Effects of Cascading Optical Processes: Part I: Impacts on Quantification of Sample Scattering Extinction, Intensity, and Depolarization

Abstract

Light scattering is a universal matter property that is especially prominent in nanoscale or larger materials. However, the effects of scattering-based cascading optical processes on experimental quantification of sample absorption, scattering, and emission intensities, as well as scattering and emission depolarization, have not been adequately addressed. Using a series of polystyrene nanoparticles (PSNPs) of different sizes as model analytes, we present a computational and experimental study on the effects of cascading light scattering on experimental quantification of NP scattering activities (scattering cross-section or molar coefficient), intensity, and depolarization. Part II and Part III of this series of companion articles explore the effects of cascading optical processes on sample absorption and fluorescence measurements, respectively. A general theoretical model is developed on how forward scattered light complicates the general applicability of Beer’s law to the experimental $\text{UV}$–vis spectrum of scattering samples. The correlation between the scattering intensity and PSNP concentration is highly complicated with no robust linearity even when the scatterers’ concentration is very low. Such complexity arises from the combination of concentration-dependence of light scattering depolarization and the scattering inner filter effects (IFEs). Scattering depolarization increases with the PSNP scattering extinction (thereby, its concentration) but can never reach unity (isotropic) due to the polarization dependence of the scattering IFE. The insights from this study are important for understanding the strengths and limitations of various scattering-based techniques for material characterization including nanoparticle quantification. They are also foundational for quantitative mechanistic understanding on the effects of light scattering on sample absorption and fluorescence measurements.

Graphical Abstract:

INTRODUCTION

Optical spectroscopic techniques that exploit material light absorption, scattering, and photoluminescence properties have been used extensively in chemical, biological, and environmental measurements for applications such as fundamental material characterization,^1–3 phototcatalysis,^4,5 industrial process control,^6,7 and quality assurance.^8–11 Fundamental theories have been developed for correlating the optical spectroscopic signals with the materials’ photophysical properties and analyte concentrations. The most well-known example is Beer’s law that correlates the $\text{UV}$–vis spectral intensity with the analyte concentrations and molar extinction coefficient. Another example is the mathematical models for correlating fluorescence signals with the fluorophore concentration, quantum yield, and molar absorptivity.^12–15 However, many of these models are developed for optically simple samples, where the cascading optical processes triggered by individual excitation photons are negligibly small. Example cascading processes include (1) the reabsorption of the fluorescence photons generated by absorbing the incident photons, which further triggers fluorescence emission, (2) the absorption of the scattered photons with and without generating fluorescence photons, and (3) the rescattering of the scattered photons followed by other possible optical processes. Indeed, numerous cascading processes can be imagined for photonically active nanoscale or larger materials, many of which are simultaneous scatterers, absorbers, and emitters.^16–18 Those cascading optical processes in combination with the practical measurement constraints (limited sampling volume, detection angles, etc.) can significantly complicate the spectral data analysis. One example is the inner filter effect (IFE) imposed by the sample light absorption. The absorption IFE introduces nonlinearity to light scattering, Raman scattering, surface-enhanced Raman scattering, and fluorescence emission intensities as a function of the analyte concentration and causes spectral distortion.^19–22

There is extensive literature on the effects of light absorption on optical spectroscopic measurements, including the mathematical models for correcting the absorption IFE on the sample scattering and fluorescence intensities.^22–26 However, the literature studying the impact of light scattering on spectroscopic measurements is sporadic. For example, questions like the general applicability of Beer’s law for experimental scattering extinction have, to our knowledge, not been critically examined. Filling this knowledge gap is essential for chemical measurements due to the emerging dominance of macromolecules, supramolecular, nanoscale, and other larger materials in chemical, biological, and materials research.

