Tailoring the Spectral Absorption Coefficient of a Blended Plasmonic Nanofluid Using a Customized Genetic Algorithm
aa r X i v : . [ phy s i c s . op ti c s ] F e b Tailoring the Spectral Absorption Coefficient of aBlended Plasmonic Nanofluid Using a CustomizedGenetic Algorithm
Junyong Seo , Caiyan Qin , Jungchul Lee , and Bong Jae Lee Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141,South Korea Center for Extreme Thermal Physics and Manufacturing, Korea Advanced Institute of Science and Technology,Daejeon 34141, South Korea * Corresponding author: [email protected] (B.J. Lee)
ABSTRACT
Recently, plasmonic nanofluids (i.e., a suspension of plasmonic nanoparticles in a base fluid) have been widely employed indirect-absorption solar collectors because the localized surface plasmon supported by plasmonic nanoparticles can greatlyimprove the direct solar thermal conversion performance. Considering that the surface plasmon resonance frequency ofmetallic nanoparticles, such as gold, silver, and aluminum, is usually located in the ultraviolet to visible range, the absorptioncoefficient of a plasmonic nanofluid must be spectrally tuned for full utilization of the solar radiation in a broad spectrum. Inthe present study, a modern design process in the form of a genetic algorithm (GA) is applied to the tailoring of the spectralabsorption coefficient of a plasmonic nanofluid. To do this, the major components of a conventional GA, such as the genedescription, fitness function for the evaluation, crossover, and mutation function, are modified to be suitable for the inverseproblem of tailoring the spectral absorption coefficient of a plasmonic nanofluid. By applying the customized GA, we obtainedan optimal combination for a blended nanofluid with the desired spectral distribution of the absorption coefficient, specifically auniform distribution, solar-spectrum-like distribution, and a step-function-like distribution. The resulting absorption coefficientof the designed plasmonic nanofluid is in good agreement with the prescribed spectral distribution within about 10% to 20%of error when six types of nanoparticles are blended. Finally, we also investigate how the inhomogeneous broadening effectcaused by the fabrication uncertainty of the nanoparticles changes their optimal combination.
Plasmonic nanofluids, which contain a suspension of plasmonic nanoparticles in a base fluid, have been proposed as effectiveworking fluids to directly convert solar radiation to thermal energy . Owing to the resonance characteristics of the localizedsurface plasmon (LSP), the absorption efficiency of the nanoparticles can be greatly enhanced with the excitation of the LSP,offering great potential in solar thermal applications. For instance, a direct-absorption solar collector (DASC) combined witha plasmonic nanofluid has drawn much attention for solar thermal energy harvesting in recent decades . Recently, Qin et al. showed how the spectral absorption coefficient of a plasmonic nanoparticle should be tuned (i.e., either uniformlyor following the solar spectrum) by engineering nanoparticle suspensions to exploit the solar radiation maximally with thegiven constraint of the total particle concentration. Therefore, the effective tuning of the spectral absorption coefficients ofplasmonic nanofluids is crucial for improving the thermal performance capabilities of DASCs.As suggested by Lee et al. , broadband absorption spectra can be designed by blending multiple types of nanoparticlesgiven that the resonance wavelength of the LSP depends on the material, size and shape of the nanoparticles. The simplest andthe most common structure is a spherical nanoparticle . Nano-spheres made with noble metals are widely utilized in variousdisciplines, such as in medical and biological applications. However, because the resonance peak of the LSP associatedwith nano-spheres mainly depends on the material properties , nano-spheres themselves may not be suitable for thermalapplications. Thus, additional types of nanoparticles, such as silica core-metallic shell and nano-rod nanoparticles,have also been considered given their potential for better controllability. For core-shell nanoparticles, the ratio between