Nonlinear electrokinetic flow near permselective membrane

NONLINEAR ELECTROKINETIC FLOW NEAR PERMSELECTIVE MEMBRANE PHAM VAN SANG (B.Eng. (Hons), HUST) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN COMPUTATIONAL ENGINEERING (CE) SINGAPORE-MIT ALLIANCE NATIONAL UNIVERSITY OF SINGAPORE 2012 Declaration I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Pham Van Sang 30 July 2012 Acknowledgements I would like to take this opportunity to express my deepest gratitude to my advisors, Prof. Kian Meng Lim, Prof. Jongyoon Han, and Prof. Jacob K. White for their invaluable advice, guidance, encouragement and continued support as well as for introducing this wonderful research topic to me. Their scientific excitement inspired me throughout my research. I feel privileged for having the opportunity to work with them. I have had the pleasure of working with many excellent scientists during my PhD study. I specially thank Prof. Boris Zaltzman for the guidance on his theory of electroosmosis instability as well as valuable advises to my study. I would like to thank Dr. Sung Jae Kim, Dr. Li Zirui and Mr. Rhokyun Kwak for valuable and fruitful discussions on the topics related to the experimental devices. The completion of this thesis would not have been possible without the strong support of my family. My wife, Thu Ha, encouraged me by bringing the best cares to our son and my mother during the years I was far away from home. My son, Hien Minh, often questions me “when are you coming back home?” which constantly reminds me to complete successfully this degree and soon return to him. I can not thank my mother, Dinh Thi Vinh, enough for everything she had done despite her difficulties and sufferings to give best education to her children. This work is supported by the Singapore-MIT Alliance, Computational Engineering Programme, National University of Singapore. i This thesis is dedicated to the memory of my beloved farther, Pham Van Vinh (1935-2008). ii Contents Acknowledgement i Summary vi List of tables ix List of figures x List of symbols xviii Introduction 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Nonlinear electrokinetic flow . . . . . . . . . . . . . . . . . . . 1.3 Numerical solution for nonlinear electrokinetic flow . . . . . . 1.4 Scope and outline of thesis . . . . . . . . . . . . . . . . . . . . 1.4.1 Scope of thesis . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Outline of the thesis . . . . . . . . . . . . . . . . . . . 10 Mathematical Model 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Governing equations for transport of ions in electrolyte . . . . 14 2.2.1 Ion transport in electrolyte . . . . . . . . . . . . . . . . 14 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 Electric double layer . . . . . . . . . . . . . . . . . . . 16 2.3.2 Solid surface . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 Membrane surface . . . . . . . . . . . . . . . . . . . . . 20 2.3 iii 2.3.4 Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3.5 Dimensionless formulation . . . . . . . . . . . . . . . . 22 2.3.6 Conclusion for chapter . . . . . . . . . . . . . . . . . 25 Numerical Methods for Electrokinetic Flow 26 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Discretization methods . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Finite volume method . . . . . . . . . . . . . . . . . . . . . . 28 3.3.1 Discretization for partial derivative terms . . . . . . . . 30 3.3.2 Source term . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.3 Temporal discretization . . . . . . . . . . . . . . . . . . 35 3.3.4 Calculation of gradient . . . . . . . . . . . . . . . . . . 36 Poisson-Nernst-Planck equations . . . . . . . . . . . . . . . . . 37 3.4.1 Discretization for the Poisson-Nernst-Planck equations 38 3.4.2 Transformation of variables . . . . . . . . . . . . . . . 39 Newton-Raphson method for system of nonlinear equations . . 43 3.5.1 Evaluation of functions . . . . . . . . . . . . . . . . . . 44 3.5.2 Jacobian matrix . . . . . . . . . . . . . . . . . . . . . . 46 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 48 3.6.1 Segregated method . . . . . . . . . . . . . . . . . . . . 49 3.6.2 Coupled method . . . . . . . . . . . . . . . . . . . . . 53 3.4 3.5 3.6 3.7 3.8 3.9 Solution method for the coupled Poisson-Nernst-Planck and Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 56 3.7.1 Segregated algorithm . . . . . . . . . . . . . . . . . . . 57 3.7.2 Coupled algorithm . . . . . . . . . . . . . . . . . . . . 58 Parallel algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.8.1 Partition domain . . . . . . . . . . . . . . . . . . . . . 60 3.8.2 Communication between processors . . . . . . . . . . . 63 3.8.3 Linear system solver . . . . . . . . . . . . . . . . . . . 65 Conclusion for chapter . . . . . . . . . . . . . . . . . . . . . 66 iv Verification and Validation for Numerical Methods 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Validation for numerical solution of the Poisson-Nernst-Planck 4.3 67 equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2.1 Charged surface in electrolyte solution . . . . . . . . . 