Compared to light absorption, which is a single-event process, because it eliminates the photons by absorption with and/or without generating fluorescent photons, scattering is far more complex in its impact on spectroscopic measurements. The fate of the scattered photons is extraordinarily complicated, even in the simplest light scattering sample, where the analytes are pure light scatterers with no absorption or emission.^27–29 In these samples, the scattered photons can be re-scattered or simply escape from the sample container (cuvette) with and without generating a signal from the instrument detector (vide infra). Further complicating the effects of light scattering is that scattering is a highly polarized process, i.e., the spatial distribution of the scattered light depends on the scatterers’ chemical composition and geometric features (size and shapes).³⁰

The goal of this series of companion studies is to develop a systematic and mechanistic understanding of the effects of cascading optical processes on experimental quantification of the sample scattering, absorption, and emission properties. The present work (Part I) focuses on the effects of scattering on experimental quantification of the sample scattering activities (molar scattering coefficients or cross-sections), scattering intensity, and depolarization. Part II is on the effects of cascading optical processes on the spectroscopic measurements for samples that contain both scatterers and absorbers, while Part III is on the cascading optical processes in fluorescent samples including the ones that contain absorbers, scatterers, and emitters. The insights from Part I are foundational for the mechanistic understanding of the effects of light scattering on sample light absorption and emission (Parts II and III).

The model analyte used for the current work (Part I) is spherical polystyrene nanoparticles (PSNPs) that are approximately pure scatterers with no significant absorption.³¹ In this case, the experimental $\text{UV}$–vis extinction spectra are the PNSP scattering extinction spectra. PSNPs of four different sizes (50, 100, 200, and 380 nm in diameter) are used for probing the size effects on the scattering measurements. The impact of the light scattering on the experimental quantification of PSNP light scattering activities, intensities, and depolarization is investigated using a combination of computational modeling and experimental measurements. The seven different spectroscopic methods (Figure 1) employed in this study are all performed with commonly accessible $\text{UV}$–vis spectrophotometers and spectrofluorometers. The conventional $\text{UV}$–vis and the $\text{ISUV}$ are conducted with a commercial $\text{UV}$–vis spectrophotometer. These two methods differ only in the detector acceptance angle (vide infra). Therefore, the comparative $\text{UV}$–vis and $\text{ISUV}$ studies provide insights into the instrument dependence of the general applicability of Beer’s law for light scattering samples.

Figure 1.

(A) Linearly polarized resonance synchronous (LPRS) and (B) integrating-sphere-assisted $\text{UV}$–vis ($\text{ISUV}$) instrumental setup used for experimentation.

Resonance synchronous (RS) spectroscopy and four different LPRS spectroscopic measurements are performed with a commercial spectrofluorometer by keeping the excitation and detection wavelength identical during the entire spectral acquisition. Since the seminal work by Pasternack and Collings on the RS detection of the light scattering intensity of aggregated porphyrins,³² the RS technique has been used interchangeably as a resonance light scattering method.^33–38 LPRS is a new variant of the RS technique. One of the key breakthroughs enabled by this LPRS method is the experimental detection and characterization of on-resonance fluorescence, a phenomenon that has been commonly mistaken as light scattering.

The conventional RS method uses nonpolarized light for excitation and detection, while both the excitation and detection photons are linearly polarized in the LPRS measurements.^39–42 By varying the combinations of excitation and detection polarizations, a total of four types of LPRS spectra (namely, LPRS VV, VH, HH, and HV) can be obtained with a spectrofluorometer equipped with excitation and detection linear polarizers (Figure 1). Spectrofluorometers with both excitation and detection polarizers are commercially available and routinely used for fluorescence anisotropy assays.^43–46 It should be noted that this LPRS method has also been referred to as polarized resonance synchronous spectroscopy and abbreviated as PRS2 in our previous work. We refer hereafter to this method as LPRS because this term better reflects the true essence of the measurement design than PRS2.

EXPERIMENTAL SECTION

Equipment and Materials.

Polybead carboxylate PSNPs were purchased from Polysciences, Inc. PSNPs with diameters of 50 nm (cat#15913-10), 100 nm (Cat #16688), 200 nm (Cat #08216-15), and 380 nm (Cat #21753-15) are abbreviated as PSNP₅₀, PSNP₁₀₀, PSNP₂₀₀, and PSNP₃₈₀, respectively. The transmission electron microscopy (TEM) images of the PSNP are shown in the Supporting Information (Figure S1). Nanopure water is used throughout for solution preparation.

