Nonlinear plastic modes in defected 2D crystals

University of Amsterdam

Faculty of Science

Institute for Theoretical Physics Amsterdam

Report Bachelor Project Physics and Astronomy

Nonlinear plastic modes in defected 2D

crystals

Author: Kevin de Vries

Student ID: 10579869

Supervisor: Edan Lerner

Second Assessor: dr. P.R. Corboz

Size: 15 EC

Conducted between: 03-04-2017 and 07-07-2017

Abstract

Crystals are solids with a highly ordered structure, but they are not perfect. There are many kinds of defects which have an effect on the mechanical properties of materials. In this thesis, dislocated two dimensional hexagonal crystals are analysed using both normal modes and nonlinear plastic modes. The goal is to be able to say something about dislocations in crystals using nonlinear plastic modes and to investigate the stability and stiffnesses of the nonlinear plastic modes. This is done by generating defected crystals and using Molecular Dynamics simulations to minimize the energies to a mechanical equilibrium. The analysis yields that there is a large difference in stiffnesses of modes between different pressures and that these stiffnesses seem to follow a general trend as the size of the line defect increases. Larger noise induced systems split up into two possible dislocation configurations at a higher pressure. Lastly, the nonlinear plastic modes seem to delocalize after collapsing systems with certain line defect sizes. The conclusions are drawn that systematically producing defected crystals in 2D is difficult and that the stiffnesses seem to follow a general trend as defect sizes increase.

Populair Wetenschappelijke Samenvatting

Als je aan kristallen denkt, dan denk je vaak aan een structuur waarin alle deeltjes netjes op hun plaats zitten en wat oneindig doorgaat. Kristallen zijn echter niet perfect. Ze bevatten namelijk bijna altijd defecten. Er zijn veel soorten defecten, zoals ontbrekende deeltjes, misplaatste deeltjes en dislocaties waarin een leegte in een kristal wordt opgevuld door omliggende deeltjes. De aanwezigheid van deze defecten heeft belangrijke gevolgen voor de fysische eigenschappen van materialen. Dislocaties zorgen er bijvoorbeeld voor dat kristallen vervormbaar kunnen zijn. De dislocaties kunnen zich namelijk makkelijker door het kristal bewegen dan de deeltjes zelf. Tijdens mijn bachelorproject heb ik onderzoek gedaan naar dislocaties in defecte kristallen in 2D. Dit heb ik gedaan door middel van zogenaamde nonlinear plastic modes. Dit zijn algemene richtingen voor alle deeltjes in een vaste stof die, als de verplaatsing vanaf de oorspronkelijke posities klein genoeg is, alle deeltjes tegelijk kunnen verplaatsen naar een volgende positie waarin de stof stabiel is. Deze non-linear plastic modes kunnen dus iets zeggen over de vervormbaarheid van een stof. Het doel van mijn onderzoek was om te onderzoeken of de nonlinear plastic modes wat kunnen zeggen over dislocaties en of de zogenaamde stevigheid van deze modes afhankelijk is van de grootte van de dislocaties.

(a) (b) (c)

In mijn onderzoek heb ik gebruik gemaakt van simulaties om de defecte kristallen te verkrijgen. Eerst cre¨eerde ik perfecte kristallen en haalde ik hier lijnen met deelt-jes van verschillende groottes uit, zoals te zien is in figuur a. Vervolgens liet ik de kristallen instorten tot op het punt waarop alle deeltjes effectief geen kracht meer op elkaar uitoefenen, zoals is te zien in figuur b. Daarna berekende ik de nonlinear plastic modes, welke er ongeveer uitzien als het voorbeeld in figuur c, en berekende ik hiervan de stevigheid.

Aan het eind van mijn project heb ik de conclusie moeten trekken dat het best moeilijk blijkt te zijn om systematisch defecte kristallen te produceren. De sim-ulaties leverden vaak onbruikbare producten op, waar rekening mee moest worden gehouden. Ook konden in sommige gevallen, door kleine willekeurige verschillen in de beginposities van de deeltjes te plaatsen, dezelfde kristallen verschillende producten opleveren. Wel blijkt uit dit onderzoek dat de stevigheid van de nonlinear plastic modes inderdaad afhankelijk lijkt te zijn van de grootte van de dislocaties.

