Lattice Resistance to Dislocation Motion at the Nanoscale
A. Dutta, M. Bhattacharya, P. Barat, P. Mukherjee, N. Gayathri, G. C. Das
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A ug Lattice Resistance to Dislocation Motion at the Nanoscale
A. Dutta † , M. Bhattacharya ‡ , P. Barat ‡∗ , P. Mukherjee ‡ , N. Gayathri ‡ , G. C. Das †† School of Materials Science and Technology,Jadavpur University, Kolkata 700 032, India ‡ Variable Energy Cyclotron Centre, 1/AF,Bidhannagar, Kolkata 700 064, India (Dated: October 26, 2018)
Abstract
In this letter we propose a model that demonstrates the effect of free surface on the latticeresistance experienced by a moving dislocation in nanodimensional systems. This effect manifestsin an enhanced velocity of dislocation due to the proximity of the dislocation line to the surface.To verify this finding, molecular dynamics simulations for an edge dislocation in bcc molybdenumare performed and the results are found to be in agreement with the numerical implementationsof this model. The reduction in this effect at higher stresses and temperatures, as revealed by thesimulations, confirms the role of lattice resistance behind the observed change in the dislocationvelocity.
PACS numbers: 61.72.Lk, 62.25.-g B and the dislocation velocity v are related as [13], v ( t ) = τ bB (1 − e − Bt/m ∗ ) (1)where τ is the applied shear load, b is the magnitude of the Burgers vector, t is the timeelapsed after the dislocation starts moving and m ∗ is the effective mass per unit length of thedislocation line [12]. The expected change in the drag coefficient B due to the introductionof the free surfaces leads to the prediction of an altered terminal velocity of dislocation, v = τ b/B .The movement of a dislocation due to the applied force per unit length of the dislocationline ( τ b ), changes the displacement field felt by the atoms in the lattice. This is resisted byan equal and opposite drag force experienced by the dislocation when it attains its terminalvelocity. In the model, we assume that the resistance to any change in the displacement fieldof an atom due to its interaction with other atoms of the lattice, contributes to the dragforce due to the lattice resistance. Cumulative contributions from all the lattice atoms givethe overall lattice resistance. Nevertheless, each atom undergoes a different change in thedisplacement field, and hence should contribute differently to this drag force. Thus, thereis need for a contribution function that can express the role of the atoms in determiningthe net lattice resistance. In order to investigate the surface effects at the nanoscale, a thinfilm is an ideal system as it provides infinitely large free surfaces with confinement along itsthickness.Consider a dislocation moving with the terminal velocity v along the positive x directionin a thin film bounded by the top and bottom surfaces in the y direction. Fig. 1(a) illustratesthe configuration of the moving dislocation with the line direction of the dislocation alongthe z axis. At an arbitrary time t the dislocation line passes through the origin O. The3osition vector r dij ( t ) of the ij th atom A ij in the lattice is given by r dij ( t ) = r cij + u ij ( t, r cij ) (2)where r cij is the position vector of A ij in the perfect crystal and u ij ( t, r cij ) is the correspondingdisplacement of the atom in the presence of the dislocation at time t . In a small time interval δt , the dislocation line proceeds by a distance δx = v δt (refer Fig. 1(b)). The net force dueto the lattice resistance is given by F ( lattice ) s = s X i = s ∞ X j = −∞ φ ij , (3)where φ ij denotes the contribution of the atom A ij to the lattice resistance. We assume asimple proportionality relation between φ ij and the change in the position vector of A ij intime δt as φ ij = κ (cid:12)(cid:12)(cid:12) r dij ( t + δt ) − r dij ( t ) (cid:12)(cid:12)(cid:12) , (4)where κ is the proportionality constant for a given loading condition. In terms of δt as theunit of time, F ( lattice ) s = B ( lattice ) s v , (5)where B ( lattice ) s = κ s X i = s ∞ X j = −∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n =1 v n − n ! ∂ n u ij ∂ ( δx ) n ! δx =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (6)A different value of the drag coefficient B ( lattice ) s is quite obvious following Eq. (6) due to thechange in the summation limits of i representing the thickness of the film. A reduction inthe lattice resistance for a thinner