CC BY
142
УДК 539.371, 004.942
Identification and space-time evolution of vortex-like motion of atoms in a loaded solid
A.I. Dmitriev1'2, A.Yu. Nikonov12, A.E. Filippov3, and V.L. Popov3
1 Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634055, Russia
2 National Research Tomsk State University, Tomsk, 634050, Russia
3 Technische Universität Berlin, Berlin, 10623, Germany
The paper studies the redistribution of internal stresses and atomic displacements in a preloaded copper crystallite using the molecular dynamics method. It is shown that relaxation within the crystallite volume is accompanied by the formation of dynamic structures in which atomic displacements produce a coherent system of vortex lines. In so doing, the displacement of atoms in neighboring vortex structures has the opposite sign of the angular velocities. The evolution of the dynamic vortex structures is analyzed using an original technique for identifying the vortex motion in the space of a vector variable with a discrete step. It is shown that a system of dynamic vortices and antivortices can propagate inside the crystallite, ensuring the transfer of stresses from the bulk of the loaded material to its unloaded periphery in order to preserve continuity. The developed technique has revealed that the lifetime of such defects depends on their size and ranges from fractions to tens of picoseconds. The simulation results correlate well with the experimental electron microscopy data on the estimation of spatial parameters and lattice curvature during strain localization in the region of elastic distortions.
Keywords: vortex motion, dynamic defects, molecular dynamics, vortex visualization, stress redistribution
DOI 10.24411/1683-805X-2018-13006
Идентификация и пространственно-временная эволюция вихреподобного движения атомов в нагруженном твердом теле
А.И. Дмитриев1,2, А.Ю. Никонов1,2, А.Е. Филиппов3, В.Л. Попов3
1 Институт физики прочности и материаловедения СО РАН, Томск, 634055, Россия 2 Национальный исследовательский Томский государственный университет, Томск, 634050, Россия
3 Берлинский технический университет, Берлин, 10623, Германия
В работе исследуются процессы перераспределения внутренних напряжений и атомных смещений в предварительно нагруженном кристалле меди. Исследования проведены с использованием метода молекулярной динамики. Показано, что на стадии релаксации в объеме кристаллита возможно формирование динамических структур, в которых атомные смещения образуют согласованную систему вихревых линий. При этом перемещение атомов в соседних вихревых структурах имеет противоположное значение угловых скоростей. Для анализа эволюции динамических вихревых структур разработана оригинальная методика идентификации вихревого движения в пространстве векторной переменной с дискретным шагом. Показано, что система динамических "вихрей" и "антивихрей" может перемещаться в объеме кристаллита, обеспечивая тем самым перенос напряжений из объема нагруженного материала к его ненагруженной периферии для сохранения сплошности. С использованием разработанной методики установлено, что время существования таких дефектов зависит от их размеров и составляет от отдельных долей до десятка пикосекунд. Результаты моделирования хорошо коррелируют с экспериментальными данными электронной микроскопии по оценке пространственных параметров и кривизны кристаллической решетки при локализации деформации в области упругих дисторсий.
Ключевые слова: вихревое движение, динамические дефекты, молекулярная динамика, визуализация вихрей, перераспределение напряжений
1. Introduction
The vortex-like and rotational motion of media is a topic of relevance across a vast number of applications, includ-
© Dmitriev A.I., Nikonov A.Yu., Filippov A.E., Popov V.L., 2018
ing dynamics of plasmas [1, 2], modon structures in geophysics [3], astrophysical flows [4, 5], the dynamics of quantum condensates [6-8], and the skyrmion magnetic
structures [9-11], to name a few. At the same time, the appearance of circular motion in such fields of modern physics as physics of strength and plasticity still remains limited. Recent studies have shown that the vortex motion including the rotational mode of deformation is a very important object that in some cases can ensure the preservation of continuity and relaxation of a loaded body [12]. Indeed, various real and virtual experiments [13-16] confirm the presence of grain boundary rotation along with other plastic flow mechanisms in nanocrystalline materials. Moreover, in the cases when the response of the loaded body becomes insufficient to provide necessary stress relaxation, the vortex-like motion becomes a precursor of the formation of a material discontinuity [17, 18].
