Experimental Confirmation of a Kibble- Zurek Scaling Law in a Nematic Liquid Crystal

Experimental confirmation of a Kibble- Zurek scaling law in a nematic liquid crystal

A thesis submitted to The University of Manchester for the degree of Master of Science (by Research) in the Faculty of Engineering and Physical Sciences

2014

Nicholas Fowler

School of Physics and Astronomy

Contents

Chapter 1: Introduction ...... 11 References ...... 13 Chapter 2: Defect formation in the early universe and condensed matter systems ...... 14 2.1 The Kibble mechanism ...... 14 2.1.1 Symmetry breaking in the early universe ...... 15 2.1.2 The Higgs field and defect formation in the early universe ...... 15 2.2 The Kibble-Zurek Mechanism ...... 16 2.2.1 Defect formation in condensed matter systems ...... 16 2.2.2 Defect annihilation in condensed matter systems ...... 18 2.3 Previous experimental work ...... 18 2.3.1 Scaling laws in liquid crystals ...... 19 2.3.1.1 Defect annihilation in liquid crystals ...... 19 2.3.1.2 Defect formation in liquid crystals ...... 20 2.3.2 Scaling laws in other condensed matter systems ...... 22 2.3.3 Scaling laws in other experimental systems ...... 24 2.4 Using a liquid crystal system to test the Kibble-Zurek mechanism ...... 24 References ...... 26 Chapter 3: Liquid crystals ...... 29 3.1 Definition of a liquid crystal ...... 29 3.2 Types of liquid crystals ...... 29 3.2.1 Thermotropic liquid crystals ...... 30 3.2.2 Lyotropic liquid crystals ...... 31 3.2.3 Liquid crystal polymers ...... 31 3.3 Liquid crystal phases ...... 32 3.4 Birefringence in liquid crystals ...... 33 3.5 Order in liquid crystals ...... 34 3.6 Phase transitions ...... 35 3.7 The Fréedericksz transition ...... 36 3.8 Defects in nematic liquid crystals ...... 37 3.8.1 Point defects ...... 37 3.8.2 The Schlieren texture ...... 38

3.8.3 Generation of defects following phase transitions ...... 40 3.8.4 Umbilic defects ...... 40 3.9 Topological classification of defects...... 41 3.10 Defect dynamics and annihilation ...... 43 References ...... 46 4. Experiment ...... 48 4.1 Overview ...... 48 4.2 Materials used ...... 48 4.2.1 ZLI-2806 ...... 48 4.2.2 4-butyl-N-[methoxy-benzylidene]-aniline (MBBA) ...... 50 4.3 Cell fabrication ...... 50 4.3.1 Glass cleaning ...... 50 4.3.2 The alignment layer ...... 51 4.3.3 Device assembly ...... 52 4.3.4 Quality control ...... 53 4.4 Experimental set up ...... 55 4.4.1 Signal generators and amplitude modulation...... 56 4.4.2 Linkham hot stage and temperature controller ...... 58 4.4.3 Nikon Optiphot-pol polarising microscope ...... 58 4.4.3 IDS uEye GigE camera ...... 59 4.5 Experimental parameters ...... 61 4.5.1 Time delay before measurement of defect density ...... 61 4.5.2 Range of ramp rates ...... 62 4.5.3 Temperature ...... 63 4.6 Frame selection ...... 65 4.7 Finding the transition point with ImageJ ...... 67 4.7.1 Mean pixel value ...... 67 4.7.2 Validating the imageJ method ...... 68 4.7.3 Measuring the threshold voltage ...... 69 4.8 Defect counting ...... 71 4.8.1 Algorithm or counting by eye? ...... 71 4.8.2 Method for counting defects ...... 73 4.9 Errors...... 74 References ...... 75 Chapter 5: Results and discussion ...... 76 3

5.1 Defect formation in ZLI-2806 ...... 76 5.1.1 Deviation from expected scaling at higher ramp rates ...... 79 5.1.3 Earlier deviation from expected scaling at lower temperatures ...... 86 5.2 Defect formation in MBBA ...... 86 References ...... 88

Chapter 6: Conclusion and future work ...... 90

Total word count: 20010

List of figures

2.1. Results from previous experiments using liquid crystals to confirm the scaling relationship between defect density and time ...... 20 2.2. String defect formation in 5CB cooled below its transition temperature ...... 21 2.3. The results from computer simulations comparing the number of defects generated at different quench rates in systems with overdamped and underdamped dynamics ...... 24 3.1. A 4’-pentyl-4-cyanobiphenyl (5CB) molecule ...... 30 3.2. A typical discotic molecule ...... 31 3.3. The molecular structure of the nematic, smetic and columnar phase ...... 32 3.4. The director in the nematic phase ...... 33 3.5. A plot of the order parameter against temperature for a nematic liquid crystal ...... 35

3.6. A capillary with homeotropic boundary conditions will produce point defects of alternating strength ...... 38 3.7. A nematic droplet produces a point defect ...... 38 3.8. The Schlieren texture ...... 39 3.9. Sliding one glass plate with respect to the other reveals that s = 1 defects in the Schlieren defects are true point defects whereas the s = 1/2 defects are the ends of a string defect ...... 40 3.10. Comparison of the Schlieren and umbilic defects under a polarising microscope ...... 41 3.11. Comparison of the geometry of a Schlieren defect and an umbilical defect ...... 41 3.12. Typical topological point defects seen in liquid crystals ...... 42 3.13. A source and a sink meet and form a dipole...... 43 3.14. A source and hyperbolic point move towards each other shortly before annihilation 43 3.15. The director field surrounding an s = +1/2 defect ...... 45 3.16. The director field surrounding an s = +1 defect ...... 45 4.1. Umbilic defects generated in ZLI-2806 after a Fréedericksz transition ...... 49 4.2. A MBBA molecule ...... 50 4.3. A cetyltrimethylammonium bromide (CTAB) molecule ...... 52 4.4. A schematic of the liquid crystal cell used in this experiment ...... 52 4.5a,b. ZLI-2806 before and after homeotropic alignment was adopted ...... 54 4.6. A visual overview of how each component use in the experiment was connected...... 56 4.7. A typical amplitude modulated wave used to generate umbilic defects ...... 57 4.8. A labelled diagram of the polarising microscope used in this work ...... 59 4.9. A visual representation of 1x subsampling both horizontally and vertically ...... 60 4.10a-f. Umbilic defect formation in ZLI-2806 during a Fréedericksz transition...... 62 5

4.11. Defect density against ramp rate at room temperature ...... 64 4.12. A plot of defect density against temperature ...... 65 4.13. A plot of threshold voltage against ramp rate ...... 66 4.14. A plot of mean pixel value against time...... 68 4.15. Voltage pulses applied one second apart from each other seen in a plot of mean pixel value against time ...... 69 4.16. The guide wave used to modulate a sine wave for calculating the threshold voltage. 70 4.17. The guide wave used for the experiment ...... 71 4.18a-d. Images of defects over a short period of time aid in their identification ...... 72 5.1. Plots of defect density against ramp rate for ZLI-2806 at different temperatures ...... 78 5.2. A summary of the scaling exponents measured at each temperature ...... 79 5.3. A plot of defect density in ZLI-2806 over time for three different ramp rates ...... 81 5.4. A plot showing how defect density is expected to scale with ramp rate if at some point a third of the defects are able to anihilate before the defect density is measured ...... 82 5.5. Individual runs showing a sharp drop in defect density and subsequent points scaling with the same exponent as previously ...... 83 5.6. The averaged result of fifteen model data runs ...... 84 5.7. A plot of defect density in ZLI-2806 against ramp rate after a shorter delay time ...... 85 5.8. Plots of defect density against ramp rate for MBBA at three different temperatures .. 88

Abstract

Umbilic defects with strength s = ±1 were generated via a Fréedericksz transition in a nematic liquid crystal with negative dielectric anisotropy. The defect density was measured as the rate at which the Fréedericksz transition occurred was varied. A scaling exponent which described how the defect density scaled with the rate at which the Fréedericksz transition occurred was calculated and found to be in good agreement with the theoretically predicted value. To our knowledge, this is possibly the first time that this scaling relationship has been confirmed experimentally in a condensed matter system.

Declaration

No portion of the work referred to in the thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning.

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Acknowledgments

Firstly, I would like to thank my supervisors Ingo Dierking and Tobias Gallas for devising such an interesting project. Thank you Ingo, for all your support over the two years. Thank you Tobias, for your stimulating lectures in non-linear physics which I’m sure will be very useful in my future research.

I would also like to thank Helen Gleeson for introducing me to liquid crystals and delivering excellent lectures over a wide range of areas in soft matter. Thanks to Cliff Jones for many interesting discussions on research-related and not so research-related topics, in particular on cell construction.

Thank you to everybody in the liquid crystals group. I really appreciate how much time and effort everybody put into creating a supportive and fun environment both inside and outside of the lab. A “special” thanks to Dave on behalf of the entire liquid crystals community for his almost daily discovery of liquid crystals that are birefringent.

Most importantly, I would like to thank my fiancée Ruth and soon to be Mother-in-law Margaret, without whom, none of this would have been possible. Thank you for your encouragement, support and love not just during my MSc but over the past six years, throughout my undergraduate degree and a brief stint in the real world.

Chapter 1: Introduction

Topological defects appear in many aspects of physics including cosmology, superfluids and liquid crystals. The Kibble-Zurek mechanism is concerned with the formation of topological defects in a system driven through a continuous phase transition at a finite rate. The name was adopted after Zurek [1] proposed that the Kibble mechanism [2], which described defect formation in the early universe, should be applicable in condensed matter systems after accounting for differences in scale. This was an exciting idea as it suggested that cosmological theories of defect formation, normally beyond the remit of experiment, could be tested.

How the initial defect density scales with the rate at which a phase transition occurs was predicted by Zurek and has since been confirmed using computer simulations [2, 3]. However, there has been no experimental confirmation of this key theoretical prediction in condensed matter systems. The aim of this work is to measure a scaling exponent which describes how the initial density of umbilic defects generated in a negative dielectric nematic liquid crystal via a Fréedericksz transition scales with the rate at which the transition occurs.

The outline of this dissertation is as follows. Chapter two introduces the Kibble mechanism and explains how it was adapted so that it also described defect formation in condensed matter systems (the Kibble-Zurek mechanism). Predictions of how defect formation and annihilation should scale in condensed matter systems are outlined and previous work attempting to provide experimental confirmation of such scaling is reviewed. The chapter concludes by suggesting how a liquid crystal system is suitable for “cosmological experiments”. Chapter three outlines relevant background information about liquid crystals, liquid crystal phases and phase transitions. It then continues to discuss the formation and annihilation of defects in nematic liquid crystals. Chapter four gives a detailed overview of the experimental system and method employed to generate umbilic defects in a nematic liquid crystal as well as how the defect density was measured. Chapter five presents the

11 results of how defect formation scales with ramp rate in two liquid crystals, ZLI- 2806 and MBBA. Measured values of the scaling exponent are compared to the theoretically predicted value and reasons for any deviation from the predicted value are discussed. A conclusion is given in chapter six along with suggestions for future work.

References

[1] W.H Zurek, Cosmic strings in laboratory superfluids and the topological remnants of other phase transitions, Acta Physica Polonica B 24(7), p1301 (1993). [2] T. W. B. Kibble, Topology of cosmic domains and strings, J. Phys. A 9, p1387 (1976). [3] A. Yates1, 2 & W. H. Zurek, Vortex formation in two dimensions: When symmetry breaks, how big are the pieces? Phys. Rev. Lett. 80(25), p5477 (1998).

Chapter 2: Defect formation in the early universe and condensed matter systems

This chapter will describe defect formation in the early universe and condensed matter systems. The chapter then introduces some universal scaling laws which are expected to be observed in both the early universe and condensed matter systems. Previous work undertaken in trying to experimentally confirm these scaling laws is then reviewed. The chapter concludes by suggesting how a liquid crystal system is suitable for “cosmological” experiments.

2.1 The Kibble mechanism

The Kibble mechanism describes the formation of topological defects following symmetry breaking phase transitions in the early universe [1]. Understanding the dynamics of topological defects in the early universe is interesting because their existence could have affected how the universe evolved. Both zero and two- dimensional defects (monopoles and domain walls) are not thought to have formed because cosmological models which include them evolve in ways that contradict present day observations of the universe [2, 3]. However, one-dimensional defects known as cosmic strings (as well as textures) have not been ruled out by such models. When cosmic strings were first proposed, it was suggested that their gravitational influence may be responsible for the large scale structure that exists throughout the universe as well as the anisotropies in the cosmic microwave background (CMB) [3]. However, increasingly precise measurements of the CMB do not support either idea [4, 5]. That said, it has been suggested that the experiments performed to date have not yet been able to completely rule out the existence of cosmic strings [6]. It has also been suggested that the presence of cosmic strings may have led to early star formation and re-ionisation of the universe [7].

