Several papers have been published in the past yearsrelated to the optical reflectance of a monolayer ofsmall particles absorbed on a flat interface. Whenparticles are very small compared to the wavelength,the particles may be treated in the dipole (or highermultipoles) approximation, and an effective-mediumapproach may be appropriate.1,2 However, when par-ticles are not small, light is scattered away from thespecular direction and one may split the reflectedoptical fields in a diffuse component and a coherentcomponent. The coherent component corresponds tothe average of the optical fields and remains in thespecular direction, whereas the diffuse component isrelated to the fluctuation of the optical fields from itsaverage and is distributed over a wide range of re-flection angles.3 In experiments it is not difficult toextract the coherent component from the whole scat-
tered light. In this case one sometimes refers to thecoherent reflectance of a well collimated beam of lightas the reflectivity. It is also not difficult to measurethe diffuse component of the scattered light. Whenparticles are large, the effective-medium approachmay no longer be valid and a scattering theory ap-proach should be pursued.4 By large particles wemean particles with radii such that the size param-eter, �2���� a, is comparable to or larger than 1. Herea is the particle radius and � is the wavelength ofradiation.
There are previous experimental studies on theadsorption of electrically charged latex particles on aflat surface based on measurements of the reflectanceof TM (transverse magnetic or “p”) polarized lightaround the Brewster angle, defined by therefractive index of the substrate (glass) and of thecontinuous medium (water) where particles were sus-pended. The reflectance of TM polarized light at thisangle is no longer zero when particles are adsorbed,and the minimum of the reflectance curve increasesand slightly shifts to other angles.5–8 In these worksit was shown that by fitting a theoretical model to theexperimental curves it was possible to retrieve accu-rately some physical properties of the adsorbed par-ticles and the kinetics of adsorption. On the otherhand, in Ref. 9 it was shown that the reflectance of alaser beam near the critical angle, in an internal-reflection configuration of a glass–water interface,changed strongly when electrically charged latex par-ticles were adsorbed on the glass surface. The sensi-tivity of these measurements to the presence of theadsorbed colloidal particles was found to be muchlarger than when measuring the reflectance of TM
M. C. Peña-Gomar is with Facultad de Ciencias Físico-Matematicas de la Universidad Michoacana de San Nicolás deHidalgo, Edificio B, Ciudad Universitaria, CP. 58030, Morelia,Michoacán, México. F. Castillo and E. Pérez are with Instituto deFísica, Universidad Autónoma de San Luis Potusí, Alvaro Obregón64, 78000 San Luis Potusí, SLP, México. A. García-Valenzuela([email protected]) is with Centro de CienciasAplicadas y Desarrollo Tecnológico, Universidad NacionalAutónoma de México, Cuidad Universitaria, A.P. 70-168, MéxicoD.F. 04510, México. R. G. Barrera is with Instituto de Física,Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México D.F., México.
polarized light close to the Brewster angle. Thisshowed the potentiality of the internal-reflection ex-perimental configuration as a sensitive tool to studythe kinetics of the adsorption process, and to deter-mine also the optical parameters of the adsorbed par-ticles. The purpose of our work here is to performthese types of experiments and to develop a reliabletheoretical model for the quantitative interpretationof the experimental results.
