COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Molecular Dynamics Simulation of Size Effect on Mechanical Properties
of Nano-Metals
D. Huang*, J. S. Zhuo
Department of Engineering Mechanics, Hohai University, Nanjing, 210098 China
Email:[email protected]
Abstract Based on molecular dynamics method, a numerical simulation scheme for analysis of mechanical properties
and behavior of nano-scale metals was proposed, in which the mutli-body potential derived from embedded atomic
method was used to describe the interatomic actions. With the present model and simulation method, the failure
process and mechanical properties of nanocrystalline nickel wires with different cross section sizes were studied.
Simulation reveals the relationships between average atomic stress, energy and crystal lattice strain in nano-scaled
metallic wires under static external load. Numerical results show that initial unstable energy, as a result of free surfaces
and surface reconstruction, takes large effects on the deformation and failure mechanism of nano wires. Energy in
exterior atoms is much higher than that in the inner ones, so that exterior atoms depart from ideal crystal lattice
positions and nano voids come into being along the surfaces firstly. The deformation process of nano wire behaves as
the expansion and connection of nano cavities from surface into inner lattices. Elastic modulus, yield and fracture
strength on definite lattice directions of nickel nano wires with different cross section sizes were obtained, and the
length size effect on mechanical properties was analyzed further. A fitted formula on elastic modulus, yield strength
and fracture strength of nickel nano wires was proposed based on numerical results, in which the elastic modulus and
yield strength of metal nano wire are both linear to the logarithm of cross section length size, but fracture strength has
an inverse relation to exponential cross section size. The formulae can predict the elastic modulus, yield strength and
fracture strength of a nanocrystalline nickel wire according to its cross section size, describe the correlativity between
mechanical properties of metallic nano wire and cross section size reasonably, and keep remarkable accordance with
simulation and experimental results.
Key words: nano metals, molecular dynamics, size effect, mechanical properties
INTRODUCTION
Much attention has been paid to nanocrystalline materials with a grain size ranging from 1 to 100 nm, because they
may provide unique mechanical, physical and chemical properties when compared with polycrystalline conventional
materials [1-3]. The studies on the mechanical properties of nanocrystalline materials, such as elasticity, strength,
hardness, ultra-plasticity, deformation, abnormal Hall-Petch relation, and so on, have been pursued from different
perspectives[4, 5]. It has been shown that nanocrystalline metals exhibit a relatively large increase in strength, keeping
accordance with the conventional technology of “minishing grain size” for larger strength. Both the theory on
mechanics of continuous medium and theory based on dislocation and microstructure evolution can not reasonably
explain the mechanism that how grain size affects the mechanical properties of materials. Different elastic properties
of many nanostructured materials have also been measured. However, because of the difficulty on nano-scaled
mechanical loading and experimental standardization, a lot of mechanical properties and behavior can not be observed
or measured easily, and detailed mechanical rules and mechanism are still not well understood. It makes nano-scaled
numerical simulation an important approach to study the mechanical properties and behavior of nanocrystals, in which
molecular dynamics simulations have become prominent as a tool for elucidating complex physical phenomena
because of the availability of accurate interatomic potentials for a range of nanocrystalline materials [6], especially
metals.
To provide insight into the grain size effect on mechanical properties of nanocrystalline metals, different experiments
and lots of simulations were performed and many useful conclusions have been proposed. Molecular dynamics
⎯ 839 ⎯
simulations found that in static uni-axial tension process, mechanical properties especially the strength of
nanocrystalline metals are much relative to boundary condition, loading direction, grain size and deformation rate [7].
In shearing deformation process, the shearing strength of nanocrystalline nickel will decrease when grain size turns
larger [8]. The tension strength of copper nano wire is also found to decrease when grain size increases [9].
Nevertheless, the essential mechanism that how grain size affects the mechanical properties of nanostructured
materials as well as conventional polycrystalline materials, and the numerical formula that can describe the
correlativity between mechanical properties and grain size are still unknown. It is suggested that nanocrystalline
materials may consist of three components: grain boundary, triple junction and grain interior. Shen [10] attempted to
gain the elastic properties of nanomaterials by considering the volume fractions of three different components.
Similarly, Broughton [11] divided the Young’s modulus of a single nano grain into three components: exterior, interior
and corner lattices and calculated the grain Young’s modulus by linearly summing up the contribution of three parts.
All these attempts made great efforts on size effect of nanomaterials from different point of view.
