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Abstract
Most of the current predictions of the radiative properties of microalgae use the homogeneous sphere approximation based on the Mie scattering theory, and the refractive indices of the model were regarded as fixed values. Using the recently measured optical constants of various microalgae components, we propose a spherical heterogeneous model for spherical microalgae. The optical constants of the heterogeneous model were characterized by the measured optical constants of microalgae components for the first time. The radiative properties of the heterogeneous sphere were calculated using the T-matrix method and were well verified by measurements. It shows that the internal microstructure has a more significant effect on scattering cross-section and scattering phase function than absorption cross-section. Compared with the traditional homogeneous models selected with fixed values as refractive index, the calculation accuracy of scattering cross-section of the heterogeneous model improved by 15%-150%. The scattering phase function of the heterogeneous sphere approximation agreed better with measurements than the homogeneous models due to the more detailed description of the internal microstructure. It can be concluded that considering the internal microstructure of microalgae and characterizing the microstructure of the model by the optical constants of the microalgae components helps to reduce the error caused by the simplification of the actual cell.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Microalgae have been recognized as a potential tool to efficiently convert the constant stream of solar energy into biofuels and biomaterials [1–4]. Some algal species such as Chlorella, Chlamydomonas, and Scenedesmus are used for biodiesel extraction because of their high content of lipids [5]. Because of the presence of hydrogenase enzymes in their cells, Chlorella and Chlamydomonas have a high potential for photosynthetic hydrogen generation [6]. To gain a better understanding of microalgae growth, reliable prediction of optical radiation transfer in microalgae suspensions is required [7]. Radiation characteristic parameters such as absorption cross-section, scattering cross-section, and scattering phase function are crucial in solving the radiation transfer equation in photobioreactors. Specifically, the scattering phase function is a significant parameter in marine optics research. It is of great significance to obtain high-precision radiation characteristics of microalgae cells, which are highly affected by their microstructure [8].
The equivalent homogeneous sphere approximation is commonly utilized in the computation and prediction of microalgae radiation properties. The homogeneous sphere allows the use of the Mie scattering theory and thus simplifies theoretical calculations of radiative properties substantially. To study the radiation transfer of Chlamydomonas reinhardtii in a torus photobioreactor, Pottier et al. [9] modeled microalgae as a homogeneous sphere and determined the optical properties of microalgae according to the in vivo pigments absorption spectra data, and calculated the spectral radiation characteristics of microalgae by Lorenz-Mie theory. To retrieve the spectral absorption and refractive indices of biofuel microalgae from experimental measurements, Lee et al. [10] approximated the spherical photosynthetic microalgae as homogeneous spheres with the same surface area as the actual spherical shape and proposed an inversion method combining Lorenz-Mie theory and a genetic algorithm. However, the morphology of the microalgae particles is not strictly spherical, and the interior is not homogeneous. Microalgae cells have irregular shapes and complex internal microstructure, which can affect the radiation properties of the microalgae. Lehmuskero et al. [11] analyzed the radiation properties of microalgae cells and showed that the radiation properties of microalgae cells are related to their shape, size, and composition. Based on the discrete dipole approximation method, Dong et al. [8] investigated the radiation characteristics of microorganisms with spine-like surface structures, which demonstrated that the external spine-like structure can significantly impact the radiation properties. Note that, the homogeneous spherical model combined with the Mie scattering theory does provide significant simplification for solving the positive and inverse problems of the radiation characteristics of microalgae, but the errors caused by the neglect of the internal microstructure are rarely analyzed.
