Optical Coherent Reflection from a Confined Colloidal Film: Modeling

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Optical Coherent Reflection from a Confined Colloidal Film: Modeling and Experiment Gesuri Morales-Luna, Augusto Garcia-Valenzuela, and Ruben G. Barrera J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b03751 • Publication Date (Web): 17 Aug 2018 Downloaded from http://pubs.acs.org on August 21, 2018

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The Journal of Physical Chemistry

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Optical Coherent Reflection from a Confined Colloidal Film: Modeling and Experiment Gesuri Morales-Luna,1 Augusto García-Valenzuela,1* Rubén G. Barrera2 1

Instituto de Ciencias Aplicadas y Tecnología, Universidad Nacional Autónoma de México, Apartado Postal 70-186, Ciudad de México 04510, México

2

Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, Ciudad de México, 01000, México *Correspondence: [email protected]; Tel.: +5255-5622-8602.

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2 ABSTRACT: We study the optical reflectivity of confined colloidal films as function of the angle of incidence in an internal reflection configuration. Two effective medium models and an extended coherent-scattering model for thin colloidal films are compared against experimental measurements with gold, latex and titanium dioxide colloids. A derivation of the coherent scattering model for confined colloidal films used in this work is presented in a comprehensive way. The model lies within the framework of multiple-scattering theory and is valid for any angle of incidence and for colloids of small or large particles compared to the wavelength of light, however, only for small and moderately small particles’ volume fraction. Reflectivity versus angle of incidence curves for an opaque colloidal film in an internal reflection configuration show the effects of two critical angles. Within the two critical angles there is a high sensitivity to the presence of colloidal particles while the volume of colloidal samples needed is in the microliter range. Upon comparing theory with experiment no model fitting was done in any case. The experimental setup and its calibration procedure are discussed. The results provide physical insight into applications involving optical properties of colloidal systems.


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3 1.

INTRODUCTION Particle suspensions, besides being intriguing physical systems by themselves, have

many different applications. Their optical properties have been an active area of research in the past1-6 and are still of current concern given the potential new applications that have surfaced in recent years7-8. In the past few decades the optical properties of colloids, has been of interest of many researchers. By colloid we mean a system of randomly located particles embedded in a homogeneous medium, also referred to as the matrix. The activity in this field has been growing in recent years due to the potential of producing colloids with exotic optical properties, using for instance, resonant particles either metallic or dielectric9-14; also, experiments together with mathematical modelling can provide means to obtain information on the physical properties of the colloidal particles15-19. Within a colloid, an incident electromagnetic (EM) field is multiple-scattered by the colloidal particles giving rise to a complex spatial structure of the total EM field, which nevertheless, it is customary and useful to split it into an average (coherent) and a fluctuating (diffuse) components20-22. While the coherent component has a well-defined direction, the diffuse one travels in all different directions. Now, when the size of the colloidal particles is very much smaller that the wavelength of the incident EM field, it is possible to assign an effective refractive index to the colloid and use it to calculate the propagation of the coherent component. On the contrary, when the size of the colloidal particles is of the order or larger that the wavelength of the incident EM field, the scattering of light is so efficient that the diffuse component may dominate over the coherent one; in addition, the particles may or may not absorb light, and one may refer to



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4 these colloids as turbid colloids. In this case, although it is still possible to assign an effective refractive index to the propagation of the coherent component in the bulk, it is not proper to use it in the traditional Fresnel’s equations for the calculation of the reflection and transmission amplitudes, due, essentially, to the nonlocal nature of the EM response23, nevertheless in some cases it could be a good approximation. An attractive alternative to overcome this difficulty is to rely on multiple-scattering theories. Our research group has focused its attention, on the use of a multiple-scattering approach for the calculation of the coherent reflection of light from a half-space of turbid colloidal suspensions23-27. Although multiple-scattering theories have been widely used for the calculation and description of the propagation properties of the diffuse component, and its consequential application to systems like clouds, rough surfaces, bulk and layered turbid colloids, and so many others, the attention to the propagation of the coherent component has not been as widespread. Since the coherent component decays readily within a turbid colloid, the most practical way to detect it is via the reflected EM field, what we call the coherent reflectance. In part, the importance of our theoretical work is that we have been interested in the calculation of the coherent reflection amplitudes in systems with a well-defined geometrical structure, that can be easily built in the laboratory and thus, one is able to perform delicate experiments to test, as accurate as possible, the validity of our theoretical results. Rigorous and exact solutions with multiple-scattering theory might be cumbersome and sometimes they do not offer enough physical insight. On the other hand, approximate models based on multiple-scattering theory, can be simple enough to


