Reproduction of Visible Absorbance Spectra of Highly Scattering Suspensions within an Integrating Sphere by Monte Carlo Simulation

Abstract

It is important to avoid the overestimation of absorption due to scattering when using absorption spectroscopy to measure scattering samples. We approached this issue by placing the sample inside an integrating sphere (IS) to collect the scattered light in all solid angles but encountered difficulty when determining the absorption coefficient from the absorbance because the light took various paths inside the IS and the sample. Therefore, by ray tracing inside the IS and the sample using Monte Carlo simulations (MC), we estimated the relationship between the absorption, scattering, anisotropy coefficients, and the measured absorbance. Scattering sample M, prepared by mixing polystyrene microspheres with trypan blue solution, and pure trypan blue solution for comparison were used as samples at various concentrations. MC reproduced the measurement results for the absorbance spectrum and its concentration dependence at 591 nm up to the measurement limit value. In addition, the saturated absorbance of sample M was lower than that of the trypan blue solution. This is because, from the distribution of distance $d$, light passed through the sample estimated by the MC, and more light with smaller $d$ was detected due to scattering for higher concentration, resulting in a smaller increase in absorbance with the absorption coefficient.

1. Introduction

UV-visible absorption spectroscopy is a measurement method that investigates the electronic structure and composition of a material from the energy transitions due to optical absorption of the atoms or molecules that make it up. Because it can evaluate the physical properties of samples in a relatively simple, safe, and nondestructive manner, it is used in a wide range of fields, including materials science such as solar cell characterization [1], medical and biological fields such as protein concentration measurement [2], cell density measurement [3], and food testing [4]. However, since the size of the particles/clusters in these samples is often comparable with the wavelength of the measured light such that the scattering is not neglected [5,6,7], the transmitted light is attenuated not only by absorption but also by scattering. To accurately evaluate the amount of absorption of such scattering samples, it is necessary to detect not only transmitted light but also scattered light to estimate the contribution of scattering to the attenuation of transmitted light.

The instrument that makes this possible is the integrating sphere (IS), which is often used for absorption spectroscopy measurements of scattering samples. Generally, the sample is placed at the entrance of the IS as shown in Figure 1a. In this arrangement, forward scattered light as well as transmitted light is collected by the IS, enabling absorbance measurement without the influence of forward scattered light [8,9,10]. However, backscattered light does not enter the IS, so the measurement cannot be said to account for all scattered light [11].

Therefore, we placed a sample inside the IS (Figure 1b) to collect the scattered light at all solid angles with the IS, which enabled us to measure absorbance by taking into account the total scattering [12]. This arrangement has been used primarily in fluorescence quantum yield measurements [13,14,15,16,17]. Although it is also used in absorbance measurements, it is not yet commonly used [18,19]. One possible reason for this is that when the sample is placed inside the IS, it is difficult to estimate the absorption coefficient ${\mu }_a$ from the measured absorbance $A$ using the Lambert–Beer law ($A={\mu }_{a}d{\log }_{10}e$) because the distance $d$ light passing through the sample does not match the thickness $h$ of the sample for the following reasons.

(1) Since the IS also collects reflected light from the sample surface, stray light that is detected without ever passing through the sample. If the sample is highly absorptive, the weak transmitted light will be buried by this reflected light [16,20,21,22,23];

(2) In the process, the IS may reach and pass through the sample again. In that case, the optical path length is longer than that of a single pass [16,24];

(3) Light leaks through apertures (entrance of irradiation and reference light to the sample) of the IS, which increases the apparent absorption [20];

(4) The IS is coated with highly reflective materials, such as BaSO4 and a special fluoropolymer, and their reflectance is greater than 90% and 95%, respectively, but not 100% in the visible region [25,26,27]. Therefore, light is slightly absorbed by the inner walls of the IS each time it reflects off these materials, which increases the apparent absorption.

Furthermore, although not directly related to placing the sample inside the IS, when measuring a scattering sample, (5) it must be considered that light does not travel straight through the sample in one direction due to scattering. In our previous work, we proposed an absorbance correction equation that reproduces the absorbance measured in the configuration shown in Figure 1b by taking into account (1) and (2) [12]. However, this correction formula does not fully take into account the effects of (3) and (5), and is not applicable to samples where multiple scattering is indicated. Therefore, we investigated a method to estimate the relationship between the absorption coefficient and measured absorbance from ray tracing using Monte Carlo simulations (MC).

