Introduction

Nanocrystalline (NC) metals have extremely high strength but low ductility restricts their applicability [1,2,3,4,5,6,7]. Introduction of a bimodal grain structure in such nanocrystalline materials is capable of enhancing strength as well as ductility [8,9,10,11,12,13]. In the case of bimodal grain structure, fine nano grains (NG) impart high strength, whereas coarse grains (CG) bestow better ductility [14]. NC Ni having bimodal grain structure shows prominent strain hardening under deformation conditions as per experimental studies reported in literature [12, 15,16,17]. Observed high uniaxial tensile strength of polycrystalline face centered cubic (FCC) metal with bimodal grain size distribution is due to lattice dislocation activities occurring in the larger grain region according to a molecular dynamics (MD) simulation based study performed by Zhang et al. [10]. Thorough understanding of time-dependent plasticity of NC metal is necessary for improving reliability during service condition as well as estimating working life according to the engineering viewpoint. Creep behavior of free-standing FCC films having bimodal grain size (i.e., 40 and 220 nm) distribution is studied experimentally at room temperature and it is reported that the dislocation activities have significant influence on creep rate [13]. During the creep process ultrafine grained NC metal becomes amorphous as per the finding reported in literature [18,19,20]. It is known that shear transformation zones (STZs), shear diffusion transformation (SDT), and free volume mechanism have significant influence on creep deformation process of amorphous materials [21,22,23,24], but understanding of the creep deformation mechanism of amorphous structure at nano-scale is still far from conclusion. However, some investigations on creep behavior of NC FCC metals and alloys having grains of equal size using molecular dynamics simulation have been reported in literature [18,19,20, 25,26,27,28], atomistic simulation study of the influence of bimodal grain size distribution on creep behavior of NC metals has not been studied so far. In this paper, for the first time, we are going to discuss the creep deformation of bimodal structured nanocrystalline metal using molecular dynamics simulation along with the effect of size of bigger grain on the creep deformation process of bimodal NC Ni. Furthermore, special attention to the underlying deformation mechanism responsible for the creep process for transformed amorphous structure from bimodal NC Ni will also be given here.

Simulation details

A cubic NC Ni with an average grain size of 4 nm having 54 number of grain is constructed by Voronoi tessellation method [29] using Atomeye software [30]. After that, a sphere of the certain diameter d is deleted from the center of this cubic specimen. Finally a spherical insertion of the same diameter constructed from single crystalline Ni is placed inside the empty region to obtain the bimodal grain structure. The three bimodal specimens having 6 nm, 8 nm, and 10 nm spherical larger grains along with 4 nm small grains contain 253,220, 253,072, and 252,911 atoms respectively in a cubic simulation box of 14 × 14 × 14 nm3 dimension. Energy minimizations or relaxations of bimodal NC Ni specimen at 900 K (0.52*Tm), 1300 K (0.75*Tm), and 1600 K (0.93*Tm) temperatures have been performed by using conjugate gradient method (where Tm is the melting temperature of pure Ni (i.e., ~ 1728 K)). The constant-temperature, constant-volume (NVT) ensemble is used to regulate temperature by Nose-Hoover method thermostat [31, 32]. Creep deformation process is also carried out at constant temperatures taking NVT ensemble and using Nosé-Hoover thermostat algorithm to control the temperature. Then, uniaxial tensile loading is applied initially along Y-direction [0 1 0] to reach 1 GPa load. Subsequently, deformation of specimen is allowed under constant uniaxial tensile load (1 GPa) using Berendsen barostat method [33] only along the Y-direction [0 1 0]. No load is applied in the other two directions (X and Z). Periodic boundary conditions are applied in all the direction during deformation simulation. In the periodic boundary condition, particles can be interacted across the boundary and it can also exit one end to re-enter the other end of the simulation box. The boundaries in x and z directions are also free to move. A time step of this simulation work is taken as 2 fs. The sectional view of NC Ni with a bimodal grain size distribution is shown in Fig. 1a,b. In this figure, a spherical shape of large grain having 8 nm is embedded inside a matrix of small grains of NC Ni specimen and colored according to Common neighbor analysis [34,35,36]. All the MD simulations work is performed using LAMMPS [37] with embedded atom method potential (EAM), which is established by Mendelev [38]. Visualization and representation of atomic snapshots of NC Ni with a bimodal grain size distribution are done using OVITO open source software [39]. Dislocation extraction algorithm (DXA) inbuilt in OVITO [40, 41] is used to obtain the dislocation distributions of the bimodal NC Ni specimen. Dislocation density [42] is taken as the sum of all dislocation length per unit volume of the specimen. Characteristic structure evaluation during the creep deformation process has been investigated through determining population fraction of full icosahedra and distorted icosahedra in terms of Voronoi Polyhedra [19, 20]. The central symmetry parameter (CSP) is used to distinguish the degree of inversion symmetry broken as compare to each atom’s local environment of specimen under the creep deformation process [43]. CSP has zero value for perfect inversion symmetry material and nonzero value for inversion symmetry is broken during plastically deformed material. CSP for each atom is defined as follows:

$$ \mathrm{CSP}={\sum}_{i=1}^{N/2}{\left|{R}_i+{R}_{i+N/2}\right|}^2 $$
(1)

where N is the number of nearest neighbors (e.g., N = 12, for the FCC lattice), Ri and Ri + N/2 are the vectors or bonds from the central atom to the six pairs of opposite nearest neighbors in the FCC lattice. The number of possible neighbor pairs (i, j) is given by N(N-1)/2 and j is equal to i + N/2 [43]. Each “equal and opposite” pair of bond vectors is added together, then the sum of two square bond vectors (i.e., |R i  + R j |2) is calculated. This calculation is carried out for smallest N/2 to evaluate CSP. Ri is generated by finding those neighbors in the distorted lattice with vectors closest in distance to the undistorted nearest-neighbor vectors. i (1, 2, …N) is used for defining the degree of inversion symmetry broken in each atom’s local environment.

Fig. 1
figure 1

Three dimensional cross-sectional view of bimodal NC Ni specimen showing (a) half and (b) full of bigger grain for better representation (colored according to common neighbor analysis (CNA))

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Evaluated free volume fraction (FVF) is defined as:

$$ \mathrm{FVF}=\frac{V_{final}-{V}_{initial}}{V_{final}} $$
(2)

where V initial is the volume of bimodal NC Ni specimen at zero time step of creep, V final is the volume of bimodal NC Ni specimen at different time steps of the creep process.

The calculation of the mean square displacement (MSD) with respect to time function equation is given below [44]:

$$ MSD=<{r}^2(t)>=\left\langle \frac{1}{N}\sum_{i=0}^N{\left({r}_i(t)-{r}_i(0)\right)}^2\right\rangle $$
(3)

where N is the total particle number in the system, t is the time period, r i (t) is the atomic position at time t and r i (0) is the initial position of atom at time zero.

Self-diffusion coefficient (D) has been calculated using following Einstein relation [45,46,47]:

$$ D=\frac{1}{6}{lim}_{t\to \infty}\frac{MSD(t)}{t} $$
(4)

