DTIC ADA546658: Molecular Dynamics Simulation of the Kinetic Reaction between Ni and Al Nanoparticles

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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
“Molecular Dynamics Simulation of the Kinetic Reaction W911NF-09-1-0214
between Ni and Al Nanoparticles”
5b. GRANT NUMBER
5c. PROGRAM ELEMENT NUMBER
611102
6. AUTHORS 5d. PROJECT NUMBER
B. Henz, T. Hawa, and M.R. Zachariah
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University of Maryland - College Park
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The views, opinions and/or findings contained in this report are those of the author(s) and should not contrued as an official Department
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14. ABSTRACT
Molecular dynamics simulations are used to simulate the energetic reaction of Ni and Al particles at the nanometer
scale. The effect of particle size on reaction time and temperature for separate nanoparticles has been considered as
a model system for a powder metallurgy system. Coated nanoparticles in the form of Ni-coated Al nanoparticles
and Al-coated Ni nanoparticles are also
analyzed as a model for nanoparticles embedded within a matrix. The differences in melting temperature and phase
15. SUBJECT TERMS
Reactive sintering, nanoparticles, simulation
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 15. NUMBER 19a. NAME OF RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE ABSTRACT OF PAGES Michael Zachariah
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301-405-4311
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Prescribed by ANSI Std. Z39.18
Report Title
“Molecular Dynamics Simulation of the Kinetic Reaction between Ni and Al Nanoparticles”
ABSTRACT
Molecular dynamics simulations are used to simulate the energetic reaction of Ni and Al particles at the nanometer
scale. The effect of particle size on reaction time and temperature for separate nanoparticles has been considered as a
model system for a powder metallurgy system. Coated nanoparticles in the form of Ni-coated Al nanoparticles and
Al-coated Ni nanoparticles are also
analyzed as a model for nanoparticles embedded within a matrix. The differences in melting temperature and phase
change behavior, e.g., the volumetric expansion of Al between Al and Ni, are expected to produce differing results for
the coated nanoparticle systems. For instance, the volumetric expansion of Al upon melting is expected to produce large
tensile stresses and possibly
rupture in the Ni shell for Ni-coated Al. Simulation results show that the sintering time for separate and coated
nanoparticles is nearly linearly dependent on the number of atoms or volume of the
sintering nanoparticles. We have also found that nanoparticle size and surface energy are important
factors in determining the adiabatic reaction temperature for both systems at nanoparticle sizes of
less than 10 nm in diameter.
Molecular dynamics simulations are used to simulate the energetic reaction of Ni and Al particles at the nanometer
scale. The effect of particle size on reaction time and temperature for separate nanoparticles has been considered as a
model system for a powder metallurgy system. Coated nanoparticles in the form of Ni-coated Al nanoparticles and
Al-coated Ni nanoparticles are also
analyzed as a model for nanoparticles embedded within a matrix. The differences in melting temperature and phase
change behavior, e.g., the volumetric expansion of Al between Al and Ni, are expected to produce differing results for
the coated nanoparticle systems. For instance, the volumetric expansion of Al upon melting is expected to produce large
tensile stresses and possibly
rupture in the Ni shell for Ni-coated Al. Simulation results show that the sintering time for separate and coated
nanoparticles is nearly linearly dependent on the number of atoms or volume of the
sintering nanoparticles. We have also found that nanoparticle size and surface energy are important
factors in determining the adiabatic reaction temperature for both systems at nanoparticle sizes of
less than 10 nm in diameter.
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“Molecular Dynamics Simulation of the Kinetic R ...
