Mixing and emulsification processes in micromixers · the flow distribution in the mixing zone by transmitted light microscopy. 2 Miscible Liquids Characteristic diffusion constants

Mixing and emulsification processes in micromixers

S. Hardt, F. Schonfeld, F. Weise, Ch. Hofmann, V. Hessel & W. Ehrfeld Institute of Microtechnology Mainz (IMM), Germany.

Abstract

Mixing and emulsification processes in micromixers developed by the Institute of Microtechnology Maim (IMM) are investigated by means of computational fluid dynamics (CFD). For both miscible and immiscible fluids the CFD results are compared to experimental data obtained with transparent micromixer made from photosensitive glass. It is found that in the case of miscible liquids the accuracy of numerical predictions is limited by artifacts from numerical diffusion. However, streamline tracking techniques offer an alternative method for computing flow patterns in micrornixers. Corresponding flow patterns comply with photographic images of the mixing zone. Subsequently, the case of immiscible liquids is investigated using volume tracking techniques. The simulations, based on a 3D model for a planar interdigital micromixer, describe the time evolution of droplet formation. Data on droplet size and the decay wavelength of the liquid lamellae are in agreement with experimental results. Both the simulations and the experiments indicate that surface tension forces play the dominant role for droplet formation. Thus, the results suggest that even in the presence of wall boundaries and shear forces the Rayleigh-Plateau instability governs the dynamics.

1 Introduction

Micromixers offer exciting new possibilities in the field of chemical process technology [l]. Typically, within such mixers fluid lamellae with a thickness of

Transactions on Engineering Sciences vol 30, © 2001 WIT Press, www.witpress.com, ISSN 1743-3533

2 1 8 Computational Methods in Multiphase Flow

several micrometers are created at larninar flow conditions. Subsequently, depending on the miscibility of the fluids, either rapid mixing by diffusion or lamellae decay, followed by formation of an emulsion or dispersion is observed.

The benefits of micromixers for chemical process technology are manifold. Due to high concentration gradients fast mixing under well defined conditions is achieved, whereas mixing is far less uniform in macroscopic turbulent mixers. Especially when mixing is superposed by chemical reaction dynamics, uniform process conditions can be mandatory in order to achieve a high selectivity and conversion [2]. Moreover, new reaction pathways can be explored with micromixers and microreactors. As an example, ignition limits are shifted due to rapid heat transfer to the channel walls. For immiscible fluids equally promising application areas open up. Emulsions formation can be achieved with a comparatively low energy input [3]. Furthermore, under certain conditions small droplets of a very uniform size distribution are found [4]. On the other hand, the effect of Ostwald ripening relates the uniformity of drop size to the stability of an emulsion 151.

At IMM various micromixers based on a number of different mixing principles have been developed over the last few years [6 ] . Fig. 1 shows diagrams of two of the most frequently used mixers, the caterpillar mixer (left side) and the interdigital mixer (right side). The caterpillar mixer relies on a split-recombine principle, where a fluid stream (entering fiom the left) is multilarninated in a microchannel containing ramp-like structures. In the interdigital mixer fluid lamellae are created by an interdigital arrangement of inlet channels with alternating feeds.

Fig. 1: Schematic design of a caterpillar mixer (left) and an interdigital mixer (right).


Conlpututior~al Methods irl Mziltiphusr Flow 2 19

A number of different fabrication technologies is employed to create the corresponding microstructures. The caterpillar mixer typically contains channels of 300 to 1000 pm width formed, e.g., by laser ablation of polymer materials. Typical dimensions of fluid lamellae created by the feed structures of an interdigital mixer are 25 to 40 pm. The technological basis for such high-aspect ratio structures is the so called LIGA process starting from deep X-ray lithography [7].

In order to perform flow distribution measurements, a couple of different interdigital mixers were fabricated by the Mikroglas company, Mainz from photosensitive glass by means of a UV-lithography process [8]. In contrast to Fig. 1, a planar design was chosen, where channels for inlet and mixing streams are arranged in one layer. The transparent cover and bottom plates allow to examine the flow distribution in the mixing zone by transmitted light microscopy.

