ISSN 10637834, Physics of the Solid State, 2009, Vol. 51, No. 9, pp. 1809–1813. © Pleiades Publishing, Ltd., 2009. Original Russian Text © Yu.A. Baimova, S.V. Dmitriev, A.A. Nazarov, A.I. Pshenichnyuk, 2009, published in Fizika Tverdogo Tela, 2009, Vol. 51, No. 9, pp. 1705–1708.
DEFECTS AND IMPURITY CENTERS, DISLOCATIONS, AND PHYSICS OF STRENGTH
Dynamics of Edge Dislocations in a TwoDimensional Crystal at Finite Temperatures Yu. A. Baimova, S. V. Dmitriev*, A. A. Nazarov, and A. I. Pshenichnyuk Institute for Metals Superplasticity Problems (IMSP), Russian Academy of Sciences, ul. Khalturina 39, Ufa, 450001 Bashkortostan, Russia * email:
[email protected] Received September 1, 2008; in final form, January 11, 2009
Abstract—The gliding and annihilation of edge dislocations in a twodimensional hexagonal lattice are investigated at different temperatures by the molecular dynamics method. The data obtained are used to determine the coefficients of the phenomenological equation of motion of dislocations that exhibit an inertia and experience retardation due to the interaction with phonons. PACS numbers: 61.72.Bb, 83.10.Rs DOI: 10.1134/S106378340909008X
1. INTRODUCTION The behavior of dislocations in crystalline materi als determines their many properties, such as the plas ticity, the internal friction, etc. [1–5]. The behavior of dislocations in solids has been widely described using the discrete dislocation dynamics method in two and threedimensional variants [6–11]. In recent works [12–14], the discrete dislocation dynamics model has been complemented by introducing disclination fields. Although the discrete dislocation dynamics method is a powerful tool for simulating the dynamics of a rather large system of dislocations, it does not completely reflect the character of the dislocation interaction, because the analytical solutions of the theory of elasticity that underlie this method cannot describe the structure of the dislocation core. The molecular dynamics method allows one to simulate systems containing a small number of dislocations and to describe the motion of dislocations with due regard for the atomic structure of their cores [15–17]. This makes it possible to thoroughly describe the disloca tion reactions (the annihilation and generation of dis locations, the formation of dislocation dipoles, and the interaction of dislocations with obstacles) in the microscopic context and with allowance made for the influence of the temperature. The molecular dynamics method provides a means for determining the phe nomenological relationships that can be subsequently used in the discrete dislocation dynamics method in order to describe more accurately the microscopic dis location phenomena. There have been attempts to use hybrid computational schemes combining the advan tages of continuum and atomistic approaches [18– 20]. Moreover, one more approach in which the for mulation of the rules describing the behavior of dislo cations (including the dislocation reactions) in the
discrete dislocation dynamics method is based on the atomic simulation has received development [21]. It is this approach that is developed in our work. The behavior of dislocations in crystals should be described by threedimensional models. However, many effects, including the instability of a uniform distribution of dislocations and the formation of vari ous dislocation structures, can be qualitatively investi gated within simpler twodimensional models (see, for example, [8, 12, 13]). We are based on the works on the discrete dislocation dynamics simulation of processes of the formation of dislocation structures in the two dimensional case [6–8], in which it was demonstrated that these processes depend substantially on the num ber of slip systems, on whether the dislocation climb is included or ignored, and a number of other factors. In these and a series of other works, the simulated medium was assumed to be isotropic, which simplifies the calculations of the interaction forces between dis locations. In this respect, it was of interest to analyze dislocations in a twodimensional isotropic hexagonal lattice. In our work, the molecular dynamics method was used to determine phenomenological regularities of the dynamics of edge dislocations in a twodimen sional hexagonal lattice with atoms interacting through the LennardJones potential. In particular, we determined the effective dislocation mass and the temperature dependence of the phonon viscosity in the course of dislocation gliding. Moreover, we obtained the temperature dependences of the lattice parameter and the elastic constants of the crystal. 2. DESCRIPTION OF THE MODEL We consider a twodimensional hexagonal lattice with a primitive cell subtended by the translation vec
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units of ε/m , σ m/ε , and ε/σ3, respectively. The temperature of a ddimensional system (in our case, d = 2) can be determined from the relationship 2 (d/2)NkBT = (1/2) Nn = 1 〈 mv n 〉 , where N is the num
P2 = 256p2
∑
y
x
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Fig. 1. Schematic drawing of two dislocations with oppo site signs in the computational cell.
