第四章 結論與未來展望
4.2 未來展望
因為研究結果顯示原子結構排列和管徑大小會影響奈米碳管之機械性 質，故可以在其他不同的原子結構排列或管徑中，對單壁碳管做拉伸的模 擬，並比較其結果是否與本研究結果接近；同時利用不同的原子排列結構 對單壁碳管做扭轉以得到剪力模數，並可與石墨板之剪力模數做比較；而 在多壁碳管層間原子作用力的研究方面，可改變不同的原子結構以及管徑 來研究其結構與管徑對於作用力之影響，並利用管徑較大的多壁碳管模型 之結果，再與石墨板作比較。並且在未來複合材料的研究中，可將奈米材 料機械性質的結果，套用於有限元素模擬時所需之奈米碳管材料模數，並 研究碳管對於複合材料機械性質的影響。
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表2.1 石墨板模型(2)、(3)受到單方向應變求得之勁度矩陣係數
C11(GPa) C12(GPa) C21(GPa) C22(GPa) MD model 2
1-layer Graphene with VDW 977.91 254.46 254.74 978.19 MD model 3
8-layers Graphene with VDW 864.29 235.16 235.21 864.34
表2.2 平面方向材料參數(E1、υ12)
E1 (TPa) ν12
Analytical solution 0.805 0.273 MD model 1
(stress method)
1-layer Graphene without VDW
0.794 0.273
MD model 2 (stress method) 1-layer Graphene with VDW
0.912 0.261
MD model 3 (stress method) 8-layers Graphene with VDW
0.795 0.272
MD model 2 (stiffness matrix method) 1-layer Graphene with VDW
0.912 0.261
MD model 3 (stiffness matrix method) 8-layers Graphene with VDW
0.795 0.272
(2004) Xiao [4] 1.153 0.195
(2007) J. Cho et al [6] 1.130 0.200 (2004) Bao WenXing (MD) [8] 1.026 -
(2006) Reddy [10] 0.671 0.428
(1970) Blakslee, O.L. (EXP) [12] 1.020 0.160
表2.3 平面方向剪力係數材料參數(G12)
G12 (TPa) Analytical solution 0.316
MD model 1
1-layer Graphene without VDW 0.318 MD model 2
1-layer Graphene with VDW 0.358 MD model 3
8-layers Graphene with VDW 0.318 (2004) Xiao [4] 0.482 (2007) J. Cho et al [6] 0.470 (2006) Reddy [10] 0.384 (1970) Blakslee, O.L. (EXP) [12] 0.440
表2.4 出平面方向剪力係數材料參數(G13)
G13 (GPa)
Analytical solution 0.209 MD model 3
8-layers Graphene with VDW 0.290 1970 Kelly [5] 0.230 2007 J. Cho et al [6] 0.268 1982 Al-Jishi, R. [11] 0.420 1970 Blakslee, O.L. (EXP) [12] 0.18~0.35
表2.5 不同凡得瓦力勢能之參數比較
Potential form Paper parameters
6
Agrawal (2006)
Dreiding
(1990) ⎥
69
Analytical Solution - 0.805 0.273 0.320 0.321(~1%)
Without Van der Waals Potential - 0.794 0.273 0.318 0.312 (1.7%)
(0.31%) 0.272 0.318 0.313 (1.6%)
Dreiding
(1990) ⎥
(28.8%) 0.253 0.409 0.410 (0.24%)
70
P.M. Agrawal
(2006) ⎥
Dreiding
(1990) ⎥
71
表2.8 利用不同凡得瓦力勢能之參數在模型(3)所得之出平面方向材料參數(G13)
Potential form Parameters
6 Analytical Solution
G13 (GPa) MD Solution
T.W. Chou
Dreiding
(1990) ⎥
表3.1 各種不同半徑之碳管模型尺寸
CNT_R (radius)
(Å)
W (Å)
θ ( ° )
L (Å)
Atom Number
σBox
(kbar)
SWCNT (5,0) 3.91 28.2614 120 85.2000 400 0.1
SWCNT (9,0) 7.05 40.3930 120 106.499- 900 0.01 SWCNT (14,0) 10.96 28.3074 120 85.2000 1120 0.1 SWCNT (26,0) 20.36 46.7021 120 106.4999 2600 0.01 SWCNT (39,0) 30.53 66.8796 120 149.0999 5460 0.01 DWCNT (5,0) (14,0) - 30.0000 90 85.2000 1520 0.01 SWCNT (3,3) 4.07 29.6150 120 86.08216 420 0.1
SWCNT (5,5) 6.78 40.127 120 98.3805 800 0.01
