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A thin-film of mechanochemically synthesized nanoparticles : an experimental and theoretical exploration… Bednar, Victor Bradley 2017

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A Thin-Film of MechanochemicallySynthesized Nanoparticles:An experimental and theoretical exploration of anabsorber compound for photovoltaic devicesbyVictor Bradley BednarBSc., Eastern Oregon University, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2017c© Victor Bradley Bednar, 2017AbstractNearly 200 years after the discovery of the photovoltaic effect, harvesting en-ergy from the sun is finally becoming a price competitive marketing option forpower generation. Government and private investments, motivated by a socialawareness of environmental issues cause by prominent power generation meth-ods, have helped create this opportunity to advance earth conscientious, greenenergy solutions. As inorganic nanoparticles in solar cell layers are one of theforefront areas of interest for solar cell research, mechanochemical material syn-thesis has been used for a scalable production of Fe2GeS4 nanoparticles carriedout via ball milling. The compound is composed of earth-abundant materials, andball milling allows for a solution free process, which minimizes chemical wastefrom material synthesis. The viability of this promising compound has been previ-ously mentioned and herein confirmed. X-Ray Powder Diffraction (XRD) showeda successful synthesis, and optical characterization confirmed favorable absorp-tion properties for solar cell implementation. New methods were implementedin doping the nanoparticles, which lead to an observable photovoltaic responsefrom a simple prototype architecture implementing the Fe2GeS4 nanoparticles.The thin film deposition of the nanoparticles used for prototype implementationshould allow for cost effective and scalable manufacturing.Since ball milling is also cost effective and scalable, an empirical model im-plementing probabilistic logic is developed and shown as capable to fit experi-mental data via measurable parameters. The eventual optimization possibilitiesfor minimizing manufacturing costs, as well as enhancement of general scientificiiunderstanding for an underrepresented branch of theory, mechanochemical solidstate reactions, motivated this work. Modeling of Fe2GeS4 production, as a solidstate chemical reaction, demonstrates a proof of principle application. Potentialapplications are not limited to mechanochemical synthesis. Extensions to otherreaction types are possible as the model utilizes chemical kinetics theory in a gen-eralized fashion. The demonstration focuses on a sigmoid trend, as observed inFe2GeS4 synthesis, though other profiles are attainable.iiiLay SummaryRising concerns about sustaining the earth and meeting the energy demandsof society both have led to great progress in using the sun to generate electricity.In this work, earth abundant resources are used to build a prototype solar cell forharvesting the sun’s energy to generate electricity. Though these materials havepreviously been investigated, new options have been realized in this research byinvestigating methods not used previously with these materials.To complement this production, with an opportunity to one day improve themanufacturing costs of such a solar cell, a mathematical model was developed.The model introduces a new approach to enhance the understanding of the me-chanical means used to create novel compounds. Both the mathematical modeland the techniques used in making a prototype show usable results with promisefor future implementations.ivPrefaceAll text within this document, unless quoted, have been written by the author.In Chapter 3, the model was developed in full by the author. The code used torun simulations, seen in Appendix A.1 was written by the author, with the ex-ception of the “RK4 Loop” which has been repurposed and edited by the authorwith the permission of the original author, Anthony A. Tovar. The code was runon a computer built by the author. The experimental and analytical works pre-sented in Chapters 4 and 5 were done by the author. Development of the researchgoals and experiments contained in this thesis were thought up by the author andimplemented under the guidance and approval of Dr. Peyman Servati.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Photovoltaics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Architectural Variations . . . . . . . . . . . . . . . . . . . 41.2.2 Economic Outlook . . . . . . . . . . . . . . . . . . . . . 8vi1.3 Absorber Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.1 Mechanochemical Synthesis . . . . . . . . . . . . . . . . 91.4 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4.1 Scanning Electron Microscopy (SEM) . . . . . . . . . . . 141.4.2 X-Ray Powder Diffraction (XRD) . . . . . . . . . . . . . . 161.5 Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.5.1 Thin Film Growth . . . . . . . . . . . . . . . . . . . . . . 191.5.2 Thin Film Deposition . . . . . . . . . . . . . . . . . . . . 201.5.3 Film Characterization . . . . . . . . . . . . . . . . . . . . 222 Motivation for Solar Cells . . . . . . . . . . . . . . . . . . . . . . . . 252.1 Environmental Solutions . . . . . . . . . . . . . . . . . . . . . . 252.2 Economic Opportunity . . . . . . . . . . . . . . . . . . . . . . . 272.3 Nanotechnology Benefits . . . . . . . . . . . . . . . . . . . . . . 292.4 Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1 Application of the Model . . . . . . . . . . . . . . . . . . . . . . 333.2 Modelling the Chemistry of Ball Milling . . . . . . . . . . . . . . 343.2.1 Interaction Model . . . . . . . . . . . . . . . . . . . . . . 353.2.2 Chemical Kinetics Model . . . . . . . . . . . . . . . . . . 373.2.3 Effective Reaction Area Terms in the Chemical KineticsModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.4 Energy Dependent Terms in the Chemical Kinetics Model 433.3 Model Understanding Examples . . . . . . . . . . . . . . . . . . 463.3.1 Interaction Model: Graph 3.1a . . . . . . . . . . . . . . . 493.3.2 Chemical Kinetics Model: Graph 3.1b . . . . . . . . . . . 493.3.3 Effective Reaction Area Terms in the Chemical KineticsModel: Graph 3.1c . . . . . . . . . . . . . . . . . . . . . 503.3.4 Energy Dependent Terms in the Chemical Kinetics Model:Graph 3.1d . . . . . . . . . . . . . . . . . . . . . . . . . 51vii3.4 Model Application Example . . . . . . . . . . . . . . . . . . . . 523.4.1 Formation Term in the Application Model . . . . . . . . . 553.4.2 Agglomeration Term in the Application Model . . . . . . 563.4.3 Size Dependent Breaking Term in the Application Model . 573.4.4 Non-linear Energy Term in the Application Model . . . . . 574 Absorber Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1 Ball Milling Synthesis . . . . . . . . . . . . . . . . . . . . . . . . 594.1.1 Mortar and Pestle Premixture . . . . . . . . . . . . . . . . 594.1.2 Ball Milling . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Absorber Characterization . . . . . . . . . . . . . . . . . . . . . 624.2.1 Scanning Electron Microscopy (SEM) Characterization . . 624.2.2 X-Ray Powder Diffraction (XRD) Characterization . . . . 665 Thin Film Application . . . . . . . . . . . . . . . . . . . . . . . . . . 745.1 Spin Coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Drop Casting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3 Film Characterization . . . . . . . . . . . . . . . . . . . . . . . . 815.4 Prototype Characterization . . . . . . . . . . . . . . . . . . . . . 826 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.1 Expansion on Previous Works . . . . . . . . . . . . . . . . . . . 896.2 Possible Limitations . . . . . . . . . . . . . . . . . . . . . . . . . 906.3 Future Implications . . . . . . . . . . . . . . . . . . . . . . . . . 91Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93A Supporting Materials . . . . . . . . . . . . . . . . . . . . . . . . . . 100A.1 Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100viiiList of TablesTable 3.1 Compound Molar Masses . . . . . . . . . . . . . . . . . . . . 46Table 3.2 Reaction Critical Size . . . . . . . . . . . . . . . . . . . . . . 50Table 3.3 Size Factor Slope . . . . . . . . . . . . . . . . . . . . . . . . . 51Table 3.4 Reaction Arctangent Width . . . . . . . . . . . . . . . . . . . 51Table 3.5 Breaking Fraction . . . . . . . . . . . . . . . . . . . . . . . . 52Table 3.6 Breaking Fraction . . . . . . . . . . . . . . . . . . . . . . . . 52ixList of FiguresFigure 1.1 Solar Cell efficiencies as listed by the National Renewable En-ergy Laboratory [12]. . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.2 Planetary mill rotations in the directions of the color coordi-nated arrows and axes of rotations. . . . . . . . . . . . . . . . 10Figure 1.3 Planetary mill mechanics: the black triple arrow path repre-sents the platform rotation, the red arrow path represents themilling jar rotation, and the white arrow path represents thesidewall launch of milling balls and material in the milling jar. 11Figure 1.4 An example schematic with components found in a Zeiss Sigmafield emission scanning electron microscope as used in this re-search. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 1.5 Incident rays from the source (blue arrow) scatter from thecrystal and head off toward the detector(violet arrows). . . . . 16Figure 1.6 Bragg’s Law states the condition (2d sinθ = nλ ) for which thedetector will receive the strongest signal from the scattered rays 17Figure 1.7 Typical current (I) versus potential (V) graph for solar cells.The curve in black represents the dark current measurementand the curve in red is the illuminated measurement. . . . . . 24Figure 2.1 A “global distribution of burden of disease attributable to 20leading selected risk factors” in units of disability-adjustedlife years (DALYs) by the World Health Organization [19]. . . 26xFigure 2.2 “2015 estimated finite and renewable planetary energy reserves(Terawatt-years). Total recoverable reserves are shown for thefinite resources. Yearly potential is shown for the renewables.”A visual representation by sphere volume size from Perez andPerez [22]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 4.1 Mortar and pestle premixing . . . . . . . . . . . . . . . . . . 60Figure 4.2 The final product . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 4.3 Visual progression of the reaction with 4.3h and 4.3i . . . . . 65Figure 5.1 SEM images of drop casts onto carbon conductive tabs withoutseparation 5.1a and with separation after settling for one hour5.1b. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 5.2 SEM images of spin coats onto glass after settling separation5.2a and 5.2b and without separation 5.2c and 5.2d. . . . . . . 77Figure 5.3 SEM images of spin coats onto Indium Tin Oxide (ITO) coatedglass after filtration 5.2a and 5.2b and heating 5.2c and 5.2d. . 78Figure 5.4 An agglomeration seen on a film post filtration at 50X. . . . . 79Figure 5.5 SEM images of drop casts onto glass without heat 5.5a andwith heat 5.5b. . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 5.6 Prototype Architecture . . . . . . . . . . . . . . . . . . . . . 84xiList of GraphsGraph 3.1 Example graphs for understanding the model. . . . . . . . . . . 48Graph 3.2 Application of the Model–the effects of formation, agglomer-ation, size dependent breaking, and non-linear energy depen-dence are all quite apparent in the changing effective reactionarea terms represented by varying radii of the elements andcompounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Graph 4.1 Identification of reactants via XRD analysis. Reference patternsfor iron (red), germanium (blue), and sulfur (yellow) have beensuperimposed onto the premix’s XRD pattern. . . . . . . . . . . 67Graph 4.2 Peaks can be windowed as in 4.2a to help visualize them or theycan be marked as done in 4.2b. . . . . . . . . . . . . . . . . . . 69Graph 4.3 Identification of the final product via XRD analysis. Referencepatterns for both a theoretical and measured XRD pattern forFe2GeS4 are shown in blue and green. . . . . . . . . . . . . . . 70Graph 4.4 Unknown peaks can be windowed for easier identification. Thishas been done for the identification of unreacted germanium(4.4a) and intermediary product iron sulfide (4.4b). . . . . . . . 72Graph 4.5 Identification of unreacted sulfur (red) in the final product viaXRD analysis. Sulfur is present in the XRD pattern from thescraped sample (red) but not the loose sample (black). . . . . . 73xiiGraph 5.1 A typical film transmission spectrum compared with a singlecrystal transmission spectrum by Platt [28]. . . . . . . . . . . . 82Graph 5.2 A typical prototype performance. RSH ∼ 1−2×104Ω ·cm RS∼50Ω · cm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Graph 5.3 A typical prototype performance using a zinc oxide windowlayer. RSH ∼ 5−10×103Ω · cm . . . . . . . . . . . . . . . . . 85Graph 5.4 Moderate phosphorous doping prototype performance RS∼ 7×103Ω · cm2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Graph 5.5 Light phosphorus doping prototype performance RSH ∼ 1−2×105Ω · cm RS ∼ 3×102Ω · cm2 . . . . . . . . . . . . . . . . . . 87Graph 5.6 Light phosphorus doping with organic stabilizers used in spincoating prototype performanceRSH ∼ 1− 3× 107Ω · cm Rs ∼1×104Ω · cm2 . . . . . . . . . . . . . . . . . . . . . . . . . . 88xiiiGlossarySEM Scanning Electron MicroscopyXRD X-Ray Powder DiffractionEDX Energy-dispersive X-rayITO Indium Tin OxideSMU Source Measure UnitxivAcknowledgmentsI am very grateful to Rowshan Rahmanian, Zenan Jiang, and Yan Wang forkindly answering questions, helping with equipment training, and for their generalgood company around the lab and office along with all other members of theFlexible Electronics and Energy Lab group. Pursuit of this research would nothave been possible without the use of equipment within the Advanced FibrousMaterials, Advanced Materials and Process Engineering Laboratory. Anita Lam,your passion for X-ray diffraction made it easy to learn. Dr. Frank Ko, yourliterary insights were greatly appreciated. Thank you, Dr. Peyman Servati forproviding this opportunity, an inquisitive environment, and financial support formy endeavors.The University of British Columbia Dance Club and others in the ballroomdance community, what a great assortment of people who listen well to eachother’s work with intense sincerity (a wonderful, happy bunch of dancing sci-entist, students, and community members).To my family, I appreciate your endless understanding with everything I chooseto pursue in life.xvDedicationTo I AM , beside me always −−my eternal gratitude .xviChapter 1IntroductionSeemeth it a small thing unto you to have eaten up the good pasture,but ye must tread down with your feet the residue of your pastures?and to have drunk of the deep waters, but ye must foul the residuewith your feet? — Ezekiel 34:18 [1]Since early times, we have known that our existence did not come without animpact to the very land that we rely on for our survival. Currently, our energydemands immensely impact our environment. In order to bring about a cessationto our energy related issues, better green energy solutions are necessary. Can weresolve these issues with solar energy harvesting? Hopefully, the explanations inthis chapter will make the possibilities clear. With an understanding of the avail-able solutions and methods used in these works, in Chapter 2, we move forward tomore specifics on why we pursue this research path and what has been done pre-viously. The focused goal of this research is to test the viability of mechanochem-ically synthesized nanoparticles in photovoltaic devices–specifically Fe2GeS4.1.1 LightOne can hardly discuss solar energy harvest without a light grasp on photons.Often things of intangible form leave those studying the phenomena of nature inawe. Light is of this category, and we owe our sight to such an existence. However,1understanding its mysteries has been an undertaking for centuries which continuesto this day.Christiaan Huygens was the first to write on the wave properties of light. SirIsaac Newton, in juxtaposition to this view, advanced optics and wrote on the cor-puscular nature of light. Thomas Young confirmed light as a transverse wave;however, it can cross the void transmitting energy in the absence of any medium.This becomes the first important piece of the puzzling experience for one dedi-cated to sunlight energy harvesting via solar cells. Wave properties of light can beunderstood using mathematical theories brought together by James Clerk Maxwelland made more usable by Oliver Heaviside.The quantity of energy can be understood through the corpuscular property oflight. These theories were advanced by Max Planck and Albert Einstein. Thisquantization becomes the second important piece of information for light energyharvesting. For a more in-depth look into the history of light, and those whohave made discoveries with a lasting impact, various physics history books forundergraduates gather such material. For example, Kuehn [2] and Shamos [3] dojust that in their books.When a photon interacts with matter, there are various options. For solar en-ergy harvesting applications, we will focus on the interactions of photons withmaterials and how it effects electrons. For starters, the photovoltaic effect.1.2 PhotovoltaicsThe photovoltaic effect is the process through which a potential is observed,across the terminals of a cell, as a result of a photon’s interacting with the material.Taking advantage of different designs, which focus on the various aspects of asolar cell, should at a minimum result in a photopotential observation. So whatare the various phenomena which allow for such an observation?First, a photon has to be absorbed by a material. Though there are other waysto draw signal from light, for example, read on some ways of how to detect anevanescent wave [4, 5]; however, we will focus on the main processes in a pho-2tovoltaic. Ignoring how or why (unless one wants to study optics and quantumfield theory), think that the energy of a photon is completely transferred to anelectron during absorption, and the photon is then no more. Let us start with asimplified idea such as an atom with one proton and one electron. In the caseof this individual atom, the energy can increase the electrical potential betweenthe electron and the nucleus of the atom. Videlicet, the spatial distance betweenthe two electrically attracted particles increases. If the photon carries enough en-ergy, the electron can become free of the nucleus (ionization). This is a simplifiedview of the photoelectric effect described by Albert Einstein–electrons freed frommaterial by the properly corresponding light energy packets (photons). In bulkmaterial, this energy has some other properties as well. For example, the lightmight penetrate some distance into the material before it is absorbed, and then theelectron might not be able to have enough energy to propagate that distance andleave the material. However, the main idea here is that material needs to absorblight and “free” electrons from their bound states.How we use these electrons is another important aspect for observing a pho-topotential. For example, in semiconducting materials, electrons spatially re-stricted by the material bonds are not free to move around enough to play a majorrole in electrical conduction. The “free” electrons do have enough energy. Thus,this energy difference between bound (valence band electrons) and “free” (con-duction band electrons) electrons forms a gap between band energies. If a photonhas enough energy to free a bound electron from a state in the valence band to astate in the conduction band, a physical distance between the location where theelectron was bound and its new “free” location is observed. This pair (a “free”electron and its bound location) are referred to as an electron-hole pair. the elec-trical potential between the pair will match that of the band gap energy betweenthe valence and conduction bands. That is to say, even if a photon imparts moreenergy on an electron, that energy will dissipate into the system (heat, photonemissions, vibrations, etc.). Thus, the observed photopotential depends on theband gap of materials involved in absorbing photons.3Nothing says these spatially separated charged structures maintain their po-tential indefinitely. Quite the opposite is often the observed case. The electronssimply recombine with their hole or the hole from another “free” electron. Thus,the terms carrier generation, carrier recombination, and carrier lifetime are usedfor describing the aforementioned phenomena. Carrier generation is a measure ofelectron-hole pair production. Carrier recombination is a measure of the pair re-combining such that a conduction band electron returns to its bound valence bandstate. Carrier lifetime is a measurement regarding the time between carrier gen-eration and carrier recombination for an electron-hole pair. For over one hundredyears, we have endeavored to optimize cells by designing architectures that putthe photovoltaic effect to use in the harvesting of solar radiation energy.Becquerel [6] writes the first account of the photovoltaic effect in 1839. Hisexperiments used a setup that would now fall in the photoelectrochemical cellcategory. Adams and Day [7] later (1877) demonstrate a solid-state setup. Finally,the current industry leader makes a partial debut, in 1946, when Ohl [8] patentsa light sensitive device made with silicon. This idea becomes a start for the firstsuccess story in the solar cell industry when Chapin et al. [9] make a six percentefficient silicon solar cell in 1954. Today, research continues to invest efforts tocontinue this story until solar cells are realized for their unequivocal superiorityin energy production.1.2.1 Architectural VariationsThe United States Department of Energy [10] categorizes photovoltaics basedon various elements and their laboratory, the National Renewable Energy Labo-ratory [11], places those into the following categories: high-efficiency crystalline,polycrystalline thin films, and emerging technologies. The current research effi-ciencies of these various types can be viewed in Figure 1.1For starters, let us look at the advantages and disadvantages of the variousarchitectures in the categories outlined by the National Renewable Energy Labo-ratory [11].4Figure 1.1: Solar Cell efficiencies as listed by the National Renewable En-ergy Laboratory [12].High-Efficiency Crystalline Solar CellsFrom Figure 1.1, it can be seen that crystalline cells lead in efficiency–especiallymultijunction cells. So, what are some of the pitfalls? Cost is an obvious bound-ary. That is to say, though they lead in industry due to years of research andmanufacturing investments, we near the limits of what can be accomplished bysuch means without yet meeting our competitive market goals to overtake otherenergy sources.Why are crystalline solar cells the most efficient? One reason has to do withthe discontinuities in materials that are not crystalline. As mentioned in Sec-tion 1.2, carrier generation, recombination, and lifetime are all important in pho-tovoltaics. Discontinuities of a material’s (not perfectly crystalline) structure leadto sinks and sources for electrons within the material. By discontinuity, I mean toexpress any disruption to a perfect lattice structure. These arise as distortions inthe lattice, to name a few: due to physical strains and stresses; chemical impuritiescausing lattice distortions; and breaks in bonds from fractures, cleavages, and ma-5terial growth edges. Therefore, carrier lifetime decreases while recombination andpossibly generation via non-photon-induced means increases. All of these lead toa decrease in utilizing photons to generated power for external connections on asolar cell. In order to successfully extract a current from a photovoltaic device,one electrode needs to supply electrons for a current and the other needs to acceptelectrons (or supply holes if one likes to think of holes as carriers). Thus, pair pro-duction with long lifetimes is desirable. Since discontinuities sequester our pairsand inhibit our photo production, keeping the material as pure and crystalline aspossible is desirable.This leads to a few of the downfalls of crystalline solar cells. Since a con-tinuous crystal structure is desirable, material boundaries, an absorber/electrodeboundary for example, lead to the same issues as discontinuities within a singlematerial. Thus, recombination, and such, at material surfaces becomes a largeissue that has been a target in improving crystalline solar cell performance. An-other issue is the cost of simply producing highly crystalline materials. Whethergrowing monocrystalline Silicon via the Czochralski process or layering a mul-tijunction cell via epitaxial growth, most methods for obtaining these extremelypure and highly crystalline materials are expensive.Finally, as these are crystalline, brittleness becomes a mechanical pitfall forthese types of solar cells. Much has been done to improve the ruggedness of solarcells. This and flexibility are partial reasons for the development of other archi-tectures along with cost and market variety to shift the monopolistic competitiontoward that of a competitive market–so as to drive down the cost of energy.Polycrystalline Thin Film Solar CellsAs previously mentioned, typical crystalline solar cells are brittle, and thislack of ruggedness and flexibility can hinder popularity for the same reason as towhy plastic is often more popular than glass. Polycrystalline thin film solar cellsaddress this issue and some others.Though Silicon is one of the most abundant earth crust elements, as previously6mentioned, crystalline and pure absorbers made from Silicon are expensive. Thincells, comparatively, use less material and can be made quickly and inexpensively.As can be seen in Figure 1.1, polycrystalline thin film cells have reached over20% efficiencies in laboratory environments. Though this only comes close tothe bottom level efficiencies as seen in crystalline solar cells, such as a singlecrystal Silicon solar cell without a sunlight concentrator, there is the large benefitof cheaper manufacturing. Therefore, the second largest market share in the solarcell industry is held by polycrystalline solar cells.So, as the cheaper and fairly efficient alternative, why are polycrystalline thinfilm solar cells not dominating the market? Well, they are still not as efficient, andtheir commercial counterparts suffer a larger disparity between their laboratoryperformance and their commercial performance when compared to the crystallineSilicon technologies. This is partially due to a number of years of research andinvestment into Silicon technologies, to which polycrystalline thin film technolo-gies are behind many years in industrial investment and a fair amount behind inresearch interests as well. This also leaves the question of the stability of thesesolar cells, as they do not have the proven reliability of crystalline Silicon solarcells. Finally, earth abundance and safety are hindrances to these types of solarcells making a large market impact. Some of the more dominate architectures,Cadmium Telluride thin cells, for example, use both rare and toxic materials.Emerging Technologies for Solar CellsMoving forward, current research tries to correct the shortcomings of both ofthe aforementioned (crystalline and polycrystalline) solar cell technologies. Thus,aims are often directed toward addressing some or all of the concerns: namely,environmental factors, stability, cost, and efficiency all can play a role in the ma-terial selection and implementation methods. It may be best to break emergingtechnologies into categories similar to those seen in Figure 1.1. These includeorganic, inorganic, and hybrid solar cells.Organic solar cells, by the very nature of their name, include carbon com-7pounds. These molecules are slightly different in electron sharing. As a com-parison with to the band states used to discuss electron levels in semiconductors,instead of valence and conduction band states for electrons to occupy, there arethe lowest unoccupied molecular orbital and highest occupied molecular orbitalrespectively. Though the mechanisms are different, carrier pairs and mobility ofthose carriers to power external devices are still necessary. Some of the benefitsinclude low production cost, various absorption spectrum selection, and possiblyuse with flexible substrates. The efficiency and stability of these types of cells arestill lacking too much. Thus, a large market impact has yet to be made.Inorganic solar cells have already made their impact. Silicon leads the industryand other leading solar cells are also made using typical semiconductor materials.advancement in the section of the industry works to capitalize every bit possibleon current technologies; however, as theoretical efficiency limits are approached,and market objectives remain unmet, alternatives become a necessity. Thus, ad-vanced materials are made using novel structures, quantum dots for example, anddifferent material selections. One of the largest obstructions to this category tendsto be environmental factors. Be it rarity or toxicity, either can drive up productioncosts and interrupt the concept of earth conscientious manufacturing. Navigating asolution to these two problems while finding a compound that still functions as anefficient and viable solar cell absorber is one main direction for current research.On that note, hybrid solar cells are recently showing some very promisingresults. Perovskite solar cells can be seen in Figure 1.1 as having exceeded 20%efficiency in the laboratory. As a type of cell that uses organic and inorganicmaterials for the absorber, spectrum selection for absorption is highly tunable.Still, the presence of Lead gives rise to toxicity concerns, and as with organicsolar cells, stability is still an issue.1.2.2 Economic OutlookSo, with many obstacles and even more options on how to succeed in solvingthe problem, research pushes forward with goals to one day make solar power have8a competitive foothold in the energy supply market.Mertens [13] writes a niceeconomic outlook expressed as a comparison to other energy forms. A similar butshorter discussion is held in Section Absorber SynthesisMy focus, in the following presented experimental work, is material synthesisand implementation of synthesized materials into solar cells. Previously, in Sec-tion 1.2.1, various architectures were referenced based on the technology used forthe absorber layers in the solar cells. Developing a method for determining goodabsorber layers is important in that the chosen material determines the cell perfor-mance possibilities. What methods we use for producing various absorber layersdetermines how close we can come to reaching the best possible cell for a givenmaterial. Using a mechanochemically synthesized nanoparticle layer falls intothe emerging technologies category. Later, it should be easier to understand thebenefits of choosing such methods, if the means themselves are first understood.1.3.1 Mechanochemical SynthesisAs might be inferred from the name, a mechanochemical reaction describesa chemical change as a result of mechanical interactions. For example, a flintstriker: by applying a suitable amount of mechanical energy to break off smallenough pieces of a material such as iron, the small pyrophoric bits ignite in air.This is true of many metals, that an unoxidized small enough particle can ignitedue to oxygen exposure. Such is one path available in ball milling. Self-sustainingreactions can take off once appropriate conditions for ignition are met. Slowsteady-state reactions [14] and other paths are also available in mechanochemi-cal synthesis. Many of these occur at lower temperatures than bulk solid-statereactions due to various properties of the reactants and conditions in the mill.9Figure 1.2: Planetary mill rotations in the directions of the color coordinatedarrows and axes of rotations.Ball Milling ParametersExcluding the chemical reaction, the simple mechanical actions occurringwithin a planetary ball mill are quite abundant, and these actions vary based onthe conditions of a given mill. Forces from pressure, impact, and friction/shear-ing are some of the main mechanical interactions occurring inside a ball mill.These forces lead to changes in the material, such as deformations, breaking, andagglomerating.A planetary mill works by rotating cylindrical jars counter to the direction oftheir rotating platform as illustrated in Figure 1.2. As a result of frictional forcesand rotation of the jars about their center, a vortex of the materials should whirlabout the center of each jar. However, the platform rotates concurrently with thejars. This leads to centripetal acceleration having two centers, the center of eachjar and the platform’s rotation center. This means each centripetal force pushesthe material differently. One would simply keep everything in a vortex shouldonly the jars spin, and the other would keep material on the outside edge farthest10Figure 1.3: Planetary mill mechanics: the black triple arrow path representsthe platform rotation, the red arrow path represents the milling jar rota-tion, and the white arrow path represents the sidewall launch of millingballs and material in the milling jar.from the platform’s rotational center if only the platform rotated. In short, sincethe acceleration vectors from the two rotations do not always align, the materialcan fly across the inside of each jar as seen in Figure 1.3. It may be easier to un-derstand these motions in terms of the fictional forces involved. Thus, it may helppicture how ball milling works to read on the centrifugal and Coriolis forces, ifthe motions do not seem intuitive. These fictional forces explain motion observedfrom a non-inertial frame. These cross collisions are a very important aspect toball milling, as they are high energy. The frequency and energy of such events canbe controlled by various parameters used to control the ball milling outcome.For the parameters of a milling jar, one can pick various volumes and materi-als. Though the main reason to used a larger volume jar is to process more materialat a time, there could be another reason. The volume ratio of the jar, milling balls,and material changes the frequency and energy of cross jar collisions as shown inFigure 1.3. With a jar more full, energy of collisions decreases due to a shorter11mean free path and more matter to absorb the impact. Jar material and millingball materials can also vary to change impact energy and chemical compatibility.Various available materials have different densities, hardnesses, and elasticities,which effects the energy of collisions. Finally, the milling balls come in varioussizes. The smaller ball sizes can reduce final material fineness size minimums, butcomes with a trade off of less energy imparted by an individual impact.Aside from the variables related to the milling jar and balls, there are a fewother setup methods to be mentioned. Though one can change the amount of ma-terial to be milled in a given run, there are ways to adjust the outcome base onthe material preparation. For example, softer materials can be made more brit-tle by cooling them. Ensuring dried sample preparation for dry milling can keepthe mill cleaner and improve product yield. Wet milling is also an option. Byincreasing the ball ratio, the frictional forces dominate, and adding fluid helpsobtain be the best fineness in this scenario. Analogous to this method, a mate-rial can be dispersed in a solid medium which will not chemically react with thesample[14, 15]. This can reduce self-agglomeration and improve final fineness.As the reactants are diluted by this process, it can also be advantageous in slowingdown the chemical reaction in order to control final product fineness.Finally, the milling machine has adjustable parameters. Planetary ball millmachines can have various rotation ratios. The Retsch PM 200, used as describedin Section 4.1.2, has a ratio of 1 : −2. Thus, jars complete two revolutions (redarrows in Figure 1.2) during the same time interval as the platform makes onerotation in the opposite direction (black arrows in Figure 1.2). Higher ratios willincrease the frequency of cross jar collisions. With a given machine, the revolu-tions per minute can be adjusted. Due to a higher centripetal acceleration and thusa larger normal force required from the jar wall, faster rotations will increase thepressure and frictional/shearing forces on a material. Also, higher rotations speedslead to increased collision speed; therefore, the impact energy will also increase.Simply put, dump more energy into the machine and you get a higher energymilling environment. Other useful functions include reversible rotations, which12can be programmed to occur at set intervals. This will cause rotational actions ofthe material inside a jar to change directions, which can cut back on buildup andimprove final fineness. Also, rotations can be paused for set intervals. This allowsthe system to cool, which can lead to a more energy efficient production in certaincircumstances. For example, a given material might break down easier when it iscooler. Cooling of hot materials can lead to fracturing, which can also assist inbreaking down materials.The variety in a selection of controllable variables such as the jar, milling balls,filling parameters, and machine settings allow for multiple applications when us-ing a ball mill. Many of these parameters were varied in search of optimal settingfor absorber synthesis.1.4 CharacterizationOnce materials have been synthesized, it is often standard procedure to char-acterize the materials in order to assure one has made the intended product. Highenergy probing allows for a verification process that is fast and simple. UsingScanning Electron Microscopy (SEM) is a fast way to view the morphology of agiven material. As various crystals have different lattice structures, the change indominate geometries can allow visual verification of a chemical change. As thematerial in question is on the nanoscale, SEM allows the diffraction limitations im-posed by light in the visible spectrum to be surpassed. Using SEM allows for morethan size and shape visualization. Some other probing methods into chemicalstructures are available; however, a more specialized piece of equipment allowsfor similar identification methods to be used with improved accuracy. Namely,analyzing data from X-Ray Powder Diffraction (XRD) gives insight into chemicalcomposition and crystal phase. With these two methods, synthesized materialswere examined. Product morphology and chemical composition, post milling re-actions, were verified in this way.131.4.1 Scanning Electron Microscopy (SEM)As previously mentioned, optically viewing a specimen will only allow fora certain resolution between two points based on a diffraction limit. Since lighthas wave properties, passing through slit or aperture, such as our pupils, causesdiffraction–a spreading of the light wave. When these waves hit a sensor, retina,in the case of our eyes, the ability to discern or distinguish between them has to dowith the amount of overlap in the intensity distributions of the waves. Even withmagnification, we eventually hit a limit of approximately 300 nanometers due tothe wavelength of visible light. For a deeper understanding and more on thesecalculations, Egerton [16] presents an easy to follow review in his first chapter”An Introduction to Microscopy.” Our concern, then, is how to observe particlessmaller than this limitation. Also, we would like to see features on these smallparticles. The solution is quite simple, use any probe with a shorter wavelengththan visible light.Electrons that have been accelerated to high enough speeds fulfill our wave-length requirements, as do ultraviolet rays, X-rays, and gamma rays for example.The wavelength can be calculated from the DeBroglie wavelength relationship,as seen in Equation 1.1. The usual physics symbols have been chosen for wave-length, λ , Planck’s constant, h, and momentum, p.λ =hp(1.1)If we use energies typically used for SEM to accelerate electrons, then thewavelength will be on the picometer scale [17]. Not only does this mean an im-provement in three orders of magnitude, but one may notice this as subatomicin comparison with A˚ngstro¨m order of magnitude for atomic spacing in crystals.Thus, we no longer need to worry as much about our wavelength in determiningour resolution like before. Instead, it becomes a matter of how focused the beamwidth can be made for the electrons, which is still dependent on the wavelength,as well as how the scanning is performed for the image we raster.14Figure 1.4: An example schematic with components found in a Zeiss Sigmafield emission scanning electron microscope as used in this research.The equipment for SEM has an electron source that supplies the probe beamwhich is controlled by electrostatic and magnetic fields. For the SEM completedin this research, a field-emission cathode in an electron gun supplies the beamlabeled as the “Source” in Figure 1.4. The aperture then clips the beam to thedesired width, 30 microns for example. Much like aperture selection in visiblelight optics for cameras, changing this will change the focusing depth of field foran image as well as the amount of energy going to the sample. Next, the beam isfocused onto the sample using magnetic and electrostatic lenses. The scan coilsmove the beam across an area of the sample in order to raster an image. Theimage is picked up by a detector of the user’s choice. Electrons scattered fromthe sample for generating an image are also accompanied by secondary electronsejected from the sample material, due to the high energy of the incoming beam.These secondary electrons can be used to examine the chemical nature of thesample. Using an Energy-dispersive X-ray (EDX) detector, one can examine thecomposition and phase of the sample. This is very similar to XRD. Thus, XRDdata analysis was the preferred chemical characterization method chosen over EDXspectroscopy in this research.15Figure 1.5: Incident rays from the source (blue arrow) scatter from the crys-tal and head off toward the detector(violet arrows).1.4.2 X-Ray Powder Diffraction (XRD)The use of XRD for material characterization includes phase identificationalong with crystallinity. Quick comparisons to large databases of standard ref-erence patterns allow for identification of samples. This is handled convenientlywith visual matching assisted with software recognition and screening. Exploita-tion of the periodic structures in crystals, the wavelength of X-rays, and Bragg’sLaw allows for such means to characterization.In 1912, the three-dimensional diffraction grating properties of a crystallinematerial were discovered to act on X-rays [18]. This has to do with the periodicstructures in crystal lattice spacing and the wavelength of X-rays. As seen in Fig-ure 1.5, scattered light from multiple points in a lattice can move in the same di-rection to be picked up by a detector. One can visualize each plane as a horizontalarray. This diffraction grating like structure will split electromagnetic waves suchthat various frequencies will scatter at different angles from the surface. Thus,monochromatic X-rays will only scatter at certain angles. The zero order mode(the first observed angle away from the path normal to the plane) will act the sameas a reflection. That is to say, the scattered light follows a path away from the16Figure 1.6: Bragg’s Law states the condition (2d sinθ = nλ ) for which thedetector will receive the strongest signal from the scattered raysplane at an angle equal to incoming light. Usually, the angle used for reflectionsis the angle of incidence, which is measured normal from the plane; however, inthe case of Bragg’s Law, the angles are with respect to the planes which containsthe scattering points and the respective observational points. So, θ = 90◦−θinc,where θ is the angle used in Bragg’s Law, and θinc is the angle of incidence usedin the law of reflection. This angle is convenient to measure, as the angle betweenlight from the source and light headed to the detector, is 2θ , as seen in Figure 1.5.To put this setup to use, a relation for when the detector will pick up strong signalsis necessary. When does the light from two sources constructively add?For the light coming from two points to add constructively, they must be inphase. Bragg’s Law defines the relationship for constructive interference using thedistance between planes containing scattering points (d), the angle of light fromeach plane (θ ), and wavelength of the light source (λ ) as seen in Figure 1.6. Aslong as the source is considered to be in phase, then the two paths must differ byan integer multiple of the source wavelength (nλ ) in order to add constructively.Since the paths are parallel from points C and D to the detector, and they bothcome from the same source, the path difference is the distance | ~AB|− | ~AC|.17| ~AB| = dsin(θ)(1.2)| ~AC| = | ~AB|cos(2θ) = dsin(θ)(1−2sin2(θ)) (1.3)Some simple trigonometry gives the magnitude of each distance in terms ofthe desired variables. Equation 1.2 uses the definition of the sine function and theright triangle formed with ~AB as the hypotenuse, a vertical side under point A, andbase along the bottom plane containing point B in Figure 1.6.Equation 1.3 usesthe definition of the cosine function and the right triangle formed with ~AB as thehypotenuse, ~CB as a vertical side, and ~CA as a base, also visible in Figure 1.6.Using the double angle identity cos(2θ) =(1−2sin2(θ)) makes for simple sub-traction of | ~AB|− | ~AC|. Thus, the path difference set as an integer multiple of thesource wavelength gives Bragg’s Law, Equation 1.4.| ~AB|− | ~AC| = dsin(θ)− dsin(θ)(1−2sin2(θ))= 2d sin(θ)∴nλ = 2d sin(θ) (1.4)Concerns might arise from the two-dimensional derivation of Bragg’s Law.However, one can always reduce two vectors in three-dimensional space to twodimensions by moving to the reference plane which both vectors share. If thethree-dimensional picture that follows from such transformations becomes un-clear, further reading might be useful. Jenkins and Snyder [18] present much ofthe necessary information in Introduction to X-ray Powder Diffractometry. A geo-metrical approach can be used looking at latices in reciprocal space and seeing thepoints lying on an Ewald Sphere. Though such depth of knowledge is useful, it isalso unnecessary for understanding how X-ray powder diffraction analysis works.All one really needs to understand is that various periodic symmetries exist in a18crystal lattice and that these will lead to increased signal intensities at given an-gles. By mapping the intensity (counts is the common measure) against the angle,while taking measurements through a range of angles, the recorded profile canbe distinctly linked to the crystal under observation. Thus, comparisons usinglarge databases full of reference patterns allows for fast, simple phase matchingcharacterization.1.5 Thin FilmsOnce material characterization confirmed desired compound synthesis wassuccessful, films which could be used in solar cell prototypes needed to be made.Though solid crystal growth is possible, and some of the leading industry tech-nologies still use these methods, thin film technology allows for less waste. Also,creating a device that is no thicker than necessary allows for a shorter transportdistance of the photon generated carriers to their respective electrodes. Ultimately,this leads to getting the best possible device. Therefore, thin film technology willbe employed in this research.Different methods can produce various thin films. One large difference ap-pears when choosing between growing a film onto a substrate versus depositingmaterial onto a substrate. The first option allows for crystalline films, whereas thesecond choice will inherently suffer from grain boundaries after deposition. How-ever, such methods can have advantages, which were a major factor in deciding todeposit films in this research. It may be easier to discuss such factors following ashort background for each method.1.5.1 Thin Film GrowthWhen growing thin films, the method builds a film from atoms. Molecularbeam epitaxy, vapor deposition, and sputtering are some examples of such meth-ods. Though these methods can be used for physical deposition and need not bechemical, in this section the focus is film growth. Films can also be grown from19solutions; however, such methods do not tend to lead to crystalline films, the mainbenefit in film growth.Molecular beam epitaxy essentially allows atomic level control of crystal growthfrom vapor phase reactants. The controlled environment leads to some of the mostcrystalline samples available for artificial crystal growth. The largest downsidecomes from the amount of time needed to grow material and the cost. The expensein this choice comes from the ultrahigh vacuum environment and the difficulty inusing the equipment.Chemical vapor deposition can similarly grow crystals from vapor phase re-actants. The chambers can be at atmospheric pressure, low-pressure, or ultrahighvacuum environments. Depending on the requirements, this can save on cost.However, the purity of grown films can suffer. The gas flow in the chamber al-lows for reactants to drift and deposit onto a substrate, as opposed to the gas beamof reactants being directed at the substrate as done in molecular beam epitaxy.Sputtering hits a target, which then ejects some material toward the substratefor deposition. What is used to hit the target will change how the deposition on thesubstrate happens. A source could be plasma, ions, inert gasses, or reactive gassesfor example. This can mean the target can travel to the substrate and hit in a vaporor liquid phase. Also, the reactive gasses allow for reactions to happen before thedeposition on the substrate. Thus, epitaxial growth is possible as well as physicaldeposition. Again, equipment complexity leads to a rise in cost for film growth.Thus, let us look at some top-down approaches to thin films as opposed to thebottom-up growth of films.1.5.2 Thin Film DepositionInstead of chemically forming a film, as in epitaxial growth, let us start with afinished product and then find a way to make a film. It has already been mentionedthat such depositions are possible with physical vapor deposition, and sputtering;however, let us look instead at some lower temperature options. For example, asuspension or colloid is prepared by using small particles of the desired compound20dispersed in a liquid. This colloid is then dispensed on the substrate for the thinfilm. Some examples include using rollers or doctor blading; as well as spray, dip,and spin coating; and drop casting.The first two methods simply spread the colloid onto a substrate mechani-cally. The thickness of applied colloid is thus controlled by the mechanics of theprocess–be it a blade, roller, or another mechanism which removes excess colloid.With the evaporation of the liquid, a film is left on the substrate.Spray coating can be done hot, such as spraying solids which have been meltedto their liquid phase. However, spray coating done at lower temperatures, suchas will be discussed, has mechanisms for film formation which differ slightly.Adhesion of the product in a colloid to the substrate requires a certain impactspeed. So, the spraying mist controls the amount of colloid dispensed, but theimpact of the spray also plays a role in the final film. Again, evaporation of theliquid leaves behind the desired film.The remaining methods rely on surface tension to determine the amount ofdispensed colloid. In dip coating, adhesion holds the colloid to the substrate asit is removed from its submersion. Then, cohesion pulls more colloid along withthe amount stuck to the surface. Thus, there is an upward flow of colloid withthe substrate rising and an outward flow where excess fluid is returning. Thestagnation point, where the flow is zero, determines the thickness. The balancingof these forces is due to the surface tension. Also, viscosity and density of thecolloid play a roll. Since viscosity is a resistance to the flow, it will affect flow ofthe colloid in all directions. Density will change the forces felt by gravity, whichsupport the flow back into the colloid reservoir.Spin coating is similar to dip coating, but the lack of enough centripetal forceholding the colloid in a lateral direction toward the center of rotation results in theloss of colloid. This is similar to how gravity pulls the colloid off of the substratein dip coating. However, the acceleration due to spinning can be many times thatof gravity, so the colloid will be pressed thinner to the substrate in comparisonwith dip coating. Also, this pressing action compresses the film. For these two21reasons along with speed and simplicity of adjustments, spin coating was one ofthe methods employed in this research along with drop casting.Drop casting is similar to spray coating, in that the amount of colloid on thesubstrate will determine the film thickness. However, this method leaves the leastuniform films. An advantage is that drop casting is quick, simple, and can makemoderately thick films in a single coat. As empirical methods lead to choosingthis method, more will be discussed in Section Film CharacterizationAs various films were made, it was important to quantify some of the differ-ences. Visual inspection allowed for qualitative and semi-quantitative measureson film uniformity and thickness. As the final objective is a working solar cellprototype, quantitative optical and electrical measurements help in ascertainingthe viability of a given film.Absorption, Transmission, and ReflectanceThe idea behind solar cell energy harvesting relies on absorbing photons fromthe sun and collecting carriers separated by such an absorption. Thus, it is a goodidea to check how much light, from the sun, a material is capable of absorbing.This can give the relevant band gap of the material in observation of the absorptiononset. It can help in determining a necessary thickness of material in order tomostly absorb the sunlight. When layering materials, it can help with deciding thelayer order based off of what light will be transmitted through a given material. Inshort, absorption, transmission, and reflection data are useful in the architecturaldesign of a solar cell.When light hits material it can scatter, transmit, or be absorbed. There arepieces of equipment that will measure transmittance, as well as spectral and dif-fuse reflections. With this, absorption is what is left over. Since reflections areelastic scattering, one can think of a nominal value coming off of with transmis-sion spectrum from 100% and the remainder is absorption. Thus, the characteristic22shape will be the same, but the value will differ.Though keeping the incident angle of zero for transmission measurements en-sures such a relation, no such perfect scenarios exist. Thus, using equipment thatmeasures both transmission and scattering is more accurate. This accuracy costmore time and money, and it serves little purpose at prototype development stages.So, characterization of films is carried out only with transmission measurements,as outlined in Section 5.3.Current Versus PotentialAs the ultimate goal of solar energy harvest is power supply from sunlight,it is good to gauge how much power is available from a solar cell. This is doneby measuring the current generated at various potentials in the influence of light.Measuring the same output in the dark can lead to useful information about asuccessful solar cell.When illuminated, the current