Diffusion

As it is known, atoms seldom leave their balance position in solids. Nevertheless, as it happens sometimes, the contacting matters can penetrate one another. Penetration of one substance into another is termed an interdiffusion. As for selfdiffusion, it takes place if the concentration of atoms of radioactive isotopes of the same substance, which was brought into some site of the sample, is flattening. Selfdiffusion of atoms in the crystal lattice is carried out by one of three mechanisms from the point of view of classical theory:

1. If there is a vacancy in some knot of the crystal lattice, one of the neighbor atoms can fulfill the jump from its standing into the vacant knot. Then the vacancy will occupy the previous place of this atom. Such transitions are similar to the movement of vacancies. Those atoms can jump into the vacancy, the kinetic energy of which is sufficient for performance of local deforming of the crystal lattice.
2. If the energy of atom oscillations is large enough, the atom can jump from its place into the space between lattice knots and turn into the interstitial atom. In this case, such a fluctuation of energy is necessary that is considerably larger than for the jump into the vacant knot, because the lattice deformation (and strain) will be much greater.
3. Lastly, the neighbor atoms can simply interchange by place. Thus, the final state does not differ from the initial state, but the strain of the lattice during the jump will be much more than for the first two termed mechanisms.

Calculations show that the contribution of jumps by the second and third mechanisms to self-diffusion is small enough; the vacancy mechanism plays the basic role. At the same time, diffusion of small impurity atoms, for example, hydrogen in platinum or carbon in iron, occurs more often by the way of jumps between the knots. Frenkel has analyzed the self-diffusion in the volume-centered crystal lattice (vcc). He supposed that diffusion is carried out by the way of jumps of atoms into vacancies. The frequency of atom jumps from one plane to another is spotted by the frequency of natural oscillations of atoms νo and geometrical factor (the number of nearest knots on the next plane). Apparently, the diffusivity increases with temperature magnification under the exponential law. Such dependence is confirmed experimentally. Impurity atoms or the radioactive atoms are in parallel planes; the distance between them is equal to half of parameter d of the crystal lattice (one pixel on the monitor screen). According to the program, atoms are disposed not in planes but in rows with different coordinate y on the screen at the same coordinate x. The filling is made up to half of the model length (and halves of screen) along x-axis, and the zero number of particles and value of concentration is set for the other half. All particles can jump to the right or to the left. In a result of the basic program work, the numbers of atoms for each coordinate are calculated, the histogram of the concentration distribution is calculated and displayed periodically in the picture canvas, and the values of the relative concentration for five columns of the histogram are represented in this window. Besides, the time, which is found as a product of the timestep with the number of the steps, and mean-squared displacements of particles relative to their initial positions are also represented in the form. It is easiest to spot the diffusivity by the Einstein’s equation through the mean-squared displacements. The program interface provides choosing of the chemical element and temperature (offered values are shown in the window of type “Memo”). Pauses in the work of the program are used for displaying the intermediate distributions of concentration, for recording the interval limits for determination of the diffusion coefficient, and for reading the values of the mean-squared displacements of particles and the time of experiment. Several basic procedures are shown below in short form: the procedure Histogram, which ensures the plotting of the concentration distribution on the

#Recommended Experiments

1. For chosen substance and parameters, calculate the diffusion coefficients for different temperatures, using values of mean-squared displacements of atoms. Let us estimate the error of diffusivity definition. Construct the dependence graph ln D on 1/T. Approximate the written values by the linear relation. Calculate the activation energy of diffusion in the relative units (Q/R) by the slope of the dependence line (taking into account scales along axis). Compare the obtained value with the tabular value noted in the window “Memo”.
2. For chosen substance and corresponding parameters (Q,d), calculate the diffusion coefficient at different temperatures, using the distribution of particles along the x-axis figured by the histogram (printed values of the relative concentration C/C0). Enter the searching boundaries of the diffusivity values into the windows Edit with inscriptions D1 and D2 so that the expression (k1-erfc) in the shown procedure changed the sign in these limits. Press the button with inscription Dif b.

This program was developed under the guidance of Professor A. Ovrutsky.