Modeling and characterization of magnetic nanoparticles intended for cancer treatment

(1)

UPTEC Q13 001

Examensarbete 30 hp

April 2013

Modeling and characterization

of magnetic nanoparticles intended

for cancer treatment

(2)

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Modeling and characterization of magnetic

nanoparticles intended for cancer treatment

Mikael Andersson

Cancer is one of the challenges for today's medicine and therefore a great deal of effort is being put into improving known methods of treatment and developing new ones. A new method that has been proposed is magnetic hyperthermia where magnetic nanoparticles linked to the tumor dissipate heat when subjected to an alternating magnetic field and will thus increase the temperature of the tumor. This method makes the tumor more susceptible to radiation therapy and chemotherapy, or can be used to elevate the temperature of the tumor cells to cause cell death. The particles proposed for this are single core and often have a size in the range of 10 nm to 50 nm. To achieve an effective treatment the particles should have a narrow size distribution and the proper size. In this work, a theoretical model for predicting the heating power generated by magnetic nanoparticles was evaluated. The model was compared with experimental results for magnetite particles of size 15 nm to 35 nm dissolved in water. The properties of the particles were characterized, including measurements of the magnetic saturation, the effective anisotropy constant, average size and size distribution. To evaluate the results from the model the AC

susceptibility and heating power were experimentally determined. The model is a two-step model. First the out-of-phase component of the AC susceptibility as a function of frequency is calculated. Then this result is used to calculate the heating power. The model gives a correct prediction of the shape of the out-of-phase component of the susceptibility but overestimates its magnitude. Using the

experimentally determined out-of-phase component of the susceptibility, the model estimation of the heating power compares quite well with the measured values.

ISSN: 1401-5773, UPTEC Q13 001 Examinator: Åsa Kassman

Ämnesgranskare: Peter Svedlindh Handledare: Eva Welinder

(3)

Karakterisering och modellering av magnetiska

nanopartiklar för cancerbehandling

Mikael Andersson

En av de stora utmaningarna för sjukvården idag är att på ett effektivt sätt behandla cancer.

Idag behandlas cancer oftast genom strålterapi, cellgifter eller kirurgi. Dessa metoder har vissa

svagheter, som att inte ha tillräckligt hög selektivitet, d.v.s. att inte endast eller huvudsakligen

de sjuka cellerna utsätts för behandlingen. För att öka selektiviteten och effektiviteten hos

bekämpningen av cancer så bedrivs mycket forskning för att utveckla nya metoder och för att

förbättra redan existerande.

En alternativ metod som har forskats på, magnetisk hypertermi, är att med hjälp av

magnetiska nanopartiklar höja temperaturen i cancercellerna för att göra cancercellerna

mindre motståndskraftiga i samband med strålterapi eller cellgifter. Temperaturhöjningen sker

genom att de magnetiska partiklarna utsätts för ett snabbt varierande magnetiskt fält med en

stor amplitud.

I detta arbete prövas en teoretisk modell för att beräkna hur mycket värme de magnetiska

nanopartiklarna genererar per sekund. Modellen prövades genom att experimentella data för

partiklarna uppmättes och utifrån datat beräknades mängden värme per sekund som

genererades av nanopartiklarna. De beräknade värdena jämfördes sedan med de

experimentellt uppmätta värdena för uppvärmingen. Modellen beräknar fram uppvärmningen

i två steg. Först beräknas en mätbar storhet som kallas urfassusceptibiliteten. Från

urfassusceptibiliteten beräknas sedan uppvärmningen. Urfassusceptibiliteten varierar med

temperatur och frekvens. Då temperaturen är relativt konstant under en behandling, så är det

mer intressant ur ett applikationsperspektiv att studera inverkan av frekvensen på

urfassusceptibiliteten. I figur 1 visas hur urfassusceptibiliteten kan variera med frekvens. I

figuren visas resultatet från en AC-susceptibilitetsmätning för 30 nm partiklar.

Figur 1. AC-susceptibiliteten som en funktion av frekvens för 30 nm partiklar från Ocean NanoTech. Mätningen

är utförd i frekvensintervallet 1 Hz till 100 kHz. Den blå linjen motsvarar ifaskomponenten och den röda linjen

100 101 102 103 104 105 -1 0 1 2 3 4 5 6 Frequency (Hz)

AC

s

us

cept

bi

lity

c

om

ponent

s (

di

m

ens

ionl

es

s)

30nm

(4)

De experimentellt uppmätta parametrarna som modellen använder för att beräkna

urfassuceptibiliteten är medelstorleken för partiklarna, deras storleksfördelning och

mättnadsmagnetisering samt den effektiva magnetiska anisotropikonstanten.

Medelstorleken och storleksfördelningen uppskattades med hjälp av bilder tagna med

transmissionselektronmikroskopi. För att analysera bilderna användes ett bildanalysprogram

som identifierade partiklarna och räknade ut deras storlek. Från storleksdatat beräknades en

medelstorlek och en standardavvikelse. För att mäta mättnadsmagnetiseringen användes en

vibrerande provmagnetometer där partiklarna utsattes för magnetfält på upp till 5 T. För att

bestämma den effektiva anisotropikonstanten utfördes en mätning av ifassuceptibiliteten med

konstant frekvens men med varierade temperatur. Mätningen utfördes med en

AC-susceptometer. Från mätningen kunde den effektiva anisotropikonstanten beräknas.

För att kunna jämföra den beräknade urfassusceptibiliteten så gjordes en mätning av

urfassusceptibiliteten med en AC-susceptometer. Mätningen utfördes med konstant

temperatur men med varierande frekvens. För att kunna jämföra den beräknade

uppvärmningen utfördes ett test där partiklarna som var lösta i vatten utsattes för ett stort

magnetfält med hög frekvens samtidigt som temperaturen mättes.

När den experimentellt uppmätta urfassusceptibiliteten jämfördes med den beräknade så

var den beräknade större, men hade rätt utseende. Detta innebär att modellen kan förutsäga

hur urfassuceptibiliteten varierar med temperatur och frekvens men inte dess storlek. Eftersom

urfassusceptibiliteten är större än den experimentellt uppmätta så är även uppvärmningen

större än den experimentellt uppmätta. För att testa hur det andra steget i modellen

överensstämmer med verkligheten så beräknades uppvärmningen från den experimentellt

uppmätta urfassusceptibiliteten och jämfördes med den experimentellt uppmätta

uppvärmningen. Modellen klarade av att beräkna ett värde som är relativt nära det

experimentellt uppmätta värdet på uppvärmningen.

I arbetet framkom också att mättnadsmagnetiseringen hos partiklarna var mycket lägre

jämfört med mättnadsmagnetiseringen för bulkmagnetit. Då urfaskomponenten för

AC-susceptibiliteten är beroende av mättnadsmagnetiseringen och uppvärmningen är beroende av

urfaskomponenten för AC-susceptibiliteten, så skulle en ökad mättnadsmagnetisering leda till

ökad uppvärmning.