A Shimadzu $\text{UV}$-2600i spectrophotometer with an ISR 2600 integrating sphere accessory (Duisburg, Germany) was used to obtain the $\text{UV}$–vis and $\text{ISUV}$ spectra, while the RS and LPRS spectra were measured using a Horiba FluoroMax 4 spectrofluorometer. The LPRS spectra were collected by equipping the spectrofluorometer with excitation and detection linear polarizers. A neutral density filter with an optical density of 2.0 ± 0.05 from 200 to 1100 nm (Thor Labs) was used for the LPRS and RS spectral acquisition.

All spectrofluorometer-based spectra were acquired with an integration time of 0.3 s and a bandwidth of 2 nm for both excitation and emission monochromators. The spectral intensity is the ratio between the signal from the sample detector and reference detector (S1/R1). The G factor spectra for correcting the polarization bias were quantified in a previous study.⁴⁷ All measurements were made at room temperature using a 1 cm Thorlabs $\text{UV}$-fused quartz fluorescence cuvette.

The LPRS spectra differing in their combination of excitation and detection polarization are referred to as LPRS VV, LPRS VH, LPRS HV, and LPRS HH. “V” refers to the linear polarization perpendicular to the instrument plane defined by the excitation source, the sample cuvette, and detector. “H” refers to the linear polarization parallel with this plane. The first letter following LPRS refers to the excitation polarization, while the second one for the detection polarization. The spectrofluorometer was operated in a resonance synchronous mode for the RS and LPRS acquisition.

Computational Modeling.

We developed a program package based on Mie theory and the Monte Carlo method to computationally simulate the light propagation path in a sample cuvette, as well as examine the effect of scattering on the measured $\text{UV}$–vis scattering extinction, intensity, and depolarization. The cuvette size is 1 cm × 1 cm, as that used in the experimental measurement, and the sample height is 1 cm. In the simulations, the incident light propagates along the $Z$ axis, and the polarization of the incident light is parallel to the $Y$ axis. The incident light beam will move forward with a predefined small distance for each step and the distance is set to 50 μm in the simulations. For each step, the beam might transmit forward with no change of propagation direction and polarization, be absorbed, or be scattered. Since there is no absorption for the PSNP, the absorption cross section of the nanoparticle is set to be zero. The probability of the beam to transmit forward or be scattered will be determined by the propagation distance, the scattering cross section of the nanoparticle, and its concentration. The propagation probability was modeled using a Monte Carlo method. When the light is scattered, the beam will propagate along a randomly chosen direction based on the Poynting vector intensity of the nanoparticle along different directions. The Poynting vector intensities of the nanoparticles of different sizes and materials at different wavelengths can be calculated using Mie theory. Once the propagating beam reaches the boundary of the cuvette, its propagation direction and polarization state will be recorded for subsequent calculations. To obtain converged results, over 2 × 10⁹ trajectories of the incident beam are used for each spectrum simulation.

RESULTS AND DISCUSSION

Scattering Activity.

Materials’ scattering activities are represented as the scattering cross-section or molar scattering coefficient, which are defined in analogy to the absorption cross-section or molar absorption coefficient for material absorption activities. For samples that can be approximated as pure absorbers, the experimental $\text{UV}$–vis spectra are the sample absorbance spectra (absorption extinction spectrum). The spectral intensity is linearly proportional to chromophore molar absorptivity and analyte concentration, as indicated by Beer’s law. In this case, the applicability of Beer’s law is dictated by the instrument linear dynamic range in the light absorbance measurement, which can be from 0.01 to as high as 4 for modest spectrophotometers, such as the one used in this work.

However, for samples that are approximately pure light scatterers, the applicability of Beer’s law of the experimental $\text{UV}$–vis spectrum is complicated by the forward scattered light. Any forward scattered light reaching the detector leads to an underestimated experimental scattering extinction $S_{\mathrm{U}\mathrm{V}}(\lambda )$ as shown with eq 1.

$$S_{\mathrm{U}\mathrm{V}}(\lambda )=-\mathrm{l}\mathrm{o}\mathrm{g}\frac{I_{\mathrm{t}}(\lambda )+\eta (\lambda )I_{\mathrm{f}\mathrm{S}}(\lambda )}{I_0(\lambda )}$$ (1)

$I_0(\lambda )$ is the incident light intensity. $I_{\mathrm{t}}(\lambda )$ is the intensity of the incident photons that pass through the samples without undergoing any light scattering. $I_{\mathrm{f}\mathrm{S}}(\lambda )$ is the intensity of forward scattered light while $\eta (\lambda )$ is the fraction of the forward scattered photons reaching the $\text{UV}$–vis detector.

