X. LIU et al.: MOLECULAR DYNAMICS SIMULATION OF Ti-6Al-4V DIFFUSION BONDING ...
365–372
MOLECULAR DYNAMICS SIMULATION OF Ti-6Al-4V
DIFFUSION BONDING BEHAVIOR UNDER DIFFERENT PROCESS
PARAMETERS
SIMULACIJA MOLEKULARNE DINAMIKE ZLITINE Ti-6Al-4V,
DIFUZIJSKO ZVARJENE PRI RAZLI^NIH PROCESNIH POGOJIH
Xiaogang Liu
*
, Lei Xu, Sun Zhang
Jiangsu Province Key Laboratory of Aerospace Power Systems, College of Energy and Power Engineering,
Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Prejem rokopisa – received: 2019-07-25; sprejem za objavo – accepted for publication: 2020-01-29
doi:10.17222/mit.2019.172
In this paper the diffusion-bonding process of a titanium alloy was investigated in detail using molecular dynamics. The
protocell model of a Ti-6Al-4V alloy was obtained by the atomic random substitution method. The mixed potential function
method, EAM (Embedded Atom Method) and Morse potential was used to deal with the Ti-Al-V ternary alloy system. And the
diffusion bonding process of Ti-6Al-4V was simulated numerically. The influence of temperature, pressure and holding time on
the diffusion bonding behavior of Ti-6Al-4V alloys was studied. The results showed that an increase of the temperature,
pressure and holding time can improve the diffusion of interfacial atoms and increase the diffusion bonding width. In addition,
the diffusion temperature has a critical value, i.e., 1100 K. When the temperature is lower than 1100 K, the interface atoms
hardly diffuse. The MSD (Mean Square Deviation) vs. t curves of Ti, Al and V atoms at different temperatures were established.
The diffusion coefficients of Ti, Al and V atoms at different temperatures were obtained by the relationship between the MSD
and the atomic diffusion coefficient, after which the Arrhenius equations of the diffusion coefficient were obtained, respectively.
Based on the diffusion coefficient equation, the diffusion activation energies of each element were deduced. Compared with the
experimental data, the rationality of the simulation results was verified.
Keywords: molecular dynamics, Ti-6Al-4V alloy, diffusion bonding, diffusion coefficient
V ~lanku avtorji natan~no opisujejo raziskavo procesa difuzijskega varjenja (spajanja) Ti zlitine z uporabo metode molekularne
dinamike. Protoceli~ni model Ti-6Al-4V zlitine so avtorji dobili z metodo naklju~ne metode nadome{~anja (substitucije). Za
obdelavo ternarnega zlitinskega sistema Ti-Al-V so uporabili me{ano potencialno funkcijsko metodo, metodo EAM (angl.:
Embedded Atom Method, ta upo{teva potencial med atomi) in Morsejev potencial. Proces difuzijskega spajanja Ti-6Al-4V so
avtorji simulirali numeri~no. [tudirali so vpliv temperature, tlaka in ~asa zadr`evanja na difuzijsko spajanje Ti-6Al-4V zlitin.
Rezultati simulacij so pokazali, da povi{anje temperature in tlaka, ter podalj{anje ~asa zadr`evanja lahko izbolj{a difuzijo med
atomi in tako pove~a {irino difuzijskega spoja. Dodatno so ugotovili, da je kriti~na temperatura 1100 K. Pod to temperaturo
medatomarna difuzija poteka te`ko (po~asi). Avtorji so ugotovili odvisnost kvadratnega korena povpre~nega odstopanja (MSD;
angl.: Mean Square Deviation) od ~asa t za Ti, Al in V atome pri razli~nih temperaturah. Difuzijske koeficiente za Ti, Al in V
atome pri razli~nih temperaturah so dolo~ili s povezavo med MSD in atomskimi difuzijskimi koeficienti oziroma dolo~ili so
njihove difuzijske koeficiente v obliki Arrheniusove ena~be. Na osnovi ena~b za difuzijske koeficiente so dolo~ili aktivacijsko
energijo za difuzijo vsakega posameznega kemijskega elementa. Na koncu so {e preverili ujemanje rezultatov simulacij z
eksperimentalnimi podatki.
