Citation |

- Permanent Link:
- http://digital.auraria.edu/AA00003307/00001
## Material Information- Title:
- Scalable dielectrophoresis of single walled carbon nanotubes
- Creator:
- Fitzhugh, William A. (
*author*) - Language:
- English
- Physical Description:
- 1 electronic file (67 pages) : ;
## Subjects- Subjects / Keywords:
- Carbon nanotubes ( lcsh )
Dielectrophoresis ( lcsh ) Carbon nanotubes ( fast ) Dielectrophoresis ( fast ) - Genre:
- bibliography ( marcgt )
theses ( marcgt ) non-fiction ( marcgt )
## Notes- Review:
- Single Walled Carbon Nanotubes (SWNTs) have attracted much attention as a candidate material for future nano-scale 'beyond silicon' devices. However industrial scale operations have been impeded by difficulties in separating the metallic and semiconducting species. This paper addresses the use of highly inhomogeneous alternating electric fields, dielectrophoresis, to isolate SWNT species in scaled systems. Both numerical and experimental methods will be discussed.
- Thesis:
- Thesis (M.I.S) University of Colorado Denver
- Bibliography:
- Includes bibliographic references,
- System Details:
- System requirements: Adobe Reader.
- General Note:
- Integrated Sciences Program
- Statement of Responsibility:
- by William A. Fitzhugh.
## Record Information- Source Institution:
- |University of Colorado Denver
- Holding Location:
- Auraria Library
- Rights Management:
- All applicable rights reserved by the source institution and holding location.
- Resource Identifier:
- 925377239 ( OCLC )
ocn925377239 - Classification:
- LD1193.L584 2015m F58 ( lcc )
## Auraria Membership |

Downloads |

## This item has the following downloads: |

Full Text |

SCALABLE DIELECTROPHORESIS OF SINGLE WALLED CARBON
NANOTUBES by WILLIAM A. FITZHUGH B.S., University of Colorado Denver, 2012 A thesis submitted to the Faculty of the Graduate School of the University of Colorado Denver in partial fulfillment of the requirements for the degree of Master of Integrated Sciences Integrated Sciences 2015 This thesis for the Master of Integrated Sciences degree by William A. Fitzhugh has been approved for the Integrated Sciences Program by Masoud Asadi-Zeydabadi, Chair Jan Mandel Randall Tagg July 24, 2015 n Fitzhugh, William A. (M.I.S.) Scalable Dielectrophoresis of Single Walled Carbon Nanotubes Thesis directed by Associate Professor Masoud Asadi-Zeydabadi ABSTRACT Single Walled Carbon Nanotubes (SWNTs) have attracted much attention as a candidate material for future nano-scale beyond silicon devices. However indus- trial scale operations have been impeded by difficulties in separating the metallic and semiconducting species. This paper addresses the use of highly inhomogeneous alter- nating electric fields, dielectrophoresis, to isolate SWNT species in scaled systems. Both numerical and experimental methods will be discussed. The form and content of this abstract are approved. I recommend its publication. Approved: Masoud Asadi-Zeydabadi DEDICATION This thesis is dedicated to my parents, for their never ending support. ACKNOWLEDGMENT This thesis would not have been possible without the generous support of Dr. Asadi, Dr. Tagg, Dr. Geyer, Dr. Golkowski, and Ryan Jacobs. Special thanks to Dr. Maron, Dept, of Chemisty, Dr. Mandel and Dr. Langou, Dept, of Applied Mathematics, and Dr. Huber and Kris Bunker, Dept, of Physics. TABLE OF CONTENTS Tables..................................................................... viii Figures ..................................................................... ix Chapter 1. Introduction............................................................... 1 1.1 Graphene............................................................ 1 1.2 Single Walled Carbon Nanotubes...................................... 4 1.3 Nanotube Absorption Spectroscopy ................................... 6 1.4 Dielectrophoresis................................................... 7 1.5 SWNT Dielectrophoresis ............................................. 8 1.6 Rectangular Waveguides and Resonators ............................. 10 1.6.1 Transverse Electric.......................................... 11 1.6.2 Transverse Magnetic.......................................... 11 1.6.3 Resonators................................................... 12 2. Theory.................................................................... 13 2.1 Accounting for Drag................................................ 13 2.2 Steady State Solution.............................................. 14 2.3 Boundary and Initial Conditions.................................... 15 2.4 Temporal Evolution................................................. 16 2.5 Slowest Decaying Eigencouple ...................................... 18 3. Numerical Methods......................................................... 20 3.1 Finite Differences ................................................ 20 3.1.1 Reshaping Tensors............................................ 20 3.1.2 Laplacian Matrix............................................. 21 3.1.3 Derivative Matrix ........................................... 21 3.2 Boundary Extrapolation ............................................ 22 3.2.1 Boundary Conditions.......................................... 24 vi 3.3 Forming A......................................................... 24 4. Numerical Results........................................................ 26 4.1 Mesh Size and Stabilty............................................ 26 4.2 T and $ in SI Units............................................... 27 4.3 Error in a Known Case............................................. 27 4.4 Parallel Plate Dielectric Waveguide .............................. 30 4.4.1 Boundary Conditions......................................... 32 4.4.2 {cp} Approximations......................................... 32 4.4.3 Forcing Exponential Decay .................................. 32 4.5 Conclusion........................................................ 34 5. Laboratory Methods....................................................... 37 5.1 Dispersion........................................................ 37 5.2 Experimental Setup................................................ 37 5.2.1 Field Intensity Distributions............................... 40 5.3 Secondary Suspension.............................................. 40 6. Experimental Results..................................................... 42 6.1 Pre and Post DEP.................................................. 42 6.2 Analysis ......................................................... 43 6.3 Conclusion........................................................ 43 6.3.1 Future Work................................................. 44 References.................................................................. 47 Appendix A. DEP Finite Difference Code............................................... 49 A.l Two Dimensions.................................................... 49 A.2 Three Dimensions.................................................. 53 vii TABLES Table 6.1 Tight binding predicted photonic energies and wavelengths for nanotube Van Hove transitions.................................................... 43 vm FIGURES Figure 1.1 Honeycomb lattice of graphene. Image from [If].......................... 2 1.2 Reciprocal Lattice of Graphene. Image from [15]......................... 3 1.3 Comparison of the tight binding approximation with ab initio calculations for the brand structure of graphene. Image from [15].................. 3 1.4 Demonstration of the graphene rolling process for a (5, 3) nanotube. Image from [4].................................................................. 4 1.5 Density of states for metallic and semiconducting SWNTs. Van Houe transitions are illustrated by arrows. Image from [8]................. 6 1.6 Absorption spectra of a thin film of SWNTs. Image from [11]........... 7 4.1 Approximated and known eigenualues for the scaled Laplacian operator. . 28 4.2 Log-log plot of minimum real eigenualue component us the maximum field intensity................................................................ 33 4.3 Log-log plot of minimum real eigenualue component us the maximum field intensity................................................................ 34 4.4 Steady state concentration profiles for (a) 6.5 mm and (b) 3.5 mm dielec- tric parallel plate umueguides stimulated at 2-45 [GHz]................ 35 5.1 Arrangement of needles in fused quartz capillaries (4mm interior diameter on the left and 8mm diameter on the right.)............................ 38 5.2 4mm capillary with gold platted needles filled with SWNT SDS solution. 39 5.3 8mm capillary with gold platted needles filled with SWNT SDS solution. 40 5.4 Time auerage intensity field distributions (Avgxy) for the (a) 8mm and (b) 4mm quartz tubing.................................................... 41 6.1 Bottom needle from 4mm capillary after four fiue minute exposures to rnicrowaues.............................................................. 