Defects in Two Dimensional Colloidal Crystals

by Lichao Yu

B.Sc., University of Science and Technology of China; Hefei, China, 2010 M.Sc., Brown University; Providence, RI, 2013

A dissertation submitted in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy in Department of Physics at Brown University

PROVIDENCE, RHODE ISLAND

May 2015 c Copyright 2015 by Lichao Yu This dissertation by Lichao Yu is accepted in its present form by Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy.

Date

Xinsheng Sean Ling, Ph.D., Advisor

Recommended to the Graduate Council

Date

John Michael Kosterlitz, Ph.D., Reader

Date

Robert Pelcovits, Ph.D., Reader

Approved by the Graduate Council

Date

Peter M. Weber, Dean of the Graduate School

iii Vitae

Lichao Yu was born in the city of Nanchang, Jiangxi, P. R. China, on July 12th 1990. He obtained his Bachelor of Science from University of Science and Technology of China in 2010, and a Master’s degree in Physics from Brown University in 2013.

iv Acknowledgements

Time ﬂies.

Five years went really fast. I still have a vivid recollection of the day I stepped into the campus of Brown. It’s sunny, warm breeze carasses my face. At that moment, I know I would, with no regret, spend the best ﬁve years of my life, here at Brown, dedicating to Physics research, the great journey of exploration. This is meant to be a wonderful experience throughout my life.

First of all, I would like to give my thanks to my advisor Professor Xinsheng Sean Ling for his professional advice and guidance on research. His patience, conscien- tiousness and dedication, has made huge impact on my development as a physicist. Besides that, he also serves as a mentor providing career advices and mental sup- ports, and a friend to share with, happiness or sadness, successes or regrets.

Second, I thank Professor John Michael Kosterlitz and Professor Robert Pelcovits for taking interests in my research and reading my Phd dissertation. They happily became my defense committees and provided valuable advices.

Then I thank my lab mates Sungcheol Kim, who guided and taught me experimental and programming techniques, Wang Miao, Helen Hanson, Xi Wang, visiting scholars and interns: Hongwen Wu, Liping Liu, Yin Di, Stephanie Huang, etc. I also wish to mention all my friends, colleagues, staﬀs and faculty who oﬀered me help

v during the past ﬁve years. I am grateful to everyone for being part of my graduate school memory.

I would also thank my family: my parents, my wife and our newborn boy Max. They support me during the dark ages and share with me all the happy moments.

Last but not least, I acknowledge the ﬁnancial support from Brown University, the Physics Department, the National Science Foundation, etc.

Thank you.

vi Abstract of “ Defects in Two Dimensional Colloidal Crystals ” by Lichao Yu, Ph.D., Brown University, May 2015

We use digital video microscopy to study the defects in two-dimensional colloidal crystals (2DCC). A crystalline solid, diﬀerent from its liquid state, preserves long- wavelength shear rigidity and broken symmetry. Questions about how shear rigidity and long-range order disappear during melting, are unresolved in terms of the com- plication of defect structures and their roles in crystal melting, especially in two dimension. Colloidal crystals (CC) serve as a promising model system to directly observe the defects under optical microscope.

In our first study, we report the effects of vacancies and interstitials on the phonon modes in a 2DCC. By applying the equipartition theorem, we extract the dispersion relation of the lattice vibrations using real-time video microscopy. We find that both longitudinal and transverse modes in the spectrum are softened by the existence of point defects.

Second, we investigate the diﬀusion process of interstitials in a 2DCC. The motion is viewed as gliding of both edge dislocations along one of the crystalline axes. The microscopic process is equivalently a point mass overcoming Peierls barrier with an exponential escaping time. We also establish a new criterion to determine the ergodicity of a defect system and discover the nonergodic behavior of di-interstitials. Contents

Vitae iv

Acknowledgments v

1 Introduction 1

2 Theory 4 2.1 Two-dimensional Melting ...... 5 2.1.1 Introduction ...... 5 2.1.2 Kosterlitz-Thouless conjecture ...... 9 2.1.3 KTHNY theory ...... 12 2.2 Lattice Dynamics in 2DCC ...... 17 2.2.1 1D atom chain ...... 17 2.2.2 Lattice dynamics ...... 18 2.2.3 Dynamical matrix and normal modes ...... 20

3 Experimental Techniques 22 3.1 Sample Cell Preparation ...... 23 3.2 Suspension Preparation and Cleaning ...... 25 3.3 Imaging with Optical Microscope ...... 30 3.4 Image Processing and Particle Tracking ...... 34

4 Eﬀect of Point Defect on Phonon Modes 38 4.1 Introduction ...... 39 4.2 Method ...... 41 4.3 Results ...... 46 4.4 3D Visualization of Phonon Bands ...... 51 4.5 Conclusion and Future Directions ...... 52

5 Microscopic Process and Nonergodicity in Defect Diﬀusion 56 5.1 Introduction ...... 57

vii 5.2 Method ...... 57 5.3 Results ...... 59 5.4 Peierls Barrier ...... 61 5.5 Nonergodicity ...... 64 5.6 Monte Carlo Simulation ...... 66 5.7 Conclusion and Future Directions ...... 67

A Kosterlitz-Thouless Transition 69 A.1 Model System ...... 70 A.2 Scaling Equations ...... 73 A.3 Real System ...... 77 A.3.1 X-Y Model ...... 77 A.3.2 2D solids ...... 78 A.4 Asymptotic Solutions ...... 79 A.5 Correlation Length ...... 86

B Diﬀusion 88 B.1 Random Walk to Diﬀusion ...... 89 B.2 Kramer’s Escape Rate ...... 91

C IDL Scripts and C++ Code 93 C.1 IDL Scripts ...... 94 C.2 IDL Source Code for Particle Tracking and Phonon Mode Analysis . 96 C.3 C++ Code for Monte Carlo Simulation ...... 121

viii List of Tables

ix List of Figures

2.1 Summary of KTHNY theory. Note the vanishing behavior of stiffness KR(T ) and Frank constant KA(T ) is not the exact solution...... 15 2.2 1D atom chain model. Atoms are interacting via ideal springs. . . . 18 3.1 Sample cell for 2DCC experiment (not to scale). Side-view (top): shaded area is the optical adhesive which bonds both substrates. In silica disk (top substrate), a channel is designed for the flow. A less than µm uniform gap between the plateau and coverslip provides space for 2DCC. Top-view (below): gray area shows the flow pattern of the colloid suspension. A very small amount of the solution travels through the plateau (inside center circle)...... 24 3.2 Cross-flow diafiltration setup (not to scale). The colloidal suspension is stored in a sealed Telfon container. The circuit is connected with Galtek Integral Ferrule fittings. The suspension is pumped out by a peristaltic pump (Fisher Scientific Variable-Flow Peristaltic Pump), providing smooth feed into the cartridge filter. The cartridge used is MidGee Microfiltration Hollow Fiber Cartridges together with Poly- sulfone membrane with 0.2 micron pore size (CFP-2-E-MM01A). A conductivity meter (Amber Scientific Model 1056 Digital Conductiv- ity Meter) is used to monitor the ionic strength. DI-water buffer solution is stored in a squeeze bottle and transferred into the reservoir continuously...... 27 3.3 Cross-flow diafiltration performance. Conductivity is measured with Amber Scientific Model 1056 Digital Conductivity Meter. Red represents the readings before dilution (refill) and blue for readings after dilution...... 28 3.4 Ion-exchange deionization setup to prepare 2DCC. The colloidal suspension is stored in a sealed Telfon container. The circuit is connected with Galtek Integral Ferrule fittings. The suspension is pumped out by a peristaltic pump (Fisher Scientific Variable-Flow Peristaltic Pump), providing smooth feed into the cartridge filter. Mixed-bed ion-exchange resin beads (Fisher Chemical Rexyn I-300 (H-OH) Beads) are stored in the ion exchange column. A flow-through conductivity meter (Am- ber Scientific Conductivity Micro Flow Cells) is used to monitor the ionic strength. Sample cell is mounted above the oil-immerse objective lens...... 29

x 3.5 Colloidal suspension under optical microscope. Top: near the circular channel. Bottom: far from the channel...... 31 3.6 Side view of sample cell showing the maximum spacing between substrates to achieve one -ayer. Solid circle represents colloid surface. Dash√ circle is the Debye layer (Electron Double Layer). D ∼ 2λD + d + 3a/2 ' 2 µm...... 32 3.7 Two-layer colloidal crystal ...... 33 3.8 Mono-layer colloidal crystal with different volume fractions and thickness...... 35 3.9 Histogram of lattice spacings and angles with equilibrium positions. Top: histogram over 200 frames. Bottom: fitting with standard Gaus- sian distribution...... 36 3.10 Fast Fourier Transform (FFT) of a cropped snapshot of 2DCC. . . . 37 4.1 Optical micrograph and delaunay triangulation diagrams of a two dimensional colloidal crystal. (a) (b) One microscope snapshot and its delaunay triangulation diagram: defect-free lattice. The inset shows ~ ~ primitive vectors (~a1, ~a2), its reciprocal lattice vectors (b1, b2), and the first Brillouin zone (not to scale) of perfect triangular lattice with standard labels of the symmetry points. (c)-(f) Lattice with point defects: mono-vacancy, mono-interstitial, both mono-vacancy and interstitial and lattice with tri-vacancy. Red and blue dots stand for 5 and 7-coordinate disclinations...... 42 4.2 Hexagonal lattice of 2DCC with equilibrium positions. The density ~ function ~r(t) is a superposition of delta functions on n sites Rn. We can measure the real-time displacement ~u(Rn) as the difference between real time position (green dots) and the equilibrium position (blue dots)...... 44 4.3 Reciprocal space of 2DCC. Two primitive vectors (~a1, ~a2), its recipro- ~ ~ cal lattice vectors (b1, b2), and the first Brillouin zone (not to scale) of perfect triangular lattice with standard labels of the symmetry points. 45 4.4 Band structure of harmonic lattice constants of 2DCC without defect. The plotting is performed from the center of the first Brillouin zone to the zone boundary (blue dash lines) in two directions. The upper curve (green) corresponds to the longitudinal modes, while lower one (red) to transverse modes. Spring constants λs(~q) are in the units of 5.80 × 10−7kg/s2...... 46 4.5 Band structure of harmonic lattice constants of 2DCC: defect-free (dots), in the presence of mono-vacancy (crosses), mono-interstitial (squares) and both point defects (blue diamonds). The inset shows the details of the ”mono-interstitial” and ”both” cases. The plotting is performed from the center of the first Brillouin zone to the zone boundary (blue dash lines) in two directions. In each individual plot, upper curve (green) corresponds to the longitudinal modes, while lower one (red) to transverse modes. For clarity, ”both” are denoted −8 2 by blue colors. Spring constants λs(~q) are in the units of 10 kg/s . . 47 4.6 Softening ratio with mono-vacancy and mono-interstitial. Green dots stand for longitudinal modes and red crosses for transverse modes. Softening ratio is defined as the ratio between λs(~q)(s = l, t). . . . . 48

xi 4.7 Band structure of harmonic lattice constants of 2D colloidal crystals: defect-free (dots), with tri-vacancy (crosses). Upper curves (green) correspond to longitudinal modes and lower ones (red) to transverse modes. The inset shows the softening ratio. Green dots stand for longitudinal modes and red crosses for transverse modes. Calculation and notations follow the same convention in Fig. 4.2...... 50 4.8 3D band structure (longitudinal mode) of 2DCC. Colormap shows the strength of harmonic spring constants...... 52 4.9 3D band structure (transverse mode) of 2DCC. Colormap shows the strength of harmonic spring constants...... 53 4.10 Evidence of transient 2D liquid droplets in 2DCC. Red and blue dots stand for 5 and 7-fold disclinations. Green loops show the liquid cluster with many disclinations...... 54 5.1 Configurations of mono-interstitial and di-interstitial. (a)-(f) are mono- interstitials and (g)-(l) are di-interstitials. Red for 5-coordinate particles and blue dots stand for 7-coordinate particles. Arrows in the inset point from 5-coordinate to 7-coordinate particles, which are perpendicular to the Burgers vector...... 58 5.2 Residence time plot (See citation: S. Kim Phd thesis)...... 62 5.3 Toy model. The zigzag line stands for Peierls potential. Defect is like a point mass located at center-of-mass of all mis-coordinate particles. The hopping process involves several attempts to overcome the barrier (the residence time), and how far (which valley) it hops to...... 63 5.4 Plot of accumulated positions of 5- and 7-coordinate particles (disclinations) for mono- and di-vacancies, mono- and di-interstitials (See citation: S. Kim Phd thesis)...... 65 5.5 Snapshot of mono-vacancy in a simulation of 2DCC...... 67 5.6 Snapshot of mono-vacancy in a simulation of 2DCC with simulated colloids and delaunay triangulation...... 68 A.1 Renormalization group flow derived from Kosterlitz-Thouless recursion relation. Each trajectory corresponds to a certain temperature with arrows indicating the direction of increasing length scale l. Con- 3 stant C(t) = −t(8πy0 − t) + O(t ) and t is the dimensionless temperature t = (T − Tc)/Tc. (I acknowledge Professor J.M.Kosterlitz for providing this figure.) ...... 76

xii Chapter One

Introduction 2

In crystalline solids, interstitials and vacancies are typical point defects. They are the keys to understand melting in lower dimension and other important phenomena in condensed matter physics. Although they have been well studied and of great importance to material science, the detailed kinetic process is poorly understood. One reason is that in atomic systems there is currently no experimental technique that allows direct observations of the atomic level.

Colloidal crystals (CC) are ordered formations of colloids at the interface of aqueous solution and air. The most widely used colloidal particles are sub-micron mono-disperse polystyrene coated with sulfate in water solution. These colloids are negatively charged with screened Coulomb potential with screening length λD (Debye length). By suppressing the ionic level of the solution, Debye length is increased to several hundred nanometers. Particles repel each other and with certain volume fraction, they form a crystal with lattice spacing about 1 µm. The most important feature of this system is the application of bright-ﬁeld microscopy and optical tweezers, which allow us to directly observe motion of each individual particle and manipulate them to ”manually” create point defects. This is an ideal model system to study a variety of physics in crystalline solids, which is inaccessible with other techniques.

In Chapter 2, we give a brief summary of Kosterlitz-Thouless-Halperin-Nelson- Young (KTHNY) theory and lattice dynamics theory. These elegant theories provide solid guidance and motivation for our experimental research. Details of the theory are in Appendix A. Simply speaking, a ”two-stage” process is predicted involving breaking-apart of dislocation and sequentially disclination pairs, which reduce the shear modulus and provide viscosity. The disassociation of dislocation pairs is closely related to the diffusion (kinetic) process of defects. The details are in Chapter 5. Lattice dynamics theory suggests the difference between our model colloidal system 3 and real atomic system and an alternative method to circumvent the difficulty of overdamped modes to access the band structure of the soften lattice.

In Chapter 3, we discuss the experimental techniques of colloidal experiment: sample cell preparation, cross-ﬂow diaﬁltration, ion-exchange process, modern microscopy, image processing, and most importantly how to achieve a two dimensional colloidal crystal (2DCC).

In Chapter 4, we present the eﬀect of point defects on phonon modes of a 2DCC. We discover that both vacancies and interstitials soften the spring constants by a certain amount depending on the wave vector. Interstitials soften signiﬁcant more than vacancies. Also the more particles are missing (vacancies) or redundant (interstitials), the lower spring constant will be for both longitudinal and transverse modes. Tri-vacancy shows some extraordinary behavior in 2DCC, which may correspond to a transient liquid droplet in solids.

In Chapter 5, we present the first experimental study of diffusion process of interstitials in 2DCC. We discover the nonergodicity and reduced dimensionality of di-interstitial, which are quite common of point defects in 2DCC system. This finding has great impact on understanding of how defects diffuse through a crystal medium and the microscopic process by which they approach equilibrium. This also provides a new pathway to directly observe the defect diffusion and ergodicity breakdown. Chapter Two

Theory 5

2.1 Two-dimensional Melting

2.1.1 Introduction

It has long been conjectured that melting in two dimension is peculiar compared to solid-liquid transition in three dimension. Since in solids long-wavelength phonons are responsible for destruction of long-range order (LRO) in lower dimensions [1], two dimension serves as a borderline in between. Power law decay of the correlation function leads to a new deﬁnition called ”Quasi-Long-Range Order” (QLRO). In real world, melting is no doubt a ﬁrst-order phase transition. A number of theories concerning melting in two dimension has been well developed, among which Kosterlitz- Thouless-Halperin-Nelson-Young (KTHNY) theory [2, 3, 4] stands out. This theory predicts a two-stage continuous transition from solid state to an isotropic liquid.

In the late thirties, Peierls and Landau predicted that there is no long-range order in system with dimension below three d < 3. Peierls gave a qualitative argument for one-dimensional atom chain and a quantitative harmonic approximation [5]. Think of a 1D atom chain with equal spacing. The interaction between atoms is short-ranged and simulated by a eﬀective spring constant k. We deﬁne the equilibrium position for each atom and its real-time deviation δ. According to harmonic approximation,

1 2 the elastic energy 2 khδ i is on the order of thermal energy kBT . The displacement of the nth atom is a sum of n − 1 displacements on the left,

n−1 X un = δi i=0 where δi follows Gaussian distribution with zero mean and standard deviation σ = p kBT/k. un can be viewed as a random walk process. One result of random walk 6

is that mean square displacement is proportional to time, or in this case, n. Thus, √ fluctuation characterized by un grows as n. In thermodynamic limit, e.g. if the system size R goes to infinity, this fluctuation will diverge, dominantly larger than the lattice spacing. Long-range order is therefore destroyed by this infinitesimal thermal vibration in 1D atom chain.

In both 1D and 2D, Peierls gave a rigorous proof [6]. We denote the displacement

ﬁeld un for nth atom. Fourier expansion of un

X ikna un = qke k

with

kBT 0 0 hqkq−k i = δkk 2 . Mωk

This is in the sense of equipartition theorem concerning each independent degree of freedom. We can calculate the mean square ﬂuctuation

π Z a 2 kBT 1 − cos kna h(un − u0) i = 2 dk 2πM π ωk − a

Note there is a cutoﬀ since the cos term will oscillate rapidly for large k (short

wavelength). In the long-wavelength region (k → 0), ωk ' ck. c is the sound speed. We can work out the integral

k T 2 n2a2 h(u − u )2i ∼ B · · ∼ n n 0 2πM na 2c2

and show the fluctuation is linearly dependent on the system size. The only assump- tion in this derivation is the long-wavelength sound speed c is finite. Besides, the interaction between atoms may not necessarily be only for nearest neighbors. Any short-range force falls off rapidly enough will give the same result. This argument 7 can be easily extended to higher dimensions. In three dimension, d3~k = k2dk. The integral turns out to be finite when n → ∞. Actually, mean square displacement

2 h(un −u0) i is proportional to temperature T . At lower temperature, the ﬂuctuation is small compared to the lattice spacing a so that LRO exists.

In two dimension, things get peculiar since d2~k = kdk. Thus,

2 h(un − u0) i ∼ log n.

In thermodynamic limit, it diverges with slow asymptotic behavior. Later on, Mer- min proved rigorously the absence of LRO in two dimension solids. Similar statement show that there is no spontaneous magnetization in 2D Heisenberg model.

The beauty of this story is the analogy to various physical systems: solids, mag- nets and superfluid helium. The analysis above also applies to the transition to superfluidity in thin films. The reduced free energy

F 1 Z F˜ = = K (∇θ)2d2x kBT 2

where K = ρs/kBT . We can do Fourier transform to q space and apply equipartition principle for each mode 1 hθ 2i = . q KΩq2

The mean square displacement can be written as an integral

X Ω Z d2q 1 hθ 2i = = ln(ΛR) q (2π)2 KΩq2 2πK ~q and diverges logarithmically as in 2D solids. In both calculation, harmonic approximation is applied. Free energy is expanded near its local minimum up to quadratic 8

order (since its ﬁrst-order derivative is zero at the minimum). In analogy to spin systems, this is usually called spin wave approximation. Similarly, spin-wave excitation destroys the LRO in two dimension.