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Approved for public release; federal purpose rights
JOURNALOFAPPLIEDPHYSICS105,124310(cid:1)2009(cid:2)
Molecular dynamics simulation of the energetic reaction between Ni
and Al nanoparticles
Brian J. Henz,1 Takumi Hawa,3 and Michael Zachariah2,3,a(cid:1)
1U.S.ArmyResearchLaboratory,AberdeenProvingGround,Maryland,21005,USA
2NationalInstituteofStandardsandTechnology,Gaithersburg,Maryland,20899,USA
3DepartmentofMechanicalEngineeringandDepartmentofChemistryandBiochemistry,
UniversityofMaryland,CollegePark,Maryland,20742,USA
(cid:1)Received 15 September 2008; accepted 12 December 2008; published online 26 June 2009(cid:2)
Molecular dynamics simulations are used to simulate the energetic reaction of Ni andAl particles
at the nanometer scale. The effect of particle size on reaction time and temperature for separate
nanoparticles has been considered as a model system for a powder metallurgy system. Coated
nanoparticles in the form of Ni-coated Al nanoparticles and Al-coated Ni nanoparticles are also
analyzed as a model for nanoparticles embedded within a matrix. The differences in melting
temperatureandphasechangebehavior,e.g.,thevolumetricexpansionofAlbetweenAlandNi,are
expected to produce differing results for the coated nanoparticle systems. For instance, the
volumetric expansion ofAl upon melting is expected to produce large tensile stresses and possibly
ruptureintheNishellforNi-coatedAl.Simulationresultsshowthatthesinteringtimeforseparate
and coated nanoparticles is nearly linearly dependent on the number of atoms or volume of the
sinteringnanoparticles.Wehavealsofoundthatnanoparticlesizeandsurfaceenergyareimportant
factors in determining the adiabatic reaction temperature for both systems at nanoparticle sizes of
less than 10 nm in diameter. © 2009 American Institute of Physics. (cid:3)DOI:10.1063/1.3073988(cid:4)
I. INTRODUCTION materials, particle size has a significant effect on the proper-
ties of the reaction product and the SHS reaction itself.8The
Nanoparticles have interesting physical properties that simulation and analysis of nanoparticle coalescence without
often vary from the bulk material. Some of these properties, the SHS reaction for like materials are extensive9–14 and in-
including increased reactivity,1 are due to the high surface volve surface passivation,10 size differences,9,11 and phase
area to volume ratio of nanoparticles. With that in mind change9 considerations.The analysis here includes all of the
nanoparticles may provide enhanced energy release rates for previously listed concerns with an additional energy release
explosive and propellant reactions.2 term from the heat of formation.
There is a considerable interest in the self-propagating The focus of this paper is to use atomistic simulation to
high-temperature synthesis (cid:1)SHS(cid:2) reactions of intermetallic model the reactive behavior of Ni–Al nanoparticles in vari-
compounds because of the associated energy release that ous configurations. Fortunately, there have been numerous
takes place3 during the alloying reaction. In addition to the efforts to determine the accurate empirical potentials for
energeticreactionobservedinthesematerialsitispossibleto simulatingtheNi–Almaterialsystem.15Priorsimulationsus-
produce structural materials that contain this energy release ingthesepotentialshaveinvestigatedthediffusionofNiand
property. Once ignited, the SHS reaction releases a large Al atoms,15 point-defect concentrations in NiAl,16 and
amount of energy in a short period of time. One significant plasticity17inadditiontomanyothermechanicalandchemi-
differencebetweenSHSandtypicalcombustionprocessesis cal properties. These efforts have primarily focused on bulk
thatthereactantsandproductsareconfinedtothecondensed materials rather than nanoparticle systems,18 even though
state.4 The SHS process has many potential applications there are many manufacturing processes that produce nano-
whereheatgenerationisrequiredandoxygenisnotavailable metersizedpowdersforSHSreactions.19Forthissimulation
or gaseous products are not desirable. These include alloy effort we have chosen a set of embedded atom method
formation, net-shape processing, propellants, and as initia- (cid:1)EAM(cid:2) parameters that reproduce reasonably well the prop-
tors. One of the compounds formed from the SHS reaction, erties of Ni,Al, and NiAl in the temperature range of inter-
and studied here, is NiAl or nickel aluminide. NiAl is an est.
important alloy because of its desirable high temperature
strength and oxidation resistance5 and the high energy of
formation.6 Recently Weihs and co-workers7 also used the
NiAl nanolaminate systems in applications of reactive weld- II. SIMULATIONAPPROACH
ing.
Notsurprisinglysincethereactioninvolvessolidstarting In this work we employ classical molecular dynamics
(cid:1)MD(cid:2) with an EAM interatomic potential to study the SHS
a(cid:2)Author to whom correspondence should be addressed. Electronic mail: reaction. The EAM is used because of its accuracy and ca-
[email protected]. pabilitytoscaleuptomaterialsystemswithover106 atoms.
0021-8979/2009/105(cid:2)12(cid:1)/124310/10/$25.00 105,124310-1 ©2009AmericanInstituteofPhysics
124310-2 Henz,Hawa,andZachariah J.Appl.Phys.105,124310(cid:2)2009(cid:1)
The MD simulations are compared with thermodynamic TABLEI. Changeinsurfaceenergyvsnanoparticlesize.
