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Experimental investigation of liquid foams by polarised light scattering technique via the Mueller matrix

Highlights

PLST was used to measure the six Mueller matrix elements of SDS foam. The liquid fraction and bubble size distribution were measured individually. The effect of foam properties on Mueller matrix was more obvious for static foam. Six independent Mueller matrix elements depended on the liquid fraction except _N_ 34. The current foam only needs five independent Mueller matrix elements.

Abstract

Experiments were conducted using well-controlled bubble size-distributed foam stabilised by Sodium Dodecyl Sulphate (SDS) solution to investigate the individual effect of liquid fraction of foam on the six independent normalised Mueller matrix elements via polarised light scattering technique. The liquid fraction, bubble size distribution and the Mueller matrix elements were measured individually during a foam free drainage process. It was found that the effect of foam properties on the Mueller matrix was more obvious when the foam was in a static state. Under the circumstance that the bubble size distribution was kept almost constant during the drainage process, the six independent normalised Mueller matrix elements showed dependence on the liquid fraction except _N_ 34, whereas _N_ 22 and _N_ 33 showed the highest sensitivity. In addition, _N_ 12 and _N_ 44 exhibited reflection symmetry in the static stage during foam draining. This suggests that only five independent Mueller matrix elements are sufficient to represent the current foam because of its axisymmetry.

Introduction

Liquid foams are randomly-packed bubbles with a small amount of liquid. They have wide applications in a variety of fields such as aerated food industry, personal care products, enhanced oil recovery, fire fighting, waste water treatment, mineral flotation and various other separation processes (Prud'homme and Khan, 1996). It is widely accepted that the quality of foam products and the efficiency of these processes involving foams are closely linked to bubble size distribution and the liquid fraction which are two of the most important parameters of foam (Narsimhan and Ruckenstein, 1986, Weaire and Phelan, 1996). Therefore, in order to better understand and control foam properties, many techniques have been developed to measure these parameters (Ekserova and Krugliakov, 1998). In terms of bubble size measurement, the following methods have been employed: photographic-image analyse, photoelectric sensor probe (Du et al., 2001), optical tomography (Fetterman et al., 2000) and magnetic resonance imaging techniques (Stevenson et al., 2010). However, these methods are either invasive to the foam or expensive. As to the measurement of the liquid fraction, the traditional conductivity method (Prud'homme and Khan, 1996) and the pressure difference method (Li et al., 2011) have been widely used. In the search for a non-invasive diagnostic technique suitable for the purpose of controlling the foam in real time, the light scattering technique appears to be a suitable candidate. Conventional light scattering methods have been used in the study of foam (Durian et al., 1991a, Durian et al., 1991b, Durian et al., 1992, Miri et al., 2010, Sun and Hutzler, 2007, Uhomoibhi and Dawson, 2000). The close-packed gas bubbles with a small amount of liquid in the foam and the large mismatch of refractive-index at the numerous gas–liquid interfaces result in multiple scattering (Vera et al., 2001). These studies have focused on light scattering but without taking the changes in polarisation of the scattered light into account. In our study, we investigate the polarised light scattering technique (PLST) that encompasses changes in polarisation as it may provide additional information about the scattering medium and/or increase the accuracy of measurements. Furthermore, PLST does not require the measurement of transport mean free path in the conventional light scattering methods which need elaborate apparatus. In our previous studies (Qian, 2012a), the polarisation scattering characteristics of foam has been studied using two types of completely polarised incident light: vertical linearly polarised light and left-handed circularly polarised light. The examination on five polarisation parameters confirmed that PLST is an effective method to examine foams (Qian, 2012b). In other words, since PLST is non-intrusive, fast and economical, it is a likely candidate for the purpose of monitoring foams. Therefore, the next stage is to further investigate the Mueller matrix (or Scattering matrix) which represents the properties of the scattering medium in light scattering. Once an accurate correlation between the Mueller matrix elements and the foam properties is obtained, this method can be used to elucidate foam microstructure on the basis of experimentally measured Mueller matrix elements followed by an inverse analysis. This has the potential to provide an improved technique for controlling processes involving foams in real time. Although the polarised light scattering technique has been successfully used to characterise several scattering media (Aslan et al., 2003, Crofcheck et al., 2005, Mengüç and Manickavasagam, 1998, Saltiel et al., 2004), only limited results about its application on foam have been reported in the literature. A Monte Carlo method had been used to simulate the light scattering processes in foams (Swamy et al., 2007, Wong and Mengüç, 2002, Wong et al., 2003). The depolarisation ratio of polarised light was found to be correlated with bubble size, bubble separation distance (Wong and Mengüç, 2002), foam thickness and polydispersity (Swamy et al., 2007). The first experimental work on micro-bubble size and gas hold-up in two-phase gas–liquid columns via elliptically polarised light scattering technique was performed in 2006 (Aslan et al., 2006). Six independent Mueller matrix elements were measured as a function of the side- and back-scattering angle. Four of them showed significant changes with changes in the bubble size. The optimum angle was determined as 120°. A few years later, Swamy et al. (2009) used the polarised light scattering technique to explore the physical characteristics of shaving foam. Three Mueller matrix elements were found to be sensitive to foam age during which bubble size and liquid fraction changed. In particular, a higher sensitivity was observed at backscattering angles between 120° and 135°. It should be noted that in their study, the individual effect of one parameter (i.e. liquid fraction or bubble size distribution) was not decoupled from the other. In these studies, experiments were compared with Monte Carlo simulations (Vaillon et al., 2004) or Mie theory results. The deviations between experimental and theoretical results might lie in inappropriate foam models or the neglect of multiple light scattering events. Then Swamy et al. (2010) used the same method to attempt to monitor the performance of a foam fractionation column. Although the enrichment ratio showed proportionality with the sum of the two matrix elements (_M_ 11+_M_ 12) which was measured at backscattering angle of 125° with different gas velocities and pH values, the bubble size distribution and the liquid fraction were not measured by employing the light scattering technique. Furthermore, the foam fractionation enrichment ratio also could not be a complete proxy for the effect of both the bubble size distribution and the liquid fraction. These previous studies can help us select specific scattering angles for optimising measurement sensitivity. However, they did not accurately measure the foam parameters, i.e. bubble size distribution and liquid fraction. In fact, the Mueller matrix should be influenced by the bubble size distribution and the liquid fraction simultaneously because they can both cause changes in the foam physical and chemical properties and consequently affect the light scattering process. Therefore, it is necessary to determine the individual effect of each foam parameter on the Mueller matrix first. By combining the results of each parameter, we are able to find the quantitative correlations between the polarised light and foam. However, none of the previous studies have investigated the independent effect of bubble size distribution and liquid fraction, so they were not able to decouple the effects caused by the different parameters. The aim in this study is to investigate the effect of liquid fraction on the Mueller matrix elements with a well-controlled bubble size distribution and this is the first time that these two foam parameters are measured simultaneously with the Mueller matrix elements obtained experimentally. Our previous study has shown that polarised light scattering technique has more utility for foams with a uniform bubble size distribution. Therefore, in the current study we first perform experiment to measure the Mueller matrix elements of foam with nearly uniform bubble size distribution. Specifically, we study the Mueller matrix associated with light scattering from the foam at a backscattering angle 135°. This angle is decided by preliminary experiments taking various determinants into consideration (Aslan et al., 2006). In Section 2 we outline the basic theoretical background of the Mueller matrix. Experiments are reported in Section 3, and the experimental results are discussed in Section 4. Finally, conclusions of this study are given in Section 5.

