Electrokinetically Driven Liquid Micro Flows

Transcription

1 8 Electrokinetically Driven Liquid Micro Flows This is page 193 Printer: Opaque this The rapid developments in micro fabrication technologies have enabled a variety of micro fluidic systems consisting of ducts, valves, pumps, and mixers to be utilized effectively for medical, pharmaceutical, defense, and environmental monitoring applications. Examples of such applications are drug delivery, DNA analysis/sequencing systems, and biological/chemical agent detection sensors on micro chips. These micro fluidic systems require seamless integration of sample collection, separation, biological and chemical detection units with fluid pumping, flow control elements, and the necessary electronics on a single micro chip. The reliability of these components are important for successful design and operation of the entire micro fluidic system. In particular, subsystems like micro valves and micro pumps with moving components are complicated to design and fabricate, and they are prone to mechanical failure due to fatigue and fabrication defects. In this chapter, we review and explore ideas of micro flow control elements using electrokinetic flow control schemes with non-moving components. We discuss the governing equations for electro-osmotic flows and develop new velocity interface and slip boundary conditions, which can be used for fast simulations of electrokinetic flows. In the second part we describe the phenomenon of dielectrophoresis, i.e., the motion of polarizable particles subject to non-uniform electric fields, and discuss relevant experiments and biomedical applications. A numerical method for simulating these particulate micro fluidic systems is discussed in section 9.3.

2 Electrokinetically Driven Liquid Micro Flows 8.1 Electrokinetic Effects Review Electrokinetic effects were first observed by Reuss in 189 in an experimental investigation on porous clay, and this was followed by experiments of Wiedmann (Probstein, 1994). In 1879, Helmholtz developed the electric double layer (EDL) theory, which related the electric and flow parameters for electrokinetic transport. The case of the EDL thickness being much smaller than the channel dimensions was analyzed by von Smoluchowski, who also derived a velocity slip condition for electroosmotically driven flows. Burgreen and Nakache presented an analysis of mixed electroosmotic pressure-driven channel flows for very thin two-dimensional channels where the channel height was comparable to the electric double layer thickness (Burgreen and Nakache, 1964). This work was followed by theoretical analysis of electrokinetic flows in thin cylindrical capillaries by (Rice and Whitehead, 1965). More recently, experimental measurements of electrokinetically driven micro flows were obtained by molecular fluorescence tagging (MFT) and micro particle image velocimetry techniques (µ-piv) (Molho et al., 1998; Paul et al., 1998). Molho et al. measured the velocity vector field in mixed electrokinetic pressure-driven micro channels and showed that Joule heating and corresponding changes in the fluid viscosity are secondary effects compared to the streamwise pressure gradients. In a separate study, Paul et al. utilized ultraviolet laser pulses to capture the flow patterns in mixed electrokinetic pressure-driven micro channel flows using a caged-dye fluorescence technique (Paul et al., 1998). A combined experimental and theoretical analysis of electroosmotic flows was presented by Cummings et al., where µ-piv was used to obtain the velocity distribution for straight channels and for crossing of two micro channels (Cummings et al., 1999). Cummings et al. also introduced the ideal electroosmosis concept, which reduces the flow field outside the electric double layer to a potential flow under some specific outer field boundary conditions (Cummings et al., 2). Also, in work by Herr et al., velocity and dispersion rate measurements were presented for electroosmotic flows through cylindrical capillaries with non-uniform surface charge distribution (Herr et al., 2). Experiments performed by a caged-dye fluorescense technique showed strong dependence of fluid velocity and dispersion rate on the surface charge. Jacobson et al. developed parallel and serial mixing mechanisms in micro capillary networks and showed that parallel mixing devices increase reliability of the micro fluidic systems (Jacobson et al., 1999). Through a series of experiments, Polson and Hayes demonstrated electroosmotic flow control (Polson and Hayes, 2). The past decade has also witnessed various numerical modeling and simulation efforts. Yang and Li have used the Debye-Hückel approximation to develop a numerical algorithm for elecrokinetically-driven liquid flows (Yang and Li, 1998). They also identified the streaming potential effects

3 8.2 The Electric Double Layer 195 and offered an explanation for deviations from the Poiseuille flow results for micro scale liquid flows (Yang et al., 1998), first reported in (Pfahler et al., 1991). Numerical simulation of micro fluidic injection using electroosmotic forces through intersection of two channels was presented by Patankar and Hu using the Debye-Hückel linearization (Patankar and Hu, 1998). A finite difference algorithm for electroosmotic and electrophoretic transport and species diffusion was developed by Ermakov et al. for two-dimensional complex geometry flow conduits (Ermakov et al., 1998). Bianchi et al. studied electroosmotically-driven micro flows in T-junctions using a finite element formulation based on the Gouy-Chapman approximation (Bianchi et al., 2). Liquid flow and forced convection heat transfer in electroosmoticallydriven micro channels were analyzed using a finite difference method (Mala et al., 1997). 8.2 The Electric Double Layer Electrokinetic phenomena can be divided into four categories (Probstein, 1994), as follows: Electroosmosis: Motion of ionized liquid relative to the stationary charged surface by an applied electric field. Electrophoresis: Motion of the charged surface relative to the stationary liquid by an applied electric field. Streaming Potential: Electric field created by the motion of ionized fluid along stationary charged surfaces (opposite of electroosmosis). Sedimentation Potential: Electric field created by the motion of charged particles relative to a stationary liquid (opposite of electrophoresis). Such electrokinetic phenomena are present due to the electric double layer (EDL), which forms as a result of the interaction of ionized solution with static charges on dielectric surfaces (Probstein, 1994). For example, a glass surface immersed in water undergoes a chemical reaction, resulting in a net negative surface potential (Cummings et al., 1999). This influences the distribution of ions in the buffer solution, as shown in Figure 8.1. The ions of opposite charge cluster immediately near the wall, forming the Stern layer, a layer of typical thickness of one ionic diameter. The ions within the Stern layer are attracted to the wall with very strong electrostatic forces as recently demonstrated by molecular dynamics studies (Lyklema et al., 1998). Immediately after the Stern layer there forms the electric double layer, where the ion density variation obeys the Boltzmann distribution, consistent with the derivation based on statistical mechanical considerations (Feynman et al., 1977). The electric potential distribution due to the