thecore radius and the shell thickness serves as a factor when tuning the absorption response , while the aspect ratio of thenano-rod performs this function . Thus, the critical question is “What would the optimal combination of various types ofnanoparticles be for the effective tuning of the spectral absorption coefficient?” Note that finding the optimal combinationof plasmonic nanoparticles is not a straightforward task due to the diversity and complexity of nanoparticles with regards totheir materials and shapes. For instance, Taylor et al. just employed the Monte-Carlo approach (i.e., random generation and1 igure 1. Schematic of the nano-sphere, core-shell, and nano-rod shapes. The design variables are the radius ( r ) for thenano-sphere, the core radius ( r c ) and the shell thickness ( t s ) for the core-shell, and the radius ( r r ) and the length ( l r ) for thenano-rod.selection) to find the optimal blending combination of core-shell particles for a nanofluid-based optical filter.In general, an inverse problem is a problem that requires the determination of the design of a system from its outputresponse. It is known that finding a proper solution to an inverse problem is often challenging because most inverse problemsare ill-posed and nonlinear . Nevertheless, if a particular solution-finding technique of an inverse problem is available, itcan be readily applied to diverse engineering fields, such as magnetic resonance imaging , combustion , and radiative heattransfer . Tailoring the absorption spectrum of a plasmonic nanofluid can also be treated as an inverse problem whendesigning a system (i.e., combination of plasmonic nanoparticles) at a given response (i.e., the desired spectral absorptioncoefficient). Because a blended combination of nanoparticles is represented with a broad range of variables, it is difficultto determine the optimal composition of a nanofluid to have the desired absorption spectrum. Therefore, a modern solutiontechnique, such as a genetic algorithm , can be employed to solve our blending problem.Genetic algorithms (GAs) have been widely employed to solve many design problems by customizing a description of anindividual chromosome (i.e., member of a population) and a fitness function (i.e., score of the chromosome) properly . Forexample, the dimensions of a tandem-grating nanostructure for a solar thermal absorber , the spectral distribution of absorp-tion coefficients for DASC , and the dimensions of a multi-layer microcylinder for a plasmonic nanojet have been optimizedbased on carefully defined chromosomes and fitness functions. Here, we also apply a GA to find the optimal combination ofplasmonic nanoparticles to achieve the desired spectral absorption coefficient of a nanofluid. To maximize the diversity of theplasmonic response of the nanoparticles, we consider two materials (gold and silver) and three types of nanoparticle shapes(nano-sphere, core-shell, and nano-rod). The target spectral absorption coefficient of the plasmonic nanofluid is first set to beeither uniform or to follow the solar spectrum . In addition, a step-function-like absorption coefficient will be designed for ahybrid solar PV/T application . Finally, how the inhomogeneous broadening effect caused by the fabrication uncertaintyof the nanoparticles changes their optimal combination is also investigated. It is well known that subwavelength-size metallic nanoparticles can support a localized surface plasmon (LSP), whose res-onance condition depends strongly on the materials, sizes, and shapes of the nanoparticles . In the present study, weconsider two materials and three shapes of nanoparticles (see Fig. 1) to diversify a number of possible blending combinations.The ranges of each design variable are carefully constrained according to the literatures . These are listed in Table 1.For a given nanoparticle, its spectral absorption efficiency, Q a ( λ ) , can be calculated by solving Maxwell’s equations. Fornano-sphere and core-shell particles, Mie-scattering theory and a modified version of it were used to determine Q a ( λ ) . Forthe nano-rod, a boundary element method (BEM) was applied to obtain the polarization- and direction-averaged absorption Table 1.