2 Miscible Liquids

Characteristic diffusion constants of liquid-liquid systems are of the order of 10.~ cm2/s, i.e. four orders of magnitude smaller than those of gaseous systems. Correspondingly, fast liquid mixing is a difficult problem even in micromixers that create fluid lamellae with a width of several micrometers. Typically micromixers are operated at laminar flow conditions, indicating an ideal application area of computational fluid dynamics (CFD) where no firther model assumptions as in the case of turbulent flows are needed.

However, the predictive power of CFD calculations is limited by artifacts from numerical diffusion when mixing time scales as characteristic for liquid-liquid systems in micromixers are studied. This is exemplified in Fig. 2, which displays simulation results of a special interdigital mixer that splits the fluid stream into two substreams . The two liquids enter the mixer from the left through 2 1 diffe-

Fig. 2: Concentration fields in an interdigital micromixer. Left: first order differencing scheme. Right: second order differencing scheme.


220 Conlpurariot~d Methods in M~rltryhu~e Flow

rent inlets. The inlet section has a rectangular cross sectional area of 500 pm X

150 pm, where one lamella has a width of approximately 24 pm. The figure shows species concentrations as a grayscale map. The CFD simulations were performed with the finite volume (FV) package CFX4 of AEA Technology, using two different discretization schemes for the concentration field. The left part of the diagram shows the results obtained with a first order hybrid differencing scheme. On the right side, the concentration field obtained with a second order upwind scheme is displayed. A zero diffusion constant was used in the simulations.

As apparent from the figure, the second order scheme represents a clear improve- ment over the first order scheme. However, both schemes fail to give a realistic description for the mixing process. The results indicate that partial mixing has occurred when the main liquid stream is split up into two substreams. This effect is clue to numerical diffusion (or equivalently: higher order discretization errors), which shows a pronounced dependence on grid structure. In the transition region between the straight section and the curved parts two sub-grids of the body-fitted multiblock structure were matched, marking a sudden transition in grid cell orientation. It is well known that discretization errors sensitively depend on the relative orientation of grid cells and velocity vectors [9] . Correspondingly, numerical diffusion is substantially enhanced at the subgrid interface.

In order to assess the importance of these results for the simulation of mixing in realistic liquid-liquid systems (with nonzero diffusion constant) it is instructive to compute typical mixing length scales by a simple dimensional analysis. Given the in1e.t velocity used in the simulations (v = 0.74 d s ) a mixing length can be derived from the characteristic mixing time scale L*/D, where L is the lamella width and D the diffusion constant. The resulting mixing length of 0.43 m, obtained with a diffusion constant of 10.' cm2/s, by far exceeds the dimensions of the micromixer. On the basis of Fig. 2 it can thus be concluded that with the computational model used a realistic description of liquid-liquid mixing in micromixers is impossible. Clearly numerical discretization errors can be reduced by choosing a smaller grid spacing. For the current investigations a grid of approx. 300000 cells was used, allowing overnight runs on a state-of-the-art workstation. Due to reflection symmetry only one quarter of the geometry had to be modeled. The grid resolution was such that a fluid lamella of 24 mm width was resolved by 4 computational cells. Taking into account the 3D nature of the model and the strength of the artifacts visible in Fig. 2, it becomes clear that realistic mixing calculations would require an increase of grid resolution by orders of magnitude.

Even if it seems extremely difficult to obtain a realistic description of liquid- liquid mixing by standard discretization techniques, a considerable amount of valuable information can be gained by an alternative approach. This approach is based exceptionally on velocity fields and does not require computation of concentration fields. The basic idea is to compute streamlines and to tag stream-


Simulation

Coriiprrtatiollal Methods it1 M~rltiphase F l o ~ , 22 1

Experiment

Fig. 3: Fluid lamellae in an interdigital micromixer as obtained from CFD simulations and transmitted light microscopy. Only a section of the mixing device is shown.

lines belonging to different liquid species with different colors. This can be done by starting the streamline integration in the inlet region of the mixer. Such a method mimics a mixing simulation with zero diffusion constant. State-of-the-art CFD postprocessing tools allow to compute streamlines with high accuracy even for body fitted grids with distorted cells.

