Quasi-Invariance of Scattering Properties of Multicellular Cyanobacterial Aggregates

Abstract

The radiative/scattering properties of cyanobacterial aggregates are crucial for understanding microalgal cultivation. This study analyzed the scattering matrix elements and cross-sections of cyanobacterial aggregates using the discrete dipole approximation (DDA) method. A stochastic random walk approach was adopted to generate a force-biased packing model for multicellular filamentous cyanobacterial aggregates. The effects of the shape and size of multicellular cyanobacterial aggregates on their scattering properties were investigated in this work. The possibility of invariance in the scattering properties of cyanobacterial aggregates was explored. The invariance interpretation intuitively represented the radiative property characteristics of the aggregates. The presented results show that the ratios of the matrix elements of cyanobacterial aggregates are nearly shape-, size-, and wavelength-invariant. The extinction and absorption cross-sections (EACSs) per unit volume exhibited shape and approximate size invariance for cyanobacterial aggregates, respectively. The absorption cross-section of aggregates is not merely a volumetric phenomenon for aggregates that exceed a certain size. Furthermore, the absorption cross-sections per unit volume are independent of the volumetric distribution of the microalgae cells. The invariance interpretation presents crucial characteristics of the scattering properties of cyanobacterial aggregates. The existence of invariance greatly improves our understanding of the scattering properties of microalgal aggregates. The scattering properties of microalgal aggregates are the most critical aspects of light propagation in the design, optimization, and operation of photobioreactors.

1. Introduction

Microalgae-based biomaterials are promising renewable energy sources that have garnered considerable attention in the pharmaceutical industry [1], environmental protection [2], carbon-neutral biofuels [3], and space food supplements [4]. It has been widely recognized that microalgae are renewable green energy sources, meaning that microalgae biomass has no net CO₂ emissions in its usage. This balance is achieved through microalgae-based carbon dioxide (CO₂) absorption in culture and CO₂ release during biomass utilization in downstream industries. Therefore, microalgae biomass is conducive to achieving carbon neutrality objectives [5,6].

Currently, the high cultivation costs of microalgae seriously limit their practical application [7]. Although we have extensive knowledge of microalgal photosynthesis and production, microalgae cultivation is affected by light intensity, nutrient concentration, pH, and temperature [8,9,10]. The light field in a microalgal suspension is one of the most influential factors. Excessive light supply can lead to photoinhibition, whereas insufficient illumination negatively impacts the photosynthesis of algal cells [11]. The interaction between algal cells and illumination can be represented by the scattering properties of microalgae cells, including the absorption, scattering, and extinction cross-sections, the scattering phase function (SPF), and the Mueller matrix elements. Therefore, the scattering properties are regarded as crucial factors that can impact light transfer within microalgae cultures.

Admittedly, previous studies measured considerable data related to the microalgal radiation characteristics. Berberoglu and Pilon [12] experimentally measured the EACSs of Anabaenavariabilis and Rhodobacter sphaeroides at wavelengths of 300–1300 nm and the SPF of microalgae at a visible wavelength. Berberoglu et al. [13,14] also reported the measured radiation characteristics of microalgae, including the absorption and scattering cross-sections, SPF, and size distribution of algae cells for more than seven different strains, such as Botryococcus braunii, Chlorococcum littorale, and Chlamydomonas reinhardtii. Accordingly, Heng and Pilon [15] experimentally studied the radiation characteristics of Nannochloropsis oculata at different growth times under a batch culture condition. Liu’s group studied the time-dependent radiation characteristics of typically green and cyanobacterial microalgae and proposed a temporal scaling law for describing the optical properties that vary with cultivation time [16,17,18]. Therefore, research on microalgal radiative properties has experienced a transition from a stationary phase to a growth phase, representing an improved understanding of microalgae growth.