Ranges of each design variable when training the surrogate modelType Design variable Range (nm)Nano-sphere Radius ( r ) 10 ∼ r c ) 5 ∼ t s ) 5 ∼ (100 − r c )Nano-rod Radius ( r r ) 6 ∼ l r ) max ( , r r ) ∼ fficiency. In this study, the open-source BEM software MNPBEM was used. In the calculations, the permittivities of gold,silver, silicon dioxide, and water (i.e., the base fluid) were used from tabulated data . As discussed by Lee et al. , if theradius (or thickness) of a metal is smaller than its mean-free-path of conduction electrons, we also must consider that the size-dependent permittivity of the metal should differ from that of the bulk metal due to electron-boundary scattering. Furthermore,a broadening effect will arise due to the modification of the permittivity. In this work, the effect of electron-boundary scatteringis neglected for simplicity, though this effect will be discussed later.With the calculated absorption efficiency of the i -th particle in water, Q a , i ( λ ) , the corresponding absorption coefficient ofthe nanofluid, α a , λ , can be expressed as : α i , λ = f i D i Q a , i ( λ ) (1)where, f i is the volume concentration of the nanoparticle and D i = p × (volume of particle) / π is the effective diameter ofeach particle . If N types of nanoparticles are dispersed together, the resulting absorption coefficient of blended plasmonicnanofluid ( α λ ) is then given by: α λ = − N ∑ i = f i ! α w , λ + N ∑ i = α i , λ (2)where, α w , λ is the absorption coefficient of the water itself. Because solar irradiance begins at approximately λ =
300 nm andthe absorption coefficient of water becomes dominant after λ = ,
100 nm, we calculate the absorption efficiency spectrum ofeach particle from 300 nm to 1,100 nm in 10 nm intervals (i.e., 81 spectral data points).In principle, Q a , i ( λ ) must be known a priori in each computation of the fitness function of a GA. Because the calculationof Q a , i ( λ ) takes about 3 min and the average number of fitness calculations in our GA numbers into the thousands, it is notfeasible to compute it every time. To reduce the computational cost, we decided to build a surrogate model to estimate theabsorption efficiency of each nanoparticle, i.e., [Input: geometry of i -th particle and λ → Output: Q a , i ( λ ) ]. To ensure theaccuracy of the surrogate model, an artificial neural network model was employed as part of a modelling technique. To trainthe neural network models, samples were composed with 2 nm intervals of the design variables and a 10 nm interval of thewavelength. Consequently, we constructed and applied accurate surrogate models with correlation values exceeding R = . Q a spectrum. The accuracyof the model was estimated with the difference between the predicted and actual Q a , i ( λ ) values at the peak location, wherethe maximum Q a , i value was achieved. As a result, the accuracy of model used in this work was found to be between 0.3%(for the nano-spheres) and 1.6% (for the core-shells and the nano-rods) on average. The genetic algorithm (GA) is a powerful method for solution processes owing to its ranges for diverse applicability for a va-riety of problems . In the world of a GA, the population consists of individuals. Each individual has its own chromosome(i.e., set of genes) and evolves along descent generations. Based on simple and bio-mimicking procedures, the population ofthe GA will evolve to obtain the best individual, which has the best chromosome, through a process of selection, crossoverand mutation. The GA can be utilized with proper modification of its gene description, fitness function, and any embed-ded algorithms or hyper-parameters. In this study, (1) descriptions of the chromosomes (or genes), (2) fitness function, (3)crossover, and (4) mutation algorithms are customized especially for solving our inverse problem, designing of the system (acombination of plasmonic nanoparticles) at the given response (the desired spectral absorption coefficient).The most significant aspect of customization is to define the chromosomes of the GA. Because the chromosomes ofindividuals must be related to a combination of plasmonic nanoparticles, we defined the chromosome to possess a set ofnanoparticles as a genes. Each gene on the chromosome has particle properties, such as the material (gold or silver), shape(nano-sphere, core-shell, nano-rod), design variables (geometric parameters), and volume concentration. A chromosome isimplemented as a list of genes. When a chromosome is created, the properties of each gene are determined randomly withintheir ranges. Note that the volume concentration is intentionally set to be less than 0.005% divided by the number of genes(i.e., the number of nanoparticle types in Eq. (2)) to match the scale of each particle’s volume fraction to 0.0001% .A fitness function is usually set to be a distance or a loss function, as the optimization process evolves to minimize a score.In this work, the fitness function is defined as the sum of square error (SSE):SSE = ∑ j = ( α λ j − α target , λ j ) (3) a r g e t S p ec t r a l A b s o r p ti on C o e ff i c i e n t , (cid:217) r _ p e c r Æ (cid:18) [ (cid:133) (cid:143) ? ] Wavelength, ª [ (cid:144)(cid:143) ] (a) TargetWater (b)
TargetWater300 500 700 900 1,100 (c)
TargetWater680 nm300 500 700 900 1,100 300 500 700 900 1,100
Figure 2.
