Dielectric Sphere Clusters as a Model to Understand Infrared

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Dielectric Sphere Clusters as a Model to Understand Infrared Spectroscopic Imaging Data Recorded from Complex Samples Ilia L. Rasskazov, Nicolas Spegazzini, P. Scott Carney, and Rohit Bhargava Anal. Chem., Just Accepted Manuscript • Publication Date (Web): 12 Sep 2017 Downloaded from http://pubs.acs.org on September 12, 2017

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Analytical Chemistry

Dielectric Sphere Clusters as a Model to Understand Infrared Spectroscopic Imaging Data Recorded from Complex Samples Ilia L. Rasskazov,† Nicolas Spegazzini,† P. Scott Carney,‡ and Rohit Bhargava∗,†,¶,§ †Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA ‡The Institute of Optics, University of Rochester, Rochester, NY 14627, USA ¶Department of Electrical & Computer Engineering, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA §Departments of Bioengineering, Chemistry, Chemical and Biomolecular Engineering, and Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA E-mail: [email protected] Abstract Understanding the infrared (IR) spectral response of materials as a function of their morphology is not only of fundamental importance but also of contemporary practical need in the analysis of biological and synthetic materials. While significant work has recently been reported in understanding the spectra of particles with well-defined geometries, we report here on samples that consist of collections of particles. First, we theoretically model the importance of multiple scattering effects and computationally

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predict the impact of local particles’ environment on the recorded IR spectra. Both monodisperse and polydisperse particles are considered in clusters with various degree of packing. We show that recorded spectra are highly dependent on the cluster morphology and size of particles but the origin of this dependence is largely due to the scattering that depends on morphology and not absorbance that largely depends on the volume of material. The effect of polydispersity is to reduce the fine scattering features in the spectrum, resulting in a closer resemblance to bulk spectra. Fourier-transform IR (FT-IR) spectra of clusters of electromagnetically coupled polymethyl methacrylate (PMMA) spheres with wavelength-scale diameters were recorded and compared to simulated results. Measured spectra agreed well with those predicted. Of note, when PMMA spheres occupy a volume greater than 18% of the focal volume, the recorded IR spectrum becomes almost independent of the cluster’s morphological changes. This threshold, where absorbance starts to dominate the signal, exactly matches the percolation threshold for hard spheres and quantifies the transition between the single particle and bulk behavior. Our finding enables an understanding of the spectral response of structured samples and points to appropriate models for recovering accurate chemical information from in IR microspectroscopy data.

Fourier transform infrared (FT-IR) spectroscopic imaging 1 is of wide interest in materials science, 2–4 forensics, 5,6 biology, 7–14 agriculture 15 and medicine. 8,16–25 Generalizing, most of these applications deal with electromagnetic waves absorption and scattering from micron-sized objects such as cells, 26,27 bacteria, 28 multiphase materials or particles with varied morphologies. 29–34 The importance of developing theoretical approaches to model and understand the spectral response of these systems 35–38 is widely recognized. A majority of theoretical works consider methods and approaches for predicting and recovering FTIR spectra of single particles. In most applications, however, chemical imaging analyses is desired from multiphasic solid samples. Such samples may be represented as a collection of particles with various morphologies and previously developed single particle approaches generally fail. Along with the complexity arising from considering particles as 3D objects, 2


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re-scattering of electromagnetic waves from one particle to others plays significant role in multi-particle systems. Consideration of these re-scattering effects necessitates making the model of electromagnetic attenuation significantly more sophisticated than the single particle approaches. To our knowledge, there is no study that has addressed this problem for IR microscopy and imaging. For single particle scattering and absorption, the electromagnetic Mie approach combined with characteristics of the experimental setup (numerical aperture, pixel size) adequately describes IR spectra recorded from single, micron-scale spherical particles. 30,31 The single particle Mie scattering model is also used to understand light scattering from biological materials. 9,32,39–41 However, for a collection of electromagnetically coupled objects, Mie theory needs to be generalized to well-known T-matrix approach 42,43 which is widely used for various types of electromagnetic scattering problems. An updated and comprehensive database of T-matrix problems and solutions is provided by Ref. 44 It is notable that collection of densely packed particles should be considered as an electromagnetically coupled system because re-scattering of light from particles and interaction of secondary electromagnetic waves with neighboring particles dramatically changes the electromagnetic response of the whole system. Compared to the single particle case, further, considering a cluster of particles increases the number of variables that are needed to describe the sample geometry; for example, the sphere size for single particle needs to be replaced by the size distribution of the cluster, the position of the sphere needs to be replaced by the distance between neighbors for cluster of particles. The increased degree of freedom for such real-life samples significantly complicates its chemical analysis. We sought here to formulate a unified approach which would be capable of predicting the recorded spectra from various types of particle clusters with different particle sizes and packing densities. We first apply the developed approach to conduct extensive numerical simulations via the T -matrix approach that are then validated with systematic experiments for poly(methyl methacrylate) (PMMA) clusters with various morphologies. We compare simulation results from single PMMA particles with various

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sizes with that from monodisperse and polydisperse PMMA aggregates. Finally, we provide general conclusions and guidelines for possible applications when considering aggregates of spheres as models for complex samples.

