Made available in DSpace on 2014-07-29T15:07:11Z (GMT). No. of bitstreams: 1 Dissertacao Lucyene Quimica.pdf: 715050 bytes, checksum: f0341d52acc4a9b488b447ca69a136f7 (MD5) Previous issue date: 2009-08-25 / A systematic study concerning the crystalline phases in the Bi2O3-rich region of the system Bi2O3-Tb4O7 was made. X-ray powder diffraction method, differential thermal analysis (DTA) and differential scanning calorimetry (DSC) were used in the study of the obtained crystalline phases. Sixteen samples with composition in the range of 4.21 to 33.33 mol% of Tb4O7 were synthesized at 800oC by solid state reaction synthesis. Three distinct crystalline phases were initially identified: a tetragonal phase in the composition range of 4.21 to 5.69 mol% of Tb4O7, a cubic phase for compositions between 6.12 and 29.03 mol% of Tb4O7, and a monoclinic one with 33.33 mol% of Tb4O7. The thermal stability of these compositions was rigorously investigated and the results shown that they are metastable phases in almost all studied range. Under thermal annealing at 500oC, tetragonal phase undergoes decomposition into two phases, monoclinic and rhombohedral ones. The cubic phase, under the same thermal annealing, decomposes partially into a rhombohedral one, except for the interval between 25.0 and 29.1 mol% of Tb4O7. In this case, there was not decomposition under thermal annealing at 500oC during 373 hours, suggesting the cubic phase is stable in this composition range. The monoclinic phase was also tested concerning its thermal stability, but no phase transition was verified. The existence of a possible composition range in the system Bi2O3-Tb4O7 with stable cubic -Bi2O3 type structure is a new and promise result, because of the high ionic conductivity presented by this phase / Um estudo sistemático das fases cristalinas na região mais rica em óxido de bismuto no sistema Bi2O3-Tb4O7 foi realizado. As técnicas de Difração de Raios X pelo método do pó, Análise Térmica Diferencial (DTA) e Calorimetria Diferencial de Varredura (DSC) foram utilizadas no estudo das fases cristalinas obtidas. Dezesseis amostras com composição entre 4,21 e 33,33 mol% de Tb4O7 foram preparadas através de síntese por reação no estado sólido a 800oC. Três fases cristalinas distintas foram inicialmente identificadas: uma tetragonal, presente nas composições entre 4,21 e 5,69 mol% de Tb4O7, uma cúbica, para as composições entre 6,12 e 29,03 mol% de Tb4O7 e uma monoclínica com composição de 33,33 mol% de Tb4O7. Essas composições foram rigorosamente testadas quanto à sua estabilidade térmicas e os resultados indicaram que elas são metaestáveis em quase todo o intervalo investigado. Sob tratamento térmico a 500oC, a fase tetragonal se decompôs em duas fases, uma romboédrica e outra monoclínica. A fase cúbica, sob o mesmo tratamento térmico, se decompôs parcialmente em uma fase romboédrica, exceto para o intervalo compreendido entre 25,0 e 29,1 mol% de Tb4O7. Neste caso não houve decomposição sob tratamento térmico a 500oC durante 373 horas, indicando que a fase cúbica é estável neste intervalo de composição. A fase monoclínica também foi testada quanto à sua estabilidade, não tendo sido verificada nenhuma transição estrutural. A existência de uma possível faixa de composição no sistema Bi2O3-Tb4O7 com estrutura cúbica do tipo -Bi2O3 estável é um resultado inédito e promissor devido ela apresentar elevada condutividade iônica

