| Computer Simulations
Lab-on-a-Chip devices for solving increasingly complex chemical and biochemical problems will require correspondingly complex fluidic designs and electrokinetic control strategies. Rather than relying solely on experimental means to design and test microfluidic structures, it seems prudent to develop computer aided design and analysis tools for prototyping and refining fluidic layouts. Ultimately these design and analysis tools should include the ability to model material transport and chemical and biochemical processes within microchips. Such tools will clearly reduce the time and expense for development of microchip devices designed to solve specific measurement problems. Computer simulations are expected to be useful in the following areas:
The computer program implements several numerical algorithms solving discrete versions of the equations which constitute the mathematical model. A finite-difference technique is used for solving the Navier-Stokes equations, convection-diffusion equations for the sample species, and equations for the electric field. The computer program incorporates the design of the channel layout (see Figure 1), the specification of the initial and boundary conditions, and simulation process itself. Features of the computer program are:
Sample transport in several basic chip elements, including a cross-intersection, T- intersection, and 90-degree turn (see Figure 2) was simulated.
CROSS. A cross intersection can be used for sample injection. An example of a gated injection sequence (simulated and obtained from an experiment) is shown on the Figure 3. Operation sequence: Sample Loading (valve closed) - Injection (valve open) - Separation (valve closed). When the valve is closed, the sample moves from channel 1 to channel 2. The buffer flows from channel 4 to 3 preventing sample penetration into the separation channel (3). During the injection phase the sample is transported into the separation channel. The injected sample plug migrates down the separation channel when the valve is closed again. Simulations were performed for the following experimental parameters: sample - Rhodamine B, neutral (µ= 0), electroosmotic mobility µeo = 4.x 10-4cm2/V µsec, channel width w = 42 µm, diffusion coefficient (RhB) D = 3 x 10-6cm2/sec.
Figure 4 demonstrates 3 pairs of images of electrokinetic focusing in the cross-intersection, simulated and experimental. The upper panels correspond to experimental images, the lower ones correspond to the simulated images. Electrokinetic focusing precedes the injection step during the pinched type of injection. Thinner sample streams (from the left to the right) correspond to larger focusing field strength. Operating parameters are the following: Substance - Rhodamine 6G, cation (µ= 1.4x10-4cm2/V sec), electroosmotic mobility µeo = 4.0 x 10-4cm2/Vsec, channel width w = 24 µ m, diffusion coefficient (R6G) D = 3 x 10-6cm2/sec.
T-INTERSECTION. The T-intersection element is useful for sample mixing, dilution, and chemical modification. An example of sample dilution (by mixing it with buffer solution) is shown on Figure 5. Three different field strengths in the mixing channel (increasing from the left to the right) were tried. Computer simulations verified that the sample mixing occurs primarily by means of diffusion, thus the concentration profile inhomogeneity across the mixing channel is determined by the diffusion time. For a fixed channel length lower field strengths (left image) result in a better sample mixing compared to the case with a higher field strength (right image). Operating parameters are: sample is Rhodamine B, channel width is 20 µ m, electroosmotic mobility is 4 x 10-4 cm2/V µsec, diffusion coefficient is 3 x 10-6 cm2/sec. Field strength in the mixing channel (going from the left panel to the right) is 100 V/cm, 200 V/cm, 500 V/cm. The field strengths in the sample and buffer channels are equal to 1/2 of that in the mixing channel. Computer simulations also showed that T-intersection has a linear mixing characteristic for both charged and neutral substances, that is, the sample concentration in the mixing channel depends linearly on the ratio between field strengths in the sample and buffer channels.
90-DEGREE TURN. Sharp turns in the channels can be a source of significant sample dispersion. Electric field and sample electrokinetic velocity vary strongly around the corners. The sample passes very quickly around the inner corner, while the outer corner is the region of stagnant flow. As a result sample molecules spend much more time in the outer parts of the corner, which leads to strong sample tailing. Computer simulation demonstrates dispersion effects caused by 90-degree turn. Figure 6 demonstrates the sample plug evolution when it passes 90-degree turn. For comparison purposes a sample plug obtained in the channel without a turn is also shown. The lower panel shows a detailed view of the concentration distribution together with the velocity vector field. Simulations were performed for the following operating parameters: Channel width w = 50 µm, electroosmotic mobility µm eo = 4 x 10-4cm2/Vµsec, electric field E = 500V/cm, separation time tsep = 0.143 sec, channel length l = 0.085 cm, Substance = Rhodamine B.
Sample mass transport in microfabricated fluidic devices is the result of the following major mass transport mechanisms: convective flow, electrophoretic transport, diffusion, and chemical reactions. Bulk convective flow of liquid arises from one or a combination of the following factors: (i) a pressure difference applied to the separation channel ends, (ii) electroosmotic flow which has its origin at the channel walls, and (iii) thermal convection due to Joule heating. Electrophoretic transport is determined by the electric field distribution which depends upon the boundary conditions and the conductivity of the solution at each point along the channel and in turn depends on the chemical composition of the solution. The latter is determined by chemical reactions taking place among the chemical species in the solution. Finally, diffusion takes place whenever a spatial non-uniformity in the composition of the solution exists. The mathematical model formulated in terms of continuous media mechanics takes into account these principal phenomena. Currently, the following simplifications are assumed: