Oberlin College
Honors Thesis
Separation of FeO and Fe3O
4
Nanoparticles Using an Inverted Linear Halbach Array
Jason Heitler-Klevans
Department of Physics and Astronomy
March 31, 2017
i
Executive Summary
Separation of FeO and Fe3O
4 Nanoparticles Using an Inverted Linear Halbach
Array
By Jason Heitler-Klevans
Magnetic nanoparticles are tiny particles that act like littlemagnetic dipoles, but
which behave significantly differently from larger magnets. When coated properly and
suspended in a fluid such aswater, these particlesmaybe used for a variety ofmedical
applications, including drug delivery, biological imaging, and cancer therapy. However,
uniformityofsizeandmagneticpropertiesisessentialfortheuseofmagneticnanoparticles
inanyof thesetechniques. Preciseandefficientseparationandpurificationmethodsare
thereforenecessaryforproperimplementationofnanoparticles.
Inthisthesis,westudythenoveluseofanarrangementofmagnetstoseparateout
two typesofnanoparticles froma suspension, one composedofmagneticFe3O4, and the
other composed of nearly non-magnetic FeO. When a fluid containing these particles is
passedabovethemagnets,themoremagneticparticlestendtosinktowardsthemagnets,
while the lessmagneticparticlespass through. Thisshouldallowus toseparate the two
typesofparticlesfromoneanotherintotwosolutionsinacontrolledfashion.
Wefirstexplainthephysicsbehindtheseparationprocess,andcontinuetodescribe
thebehaviorofourspecificnanoparticlessuspendedinfluid.Ourexperimentationexplores
theeffectofconcentrationandfieldgradientontheseparationanddevelopsmathematical
models tounderstand theoutcomes. Weconclude thatoursetupprovidesanadjustable
methodfornanoparticleseparationandoutlineamethodforevaluatingtheefficiencyofthe
device,alongwithsuggestionsforfutureimprovementsintheprocess.
ii
Acknowledgments I would like to express my deepest gratitude to the following persons for their
supportduringtheproductionofthisthesis.
Firstly,Iwouldliketothankmyadvisor,ProfessorYumiIjiri,forhertime,insight,and
supportthroughoutmytimeasherresearcherandstudent.Shehasgivensomuchtimeand
energytohelpingmegrow,learn,andsucceed,andIowehersomuch.Iwouldspecifically
liketothankherforwelcomingmeintoherlab,andforencouragingmetoberesourceful,
diligent,andthorough.
Second,Iwouldliketothankthefacultyandresearcherswhohaveproveninvaluable
in this endeavor. My gratitude to Anna Samia and her student Eric Abenojar for their
continued support in providing samples, data analysis, and information throughout this
process.IwouldalsoliketothankDougFellerforhismanyhoursofassistanceinmachining
countlesssampleholdersovertheyears.
Iwouldliketothankmygirlfriend,HelenKramer,forherlove,support,andstrategic
useofchocolate tomaintainmymoraleandenergy. Shehasstayedaconstantsourceof
comfortandcalm,andIcannotthankherenough.IwouldalsoliketothankmybrotherAri
Heitler-Klevans, forhiswillingnessto listentomeasI thoughtthroughcountlessphysics
problems,andforhissteadysupportthroughout.
Iwouldliketothankmyfriendsandfamily,specificallymyparentsandgrandparents,
fortheirunderstanding,compassion,andaffirmation.Icouldnothavewishedforamore
wonderfulgroupofpeople inmy life. Thanks tomy labpartnersEmilyHamlin,Naiyuan
Zhang, andHillaryPan, anda special thanks to theotherphysicshonors studentsKinori
Rosnow,HannonAyer,IanHunt-Isaak,JacobTurner,DahyeonLee,andKaiShinbroughfor
themanyhourswehavespenttogetherearningthismajor.
Additional thanks to the NSF DMR-1606887 grant for providing funding for this
project, along with the NSF DUE-9950606 which funded the VSM magnetometer used
extensivelyinthisthesis.
iii
Contents
Executive Summary i
Acknowledgments ii
List of Figures v
List of Tables vii
Glossary and Physical Constants viii
1 Introduction and Motivation 1
2 Background and Theory 6
2.1
2.2
Magnetism in Materials . . . . . . . . . . . . . . . . . . . . . . . .
Magnetic Separation Mechanisms . . . . . . . . . . . . . . . . . . .
6
14
3 Experimental Procedures 18
3.1
3.2
3.3
3.4
Nanoparticle Synthesis . . . . . . . . . . . . . . . . . . . . . . . . .
Structural Characterization . . . . . . . . . . . . . . . . . . . . . .
Magnetic Characterization . . . . . . . . . . . . . . . . . . . . . . .
Procedures for Nanoparticle Separation . . . . . . . . . . . . . . .
18
19
20
21
4 Results and Analysis 27
4.1
4.2
4.3
Initial Structural Characterization Results . . . . . . . . . . . . . .
Initial Magnetic Characterization Results . . . . . . . . . . . . . . .
Separation Results . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
29
31
4.3.1
4.3.2
Separations with Varying Particle Concentrations . . . . . .
Separations with Varying Field Gradient . . . . . . . . . . .
32
35
4.4 Receiver Operating Characteristic Analysis . . . . . . . . . . . . . . 39
Contents iv
5 Conclusion and Future Work 42
5.1
5.2
Conclusions
Future Work
42
43
A Mathematica Code for VSM Analysis 49
B Solidworks CAD Designs 58
v
List of Figures
2.1
2.2
2.3
2.4
2.5
Magnetic responses associated with different types of magnetic
materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetic domains of ferromagnetic materials in the absence and
presence of a saturating magnetic field . . . . . . . . . . . . . . . .
A moment-based picture of magnetic nanoparticles in the absence
and presence of a saturating magnetic field . . . . . . . . . . . . .
A plot of a normal distribution, a lognormal distribution, and a
gamma distribution to demonstrate the differences between them .
A plot of potential ROC curves demonstrating the meaning of this
form of analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
9
10
13
17
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Lakeshore 7307 Vibrating Sample Magnetometer at Oberlin College.
Diagrams of the Kel-F sample holders used for VSM measurements
and machined by Doug Feller . . . . . . . . . . . . . . . . . . . . .
The constructed linear Halbach array with 47 NdFeB permanent
magnets held together by set screws and an aluminum frame . . .
Plot of the B field along the length of the array at four different
distances from the array, as modeled by FEMMView . . . . . . . . .
Plot of the field lines along the length of the array and the
orientation of magnet magnetizations, as modeled by FEMMView . .
Picture of observable banding of magnetic Fe3O
4 nanoparticles
when placed above the array . . . . . . . . . . . . . . . . . . . . . .
Array-channel assembly design with toluene-compatible glass
channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic of separation at an optimal distance and one
complication that could arise . . . . . . . . . . . . . . . . . . . . .
20
21
22
22
23
23
24
26
Contents vi
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
TEM characterization of the particles received . . . . . . . . . . . .
XRD characterization of the particles received . . . . . . . . . . . .
Initial magnetometry results for FeO and Fe3O
4 suspensions . . . .
Magnetometry results confirming changes in particle moment by
comparing liquid samples from the FeO particles taken a month
apart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Initial magnetometry results for mixtures of FeO and Fe3O
4 with
varying concentrations . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetometry results for the filtrate solutions from the
separations performed on suspensions of varying concentration . .
Magnetometry results for the initial suspensions of FeO and Fe3O
4
mixtures separated at different field gradient . . . . . . . . . . . .
Pictures of the channel during separation for the separations
performed at varying field gradients . . . . . . . . . . . . . . . . .
Magnetometry results for the filtrate solutions from the
separations performed on suspensions separated at varying field
gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A plot of the signal shift from Fe3O
4 to FeO versus applied field
gradient after separation . . . . . . . . . . . . . . . . . . . . . . . .
A plot of potential ROC curves with one point calculated from the
25/75 mixture of FeO/Fe3O
4 . . . . . . . . . . . . . . . . . . . . . .
28
28
29
31
32
34
35
37
38
39
41
B.1
B.2
B.3
Design of plexiglass channel top for the toluene-compatible glass
channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Design of glass plate for the toluene-compatible glass channel . . .
Design of stainless steel base plate for the toluene-compatible glass
channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
59
60
vii
List of Tables
2.1 The potential results of a binary sort presented in the context of
ROC analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
4.1
4.2
4.3
4.4
4.5
Magnetic characterization results for the initial FeO and Fe3O
4
samples using Langevin analysis and assuming a lognormal
distribution of magnetic moments . . . . . . . . . . . . . . . . . .
Percentages of VSM signal attributed to the FeO and Fe3O
4 particles
for mixtures of different concentrations, using a double Langevin
analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Percentages of VSM signal attributed to the FeO and Fe3O
4 particles
for initial mixtures used in the changing field gradient experiment,
using a double Langevin analysis . . . . . . . . . . . . . . . . . . .
Percentages of VSM signal attributed to the FeO and Fe3O
4 particles
for the filtrates obtained by varying the field gradient, using a
double Langevin analysis, and with percentage shifts from initial
samples shown . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Calculated values of the false positive fraction used in ROC
analysis, as calculated from the results for the filtrates obtained by
varying the field gradient . . . . . . . . . . . . . . . . . . . . . . .
30
33
36
38
39
viii
Glossary and Physical
Constants
Boltzmann’s Constant
Permeability of Vacuum
Avogadro’s Constant
Room Temperature
Viscocity of Toluene
kB
µ0
NA
T
htoluene
=
=
=
=
=
1.38 x 10-23 J/K
4p x 10-7 N/A2 in SI or 1 in CGS
6.022 x 1023 mol-1
298 K
0.580 cP
1
Chapter 1
Introduction and Motivation
Ferrofluids are stable solutions that consist of magnetic particles withmicron to
nanometerdiameterscoatedwithasurfactantandsuspendedinafluid.Ferrofluidswere
firstdevelopedinthe1960s,andhavecontinuedtobethesubjectofextensiveresearchever
since(Franklin). Thephysicalandmagneticpropertiesofthesenanoparticlesallowfora
wide range of applications including in industrial vacuum parts, memory storage, and
wastewater remediation (Sigamaneni et al.). By far, one of themost attractive areas of
applicationofferrofluidsisinbiomedicineduetothealterabilityofananoparticle’ssurface
anditsspecificmagneticbehavior(Krishnan).
Inparticular,severalimportantareasofresearchfornanoparticleuseinbiomedicine
includetargeteddrugdelivery,improvedmagneticresonanceimaging(MRI),andaformof
cancerortumortreatmentcalledhyperthermia(Pankhurstetal.).Nanoparticlesaresmall
enoughforinsertionintothebodyandcanpassthroughmostlayersoforganicmattersuch
ascellmembranes.Duringandaftersynthesis,nanoparticlesarecoatedwithsomeformof
surfactantinordertostabilizetheparticlesinsolutionandmaythenbecoatedwithadesired
chemical,medicine,ormarker.Whenplacedinsideacellorareaofthebody,themagnetic
nanoparticlesmaybemanipulatedtomovepreciselyusingexternalmagnetic fields,such
Chapter1.IntroductionandMotivation 2
thattheexactnatureofdrugdeliverymaybemorecontrolled(Williamsetal.1,Haefeliand
Chastellian, Hayden and Haefeli). Additionally, ferrofluids can be used as a negative
contrasting agent in anMRI scan, providing clearer imaging than commercially available
materials(Malekzadehetal.).Extensiveresearchhasinvestigatedthecreationofanewform
ofimagingtechnologycalledmagneticparticleimaging(MPI).MPIallowsforaquantitative
spatialmapping ofmagnetic nanoparticles, enabling a flexible form of imagingwith the
potentialforhigherresolutionthancurrentmedicaldiagnostictechnologies(Samia2013,
Gehrcke).Magneticnanoparticlesalsohavethepotentialtoaidinhyperthermiatreatment,
amethodthatprimarilytargetscancerortumorcells. Theparticlesare injectedintothe
infectionsite,andthenanalternatingexternalmagneticfieldisusedtotransferenergyto
theparticlesintheformofheatinordertoeitherreleaseheat-sensitivedrugsintothetumor
ortodenaturethedesiredcells(Itoetal.).
