UNIVERSITY OF LJUBLJANAFACULTY OF MATHEMATICS AND PHYSICS

DEPARTMENT OF PHYSICS

SeminarSuperparamagnetic Colloids

Author: Blaz KavcicMentor: doc. dr. Igor Poberaj

Ljubljana, March 2007

AbstractThis seminar brie�y describes the general properties of colloids and then focuses on superparamagnetic

colloids. Synthesis and interactions in a magnetic �eld are discussed for such particles. At the end anexperiment with superparamagnetic colloids which con�rms certain theoretical predictions is presentedin more detail.

Contents1 Introduction 2

2 Colloids 22.1 General Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Interactions within a Colloidal Substance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Signi�cance of Colloids in Science and Industry . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Magnetic Colloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Superparamagnetic Colloids 43.1 Review of Superparamagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Synthesis of Superparamagnetic Colloids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Interactions in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 The Experiment 84.1 Motivation and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 The Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Conclusion 11

1

1 IntroductionWhat is common to smoke, milk, butter, fog, and paint? They are all colloids. The term colloid is usuallyused for both the substance itself and the particles dispersed in it and means glue-like, originating from theGreek word for glue, kolla [1]. It was �rst used by chemist Thomas Graham in 1861 to describe certainsolutions, the so-called Selmi's pseudosolutions, which have a very low rate of di�usion through a sheet ofparchment [2]. He concluded that the studied substance is a suspension of particles in a liquid medium andthat the particles were relatively large, measuring from several nanometers and up to about a micrometer.Their size is the reason they can be �ltered, unlike solutions of crystalline substances - ions and molecules -such as salt.

For several decades there was little interest in colloids, until they became important in industry andbiochemistry in the �rst half of the 20th century. Since then the interest in colloids is only growing andcolloids are used for many practical applications as well as in research laboratories as model systems.

This seminar focuses on a speci�c type of colloids, superparamagnetic colloids, and mentions their syn-thesis, behavior in a magnetic �eld, and possible applications. The general properties of colloids and themotivation for their use in general are brie�y described as well. In the last part an ongoing experiment inwhich colloids were used successfully as a model system is described in more detail.

2 Colloids2.1 General PropertiesColloid is a substance that has two components in two phases, an internal and an external phase. A dispersingmedium, the external phase, is the prevailing constituent of a colloid substance and the internal phase isdispersed in it. The two components can be in any combination of the three aggregate states, gaseous, liquidand solid, except in a gas-gas combination. Each combination has its own name: a liquid in liquid dispersionis an emulsion, gas in liquid is a foam, solid in gas is called an aerosol and so on. A few colloid substancefrom our everyday life are fog, cream, foam, smoke, paint, milk, detergents, agricultural sprays, cosmeticproducts, and motor oil are just a few examples. Arti�cial colloids are prepared by either reducing the size ofbigger particles or condensing the smaller ones until they reach the desired size. There are many laboratoryand industrial methods used to prepare them.

The particles can be monodisperse or vary a great deal in size, such as in milk (0.1 to over 20 µm [3]) orin an oil emulsion one can create by vigorous shaking. Monodisperse colloids are often desired in laboratoryexperiments because such particles have the same physical and electromagnetic properties such as mass,mobility, interaction strength etc. Because of their small size the area to volume ratio of colloids is large. Aliter of certain typical 0.1 M micellar solution has the total of 4 hectares of the interfacial area [2]! We canconclude that surface e�ects play a very important role in colloid substances and a�ect their the physicaland chemical properties signi�cantly.

See Fig.1 for some of the more interesting arti�cial products.

Figure 1: Some examples of experimentally made colloids of very interesting shapes [11]. On the right-mostphoto is an example of the much more common and widely used spherical colloids, here arranged into aperfect 2-D close-packed structure due to an external magnetic �eld.

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2.2 Interactions within a Colloidal SubstanceThere are several types of forces that govern interactions of particles inside a colloidal substance. All playan important role, though not necessarily on the same length scale and in the same way. Some may helpstabilize a colloid while others may do the opposite.