We continue to compute phase correlation to conﬁrm the absence of LRO. The phase correlation function is deﬁned as

i(θ(~x)−θ(0)) X i~q·~x G(~x) ≡ he i = hexp(i θq(e − 1))i. ~q

This average is taken with the Boltzmann factor exp(−βF ) = exp(−F˜). Write the correlation function explicitly

R Q ˜ i~q·~x ( dθq) exp(−F − iθq(e − 1)) G(~x) = q R Q ˜ ( q dθq) exp(−F )

and divide θq into real and imaginary parts. In the numerator, with q and −q cancel some odd term,

i~q·~x iθq(e − 1) = −Real(θq) · i(cos(~q · ~x) − 1) + Img(θq) · i sin(~q · ~x).

Then we can complete the squares and pick up the terms left outside the squares

" 2 2# 1 X cos(~q · ~x) − 1 sin(~q · ~x) X 1 − cos(~q · ~x) KΩq2 + = . 2 KΩq2 KΩq2 KΩq2 q q

As we compute the integral of two guassian distributions, the shift in mean doesn’t change the result. Thus the complete squares in the numerator cancels exactly the denominator. What is left is

" # 1 X 1 − cos(~q · ~x) 1 Z 1 − cos(~q · ~x) G(~x) = exp − ' exp − d2~q . KΩ q2 (2π)2K q2 q 9

The integral with ~q inside is a Bessel function

Z Λ 1 − J (q | x |) dq 0 0 q

For large x, 1 G(~x) = exp(− ln(c | x |)) ' x−1/2πK . 2πK

This is a power law decay of the correlation function that applies for all the temperature. In the following section, we will estimate the value at the transition temperature

1 (Kc = ρs/kBTc = π/2) so that the critical exponent here η = 4 . However, this cannot be true as we expect an exponentially decay in high temperature phase. We will see later that when there are singularities in the system, such as free vortices and dislocations, the quadratic free energy breaks down. Therefore, a diﬀerent picture describing vortex proliferation is demanding.

2.1.2 Kosterlitz-Thouless conjecture

In spite of increasing evidence showing the absence of LRO in 2D systems, Kosterlitz and Thouless came up with a new deﬁnition of order based on the overall response of the system rather than the correlation function [2]. This idea arose naturally from the dislocation theory of melting. In liquid phase close to its freezing temperature, dislocations can move freely to the surface with an inﬁnitesimal shear stress, while in solid state no free dislocation exists so that it resists shear.

Kosterlitz and Thouless consider vortices in x-y model. This is a simple calculation of the energy of vortices and pair interaction [7]. Consider a vortex with winding 10

number k. k θ = kφ, ~v = ∇θ = ~e . s r φ

The elastic energy is

Z Z R 1 2 2 2 dr 2 Eel = ρs (∇θ) d x = πρsk = πρsk ln(R/a). 2 a r

Another more instructive way to obtain the elastic energy is via using ∇ · (θ∇θ) = (∇θ)2 + θ∇2θ and integrating by part

1 Z E = ρ (∇θ)2d2x el 2 s 1 Z = ρ ∇ · (θ∇θ)d2x 2 s 1 Z = ρ θ~v · dΣ~ 2 s s 1 Z Z = ρ ( θ+ ~v · dΣ+ + θ− ~v · dΣ−) 2 s s s Z R 1 2 + − dr = kρs (θ − θ ) 2 a r R = πk2ρ ln( ) s a

± where dΣ = ± ~eφ. The total energy of a single vortex is

Etotal = Eel + Ecore.

If there are more than one vortex in the system, θ(~x) is the superposition of all 11

vortices. Consider a pair of vortices with winding number k1 and k2.

1 Z E = ρ d2x(~v + ~v )2 el 2 s 1 2 Z 2 = E1 + E2 + ρs d x~v1 · ~v2 Z R + − k2 = E1 + E2 + ρs(θ1 − θ1 ) dr2 r r2 R = E + E + 2πρ k k ln( ) 1 2 s 1 2 r R = πρ (k + k )2 ln( ) + U s 1 2 a pair

where pair interaction between vortices is

r U = −2πρ k k ln( ). pair s 1 2 a

When we set k1 = −k2 = 1, the ﬁrst term vanishes and pair interaction turns to

| r − r | U (r − r ) = 2πρ ln 1 2 . pair 1 2 s a

Since the energy of an isolated vortex increases with the system size R, it cannot exist at low temperatures in large system. So does free dislocation in 2D solids. However, a pair of vortices with opposite winding number have finite energy. Their effects cancel each other and leave the energy depends logarithmically on their separation. There must be some populations of vortex pairs due to thermal energy. According to Kosterlitz and Thouless, such pairs can respond to an external shear stress and produce a viscous flow, thus reducing the rigidity. In a word, there may exist a phase transition from a vortex-free state to a vortex-rich state, with unbinding of those vortex pairs.

We can easily estimate the transition temperature. In terms of the number of 12 possible locations to place the vortex in the system, the entropy is

R2 R S = k ln( ) = 2k ln( ). B a2 B a

Then the free energy is

R F = E − TS = (πρ − 2k T ) ln( ). s B a

If T < Tm = πρs/2kB, the free energy of an isolated vortex is positive. In thermodynamic limit, the system does not favor the existence of vortices. Above Tm, vortices start to appear because entropy is dominant over energy term. At the transition temperature, the largest pairs start to unbind into separate vortices. However, this is an overestimation of the critical temperature since we ignore other pairs in between. They will relax the ﬁeld and renormalize the coupling constant ρs.

πρs/ε = 2kBT, ε > 1.

The actual transition occurs at Tm lower than πρs/2kB. As we will see in the model system (Appendix A), ε is the dielectric constant.

2.1.3 KTHNY theory

Based on Kosterlitz and Thouless, we reach to a hypothetic scenario where 2D systems, including x-y model, solids and superfluids, before phase transition occurs, consist of both fluctuations and bound topological defects. Fluctuations, such as phase fluctuations, spin-wave and long-wavelength phonons, are responsible for de- stroying LRO of the system. Dissociation of topological defect pairs leads to a phase 13 transition. Specifically in 2D solids, according to K-T theory, unbinding of dislocation pairs will drive the system to the liquid state. Later on, Nelson, Halperin and Young proposed that the phase beyond K-T transition is not an isotropic liquid phase and a second-stage phase transition finally drive it to that state. There is a new intermediate state of liquid crystal, called ”hexatic” phase. Besides translational order, they argued that this phase displays orientational order with correlation function of new order parameter ψ(~r) = e6iθ(~r)

hψ∗(~r)ψ(0)i ∼ r−η6(T )

where η6(T ) = 18kBT/πKA(T ), KA(T ) is the Frank constant. Right before Tm (K-T predicted transition temperature), the system has translational QLRO and power law decay correlation function. Similarly, hexatic phase has orientational QLRO. This new type of ordering can be destroyed by isolated disclinations. Therefore, it’s straightforward to predict a follow-on transition from hexatic phase to iostropic liquid phase with neither translational nor orientational LRO.

Although there are a lot of similarities between vortices and dislocations, disclinations, the vector nature of Burgers vector (charge of dislocation) makes the problem complicate. We write down the reduced Hamiltonian of 2D solid with dislocations interacting

( ~ ~ ~ ~ ) H K X ri − rj (bi · ∆r)(bj · ∆r) NEcore H˜ = = − b~ · b~ ln | | − + T 8π i j r | ∆r | T i,j 0

4a2µ(µ+λ) where the coupling constant K = T (2µ+λ) . As calculated in the last section, the recursion relation has modiﬁed Bessel functions I0(x) and I1(x)

dy(l) K(l) = (2 − )y(l) + O(y2) dl 8π 14

dK−1(l) 3 K(l) 3 K(l) = πy2(l)eK(l)/8πI ( ) − πy2(l)eK(l)/8πI ( ) + O(y2) dl 2 0 8π 4 1 8π

And if change K∗(l) = K(l)/8π, the equation is simpliﬁed to

dy(l) = (2 − K∗)y(l) + O(y2) dl

∗−1 dK (l) ∗ ∗ = [12π2eK I (K∗) − 6π2eK I (K∗)]y2(l) + O(y2) dl 0 1

The renormalization group ﬂow is similar to K-T transition. The dislocation probability y(l) tends to zero if T < Tm, while goes to large value if T > Tm. This signals the unbinding of dislocations as predicted by K-T. And the coupling constant tends to a universal value lim K(T ) = 16π. − T →Tm

+ We examine the correlation function showing limT →Tm KA(T ) = ∞, then we have

lim η6(T ) = 0. + T →Tm

In the following hexatic phase, orientational order persists. Write the Hamiltonian with respect to disclinations

Z Z ~0 1 2 2 1 2 2 πKA X 0 ~r − r HA = KA (∇θ(~r)) d ~r = KA (∇φ) d r− s(~r)s(r~ ) ln | | +NEcore. 2 2 36 r0 r6=r0

The ﬁrst part is the smooth varying part (ﬂuctuations) in K-T theory. The rest parts are essentially the same with the Hamiltonian of x-y model

0 πKA X 0 r − r HA = − s(r)s(r ) ln | | +NEcore 36 r0 r6=r0 with the charge s(~r) deﬁned as an integer measure of disclinity at location ~r. From 15

LRO Solid Hexatic Liquid

-r/ξ -r/ξ -ηG(T) + Translational r e e

KA(T)

KR(T) 72 kBTi/π

-η6(T) -r/ξ6 Orientational constant r e

16 π Dislocation pair Disclination pair

unbinding unbinding T T Tm i

Figure 2.1: Summary of KTHNY theory. Note the vanishing behavior of stiﬀness KR(T ) and Frank constant KA(T ) is not the exact solution.

the result of K-T theory, we have

lim KA(T ) = 72kBTi/π − T →Ti

18kBTi 1 lim η6(T ) = = − − T →Ti πKA(Ti ) 4

There is a discontinuous jump of Frank constant to zero at Ti.

A summary of KTHNY theory involving two-stage melting process is shown in Fig. 2.1.

Kosterlitz-Thouless predicted a continous phase transition in most 2D systems and afterwards this theory was modiﬁed by Nelson and Halperin. Although both transitions are mathematically proved to be second-order, Nelson and Halperin [3] mentioned that other mechanisms may exist leading to a ﬁrst-order phase transition 16

[8].

Since the onset of KTHNY theory, many computer simulations have been done to prove or disprove the theory. Some work showed a significant change in the dislocation density upon transition and predicted a first-order transition. One possibility of first-order phase transition is due to the grain boundaries [9], which is more consistent with these simulations (in that grain boundary excitation will dramatically change the dislocation density). Grain boundary can be viewed as a line collection of edge dislocations. Small-angle grain boundary will twist the crystal by a small angle. In Chui’s paper [9, 10], the effective potential between grain boundaries is defined as LK h z z z z i U(z, K) = ln(sinh π ) − π coth π − ln 4πs s s s s where z is the distance between boundaries and s is the separation between dislocations on a grain boundary. He conclude that KTHNY transition will be preempted by a first-order phase transition and there will be a change from weak to strong

ﬁrst-order transitions with respect of core energy Ecore.

Another possibility is the simultaneous unbinding of dislocation and disclination [11]. Kleinert proved this possibility and predicted a first-order phase transition using mean-field theory, which is notorious dealing with fluctuation in lower dimensions. Recent large scale Monte Carlo simulation on hard disks [12] confirmed the first stage of solid-hexatic transition as a continuous one. However, they found that the second step, from hexatic to liquid, is actually a first-order transition. They observed a very narrow single hexatic phase region. The exponent of orientational correlations is far from the lower limit of −1/4 at the KTHNY transition, as this transition is preempted by a first-order transition. This is still an opening subject to test with computer simulations and experiments. The experiment techniques to be discussed 17 is a promising testing ground.

2.2 Lattice Dynamics in 2DCC

In this section, we will study the lattice dynamics in two-dimensional colloidal crystals (2DCC). Although 2DCC serves as a model system for real atomic crystals, the lattice dynamics is entirely different. Colloids are immersed in a viscous fluid, while atoms sit in free space. The theory of hydrodynamics interaction predicted that all the phonon modes in colloidal crystals are overdamped except the transverse mode in long wavelength limit (q → 0). We will briefly go through the theory in a more reader-friendly manner.

2.2.1 1D atom chain

Starting form the simplest case with one-dimensional atom chain, as shown in Fig .2.2. Consider atoms are connected by springs with spring constant k and equilibrium lattice spacing a. At any time, the displacements from their equilibrium position are

{un}. Newtonian mechanics tells us,

mu¨n = −k(un − un−1) − k(un − un+1) in free space. If atoms sit in a viscous ﬂuid, an extra term describing viscous drag adds to the equation

mu¨n = −k(un − un−1) − k(un − un+1) − γu˙n 18

Figure 2.2: 1D atom chain model. Atoms are interacting via ideal springs.

where γ is the drag coeﬃcient. By applying solution form ei(qan−ωt),

2 iqa −iqa −mω un = −k(2 − e − e )un + iωγun.

Then, the dispersion relation is

γ k qa ω2 = −iω + 4 sin2( ). m m 2

2.2.2 Lattice dynamics

In 2DCC, similarly, we can write down the Lagrangian of the system in quadratic form: 1 X 1 X L = T − V = x˙ αmx˙ α − x˙ αU αβx˙ β 2 n n 2 n np p n,α n,p,α,β

where mass matrix is identity and interaction matrix is U (2×2 matrix). Parameters α, β takes x, y as in two dimension, and n, p range from 1 − N (N is total number of particles). External force can also be represented as

1 X x˙ αW αβx˙ β. 2 n np p n,p,α,β 19

The equation of motion in Lagrangian mechanics,

d ∂L ∂L ∂W α α − α = − α + Xn dt ∂x˙ n ∂xn ∂x˙ n we get,

α X αβ β X αβ β α mx¨n + Unp xp + Wnp x˙ p = Xn . p,β p,β

α P α i~q·R~n Do Fourier transform of the coordinates xn = aq e , q

~ X i~q·Rn α X αβ β X αβ β ˜ α e (¨aq + Dq aq + Λq a˙ q − Xq ) = 0. q β β

Then for each independent wave vector ~q, the value in the parenthesis equals zero.

α Take the time Fourier transform of the normal coordinates aq (t),

2 ~˜ −ω ~aq(ω) + (Dq − iωΛq) · ~aq(ω) = Xq(ω).

~ ~˜ ~˜ Consider a rotation ~aq(ω) = NqQq(ω) and Xq(ω) = NqYq(ω),

2 ~ −1 ~˜ −ω Qq(ω) + Nq (Dq − iωΛq)Nq~aq(ω) = Yq(ω).

The normal coordinates are

s s Yq (ω) Qq(ω) = s 2 2 s (ωq ) − ω − iωλq

s and s is longitudinal and transverse. Here, ωq is the undamped frequency from

s Dynamic matrix of the spring system Dq, and λq is the frequency corresponding to

friction Λq.

To obtain the positional relaxation of the system, we calculate the autocorrelation 20

function of the normal coordinates,

Qv∗(t)Qv(0) ρv(t) = q q q v 2 |Qq (0)| +∞ +∞ 1 Z Z 0 = dωe−iωt dω0e−iω t Qv∗(ω)Qv(ω0) v 2 q q |Qq (0)| −∞ −∞ 1 Z +∞ = dωe−iωt |Qv(ω)|2 v 2 q |Qq (0)| −∞ |Y v(0)|2 Z +∞ dωe−iωt = q v 2 |Qq (0)| −∞ (ω − ω+)(ω − ω−)(ω + ω+)(ω + ω−)

s 2 2 s Here, ω+ and ω− are the complex roots of (ωq ) −ω −iωλq = 0. In a typical colloidal system,

v 2 8 λq ∼ γ/m ∼ 6πaη/m ∼ η/ρa ∼ 10 Hz

v 4 v v 2 v and ωq ∼ 10 Hz. Thus, we have ω+ ' iλq , ω− ' (ωq ) /λq . Do a simple contour integral below the real axis,

v 2 v (ωq ) v ρq (t) = A · exp(− v t) + B · exp(−λq t). λq

Second term decays faster than ﬁrst term. Exponential decay of the autocorrelation function provides evidence of positional relaxation by drag forces in the suspension.

2.2.3 Dynamical matrix and normal modes

A major diﬀerence between the lattice dynamics of colloidal crystals and that of atomic lattices is the viscous damping of the hosting ﬂuid (deionized water). Par- ticle displacements observed in the experiment are the superposition of all normal relaxation modes. Each mode is characterized by two spring constants λs(~q) and friction factor Λs(~q)(s = l, t). As we just derived, long time autocorrelation function 21

of normal coordinates, in typical colloidal crystals, exhibits an exponential decay

with factor λs(~q)/Λs(~q) [13]. Even more complicated is the lattice dynamics when long-range hydrodynamic interaction is taken into account, including the back-flow due to collective many-body effect in finite lattices. To avoid these complexities, band structure of phonon modes can be directly obtained by solving the eigenvalues of dynamical matrix, assuming that each mode of lattice vibration solely originates

from thermal energy kBT/2, i.e. equipartition principle. This approach is equivalent to Keim’s work [14].

Consider the displacement ﬁeld ~u(R~) with respect to their equilibrium positions, and its Fourier transform ~u(~q) in momentum space. According to harmonic theory [15], contribution of certain mode ~q to elastic energy is

1 X E = uα∗Dαβuβ. (2.1) 2 q q q α,β={x,y}

By applying equipartition theorem, each entry of dynamical matrix is

αβ kBT Dq = D E, (i, j = {x, y}) (2.2) α∗ β uq uq

Then, two eigenvalues λs(~q) of this 2 × 2 matrix are the spring constants, charac- terizing the phonon band structure for longitudinal (s = l) and transverse (s = t) modes. Chapter Three

Experimental Techniques 23

In this chapter, we will discuss the experimental techniques to set up a two dimensional colloidal crystal (2DCC), and the data analysis methods to process observations from optical microscope.

3.1 Sample Cell Preparation

In order to achieve single layer (monolayer) of colloids, the system is designed to conﬁne monodisperse microspheres between two glass substrates [16, 17, 18, 19]. The interface between fused silica and water has a high population of OH− groups

2− [20], which repels negative sulfate surfactant (SO4 ). The electrostatic force is so strong that particle motion on the third dimension (perpendicular to the surface) is tremendously suppressed. Under optical microscope, it is possible to quantify this suppression and compare to in-plane motion.

Sample cell designed for 2DCC experiment is shown in Fig. 3.1. On the top is the side view, where dash lines represent drilled channels for instream and outstream of colloidal suspension. Shaded area is optical adhesive providing strong chemical bonding between fused silica disk and coverslip. Thin coverslip is used with thickness within the range of working distance (W.D.) of the objective lens. Norland optical adhesive all purpose kit (NOA 60-68) is the choice of glue temporarily bonding both glasses. The adhesive is gently applied to the circular edge surface in cleanroom and cured by exposure with UV lamp (Black Ray Lamp LW UL UVL56).

Before applying the optical adhesive, both silica surfaces are ultrasonic cleaned

in aceton, isopropanol (IPA), Piranha solution (3:1 mixture of H2SO4 and H2O2) respectively, ﬁnally deionized water (DI-water) and blown dry. All the assembly 24

Figure 3.1: Sample cell for 2DCC experiment (not to scale). Side-view (top): shaded area is the optical adhesive which bonds both substrates. In silica disk (top substrate), a channel is designed for the ﬂow. A less than µm uniform gap between the plateau and coverslip provides space for 2DCC. Top-view (below): gray area shows the ﬂow pattern of the colloid suspension. A very small amount of the solution travels through the plateau (inside center circle). 25

procedures are performed in Class 100 cleanroom. It turns out that this cleaning process eﬀectively remove micron-scale (or lower) contamination.