However,allbiomedicalapplicationsofmagneticnanoparticlesrelyonuniformityof
particlesizeandmagneticproperties.Particlesofdifferentsizesandchemicalcompositions
willexhibitdifferentmagneticresponses,andinferrofluids,canthenleadtoawiderrange
offluidresponseinthepresenceofanexternalmagneticfield.Theuseoflargemacroscopic
quantities of particles with different magnetic properties can have unintended and
potentiallytoxicresultsonabody,makingtheuseofsuchfluidsunacceptableformedical
practice. Unfortunately,whilemanysynthesismethodsmayproduceorganicsolutionsof
nanoparticles with high precision, the transfer of such particles to water introduces a
numberofpotential changes in size, surfaceuniformity, and chemical composition, all of
whichmayimpactthemagneticpropertiesoftheparticles.Inaddition,chemicalvariation
withinananoparticlecanoccurwithoutavariationinparticlesize,resultinginverydifferent
magneticbehaviors.
Althoughtherearemanyresearchgroupsfocusedonsynthesizingnanoparticles,and
therearewaystoseparatebasedonsizessuchasthroughtheuseofmechanicalsieves,very
fewapproachesfocusonseparatingbasedonmagneticeffects.Wewilldiscussseveralof
themainmethodstopurifyonthebasisofmagneticpropertieshere, includingtheuseof
high gradient magnetic separation (HGMS), field flow fractionation (FFF) methods, the
differential magnetic catch-and-release (DMCR) technique, and low gradient magnetic
separation(Stephensetal.).
Chapter1.IntroductionandMotivation 3
Eachofthesetechniquesreliesontheuniquepropertiesofmagneticnanoparticlesin
order to function. Magnetic nanoparticles used in biomedicine are typically
superparamagneticasdiscussedinSection2.1;inthepresenceofasmallexternalfield,the
particlesrespondreadilyandwilltendtoaggregateintochainsofparticles.Thisallowsfor
greaterdiscriminationintheprocessofseparation,assmallerorlessmagneticparticleswill
haveanexponentially smallerprobabilityof chaining together. Inaddition,non-uniform
fields (field gradients) will induce a magnetic force, drawing the aggregated particles
towardsthesourceoftheexternalfield,whichcanbeassimpleasapermanentmagnet.
HGMSprovidesarelativelystraight-forwardapproachbyplacingtheferrofluidina
container filled with susceptible wires and then activating an electromagnet nearby
(Faraudoetal.).Thisgeneratesahigh-gradientfieldandcausesrapidparticleaggregation
around thewires,which is ideal for collectingmicron scaleparticles. Unfortunately, the
method does not allow for more fine-tuned separation or purification of magnetic
nanoparticlesuspensions,asthesizeoftheparticlesisinsufficientforseparationinthisway.
TheFFFmethodsprovidemorevarietyand finesse intomagnetic separation than
HGMS, and come in several forms such as magnetic field flow fractionation (MFFF),
quadrupolemagnetic field flow fractionation (QMgFFF), and cyclical electrical field flow
fractionation(CyFFF).FFFtechniquesrelyonthefactthatfluidflowingnearthewallsofa
capillarytunnelorchanneltendtotravelslowerthanthosetravellinginthemiddleofthe
channel(Stephensetal.).Byflowingapolydisperseferrofluidthroughsuchachanneland
applying an externalmagnetic field, larger particles or particles with a highermagnetic
moment tend to aggregatenear thewalls and flow slower than smaller or lessmagnetic
moments.Thedifferentsizesofparticlesarethencollectedatdifferenttimesfromtheoutlet
of the tube. Altering the setup by introducing a quadrupolemagnet to achieveQMgFFF
improved the efficiency of this method without drastically increasing the complexity or
changing the separation method used (Carpino et al., Orita et al.). Additionally, the
electromagneticsetupusedtomakethequadrupoleallowedforprogrammableseparation,
increasingtheflexibilityofthismethod(Williamsetal.2).CyFFFtakesanotherapproachto
thesamemethod,wrappingacapillary tubearoundanelectromagnetandcausing larger
particlestoaggregateintheportionofthetubeclosesttothemagnetic(Tascietal.).
Chapter1.IntroductionandMotivation 4
DMCRprovidesaslightlymorenuancedmethodofseparation,wheretheexternal
magnetic field is now applied horizontally rather than vertically relative to the tubes
containingtheferrofluid.Byusingamorerigorousapproachinvolvingabalanceofdragand
magnetic forces, this technique allows for more careful and precise separation of
nanoparticlesbysize(Beveridgeetal.).
Theuseof low-fieldseparationmethodsreduces thesetupcomplexityandcostof
construction, andmakes use of inter-particle interactionsmore explicitly. In a low-field
environment,differencesbetweenlargeandsmallormagneticandnon-magneticparticles
haveagreater impacton typicalparticlebehavior (Yavuzetal.,Cuevasetal.). However,
previousexperimentalsetupshavenotallowedforcomprehensiveanalysisoraquantitative
understandingoftheseparationprocessinrelationtotheory.
While all of these techniques hold interest in the area of magnetic nanoparticle
separation,noneprovideaclearbasisforindustrialuseduetolimitsonsamplesizes.For
FFF methods, the constant flow of nanoparticle suspension suggests use in commercial
separationofnanoparticles,butthistechniquerequiresslowflowratesandsmallsample
sizessuchas20µLofsamplepumpedat0.1mL/min(Carpinoetal.).CurrentDMCRresearch
still dealswith limited sample sizes and is not easily scalable. Therefore, an alternative
method for magnetic separation is clearly necessary that allows for both precision in
separating nanoparticles and scalability for industrial or at least laboratory prepartion
purposes.
Ourrecentworkemploysa linear invertedHalbacharray toachieveseparationof
magnetic nanoparticles of varyingmoments (Ijiri et al.). It iswell known that a special
arrangementofmagnetscalledaHalbacharrayresultsinanon-symmetricmagneticfield,
withonehighfieldsideandonelowfieldside.Designedin1980byKlausHalbach,thearray
takestheformofaseriesofmagnetswiththeirmagnetizationsorientedinarotatingpattern
(Halbach).WhilemostresearchthatemploysaHalbacharrayusesthehighfieldside,the
invertedsideprovidesacombinationoflowmagneticfieldandhighfieldgradient.Thefield
gradientenablestheuseofmagneticforcestodrawparticlestowardsthearray,whilethe
low field aspect allows for discrimination between higher and lower moment particles
(Hoyos et al.). This approach has the advantage of allowing for precise control over
Chapter1.IntroductionandMotivation 5
separationparameterssuchasfieldgradient,aswellasascalabledesignwiththepotential
forlargersamplesizesandindustrialuse.
Initial research has already demonstrated the ability of the array to discriminate
betweenparticleswithdifferentmagneticmomentsduetotheirparticlesize(Ijirietal.).But
whiledifferencesinsizedooftenaccountfordifferencesinmagneticmoment,otherfactors
suchaschemicalcomposition,surfaceeffects,particleshape,andthesurfactantcoatingmay
influence their magnetic behavior. We have yet to explicitly evaluate the magnetic
separatingcapacityofthedeviceorcharacterizeitsefficiency.
Inthisthesis,weinvestigatetheseparationofwüstite(FeO)andmagnetite(Fe3O4)
nanoparticlesofsimilarsizeinordertofurthercharacterizetheefficiencyandeffectiveness
ofthelinearinvertedHalbacharrayasasortingmechanism.Chapter2outlinesthetheory
behindmagnetism inmaterials, describes the properties of superparamagnetic particles,
providesamodelandjustificationforourassumptionofadistributionofmagneticmoments,
detailsthemechanicsofmagneticseparation,andoutlinesananalyticaltoolforquantifying
theefficiencyofoursortingmechanism. Chapter3providesdetailsonourmeasurement
procedures andexperimental setup, andChapter4describes the results of two seriesof
separations,oneperformedon samplesof varying concentrationsandoneperformedon
samplesundervaryingfieldgradient.Chapter5summarizesourresults,andcommentson
thepotentialforfutureresearchandinvestigationofthedevice.
6
Chapter 2
Background and Theory
This chapter presents a basic overview of the theory of magnetism in materials,
focusing on the magnetic properties of superparamagnetic materials as applied to
nanoparticles in Section 2.1. In Section 2.2, an explanation of the relevant separation
mechanismimplementedbytheHalbacharrayispresented,aswellasadescriptionofthe
proposedmethodusedtoanalyzetheefficiencyofthearrayasasortingmechanism.
2.1 Magnetism in Materials
Inordertounderstandthemagneticpropertiesofsuperparamagneticnanoparticles,
wemuststartbysummarizingthemagneticbehaviorofdifferentformsofmaterialsinthe
presenceofmagneticfields(Cullity).Additionalinformationonthemagneticpropertiesof
differentmaterialscanbefoundin(GerstenandSmith,Griffiths).
Magnetism in materials arises from the intrinsic magnetic dipole moment of
electrons,whichisaconsequenceofanelectron’sspinandorbitalangularmomentum.From
thePauliexclusionprinciple,weknowthatelectronsinthesameatomicorbitalmusthave
oppositespinsandthattheirnetangularmomentumiszero,andthereforethatthemagnetic
Chapter2.BackgroundandTheory 7
moments of paired electrons will cancel each other out. When discussing magnetic
materials,weconsideronlythemagneticcontributionofunpairedelectrons.Thesumofthe
magneticmomentsinamaterialdividedbyitsvolumeisdefinedasitsmagnetizationM,
wheremisthemagneticmomentcontributionfromeachatom,andVisthevolumeofthe
material. Themagnetizationofamaterial isdependentontemperatureandanyexternal
magneticfields.ItcanbeexpressedinrelationtotheexternalfieldHby,
wherecisknownasthemagneticsusceptibilitytensorofthematerial,whichcansometimes
reducetoasimpleconstantforisotropicmaterials.Inaddition,themagnetizationisrelated
tothemoregeneralconceptofB,theinducedmagneticfieldormagneticfluxdensity,
whereµ0isthevacuumpermeabilityconstant.Thedifferencesbetweendifferenttypesof
materialscanbeclearlyillustratedthroughtherelationshipbetweenMandH,asdepicted
inFigure2.1.Thecommonlydiscussedtypesofmagnetisminmaterialsarediamagnetism
(DM), paramagnetism (PM), ferromagnetism (FM), and anti-ferromagnetism (AFM).
Additionally, materials constructed on a small enough scale (10-6 – 10-9 m) can exhibit
superparamagnetism(SPM),withtheMvsHresponseshownbelowinFigure2.1(D).
(2.1)
(2.2)
(2.3)
Chapter2.BackgroundandTheory 8
Figure2.1:TheMvsHfieldprofilesfordiamagnetic(A),paramagnetic(B),ferromagnetic(C),andsuperparamagnetic(D)materials.Thecoercivity(Hc),theremanentmagnetization(Mrem),andthesaturationmagnetization(Msat)areindicatedfortheferromagneticgraph.
Moststablesubstancesarediamagnetic,containingprimarilypairedelectrons,and
theirresponseweaklyopposestheexternalmagneticfield.Diamagneticmaterialsgenerally
have a c of -10-6 to -10-3, whereas paramagnetic materials have a c of 10-6 to 10-1.
Paramagneticmaterialscontainunpairedelectronsthataresufficiently isolatedfromone
anothersuchthatanyexternalfieldwillaffecteachmagneticmomentindividually,tending
toalignitwiththefield.Intheabsenceofanexternalfield,thermalenergiesaresufficient
toalignthesemagneticmomentsrandomly,resultinginasamplemagnetizationofzero.
