Eindhoven University of Technology BACHELOR Characterization of the magnetic moment of superparamagnetic particles from magnetic dipole-dipole interactions Bos, A.H.

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Eindhoven University of Technology

BACHELOR

Characterization of the magnetic moment of superparamagnetic particles from magnetic dipole-dipole interactions

Bos, A.H.

Award date:

2012

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Characterization of the magnetic moment of superparamagnetic

particles from magnetic dipole–dipole interactions

Arjen Bos

MBx 2012 - 12 TU/e

Faculty of Applied Physics

supervisors: Alexander van Reenen & Arthur de Jong July 2012

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Abstract

To obtain a better insight in the magnetic behavior of superparamagnetic microparticles an accurate method to characterize magnetic properties as the susceptibilty and magnetic content are of great importance.

To determine the magnetic moment of those microsize particles we have developed a new method which analyses single particle couples. By exploiting the magnetic dipole–dipole interaction between two magnetized particles and determining their trajectories the magnetic moment, averaged over the two particles, can be determined.

Using this method we determine the magnetization curves of two types of superparamagnetic particles. As a reference for our results we use the magnetization curves from a VSM measurement which is supplied by the producer of the particles. The data from our experiments corresponded the reference data so to obtain magnetization curves from this new method is concluded succesful.

This new method offers the opportunity to obtain magnetic properties of just one pair of particles, which is not possible with conventional methods such as a VSM measurement. With this great advantage we were able to determine the magnetic moment variation between the particles at single particle level and calculate the number and size of the grains inside a magnetic particle.

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1 Introduction

Micrometer–sized magnetic particles are widely applied as carriers and labels in in–vitro diagnostics [1] as well as biosensing applications [2]. An important class of these particles are superparamagnetic particles, which consist of a poly- meric spherical matrix filled with a large number of ferromagnetic grains [3].

Superparamagnetic particles are very useful because they can be manipulated and detected from within biological fluid cells. For example magnetic particles can be detected by magneto-resistive sensors [4], Hall sensors [5] and optical detection [6].

In presence of an external magnetic field superparamagnetic particles get magnetized and experience magnetic dipole–dipole interactions. By altering the external field, particles can be directed and ordered in the fluid sample. Practi- cal uses of this are e.g. creating magnetic mini stirrers by chaining the particles together or the particles can be distributed evenly in the fluid cell to maximize the possibility of binding to one or more proteins of interest.

Magnetic properties of individual particles like their magnetic susceptibility χ and saturation magnetization MS are conventionally determined from measurements like vibrating sample magnetometry [7]. Here a large number of magnetic particles are probed and the properties of individual particles are calculated assuming they are all identical. Unfortunately this is not the A magnetic moment variation is determined by performing several case; there are often significant variations in the magnetic properties between commercially available superparamagnetic particles, even in the same batch. Therefore a VSM measurement does not provide reliable results and a different method has to be used in order to obtain accurate magnetic properties of individual particles.

In this report we present such a method which exploits the magnetic dipole–

dipole interaction between a pair of particles with which we are capable of determining the averaged magnetic moment of one pair of particles. The general idea behind this new method is to analyse two particles which are moving through a viscous fluid during dipole–dipole interaction and calculating the magnetic moment from their trajectories.

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2 Theory

As stated before the new method is based on the magnetic dipole–dipole interaction between superparamagnetic particles. To get a good understanding of this interaction we first describe the superparamagnetic particle itself and the forces acting on one.

2.1 Superparamagnetic particle

A superparamagnetic particle consists of a polystyrene matrix containing small ferromagnetic grains, see figure 1(a). The typical size of a particle is about order of micrometers and that of a grains several nanometers. The number of grains inside the particle displayed in figure 1 is heavily underestimated. Several thousands of grains inside a particle is more regular. The arrow symbolizes the magnetic moment µ of a single grain.

The grains are randomly ordered in the particle and their magnetic moment is orientated random aswell. This is due to very small size of the particles.

The magnetic anisotropy K, which scales with the volume of a particle, is to weak compared to the thermal energy k_BT to keep a steady magnetization direction. So the indivual moments of all the grains are changing all the time and they cancel each other out due to the random orientation, resulting in total magnetisation of the particle of zero.

(a) (b)

Figure 1: Schematic representation of superparamagnetic particles. In figure (a) the magnetic moments of the grains are ordered randomly, resulting in a zero magnetization of the particle itself. In figure (b) an external magnetic field is applied, which results in a net magnetized particle.

When applying an external magnetic field B an energy barrier µB has to be crossed. When this barrier is sufficiently large the moments of the grains lose their random orientation and tend to align in direction of the external field, resulting in a net magnetic dipole moment of a particle, see figure 1(b).

2.2 Interactions

A magnetized superparamagnetic particle acts like a magnetic dipole and therefore will interact with other dipoles.

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In the case there is an external magnetic field applied and the magnetic dipole moments of two particles are aligned parallel two types of interaction can be distinguished:

1. attraction of opposite magnetic poles, see figure 2(a), 2. repulsion of identical poles, see figure 2(b).

(a) (b)

Figure 2: Schematic view of magnetized particles during attractive (a) and repulsive (b) interaction.

Before discussing these interactions more specifically we should first take a look at the dominant forces acting on the particles.