Examensarbete 30 hp på civilingenjörsprogrammet

Teknisk fysik med materialvetenskap

(5)

1 Introduction

Today several methods are used to combat cancer, e.g. chemotherapy, surgery and radiation therapy [1]. These methods often have poor selectivity or involve potentially complicated surgery. This has led to an ongoing search for new as well as further development of existing methods, to replace or to be combined with existing ones. In recent years it has been suggested that magnetic nanoparticles surface-functionalized with biological probes can be attached to cancer tumors [1, 2]. If the particles are subjected to an alternating external magnetic field they can generate heat. By choosing an optimal field amplitude and an optimal frequency of the AC magnetic field the particles can generate enough heat to damage or kill the surrounding cells by elevating the temperature of the cell, while the rest of the body is more or less unaffected. This means that this suggested method could become highly selective. Elevation of temperature with the help of magnetic nanoparticles subjected to a magnetic AC field is often called hyperthermia. A current problem in magnetic hyperthermia is that the applied field and frequency must be limited to avoid inducing eddy currents in the patient receiving the treatment, yet still reach high enough heating capability. Furthermore, the particles must be biocompatible and stable in a biological environment.

Much of the research effort has so far focused on iron oxide nanoparticles, mainly F e3O4. The probable reason

for this is that F e3O4 nanoparticles are biocompatible [3]. In recent years some new nanoparticles systems have

been synthesized that are believed to be biocompatible and have different magnetic properties compared to F e3O4

[4]. Particles with larger remanent magnetization are limited by the fact that they often tend to agglomerate due to magnetic interactions. A solution to this problems is to covalently bond so-called ”polymer brushes” to the surface of the particles. By doing this the particles become sterically stabilized and thereby the nanoparticles will not agglomerate[5].

In this work a two step model for predicting the heating power of magnetic nanoparticle systems is evaluated by comparing the model to experimental results. To be able to compare the model to the experimental data, the properties of five particle systems were measured experimentally. The aim of this work was to be able to predict optimal properties for a particle system. The aim was also to determine what enhancement of properties for the tested particle systems that would yield the greatest increase in heating power.

(7)

2 Theory

2.1 Magnetic properties

2.1.1 Magnetic domains

A magnetic domain can be viewed as a group of magnetic dipoles that are coupled together by exchange interaction so that their magnetic moments are aligned in the same direction [6]. For large volumes it is not energetically favorable to have one large domain. Instead, many small domains are formed to minimize the energy of the material. These domains are separated by domain walls. Since the domain walls also cost energy to form, the material will arrive at equilibrium with a certain number of domains separated by a certain number of domain walls. The domain energy is proportional to the domain volume and the domain wall energy is proportional to the domain wall area. For very small particles it is not favorable from an energy perspective to have many domains and the particle will therefore only form one domain. This is called a single-domain particle [7]. The direction of the collective magnetic moment for a single-domain particle depends on the energy landscape of the particle. For a particle with uniaxial magnetic anisotropy in zero applied magnetic field and at a temperature of zero Kelvin there will be two energy minima and two possible directions for the magnetic moments corresponding to the two energy minima, see figure 2.1 b), H0 = 0. The magnetic energy for a magnetic particle with uniaxial anisotropy can be described by the following equation,

E = V Kef fsin2(θ) − mSHµ0cos(π − θ), (2.1)

where V is the magnetic volume of the particle, Kef f is the effective anisotropy energy constant and H is the

applied magnetic field. θ is the angle between the easy axis of magnetization and the direction of the particle magnetic moment, mS = MSV . π − θ is the angle between the direction of the magnetic field and the direction of

the particle magnetic moment [8], see figure 2.1 a). Rewriting equation 2.1 yields, E V Kef f = sin2(θ) −mSHµ0cos(π − θ) V Kef f , (2.2) HK = 2Kef f µ0MS , (2.3)

where HK is an imaginary field, called the anisotropy field, that tries to keep the magnetic moment in the particle

along the easy axis of magnetization, and MS is the saturation magnetization. By introducing the zero field energy

barrier separating the two minima for a uniaxial particle, EB = V Kef f, and combining equations 2.2 and 2.3, one

arrives at an even more simplified expression for the energy, E

EB

= sin2(θ) + 2H0cos(θ), (2.4) where H0 = H/HK. Equation 2.4 is the magnetic energy expressed in units of energy barrier EB. It can be plotted

to describe the energy landscape for different magnetic fields. To switch between the two minima, one can apply a field in the opposite direction of the particle magnetic moment; if the field is large enough the energy barrier will be erased and the moment will switch direction, see figure 2.1 b) H0= 1.

(8)

θ

Easy axis of

magnetization

Direction of applied magnetic field H

M

a)

b)

0 90 180 -2 -1 0 1 2

En

er

gy



in degrees

H´= 0 H´= 0.5 H´= 1

Figure 2.1: a) A representation of a uniaxial magnetic particle in a magnetic field applied along the easy axis of magnetization. b) Graph of equation 2.4 illustrating the energy landscape for a uniaxial magnetic particle for different H0. H0 = 0 corresponds to no applied magnetic field, while H0 = 1 corresponds to an applied magnetic field equal to the anisotropy field HK. E is in units of energy barrier, EB.

2.1.2 Superparamagnetism

If the size for a single-domain particle with uniaxial anisotropy is small enough, the thermal agitation can be large enough for the particle magnetic moment to randomly switch between the two energy minima. This can be described by a probability function and as the thermal energy increases, the odds of switching increases. In a system with many particles the probability can be viewed as a relaxation time for the system. Each superparamagnetic particle can on a larger timescale and in the absence of an applied magnetic field be viewed as non-magnetic since the particle magnetic moment will have spent the same amount of time along the two easy directions. This implies that if a short measurement time is used the particle will appear magnetic, but with a sufficiently increased measurement time the particle will appear non-magnetic [9]. This means that a system can appear either superparamagnetic or ferromagnetic for the same temperature depending on the measurement time. To determine if a system of particles is superparamagnetic or not, the temperature dependent relaxation time of the system τ is compared with the measurement time τM; if τM >> τ the system is superparamagnetic. The temperature TB corresponding to

τ = τM is often called the blocking temperature. With increasing temperature, the coercive field HC will approach

zero as the temperature approaches TB and the hysteresis loop will close. This is due to the fact that the thermal

energy prevents any stable magnetic direction [7]. HC can be viewed as the field required to bring a magnetized

material back to zero magnetization [8].

2.2 Literature study

2.2.1 Ways of heating with magnetic nanoparticles

There are several possible ways for magnetic particles to absorb energy and to transfer it as heat to the surrounding medium, e.g. by inducing eddy currents in the particle, through Brownian relaxation and N´eel relaxation (particles near the blocking temperature) or through hysteresis loss (particles far below the blocking temperature) [10]. An estimate is that about 100 kW/m3 _{should be sufficient for a successful hyperthermia treatment [11]. In Brownian}

relaxation the particles rotate in the medium in which they are disssolved to align the magnetic moment along the direction of the applied magnetic field. This is because when the field changes the system is no longer in equilibrium, and as the particles rotate to align themselves with the field to bring the system back into equilibrium, energy is

(9)

absorbed.

The average time it takes for a particle to align itself and thereby bring the system back into equilibrium is called the Brownian relaxation time, τB. For very short relaxation times the particle magnetic moments will follow the

field and the system will therefore be in equilibrium and no energy is lost. For very long relaxation times the particle magnetic moments will not change direction and the system will never have time to relax to equilibrium, which means that the system never makes a transition from a higher energy state to a lower energy state and therefore no energy is lost. In N´eel relaxation the particles relax as the particle magnetic moments change direction, see section 2.1.2. For N´eel relaxation heat is generated as the particle magnetic moment changes direction to the easy direction of magnetization closest to the magnetic field direction and thereby brings the system to equilibrium [12].

