**Optimization of Microfluidic Systems**

*-Collaborations with Prof. Bijan Mohammadi of Montpellier University, France*

**Shape-optimized micromixers for studies of protein folding kinetics**

We have applied an optimization method in conjunction with numerical simulations to minimize the mixing time of a microfluidic mixer developed for protein folding studies. The optimization method uses a semi-deterministic algorithm to find the global minimum of the mixing time by varying the mixer geometry and flow conditions. Dye quenching experiments of the original and optimized mixer designs show respective mixing times of 7 ms and 4 ms, a 40 % reduction. The new design also provides more uniform mixing across streamlines that enter the mixer. The optimized mixer is the fastest reported continuous flow mixer for protein folding.

**Figure 1. *** The goal of this project is to use a new semi-deterministic algorithm (SDA) to optimize the
performance of a microfluidic mixer for studying protein folding. The system is modeled with two dimensional FEMLAB models
and Matlab. The initial design model (shown on left) is optimized with respect to the mixer's geometry and flow rates with
the SDA (center) to discover new and novel mixer designs with faster mixing times. *

**Figure 2. ***Our original parametrically optimized design (left) and the new fully shape-optimized geometry (right)
with computation grids. The mixers are symmetric about the vertical centerline so only one symmetric half is modeled
to save computation time. The grids are finer near the intersection and along the high concentration and velocity gradient
regions of the focused stream in order to resolve them. The channels are labeled as center channel (top or North channel),
side channels (horizontal or East/West channels), and exit channel (bottom or South channel). *

**Figure 3. ***Brightfield reflection images of the fabricated mixers. Parametrically optimized design is shown on
left and current shape-optimized design is shown on right. Focal plane is approximately coincident with the inner surface
of the optical access window for each channel. *

**Figure 4. **

*Left: Predicted normalized concentration versus time. Dashed line is the parametrically optimized
mixer design and solid line is the current design. The older design takes 8 us to go from 90 % to 30 % of the initial
concentration. The current design takes only 1.2 us. *

*Right: Experimentally measured potassium iodide concentration versus time using a dye quenching technique. The current
mixer optimization improved the mixing time by approximately 3 us over the parametrically optimized design. Open circles
are the parametrically optimized design and closed circles are current shape optimized design. *

- Two dimensional models used throughout Physical quantities such as density, viscosity, diffusivity, etc. fixed System is constrained to a three-inlet, one-outlet design with perpendicular intersecting channels Minimum feature size is 2um due to contraints on photolithography Side channel widths are limited to 3um minimum width to mitigate clogging issues Maximum Reynold number is constrained to eliminate three-dimensional flow fields in the real devices Maximum channel aspect ratio (Width/Height)=1
- Mixer symmetry about a vertical centerline is maintained to reduce computation time.

Results:

We have applied a shape optimization method to the design of fast microfluidic mixers and to reduce mixing time. The optimization combines a semi-deterministic algorithm (SDA) and numerical simulations to minimize the mixing time by varying the mixer geometry and flow conditions within a specified set of constraints. The optimization reduced the expected mixing time from 8 us in the previous mixer design to 1.2 us in the shape-optimized design. This mixing time improvement was achieved solely by varying geometry and flowrates of the mixer within the minimum feature size and maximum flowrate constraints.

We experimentally measured mixing time with a dye quenching assay. We measured a reduction in mixing time from 7 us in the previous mixer design to 4 us in the current shape-optimized design. The measured mixing times fall within the estimated experimental uncertainty (a few microseconds) of the predicted mixing time.

We have used these mixers to measure the folding kinetics of a benchmark protein ACBP using Forster resonance energy transfer (FRET), and are currently using them to study protein collapse of a variety of proteins including CI2 and protein L. We have also fabricated these mixers with fused silica substrates and are measuring folding of unlabeled proteins such as cytochrome-c, lysozome, and apomyoglobin with UV-vis spectroscopy.

**Shape-optimized microchannel turns for on-chip CE**

*(link to project page)*

New corner designs have been successful in decreasing the amount of dispersion caused by a corner in CE systems. Numerical optimization can help design compensating geometries to reduce dispersion caused by the "race track" effect. The figure below shows images from a simulation that demonstrates the effect of compensating, 90 degree turn on an initially Gaussian sample. The simulations are compared to data collected in the laboratory using caged fluorescence imaging.