44 IX 6.2 Extinction spectra for SWNT SDS solution (a) before and (b) after DEP exposure 45 x 1. Introduction Consisting of a single cylindrical layer of graphene, Single Walled Carbon Nan- otubes (SWNTs) were first discovered in the early 1990s at the NEC Fundamental Research Laboratory in Tsukuba, Japan, by Sumio Iijima [9]. Since that time SWNTs have been the subject of much attention as a candidate material for future nano-scaled engineering. Many cutting edge applications have been theorized to take advantage of their remarkable physical and electrical properties. However, SWNTs exist in both semiconducting and metallic species, and all currently known manufacturing proce- dures result in a heterogeneous mixture of both types [18]. Given that the majority of SWNT based device designs require a specific electronic type, and that separation remains a highly expensive process, commercialization of these remarkable materials has been slow. 1.1 Graphene Graphene consists of a planar array of sp2 hybridized carbon atoms. This lattice structure cannot be represented by a Bravais lattice, which would require every carbon to be located at a vector, originating at any other carbon, given by: R = mdi + nda (1.1) For some lattice vectors dj, 02 and integers m,n. Instead, the graphene lattice must be represented as a lattice decorated with a basis. The unit cells of graphene are given by the Hexagonal lattice, which in two dimensional Cartesian coordinates has the following lattice vectors (for some lattice constant a). di = a( 1 /2, v/3/2) (1.2) a2 = a(- l/2,V3/2) (1.3) 1 As depicted in Figure 1.1, the solid dots correspond to the locations of the vectors R = md[ + nd2 for the lattice vectors above. Figure 1.1: Honeycomb lattice of graphene. Image from [Iff. Two carbon atoms are then added to each unit cell, creating the hexagonal shape in figure 1.1. The location of each carbon atom relative to the associated lattice point is given by the vectors [14]: * = a(0iwi) (1.4) (1.5) The final result is known as the honeycomb lattice. Using the identity bj 2nSij, the reciprocal lattice vectors, b\ and fo, can be found to be: S^fa.-L) (i.6) Â£ = -(-1.4) (1-7) a v3 Figure 1.2 presents a graph of the reciprocal lattice of graphene in this basis. The shaded area represents the first Brillouin zone and the K, K', T, and M points designate particular discrete points of important symmetry. 2 Figure 1.2: Reciprocal Lattice of Graphene. Image from [15]. The band structure of graphene is well approximated using a tight binding model [4]. In this approach, the electron wave function is calculated by the superposition of a core and nearest neighbor atomic wave functions. The bandstructure along the reciprocal space triangle YKM is depicted in figure 1.3 for both ab initio and tight binding models. Figure 1.3: Comparison of the tight binding approximation with ab initio calculations for the brand structure of graphene. Image from [15]. 3 1.2 Single Walled Carbon Nanotubes Single walled carbon nanotubes (SWNTs) consist of graphene rolled into hollow cylinders. The structure of SWNTs is expressed via the chiral vector: Ch = ma[ + na2 (1.8) Where a[ and a2 are the lattice vectors of graphene and m and n are integers. The chiral vector is one such that it circumvents the nanotube and returns to its original location. Figure 1.4 shows this process for a (5, 3) nanotube. A and A! represent the same point on the (5, 3) SWNT surface. Figure 1.4: Demonstration of the graphene rolling process for a (5, 3) nanotube. Image from Iff. Blochs theorem, rip(r) = u(r)exp(ik r), tells us that the electron wavefunctions in graphene must be a plane wave modulated by a periodic function with a periodicity given by the graphene lattice (ie. d[ and a2). Given that Ch = md[ + nap. tfj(r + Ch) = u(r + Ch)exp{ik (r + Ch)) (1.9) 4 ip(r + Ch) = u(r)exp(ik (r + Ch)) ip(r + Ch) = t/j{r)exp(ik Ch) (1.10) (1.11) Additionally, the azimuthal continuity of a SWNT requires that: ip(r + Ch) = r
(1.12) |

Full Text |

PAGE 1 SCALABLEDIELECTROPHORESISOFSINGLEWALLEDCARBON NANOTUBES by WILLIAMA.FITZHUGH B.S.,UniversityofColoradoDenver,2012 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoDenverinpartialfulllment oftherequirementsforthedegreeof MasterofIntegratedSciences IntegratedSciences 2015 PAGE 2 ThisthesisfortheMasterofIntegratedSciencesdegreeby WilliamA.Fitzhugh hasbeenapprovedforthe IntegratedSciencesProgram by MasoudAsadi-Zeydabadi,Chair JanMandel RandallTagg July24,2015 ii PAGE 3 Fitzhugh,WilliamA.M.I.S. ScalableDielectrophoresisofSingleWalledCarbonNanotubes ThesisdirectedbyAssociateProfessorMasoudAsadi-Zeydabadi ABSTRACT SingleWalledCarbonNanotubesSWNTshaveattractedmuchattentionasa candidatematerialforfuturenano-scale'beyondsilicon'devices.Howeverindustrialscaleoperationshavebeenimpededbydicultiesinseparatingthemetallicand semiconductingspecies.Thispaperaddressestheuseofhighlyinhomogeneousalternatingelectricelds,dielectrophoresis,toisolateSWNTspeciesinscaledsystems. Bothnumericalandexperimentalmethodswillbediscussed. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:MasoudAsadi-Zeydabadi iii PAGE 4 DEDICATION Thisthesisisdedicatedtomyparents,fortheirneverendingsupport. PAGE 5 ACKNOWLEDGMENT ThisthesiswouldnothavebeenpossiblewithoutthegeneroussupportofDr.Asadi, Dr.Tagg,Dr.Geyer,Dr.Golkowski,andRyanJacobs.SpecialthankstoDr.Maron, Dept.ofChemisty,Dr.MandelandDr.Langou,Dept.ofAppliedMathematics, andDr.HuberandKrisBunker,Dept.ofPhysics. PAGE 6 TABLEOFCONTENTS Tables........................................viii Figures.......................................ix Chapter 1.Introduction...................................1 1.1Graphene.................................1 1.2SingleWalledCarbonNanotubes....................4 1.3NanotubeAbsorptionSpectroscopy..................6 1.4Dielectrophoresis.............................7 1.5SWNTDielectrophoresis........................8 1.6RectangularWaveguidesandResonators...............10 1.6.1TransverseElectric........................11 1.6.2TransverseMagnetic.......................11 1.6.3Resonators............................12 2.Theory......................................13 2.1AccountingforDrag...........................13 2.2SteadyStateSolution..........................14 2.3BoundaryandInitialConditions....................15 2.4TemporalEvolution...........................16 2.5SlowestDecayingEigencouple.....................18 3.NumericalMethods...............................20 3.1FiniteDierences............................20 3.1.1ReshapingTensors........................20 3.1.2LaplacianMatrix.........................21 3.1.3DerivativeMatrix........................21 3.2BoundaryExtrapolation........................22 3.2.1BoundaryConditions.......................24 vi PAGE 7 3.3FormingA................................24 4.NumericalResults................................26 4.1MeshSizeandStabilty.........................26 4.2)-326(andinSIUnits...........................27 4.3ErrorinaKnownCase.........................27 4.4ParallelPlateDielectricWaveguide..................30 4.4.1BoundaryConditions.......................32 4.4.2 f i g Approximations.......................32 4.4.3ForcingExponentialDecay...................32 4.5Conclusion................................34 5.LaboratoryMethods..............................37 5.1Dispersion................................37 5.2ExperimentalSetup...........................37 5.2.1FieldIntensityDistributions...................40 5.3SecondarySuspension..........................40 6.ExperimentalResults..............................42 6.1PreandPostDEP............................42 6.2Analysis.................................43 6.3Conclusion................................43 6.3.1FutureWork...........................44 References ......................................47 Appendix A.DEPFiniteDierenceCode..........................49 A.1TwoDimensions.............................49 A.2ThreeDimensions............................53 vii PAGE 8 TABLES Table 6.1 Tightbindingpredictedphotonicenergiesandwavelengthsfornanotube VanHovetransitions. ............................43 viii PAGE 9 FIGURES Figure 1.1 Honeycomblatticeofgraphene.Imagefrom[14]. .............2 1.2 ReciprocalLatticeofGraphene.Imagefrom[15]. .............3 1.3 Comparisonofthetightbindingapproximationwithabinitiocalculations forthebrandstructureofgraphene.Imagefrom[15]. ...........3 1.4 Demonstrationofthegraphenerollingprocessfora ; 3 nanotube.Image from[4]. ....................................4 1.5 DensityofstatesformetallicandsemiconductingSWNTs.VanHove transitionsareillustratedbyarrows.Imagefrom[8]. ...........6 1.6 AbsorptionspectraofathinlmofSWNTs.Imagefrom[11]. ......7 4.1 ApproximatedandknowneigenvaluesforthescaledLaplacianoperator. .28 4.2 Log-logplotofminimumrealeigenvaluecomponentvsthemaximumeld intensity. ...................................33 4.3 Log-logplotofminimumrealeigenvaluecomponentvsthemaximumeld intensity. ...................................34 4.4 Steadystateconcentrationprolesfora6.5mmandb3.5mmdielectricparallelplatewaveguidesstimulatedat2.45[GHz]. ..........35 5.1 Arrangementofneedlesinfusedquartzcapillariesmminteriordiameter ontheleftand8mmdiameterontheright. ................38 5.2 4mmcapillarywithgoldplattedneedleslledwithSWNT-SDSsolution. 39 5.3 8mmcapillarywithgoldplattedneedleslledwithSWNT-SDSsolution. 