Heating through eddy currents can be substantial if the nanoparticles are subjected to a high frequency field with a large field amplitude, but this can result in problems where eddy currents are also induced in the body fluids. For magnetic nanoparticles that are close to the superparamagnetic limit at body temperature the heat generating process is generally dominated by N´eel or Brownian relaxation, while for single-domain nanoparticles that are far below the blocking temperature at body temperature the heat generating process is generally dominated by hysteresis loss [3]. For multi-domain particles it has been shown that the heat generated by domain wall motion is low compared to heat generated by N´eel relaxation or Brownian relaxation in single-domain particles [13].

2.2.2 Hyperthermia therapy and considerations for magnetic hyperthermia

Hyperthermia therapy can be combined with radiation therapy and chemotherapy and enhances the efficiency of these therapies. This is because the enzyme that is involved in repairing radiation damage in the cells is damaged by the hyperthermia treatment and the transportation mechanisms responsible for transportation of toxic elements and chemicals from the cells are disrupted by the hyperthermia treatment [14]. To avoid damage to the patient due to induced eddy currents in the body fluids, the value of Hf = 4.85 × 108A/sm has been proposed as a maximum value for safe treatment, where H is the amplitude and f is the frequency of the AC field. For short treatments a value of 109A/sm has been proposed [11]. In Germany clinical trials were conducted where a field amplitude of 4 kA/m to 5 kA/m was used with a frequency of 100 kHz, which corresponds to Hf values of 4 × 108A/sm to 5 × 108_{A/sm. The side effects of the treatment were small and could be tolerated. The temperature was elevated}

high enough so that the treatment yielded promising results, but according to the authors the distribution of the particles in the tumor has to be improved to have a more uniform heating in the tumor to improve the results of the treatment [14].

The flow of blood through many types of solid tumors is less than the flow through healthy tissue. Due to the lower flow a tumor cannot transfer away heat as well as healthy tissue can. This means that the tumors are easier to heat than normal tissue. The lower flow of blood through a tumor also means that tumors have a lower pH than normal tissue, due to the fact that the cells do not receive enough oxygen. Cells with lower pH are more affected by temperature changes [15].

2.2.3 Magnetic nanoparticles and their properties

Caruntu et al. reported synthesis of F e3O4 nanoparticles with sizes between 6.6 nm to 17.8 nm in diameter. The

magnetic properties, given by the saturation magnetization MS and anisotropy constant K measured at 300 K,

varied between sizes; the extracted MS and K values for 6.6 nm particles were 70.7 Am2/kg and 4.74 × 105J/m3

respectively, while corresponding values for 11.6 nm particles were 77.4 Am2_{/kg and 1.11 × 10}5_J/m3_{. A saturation}

magnetization of 82.5 Am2_{/kg was obtained for particles of size 17.8 nm but no anisotropy constant was calculated}

for this size since TB was above room temperature. The anisotropy constant was calculated using TB = _25kKV

B,

where kB is Boltzmann’s constant, V is the volume of the particle and TB is the blocking temperature for a

superparamagnetic particle; see section 2.1.2 for further explanation of blocking temperature. The anisotropy constant was calculated with the assumption that the particles had uniaxial anisotropy and were non-interacting [16].

Bae et al. reported synthesis of noncytotoxical CoF e2O4 and N iF e2O4magnetic nanoparticles. The size of the

CoF e2O4particles was 26.5 nm and they had a saturation magnetization of 50.8 Am2/kg. The size of the N iF e2O4

particles was 24.8 nm and they had a saturation magnetization of 36.6 Am2/kg. The authors did not report any anisotropy constant for either of particle systems [17].

(10)

2.2.4 Synthesis and surface functionalization of iron oxide nanoparticles

As mentioned in the introduction iron oxide particles have received a lot of focus due to the fact that they are biocompatible. A demand on particles that are intended for use in a medical application is that the particles are non toxic. Other demands are production of monodisperse particles and easy functionalization of the particles to ensure colloidal stability and the possibility for the particles to bind to cancer tissue. The process used to meet the above mentioned demands should preferably produce particles in large quantities with little effect on the environment.

Synthesis by Thermal Decomposition

In this method chemical decomposition of an organo-metallic complex at elevated temperature followed by an oxidation process produces monodisperse particles. The synthesis method can quite easily be upscaled for mass production of monodisperse magnetic nanoparticles but it has the disadvantage that the particles produced can generally only be dissolved in polar solvents [18]. A method where F eO(OH) is mixed with oleic acid in 1-octadecene and heated to provide an iron carboxylate salt that then decomposes in pyrolysis to form iron oxide nanoparticles is used by Ocean NanoTech. In more detail, the carboxylate salt is formed when iron oxide powder is dissolved in oleic acid and 1-octadecene when heated to about 200◦C. The solution is then heated to 320◦C while stirring. At 250◦C and higher temperatures the complex becomes unstable and at temperatures above 300◦C black magnetite nanocrystals are formed. The solution is then cooled and the nanoparticles extracted using the same methods as for quantum dots [19, 20].

Hydrothermal synthesis

The reactants are dissolved in water in a hydrothermal cell (often a Teflon-lined autoclave is used). The cell is then heated to a temperature in the range of 130◦C to 200◦C with a vapor pressure of 0.3 MPa to 4 MPa. These synthesis conditions lead to crystallization of the reactants, which form nano particles [18]. As reported by Wang et al. [21] hydrothermal synthesis can yield magnetite nanoparticles with very high MS, 85.8 Am2/kg for 40 nm

particles. But it should also be noted that using the same method the authors produced 25 nm particles that had only a saturation magnetization of 12.3 Am2/kg [21].

Sonochemical synthesis

In sonochemical synthesis ultrasound is used to create acoustic cavitations, i.e. formation, growth and collapse of bubbles in a liquid. When the bubbles implode a shock wave is formed in the gas phase, which generates a localized hot spot with extreme conditions. The transient temperature in the bubble is around 5000 K, the cooling rates is above 1010_{K/s and the pressure in the region of 1800 atm. These three extreme conditions were experimentally}

determined [18]. 10 nm F e3O4particles can be synthesized from iron(II)acetate in water with an argon atmosphere

with sonochemical synthesis, as shown by Viyaykumar et al.. The magnetic particles had very low MS, less then

1.25 Am2/kg [22], which is too low for hyperthermia treatment.

Synthesis Co-precipitation

Co-precipitation is a method where a 2:1 molar ratio of ferrous and ferric ions is mixed in a solution with low pH to produce F e3O4or γ − F e2O3nanoparticles. The process can be performed at either elevated temperatures or at

room temperature and the parameters for the method determine the shape and size of the produced nanoparticles. Some parameters that can be varied to change the shape and size of the nanoparticles are reaction temperature, pH and the type of salts used to add the ferrous and ferric ions. Particles produced by this method have MS between

30 Am2_{/kg to 80 Am}2_{/kg, which is lower than the value reported for bulk by Cullity and Graham of 98 Am}2_{/kg [8].}

This method produces particles with a broad size distribution and sometimes requires a size sorting stage after the synthesis [18].