A comparison of results of the streamline tracking method with experimental results is displayed in Fig. 3. The experiments based on transmitted light microscopy were performed using a transparent micromixer with an alternative geometry. Pure and colored water were used as liquids. Due to the experimental setup an integral information on concentrations (integrated over the transmission distance of a light beam) is obtained. This is compared to a cut through the computed streamline bundles along the symmetry plane of the mixer at half the channel height (total channel height 150 pm).

Apparently the concentration patterns match reasonably well. Details as variations of the lamellae width or streamline separation when entering the straight outlet section in the upper part of the figure are in fair accordance with the simulations. Due to CPU time constraints, only a small part of the outlet channel was modeled. The broadening of the central lamellae is presumably due to the pressure boundary condition at the outlet, which gives no realistic representation of the developing character of the flow in this region. The flow patterns shown here are characteristic for a Reynolds number range around 1000.


In combination with an evaluation of residence times for different lamellae streamline tracking can provide valuable information on mixing efficiency in different parts of the micromixer. Fig. 3 suggests that, due to small residence times and an increase in lamellae width, the mixing quality will be reduced in the central parts of the mixer. Experiments in glass mixers with two different aqueous solutions undergoing a mass transfer limited reaction yielding a colored complex confirm this conjecture.

3 Immiscible Liquids

Irnmjscible fluids form a sharp phase boundary when put in contact. Examples are emulsions, dispersions or free surface flows. In contrast to miscible phases, in general no unique, stationary flow patterns exist, as an interface may exhibit transient behavior or may be arranged in many degenerate configurations. However, in disperse systems droplet or bubble sizes are often very small compared to all other relevant length scales. This allows to employ Euler-Euler or Euler-Lagrange models of two-phase systems that, apart from the specific form of inter-phase transfer terms, neglect the spatial extension of droplets or bubbles (see, e. g., [10]), thus making multi-phase simulations numerically tractable.

Unfortunately in microfluidic systems bubble or droplet diameters are often comparable to characteristic system dimensions. Thus, transient simulations allowing to describe the time-evolution of an interface are required. Such simulation~ are still very challenging on standard workstations, so that many previous investigations were restricted to 2D models. However, the dimensions of micromixers to be studied here are such that no a-priori justification of a 2D approach seems possible.

Most suitable for modeling complex 3D free surface flows with topology changes are volume-tracking techniques [ l 1,121. The basic idea of these approaches is to define a function f describing the local volume fraction of a specific fluid. The value off is 1 within the corresponding fluid and 0 outside. The volume fraction obeys an advection equation given as

One of the disadvantages of solving Eq. (1) by standard discretization techniques is smearing of the interface by numerical diffusion. A sharp interface can be maintained when care is taken to compute the exact position of the interface within computational cells [ l 1,121.

Interface reconstruction requires considerable numerical effort. In order to keep the model numerically tractable, a somewhat simpler approach was chosen in the present work. The approach relies on a correction algorithm reversing interface


Conqxrturiorral Methods irr Multipha~e Flow 223

smearing by numerical diffusion. The algorithm determines the fluid mass which has penetrated the interface in a certain timestep, removes the mass from the wrong side of the interface and adds it to cells where fluid has been depleted. This so-called "surface-sharpening" algorithm, implemented in the finite volume solver CFX4, ensures global mass conservation. However, when the fluids form several disconnected regions, local quantities as the mass of a specific droplet may not be conserved. The volume tracking method described above is supple- mented by a continuum method for modeling surface tension [13]. Due to the large surface-to-volume ratio, surface tension plays an important role in microfluidic systems and cannot be neglected. On this basis, CFX4 was used for the simulations to be discussed in the following.