Klju~ne besede: molekularna dinamika, zlitina Ti-6Al-4V, difuzijsko spajanje (varjenje), difuzijski koeficient
1 INTRODUCTION
The Ti-6Al-4V alloy is an important manufacturing
material in the aviation field because of its excellent
comprehensive properties. At present, the titanium alloy
wide-chord hollow fan blade formed by superplastic-
forming/diffusion-bonding technology has become one
of the key technologies of advanced aero-engines.
However, the diffusion bonding zone is very narrow,
usually at the micron level. It is difficult to study the
performance of joints from a macroscopic point of view
by conventional methods. It is a satisfactory method to
study the diffusion bonding process by molecular
dynamics.
B. J. Alder et al.
1
first put forward the classic mole-
cular dynamics theory in 1957 and proved the rationality
of the theory through the "hard sphere" liquid model,
thus starting the prelude of molecular dynamics simu-
lation research. Subsequently, a large number of
researchers continued to develop the method. L. Verlet
2
proposed the frog-leaping algorithm, which has been
used today because of its fast calculation speed and
small storage capacity. C. H. Andersen,
3
D. J. Evans et
al.
4
developed the constant-pressure control method and
the temperature control method for a non-equilibrium
system to improve the molecular dynamics method.
Based on electronic theory, R. Car
5
supplemented and
explained molecular dynamics methods in principle and
successfully expanded the research field of molecular
dynamics methods. In recent years, with the rapid
Materiali in tehnologije / Materials and technology 54 (2020) 3, 365–372 365
UDK 67.017:621.039.322:544.034 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 54(3)365(2020)
*Corresponding author's e-mail:
liuxg03@nuaa.edu.cn (Xiaogang Liu)

development of computer technology, molecular dyna-
mics as an effective simulation method in the field of
computational materials has attracted the attention of
researchers. M. Grujicic
6
used the EAM potential to
simulate the martensitic transformation process of a Ti-V
alloy and analyzed the transformation mechanism of the
lattice structure. L. Long et al.
7
simulated the diffusion
process of Cu-Al heterogeneous alloys using the
molecular dynamics method, indicating that the diffusion
process was mainly caused by the diffusion of Cu atoms
into Al. The diffusion bonding process of Ti-Al hetero-
geneous alloys was simulated using LAMMPS software
and the influence of surface roughness on the diffusion
effect was also analyzed by T. Fuling.
8
L. L. Zhu
9
studied
the diffusion process of V atoms in a Ti alloy to find the
influence of the V atom concentration on the diffusion
distance and the diffusion coefficient.
In summary, a lot of research work had been done on
the structural transformation and diffusion process of
binary alloys using the molecular dynamics method.
However, the molecular dynamics simulation of the
diffusion-bonding process between ternary alloys is less
involved. In the paper the diffusion-bonding process of
the Ti-6Al-4V ternary alloy was simulated using mole-
cular dynamics. The influence of the process parameters
on the diffusion bonding width and the atomic diffusion
coefficient of alloys were analyzed in detail.
2 FUNDAMENTALS OF THE
DIFFUSION-BONDING SIMULATION BY
MOLECULAR DYNAMICS
The atoms vibrate around their equilibrium sites due
to the exchange of the chemical potential and atomic
energy that overcome the barrier migrated to the location
of the adjacent atom. The atomic migration process of
micro and macro arising from the phenomenon is called
diffusion. Diffusion-bonding technology relies on the
local plastic deformation of the material surface at high
temperature and high pressure to realize bonding on the
atomic scale by the diffusion of the bonded materials on
the surface.
10
This process contains very complex physi-
cal and chemical reactions. Therefore, it is very import-
ant to understand comprehensively the basic law of the
diffusion of metal atoms and the geometric model of
diffusion bonding.
