Electrokinetic i phenomena: electroosmotic flows Surface tension Mixing and diffusion EECE

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1 Fluids Continuum assumption Gas flows Liquid flows Governing equations Parallel flows Electrokinetic i phenomena: electroosmotic flows Surface tension Mixing and diffusion 1 Fluids Learning Objectives At the end of this section the student should be able to: define the Reynolds number and why it is important in describing flow at the microscale. describe Poiseuille flow calculate flowrate or pressure drop through channels of simple geometry describe the electrowetting effect and how the contact angle and applied voltage arerelated related describe why diffusion is important for mixing at the macroscale. 2

2 Further Reading N. T. Nguyen and S. T. Wereley, Fundamentals And Applications of Microfluidics, Second ed: Artech House, T. M. Squires and S. R. Quake, "Microfluidics: Fluid physics at the nanoliter scale," Reviews of Modern Physics, vol. 77, pp , J. R. Wlt Welty, C. E. Wicks, and R. E. Wilson, Fundamentals of Momentum, Heat, and Mass Transfer, third ed: John Wiley & Sons, J. Berthier and P. Silberzan, Microfluidics for Biotechnology: Artech House Publishers, Fluid mechanics and micro/nanofluidics It is a HUGE field! We will discuss only briefly some aspects: Basic use of fluids at macroscale Applications at microscale what is different? From statistical mechanics to continuum hypothesis Basic notions of fluid mechanics: forces, conservation laws Methods to manipulate fluids: pressure driven flows, electro osmotic flows, electrophoretic separation Typical functions: mixers, chemical analysis Scaling and dimensionless parameters 4

3 How are these two situations similar? 5 Macroscale Fluids used to carry heat around a circuit forced air heating, refrigeration used to transmit forces hydraulic lines used to create forces jet engines, paper handling systems used to transport materials leaf blowers, paints 6

4 Microscale fluids Used to carry heat around a circuit integrated circuit cooling Used to transport materials biological cell manipulation 7 Microfluidics/nanofluidics Inkjet printing heads, displays (LCDs) Microneedles for drug delivery Chemical sensing 8

5 Microfluidic devices 9 Microfluidics advantage Small size => portability Smaller required sample sizes (for analysis and separation) Greatly decreased analysis time Batch fabrication =>low cost Disposability ( one time use ) for some biological analyses 10

6 Fluid Properties A fluid is a substance that deforms continuously under the application of shear (tangential) stress, no matter how small that stress may be. The two most important parameters characterizing a fluid are: density (ρ) mass per unit volume [kg/m 3 ] viscosity (μ or η), which indicates resistance of flow. Viscosity causes shear stress when the fluid is moving; without viscosity in the fluid, there would be no fluid resistance. 11 Viscosity When a shear stress is applied to a liquid like material (e.g. water, oil, honey etc.), by for example sandwiching the liquid between two plates in relative motion, then collective atomic motions allow the liquid to flow, i.e. exhibit permanent deformation. We use a quantity called the viscosity to describe how easily the fluid flows. shearing force F shearing force F Δx = vδt y y Flow velocity, v When a shear stress is applied to a liquid like material, atomic layers flow with respect to each other resulting in permanent deformation which we call viscous flow. In this figure, the bottom plate is stationary and the top plate is being dragged at a constant velocity. Within the material, the atomic layers move with different dff velocities with respect to the bottom layer. There is therefore a velocity gradient in the material. S.O. Kasap 12

7 Viscosity Under the action of the shearing stress the atoms move with respect to each other by breaking and making bonds in an analogous way to atomic diffusion. The result is that atomic layers move with respect to each other. shearing force F Surface area A shearing force F Δx = vδt y y Flow velocity, v The velocity gradient depends on the applied shear stress. dv τ = μ dy τ = shear stress μ = viscosity μ = viscosity -1 1 μ [=] Pa s [=] kg m s dv velocity gradient dy = 13 S.O. Kasap Reynolds number The Reynolds number determines the character of the fluid flow. The Reynolds number is given by: What are the dimensions of Re? 14

