ARTICLE IN PRESS YJCIS:14167 Please cite this article in press as: C. Chassagne, D. Bedeaux, J. Colloid Interface Sci. (2008), doi:10.1016/j.jcis.2008.06.055 JID:YJCIS AID:14167 /FLA [m5G; v 1.49; Prn:1/08/2008; 15:14] P.1 (1-14) Journal of Colloid and Interface Science ••• (••••) •••–••• Contents lists available at ScienceDirect Journal of Colloid and Interface Science www.elsevier.com/locate/jcis The dielectric response of a colloidal spheroid C. Chassagne a,∗ , D. Bedeaux b a Department of Environmental Fluid Mechanics, TU Delft, Stevinweg 1, 2628 CN Delft, The Netherlands b Department of Physical Chemistry, NTNU, 7491 Trondheim, Norway article info abstract Article history: Received 25 January 2008 Accepted 28 June 2008 Keywords: Colloidal spheroids Dielectric spectroscopy Complex conductivity Electrokinetics Dipolar coefficient Double layer polarization Zeta potential In this article, we present a theory for the dielectric behavior of a colloidal spheroid, based on an improved version of a previously published analytical theory [C. Chassagne, D. Bedeaux, G.J.M. Koper, Physica A 317 (2003) 321–344]. The theory gives the dipolar coefficient of a dielectric spheroid in an electrolyte solution subjected to an oscillating electric field. In the special case of the sphere, this theory is shown to agree rather satisfactorily with the numerical solutions obtained by a code based on DeLacey and White’s [E.H.B. DeLacey, L.R. White, J. Chem. Soc. Faraday Trans. 2 77 (1981) 2007] for all zeta potentials, frequencies and κ a  1 where κ is the inverse of the Debye length and a is the radius of the sphere. Using the form of the analytical solution for a sphere we were able to derive a formula for the dipolar coefficient of a spheroid for all zeta potentials, frequencies and κ a  1. The expression we find is simpler and has a more general validity than the analytical expression proposed by O’Brien and Ward [R.W. O’Brien, D.N. Ward, J. Colloid Interface Sci. 121 (1988) 402] which is valid for κ a ≫ 1 and zero frequency.  2008 Elsevier Inc. All rights reserved. 1. Introduction and main result The equations governing the behavior of a dielectric sphere in an oscillating electric field have been solved numerically. The most widely used numerical code is due to DeLacey and White [1], whose code was later extended by Mangelsdorf and White [2]. In certain cases (for thick double layers or/and for high frequen- cies), the numerical analysis can be time consuming or sometimes impossible, but some recent studies are making progress on this topic [3]. The analytical formula we propose for the dipolar coefficient arising from a dielectric sphere in an electrolyte solution gives fast solutions for the whole range of conditions investigated. We found that this analytical formula agrees with the numerical simulations for all frequencies, zeta potentials and κ a  1 where κ is the in- verse of the Debye length and a is the radius of the sphere. The form of the analytical formula makes it possible to compare the weight of the different contributions and therefore easily predict the particle’s response upon a change in parameters. Since elec- trokinetic measurements are often used as a quality-control tool [4–6], ready-to-use formulae are useful and we will give simple expressions that correctly approximate the numerical simulations in the cases generally found in experiments. * Corresponding author. E-mail address: c.chassagne@tudelft.nl (C. Chassagne). More importantly, we will show that the form of the analyti- cal formula gives the possibility to extent our results to spheroidal particles. We define 2a p as the length along the axis of revolution of the spheroid and 2a n the maximum cross-sectional diameter normal to the axis of symmetry. We also define a = min[a p , a n ]. For a sphere we get a = a p = a n . The analytical formula we pro- pose for the frequency-dependent dipolar coefficient for a dielec- tric spheroid in an electrolyte solution in an oscillating (AC) elec- tric field is valid for all frequencies, zeta potentials and κ a  1. So far, the only analytical or numerical expressions available for the dipolar coefficients of spheroids are for zero frequency (DC electric fields) and κ a ≫ 1 [4,7–9]. We found the dipolar coefficient to be in the general case: β i = ( ˜ K 2 − ˜ K 1 + 3(1 − L i )  ˜ K ‖ + ˜ K U + ˜ K extra ‖  + 3L i  ˜ K ⊥ + ˜ K extra ⊥ ) ( 3 ˜ K 1 + 3L i ( ˜ K 2 − ˜ K 1 ) + 9L i (1 − L i )  ˜ K ‖ (a/r 0 ) 3 + ˜ K U (a/r 1 ) 3 + ˜ K extra ‖ − ˜ K ⊥ − ˜ K extra ⊥ ) , (1) where i = n and p indicate the direction normal and along the axis of symmetry of the spheroid, respectively. ˜ K 1 , ˜ K 2 , ˜ K ‖ , ˜ K ⊥ , ˜ K U are given by Eqs. (42), (48), (65), r 0 and r 1 are given by Eqs. (57), (68), L i are the depolarization factors given by Eqs. (A.6)–(A.8) using m = a p /a n in the case of spheroids. The formulae for L i can also be written for a more general type of ellipsoid, using i = p, n 1 and n 2 as the principal axes of this ellipsoid. The appropriate formulae for L i are given for example in [10]. However, the most encountered case in the experiments is the one of spheroids (n 1 = n 2 = n). 0021-9797/$ – see front matter  2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2008.06.055