# Experimental observations of the squeezing-to-dripping transition in T-shaped microfluidic junctions

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# Experimental observations of the squeezing-to-dripping transition in T-shaped microfluidic junctions

Christopher GF, Noharuddin NN, Taylor JA, Anna SL., Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Sep;78(3 Pt 2):036317.

## Soft Matter Keywords

Capillary number, shear stress, Laplace pressure, curvature

(From Paper)

## Soft Matter Discussion

The widths and heights of the channels that are the relevant geometric parameters in a T-junction break-up are shown below: Fig. 1:T-junction’s schematic diagram. Width of the channel where the continuous-phase flows at a rate Qc is denoted as wc while the width of the channel where the dispersed-phase flows at a rate Qd is denoted wd. The heights of the channels are uniform (h). b is how far downstream the droplet tip is while s is defined as the neck thickness.

The dimensionless parameters describing microfluidic droplet breakup are:

$Ca=g\frac{\mu _{c}u_{c}}{\sigma }=\frac{\mu _{c}Q_{c}}{\sigma w_{c}h}$

$\phi =\frac{Q_{d}}{Q_{c}}$

$\lambda =\frac{\mu _{d}}{\mu _{c}}$

$\Lambda =\frac{w_{d}}{w_{c}}$

where u is the average velocity and μ is the viscosity.

The dispersed-phase has to be nonwetting relative to the continuous phase to ensure stable droplet productions.

Previous studies done on the subject observed the dependency of the droplet size on the capillary number (Ca) and not on the flow rate ratio $\phi$ when the droplets generated are smaller than the channel width. The scaling models in those papers thus assume that viscous shear stress is the dominant force of the breakup agreeing to their experimental data. There is also another regime where droplets fill the channel and form “slug” or “plug” like shapes. The dominant parameter in this case is the flow rate ratio $\phi$ and not the capillary number (Ca).

In those studies, the viscous shear stress dominated regime (“dripping” regime) occur when the capillary number is relatively large (0.01 < Ca < 0.5) and when the continuous-phase channel is wider than the dispersed-phase channel. The squeezing regime occurs at low capillary number (Ca < 0.01) and when the channel widths are similar.

The authors then modeled the transition regime. They assumed that the size of the emerging droplet before the detachment is determined by the balance of the three primary forces: the capillary force resisting deformation of the interface, the viscous stress acting on the emerging droplet, and the squeezing pressure. Detachment happens once the sum of the viscous stress and the squeezing pressure exceed the capillary pressure.

The capillary force is determined by calculating the difference between the Laplace pressures at the upstream and downstream ends of the droplet multiplied by the projected area of the emerging interface bh. The mean curvature downstream of the droplet is calculated by summing up the curvatures in the cross-channel direction (2/b) and the curvature in the depth direction (2/h). The mean curvature upstream of the droplet is calculated by summing the in-plane curvature near the neck (1/b). and the curvature in the depth direction (2/h). Thus

$F_{\sigma }\approx \left[ -\sigma \left( \frac{2}{b}+\frac{2}{h} \right)+\sigma \left( \frac{1}{b}+\frac{2}{h} \right) \right]bh\approx -\sigma h$

The viscous shear force is approximated by multiplying the viscous stress acting on the emerging interface and the projected area of the emerging interface:

$F_{\tau }\approx \mu _{c}\frac{u_{gap}}{(w_{c}-b)}bh\approx \frac{\mu _{c}Q_{c}b}{(w_{c}-b)^{2}}$

The squeezing pressure is approximated from the pressure arising from the blocking of the channel by the emerging droplet. This is obtained from the characteristic pressure from a lubrication analysis for pressure-driven flow.

$F_{p}\approx \Delta p_{c}bh\approx \frac{\mu _{c}u_{gap}}{(w_{c}-b)}\frac{b}{(w_{c}-b)}bh\approx \frac{\mu _{c}Q_{c}b^{2}}{(w_{c}-b)^{3}}$

By solving for b when these three forces sum to zero, the size of the emerging droplet before the detachment is found to be:

$(1-\bar{b})^{3}=\bar{b}\times Ca$

where $\bar{b}\equiv \frac{b}{w_{c}}$

The authors’ formula thus put capillary number as the controller of the initial size of the droplet.

The final droplet length at detachment is approximated by the product of the time required for the neck to thin to zero and the velocity of the tip of the droplet. This dimensionless length is approximated to be:

$\bar{L}\approx \bar{b}+\frac{\Lambda }{{\bar{b}}}\phi$

where $\bar{L}=\frac{L}{w_{c}}$

The dimensionless volume is then:

$\bar{V}=\frac{V}{w_{c}^{2}h}\approx \bar{b}^{2}+\Lambda \phi$

The droplet volume is then predicted to be dependent on both capillary number and the flow rate ratio.

Comparing this formula with their experimental data, they found quite a good agreement. As can be seen in the figure below, the predicted volume approaches a constant value at low capillary number. As the capillary number increases, the droplet volume decreases and the slope becomes steeper. The squeezing pressure is thus the dominant factor in droplet breakup at both low capillary numbers and high flow rate ratios. The viscous stress, however, plays an important role at larger capillary number and lower flow rate. Fig. 2:Dimensionless droplet volume as a function of capillary number for a fixed viscosity ratios ($\lambda$ = 0.01) and wd=wc=150um, h=50um. Solid lines are droplet volumes from the scaling model.