In addition to the conventional violation of the crystalline order, vortex-like displacements of atomic groups can be treated as dynamic defects because of their dynamic behavior. These defects correspond to a self-consistent circular motion of a large number of material points and exist only at some stages of loading, determining the consistency of displacements in different parts of the solid body. According to Refs. [12, 18], the deformation of a heterogeneous material containing internal interfaces or free surfaces is accompanied by a collective circular motion near them. Such vortex-like patterns of elastic displacement are normally associated with the Rayleigh, Lamb, or Love surface waves, or Stoneley waves propagating along a phase boundary. For example, it was demonstrated by molecular dynamics simulations [19] that vortex structures in the velocity field are formed around grain boundaries of poly-crystals under shear loading and can provide grain boundary migration. The nature of the vortex-like motion lies in the formation of local velocity gradients and stresses tangential to them near internal or external interfaces. It can therefore be expected that circular motion takes place at different scales: from atomic to macroscopic. Psakhie et al. [20] have shown that the vortex-like motion becomes a precursor of discontinuity formation in a loaded material on the macroscale. On the other hand, the theoretical results and experimental evidence of Zhang et al. [21] indicate that the contribution of the rotational mode of deformation can significantly increase under dynamic loading, especially in nanomaterials with a high concentration of grain boundaries. Nevertheless, such a fundamental factor as the elastic circular motion in a material under dynamic loading is still not fully explained. The above suggests that revealing the role of vortex displacement in elastic energy redistribution and, as a result, in the deformation and fracture of heterogeneous or homogeneous materials near free surfaces is a particularly important problem in materials science.
Although a direct observation of such defects is impossible because of their extremely small size and short lifetime, they strongly influence the entire deformation process at both the micro- and macroscale. In view of the principal significance of the free surface, internal or external
interfaces, and dynamic nature of the considered vortex phenomena, the main approach to studying such defects is computer modeling [22]. There are difficulties in identifying such defects and especially in observing their time evolution. It is very difficult to visualize and identify the vortex motion of a discrete randomized system of a large number of particles. The most interesting task is to understand the role of vortices at different stages of the evolution. This would help us to find regularities in the motion and relate them to the general behavior of the material. We would better understand the relationship between different scale levels.
Jiang with coauthors [23] paid much attention to the question of identifying the vortex motion. They emphasize that it was a challenge to convert an intuitive description of the vortex into a formal definition. The authors reviewed existing detection methods and particularly discussed nine methods that are representatives of the state of the art, each of which has its own advantages and drawbacks. The discreteness and randomized nature of a system of moving particles raises additional difficulties in the identification. However, there are a number of approaches that allow such analysis. For example, Tordesillas et al. [24] analyzed the incompatibility of displacements of individual particles. Their approach revealed the existence of different vortices and allowed the estimation of the characteristic vortex sizes; some rules were established for the deformation and distortions in granular media.
Up to now it has been assumed that dynamic defects exist only in the active stage of loading, when forced displacement velocities are much higher than the typical time of conventional defect generation. At the same time, many fundamental studies consider the rotational mode of deformation as an important mechanism of internal stress relaxation [25]. The present paper focuses on the possibility of formation of dynamic vortex defects at the stage of internal structure relaxation in the material, i.e., after the stage of active loading. Molecular dynamics modeling is performed to study the deformation of a copper crystallite compressed to the deformation immediately preceding plastic flow. Possible vortex motion of a particle ensemble is identified using an original approach based on the vector field calculation in each point of the available space.
The paper is organized as follows. Initially, we describe the numerical model and the technique of vortex motion identification. Then, the numerical modeling results are combined with the application of the main approach, and the main features of the found vortex motion of particles are described. The final part presents conclusions of the study.
2. Model description
The initial trial structure of the model crystallite without defects is shown in Fig. 1. It has the form of a parallel-
epiped measured 21.7x21.7x3.6 nm along the [100], [010], and [001] directions that respectively contain about 150000 atoms. The crystallite was conditionally divided into three regions. Region II is a freely deformed part of the crystallite where new states of the particle ensemble were found from the Newtonian equations of motion. Atoms of regions I and III were exposed to an external influence using a two-stage loading scheme. This scheme includes an active and passive stage. The type of load at the active loading stage corresponded to uniaxial compression with the so-called string boundary conditions [26]. These conditions mean that in the [010] direction the atomic velocity projections in regions I and III are fixed, but in the directions different from [010] the velocities are determined by the corresponding atomic environment.
