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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.

406

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

U

shaped 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

Study of the Tensile Strength of a Liquid

by Molecular Dynamics Methods

V. L. Malyshev

a

,

b

, D. F. Marin

a

,

b

, E. F. Moiseeva

a

, N. A. Gumerov

a

,

c

, and I. Sh. Akhatov

a

,

d

a

Center for Micro and Nanoscale Dynamics of Disperse Systems, Bashkir State University, Ufa, Russia

b

Mavlyutov 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: victor.l.malyshev@gmail.com

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

HEAT AND MASS TRANSFER

AND PHYSICAL GASDYNAMICS

HIGH TEMPERATURE Vol. 53 No. 3 2015

STUDY OF THE TENSILE STRENGTH OF A LIQUID 407

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]

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

The critical temperature and pressure parameters

for argon are

T

c

= 150.86 K,

P

c

= 48.6 atm, so the ther

modynamic constants of the equation of state for

0.5

,

()

RT a

PVb

TVV b

=−

−+

22.5

0.4275 0.08664

,.

cc

cc

RT RT

ab

PP

==

argon are the following:

a

= 1.65037 (J

2

K

0.5

)/(mol

2

Pa),

b

= 21.7231

×

10

–6

m

3

/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

=

T

m

the minimum pressure value is

zero. Let us determine this temperature. Starting from

the equation of state, the temperature

T

m

has the value

For argon

T

m

= 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:

where

r

i

is the radiusvector of the

i

th particle and

m

i

is the mass of the

i

th particle. With the exception of

32

(3 2 2)

.

ma

TbR

−

=

2

2

()

,() (),

N

ii i

ii

d

Ur

m

dt

∂

==−

∂

rFr Fr r

P

A

V

T

>

T

m

T

<

T

m

T

=

T

m

Fig. 1.

Redlich–Kwong equation of state.

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

408

HIGH TEMPERATURE Vol. 53 No. 3 2015

MALYSHEV et al.

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

(

r

i

)

acting on

the atom

i

, which is calculated in accordance with the

interaction potential

U

(

r

N

)

, where

r

N

= (

r

1

,

r

2

,

…

,

r

N

)

is

the set of distances from the

i

th 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:

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

Δ

t

= 2

×

10

–15

s. The cutoff radius is

chosen as

r

cutoff

= 8.

0σ

, since the small values of the

cutoff radius do not describe sufficiently well the prop

erties of the system [15].

12 6

() 4 .

j

jj

Ur rr

⎡⎤

⎛⎞ ⎛⎞

σσ

=ε −

⎢⎥

⎜⎟ ⎜⎟

⎢⎥

⎝⎠ ⎝⎠

⎣⎦

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].

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

L

x

=

L

y

= 24

σ

,

L

z

= 48

σ

. 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

L

x

L

y

plane. We

denote the number of such layers as

N

bin

; in this work

it is taken as 100. The density in each layer is deter

mined according to the formula

where

N

i

is the number of particles located in the

i

th

layer. The value

i

varies from 1 to

N

bin

.

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/m

3

). 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

bin

,

i

i

xyz

NN

LLL

ρ=

L

y

L

x

L

z

Fig. 2.

Scheme of the simulation region.

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

48444032241602012 2884 36

z

,

σ

0.9

ρ

*

1

2

3

4

5

Fig. 3.

Density profile at different temperatures: (

1

)

T

* =

0.7008, (

2

) 0.9053, (

3

) 1.0048, (

4

) 1.0988, (

5

) 1.2123.

HIGH TEMPERATURE Vol. 53 No. 3 2015

STUDY OF THE TENSILE STRENGTH OF A LIQUID 409

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

Δ

t

= 1 fs and

Δ

t

'

= 2 fs.

The first trajectory is denoted as (

r

i

(

t

),

v

i

(

t

)

, and the

second is denoted as ( . Averaged over the

trajectory, the differences of the coordinates and

velocities of the first and second trajectories in the

coinciding moments of time

t = k

Δ

t = k

'

Δ

t

'

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].

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/m

3

, 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

L

x

=

L

y

=

L

z

= 43.09

σ

. 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

0.02

σ

, 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 (

0.01

σ

–0.04

σ

) 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.

'

'

((), ())

ii

tt

rv

22

1

22

1

1'

() ( () ()),

1'

() ( () ())

N

ii

i

N

ii

i

rt rt rt

N

ttt

N

=

=

Δ= −

Δ= −

∑

∑

vvv

The pressure in the system was calculated using the

method of virial sums according to the formula [24]

where

P

is pressure,

V

is volume,

T

is temperature,

N

is the number of particles, and

r

ij

and

f

ij

are the radius

vector and force of the interaction between

i

and

j

particles,

〈

...

〉

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,

<

=+

∑

1

,

3ij ij

ij

PV NT

rf

160

150

140

130

120

110

100

90

80 150012003000600 900

ρ

, kg/m

3

T

, K

Fig. 4.

Saturation curve: line – experiment, points – cal

culation by the molecular dynamics method.

10

2

10

0

10

–2

10

–4

10

–6

10

–8

1210840 62

Δ

r

2

(

t

)

Δv

2

(

t

)

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

Δ

t

= 1 fs and

Δ

t

' = 2 fs (

N

= 64000,

ρ

= 1350 kg/m

3

,

T

=85 K).

410

HIGH TEMPERATURE Vol. 53 No. 3 2015

MALYSHEV et al.

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

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

–0.9

100000 200000 300000 400000 500 000

Step

P

*

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

–0.2

Fig. 6.

Time variation of the pressure (

T

= 85 K,

ρ

=

1350 kg/m

3

,

N

= 64000).

(a)

(b)

Fig. 7.

Formation of the cavitation bubble (

T

= 85 K,

ρ

=

1350 kg/m

3

,

N

= 64000,

t

= 0.94 ns, size of the region 15

×

15

×

15 nm

3

): (a) cut of the region, (b) isosurface.

HIGH TEMPERATURE Vol. 53 No. 3 2015

STUDY OF THE TENSILE STRENGTH OF A LIQUID 411

surface (the level surface built on the density value

ρ

=

642 kg/m

3

).

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.

CONCLUSIONS

The study of the cavitation tensile strength of a

monoatomic liquid was described well by molecular

400

350

300

250

200

150

100

50

0120100 11090 130

T

, K

–P

, atm

Fig. 8.

Tensile strength of the liquid: curve is Redlich–

Kwong equation of state, points is molecular dynamics

method.

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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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 370 282 216 159 104 56 21

412

HIGH TEMPERATURE Vol. 53 No. 3 2015

MALYSHEV et al.

18. Bo, Shi,

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Translated by L. Mosina