Journal of Nano Reserach Vol. 1 (2008) pp. 31-39
online at
© (2008) Trans Tech Publications, Switzerland
Molecular Dynamics Simulations of Nanoparticle-Surface
Collisions in Crystalline Silicon
Paolo Valentini and Traian Dumitricaa
Department of Mechanical Engineering, Institute of Technology,
University of Minnesota, Minneapolis 55455, USA
a (corresponding author)
Received February 13, 2007; received in revised form April 19, 2007; accepted May 15, 2007
Keywords: molecular dynamics; nanoparticle; silicon; impact.
Abstract. We present a microscopic description for the impacting process of silicon nanospheres
onto a silicon substrate. In spite of the relatively low energy regime considered (up to 1 eV/atom),
the impacting process exhibits a rich behavior: A rigid Hertzian model is valid for speeds below 500
m/s, while a quasi-ellipsoidal deformation regime emerges at larger speeds. Furthermore, for speeds
up to 1000 m/s the particle undergoes a soft landing and creates a long-lived coherent surface
phonon. Higher speeds lead to a rapid attenuation of the coherent phonon due to a partial diamond
cubic to β-tin phase transformation occurring in the particle.
I. Introduction
Often motivated by technological applications, molecular dynamics (MD) studies of nanoparticlesurface collisions are of considerable help for understanding the fundamental physics of the impact
process [1-5]. For example, the formation of thin film through energetic cluster impact [2] has
stimulated several MD studies [3-5] focusing on small clusters (containing a few thousand atoms)
impacting with KEs of ~10 eV/atom. Systems studied have included Mo clusters impacting on Mo
(001) surfaces [3], Cu clusters on Cu (001) surfaces [4], and C60 molecules on graphite and Si
surfaces [5]. It was found that energetic impact produces impressive structural disruptions. For
instance a Mo cluster impacting with a KE of 10 eV/atom ends up being buried deep below the first
surface layer, forming a shallow crater [3].
New findings often come by exploring new regimes. Here we are concerned with impact of
relatively large nanoparticles (5 nm in radius) at relatively low KEs (up to 1 eV/atom). Such
conditions are experimentally achieved in hypersonic plasma particle deposition [6-8] and are of
interest for manufacturing microstructures with nanoparticle sprays. It was experimentally noted
that the deposited structures largely retain the grain size of the impacting nanoparticles. An
understanding of the impact process is needed to further comprehend the mechanical properties of
the resulting nanoparticulate. We provide a detailed microscopic description of the impact and show
that in spite of the low impacting KEs the collision process exhibits a surprisingly rich behavior.
The material of interest here is Si, which was the focus of several experimental studies [6,7]. We
are focusing on the understanding of Si particle adhesion and energy transfer mechanisms. The
relatively large size of the impacting particle makes it meaningful to inquire whether aspects
pertinent to macroscopic impact, such as the contact mechanics and plastic deformations, are
significant at the nanoscale.
The classical theory of collision was developed by Hertz [9]. In his model, the colliding body is
described in the lumped approximation of a perfect rigid frontally equipped with a spring-damper
system. The elastic deformation upon hertzian impact is local and it is captured by the springdamper elements, while the rigid part moves with the velocity of the center of mass. It is interesting
to see the extent to which the venerable Hertz model is relevant at the nanoscale.
Concerning the transfer mechanism of the incident KE, studies [10,11] on the impact of micronsized particles have attributed it to the occurrence of a plastic flow in the particle that has the effect
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of reducing the contact pressure. Recent nanoindentation data [12] confirmed that silicon
nanoparticles can accommodate plastic deformations. Thus, following conventional wisdom, one
conjunctures that plastic yield may occur upon impact, in agreement with macroscopic predictions
[9]. However, one concern is that hertzian theory predicts that the duration of the impact decreases
with the particle radius r0 as r06/5 [13]. Thus, the resulting strain rate experienced by the
nanoparticle could be too high for inducing plastic deformations.
This paper is organized as follows: Section II gives a description of the employed MD scheme.
Section III presents the obtained numerical results, which are summarized in Section IV.