Such a type of boundary conditions is preferable for molecular dynamics modeling of a selected crystal lattice fragment. In contrast to conventional uniaxial compression where displacements of particles in the directions different from the loading direction are equal to zero, spring boundary conditions are used and lateral expansion under compression appears also in loaded parts of the system. The rate of deformation under compression reached up to 0.5% per picosecond. To achieve this, at each time step we assigned the values of -50 m/s and 50 m/s to the Y velocity projections of atoms from regions I and III, respectively. Loading continued to the deformation immediately preceding plastic flow. The total deformation reached in the active loading stage was varied between 5 and 10% in different calculations.
At the second loading stage (passive stage) the positions of atoms along the Y axis in the loaded layers were fixed to preserve the deformation achieved at the active stage. Here we modeled lattice relaxation based on the Newtonian equations, but without any additional sources
Fig. 1. Initial structure of the model crystallite and loaded region at the active loading stage
of influence. In other words, a so-called microcanonical NVE ensemble with fixed total number of atoms, volume of the system, and its energy was applied.
Taking into account the realistic extension of the modeled fragment, we used periodic boundary conditions in the [001] direction and free boundaries in the [100] direction. To exclude the "induced effect" related to the symmetry of the ideal lattice, the copper crystallite was studied at the initial temperature 10 K. The final temperature was found 'a posteriori' using the Maxwell distribution for the atomic velocities. Detail analysis of the relaxation process was carried out by studying the evolution of atomic configurations at different points of time and for different time intervals between them.
Molecular dynamics has been adopted as the most suitable method for simulating the acoustic response to indentation. Large-scale atomic/molecular massively parallel simulator software was used as a computational tool [27]. The interaction between atoms in the copper crystallite was described using an interparticle potential reconstructed in the framework of the embedded atom method [28]. The physical correctness of the software was verified by matching the computed and experimental values of the vacancy and stacking fault formation energies. The mismatch achieved was less than 1%. The visualization of the molecular dynamics simulation data and structure analysis were carried out using the open visualization tool OVITO [29].
3. Definition of vortex motion
Observations show that there are vortex-like displacements of atoms in the modeled crystallite. As an example, Fig. 2 illustrates the displacements of atoms of the modeled crystallite for different time intervals in the plane parallel to the (001) plane. For better visibility, the line segments corresponding to the atomic displacements are enlarged 5 times. One can easily see from the framed region in Fig. 2, a, where such displacements are shown for the interval between 8.4 and 8.9 ps, that 8 vortex lines are formed in the crystallite volume. The motion of atoms in all the lines takes place around the axes parallel to [001]. The diameter of the vortex lines is about 15-20 distances between the crystal planes.
The rotation directions of the vortex lines are correlated as follows: the nearest pairs of the vortices rotate with opposite signs of the angular velocity and therefore the motion of atoms is correlated so that the continuity of the material is preserved. Our calculations have shown that such time intervals can be selected during relaxation in which the system of 8 vortex lines depicted in the framed region (Fig. 2, a) periodically changes the sign of the angular velocity. In other words, the rotation of all the 8 vortices changes synchronically. For example, Figs. 2, b and 2, c show displacements of atoms in the framed region for two
Fig. 2. Projection of atomic displacements onto the XY plane in the time interval between 8.4 and 8.9 ps. The same projections for the framed fragment A in Fig. 2, a for consecutive time points 8.4-8.9 ps (b) and 10.3-10.8 ps (c)
successive time points. It is directly seen that the rotation of atoms in Fig. 2, c is opposite to that in Fig. 2, b.
4. Identification of vortex motion
4.1. Circulation of the vortex field
The circulation of the vortex field is calculated as follows. First of all note that all positions in the data array in the plane are discrete, {xk, yk}, where the subscript k = 1, 2, ..., N numerates the nodes of the calculation lattice, N is the total number of the nodes (coinciding with real particles), and the distribution {xk, yk} is generally speaking random. The corresponding distribution of the values of the "displacement current" Jk = {dxk, dyk} is also a discrete array. For further use, it is convenient to interpolate the array into a continuous distribution J = J(x, y) over the ordered variables [x, y]: Jk = {dxk, dyk} ^ J( x, y). The area of the continuous variables [x, y] must be chosen to completely cover the area of the discrete variables {xk, yk}.