The simplest is the excluded volume force, a repulsive force present due to the fact that two (hard)particles cannot penetrate each other.

Colloidal particles may have permanent electric dipole moments, or such moments can be temporarilypresent because of the �uctuating electric charge density. Such temporary dipole will induce a dipole momentin particles in the vicinity as well. The two particles, dipoles, will attract - the force between them is the vander Waals force. If the van der Waals forces prevail in a colloid, the particles inside it will tend to aggregateand ultimately settle on the bottom or on the surface of the substance.

Often an important force is the electrostatic force, as usually some charge is adsorbed to the surface ofthe particles in a colloid. All particles adsorb a like charge so they repel. This force helps stabilizing thecolloid. Because the surface of colloidal particles is so great compared to their mass, the interaction due totheir surface charge will overcome their kinetic energy and the particles will not come close to each other.

Steric forces between particles may create a repulsive or an attractive e�ect. The surface of particlescovered with certain polymers or the dispersing mediums made of polymers that don't adsorb on the particlesdispersed in it can cause such interaction. These two possible cases are the so called steric stabilization orthe attractive depletion force.

Brownian forces (di�usion) on individual particles also play a role, as they must overcome gravity andbuoyancy and cause the dispersed particles to move in random directions. The hydrodynamic force on asmall spherical particle is the well-known Stokes' formula F = −6πηRv, where R is the size of the particleand η is the viscosity of the surrounding medium. The formula is valid for small Reynolds numbers which isalways the case when dealing with very small particles on the order of microns or less.

In a gravitational �eld the external force on the di�using particles is the di�erence of gravity and buoyancy,Fext = −mg + Fb. The density n of the particles in equilibrium is not constant and is described by the"barometric" formula n ∝ exp(−Fextz/kBT ). If there was no di�usion the external force on the particleswould cause a current jG = nv of particles moving with the terminal velocity of v = Fext/6πηR. However, inequilibrium the di�usion current jD = −D∇n due to the particle number-density gradient must just cancelthe gravitational current.

Therefore, the di�usion coe�cient of Brownian motion for colloidal particles equals D = kBT/6πηR.

2.3 Signi�cance of Colloids in Science and IndustryColloids are so interesting and important for science because they cover the so called mesoscopic region,a region between solid state and molecules spanning six orders of magnitude. They are relatively easy toproduce and the possibility of coating their surface with a speci�c adsorbent is interesting for chemistry andbiology. Such products are commercially easily available.

The colloids are also an invaluable tool for a physicist, not just because of their interesting properties,but also because they can be used as model systems for atoms and materials in statistical physics [4]. Theirsize - they are small yet big - plays an important role. They are small enough so that one can easily usemany of them and have a good statistics on measurements, also the system is quickly thermalized and readyfor another experiment because the Brownian forces quickly randomize the positions of all the particles. Itis also important colloids are big enough to observe with an optical microscope. This is a huge advantagebecause it makes the system easy to see and easy to track in real time, unlike with molecules and atoms.

2.4 Magnetic ColloidsElectrically charged colloids repel each other and are attracted or repelled by an electric �eld. One quicklyrealizes why magnetic colloids could be a much more useful tool: they can be neutral until the magnetic�eld is switched on so they do not have to be charged in the absence of a �eld to be useful when a �eld ispresent.

Another important advantage is the interaction between the colloids, which can have both positive andnegative sign, depending on the choice of the �eld. Magnetic colloids may aggregate, repel, �ow in a desireddirection when subjected to a magnetic �eld gradient and so on. It turns out that by far the most usefultype of colloids are those with superparamagnetic properties. Let us look at them in more detail.

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3 Superparamagnetic Colloids3.1 Review of SuperparamagnetismWe usually classify matter based on its magnetic properties in the presence and absence of an externalmagnetic �eld as:

- Diamagnetic: this material has no magnetic dipoles unless the �eld is present. The induced magnetiza-tion is very small and opposite to the �eld.

