Difference between revisions of "A Design for Mixing Using Bubbles in Branched Microfluidic Channels"

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"Design for mixing using bubbles in branched microfluidic channels"
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Original entry:  Donald Aubrecht,  APPHY 226,  Spring 2009
Piotr Garstecki, Michael A. Fischbach, and George M. Whitesides
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Applied Physics Letters 86(24) 244108 (2005)
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"Design for mixing using bubbles in branched microfluidic channels"<br>
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Piotr Garstecki, Michael A. Fischbach, and George M. Whitesides<br>
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Applied Physics Letters <b>86</b>(24) 244108 (2005)
  
 
== Soft Matter Keywords ==
 
== Soft Matter Keywords ==
liquid thin film, wetting, laterally heterogeneous surface
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[[Microfluidic]], [[Bubbles]], [[Laminar mixing]], [[Peclet number]]
 
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[[Image:Gang-figure1.jpg|250px|thumb|right|Figure 1.  (a) XR data for the dry (black circles) and wet (red and blue circles) obtained on the nanopatterned surface.  (b) Corresponding electron density profiles for the dry (black) and wet surfaces in the filling (red) and growing regimes (blue).]]
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[[Image:Gang-figure2.jpg|250px|thumb|right|Figure 2. Liquid adsorption within the nanowells, obtained from XR and GID measurements.  Black line represents an exponent of -0.76, while the dotted line represents an exponent of -1/3 and the dashed line is -3.4.]]
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[[Image:Garstecki-4.jpg|300px|thumb|right|Figure 1.  Schematic showing introduction of the two liquid streams and formation of air slugs.]]
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[[Image:Garstecki-3.jpg|300px|thumb|right|Figure 2.  (a) Recirculation rolls between slugs of air.  (b)-(d) Schematics indicating channel design and the paths slugs take through the branches.  As a slug enters one arm, the hydrodynamic resistance in the arm increases, cause the next slug to enter the other arm of the branch.]]
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[[Image:Garstecki-1.jpg|300px|thumb|right|Figure 3.  (a) Two fluid streams remain nearly unmixed when bubbles are not present.  (b) Introducing bubbles along with fluid streams causes the two fluids to fold into one another, aiding in diffusive mixing.  Intensity as a function of position across the channel at the specified locations are shown.]]
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[[Image:Garstecki-2.jpg|300px|thumb|right|Figure 4.  Standard deviation of intensity profiles at different positions along the channel network.  Standard deviation of 0.5 indicates unmixed streams, while a standard deviation of 0 indicates perfectly mixed streams.  The two streams are nearly homogeneous after passing through roughly 10 branches in the channel.]]
  
 
== Summary ==
 
== Summary ==
Gang, et al. present experimental work performed to verify theory on the wetting of thin filmsWorking 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 filledThe behavior of the film is studied using x-ray reflectivity (XR) and grazing incidence diffraction (GID).
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This paper details experimental work and simple supporting theory regarding mixing in microfluidic channelsFor most microfluidic systems, the Reynolds number remains small (less than 1000), so turbulence is absent and mixing only occurs via diffusion.  Typical Peclet numbers in microfluidic channels are on the order of 1e5, indicating that mixing to homogeneity requires length scales on the order of 10 meters.  These lengths are not easily achieved on microfluidic devices due to finite substrate limits for fabrication and large pressure drops in the long channels, so the authors propose a novel method of mixing that aid the diffusion processUsing bubbles to fold two liquid streams into one another, greater contact area between the fluids is created, aiding in diffusion.
  
 
== Practical Application of Research ==
 
== 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 controllableDeposition 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.
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A drawback to continuous flow microfluidics has been the inability to homogeneously mix streams on-chipThis limits the application of continuous flow microfluidics, as it is difficult to introduce additional reagents or samples to a flow.  This passive, on-chip mixing scheme open a new realm of experiments that can take advantage of introduction of precise amounts of fluid at a given spatial or temporal point in a flow.
  
== Thin Film Thickness ==
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== Microfluidic Mixing Using Bubbles ==
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>.<br>
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As shown in Figure 1, the two liquids to be mixed are combined with a stream of air at a microfluidic flow focussing junction.  The liquid streams pinch off air bubbles, which then flow downstream. This break up is mediated primarily by conservation of mass.  As the thread of air enters the junction, it restricts flow of the two outer liquidsPressure builds up in the outer liquid lines, causing the air/liquid interface to deformEventually, the interface detaches from the walls, becomes unstable, and breaks, forming a bubble [1].
  