There is an alternative for the separation between two silica substrates [21]. A patterned thin polymer (BCB from Dow Chemical, Cyclone 3022) is spin coated and patterned with reactive-ion etching (RIE) on the silica disk. However, we simpliﬁed the procedure with optical adhesive because of its chemical stability and excellent bonding with glasses. Optical glue with less viscosity can be draw by micropipette (Pipetman) and applied to silica surface. By applying gentle pressure (force on the order of the disk weight), adhesive will dispense uniformly and form a micronmeter- thick spacer. Adhesive is cured by exposure for 1 − 2 minutes with Ultraviolet lamp. The cell is then sealed with two PTFE tubings connecting the channel to other components in a ”circuit”. We will discuss the circuit in the upcoming section.

3.2 Suspension Preparation and Cleaning

Colloids used in our experiment is Thermo Scientiﬁc Latex Microsphere Suspensions 5036A/B. Its diameter is 0.36 µm and surface is coated with sulfate. Suspension is diluted by DI-water (type II water from Barnstead Easypure RF System D7031). However, colloidal crystallization puts strict constraint on Debye screening length of the solution. For colloidal suspensions, two key parameters govern the property of crystals: Debye length κ−1 and separation d. In analog to inert gas, when κ−1 < d, the electric ﬁeld is completely screened and colloids see each other as hard spheres. They don’t actually ”feel” until they touch each other. Interaction between particles is hard sphere potential, which results in a close packing structure at high density (volume fraction). This is not interesting for us because dynamics is slow. In analogy 26

to the formation of crystal in metals, when κ−1 > d, colloids mutually repel, and with appropriately high volume fraction, they form a crystal, e.g., a triangular lattice.

Debye length is inverse proportional to square root of ionic strength I(M) (equivalently conductivity σ) of the solution,

1 r eµ∗ κ−1 = √ · H . σ 4πλB

A typical volume fraction of 1% requires ionic strength on the order of 10−6M (conductivity σ ∼ 1 µs/cm). The cleaning process involves two steps: (i) Cross-ﬂow diaﬁltration, (ii) Ion-exchange deionization.

Cross-flow diafiltration helps removing organic impurities. Simply speaking, solution flows through a hollow fiber (cartridge filter) with sub-micron pores on its side wall. Large organic contaminations are filtered out along with water molecular as exhausts, while colloids remain in the solution. The setup is shown in Fig. 3.2. By continuously filling the reservoir with DI-water, the suspension is circulated through the system and getting rid of organic impurities. This procedure effectively reduces

the conductivity of stock solution (usually from 200 µS/cm to 3 − 5 µS/cm). CO2 contamination in the air is tested overnight and proven to be negligible because of good air tightness of the whole system.

After hours of cross-flow diafiltration, refilling DI-water for several times, the conductivity reaches 3 − 5 µS/cm. One example of the cleaning performance is in Fig. 3.3. The initial stock solution has volume fraction of 10%. 3× dilution gives φ = 3.33% and lower conductivity. After continuously diafiltration, the solution is condensed and conductivity rises (from blue points to next red points in Fig. 3.3). Then it drops to lower value after DI-water is refilled to replace exhaust solution. At 27

Exhaust Cartridge Filter

Conductivity Meter

Peristaltic Pump

DI Water Colloidal Suspension

Figure 3.2: Cross-flow diafiltration setup (not to scale). The colloidal suspension is stored in a sealed Telfon container. The circuit is connected with Galtek Integral Ferrule fittings. The suspension is pumped out by a peristaltic pump (Fisher Scientific Variable-Flow Peristaltic Pump), providing smooth feed into the cartridge filter. The cartridge used is MidGee Microfiltration Hol- low Fiber Cartridges together with Polysulfone membrane with 0.2 micron pore size (CFP-2-E- MM01A). A conductivity meter (Amber Scientific Model 1056 Digital Conductivity Meter) is used to monitor the ionic strength. DI-water buffer solution is stored in a squeeze bottle and transferred into the reservoir continuously. 28

Figure 3.3: Cross-flow diafiltration performance. Conductivity is measured with Amber Scientific Model 1056 Digital Conductivity Meter. Red represents the readings before dilution (refill) and blue for readings after dilution.

this stage, we found no sign of crystallization. Cross-ﬂow diaﬁltration is not enough for our cleaning purpose.

To further reduce ionic strength, we follow with the ion-exchange deionization. The experimental setup is shown in Fig. 3.4. The circuit is sealed with Galtek In- tegral Ferrule fittings. We use a flow-through conductivity meter (Amber Scientific Conductivity Micro Flow Cells). Mixed-bed ion-exchange resin beads (Fisher Chem- ical Rexyn I-300 (H-OH) Beads) are stored in the ion exchange column (Kontes) or hand-made PTFE tubings (20 cm × 1 cm). These beads are basically of milimeter- size and porous, providing large surface area. Trapping of sodium ions Na+ and

2− + sulfate ions SO4 in the suspension occurs with releasing hydrogen ions H and hydroxyl ions OH−. This continuously drives the electrolytic equilibrium of water 29

Sample Cell

Conductivity Meter Objective Lens

Ion Exchange Peristaltic Pump Column

Colloidal Suspension

Figure 3.4: Ion-exchange deionization setup to prepare 2DCC. The colloidal suspension is stored in a sealed Telfon container. The circuit is connected with Galtek Integral Ferrule fittings. The suspension is pumped out by a peristaltic pump (Fisher Scientific Variable-Flow Peristaltic Pump), providing smooth feed into the cartridge filter. Mixed-bed ion-exchange resin beads (Fisher Chemi- cal Rexyn I-300 (H-OH) Beads) are stored in the ion exchange column. A flow-through conductivity meter (Amber Scientific Conductivity Micro Flow Cells) is used to monitor the ionic strength. Sam- ple cell is mounted above the oil-immerse objective lens. molecules backwards, reducing the ionic level.

This system eﬀectively reduces the ionic strength and maintains its level for a few days. Some regions in the sample cell crystallize about one hour after introduction of the ion-exchange setup. By simple derivation from the conductivity measured experimentally, we obtain a Debye screening length κ−1 = (390 ± 10) nm. With colloid diameter 0.36 µm and lattice spacing around 1 µm, the electron double layer is suﬃciently thick for crystallization. 30

3.3 Imaging with Optical Microscope

Bright-field microscope is the simplest of all the optical microscopy illumination techniques. The contrast in the image stems from the absorption of transmitted light by some dense area of the sample. In our experiment, we use an inverted microscope (Zeiss Axiovert 135) with a 100× objective (Zeiss Plan Neofluar, N.A. = 1.3). Optical tweezers is employed to manipulate colloidal particles or clusters, for cleaning purpose, and more importantly, to create point defects. Video microscope images are recorded by a CCD camera (Sony SSC-M370), and either recorded on videotape using a Sony SVO-9500MD recorder or digitized by a frame grabber board (National Instrument NI-1409), and finally analyzed with image processing software built on Interactive Data Language (IDL).

The microscope effective magnification is 83nm/pixel. The frame rate is 30 fps and image size is 640 × 480. Instant images are recorded as JPEG files and pro- cessed by the image processing and particle tracking algorithms in Interactive Data Language (IDL), written by Crocker and Grier [22].

Fig. 3.5 is a snapshot of colloidal suspension under optical microscope. It shows the circular channel (on top). We see the circle is not as smooth in micrometer scale. In the channel is the higher population region compared to the plateau. Particles are free to travel across the boundary. At this point, colloids start to accumulate and form a crystal. The conductivity is in a high level and Debye length is quite short. Thus we observe a close-packing structure. The packing is so tight that the motion of colloids is highly conﬁned. Bottom ﬁgure is far from the channel where the population is not as high to close pack as near the channel. With current volume fraction, the goal is to reduce the suspension conductivity by cleaning process, which 31

Figure 3.5: Colloidal suspension under optical microscope. Top: near the circular channel. Bottom: far from the channel. 32

Top Substrate

a Double D Layer λD

Bottom Substrate

Figure 3.6: Side view of sample cell showing the maximum spacing between substrates to achieve one -ayer. Solid circle represents√ colloid surface. Dash circle is the Debye layer (Electron Double Layer). D ∼ 2λD + d + 3a/2 ' 2 µm.

will hopefully result in a widely-spaced triangular lattice.

Besides conductivity, we have other obstacles. One is the spacing between two fused silica substrates. The particle diameter is 0.36 µm and Debye length is on the order of 0.4 µm. Fig. 3.6 shows the side view of two layer situation. The maximum spacing between substrates to achieve one layer crystal is calculated as

√ 2λD + d + 3a/2 ' 2µm.

If the thickness of bonding material between substrates is beyond 2 µm, it will turn out to be at least two layer. This is a technical diﬃculty in experiment to apply that tiny amount of optical adhesive compared to lithograph techniques. We can estimate the volume of adhesive applied on the disk surface,

2 2 2 2 −10 3 π(Router − Rinner)D = π(1.35 − 0.7 ) × 2 × 10 m = 0.8µL which is beyond the resolution of micro pipet (ﬁnest amount for micro pipet is 1 µL). 33

Figure 3.7: Two-layer colloidal crystal 34

Besides, some adhesive types are too viscous to draw from a micro pipet. However, this is experimentally feasible in that slight pressure will push extra adhesive out of the disk and leave a thin layer while providing strong bond between substrates. For D > 2 µm, two-layer (or more) crystal forms (Fig. 3.7). And there are many ”domains” with diﬀerent orientations and grain boundaries.

Fig. 3.8 shows single layer colloidal crystals in our experiment with diﬀerent ionic strength. The spacing between two substrates is below the threshold (2 µm), so that the ionic force squeezes the extra layer of colloids, thus leaving only one layer. Although the thickness is uncontrollably depending on the pressure applied, this turns out to be feasible and reproducible.

3.4 Image Processing and Particle Tracking

With the library (in IDL) for our processing purpose, standard analysis takes the following steps: (i) select target JPEG files and read into an array in IDL using ”readjpgstack” function. (ii) crop the image by selecting partial entries of the array, and save them as new JPEG files (”crop jpeg”). (iii) use ”spretrack” routine to extract particle coordinates for each frame, and save into GDF file. GDF is a IDL- defined data file and can be read for later analysis.

”Spretrack” routine is the key part of image processing, which contains subrou- tines like bypass ﬁlter and feature function. It automatically selects particles in accordance with the input feature parameters, namely low-cut, high-cut, diameter, separation, brightness-cut, etc, and outputs a compact data ﬁle with the position information for each particle. 35

Figure 3.8: Mono-layer colloidal crystal with diﬀerent volume fractions and thickness. 36

25 30 20 25 15 20 10 15

Frequency Frequency 10 5 5 0 0 10 11 12 13 14 15 0 20 40 60 80 Lattice Constant (pixel) Angle (degree)

#, μ, σ= 625, 12.8269, 0.1421 #, μ, σ= 417, 60.0172, 0.6261 80 60 60 50 40 40 30 20 20 10 0 0 12.5 13.0 13.5 57 58 59 60 61 62 63

Figure 3.9: Histogram of lattice spacings and angles with equilibrium positions. Top: histogram over 200 frames. Bottom: ﬁtting with standard Gaussian distribution.

In Fig. 3.9, we test the histogram of lattice spacings and angles. Equilibrium positions are calculated by averaging over each position of all the frames without any defect. Lattice spacings and angles are obtained for each pair (bond) of nearest neighbors. The plot is a sample of 200 frames (280 × 200 pixels observing window, hundreds of particles). Top two figures are histograms of lattice spacings and angles. Bottom two are the Gaussian fit showing the means and standard deviations. The mean lattice spacing is 12.827 pixels, which is 1.06 µm. The mean angle is 60.017 degrees. The fitting gives clear evidence of our hexagonal lattice. Fast Fourier Transform (FFT) (Fig. 3.10) of a cropped image of the crystal also proves the hexagonal feature.

Details of IDL routines and instructions on the library can be found in Appendix C. 37

Figure 3.10: Fast Fourier Transform (FFT) of a cropped snapshot of 2DCC. Chapter Four

Eﬀect of Point Defect on Phonon Modes 39

4.1 Introduction

It has been conjectured for a long time ﬁrst by Lindemann, later Max Born, recently Granato that in three dimensional systems, point defects play a very important role in the melting process.

Lindemann criterion [23] proposed that melting occurs at the avalanche of the root-mean-square (RMS) atom displacement when it exceeds a threshold. The vibration energy is related to the temperature by the equipartition theorem. He suggested that because the amplitude of atomic vibration increases with temperature, at certain point it becomes so large that atoms start to invade the space of their nearest neighbors, initiating the melting process.

Max Born [24] derived the criterion of melting based on analysis of stability of crystal lattice against shear stress. The crystal will melt if one of its shear elastic moduli vanishes, the so-called ”rigidity catastrophe”. The shear elastic moduli are decreased with increase in temperature and point defect concentration.

Frenkel [25] considered the role of vacancies in melting. He observed that the regularity in the arrangement of atoms in a crystal begins to fall with temperature long before melting point is reached; as the latter is approached, this process is gradually accelerated, acquiring a more pronounced cooperative character. He in- terpreted sharp increase in ﬂuidity on fusion as suggesting the possibility of shift in the equilibrium position of atom on transition to the liquid state.

Granato [26] point out that melting is due to the rapid generation of thermody- namically equilibrium defects dumbbell interstitials, which drastically decrease the shear modulus at the melting point. Melting requires thermally accessible intrinsic 40 defects. The only such defects in crystals are vacancies and interstitials. Vacancies can be excluded from consideration since their formation entropy is far insuﬃcient to explain the observed latent heat of melting. Because vacancies and interstitials are the only thermally accessible intrinsic defects known from solid-state physics, interstitials should be the source of melting.

As temperature approaching the melting point, a cooperative formation of point defects takes place. The presence of defects softens the lattice, and the eﬀect ac- cumulates when the concentration of defects increases, thereby easing the insertion of more defects and introducing new local modes of vibration. At certain critical concentration, the lattice becomes unstable and collapses, i.e. a mechanical melting transition occurs.

2D solids have long been of fundamental interests to condensed matter physics as they provide testing grounds for theoretical models of 2D melting and freezing [2, 3, 4, 9, 27]. Melting involves the destruction of long-range or quasi-long-range order as well as the loss of long-wavelength shear rigidity. Many questions remain unresolved regarding the roles of crystalline defects in melting. The responsible defects include dislocations and disclinations [2, 3, 4], or vacancies and interstitials [24, 25, 26]. For these studies, 2D colloidal crystals have become the favored experimental systems for exploration due to the feasibility of observing equilibrium behavior of defects in laboratory time scales [19, 28, 29].

In this chapter, we report a quantitative study of the eﬀects of point defects on phonon modes in a two dimensional colloidal crystal (2DCC). Defect-phonon interaction is a topic of past intense studies, but mostly in the context of ”quenched disorder” that defects are considered static in their positions or hopping between two or few discrete sites. Little is known about the destructive roles played by point 41 defects in softening of the lattice. Here, we use real time positional data to extract the phonon dispersion relations [14, 30, 31] and study the eﬀects of vacancies and interstitials on these modes.

4.2 Method

The detailed experimental setup is in Chapter 3 and Ref. [19]. A 2DCC is achieved by confining 0.36 µm diameter polystyrene-sulfate microspheres within two fused silica substrates. The colloidal suspension is highly deionized by ion-exchange process, resulting in a Debye screening length κ−1 ≈ (390 ± 10) nm. The whole system is sealed with high standard of cleanliness, maintaining low conductivity of the suspension for several days. Colloid surfaces are all negatively charged. With screened Coulomb interaction between colloids, they are able to self-assemble into a monolayer hexagonal lattice. When settled, after repeated purging of the colloid through the deionizing resin matrix, the colloidal crystal system has a uniformity within 0.1 µm across the entire field (50 µm × 50 µm). Optical tweezers, constructed on an inverted microscope (Zeiss Axiovert 135) with a 100x objective (Zeiss Plan Neofluar, NA = 1.3), can be employed to manipulate individual colloidal particles to create point defects. Videomicroscope images are recorded by a CCD camera (30 fps), digitized by a frame grabber board and finally analyzed with IDL image processing software.

Optical micrograph Fig. 4.1 (a) shows a defect-free hexagonal crystal with lattice constant a ≈ 1.1 µm. Cropped ﬁeld of view contains hundreds of colloidal particles, of which the time elapsed trajectories are reconstructed using a particle tracking algorithm [22]. Standard Delaunay triangulation is applied to determine 5 and 7- 42

(a) (b)

50 Pixels = 4.15 μm

(e) (f)

Figure 4.1: Optical micrograph and delaunay triangulation diagrams of a two dimensional colloidal crystal. (a) (b) One microscope snapshot and its delaunay triangulation diagram: defect-free ~ ~ lattice. The inset shows primitive vectors (~a1, ~a2), its reciprocal lattice vectors (b1, b2), and the first Brillouin zone (not to scale) of perfect triangular lattice with standard labels of the symmetry points. (c)-(f) Lattice with point defects: mono-vacancy, mono-interstitial, both mono-vacancy and interstitial and lattice with tri-vacancy. Red and blue dots stand for 5 and 7-coordinate disclinations. 43 coordinate particles, i.e. 5, 7-fold disclinations. Fig. 4.1 (b) is the triangulation diagram of one snapshot of perfect crystal, where each particle has six nearest neighbors. Fig. 4.1 (c)-(f) show different types of point defects that can be identified as pairs of 5, 7-fold disclinations. By counting the number of missing or redundant particles in the green dashed loop, in comparison with perfect triangular lattice, we are able to keep track of vacancies and interstitials diffusion [19].

As discussed in the theory part, the major difference of lattice dynamics between colloidal crystals (model system) and atomic lattice (real system) is the viscous damping of the hosting fluid. It is reported that there is no propagating long- wavelength modes in colloidal crystals except the transverse mode in long-wavelength limit (~q → 0) [13, 32]. Particle displacements observed in the experiment are the superposition of all normal relaxation modes. Each mode is characterized by two spring constants λs(~q) and friction factor Λs(~q)(s = l, t). It is proposed that long time autocorrelation function of normal coordinates, in typical colloidal crystals, exhibits an exponential decay with factor λs(~q)/Λs(~q) [13]. Even more complicated is the lattice dynamics when long-range hydrodynamic interaction is taken into account, including the back-flow due to collective many-body effect in finite lattices.

To avoid these complexities, band structure of phonon modes can be directly obtained by ﬁnding the eigenvalues of dynamical matrix, assuming that each mode of lattice vibration solely originates from thermal energy kBT/2, i.e. equipartition principle. This approach is equivalent to Keim’s work [14]. By averaging over frames with defect-free lattice (hundreds of frames taken without applying optical tweezers), we can determine the equilibrium position of each particles. Histograms of particle positions turn out to be sharp Gaussian distributions with variance less than 0.083 µm (size of a pixel) centered at these equilibrium sites. This also assures us that there is no macroscopic drift problem in our 2DCC. 44

Figure 4.2: Hexagonal lattice of 2DCC with equilibrium positions. The density function ~r(t) is a superposition of delta functions on n sites R~ n. We can measure the real-time displacement ~u(Rn) as the diﬀerence between real time position (green dots) and the equilibrium position (blue dots).

With the equilibrium lattice deﬁned, for each frame, we can calculate the displacement ﬁeld ~u(R~) with respect to the equilibrium lattice points (Fig. 4.2). The ~ density function ~r(t) is a superposition of delta functions on n sites Rn

X ~ (2) ~ ~u(~r) = ~u(Rn)δ (~r − Rn). n

We can measure the real-time displacement ~u(Rn)

~ ~ ~u(Rn) = ~r(t; n) − Rn.

