X International Conference on Computational Plasticity
COMPLAS X
E. Oñate and D.R.J. Owen (Eds)cCIMNE,Barcelona, 2009

INFLUENCE OF TEMPERATURE AND GRAIN-SIZE ONPHASE FIELD CRYSTAL DEFORMATION SIMULATION
T. Hirouchi∗ , T. Takaki† , Y. Shibutani†† and Y. Tomita†††∗ Department of Mechanical Engineering, Osaka University

2-1, Yamadaoka, Suita, 565-0871, Osaka, Japane-mail: hirouchi@comec.mech.eng.osaka-u.ac.jp†† e-mail: sibutani@mech.eng.osaka-u.ac.jp† Graduate School of Science and Technology, Kyoto Institute of Technology

Matsugasaki, Sakyo, Kyoto, 606-8585, Kyoto, Japane-mail: takaki@kit.ac.jp††† Department of Mechanical Engineering, Fukui University of Technology

3-6-1, Gakuen, Fukui City, 910-8505, Fukui, Japane-mail: tomita@mech.kobe-u.ac.jp
Key words: Phase-ﬁeld Crystal Method, Grain Boundary, Dislocation, Deformation.
Summary. We performed deformation simulations of nanopolycrystalline metal under diﬀerent
temperature and grain-size conditions using the phase-ﬁeld crystal (PFC) method with our
proposed scheme, and demonstrated that the PFC deformation simulation could have a potentialof solving deformation problems depending on temperature and grain size.

1INTRODUCTION

It is well known that defects, such as dislocation and grain boundary (GB), have a great eﬀect
on mechanical property of metallic materials. Although behaviors of defects have been studied
by experiments and computer simulations by molecular dynamics (MD)1 , it is diﬃcult to observe
detailed dynamics at atomic level in experiments and MD simulations are limited inherently insmall time scales. Therefore, the investigation of atomic behaviors in more realistic time scale
is of great importance.
Recently, Elder et al. has developed the phase-ﬁeld crystal (PFC) method2 , which enables
one to simulate atomic behaviors in diﬀusive time scale. In our previous study3 , to express
isovolmetic state during plastic deformation of metallic material employing this method, weproposed a new numerical scheme where computational grid size was changed and conﬁrmedthat the PFC deformation simulation with our scheme could reproduce appropriate behaviors of
dislocations and GBs. However, these simulations were conducted under the limited temperatureand grain-size condition.
In this study, to reveal that the PFC deformation simulation with our scheme can reproduce
the behaviors of defects under various conditions, we performed deformation simulations of1

T. Hirouchi, T. Takaki, Y. Shibutani and Y. Tomita

nanopolycrystalline materials under diﬀerent temperature and grain-size conditions.
2

PHASE-FIELD CRYSTAL METHOD

Unlike the conventional phase-ﬁeld method, the PFC method introduces a periodic orderparameter or the phase ﬁeld φ, which represents a local-time-averaged atomic number density
and is a conserved value. To express atomic states in solid and liquid phases of metallic materials,
φ becomes periodic and constant, respectively. The free energy functional is minimized when φhas periodic proﬁle and can be expressed as



F =dV



α 2 φ4 φφ ++ (1 + ∇2 )2 φ ,242(1)

where α is proportional to the deviation of temperature from a melting temperature. Also,
following the Chan-Hilliard equation, the dynamics are described in dimensionless form by∂φ= ∇2 {αφ + φ3 + (1 + ∇2 )2 φ}.
∂t(2)In PFC deformation simulations, we employ the proposed numerical scheme in two dimensions3 .
To reproduce tensile deformation under isovolumetic condition, the grid size in x-direction increases by the displacement increment d = ε̇Δx Δt at every time step, and the grid size iny-direction decreases so as to maintain Δx Δy = Δx  Δy  . Here, Δx and Δy are initial grid
sizes, and Δx  and Δy  are grid sizes during deformation, and Δt and ε̇ are the time increment
and the dimensionless strain rate, respectively. In this methodology, since a constant strain rate
is applied to all atoms, the deformation state becomes the aﬃne deformation state.
3DEFORMATION SIMULATIONIn this section, we study behaviors of defects in nanopolycrystalline structure under diﬀerent
temperature and grain-size conditions using the PFC method with our scheme.
3.1

Computational model and condition

Fig. 1(a) shows the schematic illustration of nanopolycrystalline structure with predeﬁned
crystal orientations. Crystal orientation θ is deﬁned as shown in Fig. 1(b), and Fig. 1(c) showsa sample of phase ﬁeld proﬁle with θ = 15◦ . Fig. 1(d) shows the deﬁnition of deviation angle
ψ of the actual GB plane from the symmetrical GB plane. In Fig. 1(a), θ is represented in thecenter of hexagonal grains, and misorientaion Δθ and deviation angle ψ, which is described inparentheses, are on the GBs, respectively. GB within Δθ = 20◦ is treated as low-angle GB in thisstudy. To investigate the inﬂuence of grain size on deformation behavior, we prepare two typesof polycrystalline structure and call them Model A and B. The computational domain sizes ofModel A and B are set to be 50a × 44a(500 × 440 lattices) and 150a × 132a(1500 × 1320 lattices)
respectively, where a is the distance between atoms as shown in Fig. 1(c). Grain size of Model B
is three times as large as that of Model A. Additionally, to perform the deformation simulation
under two diﬀerent temperatures, α = −0.25 and −0.50 are employed. The temperature statecorresponding to α = −0.25 is the state near a melting temperature and higher than that of2