In estimating the optical properties of a sample by combining IS and MC, one technique [28,29,30,31,32,33,34,35,36,37,38,39,40,41,42] that is often used is to measure the diffuse reflected light (backscattered light) and total transmitted light (forward scattered light and transmitted light) of a sample by IS and link the measured results with the absorption coefficient, scattering coefficient, and anisotropy coefficient by MC.

Diffuse reflected light and total transmitted light are mainly measured by the double integrating sphere (DIS) method and the single integrating sphere (SIS) method. In optical measurements with DIS [28,29,30,31,32,33,34], the sample is sandwiched between two ISs, and the diffuse reflectance is measured with the IS in front of the sample and the total transmittance is measured with the IS behind the sample. Since the reflection and transmission properties of a scattering sample can be obtained in a single measurement, optical measurements can be performed while modifying sample conditions, such as changes in temperature [28]. On the other hand, disadvantages of the method include the loss of side scattering light and an inability to measure reflection and transmission under the same conditions because the reflected light and transmitted light are measured with a different IS [38]. SIS is a technique for measuring total transmittance by placing a sample at the entrance of the IS; diffuse reflectance is measured by placing a sample at the diffuse reflectance measurement port (the opening opposite the entrance) of the IS [35,36,37,38,39,40,41,42]. Since only one IS is used, the reflected light and transmitted light can be measured under the same conditions. However, it is difficult to measure changes in the state of the sample, such as the temperature and time dependence of optical properties, because two measurements must be performed on a single sample.

In response to these, we attempted to estimate the relationship between the measured absorbance and the optical properties of the sample (absorption coefficient, scattering coefficient, and anisotropy coefficient) by MC [18,24] when the sample is placed inside an IS. With this method, it is possible to obtain the absorption properties of a non-fluorescent microparticle suspension in a single measurement with a single IS. In this paper, model samples with known optical properties were prepared at various concentrations, and the MC, with input information on the absorption coefficient, scattering coefficient, and scattering anisotropy of the samples, reproduced the concentration dependence of the absorbance measurement results from low absorbance to the measurement limit when the samples were placed inside the IS.

2. Materials and Methods

2.1. Sample Preparation

To evaluate the accuracy of the MC simulation, an absorption-scattering sample was prepared by mixing absorbers (non-scattering in the measurement wavelength region) and scatterers (non-absorbing in the measurement wavelength region) with known optical properties, where the measurement wavelength range was 300–800 nm. A trypan blue (Sigma-Aldrich, St. Louis, MO, USA) aqueous solution was used as the absorber and an aqueous suspension of polystyrene microspheres (Polysciences Inc., Warrington, PA, USA) with a diameter of 500 nm was used as the scatterer. A pseudo-scattering sample M was prepared by mixing 120 μM of trypan blue with polystyrene microspheres at a concentration of 0.0364 spheres/μm³. A set of sample Bs was prepared with nine levels of dilution of the mixed sample M as volume fractions of 2.5, 5.0, 12.5, 25.0, 37.5, 50.0, 62.5, 75.0%, and 100% (no dilution) with purified water.

2.2. Setup

The absorption spectra were measured by placing the samples in the arrangement shown in Figure 1b on the IS built into the spectrophotometer (SolidSpec-3700DUV, SHIMADZU CORPORATION, Tokyo, Japan). A top view of the setup is shown in Figure 2. The measurement was performed using the double-beam method, with the signal light hitting the sample through aperture S and the reference light entering the IS through aperture R. The light intensity was detected with a detector attached to the bottom of the IS. The detected intensity is the intensity of the signal light hitting the sample divided by the detected intensity of the reference light [9]. This reduced the intensity fluctuations of the light source [43]. The absorbance $A$ was calculated from the detected intensity $I_0$ of the empty IS and the detected intensity $I$ when the sample was placed inside the IS, using the following equation.

$$A=-{\log }_{10}\left(\frac{I}{I_0}\right).$$

A liquid sample of 600 μL was placed in a custom-made cylindrical cell made of quartz glass (Figure 2a), and the optical cell was placed into the IS as shown in Figure 1b and Figure 2b.