2.3 Forces acting on magnetized particles in a viscous fluid

For magnetic dipoles which are moving through a fluid two dominant forces are acting on each dipole:

1. magnetic dipole–dipole interaction, due to attraction or repulsion of magnetic poles,

2. viscous drag force as they are moving through a fluid.

Consider two superparamagnetic particles as displayed in figure 3. Particle 1, which is located at r₁, is experiencing a magnetic force F^m₁ due to dipole–dipole interaction with particle 2 and a viscose drag force F^d₁. By symmetry, the same forces are exerted on particle 2 but in reverse direction.

In the next two paragraphs both forces are elaborated and mathematically expressed in terms of the magnetic moments and distances.

2.3.1 Magnetic dipole–dipole interaction

For a two particle system the magnetic dipole–dipole interaction force F^m₁ acting on particle 1, located at r₁, is given by

F^m₁ (r21, m1, m2) = 3µ0

4πr⁵ h

(m1· r21)m2+ (m2· r21)m1 (1) +(m1· m2)r21−5(m1· r21)(m2· r21)

r² r21

, (2)

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Figure 3: Two magnetized particles, located at r1and r2, having magnetic dipole moments of respectively m1 and m2, symbolized by arrows. The total distance between the particles is r = |r1− r2|.

where r₂₁ = r₂− r1 is the vector pointing from particle 2 to particle 1, m₁ is the magnetic moment of particle 1 and r = |r| is the distance between particle 1 and 2.

In case that the magnetization of the two particle are equal, m1 = m2 ≡ m, equation 2 reduces to a more compact expression. Furthermore for practical purposes F^m_i is expressed in unit vectors. This leads to the following equation for F^m₁:

F^m₁ = 3µ₀m²

4πr⁴ 2( ˆm · ˆr₂₁) ˆm + (1 − 5( ˆm · ˆr₂₁)²)ˆr₂₁ , (3) with and unit vectors defined as m = m ˆm and r = rˆr.

Let’s consider the next two cases:

1. the external magnetic field, and with that the magnetic moment of the particles, parallel to r so ˆm · ˆr = ±1;

2. and the external magnetic field perpendicular to r so ˆm · ˆr = 0.

In case 1 the magnetic force working on particle i equals F^m_1,attr= −3µ₀m²

4πr⁴ · 2 · ˆr₂₁. (4)

Note this is an attractive force as it points towards the other particle.

And in case 2, where the inner product ˆm · ˆr = 0, the magnetic force equals F^m_1,rep= 3µ0m²

4πr⁴ · ˆr21. (5)

This indicates a repulsive interaction as it points away from the other particle.

Also note that the magnetic force corresponding to attractive and repulsive interaction differs by a factor of 2.

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2.3.2 Viscous drag force

The particles are moving through a fluid and therefore they experience a viscous drag. The motion may be described by the Navier–Stokes equation;

ρ∂v

∂t + v · ∇v

| {z }

Inertia

= −∇p

| {z }

Pressure gradient

+ µ∇²v

| {z }

Viscosity

+ f

|{z}

Other forces

.

To compare the relative effect of the inertia forces to the viscous drag force, we make an estimation of the Reynolds number for our system. The Reynolds number Re = ρvL/η, in which ρ is the density of the fluid, v the speed of the object traveling through the fluid, L a characteristic length e.g. the radius of a particle and η the dynamic viscosity of the fluid. To make an approximation of Re of a particle in water we take, ρ = 10³ kg/m³, v ≈ 10⁻⁵ m/s which is determined in experiments, L = R ≈ 10⁻⁶ m which is the radius of a particle and η ≈ 10⁻³kg/(m·s). So the Reynolds number Re ≈ 10³· 10⁻⁵· 10⁻⁶/10⁻³= 10⁻⁵ 1. A Reynolds number much smaller than one means the flow is a Stokes flow and the inertia is negligible compared to the drag forces.

The hydrodynamic drag force acting on particle i with velocity v_i in case of a Stokes flow relates linearly to the speed but, ofcourse, in contrary direction;

F^d_i = −γvi= −6πηRvi (6)

The drag coefficient γ equals 6πηR, in which η is the dynamic viscosity of the fluid and R is the radius of the particle moving through the fluid.

Equation 6 does not apply on particles located closely to a physical surface. The surface will cause extra resistance. Assuming the particles only to be translating parallel to the surface, Leach et al.[9] introduced a correction factor to correct the Stokes drag on a sphere moving near a surface. For a particle at a distance s from a surface the corrected drag coefficient γc equals

γ_c(s) = γ

1 − ₁₆⁹ _R

s + ¹₈ _R

s

³. (7)

In case that the particles are in contact with the surface so s = R equation 7 becomes

γc(R) = γ ·16 9 = 32

3 πηR. (8)

So the drag force F^d_i for superparamagnetic particles moving over a surface equals

F^d_i = −32

3 πηRv_i= −32

3 πηRdr_i

dt. (9)

Assuming that both particles separate with the same speed the drag force can be expressed in the distance between the two particles r. The separation of the

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two particles dr/dt is then twice as big as the speed of one of those; dr_i/dt. So the drag force expressed in terms of the distance between the two particles r equals

F^d_i = −32 6 πηRdr

dtˆvi. (10)

2.3.3 Force balance

Now to combine both dominant forces acting on magnetized particles moving through a viscous fluid we set up a force balance. Using Newtons second law the force balance equals

ma = X

F,

= F^m_i + F^d_i.

