Nanoscale and Microscale Thermophysical Engineering, 17: 79–91, 2013Copyright © Taylor & Francis Group, LLCISSN: 1556-7265 print / 1556-7273 onlineDOI: 10.1080/15567265.2012.760692

NONEQUILIBRIUM MOLECULAR DYNAMICSSIMULATION OF NANOBUBBLE GROWTH ANDANNIHILATION IN LIQUID WATER

Yijin Mao and Yuwen ZhangDepartment of Mechanical and Aerospace Engineering, University of Missouri,Columbia, Missouri

The objective of this article is to investigate nanosized bubble growth and annihilation pro-cesses in liquid water via nonequilibrium molecular dynamics (MD) simulation and comparethe results with that obtained from the Rayleigh-Plesset equation. The TIP3P potential modelis chosen to describe the structure of the water molecule. The SHAKE algorithm is used tohold two O-H bonds and H-O-H angle as rigid, and a harmonic bond style and a Charmmangle style are applied. The results show that the hydrodynamic model based on the Rayleigh-Plesset equation is not valid for predicting nanosized bubble growth and annihilation inliquid water.

KEY WORDS: water nanobubble, molecular dynamics, growth and annihilation

INTRODUCTION

Ranging from microdevices to macroscale hydraulic machinery developments, theincreasing applications in industry require a deeper understanding of micro- or nanosizedbubble behaviors; for example, bubble cavitation [1], nucleation [2], and stability [3].Molecular dynamics (MD) simulation is a powerful method to study microscale bubblebehavior in many different situations. With regard to cavitation and bubble nucleation, Parket al. numerically simulated the hysteretic process of bubble collapse [4]. Wu and Pan con-ducted MD simulation to study the bubble nucleation rate in homogeneous liquid argon[5]. Tsuda et al. focused on bubble nucleation in liquid oxygen with different impurities[6]. Zhai et al. investigated bubble nucleation in a supercritical carbon dioxide–hexadecanesolution, which is widely used in the plastic production industry [7]. With regard to otheraspects of bubble dynamics, Lugli and Zerbetto investigated nanobubble collapse in waterand the effects of salts [8]. A stability study of helium bubbles during high energy dis-placement was carried out in order to investigate bubble damage in a fusion reactor [9].Insepov and Hassanein calculated the density and diffusion coefficient in liquid lithiumcontaining helium atoms during bubble formation by sampling data from MD simulation

Manuscript received 18 September 2011; accepted 17 December 2012.Support for this work by the U.S. National Science Foundation under grant number CBET-1066917 is

gratefully acknowledged.Address correspondence to Yuwen Zhang, Department of Mechanical and Aerospace Engineering,

E 3411 Thomas & Nell Lafferre Hall, University of Missouri, Columbia, MO 65211. E-mail: zhangyu@missouri.edu

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80 Y. MAO AND Y. ZHANG

[10]. Nagayama et al. observed bubble formation in a nanochannel with an emphasis on thenucleation phenomenon in liquid argon [11]. Okumura and Ito investigated the processesof bubble formation and contraction simulation in liquid argon by MD simulation [12], andtheir results indicated that the Rayleigh-Plesset equation is valid for nanosized bubbles gen-erated in liquid argon. Because most materials are composed of polyatomic molecules (likewater), where a more complicated force field should be accounted for, bubble behavior inliquid water should be investigated.

The objective of this article is to investigate nanosized bubble growth and annihilationprocesses in liquid water and compare the results with those obtained from the Rayleigh-Plesset equation. Sampling of the atomic movement data at the microscopic level duringthe entire process from creation to extinction is discussed. The validities of the assumptionssupporting the Rayleigh-Plesset equation are also analyzed.

In order to investigate the dynamics of a bubble, reduced MD simulations were per-formed by applying the Lennard-Jones potential. The characteristic length, energy, andmass were chosen as the Lennard-Jones diameter σ , the minimum value of the potential ε,and the atom mass m. An asterisk (∗) represents the reduced quantities such as reducedlength r∗ = r/σ , and reduced temperature T∗ = Tkb/ε, reduced pressure P∗ = Pσ/ε,reduced mass density ρ∗ = ρσ 3/m, reduced time t∗ = t (ε/m/σ 2)1/2, and reduced charge q∗= q (4πε0εσ )1/2. For the convenience of data analysis, the parameters of hydrogen atomsin water molecules were chosen as characteristic parameters during MD simulations. Alldimensionless results were converted back to dimensional for discussion.

PHYSICAL MODELS AND METHODS

Nanobubble generation and dynamic behavior are mainly affected by the thermalconductivities and viscosities of the fluid. The TIP3P model did not show a significant dif-ference in predicting either of these water properties [13, 14], although it is widely used forMD simulation. In addition, the TIP3P model is better than other three-site water molecules,such as SPC/E, for describing intermolecular potential and controlling system pressure[15]. TIP3P is also one of the most popular water models in the rigid TIP3P-CHARMM[16–18] model that specifies three interaction sites corresponding to three atoms in thewater molecule. Thus, it was chosen to describe water’s molecular structure. For the bondand angle model, the SHAKE algorithm [18] is used to hold two O-H bonds and an H-O-H angle as rigid. Figure 1 shows the typical structure of a single water molecule withthree-site.

The interaction between two water molecules in TIP3P is represented as:

Eab =∑

on ai

∑on bj

kCqaiqbj

raibj

+∑

on ai

∑on bj 4εaibj

[(σaibj

raibj

)12

−(

σaibj

raibj

)6]

, (1)

where kC is the electrostatic constant and has a value of 332.06 Å·kcal/mol; qi are the partialcharges relative to the charge of the electron; rij is the distance between charged sites; andσ ai and σ bj are the Lennard-Jones parameters. The values of the dimensionless parametersare provided in Table 1. An arithmetic average was used to calculate the Lennard-Jonespotential between oxygen and hydrogen atoms.

The particle–particle, particle–mesh method (PPPM) [19, 20] is an accurate andcomputationally efficient method for calculating interactions in MD simulation, and its

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NONEQUILIBRIUM MOLECULAR DYNAMICS SIMULATION 81

complexity is only Nlog(N) [19]. Therefore, the long-range Coulomb solver, PPPM, wasused to calculate the long-range force between the water molecules. During the computa-tion, the PPPM solver maps the water molecule’s hydrogen and oxygen atoms charge in a3D mesh and then uses 3D fast Fourier transforms to solve the Poisson equation on eachmesh. Finally, it interpolates from the grids to calculate the electric field and correspondinglong-range Coulomb force on each charged atom. A careful choice of mesh size is requiredbefore applying the PPPM approach in order to achieve comparable precision of the Ewaldsummation [21]. Here, 60 was selected as the mesh size in the x-, y-, and z-directions,based on a pair of test cases for thermal conductivity (278.15 K) calculation carried outby employing reverse nonequilibrium molecular dynamics (RNEMD [22]). In one case,the long-range electrostatic force contribution was computed with PPPM and the othercase was based on Ewald summation. It was found that when the size was set at 60, thevalue from PPPM was 0.967 ± 0.012 W/m.K, and the value for the Ewald-based case was0.954± 0.013 W/m.K. Therefore, a size of 60 is a reasonable choice. In addition, becausethe latter results were based on statistical average, the error introduced by the PPPM methodwas negligible.