Theory For simplicity, we start with electromagnetic light scattering from a single spherical particle with radius R and complex refractive index m ˜ sp located in free space with real refractive index m ˜ free . The electromagnetic response of a sphere can be represented in terms of extinction, scattering and absorption efficiencies Q which are coupled via the well-known relation:

Qext = Qsca + Qabs ,

(1)

where Qext and Qsca , according to rigorous Mie theory, can be found by plane wave decomposition of incident electromagnetic wave: 45

Qext =

Qsca

∞ 2 ! (2n + 1) Re {an + bn } , x2 n=1

∞ " # 2 ! = 2 (2n + 1) |an |2 + |bn |2 . x n=1

(2)

(3)

Here x = 2πR/λ is a dimensionless size parameter, λ is the wavelength of the incident electromagnetic wave, and n is the order of a spherical harmonic. Complex-valued Mie coefficients an and bn are explicitly defined by: 46 m ˜ 2 jn (mx ˜ i ) [xjn (x)]′ − jn (x) [mxj ˜ n (mx)] ˜ ′ , an = $ %′ (1) (1) m ˜ 2 jn (mx) ˜ xhn (x) − hn (x) [mxj ˜ n (mx)] ˜ ′ jn (mx) ˜ [xjn (x)]′ − jn (x) [mxj ˜ n (mx)] ˜ ′ bn = , $ %′ (1) (1) jn (mx) ˜ xhn (x) − hn (x) [mxj ˜ n (mx)] ˜ ′ 4


(4)

(5)

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(1)

where m ˜ =m ˜ sp /m ˜ free is the normalized refractive index of a sphere; jn and hn are spherical Bessel and spherical Hankel functions of the first kind, respectively, and prime denotes differentiation with respect to the argument in parentheses. Note that we set magnetic permeability to unity that is valid for non-magnetic dielectric PMMA spheres. Now let us turn to a system of N electromagnetically coupled spherical particles located in free space with the same refractive index m ˜ free . Here, for readers’ convenience, we provide general idea of the generalized multiparticle Mie theory. For extensive formulation and details, we refer readers to pioneering works by Mackowski 47 and Xu. 48 Each sphere in cluster is characterized by position of its center ri and dimensionless size parameter xi , where i = 1, 2, . . . , N . The incident Einc and scattered Esca fields for each particle in cluster can be expanded over vector spherical harmonics:

Eiinc =

∞ ! n 2 ! !

iEmn pimnp Y(1) mnp (kri ) ,

(6)

∞ ! n 2 ! !

iEmn aimnp Y(3) mnp (kri ) ,

(7)

n=1 m=−n p=1

Eisca

=

n=1 m=−n p=1

(τ )

where Emn are normalization coefficients, pimnp are known expansion coefficients, Ymn1 = (τ )

) (τ ) N(τ mn (kri ) and Ymn2 = Mmn (kri ) are vector spherical harmonics. For given boundary condi-

tions (for example, excitation by a plane wave), the expansion coefficient aimnp can be found as:

aimnp = ainp pimnp ,

(8)

where ain1 = ain and ain2 = bin are given by Eqs (4) and (5), and the exciting coefficient pimnp can be found from the linear equations: N ! ∞ ! ν ! 2 !

j Hmnp,µνq pjµνq = pimnp

j=1 ν=1 µ=−ν q=1

5


(9)


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where the interaction matrix H is explicitly given, for example in Ref. 49 Solving equations (8) and (9), one could calculate incident and scattered fields at each particle of the cluster via Eqs (6) and(7), and finally find Qabs , Qsca and Qext of the cluster. 50

Experimental section FT-IR spectroscopic imaging data were measured using a Spotlight 400 imaging system combined with Spectrum One FT-IR spectrometer (Perkin Elmer, Waltham, MA, USA). Spotlight 400 uses a 16 element linear array (LA) of liquid Nitrogen cooled Mercury-CadmiumTelluride (MCT) detector elements to raster scan the sample. This instrument has a 15× 0.6 NA reflective Cassegrain objective and a 6.25µm image pixel size. LA is capable of acquiring spectra with high signal-to-noise ratios (above 500:1) with 32 scans. Data for the experiment are acquired over the mid-infrared range and stored using a truncated 800-4000cm−1 region at 4 cm−1 spectral resolution with a 2 cm−1 spectral spacing. A zero filling factor of 2 was used prior to Fourier transform. For this experiment, we used PMMA microspheres with R = 2.5 − 10µm size and 1.2g/cc concentration (Cospheric LLC, Santa Barbara CA, USA). A schematic representation of IR beam interaction with cluster of PMMA spheres in experiment is shown in Figure 1. In our experiments we measure the apparent absorbance A defined as: 9,30,31