As stated before, the inertia can be neglected and this yields the following force balance for forces acting on particle i

0 = F^m_i + F^d_i. (11)

Now both forces; the magnetic dipole–dipole interaction force and the Stokes drag, are mathematically expressed in terms of the magnetic moment m and the distance r between two particles and both have been combined in one equation, the force balance. We will continue where we halted in section 2.2 with the magnetic dipole–dipole interactions repulsion and attraction.

2.4 Magnetic dipole–dipole repulsion

(a) (b)

Figure 4: Schematic representation of magnetic particles during repulsive interaction. In the figure on the left the magnetic moment of the particles is symbolized with an arrow and magnetic north and south poles are drawn. The figure on the right is a 3D image of the particles while separating from each other.

As stated in section 2.2 in case the external magnetic field and the magnetic moment of the particles, see figure 4(a), is pointed upwards ( ˆm = ˆz) while the particles align horizontally (ˆr = ˆx) the two particles repel each other due to the repulsive interaction between equivalent magnetic poles.

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Using equation 5 for the magnetic dipole–dipole interaction during repulsion and filling this in together with the Stokes drag in the force balance it yields an expression for the distance r between two particles as a function of time. The boundary condition used for solving the integral was that at t0 the particles were distanced a length of r0, so r(t=t0) = r0.

F^m₁ = −F^d₁

−3µ₀m²

4πr⁴ ˆx = −32 6 πηRdr

dtˆx

⇔ r⁴dr = 3 4

6 32

µ0m² π

dt πηR Z r

r₀

r⁴dr = 9 64

µ0m² π²ηR

Z t t₀

dt

⇔ r⁵= 45 64

µ₀m²

π²ηR(t − t0) + r⁵₀

The most important observations about this relation is that r ∝ √⁵

t and that the slope of the graph of r⁵versus t relates to the square magnetic moment m².

r(t) = ⁵ s

45 64

µ0m²

π²ηR(t − t₀) + r₀⁵ (12)

The start condition at t = 0 is r(0) = r0≥ 2R.

2.5 Magnetic dipole–dipole attraction

(a) (b)

Figure 5: Schematic representation of particles during attractive interaction. In figure (a) two magnetized particles are displayed. The arrows symbolize magnetic moments. In figure (b) several particles are formated in chains due to the attractive interaction.

As for attractive interaction a similar calculation is done as we did for repulsion. Now the magnetic field, and magnetic moment of the particles, is pointed horizontally in the ˆx direction and the north pole of one particle attracts the south poles of the particle next to it.

Using the magnetic dipole force given in equation 4 and filling this in in the

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force balance will yield the expression for r(t) during an attractive interaction.

F^m₁ = −F^d₁ 23µ0m²

4πr⁴ ˆx = −32 6 πηRdr

dtˆx

⇔ r⁴dr = −6 4

6 32

µ₀m² π

dt πηR Z r

r₀

r⁴dr = − 9 32

µ0m² π²ηR

Z t t₀

dt

⇔ r⁵= −45 32

µ0m²

π²ηR(t − t0) + r⁵₀

r(t) = ⁵ s

−45 32

µ₀m²

π²ηR(t − t₀) + r₀⁵ (13) The same r ∝ √⁵

t relation is found. But there is one remarkable difference, except for the fact that r decreases over time during attraction and increase during repulsion. The difference is that two particles approach each other √⁵

2 times faster during attraction as they separate during repulsion.

2.6 Determination of the magnetic moment m

Now we have mathematically calculated the trajectories of superparamagnetic particles during magnetic dipole–dipole interaction. With these we are able to get an expression for the magnetic moment, which is the main goal of this project.

As the trajectories for repulsion are described differently as for attraction, we also need two different expressions for the magnetic moment.

For repulsive interaction, which is described in equation 12, the magnetic moment can be expressed in a known slope a of the graph of r(t) and several physical constants;

m =

s a64

45 π²ηR

µ₀ . (14)

For attractive interaction, which is described in equation 13, the equation of the magnetic moment differs a factor √

2 with the one for repulsion;

m = s

a32 45

π²ηR

µ₀ . (15)

Until now we have only been describing the particle and its properties and motions. To get a better understanding and determine properties of the grains inside a particle, we should first take a look at paramagnetism and, how it is linked to temperature and how it paramagnetic behavior can be described.

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2.7 Paramagnetism and temperature

Paramagnetic materials have magnetic susceptibilities which strongly depends on the temperature. This susceptibility χ is suppressed when increasing the temperature T . It is possible to think of temperature as a source which destroys or randomizes the preferred orientation of the magnetic moments. To explain this the thermal energy can be compared to the magnetic energy of a dipole, which depends on the magnetic moment m and the external applied field B:

U = −~µ · ~B.

To change the orientation of the magnetic moment of one grain a barrier, which is magnetic energy, has to be crossed and a higher temperature will lead to more randomization and a decrease of the amplitude of the net magnetic moment.

The magnetization M of a paramagnet which consists of n ferromagnetic areas of which each area has a magnetic moment µ can be described using the Langevin equation [8].