8 Over a large Re range (47<Re<107 for circular cylinders), eddies are shed continuously from each side of the body, forming rows of vortices in its wake. The alternation ti leads to the core of a vortex in one row being opposite the point midway between two vortex cores in the other row, giving rise to the distinctive pattern shown in the picture. Ultimately, the energy of the vortices is consumed by viscosity as they move further down stream and the regular pattern disappears. AV Von Kármán á vortex street is a repeating pattern of swirling vortices caused by the unsteady separation of flow over bluff bodies. Von Kármán vortex street off the Chilean coast near the Juan Fernandez Islands 15 Calculation of Reynolds number For Re > 2000, flow is considered turbulent. For Re < 2000, flow is laminar. Re calculation for water in a typical microchannel: density = 1000 kg/m 3 viscosity = 10 3 Pa s d = 100 μm v = 1000 μm/s i clicker: a. Re = 10 b. Re = 1 c. Re = 0.1 d. Re = 0.01 Re calculation for a person swimming: density = 1000 kg/m 3 viscosity = 10 3 Pa s d = 2 m v = 0.5 m/s i clicker: a. Re = 10 6 b. Re = 10 4 c. Re = 10 2 d. Re = 1 16

9 Role of viscosity Reynolds number interpreted as the ratio between inertiaforces to viscous forces => viscosity dominates in laminar flow, and inertia in turbulent flow Qualitatively: viscosity is a measure of stickiness / thickness of a fluid (e.g. honey is more viscid than water) Viscosity plays a similar role in fluid mechanics as the shear modulus of elasticity in solid mechanics with strain rate replacing strain Fluids for which the shear stress is linearly related to velocity gradient are called Newtonian du τ = μ dy i clicker: which type of flow will dominate at microscale: A. Laminar B. Turbulent C. Transition region Non Newtonian behaviour paint: flows readily off the brush when it is being applied to the surface being painted, but should not to drip excessively. ketchup; whipped cream Applications: body armor dilatant fluid disperses the force of a sudden blow over a wider area. traction control viscous coupling unit to transfer power between front and rear wheels in all wheel drive systems. On high h traction road surfacing, the relative motion between primary and secondary drive wheels is the same, so the shear is low and little power is transferred. When the primary drive wheels start to slip, the shear increases, causing the fluid to thicken. As the fluid thickens, the torque transferred to the secondary drive wheels increases proportionally, until the maximum amount of power possible in the fully thickened state is transferred. 18

10 Similarity Geometrical similarity: same shape. The parameters (Π 1, Π 2,... Π N-K ) are known as similarity parameters. Similaril systems have essentially the same physics. Similarity il it is the fundamental principle underlying all testing with models. All dimensionless groups must be kept the same in the model and prototype to achieve perfect dynamic similarity. 19 Modeling & Similarity Parameters Airfoil flow Consider two airfoils which have the same shape and angle of attack, but have different sizes and are operating in two different fluids. Airfoil 1 (sea level) Airfoil 2 (cryogenic tunnel) α 1 = 5 α 2 = V 1 = 210 m/s V 2 = 140 m/s ρ 12kg/m 3 30kg/m 3 1 = 1.2 ρ 2 = 3.0 μ 1 = kg/ms μ 2 = kg/ms 300 / a = 2 = 200 m/s f ( α1,re 1, Ma1) f ( α2,re 2, Ma2 ) a 1 = 300 m/s c 1 = 1.0 m Pi products: c 2 = 0.5 m α 1 = 5 α 2 = 5 Re = Re 2 = Ma 1 = 0.7 Ma 2 = 0.7 c = c l1 l2 Since Pi products are the same arguments to the f function, we conclude that the lift coefficients will be the same as well. Dynamic similarity the basis of wind tunnel testing, where the flow around a model object duplicates and can be used to predict the flow about the full sized object. 20

11 Governing equations fluid flows aredetermined by knowledge of velocities, pressure, density, viscosity, specific heat and temperature. 1. mass conservation equation (continuity equation) 2. momentum equation (Navier Stokes equations) 21 Example Fluid flowing through a small square: ( ρ ) ρ x ρ y ρ x+δ x ρ y+δy t Δx Δ y = u Δ y + v Δx u Δy v Δx ( ρu) ( ρv) ρ + + = 0 t t t 22

