glass former in two dimensions
20Pm
20Pm 20Pm
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Dissertation
zur Erlangung des akademischen Grades des Doktors der Naturwissenschaften
an der Universität Konstanz, Fachbereich Physik vorgelegt von
Florian Ebert
Tag der mündlichen Prüfung: 13.11.2008 Referent: Prof. Dr. Georg Maret Referent: Prof. Dr. Hartmut Löwen
Konstanzer Online-Publikations-System (KOPS) URL: http://www.ub.uni-konstanz.de/kops/volltexte/2008/7160/
URN: http://nbn-resolving.de/urn:nbn:de:bsz:352-opus-71605
1 Glasses 5
1.1 Phenomenology . . . 5
1.2 Inuence of dimensionality . . . 8
1.3 Colloidal model glass formers . . . 9
2 Background 11 2.1 2D system of binary colloidal dipoles . . . 12
2.1.1 Pending water drop geometry . . . 12
2.1.2 Dipolar pair potential and interaction parameter Γ . . . 13
2.2 Mode Coupling Theory (MCT) . . . 15
2.2.1 Basic formalism . . . 16
2.2.2 MCT in 2D for hard discs . . . 18
2.2.3 Von Schweidler law . . . 18
2.3 Nonadditivity of a binary mixture . . . 19
2.4 Predicted crystal structures at T = 0 . . . 21
2.4.1 Lattice sum minimization . . . 21
2.4.2 Genetic algorithms . . . 25
3 Experiment and sample preparation 27 3.1 Colloidal suspension of super-paramagnetic spheres . . . 27
3.1.1 Super-paramagnetic particles . . . 27
3.1.2 Preparation of the colloidal suspension . . . 28 1
3.2 Experimental setup . . . 30
3.2.1 Sample holder and microscope optics . . . 35
3.2.2 Image processing . . . 40
3.2.3 Software control . . . 42
3.3 Stabilization of the monolayer and sample quality . . . 43
3.3.1 Water supply control . . . 43
3.3.2 Particle density control . . . 47
3.3.3 Setup tilt control . . . 48
3.3.4 Flatness of the interface . . . 50
3.3.5 Equilibration of the monolayer . . . 53
3.3.6 Segregation in a gravitational eld . . . 54
3.3.7 Adsorption of beads at the edge of the cell . . . 56
3.4 Data acquisition . . . 57
3.4.1 'Multipleτ' time steps . . . 57
3.4.2 Drift compensation . . . 57
4 Glassy dynamics 59 4.1 Comparison with Mode Coupling Theory . . . 60
4.2 Dynamic correlation between particle species . . . 62
4.3 Dependence on relative concentration . . . 64
4.4 Relaxation behavior for short and long delay times . . . 69
5 Local structure 73 5.1 Partial clustering of the small particles . . . 73
5.1.1 Origin of partial clustering . . . 74
5.1.2 Comparison of experiment and simulation . . . 75
5.1.3 Prepeak in the partial structure factor of small particles . . . . 76
5.1.4 Morphological analysis . . . 78 5.1.5 Dependence on interaction strength and relative concentration . 81
5.2 Local crystalline order . . . 86
5.2.1 Formation of crystallites . . . 87
5.2.2 Dependence on relative concentration . . . 90
5.2.3 Dependence of long-range order on interaction strength . . . 92
5.2.4 Local fourfold order . . . 94
6 Local Dynamics 98 6.1 Dynamical Heterogeneity . . . 98
6.2 Interpretation of dynamical heterogeneities . . . 104
Conclusions 105 Zusammenfassung 108 Outlook 112 Publications 118 A PID feed back loop 119 B Data analysis 122 B.1 Pair correlation function . . . 122
B.2 Static structure factor . . . 123
B.3 Voronoi tesselation . . . 125
B.4 Bond order correlation function . . . 126
B.5 Minkowski functionals . . . 126
B.6 Mean square displacements . . . 129
B.7 Self-intermediate scattering function . . . 130
Bibliography 132
Danksagung 138
Glasses
1.1 Phenomenology
Glasses are used as versatile materials in numerous applications of daily life, and their manufacturing techniques reach back millennia [1]. However, the rst used glasses were not made by man but were found in nature. In meteoric impacts or volcanic eruptions the extreme heating and subsequent rapid cooling forms natural glasses from molten stone. In lava ows of volcanic regions Obsidian is found, a shiny black glassy stone, used for the rst sharp blades and arrowheads by chipping o pieces.
Even nowadays, broken glass belongs to the sharpest available materials.
The tailoring of desired material properties of manmade glasses has been highly developed, and nowadays the use of glasses ranges from simple mass products like bottles, light bulbs, or window glass to sophisticated 'high-tech' applications. A well known example for such a 'high-tech' glass with specially designed properties is CERAN°R glass1 used for hot plates: it practically does not expand upon heating and therefore does not break even when quenched with cold water from tempera- tures as high as 700◦C. At the same time, CERAN°R glass does not shield heat, a necessary property for a hot plate. Examples that exploit the adjustability of the optical properties of glasses are eyeglasses, laser optics, or imaging objectives. These completely dierent applications show how widely the glass properties can be adjusted by combining dierent substances and applying dierent manufacturing techniques.
Although the methods to control the material properties are highly developed, the basic mechanisms of glassy solidication are not suciently understood so far.
In fact, the terminus 'glass' describes a more general class of materials. The most general denition is as follows (from C. A. Angell [2]): 'A glass is a condensed state of matter which has become non-ergodic by virtue of the continuous slow-down of one or more of its degrees of freedom'. This denition includes spin glasses, orientational glasses, and vortex glasses, as well as the 'classical' glasses that are characterized by
1Schott AG, http://www.schott.com/german/ (December 9, 2008).
5
0 1
Figure 1.1: Left: Viscosity η of dierent glass formers as a function of the normalized inverse temperature. The glass temperature TG is dened as the temperature where the viscosity exceeds a value of η = 1013P oise. Dierent glass formers are classied as 'strong' or 'fragile' according to their approach of the arrest. Right: The temperature dependence of the volume v or the enthalpy h is sketched for decreasing temperatures.