Somematerialscontainenoughunpairedelectrons incloseenoughproximity that
they retain a magnetization even in the absence of an external magnetic field. These
substances,oftenknownaspermanentmagnetsandoftenferromagnetic,haveamagnetic
responseas illustrated inFigure2.1 (C). In ferromagneticmaterials,neighboringatomic
magneticmomentsinteractandalignparalleltooneanother,creatingareasinthematerial
calleddomains,asdemonstratedinFigure2.2(A).Domainsforminamannerthatclosesin
theirfieldlinesinordertominimizethemagnetostaticenergyassociatedwiththeoverall
structure;otherwise, the largeB fieldgeneratedbythealignedmomentscostsadditional
energy.Intheabsenceofanexternalmagneticfield,thedomainsmaybeorientedrelatively
M
H H
M
H
M
H
M
(A)
(C)
(B)
(D)
Msat
Mrem
Hc
Chapter2.BackgroundandTheory 9
randomly,andthe ferromagneticmaterialwillhaveanetmomentequal to theremanent
magnetization, as indicated in Figure 2.1 (C) and Figure 2.2 (A). Note that the value is
dependentonthehistoryofthemagneticfieldtreatmentofthesample.Theexternalfield
required to reduce themagnetization to zero is called the coercivity fieldHc, and is also
indicatedinFigure2.1(C).Thesaturationmagnetizationisachievedwhenallofthedomain
momentsarepointed inthedirectionof theapplied field,asshowninFigure2.1(C)and
Figure2.2(B).
Figure2.2:Anaturallyoccurringferromagneticmaterialwithdomainsindicated(A),andthatsamematerial(B)inthepresenceofasaturatingfieldupwards.
Antiferromagneticmaterialsconsistofcrystalstructurescontainingplanesofatoms
with opposing magnetic moments relative to one another. Thus, otherwise magnetic
materialssuchasFeOhaveanetmagnetizationofzero,withthemagneticmomentofeach
plane canceling out those of its neighbors. We can therefore expect antiferromagnetic
materials tohave zeromagnetization. However, irregularities in the crystal structureor
surface effects may result in an imbalance of the opposing moments, with the effect of
creatingaweaklymagneticmaterial. Wewillseethisfurtherinrelationtotheresults in
Chapter4.Insomematerials,themagneticmomentcontributionfromalternatingplanesof
atomsisnotequal,leadingtoanetmagnetization,inwhatisdescribedasferrimagnetism.
Inbulk,ferrimagnetssuchasFe3O4canappeartobehavejustlikeferromagnets.
In the case of nanoparticles, another important type ofmagnetism to consider is
superparamagnetism(SPM).Onnanoormicro-lengthscales,magneticmaterialsreachan
energetic limitwhere a singlemagnetic domain is lower energy thanmultiplemagnetic
domains(BeanandLivingston).Aspreviouslystated,magneticdomainsareformedinorder
tominimizemagnetostaticenergies,butinparticleslessthanacharacteristicsize,theenergy
(A) (B)
Hsat
Chapter2.BackgroundandTheory 10
tocreatedomainsisgreaterthanthefieldenergyinherentinasingledomain.Theparticle
willthenactasoneverylargemagneticmoment;itsspecificdirectioncanbedictatedbyany
preferred direction in the particle created through the crystal structure (crystalline
anisotropy), shape (shape anisotropy), or surface (surface anisotropy), as well as the
application of an applied magnetic field. These single-domain magnetic nanoparticles
suspendedinfluidresultinasituationsimilartothatofparamagnetism,containingineffect
agroupofisolatedmagneticmomentsasshowninFigure2.3(A).However,insteadofsingle
atomicmomentsasinparamagnetism,eachmomentisaresultofasinglemagneticdomain
thatoftencontains105magneticatoms,hencethenamesuperparamagnetism.Thisleadsto
much larger values of c than those associated with paramagnetic materials, as well as
saturatingfieldsachievableinthelaboratoryasdepictedinFigure2.3(B).
Figure 2.3:Nanoparticles suspended in fluid in the absence (A) andpresence (B) of a saturatingexternalmagneticfield.Thenetmagneticmomentofeachparticleisindicatedwithanarrow.
Thesuperparamagneticresponseoftheseparticlestoanappliedmagneticfieldcan
beunderstoodintermsofstatisticalmechanicsfromadiscussionby(BeanandLivingston),
ignoring any particle-particle interactions. For a particle of magnetic momentm in a
directionq relative to the external fieldH, theparticle’s Zeemanenergywill be given as
–mH*Cos(q).ForanassemblyofsuchparticlesattemperatureT,undertheassumptionof
thermalequilibriumtherewillbeaBoltzmanndistributionofq’sovertheassembly.Thenet
magnetizationalignedinthedirectionoftheexternalfieldiscalculatedbyaveragingCos(q)
overtheBoltzmanndistribution.Thisresultsintheequation,
(2.4)
(A) (B)
Hsat
Chapter2.BackgroundandTheory 11
wherekBistheBoltzmannconstantandTisthetemperature.Thiscanberewritteninterms
oftheratioandthesubstitution,suchthatEquation2.4becomes,
Thesolutiontothisintegraltakestheformof
where L(a) is known as the Langevin function and N is the total number of magnetic
nanoparticlesinthesample.
TheLangevinfunctioniscommonlyusedtofitthemagnetizationversusappliedfield
(M vsH) curve characteristic of a superparamagnetic sample, and allows us to extract
relevant statistics by fitting the data. However, the singlemomentmodel is not always
representative of an actual fluid sample due to variations in particle size and chemical
composition,whichmay affect themagneticmoment. Severalmathematicalmethods to
modeltheparticlesintermsofadistributionofparticlediameterormagneticmomenthave
been created in the past (Chen et al., Kakay et al., Chantrell), although some havemore
physical merit than others. For example, while a Gaussian curve provides the most
straightforwardformofmodeling, itallowsfornon-physicalnegativevaluesfordiameter
andparticlemoment.TheGaussiandistributionisgivenby
whereµisthemedianandsisthestandarddeviation.
Previous researchhas focusedon theuseof a lognormaldistributionora gamma
distribution for descriptions of nanoparticle diameter due to experimental evidence that
(2.5)
(2.6)
(2.7)
Chapter2.BackgroundandTheory 12
nanoparticlesynthesestendtoleadtoaskeweddistributionofdiameters.Acomparisonof
thesedistributionswasexploredinthethesisofJan-PhilipGehrcke,whodemonstratedthat
thedifferencesbetweenthesetwodistributionsareminimalforthepurposesofmodeling
magneticnanoparticlesasshowninFigure2.4(Gehrcke).Bothdistributionscontainonly
positive values andmaybe adjusted easily, as comparedwith aGaussian distribution of
similarshapewhichcontainsnon-physicalnegativevalues,asshownbythedottedgreen
lineinFigure2.4.Thegammadistributionisgivenas
wherea is known as the shape parameter and b is known as the rate parameter. For
researchinvolvingthetimeresponseofparticlestoanexternalfield,theGammadistribution
providesmoreinsightthantheotherprobabilityfunctions.However,wehavechosentouse
thelognormaldistributionduetoitscompatibilitywithourtoolsforanalysis.Thelognormal
distributionisthenaturallogarithmofaGaussiandistributionasgivenbytheequation
whereµ is themedianands is thestandarddistribution for thecorrespondingGaussian
distribution. In previous research, a lognormal distribution in diameter for particles of
constant magnetic moment m was assumed to account for any deviation in magnetic
behavior(Fergusonetal.,Chantrell).Inthisthesis,weconsiderparticlesthatareeffectively
ofthesamediameter,butarecomposedofdifferentmaterials,andthereforeexhibitdifferent
magneticmomentsbasedonvariationsinchemicalcompositionorcrystallinestructure.In
ordertotakeallofthesepotentialcontributionsintoaccount,weassumethattheparticles
followalognormaldistributionofmagneticmoment.Fromtheseconsiderations,weseethat
Equation2.6nowtakestheformof
(2.9)
(2.8)
Chapter2.BackgroundandTheory 13
wheresisanormalizationfactorandp(m)isthelognormaldistributionofmagneticmoment
in terms of the parametersµ ands. An example comparison of thismethod is given in
AppendixA,andforanalysis,thelimitofintegrationwaschangedfrom¥to10-12toimprove
thequalityanddecrease thecomputational timeofanalysis. Thisdidnot impact the fits
obtainedasthemagneticmomentsofthedistributionwerealmostexclusivelyfoundtobe
valuesrangingfrom10-18to10-14emu.Additionaltermswereincludedinthisequation,such
as a slope term to account for the paramagnetic or diamagnetic contributions from the
carrierfluidorsampleholderrespectively.Anoffsettermwasalsousedtoaccountforany
errorincenteringasamplerelativetothemeasurementcoils.
Figure2.4:Shownisacomparisonoftheprobabilitydensityfunctionsp(x)ofanormal(dottedgreenline),alognormal(solidblueline)andagamma(dashedblackline)distribution.Asdiscussed,thereareminimaldifferencesinshapebetweenthelognormalandgammadistributions.
Duringourexperiments,wecombinedsolutionsofFeOandFe3O4nanoparticlesto
demonstrateseparationprocessesforparticlesofdifferentmagneticmoments.Inorderto
effectivelymodelsuchsolutions,weusedanalteredformofEquation2.10,suchthat
(2.10)
x
p(x)
Chapter2.BackgroundandTheory 14
where p1(m) is described by the parameters µ1 and s1 and p2(m) is described by the
parametersµ2ands2. We fit forp1(m)andp2(m)by testing theFeOandFe3O4particles
individuallyandthenfittingthedataobtainedfrommixturesoftheseparticlestofinds1and
s2.Wethendeterminedhowmuchofthesignalmeasuredforasamplewasattributedtothe
firstdistributionincomparisontothesecond.Alloftheseaspectsofourmethodforanalysis
areoutlinedinfurtherdetailinAppendixA.
2.2 Magnetic Separation Mechanisms
All of the separation methods described in Chapter 1 make use of the magnetic
propertiesofthenanoparticlesinordertodiscriminatebetweendifferentmagneticparticles
in a colloidal suspension. From basic electromagnetic field theory, we know that the
magneticforceuponanobjectisgivenbythatobject’smagneticmomentandthemagnetic
fieldgradient,asexpressedby,
wheremisthemagneticdipolemomentandBistheinducedmagneticfield(Griffiths).For
agivenfieldgradient,particleswithdifferentmagneticmomentswillexperiencedifferent
amounts of force in the direction of that gradient. However, the force experienced by
individualparticlesisnotgreatenoughtoinduceparticlestomoveinthedirectionofthe
magneticgradientintimescalesusefulforexperimentation(WilliamsandPoudel).
Fortunately,anothereffectcomesintoplay intheformof interactionsofmagnetic
nanoparticleswithoneanother, an issue for example in a ferrofluidof sufficientparticle
concentration.Suchasystemhasapotentialenergyassociatedwiththemagneticdipolar
couplingofnanoparticles,similartothatofelectricdipoles(VanReenanetal.).Thisdipole-
dipoleenergyisgivenas,
(2.12)
(2.11)
Chapter2.BackgroundandTheory 15
whereUdd is thedipole-dipolepotentialenergy,M is theparticlemagnetization,dm is the
magneticdiameteroftheparticle,andsisthedepthofthenon-magneticsurfactantusedto
suspendtheparticleinthesolventfluid.Itshouldbenotedthatthemagneticdiameterof
theparticlemaybesignificantlydifferentfromthehydrodynamicdiameteroftheparticle
dependingon thecoatingof theparticle,andpotentially thechemicalcompositionof the
particle.Giventhisenergy,whenacollectionofparticlesexperiencesthemagneticforceas
describedabove,theymaychaintogether.Therefore,minEquation2.12isnowthesumof
themagneticmomentsoftheclusterofparticles.Inthisway,particlesthatchaintogether
experienceagreaterforcethanparticlesthatdonot,leadingtoanacceleratedaggregation
effectordersofmagnitudegreaterthanthemagneticforceexperiencedbysingleparticles.
Thiscouldthenallowforseparationofparticlesbasedonchainingversuslackofchaining,
althoughotherissuesmustbeconsidered.