This thesis investigates the phenomena of phase stability in a fiber-optical MZI (Mach-Zehnder Interferometer). The MZI is a key building block of optical systems for use in experiments with both continuous-wave light and with single photons. By splitting incoming light into two beams and allowing it to interfere with itself, an interference pattern is visible at the output, and this phenomena can be used to code information. This is the operating principle in, for example, QKD (Quantum Key Distribution) experiments. This interference requires coherence that is higher than the length difference between the beams that the incoming light is split into. Particularly the phase of the beams must be equal to achieve constructive interference. If one beam is phase-shifted (with respect to the other) due to the light having traversed a longer path, only partially constructive interference is achieved. If the phase shift also varies with time this leads to a system where experiments can no longer reliably be performed. Sources of these fluctuations are thermal, acoustic or mechanical. Fiber-optical interferometers are particularly sensitive to path length fluctuations of the waveguides as the fiber-optic medium contracts and elongates with temperature, and also has a larger surface area for circulating air to mechanically disturb the waveguides than bulk optics interferometers. In this thesis, a solution to environment-induced phase drift is presented by evaluating implementations of feedback algorithms for automatic control. The algorithms PID (Proportional-, Integral-, Derivative controller) and an ICA (IncrementalControl Algorithm) have been investigated and the performance of these controllers has been compared when used with, and without, optical enclosures. The algorithms are implemented in an FPGA (Field-Programmable Gate Array) and the controller actuates an electro-optical phase modulator that can add a phase shift to one of the light beams in the MZI. This thesis shows that significant improvement in the optical stability can be achieved with active control compared to an interferometer without active phase control. / Det här examensarbetet undersöker fenomenet fasstabilitet i en fiber-optisk MZI (Mach-Zehnder-Interferometer). MZI:n är en viktig byggsten i optiska system som används till experiment med både kontinuerligt emitterande lasrar och med enskilda fotoner. Genom att dela upp inkommande ljus i två strålar och låta det interferera med sig själv så bildas ett interferensmöster vid utgången vilket kan användas för att koda information. Det här är huvudprincipen bakom, till exempel, experiment inom QKD (kvantnyckeldistribution, eng: Quantum Key Distribution). Denna interferens förutsätter en koherens (högre än längdskillnaden mellan strålarna) mellan strålarna som det inkommande ljuset är uppdelat i. Särskilt måste fasen hos de bägge strålarna vara lika för att åstadkomma fullständig konstruktiv intereferens. Om en stråle är fasförskjuten (i förhållande till den andra) på grund av att ljuset har färdats en längre sträcka så uppnås endast delvis konstruktiv interferens. Om fasförskjutningen även varierar med tiden så leder det till ett system där experiment inte längre kan pålitligt utföras. Sådana fluktuationer är orsakade av termiskt, akustiskt samt mekaniskt varierande effekter. Fiberoptiska interferometrar är särskilt känsliga mot förändringar i vågledarnas längd. Detta på grund av att det fiberoptiska mediet dras ihop respektive sträcks ut med temperaturen, samt att fibern har en större ytarea som cirkulerande luft kan påverka mekaniskt jämfört med interferometrar konstruerade av bulkoptik. I det här examensarbetet presenteras en lösning på problemet med miljöinducerad fasskift genom att utvärdera reglertekniska återkopplande algoritmer. Algoritmerna PID (Proportionell-, Integrerande-, Deriverande regulator) samt ICA (Inkrementell Regleralgoritm, eng: Incremental Control Algorithm) har undersökts och deras prestanda har jämförts med samt utan avskärmning. Algoritmerna har implementerats i en FPGA (fältprogrammerbar grindmatris, eng: Field-Programmable Gate Array) och regulatorn styr en elektrooptisk fasmodulator som kan addera en fasförskjutning till en av ljusstrålarna i MZI:n. Resultat visar att passiv avskärmning inte är tillräckligt utan behöver användas tillsammans med aktiv reglering för att uppnå stabilitet över en längre tidsperiod. Detta examensarbete visar på att en signifikant förbättring i den optiska stabiliteten kan uppnås med aktiv reglering jämfört med en interferometer utan aktiv fasreglering.