TM is the melting temperature. Curve a) results from a slower cooling rate than curve b) leading to a lower glass transition temperature. The derivatives of the curves (the ther- mal expansion coecientαp = (∂lnv∂T )p and the isobaric heat capacitycp = (∂T∂h)p) change abruptly but continuously at the glass transition (graphs from P. Debenedetti [3]).
their static amorphous structure [2]. In this work the terminus 'glass' will be always used to describe the latter, the so-called structural glasses.
Such glasses can be based on silicates, polymers, metals, and also on colloidal particles. Although these systems are totally dierent on the scale of their con- stituents, they all have common features that characterize them as a glass: 'Glasses are disordered materials that lack the periodicity of crystals but behave mechanically like solids' (P. G. Debenedetti [3]). When lowering the temperature of a uid, glass formers dynamically arrest into a non-equilibrium state while the uid structure does not change signicantly. No long-range order can be established by crystallization.
This continuous phenomenon is called the glass transition and characteristically the viscosity of the system is increased by many orders of magnitude [2, 3, 4, 5]. For several glass formers this is demonstrated in Figure 1.1, where the viscosityη is shown versus the normalized inverse temperature TG/T (with glass transition temperature TG).
A drastic continuous increase of up to 15 orders of magnitude is found in these examples. Although the curves do not follow a universal behavior in this normalized graph, they can be compared and labeled by two groups of glass formers: fragile and
strong systems2. The fragile systems approach the dynamic arrest more abruptly with a steeper η-curve than the strong glass formers. Here, the organic molecules form fragile glass formers, whereas the inorganic systems as silicon dioxide (SiO2) form strong glasses. The origin of the dierence is suspected in the individual chemical near order, but a satisfying description of the curves over the whole temperature range is still missing.
Unlike a phase transition, the glass transition does not occur at a well dened temper- ature [3]. One denition of the glass transition temperature, as used in the examples of the left graph of Figure 1.1, is the temperature where the viscosityη exceeds a value of ηG = 1013P oise, and the system can be considered solid-like. However, this value is chosen arbitrarily. Another denition of the glass transition temperature TG is given by characteristic changes of the thermodynamic quantities volume and enthalpy as illustrated in the right graph of Figure 1.1. Dilatometric measurements show that the expansion coecient α is lowered at the glass transition upon cooling. Calorimetric measurements show a characteristic step in enthalpy because degrees of freedom contributing to the heat capacity are 'frozen'. This eect is stronger in fragile than in strong glass formers. The temperature TG, where both changes happen, is dependent on the rate of cooling. The lower the cooling rate, the lower is the measured glass transition temperature TG. This is important as it emphasizes the nature of the glass transition in comparison to a phase transition: the glass transition occurs, when the system is not able to follow into the equilibrium state on the timescale of the externally applied temperature change. The relaxation timescales of the system are too slow. Thus, the glass transition is not just a property of the system but is also related to the external timescale of the temperature treatment. Therefore, information of the thermal history may be 'memorized' in a glassy system.
This phenomenon of kinetic vitrication may take place even when a phase transition into a long-range ordered state is possible and eventually may occur under appropriate conditions3; examples are one-component hard spheres [6], semicrystalline stacks of lamellar crystals in polymers [7], or binary mixtures in metal alloys [8]. They all do crystallize if the system temperature is lowered suciently slowly.
It appears obvious that disordered structure and dynamics are related. One example for a close formal connection between structure and dynamics of a glass forming system is provided by Mode Coupling Theory (MCT), a theory describing the dy- namical glass transition. There, the only input into the MCT equations is the static structure factor [9, 10]. Nevertheless, the microscopic connection between structure and dynamics is still under strong debate [11, 12, 13, 14], especially the question
2Here, systems are chosen that seem to collapse on two distinct curves, two extreme cases for strong and fragile systems. However, there are many glass formers that form curves in between.
3In fact, the example of CERAN°R glass mentioned above, earns the 'zero-expansion' property from its partially crystalline structure, a combination of micro-crystallites embedded in an amorphous glassy matrix. The expansion coecient of the crystalline regions is negative, i.e. those parts shrink upon heating. However, the glassy regions have a positive expansion coecient and therefore conventionally expand upon heating. By adjusting the volume ratio of both parts an expansion coecient of α≈0 can be reached over a suciently large temperature range.
how crystallization is connected to vitrication and dynamical heterogeneity [15, 16].
Simulations suggest that crystallization plays a key role for the glass transition [16].
H. Tanaka et al. propose that 'liquids tend to order into the equilibrium crystal, but frustration eects of locally favored short-range ordering on long-range crystalline ordering prevent crystallization and help vitrication'.
1.2 Inuence of dimensionality
The macroscopic behavior of crystalline systems sensitively depends on the dimension- ality as demonstrated by a few examples:
In 2D an intermediate phase exists between uid and crystal, the hexatic phase:
the system has no translational order while the orientational correlation is still long-range [17, 18, 19]. Such a two step melting scenario is not known in 3D. The Ising model for ferromagnetics shows a phase transition for 2D and 3D but not for 1D [20].
Another example concerning ordered phases is the existence of long-range translational invariance: It exists in 3D but not in 1D and 2D for nite temperatures. The reason for this is the energy needed for a long wavelength deformation. This energy diverges in 3D for large volumes but not in 2D and 1D what enables thermal excitation to destroy translational symmetry by long wavelength uctuations [21, 22]. A dynamical depen- dency on dimensionalitydis that the velocity autocorrelation function is dependent on delay timeτ like v(τ)∝τ−d/2. As the diusion constant is dened via the Green-Kubo relation D = 1/dR0∞dτhv(τ)v(0)i, the diusion constant is nite in 3D but diverges in 2D [23].
Nevertheless, it was found in experiments [24], simulations [25], and theory [26] that glassy systems in 2D exhibit the full range of glass phenomenology known in 3D glass formers, both in dynamics and structure. The question how dimensionality aects the glass transition is addressed in [26], where Mode Coupling Theory has been adjusted for two dimensions.