One concern is the effect of Brownianmotion,which counters the aggregation of
particles.AsimplecalculationofthethermalenergykbTincomparisontothetypicaldipole-
dipoleenergyformanymagneticnanoparticlesshowsthatthetwoenergiesareofsimilar
magnitude.WepredictbasedontheseopposingenergiesthatfortwoparticleswithUdd/kbT
greaterthan1,theparticleswilltendtochaintogether,andforUdd/kbTlessthan1theywill
tendnottochaintogether.
Inconsideringseparation,wenotethatparticlesmovinginafluidareopposedbya
drag forcedependenton the fluid’sviscosityandthesizeandspeedof theparticle. This
relationshipisgivenbyStoke’sdragforceforafluidofviscosityh,
wheredhisthehydrodynamicdiameteroftheparticleandvpistheparticle’svelocity.This
setsthetimescaleforseparationtooccurforcharacteristicclustersofsomedh.
Inordertoevaluatetheefficiencyofourseparationmethod,wehavebeguntoemploy
theReceiverOperatorCharacteristic(ROC)analysis,apracticecommonlyusedtoquantify
theaccuracyofamedicalsortingprocess,suchasatestofwhetherpatientshaveaparticular
disease(Swets).Forexample,supposeatestindicatesifasubjecthasmalariaornot.For
(2.13)
(2.14)
Chapter2.BackgroundandTheory 16
anygroupof subjects, the testwill indicate thata certainnumberare sick, anda certain
numberarehealthy. Similarly,amagneticnanoparticlesortingmechanismsortsout low
magneticmomentparticles fromhighmagneticmomentparticlesbasedonwhether they
will chain together or not. Wewill call the portion of the samplewherewe expect the
particles togo through theprocesswithout chaining the ‘Filtrate’, and theportionof the
solutionwhereweexpecttheparticlestochainthe‘Residue’.Thisputssubjectsintofour
categories,asdemonstratedinthetablebelow:
TestPositive TestNegative
ActuallyPositive TruePositive
(testedsickandaresick)
(lowmomentinthefiltrate)
FalseNegative
(testedhealthyandaresick)
(lowmomentintheresidue)
ActuallyNegative FalsePositive
(testedsickandarehealthy)
(highmomentinthefiltrate)
TrueNegative
(testedhealthyandarehealthy)
(highmomentintheresidue)
Table2.1:Givenarethefourpossiblecategoriesofsamplesgivenabinarysortingmechanism.Inparenthesesare given what each category represents for the example of a medical test and for a sort of magneticnanoparticlesofvaryingmagneticmoment(Swets).
Themostcommonlyusedrelationshipsderivedfromthese fourcategoriesarethe
sensitivity (the true positive fraction, TPF), and the specificity (which gives us the false
positivefraction,FPF).Theequationsfortheserelationsaregivenbelow:
(2.15)
(2.16)
(2.17)
Chapter2.BackgroundandTheory 17
WecangraphTPFvsFPFto
demonstratewhatareknownasROC
curves,asshowninFigure2.5tothe
right,whereseveralexamplecurvesare
depicted.Apointalongthediagonal
dashedline(TPF=FPF)wouldindicate
thatforarandommixofparticlesthe
processwouldhavenospecial
discriminationbetweenthetwo
distributionsofparticles.Thepointin
theupperleftcorner(TPF=1,FPF=0)
wouldindicateatestwithcompletediscrimination.ROCcurvesthatlieclosertothispoint
(like the red curve) demonstrate more efficient sorting processes than those with ROC
curvesclosertothediagonal(likethegreenone).
By comparing the magnetic properties, chemical composition, and particle
concentration,wecanemployROCcurveanalysistoquantifytheefficiencyofoursorting
mechanism in discriminating between two types of particles, one composed of
antiferromagneticFeO, andone composedof ferromagneticFe3O4. Separations runwith
differentparametersshouldlieatdifferentpointsalongsuchacurve,allowingustoquantify
thesortingmechanismofadevicebyfittingtheresultsofaseriesofseparationsbasedon
changingseparationparameters.Wewilldiscusshowtheexperimentsthatweperformed
couldbeusedtoconstructanROCcurveinSection4.4.
FalsePositiveFraction(FPF)
TruePositiveFraction(TPF)
Figure 2.5: Example ROC Curves going from nodiscrimination(dashed)tocompletediscrimination.
18
Chapter 3
Experimental Procedures
Thischapteroutlinestheexperimentalproceduresusedfornanoparticlepreparation,
structural andmagnetic characterization, and separation via the linear invertedHalbach
array.
3.1 Nanoparticle Synthesis
All nanoparticle solutions used in this experiment were obtained from Professor
AnnaSamiaandherstudentsatCaseWesternReserveUniversity(CWRU).Wüstite(FeO)
andmagnetite (Fe3O4)magnetic nanoparticleswere synthesized using a related process
(Samia 2016). An oleate iron precursor was created by combining iron (III) chloride
hexahydrate(FeCl3•6H2O)withsodiumoleateandasolventcomposedofdeionizedwater,
ethanol,andhexane,refluxingthemixturefor4hours.Theorganiclayerofthatmixturewas
decanted forwashing to remove byproducts, followed by drying under a vacuum for 72
hours. Asolutionoftheironoleate,oleicacid,and1-octadecenewasmixedunderargon,
heatedto100°Cfor1hour,andthenrefluxedat320°Cforanotherhour.TheresultingFeO
nanoparticles were precipitated out by centrifugation using an ethanol/toluene solvent.
Chapter3.ExperimentalProcedures
19
SomeoftheseFeOnanoparticleswereconvertedtoFe3O4byaddingtrimethylamineN-oxide
[(CH3)3NO],heatingat130°Cfor1hour,andthenheatingto280°Cforanotherhour.The
resultingFe3O4nanoparticleswereextractedfromthesolutionbycentrifugationusingan
ethanol/toluenesolvent. Theparticleswerealltransferredtotoluenesolutionsandwere
dilutedwithanhydrous toluene inanargonglovebox, such that thesolutionsofFeOand
Fe3O4nanoparticleshadthesameparticleconcentration.Desiredproportionsofparticles
were thereafter achieved by mixing corresponding proportions of solution, such that a
50/50mixtureofFeOandFe3O4particleswasobtainedbymixing1partFeOsolutionwith1
partFe3O4solution.TopreventanyadditionaloxidizationoftheFeOparticlesandtolimit
anydegradationofthesamples,thesuspensionswerekeptintheargon-filledgloveboxand
filled with argon until separation. Additionally, after separation all of the collected
suspensionswereplacedinargon-filledcontainersandstoredinafridgeat5°C.
3.2 Structural Characterization
Structuralcharacterizationsoftheparticlespre-andpost-separationwereconducted
atCWRU inorder todetermineparticle size, chemical composition, andconcentration in
solution; Transmission ElectronMicroscopy (TEM), X-RayDiffraction (XRD), and Atomic
AbsorptionSpectroscopy(AAS)werethethreeprincipalmethodsused.
TEM measurements were performed using an FEI Tecnai G2 Spirit BioTWIN
transmissionelectronmicroscopeoperatingat120kVtoimageparticlesdroppedonagrid,
and the imagingsoftware ImageJwasused toanalyze thedataandcalculate theaverage
particle diameter (Samia 2016). Additionally, TEM images allowed for qualitative
assessment of the particles, including information about the shape of the particles and
surfactantcoating.ThecrystallinecompositionoftheparticleswasdeterminedthroughXRD
measurements,whichwereperformedusingaRigakuMiniFlexX-raypowderdiffractometer
with Cu-Ka radiationl=1.54 Å. Dried and powdered samples of the nanoparticleswere
scannedfor2qvaluesof25to65°.Thetotalironconcentrationforeachliquidsamplewas
determinedusingAASmeasurements,whichwereperformedusingafastsequentialatomic
Chapter3.ExperimentalProcedures
20
absorptionspectrometerVarian220FSAAtodeterminetheconcentrationof ironineach
sample.
TheparticleconcentrationofeachsolutionwascalculatedusingtheTEMandAAS
data,alongwiththeknowndensitiesofFeOandFe3O4.
3.3 Magnetic Characterization
A Lakeshore 7307 Vibrating Sample Magnetometer was used to determine the
moment vs field profile of the samples before and after separations, and special sample
holderswereusedtocontaintheliquidsamples.Alldataweretakenatroomtemperature
forfieldvaluesrangingfrom0to1T.ApictureofthemachineusedispresentedinFigure
3.1. The data were then fit to a Langevin function as explained in Chapter 2 with the
assumptionthatthemagneticmomentsoftheparticlesineachsolutionfitintoalognormal
distribution,asexplainedinAppendixA.Forsolutionscomposedofamixtureoftwotypes
ofparticles,adoubleLangevinlognormaldistributionofmagneticmomentwasused.
Figure3.1:TheLakeshore7307VibratingSampleMagnetometeratOberlinCollegeprovidedforbyNSFDUE-9950606.
Chapter3.ExperimentalProcedures
21
Sample holders were constructed in Oberlin College's machine shop from the
material Kel-F (polychlorotrifluoroethylene), a clear elastopolymer. Kel-F is easily
machinable, does not react in the presence of toluene, and most importantly has a
diamagneticsignalintheµemurange,whichisanorderofmagnitudelessthanmostsamples
measuredintheVSM.Thedesignofthestandardliquidsampleholderusedisoutlinedin
Figure3.2(A,B),andeachholderwasfilledwith50µLofsampleusingamicropipettorand
sealedwithteflontape.However,duetothelowmagneticmomentoftheFeOnanoparticles,
forsomeofthesamples,themagneticsignalfromtheregularsampleholderwasstilllarge
enoughtoinfluencetheVSMsignal.Toaccountforthis,newsampleholders,asoutlinedin
Figure3.2(C)weremachinedsuchthatthemagneticsignalfromtheKel-Fabovethesample
wouldbalancethatfrombelowthesample,effectivelynegatingtheholder'ssignalrelative
totheinductioncoils.
Figure3.2:The sampleholder top (A), standardbottom (B), and special bottom (C)machined atOberlinCollegebyDougFeller.ThestandardsampleholderconsistedofpartsAandB.
3.4 Procedures for Nanoparticle Separation
TheinvertedlinearHalbacharrayusedinthisexperimentwasconstructedofaseries
ofNeodymium-Iron-Boridepermanentmagnetsheldwithasetofscrewsinanaluminum
case, as shown in Figure 3.3 (Poudel). The magnets were arranged such that the
magnetization of each magnet was rotated 90° relative to the preceding magnet. The
magneticfieldandfieldgradientabovethearraywerecalculatedusingFEMMviewsoftware
(Meeker),andthefieldprofileforthelow-fluxsideofthearrayisshowninFigure3.4,with
(A) (B) (C)
Chapter3.ExperimentalProcedures
22
thefieldprofileverifiedusingaLakeshoregaussmeterprobe.Thefieldlinesaredepictedin
Figure3.5asgivenbytheFEMMviewprogram.Theaveragefieldandfieldgradientvalues
fordifferentheightsabovethearraywerecalculatedforuseinseparations.
Figure3.3:TheinvertedlinearHalbacharrayusedinthisthesis,constructedof47NdFeBpermanentmagnetscasedinaluminum.
Figure3.4:AplotoftheBfieldalongthelengthofthearraymeasuredatfourheightsabovethearrayasmodeledusingFEMMView(Poudel).
200
150
100
50
0
Mod
eled
B fi
eld
valu
es (m
T)
403020100Distance along the array (magnet blocks)
At 0.13 cm from array
At 0.15 cm from array
At 0.3 cm from array
At 0.45 cm from array
Chapter3.ExperimentalProcedures
23
Figure3.5:AdiagramofthefieldlinescreatedbythelinearHalbacharrayandamagnifieddepictionfrom the grey boxed section, with the magnetizations of individual magnets indicated by blackarrows(Poudel).