In the present work, more insight was provided into the size effect on mechanical properties of nano metals. Based on
molecular dynamics method, a numerical simulation scheme for analysis of mechanical properties and behavior of
nano-scale metals was proposed, in which the mutli-body potential derived from embedded atomic method was used to
describe the interatomic actions. The failure process and mechanical properties of nanocrystalline nickel wires with
different cross section sizes were studied applying the present scheme. Simulation reveals the relationships between
average atomic stress, energy and crystal lattice strain in nano-scaled metal wires under static external load. Numerical
results show that initial unstable energy, as a result of free surfaces and surface reconstruction, takes large effects on
the deformation and failure mechanism of nano wires. Energy in exterior atoms is much higher than that in the inner
ones, so that exterior atoms depart from standard crystal lattice positions and nano voids come into being along the
surfaces firstly. The deformation process of nano wire behaves as the expansion and connection of nano cavities from
surface into inner lattices. Elastic modulus, yield and fracture strength on definite lattice directions of nickel nano
wires with different cross section sizes were obtained, and the length size effect on mechanical properties was analyzed
further. A prediction formula describing the correlativity between a certain mechanical property (for example, elastic
modulus, yield strength or fracture strength) and grain size were proposed and testified.
MOLECULAR DYNAMICS SIMULATION
Classical molecular dynamics (MD) method was used to calculate the average mechanical properties of atomic
system----nanocrystalline nickel wire. In MD simulation, the selection of interatomic potential for the system is crucial
in obtaining accurate results. The modified embedded atom method [12] is validated accurate enough in describing the
interaction in metallic crystal lattices. The multi-body potential of nickel derived from embedded atom method with
parameters tested by Voter [13] was used here and the cut-off radius of the potential function was defined as 3 times of
the nickel lattice constant, and sudden truncation was adopted, i.e. the current atom just interacts with its neighbors that
in 3 times of lattice constant (which is 0.352nm) distance. Fig.1 shows an atomistic model of nickel nano wire, the
lattice direction [100], [010] and [001] is corresponding to x , y and z axis of Descartes coordinates respectively.
Surfaces vertical to x and y directions keep free, having no neighbor atoms outside, but the surfaces vertical to
z direction are controlled repeating by periodic boundary conditions (PBC). Thus the model behaves as a nano
wire----one dimensional nanocrystalline material, and free surfaces are unrestrictive and deformable independently
under the control of Parrinello-Rahman uniform stress method [14]. In order to getting rid of the efforts of heat
activation on the motion of atoms, the system temperature keeps 0 K at all times, and a Noose-Hoover external bath
[15] is coupled with the molecular system, thereby allowing heat exchange between the system and the external bath so
as to offsetting the change of system temperature resulted from periodic boundary conditions.
With the geometrical structure in Fig. 1, we established seven models with different section sizes and simulation cells:
Model (a): 3×3×20, totally 720 atoms, cross section size d = 3a0 =1.0572nm; model (b): 5×5×20, 2000 atoms,
d = 5a0 =1.762nm; model (c): 8×8×20, 5120 atoms, d = 8a0 =2.8192nm; model (d): 10×10×20, 8000 atoms,
d = 10a0 =3.524nm; model (e): 15×15×20, 18000 atoms, d = 15a0 =5.286nm; model (f): 18×18×20, 25920 atoms,
d = 18a0 =6.3432nm; model (g): 20×20×20, 32000atoms, d = 20a0 =7.048nm; in which a0 =0.352 nm, is the lattice
constant of crystalline nickel. The strain rate is slow enough to keep static loading and all the same to each model. The
nickel nano wire model was firstly relaxed for a long time to get rid of the efforts of exterior re-construction, and
then loaded by uni-axial tension along [001] direction. A constant time step, 4 fs, is chosen for simplicity of the
algorithm. It is small enough to keep the reality of the trajectory of each atom, and large enough to keep the efficiency
of simulation. In the simulation process, system structure, arrangement of atoms on the cross section, location of
marked atoms, system energy and average stress were outputted every 100 time steps.
⎯ 840 ⎯
Figure 1: Atomistic model of nickel nano wire
RESULTS AND DISCUSSION
Seven models with different cross section sizes were simulated with the present method. We can obtain the mechanical
properties of the nickel nano wires and discuss their relativity to grain sizes according to the output results of numerical
simulations.
1. Energy and stress-strain curves Fig.2 and Fig.3 show the variation of average atomic energy and axial stress in the
tension process respectively. Energy shows the mechanical state atoms reached, and the stress-strain curve describes
the basal mechanical properties of materials. From the figures we can see that the energy-strain and stress-strain curves
of nana wires with different section sizes have the same trend, and the stress-strain curves look similar to the tension
curve of polycrystalline conventional metal.