An important dierence between 2D and 3D concerning the local density optimiza- tion has to be discussed: in 3D the local density optimized structure of four spheres is obviously a tetrahedron. However, it is not possible to completely cover space in 3D with tetrahedra, because the angle between two planes of a tetrahedron is not a submultiple of 360◦ [27]. The density optimized state with long-range order is realized by the hexagonal closed packed structure or other variants of the fcc stacking with packing fraction φ = π/√
18 ≈ 74%. The dynamical arrest in 3D is expected to be enhanced by this geometrical frustration, because the system has to rearrange its local density optimized structure to reach long-range order4. The local geometrical frus- tration scenario is dierent in 2D. There, the local density optimized structure and
4However, it is found in 3D hard sphere systems that this geometrical frustration alone is not sucient to reach a glassy state as it cannot suciently suppress crystallization [28, 29, 30], and additionally polydispersity is needed [26].
densest long-range ordered structure are identical, namely hexagonal. For the glass transition in 2D it is therefore expected that an increase of complexity is necessary to reach dynamical arrest: in simulations an isotropic one-component 2D system has been observed undergoing dynamical arrest for an inter-particle potential that exhibits two length scales, a Lennard-Jones-Gauss potential with two minima [31]. Other simula- tions showed that systems of identical particles in 2D can vitrify [16] if the mentioned local geometric frustration is created articially via an anisotropic vefold interaction potential. Alternatively, the necessary complexity can be created by polydispersity as shown in simulations [32]. The simplest form of polydispersity is bidispersity, which is realized experimentally in the system at hand [24].
1.3 Colloidal model glass formers
Colloids are particles or droplets dispersed in a continuous medium, e.g. blood, smoke, milk, or ink. Typical sizes range from100nm−10µm. If dispersed in a uid medium, the particles are subjected to Brownian motion. Colloidal suspensions belong to a class of materials often referred as Soft Matter. Further well known representatives are polymers, micro-emulsions, liquid crystals, or complex bio-materials [33]. The softness of these materials, as emerging from their small shear modulus, originates from their inherent mesoscopic length scales. The shear modulus (unit: energy/length3) of a colloidal crystal in comparison to an atomic crystal can be estimated by using the typical binding energies and the underlying length scales of the constituents [33]:
the binding energies do not dier signicantly (typically 1eV in atomic crystals, kBT ≈ 1/40eV for colloidal particles). However, the lattice constant of a colloidal crystal can be larger by a factor of10000. This results in a shear modulus of a colloidal system being smaller by up to twelve orders of magnitude corresponding quite well to experimentally observed values. Therefore, colloidal systems are extremely soft, and mechanical forces distort them easily.
Colloidal glass formers can be used as model systems to study the local phenomena of the glass formation in 2D and 3D. In experiments, colloidal glasses have the advantage over atomic systems that, besides statistical averaged information about the local structure, single particle resolution is provided by microscopy [11, 24, 34, 35].
A model system of a glass former in 3D is a suspension of colloidal hard spheres [6]:
with increasing packing fraction φ of the spheres, the viscosity is increased until the system becomes dynamically arrested at φ ≈ 58% without crystallization. The glassy dynamics has been widely studied using dynamic light scattering [6, 36, 37].
Additionally, with this model system it was possible to study many local phenomena by confocal microscopy like local structure and local dynamics [11, 35, 38].
In 2D, the binary mixture of colloidal dipoles investigated in this work is a good model system to study the glass transition [24, 26, 40, 39, 41, 42, 43]. It consists of two types of particles with dierent magnetic moments that are conned at a water-air interface by gravity. There, they are subjected to Brownian motion
in two dimensions. Trajectories of particles are obtained by video microscopy on the time and length scales relevant for the glass transition. The system exhibits all typical phenomenological features of a glass former, e.g. drastic increase of relaxation times for increasing interaction strength, no long-range order, and dynamic heterogeneities [24]. The dynamics of the system was compared with MCT and good agreement was found [26]. Partial clustering of small particles was observed [42, 44]
due to the negative nonadditivity of the dipolar binary potential. This leads to a heterogeneous distribution of particle composition which results locally in a coexisting variety of small areas with dierent underlying crystal structures. Thus, heterogenous distribution of small particles suppresses long-range order. Extended stable crystal structures for 2D binary dipoles were found in T = 0 lattice sum calculations [45, 46]
as well as by genetic algorithms [47]. Especially all locally ordered structures discussed in this work were predicted to be stable.
The idea that the disordered structure of this system is made up from particular substructures (triangular structures) was originally discussed in [39, 40]. However, in this work it is proposed that the local order originates from the tendency of the binary mixture towards crystallization.
Crystallization may be geometrically possible provided the relative concentration matches a certain crystal structure, and crystallization is not at odds with a glass transition in a binary system. A decrease in temperature can force a system into a dynamically arrested state due to the strong increase of viscosity before crystallization can establish long-range order. Since the small particles cannot reorganize fast enough, much disorder is 'frozen' in. As a consequence dierent competing crystalline structures appear, while the global structure remains amorphous.
Background
In the section 2.1 of this chapter, the basic experimental system of the 2D colloidal glass former is introduced. The concept of the colloidal monolayer at a water-air interface and the dipolar pair interaction between the colloidal particles are dened, based on the concept rst developed by K. Zahn et al. [48]. This introduction gives sucient background to understand the results of this work. The technical details concerning the experimental setup, sample preparation, and sample quality are explained separately in chapter 3.
A brief description of Mode-Coupling Theory (MCT) is given in section 2.2, one of the leading theoretical concepts to describe the dynamics of the glass transition. It is introduced here, because the dynamics of the system at hand is compared in chapter 4 with a simple glass forming model system of hard discs that was calculated using MCT.
Furthermore, this introduction is used to describe some of the general features of glass formers.
Section 2.3 describes a particularity of the interaction potential in the experimental system: The negative nonadditivity of the binary dipolar pair interaction. It becomes relevant in chapter 5 to explain the observed partial clustering of small particles. This phenomenon is responsible for the heterogeneous local concentration of small particles and therefore for the occurrence of dierent competing crystal structures.
Although the experimental system of binary 2D dipoles is never found in an equilib- rium crystalline state at low system temperatures, stable congurations at T = 0 are theoretically predicted. In fact, there are numerous of them, and they are introduced in section 2.4 to be compared with the local crystalline structures of the experimental system in section 5.2.
