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Spanish # Dislocation Motion and Deformation If you download the NetLogo application, this model is included. You can also Try running it in NetLogo Web

## WHAT IS IT?

This model allows the user to observe the effects of external forces on a close-packed 2D crystal lattice. It also gives a qualitative image of stress/strain fields around edge dislocations. An edge dislocation is a type of crystal defect that is comprised of an extra half-plane inserted into a lattice. They are an important feature of almost all materials because they facilitate deformation. In this model, an edge dislocation can be initialized within the material, and shear, tension, or compression forces can be applied to the material.

This is a Molecular Dynamics simulation, meaning an interatomic potential governs the energy and motion of each atom. Here, we are using the Lennard-Jones potential. Atoms attempt to minimize their potential energy by moving to an equilibrium distance away from the atoms surrounding them, which means that the atoms are moved in response to the force they feel from the surrounding atoms. The position and velocity of the atoms are updated every time step with the velocity Verlet algorithm.

## HOW IT WORKS

### Interatomic Forces

The Lennard-Jones potential shows that there is an equilibrium distance between two atoms where the potential energy of each atom is minimized. If the atoms are farther than this equilibrium distance, they will attract each other. If they are closer, they will repel each other. The potential is given by:

V = 4 * eps ((sigma / r)<sup>12</sup> - (sigma / r)<sup>6</sup>)

In this formula: - "V" is the potential between two particles. - "eps", or epsilon, is the depth of the potential well. - "sigma" is the distance where the inter-particle potential is zero. - "r" is the distance between the centers of the two particles.

Taking the derivative of this potential equation yields an equation for force. A negative force means the atoms repel each other and a positive force means they attract.

In this model, each atom calculates the force felt from its neighbors in a 5 unit radius. This is to reduce the number of calculations and thus allow the simulation to run faster; after this distance, the force is extremely weak, and therefore it is acceptable to ignore interactions beyond this point. These forces are summed and then divided by the particle's mass to yield an acceleration, which is passed into the velocity Verlet algorithm. This algorithm is a numerical method that integrates Newton's equations of motion. Both the velocity of an atom and the position of an atom after one time step are calculated.

### External Forces

External forces are applied by adding a term to the total LJ force felt from neighboring atoms. This force varies based on the FORCE-MODE. Atoms acted on directly by an external force are marked with a black "X". Pinned atoms, or atoms that do not move, are marked with a black dot. These atoms are excluded from updating position and velocity via the velocity Verlet algorithm.

In SHEAR mode, F-APP will be exerted left-to-right on atoms in the upper left of the material, while the bordering atoms on the bottom half of the crystal are pinned and do not move. The pinning of the bottom atoms represents the bottom of the material being clamped or otherwise held in place, so F-APP cannot simply rotate the lattice.

In TENSION mode, F-APP is exerted leftwards on the left side of the sample. The sample shape is different in tension mode to replicate the samples used in tensile testing. The atoms in the left shoulder are all acted on directly by the external force. If AUTO-INCREMENT-TENSION? is turned on, a force to "counteract" or "equalize" the forces from Lennard-Jones interactions is first applied to shoulder atoms. Then, an additional leftwards force is applied. This creates a smooth stress-strain curve and allows for deformation to occur in the neck region instead of the shoulder region. Shoulder region deformation would be negligible in a real life tensile test. In this simulation, the atoms farthest to the right (border of the right shoulder) are pinned. Note, this is the only mode in which the STRESS - STRAIN CURVE is plotted.

In COMPRESSION mode, F-APP is exerted rightwards on the left side of the sample. The bordering atoms on the right side are pinned.

To help visualize external forces exerted, the white line bookended by two arrows (or one, if in shear mode) shows where the external force is applied. The arrows point in the direction the force acts. In SHEAR and COMPRESSION, atoms within 1 unit of the force line are affected.

In this simulation, the temperature is controlled in order to reduce erratic motion from the atoms. This is done by calculating the thermal speed of the material, which entails finding the mean speed of the atoms. The thermal speed is related to temperature, so the thermal speed using the user-set system temperature is calculated. This is defined as the target speed and the current thermal speed of the material is the current speed. The target speed is divided by the current speed and the velocity of each atom is multiplied by this scaling factor.

## HOW TO USE IT

Before clicking the SETUP button, you first need to select the FORCE-MODE (SHEAR / TENSION / COMPRESSION), whether you’d like a dislocation initialized (CREATE-DISLOCATION?), and the size of the crystal (ATOMS-PER-ROW / ATOMS-PER-COLUMN). If you are in the TENSION FORCE-MODE, you will also need to select whether you'd like the force to be automatically increased or you'd like to control the force manually (AUTO-INCREMENT-TENSION?).

FORCE-MODE allows the user to apply shear, tension, or compression forces to the crystal. Refer back to the External Forces subsection of the HOW IT WORKS section for a description of how the forces are applied.

AUTO-INCREMENT-TENSION? (TENSION mode only) will automatically increase F-APP by .0005 N if the current sample length is equal to the sample length during the previous time step or if the current sample length is shorter than the sample length during the previous time step. This is useful for producing a stress-strain curve. You can start at F-APP = 0 N and allow the simulation to run until fracture occurs.