4 Electrokinetically Driven Liquid Micro Flows presence of the EDL is described by the Poisson-Boltzmann equation 2 (ψ )= 4πh2 ρ e Dζ = β sinh(αψ ), (8.1) where ψ (= ψ/ζ) is the electroosmotic potential field normalized with the Zeta potential ζ, ρ e is the net electric charge density, D is the dielectric constant and α is the ionic energy parameter given as α = ezζ/k B T, (8.2) where e is the electron charge, z is the valence, k B is the Boltzmann constant, and T is the temperature. The variable β relates the ionic energy parameter α and the characteristic length h to the Debye-Hückel parameter ω as follows: β = (ωh)2 α, where ω = 1 8πn o e = 2 z 2 λ D Dk B T. (8.3) The Debye length (λ D ) is a function of the ion density n o as given by equation (8.3). For aqueous solutions at 25 o C, the ion densities of 1 mol/m 3 and 1 mol/m 3 approximately correspond to the Debye lengths of λ D = 1 nm, and λ D =1nm, respectively (Probstein, 1994). Let us consider a two-dimensional channel and assume that the Zeta potential ζ is known, and it remains constant along the channel. Under these conditions equation (8.1) can be simplified in the following form d 2 ψ dη 2 = β sinh (αψ ), (8.4) where η = y/h and h is the half channel height. Multiplying both sides of this equation by (2 dψ dη ), and integrating with respect to η, the following relation is obtained: dψ (η) β = dη α [2 cosh(αψ ) 2 cosh(αψc )] 1 2, (8.5) where both the electric potential and its spatial gradient at point η are represented as a function of the electric potential at the channel center (i.e., ψ c = ψ η=). An analytical solution of (8.4) was obtained by (Burgreen and Nakache, 1964) in terms of an elliptic integral of the first kind. Their work presents the potential distribution as a function of the Debye length λ D and the ionic energy parameter α. It was shown in (Dutta and Beskok, 2) that

5 8.3 Near-Wall Potential Distribution 197 ψ ψ ο ζ EDL y' - Potential Distribution FIGURE 8.1. Schematic diagram of electric double layer (EDL) next to a negatively charged solid surface. Here ψ is the electric potential, ψ o is the surface electric potential, ζ is the Zeta potential, and y is the distance measured from the wall. for α 1 and λ D h the electric potential in the middle of the channel is practically zero. Hence, while ψc the last term in equation (8.5) is simplified, and using the identity cosh(p) = 2 sinh 2 (p/2) + 1, equation (8.5) can be integrated once more. This results in the following form (Hunter, 1981) : ψ (η )= 4 [ ( α ) ( α tanh 1 tanh exp αβ η )], (8.6) 4 where η is the distance from the wall (i.e., η =1 η ). In Figure 8.2, a numerical solution of the electroosmotic potential distribution as a function of various α and β values is presented. The left and right figures show the potential distribution for α = 1 and α = 1, respectively, for various values of β. Forα = 1 and β<1 the EDL is quite thick and it covers the entire channel. As the value of β is increased the electric double layer is confined to a zone near the channel walls, resulting in sharp variations in the electric potential. Comparisons of α = 1 and α = 1 cases at the same value of β show faster decay of the electroosmotic potential for increased values of α. 8.3 Near-Wall Potential Distribution The potential distribution in (8.6) can also be represented as a function of the near-wall parameter χ = y ω, where y = h y is the distance from the wall, and ω is the Debye-Hückel parameter given by (8.3). Since

6 Electrokinetically Driven Liquid Micro Flows α = 1, β = 1.9 α = 1, β = 1 α = 1, β = α = 1, β = 1 α = 1, β = α=1, β=1 α=1, β=1 α=1, β=1 α=1, β=1 α=1, β= ψ.5.5 u/u p ψ.5.5 u/u p η η FIGURE 8.2. Variation of normalized electroosmotic potential ψ across half of a channel for various values of α and β. ωh = αβ, the near-wall scaling parameter (χ), and the non-dimensional distance from the wall (η =1 η) can be represented in terms of each other (i.e., χ = αβη ). Based on this, equation (8.6) can be simplified to ψ = 4 [ ( α ) ] α tanh 1 tanh exp ( χ). (8.7) 4 It is clear that the inner layer scaling of the potential distribution is independent of β for λ D h. In Figure 8.3 the near-wall potential distribution ψ as a function of χ for several α values is presented. We observe that the electroosmotic potential decays to zero with increased χ for all these cases. The decay rate can be quantified by presenting a logarithmic plot of the electroosmotic potential in the near-wall region as a function of χ, as shown in Figure 8.3 (right). A careful examination of Figure 8.3 (right) shows exponential decay of the electroosmotic potential with slope 1 for χ>2. This result can be easily verified by expanding equation (8.7) for χ>2, where tanh (α/4) 1, and exp( χ) 1. Under these conditions ψ (χ) 4 α tanh ( α 4 ) exp ( χ). (8.8) In analogy to the 99% boundary layer thickness in fluid flow, in (Dutta and Beskok, 21) an effective EDL thickness (δ 99 ) was defined as the distance from the wall (in terms of λ D ), where the electroosmotic potential decays to 1% of its original value. The effective EDL thickness as a function of the ionic energy parameter α is presented in Table 8.1. We can calculate the value of δ 99 in terms of the η coordinates by dividing the value of δ 99 given in Table 8.1 by αβ.