M = nµL(z) (16)

Here z = µB/kBT , where kBT is the thermal energy and B the applied magnetic field.

The Langevin function L is given below

L(z) = coth(z) −1

z (17)

Now for superparamagnets the ferromagnetic-like grains have a diameter so small that they cannot consist of multiple areas with different magnetic moment orientation. Based on this it is possible to use the Langevin equation for a superparamagnet particle as well. Now n equals the number of grains inside a particle instead of the number of areas.

2.7.1 Deviation magnetic moment of the grains

In the previous section we have assumed that all grains are distributed normally within the particle and have equal magnetic moments. However, in reality, this is not the case. Deviation of the magnetization of the superparamagnetic particles from a single Langevin function is caused by this nonuniform distribution of magnetic moments of the grains [12]. In such case, the magnetization can be described by a weighted sum of Langevin functions given in equation 18, where f (µ)dµ is the fraction of grains in the particle having magnetic moments between µ and µ + dµ.

M = Msat

Z ∞ 0

L(z)f (µ)dµ (18)

The distribution function f (µ) is a log-normal distribution given as:

f (µ) = 1

√2πσµexp

− ln² µ µ₀

/(2σ²)

, (19)

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where µ₀is the median of the distribution and σ is the distribution width. The equation satisfies the normalization conditionR∞

0 f (µ)dµ = 1.

With this weighted Langevin equation we are able to determine the magnetic moment µ of a grain and how many grains are embedded in one particle, n.

Together with the magnetic moment of the particle µ it is possible to make an estimation of the volume of one grain. These are all very interesting properties, especially because the supplier of the particle usually does not supply these properties.

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3 Materials and methods

In this chapter we will explain how we recorded the movies of the particles during interaction and the materials, like particles and equipment, are described. In the methods section we discuss the analysis process of the data. Several techniques are used and we explain why we use those.

3.1 Materials

In this project two types of superparamagnetic particles have been studied:

1. Dynaparticles MyOne with a diameter of 1 µm,

2. Micromer PEG particles with a diameter 2,6 µm. The determination of these sizes is elaborated in section 3.1.2.

Both types of particles are composed of a polystyrene matrix embedded with magnetite (Fe4O3) grains. They only differ in size and magnetic content. The purchased solution was diluted a several hunderd times in de-ionized water to obtain an optimal concentration of particles in the sample. The dilution is then inserted into a small fluid cell of about 0,5 mm height and 3 mm radius. This sample with several thousands of particles is then placed under a Leica DM4000 microscope in a setup with several magnets to produce magnetic fields, see figure 6.

Figure 6: The used experimental setup. The sample with superparamagnetic particles is placed on a microscope table. The electrical steel poles which are located around the microscope table are powered with a certain current to generate a magnetic field.

A MotionPro X3 High speed camera is connected to the microscope and a small movie is made of the particle during the magnetic interactions. As stated before attractive interaction is induced when a horizontal field is applied, and repulsion occurs in case of a vertical magnetic field. The magnets which produce the magnetic fields are the two lower located magnets. An alternating current with 180 degree phase lag is applied on both electromagnets and so a horizontal field pointing from one magnet to another is generated. When the particles are aligned these two magnets are powered off and a current, without phase lag,

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is supplied on the magnets and they will produce equivalent poles and thus a vertical field in the sample.

Immediately after the magnetic field’s orientation switch the particles which are located seperate from each other due to the magnetic repulsion and this is captured on film.

3.1.1 Calibration of the magnetic field

Because we power the magnets with a certain current, a calibration of the generated magnetic field is required. The magnetic field was calibrated with a Hall probe to determine the precise strength of magnetic field in at the location of the fluid sample corresponding to a certain current applied over the electrical steel poles. The calibration graph is given in figure 7.

Figure 7: The calibration diagram of the vertical magnetic field.

3.1.2 Diameter of the particles

To determine the size of the spherical particles an experiment with dynamic light scattering is done. With dynamic light scattering the light reflected on particles from a laser is detected. The intensity of this reflected light fluctuates due to the Brownian motion of the particles. A higher rate of fluctuation indicates a higher rate of Brownian motion and thus a smaller particle. This experiment was performed using a Zetasizer Nanoseries from Malvern.

Size of MyOne particles

The results of the dynamic light scattering experiment are presented in figure 8.

The average diameter d and standard deviation σ are calculated.

The average diameter d = 1,0µm with a standard deviation of 0,2µm.

To verify these results the size of several particles has been calculated by measuring their sizes from microscope images. It is not possible to determine the absolute diameter because the we are monitoring below the diffraction limit, however we compare the diameters to estimate the variation. From a sample of 20 particles a diameter deviation of 0, 17µm is determined. This seems to

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Figure 8: Size and probability data from MyOne particles. For di ≤ 458.7 nm or di≥ 2305 nm wi= 0.

correspond with the deviation of 0/, 2µm determined with the Zetasizer.

Size of PEG particles

The results of the dynamic light scattering on PEG particles are given in figure 9. The same procedure has been performed to determine the diameter and standard deviation of the PEG particles as for the MyOne particles. And the result d ± σ = 2,6 ± 0,5 mm.

Again this is verified by counting pixels from microscope images. The result is a deviation of 0,10µm. The standard deviation is much smaller. The Zetasizer seems to determine the deviation too large. There is no clear explanation for this. To determine the exact size we should look at the particle with a larger resolution e.g. a SEM.