II. Methodology
We conducted detailed atomistic simulations to describe the nonadiabatic process of the impact of
Si nanoparticles on (001) Si surfaces. The method of investigation is classical MD, where the
motion of the Si cores is described with the environment-dependent potential proposed by Tersoff
[14], already incorporated in the computational package Trocadero [15]. The classical treatment of
Si covalent bonding allows simulating rather substantial systems. Here we are dealing with
spherical Si particles of radius r0=5 nm and containing Np=31,075 atoms, and a Si substrate with
dimensions 16.3 nm x 16.3 nm x 5.4 nm containing 72,000 atoms. An O(N) linked-cell list strategy
to search for the interacting atoms was implemented and the system was evolved in time with a 2 fs
time step with a velocity Verlet algorithm.
The dynamical treatment of the finite-size substrate requires special care. We employed artificial
absorbing boundaries [3,16] outside the region of interest. During the impact simulation the two
bottom layers (3,600 atoms) of the substrate were kept fixed in time, the next ten layers (18,000
atoms) were evolved with a damped Langevin dynamics [3] with a the friction coefficient γ=πωD/6,
where ωD is the Debye frequency for silicon, while the rest of the substrate is treated with standard
MD dynamics. Periodic boundary conditions are applied to the horizontal x and y directions (z
impact direction is vertical). In preparation for the simulations, one nanocluster has been spherically
cut from bulk Si, annealed at Tp=1000 K or Tp=300 K and optimized with a conjugate gradient
algorithm. A Si substrate has been cut from bulk along the (001) direction and a similar
annealing/optimization procedure has been applied. The procedure leads to the expected (001) Si
surface reconstruction by dimerization.
We attempted to choose simulation parameters (impact velocities, particle and substrate
temperatures) and measure quantities that are relevant for the impact experiments of Rao et al.
[6,7]. In a series of simulations, the speed of the 5 nm incident nanoparticle was taken in the range
of 550-1640 m/s. To explore the role of temperature we carried several simulations on combinations
of the cold (Tp=300 K) and hot (Tp=1000 K) particle impacting on a cold (Ts=300 K) or hot
(Ts=800 K) substrate. For a better understanding of the process we found it necessary to conduct
several simulations outside this main velocity-temperature range.
All simulations were conducted for 40 ps and were followed by structural optimizations
performed with a conjugate gradient scheme. During the simulation we have monitored a standard
set of indicators such as instantaneous particle temperature (proportional to instantaneous kinetic
energy), the pair correlation function (essentially a density-density correlation function), the particle
cohesive energy Ep, and the particle-surface adhesion Ea. When pieced together, these indicators
should converge to provide a microscopic picture for the dynamics of the impact.
III. Simulation Results
In all our simulations the impacting particles remained attached to the surface. Fig. 1 depicts, for
two incident speeds, three representative stages from the impact process: the initially undistorted
structure, the instant of maximum penetration into the substrate, and the structure after 20 ps. The
selected utmost impacting conditions vividly illustrate that even in this relatively low-energy impact
regime there is a strong qualitative dependence on the impact speed: At 550 m/s the particle
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penetrates very little into the room-temperature substrate and overall suffers little structural damage
following the impact. After the maximum penetration point, on the rebound stage, a small junction
with a neck is formed at the particle-surface interface, as it can be observed in the last sequence. On
the other hand, at an incident speed of 1640 m/s, the particle shows evident signs of structural
changes even at the maximum penetration point. The spherical shape is partially lost in favor of a
quasi-ellipsoidal shape, and a junction with a large interfacial contact surface is formed. In both
cases, the substrate appears compliant and does not show signs of significant structural deformation
or any degree of implantation with atoms from the particle.
The Contact Area and Adhesion Energy. An interesting aspect about the dynamics of the
collision process transpires from Fig. 2(a), which displays the time evolution of the contact radius,
a [17]. As the particle impinges onto the substrate, a increases up to the maximum penetration point
instant marked by arrows. Next, for the
lowest impinging speed case, a
practically saturates at the value attained
at the maximum penetration distance into
substrate. The higher impact velocity
leads to a less quasi-static behavior, as
after the maximum penetration point the
contact area slightly decreases on the
0 ps
20 ps
7.5 ps
dynamical result suggests that the contact
area is largely determined by the incident
speed and the mechanical properties of
both the particle and substrate, and to a
lesser extent by the subsequent events
that follow the maximum penetration.