68 4.2.2 Ion transport in a nanochannel . . . . . . . . . . . . . 73 Validation of numerical methods for the Poisson-Nernst-PlanckNavier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 4.3.1 4.3.2 78 Combined electroosmotic and pressure driven flows in two-dimensional microchannel . . . . . . . . . . . . . . 4.4 67 78 Electrokinetic transport in micro-nanofluidic interconnect preconcentrator . . . . . . . . . . . . . . . . . . . 80 Efficiency of methods and algorithms . . . . . . . . . . . . . . 84 4.4.1 Primitive variables vs. Electrochemical potential variables 84 4.4.2 Segregated algorithm vs. Coupled algorithm . . . . . . 86 4.4.3 Efficiency of parallel solver . . . . . . . . . . . . . . . . 88 Conclusion for chapter . . . . . . . . . . . . . . . . . . . . . 90 Electroconvective Instability near Permselective Membrane 91 4.5 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3 Current-Voltage response . . . . . . . . . . . . . . . . . . . . 94 5.3.1 Ohmic regime - Ion concentrations reduce with increasing bias voltage . . . . . . . . . . . . . . . . . . . . . . 5.3.2 94 Limiting current regime - Vanishing of ions near membrane surface . . . . . . . . . . . . . . . . . . . . . . . 96 5.3.3 Effect of extended space charge layer on the fluid flow . 98 5.3.4 Overlimiting current regime - Electroconvective instability100 5.3.5 Effect of surface wavelength on the onset of electrokinetic instability . . . . . . . . . . . . . . . . . . . . . . 107 v 5.4 Hysteretic behavior . . . . . . . . . . . . . . . . . . . . . . . . 110 5.5 Conclusion for chapter . . . . . . . . . . . . . . . . . . . . . 114 Sheared Electroconvective Instability in Electrodialysis Cell 116 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . 117 6.3 Current-Voltage response of electrodialysis cell . . . . . . . . . 118 6.3.1 Ohmic regime . . . . . . . . . . . . . . . . . . . . . . . 120 6.3.2 Limiting current . . . . . . . . . . . . . . . . . . . . . 120 6.3.3 Overlimiting current . . . . . . . . . . . . . . . . . . . 122 Sheared electroconvective instability . . . . . . . . . . . . . . . 124 6.4.1 Unidrectional electroconvective vortex flow . . . . . . . 124 6.4.2 Unchanged size vortex . . . . . . . . . . . . . . . . . . 127 6.4.3 Depletion zone formation and overlimiting current . . . 128 6.5 Propagation of the instability vortices and depletion zones . . 131 6.6 Effect of inlet flow rate and bias voltage on the vortex size . . 134 6.7 Conclusion for chapter . . . . . . . . . . . . . . . . . . . . . 135 6.4 Conclusions and Recommendation for Future Work 136 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7.2 Recommendation for future work . . . . . . . . . . . . . . . . 139 Bibliography 142 vi Thesis Summary In this thesis, we study the nonlinear electrokinetic flow near permselective membrane. We solve the full, coupled Poisson-Nernst-Planck-Navier-Stokes equations instead of using simplified models where the electrical boundary layer is represented by a set of asymptotic boundary conditions as have been done in the previous studies. To obtain numerical solution for the equations, we develop numerical methods based on the finite volume method and NewtonRaphson method. The methods are able to solve for complicated geometry domain which can be either two- or three- dimensions. We also develop a parallel algorithm for solving the equations on parallel computer systems. Based on the direct numerical simulations, we investigate the electrokinetic flow developing near a permselective membrane in contact with a quiescent electrolyte solution. Our analysis shows that, at a significantly high bias voltage applying between the membrane and the solution bulk space, an electroconvective instability occurs near the membrane surface. The instability generates a vortical flow near the membrane surface. The flow transports more ions to the membrane to promote an overlimiting current through the membrane. Our finding provides a clear picture of the mechanism for the overlimiting conduction phenomenon. More importantly, we observe hysteretic behaviors of electric current and fluid flow in the transition between the limiting and overlimiting regimes. To obtain a more complete understanding of the nonlinear electrokinetic flow near permselective membrane, we conduct direct numerical simulations for the electrokinetic flow in an electrodialysis cell of which permselective memvii branes interface with a pressure-driven electrolyte flow. An important finding we obtained is a new kind of electroconvection instability, called sheared electroconvective instability, in electrodialysis cell. Differing from the instability which occurs in quiescent electrolyte, under effect of the shear flow, the instability exhibits a unidirectional vortices flow and growing depletion zones. Interestingly, the vortices and depletion zones are not confined to the membrane surface but propagating toward downstream. Our findings are in