40 5.4 Timeaverageintensityelddistributions Avg xy forthea8mmand b4mmquartztubing. ............................41 6.1 Bottomneedlefrom4mmcapillaryafterfourveminuteexposuresto microwaves. ..................................44 ix PAGE 10 6.2 ExtinctionspectraforSWNT-SDSsolutionabeforeandbafterDEP exposure. ...................................45 x PAGE 11 1.Introduction Consistingofasinglecylindricallayerofgraphene,SingleWalledCarbonNanotubesSWNTswererstdiscoveredintheearly1990sattheNECFundamental ResearchLaboratoryinTsukuba,Japan,bySumioIijima[9].SincethattimeSWNTs havebeenthesubjectofmuchattentionasacandidatematerialforfuturenano-scaled engineering.Manycuttingedgeapplicationshavebeentheorizedtotakeadvantage oftheirremarkablephysicalandelectricalproperties.However,SWNTsexistinboth semiconductingandmetallicspecies,andallcurrentlyknownmanufacturingproceduresresultinaheterogeneousmixtureofbothtypes[18].Giventhatthemajority ofSWNTbaseddevicedesignsrequireaspecicelectronictype,andthatseparation remainsahighlyexpensiveprocess,commercializationoftheseremarkablematerials hasbeenslow. 1.1Graphene Grapheneconsistsofaplanararrayof sp 2 hybridizedcarbonatoms.Thislattice structurecannotberepresentedbyaBravaislattice,whichwouldrequireeverycarbon tobelocatedatavector,originatingatanyothercarbon,givenby: ~ R = m~a 1 + n~a 2 .1 Forsomelatticevectors ~a 1 ~a 2 andintegers m n .Instead,thegraphenelattice mustberepresentedasalatticedecoratedwithabasis.Theunitcellsofgraphene aregivenbytheHexagonallattice,whichintwodimensionalCartesiancoordinates hasthefollowinglatticevectorsforsomelatticeconstanta. ~a 1 = a = 2 ; p 3 = 2.2 ~a 2 = a )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 = 2 ; p 3 = 2.3 1 PAGE 12 AsdepictedinFigure1.1,thesoliddotscorrespondtothelocationsofthevectors ~ R = m~a 1 + n~a 2 forthelatticevectorsabove. Figure1.1: Honeycomblatticeofgraphene.Imagefrom[14]. Twocarbonatomsarethenaddedtoeachunitcell,creatingthehexagonalshape ingure1.1.Thelocationofeachcarbonatomrelativetotheassociatedlatticepoint isgivenbythevectors[14]: ~v 1 = a ; 1 2 p 3 .4 ~v 2 = a ; )]TJ/F15 11.9552 Tf 18.402 8.087 Td [(1 2 p 3 .5 Thenalresultisknownasthehoneycomblattice.Usingtheidentity ~a i ~ b j = 2 ij ,thereciprocallatticevectors, ~ b 1 and ~ b 2 ,canbefoundtobe: ~ b 1 = 2 a ; 1 p 3 .6 ~ b 2 = 2 a )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1 p 3 .7 Figure1.2presentsagraphofthereciprocallatticeofgrapheneinthisbasis. TheshadedarearepresentstherstBrillouinzoneandthe K K 0 ,,and M points designateparticulardiscretepointsofimportantsymmetry. 2 PAGE 13 Figure1.2: ReciprocalLatticeofGraphene.Imagefrom[15]. Thebandstructureofgrapheneiswellapproximatedusingatightbindingmodel [4].Inthisapproach,theelectronwavefunctioniscalculatedbythesuperposition ofacoreandnearestneighboratomicwavefunctions.Thebandstructurealongthe reciprocalspacetriangle)]TJ/F19 11.9552 Tf 136.544 0 Td [(KM isdepictedingure1.3forbothabinitioandtight bindingmodels. Figure1.3: Comparisonofthetightbindingapproximationwithabinitiocalculations forthebrandstructureofgraphene.Imagefrom[15]. 3 PAGE 14 1.2SingleWalledCarbonNanotubes SinglewalledcarbonnanotubesSWNTsconsistofgraphenerolledintohollow cylinders.ThestructureofSWNTsisexpressedviathechiralvector: ~ C h = m~a 1 + n~a 2 .8 Where ~a 1 and ~a 2 arethelatticevectorsofgrapheneand m and n areintegers.The chiralvectorisonesuchthatitcircumventsthenanotubeandreturnstoitsoriginal location.Figure1.4showsthisprocessfora ; 3nanotube. A and A 0 representthe samepointonthe ; 3SWNTsurface. Figure1.4: Demonstrationofthegraphenerollingprocessfora ; 3 nanotube.Image from[4]. Bloch'stheorem, ~r = u ~r exp i ~ k ~r ,tellsusthattheelectronwavefunctions ingraphenemustbeaplanewavemodulatedbyaperiodicfunctionwithaperiodicity givenbythegraphenelatticeie. ~a 1 and ~a 2 .Giventhat ~ C h = m~a 1 + n~a 2 : ~r + ~ C h = u ~r + ~ C h exp i ~ k ~r + ~ C h .9 4 PAGE 15 ~r + ~ C h = u ~r exp i ~ k ~r + ~ C h .10 ~r + ~ C h = ~r exp i ~ k ~ C h .11 Additionally,theazimuthalcontinuityofaSWNTrequiresthat: ~r + ~ C h = ~r .12 Comparingequations1.11and1.12,weseethattheonlywavevectorsthatare allowableunderbothoftheseboundaryconditionsarethosesuchthat: exp i ~ k ~ C h =1.13 FortheSWNTtobehavemetallically,the K wavevectorsmustbeallowed.It hasbeenshown[14]that,inthebasisdescribedbyequations3.10and3.11: K = 1 3 ~ b 1 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 ~ b 2 .14 K = 4 3 a ; 0 ; 0.15 K ~ C h = 4 3 a ; 0 ; 0 [ ma = 2 ; p 3 = 2 ; 0+ na )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 = 2 ; p 3 = 2 ; 0].16 K ~ C h = 4 3 a ; 0 ; 0 a 2 m )]TJ/F19 11.9552 Tf 11.955 0 Td [(n; p 3 m + n ; 0.17 K ~ C h = 2 3 m )]TJ/F19 11.9552 Tf 11.955 0 Td [(n .18 Forequation1.13toholdfor K ,thenforsomeinteger l =0 ; 1 ; 2 ; 3 ;::: : K ~ C h = 2 3 m )]TJ/F19 11.9552 Tf 11.955 0 Td [(n =2 l .19 5 PAGE 16 m )]TJ/F19 11.9552 Tf 11.955 0 Td [(n =3 l .20 Accordingly,ifthechiralvector n;m issuchthat n and m dierbyaninteger multipleof3,thenthe K wavevectorswillbeallowedandthenanotubewillbehave metallically.Otherwise,theSWNTwillbehaveasasemiconductor. 1.3NanotubeAbsorptionSpectroscopy Absorptionspectroscopyisthequanticationoftherelativeamountofradiation absorbedbyaspecimenasafunctionoffrequency.Inthecaseofsinglewalledcarbon nanotubes,aswellasmostcrystallinesolids,thisbehaviorisdominatedbyVanHove singularities. ThedensityofstatesDOSisdenedtobethenumberofstates,orwavefunctions,thatexistwithinagivenenergyinterval.VanHovesingularitiesrefertosharp, non-smooth,peaksintheDOS.ThedierencesintheDOSformetallicandsemiconductingSWNTsarerepresentedingure1.5. Figure1.5: DensityofstatesformetallicandsemiconductingSWNTs.VanHove transitionsareillustratedbyarrows.Imagefrom[8]. 6 PAGE 17 EachoftheVanHovetransitionsillustratedabove M 11 S 11 S 22 S 33 correspondstoaparticularchangeinenergy.Assuch,eachtransitionisassociatedwitha radiationfrequencysuchthat E = h ,where h isPlank'sconstant.Giventhatmore electronsexistwithintheVanHovesingularities,theabsorptionofSWNTspeakat thosefrequencies.Figure1.6depictsexperimentalresultsofabsorptiondatafora thinlmofSWNTscoatedonquartz.Thepeakat650 nm correspondstometallicnanotubesandthepeaksat950 nm and1700 nm correspondtosemiconducting nanotubes. Figure1.6: AbsorptionspectraofathinlmofSWNTs.Imagefrom[11]. 1.4Dielectrophoresis Dielectrophoresisistheprocessesofexploitinginhomogeneouselectriceldsto driveseparationofparticlesinasolutionbasedonthevariablepolarizabilities.For theremainderofthispaper,wewillconsiderasetofparticlesdenotedwithsubscript p dispersedinaliquiddenotedwithsubscript l Takingthedielectricfunctiontobe p;l = p;l )]TJ/F19 11.9552 Tf 12.588 0 Td [(i p;l =! ,where istheelectric permittivity, istheconductivity,and istheappliedangularfrequency,theeective 7 PAGE 18 dipolemomentofadispersedsphericalparticlecanbeexpressedas[3]: ~p =4 r 3 l Re p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l 2 l + p ~ E .21 Givenanoscillatoryelectriceld,thisproducesatimeaverageDEPforce. F DEP =4 r 3 l Re p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l 2 l + p r E 2 .22 F DEP =4 r 3 l Re CMF r E 2 .23 Where CMF istheClausiusMossottifunction.Forellipsoidalparticles,asignicantlymorecomplicatedexpressionforthetimeaverageDEPforceemerges.However, thisfunctioncanbesimpliedundertheassumptionthatthe3axesoftheparticle a;b;c arerelatedby a>>b = c therod-likeparticleassumption. 1.5SWNTDielectrophoresis Singlewalledcarbonnanotubedimensionsjustifytherod-likeparticleassumption andtheresultingtimeaveragedforceis: F DEP = d 2 l 8 l Re p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l l + p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l L r E 2 .24 WheredistheSWNTdiameter,listhelength,andLisadepolarizationfactor thatisroughly10 )]TJ/F15 11.9552 Tf 7.084 -4.339 Td [(5[3].Thehighandlowfrequencylimitsofthisexpressionyield dominatingtermsofthedielectricfunction: lim ! 0 = p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l l .25 lim !1 = p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l l .26 FormetallicSWNTsbothoftheselimitsarepositiveas p > l and p > l resultinginapositiveDEPforce,thatisaforceinthedirectionofthehighest 8 PAGE 19 inhomogeneityoftheelectriceldintensity.ForsemiconductingSWNTs,however, theoppositeistrue, p < l and p < l ,resultinginanegativeDEPforce[3][12]. Experimentalresultsverifythreeofthesefourcases.