7 8.4 Governing Equations for Electroosmotic Flows α= 1 α= 3 α= 5 α= 7 α= α= 1 α= 3 α= 5 α= 7 α= 1 ψ.5 ψ χ χ FIGURE 8.3. Electrosmotic potential distribution within the electric double layer (left) and its logarithmic scaling (right) as a function of the inner-layer scale χ = ωy. TABLE 8.1. Variation of the effective EDL thickness δ 99 and the EDL displacement thickness δ as a function of the ionic energy parameter α. The values of δ 99 and δ are given in terms of the Debye length λ D. α δ δ Governing Equations for Electroosmotic Flows Electroosmotic flow is generated when an external electric field (E) isap- plied in the presence of the EDL. This external electric field interacts with the electric double layer and creates the electrokinetic body force on the bulk fluid. The motion of ionized incompressible fluid with electroosmotic body forces is governed by the incompressible Navier-Stokes equations: ( ) v ρ f +(v )v = p + µ 2 v + ρ e E, (8.9) t where p is the pressure, v =(u, v) is a divergence-free velocity field ( v = ) subject to the no-slip boundary conditions on the walls, and ρ f is the fluid density. Here ρ e is determined from equation (8.1). The externally applied electric field can be represented by E = φ, where the electric potential (φ) is obtained from (σ φ) =, (8.1) where σ is the electric conductivity. The external electric field is subject to the insulating boundary conditions ( φ n = ) on the walls. The Zeta

8 2 8. Electrokinetically Driven Liquid Micro Flows potential is assumed to be uniform for all surfaces. The main simplifying assumptions and approximations are 1. The fluid viscosity is independent of the shear rate; i.e., Newtonian fluid is assumed. 2. The fluid viscosity is independent of the local electric field strength. This condition is an approximation. Since the ion concentration and the electric field strength within the EDL are increased, the viscosity of the fluid may be affected. However, such effects are neglected in the current analysis, which considers only dilute solutions. 3. The Poisson-Boltzmann equation (8.1) is valid. Hence the ion convection effects are negligible. 4. The solvent is continuous and its permittivity is not affected by the overall and local electric field strength. 5. The ions are point charges Numerical Formulation and Validation The numerical solution of the Poisson-Boltzmann equation (8.1) and the incompressible Navier-Stokes equations (8.9) are obtained using the spectral/hp element method (see section 9.1 and (Karniadakis and Sherwin, 1999)). It employs modal spectral expansions in quadrilateral and unstructured triangular meshes, and thus it can be used to discretize complex engineering geometries with high-order numerical accuracy. The weak (variational) form of equation (8.1) is solved via a Galerkin projection. A Newton iteration strategy for a variable coefficient Helmholtz equation is employed to treat the exponential nonlinearity in the following form (Dutta and Beskok, 2): [ 2 αβ cosh(α (ψ ) n ) ] (ψ ) n+1 = β sinh(α (ψ ) n ) αβ (ψ ) n cosh(α (ψ ) n ), where (n) denotes the iteration number. The solution from previous iteration is used for evaluation of the nonlinear forcing function, and the resulting system is solved until the residual is reduced beyond a certain level (typically 1 13 ). The numerical solution of equation (8.1) is challenging due to the exponential nonlinearity associated with the hyperbolic-sine function. In particular, for large values of α the nonlinear forcing increases rapidly for any value of β. Also, for very large values of β with α =1, similar difficulties exist. Accurate resolution of the problem requires high grid density within the EDL. A typical mesh for the α = 1 and β =1, case is presented in Figure 8.4 (right). It consists of 22 elements across the channel width, spaced in biased fashion with minimum width of.1h

9 8.4 Governing Equations for Electroosmotic Flows L 2 Error η.5.25 η ξ α=1 α= N -1 FIGURE 8.4. Exponential decay of the L 2 error norm as a function of the spectral expansion order N (Left). Sample grid used to resolve sharp electric double layer consists of 22 elements across the channel, and each element is discretized with Nth-order modal expansion per direction (Right). The quadrature points for sub-elemental discretization at select elements are also shown. Simulations are performed for β =1,. very near the walls. Once the mesh topology is fixed, the modal expansion order N is increased to resolve the problem further. For rectangular elements shown in Figure 8.4, N = 2 corresponds to a quadratic solution for ψ, typically employed in finite element discretizations. The numerical accuracy of the results is determined by using equation (8.6). In Figure 8.4 (left) the variation of the L 2 error norm as a function of the modal expansion order N is presented. The results are obtained for the mesh topology shown on the right plot. Convergence results for α =1, β =1, and α = 1, β =1, are shown. The L 2 error norm is defined as [ 1/2 Ω R (ψ ) dω] 2 L 2 = Ω dω, where Ω represents the entire flow domain. The residual of equation (8.6) is denoted by R, and it is given by R(ψ m )=ψ m 4 [ ( α ) ( α tanh 1 tanh exp αβ η )], (8.11) 4 where the superscript m denotes the numerical results. The convergence results presented in Figure 8.4 show exponential decay of the discretization error with increased N, typical of the spectral/hp element methodology (Karniadakis and Sherwin, 1999). This high resolution capability enables

10 22 8. Electrokinetically Driven Liquid Micro Flows accurate resolution of the electric double layers with substantially less number of elements compared to the low-order finite element discretizations. Figure 8.4 shows exponential convergence for both α = 1 and α = Electrokinetic Micro Channel Flows In this section mixed electroosmotic pressure-driven flows in straight micro channels are analyzed for channel heights (h) much smaller than the channel width (W ). Therefore, the flow can be treated as two-dimensional as shown in Figure 8.5. For simplicity, fully developed steady flow with no-slip boundary conditions is assumed. The streamwise momentum equation is p x = u µ 2 y 2 + ρ ee x, (8.12) where u is the streamwise velocity and E x (8.4) and (8.1) for ρ e we obtain = dφ/dx. Using equations p x = u µ 2 y 2 DE x d 2 ψ 4π dy 2. (8.13) This equation is linear and thus we can decompose the velocity field into two parts: u = u Pois + u EO, where u Pois corresponds to the pressure-driven channel flow velocity (i.e. plane Poiseuille flow), and u EO is the electroosmotic flow velocity. Analysis of equation (8.13) in the absence of pressure gradients results in balance between the viscous diffusion terms and the electroosmotic forces, which leads to the Helmholtz-Smoluchowski electroosmotic velocity u p (Probstein, 1994): u p = ζɛe x (8.14) µ Non-dimensionalizing equation (8.13), the streamwise momentum equation becomes P ξ = 2 U η 2 + d2 ψ dη 2, (8.15) where, U = u u p, and P p = µu p/h, and ξ = x/h. Here the pressure is normalized by viscous forces, rather than the dynamic head, consistent with the Stokes flow formulation (see section 2.1). In the case of zero net pressure gradient we integrate equation (8.15) to obtain (Burgreen and Nakache, 1964) U(η) =1 ψ (η). (8.16)