- Paramagnetic: the dipoles in it are randomly oriented and will at least partially align with the magnetic�eld.

- Ferromagnetic: the dipoles in it are at least partially aligned even if there is no �eld.- Ferrimagnetic: two types of dipoles are arranged in the opposite directions, one type among the two is

prevailing, so there is some magnetization present even if there is no �eld.- Antiferromagnetic: when there is no �eld the dipoles in this material are antiparallel to each other so

the net magnetization is zero.

Both the ferro- and antiferromagnetic materials become paramagnetic above the Curie and Neel temper-atures, respectively. Above those temperatures the thermal oscillations in the material are strong enough toovercome the interaction between aligned dipoles, �ip and rotate them around, which destroys the aligneddomains.

Ferro- and ferrimagnetic materials are usually de-

Figure 2: A typical hysteresis loop for materialswith di�erent magnetic properties. Bsat denotes thesaturation value when all the dipoles are aligned, Br

and Hc denote the remanence and coercivity.

scribed by the hysteresis loop, which is a diagram ofthe induced magnetization M versus the applied ex-ternal magnetic �eld of strength H for the observedmaterial. Instead of M the magnetic �eld density Bcan also be plotted (Fig.2). A typical curve showsthe saturation magnetization, the remanence and co-ercivity. After the �eld is switched o� the magneti-zation usually does not drop to zero. Its remanentvalue is called remanence and the coercivity is the�eld we would have to apply in the opposite direc-tion if we wished the magnetization to return to zeroagain. The horizontal distance between the S-shapedcurves is called hysteresis and the area of this loopdescribes the work on a magnet. This implies energylosses - the bigger the coercivity the bigger the lossesin a magnetic material.

Magnetism depends on temperature as we saw,and it also depends on the volume. At some point when the a ferro- or ferrimagnet becomes su�cientlysmall, it will become a single magnetic dipole. That is, the whole particle will be only one magnetic domain.If we make it even smaller the thermal e�ects on it will become very important. The energy needed torotate or �ip its dipole moment is easily provided by the thermal energy. At this point the particles do nothave ferromagnetic properties anymore because their magnetic moments become randomly oriented. Theybehave paramagnetically even at temperatures far below the Curie and Neel temperatures - we call this statesuperparamagnetic. The size of particles where this happens is between 4 and 20 nm, depending on thematerial [5].

The superparamagnetic particles will therefore align with the external �eld easily even in weaker magnetic�elds. They will randomize their directions and become neutral again almost immediately after the �eld isturned o� because the thermal energy will �ip the dipoles in random directions. In commercially availablemagnetic colloids these nano-sized particles are superparamagnetic at room temperature on the order ofseconds [6]. The key of superparamagnetism is therefore the particles with no memory e�ect which canbecome magnetic only when a magnetic �eld is applied. This is what makes superparamagnetic colloids souseful and so widely used.

A lot of the more speci�c information in the following section was obtained from Refs. [5] and [7].

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3.2 Synthesis of Superparamagnetic ColloidsThe synthesization of superparamagnetic colloids is far from trivial and there are several procedures full ofdetails which will be mentioned only brie�y here. There are many ways to synthesize the nanoparticles - frommetals, metal alloys, metal oxides and ferrites. Frequently used are iron oxides FeO, γ-Fe2O3 and Fe3O4

[6], they can be found in commercially available products. The processes of manufacturing the particles areusually quite demanding and their sizes must be as equal as possible.

Since they are small but one often wishes particles that are easy to see and have a large magnetization forbetter manipulation, larger composite particles, the so called mesoparticles, are usually made in the next step.A porous non-magnetic matrix is �rst prepared, or in some instances grown around the nanoparticles whichare mixed in the same suspension. When the process is complete one has particles up to 5µm big with manyuniformly distributed superparamagnetic nanoparticles inside each. This greatly increases the magnetizationand also prevents the particles within a mesoparticle from aggregating and forming ferromagnetic chunks.Such mesoparticles can be polymer- and silica-based, though recently some newer materials are considered,such as amorphous selenium.

