Transport Phenomena in Nanofluidic Channels Hirofumi Daiguji, Peidong Yang* † , Andrew J. Szeri** and Arun Majumdar** † Institute of Environmental Studies, The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Tel +81-3-5841-8587, Fax +81-3-3818-0835, E-mail daiguji@k.u-tokyo.ac.jp *Department of Chemistry, University of California, Berkeley, CA 94720-1460 **Department of Mechanical Engineering, University of California, Berkeley, CA 94720-1740 † Materials Science Division, Lawrence Berkeley National Laboratory Abstract When the Debye length is on the order of or larger than the height of a nanofluidic channel containing surface charge, a unipolar solution of counterions is generated within the channel and the coions are electrostatically repelled to maintain electrical neutrality. By locally modifying the surface charge density through a gate electrode, the ion concentration can be depleted under the gate and the ionic current can be significantly suppressed. It is proposed that this could form the basis of a unipolar ionic field-effect transistor. A pressure-gradient driven flow under such conditions can be used for ion separation, which forms the basis for electro-chemo-mechanical energy conversion. The current–potential (I-φ) characteristics of such a battery were calculated using continuum dynamics. When the Debye length of a solution is about half of the channel height, the efficiency is maximized. Keywords: nanofluidics, electrokinetic phenomena, electrical double layer, Debye length, unipolar solution 1 INTRODUCTION Ion transport in nanoscale channels has recently received increasing attention. Much of that has resulted from experiments that report modulation of ion transport through the protein ion channel, α-hemolysin, due to passage of single biomolecules of DNA or proteins [1]. This has prompted research towards fabricating synthetic nanopores out of inorganic materials and studying biomolecular transport through them [2]. Recently, the synthesis of arrays of silica nanotubes with internal diameters in the range of 5-100 nm and with lengths 1-20 µm was reported [3]. These tubes could potentially allow new devices to control the transport of ions and biomoleucles. When the tube diameter is smaller than the Debye length, which characterizes the size of the electrical double layer, a unipolar solution of counterions is created within the nanotube and the coions are electrostatically repelled. We proposed two different types of devices to use this unipolar nature of solution, i.e. ‘nanofluidic transistor’ [4] and ‘nanofluidic battery’ [5]. When the electric potential bias is applied at two ends of a nanotube, ionic current is generated. By locally modifying the surface charge density through a gate electrode, the concentration of counterions can be depleted under the gate and the ionic current can be significantly suppressed. This could form the basis of a unipolar ionic field-effect transistor. By applying the pressure bias instead of electric potential bias, a streaming current and a potential are produced. This could form the basis of an electro-chemo-mechanical battery. In this study, transport phenomena in nanofluidic channels were investigated and the performance characteristics of these devices were evaluated. 2 GOVERNING EQUATIONS In the analysis of microchannel flow, the Poisson-Boltzmann equation is often assumed for the potential of an electrical double layer [6]. This is valid when the electrical double layers of two adjacent walls do not overlap. For overlapping double layers, the Poisson-Nernst-Planck (PNP) equations and the Navier-Stokes (NS) equations are used to calculate the ionic current in nanofluidic channels. The governing equations are as follows: ∑ − = a a a en z ε ε φ 0 2 1 ∇ , (1) ( ) 0 = + ⋅ a a n J u ∇ , (2) 0 = ⋅ u ∇ , (3)               − + − = ⋅ ∑ φ µ ρ ∇ ∇ ∇ ∇ a a a en z p u u u 2 1 . (4) The electrostatic potential φ is calculated with the Poisson equation (eq 1), where ε 0 is the permittivity of vacuum, ε is the dielectric constant of medium, n a is the concentration of ions of species a, and z a e is their charge. The Nernst-Planck equation for ion species a can be written as eq 2, where J a is the particle flux due to concentration gradient and electric potential gradient, which is given by 14