measurements taken at various potentials cangive all the necessary information for power generation capabilities. Refer toFigure 1.7 for a visual indication of the important parameters to be measured.The open circuit potential (Voc) is the point where zero current is measured.This means the photopotential is equal and opposite to the applied potential at thispoint. The current measured when there is no applied potential is the short circuitcurrent (Isc). This is the largest possible current the cell can supply. If there wereno resistive losses, the power possible would be the product of the open circuitpotential and the short circuit current. By integrating the current with respect tothe potential, we can find the largest power value (Pmax). This is visualized as thelargest area box that can be drawn in the current versus potential curve as seenin Figure 1.7. The current (Imp) and the potential (Vmp) at the maximum poweroutput are important in how a cell can be implemented when supplying powerto devices. Experimental current versus potential measurements are discussed inSection 5.4.23Figure 1.7: Typical current (I) versus potential (V) graph for solar cells. Thecurve in black represents the dark current measurement and the curvein red is the illuminated measurement.24Chapter 2Motivation for Solar CellsEnvironmental and economic demands are initiating growth in solar cell tech-nology. With growing financial investments, social awareness, research, and vari-ous other forms of support much can be realized in solar cell implementation.Science must answer the call for clean energy. The health of current and fu-ture generations relies on the innovation of researchers for providing solutions tocurrent global health concerns. A quick overview regarding some of the largestareas of concern for health can be viewed in Figure 2.1. A very straight forwardapproach to solving some of these issues is to provide clean energy. Also, a quickglance shows that the developing countries clearly suffer most. Thus, economicalsolutions stand to make the largest global impact.2.1 Environmental SolutionsMany issues surrounding the general well-being of those currently living in theworld can be addressed with cheap clean energy. According to the World HealthOrganization [20],Air pollution causes 1 in 9 deaths. It is the biggest environmentalhealth crisis we face.As already mentioned, in reference to The World Health Report 2002–Reducing25Figure 2.1: A “global distribution of burden of disease attributable to 20leading selected risk factors” in units of disability-adjusted life years(DALYs) by the World Health Organization [19].Risks, Promoting Healthy Life, one can view some of the major health concernsin Figure 2.1. Currently, approximately 3 million people die each year from out-door air pollution and approximately 4.3 million from indoor air pollution [20].Thus, collectively, outdoor and indoor, air pollution can be viewed as one of thelargest burdens on life. A majority of the pollutants come from energy sources,namely fuels, in supporting various daily needs: cooking, heating, transportation,et cetera. Since many of these tasks have electrical alternatives, a solution existswhere clean affordable energy supplants the need for other energy sources. This26includes a need to replace large power stations which rely on fuels. Opinionseventually become ethics if enough people agree, and the United Nations, com-prised of 193 Member States, is pushing for this very change. As of January 1,2016, Transforming our world: the 2030 Agenda for Sustainable Development, alayout of 17 sustainability goals are in effect[21]. Along with the aforementionedhealth issues, this agenda also addresses environmental concerns for the planet.The impact left by many of the air pollutants has led to noticeable climatechanges. Also, many of the fuel sources are non-renewable. For some renewableresources, deforestation as an example, trees are not being replenished at the ratewe are using them. Awareness of the footprint left behind by our current decisionshas become a more socially necessary step in all fields; videlicet, research, busi-ness, government, and other aspects of life require earth conscientious decisions.As solar energy is diurnally abundant well beyond any foreseeable near futureneeds, visually apparent in Figure 2.2, the only environmental concerns becomethe production and implementation of solar cells.2.2 Economic OpportunityAn environmental push is not all that has motivated growth in solar energyharvesting technologies. Many modern devices requiring electrical power haveusage demands. Though battery technologies have made longer use possible whilemaintaining or decreasing size and weight, other options have helped motivatenovel solar cell applications. Combining effects from various factors: governmentefforts, industrial large-scale installations, wearable device popularity, and otherfinancial endeavors, has lead solar generated power to finally reach competitivesales values with other energy technologies.Early adoption of solar cells in devices included watches, calculators, andvarious personal devices. With more modern versions such as Eco-Drive watches,Misfit Wearables, and other such devices, solar power has not left the wearabletechnology market. Though it lags behind the novelty industry in terms of currentfinancial investment, medical wearable devices are also adding to the demand pool27Figure 2.2: “2015 estimated finite and renewable planetary energy reserves(Terawatt-years). Total recoverable reserves are shown for the finiteresources. Yearly potential is shown for the renewables.” A visualrepresentation by sphere volume size from Perez and Perez [22].for better technologies. These niche groups are an example of how every little bithelps. Their investments will continue to make an impact on the production costsassociated with solar energy harvesting technologies.On a larger scale, government policies such as various subsidy projects andtax incentives for installations all have made a noticeable pricing impact. Solarpower stations are going up around the world and project bidding has continuallydriven down the price of solar cell modules. The goals laid out in 2011, for theyear 2020, by SunShot [23] ($0.09 per kilowatt-hour for residential solar, $0.07per kilowatt-hour for commercial solar, and $0.06 per kilowatt hour for utility-scale solar localized cost of electricity pricing) have been more than 90% met asof November 2016.Large corporations pledging 100% renewable energy operations has helped in28funding such endeavors. Tesla, Panasonic, and Solar City have come together inmaking a roof that integrates solar cells. The roofs for 2017 are supposedly go-ing to be cheaper to install than a standard new roof. Competitive entrepreneurialmoves such as this will also continue to drive down the cost of solar energy har-vesting.2.3 Nanotechnology BenefitsWith so many environmental and economic incentives, emerging technologiesare everywhere in solar cell related research. In order to compete with currentsolar cell technologies, new designs must have the means to supersede existingdevices. This means passing current price per performance measures or havingthe possibility to do so.Nanotechnologies are a gateway to new solar cell performance properties. Forexample, colloidal solutions which employ nanocrystals allow for cheap deposi-tion methods on a large scale, such as roll-to-roll coatings. Nanocrystals can beband gap tuned based on their size. Also, using thin cell layers has possibilitiesin cost savings and performance enhancement for a solar cell. That is to say, lessmaterial used can mean lower costs and an appropriate layer thickness can be op-timized for a given cell structure’s architecture, which in turn improves the overallsolar cell performance.Also, nanotechnologies open possibilities in the realm of small device appli-cations. As the overall structure becomes thin enough, flexible electronic ap-plications become realizable. Likewise, the inherent nature of having nanoscalecomponents is a viable microscopic device opportunity. Thus, the mere size ofnanotechnological devices creates an avenue for implementation into novel prod-ucts.Many mechanical, electrical, and optical characteristics can benefit from nan-otechnological properties of materials. Smaller components are more thermody-namically favorable. That is due to their surface area to volume ratio increas-ing with decreasing size. For this same reason, access to photogenerated carriers29can improve, which can lead to enhanced solar cell performance. Also, at smallenough sizes, quantum mechanical properties become apparent in materials. Onebenefit of this is that it tends to improve the optical absorption characteristics ofthe material.2.4 Previous WorksWith all the opportunities and reasons for solar cell research, there is usuallynothing more than a small change from previous works in order to try and im-prove the current status of a given research project. After sorting through manydesigns in emerging technologies, I set some research boundaries. Prototype de-signs should maintain an earth conscientious design while producing a near futurepossibility of implementation. This means to me that the technology imploredshould be easily scalable and cost effective. The raw resources used should beabundant and safe. For these reasons and those given in some of the followingliterature, Fe2GeS4 was chosen as an absorber compound.This compound received attention as early as 1976 with a paper on the crystalstructure by Vincent et al. [24], and later gained some more interest in other papers[25, 26]. However, Oregon State University seems to be the first to recognize thepotential of Fe2GeS4 as a solar cell absorber material.Spies [27] recognized the potential of the material, measuring favorable val-ues for the band gap, conductivity, and Seebeck coefficient; however, sputtering afilm was not successful. Platt [28], who Spies [27] thanks in his thesis, added tothe previously mentioned data with a transmission spectrum from a single crystal,electron microprobe analysis of pellets, and theory calculations of band structuresand the absorption coefficient using Wien2K. Then, in 2011, Yu et al. [29] writea summary of the current findings supporting the use of Fe2GeS4 as a solar cellabsorber material. Pelatt [30] mentions the United States National Renewable En-ergy Laboratory’s calculation of a 21% efficiency possibility for Fe2GeS4 usingthe spectroscopic limited maximum efficiency metric. The films he sputtered suf-fered from oxygen contamination and extension of previous film characterization30included absorption and hall measurements. Finally, Ravichandran [31] wraps upthe currently available work done at Oregon State University. His theory workwas mentioned by Pelatt [30], and it supports a drift based design using Fe2GeS4as an absorber. Using sputtered films, which again suffered from oxygen contam-ination, photoelectrochemical cell measurements showed a photocurrent, but nophotopotential. A possible photopotential was extrapolated using photolumines-cence data. With these new measurements and preexisting measurements, variousdevice simulations were conducted. Other work, done under the United StatesDepartment of Energy, includes work done at Colorado State University. Fredrickand Prieto [32] report a solution synthesis preparation for a doctor bladed film. Todate, the best cell performance had a 6 mV open circuit potential and 0.3 mA·cm−2current density.Outside of the United States, this compound has also seen attention in Ko-rea. Park et al. [33] have done a mechanochemical synthesis of Fe2GeS4, leavingoptions open for possibly easy implementation of nanoparticles into a thin solarcell. There has also been a one-pot production method used to produce nanosheets[34].With the influence of these works, it seemed probable that a mechanochemicalsynthesis could alleviate delamination and stoichiometry problems arising duringsputtering. That is to say, germanium likes to alloy with many materials. Thus,high heat deposition onto metal substrates leads to delamination and the com-pound does not stick to the substrate. Also, the sputtering tended to require posttreatment to fix stoichiometry problems with the deposition. Since the stoichiom-etry is easy to control with mechanochemical synthesis and the low-temperaturedeposition of nanoparticles does not change this stoichiometry. Both problemswould appear to have a solution, which will be investigated by this research. Also,one might avoid organic compound contamination from solution synthesis. As nocurrent theory seems to model mechanochemical reactions in an experimentallyuseful way, a new model was developed in Chapter 3 with hopes to one day be ableto optimize mechanochemical processes with theoretical assistance. Experimental31works are presented in Chapters 4 and 5.32Chapter 3ModellingAn adaptable model focused on generality will be presented to address the cur-rent lack of well-defined parameters in solid state chemistry. A few sample curvesroughly matched to temporal experimental data demonstrates the feasibility of themodel. The data necessary for completely modeling the synthesis of Fe2GeS4 isin excess of the current goal. One example of a future use for such a model wouldbe optimizing the material synthesis processes through adjusting milling parame-ters to correspond with the optimal points seen in the model. Here an explanationon appropriately adjusting parameters to fit data with the real world parametersshould allow others to understand the model and implement it as best fits theirneeds. In short, the logic for why a model was written: eventual optimization pos-sibilities for minimizing manufacturing costs, as well as enhancement of generalscientific understanding for a relatively archaic system, and separation of physi-cally measurable parameters for that system.3.1 Application of the ModelA logical combination of various parameters is achieved via sigmoid functioncoefficients for terms within the system. The ternary, Fe2GeS4, to be used as anabsorber in a solar cell, has been used as an example for an application of the33model.Currently, there exist models based on mechanochemical processes from theviewpoint of thermodynamics [35], and other chemical analysis means. As well,physical models, which track the mechanics of the process [36] are plentiful; how-ever, the two ideas integrated in a meaningful and useful way still needs to emerge.This area of solid state chemical kinetics is heavily under-represented because theproblem contains so many dynamic parameters.For example, some chemical kinetics models use a mass fraction function in adifferential reaction model [37–39]. The mass fraction models are selected to fitthe reaction profile; however, this leaves many shortcomings in understanding thereaction. The many variables in an experiment (i.e. temperature, concentrations,particle sizes, and morphology) all change the same parameters for the reaction(order, rate, and induction time) [40].The goal here is to develop a model which incorporates the synthesis chem-istry, physical parameters, and the energy input for the synthesis all in one model.Individually, these can become large-scale arcane models. Unless written as awhole software program, to implement the models with a nice simple graphicaluser interface, such an approach serves little purpose for industry or experimental-ists. Thus, this goal must also maintain reasonable simplicity in the developmentof a usable model, which can allow for expansion as necessary. Such a model isdeveloped in Section 3.2, presented step by step as a demonstration in Section 3.3,applied to a proof of principle in Section 3.4. Extended application ideas for themodel are discussed in Chapter 6.3.2 Modelling the Chemistry of Ball MillingMany techniques can be utilized in modeling ball milling synthesis and havenames; videlicet, thermodynamics, fluid mechanics, classical mechanics, solidstate physics, statistical mechanics, and many other fields provide useful tools.With mathematical overlap in these fields, much can be realized through startingwith an overly simplified model. Then, additions can be made to the model as34necessary to form a usable and versatile model.For a starting point, the chemical reactions will be modeled as done in chemi-cal dynamics, where the rate of reaction is related to the product of the reactants.That is to say, the rate of change for a compound within a system is proportionalto the statistical interactions of the elements in the system which form the com-pound. Though this has past been considered irrelevant, as typically solid statereactions are governed more by the available interaction surface. Thus, diffusionis often the main rate-limiting step; however, merit can still come from this deci-sion. After all, the goal includes a generalized model, which does not only workfor solids. Also, powders can be modeled as fluids under certain circumstances,as done by Hao [41]. In milling, as nanoparticles approach molecular sizes andlocal temperatures lead to sublimation of solids, gas related chemical dynamicscannot be ignored. So, some logic to decide how close the reaction lies to eitherend of the spectrum (solids versus fluids) becomes useful.One parameter to control the statistical outcome of interactions in such a sys-tem is the coefficients used for the interactions. Sigmoid functions can work inthis way, allowing a probabilistic or fuzzy, logic approach for choosing which endof a continual logic distribution the reaction will proceed. These coefficients areintegrals of pulse functions (i.e. arctangent). Said another way, the derivative ofthe coefficient should be a pulse function. This allows a cap, such as a maximumreaction rate. For this model, this is the chemical reaction rate as a gas. Thus, themaximum rate would be the expected chemical reaction rate achieved if the appro-priate reactants collide with the correct energy. This is chemical kinetics withoutconsideration for solid properties. Then this rate will diminish as the surface areato volume ratio of a solid particle decreases. The model will be built up in thisfashion, starting from a basic interaction model.3.2.1 Interaction ModelBefore diving directly into writing differential equations for the elements withinthe system, it would be useful to address which chemical reactions will be con-35sidered in the example. According to Yu et al. [29], the enthalpy of Fe2GeS4formation from the binaries, as seen in Equation (3.3), is positive. Therefore,the ternary is more stable than the