Target absorption coefficient of the blended plasmonic nanofluid (red line): (a) uniform distribution, (b)solar-spectrum-like distribution, and (c) step-function-like distribution. The spectral absorption coefficient of water is alsoillustrated by the blue line.where, λ i is the wavelength in interest with a 10 nm interval (i.e., 300 + j nm) and α target , λ is the target absorption coefficientspectrum defined in Section 2.3. The absorption coefficient of the blended nanofluid was calculated from Eqs. (1) and (2). Toobtain Q a , i ( λ ) , the type and design variables of the nanoparticles described in the chromosome are required. The GA scoredeach individual with this SSE value and caused the population to evolve to minimize the score.After the scores of individuals are evaluated by the fitness function, the GA will prepare the population of the nextgeneration. Initially, the highest scoring individual will remain based on elitism. By default, the top 5% individuals in termsof their scores will move to the next generation. Next, the GA selects individuals as the parents of the rest (i.e., 95% of thenext generation) according to rule of natural selection. That is, individuals with better fitness values are more probable to be aparent, and a stochastic uniform selection rule is applied. For the crossover process, an offspring individual will have a genelist, which basically consists of the first parent’s genes. In addition, an arbitrary gene fragment cut from the second parentis inserted into a randomly chosen location. Finally, for the mutation process, simple one-point mutation is used. A mutatedchild will have a gene list from a parent with one point of a gene replaced by newly created nanoparticle. In this work, a 20%mutation rate is used by default. In other words, 80% of the remaining children are generated by crossover while 20% aregenerated by mutation. Hence, 162% (95% × [ × + ] %) of the total population is selected by the selection rule, and theoffspring for the next generation is born from them by following the crossover and mutation process. To demonstrate how the customized GA can effectively tune the absorption coefficient of a blended plasmonic nanofluid, weconsider three target absorption coefficient spectra, as illustrated in Fig. 2. The first two target spectra are for solar thermalapplications, especially for a direct-absorption solar collector, i.e., a uniform distribution (Fig. 2a) and a solar-spectrum-likedistribution (Fig. 2b). As reported by Qin et al. , a uniform absorption coefficient is more efficient for a highly concentratednanofluid because the heat loss can be minimized. On the other hand, a solar-spectrum-like absorption coefficient is moresuitable when the system only requires an insufficient particle concentration. For simplicity, the average of the absorptioncoefficient was set to 1 cm − considering that the magnitude of the absorption coefficient is scalable according to the particleconcentration [see Eq. (2)]. It should be noted from Fig. 2 that the absorption coefficient of water does not play much of a rolein the absorption process. Thus, the nanoparticles should be carefully designed to achieve broadband absorption, associatedwith their LSP resonances. In addition, a step-function-like absorption coefficient (Fig. 2c) is designed for hybrid solarphotovoltaic/thermal (PV/T) applications . At the zero-absorption regime of the nanofluid, incident solar irradiance willdirectly reach the PV cell and be converted into electricity. In the opaque regime of the nanofluid, the solar irradiance will beconverted to heat by the nanofluid. Here, we select the step point of the spectrum to be the bandgap of the PV cell used in ahybrid PV/T system with a high bandgap with 1.84 eV (approximately 680 nm) . Note also that a high-bandgap PV cell iswidely applied for common solar cell systems or for special purposes . The main idea when tailoring the absorption coefficient of a blended plasmonic nanofluid is to distribute the absorption peaksassociated with each type of nanoparticle along the target spectrum. It is thus expected that more types of nanoparticlesmakes the corresponding absorption coefficient a better fit to the target spectrum. Considering the productivity of a plasmonic able 2.
