Calculating Algae Growth Rates Using Nanocrystal Based Spectral Conversion Layers

(1)

Calculating Algae Growth Rates Using Nanocrystal Based

Spectral Conversion Layers

Maaike Bakker August 21, 2020

Studentnumber 11880503

Daily Supervisor dr. A. Lesage

Supervisor prof. dr. P. Schall

Second Examiner dr. R. Sprik

Course Bachelor Project Physics & Astronomy

Size 15 EC

Conducted between 30-03-2020 and 17-08-2020

Faculties Faculteit der Natuurwetenschappen, Wiskunde en Informatica UvA & Faculteit der B`etawetenschappen VU

(2)

Abstract

The process of photosynthesis in photoautotrophs, a group that includes plants, algae and cyanobacteria, is not very efficient in its energy conversion, therefore much research is focused on increasing the photosynthetic rate of these organisms. One of the possi-ble ways in which this can be done, is by tuning the incoming light spectrum so that it peaks at wavelengths which can be used most efficiently for photosynthesis by a certain photoautotroph. In this research, the possibilities and limitations of the application of a spectral conversion layer containing nanocrystals are explored. The research focuses on a spectral conversion layer of PMMA containing cesium lead halide nanocrystals. A model is created to calculate the light spectrum coming out of the layer and falling on the photoautotroph. Some properties of the layer are implemented as parameters that could be changed; thickness, nanocrystal size, volume fraction of the nanocrystals, and their photoluminescence quantum yield. For two types of algae, Synechococcus 6312 and Ulva, we tried to find parameter values that would increase the total photosynthetic rate. For Ulva, this was not possible with the options in the model. For Synechococcus 6312, an increase of 8.54 % was reached using nanocrystals of cesium lead iodide. It appears to be possible to increase the photosynthetic rate of an organism by applying a spectral conversion layer, if the emission from this layer matches the peak in the action spectrum of the organism.

Populair wetenschappelijke samenvatting (Dutch)

Planten en sommige andere organismen gebruiken licht om energie te maken voor hun groei en behoud door het proces van fotosynthese. Het blijkt echter dat dit proces niet zo effici¨ent is; heel veel energie van het zonlicht gaat verloren voor het organisme. We willen daarom onderzoeken of meer energie gebruikt kan worden in fotosynthese. Dat doen we door ervoor te zorgen dat het licht dat op het organisme valt, het juiste licht is voor dat organisme. Dan doen we door een laagje aan te brengen op het organisme, dat een deel van het licht van de zon absorbeert en dan met een andere golflengte weer uitzendt. Op die manier veranderen we het spectrum van de zon. Door zo’n laagje aan te brengen, gaat er echter ook weer energie verloren, doordat bijvoorbeeld een deel van het licht weggereflecteerd wordt. In dit onderzoek proberen we te ontdekken of het mogelijk is die verliezen klein genoeg te houden, zodat het organisme uiteindelijk met hogere snelheid aan fotosynthese kan doen.

(3)

1 Introduction

One of the most essential processes for life on earth is photosynthesis. The ability of plants, cyanobacteria and some algae to convert light energy to chemical energy is what provides humans and other heterotrophic organisms with the food they eat as well as the oxygen they breathe in.

Even though photosynthesis is a vital process for almost all life forms, it is not very ef-ficient; the photosynthetic efficiency is 5 percent [1] or lower [2]. The reasons why 95 percent of the light energy is lost are shown in figure 1. Of the total light, a large fraction of 72 percent is lost because of inefficient use of light. This loss is due to three effects: the first 47 percent gets lost due to the photons having wavelengths outside of the visible range (400 - 700 nm), which coincides with the range that is used by plants, the so called photosynthetically active radiation (PAR). The following 16 percent is lost due to incom-plete or inefficient absorption of the light by the organism and the last 9 percent gets lost because the photons have to be degraded to lower energies. The losses further decreasing the efficiency are due to the inefficiency of the chemical processes in the electron transport chain in the organisms.[1]

Figure 1: Overview of the losses in photosynthesis for an average leaf at 25◦C. Figure from [1].

Over the last few decades the world’s population has been growing dramatically and with an increased population comes an increased need for food, which in turn heightens the stress put on the farming industry. Inequality in distribution, but also shortages make starvation a big problem in many countries.[3] In an effort to resolve these issues there has been a rise in research focusing on increasing the yield and efficiency of farms across the globe. An effective way to achieve these goals would be to increase the efficiency of the pro-cess of photosynthesis itself and this has therefore been the subject of many recent studies. The focus of this research was on finding the optimal incident light for a photosynthetic

(5)

falling on the organism and the wavelength that can be used most efficiently for photo-synthesis by the organism.

To optimise the incident spectrum, something called a spectral conversion layer (SCL) could be applied on top of the organism, which changes the spectrum of the light as it passes through. This layer would contain nanocrystals that have very specific and tun-able properties, which entun-ables us to create a layer that absorbs and emits light at specific wavelengths. By doing so, the wavelengths of the incident photons can be matched to the wavelengths at which photosynthesis is most efficient in a specific organism, but applying a layer also introduces losses. The goal of this research was to find whether the incident spectrum could be changed by a SCL so that the final photosynthetic rate is higher than without the layer. For that, the gains and losses due to the SCL should be carefully weighed against one another.

To compute the gains and losses, a model was created in Python that takes an input spec-trum and calculates the percentual increase or decrease in photosynthetic rate for different characteristics of the SCL and for a specified organism. The model is specifically created for perovskite cesium lead halides (CsPbX3 with X = Cl, Br, I), focusing on cesium lead

bromide, but later also including cesium lead chloride and iodide. First, the theoretical processes used in this model are described. Then the methods that were used to calculate the spectra and percentages are explained, followed by the results for different sets of parameters. Lastly, the shortcomings of the model and their effects on the results will be discussed, resulting in a conclusion on the effect of a SCL on the photosynthetic rate.

2 Theoretical background

In this section, the relevant physical processes will be described, starting with the source of the light and following the light through the SCL to the organisms.

2.1 Solar spectrum

Every known organism capable of performing photosynthesis uses light originating from the same source, the sun. The sun’s spectrum closely resembles to that of a black body, meaning that it emits electromagnetic radiation over a broad range of wavelengths, and that the radiative energy at every wavelength is determined by the temperature of the source according to Planck’s radiation law, which describes a broad peak in the emission spectrum. Before the sunlight reaches the organisms on earth, it has to pass through the earth’s atmosphere. The atmosphere contains gases like water and carbon dioxide, that absorb light with specific energies. This causes the solar spectrum, as it falls on the organism, to show gaps at certain wavelengths, as can be seen in figure 2. This figure shows the solar spectrum for three different situations, as determined by the organisation NREL for an absolute air mass of 1.5 [4]. These spectra are used for different situations, depending on the acceptance angle of the object that is studied. In this research the direct spectrum is used, which only takes light from a small solid angle in the sky, centred around the sun, into account [5].