A simple theoretical model for the reflectance of lightfrom a monolayer of large particles adsorbed on glasssurface was proposed already in Refs. 5 and 6. Themodel takes into account the Mie scattering from thespherical particles and it is suitable for particles withlarge radii. Basically, the model in Refs. 5 and 6 ap-proximates the coherent reflected wave as the super-position of the coherent reflected wave from an isolatedmonolayer and the reflected wave from the clean glassinterface. It does not take into account, for example,the multiple reflections between the monolayer andthe glass interface. Therefore the model is valid onlyfor TM polarization near the Brewster angle since onlythen the reflection coefficient of the clean glass inter-face is practically null. Also in this model the coherentreflection amplitude from the isolated monolayer iscalculated in the single-scattering approximation,which is valid only for small angles of incidence andlow surface coverage (on the order of a few percent orless, depending on the particle’s size and refractiveindex). In Ref. 5 the validity of the model for latexparticles of refractive index 1.591 and for angles ofincidence around the Brewster angle was checkedagainst exact numerical calculations of the full electro-magnetic problem. It was found that, although differ-ences between the exact calculations and the app-roximate model could be large in some cases, estimat-ing the particle size from fitting the approximatemodel to the experimental data gave accurate resultsfor low values of the surface coverage and the particle’sradii of up to about 1000 nm. Although exact numer-ical calculations are possible, considering the timethese elaborate calculations take, it is necessary inpractice to have an approximate model to implementa fitting routine for experimental data analysis. Forexperiments in an internal-reflection configurationand near the critical angle, the reflection coefficient ofthe clean glass interface is not small and the single-scattering approximation of the coherent reflectionfrom the monolayer is not valid. Thus, to have anapproximate model in these experiments; it is neces-sary to introduce multiple-scattering effects withinthe monolayer and with the glass interface.
One of the objectives of this work is to include thesemultiple-scattering effects by extending the modelproposed in Refs. 5 and 6 to the treatment of thereflection of light from a monolayer of particles ad-sorbed on a glass–water interface, in an internal-reflection configuration. We will limit ourselves tomonodisperse systems of spheres, all with the samerefractive index. We will refer to our model as the“coherent-scattering model.”
The paper is organized as follows: In Section 2 we
derive in the single-scattering approximation the co-herent reflection from a monolayer of spherical par-ticles. Then we improve this approximation by takingpartially into account multiple scattering betweenparticles within the monolayer, and construct thecompound reflection coefficient that accounts for mul-tiple reflections between the glass interface and themonolayer. We also take into account corrections tothe reflectance due to the finite collimation of theincident optical beam. In Section 3 we compare thecoherent-scattering model with the results of our ownmeasurements. In Section 4 we compare the resultsof the coherent-scattering model with the ones ob-tained with an effective-medium model. Finally, inSection 5 we present our conclusions.
2. Theory
In Ref. 4 the coherent reflection and transmission oflight from a dilute ensemble of identical sphericalparticles of radius a, whose centers are located atrandom within a slab of width d, was addressed. Thegeometry considered is shown here in Fig. 1(a). Anincident plane wave, E0 exp�iki · r�ei, is assumed.The coherent scattered waves to the right and to theleft of the slab are plane waves, Es
� exp�iki · r�ei, andEs
� exp�ikr · r�er, respectively. In the single-scatteringapproximation, and ignoring any correlation in the po-sition of particles, the amplitudes Es
� and Es� were
found to be given by
Es� � �E0
�
cos �iS�0�, (1)
Es� � �E0
�
cos �i
sin�kzid�
dkzi Sn�� � 2�i�, (2)
where � � kd�3f�2x3�, n is 1 or 2 for TE or TM polar-ization, respectively, S1��� and S2��� are the non-zero elements of the amplitude scattering matrix,10
Fig. 1. (a) Geometry considered in the coherent reflection andtransmission of a plane wave from a thin slab of a dilute randomsystem of spherical particles; (b) geometry of the internal reflectionconfiguration for reflectivity measurements from an adsorbed frac-tional monolayer of particles.