2.1.1 Symmetry breaking in the early universe

The concept of symmetry breaking is central to particle physics theories that attempt to unify the four fundamental interactions (gravitational, electromagnetic, strong nuclear and weak nuclear). According to such theories, each fundamental interaction differentiated itself from one overarching interaction as the early universe cooled and expanded between 10-43-10-12 seconds after the Big Bang. Thinking in terms of the Hot Big Bang model, the early universe underwent symmetry breaking phase transitions as it cooled below respective critical temperatures. The phase transitions could be either first or second order (the difference between the two types is discussed in the context of liquid crystals in section 3.6) [8]. The exact number of symmetry breaking transitions that occurred in the early universe is not known but many are expected to have happened [8]. As each transition occurs, the reduction in symmetry can be seen in the re-orientation of the Higgs field – a field thought to extend throughout the entire universe.

2.1.2 The Higgs field and defect formation in the early universe

The Higgs field is a theoretical field that pervades all of space-time. W+, W- and Z bosons can interact with the field and acquire mass as a result. This process is known as the Higgs mechanism. The Higgs field is oriented such that it points in all directions equally in the symmetrical state assumed to be the initial conditions of the universe. As symmetry is broken during a phase transition, the Higgs field adopts a particular orientation determined by small fluctuations in the Higgs field. However, domains with no causal contact have no reason to adopt the same orientation. Kibble suggested that cosmological topological defects could be formed between two domains that meet during expansion when the Higgs field of one domain cannot vary smoothly into the other (topological defects are discussed in the context of liquid crystals in section 3.8) [1]. The mechanism by which defects were formed in the early universe became to be known as the Kibble mechanism. Kibble proposed that the initial density of defects was determined by the correlation length of the Higgs field below a “freeze-out” temperature (analogous to the

Ginzburg temperature, below which thermal fluctuations cannot restore symmetry [8]). Correlations cannot be established over length scales larger than the causal horizon and this provides an upper limit on the correlation length [9].

A decade after Kibble proposed the Kibble mechanism, Zurek suggested that defects should be generated in a similar way in condensed matter systems after accounting for the finite time over which the phase transition would occur as well as any differences in scale [10]. This was exciting because it suggested that condensed matter systems could provide an experimentally accessible way in which to test the predictions of the Kibble mechanism. This description of defect formation came to be known as the Kibble-Zurek mechanism.

2.2 The Kibble-Zurek Mechanism

2.2.1 Defect formation in condensed matter systems

Zurek considered how the Kibble mechanism would operate in superfluid helium as it was cooled below the freeze-out temperature in a paper published in 1993 [11]. He suggested that the density of defects generated could be determined by calculating the correlation length that is “frozen-out” during the course of the non- equilibrium phase transition. To summarise, he first proposed an expression that would describe the growth of the correlation length if the system could equilibrate. He then suggested that if the phase transition occurred slowly enough there would be sufficient time for correlations to establish on the length scale of the causal horizon of the system. Discussed later in this chapter are experimental systems where this assumption seems to be valid. However, it should be noted that it has been suggested that this assumption is not valid for all experimental systems [12]. The frozen-out correlation length according to Zurek can be written:

−1/2 (2.1) 휀 = 휀0(푡/휏푄)

16 where 휀 is the correlation length, 휏푄 is the quench rate and 푡 is the time taken for the correlation length to become comparable to the causal horizon. Using Ginzburg- Landau theory, Zurek could calculate 푡 to be

√휏0휏푄 (where 휏0 = 푣퐹⁄휀0 (푣퐹 is the Fermi velocity)) yielding an expression for how the correlation length scales with the quench rate:

1/4 휀 ∝ 휏푄 (2.2)

It is important to stress that the Eq. 2.2 and any other expressions derived from it are only applicable in condensed matter systems. In a condensed matter system, the dynamics of the medium in which defects form can be considered to be overdamped (the system returns to equilibrium without oscillating) whereas the dynamics is a cosmological system would be underdamped (oscillations occur with decreasing amplitude) [13]. As a result, the scaling exponent which describes the freeze-out correlation length in a cosmological system differs to that given in Eq. 2.2 [11]. Henceforth, any scaling laws given in this work apply to condensed matter systems only.

Knowing that the equilibrium correlation length 휀 is a reasonable estimation of domain length, the domain volume immediately after a phase transition is ~휀−3. The density of defects generated can therefore be shown to scale with the correlation length:

휀 𝜌 ∝ = 휀−2 (2.3) 휀3

where 𝜌 is the defect density. By combining Eq. 2.2 and Eq. 2.3 the initial density of defects generated in a condensed matter system can be shown to scale with quench time:

−1/2 𝜌 ∝ 휏푄 (2.4)

Zurek suggested that the scaling law given by Eq. 2.4 should be universal to all condensed matter systems. It is this scaling law that this work has attempted to validate in a liquid crystal system by comparing the predicted scaling exponent with that measured.

2.2.2 Defect annihilation in condensed matter systems

As soon as a defect is generated it experiences a force between itself and any neighbouring defects with an equal strength but opposite sign (see section 3.10). Over time, defects will be drawn towards each other until they meet and annihilate. The defect density therefore decreases with time. A relationship between the correlation length and time after defect generation can be calculated by equating the attractive force between two defects and the opposing frictional force due to the viscosity of the medium in which they exist (detail given in [14]). The result is that correlation length can be shown to scale with time:

휀 ∝ 푡1/2 (2.5)

where 푡 is the time after defect generation. Combining Eq. 2.3 and Eq. 2.5 leads to a scaling law which describes how the defect density decreases over time:

𝜌 ∝ 푡−1. (2.6)

2.3 Previous experimental work

This section will discuss the experiments that have attempted to confirm the scaling laws described in the previous section. Other scaling laws which can be derived

18 from the Kibble-Zurek mechanism investigated using liquid crystals, but are not discussed, include the size of domains formed with quench time, separation between defects over time and the ratio of positive strength defects and negative strength defects [15, 16, 17]. The focus will be primarily on the use of liquid crystals although other experimental systems are also briefly outlined. The discussion of liquid crystals will highlight important experimental considerations that were accounted for when devising the method used in this work.

2.3.1 Scaling laws in liquid crystals

2.3.1.1 Defect annihilation in liquid crystals

The first “cosmological” experiments performed using liquid crystals examined the annihilation of defects over time. In 1991, Chuang, Turok and Yurke measured 𝜌 ∝ 푡−1.02±0.09 after identifying the density of strings generated during the isotropic-nematic transition of 5CB following pressure quenches between 2.00 MPa and 4.69 MPa [18]. Figure 2.1 shows their results. The first measurements were not made until after a delay of one second. This is because defects are not easily distinguishable from each other due to the high density of strings initially generated. Only string densities between 16 mm-2 – 160 mm-2 were used in the calculation of the scaling exponent. Chuang et al. suggested that defects were hard to identify at densities above 160 mm-2 because they overlapped and therefore too few defects were counted. Noise was an issue at densities below 16 mm-2 resulting in a higher than expected defect count. Overall, the time over which scaling was observed was almost an order of magnitude.

Figure 2.1. Left: String density in 5CB over time after pressure quenches with four different magnitudes [18]. Right: Umbilic defect density in ZLI-2806 following a Fréedericksz transition [19].

The scaling law was later further confirmed by Dierking et al. in 2005 using umbilic defects generated in the liquid crystal mixture ZLI-2806 following a Fréedericksz transition [19]. Again, measurements were not made until after a time delay. Universality of the scaling law was demonstrated by varying the electric field, cell gap and temperature. The density of defects which could be generated and counted was much higher than in the experiment performed by Chuang et al. Therefore, the time over which measurements were made could exceed an order of magnitude. What is interesting is that the Fréedericksz transition is not a true phase transition and umbilic defects, unlike most other types of defect, do not exhibit a singularity in the director field (see sections and 3.7 and 3.8.4), yet scaling is observed. This suggests that umbilic defects generated in ZLI-2806 could be suitable for the confirmation of other scaling laws associated with the Kibble-Zurek mechanism.

2.3.1.2 Defect formation in liquid crystals

There has been no satisfactory confirmation of the scaling law which describes defect formation in liquid crystals given by Eq. 2.4 in section 2.2.1. This is most likely because the initial defect configuration immediately after a phase transition is somewhat hard to acquire. Firstly, as mentioned in the previous section, it can take some time before defects are distinguishable from the other defects. Secondly, the

20 time it takes for defects to form increases with quench time meaning comparisons between different quench times can be problematic.

In 1994, Bowick et al. took an alternative approach to investigating the Kibble-Zurek mechanism in a liquid crystal [20]. They cooled a drop of 5CB below its isotropic- nematic transition temperature generating nematic domains which grew before coalescing leaving behind string defects (figure 2.2). They calculated the number of strings formed per domain 푛푠 using the expression below:

퐿 (2.7) 푛 = 푠 푁휀

where 퐿 is the measured string length, 푁 is the number of domains formed before any coalescence and 휀 is the length of the domains. The domain length was estimated to be 휀 = √퐴⁄푁 , where 퐴 is the area of the image used. The experiment was repeated three times and an average value calculated 푛푠 = 0.64 ±

0.02. The group also ran a simulation found 푛푠 = 0.636 ± 0.004, in excellent agreement with the measured value.

Figure 2.2. String defect formation in 5CB cooled below its transition temperature [20].

Whilst the results were promising, the group did not have confirmation of defect formation via the Kibble-Zurek mechanism in a liquid crystal. Firstly, they could not suggest the mechanism was universal because they did not vary any external parameters such as quench time. Secondly, there were many systematic errors which did not contribute to the error given. Instead only a statistical error for three sets of data was quoted. Errors from counting the number of domains, the domain length and string length could have resulted in a large systematic error. Thirdly, the

21 simulation used for comparison initially returned a value 푛푠 = 1.564 ± 0.006 before making alterations based on observations of the experimental system. While the alterations were certainly reasonable, it couldn’t be used as confirmation of defect formation via the Kibble-Zurek mechanism.

2.3.2 Scaling laws in other condensed matter systems

The first experiments using condensed matter systems to test the formation of defect via the Kibble-Zurek mechanism used superfluid helium-4 as first proposed by Zurek [11]. The superfluid state could be achieved in helium-4 by a rapid pressure quench. During the quench the phase of the Bose-Einstein condensate wave function is chosen randomly in causal domains. The phase difference between domains with different phases causes the superfluid to flow in quantized vortex lines - analogous to string defects generated in the early universe. In 1993, Hendry et al. published results that seemed to confirm that the density of vortex lines scaled according to the Kibble-Zurek mechanism [21]. However, in 1998, the same group repeated their experiment looking to minimise vortex creation via conventional hydrodynamic ﬂow processes and found that the density of vortex lines were two orders of magnitude less than expected [22]. Karra and Rivers have also questioned the assumption that the using equilibrium correlation length to determine the defect density may not be valid for superfluid helium-4 [12].

Superfluid helium-3 was also used to test the predictions of the Kibble-Zurek mechanism. In 1995, Bäuerle et al. generated vortex lines in superfluid helium-3 rotating in a container following the absorption of a thermal neutron [23]. They measured the deficit in energy released as normal liquid helium-3 transitioned back into a superfluid, attributing the deficit energy to the formation of a network of vortex lines. They found a close agreement between the predicted and measured values of the correlation length when conducting the experiment at three different pressures. In a companion experiment, Rutuu et al. used a similar method to investigate defect formation in superfluid helium-3 [24, 25]. With a comprehensive understanding of their experimental system, the group was able to derive an

22 expression for the rate of defect formation as a function of rotation velocity utilising the same considerations as Zurek and provide experimental confirmation. They also demonstrated that the relationship scaled according to the Vachaspati-Vilenkin scaling law [26]. This was the first convincing confirmation of defect formation due to the Kibble-Zurek mechanism in a condensed matter system. However, the method by which the defects were generated did not allow the quench time to be varied and so the scaling relationship given by Eq. 2.4 could not be confirmed.

Computer simulations performed by Zurek et al. confirmed that defect formation attributable to the Kibble-Zurek mechanism should scale as predicted. Simulations were performed in one and two dimensions and with underdamped and overdamped dynamics [27]. Their results for the two dimensional system, which should describe the experimental system used in this work, is shown in figure 2.3. For a system with overdamped dynamics (akin to a condensed matter system) the exponent was -0.44 ± 0.1 and for a system with underdamped dynamics (akin to a cosmological system) the exponent was -0.60 ± 0.07 in good agreement with the predicted exponents 1/2 and 2/3 respectively. The results support the idea that condensed matter systems can be used to perform cosmological experiments because their differences only result in a slightly different scaling exponent both theoretically and in computer simulation experiments.

Figure 2.3. The number of defects generated following different quenches with different rates for a system with (a) overdamped and (b) underdamped dynamics. The number of defects were occasionally hard to identify because they were not well separated. Defects were therefore counted after some delay time. The dashed line represents an alternative method whereby the initial density of defects inferred from defect annihilation over time (as performed in superfluid helium-4 experiments [21, 22]). The scaling exponent measured was 0.79 ± 0.04 suggesting that such a method is not a reliable way to determine the initial defect density [27].

2.3.3 Scaling laws in other experimental systems

For completeness, it is worthwhile to highlight that the Kibble-Zurek mechanism has been investigated in solid state systems such as Josephson junctions and more recently ionic crystals with good agreement between experiment and theory [28, 29].

2.4 Using a liquid crystal system to test the Kibble-Zurek mechanism

The main advantage of using liquid crystal systems for cosmological experiments is that they are experimentally more accessible than other systems. Liquid crystals systems can be made many orders of magnitude wider than they are thick so that they can be considered a 2D infinite plane. The order exhibited by a liquid crystal system can be selected through choice of liquid crystal as well as influenced by choice of boundary conditions for the cell containing them. Defects can be generated by temperature or pressure quenches as well as with applied magnetic or

24 electric fields, all of which are easily controllable. Both point and string defects can be generated at scales which can be easily “seen” using a polarising microscope and often without any image processing. Their annihilation dynamics are also easy to observe and can occur over a timescale ranging from a few seconds to over a minute.

An important consideration with respect to this work is that transitions generated by the application of electric (and magnetic) fields are not true phase transitions, and furthermore, generate defects that do not exhibit a singularity in the director field which is unlike most other types of defects (see sections 3.7 and 3.8.4) [30]. The assumption throughout this work is that defects generated by such transitions are generated via the Kibble-Zurek mechanism. This assumption is supported by recent experiments confirming that defects generated via the application of an electric field annihilate in accordance with the Kibble-Zurek mechanism [19]. Another important consideration is that whilst generating and observing defects in liquid crystal system may be relatively simple, there are many experimental factors which could potentially influence the formation of defects. For example, contaminants introduced into the liquid crystal during the cell construction process, the geometry of “home-made” cells, the quality of liquid crystal alignment and any additional unwanted effects of applying an electric field across a liquid crystal. Therefore much thought has been given to how to build the experimental system and the method used to generate and measure the density of defects (see chapter four).

References

[1] T. W. B. Kibble, Topology of cosmic domains and strings, J. Phys. A 9(8), p1387 (1976). [2] Y. Zel'dovich, I. Y. Kobzarev & L. B. Okun, Cosmological consequences of the spontaneous Breakdown of Discrete Symmetry, Zh. Eksp. Teor. Fir. 67, p3 (1974) [JETP 40(1) (1974)]. [3] A. M. Srivastava, Topological defects in cosmology, Pramana – J. Phys. 53(6), p1069 (1999). [4] A. Albrecht, R. A. Battye & J. Robinson, The case against scaling defect models of cosmic structure formation, Phys. Rev. Lett. 79(24), p4736 (1997). [5] P. A. R. Ade et al. (Planck collaboration), Planck 2013 results. XXV. Searches for cosmic strings and other topological defects, arXiv:1303.5082 [astro-ph.CO] (2013). [6] O. S. Sazhina & D. Scognamiglio, Observational constraints on the types of cosmic strings, arXiv:1312.6106 [astro-ph.CO] (2013). [7] B. Shlaer, A. Vilenkin & A. Loeb, Early structure formation from cosmic string loops, arXiv:1202.1346 [astro-ph.CO] (2012). [8] A. Vilenkin & E. P. S Shellard, Cosmic strings and other topological defects (University Press, Cambridge UK, 2000). [9] T. W. B. Kibble, Some implications of a cosmological phase transition, Physics Reports 67(1) p183 (1980). [10] W.H. Zurek, Cosmoligical experiments in superfluid helium? Nature 317(10), p585 (1985). [11] W.H Zurek, Cosmic strings in laboratory superfluids and the topological remnants of other phase transitions, Acta Physica Polonica B 24(7), p1301 (1993). [12] G. Karra & R. J. Rivers, A Re-examination of Quenches in 4He (and 3He), Phys. Rev. Lett. 81(71), p3707 (1998). [13] A. Yates & W. H. Zurek, Vortex formation in two dimensions: When symmetry breaks, how big are the pieces? Phys. Rev. Lett. 80(25), p5477 (1998). [14] A. N. Pargellis, P. Finn, J. W. Goodby, P. Panizza, B.Yurke & P. E. Cladis, Defect dynamics and coarsening dynamics in smectic-C films, Physical Review A 46(12), p7765 (1992).

[15] Z. Bradač, Samo Kralj & S. Žumer, Early stage domain coarsening of the isotropic-nematic phase transition, J. Chem. Phys. 135, p024506 (2011). [16] T. Nagaya, H. Hotta, H Orihara & Y Ishibashi, Observation of annihilation process of disclinations emerging form bubble domains, Journal of the Physical Society of Japan 60(5), p1572 (1991). [17] R. Ray, Defect correlation in liquid crystal: Experimental verification of cosmological Kibble mechanism, Pramana – J. Phys. 53(6), p1087 (1999). [18] I. Chuang, N. Turok & B. Yurke, late-time coarsening dynamics in a nematic liquid crystals, Phys. Rev. Lett. 66(19), p2472 (1991). [19] I. Dierking, O. Marshall, J. Wright & N. Bulleid, Annihilation dynamics of umbilical defects in nematic liquid crystals under applied electric fields, Physical Review E 71, p061709 (2005). [20] M. J. Bowick, L. Chandar, E. A. Schiff & A. M. Srivastava, The Cosmological Kibble Mechanism in the Laboratory: String Formation in Liquid Crystals, Science 263, p943 (1994). [21] P. C. Hendry, N. S. Lawson, R. A. M. Lee, P. V. E. McClintock & C. D. H. Williams, Creation of quantized vortices at the lambda transition in liquid helium-4, J. Low. Temp. Phys. 93, p1059 (1993). [22] M. E. Dodd, P. C. Hendry, N. S. Lawson, P. V. E. McClintock & C. D. H. Williams, Non-appearance of vortices in fast mechanical expansions of liquid 4He through the lambda transition, Phys. Rev. Lett. 81, p3703 (1998). [23] C. Bäuerle, Yu. M. Bunkov, S. N. Fisher, H. Godfrin & G. R. Pickett, Laboratory simulation of cosmic string formation in the early Universe using superfluid 3He, Nature 382, p332 (1996). [24] Ruutu et al., Vortex formation in neutron irradiated-superfluid 3He as an analogue of cosmological defect formation, Nature 382, p334 (1996). [25] Ruutu et al., Defect Formation in Quench-Cooled Superﬂuid Phase Transition, Phys. Rev. Lett. 80(7), p1465 (1998). [26] T. Vachaspati & A. Vilenkin, Radiation from vacuum strings and domain walls, Phys. Rev.D 30, p2036 (1984). [27] P. Laguna & W. H. Zurek, Density of kinks after a quench: When symmetry breaks, how big are the pieces? Phys. Rev. Lett. 78(13), p2519 (1997).

[28] E. Kavoussanaki, R. Monaco & R. J. Rivers, Testing the Kibble-Zurek scenario with annular Josephson tunnel junctions, Phys. Rev. Lett. 85(16), p3452 (2000). [29] S. Ulm et al., Observation of the Kibble–Zurek scaling law for defect formation in ion crystals, Nature Communications 4, article 2290, (2013). [30] A. Saupe, Disclinations and Properties of the director field in nematic and cholesteric liquid crystals, Molecular Crystals and Liquid Crystals 21(3-4), p211 (1973).

Chapter 3: Liquid crystals

This chapter will define what is meant by a liquid crystal and summarise the types of molecule that form liquid crystal phases. The different liquid crystal phases and phase transitions are also discussed followed by an overview of defect formation and annihilation in nematic liquid crystals.

3.1 Definition of a liquid crystal

A liquid crystal is a state of matter which has thermodynamic properties that exist somewhere between a conventional liquid and a crystalline solid [1]. A liquid crystal, much like a liquid, can flow and will take the shape of any container. As with a liquid, it has no definitive shape and an approximately fixed volume which means it maintains an almost constant density. The difference between a liquid crystal and a liquid is the variation of the molecular axes of the molecules within each state. The molecular axes of molecules in a liquid crystal possess some degree of order whereas the molecular axes of molecule in a liquid vary randomly. In this respect a liquid crystal is similar to a crystalline solid. Molecules within a crystalline solid have long-range orientational and three dimensional positional order. The molecules in a liquid crystal usually exhibit orientational order and sometimes positional order. However, the degree of order in a liquid crystal is much less than that found in a crystalline solid.

3.2 Types of liquid crystals

The small degree of order which differentiates the liquid crystal phase from other phases is a result of the highly anisotropic nature of the constituent molecules [2]. There are two main ways in which a liquid crystal molecule can be anisotropic. A molecule could have an axis that is much longer than its other axes. Alternatively, parts of a molecule may have significant differences in solubility properties and form a liquid crystal with the addition of a particular solvent. Molecules with either 29 of these anisotropies tend to form liquid crystal phases within particular temperature ranges. Liquid crystals that form a liquid crystal phase within a particular temperature range are known as thermotropic liquid crystals. Liquid crystals that form a liquid crystal phase within a particular solvent concentration range and are known as lyotropic liquid crystals.

3.2.1 Thermotropic liquid crystals

A liquid crystal is a thermotropic liquid crystal if the degree of order it exhibits is dependent on its temperature. Thermotropic liquid crystals are normally long and narrow with a length: width ratio greater than 4:1 [3]. It is easily imagined, given that the molecules are sufficiently rigid how a preferred orientation or position can be adopted as a result of their geometry. In additional to geometrical and thermal considerations, electrical properties such as dipole and quadrupole interactions between molecules may promote order in a thermotropic liquid crystal.

There are three categories of thermotropic liquid crystals according the shape of the constituent molecules: calmatic, discotic and sanidic. Calmatic liquid crystals are made up of molecules with “rod-like” structure (figure 3.1), discotics have “disk- like” structure (figure 3.2) and sanidic are “lath-like” which is a structure somewhere between the two other types.

NC C5H11

Figure 3.1. 4’-pentyl-4-cyanobiphenyl (5CB), a typical calmatic molecule.

R R

Figure 3.2. A typical discotic molecule, each R denotes a side chain.

3.2.2 Lyotropic liquid crystals

Lyotropic liquid crystals form a liquid crystal phase when mixed with a specific solvent. The concentration largely determines the stability of the phase however there is some contribution from thermal processes. A lyotropic liquid crystal molecule is amphiphilic meaning it has both a hydrophilic (water-loving) and hydrophobic (water-hating) components. These components are usually situated at opposing ends of the molecule. When mixed in a polar solvent, the hydrophobic parts group so that the hydrophilic parts are in contact with the solvent.

3.2.3 Liquid crystal polymers

Liquid crystal polymers are a third type of molecule that can form a liquid crystal phase. They consist of repeating units of long, narrow and rigid molecules much like those described previously. Repeating units are connected by flexible spacers in an alternating pattern (a main chain polymer) or attached via flexible spacers to a non- mesogenic (will not form a liquid crystal phase) side group polymer. Liquid crystal polymers can be both thermotropic and lyotropic.

3.3 Liquid crystal phases

There are many different liquid crystal phases each characterised by their differing orientational and positional degrees of order. There are three common phases: nematic, smectic and columnar (figure 3.3). All three phases exhibit orientation order whilst smectic and columnar also exhibit one-dimensional and two- dimensional positional order respectively. A particular liquid crystal may be able to transition between phases as variables such as temperature, pressure or solvent concentration are varied. The work that will be discussed in subsequent chapters was performed using the nematic liquid crystal phase and so the next section will only discuss this phase. Descriptions of the smectic and columnar phase as well as many other more exotic phases can be found in introductory text books on liquid crystals.

(a) (b) (c)

Figure 3.3. (a) Molecules in the nematic phase have no positional order but tend to point in the same direction (the director). (b) Molecules in the smectic phase have one dimensional positional order (layers) and also tend to point in the same direction. The diagram shows a smectic A phase where the director is perpendicular to the layers. Molecules in the smectic C phase are positioned in the same way but the director is titled with respect to the layer normal. (c) Molecules in the columnar phase are normally discotic and form a two dimensional crystalline array.

3.3.1 The nematic liquid crystal phase

The nematic liquid crystal phase consists of molecules that tend to point in a particular direction (figure 3.4). There is no long range order between the centres of 32 mass of the molecules and so the phase does not have positional order. The nematic phase is the least ordered of all the liquid crystal phases. The direction the molecules tend to point is given by a unit vector n, known as the “director”. There is no preferred arrangement of the two ends of the molecule, even if they differ. Therefore, vectors n and –n are equivalent [4]. The direction of the director is determined by minor forces such as those exerted at the boundaries of a container. The director can also be influenced by external forces such as an applied magnetic or electric field.

Figure 3.4. Molecules in a nematic liquid crystal tend to point along the director n.