2.2 Reflection

As the light from the sun reaches the SCL, it experiences a change in the medium through which it propagates. Upon incidence, a part of the light will be reflected off the interface

(6)

Figure 2: The solar spectrum as it is observed from just outside of the earth’s atmosphere (black line), the spectrum for a flat object that receives light from one half hemisphere [6](blue line) and the spectrum as determined from the light that comes directly from the sun and a small disc around it [5](red line). The black body shape of the spectrum is very clear, it is only interrupted by the absorption lines from elements in the atmosphere. Figure obtained from the NREL [4].

and the rest of the light, which will be transmitted, now travels at a different angle due to refraction. The way in which the light is affected depends on the angle of incidence of the light, the refractive indices of the mediums on both sides of the interface and implicitly on the energy of the photons of the incident light.

For each photon energy, the reflection and transmission coefficients can be calculated using the Fresnel equations. These coefficients are defined as the intensities of the transmitted or reflected light divided by the intensity of the incident light. As the intensity is given by the power striking the interface ~S · ˆn, where ~S is the Poynting vector and ˆn the vector normal to the surface, with | ~S| = 1₂vE2

0, it is given by the equation

I = 1 2vE

2

0cos(θ), (1)

where v is the speed of light in the medium, is the permittivity of the medium and θ is the angle between the incoming light and the surface.[7]

As the medium in which the incident and the reflected waves travel are the same, the permittivity and the speed of light are the same for the incident and the reflected compo-nents. According to Snell’s law, the angle of incidence is equal to the angle of reflection, so that, using equation 1, the Fresnel coefficient for the reflection becomes

R ≡ IR II = E 2 0R E2 0I . (2)

Using the boundary conditions for the electric field, the following relations between the incident and reflected fields for the two polarization cases (parallel or perpendicular to the plane of incidence) can be determined

E_0R,k= α − β α + β E0I (3) E0R,⊥ = 1 − αβ 1 + αβ E_0I (4)

(7)

with

α ≡ cos(θT) cos(θI)

, β ≡ µ1n2

µ2n1

with µ1 and µ2 the magnetic permeability of the medium before and after the interface

respectively, and with n1 and n2 indicating the refractive indices of medium 1 and 2.[7]

In the next step, the reflection coefficients for light polarised perpendicular to the plane of incidence and for light polarised parallel to the plane of incidence can be calculated sep-arately. Because sunlight is unpolarised, the resulting reflection coefficient is the average of the two polarised reflection coefficients.

So, if the angle of incidence θI is known and the refractive indices of the mediums are

known, the reflection coefficient for different photon energies can be calculated. The reflection coefficient is dependent on the energy, because of the energy dependence of the refractive index. This relationship is empirically described by the Sellmeier equation, which gives the refractive index as a function of wavelength for a medium, characterised by a set of parameters according to the formula

n2(λ) = 1 +X i Biλ2 λ2_{− C} i . (5)

Each term in this sum represents an absorption resonance at wavelength√Ci of strength

Bi.[8] This formula is often used with only the first two or three terms, as those already

determine the refractive index for common optical glasses within an error of 5 × 10−6 [9], which is in the same order as the variation in refractive index in a glass sample due to inhomogeneities [10].

Using the Sellmeier equation and the coefficients for a certain material in the Fresnel equations, the fraction of the light that is reflected or transmitted can be calculated. These formulas are valid for every interface, but usually two cases are distinguished, which are both relevant in this research. In the transition from air to the SCL, the light transitions from an optically thin to an optically dense medium, or in other terms n1 < n2. In this

case, light rays that are refracted bend towards the normal when they are transmitted at the interface. When exiting the SCL the light transmits from an optically dense medium, the SCL, to an optically thin medium, the air or organism, so n1 > n2. This means that

the light rays bend away from the normal at the transition.

For the second case, at large angles of incidence, an interesting phenomenon takes place. Because these light rays bend away from the normal, at a large enough angle of incidence the refracted ray makes an angle of 90◦ with the normal, thereby grazing the surface of the material as can be seen in figure 3. The angle at which this happens is called the critical angle, and it is given by

θc≡ sin−1

n2

n1

. (6)

If the angle of incidence is larger than this critical angle, all the light gets reflected. This phenomenon is called total internal reflection.[7]

(8)

Figure 3: The reflection of a light ray going from an optically dense medium (e.g. glass or a polymer) to an optically thinner medium (e.g. air). The light bends away from the normal (first case). As the angle of incidence becomes larger than the critical angle (second case), total internal reflection takes place (third case). Figure from Mini Physics [11].

2.3 Absorption

The light transmitted at the interface passes through the SCL, and based on the properties of this layer, will be subject to a number of effects. The light can for example be absorbed by the materials that make up the layer; the nanocrystal material and the material of the layer itself. The light could also scatter from the nanocrystals.

The attenuation of a light beam is generally described by the Beer-Lambert law, that relates the intensity at a certain depth z inside the material to an initial intensity I0 of a

light beam and the absorption by the materials i inside the layer according to

I(z) = I0 e − N P i=1 σi Rz 0 nidz . (7)

In this formula, the absorption A is given by minus the exponent, which is indicated for a general case. The σ in this expression gives the cross section of the absorbing species, n gives the number concentration of this species and z is the depth to which the light has penetrated, as indicated above. Formula 7 can be simplified by assuming that the absorbing materials are uniformly distributed over the layer, so that the integral over dz can be replaced by a factor z. This assumption is valid as long as the size of the chunks in the material is well below the wavelength of the light.

The absorption cross section σ has units of length squared, but its meaning is best ex-plained by a probability with which a particle will absorb the light. The cross section varies with the wavelength of the incoming light and depends on the absorbing material. In the case of nanocrystals, the size of the crystal plays a role in the absorption cross section as well. The band gap energy increases with decreasing nanocrystal size. This is because the size becomes at some point smaller than the Bohr radius of the material, so that quantum confinement effects start playing a role, thereby modifying the electron wave functions that determine the electronic transitions and with that the absorption properties [12]. The absorption cross section is thus dependent on the size of the nanocrystals. 2.4 Emission

By absorption of the light by the nanocrystals, an electron-hole pair is created by exciting an electron from the valence band to a higher band. This pair will after some time relax to the band edge, after which a photon with the band gap energy is emitted as the excited

(9)

electron falls back to the valence band. The fraction of the (by this process) emitted photons to the absorbed photons is called the photoluminescence quantum yield (PLQY)

Φ = number of emitted photons

number of absorbed photons. (8)

For the type of nanocrystals considered in this research, cesium lead perovskites, the photoluminescence quantum yield can take values approaching 1 [13].

