Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 Contents lists available at ScienceDirect Colloids and Surfaces A: Physicochemical and Engineering Aspects journal homepage: www.elsevier.com/locate/colsurfa Formation of vortices in a combined pressure-driven electro-osmotic flow through the insulated sharp tips under finite Debye length effects Zhi-Yuan Sun a,b , Yi-Tian Gao a,b,∗ , Xin Yu b , Ying Liu b a State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing 100191, China Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China b a r t i c l e i n f o Article history: Received 16 February 2010 Received in revised form 24 April 2010 Accepted 27 April 2010 Available online 4 May 2010 PACS: 47.61.−k 47.57.jd 47.32.−y Keywords: Pressure-driven electro-osmotic flow Sharp tip Finite Debye length Poisson–Boltzmann model Vorticity Recirculation Finite element method a b s t r a c t Formation of vortices in an electro-osmotic flow possesses engineering applications in enhancing and controlling the microfluidic mixing. In this paper, we investigate the combined pressure-driven electroosmotic flow through the insulated sharp tips in a straight microchannel when a direct current electric field is imposed. Maximum vorticity generated near the tip back is influenced by the local Reynolds number and tip sharpness. Under a finite Debye length, the way to control the recirculation region for the single tip is discussed. Such control is applied to a pair of sharp tips which are designed as the symmetrical and asymmetrical ones in shape, or the ones staggered in position. Poisson–Boltzmann model is solved to simulate the flow by the finite element method. Results are shown to support the assumption of finite Debye length and expected to be helpful in controlling the vorticity and recirculation in the relevant microfluidic devices. © 2010 Elsevier B.V. All rights reserved. 1. Introduction Electro-osmotic flow (EOF), attributed to the interfacial effects in the micro- or nanometer scale, has become one of the nonmechanical techniques for manipulating fluids in microsystems, such as the Lab-on-Chip devices and microfuel cells [1–5]. Generally speaking, EOF is the bulk fluid motion driven by the electrokinetic force acting on the net charged ions in the diffuse layer, the outer part of an electrical double layer (EDL) [1,2]. Thickness of the EDL can be characterized by the Debye length, which is in the order of 10 nm to 1 m [3]. When such length is relatively small compared with the geometry scale (i.e., under the assumption of infinitesimal Debye length), the linear EOF theory indicates the similitude between the flow streamlines and electric field lines based on a neutral fluid with uniform density, viscosity, conductivity and surface zeta potential at the fluid–solid interface [3–5]. One feature of ∗ Corresponding author at: State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing 100191, China. Tel.: +86 1082315701. E-mail address: gaoyt@public.bta.net.cn (Y.-T. Gao). 0927-7757/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2010.04.038 the linear EOF is the irrotationality which makes the existence of vortices impossible [3–5]. Furthermore, due to the neglect of the convection term, the linear EOF can be categorized as the Stokes flow and the viscous term is balanced with the pressure term [3]. Therefore, in certain cases [3–6], the effect of EDL is represented by the Helmholtz–Smoluchowski (HS) slip velocity at all the boundaries when the EOF is governed by the Navier–Stokes equations, which admits the infinitesimal Debye length assumption [3–6]. In recent years, nonlinear electrokinetic phenomena in the microfluids have risen as the practical aspects in some investigations [7–13]. For instance, one of those refers to the concept induced-charge electro-osmosis (ICEO), which is directly related to the polarizability of the wall surface [7–9], and the vortices have been observed around the sharp tips with the symmetrical design in Ref. [10] to support the ICEO effect. However, the ICEO is not the only explanation for the nonlinear vortex dynamics. Another type of nonlinear EOF, the one we care about in this paper, is caused by the convection of the fluids and ionic species, in which the generation of vortices is also an attractive phenomenon [3,12,13]. In the early work devoted to the combined pressure-driven EOFs in the two-dimensional straight channels [14], the finite Debye length effects have been discussed and the EDL vorticity thickness has 2 Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 been defined. At the same time, utilization of the HS velocity as the matching condition between the bulk flow and EDL has been pointed to be incomplete for the mixed pressure-driven EOFs [14]. Further investigation reveals that the Reynolds number-dependent EOF is caused by the finiteness of the Debye length [3]. In such a situation, the irrotationality is broken when the vorticity advects out of the EDL and into the bulk flow [3,5]. Simulations show that the convection of vorticity has close relationships with the Reynolds numbers, Debye length, global and local geometry of the channels [3]. Convective transport of vorticity is claimed by Ref. [5] to be valuable for exploring further. With those considerations, it would be necessary to study the EOF, especially the combined pressure-driven