## 3 Results  
### Mixing Analysis  
The degree of homogeneity of the concentration profile in any given cross-section of the mixer microchannel is a measure of the full mixing of the two solutions. The normalised concentration, \(c^{*}\), is defined as:  
\[c^{*}\ =\ \frac{c-c_{min}}{c_{max}-c_{min}} \tag{4}\]  
where \(c\) is the concentration of the species in solution, and the subscripts indicate the minimum (\(min\)) and maximum (\(max\)) concentration values. Homogeneity, in terms of mixing, is characterised by the standard deviation (\(\sigma_{\text{st}}\)) of the sampling points along the concentration profile for a particular probe point [46], i.e.,  
\[\sigma_{st}\ =\ \sqrt{\frac{1}{N}\sum_{i\,=\,1}^{N}\left(c_{i}^{*}-c_{m}^{*} \right)^{2}} \tag{5}\]  
Figure 4: Concentration profiles at six positions (probe points #1, 3, 5, 7, 9, 12) across the width of the serpentine micromixer. The concentration curve at the inlet (#1, black) shows two distinct solutions with a sharp concentration gradient at the interface. The concentration curves flatten approaching the outlet and are almost completely flat near to the outlet point of the liquid jet (#12, red) indicating enhanced mixing induced by the chaotic advection effect of the serpentine geometry. where \(N\) is the number of sampling points, \(c_{i}^{*}\) is the normalised concentration at point \(i\), and \(c_{m}^{*}\) is the mean normalised concentration. The mixing efficiency (\(\eta_{mixing}\)) is thus defined as:  
\[\eta_{mixing}\ =\ 1-\ \ \sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\frac{c_{i}^{*}-c_{ m}^{*}}{c_{i}^{*}}\right)^{2}} \tag{6}\]  
The mixing efficiency was measured at each of the \(12\) probe points (shown in Figure 3a shows the simulated velocity magnitude distribution within the serpentine micromixer. The serpentine geometry causes the maximum fluid velocity to vary, moving from the centre of the microchannel towards the edges along the \(x\)-direction [43]. This is a key characteristic required to induce chaotic flow in low Reynolds number microfluidic systems, where it is essential to have transverse components of liquid flow that stretch and fold volumes of fluid over the cross-section of the microfluidic channel. This is the mechanism via which the serpentine geometry induces stirring flows, which in turn leads to a decreased average distance over which the diffusion occurs. As the geometrical and liquid parameters are constant, the chaotic flow reaches a steady state, where the stretching and folding of volumes of the fluid proceed exponentially as a function of the axial distance travelled by the fluid volume. This is related to the characteristic length which depends on the geometry of the fabricated microfluidic system: \(\Delta y=\ t\cdot exp(-\Delta x/l)\), where \(t\) is the initial transverse distance [44; 45]. The Peclet number is a physical quantity defined as the ratio of the rate of the convective mass transport to the rate of diffusion mass transport. The Peclet number is given by \(Pe=\ ul/D\), where \(l\) is the characteristic length (the distance over which the species must mix). In the present case where we have chaotic mixing occurring, the hydraulic diameter of the channel is defined as the characteristic length scale and is given by \(4A/P_{\mathrm{w}}\) where \(A\) is the cross-sectional area, and \(P_{\mathrm{w}}\) is the wetted perimeter of the channel at the cross-section. For the present simulation, due to the low diffusivity of fluorescein in water and a relatively high-velocity range of 1.89-4.72 m/s, the maximum values of Pe are around 65,000. A large Pe indicates a convective dominated mass transport, which in this case is highly dependent on the chaotic advection phenomenon in the mixer microchannel [46]. The micro-mixer areas, where the chaotic advection effect is higher, are shown by the darker blue regions within the Peclet number distribution (Figure 3b). Figure 3b shows clear periodic perturbations, characteristic of the stretching and folding of fluid volumes, which are associated with chaotic advection.