A liquid jet is known to decay into droplets by a hydrodynamic instability [14]. It suggests itself that this so-called Rayleigh-Plateau instability plays a role for droplet formation in the IMM rnicromixers. For this reason, the decay of a liquid cylinder was studied as a test case. An axial section of a liquid cylinder of Az =

2.03 cm extension was considered in a 2D axisyrnmetric model. The cylinder, consisting of water, is surrounded by air and has a radius of R = 2.3 mm. Symme- try boundary conditions were chosen at the high and low z end of the computational model. As an initial condition, a sinusoidal modulation of the cylinder radius was introduced.

In Fig. 4 the simulation results for an initial perturbation with wavelength h = 4.5 R are shown. The vectors, appropriately rescaled in each frame, indicate the direction and magnitude of fluid velocity. It is found that the cylinder does not decay in the h = 4.5R mode, but with a wavelength of 9R, which is the extent of the computational domain in axial direction. From linear stability theory for inviscid irrotational flow it is known that below h = 2nR radial modulations are damped, whereas modes with larger wavelength are unstable (for an overview,

Fig. 4.: Simulation of the decay of a liquid cylinder into droplets.


224 Conrprratiorlal Mctkoh irr Multiphase Flow

see [15]). The fastest growing mode has a wavelength of about 9R. The fact that a linearly stable mode does not decay with the original wavelength of the perturbation, but with a longer wavelength, is in accordance with experiments

Further simulations were performed for a range of wavelengths of the initial perturbation. In most cases the h = 9R mode was found to govern the dynamics. When reducing the wavelength of the perturbation below the critical value of 27tR a regime is reached where the decay occurs via coupling to excitations of longer wavelength. Artifacts from the surface sharpening algorithm were only visible when tiny satellite droplets were formed. In a few cases it occurred that fluid was transferred from a satellite droplet to a larger drop. In total the simulation results appear to comply with experimental data and linear stability theory.

Compared to the decay of a liquid cylinder at rest, the dynamics of emulsion formation in a micromixer is more complex. Due to the nonzero velocity of the liquid lamellae shear forces are induced. Furthermore, the lamellae are in contact with the channel walls when entering the mixing zone. For this reason, wetting forces are expected to play a role for the dynamics. Fig. 5 shows photographs of the emulsification process in a transparent interdigital micromixer. The liquids, silicon oil and water, to which a blue dye was added, enter the mixer through 30 inlet channels from the left. At the interface to the mixing zone the inlet channels have a cross-sectional area of 60 pm X 150 pm, separated by 50 pm walls. The cross section of the mixing zone, only a part of which is shown in the figure, is 3250 pm X 150 pm.

The photographs of the mixing zone were taken with a CCD camera, where the micromixer was arranged between the light source and the camera. In the left part of Fig. 5 a volume flow ratio of water and silicon oil of 1:1 was chosen, in the right part the ratio is 1:4. The total liquid volume flow is 800 ml/h in both cases. Even with the limited spatial resolution of the CCD camera water lamellae and droplets become visible. Whereas in the case of equal volume flows the droplets

Flow rat10 1 .l

Fig. 5 : Views on emulsion formation different volume flow ratios.

. . Flow ratio 1 :4

in a planar interdigital micromixer for


are formed by lamellae decay, ior a flow ratio of 1:4 droplet formation occurs very close to the inlets. Furthermore, it is apparent that in the case of the 1:4 flow ratio the droplets are smaller and the distance between droplets is larger than in the case of the 1:l flow ratio. It has to be mentioned that the experiments described here show water droplets dispersed in a continuous silicon oil phase. In references [3] and [4], however, the formation of silicon oil droplets using a stainless steel interdigital mixer was discussed. In the light of the results presented here this change in phase behavior may be attributed to the different wetting properties. Nevertheless, further studies with different combinations of liquids and wall materials are necessary.

In order to describe emulsion formation in the micromixer, a 3D computational model was set up. The model consists of a slice of the mixing zone of 2mm length and a lateral extension of 110 pm. It embraces (in the case of a 1: 1 flow ratio) half of a water and half of a silicon oil lamella. The planes cutting through the center lines of each lamella were defined as symmetry planes. Further use of symmetry was made by restricting the model to half the channel height (75 pm) of the mixing zone.