In the whole tension process, the average energy of nickel nana wire with larger cross section size is always higher
under the same loading conditions (see Fig. 2). This is obviously related to the exterior atoms. In the initial relaxation
phase, exterior atoms will depart from their ideal lattice location because they have no neighbors outside that can
balance the actions from interior atoms. Thus the initial energy of the physical model will higher than that of ideal
lattices, and the average energy will be higher if the ratio of exterior atoms is higher.
25
-6.0
B
D
A
-6.4
20
E
15
B
Stress (GPa)
-19
Total Energy ( 10 J)
-6.2
D
C
-6.6
E
C
F
10
G
A
F
5
-6.8
G
-7.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0
0.00
0.35
Strain
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Strain
Figure 2: Energy-strain curves in the tension process
Figure 3: Stress-strain curves in the tension process
During the initial loading period ( ε <0.11), stress increases linearly when strain turns larger, like the elastic period in
tension of polycrystalline metals, and the stress grads is higher in larger-sized grain. When strain rate reaches 0.11,
there is a stress platform just like the yielding phase of conventional metals. When strain is larger than 0.12, stress
starts to increase again, and the trend looks like the harden phase of macro metals. Compared with the tension curve of
conventional metal, we can correspondingly obtain the Young’s modulus E (the stress gradient in the linearly
increasing phase), yield strength σ s (the stress after which a platform appears on the curve) and fracture strength
σ b (the highest point of the stress curve) of nickel nana wire from the stress-strain curves. We can also find that the
yield strain is independent of cross section size, remaining ε s =0.11 constantly.
What’s more, it is shown in the stress-strain curves that yield strength and Young’s modulus of nano wire will increase
when cross section size turns larger, but the fracture strength will decrease reversely. In addition, the proportion of
length size to section size in the simulation cell also works on the numerical results, for example, the results of model
g is not so accordant with others.
With the simulation output and the stress-strain curves, mechanical properties of model a-g are listed as follows:
⎯ 841 ⎯
Table 1 Mechanical properties of nano wires with different cross section sizes
Model
D（ a0 ）
σ s （GPa）
σ b （GPa）
E（GPa）
a
3
9.2224
24.896
76.1375
b
5
12.248
23.764
107.52
c
8
13.829
22.675
125.838
d
10
14.451
22.293
135.148
e
15
15.319
21.814
147.957
f
18
13.491
19.423
127.53
g
20
13.674
19.971
130
2.2 Size effect analysis According to the data listed in table 1, formulae describing the correlativity between a
certain mechanical property and cross section size may be obtained through mathematical fitting. Fig. 4 and 5 show the
comparison between fitted curves and simulation results. It seems that the fitted function agree well with numerical
results. Furthermore, the fitted curve and function about size effect on fracture strength of nano wire is similar to the
inversely exponential relativity proposed by Liang [9].
16
Young's Modulus
140
Yielding Strength
130
Young's Modulus (GPa)
Yielding Strength (GPa)
14
120
110
12
100
10
90
80
8
2
4
6
8
10
12
14
16
70
Cross Section Size (a0)
Figure 4: Size effect on yield strength and Young’s modulus of nano wire
25.0
Numerical Results
Fitted Curve
Fracture Strength (GPa)
24.5
24.0
23.5
23.0
22.5
22.0
21.5
2
4
6
8
10
12
14
16
Cross Section Size (a0)
Figure 5: Size effect on fracture strength of nano wire
Following is the fitted formula corresponding to the curves in Fig. 4 and Fig. 5, in which the units for mechanical
properties and cross section size are GPa and a0 ( the lattice constant) respectively.
σ s = 3.7584 ln D + 5.6488, σ b = 6.39e− D / 4.6614 + 21.5493, E = 44.3129 ln D + 31.6835
(1)
Based on the simulation results, fitted curves and equations above, it is suggested that both the yield strength and
Young’s modulus are logarithmic, yet fracture strength is inversely exponential to the cross section size of nickel nano
wires. We can thereby conclude the correlativity between the mechanical properties of nano wire and its cross section
size. However, how to fix on the parameters in the formulae, and what are the key factors impacting the parameters, are
still looking forward to further study.
