Shrinking , Growing , and Bursting : Microfluidic Equilibrium Control of Water-in-Water Droplets

Byeong-Ui Moon, Dae Kun Hwang, and Scott S. H. Tsai Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, Canada Keenan Research Centre for Biomedical Science, St. Michael’s Hospital, Toronto, Canada Institute for Biomedical Engineering, Science and Technology (iBEST), a partnership between Ryerson University and St. Michael’s Hospital, Toronto, Canada Department of Chemical Engineering, Ryerson University, Toronto, Canada


Calculation of droplet volume
We calculate the segment of the droplet volume by the following equations.The volume of an undeformed spherical shape droplet is given by V = 4/3πR 3 , where R is the radius (Fig. S1 (a)), the volume of the spherical cap segment is then given by VC = 1/3 πhd′ 2 (3R -hd′), where hd′ is the height of the spherical cap segment (Fig. S1 (b)).Since the height of the discoid droplet h = 2 (R -hd′), we can calculate the discoid droplet volume, Vdiscoid = (π/12) [2D 3 -(Dhd) 2 (2D + hd)], where D and hd is the diameter of the discoid and height of the droplet, respectively, (Fig. S1 (c)). 1 We assume that the droplet height hd, is the same as the height of the channel, h.
For non-axisymmetric discoid shape droplets, we simplify the equation, Vdisk = hdA, where, A is the crosssectional area of the droplet (Fig. S1 (d)).The area is manually selected using ImageJ TM software.Electronic Supplementary Material (ESI) for Lab on a Chip.This journal is © The Royal Society of Chemistry 2016

Using tie lines to estimate the size of droplets
The model for ATPS droplet shrinking and growing is developed by utilizing tie lines.Tie lines are useful tools for estimating the exchange of water during an ATPS re-equilibrium process.To predict the final size of re-equilibrated system, we make three assumptions, as described below.First, we assume that no exchange of PEG and DEX polymers occurs during re-equilibrium.Second, we assume that the PEGb phase that enters at the second cross junction only mixes with the PEGa phase, and does not interact directly with the DEX droplets.Third, we assume that water is the only substance that enters or exits the droplets during the re-equilibrium process.
Figure S2 shows an illustration of the phase diagram for droplet shrinking.At above the binodal curve (long dashes) of a given ATPS mixture, the solution phase separates into a lower density PEG-rich phase, and a higher density DEX-rich phase.For example, an initial ATPS 1 mixture (point A on Fig. S2), which is composed of 2.43 % (w/w) PEG and 7.78 % (w/w) DEX, phase-separates into two phases. 2The equilibrated PEG phase has 6.01 % (w/w) PEG and 0.65 % (w/w) DEX (point B on Fig. S2).The equilibrated DEX phase has 0.03 % (w/w) PEG and 12.55 % (w/w) DEX (point C on Fig. S2).The equilibrated state forms a tie line indicated by BC.The tie line BC is described by the expression, y = -0.5026x + 6.3407, where x and y corresponds to DEX and PEG concentrations on the phase diagram, respectively.
In the microfluidic channel, the PEGa phase (indicated by point B in Fig. S2) mixes with the PEGb phase (indicated by point D in Fig. S2) at the second cross junction at a mixing ratio of 1:3, respectively.The mixture of the two PEG solutions leads to a new PEG phase, PEGab (indicated by point E in Fig. S2).This new continuous phase, PEGab, causes the disperse DEX phase (originally indicated by point C in Fig. S2) to become out-of-equilibrium.By mainly water-exchange the droplet DEXa phase (point C) becomes the DEXab (point F), keeping the same slope of the original tie line.The new tie line, EF, is describes the final equilibrium state of the PEG continuous phase, and DEX droplets.
The process of re-equilibrating the DEX droplet, from a lower concentration DEXa phase (point C) to a higher concentration DEXab phase (point F) is possible by water exchange out of the droplet.This is the reason why the DEX droplet shrinks when PEGb has a higher concentration than PEGa.The concentration change is calculated as M0/M = V/V0 where, M0 and V0 are the initial droplet polymer concentration and volume, respectively, and M and V are the instantaneous droplet polymer concentration and volume, respectively.
Figure S3 shows the predicted tie lines corresponding to different concentrations of PEGb.We obtain these tie lines by demanding that the slope of each tie line remain the same as that for the original ATPS 1 mixture.We observe that at below a critical concentration of PEG in the PEGab phase, the drop does not form a single discoid shape, but elongates and breaks up into several satellite droplets.Here, a drop that was first formed using DEX and PEG phases from ATPS 4 begins to dissolve after the second cross-junction, where PEGb, a 3 % (w/w) PEG solution, is introduced.Scale bar 200 µm.

Figure S2 :
Figure S2: Phase diagram of tie line used for the droplet shrinking model.2

Figure S4 :
FigureS4: Droplet elongation and satellite droplet formation.We observe that at below a critical concentration of PEG in the PEGab phase, the drop does not form a single discoid shape, but elongates and breaks up into several satellite droplets.Here, a drop that was first formed using DEX and PEG phases from ATPS 4 begins to dissolve after the second cross-junction, where PEGb, a 3 % (w/w) PEG solution, is introduced.Scale bar 200 µm.