## Design Study: Micromixers

### Flow and Mixing in Microchannels

**Laminar flow:**Re < 100 (Re = UI/v)- Diffusion is the primary mass transfer process.

**High Pe**> 100 (Pe = UI/D_{mol})- Timescale for diffusion is much slower than fluid convection

- Cumbersomely long channels are required (no "stirring" mechanism).

### Laminar Flow

- Laminar flow means that diffusion is the only mechanism to achieve mixing between parallel fluid streams. This is a slow process.
- This can also be useful.

- Diffusion length estimate:

where:D = diffusion coefficient (cm ^{2}/s)

t = time (s) - Typical values:

D (water) ~ 10^{-5}cm^{2}/s

U = 0.1 cm/s

I = channel width ~ 10 μm = 0.01 cm - How long should the channel be for the two streams to mix completely?

- Results:

I = 100 μm → L = 0.5 cm

I = 200 μm → L = 2 cm

I = 500 μm → L = 12 cm

### Microfluidic Mixer Designs

**Active Approaches**

- Applications of external forces
- Electro-osmotic
- Magnetic stirring
- Bubble-induced actuation
- Ultrasonic

- Effective in generating turbulence and/or increasing interfacial area for diffusion.
- Complex fabrication and assembly
- Electromagnetic fields and heat generation can limit compatibility with biological samples.

### Microfluidic Mixer Designs

**Passive Approaches**

- Simple design
- Good integration
- No moving parts
- 3D channel structures
- Multiple lithography steps
- Multilayer alignment

- Lamination: species to be mixed are divided into multiple streams and reassembled as alternating lamellae.
- Rotation: introduction of transverse flow component, may incorporate chaotic advection effects.

**Alternative Approach: Curved Channels**

- Transverse secondary flows due to centrifugal forces
- Shift of maximum in velocity profile toward outer wall
- Counter-rotating Dean vortices in the upper and lower halves of the channel

### Design: Placement of Splits and Expansions

**Key consideration:**want to position slits or expansions at a downstream location where at least 90° co-rotations have occurred.- Step 1: Inertial flow effects are most effective at Re > 10. For a given flow rate, the figure on the left can be used to determine the range of allowable microchannel diameters that will give Re > (i.e. the blue portion of the plot).
- Step 2: once the flow rate and hydraulic diameter are specified, Re can be calculated. Then, the scaling shown in hte figure on the right can be used to determine the ration of the radius of curvature
*R*to the minimum downstream distance L_{80}needed to achieve sufficient rotation. The splits or expansions should be positioned at or beyond point.

**Flow Analysis**

**Optimal P-SAR design:**split the microchannel at a downstream location where the transverse flow has induced simultaneous ~90° counter-rotations in the upper and lower halves of the cross-section.- Consider relative timescales associated with the axial and transverse components of fluid motion.
**Axial:**Poiseuille flow, velocity*u*(maximum centerline veloctiy)_{A}~ U_{0}**Transverse:**Dean flow, velocity scales as*u*_{D}~ Re(d/R)U_{0}

**Ratio of corresponding timescales:**- τ
_{A}/τ_{D}~*(L*_{A}/u_{A})/(L_{D}/u_{D}) = (L_{A}/R)Re - L
_{A}and L_{D}are characteristic axial and transverse length scales, and L_{D}is taken as a hydraulic diameter*d*.

- τ
- Downstream location at which a fluid element is transported across the width of the microchannel can then be estimated by setting τ
_{A}/τ_{D}~ 1, suggesting a linear scaling.

- Analysis of top view image sequence of aqueous streams (the inner stream is labeled with blue racer dye) flowing through a curved microchannel segment 200μ wide, 29μ tall, 937μ radius of curvature)
- Blue stream initially occupies 50% of the corss section upon entering the curved segment
- Downstream distance where transverse Dean effects pull the blue stream outward to occupy 80% of the channel cross section (
*L*) is determined at each value of κ (and_{80}*Re*) by analysis of digitized images

**Comparison with 3D SAR Micromixers**

**Total mixing time:**- Time required for redirecting the fluid flow from the point where the primary stream is split to the point where the individual split streams recombine
- Time for diffusive interspecies transport across lamellae upon recombination
- L
_{b}is the length required for branching - d
_{c}is the diameter of the channel - D is the diffusion coefficient
- D is the diffusion coefficient
- n is the number splits

- L

**3D SAR micromixer:**branching time increases linearly with the number of splits.**P-SAR:**branching time is essentially**independent of the number of splits**

### Design: Optimal Expansion Size

- A measure of the strength of expansion vortex effects in the horizontal plane can be inferred by considering the friction loss accompanying a sudden expansion. In engineering nomenclature, these effects can be characterized in terms of the friction loss ƒ
_{e}for the case of incompressible inviscid flow alonga streamline (i.e. losses associated with deviations from Bernoulli's equation). - For a sudden expansion ƒ
_{e}=*K*Where_{de}(u^{2}/2g)*u*is the average velocity in the narrow (inlet) channel segment, and*g*is the gravitational acceleration. *K*is an expansion-loss coefficient given by_{e}*K*where_{e}= (1 - S_{a}/S_{b})*S*and_{a}*S*are the cross-sectional areas of the narrow (inlet) and wide (outlet) segments respectively. Thus, an increase in the value of_{b}*K*corresponds to an increase in friction loss, which serves as an indication of increased expansion vortex strength._{e}- The expansion ratio chosen in the ASM design is
*S*= 1:5 (100:500 μm). Expansion ratios greater than 1:5 (indicated by red data point) yield only a minimal increase in teh value of_{a}/S_{b}*K*._{e} - Channel widths larger than 500 μm cause the channel to sag, as the material of construction is inherently soft.
- For these reasons we find an expansion ratio of 1:5 to be optimal for generating expansion vortices.

**Comparison with SHM Design**

**Stroock et al:**diffusion of a fluorescently labeled polymer tracer in an 80/20 glycerol/water soution (D=10^{-8}cm^{2}/s)**This work:**diffusion of a molecular dye (Rhodamine 6G) in water (D = 3 x 10^{-6}cm^{2}/s)- Convert Stroock data to aqueous working fluid for flow rate comparison