Validity of Lorentz–Lorenz equation in porosimetry studies Daniel Schwarz ⁎, Herbert Wormeester, Bene Poelsema IMPACT Institute, Universiteit Twente, Postbus 217, 7500 AE Enschede, The Netherlands abstract article info Available online 16 December 2010 Keywords: Ellipsometric porosimetry Effective medium approximation Ellipsometric porosimetry is a valuable tool to determine gas loading of porous materials. Usually the Lorentz– Lorenz effective medium theory is used, instead of the more accurate Bruggeman theory. In contrast to Lorentz–Lorenz, the Bruggeman model requires detailed knowledge on the constituents of the porous material. A first order perturbation of both effective medium approximations is used to analyze the difference between these models. Similar results are only found for materials with 70% porosity. Below 50% porosity, the gas load is underestimated with the Lorentz–Lorenz model. For porous silica and alumina with 50% porosity, the use of Lorentz–Lorenz leads to a systematic error of 18% of the load capacity. © 2010 Elsevier B.V. All rights reserved. 1. Introduction In ellipsometric porosimetry the loading of a porous host material with a guest material (a gas or liquid) is studied, for example the CO 2 sorption of a silica membrane [1]. In this technique the change of the dielectric function upon loading is measured. From this change, the amount of guest molecules in the host material or the material's porosity can be calculated. To do these calculations, the effect of the relative presence of the host and guest material dielectric function has to be evaluated from an effective medium approximation (EMA). Usually a Lorentz–Lorenz (also referred to as Clausius–Mossotti) approach is used, instead of a generally more accurate Bruggeman approach [2–4]. The reason for this lies in the often unknown dielectric properties of the constituents of the porous material. For example, porous silica can often not be represented as a mixture of silicon oxide and voids due to the presence of many hydrogen bonds. The SiOH material leads to a higher dielectric function than quartz [1]. If the guest material has a small dielectric constant, which is usually the case for a dilute gas, the loading of the host material will result in a small change in the total dielectric function. This means that it's expected to be possible to describe the change by a first order perturbation. This linearization is done for both effective medium theories in this article and the result is compared. A significant deviation between the two is found for low porous materials. 2. Effective medium approaches 2.1. Lorentz–Lorenz The Lorentz–Lorenz equation is derived from the Clausius– Mossotti relation, which relates the dielectric constant of spherical particles with their density N and their polarizability α [8,9]. In SI units it is: 〈ε〉-1 〈ε〉 +2 = Nα 3ε 0 ð1Þ For a mixture of several materials with polarizability α i and density N i , the contributions of the individual components are counted up to give the effective dielectric function 〈ε〉 of the mixture. This approach was originally derived by Lorentz and Lorenz to describe the optical properties of a gas, a case in which the single molecules are well separated, and do not interact. 〈ε〉-1 〈ε〉 +2 = 1 3ε 0 ∑ i N i α i ð2Þ A linearization for a dilute gas (ε g ≈1) simplifies the expression to [8]: ε g =1+ N g α g ε 0 ð3Þ For a porous material with dielectric function ε m and porosity f, the Lorentz–Lorenz is often used to describe the change in dielectric constant through the insertion of a gas with polarizability α g into the pores. The gas is assumed to fill all pore volume homogeneously, and its density N g is increased upon loading. Eq. (2) can be adapted for the two components, the solid material, and the added gas to give the Lorentz– Lorenz equation [5–7]: 〈ε〉-1 〈ε〉 +2 = 1-f ð Þ ε m -1 ε m +2 + f ε g -1 ε g +2 ð4Þ Thin Solid Films 519 (2011) 2994–2997 ⁎ Corresponding author. Tel.: +31 534893109; fax: +31 534891101. E-mail address: d.schwarz@tnw.utwente.nl (D. Schwarz). 0040-6090/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2010.12.053 Contents lists available at ScienceDirect Thin Solid Films journal homepage: www.elsevier.com/locate/tsf