Theoretical background

When polarised light is incident on the surface of one bubble, single scattering will occur. The light may be transmitted, reflected, refracted or diffracted; when polarised light is incident on a collection of bubbles (i.e. a foam), the light will be multiple scattered from one bubble to the other bubbles nearby. With every interaction on the gas–liquid interface, the light intensity is attenuated and the polarisation state is changed. This becomes a prerequisite for the use of polarised light

Experimental setup

The apparatus used for generating foam and for making the light scattering measurements is illustrated in Fig. 1. A glass foam column (5.0 cm in internal diameter; 0.25 cm in thickness; 70 cm in height) is positioned at the centre of a reservoir by a holder. At the bottom of the column, there is a 100 μL pipette tip connecting to a peristaltic pump (pump A, MasterFlex® L/S, Vernon Hills, US) for bubble generation. Two ports in the mid-section of the column (30 cm and 50 cm from the bottom

Results and discussion

The main experimental results will be presented in the next three sections. They are (1) liquid fraction and foam rising height during the foam free drainage period; (2) bubble size distributions and shape changes in the drainage period; (3) six independent Mueller matrix elements as a function of liquid fraction and thickness of bubble edge.

Conclusions

We have successfully demonstrated the quantitative measurement of the Mueller matrix elements of foams with nearly uniform-distributed bubble size. It has been shown that the liquid fraction and the foam become relatively static about 200 s after the air and the washwater are terminated. The effect of foam properties on the normalised Mueller matrix elements only manifest when the liquid fraction is less than 0.01. Specifically in the foam static stage where the liquid fraction is less than

Nomenclature

_A_ contact area of each bubble on the column wall (mm 2) _D_ thickness of bubble edge (mm) _g_ acceleration of gravity (m s−2) _h_ foam rising height (m) Δ _h_ distance between the two pressure sensors (m) _I_ voltage signal obtained from the experiment (_V_) _k_ wave number (m−1) _L_ length of bubble edge (mm) _**M**_ Mueller matrix or scattering matrix _M_ _ij_ Mueller matrix or scattering matrix elements where _i_=1, 2, 3, 4 refers to the row and _j_=1, 2, 3, 4 refers to the column _**M**_ P1 Mueller matrix of polariser-1 _**M**_ P2 Mueller matrix of polariser-2 _**M**_

Acknowledgement

The authors would like to thank Drs. Stuart Murdoch and Yiqing Xu for their useful discussions and help. Shaoyu Qian specially acknowledges the China Scholarship Council for a maintenance grant. The assistance from Dr. Paul Stevenson in the initial stage of this work is also acknowledged.