Constant-pressure MD simulations were carried out [23] by performing time inte-gration on Nose-Hoover style non-Hamiltonian equations of motion. All positions andvelocities of the atoms were generated from and NPH (isoenthalpic-isobaric ensemble)ensemble using the Verlet algorithm with the following conditions. The cubic simulationbox (6 nm × 6 nm × 6 nm) with periodic boundary conditions contained 7,200 watermolecules. For a short-range pair force, the widely used atom-atom interaction modelof CHARMM potential with long-range coulomb effect [24] was chosen to compute thepotential energy and corresponding force (contributed by the interaction of van der Waalsand Coulombic forces) by employing an additional switching function S(r) that ramps boththe energy and force smoothly to zero between an inner and outer cutoff:

E (r) ={ LJ (r) + C (r) , r < rin

S (r) LJ (r) + S (r) C (r) , rin < r < rout,S (r) LJ (r) + S (r) C (r) , r > rout

(2)

where rin and rout were set to be 20σ and 25σ , respectively. Other functions in Eq. (2) are

Table 1 TIP3P water model parameters

Items REAL units Lennard-Jones units

mO 15.9994 (g/mol) 15.8724mH

∗ 1.008 (g/mol) 1.0qO −0.834e −112.0qH 0.417e 56.0

εO-O 0.1521 Kcal/mol 3.30652σO-O 3.1507 Å 7.87617ε∗

H-H 0.0460 Kcal/mol 1.0σ ∗

H-H 0.4000 Å 1.0R0

OH 0.9572 Å 2.39292θHOH 104.52◦ 104.52◦

∗Characteristic dimensions.

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82 Y. MAO AND Y. ZHANG

LJ(r) = 4ε

[(σ

r

)12 −(σ

r

)6]

(3)

C (r) = fdamp (r)Cqiqj

εr(4)

S (r) =[r2

out − r2]2[

r2out + 2r2 − 3r2

in

]2

[r2

out − r2in

]3 , (5)

where σ and ε are determined by atom type, and f damp(r) in Eq. (4) is determined notonly by the distance between atoms but is also related to the G-Ewald parameter, which isadjusted through mesh size based on desired precision.

A harmonic bond model [24] was used to compute the bond force between oxygenand hydrogen atoms and the potential energy contribution was calculated:

E = K(r − r0)2 (6)

The CHARMM angle style [24], which considers the potential energy contribution fromthe alteration of bond angle from balanced θ0 with an additional Urey-Bradley term, waschosen as the angle model; that is,

E = K(θ − θ0)2 + KUB(r − rUB)2, (7)

where K, θ0, KUB, and rUB are coefficients defined for each angle type.The system pressure, after 20,000 time steps (each time step was 0.005t∗ or 0.46 fs)

equilibrated through a Langevin thermostat, was well controlled to be 0 (1 bar; for P =1 bar, σ = 0.4 Å, ε = 0.0460 Kcal/mol, the reduced pressure was P∗ = Pσ/ε = 2 × 10−10,which is extremely close to 0), and initial temperature is equilibrated to 12.0 (278.15 K).Eighty molecules near the center of the simulation box were selected and instantaneouslyheated to 132 (3059.65 K) by velocity rescaling. The temperature in the domain includingthe heated atoms was much higher than the boiling point, which is 16.11 (373.5 K) at 1 atm.During the latter 11,000 time steps, the atomic movement of the system was recorded untilbubble extinction.

The volume and radius of the bubble were estimated using the following method.First, all sides of the simulation cell were divided into 10 sections so the simulation box wasdivided into 1,000 small cubes. The side length �L∗ of the small cube fluctuated between14.87 (5.95 Å) and 14.95 (5.98 Å) during the MD simulations. The average density was cal-culated for each small cube over 100 time steps, which is much smaller than the timescaleof the bubble dynamics (11,000 steps) and much larger than that of the atomic dynamics(∼20 steps). The vapor region was recognized as the one in which the average density inthe small cube was less than the critical density ρc

∗ = 6.83 × 10−4 (320 kg/m3) [25], andthe liquid region was the one in which the average density was greater than ρc

∗. The bubblevolume was calculated as the sum of the volumes of the vapor regions. Because the bubblewas fairly spherical in shape, the effective bubble radius R∗ was estimated from the bubblevolume Vbub

∗ using the following equation:

R∗ =(

3

4πV∗

bub

)1/3

. (8)

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NONEQUILIBRIUM MOLECULAR DYNAMICS SIMULATION 83

For MD simulations, the statistical results based on multiple cases were more meaningfulthan the single cases. In order to improve the statistical accuracy, 15 MD simulations withrandomly chosen initial molecule configurations (initial atom positions and velocities) wereperformed. The final results were based on averaging over 15 cases and the error bar wascalculated by the standard deviation of R∗ from the 15 cases.