The commonly used homogeneous sphere model is usually adequate to predict absorption [12,13], but it leads to underestimation in algae extinction and backscattering [14]. Quirantes and Bernard [12] established two models for microalgae particles: eccentric coated sphere and coated spheroid, whose optical constants were set to a fixed value, and calculated the scattering properties of algal particles by the T-matrix method. Compared with the simplified homogeneous model, the results showed that the extinction factor and scattering factor were not sensitive to the particle structure, but the particle structure had a significant influence on the backscattering factor. Organelli et al. [14] applied the coated sphere model to predict the extinction and backscattering coefficients of ocean particles. The optical constants of the model were set to a fixed value. The results showed that the homogeneous sphere model failed to simultaneously reproduce the measured particle extinction coefficient and backscattering coefficient and the coated sphere has a larger backscattering, which is equivalent to the sum of a homogeneous sphere and several small spheres with the same refractive index. Note that the coated sphere is more accurate than the homogeneous sphere model. Kandilian et al. [15] simulated Chlorella as polydispersed coated spheres. The optical constants of the model were determined by the subtractive Kramers–Kronig relation combined with the anchor refractive index. The results showed that the homogeneous sphere approximation could significantly overestimate the absorption cross-section, while the coated sphere approximation of Chlorella could modify this problem. The results also illustrated that some microalgae which have thick cell wall with high refracting cannot be described as homogenous spheres, implying that other organelles must be included in the microalgal model. Therefore, the heterogeneity of the microalgae must be taken into account at the organelle level. Kitchen and Zaneveld [16] modeled phytoplankton as a three-layer concentric sphere model to investigate optical properties in the ocean. The outermost layer represented the cell wall with a high refractive index, the middle layer represented the chloroplast structure with absorption and relatively low refractive index, and the central nucleus represented the cytoplasm with a low refractive index and non-absorption. Each structure was given a refractive index value that was constant. The results showed that compared to the low-index homogeneous sphere model, the three-layer model produced greater backscattering and extinction. Based on the T-matrix method, Bhowmik and Pilon [17] proposed a more detailed cell model to study spherical heterogeneous microalgae cells represented by Chlamydomonas reinhardtii. The model consisted of intracellular compartments including cell wall, cytoplasm, chloroplast, nucleus, mitochondria, etc. Although the model takes into account the organelles of Chlamydomonas reinhardtii in detail, the refractive index of the microalgae model and the surrounding medium water was simply defined as a fixed value independent of wavelength. In fact, the refractive index of microalgae changes with wavelength, especially water.
It is remarkable that, due to the constraints of the existing understanding of the optical properties of intracellular organelles, few works have investigated the effects of the internal microstructure of microalgal cells on radiation properties. In particular, the optical constants of the internal microstructure are rarely reported and are usually considered to be fixed values that do not change with wavelength. The optical constants mentioned in this article mainly refer to the refractive index and absorption index. In addition, microalgae models constructed in previous studies often lacked relevant microscopic images as a basis. This study aims to establish a theoretical model with consideration of internal heterogeneity, to more precisely forecast the radiation properties of microalgae. According to transmission electron microscope images, the typical spherical microalga Chlorella sp. was modeled as a heterogeneous model with multiple internal spheres to represent its internal microstructure, including the cell wall, nucleus, pyrenoid, and vacuole. More importantly, the wavelength-dependent optical constants of the microalgae model were determined by the results of previous experiments. The multiple-sphere T-matrix method, which is a numerically precise and quick method for models with multiple internal spheres, was used to solve the radiation characteristics of the heterogeneous sphere, and the results were compared to the equivalent homogeneous model (coated sphere model). The improved transmission method and LISST-VSF were utilized to determine the radiative characteristics of the Chlorella sp., and the experimental results were used to confirm the suggested model. Additionally, the traditional homogeneous coated sphere models assigned with fixed values of refractive index were presented to evaluate the characterization method which considers the optical constants of various components of microalgae.
2. Methods
2.1 Cell model with internal microstructure
Chlorella sp. has high CO2 tolerance and is a typical CO2 fixation and lipid accumulation green algal. The Chlorella sp. was cultivated in an airlift flat-panel polymethyl methacrylate photobioreactor (Shanghai Guangyu Biological Technology Co., Ltd.). The microscopic images were obtained by a biological microscope ((UB203i-5.0 M, Chongqing, China) with a CCD camera and analyzed by the ImageView to obtain the size of the microalgae. Figure 1 shows the micrograph and cell size distribution of Chlorella sp. It can be seen that Chlorella sp. is near-spherical and unicellular, and the equivalent diameter of the cell is about 4.57 µm.