12 Conservation of mass Mass balance over a general control volume: da n θ arbitrarily chosen control volume of fixed position and shape. thus there is no net flux of fluid. v ρ not necessarily constant. (ρ inside c.v. must change to balance mass flux in or out) rate of mass efflux from control volume ( ρv)( dacosθ ) ρ( v n) da ρ( v n) c.s. rate of mass flow into control volume da + rate of mass accumulation within control volume t c.v.. ρ dv +ve, net efflux of mass ve, net influx of mass 0, mass within control volume is constant = 0 imbalance in mass flux in or out is compensated by accumulation of mass inside. 23 Continuity equation = 0 for steady flow. (properties of flow field are invariant with time) Continuity equation: for incompressible flow: for incompressible fluid: ρ ρv + = 0 t v = 0 ρ = constant applies to unsteady, threedimensional flow therefore for incompressible fluid: ρ = 0 and ρ = 0 tt Alternative way to write the Continuity equation: Dρ + ρ v = 0 Dt since D = + u + v + w Dt t x y z 24

13 Conservation of linear momentum Closed systems: conservation of linear momentum in mechanics r d m For a single particle: r ( ) v r r m a = F i = i dt F i i For a solid rigid: d ( mv ) d mv r dt C r = F i ext, i 25 Conservation of momentum sum of external net rate of linear time rate of change forces acting on = momentum + of linear momentum control volume efflux within the c.v. F= v v n + ρvdv cs.. t cv.. ρ ( ) da Navier Stokes equation: Dv 2 ρ = ρ g P + μ v Dt applies for: - incompressible flow - constant viscosity 26

14 Navier Stokes equation 27 Plane Poiseuille flow h/2 y x u u u u p u u u ρ + u + v + w = + μ t x y z x x y z 28

15 Plane Poiseuille flow 29 Flow in a rectangular duct To find ΔP/ΔL as a function of flow rate: let w = width of channel, integrate velocity to find volumetric flowrate: Flow (rectangular duct) Pressure drop can be applied in microchannels for h << w 30

16 Poiseuille Flow: steady, laminar, incompressible flow through a circular pipe of constant cross section exact solution to the Navier Stokes equation in this case: 2 2 vz vz vθ vz vz p 1 vz 1 vz v z ρ + vr + + vz = + μ r t r r θ z z r r r r θ z 31 Poiseuille Flow: steady, laminar, incompressible flow through a circular pipe of constant cross section exact solution to the Navier Stokes equation in this case: 32

17 2R Q 1 L L R Q 2 i clicker: If the ΔP across both of these tubes is the same, and their lengths are the same, what is the ratio of the flowrates through these tubes? A. Q 1 = 2 Q 2 B. Q 1 = 4 Q 2 C. Q 1 = 8 Q 2 linkto video fromclass: D. Q 1 = 16 Q :45 Poiseuille flow 33 Hydraulic Diameter One method for approximating the flows through various geometries uses the hydraulic diameter. D h = 4 Cross Sectional Area Wetted Perimeter 34

18 Isotropic and Anisotropic Etching Pure silicon crystals are not isotropic in their properties due to nonuniform distribution of atoms at their interior. Such anisotropic properties are representedby three distinct planes: The (111) planemakes an angle of with the (100) plane. Corresponding to these (3) planes are 3 distinct directions in which etching takes place: <100>, <110> and <111>. The <100> is the easiest direction for etching, and the <111> is the hardest direction for etching. Tai Ran Hsu, Microsystems Design and Manufacture 35 Anisotropic etching of silicon Alkaline li chemicals with ph > 12 for anisotropic etching. Popular anisotropic etchants are: KOH ( potassium hydroxide) EDP (ethylene diamine and pyrocatechol) TMAH (tetramethyl ammonium hydroxide) hd id) Hydrazine M. Madou, MEMS 36