Results and discussion

Simulated creep curves for three bimodal NC Ni specimens having bigger grains of different size (e.g., ~ 6, 8, and 10 nm) for 900 K, 1300 K, and 1600 K creep temperatures are presented in Fig. 2a-c to investigate the influence of size of bigger grain on creep curves of bimodal NC Ni. It is observed from Fig. 2a and b that secondary creep regime for bimodal NC Ni specimen are widened with the increase of size of bigger grain for 900 K and 1300 K creep temperatures. Creep curves for bimodal NC Ni having larger coarse grain are found to be shifted toward lower strain in the tertiary creep regime for 900 K and 1300 K creep temperature cases (refer Fig. 2a and b). So it can be inferred that creep resistance is increased with increase of bigger grain size significantly for bimodal NC Ni specimens for moderate creep temperatures. On the other hand, creep curves of bimodal NC Ni are not found to be influenced by variation of size of bigger grains ranging from 6 to 10 nm during creep deformation process occurring at 1600 K (refer Fig. 2c). This is because faster diffusion at such high creep temperature takes place and consequently quick amorphization of the entire specimen occurs. Negative creep phenomenon is also observed at a later stage (in the tertiary creep regime) of the creep deformation process taking place at 1600 K. Negative creep normally takes place when internal stress of materials become higher than applied external stress as it causes a decrease in creep rate [20, 48]. Secondary creep regime is found to be reduced with the increase of creep temperatures (refer to Fig. 2a-c) as the diffusion process is faster at higher temperature, which in turn accelerates the creep process.

Fig. 2
figure 2

Simulated creep curves of bimodal NC Ni for (a) T = 900 K, (b) T = 1300 K, and (c) T = 1600 K creep temperatures

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Atomic snapshots showing dislocation distribution for bimodal NC Ni having a larger grain of three different sizes after 3 ps during creep deformation occurring at 900 K are represented in Fig. 3a-c. The different types of dislocations, such as Perfect dislocation (blue), Shockley partial dislocation (green), Stair-rod dislocation (pink), Hirth partial dislocation (yellow), Frank partial dislocation (cyan), and other dislocation (red) are observed in all three specimens. According to the Burgers vector family, different types of dislocations are defined [49,50,51], such as 1/2 <110> for Perfect dislocation, 1/6 <112> for Shockley partial dislocation, 1/6 <110> for Stair-rod dislocation, 1/3 <001> for Hirth partial dislocation, and 1/3 <111> for Frank partial dislocation. It is observed from Fig. 3 that the dislocations are found to be more in the region of smaller grain as compared to the region of larger grain for all the cases of bimodal NC Ni during the creep process. Consequently, at the initial stage of the creep process, the smaller grain favors deformation, while coarse grain provides better creep resistance. The evaluated dislocation density plots for bimodal NC Ni during the simulated creep process at three different temperatures (i.e., 900, 1300, and 1600 K) have been presented in Fig. 4a-c. Dislocations are found to be present a longer period in the bimodal NC Ni having larger grain during the creep deformation process. It is because formation of the amorphous structure is delayed by the presence of larger coarse grain in bimodal NC matrix. It is also observed that dislocation density is present a much longer period and higher quantity at lower creep temperature (i.e., 900 K) as compared to higher creep temperature (i.e., 1600 K). The dislocation density of this bimodal structure is found to be decreased exponentially during the creep process for all the cases studied here, as each specimen is transformed to amorphous structure due to diffusion and accumulation of atoms in grain boundary.

Fig. 3
figure 3

Dislocation distribution snapshots of bimodal NC Ni having (a) d = 6 nm, (b) d = 8 nm, and (c) d = 10 nm bigger grains after 3 ps during creep deformation occurring at 900 K

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Fig. 4
figure 4

Plots of evaluated dislocation density vs. time for three different larger grain sizes of bimodal NC Ni at (a) T = 900 K, (b) T = 1300 K, and (c) T = 1600 K creep temperatures