In the molecular dynamics model, atoms are usually
approximated as particles or compact pellets. The motion
state of any particle can be obtained by integrating
Newton’s motion equation,
11
where the differential
equation of Newton’s second law is expressed as:
m
r
t
Vrr r
i
i
i n
d
d
2
2 12
=−∇ ( , ,..., ) (1)
where m
i
and r
i
are the mass and position parameters of
the number i atom, respectively. V(r
1
, r
2
,... r
n
)i st h e
potential function of the atom.
The trajectory of the system can be described by
calculating the particle’s velocity and position according
to Equation (1). By averaging the calculated results of
the system with a statistical method, the required macro-
scopic parameters such as temperature, pressure and
stress can be further obtained. In addition, the micro-
scopic structure of the particles can also be investigated
intensively.
In molecular dynamics simulations the interaction
between different atoms is expressed by the potential
function, which defines the relationship between atomic
potential energy and the coordinates.
12
The main
potential functions are the Lennard-Jones(L-J/n-m)
potential, the Morse potential, the Tersoff-Brenner
potential, the Embedded Atom Method (EAM) potential,
the Sutton-Chen potential and so on.
6
Among them, the
Morse potential is a typical pairing potential function,
which is widely used in metals and alloys. It has a high
accuracy in simulating metallic systems with FCC or
HCP structures.
13
The Morse potential function is
expressed as follows in Equation (2):
14
[][] {} VD rr rr
ij ij ij
=−− −−−
000
22 exp ( ) exp ( )  (2)
where D
0
has an energy dimension to describe the
strength of the interactions between atoms. The r
0
is the
mean radius and  is the normalized coefficient.
The EAM is a typical multi-body potential function.
It takes into account the free electron gas effect of each
atom’s local region. The gel model is used to simplify
the description of the complex environment around the
atoms.
15,16
It can be expressed as:
UF r
i
i
ij
ji i
total
=+ ∑∑ ∑
≠
() () 
1
2
(3)
ii j
ji
fr =
≠
∑ () (4)
where F represents the atoms’ embedded potential,
which is a function of  i
. The function  is the short-
range pair potential  i
is the function of the density of
electrons in an atom. The first item at the right-hand end
of Equation (4) is the energy required to embed atoms in
the background of an electron mass. The second item is
the traditional pairing potential that describes the repul-
sion between atoms in a solid system. According to
Equations (3) and (4), it can be deduced that the
interaction force between the atoms i and j is expressed
as:
[] {} FFFf rr
ij i j ij ij
=++ '( ) '( ) '( ) '( )  (5)
At present, the EAM potential function of Ti-Al-V
ternary alloys has not been systematically studied. In the
paper, the mixed potential method is used. The EAM
potential is used for the atoms Ti-Al and Al-V, while the
Morse potential is used for the atoms Ti-V.
The process parameters such as holding temperature
and pressure have an important influence on the quality
X. LIU et al.: MOLECULAR DYNAMICS SIMULATION OF Ti-6Al-4V DIFFUSION BONDING ...
366 Materiali in tehnologije / Materials and technology 54 (2020) 3, 365–372

of the diffusion bonding. The selection of the appropriate
temperature- and pressure-control methods is an im-
portant step in the molecular dynamics simulation of
diffusion bonding. In the paper, the Nose-Hoover
thermostat method was used to control the temperature
in the simulation process.
4
The basic principle of this
method is to introduce coordinate variables to realize the
energy exchange between the atoms and the external
thermostat. The Lagrange equation of the energy is
expressed as:
L
m
sr Ur
M
s
g
s
Nose
i
i
i
N
N
=−+ ⋅ −
=
∑
22
22
1
2
()
 ln
 (6)
When the system temperature agrees with T > T
c
, the
atoms in the system transfer energy to the external
thermostat and the velocity of the atoms decreases.
Otherwise the system absorbs energy and the velocity of
the atoms increases accordingly.
In a molecular dynamics simulation, the pressure of
the system varies due to the collision and extrusion
caused by the atomic vibration and the free movement.
However, the Parrinello-Rahman control method is used
to control the pressure conditions so as to maintain the
constant pressure state of the system.
17
3 NUMERICAL SIMULATION OF THE
DIFFUSION-BONDING PROCESS OF Ti-6Al-4V
3.1 Modeling
In molecular dynamics, each atom in the initial
model can be arranged in a given lattice coordinate.