Fig. 1. Micrograph of (a) Chlorella sp., and equivalent cell size distribution of (b) Chlorella sp. diameter
To better consider the internal microstructure of the microalga, the Chlorella sp. is modeled as a heterogeneous sphere model as depicted in Fig. 2. The smaller spherical structures inside represent organelles nucleus, pyrenoid, and vacuole, respectively. The chloroplast was not treated as a separate sphere, which is attributed to its irregular shape and a large proportion of the intracellular volume. The absorption spectra of the chloroplast were expressed by the cytoplasm. In addition, as shown in Fig. 2(c), a homogeneous model (coated sphere model) with the same composition and volume was considered to compare the influence of internal microstructure on the radiation characteristics of microalgae. The thickness and components of the cell wall were identical to those of the heterogeneous model. The inner core of the homogeneous model is equivalent to the internal microstructure of the heterogeneous model, including the cytoplasm, nucleus, pyrenoid and vacuole. The volume fractions of individual compartments in the heterogeneous and homogeneous models are shown in Table 1.
Fig. 2. Transmission electron micrograph of (a) Chlorella [20], and the schematics of (b) heterogeneous model and (c) equivalent homogeneous model of spherical eukaryotic microalgal cell Chlorella sp.
Table 1. Volume fractions of individual compartments in the heterogeneous and homogeneous models
Because it is challenging to experimentally acquire information on the optical characteristics of microalgae organelles, the compartments of the model were viewed as a mixture of the primary components of Chlorella sp. in order to define the optical constants of the model, which is a convenient method to approximate the actual microalgae from the perspective of the composition. The microalgae components considered in the model are water, protein, carbohydrate, and lipid, and the types of photosynthetic pigments include chlorophyll a and chlorophyll b. The external medium is water. Therefore, the optical properties of the model are closely related to those of the microalgae components. The optical constants of the main components of Chlorella sp. such as protein, carbohydrate, and lipid were measured [21], as shown in Fig. 3.
Fig. 3. (a) Refractive index and (b) absorption index of the main components of microalgae including water, protein, carbohydrate, lipid, and photosynthetic pigments (chlorophyll a and chlorophyll b).
It can be seen in Fig. 3(a) that the refractive indices of protein, carbohydrate, and lipid are higher than those of water (the values of water are from Ref. [22]) and decrease with the increase of wavelength. In the wavelength range of 350-750 nm, the refractive indices of protein, carbohydrate, and lipid vary in the range of 1.56-1.50, 1.57-1.50, and 1.50-1.48, respectively, which agree well with the values 1.57 ± 0.01 for protein, 1.55 ± 0.02 for carbohydrate, and 1.47 ± 0.01 for lipid reported in the Ref. [23]. Compared with other components, the content of the pigment is very small, and the refractive index of the lipid is very close to the refractive index of pigment 1.50 ± 0.04 reported in Ref. [23], so the refractive index of the lipid is selected as the refractive index of pigment.
The absorption index ${k_{i,\lambda }}$ of each pure pigment is derived from [24]
where $E{a_{i,\lambda }}$ (m2/mg) is in vivo spectral mass absorption coefficient of pigment i, taken from [25]. For pure pigments, ${C_i}$ (kg/m3) is the density of photosynthetic pigments. As shown in Fig. 3(b), the absorption peaks of chlorophyll a are around 440 and 670 nm, and the absorption peaks of chlorophyll b are around 470 and 650 nm, which correspond to those of Ea. Photosynthetic pigments, as the main photosensitive components of microalgae, have a much higher absorption index than other components. In the researched wavelength range, protein, carbohydrate, and lipid also exhibit a certain degree of absorption, with absorption indices between 0.003 and 0.01. Therefore, the absorption of the main components should not be ignored.The heterogeneous model contains five compartments, namely the cell wall, cytoplasm, nucleus, pyrenoid and vacuole, represented by subscript i. The effective complex refractive index mi of each compartment can be calculated according to the optical constants of the components combined with the Maxwell-Garnett mixing rule [26,27]
Fig. 4. (a) The refractive index and (b) absorption index of each structure of the heterogeneous model and the homogeneous model calculated by Eq. (2).