3.1.3 Microscope pixel-distance calibration

The precise microscope resolution was determined by imaging a small calibration disk from thorlabs. The image used, with 500× magnification, of this disk is displayed in figure 10.

The distances between line 0 and line 1 is 100 µm. To determine the pixel- distance relation the distance between the two line, in pixels, is counted. Point 1 was located on pixel 719 ± 1 and point two on pixel 290 ± 1. So the total distance in pixels is 719 − 290 = 429 ± 2. The corresponding resolution is 4,29 ± 0,02 pixel/µm.

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Figure 9: Size and probability data from PEG particles. For di ≤ 1484 nm or di≥ 5560 nm wi= 0.

Figure 10: Microscope shot of the calibration disk at a 500× magnification.

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3.2 Methods

The general procedure of the new method is described in this chapter. To process of using the recorded film to data from which the magnetic moment m of the particle can be determined is elaborated. We explain the algorithm used to locate the particles from the images and calculate their exact center.

From their trajectories the distance r(t) between two particles of interest can be calculated. And with a fit of r versus t the magnetic moment is determined.

For a more detailed description of what is mentioned in this chapter, including elaboration of the matlab scripts, see appendix A.2.

3.2.1 Locating the particles

With the camera used in the setup shown in figure 6 a small movie is recorded of particles during repulsive interaction. This film is converted into a sequence of images, preparing to analyse them with matlab. After conversion, the images are cropped to a region of interest including (preferably) two particles, for an example see figure 11(a).

The following procedure is done for every image (frame) of the movie:

1. A convolution is performed with a template image of a particle, see figure 11(b). This template image is a circular image of only one particle which is prepared before processing the image sequence.

2. In this convoluted image is searched for values above a certain threshold value defined by the user, symbolized by the black dots in the figure. This arbitrary threshold factor is defined by the user. A flexible way of selecting points is used, involving the mean and standard deviation of the image:

find (value > mean + std*threshold_factor).

3. All found points are divided in clusters, each cluster represents one single particle.

4. A weighted average is taken of all points in one cluster to determine its center, and therefore the center of the particle. The weightfactor is the actual value of that pixel. So pixels with a greater value, which are closer to the center, have a larger weightfactor. A white plus sign with an index number is plotted for every detected particle.

5. Calculate the distance, center to center, between the two particles in the image by Pythagoras. Or, if there are more particles in the image, calculate the distance between the two particles of interest.

The procedure is to repeat step 1 to 5 for every frame in the image sequence.

After this the distance between two particles of interest will be known at every timestep. So actually the data obtained is the time dependend distance r(t), which is exactly what is needed to determine the magnetic moment.

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(a) (b)

Figure 11: Both image (a) and (b) are the same particles. Image (a) is the raw data as it is recorded with the camera. Image (b) is the convoluted image of image (a) and here the particles are solid circles, which makes it easier to determine their exact center. In image (b) black dots, points which meet the criterion, and white plus marker, which are the centers of the particles, are plotted aswell.

3.2.2 Fitting r(t)

Two different fitting methods are used to determine the magnetic moment of a particle couple from which r(t) is known.

First, the raw data is fitted with a fifth root curve, see figure 12(a), because of r ∝ √⁵

t relation, see equation 12 and 13. In the ideal case, that there are sufficient data points to fit the fifth root curve, no problems will be experienced with this fit. However when there happens to be a lack of data points, especially in the steep part, the accuracy of the fit will be lost.

A lack of data in the steep part, which is the start-up phase of the repulsion, occurs often. This might be due to several reasons e.g. when an image with very closely located particles is convoluted the particles are likely to overlap which makes it impossible to part them in clusters and determine their centers.

Another reason is that in the start-up phase of repulsion the particles separate the fastest so it is hard to capture this on film.

To prevent this loss of accuracy a linear fit of r⁵ versus t, see figure 12(b), is made. Now over the whole range the fit will be linear and because of this, no loss of accuracy will be experienced when there is a small loss of data points.

Now the magnetic moment can be determined with the slope of the fit, see equation 14. So by filling in the slope from the fit a, the dynamic viscosity of the fluid η and the radius of the particle R, the averaged magnetic moment of this particle couple at a certain strength of external field is found.

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(a) A non linear fit of r versus t. (b) A linear fit of r⁵versus t.

Figure 12: Two different fitting methods are used to determine the magentic moment.

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4 Results and discussion

As described in the previous section experiments were performed on the breaking of two particle clusters using several magnets to produce magnetic fields which induce attraction or repulsion between the particles. In this section the results from these experiments will be discussed:

• the averaged magnetic moment m of a particle couple was determined,

• magnetization curves are made,

• the magnetic moment variation in the particles is calculated

• and the total number of grains and their size is determined.

Only data from repulsive interaction between the particles is processed because it is more efficient. Particles split up instantly when the external field switches orientation and so a movie of just a few seconds can capture multiple couples splitting up. On the other hand when switching to a horizontal field, which induces attraction, the particles do not start attracting to each other immediatly.

They must be close enough to ‘feel’ each other.

Later on, in section 4.2, we will show that magnetic moments obtained from repulsive interaction are just as accurate as those obtained from attractive interaction. So we choose for the most efficient way of experimenting: analysing repulsion only.