The liquid channel used in this experiment was constructed to allow fluid flow
horizontallyacrossthelengthofthearrayandwasoffsetverticallyfromthearrayusingtwo
Standa 092354 stages. The vertical stageswere zeroed relative to the array before each
separation.Thechannelwasconstructedofametalbaseplate,aVitonAgasket,aglassplate,
andaplexiglassplate,withplastic tubingallowing for fluid topass through.TheVitonA
gasket and glass plate were used due to their resistance to toluene. Thesematerials fit
together inamannerdemonstrated inFigure3.7,andareheldtogetherbyusingasetof
screwsandnuts.MoredetailsontheconstructionofthechannelaregiveninAppendixB.
Mineral oil was used to create a uniform index of refraction between the glass and the
plexiglass,sothatqualitativeobservationscouldbemadeabouttheparticleactivityinthe
channel.
Figure3.6:Thetoluene-compatibleglasschannelontopoftheHalbacharrayduringaseparation.Distinctaggregationoftheparticlesatspecificpointsalongthearrayisindicatedbydarkbands.
Chapter3.ExperimentalProcedures
24
Figure3.7:Array-channeldesignwithtoluene-compatibleglassandPlexiglasschannel
During the construction of the channel, the screwswere tightenedwith a torque
metertoavoidexcessstresstotheglassplateandtoprovideconstantpressureonthegasket
in order to create a liquid-tight seal. The inner 12 screws were tightened to a uniform
maximumtorqueof2.5inchpounds,andtheouter6screwsweretightenedtoauniform
maximumtorqueof1.5inchpounds.Afterconstruction,thechannelwasfilledwithtoluene
tocheckforleaksandpreparethechannelforeachnanoparticleseparation.Allfluidswere
pumpedintothechannelusingaHarvardApparatusSyringepump11Elite.
Thepreparedchannelwasfilledwithnanoparticlesuspensionandwasthenplaced
overthearray.Morenanoparticlesuspensionwaspumpedatachosenpumprateandthe
channelwasraisedverticallyoverthearrayatachosenheight.Thefirst120µLofsuspension
wasseparatedtoavoidobtainingparticlesthatdidnotexperiencethemagneticforcefrom
thearray.Alltheremainingfluidwascollectedinsmallvials.Thefluidcollectedwhilethe
nanoparticlesuspensionwaspumpingatthechosenpumpratewasdubbedthe“filtrate.”
Toluene was then pumped at the chosen pump rate, and the nanoparticle suspension
obtainedwasdubbed the “intermediate” separation. The channelwas removed from the
arrayandtheremainingsuspensionwaspumpedoutatahighpumprateanddubbedthe
Chapter3.ExperimentalProcedures
25
“residue.”Adiagramof theexpectedparticlebehaviorduringeachseparation isgiven in
Figure3.8.
Inpreparationfortheseparationsperformedwithvaryingparticleconcentrations,
fivesuspensionswerecreatedbycombiningsamplesofFeOandFe3O4,suchthattherewas
onesuspensioneachof100%FeO,75%FeOand25%Fe3O4,50%FeOand50%Fe3O4,25%
FeOand75%Fe3O4,and100%Fe3O4,withpercentagesbasedonparticlenumber.Allfive
separationswererunwiththesameparametersofheightabovetheHalbacharrayandpump
ratethroughthechannel.Theheightofthechannelwaschosentobe6.04mmabovethe
array, such that theparticleswouldexperiencea2.7T/mfieldgradient.Theseparations
wererunwithapumprateof15.3µL/minbasedontheestimatedtimeitwouldtakeagroup
ofthreeFe3O4particlesofmedianmagneticmomenttosinktothebottomofthechannel.
In preparation for the separations performed at varying field gradients, four
suspensionswerecreatedsuchthateachwascomposedof50%FeOnanoparticlesand50%
Fe3O4nanoparticles.Theseparationswererunat5.08mm,4.13mm,3.18mm,and2.22mm
abovethearray,whichcorrespondedtofieldgradientsof4.7T/m,10.6T/m,23.2T/m,and
48.1T/mrespectively.TheseaveragefieldgradientswerecalculatedfromtheFEMMView
modelshowninFigure3.4(Poudel).Thepumprateforeachseparationwasbasedonthe
estimatedtimeitwouldtakeagroupofthreeFe3O4particlesofhighmagneticmoment(90th
percentile)tosinktothebottomofthechannel.Usingthismodel,thepumprateincreased
withthefieldgradientduetotherelatedincreaseofthemagneticforceasgivenbyequation
2.12.ThishigherpumpratewasusedbasedoffofobservationsmadeinSection4.3.1,and
willbediscussedinmoredetailinSection4.3.2.
Chapter3.ExperimentalProcedures
26
Figure3.8:Aschematicoftheseparationprocessatanoptimaldistanceabovethearray,alongwithpotentialcomplicationsthatcouldarise.(a)showsadepictionofthechannelplacedatanoptimaldistanceabovethearray,(b)showssuspensionbeingpumpedintothechannel,and(c)showstheideal behavior of the suspension during separation. Here the larger red dots denote particles ofhighermagneticmoment(Fe3O4),andthesmallerbluedotsrepresentparticlesof lowermagneticmoment(FeO).(d)showsapossiblecomplicationthatcouldarise,withsomehighmomentparticlesflowingthroughthechannelunimpededandsomelowmomentparticlesaggregatingtothebottomofthechannel(Poudel).
27
Chapter 4
Results and Analysis
Theresultsfromstructuralandmagneticcharacterizationsofmagneticnanoparticle
suspensionsandresultsofseparationexperimentsarepresentedandinterpretedhere in
sections4.1–4.3.Insection4.4,theuseofreceiveroperatingcharacteristicanalysisinthe
quantificationoftheefficiencyofoursortingmechanismisdiscussed.
4.1 Initial Structural Characterization Results
Figure 4.1 shows typical transmission electron microscopy images taken by
collaborators at CWRU. From the analysis software ImageJ, the FeO nanoparticleswere
foundtohaveadiameterof22.0±1.4nm,andtheFe3O4nanoparticleswerefoundtohavea
diameterof22.2±1.4nm. Thediameterof22nmwasusedinallanalysisforthesetwo
samples,asthediameteroftheFe3O4particlesarewellwithinthestandarddeviationofthe
diameterof theFeOparticles. Anadditional sampleofFe3O4nanoparticleswasused for
experimentationintogradienteffects,andtheseparticleswerefoundtohaveadiameterof
16.76±0.90nm.
Chapter4.ResultsandAnalysis 28
Figure 4.1: The TEM image of the FeO (A), the larger Fe3O4 (B), and the smaller Fe3O4 (C)nanoparticles,with themeasured diameters 22.0± 1.4 nm, 22.2± 1.4 nm, and 16.76± 0.90 nmrespectively.
FromtheX-raydiffractionresultspresentedinFigure4.2,theFeOnanoparticleswere
determinedtobe94%FeOand6%Fe3O4,thelargerFe3O4nanoparticlesweredetermined
tobe8%FeOand92%Fe3O4,andthesmallerFe3O4nanoparticlesweredeterminedtobe
6%FeOand94%Fe3O4.
Figure4.2:XRDspectra for theFeOand largerFe3O4nanoparticles (A)and for the smallerFe3O4nanoparticles (B). Thepointson thebottomof each figure indicate expectedpeak locations andrelativeintensities.
AtomicAbsorptionSpectroscopyanalysisindicatedthattheFeOsamplehadaninitial
Feconcentrationof21.1mg/mL,thatthelargerFe3O4samplehadaninitialconcentrationof
16.5mg/mL,andthatthesmallerFe3O4samplehadaninitialconcentrationof15.7mg/mL.
Allthesamplesweredilutedsuchthateachhadaparticleconcentrationofapproximately
2x1015particles/mL.
(A) (C)(B)
(A) (B)
Chapter4.ResultsandAnalysis 29
4.2 Initial Magnetic Characterization Results
VibratingSampleMagnetometrymeasurementswereperformedoneachsampleas
is, with the field profiles shown below in Figure 4.3. The Fe3O4 measurements were
performedontwoseparatesamples,withaveragenanoparticlediametersofeither22or17
nm,asindicatedbyTEManalysis;thedataareconsistentwiththeexpectedLangevinsignal
forsuperparamagneticnanoparticlesofslightlydifferentsizes.Twofluidsampleswerealso
takenfromtheFeOsolution,one frombeforetheexperimentationbeganandtheothera
month later; unfortunately, the differences in the signals of these samples indicate that
changeshavetakenplaceintheoriginalFeOsuspensionovertime.
Figure4.3:TheVSMdata(dots)andfitcurves(lines)fortheFeOandFe3O4samples,withtheFeOsamplesbeing thesamesizeand theFe3O4samplesbeingofdifferentsizes, corresponding to thelabelsusedinTable4.1.Theslopesaretheresultsofparamagneticordiamagneticcontributionstothesamplefromthetoluenesolutionorthesampleholderrespectively.
The moment versus field curves in Figure 4.3 were fit via the single Langevin
lognormalmomentdistribution(SLLMD)analysis,withresultsgiven inTable4.1, further
confirmingthequalitativeobservationsjustdescribed.AsdiscussedinChapter2,thenatural
Chapter4.ResultsandAnalysis 30
logarithmofµvalueisequaltothemedianofthelognormaldistribution,whilethesvalue
indicates the standard deviation of the lognormal distribution. The median magnetic
moment can therefore be calculated from the value ofµ, while the relative size of each
distributioncanbeinterpretedbycomparingsvaluesfordifferentsamples.
Sample µvalue svalue LowMoment(20thpercentile)
MedianMoment(50thpercentile)
HighMoment(90thpercentile)
FeO(1/9/17) -39.2 1.8 2.0x10-18emu 9.9x10-18emu 1.1x10-16emu
FeO(2/8/17) -38.9 1.7 2.8x10-18emu 1.3x10-17emu 1.4x10-16emu
Fe3O4(1/9/17) -33.3 1.9 7.0x10-16emu 3.4x10-15emu 4.0x10-14emu
Fe3O4(3/8/17) -34.7 0.2 7.6x10-16emu 8.6x10-16emu 1.1x10-15emu
Table4.1:Thefitvaluesforthelognormaldistribution(asdefinedbyµands)arepresented,aswellas themagnetic particlemoments corresponding to different percentiles of the distribution. Thesamples are labeled by chemical composition and date measured. The two FeO samples arecomposed of particles of the same size (22.0 nm), while the two Fe3O4 have different averagediameters(22.2nmand16.8nmrespectively).
ThedifferencesbetweentheµandsvaluesforthetwodifferentFe3O4samplescan
beattributedtothedifferenceindiameterandtheassociatedvariationofdiameterbetween
the two. The smaller particles have a smaller moment per particle, and the smaller
distributionofsizescorrespondswithasmallerdistributionofmagneticmoments.Asseen
in Table 4.1, the later (2/8/17) data on the FeO particles resulted in a higher median
moment,aswellasaslightchangeinthestandarddeviationofthedistributionrelativeto
the earlier (1/9/17) data. Such changes must be taken into consideration during data
analysis and imply that the particlesmayhave a limited lifetime of utility. This issue is
apparentinSection4.3.2,particularlyinFigure4.7andisdiscussedfurtherinthatsection.
To ensure that the observed differences over time resulted from a change in the
sampleratherthansomeerrorassociatedwiththeVSM,severaltestsforreproducibilityof
signalwereperformed.AcomparisonoftwodatarunsoftheFeOsampledon02/08/17and
onefromtheFeOsampledon01/09/17isshowninFigure4.4.Asindicatedbythefigure,
Chapter4.ResultsandAnalysis 31
thedifferencesbetweentheVSMdataacquiredfromthesamesampleareminimalcompared
withtheVSMdataacquiredfromthetwodifferentsamples.
Figure4.4:TheVSMdata(dots)andfitcurves(lines)fortheFeOsample,withonesampletakenfromtheoriginalsolutioninJanuaryandtheothersampletakenfromtheoriginalsolutioninFebruary.TwodatarunsfromthesecondsamplearedisplayedtoindicatethereproducibilityoftheVSMdataforindividualsamplesascomparedtothedifferencesbetweensamples.