Optimal combination of plasmonic nanoparticles for the desired spectral absorption coefficient of the nanofluid.The design variables are r for the nano-sphere, ( r c , t s ) for the core-shell, and ( r r , l r ) for the nano-rod.Particle type Property Uniform Solar-spectrum-like Step-function-likeIndex distribution distribution distribution f i × f i × f i × f i × f i × f i × N ) should not be excessive. Given that plasmonic nanofluids with 3 to 5 typesof nanoparticles have been experimentally demonstrated , N is limited to 6 or less.Although not shown here, a higher value of N can achieve a smaller root-mean-square error (RMSE), defined as RMSE = p SSE /
81. Henceforth, we discuss the optimal blending combination case of N =
6, which can retain the smallest RMSEvalue (i.e., the closest absorption coefficient spectrum to the target). The detailed dimensions as well as the locations of themajor absorption peaks of each type of nanoparticles are listed in Table 2.Figure 3a shows the absorption coefficient of a blended plasmonic nanofluid for the target spectrum with a uniformdistribution. The designed plasmonic nanofluid results in a RMSE value of 0.099 cm − , which is less than 10% of thetarget absorption coefficient (i.e., 1 cm − ). The absorption peaks of each type of nanoparticle are well distributed along thevisible and near-infrared spectral regions for broadband absorption. Although the and the resulting absorption peak is often confined to a narrow spectral range (i.e., Au: 500 ∼
540 nm and Ag: 380 ∼ p ec t r a l A b s o r p ti on C o e ff i c i e n t , (cid:217) (cid:18) [ (cid:133) (cid:143) ? ] Wavelength, ª [ (cid:144)(cid:143) ]300 500 700 900 1,1001.51.00.50.0 (a) (b)(c)1.5 ª r (cid:228) r{{ (cid:133)(cid:143) ?5 ª r (cid:228) tut (cid:133)(cid:143) ?5 ª r (cid:228) tr{ (cid:133)(cid:143) ?5 Figure 3.
Absorption coefficient of a blended plasmonic nanofluid for the target spectrum: (a) uniform distribution, (b)solar-spectrum-like distribution, and (c) step-function-like distribution. The effect of water is illustrated by the blue dottedline.To follow the solar spectrum, the absorption peaks should be confined to the major spectral regions of solar radiation (i.e.,from 400 to 700 nm and from 800 to 900 nm). It can be observed that the − (about twice that the uniform case)for the solar-spectrum-like distribution, the designed absorption coefficient reasonably follows the solar spectrum except forwavelengths between 300 and 400 nm.Finally, we also demonstrate the blended plasmonic nanofluid for the step-function-like distribution in Fig. 3c. As in thecase of the solar-spectrum-like distribution, the designed absorption coefficient captures the features of the target spectrum. able 3. Optimal combination of plasmonic nanoparticles considering inhomogeneous broadening due to polydispersednanoparticles in reality. The notation of the design variables follows that in Table 2.Property f i × − in the wavelengths between 300 and 680 nm,mainly due to intrinsic absorption by silver and gold. The resulting RMSE value is 0.209 cm − , which is slightly less thanthat in Fig. 3b. Interestingly, Table 2 reveals that the volume fractions of the , its resonance condition is polarization-dependent due to its geometrical anisotropy.Hence, the polarization-averaged absorption efficiency becomes less significant as compared to the geometrically isotropiccore-shell structure. The present optimization results clearly indicate that the core-shell nanoparticle is superior to the nano-sphere and the nano-rod structures in terms of the tunability of the LSP resonance condition as well as the enhanced absorptionefficiency associated with the LSP.Thus far, we have demonstrated how to achieve broadband absorption by blending nanoparticles made of noble metals(such as Au and Ag), which usually exhibits sharp resonance peaks. In reality, however, there could be many factors that giverise to a broadening effect, such as the electron-boundary scattering effect when the characteristic size of the metal is smallerthan the mean-free-path of electrons or inhomogeneous broadening due to a non-uniform size distribution of the nanoparticles.Because the electron-boundary scattering effect occurs only for the core-shell structure with an extremely thin metallic shell ,for instance, its effect may not be prominent as compared to the inhomogeneous broadening that occurs inevitably due topolydispersed nanoparticles . Here, we examine how inhomogeneous broadening occurring in reality can affect the optimalcombination of plasmonic nanoparticles by applying a randomized distribution of design variables. To do this, a Gaussiandistribution with the mean value of the design variable and a standard deviation of 10% of the mean is assumed. For instance, S p ec t r a l A b s o r p ti on C o e ff i c i e n t , (cid:217) (cid:18) [ (cid:133) (cid:143) ? ] Wavelength, ª [ (cid:144)(cid:143) ]300 500 700 900 1,100 ª r (cid:228) r{y (cid:133)(cid:143) ?5 Figure 4.
