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Strong scattering properties are obtained for a monolayer of randomly packed polystyrene microspheres. This gives rise to structural colors in transmission. For a sphere diameter between 0.5 and 1 micron, light is mainly scattered in the forward direction. Consequently, in-plane multiple scattering can be neglected when spheres are not too close to each others. This allows one to use a single scattering approximation to reproduce transmission spectra of the system. The film color is dependent on the sphere size, but also on the observation angle. This angular dependant color is reproduced taking into account multiple scattering between spheres. These films can be useful when low reflection is needed.
©2008 Optical Society of America
1. Introduction
Three-dimensional random media are known to produce color effects, as is the case for colloidal dispersion such as colloidal gold and silver nanoparticles, already used by Romans to color glasses, and Christiansen filters, usually made of quartz particles within an aqueous medium [1]. Random media are also the origin of the color of the sky. They have been extensively studied from the point of view of light localization and more recently as a feedback mechanism when introduced into the gain medium of a laser [2]. Due to multiple scattering, photons propagating inside the gain medium can induce stimulated emission before leaving the cavity [3].
In a scattering media, the transport of light is characterized by its mean free path. This path must be smaller than the sample thickness in order to localize the light and thus obtain good scattering and gain [4]. For this reason, single layers of random scatterers are considered less interesting than colloidal solutions [2] or powders [5]. To our knowledge, the only optical effect described for single layer of dielectric particles (110nm diameter silica spheres) with a random packing is high performance antireflection coating [6].
In this paper, it is shown that single layers of randomly packed microspheres can display strong structural colors. Due to scattering, 5% transmission is obtained at some wavelength even for a single layer. This system is interesting because it allow us to investigate different scattering regimes, from the isolated sphere approximation (ISA) in the case of a dilute deposition to the more complete single scattering approximation (SSA) for a dense layer, whereas multiple scattering theory (MST) is needed to describe the optical properties of a dense film observed at an angle of 45°.
2. Experimental procedure
The following process was used to make random layers of polystyrene spheres: after cleaning, glass substrates were functionalized by adsorbing dendrimeric molecules to make the surface positively charged. Negatively charged polystyrene beads [purchased from microParticles GmbH and Bangs Laboratories, Inc.] with a diameter between 508 to 870nm, were prepared in an ethanol solution. The glass substrate is then left for 24 hours in the bead solution. On removal the glass is rinsed with ethanol. Negatively charged polystyrene particles are adsorbed onto the positively charged surface due to electrostatic forces, as described in [6,7]. The beads adhere on first contact with the surface and give a random distribution. Beads not adhering to the surface are removed during the rinsing process.
Spectroscopic measurements were taken in transmission using a Nikon microscope. The sample is illuminated at a numerical aperture (NA) of 0.1 and the image is projected onto a 400µm fiber attached to an Ocean Optics CHEM2000-UV-VIS Spectrometer.
Fig. 1. SEM image of a layer of randomly packed polystyrene spheres (diameter 725nm) on dendrimeric molecules. (a) Particle solution concentration of 2.5% in weight per volume (w/v). (b) Particle solution concentration of 0.1% w/v. Inset: Fourier transform of the image presenting a ring representative of a uniform mean distance between beads.
3. Experimental results
A dense layer of spheres with a random packing is obtained with the procedure previously described, see Fig. 1. Inset is the Fourier transform of the image obtained by scanning electron microscopy (SEM). It shows a ring. The diameter of the ring indicates that the mean distance between the beads r̄ is around 1µm. r̄ can be tuned by changing the bead concentration in the solution (cf. Fig. 1(b)) which changes the filling factor η (η=Nπd 2/4 where N is the number of spheres per unit area of diameter d). The maximum filling factor obtained experimentally was around 0.42. The theoretical maximum for a random system is given by the “jamming limit” η max=0.55 [8].
Figure 2 shows white light transmission images for different randomly packed particle layers. A blue color is seen for 725nm diameter beads, whereas red and yellow are obtained for 590 and 508nm diameter beads, respectively. These colors are observed for dense layers (η≈0.4). On decreasing η, the color contrast becomes weaker (cf. Fig. 2(a)). At the other limit, i.e. for an ordered close-packed single layer (η max=0.91), the color vanishes.
Fig. 2. Dense layer of randomly packed polystyrene spheres illuminated in transmission for different particle diameter d and filling factor η: (a) d=725nm, η=0.30; (b) d=725nm, η=0.40; (c) d=590nm, η=0.40; (d) d=508nm, η=0.40.
Fig. 3. Specular transmission in the visible for dense layers of different particle diameters against size parameter x.