binaries. In the spirit of this statement, a twostage reaction from the base elements to these binaries (Equations (3.1) and (3.2))followed by the aforementioned reaction will be used.Chemical Reaction FormulasFe2++S2− FeS (3.1)Ge4++2S2− GeS2 (3.2)2FeS+GeS2 Fe2GeS4 (3.3)Instead of starting with chemical kinetics, first look at the continuous model ina discretized manner in order to understand the stoichiometric balancing. A briefinspection of the reactants in Equations (3.1) to (3.3) will lead to Equations (3.4)to (3.9). In detail: the forward reaction in Equation (3.1) can be thought of as oneof each reactant is lost to form one product. So, by bringing the two reactants to-gether, one of each is lost and the product is formed. These interaction terms (theproduct of two elements of the system) are seen as the negative terms (-[Fe][S]) inthe rates given in Equations (3.4) and (3.6). Next, in Equation (3.2), one Germa-nium ion and two Sulfur ions are lost to form one product. These product terms areunlike the previous ones in that Sulfur loses two ions. Thus, the interaction term(-[Ge][S]) has a coefficient of one for Germanium Equation (3.5)), and receives acoefficient of two for Sulfur (Equation (3.6)). Using this logic, it becomes appar-ent how to build an interaction model (Equations (3.4) to (3.9)) from the chemicalreaction formulas (Equations (3.1) to (3.3)).36Interaction Model Equations∂Fe∂ t=−[Fe][S] (3.4)∂Ge∂ t=−[Ge][S] (3.5)∂S∂ t=−[Fe][S]−2[Ge][S] (3.6)∂FeS∂ t= [Fe][S]−2[FeS][GeS2] (3.7)∂GeS2∂ t= [Ge][S]− [FeS][GeS2] (3.8)∂Fe2GeS4∂ t= [FeS][GeS2] (3.9)3.2.2 Chemical Kinetics ModelThe equations of the interaction model need some extra input for chemicalkinetics. This information comes from the field of chemical dynamics. Chemicaldynamics studies the mechanisms behind reaction rates. The approach is to useconcentrations. This is typically applied to fluids (gasses and liquids); however,as previously mentioned, powders can be viewed as a fluid for some applications.One of the standard views is to think of an ideal gas reaction based on collisionprobabilities. If a molecule exists in a given volume, the odds of finding it in thegiven space is the quotient of the molecule volume over the entire space volume.This is essentially concentration if the units used to represent the molecule andspace are chosen properly and the volume of an individual molecule is small incomparison to that space. Combine this with basic statistics for a coincident eventof two probabilistic events (i.e. finding molecule A in space C out of volume D atthe same time as finding molecule B in space C out of volume D) and one realizes37the event occurs with the product of the concentrations (ACD × BCD ). In order toproperly treat such cases, this becomes a statistical mechanics problem where theelectrons in the bonds become the focus. This is one reason, only the simplestreactions can be modeled computationally for reaction rates.“One of the observations regarding the study of reaction rates is thata rate cannot be calculated from first principles. Theory is not de-veloped to the point where it is possible to calculate how fast mostreactions will take place. For some very simple gas phase reactions,it is possible to calculate approximately how fast the reaction shouldtake place, but details of the process must usually be determined ex-perimentally. Chemical kinetics is largely an experimental science[42].”With this in mind, though the equations remain empirical, efforts are madeto organize coefficients in a fashion to allow for ease of use and understandingwhen matching the parameters with experimental measurements. As well as inthe previous model, chemical kinetics relates the rate of a chemical reaction tothe rates of all the reactants by the negative inverse of their chemical reactionformula coefficient and the positive inverse for the products. Though this is doneas shown in Equations (3.10) to (3.12), the reason to start with an interactionmodel is hopefully apparent. Correctly using these three reactions which all havecoupled rates can become an accidental mess pretty quick.Chemical Reaction Equation (3.1) Rate =−∂Fe∂ t=−∂S∂ t=∂FeS∂ t(3.10)Chemical Reaction Equation (3.2) Rate =−∂Ge∂ t=−12∂S∂ t=∂GeS2∂ t(3.11)38Chemical Reaction Equation (3.3) Rate = (3.12)− 12∂FeS∂ t=−∂GeS2∂ t=∂Fe2GeS4∂ tAnother variation from the interaction model was the change in the ordersof the reactions and the addition of rate coefficients. The reaction orders comedirectly from the probability view mentioned previously. This is not the case forall reactions, but for many elementary reactions with no intermediary steps, thisis a good starting guess. This is the reason for modeling the example reaction asmultiple elementary reactions. If modeled as a reaction from the starting elementsto the final product, then an intermediary step could contain unknown information,but this could not be gained by skipping these steps. This is less of a concern whenlisting all the steps as elementary reactions.The other concern to be addressed before moving on is the reason for using thethree reactions chosen. For justification of this decision: the previously mentionedreference by Yu et al. [29]; along with experimental observations where FeS isfound in the incomplete reactions of the product; and similar findings by Parket al. [33] who previously synthesized the compound via ball milling, all supportsuch a model. There are other possible paths, but upon further investigation, theycontained less stable derivatives compared to the ones used in these steps.As a final note, coefficients for the reaction rates function as a fitting parame-ter in Equations (3.13) to (3.18). These are three separate coefficients as we havethree reaction rates (Equations (3.10) to (3.12)), and they may or may not hap-pen at the same speed. As stated, these formulas get used empirically in the longrun, which includes needing to know how fast a reaction occurs while limiting thevariable parameters involved in the reaction. Though these coefficients have somepre-existing empirical relations that can be correlated to physical parameters, de-velopment of this model is independent of such approaches. The reason being thatthe current models simply relate each coefficient to an exponential or product of39exponential functions, as that is the form of the solution to the first order reaction.Then, they leave the coefficient in front as a dynamic fitting parameter. Again,that really only holds significant meaning for certain first order reactions.“As we consider a few types of solid state reactions, we will see thatthere is no simple interpretation of k possible in some instances [42].”Chemical Kinetics Model Equations∂Fe∂ t=−kFeS[Fe][S] (3.13)∂Ge∂ t=−kGeS2[Ge][S]2 (3.14)∂S∂ t=−kFeS[Fe][S]−2kGeS2[Ge][S]2 (3.15)∂FeS∂ t= kFeS[Fe][S]−2kFe2GeS4[FeS]2[GeS2] (3.16)∂GeS2∂ t= kGeS2 [Ge][S]− kFe2GeS4[FeS]2[GeS2] (3.17)∂Fe2GeS4∂ t= kFe2GeS4[FeS]2[GeS2] (3.18)As an alternative to trying to make this type of form which comes from the so-lutions, where the meaning behind the model quickly gets lost in the mathematicsfor most people, instead break up the coefficients in a more direct way. For thosecomfortable with differential equations, this will remain as clear as the interactionmodel. Another reason is to be more explicit in the dependencies in a way that fitswell to the experimental data.For the typical approach to the problem, one notes the steady state solution–theratios of the product over the reactants is constant at thermodynamic equilibrium.40This is the thermodynamic approach, to which one can relate this to Gibbs freeenergy. Assume a Boltzmann distribution of your molecules to correlate with thespeed of the molecules. Then, you get the typical Gibbs energy solution, whichthen takes the form of the Arrhenius equation if you relate the change in Gibbsfree energy to the change in enthalpy.To this end, energy dependence is one major parameter. In favor of not statingenergy as a proportionality by using a dynamic coefficient and a Boltzmann energydistribution, stick with a probability approach. This will still take on the form ofmultiplying the coefficients where large scale problem is described by variouscoefficients, but in a normalized (in the range of 0-100%) manner.3.2.3 Effective Reaction Area Terms in the Chemical KineticsModelIn mechanochemical processes, such as ball milling, materials are made toreact through an application of mechanical force. The molecules collide and achemical change occurs on the outer surface of the particles or at dislocations.Even in the event of a self-sustained thermally driven process, it happens in a waythat the reaction locations can be considered to occur over some functional area.The example case, as an exothermic reaction releasing energy during the forma-tion of Fe2GeS4, seen in Equation (3.3), proceeds in such a way. Since the chem-ical reaction speed maximum is already implemented through other coefficients,the next focus on size dependence is a concept of effective area dependence.To this end, a coefficient that increases as more available molecules in theparticle can react will be an effective reaction area related term. The range ofarctangent is from −pi2 to pi2 . So, the sigmoid normalized coefficient, as shown inEquation (3.19), will be arctangent. Other sigmoid options are viable, this is oneexample. By dividing the arctangent function by pi and adding one-half, the rangeof goes from zero to one, just as desired. For simplicity of the example, imaginea particle as a sphere. Thus, the surface area is proportional to the particle radius(Ri), which decreases at some rate (kiR) as milling proceeds (Equation (3.20)).41At some critical effective reaction area (Ricrit ), the induction period begins, thecoefficient as a function of the particle radius, reaches an inflection (the slope goesfrom positive increasing to positive decreasing) and a value of one-half. Howquickly the value changes from near zero to near one depends on the set width(Wi). In the limit of Wi→ ∞ this becomes instantaneous (the Heaviside function).This allows insight to size dependence of the reaction. If the reaction is heavilydependent on some critical particle size, the value becomes immense. However,if the reaction is continually taking place, but simply occurs more efficiently atsome critical size, the value will be lower. This allows us to easily fit the model toempirical data from particle size and to understand the dependence.Ridep =12+arctan(Wi(Ricrit −Ri))pi(3.19)dRidt= kiR (3.20)As a side note, Ri need not be a radius, it was merely convenient notation.All that really matters is picking some variable which can easily be correlated toparticle size and effective reaction area–so as to readily understand the size de-pendence of the reaction. Another notation convenience was keeping all the sameki reaction coefficients in Equations (3.21) to (3.26) as listed in Equations (3.13)to (3.18). This can be taken a few ways. Preferably, interpret the dependen-cies coming out of the coefficient. In other words, the ki reaction coefficientsin Equations (3.13) to (3.18) contain the information of the ki (which should bek′i) reaction coefficients in Equations (3.21) to (3.26) such that ki = Ridepk′i. Inthis fashion, one can continue understanding more about individual dependenciesfor a reaction, instead of all dependencies lumping into one dynamic coefficient.The final goal, if achievable, is an expression where all dependencies are listed.Then, a fitting parameter (ki) becomes nothing more than a normalizing constant,unvarying with any reaction parameters, because all dynamic dependencies areclearly listed in other terms.42Effective Reaction Area Terms in the Chemical Kinetics Model Equations∂Fe∂ t=−RFedepRSdepkFeS[Fe][S] (3.21)∂Ge∂ t=−RGedepRSdepkGeS2[Ge][S]2 (3.22)∂S∂ t=−RFedepRSdepkFeS[Fe][S] (3.23)−2RGedepRSdepkGeS2[Ge][S]2∂FeS∂ t= RFedepRSdepkFeS[Fe][S] (3.24)−2RFeSdepRGeS2depkFe2GeS4[FeS]2[GeS2]∂GeS2∂ t= RGedepRSdepkGeS2[Ge][S] (3.25)−RFeSdepRGeS2depkFe2GeS4[FeS]2[GeS2]∂Fe2GeS4∂ t= RFeSdepRGeS2depkFe2GeS4[FeS]2[GeS2] (3.26)3.2.4 Energy Dependent Terms in the Chemical KineticsModelThe final step, of this model, is to add in energy dependence. So far, reactionprobability based on collisions, and availability of those reactions based on aneffective reaction area are included. What of energy? Temperature, mechanicalinput, and other energy influences need their place in many chemical reactions.43Think about the system to apply this as a final logic coefficient. For the examplecase of ball milling, consider energy for when the particles collide. Particles canfracture, break into smaller particles, or react to form new particles.Since the particle could fracture or break. Think of fractures as storing energy,until some critical amount of energy leads to a moment when the particle shatters.This is a step function. However, more than one logic step like in the previousarctangent selection is desired. Particles that have broken need a possibility tocontinue with size reduction. This staircase type of function can be built as shownin Equation (3.28).Ridep =12+arctan(Wi(Ricrit −Ri))pi(3.27)∂Ri∂ t= (3.28)(dEi(t)dtni!!(ni−1)!!)ln(Bi)Ri sinni (piEi(t))To understand the parameters: Ei(t) is the time-dependent function for whenparticles break (thus it relates to energy input required to break a particle); Bi isthe amount left after each break (i.e. 23 would mean each break would leave theparticles at a size of 23 their current value); Ri is a size factor; and ni is the width,which must be even and greater than zero, just as Wi for arctangent, the limit asni→ ∞ will make the transitions instantaneous (vertical steps), and lower valuesrepresent relaxed transitions. Next, look closer at these functions.The integral of an even powered sine function gives a function plus a sumof sine functions with increasing frequencies and decreasing coefficients. Note,the coefficient in front of the first term is (ni−1)!!ni!! that integration by parts lower-ing the sine function power by two at a time creates. For example, ni = 6 =⇒(6−1)(4−1)(2−1)(6)(4)(2) =516 . Evaluate at one-half of a period (Ei(t)), all the sine termscontribute nothing, so(dEidt)−1 (ni−1)!!ni!!is the contribution to each step, where44(dEidt)−1comes from anticipating a substitution needed for the reverse chain ruleto work. The lowest frequency term in the sine sum is twice that of the original.For this reason Ei(t) sets the period for breaking events.To summarize, the sine functions are the reason for a staircase. The changein each step is(dEidt)−1 (ni−1)!!ni!!–invert this to normalize each step as unity. Then,control the change by some fraction of the current size factor (ln(Bi)Ri). Thisallows a coupling of energy to a physical parameter. Together, they impact theprobability of a reaction by the sigmoid logic function of choice. Now, all majorparameters are coupled into one model: reactivity, thermodynamic and kineticfactors; solid state availability to react, currently as an effective reaction area; andenergy.Energy Dependent Terms in the Chemical Kinetics Model Equations∂Fe∂ t=−RFedepRSdepkFeS[Fe][S] (3.29)∂Ge∂ t=−RGedepRSdepkGeS2[Ge][S]2 (3.30)∂S∂ t=−RFedepRSdepkFeS[Fe][S] (3.31)−2RGedepRSdepkGeS2[Ge][S]2∂FeS∂ t= RFedepRSdepkFeS[Fe][S] (3.32)−2RFeSdepRGeS2depkFe2GeS4[FeS]2[GeS2]45∂GeS2∂ t= RGedepRSdepkGeS2[Ge][S] (3.33)−RFeSdepRGeS2depkFe2GeS4[FeS]2[GeS2]∂Fe2GeS4∂ t= RFeSdepRGeS2depkFe2GeS4[FeS]2[GeS2] (3.34)3.3 Model Understanding ExamplesFor some visual clarity, it may help to look at some graphs with various inputparameters as the model evolves. First, look at inputs connected to all models,for example, the molar masses (Table 3.1). All models used the initial conditionswhere the beginning number of moles for each molecule was calculated such thatthere would be a stoichiometrically sufficient amount based on producing fivegrams of Fe2GeS4 from Iron, Germanium, and Sulfur. No Iron Sulfide, Germa-nium Sulfide, or Fe2GeS4 were initially present. Also, the jar use for milling isconsidered to be the volume of the system (50 mL).Molecule Molar Mass (g/mol)FeGeSFeSGeS2Fe2GeS455.845072.640032.065087.9100136.7700312.5900Table 3.1: Compound Molar Masses46Amount (Mass Fraction) (min)0 2 4 6 8 10LegendIronGermaniumSulfurIron SulfideGermanium SulfideFe2GeS4(a) Interaction Model–conservation of mass is observed along with the series of reactions.Amount (Mass Fraction) (s)0 0.02 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 1LegendIronGermaniumSulfurIron SulfideGermanium SulfideFe2GeS4(b) Chemical Kinetics Model–with the addition of probabilities and reaction rates, thereactions now proceed at various orders.47Radius (nm)02004006008001000Amount (Mass Fraction) (hr)0 2.5 5 7.5 10 12.5 15LegendIronGermaniumSulfurIron SulfideGermanium SulfideFe2GeS4Radius FeRadius GeRadius SRadius FeSRadius GeS2(c) Effective Reaction Area Terms in the Chemical Kinetics Model–this term has intro-duced size dependence (Ricrit ) into the reaction rates.Radius (nm)02004006008001000Amount (Mass Fraction) (hr)0 2.5 5 7.5 10 12.5 15LegendIronGermaniumSulfurIron SulfideGermanium SulfideFe2GeS4Radius FeRadius GeRadius SRadius FeSRadius GeS2(d) Energy Dependent Terms in the Chemical Kinetics Model–the size reduction can nowbe controlled with an energy term (Ei(t)) to relate the model to ball milling.Graph 3.1: Example graphs for understanding the model.483.3.1 Interaction Model: Graph 3.1aThe volume is not yet considered in the interaction model, so the time for thereaction time depends solely on the amount of desired product. Most of the fivegrams of material will become final product after ten minutes. The most importantpart to note at this point is the similar curves. All of the base elements reduceexponentially. The binary compounds increase for a while and then also decreasewith some exponential degree. The ternary compound follows a sigmoid shapedproduction.3.3.2 Chemical Kinetics Model: Graph 3.1bThe chemical kinetics model introduced using concentration. So the initialmoles were divided by the system volume and used as initial starting concentra-tions. Though it might be better to define an effective concentration as in somemodels [41], this is a demonstration model focusing on simplicity. Also, similarresults can be realized with the size dependence parameters. The rate coefficientswere chosen to put most of the reaction in a time of less than one second. This isa reaction that should be explosive under the proper gas conditions, so the reac-tion should be achievable in a short time period. The enthalpy of formation order(Fe2GeS4 >FeS>GeS) is also used partly as