混相流 vol.32 no

* 2017.10.24 ** 671-2280 2167 TEL: (079)267-4848 FAX: (079)267-4830 E-mail: [email protected]

Molecular Dynamics Simulation on Stability of Vapor Nanobubbles

ITOH Kazuhiro KISA Yuto YAMAMOTO Takuji MAEDA Kouji

Abstract We investigate the vapor bubble stability in liquid argon and water using a molecular dynamics simulation. The Lennard–Jones interparticle interaction potential is used to simulate the interaction forces between molecules. The Stillinger’s cluster criterion is employed to classify the vapor molecules evaporated from the bulk liquid. Using this criterion, the vapor molecules are determined to have no neighboring molecules within a 1.23 to 1.32σ radius, where σ is the interaction radius in the L–J potential. The pressure of vapor and liquid phase can be calculated from the virial equation of sate. The stability of bubble is disscussed applying the Young–Laplace equation. The spherical bubble shape is maintained, when the liquid pressure takes the negative value. The thickness of vapor–liquid interface and the number of molecules across vapor–liquid interface are not proportional to the size of bubbles. Keywords: Molecular Dynamics Simulation, Nanobubbles, Lennard–Jones Fluid,

Young–Laplace equation, Phase Equilibrium 1.

1 mR

PV PL

Young – LaplaceY–L

RPP LV

2 (1)

Epstein [1]s

Y–L[2-4]

PV

0.6 1.6 [5]

Wang [6] Zhukhoviiskii [7]

Stillinger [8] cluster criterion

rc 1.6

Y–L

混相流 32 巻１号（2018） 43

- 4 -

* 2017.10.24 ** 671-2280 2167 TEL: (079)267-4848 FAX: (079)267-4830 E-mail: [email protected]

Molecular Dynamics Simulation on Stability of Vapor Nanobubbles

ITOH Kazuhiro KISA Yuto YAMAMOTO Takuji MAEDA Kouji

Abstract We investigate the vapor bubble stability in liquid argon and water using a molecular dynamics simulation. The Lennard–Jones interparticle interaction potential is used to simulate the interaction forces between molecules. The Stillinger’s cluster criterion is employed to classify the vapor molecules evaporated from the bulk liquid. Using this criterion, the vapor molecules are determined to have no neighboring molecules within a 1.23 to 1.32σ radius, where σ is the interaction radius in the L–J potential. The pressure of vapor and liquid phase can be calculated from the virial equation of sate. The stability of bubble is disscussed applying the Young–Laplace equation. The spherical bubble shape is maintained, when the liquid pressure takes the negative value. The thickness of vapor–liquid interface and the number of molecules across vapor–liquid interface are not proportional to the size of bubbles. Keywords: Molecular Dynamics Simulation, Nanobubbles, Lennard–Jones Fluid,

Young–Laplace equation, Phase Equilibrium 1.

1 mR

PV PL

Young – LaplaceY–L

RPP LV

2 (1)

Epstein [1]s

Y–L[2-4]

PV

0.6 1.6 [5]

Wang [6] Zhukhoviiskii [7]

Stillinger [8] cluster criterion

rc 1.6

Y–L

Japanese J. Multiphase Flow Vol. 32 No. 1（2018）44

jiji ji

jiL

BL

LL r

rV

TkVNP

liquid, ,,6

1 (3)

liquidvapor ,

,vapor, ,

,61

ji ji

ji

jiji ji

jiV

BV

VV r

rr

rV

TkVNP

(4) N V L

V T3

[2,4]3

4

VL

LA

LL N

MNV (5)

M NA L VV

VL

VV

3.3.1 Y–L

1.5 2.1 nmFig. 2

NVEt = 0.2 ps t = 20 ps

Case 1

Case 3 20 psCase 2

Case 2 Fig.

3 4 PV

3 PL

41 PV’

Y–L 2g/RPV PL

PL PV

Y–L PL

Y–L

(a) Case 1 (left: t = 0.2 ps, right: t = 20 ps)

(b) Case 2 (left: t = 0.2 ps, right: t = 20 ps)

(c) Case 3 (left: t = 0.2 ps, right: t = 20 ps)

Fig. 2 Snapshots of cross-sectional view for Ar.

Y–L

D

RPPD LV

2 (6)

0 4 8 12 16 20-25

-20

-15

-10

-5

0

5

Time [ps]

Pres

sure

[MPa

]

Case 2 PV by eq. (4) PL by eq. (3) PV' by equation of state

Fig. 3 Variation of pressure for Ar.

D Fig. 4 Park [13]

Tolman6

10.4 mN/m[13,14]

6

Case 1 DCase 2 Case 3 D

Case 3500 ps D

[11]

-5 0 5 10 15 20-30

-20

-10

0

10

20

30

Time [ps]

D [M

Pa]

NVT const. NVE const.