On a macroscopic scale, a hydrodynamic equation, namely, the Rayleigh-Plessetequation, which is applicable for an incompressible and inviscid fluid, can be used todescribe expansion and contraction of a bubble:

R∗R∗ + 3

2R∗2 = 1

ρ∗l

[P∗ (

R∗) − P∗∞ − 2S∗

R∗

], (9)

where P∗(R∗) is the pressure at the bubble surface, P∞∗ is the pressure far from the bubblein the liquid region, and S∗ is the reduced surface tension of the bubble. The initial conditionfor Eq. (9) is. It is difficult to precisely determine P∗(R∗) because the radius of the generatedbubble is only a few times the molecular diameter.

In this work, the molecules constituting the bubble surface were used to estimate thesurface pressure of the water bubble:

P∗ (R∗) = ρ∗

c

3

1

Ns − Nc

{∑Ns

i=1m∗

i r2i +

[∑Ns

n=1

(ri · fi

)]b,a,s,k

}, (10)

where Ns is the molecule number within Router of the MD simulation cell (see Figure 2) andcan be simply calculated by

Ns = V∗ρ∗l , (11)

where the density at the edge of the effective bubble is assumed to be liquid density, andV∗ is the volume of the spherical shell, which can be estimated by

V∗ = 4π[R∗

outer3 − (

R∗outer − d∗

m

)3]/3. (12)

Nc in Eq. (10) is the molecule number within the inner spherical region, as shown inFigure 2, and its contribution to pressure will not be accounted for. The subscripts b, a,s, k, in Eq. (10) represent bond and angle interactions, the internal constraint force applied

Figure 1 Water molecule illustration (color figure available online).

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84 Y. MAO AND Y. ZHANG

Figure 2 Molecule number estimation within the thickness of one monomolecular layer (color figure availableonline).

to atoms from the SHAKE algorithm, and long-range force contribution, respectively; d∗mis the diameter of a single water molecule.

RESULTS AND DISCUSSION

All MD simulations in this article were performed using LAMMPS, which is an opensource software developed by Sandia National Laboratories. Figure 3 shows snapshots ofthe bubble formation and annihilation using Visual Molecular Dynamics [26] software.These snapshots show that the molecules within 10% thickness of the simulation box inthe depth direction. It can be seen that the heated molecules scattered the neighboringnonheated molecules until a bubble was created, and then the bubble was cooled and com-pressed by the surrounding subcooled liquid later. Physically, the molecules in the center(shown in white) were rescaled to the desired temperature at time 0, and heat was trans-ferred to the surrounding molecules along the temperature gradient by collision betweenthe hot molecules and the cold ones. As a result, a bubble was generated as shown duringthe period from time 0.368 to 2.162 ps. A 0.368-ps time lag existed before a clear bubbleemerged. After time 2.162 ps, the bubble was compressed by the surrounding cold liquidmolecules, causing it to become smaller and smaller until it finally disappeared.

The density profile within the 10% slab of the simulation box in the depth directionwas computed by uniformly partitioning the box into 1,000 small cubes whose densitywas obtained by averaging the 15 cases (see Figure 4). At time 0 ps, the density was notuniformly distributed but fluctuated around 1,000 kg/m3, which is the density of liquidwater at a pressure of 1 bar. It can be observed that bubble expanded from 0 to 2.162 psannihilation began during the latter period. The density increased to 1,000 kg/m3 again at5.06 ps. To demonstrate the heat spreading around the bubble, the temperature distribution

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NONEQUILIBRIUM MOLECULAR DYNAMICS SIMULATION 85

Time 0 ps Time 0.368 ps Time 0.644 ps

Time 0.874 ps Time 1.242 ps Time 2.162 ps

Time 3.082 ps Time 3.956 ps Time 5.06 ps

Figure 3 Snapshots of the bubble during the period from 0 to 5.06 ps (color figure available online)

was also computed and is shown in Figure 5. At the beginning, it can be seen that theregion around the heated zone had almost the same temperature. Within a very short period(0.368 ps), a clear temperature difference around the hot bubble was observed due to therapidity of the heat spreading to the surrounding cold liquid.