tors w1 = (a, 0) and w2 = (a/2, a 3/2 ), where a is the lattice parameter. The lattice sites are numbered by integer numbers m, n. The radius vector of the site m, n has the form rm, n = mw1 + nw2. The lattice can be subjected to homogeneous strain of the general form with the components εxx, εyy, and εxy. The generating vectors of the strained lattice are written in the form p1 = w1 + w1H and p2 = w2 + w2H, where the coeffi cients of the matrix H are represented as h11 = εxx, h12 = h21 = εxy, and h22 = εyy. In an ideal crystal, all lattice sites are occupied by atoms. For the atom m, n we introduce the vector of its displacement dm, n = (dx, m, n, dy, m, n) from the lattice site. As a result, the radius vector of the atom m, n can be written in the form rm, n = mp1 + np2 + dm, n. Since the aim of our work did not involve consider ation of a specific material, we chose the classical Len nardJones interatomic pair potential. In order to avoid a jump in the energy by cutting off the potential, it was supplemented by a polynomial function, which provided a smooth vanishing of the potential; that is, 12
6
⎧ ε [ ( σ/r ) – ( σ/r ) ], 0 < r < r LJ , (1) ⎪ ϕ = ⎨ A(r – r ) 5 + B(r – r ) 4 + C(r – r ) 3 , r ≤ r < r , c c c LJ c ⎪ ⎩ 0, r ≥ r c , where rc is the cutoff radius and rLJ is the cutoff radius of the LennardJones potential. In potential (1), the coefficients of the polynomial function were deter mined from the continuity conditions for the function ϕ and its first and second derivatives at the point r = rLJ. Moreover, it is evident that, at r = rc, the following conditions are satisfied: ϕ(rc) = 0, ϕ'(rc) = 0, and ϕ''(rc) = 0. Potential (1) vanishes at r = σ and reaches a minimum ϕmin = ε at r = 21/6σ. The distance, the energy, and the mass are mea sured in units of σ, ε, and m, respectively, where m is the atomic mass. In this case, the velocity, the time, and the stress and the elastic moduli are measured in
ber of atoms in the system, kB = 1.38 × 10–23 J/K is the Boltzmann constant, and 〈…〉 means the averaging over the time. Hereafter, the temperature will be mea sured in units of ε/kB. Without loss of generality, it was assumed that σ = 1, ε = 0.25, and m = 10. The cutoff radii were as fol lows: rLJ = 2.4 and rc = 3.9. For this potential, the equilibrium lattice parame ter at zero temperature is a0 = 1.1117. For the crystal with the equilibrium lattice parameter at T = 0, the Poisson’s ratio and the shear modulus were deter mined to be ν0 = 1/3 and μ0 = 7.103. The computational cell in the form of a regular rhombus included 256 × 256 atoms; i.e., the cell vol ume was determined by the vectors P1 = 256p1 and P2 = 256p2. We used the periodic boundary conditions. The hexagonal lattice under consideration exhibits an isotropy property and has three slip systems. A pair of edge dislocations with opposite signs located in one slip plane was introduced through the relative dis placements of atomic planes (Fig. 1). After relaxation of the system, we investigated the dynamics of disloca tions at a specified temperature. In this case, the initial distance between the dislocations was equal to 110 interatomic distances. In the system, zero external stresses were maintained by changing the strain of the computational cell. At the initial instant of time, the temperature was introduced by displacing the atoms from their equilib rium positions, so that the initial velocities of atoms were equal to zero. The parameters under investiga tions were determined after the system reached ther modynamic equilibrium, as can be judged from the absence of the drift (with time) of microscopic param eters, such as the strain tensor components for the computational cell (or acting stresses), as well as the kinetic (or potential) energy of the computational cell. 3. RESULTS OF THE SIMULATION The temperature is a significant factor determining the dislocation dynamics. First and foremost, an increase in the temperature leads to an increase in the phonon temperature, which retards the dislocation gliding but facilitates the dislocation climb due to the increase in the concentration and the mobility of point defects. The temperature also affects the elastic con stants of the material and, hence, the dislocation interaction forces determined by the elasticstress fields. Below, we evaluate the influence of the temperature on the lattice parameter and the elastic constants of
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the twodimensional crystal and then study the gliding of a pair of dislocation with opposite signs in the com putational cell at different temperatures in order to determine the effective dislocation mass and the phonon viscosity.