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5 be attractive for their use in the analysis of actual experimental data and to provide physical insight into applications involving optical properties of colloidal systems. The aim of this paper is to provide simple tools to analyze the optical reflectivity of a colloidal thin film confined between to homogenous media. We refer to a thin film of a turbid colloid whenever the reflectivity depends on the thickness of the film, and this happens when the coherent component of light reaches the bottom interface of the colloidal film and reflects back out of the film. In practice, working with a thin film of colloidal samples will usually require only small amounts of the sample. In this work we extend the previously developed Coherent Scattering Model (CSM)24 to a confined thin colloidal film. In particular, the interfaces with an incidence and transmission medium, different from that of the colloidal matrix are introduced, providing new expressions, essential for modelling actual experiments and practical applications. The derivation of the coherent reflection coefficient is presented and its predictions are compared with our own experimental data, as well as with the predictions of the wellknown Maxwell Garnett’s (MG)28 and van de Hulst’s (vdH) effective medium models29. The vdH EMM can be used in some cases to predict the reflectivity from turbid colloids of particles comparable to the wavelength of light26,30. However, in Reference25 it was shown that although an effective index of refraction can be properly defined in case the size of the colloidal particles is comparable or even larger than the wavelength of incident radiation, it is not strictly legitimate to use it, in these cases, in the calculation of Fresnel’s reflection and transmission coefficients23-25. The use of MG model is only justified when the size of the colloidal particles is very small with respect to the wavelength of the



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6 incident radiation, whether their absorption is important or not. This is not the case in all examples here, where scattering by the particles actually is very important. However, we will also include a comparison of our results with the MG model in cases where losses are due only to scattering due to the widespread and naive use of this model in all possible situations. We expect that the reflectivity from a thin film of a colloidal sample provides more precise information about the colloidal suspension than the reflectivity from a half-space of the same colloid. Also, comparing theoretical models with experiments with confined colloidal films is a more stringent test on the theories than for a colloidal half-space. On the one hand, this is because in addition to the coherent reflection coefficient at the interface between the incidence and colloidal media, the coherent reflection coefficient from inside the colloidal medium, at the interface with the transmission medium, is also involved. On the other hand, there are contributions to the net reflectivity of a thin colloidal film from light that was transmitted into the colloidal medium and propagated within it, either after one or multiple reflection at the interface with the transmission medium. These contributions, besides depending on the real part of the effective refractive index of the colloidal sample, can depend more strongly on its imaginary part (which is directly related to the extinction coefficient of the colloid). This dependence can be particularly strong in an internal reflection configuration in which the incidence medium has a refractive index larger than that of the colloidal film, and this in turn, has a larger refractive index than that of the transmission medium. In this case there will be two critical angles. The angular location of the second critical angle will be depend on the real


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7 part of the effective refractive index of the colloidal film, and in between both critical angles, the reflectivity will be a strong function of the extinction coefficient within the colloidal medium. We chose this configuration to perform the experiments and compare with the theory. This paper is organized as follows. In section 2 methods are presented, the formulae for the Maxwell Garnett effective medium model and the effective medium model for dilute colloids based on the so-called van de Hulst effective refractive index are provided. Also, our multiple-scattering approximation for the coherent reflection of light by a confined colloidal film is derived. Our experimental methodology is described and in section 3 a comparison of our experimental reflectivity results with the three models considered in section 2, is presented. Finally, in section 4, a brief discussion of the results and possible applications, is bestowed together with our conclusions.

