ISSN 0018151X, High Temperature, 2015, Vol. 53, No. 3, pp. 406–412. © Pleiades Publishing, Ltd., 2015.
Original Russian Text © V.L. Malyshev, D.F. Marin, E.F. Moiseeva, N.A. Gumerov, I.Sh. Akhatov, 2015, published in Teplofizika Vysokikh Temperatur, 2015, Vol. 53, No. 3,
pp. 423–429.
HEAT AND MASS TRANSFER
AND PHYSICAL GASDYNAMICS
Study of the Tensile Strength of a Liquid
by Molecular Dynamics Methods
V. L. Malysheva, b, D. F. Marina, b, E. F. Moiseevaa, N. A. Gumerova, c, and I. Sh. Akhatova, d
a Center
for Micro and Nanoscale Dynamics of Disperse Systems, Bashkir State University, Ufa, Russia
Institute of Mechanics, Ufa Scientific Center, Russian Academy of Sciences, Ufa, Russia
c
Institute for Advanced Computer Studies, University of Maryland, College Park, MD, USA
d North Dakota State University, Fargo, ND, USA
email: [email protected]
b Mavlyutov
Received February 14, 2014
Abstract—The cavitation tensile strength of a liquid for simple materials by the example of argon has been
studied using molecular dynamics methods. Results on the negative tensile pressure have been obtained
within the temperature range from 85 to 135 K. The tensile strength of liquid argon organization has been
studied theoretically using the Redlich–Kwong equation of state. These approaches are in good agreement.
Comparison with the earlier results of other authors has been performed. The test of the determination of the
tensile pressure by molecular dynamics methods for homogeneous systems will make it possible to analyze
qualitatively the cavitation strength in multicomponent systems as well as during consideration of heteroge
neous nucleation, where the theoretical studies are extremely troublesome.
DOI: 10.1134/S0018151X15020145
INTRODUCTION
According to the theory of the origin of cavitation,
it is considered that cavities (bubbles) are formed
when the local pressure in a liquid decreases to the
pressure of the saturated vapor. When a cavity in a homo
geneous liquid is formed, the continuity of the liquid
should be broken; therefore, the necessary pressure is
determined not by the pressure of the saturated vapor but
by the tensile strength of the liquid at the given tempera
ture [1]. The absolute value of the maximum negative
pressure that can be applied to the liquid is taken as the
upper boundary of the strength of the liquid.
There exist different theories describing the deter
mination of the tensile strength of the liquid. One of
the first is the theory of the internal pressure, which is
based on estimating the force of the intermolecular
interactions inside the system. This theory gives a
pressure value of –1700 atm for liquid argon. The sec
ond approach proposed by Temperly in 1947 is based
on consideration of the vanderWaals equation of
state [2]. This equation describes well the behavior of
the gas; however, it gives essential deviations from the
experimental data when describing the liquid state. On
the basis of this method, in 1975 Trevena calculated
the strength of the liquid for simple materials, such as
argon, oxygen, and nitrogen [3]. The calculated pres
sure for argon was –130 atm. The third theory elabo
rated by Fisher in 1948 [4] is based on the classical
nucleation theory [5]. The value calculated according
to this method was –190 atm for argon.
The tensile strength of the liquid was also studied
by experimental methods. Many researchers used the
method elaborated by Berthelet [6] in 1850. Later
Briggs [7] proposed to use Ushaped capillaries, which
were drawn directly before the experiment for produc
ing the “pure” surface of the capillary. However, there is
a difficulty in determining the cavitation strength. For
example, the tensile strength of water in a glass Berthe
let tube is –50 atm, and in a steel tube it is –13 atm.
According to the results of Beams’s experiments [8],
the cavitation strength for liquid argon was –12 atm.
Thus, the tensile strength of the liquid depends strongly
on the wall material, the quality of the surface purifica
tion, the presence of the gas and impurities in the liquid,
the purity of the experiment, and other factors.
Among recent works in this field are [9] and [10]. In
[9] the studies of explosive cavitation in superheated
liquid argon are compared with the homogeneous
nucleation theory. In [10] different theories on the ori
gin of cavitation, experimental results, and computer
simulation methods for liquid argon are compared.
