Elektrotehniški vestnik 75(1): 82-84, 2008 Electro technical Review, Ljubljana, Slovenija Linearized Poisson Boltzmann theory in cylindrical geometry Klemen Bohinca'6, Aleš Igliča, Tomaž Slivnika a Laboratory of Physics, Faculty of Electrical Engineering, Tržaška 25, SI-1000 Ljubljana, Slovenia b College for Health Studies, University of Ljubljana, Poljanska cesta 26a, 1000 Ljubljana, Slovenija E-pošta: klemen. bohinc@fe.uni-lj.si Abstract. The linearized Poisson-Boltzmann theory was applied to the system of a charged cylindrical surface in contact with an electrolyte solution composed of monovalent counter- and co-ions. An analytical solution for the spatial dependence of the electric potential and the concentration of counter and co-ions inside the charged tube was obtained. Key words: electric potential, cylindrical geometry © © © © © © © © © © © © ® © © © © © © © © © © © © © © 0 © © © © © © © 1 Introduction In some biological systems, the walls of membrane nanotubes are charged and in contact with the electrolyte solution. Due to the electric charge of the nanotube wall, counterions of the electrolyte are accumulated near the charged wall while coions of the electrolyte are depleted from the region near the wall. At inner (Fig. 1) and outer (Fig. 2) surfaces of the charged tube the electric double layers of cylindrical geometry are therefore formed [1; 2]. Figure 1. Schematic presentation of a cylindrical electric double layer inside the tube with radius n and surface charge density Oi. The counterions are accumulated near the charged surface while the coions are depleted from the charged surface. Received 19 December 2007 Accepted 2 April 2008 Figure 2. Schematic presentation of a cylindrical electric double layer outside the tube with radius r0 and surface charge density o0. The counterions are accumulated near the charged surface while the coions are depleted from the charged surface. The constituents of the biological membranes may carry ionized or polar groups [2]. The ionized groups contribute to the surface charge density of the inner and outer surfaces of the membrane nanotube. The examples of such membrane molecules with charged groups are lipids, glycolipids, or proteins. The molecules which are bound to or absorbed onto the membrane surface may also contribute to the surface charge density of nanotube surfaces [3]. A widely used description of the electric double layer is given by the Poisson-Boltzmann (PB) theory [4; 5; 6] in which the ions are treated as point charges in a dielectric continuum enclosed by a uniformly charged surface. For monovalent salt, the predictions of the PB theory are found to agree well with experiments and simulations [7]. A quite useful approximation to the PB theory is the linearized PB theory (LPB) [8], where the electrostatic energy is assumed to be small compared to the thermal energy. In the present paper we consider the charged micro or nano tube in contact with an electrolyte solution composed of counterions and coions. We study the inner charged surface of the tube in a contact with the inner electrolyte solution and the outer charged surface of the tube in a contact with the outer electrolyte solution. The potential and the concentration profiles are calculated. 2 Theory We consider the charged tube in contact with a solution of symmetric monovalent electrolyte. The electric double layer inside nanotube (concave case) and outside nanotube (convex case) are studied. v\ is the radius of the inner tube, while rQ is the radius of the outer tube. The length I of the tube is assumed to be much larger than the radius. The system of the outer surface and outer electrolyte solution is electrically neutral. Also the system of the inner surface and inner electrolyte solution is electrically neutral. If the electrostatic energy is small compared to the thermal energy , \e$/kT\ << 1, where $ is the electrostatic potential, then the linearized Poisson Boltzmann (LPB) equation can be used A^ = k2^ (1) where ^ = is the reduced electrostatic potential and ft-1 = ^%nl°Na18 Debye length, e is the dielectric constant of the solution, eo is the permittivity of vacuum, T is the temperature and k is the Boltzmann constant, Na is the Avogadro number, nd is the bulk counterion concentration, eo is the elementary charge, A is the Laplace operator and r is the radial coordinate. In the case of uniformly charged and very long cylinder the three-dimensional problem reduces to one-dimensional problem, i.e. the differential equation (1) for the potential becomes Multiplying Eq. (2) by r2 and performing the derivation we get rV'(r) + rV(r) - k2 r2 V(r) = 0, (3) where the symbol ' denotes derivation with respect to the radial coordinate r. The solution of differential Eq. (2) is [9] \P(r) = AI0 (K,r)+B K0 (nr) (4) d 9(r) dr 1 = 0 (5) d 9(r) dr 1 v0eo eeokT (6) where oQ is the surface charge density of the outer surface of the tube. The first boundary condition (Eq. 5) gives A = 0, while from the second boundary condition (Eq. 6) the constant B was calculated. The electrostatic potential outside the tube is [7] 9(r) = ooe0 K0(hior) kTee0Ko ifi(ft0r0) (7) where nQ = ^d 18 the electrolyte strength far from the charged tube and K\ is the modified Bessel function of order 1. Inside the tube we consider electric double layer being in contact with the concave surface of radius rf, where the inner part of the tube is in contact with the electrolyte solution. The boundary conditions inside the tube are dfr(r) dr dfr(r) dr 1 r=0 — 0 o eeo kT (8) (9) where &i is the surface charge density of the inner surface of the tube. The first boundary condition (Eq. 5) gives B — 0, while from the second boundary condition (Eq. 6) the constant A was calculated. The electrostatic potential inside the tube becomes 9(r) = Q-je p Ip(Kjr) kTee0Ki h^n) ' (10) where Ki = , is the electrolyte strength along the axis of the tube and 1\ is the modified Bessel function of order 1. The concentration of counterions nct is given by the Boltzmann distribution ftct = nd£ (ii) where /o and K0 are the modified Bessel functions of order 0. The constants A and B are determined by boundary conditions. In a convex case the outer charged surface of the tube with the radius rQ is in a contact with the outer electrolyte solution. The boundary conditions outside the tube are where nd is the bulk concentration of counterions. In the LPB theory the exponent in Eq. (11) is expanded in electrostatic potential up to the first odrer. The concentration of counterions outside the charged tube thus becomes ct