Therefore, the initial model can be accurately established
by knowing the coordinates of the lattice arrangement
and the kinds of atoms located at each coordinate. Using
the MATLAB program, a series of coordinate sites
arranged strictly according to the lattice structure were
generated. By calculating the percentage content of each
constituent atom, the coordinate sites were randomly
assigned to the atomic number according to the corres-
ponding percentage by using the random substitution
method. Thus, the proto-cell model was generated. The
molecular dynamics model of an alloy system with any
given structure and size could be generated by the
method.
For Ti-Al-V ternary alloys, the numbers of Ti, Al and
V atoms were set to 1, 2 and 3, respectively. The initial
three-dimensional model of the Ti-6Al-4V was set as an
integrity BCC (body-centered cubic) structure. The axis
x, y and z correspond to the orientations of [100], [010]
and [001], respectively. The system of the proto-cell
model was an NVT ensemble. The corresponding Ti
atoms were randomly replaced by 10 a/% Al and 3.6 a/%
V, respectively. The scale of the proto-cell model was
25×25×18 (the Ti lattice constant is 0.328 nm) and the
total number of atoms was 43, 236. Among them, there
were 37,154 Ti atoms, 4516 Al atoms and 1566 V atoms.
The final atomic model was shown in Figure 1. In order
to observe clearly the upper and lower models of the
contact surface they were distinguished by different
colors. The pressure loads were applied alone the z
direction of the boundary layer. Considering the scale
effect, the periodic boundary conditions were set in the x
and y directions and contractive boundary conditions
were set in the z direction.
3.2 The influence of process parameters on the diffu-
sion bonding width
3.2.1 Temperature
The melting point of Ti-6Al-4V alloy is about
1900 K. The temperature is an important factor that
mainly affects the diffusion behavior of the atoms in the
interface region for the diffusion-bonding process of
materials. A higher holding temperature leads to a plastic
yield at the diffusion-bonding interface, which will
expand the contact area and promote the diffusion of
atoms.
Figure 2 shows the atom distribution on the [010]
crystal plane after complete diffusion at different holding
temperatures. The atoms of the upper half model were
set to white. The red, blue and yellow atoms were Ti, Al
and V, respectively. As can be seen from Figure 2, the
atomic diffusion distance increases, and the contact
interface diffusion becomes more sufficient with an
increase of the system temperature (from 1000 K to
1183 K) under the conditions of constant pressure and
holding time. During the diffusion process, the Ti matrix
atoms near the interface migrate significantly. The
concentration of V atoms in the alloy is slightly lower
X. LIU et al.: MOLECULAR DYNAMICS SIMULATION OF Ti-6Al-4V DIFFUSION BONDING ...
Materiali in tehnologije / Materials and technology 54 (2020) 3, 365–372 367
Figure 1: Ti-6Al-4V initial 3D model diagram in the direction of
[001] (Green and pink are titanium atoms, blue and dark red are
aluminum atoms, black and yellow are vanadium atoms)

than that of the Al atoms. It can also be seen from Fig-
ure 2 that the atoms on the interface hardly diffused
below 1100 K. It is indicated that the holding temper-
ature should be higher than a critical value in order to
obtain good quality of the diffusion-bonde jdoints.
The concentration distribution of different elements
along the z direction was obtained, as shown in Figure 3.
Because the initial content of V was very small and the
content of the diffusion-bonding layer was less, the
atomic concentration of Ti-V was summed up. In Fig-
ure 3, the Ti-V1 and Al1 were model atoms below the
contact surface, while Ti-V2 and Al2 were model atoms
above the contact surface. It was considered that the
region where the concentration of Ti-V atoms changes
more than 5 % is the diffusion bonding layer, as shown
by the black dotted line region. It can be seen from Fig-
ure 3 that the width of the diffusion-bonding layer
increases obviously with the increase of the holding
temperature.