Then we compute Fourier transform ~u(~q) in the momentum space. ~q is chosen ~ ~ as certain linear combination of the reciprocal primitive vectors (b1, b2), as shown in Fig. 4.3,

~ X ~ −i~q·Rn ~u(~q) = ~u(Rn)e . n According to harmonic theory [15], contribution of certain mode ~q to elastic energy 45

2 a1 b a2

Figure 4.3: Reciprocal space of 2DCC. Two primitive vectors (~a1, ~a2), its reciprocal lattice vectors ~ ~ (b1, b2), and the ﬁrst Brillouin zone (not to scale) of perfect triangular lattice with standard labels of the symmetry points. is 1 X E = u∗(~q)D (~q)u (~q). (4.1) 2 i i,j j i,j={x,y}

By applying equipartition theorem, each degree of freedom each entry of dynamical matrix is [14]

∗ −1 Di,j(~q) = kBT hui (~q)uj(~q)i , (i, j = {x, y}). (4.2)

Then, two eigenvalues λs(~q) of this 2 × 2 matrix are the spring constants, charac- terizing the phonon band structure for longitudinal (s = l) and transverse (s = t) modes. The frequency is related as,

r λ (~q) ω (~q) = s . s m

Results are averaged over three equivalent hexagonal axes. 46

Figure 4.4: Band structure of harmonic lattice constants of 2DCC without defect. The plotting is performed from the center of the ﬁrst Brillouin zone to the zone boundary (blue dash lines) in two directions. The upper curve (green) corresponds to the longitudinal modes, while lower one −7 2 (red) to transverse modes. Spring constants λs(~q) are in the units of 5.80 × 10 kg/s .

4.3 Results

Fig. 4.4 shows the λs(~q) of defect-free lattice measured in the reciprocal space, from the center of the ﬁrst Brillouin zone Γ to its zone boundary M, then to K (symmetry points), ﬁnally back to center Γ. These high-symmetry directions of the reciprocal lattice are illustrated in Fig. 4.3. The top branch is longitudinal and the bottom one is transverse mode because longitudinal mode has a dominant spring restoration force than transverse one.

Next ﬁgure is for comparison. In each plot of Fig. 4.5, top branch (green) is longitudinal and the bottom one (red) is transverse mode. Speciﬁcally, in the defect- free lattice (dots), spring constants show similar dependence on wave vector ~q with 47

3.48 −3 x 10 7 Defect-free 6 Mono-vacancy 5 Mono-interstitial 2.90 Both 4 3 2 1 2.32

0 2 4 6 8 10 )

2 q a

kg/s 1.74 -8 (q) (10 s 1.16

0.58

0 0 1 2 3 4 5 6 789 10

q a

Figure 4.5: Band structure of harmonic lattice constants of 2DCC: defect-free (dots), in the presence of mono-vacancy (crosses), mono-interstitial (squares) and both point defects (blue diamonds). The inset shows the details of the ”mono-interstitial” and ”both” cases. The plotting is performed from the center of the ﬁrst Brillouin zone to the zone boundary (blue dash lines) in two directions. In each individual plot, upper curve (green) corresponds to the longitudinal modes, while lower one (red) to transverse modes. For clarity, ”both” are denoted by blue colors. Spring −8 2 constants λs(~q) are in the units of 10 kg/s .

a previous study, although the system size and particle pair interaction are diﬀerent

[14]. For longitudinal mode, λs(~q) reaches its maximum value near M point, while that of transverse mode has a maximum at point K. Harmonic restoring force is always the strongest near the zone boundary. We repeat the same calculation with smaller observing and time window, those tiny oscillations of the dispersion curve are enhanced. We believe that this stems from insuﬃciency of statistics. Computation with less statistics results in a coarser dispersion curve. Further testing with more statistics needs to verify this point.

Once a point defect is introduced by optical tweezers, it starts to diﬀuse as a 48

1 mono-vacancy 0.9

0.8

0.7

0.6

0.5

0.4

SofteningRatio 0.3

0.2

Longitudinal 0.1 Transverse

0 0 1 2 3 4 5 6 7 8 9 10 ϙ Ϣ Ϡ ϙ q a

Longitudinal mono-interstitial 0.9 Transverse

0.8

0.7

0.6

0.5

0.4

SofteningRatio 0.3

0.2

0.1

0 0 1 2 3 4 5 6 7 8 9 10 ϙ Ϣ Ϡ ϙ q a

Figure 4.6: Softening ratio with mono-vacancy and mono-interstitial. Green dots stand for longitudinal modes and red crosses for transverse modes. Softening ratio is defined as the ratio between λs(~q)(s = l, t). 49 quasiparticle. We collect the whole frames (> 300 frames, 10 seconds for each type studied) in the presence of a certain type of point defect in a rectangular area (∼ 25 µm × 17 µm). As a control group, spring constants are calculated for defect-free lattice only in the same region (dots in Fig. 4.5). Meanwhile, for the experimental group with mono-vacancy (crosses), mono-interstitial (squares) or both point defects (blue diamonds) respectively, we perform the same calculation within this subsystem. There are two motivations: one is to quantify the effect with a constant defect density (number of point defects divided by the area); the other is to circumvent the influence of nearby defects outside the window of interest (There appears no other defect within twice the scale of the window). This may underestimate the effect of point defects on phonon modes because the displacement field produced by a point defect, probably decays so slowly that particles outside the region still feel its existence. In this framework, we observe that both longitudinal and transverse modes are softened by vacancies and interstitials. In Fig. 4.5, qualitatively, with mono-vacancy number density ρ = 1/(25 × 17 µm2), both spring constants decrease by less than one half, while those of mono-interstitial with the same defect density reduce by one order of magnitude. Moreover, with both mono-vacancy and mono- interstitial collaboration (blue squares), spring constants are close to the interstitial case. This suggests that vacancies have less effect on the phonon modes, especially with an interstitial nearby.

In Fig. 4.6, we plot ~q-dependent softening ratio, defined as the ratio between spring constants λs(~q) of defect-rich and defect-free system (control group). We find that for mono-interstitial, of which softening phenomenon is dominant, longitudinal modes are suppressed less than transverse ones, especially in long-wavelength region (~q → 0). Tiny oscillations come from insufficiency of statistics, as mentioned before. In constrast, softening ratios are approximately 0.5 for both modes in vacancy case. 50

1 Longitudinal Defect-free 0.8 Transverse

2.5 Tri-vacancy 0.6

0.4 Softening Ratio

0.2

2 0 0 2 4 6 8 10 q a ) 2

1.5 kg/s -8 (q) (10

s 1 휆

0.5

0 0 1 2 3 4 5 6 7 8 9 10 q a 훤 훭 훫 훤 Figure 4.7: Band structure of harmonic lattice constants of 2D colloidal crystals: defect-free (dots), with tri-vacancy (crosses). Upper curves (green) correspond to longitudinal modes and lower ones (red) to transverse modes. The inset shows the softening ratio. Green dots stand for longitudinal modes and red crosses for transverse modes. Calculation and notations follow the same convention in Fig. 4.2.

This further quantiﬁes the previous observation.

Band structure of harmonic lattice constants with one tri-vacancy is shown in Fig. 4.7. Tracing back to Fig. 4.1 (f), we can locally identify three missing particles and it appears to be an edge dislocation pair. Tri-vacancy can be treated as three independent mono-vacancies tightly bound together. Work by simulation predicts strong attraction interaction between vacancies [33]. Thus, the observed particle displacements are the superposition of three mono-vacancies. Fig. 4.6 shows a tri-vacancy with defect density ρ = 1/(25 × 17 µm2), which softens the phonon vibration spring constants by approximately one order of magnitude. Similar to mono-interstitials, in the inset of Fig. 4.7, transverse phonons in tri-vacancy system are softened more than longitudinal ones for all ~q measured. 51

Here, we show that band structures of both defect-free lattices (control groups) are slight diﬀerent (dots in Fig. 4.5 and Fig. 4.7). Essentially, this calculation is very sensitive to the quality of the lattice, especially near the Brillouin zone boundary [14]. Besides, ergodicity of microstates is not well satisﬁed because of limited statistics. Within our observation window, point defects studied may experience a portion of the entire phase space. This casts light on prospective direction of study on improving this band structure approach and exploring new features introduced by other defect types.

The effect of tri-vacancy on the phonon modes demands special attention. We observe that, (data not shown, but available upon request), immediately after the tri-vacancy is created by using optical tweezers, the structure is locally disordered, reminiscent of a 2D liquid droplet. Removing three particles from a local region causes the system to cross into the liquid side of the solid-liquid phase boundary. After dozens of frames (less than a second), the disordered local structure merges with the rest of the ordered lattice, leaving behind a pair of loosely bound fluctuating edge dislocations. This direct observation leaves very little room for speculation: the drastic softening effects are caused by these dislocation pairs, as in fact theoretically predicted long ago by Kosterlitz and Thouless in their landmark paper [2]. Disloca- tions in 2D solids, even when they are bound, can relax the strain field significantly, and ultimately cause melting when many such pairs exist.

4.4 3D Visualization of Phonon Bands

Fig. 4.8 and Fig. 4.9 are 3D visualizations of band structure of our 2DCC. Spring constants are calculated the same way as before. The only diﬀerence is that ~q is no 52

Figure 4.8: 3D band structure (longitudinal mode) of 2DCC. Colormap shows the strength of harmonic spring constants. longer along symmetry axes. ~q are selected as a 30 × 30 grid ranging from qa = −4 to qa = 4. There is a finite-size effect that the calculation framework breaks down near q = 0. There is a six-fold symmetry except for those areas with this effect. Colormap is same across two plots. We see longitudinal modes are more stiff.

4.5 Conclusion and Future Directions

An interesting discovery is in Fig. 4.10, where we observed an evidence of transient 2D liquid droplets in 2DCC. This is called, in our dictionary, tri-vacancy, which contains three particles missing at the same time. In our previous study, ti-vacancy softs the lattice much more than mono-vacancy. Fig. 4.10 shows 20 consecutive video images (fps= 30). It has been proposed that a cluster of vacancies can form a nucleus droplet leading to a liquid state. This transient liquid state is unstable in crystalline lattice. With the diﬀusion and vibration of nearby lattice, they quickly 53

Figure 4.9: 3D band structure (transverse mode) of 2DCC. Colormap shows the strength of harmonic spring constants. disappear and reach the equilibrium state. Dislocation pair is the most frequent conﬁguration of ti-vacancy, so as the most stable one (most probable). This process is irreversible. We never observe tri-vacancy turns back to cluster of dislocations. This model colloidal system may be applied to future study of co-existing states and phase transition between liquid and solid. With optical tweezers, one can locally create a ”hole”, e.g. a cluster of vacancies, and decrease local density. This, in phase diagram, is eﬀectively crossing the solid-liquid phase boundary along the density dimension.

In conclusion, we analyze real-space microscopic images and use positional data to compute the band structure of a two dimensional colloidal crystal and study the effects of point defects on spring constants. We conclude that interstitials are more efficient in softening the lattice than vacancies. Furthermore, we observe the effect of both types (mono-vacancy and mono-interstitial) coexistence and tri-vacancy (three vacancies strongly binding). Optical tweezers serve as a powerful tool to create lower- 54

Figure 4.10: Evidence of transient 2D liquid droplets in 2DCC. Red and blue dots stand for 5 and 7-fold disclinations. Green loops show the liquid cluster with many disclinations. density liquid droplets. However, they are unstable in solids and the coexistence phase is meteoric. According to the phase diagram, this process corresponds to an isothermal path across the phase boundary, along which local liquid region and the bulk solid phase merge into a less dense defective crystal. In this aspect, optical tweezers can be employed to continously grabbing colloids out of the perfect crystal, leaving behind a locally sparse liquid droplet. The more defects created, the larger the liquid regime will be (tri-vacancy liquid droplet turns out to be much larger than that of mono-vacancies). Thus longer equilibration time allows detail study of solid-liquid transition. This provides new insights on the importance of point defects and optical tweezers in the future study of 2D melting. In future study, more experiments and computer simulations on two dimensional colloidal crystals can hopefully be carried out to witness melting process in real time.

We want to acknowledge useful discussions with J.M. Kosterlitz and C. Reich- 55 hardt, and the ﬁnancial support from NSF (Grants No. DMR-1005705 and No. DMR-9804083). Chapter Five

Microscopic Process and Nonergodicity in Defect Diﬀusion 57

5.1 Introduction

There are longstanding questions about how defects diﬀuse through crystalline lattice and the microscopic process by which they approach thermal equilibrium. In this chapter, we study the interstitial diﬀusion using a two-dimensional colloidal crystal (2DCC) as a model system. We also propose a new criterion in determining the ergodicity of a defect system and use this criterion to analyze the nonergodicity behavior of point defects in colloidal system.

5.2 Method

The detailed experimental setup is in Chapter 3 and [19]. Optical tweezers are applied to create point defects by trapping a particle. The optical trap places per- turbation on the lattice and accidentally increase or decrease the regional density of particles. When there are one or more extra particles, this is an interstitial. In Fig. 5.1, we show the triangulation of two species of interstitials. (a)-(f) are mono- interstitials, namely there is only one extra particle. You can count the total number in the green box and compare to perfect hexagonal lattice. When an interstitial is introduced to the system, nearby particles adjust their positions to accommendate the ”newcomer”, and as a result no long 6-fold coordinated. We count the nearest neighbors for each particle and label the disclinations as red and blue dots. Red for 5-coordinate particles and blue dots stand for 7-coordinate particles. Each pair of 5- and 7-fold disclinations is an edge dislocation. Thus an interstitial can be viewed as a pair of dislocations parallel to each other (Sometimes the conﬁguration is more complicated as in Fig. 5.1 (e) (f) (k) (l)). 58

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

Figure 5.1: Conﬁgurations of mono-interstitial and di-interstitial. (a)-(f) are mono-interstitials and (g)-(l) are di-interstitials. Red for 5-coordinate particles and blue dots stand for 7-coordinate particles. Arrows in the inset point from 5-coordinate to 7-coordinate particles, which are perpendicular to the Burgers vector. 59

The histogram of these various configurations show that the most probable states are Fig. 5.1 (a) (b) (g) (h) (i). We will see in the next section, most of them are equivalent. We notice some distorted lattice like Fig. 5.1 (d) (j). The extra pair of dislocations is due to the sensitivity of our defect-finding algorithm. Because the routine to determine nearest neighbors is quite sensitive to positions, small displacement of particles (which comes from thermal vibration) may end up with different configurations.

In order to track the motion of point defect, we deﬁne the center of mass as the average position of its disclinations, equivalently, of the dislocations. Since the extra particle is always indistinguishable from others, a better description of interstitial is a ”quasi-particle” with mis-coordinated particles, whose position is at the center-of- mass. With that settled, it becomes a simple diﬀusion problem of a classical point mass. Moreover, the symmetry of dislocations can be studied in the center-of-mass coordinate.

5.3 Results

First result is the overall diﬀusion coeﬃcient of interstitials [34], compared to vacancy results from [21]. Time-elapsed trajectories are constructed from 900 frames for mono-interstitials and 300 frames for di-interstitals. By calculating the Mean Square

2 Displacement (MSD) and linear ﬁtting hx (t)i ∼ t, we obtain Dmono = 10.95 ± 0.04

2 2 µm /s and Ddi = 7.53 ± 0.05 µm /s. MSD of both interstitials grows linearly with time in accordance with the random walk property. In comparison, diﬀusion

2 2 constant of vacancies are Dmono = 3.60 µm /s and Ddi = 4.50 µm /s [21]. The ﬁrst observation is: interstitials diﬀuse faster than vacancies. 60

Second, similar memory effect is found in interstitials to vacancies, especially in di-interstitials. Stronger memory effect in di-interstitials makes its diffusion quasi- one-dimension, which will significantly reduce the measured diffusion constant (by a factor of 2). The MSD is thus usually written as

hx2(t)i = dD?t

where d is the dimensionality (See Appendix B). D? describe the microscopic process of diﬀusion. In the classical model of random walk,

a2 D? = . τ

Let’s think about the diffusion coefficients of mono- and di-interstitials. Getting rid of the dimension factor of 2 for mono-interstitials (it diffuses in two-dimensional

? 2 ? 2 space), Ddi = 7.53 µm /s and Dmono = 5.48 µm /s. There must be some internal diﬀerence of the microscopic process between both interstitials.

Diﬀusion coeﬃcient is a measure of how long are the time step τ and length step a. In our practice, we plot the histogram of hopping distance |R~| and residence time t of both interstitials in Fig. 5.2 [34]. Hopping distance is measured from displacement of center-of-mass of consecutive frames (Fig. 5.2 (a) (b)). We notice peaks in the |R~| histogram, which are multiple integers of half the lattice constant

a = a0/2. Residence time is deﬁned as the waiting time for the defect to move by

at least a (we set up a threshold of a0/4 to distinguish between hopping events and local ﬂuctuations). Time unit is 1/30 second. The picture is clearly illustrated in our toy model in Fig. 5.3. With thermal activation, the defect attempts several times to escape the potential well (barrier) before ﬁnally hops to its nearby minima. The histogram plot is consistent with the model. With highest probability, it will stay 61

(reside) in the well. The next peak is at a0/2, meaning one of the dislocations glide

by one lattice spacing a0 (thus the center of mass moves by one half). As the most probable conﬁguration for interstitial is dislocation pairs, its diﬀusion is dominated by ”binding-unbinding” of these pairs. The periodic potential for dislocations is Peierls barrier.

5.4 Peierls Barrier

We should ﬁrst discuss the Peierls-Nabarro barrier (Peierls potential). Peierls potential is an energy barrier for edge dislocation to overcome in order to glide. Frenkel (1926) [35] ﬁrstly attempted to estimate the theoretical shear stress by considering two adjacent arrays of atoms subjected to external shear. It turns out to be related

to shear modulus τm = Gb/2πa. And this value is way more than the experimental measurement of minimal shear stress to produce plastic deformation. Later, Peierls (1940) and Nabarro (1947) [36] redid the calculation using elastic theory. They

both showed that experimental stress (Peierls stress σP ) is a sensitive function of

−3 core structure, and determined by crystal bonding and structure. σP ∼ 10 when γ ∼ 0.3. Huntington (1955) [37] argued the symmetry of two conﬁgurations when dislocation glides. The period of the potential should be b/2. He estimated the amplitude of potential barrier as:

µa2(3.28 − 0.98σ) π B ' exp(− ) 16(1 − σ) 1 − σ

for lattice spacing equals plane spacing. σ is the Poisson ratio of the lattice and µ is the shear modulus. This result gives a numerical value for the barrier height. 62

(a) (b)

R ș 2 µm 2 µm

1 (c) mono-interstitial (d) த = 0.040 sec 3 3 10-1 Hist. 10-2 2 2 (µm) (µm) R R 10-3 0 2 4 6 8 1 1 Residence Time (1/30 sec.)

0 0

0 2 4 6 8 0.00 0.08 0.16 0.24 0.32 Time (sec) Hist.

1 (e) di-interstitial (f) த= 0.024 sec 3 3 10-1 Hist.

-2 2 2 10 (µm) (µm)

R R 10-3 0 2 4 6 8 1 1 Residence Time (1/30 sec.)

0 0

0 2 4 6 8 0.00 0.08 0.16 0.24 0.32 Time (sec) Hist.

Figure 5.2: Residence time plot (See citation: S. Kim Phd thesis). 63

Figure 5.3: Toy model. The zigzag line stands for Peierls potential. Defect is like a point mass located at center-of-mass of all mis-coordinate particles. The hopping process involves several attempts to overcome the barrier (the residence time), and how far (which valley) it hops to.

As we discussed, interstitial can be viewed as two dislocations gliding on the same crystalline axis. The energy barrier for interstitial to move along the gliding plane (axis), is actually Peierls barrier. And the hopping process of defect center-of-mass is discrete. To describe the microscopic process of defect diﬀusion, we propose a toy model (Fig. 5.3).

The average residence time τ in Fig. 5.2 is inverse the Kramer’s escape rate (See Appendix B) τ = τR ∼ τeβh

β = 1/kBT . The escape time, or residence time, follows a binomial distribution with the parameter equal to 1/R. R is the hopping/escaping rate in Kramer’s derivation (see Appendix B). Therefore it follows an exponential distribution. We ﬁt the histogram of the log of residence time with a straight line (Fig. 5.2 (d) (f) insets).