Contents

1 Introduction 4

2 Basic terminology and definitions 5

3 Harmonic Approximation 5

4 Nonlinear Plastic Modes 6

5 Shear force response 7

6 Simulation Methods 8

7 Analysis and Results 10

7.1 Perfect crystals . . . 11 7.2 Crystals with noise . . . 19

8 Discussion 22

1

Introduction

Crystals are highly ordered solids where the particles are arranged to form a crys-tal structure. Although cryscrys-tals are highly ordered, they are generally not perfect. There are many kinds of defects which disrupt the structure of a crystal. These defects include but are not limited to point defects, line defects, planar defects and dislocations. The presence of defects plays a large role in the mechanical properties of many materials. For example, the plastic deformation of a crystal is caused by the movement of dislocations within the crystal as a result of external influences such as forces being applied to the crystal or by adding heat to the system [1]. Because of the effects of defects on mechanical properties of materials, the study of defects and dislocations in crystals is an important part of materials science.

In this thesis dislocations in defected two dimensional crystals will be analysed using nonlinear plastic modes. The defected crystals will be simulated using Molecular Dynamics. Nonlinear plastic modes are collective displacement directions (modes) which are indicative of the geometry and spatial structure of plastic instabilities, which have been used in the study of structural glasses to some degree of success [2, 3]. The goal of this thesis is to investigate if the behaviour of dislocations can be described or predicted using nonlinear plastic modes and to investigate the stability and stiffnesses of these nonlinear plastic modes. The expectation is to find a gen-eral trend according to which the stiffnesses behave as the sizes of the defects increase. The thesis will be structured in the following way. First a description will be given about both normal modes and nonlinear plastic modes which will be used in the analysis. The method used to find nonlinear plastic modes in the vicinity of disloca-tions will also be discussed. Afterwards, the simulation of the defected crystals will be described, followed by the analysis along with the results. Lastly, the results will be discussed and suggestions for future research will be discussed.

2

Basic terminology and definitions

Before describing the underlying theory, the basic terminology and definitions will be discussed.

The potential of the system will be described as the sum over pairwise interaction potentials given as

U =X

i<j

ϕ(rij). (1)

The derivatives of the potential will be expressed as U(n) with n ∈ Z+ where each nth derivative is described as a rank-n tensor of derivatives of U . The derivatives can be written as a variety of equivalent expressions. For example, the dynamical matrix M can be written as U00 = Mij = ∂

2_U

∂~xi∂~xj, where ~xi denotes the coordinate

vector of the ith particle. Higher order tensors will have similar expressions.

In cases where tensoric notation is relevant, it will be used in an expression according to tensoric notation convention, otherwise it will be implicit. Tensor contractions will be denoted by ·, : and .: for single, double and triple contractions respectively. For example, Mijzizj will be written as M : ~z~z. Repeated indices are to be understood

as summed over. Multiple contractions with only the same vector will sometimes be denoted with •~z(n)where n indicates how many contractions with the same vector ~z are done.

All simulated systems analysed in this report will be in mechanical equilibrium unless stated otherwise. This means that the net force on all particles will be approximately equal to zero. Because of the numerical nature of simulations the numerical zero used in this report will be any number lower than 10−9.

3

Harmonic Approximation

The harmonic approximation is done by taking the Taylor expansion of the potential energy and truncating it after the second expansion coefficient. This results in the following expression U (~x) ' U0+ ∂U ∂~x    ~ x=~x(0)· δ~x + 1 2δ~x· ∂2_U ∂~x∂~x    ~ x=~x(0)· δ~x, (2)

where δ~x is the displacement about the mechanical equibibrium position and U0 is

the energy at mechanical equilibrium. Because the system is in mechanical equilib-rium, the first expansion coefficient ∂U_∂~x

~

x=~x(0) drops out of the equation. By defining

U (δ~x) = U (~x) − U0 the expression that is left is

U (δ~x) ' 1

Diagonalizing M yields the normal modes ˆΨm of the system. The eigenvalues λm

associated with these normal modes can be related to the vibrational frequencies of the system. The eigenvalues are equivalent to the stiffnesses associated with the normal modes and can be calculated using λm= M : ˆΨmΨˆm

4

Nonlinear Plastic Modes

Nonlinear plastic modes are collective displacement directions that are indicative of the spatial structure and geometry of imminent plastic instabilities.

