Home Search Collections Journals About Contact us My IOPscience Temperature sensitivity of void nucleation and growth parameters for single crystal copper: a molecular dynamics study This content has been downloaded from IOPscience Please scroll down to see the full text 2011 Modelling Simul Mater Sci Eng 19 025007 (http://iopscience.iop.org/0965-0393/19/2/025007) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 134.148.10.13 This content was downloaded on 18/04/2017 at 09:57 Please note that terms and conditions apply You may also be interested in: Effect of material damage on the spallation S Rawat, M Warrier, S Chaturvedi et al Molecular dynamics simulations of near-surface Fe precipitates in Cu under high electric fields Simon Vigonski, Flyura Djurabekova, Mihkel Veske et al Molecular dynamics simulations of micro-spallation of single crystal lead Meizhen Xiang, Haibo Hu, Jun Chen et al Excitation of characteristic modes of a crystal during solid fracture at high tensile pressure S Rawat, M Warrier, D Raju et al Effects of orientation and vacancy defects on the shock Hugoniot behavior and spallation of single-crystal copper Enqiang Lin, Huiji Shi and Lisha Niu Dislocation detection algorithm for atomistic simulations Alexander Stukowski and Karsten Albe Molecular dynamics study on the grain boundary dislocation source in nanocrystalline copper under tensile loading Liang Zhang, Cheng Lu, Kiet Tieu et al A molecular dynamics study of void interaction in copper S Z Xu, Z M Hao and Q Wan Structure identification methods for atomistic simulations of crystalline materials Alexander Stukowski IOP PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING Modelling Simul Mater Sci Eng 19 (2011) 025007 (20pp) doi:10.1088/0965-0393/19/2/025007 Temperature sensitivity of void nucleation and growth parameters for single crystal copper: a molecular dynamics study S Rawat1 , M Warrier2 , S Chaturvedi2 and V M Chavan1 Refueling Technology Division, Bhabha Atomic Research Centre, Mumbai 400 085, India Computational Analysis Division, Bhabha Atomic Research Centre, Visakhapatnam 530 012, India E-mail: alig.sunil@gmail.com Received 28 May 2010, in final form 23 January 2011 Published 28 February 2011 Online at stacks.iop.org/MSMSE/19/025007 Abstract The effect of temperature on the void nucleation and growth is studied using the molecular dynamics (MD) code LAMMPS (Large-Scale Atomic/Molecular Massively Parallel Simulator) Single crystal copper is triaxially expanded at × 109 s−1 strain rate keeping the temperature constant It is shown that the nucleation and growth of voids at these atomistic scales follows a macroscopic nucleation and growth (NAG) model As the temperature increases there is a steady decrease in the nucleation and growth thresholds As the melting point of copper is approached, a double-dip in the pressure–time profile is observed Analysis of this double-dip shows that the first minimum corresponds to the disappearance of the long-range order due to the creation of stacking faults and the system no longer has a FCC structure There is no nucleation of voids at this juncture The second minimum corresponds to the nucleation and incipient growth of voids We present the sensitivity of NAG parameters to temperature and the analysis of double-dip in the pressure–time profile for single crystal copper at 1250 K Introduction Solid fracture under high strain-rate deformation is of interest for high velocity impact and penetration problems It is known that ductile fracture in solids takes place through the nucleation, growth and coalescence of voids Many researchers have studied the process of material failure Belak’s study [1] of nucleation and growth of voids in polycrystalline copper showed that a void at a grain junction yields at a lower strain than that at grain center In the study of growth of pre-existing voids under isotropic tension in copper, Belak and Minich [2] fit their MD results with the DFRACT model and found that the growth exponent is three times higher than that expected from experiment Rudd and Belak [3] in polycrystalline copper under triaxial expansion found that the voids nucleate at weaker junctions and this void 0965-0393/11/025007+20$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al nucleation does not take place at all available junctions Traiviratana et al [4] estimated the void size dependence of the stress threshold for the dislocation emission in mono-crystalline and bi-crystalline copper Seppala et al [5] studied the effect of stress triaxiality on the growth of pre-existing voids in single crystal copper Seppala et al [6] in their coalescence study estimated the inter-void ligament distance to be one void radius to coalesce the voids Wang and co-workers have studied the instabilities in perfect lattice at finite deformation using molecular dynamics simulation [7] Some studies focus on the variation of bulk properties of copper with temperature using MD [8, 9] Kanel [10] performed experiments on aluminum AD1 with different temperatures and found that the spall strength decreases with increase in temperature Experiments have been carried out at strain rates as high as 1010 s−1 [11] Laser blow off experiments reportedly have strain rates of the order 107 –109 s−1 [12–14] Molecular dynamics simulations of uniform expansion have also been performed at 1010 s−1 [5] Hydrodynamic simulations of fracture at the macro-scales divide the region of interest into cells and require (a) an equation of state (b) a model to predict the strength of the material (i.e variation of yield strength and shear modulus of the material) and (c) a dynamic fracture model (e.g void nucleation and growth model, the results of which depend on the temperature and pressure values in the cell For a given cell in the hydrodynamic simulations, the temperature and pressure are constant at any given time In this paper we focus on (c), the void nucleation and growth (NAG) model We have carried out molecular dynamics simulations of single crystal copper deformed triaxially at × 109 s−1 strain rate at temperatures of 300, 600, 800, 1000 and 1250 K We choose to perform constant temperature MD simulations since the ‘nucleation and growth (NAG)’ model [15] (described in section 2) is used to determine the number of voids nucleating in each cell of a macroscopic hydrodynamic simulation which has an assigned temperature and pressure at that instant This paper has two objectives The first is to check if the NAG model remains applicable for single crystals, even though it was originally developed for polycrystalline materials with preexisting voids at much lower strain rates We show that the nucleation and growth of voids at the atomistic scale follows the NAG model Quantitatively, the values of the parameters are very different due to the fact that we are dealing with perfect crystals The second objective is to study the dependence of the NAG parameters (Pn0 : the ambient pressure threshold for void nucleation, P1 : pressure sensitivity for nucleation, N˙0 : nucleation rate at threshold, Pg0 : threshold for void growth, η: material viscosity) on temperature and to understand that dependence in physical terms We find that there is a steady decrease in the void nucleation and growth thresholds for single crystal copper with an increase in temperature The typical pressure–time profile for such a simulation shows that the magnitude of pressure increases as the system is triaxially expanded and after a time point there is a turn around of the pressure This occurs around a pico-second after the nucleation and incipient growth of voids since void formation relaxes the built-up stress This turn around is followed by a rapid reduction in the magnitude of pressure due to the exponential growth of voids Finally, the rapid reduction in the magnitude of pressure stops and it begins to fluctuate around a small negative value corresponding to the pressure threshold for void growth (Pg0 ) We carry out MD simulations of this triaxial expansion at 300, 600, 800, 1000 and 1250 K for single crystal copper The above described pressure–time profile is observed in all cases with a steady decrease in Pn0 and Pg0 with increase in temperature (figure 1) Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al 20 T = 300 K -20 Pressure (kbar) -40 -60 -80 -100 -120 -140 -160 -180 10 12 14 16 18 20 22 24 26 Time (ps) Figure Internal pressure as a function of time for single crystal copper deformed triaxially at high strain rate (5 × 109 s−1 ) with 300 K In all cases except at 1250 K, we observe a single minimum (turn around) in the pressure– time profile However, in the 1250 K case, we observe two minima in the pressure–time profile We show that the first minimum corresponds to disappearance of long-range order in the system due to creation of stacking faults and not due to void nucleation The second minimum corresponds to the nucleation and incipient growth of voids We want to study the fracture properties of solids at high strain rate As a first step, we are presenting the void nucleation and growth parameters for single crystal copper Later on this work will be used to study the nucleation and growth of voids at macro-scale using a macro-scale model [16] This paper is organized as follows: In section 2, we describe the macroscopic nucleation and growth (NAG) model In section 3, we discuss the MD simulation and the post-processor for calculating the nucleation and growth of voids from the data obtained by the simulations In section 4, the best-fit nucleation and growth parameters obtained are presented at different temperatures In this section, we analyze the double-dip in the pressure–time profile at 1250 K Limitations of the work are mentioned in section and conclusions are presented in section Nucleation and growth (NAG) model The nucleation and growth model (also known as the DFRACT model) developed at Stanford Research Institute [15, 17] is a micro-physical model which describes the fracture processes that occur as a result of nucleation and growth of voids in ductile materials 2.1 Nucleation When the tensile pressure Ps in the solid material exceeds the nucleation threshold Pn0 of the material, new voids are created The rate of nucleation is given by Ps − Pn0 , Ps > Pn0 , N˙ = N˙0 exp (1) P1 N˙ = 0, Ps Pn0 , (2) Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al where N˙0 is the threshold nucleation rate and P1 is the pressure sensitivity for nucleation Both N˙0 and P1 are material constants Ps is the tensile pressure and not the average pressure in the system The volume of the voids nucleated in a time interval, t, is given by Vn = 8π N˙ tRn3 , (3) where Rn is a material parameter known as nucleation size parameter 