analyses in order to provide validation of the simulation re-
Nanoparticleradius (cid:2)E
sults and assess the expected energy release. surf
(cid:1)nm(cid:2) (cid:1)kJ/mol(cid:2)
The MD simulation was conducted using the LAMMPS
software package.20 For the Ni–Al interactions the Finnis– 3 (cid:1)18.35
Sinclair EAM potential21 with parameters from Angelo 5 (cid:1)11.41
et al.22 was used.The Finnis–Sinclair EAM potential allows 10 (cid:1)6.17
for nonsymmetric embedding potential terms, potentially
providing improved accuracy for metallic alloys.23 In addi-
tion to the parameters for NiAl from Angelo et al.22 other H =(cid:1)0.5(cid:2)(cid:1)H (cid:2)+(cid:1)0.5(cid:2)(cid:1)H +H (cid:2)
reac Al,fusion Al,600 K Ni,600 K
authors have also developed parameters for the Ni–Al
=11.85 kJ/mol. (cid:1)3(cid:2)
system16 that may also be described by using the Finnis–
Sinclair EAM. This enthalpy result includes the enthalpy of solid Ni and
Threeprimarynanoparticlesizesconsideredinthiswork liquid Al.6 The Al nanoparticle is assumed to be liquid be-
from smallest to largest are nanoparticles with 1289, 5635, cause for small nanoparticles the melting temperature is
and 36523 atoms each, which correspond approximately to known to be appreciably below the bulk melting
3,5,and10nm,respectively.Therangeofsizeswaschosen temperature.24 Additionally, for the EAM potential used
because it represents nanoparticles that may be produced in here22 the aluminum is liquid for these nanoparticle sizes at
the laboratory and which offers reasonable computational 600 K. The choice of initial temperature will have a nearly
time to conduct parametric studies. For the largest system lineareffectontheadiabatictemperatureaslongasthetem-
studied, the 10 nm diameter nanoparticle energetic reaction peratureisbetweenthemeltingtemperatureoftheAlandNi
simulation requires (cid:5)2 days and 64 processor cores to com- nanoparticles. This linear effect has been observed in
plete a few nanoseconds of simulated time on 3.0 GHZ INTEL experiments25 and is a reasonable assumption so long as the
WOODCREST processors. heat capacities of the solid phases of Ni and NiAl are rela-
tively insensitive to temperature in the ranges studied.
For the products of the SHS process the enthalpy calcu-
lationmusttakeintoaccountthecontributionsfromthemelt-
III. THERMODYNAMICANALYSIS OF SEPARATE
ing of the nickel and the NiAl nanoparticle, enthalpy of for-
NANOPARTICLES
mation for the NiAl alloy, and changes in surface energy.
The separate nanoparticle system is used as a model for The first of these, the enthalpies of melting for Ni and NiAl,
powder metallurgy systems where Ni and Al particles are is experimentally determined to be 17.2 and 31.4 kJ/mol,
compressed into a structural component. In addition to me- respectively. The enthalpy of mixing for Ni andAl has gar-
chanical properties, the structural component will contain nered close scrutiny in the experimental community with a
stored energy for future release through a SHS reaction. A wide range of reported values. The enthalpy of formation
thermodynamicanalysisoftheSHSreactionfortheseparate that is used here is approximately in the middle of the re-
NiandAlnanoparticlesystemsisusedheretodeterminethe ported values at about (cid:1)65 kJ/mol.6,26,27
expected trends and data points for simulation validation. In The last contribution to the enthalpy of the products re-
thethermodynamicanalysisweareinterestedindetermining sults from the change in surface energy due to the reduced
the system parameters of the Ni–Al nanoparticle system that total surface area of the combined nanoparticle.28 The con-
contribute to the combustion temperature and reaction time. tribution to the change in system energy from the change in
Here we have assumed an adiabatic process so that energy surface area is given as
released to the surroundings can be ignored. This is a good
(cid:2)E =(cid:3) a −(cid:1)(cid:3) a +(cid:3) a (cid:2). (cid:1)4(cid:2)
approximation since the reaction occurs on relatively short surf NiAl NiAl Ni Ni Al Al
timescalesandthenanoparticlesareexpectedtobeincluded InEq.(cid:1)4(cid:2),a ,a ,anda arethesurfaceareaoftheNiAl,
NiAl Ni Al
inamuchlargersystemwheretheoverallsurfacetovolume Ni, and Al nanoparticles, respectively. For the 3, 5, and 10
ratio is small, limiting convective and radiative heat loss. nm Al nanoparticles the reactant surface area is computed
The validity of this assumption is explored in Secs. V and from the Gibbs surface29 as 36.32, 98.17, and 343.7 nm2,
VI. The SHS reaction of an equimolar Ni andAl mixture is respectively. For the associated Ni nanoparticles the surface
written as area is 27.15, 73.59, and 257.87 nm2, respectively. The sur-
0.5Ni+0.5Al→Ni Al . (cid:1)1(cid:2) faceenergyis(cid:5)1115 mJ/m2forAland2573 mJ/m2forNi
0.5 0.5 at 600 K.30 The surface area of the sintered NiAl nanopar-
Inordertocomputetheadiabatictemperatureforthesynthe- ticles is 50.77, 137.18, and 480.25 nm2 for the 3, 5, and 10
sis reaction the enthalpy of the products and reactants must nm nanoparticle cases, respectively. In the experimental
be equal, analysis of the free surface energy of NiAl near its melting
H (cid:1)T (cid:2)=H (cid:1)T (cid:2). (cid:1)2(cid:2) point, the free surface energy has been reported as
prod ad reac 0 1400 mJ/m2.31 The approximate change in energy versus
Assuming that the reaction begins with the reactants at 600 nanoparticle size is tabulated in Table I. In Table I the trend
K, above the simulated melting temperature of theAl nano- is for a lower surface energy contribution to the reaction as
particles, the enthalpy of the reactants is computed as the nanoparticle size increases. Intuitively, one may expect
124310-3 Henz,Hawa,andZachariah J.Appl.Phys.105,124310(cid:2)2009(cid:1)
TABLEII. Computedadiabatictemperaturevsnanoparticleradiusinclud-
ingcontactofflatsurfacesorinfinitelysizedspheres.