Nonlinear plastic modes emerge from the following theoretical framework. The po-tential energy U can be expanded by taking its Taylor expansion to get the expression

δUzˆ(s) = U (s) − U0 = X n>1 U(n)• ˆz(n) n! s n_{, (4)}

where the first Taylor expansion coefficient is equal to zero due to the analysed sys-tems being in mechanical equilibrium. This is the variation of the potential energy δUzˆ(s) when displacing the particles of a system about its initial state ~x0 along a

general collective displacement direction ˆz according to δ~x = sˆz. These collective displacement directions are not necessarily eigenmodes of M.

When s is small, equation (4) can be truncated to get the expression δUzˆ(s) ' 1 2κzˆs 2₊1 6τzˆs 3 ₍₅₎

where κzˆ ≡ M : ˆz ˆz is defined as the stiffness and τzˆ ≡ U000 .: ˆz ˆz ˆz is defined as

the asymmetry associated with the collective displacement direction ˆz. From this truncated form two stationary points of the energy variation can be found which correspond to a minimum for s = 0 and a maximum for s? = −2κ_τzˆzˆ. The energy

variation at the maximum distance s? can be defined as a ‘barrier’ function

b(ˆz) ≡ 1 2κzˆs 2 ?+ 1 6τˆzs 3 ? = 2κ3_ˆz 3τ2_zˆ. (6) Nonlinear plastic modes are modes ˆπ for which this barrier function attains a local minimum, which means that the modes ˆπ satisfy _∂~∂b_z~z=ˆπ = 0 and all eigenvalues of

∂2_b

∂~z∂~z

~z=ˆπ are non-negative.

Because the barrier function is invariant to the magnitude of its vector argument, partial derivatives of the barrier function with respect to ~z can be meaningfully taken to get the gradient

∂b ∂~z = 4 κ2_zˆ τ2_zˆ  M · ~z −κˆz τzˆ U000: ~z~z  . (7)

Since _∂~∂b_z is equal to zero when ~z = ˆπ, nonlinear plastic modes can also be defined as solutions ˆπ to the nonlinear equation

M · ˆπ = κzˆ τ_zˆU

000

: ˆπ ˆπ. (8) It should be noted that only modes ˆπ which correspond with a small b take the system over saddle points in the potential energy and indicate imminent plastic instabilities. This is because a small b means that s? is small, which means that equation (5) is a

good approximation of the actual energy variation given in equation (4). [3]

5

Shear force response

In order to find nonlinear plastic modes in the vicinity of a dislocation, an initial guess can be calculated using strain coupling. This is done through calculating the force response which results from applying a shear to the system when rotated over an angle θ. An intuitive explanation would be that the force response is larger around dislocations, due to the particles there having a larger freedom of movement. This causes the methods used to calculate nonlinear modes to converge to modes in the vicinity of a dislocation.

In order to couple a nonlinear plastic mode with a deformation the derivative of the stiffness of a system with respect to the strain dκ_dγ can be found, which is given by

dκ dγ ≈ −

U000 .: ˆπ ˆπ ˆπ_∂γ∂~∂2U_x · ˆπ

κ . (9)

Here the only expression dependent on the deformation is _∂~∂_x∂γ2U , which is the shear force response. The shear force response is given by

∂2U ∂~x∂γ = sin 2θ X α  ϕ00 r2 − ϕ0 r3  ( ~R · ˆx)2− ( ~R · ˆy)2 2 R +~ (10) cos 2θX α  ϕ00 r2 − ϕ0 r3  ~ R · ˆxˆy · ~R ~R

= sin 2θ ~Ξpure+ cos 2θ ~Ξsimple, (11)

6

Simulation Methods

Figure 1: A plot of the Lennard-Jones potential used for the simulations.

