4 folds; (3) reprise of the 2nd BT. The *n* times of BT forms 2*<sup>n</sup>* folds, meaning the diffusion distance among every

Precision Micro Machining Methods and Mechanical Devices 71

In order to confirm the flow distribution directly, we introduced cure liquid silicone rubbers to cut them when they totally cured in 8 hours, and observed cross sections. The fabricated BT structure was sealed with the flat aluminum plate, and two inlets were connected to syringes, providing the silicone rubbers. Rotating a handle moves syringes translational to push the silicone rubbers out. We used the white silicone rubber TSE3504 (Momentive Performance Materials Inc.; viscosity: 10 Pa s at 23C, specific gravity: 1.22 at 23C, demold time: 8 hours at 25C) for one inlet, and for the other, the blue colored master ME50‐M (Momentive Perform‐ ance Materials Inc.; viscosity: 800 Pa s at 23C) was blended into TSE3504 with 1 wt% for

The distribution of vertical cross‐sectional views was also investigated numerically by using ANSYS CFX. A control volume was the tetrahedral shape and the length of each side was 0.2 mm at the maximum. The boundary conditions were: 6 mm/s flow rate forthe inlets; 0 Pa static pressure for the outlets; and 0 mm/s flow rate for the channel wall. The solution viscosity and

Both the experimentalresults and numerical simulation results are displayed in Figure 22. The top line is the cured silicone rubber. The middle and the bottom line are the experimental and

the simulation results as cross‐sectional views in the flow direction, respectively.

layer results in 1/2*<sup>n</sup>* to be the exponential acceleration of fluid mixing.

specific gravity were set to 10 Pa s and 1.0, respectively.

#### *4.3.1. Prototype*

A miniature BT device for mixing high viscosity liquids was prototyped as a mixer itself, not as a mold herein. The aluminium alloy plate was fabricated by using the machining process with a 2 mm diameter end mill, along with the design illustrated in Figure 20. The fabricated structure is shown in Figure 21.

**Figure 20.** Illustration of three-dimensional BT mixer:

**Figure 21.** Prototype BT mixer made of hard aluminium alloy

In Figure 20, the side, top, and cross‐section view are shown in upper, middle, and lowerlevel, respectively. The dotted lines indicate the replication points in the flow channel. The far left ofthe channel illustration is connected to two separate inlets. Two colors representthe different fluids mixed by BT ideally: (1) 1st BT forming 2 folds; (2) 2nd BT forming 4 folds; (3) reprise of the 2nd BT. The *n* times of BT forms 2*<sup>n</sup>* folds, meaning the diffusion distance among every layer results in 1/2*<sup>n</sup>* to be the exponential acceleration of fluid mixing.

#### *4.3.2. Preliminary experiment*

**4.3. Miniature BT device for high viscosity fluid mixing**

of micro BT mixer.

Biomedical Engineering

70

(BT) is feasible.

*4.3.1. Prototype*

structure is shown in Figure 21.

**Figure 20.** Illustration of three-dimensional BT mixer:

We have also developed the miniature BT mixer, which was a scale‐up model of the mass‐ producible micro BT mixer, to make it possible that the mixing methodology we proposed was available for multi‐scale structures [35].With the mass‐producible mold design, its shortlength and high mixing ratio characteristics, the miniature BT mixer would feasible for commercial use of treating high viscosity solutions in mixing/heating/chemistry. Herein, we developed the miniature fluid channel structure along with the basic guideline just like as the development

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

High viscosity solution forms the laminar flow when flowing inside the miniature fluid channel with tens of mm2 passage area. In the food industry, oils or flavouring materials need to be mixed smoothly and homogeneously. The same need is for material industries like mixing resins or other chemicals. The viscosity of foods has the distribution of 0.1 – 100 Pa∙s, which is 100 – 100,000 times bigger than that of water. Just like the water solutions in a simple micro channel, a simple macro channel cannot mix food solutions, therefore baker's transformation

A miniature BT device for mixing high viscosity liquids was prototyped as a mixer itself, not as a mold herein. The aluminium alloy plate was fabricated by using the machining process with a 2 mm diameter end mill, along with the design illustrated in Figure 20. The fabricated

In order to confirm the flow distribution directly, we introduced cure liquid silicone rubbers to cut them when they totally cured in 8 hours, and observed cross sections. The fabricated BT structure was sealed with the flat aluminum plate, and two inlets were connected to syringes, providing the silicone rubbers. Rotating a handle moves syringes translational to push the silicone rubbers out. We used the white silicone rubber TSE3504 (Momentive Performance Materials Inc.; viscosity: 10 Pa s at 23C, specific gravity: 1.22 at 23C, demold time: 8 hours at 25C) for one inlet, and for the other, the blue colored master ME50‐M (Momentive Perform‐ ance Materials Inc.; viscosity: 800 Pa s at 23C) was blended into TSE3504 with 1 wt% for distinction.