11 8.5 Electrokinetic Micro Channel Flows 23 Anode ψ = ψ =1 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx ψ =1 E ψ = Cathode FIGURE 8.5. Schematic view of mixed electroosmotic pressure-driven flow channel. The inlet and exit portions of the channel have negligible electroosmotic effects. Figure 8.2 (right, vertical axis) shows the velocity variation in pure electroosmotic flows for various values of α and β. As shown in the figure, in the limit of small Debye lengths the electroosmotic potential ψ decays very fast within the thin electric double layer and a uniform plug-like velocity profile is obtained in most of the channel. The plug flow behavior has been observed in various experiments (Paul et al., 1998; Herr et al., 2). For the mixed electroosmotic pressure-driven flows, the superposition principle for linear equations is used to obtain the following non-dimensional velocity profile (Burgreen and Nakache, 1964) U(η) = 1 dp ( 1 η 2 ) +1 ψ (η), (8.17) 2 dξ where dp dξ corresponds to the pressure gradient in the mixed electroosmotic pressure-driven flow regime. Substituting the solution for ψ from equation (8.6) we obtain an analytical formula for the velocity distribution. In Figure 8.6 velocity profiles under various pressure gradients are shown. The case for dp = corresponds to a pure plug-like flow, and the cases dp < and dξ dp dξ >, corresponds to flow with favorable and adverse pressure gradients, respectively. In order to obtain the mass flowrate, we integrate the velocity and the electroosmotic potential distribution across the channel (see equation (8.17)). This can be cumbersome in the η-coordinate system, where ψ is a function of both α and β, but in the χ-coordinate system, ψ is only a function of α. In (Dutta and Beskok, 21) the electric double layer displacement thickness was defined δ in analogy with the boundary layer displacement thickness in fluid mechanics in the following form: δ = ˆχ ψ dχ, (8.18) dξ

12 24 8. Electrokinetically Driven Liquid Micro Flows where ˆχ is a large enough distance to include variations in ψ as observed from Figure 8.3. For example, ˆχ 1 is sufficient to define accurately the δ. Typical values of δ as a function of α are presented in Table 8.1. The physical meaning of δ is that it expresses the volumetric flowrate defect due to the velocity distribution within the EDL. Integration of the ψ term in equation (8.17) is performed by using equation (8.18), where 1 1 ψ dη =2 1 ψ dη = 2 ˆχ αβ ψ dχ = 2δ αβ. The resulting volumetric flowrate per channel width, normalized by u p h, becomes Q = 2 dp ) (1 3 dξ +2 δ. (8.19) αβ Since most of the micro fluidic experiments are performed by imposing a certain amount of pressure drop along the micro channel, we use equation (8.19) to correlate the volumetric flowrate with the imposed pressure drop. Also, for applications with specified volumetric flowrate, one can always obtain the resulting pressure variation along the channel. The shear stress on the wall for the mixed electroosmotic pressure-driven flow region is found by differentiating (8.17) with respect to η, and utilizing equation (8.5), which results in τ w = β 2 cosh(α) 2 cosh(αψc ) dp α dξ. (8.2) This is an implicit exact relation under the assumptions of our analysis, which require ψc. Assuming that ψc = (valid for α 1 and λ D h), the following approximate relation can be found: β τw dp = 2 cosh(α) 2 α dξ. (8.21) The first term on the right-hand-side is due to the variation of velocity within the EDL, while the second term is due to the parabolic velocity profile. The shear stress in the mixed electroosmotic pressure-driven flow region is enhanced due to the presence of the EDL. The aforementioned analytical results can be used to validate the numerical computations. In order to eliminate the channel entrance effects, a parabolic velocity profile at the inlet with a maximum inlet velocity of U in = u in /u p is specified. It is possible to generate the desired pressure gradients in the mixed electroosmotic pressure-driven zone using specific

13 8.5 Electrokinetic Micro Channel Flows 25 values of U in. The numerical simulations are performed for Re =.5, where Re is based on the average channel velocity and the channel halfheight. In the results that follow, the streamwise electric field strength and the EDL properties are constant at α = 1 (corresponding to ζ =24mV ) and β =1,. Therefore, the Debye length in the simulations is about one-hundredth of the channel half-height. The entire flow domain including the EDL is resolved in the simulation. Figure 8.6 presents the non-dimensional pressure distribution along the channels for various values of U in. The numerical algorithm specifies zero gauge pressure at the channel outlet. Therefore, all numerical results show zero gauge pressure at the exit. The entrance and exit portions of the channels are purely pressure-driven, and the electroosmotic forces are present only for 3.1 ξ 6.2. The effective electric field is in the positive streamwise direction. Using equation (8.19), we estimate the theoretical value of U in which results in a desired pressure gradient in the mixed region. For example, the theoretical value of U in =1.485 for α = 1 and β =1, gives zero pressure gradient in the mixed electroosmotic pressure-driven flow region. The corresponding velocity profiles across the channel at ξ = 4.5 are presented in Figure 8.6 (right plot). A plug-like velocity profile is observed for U in =1.485, as predicted by the theory. Setting U in =2.5 corresponds to a favorable pressure gradient case, which is a combination of a plug-like flow with a parabolic profile in the bulk of the channel. The corresponding pressure variation shown on the left indicates significant pressure drop at the entrance and exit portions of the channel. However, in the mixed zone, the pressure drop is relatively low due to the electroosmotic pumping. P * U in =2.5 U in =2. U in =1.485 U in =1. U in = U in = U in = U U in = U in = ξ U in = η FIGURE 8.6. Normalized pressure and velocity distribution in a mixed electroosmotic pressure-driven channel for various values of U in. The case U in =1.485 corresponds to the plug-like flow. The electroosmotic forces are present only for 3.1 ξ 6.2. Simulation results are for α =1,β =1, and Re =.5.