The defected crystals are simulated using Molecular Dynamics where the pairwise potential used to describe the interactions between particles is a Lennard-Jones po-tential with additional terms which make the popo-tential smoother. The popo-tential is described by ϕ(r) =σ r 12 −σ r 6 + 0.0006201261686784r σ 4 − 0.00970155098112r σ 2 + 0.040490237952 (12)

where r is the interatomic distance and σ is the finite distance where the potential is equal to zero. The crystals are only made up of a single type of particle, which means that σ is equal to two times the radius of the particles. This potential is chosen, because it has both an attractive term and a repulsive term. A plot of the potential can be seen in figure 1.

The dislocated crystals are created by first creating a perfect hexagonal crystal with a line defect of m particles in size and then using overdamped dynamics, where particles are displaced according to

~

xi(t + ∆t) = ~xi(t) + ∆t · Fi. (13)

Here Fi = P j6=i

−∇ϕ(r_ij) is the net force on the ith particle in the system and ∆t is the time step used during the overdamped dynamics. After the ratio between the average force per contact and the average net force per particle becomes smaller than 10−2, an energy minimization using a conjugate gradient algorithm is used until the system reaches mechanical equilibrium. An example of a system before and after this process can be seen in figure 2.

(a) (b)

Figure 2: (a) A generated N = 502_{crystal with a line of particles taken out. (b) The crystal}

after collapsing the system to mechanical equilibrium through energy minimization.

Systems of size N = 502, 642, 802, 1002 and 1402 are investigated. The systems are investigated with line defects between mmin < m <

√ N

2 , where mmin is the largest

line defect where the energy minimization does not result a system with two separate dislocations. Systems where m ≥

√ N

2 are not analysed due to periodic boundary

conditions. The densities ρ = N_V at which these systems are initialized are chosen to be ρ = 0.840 and ρ = 0.822, because these densities correspond to pressures P ' −Pmin and P ' −1₂Pmin respectively, where Pmin is the minimum of the

pressure as a function of the density of the system. The pressure is defined as P =X

i<j

F (rij)rij

d¯V , (14)

where d¯ is the dimension of the system and V is the volume of the system. A figure of the pressure as a function of the density is provided in figure 3.

Figure 3: The pressure for any system size as a function of the density ρ of the system.

The same systems, except for N = 1402, are also investigated by adding a random displacement smaller than 10−3 times the space between two particles in a perfect crystal to each particle, while keeping the center of mass of the system in place, before minimizing the energies. Per system with line defect size m, 10 samples with different noise realizations are created.

7

Analysis and Results

The normal modes of the system were calculated using two possible methods. The first method was to diagonalize the dynamical matrix M. The second method was to use a conjugate gradient minimization method to calculate the normal mode with the smallest associated eigenvalue.

Nonlinear plastic modes were calculated using the same conjugate gradient method used to calculate normal modes to minimize the barrier function given in equation (6) over directions ~z. The initial vector, over which b(~z) was minimized, was

calcu-lated by using the shear force response in equation (10) to calculate the nonaffine displacement field ~v ≡ d~x dγ = −M −1_· ∂2U ∂~x∂γ. (15) The initial vector is taken to be the normalized nonaffine displacement field ˆz0 = _||~~v_v||

[2, 3, 4, 5]. When plotting the stiffnesses κ of the nonlinear plastic modes, the initial vector used to minimize b(ˆz) was calculated using the shear force response at an angle of θ = 0 unless stated otherwise.

7.1

Perfect crystals

Figure 4: Plot of the lowest eigenvalues per line defect size m of a perfect crystal with N = 1002 _{and ρ = 0.84}

First the normal modes and nonlinear plastic modes of the perfect crystals were cal-culated. After calculating these modes, the associated stiffnesses κ were calcal-culated.