The time constants from fitting are τ mono = 0.040 seconds and τ di = 0.024 seconds. Di-interstitials, with shorter residence time τ, turns out to have a larger D?. We can roughly calculate the product of τ and D?. It’s 0.219 µm2 for mono-interstitial and 0.181 µm2 for di-interstitial. They are very close. This explains our observation and the discrepancy: mono-interstitial diffuses in 2D with longer residence time; di-interstitial diffuses in quasi-1D while having shorter residence time. 64

The unresolved question is, why they exhibit diﬀerent residence times while the Peierls barrier for dislocations retain the same form. Shear modulus and Poisson ratio are bulk properties of the lattice. It has been suggested that the excess strain caused by extra particle in di-interstitials may suppress the barrier height, thus making the hopping more frequent.

5.5 Nonergodicity

In Fig. 5.4 [34], we plot the cumulative positions of 5- and 7-coordinate particles for each frame, with respect to its center-of-mass position. On the top are vacancies, and bottom are interstitials. Colors stand for different configurations of each defect. Although they constitute different portions of the plot, in an ergodic system, their orientations should fluctuate with thermal energy with equal probability along the six directions (three axes) of the lattice. The histogram plot is an illustration of a spontaneous breaking of the 6-fold symmetry with the presence of point defects.

Optical tweezers causes disturbance to the lattice. It accidentally create some point defects. At the beginning, it’s usually a large defect region. After several frames (1/30 seconds), it becomes a single point defect. This manifests the spontaneous breaking of the rotational symmetry in 2DCC. For the rest hundreds of frames, we observe the full restoration of that six-fold symmetry in mono-vacancy, di-vacancy and mono-interstitial (Fig. 5.4 (a)-(f)). This is a clear evidence and new representation of ergodicity in a defect system. Ergodicity means it has experienced all possible microstates in phase space and all accessible microstates are equiprobable over a period of time. Our observation time window is long enough to capture the ergodicity of the ﬁrst three kinds of defects. However, we discover that di-interstitial 65

5-fold disclinations 7-fold disclinations (a) SV (b) SV 9 9 9 9 V4 V4

0 0 y (µm) y (µm) ï ï

ï ï mono-v mono-v. ï ï ï ï ï 0 ï ï ï 0 x (µm) x (µm) (c) SDa (d) SDa SDb SDb ' ' 'G 'G

0 0 y (µm) y (µm) ï ï

ï ï di-v. di-v. ï ï ï ï ï 0 ï ï ï 0 x (µm) x (µm) , (f) , (e) , , ,G ,G I4 I4

0 0 y (µm) y (µm) ï ï

ï ï mono-i. mono-i. ï ï ï ï ï 0 ï ï ï 0 x (µm) x (µm) (g) ', (h) ', ',G ',G ',G ',G ',G ',G

0 0 y (µm) y (µm) ï ï

ï ï di-i. di-i. ï ï ï ï ï 0 ï ï ï 0 x (µm) x (µm)

Figure 5.4: Plot of accumulated positions of 5- and 7-coordinate particles (disclinations) for mono- and di-vacancies, mono- and di-interstitials (See citation: S. Kim Phd thesis). 66 does not recover the underlying six-fold symmetry (Fig. 5.4 (g) (h)). Di-interstitial is nonergodic in our laboratory time window. This has something to do with the quasi-one-dimensional behavior, or say memory eﬀect.

5.6 Monte Carlo Simulation

The motion of defect is driven by random kicks of water molecules. The system is connected to the thermal bath, approaching equilibrium. Monte Carlo (MC) method can be applied to simulate the thermal equilibrium configuration of clean lattice (without defects) and even with a single point defect (vacancy or interstitial). In contrast to Brownian Molecular Dynamics, the randomness of the system is introduced in a form of random selection of particle and random displacement of that particle. We use Metropolis algorithm to differentiate the valid steps. The transition probability between steps is determined by the ratio of their configuration energy, which is computed by summing up all the interaction energy between particle pairs. In our analysis, the energy is dominated by Yukawa potential, e.g. screened Coulomb potential. Only nearest neighbor interaction is considered for simplicity.

Here goes the selection algorithm. In every step, we randomly select a particle in the system and ask for a random move. By computing the energy change because of the small displacement, we can obtain the ratio. If the resulting energy is lower than before, the algorithm accepts the step and continues the loop. If not, then draw another random number from our pseudo generator and compare with the ratio. If less, we roll back to the conﬁguration of previous step. It’s the standard Metropolis method, to random sampling the Markov Chain of our unknown probability distribution. 67

Figure 5.5: Snapshot of mono-vacancy in a simulation of 2DCC.

A typical snapshot of the system is in Fig. 5.5 and Fig. 5.6. It turns out with 108 iterations, which runs hourly on the local machine, that we observe the similar lattice vibration with experiment, but lower jumping rate of the defects. Improving convergence rate of Monte Carlo algorithm is always helpful to shorten the running time. Also there are kinds of variance reduction techniques to improve the eﬃciency of simulation, such as antithetic variable method, control variance method and low discrepancy sequence (to replace the pseudo random number sequence).

5.7 Conclusion and Future Directions

First, more statistics should be collected to verify the diﬀusion constant measurement in our laboratory time window. This means more events (vacancies and interstitials) and longer time. We may be able to observe the restoration of six-fold symmetry in di-interstitials. Di-interstitial has strong memory eﬀect. 68

Figure 5.6: Snapshot of mono-vacancy in a simulation of 2DCC with simulated colloids and delaunay triangulation.

Second, dimensionality is vital to diffusion constant analysis. One way is to exclude the effect of dimension by dividing a effective dimension factor (1 < d∗ < 2). With that, we can compare the microscopic hopping process over the potential barrier. Hexagonal lattice is trickier than rectangular lattice in the theory paper [35, 36, 37]. Further efforts are expected to verify the escape toy model proposed in the previous section.

Also computer simulation is becoming popular and powerful tools to assist the analysis and understanding of experimental observations. One question related to Monte Carlo simulation is the convergence speed. A faster collective-move ”event- chain” MC algorithm was recently applied to study two-step melting, providing a guideline for hard-disk solutions for large-scale computer simulation [12]. Appendix A

Kosterlitz-Thouless Transition 70

A.1 Model System

Since in 2D solids and x-y model, both dislocations and vortices interact via a loga- rithmic potential, and they both carry ”charge” (Burgers vector of dislocations and winding number of vortices), it is natural to study a model system with 2D Coulomb gas with same pair interaction. We assume the core energy Ecore is large compared to temperature. This means that fugacity y = e−βµ is a small quantity. We will see in a moment calculation is expanded to the dominant order of y. The calculation below is based on Kosterlitz-Thouless paper [2].

The hamiltonian of the model system is

1 X H(~r , ... ~r ) = U(| ~r − ~r |) 1 N 2 i j i6=j with

2 ~ri − ~rj U(r) = U(| ~ri − ~rj |) = 2q ln | | +2µ r0 where q is the unit charge and r0 is equivalent to a in previous discussion. We follow the same notation with Kosterlitz-Thouless paper [2] for convenience. Using this pair interaction as entry of the Boltzmann factor, we can evaluate the mean square separation between individual charges

∞ R e−βU(r)2πrdr · r2 R ∞ e3−2βq2 dx hr2i = r0 = r 2 1 . R ∞ e−βU(r)2πrdr 0 R ∞ e1−2βq2 dx r0 1

If 1 < βq2 < 2, the integral in the numerator diverges. Thus hr2i → ∞. We can work out the integrals if βq2 > 2,

βq2 − 1 hr2i = r 2 . 0 βq2 − 2 71

The polarizability is proportion to the mean separation of dipoles. Therefore, we

1 2 predict a phase transition at kBTc = 2 q . Above Tc, the closely bound dipole will dissociate and form a 2D plasma of opposite charged particles. This leads to a conducting state.

The physical picture is clear if we calculate the mean square separation d between dipole pairs. The probability of ﬁnding a pair within area A is inversely proportional to d2

RR −1 2 A exp(−βU(| ~ri − ~rj |)) π 2 h(d ) i = RR = 2 2 y + O(y ) wholespace exp(−βU(| ~ri − ~rj |)) r0 (βq − 1)

where y is the fugacity. We expand this to the leading order of y. This is a dilute gas cloud with mean separation between pairs much larger than the pair size when

µ/kBT 1. Only for those pairs with | r+ −r− |> d, their energy should be modiﬁed by pairs in between. The larger the pair size r, the more it is screened by other pairs. Kosterlitz and Thouless applied an iterated mean ﬁeld approximation to solve this coupled problem.

We introduce a dielectric constant ε(r) so that eﬀective pair interaction is

Z 0 2 2 dr 2q r Ueff (r) = 2q 0 0 + 2µ = ln + 2µ. r ε(r ) ε(r) r0

Then linear response theory is used to determine the polarizability per pair at r

∂ 1 p(r) = q lim hr cos θi = βq2r2. E→0 ∂E 4π

In the Boltzmann factor, an energy with external ﬁeld E is used

U(r) = Ueff (r) + Eqr cos θ. 72

Although this expression diﬀers by a prefactor of 2π with Kosterlitz’s original paper, it will not change the ﬁnal conclusion. The pair density between r and r + dr is calculated with the same argument

2 − 2βq y2 r ε(r) dn(r) = 4 2πrdr. r0 r0

Therefore, the change of dielectric constant between r and r + dr is

dε(r) = 4πdχ(r) = 4πp(r)dn(r).

2βq2 If we change variables x = ln(r/r0), v = (r) − 4, the diﬀerential equation becomes

dv = −πy2(v + 4)2e−xv dx

2 with boundary conditions v(x = 0 ⇐⇒ r = r0) = 2βq − 4 and v(x = ∞ ⇐⇒ r =

2 ∞) = 2βq /εbulk − 4, since ε(r0) = 1 and ε(∞) = εbulk. We take v ∼ v(0) and rescale two variables dv˜ = −e−x˜v˜. dx˜

There are two classes of solution depending on intial condition ofv ˜(0). Ifv ˜(0) >

v˜c(0), v(∞) > 0. Ifv ˜(0) ≤ v˜c(0), v(∞) → −∞. This shows a transition to a conductive state (ε = 0) with dipole pairs unbinding.

v˜c(0) ≈ 1.3. We work out the Tc as

2 q −1 = 2[1 − v˜c(0)πyc] ≈ 2(1 + 1.3πyc) kBTc

yc is the fugacity at Tc. Since this is an expansion over small fugacity, the leading order is equivalent to our estimation, which is an upper bound. 73

Another result from asysmptotic matching is the critial behavior of ε(T ) as temperature approach Tc from below.

2 r ! βq T − − 2 ∼ exp − ln ,T → Tc ε(T ) Tc − T

Later on, Young showed that the scaling equations of Kosterlitz x-y model can be derived with an alternative method [38]. The fundamental theories are identical. The difference lies in the unnecessary approximation made by Kosterlitz and Thouless. This difference is found to be so small that Kosterlitz and Thouless theory is self- consistently justified. We will discuss Young’s renormalization group method in the next section.

A.2 Scaling Equations

Start from the pair interaction

~ri − ~rj U(| ~ri − ~rj |) = 2πJ ln | | +2µ. r0

The partition function associated with this energy is

Z ∞ Z 2 2 X 1 Y d ~rid ~rj Z = δψ(r) e−βU(|~ri− ~rj |). (n!)2 (r )2 0 Ω2n i 0

Expand to the ﬁrst order O(y),

Z 2 2 d ~r −βU(r) 2 Z = 1 + y 2 e + o(y ). (r0) 74

As a reminder, the number of dipoles between r and r + dr is

2 y −βU(r) n(r)dr = 4 e 2πrdr. (r0)

Introduce ε(r) = 1 + 4πχ(r), thus dε(r) = 4πdχ(r). The change in susceptibility is the product of polarizability per pair and number of pairs

2 1 2 y −βU(r) dε(r) = 4πdχ(r) = 4π · βπJr · 4 e 2πrdr. 4π (r0)

The introduction of dielectric constant will modify the potential because of screening eﬀect. We change variable l = ln(r/r0) and scaling K(l) = βJ/ε(l),

Z r Z l ∗ 2πβJ 0 0 0 βU (r) = 0 0 dr = 2πK(l )dl . a r ε(r ) 0

With new variable,

4 r ∗ ∗ dε(l) = 2π2βJy2 e−βU (l)dl = 2π2βJy2e2le−βU (l)dl r0

and deﬁne

l 1 ∗ 1 Z y(l) = y √ e2le−βU (l) = y √ e2l exp[− 2πK(l0)dl0]. 2π 2π 0

It’s easy to derive the recursion relation of y(l) and K(l)

dy(l) = (2 − πK(l))y(l) dl

dK−1(l) = 4π3y2(l) dl

These are exactly the same scaling equations of Kosterlitz without making the 75

approximation from U to K [38].

The physical understanding of the relation is as below. When we rescale the system from a to , say 5a, some vortex pairs with size small than 5a will vanish, reducing the stiﬀness. Thus K−1(l) is always increasing with l. In contrast, y(l) (y(l) is positive) depends on the value of K−1(l). If the density of vortices decreases with increasing l, according to the deﬁnition, y(l) will also decrease. This is what

−1 we expect when K (l) < π/2 (below the transition temperature T < Tc). There is transition occuring since y(l) start to grow. This is a sign of vortex unbinding.

In Fig. A.1, a solution of the recursion relation is plotted in K−1(l) − y(l) plane. Red line is the starting points of the ﬂow. The equation of the red line is from the relation between y and K,

E y(l) ∼ y = exp − core ∼ eK(l). kBT

−1 Transition occurs at T = Tm with ﬂow pointing to K = π/2. The ﬂow below Tm will tend to zero dislocation probability y and those above will tend to large y and vanishing K. The sytem will lose the rigidity and incur a dislocation proliferation.

With the recursive relation, we can also derive the unusual divergence behavior of the correlation length ξ(T )

b0 ξ(T ) ∼ exp √ . T − Tc

This is a much faster divergence than usual type and can be understood in a simple manner. Since the phase field is significantly modified by nearby free vortices rather √ than bound pairs, they remain correlated beneath a typical distances 1/ nv. Density 76

0.10

0.08

0.06 y

II C < 0, 0 < t < 8 Πy 0.04 0

0.02

III C>0, t > 8 Πy 0 I C>0, t<0 0.00 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 K

Figure A.1: Renormalization group ﬂow derived from Kosterlitz-Thouless recursion relation. Each trajectory corresponds to a certain temperature with arrows indicating the direction of in- 3 creasing length scale l. Constant C(t) = −t(8πy0 −t)+O(t ) and t is the dimensionless temperature t = (T − Tc)/Tc. (I acknowledge Professor J.M.Kosterlitz for providing this ﬁgure.) 77

of free vortices is proportional to the core energy versus temperature. Therefore,

s 1 E ξ(T ) ∼ √ ∼ exp core nv kBT

can explain where the exponential dependence comes from. And the diﬀerence lies in the correction of renormalization.

A.3 Real System

A.3.1 X-Y Model

In x-y model, we can separate the order parameter into two parts,

φ(~r) = ψ(~r) + φ(~r).

Spin-wave ﬂuctuation will destroy LRO in two dimension. If we ignore vortices, correlation function is a power law decay

− kB T h~si · ~sji = hexp(i(φi − φj))i ∼ | ri − rj | 4πJ .

Apart from spin-wave, we consider only vortex part φ(~r), which deﬁnes the local minima of energy in the presence of vortices. The Hamiltonian

1 X X ~ri − ~rj H = U(| ~ri − ~rj |) = 2πρs qiqj ln | | +2µ 2 r0 i6=j i6=j 78

2 2 with 2πρs ↔ q and µ = π ρs in the model system. So we have the equation of Tc,

2πρ 2πρ s ≈ 2 1 + 1.3π exp − s . kBTc kBTc

Solve for 2πρs/kBTc ≈ 1.12,

kBTc ≈ 0.88πρs.

A.3.2 2D solids

We divide the displacement ﬁeld into phonon and dislocation part ~u(~r) = ~v(~r)+~u(~r). Long-wavelength phonon modes destroy the LRO and dislocation unbinding causes the transition to liquid phase. Because of the vector nature of Burgers vector, there is an additional term in the pair interaction

Kb2 r 1 Upair(r) ≈ ln | | − cos 2θ + 2µ. 4π r0 2

The main task is to find an analogy to the dielectric constant ε(r) in the model system. Kosterlitz and Thouless examined the effect of dislocation dipoles on the stress function χ(r) and find

Z Z K 2 0 0 0 1 2 0 0 0 χ(~r) = d r η(~r )g(~r−~r ) = −1 d r η(~r )g(~r−~r ). 1 − K(C1 + C2) K − (C1 + C2)

Thus the eﬀect of dislocation pairs is clear, renormalizing the stiﬀness K−1 to K−1 −

(C1 + C2),

ε(r) = 1 − K(C1(r) + C2(r))

where the two constants 1 c (r) = − βb2nhr2i 1 4 79

1 c (r) = βb2nhr2 cos 2θi. 2 8

The average is taken with the Boltzmann factor exp(−βUeff ) deﬁned as in the model system.

A.4 Asymptotic Solutions

Use linear response theory to calculate the polarizability per pair at location ~r. The eﬀective energy under external electric ﬁeld E is

2q2 r Ueff (r) = ln + 2µ + Eqr cos θ ε(r) r0

and the polarizability is deﬁned as

∂ p(r) ≡ q lim hr cos θi. E→0 ∂E

We can work out the average with the Boltzmann factor e−βUeff

2π r+dr R dθ0 R r0dr0(r0 cos θ0) exp[−βU (r0) − βEqr0 cos θ0] f(E) hr cos θi = 0 r eff ≡ R 2π 0 R r+dr 0 0 0 0 0 g(E) 0 dθ r r dr exp[−βUeff (r ) − βEqr cos θ ]

and do the partial derivative before evaluating the complicate integral

∂ f(E) f 0g − g0f p = q lim = q lim . E→0 ∂E g(E) E→0 g2

It’s obvious that f(E) = 0 since the integral over θ is zero.

2 0 2 Z r+dr r0 −2βq /ε(r ) r −2βq /ε(r) g(E = 0) = 2πy2 r0dr0 = 2πy2 rdr r r0 r0 80

2 0 2 Z 2π Z r+dr r0 −2βq /ε(r ) 1 r −2βq /ε(r) f 0(E = 0) = βqy2 cos2 θ0dθ0 r03dr0 = βqy2 r3dr 0 r r0 2 r0

Then the approximate polarizability is

f 0(0) 1 p ∼ q = βq2r2. g(0) 4π

dv −xv Solution of dx = −e can be divided into two catergories according to their initial condition

v(∞) ≥ 0, v(0) ≥ vc(0)

v(∞) → ∞, v(0) > vc(0)

We can change variable z = 1/xv, w = z − ln(x/v) = xv − ln(x/v) to move the singularity to the origin. We try to derive the new diﬀerential equation with respect to z and w dw 1 v d x 1 −x0v + v0x = − − = − − dz z2 x dz v z2 xv

where the derivative 0 is for z. From the deﬁnition of z and the old equation,

1 dz 1 1 = − + e−xv v dx x2v2 xv3

1 dz 1 1 = − + e−xv x dv x2v2 vx3

Then the new equation is

dw 1 1 ew + 1 1 1 w = − + · = − + coth . dz z2 z ew − 1 z2 z 2

This equation doesn’t have analytical solution. For w(z) ≥ 0 corresponding to

solution class y(0) ≥ yc(0), we have three regions (Boundary layers) with diﬀerent singularities: w → 0+, z → 0; w → ∞+, z → 0; w → ∞+, z → ∞. For each region, 81

we can do approximation with coth(w/2). In region 1,

dw 1 2 = − + . dz z2 wz

Changing the variable 2t = 1/z − w,

dt 1 1 dw 1 = (− − ) = − dz 2 z2 dz wz

dz = −wz = 2tz − 1 dt

General solution of this is et2 . Thus the full solution z(t) = et2 f(t), where et2 f 0(t) = 1. √ Z t1 t2 −s2 π t2 z(t) = e e ds = e (erf(t1) − erf(t)) t 2 where erf(x) is the error function. The constant from integration is represented by

+ the upper limit t1. In region 1, as z → 0 , t → +∞. The choice of t1 is consistent with the property of the asymptotic region

+ t1 → ∞, v(0) → vc (0).