Case1Case2Case3

Fig. 4 Time variation of D in eq. (6).

3.2

t = 0.4, 10, 20 psFig. 5 NL*

13

NL* NL0* Case 1

Case 3

0 1 2 3 4 500.20.40.60.8

1

Case 2 0.4 ps 10 ps 20 ps

0 1 2 3 4 500.20.40.60.8

1


N* L

[-]

0 1 2 3 4 500.20.40.60.8

1


r [nm]

Fig. 5 Number density around vapor-liquid interface.

Fig. 6NL* 0.1NL0* 0.9NL0*

10 90 thickness[15]

0 4 8 12 16 200.4

0.6

0.8

1

1.2

1.4

Time [ps]

[nm

]

Case 2 Case 3

Fig. 6 Variation of interface thickness.

混相流 32 巻１号（2018） 45

jiji ji

jiL

BL

LL r

rV

TkVNP

liquid, ,,6

1 (3)

liquidvapor ,

,vapor, ,

,61

ji ji

ji

jiji ji

jiV

BV

VV r

rr

rV

TkVNP

(4) N V L

V T3

[2,4]3

4

VL

LA

LL N

MNV (5)

M NA L VV

VL

VV

3.3.1 Y–L

1.5 2.1 nmFig. 2

NVEt = 0.2 ps t = 20 ps

Case 1

Case 3 20 psCase 2

Case 2 Fig.

3 4 PV

3 PL

41 PV’

Y–L 2g/RPV PL

PL PV

Y–L PL

Y–L

(a) Case 1 (left: t = 0.2 ps, right: t = 20 ps)

(b) Case 2 (left: t = 0.2 ps, right: t = 20 ps)

(c) Case 3 (left: t = 0.2 ps, right: t = 20 ps)

Fig. 2 Snapshots of cross-sectional view for Ar.

Y–L

D

RPPD LV

2 (6)

0 4 8 12 16 20-25

-20

-15

-10

-5

0

5

Time [ps]

Pres

sure

[MPa

]

Case 2 PV by eq. (4) PL by eq. (3) PV' by equation of state

Fig. 3 Variation of pressure for Ar.

D Fig. 4 Park [13]

Tolman6

10.4 mN/m[13,14]

6

Case 1 DCase 2 Case 3 D

Case 3500 ps D

[11]

-5 0 5 10 15 20-30

-20

-10

0

10

20

30

Time [ps]

D [M

Pa]

NVT const. NVE const.

Case1Case2Case3

Fig. 4 Time variation of D in eq. (6).

3.2

t = 0.4, 10, 20 psFig. 5 NL*

13

NL* NL0* Case 1

Case 3

0 1 2 3 4 500.20.40.60.8

1


0 1 2 3 4 500.20.40.60.8

1


N* L

[-]

0 1 2 3 4 500.20.40.60.8

1


r [nm]

Fig. 5 Number density around vapor-liquid interface.

Fig. 6NL* 0.1NL0* 0.9NL0*

10 90 thickness[15]

0 4 8 12 16 200.4

0.6

0.8

1

1.2

1.4

Time [ps]

[nm

]

Case 2 Case 3

Fig. 6 Variation of interface thickness.


[14,16]Case 2 Case 2 Case

3 t = 12 20 ps

Table 3Case 3

30% Case 290% Case 2

Table 3 Mean value of and R between 12 and 20 ps. Mean [nm] Mean R [nm]

Case 2 1.09 1.18 Case 3 0.87 3.51

NLV

NVL Fig. 7 0.9NL0* 0.1NL0* NLV

0.9NLV 0.1NLV NVL

0.9NL0*

0.1NL0*

0 4 8 12 16 200

10

20

30Case 2 0.1NLV, 0.9NLV

0.1NVL, 0.9NVL

0 4 8 12 16 200

20

40

60

80

NLV

, N

VL [

num

ber]

Case 3

Time [ps]

0.1NLV, 0.9NLV 0.1NVL, 0.9NVL

Fig. 7 Variation of NLV and NVL.

0.9NL0*

Case 2 Case 3NLV NVL

Table 3 Case 3Case 2 9 0.9NL0*

4

3.3

L–J Table 2Fig. 8

1.3 1.6 nm

ArFig. 8

(a) Case 4 (273 K, left: t = 0.2 ps, right: t = 20 ps)

(b) Case 5 (300 K, left: t = 0.2 ps, right: t = 20 ps)

(c) Case 6 (640 K, left: t = 0.2 ps, right: t = 20 ps)

Fig. 8 Snapshots of cross-sectional view for water.