For the systems with N = 1002 however, some systems did not converge to any non-linear plastic modes. For these systems the lowest eigenvalue of M was always the only negative eigenvalue. In figure 4 can be seen which systems had negative eigen-values. This meant that those systems were stuck on saddle points of the potential energy. For these systems the particles were displaced along the normal mode Ψ with the negative eigenvalue according to δ~x = s ˆΨ where after every step the change in potential energy was calculated.

Figure 5: Variation of the potential in the direction of a normal mode with a negative eigenvalue for a system of N = 1002 with m = 39.

From the variation of the potential shown in figure 5 the saddle point and the minima of the energy of one system can be discerned. The systems stuck on saddle points were healed by displacing the system closer to one of the minima and minimizing the energy of the system again.

(a) (b)

(c) (d)

(e)

Figure 6: (a-e) Lowest eigenvalues of M for system sizes N = 502, 642, 802, 1002 and 1402 respectively with ρ = 0.84

The lowest eigenvalues of the systems with ρ = 0.84 are presented in figure 6. For smaller systems, the stiffnesses seem to vary continuously with dislocation size, which can be seen in figures 6a and 6b. For these systems there are relatively few m where the stiffnesses deviate much and they generally start deviating from m where the geometry of the dislocation starts to change. When the system size becomes larger, the stiffnesses start to become small at m ' 28. This can be seen in figures 6c-6e. The eigenvalues of systems sizes N = 1002 and 1402 are also plotted with the y-axis on a logarithmic scale, which can be seen in figure 10. For the system with N = 1002when the eigenvalues become small, it starts to become smaller as m increases, although it becomes less predictable. For system sizes 1402 the eigenvalues start behaving largely the same way, although they do not become smaller as m increases.

(a) (b)

(c) (d)

(e)

Figure 7: (a-e) stiffnesses associated with nonlinear plastic modes for system sizes N = 502, 642, 802, 1002 and 1402 respectively with ρ = 0.84

The nonlinear plastic modes of the systems with ρ = 0.84 are presented in figure 7. The nonlinear plastic modes of the system largely resemble the normal modes of the system given in figure 6, but with a few differences. The stiffnesses of the nonlinear plastic modes start to fall linearly with m at N = 802 _{and start falling exponentially}

with m for N = 1002 and 1402 until κ becomes small. When the stiffnesses become small, the nonlinear plastic modes start to delocalize from one dislocation to both. This can be seen in figure 12. The stiffnesses of systems sizes N = 1002 and 1402 are also plotted with the y-axis on a logarithmic scale, which can be seen in figure 11. When the stiffnesses become small, they start to behave the same way as the lowest non-zero eigenvalues of M at the same m.

(a) (b)

(c) (d)

(e)

Figure 8: (a-e) Lowest eigenvalues of M for system sizes N = 502_{, 64}2_{, 80}2_{, 100}2 _{and 140}2

respectively with ρ = 0.822

The lowest eigenvalues of the systems with ρ = 0.822 are presented in figure 8. For all systems the stiffnesses seem to remain approximately constant regardless of the size of the system and the line defect size m. For N = 1402 the stiffnesses start to become unstable when m becomes larger than 60.

(a) (b)

(c) (d)

(e)

Figure 9: (a-e) stiffnesses associated with nonlinear plastic modes for system sizes N = 502, 642, 802, 1002 and 1402 respectively with ρ = 0.822

The nonlinear plastic modes of the systems with ρ = 0.822 are presented in figure 9. In all systems the stiffnesses tend to form a curve as m increases. As the system size increases, the curves become more distorted and the stiffnesses become less pre-dictable. In addition the stiffnesses also become lower in general as the system size increases.

There are several problems with the data which could explain the distortions and unpredictabilities in the stiffnesses. For example, the nonlinear plastic modes used to plot the stiffnesses are calculated with a shear response at an angle of θ = 0. By

calculating nonlinear plastic modes at different angles, sometimes multiple different modes can be found. Two examples of stiffnesses as a function of θ is illustrated in figure 13 for a system of N = 802 particles. It is possible for one system to converge to multiple modes, but sometimes only one mode can be found. This adds some uncertainty to the results. Some discontinuous behaviour is also due to the systems converging to dislocations with different geometries, which seems to happen after a certain m. After a transition, the system typically does not converge to the previous form again.