the logic in selecting rate constants,along with the observed formation order in X-Ray diffraction phase identifica-tion performed on a temporal basis [33]. Therefore, 100 Lmol−1s−1 was used forKFeS, 50 L2mol−2s−1 for KGeS2 , and 1000 L2mol−2s−1 for KFe2GeS4 . Again, thesenumbers should be measured experimentally, but the purpose here is a demon-stration. Most of the five grams of material will become final product after onesecond, the most important part to note is now the varying curves. All of the baseelements follow different degrees of reducing exponentially. Sulfur having thehighest degree and Germanium the lowest. This is expected, due to the order ofthe reactions and their temporal formation order. The binary compounds increasefor a while and then also decrease with different exponential degrees, as they arealso of different reaction orders. Finally, all of this information is absorbed into49the degree with which the final product can be produced, as it depends on all re-actions. The graph is asymmetrically presented to show that the induction periodand the inflection point, where it switches to asymptotic behavior, all happens inless than two-hundredths of a second.3.3.3 Effective Reaction Area Terms in the Chemical KineticsModel: Graph 3.1cWith effective reaction area dependence, set a size-related parameter for whichthe reaction will occur. Here was used an initial size of one micron and a criticalsize of 100-200 nanometers for Ricrit (Table 3.2). As there were formations ofparticles in a range from 20-200+ nanometres seen in the milling synthesis, aswell as from the paper by Park et al. [33], this should be reasonable. The slope ofR (kiR) was set to make the reaction start around eight hours finish at twelve hours(Table 3.3). The width (W ) was set to make an early appearance of Iron Sulfidesometime after four hours, and of an easily noticeable amount around ten hours.Also, keeping Germanium Sulfide from showing and having a reasonable amountof the ternary compound by ten hours were set with the widths (Table 3.4). Theseparameters were also chosen to match with X-Ray diffraction phase identificationperformed on a temporal basis [33].Ricrit Critical Size (nm)RFecritRGecritRScritRFeScritRGeS2crit150.0150.0100.0200.0200.0Table 3.2: Reaction Critical Size50kiR Slope (nm/s)kFeRkGeRkSRkFeSRkGeS2R−(1000.0−RFecrit )/(10.0×3600)−(1000.0−RGecrit )/(12.0×3600)−(1000.0−RScrit )/(8.0×3600)−(1000.0−RFeScrit )/(12.0×3600)−(1000.0−RGeS2crit )/(12.0×3600)Table 3.3: Size Factor SlopeWi Width (rad/m)WFeWGeWSWFeSWGeS1000.001000.000.500.010.01Table 3.4: Reaction Arctangent Width3.3.4 Energy Dependent Terms in the Chemical KineticsModel: Graph 3.1dSimilar to how size dependence has been set, the energy parameters are alsomostly a proof of principle demonstration since there has not been enough appro-priate data collected for good statistical matching. The breaking fraction (Bi) wasset based on material hardness, such that softer materials break more (Table 3.5).The step width (ni) was set to ten for a relaxed parameter. Finally, timespan for thesteps was set with a breaking energy time (Ei(t)) as a linear function of time witha coefficient for maintaining the same time as the size factor slope, kiR (Table 3.6).Each coefficient comes from an analytical solution of the size factor functions,Ri, which have a rate of change dependent on their own value; therefore, the nat-ural logarithm appears. Yet, again one can see that with a little effort temporalmatching is quick and simple.51Ei(t) Energy FunctionEFe(t)EGe(t)ES(t)EFeS(t)EGeS(t)ln(RFecrit /1000.0)10.0×3600ln(BFe)tln(RGecrit /1000.0)12.0×3600ln(BGe)tln(RScrit /1000.0)8.0×3600ln(BS) tln(RFeScrit /1000.0)12.0×3600ln(BFeS)tln(RGeS2crit /1000.0)12.0×3600ln(BGeS) tTable 3.5: Breaking FractionBi FractionBFeBGeBSBFeSBGeS0.700.800.600.800.80Table 3.6: Breaking Fraction3.4 Model Application ExampleWith the basic outline in place, any reaction can be fit given the appropri-ate data. Fitting parameters such as the various logical widths and inflectionpoints (critical parameters) yield information on the system behavior. For exam-ple, as to how mechanical and chemical interactions lead to a ternary compound(Fe2GeS4) synthesis during ball milling. Where do the major dependencies ofthe reaction lay? Mechanically induced self-propagating reactions (MSR), suchas the aforementioned ternary compound’s formation, follow many interlinkedprocesses [43]. These processes can be categorized to energy, interactions, andmorphology for elements in the system [40].52As all of these categories have already been placed in the model, all that re-mains is building an example. Looking at the examples for understanding, Sec-tion 3.3, nothing in the chemical kinetics should change. So, to expand more onthe model’s flexibility, a demonstration will be made using formation (Fi j) andagglomeration (Ai), as well as size dependent breaking (Bi(Ri)), and non-linearenergy dependence (Ei(t)). The result can be viewed in Graph 3.2.Radius (nm)02004006008001000Amount (Mass Fraction) (hr)0 2.5 5 7.5 10 12.5 15LegendIronGermaniumSulfurIron SulfideGermanium SulfideFe2GeS4Radius FeRadius GeRadius SRadius FeSRadius GeS2Radius Fe2GeS4Graph 3.2: Application of the Model–the effects of formation, agglomera-tion, size dependent breaking, and non-linear energy dependence areall quite apparent in the changing effective reaction area terms repre-sented by varying radii of the elements and compounds.Model Application EquationsBi(Ri) =(1+2arctan(WiB(Ri−RBcrit ))2pi(3.35)× (Bimin−Bimax)+BimaxEi(t) = 3√WiE (t− ticrit ) (3.36)53Ai =(1+2arctan(WiA(RAcrit −Ri))2pi([i]MiDi)2Ri (3.37)R f ormi j =a√RaiDiDi j+RajD jDi j(3.38)R f ormi jmax =(1+2arctan(WHVY (R f ormi j −Ri j))2pi(3.39)[i]exist =(1+2arctan(WHVY ([i]−Xi[Fe2GeS4] f inal))2pi(3.40)Fi j = [i]exist [ j]existR f ormi jR f ormi jmaxRi jdepki j (3.41)Ridep =12+tan−1(Wi(Ricrit −Ri))pi(3.42)dRidt= [i]existAi (3.43)+(dEi(t)dtni!!(ni−1)!!)ln(Bi)Ri sinni (piEi(t))dRi jdt= [i j]existAi j +[i j]existFi j (3.44)+(dEi j(t)dtni j!!(ni j−1)!!)ln(Bi j)Ri j sinni j(piEi j(t))dFedt=−RFedepRSdepkFeS[Fe][S] (3.45)dGedt=−RGedepRSdepkGeS2 [Ge][S]2 (3.46)54dSdt=−RFedepRSdepkFeS[Fe][S] (3.47)−2RGedepRSdepkGeS2[Ge][S]2dFeSdt= RFedepRSdepkFeS[Fe][S] (3.48)−2RFeSdepRGeS2depkFe2GeS4 [FeS]2[GeS2]dGeS2dt= RGedepRSdepkGeS2[Ge][S] (3.49)−RFeSdepRGeS2depkFe2GeS4[FeS]2[GeS2]dFe2GeS4dt= RFeSdepRGeS2depkFe2GeS4[FeS]2[GeS2] (3.50)3.4.1 Formation Term in the Application ModelTo understand how the formation of new particles from the reactions of otherelements in the system have been modeled, look at each term of Equation (3.41).Many of the terms are conditional and/or weighted logic.The test of element existence ([i]exist) could be written as a Heaviside function;however, using arctangent with a large width (WHVY ) keeps the equation contin-uous and more flexible. Also, using a difference, instead of just the element inquestion allows the setting of an effective zero. That is to say, zero can becomeproportional to the amount of stuff involved in the reaction. Using some frac-tion (Xi) of the final amount of product ([Fe2GeS4] f inal) allows such an approach.Thus, formation of new particles relies on existence of the elements required to55produce the new compound.The largest possible new formation (R f ormi jmax) would be two particles comingtogether and reacting to form a new particle of weighted size (R f ormi j) from itsconstituent parts. So, the elements are weighted against their current density (Diand D j) to that of the new compound density (Di j). The size growth rate anddensities used depends on the particle dimensions (a), such that one could use1-D, 2-D, or 3-D growth. In this application, a = 3, and the growth is by volume.As formation should not lead to growing larger than this weighted size, a logicalcap is put in place (R f ormi jmax).Finally, the formation is in proportioned to the maximum, based on the ratewith which the chemical reaction is proceeding (Ri jdepki j). This way, the forma-tion does not occur just because the constituents for a given product are present,the formation rate depends as well on the reaction probabilities.3.4.2 Agglomeration Term in the Application ModelParticles colliding with themselves can lead to agglomeration. The interactionprobability is calculated as a volume ratio product and the amount is set by a logiccurve based on a critical size.To represent the physical collisions of particles and not the individual ele-ments, a volume ratio is calculated using the concentration value, molar mass,and density ([i]MiDi ). This is the ratio of volume to a given reactant or product outof the system volume. Thus, the probability of self-interaction is taken as a prod-uct of each ratio. The assumption is that it represents the probability of findingthe element and probabilities are equal everywhere.The logic curve is to account for variance in agglomeration based on size. Ingeneral, as the particles get smaller more agglomeration is observed. Also, abovea critical size (RAcrit ) nothing mentionable is observed. Therefore, the smallnessat which particles start to agglomerate can be set as well as how relaxed the actionis to this parameter by the width (WiA). Again, arctangent has been chosen for thelogic function, but any sigmoid logic would work.56Finally, the growth is in proportion to current size (Ri). That way, at most,two particles can make a size proportional to their total sum. A look at formationreveals this as the same logic seen in Equation (3.38). As each density would bethe same, a√2 is the only factor left out. This has been done intentionally sinceR is an effective reaction area to be decided based off observable data. Thus,differential equation proportionality order is maintained, and the constant can beabsorbed into the logic function.3.4.3 Size Dependent Breaking Term in the Application ModelThe breaking functions for ball milling are quite abundant. In order to breaka crystal, bond layers must be pulled apart past their elastic limit [44]. Sinceenergy stored from material strains is around dislocations, corners, and such; andtherefore, not uniformly distributed, breaking can be quite unpredictable.Therefore, to capture prevalent trends, think of the impacts in a mill (or anycollision). The contact point of an applied force is at the surface and then themechanical energy can propagate through the material by various means suchas deformations, vibrations, heat, etc. Since the available surface (contact area)and the volume (energy storage zone for fractures) change in ratio as the particlereduces in size, so does the relative size of a crack of given depth from an impact.Using this logic, breaking fractions vary from a minimum (Bimin) to maximum(Bimax) amount as the particles reduce in size. Again this logical curve has aninflection size (RBcrit ) about which the minimum and maximum break fractionsare centered. There is also a width (WiB) to account for how sharp the changeoccurs between the two amounts.3.4.4 Non-linear Energy Term in the Application ModelFor much of the same reasons used in determining the breaking, the energybetween breaks varies with milling time. Collisions not only break materials, butalso, store energy and cause dislocations.As milling continues, a particle decreases in size. Surface area and dislo-57cations increase, energy storage locations; however, the overall available space,volume, for energy to store decreases. At some point, the size becomes smallenough that the particles flow like a fluid and the glancing collisions lead to lessbreaking. For describing this increase and decrease, a monotonically decreasingfunction with an increasingly negative slope that reaches and inflection where theslope begins decreasing is chosen.By using a cube root function, the time for when an inflection occurs is se-lected (ticrit ) just as other critical parameters have been selected for logic functions.There is also a width (Wie) to control the breadth of the change. Thus, over thetimespan for experiment, the time between breaks is seen to decrease and thenincrease in Graph 3.2.58Chapter 4Absorber SynthesisPrevious syntheses have been done using tube furnaces, solution syntheses,and ball milling methods. In an effort to keep with green synthesis procedures,ball milling was the synthesis choice for this experiment.4.1 Ball Milling SynthesisPrior to material loading for ball milling synthesis, a premixture of dry ele-ments was prepared via mortar and pestle. Then, various milling parameters werevaried in multiple mechanochemical syntheses using the ball milling machine.4.1.1 Mortar and Pestle PremixtureStandard wash procedures were used to prepare the mortar and pestle.1. Wash the agate mortar and pestle with soap and water and leave in the air todry.This step is to remove bulk debris from the surface of the agate.2. Wipe the mortar and pestle with acetone and leave in the air to dry.This step is to remove materials that are insoluble in water from thesurface.59Figure 4.1: Mortar and pestle premixing3. Rinse the mortar and pestle with methanol and leave in the air to dry.This step is to remove the acetone.4. Rinse the mortar and pestle with deionized water and leave in the air to dry.This step is to remove the methanol.Following the preexisting literature by Park et al. [33], measures were takento prepare a premixture of the elements iron (Fe), germanium (Ge), and sulfur (S)in a ratio of 2 : 1.5 : 4 in order to synthesize Fe2GeS4.First, a bulk crystal of pure germanium was ground into powder using theclean mortar and pestle. The powder was then transferred into a jar for storageand the mortar and pestle cleaned according to the previously outline steps.To prepare approximately 20 grams of premix, some of the germanium pow-der, along with iron powder, and sulfur crystals were all mixed, in a ratio of2 : 1.5 : 4, using the clean mortar and pestle as seen in Figure 4.1. Upon visuallyreaching a feed size of submillimeter for the powder, the premix was transferredinto a jar for storage.60Figure 4.2: The final product4.1.2 Ball MillingUsing 5 : 8 scaling, due to using a 50 milliliter jar in comparison to the 80-milliliter jar used by Park et al. [33], previously used mass ratios of milling ballsand premix were maintained for the first synthesis. After following jar checkingprocedures, the premix was left to mill for 12 hours continuously. Characteriza-tion later revealed the reaction as a success.For the first run, approximately 16 grams of 10-millimeter and 16 grams of5-millimeter Zirconium Oxide balls were used in a 50-milliliter capacity steel jar.Approximately 3.2 grams of premix was added to the steel jar with the ZirconiumOxide balls. A counterbalance using a 50-milliliter capacity steel jar with approx-imately 24 grams of steel balls was also prepared for use in a Retsch PM 200planetary ball mill.The planetary ball mill was set up with the rotation speed setting at 550 revo-lutions per minute. Following the manual for long milling operations, jar clampsand security were checked at 3 minutes, 1.05 hours, and 3.9 hours prior to runningthe mill continually for 12 hours. The final product was a matte grey-black as seenin Figure 4.2.Subsequent milling runs varied the milling time, rotation directions changingat various intervals, and the milling ball to premix weight ratios. Using 30 minute61rotation direction change intervals seemed to significantly cut down on agglomer-ation deposits of the powder to the side walls of the milling jar. It was also foundthat the reaction would reach completion even if milling was halted for large inter-vals. For example, one run milled for 7 hours and then halted for approximately13.5 hours before finishing the synthesis with 5 more hours of milling time.Further plans for using the ball mill were halted due to a safety slider needingrepair work. Thus, synthesis was never completed using electronic grade purityelements. Also, final fineness was lacking in uniformity. This was likely due tothe mixed use of 5 and 10-millimeter diameter milling balls; however, confirmingsuch hypothesis will need to be left for future work.4.2 Absorber CharacterizationThe quickest verification of successful compound synthesis, post ball milling,is visual confirmation. When checking the product during various milling exper-iments, it became clear that the visual change from a metallic gray to a mattegray-black appearance was the simplest confirmation of a chemical reaction. Thischange is quite apparent if the material in Figure 4.1 is compared to the prod-uct seen in Figure 4.2. However, oxidation of metals is common in ball milling.Seeing as Hematite is black in appearance and is an iron oxide, some other con-firmation was necessary in order to validate the final product.4.2.1 Scanning Electron Microscopy (SEM) CharacterizationOne way to rule out Hematite is a closer visual inspection of the product. Dis-cussed in Section 1.4.1, SEM is a good way to investigate size and morphologyof very small materials. As Hematite crystals are in the trigonal system, hexag-onal scalenohedral class, and the desired product is in the orthorhombic system,dipyramidal class, a visual distinction is easy enough by the difference in prevalentgeometries. The same general observation is also true of the reactants in compari-son with the final product. The change in observable geometries allows for a quick62visual observation of the reaction. As can be seen in Figure 4.3 the formation ofplate-like structures gives a visual signal that the reaction has commenced.The ten thousand times magnification level images for the reaction initiation(Figure 4.3j) and the reaction near completion (Figure 4.3k) show the beginningof platelets forming and nearly all of the material in stacked plate-like geometries.The organization of Figures 4.3a, 4.3b, 4.3d to 4.3i and 5.5a is such that the samesample location is viewable at different magnifications by column and differentsamples are in different columns. For example: Figures 4.3a, 4.3b and 5.5a are allat one hundred times magnification, with each image as a different sample; andFigures 4.3a, 4.3d and 4.3g are all the same sample at one hundred, one thousand,and ten thousand times magnification respectively.As stated, this is a quick visual indication. The final compound has an olivinestructure and we can see the change in morphology into plate-like structures, in-dicating the possibility of an olivine compound. For a more accurate verificationof the material, chemical composition, and phase is analyzed using XRD.63(a) The reactants at 100X (b) Starting to react at 100X (c) Near final product at 100X(d) The reactants