Recommendation for the future work

Chapter 5 Concluding Remarks

5.2 Recommendation for the future work

Based on the studies presented in this thesis, several important future works are outlined as follows:

1. To study more detailed collision dynamics of the pair droplets considering the followings, but not limited to:

i. Size effects of the two droplets;

ii. Relative sizes effects of the two droplets;

iii. Parametric effects of the inter-atomic potential between droplet and background gas;

iv. Rotational effects of the colliding droplets;

v. More realistic intermolecular potential models such as REBO [Brenner et. al., 2002] for hydrocarbon droplets;

2. To study multi-droplet collision dynamics under various kinds of test conditions, for example, three-droplet collision;

3. To study the droplet-solid collisions under various kinds of conditions;

References

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M. and Zhao. Z., Wetting effects on the spreading of a liquid droplet colliding with a flat surface: experiment and modeling., Physics Fluids, vol.7, pp.236-247, 1995.

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13. Hu, Z. L., Bruintjes, R. T. and Betterton, E.A., Sensitivity of cloud droplet growth to collision and coalescence efficiencies in a parcel model. Journal of Atmospheric Sciences, vol.15, pp.2502-2515, 1998.

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Computational Physics, vol.198, pp.628–644, 2004.

15. J.-S. Wu and K.-C. Tseng, Concurrent DSMC Method Using Dynamic Domain Decomposition, The 23rd International Symposium on Rarefied Gas Dynamics, Whistler Conference Centre Whistler, British Columbia, July, pp.20-25, 2002.

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17. Karypi. G. s, Schloegel. K. and Kumar. V., ParMetis, University of Minnesota, Department of computer Science, September, 1998.

18. Lennard-Jones J. E., The determination of molecular fields. I. from the variation of the viscosity of gas with temperature., Proc. Roy. Soc. (Lond), vol.106A, pp.441, 1924.

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experimentally established coalescence efficiencies and fragment size

distributions in breakup. Journal of Atmospheric Sciences, vol.39, pp.1591-1606, 1982.

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21. Murad, S. and Law, C. K., Molecular simulation of droplet collision in the presence of ambient. Molecular Physics, vol.96, pp.81-85, 1999.

22. Nicol. D. M. and J. Saltz. H. et al., Dynamic Remapping of Parallel Computations with Varying Resource Demands, IEEE Transactions on Computer, vol.37, pp.1073-1087, 1988.

23. Pan, K. L. and Law, C. K., On the dynamics of head-on droplet collision:

experiment and simulation. The 42^nd AIAA Aerospace Sciences Meeting &

Exhibit, Reno, USA, Paper Number 1159 (January, 2004).

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AIAA-2005-0352 (January, 2005).

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26. Plimpton. S., Fast Parallel Algorithms for Short-Range Molecular Dynamics, J.

of Computational Physics, vol.117, pp.1-19, 1995.

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impingment of liquid droplets in spraying processes., Metall. Trans. B, vol.22, pp.901, 1991.

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Tables

* System of Units Used in Soft-Sphere Molecular Dynamics Programs * Fundamental Quantities:

Mass m = mass of one atom Length σ

Energy ε Time σ m/ε

Derived Quantities:

Adiabatic compressibility k_s^* =k_sε/σ³

Configurational internal energy U_c^* =U_c/N_ε = u^* = u/N_ε Density ρ^* =Nσ³/V

Force F^* =Fσ/ε Heat capacity C_v^* =C_v/Nk Radial position r^* =r/σ Pressure P^* =Pσ³ /ε Temperature T^* =kT/ε Thermal pressure coefficient γ_v^* =γ_vσ³/k Total energy E^* =E/Nε

Velocity υ^* =υ m/ε

Table. 1 nondimensionalize

Temp.(T*) Density (ρ*) No. of Link-cells

condensed 0.7 0.7 75*75*75

vaporized 1.1 0.7 75*75*75

supercritical 0.7 0.7 39*39*39

Table. 2 Simulation conditions for three different cases via PCMD code (condensed, vaporized and supercritical states).

(a)

(b)

(c)

(d)

Figures

Fig. 1. 1 Terminology of possible droplet-droplet collision outcome, (a) bounce, (b) coalescence, (c) disruption and (d) fragmentation.

(Fig of Pan, K. L. and Law, C. K., The 42^nd AIAA Aerospace Sciences Meeting & Exhibit.)