**Figure 1. ***Simulation (top) and bleached fluorescence imaging (bottom) showing the dispersion of an initially
Gaussian analyte band traveling through a compensating turn. *

**Optimization of channel injections**

We have developed a systematic, experimentally-validated method of designing electrokinetic injections for on-chip capillary electrophoresis applications. This method can be used to predict point-wise and CCD-imaged electrophoregrams using estimates of species mobilities, diffusivities and initial sample plug parameters. A simple Taylor dispersion model is used to characterize electrophoretic separations in terms of resolution and signal-to-noise ratio. Detection convolutions using Gaussian and Boxcar detector response functions are used to relate optimal conditions for resolution and signal as a function of relevant system parameters including electroosmotic mobility, sample injection length, detector length scale, and the length-to-detector. Analytical solutions show a tradeoff between signal-to-noise ratio and resolution with respect to dimensionless injection width and length to the detector. In contrast, there is no tradeoff with respect to the Peclet number as increases in Peclet number favor both SNR and R. We validate our model with quantitative epifluorescence visualizations of electrophoretic separation experiments in a simple cross channel microchip. For the pure advection regime of dispersion, we use numerical simulations of the transient convective diffusion processes associated with electrokinetics together with an optimization algorithm to design a voltage control scheme which produces an injection plug that has minimal advective dispersion. We also validate this optimal injection scheme using fluorescence visualizations. These validations show that optimized voltage scheme produces injections with a standard deviation less than one-fifth of the width of the microchannel.

**(Mouseover figure to begin movie)**

**Figure 1. ***Movie of optimized injection.*

**References:**

*Protein Mixers:*

1. Ivorra, B. Mohammadi, B., D.H., Hertzog, J.G. Santiago, “Semi-deterministic and
Genetic Algorithms for Global Optimization of Microfluidic Protein Folding
Devices,” in press, *International Journal
for Numerical Methods in Engineering*, 2005.

2. Alexis-Alexandre, G., B. Mohammadi, J.G. Santiago, and R. Bharadwaj,
“Microfluidic Flow Simulations: Stacking One-Dimensional Study,” * Houille Blance-Revue Internationale De Leau*,
No. 5, pp. 18-23, 2003.

3. Mohammadi, B. and J.G. Santiago, “Incomplete Sensitivities in Design and Control
of Fluidic Channels,” *Computer Assisted
Mechanics and Engineering Sciences,* No. 10, pp. 201-210, 2003

Optimized Injection Work:

4. Bharadwaj, R., J.G. Santiago, and B. Mohammadi, "Design
and Optimization of On-Chip Capillary Electrophoresis," * Electrophoresis*, Vol. 23, pp.
2729-2744, 2002

Optimized Turns for on-chip CE

5. Mohammadi,
B., Molho, J.I., and J.G. Santiago, “Incomplete sensitivities for the design of
minimal dispersion fluidic channels,” * Computer Method in Applied Mechanics and Engineering*, Vol. 192, No.
37-38, pp. 4131-4145, 2002

6. Mohammadi,
B. and J.G. Santiago, “Simulation and Design of Extraction and Separation
Fluidic Devices,” *Mathematical Modelling and
Numerical Analysis*, Vol. 35, No. 3, pp. 513-523, 2002

7. Molho,
J.I., A.E. Herr, B.P. Mosier, J.G. Santiago, T.W. Kenny, R.A. Brennen, G.B.
Gordon, B. Mohammadi, “Optimization of Turn Geometries for On-Chip
Electrophoresis,” *Analytical Chemistry*,
Vol. 73, No. 6, pp. 1350-1360, 2001__ __

EKI and Stacking

8. Lin, H., R. Bharadwaj, J.G. Santiago and B. Mohammadi, "A High Fidelity Electrokinetic Flow Model for the Prediction of Electrophoregrams in On-chip Electrophoresis Applications," published in the Proceedings of the International Mechanical Engineering Congress and Exposition, Orlando, FL, IMECE2005-79439, November 5-11, 2005.

9. Alexis-Alexandre, G., B. Mohammadi, J.G. Santiago, and R. Bharadwaj, “Microfluidic flow simulations: Stacking one-dimensional study,” Houille Blance-Revue Internationale De Leau, No. 5, pp. 18-23, 2003.

10. Bharadwaj, R., J.G. Santiago, and B. Mohammadi, " Investigation of Dispersive Effects in Field Amplified Sample Stacking," Second Gordon Research Conference on the Physics and Chemistry of Microfluidics, Big Sky, MT. USA, August 24-29, 2003

11. 2004, Shape Optimization in Fluid Mechanics, Bijan Mohammadi and Olivier Pironneau, Annual Review of Fluid Mechanics, Vol. 36: 255-279.

12. 2002, Applied Shape Optimization for Fluids, Bijan Mohammadi and Olivier Pironneau, Oxford University Press.