[12]usedamicroelectrode arrayanda20voltpeak-to-peak 10[ V ]ACsignalovera10microngapatvarying frequenciestodeterminetheDEPforcesactingonSWNTsdispersedinsolutionusing variousconcentrationsofthesurfactantSDBS.InthecaseofmetallicSWNTs,their resultsconrmedthetheoreticallypredictedpositiveDEPforcesatallfrequencies.In thesemiconductingSWNTs,theyfoundthatthesurfactantinducedasmallsurface chargethatenabledpositiveDEPforcesatlowfrequency.However,athighfrequencythesemiconductingSWNT/SDBSmicellesrevertedbacktothetheoretically predictednegativeDEPforce. ThefrequencyatwhichsemiconductingSWNTstransitionfrompositivetonegativeDEPforcewastermedthecriticalfrequency.ForSDBSconcentrationsof0 : 1% byweight,thisfrequencywasfoundtoberoughly1[ MHz ].Insummary,alternating electriceldswithfrequencieslessthan1[ MHz ]causedpositiveDEPforcesinboth metallicandsemiconductingSWNTsandaccordinglyseparationdidnotoccur.As theappliedeldfrequencyincreasedabove1[ MHz ],themetallicSWNTscontinued toexperiencepositiveDEPforceswhilethesemiconductingSWNTstransitionedto negativeDEPforces. Themaximumelectriceldstrength,whichoccurreddirectlybetweenthemicroelectrodes,neededtocauseseparationwasdeterminedbyenergeticallycomparing theDEPforcewithBrownianmotion.ElectrohydrodynamicalforceswerenotconsideredastheSWNTsizewouldindicatethatBrownianmotionwouldbetheprimary competingforce[12]. Denotingthepotentialenergychangebetweenmaximumelectriceldandminimumeldatfardistances,theDEPescapepotentialwasfoundtobe: 9 PAGE 20 U DEP = )]TJ/F19 11.9552 Tf 10.494 8.087 Td [(d 2 l 8 l Re f CMF g E 2 max .27 Settingthisequaltothethermalenergyyieldsamaximumelectriceldof[12]: E max s 8 kT d 2 l l Re f CMF g 10 6 [ V=m ].28 1.6RectangularWaveguidesandResonators Herewewillderivetheeigenmodesforrectangularwaveguidesand,byextension,rectangularresonators.Waveguidesaredesignedsuchthatpropagatingwaves destructivelyinterfereatboundarieswhichwouldotherwisebecomesourcesofattenuation.Maxwell'sequationsareusedtoderivethewaveequationsforbothelectric andmagneticelds. c 2 r 2 )]TJ/F19 11.9552 Tf 15.265 8.088 Td [(@ 2 @t 2 ~ E =0.29 c 2 r 2 )]TJ/F19 11.9552 Tf 15.264 8.088 Td [(@ 2 @t 2 ~ H =0.30 Formetallicwavegudies,theboundaryconditionsarethatofanyconductor:the tangentialcomponentoftheelectriceldandthenormalcomponentofthemagnetic eldmustbezero. E ~r;t ^ n =0.31 H ~r;t ^ n =0.32 Ifweassumethattheelectromagneticwaveistravelinginthe z direction,witha wavefrontmodulationfactortoensureproperboundaries ~ E ~r;t = E 0 ~ F x;y e i k z z )]TJ/F20 7.9701 Tf 6.587 0 Td [(!t and ~ H ~r;t = H 0 ~ G x;y e i k z z )]TJ/F20 7.9701 Tf 6.587 0 Td [(!t ,thensubsitutionintoequations1.29and1.30 yeilds: 10 PAGE 21 h @ 2 F x @x 2 + @ 2 F x @y 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k 2 z ; @ 2 F y @x 2 + @ 2 F y @y 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k 2 z ; 0 i = )]TJ/F19 11.9552 Tf 10.494 8.087 Td [(! 2 c 2 h F x ;F y ; 0 i .33 h @ 2 G x @x 2 + @ 2 G x @y 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k 2 z ; @ 2 G y @x 2 + @ 2 G y @y 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(k 2 z ; 0 i = )]TJ/F19 11.9552 Tf 10.494 8.088 Td [(! 2 c 2 h G x ;G y ; 0 i .34 Thesolutionstothissystem,forawaveguideofwidth a onthe x -axisandheight b onthe y -axis,canbeshowntobe[1][10]: 1.6.1TransverseElectric E x = iE 0 n b cos m a x sin n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .35 E y = )]TJ/F19 11.9552 Tf 9.298 0 Td [(iE 0 m a sin m a x cos n b y e i !t )]TJ/F20 7.9701 Tf 6.587 0 Td [(k z z .36 H x = H 0 m a sin m a x cos n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .37 H y = H 0 n b cos m a x sin n b y e i !t )]TJ/F20 7.9701 Tf 6.587 0 Td [(k z z .38 H z = H 0 k 2 z + 2 =c 2 k z cos m a x cos n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .39 Here m and n representthemodenumberinthe x and y directions,respectively. k z = q 2 c 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( m a 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( n b 2 istheeectivewavenumberinthe z direction. 1.6.2TransverseMagnetic E x = )]TJ/F19 11.9552 Tf 9.299 0 Td [(E 0 m a cos m a x sin n b y e i !t )]TJ/F20 7.9701 Tf 6.587 0 Td [(k z z .40 E y = )]TJ/F19 11.9552 Tf 9.299 0 Td [(E 0 n b sin m a x cos n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .41 11 PAGE 22 E z = E 0 k 2 z + 2 =c 2 k z sin m a x sin n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .42 H x = iH 0 n b sin m a x cos n b y e i !t )]TJ/F20 7.9701 Tf 6.587 0 Td [(k z z .43 H y = )]TJ/F19 11.9552 Tf 9.298 0 Td [(iH 0 m a cos m a x sin n b y e i !t )]TJ/F20 7.9701 Tf 6.586 0 Td [(k z z .44 1.6.3Resonators Resonatorsareconstructedbycappingeitherendofthewaveguidesuchthat reectionsbuidstandingwaves.TheTE/TMmodestructureispreserved,whilethe e i !t k z z termsbecome2 cos !t cos k z z 12 PAGE 23 2.Theory Aswasshowninsection1.5,anenergeticapproachtocalculatingtheelectriceld strengthnecessaryforsuccessfulsortingofSWNTspredictedthattheeldwouldneed topeakatavalueontheorderof10 6 [ V m ].ThiswasfoundbydeterminingaDEP escapepotentialthatwaslargerthanthethermalenergythatthesolutioncould provide.Clearly,thisisasucientconditionforDEPtooccur.However,inthecase ofhighdragsystems,thisconditionmaynotbenecessaryasnon-conservativeforces candominate. Auidicforcemodelofthesystemwouldprovidemoreinsightintotheeld requirements.DeningthehighfrequencytimeaverageDEPforce,Browniandiusive force,anddragforcetobe: F DEP = d 2 l 8 p )]TJ/F19 11.9552 Tf 11.956 0 Td [( l r E 2 .1 F dif = )]TJ/F15 11.9552 Tf 15.675 8.088 Td [(1 N A r .2 F drag = )]TJ/F15 11.9552 Tf 9.298 0 Td [(6 av .3 Where N A isAvogadro'snumber, isthemolarchemicalpotential, a isthe Stoke-Einsteinradius, istheviscosityofthemedium,and v isvelocity. 2.1AccountingforDrag Takingadvantageoftheknown -polarityofeachforce,thetimeaveragedform ofNewton'ssecondlawgives: d 2 l 8 p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l r E 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 av )]TJ/F15 11.9552 Tf 18.331 8.088 Td [(1 N A r = m a .4 Equation2.4canbefurthersimpliedbyassumingthattheparticlesquickly achieveaterminaldriftvelocity: 13 PAGE 24 d 2 l 8 p )]TJ/F19 11.9552 Tf 11.955 0 Td [( l )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 av t )]TJ/F15 11.9552 Tf 18.331 8.087 Td [(1 N A r =0.5 v t = [ d 2 l 8 p )]TJ/F19 11.9552 Tf 11.956 0 Td [( l r E 2 )]TJ/F17 7.9701 Tf 17.405 4.707 Td [(1 N A r ] 6 a .6 Finallyusingtherelationbetweenchemicalpotentialandmolefraction, r = RT x p r x p ,andmolefractionandmolarconcentrationprole, C p = x p C v t =[)]TJ/F22 11.9552 Tf 22.992 0 Td [(r E 2 )]TJ/F15 11.9552 Tf 15.492 8.087 Td [( C p r C p ].7 Where)-347(= d 2 l 48 a p )]TJ/F19 11.9552 Tf 12.285 0 Td [( l and= RT N A 6 a .Usingtherelationbetweendiusive ux,molarconcentrationprole,andvelocity, J = C p v t ,andthecontinuityequation, @C p @t + r J =0 @C p @t +)]TJ/F22 11.9552 Tf 19.076 0 Td [(r C p r E 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( r 2 C p =0.8 Thisequationisthequasi-linearheat,orthediusion-advection,equation.In thelimitthattheelectricelddisappears,thisreducestothelinearhomogenousheat equation,acommonmodelfornoninteractingdiusiveprocesses. 2.2SteadyStateSolution Thesteadystatesolutioncanbefoundbysetting @C p @t =0.Clearlythisleadsto theidentity: )]TJETq1 0 0 1 266.836 204.283 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 266.836 193.093 Td [( r [ C p r E 2 ]= r 2 C p .9 Usingtheproductruleonascalar-vectorproduct: r [ ~ A ]= r ~ A + r ~ A andtherelation rr = r 2 ,thisfurthersimpliesto: r C p r E 2 + C p r 2 E 2 = )]TJ/F22 11.9552 Tf 9.08 8.201 Td [(r 2 C p .10 14 PAGE 25 Guessingthesolution C p ~r = C 0 e )]TJETq1 0 0 1 317.913 714.45 cm[]0 d 0 J 0.359 w 0 0 m 5.203 0 l SQBT/F18 5.9776 Tf 317.913 709.332 Td [( E 2 ~r ,theconcentrationproleLaplacian becomes: r 2 C p = r )]TJETq1 0 0 1 269.048 639.093 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 269.048 627.904 Td [( C 0 e )]TJETq1 0 0 1 298.426 643.033 cm[]0 d 0 J 0.359 w 0 0 m 5.203 0 l SQBT/F18 5.9776 Tf 298.426 637.915 Td [( E 2 ~r r E 2 = )]TJETq1 0 0 1 367.773 639.093 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 367.773 627.904 Td [( r C p r E 2 .11 Clearly,asaresultofequation2.11,theguesssatisesequation2.9andaccordinglythesolutionholdsforanyadequatelycontinuouselectriceldintensity. C steadystate ~r = C 0 e )]TJETq1 0 0 1 368.337 540.637 cm[]0 d 0 J 0.359 w 0 0 m 5.203 0 l SQBT/F18 5.9776 Tf 368.337 535.519 Td [( E 2 ~r .12 2.3BoundaryandInitialConditions Foragivencontainmentchamberdescribedbythedomainwithrigidwalls, @ ,theboundaryconditionsareoftheNeumanntype.Specically,theboundaries inhibitanyconcentrationuxtopassthroughthem.Intermsofthenormalvector, ^ n ,thisisexpressedas: ^ n J ~r = C p ~r ^ n ~v t ~r =0:[ 8 ~r@ ].13 Substitutinginequation2.7expressesthisconditionasarequirementofthe concentrationprole. [ C p )]TJ/F22 11.9552 Tf 7.315 0 Td [(r E 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( r C p ] ^ n =0.14 )]TJ/F19 11.9552 Tf 7.314 0 Td [(C p @E 2 @ ^ n = @C p @ ^ n .15 Where @f @ ^ n ^ n r f .Consideringthesteadystatesolutiondeterminedinthelast section,itisclearthatthisboundaryconditionisinnatelysatised. r C steadystate = )]TJETq1 0 0 1 320.597 101.887 cm[]0 d 0 J 0.478 w 0 0 m 8.454 0 l SQBT/F15 11.9552 Tf 320.597 90.697 Td [( C steadystate r E 2 .16 15 PAGE 26 ! )]TJ/F19 11.9552 Tf 7.314 0 Td [(C steadystate @E 2 @ ^ n = @C steadystate @ ^ n .17 Anyphysicallyrealizableinitialconditioncouldbeusedwiththisapproach.However,fortheremainderofthispaper,onlyinitialconcentrationprolesoffullydispersedSWNT'swillbeconsidered. C p ~r;t =0= C i : 8 ~r .18 Lastly,conservationofmassgivesthenalconditiononthesystem.Forasystem consistingof N SWNT numberofSWNT'sdispersedinthesolution: F t = Z C ~r;t dV = N SWNT : 8 t .19 Insummary,thesystemhasbeenreducedtothepartialdierentialequationand conditions: @C p @t +)]TJ/F22 11.9552 Tf 19.076 0 Td [(r C p r E 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( r 2 C p =0.20 C p ~r;t =0= C initial : 8 ~r .21 C p ~r;t !1 = C 0 e )]TJETq1 0 0 1 343.84 257.292 cm[]0 d 0 J 0.359 w 0 0 m 5.203 0 l SQBT/F18 5.9776 Tf 343.84 252.174 Td [( E 2 ~r : 8 ~r .22 F t = Z C p ~r;t d~r = N SWNT : 8 t .23 2.4TemporalEvolution Inordertoevaluatethetemporalevolutionofthissystem,anoperator A must beconstructedsuchthat: @C @t = )]TJ/F19 11.9552 Tf 9.298 0 Td [(AC .24 16 PAGE 27 ! AC )]TJ/F22 11.9552 Tf 7.314 0 Td [(r C p r E 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( r 2 C p .25 Theconstructionofsuchanoperatorwillbenumericallyconsideredinthesubsequentchapter.Intermittently,considerationoftheeigencouplesoftheoperator, n ;v n ,allowsforanexpansionofequation2.24inaneigenbasis. C = ae )]TJ/F20 7.9701 Tf 6.587 0 Td [(At .26 C = X i a i v i e )]TJ/F20 7.9701 Tf 6.586 0 Td [( i t .27 Forsomesetofconstants f a i g .Giventhepresenceofrstorderderivativesin equation2.20,AisnotguaranteedtobeHermitian.Asadirectresultofthis,complex eigenvaluesmustbeconsidered. j = j + i j .28 C = X i a i e )]TJ/F20 7.9701 Tf 6.587 0 Td [( i t e )]TJ/F20 7.9701 Tf 6.586 0 Td [(i i t v i .29 Thevaluesof i areknowntobenon-negative,asanegativevaluewouldleadto exponentialgrowthandaviolationoftheconservationofmass.Clearly,thedecay ofthesolutiontowardsanyeigenfunctionisdominatedbytheterm a i e )]TJ/F20 7.9701 Tf 6.587 0 Td [( i t .Unfortunately,thevaluesof a i arenotgivenbytheinnerproductastheeigenfunctionsare notknowntobeorthogonal: a i 6 = h C j v i i .30 However,thevaluesofthecoecients f a i g canbeexpressedaslinearfunctions ofthefunction C .Inotherwords, 8 a i ;v i ; 9 w i suchthat: 17 PAGE 28 a i = h w i j C i .31 2.5SlowestDecayingEigencouple Inordertodeterminethelongtermbehaviorofequation2.20,attentionmustbe focusedontheeigenvaluewiththesmallestrealpart.Thisstemsfromitdecayingat theslowestrateandthereforedominatingastimegoestoinnity. Consideringtheexpressionoftheconcentrationinthe N dimensionaleigenbasis of A : C = N X i a i v i = N X i v i a i .32 C =[ v 1 ;v 2 ;:::;v n ][ a 1 ;a 2 ;:::;a N ] T = Ua .33 Where U isdenedtobethematrixmadefromcolumnsoftheeigenfunctions and a isacolumnvectorwithcomponents f a i g .Substitutinginequation2.25forthe elementsof a ,equation2.27isexpressedintermsofamatrix W withrowvectors givenby w i C = UW C .34 W = U )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 .35 Whilehavingtherelationinequation2.35isextremelyuseful,directcalculation oftheinverseoftheconjugatedeigenfunctionmatrixishugelyexpensivecomputationally.Instead,focusingonlyonobtainingthedominantterm a 1 andtherefore w 1 cangreatlyreducethiscost. W [1 ; 0 ;:::; 0] T = U )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 e 1 = w 1 .36 18 PAGE 29 ! U w 1 = e 1 .37 Restrictingconsiderationtotherst m eigen-functionstheeigenfunctionscorrespondingtothe m eigenvalueswithlowestmagnituderealcomponent,thiscanbe approximatedintermsof U m =[ v 1 ;v 2 ;:::;v m ]: U m w 1 = e m 1 .38 Ofcourse, N dimensionalvector w 1 cannotbefounduniquelyviaequation2.38. Howeveran m dimensionalapproximation, w 1 = U m z ,canbefoundbysolving: U m U m z = e m 1 .39 Inconclusion,theeigen-expandeddecayratesoftheconcentrationtowardsthe steadystatesolutioncanbeboundedtermwiseasfollows. C = X i a i e )]TJ/F20 7.9701 Tf 6.587 0 Td [( i t e )]TJ/F20 7.9701 Tf 6.586 0 Td [(i i t v i .40 a i e )]TJ/F20 7.9701 Tf 6.587 0 Td [( i t e )]TJ/F20 7.9701 Tf 6.586 0 Td [(i i t v i e )]TJ/F20 7.9701 Tf 6.587 0 Td [( i t jj v i jjjj w i jj e )]TJ/F20 7.9701 Tf 6.586 0 Td [( i t jj v i jjjj U m z i jj .41 19 PAGE 30 3.NumericalMethods Inchapter2,theproblemofsolvingfortherateofapproachofconcentrationtowardsthesteadystatewasdiscussedindetailusingtheeigenexpansionofthesolution intermsofanoperator A suchthat @C @t = AC .Inthischapter,Awillbeconstructed usingthenitedierencemethodforinteriorpointsandpolynomialextrapolationon theboundaries.Thismethodisoutlinedin[6].Additionallyupwindingmethodscan beusedtoenhancestability[5]. 3.1FiniteDierences Themethodofnitedierencesconsistsofapproximatingthecontinuousconcentrationfunctionasadiscretemesh.Forameshinwhichthereare m n p mesh pointsinthex,y,andzdirections,thisisrepresentedbyassumingaunitarydistance betweenthemeshpoints: C x;y;z C i;j;k :1 i m; 1 j n; 1 k p .1 3.1.1ReshapingTensors Giventhatbothconcentration, C ,andelectriceldintensity, E 2 ,tensorsare threedimensional,operatorsactingonthesetensorsmustbefourdimensional.To avoidaddedcomplications,these m n p tensorsarereshapedas mnp 1column vectors. C i;j;k 0 B B B B B B B B B B B B B B B B B B B B B @ C 1 ; 1 ; 1 . C m; 1 ; 1 C 1 ; 2 ; 1 . C 1 ;n;p . C m;n;p 1 C C C C C C C C C C C C C C C C C C C C C A .2 20 PAGE 31 3.1.2LaplacianMatrix ThematrixrepresentationoftheLaplacianoperator, r 2 ,canbefoundviaa centraldierenceapproximation.Equation3.3givesthevepointcentraldierence formulaforanequidistantmeshwithstepsize h =1. @ 2 A i @x 2 )]TJ/F15 11.9552 Tf 26.039 8.088 Td [(1 12 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2+ 4 3 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(5 2 A i + 4 3 A i +1 )]TJ/F15 11.9552 Tf 16.077 8.088 Td [(1 12 A i +2.3 ThefollowingMATLABcodeconstructsthethreedimensionalLaplacianmatrix forapplicationontoareshaped m n p tensor.The kron functionistheKronecker tensorproductusedtodoblockmatrixmultiplications.Avepointformulaisused forfourthorderaccuracy. LX=spdiagsonesm,1 [-1/124/3-5/24/3-1/12],-2:2,m,m; LY=spdiagsonesn,1 [-1/124/3-5/24/3-1/12],-2:2,n,n; LZ=spdiagsonesp,1 [-1/124/3-5/24/3-1/12],-2:2,p,p; LY expanded=kronLY,speyem,m; LZ expanded=kronLZ,speyem n,m n; L=kronspeyen p,LX+kronspeyep,p,LY expanded+LZ expanded./h; 3.1.3DerivativeMatrix Asimilarapproachcanbeusedtondthederivativematrices, @ x A @ y A @ z A : @A i @x )]TJ/F15 11.9552 Tf 26.04 8.088 Td [(1 12 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(2+ 2 3 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 A i +1+ 1 12 A i +2.4 X=spdiagsonesm,1 [-1/122/30-2/31/12],-2:2,m,m; 21 PAGE 32 Y=spdiagsonesn,1 [-1/122/30-2/31/12],-2:2,n,n; Z=spdiagsonesp,1 [-1/122/30-2/31/12],-2:2,p,p; Y expanded=kronY,speyem,m; Z expanded=kronZ,speyem n,m n; DX=kronspeyen p,X; DY=kronspeyep,p,Y expanded; DZ=Z expanded; 3.2BoundaryExtrapolation TheaboveexpressionsfortheLaplacianandderivativematricessuerfromthe needof'ghost'pointstobewelldenedatboundariesie. @ x A A )]TJ/F20 7.9701 Tf 6.586 0 Td [(A G )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 2 Theseghostpointscanbeeliminatedbytheuseofforward/backwardsdierence approximations. @A i @x 1 )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(3 2 A i +2 A i 1 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 A i 2.5 %Fourthordercentraldifferenceapproximation. X=spdiagsonesm,1 [1/12-2/302/3-1/12],-2:2,m,m; Y=spdiagsonesn,1 [1/12-2/302/3-1/12],-2:2,n,n; Z=spdiagsonesp,1 [1/12-2/302/3-1/12],-2:2,p,p; %Boundaryextrapolationsecondorderforward/centralapproximation. X:2,1:4=[-3/22-1/20;01-21]; Xend-1:end,end-3:end=[1-210;01/2-23/2]; Y:2,1:4=[-3/22-1/20;01-21]; Yend-1:end,end-3:end=[1-210;01/2-23/2]; Z:2,1:4=[-3/22-1/20;01-21]; Zend-1:end,end-3:end=[1-210;01/2-23/2]; 22 PAGE 33 Y expanded=kronY,speyem,m; Z expanded=kronZ,speyem n,m n; DX=kronspeyen p,X; DY=kronspeyep,p,Y expanded; DZ=Z expanded; Forwardsandbackwardsdierenceapproximationsarealsobeusedintherst, second,secondtolast,andlastrowsoftheLaplacianmatrices. @ xx A x i 2 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(5 A i +1 +4 A i +2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(A i +3 .6 @ xx A x i 2 A i )]TJ/F15 11.9552 Tf 11.955 0 Td [(5 A i )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 +4 A i )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(A i )]TJ/F17 7.9701 Tf 6.587 0 Td [(3 .7 0 B B B B B B B B B B B @ )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 = 24 = 3 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = 12 ::: 0 4 = 3 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 = 24 = 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = 12 ::: . ::: )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = 124 = 3 )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 = 24 = 3 0 ::: )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 = 124 = 3 )]TJ/F15 11.9552 Tf 9.299 0 Td [(5 = 2 1 C C C C C C C C C C C A 0 B B B B B B B B B B B @ 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(54 ::: 0 02 )]TJ/F15 11.9552 Tf 9.299 0 Td [(54 ::: . ::: 4 )]TJ/F15 11.9552 Tf 9.299 0 Td [(520 0 ::: 4 )]TJ/F15 11.9552 Tf 9.299 0 Td [(52 1 C C C C C C C C C C C A .8 LX=spdiagsonesm,1 [-1/124/3-5/24/3-1/12],-2:2,m,m; LY=spdiagsonesn,1 [-1/124/3-5/24/3-1/12],-2:2,n,n; LZ=spdiagsonesp,1 [-1/124/3-5/24/3-1/12],-2:2,p,p; LX:2,1:4=[2-540;02-54]; LXend-1:end,end-3:end=[4-520;04-52]; LY:2,1:4=[2-540;02-54]; LYend-1:end,end-3:end=[4-520;04-52]; LZ:2,1:4=[2-540;02-54]; 23 PAGE 34 LZend-1:end,end-3:end=[4-520;04-52]; LY expanded=kronLY,speyem,m; LZ expanded=kronLZ,speyem n,m n; L=kronspeyen p,LX+kronspeyep,p,LY expanded+LZ expanded; 3.2.1BoundaryConditions Itisimportanttonotethattheuseofforwards/backwardsdierencemethods attheboundariesisnotalternativestoboundaryconditions.Thegeneralcaseis complicatedbythefactthattheboundariesoftheelectriceldandtheboundariesof theconcentrationarenotthesame.Asanexample,consideranelectriceldpassing throughatransparentcontainer.ThewallsofthecontainerwouldenforceaNeumann conditionontheconcentration,butnotontheelectriceld.Theboundaryconditions ontheelectriceldwouldbedeterminedbysomesetupexternaltothecontainer. Foraknownelectriceld,withboundariesoutsidethechamber,theabovecode wouldallowforapproximationsofthederivativesandLaplacianusingonlyinterior points.However,boundaryconditions,DirichletorNeumann,mustbespeciedfor theconcentrationprole.Theseconditionsareaddressedonacase-by-casebasis. 