Puntos críticos en una placa circular

boundary condition, such as electrokinetic-driven fluid flow [145] .

Figure 2.2: Poiseuille parabolic fluid velocity flow profile as a function of xand yin the microfluidic channel utilised in chapters 5 and 6, assuming an incompressible fluid with non-slip boundary conditions. Channel dimensions are a width of 150 µm and height of 70 µm (i.e. = 0.47), with a flow rate of 77 µl hr−1. Velocities plotted are νx, y=0 and

νx=0, y.

2.3

Hydrodynamic Focusing

After reviewing the velocity profiles in a pressure driven Poiseuille flow, it is appa- rent that any particle suspended within microfluid flow can travel at a large range of velocities. For delivering particles to an optical beam for sorting, trapping, photopo- ration etc. it is desirable to firstly have all particles follow similar paths within the fluid, such that they all enter the beam at the same location, and they do this with similar velocities. One method could be to use fluid channels that are similarly sized to the particles themselves, but in practice this is normally prohibited due to the par- ticles attaching to the channel walls, blocking the path of further particle flow. The second option is the use of a positioning technique to place all particles within the same laminar flow stream. There are a number of methods for achieving this, inclu- ding electrokinetic and optical techniques, however the method utilised in this thesis along with numerous other works is hydrodynamic focusing. [18,107,110,151,152]

Hydrodynamic focusing is the use of twobuffer fluid streams to squeeze and thus

narrow a third stream, containing the sample, down the centre of a flow channel. By controlling the relative flow rates of the three fluids, the central sample flow can be squeezed to a width as required, or translocated from side to side. This allows the

2.3. Hydrodynamic Focusing 37

positioning of the sample fluid into a restricted range in the Poiseuille flow profile, at least in one dimension. Figure 2.3 shows the hydrodynamic focusing of a dye between two flows ofDIH₂O, for varying ratios of the buffer and sample flows.

Figure 2.3: Hydrodynamic focusing in a PDMS microfluidic chip. Flow is upwards in the

images, for flow ratios of (a) 1:1:1 (b) 2:1:2 (c) 4:1:4 (d) 8:1:8 (e) 16:1:16 (f) 32:1:32 of the three respective input channels (widths 50µm). Here, blue food colouring is focused by two buffer water flows. The outlet channels are missmatched (100 µm, 50 µm) in width such that if sufficiently narrowed, all the dye flows to the left hand outlet. The central channel region is 300µm in length and 150 µm wide. On the left is an integrated optical fibre that is not in use in this demonstration.

The width of the focused flow stream can be theoretically determined, and Lee

et al.[150] derived the following equations for this purpose

wf =

w

γ(1 +α) (2.8)

where wf is the width of the focused sample stream in a channel of width w and

height h, and α = Qs+Qb

w×h with Qs, b being the volumetric flow rates of the sample

and each of the buffer streams. γ is to be determined simultaneously with wf as

defined by γ = νˆf ˆ νo = 1− 192h π5_w f P∞ `=0 <sub>(2</sub>n+1)1 5 sinh[(2`+1)πwf/2h] cosh[(2`+1)πw/2h] 1−192h π5<sub>w</sub> P∞ `=0 tanh[(2`+1)πw/2h] (2`+1)5 (2.9) where ˆνf, o are the mean velocities of the focused stream and the overall output

2.3. Hydrodynamic Focusing 38

stream respectively. The widths of the focused streams in Figure 2.3 were measured and are plotted along with the theoretically predicted values in Figure 2.4, showing excellent agreement.

Figure 2.4: Experimental and theoretical determinations of the relative reduction in

sample fluid width by the use of hydrodynamic focusing, for different ratios of sample and buffer flow rates, for the chip dimensions utilised in Chapters 5 and 6.

Focusing a sample in a second dimension, so as to restrict its position in bothx

andy position, is indeed also possible. There is a discrepancy in the literature as to

whether to call this 2D or 3D hydrodynamic focusing, arising from the requirement for three dimensional microfluidic structures to be fabricated in order to restrict a sample in two dimensions [153]. 3D hydrodynamic focusing strictly is containing a sample within a spheroid or similar shape, surrounded and contained using a buffer fluid. The closest example of this is by Linet al. [154] where particles are contained

within a “hydrodynamic vortex trap”, where a recirculation of the fluid occurs in close proximity to a sudden expansion in the channel width. There are a number of different designs in the literature for obtaining 2D hydrodynamic focusing [153–158]. An interesting design makes use of chevron shaped structures on the top and bottom of the channel after a 1D hydrodynamic focusing junction, which acts to pull the buffer streams in on top and from below the sample, thus restricting it in the vertical plane as well.

2D hydrodynamic focusing was not employed for any of the experimental chap- ters. Many fabrication challenges were found in creating 3D structures within PDMS, particularly with the two restrictions of maintaining the coverslip as the base of channel, and having the sample focused within 50 µm of the coverslip for