S1�� � 0� � S2�� � 0� � S�0�, f is the volume fractionoccupied by the spheres, �i is the angle of incidence,kz
i � k cos �i, k is the wavenumber in the mediumsurrounding the spheres, which we will call the“matrix” medium, and x � ka, where a is the radius ofthe particles. The coherent reflection and transmis-sion coefficients from a slab of randomly placed par-ticles of thickness d �� 1�k follow from theseexpressions:
ts � 1 ��
cos �iS�0�, (3)
rs � ��
cos �iSn�� � 2�i�, (4)
The derivation leading to Eqs. (3) and (4) assumesthat the center of the spheres is contained within aslab between z � 0 and z � d. This means that halfa sphere may lay outside the slab from either side. Wecan obtain the reflection and transmission coeffi-cients of the coherent wave from a monolayer of par-ticles from the previous expressions in the followingmanner. Consider N particles inside a slab of volumeAd, where A is the area of the slab in the x–y planeand d is the thickness of the slab along the z axis,recalling that f � �N�V��4�3��a3, we have that
� �32
f
x3 kd �2�
x2 ,
where � � �N�A��a2. Since the right-hand side of theequation does not depend on d, we can take the limitsd → 0 and N, A → while keeping the surface num-ber density N�A � s constant. In this limit, � is thesurface coverage and the plane passing through thecenter of all the particles contained in the (partiallycovered) monolayer is the plane z � 0. This may becalled the plane of the monolayer.
Now, we propose an approximation that includes,in an average way, multiple-scattering effects amongthe particles, by considering that the field driving thescattering process is the transmitted wave ratherthan the incident one. Then we can write
tsEi � Ei �
�S�0�cos �i
tsEi, (5)
rsEi � �
�Sn�� � 2�i�cos �i
tsEi. (6)
Solving this pair of equations we obtain the followingtransmission and reflection coefficients for the coher-ent wave:
ts �cos �i
cos �i � �S�0�, (7)
rs � ��Sn�� � 2�i�
cos �i � �S�0�. (8)
Note that these coefficients behave correctly at graz-ing incidence. Therefore, as long as the surface cov-erage of the spheres is small, that is, � �� 1, one canexpect that the previous formulas provide a good ap-proximation at any angle of incidence, including graz-ing incidence. Now, if we assume that the plane of themonolayer is at z � z0, then the reflection coefficientis multiplied by the phase factor exp�2ikz
iz0�.Let us now consider the case of interest in this
paper, where the base of a glass prism of index ofrefraction n1 coincides with the plane z � 0 and isimmersed in a matrix of index of refraction nm. Thenlet us suppose that a monolayer of spherical particlesof radius a is adsorbed on the base of the prism, asdepicted in Fig. 1(b). The plane of the monolayer, asdefined above, is now at z � a. If a plane wave is nowincident from the prism side to the interface at anangle �i (from the normal), a plane wave will be trans-mitted outside the prism into the matrix, making anangle �m. Both angles are related through Snell’s law,n1 sin �i � nm sin �m. The transmitted wave is re-flected back and forth between the monolayer and thebase of the prism; thus the reflected coherent wavefrom the base of the prism contains the superpositionof all the coherent waves reflected from the mono-layer and transmitted back into the prism. This isanalogous to the multiple reflections within a homo-geneous slab between two homogeneous mediums.The resulting compound reflection coefficient for thecoherent wave is given by
rCSM �r12��i� � rs��m�
1 � r12��i�rs��m�, (9)
where r12 is the Fresnel reflection coefficient betweenthe prism and the matrix medium without particles,and rs is given by
rs��m� � ��Sn�� � 2�m�
cos �m � �S�0�exp�2ikz
ma�, (10)
with kzm � k0nm cos �m � k0�nm
2 � n12 sin �m, S�0� and
Sn�� � 2�m� are evaluated for the particles sur-rounded by the liquid of index of refraction nm, and�m � sin�1��n1�nm� sin �i�. Thus Eq. (9) is the plane-wave reflection coefficient for the prism–liquid inter-face when a fraction of a monolayer is adsorbed on it,at the liquid interface. When the angle of incidence islarger than the critical angle of the prism–matrix in-terface, �c � sin�1�n1�nm�, �m becomes complex. There-fore the average field exciting the particles is actuallyan evanescent wave. This poses no problem in ourformulation in terms of the elements of the amplitudescattering matrix because the mathematical proce-dure remains valid for an exciting field with a com-plex wave vector.