Nanoparticleradius T
ad
(cid:1)nm(cid:2) (cid:1)K(cid:2) FIG.1. (cid:1)Coloronline(cid:2)IllustrationofsinteringprocessshowingtheliquidAl
nanoparticlefirstcoatingthesolidNinanoparticleandthencompletealloy-
3 2115
ingaftertheNinanoparticlehasmelted.
5 1920
10 1772
(cid:4) 1599 coating the solid Ni nanoparticle while forming some Ni–Al
bonds on the surface (cid:3)(cid:1)b(cid:2)–(cid:1)d(cid:2)(cid:4). Next, the alloying process
proceeds with the Ni nanoparticle being heated above its
thisbecausethesurfaceareatovolumeratioisalsodecreas-
meltingpointandbecomingliquidsothatmixingmayoccur
ing with increasing particle size and therefore has less influ- (cid:1)e(cid:2). The formation of Ni–Al bonds beyond the interfacial
enceonthesinteringprocess.Withtheenthalpyofformation
surfacerequiresdiffusionofAlintotheNinanoparticleorNi
forNiAlaround(cid:1)65kJ/mol,thesurfaceenergycontribution
into the liquid Al. Either of these processes is possible but
tothechangeinenthalpyforcoalescenceof10nmdiameter
since diffusion is a relatively slow process in solid materials
nanoparticles is less than 10% of the total enthalpy change.
it is expected that the Ni nanoparticle must melt before the
This means that even at relatively small nanoparticle sizes,
coalescence process proceeds appreciably.
e.g., 10 nm, the effect of nanoparticle size on energy release
The nanoparticle sintering process is driven by two
is minimal.
sources of energy as previously discussed. The first of these
With the preceding discussion it is possible to take into
is a decrease in surface area that lowers the total surface
account many of the sources of enthalpy change in the reac-
energy of the system.This energy release mechanism is also
tion products including phase and surface area changes.The
observed in the sintering of homogeneous material systems
enthalpy of the products is now estimated as
(cid:6) such as silicon nanoparticles.28,33 The second source of en-
H =H +(cid:2)H + Tad C (cid:1)T(cid:2)dT ergy is from the reactive synthesis that occurs initially at the
prod form,NiAl surf p,NiAl
interface between the nanoparticles and later throughout the
298 K
entire system. The energy release from the surface sintering
+H . (cid:1)5(cid:2)
melt,Ni isproportionaltothesurfaceareaoftheNinanoparticlethat
The heat capacity of solid and liquid NiAl is given by is coated byAl and in the whole system to the total number
Kubaschewski.32 For the 3 nm case, assuming the NiAl of Ni and Al atoms. Additionally, with the temperature in-
nanoparticle melting temperature to be about 1350 K or the crease there is a decrease in the viscosity of the liquid alu-
melting point of a similarly sized Ni nanoparticle, it is pos- minum that will affect the predicted coalescence time.
sible to compute the adiabatic reaction temperature (cid:1)Table The coalescence of nanoparticles in the liquid and solid
II(cid:2).NoticeinTableIIthatifnosurfaceenergycontributionis phases has been examined extensively.9–11 These studies are
considered, i.e., infinitely large spheres, the final adiabatic primarily concerned with the coalescence of two liquid or
temperature is computed to be 1599 K. In the simulation two solid nanoparticles. The analysis of the Ni–Al system
section we will observe that these results are reasonable and
requires considering the coalescence of a liquidAl nanopar-
accurately predict the simulated increase in temperature at- ticleandasolidNinanoparticle.Lewisetal.9consideredthe
tributable to the contribution from the surface energy.