It is straightforward to deduce eqs 2 and 3 for predicting $I_{\mathrm{t}}(\lambda )$ as a function of scatterer concentration C and scattering activities in molar scattering coefficient ${\epsilon }_{\mathrm{s}}(\lambda )$. $S_{\mathrm{T}}(\lambda )$ in eqs 2 and 3 is the sample theoretical scattering extinction, and $l$ is the cuvette path length in the $\text{UV}$–vis measurements.

$$I_{\mathrm{t}}(\lambda )=I_0(\lambda ){10}^{-{\mathrm{S}}_{\mathrm{T}}(\lambda )}$$ (2)

$$S_{\mathrm{T}}(\lambda )=C{\epsilon }_{\mathrm{s}}(\lambda )l$$ (3)

Combining eqs 1 and 2 leads to eq 4, showing that the experimental $S_{\mathrm{U}\mathrm{V}}(\lambda )$ can be used for determining the scatterer concentration or scattering activity only when $\eta (\lambda )I_{\mathrm{f}\mathrm{s}}(\lambda )$ is negligibly small in comparison to the intensity of transmitted incident photons. Under this case, $S_{\mathrm{U}\mathrm{V}}(\lambda )$ is equivalent to the Beer’s-law-abiding (BLA) theoretical scattering extinction $S_{\mathrm{T}}(\lambda )$ defined in eq 3.

$$S_{\mathrm{U}\mathrm{V}}(\lambda )=-\mathrm{l}\mathrm{o}\mathrm{g}\frac{I_0(\lambda ){10}^{-{\mathrm{S}}_{\mathrm{T}}(\lambda )}+\eta (\lambda )I_{\mathrm{f}\mathrm{S}}(\lambda )}{I_0(\lambda )}$$ (4)

Experimental demonstration of the interference of the forward-scattered light with scattering extinction spectra is the comparison of the $\text{UV}$–vis and $\text{ISUV}$ scattering extinction spectra acquired with the same sample (Figure 2). The $\text{UV}$–vis and $\text{ISUV}$ spectra are both acquired with the same double-beam-in-time spectrophotometer, differing only in the detector acceptance angles. The latter is defined as the angle between the propagation direction of the incident light and the maximum angle along which the forward scattered light can be collected by the detector. Assuming that all forward scattered photons are collected with the integrating sphere used in $\text{ISUV}$ measurements (Figure 1), we can parameterize $S_{\text{ISUV}}(\lambda )$ intensity using eq 5 by replacing $\eta (\lambda )$ in eq 4 with unity. Since the acceptance angle of conventional $\text{UV}$–vis measurement is significantly smaller than that of $\text{ISUV}$ measurements, the $\text{UV}$–vis intensity is invariably higher than the $\text{ISUV}$ intensity of the scattering samples.

$$S_{\mathrm{I}\mathrm{S}\mathrm{U}\mathrm{V}}(\lambda )=-\mathrm{l}\mathrm{o}\mathrm{g}\frac{I_0(\lambda ){10}^{-{\mathrm{S}}_{\mathrm{T}}(\lambda )}+I_{\mathrm{f}\mathrm{S}}(\lambda )}{I_0(\lambda )}$$ (5)

Figure 2.

Comparison of (black) $S_{\text{UV}}$, (red) $S_{\text{ISUV}}$, and (blue) the difference spectrum between $S_{\text{UV}}$, and $S_{\text{ISUV}}$ obtained with (A) PSNP₅₀, (B) PSNP₁₀₀, (C) PSNP₂₀₀, and (D) PSNP₃₈₀ sample.