The emission spectrum shows a sharp peak and both the peak energy and full width at half maximum (FWHM) of this peak are determined by the type of nanocrystal, the temperature, and the size of the nanocrystals. The peak energy corresponds closely to the energy of the band gap, which depends on the temperature and on the size of the nanocrystal. In this research, temperature is taken to have a constant value of 293 K, so the band gap energy and the broadness of the emission peak are calculated for this temperature. However, as the nanocrystals change in size, the band gap energy Eg shifts

because of size dependent quantum confinement effects as described by ∆Eg = ~

2_π2

2m∗_d2, (9)

where d is the radius of the nanocrystal and m∗ is the reduced carrier mass [14]. This is the effective mass of the exciton, the bound state of an electron and a hole, which is calculated using

m∗ = memh me+ mh

, (10)

with me and mh the reduced masses of the electron and hole, respectively. The radius

of the nanocrystal is actually used to describe spherical particles, but as it turns out, the equations for size dependency are also applicable for cubic nanocrystals, where then the edge length is used for d.[15] This shift in the band gap energy is also visible in the absorption spectrum, so this size dependency causes a blueshift in both the absorption and the emission spectrum with decreasing size.

Another factor affecting the position of the emission peak is the Stokes shift, which is also size dependent. The Stokes shift is the difference in the emission peak energy and the absorption peak energy of the same electronic transitions, for example the band gap. The emission peak is usually slightly redshifted in comparison to the absorption peak, due to some energy losses in the relaxation before the emission of a photon. The Stokes shift increases with decreasing nanocrystal size.[16]

2.5 Actions

The light reaches the photosynthetic organism after it exits the SCL. The photosynthetic rate of the organism can be determined for incident lighting with different specific photon energies and it is expressed either as a number of moles of carbon dioxide that is taken up or as a number of moles of evolved oxygen at a certain wavelength, which can be given in a so-called action spectrum [17]. Both are an indication of the photosynthetic rate, since the process of photosynthesis is in general given by

6 CO2 + 6 H2O light

−−−→ C₆H12O6 + 6 O2 (11)

so that carbon dioxide and water are taken up and glucose and oxygen are produced. An action spectrum is often given as a relative spectrum; not comparing specific numbers,

(10)

but showing the action at incident lighting with a certain energy relative to the maximum action. these relative action spectra allow us to compare lighting of different energies with each other.

The action of an organism under specific lighting depends on two factors: the absorption of the photons by the photosynthetic pigments in the organism and the efficiency with which the absorbed light can be used for photosynthesis. The last factor is described by the photosynthetic quantum yield specific for an organism, which is defined as the photo-synthetic rate (again expressed in a number of moles of carbon dioxide taken up or oxygen evolved) per number of absorbed incident photons [17]. In 1970, McCree measured and calculated absorptions, photosynthetic quantum yield and actions of a large number of crop plants, creating the so-called McCree curves shown in figure 4 that are still widely used for calculations of photosynthetic rates. It is important to note that these spectra are determined for crop plants. For other types of organisms, their shapes can differ sig-nificantly, mostly because of different pigment compositions.

However, from these spectra, it can already be seen that not all light is used efficiently by the organism. From the middle graph in figure 4 it becomes clear that shining red light on the crop plants results in a higher photosynthetic rate than if blue light is used.

(11)

Figure 4: The McCree curves, showing the average of absorptance, action and quantum yield of a number of crop plants. The shown values are relative values for the quantities at specific wavelengths in nanometers. Figure from McCree [17].

3 Methods

In the next section, the numerical methods used to create the model in Python will be described, again roughly following the order in which the light passes through the SCL and

(12)

reaches the organism. This order is also used in the model itself; for each step a function is defined that calculates an output spectrum for a given input spectrum and for specific values of a number of parameters.

3.1 Incident light

The model we created gives a choice for an input spectrum; it starts with either the AM1.5 solar spectrum as determined by NREL or with an artificial spectrum. This second option is included to be able to compare results from the model more accurately to future experimental results. In experiments, lighting is often done artificially by a number of coloured LEDs. If an artificial light spectrum is wanted, the user has to specify the peak energy, the full width at half maximum and the total number of emitted photons for a number of LEDs of their choice. To calculate the incident spectrum, the individual LEDs were modeled as Lorentzian peak functions which were added together to create the spectrum of all the LEDs. An example of an artificial spectrum with three LEDs is shown in figure 5.

The results of this research, however, are all based on the solar AM1.5 spectrum in figure 6, which is the same as the spectrum in figure 2, but now has photon energy on the x-axis instead of wavelength.

(a) An experimental setup for measurements on the growth of photosynthetic organisms used by Luim-stra et al (2018) [18]. Here blue, orange, and red LEDs are used separately.

(b) An example of an artificial spectrum created by the model as it would be for a blue, an orange and a red LED, each showing a peak at 2.76 eV, 1.98 eV, and 1.88 eV respectively. All have a full width at half maximum of 0.10 eV and the total number of photons emitted per peak is 70 µmol m−2s−1 Figure 5: Artificial lighting

3.2 Reflection off the top layer

To calculate what fraction of the light is reflected off the top layer, equations 3 and 4 are substituted in equation 2 to obtain the parallel and perpendicular reflection coefficients, which then give the final reflection coefficient R = Rk+R⊥

2 .

To obtain a percentage of reflected light, the values for α and β in the equations have to be found. For the reflection off the top layer, the angle of incidence θI is assumed to be

(13)

Figure 6: The solar spectrum, calculated with data from NREL. [4] .

Obtaining β on the other hand is a little more complicated, since it depends on the re-fractive indices of the materials before and after the plane of incidence, and those are in turn dependent on the wavelength of the incident light.

As the light rays travel from air to the SCL, the symbol n1 is used for the index of

refrac-tion of air and n2 for the index of refraction of the layer material, for which polymethyl

methacrylate (PMMA) has been chosen. The index of refraction of air changes by less than 5 · 10−5 in the range of wavelengths considered here at a fixed temperature, so n1

is taken to have a constant value of 1. The refractive index of PMMA, however, changes much more dramatically with wavelength. This dependence is described by the Sellmeier equation (equation 5) with B = 1.1819 and C = 1.1313 · 104 nm2, only using the first term in the sum [19][20]. With this relation, shown in figure 7, the refractive index for any wavelength inside the visible range can be determined, so β can be calculated, and consequently the reflection coefficient.