4.3 Separation Results
As described in the Chapter 3, two series of separationswere performed, one of
varying concentrationsofFeOandFe3O4particlesat a single fieldgradient, andoneof a
50/50mixture of FeO and Fe3O4 particles at different field gradients. The larger Fe3O4
particles were used in the separations of varying concentration and the smaller Fe3O4
particleswereused intheseparationsofvaryingfieldgradient,withresultsasdescribed
belowinSection4.3.1and4.3.2.
Chapter4.ResultsandAnalysis 32
4.3.1 Separations with Varying Particle Concentrations
Initial VSM measurements of the suspensions of FeO and Fe3O4 with different
concentrations were taken as shown in Figure 4.5. The data were fit using the double
Langevinlognormalmomentdistribution(DLLMD)analysisasdescribedinequation2.11
with extracted information listed in Table 4.2. The percentages of VSM signal were
determinedbydividingthescalefactorassociatedwitheachdistributionbythesumofthe
scalefactors(s1+s2),asoutlinedindetailinAppendixA.Asindicatedbytheinformation
found in Table 4.2, the percentages of VSM signal change vary with the changes in
concentration,with theFe3O4particles’ signalsdominating theshapeofVSMdatadue to
theirvastlygreatermomentvaluesascomparedtotheFeOparticles.
Figure4.5:TheVSMdata(dots)andfitcurves(lines)fortheinitialsuspensionsmadeofdifferentconcentrations of FeO and Fe3O4. The legend indicates the composition of each solution bypercentagesofFeO/Fe3O4inthatorder,aswellasthedateeachseparationwasperformed.
Chapter4.ResultsandAnalysis 33
MixtureofFeO/Fe3O4 PercentofVSMsignalattributedtoFeO
PercentofVSMsignalattributedtoFe3O4
100%FeO(2/8/17) 100% 0%
75%FeOand25%Fe3O4(1/25/17) 22.9% 77.1%
50%FeOand50%Fe3O4(1/11/17) 1.0% 99.0%
25%FeOand75%Fe3O4(1/31/17) 0% 100%
100%Fe3O4(1/9/17) 0% 100%
Table4.2:Thepercentagesof theVSMsignalattributedtothetwotypesofparticle,basedontheDLLMDanalysisdescribedinAppendixA.Thedatethedataweretakenisgiveninparentheses.
Unfortunately,someofthedatashowedchangesthatwerenotcompatiblewiththis
form of analysis, specifically themixture of 25% FeO and 75% Fe3O4. From qualitative
observationsofthesuspensions,wefoundthatallthesamplescontainingFe3O4hadparticles
thatsettledtothebottomofthecontaineroverthecourseofacoupleofhours.Additionally,
uponvigorousshakingofthesamples(throughsonication)macroscopicclumpsofparticles
wereobserved,suggestingthattheFe3O4particleswerenolongerfullydispersedinthefluid.
Thismayhave led togroupsofparticleswitha largermagneticmoment than thatof the
original Fe3O4 distribution, resulting in the relatively poor fit in the analysis of certain
samples.Notably,the25%FeO/75%Fe3O4mixturewasmeasuredandseparatedafterall
theothermixturescontainingthefirstbatchofFe3O4nanoparticles.Wehypothesizedthat
thesettlingbehaviormayhavebeenaresultofincompletesurfactantcoatingoftheparticles.
Thiscoupledwiththehighmomentvaluefor22nmparticlescouldhaveledtotoohigha
dipole-dipoleenergy,resultinginirreversiblyclumpedparticles.ThesecondbatchofFe3O4
particlesweresynthesizedwithasmalleraveragediametertocounteractthisproblemand
wereusedinallsubsequentseparation.
Thefiltratesobtainedafterseparationweremeasured,andDLLMDanalysisindicated
thatalmostallthesignalfromthefiltratesamplesresultedfromtheFeOparticles.Thefits
andVSMdataforthefiltratesamplesoftheseseparationsaredisplayedinFigure4.6.The
lack of signal like that of the Fe3O4 particles implies that nearly all the Fe3O4 particles
aggregatedtogetherandfelltothebottomofthechannel.However,thesettlingbehaviorin
suspensionscontainingFe3O4particleswasobservedtotakeplaceontimescalessmaller
Chapter4.ResultsandAnalysis 34
thanthedurationofeachseparation.ThisimpliesthatthebehavioroftheFe3O4particles
may not have resulted solely due to their larger magnetic moment relative to the FeO
particles,butratherduetosomedifferenceintheircoatingthatmadethemlesssolublein
toluene,asdiscussedearlier.
Figure4.6:TheVSMdata(dots)andfitcurves(lines)forthefiltratesoftheFeOandFe3O4mixtureseparations.ThelegendindicatesthecompositionofeachsolutionbypercentagesofFeO/Fe3O4inthat order, aswell as the date each separationwas performed. Wewere unable to fit the Fe3O4(green)curveduetohighcontributionstothesignalfromthesampleholder.
Despitethis,wecanconcludefromtherelativeuniformityofthefiltratesthatvarying
the concentrationof the solutiondoesnothavea significant impacton theoutcomeof a
separation.Wethereforeturntoaseriesofseparationsbasedonvaryingtheappliedfield
gradient to further understand the impact of field gradient and pump rate on particle
behaviorinoursetup.
Chapter4.ResultsandAnalysis 35
4.3.2 Separations with Varying Field Gradient
Forthisseriesofexperiments,thesmaller(16.76nm)Fe3O4particleswereuseddue
totheirreversibleclumpingbehaviorobservedinthelarger(22.2nm)Fe3O4particles.In
addition, the pump ratewas calculated using the highmoment particles rather than the
median moment particles to minimize the non-magnetic field based settling behavior
observed for the first batch of Fe3O4 suspensions as just discussed. Initial VSM
measurementsofthe50%FeO/50%Fe3O4suspensionswithvaryinggradientsweretaken
withtheVSMdataandfitsdisplayedinFigure4.7andtheextractedinformationshownin
Table 4.3. As before, the percentages of VSM signalwere determined using themethod
describedinAppendixA.
Figure4.7:TheVSMdata(dots)andfitcurves(lines)forinitialsuspensionscomposedof50%FeOand50%Fe3O4.Theseparationswereperformedonthedatesindicatedinthelegend.
Chapter4.ResultsandAnalysis 36
FieldGradient PercentofVSMsignalattributedtoFeO
PercentofVSMsignalattributedtoFe3O4
4.7T/m(3/14/17) 41.5% 58.5%
10.6T/m(3/12/17) 41.8% 58.2%
23.2T/m(3/21/17) 38.1% 61.9%
48.1T/m(3/23/17) 24.2% 75.8%
Table4.3:Thepercentagesof theVSMsignalattributedtothetwotypesofparticle,basedontheDLLMDanalysisdescribedinAppendixA.Thedatethedataweretakenisgiveninparentheses.
DifferencesintheshapeofthecurvesdisplayedinFigure4.7indicateddifferencesin
thesamplesbeingmeasured.ThiswasconfirmedinthecalculationsdisplayedinTable4.3
from the signal percentages for the FeO and Fe3O4 distributions. The suspensions were
createdatthesametime,butweremeasuredonthedatesindicatedinthefigureandtable.
Atrendcanbeobservedbetweenthedateasamplewasmeasuredandtheamountthatits
VSM signal matches the curve associated with Fe3O4 particles. The XRD measurements
indicatedthattheFe3O4samplewasnearlycompletelyoxidizedandnosettling/clumping
behaviorwasobserved.ThisimpliesthatthemagneticpropertiesoftheFeOnanoparticles
werechangingoverthecourseoftwoweeks,mostlikelyduetosomeslowoxidationprocess
orchangeinthesurfacecharacteristics.ThischangeislikelysimilartothechangesinVSM
signalfoundinSection4.2.
Inproceedingtothenlookatperformingseparationswiththearray,weaccounted
forthisshiftbyusingtherelativechangeintheFeOsignalbetweeneachinitialsampleand
itscorrespondingfiltrate,whichwerepreparedandmeasuredonthesameday.Qualitative
imagesofeachseparationaregiveninFigure4.8(A-D)andclearlyshowthatthearrayis
removinganincreasingquantityofparticlesasthefieldgradientisincreased.
Chapter4.ResultsandAnalysis 37
Figure4.8:Imagesfromseparationsatfieldgradientsof4.7T/m(A),10.6T/m(B),23.2T/m(C),and48.1T/m(D).Nobandingisobservedin(A)or(B),faintandunevenbandingisobservedin(C),anddistinctiveevenbandingisobservedin(D).
ThefiltratesobtainedafterseparationweremeasuredasshowninFigure4.9,andthe
results of DLLMD analysis are displayed in Table 4.4. The fits and VSM data for these
separationsaredisplayed inFigure4.9. TheVSMsignalpercentagescalculated fromthe
DLLMDscalefactorsindicateastheappliedfieldgradientincreased,sodidthepercentage
ofthescalefactorassociatedwiththeFeOdistributionrelativetothesumofthescalefactors,
asshowngraphicallyinFigure4.10.ThisindicatesthatmoreFe3O4particleswereretained
inthechannelasthechannelwasbroughtclosertothearray.
(A)
(C)
(B)
(D)
Chapter4.ResultsandAnalysis 38
Figure4.9:TheVSMdata(dots)andfitcurves(lines)forthefiltratesofthesuspensionscomposedof50%FeOand50%Fe3O4,separatedunderdifferentfieldgradients.Theseparationswereperformedonthedatesindicatedinthelegend.
FieldGradient PercentofVSMsignalattributedtoFeO
PercentofVSMsignalattributedtoFe3O4
PercentchangeofFeOsignal
4.7T/m(3/14/17) 42.5% 57.5% 1.0%
10.6T/m(3/12/17) 44.4% 55.6% 2.6%
23.2T/m(3/21/17) 41.2% 58.8% 3.1%
48.1T/m(3/23/17) 34.6% 65.4% 10.4%
Table4.4:Thepercentagesof theVSMsignalattributedtothetwotypesofparticle,basedontheDLLMD analysis described in Appendix A, as well as the percentage change of the distributionsrelativetotheinitialsamples.Thedatethedataweretakenisgiveninparentheses.
Chapter4.ResultsandAnalysis 39
Figure4.10:AplotofthepercentageshiftofsignalfromFe3O4toFeOvs.appliedfieldgradient.Alinearfitisindicatedbytheblackline.
4.4 Receiver Operating Characteristic Analysis
AssuggestedinChapter2,weexpectedtoperformROCanalysisontheresultsofour
separationsbymeasuringthenumberofFeOandFe3O4particlesbeforetheseparationand
inthefiltrateaftertheseparation.AsindicatedbyTable2.1,wecouldobtaintheTPF(true
positivefraction)usingEquation2.15byconsideringthesenumbersfortheFeOparticles,
andsimilarlywecouldobtaintheFPF(falsepositivefraction)usingEquations2.16and2.17
by considering these numbers for the Fe3O4 particles. Unfortunately, we are unable to
performthisanalysisatthistime,astheVSMdataobtainedforthetwoseriesofseparations
alonehaveprovedinsufficientasdetailedmorecompletelybelow.
Fortheseparationsofsolutionswithvaryingparticleconcentrations,thesimilarities
inVSMsignalfromthefiltratesdidnotprovideenoughinformationtodeterminerelative
changesintheFeOparticleconcentration.Morecritically,alltheVSMdatafromthefiltrates
ofthisseriesindicatedthatessentiallynoFe3O4particlesremainedinsolution,whichwould
meanthattheFPFwouldequalzero.ThispreventstheconstructionofanROCcurve,asall
theassociateddatapointsforthisserieswouldbelocatedalongthey-axisforgraphsofTPF
versus FPF. While such a curve would reflect the significant clumping and sample
degradation,itwouldnotindicatethedeviceperformance.