SWCNT (8,8) 10.85 25.4050 120 86.08216 1120 0.01 SWCNT (15,15) 20.34 47.3670 120 110.6780 2700 0.01 SWCNT (22,22) 29.83 66.1790 120 172.1658 6160 0.01 DWCNT (3,3) (8,8) - 30.0000 90 86.0822 1540 0.01
73
表3.2 利用各種條件下單壁奈米碳管模型 A、B 所得之軸方向材料參數
With van der Waals potential Without van der Waals potential
R (Å)
Area of BOX
(Å2)
Area of CNT (Å2)
E1
(TPa) ν12 R
(Å)
Area of BOX
(Å2)
Area (Å2)
E1
(TPa) ν12 Zig-zag (5,0) 2.024 694.52 43.247 0.6817 0.302 2.025 694.54 43.266 0.6837 0.304
Zig-zag (9,0) 3.547 1454.10 75.783 0.7758 0.276 - - - - -
Zig-zag (14,0) 5.491 701.45 117.3 0.7889 0.272 5.495 701.43 117.4 0.7907 0.272 Zig-zag (26,0) 10.178 1916.00 217.44 0.7918 0.272 10.185 1916.00 217.57 0.7936 0.272
Zig-zag (39,0) 15.265 3901.72 326.041 0.7920 0.272 - - - - -
Armchair (3,3) 2.08 760.65 44.34 0.7393 0.2822 2.08 760.66 44.34 0.741 0. 286
Armchair (5,5) 3.408 1315.10 72.805 0.7794 0.2752 - - - - -
Armchair (8,8) 5.43 624.59 116.05 0.7885 0.2732 5.44 624.57 116.13 0.7902 0.2735 Armchair (15,15) 10.17 1943.10 217.25 0.7912 0.273 10.17 1943.20 217.25 0.7930 0.273
Armchair (22,22) 14.915 3754.48 318.542 0.7917 0.273 - - - - -
74
DWCNT(14,0) (5,0) With VDW
Ro=5.46
Ri=2.01 - - - 160.753 0.756 0.2234 (outer tube)
0.2340 (inner tube)
表3.3(b) (3,3) (8,8)雙壁奈米碳管模型與單壁奈米碳管模型軸方向材料參數之比較
DWCNT(8,8)(3,3) With VDW
Ro=5.35
Ri=2.07 - - - 155.715 0.7985 0.2639 (outer tube)
0.2920 (inner tube)
圖2.1 石墨板結構[]
B A
A
6.8 Å 1.42 Å
B A
A
6.8 Å 1.42 Å
圖2.2 石墨板 A-B 疊層結構
0 2
C O
B
A
Graphite sheet Simplified model
C O
Graphite sheet Simplified model 圖2.5 簡化石墨模型
θ
Length variation Angle variation
a
Length variation Angle variation
a
A
圖2.11 碳原子之距離對應凡得瓦勢能與凡得瓦力
α
β α
β
圖2.12 單顆碳原子相對於相鄰層分為
α
與β
兩種平衡位置d
00 j
j
x
x −
d
00 j
j
x
x −
圖2.13 原子位置改變關係
Distance (Å)
Energy(J) force(GN)
2 3 4 5 6 7 8 9 10
-4E-22 -2E-22 0 2E-22 4E-22 6E-22 8E-22
-3E-21 -1.5E-21 0 1.5E-21 3E-21 4.5E-21 6E-21
Energy(J) force(GN)
L
0h
W0
3.4 Å
1 3 2
L
0h
W0
3.4 Å
1 3 2
1 3 2
圖2.14 單層石墨板不考慮凡得瓦力(模型 1)
1 3 2
1 3 2
L
0h
W0
3.4 Å
圖2.15 單層石墨板且考慮凡得瓦力(模型 2)
3.4 Å
A B L
0h
1 3 2
1 3 2
圖2.16 八層石墨板考慮凡得瓦力(模型 3)
time (ps)
time (ps)
Tenperature(K)
time (ps)
1-dir.pressure(kbar)
time (ps)
2-dir.pressure(kbar)
time (ps)
3-dir.pressure(kbar)
time (ps)
time (ps)
Tenperature(K)
time (ps)
1-dir.pressure(kbar)
time (ps)
2-dir.pressure(kbar)
time (ps)
3-dir.pressure(kbar)
圖2.19 分子動力學模擬之石墨板(模型 1)
Model 1 Model 2 Model 3
Model 1 Model 2 Model 3
圖2.20 MD 石墨結構(模型 1-3)施以單軸拉伸應力
time (ps)
time (ps)
Temperature(K)
time (ps)
1-dir.pressure(Kbar)
time (ps)
2-dir.pressure(Kbar)
time (ps)
3-dir.pressure(Kbar)
Model 2 model 3 圖2.22 MD 石墨結構(模型 2-3)施以單軸拉伸應變
Undeformed model
γ
τ12
deformed model LB
WB
2
1 3
Undeformed model
γ
τ12
deformed model
γ
τ12
deformed model LB
WB
LB WB
2
1 3
2
1 3
圖2.23 MD 單層石墨(模型 1)施結構施以剪應力
Model 2 model 3 圖2.24 MD 單層石墨(模型 2-3)施結構施以剪應力