19 Isotropic etching Isotropic etchants etch in all crystallographic directions at the samerate: Usually acidic (HNA i.e. HF, HNO3 and CH3COOH) M. Madou, MEMS 37 Planar silicon microneedle J. Chen, K. D. Wise, J. F. Hetke, and S. C. Bledsoe, Jr., "A multichannel neural probe for selective chemical delivery at the cellular level," Biomedical EECE Engineering, IEEE Transactions on, vol. 44, pp ,

20 Molded polysilicon microneedles D. V. McAllister, M. G. Allen, and M. R. Prausnitz, "Microfabricated Microneedles for Gene and Drug EECE Delivery," Annual Review of Biomedical Engineering, vol. 2, pp , Pressure drop in microneedle Determine the pressure drop for a microneedle if it is used to deliver water at a flow rate of 100 μl/min. (circular channel: 200 μm diameter, 2 mm length)

21 Pressure drop in normal syringe Determine the pressure drop through a syringe if it is used to deliver water at a flow rate of 100 μl/min. (inner radius 3.3 x 10 4 m, length 37 mm ) 41 Flow Resistance Analogy to electrical resistance: R = Δp Resistance q Viscosity isa complex function Rate of Strain Temperature 8μL R= 4 π R Pressure (not important for liquids) Newtonian fluid μ is not a function of strain rate (du/dy) Non Newtonian Newtonian Fluid everything else for circular pipes, where R = radius 12μL R = for rectangular channels (w >> h) 3 wh 12μL h nπw R = 1 tanh wh w π n h 1 42

22 Microfluidic resistors Depending on the number of side channels that are open, the circuit becomes more or less resistive; since all the channels are made with rectangular cross sections, the flow resistances can be predicted. Lab Chip, 2009, 9, DOI: /B806803H 43 Microfluidic resistance The basic design principle is that the resistance of a microfluidic channel scales according to the length of the channel, as predicted by the equation to estimate flow resistance R in a rectangular channel with low aspect ratio: where µ is the dynamic viscosity of water, L is the length of the channel and w and h are the shorter and larger cross sectional dimensions i of the channel, respectively. The lengths (L) of the microfluidic resistor (µfr) channels in our device are 4.1 mm, 8.1 mm, 17.3 mm, and 31.3 mm; their flow resistances are termed R 1, R 2, R 3, and R 4, respectively. Thus, turning on and off the 4 resistor valves (a total of 16 unique combinations) the total length of this µfr circuit can be tuned between 0 mm and any of the 15 combinations of channel lengths up to ( =) 60.8 mm (spanning a factor of 15). In our simplified model, the flow rate can be calculated by dividing the change in pressure across the device by the total fluidic resistance: where R 0 denotes the resistance of the system outside of the µfrsand V R1, V R2, V R3, V R4 denotes the state of the resistor valves (open = 0 and closed = 1). Lab Chip, 2009, 9, DOI: /B806803H 44

23 Flow rate control Flow rate control is achieved by selectively closing valves to divert fluid through paths of defined lengths (termed resistor channels ). The valves and resistor channels are modeled as switches (e.g. VR1) and electrical resistors (e.g. R1), respectively, in a circuit diagram, shown in (a) with all valves open, i.e. {V R1, V R2, V R3, V R4 } = {0000}; note that an open valve is equivalent to a closed switch. Each downstream path is designed to be twice as long (and thus twice as resistant) as the upstream path. Thus, combinations of the 4 resistance channels enable 16 discrete flow rates. Valve states t are represented as 0 = open/bypass and 1 = closed. Resistor values are shown as multiples of the length of channel R 1. (b,c) The highest resistance setting has all valves closed {1111} as shown in (b) overview and (c) close up images of resistors R 1 and R 2. Arrows denote direction and relative volume of fluid flow. (d,e) All valves are open for the lowest resistance setting as demonstrated in (d) overview and (e) close up images. (f) Measurements of flow rates are plotted against the inverse equivalent resistance path length (normalized to include the contribution of R 0 ) for various valve states at inlet driving pressures of 0.5, 1, 2, and 3 psi. Linear fits at each driving pressure demonstrate that the system behaves as predicted over the ranges of driving pressures and channel lengths tested. (g) The values for each fit's slope, a, are plotted against the driving pressure P and fit to a linear curve. Standard deviations of the flow rate measurements were less than 3 µl/min. Lab Chip, 2009, 9, DOI: /B806803H 45 Scaling in fluid mechanics The volumetric flowrate in a pipe, laminar flow: i clicker: Q scales with A. R 2 B. R 3 C. R 4 The average velocity and cross sectional area are related to the volumetric flowrate: The pressure gradient (pressure drop per unit length) is: i clicker: ΔP/L scales with A. R 11 B. R 2 C. R 3 46