Fig. 1. 2 Schematic of different droplet collision regimes as function of Weber number (We) and impact number (b).

Fig. 2. 1 Cartesian frame

X Y

Z

O

r

atom i

Fig. 2. 2 Lennard-Jones (LJ) pairwise intermolecular potential

Repulsive

Attractive force

O

(distance between atoms)

X Y

**potential u*(r)**

**force f*(r)**

Fig. 2. 3 Water molecules i and j.

H

₄

H

₃

H

₂

O

₅

O

r

₅₆

i

j

-2q -2q

q q

M

₇

M

₈

r

=0.09572 nm r

=0.024994 nm

∠HOH=104.52

Fig. 2. 4 periodic boundary conditions

(a)

(b)

(c)

Fig. 2. 5 (a.)all pair, (b)cell link, and (c)Verlet list methods

Fig. 2. 6 Verlet list

Fig. 2. 7 Verlet +Cell link

Fig. 2. 8 Surface tension concept by Tabor [1991]

Start

velocities of atoms in the neighbor list

Compute force

Send force data to other atoms in the neighbor list

Integrate to update positions & velocities of atoms

Apply boundary conditions

Fig.3. 1 Proposed flow chart for parallel molecular dynamics simulation using dynamic domain decomposition.

(a)

(b)

(c)

Fig. 3. 1 Evolution of domain decomposition for large problem size using 25 processors at start and final. (a) condensed state; (b) vaporized state; (c) supercritical state.

0 50000 100000

Atoms number per CPU. upper threshold value (+20%).

lower threshold value (-20%).

(a)

Atoms number per CPU. upper threshold value (+20%).

lower threshold value (-20%).

(b)

Atoms number per CPU. upper threshold value (+20%).

lower threshold value (-20%).

(c)

Fig.3.3 Distribution of the number of atoms in each processor as a function of simulation time steps (25 processors). (a) condensed state; (b) vaporized state; (c) supercritical state.

0 10 20 30 40 50

Number of Processors

0 10 20 30 40 50

Speedup

Ideal speedup Condensed Vaporized Supercritical

Vaporized (non-repartition **)

Sym. Data*

* 1million L-J atoms

** non-repartition after 30,000 steps

Fig.3.4 Parallel speedup as a function of the number of processors for three different test cases (condensed, vaporized and supercritical states).

0 500 1000 1500

Relative Velocity(m/sec) 0

25 50 75 100

b (A)

Droplet-Collision under V acuum:

Direct Coalescence Strecting Separation Shattering

Fig. 4. 1 Distribution map of various regimes under vacuum ambient.

0 500 1000 1500 Relative Velocity (m/sec)

0 25 50 75 100

b (A)

Droplet-Collision under 0.055atm ambient gas:

Direct Coalescence Stretching Coalescence Stretching Separation Shattering

Fig. 4. 2 Distribution map of various regimes under low pressurized ambient (~0.055 atm).

0 500 1000 1500 Relative Velocity (m/sec)

0 25 50 75 100

b (A)

Droplet-Collision under 0.55atm ambient gas:

Direct Coalescence Stretching Coalescence Stretching Separation Shattering

Fig. 4. 3 Distribution map of various regimes under high pressurized ambient (~0.55 atm).

40 80 120 160 200 Time-Step

0 0.02 0.04 0.06 0.08

Evaporation Rate (%)

Evaporation rate:

Vacuum

Low pressurized ambient (0.055 atm) High pressurized ambient (0.55 atm)

Fig. 4. 4 The evaporation rate of simulation case which b=0.25 V=250 m/s, under vacuum, low pressurized ambient and high pressurized ambient, respectively.

(a) (b)

Fig. 4.5 Head-on (b= 0) droplets pair collision initial setup, (a) y-z plane without vapor ambient, (b) x-z plane without vapor ambient, (c) y-z plane under vapor ambient, (d) x-z plane under vapor ambient

(a)

(b)

Fig. 4.6 Non-head-on (ex; b= 0.5) droplets pair collision initial setup, (a) x-y plane without vapor ambient, (b) x-y plane under vapor ambient.

(a) (b)

(e) (f)

Fig. 4.7 Snapshot of droplet pair collision under vacuum b=0 V=1250 m/s, at (a)10ps, (b)50ps, (c)75ps, (d)100ps, (e)125ps, (f)175ps.