The mutual correspondence of the surface J(x, y) and the discrete set Jk = {dxk, dyk} can be controlled visually.
As an example, a particular implementation of the absolute value of the current J(x, y) = |J(x, y)| is depicted in Fig. 3, a. For further use, it is convenient to plot the value J(x, y) in the form of a planar density distribution shown as a grayscale density map in Fig. 3, b, which clearly illustrates the fine structure of the current including the minima of the current J(x, y).
Mathematically, the points where J(x, y) turns to zero are indicators of probable positions of vortices. In some cases, these points correspond to saddle configurations of the current and hence they are not centers of the vortices. So, to find real vortices, we need to calculate the circulation of current around such points. However, since the field J(x, y) is defined as an interpolation of a randomized data array, exact zeros of the current J(x, y) cannot be reached. Taking this into account, it is convenient first to determine the circulation around every point of the regular array and only then to interpret the results obtained.
The circulation of displacements around all points ofthe array can be calculated by defining the radius of its contour R, which must belong to intermediate scales between the main characteristic distances of the system. The minimum distance is the mean distance between the nearest neighbors ofthe array {xk, yk}, hence R >> mean(min | rk - rk-1). The maximum distance is the scale comparable to the complete system R << L, where L is the linear size of the system. In all the cases, when the contour formally extends beyond the regular array [x, y], where data about J(x, y) are absent, the contour must be plotted along the corresponding boundary from one of the points of its intersection with the boundary to another.
Numerically, the contour is a discrete data array with radius R and angle 0 <an < 2n, where the subscript n = 1, 2,..., Na, and the number of segments must be much larger than the unit Na >> 1 but is limited by the volume of avail-
able numerical information Na << N. Each value of n = = 1,2, ..., Na corresponds to the rotation of the radius-vector dR(n) = {dRx (n), dRy (n)} with the projections dRx (n) = R[cos an - cos an-1] and dRy (n) = R[sin an -
-sin an-1].
Each point of the contour
R(n) = [Rx(n), Ry (n)], (1)
Rx (n) = R cos an and Ry (n) = R sin an, plotted around every point [x, y] of the system, i.e., each point with the coordinates xcont = x + R cosan and ycont = y + R sin an can be associated with the nearest point of the regular array [x, y] (let us call it [ xn, yn ]) with the respective value of the current J( xn, yn).
As a result, for every point of the array [x, y] we have the set of the contour points [xn, yn], current vector J(xn, yn), and rotation vector R(n) according to Eq. (1). Now the circulation of the vector J (xn, yn) around such a contour can be calculated according to a formal definition:
C =£ J(xn, yn) • R(n), (2)
n=1
where symbol • means the scalar product.
Performing this operation for each point of the regular array [x, y] produces a 2-dimentional surface of the circulation C = C(x, y). The initial problem is thus generally solved. Finally, the same operations can be done for all time points of preliminary saved data for {xk, yk} and respectively for all other possible displacement currents
Jk = {dxk ,dyk}-
Now the study is reduced to the extraction of quantitative and visual information from consequently calculated distributions of the circulation C = C(t; x, y), which depend on the time t and regular coordinates [x, y].
4.2. Identification of vortex centers
First of all, we must formalize the calculation of the nominal positions of the centers of vortices and antivortices. Formally, we must define the positions of (all) local extrema of the function C = C(t; x, y) for each time point.
However, the direct use of this procedure in practical calculations can lead to artifacts related to some degree of randomization.
The above problem can be avoided if we make preliminary spatial smoothing of the function C = C(t; x, y) for the scales smaller or close to the characteristic scales of vortices. It is convenient to perform smoothing by a Gaussian convolution. Particularly, we can perform the fast Fourier transform C(t; x, y) ^ C(t; qx, qy), multiply the obtained distribution by the Gaussian
G (qx, qy ) = exp( - (qx2 + q')/A 2) with a given width A in the Fourier space, define the new function C(t; qx, qy) = C(t; qx, qy )Gq, qy), and perform the inverse transform C(t; qx, qy) ^ C(t; qx, qy).
Numerical check shows that for the absolute majority of the C(t; x, y) configurations such smoothing preserves all properties of the original distribution if A is correctly chosen. It allows one to uniquely define the position of maxima and minima (i.e., centers of vortices and anti-vortices) correctly corresponding to the structure of the displacement current (see Fig. 4).