coalescenceofaliquidandasolidgoldnanoparticle.Thisis
similar to the situation here except that the material system
IV. THE COALESCENCE PROCESSES considered was homogeneous.
In Ref. 9, the authors were able to simulate two phases
ForNiandAlnanoparticlestheSHSreactionconsistsof
occurring simultaneously for a single material by choosing
two processes, namely, coalescence and alloying. In this
the size of each nanoparticle such that at a specific tempera-
work we have considered the coalescence of a two nanopar-
ture the phase of the nanoparticles is different. Lewis found
ticlesystemwithanAlandaNinanoparticlewithanatomic
that coalescence proceeded in two stages. First, the contact
ratio of unity. A complete SHS reaction of this system will
areawasmaximized,andsecond,“sphericization”tookplace
result in a single NiAl nanoparticle. The MD simulations
driven by surface diffusion. The first stage is much faster
used to work model adiabatic conditions with a constant
than the second and is very similar to the process observed
number of atoms and total system energy. The purpose of
here where the Al nanoparticle maximizes the contact area
thesesimulationsistoanalyzetheeffectofnanoparticlesize
on sintering time, adiabatic combustion temperature, and to andpartiallycoatstheNinanoparticle.Inthiscasethereisan
visualize the process. The assumed process is illustrated in addeddrivingforceinadditiontothesurfaceenergy,specifi-
Fig. 1. In Fig. 1 the nanoparticles are initially in contact at a cally the energy release in forming of Ni–Al bonds as com-
point (cid:1)a(cid:2) and theAl nanoparticle is larger than the Ni nano- pared to the Al–Al and Ni–Ni bonds. During the second
particle because of the longerAl–Al bond length.The simu- stage the atoms in the two nanoparticles diffuse and rear-
lations are initialized at 600 K so that theAl nanoparticle is range until the system becomes a single spherical nanopar-
liquidandtheNinanoparticleissolid.InFig.1thesintering ticle.ThisstageisdrivenstronglybytheformationofNi–Al
process proceeds with the liquid Al nanoparticle initially bonds and is expected to occur on a much shorter time scale
124310-4 Henz,Hawa,andZachariah J.Appl.Phys.105,124310(cid:2)2009(cid:1)
2a
2m)500 Nickel
Al r n Aluminum
Al θ Ni a(400
D e
r r
A
Ni ce300
a
v rf
vAl Su200
Ni d
e
s
FIG.2. Illustrationofparametersusedinanalyticalmodelofreactivecoa- o100
p
lescenceofNiandAlnanoparticles. x
E
0
0 2 4 6 8 10
thanfortwonanoparticlesofthesamematerial.Theanalyti- (a) CentertoCenterDistance(nm)
cal model and MD simulation results shown in Secs. V and
650
VI will explore this assumption.
)
2m
n
a(600
e
V.PHENOMENOLOGICALMODELOFNANOPARTICLE Ar
REACTIVE SINTERING e
c
a550
rf
Togainfurtherinsightwehavedevelopedaphenomeno- u
S
logical model for the reactive sintering of Ni and Al nano- d
particles. The model includes energy release from surface se500
o
p
energy, bond formation, and viscous dissipation through de- x
E
formation. Frenkel34 developed a model for the coalescence
450
0 2 4 6 8 10
of two homogeneous nanoparticles; however his model did
(b) CentertoCenterDistance(nm)
not account for any phase change, kinetic sintering, or het-
erogeneous materials. Here we extended Frenkel’s model FIG.3. (cid:1)a(cid:2)PlotofexposedNiandAlnanoparticlesurfaceareasasafunc-
thatconsidersthecoalescenceoftwoliquiddropstoconsider tionofdistancebetweennanoparticlecenters.(cid:1)b(cid:2)Plotoftotalexposedsur-
face area as a function of distance between nanoparticle centers. These
the coalescence of a liquid and a solid drop with reactive
resultsassumeaNinanoparticleofradius4.53nmandanAlnanoparticleof
synthesis. 5.23 nm. Notice that the total exposed surface area is monotonically de-
The analytical model is initialized with the Al and Ni creasing,indicatingthatthesurfaceenergyisalsodecreasingmonotonically.