4.1 Determining magnetic moment of particle couples

Using the setup shown in figure 6 a small movie is recorded during the magnetic field’s orientation switch from horizontal to vertical, inducing repulsion so the particles split up. Running the scripts locating the particle will supply us with r(t) and as mentioned before two different fit curves are used to obtain the magnetic moment of the particle couple of interest.

The same procedure has been followed but instead of inducing repulsion the external field is switched from a vertical to a horizontal orientation and the particles start attracting each other.

Results from these experiment are given in figure 13 and 14. The two fits, mathematically described in section 3.2.2, of a repulsive interaction are given.

The r ∝ √⁵

t behavior is clearly visible in the left graph. Of these same particles attraction has been analysed. The fits from this experiment are given in figure 14. Again the fifth root relation is visible.

The magnetic moment determined from these fits are given in table 1.

The magnetic moment determined from the attractive process are slightly higher than those determined from repulsion. However the values from attraction and repulsion, both from linear and non linear fitting still overlap taken in account the errors. Although all four methods are applicable determining the magnetic moment of a pair of particles, we’ll focus on the repulsive part. As stated before

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(a) (b)

Figure 13: Fits of the experimental data recorded during repulsion. In figure (a) the distance r between two particles during repulsive interaction is plotted versus t. In figure (b) the relation is made linear, r⁵ has been plotted versus t.

(a) (b)

Figure 14: Fits of the experimental data recorded during attraction. In figure (a) the distance r between two particles during repulsive interaction is plotted versus t. In figure (b) the relation is made linear, r⁵ has been plotted versus t.

Linear fitting method (10⁻¹⁴ Am²)

Non linear fitting method (10⁻¹⁴ Am²)

Repulsion 1,90 ± 0,03 1,92 ± 0,03

Attraction 1,96 ± 0,04 1,96 ± 0,03

Table 1: Magnetic moment of one pair of particles determined during attraction and repulsion with a linear and a non linear fitting method.

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analysing repulsion is more efficiently because the particles separate simultane- ously.

4.2 Magnetization curve

In the previous section the magnetic moment of one particle couple was determined at a certain strength of external magnetic field. The magnetization curve is the relation between the magnetic moment and the applied external field. So to determine this relation the experiment from the previous paragraph is repeated for different strengths of magnetic fields. In the curve the magnetic moment is plotted versus the external magnetic field and gives an impression of the magnetic behavior of the particles. A magnetization curve has been made for both PEG and MyOne particles. As stated before, only data from repulsion is analysed.

4.2.1 PEG

Figure 15: The magnetization curve of PEG particles. The Langevin fit of the data (black) aswell as the reference VSM measurement (purple and green) is plotted.

In figure 15 the magnetic moment of PEG particles is plotted versus applied magnetic field.

The blue circles are data points calculated from the linear fit of r⁵ versus t. A Langevin fit (the black dashed line — see section 2.7 for information about the Langevin equation) has been used to fit these data points and this is compared to a reference measurement with a VSM, the purple dashed line. This VSM line does not correspond correctly with the fit from the data. Probably this is due to the fact that while calculating the magnetic moment from the VSM

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experiments the diameter of a particle was assumed to be exactly 3 µm, as the supplier claims it to be. However with dynamic light scattering we have determined the diameter to be 2,6 µm (section 3.1.2).

Now the magnetic moment m is proportional to the volume, so m ∝ R³and we correct the VSM with a factor (2,6/3)³ ≈ 0,65. The corrected reference data seems to correspond almost perfectly with the experimental data. The small deviation might be caused by a small batch difference and can be corrected with taking a larger sample.

The data and Langevin fit obtained with the non linear fitting method is given in appendix B. Compared with the results from linear fitting there are no strong deviations. Magnetic moments determined with the linear fitting method are more accurate, so we prefer to use the linear fitting method over the non linear one.

4.2.2 MyOne

As for the PEG particles a magnetization curve has been measured for the MyOne particles, see figure 16. Again the black curve is the Langevin fit of the experimental data and the purple one is VSM data.

Figure 16: The magnetization curve of MyOne particles. The data has been fitted using a Langevin equation (black) and is compared to a VSM measurment (purple).

The calculated data corresponds almost perfectly with the reference data from the VSM measurement. However the shapes of the curves slightly differ. Our measured curve seems flatter than our reference. This might be due to the lack of data points at higher magnetic field. Saturation isn’t reached yet and could change the shape of the curve and taken in count the error margin of the fit it

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could be possible that the data curve still overlaps with the reference.

4.3 Particle variation

Apart from calculating the magnetization curves from the MyOne and PEG particles, this method can provide us an estimation for the mutual particle variation in the magnetic moment. By measuring the magnetic moment of several particle couples all at the same strength of magnetic field a variation will appear due to variation of magnetic content embedded in the particles.

Actually we are determining a variation per pair of particles. The calculated moment, which is averaged over two particles, is given by mav =√

m1m2. Taken in account that the standard deviation in m1and m2are equal, so σm₁= σm₂ = σman expression can be found for the deviation of one particle σmas a function of the measured deviation σm_av.

The relative error in σm₁m₂ is given below.

