Influence of Gas on the Rupture Strength of Liquid: Simulation by the Molecular Dynamics Methods

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1 ISS 008-5X, High Temperature, 206, Vol. 54, o. 4, pp Pleiades Publishing, Ltd., 206. Original Russian Text V.L. Malyshev, D.F. Mar in, E.F. Moiseeva,.A. Gumerov, 206, published in Teplofizika Vysokikh Temperatur, 206, Vol. 54, o. 4, pp SHORT COMMUICATIOS Influence of Gas on the Rupture Strength of Liquid: Simulation by the Molecular Dynamics Methods V. L. Malyshev a, b, d, *, D. F. Mar in b, d, E. F. Moiseeva a, b, d, and. A. Gumerov b, c a Ufa State Petroleum Technological University, Ufa, Russia b Center for Micro- and anoscale Dynamics of Dispersed Systems, Bashkir State University, Ufa, Russia c University of Maryland, Institute for Advanced Computer Studies, College Park, MD, USA d Skoltech Center for Design, Manufacturing and Materials, Skolkovo Innovation Center, Moscow, Russia * victor.l.malyshev@gmail.com Received February 6, 205 Abstract The cavitation rupture strength of liquid is studied by the molecular dynamics methods for simple materials by the example of argon in the presence of neon. Results on negative rupture pressure are obtained for different concentrations of the gas dissolved in the system. A linear dependence between these parameters is established. DOI: 0.34/S0085X ITRODUCTIO Investigation of the cavitation phenomenon, which plays a key role in hydrodynamics (high-speed flows in channels, motion of ship propellers, mechanical devices applied in medicine and biotechnologies, etc.), is closely related to rupture analyses of liquids, which are generally performed experimentally [ 3] or in applying the conventional theory of nucleation kinetics [4 6]. A new investigation tool implying molecular-dynamics (MD) calculations is a significant advance in research. Currently, nucleation in liquids is being intensively studied by this method throughout the world. Let us mention the main studies on this subject area. At the end of the last century, T. Kinjo and M. Matsumoto [7] investigated the kinetic limit of stability of a liquid under negative pressures by the MD method. The single-bubble nucleation rate was calculated, and the results were compared with the classical nucleation theory. In 2000, A.A. Bogach and A.V. Utkin [8] presented a study on determining the strength of water upon pulsed tension, which arises when compression pulses are reflected from a free surface of the material under study. It was shown that the increase in the compression pulse amplitude makes it possible to reduce the water strength from 46 to 22 MPa. The deformation rate affects only slightly the liquid strength. Russian researchers V.G. Baidakov and S.P. Protsenko [9] investigated the P, ρ, and T properties of a Lennard-Jones fluid in the metastable state at the liquid gas phase transition by the MD methods. It was shown that the phase separation arises near the spinodal at supersaturations. It was also noted that the results of solving the problem of nucleation kinetics in MD models are very sensitive to the interatomic-interaction potential cutoff; therefore, one should take into account the influence of particles spaced by a distance of no less than (5 6)σ, where σ is the characteristic interatomic distance. The influence of degassing and temperature effects on the formation of water nanobubbles on a mica surface was experimentally investigated in [0]. It was stated that gas dissolved in a liquid affects significantly the formation of nanobubbles on the surface, while the temperature of the liquid affects the nanobubble density and growth rate. M. Matsumoto and K. Tanaka [] performed a numerical analysis of the properties of an argon vapor nanobubble. It was found that the vapor density and pressure in the bubble are independent of its radius. The liquid surrounding the bubble is in the strongly tensed state. The surface tension calculated from the Laplace equation depends weakly on the bubble radius. In 2008, Japanese researchers [2, 3] proposed two approaches for tracing the dynamics of change in the bubble volume. The growth of cavitation bubbles and their radius were analyzed. The process of nucleation and growth of cavities in tensed Lennard- Jones systems was simulated under the guidance of G. orman [4, 5]. The nucleation rate was determined as a function of pressure and temperature. The initial stage of growth of a spherical cavity was simulated, and the dependence of the growth rate on pressure along two isotherms was determined. A kinetic model of liquid rupture under tension with a constant rate was proposed. In 202, K. Ando, A.Q. Liu, and C.-D. Ohl [6] experimentally determined the rupture strength of water. Impurities were removed from the 607