one, under the finiteDebye-length assumption for the similar sharp-tip design as in Refs. [9,10]. Appearance of vortices around the sharp tips is explained by the ICEO mechanism and considered to be valuable for enhancing the mixing in the microfluidic devices under the direct current (DC) electric field [7,8]. Meanwhile, the microchannels with those tips are similar to the ones with different types of roughness, such as the rectangular [15,16] and wave-like (or curve-like) [17,18] roughness, but the generation of vortices for them has been found to be scarce [15–18]. In this paper, we will mainly focus on the combined pressure and electrokinetic effects on the generation of vorticity around the tip back and recirculation region behind the tip. Pressure-driven Poiseuille flow with the parabolic velocity profile will be given at the inlet of the microchannel, and the zeta potential will be distributed uniformly on the surface of the insulated sharp tips, while other sections of the channel will have the zero potential. Convection term will be considered and may have to be balanced with the pressure gradient, which brings complex cases for the convective transport within the scale of the finite EDL. Additionally, one will be able to refer to Refs. [14,19,20] concerning the combined pressure-driven EOFs, including the cases in the square and wavy microchannels. Moreover, we will have to emphasize that the thermal effects (Joule heating) will be neglected in our study to avoid the additional influence on the dynamics of vortices [5]. Poisson–Boltzmann (PB) model will be employed to describe the EOF in our framework and the finite element method (FEM) will be used to solve such model. PB model has its accuracy up to the high ionic concentration or for the smaller EDL thickness compared with the channel scale [21]. Simulations of such model have been used to describe the complex electrokinetic cases, e.g., the electrokinetic transport in the microchannels with random roughness [22] and the electrokinetic interactions in the microscale cross-slot flow [11]. Refs. [23,24] have solved the PB-typed model with the lattice Poisson–Boltzmann method (LPBM) to investigate the mixing enhancement in the microfluidics driven by both the pressure gradient and electrokinetic forces. Further studies show that the LPBM (including the coupled ones) is effective in simulating those electrokinetic flows [25,26]. Actually, the linearized PB model for the small zeta potential has been presented in Refs. [3,14] to discuss the finite-Debye-length effects and convective transport in the EOF. On the other hand, we will use the PB model with the Boltzmann distribution in order to accord with the framework of Refs. [3,27], which shows that the vorticity can be transported to the bulk region by employing such distribution instead of solving the ion transport equation. Goals of this paper will be to clarify the following aspects: (1) how to influence the vortices generated around the tip back by the Reynolds number and tip sharpness in the combined pressuredriven EOF; (2) how to reduce the recirculation region generated behind the sharp tip based on the infiniteness of Debye length; (3) how to control the recirculation regions for a pair of sharp tips designed in three typical manners. Question (1) will be mainly focused on the small-scale vortices while Questions (2) and (3) will be concerned about the large-scale eddies which can influence the Fig. 1. Sketch of the microchannel with an isolated sharp tip. flow structures. With those aims, the structure of this paper will be arranged as follows: Geometry of the model and mathematical formulation will be presented in Section 2. Numerical results and discussions will be provided in Section 3 to investigate the formation of vortices and control of recirculation under the effects of finite Debye length. Finally, our summary will be addressed in Section 4. 2. Formulation Consider a straight finite-length channel with an isolated sharp tip, which is located between two electrodes which produce an electric field directed along the x-axis (see Fig. 1). The channel is assumed to be sufficiently long upstream and downstream, while the sharp tip is discussed in the regions far from the inlet and outlet (where the channel length L H). Such channel is filled with an incompressible Newtonian electrolyte of uniform dielectric constant r , dynamic viscosity and density . When the externally applied electric field is generated by the electrodes placed at the channel inlet and outlet, the distribution of the external electric potential is described by the Laplace equation [3], ∇ 2 = 0. (1) For the boundary conditions, at the inlet and outlet of the channel is specified ( in = 0 , out = 0) and satisfies n · ∇ = 0 at the wall, where n is the unit normal vector. When the electrolyte contacts with the solid surface (tip surface), the surface charge causes rearrangement of both counter-ions and co-ions in the liquid phase nearby, which leads to the formation of the EDL (we assume that the EDL is formed along the sharp tips uniformly) [1–3]. The EDL thickness can be characterized by the Debye length [1–4], = 2z 2 e2 n0 0 r kB T −1/2 , (2) where z is the valance of ion, e is the charge of an electron, n0 is the bulk ionic number concentration, 0 is the permittivity of vacuum, kB is the Boltzmann constant and T is the absolute temperature. For a 1:1 symmetric electrolyte, if we consider the very slow quasisteady-state-electrokinetic flow and neglect the ionic convection within the EDL (the flow in the bulk region is considered to be electrically neutral), the conservation equations for the positive and negative ionic species reduce distribution in such to the Boltzmann a scale [3,21], n = n0 exp −ze /(kB T ) , where is the inherent electric potential in the EDL. Therefore the net charge density e in the EDL can be expressed as [1,3,4] e = −2zen0 sinh ze kB T . (3) According to the theory of electrostatics, within the EDL, the electric potential obeys the Poisson equation, ∇ 2 = −e /(0 r ) [3,4,11]. Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 3 Substituting Eq. (3) into the Poisson equation yields the nonlinear PB equation [3,11,21,22,28] for the EOF, ∇2 = 2zen0 0 r ze kB T sinh . (4) Eq. (4) is a type of the nonlinear partial differential equations (NPDEs). Such NPDEs have played an important role in various fields of science and technology, and their solutions have been investigated by symbolic computation [29]. Normally, the electric field due to the EDL (−∇ ) is sufficiently larger than the externally applied electric field (−∇ ), which indicates that the influence of the applied electric field on the ionic distribution can be neglected [3,11]. Eq. (4) is subjected to the following boundary conditions: = at the sharp tip, where is the zeta potential (variation of the zeta potential with the concentration can be referred to Ref. [30]); = 0 at other sections of the channel. For the quasi-steady-state EOF, all other electromagnetic forces can be neglected compared with the static electric force [22], so the body force f is induced by the electric filed −∇ on the net charge density in the EDL region as f = −e ∇ . Incompressible fluid flow in the steady state is described by the continuity equation and Navier–Stokes equation as [3–6,11,22,28] ∇ · u = 0, (5a) u · ∇ u = −∇ p + ∇ u + f, 2 (5b) where u is the velocity vector and p is the pressure. Eqs. (5a) and (5b) are subjected to the following boundary conditions: the parabolic velocity profile ux (y) = U0 (3/2 − 6y2 /H 2 ) at the inlet, where U0 is the average velocity (while the influence of the spatial gradient in zeta potential on the assumed parabolic velocity distribution upstream is neglected since the channel is sufficiently long, and some revisions for such boundary conditions will be mentioned in the Remark of Section 3.1); no slip condition at the channel walls; normal stress at the outlet, i.e., pout ≈ 0. Introduce the following dimensionless parameters: x∗ = p∗ = x , H p U02 y∗ = , y , H u∗ = ∗ = , ∗ = ze , kB T u , U0 (6a) Fig. 2. Sketch of the dimensionless boundary conditions. solutions for the flow field. This solving process can save plenty of computational resources since the variables are not coupled very intensively [28]. The non-uniform grid applied to discretize the domain in the bulk flow is extremely fine, while the finer case is employed in the near-sharp-tip region, where the EDL locates. The convergence criterion is chosen with the relative tolerance to be 1.0 × 10−6 and maximum number of iterations to be 100. Consider the microchannel with the height H = 100 m, the height and width of the sharp tip for the standard model shown in Fig. 1 being h = 30 m and d = 40 m, respectively. The two sides of the tip are arranged symmetrically. In the simulation, KCl solution (valence |z| = 1) is used as the electrolyte, the physical properties of which contain = 1.0 × 103 kg m−3 , = 0.9 × 10−3 kg m−1 s−1 and r = 80. The physical constants include 0 = 8.854 × 10−12 C2 J−1 m−1 , kB = 1.38 × 10−23 J K−1 and e = 1.6 × 10−19 C. The temperature is chosen as T = 298 K. The bulk ionic number concentration is considered as n0 = 1.0 × 10−6 M with the zeta potential = 0.2 V [30] for the initial simulation. With those parameters, the characteristic length of the Debye length is estimated to be 307 nm [the dimensionless parameter K∼325.3 correspondingly, and other dimensionless parameters can be confirmed via Eqs. (6a) and (6b)], and the size of the initial layer thickness (thickness of the first element layer, that is, the layer adjacent to the corresponding boundary) within the EDL is chosen as 5 nm. The numbers of the nodes and meshes are 165,505 and 18,649, respectively. 3.1. Development of vortices with Reynolds number Re = HU0 , K = −1 H, G= 2zen0 U02 , (6b) where ∗ represents the dimensionless parameters, K is the electrokinetic parameter and Re is the average reference Reynolds number. Then Eqs. (1), (4), (5a) and (5b) can be non-dimensionalized by those parameters as ∇ 2 ∗ = 0, ∇ 2 ∗ 2 = K sinh( (7) ∗ ), (8) ∗ ∇ · u = 0, 1 2 ∗ ∇ u + G sinh( u∗ · ∇ u∗ = −∇ p∗ + Re (9) ∗ )∇ ∗ . (10) The corresponding dimensionless boundary conditions for Eqs. (7)–(10) are illustrated in Fig. 2. 