molecular dynamics simulation of vitrification and plastic deformation of a two-dimensional...

M. J . KOTELYANSKII et al.: Molecular Dynamics Simulation of Vitrification 25

phys. stat. sol. (b) 166, 25 (1991)

Subject classification: 61.25 and 66.10; 62.20; 64.60

Institute of Chemical Physis, Academy of Sciences of the USSR, Moscow')

Molecular Dynamics Simulation of Vitrification and Plastic Deformation of a Two-Dimensional Lennard-Jones Mixture

BY M. J. KOTELYANSKII, M. A. MAZO, E. F. OLEYNIK, and A. G. GRIVTSOV~

A computer model of a two-dimensional atomic glass is generated by molecular dynamics. This model consists of 500 Lennard-Jones particles of two different diameters. A volume-temperature diagram is obtained. The bend found on this diagram at T = 0.288 is interpreted as the glass transition. Density fluctuations and changes in the particles coordination numbers at the glass transition are studied by means of Voronoy polygons. The spatial correlation functions for particle displacements in the liquid and glassy states are analyzed. It is found that the glass structure is always a frozen liquid structure. Below the glass transition particle displacements from their equilibrium positions become spatially correlated, and the correlation length is greater than the model's size. Unusual roton-like collective vibrational modes are found in the glass. The simulation of shear deformation shows that these modes may play an important role in the plastic deformation. No correlation is found between the diffusional mobility of particles and their local free volumes. This fact contradicts the main physical concept upon which a free volume approach is based.

Ein Computermodell eines zweidimensionalen atomaren Glases wird mit Molekulardynamik generiert. Das Modell besteht aus 500 Lennard-Jones Teilchen mit zwei verschiedenen Durchmessern. Ein Volu- men-Temperaturdiagramm wird erhalten, in. dem die Krummung bei T = 0,28~ als Glasubergang interpretiert wird. Dichteschwankungen und Anderungen der Koordinationszahlen der Teilchen beim Glasubergang werden mit Voronoy-Polygonen studiert. Die raumlichen Korrelationsfunktionen der Teilchenverschiebung im flussigen und glasartigen Zustand werden untersucht. Es wird gefunden, daB die Glasstruktur immer eine eingefrorene Flussigkeitsstruktur ist. Unter dem Glasubergang werden die Verschiebungen der Partikel weg von den Gleichgewichtspositionen raumlich korreliert, und die Korrelationslange ist groBer als die ModellgroBe. Uniibliche rotationsahnliche kollektive Eigenschwin- gungen werden im Glas gefunden. Die Simulation von Scherung zeigt, daB die Eigenschwingungen eine wichtige Rolle in der plastischen Deformierung haben konnten. Es kann keine Korrelation gefunden werden zwischen Diffusionsmobilitat von Teilchen und ihrem freiem Volumen. Dies widerspricht dem grundlegenden physikalischen Modell, das auf dem Konzept des freien Volumens basiert.

1. Introduction

The interest in the structure and properties of amorphous solids has increased remarkably during the past ten years. There are at least two reasons for this growth. The first is connected with the limitations of most crystal-physical approaches, generally based on the concept of translational symmetry of lattices, which loses its predictive power for the glassy state of matter. The need for new approaches capable of adequately describing the structure and properties of amorphous solids at micro- and macroscopic levels has become of interest for physicists all over the world.

') U1. Kosygina 4, SU-117977 Moscow, USSR.

26 M. KOTELYANSKII, M. A. MAZO, E. F. OLEYNIR, and A. G. GRIVTSOV

The other reason has a technical origin. Many new classes of amorphous materials such as metallic glasses, amorphous engineering plastics, amorphous ceramics, and other glassy substances have been developed recently. Many of them appear conducive to advanced technologies. The search for structure-to-properties relationships in amorphous solids is therefore important to modern applied materials science.

Due to the absence of analytical theories which can describe the behavior of amorphous solids under different circumstance, computer-simulation methods have become essential to this analysis.