Using the described procedure, we can both visualize the positions of all vortices in the system and automatically calculate their total number for all time points t. We can also determine the total circulation in the system Cs = J5 C(t; x, y) with the complete area S where the field J(x, y) is defined. The conservation or variation, if it takes place, of the circulation balance can be determined for uniaxial compression or compression with shear, respectively.
A direct observation of the appearance and disappearance of vortices on numerical visualizations of spatially distributed circulations have shown that even in the cases when the total value of Cs for the whole system is equal to zero, Cs = J5C (t; x, y) = 0, its partial values for separate quadrants of the system can differ from zero. In other words, vortices and antivortices are generated separately in different parts of the system, but annihilate near its center.
Y 150 100
50
0
f * % ь T 4
Fig. 3. Surface profile of the calculated absolute value of current J(x,y) (a) and its projection onto the (x,y) plane plotted as a grayscale map (b)
Y
150
100
50 0
Ж I
ИМ
0
50 100 150 200 X
Y
150
100
50 0
Ш
■Il
ÎÈ^izE; ; : i :
IM
Ü Uli
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50 100 150 200 X
Fig. 4. Spatial distribution (a) of the circulation of field С after smoothing by a Gaussian convolution and positions of vortex and antivortex centers in the corresponding time point (b). The light shades hereinafter denote clockwise circulation (vortices), dark shades denote counterclockwise circulation (antivortices)
This process can be quantitatively described by calculating the partial value of the total circulation C(t; x, y) for each quadrant:
Q4 =JS4 c ; x, У )
and by controlling their instant balance and mutual compensation at the moments of vortex annihilation in different parts of the system. The calculation results for a typical implementation of the time-space evolution of the system are represented in Fig. 5. The above described arguments, which relate to the nonbalanced appearance of vortices and antivortices in different parts of the system, their mutual compensation in different parts of the system, and finally annihilation in particular time points, are visualized by the
curves C2 = C2(t) (Figs. 5, a and 5, b) and C^4 = C^(t) (Figs. 5, c and 5, d).
The most valuable result of the described approach is perhaps the existence of the "scalar" but absolutely transparent language for the description of complex rearrangements in the compressed system, which is based on describing the position of the vortex centers, their total number C2 (t), and the number of vortices in separate parts of the system (t).
The proposed approach provides an additional "bonus". If the positions of the vortex centers are known, their motion can be tracked from the moment of their emergence to meeting and annihilation. In other words, the vortex motion can be described in a "static manner" as a map of their
15
£10-
! I
20
Time step
0
10
11 M
20
Time step
S
1С
30
40
20
Time step
Fig. 5. Time dependence of total circulation (a) and in a separate quadrant (b). Time variation of the total number of vortices and antivortices (c) and their difference in the chosen quadrant (d)
120 160 200 X
120 160 200 X
Fig. 6. Motion trajectories of some vortices and antivortices for the time intervals 0.5-5.5 ps (a), 5.5-9.5 ps (b). The zero time hereinafter denotes the beginning of the relaxation stage
trajectories on the plane of the specimen cross section. It excludes long qualitative speculations and provides a quantitative language for further description of very different systems of this type.
As a result, we obtain a simple algorithm for processing experimental or numerical results for the system under external pressure, which includes:
(i) interpolation of the distribution of discrete particles and their displacements to a regular data array,
(ii) calculation of the circulation for every point of the obtained array using a fixed radius comparable with the characteristic scale of visually observed vortices,
(iii) determination of all extrema (after preliminary smoothing of the circulation, if it is necessary) that define the positions of vortices and antivortices,
(iv) calculation of the total number of the circulation extrema, their balance in the whole system, and their mutual redistribution in its parts,
Fig. 7. Fine structure of vortex evolution for a fragment of the upper quadrant of crystal lattice in the time interval 6.4-6.9 ps
(v) determination of the spatial motion trajectories of vortices and antivortices from the moment of their emergence to annihilation, and finally derivation of a complete description of the system vortex state in a closed static form.