The last phase in synthesization is very important too - capping and an optional functionalization. Thecap around the bead is usually made as a monolayer of some substance, often amphiphilic, which means itexhibits both hydrophilic and hydrophobic properties and safely seals the inside of the beads. There aremany types of surface coatings for the nanoparticles, from silica to metallic and polymer coatings. Theappropriate coating must be chosen depending on what the colloids will be used for, what material they willbe embedded in, will they be biocompatible if they are to be used in medicine, and so on.

If we deal with the composite mesoparticles, we can also coat their surface in a way that functionalizesthem. The term means that we attach certain molecules to them, like reagents or antibodies, which willselectively bind certain other molecules, cells, or antigens. This opens the possibility of separation and manyother uses in chemistry, biology and medicine.

There are several companies producing superparamagnetic beads commercially and for research purposes,among them Micromod, Bangs Laboratories, Estapor and Dynal [7]. One of the most popular is Dynal whosebeads are the most popular [7] for laboratory use (Fig.3), as Ref.6 put it. These colloids were used in theexperiment described in Sec.4 as well and we can take a closer look at then. The measurements for Dynalbeads cited here were performed by G. Fonnum et. al [6]. Dynal makes spherical monodisperse beads inthree sizes (1, 2.8 and 4.5 µm), which deviate from the mean by less than 2 %. They have a density of about1.5 g/cm3 and contain between 100 and 250 mg/g of iron oxides, which are, according to measurements,more than 99% in the superparamagnetic state. Its nanoparticles are about 7.5 nm big and the saturationmagnetization is between 11 and 24 Am2/kg. The reason for such di�erences in iron content, which isespecially high in the smallest beads, is due to the fact that smaller particles have smaller mobility.

Figure 3: Hysteresis curve of the 2.8 µm Dynal magnetic beads, the small inset image shows the curve for�elds up to 10 mT - the beads trully have superparamagnetic properties. Top right is the SEM photo ofthe bead, the granule structure magni�ed on the bottom right photo are actually the superparamagneticnanoparticles. [6]

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3.3 Interactions in a Magnetic FieldMagnetic force and energy: Let us look at how superparamagnetic colloids behave in a magnetic �eldfor a few simple examples. This will help us understand why magnetic colloids have proven so useful.

Our coordinate system is centered on a magnetic dipole with a magnetic moment ~µ1, while ~r and r arethe radius vector and its magnitude. A mesoscopic superparamagnetic bead of volume V0 can be representedby such magnetic dipole, but let this treatment be completely general at �rst. The magnetic �eld ~B1 of themagnetic dipole is well known:

~B1(~r) =µ0

4π

3~r(~µ1 · ~r)− r2~µ1

r5(1)

We now take a second magnetic dipole, ~µ2, and put it at point ~r in the magnetic �eld of the �rst dipole.The potential energy of the two dipoles, also called the interaction energy, is calculated with the standardexpression,

Eint = −~µ2 · ~B1 (2)For the magnetic �eld of the �rst dipole (2), the expression for the interaction energy becomes:

Eint =µ0

4π

((~µ1 · ~µ2)

r3− 3(~µ1 · ~r)(~µ2 · ~r)

r5

)(3)

From (3) one can also derive the force exerted on a magnetic dipole in an arbitrary magnetic �eld,replacing with ~µ and ~B:

~F = −∇Eint = (~µ · ∇) ~B (4)The magnitude µ of the magnetic moment of a dipole, for example our superparamagnetic bead, depends

on the volume of the bead, the di�erence of susceptibilities ∆χ of the bead and the surrounding medium,and of course the density and the direction of the external magnetic �eld ~B:

~µ =V0∆χ

µ0

~B (5)

The force on the bead is therefore:

~F =V0∆χ

µ0( ~B · ∇) ~B (6)

It is important to note that in superparamagnetic particles the dipole moment will in most practicalcases always be aligned with the externally applied �eld.