Kuksin and Norman considered the phase transitions
of the first kind and mechanism of the destruction of
liquids [11, 12]. In their works the homogeneous
nucleation of bubbles in the liquid was formulated and
the growth of the formed cavities was studied using the
molecular dynamics methods.
The results of the measurement of the tensile
strength of the liquid show that quite high tensile stress
can exist in it. However, the measurement results have
a large scatter in works of different authors and in
those of the same group of experimenters. The scatter
of measurement results for the same liquid makes it
possible to suppose that regions of lowered and vari
406
STUDY OF THE TENSILE STRENGTH OF A LIQUID
able strength are formed in it, in which the breakup
occurs. These can be places of weaker adhesion of the
liquid to the container walls or “weak spots” in the liq
uid itself. The experimental studies make it possible to
assume that the appearance of the “weak spots” is due
to the presence of the impurities and the smallest gas
bubbles in the liquid. Thus, qualitative experimental
study of homogeneous nucleation requires expensive
equipment and highquality purification of the liquid.
The theoretical descriptions are based on approximate
models introducing their own errors in the results.
Therefore, study of this process on the molecular level
using computer simulation is the most available for
this type of problems.
This work presents the results of the calculation of
homogeneous nucleation in liquid argon in the
absence of any impurities using the molecular dynam
ics method. The theoretical calculation of the tensile
strength of the liquid was implemented according to
the Temperly method for the Redlich–Kwong equa
tion of state. Two approaches and the earlier results of
other authors were compared. The simulation results
show that molecular dynamics methods are an effi
cient means for solving similar problems. The experi
mental database is very limited, and the continuum
models mainly describe only the properties of simple
systems. Molecular dynamics methods make it possi
ble to simulate a wide range of problems associated
with homogeneous and heterogeneous nucleation.
Thus, the results obtained in this work are the basis for
solving more difficult problems associated with cavita
tion effects in a complex system.
The Redlich–Kwong Equation of State
Many works associated with the theoretical study
of the tensile strength of liquids are based on the van
derWaals equation of state [2]. However, it does not
describe very well the liquid state of the material [13].
In this work the twoparameter Redlich–Kwong equa
tion of state is considered [14]
,
P = RT − 0.5 a
V − b T V (V + b)
where P is pressure, R is the universal gas constant, T is
temperature, V is the molar volume, and a and b are
gas parameters, which are calculated according to the
formula
2
2.5
0.4275R Tc
0.08664 RTc
, b=
.
Pc
Pc
The critical temperature and pressure parameters
for argon are Tc = 150.86 K, Pc = 48.6 atm, so the ther
modynamic constants of the equation of state for
a=
407
P
T > Tm
T = Tm
V
T < Tm
A
Fig. 1. Redlich–Kwong equation of state.
argon are the following: a = 1.65037 (J2 K0.5)/(mol2 Pa),
b = 21.7231 × 10–6 m3/mol.
The characteristic form of the Redlich–Kwong
equation is given in Fig. 1. The theoretical value of the
strength of the liquid is determined as the value of the
pressure at the point of the minimum for the corre
sponding isotherm. It is noted in the figure that at the
temperature T = Tm the minimum pressure value is
zero. Let us determine this temperature. Starting from
the equation of state, the temperature Tm has the value
a(3 − 2 2)
32
.
Tm =
bR
For argon Tm = 135 K. Thus, the possible range of
the negative pressures is within temperatures from 83.8 K
(the melting temperature of argon) to 135 K. Table 1
shows the results of the tensile strength of the liquid for
argon obtained by different methods for the tempera
ture T = 85 K and the results calculated using the
Redlich–Kwong equation of state.
Such a scatter of values is possible due to the insuf
ficient highquality purification of the surface, the
presence of the gas and impurities in the liquid, the
purity of the experiment, and inaccuracy of the
applied models and equations of state.
THE MATHEMATICAL MODEL
In the molecular dynamics method, the positions
of molecules are determined from the solution of the
classical equations of motion:
2
d ri F(ri )
=
, F(ri ) = − ∂ U (r N ),
2
mi
∂ri
dt
where ri is the radiusvector of the ith particle and mi
is the mass of the ith particle. With the exception of
Table 1. Comparison of the results of the tensile strength of liquid argon
Temperly theory
Fisher theory
Beams experiment
Internal pressure
Redlich–Kwong equation
–130 atm
–190 atm
–12 atm
–1700 atm
–370 atm
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408
MALYSHEV et al.