In region 2 and 3 (w → +∞), we approximate

dw 1 1 = − + dz z2 z

1 w(z) = + ln z + const. z

with const. = w − ln z − 1/z = − ln(x/v) − ln(1/xv) = 2 ln v(x). We can determine the constants in two region from their asymptotic behavior. In region 2, z → 0+, x → ∞, we take 2 ln v(∞); In region 3, z → ∞, x → 0, we take 2 ln v(0). 82

With all these solutions set up, we can do asymptotic matching at intermediate region, say w(z) = 2. The first task is to find the intersection with respect to variable t. We use Taylor expansion to find asymptotic series of the intersections of √ 1 π t2 z(t) = 2(t+1) and z(t) = 2 e (erf(t1)−erf(t)). Fig shows the plot of two functions.

The ﬁrst solution is between region 1 and 2. The leading order is obvious t → ∞.

If we set t = t1, z(t = t1) → 0. To the next order, t = t1 + δ, δ ∼ o(t1),

t1 1 2 Z 2 = e(t1+δ) e−s ds. 2(t1 + 1 + δ) t1+δ

We use the asymptotic series for x 1,

√ Z x −x2 −s2 π e 1 −2 e ds = − (1 − 2 + o(x )) 0 2 2x 2x

and keep the leading order,

" 2 2 # −t1+δ −t1 2t1δ 1 2 e e 1 e = e(t1+δ) − = − 2(t1 + 1 + δ) 2(t1 + δ) 2t1 2(t1 + δ) 2t1

e2t1δ 1 1 = ∼ 2 t1 (t1 + δ)(t1 + δ + 1) t1

So we have δ = − ln t1 , and one intersection is 2t1

ln t1 t = t1 − . 2t1

Do asymptotic matching at this intersection to determine the integration constant v(∞), 1 2 ln v(∞) = w − − ln z = −2t − ln z(t) z 83

and √ −t1 v(∞) = e 2t1.

Another intersection lies between region 2 and 3. Numerical solution for

√ 2 1 π tc = e (1 − erf(tc)) 2(tc + 1) 2

is tc = −0.843. The second order correction is not as important. But with the same

logic, t = tc + δ, we can ﬁnd

2 2 t + 1 etc −t1 δ = c · . −2tc − 1 t1

This process requires a little bit algebra since we set the upper limit to exact value

t1 instead of ∞. The asymptotic behavior of ln v(∞) is

s 1 1 r T ln v(∞) ∼ −t1 + ln t1 ∼ − log ∼ − log . 2 v(0) − vc(0) Tc − T

− So when T → Tc ,

! q2 r T − 2 ∼ exp − log . kBT ε(T ) Tc − T

2 At the critical point, q /kBT ε(T ) − 2 ∼ 0 and the susceptibility does not diverge!

We can also do matching at t = tc is straightforward

1 1 2 ln vc(0) = w − − ln z(tc) = −2tc + ln = −2tc ln 2(tc + 1) z(tc) z(tc) and

p −2t v˜c(0) = 2(tc + 1)e c ∼ 1.3. 84

We put the ”tilde” back to the solution and get the K-T transition temperature

2 2 2q 2 2q vc(0) = − 4 = (2πycβq )v ˜c(0) =v ˜c(0)πyc kBTc kBTc

2 q −1 = 2[1 − v˜c(0)πyc] ≈ 2(1 + 1.3πyc) = 2 + O(yc) kBTc

In another aspect, temperature is approaching Tc from above (v(0) ≤ vc(0)). The boundary layer lies at z → ∞, w → −∞. As before, in region 1,

dw 1 2 = − + . dz z2 wz

Changing the variable 2t = 1/z − w and we get the full solution

∞ √ 2 Z 2 π 2 z(t) = et ( e−s ds + K0) = et (1 − erf(t) + K) t 2

where K ≥ 0. This is diﬀerent with T < Tc case where K < 0 and we use t1 to replace ∞. The reason to pick this positive constant is that when t → +∞, w ∼ −2t ∼ −∞, meanwhile z → +∞. This requires we choose a positive term Ket2 driving z to +∞ when t → +∞. On the other hand, in the boundary layer w → −∞, coth w/2 ∼ −1

1 w(z) = − ln z + const. z

1 with const. = w + ln z − 1/z = − ln(x/v) + ln(1/xv) = −2 ln x. To make z = xv(x) going to inﬁnity, in this case, we choose x = X to be the root of v(x) = 0. With ﬁnite X and v(X) = 0, z(x = X, v = 0) → ∞ and w(x = X, v = 0) = xv−ln(x/v) → −∞. We can do asymptotic matching at w = ±2 and obtain the following relation

r p T + ln X ∼T →Tc − ln(vc(0) − v(0)) ∼ ln T − Tc 85

and r T + X ∼T →Tc exp ln T − Tc

We will see shortly why this length scale is important.

We can estimate the conductivity of the system right after the unbinding pairs start to carry charge. First estimate the density of pairs, of which the size is below R 2 2 2−2βq − 2βq R Z R Z R 2 ε(r) 1 − y r π r0 n(R) = dn(r) = 4 2πrdr = 2 · 2 . r0 r0 r0 r0 r0 βq − 1 Here, we approximately take ε(r) to be 1. Thus the change of dissociated pairs (conducting charges) δn(T ) is

dn δn(T ) = − dR ∼ −2dR ∼ −dX dR

and relative change over the total density of charges n

δn(T ) r T ln ∼ − exp ln . n T − Tc

+ The conclusion is that as T → Tc ,

δn(T ) ∼ 0 n

dp δn(T ) ∼ 0, for any integer p. dT p n

With derivative of any order is zero, the DC conductivity above the transition point is a ”ﬂat function”. 86

A.5 Correlation Length

This calculation is from famous Chaikin and Lubensky textbook [? ]. The recursion relation of y(l) and K(l) dy(l) = (2 − πK(l))y(l) dl dK−1(l) = 4π3y2(l) dl

We change variable K(l) = K∗(l) = 2(1 − x(l))/π, x(l) = 1 − π/2K(l). With dx/dK−1 = 2(1 − x2)/π ∼ 2/π, dx = 8π2y2 dl dy = 2xy dl

With simple algebra, we can cancel the variable on the right.

d(x2) = 4π2 d(y2)

1 y2 = (x2 + C) 4π2

The ﬁxed point (x∗ = y∗ = 0, k = k∗ = 2/π) corresponds to C = 0. We can write

2 C = b (T −Tc) as a linear dependence. For T < Tc and C < 0, with useful integration

R du 1 1−u u2−1 = 2 ln 1+u , the solution is

√ 1 1 − u(l) 1 + u(0) 2l −C = ln · 2 1 + u(l) 1 − u(0) where √ √ √ 1 − D exp(−4 −Cl) √ x(l) = −Cu(l) = − −C 0 √ ∼ − −C 1 + D0 exp(−4 −Cl) 87

when l → ∞. For T > Tc and C > 0, we can integrate out

Z x(l) dx 1 x(l) x(0) √ √ √ 2l = 2 = arctan − arctan . x(0) x + C C C C √ If we choose x(l∗) C, π 2l∗ = √ . C

Thus the correlation length

0 ξ ∗ b = el = exp √ . a T − Tc

This is one of the main results of K-T theory, ν = 0.50. In comparison with Halperin and Nelson’s result for solid phase, ν = 0.3696 is diﬀerent from this analogy [3, 4]. This diﬀerence is mainly due to the situation when two Burgers vector add up to another unit vector in triangular lattice, thus adding a correction term to the recursion relations. Appendix B

Diﬀusion 89

B.1 Random Walk to Diﬀusion

Classic random walk describes a discrete process. Brownian motion W (t) (so called Wiener process) is a continuous time stochastic version with the following properties: (1) W (0) = 0 (2) W (t) has independent increments with W (t) − W (s) ∼ N(0, t − s) for t > s They are connected internally. Starting from the simplest random walk model, we can derive the complicated equation describing diﬀusion process.

Consider an one-dimensional (1D) random walk with step size a and time step τ, and non-zero drift δ. The probability of ﬁnding the walker at x = ai, t = nτ is related to previous time,

1 + δ 1 − δ P (ai, nτ) = P (a(i − 1), (n − 1)τ) + P (a(i + 1), (n − 1)τ). 2 2

This is a Markov process. The probability of jumping to right site is (1 + δ)/2. We can work out the equation by grouping components of same entry and take the limit of inﬁnitesimal a and τ. We obtain the diﬀerential equation similar to Fokker-Planck equation, ∂P a2 ∂2P aδ ∂P = − ∂t 2τ ∂x2 τ ∂x

2 with D0 = a /2τ and vD = aδ/τ.

We can estimate the spread of the probability distribution, or in other words, σ in Gaussian distribution. For 1D random walk,

n X x(n) = lia i=1 90

with li = ±1. It’s a stochastic process. The mean square spread at t = nτ

n X a2 hx2(t)i = h( l a)2i = na2 = 2 · · nτ = 2D t i 2τ 0 i=1

With finite drift δ, the effect got smeared out when taking the ensemble average and we get the same result. On the other hand, we can solve the partial differential equation with initial condition P (x, 0) = δ(x),

2 1 − (x−vDt) P (x, t) = √ e 4D0t . 4πD0t

Starting with a delta function, the probability density spreads out with a speed of

2D0, and a translational motion of the mean.

∂P (x,t) ~ From Fick’s law ∂t + ∇ · j(x, t) = 0, we can obtain the diﬀusion equation directly ∂n = D∇2n. ∂t

The Green function of this equation G(x, t) and its Laplace and Time Fourier Trans- form is [7] −1 − izG~ (~q, z) + q2DG~ (~q, z) = 0.

Solve for 1 G~ (~q, z) = . −iz + q2D

Do inverse transformation

Z d Z d 2 d q i~q·~x d q i~q·~x −q2Dt 1 − x G(~x,t) = e G(~q, t) = e e = √ e 4Dt (2π)d (2π)d 4πDt

This is a Gaussian distribution with zero mean and variance 2Dt (d = 1). This result can be extended to any dimension d. 91

B.2 Kramer’s Escape Rate

This section introduces Kramers’ escape rate problem in his seminal paper on thermally activated barrier crossing [39]. Consider a collection of Brownian particles initially reside in a deep potential well. The potential proﬁle U(x) consists a deep well (with source) at x = a, a sink at x = b and an energy barrier at x = c. Fokker-Planck equation gives the evolution of particle density (probability),

∂P (x, t) ∂2P (x, t) ∂P (x, t) = D − v . ∂t ∂t2 ∂x

In the form of external force,

∂P (x, t) ∂ ∂P (x, t) = D − Γf(x)P (x, t)). ∂t ∂x ∂t

∂P (x,t) In steady state ( ∂t = 0), the net ﬂow rate (current) is

∂P dU J = − − β P ∂x dx

where D = kBT/β = Γ/β. Integrate J(x) over interval (a, b),

P (a)eβU(a) − P (b)eβU(b) J = . R b βU(x) a e dx

If we set J = 0, we get the detail balance condition of equilibrium,

P (a) = eβ(U(b)−U(a)). P (b)

Because of the immediate removal feature at x = c, P (b) = 0.

The current J is the conditional probability of escape rate given that the particle 92

is inside the potential well at x = a. Thus, the escape rate R = J/nA. nA is a measure the probability of ﬁnding a particle inside the well. It’s actually the density at x = a (or in proximity). We can expand the potential around point x = a,

0 −βU(x) 0 −βU(a)− 1 βU 00(x−a)2 P (x) = C e ' C e 2 .

We can work out the integral and obtain the escape rate, which is the Kramer’s escape rate formula, Γ R = U 00(a)U 00(b)e−β(U(c)−U(a)). 2π

The escape rate depends exponentially on the barrier height U(c) − U(a). Appendix C

IDL Scripts and C++ Code 94

C.1 IDL Scripts

Below routines are for particle tracking with standard library:

Adjust the window size to ﬁt our frame size:

1 Window, xsize = 640, ysize = 480 2 3 s e t p l o t , ”win”

First step is to test image quality and select tracking parameters:

1 a = readjpgstack(’fg’, 0, 100) 2 a1 = a (∗ , ∗ , 0) 3 4 ; I n v e r s e and show the image 5 temp = 255b − a1 6 tvscl , temp 7 8 ;Apply bypass filter with low and high pass parameter 1 and 7 p i x e l s 9 b = bpass(temp, 1, 7) 10 11 ; Find f e a t u r e s and compare the results 12 f = f e a t u r e (b , 9) 13 f 0 = fover2d ( a , f ) ; or use / big 14 plot , f (2 , ∗) , f (3 , ∗) , psym = 6 15 16 ;Use alternative keyword ”mass” to improve the selection 17 f = feature(b, 9, mass = 100) 18 p l o t h i s t , f (0 , ∗) mod 1

Then do bundle tracking with the ”best” parameters found above: 95

1 spretrack1, ’fg’, bplo = 1, bphi = 7, dia = 9, mass = 200, start f r a m e = 0 , stop frame = 100, /jmulti, /invert 2 ;This will generate a gdf file called ”pt.fg0 10 0 ” : 3 4 pt = r e a d gdf(’pt.fg0 10 0 ’ ) 5 p = pt (∗ , where(pt(5, ∗) eq 0)) ; get frame 0 6 t = track(pt, 5, goodenough = 10, memory = 5) 7 8 ;Save the trajectory for f u t u r e use 9 w r i t e gdf, t, ’tt.crop0 311’ ; write track trajectory to file 10 t = r e a d gdf(’tt.crop0 311 ’ ) 11 12 ;Compute the average (equilibrium) positions for each site across all available frames and obtain the tilted angle of the lattice ($0.4125 $ ) 13 14 equipos , t , 0 , 311 15 equip = r e a d gdf( ’equip0 311 ’ ) 16 17 ; Feed $t$ and $equip$ array into the routine to calculate dispersion curve with certain wavenumbers 18 vdispersion , t, equip, 0, 311, 0.4125, wavenumber = 50

The following is to plot the autocorrelation function g(r) and g6(r):

1 ;Plot to see the crystal 2 cgScatter2D, p(0, ∗ ) , p ( 1 , ∗ ), psym = 6, Symsize = 0.3 3 gr = ericgr2d(p, rmax = 60) ; $60 pixels = 5 \mu m$ 4 make movie1 , pt 5 6 ;For a single frame 7 res = g6(pt,0) ; frame 0 8 plot, res(0, ∗ ) , r e s ( 1 , ∗ ) ; gr 96

9 plot, res(0, ∗ ) , r e s ( 2 , ∗ ) ; g6r 10 plot , r e s ( 0 , ∗ ) , r e s ( 2 , ∗ ), /ylog, yrange = [0.001,1] ; semi log plot 11 12 ;Also calculate dislocation density in the frame 13 d i s l o c a t i o n density , pt, 0, 8920

C.2 IDL Source Code for Particle Tracking and

Phonon Mode Analysis

Spretrack.pro

1 ; Options related to feature & bpass: 2 ; bplo,bphi,dia,sep,min,mass 3 ; 4 ; img= the image to be examined 5 ; b = bpass(img,bplo,bphi) 6 ; f = feature(b,dia,sep,min=min,masscut=mass) 7 ; 8 ; Options related to the image: 9 ; /invert = invert the image before bpass 10 ; /field = process even & odd fields separately (only for CCD cameras ) 11 ; /first = only process first & second frames from each file 12 ; ( u s e f u l for a quick check) 13 ; fskip = regrid the time stamp (useful for time−l a p s e ) 14 ; ( this option mostly unused) 15 ; 16 ; Options related to file type: 17 ; / gdf = use r e a d g d f 18 ; / t i f f = use r e a d t i f f 97

19 ; /multi = use readtiffstack (multiple −image tiff files) 20 ; / noran = use read noran (Noran Oz confocal files) 21 ; / nih = use read nih (NIH Image files) 22 ; /jmulti = use readjpgstack (multiple image jpeg files) 23 ; 24 ; Additional options: 25 ; /quiet = supress printing messages 26 ; thresh = use a threshold (of the specified value) before bpass 27 ; /nofix =don’t try to fix a bug with noran files 28 ; prefix =”pt.” by default 29 ; 30 ; Add option to read sequential jpg files , and background substraction 31 ; function , and s t a r t f r a m e , stop frame option 32 33 pro spretrack , stk , $ 34 bplo=bplo ,bphi=bphi,dia=dia ,sep=sep ,min=min,mass=mass, $ 35 invert=invert,field=field ,first=first ,fskip=fskip, $ 36 gdf=gdf, tiff=tiff ,noran=noran,nih=nih,multi=multi , $ 37 quiet=quiet ,thresh=thresh ,nofix=nofix ,prefix=prefix , $ 38 s t a r t frame=start f r a m e , stop frame=stop frame ,jmulti=jmulti , $ 39 back=back,avi=avi , option=option , nobpass=nobpass 40 41 msg=’ D e f au l t s : ’ 42 i f ( not keyword set(bplo)) then begin 43 bplo = 1 & msg=msg+” bplo=1” 44 e n d i f 45 i f ( not keyword set(bphi)) then begin 46 bphi = 5 & msg=msg+” bphi=5” 47 e n d i f 48 i f ( not keyword set(dia)) then begin 49 dia = 9 & msg=msg+” dia=9” 50 e n d i f 51 i f ( not keyword set(sep)) then begin 98

52 sep = dia+1; this is what feature uses as a default 53 msg=msg+” sep−unset ” 54 e n d i f 55 i f ( not keyword set(min)) then begin 56 min = 0 57 msg=msg+” min−unset ” 58 e n d i f 59 i f ( not keyword set(mass)) then begin 60 mass = 0 61 msg=msg+” mass−unset ” 62 e n d i f 63 prefixflag=1; means user set prefix 64 i f ( not keyword set(prefix)) then begin 65 p r e f i x=’ pt . ’ 66 prefixflag = 0; means user didn’t set prefix 67 e n d i f 68 69 i f ( not keyword set(quiet)) then begin 70 s l e n = s t r l e n (msg) 71 i f (slen lt 10) then msg=msg+” no default values used, all user− d e f i n e d ” 72 print,”starting epretrack...” 73 print , msg 74 e n d i f 75 76 ; s e t frame numbers 77 i f not keyword set ( s t a r t frame) then start f r a m e = 0 78 i f not keyword set ( stop frame) then stop frame = 0 79 80 i f (size(stk,/type) eq 7) then begin 81 i f not keyword set(jmulti) and not keyword set(avi) then begin 82 filen = findfile(stk) 83 i f filen(0) eq ’’ then message,”No file ’”+stk+”’ found” 99

84 n f i l e s = n elements(filen) 85 e n d i f 86 i f keyword set(jmulti) then begin 87 f i l e n = stk 88 n f i l e s = 1 89 e n d i f 90 i f keyword set(avi) then begin 91 f i l e n = stk 92 n f i l e s = 1 93 e n d i f 94 u s i n g f i l e s = 1 95 e n d i f else begin 96 n f i l e s = 1 97 u s i n g f i l e s = 0 98 e n d e l s e 99 100 ; fskip is the frame/field # increment during time lapse video 101 i f not keyword set(fskip) then fskip=1 102 rep = 1 103 104 for j =0, n f i l e s −1 do begin 105 106 i f (usingfiles eq 1) then begin 107 i f ( keyword set(jmulti)) then begin 108 print, ’reading image stack: ’+ffilename(filen(j),start f r a m e )+’ to ’+ffilename(filen(j),stop frame ) 109 e n d i f else begin 110 print,’reading image stack: ’+filen(j) 111 e n d e l s e 112 113 i f ( keyword set(avi)) then stk = i readvideo(filen(j)) 114 i f ( keyword set(gdf)) then stk=read gdf(filen(j)) 115 i f ( keyword set(tiff)) then stk=read tiff(filen(j)) 100