Static magnetic �eld: Let an external magnetic �eld ~B be present in the z-direction. We can use thecylindrical coordinate system where the azimuth angle is not important at the moment. Angle θ is the anglebetween the �eld and the position vector ~r between the beads (Fig.4).

Figure 4: The situation in the cylindrical coordinate system.

The external �eld is of course so strong that the magnetic moments stay aligned with it at all times, so~µ1 · ~µ2 = µ1µ2. The interaction energy in this coordinate system is:

Eint =µ0

4πr3

(µ1µ2 − 3µ1µ2cos

2θ)

(7)

6

If we con�ne the dipoles to the xy plane, the interaction energy between them (9) reduces to only therepulsive term:

Eint =µ0

4π

µ1µ2

r3(8)

The interaction is purely repulsive and falls of as r3 (Fig.5.a). If the dipoles are not con�ned and arefree to move in 3-D space, the angle θ can vary. At some point, the term in brackets in (7) changes sign(at θ = 54.6◦) and the interaction becomes attractive. The attraction is strongest when the dipoles are ontop of each other, forming a dimer aligned with the external �eld. The interaction between the dipoles fora �xed r is shown in Fig.5.b.

Figure 5: a) Interaction energy between two magnetic dipoles in a magnetic �eld perpendicular to the planethey are con�ned to. b) Interaction energy depending on the angle between the �eld and ~r for the twodipoles for at a �xed distance r.

Rotating magnetic �eld: Let us again have the dipoles con�ned in two dimensions and the externalmagnetic �eld ~B lies in the xy plane. The magnetic dipole moments of the particles also lie in this planeand are aligned with the �eld at all times. To make it simple, let ~r between the dipoles be aligned with thex-axis.

If the �eld rotates in the plane then at some points in time it will be perpendicular to the ~r connectingthe dipoles so the dipoles repel, while at other points in time it will be parallel so the dipoles attract, asdescribed for the static case. If we now use θ for the azimuthal angle then the interaction is described by (7)for any angle the magnetic �eld is pointing at. θ now changes in time as the �eld rotates so we need to �ndout what the average interaction is for one revolution in the xy-plane. One must therefore �nd an averagevalue of the term in brackets in (7) over the whole circle between 0 and 2π. The average of the square ofcosine is 1/2. We can write:

Eint =µ0

4π

µ1µ2

r3(1− 3/2) = −µ0

8π

µ1µ2

r3(9)

The interaction is negative and the magnetic dipoles (or our beads) attract. There are several factorsthat determine the adequate frequency to achieve the desired attractive e�ect, among them viscosity of themedium. Experimentally, a frequency of about 100 Hz is adequate.

Magnetic �eld gradient: If the magnetic beads are put into a magnetic �eld gradient, the magneticforce exerted on them will behave according to (6). They will move either in direction of the weakest orstrongest �eld, depending on the ∆χ, the magnetic susceptibilities of the medium and the colloidal particles.We can therefore control the direction of motion of colloids with the proper selection of the medium andparticles. One can get a very inhomogeneous magnetic �eld by using tapered magnets or electromagnetswith tapered inner cores, possibly made of soft iron. The idea of magnetic tweezers comes to mind, thoughit would not be possible to selectively move a single particle.

Interactions summary: The conclusion from these simple derivations is important and in practicevery useful! If we make a two-dimensional sample with superparamagnetic colloids and apply a magnetic�eld perpendicular to the system, the colloids inside will repel each other. If the �eld is in the plane of thesystem and rotating, the colloids attract each other. With simple �elds we can therefore control the sign andstrength of the interaction. Of course the rotation must be quick enough so that the colloids only experience

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the average interaction. Unfortunately it is not possible to create an isotropic attraction in 3-D space thisway.