11
2.1 2D system of binary colloidal dipoles
2.1.1 Pending water drop geometry
The system consists of a suspension of two kinds of spherical and super-paramagnetic colloidal particles A and B with dierent diameters (dA = 4.5µm, dB = 2.8µm) and magnetic susceptibilities (χA ≈ 10 · χB). Due to their high mass density of ρ ≈ 1.5g/cm3, they are conned by gravity to a water-air interface formed by a pending water drop suspended by surface tension in a top sealed cylindrical hole (6mm diameter, 1mm depth) in a glass plate. This basic setup is sketched in Figure 2.1.
A magnetic eld H is applied perpendicularly to the water-air interface inducing a magnetic moment M = χH in each particle leading to a repulsive dipole-dipole pair interaction.
The set of particles is visualized by video microscopy from below the sample and is recorded by an 8-bit CCD camera. The gray scale image of the particles is then analyzed in situ with a computer. The eld of view has a size of ≈1mm2 containing typically 3 × 103 particles, whereas the whole sample contains about up to 105 particles1. Standard image processing is performed to get size, number, and positions of the colloids. A computer controlled syringe, driven by a micro stage, controls the volume of the droplet to reach a completely at surface. To achieve a horizontal interface, the inclination of the whole experimental setup has to be aligned. This inclination is controlled actively by micro-stages with a resolution of ∆α ≈ 1µrad.
After typically several weeks of adjustment and equilibration this provides best equilibrium conditions for long-time stability. During data acquisition the images are analyzed with a frame rate down to 10Hz. Trajectories of all particles in the eld of view can be recorded over several days providing the whole phase space information.
The thermal activated 'out of plane' motion of the particles is expected to be in the range of a few tens of nanometer. Thus, the ensemble is considered as ideally two dimensional.
Information on all relevant time and length scales is available, an advantage com- pared to many other experimental systems. Furthermore, the pair interaction is not only known, but can also be directly controlled over a wide range. For all typical experimental particle distances the dipolar interaction is dominant compared to other interactions between particles like van der Waals forces or surface charges [49].
1These numbers are valid for the binary case. In a one-component sample much higher densities can be reached of up to 10000 particle in the eld of view.
H
air water glass cell in side view
T
x,y
Figure 2.1: Left: Super-paramagnetic colloidal particles conned at a water-air interface due to gravity. The curvature of the interface is actively controlled to be completely at, and the system is considered to be ideally two dimensional. A magnetic eld H perpendicular to the interface induces a magnetic moment Mi in each bead leading to a repulsive dipolar pair interaction. Right: Induced dipole moments Mi tilted with an angle θ with respect to the magnetic eld H. The potential is repulsive for θ > 54.7◦. For lower values of θ it is attractive.
2.1.2 Dipolar pair potential and interaction parameter Γ
The external magnetic eld H induces a dipole moment Mi = χiH in each particle indexed with i where χi is the individual susceptibility of a particle.
The dipolar interaction between two spheres is now regarded as depicted in the right sketch of Figure 2.1. The magnetic spheres are homogeneous, and therefore their magnetic eld induction B1(x) at distance x is that of a point dipole which is
B1(x) = µ0
4π
3(M1·x)x−(x·x)M1
x5 (2.1)
with the permeability of vacuum µ0. This is not only true asymptotically for large x but also close to the sphere as the geometry of a homogeneous magnetized sphere has no higher multipoles [50]. The interaction energyEmagn of two arbitrary oriented point dipoles M1 and M2 with connection vector r is given by
Emagn = −1
2B1·M2 (2.2)
= −µ0 8π
3(r·M1)(r·M2)−r2(M1·M2)
r5 . (2.3)
The factor 12 results from the fact that induced dipole moments are regarded [50].
Both induced dipole moments are parallel to the external eld, which simplies the expression to
Emagn = µ0M1M2 8π
(1−3 cos2θ)
r3 (2.4)
where θ is the angle between magnetic eld and connection vector r as indicated in Figure 2.1. The angular dependence of Emagn shows that the interaction potential is not always repulsive. It vanishes for θ ≈ 54.7◦, and for smaller angles the potential becomes attractive.
Assuming that the magnetic eld is always perpendicular to the connection vector r (particles are conned to a plane perpendicular to the eld), the expression further reduces to
Emagn = µ0
8π · M1M2
r3 . (2.5)
Counterpart of the potential energy Emagn is thermal energy kBT which generates Brownian motion. Thus, a dimensionless interaction parameter Γis introduced by the ratio of potential versus thermal energy:
Γ = γ· Emagn kBT = µ0
4π · H2 ·(πn)3/2
kBT (ξ·χB+ (1−ξ)χA)2 (2.6)
∝ 1
Tsys. (2.7)
Here, ξ = NB/(NA + NB) is the relative concentration of the small species with NA big and NB small particles and n is the area density of all particles. In this expression for Γ, the two dierent susceptibilities χA and χB are averaged to an eective susceptibility weighted with the relative concentration ξ. The magnetic energy is thus calculated as if the two particle species were identical with this average susceptibility2.
The denition of Γ is adjusted for reasons of tradition by several factors that are accumulated in the constant γ in equation 2.6: The factor of 12 was omitted andπ3/2 is added. Setting r =n−1/2 implies a square arrangement of dipoles in the plane. Other
2Another denition Γ?= n3/2k(χAH)2
BT for the interaction strength is often used where only the big particles are considered and the contribution of the small particles is neglected [42, 44, 51] (further- more, in this denition the factor of µ4π0 ·π2/3 is not included). For the used experimental parameters of χA/χB ≈0.1 and relative concentrations between 0 < ξ <0.5, the values of the denition used in this work only deviate 0−16% in comparison to the denition Γ? where the small particles are neglected.
crystalline patterns or an amorphous arrangement would lead to another prefactor3. Only the interaction between two neighboring particles is taken into account although the dipolar potential is long-range. Consideration of the interaction of all particles leads to a Madelung constant for a crystalline pattern and a corresponding factor for an amorphous arrangement.