2.3. Optical Properties of Samples

To reproduce the absorbance measurement results by MC, information on the absorption coefficient of the absorber (trypan blue); scattering coefficient of the scatterer (polystyrene microspheres); and scattering anisotropy (the expected value of the cosine of the scattering angle) of sample M was needed. These optical properties were obtained by the following procedure, respectively. The absorption coefficient of the absorbent was obtained by the same procedure used in [12], where the absorbent, a trypan blue solution, was placed in a rectangular optical cell (made of quartz glass) with a 10 mm optical path length and measured in the arrangement shown in Figure 1a. The measurement results are shown in Figure 3. From the absorption spectrum (left), the concentration–absorption relationship at the absorption peak wavelength of 591 nm was extracted (right), and a linear fit was performed to obtain the relationship between the concentration and the absorption coefficient of trypan blue. The scattering and anisotropy coefficients of polystyrene microspheres were then calculated from the complex refractive index and diameter of the polystyrene microspheres based on the Mie scattering theory. For the calculation of the Mie scattering theory, we used the program “wiscombe-mie-code” written in Fortran, which was installed from [44]. The values of the real part of the complex refractive index of polystyrene microspheres were taken from [45]. The imaginary part was set to 0, as the absorption of visible light is negligible [46]. The input parameters used in the Mie scattering theory calculations and the calculated optical properties are shown in

Table 1. Optical properties of polystyrene microspheres (diameter: 500 nm) irradiated with 591 nm light are calculated by a numerical algorithm based on Mie scattering theory. The scattering efficiency, absorption efficiency, and anisotropy coefficient of polystyrene microspheres were calculated by a numerical calculation program based on Mie scattering theory using the wavelength of irradiated light, particle diameter and complex refractive index of polystyrene microspheres as input parameters.

Input Parameters				Output Parameters
Wavelength [nm]	Diameter [nm]	Refractive Index		Scattering Efficiency ${\mathit{\mu }}_{\mathit{s}}$	Absorption Efficiency ${\mathit{\mu }}_{\mathit{a}}$	Anisotropy Efficiency $\mathit{g}$
		Real	Imaginary
591	500	1.585	0.000	0.8346288	0.0000000	0.8394426

.

2.4. Monte Carlo Simulation (MC)

The behavior of the photon inside the IS and the sample was simulated in three dimensions by ray tracing with MC to estimate the measured absorbance $A$. MC was performed by an executable program (written in Fortran) that reproduced the actual measurement setup (Figure 2), including the size of the IS, the shape of the sample and its arrangement inside the IS, and the optical properties of each sample (absorption coefficient ${\mu }_a$, scattering coefficient ${\mu }_s$, anisotropy coefficient $g$ obtained in the procedure in Section 2.3) as input parameters. The behavior of the photons inside the IS and the sample was simulated in three dimensions by MC. Following the flowchart shown in Figure 4, signal light (incident through the aperture S in Figure 2) and reference light (incident through the aperture R in Figure 2) simulations were performed for 50,000 photons. This number of photons was set to a sufficient number of simulations so that the standard deviation of the simulation results was sufficiently small (Appendix A). The photon is initially given a weight $W=1$, and each time there is absorption by the inner walls of the IS or by the sample, $W$ is attenuated by an amount equivalent to the amount of absorption. For example, when traveling a distance p in an absorbing medium with absorption coefficient ${\mu }_a$, $W$ is attenuated from $W\to W{e}^{-{\mu }_{a}p}$. The direction in which photons are reflected by the inner walls of the IS, whether they are reflected at the sample surface, and the direction of scattering by scatterers inside the sample were determined by random numbers based on the probability density distribution of each physical phenomenon [47]. The details of the simulation are as follows.

2.4.1. Photon Initialization

The initial position of the photon at the aperture of the IS was determined by a function based on the Gaussian distribution which assumed that the incident light had an intensity distribution that followed a Gaussian distribution. The spread of the incident light (dispersion of the Gaussian distribution) was set so that the saturated absorbance obtained from the measurement results matched the saturated absorbance obtained from the MC. This is because the spread of the incident light (dispersion of the Gaussian distribution) is reflected in the measurement limit of absorbance [12]. $W$ was also initialized to $W=1$.

2.4.2. Inside the IS

A photon travels straight until it collides with the inner walls of the IS or the surface of the sample, and when the photon collides, an event is generated that corresponds to the point of impact. If a photon reaches the inner wall of the IS, it is reflected by the inner wall. The direction of reflection is determined randomly, assuming that the reflection has equal probability in all directions in the range of the elevation angle 0~π/2 and azimuth angle 0~2π. $W$ is reduced by a small amount of absorption by the inner wall and is changed to $W\to R_{i}W$. $R_i$ is the reflectance of the inner wall, which was determined by measuring the relative reflectance to the standard white plate with the diffuse reflectance measurement of the IS. When it reaches the surface of the sample, it is determined whether it is reflected or refracted on the sample surface by a random number according to the reflectance of that surface. The reflectance at the sample surface was derived from Fresnel’s equation. The reflectance was calculated from the angle of incidence on the sample under the following conditions: the incident light was unpolarized, the refractive index of the sample was 1.46 (quartz glass), which is the refractive index of the optical cell, and the refractive index outside the sample (inside the IS) was 1 (air). If it was transmitted, it was then processed inside the sample (Section 2.4.3). If it was reflected, the direction of travel was changed to the direction of specular reflection and sent straight inside the IS again.