X. LIU et al.: MOLECULAR DYNAMICS SIMULATION OF Ti-6Al-4V DIFFUSION BONDING ...
368 Materiali in tehnologije / Materials and technology 54 (2020) 3, 365–372
Figure 2 Atomic diffusion results at different temperatures (P = 3.5 MPa, t = 400 ps): a) T = 1000 K, b) T = 1100 K, c) T = 1183 K
Figure 4: Atomic diffusion results at different pressure (T = 1183 K, t = 400 ps): a) P = 2.5 MPa, b) P = 3.5 MPa, c) P = 4.5 MPa
Figure 3: Atomic concentration distribution of Ti-Al-V after diffusion bonding at different temperatures (P = 3.5 MPa, t = 400 ps): a) T = 1000 K,
b) T = 1100 K, c) T = 1183 K

3.2.2 Pressure
A higher pressure can promote the plastic defor-
mation of the interface, reduce the recrystallization
temperature of the interface, and thus improve the
quality of the diffusion bonding.
Figure 4 shows the atom distribution on the [010]
crystal plane after complete diffusion under different
pressures. It can be seen from Figure 4a that the atomic
migration occurred obviously at the interface of the
model at P = 2.5 MPa. When the pressure increased from
2.5 MPa to 4.5 MPa, the diffusion distance of the atoms
along the depth direction increases and the diffusion
becomes more sufficient.
Figure 5 shows the concentration distribution of
Ti-Al-V along the z direction after diffusion bonding
under the condition of different pressures. It can be
found that the width of the diffusion-bonding layer
increases obviously with the increase of the pressure
when the other process parameters are constant.
4 SUMMARY OF THE INFLUENCE OF THE
PROCESS PARAMETERS ON THE DIFFUSION
BONDING WIDTH
In order to investigate the influence of the process
parameters on the diffusion-bonding width, the diffu-
sion-bonding process of Ti-6Al-4V under different
pressures, temperatures and holding times were simul-
ated by molecular dynamics. A total of 48 cases were
obtained, as shown in Table 1.
According to the simulation results for the diffusion-
bonding width in Table 1, the three-dimensional curved
surfaces of the Ti-6Al-4V diffusion bonding width
varying with pressure, temperature and holding time
were obtained by polynomial fitting with MATLAB
software, as shown in Figure 6. As can be seen from
Figure 6, the pressure and holding time have little effect
on the diffusion-bonding width below 1100 K. There-
fore, for the ideal model, the diffusion-bonding tem-
perature of Ti-6Al-4V should be greater than 1100 K.
When the holding time was over 800 ps with constant
pressure, the diffusion-bonding width increases first and
X. LIU et al.: MOLECULAR DYNAMICS SIMULATION OF Ti-6Al-4V DIFFUSION BONDING ...
Materiali in tehnologije / Materials and technology 54 (2020) 3, 365–372 369
Figure 5: Atomic concentration distribution of Ti-Al-V after diffusion bonding at different pressures (T = 1183 K, t = 400 ps): a) P = 2.5 MPa,
b) P = 3.5 MPa, c) P = 4.5 MPa
Table 1: Influence of process parameters on the diffusion-bonding
width
Holding
time /ps
Temperature
/K
Pressure
/MPa
Diffusion bong
width /nm
No.
200
1283
4.5 2.971 1
3.5 2.324 2
2.5 1.797 3
1183
4.5 1.865 4
3.5 1.434 5
2.5 1.288 6
1100
4.5 1.370 7
3.5 0.912 8
2.5 0.581 9
1000
4.5 0.764 10
3.5 0.752 11
2.5 0.214 12
400
1283
4.5 4.407 13
3.5 3.130 14
2.5 2.825 15
1183
4.5 3.503 16
3.5 2.646 17
2.5 2.123 18
1100
4.5 1.689 19
3.5 1.31 20
2.5 0.744 21
1000
4.5 0.962 22
3.5 0.880 23
2.5 0.411 24
800
1283
4.5 6.438 25
3.5 5.899 26
2.5 5.212 27
1183
4.5 4.871 28
3.5 4.258 29
2.5 3.241 30
1100
4.5 2.301 31
3.5 1.721 32
2.5 1.241 33
1000
4.5 1.322 34
3.5 1.010 35
2.5 0.872 36
1200
1283
4.5 9.725 37
3.5 8.833 38
2.5 7.838 39
1183
4.5 6.335 40
3.5 5.710 41
2.5 4.968 42
1100
4.5 2.762 43
3.5 2.471 44
2.5 1.861 45
1000
4.5 1.738 46
3.5 1.532 47
2.5 1.291 48

then decreases slightly with an increase of the holding
temperature. The optimum holding temperature is about
1200 K, which is consistent with the actual temperature
used in the Ti-6Al-4V diffusion bonding. In addition, it
is clear that the influence of pressure on the diffusion
bonding width became more and more significant with
the increase of the holding temperature for a constant
holding time. Therefore, in the actual diffusion-bonding
process of titanium alloy, the full diffusion of atoms in
the interface region can be promoted by increasing the
holding temperature to 1200 K, step by step, and
increasing the pressure properly.