The distribution of vertical cross‐sectional views was also investigated numerically by using ANSYS CFX. A control volume was the tetrahedral shape and the length of each side was 0.2 mm at the maximum. The boundary conditions were: 6 mm/s flow rate forthe inlets; 0 Pa static pressure for the outlets; and 0 mm/s flow rate for the channel wall. The solution viscosity and specific gravity were set to 10 Pa s and 1.0, respectively.

#### *4.3.3. Results*

Both the experimentalresults and numerical simulation results are displayed in Figure 22. The top line is the cured silicone rubber. The middle and the bottom line are the experimental and the simulation results as cross‐sectional views in the flow direction, respectively.

**4.4. Conclusion**

cross‐sectional phases.

**Author details**

**References**

44‐50.

243‐250.

1994;43(1) 35‐38.

Annals of the CIRP 2005;54(1) 321‐324.

Eiji Shamoto, Norikazu Suzuki, Takashi Kato and Burak Sencer

Department of Mechanical Science and Engineering, Nagoya University, Nagoya, Japan

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In order to meet the demand of mixing fluids for multi‐scale mixers, We have developed a novel methodology to fabricate three‐dimensional passive‐type mixers based on the baker's transformation. We aimed at fabricating BT structure with isovolumetric change without any separation/joining process of two channels for the sake of mass‐procucing BT mold structures by utilizing precision cutting techniques. Two scales of BT mixers, one for microfluidic analytical system and the other for mixing high viscosity fluids in food processing or resin blending, were developed. We performed laboratory experiments for evaluating the mixing performances of two prototypes. The confocal microscopy for 10 cycles (10.4 mm in length) of microfluidicBTshowedthatthe significant potentialfor mixing FITC solution andIgGsolution more efficiently and rapidly than in a microchannel without BT structures, and mixing rate was more than 70‐fold faster for FITC solution and 900‐fold faster for IgG solution. The comparison of numerical analysis and the experimental result of mixing two colored silicone rubbers by using prototyped miniature BT mixer (3 cycles) showed good similarities in their

Precision Micro Machining Methods and Mechanical Devices 73

**Figure 22.** Results of four cycles of BT procedures

In Figure 23, the experimental folding results in the first three cycles (center column) were compared with the ideal BT illustration (left column) and the numerical analysis (right column).

The experimentalresults show relatively good similarities with the numerical analyticalresults of two, four, and eight layers folded inside flow passages. Meanwhile, compared with the ideal BT illustration, there are not small differences, especially in its marginal regions. The every folded edge wasn't processed perfectly because of the shear resistance decreasing the flow rate. The mixing performance, however, relies mainly on interlayer distances near the center of flow path. Therefore, the proposed miniature BT mixer is able to achieve high mixing performance even though it doesn't show the ideal BT mixing.

**Figure 23.** Comparison among cross-sections after each BT cycle

#### **4.4. Conclusion**

In order to meet the demand of mixing fluids for multi‐scale mixers, We have developed a novel methodology to fabricate three‐dimensional passive‐type mixers based on the baker's transformation. We aimed at fabricating BT structure with isovolumetric change without any separation/joining process of two channels for the sake of mass‐procucing BT mold structures by utilizing precision cutting techniques. Two scales of BT mixers, one for microfluidic analytical system and the other for mixing high viscosity fluids in food processing or resin blending, were developed. We performed laboratory experiments for evaluating the mixing performances of two prototypes. The confocal microscopy for 10 cycles (10.4 mm in length) of microfluidicBTshowedthatthe significant potentialfor mixing FITC solution andIgGsolution more efficiently and rapidly than in a microchannel without BT structures, and mixing rate was more than 70‐fold faster for FITC solution and 900‐fold faster for IgG solution. The comparison of numerical analysis and the experimental result of mixing two colored silicone rubbers by using prototyped miniature BT mixer (3 cycles) showed good similarities in their cross‐sectional phases.