2 608 MALYSHEV et al. liquid by the microhydrodynamics methods. The tension was performed using a laser system. The cavitation strength limit for water at room temperature was 600 atm. The homogeneous and heterogeneous nucleations of bubbles in Lennard-Jones liquids were studied under the guidance of V.G. Baidakov from the MD [7] and experimental [8, 9] points of view. It was noted in [7] that the nucleation rates calculated based on the MD approach and within the classical nucleation theory differ by 8 20 orders of magnitude. The homogeneous and heterogeneous nucleations of bubbles in an ethane nitrogen solution were investigated in experimental studies [8, 9]. The experimental and theoretical results were found to be in satisfactory agreement. Possible reasons for the difference in the results were discussed. In [20], the cavitation rupture strength of liquid was analyzed for simple materials by the example of argon without any impurities. Results on the negative rupture pressure were obtained in the temperature range from 85 to 35 K. In this paper, we report the results of calculating the homogeneous nucleation in liquid argon in the presence of dissolved gas by the molecular dynamics method. The dependence of the rupture strength of liquid on the dissolved gas concentration was calculated, and their relationship was determined. MATHEMATICAL MODEL Within the molecular dynamics method, the molecular positions are determined by the classical motion equations 2 d ri F( ri) = = 2, Fr ( i ) U ( ri ), dt mi ri where r i and m i are the radius vector and mass of the ith particle, respectively. Except for the simplest cases, this system of equations is solved numerically according to a chosen algorithm (Velocity Verlet method, leapfrog, etc.). However, first of all, one must calculate force F(r i ) exerted on atom i based on the interaction potential U( r i ), where r i = (r, r 2,, r ) is the set of distances from the ith particle to all the other particles. Argon and neon atoms are nonpolar; therefore, the Lennard-Jones potential, cut off at distance r cutoff to reduce the calculation time, is chosen as a potential function. When studying multicomponent systems, the choice of scale units, for which σ* = Å, ε* = 0 2 J, and m* = u, is most appropriate for passing to dimensionless values. Liquid and gas atoms will be indicated by subscripts l and g, respectively. In this case, the dimensionless potential parameters are equal to (σ ll, σ gg ) = (3.405, 2.705) and (ε ll, ε gg ) = (.653, 0.49). The parameters of interaction between argon and neon particles are calculated according to the Lorentz Berthelot rule: σ lg = (σ ll + σ gg )/2, ε ll = εε ll gg. The integration time step is Δt = s. The dimensionless argon and neon atomic masses are m Ar = 39.9 and m e = The potential cutoff radius is chosen to be r cutoff = 8.0σ because small cutoff radii describe the properties of the system insufficiently well [2]. The simulation region is a cube, the sizes of which depend on the specified density and number of particles. Boundary conditions periodic in all directions are used. A number volume temperature (VT) ensemble is considered for simulating the statistical properties of the system. Temperature is maintained constant using the Berendsen thermostat [22]. The simulation is performed using a special data structure developed by authors; the molecular dynamics method is implemented by graphical processors using the VIDIA CUDA technology. Specific features of construction of the data structure and verification of the algorithm were reported in [23]. SIMULATIO OF THE CAVITATIO STREGTH OF LIQUID I THE PRESECE OF GAS Let us consider the process of simulation of the cavitation strength of a liquid with dissolved gas by the molecular dynamics methods. The simulation region is a cube with the sizes L x = L y = L z = 46 Å containing particles. At the initial instant, particles are distributed uniformly over the entire simulation region. Let us introduce the concept of gas particle concentration: c e g = e + Ar 00%, where c g is the concentration and e and Ar are the number of neon and argon molecules, respectively. Pressure in the system is decreased quasi-statically by extending the region. Every 5000 steps the simulation region is increased in all directions by the value x = y = z = Å; intermolecular distances are also increased by the value corresponding to the region extension. It was shown in [8, 5, 20] that the extension rate affects the liquid strength only slightly. Such a weak dependence of the spallation strength on the extension rate is due to the strong dependence of the nucleation rate on pressure [5]. We investigate the rupture strength of a liquid at a temperature of 85 K (when argon is the liquid material and neon is the gas because its critical temperature is 44.4 K). Therefore, the system is a liquid with gas dissolved in it. According to the MD method, the pressure for this system can be calculated from the formula P = T + Σi< jrf ij ij, () V 3V HIGH TEMPERATURE Vol. 54 o