This paper is concerned with molecular dynamics (MD) simulations of the behavior of two-dimensional (2D) atomic liquids and glasses consisting of 500 particles of two different diameters, and interacting via Lennard-Jones potentials.

We shall study the particle dynamics in the liquid and glassy states, the plastic flow of the system under external shear force, and the structure of the system in terms of Voronoy polygons (VP).

2. Description of the Model and Simulation Algorithm

To generate the 2D amorphous systems 500 particles (239 A and 261 B particles) of different diameters are placed in a square cell with periodical boundary conditions. The interaction potential U i j between particles i and j is given by

I 0; ri j > 3.50,

where E is the depth of the potential, cAA = 0.2329 nm, nBB = 0.3419 nm, and oAB = 0.5 (cAA + oAA) are the parameters of the potential which reflect the diameters of the particles, and r i j is the distance between the i-th and j-th particles.

It has been shown [l] that such a mixture may exhibit a disordered solid state. We have used the following parameters: E = 1, particle masses MA = 15, MB = 30 (atomic-mass units), time in ps, and distances in nm. For the A-A interaction, the characteristic time was about 2 ps.

The temperature ( T ) of the system was determined as an average of the kinetic energy per degree of freedom. The pressure P was determined as a time average of

2

- c 1 ( x j k - X i k ) F i j k k = l i j

P = 2 v

7

where F i j k is the k-th component of the interaction force between the i-th and the j-th particle, xik - xjk the difference in the k-th component of their radius vectors, K the total kinetic energy of the system, and V the computational cell area [2].

We used periodical boundary conditions to simulate the glass transition, particle diffusion, and motion in the liquid and glassy states.

For the simulation of the system behavior under external mechanical shear stresses, we used nonperiodical boundary conditions, which we call “rigid walls”. In this case, particles

Molecular Dynamics Simulation of Vitrification and Plastic Deformation 27

with coordinates xl, such that Ixl - 11 -= 4 . 5 ~ or x1 < 4.56 where 1 is the computational cell size, were located in the “rigid wall”. Their x1 coordinates were fixed, and velocities in the x,-direction were equal to zero. Those particles were allowed to move only in the x,-direction. Periodical boundary conditions along the x,-direction were preserved. External shear forces were applied to all particles in the “rigid walls”. After changing boundary conditions from periodical to “rigid walls” the system was relaxed during 30 ps.

Newton equations of motion were solved numerically with the aid of a third-order predictor-corrector algorithm [3], with time step h = 0.07 ps. It provides energy conservation with an accuracy of 0.01% over 1000 time steps.

3. The Vitrification Procedure

A system at T = 0.558, computational cell area V = 54 nm’, and zero pressure, was chosen as the initial state for cooling to the glassy state. Such a system was fluid because the packing coefficient was high (78.5%) and the particles were quite mobile. Fig. 1 shows mean-square displacements ( r ’ ) averaged over all particles in this state as a function of time. Diffusion coefficients (the slopes of lines in Fig. 1) were D = 2.4 x lo-’ nm2/ps and D = 1.9 x lo-’ nm2/ps, which are reasonable for simple liquids.

The quenching of the fluid was performed by reducing each particle’s velocity by a factor of 0.9997 every time step. This procedure provides a cooling rate of 0.333 x lo3 q‘ps, which would correspond to a quenching of a real metal alloy at a rate of about 10” K/s.

During the cooling, zero pressure was approximately maintained by scaling the particle coordinates and the cell size by a factor a = 1 + bP, depending on pressure, every 100-th step; P was averaged over the previous 100 steps.

We performed two MD runs, with different b (0.05 or 0.015), and the results were found to be the same.

Fig. 1. Mean-square displacements ( r ’ ) vs. time t for fluid at T = 0.55~. A-particles, x B- particles, 0 averaged over all particles

I

2% M. KOTELYANSKII, M. A. MAZO, E. F. OLEYNIK, and A. G. GRIVTSOV

a 10 a20 a30 I

T/& -

Fig. 2. Volume-temperature dependence during vitrification. The glass transition occurs at T = 0.288

0

4. The Glass Transition

Fig. 2 shows the volume-temperature diagram for the system cooling. At T = 0.28~, the curve shows a sharp bend. Further, no diffusion or structural rearrangements are observed below T = 0.288. We have therefore concluded that our system exhibits a glass transition at T, = 0.288.