Figure 7: The refractive index of PMMA as a function of the wavelength. Shown in red are the data points from [20] and in green is the line associated with the Sellmeier equation and its parameter values for PMMA.

(14)

3.3 Absorption

Once inside the SCL a fraction of the light will be absorbed by the nanocrystals or by the PMMA. Eventually, we are interested in the light spectrum exiting the SCL and to find this it is necessary to consider a few different processes at the same time. Inside the SCL, namely, the light has a chance to be absorbed by both the nanocrystals and the PMMA at every depth z in the layer. At the same time, part of the light absorbed by the nanocrystals will be re-emitted, thereby adding to the total light at each point z. This emitted light can in principle be reabsorbed by the neighbouring nanocrystals and the material.

To calculate the absorption at each wavelength, the Beer-Lambert law from equation 7 is used with two terms in the exponent; one for the absorption by the nanocrystals and another for the absorption by the PMMA. Inclusion of emission in this formula would give a depth-dependent factor that increases the total light at any given depth. There are, however, two problems with this. Firstly, this emission factor is dependent on the absorption at a depth z, but it also contributes to the total light at this depth, thereby also influencing the absorption again. Secondly, the absorption is calculated for a certain wavelength, but the emission is not at the same wavelength as this absorption. To solve this exactly, it would therefore be necessary to solve a set of coupled equations.

These factors would make the calculation of the spectrum really complicated. Therefore, we assume in this research that the processes of absorption and emission can be separated, so first the absorption by the PMMA and the nanocrystals is calculated and then the emis-sion is added separately, calculated from the total absorption by the nanocrystals. It is assumed that this is a reasonable approximation, because the absorption at the energy of the emission peak is relatively low compared to the absorption at higher energies and also because the absorption by the PMMA is in general quite low, so that the fraction of the light that is reabsorbed will be small.

3.3.1 Absorption by the nanocrystals

To be able to calculate the absorption by the nanocrystals, the absorption cross section needs to be known.

For the absorption cross section of the cesium lead halide, we started with an experimental absorption spectrum of CsPbBr3, shown in figure 8a. In this spectrum, the optical density

of a cesium lead bromide sample for different photon energies is shown. As optical density is calculated with the total incident and exiting intensities, this quantity heavily depends on the parameters of the sample that is used. As these were unknown, this spectrum in itself could not be used to calculate the cross sections, but it was useful for giving a general idea of the shape of the spectrum.

To find the cross section, we used data from De Roo et al. (2016) [21]. In their research they calculated the intrinsic absorption coefficient µ, which is defined for a wavelength i as

µi ≡

ln 10A

f L , (12)

where A is the absorbance of the nanocrystal, f is the fraction of the volume occupied by the nanocrystals, and L is the optical path length [21]. The experimental results for absorbance were thus converted to the more general quantity of the intrinsic absorption

(15)

coefficient using equation 12. For it to be useful in this research, it had to be converted to the absorption cross section σλ, which was achieved by using the relation between the

molar attenuation coefficient i and the intrinsic absorption coefficient given by

λ=

Na

ln 10µλd

3_, ₍₁₃₎

where Na is Avogadro’s constant and d is the edge length of the nanocrystal. Then using

the relation

σλ=

ln 10 Na

λ (14)

we find that the intrinsic cross section and the molar attenuation coefficient are related by

σλ = µλd3. (15)

This equation was used to calculate the absorption cross sections in the relevant wavelength range. We then find that the shift in band gap energy due to quantum confinement effects changing as the size of the nanocrystal changes, has to be taken into account. This shift was included in the model by adding a term

∆E(d) = ~ 2_π2 2m∗_d2 + E15nm (16) with E15nm= ~ 2_π2 2m∗(15 nm)2 (17)

where m∗ is the reduced carrier mass of the electrons and holes in the nanocrystal and where d is once again the size of the nanocrystal [14]. The term E15nm is included to

compensate for the size of the nanocrystals on which the band gap in the experiment from De Roo et al (2016) was done. This resulted in the cross sections in figure 8b.

(a) Experimentally determined spectrum for the optical density of CsPbBr3.

(b) The absorption cross sections of CsPbBr3 as calculated from the intrinsic absorption coefficients in [21]. Spectrum calculated for a nanocrystal with edge size d = 10 nm.

(16)

3.3.2 Absorption by the PMMA

To find the absorption by the PMMA, a slightly different approach was used. From Al-Taa’y et al (2015) [22], data for the extinction coefficient kλwas taken. With the extinction

coefficient the absorption coefficient can be calculated using αλ =

4πkλ

λ . (18)

Since the absorption coefficient is simply the cross section times the number density of the nanocrystals, α = σn, the total absorption can then be calculated by combining the absorption cross sections from the nanocrystals and the absorption coefficient of the PMMA in equation 7.

3.4 Emission

After the absorption in the entire SCL has been calculated, the emission is added sep-arately. In order to calculate the emission, the number of photons that is absorbed by the nanocrystals has to be determined. This is done by taking the fraction of the total absorbed intensity due to the absorption by the nanocrystals and converting this to a number of photons using the photon energy. The number of emitted photons could then be calculated by multiplying the number of absorbed photons with the photoluminescence quantum yield of the nanocrystals.

Figure 9: Experimental data for the emission of CsPbBr3, fitted to a Lorentzian peak function

as in equation 19.

To find the shape of the emission spectrum, experimental data of the emission of CsPbBr3

was used, which is shown in figure 9. From fitting the data to different peak functions, we see that the peak in emission in photon energy most closely resembles a Lorentzian, which is described by f (x) = 1 πγ γ2 (x − x0)2+ γ2 , (19)

where x0 indicates the peak position and γ is an indication of the width of the peak. Using

(17)

maximum of this emission peak, the spectrum can be characterised. With equation 9, the shift of the band gap with edge size can be determined. To include the size dependence of the Stokes shift, data from [16] is used. The calculated difference in Stokes shift due to the size is subtracted from the peak energy, so that the energy difference in the absorption peak and the emission peak is the right Stokes shift for the size of the nanocrystal. Then, to calculate the height of the emission peak, the spectrum of the absorption by the nanocrystals is converted to a number of photons, which is integrated over the whole energy range to find the total number of absorbed photons. This could then be used to find the total number of emitted photons by multiplication with the quantum yield, which is assumed to be constant over the spectrum. Lastly, to find the emission spectrum, the integral over a standard Lorentzian is used, which is equal to πγ. Dividing the total number of emitted photons by this integral gives the amplitude of the peak.