14 26 8. Electrokinetically Driven Liquid Micro Flows The adverse pressure gradient case of U in =.5 isanelectrokinetically driven micro pump. For this case, the inlet and the exit pressures are the same, corresponding to a laboratory-on-a-chip device that is exposed to atmospheric pressure at both ends. The entire flow is driven by the electrokinetic forces, which overcome the drag force within the entire channel system. The pressure drop at the inlet and exit portions of the channel (ξ 3.1 and ξ 6.2) is due to the shear stress. A micro pump must be able to raise the system pressure in order to drive the flow. The electroosmotic pump is doing precisely this. The net pressure gradient is positive within the pump, as shown in Figure 8.6. Here we note that In purely electroosmotic system, plug-like velocity profiles with zeropressure gradient will be obtained. In the case of U in =.5, the adverse pressure gradient is present to overcome the pressure drop at the inlet and exit sections. Therefore, any mixed flow system should exhibit a behavior similar to the simulation results presented in Figure 8.6. The velocity profile for this case indicates combination of plug and adverse pressure gradient channel flow behavior, and the net volumetric flowrate is positive, as shown in Figure 8.6. P * Zero net flow ξ U η FIGURE 8.7. Pressure build-up along the micro channel for zero net flow (left); and corresponding velocity distribution across the channel (right). For a closed system it is possible to create large pressure gradients using electroosmotic forces. This can be used for micro scale actuation of micro pistons or micro bellow mechanisms. Such a configuration is simulated by closing the exit of the channel. Due to the presence of electroosmotic forces the pressure rises linearly within the electroosmotic region, as shown in

15 8.6 EDL/Bulk Flow Interface Velocity Matching Condition 27 Figure 8.7. This pressure rise is accompanied by the electroosmotic flow near channel walls and a reverse flow in the middle of the micro channel as shown in Figure EDL/Bulk Flow Interface Velocity Matching Condition In this section, we present the velocity matching condition between the electric double layer (EDL) and the bulk flow regions for mixed electroosmotic pressure-driven flows. The interface velocity condition is important in order to assess the interaction of high vorticity fluid within the EDL with vorticity of the bulk flow under various conditions. The commonly accepted interface matching condition described by the Helmholtz-Smoluchowski velocity 8.14 is not adequate to describe this interaction properly for very small EDL thickness. A similar situation also exists in regions of the domain with complex-geometry, where the EDL thickness is comparable to the radius of curvature of the domain. In such cases, the velocity near the surface should be decomposed into two components: one due to the electroosmotic effects, and the other due to the pressure-driven bulk flow. Utilization of the Helmholtz-Smoluchowski velocity 8.14 as the matching condition at one Debye length (λ D ) away from the wall is incomplete for the following two reasons: First, such a matching condition should be implemented at the effective EDL thickness (δ 99 λ D ), which is considerably larger than the Debye length predicted by the inverse Debye-Hückel parameter. Second, a limitation arises due to the variation of bulk velocity across the small but finite EDL. If we examine the velocity distribution at the edge of the EDL in Figure 8.8, it is clear that the matching velocity changes with the velocity gradient of the bulk flow region. Hence, the appropriate velocity matching condition (u match ) at the edge of the EDL (y = δ 99 λ D ) should become u match = λδ 99 u y w + u p, (8.22) where u y w corresponds to the bulk flow gradient evaluated at the wall. The appropriate matching distance is taken to be the effective EDL thickness (δ 99 λ D ). Equation (8.22) in normalized form becomes U match = ( δ99 αβ ) U η w + U p. (8.23)

16 28 8. Electrokinetically Driven Liquid Micro Flows Here, the first term in equation (8.23) corresponds to a Taylor series expansion of bulk flow velocity at the edge of the EDL from the wall. Equation (8.23) is analogous to slip velocity in rarefied gas flows given in section 2.3. The analytical results in Figure 8.8 are for α = 1 and β =1,, which corresponds to.1 mm buffer solution in 6 µm glass channel, with δ 99 =4.58λ D =.137 µm. It is noteworthy that for finite Debye layers with large bulk flow gradients, the velocity matching conditions using equation (8.23) results in considerable deviations from the Helmholtz-Smoluchowski prediction. 1 δ 99 /(αβ) 1/2 dp * dξ = dp * dξ =..75 dp * dξ =1.862 U η FIGURE 8.8. A magnified view of the velocity distribution in a mixed electroosmotic/pressure-driven flow near a wall for α = 1,β = 1,. Extrapolation of the velocity using a parabolic velocity profile with constant slip value U p on the wall are shown by the solid lines. The analytical solution is shown by the dash lines. 8.7 Electroosmotic Slip Condition The electroosmotic forces are concentrated within the EDL, which has an effective thickness in the order of 1 nm to 1 nm. On the other hand, the micro-channels utilized in many laboratory-on-a-chip applications have a typical height of 1 µm to 1 µm. This two- to five- orders of magnitude difference in the EDL and the channel length scales presents a great challenge to numerical simulation of electroosmotically driven micro flows. Therefore, it is desired to develop a unified slip condition, which incorporates the EDL effects by specifying an appropriate velocity slip condition at the wall. Examining Figure 8.8 and equation (8.23), we observe that the bulk velocity

17 8.7 Electroosmotic Slip Condition 29 extended up to the wall has a constant slip value equivalent to u p. Hence, the appropriate slip condition at the wall is the Helmholtz-Smoluchowski velocity u p, even for finite EDL thickness conditions. For a general numerical algorithm, implementation of slip velocity u p at the wall results in overprediction of the volumetric flowrate, since the velocity distribution within the EDL is neglected. This flowrate error can be corrected by subtracting 2δ / αβ (in nondimensional form) using the EDL displacement thickness δ given in Table 8.1. For engineering applications with α = 1 and β =1,, corresponding to.1 mm buffer solution in a 6 µm glass channel, the error in the conservation of mass due to this slip condition is about 4.5%. With regards to the errors in the momentum equation, neglecting the shear stresses due to the velocity distribution within the EDL, given by equation (8.21), results in gross error. However, for steady Stokes flows, the total drag force can be predicted using a control volume analysis and imposing Newton s second law within the entire control volume. This requires proper inclusion of the electrokinetic body forces. For example, the drag force can be predicted in numerical simulations during the post-processing stage, by first solving the flow system with the slip condition and subsequently calculating the overall drag by approximating the electroosmotic body forces concentrated on the domain boundaries. This is presented in more detail below Approximate Evaluation of Drag Force due to Electroosmotic Effects The drag force acting on a control volume due to the electrokinetic effects can be expressed as F B = ρ eedv. (8.24) CV Substituting ρ e from the Poisson-Boltzmann equation and E = φ, we obtain [ φ F B = ɛ 2 ψ CV n e n + φ l e l + φ ] s e s dv, (8.25) where n, l, and s are the normal, streamwise, and spanwise coordinates, respectively, and dv = dndsdl. This volume integral is complicated to evaluate. However, some simplifications can be made when λ D /h 1. Also, for a general complex geometry, we further assume that the radius of curvature R is much larger than the Debye length λ D. This last condition is required to exclude application of this simplified procedure in the vicinity of sharp corners. Based on these assumptions, 2 ψ can be approximated to be d2 ψ dn. 2 Also, φ n across the entire EDL, which is approximately valid due to