film is indicative of an enhanced velocity of dislocation.Similar changes are expected due to the variation in position of the dislocation line alongthe film thickness. MD simulations are carried out to verify such effects.A typical simulation starts with a virtual freestanding thin film of single crystal bccmolybdenum created using the Finnis-Sinclair potential [15]. The simulation cell is shownin Fig. 2 with its x , y and z axes along < > , < ¯101 > , and < > directions respectively.The crystal dimensions along the x and z Cartesian directions are 10.76 nm and 3.85 nmrespectively, whereas the y dimension representing the film thickness is varied to study thesurface and size effects. An edge dislocation is introduced at the centre of the film with4islocation line along the z axis and Burgers vector a< > /2 along the x direction wherethe lattice constant a =0.31472 nm. Periodic boundary conditions are imposed on all thethree directions, however, the boundaries are sufficiently extended along the y directionso that free top and bottom surfaces can be created and interactions among the periodicimage films can be eliminated. The dislocation core is identified by specifying a centro-symmetric deviation parameter window [16] of width 0.024-0.1 nm . The system is theninitialized at 300 K temperature. Precisely calculated forces are applied to the atoms ofthe top and bottom surfaces of the film with directions parallel and antiparallel to theBurgers vector respectively so that a shear stress of 250 MPa can be produced. Followinga time lag, this applied stress is transmitted to the dislocation line, which in turn attainsa terminal velocity in several femtoseconds [16] following Eqn. (1). Constant temperatureis maintained by implementing the Nos´e-Hoover thermostat [17, 18]. Trajectories of allthe atoms are calculated at a time step of 0.5 fs. Positions of the dislocation core arerecorded with respect to time and thus the dislocation velocity is extracted. Simulations areperformed in two ways, case I: by reducing the film thickness equally about the dislocationline and case II: by varying the position of the dislocation line at different depths beneaththe top surface of a film of fixed thickness.Figure 3(a) shows the variation in the velocity of edge dislocation as a function of filmthickness. The MD simulations clearly exhibit a significant increase in this velocity whenthe film thickness is reduced from ∼
70 nm to 8.5 nm. However, at higher thicknesses thevelocity of dislocation attains a constant value of ∼
728 m/sec. A significant rise of 46% inthis velocity for the film of 8.5 nm thickness under the same loading conditions is noteworthyin this context. A rising trend in the dislocation velocity is also observed as the dislocationline is brought closer to the top free surface in a film of 35.2 nm thickness (refer Fig. 3(b)).The results obtained from the MD simulations establish the effects of surface and size onthe velocities of dislocations in thin films. These results have been used as a tool to separateout the contribution of lattice resistance from the overall drag. Equation (5) enables usto express the net drag coefficient B as the sum of the drag coefficients due to latticeresistance B ( lattice ) s and the remaining part B ′ due to other drags. B values are calculatedusing the velocities of dislocation extracted from MD simulations for two widely differentfilm thicknesses. The ratio of the drag coefficients corresponding to the lattice resistance forthese two film thicknesses is evaluated and then used to separate out the constant part B ′ B . The net drag coefficient B for any arbitrary film thickness is obtained by combiningthe calculated values of B ′ and B ( lattice ) s so that the respective velocity of dislocation canbe evaluated. In order to perform these calculations, the position vectors of the atoms aredetermined by superposing the following elastic displacement fields of an edge dislocation[12] on a perefect bcc crystal; u x ( x, y ) = b π [ tan − yx + xy − ν )( x + y ) ] , (7) u y ( x, y ) = − b π [ 1 − ν − ν ) ln( x + y )+ x − y − ν )( x + y ) ] , (8) u z ( x, y ) = 0 , (9)where the value of the Poisson’s ratio ν is 0.3 [19]. The change in the displacement fieldsof atoms due to the incremental change δx in the dislocation line position can be found todecay rapidly with the distance from the dislocation line as compared to the displacementfield itself. Hence, the interactions among the