5. Structure and Its Changes at T,

The language of Voronoy polygons (VP) provides a convenient way of characterizing the structure of condensed disordered systems [4]. Particle coordination numbers may be defined as the number of V P faces constructed around each particle. The volume of each V P (area, for the 2D case) is the partial or local volume of the system belonging to each particle. The volume of each V P includes a local “free volume”, i.e., the V P volume excess in relation to the corresponding ideal crystal.

Due to the equality of all VP volumes in a perfect crystal, one may use to characterize the local free volume in a disordered solid the entire free volume of each VP without subtracting the perfect-crystal part. However, to compare qualitatively the volumes of each VP, we should normalize these values, because we are dealing with a mixture of particles of different sizes.

We have used the normalized local VP volumes of the particles as a measure of the local free volume, or local density of each particle. In doing so, we hope to correlate this free volume with the mobility of each particle.

Table 1 shows the coordination number of A and B particles in the inital state of the system obtained by different cooling paths. The number of VPs in Table 1 is smaller than the total number of particles due to difficulties in defining correctly the VP for particles located near the cell border. Table 1 shows several interesting structural features of the investigated system.

Molecular Dynamics Simulation of Vitrification and Plastic Deformation 29

Table 1

way of temperature sort coordination number quenching

(4 5 6 7 8

0.548 A 0 101 12 1 0 B 0 4 91 93 3

I 0.30~ A 0 107 69 0 0 B 0 0 91 96 0

0.05.5 A 0 98 81 0 0 B 0 0 86 101 0

11 0.208 A 0 111 70 0 0 B 0 4 76 108 1

0.1 08 A 0 114 53 0 0 B 0 2 81 113 0

1. Only coordination numbers 5, 6, 7 exist. The coordination numbers depend upon particle size. All pentagons are located around small particles A, while heptagons are found around the larger particles B. The amount of six-coordinated particles is about 40 to 50%, and they are equally distributed through particles A and B.

2. This coordination number distribution remains unchanged when the system passes through T,.

3. The behavior of local density (free volume) normalized with respect to the particle sizes also looks intriguing. Fig. 3 shows distributions of particles over VP areas S,, (Fig. 3 a), V P areas divided by the particle diameter Svp/gii (Fig. 3 b), and normalized Sv,/ai values (Fig. 3c) for the liquid system at T =0.34~. The distribution functions in Fig. 3a and c show two pronounced peaks corresponding to VP belonging to A and B particles. The distribution in Fig. 3b has only one peak. That provides a possibility for a quantitative

Fig. 3. Distribution of particles over a) VP area Svp, b) 6 = Svp/aii, c) SVp/af

30 M. KOTELYANSKII, M. A. MAZO, E. F. OLEYNIK, and A. G. GRIVTSOV

t 4uI Fig. 4. Distribution of particles over S = Svp/ai in glass at T = 0.05~ (solid line) and in liquid at T = 0.336 (dashed line)

comparison of particle V P volumes, which is quite complicated with the bimodal distributions shown in Fig. 3a and c. Distribution functions for V P over the parameter 6 = SVp/o are similar for particles A and B, both in the liquid and glassy states. Therefore we will use this parameter 6 as a measure of local particle free volume or local density.

Fig.4 shows the distributions N(6,) for different temperatures of the system. All VP volumes and their distributions were measured for both instantaneous (I) and frozen (F) structures, corresponding to a potential energy minimum [ 5 ] . All distributions appear to be the same for I and F structures. Therefore, we can conclude that the thermal motion of the particles does not disturb significantly the free volume distribution in the system.