3.3FormingA Giventhevectorsresultingfromreshapingtheconcentrationandelectriceld intensitytensors, C and E 2 ,theoperatorcanberepresentedusingtheLaplacianand derivativematrices: AC =)]TJ/F22 11.9552 Tf 19.74 0 Td [(r C r E 2 +)]TJ/F19 11.9552 Tf 19.076 0 Td [(C r 2 E 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [( r 2 C .9 AC =\050 @ x E 2 @ x C + @ y E 2 @ y C + @ z E 2 @ z C +)]TJ/F19 11.9552 Tf 26.285 0 Td [(C r 2 E 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( r 2 C .10 A =\050 D @ x E 2 x + D @ y E 2 y + D @ z E 2 z +)]TJ/F19 11.9552 Tf 26.284 0 Td [(D LE 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( L .11 24 PAGE 35 Where D @ x E 2 ;D @ y E 2 ;D @ z E 2 arethediagonalizedvectorsresultingfromtheapplicationofthecentraldierencederivativematricesontheelectriceldintensity vector, x ; y ; z aretheup-windingderivativematrices, D LE 2 isthediagonalized vectorfromthecentraldierenceLaplacianappliedtotheelectriceldintensity,and L istheup-windingLaplacianmatrix. 25 PAGE 36 4.NumericalResults TheMITElectromagneticEquationPropagationFDTDsoftware,orMEEP,numericallysolveselectromagneticsproblemsinwhichdielectricsandconductorsare welldened[17].Theoutputie. E H E 2 H 2 isreturnedonanequidistantmesh ofthesetspatialdimension-3Dplusonetemporaldimension.Thisoutputis particularlywellconstructedforuseinthemethodoutlinedinchapter3. 4.1MeshSizeandStabilty IfweconsidertheDEPdiusionequationintermsofunitsofvoltageVolts[V],timeSeconds-[S],concentrationarbitrary-[C],anddistantunity-[D]. @C @t = r 2 C )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F22 11.9552 Tf 7.314 0 Td [(r E 2 r C )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F19 11.9552 Tf 7.314 0 Td [(C r 2 E 2 .1 [ C ][ S ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 =[][ D ] )]TJ/F17 7.9701 Tf 6.586 0 Td [(2 [ C ] )]TJ/F15 11.9552 Tf 11.946 0 Td [([] [ D ] )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 [ V ] 2 [ D ] )]TJ/F17 7.9701 Tf 6.586 0 Td [(2 [ D ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 [ C ] )]TJ/F15 11.9552 Tf 11.945 0 Td [([][ C ][ D ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 [ V ] 2 [ D ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 .2 []=[ D ] 2 [ S ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(1 .3 []=[ D ] 4 [ V ] )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 [ S ] )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 .4 Equation4.1speciesthediusionandconvectioncoecientstobe a =and b =)]TJ/F19 11.9552 Tf 20.881 0 Td [(@ E 2 respectively,where is x y ,or z .Thestabilityrequirementformesh sizeisgivenby[13]tobe: h << a b .5 Giventheequidistantmesh h x = h y = h z = h equation4.5reducestothe requirement: h<< )]TJ 37.177 16.289 Td [(1 max i; @ E 2 i .6 26 PAGE 37 Expressingthisasarequirementonthenumberofnodesforrectangulardomain withsidesgivenby L x ;L y ;L z : [ n;m;p ]=[ L x h x ; L y h y ; L z h z ]= 1 h [ L x ;L y ;L z ].7 [ n;m;p ] [ L x ;L y ;L z ] )-167(max i @E 2 i .8 4.2 )]TJ/F16 11.9552 Tf 11.797 0 Td [(and inSIUnits Herethevaluesof)-274(andwillbecalculatedforametallicnanotubeofdiameter, d 1 : 2[ nm ],andlength, l 1000[ nm ]notethatthesedimensionsarebasedon themanufacture'sclaimsoftheSWNTsusedintheexperimentalsection.Metallic nanotubeshavebeenestimated[2][12]tohaveaneectiverelativedielectricconstant ontheorderof10 3 -10 4 .Toensurestability, r =10 4 willbeusedasitcorresponds toalarge)-323(andthereforesmallerstepsize.TheStokes-EinstienradiusforaSWNT wasgivenintermsof d and l as[16]: a = 1 2 l ln l=d +0 : 32 85[ nm ].9 )-278(= d 2 l 8 met )]TJ/F19 11.9552 Tf 11.955 0 Td [( H 2 O 6 a 3 : 49 E )]TJ/F15 11.9552 Tf 11.955 0 Td [(23 m 3 V 2 S .10 = RT N A 6 a 2 : 85 E )]TJ/F15 11.9552 Tf 11.955 0 Td [(12 m 2 S .11 4.3ErrorinaKnownCase Inthecaseofaconstantelectriceld,theDEPnitedierenceoperator A reduces tothescalednegativeLaplacian, A = )]TJ/F15 11.9552 Tf 9.299 0 Td [( r 2 .Resultinginwellknowneigenvalues. Equation4.13givestheeigenvaluesforthetwodimensionLaplacianonarectangular domainwithNeumannboundaryconditions.NotethattheDEPboundarycondition isinfactthezerouxNeumanncondition: r C = )]TJETq1 0 0 1 369.539 76.661 cm[]0 d 0 J 0.478 w 0 0 m 6.116 0 l SQBT/F17 7.9701 Tf 369.539 69.55 Td [( C r E 2 =0. 27 PAGE 38 A !)]TJ/F15 11.9552 Tf 24.575 0 Td [( r 2 .12 = n=L x 2 + m=L y 2 : n;m =0 ; 1 ; 2 ; 3 ;::: .13 UsingthetwodimensionalcodegiveninappendixA,withtheLaplacianboundary conditionssetasdescribedabove,thedriver constant test givenbelowresultsinthe smallest20realeigenvaluecomponents.Together,thesevalueshavealessthanone percentrelativeerrorwhencomparedtotheknownvaluesresultingfromequation 4.13.Theresultsaregiveningure4.1. Figure4.1: ApproximatedandknowneigenvaluesforthescaledLaplacianoperator. %Testsdep fd 2dwhenintensityisconstantandtheDEPequationreduces %tothescaledLaplaceequation:AC=-Phi LaplacianC 28 PAGE 39 % %Knownsolutionislambda i=Phi-1 n Pi/L+m Pi/L %forn,m=0,1,2,...,N-1 %TestisonaNxNmeshwithstepsizehlength,L=h N N=30;%Defaultmeshsize h=0.01;%Defaultstepsize gamma=3.49E-23; phi=2.85E-12; %Finddep fd 2destimatedeigenvalues A=dep fd 2donesN,N,gamma,phi,h; D=eigsA,20,sr; R=realD; %Calculateknowneigenvalues; L=[]; L1=[]; fori=0:N-1 lambda=i pi/N h; L1=[L1lambda]; end fori=1:N L=[LL1i ones,N+L1]; end L=sortL; L=phi. L; L=L:N; SE=normR-L:20/normR; 29 PAGE 40 fprintfRelativeerrorinthesmallest20components:%8.2E n n,SE X=linspace,20,20; figure plotX,L:20,X,R, titleSmallest20realeigenvaluecomponents legendKnownvalues.,Approximatedvalues.,Location,Northwest; >> constant test Relativeerrorinthesmallest20components:2.71E-03 4.4ParallelPlateDielectricWaveguide Consideradielectricparallelplatewaveguidewithwallgap a thatisdesignedto propagateinthe z axis.Choosing a suchthatsinglemode, TE 10 ,propagationoccurs forfrequenciesaround2 : 45[ GHz ]wouldresultin a = 1 2 p 2 : 45 E 9 6 : 5[ mm ]forthe dielectricvalueoftheDEPsolution.ThefollowingMEEPcodesimulatesthissystem undertheexcitationofaninternalmicrowavelinesource. set!geometry-lattice makelatticesize9.553no-size set!geometry list makeblock center00 sizeinfinityinfinityinfinity materialair 30 PAGE 41 makeblock center00 size6.5infinityinfinity materialmakedielectricepsilon80 set!sources list makesource srcmakecontinuous-srcfrequency0.0082 componentEz center6.50 set!pml-layers list makepml thickness1.0 set!resolution10 run-until1000 at-beginningoutput-epsilon to-appended"power"at-every1output-dpwr 31 PAGE 42 4.4.1BoundaryConditions Forthe TE 10 propagationmode,theelectriceldtakestheformgivenbyequation 1.35.Usingthedoubleangleidentityresultsintheelectriceldintensityofequation 4.15. ~ E = E 0 sin a x ~e x .14 E 2 = E 0 2 )]TJ/F19 11.9552 Tf 11.955 0 Td [(cos 2 a x .15 @ x E 2 j ;a = E 0 2 a sin 2 a x =0.16 Clearly,asaresultofequation4.16,theboundaryconditionfortheconcentration proleisagaintheNeumanncondition @C @~n =0. 4.4.2 f i g Approximations TheMEEPintensityoutputcanbescaledtond,using dep fd 2 d givenin AppendixA,theminimumrealeigenvaluecomponentforvariousintensitymaximum magnitudes.Theresultsofvaryingtheintensitypeakfrom10 )]TJ/F17 7.9701 Tf 6.586 0 Td [(5 V 2 m 2 to10 10 V 2 m 2 are plottedingure4.2. 4.4.3ForcingExponentialDecay Asopposedtopropagatingwaves,reducingthewallseparationtobelowthe distancecorrespondingtothecutoof2 : 45[ GHz ]leadstoexponentiallydecaying eldsinthe z direction.TheonlychangetotheMEEPcodetoreectthisisgiven below.Again,theresultsofvaryingtheintensitypeakfrom10 5 V 2 m 2 to10 10 V 2 m 2 are plottedingure4.3. set!geometry 32 PAGE 43 Figure4.2: Log-logplotofminimumrealeigenvaluecomponentvsthemaximumeld intensity. list makeblock center00 sizeinfinityinfinityinfinity materialair makeblock center00 size3.5infinityinfinity materialmakedielectricepsilon80 set!sources list makesource 33 PAGE 44 srcmakecontinuous-srcfrequency0.0082 componentEz center3.50 Figure4.3: Log-logplotofminimumrealeigenvaluecomponentvsthemaximumeld intensity. 4.5Conclusion Figures4.2and4.3clearlydemonstratethattheslowestlargesttimeconstant, givenby slow = )]TJ/F17 7.9701 Tf 6.586 0 Td [(1 min ,becomesprogressivelyfasterastheintensitymaxincreasesin magnitude.Accordingly,thosesystemswithhigherintensitymaximumareexpected todecaytothesteadystatefasterthanasystemwiththesamegeometrybutweaker max.Further,theslightincreaseintheeigenvaluesofgure4.3overgure4.2indicate thatasharpergradientinthiscase,resultingfromanexponentialdecaydoescause anincreaseintherateofdecaytosteadystate. 34 PAGE 45 aAboveCuto bBelowCuto Figure4.4: Steadystateconcentrationprolesfora6.5mmandb3.5mmdielectricparallelplatewaveguidesstimulatedat2.45[GHz]. Additionally,seegure4.4,thesteadystatesolutioninthecaseofanexponentiallydecayingelectriceldintensityhasamuchsharperpeakthenthatofthe non-decayingcase.Giventhattheexponentialdecayobtainsahigherpeakconcentrationandachievesitfasterthanthenon-decayingcase,itcanbeconcludedthat intensitygeometriesthatapproachsingularitiesaresuperiorindrivingSWNTDEP. Lastly,linearlyextrapolatingthelineingure4.3tothebreakdownintensityof air,approximately10 13 V 2 m 2 whichwouldhaveviolatedequation4.6tosolvedirectly, wouldsuggestaminimumrealeigenvaluecomponentofroughly min 10 )]TJ/F17 7.9701 Tf 6.587 0 Td [(2 .This 35 PAGE 46 resultsinatimeconstantontheorderofjustafewminutes. 