Absorption coefficient of a blended plasmonic nanofluid considering inhomogeneous broadening due topolydispersed nanoparticles in reality. The effect of water is also illustrated by the blue dotted line. he core radius of core-shell particle ( r c ) is treated as a random variable following a normal distribution, N ( r c , ( . r c ) ) . In thecalculation, 100 random particles following a Gaussian distribution were calculated using the surrogate neural network modeland their absorption spectra were averaged to determine the broadened absorption spectrum of randomized nanoparticles.Figure 4 shows the absorption coefficient of a blended plasmonic nanofluid considering inhomogeneous broadening dueto polydispersed nanoparticles in reality. It is remarkable that we can achieve an even lower RMSE value (i.e., 0.097 cm − )than the previous blending result (RMSE = .
099 cm − ) with only five types of nanoparticles if inhomogeneous broadeningis taken into account. The optimal combination of plasmonic nanoparticles considering inhomogeneous broadening is listedin Table 3. As noted in Table 3, the absorption peaks of each type of nanoparticle are well distributed, spanning the entirespectral region of interest, and the inhomogeneous broadening causes the absorption coefficient of the blended plasmonicnanofluid to be more uniform. Similarly, it is also expected that the electron-boundary scattering effect eventually makes thedesigned spectrum more uniform, possibly leading to a reduction in the required number of nanoparticle types. It should benoted that the optimal combination in Table 3 is wholly different from that in Table 2, suggesting that the customized GA isvery effective at finding the solution under any given constraint. We have employed a customized genetic algorithm to tailor the spectral absorption coefficient of a blended plasmonic nanofluidmade of nano-sphere, core-shell, and/or nano-rod structures. The chromosome description, fitness function, crossover and mu-tation process in a conventional GA were customized to be suitable for the inverse problem of finding the optimal combinationof plasmonic nanoparticles for the prescribed distribution of the absorption coefficient. In addition, neural network modelsestimating the absorption coefficient of a plasmonic nanoparticle were constructed and coupled with the customized GA to re-duce the computational cost of the optimization process. In this work, three different target absorption coefficients, specificallya uniform distribution, solar-spectrum-like distribution and step-function-like distribution, were considered. The resulting ab-sorption coefficient of a designed plasmonic nanofluid was in good agreement well with the prescribed spectral distributionwithin about 10% to 20% of error when six types of nanoparticles were used. Finally, we also considered inhomogeneousbroadening mainly due to polydispersed nanoparticles during the optimization process. It was found that we can achieve aneven lower RMSE value (i.e., 0.097 cm − ) than in the previous blending result (RMSE = .
099 cm − ) with fewer types ofnanoparticles if inhomogeneous broadening is considered. The design methodology proposed here will facilitate the futuredevelopment of a direct-absorption solar collector using a blended plasmonic nanofluid. Data availability
All data that support the findings of this study are available from the corresponding author upon request.
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This research was supported by the Basic Science Research Program (NRF-2019R1A2C2003605) and by the Creative Mate-rials Discovery Program (NRF-2018M3D1A1058972) through the National Research Foundation of Korea (NRF) funded bythe Ministry of Science and ICT. This research was also supported by the Korea Institute of Energy Technology Evaluation andPlanning (KETEP) and the Ministry of Trade, Industry & Energy (MOTIE) of the Republic of Korea (No. 20172010000850).
Author contributions
Data preparation and customizing genetic algorithm were driven by J.S. under the supervision of B.J.L and J.L. The basicmethod for use of MNPBEM software was assisted by C.Q.
Competing interests