Consistent with the proceeding theoretical consideration, the intensities of the PSNP $S_{\mathrm{U}\mathrm{V}}(\lambda )$ spectra are all significantly higher than those of their respective $S_{\text{ISUV}}(\lambda )$ counterparts (Figure 2A–D). In contrast, $A_{\mathrm{U}\mathrm{V}}(\lambda )$ and $A_{\mathrm{I}\mathrm{S}\mathrm{U}\mathrm{V}}(\lambda )$ spectra obtained with KMnO₄, a small molecular chromophore that is approximately a pure light absorber, are the same (Figure S2). This control measurement confirms that the difference between $S_{\text{UV}}(\lambda )$ and $S_{\text{ISUV}}(\lambda )$ spectra of the PSNP samples is due to interference of forward scattering light to $\text{UV}$–vis extinction measurements.

While the detector acceptance angle is usually small for conventional $\text{UV}$–vis spectral measurements with commercial spectrophotometers, it cannot be infinitesimal small but must have a finite value. Therefore, one should always consider the possibility of the interference of forward scattered light in the acquisition and interpretation of $\text{UV}$–vis spectra of scattering samples.

It is currently impossible to determine the $\eta (\lambda )$ value because it depends not only on the instrument acceptance angle, but also on the spatial distribution of the forward scattered light. Fruitfully, however, general guidelines ensure the following condition $I_0\left(\lambda \right){10}^{-{\mathrm{S}}_{\mathrm{T}}\left(\lambda \right)}\gg \eta \left(\lambda \right)I_{\mathrm{f}\mathrm{S}}(\lambda )$, thereby allowing $\text{UV}$–vis quantification of the NP scattering activities to be stipulated based on both theoretical conditions and computational modeling. First, whenever possible, one should use $\text{UV}$–vis spectrophotometers with small detector acceptance angles to minimize the $\eta (\lambda )$ value. Second, which is more important and more convenient to implement in practical applications, one should use samples with low concentration (thereby low theoretical scattering extinction) so that $I_0\left(\lambda \right)-{10}^{-{\mathrm{S}}_{\mathrm{T}}(\lambda )}$ can be significantly larger than $I_{\mathrm{f}\mathrm{S}}(\lambda )$.

The interference of forward scattered light to $\text{UV}$–vis measurement is shown with both computationally modeled (Figure A,B) and experimental measurement data (Figure 3C,D). The BLA theoretical extinction of the experimental samples is determined using the procedures described in the Supporting Information (Figure S4). The modeled data showed that threshold scattering extinction decreases with both increasing detector acceptance angle (Figure 3A) and increasing particle sizes (Figure 3B), which are consistent with the experimental results shown in Figure 3C,D, respectively. With its large detector acceptance angle, the $\text{ISUV}$ intensity deviates significantly from Beer’s law even when the samples’ conventional $\text{UV}$–vis extinction is very low (~0.2) (Figure 3C).

Figure 3.

(A) Correlation between the simulated experimental extinction $S_{\mathrm{U}\mathrm{V}}(\lambda )$ as a function of the detector acceptance angles and the sample theoretical extinction $S_{\mathrm{T}}(\lambda )$. The size of the simulated PSNP is 50 nm in diameter. (B) Effects of PSNP sizes on the deviation between the simulated experimental extinction and theoretical extinction. The detector acceptance angle used in the simulation is 2°. (C) Experimental $\text{UV}$–vis and $\text{ISUV}$ extinction as a function of concentration for PSNP₂₀₀. The as-acquired spectra are shown in Figure S2. (D) Experimental $\text{UV}$–vis extinction as a function of PSNP theoretical or BLA extinction. The linear lines in the plots are for guiding views.

The larger PSNPs are, the more significant the interference of the forward scattered light (Figure 3). When the PSNP concentration is low, all experimental scattering extinction is equal to its theoretical value. However, the upper BLA scattering extinction limit for PSNPs decreases with increasing nanoparticle sizes. It is somewhere between 4 and 5 for PSNP₅₀, but significantly lower than 3 for PSNP₃₈₀ (Figure 3D). While it is impossible to estimate the upper BLA scattering extinction limit for $\text{UV}$–vis quantification of scatterers’ concentration or scattering activities, we believe that Beer’s law is applicable for samples with experimental scattering extinction below 2 for samples with scatterers smaller than 400 nm in diameter. This conclusion is supported by the computational simulations conducted on the experimental light scattering extinctions (Figure 3B) performed with a detector acceptance angle of 2° and for the PSNPs of 50, 100, 200, and 380 nm, respectively. The acceptance angle in commercial $\text{UV}$–vis spectrophotometers is usually small (<2°) because of the relatively long distance between the cuvette and the detector (Figure S5). Further research is needed for evaluating the BLA-scattering extinction for scatterers with larger sizes.