Γ can be interpreted as an inverse temperature Tsys and is the thermodynamic control parameter of the system. The interaction strength is externally controlled only by means of the magnetic eld H. All other parameters in equation 2.6, as the thermal sample temperature T, particle density n, and magnetic susceptibilities χi, are kept constant during the experiment. The system temperature can be changed quasi instantaneously and homogenously over the whole sample due to the direct control via the external magnetic eld.
2.2 Mode Coupling Theory (MCT)
Mode Coupling Theory (MCT) is a microscopic theory to describe the glass transi- tion and was rst introduced by Bengtzelius, Götze, and Sjölander in 1984 [9]. It makes numerous predictions that were largely successfully checked by experiments and simulations [52, 53]. The main success is that it predicts a dynamical bifurcation as a function of the thermodynamic control parameter: the long-time dynamics changes from ergodic to non-ergodic, while thermodynamic or structural quantities do not be- come singular. This describes the characteristic behavior of experimental glass formers:
The viscosity shows a drastic increase while structural or thermodynamic quantities are almost not aected (e.g static structure factors or isothermal compressibility). MCT therefore states that the glass transition is purely dynamical in nature, an important point where it diers from other interpretations of the glass transition, e.g. ideas where crystallization plays a key role for the dynamic arrest [16]. Although in MCT struc- tural quantities do not become singular at the glass transition, dynamics and structure are formally closely related. In fact, the static structure factor is the only quantity that enters the MCT equations that generate the dynamical density correlation func- tions. Thus, the information if a system is ergodic or non-ergodic is already uniquely contained in the static structure factor.
MCT has indeed some signicant predictive power for the physics of the glass transi- tion as discussed in [10] and [54]. Some of the successes are: MCT predicts distinct values like packing fractions, where dynamical arrest occurs, power law exponents of the α- andβ-decays and their direct relation, the divergence of the α-relaxation time, and non-ergodicity parameters fq. It also predicts the time-temperature superposition of the α-process: the correlators collapse onto a non exponential master function.
3In this denition ofΓit is not considered that the underlying structure might change when altering the magnetic eld, e.g. when the system is undergoing a phase transition.
A well known failure of MCT is the predicted value of the critical packing fractionφd=3c in a 3D hard sphere system: the experimental value is determined as φd=3c ≈ 0.58 [6]
whereas MCT predicts a value of φM CT ≈ 0.51 [55]. However, the predicted critical packing fraction sensitively depends on certain features of the structure factors used as the input for the MCT equations, especially on the height of the main peak. Dierent methods can be used to determine the input structure factors like simulation, Liquid integral equation theory, or even experiments. However, dierent approximate methods lead to dierent results of the critical packing fraction. Thus, the failure to predict the exact value of φd=3c may partially lie in the approximative methods used to determine the input structure factors.
2.2.1 Basic formalism
As the mathematical description of MCT is very extended and complex, it is not intended to give a sucient introduction here. For a deeper understanding see review articles and introductions [10, 26, 54, 56, 57] and the references therein. Here, only the basic equations, ideas and consequences are presented. Emphasis is put on the parts that concern the comparison with the experimental system at hand.
In the following the MCT equations for a one-component liquid are shown dening the collective particle correlator Φq(τ)of density uctuations with wave vector q. The elaborate description of this is given in [26, 58].
MCT provides equations of motion for the collective and the tagged particle correlators of density uctuations. These quantities are analogous to the intermediate scattering function or its self part respectively (see appendix B.7). For a system with Brownian dynamics the equation of motion is
γq˙Φq(τ) + Φq(τ) +
Z ∞
0 dτ0mq(τ −τ0) ˙Φq(τ0) = 0 (2.8) with the memory kernel mq(τ) and a characteristic microscopic timescale γq. These equations of motion are derived from the Langevin- or Smoluchowski-equation respec- tively, by projection onto the slowly varying variables of the system, i.e. the density uctuations [10, 54]. The memory kernel can be interpreted as a generalized friction coecient that describes the uctuating stresses of the system. If the memory kernel is set to zero, a simple dierential equation is left that has an single exponential de- cay as solution for Φq(τ), the solution of free diusion. Thus, the information about interaction of particles and retardation of the dynamics is contained in this integral.
To solve equation 2.8, in MCT it is assumed that the main contribution at long times are given by density pair uctuations. With this approximation the memory kernel is
then written as
mq(τ) =
Z ddk
(2π)dV(q,k,p)Φk(τ)Φp(τ) (2.9) where the vertices V(q,k,p) express the overlap of uctuating stresses with the pair density modes. They are determined by the equilibrium structure of the system, by
V(q,k,p) = n 2
SqSkSp
q4 [q·kck+q·kcp]2δ(q−k−p) (2.10) where Sq is the static structure factor and n the number of particles per unit d- dimensional volume. The direct correlation function cq is uniquely related to the pair correlation function via the Ornstein-Zernike equation
h(r) = c(r) +n
Z
ddr0c(r0)h(r−r0) (2.11)
where h(r) ≡ g(r) −1 [23]. As the pair correlation function g(r) is connected via Fourier transform to the static structure factor Sq, equation 2.10 is only dependent on the latter. Thus, the solution of Φq(τ) in equation 2.8 only depends on the static structure factor.
The vertices V(q,k,p) are nonzero only for values that fulll the relation q = k+p which expresses the assumption that only density pair uctuations contribute that are coupled like this. The name 'Mode Coupling Theory' originates from this approximation. Static three-point correlations do not explicitly enter the memory kernel of MCT.
Here, only the equation of motion for the collective density correlatorΦq(τ)was shown.
The equations of the self part Φ(s)q (τ) are almost analogous. There, the integral over the memory function is not only a functional of the self part Φ(s)q (τ) but additionally depends on the collective part Φq(τ). Furthermore, the mean square displacement h∆r(τ)2ican be calculated directly from the MCT equations in the small wave vector limit q →0.
Note, that the interaction potential does not explicitly enter the MCT equations, only indirectly via the static structure factor.
The only direct dependence of MCT on the dimensionality is found in the integration element of the memory kernel in equation (2.9) which is (2π)−dddk ∝ (2π)−dkd−1dk. Further dependence on dimensionality only enters indirectly via the input static structure factors.