It can be seen from Fig. 4(a) that the refractive index of the internal microstructure of the model tends to decrease with the increase of wavelength, which has the same trend as the main components of Chlorella sp. The refractive indices of the nucleus, cytoplasm and cell wall vary in the range of 1.42-1.47, 1.35-1.38, and1.37-1.41, respectively, which are close to the ranges reported in Ref. [29]: 1.36-1.53 for the cell wall, 1.36-1.44 for the nucleus, and 1.34-1.39 for the cytoplasm. As shown in Fig. 4(b), in the visible spectrum, the cytoplasm and the core of the homogeneous model have much higher absorption indices than other model compartments, and the absorption peaks correspond to those of the pigment. It is because of the strong absorption of pigment which is only distributed in the cytoplasm of the heterogeneous model and the core of the homogeneous model when modeling the microalgae cell.
Another simple way to characterize the optical constants of microalgae models is to select fixed values based on experience. However, this method may lead to significant errors in the theoretical calculation of radiation characteristics without experimental results as a reference. In order to evaluate the two characterization methods for the optical constants of microalgae models, three cases for the homogeneous coated sphere (Case1, Case2, and Case3) with the same volume and pigment concentration as the heterogeneous model were considered. The refractive indices for the three cases were chosen as ${n_{\textrm{wall},1}} = 1.38$ and ${n_{\textrm{core},1}} = 1.36$, ${n_{\textrm{wall},2}} = 1.40$ and ${n_{\textrm{core},2}} = 1.38$, ${n_{\textrm{wall},3}} = 1.45$ and ${n_{\textrm{core},3}} = 1.40$, respectively, according to the reported values [29]. For the case where only the absorption of pigments was considered, the absorption index of a homogeneous sphere can be calculated by [10]
2.2 T-matrix method
The random orientation absorption Cabs,λ and scattering Csca,λ cross-sections of the heterogeneous Chlorella sp. model were calculated using the multiple sphere T-matrix method for the case of several internal spheres [30]. The T-matrix method superposes the scattered electromagnetic field from each spherical compartment to estimate the scattered electromagnetic field from the entire heterogeneous sphere [31]. This method expresses the transformed fields in the model by a superposition of vector spherical wave function (VSWF) expansions. The interaction equations were established using the orthogonality relations for the VSWFs and the continuity relations on the electric and magnetic fields. The equations were then solved using the biconjugate gradient method iteratively. The multiple sphere T-matrix code (MSTM-v3.0) developed by Mackowski was used in this paper [30]. By operating this code, the total absorption factor Qabs,λ and extinction factor Qext,λ of the outermost sphere of the heterogeneous model, i.e. the cell wall, as well as the scattering matrix elements Sij(θ) can be calculated, where θ represents the scattering angle. The total scattering factor Qsca,λ in the superimposed scattering field can be obtained from ${Q_{\textrm{sca},\lambda }} = {Q_{\textrm{ext},\lambda }} - {Q_{\textrm{abs},\lambda }}$. Furthermore, the random orientation scattering and absorption cross-sections of the microalgae models were obtained from the calculated scattering Qsca,λ and absorption Qabs,λ efficiency factors according to ${C_{\textrm{sca},\lambda }} = {Q_{\textrm{sca},\lambda }} \cdot \pi {r^2}$ and ${C_{\textrm{abs},\lambda }} = {Q_{\textrm{abs},\lambda }} \cdot \pi {r^2}$, where r is the radius of the microalgae model. The radii ri of each compartment were calculated based on the average diameter of the microalgae and the volume fraction of the cell compartments. The spatial coordinates of each compartment were determined using random numbers. The value of the “fixed_or_random_orientation” item in the code was set to 1 to calculate the radiative characteristic parameters of random orientation, thereby eliminating the influence of the random position of the nucleus, pyrenoid and vacuole inside the heterogeneous model. The spatial coordinates of the centers of the cell wall and cytoplasm are both 0. The scattering phase function S11 is normalized so that
