New Computational Technique for Evaluating Deformation ofNanocrystalline Metal using Phase Field Crystal Method
Tomoyuki Hirouchi1, Tomohiro Takaki2, Yoshihiro Tomita1
1

Graduate School of Engineering, Kobe University,
1-1, Rokkodai, Nada, Kobe, 657-8501, Japan, hirouchi@solid.mech.kobe-u.ac.jp;
2
Graduate School of Science and Technology, Kyoto Institute of Technology,
Matsugasaki, Sakyo, Kyoto, 606-8585, Japan
ABSTRACTThe phase field crystal (PFC) method is anticipated as a new multiscale method, because thismethod can reproduce physical phenomena depending on atomic structures in metallic materials
on the diffusion time scale. Although the PFC method has been applied to some phenomena,
there are few studies related to evaluations of mechanical behaviors of materials by appropriate
PFC simulation. In a previous work using the PFC method, tensile deformation simulations havebeen performed under conditions where the volume change was unavoidable during plastic
deformation. In this study, to reproduce deformation under isovolumetic conditions, we proposeda novel numerical technique for PFC deformation simulation. Moreover, to evaluate thecontribution of the driving force α (temperature) on deformation behaviors, we performed thetensile deformation simulations of nanopolycrystalline structures with different temperatures.
The results confirmed that the resitence to dislocation motions increased by lowering
temperature. As a results, grain rotation and grain boundary migration were restrained.
1. Introduction
It is of great importance to predict macroscopic properties of industrial metallic materials
depending on defects in microstructures to improve material properties and/or create novel
functions of such materials. Therefore, the development of a multiscale model bridging amolecular description and a continuum field theory is an urgent need. Recently, the phase field
crystal (PFC) method [1] has been developed as a new multiscale numerical method that operates
for atomic length scale with diffusive time scale. The PFC model introduces a periodic orderparameter, which represents a local-time-averaged atomic density field to express regulararrangements of atoms in a solid phase. Therefore, this model inherently includes features ofcrystal systems, elasticity, plasticity and multiple orientations. Although, the PFC method has
been applied to some phenomena, there are few studies on appropriate deformation simulation
using this method. In previous PFC simulations of tensile deformation [1, 2], the volume ofpolycrystalline structure increases during plastic deformation simulation due to the difficulty insetting the boundary conditions appropriately. In this study, we propose a novel numerical
technique for deformation simulation by the PFC method. By employing this technique, thedeformation behaviours of nanopolycrystalline structures are studied at different temperatures.

2. Phase Field Crystal Method
The free energy functional in the PFC method is minimized by a periodic order parameter or thephase field φ, which is defined as the local-time-averaged atomic density and is a conserved
value [1]. Therefore, this model can reproduce the dynamics based on individual atoms on thediffusion time scale that is much larger than the atomic vibrational time scale. In the steady solid
phase of metallic materials, atomic locations are independent of time and become regulararrangements. Therefore, to express these periodic states, the phase field φ must have periodic
profiles. Here, the local maxima of the phase field φ correspond to atomic positions. On the otherhand, since atomic positions in the liquid phase become random depending on time, the phasefield φ is averaged and becomes a constant value φ0. From the phase field φ defined above, thedimensionless free energy functional F is given by [1]

⎡α2 ⎤
φ4 φ
F = ∫⎢ φ2 ++ 1 + ∇ 2 φ ⎥dV ,
4 2
⎣2
⎦

()

(1)

where α is the driving force, which is proportional to a distance in temperature from a melting
point, T - Tm. Here, T is a temperature and Tm is a melting point. Since φ is the time-averaged
density and a conserved value, the Chan-Hilliard equation provides the evolution of φ as[) ]2δF∂φ= ∇2= ∇ 2 αφ + φ 3 + 1 + ∇ 2 φ .
∂tδφ((2)

Fig. 1(a) schematically shows the proposed numerical scheme for PFC deformation simulation intwo dimensions. To reproduce tensile deformation under isovolumetic condition, the grid size in
x-direction increases by the displacement increment d = ε& ∆x∆t at every time step, and the gridsize in y-direction decreases so as to maintain ∆x∆y = ∆x’∆y’. Here, ∆x and ∆y are initial gridsizes, and ∆x’ and ∆y’ are grid sizes changed during deformation, and ∆t and ε& are the timeincrement and dimensionless strain rate, respectively. In this methodology, since a constantstrain rate is applied to all atoms, the deformation state becomes the affine deformation state, andperiodic conditions can be used as boundary conditions.
3. Deformation Simulation
The deformation behaviors of a nanopolycrystalline structure are simulated by the PFC methodwith our technique. Before deformation simulation, a regular hexagonal grain structure withpredetermined orientations is prepared by solidification simulation.
3.1 Computational model and conditions
Fig. 1(b) is a schematic illustration of the desired nanopolycrystalline structure with regularhexagonal grains of which a distance between two sides is approximately 14 atoms. The
predefined crystal orientations of each grain are shown at the center of each grain. The computa-