3. Results and discussions Simulation of the pressure-driven EOFs in our model is carried out by the 2D code of the COMSOL Multiphysics 3.4 (COMSOL Inc.), which is based on the FEM. The code solves Eqs. (7) and (8) for the steady external electric field and inherent electric field within the EDL. Then the solutions are applied to Eqs. (9) and (10) to derive the In the corner region of the microchannel, there exists transport of the vorticity from the EDL region to the bulk region, which is considered to be caused by the convective fluid motion [3]. Such transport becomes remarkable with the Reynolds number increasing [3]. In the present work, first we will investigate how the Reynolds number influences the vortices induced by the combined effects of the pressure and electroosmosis. We fix 0 to be 2.3 V (the dimensionless 0∗ ∼11.5) in order to provide an external electric field approximately 1 × 104 V/m, and control the Reynolds number to be less than 20 (Ref. [3] has discussed the vorticity transport near the channel corners up to Re = 100, which is considered to be difficult to achieve experimentally [5]). Fig. 3 shows the velocity streamlines and vorticity contours for the maximum Re = 0.02, 0.05, 0.66 and 0.98 (appearing around the tip), respectively. It is evident that with the low Reynolds number increasing, the vorticity contour around the tip is pushed directly to the downstream region, and a recirculation region (the eddy) is formed behind the tip. As seen in Fig. 3(a), when Re is relatively small, the flow exhibits the EOF feature around the tip; as the eddy being generated, there exists a flow layer between the recirculation region and channel wall [see Fig. 3(c) and (d)], which is due to the finite EDL and different from the case for the pure pressure-driven flow (details of which will be discussed in Part 3.3). 4 Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 Fig. 3. Numerical results for the vorticity contours (left column) and velocity streamlines (right column) for four Reynolds numbers (flow direction: from the left to the right): (a) Re = 0.02; (b) Re = 0.05; (c) Re = 0.66; (d) Re = 0.98. With respect to the high Reynolds number (less than 20), the recirculation region expands (the scale of the downstream region is selected in order to avoid the eddy crossing the outlet boundary), and the maximum positive value of vorticity max is calculated to occur at the tip back. Fig. 4 illustrates the variation of max with the Reynolds number for the pressure-driven EOF and pure pressuredriven flow as comparison. As the Reynolds number increasing, max for the whole comparison is experiencing three phases: when Re ≤ 2, max for the pressure-driven EOF keeps constant while max for the pure pressure-driven flow increases nearly linearly; when 2 ≤ Re ≤ 8, max for both cases are almost equivalent; when Re ≥ 8, the deviation between the two increases gradually. The variation could be understood as the following: in the stage with the relatively low Reynolds number, the EOF dominates the vorticity generation, i.e., the vorticity contributed by the pressure gradient is finite; as the Reynolds number increasing, especially up to the high value, the absolute value of vorticity is mainly influenced by the pressure-driven flow. However, the deviation at the relative high Reynolds number indicates the more remarkable vorticity transport within the EDL. Furthermore, Fig. 5 shows the vorticity contours for three groups of Reynolds numbers Re = 1.7, 5.7 and 12.1 around the wall of tip back with the EDL scale. In the view of such microscale, the vorticity contours are pushed to an expanded region with the Reynolds number increasing, and the tails of vorticity approach to the bulk flow region. Ref. [3] attributes the Reynolds-number-related vorticity transport to the effect of convective term in taking the curl of the Navier–Stokes equation. In addition, efforts have been made to capture the Moffatt eddies, a sequence of weak eddies generated near the corners in the pure viscous flow as the Reynolds number approaches to zero [31,32]. The angle of the corner in our model, as seen in Fig. 1, is 123.7◦ , which does not exceed the critical angle 2critical ≈ 146.3◦ [31], and our angle admits the Moffatt eddies in the pure pressure- Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 Fig. 4. Variation of max with the Reynolds number for the pressure-driven EOF (square plot) and pure pressure-driven flow (round plot). 5 driven flows. We use the same numerical approach as the one employed above and apply a refined local grid refinement to simulate the Moffatt eddies at the tip corners only considering the pressure-driven flow. Two illustrative results are shown in Fig. 6(a) and (b), and the corresponding pressure-driven-EOF cases have been provided in Fig. 3(a) and (d) previously. Sizes of Moffatt eddies in those two figures are ∼1.63 × 0.98 m2 and ∼0.67 × 0.44 m2 , respectively, which are much smaller than the scale of the channel tip { Note that the Moffatt eddy can also exist at Re ≈ 1, as seen in Fig. 6(b), which accords with the results in Ref. [32]}. Moffatt eddies at the back corner of the tip for Re = 0.005 are similar to the ones at the front corner in Fig. 6(a), and a recirculation region has replaced the back corner vortex for Re = 0.90. As the corner is approached, successive eddies are of the quickly decreasing size for our angle [31], and it is hard to capture the smaller neighboring eddies with reasonable resources. Furthermore, for our case of pressure-driven EOFs, the weak Moffatt eddies can be greatly influenced by the electrokinetic effects. Numerical simulation shows that in the region of Moffatt eddies, the velocity filed is of the order 10−5 to10−4 m s−1 for the pressure-driven EOFs, while 10−9 to10−8 m s−1 (Re = 0.005) or 10−7 to10−6 m s−1 (Re = 0.90) for the pure pressure-driven flow. Such difference indicates that the Moffatt eddies illustrated in Fig. 6(a) and (b) can vanish and be neglected in our discussions Fig. 5. Vorticity contours around the wall of tip back with the EDL scale for (a) Re = 1.7; (b) Re = 5.7; (c) Re = 12.1. Fig. 6. Moffatt eddy at the front corner of the tip in the pure pressure-driven flow with the Reynolds number (a) Re = 0.005; (b) Re = 0.90 (the streamlines are arranged uniformly). 