3.5 Back reflection

After the light has already passed through the SCL, it reaches the interface from the layer to the organism or the air. There, the reflections have to be considered carefully, as the assumptions made at the first interface are no longer valid. The first assumption there was that the light entered the SCL under an angle of incidence of zero degrees. At this second interface, there will be a distribution of angles of incidence. The purely transmitted light still incides perpendicular to the interface, but light emitted by the nanocrystals can incide at any angle, since the emission by the nanocrystals is omnidirectional. To take this into account, the transmitted and emitted light was considered separately, using an angle of incidence of zero for the first and an assumed flat distribution of angles for the second. An important note here is that only half of the emitted light is considered at this interface, as half of the emission by the nanocrystals will be upwards, and is for the sake of simplicity considered lost in this research.

Another difference between the first and the second interface arises due to a difference in indices of refraction. At the second interface, light travels from an optically denser medium to an optically thinner one, which means that light inciding at a large enough angle is completely reflected. In practice this means that only light inciding at small angles is transmitted as can be seen in figure 10, where the reflection coefficient R is shown for different angles at λ = 500 nm, calculated using equations 2, 3 and 4. Because of this, a big fraction of the emitted light gets lost.

After consideration of these reflections, the two spectra are added to give the spectrum as it is changed by the SCL.

3.6 Photosynthetic rate

The light then enters the organism. For two organisms, Synechococcus 6312 and the Ulva species, the relative action spectra were taken from Lemasson et al (1973) [23] and Gor-ton (2010) [24]. To calculate the spectral photosynthetic rates, the light spectrum was multiplied by the action spectrum. Eventually, what we were interested in was either an increase or decrease of the total photosynthetic rate of the organism as a result of the introduction of the SCL. To find this, the rate was calculated with and without the SCL, using the action spectra and the spectrum coming out of the SCL in the first case and the action spectrum and the solar spectrum in the second. By integrating over the resulting spectra, a total photosynthetic rate was found and it was thereby possible to find a per-centual increase or decrease due to the SCL.

(18)

Figure 10: Reflection coefficient calculated for different angels of incidence at a wavelength of 500 nm. It is clear that at angles larger than the critical angle θc= 69.5◦, the reflection coefficient

is 1, so that there is total internal reflection.

4 Results

The goal of this research was to find out whether it was possible to increase the total photosynthetic rate of an organism by introducing a spectral conversion layer. This layer can, however, be made in different ways. For example, the thickness and number density of nanocrystals inside the layer can be varied, and the nanocrystals themselves can be changed as well: the size, type and quantum yield of the nanocrystals are all parameters in the model. We started by looking at the effects of a SCL with nanocrystals of CsPbBr3,

after which we expanded the research to include CsPbI3.

One of the first things we can see, is that changing the thickness and the density of the nanocrystals changes the photosynthetic rate in the same manner, because they both in-fluence the fraction of the light that is absorbed. This can be seen in figure 11, where for a SCL with nanocrystals of CsPbBr3 that have an edge size of 10 nm the thickness of the

layer and the volume fraction of the nanocrystals is changed.

It is clear that the graphs and the percentual absorption change in a similar way if the thickness of the layer is increased by a certain factor as if the volume fraction would be increased by the same fraction. This is due to the fact that both the volume fraction and the thickness appear in the exponent in the Beer-Lambert law. Therefore, only the percentual absorption will be used as a parameter, instead of the thickness and the vol-ume fraction separately, since the absorption is also easily measured in experiments. The aforementioned graphs show that absorptions in a range of 1.7 % to 31.3% can be reached. It is unlikely that much higher percentages are possible, as the lower energy part of the spectrum is not absorbed at all. The fraction of light that is absorbed by the nanocrys-tals in the considered range can only be changed significantly if a shift in the absorption spectrum is realised. This is possible if, for example, a different type of nanocrystal than CsPbBr3 is used or if the crystals have a different edge size.

(19)

(a) Volume fraction = 0.01, thickness = 1µm, absorption = 1.7% (b) Volume fraction = 0.01, thickness = 5µm, absorption = 7.3% (c) Volume fraction = 0.01, thickness = 10µm, absorption = 12.5% (d) Volume fraction = 0.05, thickness = 1µm, absorption = 7.4%

(e) Volume fraction = 0.05, thickness = 5µm, absorption = 21.5% (f ) Volume fraction = 0.05, thickness = 10µm, absorption = 27.3% (g) Volume fraction = 0.10, thickness = 1µm, absorption = 12.7% (h) Volume fraction = 0.10, thickness = 5µm, absorption = 27.3%

(i) Volume fraction = 0.10, thickness = 10µm, absorption = 31.3%

Figure 11: The percentual absorption as determined by the volume fraction of nanocrystals CsPbBr3 in the SCL and the thickness of the layer is shown. It is clear that there is a symmetry

along the diagonal; the graphs on the first row for example are similar to the graphs in the first column.

This means that there are three remaining parameters: the size of the nanocrystals, the photoluminescence quantum yield of the nanocrystals and the fraction of the light that is absorbed by the nanocrystals.

When the percentual absorption by the nanocrystals is increased, keeping the size and photoluminescence quantum yield fixed at 10 nm and 0.80 respectively, the emission peak becomes more pronounced, as more light from the high energy part of the spectrum is absorbed, resulting in a higher emission. This effect becomes visible when comparing fig-ure 12a to figfig-ure 12b and figfig-ure 12c to figfig-ure 12d. Figfig-ures 12a and 12c show the spectra inciding on Synechococcus 6312 and on Ulva respectively, with and without SCL, for a 12.7 % absorption. In figures 12b and 12d, the same spectra are shown, but with 32.9 %

(20)

absorption. For both algae species, the emission peak in the spectrum with SCL is no-ticeably higher with high absorption, whereas the intensity at higher energies than the emission becomes lower.

The effect these changes have on the photosynthetic rate becomes clear by comparing the peak position in the incident spectra to the peak position in the action spectra, shown in green in figures 12a to 12d. In case of the Synechococcus 6312, an increase in absorption takes away light energy from the parts of the spectrum where it can be used efficiently and heightens the emission peak, which is positioned exactly where the action is lowest. This results in an increase in the negative change in photosynthetic rate due to the SCL as the percentual absorption increases. This can be seen in figure 12e, where the change in photosynthetic rate due to the SCL is plotted for different values of the percentual absorption. For the Ulva species, this effect is even more dramatic, as the largest peak in the action spectrum is positioned in the part where most light is absorbed.