2.3 Experimental measurement of radiation properties of microalgae
The model of the optical radiation transmission system of microalgae suspension is shown in Fig. 5. Traditional radiation transmission models often only consider first-order transmission and ignore higher-order transmission. This article uses an improved transmission method to measure the absorption coefficient α and scattering coefficient β of microalgae samples. The improved transmission method introduces second-order and higher-order transmission terms [32,33]. The errors caused by the neglect of higher-order transmission terms were analyzed in Ref. [32]. The absorption coefficient α and the extinction coefficient β of the microalgae suspension sample were calculated by [32]
Fig. 5. Schematic of the optical radiation transmission system of microalgae suspension.${R_1}$ and ${R^{\prime}_1}$ are the reflectance of layer1 from the incident surface and the non-incident surface, respectively.${R_3}$ and ${R^{\prime}_3}$ are the reflectance of layer3 from the incident and non-incident sides, respectively. T1 and T3 are the transmittance of layer1 and layer3 (glass), respectively.${t_2}$ is the transmissivity of microalgae suspension layer2 [32].
The transmittances ${T_{\textrm{EXP}}}$ and ${T_{\textrm{h},\textrm{EXP}}}$ of the microalgae sample were obtained by the RC2-DI spectroscopic ellipsometer (J.A. Woollam Co., Inc., USA) and the RTC-060-IG integrating sphere (Labsphere, Inc., USA) with the Omni-DR830-SDU measuring system (Zolix Instruments Co., LTD., China) measured normal-normal transmittance and normal-hemispherical transmittance of microalgae samples. As shown in Fig. 6, to measure the scattering phase functions of microalgae, we used a LISST-VSF multi-angle polarized light scattering meter (produced in Sequoia Scientific, Inc., USA). LISST-VSF is a commercial multi-angle polarized light scattering instrument. It is the world’s first autonomous, submersible instrument for measuring the volume scattering function (VSF) of water with depolarization. It measures P11, P12 and P22 elements of Mueller matrix of water, as well as beam attenuation and scattering. The LISST-VSF employs a single 515 nm TE-cooled diode laser and two photomultiplier tube (PMT) detectors. One PMT measures scattered light into perpendicular polarization and the other into parallel polarization. The angular range of the LISST-VSF covers from 0.1 to 150° at wavelength 515 nm. The eyeball was submerged in the sample. The laser source emitted from the laser housing to the ring detector in the nearforward housing to measure the scattering less than 15°. For larger angles from 15-150°, the eyeball rotates to view scattering along the laser beam at different angles.
3. Results and discussion
Figure 7 shows the random orientation spectral absorption cross-section and scattering cross-section of the randomly oriented heterogeneous model and the homogeneous models obtained by the T-matrix code from 350-750 nm, as well as the scattering phase function at 515 nm. Experimental results in the wavelength range of 400-750 nm are also shown in Fig. 7.
Fig. 7. The orientation-averaged (a) absorption cross-section, (b) scattering cross-section, and (c) scattering phase function at the wavelength of 515 nm of the randomly oriented heterogeneous model and homogeneous models calculated by the T-matrix code. The equivalent homogeneous model without internal microstructure was equivalented from the heterogeneous model. Case1, Case2, and Case3 are the three cases for the traditional homogeneous coated sphere approximation with fixed values of refractive index which are ${n_{\textrm{wall},1}} = 1.38$ and ${n_{\textrm{core,1}}} = 1.36$ for Case1, ${n_{\textrm{wall},2}} = 1.40$ and ${n_{\textrm{core},2}} = 1.38$ for Case2, ${n_{\textrm{wall},3}} = 1.45$ and ${n_{\textrm{core,3}}} = 1.40$ for Case3, respectively.