In addition to the progress in experimental measurements, previous studies have also analyzed the scattering properties of microalgae by conducting theoretical studies. The Lorentz–Mie theory is typically employed to estimate the scattering properties of spherical unicellular microalgae cells [19,20]. The effects of the internal or external structures of cells on microalgal scattering properties were studied by Dauchet et al. [21] and Dong et al. [22]. Bhowmik and Pilon [23] further studied whether the optical properties of unicellular photoautotrophic spherical-shaped algae cells can be described using a single optical homogeneous-appropriate spherical cell or a two-layer spherical model using the T-matrix calculation approach. They verified that the homogeneous spherical-shaped framework could accurately predict the EACSs of internally non-uniform microalgae. Heng et al. [24] and Kandilian et al. [25] reported the optical properties of regular and irregular colonies of spherical monomers and their effective particle approximation frameworks. Lee and Pilon [26] studied the absorption and scattering of long and randomly oriented sphere chains to approximate some realistic cyanobacteria. These studies found that the absorption and scattering cross-section per unit length of chain-shaped microalgae could be determined by using infinitely long cylinder solutions. Ma et al. [27] investigated the spectral radiation characteristics of multicellular cyanobacteria using different cell shapes and equivalent optical models. It was discovered that the volume-equivalent homogeneous cylinder model provided good predictions on EACSs cross-sections and polarization-dependent Mueller matrix elements of multi-cellular filamentous cyanobacteria. These results show that microalgal radiation characteristics could be determined by employing optically homogeneous models for both spherical microalgae and multicellular filamentous cyanobacteria. Hoeniges et al. [28] studied the scattering and absorption characteristics of fractal cell colonies of sphere-shaped Botryococcus braunii and the effective coated sphere approximation. However, the scattering properties of multicellular filamentous cyanobacterial aggregates remain an important problem yet to be studied. This study attempted to explore the spectral scattering properties of cyanobacterial aggregates with different shapes, specifically the effect of the scattering properties of multicellular, long chain-like cyanobacteria aggregates on explaining and understanding microalgal growth.

This study investigated the scattering properties and provides theoretical methods for cyanobacteria aggregates, which are important to comprehend and measure how light interacts with the aggregation of photosynthetic microalgae. The DDA numerical method was adopted to report the spectral scattering properties of multicellular filamentous cyanobacterial aggregates. The effects of the shape and size of the microalgae aggregates on the scattering matrix elements and cross-sections were studied. To interpret the results, the cross-section per unit volume was introduced to represent the characteristics of the microalgal aggregates. In addition, the internal electric fields within the microalgal aggregates were presented to explain the physical mechanism behind the variation in the cross-section. This research will improve the understanding of the scattering properties of microalgal aggregates and advance the investigation of the radiation characteristics of multicellular filamentous cyanobacterial aggregates.

2. Models of Photoautotrophic Cyanobacteria Aggregates

2.1. Aggregates of Cyanobacteria

Nostoc sp. was purchased from the FACHB freshwater algae species bank, which is located in Wuhan, China. Nostoc sp. was cultivated in a bioreactor with a working volume of 100 mL. The bioreactor was put into the incubator under cultivation conditions of 12 h of darkness followed by 12 h of light, and it was exposed to fluorescent light bulbs with an illumination of 2000–2500 lx. The temperature of the cultivation environment was maintained at 25 °C. The BG11 medium was used for the cultivation of Nostoc sp.

Microalgae are photosynthetic microorganisms that grow in natural environments or artificial photobioreactors. Microalgae cultures form nonmotile colonies under certain environmental conditions [29]. For example, Ratcliff et al. [30] demonstrated the multicellular complexity of C. reinhardtii, which develops from a single cell. Some microalgae strains secrete exopolysaccharides, which are viscous substances coating the surfaces of microalgal cells, causing their aggregation into colonies [31]. Moreover, microalgae aggregation occurs at high cell concentrations, owing to the electrostatic force between the cells [32].

Figure 1 shows the micrographs of Nostoc sp. at different magnifications. As shown, the Nostoc sp. is 2–4 μm in diameter and more than 40 μm in length, and the cells are approximately oval-shaped. Cyanobacteria are usually multicellular with diverse cell shapes, such as spherical, oval, and cylindrical [26].

2.2. Aggregates Model Establishing

Microalgal cells typically form aggregates during cultivation. The fractal model is typically used to simulate a spherical and unicellular microalgae colony. Cyanobacteria, however, are multicellular species, and their shape consists of long linear or curved chains. The shape of cyanobacteria is more likely to resemble curved fibers when details are ignored. We used a stochastic model to generalize the force-biased packing method for multicellular filamentous cyanobacteria in the form of a spherical chain, which can take into account cyanobacteria chain details. This approach is based on the random walk method as follows [33]:

(1)$P=\left\{p_0,\dots ,p_n\right\},p_i=\left(x_i,{\mu }_i,r_i\right)$

where $P$ is the sequence of random walk points. Here, it should be understood that the point of random walk is related to its radius and current direction. The initial coordinate, $x_0$, is distributed in a cubic uniformly, the radius, $r$, and the number of samples, $n$, can be fixed for the entire system. The orientation can be generated from the $\beta$ orientation distributions as follows [34]:

(2)$p\left(\theta ,\varphi |\beta \right)={\displaystyle \frac{\beta \sin \theta }{4\pi {\left[1+\left({\beta }^2-1\right){\cos }^2\theta \right]}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$

where $\left(\theta ,\varphi \right)$ are the polar coordinates of the orientation, ${\mu }_0$. The global parameter, $\beta$, describes the features of orientation distributions, and $\beta =1$ was selected for isotropic orientation distribution. The bending characteristics of cyanobacterial aggregates are indirectly controlled by two parameters, ${\kappa }_1$ and ${\kappa }_2$, in the multivariate von Mises–Fisher distribution as follows [33]:

(3)$f\left(\mu |{\mu }_1,{\kappa }_1,{\mu }_2,{\kappa }_2\right)=c\left({\mu }_1,{\kappa }_1,{\mu }_2,{\kappa }_2\right)\exp \left({\kappa }_1{\mu }_1^T\mu +{\kappa }_2{\mu }_2^T\mu \right)$

where ${\mu }_1$ and ${\mu }_2$ are the two preferred directions, and ${\kappa }_1$ and ${\kappa }_2$ denote the reliability parameters that control bending. The factor $c\left({\mu }_1,{\kappa }_1,{\mu }_2,{\kappa }_2\right)$ was used for normalization. Here, the preferred direction is determined based on the initial chain direction and the direction of the previous point. Therefore, ${\kappa }_1$ and ${\kappa }_2$ describe the reliability of the main orientation and last orientation of the fiber, respectively, and hence specify the bending smoothness. The cell aggregate packing applies a repulsion force to avoid crossing chains and a recovery force to ensure that the chain structure obtains a hard-core aggregate system [33]. The final formation of aggregates by the random walk approach is obtained as a balance between the repulsive and restorative forces. The actual aggregate model generation in this process will be stopped by a criterion in which the total force strength is reduced to a certain limit. The resulting cell chains can be used as a model of cyanobacterial aggregates for subsequent calculations.

Figure 2 shows the model realizations using input parameters for the quantity of cyanobacteria components, N = 121, with an isotropic distribution of cyanobacteria orientation. The number of cyanobacteria objects in the aggregates was chosen based on the required volume fraction. Additionally, the realized volume fraction for each implementation is provided below.

The Gaussian random sphere’s radius vector in spherical dimensions is defined as follows [35]:

(4)$\mathbf{r}\left(\vartheta ,\phi \right)={\displaystyle \frac{a\exp \left[s\left(\vartheta ,\phi \right)\right]}{\sqrt{1+{\sigma }^2}}}{\mathbf{e}}_r$

(5)$s\left(\vartheta ,\phi \right)=\sum _{l=0}^{\infty }\sum _{m=-l}^l{s}_{lm}{Y}_{lm}\left(\vartheta ,\phi \right)$

where $a$ and $\sigma$ are the mean and relative standard deviation of the radius vector, $\left(\vartheta ,\phi \right)$ are the spherical coordinates, $s$ is the so-called logradius, ${\mathbf{e}}_r$ is a unit vector pointing outward in the direction $\left(\vartheta ,\phi \right)$, $Y_{lm}$ is the orthonormal spherical harmonic, and $s_{lm}$ is the Gaussian random variable with zero means. The real and imaginary parts of a Gaussian random variable with zero mean and variance can be independently expressed as follows [35]:

(6)$\mathrm{Var}\left[\mathrm{Re}\left(s_{lm}\right)\right]=\left(1+{\delta }_{m0}\right){\displaystyle \frac{2\pi }{2l+1}}\ln \left(1+{\sigma }^2\right)C_l,$

(7)$\mathrm{Var}\left[\mathrm{Im}\left(s_{lm}\right)\right]=\left(1-{\delta }_{m0}\right){\displaystyle \frac{2\pi }{2l+1}}\ln \left(1+{\sigma }^2\right)C_l, l=\mathrm{0,1},\dots ,\infty , m=\mathrm{0,1},\dots ,l,$

where ${\delta }_{m0}$ is the Dirac delta function, and $C_l$ is the nonnegative weight of degree $l$ in the Legendre polynomials. Moreover, the covariance function, ${\sum }_s\left(\gamma \right)$, describes the auto-covariance of the random variables $s\left({\vartheta }_1,{\phi }_1\right)$ and $s\left({\vartheta }_2,{\phi }_2\right)$, which is given by a series of Legendre polynomials as follows [35]:

(8)${\sum }_s\left(\gamma \right)=\ln \left(1+{\sigma }^2\right)\sum _{l=0}^{\infty }{C}_l{P}_l\left(\cos \gamma \right)$

where $\gamma$ represents the angular distance between the two directions of $\left({\vartheta }_1,{\phi }_1\right)$ and $\left({\vartheta }_2,{\phi }_2\right)$. For the Legendre polynomials to be valid, the following is required:

(9)$\sum _{l=0}^{\infty }{C}_l=1$

Practically, the infinite series representations in Equations (5)–(9) generally need to be truncated at some degree by $l_{\max }$. Thus, the Gaussian random sphere can be fully determined by the mean radius, $a$, and the Legendre polynomial coefficient, $C_l$. In addition, the conventional power-law covariance function for the representation $C_l$ can usually be used as follows [36]:

(10)$C_l={\displaystyle \frac{\tilde{C}}{l^{\nu }}}, l=\mathrm{2,3},\dots ,l_{\max }$

where $C_0=C_1=0$, and $\tilde{C}={\left[{\displaystyle \sum _{l=2}^{l_{\max }}}{l}^{-\nu }\right]}^{-1}$ is the normalization constant. Therefore, the Gaussian random sphere can be determined using the relative standard deviation, $\sigma$, power-law index, $\nu$, and truncated degree, $l_{max}$. The power-law covariance function was chosen because it facilitates fine control of model size and translation variation, and it has been widely used to model natural cells of various shapes.