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Sectional view of atomic snapshots colored as per CSP along with the corresponding radial distribution function (RDF) of bimodal NC Ni specimen having 8 nm of larger grain size during the creep process for 900 K and 1600 K creep temperatures are presented in Fig. 5 as representative examples. It is observed that the grain boundary diffusion mechanism is more dominant in the region having smaller grains (i.e. ~ 4 nm) compared to the larger grain region. This actually causes initiation of amorphous structure formation in the region of smaller grains and, afterward the amorphization process moves toward the interior of larger grain from the grain boundaries between coarser grain and smaller grain with the progress of creep deformation. In another perspective, with respect to creep temperatures, amorphous structure formation is initiated earlier in the case of 1600 K compared to 900 K. It is observed that the specimen loses crystallinity with the progress of creep deformation and becomes completely amorphous after 120 ps for 1600 K and 225 ps for 900 K, as evident from Fig. 5d and e respectively. It may be due to the fact that grain boundary diffusion is faster in the case of higher creep temperature. It is also observed that coarser grain is further increased in size at the cost of finer grain initially up to 30 ps time for 1600 K, 120 ps time for 900 K. After that, amorphization causes reduction of grain size of bigger grain, as evident from Fig. 5c and d. Moreover, first peak height (or intensity) of RDF plots for bimodal NC Ni specimen is found to be increased with increase of creep temperature, as evident from Fig. 5. This is due to the fact that larger grain is further increased in size initially at higher creep temperature as compared to lower creep temperature and consequently overall volume fraction of crystalline structure is enhanced at the initial period in the case of higher creep temperature. Vector plots of atomic displacements for bimodal NC Ni specimen having 10 nm of larger grain size during the creep process at 900 K after 180 ps and 270 ps time step are represented in Fig. 6. Atomic diffusion is found to occur from grain boundaries to the interior of bigger grain for bimodal NC specimen and randomizes the atomic arrangement inside the bigger grain as creep deformation progresses.

Fig. 5
figure 5

Radial distribution function plots with atomic snapshots of NC Ni specimen having 8 nm coarser grain colored as per Centro-symmetry parameter at (a) initial stage and after (b) 3 ps, (c) 30 ps, (d) 120 ps, (e) 225 ps, and (d) 300 ps time periods during creep deformation occurring at 900 and 1600 K

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Fig. 6
figure 6

Vector plots of atomic displacements for bimodal NC Ni specimen having 10 nm of larger grain size during the creep process at 900 K after 180 ps and 270 ps time step

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The evaluated free volume fraction (also called active volume) vs. time plots for bimodal NC Ni during the creep process at three different temperatures (i.e., 900, 1300, and 1600 K) are presented in Fig. 7a-c. The free volume concept has been introduced and effectively utilized for simulating the diffusion and deformation process of metallic glass subjected to external load by Spaepen [52]. In this study from Fig. 7, free volume fraction is observed to increase very sharply at the initial stage, after that increment rate slows down, and then almost becomes constant with the progress of the creep process. It is also observed that free volume fraction curves are found to be shifted downward with increasing the size of bigger grain present in bimodal NC Ni specimen but at 300 ps, all curves converge to the same point (refer Fig. 7). Critical investigation of creep curve plots (refer Fig 2a-c) and corresponding active volume plots (refer Fig. 7) shows that change in active volume has significant influence on creep behavior. This may be due to the fact that generated active volume during deformation process can cause strain softening and strain localization [53]. Creep rate vs. time curves for bimodal NC Ni specimens having three different bigger grains size at 900 K, 1300 K, and 1600 K creep temperatures are presented in Fig. 8a-c. It is observed from Fig. 8a and b that the creep rate of secondary creep regime for 900 K and 1300 K creep temperatures becomes slower as the size of the bigger grain of bimodal NC specimen increases. Voronoi analysis is performed to study the polyhedra environment during the creep deformation process for bimodal NC Ni specimens and the variation of population fraction of Voronoi polyhedra during the creep process occurring at 900 K, 1300 K, and 1600 K have been presented in Figs. 9, 10, and 11 respectively. Variation in population of full icosahedra (i.e., <0, 0, 12, 0>) is very minimum and almost negligible compared to that of distorted icosahedra. Population of distorted icosahedra having voronoi indices <0, 2, 8, 5>, <0, 2, 8, 2 >, and <0, 1, 10, 2 > are found to reach maximum value around 150 ps to 180 ps (refer Fig. 9a), 210 ps to 270 ps (refer to Fig. 9b), and 270 ps (refer to Fig. 9c) time periods for different bimodal NC Ni specimens having 6 nm, 8 nm, and 10 nm coarse grain respectively. Interestingly, after secondary creep regime creep rate increases to its maximum value in the above mentioned time range for respective bimodal NC specimens for the creep process occurring at 900 K (refer to Fig. 8a). Similar correspondence are seen between creep rate vs time plots (refer to Fig. 8b and c) and variation in population of distorted icosahedra (refer to Figs. 10 and 11) during the creep process taking place at 1300 K and 1600 K. At this juncture, it can be inferred that the presence of <0, 2, 8, 5>, <0, 2, 8, 2 >, and <0, 1, 10, 2 > Voronoi polyhedra has a significant influence on creep behavior in the tertiary creep regime. In the case of creep process occurring at 1600 K, the population of Voronoi polyhedra vs. creep time plots are found to exhibit the same trend for all three specimens (refer to Fig. 11) and the deviation in the nature of creep curve of all three bimodal NC specimens is negligible for such high creep temperature (refer to Fig. 2c) as well. To sum up, variation in distorted Voronoi polyhedra population is an influencing factor for the creep process of such bimodal NC specimens when it converts to amorphous structure. This can be accounted for by the fact that the structures with a lower degree of local fivefold symmetry (LFFS), such as <0, 2, 8, 5>, <0, 2, 8, 2 >, and <0, 1, 10, 2 > clusters are capable of accommodating larger plastic deformation in the case of metallic glass [53, 54]. Actually, the lower degree of LFFS such as <0, 2, 8, 5>, <0, 2, 8, 2 >, and <0, 1, 10, 2 > clusters increases the defects and favors plastic deformation in comparison with the <0, 0, 12, 0 > full icosahedra [53].