(a) (b)

(e) (f)

Fig. 4.8 Snapshot of droplet pair collision under vacuum, b=0, V=1375 m/s, at (a)10ps, (b)40ps, (c)75ps, (d)150ps, (e)200ps, (f)300ps.

(a) (b)

(e) (f)

Fig. 4.9 Snapshot of droplet pair collision under vacuum, b=0, V=1500 m/s, at (a)10ps, (b)30ps, (c)75ps, (d)100ps, (e)175ps, (f)300ps.

(a) (b)

(e) (f)

Fig. 4.10 Snapshot of droplet pair collision under low pressurized ambient (~0.055 atm), b=0, V=1250 m/s, at (a)10ps, (b)40ps, (c)75ps, (d)100ps, (e)125ps, (f)200ps.

(a) (b)

(e) (f)

Fig. 4.11 Snapshot of droplet pair collision under low pressurized ambient (~0.055 atm), b=0, V=1375 m/s, at (a)10ps, (b)40ps, (c)75ps, (d)100ps, (e)150ps, (f)250ps.

(a) (b)

(e) (f)

Fig. 4.12 Snapshot of droplet pair collision under low vapor ambient, b=0, V=1500 m/s, at (a)10ps, (b)40ps, (c)60ps, (d)90ps, (e)150ps, (f)250ps.

(a) (b)

(e) (f)

Fig. 4.13 Snapshot of droplet pair collision under high vapor ambient, b=0, V=1250 m/s, at (a)10ps, (b)40ps, (c)75ps, (d)100ps, (e)125ps, (f)200ps.

(a) (b)

(e) (f)

Fig. 4.14 Snapshot of droplet pair collision under high vapor ambient, b=0, V=1375 m/s, at (a)10ps, (b)40ps, (c)75ps, (d)100ps, (e)250ps, (f)325ps.

(a) (b)

(e) (f)

Fig. 4.15 Snapshot of droplet pair collision under high vapor ambient, b=0, V=1500 m/s, at (a)10ps, (b)40ps, (c)60ps, (d)90ps, (e)150ps, (f)250ps.

(a) (b)

(e) (f)

Fig. 4.2 Snapshot of droplet pair collision under vacuum, b=0.25, V=250 m/s, at (a)10ps, (b)50ps, (c)100ps, (d)200ps, (e)250ps, (f)375ps. This case is classified in Coalescence regime.

(a) (b)

(e) (f)

Fig. 4.3 Snapshot of droplet pair collision under vacuum, b=0.625, V=1000 m/s, at (a)10ps, (b)40ps, (c)75ps, (d)100ps, (e)150ps, (f)250ps. This case is classified in Stretching Separation regime.

(a) (b)

(e) (f)

Fig. 4.4 Snapshot of droplet pair collision under vacuum b=0.25, V=1375 m/s, at (a)10ps, (b)40ps, (c)75ps, (d)100ps, (e)150ps, (f)250ps. This case is classified in Shattering regime.

(a) (b)

(e) (f)

Fig. 4.5 Snapshot of droplet pair collision under low pressurized ambient,

b=0.25, V=250 m/s, at (a)10ps, (b)50ps, (c)100ps, (d)200ps, (e)250ps, (f)375ps. This case is classified in Coalescence regime.

(a) (b)

(e) (f)

Fig. 4.20 Snapshot of droplet pair collision under low pressurized ambient, b=0.25, V=750 m/s, at (a)10ps, (b)40ps, (c)75ps, (d)150ps, (e)250ps, (f)500ps. This case is classified in Stretching Coalescence regime.

(a) (b)

(e) (f)

Fig. 4.21 Snapshot of droplet pair collision under low pressurized ambient, b=0.625, V=1000m/s, at (a)10ps, (b)40ps, (c)75ps, (d)100ps, (e)150ps, (f)250ps.

This case is classified in Stretching Separation regime.

(a) (b)

(e) (f)

Fig. 4.22 Snapshot of droplet pair collision under low pressurized ambient, b=0.

25, V=1375m/s, at (a)10ps, (b)40ps, (c)75ps, (d)100ps, (e)150ps, (f)250ps. This case is classified in Shattering regime.