All the phenomena described here are easily observed in real systems, such as the system described inSec.5.3. We have found that in practice, where the system is never ideal, the individual beads start revolvingaround the z-axis, which can be a problem if the �elds are generated with function generators. However, therevolving does not occur if instead of rotating the �eld in one direction we reverse the direction of rotationfor every full circle. For our experiments we use a system which can produce such �elds.

4 The Experiment4.1 Motivation and TheoryLet us look at the following situation. A sort of a self-organization process can produce interestingly shapedstripe phases in many two-dimensional systems. Magnetic and polymer �lms, Langmuir and lipid monolayers,and liquid crystals are among them [8]. The studying of such systems is motivated by potential applicationsof such materials in nanolitography and nanoelectricity, to name two examples. As the source continues,the formation of such striped ordered phases in which dipole particles form two bonds with their nearestneighbors - like assembling a chain - is usually believed to be the competition of potentials between particlesat two length scales, between a short-range attractive force and a long-rage repulsion.

Simulations have shown that the model successfully represents such systems, while later simulationsalso showed that similar phases may also form if the potential is only repulsive, as long is it also has twocharacteristic distances for interaction [9]. This property is characteristic for the so called core-corona systemswith two characteristic lengths for a hard core and a softer corona. One could imagine this by taking theparticles inside the material to be balls with a very hard core and a softer outer part. Such core-coronasystems are many polymer systems, for example diblock copolymers, dendritic and hyper-branched starpolymers [8].

The same type of interactions are also characteristic for core-softened �uids (CSF) which seem to be goodmodels for explaining phenomena like liquid-liquid critical points and negative thermal expansion seen inwater, liquid carbon and liquid phosphorus [9]. This section discusses the core-corona theory in more detail,as the motivation for the experiment described in the further sections.

The theoretical model for a core-corona system is usually a two-dimensional system and a radially sym-metric potential with in�nite hard-core repulsion at the hard-core radius of involved particles σ0 and anadditional �nite repulsive square (or similar) "shoulder" which extends to the radius of the soft corona, σ1

[8]:

U(r) =

∞, r < σ0

U0, σ0 ≤ r < σ1

0; r ≥ σ1

(10)

For high temperatures or very high densities the e�ective radius of particles is σ0 because particle eitherposses enough kinetic energy to jump on the potential shoulder or are pushed close by the pressure of otherparticles if the density is high. If the temperature is low the particles can never overcome the barrier of thesoft shoulder and the e�ective radius is σ1. We are interested in the behavior somewhere in between theextremes, which is the case in the studied materials. Another type of potential used is a combination of aLennard-Jones potential with an additional long-range repulsion term [9 - 10]:

U(r) = 4ε

[(σ

r

)12

−(σ

r

)6]

+ ε′(σ

r

)3

(11)

The σ and ε are the distance and energy parameters of the Lennard-Jones and the third term is theadditional energy parameter. This term is repulsive of the form 1/r3 (see Eqn.8) if the dipoles are allperpendicular to the �uid plane, as discussed in Sec.4.3. This potential has a slightly more complex shapeand is actually the intermolecular potential in core-softened �uids [9]. See Fig.6 for the diagram of (10) and(11).

The �lling fraction ρ∗, usually referred to as density, is introduced, and a reduced temperature T ∗ =kbT/ε, which tells the importance of kinetic energy due to thermal e�ects compared to the energy of theinteraction between particles. The temperature is therefore considered to be low if the interaction energy εbetween particles is large.

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Figure 6: The model potentials given by Eqns. (10) and (11), used by the Refs. [8] and [9, 10], respectively.

Such a system is then simulated using standard procedures such as Monte Carlo with periodic boundaryconditions and an appropriate time step and initial conditions. The temperature and density are then variedand the results may be checked for di�erent number of particles in the model system. References [8-10] discussthe details and have published the results presented next. Similar simulations have also been performed bydr. Primoz Ziherl.