nanoparticles in contact at a point. The distance from nano-
particle center to center is equal to the sum of the respective E =(cid:3)s S +(cid:3)l S . (cid:1)6(cid:2)
surf Ni Ni,exposed Al Al,exposed
radii, denoted as D in Fig. 2. The sintering process initially
proceeds by the liquid Al nanoparticle coating the solid Ni The exposed area of the Ni nanoparticle can be written as
nanoparticle,asillustratedinFig.1.Duringthisphaseofthe S =S −2(cid:6)r v =4(cid:6)r2 −2(cid:6)r v , (cid:1)7(cid:2)
Ni,exposed Ni Ni Ni Ni Ni Ni
sintering process, two sources of energy release are occur-
ring. The first of these is related to the decrease in surface where
area and proportional to the respective surface tension val- v =r (cid:3)1−cos(cid:1)(cid:5)(cid:2)(cid:4). (cid:1)8(cid:2)
Ni Ni
ues. The second source of energy release is from the forma-
tion of Ni–Al bonds at the interfacial region. Figure 2 is an Initially during the sintering process the Ni nanoparticle is
illustration of the geometric parameters used to model the assumed to remain in the solid phase, thus maintaining a
coalescence time. constantradius.Thisassumptionisreasonablebecauseofthe
InFig.2,2aisthediameterofacirclecircumscribedby higher melting temperature of the Ni nanoparticle.
the contact circumference of the two nanoparticles. v and The exposed surface area of theAl nanoparticle is writ-
Al
v are the distances from the Al and Ni nanoparticle sur- ten as
Ni
facestothesurfaceofthecontactcircle,respectively.(cid:5)isthe S =S −2(cid:6)r v =4(cid:6)r2 −2(cid:6)r v , (cid:1)9(cid:2)
Al,exposed Al Al Al Al Al Al
contact angle as measured from the center of the Ni nano-
particleandrangesfrom0to(cid:6)radians.Inordertomodelthe where
(cid:7)
changeinenergyofthecoalescingnanoparticlesystem,three v =r − r2 −a2, (cid:1)10a(cid:2)
energy change mechanisms must be considered. These Al Al Al
mechanismsareenergyreleaseduetochangeinsurfacearea, (cid:7)
a= v (cid:1)2r −v (cid:2). (cid:1)10b(cid:2)
energyreleaseduetoenergeticreactionsattheinterface,and Ni Ni Ni
energy loss due to viscous dissipation. The rate of energy TheradiusoftheAlnanoparticleiscomputednumericallyby
change due to all three must balance at all times. using conservation of volume for the Al nanoparticle. The
The first energy term considered, namely, the surface exposed surface area of each nanoparticle versus the center-
energy of the nanoparticle system, is simply the surface ten- to-centerdistanceisplottedinFig.3(cid:1)a(cid:2).Noticethatalthough
siontimesthetotalexposedsurfacearea.Thisenergytermis thesurfaceareaoftheAlnanoparticleincreasesduringmost
writtenasasumoftheAlandNinanoparticlecontributions. ofthecoalescenceprocessthecombinedtotalsurfaceareaof
124310-5 Henz,Hawa,andZachariah J.Appl.Phys.105,124310(cid:2)2009(cid:1)
300 620
600 Surf.Area,MD
250
Surf.Area,Model
2m) 2m)580
n200 n
( (
ea ea560
Ar150 Ar
ct e540
a c
ont100 urfa520
C S
50
500
00 2 4 6 8 10 4800 20 40 60 80
CentertoCenterDistance(nm) Time(ps)
FIG.4. Contactorinterfaceareaasafunctionofcenter-to-centerdistance. FIG.5. TotalsystemsurfaceareavstimefrommathematicalmodelandMD
The contact area is increasing as the nanoparticles move closer together simulations for the sintering of 10 nm diameter nanoparticles, where the
(cid:1)righttoleftonx-axis(cid:2). finalsurfaceareaoftheNiAlnanoparticleis(cid:5)480 nm2.
(cid:6) (cid:8) (cid:9)
the Ni and Al nanoparticles decreases monotonically dE rAl,0 8 dD 2
throughout the entire coalescence process. In Fig. 3 the viscous=2(cid:10) (cid:9)2(cid:1)4(cid:6)r2(cid:2)dr= (cid:6)r3 (cid:10) , (cid:1)14(cid:2)
dt 3 Al,0 dt
center-to-centerdistanceneverreacheszerobecausethecoa- 0
lescence is considered complete once the Ni nanoparticle is where (cid:10)is the viscosity of liquid aluminum and r is the
Al,0
completely enveloped by theAl nanoparticle.
initial radius of theAl nanoparticle.