0 ps
3 ps
20 ps
A broader view about the role of the
impacting velocity emerges from Fig.
Fig. 1. Three instants of the deposition of a 5-nm Si
2(b), which plots the final contact radius
particle under (a) slow-cold, at vp = 550 m/s and Tp = a as a function of the incident speed
Ts = 300 K, and (b) fast-hot conditions, at vp = 1640
under room temperature conditions. To
m/s and Tp = 1000 K, Ts = 800 K. The middle instant
elucidate the speed dependence it was
corresponds to a maximum penetration.
rewarding to perform additional low KE
simulations, outside our main vp range of interest. After a careful analysis we concluded that for
speeds below 500 m/s the impact largely follows the hertzian predictions: The measured contact
1640 m/s
550 m/s
Fig. 2. (a) Time-dependence of the contact radius a for the two simulations of Fig. 1.
Down arrows mark the maximum penetration instant. (b) Final a values under
different vp and room-temperature conditions.
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Fig. 3. (a) Final particle-substrate adhesion Ea and (b) particle cohesive Ep energies under
different vp and various temperature conditions.
radii (empty circles) fits into the typical vp2/5 dependence, which assumes a rather localized particle
deformation at the maximum penetration point. (The free term correction reflects the atomistic
aspect of the contact of nonconforming shapes.). Additionally, the dynamically measured a values
are very close to the predictions (shown with empty triangles) made with the formula [9] a=π
pmr0/2Y, where pm represents the mean contact stress at the maximum penetration and Y=65 GPa is
the computed [18] lumped particle-substrate modulus. A qualitative scaling change can be seen in
the a data above the 500 m/s, where the hertzian predictions become less valid. The better
agreement with the stronger power vp1/2 scaling suggests the impact enters into an extended
deformation regime. As noticed in Fig. 1(b) the particle exhibits a quasi-ellipsoidal deformation at
the maximum penetration point, similar with the experimental observation on the macroscopic
impact of gel balls [19].
In discussing the energetics of the bound system Es-p it is instructive to distinguish between the
particle-substrate adhesion energy Ea and the individual cohesive energies. For this propose,
following the concluding relaxation we have artificially detached the particle and separately
computed the particle Ep and substrate Es cohesive energies. Thus, Es-p = Ea + Ep+ Es. The
energetically favorable aspect of the deposition process is portrayed in the linear vp trend of the
obtained adhesion energy Ea, as presented in Fig. 3(a). A higher impact speed causes a larger
contact surface, thus “eliminating” the less energetically favorable surface. We notice that the
thermal conditions are secondary and Ea is mainly determined by the incident kinetic energy.
Collision Modes. We next turn to a detailed analysis of the dynamics of the impact, aiming to
identify characteristic collision modes. Particularly useful for this propose is the Ep data presented
in Fig. 3(b), where a threshold-like dependence on the particle speed can be noted. For collision
speeds of up to ~1000 m/s, the Ep variation with the nanoparticle speed exhibits a plateau. Beyond
the ~1000 m/s value there is a rapid Ep increase that clearly indicates that the nanoparticle suffered
a significant structural change after landing. Although a clear-cut critical speed value was not
identified, our data suggest that function of the incident speed there are two rather distinct regimes
of impact, having very different consequences on the final structure of the nanoparticle. Again, the
thermal conditions appear to influence the impact outcome very little.
To better understand these two distinct collision modes we have concentrated on the dynamical
features exhibited by two extreme cases, namely a slow-cold impact at vp=550 m/s with Tp=Ts=300
K, and a fast-hot impact vp=1640 m/s with Tp=1000 K and Ts=800 K. To gain insight about the
transfer of the incident kinetic energy we have plotted in Fig. 4 the instantaneous speed of the
center of mass of the nanosphere. Fig. 4a, referring to the 550 m/s impaction, indicates the creation
of a transversal coherent phonon mode with a small dephasing time. The obtained data calls for
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Fig. 4. Particle velocity vp as a function of time. The initial impacting velocities
were (a) 550 m/s and (b) 1640 m/s. Arrows mark the maximum penetration instant.