(b) 250
3.2. Dislocation Gliding at Different Temperatures As an example, Fig. 2a shows the typical time dependences of the coordinates x for the pair of the opposite dislocations under consideration at different temperatures according to the molecular dynamics data. It can be seen from Fig. 2a that, as the tempera ture increases, the time it takes for the dislocations to PHYSICS OF THE SOLID STATE
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3.1. Temperature Dependence of the Lattice Parameter and the Elastic Constants With the aim of calculating the temperature depen dence of the lattice parameter, the size of the compu tational cell was changed using the Parrinello–Rah man method for simulating zero external stresses. The volume strain of the cell ε = (εxx + εyy)/2 was used to calculate the lattice parameter a = a0(1 + ε) at the specified temperature. In the lowtemperature range T ≤ 0.02 under investigation, it was revealed that the variation of the lattice parameter with the temperature is described with a high accuracy by the linear depen dence (2) a = a 0 + αT, where α = 0.3348. For the calculations of the shear modulus μ at the specified temperature, the shear strain εxy of the com putational cell was chosen so that the acting shear stress was equal to the specified low stress σxy = 0.01. Then, from the Hooke law, we have μ = σxy/εxy. At low temperatures T ≤ 0.02, the linear temperature depen dence of the shear modulus is valid with a high accu racy; that is, (3) μ = μ 0 + βT, where β = –42.1. In order to calculate the bulk modulus at the spec ified temperature, the volume strain of the computa tional cell ε = (εxx + εyy)/2 was chosen so that the vol ume stress σ = (σxx + σyy)/2 was equal to the specified low stress σ = 0.01. In this case, the shear components of the strain and the stress were equal to zero. Then, the bulk modulus is given by the formula B = σ/ε. It was revealed that, at low temperatures T ≤ 0.02, the bulk modulus is represented by the expression (4) B = B 0 + γT, where B0 = 28.41 and γ = –215.0. The Poisson’s ration can be calculated using rela tionships (3) and (4) from the following expression: 2B – 2μ . (5) ν = 2 ( 2B + μ )
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500 1000 1500 t
Fig. 2. (a) Time dependences of the coordinates x for the pair of the opposite dislocations under consideration at the temperatures T1 = 3 × 10–4, T2 = 0.0096, and T3 = 0.015 and (b–d) dislocation motion trajectories obtained by numerical integration of Eq. (6) at M = 10.95 and different phonon viscosities B.
annihilate increases and the dislocation motion becomes more chaotic due to the increase in the influ ence of thermal lattice vibrations. The results similar to those presented in Fig. 2a were obtained at different temperatures in the range 0 ≤ T ≤ 0.02, and the time to the dislocation annihila tion tann was determined for each experiment (the total number of experiments was approximately equal to 200). 4. PHENOMENOLOGICAL DESCRIPTION OF THE DISLOCATION GLIDING The phenomenological equation describing the dislocation gliding has the form (6) Mx·· + Bx· = F, where x(t) determines the position of a dislocation at the instant of time t, M is the effective dislocation mass, which is assumed to be independent of the tem perature, B(T) is the temperaturedependent phonon viscosity [2], and F = bτ is the Peach–Koehler force acting on the dislocation per unit length (b = a is the magnitude of the Burgers vector and τ is the shear stress acting in the slip plane of the dislocation under consideration). The shear stress τ was calculated using the known solution of the theory of elasticity for the field of stresses induced by a straight wall formed by edge dislocations [1] (the calculations are described in detail in [22]). In order to determine the effective dislocation mass M entering into the phenomenological equation (6), the annihilation time of two dislocations at zero tem perature was obtained by the molecular dynamics method. In this case B = 0, and the quantity M was determined from the condition that the integration of Eq. (6) leads to the annihilation time identical to that obtained using the molecular dynamics calculations.
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with the annihilation times obtained by the molecular dynamics method, we determined the temperature dependences of the phonon viscosity B(T), which are shown in Fig. 3 without regard (curve 1) and with due regard for (curve 2) the temperature dependences of the elastic constants.
0.3 B
0.04
B
0.2
0.02 0
0.005 T
0.1
5. CONCLUSIONS
1 2 0
0.005
0.010 T
0.015
0.020
Fig. 3. Temperature dependences of the phonon viscosity (1) without regard and (2) with due regard for the temper ature dependences of the elastic constants. The inset shows the initial portions of the dependences B(T) on an enlarged scale.