2.2 Growth The existing voids grow if the tensile pressure in the material exceeds the threshold for void growth Pg0 The new void volume is given by Ps − Pg0 Vg = Vv0 exp t (4) η where Vv0 is the void volume at the beginning of the time interval and η is the material viscosity The total void volume due to nucleation and growth of voids at the end of the time interval is given by Ps − Pg0 Vv = Vv0 exp (5) t + Vn η Computational technique 3.1 Initial set up We have simulated isotropic tension in single crystal copper using the molecular dynamics code, LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [18] A cubic simulation cell containing a certain number of atoms N (4 × 106 atoms, 100 × 100 × 100 unit cells) is created by replicating FCC unit cells along the X-, Y - and Z-axes The number of atoms is chosen for our study such that any further increase does not change the results (figure 2) The embedded-atom method [19, 20] with parameters obtained by Foiles et al [21] is used for our simulation The equations of motion are integrated with a time step of femtosecond (fs) using the velocity-Verlet algorithm since our results not change for timesteps 1 fs Periodic boundary conditions are used in all three directions for this study The thermal velocities of all the atoms have been initialized using a Gaussian distribution at a temperature of 300 K To relax the system to minimum energy, NPT simulation is performed at bar and 300 K Nose–Hoover thermostat and barostat [22, 23] have been used to control the temperature and pressure After the pressure and temperature stabilize, the barostat is turned off and the high strain rate (5×109 s−1 ) triaxial expansion is applied at constant temperature This is done by rescaling the atomic positions within the simulation domain which is extended in all three directions as follows: L(t) = L(0)(1 + ˙ × t), (6) where L is the instantaneous length of one side of the system, ˙ is the engineering strain rate and t is the elapsed time The rescaling is carried out every 0.1 ps because rescaling every time step would hinder the dynamics of atoms By deforming a box of length L as explained in equation (6), the interatomic distance changes by dL (e.g L = 361.0 Å and dL = 0.0018 Å) and would allow the atoms a couple of oscillations to equilibrate to the new pressure conditions Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al 0.25 Void volume fraction 0.2 0.15 0.1 N = 108000 N = 500000 N = 864000 N = 2916000 N = 4000000 N = 10976000 0.05 16 18 20 22 Time (ps) 24 26 28 Figure Void volume fraction as a function of time for domains of different sizes Note that for N > 916 000, the void volume fraction does not change 3.2 Post-processing A post-processor is developed to calculate the void volume at a desired time The whole simulation domain is divided into small cubic cells and the size of the cubic cell is taken such that it should be 1.01 times the unit cell size Due to atomic vibrations, the unit cell may transiently become empty This is why we use the size of cubic cell which is slightly (∼1%) greater than the unit cell size The void nucleation size parameter is considered as one small cubic cell in our study As the distension in the material increases, the void size also increases This increase in void size includes the distension caused by the application of tension every few time steps and also the actual growth of the void Therefore, we remove this distension effect to obtain the void volume due to the actual growth of the void At each time, the post-processor counts the empty cells in the simulation domain and by multiplying these empty cells with the initial size of the mesh, the total void volume is obtained To get the void volume fraction at a particular time, the total void volume at that time is divided by the initial volume of the domain, not by the extended volume of the domain which is changing instantaneously due to expansion Results and discussion 4.1 Nucleation and growth parameters at different temperatures 4.1.1 Pressure–time profile Internal pressure for single crystal copper triaxially deformed at × 109 s−1 strain rate at different temperatures is shown in figure It is seen that the mean stress in the system increases continuously up to a time point The triaxial expansion pulls the atoms away from their equilibrium positions leading to an increase in the pressure of the system and the increased volume causes a drop in the density of the system Further expansion leads to bond breaking and the nucleation of voids which are nothing but the small disordered regions [1] nucleated due to the lattice instability as discussed by Wang et al [7] We note that more than a single void nucleates within the volume being simulated This nucleation of voids causes a reduction in pressure due to relaxation of the built-up stress This turn around Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al 20 A -20 D Pressure (kbar) -40 1250 K E -60 F G 1000 K -80 -100 800 K -120 C 600 K -140 300 K -160 B -180 10 12 14 16 18 20 22 24 26 Time (ps) Figure Internal pressure for single crystal copper triaxially deformed at (5 × 109 s−1 ) strain rate at different temperatures The points A, B, C and D in the pressure–time profile for 300 K corresponds to the time points for which the radial distribution function (RDF) is calculated, while for 1250 K the RDF calculations are performed