Investigation of diffusion mechanisms in ideal two-dimensional blocks of defect-free crystals Ni, Ni3Al and al by computer modeling method

The main result of the phenomenon of diffusion is the mass transfer, i.e. thermally activated movement of atoms in the crystal lattice. In real crystals, the diffusion processes are affected by the presence of defects of the crystal lattice, such as vacancies, dislocations, dislocation loops, stacking faults, phase boundaries, block boundaries and grains [1]. The presence of imperfections should affect the activation temperature of diffusion. Obviously, in each case of imperfections some mechanism or set of mechanisms must exist, which would create conditions for thermally activated mass transport. One of the major factors affecting the diffusion processes and therefore such parameters as the rate of atom diffusion, diffusion coefficient, the diffusion activation energy, is the presence of local free volumes in the crystals, which may occur near the various types of defects of the crystal lattice.

Real experiments allow us to study diffusion in metals and alloys, as a rule, by initial and final states of the structure, which gives only an indirect picture of the various mechanisms of diffusion.

For a visual and more detailed study of the mechanisms of diffusion is now more intensively used computer modeling to track the trajectory of atoms, and receive a detailed picture of individual diffusion mechanisms taken in dynamics [2-5]. This method is in addition to the known experimental and theoretical research methods, often acting as a liaison between the two. The computer model can serve as a means of testing theoretical insights and, conversely, to explain or predict phenomena not previously lit in theory and experiment to the full.

As a method of computer modeling the method of molecular dynamics was chosen. Its main advantages over other methods of computer modeling as applied to condensed matter physics lie in the fact that the atoms in it are not tied to nodes of the ideal crystal lattice, which allows one to model the phenomena associated with de-crystallization of the structure and the displacement of atoms. The choice of the two-dimensional model is justified, first of all, because in tridimensional crystals diffusion processes are implemented along the close-packed directions, which are located in the planes{111} of the face-centered cubic (FCC) lattice. At the same time, according to the author [6], a simple model has its advantages: the simpler the model, the less the possibility of erroneous conclusions. Depending on the task a computer experiment can be simplified by using a sequence of different approaches to the study of materials, ranging from simple two-dimensional systems with the transition to two-dimensional layered systems, and then to three-dimensional structures.

Two-dimensional crystal is like a sweep of such planes in tridimensional material. Such structures are realized in nano-structured thin-covered materials, which recently received more and more increasing attention due to their possible use as intelligent materials with new properties. The experiments were carried out using the program [7].

Outside the designed-basis block crystal is repeated with periodic boundary conditions. Then, using the method of molecular dynamics a computer experiment takes place. The molecular dynamics method is that, by using Newton's equations for the motion, and knowing the strength of interaction, velocity of the particles and their displacement with a small step of integration is calculated. The interaction between the atoms in the crystal is given by the respective functions of the Morse potential, the parameters of which were calculated from experimental data of overall modulus, sublimation energy and lattice parameter [8].

(1)

where r - the distance between the atoms;

λ, β, D - experimental parameters.

In this method limits of the amount of the design cell were about 10³-10⁶ atoms. From the macroscopic point of view, it is extremely small. So that the results can be extended to macro-volume, boundary conditions that allow an approximation to "stitch" the design cell to the outside volume were put on a design-biased block.

Preliminary (primary) relaxation was performed using the method of molecular dynamics. The initial rates of atomic vibrations were set equal to zero, corresponding to an initial temperature of 0 K. During relaxation the cell temperature increased and reached the level at which there was stabilization of the kinetic energy, and the temperature rise stopped. After stabilization of the temperature the cell was subjected to ultra-fast cooling. All the atoms’ rates were periodically leveled to zero (when oscillations of kinetic energy reached the maximum) as long as the atoms occupied the positions of equilibrium, and an increase in temperature associated with relaxation phenomena could no longer be observed. When starting the main experiment it was believed that the established structure of the designed cell is stable at temperatures close to absolute zero.

The paper evaluated the diffusion parameters of the main components that make up the composite structure under study.