agreement with experimental results obtained by Rhokyun Kwak [1] in collaboration. This agreement allows us to gain a fuller understanding of the mechanism for overlimiting conductance in the electrodialysis cell. viii 6.6 Effect of inlet flow rate and bias voltage on the vortex size 4.5 3.5 Vortex height (µm) Vortex height (µm) 3.5 2.5 0.5 0.55 0.6 0.65 V(volt) (a) 0.7 0.75 2.5 1.5 U(mm) (b) Figure 6-18: Vortex height increases with increasing bias voltage (a), and decreases with increasing inlet flow rate (b). We define the height of an instability vortex by the distance from the membrane surface to the highest point of the vortex streamlines. The vortex height is determined by fluid velocity in the vortex, and the inlet flow. An increase in bias voltage generates a higher pressure and larger space charge in the extended space charge layer; thus, it induces a faster rotation velocity in the instability vortex. The faster flow drives more fluid upstream; as a result, the vortex is enlarged. Figure 6-18a displays the vortex height at different bias voltages. The vortex height seems to increase linearly with bias voltage. When a bias voltage is applied between the membranes, ions in the pressuredriven flow are driven through the membranes by the induced electric field to generate a certain electric current. At the same bias voltage, a higher inlet flow rate drives more ions into the channel, as a result, the depletion zones are reduces. The rotation velocity of the vortices decreases accordingly. Therefore, the vortices are reduced in height. The effect of inlet flow rate on the vortex height is displayed in Fig. 6-18b, where vortex height at V = 22 is measured with different inlet flow rates: 60U0 , 80U0 , 120U0 , and 240U0 . It shows clearly that the vortex height is decreased with increasing inlet flow rates. 134 When the inlet flow rate is significantly large, the extended space charge layer is suppressed because the amount of ions carried in by the inlet flow exceeds the amount of ions passing though the membrane. Due to the absence of the extended space charge layer, the instability disappears from the channel which means that the vortex height reduces to zero (no vortex appears in the channel). 6.7 Conclusion for chapter We conducted simulations for ion transport in a typical electrodialysis cell. Our simulation results showed new kind of electroconvective instability occurring in electrodialysis cell, which is called the sheared electroconvective instability. The pressure-driven flow eliminates unfavorably-directed vortices and thrusts favorably-directed vortices along the membrane surface. The instability accelerates fluid flow near the membrane surface to transport the fluid, which has high ion concentration, to the membrane to produce an overlimiting current passing through the membrane. Due to the confining effect of the shear flow, the instability vortices cannot expand entirely over the domain as occurred in quiescent solution. Therefore, there is no current jump in the transition between the limiting and overlimiting current regimes; the hysteresis behavior does not appear in the I-V curve. It was also shown that the oscillating current observed in the over-limiting current resulted from of the propagation of depletion zones along the membrane surface. Our simulation results are in good agreement with experiment results. The agreement allows us to conclude that the appearance of instability vortices is the mechanism for the over-limiting conductance in the electrodialysis cell. 135 Chapter Conclusions and Recommendation for Future Work 7.1 Conclusions In the present thesis, we studied the nonlinear electrokinetic flow near a permselective membrane. By solving directly the coupled Poisson-Nernst-PlanckNavier-Stokes equations, we obtained the numerical solutions for the ion transport near the permselective membrane. Our study contributes to a detail understanding of the electroconvective instability developing over permselective membranes. The agreement between our simulation result and the experimental results allowed us to suggest a mechanism for the overlimting conductance in electrodialysis cell. To solve the Poisson-Nernst-Planck and Navier-Stokes equations, we developed numerical methods based on the finite volume method and the NewtonRaphson method. To improve the convergence rate of the Poisson-NernstPlanck equations solver, we used a transformation of variables which relates logarithmically the ion concentration and the electric potential. The Navier- 136 Stokes equations were solved by a coupled method which uses the Rhie-Chow interpolation to produce an equation for pressure from the continuity equation. Linear algebraic equations were solved using the PETSc library which implements the Krylov-subspace method. In order to couple the Poisson-Nernst-Planck and Navier-Stokes equations, we developed two coupling algorithms. In the first algorithm, the segregated algorithm, the sets of equations are solved separately. Starting with solving the