5. Modeling results
The proposed approach to identifying dynamic vortices has been tested using a time-space analysis of defect motion in an originally ideal copper crystallite at the active stage of uniaxial loading. Some examples of vortex and antivortex trajectories from the moment of their appearance to annihilation are presented in Fig. 6. In Fig. 6, a the vortex center trajectories are shown in the interval from 0.5 to 5.5 ps which starts exactly from the beginning of relaxation. It is easily seen how the diagonal vortex-anti-vortex pairs are generated in the regions away from the center of the crystallite. Later they move towards the center and annihilate. The defects from the left side of the crystallite annihilate with the opposite-sign defects from the right side. The trajectories of the dynamic defects in the interval between 5.5 and 9.5 ps are shown in Fig. 6, b. There is one dynamic defect (type 1 defect) in each quadrant which is generated in the regions maximally distant from the center and moves gradually to the central region. In contrast to the previous time interval, the annihilation of such defects takes place in the vertical, not horizontal, direction. The dynamic defects from the bottom part of the figure annihilate with the defects from the upper half. Another type of defect (type 2 defect) is generated in the bulk of the crystallite and moves to its periphery. According to the modeling results, type 2 defects disappear not because of annihilation, but because they escape through the open surface.
It should be noted that in some cases the vortices and antivortices moving in the bulk of the crystal lattice can elongate and split into two ones. Later, one of the centers disappears and the whole vortex "jumps" into a new point in space. This effect looks like "quantum tunneling". The corresponding parts of the trajectories are shown by the
dashed lines (see intervals A and B in Fig. 6, b). The fine structure of such dynamic transformations is presented in Fig. 7. The moments of deformation and fast spatial motion of the vortex cores are marked by the arrows.
Now let us analyze the evolution of the total number of vortices and antivortices. Generally, we find that there is often only one vortex in each quadrant, or two of them with different signs of circulation. The corresponding data are represented in Fig. 8 showing the time dependences of the number of vortices and antivortices (Fig. 8, a) and their difference (Fig. 8, b). It is seen that the difference between their numbers is normally equal to 0 or 1, and rarely reaches 2 or more. The number of vortices is obviously limited by the small size of the system. Their deformation and core splitting can influence the formally calculated number of vortices (probably with higher original circulation). However, taking into account these limitations, it is possible to make general conclusions about the vortex motion of atoms at the active loading stage.
Vortex motions promote the transfer of internal stresses/ displacements from the crystallite bulk to the open surfaces. The stresses redistribute either because of the formation of solitary defects or due to neighboring defects of different signs which rotate like "bearings" and transfer the stresses/ displacements in a zigzag manner. This process preserves continuity of the system and is schematically shown in Fig. 9, a.
Comparison of atomic trajectories in the bulk and near the open surface also illustrates the redistribution of stresses/ displacements. In particular, the atomic trajectories in the bulk of the crystallite for two consecutive time points are shown in Fig. 9, b. One can see that the atom moves almost cyclically along a curve around its equilibrium position. The initial position of the atom is marked by the circle. In contrast to the internal atoms, the same trajectories for the atoms close to the open surface (see Fig. 9, b) are more extended and preferably oriented in the X direction of the main transfer of stresses/displacements from the center to the periphery of the material.
H--1-r
10 15 20 Time, ps
10 15 20 Time, ps
Fig. 8. Time dependences of the total number of vortices (a) and their difference inside an arbitrarily chosen quadrant (b)
£ ^
0
Fig. 9. Atomic displacements in the central crystallite fragment in the time interval 12.2-12.4 ps (a). Trajectories of an atom for two consecutive time points (b): from 24 to 28 ps (upper plot) and from 31 to 35 ps (lower plot), and of an atom located near an open surface (c) for the intervals from 26 to 30 ps (upper plot) and from 30 to 35 ps (lower plot)
A detailed analysis revealed a definite correlation between the vortex and antivortex sizes and their life cycle duration. It has been established for a particular system that the sizes of the observed vortices vary from 5-6 to 20-25 interplanar spacings. In this case, the vortex size is used to mean a half of the distance between two neighboring defects. Close defects with the distance between their centers smaller than 15 interplanar spacings are treated as a solitary defect with a split.
It has been found that larger size defects live longer. Their maximum life duration is about 10 ps. The position of such a defect is not fixed and changes in space during this time interval. Small size defects usually exist for fractions of picoseconds and form a pair with an opposite-sign defect, as it is shown in Fig. 10.