at A, E, F and G time points in pressure is followed by a rapid reduction in pressure due to the exponential growth of voids Finally, the rapid reduction in pressure stops and it begins to fluctuate around a small negative value corresponding to the pressure threshold for void growth (Pg0 ) As the system is triaxially deformed with different temperatures, the turn around of pressure takes place not only at lower values of pressure but also at earlier times with an increase in temperature Therefore, the increase in temperature lowers the magnitude of pressure threshold for void nucleation It can also be seen that there is a single dip in the pressure–time profile for all temperatures except 1250 K, which has two minima The turn around of pressure in figure at 300 K is taking place at 168 kbar which is closer to that found by Seppala et al [5] in their study of triaxial expansion of single crystal copper at 109 s−1 strain rate The void volume fractions at the above temperatures are plotted in figure Note that the void volume fraction becomes significant only after the time at which pressure turns around (figure 3) The void volume fraction increases exponentially due to the growth of voids up to a time point and then increases linearly The onset of the linear region of the void volume fraction versus time graph corresponds to the pressure threshold for void growth (Pg0 ) For the case of 1250 K which has two minima in the pressure–time profile, the significant contribution to the void volume fraction is only after the second minimum in the pressure–time profile The void volume fraction versus time graph can be divided into the following regions: • the initial part of the curve belongs to the region where the rate of void volume generated by nucleation dominates over the void volume generated by growth • the region where void volume grows exponentially with time Here void volume generated by growth dominates over that by nucleation • the linear region, where the magnitude of tensile pressure in the system is no longer greater than that of the pressure threshold for void growth The increase in void volume fraction is due to coalescence The snapshots of void distribution at different times are shown in figure Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al 0.25 Void volume fractio 0.2 0.15 300 K 0.1 600 K 800 K 0.05 1000 K 1250 K 0 10 12 14 16 18 20 22 24 26 Time (ps) Figure Void volume fraction for single crystal copper deformed triaxially at high strain rate (5 × 109 s−1 ) at different temperatures Many independent voids and void clusters are created under the triaxial deformation of single crystal copper at × 109 s−1 strain rate The number of independent voids and void clusters as a function of time are shown in figure These are the voids which contribute to the void volume fraction plotted in figure at 300 K It is seen that the void volume fraction (figure 4) corresponds to nucleation and growth of many voids 4.1.2 NAG parameters For a given (assumed) set of NAG parameters, the known pressure– time profile can be used to determine the number of nuclei and the total void volume due to nucleation and growth as a function of time The total void volume obtained from the NAG model (equation (5)) is then compared with the post-processed MD output An overall minimization procedure is then used to determine best-fit NAG parameters (Pn0 , P1 , N˙0 , Pg0 and η) Note that we have not used equation (3) for the void volume (Vn ) nucleated in t time interval In our results, we have used N˙ tRn3 for the void volume (Vn ) nucleated in t time interval The relative error (φ) at any point is defined as Vpp φ =1− , (7) VNAG where Vpp is the total void volume obtained by post-processor and VNAG is the total void volume obtained by the NAG model The square root of average squared relative error () is defined as M i=1 φ = , (8) M where M is the total number of data points The best-fit NAG parameters obtained for single crystal copper at different temperatures are presented in table From table 1, it can be seen that • The threshold for void nucleation (Pn0 ) is much higher for single crystal copper than that for polycrystalline copper This is reasonable, because in polycrystalline copper, there Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al Figure Snapshots of void distribution for single crystal copper triaxially deformed at 5×109 s−1 strain rate with 300 K (a) 17.7 ps, (b) 17.9 ps, (c) 18.3 ps, (d) 18.5 ps, (e) 18.7 ps, (f ) 18.9 ps, (g) 19.3 ps and (h) 21.9 ps This image is shown at one x–y cross-section through the domain (This figure is in colour only in the electronic version) are weak points, e.g grain junctions, which can lead to void nucleation at lower tensile pressures [1] This difference in nucleation thresholds may also be the result of differences in the strain rates used in experiments and MD simulations Our void nucleation threshold (160 kbar at 300 K) is very close to the spall strength (156 kbar) measured by Moshe et al Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al 800 700 Independent voids Void clusters Number of voids 600 500 400 300 200 100 17 17.5 18 18.5 19 Time (ps) 19.5 20 20.5 Figure Number of independent voids and void clusters as a function of time for single crystal copper uniformly deformed at × 109 s−1 strain rate with 300 K • • • • [24] under laser shock experiments