Chapter4.ResultsandAnalysis 40
Fortheseparationsofsolutionswithvaryingappliedfieldgradient,thereductionin
Fe3O4-likesignaldoesprovideenough informationtocalculate theFPFbycomparingthe
initialandfinalsamples.TheFPFvaluesaregiveninTable4.5andwouldprovideenough
informationtoformacurvegiventheTPFvalues.However,theVSMdataaloneprovideno
insightintothenumberofFeOparticlesthataggregatedduringsolutionasitisdifficultto
retrievetheresidueportionwithoutachangeinconcentrationduetotheneedforadditional
solvent.Additionally,acquiringaggregatedparticlesproveddifficult,asoncetheparticles
hadcomeoutofsolutionwehadnodirectwayofre-agitatingthembackintosuspension
suchas through sonication. This thereforeprevented the calculationof theTPFwithout
furthercorrelatinginformation.
FieldGradient FPF(filtrateFe3O4%/initialFe3O4%)
4.7T/m(3/14/17) 0.98
10.6T/m(3/12/17) 0.96
23.2T/m(3/21/17) 0.95
48.1T/m(3/23/17) 0.86
Table4.5:Thefalsepositivefraction(thepercentofFe3O4particlesthatendupinthefiltrate)fortheseriesofseparationsperformedatvaryingfieldgradients.
However, provided additional data about the number of particles in the initial
solutions and the filtrates through XRD and AAS measurement, an ROC curve could be
constructedinthiscase,giventheexperimentalresultswehavedemonstrateduptonow.
We were able to calculate the TPF and FPF for one of the separations from such
measurementsprovidedbyour collaborators atCWRU, as indicatedby thebluepoint in
Figure 4.11, but encountered problems with the other due to changes in the particle
concentration.Completingsuchanendeavorwouldbevaluableinquantifyingtheefficiency
oftheHalbacharrayasasortingmechanism,andweareworkingonanewsetofseparations
topursuethatgoal.
Chapter4.ResultsandAnalysis 41
Figure4.11:Figure2.5ofexampleROCcurveswiththeadditionoftheTPFandFPFpointcalculatedfromtheXRDandAASmeasurementsofthe25%FeO/75%Fe3O4mixture.
FalsePositiveFraction(FPF)
TruePositiveFraction(TPF)
42
Chapter 5
Conclusions and Future Work
5.1 Conclusions
Particle uniformity is essential for the application of magnetic nanoparticles to a
varietyofdevelopingbiomedicaltechnologiesandprocedures. Thisthesishasworkedto
characterizetheabilityofalinearinvertedHalbacharraytoseparateoutnanoparticlesof
differing magnetic moments under two regimes, one of varying particle concentration
separatedatasinglefieldgradient,andoneofasingleparticleconcentrationseparatedat
varying field gradients. Wehavedevelopedquantitativemethods for analyzing thedata
obtainedbeforeandaftereachseparationbasedonanassumedlognormaldistributionof
magneticmoments. Wehavealsoshownhowthesemethodsmaybeused toapplyROC
analysistoquantifytheuseoftheHalbacharrayasasortingmechanism.
Ourresultsfromvaryingparticleconcentrationindicatedthatasexpected,particle
behaviorinthepresenceoftheHalbacharrayisaffectedprimarilybythefieldgradientand
pump rate experienced by the particles, and not the number of particles present in the
channel.However,ourresultsarehinderedbythechangesofmagneticsignalobservedin
Chapter5.ConclusionsandFutureWork 43
the FeO suspension and the settling behavior observed in the Fe3O4 suspension; these
sampleissuescloudfurtherinterpretationofthedata.
Our results from varying field gradient indicated a positive linear relationship
betweenthepercentofFe3O4particlesthataggregatedoutofthefiltrateandanincreasein
thefieldgradient,whichwasadditionallyconfirmedqualitatively.Theseresultsbeginthe
process of a more complete characterization of the array as a sorting mechanism.
Unfortunately,theinformationextractedthroughouranalysismethodswasinsufficientfor
constructinganROCcurve,andfurtherstructuralcharacterizationoftheresultingsolutions
isnecessary.
5.2 Future Work
Thesamplesusedinthisthesisunderwentstructuralcharacterizationthroughthe
assistanceofourcollaboratorsatCWRU,butchanges intheparticleconcentrationofour
samplespreventedusfrommakingfulluseofthisanalysis. Subsequentexperimentation,
structuralcharacterization,anduseofROCanalysiswillprovide further insights into the
efficiencyoftheinvertedlinearHalbacharrayasananoparticlesortingmechanism.
Fromtheresultsofourexperimentalefforts,weconcludethattherearethreefactors
impedingtheclarityofourcharacterizationofthearray:thechangeinthenanoparticlesover
time, the physical interpretation of our DLLMD analysis, and the nature of the channel
design.
AsevidencedinChapter4,changesinthenanoparticlesamples,whichwereheldin
anhydrous toluene inanargongloveboxuntiluse, occurredover the spanof a coupleof
weeks. This implies that the uniformity of the magnetic properties of a suspension of
nanoparticlesmay apply for a limited time, decreasing the utility of any single batch of
nanoparticles.Whilethismaynotposeaproblemforone-timeusebiomedicalprocesses,it
isanimpedimenttoindustrialsynthesis,purification,andcharacterizationofnanoparticles.
Particlesthataresoprocessedmaynotarriveattheirdestinationinthesamestate,which
could be problematic for applications sensitive to changes in the particles magnetic
Chapter5.ConclusionsandFutureWork 44
properties,suchastherapeutichyperthermia.Therefore,wemayneedtoevaluatethearray
withparticlespreparedinadifferentfashion.
Currently, our method for analyzing a sample containing two types of particles
requiresmeasurementsandanalysisofeachtypeindividually.Thisstrategydoesnotallow
forchangesoftheparticlesovertime,andemploysthesocalled“scalefactors”.Inthesingle
moment Langevinmodel, the normalization constants consist of the number of particles
multipliedbythesinglemagneticmoment.However,thescalefactorcannotbeequivalentto
thesenormalizationconstants,asinourmodeltheparticlemagneticmomentisnolonger
consideredaconstant.Furtheranalysisoftheimpactofthescalefactoronthemodelmay
lead to a physical conclusion, and may help clarify the difficulties encountered when
attemptingtocalculatetheTPFfortheseriesofseparationsperformedwithvaryingfield
gradients.
AsmentionedinSection4.4,aggregatedparticlesremaindifficulttoacquirefromthe
channelwallwithoutchangesinconcentration. Wethereforesuggestthat improvements
might be made to increase particle recovery following the separation processes. One
possibilitywouldbetoreplacethecurrentbaseplatewithadifferentmaterial,ideallyone
thatwouldfacilitateparticlerecovery.Anotherpossibilitywouldbetoredesignthechannel
insuchawaythatitcouldbeplacedinasonicatingbath,energizingtheparticlesandre-
suspendingtheminthecarrierfluid.
45
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49
Appendix A
Mathematica Code for VSM
Analysis
Thefollowingisthetemplateofcodeusedtoanalyzethemagnetometrydatausinga
modifiedLangevinequationmodeldescribedinChapter2.ItwaswritteninMathematica,a
computationalsoftwareprogram,andtheinputswherechangedtomatcheachsetofVSM
data. Presentedbeloware both the single Langevin and thedouble Langevin lognormal
momentdistributionmodel(SLLMDandDLLMDrespectively),aswellasasingleLangevin,
singlemomentmodel(SLSM)forcomparison.Theassumptionofalognormaldistribution
inmomentforeachsamplewasbasedonpreviousdatarelatedtothesynthesisofmagnetic
nanoparticles(Kakayetal.),andisincorporatedintotheLangevinequationintheSLLMD
andDLLMDmodels.TheDLLMDmodelwasusedtoanalyzesolutionscontainingamixture
oftwodifferenttypesofparticles.ASLLMDanalysiswasperformedoneachofthesolutions
usedinthemixture,andtheresultsofthatanalysiswereusedintheDLLMDmodel.
AppendixA.MathematicaCodeforVSMAnalysis 50
Eachanalysistookthesamebasicform,beginningwithaplotofboththedatasetand
thepotentialfitfunction.Thismethodwasusedtoobtainstartingvaluesforeachvariable,
whichwerethenpluggedintotheFindFitfunction,whichusedc2analysistofindthevalues
thatbest fit thedata. Theaccuracyof the fitwas thenconfirmedbyplotting thenew fit
functionwiththedataset,andanyrelevantdatawereextractedfromthefitvalues.Forthe
SLLMDmodel,thefitvariablesweres,µ,s,theslope,andtheoffset,wheresrepresentsthe
scalefactor,andµisthemeanandsisthestandarddeviationofthenormaldistributionon
whichthelognormaldistributionisbased.Thesloperepresentsthemagneticcontribution
ofthesampleholderorcarrierfluidandtheoffsetaccountsforanyerrorsincenteringthe
samplebetweenthedetectioncoilsoftheVSM.Themedianmagneticmomentofthesample
wasdeterminedbytakingthemedianofthelognormaldistributiondefinedbyµands.The
magneticmomentofdesiredportionsofthedistribution,namelythe20thand90thpercentile,
wereobtainedinasimilarmanner.
FortheDLLMDmodel,thefitvariablesweres1,s2,theslope,andtheoffset,wheres1
and s2 represent the scale factor for distribution 1 and distribution 2 respectively. This
model requires SLLMD analysis on the two types of particles used, with µ1 and s1
correspondingtothefitvaluesforthefirstdistribution,andµ2ands2correspondingtothe
fitvaluesfortheseconddistribution.ThepercentageofVSMsignalfromeachtypeofparticle
wasobtainedbyfindingthepercentageofeachscalefactorincomparisontothesumofthe
scalefactors.