From this we can conclude that higher absorption rates are not efficient if the position of the peak in emission does not coincide with the position of the peak in the action spectrum. To change the position of the emission peak, the size of the nanocrystals can be altered. Figures 13a and 13c show the spectra of the light inciding on the algae for a nanocrystal of 5 nm and figures 13b and 13d for a nanocrystal of 15 nm. The emission peak en-ergy for smaller nanocrystals is clearly higher than for bigger nanocrystals. This change in peak position works favourably for the Synechococcus 6312, where the emission peak shifts towards the peak in the action spectrum as the nanocrystal size increases, but the opposite happens for Ulva. There, the emission peak actually shifts away from the action peak. The increase in overlap for Synechococcus 6312 results in a favourable change in photosynthetic rate due to the SCL as the nanocrystal size increases, whereas the change in photosynthetic rate becomes increasingly negative for Ulva, as shown in figure 13e. Finally, we see that increasing the photoluminescence quantum yield heightens the emis-sion peak. This becomes clear by comparing figure 14a to 14b, and 14c to 14b, where in the figures on the left, the quantum yield is 0.60 and in the figures on the right, it is 0.95. An increase in photoluminescence quantum yield results in more emission for the same percentual absorption, thereby heightening the emission peak, but not changing the rest of the spectrum. As a results, an increasing photoluminescence quantum yield always results in a more favourable change in photosynthetic rate due to the SCL, which is shown in figure 14e.

These results show that the introduction of a SCL containing CsPbBr3nanocrystals is not

able to increase the overall photosynthetic rate in an organism according to this model. This is mainly due to a mismatch in the positions of emission peak and the peak in the action spectrum. By changing the nanocrystal size, this position can be changed, thereby affecting the change in photosynthetic rate. However, the effect of changing the size of the nanocrystal is too small to shift the peaks enough to actually increase the photosynthetic rate. The effect of the percentage of the light that is absorbed by the nanocrystals is rather big, and it turns out that an increased absorption is counter effective if the peak position is not exactly right. The effect of an increase in photoluminescence quantum yield is small, but always positive; if the quantum yield increases, the loss due to the SCL decreases.

(21)

for which the photosynthetic rate increases as a result of the SCL. To this end, the other cesium lead halides were considered; cesium lead iodide and cesium lead chloride, which respectively have emission peaks at lower and higher energies than cesium lead bromide. The approach with these nanocrystal materials was a bit different than with cesium lead bromide. Instead of finding the absorption and emission spectra of these crystals, the ones from CsPbBr3 were taken and slightly adapted. The cross sections from CsPbBr3

were taken, but the whole spectrum was shifted to match the band gap energy of the new material and the same was done for the emission spectrum. A final adaptation that was made was a correction in the Stokes shift, as this shift is significantly smaller in CsPbI3

and CsPbCl3 than in CsPbBr3 [25].

Similar tests of the parameters were done for these types of nanocrystals, only now start-ing by findstart-ing the optimal size of the nanocrystal and continustart-ing from there. This turned out only to work with CsPbI3, as its emission peak is around 2.0 eV, whereas the emission

from CsPbCl3 peaks at energies higher than 3.0 eV, thereby falling outside of the range

of the action spectra.

Because of this, we tried to find an optimal photosynthetic rate with nanocrystals of CsPbI3. This process is shown in figure 15. Keeping the values for the absorption and

photoluminescence quantum yield fixed, the size of the nanocrystals was changed. For the Synechococcus 6312, there was a clear, positive peak around 4 nm, as shown in figure 15a. It turns out that with this size of the nanocrystal, an increase in photosynthetic rate can be achieved, because with this size the emission and absorption peak around 2.1 eV, which overlaps with the (broad) peak in the absorption spectrum of the Synechococcus 6312. For Ulva, no such peak was found and the search was therefore continued solely for the Synechococcus 6312. As the optimal size was found, we fixed this value and tried to find the optimal percentual absorption. Figure 15b shows that there is an optimal percentual absorption at 35.8 %.

Interesting here is that the change in photosynthetic rate decreases as the absorption is more than this 35.8 %. This is probably due to the absorption of photons with energies just above the emission energy, for which the actions are nonzero. As the percentual absorption increases, the photons with much higher energy than the emission peak will be absorbed, which contributes positively to the photosynthetic rate, as the energy is transferred from a part of the spectrum where the action is zero to the part where the energy can be used efficiently. At the same time, photons with energies just above the energy of the emission peak will be absorbed and thereby taken from the part of the spectrum where the action is nonzero, an effect that contributes negatively to the total photosynthetic rate. As the absorption increases, at some point most of the high energy photons will be absorbed, so that the positive effects of this absorption can no longer balance out the negative effects of the absorption of the lower energy photons. The change in photosynthetic rate due to the SCL then starts to decrease with higher absorption.

Keeping the value for optimal absorption fixed as well, the optimal photoluminescence quantum yield was sought. The change in photosynthetic rate becomes positive for a quantum yield of 0.70 and increases further for higher quantum yields. In figure 15d, the spectra for the optimal values are shown.

(22)

(a) Absorption = 12.7 % Synechococcus 6312 (b) Absorption = 32.9 % Synechococcus 6312 (c) Absorption = 12.7 % Ulva (d) Absorption = 32.9 % Ulva

(e) The percentual change in photosynthetic rate with the percentage of light absorbed by the nanocrystals for both the Synechococcus 6312 and the Ulva species.

Figure 12: Photosynthetic rate for absorption by the nanocrystals. Figure 12a shows the spectra at an absorption of 12.7 % for light inciding on the Synechococcus 6312 with and without the SCL. Figure 12b shows the same, but for a percentual absorption of 32.9 %. The same spectra for the Ulva species are shown in figures 12c and 12d. In these plots, the action spectra for the algae are shown in green and the region of interest is indicated by the vertical lines. In figure 12e the effects of the change in absorption on the change in photosynthetic rate with SCL as compared to the

(23)

(a) Size = 5 nm Synechococcus 6312 (b) Size = 15 nm Synechococcus 6312 (c) Size = 5 nm Ulva (d) Size = 15 nm Ulva

(e) The percentual change in photosynthetic rate with the size of the nanocrystal.

Figure 13: Photosynthetic rate for nanocrystals size. Figure 13a shows the spectra of the light inciding on the Synechococcus 6312 with and without the SCL, for an nanocrystal size of 5 nm. Figure 13b shows the same, but for a size of 15 nm. The same spectra for the Ulva species are shown in figures 13c and 13d. In these plots, the action spectra for the algae are shown in green and the region of interest is indicated by the vertical lines. In figure 13e the effects of the change in size on the change in photosynthetic rate with SCL as compared to the photosynthetic rate without SCL are shown. Spectra and values are calculated for a percentual absorption of ∼ 27.4% and a photoluminescence quantum yield of 0.80.