on copper foils with thicknesses in the range 1–10 µm and strain rate in the range (1.5–4.5) × 108 s−1 Moshe et al used polycrystalline copper for their study However, with a thickness of only a few micrometers, it is possible that in the axial direction, it could still be a single crystal, although that is not certain Many workers [25–29] have performed experiments on single crystal copper The spall strength measured by Tonks et al [25] for 1 0 single crystal copper is 73.0 kbar at 9.8 × 106 s−1 strain rate The spall strength measured by Kanel et al [26–28] for single crystal copper is in the range 33–45 kbar for an impact velocity of 660 ± 20 m s−1 McQueen and Marsh [29] performed experiments on single crystal copper for the shock pressure in the range 300–600 kbar and found that the spall strength of single crystal copper is in the range 50–150 kbar The differences in these experiments and simulations could be due to (a) the fact that single crystals used in experiments have defects and impurities and (b) strain rates in the experiments are different compared with the simulations The value of the pressure sensitivity parameter for nucleation (P1 ) is lower for single crystals, which means that the nucleation rate in single crystals is much more sensitive to pressure than that for polycrystalline material The material viscosity of single crystal is very low compared with polycrystalline material This signifies that growth of the nucleated voids takes place very rapidly once the nucleation threshold is crossed There is a monotonic decrease in the values of the void nucleation threshold (Pn0 ) and void growth threshold (Pg0 ) with an increase in temperature The enhanced atomic motions and increased average interatomic separation at higher temperature are responsible for decrease in the thresholds for void nucleation and growth The nucleation threshold (Pn0 ) is very high compared with the growth threshold (Pg0 ) for single crystal copper, while thresholds for nucleation and growth of the voids for polycrystalline copper [15] are the same For aluminum, the nucleation threshold is lower than the growth threshold, while in the case of mild steel the nucleation threshold is five times higher than the growth threshold [15, 16] Therefore, it is not necessary that the nucleation threshold should be equal to the growth threshold Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al Table Comparison of best-fit NAG parameters obtained for single crystal copper (N = × 106 atoms) with the experimental values for polycrystalline copper (Pn0 = −5.0 kbar, P1 = −2.0 kbar, N˙0 = 2.8 × 1018 m−3 s−1 , Pg0 = −5.0 kbar, η = −7.5 × 10−5 bar s and Rn = 1.0 × 10−6 m) [15] The value of nucleation size parameter (Rn ) in our simulation is 3.64 Å, which is the length dimension of the small cubic cell T (K) 300 600 800 1000 1250 NAG parameters Single crystal copper Pn0 (kbar) P1 (kbar) N˙0 (m−3 s−1 ) Pg0 (kbar) η (bar s) −160.4 −0.167 7.1 × 1018 −21.2 −3.4 × 10−8 0.19 Pn0 (kbar) P1 (kbar) N˙0 (m−3 s−1 ) Pg0 (kbar) η (bar s) −124.0 −0.366 3.1 × 1020 −18.5 −2.8 × 10−8 0.14 Pn0 (kbar) P1 (kbar) N˙0 (m−3 s−1 ) Pg0 (kbar) η (bar s) −113.7 −0.119 3.1 × 1018 −16.0 −2.5 × 10−8 0.16 Pn0 (kbar) P1 (kbar) N˙0 (m−3 s−1 ) Pg0 (kbar) η (bar s) −80.0 −0.398 1.0 × 1017 −12.2 −2.2 × 10−8 0.24 Pn0 (kbar) P1 (kbar) N˙0 (m−3 s−1 ) Pg0 (kbar) η (bar s) −58.8 −0.156 4.6 × 1016 −10.0 −2.3 × 10−8 0.11 As far as P1 and N˙0 are concerned, multiple combinations of these parameters could yield a similar level of match between NAG and MD results Hence it will not be appropriate to read physical meaning into the variations in N˙0 and P1 The comparison of void volume fraction VNAG obtained by the NAG model using the above parameters (table 1) and that by post-processor, VPP , for temperatures 300, 600, 800, 1000 and 1250 K, is shown in figure The comparison of relative error for single crystal copper at different temperatures is shown in figure 4.2 Analysis for 1250 K and its comparison with 300 K case Internal pressure as a function of time for 300 and 1250 K is shown in figure To analyze the double-dip in the pressure–time profile for 1250 K, we have calculated the radial distribution function (RDF), structure factor, centro-symmetry parameter and common neighbor analysis (CNA) For comparison, the same calculation has been done for 300 K 4.2.1 Radial distribution function We have taken a small volume (∼76 Å3 ) from the center of the simulation domain to compute the RDF because large computation time is required for computing the RDF for the whole simulation domain For 300 K, the RDF has been calculated 10 Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al 0.18 0.16 0.16 300 K 0.14 600 K 0.12 Void volume fraction Void volume fraction 0.14 0.12 0.1 NAG model Post-processor 0.08 0.06 0.1 NAG model Post-processor 0.08 0.06 0.04 0.04 0.02 0.02 0 16 17 18 19 20 21 22 23 24 25 14 15 16 17 18 Time (ps) 0.18 0.2 0.16 0.18 0.12 0.1 NAG model Post-processor 0.08 20 21 22 23 24 25 0.16 800 K Void volume fraction Void volume fraction 0.14 19 Time (ps) 0.06 0.14 1000 K 0.12 NAG model Post-processor 0.1 0.08 0.06 0.04 0.04 0.02 0.02 0 13 14 15 16 17 18 19 20 21 22 23 24 25 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Time (ps) Time (ps) 0.12 Void volume fraction 0.1 1250 K 0.08 NAG model Post-processor 0.06 0.04 0.02 10 11 12 13 14 15 Time (ps) 16 