T. Hirouchi, T. Takaki, Y. Shibutani and Y. TomitayTensile directionA: 44a ( 440 lattices)
B: 132a (1320 lattices)yx

16.8 44.8
15.7
12.6(6.8)(19.6)(21.1)
25.7
1.64
57.4
(2.0)
16.1
18.5
21.7
(19.6)
(6.7)
(8.3)
15.5
26.4
19.1
(19.7)(11.6)
16.1
22.8
26.0
(2.9)
(0.5)
(1.1)
18.8
35.2
41.9
(9.7)
25.1
18.4
22.8
(12.2)
(8.9)
(6.7)
14.0
23.3
0.32
(1.8)
(0.3)
20.9
18.3
13.9
(19.2)(20.5)(18.4)
15.5
21.2
18.6
(7.4)
26.2
23.6
23.1
(1.7)
(3.0)
(0.2)
19.7
44.8
21.2
(18.6)
(8.0)

41.7
25.8
(1.2)

4.69
23.0
(6.8)

41.7

Δθ

CrystalOrientation

19.9(8.3)

1.64

AtomicArrangement

14.3(21.2)

15.9
11.2(19.7)

Actual grain boundaryx

Periodic boundary conditions in all direction

19.3
(4.5)

35.2
29.5
(4.5)

(b)

Approximately
A: 14atoms (1)B: 42atoms (3)grain Ia : Distance between atoms

grain II

16.5
(17.1)

21.2
20.5
(1.5)

A: 50a (500 lattices)
B: 150a (1500lattices)y

Crystal orientationyθ IIϕ

Symmetric grain boundaryx(a)

θI

x(c)

(d)

Figure 1: (a) Nanopolycrystalline model with predeﬁned crystal orientations. (b) Deﬁnition of crystal
orientation. (c) A sample of phase ﬁeld proﬁle with θ = 15◦ . (d) Deﬁnition of deviation angle ψ.

α = −0.50. To use the same initial structure for α = −0.25 and −0.50 in the deformation
simulation, the structure is formed by solidiﬁcation simulation with α = −0.25 and following
the solidiﬁcation simulation, a relaxation simulation with α = −0.50 is performed to create theinitial structure for α = −0.50. The obtained initial structures for Model A and B are shownin Fig. 2(a). Tensile deformation simulations are performed under the following condition,d = 4.0 × 10−9 and Δt = 3.00 × 10−3 (α = −0.25), 2.55 × 10−3 (α = −0.50), which corresponds todeformation condition under realistic strain rates of approximately 100 s−1 to 101 s−1 . Periodic
boundary conditions are also adopted in all direction.
3.2

Numerical results and discussions

Fig. 2 shows deformation behaviors of nanopolycrystalline structure. In Model A (α =−0.25), as the strain increases, grain coalescence by grain rotation occur and grain growth resultsfrom curvature-driven GB motions. On the other hand, in Model A (α = −0.50), although grainrotation and GB migration are also observed, they aren’t remarkable comparing to the case of
α = −0.25 due to lower temperature. Alternatively, the migrations of dislocations comprisingof low-angle GB contribute to the plastic deformation (Circle A in Fig. 2(b)). In Model B(α = −0.25), due to larger grain size, grain rotation becomes harder to occur. By contrast,
GB migration becomes dominant deformation mechanism. These GB migrations are mainly
observed at GBs that aren’t parallel to the tensile direction. Interestingly, symmetric low-angle
GBs (ψ < 5◦ ) displace greatly (GB B in Fig. 2(c)), whereas high-angle GBs migrate hardly. InRef.[4], under high temperature, these low-angle GB motions are induced by the shear stress andhigh-angle GB motions are interrupted by GB sliding occurred during deformation. We can’t,however, reveal clearly the detailed mechanism. In Model B (α = −0.50), more dislocations moveinto grains from low-angle GBs and the above GB migrations are also observed. Comparing theresults of Model A with B, we can consider that grain boundary property aﬀects strongly on
deformation behaviors of the nanopolycrystalline structure with larger grain size. Thus, we
can conﬁrm that the PFC deformation simulation with our scheme can reproduce deformation
behaviors of nanopolycrystalline structure as shown in experiments and MD simulations.

3T. Hirouchi, T. Takaki, Y. Shibutani and Y. Tomita

Model A ( α = -0.25)
0.041

0.021

Free energy

Model A ( α = -0.50)
0.012-0.016

Free energy

A

Model B ( α = -0.25)

B

0.041

0.021

Free energy

Model B ( α = -0.50)
0.012-0.016

Free energy(a) ε = 0.000(b) ε = 0.033(c) ε = 0.066(d) ε = 0.100

Figure 2: Deformation behaviors in nanopolycrystalline structure.

4

CONCLUSIONS

We conducted the PFC deformation simulations of nanopolycrystalline structure under different temperature and grain-size conditions. As a result, we could conﬁrm that by loweringtemperature and increasing grain size, grain rotation and GB migration became harder to occur and behaviors of dislocation and GB changed. Moreover, these deformation behaviors werein qualitative agreement with the results in experiments and MD simulations. Thus, we concluded that the PFC deformation simulation with our scheme could have a potential of solving
deformation problems depending on temperature and grain size.
REFERENCES[1] M.A. Meyers, A. Mishra and D.J. Benson. Mechanical properties of nanocrystalline materials. Prog. Mater. Sci., 51, 427–556, (2006).
[2] K.R. Elder and M. Grant. Modeling elastic and plastic deformations in nonequilibrium
processing using phase ﬁeld crystals. Phys. Rev. E, 70, 051605-1–051605-18, (2004).
[3] T. Hirouchi, T. Takaki and Y. Tomita. Development of numerical scheme for phase ﬁeld
crystal deformation simulation. Comp. Mater. Sci., 44, 1192–1197, (2009).
[4] J.W. Cahn, Y. Mishin and A. Suzuki. Coupling grain boundary motion to shear deformation. Acta Mater., 54, 4953–4975, (2006).
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