Figure 6 shows the calculated pressure at the surface of the generated bubble. Theresults from Nc = 0, 3, 5 were not significantly different, which means that the pressureestimated from Ns = 20 is reliable. It can be seen that the pressure value reached 1,450 Mpain 0.025 ps and then decreased relatively slowly to atmospheric pressure during the follow-ing period from 0.025 to 0.375 ps. The fluctuation period from 0.375 to 5 ps indicates thatthe pressure is decreased to the system pressure of 1 atm again. Interestingly, the pressuredramatically increased from 0.1 to 1,450 Mpa within a very short time, 0.025 ps, afterthe system was heated. Indeed, a comparison of time evolution between Figure 6 and thebubble behavior history shown in Figure 4 (or Figure 5) indicated that the bubble wasstill expanding even though the pressure was reduced to atmosphere pressure. Later, thispressure history was used to predict the evolution of the bubble radius using Eq. (9).

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86 Y. MAO AND Y. ZHANG

Time 0 ps Time 0.368 ps Time 0.644 ps

Time 0.874 ps Time 1.242 ps Time 2.162 ps

Time 3.082 ps Time 3.956 ps Time 5.06 ps

Figure 4 Density profile from 0 to 5.06 ps (color figure available online).

Figure 7 shows the time history of the calculated bubble radius. The hydrodynamicsline was obtained by using the hydrodynamic model (Rayleigh-Plesset equation). It wasobserved that if a nonzero surface tension was used, the span of the bubble radius evolutioncurve could not fit the one obtained from MD simulation. Indeed, even if a decent piecewisefunction of surface tension (S(R)) was used in the hydrodynamic model, the radius of bub-ble decreased to zero within a much shorter time than that obtained from MD. Therefore,in order to cover the whole period of bubble behavior, the surface tension was set to 0.The error bar line (MD) represents the statistical result obtained from the 15 MD cases; thecurve fitting line was fitted based on MD simulation results, which will be discussed later. Itcan be seen that the bubble immediately grew after heating and reached a maximum radiusduring the 4.6-ps-long period. In other words, the bubble achieved a maximum radius of0.6 nm at t =1.69 ps and disappeared at t = 4.6 ps. It is worth mentioning that the lower fluc-tuation in the radius after t = 4.6 ps was due to the density fluctuations in the liquid, whichmeans that the bubble size was zero during the period after 4.6 ps. Thus, the important Rvalues were those shown before t = 4.6 ps. The radius predicted by the Rayleigh-Plessetequation was obviously larger than that from MD simulation throughout the entire process

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NONEQUILIBRIUM MOLECULAR DYNAMICS SIMULATION 87

Time 0 ps Time 0.368 ps Time 0.644 ps

Time 0.874 ps Time 1.242 ps Time 2.162 ps

Time 3.082 ps Time 3.956 ps Time 5.06 ps

Figure 5 Temperature profile from 0 to 5.06 ps (color figure available online).

(roughly two times larger). Various possible surface tension functions were tested in orderto match the variation in radius obtained from MD simulation. However, as reported above,a possible surface tension function could not be found.