(24)

(a) PLQY = 0.60 Synechococcus 6312 (b) PLQY = 0.95 Synechococcus 6312 (c) PLQY = 0.60 Ulva (d) PLQY = 0.95 Ulva

(e) The percentual change in photosynthetic rate with the photoluminescence quantum yield of the nanocrystal.

Figure 14: Photosynthetic rate for photoluminescence quantum yield. Figure 14a shows the spectra of the light inciding on the Synechococcus 6312 with and without the SCL, for a quantum yield of 0.60. Figure 14b shows the same, but for a photoluminescence quantum yield of 0.95 The same spectra for the Ulva species are shown in figures 14c and 14d. In these plots, the action spectra for the algae are shown in green and the region of interest is indicated by the vertical lines. In figure 14e the effects of the change in quantum yield on the change in photosynthetic rate with SCL as compared to the photosynthetic rate without SCL are shown. Spectra and values are

(25)

(a) Size dependence of change in photosynthetic rate, at fixed absorption of ∼ 21.4 % and at a PLQY of 0.80.

(b) Change in photosynthetic rate with percentual absorption by the nanocrystal at optimal size 4 nm and with a PLQY of 0.80.

(c) PLQY dependence of the change in photosyn-thetic rate at optimal absorption of 35.8% and op-timal size 4 nm.

(d) The optimal outcoming spectrum and the ac-tion spectrum of Synechococcus 6312.

Figure 15: Finding the optimal parameters for a SCL containing CsPbI3 nanocrystals on the

Synechococcus 6312.

5 Discussion

In the creation of this model, many processes have been neglected or approximated, thereby possibly impacting the results.

The first, important approximation considered the processes inside the SCL. At the bot-tom interface, only light that came directly through the material or that was emitted once was taken into account. Processes like re-absorption and emission, but also scatterings and back-reflections have all been neglected. The way in which the emission and absorption was calculated is not exact either. To come up with a spectrum that closely resembles the spectrum coming through a real SCL, a simulation would be necessary. This would enable inclusion of re-absorption processes and would make the calculation of the total intensity at the bottom of the layer more complete, as in that case, at each point all the light from all sources can be taken into account. The effects of reflections can be studied more accurately then as well. In this research, all light reflected from the bottom interface is considered lost. However, as this light is reflected back into the layer, it can again be absorbed and emitted. An overview of the losses in the SCL is given in figure 16, where

(26)

we can see that 28, 7 % of the light that enters the SCL gets lost and that reflections and directionality of emission account for 68, 6 % of these losses (19, 7 % of the total light ends up there). So a large fraction of the light that is considered lost in this research in principle remains in the system and can still be absorbed by the organism.

Figure 16: An overview of the losses in the SCL for the case where the photosynthetic rate is optimised with nanocrystals (NCs) of CsPbI3.

Another approximation considering the reflections is made in the use of the indices of refraction. For the reflection off the top layer, as well as the reflection off the interface between the SCL and the organism, the refractive index of PMMA is used, even though the SCL contains nanocrystals as well. As long as the volume fraction of nanocrystals in the material is not too large, the index of refraction will not be affected significantly by their presence. However, as this fraction becomes bigger, the refractive index should be calculated more precisely, using the Bruggeman effective medium approximation, which calculates optical constants for mixed mediums using their volume fractions and the opti-cal constants of the individual components [26]. When higher absorption fractions by the nanocrystals are wanted, it might become necessary to increase the volume fraction of the nanocrystals, which then affects the index of refraction of the SCL.

Similarly, for the index of refraction of the organisms, an average value for organic mate-rial was taken in this research. In reality, this value will differ greatly with the organism considered.

Furthermore, the use of the action spectra in this research is not completely valid. An action spectrum actually shows the photosynthetic response of an organism to light of one specific wavelength. With the sun as a light source, organisms will actually receive light with many different wavelengths at once, which influences the composition of the photosynthetic apparatus of the organisms, so that the photosynthetic response might not just be the sum of the responses to individual wavelengths. This is called the Emmerson

(27)

effect. Plants are, for example, able to shift the fractions of the different photosynthetic pigments they contain, thereby changing the wavelengths which they can absorb. Some organisms are also able to change the fraction of light harvesting systems associated with photosystem I to photosystem II, to increase the efficiency with which light can be used. This means that the photosynthetic organism changes its susceptibility to certain wave-lengths based on the light it receives.[27] It is therefore really complicated to predict how an organism will respond to a spectrum, and it is not as straightforward as a multipli-cation of the incoming spectrum with the action spectrum, as was done in this research. Comparisons with experiments have to be made to find whether the model on this part is acceptable.

Also, for the calculations with CsPbI3, many assumptions have been made. It is assumed

that the absorption cross sections are exactly the same as for CsPbBr3, with the only

implemented difference being the energy at which the band gap and thereby also the rest of the absorption features are appearing. The shape of the graph showing the cross section at different photon energies appears to be rather similar for the cesium lead halides [25], but the exact values may differ. However, as we used percentual absorption to find the change in photosynthetic rate, this might not be relevant. Furthermore it is assumed that the emission peaks of the cesium lead halides have similar shapes. This seems reasonable, as for both CsPbBr3 and CsPbI3 the emission comes from the same electronic transition.

Lastly, it is assumed that the band gap and Stokes shift change in the same way with nanocrystal size for the different types of nanocrystals. This might well be incorrect, as the different halogens are associated with different quantum confinement effects and size regimes. For future research on this topic, it is therefore essential to get a more accurate view on the size dependence of absorption and emission for other types of nanocrystals. The research could also be expanded to include other compositions of nanocrystals than the three described here. With that, and by changing the size of the nanocrystal, the emis-sion peak can be positioned at any point in the spectrum, so that it can be created at the optimal position for the photosynthetic rate, thereby making it easier to increase this rate. Lastly, it is important to keep the ultimate goal of the research in mind; increasing of the photosynthetic rate so that crop growth is more efficient. This means that the SCL has to fulfil many more requirements than discussed here. This research aimed to explore the possibilities of a SCL, to find whether it could increase the photosynthetic rate. For actual appliance of the SCL on organisms, it should have no environmental or biological impact and the production process should not be too expensive or difficult. These are, however, all outside the scope of this research.

6 Conclusion

Due to the many assumptions and approximations in this research, caution is required when drawing any conclusions. Overall, a SCL on top of an organism has the potential to increase the total photosynthetic rate, but in order to do so it is essential to find the type of nanocrystals that emit photons with energies that match the action spectrum of the organism in question.