A = − log10

&

I I0

'

,

(10)

where I0 is the intensity of IR beam and I is intensity which is detected directly in forward direction. Thus, the power which is recorded by detector in the absence of PMMA particles is I0 G, where G is illuminated area of the detector. In the presence of dielectric spheres, incident beam might be scattered, absorbed and transmitted directly to detector. In this case, due to the power conservation law: 9,31

6


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IR beam, I0G

multiple scattering, I0!sca g1

gi

g2 gN absorption, I0!abs

transmitted light, IG

detector G

Figure 1: Schematic representation of infrared beam scattering from a cluster of spheres.

I0 G = IG + I0 σabs + I0 σsca ,

(11)

where σabs = gc Qabs and σsca = gc Qsca are absorption and scattering cross sections, respecN N ! ! tively. Here gc = gi = πRi2 is the overall geometric cross-section of the cluster. i=1

i=1

Results and discussion Although FT-IR microspectroscopy spectra from single particles have been reported, it is insightful to start with consideration of spectra from isolated PMMA spheres before analyzing multi-particle clusters. Figure 2 shows the numerically obtained extinction, absorption and scattering efficiencies of single PMMA microspheres of different sizes. The extinction and scattering spectra significantly depend on radius R, as expected, showing a dramatic dependence on R. We especially note the short-range (ripples) and long-range (wiggles) oscillations in extinction and scattering spectra (at 2000 − 4000 cm−1 region) that become 7



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pronounced with increasing radius. A detailed analysis of these oscillations can be found in Refs. 31,51 The pattern of the absorption features changes less dramatically, it is largely similar for spheres with different sizes and in a good agreement with bulk PMMA absorption spectra. 30 We note that the total extinction is largely dominated by scattering and the spectra for larger spheres tend to change less with increasing size. From this analysis, we can hypothesize that the recorded absorption spectrum from multi-sphere clusters for larger sizes will not significantly differ from the corresponding spectrum of a single sphere. Due to strong size-dependence of the scattering pattern, however, we expect to that significant differences between mono- and polydisperse clusters spectra will likely arise from scattering.

Figure 2: Simulated extinction, absorption and scattering spectra (Qext , Qabs and Qsca respectively) for single PMMA spheres of different radius R. We next consider electromagnetically coupled clusters of PMMA spheres. As mentioned 8


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previously, there are two geometrical parameters of the cluster which define the behavior of its electromagnetic response: (i) sizes of particles within the cluster, which generally classify the cluster as monodisperse with particles of the same size, or polydisperse with particle sizes that obey a size distribution function; (ii) volume fraction of particles, φ, within the simulation box, which generally classifies the cluster as sparse or low φ or dense with a large φ. To incorporate both of these aspects and reveal the impact of φ on recorded IR spectra, we use a T -matrix approach to calculate the Qabs and Qsca for the cluster of PMMA particles 52 with different configurations. First, we consider two cases of cluster morphology: monodisperse and polydisperse. For monodisperse clusters, we set Ri = const = 6.5µm since it is approximately the dominant frequency in biomedical spectra. For polydisperse samples, we set normal distribution for radii Ri with mean value ⟨Ri ⟩ = 6.5µm and standard deviation 1.3µm to illustrate the predictions. For polydisperse cluster in this case, 99.9% of spheres have radii from 2.6µm to 10.4µm, which covers the range of the experimentally measured clusters of PMMA spheres. We consider a cluster of randomly distributed particles within the box of fixed volume Vbox = 450 × 450 × 20µm3 . Second, in order to reveal concentrationdependent spectral features, we vary the overall volume of spherical particles within the box for both monodisperse and polydisperse samples. To characterize the concentration of particles, we use parameter φ = Vsp /Vbox , where Vsp is the overall volume of N spheres comprising the cluster. We will refer to parameter φ as to the volume fraction. Figure 3 shows results of simulations for described clusters of PMMA spheres with different volume fraction φ from 0.005 to 0.450 and for both morphologies - monodisperse and polydisperse. There are several important features that arise from this comparative picture. First, the absorption spectra for fixed φ does not depend on cluster morphology. Moreover, with increasing of φ, the absorption spectrum gradually changes: characteristic peaks in 750cm−1 − 1750cm−1 region and at 2950cm−1 become more pronounced as for single particle with increasing size (see Figure 2). Second, as expected, the scattering pattern significantly changes with transition from monodisperse to polydisperse aggregates for a fixed φ. Notably,

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Figure 3: Simulated extinction, absorption and scattering spectra (Qext , Qabs and Qsca , respectively) for monodisperse and polydisperse clusters of PMMA spheres for different values of volume fraction φ.