However, in the experiment one is not using aplane wave. Instead one uses a well-collimated beam.Most laser beams have a Gaussian intensity profile.The reflectance of a Gaussian beam can be calculated
from the plane-wave reflection coefficient, r, as11
R��i� ��0k1
�2��
0
��2
�r����2 exp���0k1�2
2 �� � �i�2 d�,
(11)
where �0 is the beam’s waist radius and k1 is thewavenumber of light in the incident medium (in thiscase the prism). For well-collimated beams, the limitsof this integral can be extended to with negligibleerrors. The effect of a finite �0 is stronger where thederivative of |r���|2 is largest. In general, this is nearthe critical angle between the prism and the externalmedium. In our experiments a semicylindrical prismis used. The laser beam is focused upon the entranceto the prism due to the curvature of its surface. Thenthe cross section of the laser beam actually becomeselliptical. The value of �0 that should be used withEq. (11) is the semiaxis of the ellipse along the planeof incidence. From geometrical considerations we mayapproximate �0 inside the prism as ��n1Y���0
out,where Y is the radius of the semicylindrical prismand �0
out is radius of the beam outside the prism. Inour experiments �0 is smaller than �0
out, which
means that the laser beam is less collimated, thususing Eq. (11) to calculate accurately the reflectancebecomes essential.
3. Measurements
To test our model against experimental data we mea-sured the coherent reflectance of a laser beam, atdifferent angles of incidence, from the base of a semi-cylindrical prism with a monolayer of latex particlesadsorbed onto it. Our experimental setup is shown inFig. 2. The half-cylinder prism �n1 � 1.51� is insertedlaterally on the cylindrical container and mounted ontop of a high-precision goniometer to measure andcontrol the angle of incidence. A linearly polarizedHe–Ne laser beam �� � 0.6328 �m� of Gaussian crosssection with a diameter of approximately 0.81 mm isreflected from the base of the prism. Then the angleof incidence is adjusted by rotating the goniometer,and the reflected optical power is collected by a siliconphotodetector and a digital voltmeter. First, the con-tainer was filled with de-ionized water and a plot ofthe reflectance as a function of the angle of incidenceabout the critical angle, �c � sin�1�nm�n1�, was regis-tered. This curve was used to adjust the value of �0and correct any systematic error in the angle of inci-dence �i. Then the water was replaced by a dilutecolloidal suspension of electrically charged latex par-ticles. The particles adsorbed gradually over time onthe base of the prism. After approximately 10–15min, the colloidal suspension was replaced by de-ionized water, stopping the adsorption process. A sec-ond reflectance plot over the same angular range asthe previous one for water was registered. The con-tribution of diffuse light to the reflectivity signal waschecked to be negligible. After each experiment theprism was dismounted from the container and its
base was inspected with an optical microscope to en-sure that a monolayer had been formed.
In Figs. 3(a)–3(c) we show plots of the measure-ments of the reflectivity as a function of the angle ofincidence around the critical angle for both a cleanprism surface in water and after the monolayer of par-ticles had been adsorbed. The particle sizes in the ex-periments corresponding to Figs. 3(a), 3(b), and 3(c)were 137, 256, and 344 nm, respectively. In these fig-ures we also show micrographs taken from the base ofthe prism after the experiments. An estimation of thesurface coverage from the micrographs proved to beunreliable. The reason is that an unknown amount ofparticles was desorbed from the prism surface whenexposed to air while the prism was taken to the op-tical microscope. Therefore, to compare theory withexperiment, we left � as an adjustable parameter. Inthe figures we also plot a reflectance curve calculatedwith our coherent-scattering model. The refractiveindices used for the prism, for water, and for theparticles are 1.515, 1.331, and 1.59, respectively.Also, for the three graphs in Fig. 3 we used �0� 18.6 �m and � � 0.6328 �m. The value of � wasadjusted to best fit the experimental data. We see inthe figures that the theoretical curves reproducefairly well the experimental data, although some dis-crepancies between theory and experiment can beappreciated. These discrepancies may be due to theformation of particle aggregates on the surface thatare not accounted for in the model. The formation ofaggregates on the prism surface can be appreciated inthe micrographs taken after the optical measure-ments.