For Ulva, no increase could be found, as the emission peaks of the types of nanocrystals used in this research could not match the action spectrum of this organism. In the case

(28)

of the Synechococcus 6312, an emission peak at 2.11 eV turned out to be ideal. This emission peak was found in this research by using nanocrystals of modelled CsPbI3 with

an edge size of 4 nm. It was found that a photoluminescence quantum yield of 0.70 was the minimum for the SCL to have a positive effect on the photosynthetic rate. With op-timal absorption and photoluminescence conditions, the photosynthetic rate increased by 6.87 % using this type of SCL.

7 Acknowledgements

This project has been long and difficult, but also very interesting and fulfilling for me. A lot of people have helped me throughout this project, each in their own way, and I would like to express my appreciation for all of it. More specifically I would like to thank the people in the quantum dot meetings for their feedback, which has proved to be very helpful, and I would also like to thank my daily supervisor Arnon Lesage who has been very patient and kind, and whose suggestions and feedback were invaluable.

References

[1] Oakley, D. O., Rao, K. K. 1999. Photosynthesis. Sixth Edition. Cambridge: Cambridge University Press.

[2] Blankenship, R.E., Tiede, D.M., Barber, et al. (2011). Comparing Photosynthetic and Photovoltaic Efficiencies and Recognizing the Potential for Improvement. Science, 332, 805 - 809.

[3] Food Aid Foundation. World Hunger Statistics. (2019). https://www. foodaidfoundation.org/world-hunger-statistics.html

[4] NREL. Reference Air Mass 1.5 Spectra. (no date). https://www.nrel.gov/grid/ solar-resource/spectra-am1.5.html

[5] PV Lighthouse. The direct standard spectrum (AM1.5d) (no date). https: //www2.pvlighthouse.com.au/resources/courses/altermatt/The%20Solar% 20Spectrum/The%20direct%20standard%20spectrum%20(AM1-5d).aspx

[6] PV Lighthouse. The global standard spectrum (AM1.5g) (no date). https: //www2.pvlighthouse.com.au/resources/courses/altermatt/The%20Solar% 20Spectrum/The%20global%20standard%20spectrum%20%28AM1-5g%29.aspx

[7] Griffiths, D. J. 2013. Introduction to Electrodynamics. Fourth Edition. Pearson Edu-cation.

[8] Sellmeier, W. (1872). Ueber die durch die Aetherschwingungen erregten Mitschwingun-gen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien (II. Theil). Ann. Ph. Ch. 223(11), 386–403

[9] Ohara. Optical Properties. (no date). http://oharacorp.com/o2.html [10] Ohara. Guarentee of Quality. (no date). http://oharacorp.com/o7.html

(29)

[11] Mini Physics. Total Internal Reflection. (December 25, 2015). https://www. miniphysics.com/total-internal-reflection-2.html

[12] Lin, J., Gomez, L., De Weerd, C., Fuyiwara, Y., Gregorkiewicz, Suenaga, K. (2016). Direct Observation of Band Structure Modifications in Nanocrystals of CsPbBr3

Per-ovskite. Nano Lett., 16, 7198 - 7202.

[13] Dutta, A., Behera, R.K., Pal, P., Baitalik, S., Pradhan, N. (2019). Near-Unity Photoluminescence Quantum Efficiency for All CsPbX3 (X=Cl, Br, and I) Perovskite Nanocrystals: A Generic Synthesis Approach. Angew. Chem. Int. Ed., 58, 5552 - 5556. [14] Protescu, L., Yakunin, S., Bodnarchuk, M. (2015). Nanocrystals of Cesium Lead Halide Perovskites (CsPbX3, X = Cl, Br, and I): Novel Optoelectronic Materials Show-ing Bright Emission with Wide Color Gamut. Nano Lett., 15, 3692-3696

[15] Lin, J., Gomezm L., De Weerdm C., Fujiwara, Y., Gregorkiewicz, T., Suenaga, K. (2016). Direct Observation of Band Structure Modifications in Nanocrystals of CsPbBr3 Perovskite. Nano Lett., 16(11), 7198-7202

[16] Brennan, M. C., Herr, J. E., Nguyen-Beck, T. S., et al. (2017). Origin of the Size-Dependent Stokes Shift in CsPbBr3 Perovskite Nanocrystal. J. Am. Chem. Soc. 139, 12201-12208

[17] McCree, K.J. (1970). Action spectrum, absorptance and quantum yield of photosyn-thesis in crop plants. Agric. Meteorol., 9, 191-216

[18] Luimstra, V. M., Schuurmans, J. M., Verschoor, A. M., Hellingwerf, K. J., Huisman, J., Matthijs, H. C. P. (2018). Blue light reduces photosynthetic efficiency of cyanobac-teria through an imbalance between photosystems I and II. Photosynth. Res., 138, 177-189

[19] Polyanskiy, M. Optical Constants of Plastics: PMMA - poly(methyl acrylate). (no date). Refractive Index Info. https://refractiveindex.info/?shelf=3d&book= plastics&page=pmma

[20] Sultanova, N., Kasarova, S., Nikolov, I. (2009). Dispersion Properties of Optical Poly-mers. Acta Phys. Pol. A., 116, 585-587

[21] De Roo, J., Ib´a˜nez, M., Geiregat, P., Nedelcu, G., Walravens, W., Maes, J., Martins, C., Van Driessche, I., Kovalenko, M. V., Hens, Z. (2016). Highly Dynamic Ligand Bind-ing and Light Absorption Coefficient of Cesium Lead Bromide Perovskite Nanocrystals. Acs Nano, 10, 2071-2081

[22] Al-Taa’y, W. A., Oboudi, S. F., Yousif, E., Nabi, M. A., Yusop, R. M., Derawi, D. (2015). Fabrication and Characterization of Nickel Chloride Doped PMMA Films. Advances in Materials Science and Engineering, 2015, 913260

[23] Lemasson, C., Tandeau de Marsac, N., Cohen-Bazire, G. (1973). Role of Allophyco-cyanin as a Light-Harvesting Pigment in Cyanobacteria. Proc. Nat. Nacad. Sci. USA., 70, 3130-3133

[24] Gorton, H.L. (2010). Biological Action Spectra. http://photobiology.info/ Gorton.html

(30)

[25] Diroll, B. T., Zhou, H., Schaller, R. D. (2018). Low-Temperature Absorption, Pho-toluminescence, and Lifetime of CsPbX3 (X = Cl, Br, I). Nanocrystal. Adv. Funct. Mater., 28, 1800945

[26] Bruggeman, D. A. G. (1935). Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mis-chkörper aus isotropen Substanzen. Ann. Phys., 24(7), 636-664

[27] Blankenship, R.E. 2002. Molecular Mechanisms of Photosynthesis. Second Edition. Saint-Louis: Wiley-Blackwell.

Calculating Algae Growth Rates Using Nanocrystal Based Spectral Conversion Layers