18 21 8. Electrokinetically Driven Liquid Micro Flows the small EDL thickness and the no-penetration boundary condition of the externally applied electric field on the surfaces. This enables us to separate the volume integral in equation (8.25) into the following two components: [ ] 2h W L F B = ɛ d 2 ψ dn 2 dn [ φ l e l + φ ] s e s dlds, (8.26) where L and W are the streamwise and spanwise length of the domain, respectively. Also, for a general geometry we assume the separation distance between the two surfaces to be 2h. The second integral in the above equation can be obtained, during the post processing stage, from the solution of the electrostatic problem. Numerical evaluation of the first integral requires resolution of the EDL region, which requires enhanced near wall resolution and results in the numerical stiffness. However, this integral can be evaluated analytically in the following form: 2h d 2 ψ dn 2 dn = 2h d dψ dn = ζ h [ dψ =2 ζ dη 1 h Hence, the entire drag force can be evaluated as 2ɛ ζ β W L 2cosh(α) 2 h α ] 1 β 2cosh(α) 2. (8.27) α [ φ l e l + φ ] s e s dlds. (8.28) This approach leads to acceptable accuracy since the velocity profiles are approximated reasonably well using the slip condition. Since the EDL and corresponding electroosmotic body forces are not fully resolved, the numerical stiffness of the problem is reduced. The pressure drop in the system is imposed by either the inlet and outlet boundary conditions or the specified flowrate. The drag force due to the electrokinetic effects can be calculated during the post processing stage, under the approximation of decoupling the directions of the electroosmotic and externally applied electric field potentials. This approach is valid for λ D /h 1 and λ D /R Complex Geometry Flows The electroosmotic forces can be selectively applied for flow control in complex micro geometries, by using different materials (conductors and insulators) on various portions of the device surface or by having variations in the Zeta potential (ζ), either due to the contamination on the capillary walls, variations in the wall coating, or gradients in the buffer ph (Herr et al., 2). In this section, numerical simulation results are presented to demonstrate flow control in complex micro geometries.

19 8.8 Complex Geometry Flows 211 The simulation results presented here are performed for a dielectric material of ζ =25.4 mv and the channel height of h =6µm, corresponding to α =1,β =1,. The electroosmotic potential distribution (ψ ) and the externally imposed electric field potential (φ) are obtained by solution of equations (8.1) and (8.1), respectively. The magnitude of the externally imposed electric field E o in dimensional form corresponds to 285 V/cm. The buffer solution is water and the ion concentration density is n o =.1 mm (where M stands for Mole per liter). The Reynolds number based on the average channel velocity is Re =.5. These sets of simulation parameters are selected to match recent micro channel experiments (Paul et al., 1998; Cummings et al., 1999). This is practically Stokes flow with electroosmotic body forces. In the absence of severe geometric complexities, the inertial forces are negligible and the flow characteristics are insensitive to the Reynolds number, when the simulation results are normalized according to the Stokes flow limit. Therefore, parametric studies as a function of the Reynolds number are not necessary. Of course, this is not true if the local Reynolds number Re 1; for such cases, fluid inertia also plays a major role in micro fluidic transport. All the numerical simulations are performed by specifying the flowrate at the chanel inlet. This can be achieved by regulating the flowrate with a valve for flow control. The combined effects of pressure, viscous, and electroosmotic body forces on flow through complex micro geometries are studied Cross-Flow Junctions The cross-junction of two micro channels has many important applications in electrophoretic separation (Polson and Hayes, 2; Culbertson et al., 2), serial and parallel mixing (Jacobson et al., 1999), and speciestransport control devices (Cummings et al., 1999). In this section, electroosmotic forces in a cross-flow junction are applied to demonstrate the flow behavior as a function of the magnitude of the applied electric field strength. The simulation results are shown in Figure 8.9. At the entrance of the channels A and B normalized volumetric flowrate of Q A = Q B = , corresponding to a pure plug-like flow, is imposed. This volumetric flowrate can be obtained from Q = Q ) (1 u p h =2 δ, (8.29) αβ where δ is the flowrate defect due to the EDL. Using Table 8.1 for α =1, δ = In Figure 8.9, results of a scalar transport equation are presented in order to demonstrate the possibility of species transport control. The scalar

20 Electrokinetically Driven Liquid Micro Flows transport equation is given by θ 1 +(v θ) = t Pe 2 θ, (8.3) where θ is the scalar quantity and Pe is the Peclet number. For heat transfer, θ becomes a non-dimensionalized temperature and the Peclet number is based on the Reynolds and Prandtl numbers (Pe = Re Pr). For species transport, θ is a normalized species concentration density, and Pe is based on the Reynolds and Schmidt numbers (Pe = Re Sc). Here, we assumed that the electric field does not interact with the species transport. Therefore, the species are electrically neutral and they do not alter the electrokinetic effects within the buffer solution. In the following results, channel A has scalar concentration density of unity and channel B has scalar concentration density of zero. The normalized vertical electric field strength, E ver /E o, was kept at unity, while the normalized horizontal electric field strength, E hor /E o, was varied to obtain different amounts of high scalar concentration fluid of channel A in the exit channel C. For the scalar transport equation Pe = 5 is used in order to minimize the molecular diffusion effects in the species transport equation. This value is rather high for applications but it is used here to enhance visually the flow control concepts presented. Figure 8.9 (left) shows the streamlines and velocity vectors for equal horizontal and vertical electric field strength. As expected, electroosmotic plug-like velocity is observed in all four channels, except very close to the junction. The normalized discharge in channels C and D is Q = The liquid in channel A exits from the channel D, while the liquid in channel B exits from channel C. These results agree qualitatively with the experimental µ-piv results (Cummings et al., 1999) for equal horizontal and vertical electric field strengths presented in Figure 8.1. The velocity vectors in Figure 8.9 are shaded according to the scalar concentration density field. Species transport results for the biased electric field strength of E hor = 2E ver are shown in the right plot of Figure 8.9. In this case, the volumetric flowrate, geometry, and the surface conditions are identical to the case presented in the left plot. The biased electric field causes fluid pumping from channel A to channel C as demonstrated by the scalar density shading of the velocity vectors and the streamlines in the right figure. Depending on the relative magnitude of E hor /E ver, the amount of fluid pumped from channel A to channel C can be controlled. We have shown that variation of the E hor /E ver ratio controls the flowrate in channels C and D linearly; this is demonstrated in Figure This linear behavior is very important in flow/species control in micro channel junctions. The results also show that it is possible to block the flow in channel D completely, if E hor /E ver =2.8 is applied. In summary, locally imposed electroosmotic forces in the Stokes flow