Fig. 3. Experimental data of the reflectance of a TM polarized He–Ne laser beam from a clean glass–water interface (open circles) andwith an adsorbed monolayer of latex particles (full circles). The radii of the particles are (a) 137, (b) 256, (c) 344 nm. The refractive indexof the glass, water, and particles is 1.515, 1.331, and 1.59, respectively. Theoretical curves with the coherent-scattering model (full curve)and with the effective-medium model (dotted curve) are also plotted. The values of the surface coverage used to adjust the scattering-theorymodel to the experimental data are (a) 0.08, (b) 0.03, and (c) 0.016. Microscopic images from the base of the prism after the experimentsare also shown.
It is instructive to compare the results for the reflec-tance obtained with the coherent-scattering modelwith the one obtained with an effective-mediummodel. Effective-medium models (EMM) have beenused for monolayers of particles on a flat substratewhen particles are small compared to the wavelengthof radiation;1,2,8 however, there have been attempts touse them when particles are not small (see, for ex-ample, Ref. 12). We may think of replacing the ran-dom monolayer of particles by an artificial film witheffective properties. A simple model may be to con-sider a film of thickness equal to the diameter of theparticles, 2a, and an effective refractive index. In abulk, dilute colloidal system and when the colloidalparticles are not necessarily small with respect to thewavelength of the incident beam, a well-validatedeffective index of refraction is the one first derived byvan de Hulst,13 and given by neff � nm�1 � i�S�0��,where � � 3f�2x3.4,13 We have added a tilde on top ofneff just to remark that it is a complex quantity. Inthe case of a monolayer one might (daringly) assumethat the effective index of refraction is the same as inthe bulk, thus one should simply replace in the ex-pression for neff the volume fraction of particles f inthe effective film by �2�3��, where � is the surface co-verage. Therefore the reflectance in this effective-medium model is calculated using the well-knownexpression for reflection coefficient in a three-layeredmedia,
rEMM �r12 � r23 exp�2ikz2d�
1 � r12 r23 exp�2ikz2d�. (12)
Here r12 and r23 are the Fresnel reflection coefficientsfor the glass–film interface and the film–water inter-face, respectively, kz2 � k0�neff
2 � n2 sin2 �i�1�2 , andd � 2a. Evaluation of the Fresnel reflection coeffi-cients is done using the van de Hulst’s effective re-fractive index, neff. In Figs. 3(a)–3(c) we also plot thereflectance curve predicted by the effective-mediummodel just described, assuming the same value of �used to adjust the coherent-scattering model. We cansee that the effective-medium model also reproducesthe experimental data relatively well. The differencesbetween the experimental data and the effective-medium model are similar to those with the coherent-scattering model, even though both models do notcoincide exactly. For larger surface coverage, largerparticle radii, or larger refractive-index contrast thanthose in our experiments, the reflectance curvesaround the critical angle predicted by the effective-medium model differ more noticeably from those pre-dicted by the coherent-scattering model.