As shown in Fig. 7(a), the absorption cross-sections predicted by the heterogeneous and homogeneous models have the same absorption peaks as chlorophyll a (around 435 and 676 nm) and chlorophyll b (around 475 and 650 nm) [10,25] and show good agreement with measurements. The relative differences in absorption cross-section averaged around the absorption peaks (in 400-480 nm and 650-680 nm) between the heterogeneous model and measurements were about 9%. The absorption cross-sections of the three cases (Case1, Case2, and Case3) for the traditional homogeneous coated sphere have the same trend as the heterogeneous model, but the values around the absorption peaks are larger (3%-8%) than those of the heterogeneous model. Note that the absorption cross-section is not visibly affected by the refractive index selection. In addition, considering the optical constants of microalgae components, the relative difference in the absorption cross-sections predicated by the T-matrix method between the equivalent homogeneous and the heterogeneous models was smaller than 3%. It shows that when the optical constants of the models are defined using the same contents of the microalgae components, the internal microstructure has no significant effect on the absorption cross-section. However, in 500-600 nm, there is a relatively large gap in absorption cross-section between theoretical predictions and measurements. Since the absorptivity of various microalgae components was considered, the absorption cross-section of the heterogeneous model can maintain relatively high values in the band where photosynthetic pigment absorption is weak, and agreed well with measurements than traditional homogeneous coated sphere models. In the calculations, only two main photosynthetic pigments (chlorophyll a and chlorophyll b) with relatively high content were considered. In fact, microalgae contain a variety of accessory pigments with varied absorption spectra, including photosynthetic carotenoids (PSC) and photoprotective carotenoids (PPC) [10,25]. Therefore, an accurate understanding of the types and contents of pigments is essential to improve the calculation accuracy of the radiation characteristics of microalgae.
As depicted in Fig. 7(b), the theoretical predictions of Case1, Case2, and Case3 of the traditional homogeneous coated sphere did not agree well with the measured scattering cross-section. The three cases of the traditional homogeneous model used a fixed value as the refractive index of the cell wall and core, which is a currently used method to approximate the actual microalgae. The relative differences for the three cases in scattering cross-section, averaged over the wavelength range of 400-750 nm, were 32%, 66%, and 167%, respectively. Additionally, the relative difference increased significantly with the increase of the refractive index of the traditional homogeneous model. For the equivalent homogeneous model whose optical constants were determined by microalgae components, in 400-750 nm, the predicted scattering cross-section was relatively well coincident with the experimental value with an average relative difference of 26%. However, the relative difference between the equivalent homogeneous model and the measured values of scattering cross-section reached up to 60% in the wavelength range of 400-500 nm. Unlike homogeneous models, the heterogeneous model has the highest accuracy in scattering cross-section. The relative difference in scattering cross-section averaged over the wavelength range of 400-750 nm between the heterogeneous model and measurements was 17%. It attributes to the consideration of microalgae components and internal microstructure, which makes the theoretical model have the most similar characteristics to the actual microalgae than homogeneous models. In addition, there was a relatively large gap between the heterogeneous model and experimental values from 400 to 500 nm with an average relative difference of 37%, which is much smaller than homogeneous models. The scattering cross-section errors between the heterogeneous model and experimental values can be attributed to the fact that there are still deviations between the heterogeneous model and actual microalgae cells. Compared with the traditional homogeneous models (Case1, Case2, and Case3) without considering the microalgae components and internal microstructure, the theoretical calculation accuracy of the scattering cross-section of the heterogeneous model improved by 15%-150%. Additionally, compared with the equivalent homogeneous model that does not consider the internal microstructure, the calculation accuracy of the heterogeneous model at 400-500 nm improved by 23%. These results show that, in the absence of experimental values as a reference, determining the refractive index of the microalgae model by the selection of fixed value will cause large errors in the theoretical calculation of the scattering cross-section. It is more reasonable to define the optical constants of the cell model by the optical constants of the microalgae components and to consider the internal microstructure, which helps to reduce the error caused by the simplification of the actual microalgae cells when modeling.