It should be noted that, naturally, the initial positions of two dislocations in the periodicity cell were taken identical to those used in the molecular dynamics cal culations, namely, x0 = ±56a, and the initial velocities of dislocations were equal to zero. For the mass M = 10.95 determined by numerical integration of Eq. (6), we calculated the trajectories of dislocation motion at different phonon viscosities B. The examples of the results obtained in these calculations when the elastic constants were assumed to be independent of the tem perature are presented in Figs. 2b–2d. The time of the first dislocation collision was taken to be equal to the annihilation time. Let us dwell in more detail on a comparison of the dislocation dynamics obtained from the molecular dynamics simulation and the calculations from the phenomenological equation (6). The main difference between the dislocation trajectories shown in Fig. 2a (molecular dynamics calculations) and Figs. 2b–2d (phenomenological calculations) is that the disloca tions in the molecular dynamics calculations annihi late after the first collision even at zero temperature, i.e., at B = 0, whereas Eq. (6) describes the damping (at B > 0) oscillatory process for the pair of the dislo cations under consideration with their multiple colli sions. This difference is explained by the fact that Eq. (6) ignores the microscopic structure of the disloca tion cores and the possibility of annihilating the dislo cations upon their collision. In the discrete dislocation dynamics calculations, this problem is overcome under the assumption that the dislocations drawn close to each other annihilate and are removed from the system. We calculated the annihilation times for the pair of dislocations from Eq. (6) for different phonon viscos ities B without regard and with due regard for the tem perature dependences of the elastic constants accord ing to relationships (3)–(5). By comparing these data
Thus, the gliding and annihilation of a pair of edge dislocations in the twodimensional crystal at different temperatures were investigated by the molecular dynamics method. The technique was developed for determining the coefficients of the phenomenological equation that describes the gliding of the dislocation under the action of the shear stress τ(x); in this case, the disloca tion having the effective mass M experiences viscous retardation due to the interaction with phonons. The model parameters were fitted by comparing the anni hilation times of a pair of dislocations with the corre sponding data obtained using the molecular dynamics calculations. The dislocation mass M and the temper ature dependence of the phonon viscosity B(T) were determined in the lowtemperature range 0 ≤ T ≤ 0.02. According to the theoretical estimates [2], the phonon viscosity is characterized by dependences B ~ T3 at low temperatures and B ~ T at high temperatures. Our results (Fig. 3) are in agreement with the theoret ical data. The cubic dependence B(T) is observed at T < 0.004, and the linear dependence occurs at T > 0.006. With allowance made for the influence of the tem perature on the elastic constants, it is possible to eval uate the influence of the temperature on the disloca tion interaction force. The temperaturedependent factor determining the Peach–Koehler force has the form μ/(1 – ν) [1]. By using relationships (3)–(5), we find that this factor at the maximum temperature T = 0.02 under investigation is equal to 9.309, whereas the corresponding factor at T = 0 amounts to 10.653. Therefore, the shear stress at T = 0.02 appears to be lower than that at zero temperature by 12.6%. The inclusion of the influence of the temperature on the elastic constants resulted in a change in the esti mate of the phonon viscosity (Fig. 3), i.e., in a decrease in the estimates by approximately 20%. The purpose of further investigation is to use the data obtained on the phenomenological properties of dislocations for the development of the discrete dislo cation dynamics method. ACKNOWLEDGMENTS This study was supported by the Russian Founda tion for Basic Research (project nos. 090800695a, 070812152, and 080291316IND_a), the Acad emy of Sciences of the Republic of Bashkortostan
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(grant “Povolzhie” no. 40/61P), and the Intel Foun dation (Magnitogorsk). REFERENCES 1. J. Hirth and J. Lothe, Theory of Dislocations (McGraw Hill, New York, 1968; Atomizdat, Moscow, 1972). 2. T. Suzuki, H. Yosinaga, and S. Takeuti, Dislocation Dynamics and Plasticity (Syokabo, Tokyo, 1986; Mir, Moscow, 1989). 3. A. N. Orlov, Introduction to the Theory of Defects in Crystals (Vysshaya Shkola, Moscow, 1983) [in Russian]. 4. L. M. Zubov, Nonlinear Theory of Dislocations and Dis clinations in Elastic Bodies (Springer, Berlin, 2001). 5. Dislocations in Solids, Ed. by E. R. N. Nabarro (Elsevier, Amsterdam, 1979–2008), Vols. 1–14. 6. E. van der Giessen and A. Needleman, Modell. Simul. Mater. Sci. Eng. 3, 689 (1995). 7. A. N. Gullouglu and C. S. Hartley, Modell. Simul. Mater. Sci. Eng. 1, 1 (1992). 8. B. Bako, I. Groma, G. Gyorgyi, and G. Zimanyi, Comput. Mater. Sci. 38, 22 (2006). 9. R. J. Amodeo and N. M. Ghoniem, Phys. Rev. B: Con dens. Matter 41, 6958 (1990). 10. M. C. Fivel, C. R. Phys. 9, 427 (2008). 11. A. Hartmaier and P. Gumbsch, J. Comput.Aided Mater. Des. 6, 145 (1999).
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Translated by O. BorovikRomanova