Data Import(*These are instructions for fitting magnetometry data.*)
Import[
"C:\\Users\\moment\\Documents\\Your stuff\\Your folder (optional)\\Your file.xlsx"]
(*Specify the data file and import it. The data works better if you
import it from an Excel file, which is why "Your file" ends in .xlsx .*)
(*data*) = (*Insert the data set just imported here. Make sure there isn't an
extra set of curly brackets around it because Mathematica will not accept.*);
k = 4.1 * 10^-14; (*This is your value of KbT, the thermal energy,
which we say is constant for this experiment. Before running any of the fits,
make sure to store this value.*)
Single Moment ModelLangevin[h_] := n * m * Cothm * h k - k m * h
(*This is the Langevin equation as defined in Chapter 2,
where n means the number of particles, m is the magnetic particle moment, and the
slope adjusts for any paramagnetic or diamagnetic signal from the sample holder.*)
Show[Plot[Langevin[h] + (*slope*) * h, {h, -10 000, 10000}], ListPlot[(*data*)]]
(*Change your guesses of n, m, and the slope,
along with whatever the name of the dataset is for data. This will
help you eyeball the fit so you have starting guess values later.*)
FindFit(*data*), n * m * Cothm * h k - k m * h + l * h,
{{n,(*starval*)}, {m,(*startval*)}, {l,(*startval*)}}, h
(*Plug in your starting guess values for n, m, and l wherever it says startval,
and the name of your dataset where it says data. It's going to give you some error
messages and take some time 5-15 minutes, more if you have bad startvals,
but eventually it will pop out the best fit values for the three parameters.*)
ShowPlot(*n*) * (*m*) * Coth(*m*) * h k - k (*m*) * h + (*slope*) * h,
{h, -10 000, 10000}, ListPlot[(*data*)]
(*Same thing you did before for eyeballing the fit,
only now you plug in the values that you just got for the
parameters. This is just so you can see how well the fit was approximated.*)
Log-normal Moment Distribution ModelpdfLangevin(*I*)[h_?NumericQ] :=
NIntegratePDF[LogNormalDistribution[(*μ*), (*σ*)], m] *
Cothm * h k - k m * h, m, 0, 10-12
(*Adjust the name of this function if you are working with more than one set
of data in one notebook by changing the I to the appropriate identifying letter*)
Show[Plot[(*s*) * pdfLangevin[h] + (*slope*) * h, {h, -10 000, 10 000}],
ListPlot[(*data*)]]
(*Change your guesses of s, μ, and σ, along with whatever the name of
the dataset is for data. This will help you eyeball the fit so you have
starting guess values later. Here, s means saturation value or scale factor,
μ is the mean of the normal distribution that the lognormal fit is derived from,
and σ is the standard deviation of this normal distribution,
and controls the variance.*)
FindFit(*data*), s * NIntegrate
PDF[LogNormalDistribution[μ, σ], m] * Cothm * h k - k m * h, m, 0, 10-12 + l * h,
{{s,(*startval*)}, {μ,(*startval*)}, {σ,(*startval*)}, {l,(*startval*)}}, h
(*Plug in your starting guess values for s, μ, and σ wherever it says startval,
and the name of your dataset where it says data. It's going to give you some error
messages and take some time 5-15 minutes, more if you have bad startvals,
but eventually it will pop out the best fit values for the three parameters.*)
ShowPlot(*s*) * NIntegratePDF[LogNormalDistribution[(*μ*), (*σ*)], m] *
Cothm * h k - k m * h, m, 0, 10-12 +
(*slope*) * h, {h, -10 000, 10 000}, ListPlot[(*data*)]
(*Same thing you did before for eyeballing the fit,
only now you plug in the values that you just got for the
parameters. This is just so you can see how well the fit was approximated.*)
Median[LogNormalDistribution[(*μ*), (*σ*)]]
Probability[x <(*moment*), x LogNormalDistribution[(*μ*), (*σ*)]]
(*For checking the median of your distribution and what
percentile various moment values are. Useful once you start calculating
separation parameters in the Excel file. Values should be in emu.*)
Log-normal Moment Distribution Double Langevin ModelpdfLangevin(*I1*)[h_?NumericQ] :=
NIntegratePDF[LogNormalDistribution[(*μ1*), (*σ1*)], m] *
Cothm * h k - k m * h, m, 0, 10-12
(*Adjust the name of this function by changing the I1 and
I2 to the appropriate identifying letter,
and plug in the μ and σ values that match the two distributions you are combining,
distribution 1 and distribution 2 respectively.*)
pdfLangevin(*I2*)[h_?NumericQ] :=
NIntegratePDF[LogNormalDistribution[(*μ2*), (*σ2*)], m] *
Cothm * h k - k m * h, m, 0, 10-12
2 MathmaticaFitTemplate.nb
Show[Plot[(*s1*) * pdfLangevin(*I1*)[h] + (*s2*) * pdfLangevin(*I2*)[h] +
(*slope*) * h +(*offset*), {h, -10 000, 10 000}], ListPlot[(*data*)]]
(*Change your guesses of s1, s2, the slope and offset,
along with whatever the name of the dataset is for data. This will
help you eyeball the fit so you have starting guess values later. Here,
s means saturation value or scale factor,
μ is the mean of the normal distribution that the lognormal fit is derived from,
and σ is the standard deviation of this normal distribution, and controls the variance.*)
FindFit(*data*), s1 * NIntegrate
PDF[LogNormalDistribution[μ1, σ1], m] * Cothm * h k - k m * h, m, 0, 10-12 +
s2 * NIntegratePDF[LogNormalDistribution[μ2, σ2], m] * Cothm * h k - k m * h,
m, 0, 10-12 + l * h + g, {{s1,(*startval*)},
{s2,(*startval*)}, {l,(*startval*)}, {g,(*startval*)}}, h
(*Plug in your starting guess values for s1, s2, the slope,
and the offset wherever it says startval, and the name of your dataset where it
says data. It's going to give you some error messages and take a minute or two,
and eventually it will pop out the best fit values for the three parameters.*)
ShowPlot(*s1*) * NIntegratePDF[LogNormalDistribution[(*μ1*), (*σ1*)], m] *
Cothm * h k - k m * h, m, 0, 10-12 +
(*s2*) * NIntegratePDF[LogNormalDistribution[(*μ2*), (*σ2*)], m] *
Cothm * h k - k m * h, m, 0, 10-12 + (*slope*) * h +
(*offset*), {h, -10 000, 10 000}, ListPlot[(*data*)],
AxesLabel → {"Field (Oe)", "Moment (emu)"}
(*Same thing you did before for eyeballing the fit,
only now you plug in the values that you just got for the
parameters. This is just so you can see how well the fit was approximated.*)
Percent of VSM signal from Langevin 1
s1
s1 + s2
(*This should give you approximately the amount
of signal the VSM receives from the 1st distribution*)
Percent of VSM signal from Langevin 2
s2
s1 + s2
(*This should give you approximately the amount
of signal the VSM receives from the 2nd distribution*)
s1
s2
(*This gives a ratio of the two distributions*)
Example and ComparisonBelow is a comparison of the two methods, the first modeling an example set of data by assuming a
single magnetic moment, and the second modeling the same data set by assuming a lognormal distribu-
MathmaticaFitTemplate.nb 3
tion of magnetic moments.
Single Moment Langevin
Langevin[h_] := 5.5 × 1012 * 10-15 * Coth10-15 * h k - k 10-15 * h
ShowPlotLangevin[h] - 8 × 10-8 * h, {h, -10 000, 10000}, ListPlot[Fe3O4orig]
-10 000 -5000 5000 10 000
-0.004
-0.002
0.002
0.004
FindFitFe3O4orig, n * m * Cothm * h k - k m * h + l * h,
n, 5.5 × 1012, m, 10-15, l, -8 × 10-8, h
n → 1.58678 × 1012, m → 3.17244 × 10-15, l → 3.46227 × 10-9
ShowPlot1.5867814037991392`*^12 * 3.17244122984884`*^-15 *
Coth3.17244122984884`*^-15 * h k - k 3.17244122984884`*^-15 * h +
3.462266511992397`*^-9 * h, {h, -10 000, 10000}, ListPlot[Fe3O4orig]
-10 000 -5000 5000 10 000
-0.004
-0.002
0.002
0.004
Lognormal Moment Distribution Langevin
pdfLangevinfe304[h_?NumericQ] :=
NIntegrate
PDF[LogNormalDistribution[-34, 1], m] * Cothm * h k - k m * h, m, 0, 10-12
4 MathmaticaFitTemplate.nb
ShowPlot0.0055 * pdfLangevinfe304[h] - 5 × 10-8 * h, {h, -10 000, 10000},
ListPlot[Fe3O4orig]
-10 000 -5000 5000 10 000
-0.004
-0.002
0.002
0.004
FindFitFe3O4orig,
s * NIntegratePDF[LogNormalDistribution[μ, σ], m] * Cothm * h k - k m * h,
m, 0, 10-12 + l * h, {s, 0.0055}, {μ, -34}, {σ, 1}, l, -5 × 10-8, h
NIntegrate: The integrand -4.1×10-14
hm+ Coth2.43902×1013 hm
ⅇ-1
2Power[2] Power[2]
m 2π σ
m > 0
0 True
has evaluated to
non-numerical values for all sampling points in the region with boundaries 0,1
1000000000000.
NIntegrate: The integrand -4.11514×10-18
m+ Coth2.43005×1017 m
ⅇ-1
2Power[2] Power[2]
m 2π σ
m > 0
0 True
has evaluated to
non-numerical values for all sampling points in the region with boundaries 0,1
1000000000000.
NIntegrate: The integrand -4.25832×10-18
m+ Coth2.34834×1017 m
ⅇ-1
2Power[2] Power[2]
m 2π σ
m > 0
0 True
has evaluated to
non-numerical values for all sampling points in the region with boundaries 0,1
1000000000000.
General : Further output of NIntegrate::inumr will be suppressed during this calculation.
s → 0.00547502, μ → -33.3226, σ → 1.91298, l → -5.19952 × 10-8
MathmaticaFitTemplate.nb 5
Show
Plot0.0054750237751093065` * NIntegratePDF[LogNormalDistribution[-33.32256712064471`,
1.9129827701272757`], m] * Cothm * h k - k m * h, m, 0, 10-12 -
5.199524268171383`*^-8 * h, {h, -10 000, 10000}, ListPlot[Fe3O4orig]
-10 000 -5000 5000 10 000
-0.004
-0.002
0.002
0.004
Median[LogNormalDistribution[-33.32256712064471`, 1.9129827701272757`]]
3.37437 × 10-15
Probabilityx < 4 × 10-14,
x LogNormalDistribution[-33.32256712064471`, 1.9129827701272757`]
0.901921
6 MathmaticaFitTemplate.nb
Comparison
In[198]:= ShowListPlot[Fe3O4orig, PlotStyle → {Blue}, PlotMarkers → {Automatic, Tiny}],
Plot1.5867814037991392`*^12 * 3.17244122984884`*^-15 *
Coth3.17244122984884`*^-15 * h k - k 3.17244122984884`*^-15 * h +
3.462266511992397`*^-9 * h, 0.0054750237751093065` *
NIntegratePDF[LogNormalDistribution[-33.32256712064471`, 1.9129827701272757`], m] *
Cothm * h k - k m * h, m, 0, 10-12 - 5.199524268171383`*^-8 * h,
{h, -10 000, 10000}, PlotStyle → {{Black, Thickness[0.004], Dashing[0.015]}, {Purple}},
PlotLegends → Placed[{"Single Moment", "Lognormal Moment Distribution"}, {0.25, 0.75}],
AxesLabel → {"Applied Field (Oe)", "Moment (emu)"},
Frame → True
Out[198]=
●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●
●
●
●
●
●
●
●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
●●●●●●●●●
●
●
●
●
●●●●●●●●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Single Moment
Lognormal Moment Distribution
-10 000 -5000 0 5000 10 000
-0.004
-0.002
0.000
0.002
0.004
Applied Field
Moment (emu)
As can be seen in the graph above, the Lognormal Moment Distribution model (in purple) matches the
data (the blue dots) better than the Single Moment model (the dashed black curve). The Single Moment
model gives a slightly higher magnetic moment of 3.17 x 10-15 emu versus the Lognormal Moment
Distribution model, which gives a median magnetic moment of 3.37 x 10-15 emu and a σ value of 1.9.
MathmaticaFitTemplate.nb 7
58
Appendix B
Solidworks CAD Designs
FiguresoftheCADdesignsofthealuminumchannelbase,toluene-compatibleglass
plate, and plexiglass plate used during the experimental section of this thesis have been
includedinthisAppendix.
Figure B.1: Design of the plexiglass top for the toluene-compatible channel, designed byEmilyHamlin,withholesfor18aluminumscrewstoholdthesystemtightandpreventleaksofnanoparticlesolutions.
AppendixB.SolidworksCADDesigns 59
FigureB.2:Designoftheglasscomponentforthetoluene-compatiblechannel,designedbyEmilyHamlin,withtwoholesforplastictubing.ThispiecewasheldinplacebytheplexiglasspieceshowninFigureB.1.
AppendixB.SolidworksCADDesigns 60
FigureB.3:Designofthestainlesssteelbottomplateusedinthetoluene-compatiblechannel,designedbyChetanPoudelinsuchawaythattheglassplatefitstightly. Screwsholdthebottomplatewiththetopplateandholdtheglassplateinthemiddlefirmlyinplace.
11.00
1.45
0.2
5
10.00
0.80 0
.20
ISOMETRIC VIEW
0.5
6 0
.56
14 x clearance holes for 6-32
0.1
6 0
.16
0.8
0
R0.
10
0.2
0
clearance hole for 6-32
0.25
recess for 6-32 socket head
clea
ranc
e 0.
15 0.80
0.25 2.10 2.10 2.10 2.10 2.10 0.25
recess for 6-32 socket head
scale 2:1
TOP VIEW
BOTTOM VIEW
SIDE VIEW
bottom plate drawingWEIGHT:
A4
SHEET 1 OF 1SCALE: 1:2 unless otherwise mentioned
DWG NO.
TITLE:
REVISIONDO NOT SCALE DRAWING
MATERIAL: Stainless Steel 316
DATESIGNATURENAME
DEBUR AND BREAK SHARP EDGES
FINISH:UNLESS OTHERWISE SPECIFIED:DIMENSIONS ARE IN MILLIMETERSSURFACE FINISH:TOLERANCES: LINEAR: ANGULAR:
Q.A
MFG
APPV'D
CHK'D
DRAWN By Chetan PoudelJuly 3, 2013