2.4.3. Inside the Sample

The light that enters the sample travels straight for a distance $s$ until it hits the scatterer and is scattered by the scatterer. $s$ is determined by a random number based on the probability $p_{dis}\left(s\right)=1-e^{-{\mu }_{s}s}$ of traveling a distance $s$ without ever hitting a scatterer. If it reaches the sample surface before reaching the scatterer, it travels to the boundary (distance $p$) and determines whether it is reflected or refracted on the boundary by a random number using the same procedure as in Section 2.4.1. If the photon is refracted, it exits the sample and returns to the IS’s internal processing (Section 2.4.1). In the case of reflection, the direction of travel is changed to the direction of specular reflection, and the travel distance $s$ to the scatterer is changed to $s\to s-p$ where $s$ is subtracted from $p$, and the sample is moved straight through the sample again. If it reaches the scatterer, it is scattered and changes its direction of travel. The scattering direction is determined by a random number based on the scattering phase correlation number $p_{sca}\left(\cos \theta \right)$. $p_{sca}\left(\cos \theta \right)$ is given by the Henyey–Greenstein function, an approximation of the Mie scattering theory [47], expressed by

$$p_{sca}\left(\cos \theta \right)=\frac{1-g^2}{2\left(1+g^2-2g\cos \theta \right)}.$$

(1)

Each time the photon travels inside the sample, it is absorbed from the absorbing medium, so the weight of the photon is changed to $W\to W{e}^{-{\mu }_{a}d}$ for the travel distance $d$.

2.4.4. End of Photon Processing and Estimation of Absorbance

The three end conditions of the simulation for a single photon are: the photon reaches the detector and is detected, reaches the aperture, exits the IS and W is negligibly small ($W<W_{end}=1.0\times {10}^{-6}$). If a photon is detected, the current $W$ is added to the sum of the $W$ of the detected photons, $W_{total}$. The average value of $W_{total}$, $W_{total}/N$, corresponds to the ratio of the detected intensity to the incident light intensity ($W=1$), i.e., the measured transmission $\overline{T}$. Because of the double-beam system, $\overline{T}$ is expressed as $\overline{T}={\overline{T}}_s/{\overline{T}}_R$, where ${\overline{T}}_s$ is the incident light on the sample divided by ${\overline{T}}_R$, the reference light. Thus, the measured absorbance $A$ is calculated by

$$A=-{\log }_{10}\overline{T}=-{\log }_{10}\left(\frac{{\overline{T}}_s}{{\overline{T}}_R}\right).$$

(2)

3. Results

3.1. Absorbance Measured When a Sample Is Placed inside the IS

Figure 5 shows the absorbance measurement results for the setup in Figure 2. A broad absorption peak at 591 nm was observed for both the pure trypan blue solution (non-scattering sample), and the sample M (scattering sample) which was mixed with polystyrene microspheres as scatterers. In general, according to the Lambert–Beer law, absorbance and concentration should have a linear relationship, but when the sample is placed inside the IS, the relationship becomes nonlinear. In addition, sample M, which contained scatterers, and showed a smaller measurement limit than the pure trypan blue solution. Even in the case of pure trypan blue solution, the saturation absorbance is about Abs. = 1.5 which is less than the instrument limit (Abs. = 5).

3.2. Reproduction of Absorbance Measurement Results by MC

Next, the absorbance estimated by the MC was compared to the measured absorbance. Figure 6a and Figure 6b compare the absorbance estimated by MC with the measurements shown in Figure 5a and Figure 5b, respectively. Absorption spectrum estimates by MC were performed every 10 nm in the wavelength range 400~700 nm for concentrations of 9.0, 30.0, and 120.0 μM of trypan blue. Figure 6 shows that MC can reproduce the absorbance measurement results when the sample is placed inside the IS for both non-scattering and scattering samples.

4. Discussion

Sample M (the scattering sample), which was a trypan blue solution (non-scattering sample) mixed with non-absorbing scatterers, had the same absorption coefficient if the absorber, trypan blue, was at the same concentration. Therefore, it was expected that there would be no significant difference in the concentration dependence of the measured absorbance of the two when the absorbance measurement was performed considering total scattered light by placing a sample inside the IS. However, as is shown in Figure 6, there were differences in the shapes of the absorption peaks and saturation absorbance depending on the presence or absence of scattering. We considered this to be due to the fact that the distance that light travels inside the sample changed with the presence of scattering.

