Timescale and spatial distribution of local plastic events in a two-dimensional microfluidic crystal

When a microfluidic crystal consisting of a concentrated emulsion flows in a convergent channel, the boundary conditions enforce a sequence of droplet rearrangements, also known as T1 events. At low flow rates, these T1 events are shown to be periodic both in space and in time, giving rise to a surprisingly ordered flow pattern. At high flow rates, however, the order is lost. To understand the transition from order to disorder, this paper examines the timescale and the spatial distribution of T1 events during the flow of a monolayer of monodisperse droplets within a concentrated emulsion confined in a convergent tapered microchannel. We show that the duration of a single T1 event consists of three distinct regimes with two transitions upon an increase in the applied flow rate. The first transition can be understood as a change in the forces that dominate during the T1 event, and the second transition can be associated with the emulsion transitioning from a solidlike state to a liquidlike state. Our results suggest that the loss of order in the flow of the concentrated emulsion, or the microfluidic crystal, is directly associated with the solidlike to liquidlike transition of the emulsion. Practically, our results are significant in understanding the relationship between the macroscopic property and the microscopic flow structures of the emulsion, as well as in guiding the design of flow control elements in microfluidic devices.

Figure 1

(a) Progression of a T1 event. The green circles highlight a converging pair of droplets. The red circles highlight a diverging pair. The time interval between two snapshots is 0.05 s. (b) Microscopic image of the emulsion flowing in the tapered microchannel. $x=0$ is set as the left boundary of the field of view shown. The blue dot represents the center of the five-edge Voronoi cell, which approximates the location of a T1 event. (c) Definition of control volume (CV). The blue markers (overlaid on top of each other) represent the locations of T1 events after 2000 drops advected through the field of view. Each red dashed box highlights a CV that has a width of 5 droplet diameters.

Figure 2

(a),(b) Voronoi tessellation during a T1 event at (a) $\mathrm{Ca}=4.9\times {10}^{-7}$ and (b)$\mathrm{Ca}=2.1\times {10}^{-3}$, respectively. The green lines represent the Voronoi cell edges. The red line segment represents the growing edge shared by a pair of converging drops during the T1 events. The blue dots represent the centers of five-edged Voronoi cells. (c)–(e) Length of the growing edges as a function of time for (c) ${10}^{-7}<\mathrm{Ca}<{10}^{-6},$ (d) ${10}^{-6}<\mathrm{Ca}<{10}^{-3},$ and (e) ${10}^{-3}<\mathrm{Ca}<{10}^{-2}$. Each set of symbols represents the measurement of one T1 event at a given Ca.

Figure 3

(a) Representative distribution of dimensionless T1 duration $T^{\prime }/\langle T^{\prime }\rangle$ at various Ca. Each distribution consists of a batch of $T^{\prime }$ measurements of $>200$ T1 events within one CV at a given flow rate $Q$. $T^{\prime }$ was normalized by $\langle T^{\prime }\rangle$, the value of $T^{\prime }$ averaged over $>200$ measurements. The dashed line is a visual guide only. (b) Log-log plot showing $T^{\prime }$ as a function of Ca. Each data point consists of a batch of $T^{\prime }$ measurements of >200 T1 events within one CV at a given flow rate $Q$. The vertical and the horizontal error bars represent the standard deviation in $T^{\prime }$ and that in Ca for all $>200$ measurements within one CV at one value of $Q$, respectively (see Sec. 2f). The three dashed lines have logarithmic slopes of $-0.14, -0.61$, and $-0.37$, respectively.

Figure 4

(a) Scaled T1 event count $P_{\mathrm{T}1}$ as a function of Ca. (b) Time between two successive T1 events $d{T}^{\prime }$ as a function of Ca in a logarithmic scale. The results of $\langle T^{\prime }\rangle$ in Fig. 3 are also shown for a direct comparison. The dashed line shows a fit to the $d{T}^{\prime }$ data. It has a logarithmic slope of −0.86. (c)–(e) Distribution of event onset time $t_{\mathrm{T}1}$ normalized by $\langle T^{\prime }\rangle$ at (c) $\langle \mathrm{Ca}\rangle =2.5\times {10}^{-7}$, (d) $\langle \mathrm{Ca}\rangle =2.4\times {10}^{-5}$, and (e) $\langle \mathrm{Ca}\rangle =1.4\times {10}^{-3}$, respectively.

Figure 5

(a)–(c) Spatial distribution of T1 events for Ca ranges of (a) $1.1\times {10}^{-7}<\mathrm{Ca}<1.6\times {10}^{-6}$ (within regime 1), (b) $8.5\times {10}^{-6}<\mathrm{Ca}<6.3\times {10}^{-5}$ (within regime 2), and (c) $7.9\times {10}^{-4}<\mathrm{Ca}<8.5\times {10}^{-3}$ (within regime 3), respectively. The Ca range is measured from the left edge ($x=0$) to the right edge ($x=x_{\max }$) of the channel. To obtain the spatial distribution, we generated $10\times 10\mu \mathrm{m}$ mesh grids in the field of view and counted the number of T1 events that occurred within each grid. The location of each marker represents the center of a grid. The color of the markers represents the number of T1 events that occurred in the corresponding grid. The white space between the markers has no recorded T1 events. Each plot was generated by tracking a total of 2000 drops advected through the field of view. The dashed box marks the boundaries of the control volume (CV).

Figure 6

(a)–(c) Trajectories of 2000 droplets advected through the channel for (a) $1.1\times {10}^{-7}<\mathrm{Ca}<1.6\times {10}^{-6}$ (within regime 1), (b) $8.5\times {10}^{-6}<\mathrm{Ca}<6.3\times {10}^{-5}$ (within regime 2), and (c) $7.9\times {10}^{-4}<\mathrm{Ca}<8.5\times {10}^{-3}$ (within regime 3), respectively. Each marker represents the instantaneous position of the centroid of one droplet. The insets are the time-averaged, $x$ component of droplet velocity ($\overline{U_x}$) across the width of the channel calculated at the locations marked by the red and green boxes, respectively. $\overline{U_x}$ is normalized to ${\overline{U_x}}_{\max }$, the maximum $x$ component of velocity observed in the constriction. The rearrangement zones are indicated by blue arrows in (a) and (b). (d)–(f) Snapshots of successive T1 events at (a) $\langle \mathrm{Ca}\rangle =2.5\times {10}^{-7}$ (regime 1), (b) $\langle \mathrm{Ca}\rangle =1.2\times {10}^{-5}$ (regime 2), and (c) $\langle \mathrm{Ca}\rangle =1.4\times {10}^{-3}$ (regime 3), respectively. The blue arrows indicate the direction along which successive T1 events occur. For T1 events along the 60 degree slip plane, the converging (diverging) drops are colored in green (red). For T1 events along the direction of imposed flow, the converging (diverging) drops are colored in yellow (purple). For both (d) and (e), each sequence shows five successive T1 events along the slip plane. For (f), the sequence shows five successive T1 events along the slip plane and the flow direction, respectively.

Figure 7

$d{T}^{\prime }, T^{\prime }$, and De as a function of Ca. $\mathrm{C}{\mathrm{a}}_{1-2}, \mathrm{C}{\mathrm{a}}_{2-3}$, and $\mathrm{C}{\mathrm{a}}_{S-L}$ represent the transition between regimes 1 and 2, the transition between regimes 2 and 3, and the solidlike to liquidlike transition when $\mathrm{De}=1$, respectively.