Functionalization of magnetic nanoparticles

As mentioned above, surface functionalization of magnetic nanoparticles is essential to ensure colloidal stability of the particle system. Since nanoparticles have a large area to volume ratio and often have hydrophobic surfaces, the particles tend to form clusters to minimize the interface to water. The particles are often functionalized by small bio-molecules, surfactants, polymers and silica, but there are of course many other alternatives for surface

(11)

functionalization [18]. Since the particles that were used in this study are from Ocean Nanotech this work focuses on the surface functionalization of their particles. The particles from Ocean Nanotech are synthesized by thermal decomposition in oleic acid, as mentioned above. After the nanoparticle formation, the particles are covered with oleic acid and are hydrophobic. To make the particles water soluble, one can either replace the oleic acid or build a second layer on top of the oleic layer. To build on the oleic layer a amifiphilic polymer can be used, e.g. PMO (copolymer of poly(maleic acid)). To replace the entire oleic layer, PEI (polythylenamine) can be used; PEI replaces the oleic acid by substituting itself with the oleic acid. This can be done at room temperature by dissolving PEI and the nanoparticles in chloroform and letting the reaction take place for 48 h [20]. To have a carboxcylic acid functionalization, particles surface functionalized with PEI can be treated with a solution of poly(acrylic acid) and methanol to form a surface of carboxcylic acid [23].

(12)

3 Development of a computer simulation program

A computer simulation program that calculates the energy loss by adopting a linear response theory (LRT) model for particles that exhibit superparamagnetic relaxation at room temperature was written using matlab. The equation

PSP M= µ0πf χ

00

H2, (3.1)

was used to calculate the energy loss [7]. χ00 was calculated for an ensemble of immobilized uniaxial particles with random orientations, using a volume distribution described by a lognormal function. The out-of-phase component of the AC susceptibility can be calculated by solving the integral,

χ00(ω) = Z v 1 3χk ωτk 1 + (ωτk)2 GV(V )dV, (3.2)

where χ_k≈ µ0MS2V /(kBT )−µ0MS2/K , τk= τ0eKef fV /(kBT ), ω = 2πf and GV(V ) is the particle volume distribution

[24]. Combining equations 3.1 and 3.2 gives,

PSP M(ω) = µ0πf H2 Z V 1 3χk ωτk 1 + (ωτk)2 GV(V )dV. (3.3)

Since constructing equipment that uses very large fields is impractical, an AC field amplitude Hlim is introduced,

where Hlim is the maximum field that the equipment can produce, H ≤ Hlim. The product between frequency

and magnetic field has also to be limited to 4.85 × 108_{A/sm to avoid heating of body fluids, see section 2.2.2. This}

means that after Hf = 4.85 × 108_{A/sm has been reached, H must decrease to allow larger values of f , see figure}

3.1.

As can be seen from equation 3.2, the out-of-phase component of the AC suscptibility χ00 for a monodisperse particle system where all particles have the same volume and anisotropy constant will have its maximum amplitude for the frequency ωτk = 1. Since it is not possible to synthesize a completely monodisperse particle system,

there will be a distribution of relaxation times leading to a distribution of χ00 components, where each component is characterized by a volume-dependent relaxation time τ_k. χ00 for the system will be the sum of all of these components. This means that a narrow volume distribution will have a larger peak magnitude for χ00 than a broad volume distribution. To have the largest possible χ00 peak magnitude, two conditions need to be met. The volume distribution GV(V ) should be as narrow as possible and the volume V , implying a relaxation time fulfilling the

condition ωτ|= 1, should be equal to the modal volume of GV(V ), where the modal volume is defined as the most

common volume in the volume distribution.

The maximum PSP M occurs for a particle system characterized by a narrow size distribution centred around a

mean volume V∗ when H = Hlim, ω/(2π) = 4.85 × 108/Hlimand ωτk(V∗) = 1. Since τk= τ0eKef fV /kBT, assuming

that the equipment used for hyperthermia treatment has fixed values for Hlimand f , one should thus use a particle

system with an effective anisotropy constant Kef f and a narrow size distribution characterized by a mean particle

(13)

Figure 3.1: Graph of how the magnetic field has to decrease to allow higher frequencies and still not break the limit Hf = 4.85 × 108A/sm. Hlim= 15 kA/m. It should be noted that the field has a constant value of 15 kA/m until it

reaches the point f = 4.85×10_H 8

lim from where it decreases linearly (note that the graph has a logarithmic x-axis) with

(14)

4 Experimental techniques

In this chapter the different measurement techniques that were used in this work are introduced. Some more common techniquesare discussed very briefly and some less common include some theory.

4.1 The magnetic nanoparticles

The samples in all measurements were nanoparticles of different sizes disolved in deionized water. The concentra-tion of iron in the ferrofluid was 5 mg/mL. With the assumpconcentra-tion that the entire particle is made of F e3O4, the

concentration of magnetite in the ferrofluid is about 6.9 mg/mL. If not otherwise stated, the concentration used in the measurements is the concentration mentioned above. Five particle sizes were chosen for this study, 15 nm, 20 nm, 25 nm, 30 nm and 40 nm [25]. This choice was based on simulation results using the theory presented in chapter 3 and the results presented by Lundgren [26].

4.2 Size and size distribution of the magnetic nanoparticles

4.2.1 Dynamic Light Scattering (DLS)

To measure the size distribution a Zetasizer Nano ZS was used. A sample studied using this technique consisted of 1000µL deionized water and 2 µL of ferrofluid. To measure the size distribution the Zetasizer Nano uses a method called dynamic light scattering (DLS), which is sometimes called photon correlation spectroscopy (PCS), which relates the movement of particles in a liquid to the size distribution. The particles will exhibit random motion due to collisions with molecules in the liquid, often refered to as Brownian motion. The instrument relates the Brownian motion to the size distribution by illuminating the sample with a laser and detecting the scattering of the light by the particles. For very short time intervals the particles in the liquid can be viewed as stationary and the light scattered from the particles will form a speckled pattern, with light and dark areas. The speckled pattern is formed due to constructive and destructive interference of the scattered light. For a measurement a time interval later, the speckled pattern will have changed due to the random motion of the particles. The light will still interfere constructively and destructively, but due to the random motion of the particles some of the bright areas will now be darker or brighter and some of the dark areas will be brighter or darker. This means that the intensity has changed. The Zetasizer Nano measures the fluctuation in intensity with time and correlates this to the movement of the particles. Since small particles will move faster than large particles, the intensity for small particles will fluctuate faster than for large particles. To measure the change in intensity one needs something to compare with. The instrument measures the intensity at t0, t1, t2, etc. and then compares how well the measured data at later

times correlates with the data measured at t0. The correlation between the data measured at t0 and t1 is higher

than the correlation between the data at t0 and t2 since the correlation will decrease with increasing time due to

the random motion of the particles. Plotting the correlation against time will give a graph with a decaying function (correlation function) as can be seen in figure 4.1. Note that for small particles the correlation function decays faster than for large particles. The Zetasizer Nano ZS software then calculates the diffusion coefficient D from the plot (D is lifetime of the exponential decay), by using the Stokes-Einstein equation,

d(H) = kBT

3πηD, (4.1)

where d(H) is the hydrodynamic diameter of the particle, kBis Boltzmann’s constant, T is the absolute temperature,

(15)

one should note that larger particles scatter more light than smaller particles. This means that the signal from the light scattered by the small particles might drown in the signal from the light scattered by the large particles. According to the Rayleigh approximation, the intensity of the scattered light from 50 nm particles is 106 _times

larger than for 5 nm particles, I ∝ d6_{[27, 28]. To determine if a sample is monodisperse or not, a value called the}

poly dispersive index (PDI) is calculated from the natural logarithm of the correlation function. A value of 0.1 is considered monodisperse according to Beckman Coulter, Inc. [29], while according to M¨uller and Schuhmann a value of 0.03-0.06 is considered monodisperse and a value of 0.1-0.2 is considered a narrow distribution [30].