Transmission spectra of films with different bead sizes are compared in Fig. 3. These spectra have been measured in the visible wavelength range (λ=400–800nm) as a function of the size parameter x (x=πd/λ). The overlapping curves show that the effect is scalable, i.e. one obtains the same effect for another frequency range with the appropriate particle size.
In case of non-absorbing homogeneous thin films, high reflectivity R is obtained only when the transmission T is low (R+T=100%). In the present case, i.e. non-absorbing inhomogeneous thin films, both R and T are lower than 10% at some wavelengths, which implies that this monolayer is a very efficient scatterer. Due to this high scattering, the observed color disappears on increasing the NA of illumination or collection. For this reason, care was taken to keep the NA of the system low while measuring spectra (NA=0.1).
4. Theoretical study
In order to explain these optical properties, we look to scattering theory. Consider a system of identical dielectric spheres where r̄ is large enough in order to neglect multiple scattering. The amount of light transmitted is then calculated through the SSA. A slab of particles, in a (x, y) plane, is illuminated at normal incidence by a plane wave propagating in the z direction. In far field (z≫x+y), the intensity I transmitted by the sphere layer at a point M(x, y, z) is obtained by squaring the modulus of two set of waves: a plane wave of amplitude E 0 and wave vector k=2π/λ incident onto the layer, and spherical waves of amplitude E s scattered by each particle [9]:
where S(0) is the amplitude function of the scattering particle at the angle θ=0. S(0) is calculated using Mie theory [9–11]:
with a n and b n the Mie coefficients of order n. After integration of I over the area of an arbitrary detector, the transmission T SSA is given by [9]:
T SSA can be rewritten in the form:
Eq. (4) is key to calculate the optical effects caused by a slab of scattering particles. If the surface coverage is low (η≪1 and η≪x 2), the third term in Eq. (4) becomes negligible, giving:
C SCA, the cross section of the particle, has the dimension of an area and is defined by [9–11]:
Each isolated sphere can be seen as if it removes an intensity comprised in a spot of area C SCA from the incident beam. The ISA (Eq. (5)) is commonly used to derive Beer-Lambert’s law in the case of a dilute single layer of scatterers. The ISA should not be overused since the third term in Eq. (4) is important in other regimes.
Multiple scattering between spheres should not be neglected when either particles are too close to each other or in-plane scattering increases. In such cases, T SSA can become higher than one when the second term in Eq. (4) is smaller than the third one. The MST is afterwards needed to calculate scattering parameters. The equation for coherent transmission in the case of the MST is similar than for SSA [9,11,12]:
with S(0)MST the amplitude function of the scattering particle in the forward direction taking into account multiple scattering within the layer. S(0)MST can be calculated for a cluster of spheres using public domain codes [13]. The value for a single bead is then derived from it and used in Eq. (7).
Equivalent equations can be set up in the reflection geometry by looking at the backscattered components S(π)SSA and S(π)MST.
5. Comparison between experiments and theory
The three different regimes previously described (ISA, SSA and MST) are now compared with experimental results. It should be noted that neither the effect of the substrate, nor the presence of small clusters of 2–3 aggregated beads (cf. Fig. 1) have been taken into account in the following calculations. For clarity, results with different filling factors are presented here for only one bead size (d=725nm).
Fig. 4. Comparison between experimental (solid line) and theoretical spectra for a dense layer (η=0.4) of 725nm polystyrene spheres. Theoretical data are calculated using SSA for a standard deviation of particle size σ=0nm (dotted line) and σ=20nm (dashed line).
In order to quantify the agreement between theory and experiment, the average deviation δ (calculated from 400 to 800nm) between two transmission spectra is introduced and defined by:
with T 1 and T 2 the two transmission results compared for L different points.
In Fig. 4, experimental and theoretical results for η=0.40 are compared. Eq. (5) (ISA) is not used here since the transmission can be negative for such a dense layer (NC SCA>1). The dotted curve is obtained from Eq. (4) (SSA) for a layer of 725nm polystyrene beads with a refractive index of 1.59. Theory (dotted curve) and experiment (solid curve) are in good agreement (δ=1.9%). A standard deviation σ of 20nm is representative of the size variation of the present system. To take into account σ, T SSA is calculated for different bead size. Each result is weighted based on σ and summed. Figure 4 shows that such a polydispersity does not influence the result except smoothing slightly the obtained spectra. The following results are all calculated with σ=20nm.
In Fig. 5, experimental spectra are compared with those calculated with ISA and SSA at low filling factors. With η=0.19, δ between experimental transmission and SSA is lower than 2% whereas it is higher than 13% between experimental and ISA values. Again, SSA and experiments show good agreement in specular transmission for such layers. The ISA error is due to the third term in Eq. (4). For polystyrene spheres with sizes x>1, this term lies in a range of η 2 to 5η 2. Consequently, δ is lower than 2% for η=0.06.