It is interesting to compare the local structure of our system to those proposed by Srolovitz et al. [6]. They introduced three types of defects. n-, p-, and t-type. The method of potential energy minimization was used in their simulation. By using the atomic level stress tensor invariants proposed in [6] we could not detect any atomic clusters with a structure similar to defects of the n- or p-types in the glassy state of our system. We checked all atoms in our system and we found that those atoms that could belong to n or p-type defects (by the values of their invariants) do not form clusters but are homogeneously distributed throughout the system. Further, two neighboring atoms often have characteristics corresponding to n- and p-type defects. A similar conclusion was made in [7], where M D simulations of amorphous iron were performed. It was found that n- and p-type defects are unstable when thermal motions exist in a system.

6. The Correlation between Diffusion and Local Density

The most common approach towards a description of the temperature and pressure dependence of the viscosity and diffusion in liquids is the free volume approach [8]. Usually, the free volume is based on a macroscopic representation of the sample free volume [9]. The only approach which takes into account the microscopic features of free volume was developed by Cohen and Grest [lo]. These authors were able to explain the empirical equations of Doolittle [ll] and Hildebrandt [12] for the diffusion and the viscosity of liquids.

Molecular Dynamics Simulation of Vitrification and Plastic Deformation 31

The main idea of the Cohen-Grest (CG) model is the division of all particles into two groups: liquid- and solid-like, depending on the volume of the cage, formed by the particle’s closest neighbors. Particles can form clusters of the liquid- or solid-like type. The portion of particles in each cluster depends on the temperature.

Particle diffusion occurs due to an activationless exchange of free volume between liquid-like cages. This exchange may produce a hole sufficiently large for a particle to jump into it. The free volume exchange process between solid-like cages is difficult, due to the presence of activation barriers. The barriers are higher for smaller cages. An important conclusion from this picture is that solid-like particles spend a lot of time within their cages and do not really take part in macroscopic mass transfer. Macroscopic diffusion in a sample occurs mainly through liquid-like clusters.

This model of diffusion cannot be verified experimentally; this, however, can be done by MD simulation.

The validity of the Doolittle equation [ll] for systems consisting of 864 particles was confirmed by Kimura and Yonezawa [13] through MD simulations of the Lennard-Jones system.

In this paper, we try to correlate the mobility of particles and their local, normalized volume of VP.

Fig. 5 shows the time dependence of particle displacements as a function of 6 = Svpi/ai, for the fluid system at T =0.33&. Each point on the figure corresponds to one particle. The displacement of each particle was measured as

t

AR: = l/t (Ri(z) - Ri(0))’ dz , 0

(3)

where Ri(z) is the position of the i-th particle at time z, and R(0) is its initial position. If the particle only oscillates about its equilibrium position, AR?(t) does not increase with time. If it diffuses, ARZ(t) increases.

Fig. 6 shows the relation between AR:(t) and cage volume, according to the CG model. The fluidity of the system at this temperature, which is slightly above T,, is assessed by

calculating macroscopic diffusion coefficients (Fig. 7). The diffusion coefficients appear to be D, = 0.072 x lo-’ nm2/ps and D , = 0.068 x

The simulation results presented in Fig. 5 show conclusively that there is no correlation between local V P volume and the particle diffusional mobility. Highly mobile particles exist at small and large 6. Particles in cells of different sizes do not show noticeably different displacements over 115 ps. Nevertheless, as time increases, particles exhibit quite large displacements, comparable to the square of the particle diameter. The mean-square particle displacement is linear in time (Fig. 7). However, mobile particles exist in the whole range of local free volumes, i.e. among liquid-like and solid-like cages. An analysis of Fig. 5 does not provide the possibility of extracting the critical value of local free volume 6,, which would enable us to divide all particles of the system into solid-like and liquid-like types.

These results contradict the CG model. And this contradiction is related to the correlation between particle mobility and its local free volume. Macroscopic mobility, which is reflected by diffusion coefficients, correlates with the total amount of free volume (density) of the system. In [13], the correlation between total sample free volume and mobility averaged over all particles of the system was also found by MD simulation.

nm2/ps.