Figure 4.1: Graph showing the correlation between the speckled patterns in a DLS measurement.

4.2.2 Transmission Electron Microscopy (TEM)

A TEM model JEM 2100 from Jeol was used to measure the size of the particles. A drop of a diluted ferrofluid was placed on a copper/carbon TEM grid. The ferrofluid was diluted so that less than a monolayer of particles formed on the TEM grid. The grid was then dried under a lamp until the liquid disappeared. Thereafter the grid was mounted on a sample holder and inserted into the TEM. The image processing program Image J was used to analyze the TEM pictures. Size data was collected from several pictures, implying that hundreds of particles were used to create a size distribution. It should be mentioned that for Image J to be able to recognize the particles they must be well separated and often some kind of image filter must be used to improve the contrast between the particles and the background. While the filters let Image J identify the particles there is also a risk that the filters change the particle size while enhancing the contrast. Since the particles must be well separated to be detected, particles of a size that tend to agglomerate more will be underrepresented compared to other sizes. The diameter of the particles was determined from the pictures where image J measures the longest and the shortest axis of the particle and presents the average of two axes as a diameter. To calculate the particle volume, a spherical particle shape is adopted using the average of the longest and shortest axes as diameter. This means that a spherical and a non-spherical particle can have the same calculated diameter and volume even though the real volumes of the two particles differ.

4.3 Magnetic measurements

4.3.1 Dynomag AC susceptometer

A DynoMag AC susceptometer from Imego was used to measure the N´eel and Brownian relaxation times. The Dynomag is a susceptometer that can measure the AC susceptibility from 5 Hz to 200 kHz. At 1 kHz the instrument

(16)

As can be seen in figure 4.2, the equipment consists of three coils, one excitation coil and two oppositely wound pickup coils. An AC source is connected to the excitation coil that feeds the coil with an alternating current. The alternating current through the excitation coil creates a magnetic field that is sensed by the pickup coils; if no sample is inserted and with identical but oppositely wound pickup coils the induced voltage will be zero. If the sample, which is centered in one of the pickup coils, is magnetic, it will be magnetized and the two pickup coils will register different signals. The time varying magnetization of the sample dM/dt can be determined from the induced voltage,

∆V = µ0N Aα

dM

dt , (4.2)

where N is the number of turns in one pickup coil (same number of turns in both coils), A is the cross section area in the pickup coils and α is the magnetic coupling factor. α can be determined through calibration of the system using a sample of a material with known magnetic properties. From dM/dt the AC susceptibility can be determined from the expression,

dM dt = (χ

00_{− iχ}0_)ωH

0, (4.3)

where H0 is the field amplitude. The instrument relies upon the fact that both pickup coils are identical, which

of course they are not. This will give an error in the measurements that can be taken care of by calibrating the instrument. The instrument also has some problems close to its resonance frequency that may disturb the measurement data. For weakly magnetic samples, the signal from the sample may even be buried by this disturbance. This of course only happens in a certain frequency interval where the resonance frequency of the instrument is found [31].

Figure 4.2: Schematic picture of Imegos AC susceptometer DynoMag showing the excitation coil, pickup coils, sample, AC source and the lock in amplifier. The picture is adapted from ref. [31].

Brown relaxation

For samples that exhibit a Brownian relaxation peak, the hydrodynamic diameter can be calculated from

D3_hyd= 2kBT τB

πη , (4.4)

where τB= 1/2πfB and fB is the frequency where the out-of-phase component of the AC susceptibility exhibits a

maximum.

N´eel relaxation

A superabsorbent polymer was added to the solution to suppress Brownian relaxation. The polymer absorbs the water and forms a dry substance, which implies that Brownian relaxation is suppressed. An alternative to immobilize the particles is to place a drop of ferrofluid on a piece of paper and let it dry. The paper can then be inserted in the measurement vial [32]. By suppressing the Brownian relaxation, the AC susceptibility will be

(17)

governed by N´eel relaxation. When low concentrations of nanoparticles are used, the resonance frequency of the Dynomag instrument can be seen in the results for frequencies larger than 100 kHz. Therefore, results where the AC susceptibility starts to change rapidly above 100 kHz and often towards negative values, and both the in-phase and out-of-phase components of the AC susceptibility can therefore contain errors.

4.3.2 Physical Properties Measurement System (PPMS)

A PPMS system from Quantum Design was used to measure magnetization versus temperature and magnetic field and AC susceptibility versus temperature. The sample volume used in these measurements was 30µL. The superconducting magnet used in this setup has a maximum field of 9 T and the temperature of the sample can be varied between 1.9 K to 400 K, but for the AC susceptibility option the maximum temperature is about 350 K. AC susceptibility measurements with PPMS

The AC susceptibility for the particles of size 15 nm, 20 nm and 25 nm was measured as a function of temperature for temperatures between 8 K to 260 K. The measurements were performed for four different frequencies, 51 Hz, 510 Hz, 5100 Hz and 10 000 Hz, using an AC magnetic field with an amplitude of 796 A/m. For the sizes 30 nm and 40 nm, the susceptibility was measured for temperatures between 8 K to 72 K. For all of the above-stated AC susceptibility measurements in the PPMS, the measurement started at the highest temperature and ended at the lowest temperature. One additional measurement was made for the 25 nm particles where the particles were immobilized with a water-absorbing polymer. This made it possible to look at N´eel relaxation as a function of temperature above the melting temperature for the solvent, which was deionized water. For this measurement the N´eel relaxation was studied in the temperature range of 220 K to 352 K, using a sample volume of 20µL.

The AC susceptibility measurements were used to calculate K and Kef f, as described below. As the temperature

decreases towards zero Kelvin, the in-phase component of the AC susceptibility approaches a frequency-independent value, see figure 4.3. This frequency-independent value corresponds to the intra-potential-well response of the particle magnetic moments. By performing measurements at different frequencies while decreasing the temperature towards zero Kelvin, the frequency independent value can be extracted. The measurements can of course not be made at zero Kelvin but by extrapolating the measured data, the value for zero Kelvin can be calculated. From the frequency-independent value of the AC susceptibility, K for uniaxial particles can be calculated [33] as,

K = µ0M

2 S(0)

3χ⊥

, (4.5)

where χ⊥ is the perpendicular AC susceptibility and is equal to the frequency independent in-of-phase component

of the AC susceptibility at zero Kelvin. A method to determine Kef f is to use the Vogel-Fulcher law, which takes

into account interactions between the particles influencing the dynamic properties of the system,

τ (Tf) = τ0e

Keff V

kB (Tf −T0)_, _(4.6)

where Tfis the freezing temperature and T0is a temperature relating to the particle interactions. Tf is often defined

as the cusp temperature in an in-phase susceptibility versus temperature measurement. By using several different frequencies one can plot log(τ /τ0) vs 1/(Tf− T0) and obtain a straight line from which Kef f can be calculated [33].

