Electroosmotic Flow in a Nanopore

A nanopore is simply an orifice in an impermeable membrane separating two reservoirs filled with an electrolyte. The prefix 'nano' usually signifies that the diameter of the orifice is in the range of one to a few hundred nanometers. A nanopore is perhaps one of the simplest "molecular machines'' that determine cellular function. Protein nanopores embedded in lipid membranes mediate many activities of living cells. Examples include signal transduction in nerve cells, the controlled transport of water in and out of cells, movement of macromolecules such as DNA, RNA and proteins between the cytoplasm and various intracellular bodies (nucleus, mitochondria) etc. Recently, "biomimetic" nanopores have been utilized in single molecule detection and manipulation. One of the technologies being actively pursued in this area is ultra-rapid DNA sequencing using nanopores. Furthermore, the properties of single nanopores underlie the behavior of membranes. Membrane science is a subject unto itself and has applications in living systems (e.g. kidney dialysis membranes) as well as various technologies ranging from water purification to fuel cells.;The walls of the nanopore as well as the surface of the supporting membrane under most circumstances have a static charge due to adsorption and dissociation of chemical groups in the presence of the surrounding electrolyte. The copious supply of ions of either sign in the surrounding electrolyte thus creates a thin "Debye Layer'' (DL) or "Electric Double Layer'' (EDL) of mobile counterions that shield the free charges on the solid fluid interface. Electrical forces originating in this EDL determine fluid flow through the pore when pressure or voltage differences are applied across it. The resulting flow field is very important in determining the transport of small molecules or polymers through the pore, especially when the particles and polymers are uncharged. This dissertation addresses the problem of electroosmotic pumping of fluid through a nanopore that traverses a charged, dielectric membrane. The continuum equations of hydrodynamics and electrostatics are presumed to apply. Such an assumption would be reasonable for pores greater than about 5 nm in size which is the case for most synthetic membranes. On the other hand, nanopores formed by proteins on lipid membranes are much smaller---a few water molecules across. Our basic approach is not likely to be accurate in such cases except perhaps for providing qualitative answers.;The strength and structure of the flow depends on several parameters including the geometry of the nanopore, external voltage, concentration of the electrolyte solution and the properties of the membrane material, such as polarity and surface charge density. Under the weak field assumption, perturbation methods can be utilized for theoretical modeling of the flow field in the linear regime. The reciprocal theorem gives the flow rate generated by an applied voltage in the form of an integral over the fluid domain. The integral involves (1) the equilibrium double layer charge density (rho0) (2) the flow field in the given geometry with no charge effects but a unit applied pressure drop (G) (3) the harmonic function corresponding to the solution of the Ohmic problem where the electrolyte is regarded as a homogeneous conductor (chi). For a circular hole in a membrane of zero thickness (1)-(3) are known, and therefore, the integral may be evaluated by quadrature. When the membrane thickness is non-zero, the functions (1)-(3) do not have known analytical forms. An approximate evaluation of the flow rate is however possible by "patching'' the zero-thickness solution with the analytical solution of electroosmotic flow in a cylindrical channel and exploiting the constraints of continuity of current and fluid flux. The approach works well except when the EDL thickness is large compared to the pore size or membrane thickness. The cause of the failure is attributed to the "spilling'' of charge from the pore and a method for correcting the theory to include its effect is demonstrated. When the intrinsic surface charges on the membrane are neutralized, eddies with length scale of a few nanometers could arise within the pore due to induced charge electroosmosis. The flow strength in this case is found to be proportional to square of the applied voltage. The shape of the eddies is found to be dependent on the geometry of the pore.;Numerical solvers to solve the full governing system of equations are developed by utilizing the finite volume solver modules available in the OpenFOAM software package. The solvers are able to compute the coupled electrostatics, ionic transport and Stokes flow equations. Numerical results agree with theoretical models in the corresponding parameter regimes.