Poisson-Nernst-Planck equations, concentrations and electric potential are obtained and used to calculate electric body force. The Navier-Stokes equations are then solved with this body force. Velocity field from the solution of the Navier-Stokes equations is then used to calculate the convection term of ion fluxes which are then substituted into the Poisson-Nernst-Planck equations. The process is repeated until convergence reaches for all variables. In the second algorithm, the coupled algorithm, the Poisson-Nernst-Planck and Navier-Stokes equations are solved simultaneously. The equations are discretized and assembled into a Jacobian matrix. The Jacobian matrix is much larger than that generated by the segregated algorithm. It was found that the segregated algorithm takes more iterations to reach convergence for all equations, but it takes shorter computation time and requires less memory requirement than the coupled algorithm. On the other hand, the coupled algorithm is able to reach convergence in fewer iterations.However, the coupled algorithm has a massive memory requirement, and it takes much longer computation time compared to the segregated algorithm. Depending on size of problem (determined by the number of control volumes in the computational mesh), the appropriate algorithm for simulation should be chosen. We developed a parallel algorithm to execute the numerical method on a parallel computer system. However, efficiency of the parallel solver is problemdependent. It was found that the system of linear equations generated by discretization of Poisson-Nernst-Planck equations could be solved efficiently in parallel using the Krylov subspace method if the number of partitions is 137 less than 8. The numerical methods were verified though four case studies where our numerical solutions are in good agreements with analytical solutions and numerical solutions published in literature. The agreements indicate the high accuracy of our numerical methods. Using the numerical numerical methods, we obtained numerical solution for the nonlinear electrokinetic flow developed near a permselective membrane in contact with a quiescent electrolyte solution. The result demonstrated the occurrence of the electroconvective instability at a significantly high voltage applied at the membrane. This result contributes a numerical validation of the experimental results in literature. Based on the electroconvective instability, we explained the mechanism for the change of conductivity in I-V curve. We shown that the vortical flow generated by the instability transport more ions to the membrane surface to promote an overlimiting current passing through the membrane. More importantly, we observed a hysteretic behavior in the transition between the limiting and overlimiting regimes. The hysteresis is characterized by a significant difference between flow pattern in limiting and over-limiting regime. The hysteresis is caused by the redistribution of ion concentration in overlimiting regime and the formation of a large depletion zone. The role of electroconvective instability flow is stated through the hysteresis, involving the mixing of the diffusion layer, generation of the depletion zones, amplification of the electric field, and maintaining the overlimiting current regime at even lower critical value voltage. In order to obtain a more complete understanding of the electrokinetic flow developed near permselective membrane. The numerical solution for electrokinetic flow in electrodialysis cell was also carried out. It was found that electroconvective instability occurs in the electrodialysis cell of which the permselective membranes interface with a shear flow. The shear flow makes the electroconvective instability different from the instability occurring in the quiescent electrolyte. The instability exhibits unidirectional vortices and grow138 ing depletion zones. Interestingly, the vortices and depletion zones propagate along membrane surface. Importantly, we observed that when the instability occurs, current passing through the membrane exceeds the limiting value. A similar observation was also obtained in experiment. These observations allowed us to state conclusively that the overlimiting conductance in the electrodialysis cell is not a result of neither chemical effects (water splitting) nor electrostatic effects in microscale system (surface, conduction and electroosmotic flow) but is a result of the electroconvective instability. We also found that, due to the effect of the shear flow in the electrodialysis cell, the instability vortices are not able to expand entirely over the domain like in the quiescent case. The current increases gradually with the voltage applied. Thus, there is no jump in the I-V curve, and no hysteretic behavior was observed in the electrodialysis cell. 7.2 Recommendation for future work Some important contributions of the present study were mentioned above. However, some unexplored problems still remain and need further studies. These problems include: 1. Throughout