3.4 Birefringence in liquid crystals

Liquid crystals are birefringent due to their anisotropic nature. The refractive index of a birefringent material depends on propagation direction and polarisation of incident light. A uniaxial birefringent material such as a nematic liquid crystal has two distinct refractive indices. The optic axis is defined as the axis wherein all perpendicular directions are optically equivalent. In the case of a nematic liquid crystal, the optic axis coincides with the director. Components of light travelling through the material parallel and perpendicular to the optic axis have different refractive indices, 푛푒 and 푛표 respectively, where the subscripts denote “extraordinary” and “ordinary”. The components of light therefore travel at different velocities and become out of phase. After the two components exit the

33 material and recombine, the polarisation state will have changed due to the phase difference. Birefringence ∆푛 can therefore be quantified as the difference between the two refractive indices:

∆푛 = 푛푒 − 푛표. (3.1)

The fact that liquid crystals are birefringent allows them to be easily studied using polarising microscopy. A liquid crystal placed between two crossed polarisers will cause the initially linearly polarised light to become elliptically polarised. Some component of the elliptically polarised light will then be able to pass through the second polariser. Interference between the transmitted components of the ordinary and extraordinary ray can result in birefringence colours. Normally regions of differing brightness and colour are seen due to the non-uniformity of the order and birefringence value throughout a liquid crystal sample. Polarising microscopy is therefore an excellent method for determining the phase of a liquid crystal.

3.5 Order in liquid crystals

Each liquid crystal phase can be defined by the degree of order it exhibits which in turn describes the symmetry of the phase. The degree of order of a phase can be described statistically by the order parameter 푠 [5]:

1 (3.2) 푠 = 〈3푐표푠2휃 − 1〉 2

where 휃 is the angle between the long axis of each molecule and the director and the brackets denote an average over all molecules. The order parameter is equal to zero when describing a purely symmetrical state such as an isotropic liquid and equal to unity when describing a perfect crystal [4]. The value of the order

34 parameter changes with variables such as temperature or pressure [2]. Figure 3.5 shows how the order parameter for a nematic liquid crystal typically varies with temperature. Significant changes in the order parameter normally result as a liquid crystal undergoes a phase transition.

Figure 3.5. At lower temperatures, a nematic liquid crystal tends to have an order parameter with a value between 0.7 - 0.8. As temperature increases, the order parameter falls to a value between 0.3 - 0.4 just before the nematic-isotropic transition temperature. The order parameter then drops abruptly to zero as the transition temperature is reached [6].

3.6 Phase transitions

Phase transitions in liquid crystals are normally either first or second order. A discontinuity in the first derivative of free energy with respect to a thermodynamic variable is known as a first order phase transition. A second order transition is continuous in the first derivative of free energy and discontinuous in the second derivative. Transitions induced by temperature quenches as well as pressure quenches are usually first order. Experimental observations of nematic liquid crystals reveal that the order parameter decreases monotonically as temperature is increased before abruptly dropping to zero as the nematic-isotropic transition occurs [5]. Second order transitions in liquid crystals may involve an applied external magnetic or electric field. An applied field can alter the first order nature of the nematic-isotropic transition provided the magnitude of the applied field exceeds a critical value. The applied field acts to promote order in the isotropic

35 phase which effectively smooths out the discontinuity in the first order derivative of free energy that characterises the first order transition [7].

3.7 The Fréedericksz transition

A Fréedericksz transition occurs when an external magnetic or electric field is applied across a liquid crystal resulting in the deformation of a uniform director field (a vector field which describes the orientation of the director at any given point in space). It is not a phase transition because all the molecules in the bulk of the liquid crystal move in the same way with respect to each other and therefore the degree of order is unchanged [2].

As the magnitude of an applied field is increased past a threshold value it may cause the director field to twist and eventually align with the field. How an individual liquid crystal moves in the presence of an applied electric or magnetic field is determined by its relative permittivity or relative permeability anisotropy respectively. The relative dielectric anisotropy ∆휀 of a liquid crystal is given by:

∆휀 = 휀∥ − 휀⊥ (3.3)

where 휀∥ and 휀⊥ are the relative permittivities parallel and perpendicular to the long molecular axis, respectively. Permittivity is directly related to electrical susceptibility which is a measure of a dielectric’s ability to polarise in response to an electric field. Therefore, a molecular axis with a larger relative permittivity will have a larger polarisation ability and align with an applied electric field. When an electric field is applied to a liquid crystal with a positive dielectric anisotropy the long molecule axis will align with the field. Conversely, a liquid crystal with a negative dielectric anisotropy will align with its long molecular axis perpendicular to an applied electric field.

3.8 Defects in nematic liquid crystals

Defects are regions which break the local symmetry in an ordered medium [8]. Defects can be generated in liquid crystals following a quench of a thermodynamic variable, due to effects from applied fields such as the Fréedericksz transition, due to inhomogeneities at the surface of the liquid crystal or geometrical confinements. In principle, a defect can be a point, string or sheet. However, sheet defects are unstable in the nematic phase and therefore not seen [4]. This section will outline the different types of defects seen in the nematic phase and how they can be described using topology. It then continues to explain how they are generated, their subsequent dynamics and eventual annihilation.

3.8.1 Point defects

There are two types of point defects, “hedgehogs” and “boojums”, which are formed in the bulk and the surfaces of the liquid crystal respectively. Hedgehogs are so named because of their “spiky” radial geometry. Initially, it was thought they should appear following a symmetry breaking transition such as the nematic- isotropic transition but they are rarely observed. It has been suggested the low probability of hedgehog formation in the bulk of a liquid crystal is due to the geometry surrounding a defect being very hard to acquire from random initial conditions [9]. Observations of point defects have been seen 1-2 seconds after a quench as a result of interactions between strings or collapsing loops [3C]. Boojums are much more commonly observed. The inspiration for the unusual name came from “The Hunting of the Snark”, a poem by Lewis Carol. They usually form as a result of inhomogeneities at the liquid crystal surface or as a result of confining the liquid crystal to a specific geometry such as a capillary tube with homeotropic boundary conditions (figure 3.6) or in a planar configuration between hybrid boundary conditions [10]. Nematic droplets also exhibit point defects as a result of their geometry [11] (figure 3.7).

Figure 3.6. A capillary with homeotropic boundary conditions will produce point defects of alternating strength [10]. Figure 3.7. A nematic droplet produces a point defect in the centre due to geometrical restrictions [11].

3.8.2 The Schlieren texture

The Schlieren texture is seen when a nematic liquid crystal is placed between glass plates with a separation in the region ~1-50μm and viewed under a polarising microscope. It consists of dark points connected by either four thin brushes or two thicker brushes. The dark brushes are not string defects. They appear where the director field is parallel or perpendicular to a plane of polarisation. Figure 3.8 shows a typical Schlieren texture and has identified the two different types of points by labelling their respective strength (see section 3.9). The defect strength is equal to a quarter of the number of dark brushes [3].

Figure 3.8. The Schlieren texture, the strength of the defects have been labelled [12].

The two types of point are actually quite different. Points with two brushes constitute the ends of stable defect strings connecting the two glass plates. The string connecting two dark points is revealed when sliding one glass plate with respect to the other (figure 3.9) [13]. The director is parallel to the bounding glass plates. This can be demonstrated by rotating the sample by 180 degrees and noticing that the director n becomes its equivalent state –n. The s = 1/2 points are not defects because the director field surrounding the s = 1/2 points is continuous. In contrast, the s =1 points are defects. The points where four brushes meet are boojums and are connected by a core where the director is perpendicular to the plane of the polarisation (also known as an “escape in the third dimension”) [14]. Again, this can be seen by sliding one glass plate with respect to the other (figure 3.9) [13]. The boojums remain in place but there is no discontinuous core connecting the two.

Figure 3.9. Sliding one glass plate with respect to the other reveals that s = 1 defects in the Schlieren defects are true point defects whereas the s = 1/2 defects are the ends of a string defect [13].

3.8.3 Generation of defects following phase transitions

Both first order and second order phase transitions can generate defects but the mechanism by which this happens is slightly different in each case. During a first order transition, such as the isotropic-nematic transition, germs of broken symmetry are formed as the temperature drops below the critical value [15]. Each germ may adopt a different director orientation before expanding and forming a domain until it meets other domains. Where two domains meet the director field either smoothly varies from one into the other or becomes discontinuous and a string defect is generated. In a second order phase transition, the change of phase occurs at the same time throughout the sample [16]. As the transition progresses, local domains of liquid crystal arbitrarily select an orientation of the director. However, unless the transition occurs infinitesimally slowly, each domain does not have sufficient time to communicate its preferred orientation with other domains. As a result, domains with differing director orientations form, separated by a network of string defects.

3.8.4 Umbilic defects

Umbilic defects are points surrounded by four brushes and look much like the s = 1 Schlieren defects when viewed under a polarising microscope (figure 3.10 and 3.11). 40

Umbilic defects are produced when an external electric or magnetic is applied across a nematic liquid crystal with particular boundary conditions [17]. For example, umbilic defects appear when an electric field is applied across a nematic with a negative dielectric anisotropy (Δε < 0) under homeotropic boundary conditions. The larger relative permittivity of the molecular axis perpendicular to the long molecular axis causes molecules in the bulk of the sample to twist as the electric field is increased past a threshold. Molecules at the surface remain anchored by the homeotropic boundary conditions and do not re-orientate. Defects are formed where molecules get trapped between two domains of re-orientating molecules and remain in their upright position. When viewed from above they appear to be defects but there is no singularity present (see figure 3.11).

Left: Figure 3.10. A comparison of the Schlieren and umbilic defects seen under a polarising microscope [18]. Right: Figure 3.11. Comparison of the geometry of a Schlieren defect and an umbilical defect. The solid lines represent the director orientation (adapted from [18]).

3.9 Topological classification of defects

The topological defects that are usually seen in nematic liquid crystals are summarised in figure 3.12. The lines represent the director field surrounding the defect. Normally, a direction would be indicated by adding an arrow to each line. However, as explained previously, the molecules in a liquid crystal can point in

41 either direction in the director plane and so no arrow is required. The brushes that surround a defect can be visualised by laying a set of axes over each diagram to represent the directions of the crossed polarisers. Where the molecules are orientated in a direction parallel to the axes is where the dark brushes will occur.

Figure 3.12. Typical topological point defects seen in liquid crystals (adapted from [19]).

The strength, also known as winding number or topological charge, is given below each topological defect. The strength can be calculated by following an anticlockwise trajectory around the defect and noting how many times the director rotates by 2π [20]. The sign of the strength corresponds to the direction the director rotates as you do this. By convention a director that rotates anticlockwise is given a positive strength.

The strength of a defect depicts information about how the defect was generated and how it may evolve over time. For example, the net strength (or more naturally, the topological charge) must be conserved as defects are generated. Looking at the Schlieren texture reveals that there are as many s = +1 defects as there are s = -1. The same can be said for the s = +1/2 and s = -1/2 defects. The next section will outline how the defect strength can be used to describe a defects dynamics, its interaction with other defects and eventual annihilation.

3.10 Defect dynamics and annihilation

Defects undergo various dynamic processes following their generation. They may experience attractive or repulsive forces between themselves and other defects, may decay into two other types of defect or even annihilate completely.

Defects with the same modulus strength are said to be topologically equivalent [21]. The energy required for topologically equivalent defects to interact is much less than for non-equivalent defects. A defect may be deformed into another type of topological defect of equal strength. Two defects of equal strength may interact by combining into a new defect with strength equal to the sum of their strengths. Figure 3.13 shows how source and a sink both with s = +1 come together to form a dipole with s = +2. Defects with opposite strength will experience an attractive force between them causing them to move towards each other until they meet and annihilate. Figure 3.14 shows a source and a hyperbolic point moving towards each other shortly before annihilation. Defects can also decay into other defects provided that the sum of the strengths of the resulting defects equals the strength of the decaying defect.

Left: Figure 3.13. A source and a sink meet and form a dipole. Right: Figure 3.14. A source and hyperbolic point move towards each other shortly before annihilation. (Both figures adapted from [19]).

Not all observations of defects can be explained by solely referring to their strengths. For example, defects with s > 1 are rarely seen even though topologically

43 permitted. Furthermore, s = ±1 Schlieren defects never decay into two s = ±1/2 defects. However, these observations can be explained by considering the free energy of a defect. A defects free energy per unit length 퐹퐿 can be calculated by considering the increase in Helmholtz free energy due to deviations in the orientational order from a uniformly aligned nematic [13]. What results is a simple but powerful relationship between the free energy per unit length and defect strength:

2 퐹퐿 ∝ 푠 . (3.4)

Defects with s = ±2 are not seen because they have four times the free energy density of s = ±1 defects. If an s = ±2 defect were to be generated it would quickly decay into two s = ±1 defects. Following the same logic suggests that all s = ±1 defects should eventually decay into s = ±1/2. However, counting each type of defect in the Schlieren texture reveals there are roughly as many s = ±1 defects as there are s = ±1/2 and no decay. The reason for this is because there exists a lower energy state whereby the s = ±1 defect “escapes into the third dimension”. This happens when molecules move into the defect core. The s = ±1 defect is topologically unstable and will always prefer to escape into the third dimension. The director field surrounding a defect must not be able to be continuously deformed for a defect to be topologically stable [22]. Only defects with half integer strength are topologically stable. This can be appreciated visually by comparing the geometry surrounding two defects with integer and half integer strength (figure 3.15 and 3.16).