3 IFLUECE OF GAS O THE RUPTURE STREGTH 609 where P is pressure, V is volume, T is temperature, is the number of particles, and r ij and f ij are, respectively, the radius vector and interaction force between the ith and jth particles. The second term is a virial part of the pressure. The summation is over all the particles contained in the system. Since a mixture of argon and neon is under consideration, there are pair interactions between particles (argon argon, neon neon, and argon neon). For convenience related to dimensionless variables, this virial expansion is partitioned into three terms: P Ar Ar Ar e e = rf, P = rf ij ij, 3V 3 ij ij e V i= j= i+ i= j= i+ P Ar e = 3V Ar e i= j= where P Ar and P e are the virial parts of pressure formed by argon and neon, respectively, and P Ar e is the virial part of pressure arising between argon and neon particles. Figure shows the time dependences of the dimensionless pressure components P Ar, P e, and P Ar e at the gas concentration c g = 0% and temperature T = 85 K. Points indicate the pressure values calculated at certain instants, and solid lines are the result of averaging the obtained data. Parameter t* in Fig. corresponds to each thousandth step of the algorithm. While the region is extended, pressure in the system decreases to some minimum value and then increases with a sharp jump. At this instant a bubble is formed in the system. According to formula (), the negative pressure with a maximum magnitude in dimensional quantities is P 288 atm and is achieved at approximately the 25000th step (t 0.43 ns). Figure 2 shows the cuts of the regions along the xz plane during bubble nucleation. The results of calculating the rupture strength of the liquid at temperature T = 85 K, the maximum size of the system at which the bubble nucleation occurs, and the minimum density of the system before the occurrence of a bubble are given in the table for the gas (neon) concentrations c g = 0, 3, 7, 0, 5, and 20%. Figure 3 shows the results of determining the cavitation rupture strength of the liquid (see table) in graphical form. Points indicate MD calculation values. The obtained data can be interpolated. The dependence of the volume rupture strength of the liquid on the dissolved-gas concentration was found to be linear. The results obtained at some other temperatures confirmed the linear dependence with a similar slope. This result can be explained based on the considerations reported in [24]. To induce significant changes rf ij ij, Fig.. Time distribution of pressure in the (a) argon argon, (b) argon neon, and (c) neon neon systems. in the growth or collapse of cavities, the impurity should change significantly physical properties such as viscosity, density, and surface tension, as well as the thermophysical properties of the material. To this end, (а) (b) (c) HIGH TEMPERATURE Vol. 54 o

60 MALYSHEV et al. (а) (b) (c) (d) Fig. 2. Formation of a bubble in the liquid: (a) t 0 = 0.43 ns, (b) t 0 + 20 ps, (c) t 0 + 40 ps, and (d) t 0 + 60 ps.

the amount of impurity should be so large that it becomes a component part of the system rather than impurity.

considered). Complete gas dissolution in the liquid makes it possible to retain its high rupture stability.

Thus, it should be noted that low dissolved-gas Rupture strength of the liquid, maximum size, and minimum density of the system before the occurrence of a bubble at different dissolved-gas