The second source of energy release, namely, the reac-
By conservation of energy the rate of coalescence can
tive synthesis term, is considered by assuming a constant
now be computed,
surface density of the Ni nanoparticle and the transient con-
dE dE dE
tact area of the Ni–Al interface, viscous= surf+ reactive, (cid:1)15a(cid:2)
dt dt dt
E =(cid:7) a V . (cid:1)11(cid:2) (cid:8) (cid:9)
reactive Ni,surface interface bond energy 8 dD 2 d
(cid:6)r3 (cid:10) = (cid:3)(cid:3)s S +(cid:3)l S (cid:4)
Thesurfacedensityterm(cid:7) isproportionaltothenum- 3 Al,0 dt dt Ni Ni,exposed Al Al,exposed
Ni,surface
berofNi–Albondsatthecontactinterface.Thesurfaceden- d
sityandbondenergytermsVbond energycanbecombinedinto + dt(cid:3)2(cid:8)(cid:6)vNirNi(cid:4). (cid:1)15b(cid:2)
asingleconstantthatdefinestheenergyreleaseperunitarea
of interface, AfterwritingEq.(cid:1)15b(cid:2)intermsofd(cid:5)/dtandsimplifyingthe
right and left hand sides we find that Eq. (cid:1)15b(cid:2) is only lin-
(cid:8) =(cid:7) V . (cid:1)12(cid:2) early dependent on d(cid:5)/dt. Even with this simplification, Eq.
density Ni,surface bond energy (cid:1)15b(cid:2) is most easily solved numerically using an iterative
solver. In order to solve Eq. (cid:1)15b(cid:2) we need some physical
The interfacial contact area is a function of the distance be-
properties of Al, Ni, and NiAl. The dynamic viscosity of
tween nanoparticle centers (cid:3)Fig. 4(cid:4). The interfacial area in-
bulk molten Al at the melting temperature is about (cid:10)=1.3
creasesmonotonicallyupuntiltheNinanoparticlesurfaceis
(cid:11)10−3 Pas.35 Based on the comparison of the configura-
completelycovered.Thisresultisexpectedsincethereactive
tional energy in MD simulations of separate nanoparticles
energy term is negative, or releases energy during the entire
and Al-coated Ni nanoparticles the energy release per unit
process, in addition to the minimization of surface energy
area (cid:8) is estimated to be 20.7 eV/nm2.This number is
that is driven by the surface tension of Ni andAl.The inter- density
computed by subtracting the system energy of an Al-coated
facial area is written as
Ni nanoparticle system from the energy of a system with
separatenanoparticlesanddividingbytheinterfacialsurface
ainterface=2(cid:6)vNirNi, (cid:1)13(cid:2) area.This method results in the net change in energy during
coatingoftheNisurfacewithAlsincesomeAl–Albondsare
where v is a function of (cid:5)as given in Eq. (cid:1)8(cid:2). lost during the coating process while some Ni–Al bonds are
Ni
The third energy term represents the viscous dissipation formedattheinterface.BynumericallysolvingEq.(cid:1)15b(cid:2)we
due to deformation of theAl nanoparticle. This viscous dis- areabletocomputethecontactangle(cid:5)asafunctionoftime
sipation is a function of the viscosity in the liquidAl nano- andrelatethistototalexposedsurfaceareaofthecoalescing
particle and the rate of deformation. The extent of the vis- nanoparticles. This result is presented in Fig. 5 along with
cous flow can be specified by the decrease in distance the comparison to the MD simulation results.
between the center of each drop and the surface of contact Although qualitatively the results in Fig. 5 show similar
with the Ni nanoparticle. A velocity gradient (cid:9)can be de- trends the absolute rate of coalescence is slightly under pre-
fined as (cid:3)(cid:1)d/dt(cid:2)D(cid:4)/r . The energy dissipated in the whole dicted by the model. This difference can be attributed to the
Al
body per unit time is therefore approximately obvious simplicity of the model and more specifically to the
124310-6 Henz,Hawa,andZachariah J.Appl.Phys.105,124310(cid:2)2009(cid:1)
2400
2200
2000
K)1800
(
FIG.6. (cid:1)Coloronline(cid:2)CrosssectionalviewfromMDsimulationsofNi/Al e1600
r
nanoparticlesinteringprocessshowingthestartofthesecondstageofcoa- atu1400
lmesacxeinmciezawtiohne.reAdluifmfuinsiuomnaistotmhesadrreivbilnugefaonrdceniacskeolpaptoosmedsatroecreodn.tact area per1200 PredictedTad
m
e1000
T Al ,Ni
difficulty in obtaining accurate material parameters. For in- 800 Al1289,Ni1289
5635 5635
stance, it is difficult to compare the viscosity of a nanopar- 600 Al36523,Ni36523
ticletothebulkmaterial,12andsincethecoalescencetimeis 4000 500 1000 1500
linearly dependent on the viscosity a change in viscosity is Time(ps)
directly proportional to a change in modeled coalescence
FIG.8. (cid:1)Coloronline(cid:2)Temperaturevstimeinthesinteringofnanoparticles
time.Additionally, the energy release per unit area term as-
withaNi:Alratioof1:1.Thesubscriptsinthelegendrefertothenumberof
sumes that the net change in energy due to the addition of atomsofeachmaterialandcorrespondtonanoparticlesofdiameterof(cid:5)3,
Ni–Al bonds at the interface is a constant value. This is 5, and 10 nm. The color coded dashed lines are the computed adiabatic
likely not completely accurate since fewerAl bonds must be temperaturesfromthethermodynamicanalysis.Theblackdashedlineisthe
predictedtemperatureforcoalescenceofbulkAlandNi.