Figure 3 illustrates the Gaussian-shaped microalgal aggregate system. The results are obtained by combining Gaussian random spheres and cyanobacterial aggregates of random walks. The Gaussian random sphere was generated using the same random sequence of numbers; therefore, the shapes’ differences result from different weightings to the spherical harmonics coefficients, $s_{lm}$. The random walks of the realization of the packed microalgae system were the same for the two different Gaussian sphere cases. Therefore, all the differences in the cases Figure 3a,b are mainly due to the external shape of the Gaussian sphere.

3. Scattering Theory and Methods

3.1. Lorenz–Mie Theory Review

Lorenz–Mie scattering is regarded as an important solution to Maxwell’s equations for optically homogenous spheres. It is crucial to understand the scattering by spherical particles. Here, the external shape of the cyanobacterial aggregates was selected and considered to be approximately spherical. Therefore, the Lorenz–Mie scattering theory can be adopted to analyze and calculate the scattering/optical properties of microalgal aggregates. According to the electromagnetic scattering theory [37], the incident field, and scattered field can be expressed using vector spherical harmonics for a homogenous sphere. The coefficient of the external scattering field can be expressed as follows:

(11)$a_n={\displaystyle \frac{m{\psi }_n\left(mx\right){\psi }_n^{\prime }\left(x\right)-{\psi }_n\left(x\right){\psi }_n^{\prime }\left(mx\right)}{m{\psi }_n\left(mx\right){\xi }_n^{\prime }\left(x\right)-{\xi }_n\left(x\right){\psi }_n^{\prime }\left(mx\right)}}$

(12)$b_n={\displaystyle \frac{{\psi }_n\left(mx\right){\psi }_n^{\prime }\left(x\right)-m{\psi }_n\left(x\right){\psi }_n^{\prime }\left(mx\right)}{{\psi }_n\left(mx\right){\xi }_n^{\prime }\left(x\right)-m{\xi }_n\left(x\right){\psi }_n^{\prime }\left(mx\right)}}$

The expression for the coefficients of the internal electromagnetic field can be written as follows:

(13)$c_n={\displaystyle \frac{m{\psi }_n\left(x\right){\xi }_n^{\prime }\left(x\right)-m{\xi }_n\left(x\right){\psi }_n^{\prime }\left(x\right)}{{\psi }_n\left(mx\right){\xi }_n^{\prime }\left(x\right)-m{\xi }_n\left(x\right){\psi }_n^{\prime }\left(mx\right)}}$

(14)$d_n={\displaystyle \frac{m{\psi }_n\left(x\right){\xi }_n^{\prime }\left(x\right)-m{\xi }_n\left(x\right){\psi }_n^{\prime }\left(mx\right)}{m{\psi }_n\left(mx\right){\xi }_n^{\prime }\left(x\right)-{\xi }_n\left(x\right){\psi }_n^{\prime }\left(mx\right)}}$

where $x$ is the size parameter, and $m$ represents the complex optical constants of the sphere medium. The Riccati–Bessel functions are expressed as follows:

(15)${\psi }_n\left(x\right)=x{j}_n\left(x\right),{\xi }_n\left(x\right)=x{h}_n\left(x\right)$

where $j_n\left(x\right)$ and $h_n\left(x\right)$ are spherical Bessel and Hankel functions, respectively. Observing the expressions for the coefficients of the internal and external electromagnetic fields, it can be seen that the denominators of $a_n$ and $d_n$ are identical, as well as the denominators of $b_n$ and $c_n$. This indicates that the internal field and scattered field modes are resonant for the same parameters of $x$ and $m$. This can be used to analyze and understand the relationship between the internal and external scattered electric fields. The scattering cross-section can be obtained by integrating the scattered power over all directions, and the extinction cross-section can be obtained using the optical theorem as follows [37]:

(16)$C_{sca}={\displaystyle \frac{2\pi }{k^2}}\sum _{n=1}^{\infty }(2n+1)({\left|a_n\right|}^2+{\left|b_n\right|}^2)$

(17)$C_{ext}={\displaystyle \frac{2\pi }{k^2}}\sum _{n=1}^{\infty }(2n+1)\mathrm{Re}(a_n+b_n)$

where $k$ denotes the wavenumber. The abovementioned formulas provide an understanding of individual particles’ radiative/optical properties. We have reviewed the Lorenz–Mie theory above, although the calculation of the theory has been directly demonstrated once. The analytical solution of the Mie theory has high computational efficiency compared to other numerical methods, especially for large particles or microalgal aggregates. Moreover, the Lorenz–Mie theory is essential for understanding the scattering properties of cellular particles and their aggregates.