36 PAGE 47 5.LaboratoryMethods TheuidicforcenatureofdielectrophoresisofCarbonNanotubeswasinvestigatedusingathreestepprocess.First,superpuriedplasmadischarge > 95% w=v SWNTbymanufacturer'sclaimnanotubeswerepurchasedfromNanotintegrisand suspendedina0 : 25% w=v aqueoussolutionofSDS.Thisconcentrationisjustabove criticalmicelleconcentration.Aftersuccessfuldispersion,theaqueousSWNT-SDS solutionwasplacedinacylindricalquartzcapillaryeectivelyformingadielectric waveguide.Propagationmodeswereexcitedusinggoldcoatedeldconcentratorsand a2.45[GHz]source.Lastly,thedepositednanotubeswereresuspendedandoptically evaluated. 5.1Dispersion AnaqueoussolutionoftheanionicsurfactantsodiumdodecylsulfateSDSwas preparedataconcentrationof0 : 25% w=v .ThecriticalmicelleconcentrationofSDS rangesfrom6-8mMorroughly0 : 17 )]TJ/F15 11.9552 Tf 11.254 0 Td [(0 : 24% w=v .Singlewallcarbonnanotubeswere dispersedinthissolutionataconcentrationof0.06mg/mL. TheSWNT/SDSaqmixturewasinitiallybathsonicatedtwenty-threetimes witheachrepetitionconsitingof8minutes,totalling3hoursand4minutes.The bathsonicationwasimmediatelyfollowedbythirtyminutesofhornsonication. Afterasmallreferencesamplewastakenfromthisinitialdispersion,theremainderwascentrifugedinaHemleZ360kat13,000rpmfor90minutes.Thetop80%of thesupernatantwascarefullydecantedandstored. 5.2ExperimentalSetup Anelectromagneticsinglemoderesonancechamberwasconstructedusinga shortedaluminumwaveguidedesignedfor2.45[GHz]waves.Atoneend,amagnetronaperturewasplacedataquarterwavelengthfromthebackwall.Thisallowed backwardstravellingwavestomoveaquarterwavelengthtothebackwall,undergo ahalfwavelengthreection,andtravelbackaquarterwavelength,endinginafull 37 PAGE 48 wavelengthpathlengthallowingconstructiveinterferencewiththeforwardtravelling wave.Asecondaperturewasplacedatahalfwavelengthfromtheforwardwall. Giventhatthislocationisatanantinodeofthegeneratedstandingwave,maximum electriceldamplitudeisexperiencedatthisaperture. Themagnetron,bydesign,operatesasadiode-soavoltagedoublerisusedto maximizetheeciency.Arelayisusedtopulsatepowertothemagnetronandhence controlthetimeaveragepowerdeliveredtotheresonantcavity. TheSWNT-SDSaqdispersionisplacedinafusedquartzcapillaryalongwith twoaxialgoldplatednickelneedlescreatingasmallgap.Thisisdisplayedinthe gurebelow. Figure5.1: Arrangementofneedlesinfusedquartzcapillariesmminteriordiameter ontheleftand8mmdiameterontheright. Giventhehighrelativedielectricpermittivityofthesolution,thearrangement createsadielectricwaveguide.Theneedlesconcentrateanyappliedeldtoaregion muchsmallerthanthewavelengthoftheappliedeld.Thisisknowntocreatean approximatelysphericalscatteredwave-inturn,excitinganypossiblemodeswithin 38 PAGE 49 Figure5.2: 4mmcapillarywithgoldplattedneedleslledwithSWNT-SDSsolution. thecapillary. Unlikerectangularwaveguides,withthesimplepropagationmodesgiveninchapter1,theeectivewavefrontsofacylindricalwaveguidearedependentonBessel's functions.Despitethecomplexityofthegeneralcase,thecutofrequencyforthe lowestordermodes TE 11 aregivenby[1]: f c = 1 : 8412 2 r r 1 .1 f 4 mm c 4 : 88[ GHz ]5.2 f 8 mm c 2 : 44[ GHz ]5.3 Asaresultofthesecutofrequencies,whenexposedtomicrowave2 : 45[ GHZ ] frequencies,modesareexcitedinthe8 mm capillary,whereasthewavesexponentially decayinthe4 mm capillary. 39 PAGE 50 Figure5.3: 8mmcapillarywithgoldplattedneedleslledwithSWNT-SDSsolution. Eachcapillaryisinsertedintothesecondaryapertureandsubjectedtopulsed microwaves.Themagnetronispoweredin25 S pulsesata13[ kHz ]repetitionrate. 5.2.1FieldIntensityDistributions MEEPwasusedtosimulatethissetupandresultingelectriceldintensityproles aredepictedingure5.4.Thehighercontrastinthesubgurebindicatesthatthe 4[ mm ]capillaryexperiencedasharpergradientthanthatofthe8[ mm ]capillary. 5.3SecondarySuspension Afterfourcycles,eachveminutesinduration,ofpulsedmicrowaves,thegold needlesareremovedfromthesolution.AnySWNT'sdepositedontheneedlesare removedbyplacingtheneedlesinasmallplastictubewithaqueoussodiumdodecyl sulfonateandbathsonicatedfor24minutes. Asampleoftheresultingsolutionwastakenandcentrifugedinasimilarmethod totheinitialdispersion.Thissamplethenunderwentabsorptionspectroscopytoevaluatemetallicenrichment.Adualbeamspectrophotometerisusedwithanaqueous SDSbackgroundtoremoveanyabsorptionnotduetothenanotubes. 40 PAGE 51 aIntensityProlein8[ mm ]Quartz bIntensityProlein4[ mm ]Quartz Figure5.4: Timeaverageintensityelddistributions Avg xy forthea8mmand b4mmquartztubing. 41 PAGE 52 6.ExperimentalResults TheopticalabsorptionspectraofSingleWalledCarbonNanotubescanbepredictedusingatightbindingmodelasoutlinedin[19].Itwasdeterminedthatthe spacingoftheVanHovesingularitieswasgivenbytheratio1:2:3formetallic SWNT'sand1:2:4forsemiconductingSWNT's.Further,thelowestorderVan Hovesingularityforeachspecieswasfoundtobe: E M 11 = 6 0 a cc d .1 E S 11 = 2 0 a cc d .2 Where 0 2 : 9[ eV ]istheenergyvalueofthehoppinginteraction, a cc 0 : 143[ nm ]isthenearestneighborcarbon-carbondistance,and d =0 : 9 )]TJ/F15 11.9552 Tf 11.764 0 Td [(1 : 7[ nm ]is theSWNTdiameterrangeaccordingtothemanufacturer.Theserelationsleadto thetransitionenergiesgiveninthetable6.1. The S 11 transitionliesoutsideoftherangeofmostspectrophotometersandthe M 22 and S 33 havesignicantoverlap.Asaresult,the M 11 and S 22 arethemost wellsuitedforanalysis.Giventhe900[ nm ]limitofthespectrophotometerandthe interferenceofhigherenergytransitionsbelow 600[ nm ],attentionwillbefocused ontherange600 )]TJ/F15 11.9552 Tf 11.955 0 Td [(900[ nm ]. Giventhedataintable6.1,the M 11 isexpectedtodominateintheregionof approximately600 )]TJ/F15 11.9552 Tf 10.776 0 Td [(650[ nm ].Conversely,the S 22 transitionshoulddominatedinthe 850 )]TJ/F15 11.9552 Tf 11.982 0 Td [(900[ nm ]region.Theintermittentregion,650 )]TJ/F15 11.9552 Tf 11.982 0 Td [(850[ nm ],willbeinuencedby bothofthesetransitions. 6.1PreandPostDEP Aftertheinitialsonicationandcentrifugationaswasdescribedinchapter5,the absorptiondataingure6.2awastaken. AfterthefourveminuteDEPcycles,theneedleswereremovedfromthesolution andallowedtodry.Novisibledepositionhadoccurredontheneedlesplacedinthe 42 PAGE 53 Table6.1: Tightbindingpredictedphotonicenergiesandwavelengthsfornanotube VanHovetransitions. TransitionFormulaEnergyWavelength E M 11 6 0 a cc d 1 : 464 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 : 765[ eV ]448 )]TJ/F15 11.9552 Tf 11.955 0 Td [(847[ nm ] E M 22 12 0 a cc d 2 : 928 )]TJ/F15 11.9552 Tf 11.955 0 Td [(5 : 529[ eV ]224 )]TJ/F15 11.9552 Tf 11.955 0 Td [(423[ nm ] E S 11 2 0 a cc d 0 : 488 )]TJ/F15 11.9552 Tf 11.955 0 Td [(0 : 922[ eV ]1344 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2541[ nm ] E S 22 4 0 a cc d 0 : 976 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : 844[ eV ]672 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1270[ nm ] E S 33 8 0 a cc d 1 : 952 )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 : 688[ eV ]336 )]TJ/F15 11.9552 Tf 11.955 0 Td [(635[ nm ] 8 mm capillary;however,aclearlyperiodicdepositionpatternoccurredinthe4 mm capillarygure6.1.NotethatsphericalwavespropagateasBessel'sfunctions,not sinusoids,soanincreasingwavelengthisexpectedasthewavemovesfromthepoint towardsthebaseoftheneedle. Theneedleswerecutandplacedinabathofaqueoussodiumdodecylsulfonate andbathsonicateduntilthedepositedhadvisiblybeenremovedfromthesurface oftheneedlesminutes.Theneedleswerethendiscardedandthesolutionwas furtherbathsonicatedfor22eightminutecyclestotalling3hoursand4minutes. Opticalextinctiondatawasthentakenforthispost-DEPsamplegure6.2. 6.2Analysis Thepost-DEPdropinthe S 22 and S 33 intothenegativeabsorptionrangeindicates thatthefreemicellesinthebackgroundsolutionwereabletoabsorbmoreinthat regionthattheremainingsemiconductingSWNTs.Theabsenceofthe M 11 peakfrom 448 nm toroughly725 nm isbelievedtobetheresultofthesmallerdiameter,and thereforemoredense,SWNTsbeinglteredoutduringthecentrifugationprocesses. 6.3Conclusion Comparisonofthedatapresentedingure6.2showsasharpdecreaseinthe semiconductingdominatingregion+[ nm ],anoticeabledecreaseinthe650 )]TJ0 g 0 G/F15 11.9552 Tf -212.555 -18 Td [(43 PAGE 54 Figure6.1: Bottomneedlefrom4mmcapillaryafterfourveminuteexposuresto microwaves. 850[ nm ]regioncorrespondingtoboth M 11 and S 22 transitions,andonlyasmall decreaseinthe M 11 dominatedregionof600 )]TJ/F15 11.9552 Tf 9.688 0 Td [(650[ nm ].InaccordancewithBeer'slaw, thisindicatesthatthesemiconductingconcentrationwasdecreasedinmoredrastic mannerthanthemetallic. 6.3.1FutureWork Theworkdemonstratedhereclearlyrepresentsasignicantsteptowardslarge scaleSWNTseparation,howevermanydicultiesstillremain.Depositiondidnot increasesubstantiallywithlongerexposuretimes.Further,whilethecontentofthe depositednanotubeswashighlymetallic,theremainingsolutionwasnotofsignicantlyincreasedsemiconductingcontent.Thisindicatesthatonlyasmallpercentage ofthemetallicnanotubesweredeposited. Giventhatthesolutionhadpre-centrifugetotalSWNTconcentrationof0 : 06 mg mL thatapproximatelyonethirdofnanotubesaremetallic,andthatthe4 mm capillary heldroughly1 mL ofsolution,itcanbeassumedthat < 20 g possiblymuchlessdue tothecentrifugeofmetallicnanotubeswereinthesystemtobeginwith.Fromhere 44 PAGE 55 aPre-DEP bPost-DEP Figure6.2: ExtinctionspectraforSWNT-SDSsolutionabeforeandbafterDEP exposure. 