brokentoformnewNi–Albondsduringtheinitialcontactof
the nanoparticles. However, the deviation in this energy re-
lease term is likely to be minimal. The deviation of the Ni–Al bond formation. This period lasts about 50 ps in this
model time from the simulation results at about 50 ps is due simulation,asnotedinFigs.6and7.Betweenstages1and2
to the switch from stage 1 to stage 2 in the kinetic coales- the driving forces associated with the surface energy are
cenceprocess,asdescribedbyLewisetal.9Asdescribedby counteracted by a resistance to flow in the Al nanoparticle,
Lewis, during stage 2, surface diffusion is the predominant causing the coalescence process to slow down dramatically.
factorincontinuedcoalescenceandisamuchslowerprocess During stage 2, lasting about 450 ps, the surface area is not
than contact area maximization. The actual simulation re- changing so that energy release from the surface energy
sults, as compared with the illustration in Fig. 1, of the ob- terms has ceased to contribute to the change in system po-
served coalescence process are given in Fig. 6. tential energy. The subsequent energy release is entirely at-
In Fig. 6 each of the steps in the coalescence process is tributabletotheformationofNi–Albonds.Thisstagelastsa
shown with plots from a MD simulation of the coalescence much longer time than the initial nanoparticle coalescence
of 10 nm diameterAl and Ni nanoparticles. The correlation stage and is governed by the material diffusion coefficients.
ofthesinteringstagestothereactiontemperatureandtimeis Initially at stage 2 the Ni nanoparticle is still solid and the
illustratedinFig.8forthesinteringofseparate10nmdiam- formation of Ni–Al bonds is only possible by Al diffusing
eter nanoparticles. In the initial step the liquid Al nanopar- intotheNicoreorNionthesurfaceofthecoremeltingand
ticle, blue atoms in Fig. 6, has melted and is spherical in diffusingawayfromtheinterface.Thisprocessproceedsun-
shape.ThesolidNinanoparticle,redatoms,haslargefaceted til the Ni core has reached its melting point and mixing of
sides and is a single crystal, a typical configuration for a the remaining Ni and Al atoms occurs more rapidly, driven
crystalline nanoparticle at low temperatures. During stage 1 by the enthalpy of formation of NiAl. From stage 2 until
theAl nanoparticle is attracted to the Ni surface because of complete alloying has occurred, taking (cid:5)400 ps, diffusion
the dual driving forces of surface energy minimization and and mixing of Ni and Al atoms are the primary driving
forces.
1800
1600 Completion(500ps+) VI. MD SIMULATION RESULTS OF SEPARATE
NANOPARTICLE REACTIVITY
K)1400
( We have previously predicted the adiabatic temperature
e
ur1200 and sintering time for the reactive sintering process of sepa-
at
er1000 rateequimolarnanoparticlesofAlandNi.InFig.8,theMD
mp simulationresultsfortheequimolarnanoparticlesareplotted
Stage2(50ps-500ps)
e 800 alongwiththecomputedadiabatictemperatureforeachcon-
T
sidered particle size.
600
Stage1(0-50ps) From Fig. 8 it is apparent that the predicted adiabatic
400 temperature is in close agreement with the simulated tem-
0 200 400 600 800 1000
Time(ps) perature. The variability of the computed temperature arises
from the wide range of experimental results for the surface
FIG.7. (cid:1)Coloronline(cid:2)Timevstemperatureplotforsinteringofseparate10 tension for liquid Al and solid Ni, the reported enthalpy of
nmdiameterAlandNinanoparticles.Thevariousstagesofthecoalescence
formationforNiAl,andtheassumedmeltingtemperaturefor
processesaredenotedonthecurveincludingthefinalcompletionstagethat
occursaftertheNinanoparticlehasmelted. theNiandNiAlmaterialsatthisscale.Eachoftheseexperi-