(a) (b)

(e) (f)

Fig. 4.23 Snapshot of droplet pair collision under high pressurized ambient, b=0.25, V=250 m/s, at (a)10ps, (b)50ps, (c)100ps, (d)200ps, (e)250ps, (f)375ps. This case is classified in Coalescence regime.

(a) (b)

(e) (f)

Fig. 4.24 Snapshot of droplet pair collision under high pressurized ambient, b=0.25, V=750 m/s, at (a)10ps, (b)40ps, (c)75ps, (d)150ps, (e)250ps, (f)500ps. This case is classified in Stretching Coalescence regime.

(a) (b)

(e) (f)

Fig. 4.25 Snapshot of droplet pair collision under high pressurized ambient, b=0.625, V=1000m/s, at (a)10ps, (b)40ps, (c)75ps, (d)100ps, (e)150ps, (f)250ps.

This case is classified in Stretching Separation regime.

(a) (b)

(e) (f)

Fig. 4.26 Snapshot of droplet pair collision under high pressurized ambient, b=0.25, V=1500m/s, at (a)10ps, (b)40ps, (c)75ps, (d)100ps, (e)150ps, (f)250ps. This case is classified in Shattering regime.

(a) (b)

(e) (f)

Fig. 4.27 Snapshot of droplet pair collision under high pressurized ambient, b=0, V=10 m/s, at (a)25ps, (b)250ps, (c)500ps, (d)750ps, (e)1000ps, (f)1250ps. This case is classified in Bounce regime.

(a) (b)

(e) (f)

Fig. 4.28 Snapshot of droplet pair collision under high pressurized ambient, b=0, V=30 m/s, at (a)25ps, (b)250ps, (c)500ps, (d)750ps, (e)1000ps, (f)1250ps. This case is classified in Bounce regime.

(a) (b)

(e) (f)

Fig. 4.29 Snapshot of droplet pair collision under high pressurized ambient (0.55 atm, T=324K ), b=0, V=10 m/s, at (a)25ps, (b)250ps, (c)500ps, (d)750ps, (e)1000ps, (f)1200ps. This case is classified in Bounce regime.

(a) (b)

(e) (f)

Fig. 4.30 Snapshot of droplet pair collision under high pressurized ambient (0.55 atm, T=324K ), b=0, V=30 m/s, at (a)25ps, (b)250ps, (c)500ps, (d)750ps, (e)1000ps, (f)1250ps. This case is classified in Bounce regime.

⁰ ⁴⁰⁰⁰ Size of Fragments⁸⁰⁰⁰ ¹²⁰⁰⁰ ¹⁶⁰⁰⁰ ²⁰⁰⁰⁰

Fig. 4.31 Snapshot of density contour and clusters size distribution under vacuum, b=0, V=1250 m/s, at (a)25ps, (b)75ps, (b)150ps. This case is classified in Stretching Coalescence regime.

⁴⁰⁰⁰ Size of Fragment⁸⁰⁰⁰ ¹²⁰⁰⁰ ¹⁶⁰⁰⁰ ²⁰⁰⁰⁰

Fig. 4.32 Snapshot of density contour and clusters size distribution under low pressurized ambient, b=0, V=1250 m/s, at (a)25ps, (b)75ps, (b)150ps. This case is classified in Stretching Coalescence regime.

⁴⁰⁰⁰ Size of Fragment⁸⁰⁰⁰ ¹²⁰⁰⁰ ¹⁶⁰⁰⁰ ²⁰⁰⁰⁰

Fig. 4.33 Snapshot of density contour and clusters size distribution under high pressurized ambient, b=0, V=1250 m/s, at (a)25ps, (b)75ps, (b)150ps. This case is classified in Stretching Coalescence regime.

⁰ ⁴⁰⁰⁰ Size of Fragments⁸⁰⁰⁰ ¹²⁰⁰⁰ ¹⁶⁰⁰⁰ ²⁰⁰⁰⁰

Fig. 4.34 Snapshot of density contour and clusters size distribution under vacuum, b=0, V=1500 m/s, at (a)25ps, (b)75ps, (b)150ps. This case is classified in Shattering regime.

⁴⁰⁰⁰ Size of Fragment⁸⁰⁰⁰ ¹²⁰⁰⁰ ¹⁶⁰⁰⁰ ²⁰⁰⁰⁰

Fig. 4.35 Snapshot of density contour and clusters size distribution under low pressurized ambient, b=0, V=1500 m/s, at (a)25ps, (b)75ps, (b)150ps. This case is classified in Shattering regime.