Case 4 t = 20 ps

Case 5

Case 6 Case 4 Case 5 PV, PL

Fig. 9 100MPa

PL

Ar PV

0 4 8 12 16 20

-100

-50

0

Case 4 PV, PL

0 4 8 12 16 20

-100

-50

0

Time [ps]

Pres

sure

[MPa

]

Case 5 PV, PL

Fig. 9 Variation of pressure for water.

4.

L-JStillinger cluster

criterion

1.4 nm

Y L

NVL, NLV

[1] Epstein, P. S. and Plesset, M.S., On the Stability of Gas Bubbles in Liquid-Gas Solutions, J. Chem. Phys. Vol. 18 (11) 1505–1509 (1950).

[2] Matsumoto, M., How can Molecular Simulations Contribute to Thermal Engineering, J. Therm.

Sci. Techol., 3(2) 309–318 (2008). [3] Tsuda, S., Takagi, S. and Matsumoto, Y., A

Study on the Growth of Cavitation Bubble Nuclei using Large-scale Molecular Dynamics Simulations, Fluid. Dym. Res. Vol. 40 606–615 (2008).

[4] Matsumoto, M. and Tanaka, K., Nano Bubble - Size Dependence of Surface Tension and Inside Pressure, Fluid Dynamic Res., Vol. 40(7-8), 546–553 (2008).

[5] Maruyama, S. and Kimura, T., A Molecular Dynamics Simulation of a Bubble Nucleation on Solid Surface, Proc. 5th ASME-JSME Thermal. Eng. Joint Conf., AJTE99-6511 (1999).

[6] Wang, Z. J., Valeriani, C. and Frenkel, D., Homogeneous Bubble Nucleation Driven by Local Hot Spots, J. Phys. Chem. B, Vol. 113, 3776–3784 (2009).

[7] Zhukhovitskii, D. I., Molecular Dynamics Study of Nanobubbles in the Equilibrium Lennard-Jones Fluid, J. Chem. Phys. Vol. 139, 164513 (2013).

[8] Stillinger Jr. F. H., Rigorous Basis of the Frenkel-Band Theory of Association Equilibrium, J. Chem. Phys. Vol. 38, 1486–1494 (1963).

[9] Matsumoto, M., Miyamoto, K., Ohguchi, K. and Kinjo, T., Non-equilibrium Vapor Condensation on Shrinking Bubble Surface, Trans. Japan Soc. Mech. Eng., Ser. B, Vol. 68(671), 1890–1897 (2002).

[10] Yamamoto, T. and Matsumoto, M., Yamamoto, Initial Stage of Nucleate Boiling, J. Thermal Sci. Technol., Vol. 7(1), 334–349 (2012).

[11] Itoh, K., Kisa, Y., Yamamoto, T. and Maeda, K., Molecular Dynamics Study on Phase Equilibrium around Vapor bubbles in Low-Density Liquid Argon, J. Molecular Liquids, Vol. 230, 322–328 (2017).

[12] Japan Soc. Mech. Eng., ed., Mechanical Engineers’ Handbook, 6 (Comp. Mech.), Sec. 10.22, JSME, Tokyo (2007).

[13] Park, S. H., Weng, J.G. and Tien, C.L., Molecular Dynamics Study on Surface Tension of Microbubbles, Int. J. Heat Mass Trans., Vol. 44, 1849–2189 (2001).

[14] Yaguchi, H., Yano, T. and Fujikawa, S., Molecular Dynamics Study of Vapor-Liquid Equilibrium State of an Argon Nanodroplet and Its Vapor, Tran. Japan Soc. Mech. Eng., Ser. B, Vol. 75(752), 658–667 (2009).

[15] Kon, M., Kobayashi, K. and Watanabe, M., Method of Determining Kinetic Boundary Conditions in Net Evaporation/Condensation, Phys. Fluids, Vol. 26, 072003 (2014).

[16] Kobayashi, K., Sasaki, K., Kon, M., Fujii, H. and Watanabe, M., Kinetic Boundary Conditions for Vapor–Gas Binary Mixture, Microfluid Nanofluid, 21:53 (2017).

混相流 32 巻１号（2018） 47

[14,16]Case 2 Case 2 Case

3 t = 12 20 ps

Table 3Case 3

30% Case 290% Case 2

Table 3 Mean value of and R between 12 and 20 ps. Mean [nm] Mean R [nm]

Case 2 1.09 1.18 Case 3 0.87 3.51

NLV

NVL Fig. 7 0.9NL0* 0.1NL0* NLV

0.9NLV 0.1NLV NVL

0.9NL0*

0.1NL0*

0 4 8 12 16 200

10

20

30Case 2 0.1NLV, 0.9NLV

0.1NVL, 0.9NVL

0 4 8 12 16 200

20

40

60

80

NLV

, N

VL [

num

ber]

Case 3

Time [ps]

0.1NLV, 0.9NLV 0.1NVL, 0.9NVL

Fig. 7 Variation of NLV and NVL.