modeling the resulting oscillatory motion as an under-damped second-order system, i.e., vp(t) ~
exp(-ζ ωn t) cos(ωn(1- ζ2)1/2 t + φ). Here ωn is the system's natural frequency, φ a phase factor, and ζ
is the dephasing parameter. Fitting to the atomistic data we obtained ζ =0.04. Thus, at this speed
most of the incident KE is transferred to the coherent transversal vibrations of the particle. Fig. 4b
corresponding to an 1640 m/s incidence, indicates the creation of the same type of mechanical
surface oscillations but the energy dissipation is much more efficient. This is reflected by the
increase in the dephasing parameter, which now becomes ζ =0.37.
We have further characterized the dynamics of the collision by monitoring the instantaneous
particle temperature Tp in Fig. 5 and the spatial T distribution across the system, in Fig. 6. Overall,
in the low energy regime Tp is simply enhanced to about 400 K. The spatial decomposition of Fig.
6a reveals that at the touchdown instant only small portion of the interface material reaches a
temperature of about 550 K. Next, Tp is seen to rise rather uniformly, meaning that part of the
initial incident kinetic energy is converted into thermal energy that moderately heats up the system.
However, a very different dependence is observed in the fast impact, Figs. 5 and 6b. After the
touchdown instant there is a sharp Tp rise to 1450 K. This is followed, during the next few ps, by an
ample Tp increase up to a maximum of 1600 K. For
the remaining 30 ps of the simulation Tp slowly
drops, as the system is driven towards equilibrium by
the Langevin dynamics of the substrate. Fig. 6b
1640 m/s
clearly indicates that the early global Tp peak can be
attributed to the localized increase of temperature at
the interface, which can reach values up to 2400 K. A
distinguished behavior is seen during the next few ps:
550 m/s
Instead of a homogenization, the 2500 K front
expands into the nanoparticle only, specifically in the
conical region corresponding to where the
amorphization is found at the end of the simulation as
Fig. 5. Time evolution of particle
seen in Fig. 1(b). As in the slow impact case, a rather
temperature Tp for two vp .
uniform T distribution can be seen towards the end of
the simulation.
Particularly relevant are the results for the time evolution for the contact stress pm, shown in Fig.
7. Again, there are obvious differences in the two impact cases: While pm exhibits large oscillations
for the slow impact, in the fast collision the first pressure peak is quickly dampened and, shortly
after the impact, the stress at the interface becomes zero. In both cases, very large pressures are
reached, around 13 GPa for the slow-elastic regime, and around 18 GPa for the fast-damped one.
Such stresses are amply within the range of pressures capable of determining a phase transition in
silicon, i.e., 10-15 GPa for the cubic diamond β-tin phase transformation [20,21].
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Microscopic Mechanism for Inelastic Collision. What is the microscopic mechanism causing the
above-threshold behavior? To answer this question we have analyzed in detail the structural
dynamics of our system. We found that the extreme pressures reached at the nanoparticle-substrate
interface provide the explanation
for why such a localized
increase in temperature takes
place only in the nanosphere,
leaving the substrate almost
unaffected. Specifically, the
nanosphere responds to the high
during the initial compression
stage of the impact with creation
of a new β-tin phase in the most
strained regions. This way, a
large amount of the elastic
energy accumulated during
compression is not returned as
elastic energy for the particle to
relaxation is accompanied by a
KE release, as noted in Fig. 5.
To better illustrate this
mechanism we have performed
an additional simulation with the
Fig. 6. Distributed instantaneous T. The impact parameters
particle impacting with 2200 m/s
in (a) and (b) correspond to those of Fig. 1 (a) and (b),
under very low temperature
conditions (10K for both the
particle and the substrate). The evidence for a dynamical structural transition is presented in Fig. 8,
where the fraction of nanoparticle's atoms having 4, 5, 6, and 7 nearest neighbors of the
nanoparticle is plotted for the first 20 ps of the impact. Almost 90% of the atoms are initially 4-fold
coordinated, as expected since most of the particle has a cubic diamond structure; however, during
the impact, the fraction of atoms having a higher coordination number (up to 7) increases rapidly,
and at around 20 ps more than 40% of the particle is made up of atoms with a coordination number
between 5 and 7, with less than 15% of the atoms having only 4 nearest neighbors. Note that the βtin phase is characterized by a coordination number of 6.