(a) (b)

Figure 10: (a-b) Lowest eigenvalues of M for system sizes N = 1002 _{and 140}2 _respectively

with ρ = 0.84 where the scale of the y-axis is logarithmic.

(a) (b)

Figure 11: (a-b) Stiffnesses of nonlinear plastic modes for system sizes N = 1002 _{and 140}2

(a) (b)

Figure 12: Nonlinear plastic modes calculated for systems with N = 1002_{. (a) is a localized}

mode calculated for a system with m = 22. (b) is a delocalized mode calculated for a system with m = 28.

(a) (b)

Figure 13: Stiffnesses of nonlinear plastic modes of a system with N = 802_{, ρ = 0.84 and}

with m = 18 for (a) and m = 32 for (b) as a function of the angle θ from which the shear force response is calculated.

7.2

Crystals with noise

When a small random displacement was added to the systems, the smaller systems mostly did not converge to different configurations. At systems with N = 1002 parti-cles however, the systems consistently converged to one of two possible configurations at densities of ρ = 0.84. The first configuration, named type A, is characterized by two dislocations with slanted arrow-like shapes as is illustrated in figure 14a. The sec-ond configuration, named type B, is characterized by two dislocations with straight arrow-like shapes, which is illustrated by figure 14c. Starting from a certain m, the systems eventually converged to configurations where for both the type A and type B dislocations the arrow-like shapes have collapsed. The collapsed arrow-like dislo-cations are shown in figures 14b and 14d. The type A dislodislo-cations have the tendency to move mostly vertically through the crytal, while type B dislocations tend to move primarily horizontally through the crystal.

(a) (b)

(c) (d)

Figure 14: (a) A dislocation of type A, characterized by a slanted arrow-like shape. (b) A dislocation of type A where the arrow-like shape has collapsed. (c) A dislocation of type B, characterized by a straight arrow-like shape. (d) A dislocation of type B where the arrow-like shape has collapsed.

(a) (b)

Figure 15: For systems with N = 1002 and ρ = 0.84: (a) Lowest eigenvalues plotted for type A and type B dislocations. (b) Stiffnesses of nonlinear plastic modes plotted for type A and type B dislocations.

The lowest eigenvalues and stiffnesses of the nonlinear plastic modes of the systems with N = 1002 particles and densities ρ = 0.84 are illustrated in figure 15. The plots of the stiffnesses of the type B dislocations are largely the same as with the systems without added noise. The plots of the stiffnesses of the type A dislocations show that the stiffnesses of type A dislocations become smaller earlier than the stiffnesses of type B dislocations. At m = 43 the type A systems are too close to the end of the box, which means that due to boundary conditions these systems were considered defective from m = 43 onward. Besides the difference in the magnitude of the stiff-nesses, the plots of the stiffnesses of type A and type B look somewhat alike. Because of this the same things can be said about these systems as could be said about the systems in figures 6 and 7.

At m ' 17 for type A and m ' 28 for type B dislocations the dislocations become shaped as the dislocations given in figures 14b and 14d. At these m the stiffnesses of the nonlinear plastic modes also start to become much smaller. The visualizations of the nonlinear plastic modes of these systems shown in figure 17 reveal that the modes become delocalized at these m. The stiffnesses and lowest eigenvalues of both types have also been plotted with the y-axis on a logarithmic scale, which is shown in figure 16. From this the differences between the types when the stiffnesses are small can be found. Stiffnesses of type A dislocations tend to stay approximately of the same magnitude, while stiffnesses of type B tend to become smaller as m increases. At densities ρ = 0.822 none of the system sizes produced enough samples that were different from the systems without noise. This made an analysis of the different dis-location types meaningless for these systems. Also, not for all m were two different dislocation types found, because only a finite amount of samples per noise realization could be created.

