# Formation of Monodisperse Bubbles in a Microfluidic Flow-Focusing Device

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Original entry: Tony Orth, APPHY 226, Spring 2009

G. Whitesides, H. Stone et al., Applied Physics Letters Vol.85 No.13 (2004) [1]

### Brief Summary

This paper reports one of the first devices to produce monodisperse bubbles for microfluidic applications. Droplet-based microfluidics have huge potential in medical diagnostics, chemistry, drug discovery and beyond. The physical understanding of the process by which bubbles or droplets are created in various microfluidic geometries is critical to the development of such methods. The authors of this paper demonstrate the creation of gaseous bubbles surrounded by a liquid phase, with a polydispersity of less than 2% and at rates of $10^5$ per second. Some scaling relations are presented in an attempt to develop an intuition of the factors affecting the volume of the bubbles produced and the frequency behaviour of the bubble formation.

Figure 1: Schematic of the flow focusing device.
Figure 2: Scaling of the volume of bubbles as a function of flow rate and viscosity. $V_b$ is the volume of the bubble, $\mu$ the viscosity of the liquid. The product $q\mu$ is scaled by $L/h^4$ to yield units of pressure, where $L$ is the length of the outlet channel and $h$ is its height. A bifurcation occurs at around $q \mu (L/h^4) = 45 kPa$
Figure 3: a) Frequency of the bubble formation as a function of flow rate The solid line is a fit to $f\propto pq$; in this plot the ratio $p/q = const.$. b) Volume fraction (and radius of bubbles) of the resulting foam in the outlet channel as a function of flow rate. The bubble radius is the horizontal line (ie. bubbles are monodisperse). The plot shows how the volume fraction of the foam can be controlled independently from the bubble radius since $V_b \propto p/q$

### Capillarity Phenomena

The basic idea behind the formation of bubbles in the geometry investigated is simple enough. A gaseous phase (nitrogen) is set to flow through a central channel and comes to a first orifice of width $W_g$. This orifice is met by two perpendicularly flowing channels of a liquid phase. Downstream of this, there is a second, smaller orifice of width $W_{or}$. A pressure is applied to the central channel so that a thread of gas reaches out of the second smaller orifice. The elongated liquid-gas interface carries an increasingly large energy which leads to a Rayleigh-Plateau-like instability [2]. At a critical and well defined surface energy, the gaseous thread breaks into a bubble which is then carried downstream. The geometry is depicted in figure 1. Note that this statement is potentially misleading, as the authors go on to point out that the surface tension plays only a minimal role in the dynamics. The "surface energy" that the authors refer to is presumably a conglomeration of interfacial pressures, shears and surface tensions.

The scaling of the bubble formation is quite interesting. For a fixed gas pressure $p$ there are two distinct regimes of bubble formation, called "period-1" and "period-2" The period-1 behaviour produces sequential monodisperse bubbles of gas. Once the fluid flow rate is raised above a certain critical value, two sizes of bubbles are produced in a single cycle. Each size of bubble is relatively monodisperse in itself ($<5%$) which suggests that the period-2 cycle is still stable. This bifurcation is shown in figure 2; the volume of bubbles in both regimes are found to scale inversely with the product $q\mu$ ($\mu$ the viscosity of the liquid) which is said to suggest that the capillary number ($Ca\propto q\mu/\gamma$) may be the crucial nondimesional parameter which describes the pinching off of the bubbles. However, upon varying the amount of surfactant in the liquid by a factor of 2 ($\gamma=37-72 mN/m$), the authors did not observe any significant change in the volume of the bubbles. This is quite strange since intuitively one would attribute the pinch off of the bubble to a high surface tension energy (as in the Rayleigh-Plateau instability). To explain this puzzling result, the authors offer the following explanation.

The volume of the bubble is $V_b=q_{gas}\tau$ where $q_{gas}$ is the gas flow rate and $\tau$ is the time it takes for the bubble to pinch off after entering the small orifice. From some to-be-published data (at the time of this letter), the authors state that $\tau\propto 1/q$. Also, $q_{gas}\propto p/\mu$ for a viscously resistive Poiseuille flow. Combining these two relations give $V_b \propto p/q\mu$ which is corroborated by the results in figure 2 (note also the inset - the linear relationship between $V_b$ and $p$).

The authors conclude by touching on the reason for the bifurcation form period-1 to period-2 behaviour. They hypothesize that this transition is the product of competing Laplace pressure of the tip and pressure from the liquid flow around the thread. This bifurcation is surely the most physically rich portion of the letter and is investigated in more detail in a later paper [3], where additional bifurcations are observed.