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<p>We explore the effect of size on the phase stability of nanosystems by comparing calculated trends with the annealing behavior of nanoparticles (NPs) initially in a core-shell configuration. The NPs are characterized using a variety of transmission electron microscopy (TEM) techniques.</p> <p>We first theoretically consider the equilibrium within a Au-Pt NP of a given size. When considering the contribution of surface and interface energies, we note the appearance of a restricted composition range of the phase diagram over which the liquid and solid phases cannot coexist in a core-shell configuration. A critical radius of ",42 nm is identified below which the NP is single-phased for any composition. It is demonstrated that both branches of the miscibility gap of the Au-Pt phase diagram shift towards the Au-rich composition with increasing curvature. The magnitude of the shift is found to be strongly correlated with the coupling of nonlinear terms entering the Gibbs energy. The main contribution to the shift arises from the composition-dependent surface energy, calculated by considering the selective adsorption of Au to the surface, evaluated using the available thermodynamic properties of the Au-Pt system.</p> <p>An array of TEM-related analytical methods were developed or adapted for the characterization of individual NPs. In particular, chemical maps with quantitative information from a NP with a spatial resolution of '" 1.2 nm could be achieved, with their corresponding error analysis. We introduce an algorithm to retrieve the radial elemental composition from the projected chemical map of a NP if a spherical symmetry can be assumed and test it with NPs of known structures. We also present a technique to determine the composition of a NP having one of the elements depleting during analysis, and test it experimentally with 5-20 nm Au-Ag NPs. Typically, for every Ag characteristic X-ray detected, one Ag atom is lost to knock-on damage. We discuss the detection limit of the method as a function of NP size and composition.</p> <p>We follow the structural evolution of a ",20 nm Au(core) Pt(shell) NP during annealing at various temperatures between 300 and 800 °e. At low temperatures, interdiffusion occurs between the core and the shell, while at temperatures abovt: ",600 °e, the configuration evolves towards one composed of Au- and Pt-rich spherical caps, separated by a relatively fiat interface. We could measure a 5-10% shift in the composition of each phase with respect to the bulk phase diagram that we assigned to capillarity effect. The shift agrees qualitatively with the calculated trends. The ratio of the surface to the interface energy is measured directly from a TEM micrograph of a segregated NP and is in close agreement with the calculated ones.</p> <p>This work contributes to the understanding of the phase stability of binary NPs. The prospect of extending these studies to NPs of other bimetallic systems while probing their properties seems promising, especially in view of their catalytic, magnetic and optical potential.</p> / Doctor of Philosophy (PhD)

Ferroelectric and ferromagnetic materials have been extensively studied for potential applications in sensors, actuators and transducers. Highly electrostrictive (1-x)Pb(Mg_1/3Nb_2/3)-xPbTiO₃ (PMN-xPT) and highly magnetostrictive Fe-xat.%Ga are two such novel materials. Both materials systems have chemical disorders and structural inhomogeneity on a microscale, giving rise to an interesting diversity of crystal structures and novel macroscopic physical properties, which are dependent on thermal and electrical histories of the crystals. In this thesis, I have to investigated phase transformations in these two systems under thermal and field (electric/magnetic) histories, using x-ray and neutron scattering techniques. In PMN-xPT crystals, x-ray and neutron diffractions were performed along the different crystallographic orientations and for different thermal and electrical histories. Various intermediate monoclinic (M) phases that structurally “bridge” the rhombohedral (R) and tetragonal (T) ones across a morphtropic phase boundary (MPB) have been observed. Systematic investigations of (001) and (110) electric (E) field-temperature phase diagrams of PMN-xPT crystals have demonstrated that the phase stability of PMN-xPT crystals is quite fragile: depending not only on modest changes in E (≤ 0.5kV/cm), but also on the direction along which E is applied. Structurally bridging monoclinic Mc or orthorhombic (O) phases were found to be associated with the T phase, whereas the monoclinic Ma or Mb phases bridged the Cubic (C) and R ones. In addition, neutron inelastic scattering was performed on PMN-0.32PT to study the dynamic origin of the MPB. Data were obtained between 100 and 600 K under various E applied along the cubic [001] direction. The lowest frequency zone-center, transverse optic phonon was strongly damped and softened over a wide temperature range, but started to recover on cooling into the T phase at the Curie temperature (TC). Comparisons of my findings with prior ones for PMN and PMN-0.60PT suggest that the temperature dependence and energy scales of the soft mode dynamics in PMN-xPT are independent of PT concentration below the MPB, and that the MPB may be defined in composition space x when TC matches the temperature at which the soft mode frequency begins to recover. High-resolution x-ray studies then showed that the C–T phase boundary shifted to higher temperatures under E by an expected amount within the MPB region: suggesting an unusual instability within the apparently cubic phase at the MPB. In Fe-xat.%Ga alloys, the addition of Ga atoms into the b.c.c. α-Fe phase also results in diversity of crystal structures and structural inhomogeneity, which are likely the source of its unusual magneto-elastic properties. I have carefully investigated decomposition of Fe-xat.%Ga alloys subjected to different thermal treatments by x-ray and neutron diffraction for 12 < x < 25. Quenching was found to suppress the formation of a DO₃ structure in favor of a high-temperature disordered bcc (A2) one. By contrast, annealing produced a two-phase mixture of A2 + DO₃ for 14 < x < 20 and a fully DO₃ phase for x = 25. A splitting of the (2 0 0) and (0 0 2) Bragg peaks observed along the respective transverse directions indicated that Fe-xat.%Ga –crystals' are composed of several crystal grain orientations (or texture structures), which are slightly tilted with respect to each other. In order to investigate the local structural distortions and heterogeneities, neutron diffuse scattering was performed on Fe-x%Ga alloys for different thermal conditions. Diffuse scattering around a (100) superlattice reflection was found for 14 < x < 22 in the furnace-cooled condition, indicative of short-range ordered DO₃ nanoprecipitates in an A2 matrix. This diffuse intensity had an asymmetric radial contour and an off-centering. Analysis (x=19) revealed two broad peaks with c/a–1.2: indicating that the DO₃-like nanoprecipitates are not cubic, but rather of lower symmetry with a large elastic strain. The strongest diffuse scattering was observed for x=19, which correspondingly had maximum magnetostriction: indicating a structural origin for enhanced magnetostriction. / Ph. D.

