ARTICLE IN PRESS Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287 www.elsevier.com/locate/jqsrt A Monte Carlo ray tracing study of polarized light propagation in liquid foams J.N. Swamya,Ã, Czarena Crofchecka, M. Pinar Mengu¨c- b,ÃÃ aDepartment of Biosystems and Agricultural Engineering, 128 C E Barnhart Building, University of Kentucky, Lexington, KY 40546, USA bDepartment of Mechanical Engineering, 269 Ralph G. Anderson Building, University of Kentucky, Lexington, KY 40506, USA Received 10 July 2006; accepted 28 July 2006 Abstract A Monte Carlo ray tracing scheme is used to investigate the propagation of an incident collimated beam of polarized light in liquid foams. Cellular structures like foam are expected to change the polarization characteristics due to multiple scattering events, where such changes can be used to monitor foam dynamics. A statistical model utilizing some of the recent developments in foam physics is coupled with a vector Monte Carlo scheme to compute the depolarization ratios via Stokes–Mueller formalism. For the simulations, the incident Stokes vector corresponding to horizontal linear polarization and right circular polarization are considered. It is observed that bubble size and the polydispersity parameter have a significant effect on the depolarization ratios. This is partially owing to the number of total internal reflection events in the Plateau borders. The results are discussed in terms of applicability of polarized light as a diagnostic tool for monitoring foams. r 2006 Elsevier Ltd. All rights reserved. Keywords: Polarized light scattering; Liquid foams; Bubble size; Polydispersity; Foam characterization; Foam diagnostics; Cellular structures 1. Introduction Liquid foams are random packing of bubbles in a small amount of immiscible liquid [1] and can be found in a wide variety of applications. Many of the processed foods like chocolate bars, bread, cakes, beer, whipped cream, etc. are essentially foams [1]. In contrast, specially designed detergent foams are being developed to decontaminate areas exposed to biological weapons. Owing to their low density and unique rheological properties, foams have a wide range of applications. It is widely accepted that the quality of foam products and the efficiency of processes involving foams are largely dependent on foam structure. Hence, there is a well- motivated need to develop techniques to monitor foam properties (density and consistency) during production and stabilization. ÃCorresponding author. Tel.: +1 859 257 3000; fax: +1 859 257 5671. ÃÃAlso to be corresponded. Tel.: +1 859 257 6336; fax: +1 859 257 3304. E-mail addresses: [email protected] (J.N. Swamy), [email protected] (M.P. Mengu¨c-). 0022-4073/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2006.07.022 ARTICLE IN PRESS 278 J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287 Nomenclature A plateau border area E number of edges I stokes intensity L length of edges/borders P degree of polarization PM phase matrix p polydispersity parameter Q tendency for horizontal polarization R radius S scattering matrix or minimized surface area TIR total number of total internal reflections in borders TSE total number of scattering events U tendency for +451 linear polarization Greek symbols Y scattering angle F gas or liquid fraction d turning angle of Plateau borders m moment of a distribution s standard deviation f vertical opening angle Subscripts b bubble f foam g gas l liquid p Plateau border t total Foams are continuously evolving systems far from equilibrium, which require measurement using non- intrusive diagnostic techniques. Being cellular structures, foams exhibit a highly multiple scattering behavior resulting in a familiar white appearance. This limits the utility of conventional measurement techniques like photography, video imaging, etc. Optical tomography has been used with some success to study the topology of three dimensional foams [2]. However, such techniques require complex image reconstruction algorithms, which are computationally expensive, limiting its utility as a research tool. Several studies have been conducted in the past to investigate multiple scattering of light in foams using a diffusion approximation [3–7]. Using such techniques, Durian et al. [3] studied the transient behavior of foam due to coarsening and drainage. In this method, foams are modeled as air bubbles separated by liquid films and a model for photon transport based on random walk is applied. The resulting photon transport mean free path is correlated with the foam microstructure. Thus, the validity of correlations is largely dependent on the mechanism underlying the random walk. However, it is suggested that implementing the rules from geometric optics for transmittance and reflectance of light results in a persistent random walk [6]. The premise of the current approach is that cellular structures like foam are likely to alter the polarization of the incident radiation due to successive scattering events. Thus, monitoring polarization changes in addition to the attenuated intensity can lead to additional information about foams. If the properties of the foam layer ARTICLE IN PRESS J.N. Swamy et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 277–287 279 can be related to the changes in polarization of the incident light, an intelligent diagnostic scheme can be developed to monitor foam properties. The changes in polarization due to multiple scattering events, assuming foam to be an emulsion of spherical bubbles, has been investigated earlier using a geometric optics approximation [8,9]. However, in these studies only a few or non-contact bubbles were considered and hence did not include the scattering effects due to the Plateau borders and vertices. In the current work, a similar approach is utilized with a more realistic physical foam model, where the effects of the Plateau borders are included. 2. Theoretical background and problem formulation 2.1. Overview The main goal of this study is to provide a theoretical understanding of the interaction of light with foam using a geometric optics model coupled with a statistical model for foam structure using a Monte Carlo scheme. Recent developments in foam physics and availability of tools like surface evolver have led to theoretical calculations of the geometric properties of three dimensional polydisperse foams [10,11]. This has lead to significant improvements in the overall understanding of foam structure. A temporal approach is used for ray tracing considering the computational expense for a spatial approach. In a temporal approach, the foam microstructure is generated as the photon travels through it, based on relevant statistics available from theoretical calculations. A bubble size distribution (or cell size distribution; the terms bubbles and cells are used interchangeably here) is assumed to start with; based on this the statistics for shape and structure are computed to provide for the geometrical constraints required for ray tracing simulations. The bubbles/cells are modeled as polyhedra with F faces with finite film curvature and E edges constrained by the rules of Euler and Plateau. Further, to simplify the ray tracing calculations, it is assumed that a given film has a constant mean curvature (i.e., a spherical cap) following the isotropic plateau polyhedra theory [10]. A polarized laser beam is assumed to impinge normally on a cylindrical system (containing the foam), at the circumferential surface and the scattered signal is assumed to be collected on the same plane. The medium temperature is assumed low, such that there is no radiative emission. A geometric optics approach is used instead of a more detailed physical optics analysis, given that the wavelength of light is much shorter than the bubble sizes considered (Rb 40.5 mm). The effects of dependent scattering, diffraction, and interference phenomena are neglected. The typical thickness of Plateau borders in foams varies from 0.1 to 2 mm depending on their liquid content [1]. Hence, with exception of extremely dry foams (Fgo0.02), the clearance between bubbles is at least one-half of the wavelength making the neglect of dependent scattering effects reasonable [8]. Stokes–Mueller formalism is used to track the changes in intensity and polarization via Vector Monte Carlo method presented by Vaillon et al. [12]. The elements of the scattering matrix for individual reflection/ transmission events follow Fresnel equations. 2.2. Physical model for foam Very dry foam is made up of cells that are considered to be polyhedral in shape; while very wet foams are made up of spherical bubbles. For most practical applications, foams are neither completely dry, nor completely wet. Considering the differences between the two extremes, the overall foam structure is difficult to model. However, recent developments have led to interesting geometric correlations which relate the structural parameters to the number of faces and volume [10]. Some of the numerical experiments conducted by Kraynik et al. [11] have revealed very useful statistics of random polydisperse foam, utilized here. All the topological and geometric parameters are expressed in terms of the polydispersity, p, based on the Sauter mean diameter. Fig. 1 illustrates the simplified microstructure of foam, used in this work, where three bubbles (B1–B3) come together to form a Plateau border. Fig. 1(a) illustrates