21 8.8 Complex Geometry Flows 213 D D E o E o A C A C E o 2 E o B B FIGURE 8.9. Streamlines and velocity vectors for electroosmotic flow in a cross-flow junction. The liquid in the horizontal and vertical channels have two different scalar concentrations. The shading imposed on the velocity vectors identifies the scalar concentration density. Left: E hor = E ver = E o; Right: E hor =2E ver =2E o. regime enable linear flowrate control in the branches of a micro channel network system. This linear response can be utilized in the design of various electroosmotically actuated micro pump/valve systems and flow switches. For example, the cross-flow junction geometry presented above can be used for dispensing precise amount of fluid from one channel to another in the absence of valves or pumps with moving components Array of Circular and Square Posts Measurements of electroosmotic flow in arrays of circular and square posts have been obtained by Cummings in the absence of externally imposed pressure gradients, by maintaining zero elevation difference between the upstream and downstream reservoirs (Cummings, 21). The micro fluidic system consisted of uniformly distributed post arrays that are isotropically etched in glass with a thermally bounded glass coverslip. The circular posts have a diameter of 93 µm and the square post dimensions are 14 µm, with center-to-center separation of 2 µm. Phosphate-buffered saline solution of 1 mm, resulting in buffer ph =7.7, is used. An external electric field was applied in various angles to the post arrays, and the electric field value was kept low to avoid particle dielectrophoresis. Micro PIV measurements of the velocity field were performed and the results were presented in the form of simulated interferogram (Cummings, 21). Figure 8.12 shows an electrokinetic speed field in an array of circular posts at 45 o angle with respect to the applied electric field of 2 V/mm.

22 Electrokinetically Driven Liquid Micro Flows FIGURE 8.1. The µ-piv velocity vector measurements of electroosmotic flow in a cross-flow junction, where E hor = E ver. (Courtesy of E. Cummings) The flow is from lower left toward upper right. Lines of constant gray scale are contours of constant speed. The magnitude of speed at any point can be estimated by counting and interpolating the fringes starting at a stagnation point. The interferogram fringe spacing in the figure corresponds to 24.5 µm/s, and the stagnation points on each post is at 45 o and 225 o from the horizontal axis, aligning with the applied electric field. Figure 8.13 shows electrokinetic streamwise velocity field in an array of square posts. The electric field is applied from left to right at a value 1 V/mm, creating flow in the electric field direction. The interferogram fringe spacing in the figure corresponds to 9.8 µm/s. The uniform flow between the top and bottom posts is pure electroosmotic flow. Two-dimensional flow is observed in the region between the two posts, where the flow expands and contracts; the velocity contours are symmetric in this zone. In the flow examples given above, the EDL is infinitesimally small compared to the flow dimensions, and the total pressure is constant at the entry and exit ports. Hence, the flow conditions obey the ideal electroosmosis requirements of (Cummings et al., 2), which states that The flow field outside the EDL is proportional to the external electric field. Therefore, the bulk flow is potential flow. The velocity and speed contours presented in Figures 8.12 and 8.13 closely follow the numerical solution of potential flow past circular and square array posts, as rigorously demonstrated in (Cummings, 21). On the other hand, the bulk flow is confined between two plates (substrate and the top glass cover) with very small separation gap. Hence, the experimental conditions resemble Hele-Shaw flow (Batchelor, 1998), where the Stokes flow mimics the potential flow. Pointwise pressure measurements (with an exception of the inlet and exit post pressures) are not

23 8.9 Dielectrophoresis Percentage of Flow Channel C Channel D E hor /E ver FIGURE Flow control through cross-junction by electroosmotic forcing. The variation of the channel exit flowrate as a function of the horizontal to the vertical electric field strength. Equal amounts of flowrate are specified at the entrance of both channels (sections A and B). available. The Reynolds number calculated using the flow speed, channel, dimensions, and the bulk flow viscosity is very low, corresponding to Stokes flow. Further increase in the applied electric field could have resulted in increased Reynolds number. Hence, deviations from the potential flowlike behavior could have been observed, provided that there are such effects. However, further increase in the applied electric field resulted in onset of dielectrophoretic transport, which we will examine in the next section. 8.9 Dielectrophoresis Dielectrophoresis is the motion of polarizable particles that are suspended in an electrolyte and subjected to a spatially non-uniform electric field (Pohl, 1978). The particle motion is produced by the dipole moments induced on the particle and the suspending fluid due to the non-uniform electric field. When the induced dipole moment on the particles is larger than that of the fluid, the particles move toward the regions of high electric field density. This is known as the positive dielectrophoresis. In the case of the fluids being more polarizable than the particles, the particles move away from the high electric field density, which is known as the negative dielectrophoresis (Cummings and Singh, 2). The time average dielectrophoretic force is given by F DEP =2πr 3 ɛ m Re[K(ω)] E rms 2, (8.31)