Left: Figure 3.15. The director field surrounding an s = +1/2 defect. A significant amount of energy would be required to re-orientate the molecules so that they could escape into the defect core [19]. Right: Figure 3.16. The director field surrounding an s = +1 defect. The molecules do not need to re- orientate in order to escape and so the energy requirement is much less [19].

References

[1] I. Dierking, Textures of Liquid Crystals (Wiley-VCH, Weinheim, 2003). [2] P. J. Collings, Liquid crystals – Nature’s delicate phase of matter 2nd ed. (Princeton University Press, Oxfordshire, 2002). [3] P. J. Collings & Michael Hird, Introduction to Liquid Crystals: Chemistry and Physics, (Taylor and Francis Ltd. London, 1998). [4] P. G. De Gennes & J. Prost, The Physics of Liquid Crystals 2nd Ed. (Oxford University Press, Oxford 2008). [5] E. F. Gramsbergen, L. Longa & W. H. de Jeu, Landau theory of the nematic- isotropic phase transition, Physics Reports 135(4), p195 (1986). [6] B. Senyuk, [http://dept.kent.edu/spie/liquidcrystals/maintypes2.html: Accessed 4th November 2014]. [7] C. Fen & M. J. Stephen, Isotropic-Nematic Phase Transition in Liquid Crystals, Phys. Rev. Lett. 25, p500 (1970). [8] M. Kleman, Defects in liquid crystals, Rep. Prog. Phys. 52(5), p555-654 (1989). [9] M. Hindmarsh, Where Are the Hedgehogs in Quenched Nematics? Physical Review Letters E 75(13), p2502 (1995). [10] M. Kleman and O. D. Lavrentovich, Topological point defects in nematic liquid crystals, Philosophical Magazine 86(25–26), p4117 (2006). [11] R. B. Meyer, Point Disclinations at a Nematic-lsotropic Liquid Interface, Molecular Crystals and Liquid Crystals 16, p355 (1972). [12] B. Senyuk, Liquid Crystals: a Simple View on a Complex Matter, URL: http://dept.kent.edu/spie/liquidcrystals/textures1.html, Accessed: 4th July 2014. [13] M. Kleman & O. D. Lavrentovich, Soft Matter Physics: An Introduction, (Springer-Verlag, New York, 2003). [14] W. F. Brinkman & P. E. Cladis, Defects in liquid crystals, Physics Today 35(48), p2 (1982). [15] W. H. Zurek, Cosmological experiments in condensed matter systems, Phys. Rept. 276, p177 (1996). [16] R. H. Brandenberger and A-C. Davis, Formation of topological defects in a second order phase transition, Physics Letters B 332, p305 (1994).

[17] I. Dierking, O. Marshall, J. Wright & N. Bulleid, Annihilation dynamics of umbilical defects in nematic liquid crystals under applied electric fields, Physical Review E 71, p061709 (2005). [18] I. Dierking, M. Ravnik, E. Lark, J. Healey, G. P. Alexander & J. M. Yeomans, Anisotropy in the annihilation dynamics of umbilic defects in nematic liquid crystals, Physical Review E 85, p021703 (2012). [19] H. R. Trebin, Classification of defects in liquid crystals, Defects in Liquid Crystals: Computer Simulations, Theory and Experiments, p1-16 (Kluwer Academic Publishers, Netherlands, 2001). [20] M. V. Kurik & O. D. Lavrentovich, Defects in liquid crystals: homotopy theory and experimental studies, Sov. Phys. Usp. 31, p196 (1988). [21] H. R. Trebin, The topology of non-uniform media in condensed matter physics. Advances in Physics 31(3), p195 (1982). [22] L. M. Michel, Symmetry defects and broken symmetry. Configurations hidden symmetry, Rev. Mod. Phys. 52(3), p617 (1980).

4. Experiment

This chapter describes the creation of the experimental system and method employed to generate umbilic defects in a nematic liquid crystal as well as measure the defect density.

4.1 Overview

A liquid crystal with negative dielectric anisotropy was placed under homeotropic boundary conditions in a “home-made” cell. The width of the cell was over three orders of magnitude wider than it was thick. The experimental system could therefore be considered to be a 2D infinite plane. A CMOS camera was used to capture the formation of umbilic defects produced by a voltage ramp. The density of defects after a short time delay was measured in a 0.243 mm2 area. By varying the voltage ramp i.e. the quench time, the scaling exponent for the following universal scaling law could be found and compared to the predicted value (훼 = 1/2):

훼 𝜌 ∝ 휏푄 (4.1)

where 𝜌 is the defect density, 휏푄 is the quench time and 훼 is the scaling exponent.

4.2 Materials used

4.2.1 ZLI-2806

The liquid crystal mixture used in this experiment, ZLI-2806, was obtained from Merck Group [1]. It has the following phase sequence on cooling:

퐼푠표 ↔ 푁 ↔ 푐푟푦푠푡 100.5℃ −20℃

The mixture has a negative dielectric anisotropy ∆휖 = −4.8 (푇 = 20℃, 푓 = 1kHz) [2]. This means that ZLI-2806 will produce umbilic defects during a Fréedericksz transition when contained in a cell (separation ~1-50µm) with homeotropic boundary conditions.

Figure 4.1 shows a typical umbilic defect texture seen 0.25 seconds after a Fréedericksz transition. Umbilic defects generated in ZLI-2806 have already been used to confirm the scaling behaviour of defect annihilation as well as the anisotropy between annihilation dynamics of umbilic defects with opposite strengths [2, 3]. It is advantageous to use umbilic defects when investigating defect formation because only 푠 = ±1 defects are produced and they can be easily identified without any image processing. The defect density is simply acquired by counting the number of defects in an area. In experiments where strings are generated, the defect density is calculated by measuring string lengths which can result in a significant error when measuring particularly high or low defect densities. Another advantage is that once generated an umbilic defect will not decay into two or more defects. This is because the core of the defect is an “escape in the third dimension” and therefore topologically stable as discussed in section 3.10.

50 µm

Figure 4.1. Umbilic defects generated in ZLI-2806 after a Fréedericksz transition. There is good contrast between the defects/ brushes and the rest of the texture.

4.2.2 4-butyl-N-[methoxy-benzylidene]-aniline (MBBA)

MBBA (figure 4.2) was also used to support the universality of the defect formation scaling law. It has a negative dielectric anisotropy ∆휖 = −0.56 (푇 = 25℃) [4]. Its phase sequence on cooling and structural formula are:

퐼푠표 ↔ 푁 ↔ 푐푟푦푠푡 ~43℃ 17℃

CH3O CH N C4 H9

Figure 4.2. A MBBA molecule.

4.3 Cell fabrication

Soda lime glass of thickness 1.1mm coated with a 20nm indium tin oxide (ITO) layer of resistivity between 80-100Ω/sq. was supplied by VisionTek Systems Ltd [5]. The glass was cut into plates approximately 15mm x 25mm. Each plate was then cleaned and spin coated with an alignment layer. Two glass plates were held together with UV curing glue containing 10µm spacer beads.

4.3.1 Glass cleaning

It was vital that any glass used was thoroughly cleaned so that no dust, glass fragments or contaminants of any kind were present in the cell. This is because contaminants could alter the properties of the liquid crystal in an unknown way. Any dust or glass fragments within the cell could provide a nucleation site where defects could form independently of the applied electric field.

After the glass plates had been cut, any glass fragments stuck to the plates were removed by washing each glass plate by hand in soap and water and drying with a paper towel. The ITO coated side of each glass plate was identified using a multimeter before being placed into a Teflon holder. The Teflon holder was then placed into a beaker containing a solution of ~ 1% Decon 90 in deionised water and placed in an ultrasonic bath for 20 minutes at 60°C. The Decon 90 solution was drained away and replaced with more deionised water before further sonication for 20 minutes at 60°C. This process was repeated once more. Afterwards, the Teflon holder was transferred to a beaker containing methanol, covered with aluminium foil and left overnight. Before the alignment layer was applied, each cell was dipped in deionised water to remove any residue from the solvent and dried using a high pressure stream of nitrogen.

4.3.2 The alignment layer

A liquid crystal is aligned homeotropically when the long axes of the molecules are perpendicular to the glass surface. Homeotropic alignment can be achieved using a layer of surfactant molecules. The surfactant molecules adhere to the glass so that it is energetically favourable for liquid crystal molecules to adsorb with their long axes perpendicular to the glass surface.

The alignment layer used was a solution of 0.5% cetyltrimethylammonium bromide (CTAB, figure 4.3) in deionised water. The solution was kept at 60°C whilst being stirred using a magnetic mixer for at least an hour to ensure the CTAB was completed dissolved. When CTAB is coated onto glass plates, its molecular head becomes anchored to the glass plates via polar interactions whilst the long hydrocarbon tail couples with nearby liquid crystal molecules. Homeotropic alignment is transmitted through the bulk of the liquid crystal by the liquid crystal molecules anchored perpendicular to the glass surfaces.

- Br CH3

N+

CH3 CH3

Figure 4.3. A cetyltrimethylammonium bromide (CTAB) molecule.

The alignment layer was spun onto the ITO side of each glass plate for a minute at 30 revolutions per second. The thickness of the layer was of the order of a micron. Afterwards, each glass plate was placed under a clean glass petri dish to minimise any dust landing on the surfaces whilst the remaining water evaporated from the glass plates at room temperature.

4.3.3 Device assembly

Small dots of UV curing glue containing 10 µm spacer beads were placed at equal intervals across both sides of one glass plate and then joined together to produce an equal distribution of glue across each side. The other glass plate was then carefully placed as shown in figure 4.4, leaving enough room on each side to attach wires with indium solder.

Glass UV glue with 10µm Liquid crystal spacer beads Wire CTAB alignment layer

ITO layer

Figure 4.4. A schematic showing how the liquid crystal cells used in this experiment were assembled.

Observing the interference patterns created by light reflecting from both glass surfaces provided a good way to check the separation of the two glass plates before curing the UV glue. One thick band across the glass surface would indicate a

52 constant separation between the glass plates. However, in practice more than one dark band was always observed. A number of parallel bands indicated that one of the axes of a glass plate was tilted with respect to the other whereas curved bands indicated that both axes were tilted. The interference patterns and therefore the separation of the glass plates could be manipulated by applying pressure with a pair of tweezers to one of the glass plates. The glass plates were manipulated until there were no more than four dark bands. Once acceptable interference patterns were obtained, the cell was placed under a UV light source to cure the glue. Only the areas that were glued were exposed to light by shielding most of the cell with a piece of thick black card. The cell was left to cure for 20 minutes before being flipped and left to cure for another 20 minutes.

4.3.4 Quality control

The cell gap across the cell, the quality of the homeotropic alignment, the transition temperature of the liquid crystal after filling and defect memory effects were all tested to confirm the quality of a cell. The cell gap was measured by considering the cell to be a Fabry-Pérot interferometer and analysing its reflectance spectrum [6]. The cell gap is given with an error of ±0.1 µm by the equation:

푛 휆1휆2 푑 = ( ) 2 휆2 − 휆1 (4.2)

where 푑 is the cell gap, 푛 is the number of peaks between two maxima 휆1 and 휆2. A computer program written by a previous member of the group automatically calculated the cell thickness given the wavelengths of two maxima and the number of peaks between them [6]. The cell gap was measured in the centre of the cell as well as each corner. Cells typically had a separation between 11 – 12 µm. Cells which had a cell gap variance of more than 0.4 µm were discarded.

Cells with acceptable cell gaps were filled with liquid crystal by capillary action. Cells

53 containing ZLI-2806 were then sealed with UV glue and cured with the liquid crystal shielded with thick black card. Cells containing MBBA were sealed differently because MBBA is very sensitive to light and would break down if exposed to UV light. Cells containing MBBA were instead sealed using Alradite epoxy resin, baked at 60oC for at least 4hrs to encourage the resin to cure to completion and then left overnight. Once sealed, the cells were placed under a polarising microscope to check the quality of the homeotropic alignment. It was noticed that there was no homeotropic alignment in cells containing ZLI-2806 until the cell was heated above ~65oC. At approximately this temperature, homeotropic alignment was suddenly adopted and remained even after cooling below this temperature. Figure 4.5a shows the texture seen at room temperature prior to any heating. Figure 4.5b shows the dark image seen after heating above the transition temperature and cooling to ~28oC. A similar effect was seen in cells containing MBBA before being sealed. For a short amount of time after filling, a purple texture with a similar structure to that in figure 4.5a was observed. Homeotropic alignment could then been seen sweeping across the liquid crystal until the entire field of view was dark. It is possible that homeotropic alignment was not initially fully adopted due to the nature of filling a cell by capillary action. It is known that liquid crystal molecules may adopt a tilt with respect to the anchoring direction when filled depending on the direction and speed of the flow [7]. Homeotropic alignment was verified in both types of cell by rotating the sample between crossed polarisers and observing no variance in light intensity.