The absorbance is generally proportional to ${\mu }_{a}d$ according to the Lambert–Beer law, which is the product of the absorption coefficient ${\mu }_a$ and the distance $d$ that light passes through the sample until it is detected. Therefore, the smaller the value of $d$, the smaller the amount of the increase in absorbance $A$ with respect to the increase in absorption coefficient ${\mu }_a$. Based on the above, the reason why the increase in absorbance relative to the increase in concentration (increase in ${\mu }_a$) becomes smaller as the concentration (absorption) of the sample increases is considered as follows: light with relatively small $d$ is detected more often as the concentration of the sample increases. In order to verify this, we estimated the distribution of $d$ of the detected light for signal light (incident light from the aperture S shown in Figure 2) by MC. Figure 7 is a histogram of $d$ by the MC showing what percentage of the photons detected were classified into each optical path length range. The region $L_{mid}$ is the range of $d$ that can be taken when signal light travels straight through the sample, and it is classified into region $L_{long}$ or $L_{short}$ if it is longer or shorter than $L_{mid}$, respectively. If the photon does not transmit through the sample due to reflection on its surface, the photon is classified as $d=0$.

For non-scattering samples, there are no photons classified in the region $L_{short}$. This is because light does not exit on its way through the sample by scattering as the non-scattering sample has no scatterers. Therefore, light travels straight through the sample if it is not reflected at the sample’s surface. In the case of a high-absorption sample, since the transmitted light becomes weak, the light of $L_{long}$, $L_{mid}$ is buried in the light of $d=0$. For light with $d=0$, $A$ does not increase even if ${\mu }_a$ increases and the increase in absorbance plateaus. This is also explained in the absorbance correction formula when the sample is placed inside the IS, as shown in the previous paper [12].

On the other hand, in the case of the scattering sample, there are photons contained in $L_{short}$, and the ratio increases as the concentration (absorption) of the sample increases. It is suggested that the presence of photons classified as $L_{short}$ makes the $d$ of the scattering samples smaller than that of the non-scattering samples in the high-concentration (high-absorption) region, resulting in lower saturation absorbance. The reason why the ratio of $L_{short}$ is higher at higher concentrations is that the amount of light exits the sample while traveling straight through the sample due to scattering increases as the concentration of the scatterer increases. At the same time, there is also an increase in the optical path length due to scattering, but when $d$ is sufficiently long in the high-absorption region, the intensity is so weak that almost no light is absorbed.

5. Conclusions

Absorbance measurements were performed considering the effect of scattering by placing the sample inside the IS and using it to collect scattered light at all solid angles. Using this method, the distance $d$ that light passes through the sample before it is detected does not match the thickness of the sample due to the repeated scattering of light on the inner walls of the IS, which takes various optical paths inside the IS, and in the case of a scattering sample, takes various paths inside the sample due to scattering. Therefore, it is difficult to quantitatively evaluate the absorption coefficient ${\mu }_a$ from the Lambert–Beer law ($A={\mu }_{a}d{\log }_{10}e$). Therefore, we estimated the relationship between the measured absorbance and absorption coefficient by simulating the optical path tracking inside the IS and inside the sample, which reflected the actual optical setup by MC.

For the scattering sample M with known optical properties and the non-scattering sample for comparison, absorbance measurements were performed with the sample inside the IS, respectively, and the measured absorbance estimated by the MC was compared with the measured results. A trypan blue solution was used for the non-scattering sample, and sample M was made by mixing it with polystyrene microspheres as scatterers. The MC results reproduced the experimental results well, reproducing the shape of the absorption spectrum and the behavior of absorbance with respect to sample concentrations at the peak absorption wavelength (591 nm) for both non-scattering and scattering samples.

In the absorbance measurements inside the IS, the scattering samples showed a smaller saturation absorbance than the non-scattering samples. We consider that this is because, in the case of scattering samples, $d$ decreases with increasing sample concentrations, resulting in a smaller increase in absorbance with increasing sample concentrations, because the absorbance depends on the product of the concentration of the sample and $d$ according to to the Lambert–Beer law. This was verified by estimating the $d$ distribution of the detected light by MC. The simulation results showed that for the scattering sample, the distribution of $d$ of the detected light is biased toward smaller values as the concentration (absorption) of the sample increases, compared to the non-scattering sample. This is thought to be because the higher the concentration, the more scatterers there are, increasing the probability that light will travel out of the sample by scattering on its way straight through the sample.