this thesis, we modeled the permselective membrane by a homogeneous surface which is impermeable to co-ions and contains a uniform counter-ion concentration. Using this membrane model, the computational mesh only requires refinement in the normal direction of membrane surface; hence, the simulation computational cost can be saved. In fact, the membrane is a heterogeneous structure in which counter-ions distribute inconstantly. To achieve a more realistic simulation, a recommendation is to model the permselective membrane by a heterogeneous surface which is composed of ion-exchange spots embedded in a nonconducting material matrix. The difference in electrical properties of the spots and nonconducting material causes an unsmooth 139 electric potential and ion concentrations at the contacting surface between the materials. Thus, the computational mesh must be refined at that surface to capture rapid changes of the electric potential and ion concentrations. 2. Actual permselective membranes are nano-porous structures. The most accurate simulation for the electrokinetic flow near permselective membrane is to model the permselective membrane directly by a series of nano-pores. Hence, future work should be done in examining the existence of electroconvective instability over such kind of membrane model. 3. The problems we have studied in this thesis are in two-dimensional space. The third dimension was assumed to be infinitely long. In experimental devices, the third dimension ranges from tens of µm to several mms. Recently, some experiments have shown that the ion concentration polarization developed near permselective membrane is also affected by geometrical confinement [61]. Therefore, the electrokinetic flow in the third dimension may affect significantly the conductance of the membrane. Hence, a recommendation for future work is to include the third dimension in the simulation. 4. Nano-slit and series of nanochannels share the permselectivity with the membrane. A difference between them is flow of fluid through the nanoslit and series of nanochannels. Therefore, studying effect of the flow on the electroconvective instability occurring near the nano-slit and series of nanochannels is another avenue for future work to obtain a more complete understanding of the transport phenomena near permselective surfaces. 5. We developed numerical methods to solve for the coupled Poisson-NernstPlanck-Navier-Stokes equations. The methods are able to solve for threedimensional meshes, and to execute on parallel computer systems. How- 140 ever, as it has been shown in chapter 4, the Krylov-subspace method used to solve in parallel the system of linear equations is inefficient for a large number of partitions. The parallel solver becomes inefficient when the number of partitions is large. Therefore, further studies should be focused on improving the efficiency of the parallel solver. A recommendation for improvement of the linear system solver is to develop a parallel solver based on the multigrid method. 6. The rapid change of ion concentrations and electric potential near membranes surfaces requires computational mesh to be strongly refined near the surfaces. Therefore, another recommendation for future studies is to develop an algorithm which can estimate the error over the mesh and automatically refines the control volumes which have large error. 141 Bibliography [1] R. Kwak and J. Han, “Microscale electrodialysis: Concentration profiling and vortex visualization,” Desalination (Accepted), 2012. [2] S. R. Mathur and J. Y. Murthy, “A multigrid method for the poissonnernst-planck equations,” International Journal of Heat and Mass Transfers, vol. 52, no. 131, pp. 4031–4039, 2009. [3] H. Daiguji, P. Yang, A. Szeri, and A. Majumdar, “Ion transport in nanofluidic channels,” Nanoletters, vol. 4, no. 1, pp. 137–142, 2009. [4] Y. Wang, K. Pant, Z. Chen, G. Wang, W. F. Diffey, P. Ashley, and S. Sundaram, “Numerical analysis of electrokinetic transport in micronanofluidic interconnect preconcentrator in hydrodynamic flow,” Microfluid Nanofluid, vol. 7, pp. 683–696, 2009. [5] R. F. Probstein, Physicochemical hydrodynamics: An introduction. Butterworths, 1989. [6] S. J. Kim, S. H. Ko, K. H. Kang, and J. Han, “Direct seawater desalination by ion concentration polarization,” Nature Nanotechnology, vol. 5, pp. 297–301, 2010. [7] S. S. Dukhin, “Electrokinetic phenomena of the second kind and their applications,” Advanced Colloid Interface Science, vol. 35, pp. 173–196, 1991. 