Rapid, Self-driven Liquid Mixing on Open-Surface Microfluidic Platforms

Self-driven surface micromixers (SDSM) relying on patterned-wettability technology provide an elegant solution for low-cost, point-of-care (POC) devices and lab-on-a-chip (LOC) applications. We present a SDSM fabricated by strategically patterning three wettable wedge-shaped tracks onto a non-wettable, flat surface. This SDSM operates by harnessing the wettability contrast and the geometry of the patterns to promote mixing of small liquid volumes (µL droplets) through a combination of coalescence and Laplace pressure-driven flow. Liquid droplets dispensed on two juxtaposed branches are transported to a coalescence station, where they merge after the accumulated volumes exceed a threshold. Further mixing occurs during capillary-driven, advective transport of the combined liquid over the third wettable track. Planar, non-wettable “islands” of different shapes are also laid on this third track to alter the flow in such a way that mixing is augmented. Several SDSM designs, each with a unique combination of island shapes and positions, are tested, providing a greater understanding of the different mixing regimes on these surfaces. The study offers design insights for developing low-cost surface microfluidic mixing devices on open substrates.


of the manuscript)
In a parametric study involving multiple variables, an interaction plot displays the extent to which the effect of one factor (independent variable) on the response variable (the dependent variable) changes depending on the level of the other factor(s). For example, a 2-dimensional interaction plot would explain whether the variation of a dependent variable z(x,y) -where x and y are the independent variables -with x (or y) is influenced by values of the other variable y (or x) at which the z data are reported. The lines in each subplot in Figure 3 of the manuscript were generated using a least squares regression analysis built in the DOE software (JMP, SAS®). In general, for a set of data points {( , , ): ∈ , ∈ , ∈ }, with one dependent variable (z) and two independent variables (x and y), a linear expression, considering a first-order interaction between x and y can be derived and fit to the dataset using the following expression (similar to the method outlined by Jaccard et al. 1 ) Eq. E1, when rearranged, becomes The first and second parenthesis terms of Eq. E2 represent the intercept and slope of the z(x) line, respectively, and e denotes a residual term, which is essentially the difference between the actual values of z (i.e. from the data) and the expected values of z (i.e. from least-squares fitting).
The coefficients 0 , 1 , 2 , 3 and e are calculated during the least-squares regression analysis, 2 and depending on their values, it is possible to determine if there is a first-order interaction between the independent variables x and z.
To demonstrate a relevant example of how a first-order interaction between two independent variables can influence a dependent variable, let us consider a simple case where the area ratio (α) and constriction ratio (δ*) are the only independent variables that can influence the mixing efficiency (η); in doing so, we neglect any influences from the shape or orientation (θ) of the superhydrophobic island. By plotting η against α for a constant value of δ*, one can write Eq. E2 as To fully understand how η is influenced by the interaction between α and δ*, let us choose two constant values for δ*, a low value and a high value, * and ℎ * , respectively. The two lines are then represented by the following equations The difference in slopes of these two lines is [ 3 ( ℎ * − * )]. If there is no interaction between α and δ*, 3 = 0 and the two curves are parallel with slopes equal to 1 and have a y-intercept offset of 2 ( ℎ * − * ). However, if there are interactions present, 3 ≠ 0; hence the slopes of (= 1 + 3 * ) and ℎ (= 1 + 3 ℎ * ) differ.
It is important to note that the above example only considers first-order interactions between two independent variables. A least-squares regression analysis, which considers interactions between three or more independent variables, is more complicated to describe here. Figure 3 in the manuscript considers that there are first-order interactions between four independent variables (α, δ*, shape, and θ). A general form describing how η varies with α, while considering interactions between α, δ*, shape, and θ may be written as: where the slope (m) and η-intercept (B) are given by = 0 + 2 * + 3 ( ℎ ) + 4 + 8 ( ℎ ) * + 9 * + 10 ( ℎ ) (E7) The least-squares regression analysis for the four independent variables (α, δ*, shape, θ) was performed using the JMP statistical software (SAS®), which generated the sub-plots in Figure   3 of the manuscript. Further details on interactions and regression analysis for three or more independent variables are given in Dawson and Richter 3 and Aiken et al. 4

Figure S4: Superhydrophilic wedge track (grey shape) with a superhydrophobic island (white). xend is common for both the control case (no island) and SDSM with a superhydrophobic island, and is approximately 1 mm from the far edge of CReservoir. xu and xd are locations on a SDSM having a superhydrophobic island, and are approximately 500 μm upstream and downstream from the island edges, respectively. A pixel analysis was carried out at each location every millisecond beginning with t = 0 ms, and up to t = Nt ms.
Mixing homogeneity (σk) and mixing efficiency (ηk) values were calculated for every millisecond (k th instant of time) of transport on the SDSM. Since the SDSM displayed some non-uniformities in TiO2 particle distribution, spatial heterogeneity to transmitted light on the substrate was observed even on a dry track. Regions with higher concentrations of coating nanoparticles appeared darker than regions of lower concentrations. Therefore, the mixing homogeneity on a wet track was calculated from the pixel information of a transient image after normalizing the local pixel intensity values with the respective pixel intensity values on the dry substrate. The mixing efficiency at a particular time (k) for a particular location (xu, xd, and xend) was calculated using the following algorithm ( Figure S4): 1. Identify the pixel intensity at a particular pixel point (j th pixel) before and after complete mixing ( ,0 and ,∞ , respectively).