50 µm 50 µm

Figure 4.5a,b. ZLI-2806 before and after homeotropic alignment was adopted after heating above ~65oC.

It is possible that the UV glue or Araldite used to seal the cell could contaminate the liquid crystal if not fully cured. A contaminated liquid crystal cell is most easily recognised by a reduction in the transition temperature of the liquid crystal. The transition temperature of the liquid crystal in a sealed home-made cell was compared to the transition temperature measured in an unsealed commercially made cell. Cells with a transition temperature within 0.1°C of the transition temperature measured in the commercial cell were considered acceptable.

The final quality check involved observing where defects were formed. The experiment required that any defects formed as a result of a Fréedericksz transition and not due to the construction of the cell. Defects repeatedly forming in the same places could be nucleating from inhomogeneities at the surfaces of the glass plates. Sequences of defect formation at different ramp rates were captured before and after holding the liquid crystal above its transition temperature for two hours. The cell was considered acceptable if defects formed in different positions after each ramp.

4.4 Experimental set up

This section will describe each component of the experiment and how they were calibrated. Figure 4.6 gives a visual overview of how each component was connected.

TTi TGA12101 Linkham TMS- IDS uEye GigE Computer 91 CMOS camera

Video capture: temperature IDS uEye Guide wave controller Cockpit

Video analysis: ImageJ v1.48 HP 3312A Liquid crystal Nikon Defect cell on Optiphot-pol counting: Linkham hot polarising UTHSCSA stage microscope Carrier wave ImageTool 3.0

Figure 4.6. A guide wave produced by a TTi TGA12101 signal generator is used to modulate a carrier wave produced by a HP 3312A signal generator. The modulated voltage ramp is then applied across a liquid crystal cell placed between crossed polarisers and held at a constant temperature. The resulting defect formation is captured using a CMOS camera and analysed on a computer using ImageJ. The defects are then counted using ImageTool.

4.4.1 Signal generators and amplitude modulation

Two signal generators were used in order to produce amplitude modulated (AM) voltage ramps. Amplitude modulation is achieved by using the amplitude of a guide wave to vary the amplitude of a much higher frequency carrier wave. In this experiment a sine wave (푓 = ~1 kHz, V푝푝 = 16 V) was produced by a HP 3312A signal generator and modulated using DC ramps that varied between ~0.4 V/s – 40 V/s. The voltage ramps were produced using TTi Waveform Manager Plus v4. In Waveform Manager, waveforms can be created by stating a mathematical expression for how the voltage should vary over time. The waveform would appear as a graph with arbitrary units for both voltage and time. Waveforms were uploaded to a TTi TGA12101 signal generator via a R232 serial connection. The frequency at which the waveforms were repeated by the signal generator can be calculated with the following equation:

푓푆퐺 (4.3) 푓 = 푊 푁

56 where 푓푊 is the frequency at which the waveform is repeated, 푓푆퐺 is the frequency set on the signal generator and 푁 is the length of the waveform in arbitrary units. Essentially, the frequency of the signal generator determined the rate at which the voltage at each arbitrary time unit was generated. For example, a waveform uploaded to the signal generator with a length of 1000 arbitrary time units with the frequency on the signal generator set to 1 kHz would be repeated at a frequency of 1 Hz. Figure 4.7 represents how the amplitude of the carrier wave typically varied over time during a voltage ramp. The ramp did not begin immediately. The reason for this is discussed later in section 4.7.3. This was achieved by using the sequencing function of the signal generator to generate a guide wave with no amplitude followed by the DC ramp guide wave. The sequencing function permitted up to 1024 waveforms to be generated in a pre-determined sequence.

Voltage

Time

Figure 4.7. A typical amplitude modulated wave used to generate umbilic defects without inducing electroconvection. The graph is not to scale.

The amplitude modulated ramp was used to prevent electroconvection which was seen during preliminary experiments which simply used a DC ramp. Electroconvection occurs when liquid crystal molecules and ion impurities follow a low frequency electric field creating a flow in the bulk of the liquid crystal. This flow could potentially affect defect formation and therefore needed to be prevented. Using a high frequency carrier wave prevented electroconvection because the electric field oscillated too quickly for the liquid crystals molecules and ion 57 impurities to follow. Another issue with applying DC fields is that ions may build up on one side of the liquid crystal and ultimately reduce electric field across the liquid crystal in an indeterminable way. The actual ramp rate experienced by the liquid crystal would therefore not be the ramp rate produced by the signal generator. Also, the difference between the two would increase as the liquid crystal was subjected to a DC field for more time.

4.4.2 Linkham hot stage and temperature controller

A Linkham hot stage with a platinum resistance thermometer connected to Linkham TMS-91 temperature controller was used to keep the liquid crystal at a particular temperature, precise to 0.1°C. A liquid crystal cell was clamped to the heating element in order maximise their contact. A screw on lid minimised any heat flow by convection.

4.4.3 Nikon Optiphot-pol polarising microscope

A diagram for the polarising microscope used is shown in figure 4.8. The microscope was set up as standard for transmitted light experiments. The hot stage containing the liquid crystal cell was placed on the stage in between two inbuilt polarisers. The 10x objective was used throughout.

Figure 4.8. A labelled diagram of the polarising microscope used in this work [8].

4.4.3 IDS uEye GigE camera

The camera was clamped to the vertical tube on the top of the microscope where 86% of the total light intensity was redirected from the observation tube. The field of view of the camera was smaller than that of the microscope. It was found to be 0.243 mm2 by focusing on a graticule with 0.01 µm spacing, capturing a high resolution image and measuring the pixel: micrometre ratio. The camera was attached to a computer via an Ethernet cable. Software developed by IDS (uEye Cockpit) was used to control the camera parameters as well as capture images and video [9].

The video captured for this experiment needed to have a high frame rate so as to not lose any defects annihilating between frames. A high frame rate can be achieved by increasing the pixel clock. The pixel clock determines the speed at which the sensor pixels in the CMOS can be read. However, as the pixel clock is increased so does the bandwidth requirement for data transmission. Exceeding the total bandwidth available results in some of the frames being dropped because they

59 cannot be processed fast enough. For this experiment it was critical that the number of dropped frames was kept to a minimum so that defect densities at equivalent times could be compared.

Two camera parameters were adjusted in order to maintain both the frame rate and bandwidth required at acceptable levels. Reducing the exposure time reduced the time over which the sensor pixels captured data allowing for a faster frame rate. However, reducing the exposure time also reduced the brightness of the video. A compromise was found between maintaining a high frame rate whilst still being able to identify defects easily. Reducing the resolution also reduced the bandwidth requirement. Subsampling was used to reduce the resolution but not the field of view of the camera. Subsampling works by skipping multiple sensor pixels when reading out image data as shown in figure 4.9. Subsampling by a factor 푛 means only each 푛푡ℎ pixel is read. Obviously, reducing the resolution in this way reduces the quality of the image. 4x subsampling both horizontally and vertically were found to provide a good compromise between frame rate and image quality.

Figure 4.9. A visual representation of 1x subsampling both horizontally and vertically. Every alternate pixel is read allowing for an increased frame rate whilst not restricting the field of view. However, the quality of the image will be reduced.

The uEye Cockpit software recorded videos as a sequence of JPEG format images and then combined them into a video in AVI format. Prior to recording, the quality setting of the JPEG encoder could be chosen. As a video was captured, the uEye software displayed the current frame rate and the total number of dropped frames. Many videos were recorded at different quality settings and frame rates in order to determine the highest stable frame rate possible. It was found that <1% of total

60 frames were dropped when recording ~1000 frames at 100 frames per second and encoding at 25% JPEG quality. Sequences recorded above 100 frames per second (up to 200 frames per second) with <1% dropped frames was possible but only by reducing the resolution below an acceptable level for easy defect identification.

4.5 Experimental parameters

In theory, a universal scaling may hold over many orders of magnitude. However, in practice, the range over which a universal scaling law holds will be limited by various experimental parameters. This section will discuss how various parameters, which affected the range of ramp rates over which scaling behaviour could be observed, were chosen.

4.5.1 Time delay before measurement of defect density

Section 3.7 explained how a Fréedericksz transition occurs once an electric field above a threshold voltage is applied across a liquid crystal. However, there is some time delay after the threshold voltage is reached before any defects formed become well defined. Therefore, any measurement of the defect density had to be made after some time delay. From here on “well defined” refers to defects that can be easily identified by eye. Figure 4.10 shows defect generation at 10 V/s over a period of ~1 second. Defects shown in 4.10e and 4.10f are considered to be well defined. It was decided that this time delay should be the same for each ramp rate so as to compare defect densities at equivalent times. This was to take into account any defect annihilation. After formation, defects will move towards each other and annihilate as discussed in section 3.10. The density of defects generated in ZLI-2806 has been shown to scale inversely with time (𝜌 ∝ 푡−1) [2]. As long as the time delay before defect density is measured is kept the same, the measured defect density will be the same fraction of the initial defect density at formation. Therefore, the measured defect densities would be expected to scale with the same scaling exponent as the initial defect densities.

50 µm 50 µm 50 µm

Figure 4.10a-f. Umbilic defect formation in ZLI-2806 during a Fréedericksz transition.

It was noticed that the time taken for defects to become well defined varied with ramp rate. The higher the ramp rate the shorter the time delay before defects became well defined. This placed a constraint on the range of ramp rates that could be tested. Shortening the delay time meant defects generated at lower ramp rates did not have sufficient time to become well defined so they could be measured. Waiting longer before measurement would allow defects to have annihilated to such low numbers that they are no longer accurately described by a scaling law. The next section will discuss how the range of ramp rates with an agreeable delay time was determined.

4.5.2 Range of ramp rates

In principle, the predicted scaling law that describes defect formation should hold for all ramp rates. Ideally, an experiment that attempts to confirm the scaling exponent would want to maximise the range over which scaling was observed. However, differences between the system for which the prediction was made and the experimental system places upper and lower limits on the scaling behaviour. For particularly low ramp rates, where a very low number of defects are generated, the defects observed in the field of view of the camera may not be representative of the defect density across the liquid crystal. Taken to its extreme, it is conceivable that

62 defects could be generated with separations larger than the width of the field of view of the camera and so no defects would be observed. An upper limit must exist due to the fact that only a finite amount of defects can be generated in a finite area as the ramp rate tends towards infinity. With these considerations in mind, it was decided to find the slowest ramp that would repeatedly produce ten defects and increase the ramp rate in equal intervals so that two orders in magnitude were tested in total. Then the shortest delay time before measurement of the defect density was chosen so that as many of the ramp rates were included as possible. It was considered acceptable to choose a delay time whereby defects were still not fully defined at lower ramp rates because the low number of defects generated meant that defects were easily identifiable. Overall, it was decided to conduct the experiment between 0.44 V/s - 24.41 V/s after a delay of 0.25 seconds.

4.5.3 Temperature

ZLI-2806 has a wide nematic phase temperature range (-20°C – 100.5°C) and so it was considered whether temperature would affect defect formation and/or any scaling behaviour. It was noticed that the time taken for defects to form decreased as temperature increased. As discussed in the previous section, if defects take longer to form then there must be a larger delay time before measurement of the defect density, reducing the range of ramp rates that can be compared. Therefore, temperatures close to the transition temperature would be the most suitable for confirming any scaling behaviour. It was also noticed that the density of defects was much larger at lower temperatures than at temperatures closer to the transition temperature. Therefore, it was expected that the upper limit on any scaling behaviour would be reached earlier at lower temperatures. The upper limit can be seen in figure 4.11 which shows defect formation after a delay of 0.85 seconds at room temperature (T-TNI = -77.8°C). The error bars represent a 5% error in counting defects (errors are discussed later in section 4.9). Again, it would be optimal to make measurements of defect density at temperatures close to the transition temperature in order to maximise the range over which scaling could be observed.

Figure 4.11. Defect density against ramp rate at T-TNI = -77.8°C (room temperature). An upper limit of defect formation due to the finite size of the liquid crystal is apparent after ramp rates above ~7 V/s.

Defect formation at room temperature did not scale with ramp rate as expected. A scaling exponent of 훼 = 1⁄4 was measured for ramp rates between ~1 – 7 V/s. The reason for this was unclear and so an experiment was conducted to determine whether defect formation depended on temperature. Defect formation during voltage ramps of 1.6 V/s and 114 V/s were recorded at eight different temperatures. The shortest possible delay time before measurement of the defect density was found to be 0.75 seconds after a 1.6 V/s ramp and 0.4 seconds after a 114 V/s ramp. The results are shown in figure 4.12 and clearly demonstrate a temperature dependence on defect formation between temperatures T-TNI = -75°C and -45°C. The defect density remains approximately constant at temperatures above T-TNI = -45°C suggesting little, if any, temperature dependence. It was therefore decided to use three temperatures in this region in future experiments.