24 Electro osmotic flow Idea: control the fluid motion in microchannels through the application of an electric field in the desired flow direction Operating principle: the applied electric field causes a body force on charged particles withinthethe fluid => they willviscously dragthe surrounding fluid along with them Unlike pressure driven flow (Poiseuille flow), electro osmotic flow results in an almost perfectly flat velocity profile > advantageous for chemical detection devices 47 Electrokinetic effects study of the motions of ionized particles or molecules and their interactions with electric fields and the surrounding fluid electro osmosis fluid moves relative to a stationary charged or conducting surface through application of an electric field. electrophoresis charged species in a fluid are moved by an electric field relative to the fluid molecules. dielectrophoresis dielectric particles experience a net force in a spatially nonuniform electric field. 48

25 Electrical double layer At the solid liquid interface in an aqueous electrolyte solution, there is an accumulation of the charge carriers. The charge carriers located at the surface are compensated by counter ions that are partly fixed (Stern layer) and partly in a diffuse distribution. shear plane boundary between compact layer and diffuse layer. Example: in glass there are unbalanced SiO terminations attraction of positive ions. 49 Zeta Potential ζ is the zeta potential ti The zeta potential is the potential value at the outside of the immobilized ions (shear plane). Only non immobilized ions participate in electrokinetic effects. The Debye length λ D is a characteristic length of electrostatic screening (diffuse layer). In this layer, ion density is thus unbalanced. Debye length depends not directly on the surface charges, but on the electrolyte characteristics (ion charge and density). Surface charges (i.e. zeta potential) depend onthe ph ofthe solution. 50

26 Debye length where ε 0 is the vacuum permittivity ε0ε rkt ε B r is the relative permittivity D N 2 A is Avogadro s number 2NeI A e is the elementary charge k B isthe Boltzmannconstant T is the temperature 1 I is the ionic strength 2 I = cz i i c 2 i is the concentration of ion i z i is the valence of the ion i clicker: Debye length A. increases with increasing ion concentration B. decreases with increasing ion concentration λ = 51 Electro osmosis (Electro osmotic flow, EOF) a flow is generated by the mobile ions (diffuse layer) at the interface by an applied electric field. 52

27 EOF On a plane surface, ψ (potential) decreases exponentially within the double layer and vanishes far from it. Outside of the double layer, where potential is zero: u ε 0 = ε0ε rζe μ (Helmholtz Smoluchowski equation) 53 Comparison: EOF and Pressure driven flow Electro-osmotic flow (round capillary) Speed Pressure driven flow (round capillary) Speed εζe εζ V Qeo = π r = π r μ μ l εζ E εζ V veo = = μ μ l 4 π r p Qp = 8μ l 2 r p vp = 8μ l 2 2 EECE 300 Comparison 2011 of plug dispersion in pressure driven flow and electro-osmotic flow 54

28 Electro osmotic (EO) flow In most microfluidic devices h>>λ D => a flat velocity profile (compared to the parabolic one for pressure driven flow) EO fluid velocity is independent of the channel geometry (channel width)! (unlike pressure driven flows, where the smaller the channel dimensions, the larger the required dp/dx to achieve the same average velocity εϕ E η y εϕ E 1 η ε kt B i w x D w x ( ) = e λ 1 λ = ( ) 2 D ni0 zq i e u y 55 Visualization of EO Electro osmotic flow: the applied electric field pulls very thin plates of charge near the walls along the microchannel > they viscously drag the bulk fluid with them => a characteristic plug flow pattern 56