2. METHODS 2.1. Theoretical 2.1.1 Effective Medium Models. Here the formulas to calculate the reflectivity from a confined colloidal film with two effective medium models (EMM) are provided. Let us consider a plane wave incident into a colloidal slab. The colloid is regarded as a system of randomly located, non-magnetic, identical spherical particles of radius

and refractive index , surrounded by a homogeneous medium of refractive index , referred to as the matrix, see Figure 1a. So we regard this medium as an effective medium with an effective refractive index , see Figure 1b. ACS Paragon Plus Environment


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Figure 1. a) Illustration of a slab of a colloidal suspension confined between two homogeneous half-spaces. b) The canonical three media problem with an effective medium. 2.1.2 Maxwell Garnett Formulae. The most common Effective Medium Model (EMM) used for colloidal suspensions is that of Maxwell Garnett (MG)28,29. This EMM has been used since long and in many cases outside its range of validity. It is known to be valid when the particles size is very small compared to the wavelength of radiation, and if the volume fraction occupied by the particles, f, is small or moderately small. The MG-EMM expression for the effective electric permittivity of a colloidal suspension is given by, 2 2

, #1 2


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where is the permittivity of the matrix medium, is the permittivity of the particles and is the volume-filling fraction of the particles. Since we are assuming the particles to be nonmagnetic the MG refractive index is obtained from the effective permittivity as,

. #2

2.1.3 van de Hulst’s Effective Medium Model. The other effective medium model that we will consider, is the one referred to, as the van de Hulst’s effective refractive index (RI)31, given by,

1

3 #0% , #3 2 ! "

where ! is the particles’ size parameter in the matrix, ! &' , &( 2)/+ is the wave number in the vacuum, + being the vacuum wavelength, is the refractive index of the

matrix medium surrounding the particles, is the radius of the particles, is the volume

filling fraction and # 0 is the forward scattering amplitude of the particles inside the matrix. The so-called van de Hulst’s effective RI has been derived by several authors32-34, but the simplest derivation is found in van de Hulst’s book31, and that is why some authors attached to it his name. Eq. (3) can be used with dilute colloids only, but with particles of size small or large compared to the wavelength. In the latter case, its use is restricted to



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10 the calculation of the propagation features of light in the bulk, but due to the nonlocal nature of the effective permittivity and the unexpected presence of a non-local magnetic permeability, it turns out to be non-legitimate to use the effective vdH index of refraction in the standard formulas of continuous electrodynamics23. The main appeal of vdH EMM is its simplicity, and for that reason we will compare its unrestricted use in Fresnel relations with the predictions of our multiple-scattering model and with experimental data. 2.1.4 Reflectivity with an Effective Medium Model. Within an effective-medium approach, a colloidal layer is considered as a homogeneous medium with an effective RI . We can regard the confined colloidal layer of thickness ℎ, as the canonical threemedia problem for the reflection and transmission of light solved in many text books (see for instance35,36). Thus, to calculate the reflectivity of a confined colloidal film with an effective medium model we shall use the following formula, -."

-. -" / 0123 4 , #4 1 -. -"/ 0123 4

where -06 are given by the Fresnel reflection coefficients35,36 for the corresponding polarization of light at the medium i/medium j interface. These coefficients take the form, -06

&07 &67 , #5 &07 &67

for s (or TE) polarization and,


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11 -06

6 &07 0 &67 6 &07 0 &67

, #6

for p (or TM) polarization, where take the value of 1 or 2 and : take the values of 2 or 3. In the above expressions, &;7 &( ; . sin ?0 . For we should use the effective refractive index, given in Eq. (2) for the MG-EMM, or given in Eq.

(3) for the van de Hulst’s EMM. ?@ is the incidence angle and &( 2)/+, is the wave

number in the vacuum, + being the vacuum wavelength.