In conjunction with the 2D simulations discussed above it was found that when no initial modulations of the radius were introduced, the numerical noise was not sufficient to induce a decay into droplets. For this reason, in the 3D model random velocity fluctuations were superposed to the fixed velocity at the inlet. Due to the randomness of the fluctuations, no further length scale is introduced which could distort the dynamic evolution of the system. Since wetting forces are expected to be of major importance, the contact angle of the system silicon oiVwater needed to be determined. With the measuring equipment available at IMM no reliable measurement of the contact angle was possible. As it was found that silicon oil has far better wetting properties than water for the glass material chosen, an angle of 30" was used in the simulations.

In Fig. 6 the shapes of the liquid interfaces corresponding to one water lamella as obtained from the simulation are displayed. The total volume flow was chosen 600 ml/h in both cases. In the case of the 1 : 1 flow ratio a water lamella is fomied, which, by action of wetting forces, detaches from the channel walls and assumes

Fig. 6: Simulation results on breakup of the liquid-liquid interface


226 Computatiot~al Methods ill Mltltiphase Flow,

a circular cross section. The lamella decays into droplets which again touch the upper and lower channel wall. Subsequently, the droplets merge with neighboring droplets to the right and to the left. The contact areas to the walls and to neighboring droplets are visible as elliptical holes in the liquid interface. In agreement with experimental data, droplet formation occurs close to the inlets in the case of the 1:4 flow ratio. In general, droplets are smaller and coalescence with neighboring droplets occurs less frequently in this case. It should be noted that details of the coalescence mechanism are not likely to be realistically described by the simulations, as in the model the time evolution of different lamellae is assumed to be identical. In practice, neighboring lamellae do not decay in a completely identical manner, and coalescence can be avoided when a droplet is formed in between two neighboring droplets. However, the experiments indicate that in the case of equal volume flows droplets merge further downstream, whereas coalescence is avoided in the case of the 1 :4 flow ratio.

Experiments are ongoing and a final analysis of data is not yet available. Nevertheless, a first quantitative comparison of simulations and experiments shows rather good agreement. From the experiments for equal volume flows (300 mVh per liquid) a droplet diameter of 236 pm, and a distance between subsequent droplets of 470 pm is obtained. Due to rapid coalescence with neighboring drops the diameter could be determined only for one droplet being formed in the sirnulations, where exactly the experimental value of 236 pm was obtained. The simulated droplet distance was determined as 466 and 603 pm, respectively. Relating these numbers to the average radius of the circular lamella cross section, one obtains 9.32R and 12.06R, with R = 50 pm. The corresponding value taken from experimental data is 9.4R, where the lamella radius is identical to the computed value. Thus, the decay wavelength is quite close to the wavelength of the most rapidly growing perturbation of the Rayleigh-Plateau instability, which is 9.01R. It is therefore natural to assume that a similar mechanism is responsible for droplet formation by lamella decay in micromixers: Formation of cylindrical liquid volumes induced by wetting forces followed by minimization of the interfacial area driven by surface tension.

Summary and Outlook

Flow distributions of miscible and immiscible liquids in micromixers were studied both experimentally and by computer simulations. In the case of miscible liquids, a FVM based solution of the advection-diffusion equation for the concentration fields failed to give a realistic description of the mixing process. As an alternative, streamline tracking techniques were used to determine the flow patterns in an interdigital micromixer. The simulation results match the corresponding experimental data reasonably well. Streamline tracking represents a comparatively simple method to obtain qualitative informations on liquid-liquid mixing in micromixers.