4.1 The influence of temperature on the diffusion
coefficient
In a molecular dynamics simulation, the Mean
Square Displacement (MSD) is normally used to analyze
the change of the atomic trajectory with time.
18
The
dynamic information, such as the diffusion coefficient of
the system, can be obtained. The main idea is to obtain
instantaneous coordinates of the particles and solve the
differential equations by recording the dynamic evolu-
tion process of particles at a certain temperature, so as to
determine the MSD of the system. The Einstein function
relationship between the MSD and the correlation time t
is shown as follows in Equation (7):
MSD t
N
rt r
ii
i
N
() () () =−
=
∑
1
0
2
1
(7)
where N is the number of atoms in the simulation
system, r
i
(t) is the position vector of the atom i at time
t, and r
i
(0) is the initial position vector of the atom i.
When the system runs for long enough, the MSD limit
formula can be expressed as:
lim ( )
t
MSD t c D
→∞
=+ 2 (8)
where,  is the dimension of the system, D is the
diffusion coefficient, and c is a constant.
Equation (8) shows that in a three-dimensional
system,  = 3, the diffusion coefficients of various
elements can be obtained from the slope of the corres-
ponding MSD distribution curve vs. time. Figure 7
shows the distribution curve for the MSD of Ti, Al and V
under 1000-1500 K. It can be seen from the figure that
the MSD of the Ti, Al and V atoms shows a similar
trend, i.e., it increases gradually with the increase of the
temperature and shows an approximate linear relation-
ship with the holding time. This indicates that the higher
the temperature is, the higher the transition frequency of
the Ti, Al and V atoms is at the interface.
The diffusion coefficients of various atoms under
different temperatures were obtained by fitting data from
the MSD vs. time in Figure 7, as listed in Table 2.Itcan
be found that the diffusion coefficient of a Ti atom is
relatively high and that of an Al atom is the smallest at
the same temperature. When the holding temperature is
lower than 1100 K, the diffusion coefficients of the Ti,
X. LIU et al.: MOLECULAR DYNAMICS SIMULATION OF Ti-6Al-4V DIFFUSION BONDING ...
370 Materiali in tehnologije / Materials and technology 54 (2020) 3, 365–372
Figure 7: Distribution curve of the MSD with the holding time at different temperatures of the atoms Ti, Al and V
Table 2: Diffusion coefficients of Ti, Al and V atoms under different temperatures
T (K) 1000 1100 1183 1250 1300 1400 1500
DTi
(10
–14
m
2
/s) 1.7×10
–5
6×10
–4
4.1 5.1 12.6 50.4 83
DAl
(10
–14
m
2
/s) 3.5×10
–6
1.6×10
–4
2 4.2 8.4 20.8 48
DV
(10
–14
m
2
/s) 2.3×10
–5
6.7×10
–4
2.8 4.8 9.2 28 90
Figure 6: Variation of the diffusion-bonding width with the process
parameters

Al and V atoms are so small that the order of magnitude
is about 10
–19
m
2
/s. It shows that the Ti, Al and V atoms
are hardly diffusing below 1100 K. This is consistent
with the previous conclusion.
