H YDRODYNAMIC FOCUSING

∂ . (3-9)

Therefore, for non-steady or continually changing state diffusion, one has to use Fick's second law,⁹⁰

which is obtained by combining equation (3-8) and (3-9). In order to fully describe local concentration and concentration changes, the Navier-Stokes equation or – for purely laminar flow – the Stokes equation has to be additionally considered since it is coupled to equation (3-10). In general, the resulting system of coupled differential equations is very complicated and cannot be analytically solved. To describe the physics of the microflow inside the microchannel system, finite element simulations are therefore used (chapter 2.3).

3.5. Hydrodynamic focusing

Laminar flow conditions in microchannels force mixing to occur purely by diffusion.

This can be desirable or not, depending on the application. However, unique flow phenomena on microscales can be leveraged to design and fabricate microfluidic devices according to experimental needs. For analyzing DNA compaction under highly defined conditions, hydrodynamic focusing devices are used. Hydrodynamic focusing features crossed (micro) channel geometry with three inlets and one outlet. Solutions injected in the microfluidic device meet at the confluence of the microchannels. The

3. Microfluidics

Figure 3-2: (a) Finite element simulations of the velocity field u (top) and the concentration of side channel component cside (bottom) illustrating the effect of laminar mixing. (b) Line profiles of u and cside at annotated positions along the main channel.

main stream is hydrodynamically focused by the influx of the solutions injected into the two side channels and fluid elements are accelerated.¹⁰³ In addition to hydrodynamic focusing of a liquid stream, using two miscible liquids in crossed microchannels allows for diffusive mixing of these liquids. This opens a wide field of interesting experiments.56, 60, 104 The details of subsequent mixing between fluids vary greatly depending on the geometry of the outlet channel. In what follows, two fundamental mixing scenarios are discussed.

In Figure 3-2a, flow conditions in a hydrodynamic focusing device with an outlet channel of uniform width are calculated using 2D finite element simulations.

Simulations describe the velocity field u (top) and the concentration cside (bottom) of the side channel component evolving when aqueous solutions with identical viscosities ηmain = ηside ≈ ηwater are injected in the main and side channels with a fixed initial velocity u0 = 0.7mm^.s^-1. Both side channel solutions contain small molecules of hydrodynamic radius Rside = 0.45nm with a diffusion constant Dside = 5·10⁻¹⁰m²s^-1.

3. Microfluidics

Figure 3-3: (a) Finite element simulations of the velocity field u (top) and the concentration of side channel component cside (bottom) illustrating the effect of fast mixing in a device with a narrowed region in the outlet channel. (b) Line profiles of u and cside at annotated positions along the main channel.

These values are comparable to those of small biologically relevant molecules such as glucose. cside ranges from 0 to 1.0. Owing to the symmetry of the microfluidic device, it is sufficient to simulate half of the device.

Lines crossing the channels quantify values of u and cside at different x positions 1, 2, and 3 (Figure 3-2b). The mixing in the device shown in Figure 3-2a is a consequence of the laminar flow and the diffusion within the channel. A continuous increase in cside at the channel center (y = 0) is recorded. cside will approach a constant value at some further point x when side and main components are completely mixed. Velocity profiles of this device approach a steady-state value at some distance from the confluence area.

Taking advantage of the highly defined laminar flow conditions along the outlet channel, diffusive mixing establishes stable concentration gradients. Introducing distinct chemical reagents into the injected solutions, concentration gradients in devices having such geometry can be utilized to study non-equilibrium structure formation in the intermixed region. Each observation point contains a different concentration of reactants. Thus, a wide variety of chemical environments can be easily created.

3. Microfluidics

Temporal resolution of particular time points in the reaction can be achieved by observing the state of the reaction at different positions along the outlet channel because distance in the flow direction translates into a time of reaction.103, 105, 106

Consequently, interaction dynamics depending on the concentration distribution can be spatially separated. Remarkably, non-equilibrium dynamics are thus accessible in steady-state flow.^59-61

Besides establishing highly defined concentration gradients, rapid mixing of different solutions is often desired, e.g. to quantify time-evolved properties of a fully mixed ensemble. Figure 3-3a shows the geometry of a hydrodynamic focusing device with this ability. The inclusion of an abrupt decrease in the width of the interaction channel creates markedly different flow conditions than those in a straight channel having a uniform width. Finite element simulations of the second device have been performed using identical initial velocities and concentrations. However, the scale of u is 10-fold higher for the device in Figure 3-3a due to the rapid increase in u in the narrow channel region. Line profiles at x positions close to the beginning of the channel step (position 1) exhibit extremely steep u and cside values moving from the center of the channel outward along the y-axis. The fast mixing induced by the strong reduction of the distance, which molecules have to cover by diffusion, results in a nearly constant value of cside at positions after the confined region (i.e. position 3).

The two scenarios only serve as examples. Interactions within microchannels can also be controlled by variables such as relative flow velocity and further geometric modifications. The width of the focused stream can be adjusted by the ratio of flow rates of the main channel and the side channels. Increasing the flow rate of the side channels leads to a narrowing of the center stream. Clearly, a wide variety of parameters are sufficiently malleable to enable investigations of real-time dynamic interactions. Furthermore, these parameters may be tailored to address specific experimental considerations.

An additional, important benefit of utilizing hydrodynamic focusing devices to study complex fluids results from the ability to introduce an additional hydrodynamic stress due to the sudden increase of the overall flow rate in the crossing region. Fluid elements become extended along the flow direction, as do the polymer molecules within the flow. Providing a sufficiently large rate of extension ε_& compared to τP-1

(high Wi), polymer molecules are stretched and aligned during the self-assembly process. This improves the formation of liquid-crystalline phases, which allows for a significantly enhanced characterization by X-ray diffraction.^59-61 Thus, additional information regarding the liquid-crystalline order of the complex fluid is obtained.

However, biomaterials are typically sensitive to the high energy of X-rays and can rapidly degrade when exposed to a beam for long periods of time. Measurements in

3. Microfluidics

microdevices eradicate this concern, because while the X-ray beam remains at a fixed position, the materials within the device continue to flow. As a consequence, each individual molecule is exposed to the X-ray beam only for a very short time, and the detector records an ensemble average instead of a time average.