17 18 19 Figure Comparison of void volume fraction obtained by the NAG model and that obtained by post-processor for single crystal copper deformed triaxially at high (5 × 109 s−1 ) strain rate with (a) 300 K, (b) 600 K, (c) 800 K, (d) 1000 K and (e) 1250 K for A (0 ps), B (18 ps), C (19 ps) and D (20 ps) time points in the pressure–time profile (figure 3) and is shown in figure The RDF at ps shows long-range order After 18 ps, the RDF shows short-range order This short-range order is the result of stacking faults and the creation and growth of the voids due to which the system becomes disordered For 1250 K, the radial distribution is calculated for A (0 ps), E (8 ps), F (10 ps) and G (12 ps) time points and is shown in figure 10 Note that 8, 10 and 12 ps are the time points which covers the double-dip in the pressure–time profile (figure 3) For ps, the RDF shows the long-range order But the RDF shows disappearance of longrange order after ps which corresponds to the first minimum in the pressure–time profile 11 Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al 0.6 300 K 600 K 800 K 1000 K 1250 K Relative error (φ) 0.5 0.4 0.3 0.2 0.1 10 12 14 16 18 Time (ps) 20 22 24 26 Figure Comparison of relative errors for single crystal copper deformed at different temperatures Radial distribution function D (20 ps) C (19 ps) B (18 ps) A (0 ps) 10 12 14 16 18 20 22 24 26 28 30 r (A ) Figure Radial distribution function as a function of distance at different time points The plots for 18, 19 and 20 ps are shifted along the Y -axis for easy comparison The points A, B, C and D correspond to the time points in figure T = 300 K If we see the void volume fraction in figure 4, then it becomes significant only after 12 ps onwards This means that the first minimum in the pressure–time profile corresponds to the loss of long-range order and the second minimum results from the void nucleation and their incipient growth 4.2.2 Structure factor Experimentally, one can get the structure factor by x-ray diffraction that can be used to identify the structure For allowed reflections, the structure factor will be non-zero and will be zero for forbidden reflections Using the reciprocal lattice vector corresponding to FCC, BCC and HCP, we can calculate the structure factor, S(k), for FCC, 12 Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al Radial distribution function G (12 ps) F (10 ps) E (8 ps) A (0 ps) 10 12 14 16 18 20 22 24 26 28 30 r (A0) Figure 10 Radial distribution function as a function of distance at different time points The plots for 8, 10 and 12 ps are shifted along the Y -axis for easy comparison The points A, E, F and G correspond to the time points in figure T = 1250 K BCC and HCP as follows: 2 i cos (k.ri ) + i sin (k.ri ) S(k) = N (9) where k is the reciprocal lattice vector and ri is the position vector of the ith atom N is the total number of atoms in the system The calculation of the structure factor will not only provide another check for the existence of the crystalline order but also indicates the kind of crystal structure (FCC/BCC/HCP) The structure factor for FCC, BCC and HCP is calculated for 300 and 1250 K and is shown in figure 11 For 300 K, the structure factor for FCC rapidly drops after 18 ps as shown in figure 11(a) This indicates that up to 18 ps, the system pertains the FCC structure and after that it becomes disordered This is due to the nucleation and incipient growth of the voids In the case of 1250 K, the structure factor for FCC starts decreasing after ps and vanishes after 11 ps (figure 11(d)) This means that the first minimum in the pressure–time profile corresponds to the order to disorder transition However, this does not correspond to the nucleation and incipient growth of voids as we have mentioned in section 4.2.1 and as observed in figure In figures 11(b), (c), (e) and (f ), very small value of structure factor signifies that the loss of FCC structure in both cases does not correspond to the creation of BCC and HCP structures 4.2.3 Centro-symmetry parameter (CSP) To study the defects created in our system due to triaxial deformation at high strain rate, we have calculated the CSP for 300 and 1250 K The CSP for an atom is calculated [30] as follows: CSP = |Ri + Ri+6 |2 , (10) i=1,6 where Ri and Ri+6 are the vectors which corresponds to the six pairs of opposite nearest neighbors 13 Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al 0.0004 0.9 BCC, 300 K 0.00035 0.8 0.0003 Structure factor Structure factor 0.7 0.6 0.5 FCC, 300 K 0.4 0.00025 0.0002 0.00015 0.3 0.0001 0.2 5e-05 0.1 0 10 12 14 16 18 20 22 24 26 10 12 14 16 18 20 22 24 26 Time (ps) Time (ps) 0.0012 0.9 0.8 HCP, 300 K 0.7 0.0008 Structure factor Structure factor 0.001 0.0006 0.0004 FCC, 1250 K 0.6 0.5 0.4 0.3 0.2 0.0002 0.1 0 10 12 14 16 18 20 22 24 26 Time (ps) 12 14 16 18 20 18 20 0.0014 BCC, 1250 K 0.0006 0.0012 0.0005 0.001 Structure factor Structure factor 0.0007 10 Time (ps) 0.0004 0.0003 0.0008 0.0006 0.0002 0.0004 0.0001 0.0002 HCP, 1250 K 0 10 12 14 16 18 Time (ps) 20 10 12 14 16 Time (ps) Figure 11 Structure factor as a function of time for single crystal copper For 300 K: (a) FCC, (b) BCC, (c) HCP and for 1250 K: (d) FCC, (e) BCC and (f ) HCP Zhao’s CSP limits [31] for single crystal copper to distinguish between different defects are given by • • • • Perfect lattice: 0.5 < CSP Partial