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>.<br>
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Once in the device, sequential air bubbles alternate arms of the branched channels they flow into (see Figure 2).  As an air bubble enters on branch, it causes the hydrodynamic resistance in that side of the branch to increaseThis causes a greater percentage of the new incoming flow to be diverted down the other branch.  This causes the next air bubble to flow through the branch that does not contain the first bubble.  As the authors mention in the paper, this is an additive effect and will work for any number of droplets in a series of branches.  The incoming droplet will end up being passively steered by the majority of the fluid streamlines to the channel with the smallest hydrodynamic resistance.
  
In figure 2, we see that the experimental data falls somewhere between the two theoretical extremesOpen 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).<br>
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In the case where no bubbles are present, the fluidic resistances of the two arms are equivalent and the fluid streamlines will split evenly left and rightThis means that if the two inlet fluids are introduced at equal flow rates, the fluid that starts on the left of the center of the main channel will always take the left branch, while the fluid on the right of center will always take the right branch (see Figure 3(a)).  The only mixing that occurs is when the two fluids are in contact with each other just before each branch point.  Thus, diffusion will take a long distance to homogeneously mix the two.
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However, the presence of the bubbles leads to the aforementioned variations in fluidic resistances and thus the fluid streams do not split down the middle at each branch. At times, more liquid flows to the left branch, so some of the liquid that starts on the right half of center ends up in the left branch, and vice versa (Figure 3(b)).  In this manner, the two fluids are folded into each other every time they pass one of the branch points. This creates more interfaces between the two fluids and decreases the characteristic length over which diffusion has to occur.  The other way to think of this is that the gradients are increased (since length to diffuse is decreased), leading to a greater driving force for diffusive mixing.
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To characterize their experimental results a little, the authors prepare a simple theoretical model to estimate the number of branch points required to completely mix the two fluids.  The authors compare the average distance between two interfaces with the diffusion length scale.  Assuming each branch point doubles the interface area between the two fluids and the overall increase is exponential in the distance travelled downstream, the average distance between interfaces is defined as <math>d_{inter}=w2^{l/a}</math>.  <math>w</math> is the channel width, <math>l</math> is the length travelled downstream, and <math>a</math> is the arc length of the arm in the branching section.  The diffusion length scale is <math>d_{diff}=(tD)^{1/2}</math>, where <math>t</math> is the time it takes to traverse a certain length of channel <math>\left(t=lw^2/Q\right)</math>, and <math>D</math> is the diffusivity. Homogeneity is assumed to be achieved when <math>d_{inter}=d_{diff}</math>.  Equating the two expressions, solving for <math>l/a</math> and plugging in typical values gives <math>l/a</math> on the order of 10.  From Figure 4, this prediction is in nice agreement with experimental evidence, as the standard deviation drops nearly to zero after the tenth branch, indicating near homogeneity across the channel.
  
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.
 
  
 
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[1] I.M. Tidswell, T.A. Rabedeau, P.S. Pershan, and S.D. Kosowsky, Physical Review Letters <b>66</b>, 2108 (1991)
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[1] P. Garstecki, H.A. Stone, and G.M. Whitesides, Physical Review Letters <b>94</b> 164501 (2005)
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written by Donald Aubrecht

Latest revision as of 02:48, 28 November 2011

Original entry: Donald Aubrecht, APPHY 226, Spring 2009

"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

Microfluidic, Bubbles, Laminar mixing, Peclet number

Figure 1. Schematic showing introduction of the two liquid streams and formation of air slugs.
Figure 2. (a) Recirculation rolls between slugs of air. (b)-(d) Schematics indicating channel design and the paths slugs take through the branches. As a slug enters one arm, the hydrodynamic resistance in the arm increases, cause the next slug to enter the other arm of the branch.
Figure 3. (a) Two fluid streams remain nearly unmixed when bubbles are not present. (b) Introducing bubbles along with fluid streams causes the two fluids to fold into one another, aiding in diffusive mixing. Intensity as a function of position across the channel at the specified locations are shown.
Figure 4. Standard deviation of intensity profiles at different positions along the channel network. Standard deviation of 0.5 indicates unmixed streams, while a standard deviation of 0 indicates perfectly mixed streams. The two streams are nearly homogeneous after passing through roughly 10 branches in the channel.