The stability of interfaces in a composite material on the processes of thermal activation depends directly on the diffusion characteristics of individual components. The beginning of diffusion processes is associated with the emergence of self-organized collective displacements of atoms, leading to structural changes in the material. The emergence of collective atomic displacements from chaos, corresponding to dynamic vibrations of atoms is random. The probability of this state depends on the time of the computer experiment during which the material is held at a given temperature. With the increase in the length of time of the computer experiment random events turn into the state of certain statistical regularities. However, in solving a number of problems the main parameter of the study is to detect primary change in the system. In such cases, the time can be restricted to a small interval for impulse heating of the system up to a certain temperature.

For ideal two-dimensional blocks of defect-free crystals, Ni, Ni₃Al and Al onset temperature diffusion processes were found to be 1920K, 1700K and 1150K, respectively, much higher than the melting temperature of these materials.

Figure 1. Atomic displacements in ideal defect-free two-dimensional blocks of crystals a) Ni; b) Ni₃Al; c) Al

In all cases at first stages the ring mechanism of atom diffusion in triangles, quadrilaterals, pentagons and hexagons of the nearest neighbors could be observed (Fig. 1a, b, c). In this case, there was the mass transfer of the material, but structural changes in the system did not occur. In inter-metallic compound, migrations of Ni-atoms, on sub-lattice Ni correspond to the diffusion ring mechanism in hexagon of the nearest neighbors (Fig. 1 b), which does not result in the violation of order in the system. With the increasing of temperature, movements of atoms by circular trajectories, when Frenkel pair appears and annihilates in the system, are added to the given diffusion mechanisms. If there is no annihilation of Frenkel pairs, the trajectory of movements is a broken line, at the ends of which there is a Frenkel pair - vacancy and interstitial atom (Fig. 2a). The diffusion coefficient of this dramatically increases (1,642 10^-11 m²/s). In inter-metallic compound Ni₃Al such processes create disorder areas in which there may be observed buds and clusters of new phases of the system Ni - Al (Fig. 2 b).

Figure 2. Atomic displacements (a), phase composition (b) under pulsed heating of the crystal Ni₃Al to 1800K

It is enough just to introduce the vacancy as the temperature of the onset of diffusion processes is reduced drastically to 1600K, 1500K, 800K for Ni, Ni₃Al and Al, respectively. This is below the melting temperature. The main mechanisms of diffusion in this case are the crowdions mechanism, and the movement of atoms along the broken line by the vacancy mechanism. When approaching the melting point again, to the two above-mentioned mechanisms of migration, the mechanism of forming and annihilation of Frenkel pairs is added. In this case, the paths of the moving atoms have a larger extent and are closed lines if Frenkel pairs annihilate. When annihilation in computer experiment did not occur, visualizers showed the presence of Frenkel pairs. In all materials investigated the total energy of the crystal after the procedure to quench 0K increases, when Frenkel pairs are saved, and decreases after the completion of their annihilation. In inter-metallic compound, the total energy of the crystal increases due to the presence of the accompanying diffusion process of the superstructure destruction.

Figure 3. Atomic displacements after a) introducing vacancy into crystal Ni; b) introducing vacancy into crystal Ni₃Al; c) introducing vacancy into crystal Al

Introduction of a bi-vacancy into the crystal lattice causes a decrease of critical temperature of the diffusion process onset to the level of 950K (D=0,882 10^-11 m²/s), 900Ê(D=0,814 10^-11 m²/s) and 250Ê (D=0,915 10^-11 m²/s) for Ni, Ni₃Al è Al, respectively (Fig. 4a).

1 - Migration of bi-vacancy in a broken path, 2 - Frenkel pair

Figure 4. Atomic displacements after introducing the bi-vacancy into a crystal a) Ni; b) Ni₃Al; c) Al

The main mechanisms of diffusion in this case are either migrating of bi-vacancies in a broken path or bi-vacancies transformation into a complex consisting of an interstitial atom and three closely spaced vacancies, subsequent recovery of the bi-vacancies from the complex after the hardening process. In the alloy Ni₃Al, the complex creates a region of disorder during migration process.

With the increasing of the time for the computer experiment, the difference in the distribution of the trajectories of atomic displacements within the units of the crystal under study is due to the fact that the investigated processes develop at random, so their own elements of diffusion of atoms in the crystal at the discrete time interval of pulse heating begin to develop.

Thus, the main diffusion mechanisms are crowdions, collective movement of atoms along closed paths, mechanism for moving the interstitial atom along a closed path to annihilation with a vacancy in the dynamic formation of Frenkel pair and transformation of bi-vacancies into complex.

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