Computational Methods in Multiphase Flow 227

Subsequently, binary systems of immiscible liquids were studied. The Rayleigh- Plateau instability of a liquid cylinder was chosen as a test case for the computational approach of volume tracking. The simulation results obtained comply with results of a linear stability analysis. On the basis of the volume tracking method a 3D model for a planar interdigital micromixer was set up. Parallel to the modeling photographic images of lamella decay and emulsion formation in a glass mixer were taken. The simulations and the experiments agree reasonably well both qualitatively and quantitatively. The observed ratio of decay wavelength and lamella radius suggests that, despite the breakdown of many assumptions used in the corresponding linear stability analysis, the decay is driven by the same mechanisms as responsible for the Rayleigh-Plateau instability. The analysis will be continued with further simulations and experiments. The goal is to create a map in the spirit of a phase diagram, which allows to extract characteristic state variables such as droplet diameters for a range of operation, geometry and material parameters.

In the future, such detailed studies should allow to control the emulsification process in micromixers and create tailor-made emulsions and dispersions with well defined properties. The field of possible technological applications is broad and ranges from self-organizing nanosystems to extraction processes.

Acknowledgments

The authors wish to thank A. Freitag and T. Dietrich from Mikroglas AG, Maim, for their help during the discussions on the design of the glass micromixers and for realization of these devices.

References

[ l] Ehrfeld, W., Hessel, V. & Haverkamp, V. Ullmann's Encyclopedia of Industrial Chemistry, 6Ih Edition, Wiley-VCH: Weinheim, 1999.

[2] Richter, T., Ehrfeld, W., Gebauer, K. et al. Metallic Microreactors: Components and Integrated Systems. Proc. of IMRET2: 2nd ~ n t . Con$ on Microreaction Technology, ed. W. Ehrfeld et al., AIChE, p. 146, 1998.

[3] Bayer, T., Heinichen, H. & Natelberg, T. Emulsification of silicon oil in water - comparison between a micromixer and a conventional stirred tank. Proc. of IMRET4: 4th Int. Conf on Microreaction Technology, ed. I. Rinard, AIChE, p. 167,2000.

[4] Haverkamp, V., Ehrfeld, W., Gebauer, K. et al. The potential of micromixers for contacting of disperse liquid phases. Fresenius Journal ofAnalytica1 Chemistry, 364, p. 617, 1999.

[5] Lagaly, G., Schulz, 0. & Zimehl, R. Dispersionen und Emulsionen, Steinkopff: Darmstadt, 1997.


[6] Lowe, H., Ehrfeld, W. & Hessel, V. Micromixing Technology. Proc. of IMRET4: 41h Int. Con$ on Microreaction Technology, ed. I . Rinard, AIChE, p. 3 1, 2000.

[7] Ehrfeld, W. & Miinchrneier, H. Three dimensional microfabrication using synchrotron radiation. Nucl. Instr. and Methods in Physics Research, A303, p. 523, 1991.

[g] Freitag, A., Dietrich, T.R., Scholz, R. & Hessel, V. Glass as a Material for Microreaction Technology. Proc. of Micro. Tec2000: World Microtechnolo- gies Congress, ed. P. Knoll, VD1 Verlag: Berlin, p. 355, 2000

[9] Noll, B. Numerische Stromungsmechanik, Springer: Berlin, 1993. [l01 Sadhal, SS. , Ayyaswamy, P.S. & Chung, J.N. Transport Phenomena with

Drops and Bubbles, Springer: New York, 1997. [ l l ] Hirt, C.W. & Nichols, B.D. Volume of fluid method for the dynamics of free

boundaries. Journal of Comp. Phys., 39, p. 201, 198 1. [l21 Rider, W.J. & Kothe, D.B. Reconstructing volume tracking. Journal of

Comp. Phys., 141, p. 112, 1998. [l31 Brackbill, J.U., Kothe, D.B. & Zemach C. A continuum method for

modeling surface tension. Journal of Comp. Phys., 100, p. 335, 1992. [l41 Rayleigh, Lord, J.W.S. The Theory ofsound, Vol. 11, 2nd ed., Dover: New

York, 1945 [I 51 Eggers, J., Nonlinear dynamics and breakup of free surface flows. Rev. Mod.

Phys., 69(3), p. 865, 1997


Documents

Mixing and emulsification processes in micromixers · the flow distribution in the mixing zone by transmitted light microscopy. 2 Miscible Liquids Characteristic diffusion constants