O cálculo do equilíbrio de fases é um problema de grande importância em processos da engenharia, como, por exemplo, na separação por destilação, em processos de extração e simulação da recuperação terciária de petróleo, entre outros. Mas para resolvê-lo é aconselhável que se estude a priori a estabilidade termodinâmica do sistema, a qual consiste em determinar se uma dada mistura se apresenta em uma ou mais fases. Tal problema pode ser abordado como um problema de otimização, conhecido como a minimização da função distância do plano tangente à energia livre de Gibbs molar, onde modelos termodinâmicos, de natureza não convexa e não linear, são utilizados para descrevê-lo. Esse fato tem motivado um grande interesse em técnicas de otimização robustas e eficientes para a resolução de problemas relacionados com a termodinâmica do equilíbrio de fases. Como tem sido ressaltado na literatura, para proporcionar uma completa predição do equilíbrio de fases, faz-se necessário não apenas a determinação do minimizador global da função objetivo do teste de estabilidade, mas também a obtenção de todos os seus pontos estacionários. Assim, o desenvolvimento de metodologias para essa tarefa desafiadora tem se tornado uma nova área de pesquisa da otimização global aplicada à termodinâmica do equilíbrio, com interesses comuns na engenharia química e na engenharia do petróleo. O foco do presente trabalho é uma nova metodologia para resolver o problema do teste de estabilidade. Para isso, usa-se o chamado método do conjunto gerador para realizar buscas do tipo local em uma rede de pontos previamente gerada por buscas globais efetuadas com uma metaheurística populacional, no caso o método do enxame de partículas.Para se obter mais de um ponto estacionário, minimizam-se funções de mérito polarizadas, cujos pólos são os pontos previamente encontrados. A metodologia proposta foi testada na análise de quatorze misturas polares previamente consideradas na literatura. Os resultados mostraram que o método proposto é robusto e eficiente a ponto de encontrar, além do minimizador global, todos os pontos estacionários apontados previamente na literatura, sendo também capaz de detectar, em duas misturas ternárias estudadas, pontos estacionários não obtidos pelo chamado método de análise intervalar, uma técnica confiável e muito difundida na literatura. A análise do teste de estabilidade pela simples utilização do método do enxame de partículas associado à técnica de polarização mencionada acima, para a obtenção de mais de um ponto estacionário (sem a busca local feita pelo método do conjunto gerador em uma dada rede de pontos), constitui outra metodologia para a resolução do problema de interesse. Essa utilização é uma novidade secundária deste trabalho. Tal metodologia simplificada exibiu também uma grande robustez, sendo capaz de encontrar todos os pontos estacionários pesquisados. No entanto, quando comparada com a abordagem mais geral proposta aqui, observou-se que tal simplificação pode, em alguns casos onde a função de mérito apresenta uma geometria mais complexa, consumir um tempo de máquina relativamente grande, dessa forma é menos eficiente. / The phase equilibrium calculation is an important problem in engineering processes, for example, in the separation by distillation, in extraction processes, and simulation of the tertiary petroleum recovery, among others applications. But, before this calculation is advisable a phase stability analysis, which will determine if a given mixture presents one or more phases. Such stability analysis can be approached as an optimization problem, called the minimization of the tangent plane distance function to molar Gibbs free energy, where thermodynamic models, with a non-convex and also non-linear nature, are used to describe this problem. Such fact has been motivating a large interest in robust and efficient techniques for optimization of problems related with the phase equilibrium thermodynamics. As has been highlighted in the literature, in order to provide a complete prediction of the phase equilibrium, the determination