⁴⁰⁰⁰ Size of Fragment⁸⁰⁰⁰ ¹²⁰⁰⁰ ¹⁶⁰⁰⁰ ²⁰⁰⁰⁰

Fig. 4.36 Snapshot of density contour and clusters size distribution under high pressurized ambient, b=0, V=1500 m/s, at (a)25ps, (b)75ps, (b)150ps. This case is classified in Shattering regime.

0 100 200 300 400 500

Fig. 4.37 Measurements of largest fragment of droplet pair collision under vacuum, b=0.25, V=250 m/s, (a)Number of atoms, (b)Vibrational temperature (k), (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Coalescence regime.

0 100 200 300 400

Fig. 4.38 Measurements of largest fragment of droplet pair collision under

vacuum, b=0.625, V=1000 m/s, (a)Number of atoms, (b)Vibrational temperature (k), (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Stretching Separation regime.

0 100 200 300

Fig. 4.39 Measurements of largest fragment of droplet pair collision under

vacuum, b=0.25, V=1375 m/s, (a)Number of atoms, (b)Vibrational temperature (k), (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Shattering regime.

0 100 200 300 400 500

Fig. 4.40 Measurements of largest fragment of droplet pair collision under low pressurized ambient, b=0.25, V=250 m/s, (a)Number of atoms, (b)Vibrational temperature (k), (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Coalescence regime.

0 200 400 600

Fig. 4.41 Measurements of largest fragment of droplet pair collision under low pressurized ambient, b=0.25, V=750 m/s, (a)Number of atoms, (b)Vibrational temperature (k), (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Stretching Coalescence regime.

0 100 200 300 400

Fig. 4.42 Measurements of largest fragment of droplet pair collision under low pressurized ambient, b=0.625, V=1000m/s, (a)Number of atoms, (b)Vibrational temperature (k), (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Stretching Separation regime.

0 100 200 300

Fig. 4.43 Measurements of largest fragment of droplet pair collision under low pressurized ambient, b=0.25, V=1375m/s, (a)Number of atoms, (b)Vibrational temperature, (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Shattering regime.

0 100 200 300 400 500

Fig. 4.44 Measurements of largest fragment of droplet pair collision under high pressurized ambient, b=0.25, V=250 m/s, (a)Number of atoms, (b)Vibrational temperature (k), (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Coalescence regime.

0 200 400 600

Fig. 4.45 Measurements of largest fragment of droplet pair collision under high pressurized ambient, b=0.25, V=750 m/s, (a)Number of atoms, (b)Vibrational temperature (k), (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Stretching Coalescence regime.

0 100 200 300 400

Fig. 4.46 Measurements of largest fragment of droplet pair collision under high pressurized ambient, b=0.625, V=1000m/s, (a)Number of atoms, (b)Vibrational temperature (k), (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Stretching Separation regime.

0 50 100 150 200 250

Fig. 4.47 Measurements of largest fragment of droplet pair collision under high pressurized ambient, b=0.25, V=1500m/s, (a)Number of atoms, (b)Vibrational temperature, (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Shattering regime.

0 200 400 600

Fig. 4.48 Measurements of largest fragment of droplet pair collision under high pressurized ambient, b=0, V=30m/s, (a)Number of atoms, (b)Vibrational

temperature (k), (c)Rotational energy, (d)Angular momentum, respectively. This case is classified in Bouncing regime.

50 60 70 100

1000 10000

100 1000 10000

Atoms distributed at various time (PS):

25 250 500 750 1000 1250

Fig. 4.49 The atoms distributions in X direction of bounce case under high pressurized ambient, b=0, V=10 m/sec.

50 60 70 100

1000 10000

100 1000 10000

Atoms distributed at various time (PS):

25 250 500 750 1000 1250

Fig. 4.50 The atoms distributions in X direction of bounce case under high pressurized ambient, b=0, V=30 m/sec..

50 60 70

Position of X-Axis

100 1000 10000

Number of atoms

100 1000 10000

Atoms distributed at various time (PS):

25 250 500 750 1000 1250

Fig. 4.51 The atoms distributions in X direction of bounce case under high pressurized ambient (0.55 atm, T=324 K), b=0, V=10 m/sec.