45 PAGE 56 itisobviousthatifonlyafractionofthemetallicnanotubesaredeposited,thenthe yieldofthissystemisquietlow. Finitedierenceswasusedinordertogaininsightintothebehaviorofthesystem inthepresenceofeldsapproachingsingularities.Giventheinherentweaknessof nitedierencestodescribesingularities,furthernumericalstudyofthesystemwould greatlybenetfrommoreaptmodelssuchasniteelement. 46 PAGE 57 REFERENCES [1] C.A.Balanis AdvancedEngineeringElectromagnetics,ed.2 ,Wiley,2012. [2] L.Benedict,S.Louie,andMCohen Staticpolarizabilitiesofsingle-wall carbonnanotubes ,Phys.Rev.B,52,8541,1995. [3] S.Blatt DielectrophoresisofSingleWalledCarbonNanotubes ,Dissertation KasrlsruheInstituteofTechnology,2008. [4] J.C.Charlier,X.Blase,S.Roche ElectronicandTransportPropertiesof Nanotubes ,Rev.ofMod.Phy.,79,2007. [5] R.Courant,E.Isaacson,M.Rees Onthesolutionofnonlinearhyperbolic dierentialequationsbynitedierences ,Comm.PureAppl.Math.5,243-255, 1952. [6] F.Gibou,R.Fedwik AFourthOrderAccurateDiscretizationfortheLaplace andHeatEquationsonArbitraryDomainswithApplicationstotheStefanProblem ,J.Comp.Phy.202.2,577-601,2005. [7] H.T.Ham,Y.S.Choi,I.J.Chung AnExplanationofDispersionStatesof Single-walledCarbonNanotubesinSolventsandAqueousSurfactSolutionsUsing SolubilityParameters ,J.Col.Int.Sci.286,216-223,2005. [8] S.A.Hodge,M.K.Bayazit,K.S.Coleman,andM.S.P.Shaffer Unweavingtherainbow:areviewoftherelationshipbetweensingle-walledcarbon nanotubemolecularstructuresandtheirchemicalreactivity ,Chem.Soc.Rev., 41,2012. [9] S.Iijima HelicalMicrotubulesofGraphiticCarbon ,Nature543.6348,1991. [10] C.Johnk EngineeringElectromagneticFieldsandWaves,ed.2 ,Wiley,1988. [11] A.Hartschuh,H.N.Pedrosa,J.Peterson,L.Huang,P.Anger,H. Qian,A.J.Meixner,M.Steiner,L.Novotny,andT.D.Krauss SingleCarbonNanotubeOpticalSpectroscopy ,ChemPhysChem,6,2005. [12] R.Krupke,F.Hennrich,M.M.Kappes,andH.V.Lohneysen Surface ConductanceInducedDielectrophoresisofSemiconductingSingle-WalledCarbon Nanotubes ,NanoLetters,Vol.4,No.8,2004. [13] E.MajchrzakandL.Turchan Thenitedierencemethodfortransient convection-diusionproblems ,ScienticResearchoftheInstituteofMathematics andComputerScience,Vol.11,No.1,2012. 47 PAGE 58 [14] M.P.Marder CondensedMatterPhysics ,JohnWiley&Sons,2010. [15] A.N.Mina,A.A.Awadallah,A.H.Phillips,andR.R.Ahmed SimulationoftheBandStructureofGrapheneandCarbonNanotube ,J.Phys.:Conf. Ser.,343,2012. [16] N.Nair,W.Kim,R.Braatz,andM.Strano DynamicsofSurfactantSuspendedSingle-WalledCarbonNanotubesinaCentrifugalField ,Langmuir, 24,1790-1795,2008. [17] ArdavanF.Oskooi,DavidRoundy,MihaiIbanescu,PeterBermel, J.D.Joannopoulos,andStevenG.Johnson MEEP:AexiblefreesoftwarepackageforelectromagneticsimulationsbytheFDTDmethod ,ComputerPhysicsCommunications181,687702,2010. [18] J.Ouellette BuildingtheNanofuturewithCarbonTubes ,IndustrialPhysicist, Vol.8,Iss.6,2002. [19] M.Ouyang,J.Huang,andC.Lieber FundamentalElectronicProperties andApplicationsofSingle-WalledCarbonNanotubes ,Acc.Chem.Res.,35, 1018-1025,2002. [20] B.R.PriyaandH.J.Byrne InvestigationofSodiumDodecylBenzeneSulfonateAssistedDispersionofDebundlingofSingle-WallCarbonNanotubes ,J. Phys.Chem.C,112,332-337,2008. 48 PAGE 59 APPENDIXA.DEPFiniteDierenceCode Thecodelistedherewasusedthroughoutthisthesistondtheoperator A such thattheDEPequation2.20holds.Itisprovidedinbothtwoandthreespatial dimensionalcoordinatesystems. A.1TwoDimensions function[A]=dep fd 2dintensity,gamma,phi,h %Usesmethodoffinitedifferencestobuildthe2DDEPoperator: %dC/dt=-AC % %INPUT:intensity-2DEtensor %gamma-Advectionconstant %phi-Diffusionconstant %h-intensitytensorstep-sizescale % %OUTPUT:A-operatorsuchthatdC/dt+AC=0 [m,n]=sizeintensity;%Savetensordimensions intensity=reshapeintensity,[],1;%Formvectorfrom3Dtensor %Laplacianmatrix L=laplacian 2dm,n,h; %DerivativeMatrice [DX,DY]=derivative matrices 2dm,n,h; %Findelectricintensityderivativeandlaplacianvectors; EDXE=DX intensity; EDYE=DY intensity; 49 PAGE 60 ELE=L intensity; %Diagonalize DEDXE=spdiagsEDXE,0,m n,m n; DEDYE=spdiagsEDYE,0,m n,m n; DELE=spdiagsELE,0,m n,m n; %NeumannBCDerivativeMatrix [CDX,CDY]=derivative matrices 2dm,n,h; %Xboundaryconditions fori=0:n-1 j=1+i m; k=m+i m; %Clearboundaryrows CDXj,:=zeros,m n; CDXk,:=zeros,m n; %Loadnofluxrequirement CDXj,j=EDXEj; CDXk,k=EDXEk; end %Yboundaryconditions fori=1:m j=m n-i+1; %ClearBoundaryRows CDYi,:=zeros,m n; CDYj,:=zeros,m n; %LoadData CDYi,i=EDYEi; CDYj,j=EDYEj; 50 PAGE 61 end %Calculateoperator A=gamma. DEDXE CDX+DEDYE CDY+DELE-phi. L; function[DX,DY]=derivative matrices 2dm,n,h %Returnsthemnxmnderivativematricesincartesiancoordinates. %Uses4thordercentraldifferenceapproxinrows3:end-2,2ndorder %centraldifferenceapproxinrows2andend-1,2ndorderforward %differenceinrow1,andsecondorderbackwardsdifferenceinrowend. ifm < 5 jj n < 5 errorm,n,mustbegreaterthanorequalto5 end X=spdiagsonesm,1 [1-808-1],-2:2,m,m; Y=spdiagsonesn,1 [1-808-1],-2:2,n,n; %Boundaryextrapolationndorderforward/central X:2,1:4=[-1824-60;-6060]; Xend-1:end,end-3:end=[0-606;06-2418]; Y:2,1:4=[-1824-60;-6060]; Yend-1:end,end-3:end=[0-606;06-2418]; Y expanded=kronY,speyem,m; DX=kronspeyen,X./ h; DY=Y expanded./ h; end 51 PAGE 62 n end f listing g n begin f lstlisting g function[L]=laplacian 2dm,n,h %ReturnsthemnxmnLaplacianmatrixincartesianCoordinates %Uses4thordercentraldifferenceapproxinrows3:end-2,2ndorder %centraldifferenceapproxinrows2andend-1,2ndorderforward %differenceinrow1,andsecondorderbackwardsdifferenceinrowend. ifm < 5 jj n < 5 errorm,nmustbegreaterthanorequalto5 end LX=spdiagsonesm,1 [-1/124/3-5/24/3-1/12],-2:2,m,m; LY=spdiagsonesn,1 [-1/124/3-5/24/3-1/12],-2:2,n,n; %LX=spdiagsonesm,1 [1-21],-1:1,m,m; %LY=spdiagsonesn,1 [1-21],-1:1,n,n; %UpperBoundaryExtrapolation LX:2,1:5=[-11000;1-2100]; LY:2,1:5=[-11000;1-2100]; %LowerBoundaryExtraptolation LXend-1:end,end-4:end=[001-21;0001-1]; LYend-1:end,end-4:end=[001-21;0001-1]; LY expanded=kronLY,speyem,m; L=kronspeyen,LX+LY expanded./h; end 52 PAGE 63 A.2ThreeDimensions function[A]=dep fd 3dintensity,gamma,phi,h %Usesmethodoffinitedifferencestocalculateasteadystateconcentrationprofile %resultingfromaknownelectricfield. % %INPUT:intensity-3DEtensor %gamma-Advectionconstant %phi-Diffusionconstant %h-Stepsizeresolutionofintensitytensor. % %OUTPUT:A-operatorsuchthatdC/dt+AC=0 [m,n,p]=sizeintensity;%Savetensordimensions intensity=reshapeintensity,[],1;%Formvectorfrom3Dtensor %Laplacianmatrix L=laplacian 3dm,n,p,h; %DerivativeMatrice [DX,DY,DZ]=derivative matrices cartesianm,n,p,h; %Findelectricintensityderivativeandlaplacianvectors; EDXE=DX intensity; EDYE=DY intensity; EDZE=DZ intensity; ELE=L intensity; %Diagonalize DEDXE=spdiagsEDXE,0,m n p,m n p; 53 PAGE 64 DEDYE=spdiagsEDYE,0,m n p,m n p; DEDZE=spdiagsEDZE,0,m n p,m n p; DELE=spdiagsELE,0,m n p,m n p; %ConcentrationDerivativeMatricesNeumannBoundary CX=sparseDX; CY=sparseDY; CZ=sparseDZ; %Xboundaryconditions fori=0:n p-1 j=1+i m; k=m+i m; %Clearboundaryrows CXj,:=zeros,m n p; CXk,:=zeros,m n p; %Loadnofluxrequirement CXj,j=EDXEj; CXk,k=EDXEk; end %Yboundaryconditions fori=0:p-1 forj=1:m k=j+i m n; CYk,:=zeros,m n p;%ClearBoundaryRow CYk,k=EDYEk;%LoadData end forj=+m n-1:m n k=j+i m n; CYk,:=zeros,m n p;%ClearBoundaryRow CYk,k=EDYEk;%LoadData 54 PAGE 65 end end %Zboundaryconditions fori=1:m n j=m n p-i; %ClearBoundaryRows CZi,:=zeros,m n p; CZj,:=zeros,m n p; %LoadData CZi,i=EDZEi; CZj,j=EDZEj; end %Calculateoperator A=gamma DEDXE CX+DEDYE CY+DEDZE CZ+DELE-phi L; end function[DX,DY,DZ]=derivative matrices cartesianm,n,p,h %Returnsthemnpxmnpderivativematricesincartesiancoordinates. %Uses4thordercentraldifferenceapproxinrows3:end-2,2ndorder %centraldifferenceapproxinrows2andend-1,2ndorderforward %differenceinrow1,andsecondorderbackwardsdifferenceinrowend. %Allapproximationsusestepsizeh. ifm < 5 j n < 5 j p < 5 errorm,n,pmustbegreaterthanorequalto5 end X=spdiagsonesm,1 [1/12-2/302/3-1/12],-2:2,m,m; 55 PAGE 66 Y=spdiagsonesn,1 [1/12-2/302/3-1/12],-2:2,n,n; Z=spdiagsonesp,1 [1/12-2/302/3-1/12],-2:2,p,p; %Boundaryextrapolationndorderforward/central X:2,1:4=[-3/22-1/20;01-21]; Xend-1:end,end-3:end=[1-210;01/2-23/2]; Y:2,1:4=[-3/22-1/20;01-21]; Yend-1:end,end-3:end=[1-210;01/2-23/2]; Z:2,1:4=[-3/22-1/20;01-21]; Zend-1:end,end-3:end=[1-210;01/2-23/2]; Y expanded=kronY,speyem,m; Z expanded=kronZ,speyem n,m n; DX=kronspeyen p,X./h; DY=kronspeyep,p,Y expanded./h; DZ=Z expanded./h; end function[L]=laplacian 3dm,n,p,h %ReturnsthemnpxmnpLaplacianmatrixincartesianCoordinates %Uses4thordercentraldifferenceapproxinrows3:end-2,2ndorder %centraldifferenceapproxinrows2andend-1,2ndorderforward %differenceinrow1,andsecondorderbackwardsdifferenceinrowend. %Allapproximationsusestepsizeh. ifm < 5 j n < 5 j p < 5 errorm,n,pmustbegreaterthanorequalto5 end 56 PAGE 67 LX=spdiagsonesm,1 [-1/124/3-5/24/3-1/12],-2:2,m,m; LY=spdiagsonesn,1 [-1/124/3-5/24/3-1/12],-2:2,n,n; LZ=spdiagsonesp,1 [-1/124/3-5/24/3-1/12],-2:2,p,p; %UpperBoundaryExtrapolation LX:2,1:5=[00000;1-2100]; LY:2,1:5=[00000;1-2100]; LZ:2,1:5=[00000;1-2100]; %LowerBoundaryExtraptolation LXend-1:end,end-4:end=[001-21;00000]; LYend-1:end,end-4:end=[001-21;00000]; LZend-1:end,end-4:end=[001-21;00000]; LY expanded=kronLY,speyem,m; LZ expanded=kronLZ,speyem n,m n; L=kronspeyen p,LX+kronspeyep,p,LY expanded+LZ expanded./h; end 57 |