(σm₁m₂)²

(m₁m₂)² = 2 σ²_m

m², taking m1= m2≡ m, (σ_m₁_m₂)²

m⁴ = 2 σ²_m m², so (σm₁m₂)²

m² = 2 σ²_m.

Taking the square root of this will give us an expression for the standard deviation of m₁m₂:

σ_m₁_m₂

m =√

2 σ_m.

The relative error in σ^√_m₁_m₂ is given below;

σ^√m₁m₂ = σm₁m₂

∂

∂mm

√mm = 1 2

σ√m1m2

mm =1 2

σm1m2

m ,

Combining these gives an expression for σm in terms of σm_av: σ^√m₁m₂ =

√ 2 2 σm, σ_m=√

2 σ^√_m₁_m₂ =√ 2 σ_m_av.

So the variation in the magnetic moment of one particle is the variation of the couple multiplicated with a factor√

2.

4.3.1 PEG

A histogram of the magnetic moments determined at about 13 mT is displayed in figure 17. Assuming that the magnetic moment is distributed normally, a

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Gaussian fit has been made throught the data and the mean and standard deviation of this sample of 44 measurements has been determined.

Figure 17: A histogram of the variation of the magnetic moment of PEG particles. The red line is a Gaussian fit of this sample of 44 measurements.

The mean magnetic moment µm equals (2,1 ± 0,4) · 10⁻¹⁴ Am². The relative variation is characterized by the ratio between the standard deviation σ and µm

which equals 0,4/2,1 = 17%.

So the PEG particle variation of the magnetic moment per particle equals√ 2 · 17% = 24%. The supplier does not provide information about the particle variation and there are no other reference values avaible. So unfortunately we are not able to verify these results.

4.3.2 MyOne

In figure 18 a histogram of the magnetic moment of particles determined at 13 mT is given. A Gaussian function has been fitted with the data and again the mean and standard deviation of this sample is calculated. The sample consisted of 92 measurements.

The mean and standard deviation are determined µm± σ = (6,7 ± 0, 7) · 10⁻¹⁵ Am². This results in a relative variation of 0,7/6,7 = 12%.

Per particle the variation equals√

2 · 12% = 17%. As with the PEG particles there is no reference avaible and we are not able to verify these results. However, taken in mind that the PEG particles are of poorer quality than the MyOne particles a 17% variation is very reasonable.

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Figure 18: A histogram of the variation of the magnetic moment of MyOne particles. A Gaussian fit, the red line, has been made. The sample included 92 measurements.

4.4 Grain moments and sizes

In section 4.2 the magnetization curves were compared to reference measurements to check their reliability. The Langevin curve (see equation 17) fitting allows us to determine the number of grains n, their magnetic moment µ aswell as the radius of a grain. The grains consist of magnetite, Fe3O4, and because they are smaller than the magnetic domain wall, which is 80 nm [10], they may be considered single domain areas. For this reason the volume of a grain is given as the ratio between the magnetic moment and the saturation magnetization:

Vgrain= µ Msat

. (20)

For magnetite the saturation magnetization Msat= 6,2 · 10⁵J m⁻³ T⁻¹ [11].

The summed weighted Langevin curve, see equation 19 , has been used to fit the magnetization curve of the PEG and MyOne particles. Advantages of this summed Langevin fitting is that it provides several properties of the grains inside the particles. With the fitting, the grain moment distribution is determined and with this the mean volume and radius of a grain can be calculated using equation 20.

The second useful variable which can be determine by fitting with the summed Langevin equation is the total number of grains which lay inside one particle n.

The volume of a grain together with the total amount of grains will provide the total volume occupied by the grains and with that the filling factor of magnetic content in a particle. These properties are given in table 2.

A typical grain size is about 6 to 12 nm, so this corresponds with calculated values for grains in the MyOne and PEG particles.

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MyOne PEG radius r 6,3 nm 9,6 nm µ (10⁻¹⁸ Am²) 0,66 2,2

number n 6, 7 · 10⁴ 1, 1 · 10⁴ Filling factor 27% 0,4%

Table 2: Grain properties of MyOne and PEG particles

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5 Conclusion

Here, we have studied the analysis of couples of superparamagnetic particles.

We have been able to determine the magnetization curve of these. Two types of particles were investigated:

• Micromer PEG particles — 2,6 µm diameter,

• Dynabeads MyOne particles — 1,0 µm diameter.

By exploiting the magnetic dipole–dipole interaction of the superparamagnetic particles, especially repulsion, it is possible to determine the magnetic moment of one pair of particles. This magnetic moment depends on the applied external magnetic field and a magnetization curve, the curve of the magnetic moment versus the applied field, has been made for both types of particles.

The magnetization curves have been compared to those made with a VSM measurement. They corresponded almost perfectly and we conclude our new method is succesful for determination of the magnetic moment of two particles.

Compared to the VSM measurement the method described in this report has a great advantage. The averaged magnetic moment of a pair of particles has been determined. With a VSM measurement this is not possible. The magnetic moment is determined of a large number of particles.

Therefore, with a VSM measurement it is also impossible to obtain an estimation for the deviation or variation in the magnetic moment of these particles. And with our new method it is. By applying statistics on a sample of particle couples it is possible to calculate the standard deviation of the magnetic moment, and therefore the magnetic content, of a superparamagnetic particle.