Ly
Lx
Lz
In this work the variables are nondimensionalized
according to [16]. The simulation region is a parallel
epiped, the sizes of which are determined from the
given density and the number of particles. The peri
odic boundary conditions are applied to the system in
all directions. The NVT ensemble (Number Volume
Temperature ensemble) is considered for the simula
tion of the statistical properties of the system in which
the kinetic energy of molecules is fixed. The mainte
nance of constant temperature is implemented using a
Berendsen thermostat [17].
Fig. 2. Scheme of the simulation region.
ρ*
0.9
1
0.8
2
3
4
0.7
0.6
5
0.5
0.4
0.3
0.2
0.1
0
4
8 12 16 20 24 28 32 36 40 44 48
z, σ
Fig. 3. Density profile at different temperatures: (1) T* =
0.7008, (2) 0.9053, (3) 1.0048, (4) 1.0988, (5) 1.2123.
the simplest cases, this system of equations is solved
numerically according to the chosen algorithm (the
velocity Verlet method, leapfrog, etc.). However, first of
all it is necessary to calculate the force F(ri) acting on
the atom i, which is calculated in accordance with the
interaction potential U(r N), where r N = (r1, r2, …, rN) is
the set of distances from the ith particle to all other
particles.
The LennardJones potential is chosen as the
potential function, since the shortrange interaction
for simple monoatomic molecules is considered by the
example of argon:
⎡⎛ ⎞12 ⎛ ⎞6 ⎤
U (r j ) = 4ε ⎢⎜ σ ⎟ − ⎜ σ ⎟ ⎥ .
⎢⎣⎝ r j ⎠
⎝ r j ⎠ ⎥⎦
The parameter values of the LennardJones poten
tial interaction are chosen as follows: σ = 3.4 Å,
ε = 1.64 × 10–21 J that corresponds to argon. The mass
of the molecule m = 66.4 × 10–27 kg, and the integra
tion time step Δt = 2 × 10–15 s. The cutoff radius is
chosen as rcutoff = 8.0σ, since the small values of the
cutoff radius do not describe sufficiently well the prop
erties of the system [15].
CODE VERIFICATION
The free dynamics of the vapor–liquid argon
medium is considered for verification of the molecular
dynamics calculations.
The simulation region is a parallelepiped with sizes
Lx = Ly = 24σ, Lz = 48σ. Liquid argon is placed in the
center with the dimensionless parameters ρ* = 0.7 and
T* = 0. The scheme of the region is shown in Fig. 2.
The temperature of the system is kept constant by
the algorithm of the velocity correction (the Ber
endsen thermostat) during the first 50000 steps. Then
for 25000 steps the system is in the absence of thermo
statting for the establishment of the equilibrium state. For
the following 125000 steps (with an interval of 100 steps),
the macroscopic properties (e.g., density) are selected,
which then are averaged out over the whole set of the
particles.
To determine the density, the simulation region is
divided into layers parallel to the LxLy plane. We
denote the number of such layers as Nbin; in this work
it is taken as 100. The density in each layer is deter
mined according to the formula
ρi =
N i N bin
,
Lx Ly Lz
where Ni is the number of particles located in the ith
layer. The value i varies from 1 to Nbin.
Figure 3 shows the density profiles at different tem
peratures of the system. To make a symmetrical pic
ture, every 500 steps the particles were centered with
respect to the parallelepiped center with conservation
of the intermolecular distances.
Starting from the built distributions (Fig. 3), it is
possible to determine the density of liquid argon from
the central part of the plot and gaseous argon from the
extreme regions. By calculating in such way the density
at different temperatures, it is possible the build the sat
uration curve. The initial density of argon located in the
parallelepiped ρ* = 0.7 (ρ = 1182 kg/m3). The temper
ature T* varies within the range from 0.7 to 1.25 (from
85 to 150 K, respectively, in dimensional quantities).
Figure 4 shows the obtained results in dimensional
quantities.