The existence of such a surface tension function was further studied by assuming thatthe Rayleigh-Plesset equation is still valid in liquid water. The bubble radius time historyfunction R(t) was obtained based on the radii from MD simulations by curve fitting, andthen the following polynomial equation with R2 = 99.97% was obtained:

R (t) =1.053 × 10−7t5 − 1.289 × 10−5t4 + 1.272 × 10−3t3 − 8.778 × 10−2t2

+ 2.258t − 3.415(13)

By substituting the fitted equation of R(t) into the Rayleigh-Plesset equation, both S(t)and S(R) were obtained and are plotted in Figure 8. Figure 8a shows the variation in sur-face tension with time. Figure 8b shows that the surface tension varied with corresponding

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88 Y. MAO AND Y. ZHANG

Figure 6 Pressure at the surface of the bubble (color figure available online).

Figure 7 Bubble radius variation in water (S refers to the sample point) (color figure available online).

radius. Both figures show that a nonphysical surface tension function (nonpositive sur-face tension) should be applied to reproduce results from MD simulation. A closer lookat Figure 8a indicates that, during the time period from 0 to ∼0.1 ps, a surface tensionof 0 should be applied and then a positive surface tension is used from ∼0.1 to ∼0.3 ps.

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NONEQUILIBRIUM MOLECULAR DYNAMICS SIMULATION 89

Figure 8 Surface tension profiles obtained by assuming the validity of the hydrodynamics equation (color figureavailable online).

Finally a long nonpositive value is used to drag the expanding bubble down. It can be seenfrom Figure 8b that there exists a pair of σ corresponding to one radius due to expandingand compressing. The surface tension σ corresponding to one radius does not exactly over-lap but has similar trends, which makes this method acceptable for finding possible S(R).Figure 8 indicates that a reasonable surface tension function S(R), which can reproduce theresults of MD simulation, does not exist.

From a physics aspect, the surface tension in a water bubble is contributed to by amore complicated force field than monatomic argon liquid. In fact, the atomic structure of awater molecule consists of two positively charged hydrogen atoms joined to one negativelycharged oxygen atom with a bond angle of 104.52◦. This unique composition causes oneside of the molecule to have a positive charge and the area in the opposite direction to have anegative charge [27]. Therefore, water molecules are attracted to each other and form strongmolecular bonds as a result of the polarity of charge in a single molecule. Comparing argonliquid and water liquid, the main difference is that a water molecule includes electrostaticforce, bond, and angle interactions. In other words, it has a more complicated force field.Considering the fact that there are many other possible water models existing in the liter-ature (three-site, four-site, five-site, rigid, flexible, etc.), the surface tension contributed bythis complicated force field should be further investigated for liquid water at a microscalein order to further verify the hydrodynamic equation. Thus, based on the current results forthe TIP3P model, it can only be concluded that predicting water bubble dynamics using theRayleigh-Plesset equation does not match predictions from MD simulation, which are dif-ferent from that for growth and annihilation of a nanosized bubble in the monatomic liquidargon.

CONCLUSIONS

Nanobubble formation and annihilation processes were simulated in polyatomicmolecule liquid (water) through nonequilibrium MD simulation. After equilibration for20,000 time steps, 80 molecules near the center of the simulation box were selected andinstantaneously heated to 3059.65 K by velocity rescaling. During the later 11,000 timesteps, the atomic movement of the system was observed until bubble extinction. In orderto improve the statistical accuracies, 15 MD simulations with different initial molecule

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90 Y. MAO AND Y. ZHANG

configurations were performed. The pressure at the surface of the nanosized bubbleimmediately increased to 1,450 Mpa from 0.1 Mpa within 0.025 ps and later decayed to1 bar within 0.4 ps. The bubble fully expanded to R = 0.6 nm at 1.69 ps and disappearedat 4.6 ps. A time lag of about 0.368 ps was observed before a clear bubble emerged afterheating the molecules. The density profile within the heated simulation box was recordedto show the evolution of density distribution and a clear bubble capture. The results, basedon the TIP3P model, showed that the hydrodynamic model based on the Rayleigh-Plessetequation is not valid in predicting nanosized bubble growth and annihilation in liquid water.Because different force fields can lead to different surface tensions on the bubble, and con-sidering the existence of many possible water models, further investigation on the validityof the Rayleigh-Plesset equation should be carried out.

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