2.2.2 MCT in 2D for hard discs
The system used by Bayer et al. in [26] consists of two-dimensional hard discs with packing fraction φ and will be used for comparison with the experimental system at hand in chapter 4.
The structure factors are calculated using the Ornstein-Zernike equation with a modied hypernetted chain approximation as closure relation. Unlike in real 2D monodiperse systems, crystallization is prohibited by the method itself. It is assumed that this is a reasonable model for polydisperse hard discs, at least on a qualitative level.
The MCT equations reveal a bifurcation in dynamics where the system has an ergodic- nonergidic transition at a critical packing fractionφd=2c ≈0.697. The density correlators do not decay to zero for all times analogous to a hard sphere system in 3D, where MCT predicts an arrest at φd=3c ≈0.51 [55]. The predicted packing fraction in 2D is higher but they only dier by 5% if both critical packing fractions are scaled with the corre- sponding fraction of random close packing (φd=2rcp ≈0.84, φd=3rcp ≈0.64) [26, 59, 60].
2.2.3 Von Schweidler law
As mentioned above, MCT predicts power law behavior for the long-timeα-relaxation before the system nally relaxes via a Kohlrausch decay Φq(τ) ∝ exp(−τ /τ0)β. The scale free decay of the correlation function
Φq(τ) =fq−aτb (2.12)
is known as the von Schweidler law, and the exponentbis the von Schweidler exponent being independent of temperature or packing fraction respectively (constant a > 0, non-ergodicity parameter fq). This decay behavior is a general observation in glass formers [4].
Figure 2.2 shows MCT data of the hard disc system introduced above [26]. The data is close to, but below the glass transition with ²= φcφ−φc =−10−2. To show the powerlaw decay, the data is displayed in the form of the normalized self-intermediate scattering function P(q, τ) in Gaussian approximation for q=π. This function is dened as
P(q, τ) = log10
¯¯
¯¯
¯
FS(q, τ)−fq 1−fq
¯¯
¯¯
¯ (2.13)
where FS(q, τ) is the conventional self-intermediate scattering function, and fq = FS(q, τmax) is the plateau height with τmax being the time in the point of inection of the mean square displacement (for details see appendix B.7). The second part clearly follows a straight line revealing the von Schweidler-law of equation 2.12 with an exponent of b= 0.73.
Figure 2.2: Normalized intermediate scattering functions of a hard disc system cal- culated from MCT in 'Gaussian approximation' (see appendix B.7, raw data ob- tained from [58]). The packing fraction is close to, but below the glass transition (² = φcφ−φ
c = −10−2). The curve is divided in a fast and slow part at the time of maximum stretching marked by the discontinuity (dened by the time of inection in the mean square displacement). The fast part follows a stretched decay, whereas the slow part shows power law behavior before converging into an stretched exponential de- cay. A power law with exponent 0.73 is plotted as the dashed line in the slow part. This scale free relaxation behavior is known as the 'von Schweidler' law.
2.3 Nonadditivity of a binary mixture
The eective hard core diameters of particles in a binary mixture can be dened with respect to the pair-interaction with another particle. Unlike for real hard spheres, this eective diameter can be dependent on the vicinity of the particle, i.e. if the particle is next to a particle of same type or dierent type. To quantify this behavior, the nonadditivity parameter ∆for a binary suspension4 is dened as
∆ = 2σ12
(σ11+σ22)−1 (2.14)
4Originally, the nonadditivity parameter is dened as ∆ = 2σ˜ 12−(σ11+σ22). Here, we use the normalized version to avoid real dimensions and a dependency on the external eld. The interpretation of this quantity is not aected by this.
with the Barker-Henderson eective hard core diameters σij =
Z ∞
0 dr{1−exp[−vij(r)/kBT]} (2.15)
where vij is the interaction potential between two particles of species i and j, Boltz- mann's constant kB, and absolute temperature T [61].
In the case of the binary mixture of 2D dipoles this quantity ∆ can be calculated an- alytically for a given ratio of susceptibilities. The interaction potential between two beads with susceptibilities χi and χj (see section 2.1.2) is given by
vij = χiχjH2
r3 (2.16)
whereHis the strength of the magnetic eld. Inserting this potential into equation 2.15 and using the substitution u(r) = r·(kBT /χiχjH2)1/3 for the integral leads to
σij =a·b·(χiχj)1/3 (2.17)
where a:= (H2/kBT)1/3 and b :=R0∞du{1−exp[−1/u3]}=const. Equation 2.14 for this system is then expressed by
∆(m) = 2m1/3
1 +m2/3 −1 (2.18)
withm=χ2/χ1. The nonadditivity parameter∆(m)is negative for all values ofm 6= 1 as shown in Figure 2.3 (blue curve). The graph is reciprocally symmetric around 1, i.e. ∆(m) = ∆(1/m). A comparison with a repulsive coulomb potential (vij ∝1/r) is shown (yellow curve). There, the negative nonadditivity is even stronger [62].
The negativity of the nonadditivity parameter means that the eective particle size is dependent on the surrounding of the particle: a small particle is eectively smaller in the vicinity of a big particle than in the vicinity of another small particle. This has consequences on the equilibrium structure of the mixture which will be investigated in the system at hand in section 5.1. There, no phase separation is observed, but a partial clustering of the smaller species while the big particles are homogenously distributed.
Figure 2.3: Normalized nonadditivity parameter ∆for a binary mixture of dipoles (blue curve). The dashed lines indicate the value of ∆for the spheres used in the experiment (χA = 6.22·10−11Am2/T and χB = 6.6·10−12Am2/T). ∆ is plotted also for a binary mixture with repulsive coulomb potential for comparison (yellow curve). There, m is the ratio of the corresponding charges.
2.4 Predicted crystal structures at T = 0
The stable ground state atT = 0 of a one-component dipolar system in 2D is a hexag- onal lattice. The question arises how this scenario changes when a second species is added with a dierent magnetic moment. Indeed, a huge variety of binary crystal struc- tures is predicted to be stable as described in the following. Two dierent approaches were used: i) selection of promising candidate structures and minimization of lattice sums with respect to the Gibbs free energy [45] and ii) genetic algorithms searching for candidate structures with subsequent minimization of the lattice sums free energy [47].