at 1,000X (e) Starting to react at 1,000X (f) Near final product at 1,000X(g) The reactants at 10,000X (h) Starting to react at 10,000X (i) Near final product at 10,000X64(j) Starting to react at 10,000X(k) Near final product at 10,000XFigure 4.3: Visual progression of the reaction with 4.3h and 4.3ireproduced larger for better viewing of final changes during the reaction.654.2.2 X-Ray Powder Diffraction (XRD) CharacterizationEvery crystal has a distinct set of constructive peaks when an X-ray scattersat various angles. The cause for this observation is briefly introduced in Sec-tion 1.4.2. A quick look at some various databases and one can visually matchtheir samples. With the aid of computer software, which compares selected peakswith those stored in databases of the user’s choice, it can be overwhelming atfirst. However, with some understanding of the different types of reference sam-ples such as: theoretical versus measured; or verified as a valid reference versusunverified; the system becomes navigable. With some experience on tuning thesearches and setting the peak patterns, the task becomes even more manageable.For example, peak matching is visualized in Graph 4.1 by the color of eachelements’ lattice spacing reference pattern being superimposed onto the premixsample’s diffraction count pattern. As almost all peaks are accounted for, thepremix is fairly pure.66Graph 4.1: Identification of reactants via XRD analysis. Reference patterns for iron (red), germanium (blue),and sulfur (yellow) have been superimposed onto the premix’s XRD pattern.67Some may find it more appealing to remove the background as has been donein Graph 4.2 in comparison with the raw pattern shown in Graph 4.1. The peakslabeled 2998, 2765, and 1798 at 2θ angles greater than 65◦ are not really uniden-tified peaks. They are most likely the Kα2 doubles of their adjoining Kα1 signal.This is discernible by their shape and relative intensity. The signals from the Kα2radiation gives approximately half the intensity of the Kα1 and the shape of thesignal is the same. These two signals are less superimposed at larger angles, andthe dual peaks become more apparent. This is not the case for the peak near 36◦labeled 842. This peak likely belongs to some contaminate. The impurity couldbe present due to the raw elements not being pure or because of contaminationduring the preparation of the premix.Similarly, we can look at the XRD pattern of a nearly complete reaction andverify the chemical composition and phase of our final product. As seen in Graph 4.3,the measured pattern is a good match with the reference patterns–confirming asuccessful synthesis.68(a) Peak windowing(b) Peak labelingGraph 4.2: Peaks can be windowed as in 4.2a to help visualize them or theycan be marked as done in 4.2b.69Graph 4.3: Identification of the final product via XRD analysis. Reference patterns for both a theoretical andmeasured XRD pattern for Fe2GeS4 are shown in blue and green.70The unmatched peaks in Graph 4.3 have been resolved in Graph 4.4. Thereaction is revealed to be incomplete by the remaining presence of germanium.Also, a reaction path is revealed by the presence of the intermediary product,iron sulfide. Scraping the sample from the wall of the milling jar also revealsunreacted sulfur not present in the loose product. This has been made clear inGraph 4.5, where the windowed peak around 23.2◦ shows sulfur in the scrapedsample, but not the loose sample. Such observations were useful in adjustingmilling parameters to have a higher amount of reaction completeness.71(a) Germanium (purple) peak identification(b) Iron sulfide (reddish brown) peak identificationGraph 4.4: Unknown peaks can be windowed for easier identification. Thishas been done for the identification of unreacted germanium (4.4a) andintermediary product iron sulfide (4.4b).72Graph 4.5: Identification of unreacted sulfur (red) in the final product via XRD analysis. Sulfur is present inthe XRD pattern from the scraped sample (red) but not the loose sample (black).73Chapter 5Thin Film ApplicationMany methods for thin film deposition exist and a few have been discussed inSection 1.5. By using nanoparticles synthesized via ball milling, as outlined inChapter 4, various methods for colloid suspensions were tested and implementedin film depositions.As the overall focus of work has an underlying theme of green energy, notall methods that are chosen necessarily correlate with those which are most com-monly used. The stabilizers and solvents often used to make a liquid for suspend-ing ceramic nanoparticles in a colloid are abundant with heavy, polar, long chainorganic molecules. Many of these are toxic. So, many attempts were done witha simple suspension in water and/or ethanol. As these eventually settle, agitationis necessary, and viscosity was often not correct for spin coating thick enoughlayers. Thus, for many films drop casting was used.5.1 Spin CoatingBased on the various literature, early spin coating experiments were done withhigh weight ratio to allow for quick visual analysis of the results. That is to say,higher weight concentrations tend to yield thicker single coat films, which madeidentification of issues doable by unaided sight or optical microscopy. Spin speeds74and times were varied in order to see if an optimum combination was possible.Various papers have similar explanations of spin speed and spin time. For exam-ple, Zhao and Marshall [45] show a nonlinear decrease of thickness with relationto time, and Tyona [46] shows a nonlinear decrease of thickness with relation tospin speed.With this in mind, extremes were tested to see various impacts on the final film.A major factor in determining the thickness is initially set by the amount of spin-off material, which leaves the substrate as waste. Thus, some initial films wouldspin for only one or two seconds at a given speed (2,000 rpm for example) beforeslowing down for evaporation of remaining liquid (1,000 rpm for example). Thislead to a clear understanding of acceleration and speed choices for films. It wasrealized that for water or ethanol suspensions, acceleration values which were toolow lead to nonuniform films, and speed values that were above 3000 rpm madelittle difference in making the films any thinner. Somewhere around 5,000-7,000rpm, thicker films would start to break. Though various weights were tested, nouniform, continuous, single coat films could be made at speeds above 500rpm. Asobserved by Wang et al. [47], speeds that are too high result in a sparse dispersionof the nanoparticles. However, lowering the speed below 300 leads to no wastespin-off. Thus, continuous, uniform films could be obtained at around 200 rpm,but this only leads to a slight improvement in film uniformity and nanoparticleorganization in comparison with a drop cast.With a basic understanding to some of the limitations for spin coating thenanoparticles, various measures were taken to improve the suspension. A com-parison of a simple drop cast of the unsettled solution versus some separation ofnanoparticles after settling revealed a path to improving film quality. As can beseen in Figure 5.1, simply leaving the suspension for some time after agitation andpulling off the remaining unsettle suspension after some time can greatly reducethe presence of larger particles. During slower spin coats these larger particlescause agglomerations that disrupt the film uniformity. At higher speeds, the largerparticles tend to be drawn off the substrate causing comet streaks in the film.75(a) Without settling at 100X (b) After setting separation at 100XFigure 5.1: SEM images of drop casts onto carbon conductive tabs withoutseparation 5.1a and with separation after settling for one hour 5.1b.Putting forward a slightly more quantitative comparison: measuring a settlingtime of one hour and separating a suspension in ethanol for comparison of effectson spin coating. Both spin coatings were performed with the same accelerationand ran for twenty seconds at 2,000rpm. As can be seen in Figure 5.2, bothhigh magnification (25,000X) images, Figures 5.2a and 5.2c, show a fairly similardistribution of nanoparticles; however, a clear distinction, and advantage, of theseparation after settling for and hour is apparent in Figures 5.2b and 5.2d. Atlower magnification (2,500X), one can clearly see large particles, over one micron,which would disrupt the ability to make a submicron film. Thus, it was decided totry filtration, a standard practice in spin coating film colloid preparation.Paying note to quality issues observed in Figure 5.2, the discontinuity betweennanoparticles has been well investigated, and causes are attributed to spin speedsthat are too high [47]. Filtration with a 0.45-micron syringe filter was performedon a solution pulled from a 30-minute separation in ethanol. The solution ap-peared clear after filtration. Two step spin coating with thirty seconds at 500 rpmfollowed by thirty seconds at 2,000 rpm was used to deposit a film onto an IndiumTin Oxide (ITO) coated glass substrate. There were no large differences in filmquality when deposited immediately after filtration versus four days later. Thus,76(a) After settling separation at 25,000X (b) After settling separation at 2,500X(c) Without settling at 25,000X (d) Without settling at 2,500XFigure 5.2: SEM images of spin coats onto glass after settling separation 5.2aand 5.2b and without separation 5.2c and 5.2d.agitation seems sufficient for suspension preparation, once large particles are re-moved, as agglomeration in ethanol or water does not appear to happen at a fastrate.Heating the solution to try and create a higher weight percent of nanoparticleswas performed at 50◦C for varying times in an oven over two days, but with theevaporation seeming too slow, the temperature was raised to 60◦C. This causedcrystals to form while cooling. Those crystals were large and flat, as can be seen inFigure 5.3d. A ten coat film was made following the same single coat procedures,but adding a one minute heating of the film at 100◦C on a hot plate between each77(a) After filtration at 10,000X (b) After filtration at 100X(c) After filtration and heating at 10,000X (d) After filtration and heating at 100XFigure 5.3: SEM images of spin coats onto ITO coated glass after filtration5.2a and 5.2b and heating 5.2c and 5.2d.coat. This was done to ensure complete evaporation of the liquid and help secureeach layer. A comparison of the film without the crystals and with them has beenpresented in Figure 5.3. Though one might believe them to also be quite thick;however, the films were still fairly transparent. Thus, this method was put on holdas it does not allow for thick enough films with the current product available fromthe outlined ball milling procedures in Chapter 4. Future plans for filtration ofbetter fineness consistency product will commence once the ball milling machinereturns to an operational status.Seeing as even particle agglomerations could not lead to as significant ab-78Figure 5.4: An agglomeration seen on a film post filtration at 50X.sorption of light, as expected, at least 500-1,000 nanometer films would be moredesirable. Wang et al. [47] mention contaminations such as dust causing nucle-ation sites for agglomerations. A similarly identified agglomeration is pointed outin Figure 5.4. However, such agglomerations are still too transparent. With a needfor thicker films, but an inability to create high enough particle weights post filtra-tion, drop casting seemed like a temporary alternative to try in order to test someprototype architectures.5.2 Drop CastingDrop casting was performed with various parameters in order to observe whatwould lead to the most uniform surface drop cast film. One example was thedifference in weight ratios of product and liquid. Though the weight ratios seemedto make one of the larger impacts, other variables included the amount deposited,substrate temperature, and liquid for the suspension.79(a) Without heat at 500X (b) 100◦C heating at 500XFigure 5.5: SEM images of drop casts onto glass without heat 5.5a and withheat 5.5b.Higher weight ratios of product in liquid had a tendency to decrease film uni-formity. Agglomerations would form in clusters on the film. While weight ratiosmuch below 5% had a tendency to not fully coat the substrate. Instead, small is-land films would form on the substrate. Using water instead of ethanol seemed tohelp with having fewer agglomerations or islands. This could be due to a higherdipole moment as well as the density of water. This was one reason ethanol waschosen for separation of larger particles. The settling rate was significantly higherin ethanol when compared to water.It can be seen in Figure 5.5 that heating the substrate does seem to cause graingrowth during the drop cast. Though the thicknesses of the films are different, itis clear that heating helped with improving film density and crystallization. Fromthis observation, doctor blading onto a heated substrate may be a nice option forgood film quality.In order to observe the difference in the amount deposited, multiple films weremade at the same 5% weight ratio. The number of drops were counted for refer-ence and later used to try and consistently make similar films. Ten drops arearound half a milliliter. This was found to give good films for a 2 cm x 2 cmsubstrate. If just enough to coat the substrate was used, the meniscus shape was80apparent in the film–thinner in the middle and thicker on the edges. Likewise,but to a lesser extent, a bulging amount would lead to a raised oval shape in thecenter. Thus, something in-between was found to be a good visual indicator forthe amount to dispense. As stated this was around half a milliliter.5.3 Film CharacterizationInitially, visual inspection along with optical magnification was used to in-spect uniformity of films. For the sake of quick measurement, simple transmis-sion measurements were taken on film samples. Since the product is fairly matte,reflections are not necessarily a large issue. Also, the incident angle was keptnear zero. Thus, characteristically, the absorption spectrum is going to be closeto 100−transmission. Transmission data has been reported as that is what wasmeasured.Using a xenon lamp sourced to a monochromator provided a selectable wave-length beam for transmission data. Transmission data was collected using an en-ergy/power meter. Instruments were connected through a Source Measure Unit(SMU) and LabTracer 2.0 was used to control the data collection.Background readings were collected through a blank glass slide and subtractedfrom the film readings. Measurements were taken in groups of three and averaged.The line was smoothed by a running three-point average. An example of a typicaltransmission profile for a film of Fe2GeS4 can be viewed in Graph 5.1. As can beseen in the comparison to the single crystal referenced spectrum [28], the onsetrate is similar (the slopes are parallel). However, there are two, one that matchesthe onset of the previous work and a second slope. Previous theoretical absorptioncalculations show a similar profile [30]. This is likely due to the two band gapenergies around 1.5 eV and a higher energy band gap of 2.6 eV as suggestedby Ravichandran [31]. Here we see similar values around 1.6 eV and 2.0 eVfrom the onsets. The shift in values could be due to the size of the nanoparticles,crystallinity differences, as well as impurities.81TansmissionMeasuredReproducedTransmission (A.U. Reproduced) % (Measured) (nm)300 400 500 600 700 800 900 1000Graph 5.1: A typical film transmission spectrum compared with a singlecrystal transmission spectrum by Platt [28].5.4 Prototype CharacterizationUsing the same SMU and software, current versus potential measurementswere taken for various prototype architectures. For light measurements, the xenonlamp light was passed through a filter in order to simulate a natural sunlight spec-trum (AM 1.5). Dark measurements were taken with lights off at night to min-imize any light contamination. Shielding the sample with a dark cover in thisatmosphere made little difference to the measurements. Since adding and remov-ing the shield added the risk of bumping the connections to the electrodes, usingthe shield was determined as unnecessary in these conditions. A typical currentversus potential prototype measurement is shown in Graph 5.2. With a quick cal-culation of the series and shunt resistances, it is clear that the series resistance istoo high in all the devices. More on this will be discussed in Section 6.2. Where82Sample 4Device 9 LightDevice 9 DarkCurrent Density (mA⋅cm-2)−10−50510Voltage (V)−1 −0.5 0 0.5 1Graph 5.2: A typical prototype performance. RSH ∼ 1−2×104Ω ·cm RS ∼50Ω · cm2possible, the shunt resistance (RSH) has been calculated using the inverse of theslope at zero potential (V ) divided by an estimated device thickness of 50-100microns using the dark measurements. Using the second semi-linear region, afterthe start of the exponential rise in current, the series resistance (RS) is calculatedfrom the inverse of the slope; again, when possible, using the dark measurements.Since the top electrodes, which had been evaporated onto the device, did notmake good electrical contact, many of the first set of readings were done withan electrical probe resting on the back of the device. This does not allow foran accurate representation of the device area. So, contacts were painted over theevaporated contacts. This allowed for a proper connection to the side of the device83Figure 5.6: Prototype Architecturewith an electrical probe. As a means to observe phosphorus doping (which has yetto be done with this compound in any literature) and check it as a viable meansof creating a solar cell with Fe2GeS4, various control experiments were set up.In describing these various setups, refer to Figure 5.6 for a visual on the variouslayers.For all devices, the transparent electrode was ITO on a glass substrate. Withonly Fe2GeS4 as a p-type layer and silver as a back electrode, devices showeda resistor (linear) current versus potential curve as was expected. Adding a zincoxide layer as an n-type window layer produce expected diode behavior as seenin Graph 5.3. The zinc oxide layers were prepared using a sol-gel process as de-scribed by Chou et al. [48]; however, spin coating was performed at 1000 rpm for30 seconds followed by 60s at 2500 rpm to provide a thicker film. These architec-84Sample 7Device 5 LightDevice 5 DarkCurrent Density (mA ⋅ cm-2)−10−50510Voltage (V)−1 −0.5 0 0.5 1Graph 5.3: A typical prototype performance using a zinc oxide windowlayer. RSH ∼ 5−10×103Ω · cmtures showed little