3.2. Scattering by Particles

According to the electromagnetic theory, the incident and scattered fields can be related to the amplitude scattering matrix $\mathbf{S}$ for an arbitrary particle as follows [37]:

(18)${\mathbf{I}}_s={\displaystyle \frac{1}{k^2{r}^2}}\mathbf{S}{\mathbf{{\rm I}}}_i$

where ${\mathbf{I}}_i$ and ${\mathbf{I}}_s$ represent the incident and scattering Stokes vector parameters of the incident light field, respectively, $r$ represents the distance from the scatterer to observation points, and $k$ is the wave number. The amplitude scattering matrix is block-diagonal with $S_{11}$, $S_{12}=S_{21}$, $S_{22}$, $S_{33}$, $S_{34}=-S_{43}$, and $S_{44}$ as non-zero elements for an arbitrary sphere, and $S_{11}$ and $S_{22}$ are equal, as are $S_{33}$ and $S_{44}$, which are a result of non-diagonal elements in the scattering amplitude matrix equal to zero [37]. The SPF, $\Phi \left(\cos \Theta \right)$, can be directly obtained using the amplitude scattering matrix element $S_{11}$ as follows [37]:

(19)$\Phi (\cos \Theta )={\displaystyle \frac{4\pi S_{11}}{{\mathrm{k}}^2{C}_{sca}}}$

This demonstrates how likely it is that light propagation in the solid angle, $\mathrm{d}{\Omega }^{\prime }$, around the direction, ${\mathbf{s}}^{\prime }$, will scatter into the solid angle, $\mathrm{d}\Omega$, around the direction, $\mathbf{S}$. For unpolarized incident waves, the scattering matrix ratio, $-S_{12}/S_{11}$, represents the linear polarization degree of the scattered light. Moreover, the element ratio, $S_{22}/S_{11}$, is equal to 1 for a perfect sphere and describes the non-spherical nature of the scattering grains [38].

3.3. DDA Method

The scattering properties of microalgal aggregates are studied by using the DDA method, which is a widely used approach to find field solutions for the scattering of arbitrarily shaped particles or aggregates [27,39,40]. The scatterer was divided into discrete dipoles to represent particles, or aggregates. The electric field values were calculated using a group of linear equations based on approximation. The DDA solution for biological particles is more accurate owing to the relatively complex refractive index that is close to unity in water [41]. Therefore, the DDA method has innate strength when calculating the scattering properties of microalgal colonies. In our DDA calculations, the ADDA code released by Yurkin et al. was used [42,43]. ADDA can run in parallel via MPI implementation, thereby reducing the calculation time. Furthermore, Inzhevatkin and Yurkin [44] reported using the Wentzel–Kramers–Brillouin method and its improved method as the initial guess in the DDA iterative solution process, which allowed us to accelerate the computational efficiency with the same accuracy. These features are beneficial in the calculation of the scattering properties of microalgal aggregates. Figure 4 shows the scattering matrix elements of the cubical microalgae aggregate model (Figure 2a) for different dipoles per lambda (dpl) values. The refractive index is 1.021, and the absorption index is 1.12 × 10⁻⁵. The matrix elements are orientation averaged over five different orientations. The size of the aggregate along the x direction is 16.8 μm. Evidently, the elements converged for dpl = 6 compared with those for dpl = 12. The value of dpl was larger than 6 in the following simulation to ensure convergence. The scattering characteristics were orientation averaged over five different orientations for each realization. The lattice dispersion relation (LDR) for the polarizability of cubic dipoles was used in the calculations.

3.4. Optical Constants

The major metabolite compositions of microalgae include proteins, carbohydrates, and fats [45]. The effective complex optical constants of microalgae can be estimated using the effective medium approximation, for example, the Maxwell–Garnett model [46] and Bruggeman’s approach [47]. In addition, the experimental refractive index of microalgae ranges from 1.004 to 1.167 because of the compositional variation of the components [19,45,48]. Moreover, the absorption index of the imaginary part of microalgae can be calculated according to the concentration of pigments in algae cells. Previous studies indicated that scattering is mainly influenced by the real-part mismatch of the optical constants between the aggregates and the peripheral aqueous media, while absorption originates from the absorption index [24,26,49]. Therefore, variations in the refractive index indicate differences in scattering.