6 Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 Fig. 7. Dimensionless velocity profiles at different channel cross-sections. The separation distances between those cross-sections and the central line of sharp tip vary from 100 m (the first parabolic curve from the left to the right) to 30 m (the last distorted parabolic curve from the left to the right) with the step about 11.7 m. Fig. 8. Comparison of Cases A (40 m, 30 m) and B (20 m, 30 m) on the variation of max with the Reynolds number for the pressure-driven EOF. based on the pressure-driven EOFs although the Reynolds number is very small. Remark on the distribution of inlet velocity As mentioned above, we have employed the parabolic velocity distribution at the inlet in our numerical simulations. In fact, the spatial gradient in zeta potential can influence the assumed velocity distribution in the finite region near the sharp tip upstream. For instance, Fig. 7 shows the velocity profiles for the upstream cross-sections adjacent to the sharp tip with the same physical conditions of the flow in Fig. 3(d). We notice that the velocity profiles keep the parabolic shape when the separation distance is beyond about 65 m, i.e., the assumption that the channel is sufficiently long can be approximated in such a situation. Further, if we have not chosen a channel long enough upstream, the distorted parabolic velocity profiles in Fig. 7 (e.g., the last two ones from the left to the right) can be selected to revise the boundary conditions for Eqs. (5a) and (5b) in the actual simulations. Additionally, as the Reynolds number and zeta potential on the tip increase, the channel with longer separation distance between the inlet and sharp tip should be used to satisfy our assumption. More works based on the long-channel-length assumption can be found in Refs. [6,12,13,32,33], in which the influence induced by the non-uniform zeta-potential gradient on the inlet velocity is negligible under such assumption. increasing, max , which is mainly affected by the pressure gradient, increases faster for the sharper tip. Fig. 9 provides us with the more detailed images at the location where max is induced. We notice that the vorticity around the tip back for Case B expands to wider regions along both tangential and normal directions to the tip wall. At the same time, the normalized velocity arrows indicate the larger convective transport region for Case B, where the fluid moves to the bulk flow along the normal direction to the tip wall. Through above analysis, it can be inferred that keeping appropriate Reynolds number and tip sharpness is necessary for controlling the maximum vorticity around the tip back. If max is required to be finite and not to be very sensitive to the Reynolds number, relatively small Reynolds number and selected sharp tip (less sharper) might be the good choices for such pressure-driven EOFs. 3.2. Effect of tip sharpness on vorticity formation 3.3.1. Effect of externally applied electric field For the outer boundary of the EDL without irrotationality, the HS formula Global geometry (large-scale configuration, e.g., channel crosssection) and local geometry (small-scale configuration, e.g., roughness) have considerable influence on the EOFs including the pressure-driven ones [3,10,20,22,28]. Numerical simulations in Ref. [3] reveal that the vorticity leakage (convective transport) can be reduced remarkably by increasing the corner radius of the channel at even high Reynolds number. This part will be devoted to the comparison of the vorticity generation for two different sharpness of the channel tips. The shape parameters (d, h) are chosen as A: (40 m, 30 m) and B: (20 m, 30 m), respectively. Fig. 8 gives two curves to compare the variation of max with the Reynolds number between the two cases. The flow condition for the sharper tip (Case B with apex angle 36.9◦ while A with 67.4◦ ) is more likely to be dominated by the pressure-driven flow effect. The first stage for Case B, i.e., max being kept as almost constant, lasts relatively shorter than that for Case A, and such constant approaches to a smaller value for the sharper tip. As the Reynolds number 3.3. The way to reduce recirculation region under finite Debye length The vorticity generated around the wall of tip back has been investigated in above two parts. In fact, the finite Debye length is able to influence the recirculation region behind the tip when the Reynolds number is relatively high, which will be discussed in this part. u =− 0 r E , (11) gives the effective slip velocity u at the electrolyte–solid interface induced by the tangential electric field E [7]. Furthermore, the flux of vorticity Jω parallel to the channel wall within the finite EDL can be calculated as Jω = 0.5u2 [3]. If large electric field is applied, the corresponding u can be derived to pull the