0.9NL0*

Case 2 Case 3NLV NVL

Table 3 Case 3Case 2 9 0.9NL0*

4

3.3

L–J Table 2Fig. 8

1.3 1.6 nm

ArFig. 8

(a) Case 4 (273 K, left: t = 0.2 ps, right: t = 20 ps)

(b) Case 5 (300 K, left: t = 0.2 ps, right: t = 20 ps)

(c) Case 6 (640 K, left: t = 0.2 ps, right: t = 20 ps)

Fig. 8 Snapshots of cross-sectional view for water.

Case 4 t = 20 ps

Case 5

Case 6 Case 4 Case 5 PV, PL

Fig. 9 100MPa

PL

Ar PV

0 4 8 12 16 20

-100

-50

0

Case 4 PV, PL

0 4 8 12 16 20

-100

-50

0

Time [ps]

Pres

sure

[MPa

]

Case 5 PV, PL

Fig. 9 Variation of pressure for water.

4.

L-JStillinger cluster

criterion

1.4 nm

Y L

NVL, NLV

[1] Epstein, P. S. and Plesset, M.S., On the Stability of Gas Bubbles in Liquid-Gas Solutions, J. Chem. Phys. Vol. 18 (11) 1505–1509 (1950).

[2] Matsumoto, M., How can Molecular Simulations Contribute to Thermal Engineering, J. Therm.

Sci. Techol., 3(2) 309–318 (2008). [3] Tsuda, S., Takagi, S. and Matsumoto, Y., A

Study on the Growth of Cavitation Bubble Nuclei using Large-scale Molecular Dynamics Simulations, Fluid. Dym. Res. Vol. 40 606–615 (2008).

[4] Matsumoto, M. and Tanaka, K., Nano Bubble - Size Dependence of Surface Tension and Inside Pressure, Fluid Dynamic Res., Vol. 40(7-8), 546–553 (2008).

[5] Maruyama, S. and Kimura, T., A Molecular Dynamics Simulation of a Bubble Nucleation on Solid Surface, Proc. 5th ASME-JSME Thermal. Eng. Joint Conf., AJTE99-6511 (1999).

[6] Wang, Z. J., Valeriani, C. and Frenkel, D., Homogeneous Bubble Nucleation Driven by Local Hot Spots, J. Phys. Chem. B, Vol. 113, 3776–3784 (2009).

[7] Zhukhovitskii, D. I., Molecular Dynamics Study of Nanobubbles in the Equilibrium Lennard-Jones Fluid, J. Chem. Phys. Vol. 139, 164513 (2013).

[8] Stillinger Jr. F. H., Rigorous Basis of the Frenkel-Band Theory of Association Equilibrium, J. Chem. Phys. Vol. 38, 1486–1494 (1963).

[9] Matsumoto, M., Miyamoto, K., Ohguchi, K. and Kinjo, T., Non-equilibrium Vapor Condensation on Shrinking Bubble Surface, Trans. Japan Soc. Mech. Eng., Ser. B, Vol. 68(671), 1890–1897 (2002).

[10] Yamamoto, T. and Matsumoto, M., Yamamoto, Initial Stage of Nucleate Boiling, J. Thermal Sci. Technol., Vol. 7(1), 334–349 (2012).

[11] Itoh, K., Kisa, Y., Yamamoto, T. and Maeda, K., Molecular Dynamics Study on Phase Equilibrium around Vapor bubbles in Low-Density Liquid Argon, J. Molecular Liquids, Vol. 230, 322–328 (2017).

[12] Japan Soc. Mech. Eng., ed., Mechanical Engineers’ Handbook, 6 (Comp. Mech.), Sec. 10.22, JSME, Tokyo (2007).

[13] Park, S. H., Weng, J.G. and Tien, C.L., Molecular Dynamics Study on Surface Tension of Microbubbles, Int. J. Heat Mass Trans., Vol. 44, 1849–2189 (2001).

[14] Yaguchi, H., Yano, T. and Fujikawa, S., Molecular Dynamics Study of Vapor-Liquid Equilibrium State of an Argon Nanodroplet and Its Vapor, Tran. Japan Soc. Mech. Eng., Ser. B, Vol. 75(752), 658–667 (2009).

[15] Kon, M., Kobayashi, K. and Watanabe, M., Method of Determining Kinetic Boundary Conditions in Net Evaporation/Condensation, Phys. Fluids, Vol. 26, 072003 (2014).

[16] Kobayashi, K., Sasaki, K., Kon, M., Fujii, H. and Watanabe, M., Kinetic Boundary Conditions for Vapor–Gas Binary Mixture, Microfluid Nanofluid, 21:53 (2017).


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