Also numerical evaluation of the reflectance usingEqs. (9) and (12) show that the coherent-scatteringmodel and the effective-medium model predict quitedifferent contributions of the colloidal particles to thereflectance at other angles of incidence. To illustratethis, we consider two examples: (i) Latex particleswith refractive index np � 1.6, radius a � 350 nm,
and assuming a surface coverage fraction � � 0.05and (ii) TiO2 (rutile) particles with refractive indexnp � 2.73, radius a � 200 nm, and assuming � �0.02. Both examples were considered in an internalreflection configuration with a glass–water interfaceand with TE (transverse electric or “s”) polarized lightof wavelength � � 632.8 nm. In Fig. 4 we plot therelative difference in TE reflectance due to theadsorbed particles predicted by both models in�R � �Rmodel � Rs��Rs, where Rmodel is calculatedwith either the coherent-scattering or the effective-medium model, and Rs is the reflectance of the cleanglass–water interface. It can be appreciated in Fig. 4that both models predict quite different values of �Rat most angles of incidence. These differences in-crease for larger surface coverage. In the inset of thefigures we plot the difference of �R predicted by bothmodels. For angles of incidence just before the criticalangle of the glass–water interface, and for higherangles of incidence, the curves of �R versus �i havesimilar shapes although different magnitudes. This
Fig. 4. Plots of the change in reflectance, �R � (Rmodel � Rs)�Rs,of a glass–water interface (refractive indices 1.515 and 1.331, re-spectively) due to the adsorption of (a) latex and (b) TiO2 (rutile)particles of radii 500 and 200 nm, respectively. A wavelength of632.8 nm was assumed. The surface coverage fraction was as-sumed to be (a) 5% and (b) 2%. The full curve is with the coherent-scattering model and the dashed curve is with the effective-medium model.
was found to be true also for TM polarization. Thismeans that the reflectance using the effective-medium model may be adjusted reasonably well tothe reflectance predicted by the coherent-scatteringmodel in an internal-reflection configuration andaround the critical angle, but it would be necessary touse a smaller value of the surface coverage in theeffective-medium model. Further experiments arenecessary to see whether the coherent-scatteringmodel remains accurate in these cases.
Even though the effective-medium model proposedhere seems to predict consistently the reflectancecurves around the critical angle for small enoughsurface coverage, we must point out its heuristiccharacter. In particular, the thickness of the effectivefilm was chosen to be 2a, without a clear physicaljustification. We have checked that other choices ofthis “effective” thickness will weaken the accuracy ofthe effective film model around the critical angle, atleast in the case of the experiments presented here.Therefore, in general, we could not advise with con-fidence the use of an effective film model to estimatethe surface coverage factor, or any other parameter ofthe particles for that matter. At other angles of inci-dence the effective-medium model simply should notbe used.
5. Conclusions
We developed a simple coherent-scattering model forlight reflection from a monolayer of large particlesadsorbed on a flat interface. The model is valid for allangles of incidence in an internal or external reflec-tion configuration as long as the surface coverage issmall compared to one. By large we mean particleswith radii comparable to the wavelength of the inci-dent radiation. We performed experimental measure-ments of the optical reflectance from a glass–waterinterface in an internal reflection configuration aro-und the critical angle. Electrically charged latex par-ticles were adsorbed on the glass–water interface,forming a monolayer with small values of the surfacecoverage. The coherent-scattering model could repro-duce well the experimental data. Only the value ofthe surface coverage was adjusted to best fit the ex-perimental curve.
Additionally, we compared the coherent-scatteringtheory model with an heuristic effective-mediummodel and found that, in general, they predict quitedifferent contributions to the reflectance from thepresence of an adsorbed monolayer of large particles.However, around the critical angle in an internal-reflection configuration, both models predict similarresults at low enough values of the surface coverage.At other angles of incidence, and for large particles,the effective-medium model should not be used at all.For larger surface coverage more experiments are
necessary to check the predictions presented in thiswork.
We are grateful to Ma. Lourdes González-Gonzálezand Asur Guadarrama-Santana for technical sup-port. We also acknowledge financial support from Di-rección General de Asuntos del Personal Académicofrom Universidad Nacional Autónoma de Méxicothrough grants IN112905 and IN108402-3, PROMEP,Proyecto de Intercambio px-230 UNAM-UASLP, andConacyt 36464-E. M.C. P.-G. acknowledges a fellow-ship from Consejo Nacional de Ciencia y Tecnología(México). The authors also thank J.L. Sánchez(UASLP) for Fig. 2.
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