2.1.5 Multiple-Scattering Model for the Reflectivity of a Confined Colloidal Film. In this section, we first obtain a multiple-scattering approximation for the coherent reflection and transmission amplitudes of a dilute layer of randomly-located particles in a boundless matrix. This layer will be referred as a “free” colloidal film. A brief derivation of this result was given in Reference24, but here we derive it in a rather more comprehensive way. Then, we extend this result in order to construct an approximate expression for the coherent reflection of light for a confined colloidal film, which takes account of the multiple reflection between the layer of colloidal particles and the interfaces with the incidence and transmission media. 2.1.6 Single-Scattering Coefficients of a Thin Colloidal Film. In order to calculate the reflectance and transmission amplitudes of a colloidal thin film of finite thickness ℎ,

we first calculate these amplitudes for an extremely thin film of thickness A A → 0 and then use these results to construct an expression for a film of finite thickness. Let us



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12 DE @ ⋅ -E LFM/̂0 is suppose a plane electromagnetic wave given by CDE @ -E, F C( expJ&

incident to a thin colloidal layer in a boundless matrix medium. In the latter expression /̂0

DE @ are the polarization and incident wave vectors, respectively. Let us assume the and & center of the particles are distributed randomly with a uniform probability density between two planes, O A ⁄2 and O A ⁄2, defining a slab of thickness A, as depicted in Figure 2. Since the center of any particle should be in-between the two planes, but, one half of a particle may lie outside them, as illustrated in Figure 2. These planes do not correspond to the physical interface, and we will call them probability interfaces. Let us also assume that the layer’s width d is extremely small compared to the wavelength of light, +. On the one hand, note that this do not mean that the particles radius must be

very small compared to +. On the other hand, we will disregard the impenetrability of the

particles, by ignoring the possibility of any two particles of finite radius , intersecting each other (valid in dilute limit); in other words, we will assume the particles to be point particles, but with an amplitude scattering matrix given by the one corresponding to a particle of radius (point scatterers). In Reference24 it was shown that, within the single-scattering approximation, the configurational-average of the fields scattered by the particles in the colloidal film of a thickness A ≪ +, are given by, DE @ ⋅ -E]; for O > 〈CDE S -E〉V;WX

CDEUS exp[& U


A #7 2

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13 A DE b ⋅ -E]; for O < #8 〈CDE S -E〉V;WX

CDEaS exp[& a 2

DE b is the wavevector in the specular direction, and where & CUV eC( A

& #0, #9 cos ?0

and CaV eC( A

& # ) 2?0 , #10 cos ?0 6

where e 3/2! " , where ! & is the size parameter, where & is the wave number in the incidence medium [& 2)/+ ], f is the volume-filling fraction of the particles in the colloidal layer, is the particles’ radius and A is the thickness of the colloidal layer

taken as the distance between the probability interfaces. The sub-index : takes the value of 1 or 2 for an incident wave with s or p polarization, with respect to the plane of incidence (/̂0 hi for s and /̂0 cos ?0 hj sin ?0 h7 for p). In the latter expressions, the #? functions are the elements of the amplitude scattering matrix of an isolated

particle in the matrix medium as defined by Bohren and Huffman in their book37. The amplitude scattering matrix relates the scattering field in the far-zone with an incident plane wave as, V CWb,∥ exp[&-] # ? #" ? C0∥ k V n

o q k n , #11 CWb,m &- #p ? #. ? C0m

where the parallel and perpendicular components of the electric fields refer to the scattering plane defined by the incidence and scattering directions, while ? is the angle



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14 between them. For spherical particles, #" ? #p ? 0 and #. ? and # ? are

calculated with the so-called Mie-solution37. The function #0 in Eq. (9) means

#. ? 0 # ? 0 and is referred to as the forward scattering amplitude.

Figure 2. The centers of the particles are within the planes -d/2

Optical Coherent Reflection from a Confined Colloidal Film: Modeling

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