32 M. KOTELYANSKII, M. A. MAZO, E. F. OLEYNIK, and A. G. GRIVTSOV

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* * * * * * ** ** * * * * * * * * * ; * * * 2 * + * * ***** 5 r f*.& * ** * * *

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* 50.2~~ * * * * * * * * 1 * * * * * 20 * 2 * ** * * * * * * * * * *

28.7~s

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74.3ps

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Fig. 5. Particle square displacements AR2 vs. their free volumes 6. Each point corresponds to one particle

The mobility of particles with relatively small hi is also not connected to the redistribution of local free volume. We verified the local free volume of particles at t = 0 and t = 115 ps and have found that the maximum relative deviation of hi observed in our simulations was less than 7%. The process of particle diffusion is not accompanied by a significant redistribution of cage volumes. The main reason for this is the non-jump mechanism of particle diffusion. The diffusion occurs through cooperative rearrangements of several particles simultaneously and is independent of atomic level density fluctuations. This picture disagrees with the classical hole or free volume models of diffusion.

Molecular Dynamics Simulation of Vitrification and Plastic Deformation 33

4 6-

Fig. 6. The correlation of particle mobility and its free volume expected from Cohen-Grest model, 6, is the critical value of free volume, dividing solid- and liquid-like particles

7. Thermal Motions

The thermal motion of the particles in the system should change its character when the system undergoes a glass transition. Particle displacements should become more spatially correlated in the glassy state.

To visualize that change we used the spatial correlation function for displacements in the following form:

where ui(t) is the displacement of the i-th particle from its equilibrium position at time t, (), is the average over those pairs of particles having distances equal to i-, and (), denotes a time average. In the liquid, F(r ) was averaged over 50ps. In the glass it was averaged

0 20 40 60 80 ; tlps i _f

Fig. 7. Mean-square displacements ( r ’ ) vs. time for T = 0.338 o A-particles, x B-particles

7

3 physica (b) 16611

34 M. KOTELYANSKII, M. A. MAZO, E. F. OLEYNIK, and A. G. GRIVTSOV

I 0 05 10 25 2.0 2.5 30

rlnrni - Fig. 8. Space correlation function of displacements F(r ) for liquid (0) and glass ( x). a) For periodical boundary conditions, b) for “rigid walls”

over 13 ps. Fig. 8 shows the correlation functions F(r ) for a fluid at T = 0.558 and for a glass at T 0.0% (for the cell with periodical boundary conditions). In the fluid, F(r) vanishes at distances of about 1.8 nm. In the glass it does not vanish at distances comparable to the cell size (6.9 nm). Fig. 9 a, b show the instantaneous displacement fields in the fluid and glassy states for the cell with periodical boundary conditions. In the liquid state (Fig. 9a), there are no correlations between particle displacement directions. On the contrary, the glassy state shows strong correlations between displacement directions (Fig. 9 b). Due to this correlation, the glassy state should be sensitive to changes in boundary conditions. We performed MD simulations of a glass at T = 0.0% and a fluid at T = 0 . 5 5 ~ with “rigid walls”. After changing the boundary conditions, the system was relaxed for 13 ps. Fig. 8b shows the spatial correlation functions F(r) for the fluid and glassy states with “rigid walls”.

These F(r) are qualitatively similar to those shown in Fig. 8a. For the liquid state they vanish, while for the glass they do not.

In a glass one can see large vortices covering almost the entire cell. Similar vortices are observed with periodical boundary conditions (see Fig. 10b and 9b), although their configuration is different. Thermal motions in the vortex look like the rotational collective vibrations around the vortex center with a time period of about 14 ps. We performed MD runs for this structure and different random initial velocities, starting from the same initial configuration corresponding to a minimum of potential energy, but such vortices had always the same centers. We suppose that this singularity of displacement fields is connected with structural defects. According to [ 141, similar mobility modes may exist around the dislocations in a crystal. In [15], Rehr and Alben speculated that the minimum of the dispersion curve observed in amorphous systems may be connected to some roton-like mobility modes.