Summary

This paper details experimental work and simple supporting theory regarding mixing in microfluidic channels. For most microfluidic systems, the Reynolds number remains small (less than 1000), so turbulence is absent and mixing only occurs via diffusion. Typical Peclet numbers in microfluidic channels are on the order of 1e5, indicating that mixing to homogeneity requires length scales on the order of 10 meters. These lengths are not easily achieved on microfluidic devices due to finite substrate limits for fabrication and large pressure drops in the long channels, so the authors propose a novel method of mixing that aid the diffusion process. Using bubbles to fold two liquid streams into one another, greater contact area between the fluids is created, aiding in diffusion.

Practical Application of Research

A drawback to continuous flow microfluidics has been the inability to homogeneously mix streams on-chip. This limits the application of continuous flow microfluidics, as it is difficult to introduce additional reagents or samples to a flow. This passive, on-chip mixing scheme open a new realm of experiments that can take advantage of introduction of precise amounts of fluid at a given spatial or temporal point in a flow.

Microfluidic Mixing Using Bubbles

As shown in Figure 1, the two liquids to be mixed are combined with a stream of air at a microfluidic flow focussing junction. The liquid streams pinch off air bubbles, which then flow downstream. This break up is mediated primarily by conservation of mass. As the thread of air enters the junction, it restricts flow of the two outer liquids. Pressure builds up in the outer liquid lines, causing the air/liquid interface to deform. Eventually, the interface detaches from the walls, becomes unstable, and breaks, forming a bubble [1].

Once in the device, sequential air bubbles alternate arms of the branched channels they flow into (see Figure 2). As an air bubble enters on branch, it causes the hydrodynamic resistance in that side of the branch to increase. This causes a greater percentage of the new incoming flow to be diverted down the other branch. This causes the next air bubble to flow through the branch that does not contain the first bubble. As the authors mention in the paper, this is an additive effect and will work for any number of droplets in a series of branches. The incoming droplet will end up being passively steered by the majority of the fluid streamlines to the channel with the smallest hydrodynamic resistance.

In the case where no bubbles are present, the fluidic resistances of the two arms are equivalent and the fluid streamlines will split evenly left and right. This means that if the two inlet fluids are introduced at equal flow rates, the fluid that starts on the left of the center of the main channel will always take the left branch, while the fluid on the right of center will always take the right branch (see Figure 3(a)). The only mixing that occurs is when the two fluids are in contact with each other just before each branch point. Thus, diffusion will take a long distance to homogeneously mix the two.

However, the presence of the bubbles leads to the aforementioned variations in fluidic resistances and thus the fluid streams do not split down the middle at each branch. At times, more liquid flows to the left branch, so some of the liquid that starts on the right half of center ends up in the left branch, and vice versa (Figure 3(b)). In this manner, the two fluids are folded into each other every time they pass one of the branch points. This creates more interfaces between the two fluids and decreases the characteristic length over which diffusion has to occur. The other way to think of this is that the gradients are increased (since length to diffuse is decreased), leading to a greater driving force for diffusive mixing.

To characterize their experimental results a little, the authors prepare a simple theoretical model to estimate the number of branch points required to completely mix the two fluids. The authors compare the average distance between two interfaces with the diffusion length scale. Assuming each branch point doubles the interface area between the two fluids and the overall increase is exponential in the distance travelled downstream, the average distance between interfaces is defined as <math>d_{inter}=w2^{l/a}</math>. <math>w</math> is the channel width, <math>l</math> is the length travelled downstream, and <math>a</math> is the arc length of the arm in the branching section. The diffusion length scale is <math>d_{diff}=(tD)^{1/2}</math>, where <math>t</math> is the time it takes to traverse a certain length of channel <math>\left(t=lw^2/Q\right)</math>, and <math>D</math> is the diffusivity. Homogeneity is assumed to be achieved when <math>d_{inter}=d_{diff}</math>. Equating the two expressions, solving for <math>l/a</math> and plugging in typical values gives <math>l/a</math> on the order of 10. From Figure 4, this prediction is in nice agreement with experimental evidence, as the standard deviation drops nearly to zero after the tenth branch, indicating near homogeneity across the channel.



[1] P. Garstecki, H.A. Stone, and G.M. Whitesides, Physical Review Letters 94 164501 (2005)


written by Donald Aubrecht