Several properties of the grains, which are embedded in the particle, like grain size and magnetic moment, are determined with weighted Langevin curve fitting.

With these properties information like the total volume of the grains and the number of grains in a particle could be calculated.

In conclusion, the method we developed in this project has been proved succesful in producing a magnetization curve for magnetized superparamagnetic particles, which was the primairy goal. And previously impossible determinable properties like the number and size of grains inside one particle can now be calculated.

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References

[1] E. Pollert, K. Kn´ıˇzek, M. Maryˇsco, P. Kaˇspar, S. Vasseur, E. Duguet, 2007, J. Magn. Magn. Mater. 316 (2), 122–125.

[2] J.A. Abels, F. Moreno–Herrero, T. van der Heijden, C. Dekker, N.T.

Dekker, 2005, Biophys. J. 88.

[3] G. Fonnum, C. Johansson, A. Molteberg, S. Mørup, 1997, J. Magn. Magn.

Mater. 5–14.

[4] M.W.J. Prins, M. Megens, 2007, Chapter in Encyclopedio of Materials:

Science and Technology. Elsevier, 1–6.

[5] P.A. Besse, G. Boero, M. Demierre, V. Pol, R. Popvic, 2002, Appl. Phys.

Lett. 4199.

[6] S.P. Mulvaney, R.L. Cole, M.D. Kniller, M. Malito, C.R. Tamanaha, J.C.

Rife, M.W. Stanton, L.J. Whitman, 2007, Biosens. Bioelectron, 191–200 [7] G. Fonnum, C. Johansson, A. Molteberg, S. Mørup, E. Aknes, 2005, J.

Magn. Magn. Mater. 41.

[8] H. Swagten, 2008, Course syllabus Magnetism & Magnetic Materials [9] J. Leach, H. Mushfique, S. Keen, R. Di Leonardo, G. Ruocco, J.M. Cooper,

M.J. Padgett, 2009, Phys. Rev. E 79, Comparison of Fax´en’s correction for a microsphere translating or rotating near a surface.

[10] R.F. Butler & S.K. Banerjee, 1975, Theoretical single-domain size range in magnetite.

[11] C.E. Housecroft & A.G. Sharpe, 2005, Inorganic chemistry 2nd Ed. Pear- son.

[12] G. Mihajlovic, K. Aledealat, P. Xiong, S von Moln´ar, M. Field, 2007, Appl.

Phys. Lett 91, Magnetic characterization of a single superparamagnetic bead by phasesensitive micro-Hall magnetometry.

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A matlab scripts

A.1 The protocol

The protocol for tracking particles in an image is as following:

1. Crop the image to a region of interest only including the particles. This can be done by using

• cutregion.m, which simply cuts out the ROI of a sequence of images or

• cutregion_backwards.m, which does exactly the same but names the image in reverse order. This is useful when repulsive interaction is analysed,

2. Load the first image (read_image.m) and define a template particle (template_particle.m) used in the convolution,

3. Perform the convolution (convolution.m),

4. Search for values in the convoluted image im_conv above a certain threshold. This threshold depends on the standard deviation of the image and an user define factor, threshold_factor,

5. Part all found locations in clusters, each cluster is a particle, and calculate the center of each particle by taking the weighted mean of the corresponding cluster.

The centers then are stored and step 3 to 5 is repeated for every image in the sequence.

The output of the script (particle_tracking.m), which includes looping through step 2 to 5, will be the locations of the center (in pixels) of each particle in time and the number of particles found.

An option to run the script data_analysis.m at the end of

particle_tracking.m is provided. It will use the locations of the particles to determine the desired magnetic moment m. See subsection A.2 for more information about this script.

A.2 Data analysis

The script data_analysis can be run at the end of the particle tracking script to analyse the output of this. It takes all the positions of the particles (which is the output of particle_tracking) and calculates the distance r (in meters) between two particles taking in account the resolution.

A weighted linear fit of r⁵ versus the time t is made with lscov. Weightfactors are taken 1/σ²_i, with σ_i the error in data points (the distance r, or r⁵depending on if the non linear or linear fitting method is used) i. The slope of the fit is used to determine the magnetic moment m. As explained in section 2.3.1 this actually is the average of the magnetic moments of the two particles.

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To check the value of m a fifth root fit of r versus t is made aswell. The command nlinfit is used to fit the data on a custom equation.

The calculated data, like particle position, distances and calculated magnetic moments, is printed in a file which is saved in the on the disk, optionally together with the convoluted images.

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B Data from non linear fitting

In this appendix the magnetization curves, Langevin fits and reference VSM fit from MyOne and PEG particles obtained from the non linear fitting method are given. For the magnetization curve of the PEG particles, see figure 19, and the magnetization curve of the MyOne particles is given in figure 20.

The graphs corresponding to the non linear fitting method are very similar to those corresponding to the linear fitting method (figures 19 and 20).

Figure 19: The magnetization curve of PEG particles. The Langevin fit of the data (black) aswell as the reference VSM measurement (purple and green) is plotted.

Figure 20: The magnetization curve of MyOne particles. The Langevin fit of the data (black) aswell as the reference VSM measurement (purple) is plotted.

Eindhoven University of Technology BACHELOR Characterization of the magnetic moment of superparamagnetic particles from magnetic dipole-dipole interactions Bos, A.H.