Figure 4.12. The temperature dependence on defect formation seems to disappear at temperatures above T-TNI = -45°C.

4.6 Frame selection

This section will outline the method used to select the frame used to measure the defect density. The selected frame 퐹 can be calculated simply:

퐹 = 푅(푇 + 휏) (4.4) where 푅 is the frame rate, 푇 is time elapsed in a recorded video before a Fréedericksz transition occurs and 휏 is the time delay before the defect density is measured. Initially, during preliminary experiments, 푇 was found by calculating the time taken for a ramp to reach the threshold voltage measured using a photodiode clamped to the vertical tube on the top of the microscope. The output voltage of the photodiode was directly proportional to the intensity of the light entering the photodiode. An electric field with frequency 푓 = ~1 kHz was applied across a liquid crystal cell. The peak to peak voltage was increased in small increments until a difference in intensity was measured signalling the occurrence of a Fréedericksz

65 transition. However, after collecting some unexpected results it was noticed that the threshold voltage varied with ramp rate (figure 4.13).

Figure 4.13. The voltage at which a Fréedericksz transition increases as ramp rate is increased.

Figure 4.13 shows how as the ramp rate is increased, the threshold voltage approaches the threshold voltage measured with an electric field with frequency 푓 = ~1 kHz. The error bars represent the standard error of the mean calculated from three repeats of data. It is thought that variance in threshold voltage is due to the response time of the liquid crystal. The response time of a liquid crystal is the time taken for molecules to align with an electric field. Commercially used liquid crystals typically have a response time in the region ~1-20 ms. The larger the ramp rate the higher the voltage is by the time the liquid crystals molecules have re-aligned themselves and therefore a higher threshold voltage is measured.

The data used to calcuate the threshold voltage and produce figure 4.13 was not collected using the photodiode method outlined at the beginning of the section. It was decided instead to measure the threshold voltage directly from each video of defect formation so that no synchronisation between a device measuring the threshold voltage and the video recorded was required. The next section will explain

66 how the threshold voltage for each data run was obtained using ImageJ.

4.7 Finding the transition point with ImageJ

ImageJ is a Java based program used for image processing. It was developed and made freely available by the National Institute of Health [10]. In this work, imageJ v1.48 was used to determine the frame at which a Fréedericksz transition begins by comparing the mean pixel value with the background mean pixel value.

4.7.1 Mean pixel value

A video in AVI format was uploaded into ImageJ as a sequence of JPEG images. A pre-written plugin that comes as part of the basic installation of ImageJ was used to measure the mean pixel value of each JPEG image. Pixel value represents the “intensity” of a pixel. For a greyscale image, a pixel value is simply the brightness of a pixel whereby 0 usually denotes black and 255 denotes white. For a coloured image, each perceivable colour is defined by a vector in three-dimensional colourspace. One of these dimensions normally defines the brightness of a pixel and the others define the hue and saturation. Figure 4.14 shows how the mean pixel value varied over time during a Fréedericksz transition generated in ZLI-2806 during a voltage ramp. The plot is much like a plot of light intensity measured by a photodiode against time. Although, the resolution is limited by the frame rate of the camera but it is sufficient for determining the threshold voltage.

Figure 4.14. 1 - The average background mean pixel value can be calculated before the voltage ramp exceeds the threshold. 2 – The threshold voltage is exceeded causing a rapid increase in the mean pixel value. 3 – The mean pixel value continues to increase but at a lower rate as the defects become more defined. 4 – Defect annihilation reduces the number of dark brushes causing the mean pixel value to increase slightly. 5 – The voltage ramp ends so the liquid crystal quickly re-adopts homeotropic alignment and the mean pixel value returns to the background level.

4.7.2 Validating the imageJ method

An experiment was used to test the validity of using imageJ in this way before calcualting threshold voltages. Five 40 ms square voltage pulses were applied across a sample of ZLI-2806 one second apart from each other whilst recording a video and ensuring that no frames were dropped. The frame rate was 100.54 frames per second. Figure 4.15 shows the resulting intensity-time plot with the time at the peak of each intensity pulse labelled. Each peak is shown to be exactly one second apart precise to the nearest 0.01 seconds – the time resolution of the camera.

Figure 4.15. Voltage pulses applied one second apart from each other can be seen in a plot of mean pixel value against time.

4.7.3 Measuring the threshold voltage

In order to obtain a value for the threshold voltage, the frame number at the start of the ramp and the start of the transition needed to be measured. The time difference between these two points could then be calculated and therefore the voltage at the point of transition. Figure 4.16 shows the waveform that was used to modulate a sine carrier wave (푓 = ~1 kHz, V푝푝 = 16 V). No voltage was applied for two seconds before a 40 ms square pulse with a voltage of 16 V was applied. After the pulse, another two seconds with no applied voltage was followed by a voltage ramp which continued until it reached 8 V.

Voltage

Time

Figure 4.16. The guide wave used to modulate a sine wave. Time periods with no applied voltage sit between the voltage pulse and voltage ramp in order to calculate the background mean pixel value.

The frame number of the start of the ramp and the start of the transition were measured relative to the peak of intensity produced by applying the square pulse. This meant that that there was no need to synchronise the camera and its software with the signal generator. The time where no voltage was applied before the ramp started was used to measure the mean pixel value when no field is applied, that is, the background pixel intensity. The transition point could then be said to occur when the mean pixel value of a frame exceeded the average background pixel value by at least five standard deviations. The threshold voltage could then be calculated by finding how many frames were recorded between the start of the ramp and the transition point, converting this into the time elapsed and working out what the voltage must have been from the ramp rate. After confirming that the threshold voltage does indeed increase with ramp rate, data was captured using a waveform without the pulse (figure 4.17). This was because the frame selected to measure the defect density could be found without knowing exactly when the voltage ramp began.

Voltage

Time

Figure 4.17. The guide wave used for the experiment did not require a voltage pulse.

4.8 Defect counting

This section will explain how defects were counted. It begins with a brief justification of the decision to count defects by eye as opposed to using an algorithm. It then continues to describe how defects were counted and the methods used to minimise errors.

4.8.1 Algorithm or counting by eye?

Previous experiments using liquid crystals to test scaling exponents have either used an algorithm to count defects or counted them by eye. The obvious advantage of using an algorithm is that it the counting process is repeatable and not vulnerable to human error. The disadvantage is that producing a reliable algorithm may not be possible. Performing some kind of image processing can make it easier for an algorithm to work but care has to be taken not to lose any important features in the process.

An algorithm could have been used to identify defects based on their surrounding brushes. Another approach could have been to look at the space between defects

71 and infer where defects must be. However, it was felt that the low resolution of the videos would make it hard to distinguish defects in certain situations. For example, some of the brushes that surrounded defects were formed close to each other and looked as though a defect sat between them when in fact there wasn’t (figure 4.18a). Two defects could be generated very close together and with short brushes surrounding them on one side which made them look as though they were just the one defect (figure 4.18b). Also, some defects were close to annihilating and appeared as a defect with a large core and with only two brushes (figure 4.18c). Another issue was that at low ramp rates the defects were not fully defined by the time measurement of the defect density needed to be made (figure 4.18d).

Figure 4.18a. Two brushes near each other look Figure 4.18b. Two defects about to annihilate like an umbilic defect. Subsequent frames look as though they are one defect. However, suggest it is not a defect because a single frames beforehand reveal that there are umbilic defect cannot annihilate without actually two defects on the edge of annihilation. meeting another defect with opposite strength.

Figure 4.18c. What looks like a single defect with Figure 4.18d. The defects are yet to become two brushes can be shown to be two defects well defined. Looking at the same defects two just before they annihilate. In this situation both frames later assists in their identification. defects are counted.

Counting by eye could overcome all the issues discussed so far. In addition to simply counting defects in the selected frame, defects could be identified by observing the

72 dynamics of the defects prior to the selected frame as well as just after. For example, it could be determined whether brushes originated from a defect or if they were just two brushes formed close to each other. Defects just about to annihilate could be seen moving towards each other in the frames beforehand. Less well- defined defects could be identified by their dynamics, their interaction with other defects and following their evolution as they become increasingly well defined. However, counting defects by eye is liable to human error, the next section describes the steps that were taken to minimise such errors.

4.8.2 Method for counting defects

ImageTool 3.0 developed by the University of Texas Health Science Center was used to keep count of the number defects in each image [11]. After loading an image, defects could be counted by clicking on them, which would place a small red dot over the defect and return the total number of clicks so far. This approach prevented any defects from being counted twice as well as any errors calculating the total number of defects. In addition, the following rules were devised in order to be as consistent in counting defects as possible:

 Defects were counted in the same way each time (five sweeps at different heights from left to right).  When unsure if a defect exists, the frames just before and after the selected frame were cycled through to observe the defects dynamics.  After the initial counting process a second check was performed to look for any missed defects.  Any interruptions during the counting process meant it had to be restarted.  Two defects about to annihilate are both counted as defects.  At least two brushes must been observed coming from the core of a defect at the edge of the field of view in order for it to be counted.

4.9 Errors

Sources of error in the experimental method concern the defect counting process and the selection of the frame used to measure the defect density. The error associated with counting defects was estimated by asking five other members of the group to individually measure the total number of defects in the same image. Beforehand, they were informed of what constituted a defect and the rules used for counting set out in the previous section. Their totals were averaged and the percentage standard deviation calculated to be 4.2% suggesting that an error of 5% was a reasonable estimate. The number of dropped frames was the main contributor to the error associated with frame selection. As mentioned previously, any recordings where more than 1% of the total frames were dropped were discarded and the data run repeated. The number of defects over time was recorded after two voltage ramps with rates 0.44 V/s and 24.4 V/s respectively to attempt to quantify the error associated with dropped frames. It was found that during the 2.5 seconds after the slow voltage ramp that defects annihilated at an average rate of 0.08 defects per frame. The largest number of dropped frames for such a ramp was ten which meant that, at most, only one defect was not counted due to dropped frames. During the 0.25 seconds after the faster ramp, defects annihilated at an average rate of 2.5 defects per frame. For this ramp rate, a maximum of one frame was dropped. In both cases the total possible error was less than the error associated with counting defects. It should also be highlighted that the majority of dropped frames accrued immediately after the camera began recording and much before the transition point where the delay time was measured from. Therefore, in the majority of data runs the error associated with dropped frames would have been considerably less than the error associated with counting defects and quite often non-existent. It was therefore decided to not include this error in the results.

References

[1] Merck Group [http://www.merckgroup.com/en/index.html: Accessed 28th July 2014].

[2] I. Dierking, O. Marshall, J. Wright & N. Bulleid, Annihilation dynamics of umbilical defects in nematic liquid crystals under applied electric fields, Physical Review E 71, p061709 (2005). [3] I. Dierking, M. Ravnik, E. Lark, J. Healey, G. P. Alexander & J. M. Yeomans, Anisotropy in the annihilation dynamics of umbilic defects in nematic liquid crystals, Physical Review E 85, p021703 (2012). [4] E. J. Sinclair & E. F. Carr, Flow patterns and molecular alignment in a nematic liquid crystal due to electric fields, Mol. Cryst. Liq. Cryst. 37(1), p303 (1978). [5] VisionTek System Ltd. [http://www.visionteksystems.co.uk: Accessed 8th July 2014]. [6] H. G. Yoon, Chiral liquid crystal studies: Fitting theroectical models to optical data, unpublished thesis, The University of Manchester (2008). [7] X-D Mi & D-K Yang, Capillary filling of nematic liquid crystals, Phys. Rev. E 58, p1992 (1998). [8] Nikon, Polarizing Microscope OPTIPHOT-POL Instructions [http://earth2geologists.net/Microscopes/documents/Nikon_Optiphot_Pol_instruct ions.pdf: Accessed 8th July 2014]. [9] IDS, uEye Cockpit [http://en.ids-imaging.com/ueye-cockpit.html: Accessed 8th July 2014]. [10] National Institute of Health, imageJ [http://imagej.nih.gov/ij/: Accessed 10th July 2014]. [11] University of Texas Health Science Center, ImageTool 3.0 [http://compdent.uthscsa.edu/dig/itdesc.html: Accessed 10th July 2014].

Chapter 5: Results and discussion

This chapter presents the results of how defect formation scales with ramp rate in both ZLI-2806 and MBBA. Measured values of the scaling exponent are compared with the predicted value of 훼 = −1/2. Reasons for any deviation from the predicted value are then discussed.

5.1 Defect formation in ZLI-2806

Ramp rates with magnitudes spaced equally between 0.44 V/s - 24.41 V/s were used to generate umbilic defects in ZLI-2806 via a Fréedericksz transition. The density of the defects generated was measured after a delay of 0.25 seconds from the beginning of the transition. Scaling behaviour was expected to take the form:

훼 𝜌 ∝ 휏푄 (5.1)

where 𝜌 is the defect density, 휏푄is the quench time and 훼 is a scaling exponent [1]. The ramp rate 푅 is inversely proportional to the quench time so that:

𝜌 ∝ 푅−훼. (5.2)

Taking logs on both sides results in an equation whereby the scaling exponent can be determined from the gradient of a log-log plot of the defect density against ramp rate:

푙표푔(𝜌) = −훼푙표푔(푅) + 푐 (5.3)

where −훼 is the gradient of the log-log is plot and a constant 푐 is the intercept.