adjacent fluid in the bulk flow to counteract the opposite velocity induced by the recirculation flow. We have carried out numerical simulation to study such phenomenon and some results have been sketched in Fig. 10. With 0 increasing (the externally applied electric field increases), the recirculation region for the same Reynolds number is shrinking and the front stagnation point (where the flow reattaches to the upper horizontal wall) is moving along the wall to the lower position. We define the distance between the front and back stagna- Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 7 Fig. 9. Vorticity contours and velocity arrows (normalized) around the wall of tip back with the Reynolds number Re = 5.7 for (a) Case A; (b) Case B. Fig. 10. Streamlines of the recirculation region behind the sharp tip (flow direction: from the left to the right) at Re = 1.1 for (a) 0 = 2.3 V (0∗ ∼11.5); (b) 0 = 17.3 V (0∗ ∼86.5); (c) 0 = 34.5 V (0∗ ∼172.5). Fig. 11. Variation of the dimensionless number with the applied voltage 0∗ for (a) three Reynolds numbers Re = 1.1, 2.1 and 3.2 when the zeta potential is fixed as ∗ = 7.8; (b) three zeta potentials ∗ = 0.97, 1.95 and 2.92 with Re = 1.08. 8 Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 Fig. 12. Variation of with (a) the dimensionless zeta potential ∗ = ze/(kB T ); (b) the bulk concentration n0 for Re = 2.2 and 0 = 92 V (0∗ ∼460). tion points (where the flow reattaches to the lower horizontal wall) as the reattachment length LR . A dimensionless number quantifying the recirculation region can be introduced as the reattachment length divided by the tip height, i.e., = LR /h. Fig. 11(a) illustrates the numerical results on the variation of with the applied electric field (represented by 0∗ ) for three different Reynolds numbers. For the same Reynolds number, decreases monotonously with the applied electric field increasing (when 0 increases to certain value, the recirculation region becomes not visible, and is not recorded beyond such value in our investigation). For larger Reynolds number, controlling the recirculation region to the same level requires the higher applied voltage to provide the driven flow near the EDL. In order to further study the variation of with 0∗ , Fig. 11(b) illustrates the similar conditions to the square plots in Fig. 11(a) for three relative small zeta potentials when the Reynolds number is fixed as ∼1.08. Results reveal that in the range of ∗ = 0.97 − 7.8 ( = 25–200 mV), the ∼0∗ variation holds the similar features describing the reduction of recirculation. For the smaller zeta potential [e.g., the circle plots in Fig. 11(b)], larger applied electric field is necessary to reach the same decrement of . Meanwhile, those discussions indicate the consistency of our numerical simulation within the zeta potentials of order 101 to102 mV. 3.3.2. Effects of zeta potential and bulk concentration Due to the Boltzmann distribution within the finite EDL, the zeta potential and bulk concentration can influence the net charge density, which could act the similar effect on reducing the recirculation region behind the tip. Fig. 12(a) shows the variation of with the dimensionless zeta potential ∗ for Re = 2.2 and 0 = 92 V (the dimensionless 0∗ ∼460). With increasing, decreases almost as the linear manner. For small value of ∗ , Eqs. (3) and (4) can be linearized, which indicates that the net charge density is proportional to the internal electric potential distribution within the EDL. And the increasing net charge density leads to the corresponding increase of the body force under the given electric field, which results the growth of flow velocity for the finite EDL, if the convective term in Eq. (5b) is considerably weak. Such growth induces larger velocity field near the EDL to counteract the recirculation region behind the tip. This process provides a simple explanation for the variation shown in Fig. 12(a). With regard to the effect of the bulk concentration, the variation of with n0 is illustrated in Fig. 12(b). When n0 is relatively small (about 10−8 to10−6 M), decreases rapidly with n0 increasing, and in such range, the recirculation region can be effectively controlled by selecting appropriate concentration. When n0 is higher than approximately 0.5 × 10−5 M, keeps almost constant with n0 increasing, and it is limited to control the recirculation region by changing the concentration. We will give a simplified understanding of such variation by virtue of the analytic solution for the EOF near the parallel plate [14,34]. For the two-dimensional parallel plates, if the zeta potential is assumed to be constant along the channel surface, Eq. (4) can be simplified and its analytic solution has been obtained as [14,34] ∗ ( ∗ )= 4 tanh−1 tanh ˛ ˛ 4 exp(− ˛ˇ ∗ ) , (12) where ∗ is the normalized distance from the wall, the parameters ˛ = ze/(kB T ) and ˇ = 8 n0 z 2 e2 H 2 /(0 r kB T˛). From Formulae (3) and (12), we can derive the following relation, e ∼(2ze)n0 sinh 4 tanh−1 tanh ˛ 4 exp(− √ n0 ∗ ) , (13) 1/2 where = [8 z 2 e2 H 2 /(0 r kB T )] . Further, if we assume that u∼e is applied to the velocity field near the finite EDL to counteract the recirculation flow, the variation trend of u with n0 can be estimated at the same positions near the EDL when other physical conditions are fixed. Numerical calculations of Expression (13) reveal that close to the wall, the net charge density increases quickly to a balanced value with the bulk concentration increasing, which is sketched in Fig. 13 (in fact, if n0 rises to enough high level, e will present the decreasing trend, which is not considered in our study). Correspondingly, the simplified parallel-plate case might provide a local description on the variation of flow condition with n0 (similar to that for e ) within the finite-EDL framework, Fig. 13. Estimation of e varying with n0 at = 0.2 V for curve) and 4 × 10−4 (the lower curve). ∗ = 3 × 10−4 (the upper Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 9 Fig. 14. (a) Variation of with 0∗ for the single tip and symmetrical tip pairs at the inlet U0 = 0.2 m/s; (b) reduction of the recirculation for the symmetrical tip pairs with Re = 1.6 and 0 = 46 V (0∗ ∼230). and the change of recirculation region shown in Fig. 12(b) is experiencing the opposite process. In other words, the fast decrease for is induced by the corresponding increase for the fluid velocity of flow layer between the recirculation and tip wall. When such velocity steps into the balanced region (varying little with n0 ), keeps balanced as illustrated in Fig. 12(b). Therefore, appropriate control of the externally applied electric field and zeta potential can bring us the similar effects on reducing the recirculation region behind the tip, while changing the bulk concentration in the balanced region is not a very effective choice. 3.4. Control of recirculation for a pair of sharp tips In above parts, we have discussed the vorticity generation with different Reynolds numbers and tip sharpness for a single tip. Based on the finite EDL, the way to control the recirculation through varying the externally applied electric field, zeta potential and bulk concentration has also been investigated. In the applications of microfluidic systems, a pair of symmetrical sharp tips in a straight channel are designed with the intention to enhance the nonlinear effects [9], and the micromixer fabricated incorporating a mixing section comprised of four symmetrically opposed corner pairs can effectively improve the mixing in the EOF [10]. Thereby, in this part we will give a brief study on controlling the recirculation for a pair of sharp tips, which are designed as the symmetrical and asymmetrical ones in shape, or the ones staggered in position. 3.4.1. Effect of applied electric field and control of recirculation for a pair of symmetrical tips Similar to the discussion in Part 3.3.1, first we will carry out brief numerical simulation aiming to investigate how the applied electric field influences the recirculation region for a pair of symmetrical tips. Fig. 14(a) compares the variation of with 0∗ between the single-tip and tip-pair cases when the inlet U0 is set to be the same value. It can be inferred that the maximum cell reference Reynolds number (appears around the sharp tip) for the tip-pair case is larger than that for the single one, and such condition is due to the shorter distance between the two tip apexes, which results in the higher velocity for the fluid passing the tips. The variation of with 0∗ for higher Reynolds number accords with that shown in Fig. 11(a) for the single tip, which indicates that a nonlinear increase in 0∗ is needed to reduce the recirculation region to the same level as the Reynolds number increasing (the absolute value of slope for the curve with higher Reynolds number is smaller). Fig. 14(b) illustrates the symmetrical reduction of the recirculation regions for the tip pair. In fact, for the symmetrical sharp tips, the asymmetrical recirculation can be generated if each tip has the unequal zeta potential distribution on its surface, which can be viewed as the combined effects discussed in Part 3.3.2 for the single tip. Fig. 15(a) provides such an example, in which the stronger zeta potential for the up tip reduces the recirculation to a smaller region than that for the down tip. Furthermore, for those two recirculation regions, one can be eliminated completely while another one still exists if the zeta potential for each tip is well selected. 3.4.2. Control of recirculation for a pair of asymmetrical tips in shape and staggered tips in position Regarding to a pair of sharp tips asymmetrically in shape, we adopt the two tips (Cases A and B) introduced in Part 3.2 and arrange them in apex-opposed position. The numerical results showing the recirculation control are provided in Fig. 15(b). If the two tips have the same zeta potential (e.g., both at 0.1 V), the recirculation region for the shaper tip (the up one) is larger. Enhancing up to 0.25 V, the recirculation can be reduced to almost the same level as the one for the down tip by virtue of the zeta-potential effect investigated in Part 3.3.2. Fig. 15(c) and (d) illustrates the application of recirculation control for a pair of sharp tips staggered in position (the distance between symmetrical axes of both tips is 10 m). From Fig. 15(c), it can be found that the recirculation region for the back tip is visibly larger than that for the front one since the jet-like flow passing the two tips is directed downwards and the local Reynolds number for the back tip increases (the situation for the pure pressure-driven flow has the similar feature). The difference between those two recirculation regions can be reduced if we increase up and decrease down , which is shown in Fig. 15(d). Through such control, the flow layer between the recirculation region and tip wall is increasing for the up tip and decreasing for the down tip. In this part, we have briefly demonstrated some examples of the recirculation control. Further applications are under investigation and will be presented in a future publication. 