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the characteristic ripples and wiggles of single spheres are highly suppressed for polydisperse clusters. This smoothing likely arises from the averaging of electromagnetic response from various sizes of particles, which leads to flattening of the scattering spectrum. We compare these computationally obtained insights with experimental results. First, we start with considering an image with single and multiple scattering effects in a sample as shown in Figure 4. We consider different regions of the sample highlighted by colored squares and plot corresponding apparent absorbance A for each region. It can be easily seen that spectral data recorded in different regions of the sample significantly changes for different density of spheres within the box region. Starting from point (b), which is dominated by the background, and moving towards the center of the sample that is the densest part, the apparent absorbance A increases and spectra acquire the shape of absorption spectra for the bulk sample. This is likely the result of minimizing of scattering in forward direction, such that absorption becomes predominant in the case of scanning a densely packed region of the sample. Moreover, curves for (b)-(f) pixels significantly differ from spectra of isolated spheres depicted in the (a) pixel. These data clearly point to the local surrounding of particle as well as interaction within the whole cluster of multiple particles should be taken into account to adequately describe a dense sample’s IR spectrum. Let us consider a sample with various volume fractions φ of PMMA spheres within the fixed scanning area. To get the apparent absorbance of a collection of spheres, we scan a large 450 × 450µm2 area pixel-by-pixel and define the clusters’ A by averaging Ai from each pixel of the scanned area. We consider experimental samples with various volume fraction φ from ≈ 0.043 for sparse sample regions to ≈ 1 for densely packed spheres. Figure 5 shows normalized A for each recorded spectrum at certain wavenumbers that correspond to peaks in absorption spectra of bulk slab of PMMA. It can be seen that around φ ≈ 0.18, the recorded apparent absorbance A for sphere clusters changes only gradually with increasing of φ. For sparsely packed clusters with φ < 0.18, the apparent absorbance A dramatically depends on the number of particles per scanned area. This is an interesting result as one could define

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Figure 4: Apparent absorbance measured for an isolated PMMA sphere of R ≈ 10µm (a) and across various regions of PMMA aggregates (b)-(f). Pixel size is 25 × 25µm2 . Each plot is normalized to its maximum value. φ ≈ 0.18 as a threshold point that distinguishes sparsely and densely packed clusters to define two major regions of sparsity of samples affecting the recorded data. Interestingly, the obtained value matches the percolation threshold for 3D hard (non-overlapping) spheres 53 which explains that recorded A for clusters with φ > 0.18 are very close to the bulk spectrum. This finding is very important for chemical analysis in FT-IR microspectroscopy due to its universality: one could neglect the morphology of the target as long as target is packed densely enough or for thicker samples. As it shown in Figure 5, decreased spatial and spectral resolution don’t affect the pattern of recorded spectrum for dense targets.

Conclusion Here we have extensively studied FT-IR spectra of clusters of electromagnetically coupled dielectric spheres. We show that sparsity of the cluster and particles size distribution determine the pattern of the sample’s absorption spectrum. There is an essential difference 12


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Figure 5: (a) Extinction coefficient Im [m] ˜ for a slab of PMMA; (b) Normalized apparent absorbance A as a function of volume fraction φ recorded at various wavelengths; (c) PMMA microspheres images acquired in the reflection-absorption mode onto reflective low-emission (Low-E) glass slides for IR imaging; (d) Apparent absorbance A measured for targets with (1) bulk, (2) dense, and (3) sparse configurations. Spectra are provided for various setup parameters: scans with pixel size 25 × 25µm2 and 50 × 50µm2 ; spectral resolution 4cm−1 and 12cm−1 .

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between monodisperse and polydisperse clusters extinction spectra that results in a strong suppression of wiggles (long-range oscillation) in the case of polydisperse cluster. For concentration dependence, the overall apparent absorption A as well as Qext increase with increasing of particles number density per scanning area; however, spectral features change little for volume fractions larger than a threshold value of φ ≈ 0.18. This result implies that the spectra of complex samples that are sufficiently packed can be simplified to model as a collection of particles. For sparse samples, single particle or specific morphologies would have to be modeled. By providing these regimes and pointing to a simplified understanding of material response, this work enables both an understanding of spectral origin as well as directions to pursue to extract information from complex samples.

Acknowledgement This work made use of the Illinois Campus Cluster, a computing resource that is operated by the Illinois Campus Cluster Program (ICCP) in conjunction with the National Center for Supercomputing Applications (NCSA) and which is supported by funds from the University of Illinois at Urbana-Champaign. The research is based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via AFRL contract number FA8650-16-C-9108. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.

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Dielectric Sphere Clusters as a Model to Understand Infrared

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