periodic image dislocations are not expectedto affect the results significantly. Thus, the numerical computations have been done takinginto account 200 atoms on both sides of the dislocation line in each row along the x direction.Number of rows and the position of the origin are varied according to the configurations incase I and case II respectively. The results of the calculations are shown in Fig. 3(a) and(b) in the form of solid lines. The model is found to reproduce the trends observed in MDsimulations.Dislocation velocity at applied shear stress τ and temperature T is empirically given as v ∼ τ m exp ( − Q/kT ) where m is the stress exponent [13, 20]. However, with rise in stress andtemperature, the phonon drag becomes the predominant mechanism governing the dynamicsof dislocations [21]. Since this phonon drag primarily constitutes B ′ , the size effect on thevelocity of dislocation should diminish at higher temperatures and applied stresses. Figure4 represents the velocities of the edge dislocation obtained from the MD simulations for6hree different film thicknesses at three different applied loads as a function of temperature.The size effect on this velocity is noticeable only at 250 MPa shear stress and disappears athigher stresses of 500 MPa and 1000 MPa for the entire range of temperature studied. TheMD simulations performed at 250 MPa stress show a decrease in the velocity of dislocationwith increasing temperature. The reduction in the dispersion of the velocity of dislocationwith thicknesses of thin films at higher stresses and temperatures is supportive of the factthat lattice resistance is the key factor behind the observation of the effects under discussion.This letter reports a pronounced change in the velocity of a dislocation due to the presenceof a free surface in the proximity of the dislocation line in a finite nanoscale crystallinesolid. This effect has been attributed to the altered lattice resistance in different systemconfigurations. A model following an unconventional approach to the lattice resistance hasbeen developed that serves as a tool for explaining these observations. The fundamental ideasas well as the proposed model have been verified through the MD simulations of an edgedislocation in bcc molybdenum. Similar studies are yet to be done for more complex types ofdislocations in different crystal structures and this largely simplified model provides amplescope of necessary modifications to suite these special cases. Development of a generalizedmodel, especially the one covering up to the bulk regime needs further intensive study wherethe ideas as presented here, can provide a fundamental framework for the understanding ofthe mechanism of the lattice resistance along with the associated dislocation dynamics.The authors thank Dr. Wei Cai for technical suggestions regarding the use of the MD++molecular dynamics package. [1] Steffen Brinckmann, Ju-Young Kim, and Julia R. Greer, Phys. Rev. Lett. , 155502 (2008).[2] A. K. Schmid, N. C. Bartelt, J. C. Hamilton, C. B. Carter, and R. Q. Hwang, Phys. Rev.Lett. , 3507 (1997).[3] L. Nicola, E. Van der Giessen, and A. Needleman, J. Appl. Phys. , 5920 (2003); L. Nicola,E. Van der Giessen, and A. Needleman, Thin Solid Films , 329 (2005).[4] H. 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50 100 150 200 250 300 350600700800900100011003 6 9 12 15 18850900950100010501100 Film thickness (nm) (a)(b) D i s l o ca t i on V e l o c i t y ( m / s ) Depth of dislocation core from top surface (nm) d f case Icase II d c . n m FIG. 3: (color online). (a) The dislocation velocity obtained from MD simulations is plotted(circles) as a function of film thickness d f (case I). Here the dislocation line is equidistant from boththe free surfaces of the film as illustrated schematically in the inset. (b) The dislocation velocityextracted from MD simulations is presented (squares) for different depths of the dislocation linefrom the top surface ( d c ) of the film of 35.2 nm thickness (see case II in the inset). The solid linesrepresent the output of numerical calculations based on the model in both (a) and (b). Shear stressand temperature are 250 MPa and 300 K respectively. Error bars for MD simulation results asindicated in both the plots are due to the randomness of the initial velocities and positions of theatoms in the simulation cell.
150 200 250 300 350 400 450 50080010001200140016001800 D i s l o ca t i on V e l o c o t y ( m / s ) Temperature (K)