In equation (1), the minimum value of yield strength and Young’s modulus are 5.6488 GPa and 31.6835 GPa
respectively, when D= a0 , i.e. when there is only one lattice on the cross section----which is out of the question in fact,
⎯ 842 ⎯
because most atoms will be free and move without any restriction. The minimum value of fracture strength is 21.5493
GPa, when section size goes to infinite as an ideal crystal. This value is just close to the theoretical strength of a nickel
lattice [10], keeping accordance with Griffith’s fracture criterion of ideal crystals. On the other hand, the fracture
strength of a nickel nano wire will not higher than 27.94 GPa (when D = 0, which is also impossible).
CONCLUSIONS
In this paper, we have provided a generic molecular dynamics modelling of nanocrystalline metal, and applied it to the
simulation of axial tension process of nickel nano wires. Simulation shows that exterior atoms take large effects on the
mechanical behavior and properties of nanostructured materials. The average energy in nano wires with larger
exterior-lattice ration is higher. The stress-strain curves of metallic nano wire are similar to that of the polycrystalline
conventional metals, and can be divided into three parts: a linearly increasing phase, a short yielding phase, and a
harden phase. The yield strain of metallic nano wire is independent of cross section size, for example, the yield strain
of all the nickel nano wires in the present work remains 0.11.
Mechanical properties including yield strength, fracture strength and Young’s modulus of nickel nano wires with
different cross section sizes were obtained and compared here. Numerical analysis shows that both yield strength and
Young’s modulus of a nano wire take a logarithmic correlativity to cross section size, and fracture strength is inversely
exponential versus section size yet. Compared with experiments and early simulation results, it is shown that the
present scheme is appropriate for nanocrystalline metals, and the fitted prediction formula works perfectly on
revealing the correlativity between mechanical properties of metallic nano wires and their cross section size. Finally,
the further study on analysis of parameters in the proposed prediction formulae is still significant and promising.
Acknowledgements
The support of the National Natural Science Foundation under Grant No. 10572125 and the Start-up Foundation of
Hohai University is gratefully acknowledged.
REFERENCES
1. Kong J, Franklin NR, Zhou C et al. Nanotube molecular wires as chemical sensors. Science, 2000; 287: 622-625.
2. Yu MF, Lourie O, Dyer MJ et al. Strength and breaking mechanism of multiwalled carbon nanotubes under
tensile load. Science, 2000; 287: 637-640.
3. Gao H, Huang Y, Abraham FF. Continuum and atomistic studies of intersonic crack propagation. Journal of the
Mechanics and Physics of Solids, 2001; 49: 2113-2132.
4. Ma X, Yang W. MD simulation for nanocrystals. Acta Mechanica Sinica, 2003; 19(6): 485-507.
5. Lu L, Shen Y, Chen X et al. Ultrahigh strength and high electric conductivity in copper. Science, 2004; 304:
422-426.
6. Liu WK, Karpov EG, Zhang S et al. An introduction to computational nanomechanics and materials. Computer
Methods in Applied Mechanics and Engineering, 2004; 193: 1529-1578.
7. Hu SY, Ludwig M, Kizler P et al. Atomistic simulations of deformation and fracture of α -Fe. Modelling
Simulation in Material Science and Engineering, 1998; 6: 567-586.
8. Horstemeyer MF, Baskes MI. Atomistic finite deformation simulation: a discussion on length scale effects in
relation to mechanical stresses. Journal of Engineering Materials and Technology, T-ASME, 1999; 121:
114-119.
9. Liang HY, Wang XX, Wu HA et al. Molecular dynamics simulation of length scale effects on tension nano
crystalline copper wire. Acta Mechanica Sinica, 2002; 34(2): 208-215.
10. Shen TD, Koch CC, Tsui TY et al. On the elastic moduli of nanocrystalline Fe, Cu, Ni and Cu-Ni alloys prepared
by mechanical milling/alloying. Journal of Materials Research, 1995; 10(11): 2892-2896.
11. Broughton JQ, Meli CA, Vashishta P et al. Direct atomistic simulation of quartz crystal oscillators: bulk
properties and nanoscale devices. Physical Review B, 1997; 56(2): 611-618.
12. Baskes MI. Application of the embedded-atom method to covalent materials: a semiempirical potential for
silicon. Physical Review Letters, 1987; 59: 2666-2669.
13. Voter AF, Chen SP. Accurate interatomic potentials for Ni, Al, and Ni3Al. Materials Research Society
⎯ 843 ⎯
Symposium Proceeding, 1987; 82: 175-182.
14. Parrinello M, Rahman A. Polymorphic transitions in single crystals: a new molecular dynamics method. Journal
of Applied Physics, 1981; 52(12): 7182-7190.
15. Hoover WG. Molecular dynamics. Springer-Verlag, Berlin, Germany, 1986.
⎯ 844 ⎯