29 Electrophoretic separation Assume: in addition to the background ions (ensuring the necessary local charge), there are other ions, of different chemical species (in relatively small amounts) The applied electric field will affect them differently than the background ions => depending on their size and charge, they will move relative to the bulk fluid (relative motion called drift) Electrophoresis the backbone of chemical analysis in microfluidic devices. 57 Electrophoresis Electrophoretic mobility quantitative characterization of the effect of E x on low concentration ions u u = ep =velocity of the ion relative to the bulk fluid μ E ep ep x E x = applied electric field Typical microfluidic device used in electrophoretic separation: Typical dimensions: μm wide 10 30μm deep Usually etched in glass 58

30 Chemical analysis system 1. Channels are filled with a buffered electrolyte (stable ph) 2. A plug of the sample is introduced into one of the entry ports of the shorter channel 3. Voltage applied across short channel => sample fills the short channel 4. Voltage applied across the separator column => sample moves in separation channel 59 Chemical analysis system (2) 5. As the sampled fluid is carried through the separator channel, different ions will travel with different speeds (different electrophoretic mobilities) relative to the bulk fluid motion => separate into distinct bands alongthe separation column 6. Detection of bands by optical techniques (e.g. fluorescence) REMARK: the flat velocity profile in EO greatly enhances the effectiveness of the optical dt detectionti Microfluidic systems for chemical analysis = Lab on a chip devices/miniaturized total analysis systems (μ TAS) T separation ~sec separation In total: ( ) v= v + μ E = μ + μ E eo ep eo ep 60

31 Surface energy Surface energy results from imbalance of forces on surface molecules. Surface molecules subjected to imbalanced attraction. Interior molecules subjected to balanced attraction. Molecules in the bulk of a liquid have interaction with all neighboring molecules (organic liquids: van der Waals interactions; polar liquids: hydrogen bonds). At the interface they have half the number of interactions. Energy must be added to molecules to bring them to the surface. 61 Surface energy minimization The surface acts to minimize energy. Energy Area Surface acts to mimize area. - Why do liquids look wet? Their surface area is minimized by being smooth. - Why are bubbles spherical? A sphere has minimum surface area for a given volume. E Force = =γ A dl dl F 62

32 Surface tension Relation of surface tension to pressure across an interface: Split the bubble in half gas liquid Force due to surface tension Force due to pressure imbalance 63 Bubble valve Surface tension allows a bubble to oppose pressure. r 1 r 2 P 1 P 2 P inside Surface energy minimization. Pushing the bubble into a converging channel increase surface area. Energy Bubble position along channel 64

33 Surface tension effects Surface tension: intensity of molecular attraction per unit length along any line in the surface. Young s equation describes the balance of force between the liquid-solid, liquid-vapor, and solid-vapor interfacial surface energies of a droplet on a solid surface. γ [=] N/m or Joule/m 2 Perfectly wetting surface θ = 0 Perfectly non-wetting surface θ = 180 polystyrene water 86 glass water 14 silicone water Hydrophilic vs. hydrophobic 66

34 Capillary Forces In a capillary, the surface tension force tending to draw liquid into the channel (assuming a round channel) is The gravitational force on the rising column of liquid of height h is 67 Capillary forces Equating these two forces gives the maximum rise in the height of fluid in capillary against gravity Since tubes have different inner diameters, water rises higher in the smaller tubes due to the capillary forces 68

35 Electrowetting Application of a voltage between the droplet and a counterelectrode underneath the insulator reduces the solid liquid interfacial energy, leading to a reduction in θ and improved wetting of the solid by the droplet. 69 A microfluidic network along which drops are driven H. Moon and C. J. Kim, "Electrowetting: Thermodynamic Foundation and Application to Microdevices," in Microfluidic Technologies for Miniaturized Analysis Systems, S. Hardt and F. Schönfeld, Eds.: Springer US, 2007, pp

36 Electrowetting examples Electrowetting effect (in air). The droplets are approximately 700 nl in volume (about mm diameter) and are surrounded by silicone oil Electrowetting flow on 2D array Top view of programmable flow on a 2 D electrode array. 72

37 Electrowetting flow on 2D structure Top view of flow on a ring structure Electrowetting droplet splitting and merging KCl droplet is split and merged across several electrodes. The volume of each droplet tis about t μl. L 74