difference in performance compared with using a phosphorusdoping for the n-type layer.To test various doping levels using phosphoric acid, which has been success-fully demonstrated on silicon wafers by Kim et al. [49], various acid concentra-tions were prepared. A moderated doping of 1 ppt ratio of phosphorus to Fe2GeS4and a light doping of 1 ppm were both used. The diode response is sharper with amoderate doping as seen in Graph 5.4 in comparison to lighter doping, where thecurrent onset is more gradual as seen in Graph 5.5.To test the use of polyethylene glycol for stabilizing the colloid suspensionand helping to bind the nanoparticles to the ITO layer, a solution was prepared85Sample 1Device 18 LightDevice 18 LightDevice 18 DarkCurrent Density (mA ⋅ cm-2)−0.02500.050.10.1250.15Voltage (V)0 0.5 1 1.5 2Graph 5.4: Moderate phosphorous doping prototype performance RS ∼ 7×103Ω · cm2dissolved in ethanol. Rinses with deionized water, acetone, and isopropyl alcoholfollowed by hot plate heating were used to minimize contamination of the organiccompound in the Fe2GeS4 nanoparticle film layer. Similar device performance aspreviously seen is shown in Graph 5.6.86Sample 2Device 18 LightDevice 18 DarkCurrent Density (mA ⋅ cm-2)−0.05−0.02500.050.10.125Voltage (V)−0.025 0 0.025 0.05 0.075 0.1Graph 5.5: Light phosphorus doping prototype performance RSH ∼ 1−2×105Ω · cm RS ∼ 3×102Ω · cm287Sample 7Device 18 LightDevice 18 DarkCurrent Density (mA ⋅ cm-2)−0.0500.050.10.15Voltage (V)0 0.5 1 1.5 2 2.5 3Graph 5.6: Light phosphorus doping with organic stabilizers used in spincoating prototype performanceRSH ∼ 1−3×107Ω ·cm Rs∼ 1×104Ω ·cm288Chapter 6ConclusionTo the knowledge of the author, current existing literature does not containany mechanochemical models which integrate reaction chemistry and mechan-ics without using abstract fitting parameters (dynamic fitting coefficients with nodirect correlation to measurable factors within a process). Also, the successfulimplementation of mechanochemically synthesized Fe2GeS4 nanoparticles into asolar cell prototype has yet to be realized. Successes of preexisting works, whichmotivated these research endeavors, have been discussed in Section Expansion on Previous WorksIn Chapter 3, a new model for mechanochemical synthesis is presented. Itis found that this model is simple to use and easy to understand, which was theoriginal goal. In Chapter 4, the feasibility and flexibility of Fe2GeS4 nanoparticlemechanochemical synthesis were confirmed. Various reaction boundaries weretested (milling parameters discussed in Chapter 4); however, more research isnecessary to discern the full depth of options available while still achieving asuccessful synthesis. Chapter 5 expands on the successes achieved by Orefuwaet al. [50]. Their work presents a 6 mV open circuit potential and 0.3 mA·cm−2short circuit current density. In Section 5.4, a photoresponse is observed with an89open circuit potential many orders above that (between 1.5 and 2 V) and the dioderesponse is sharper. However, the short circuit current is still small. A special noteof significant accomplishments at this point go to the doping of a nanoparticlelayer for the n-type layer. Not only is this a first time for this compound, but amethod which may prove useful for other future solar cell designs.6.2 Possible LimitationsIn order to improve the model for use from its current generality, applicationto more specific experiments would make for a good opportunity to check theoptimizing potential of such a model. Also, as it is designed for empirical fitting,it may take significant effort to produce a working, first principles example usingthe model. As for the experimental work, the synthesis could be matched withmore chemical composition characterizations. This would provide more data forthe model, as well as confirm various issues reported in preexisting works. It ispossible that the Fe2GeS4 nanoparticles are heavily oxygen contaminated. Suchimpurities can help explain the small short circuit current observed in the finalprototype.Other issues, which could be plaguing the cells performance, mostly centeron the thin film layer and the boundaries between layers. As can be seen in thevalues reported with Graphs 5.2 to 5.6, all of the shunt resistances are above 103Ω ·cm and none of the series resistances are below 10Ω · cm2. At shunt resistancevalues below 103Ω · cm, significant power loss becomes observable as the photongenerated current is lost to the causes of the low resistance. Likely causes arematerial discontinuities. This can even be seen to drag down the open circuitpotential as seen in graphs 5.2, 5.3 and 5.5 in comparison to Graphs 5.4 and 5.6where the shunt resistance is higher. However, the real issue seems to be thehigh series resistance, as seen in all Graphs 5.2 to 5.6. A frequent issue causingthis high value is often poor contact with the electrodes. This leads to a poorextraction of the light generated carriers from the inner layers, which ultimatelyhas reduced the short circuit current to almost nothing in these devices. Another90issue could be the extremely thick devices, which also lowers the probability ofcarrier collection by the electrodes. In Section 4.3 of their site info, Honsberg andBowden [51] have some nice explanations of these issues along with interactivegraphs to better grasp the effects of series and shunt resistance along with theirimpact on solar cell efficiency.Many of the reasons for these resistance issues comes not only from the ma-terials chosen, and how well they integrate with one another, but also the processof building a prototype. Without any high-temperature sintering, the films suf-fer from discontinuities caused by partially amorphous nanoparticles and poorconnections between the particles. Discontinuities also exist from a lack of anysurface passivation, which could also alleviate some of the oxygen contamina-tion [32]. Finally, the film thickness and doping levels are far from optimized. Ithas been theoretically found, by Ravichandran [31], that for an optimal drift-cellconfiguration, something around a 750-nanometer thickness would be optimal.As the prototypes were more than 10 fold this thickness, collection probabilityfor carriers was greatly decreased. Finally, the back electrode connection needsimprovement as was demonstrated in Section Future ImplicationsWith a working demonstration for a new theoretical mechanochemical synthe-sis model completed, options open up for expansion both within mechanochem-istry and other fields. The final energy dependent terms could be tailored to anexperiment by replacing the time with whatever dependence is necessary to ac-curately model size reduction. The amount of particle breaking could also besimilarly tailored. To name a few options, current models pick up some fracturelevel (but remaining the same size), or changing size (decreasing if breaking andincreasing if coalescing) by a matrix path [52], or neural network paths [53], orsome Monte Carlo generated random option picks from a weighted list. Sinceall of these and many other models are already in place in the market, and thismodel has been generalized, it has potential application in various fields which91study many types of syntheses. Not only do mechanochemical applications bene-fit from being able to use this model, but it is possible to apply theories for granularinteractions and extend them to fluids based on concepts concerning viscosity. Inshort, energy input (in this case mechanical) can be paired with chemical dynam-ics using logic functions to model a system more completely.As for the possibilities open to further work with Fe2GeS4 as a solar cell ab-sorber, enough promise has been shown to rationalize investing more into investi-gating the use of this compound. Not only has surface passivation been shown tobe effective [32], as well as hydrogen sulfide gas environment heat treatments toremove oxygen and improve nanoparticle crystallization [33], but work presentedin Section 5.4 shows the possibility of doping the nanoparticles. The inability todo so successfully had previously prevented successful drift-cell configurations[31]. Improvements to the mechanochemical synthesis can likely benefit frommilling optimization. That includes using a solid diluent [14] to reduce reactionspeed, as would be predicted by the model in Chapter 3, the reaction can be slowedby hindering collisions between reactants. This is one way to improve final par-ticle size and uniformity. With investments into good nanoparticle production,better results should be realizable for thin film deposition. Many methods existfor depositing nanoparticle. Proper optimization starts with stabilizing a colloidsuspension. Though various means were explored for doing so, ultimately it wasdecided to move forward with a prototype without optimizing a film in order tovalidate further research into using Fe2GeS4 nanoparticles for a solar cell absorberlayer. With the positive results, it would now be appropriate to extend the workdone in Chapter 5. As suggested in Section 5.2, doctor blading may be a niceoption for this compound. Such an option keeps with the theme of green manu-facturing (low waste) and easy scalability. 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ISSN 2168-9806.doi:10.4172/2168-9806.1000106. → pages 9199Appendix ASupporting MaterialsA.1 CodeThis code generates a column printing of all the desired parameters for gen-erating a graph from a data file using fourth order Runge-Kutta for numeric ap-proximation. Parameters for changing the output match the parameters listed inChapter 3. The differential equations in the ”Define the ODE” section match thefinal set of differential equations from Section 3.4.#include <s t d i o . h>#include <math . h>#include <s t d l i b . h>/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ //∗ Parameters you may want to change ∗ //∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /#define N 11 /∗ Number o f F i r s t Order Equat ions ∗ /#define DELTA T 0.001 /∗ Stepsize i n t ∗ /#define INITIAL Y3 0 /∗ I n i t i a l I r on Su l f i de ∗ /#define INITIAL Y4 0 /∗ I n i t i a l Germanium Su l f i de ∗ /#define INITIAL Y5 0 /∗ I n i t i a l Compound ∗ /#define INITIAL Y6 1∗1.0/pow(10 .0 ,6 ) /∗ I n i t i a l Pa t i c l e Radius ∗ /#define INITIAL Y7 1∗1.0/pow(10 .0 ,6 ) /∗ I n i t i a l Pa t i c l e Radius ∗ /100#define INITIAL Y8 1∗1.0/pow(10 .0 ,6 ) /∗ I n i t i a l Pa t i c l e Radius ∗ /#define INITIAL Y9 1∗1.0/pow(10 .0 ,6 ) /∗ I n i t i a l Pa t i c l e Radius ∗ /#define INITIAL Y10 1∗1.0/pow(10 .0 ,6 ) /∗ I n i t i a l Pa t i c l e Radius ∗ /#define MFe 55.8450 /∗ Fe g / mol ∗ /#define MS 32.0650 /∗ S g / mol ∗ /#define MGe 72.6400 /∗ Ge g / mol ∗ /#define MFeS 87.9100 /∗ FeS g / mol ∗ /#define MGeS2 136.7700 /∗ GeS2 g / mol ∗ /#define MFe2GeS4 312.5900 /∗ Fe2GeS4 g / mol ∗ /#define PFe2GeS4 5.0 /∗ Produced Fe2GeS4 g ∗ /#define KFeS 10.0 /∗ FeS Rate Lmol−1s−1 ∗ /#define KGeS2 10.0 /∗ GeS2 Rate L2mol−2s−1 ∗ /#define KFe2GeS4 10.0 /∗ Fe2GeS4 ra te L2mol−2s−1 ∗ /#define Vol 0.050 /∗ Jar Volume L ∗ /#define RFecr i t 200∗1.0/pow(10 .0 ,9 ) /∗ C r i t i c a l Radius ∗ /#define RGecri t 200∗1.0/pow(10 .0 ,9 ) /∗ C r i t i c a l Radius ∗ /#define RScr i t 200∗1.0/pow(10 .0 ,9 ) /∗ C r i t i c a l Radius ∗ /#define RFeScr i t 200∗1.0/pow(10 .0 ,9 ) /∗ C r i t i c a l Radius ∗ /#define RGeS2crit 200∗1.0/pow(10 .0 ,9 ) /∗ C r i t i c a l Radius ∗ /#define WFe 1.0∗pow(10 .0 ,12 ) /∗ Arctangent Slope ∗ /#define WGe 1.0∗pow(10 .0 ,12 ) /∗ Arctangent Slope ∗ /#define WS 1.0∗pow(10 .0 ,12 ) /∗ Arctangent Slope ∗ /#define WFeS 1.0∗pow(10 .0 ,10 ) /∗ Arctangent Slope ∗ /#define WGeS2 1.0∗pow(10 .0 ,10 ) /∗ Arctangent Slope ∗ /#define BFe −log ( 0 . 7 0 ) /∗ Break Frac t i on ∗ /#define BGe −log ( 0 . 4 0 ) /∗ Break Frac t i on ∗ /#define BS −l og ( 0 . 8 5 ) /∗ Break Frac t i on ∗ /#define BFeS −l og ( 0 . 3 0 ) /∗ Break Frac t i on ∗ /#define BGeS2 −log ( 0 . 3 5 ) /∗ Break Frac t i on ∗ /#define EFe log ( . 2 ) / (6.0∗3600∗ log ( 0 . 7 0 ) ) /∗ E Breaking Coe f f i c i e n t ∗ /#define EGe log ( . 2 ) / (9.0∗3600∗ log ( 0 . 4 0 ) ) /∗ E Breaking Coe f f i c i e n t ∗ /#define ES log ( . 2 ) / (3.0∗3600∗ l og ( 0 . 8 5 ) ) /∗ E Breaking Coe f f i c i e n t ∗ /#define EFeS log ( . 2 ) /(12.0∗3600∗ log ( 0 . 3 0 ) ) /∗ E Breaking Coe f f i c i e n t ∗ /#define EGeS2 log ( . 2 ) /(11.0∗3600∗ l og ( 0 . 3 5 ) ) /∗ E Breaking Coe f f i c i e n t ∗ /#define nFe 10 /∗ Sine Funct ion Power ∗ /#define nGe 10 /∗ Sine Funct ion Power ∗ /#define nS 10 /∗ Sine Funct ion Power ∗ /101#define nFeS 10 /∗ Sine Funct ion Power ∗ /#define nGeS2 10 /∗ Sine Funct ion Power ∗ //∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ //∗ Double f a c t o r i a l f r a c t i o n n ! ! / ( n−1) ! ! ∗ //∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /double norm (double z ){double i , num, den ;num=1.0 ;for ( i =z ; i >=1; i −=2){num ∗= i ;}den =1.0 ;for ( i =z−1; i >=1; i −=2){den ∗= i ;}return num/ den ;}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ //∗ Define the ODE ∗ //∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /double f (double x , double y [ ] , i n t i ) {double RFedep , RGedep , RSdep , RFeSdep , RGeS2dep , RFeScoef , RGeS2coef ,RFe2GeS4coef ;RFedep = 0.5+ atan (WFe∗( RFecr i t−y [ 6 ] ) ) / M PI ;RGedep = 0.5+ atan (WGe∗( RGecrit−y [ 7 ] ) ) / M PI ;RSdep = 0.5+ atan (WS∗( RScr i t−y [ 8 ] ) ) / M PI ;RFeSdep = 0.5+ atan (WFeS∗( RFeScrit−y [ 9 ] ) ) / M PI ;RGeS2dep = 0.5+ atan (WGeS2∗( RGeS2crit−y [ 1 0 ] ) ) / M PI ;RFeScoef = RFedep∗RSdep ;RGeS2coef = RGedep∗RSdep ;RFe2GeS4coef = RFeSdep∗RGeS2dep ;i f ( i ==0) return102−RFeScoef∗KFeS∗y [ 0 ]∗ y [ 2 ] ; /∗ dFe / d t ∗ /i f ( i ==1) return−RGeS2coef∗KGeS2∗y [ 1 ]∗ y [ 2 ]∗ y [ 2 ] ; /∗ dGe / d t ∗ /i f ( i ==2) return−RFeScoef∗KFeS∗y [ 0 ]∗ y[2]−2∗RGeS2coef∗KGeS2∗y [ 1 ]∗ y [ 2 ]∗ y [ 2 ] ; /∗ dS / d t ∗ /i f ( i ==3) returnRFeScoef∗KFeS∗y [ 0 ]∗ y[2]−2∗RFe2GeS4coef∗KFe2GeS4∗y [ 3 ]∗ y [ 3 ]∗ y [ 4 ] ; /∗ dFeS / d t ∗ /i f ( i ==4) returnRGeS2coef∗KGeS2∗y [ 1 ]∗ y [ 2 ]∗ y [2]−RFe2GeS4coef∗KFe2GeS4∗y [ 3 ]∗ y [ 3 ]∗ y [ 4 ] ; /∗ dGeS2 / d t∗ /i f ( i ==5) returnRFe2GeS4coef∗KFe2GeS4∗y [ 3 ]∗ y [ 3 ]∗ y [ 4 ] ; /∗ dFe2GeS4 / d t ∗ /i f ( i ==6) return−y [ 6 ]∗ ( norm ( nFe ) )∗EFe∗(BFe)∗pow( s in ( M PI∗x∗EFe) ,nFe ) ; /∗ dRFe / d t ∗ /i f ( i ==7) return−y [ 7 ]∗ ( norm (nGe) )∗EGe∗(BGe)∗pow( s in ( M PI∗x∗EGe) ,nGe) ; /∗ dRGe/ d t ∗ /i f ( i ==8) return−y [ 8 ]∗ ( norm (nS) )∗ES∗(BS)∗pow( s in ( M PI∗x∗ES) ,nS) ; /∗ dRS/ d t ∗ /i f ( i ==9) return−y [ 9 ]∗ ( norm (nFeS) )∗EFeS∗(BFeS)∗pow( s in ( M PI∗x∗EFeS) ,nFeS) ; /∗ dRFeS / d t ∗ /i f ( i ==10) return−y [ 1 0 ]∗ ( norm (nGeS2) )∗EGeS2∗(BGeS2)∗pow( s in ( M PI∗x∗EGeS2) ,nGeS2) ; /∗ dRGeS3/ d t ∗ /else return 0;103}/∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ //∗ Main Funct ion ∗ //∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ /i n t main ( void ){double t , y [N] , Produced ;i n t count ;void runge4 (double x , double y [ ] , double step ) ; /∗ Runge−Kut ta f unc t i on ∗ /Produced=PFe2GeS4 /MFe2GeS4;y [0]=2∗Produced / Vol ; /∗ i n i t i a l Y [ 0 ] ∗ /y [ 1 ] = Produced / Vol ; /∗ i n i t i a l Y [ 1 ] ∗ /y [2]=4∗Produced / Vol ; /∗ i n i t i a l Y [ 2 ] ∗ /y [ 3 ] = INITIAL Y3 ; /∗ i n i t i a l Y [ 3 ] ∗ /y [ 4 ] = INITIAL Y4 ; /∗ i n i t i a l Y [ 4 ] ∗ /y [ 5 ] = INITIAL Y5 ; /∗ i n i t i a l Y [ 5 ] ∗ /y [ 6 ] = INITIAL Y6 ; /∗ i n i t i a l Y [ 6 ] ∗ /y [ 7 ] = INITIAL Y6 ; /∗ i n i t i a l Y [ 7 ] ∗ /y [ 8 ] = INITIAL Y6 ; /∗ i n i t i a l Y [ 8 ] ∗ /y [ 9 ] = INITIAL Y6 ; /∗ i n i t i a l Y [ 9 ] ∗ /y [10 ]= INITIAL Y6 ; /∗ i n i t i a l Y[ 10 ] ∗ /count=−1;p r i n t f ( ” Time\ t I r o n \tGermanium\ t S u l f u r \ t I r o n S u l f i d e\tGermanium S u l f i d e\tFe2GeS4\tS / Fe\ tS /Ge\tRFe\tRGe\tRS\tRFeS\tRGeS2\tRFedep\tRGedep\tRSdep\tRFeSdep\tRGeS2dep\n ” ) ;/∗ column labe l s ∗ /p r i n t f ( ”%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \n ” ,0 .0 , y [ 0 ]∗MFe∗Vol / PFe2GeS4,y [ 1 ]∗MGe∗Vol / PFe2GeS4, y [ 2 ]∗MS∗Vol / PFe2GeS4, y [ 3 ]∗MFeS∗Vol / PFe2GeS4,y [ 4 ]∗MGeS2∗Vol / PFe2GeS4, y [ 5 ]∗MFe2GeS4∗Vol / PFe2GeS4,( y [ 2 ] + y [3]+2∗ y [4]+4∗ y [ 5 ] ) / ( y [ 0 ] + y [3]+2∗ y [ 5 ] ) , ( y [ 2 ] + y [3]+2∗ y [4]+4∗ y [ 5 ] ) / ( y [ 1 ] + y[ 4 ] + y [ 5 ] ) ,y [ 6 ]∗pow(10 .0 ,9 ) , y [ 7 ]∗pow(10 .0 ,9 ) , y [ 8 ]∗pow(10 .0 ,9 ) , y [ 9 ]∗pow(10 .0 ,9 ) , y [10 ]∗pow(10 .0 ,9 ) ,1040.5+ atan (WFe∗( RFecr i t−y [ 6 ] ) ) / M PI ,0 .5+ atan (WGe∗( RGecrit−y [ 7 ] ) ) / M PI ,0 .5+ atan (WS∗( RScr i t−y [ 8 ] ) ) / M PI ,0.5+ atan (WFeS∗( RFeScrit−y [ 9 ] ) ) / M PI ,0 .5+ atan (WGeS2∗( RGeS2crit−y [ 1 0 ] ) ) / M PI ) ;/∗∗∗∗∗∗∗∗∗∗∗∗∗ //∗ Time Loop ∗ //∗∗∗∗∗∗∗∗∗∗∗∗∗ /for ( t =0; t<54000; t +=DELTA T) {i f ( y [0]<0){y [ 0 ] = 0 ;}i f ( y [1]<0){y [ 1 ] = 0 ;}i f ( y [2]<0){y [ 2 ] = 0 ;}i f ( y [3]<0){y [ 3 ] = 0 ;}i f ( y [4]<0){y [ 4 ] = 0 ;}i f ( y [5]<0){y [ 5 ] = 0 ;}i f ( y [6]<0){y [ 6 ] = 0 ;}i f ( y [7]<0){y [ 7 ] = 0 ;}i f ( y [8]<0){y [ 8 ] = 0 ;}105i f ( y [9]<0){y [ 9 ] = 0 ;}i f ( y [10]<0){y [1 0 ]= 0 ;}runge4 ( t , y , DELTA T) ;count ++;i f ( count ==10000){p r i n t f ( ”%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \ t%l f \n ” , t /3600 , y [ 0 ]∗MFe∗Vol / PFe2GeS4,y [ 1 ]∗MGe∗Vol / PFe2GeS4, y [ 2 ]∗MS∗Vol / PFe2GeS4, y [ 3 ]∗MFeS∗Vol /PFe2GeS4 ,y [ 4 ]∗MGeS2∗Vol / PFe2GeS4, y [ 5 ]∗MFe2GeS4∗Vol / PFe2GeS4,( y [ 2 ] + y [3]+2∗ y [4]+4∗ y [ 5 ] ) / ( y [ 0 ] + y [3]+2∗ y [ 5 ] ) , ( y [ 2 ] + y [3]+2∗ y[4]+4∗ y [ 5 ] ) / ( y [ 1 ] + y [ 4 ] + y [ 5 ] ) ,y [ 6 ]∗pow(10 .0 ,9 ) , y [ 7 ]∗pow(10 .0 ,9 ) , y [ 8 ]∗pow(10 .0 ,9 ) , y [ 9 ]∗pow(10 .0 ,9 ) , y [10 ]∗pow(10 .0 ,9 ) ,0.5+ atan (WFe∗( RFecr i t−y [ 6 ] ) ) / M PI ,0 .5+ atan (WGe∗( RGecrit−y [ 7 ] ) ) /M PI ,0 .5+ atan (WS∗( RScr i t−y [ 8 ] ) ) / M PI ,0.5+ atan (WFeS∗( RFeScrit−y [ 9 ] ) ) / M PI ,0 .5+ atan (WGeS2∗( RGeS2crit−y[ 1 0 ] ) ) / M PI ) ;count =0;}}return 0;}/∗∗∗∗∗∗∗∗∗∗∗∗ //∗ RK4 Loop ∗ //∗∗∗∗∗∗∗∗∗∗∗∗ /void runge4 (double x , double y [ ] , double step ) {double h=step / 2 . 0 ; /∗ the midpoin t ∗ /double t1 [N] , t2 [N] , t3 [N ] ; /∗ temporary storage ar rays ∗ /double k1 [N] , k2 [N] , k3 [N] , k4 [N ] ; /∗ f o r Runge−Kut ta ∗ /i n t i ;for ( i =0; i<N; i ++) t1 [ i ]= y [ i ]+0 .5∗ ( k1 [ i ]= step∗ f ( x , y , i ) ) ;106for ( i =0; i<N; i ++) t2 [ i ]= y [ i ]+0 .5∗ ( k2 [ i ]= step∗ f ( x+h , t1 , i ) ) ;for ( i =0; i<N; i ++) t3 [ i ]= y [ i ]+ ( k3 [ i ]= step∗ f ( x+h , t2 , i ) ) ;for ( i =0; i<N; i ++) k4 [ i ]= step∗ f ( x+step , t3 , i ) ;for ( i =0; i<N; i ++) y [ i ]+=( k1 [ i ]+2∗k2 [ i ]+2∗k3 [ i ]+ k4 [ i ] ) / 6 . 0 ;return ;}107


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