116 i f ( keyword set(multi)) then stk=readtiffstack(filen(j),start f r a m e =s t a r t f r a m e , stop frame=stop frame ) 117 i f ( keyword set(jmulti)) then stk=readjpgstack(filen(j),start f r a m e =s t a r t f r a m e , stop frame=stop frame ,/option) 118 i f ( keyword set(nih)) then stk=read nih(filen(j)) 119 i f ( keyword set(noran)) then begin 120 stk=read noran(filen(j) ,/lomem) 121 i f ( not keyword set(nofix)) then begin 122 nnoranfix=n elements(stk(0,0, ∗ )) 123 foo=f l t a r r ( nnoranfix −1) 124 ; next b i t from ” imagecor2 ” 125 i f (nnoranfix lt 200) then begin 126 for i=1,nnoranfix −1 do begin 127 foo ( i −1)=correlate(stk( ∗ , ∗ , i −1) , stk ( ∗ , ∗ , i ) ) 128 endfor 129 temp=min ( foo , minorfix ) 130 e n d i f else begin 131 for i=nnoranfix −100,nnoranfix −1 do begin 132 foo ( i −1)=correlate(stk( ∗ , ∗ , i −1) , stk ( ∗ , ∗ , i ) ) 133 endfor 134 temp=min ( foo ( nnoranfix −100:nnoranfix −2),minorfix) 135 minorfix=minorfix+nnoranfix −100 136 e n d e l s e 137 e n d i f 138 e n d i f 139 e n d i f else begin 140 print,’using the image array given to me’ 141 e n d e l s e 142 143 i f keyword set(invert) then stk = 255b−stk 144 i f keyword set(thresh) then stk = (stk > thresh ) 145 146 ns = n elements(stk(0,0, ∗ )) 101

147 i f ns gt 200 then rep = 25 148 149 s s=s i z e ( stk ( ∗ , ∗ , 0 ) ) 150 sx = s s ( 1 ) 151 152 i f keyword set(first) then ns = 1 ;handy for a quick looksee... 153 154 r e s = f l t a r r ( 6 ) 155 i f keyword set(field) then begin 156 for i = 0 , ns−1 do begin 157 i f (((i+1) mod rep eq 0) and not keyword set(quiet)) then begin 158 print,’processing fields of frame’$ 159 +strcompress(i+1)+’ out of’+strcompress(ns)+’.... ’ 160 e n d i f 161 ; the next five lines are from J Crocker’s ”fieldof” 162 sz=size(array) & img=reform(stk( ∗ , ∗ , i ) ) 163 i f keyword set(back) then img = bytscl(abs(img − back ) ) 164 f=0 & ny2=f i x ( ( sz ( 2 )+(1− f))/2 ) & rows=indgen(ny2) ∗2 + f 165 e v e n f i e l d = img (∗ , rows ) 166 f=1 & ny2=f i x ( ( sz ( 2 )+(1− f))/2 ) & rows=indgen(ny2) ∗2 + f 167 o d d f i e l d = img (∗ , rows ) 168 im = bpass(evenfield ,bplo,bphi,/field) 169 f = f e a t u r e ( im (2∗ bphi : sx −2∗bphi , ∗ ) ,dia ,sep,min=min, $ 170 mass=mass,/field,quiet=quiet) 171 i f f ( 0 ) ne −1 then begin 172 f ( 0 , ∗ , ∗ ) = f ( 0 , ∗ , ∗ ) +2∗bphi 173 nf = n elements(f(0, ∗ )) 174 res = [[res],[f,fltarr(1,nf)+2∗ i ∗ f s k i p ] ] 175 e n d i f 176 im = bpass(oddfield,bplo,bphi,/field) 177 f = f e a t u r e ( im (2∗ bphi : sx −2∗bphi , ∗ ) ,dia ,sep ,mass=mass, $ 178 min=min , / f i e l d , q u i e t=q u i e t ) 179 i f f ( 0 ) ne −1 then begin 102

180 f ( 0 , ∗ , ∗ ) = f ( 0 , ∗ , ∗ ) +2∗bphi 181 f ( 1 , ∗ ) = f ( 1 , ∗ )+1 182 nf = n elements(f(0, ∗ )) 183 res = [[res],[f,fltarr(1,nf)+2∗ i ∗ f s k i p +1]] 184 e n d i f 185 endfor 186 e n d i f e l s e begin 187 f o r i = 0 , ns−1 do begin 188 if (((i+1) mod rep eq 0) and not keyword set(quiet)) then begin 189 print , ’ p r o c e s s i n g frame ’+$ 190 strcompress(i+1)+’ out of’+strcompress(ns)+’.... ’ 191 e n d i f 192 i f keyword set(back) then img = bytscl(abs(stk( ∗ , ∗ , i ) − back ) ) e l s e img = stk ( ∗ , ∗ , i ) 193 i f keyword set(nobpass) then im = img else im = bpass(img,bplo, bphi ) 194 f = feature(im,dia,sep,mass=mass,min=min,quiet=quiet) 195 nf = n elements(f(0, ∗ )) 196 i f ( f ( 0 ) ne −1) then res = [[res],[f,fltarr(1,nf)+i ∗ f s k i p ] ] 197 endfor 198 e n d e l s e 199 200 i f ( not keyword set(nofix) and keyword set(noran)) then begin 201 nn=n elements(res( ∗ , 0 ) ) 202 r e s (nn−1 ,∗)=r e s (nn−1 ,∗) + (nnoranfix −minorfix −1) 203 r e s (nn−1 ,∗)=r e s (nn−1 ,∗) mod nnoranfix 204 s=s o r t ( r e s (nn−1 ,∗)) 205 r e s=r e s (∗ , s ) 206 e n d i f 207 208 i f ( u s i n g f i l e s eq 1) then begin 209 wname = p r e f i x+f i l e n ( j ) 210 i f keyword set(jmulti) then wname = wname + strtrim(string( 103

s t a r t frame) ,2)+’ ’ + strtrim(string(stop frame ) ,2) 211 e n d i f e l s e begin 212 i f ( p r e f i x f l a g eq 0) then begin 213 wname = ” pretrack . gdf ” 214 print,”I wasn’t given a file name, so writing data” 215 print , ” to default f i l e : ” ,wname 216 e n d i f e l s e begin 217 wname = p r e f i x 218 e n d e l s e 219 e n d e l s e 220 i f ( not keyword set(quiet)) then $ 221 print,’writing output file to:’ + wname 222 w r i t e g d f , r e s ( ∗ , 1 : ∗ ) ,wname 223 endfor 224 225 end

Equipos.pro

1 pro Equipos, tt, startf , stopf, xmin, xmax, ymin, ymax 2 on error , 2 3 4 PI=3.14159 5 6 framenumber = stopf −s t a r t f +1 7 partnumber = max(tt(6, ∗ ) )+1 8 equipos = fltarr(2, partnumber) 9 count = intarr(partnumber) 10 filename = ’equip’+strtrim(string(startf),2)+’ ’+strtrim(string( s t o p f ) ,2) 11 12 13 ntt = n elements(tt(0, ∗ )) 104

14 for i = 0 , ntt −1 do begin 15 equipos[0,tt[6,i]] +=tt[0,i] 16 equipos[1,tt[6,i]] +=tt[1,i] 17 count [ t t [ 6 , i ] ] +=1 18 endfor 19 20 21 coun3=0 22 for i=0, partnumber−1 do begin 23 equipos[0,i] /=count[i] ;equilibrium position 24 equipos[1,i] /=count[i] 25 i f equipos[1,i] lt ymin or equipos[1,i] gt ymax or equipos[0,i ] l t xmin or equipos[0,i] gt xmax then begin ;boundary 26 equipos[0,i] =10000 ;set boundary to 10000 as a flag 27 coun3 +=1 28 e n d i f 29 endfor 30 31 equi = fltarr(2, partnumber−coun3 ) 32 c = intarr(partnumber−coun3 ) 33 ; eluminate boundary particles 34 coun2 =0 ; coun2==partnumber−coun3 35 for i = 0, partnumber−1 do begin 36 i f equipos[0,i] ne 10000 then begin 37 equi[0,coun2]=equipos[0,i] 38 equi[1,coun2]=equipos[1,i] 39 c ( coun2 ) = count ( i ) 40 coun2 +=1 41 e n d i f 42 endfor 43 44 coun1 = 0 45 for i = 0 , coun2−1 do begin ; eluminate repeated particles 105

46 i f equi[0,i] eq 0 then continue 47 tempx = equi [ 0 , i ] ∗ c ( i ) ; Sum 48 tempy = equi [ 1 , i ] ∗ c ( i ) 49 tempc = c ( i ) 50 for j = i+1, coun2−1 do begin 51 i f sqrt((equi[0,i]− equi(0,j))ˆ2 + (equi[1,i]− equi(1,j))ˆ2) lt 8 then begin 52 tempx += equi [ 0 , j ] ∗ c ( j ) 53 tempy += equi [ 1 , j ] ∗ c ( j ) 54 tempc += c ( j ) 55 equi [ 0 , j ] = 0 ; f l a g 56 equi [ 1 , j ] = 0 57 coun1 +=1 58 e n d i f 59 endfor 60 equi[0,i] =tempx/tempc 61 equi[1,i] =tempy/tempc 62 endfor 63 64 equ = fltarr(2, partnumber−coun3−coun1 ) 65 coun4 = 0 66 for i = 0, partnumber−coun3−1 do begin 67 i f equi[0,i] ne 0 then begin 68 equ[0,coun4]=equi[0,i] 69 equ[1,coun4]=equi[1,i] 70 coun4 +=1 71 e n d i f 72 endfor 73 print,partnumber, coun3, coun2, coun1, coun4 74 w r i t e gdf ,equ,filename 75 76 coun = 0 77 tang = 0 106

78 for i=0, partnumber−coun3−coun1−1 do begin ;find tilt angle 79 i f equ[1,i] gt ymin and equ[1,i] lt ymax and equ[0,i] gt xmin and equ[0,i] lt xmax then begin 80 nn = Findnn ( equ [ 0 , ∗ ] , equ [ 1 , ∗ ] , i , 1) 81 i f nn eq 10000 then continue 82 tang +=(equ [ 1 , nn]−equ[1,i ])/(equ[0,nn]−equ [ 0 , i ] ) 83 coun +=1 84 e n d i f 85 endfor 86 87 theta1 = atan(tang/coun) ;average 88 theta2 = theta1 + PI/6 89 print, theta1, theta2 90 91 sd=p l o t ( equ [ 0 , ∗ ] , equ [ 1 , ∗ ] , linestyle=’ ’,symbol=’d’,symsize=1) 92 93 end

VDispersion.pro

1 ;Name: VDispersion 2 ;Purpose: Draw the dispersion relation in first Brillouin Zone in K space 3 ;Input: tt pointer for the file ”tt.DS0 100”, startframe and stopframe 4 ; 5 ; Hisotry : 6; CreatedbyLichaoYu 7 ; 8 9 pro VDispersion, tt, equipos, startf , stopf , theta1, space, wavenumber= wavenumber, option=option , testmode=testmode , plot=plot 10 on error , 2 107

11 12 i f not keyword set(testmode) then testmode=0 ; in test mode, default filename become ”testmode”, in avoid of naming confliction. 13 i f not keyword set(option) then option=0 ;option = 0, plot frequency; option = 1, plot frequency square(band structure) 14 i f not keyword set(plot) then plot=0 15 16 filename = ’testmode.eps’ 17 dia = 0.36 18 PI = 3.14159 19 constant = 4.78046 ; constant=kb∗T∗2/sqrt3= 4.78∗10E−21 Joule . 20 ; i f 1 pixel=0.083um, constant= 6.93927∗10E−7 Joule per m2==N per m 21 ; itseemsweneedtoaddaradiusthicknessto modify the unit of shear modulus to N/m2=Pa 22 ; i f diameter=0.36um, we get the final result to be : 1.936 Pa 23constant2=1 ; constant2=Kb∗T= 4.14∗E−21 Joule 24 25 26 framenumber = stopf −s t a r t f +1 27 partnumber = n elements(equipos[0 , ∗ ]) 28 shear = dblarr(framenumber, partnumber) 29 30 31 i f not keyword set(wavenumber) then wavenumber=partnumber 32 33 34 ufourierX1 = make array(framenumber , wavenumber, /complex) 35 ufourierY1 = make array(framenumber , wavenumber, /complex) 36 ufourierX2 = make array(framenumber , wavenumber, /complex) 37 ufourierY2 = make array(framenumber , wavenumber, /complex) 38 ufourierX3 = make array(framenumber , wavenumber, /complex) 108

39 ufourierY3 = make array(framenumber , wavenumber, /complex) 40 D1 = make array(2,2, wavenumber, /complex) 41 D2 = make array(2,2, wavenumber, /complex) 42 D3 = make array(2,2, wavenumber, /complex) 43 freq1 = fltarr(2, wavenumber) 44 freq2 = fltarr(2, wavenumber) 45 freq3 = fltarr(2, wavenumber) 46 47 f = f l t a r r (6∗ wavenumber) 48 wav = f l t a r r (6∗ wavenumber) 49 wf = f l t a r r (2 , 6∗wavenumber) 50 51 52 for i= startf , stopf do begin ;Fourier Transform 53 print , i 54 p = t t (∗ ,where(tt(5, ∗ ) eq i ) ) 55 np = n elements(p(0, ∗ )) 56 57 58 for q=1, wavenumber do begin ;choose discrete q array 59 60 for j =0, np−1 do begin 61 62 f l a g =0 63 for k=0, partnumber−1 do begin 64 i f sqrt((equipos[0,k]−p(0,j))ˆ2 + (equipos [ 1 , k]−p(1,j))ˆ2) lt 6 then begin 65 eq1=k 66 f l a g =1 67 break 68 e n d i f 69 endfor 70 i f flag ne 1 then continue 109

71 wavex1 = q∗ s i n ( PI/3−theta1 ) ∗2∗PI/sqrt(3)/space/ wavenumber ;M d i r e c t i o n 72 wavey1 = q∗ cos ( PI/3−theta1 ) ∗2∗PI/sqrt(3)/space/ wavenumber ;M d i r e c t i o n 73 74 wavex2 = s i n ( PI/3−theta1 ) ∗2∗PI/sqrt(3)/space − q∗ s i n ( PI/6+theta1) ∗2∗PI/3/space/wavenumber ; M−>K d i r e c t i o n 75 wavey2 = cos ( PI/3−theta1 ) ∗2∗PI/sqrt(3)/space + q∗ cos ( PI/6+theta1) ∗2∗PI/3/space/wavenumber ; M−>K d i r e c t i o n 76 77 wavex3 = q∗ s i n ( PI/6−theta1 ) ∗4∗PI/3/space/wavenumber

;K direction 78 wavey3 = q∗ cos ( PI/6−theta1 ) ∗4∗PI/3/space/wavenumber

;K direction 79 80 ux = p (0 , j )−equipos[0,eq1] 81 uy = p (1 , j )−equipos[1,eq1] 82 u f o u r i e r X r e a l 1 =ux∗ cos ( wavex1∗ equipos [0 ,eq1]+wavey1∗ equipos[1,eq1]) ;M direction 83 ufourierXimg1 =−ux∗ s i n ( wavex1∗ equipos [0 ,eq1]+wavey1∗ equipos[1,eq1]) 84 u f o u r i e r Y r e a l 1 =uy∗ cos ( wavex1∗ equipos [0 ,eq1]+wavey1∗ equipos[1,eq1]) 110

85 ufourierYimg1 =−uy∗ s i n ( wavex1∗ equipos [0 ,eq1]+wavey1∗ equipos[1,eq1]) 86 ufourierX1 ( i , q−1) += complex(ufourierXreal1 , ufourierXimg1) 87 ufourierY1 ( i , q−1) += complex(ufourierYreal1 , ufourierYimg1) 88 89 u f o u r i e r X r e a l 2 =ux∗ cos ( wavex2∗ equipos [0 ,eq1]+wavey2∗ equipos[1,eq1]) ;M−>K d i r e c t i o n 90 ufourierXimg2 =−ux∗ s i n ( wavex2∗ equipos [0 ,eq1]+wavey2∗ equipos[1,eq1]) 91 u f o u r i e r Y r e a l 2 =uy∗ cos ( wavex2∗ equipos [0 ,eq1]+wavey2∗ equipos[1,eq1]) 92 ufourierYimg2 =−uy∗ s i n ( wavex2∗ equipos [0 ,eq1]+wavey2∗ equipos[1,eq1]) 93 ufourierX2 ( i , q−1) += complex(ufourierXreal2 , ufourierXimg2) 94 ufourierY2 ( i , q−1) += complex(ufourierYreal2 , ufourierYimg2) 95 96 u f o u r i e r X r e a l 3 =ux∗ cos ( wavex3∗ equipos [0 ,eq1]+wavey3∗ equipos[1,eq1]) ;K direction 97 ufourierXimg3 =−ux∗ s i n ( wavex3∗ equipos [0 ,eq1]+wavey3∗ equipos[1,eq1]) 98 u f o u r i e r Y r e a l 3 =uy∗ cos ( wavex3∗ equipos [0 ,eq1]+wavey3∗ equipos[1,eq1]) 99 ufourierYimg3 =−uy∗ s i n ( wavex3∗ equipos [0 ,eq1]+wavey3∗ equipos[1,eq1]) 100 ufourierX3 ( i , q−1) += complex(ufourierXreal3 , ufourierXimg3) 111

101 ufourierY3 ( i , q−1) += complex(ufourierYreal3 , ufourierYimg3) 102 103 104 endfor 105 106 endfor 107 endfor 108 109 for q=1, wavenumber do begin ;dynamical matrix 110 D1( 0 , 0 , q−1) = mean(conj(ufourierX1(∗ , q−1) ) ∗ ufourierX1(∗ , q−1) ) ;M d i r e c t i o n 111 D1( 0 , 1 , q−1) = mean(conj(ufourierX1(∗ , q−1) ) ∗ ufourierY1(∗ , q−1) ) 112 D1( 1 , 0 , q−1) = mean(conj(ufourierY1(∗ , q−1) ) ∗ ufourierX1(∗ , q−1) ) 113 D1( 1 , 1 , q−1) = mean(conj(ufourierY1(∗ , q−1) ) ∗ ufourierY1(∗ , q−1) ) 114 print , D1( ∗ , ∗ , q−1) 115 hes = ELMHES(D1( ∗ , ∗ , q−1) ) 116 i f option eq 0 then freq1[ ∗ , q−1] = sqrt(constant2/HQR(hes)) else f r e q 1 [ ∗ , q−1] = constant2/HQR(hes) 117 118 D2( 0 , 0 , q−1) = mean(conj(ufourierX2(∗ , q−1) ) ∗ ufourierX2(∗ , q−1) ) ;K −> M d i r e c t i o n 119 D2( 0 , 1 , q−1) = mean(conj(ufourierX2(∗ , q−1) ) ∗ ufourierY2(∗ , q−1) ) 120 D2( 1 , 0 , q−1) = mean(conj(ufourierY2(∗ , q−1) ) ∗ ufourierX2(∗ , q−1) ) 121 D2( 1 , 1 , q−1) = mean(conj(ufourierY2(∗ , q−1) ) ∗ ufourierY2(∗ , q−1) ) 122 hes = ELMHES(D2( ∗ , ∗ , q−1) ) 123 i f option eq 0 then freq2[ ∗ , q−1] = sqrt(constant2/HQR(hes)) else f r e q 2 [ ∗ , q−1] = constant2/HQR(hes) 124 125 D3( 0 , 0 , q−1) = mean(conj(ufourierX3(∗ , q−1) ) ∗ ufourierX3(∗ , q−1) ) ;K 112

d i r e c t i o n 126 D3( 0 , 1 , q−1) = mean(conj(ufourierX3(∗ , q−1) ) ∗ ufourierY3(∗ , q−1) ) 127 D3( 1 , 0 , q−1) = mean(conj(ufourierY3(∗ , q−1) ) ∗ ufourierX3(∗ , q−1) ) 128 D3( 1 , 1 , q−1) = mean(conj(ufourierY3(∗ , q−1) ) ∗ ufourierY3(∗ , q−1) ) 129 hes = ELMHES(D3( ∗ , ∗ , q−1) ) 130 i f option eq 0 then freq3[ ∗ , q−1] = sqrt(constant2/HQR(hes)) else f r e q 3 [ ∗ , q−1] = constant2/HQR(hes) 131 132 endfor 133 134 135 136 for q=1, wavenumber do begin 137 wav [ 2 ∗ q−2] = q∗2∗PI/sqrt (3)/wavenumber 138 wav [ 2 ∗ q−1] = wav [ 2 ∗ q−2] 139 f [ 2 ∗ q−2] = freq1[0, q−1] 140 f [ 2 ∗ q−1] = freq1[1, q−1] 141 142 wav [ 2 ∗ wavenumber+2∗q−2] = 2∗PI/sqrt(3) + q∗2∗PI/3/wavenumber 143 wav [ 2 ∗ wavenumber+2∗q−1] = wav [ 2 ∗ wavenumber+2∗q−2] 144 f [ 2 ∗ wavenumber+2∗q−2] = freq2[0, q−1] 145 f [ 2 ∗ wavenumber+2∗q−1] = freq2[1, q−1] 146 147 wav [ 4 ∗ wavenumber+2∗q−2] = 2∗PI/sqrt(3) + 2∗PI/3 + 4∗PI/3 − q∗4∗ PI/3/wavenumber 148 wav [ 4 ∗ wavenumber+2∗q−1] = wav [ 4 ∗ wavenumber+2∗q−2] 149 f [ 4 ∗ wavenumber+2∗q−2] = freq3[0, q−1] 150 f [ 4 ∗ wavenumber+2∗q−1] = freq3[1, q−1] 151 152 endfor 153 154 113

