Electronic Supplementary Information
Derivation of Mathematical Model for MATLAB calculations
The volume of a constricted droplet can be approximated by summing the volumes of a
trapezoidal prism and two half-ellipsoids, according to the following figure and equation (E.1).
lb
Δl
lf
h
Rf
Rb
Figure E1. Dimensions of geometric droplet model used for MATLAB calculations.
, (E.1)
where Δl = lf - lb. lf and lb are distances inside the channel with respect to the location of the trap
opening. That is, the trap inlet is defined as 0 and the end of the trapezoidal trap region is 150.
From the geometry of the trapezoidal portion of the trap, a linear equation can be written
to find the radii Rf and Rb at any point “x” along the trap length. This equation is derived by first
calculating the slope of the slanted side of the trapezoid. Then, using the equation of a straight
line for the slanted side to calculate the width of the channel at a particular x position, the droplet
radii Rf and Rb may be calculated. For example, for a trap with a 20 µm opening, the radius “r”
of a 50 μm droplet (in µm) at any “x” can be found from:
(E.2)
Therefore, the radii of the 2 hemispheres at the ends of the droplet can be expressed in terms of
their position inside the channel—Rf and Rb can be calculated from lf and lb, respectively. By
defining the volume of the droplet of interest, equation E.1 can be rewritten as a quadratic
equation to solve for lf. Choosing the correct root for lf from the solution of this quadratic,
equation E.2 can again be used to solve for Rf, and the Laplace pressure on the droplet at this
point can be calculated, according to equation 5.
A MATLAB program was written to calculate the Laplace pressure on a droplet of fixed
volume at many different values for the position lb. The value of lb was increased in small
increments to simulate the forward movement of a droplet inside the trap, and the program
calculates the geometry of the droplet for each value of lb, enabling calculation of the Laplace
pressure at each lb position. This provides a description of Laplace pressure on the droplet as it
moves forward into the trap.
Assumptions for MATLAB model of Laplace pressure and CFD model
1. Volume overestimation
The assumption of part of the droplet volume as a trapezoidal prism results in an overestimation
of this part of total droplet volume, due to the fact that a thin lubrication layer of continuous
phase surrounds the droplet, and thus the droplet volume does not occupy the corners of the
microchannel, as shown in Figure E2 below. Since a larger droplet volume results in the
development of a larger Laplace pressure (as shown in Figure E1), the Laplace pressure
calculated from this model is higher than the true value.
Figure E2. a.) Location of cross-section of the Laplace trap. b.) Cross-section of a droplet in the Laplace trap,
showing spaces in the corners of the microchannel where the volume is not occupied by the droplet, leading to
volume overestimation in the MATLAB model.
2. Disregard of friction forces
The CFD-ACE model of droplet movement into the Laplace trap takes into account surface
tension forces, but neglects frictional drag forces of the droplet along the microchannel wall and
fluid drag force between the droplet and its surrounding oil lubrication layer. Once a droplet
moves far enough into the trap to become a plug, recirculating flows within the droplet form, due
to contact of the droplet with microchannel walls [1]. By combining the data from the
MATLAB calculations and the CFD-ACE model, the contribution of frictional force can be
obtained from the following force balance expression:
(E.3)
Additional results of CFD-ACE Simulation
The effect of varying hydrostatic pressure applied in the device on the movement of a 50 μm
diameter droplet into a 20 μm diameter trap was studied using computational fluid dynamics
software (CFD-ACE, ESI Group Inc., Paris). The inlet hydrostatic pressure was varied between
250 Pa and 2000 Pa in increments of 250 Pa, and the position of the front of the droplet was
tracked over time. As shown below in Figure E3, a larger inlet pressure causes the droplet to
move more rapidly into the Laplace trap. In addition, the stopping position of the droplet in the
trap is dependent on the inlet pressure as well. As the inlet pressure is increased from 250 to
1750 Pa, the droplet stopping position moves farther into the trap, until at an inlet pressure of
2000 Pa, the hydrostatic force overcomes the Laplace pressure force and frictional forces
completely and moves out of the trap.
255
Front of droplet position (um)
235
215
195
250 Pa
175
500 Pa
750 Pa
1000 Pa
155
1250 Pa
1500 Pa
135
1750
2000 Pa
115
0
0.05
0.1
0.15
0.2
0.25
0.3
time (s)
Figure E3. Results from a CFD-ACE simulation tracking the movement of the front of a 50 μm diameter droplet
with time, as different pressures are applied to the device at the inlet. As more pressure is applied at the inlet, the
droplet moves more rapidly into the trap and stops at a later point in the trap. The droplet position 260 μm reflects
the case in which the droplet completely leaves the trap. Thus, when 2000 Pa of pressure were applied at the device
inlet, the droplet did not stop in the trap, but was pushed through.
1.
Handique K, Burns MA: Mathematical modeling of drop mixing in a slit-type
microchannel. Journal of Micromechanics and Microengineering 2001, 11:548.