The simulations [8 - 10] showed (Fig.7 and Fig.8) what has been predicted and is observed in the abovementioned materials, for example the CSF. At relatively low temperatures and very low densities the particlesare randomly distributed on the plane, and as the density is increased a triangular structure forms. If thedensity is very high a triangular close packed structure occurs. For densities between these two, manyinteresting mesophases with no long-range order are formed. With increasing density dimers start forming,then trimers and �nally more complex phases named "labyrinthine". As the density nears close packingpolygons are also observed.

Figure 7: Simulation results [10] for a range of densities at T ∗ = 0.005. The triangular, dimer, labyrinthineand ultimately close packing patterns are clearly visible.

Figure 8: Radial distribution function g(r) at three densities (separated by 4 units vertically) at T ∗ = 0.005[8].

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The results for similar potentials also suggest that the mesophases are quite indi�erent to small changesin the potential and also to the size of two characteristic lengths, as long as they are of the same order. Itis also suggested that such potentials with an additional small attractive component may cause both thestriped phases and liquid-liquid �rst-order transitions [8]. The mentioned sources discuss in detail thesephenomena and their properties such as correlation functions and length and size distribution of stripes andclusters.

The complexity of such seemingly simple �uids caused by the competition of two length scales of therepulsion interaction is quite stunning, although not so hard to understand. Let us picture a system of highdensity. If particles are ordered in a triangular structure they will have six nearest neighbors each and willfeel the soft-shoulder repulsion of all of them. A more energy-e�cient situation for potentials mentioned inthis section is that a particle moves even closer to only two of the neighbors and thus the stripes or pairs areformed. The particle will experience a stronger repulsion with the two neighbors but the interaction energywill be smaller than that with six neighboring particles at slightly bigger distances.

As the authors of the past simulations and theoretical works realize, a more systematic research is neededon real systems. More research and experiments are necessary to understand the phenomena and perhapsput them to use some day. The key word is self-assembly and organization, which could be used for examplein nanochip production [8].

Since it is hard to study such systems on polymers and CSF because the particles (molecules) in themare very small, colloids come to mind. They are easy to track and have proven to be good model systemsfor atomic scales. Until now nobody attempted to use them for the speci�c issue presented here, whichwas the main motivation for a current experiment. The experiment and its �rst results, which con�rm thesimulations, will be explained in more detail.

4.2 The IdeaThe idea is to study a model system of the interesting CSF and polymers which exhibit striped phases withsuperparamagnetic colloids, which we decided to do in our laboratory.

With some care one can prepare an almost two-dimensional system on a microscopic slide. With themagnetic �eld in perpendicular direction one induces a repulsive interaction and can also control the temper-ature by tuning the external �eld and thus increasing the dipole interaction between particles in the system,thus making the ε big compared to kbT .

The density can be controlled quite easily during specimen preparation, and can also be increased duringthe experiment with a magnet providing a magnetic �eld gradient which pulls the particles in the specimentogether.

A more tricky part is the correct shape of the potential, which unfortunately we are not able to producein a completely two-dimensional potential. An interesting idea is making the system slightly thicker, by afraction of the colloids' size. This way two colloids could move just slightly on top of each other, up to thepoint somewhere below the critical angle θ = 54.6◦ as mentioned in Sec.3.3, which makes their repulsiveinteraction slightly smaller and thus �atten out the potential curve. This way the desired analogy of the softshoulder could be achieved. As it turned out this method works.

4.3 Experimental SetupFor colloid experiments in general we use an optical microscope coupled with optical tweezers. This allowsus to manipulate individual particles or larger groups by the use of optical traps. The more interesting com-ponent of our setup is a unique magnetic system designed by dr. Jurij Kotar, which is used for manipulationof magnetic colloids and can be attached to the microscope (Fig.9).

The system consists of six Helmholtz solenoids arranged in pairs all around the specimen observed. Thecurrent in each solenoid produces a magnetic �eld and each is controlled individually by a Direct DigitalSynthesis (DDS) circuit which lets us digitally create arbitrary waveforms and frequencies using a sourcewith a single frequency. Typical maximum useful currents are 4 A at 100 Hz and 1 A at 1000 Hz, wherethe performance is limited by solenoid inductance and Joule heating. The maximum strength of the �elds isbetween 10-20 mT.