2.4.1 Lattice sum minimization
The system of L. Assoud and coworkers [45] is comparable to the experimental system in this work: it consists of two dierent species of particles in 2D with dipolar pair interaction. However, the calculations were made for temperature T = 0 for many dierent susceptibility ratios. To compare the stability of dierent binary crystal structures they have to be dened by a set of parameters. The primitive cell of a crystal is uniquely dened by two lattice vectors a and b = aγ(cosθ,sinθ),
with θ being the angle between vectors a and b and γ is the aspect ratio of their lengths. Furthermore, the position vectors of the particles inside the above described parallelogram have to be dened. With these denitions the Gibbs free energy of an extended lattice is calculated for a xed relative concentration at temperature T = 0 and a xed dipole ratio m = χB/χA. A common tangent construction is then used to optimize the parameters a, γ, θ, and the positions of the given particles inside the parallelogram. Note, for this method an initial 'smart guess' of the crystal structure is necessary, and therefore an unbiased search in parameter space is dicult.
The results are shown in Figure 2.4 where the graph shows the optimized lattice structures for a given relative concentration of small particles X = NNB
A+NB (with NA big and NB small particles) and the dipole ratio m. The corresponding structures are depicted above. A large number of dierent crystal structures are found to be stable.
With increasing asymmetry in dipole ratio m and increasing relative concentration X the diagram becomes more complex. For a low asymmetry the system develops towards the one-component case with the stable triangular conguration, in particular the structures T(AB2), T(A), T(B), and T(A2B). The experimentally accessible region in this work is a line in the graph at constant susceptibility ratio m ≈ 0.1 (dashed yellow line). On this line a large number of dierent crystal structures are predicted to be stable (highlighted by yellow boxes in the pictographs), in particular the square lattice S(AB) at X = 0.5. All the structures lying below S(AB) are combinations of this square lattice with triangular patterns of species A. The structures above S(AB) become increasingly complex.
1
2
3
4
T(A)B6 Re(A)B4
P(A)AB4
T(B) S(A)B4
T(A)B4
Rh(A)B2 Rh(A)AB4 T(A)
Rh(A)AB3 Rh(A)AB2 S(AB) Rh(A)A2B2
Rh(A)AB T(A2B) Re(A)A2B Rh(A)A3B T(AB2)
experimental parameter
Figure 2.4: Lattice sum calculations at T = 0 predict stable binary crystalline struc- tures. Top: The pictographs show crystalline structures with their primitive cells. They correspond to the parameters in the graph (see notations). Bottom: Stable crystalline structures for dierent susceptibility ratios m= χB/χA and relative concentrations X are indicated by lines. The gray box covers an unknown region. The symbol # (∗) denotes continuous (discontinuous) transitions.
The experimentally accessible region is a line with constant m ≈ 0.1 (yellow dashed line) as only two types of colloidal particles were available. The structures on the dashed line are highlighted with yellow boxes in the pictographs. In this region lies the square lattice structure (C3) and the triangular lattice (E2). Furthermore, there are several combinations of both, namely (D3) (D4) and (C4). The graph and pictures were taken from L. Assoud [45] and were rearranged with author's permission.
A
m=0.003 m=0.018
m=0.06 m=0.18
B C D
2
3
4
5
6
7
8
9 1
[=1/3 [=1/7
[=1/5
[=1/2
[=2/3
[=5/7
[=7/9
[=4/5
[=6/7
Figure 2.5: Genetic algorithms predict stable crystalline structures for a binary dipolar pair potential. The relative concentration ξ of small particles is increased from top to bottom (indices 1-9), and the susceptibility ratio m is decreased from left to right (indices A-D). The susceptibility ratios of these structures are lower than those of the lattice sum minimizations shown in Figure 2.4 but have a certain overlap. The experimental value m=χB/χA≈0.1 is within this overlap and corresponds best to the structures of the rst column (m = 0.18, highlighted with yellow boxes). Pictographs are taken from J. Fornleitner [47] and were rearranged with author's permission.
2.4.2 Genetic algorithms
An alternative approach using Genetic Algorithms (GA) to nd equilibrium crystal structures was used by J. Fornleitner and coworkers [47]. There, the task of making 'smart guesses' on possible crystal structures is performed by the algorithm itself.
No preselected candidate structures have to be used which allows for an unbiased search. The optimization techniques are modeled after the natural evolution process.
A random 'population' of crystal structures is generated by the algorithm where every individual crystal structure is dened by random parameters of the primitive cell (analogous to parameters in subsection 2.4.1). These parameters are binary coded and are considered as the 'genes' that have to be optimized with respect to free energy. Certain biological processes in evolution are mimicked like mating, mutation and 'survival of the ttest' to nd the optimum structure. 'Fitness' in this context of crystal structures is quantied by the free energy of an individual structure and is the measure how probable the survival of that individual and its genes is. As mating and mutation is performed by crossing and altering the bits of the parental 'genes', changes are made in dierent orders of magnitude. This ensures an ecient exploration of parameter space. However, this also means that the energetic minimum is not generally reached exactly due to the nite length of the binary gene coding.
After convergence a 'steepest descent method' in energy landscape as in section 2.4.1 has to be applied to relax the structures to the exact minimum.
The dipole ratios m used in [47] are lower (0.003 > m > 0.18) than in [45]
(0.08 > m > 1) but have a certain overlap for comparison. Fortunately, the experi- mental ratio of m ≈ 0.1 lies in this overlap region. All crystal structures predicted in [45] for this region were also found by GA.
The used pair potential for GA calculations is approximately dipolar. Small deviations from 1/r3-behavior are only relevant for very high packing fractions as shown in [47].
The results are shown in the pictographs of Figure 2.5. Again, a huge variety of structures is found with increasing complexity for higher relative concentrations of small particles. Very exotic structures like the ones in line D evolve: structures where dimers of small particles sit next to a trimer and a monomer (D7) or complex star-congurations (D9). These structures are hard to predict by 'smart guesses' because their unit cells are very complex.
The parameters closest to the experimental ones are those of column A. Again, the square lattice is found to be stable (A4) and all combinations of that with triangular lattices of type A particles (A1-A3).