(18)

0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 2 2.5 3 Temperature (K) In -p h a s e c o m p o n e n t (d im e n s in le s s )

In-phase component versus temperature for 15nm

51 Hz 510 Hz 5100 Hz

Figure 4.3: The in-phase component of the AC susceptibility versus temperature for 15 nm particles. The different curves correspond to different frequencies; 51 Hz, 510 Hz and 5100 Hz.

Magnetization versus magnetic field

Measurements of the magnetization as a function of magnetic field were performed at 10 K and 300 K by varying the magnetic field between 5 T to −5 T and −5 T to 5 T while measuring the magnetization. During cooling to 10 K the sample was exposed to a field of 5 T, meaning that the sample was field cooled. The measurement was performed with the Vibrating Sample Magnetometer (VSM) option for the PPMS, where the sample vibrates with a frequency of 40 Hz in the magnetic field produced by the superconducting magnet and induces a voltage in the pickup coil, which is measured and translated to a magnetization. When the sample reaches magnetic saturation the influence from the diamagnetic background can be seen, see figure 4.4. Sources for the diamagnetic background can, for example, be the plastic sample vial. The diamagnetic background varies linearly with the magnetic field. The reason why the magnetization curve exhibits a negative slope at large fields (figure 4.4 a)) is that diamagnetic materials have a negative susceptibility. By performing a measurement on an empty sample vial and fitting a linear function to the experimental curve, the diamagnetic background can be calculated and subtracted from magnetization curves recorded with the sample vial filled with ferrofluid, see figure 4.4 b).

Figure 4.4: a) Magnetization versus magnetic field curve at 300 K without the diamagnetic background being subtracted. b) Same magnetization versus magnetic field curve but where the diamagnetic background has been subtracted.

Magnetization versus temperatures

Measurements of magnetization as a function of temperature were performed using the VSM option for the PPMS with an applied field of 1 T. The temperature interval measured was 10 K to 250 K. Also here the diamagnetic background has to be subtracted, but the background will be more or less constant since the field is constant and the diamagnetic background varies little with temperature. One can use the diamagnetic background extracted from the magnetization versus magnetic field results described above and subtract it from the magnetization versus temperature curve. From the magnetization versus temperature curves one can see if there are any changes in

(19)

magnetic ordering in the measured temperature interval. This is not expected for F e3O4 nanoparticles.

4.4 Heating Measurements

To test the heating capacity, the particles were subjected to a magnetic AC field with a large amplitude. The measurements were performed using the DM100 instrument from nanoScale Biomagnetics. Two frequencies were tested, 229 kHz and 828 kHz, and for both frequencies the same amplitude for the AC field was used, 18.46 kA/m. The frequencies correspond to the lowest and the highest frequency allowed by the equipment. The sample was 1 mL of 20 nm ferrofluid in a glass vial. The vial was placed in a vacuum dewar to thermally isolate the sample and the temperature was measured with a fiber optic sensor that was placed inside the glass vial. The sensor measured the temperature every 200 ms with a resolution of 0.2◦C. The particles in the sample were free, meaning that Brownian relaxation was dominant. The measurement was performed four times for both frequencies to ensure reliability of the measurement data. Between each measurement the sample was allowed to reach thermal equilibrium [34]. To be able to compare the results between different measurement systems, the quantity Intrinsic Loss Power (ILP) was used. ILP = SP A/f H2

0 where SPA (specific power absorption) is the amount of generated heat expressed in

units of W/g and SP A = PSP M/ρ(F e3O4), where ρ is the density and PSP M is the heating power per unit volume.

The reason for using the ILP quantity was that systems that use different frequencies and field amplitudes can be compared. The quantity is far from perfect from a hyperthermia point of view since if one measures the ILP for one frequency one can only apply it to that frequency because the ILP is a function of the out-of-phase component of the AC susceptibility that is a function of frequency. One cannot simply compare the ILP value for two systems measured at different frequencies and say which system is most suited for hyperthermia therapy. ILP values work well if one wants to compare results from different systems at the same frequency but at different field amplitudes. To be able to compare results from different heating measurements the test conditions should be the same since not only does the generated heat change with increasing amplitude and frequency of the magnetic field, but the heat conductivity and heat capacity of the solution will also affect the measurement results. Therefore it is questionable to compare results obtained for immobilized particles with results obtained for free particles. At least two possibilities exist that allow measurements on immobilized particles in water. One can either attach the particles to a surface or one can increase the hydrodynamic radius of the particles so that N´eel relaxation becomes dominant. These two immobilization methods would, of course, lead to changes in the system that might affect the measurement results, e.g. if the particles were bound to a surface the distribution of the particles in the solution would change.

(20)

5 Results and discussion

5.1 Size and size distribution

Mean size and size distribution by TEM

A TEM picture for each particle size is shown in figure 5.1. The data extracted from several TEM pictures for each particle size is presented in table 5.1 as a mean particle diameter and as a standard deviation for a volume distribution. The data is also presented as histograms for each particle size in figure 5.2 a) to e). In figure 5.2 f) a comparison between the experimentally determined volume distribution for the 15 nm particle system and a lognormal distribution, generated using the data presented in table 5.1 for the same particle system, is shown. As can be seen in this figure, a lognormal distribution gives a good description of the distribution of particle sizes in the ferrofluid. In the TEM pictures in figure 5.1 a), b) and c) it can be seen that the 15 nm to 25 nm particles appear to be spherical, implying that the contribution from shape anisotropy is small for these particles. This would mean that the anisotropy is mainly determined by magnetocrystaline and magnetoelastic contributions. Moreover, it can be seen that a second peak in the distribution of particle sizes appears around 7 nm for 25 nm and 30 nm particles in the histograms in figure 5.2 c) and d). The two particle sizes can also be seen in the TEM pictures in figure 5.1 c) and d). It should also be noted that the labeled sizes do not agree well with the mean sizes calculated from the TEM pictures for particle sizes 20 nm, 30 nm and 40 nm, as can be seen in figure 5.2 b), d) and e) as well as in table 5.1. To avoid confusion the particle systems are still referred to using the labeled sizes.

Table 5.1: Mean diameter and standard deviation of the particle volume for the five particle systems. It should be noted that the mean diameter and standard deviation have been calculated from the main peaks in figure 5.2, implying e.g. that the particles with sizes around 7 nm and 25 nm in figure 5.2 a) have been excluded from the analysis of the 15 nm particles.