Figure 1. (a) Numerical scheme of tensile deformation simulation in two dimensions and (b)desired nanopolycrystalline structure with regular hexagonal grains and crystal orientations.
tional domain size is set to be 50a × 44a (500 × 440 lattices) and φ0 = -0.285 is selected. Here, a is
a distance between nearest atoms. To perform the deformation simulations under two different
temperatures, α = -0.25 and -0.50 are employed. The temperature corresponding to α = -0.25 ishigher than that of α = -0.50. The initial structure is formed by solidification simulation thatstarts from nuclei with predefined crystal orientations. As a result of solidification simulation
with α = -0.25, the initial structure shown in Fig. 2(a) is obtained. To prepare same atomicstructure for α = -0.50, after obtaining the structure shown in Fig. 2(a), further relaxation
simulation is performed by changing to α = -0.50. The computed structure is shown in Fig. 3(a).
Tensile deformation simulations are performed under a constant dimensionless strain rate ε& of
1.84 × 10-6, which corresponds to realistic strain rates of approximately 100s-1 to 101s-1, and use
the dimensionless time step ∆t = 3.0 × 10-3. Periodic conditions are used in all directions.
3.2 Numerical results and discussion
Fig. 2 indicates the deformation states of the nanocrystalline structure for the case of α = -0.25.
As the strain increases, grain rotations occur and cause the motions of dislocation at grainboundaries, such as the dislocation shown in the circle of Fig. 2(a). It is also observed that grain
rotation occurs between two neighboring grains with a relatively small misorientation (arrows inFigs. 2(a)(b)). As a result of coalescence due to grain rotation, the grain boundary between
integrated grains disappears and the grain boundary migration driven by curvature is accelerated,such as the grain boundary in the circle of Fig. 2(c). With further deformation, grain rotations
and grain boundary migrations occur continuously in other grains (Figs. 2(d)(e)).

Figure 2. Deformation process of nanocrystalline structures in the case of α = -0.25.

Figure 3. Deformation process of nanocrystalline structures in the case of α = -0.50.
Fig.3 shows the deformation states in the case of α = -0.50 that is a lower-temperature state thanthe former. The comparison between Figs. 2(a)-(c) and Figs. 3(a)-(c) suggests that the dislocation
motion at grain boundary becomes slower than those at higher temperature. Moreover, grain
rotation is harder to occur. As a result, grain coalescence observed in the former can’t beconfirmed. With further deformation as shown in Figs. 3(d)(e), unlike the former, it can beconfirmed that grain boundary migration is restrained. These results clarify that the PFC
deformation simulation with our technique can reproduce the increase of the resistance todislocation motion by lowering temperature.
4. Conclusion
To express deformation under isovolumetic condition, we proposed the novel numerical scheme
for PFC deformation simulation. Moreover, to evaluate the influence of the driving force α ondeformation behaviours, we performed the tensile deformation simulation of nanopolycrystalline
models at two different temperatures. At higher temperature (α = -0.25), the intergranular
deformation such as grain rotation, grain boundary migration and grain growth became thedominant deformation mechanism. On the other hand, at lower temperature (α = -0.50), unlike
the former, the resistance to dislocation motions at grain boundary increased by lowering
temperature. As a result, grain rotation and grain boundary migration were restrained.
Acknowledgements
Financial support from the Ministry of Education, Culture, Sports, Science and Technology ofJapan through a Grant-in-Aid for Scientific Research is highly appreciatedReferences[1] K. R. Elder and M. Grant, “Modeling elastic and plastic deformations in nonequilibrium
processing using phase field crystals”, Physical Review E, 70, 051605 (2004).
[2] T. Hirouchi, T. Takaki and Y. Tomita, “Phase Field Crystal Simulation during Plastic
Deformation of Nanopolycrystal Metal“, Proceedings of Third Asian-Pacific Congress on
Computational Mechanics (CD-ROM), MS31-6-3, 1-8 (2007).