dislocations: 0.5 < CSP < 3.0 Stacking faults: 3.0 < CSP < 16.0 Surface atoms: CSP > 16.0 The CSP filter for single crystal copper given by Zhao et al [31] is for K temperature This does not take into account the effect of temperature Therefore, for non-zero temperatures, it is necessary to determine corrected values, since particle displacements from their mean 14 Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al 0.9 Fraction of atoms 0.8 300 K 0.7 0.6 Perfect lattice Dislocations and stacking faults Surface atoms 0.5 0.4 0.3 0.2 0.1 0 10 15 Time (ps) 20 25 Figure 12 Centro-symmetry parameter as a function of time for single crystal copper triaxially deformed at high strain rate (5×109 s−1 ) with 300 K Perfect lattice: CSP < 3.4, partial dislocations and stacking faults: 3.4 < CSP < 16.0, surface atoms: CSP > 16.0 0.9 1250 K Fraction of atoms 0.8 0.7 0.6 Perfect lattice Partial dislocations and stacking faults Surface atoms 0.5 0.4 0.3 0.2 0.1 0 10 12 Time (ps) 14 16 18 20 Figure 13 Centro-symmetry parameter as a function of time for single crystal copper triaxially deformed at high strain rate (5 × 109 s−1 ) with 1250 K Perfect lattice: CSP < 3.4, partial dislocations and stacking faults: 3.4 < CSP < 16.0, surface atoms: CSP > 16.0 positions change with temperature Firstly, we have verified using CNA (section 4.2.4) and structure factor (section 4.2.2) calculations at 300 K that the domain is a perfect crystal with an FCC structure For this case, we have determined the threshold value of CSP for perfect lattice at 300 K, after performing NPT MD for single crystal Cu at atm pressure This is found to be 3.4 Å2 With this new threshold value for perfect lattice and using CSP limits 3.4 < CSP < 16.0 for partial dislocations and stacking faults and CSP > 16.0 for surface atoms, the CSP is computed for 300 and 1250 K and is shown in figures 12 and 13, respectively 15 Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al If we compare CSP for both cases, then it can be seen that • At t = ps, the lattice for 300 K is free from any defect, while for 1250 K, the partial dislocations and stacking faults are present at t = ps The reason for the presence of defects is that CSP has limitations due to thermal vibrations • Stacking faults and dislocations for 300 K come into the picture significantly after some time point (∼10 ps) and suddenly increase after 18 ps which is the time at which pressure turns around (figure 3) Note that there are no defects at the start of the simulation for this case For 1250 K, the stacking faults and dislocations are already present in the system at this temperature and as the triaxial expansion is applied, the defect density increases linearly and flattens off after 10 ps which corresponds to the time at which maximum occurs between two minima in the pressure–time profile (figure 3) • Surface atoms for 300 K are observed after 18 ps This is the time at which pressure turns around Our post-processor also indicates the void volume fraction becomes significant after this time point (figure 4) For 1250 K, surface atoms appear after 12 ps This indicates that the second minimum in figure corresponds to the nucleation and incipient growth of voids After this analysis, we can conclude that the number and type of different defects created are affected by temperature The presence of surface atoms for 1250 K after 12 ps (second minimum) indicates that nucleation and incipient growth of voids takes place around this time but not corresponding to the first minimum in the pressure–time profile (figure 3) Note that the deviations of atom positions from the mean (equation (10)) can change the CSP limits and at high temperature, these deviations increase and the ranges change For example, the maximum CSP for perfect crystal at 1250 K is 14.9 Å2 from our simulations, which makes it within the range of partial dislocations However, structure factor calculations show that lattice is FCC at t = at 1250 K Therefore, we conclude that CSP is not a good indicator for differentiating perfect lattice and partial dislocations at high temperatures 4.2.4 Common neighbor analysis During the triaxial deformation of single crystal copper at high strain rate, numerous particle configurations are created that have different structural characteristics In order to identify these structures, the CNA [32, 33] method is very useful We have computed the CNA pattern for each atom in the system for both 300 and 1250 K The time evolution of fraction of atoms belonging to each crystal structure is shown in figures 14 and 15 The CNA method shows that • Loss of FCC structure at 1250 K does not result in the BCC, HCP and icosahedral structures, but results in an unknown structure (possibly amorphous) This method also supports the loss of FCC structure at a time corresponding to the first minimum in the pressure–time profile (figure 3) • For 300 K, the system is perfect FCC crystal corresponding to t = ps, while for 1250 K only about 24% atoms belong to the FCC crystal structure Note that CNA analysis has also limitations due to thermal vibrations • For 300 K, most of the atoms belong to an unknown structure corresponding to t = 19 ps, while for 1250 K, all the atoms are part of the unknown structure after t = 10 ps Thus, the analysis by CNA and CSP methods shows that for 1250 K, 100% atoms belonging to stacking faults at