Thus, short time intervals and small defect sizes allow the defects to be treated as a "support tool" that helps the "main" vortices and antivortices to perform the redistribu-
Fig. 10. Atomic displacements in the central fragment ofthe modeled crystallite in the time interval 12.5-12.7 ps
tion of stresses and displacements in order to preserve the material continuity.
6. Conclusions
Although recent data indicate that rotational deformation occurs at different scales (from atomic to macroscopic), this study focused on some aspects of a collective rotational motion of atoms. Molecular dynamics was used to model the evolution of a preloaded copper crystallite at the stage of relaxation. The results demonstrate the possibility to generate a coordinated vortex-like motion of atoms with the opposite signs of circulation in conjugated regions and alternating directions of rotation over time. In view of their dynamic behavior, such structures can be treated as dynamic defects. Despite the fact that these structures are within the elastic range, they can lead to non-elastic consequences and induce the formation of conventional static defects in the crystal lattice. Such defects are formed, in particular, by increasing the duration of the active loading stage (not considered in this paper), mainly in the zone of conjugation of loaded layers I and III with a free surface.
The experimental detection of vortex dynamic defects is objectively difficult because of their extremely small sizes and very short lifetime. However, recent electron microscopy studies of deformation localization in the region of elastic distortions during torsion in Bridgman anvils of nickel have proved the validity of numerical modeling results [30]. This suggests that numerical simulation methods are an effective way to study the role of circular movement in a loaded solid. There are, however, technical difficulties in the practical identification of vortex motion due to a discrete distribution of the vector field on randomized arrays in space. In this study, we proposed an original approach to solving the problem. It is based on the interpolation of the field to a regular array, calculation of the circulation in every point of the array, and then the use of a
Gaussian convolution to smooth the results. This technique allows both visualizing the vortex motion of particles in space and tracking its evolution in time.
The proposed approach was used to determine the characteristic sizes of vortices and antivortices (circulations of atomic displacements of opposite signs) and their lifetime duration. Possible vector field transformations in space and preferable directions of motion of vortex cores were determined. It was found that the vortex-antivortex pairs can annihilate if they approach close to each other. Some effects of vortex core splitting and instant vortex "jumps" to new positions at large vector field deformations were observed. In the case when a number of relatively close vortices with opposite circulations appear in the proximity of each other, an additional effect of a "roller" transfer mechanism of atomic displacements can arise. In this case, particle displacements along boundaries between vortices become correlated, and the self-consistent flow of particles transfers internal stresses from the bulk of the crystal to the open boundaries on its periphery.
Limitations of the proposed approach are the following. The accuracy of identifying the vortex motion is limited by the natural step between the points of the array corresponding to the particles for which the vector variable must be applied. The Gaussian convolution is also limited by itself and makes it impossible to identify small-sized and short-living dynamic defects. Increasing the algorithm accuracy in this limit sometimes leads to "phantom" centers of collective rotational motion with a nonzero but extremely low angular velocity. The last problem implies further modification of the proposed approach to parameterize it correctly depending on the angular velocity in a wide range of its values. The specific choice of boundary conditions also influences the value of the spatially distributed circulation obtained in the close vicinity of the crystallite boundaries.
Acknowledgments
The work was financially supported by the Fundamental Research Program of the State Academies of Science for 2013-2020 (Project Ш.23.2.4). Molecular dynamic modeling was carried out with the financial support of RSF Grant 17-19-01374. The supercomputer of Tomsk State University was used in the framework of the TSU competitiveness improvement program. The development of the algorithm to visualize the circular motion of particles in space was supported by the German Academic Exchange Service (DAAD). We express our gratitude to Prof. S.G. Psakhie for a fruitful discussion of the results.
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Поступила в редакцию 19.02.2018 г.
Сведения об авторах
Andrei I. Dmitriev, Dr. Sci. (Phys.-Math.), Leading Researcher, ISPMS SB RAS, Prof., TSU, dmitr@ispms.tsc.ru Anton Yu. Nikonov, Cand. Sci. (Phys.-Math.), Researcher, ISPMS SB RAS, Researcher, TSU, anickonoff@ispms.tsc.ru Alexander E. Filippov, Dr. Sci. (Phys.-Math.), Prof., Technische Universität Berlin, Germany, filippov_ae@yahoo.com Valentin L. Popov, Dr. Sci. (Phys.-Math.), Prof., Technische Universität Berlin, Germany, v.popov@tu-berlin.de
