Separation of FeO and Fe O Nanoparticles Using an …

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .


4 particles

for initial mixtures used in the changing field gradient experiment,

using a double Langevin analysis . . . . . . . . . . . . . . . . . . .


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.).


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.).


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


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)


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


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


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


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)


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)


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)


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)


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)


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)


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


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.


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)


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





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.


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


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.


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)


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


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,


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.


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.


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


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.


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.


FieldGradient PercentofVSMsignalattributedtoFeO


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.


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)


Figure4.9:TheVSMdata(dots)andfitcurves(lines)forthefiltratesofthesuspensionscomposedof50%FeOand50%Fe3O4,separatedunderdifferentfieldgradients.Theseparationswereperformedonthedatesindicatedinthelegend.

FieldGradient PercentofVSMsignalattributedtoFeO


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.


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.


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.


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*)]




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)"}




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


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.


m+ Coth2.43005×1017 m

ⅇ-1

2Power[2] Power[2]

m 2π σ

m > 0

0 True

has evaluated to


1000000000000.


m+ Coth2.34834×1017 m

ⅇ-1

2Power[2] Power[2]

m 2π σ

m > 0

0 True

has evaluated to


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


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


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.


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

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