155 wffilename = ’wf.DISP’+strtrim(string(startf) ,2)+’ ’+strtrim(string( stopf) ,2)+’ ’+strtrim(string(wavenumber) ,2)+’.txt ’ 156 wf [ 0 , ∗ ] = wav 157 wf [ 1 , ∗ ] = f 158 w r i t e text ,wf,wffilename 159 160 i f (plot) then begin 161 i f not testmode then filename = ’DispersionRelation ’+strtrim(string (startf) ,2)+’ ’+strtrim(string(stopf) ,2)+ $ 162 ’ wavenumber=’+strtrim(string( wavenumber) ,2)+’ .eps ’ 163 164 t h i s D e v i c e = !D.NAME 165 s e t p l o t , ’ ps ’ 166 ! p . f o n t=0 167 ; ! p . t h i c k =1.4 168 ; ! p . c h a r s i z e =1.6 169 device , file=filename ,/encapsulate ,/color ,bits p e r pixel=8,/helvetica 170 171 172 cgplot , wav, f, xstyle=1, ystyle=1, psym=1, symsize=1.3, symcolor=’ red’, xtitle=’q a’, ytitle=’Frequency’ 173 plots , [ 2 ∗ PI/sqrt(3) ,2∗ PI/sqrt(3)] ,[min(f),max(f)],color=fsc c o l o r ( ’ Slate Blue’),linestyle=2,/data 174 plots , [ 2 ∗ PI/sqrt(3)+ 2∗PI /3 ,2∗ PI/sqrt(3)+ 2∗PI/3] ,[min(f),max(f)], c o l o r=f s c color(’Slate Blue’),linestyle=2,/data 175 176 177 178 device , / c l o s e 179 s e t plot ,thisDevice 180 e r a s e 181 ! p . f o n t=−1 114

182 ! p . multi =[0 ,1 ,1] 183 184 case !version.os of 185 ’ Win32 ’ : begin 186 i f !version.arch eq ’x86’ then cmd = ’”c: \ Program Files \ Ghostgum\ gsview \ gsview32.exe ”’ else cmd = ’ ”c : \ Program F i l e s \Ghostgum\ gsview \ gsview64.exe ”’ 187 spawn , [ cmd , filename ] , / log output ,/noshell 188 end 189 ’darwin’: spawn,’gv ’+filename 190 else : spawn,’gv ’+filename 191 endcase 192 193 e n d i f 194 end

Crop jpeg.pro

1 ; Name Crop jpeg 2 ; 3 ;Purpose: To crop jpg files to the window interested in 4 ; 5 ; 6 ; 7 ;History: Created by Lichao Yu 8 ; 9 function ffilename1 ,imagefile ,index 10 11 i f not keyword set(index) then index=0 12 13 i f (index eq 0) then return , imagefile+’ 00000 ’ else $ 14 i f (index lt 10) then interval = ’ 0000 ’ else $ 15 i f (index lt 100) then interval = ’ 000 ’ else $ 115

16 i f (index lt 1000) then interval = ’ 00 ’ else $ 17 i f (index lt 10000) then interval = ’ 0 ’ else $ 18 i f (index lt 100000) then interval = ’ ’ 19 20 filename = imagefile+interval+strtrim(string(index) ,2) 21 22 return , filename 23 end 24 25 26 Pro Crop jpeg , pt 27 on error , 2 28 29 NF = n elements(pt[0 ,0 , ∗ ]) 30 x = n elements(pt[ ∗ , 0 , 0 ] ) 31 y = n elements(pt[0 , ∗ , 0 ] ) 32 window,0,xsize=x, ysize=y 33 34 for i =0, nf −1 do begin 35 a0 = pt [ ∗ , ∗ , i ] 36 t v s c l , a0 37 filename=ffilename1(’crop’, i) 38 image = cgsnapshot(/jpeg, filename=filename, /nodialog) 39 40 endfor 41 42 end

checkequi.pro

1 pro checkequi, equip, xmin, xmax, ymin, ymax 2 on error , 2 3 PI = 3.14159 116

4 5 filename = ’checkequi.eps’ 6 thisDevice = !D.NAME 7 s e t p l o t , ’ ps ’ 8 ! p . f o n t=0 9 ;!p.thick=1.4 10 ; ! p . c h a r s i z e =1.6 11 device , file=filename ,/encapsulate ,/color ,bits p e r pixel=8,/helvetica 12 13 partnumber = n elements(equip[0 , ∗ ]) 14 spacing = f l t a r r (6∗ partnumber) 15 angle = f l t a r r (4∗ partnumber) 16 17 coun1 = 0 18 coun2 = 0 19 positionX = equip [ 0 , ∗ ] 20 positionY = equip [ 1 , ∗ ] 21 triangulate, positionX, positionY, tr 22 ntr = n elements(tr(0, ∗ )) 23 for i =0, ntr −1 do begin 24 i f (positionX(tr(0,i)) gt xmin) and (positionX(tr(0,i)) lt xmax ) and (positionY(tr(0,i)) gt ymin) and (positionY(tr(0,i)) l t ymax) $ 25 and (positionX(tr(1,i)) gt xmin) and (positionX(tr(1,i)) lt xmax) and (positionY(tr(1,i)) gt ymin) and (positionY(tr(1,i ) ) l t ymax) $ 26 and (positionX(tr(2,i)) gt xmin) and (positionX(tr(2,i)) lt xmax) and (positionY(tr(2,i)) gt ymin) and (positionY(tr(2,i ) ) l t ymax) $ 27 then begin 28 len1 = sqrt((positionX(tr(0,i))−positionX(tr(1,i)))ˆ2 + ( positionY(tr(0,i))−positionY(tr(1,i)))ˆ2) 29 spacing(coun1)=len1 117

30 coun1 +=1 31 len2 = sqrt((positionX(tr(2,i))−positionX(tr(1,i)))ˆ2 + ( positionY(tr(2,i))−positionY(tr(1,i)))ˆ2) 32 spacing(coun1)=len2 33 coun1 +=1 34 len3 = sqrt((positionX(tr(0,i))−positionX(tr(2,i)))ˆ2 + ( positionY(tr(0,i))−positionY(tr(2,i)))ˆ2) 35 spacing(coun1)=len3 36 coun1 +=1 37 cosine = abs(((positionX(tr(1,i))−positionX(tr(0,i))) ∗( positionX(tr(2,i))−positionX(tr(0,i))) $ 38 +(positionY(tr(1,i))−positionY(tr(0,i))) ∗( positionY(tr(2,i))−positionY(tr(0,i)))))/ len1 / len3 39 angle(coun2) = acos(cosine) ∗180/ PI 40 coun2 +=1 41 cosine = abs(((positionX(tr(0,i))−positionX(tr(1,i))) ∗( positionX(tr(2,i))−positionX(tr(1,i))) $ 42 +(positionY(tr(0,i))−positionY(tr(1,i))) ∗( positionY(tr(2,i))−positionY(tr(1,i)))))/ len1 / len2 43 angle(coun2) = acos(cosine) ∗180/ PI 44 coun2 +=1 45 e n d i f 46 endfor 47 space = fltarr(coun1+1) ;eluminate zero term in s vector to plot histogram 48 ang = fltarr(coun2+1) ;eluminate zero term in s vector to plot histogram 49 50 tem=0 51 for j = 0 , coun1 do begin 52 i f spacing(j) ne 0 then begin 118

53 space(tem)=spacing(j) 54 tem +=1 55 e n d i f 56 endfor 57 58 tem=0 59 for j = 0 , coun2 do begin 60 i f angle(j) ne 0 then begin 61 ang ( tem ) = angle ( j ) 62 tem +=1 63 e n d i f 64 endfor 65 66 ! p . multi =[0 ,2 ,2] 67 cghistoplot, space, xtitle=’Lattice Constant (pixel)’,ytitle=’ Frequency ’ , binsize=0.01, mininput=10, maxinput=15, CHARSIZE =1.1 68 cghistoplot , ang, xtitle=’Angle (degree)’,ytitle=’Frequency’, binsize=0.1, mininput=0, maxinput=90, CHARSIZE=1.1 69 histogauss, space, A, CHARSIZE=0.8 70 histogauss, ang, B, CHARSIZE=0.8 71 72 device , / c l o s e 73 s e t plot ,thisDevice 74 e r a s e 75 ! p . f o n t=−1 76 ! p . multi =[0 ,1 ,1] 77 78 case !version.os of 79 ’ Win32 ’ : begin 80 i f !version.arch eq ’x86’ then cmd = ’”c: \ Program Files \Ghostgum\ gsview \ gsview32.exe ”’ else cmd = ’ ”c : \ Program Files \Ghostgum\ gsview \ gsview64.exe ”’ 119

81 spawn,[cmd,filename],/log output ,/noshell 82 end 83 ’darwin’: spawn,’gv ’+filename 84 else : spawn,’gv ’+filename 85 endcase 86 87 end

dislocation density.pro

1 ;+ 2 ; Name: dislocation d e n s i t y 3 ; Purpose: calculate dislocation pairs density for certain observe window 4 ; Input: dislocation density , pt, startf , stopf, field , exceptions 5 ; Example: dislocation density ,pt,0,60,field=[200,300,200,300] 6 ; 7 ; Exceptions keyword tells the program to skip certain ammount of frames 8 ;− 9 10 Pro d i s l o c a t i o n density, pt, startf, stopf, field = field 11 on error , 2 12 13 i f ( not keyword set(field)) then begin 14 wxmin = 0 15 wxmax = 480 16 wymin = 0 17 wymax = 600 18 e n d i f 19 20 i f n elements(field) eq 4 then begin 21 wxmin = float ( f i e l d [ 0 ] ) 120

22 wxmax = float ( f i e l d [ 1 ] ) 23 wymin = float ( f i e l d [ 2 ] ) 24 wymax = float ( f i e l d [ 3 ] ) 25 e n d i f 26 27 filename = ’dislocation d e n s i t y ’+ $ 28 strtrim(string(startf),2)+’ ’+ $ 29 strtrim(string(stopf),2)+’ [ ’+ $ 30 strtrim(string(wxmin),2)+’,’+$ 31 strtrim(string(wxmax),2)+’,’+$ 32 strtrim(string(wymin),2)+’,’+$ 33 strtrim(string(wymax),2)+’].txt’ 34 35 nf = s t o p f − s t a r t f + 1 36 r e s = i n t a r r (2 , nf ) 37 38 for i = startf, stopf do begin 39 ; showing p r o g r e s s 40 i f i mod 100 eq 0 then print, ’working on frame ’ + strtrim(string(i) ,2) ; 41 42 ptc = pt (∗ ,where(pt(5, ∗ ) eq i ) ) 43 n f f = n elements(ptc(0, ∗ )) 44 r e s (1 , i − s t a r t f ) = i 45 46 t r i a n g u l a t e , ptc ( 0 , ∗ ) , ptc ( 1 , ∗ ), tr, conn=con 47 for j =0, nff −1 do begin 48 i f ptc(0,j) lt wxmax and $ 49 ptc (0 , j ) gt wxmin and $ 50 ptc (1 , j ) l t wymax and $ 51 ptc(1,j) gt wymin then begin 52 i f con [ j +1] − con[j] gt 6 then res(0,i − startf) += (con[j+1] − con [ j ] − 6) ; 121

53 e n d i f 54 endfor 55 56 endfor 57 58 w r i t e text ,res ,filename 59 cgPlots , r e s ( 1 , ∗ ) , r e s ( 0 , ∗ ) ,symsize=0.5, psym=SymCat(16) ; 60 ; p l o t hist , res(0, ∗ ), binsize = 5, /fit 61 62 63 end

C.3 C++ Code for Monte Carlo Simulation

1 2 //// 3 // main . cpp 4 // 2DCCMC 5 // 6 // Created by LICHAO YU on 05/20/13. 7 // Copyright (c) 2014 LICHAO YU. All rights reserved. 8 // This is a simple skeleton of MC of 2DCC 9 // 10 #include $11 #include$ ~~12 #include 13 #include 14 #include ~~15 #include 16 #include 17 #include 122~~~~

18 #include 19 using namespace std // 20 #define N 40 // system size N∗N 21 #define Steps 1E10 22 float pt [N∗N ] [ 2 ] ; // 1st col=x, 2nd col=y coordinates. row = particle index 23 24 #define debye 0 . 5 // Debye length is half the lattice spacing. In reality , a=1.1 um, debye = 0.39+0.18 = 0.57 um. 25 #define stepsize 0.05 26 #define pi 3.14159265 27 #define b sqrt(3.0)/2 // lattice spacing a =1, b = sqrt(3)/2 28 29 void Initialize( int i n i t ) 30 { 31 int row , c o l ; float phi , r ; 32 for ( int i = 0 ; i < N∗N; i++) 33 { 34 row = i /N; 35 c o l = i%N; 36 i f (row%2) pt[i][0] = col + 0.5; 37 else pt[i][0] = col; 38 pt [ i ] [ 1 ] = b ∗ row ; 39 } 40 41 i f ( i n i t == 1) return ; 42 43 srand ( ( unsigned int )time(NULL)) ; 44 for ( int i = 0 ; i < N∗N; i++) 45 { 46 phi = ( ( float ) rand ( ) /(RAND MAX+1) ) ∗ 2 ∗ pi ; 47 r = ( ( float ) rand ( ) /(RAND MAX+1) ) ∗ 0 . 0 5 ; 48 pt [ i ] [ 0 ] += r ∗ cos ( phi ) ; 123

49 pt [ i ] [ 1 ] += r ∗ s i n ( phi ) ; 50 } 51 } 52 53 float Yukawa( int r1 , int c1 , int r2 , int c2 ) // repulsive Yukawa potential , consider only NN interaction 54 { 55 int index1 = r1 ∗N+ c1, index2 = r2 ∗N + c2 ; 56 float x1 = pt[index1][0], y1 = pt[index1][1], x2, y2; 57 58 i f ( r2 == −1){ x2 = pt[index2 + N∗N][0]; y2 = (pt[index2 +N∗N ] [ 1 ] − N∗b) ; } 59 else i f ( r2 == N) { x2 = pt[c2][0]; y2 = (pt[c2][1] +N∗b) ; } 60 else i f ( c2 == −1){ x2 = (pt[index2 + N][0] − N); y2 = pt[index2 + N ] [ 1 ] ; } 61 else i f ( c2 == N) { x2 = (pt[index2 − N][0] +N); y2 = pt[ index2 − N ] [ 1 ] ; } 62 else {x2 = pt[index2][0], y2 = pt[index2][1]; } 63 64 float dis = sqrt( pow(x1 − x2,2) + pow(y1 − y2 , 2) ) ; 65 return exp(− dis/debye)/dis ; // reduced−temperature incorporate the p r e f a c t o r 66 } 67 68 float Energy ( int index ) // calculate total interaction energy of a particle with its NNs 69 { 70 float e ; 71 int row , c o l ; 72 row = index /N; 73 c o l = index%N; 74 i f ( row%2) { 124

75 e=(Yukawa(row, col, row−1, col) + Yukawa(row, col, row−1, c o l +1) + Yukawa(row, col , row, col −1) 76 + Yukawa(row, col, row, col+1) + Yukawa(row, col, row+1, col) + Yukawa(row, col , row+1, col+1)); 77 } 78 else { 79 e=(Yukawa(row, col, row−1, col −1) + Yukawa(row, col , row−1, col) + Yukawa(row, col, row, col −1) 80 + Yukawa(row, col, row, col+1) + Yukawa(row, col, row+1, col −1) + Yukawa(row, col , row+1, col)); 81 } 82 return e ; 83 } 84 85 int main ( ) 86 { 87 ofstream m y f i l e ; 88 myfile.open(”data.txt”); 89 90 // initialize 91 int init , boundary; float Temperature ; 92 cout << ”Choose initial condition: 1 for perfect lattice , 2 for small d e v i a t i o n ” << endl ; 93 c i n >> i n i t ; 94 I n i t i a l i z e ( i n i t ) ; 95 cout << ”Choose Boundary condition: 1 for default periodic , 2 for fixed boudnary” << endl ; 96 c i n >> boundary ; 97 cout << ”Set Temperature (unit: 0.001)” << endl ; 98 c i n >> Temperature ; 99 Temperature ∗= 0 . 0 0 1 ; 100 // S t a r t 101 ti me t s t a r t time = clock(); 125

102 103 // Monte Carlo 104 int index, flag = 1; 105 float dE, ran, phi, r, oldx, oldy; 106 107 srand ( ( unsigned int )time(NULL)) ; 108 for ( int i = 0 ; i < Steps ; i++) { 109 i f ( i == ( int ) ( f l a g ∗ Steps ∗ 0 . 0 1 ) ) { 110 cout << f l a g << ”%” << endl ; 111 f l a g ++; 112 } 113 114 i f (boundary == 1) { 115 do{ 116 index = rand ( ) %(N∗N); 117 }while ( index == 0 | | index == N−1 | | index == (N−1)∗N | | index == N∗N − 1) ; 118 } 119 i f (boundary == 2) { 120 do{ 121 index = rand ( ) %(N∗N); 122 }while ( index /N < 1 | | index /N > N−2 | | index%N <1 | | index%N > N −2) ; 123 } 124 125 dE = −Energy(index); 126 oldx = pt [ index ] [ 0 ] ; 127 oldy = pt [ index ] [ 1 ] ; 128 // create a random displacement trial 129 phi = ( ( float ) rand ( ) /(RAND MAX+1) ) ∗ 2 ∗ pi ; 130 r = ( ( float ) rand ( ) /(RAND MAX+1) ) ∗ 0 . 0 5 ; 131 pt [ index ] [ 0 ] += r ∗ cos ( phi ) ; 132 pt [ index ] [ 1 ] += r ∗ s i n ( phi ) ; 126

133 134 dE += Energy ( index ) ; 135 i f (dE > 0) { 136 ran = ( ( float ) rand ( ) /(RAND MAX+1) ) ; 137 i f ( ran > exp(−dE/Temperature)) { 138 pt [ index ] [ 0 ] = oldx ; 139 pt [ index ] [ 1 ] = oldy ; // reject the move 140 } 141 } 142 } 143 144 for ( int i = 0 ; i < N∗N; i++) 145 m y f i l e << pt [ i ] [ 0 ] << ’’ << pt [ i ] [ 1 ] << endl ; 146 m y f i l e . c l o s e ( ) ; 147 float time1 = ( float ) ( c l o c k ( ) − s t a r t t i m e ) / CLOCKS PER SEC; 148 cout << ”Total time cost is ” << time1 << ” seconds ” << endl ; 149 return ; 150 } 127

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