Moreover, the optical constants can be approximately considered wavelength-independent constants in the visible spectral range, as observed experimentally and theoretically [45,48]. This indicates that the minor difference in the refractive index does not affect the conclusion of the scattering properties of microalgae. Here, the spectral dependent optical constants of microalgae were selected [19] to make the calculated results similar to the experimentally measured values. The spectral optical constants of the non-absorbing culture medium surrounding the cell can be determined using the Cauchy dispersion relation as follows [28]:

(20)$n_{\lambda }=A_1+{\displaystyle \frac{A_2}{{\lambda }^2}}+{\displaystyle \frac{A_3}{{\lambda }^4}}$

where the constants in the expression are $A_1$ = $1.32711$, $A_2$ = 2.6 × 10⁻³ μm², and A₃ = 5 × 10⁻⁵ μm⁴.

Figure 5 shows the spectral complex refractive index of microalgae cells and the refractive index of the culture medium. As shown, the refractive and absorption indices of the microalgae ranged from 1.344 to 1.351 and 2.118 × 10⁻⁵ to 1.647 × 10⁻³, respectively. The absorption peaks in the imaginary part of the complex optical constants correspond to in vivo chlorophyll a and chlorophyll b in the microalgae cells [50]. In addition, the small dips in the refraction index corresponding to the peaks, caused by chlorophyll a, can be attributed to oscillator resonance theory [37]. The spectral optical constants of the culture medium decreased monotonically from 1.348 to 1.332.

4. Results and Discussion

4.1. Size and Shape Invariance of Scattering Elements

The scattering matrix elements could be employed in applications that consider the polarization of scattered radiation, such as remote sensing and phytoplankton identification. Figure 6 shows the Mueller scattering matrix element ratios for spherical, cubic, and Gauss sphere-shaped aggregates as functions of scattering angles using the DDA method. The same kind of Mueller matrix element ratios were predicted for 16.6 and 33.4 μm of microalgae aggregates for each shape. As shown, the Mueller matrix element ratios, −S₁₂/S₁₁, S₃₃/S₁₁, and S₄₄/S₁₁, were identical for all shapes and sizes. The linear polarization degree of the algae aggregates, −S₁₂/S₁₁, reached 1 at a 90° scattering angle and was equal to 0 at 0° and 180° scattering angles. The Mueller matrix element ratios, S₂₂/S₁₁ and S₃₄/S₁₁, were similar for all aggregates, approximately equal to 1 and 0 for all scattering angles, respectively. The Mueller matrix element ratio, S₃₃/S₁₁, dropped from 1 to −1 when the scattering angle was changed from the forward to the backward direction, as did the element ratio S₄₄/S₁₁. The different Mueller matrix element ratios shown in Figure 6 for microalgae aggregates are similar to those for an individual sphere [51]. Moreover, no resonance peaks appeared in the Mueller matrix element ratios of the multicellular filamentous cyanobacterial aggregates. These results are consistent with the results discovered by Mackowski and Mishchenko [52], which illustrated that aggregates caused the damping of oscillations in the matrix elements. In addition, the disappearance of the oscillation was also attributed to the orientation averaging of the aggregates. Furthermore, the ratio, S₂₂/S₁₁, deviated from unity was also observed by Mishchenko et al. [53], which can be used as an indicator of the nonsphericity of the aggregates.

The S₁₁ element exhibits a significant difference at the forward scattering angle, which increases as the aggregate size increases [54] (see the Supplementary Materials). In addition to the increase in the forward direction of S₁₁, the scattering matrix elements can be approximately considered as shape- and size-invariant for microalgae aggregates.

The Mueller scattering matrix results are close to the Mie theory results. The reason for this phenomenon is that the relative complex refractive index of cell particles in water is very close to 1, and the volume-equivalent size parameter of cell aggregates ranges from 70 to 550, classifying them as optically large particles. Consequently, the Mueller scattering matrix of complex cell aggregates can be theoretically approximated using Mie theory, which greatly simplifies the calculation of the scattering matrix. This is presented as a quasi-invariance of the scattering matrix for microalgal aggregates.

4.2. Size and Shape Invariance of Cross-Sections

Figure 7 demonstrates the EACSs per unit volume of the colonies of microalgae aggregates for different sizes as 16.6, 33.4, and 66.5 μm. The cross-sections per unit volume of microalgae aggregates were defined as the cross-sections divided by the volume of cell aggregates. Figure 7a indicates that the extinction cross-section presents an increasing trend with the increase in wavelength for all aggregate sizes. The extinction cross-section displays variations for different aggregate sizes; however, all variations lie within the interval of 0.08 to 0.18. Despite some differences, the extinction cross-sections were remarkably similar. The absorption cross-section almost overlaps in the spectral regions of 510 to 650 nm and 680 to 750 nm, while the differences are relatively significant at absorption peaks around 450 nm. The non-overlapping of the cross-sectional parameters implies the presence of multiple scattering that cannot be ignored [55]. The absorption cross-sections corresponding to different sizes do not overlap, indicating that the absorption cross section is no longer a purely volumetric phenomenon [24,25,56]. This is attributed to the larger size of the aggregates and the absorption peaks in the absorption index, which are no longer simple values. The EACSs of the microalgae aggregates can be considered approximately size invariant when overlooking some details. Moreover, the larger the microalgae aggregate size, the smaller the absorption cross-section, which indicates that an increase in microalgae aggregate size will reduce light utilization efficiency.