50 60 70

Position of X-Axis

100 1000 10000

Number of atoms

Atoms distributed at various time (PS):

25 250 500 750 1000 1250

100 1000 10000

Fig. 4.52 The atoms distributions in X direction of bounce case under high pressurized ambient(0.55 atm, T=324 K), b=0, V=30 m/sec..

Fig. 4.53 The variation of surface tensions distributions of coalescence case under low pressurized ambient (0.055 atm, T=216 K), b=0.25, V=750 m/sec..

T=2.5 ps

T=5 ps

T=7.5 ps

T=10 ps

T=50 ps

T=150 ps

T=250 ps

T=500 ps

Fig. 4.54 The variation of surface tensions distributions of separation case under low pressurized ambient (0.055 atm, T=216 K), b=0.625, V=1000 m/sec..

T=10 ps T=40 ps

T=75 ps T=100 ps

T=250 ps T=150 ps

Fig. 4.55 The variation of surface tensions distributions of bouncing case under low pressurized ambient (0.055 atm, T=216 K), b=0, V=30 m/sec..

T=250 ps

T=750 ps

T=1000 ps

Autobiography

Yu-Lin Hsu (許祐霖)

Phone：+886-3-5712121-55175 E-mail：[email protected]

Education：

1999-Pressent PhD Program in Mechanical Engineering,

National Chiao-Tung University, Hsin-chu, Taiwan The research emphasized on numerical methods of the Parallel Cellular Molecular Dynamics simulation (PCMD) code and analysis the Droplet-Droplet collision dynamics by using the PCMD code.

1997-1999 M. S. in Mechanical Engineering,

Chung-Yuan Christian University, Chung-Li, Taiwan The research emphasized on numerical analysis by using the Molecular Dynamics simulation method for studying the rupture process of thin polymer solution films.

1995-1997 B. S. in Mechanical Engineering,

Yuan-Ze College of Engineering, Chung-Li, Taiwan

Language：

Mandarin Chinese, Taiwanese and English

List of Publications

Referred Papers:

1. J.-S. Wu, and Y.-L. Hsu, “Derivation of VSS model Parameters in Direct Simulation Monte Carlo Method Using Quantum Chemistry Computation,”

Japanese Journal of Applied Physics, Vol. 42, pp.7574-7575, 2003.

2. J.-S. Wu, Y.-L. Hsu and Y.-M. Lee, “Parallel Implementation of Molecular Dynamics Simulation for Short-Ranged Interaction,” Computer Physics Communications, Vol. 170, pp. 175-185, 2005.*

3. Y.-L. Hsu, C.-H. Chiang, Y.-M. Lee and J.-S. Wu, “MD Simulation of Two Colliding Nanoscale Argon Droplets Under Vacuum and Pressurized

Environments,” 2006 (Preparing).*

4. Y.-L. Hsu, C.-H. Chiang, Y.-M. Lee and J.-S. Wu, “MD Simulation of Two Bouncing Nanoscale Argon Droplets,” 2006 (Preparing).*

International Conference Papers:

1. Y.-L. Hsu, J.-S. Wu, C.-H. Chiang and Y.-M. Lee, “MD Simulation of Two Colliding Nanoscale Argon Droplets Under Vacuum and Pressurized

Environments,” 25^th Internal Symposium on Rarefied Gas Dynamics, July 21- 28, 2006, Saint-Petersburg, Russia.*

National Conference Papers:

1. J.-S. Wu, Y.-L. Hsu and U.-M. Lee, “平行化分子動力學模擬方法之發展 (Parallel Cellular Molecular Dynamics Simulation for Short-Ranged

Interaction),” The 11^th National Computational Fluid Dynamics Conference (0405), 2004.*

2. J.-S. Wu, J.-S. Wu, U.-M. Lee and Y.-L. Hsu, “Argon 離子轟擊引致銅濺現象 之分子動力學模擬(Molecular Dynamics Simulation of Sputtering of Copper Atoms Due to Argon Ion Bombardment),” The 11^th National Computational Fluid Dynamics Conference (1504), 2004.

(* are those presented in this thesis)

在文檔中應用平行化分子動力學模擬法於液滴‐液滴間碰撞力學之研究 (頁 88-164)