Fig. 7
figure 7

Plots of evaluated free volume fraction vs. time of three different bigger grain sizes of bimodal NC Ni for (a) T = 900 K, (b) T = 1300 K, and (c) T = 1600 K creep temperatures during the creep process

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Fig. 8
figure 8

Creep rate vs. time plots for three different larger grain sizes of bimodal NC Ni at (a) T = 900 K, (b) T = 1300 K, and (c) T = 1600 K

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Fig. 9
figure 9

Population fraction of Voronoi polyhedra vs. time plots for (a) d = 6 nm, (b) d = 8 nm, and (c) d = 10 nm bigger grain sizes of bimodal NC Ni at T = 900 K creep temperatures during the creep process

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Fig. 10
figure 10

Population fraction of Voronoi polyhedra vs. time plots for (a) d = 6 nm, (b) d = 8 nm, and (c) d = 10 nm bigger grain sizes of bimodal NC Ni at T = 1300 K creep temperatures during the creep process

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Fig. 11
figure 11

Population fraction of Voronoi polyhedra vs. time plots for (a) d = 6 nm, (b) d = 8 nm, and (c) d = 10 nm bigger grain sizes of bimodal NC Ni at T = 1600 K creep temperatures during the creep process

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Atomic configurational snapshots of bimodal NC Ni having 6 nm bigger grain size after 165 ps, 168 ps, 171 ps, and 174 ps time steps indicating shear diffusion transformation (SDT) during creep deformation process at 900 K are represented in Fig. 12a-d. SDT is found to start at the initial stage of tertiary creep regime for bimodal NC Ni having 6 nm bigger grain size specimen at 900 K creep temperature and the amorphization of bimodal NC Ni specimen is also intensified at that time. It is observed that stress-driven shear diffusion transformation (SDT) mechanism is operative when bimodal NC Ni specimens become amorphous; it is also found in previously reported literature that SDT is responsible for deformation of amorphous materials [24, 55, 56]. It is observed from Fig. 12 that SDT also helps to create active volume during creep deformation process.