Scattering Intensity.

Scattering intensity is one key scattering-based technique for material characterization, including molecular weight and macromolecular concentration determination.^48–51 Since scattering is a highly anisotropic process, the experimental scattering intensity depends on the instrument setup, including the excitation and detection geometry and sample volumes. In this work, we explore the concentration dependence of the PSNP scattering intensity detected with the RS method performed with spectrofluorometers. The excitation and detection beams in the RS measurement are perpendicular to each other, which is the most used geometry for scattering detection. Therefore, the learning from the RS studies is applicable to many scattering-based techniques. Since PSNPs are approximately pure light scatterers in the studied wavelength region (300 to 800 nm), their RS spectra are indeed light scattering intensity spectra. Hereafter, we use RS and scattering intensity interchangeably to facilitate discussion.

The scattering extinction or the PSNP concentration dependence of scattering intensity spectra is extraordinarily complicated (Figure 4). At low PSNP concentration, the scattering intensity increases with increasing PSNP concentration, and the shape of the RS spectra remains approximately constant (Figure 4A–D); however, further increasing the PSNP concentration introduces RS spectral distortion. The correlation between the scattering intensity and the PSNP concentration represented with the sample theoretical scattering extinction is very complex (Figure 4E–H). There is an initial induction period where PSNP scattering intensity increases slowly with PSNP extinction. Such an induction period is especially evident in Figure 4G,H, where sample extinction is relatively low. However, when the PSNP concentration is higher than a threshold value, further increasing PSNP concentration reduces the scattering intensity. The latter is due to the IFE induced by light scattering, the light loss along the photon path length before reaching the instrument sampling volume (vide infra).

Figure 4.

Scattering intensity spectra of (A) PSNP₅₀, (B) PSNP₁₀₀, (C) PSNP₂₀₀, and (D) PSNP_380. (E–H) Scattering intensity at the specified wavelengths as a function of the BLA extinction, the data obtained with PSNP₅₀, PSNP₁₀₀, PSNP₂₀₀, and PSNP₃₈₀ are color-coded in the legend.

The particle-size dependence of experimental correlation between PSNP scattering intensity and scattering extinction is in excellent qualitative agreement with the computational modeling (Figure S6). The simulated scattering intensity initially increases with increasing PSNP extinction but decreases when the sample extinction is higher than a threshold value. Such threshold extinction also increases with increasing PSNP sizes, as what has been observed experimentally (Figure 4E–H).

The correlation between the scattering intensity and the scatterers’ concentration can be understood by taking practical measurement constraints into consideration. Scattering intensity is detected 90° relative to the incident beam at one specific sampling point that is typically at the cuvette center. While the number of scattered photons invariably increases with increasing PSNP concentration, as shown in the $\text{UV}$–vis extinction measurements (Figure 3), the number of the scattered photons that can reach the detector, thus producing the scattering signal, depends on the spatial distribution of the scattered photons and instrument geometry. Indeed, existing scattering intensity measurements including the RS method are all spatially selective under-sampling techniques, only a fraction of the scattered photons in sampling volume is detected. The effects of the instrument constraints on the scattering intensity measurements are shown schematically in Figure 5, where the two pin-holes in the detection chambers represent the collective effects of the lens, mirrors, monochromator slits, and other instrumental components that can limit the photons reaching the detector. Such pin-hole effects are operative in many spectral acquisitions performed with spectrofluorometers.³¹ For signal generation, the scattered photons must propagate sidewise out of the cuvette. However, increasing PSNP concentration increases both light scattering and scattering IFE. When the scattering IFE is too strong, further increasing PSNP concentration reduces scattering intensity, as shown both experimentally (Figure 4) and computationally (Figure S6).

Figure 5.