Fig. 7. Time evolution of the average
contact stress pm.
Fig. 8. Time evolution of the number of
neighbors for impact at vp=2200 m/s.
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The precise evidence of an ephemeral β-tin phase is given in Fig. 9 showing the radial
distribution function g(r) obtained from the nanoparticle structure at t = 3.9 ps after a structural
optimization. In the insert we have also plotted the g(r) for the perfect infinite β-tin crystal
computed based on the lattice parameters of Ref. [22]. Aside from the first peak caused by the
residual original structure, located at roughly 2.35 Å (i.e., the nearest neighbor distance in the Si
cubic diamond lattice), the next spikes well correspond to those obtained from the perfect infinite βtin crystal.
Fig. 9. The radial distribution function g(r) for a 5 nm particle at the 3.9 ps
maximum penetration instant. The vertical lines correspond to the bulk β-tin phase.
As the simulation progressed, we have found that the rebounce stage produces structural
amorphization, leading to a further release of kinetic energy. In fact, hardly any trace of the β-tin
phase is found at the end of the simulation. The observed amorphization during the rapid upward
bouncing motion appears in agreement with the amorphization noted in bulk silicon during a rapid
retraction of the nanoindentor [23,24].
IV. Summary
We carried out MD simulations using the Tersoff potential to study the deposition process of a 5 nm
Si nanoparticle onto a (001) Si surface. Our goal was to gain insight into the particle adhesion and
the mechanism of dissipation in the KE range of below 1 eV/atom. All simulations clearly indicated
the deposition speed as the important parameter and the particle and/or substrate temperature
conditions as secondary.
We found it rewarding to examine the same aspects as in a macroscopic impact [9,19]. In all our
simulations the particle remained attached to the surface. We quantified the degree of adhesion
through the microscopic contact radius a, and revealed some important regularities: The contact
radius at the end of the simulation is equal with the contact radius initially attained at the maximum
penetration point. For velocities below 500 m/s, the obtained scaling a ~ vp2/5 is in agreement with
hertzian predictions. Above this speed the scaling law changes to a ~ vp1/2 as the incident particle
underwent extensive ellipsoidal changes during the impact. Next the particle-surface adhesion
energy showed a linear scaling Ea ~ vp.
Regarding the aspect of energy dissipation we identified two distinct regimes: For speeds below
~1000 m/s the impact is mainly elastic, and the nanoparticle structure and shape are well preserved.
The impacting particle produced a transversal coherent phonon mode with a small dephasing time.
For a more violent impact, above ~1000 m/s, there is an efficient transfer of the incident KE into the
heating of the particle. A distinguished nanoscale feature was revealed: in the inelastic collision the
KE transfer is not mediated by a plastic yield as in the macroscopic case but by an ephemeral cubic
diamond to β-tin phase transformation occurring in the nanoparticle during the initial stages of the
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Beside the fundamental aspect of describing at the microscopic level the impact in a new velocity
- nanosize regime, our study contributes towards the understanding of nanoparticulate materials,
especially by way of a hierarchical multiscale modeling anchored on the presented atomistic data.
We thank P.H. McMurry, S.L. Girshick, J.V.R. Heberlein, M. Cococcioni, and W.W. Gerberich for
useful discussions. This work was supported by NIRT-NSF CTS-0506748 and by the MRSEC-NSF
under Award Number DMR-0212302. Computations were carried out at the Minnesota
Supercomputing Institute.
C.L. Cleveland and U. Landman, Dynamics of cluster-surface collision, Science 257 (1992)
H. Haberland, Z. Insepov and M. Moseler, Thin film growth by energetic cluster impact
(ECI): comparison between experiment and molecular dynamics simulations, Z. Phys. D 26
(1993) 229-232.
H. Haberland, Z. Insepov and M. Moseler, Molecular-dynamics simulation of thin-film
growth by energetic cluster impact, Phys. Rev. B 51 (1995) 11061-11067.