Figure 7(c) shows the scattering phase function at 515 nm for the heterogeneous model and the homogeneous models, as well as the experimental results. For both models and experimental measurements, the scattering phase functions had a strong peak in the scattering angle of 0°. It is due to the huge size of the microalgal cell in comparison to the wavelength. For the forward scattering angle between 0° and 10°, the scattering phase functions of the heterogeneous and homogeneous models were nearly the same and were consistent with the experimental values. For the scattering angle greater than 10°, the scattering phase functions of the models began to oscillate due to the coherence of electromagnetic waves. For backscattering (angle >90°), the scattering phase function of the heterogeneous model with internal microstructure had a smaller amplitude and stronger backscattering, which was closer to the experimental value. In addition, the backscattering for both traditional and equivalent homogeneous models was much smaller than the measurements. It is due to the more refined description of the internal microstructure of microalgae in the heterogeneous model. There were spherical structures with smaller sizes but higher refractive index inside the heterogeneous model, which perform stronger inhomogeneity and destroy the oscillation to some extent, and thus enhance the backscattering [14,17]. For the three cases (Case1, Case2, and Case3) of the traditional homogeneous coated sphere, due to the neglection of the internal microstructure, the scattering phase functions show a higher amplitude than that of the heterogeneous model. In addition, the scattering phase functions of the traditional homogeneous models show a growing trend as the refractive index increases. Although the scattering phase function of Case3 was closer to the experimental value, the scattering cross-section of Case3 deviates significantly from the measurements, as shown in Fig. 7(b). It indicates that characterizing the optical constants of the internal microstructure of the model based on microalgae components helps to improve the calculation accuracy of the scattering phase function, rather than randomly choosing fixed values as the optical constants for the microalgae model.
4. Conclusion
Based on the optical constants of the main components of Chlorella sp. and the Maxwell-Garnett mixing rule, a novel heterogeneous model of microalgae cells was proposed. It is the first time to characterize the microalgae microstructures using the optical constants of microalgae components. The radiative properties of the heterogeneous model were obtained using the T-matrix method and compared with the homogeneous models and measurements. The results show that the heterogeneous model characterized by the optical constants of microalgae components can bring a much higher accuracy of theoretical predictions of radiative properties than the traditional method that uses fixed values as the refractive index of models. The theoretical prediction accuracy of the scattering cross-section of the heterogeneous model was 15%-150% higher than the traditional homogeneous coated sphere models. Compared with the equivalent homogeneous model without considering the internal microstructure, the average relative difference in scattering cross-section between the heterogeneous model and measurements decreased by 23% in 400-500 nm. The absorption cross-section seems less affected by the internal microstructure, and the relative difference in absorption cross-section between the heterogeneous and the homogeneous models was 3%-8%. In addition, the scattering phase function of the heterogeneous model agreed better with measurements than the traditional and equivalent homogeneous models due to the more detailed description of the internal microstructure of microalgae. The results show that characterizing the model optical constants based on the optical constants of microalgae components can reduce the error caused by the simplification of the actual cell microstructure and helps to obtain a more accurate theoretical prediction of radiative properties. Choosing fixed values as the refractive index of microalgae models will produce non-negligible errors. On the contrary, the combination of the microalgae components makes the heterogeneous model achieves the approximation of the actual microalgae cells at the morphological and compositional levels. In the theoretical calculation, the external environment of microalgae is regarded as pure water, the influence of the culture medium is not fully considered, and the estimation of photosynthetic pigments is simplified, resulting in the difference between the prediction and experimental values. In order to improve the theoretical calculation accuracy of radiation characteristics, it is necessary to consider the internal microstructure of the microalgae and comprehensively consider the related factors of the microalgae itself and the external environment.
Funding
National Natural Science Foundation of China (52106080).
Disclosures
The authors declare no competing interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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