38 Electrowetting droplet mixing Top view of 2 electrode mixing at 1 Hz switching speeds (slow mixing). Top view of a colorimetric glucose assay with 1 μl sample and reagent droplets Practice Problem Electrowetting on an Insulated Electrode An electrolyte droplet is placed on an insulated electrode. The initial contact angle is θ = 120. The surface tension of the electrolyte is 72 mn/m. The electrolyte is a 10 6 M NaCl solution with a relative dielectric constant of 80. The dielectric has thickness d and dielectric constant ε. Find an expression for the minimum voltage needed to make the surface hydrophilic. 76

39 Electrowetting on insulated electrode 77 Summary Surface tension effects calculation of contact angle capillary forces hydrophilic vs. hydrophobic thermocapillary effect electrowetting 78

40 Mixing of fluids construction pharmaceutical manufacturing kitchen chemistry lab 79 Mixing Mixing of the fluid flowing through microchannels is important for homogenization of solutions of reagents used in chemical reactions biological processes such as cell activation, enzyme reactions, and protein folding require mixing of reactants for initiation biochemistry, drug delivery, sequencing or synthesis of nucleic acids. 80

41 Strategies to Achieve Good Mixing of Fluids Reduction of characteristic length L: Formation of thin fluid layers Stretching of fluid layers Folding of fluid layers Examples:... EECE slide 300 courtesy 2011 of Prof. Stoeber, ME/EECE, UBC 81 Laminar flow, low Re high degree of laminarity implies that the streamlines are locally parallel streamlines are locally parallel. 82

42 Laminarity of microflows Microfluidics for Biotechnology, J. Berthier and P. Silberzan 83 Stokes equation μ μ 2 V = P Reversible fluidic motion? Note that the Stokes equation has no time term, and so all motion is symmetric in time. This means that if forces are reversed, the motion of the fluid is also reversed 84

43 linkto video fromclass: 0 7:45 Poiseuille flow; 13:18 reversibility 85 Mixing in Microfluidics laminar flow no turbulence mixingthrough diffusion across layer interface Péclet number: for fast mixing, generate small diffusion lengths decrease mixing path, increase contact surface layering of fluids stretching and folding of fluid layers 86

44 Diffusion 1-D Diffusion distance: Diffusion time: The diffusion coefficient as a function of temperature: D is the diffusion coefficient D 0 is the maximum diffusion coefficient (at infinite temperature) E A is the activation energy for diffusion T is the temperature R is the gas constant 87 Diffusivity 88

45 Parallel lamination micromixer (a) basic T-mixer, (b) Y-mixer, (c) concept of parallel lamination, (d) hydraulic focusing NGUYEN, N.-T. & WU, Z. (2005) Micromixers - a review. Journal of Micromechanics and Microengineering, 15, R1-R Mixing: practice problem In order to mix ethanol completely l with water in a parallel l micromixer i with two inlets, what is the required length of the mixing channel? (channel width is w, channel height is h, diffusion coefficient is D, and flowrate of both ethanol and water is each Q) 90

46 Mixer: parallel lamination A subdivision i i of each stream into n laminae leads to mixing i that t is faster by a factor of n 2. The above mixer has to be designed with more lamination layers. In the new design, the channel length should be shorter by a factor of 3. In how many layers should each stream be separated? (Assume unchanged geometry: channel width is w, channel height is h, diffusion coefficient is D, and flowrate of both ethanol and water is each Q) 91 Micromixing Passive Mixing Continuous diffusion length reduction through channel geometry Lamination F.G. Bessoth, A.J. demello and A. Manz, Microstructure for efficient continuous flow mixing, Analytical Communications,, vol. 36,,pp pp , M.S. Munson and P. Yager, A Novel Microfluidic Mixer Based on Successive Lamination, Proceedings of the µtas 2003 Symposium, Squaw Valley, CA, U.S.A., pp ,

47 Summary Mixing and Diffusion Peclet number relates diffusion time to convection time Diffusion coefficient is a function of the solute, solvent, temperature at low Re: laminar flow no turbulence mixingthrough diffusion across layer interface implications for mixing strategies for mixing within microfluidic channels 93