We also performed an MD simulation of a two-dimensional A-particle hexagonal crystal with a dislocation dipole and “rigid wall” boundary conditions at T = 0.058. Fig. 11 shows the vortices in the vicinity of the dislocation core in the crystal. The cause of the vortex-like

Molecular Dynamics Simulation of Vitrification and Plastic Deformation 35

Fig. 9. Instantaneous displacement fields for systems with periodical boundary conditions. a) liquid, b) glass. Displacements are ten times expanded

3*

36 M. KOTELYANSKII, M. A. MAZO, E. F. OLEYNIK, and A. G. GRIVTSOV

b Fig. 10. Instantaneous displacement fields for systems with “rigid walls” boundary conditions. a) liquid, b) glass. Displacements are 10 times expanded

Molecular Dynamics Simulation of Vitrification and Plastic Deformation 37

Fig. 11. Displacement field for the hexagonal crystal with dislocations. Displacements are 10 times expanded

thermal motion in the glass may be a structural defect, but the definition of these defects in a glass, in structural terms, is much more difficult than in a crystal. The behavior of those defects during plastic deformation and the stabilization of vortices in the undeformed system and under deformation leads us to believe that the origin of the vortices has a structural nature. This assumption is also supported by results of static simulation of the deformation of a 2D model of amorphous iron performed by Maeda and Takeuchi [16]. They observed vortex-like particle rearrangements about the holes during the elastic deformation.

Fig. 12. Dependence of strain on time in the shear deformation experimens. ~ first, - - - second, and . . . . . . third run. The fast nonlinear increase of strain occurs after about 7 ps from the beginning of the loading

38 M. KOTELYANSKII, M. A. MAZO, E. F. OLEYNIK, and A. G. GRIVTSOV

a

Molecular Dynamics Simulation of Vitrification and Plastic Deformation 39

Fig. 13. Displacement fields during the third shear deformation run at a) 1.8, b) 7.2, c) 11.8 ps

8. Plastic Deformation

Recently, the nature of plastic deformation in disordered solids has become of great interest

We performed a simulation of shear deformation for a glassy system with rigid walls. Shear stresses were simulated by external forces applied to the particles in the rigid walls. Particles in the opposite walls exerted the same forces in the opposite direction. Three experiments were carried out for the same initial configuration, but different time dependences of the external force. In the first and second runs the force F increased within 10 ps from zero to F,,, which was equal to 10&/nm in the first run and 34nm in the second. The mean value of the interparticle forces was about 30~/nm. In the third experiment, the external force increased with times as F = 0.002~ The force is quite larger than the critial shear stress z,,, = G/27c(G is the shear modulus) that was estimated for a perfect crystal by Frenkel.

1171.

The shear strain was determined as

where U , an U2 are the displacements of the rigid walls, and 1 is the cell size. Fig. 12 shows the curves of shear strain y in all runs as a function of experiment time. It can clearly be seen that the growth of the strain occurs nonlinearly. The transition of the system to a high strain rate may be interpreted as a transition to macroscopic plastic flow.

40 M. KOTELYANSKII, M. A. MAZO, E. F. OLEYNIK, and A. G. GRIVTS~V

Fig. 13 shows displacement fields at different times during the third experiment. Up to 7 ps after starting, one can see vortices similar to the initial displacement field in Fig. lob, the rearrangement of which takes place at 7 ps when the transition to plastic flow begins. After the beginning of plastic flow there exists only one vortex which decays to smaller ones as the force increases.

The largest structural distortions are localized at the centers of the vortices and between them, in agreement with our speculations that the location of vortices in a cell has structural origin.