of the global minimizer of the objective function related with the stability test is not enough. Is also necessary the obtaining of all its stationary points. Thus, the development of methodologies for this challenging task became a new research area of global optimization applied to the phase equilibrium thermodynamics, with common interests in chemical and petroleum engineering. The focus of the present work is a new methodology to solve the stability test problem. For such, here it is used the called Generating Set Search method, which accomplishes local searches in points of a net previously generated by an evolutionary methaheuristic, the Particle Swarm algorithm. To obtain more than one stationary point, polarized merit functions are minimized, where the poles of such merit functions are the points previously found. The methodology proposal was tested in the analysis of fourteen polar mixtures, previously considered in the literature. The results showed that the proposed method is robust and efficient, being capable to find all stationary points previously indicated in the literature, besides the global minimizers. For two ternary mixtures, it was noticed that this method detected some stationary points not obtained before by the interval analysis method, a reliable technique very diffused in the literature. The stability analysis using only the Particle Swarm algorithm together with the polarization technique (without the local search done by the Generating Set Search method) constitutes other methodology for the resolution of the problem considered here. That use is a secondary novelty of this work. This simplified methodology exhibited also a significant robustness, being capable to find all the stationary points. However, when compared with the more general approach proposed here, in some cases, where the merit function presents a more complex geometry, it was observed that such simplification can consume a relatively large computational time. Therefore, the simplified methodology is less efficient.

The perovskite family of materials is extremely large and provides a template for designing materials for different purposes. Among them, hybrid organic-inorganic perovskites (HOIPs) are very interesting and have been recently identified as possible next generation light harvesting materials because they combine low manufacturing cost and relatively high power conversion efficiencies (PCEs). In addition, some other applications like light emitting devices are also highly studied. This thesis starts with an introduction to the solar cell technologies that could use HOIPs. In Chapter 2, previously published results on the structural, electronic, optical and mechanical properties of HOIPs are reviewed in order to understand the background and latest developments in this field. Chapter 3 discusses the computational and experimental methods used in the following chapters. Then Chapter 4 describes the discovery of several hybrid double perovskites, with the formula (MA)$_2$M$^I$M$^{III}$X$_6$ (MA = methylammonium, CH$_3$NH$_3$, M$^I$ = K, Ag and Tl, M$^{III}$ = Bi, Y and Gd, X = Cl and Br). Chapter 5 presents studies on the variable presure and temperature response of formamidinium lead halides FAPbBr$_3$ (FA = formamidinium, CH(NH$_2$)$_2$) as well as the mechanical properties of FAPbBr$_3$ and FAPbI$_3$, followed by a computational study connecting the mechanical properties of halide perovskites ABX$_3$ (A = K, Rb, Cs, Fr and MA, X = Cl, Br and I) to their electronic transport properties. Chapter 6 describes a study on the phase stability, transformation and electronic properties of low-dimensional hybrid perovskites containing the guanidinium cation Gua$_x$PbI$_{x+2}$ (x = 1, 2 and 3, Gua = guanidinium, C(NH$_2$)$_3$). The conclusions and possible future work are summarized in Chapter 7. These results provide theoreticians and experimentalists with insight into the design and synthesis of novel, highly efficient, stable and environmentally friendly materials for solar cell applications as well as for other purposes in the future.