4 60 MALYSHEV et al. (а) (b) (c) (d) Fig. 2. Formation of a bubble in the liquid: (a) t 0 = 0.43 ns, (b) t ps, (c) t ps, and (d) t ps. P, atm c g, % Fig. 3. Dependence of the rupture strength of the liquid on the dissolved-gas concentration. the amount of impurity should be so large that it becomes a component part of the system rather than impurity. In the experimental results of [25], the capillary constant and surface tension for argon helium and argon neon mixtures depend weakly on the dissolved-gas concentration (concentrations below 3% were considered). Complete gas dissolution in the liquid makes it possible to retain its high rupture stability. It was shown theoretically in [26] that the strength of pure water saturated with air decreases by less than 0.5%. Thus, it should be noted that low dissolved-gas Rupture strength of the liquid, maximum size, and minimum density of the system before the occurrence of a bubble at different dissolved-gas concentrations с g, % P, atm L, Å ρ, kg/m concentrations affect only slightly the rupture strength of a liquid. However, the electric field strength was found in [27, 28] to decrease significantly with an increase in the dissolved gas concentration, which is caused by the formation of cavitation voids. COCLUSIOS It was shown that the pressure in multicomponent systems can be determined by the molecular dynamics method. The dependence of the rupture strength of liquid argon on the concentration of gaseous neon dissolved in the system was determined and found to be linear. ACKOWLEDGMETS This work was supported by Russian Foundation for Basic Research (grant no mol_a). REFERECES. Zel dovich, Ya.B., Zh. Eksp. Teor. Fiz., 942, vol. 2, no., p Skripov, V.P., Metastabil naya zhidkost (Metastable liquid), Moscow: auka, Volmer, M., Kinetik der Phasenbildung (Kinetics of phase formation), Dresden: Steinkopff, Beams, J.W., Phys. Fluids, 959, vol. 2, no., p.. 5. Bertholet, M., Ann. Chim. Phys., 850, vol. 30, p Briggs, L.J., J. Appl. Phys., 950, vol. 2, p Kinjo, T. and Matsumoto, M., Fluid Phase Equilib., 998, vol. 44, p Bogach, A.A. and Utkin, A.V., J. Appl. Mech. Tech. Phys., 2000, vol. 4, no. 4, p Baidakov, V.G. and Protsenko, S.P., High Temp., 2003, vol. 4, no. 2, p Zhang, X.H., Zhang, X.D., Lou, S.T., Zhang, Z.X., Sun, J.L., and Hu, J., Langmuir, 2004, vol. 20, p Matsumoto, M. and Tanaka, K., Fluid Dyn. Res., 2007, vol. 40, p Sekine, M., Yasuoka, K., Kinjo, T., and Matsumoto, M., Fluid Dyn. Res., 2007, vol. 40, p HIGH TEMPERATURE Vol. 54 o

5 IFLUECE OF GAS O THE RUPTURE STREGTH 6 3. Tsuda, S., Takagi, S., and Matsumoto, Y., Fluid Dyn. Res., 2008, vol. 40, p Bazhirov, T.T., orman, G.E., and Stegailov, V.V., J. Phys.: Condens. Matter, 2008, vol. 20, no., p Kuksin, A.Yu., orman, G.E., Pisarev, V.V., Stegailov, V.V., and Yanilkin, A.V., High Temp., 200, vol. 48, no. 4, p Ando, K., Liu, A.Q., and Ohl, C.-D., Phys. Rev. Lett., 202, vol. 09, p Baidakov, V.G. and Bobrov, K.S., J. Chem. Phys., 204, vol. 40, p Baidakov, V.G. and Pankov, A.S., Heat Mass Transfer, 205, vol. 86, p Baidakov, V.G. and Pankov, A.S., Heat Mass Transfer, 205, vol. 86, p Malyshev, V.L., Mar in, D.F., Moiseeva, E.F., Gumerov,.A., and Akhatov, I.Sh., High Temp., 205, vol. 53, no. 3, p Wang, D., Zeng, D., and Cai, Z., J. Chongqing Univ. (Engl. Ed.), 2002, vol., no. 2, p Berendsen, H.J.C., Postma, J.P.M., van Gunsteren, W.F., Di ola, A., and Haak, J.R., J. Chem. Phys., 984, vol. 8, no. 8, p Malyshev, V.L., Mar in, D.F., Moiseeva, E.F., Gumerov,.A., and Akhatov, I.Sh., Vestn. izhn. ovgorod. Gos. Univ., 204, no. 3, p Knapp, R.T., Daily, J.W. and Hammitt, F.G., Cavitation, ew York: McGraw-Hill, Kaverin, A.M., Andbaeva, V.., and Baidakov, V.G., Russ. J. Phys. Chem. A, 2006, vol. 80, no. 3, p Kuper, C.G. and Trevena, D.H., Proc. Phys. Soc., London, Sect. A, 952, vol. 65, p Kupershtokh, A.L., Vychisl. Metody Program., 202, vol. 3, p Kupershtokh, A.L., Comput. Math. Appl., 204, vol. 67, no. 2, p Translated by A. Sin kov HIGH TEMPERATURE Vol. 54 o

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