To understand the nature of our 2D experimental system at dipole ratio m ≈ 0.1, a result from both calculation methods has to be discussed: for low values of m and for values of m close to 1 the sub-lattice of A-particles is hexagonal. Only for a certain intermediate region the small particles are able to inuence the hexagonal structure of type A. If m is close to 1, particle species A and B are almost identical, and a hexagonal structure is likely (see Figure 2.4). If m is very small, species B
particles do not signicantly deform the hexagonal order of species A (lines C and D in Figure 2.5). The small particles have to arrange in the potential dips that are made up by the hexagonal order of the big particles. This result is important as the dynamics of a hexagonal one-component dipolar crystal is trivially not glassy. From these predictions the complex correlated glassy structure and dynamics of both species is only expected for an intermediate dipole ratio m, i.e. that region where the used experimental value m ≈0.1 is located.
Furthermore, in this intermediate region there is competition of many dierent stable crystal structures, and therefore it is expected that reaching equilibrium in experiment for high interaction strengths is dicult. Although there is no quantitative comparison of the crystal free energies, the energetic driving force towards the globally equilibrated state is low when areas are locally in dierent stable crystalline congurations. The prediction of several stable crystalline congurations is important to understand the interplay of crystallinity and glass transition in the system at hand.
How these predictions compare with the experimentally found structures is discussed in section 5.2. There, good agreement with the predictions is found for local crystalline arrangements.
Experiment and sample preparation
This chapter provides detailed insight into the experimental techniques, sample prepa- ration, sample properties, and the specics of the new constructed setup. The necessary requirements for sucient data acquisition is explained, and the quality of the sample is discussed.
The details of this chapter are not necessary to understand the results of this work in the subsequent chapters as the principle experimental setup was already briey introduced in section 2.1.
3.1 Colloidal suspension of super-paramagnetic spheres
3.1.1 Super-paramagnetic particles
The colloidal particles used in this work are commercially available1. They are porous polystyrene spheres doped with domains of magnetite (F e2O3) [63]. The surface of the beads is sealed with a thin layer of epoxy. These magnetic domains have a size of typically ten nanometers, small enough for thermal energy to overcome the magnetic coupling forces. Thus, magnetic moments are distributed randomly. If no external magnetic eld is applied, the total magnetization is zero. Thus, the material exhibits no remanence, a characteristic property of paramagnetic materials. The prex 'super' originates from the very strong magnetization when a magnetic eld is applied. The magnitude of the susceptibility is comparable to that of ferromagnetic materials. Two types of particles were used2:
1Dynabeads, Dynal Biotech GmbH, M-450 EPOXY, http://www.invitrogen.com (December 9, 2008).
2Information on both, mass densities and diameter, of the small particles were taken from the manufacturer. Magnetic susceptibilities were obtained by SQUID measurements (Group Prof. Schatz,
27
species A (big) B (small)
diameter 4.5±0.05µm 2.8µm
mass density 1.5g/cm3 1.3g/cm3 susceptibility 6.22·10−11Am2/T 6.6·10−12Am2/T
In Figure 3.1 slices of both types of beads are shown visualized with transmission elec- tron microscopy (TEM)3. Particles were embedded in an epoxy matrix to cut slices of 80−100nmthickness with a microtome diamond blade4. No contrast agent was used, as the contrast between polystyrene, epoxy and magnetite is sucient. All pictures show the magnetic content as black akes with typical cross sections of5−20nm. The upper pictures show representative examples of slices from big particles. The magnetic content is distributed homogenously and the shape of the big spheres is spherical. The small particles can have stronger deviations in shape and magnetic doping as seen in the bottom pictures of Figure 3.1. The left picture of a small bead shows strong irreg- ularities in magnetic content and shape. The right picture reveals a slight gradient in doping with a higher concentration at the edge.
The big particles are therefore expected to be monodisperse in size and magnetic mo- ment whereas the small particles might have higher polydispersity5.
3.1.2 Preparation of the colloidal suspension
The big particles are supplied by the manufacturer in pure water solution, while the small particles are provided as powder. To obtain a binary mixture with the desired relative concentration ξ of small particles and also the right absolute concentration of particles, both suspensions have to be prepared and characterized separately:
Suspension of big particles: The provided solution of big particles6 is diluted with desalted water. The parameters for dilution were estimated best by the color of the suspension: if the absolute number of big particles in the eld of view in experiment is ≈ 2000, the suspension has a very transparent light brown color. To prevent aggregation of the spheres, sodium dodecyle sulfate (SDS) is added until a concentration of c ≈ 0.9×cCM C is reached where cCM C is the critical micelle concentration (CMC). SDS is an anionic surfactant that covers
University of Konstanz [40, 41]) and they may vary between batches. It was found that the result of the SQUID measurements are sensitive to the particle density. The diameter of the big spheres was determined microscopically by measuring the length of several hundred particle chains in an in-plane magnetic eld.
3A transmission electron microscope Libra 120 (Zeiss, Oberkochen) was used with acceleration voltage120kV (Nanolabor, University of Konstanz).
4Leica EMUC6.
5Big particles have3%polydispersity in size (manufacturer information). No information is pro- vided for the small particles.
6The susceptibility of dierent supplied solutions may vary. The solution used in this work was always the same, labeled as 'Batch 8'.
1Pm 200 nm
1Pm 500 nm
Figure 3.1: Transmission electron microscopy (TEM) of sliced big (top) and small (bottom) super-paramagnetic spheres. TheF e2O3 clusters can be seen as dark akes in both types of particles. The diameters do not correspond to the real particle diameters, because spheres were not necessarily cut at the equator. The large stripes are due to heating from the electron beam.
the bead surface, with its polar end directing towards the solvent away from the sphere. This sterically stabilizes the colloidal suspension. Without SDS lots of particles form aggregates due to van der Waals attraction. SDS must be not older than one year, because decomposition strongly reduces the stabilizing properties. To avoid the growth of bacteria, the poisson Thimerosal (1µl/ml of a solution with 2% content) is added. Sedimentation and aggregation of the beads is avoided by storing the prepared suspension under permanent rotation and weak ultrasonic treatment. It takes approximately two days before the SDS has suciently stabilized the colloidal particles and almost no aggregates are found in the sample.