The saturation curve for argon was also determined
by other authors [18, 19]. However, it is complicated to
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STUDY OF THE TENSILE STRENGTH OF A LIQUID
perform a comparison with their results, since quite
often only figures are presented without tables and
data. In this respect, Fig. 4 shows the calculation
results by molecular dynamics method and experi
mental data from [20]. It can be seen from the plot that
the numerical calculation results are in good agree
ment with the experiment.
The dynamics of the system consisting of many
particles has stochastic properties that are described in
detail in [21, 22]. We consider two trajectories calcu
lated from the same initial conditions but with differ
ent numerical integration steps Δt = 1 fs and Δt' = 2 fs.
The first trajectory is denoted as (ri(t), vi(t), and the
second is denoted as ((ri'(t), v 'i (t)). Averaged over the
trajectory, the differences of the coordinates and
velocities of the first and second trajectories in the
coinciding moments of time t = kΔt = k'Δt'
∑ (r (t) − r '(t)) ,
Δ v (t ) = 1
N
∑ (v (t) − v '(t))
2
2
i
T, K
160
150
140
130
120
110
100
90
80
0
600
300
900
1200
1500
ρ, kg/m3
Fig. 4. Saturation curve: line – experiment, points – cal
culation by the molecular dynamics method.
N
Δ r (t ) = 1
N
2
409
i
i =1
N
i
102
2
Δr2(t)
Δv2(t)
100
i
i =1
10–2
are given in Fig. 5.
It is seen well from the plot that the “time of mem
ory” of the systems is on the order of 5 ps, which is in
good agreement with [22].
10–4
10–6
10–8
SIMULATION OF THE CAVITATION
STRENGTH OF THE LIQUID
We consider the molecular dynamics simulations of
the cavitation strength. The temperature and the den
sity corresponding to the liquid state of argon are cho
sen according to the saturation curve (Fig. 4). The ini
tial density is ρ = 1350 kg/m3, and the temperature of
the system is T = 85 K. The simulation region is a
cube, the sizes of which are determined from the given
density and the number of particles. In this case the
region has sizes Lx = Ly = Lz = 43.09σ. At the initial
moment of time, the particles are distributed uni
formly over the total simulation region. The tempera
ture in the system is kept by the Berendsen thermostat,
which is used at each time. Each 5000 steps the simu
lation region is increased in all directions by the value
of 0.02σ, and also the intermolecular distances are
increased by the value corresponding to the tension of
the region. In [12, 23] the spall strength of simple liq
uids was studied. The destruction model used in [23]
contributes changes to the value of the spalling
strength of hexane by 20%. The calculations show that
for different tension velocities (0.01σ–0.04σ) the ten
sile pressure remains the same. Thus, it is possible to
note the weak dependence of the maximally reachable
pressure on the tension velocity that was also noted in
[12]. The calculations show that the value of the cavita
tion strength depends weakly on the tension velocity.
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0
2
4
6
8
10
12
t, ps
Fig. 5. Normalized averaged recession of coordinates and
velocities on two trajectories (the logarithmic scale on the
ordinate) calculated from the identical initial conditions with
steps Δt = 1 fs and Δt' = 2 fs (N = 64000, ρ = 1350 kg/m3,
T = 85 K).
The pressure in the system was calculated using the
method of virial sums according to the formula [24]
PV = NT + 1
3
∑r f
ij ij
,
i< j
where P is pressure, V is volume, T is temperature, N
is the number of particles, and rij and fij are the radius
vector and force of the interaction between i and
jparticles, 〈...〉 is time averaging. The summation is
performed over all atoms located in the system, and
the interaction force is calculated according to the
LennardJones potential. The pressure is calculated
every 2000 steps. For higher accuracy the pressure is
calculated without using the cutoff radius, since it is
very sensitive to its value.
The complexity of the calculation algorithm of the
pair interaction by the direct calculation method is
O(N 2), where N is the number of particles. Therefore,
410
MALYSHEV et al.
P*
(a)
–0.2
–0.3
–0.4
–0.5
–0.6
–0.7
–0.8
–0.9
100000 200000 300000 400000 500000
Step
Fig. 6. Time variation of the pressure (T = 85 K, ρ =
1350 kg/m3, N = 64000).
