# A Design for Mixing Using Bubbles in Branched Microfluidic Channels

"Design for mixing using bubbles in branched microfluidic channels" Piotr Garstecki, Michael A. Fischbach, and George M. Whitesides Applied Physics Letters 86(24) 244108 (2005)

## Soft Matter Keywords

liquid thin film, wetting, laterally heterogeneous surface

## Summary

Gang, et al. present experimental work performed to verify theory on the wetting of thin films. Working with a nanopatterned silicon surface and a film of methyl-cyclohexane (MCH), the authors should that film thickness has two different power law dependences on chemical potential offset from the bulk liquid-vapor equilibrium. These two dependences delineate two regimes of film growth. The first regime is characterized by constant film thickness on the surface and filling of the nanowells, while the second regime is characterized by growth of the surface film after the wells have been completely filled. The behavior of the film is studied using x-ray reflectivity (XR) and grazing incidence diffraction (GID).

## Practical Application of Research

Though not all results from this paper fully explained by theory, the authors have shown that film thickness on a heterogeneous surface is controllable. Deposition of coatings will benefit from this work, and it is feasible that nanopatterning a surface before deposition can help give an extra bit of control over the liquid thin film.

## Thin Film Thickness

With their patterned silicon/MCH system, the authors set out to validate theories on the dependence of liquid thin film thickness, <math>d</math>, on chemical potential difference, <math>\Delta \mu</math>. From previous work in the Pershan lab and others, the thickness of a film on a flat surface with van der Waals absorption, the authors expect a power-law of <math>d \propto \Delta \mu^{-1/3}</math>. However, for a surface containing isolated, infinitely deep parabolic cavities, the power law is expected to be roughly <math>d \propto \Delta \mu^{-3.4}</math>.

From XR data taken on their samples, the authors calculate the electron density, <math>\rho</math>, in the well and on the flat surface (see figure 1). They then introduce the areal adsorption, <math>\Gamma</math>, which is defined as: <math>\Gamma = \langle\int_0^{\infty}\left(\rho(z) - \rho_{dry}(z)\right)dz\rangle</math>. For van der Waals adsorption on flat surfaces, <math>d = \Gamma/\rho</math>, and <math>\Gamma(\Delta T) \propto d \propto \Delta T^{-1/3}</math>. The chemical potential difference is proportional to the temperature difference, according to the following relationship: <math>\Delta \mu = \frac{\partial \mu_0}{\partial T}\Big|_p\Delta T</math> [1]. Thus, we expect <math>d \propto \Delta \mu^{-1/3}</math>.

In figure 2, we see that the experimental data falls somewhere between the two theoretical extremes. Open squares are data derived from XR measurements, while filled circles are data derived from GID measurements. The black line represents an exponent of -0.76, while the dotted line is an exponent of -1/3 (theorized for a completely flat surface) and the dashed line represents an exponent of -3.4 (theorized for a surface with isolated, infinitely deep parabolic wells).

The authors conclude that the presence of the nanowells modifies the wetting behavior expected for flat surfaces. Additionally, as the wells fill, the <math>\Delta \mu</math> dependence of liquid adsorption is significantly weaker than predicted for deep parabolic wells. They suggest that finite-size effects have an considerable impact on the wetting behavior of nanoscale wells, which theory predicts should have only depended on well shape.

[1] I.M. Tidswell, T.A. Rabedeau, P.S. Pershan, and S.D. Kosowsky, Physical Review Letters **66**, 2108 (1991)