Fig. 5. Comparison between calculated (either using SSA or ISA) and experimental spectra of a randomly ordered layer of 725nm diameter polystyrene spheres for different filling factors.
Multiple scattering events do not significantly affect transmission in the previous cases. For the dense case shown in Fig. 4, δ between MST and SSA is around 0.5% (data not shown). This is due to the fact that in this regime, the scattering occurs mainly in the forward direction. It also means that the presence of small clusters of aggregated spheres does not significantly influence the scattering properties of the film.
Eq. (4) has been obtained by summing incident and forward scattered waves. Theoretically, a minimum in transmission is obtained when these two waves are of similar amplitude and opposite phase [14]. In the present experiment, transmission is as low as 5% at some frequency ranges (cf. Fig. 4), producing strong color. On increasing η up to the closely packed case, the amplitude of the scattered wave increases which explains that the color effect vanishes.
An additional effect observed for films with a dense layer of spheres (η≈0.40) is a color change observed in transmission when the film is tilted. For example, the red color observed with 590nm beads gradually changes to yellow at 45°, and from blue to violet with 725nm diameter beads, as shown in Fig. 6.
In order to calculate the observed phenomena with the SSA, a tilt angle of 45° is first considered equivalent to a projection of the bead layer in a plan perpendicular to the incident light. η is then increased by a factor of √2 and implemented in Eq. (4), see Fig. 7. The trend of the SSA spectra mainly reproduces the experimental curve. From 400 to 500nm, δ=1.2% whereas δ is higher than 5% between 500 and 800nm.
Fig. 7. Comparison between experimental and theoretical spectra for a dense layer (η=0.4) of 725nm diameter polystyrene spheres observed at 45°. Theoretical data are calculated using either SSA or MST. Inset: Polar diagram of the intensity scattered by a 725nm polystyrene sphere at λ=425nm (blue dashed line) and λ=775nm (red solid line). The forward direction is represented by an angle θ=0°.
SSA gives a good fit at short wavelength since for a given sphere, scattering in the forward direction S(0) is larger at short wavelengths (cf. inset of Fig. 7). The amount of light scattered at an angle θ≠0 is then relatively low and so is multiple scattering. SSA with an increase of η is sufficient to reproduce the experimental spectrum here. Each sphere behaves independently and the tilt angle configuration corresponds to a projection of all the beads into a plan perpendicular to the incident light for this range.
On the other hand, SSA is less accurate at longer wavelengths: S(0) is lower, the scattering pattern is less unidirectional (cf. inset of Fig. 7) and light is scattered in a cone of solid angle increasing with λ. Beads are no longer laying in a single plane perpendicular to the incident light, and light scattered even at a small angle can irradiate neighboring spheres. MST should be used for a better precision. To do so, a cluster of around 100 randomly packed spheres is designed (with η 0=0.40 and r̄=1µm). Spheres are not in contact between each other. The scattering amplitude of the cluster tilted at 45° is then calculated taking into account multiple scattering events, and S(0)MST is afterwards derived from it. Finally, Eq. (7) is used with η=√2η 0=0.57 and dashed curve of Fig. 7 is obtained. This gives good agreements with experiments (δ=1.1%).
Since such calculations are time consuming [15], it is necessary to previously check if it is necessary to use MST. The scattering diagram depicted in the inset of Fig. 7 gives then important information. From it, it is understandable that multiple scattering at φ=45° is more significant at 775nm than at 425nm. In general, multiple scattering will be more important when the tilt angle is increased, even at small wavelength. Indeed, from our data, δ between SSA and MST is higher than 6% for φ=60° at λ=425nm.
6. Conclusion
In this paper, we have investigated the optical properties of random single layers of polystyrene spheres with different filling factors η and particle diameters d. Taking advantage of electrostatic interaction between beads and functionalized surfaces, random structures have been obtained with negligible bead aggregates. The scattering of light from dense bead layers (η≈0.4) gives rise to unexpected colors in transmission. The color can be modified by changing d and also by observing the film at an angle φ.
These properties have been explained in terms of scattering and have been calculated theoretically. The strong colors observed for φ=0 are due to interference between incident and scattered light and are reproduced with a single scattering approximation. Such approximation is less accurate once the film is tilted, and multiple scattering between spheres has to be considered to describe the angular color dependency. The various types of films can be reproduced homogeneously at a large scale and the denser ones could be used as filters with low reflection.
Acknowledgments
The authors appreciate the help of L. A. Dunbar for useful discussions. This project was supported by the COST action P11 “Advances in Physics and Technology of Photonic Crystals”.
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