142 [8] S. S. Dukhin, N. A. Mishchuk, and P. B. Takhistov, “Electroosmosis of the second kind and unrestricted current increase in a mixed ionite monolayer,” Kolloidn, vol. 51, pp. 616–618, 1989. [9] I. Rubinstein and B. Zaltzman, “Electroosmotic slip of the second kind and instability in concentration polarization at electrodialysis membranes,” Mathematical Models and Methods in Applied Sciences, 1999. [10] L. F. Cheow, S. H. Ko, S. J. Kim, K. H. Kang, and J. Han, “Increase of sensitivity of elisa using multiplexed electrokinetic preconcentrator,” Analytical Chemistry, vol. 82, no. 8, pp. 3383–3388, 2010. [11] V. Liu, Y. A. Song, and J. Han, “Capillary-valve-based fabrication of ionselective membrane junction for electrokinetic sample preconcentration in pdms chip,” Lab on a Chip, vol. 10, pp. 1485–1490, 2010. [12] S. J. Kim, Y. C. Wang, J. H. Lee, H. Jang, and J. Han, “Concentration polarization and nonlinear electrokinetic flow near a nanofluidic channel,” Physical Review Letters, vol. 99, no. 044501, 2007. [13] I. Rubinstein and L. Shtilman, “Voltage against current curves of cation exchange membranes,” Journal of the Chemical Society and Faraday Trans. II, vol. 75, pp. 231–246, 1979. [14] N. A. Mishchuk, T. Heldal, T. Volden, J. Auerswald, and H. Knapp, “Micropump based on electroosmosis of the second kind,” Electrophoresis, vol. 30, pp. 3499–3506, 2009. [15] D. J. Laser and J. G. Santiago, “A review of micropumps,” Journal of Micromechanics and Microengineering, vol. 14, no. 6, pp. R35–R64, 2004. [16] M. U. Kopp, H. J. Crabtree, and A. Manz, “Developments in technology and applications of microsystems,” Current Opinion in Chemical Biology, vol. 1, no. 3, pp. 410–416, 1997. 143 [17] I. Rubinstein and B. Zaltzman, “Electroosmotically induced convection at a permselective membrane,” Physical Review E, vol. 62, no. 2, pp. 2238– 2251, 2000. [18] V. G. Levich, Physicochemical hydrodynamics. Prentice-Hall, 1962. [19] G. Yossifon, P. Mushenheim, Y. C. Chang, and H. C. Chang, “Eliminating the limiting-current phenomenon by geometric field focusing into nanopores and nanoslots,” Physical Review E, vol. 81, no. 046301, 2010. [20] G. Yossifon, P. Mushenheim, Y. C. Chang, and H. C. Chang, “Nonlinear current-voltage characteristics of nanochannels,” Physical Review E, vol. 79, no. 046305, 2009. [21] F. Maletzki, H. W. Rosler, and E. Staude, “Ion transfer across electrodialysis membranes in the overlimiting current range: stationary voltage current characteristics and current noise power spectra under different conditions of free convection,” Journal of Membrane Science, vol. 71, pp. 105–115, 1992. [22] G. Yossifon and H. C. Chang, “Selection of nonequilibrium overlimiting currents: Universal depletion layer formation dynamics and vortex instability,” Physical Review Letters, vol. 101, pp. 254501–4, 2008. [23] S. M. Rubinstein, G. Manukyan, A. Staicu, I. Rubinstein, B. Zaltzman, R. G. H. Lammertink, F. Mugele, and M. Wessling, “Direct observation of a nonequilibrium electroosmotic instability,” Physical Review Letters, vol. 101, pp. 236101–4, 2008. [24] R. J. Yang, L. M. Fu, and C. C. Hwang, “Electroosmotic entry flow in a microchannel,” Journal of Colloid and Interface Science, vol. 244, pp. 173–179, 2001. [25] M. Kato, “Numerical analysis of the nernstˆaplanckˆapoisson system,” Journal Theoretical Biology, vol. 177, pp. 299–304, 1995. 144 [26] J. Hsu, Y. Kuo, and S. Tseng, “Dynamic interactions of two electrical double layers,” Journal Colloid Interface Science, vol. 195, pp. 388–394, 1997. [27] E. Samson, J. March, , and K. A. Snyder, “Calculation of ionic diffusion coefficients on the basis of migration test results,” Materials and Structures, vol. 36, pp. 156–165, 2003. [28] E. Samson, J. March, J. L. Robert, and J. P. Bournazel, “Modelling ion diffusion mechanisms in porous media,” International Journal for Numerical Methods in Engineering, vol. 46, no. 12, pp. 2043–2060, 1999. [29] D. L. Scharfetter and H. K. Gummel, “Large-signal analysis of a silicon read diode oscillator,” IEEE Trans. Electron Devices, vol. 16, pp. 64–77, 1969. [30] P. A. Farrel and E. C. Gartland, “On the schargetter-gummel discretization for drift-diffusion continuity equations,” Computational methods for boundary and interior layers in several dimensions, pp. 51–79. [31] E. A. Demekhin, V. S. Shelistov, and S. V. Pollyanskikh, “Linear and nonlinear evolution and diffusion layer selection in electrokinetic instability,” Physical Review E, vol. 84, no. 036318, 2011. [32] J. D. Anderson, Computational fluid dynamics a basic with applications. McGraw-Hill, 1995. [33] J. H. Feriger and M. Peric, Computational methods for fluid dynamics. Springer, 2001. [34] M. Darwish, J. Sraij, and F. Moufalled, “A coupled finite volume solver for the solution of incompressible flow on unstructured grid,” Journal of Computational Physics, vol. 288, pp. 180–201, 2009. 