1.7 Å 1.7 Å
3.4 Å 3.4 Å 3.4 Å 3.4 Å 3.4 Å 3.4 Å 3.4 Å
1
2 3
1.7 Å 1.7 Å
3.4 Å 3.4 Å 3.4 Å 3.4 Å 3.4 Å 3.4 Å 3.4 Å
1.7 Å 1.7 Å
3.4 Å 3.4 Å 3.4 Å 3.4 Å 3.4 Å 3.4 Å 3.4 Å
1
2 3
1
2 3
圖2.25 MD 多層石墨結構(模型 3)施以出平面剪應力
圖 3.1 以向量表示石墨捲曲方式說明圖[30]
angle variation
Potentialenergy(kcal/mol)
-20 -10 0 10 20
0 0.5 1 1.5 2 2.5 3
Harminic potential
One-term trigonometric potential
圖 3.2 勢能函數之曲線擬合(curve fitting)曲線圖
A
D
t
D
t
1 2 3
W
θ
1 2 3
1 2 3
W
θ
1 3 2
D0
L0
Big Box
originalBox
1 3 2
1 3 2
D0
L0
Big Box
originalBox D0
L0
Big Box
originalBox
圖 3.4 單壁奈米碳管模擬室
time (ps)
time (ps)
Tenperature(K)
time (ps)
X-dirpressure(kbar)
time (ps)
Y-dirpressure(kbar)
time (ps)
Z-dirpressure(kbar)
time (ps)
time (ps)
Temperature(K)
time (ps)
X-dirpressure(kbar)
time (ps)
Y-dirpressure(kbar)
time (ps)
Z-dirpressure(kbar)
Diamiter (Å)
Young'sModulus(TPa)
0 5 10 15 20 25 30 35
0.65 0.7 0.75 0.8 0.85
Ezigzag Earmchair
圖 3.7 SWCNT 在原子排列結構不同時單壁碳管對於管徑大小改變之楊氏
模數
Do
Di 2
W 1
t t
90°
Do
Di 2
W 1
t t
90°
Zigzag (14,0) (5,0) Periodic BC
Periodic BC
85.02 Å
Zigzag (14,0) (5,0) Periodic BC
Periodic BC
85.02 Å
圖 3.8 雙壁奈米碳管(5,0)(14,0)分子動力模擬下之結構
time
time (ps)
Temperature(K)
time (ps)
X-dirpressure(kbar)
time (ps)
Y-dirpressure(kbar)
time (ps)
Z-dirpressure(kbar)
圖 3.9 雙壁奈米碳管(5,0)(14,0)在不受力狀態下之(a)能量變化圖、(b) 溫度變化圖、(c)1 方向壓力變化圖、(d)2 方向壓力變化圖、(e)3 方
向壓力變化圖
time (ps)
time (ps)
Temperature(K)
time (ps)
X-dirpressure(kbar)
time (ps)
Y-dirpressure(kbar)
time (ps)
Z-dirpressure(kbar)
(a) b)
圖
(a) (5,0)(14,0) (b) (9,9)(14,14) (
3.11 軸向非連續性之雙壁碳管
(a) To
圖 3.12 雙壁碳管結構未受束縛的平衡狀態 tal energy-time (b) Temperature-time
Pullout force Frozen 1, 2, 3 direction
Pullout Length
Pullout force Frozen 1, 2, 3 direction
Pullout force Frozen 1, 2, 3 direction
Pullout Length
圖 3.13 非週期性雙壁碳管之內管沿軸方向平移拉出結構圖[]
time (ps)
Temperature(K)
25 50 75 100
-5 0 5 10 15 20 25 30
time (ps)
Totalenergy(kcal/mol)
25 50 75 100
4660 4680 4700 4720 4740 4760 4780 4800 4820 4840 4860
Pullout Length (Å)
Pulloutforce(nN)
0 20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
最高點
最低點 最高點與最
低點平均
A
B C D
Pullout Length (Å)
Pulloutforce(nN)
0 20 40 60 80 100
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
最高點
最低點 最高點與最
低點平均
A
B C D
圖 3.14 非週期性雙壁碳管之拉出碳管之力對應拉出平移長度
A
rcutoff
A
rcutoff rcutoff
(a)
rcutoff
A
rcutoff rcutoff
A
(b)
rcutoff
A
rcutoff rcutoff
A
(c)
圖 3.15 雙壁碳管內管逐漸拉出觀察圖
theta
posi_z(Å)
0 100 200 300
-6 -4 -2 0 2 4 6
theta
posi_z(Å)
0 100 200 300
-6 -4 -2 0 2 4 6
Point A Pullout L=0Ǻ Point B Pullout L=12.58Ǻ
theta
posi_z(Å)
0 100 200 300
-6 -4 -2 0 2 4 6
theta
posoi_z(Å)
0 100 200 300
-6 -4 -2 0 2 4 6
Point C Pullout L=23.14Ǻ Point D Pullout L=37.92Ǻ
圖 3.16 雙壁碳管拉出內管時之內外管結構之相對位置
1 2
fix 1、2 、3 dir. 3 ( both inner and outer tube ) fix 1、2 、3 dir.