3.5. Effectiveness of Eqs. (3) and (4) In this part, the assumption that the ionic convection is negligible within the EDL will be briefly examined with the scale analysis [35,36] on the following Nernst-Planck equations [12,21,22,26]: J = −D0 ∇ n − zeD0 n ∇ kB T ∂n + ∇ · J = 0, ∂t + nu, (14) (15) 10 Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 Fig. 15. Sketches of controlling the recirculation regions for (a) a pair of the symmetrical tips with U0 = 0.2 m/s, 0 = 34.5 V, up = 0.25 V and down = 0.1 V; (b) a pair of the asymmetrical tips with U0 = 0.2 m/s, 0 = 23 V, up = 0.25 V and down = 0.1 V; (c) a pair of the staggered tips with U0 = 0.2 m/s, 0 = 23 V and up = down = 0.2 V; (d) a pair of the staggered tips with U0 = 0.2 m/s, 0 = 23 V, up = 0.26 V and down = 0.06 V. where J is the ionic flux, n is the number density of the ionic species in the EDL and D0 is the ion diffusivity. Such equations are employed to describe the ionic transport in the electrokinetic flows [12,21,22,26]. Using Dimensionless Parameters (6) and n∗ = n/n0 , we can non-dimensionalize Eqs. (14) and (15) for the quasi-steadystate case as ∇ 2 n∗ + ∇ · (n∗ ∇ ∗ )− HU0 ∇ · (n∗ u∗ ) = 0. D0 (16) If we consider the direction along the Debye length within the EDL, we have x∼y∼ and ∼, the scale of the diffusion term in Eq. (16) is given by ∇ 2 n∗ ∼nH 2 /(n0 2 ), where n is the ionic density difference. By the similar approach, the electrochemical migration term appears to have ∇ · (n∗ ∇ ∗ )∼nH 2 ze/(n0 2 kB T ) and the ionic convection term presents ∇ · (n∗ u∗ )∼nHu/(n0 U0 ), where u is the characteristic flow speed along the normal direction against the tip wall within the EDL. Thereby in a scale sense, Eq. (16) leads to the scale comparison as 1∼ ze u ∼ . kB T D0 (17) In our discussions, varies in the range of approximately 0.05–0.2 V, and such values lead to ze/(kB T )∼(1.9–7.8) in Formula (17). Additionally, the Debye length is of the order ∼102 nm (about 70–300 nm), and u is estimated to be ∼(10−4 to10−3 ) m s−1 from the numerical results within the EDL. Using those approximate values and D0 = 2 × 10−9 m2 s−1 for KCl, we can derive u/D0 ∼(10−2 to10−1 ), which indicates that the convective transport of the ions in the EDL can be neglected (with errors not beyond about 10% compared with the diffusion term) if the Reynolds number, applied electrical field and zeta potential are appropriate in most of our calculations. In fact, Eqs. (3) and (4) are employed to describe the ion distribution in Refs. [3,11,13,27] when the vortices are existing. Other NPDEs can be found, e,g, in Refs. [29,37,38]. 4. Summary Motivated by the finite-Debye-length assumption, we have studied the combined pressure-driven EOF through the insulated sharp tips in a straight microchannel by numerically solving Eqs. (7)–(10). The maximum vorticity generated around the tip back has been investigated and the way to control the recirculation region (eddy) behind the tips (single one and a pair) has been provided for the relatively high-Reynolds-number flow, as seen in Figs. 10 and 15. Our simulation and analysis have revealed the following results: • Maximum vorticity generated around the tip back experiences three stages with the Reynolds number increasing, as seen in Fig. 4. In the first two stages, features of the EOF and pressuredriven flow dominate the vorticity generation, respectively. In the third stage, the vorticity transport within the finite EDL becomes remarkable for the high-Reynolds-number flow. For the sharper tip, effects of the pressure-driven flow are more obvious and the vorticity gets stronger around the tip back, as seen in Fig. 8. • At the high Reynolds number, recirculation region behind the tip can be reduced through the change of the externally applied electric field, zeta potential and bulk concentration, as seen in Figs. 11 and 12. Control of the recirculation by the bulk concentration is not so effective if the concentration varies in the balanced range [with n0 > 0.5 × 10−5 M in Fig. 12(b)]. • For a pair of the sharp tips designed in three manners (symmetrical and asymmetrical ones in shape, staggered ones in position), recirculation behind each tip can be controlled if the selected zeta potentials are distributed, as seen in Fig. 15. Correspondingly, flow layer between the recirculation region and tip wall can be visible during such control. Under the finite-Debye-length assumption, numerical results and discussions could be expected to be helpful in understanding Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11 the nonlinear convective EOFs, and further applied to the microfluidic devices for the purpose of controlling the mixing. Acknowledgements We express our sincere thanks to Editor N. Furlong and the Referees for their valuable suggestions. We are also very grateful to all the members of our discussion group for their beneficial comments. 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