Figure 8 shows the curves of the diffusion
coefficients of Ti, Al and V in exponential coordinates at
the corresponding temperatures. In Figure 8, T

= 1155
K is the transition temperature from phase  to phase  in the Ti-6Al-4V alloy. It can be seen from the figure that
the diffusion coefficients of the above three atoms
increase obviously near the phase-transition temperature.
The main reason is that the BCC structure has more
interstitial space than the HCP structure. The diffusion
rate of the atoms in the BCC crystal structure is usually
higher than that in other crystal structures.
The diffusion coefficient D of the elements depends
on the diffusion activation energy Q and the temperature
T. Q is defined as the smallest energy that atoms possess
when they transit. The Arrhenius equations is satisfied
among D, Q and T, as shown in Equation (9):
DD
Q
Rt
=−
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 0
exp (9)
where D
0
is the diffusion-coefficient factor, R is the gas
constant, which generally takes .
The Arrhenius equations were fitted to the diffusion
coefficients and the temperatures of the Ti, Al and V
atoms (Table 2), respectively, which are shown as:
D
Ti
= 4.38×10
–9
exp(–13593/T) (10a)
D
Al
= 3.52×10
–9
exp(–14230/T) (10b)
D
V
= 1.26×10
–8
exp(–15769/T) (10c)
where the dimensions of the diffusion coefficient and
temperature are m
2
/s and K, respectively.
According to Equation (10), the diffusion activation
energies of the corresponding atoms in the temperature
range from 1183 K to 1500 K can be obtained and listed
in Table 3, as well as the experimental values in the
literature. It can be seen that the diffusion activation
energy calculated in the paper shows good agreement
with the experimental values in the literature.
19–21
Table 3: Comparison of the calculated diffusion activation energies of
Ti, Al and V with the experimental values
Para-
meter
Ti Al V
Experi-
mental
data
Simu-
lation
results
Experi-
mental
data
Simu-
lation
results
Experi-
mental
data
Simu-
lation
results
(eV)
1.49
1.19
1.29
1.23
1.53
1.36
1.35 1.53 1.68
5 CONCLUSION
The diffusion-bonding process of the Ti-6Al-4V alloy
was numerically simulated using the molecular dynamics
method. The influence of various process parameters on
the diffusion-bonding behavior was studied. The effects
of the different holding temperatures on the diffusion
behavior of the Ti, Al and V atoms were also investi-
gated. The main conclusions are as follows:
1. Based on the fundamentals of molecular dynamics,
the mixed potential function method (the EAM potential
and the Morse potential) was used to deal with the
potential of the Ti-Al-V ternary alloy system. The
diffusion-bonding process of the Ti-Al-V ternary alloy
was numerically simulated and good results were
obtained. The simulation results showed that the Ti-atom
diffusion was dominant in the ternary alloy system.
2. The influence of temperature, pressure and holding
time on the diffusion-bonding width was studied. The
general trend was that the diffusion-bonding width
increases with the appropriate increase of temperature,
pressure and holding time, thus improving the quality of
the diffusion joints.
3. The atoms at the interface hardly diffused below
1100 K. When the holding time was long enough, the
diffusion-bonding width increases first and then de-
creases slightly with the increase of the temperature. The
optimum diffusion-bonding temperature of Ti-6Al-4V
alloy was about 1200K.
4. The diffusion coefficients of the Ti, Al and V
atoms change sharply near the transition temperature of
the  phase to the  phase. The Arrhenius diffusion
equation of for the Ti, Al and V atoms was established
based on the simulation results. The diffusion activation
energies of the three types of atoms were deduced from
the diffusion equation, which was in good agreement
with the experimental results in the literature.
Acknowledgement
Financial support by the Fundamental Research
Funds for the Central Universities (No. NS2016022) and
X. LIU et al.: MOLECULAR DYNAMICS SIMULATION OF Ti-6Al-4V DIFFUSION BONDING ...
Materiali in tehnologije / Materials and technology 54 (2020) 3, 365–372 371
Figure 8: Distribution of the diffusion coefficients of the Ti, Al and V
atoms at different temperatures

National Science and Technology Major Project
(2017-IV-0012-0049) are gratefully acknowledged.
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