By using the right pre-programmed current sequence we can create a static magnetic �eld in any directionin 3D space, simple rotating �elds, �elds with ellipsoidal magnitudes and other more complicated time-dependent sequences. For the experiment, we used the 1-micron superparamagnetic particles produced byDynal, whose properties have been described in Sec.3.2. The sketch of the specimen examined is in Fig. 10.

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Figure 9: The optical microscope used in the experiment, the magnetic system, and a closeup of the magneticsystem mounted on the microscope. The 6 solenoids placed around the specimen are visible.

Figure 10: Sketch of the side view of the specimen studied.

4.4 ResultsWe look at and record the behavior of the system after the �eld has been turned on. We must do sofor di�erent densities and di�erent temperatures (�eld strengths) to get as complete a phase diagram aspossible. We also wish to know if the potential in our system really is of the shape we predicted. Theexperiment is still running because the analysis of the data is not yet complete, but the results and somepreliminary calculations are quite fascinating. We observed in several samples what has been predicted, fromthe triangular structures at low and high densities to the complex mesophases in between. The photos showsome of the observed phases on Fig.11. It is easy to see the dimers, trimers, longer snakes and labyrinthineformations. Simulations made by dr. Jure Dobnikar for our setup agree with these results as well.

We are currently working on the pair correlation functions g(r), �lling fractions and determination ofthe potential shape. The main trouble is that particles must be tracked very accurately so that their exactcoordinates can be obtained. The preliminary result shows that the potential is as expected. The g(r)functions also look well and di�erent for each density, but we are in the phase of optimizing our analysistools before we can be satis�ed with our results. We also wish to gather some more data to improve thestatistics in the samples for potential measurements, which are done between two isolated particles and needvery accurate detection. Final results will hopefully be published soon.

5 ConclusionThis seminar barely scratched the surface of the ocean of colloidal science. The colloids are already widelyused and seem to have a bright future, especially if some of the possible applications that are currently devel-oped and that work in the laboratories can be made to work in the real world. The possible applications arefor example in ferro�uids, magnetic separation, guided drug delivery, MRI, microrheology and micro�uidics.With so many people working in this �eld we can be sure to see success and new ideas, for years to come asit seems at the moment.

A physicist can bene�t from colloidal atomic model systems as it was again shown in the last partof the seminar on an experiment that has not been attempted before. The results seem to agree with ourexpectations very well which encouraged us to proceed and hopefully work on new experiments in the future.

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Figure 11: The experimental data for a range of densities. A rich set of mesophases is observed, as predictedby the simulations. The results are very much like the ones presented in Fig. 9.

References[1] Resource Library, Merck website, main page:

http : //www.mercksource.com/pp/us/cns/cns_home.jsp, cited on Feb 22, 2007

[2] Max Planck Institute, Colloid Department website:http : //www.mpikg− golm.mpg.de/kc/what_is_a_colloid, cited on Feb 22, 2007

[3] http : //www.du.edu/ jcalvert/phys/colloid.htm, cited on Feb 23, 2007

[4] S. Suresh, Nature Materials 5, 253 (2006)

[5] U. Jeong, X. Teng, Y. Wang, H. Yang and Y.Xia, Adv. Mater. 19, 33 (2007)

[6] G. Fonnum, C. Johansson, A. Molteberg, S. Morup and E. Aksnes, J. Magn. Magn. Mater. 293,41 (2005)

[7] N. Pamme, Lab Chip 6, 24 (CRC, New York, 2006)

[8] G. Malescio and G. Pellicane, Phys. Rev. E 70, 021202 (2004)

[9] P. J. Camp, Phys. Rev. E 71, 031507 (2005)

[10] P. J. Camp, Phys. Rev. E 68, 061506 (2003)

[11] A. van Blaaderen Nature 439, 545 (2006)

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