CREATE-DISLOCATION? allows the user to initialize an edge dislocation in the top half of the lattice. It is not how the dislocation would exist within the material in an equilibrium state. Although the dislocation may exist in a metastable state within this simulation, that is due to the pinning of atoms. In an unpinned material, the dislocation would usually propagate out in lattices of these nano sizes due to surface effects and the Lennard-Jones potential governing atomic motion (however, this propagation is temperature dependent so at low enough temperatures it may be possible to form a metastable dislocation).

ATOMS-PER-ROW sets the number of atoms horizontally in the lattice, while ATOMS-PER-COLUMN sets the number of atoms vertically in the lattice.

These are all of the settings that need to be selected prior to SETUP. The other settings can also be selected before SETUP, but they are able to be changed mid-simulation, while the aforementioned settings are not. The functions of the other settings are listed below.

LATTICE-VIEW provides three options for observing the crystal lattice. LARGE-ATOMS shows atoms that nearly touch each other in equilibrium which helps to visualize close packing. SMALL-ATOMS shows atoms with a reduced diameter which can be used with links to see both atomic movement and regions of tension and compression in the crystal. HIDE-ATOMS allows the user to hide the atoms completely and only use the links to visualize deformation within the material.

SYSTEM-TEMP sets the temperature of the system.

F-APP is the external applied force on the material. It is the total force applied, not the individual force on each atom. The actual numbers are arbitrary, since the Lennard-Jones force has not been calibrated to model a real material. The units are in Newtons.

DELETE-ATOMS allows the user to delete atoms by clicking on them in the display. If the button is pressed, clicking on the View will delete atoms. If it is not pressed, clicking will do nothing.

UPDATE-ATOM-COLOR? controls whether the color of the atoms is updated. The color is an indicator of the potential energy of each atom from Lennard-Jones interactions. A lighter color means the atom has a higher potential whereas a darker color indicates a lower potential.

The monitor EXTERNAL FORCE PER FORCED ATOM displays the individual force that each atom directly being influenced by the external force is experiencing (in the case of TENSION mode when AUTO-INCREMENT-TENSION? is on, this is the average individual force). CURRENT F-APP is the total external force on the sample (in SHEAR, COMPRESSION, and TENSION without AUTO-INCREMENT-TENSION?, this is the same as the F-APP slider value). SAMPLE LENGTH is the length of the sample from end to end. It is in terms of the equilibrium interatomic distance between two atoms (rm). The stress-strain curve only works for TENSION mode and can be used with AUTO-INCREMENT-TENSION?.

## THINGS TO NOTICE

SHEAR Mode

As the material deforms, how does the edge dislocation travel?

Where are the areas of tension and compression around the edge dislocation?

TENSION Mode

When the material deforms, does it do so randomly or are there observable preferences for deformation in the material?

How does the stress-strain curve correspond to elastic deformation within the material?

How does changing the temperature affect the deformation patterns within the material?

COMPRESSION Mode

When the material deforms, are there sections of atoms that maintain their original shape? Or is the deformation completely random?

## THINGS TO TRY

In SHEAR mode, initialize an edge dislocation and apply the smallest force possible to deform the material. Does the material continue to deform after the edge dislocation propagates out? What force is require to deform the material after the initial edge dislocation has propagated out?

In TENSION mode, samples with larger numbers of atoms per row and smaller numbers of atoms per column are generally best for observing deformation/fracture (For example, 18 atoms per row and 13 atoms per column). Increasing the number of atoms per column also works well; you just want to maintain a sample with a long "neck" area. To produce a stress-strain curve, set F-APP to 0 N and turn AUTO-INCREMENT-TENSION? on. While running the simulation at a lower temperature creates a smoother stress-strain curve, the slip behavior differs for low and high temperatures, so it is worthwhile to run the simulation with both.

While running the simulation, pay attention to the different directions of links. Are tension and compression concentrated in certain areas? Do they differ in different directions?

Create vacancies with the sample in tension and compression by deleting atoms. Does this change how the material deforms?

Create a second edge dislocation in the shear mode by deleting atoms. How does this change material deformation?

## EXTENDING THE MODEL

Add a slider to vary eps (epsilon). How does changing eps affect deformation? Are smaller or larger forces needed to deform the material as eps increases? Does the material observably deform differently?

Color the atoms according to a different property than their potential energy. Suggestions include according to the magnitude of force felt or the direction of the net force on each atom.

Apply forces in different directions than the ones provided. Does the material deform in the same way? Why or why not?

(Advanced) Add in another type of atom. Give this atom different properties, such as a different mass, or a different radius. You will need to store a separate eps and sigma for each atom and their different types of interactions. For example, if you have A and B atoms, you will need to have an A-A eps and sigma, an A-B eps and sigma, and a B-B eps and sigma.

## RELATED MODELS

The Lennard-Jones model

## NETLOGO FEATURES

When a particle moves off of the edge of the world, it doesn't re-appear by wrapping onto the other side (as in most other NetLogo models). We would use this world wrapping feature to create periodic boundary conditions if we wanted to model a bulk material.

Link coloring uses the transparency feature in order to create greyish shades. In order to make links in tension (yellow) and compression (red) that are very close to equilibrium (grey) visually close to equilibrium, they are more transparent in order to produce a grey hue. The farther away a link is from equilibrium, the more opaque it is.

## HOW TO CITE

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