Figure 8 shows the EACSs per unit volume of the colonies of microalgae aggregates for different shapes, i.e., spherical, cubical, and Gaussian. As shown, the variation trend of the extinction cross-section is consistent for different shapes, and they nearly overlap over the PAR region. The absorption peaks of the absorption cross-sections of different shapes correspond completely and overlap perfectly in the PAR spectral region. This confirms previous results reported by the authors of [24,25,56], illustrating that absorption is a volumetric phenomenon. Therefore, the EACSs per unit volume are independent of the microalgal aggregate shape, which we call the shape invariance of the EACSs. This implies that there is no need to consider the aggregate shape’s effect on the cross-sections’ scattering properties.

Figure 9 presents the EACSs per unit volume of the colonies of microalgae aggregates with equivalent volumes for different shapes. As shown, the variation trend of the extinction cross-section was similar for different aggregate shapes, but they did not overlap over the PAR region. This can be attributed to extinction not being a volumetric phenomenon. Nevertheless, the absorption peaks of the absorption cross-sections correspond completely for different shapes and nearly coincide in the PAR spectral region. Therefore, the absorption cross-sections can be considered independent of the volumetric distribution of the microalgae cells. This implies that the Lorenz–Mie theory can be employed to calculate the absorption cross-sections of microalgal aggregates under the volumetric phenomenon assumption.

To summarize, the EACSs per unit volume of the microalgal aggregates are shape- and approximate size-invariant. The absorption cross-section is not a fully volumetric phenomenon when the aggregate size is sufficiently large and the absorption peaks in the absorption index are considered.

4.3. Wavelength Invariance of Scattering Matrix Elements

As shown in Figure 10, scattering matrix elements of the colonies of spherically shaped microalgal aggregates. A previous section of this work validated that the Mueller matrix element ratios are independent of the shape and size of the aggregates. As shown, apart from S₁₁, the ratios of the scattering matrix element are also nearly wavelength-invariant. However, element S₁₁ is proportional to the SPF, which can be approximately independent of wavelength in radiative transfer studies of microalgae [57]. The main difference in S₁₁ over the PAR wavelength is the forward scattering. In addition, the microalgal asymmetry factor has been widely reported to be approximately 0.97 [13,58].

Furthermore, the light scattering in microalgae suspensions can be described using the H-G phase function to achieve reasonable accuracy [59]. The linear polarization degree of the scattered electromagnetic field, −S₁₂/S₁₁, which resembles a parabola that opens downward, is equal to 0 in both the forward and backward orientations and achieved 100% in the direction of the 90° scattering angle for all wavelengths. The Mueller matrix element ratio, S₂₂/S₁₁, was found to be nearly 100% for all scattering angles at the photosynthetically active radiation spectral range (equivalent to size parameters). The Mueller matrix element ratio, S₃₄/S₁₁, can be considered to be approximately zero for all scattering angles and wavelengths. Moreover, the main diagonal element ratios in Figure 10d,f varied from 1 to −1 for the entire scattering directions, which can be considered identical for all wavelengths. These results are similar to those of individual spheres at all wavelengths, which, due to the aggregates, can be considered directionally independent. Combining the previously obtained conclusions on the invariance of the scattering element ratios with respect to shape and size, the scattering matrix element ratios are nearly shape-, size-, and wavelength-invariant for microalgal aggregates.

4.4. Internal Electric Field of Microalgae Aggregates

The electromagnetic fields inside the aggregates were calculated in order to better understand and interpret the non-volumetric phenomena and large concavities exhibited by the absorption cross-section of the aggregates. Figure 11 shows the real value of the total electric field of the colonies of spherical-shaped microalgae aggregates, with a diameter of 66.5 μm on the z = 0 plane for the upper panel and y = 0 plane for the lower panel. The input parameters were the complex refractive indices at wavelengths of 440, 445, and 450 nm. As shown, the internal electric field of the aggregates at 445 nm shows a clear difference compared to that at 440 and 450 nm. Typical interference peaks at the periphery of the microalgal aggregates were observed. For fields at 445 nm, the internal field is relatively neglectable, except near the surface, which leads to a large dip in the absorption cross-section. This can be attributed to the resonance and non-resonance phenomena of the electromagnetic fields around the absorption peaks [60]. The presented internal field results confirm the cross-section oscillation results in the 440–450 nm spectral range and explain the rapid decline and increase in the absorption cross-section between 440 and 450 nm.