145 [35] C. M. Rhie and W. L. Chow, “A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation,” AIAA Journal, vol. 21, pp. 1525–1532, 1983. [36] X. Jin, S. Joseph, E. N. Gatimu, P. W. Bohn, and N. R. Aluru, “Induced electrokinetic transport in micro-nanofluidic interconnect devices,” Langmuir, vol. 23, no. 1, pp. 13209–13222, 2007. [37] A. Prhol and M. Schmuck, “Convergent finite element discretization of the navier-stokes-nernst-planck-poisson system,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 44, no. 03, pp. 531–571, 2010. [38] D. C. Grahame, “The electrical double layer and the theory of electrocapillarity,” Chemical Reviews, vol. 41, p. 441ˆa501, 1947. [39] J. E. Dennis and R. B. Schnabel, Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall, Inc. , Englewood Cliffs, NJ, 1983. [40] M. Pernices and H. F. Walker, “A newton iterative solver for nonlinear systems,” Journal on Scientific and Statistical Computing, vol. 19, pp. 302–318, 1998. [41] H. Jasak, Error analysis and estimation for the finite volume method with applications to fluid flows. Phd thesis, Imperial College of Science, Technology and Medicine, London. [42] S. V. Patankar, Numerical heat transfer and fluid flow. McGraw-Hill, 1980. [43] D. Brennan, The numerical simulation of two-phase flows in settling tanks. Phd thesis, Imperial College of Science, Technology and Medicine, London. 146 [44] C. Woodford and C. Phillips, Numerical methods with worked examples. Springer, 1997. [45] J. C. Meza, “Newton’s method,” Lawrence Berkeley National Laboratory Berkeley, California 94720. [46] G. Karypis and V. Kumar, “Multilevel k-way partitioning scheme for irregular graphs,” Journal of Parallel and Distributed Computing, vol. 48, no. 1, pp. 100–118, 1998. [47] G. Karypis and V. Kumar, “Metis: A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices, version 4. 0. technical report, university of minnentosa, department of computer science and engineering,” 1998. [48] D. Fritzsche, A. Frommer, and D. B. Szyld, “Extensions of certain graphbased algorithms for preconditioning,” 2007. [49] P. Heres and W. Schilders, “Krylov subspace methods in the electronic industry,” in Proceedings 13th European Conference on Mathematics for Industry, Eindhoven, The Netherlands, June 21-25, 2004. [50] Y. Saad, Iterative method for sparse linear system (Second edition). Society for Industrial and Applied Mathematics, 2003. [51] S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, “Efficienct management of parallelism in object oriented numerical software libraries,” in Modern Software Tools in Scientific Computing (E. Arge, A. M. Bruaset, and H. P. Langtangen, eds.), p. 163ˆa202, Birkhauser Press, 1997. [52] S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, “Petsc home page.” http://www. mcs. anl. gov/petsc, 2008. [53] D. C. Grahame, “Diffuse double layer theory for electrolytes of unsymmetrical valence types,” Journal Chemical Physic, vol. 21, no. 6, pp. 1054– 1060, 1953. 147 [54] P. Dutta and A. Beskok, “Analytical solution of combined electroosmotic/pressure driven flows in two-dimensional straight channels: Finite debye layer effects,” Analytical Chemistry, vol. 73, pp. 1979–1986, 2001. [55] R. J. Hunter, Zeta potential in colloid science: Principle and Applications. Academic Press: New York, 1981. [56] J. H. Masliyah and S. Bhattacharjee, Electrokinetic and colloid transport phenomena. Willey-Interscience, Hoboken, New Jersey, 2006. [57] I. Rubinstein, E. Staude, and O. Kedem, “Electroconvection at an electrically inhomogeneous permselective interior,” Desalination, vol. 69, pp. 101–114, 1988. [58] I. Rubinstein, “Electroconvection at an electrically inhomogeneous permselective interior,” Physic of Fluid, vol. 3, no. 10, pp. 2301–2309, 1991. [59] T. Pundik, I. Rubinstein, and B. Zaltzman, “Bulk electroconvection in electrolyte,” Physical Review E, vol. 72, no. 061502, 2005. [60] V. V. Nikonenko, N. D. Pismenskaya, E. I. Belova, P. Sistat, P. Huguet, G. Pourcelly, and C. Larchet, “Intensive current transfer in membrane systems: modelling, mechanisms and application in electrodialysis.,” Advances in Colloid and Interface Science, vol. 160, no. 1-2, pp. 101–123, 2010. [61] E. V. Dydek, B. Zaltzman, I. Rubinstein, D. S. Deng, A. Mani, and M. Z. Bazant, “Overlimiting current in a microchannel,” Physical Review Letters, vol. 107, no. 118301, 2011. [62] S. P. Sutera and R. Skalak, “The history of poiseuille’s law,” Annual Review of Fluid Mechanics, pp. 1–19, 1993. [63] I. Rubinstein, B. Zaltzman, J. Pretz, and C. Linder, “Experimental verification of the electroosmotic mechanism of overlimiting conductance 148 through a cation exchange electrodialysis membrane,” Russian Journal of Electrochemical, vol. 38, no. 8, pp. 853–863, 2002. [64] S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, “Petsc users manual,” Tech. Rep. ANL-95/11 - Revision 3. 2, Argonne National Laboratory, 2011. [65] C. Geuzaine and J. F. Remacle, “Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities,” International Journal for Numerical Methods in Engineering, vol. 79, no. 11, pp. 1309–1331, 2009. [66] L. H. Olesen, AC electrokinetic micropumps. Phd dissertation, Technical University of Denmark, Department of Micro and Nanotechnology. [67] K. T. Chu, Asymtotic analysis of extreme electrochemical transport. Phd dissertation, Massachusetts Institute of Technology, Cambridge. [68] O. Ubbink, Numerical prediction of two fluid systems with sharp interfaces. Phd thesis, Imperial College of Science, Technology and Medicine, London. [69] H. C. Chang and G. Yossifon, “Understanding electrokinetic at the nanoscale: a perspective,” Biomicrofluidics, vol. 3, no. 012001, 2009. [70] H. C. Chang, G. Yossifon, and E. A. Demekhin, “Nanoscale electrokinetics and microvortices: How microhydrodynamics affects nanofluidic ion flux,” Annual Review of Fluid Mechanics, vol. 44, pp. 401–26, 2012. [71] J. M. Acosta, Numerical algorithm for three dimensional computational fluid dynamic problems. Phd thesis, Universidad Polit Icnica de Catalunya, Terrasa. 