Schematic illustration of the IFE induced by light scattering in the scattering intensity spectrum. The IFE by the light scattering refers to the scattered photon leakage from the cuvette before reaching the sampling volume.

It is emphasized that the IFEs induced by sample absorption and scattering on scattering intensity measurements are drastically different. The absorption IFE is an intrinsic photophysical phenomenon that invariably reduces the sample scattering intensity.⁵² In contrast, the scattering IFE is due to the change of the spatial distribution of the scattered light, which affect their probability to reach the detector. The significance of the scattering IFE can vary significantly depending on the instrument geometry, spatial distribution of the scattered light, and the sample concentration. Attempts to develop a generally applicable mathematical model for correcting the scattering IFE on scattering intensity measurement have not been successful due to the difficulties in parameterization of the measurement configurations.

Scattering Depolarization.

Scattering depolarization, or interchangeably scattering anisotropy, is another scattering property.⁵² Many earlier studies evaluated light scattering depolarization $P(\lambda )$ using lasers for excitation, allowing quantification of scattering depolarization at individual laser wavelengths. The advent of the LPRS method enables one to readily acquire the scattering depolarization spectrum across the entire instrument wavelength range (eq 6).⁴⁷

$$P(\lambda )=G(\lambda )\frac{I_{\mathrm{V}\mathrm{H}}(\lambda )}{I_{\mathrm{V}\mathrm{V}}(\lambda )}$$ (6)

$I_{\mathrm{V}\mathrm{H}}(\lambda )$ and $I_{\mathrm{V}\mathrm{V}}(\lambda )$ in eq 6 are the sample LPRS spectra acquired using excitation and detection polarization combination of $\text{VH}$ and $\text{VV}$, respectively. $G(\lambda )$ is the instrument $G$-factor spectrum and is experimentally quantified using dissolved small molecular fluorophores.⁴⁷

Like what was observed with the scattering intensity spectra (Figure 4), PSNP LPRS $\text{VV}$ and $\text{VH}$ spectra (Figures 6, and S7) also exhibit the spectral distortion and nonlinearity between spectral intensity and PSNP concentration when PSNP concentration is high. This observation is not surprising because LPRS also measures scattering intensity. It differs from the RS method only in the polarization of the excitation and detection light (Figure 1). As a result, LPRS measurements also suffer from the scattering IFE depicted schematically in Figure 5.

Figure 6.

(A) LPRS VV and (B) LPRS VH spectra; (C) scattering depolarization spectra as a function of the PSNP₅₀ scattering extinction. (D–F) Scattering depolarization spectra as a function of the scattering extinction of PSNP₁₀₀, PSNP₂₀₀, and PSNP₃₈₀, respectively. Insets show the correlation between scattering depolarization at 400 nm as a function of PSNP extinction.

The PSNP concentration dependence of LPRS $\text{VV}$ and $\text{VH}$ spectra differs significantly (Figure 6A,B). The LPRS VH intensity increases faster and has a longer increasing period than LPRS $\text{VV}$, before reaching the turning point where the spectral intensity decreases with increasing PSNP concentration (Figures 6 and S7). The difference between the $\text{VV}$ and $\text{VH}$ intensity increase is due to cascading scattering depolarization, and the degree of scattering depolarization increases with further light scattering. Indeed, the experimental scattering depolarization calculated using eq 6 increases with increasing PSNP scattering extinction before reaching a plateau depolarization.

Scattering depolarization is a fundamental scatterer’ property that is related to the scatterers’ size, shape, and chemical compositions.³⁰ However, the experimental quantification of the intrinsic scatterers’ scattering depolarization is challenging because of the multiple light scattering that can occur in essentially any practical samples. It is known that multiple scattering invariably enhances the sample scattering depolarization, i.e., increasing the homogeneity of the spatial distribution of the scattered photons.³⁹ However, this work demonstrates that the maximum sample depolarization that can be achieved is a finite value less than unity, even with samples having exceedingly large scattering extinctions (e.g., >15 as shown in Figure 6D). In other words, regardless of the degree of multiple scattering, it is impossible to achieve isotropic scattering in practical scattering depolarization measurements. Another new insight from this study is that the smaller the NP, at given incident wavelength, the smaller the threshold scattering extinction for the sample scattering depolarization to reach its plateau depolarization. As an example, the threshold of the BLA scattering extinction for the sample to reach its plateau depolarization is ~3, ~5, ~8, and ~15 for PSNP₅₀, (B) PSNP₁₀₀, (C) PSNP₂₀₀, and (D) PSNP₃₈₀, respectively.