Size according to label (nm) Mean diameter (nm) Standard deviation (m3₎

15 14.66 6.367 × 10−25

20 16.57 5.990 × 10−25

25 24.97 2.348 × 10−24

30 24.91 1.862 × 10−24

40 35.68 7.105 × 10−24

Size and size distribution by DLS

The hydrodynamic volumes taken from the specification sheets of the different particle sizes [25] are listed in table 5.2 together with similar data extracted from the DLS measurements. As can be seen in table 5.2, all particle systems exhibit a narrow size distribution since their PDI is below 0.2. It can also be seen that the 25 nm and 30 nm particles have a different hydrodynamic diameter, which is unexpected since, according to the statistics from the TEM pictures, they are close in size. This could possibly be explained by that the particles of size 30 nm are less spherical and more ellipsoidal in shape. When the diameter is extracted from the TEM pictures it is extracted as the average of the longest and shortest axes, as explained in section 4.2.2. This implies that two different shapes could have the same average of the longest and shortest axes and could be presented as the same size in the statistics. This has not been confirmed from the TEM pictures and is just a possible explanation. The other explanation is that for some reason the larger particles in the TEM pictures for the 30 nm particle system have not been detected by the image analysis program and thereby the the average is smaller, though this seems very unlikely.

(21)

Figure 5.1: TEM pictures for the different particle sizes. a) 15 nm, b) 20 nm, c) 25 nm d), 30 nm and e) 40 nm. For c) and d) two separate sizes can be seen. This can also be seen in the histograms presented in figure 5.2.

(22)

a)

b)

e)

c)

d)

f)

8 10 12 14 16 18 20 22 24 26 0 5 10 15 20 25 Diameter (nm) Amo u n t o f p a rt icl e s (% )

15nm

20nm

5 10 15 20 25 30 35 0 5 10 15 20 25 Diameter (nm) Amo u n t o f p a rt icl e s (% )

25nm

5 10 15 20 25 30 35 40 45 0 5 10 15 20 25 Diameter (nm) Amo u n t o f p a rt icl e s (% )

30nm

40nm

10 11 12 13 14 15 16 17 18 19 20 0 2 4 6 8 10 12 14 Diameter (nm) Amo u n t o f p a rt icl e s (% )

15nm

Exp Lognormal

Figure 5.2: Histograms for different particle sizes. a) 15 nm, b) 20 nm, c) 25 nm d), 30 nm and e) 40 nm. The data was extracted from TEM pictures using the image analysis software Image J. For c) and d) two separate size distributions can be seen. In f) a comparison between the volume distribution extracted from the TEM pictures for the 15 nm particle system and a lognormal volume distribution generated using the data for the same system presented in table 5.1 is shown.

(23)

Table 5.2: Mean hydrodynamic diameters for the different particle sizes. Dhyd,specwas taken from the specification

sheet for each paricle size [25], while Dhydand Dhyd,V were extracted from the results using a number and a volume

distribution of the particle sizes, respectively. The table also includes the measured poly dispersive index (PDI). Size according to label (nm) Dhyd,spec(nm) Dhyd(nm) Dhyd,V(nm) PDI

15 22.39 18.75 22.54 0.200

20 23.45 20.18 26.29 0.142

25 31.90 29.18 35.94 0.141

30 35.7 34.21 40.99 0.083

40 40.90 37.94 44.80 0.059

5.2 Results from magnetic measurements

5.2.1 Magnetization versus temperature and magnetization versus magnetic field

Magnetization as a function of temperature

Magnetization versus temperature curves for each particle size are presented in figure 5.3. As can be seen in figure 5.3 d) and e) there is a cusp in the magnetization versus temperature curve at approximately 200 K for the 30 nm and 40 nm particle systems. The cusp can be explained by these particles containing both magnetite and wüstite (F eO); wüstite orders antiferromagnetically around 200 K [35]. The magnetization versus temperature curve for an antiferromagnetic material can be described as follows. At temperatures above the Néel temperature, which is the magnetic ordering temperature, the magnetizaztion increases with decreasing temperature. The magnetization exhibits a cusp at the Néel temperature and on further reduction of the temperature the magnetization decreases. Assuming that the magnetization due to magnetite is saturated and weakly temperature dependent, this means that the contribution coming from wüstite will be most apparent at temperatures close to 200 K. A comparably large content of wüstite for the two largest sizes can also explain the much lower magnetization seen in figure 5.3 d) and e), since the antiferromagnetic contribution to the measured magnetization is very small in comparison to the contribution coming from ferrimagnetic magnetite. For the smaller particle sizes, see figure 5.3 a), b) and c), there is no cusp in the magnetization versus temperature curve at ∼ 200 K, and the magnetization is much larger when compared to that of 30 nm and 40 nm particles. However, at the lowest temperatures there is a slight increase in magnetization with decreasing temperatures, indicating a paramagnetic or superparamagnetic contribution to the measured magnetization.

Magnetization as a function of magnetic field

The magnetization versus magnetic field curves for particle sizes 15 nm, 20 nm and 25 nm at 300 K and 10 K are presented in figure 5.4, while the corresponding results for particle sizes 30 nm and 40 nm at the same temperatures are presented in figure 5.5. As can be seen in figures 5.4 and 5.5, all of the particle systems have closed hysteresis loops at 300 K for a measurement time τM of a few seconds. This means that τ < τM and that the particle system is

superparamagnetic for the given temperature and measurement time. For measurements at 10 K the hysteresis loop is open for all of the particle systems, τ > τM, implying that the direction of the particle magnetic moments with

decreasing field will become blocked along the easy magnetization directions closest to the magnetic field direction. This is in good agreement with theory, see section 2.1.2. The saturation magnetization in figures 5.4 and 5.5 is much lower than reported values for bulk, 510 kA/m for 0 K and 480 kA/m for 293 K [8]. Moreover, if one compares the MS

value presented by Caruntu et al. with the MS values measured for the particles from Ocean NanoTech, presented

in table 5.3, the MS values for the particles from Ocean NanoTech are much lower. This leads to the conclusion

that the ferrimagnetic order within the nanoparticles is much disturbed and that the measured magnetization will not saturate until much larger fields than those used in the present study. Furthermore, for the largest particles, the small values for the magnetization are also explained by the fact that the particles contain both magnetite and w¨ustite. As can be seen figure 5.3 c), d), e) and f), the compensation for the diamagnetic background was not sufficient. This is probably because the measurement vials are not identical and the background measurement was done for one vial with the asumption that the difference between the vials was negligible. The trends seen in the magnetization curves will not change, but the value for magnetic saturation will increase with better compensation for the background.

(24)

a)

b)

e)

c)

d)

0 50 100 150 200 250 230 240 250 260 270 280 290 V o lu m e m a g n e ti z a ti o n ( k A /m ) Temperature (K)

15nm

20nm

25nm

30nm

40nm

Figure 5.3: Magnetization versus temperature at 1 T for the different particle sizes. a) 15 nm, b) 20 nm, c) 25 nm d), 30 nm and e) 40 nm. Note the much smaller magnetization and the cusp in the magnetization curve in the temperature range 200 K to 250 K for the two largest particle sizes.

(25)

a)

b)

e)

c)

d)

f)

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000 -300 -200 -100 0 100 200 300 V o lu m e m a g n e ti z a ti o n ( k A /m )

Magnetic field (kA/m) 15nm 10K -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 -300 -200 -100 0 100 200 300 V o lu m e m a g n e ti z a ti o n ( k A /m )

Magnetic field (kA/m) 25nm 300K

Figure 5.4: Magnetization versus magnetic field for different particle sizes and different temperatures. a) 15 nm and 10 K, b) 15 nm and 300 K, c) 20 nm and 10 K, d) 20 nm and 300 K, e) 25 nm and 10 K and f) 25 nm and 300 K.