O cálculo do equilíbrio de fases é um problema de grande importância em processos da engenharia, como, por exemplo, na separação por destilação, em processos de extração e simulação da recuperação terciária de petróleo, entre outros. Mas para resolvê-lo é aconselhável que se estude a priori a estabilidade termodinâmica do sistema, a qual consiste em determinar se uma dada mistura se apresenta em uma ou mais fases. Tal problema pode ser abordado como um problema de otimização, conhecido como a minimização da função distância do plano tangente à energia livre de Gibbs molar, onde modelos termodinâmicos, de natureza não convexa e não linear, são utilizados para descrevê-lo. Esse fato tem motivado um grande interesse em técnicas de otimização robustas e eficientes para a resolução de problemas relacionados com a termodinâmica do equilíbrio de fases. Como tem sido ressaltado na literatura, para proporcionar uma completa predição do equilíbrio de fases, faz-se necessário não apenas a determinação do minimizador global da função objetivo do teste de estabilidade, mas também a obtenção de todos os seus pontos estacionários. Assim, o desenvolvimento de metodologias para essa tarefa desafiadora tem se tornado uma nova área de pesquisa da otimização global aplicada à termodinâmica do equilíbrio, com interesses comuns na engenharia química e na engenharia do petróleo. O foco do presente trabalho é uma nova metodologia para resolver o problema do teste de estabilidade. Para isso, usa-se o chamado método do conjunto gerador para realizar buscas do tipo local em uma rede de pontos previamente gerada por buscas globais efetuadas com uma metaheurística populacional, no caso o método do enxame de partículas.Para se obter mais de um ponto estacionário, minimizam-se funções de mérito polarizadas, cujos pólos são os pontos previamente encontrados. A metodologia proposta foi testada na análise de quatorze misturas polares previamente consideradas na literatura. Os resultados mostraram que o método proposto é robusto e eficiente a ponto de encontrar, além do minimizador global, todos os pontos estacionários apontados previamente na literatura, sendo também capaz de detectar, em duas misturas ternárias estudadas, pontos estacionários não obtidos pelo chamado método de análise intervalar, uma técnica confiável e muito difundida na literatura. A análise do teste de estabilidade pela simples utilização do método do enxame de partículas associado à técnica de polarização mencionada acima, para a obtenção de mais de um ponto estacionário (sem a busca local feita pelo método do conjunto gerador em uma dada rede de pontos), constitui outra metodologia para a resolução do problema de interesse. Essa utilização é uma novidade secundária deste trabalho. Tal metodologia simplificada exibiu também uma grande robustez, sendo capaz de encontrar todos os pontos estacionários pesquisados. No entanto, quando comparada com a abordagem mais geral proposta aqui, observou-se que tal simplificação pode, em alguns casos onde a função de mérito apresenta uma geometria mais complexa, consumir um tempo de máquina relativamente grande, dessa forma é menos eficiente. / The phase equilibrium calculation is an important problem in engineering processes, for example, in the separation by distillation, in extraction processes, and simulation of the tertiary petroleum recovery, among others applications. But, before this calculation is advisable a phase stability analysis, which will determine if a given mixture presents one or more phases. Such stability analysis can be approached as an optimization problem, called the minimization of the tangent plane distance function to molar Gibbs free energy, where thermodynamic models, with a non-convex and also non-linear nature, are used to describe this problem. Such fact has been motivating a large interest in robust and efficient techniques for optimization of problems related with the phase equilibrium thermodynamics. As has been highlighted in the literature, in order to provide a complete prediction of the phase equilibrium, the determination of the global minimizer of the objective function related with the stability test is not enough. Is also necessary the obtaining of all its stationary points. Thus, the development of methodologies for this challenging task became a new research area of global optimization applied to the phase equilibrium thermodynamics, with common interests in chemical and petroleum engineering. The focus of the present work is a new methodology to solve the stability test problem. For such, here it is used the called Generating Set Search method, which accomplishes local searches in points of a net previously generated by an evolutionary methaheuristic, the Particle Swarm algorithm. To obtain more than one stationary point, polarized merit functions are minimized, where the poles of such merit functions are the points previously found. The methodology proposal was tested in the analysis of fourteen polar mixtures, previously considered in the literature. The results showed that the proposed method is robust and efficient, being capable to find all stationary points previously indicated in the literature, besides the global minimizers. For two ternary mixtures, it was noticed that this method detected some stationary points not obtained before by the interval analysis method, a reliable technique very diffused in the literature. The stability analysis using only the Particle Swarm algorithm together with the polarization technique (without the local search done by the Generating Set Search method) constitutes other methodology for the resolution of the problem considered here. That use is a secondary novelty of this work. This simplified methodology exhibited also a significant robustness, being capable to find all the stationary points. However, when compared with the more general approach proposed here, in some cases, where the merit function presents a more complex geometry, it was observed that such simplification can consume a relatively large computational time. Therefore, the simplified methodology is less efficient.