The experimental correlation between the PSNP scattering depolarization and scattering extinction is in excellent qualitative agreement with the computational simulations (Figure 7). The latter also showed that the larger the PSNPs, the slower the sample depolarization increase with the increasing PSNP scattering extinction, and the larger the threshold extinction for the experimental PSNP scattering depolarization to reach the plateau depolarization. As an example, when the sample scattering extinction is increased to 7, which is the maximum extinction we simulated computationally, the highest scattering depolarization is 0.96, 0.93, 0.82, and 0.45, respectively. Only PSNP₅₀, the smallest of the PSNPs, reached the approximate saturation depolarization when the scattering extinction is 7. We attribute the difference in the maximum depolarization of the simulated and experimental values to the instrumental parameters, which has not been reliably modeled in the simulations.

Figure 7.

Simulated experimental PSNP scattering depolarization as a function of the PSNP BLA scattering extinction for (A) PSNP₅₀, (B) PSNP₁₀₀, (C) PSNP₂₀₀, and (D) PSNP₃₈₀.

Mechanistically, the presence of the plateau depolarization is due to polarization dependence of the scattering IFE. Since the light propagation direction must be perpendicular to its polarization direction, the likelihood of the vertically polarized photons leaking out of the cuvette before reaching the sampling volume must be different from the horizontally polarized light responsible for the LPRS VH signal (Figure 1). In other words, even with hypothetically perfectly isotropic incident light, the light reaching the sampling volume is anisotropic, making the experimental depolarization smaller than unity.

Experimental confirmation of the polarization dependence of the scattering IFE is performed with the LPRS HV and HH measurements (Figure S8). Since the polarization directions of V and H of the detection polarizer are both perpendicular to the excitation polarization of H (Figure 1), the theoretical LPRS HV and HH intensity must be the same for H polarized excitation photons. In other words, the observed difference between the experimental LPRS HV and HH signal (Figure S8) must be due to the polarization dependence of the scattering IFE, i.e., the difference in the fraction of the H- or V-polarized scattered light leaking out of the cuvettes before reaching the sampling volume.

CONCLUSIONS

Even with the simplest light scattering samples that contain only scatterers with no absorbers or emitters, experimental quantification of the scatterers’ scattering activities, intensity, and depolarization can be highly complicated. Interference of forward scattered light can compromise the applicability of Beer’s law for using experimental $\text{UV}$–vis extinction spectrum to calculate the analyte molar scattering activities or scatterers’ concentration. The maximum experimental scattering extinction beyond which Beer’s law is problematic depends on the scatterers’ sizes and the instrument detector acceptance angle. The correlation between the scattering intensity and scatterers’ concentration is also strongly particle-size dependent, with no robust linearity between scatterers’ concentration. A strong scattering IFE appears when the scattering extinction is high, which causes decreased scattering intensity with increasing scatterers’ concentration. Scattering depolarization increases with increasing PSNP concentration, but no isotropic scattering can be achieved even when the sample is extraordinarily optically dense (OD = 15). The insights from this study are important for mechanistic understanding of the strength and limitation of the various light scattering-based parameters (extinction, intensity, and depolarization) for material characterization including nanoparticle quantifications. It is also foundational for Part II and III of this companion work, which will be on the effects of light scattering on light absorption and fluorescence emission, respectively.

Supplementary Material

Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.analchem.2c03917.

TEM images of the PSNP particles; A_UV(λ) and A_ISUV(λ) of KMnO₄; S_UV(λ) and S_ISUV(λ) of PSNP₂₀₀; determination of the BLA theoretical extinction, estimation of the detector acceptance angle in UV–vis spectrophotometers; simulated scattering intensity as a function of scattering extinction; PSNP LPRS VV and VH spectra; and PSNP LPRS HH and HV spectra (PDF)