(26)

a)

b)

c)

d)

-4000 -3000 -2000 -1000 0 1000 2000 3000 4000 -300 -200 -100 0 100 200 300 V o lu m e m a g n e ti z a ti o n ( k A /m )

Magnetic field (kA/m) 40nm 300K

Figure 5.5: Magnetization versus magnetic field for different particle sizes and different temperatures. a) 30 nm and 10 K, b) 30 nm and 300 K, c) 40 nm and 10 K and d) 40 nm and 300 K.

(27)

5.2.2 AC susceptibility measurements

AC susceptibility at different temperature

The results for the AC susceptibility components versus temperature measurements are presented in figure 5.6. Figure 5.6 a), b), c) and d) shows how the AC susceptibility components for particle sizes 15 nm and 20 nm vary with temperature in the temperature range 8 K to 260 K. The results for the AC susceptibility components versus temperature for a particle size of 25 nm are shown in figure 5.6 d) and e) for the temperature range 220 K to 352 K. The anisotropy constants for the five different sizes were calculated from the frequency independent value of in-phase component of the AC susceptibility at zero Kelvin, as described in section 4.3.2; the calculated anisotropy constants are presented in table 5.3. For the 15 nm particles, the anisotropy constant could also be calculated from the frequency dependent cusp temperatures of the in-phase component of the AC susceptibility, see figure 5.6 a). The best fit using the Vogel-Fulcher law, equation 4.6, was obtained for T0= 0, which indicates that the

interaction between the particles is small; the value for Kef f is presented in table 5.3. There is a difference between

the extracted values of Kef f and K. This is discussed further in section 5.4. From table 5.3 it can be seen that the

anisotropy constant is smaller for smaller particle sizes, except for 20 nm particles, which exhibit the smallest value for the anisotropy constant. The trend where smaller sizes have a lower anisotropy constant can be explained by the fact that smaller sizes generally have fewer defects that contribute to the magnetic anisotropy. Particles with smaller sizes also appear to be more spherical, meaning that they will have a smaller contribution from shape anisotropy. Some studies have reported an increase in magnetic anisotropy for smaller particle sizes and have explained this with the fact that for very small particles the surface spin disorder increases due to incomplete coordination for surface atoms [16]. If one compares the extracted K value for 15 nm particles to the value reported by Caruntu et al. for 11.6 nm particles, the value obtained in this work is about 2.5 times smaller. It can also be seen in figure 5.6 that the peak for the out-of-phase component of the AC susceptibility appears at higher temperatures for higher frequencies. This is in good agreement with the theory presented in chapter 3.

The second peak in the size distribution appearing for particles labeled 25 nm and 30 nm in the size range 5 nm to 10 nm, see figure 5.2 c) and d), implies that one can expect a second frequency dependent out-of-phase component peak that occurs at lower temperatures, since according to theory the N´eel relaxation time is much shorter for smaller particles. In figure 5.7, the out-of-phase component of the AC susceptibility in the temperature interval 8 K to 100 K is shown for the 15 nm, 20 nm and 25 nm particles; the different curves for each particle size correspond to different frequencies of the AC field (51 Hz, 510 Hz and 5100 Hz). A well-defined frequency dependent peak can be seen in figure 5.7 c) for the 25 nm particles, which is in good agreement with the experimentally determined size distribution. A low temperature frequency dependent peak in the out-of-phase component of the AC susceptibility was also observed for 15 nm and 20 nm particles, see figure 5.7 a) and b), in agreement with the statistics extracted from the TEM pictures. The reason why the low temperature peak for 15 nm particles appears less distinct is that it overlaps with the main peak of the out-of-phase component of the AC susceptibility.

Table 5.3: Anisotropy constants Kef f and K, saturation magnetization at 10 K, MS(10), and at 300 K, MS(300).

Size according to label(nm) Kef f (J/m3) K (J/m3) MS(10) (kA/m) MS(300) (kA/m)

15 1.55 ∗ 104 _{5.50 ∗ 10}4 ₃₀₆ ₂₈₄

20 - 5.33 ∗ 104 ₂₈₉ ₂₄₁

25 - 6.83 ∗ 104 ₂₇₁ ₂₄₇

30 - 24.9 ∗ 104 ₁₉₁ ₁₈₉

(28)

a)

b)

e)

c)

d)

f)

0 50 100 150 200 250 0 5 10 15 Temperature (K) In -p h a s e c o m p o n e n t (d im e n s in le s s )

15nm

51 Hz 510 Hz 5100 Hz 10000 Hz 0 50 100 150 200 250 0 0.5 1 1.5 Temperature (K) O u t-o f-p h a s e c o m p o n e n t (d im e n s in le s s )

15nm

51 Hz 510 Hz 5100 Hz 10000 Hz 0 50 100 150 200 250 0 5 10 15 Temperature (K) In -p h a s e c o m p o n e n t (d im e n s in le s s )

20nm

51 Hz 510 Hz 5100 Hz 10000 Hz 0 50 100 150 200 250 0 0.5 1 1.5 Temperature (K) O u t-o f-p h a s e c o m p o n e n t (d im e n s in le s s )

20nm

51 Hz 510 Hz 5100 Hz 10000 Hz 220 240 260 280 300 320 340 0 5 10 15 Temperature (K) In -p h a s e c o m p o n e n t (d im e n s in le s s )

25nm

51 Hz 510 Hz 5100 Hz 10000 Hz 220 240 260 280 300 320 340 0 0.5 1 1.5 Temperature (K) O u t-o f-p h a s e c o m p o n e n t (d im e n s in le s s )

25nm

51 Hz 510 Hz 5100 Hz 10000 Hz

Figure 5.6: AC susceptibility components versus temperature. The different curves correspond to different frequen-cies of the AC magnetic field; 51 Hz, 510 Hz, 5100 Hz and 10 000 Hz. Since the temperature is below the freezing point of the solvent (deionized water) only N´eel relaxation will be present. a) in-phase component for 15 nm par-ticles, b) out-of-phase component for 15 nm parpar-ticles, c) in-phase component for 20 nm parpar-ticles, d) out-of-phase component for 20 nm particles, e) in-phase component for 25 nm particles and f) out-of-phase component for 25 nm particles.

(29)

a)

b)

c)

0 10 20 30 40 50 60 70 80 90 100 0 0.02 0.04 Temperature (K) O u t-o f-p h a s e c o m p o n e n t (d im e n s in le s s )

15nm

51 Hz 510 Hz 5100 Hz 0 10 20 30 40 50 60 70 80 90 100 0 0.02 0.04 Temperature (K) O u t-o f-p h a s e c o m p o n e n t (d im e n s in le s s )

20nm

51 Hz 510 Hz 5100 Hz 0 10 20 30 40 50 60 70 80 90 100 0 0.02 0.04 Temperature (K) O u t-o f-p h a s e c o m p o n e n t (d im e n s in le s s )

25nm

51 Hz 510 Hz 5100 Hz

Figure 5.7: Out-of-phase component of the AC susceptibility versus temperature. The different curves correspond to different frequencies of the AC magnetic field; 51 Hz, 510 Hz and 5100 Hz. Since the temperature is below the freezing point of the solvent (deionized water), only N´eel relaxation will be present. a) 15 nm particles, b) 20 nm particles and c) 15 nm particles.