9. Discussion

It is interesting to compare our results with a recent study performed by Deng and coworkers [18]. They performed M D simulation of a 2D mixture of Lennard-Jones particles corresponding to a Cu-Zr alloy, which can be obtained in the amorphous state from a real experiment. The particle diameter ratio in [18] was about 1.2. While that system is closer to reality, in the 2D case it readily makes a hexagonal crystal, because of the relatively low diameter difference. Our system is disordered. It may be for this reason that in [18] the amount of five- and seven-coordinated particles depends strongly on temperature and coordination number, and is not closely connected to the particle diameter. In our system, however, the glass structure is essentially a frozen liquid structure, and coordination numbers are independent of temperature.

Our results show that there is not a strong correlation between local plastic deformation and coordination number and local free volume at any sample point. It is connected with the centers of rotons. Deng et al.’s structure is closer to crystalline with defects and all local fluctuations of free volume and coordination number are localized in the defects; in our system, all of these fluctuations are distributed throughout the whole sample.

This shows that mixtures with different diameter ratios can exhibit a quite different behavior. We believe that although our system has no natural analog, it is closer to an amorphous, topologically disordered structure, while the system investigated in [ 181 in two-dimensions is closer to the crystal. We believe that both approaches are quite useful and reflect the existing diversity of real disordered systems: completely and partially disordered.

Now we would like to comment on the diffusion and local volume results. There are arguments [19] supporting that the low mobility of some liquid-like particles with a relatively large amount of free volume may be explained by the fact that they exist within solid-like clusters. This is not the case, however, because the total fraction of solid-like particles is very low in a fluid at T = 0.33s. Further, low-mobility liquid-like particls are within liquid-like surroundings. Instead, from Fig. 5 it can be readily seen that there are highly mobile particles with a relatively low free volume. We therefore conclude that there is no correlation between particle diffusion mobility and its free volume.

There is a lot of experimental data supporting the hypothesis that viscosity and diffusion in liquids and glasses are controlled by the free volume. This may be connected with the fact that real experiments deal with the average characteristics of macroscopic system containing about particles. This relationship becomes probably valid for parameters averaged over some amount of particles. For our M D system, containing 500 particles, we

Molecular Dynamics Simulation of Vitrification and Plastic Deformation 41

obtained relationships between diffusion and density similar to the macroscopic ones. We suppose that this amount is close to ten or hundred particles. It is interesting to find this value from M D simulation and we will try to do it in our M D simulations of three- dimensional systems, which are now in progress.

10. Conclusion

What conclusions can we draw from the M D simulation of a two-dimensional mixture of Lennard-Jones particles of different diameters?

First it should be pointed out that such a system is stable in an amorphous state. Further, in contrast to a system containing only one species, there is no trend to crystallization.

While the mobility significantly decreases, no significant structural changes were observed during the glass transition.

The study of diffusion in a fluid just above < shows no correlation between the diffusional mobility of a particle and its VP area. This contradicts the Cohen-Grest theory and does not allow to distinguish between solid-like and liquid-like particles. It should be mentioned that this result does not contradict the density dependence of the diffusion coefficient.

Due to the increase of shear stiffness during the glass transition, thermal motions become strongly spatially correlated. Unusual vortex vibrational modes are observed, which apparently are connected with the defects of the amorphous structure and play a significant role in plastic flow.

An attempt to analyse the defects in the way described in [6] showed that p- and t-type atoms do not form clusters and we could not distinguish such “defects”. Plastic dcformation is localized, and it occurs at the centers of vortices, or between them.

All results reported here deal with a 2D system; we hope that similar simulations of 3D mixtures, which are now in progress, will provide additional significant information.

Acknowledgements

The authors would like to thank Prof. A. Argon for fruitful discussions of this work. We are also grateful to Dr. V. Bulatov and Dr. A. Gusev for their interest in our work and for interesting discussions. We are also indebted to M. M. Frank-Kamenetskii, A. Romanenko, Dr. G. G. Malenkov, and their coworkers for valuable help with progamming aspects of this work. Finally, we would like to thank Dr. J. de Pablo for discussions of this report and significant help with its writing.

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42 M. KOTELYANSKII et al.: Molecular Dynamics Simulation of Vitrification

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(Received April 20, 1990; in revised form April 8, 1991)