(a) (b)

Figure 16: For systems with N = 1002 _{and ρ = 0.84 with the scale of y-axis being}

loga-rithmic: (a) Lowest eigenvalues plotted for type A and type B dislocations. (b) Stiffnesses of nonlinear plastic modes plotted for type A and type B dislocations.

(a) (b)

Figure 17: Nonlinear plastic modes calculated for systems with N = 1002_{. (a) is a}

de-localized mode calculated for a system with a type A dislocation and m = 22. (b) is a delocalized mode calculated for a system with a type B dislocation and m = 28.

8

Discussion

From the analysis of the perfect crystals the following can be said. For larger systems with ρ = 0.84 both the lowest eigenvalues and the stiffnesses of the nonlinear plastic modes seem to go towards zero at lower m. In the case with nonlinear plastic modes the stiffnesses even seem to start linearly go to zero and even seem be exponentially going to zero at larger systems.

Systems with ρ = 0.822 seem to have more predictable data. The lowest eigen-values seem to stay approximately constant regardless of system size and m and the stiffnesses of the nonlinear plastic modes seem to generally form a curve. At larger system sizes though, the stiffnesses start to become more unpredictable and erratic, which distorts the plot of the stiffnesses. From this can be discerned that the nonlin-ear plastic modes are very dependent on the pressure within the crystal. Doing an expanded nonlinear mode analysis at densities similar to ρ = 0.822 might yield more conclusive results for future research.

From the analysis of the crystals with noise it becomes clear that most investigated system sizes are still too robust to converge to different dislocations. It has also be-come clear that systems generally converge to one of two types of dislocations. The stiffnesses of the nonlinear modes also show that there seems to be little difference between the stiffnesses of both types except that type A dislocations generally have lower stiffnesses and go towards zero earlier than type B dislocations. Starting from a certain line defect size m the nonlinear plastic modes also start to delocalize for any system which is large enough. The question as to why that happens exactly also still has to be answered.

Since nonlinear plastic modes have not yet been used to analyse dislocations in crys-tals, there is little literature with which the results can be compared, but there are still some conclusions which can be drawn. Systematically creating defected crys-tals in 2D has proven to be somewhat difficult. The systems can converge to saddle points and they can also converge to defective systems starting from some initial conditions. Adding a small amount of noise to a system also causes larger systems to converge to multiple possible configurations. However, the stiffnesses do seem to behave according to a general trend as the defect size increases and behave uniquely but consistently at different pressures.

There are a lot of ways in which the research can be expanded. Analysing larger systems would probably open the possibility to analyse the different dislocation types further. Researching systems at pressures between the pressures used in this thesis or larger might also yield more useful results. The most obvious next step would be expanding the research to dislocated three dimensional crystals.

9

Conclusion

Both a normal mode analysis and nonlinear plastic mode analysis have been done for multiple system sizes, densities and with different noise realizations.

The relation of the stiffnesses and lowest eigenvalues of M with the defect size m seems to be heavily dependent on both system size and the pressure in the system. The stiffnesses of the nonlinear plastic modes seem to go towards a small value earlier as system size increases at higher pressure. At lower pressure the lowest eigenvalues remain approximately constant and the stiffnesses go slowly toward a small value regardless of system size.

Multiple noise realizations reveal that systems converge to two different dislocation types. The difference between the two dislocation types is mostly that the stiffnesses of one type are larger than the stiffnesses of the other type and the m at which the nonlinear plastic modes delocalize.

From the analysis can be concluded that systematically creating defected crystals in 2D is difficult due to issues where systems converge to saddle points or become defective and due to larger systems converging to different possible configurations with different dislocation types. It can also be concluded that the stiffnesses of the modes do seem to follow a general trend as m increases.

For further research the most obvious recommendation is to expand the research to dislocated three dimensional crystals. Otherwise, analysing larger system sizes and systems with different pressures could also yield better results.

Acknowledgements

I would like to thank my supervisor for this project, Dr. Edan Lerner, with whom I have had many fruitful discussions and meetings and from whom I always left with more ideas about what to do with my project than I could manage to explore.

References

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