5. Conclusions

On the basis of the results presented, rather than emphasizing the variation in the scattering properties of microalgal aggregates with size, shape, and wavelength, this study explored the possibility of invariance in the scattering properties. The scattering matrix element ratios of the cyanobacterial aggregates were nearly shape-, size-, and wavelength-invariant. The EACSs per unit volume are shape-invariant and approximately size-invariant for cyanobacterial aggregates. In addition, the absorption cross-section of the aggregates was independent of the spatial distribution of the microalgal cells. Furthermore, the absorption cross-section is not solely a volumetric phenomenon when the aggregates exceed a certain size. The invariant interpretation of the scattering properties of microalgal aggregates allows us to understand the scattering properties of cyanobacterial aggregates intuitively. The shape invariance of the scattering properties of the microalgal aggregates implies that the agglomeration process of algae and the motion of algal cells within the aggregates do not affect the light utilization of the cellular aggregates. The scattering characteristics of cyanobacterial aggregates are crucial for quantifying the interactions between light and photosynthetic microalgae.

Author Contributions

Conceptualization, C.M. and Y.W.T.; methodology, C.M.; software, C.M.; validation, C.M.; formal analysis, C.M. and Q.L.; resources, C.M. and Y.W.T.; data curation, C.M.; writing—original draft preparation, C.M.; writing—review and editing, C.M., Q.L. and Y.W.T.; supervision, Y.W.T.; project administration, C.M. and Y.W.T. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The data are available from the corresponding author on reasonable request.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures

Figure 1. Micrographs of Nostoc sp. (a) 40× and (b) 100× colonies resembling aggregates.

Figure 2. Realizations for packed microalgae aggregates. The following are typical parameters: number of samples, [Forumla omitted. See PDF.] = 30, for each chain and bending parameters, [Forumla omitted. See PDF.] = 5 and [Forumla omitted. See PDF.] = 5, are the dependability metrics in the desired directions. The volume fractions are (a) 37.9%, and (b) 44.8%.

Figure 3. Realizations for Gaussian spherical-shaped microalgae aggregates systems with a power-law index of [Forumla omitted. See PDF.] = 3.5, a truncated degree of [Forumla omitted. See PDF.] = 50, and a relative standard deviations of radial distances [Forumla omitted. See PDF.] = 0.05 and 0.10 (from left to right). The common parameters of microalgae aggregates are the same as in the previous, and volume fractions are (a) 45.5% and (b) 45.4%.

Figure 4. Convergence verification of DDA solutions for dpl value, (a) S11, (b) −S12/S11, (c) S22/S11, (d) S33/S11, (e) S34/S11, (f) S44/S11.

Figure 5. Spectral complex refractive index of microalgae cells, (a) spectral refractive index, (b) spectral absorption index.

Figure 6. Scattering matrix elements of spherical, cubical, and Gauss sphere-shaped aggregates for different sizes, (a) −S12/S11, (b) S33/S11, (c) S44/S11, (d) S22/S11, (e) S34/S11.

Figure 7. Cross-sections per unit volume of the colonies of microalgae aggregates for different sizes, (a) extinction cross-section, (b) absorption cross-section.

Figure 8. Cross-sections per unit volume of the colonies of microalgae aggregates for different shapes, (a) extinction cross-section, (b) absorption cross-section.

Figure 9. Cross-sections per unit volume of the colonies of microalgae aggregates with equivalent volume, (a) extinction cross-section, (b) absorption cross-section.

Figure 10. Mueller matrix elements of colonies of microalgae aggregates, (a) S11, (b) −S12/S11, (c) S22/S11, (d) S33/S11, (e) S34/S11, (f) S44/S11.

Figure 11. Internal fields of colonies of microalgae aggregates: upper panel for z = 0 plane and lower panel for y = 0 plane. Incident linearized polarized light at the propagation of z direction, (a,d) 440 nm, (b,e) 445 nm, and (c,f) 450 nm.

Figure 11. Internal fields of colonies of microalgae aggregates: upper panel for z = 0 plane and lower panel for y = 0 plane. Incident linearized polarized light at the propagation of z direction, (a,d) 440 nm, (b,e) 445 nm, and (c,f) 450 nm.

Supplementary Materials

The following supporting information can be downloaded at https://www.mdpi.com/article/10.3390/photonics12020142/s1: Figure S1: The scattering matrix element S₁₁ for spherical, cubic, and Gauss sphere shaped microalgae aggregates as functions of scattering angles by using the DDA method.

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