t = 10 ps by CSP analysis correspond to an unknown structure (CNA analysis) 16 Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al 0.9 300 K Fraction of atoms 0.8 0.7 FCC HCP BCC ICOSAHEDRAL UNKOWN 0.6 0.5 0.4 0.3 0.2 0.1 0 10 12 14 16 Time (ps) 18 20 22 24 26 Figure 14 Fraction of atoms as a function of time Common neighbor analysis has been done to identify the local crystal structure around an atom during triaxial deformation of single crystal copper at × 109 s−1 strain rate with 300 K FCC (CNA = 1), HCP (CNA = 2), BCC (CNA = 3), icosahedral (CNA = 4), unknown (CNA = 5) 0.9 Fraction of atoms 0.8 1250 K 0.7 FCC HCP BCC ICOSAHEDRAL UNKOWN 0.6 0.5 0.4 0.3 0.2 0.1 0 10 12 Time (ps) 14 16 18 20 Figure 15 Fraction of atoms as a function of time CNA has been done to identify the local crystal structure around an atom during triaxial deformation of single crystal cooper at × 109 s−1 strain rate with 1250 K FCC (CNA = 1), HCP (CNA = 2), BCC (CNA = 3), icosahedral (CNA = 4), unknown (CNA = 5) We have calculated structure factor, RDF, CSP and CNA for single crystal copper at 300 K and 1250 K All these calculations show that the system under study is perfect FCC crystal at 300 K at t = ps For 1250 K, the structure factor calculation shows that it is FCC at t = but CNA analysis does not show perfect FCC at this time and CSP analysis shows the presence of defects The reason for this is that both CSP and CNA analysis have limitations due to atomic vibrations and that is why the analysis shows defects at t = for 1250 K 17 Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al -12.8 Potential Energy (MeV) -13 T = 300 K -13.2 -13.4 -13.6 -13.8 -14 -14.2 10 12 14 16 Time (ps) 18 20 22 24 26 Figure 16 Potential energy of the system as a function of time N = × 106 atoms -12.2 1250 K Potential energy (MeV) -12.4 -12.6 -12.8 -13 -13.2 -13.4 10 12 Time (ps) 14 16 18 20 Figure 17 Potential energy of the system as a function of time N = × 106 atoms 4.2.5 Potential energy of the system The potential energy of the system as a function of time for 300 and 1250 K is shown in figures 16 and 17, respectively For 300 K, it can be seen that the potential energy of the system increases (negative value decreases) as the density decreases due to expansion of the system The potential energy peaks at 18 ps and after that drops suddenly which is indicative of the creation and enhancement of the voids The effective two-atom interaction becomes stronger and the bond length shortens once the voids are nucleated which results in the rapid drop in potential energy For 1250 K, the potential energy of the system also increases with time as mentioned above but peaks only after 12 ps, not corresponding to ps (first minimum in figure 3) This also indicates that the nucleation of voids does not take place corresponding to the first minimum in the pressure–time profile 18 Modelling Simul Mater Sci Eng 19 (2011) 025007 S Rawat et al Limitations of this work (i) The DFRACT model used for this study does not include the coalescence of the voids (ii) The best-fit NAG parameters presented here are for perfect crystal copper, while experimental single crystals contains various kinds of defects Therefore, before comparing these results with the experimental data, a systematic study of effect of impurities and defects on the void nucleation threshold should be performed Conclusion We have studied the effect of temperature on the nucleation and growth parameters for single crystal copper deformed at high strain rate The best-fit NAG parameters have been obtained for different temperatures We find a systematic reduction in the magnitude of nucleation and growth thresholds with rise in temperature However, as melting point (the melting temperature of copper 1358 K [34]) is approached, we observe a curious double-dip in the pressure–time profile Analysis of this double-dip by structure factor, RDF, CSP and CNA shows that the first minimum corresponds to the loss of the long-range order due to the creation of stacking faults and an unknown structure and the system no longer has a FCC structure There is no nucleation of voids at this juncture The second minimum corresponds to the nucleation and incipient growth of voids In this paper, we have shown (1) the DFRACT model for nucleation and growth of voids for polycrystalline macroscopic sample at strain rates of (104 –106 s−1 ) is also valid at high strain rates (5×109 s−1 ) at the atomistic scales for perfect crystals, (2) the effect of temperature on the nucleation and growth parameters of the DFRACT model for crystal copper and (3) analysis of the structural changes of the crystal subjected to high strain rate at different temperatures The detailed analysis of the void sizes, their morphology and their temperature dependence is our future work Acknowledgments The authors would like to thank the 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copper is triaxially expanded at × 109 s−1 strain rate keeping the temperature constant... tension in single crystal copper using the molecular dynamics code, LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [18] A cubic simulation cell containing a certain number... (b) a model to predict the strength of the material (i.e variation of yield strength and shear modulus of the material) and (c) a dynamic fracture model (e.g void nucleation and growth model,