M. Moseler, O. Rattunde, J. Nordiek and H. Haberland, On the origin of surface smoothing
by energetic cluster impact: molecular dynamics simulation and mesoscopic modeling,
Nucl. Inst. Meth. B 164-165 (2000) 522-530.
K. Beardmore, R. Smith and R. P. Webb, Energetic fullerene interactions with Si crystal
surfaces, Modelling Simul. Mater. Sci. Eng. 2 (1994) 313-328.
N.P. Rao, H.J. Lee, M. Kelkar, D.J. Hansen, J.V.R. Heberlein, P.H. McMurry and S.L.
Girshick, Nanostructured materials production by hypersonic plasma particle deposition,
Nanostruct. Mater. 9 (1997) 129-132.
N.P. Rao, N. Tymiak, J. Blum, A. Neuman, H.J. Lee, S.L. Girshick, P.H. McMurry and J.
Heberlein, Hypersonic plasma particle deposition of nanostructured silicon and silicon
carbide, J. Aerosol Sci. 29 (1998) 707-720.
F. Di Fonzo, A. Gidwani, M.H. Fan, D. Neumann, D.I. Iordanoglou, J.V.R. Heberlein, P.H.
McMurry, S.L. Girshick, N. Tymiak, W.W. Gerberich and N.P. Rao, Focused nanoparticlebeam deposition of patterned microstructures, Appl. Phys. Lett. 77 (2000) 910-912.
K.A. Johnson, Contact Mechanics, first ed., Cambridge, Cambridge, 1985.
L.N. Rogers and J. Reed, The adhesion of particles undergoing an elastic-plastic impact with
a surface, J. Phys. D 17 (1984) 677-689.
M. Xu and K. Willeke, Right-angle impaction and rebound of particles, J. Aerosol Sci. 24
(1993) 19-30.
W.W. Gerberich, W.M. Mook, C.R. Perrey, C.B. Carter, M.I. Baskes, R. Mukherjee, A.
Gidwani, J. Heberlein, P.H. McMurry and S.L. Girshick, Superhard silicon nanospheres, J.
Mech. Phys. Sol. 51 (2003) 979-992.
L.D. Landau and E.M. Lifshitz, Theory of Elasticity, first ed., Pergamon, Oxford, 1986.
J. Tersoff, New empirical approach for the structure and energy of covalent systems, Phys.
Rev. B 37 (1988) 6991-7000.
R. Rurali and E. Hernandez, Trocadero: a multiple-algorithm multiple-model atomistic
simulation program, Comp. Mat. Sci. 28 (2003) 85-106.
Journal of Nano Reserach Vol. 1
Y. Hu and S.B. Sinnott, Constant temperature molecular dynamics simulations of energetic
particle–solid collisions: comparison of temperature control methods, J. Comp. Phys. 200
(2004) 251-266.
We computed a based on the microscopic definition given in M. Vergeles, A. Maritan, J.
Koplik and J.R. Banavar, Adhesion of solids, Phys. Rev. E 56 (1997) 2626-2634.
P. Valentini and T. Dumitrica, Microscopic theory for nanoparticle-surface collisions in
crystalline silicon, Phys. Rev. B 75 (2007) 224106-9.
Y. Tanaka, Y. Yamazaki, and K. Okumura, Bouncing gel balls: Impact of soft gels onto
rigid surface, Eurphys. Lett. 63 (2003) 146-152.
K. Gaal-Nagy, P. Pavone and D. Strauch, Ab initio study of the beta-tin -> Imma -> sh
phase transitions in silicon and germanium, Phys. Rev. B 69 (2005) 134112-11.
M.I. McMahon and R.J. Nelmes, New high pressure phase of Si, Phys. Rev. B 47 (1993)
H. Balamane, T.Halicioglu and W.A. Tiller, Comparative study of silicon empirical
interatomic potentials, Phys. Rev. B 46 (1992) 2250-2279.
V. Domnich, Y. Gogotsi and S. Dub, Effect of phase transformations on the shape of the
unloading curve in the nanoindentation of silicon, Appl. Phys. Lett. 76 (2000) 2214-2216.
C.F. Sanz-Navarro, S.D. Kenny and R. Smith, Atomistic simulations of structural
transformations of silicon surfaces under nanoindentation, Nanotechnology 15 (2004) 692697.
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