24 Electrokinetically Driven Liquid Micro Flows FIGURE Electrokinetic speed contours in an array of circular posts at 45 o with respect to the applied electric field of 2 V/mm. (Courtesy of E. Cummings) where E rms is the rms electric field, r is the particle radius, ɛ m is the dielectric permittivity of the medium, ω is the electric field frequency and Re[K(ω)] indicates the real part of the Clausius-Mossotti factor (K(ω)), which is a measure of the effective polarizability of the particle, given by (Morgan et al., 1999) K(ω) = (ɛ p ɛ m) (ɛ p +2ɛ m), (8.32) where ɛ p and ɛ m are the complex permittivities of the particle and the medium, respectively. The complex permittivity is defined by ɛ = ɛ 1 σ ω, (8.33) where ɛ is the permittivity, and σ is the conductivity of the dielectric medium. Ignoring the Brownian motion, the buoyancy force, and the motion of the buffer solution, the equation of motion for a suspended particle can be written as dv m p dt = F DEP F Drag, where F Drag is the instantaneous drag force acting on the particle. For particle sizes smaller than 1 µm in buffer solutions with viscosity close to

25 8.9 Dielectrophoresis 217 FIGURE Electrokinetic streamwise velocity contours in an array of square posts. The electric field is from left to right (1 V/mm). (Courtesy of E. Cummings) that of water, the Reynolds number based on the particle size is smaller than unity. Hence, the inertial effects on particle motion can be neglected. If we assume dilute solution so that particles do not interact and spherical particles with radius r, we can use Stokes formula for the drag force (see also section 9.3): F Drag =6πµrV, where µ is the dynamic viscosity and V is the velocity of the particle. Since the inertial effects are negligible, one can assume that instantaneous velocity of a particle V is proportional to the instantaneous dielectrophoretic force. This results in the dielectrophoretic mobility of a particle given by (Morgan et al., 1999) V = r2 ɛ m Re[K(ω)] E rms 2. (8.34) 3µ Since the surface area of the particle is proportional to r 2, the dielectrophoretic particle velocity is proportional to the surface area of the particle. Further examination of equation (8.34) also reveals that the particle velocity is determined by the square of the rms electric field. Therefore, dielectrophoresis can be maintained by either DC or AC electric fields.

26 Electrokinetically Driven Liquid Micro Flows The positive and negative dielectrophoresis (i.e., motion of particles towards or away from the large electric field gradients) are obtained when Re[K(ω)] > and Re[K(ω)] <, respectively. These properties of dielectrophoresis enable highly controlled selective micro fluidic particle/cell separation methodologies. FIGURE A 5 khz AC electric signal induces electric polarization on human leukemia cells and moves them to the center of four spiral electrodes, while the normal cells are trapped on the electrode surfaces. (Courtesy of P. Gascoyne and X. Wang) Manipulation of very small (sub-micron) particles with dielectrophoresis in reasonable time requires large electric field gradients. This is because the dielectrophoretic mobility is proportional to the surface area. Large electric field results in Joule heating. However, further expected miniaturization of device components will reduce Joule heating effects considerably. This is because large electric field gradients can be achieved with relatively smaller potential differences between the electrodes. Analysis of Joule heating effects in dielectrophoresis can be found in (Morgan et al., 1999). In the rest of this chapter, we will concentrate on various biomedical and micro fluidic applications of dielectrophoresis. Green and Morgan have shown for the first time that it is possible to separate a population of nano particles (latex beads of 93 nm) into two subpopulations due to the differences in their dielectrophoretic properties, by using micro fabricated electrode arrays (Green and Morgan, 1997). This initiated many applications of separation of small, yet similar-size particles with different biological prop-

27 8.9 Dielectrophoresis 219 erties ranging from chromosomes, viruses, DNA, to macromolecules. For example, Gascoyne et al. was able to separate human breast cancer cells from blood by using (AC electric field) dielectrophoretic separation (Gascoyne and Wang, 1997). Their technique utilized frequency dependence of dielectrophoretic (DEP) properties of blood and cancer cells and worked in the following sequence: first, trapping and accumulation of both blood and cancer cells on micro fabricated dielectric affinity chambers (electrodes) using DEP collection at 5 khz; second, reducing the DEP collection to 5 khz, where the blood cells are released and only the cancer cells are trapped on the electrodes. This is followed by washing the released blood cells with pressure-driven flow, where the blood cells are convected downstream, while the cancer cells remained on the electrode tips (Gascoyne and Wang, 1997). In a somewhat different and more recent design, Gascoyne and Wang used four spiral micro fabricated electrodes for dielectrophoretic separation of human leukemia cells from the normal cells. Figure 8.14 shows the leukemia cells concentrated toward the center of four spiral electrodes. In this design the normal cells are trapped on the electrode surfaces and human leukemia cells are washed toward the center. Several other applications of dielectrophoresis can be found in (Gascoyne et al., 1992; Markx and Pethig, 1995; Markx et al., 1996; Fiedler et al., 1998) and in (Cheng et al., 1998; Morgan et al., 1999). In a series of papers Gascoyne and coworkers have also utilized combined dielectrophoretic/gravitational field flow fractionation for cell separation on micro fabricated electrodes (DeGasperis et al., 1999; Yang et al., 1999; Yang et al., 2). The gravitational field flow fractionation utilizes balance between the vertically applied dielectrophoretic forces and the gravitational forces for levitation of different particles to different heights in a miniaturized channel flow system. The bulk flow is pressure driven in the axial direction and it splits into two different outlet ports at a desired channel height, separating the heavier particles in the bottom exit port from the lighter ones in the top exit port. This particle separation system is free from any moving mechanical components. Most of the dielectrophoretic applications utilize AC electric fields. However, as we have stated earlier in the chapter, it is possible to utilize a DC electric field. In this case the Clausius-Mossotti factor given by equation (8.32) is real and there is no frequency dependence in the electrophoretic force. Cummings and Singh built arrays of insulated circular and square posts without embedded electrodes (as shown earlier in Figures 8.12 and 8.13). The flow is driven by electrodes outside the post arrays with DC electric field in a desired angle to the post row orientation (Cummings and Singh, 2). Under a weak electric field, dielectrophoretic effects are overwhelmed by the electrokinetic effects and diffusion, since dielectrophoresis is second-order effect in the applied electric field. When the electric field was increased, two additional distinct flow phenomena are observed. The

28 22 8. Electrokinetically Driven Liquid Micro Flows FIGURE Particle fluorescence image of filamentary (upper) and trapping (lower) dielectrophoresis. Regions of high particle concentration emit intense fluorescence. The flow is from top to bottom produced by an applied field of 25 V/mm and 1 V/mm, for the upper and lower figures, respectively. The circular posts have diameter of 33 µm with center separation of 63 µm. (Courtesy of E. Cummings)