The results for defect formation at three different temperatures are shown in figure

5.1. The first data point in the results at temperatures 푇 − 푇푁퐼 = -19.5°C and - 29.5°C were not included in the results after consistent difficulty identifying defects. Five data runs were performed for each temperature before performing another ten data runs. Each data point constitutes the average defect density over the fifteen runs. The standard error (𝜎⁄√푁, where 𝜎 is the standard deviation and 푁 is the number of number of repeats) was calculated for each data point and found to be smaller than the symbols used to plot the points suggesting that statistical errors were successfully minimised. The error bars represent the 5% error associated with counting defects discussed in the previous chapter.

The dotted line on each graph represents the expected scaling (훼 = −1/2). The dotted line was positioned so that it sat between the error bars of as many points as possible. A linear fit was then applied to those points so that the scaling exponent could be determined. The resulting fit is the green line in each of the graphs. The first point in the 푇 − 푇푁퐼 = -19.5°C results was not included in the fit because it deviated significantly from the expected scaling. The reason for the deviation is not clear. The standard deviation of the fifteen runs for that point is similar to the standard deviation of each other point suggesting that it is not a statistical anomaly. Furthermore, the lower than expected defect density remained even after performing runs at different temperatures and ramp rates. Scaling deviated from that expected at higher ramp rates with a scaling exponent of approximately 훼 = −1/3. A linear fit of these points is given by the red line.

Figure 5.1. Defect density against ramp rate at three different temperatures. As temperature decreases, the order of magnitude over which the expected scaling behaviour is observed also decreases. The expected scaling behaviour deviates from that expected at approximately the same

defect density each time. The deviation itself also scales but with an exponent 훼 = −1/3.

The scaling exponents measured at each temperature are summarised in figure 5.2. The experimental data is in good agreement with the predicted scaling behaviour. However, the range of voltage ramps over which the predicted scaling was observed was only one order of magnitude for 푇 − 푇푁퐼 = -9.5°C and much less for the other two temperatures. The next section will discuss the reason for the deviation from the expected scaling behaviour in hope of extending the order of magnitude over which scaling can be said to occur.

Figure 5.2. A summary of the scaling exponents measured at each temperature. Defects are

−1/2 generated according to the scaling law 𝜌 ∝ 휏푄 as proposed by Zurek [1].

5.1.1 Deviation from expected scaling at higher ramp rates

A deviation from the expected scaling behaviour was anticipated at higher ramp rates as discussed in section 4.4.2. However, the defect density has not plateaued having reached an upper limit due to the finite size of the area in which the defects are generated. Comparing the three results, it seems that the scaling deviates from that expected at approximately the same defect density each time. It was noticed that there was very little, if any, defect annihilation by the time measurement of the 79 defect density was made during ramps before the deviation. Therefore, it could be that the deviation exists because defects were generated at such a density that their separation was small enough for a significant proportion to have annihilated before measurement of the defect density was made.

Umbilic defects generated in ZLI-2806 have been shown to annihilate inversely with time according to the scaling law [2]:

𝜌 ∝ 푡−1. (5.4)

Provided that the scaling law holds, the same percentage of defects should annihilate in the time before the defect density is measured. However, this scaling law was confirmed after generating defects in ZLI-2806 with a high frequency square wave electric field and defects were only counted after one second from when they were generated. Conversely, in this work, defects were generated using an amplitude modulated ramp electric field and defects were counted 0.25 seconds after generation. It is therefore possible that defects do not scale according to Eq. 5.4. To ascertain whether this was the case, the density of defects was measured over time for three different ramp rates. The results shown in figure 5.3 constitute the average of three data runs. The results show that defect density does not scale according to Eq. 5.4 but with an exponent of approximately -1/3. The dotted line only includes defects counted after one second. The exponent of the dotted line agrees with previous experiments that only counted defects after at least one second from when they were generated [2, 3]. It is not known why the defect density doesn’t scale with time according to Eq. 5.4 until after some time. It could be that both formation and annihilation processes are present at first before formation ceases and leaves only annihilation.

Figure 5.3. Defects initially annihilate according to 𝜌 ∝ 푡−1/3. After one second, the defect density seems to scale according to 𝜌 ∝ 푡−1 in agreement with previous experiments using liquid crystals.

Because defects annihilate according to the same scaling law in the time before the defect density is measured, the same proportion of defects will have annihilated each time. However, this would only be the case for defects generated at higher densities because defects generated at low densities would not have had time to begin annihilating. On a log-log plot of defect density against ramp rate, such annihilation would be represented by a downwards translation of the measured defect density. To illustrate, figure 5.4 shows how the defect density would scale if at some ramp rate defects were generated with sufficient density so that a third of the defects annihilated in the time before measurement. There is a sharp drop in defect density followed by all subsequent points scaling as beforehand. However, this is not what is seen in the experiment results. Although, the experimental results constitute an average of fifteen runs and therefore any sharp drops that could exist in individual runs could have been “smoothed out” in the averaging process. Sharp drops in defect density can be seen in roughly half (47%) of all experimental runs when viewed individually. In some of these runs, data points following a sharp drop in defect density can be seen to scale as expected. Figure 5.5 includes two examples

81 of data runs performed at T − TNI = -9.5°C where such behaviour was observed. It is possible that this behaviour is not seen in all runs because of the statistical variation in the number of defects generated each time.

Figure 5.4. At a particular defect density a third of the total number of defects have annihilated according the scaling law 𝜌 ∝ 푡−1. At this point, there is a sharp drop in defect density followed by all subsequent points scaling as before.

Figure 5.5. Some individual runs show the sharp drop and subsequent points scaling with the same exponent as the preceding points as described in figure 5.3. This behaviour is likely not seen in all data runs due to the statistical variation in the number of defects generated. However, such statistical variation also means it is difficult to conclude anything meaningful from an individual run.

The effect of the averaging the data was investigated by modelling fifteen data runs. The first three points in each run of the model were chosen so that they always scaled with an exponent 훼 = 1/2. The point at which the sharp drop occurred in the subsequent ten points was determined at random for each run. The results are shown in figure 5.6. There is no sharp drop in the defect density as ramp rate is increased. Scaling exponents are measured in the same way as they were for the experimental data and agree with those measured experimentally. This supports the idea that the deviation from the expected scaling behaviour is a result of significant defect annihilation occurring at different ramp rates in the time before the defect density is measured.

Figure 5.6. Fifteen runs were generated so that the first three points always scaled as expected. Then at a randomly determined point, one third of the defects annihilated. The above plot shows the result of averaging the fifteen runs. The scaling exponents match what is measured experimentally.

In order to acquire experimental confirmation, the seven and eight data points that made up the deviation in the results for T − TNI = -9.5°C and -19.5°C respectively were re-analysed with a shorter delay time of 0.1 seconds. If the delay time was the main reason for a deviation then it was expected that after a shorter delay time

84 these points would initially scale with an exponent α = -1/2 before deviating at approximately the same ramp rate. The results are shown in figure 5.7 and agree with the predicted behaviour.

Figure 5.7. Re-analysing the data at a shorter delay time has extended the order of magnitude over which defect formation in ZLI-2806 can be described by the scaling law proposed by Zurek. The defect density initially scales with the expected exponent 훼 = -1/2 before deviating at approximately the same defect density. It is expected that scaling at even higher ramp rates could be demonstrated if the data was re-analysed after an even shorter delay time until the upper limit on defect formation (due to the finite size of the system) was obtained. 85

5.1.3 Earlier deviation from expected scaling at lower temperatures

Comparing the three results given in figure 5.1, it seems that the lower the temperature of the liquid crystal, the higher the density of defects generated and therefore the earlier the scaling deviates from the expected scaling exponent. This seems to contradict the results acquired when considering the temperature range over which to perform the experiment (see section 4.5.3). Previously, it was suggested that the number of defects generated would level out as the temperature of the liquid crystal approached the transition temperature. All three temperatures used for the experiment were in the range where the defect density had levelled out and so it was expected that each ramp rate would generate similar number of defects regardless of temperature. However, only one data run was performed to produce the defect density against temperature plot which was likely not sufficient to observe a temperature dependence on defect formation at temperatures close to the transition temperature. The higher number of defects formed at lower temperatures could be due to the viscosity of the liquid crystal increasing as temperature is reduced. Correlations between molecules in a more viscous liquid form at a lower rate perhaps resulting in a lower equilibrium correlation length which would in turn result in a higher number of defects frozen out between after each correlation length.

5.2 Defect formation in MBBA

The experiment was repeated with MBBA in order to demonstrate the universality of the defect formation scaling law. The contrast between the defects generated in MBBA and the rest of the texture was less good than in ZLI-2806. In order to ensure defects were easily identified the resolution of the camera was increased to 800x600, the quality setting of the JPEG encoder was raised to 50% and the gain was increased to 10x. To account for the increase in image quality the frame rate was reduced to 50 frames per second. The lower frame rate was not considered an issue because less defects were generated in MBBA compared to in ZLI-2806 at similar ramp rates and so defect annihilation occurred over a longer timescale.

The transition temperature of MBBA is ~43°C, much lower than the transition temperature of ZLI-2806. As a result, defects generated in MBBA took much longer to become well-defined than defects generated in ZLI-2806 at temperatures close to its transition temperature. Therefore, a larger delay time before measurement of the defect density was required which in turn reduced the range of ramp rates which could be compared.

Figure 5.8 shows the results of defect formation in MBBA at three different temperatures after a delay time of 1.25 seconds from the beginning of the Fréedericksz transition. Each data point constitutes an average of five data runs. The results for T − TNI = -2°C are comparable to those acquired for ZLI-2806. At first, the data scales with an exponent 훼 = -1/2 before deviating with an exponent of 훼 = - 1/3. Again, some individual runs show a clear drop in defect density with subsequent points scaling with an exponent of approximately 훼 = -1/2. The drop in defect density has remained in the averaged data but the data only constitutes five repeats so will not have been completely “smoothed out”.

The results for T − TNI = -6°C and -9.4°C scale as expected before the defect density plateaus. The plateau in defect density is likely a result of the large delay time before the measurement of defect density and relatively low numbers of defect generated. With such a large delay time, defects have sufficient time to annihilate to a point where any remaining defects are well separated and would require much more time to annihilate further. As with ZLI-2806, as the temperature is reduced the number of defects generated increases and the deviation occurs at a lower ramp rate.

Whilst the order of magnitude over which scaling can be observed in MBBA is less that in ZLI-2806, it is believed that these results provide the first step in confirming the universality of the scaling law which describes defect formation.

Figure 5.8. Defect density against ramp rate for MBBA at three different temperatures. 88

References

[1] W.H Zurek, Cosmic strings in laboratory superfluids and the topological remnants of other phase transitions, Acta Physica Polonica B 24(7), p1301 (1993). [2] I. Dierking, O. Marshall, J. Wright & N. Bulleid, Annihilation dynamics of umbilical defects in nematic liquid crystals under applied electric fields, Physical Review E 71, p061709 (2005). [3] I. Chuang, N. Turok & B. Yurke, late-time coarsening dynamics in a nematic liquid crystals, Phys. Rev. Lett. 66(19), p2472 (1991).

Chapter 6: Conclusion and future work

The aim of this work was to provide experimental confirmation of a scaling law which describes how defect formation scales with quench time in a condensed matter system. A scaling exponent was measured in two liquid crystal systems and found to be in good agreement with the theoretically predicted value. To our knowledge, this is possibly the first time this has been done in a condensed matter system. These results support the Kibble-Zurek scenario for defect formation in condensed matter systems and perhaps the early universe.

This work has made some progress in validating the universality of the scaling law by comparing defect formation over a range of temperatures as well as using two different liquid crystals. Interestingly, defects generated in ZLI-2806 at room temperature did not seem to scale as expected whereas those generated in MBBA at room temperature did. Although, it should be made clear that the data used to calculate the scaling exponent for ZLI-2806 at room temperature consisted of only one data run and is therefore not necessarily a reliable result. Future work could involve looking at defect formation in liquid crystals at low temperatures because there is no theoretical reason, as far as we know, for defect formation at low temperatures not to scale as expected. The universality of the scaling law could be further investigated by varying cell thickness.

The annihilation of defects over time was investigated in order to justify counting defects a short time after they were generated. Unexpectedly, it was found that defects did not initially annihilate according to the theoretically predicted and experimentally confirmed scaling law 𝜌 ∝ 푡−1. Defect density was found to scale with time with an exponent of -1/3 in the first second after defect generation before scaling with the expected exponent of -1 from one second onwards. Future work is needed to explain this result.