(b)
significant computing resources are required for cal
culating large systems. In this work we used a special
data structure elaborated by the authors and a hetero
geneous workstation with two 6core CPU Intel Xeon
5660 2.8 GHz (a total of 12 physical cores and 12 vir
tual cores using the Hyper–Threading technology),
32 GB RAM, and four GPU NVIDIA Tesla C2075
with 6 GB RAM. The code on GPU was written using
the NVIDIA CUDA technology, and the parallel
working of several GPU was implemented using the
OpenMP technology. Calculations were performed
using numbers with a floating point of double accu
racy. The comparison with the package LAMMPS was
also performed concerning the performance. The sim
ulator was chosen in which a method is implemented
that makes it possible to reduce the computing com
plexity of the whole algorithm and the GPU. The
comparison was performed with the same physical
parameters and the uniform distribution of molecules
over the region. The package USER–CUDA was used
for calculating on the GPU in LAMMPS. The compar
ison was performed on identical computing stations
described above. The comparison showed that the per
formance of the program elaborated by the authors is up
to three times higher than the power of LAMMPS on
one GPU. A detailed description of the molecular
dynamic acceleration using GPU can be found in
[25]. The calculation results for different sizes of sys
tems showed that the calculation of the pressure has no
essential deviations for the sizes of systems from N =
8000 to N = 512000 particles. Therefore, in this work
the value of the number of particles N = 64000 was
chosen for the sake of convenience of the calculation
and larger visibility.
The plot of the pressure of the system as a function
of time is shown in Fig. 6. Under the tension of the
region, the pressure in the system drops to a certain
Fig. 7. Formation of the cavitation bubble (T = 85 K, ρ =
1350 kg/m3, N = 64000, t = 0.94 ns, size of the region 15 ×
15 × 15 nm3): (a) cut of the region, (b) isosurface.
minimum value showing the maximum absolute neg
ative pressure that the liquid can endure. Since the
pressure has some fluctuations, a certain time domain
containing the minimum value is chosen, over which
the averaging is performed for determining the cavita
tion strength. In Fig. 6 the maximum absolute negative
pressure is P * ≈ –0.88, which is reached at approxi
mately the 330000th step (t ≈ 0.66 ns). At the subse
quent moments of time, the pressure has a sharp step
and goes into the stationary region. The cavitation
bubble is formed at the moment of a sharp increase in
the pressure in the system. Figure 7 shows the cut of
the region, in which the bubble is located, and its iso
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STUDY OF THE TENSILE STRENGTH OF A LIQUID
Table 2. Value of the tensile strength of liquid argon at dif
ferent temperatures obtained by the molecular dynamics
method (MD) and from the Redlich–Kwong equation of
state (RK)
T, K
85
93
100
107
115
123
130
–P, MD, atm 367
292
229
179
119
75
32
–P, RK, atm
282
216
159
104
56
21
370
surface (the level surface built on the density value ρ =
642 kg/m3).
Thus, it is possible to determine the cavitation
strength of the liquid by calculating the point of the
minimum on the pressure plot. The pressure values for
other temperatures were calculated analogously. Table 2
shows the results obtained using the molecular
dynamics method and from the Redlich–Kwong
equation of state (according to the Temperly method).
Figure 8 shows the results of determination of the
cavitation strength obtained by the molecular dynam
ics method and from the Redlich–Kwong equation of
state (Table 2). It is possible to note that the results of
molecular dynamics and the equation of state are in
good agreement. The obtained results show that it is
possible to use the Redlich–Kwong equation of state
in simple systems for determining the tensile strength
without using moleculardynamic simulation.
411
dynamics methods even for small systems (with the
number of molecules from 8000). In the given
approach the decrease in the pressure in the system
was reached due to the tension of the simulation
region and the tension rate does not play an essential
role. Good agreement between the molecular
dynamics results and the theoretical results on the
basis of the Redlich–Kwong equation of state was
obtained. The moleculardynamic approach will
make it possible to determine the tensile pressure in
different multicomponent and heterogeneous sys
tems, in which it is already impossible to use the
equation of state.
ACKNOWLEDGMENTS
This work was supported by the Ministry of Educa
tion and Science of the Russian Federation, project
no. 11.G34.31.0040.
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Translated by L. Mosina
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