149 [...]... local electric field, and the resulting fluid flow near the permselective membrane, which is a numerically expensive, coupled, and multiscale problem Finding a numerical solution for such problem is a challenging task but desirable to gain understanding of the electrokinetic phenomena occurring near permselective membranes 1.3 Numerical solution for nonlinear electrokinetic flow In an electrochemical system,... the membrane We also investigate the effect of the instability flow on the ion transport To obtain a more complete understanding of the nonlinear electrokinetic flow near the permselecive membrane, in chapter 6, we study the electrokinetic flow in an electrodialysis cell where the permselective membranes are interfaced with a pressure-driven electrolyte solution We examine effect of the shear flow on the electrokinetic. .. Conduct direct numerical simulation for the electrokinetic flow developed near a permselective membrane located in a quiescent electrolyte solution Investigate the electroconvetive instability in the system, and explain the mechanism for overlimiting current passing through the permselective membrane • Conduct direct numerical simulation to investigate the nonlinear electrokinetic flow in electrodialysis system... systems, this assumption may not be satisfied in practice To accurately analyze such systems, it is necessary to solve the full nonlinear electrokinetic model instead of the simplified models 1.2 Nonlinear electrokinetic flow Permselective materials such as nanochannels and nanoporous membrane consist of many nano-size pores The nano-size leads to an overlapping of the electric double layers in the pores Consequently,... flow on the electrokinetic flow developed near the membranes We will show agreements between our simulation results and the experimental results Based on the agreements, we suggest a mechanism for overlimiting conductance phenomenon in the electrodialysis cell Finally, in chapter 7, we conclude with remarks on the nonlinear electrokinetic flow near permselective membrane and its effect on the ion transport... [5], followed by the classical Helmholtz-Smoluchowski formula for the electroosmotic slip velocity [5], the nonlinear electroosmotic slip velocity for a curved, permselective granule suggested by Dukhin [7, 8], and the nonlinear electroosmotic slip velocity for arbitrary geometry permselective membranes suggested by Rubinstein and Zaltzman [9] A common assumption used in these studies is the electroneutrality... system of linear equations resulting from the discretization process 9 in the Newton-Raphson method is accurately solved by using the Portable, Extensible Toolkit for Scientific Computation (PETSc) library package By conducting direct numerical simulation for the full problem, the results obtained in this present study may provide a clearer picture of nonlinear electrokinetic flow near permselective membrane. .. to be conducted through the membrane The counter ions accumulated in the membrane induce a significant change in ion concentration near the membrane Such change was numerically studied by Shtilman and Rubinstein [13] When the voltage applied to a permselective membrane exceeds a critical value, the electric double layer at the membrane surface becomes unstable exhibiting via the development of an extended... of nanoslit and permselective membrane The mechanism relating fluid flow and ionic current was suggested by Dukhin [7, 8] to be electroconvection Later, Rubinstein and Zaltzman demonstrated that, at significantly high voltage, electroosmosis instability develops near permselective membrane surface and generates a vortical flow The flow 5 mixes the diffusion layer and carries more ions to the membrane to produce... also discussed in this chapter In chapter 5, we study the electrokinetic flow developed near a permselective membrane interface with a quiescent electrolyte solution We examine the development of electrokinetic flow, and investigate the effect of the electroconvective flow on ion distribution in the system and the electric current passing through the membrane We provide details in the formation and development . understanding of the nonlinear electrokinetic flow near permselective membrane, we conduct direct numerical simulations for the electrokinetic flow in an electrodialysis cell of which permselective mem- vii branes. NONLINEAR ELECTROKINETIC FLOW NEAR PERMSELECTIVE MEMBRANE PHAM VAN SANG (B.Eng. (Hons), HUST) A THESIS SUBMITTED FOR THE. . . 139 Bibliography 142 vi Thesis Summary In this thesis, we study the nonlinear electrokinetic flow near permselective membrane. We solve the full, coupled Poisson-Nernst-Planck-Navier-Stokes equations