Fig. 12
figure 12

Atomic snapshots indicating shear diffusion transformation (SDT) in transient creep regime

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Self-diffusion (D) vs temperature (T) plots of three different regions for bimodal NC Ni having 6, 8, and 10 nm of larger grains are presented in Fig. 13a-c, which are calculated from the slope of the mean square displacement (MSD) curves only in the linear region obtained for different temperature as shown in Fig. 14. A representative MSD simulation was carried out for bimodal NC Ni having 6 nm larger grain at 900 K for 2000 ps time period and corresponding MSD vs. time plot is presented in Fig. 15. The value of calculated diffusion coefficient for this representative case is almost the same for the MSD calculation performed for the 300 ps and 2000 ps time period as the slope of linear region of MSD curve is not varied in the MSD calculation time period from 300 ps to 2000 ps (evident from Fig. 15). Self-diffusion of atoms in an MD simulation is determined via the Einstein relations, which include tracking atomic displacements or velocities of atoms [57, 58]. Inner portion of larger grain is designated as region 1 and grain boundary between the larger grain and smaller grain is considered as region 2 whereas region 3 indicates the region having smaller grains. It is observed that self-diffusion in these three different regions of specimen are increased as temperature increases. It is observed from Fig. 13a-c that maximum self-diffusion happens in the region with smaller grains (i.e., region 3), while minimum occurs in the inner part of the larger grain (i.e., region 1) for bimodal NC Ni specimens. Moreover, self-diffusion plots for bimodal NC Ni specimen is shifted upward from region 1 (i.e., large grain) to region 3 (i.e., smaller grain) with increasing temperature, as evident from Fig. 13a-c. Activation energies of creep and diffusion processes for bimodal NC Ni specimen are calculated from the slope of the ln minimum creep rates and diffusivities versus 1/T curves as shown in Fig. 16a and b. Obtained activation energies for creep of bimodal NC Ni having coarser gains of 6 nm, 8 nm, and 10 nm sizes are equal to 0.33 eV, 0.12 eV, and 0.05 eV respectively, while activation energies for self-diffusion of bimodal NC Ni having 6 nm, 8 nm, and 10 nm is equal to 0.96 eV, 0.90 eV, and 0.80 eV respectively. In this investigation, activation energy of creep for bimodal NC Ni having 6 nm, 8 nm, and 10 nm large grains is found to be less compared to the activation energy of self-diffusion for the same system. This difference can be attributed to the fact that activation energy for the creep process is calculated when external load is applied on the specimen, whereas activation energy for the self-diffusion process is obtained without any external applied load. It is also observed that activation energies of bimodal NC Ni are decreased with the increase in size of larger grains for both the creep process and self-diffusion process.

Fig. 13
figure 13

Self-diffusion coefficient (D) plots of bimodal NC Ni at three different regions with respect to temperature for (a) d = 6 nm, (b) d = 8 nm, and (c) d = 10 nm bigger grains

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Fig. 14
figure 14

Mean square displacement (MSD) vs. time plots for different temperatures and different larger grain sizes of bimodal NC Ni at three different regions and whole specimen

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Fig. 15
figure 15

MSD vs. time plot for whole bimodal NC Ni specimen having 6 nm larger grain size (calculated at 900 K)

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Fig. 16
figure 16

(a) logarithmic of minimum creep rate vs 1000/temperature(K) for a constant 1 GPa applied load, (b) logarithmic of D for whole specimen of bimodal NC Ni vs. 1000/temperature

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Conclusions

In summary, it can be inferred that creep property of bimodal NC Ni specimens is enhanced with the increase in size of larger grain. Maximum self-diffusion happens in the smaller grains region, while minimum self-diffusion occurs in the larger grain region of bimodal NC Ni specimens. The Coble creep mechanism is significantly dominant for NC Ni having bimodal grain distribution during nanocrystalline phase, while the shear diffusion transformation mechanism is operative after bimodal NC Ni transforms to amorphous structure. Free volume fraction curves for bimodal NC Ni are observed to be shifted downward with increasing the size of bigger grain during the creep process but converged after 300 ps. Dislocations for bimodal NC Ni are found to be present for a longer period with increasing size of the bigger grain during the creep process.