( both inner and outer tube )
1 2
3 1 2
fix 1、2 、3 dir. 3 ( both inner and outer tube ) fix 1、2 、3 dir.
( both inner and outer tube )
(a)
angle rotate angle rotate
(b)
圖 3.17 雙壁碳管做內管剛體旋轉之模型
rotational angle (rad)
Torque(nN-Å)
-0.1 0 0.1 0.2 0.3 0.4
-5.0E-04 -4.0E-04 -3.0E-04 -2.0E-04 -1.0E-04 0.0E+00 1.0E-04 2.0E-04 3.0E-04 4.0E-04 5.0E-04
(a)
rotational angle (rad)
Totalinterlayervdwenergy(kcal/mol)
-0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 -627.003
-627.002 -627.001 -627 -626.999 -626.998 -626.997 -626.996 -626.995 -626.994 -626.993 -626.992
(b)
圖 3.18 雙壁碳管(5,0) (14,0)在平衡條件(A)下之(a) 扭轉力變化圖 (b)
層間凡得瓦勢能變化圖
x, y (Å)
z(Å)
-0.0001 -5E-05 0 5E-05 0.0001
-50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0
x-z y-z axis-z
(c)
x, y (Å)
z(Å)
-0.0001 -5E-05 0 5E-05 0.0001
-50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0
x-z y-z axis-z
(d)
圖 3.18 雙壁碳管(5,0) (14,0)在平衡條件(A)下之
(c) 內管中心軸 (d)外管中心軸
rotational angle (rad)
Torque(nN-Å)
0 0.1 0.2 0.3 0.4
-8.0E-04 -6.0E-04 -4.0E-04 -2.0E-04 0.0E+00 2.0E-04 4.0E-04 6.0E-04 8.0E-04
(a)
rotational angle (rad)
Totalinterlayervdwenergy(kcal/mol)
0 0.1 0.2 0.3 0.4
-623.115 -623.11 -623.105 -623.1 -623.095 -623.09 -623.085 -623.08
(b)
圖 3.19 雙壁碳管(5,0) (14,0)在平衡條件(B)下之 (a) 扭轉力變化圖 (b)
層間凡得瓦勢能變化圖
x, y (Å)
z(Å)
-1E-09 0 1E-09
-40.0 -20.0 0.0 20.0 40.0
x-z y-z axis-z
(c)
x, y (Å)
z(Å)
-1E-13 -5E-14 0 5E-14 1E-13
-50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0
x-z y-z axis-z
(d)
圖 3.19 雙壁碳管(5,0) (14,0)在平衡條件(B)下之
(c) 內管中心軸 (d)外管中心軸
rotational angle (rad)
Torque(nN-Å)
0 0.05 0.1 0.15
-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02
(a)
rotational angle (rad)
Totalinterlayervdwenergy(kcal/mol)
0 0.05 0.1 0.15
-864.96 -864.95 -864.94 -864.93 -864.92 -864.91 -864.9 -864.89 -864.88 -864.87 -864.86 -864.85
(b)
圖 3.20 雙壁碳管(9,9) (14,14)在平衡條件(B)下之(a) 扭轉力變化圖 (b)
層間凡得瓦勢能變化圖
x, y (Å)
z(Å)
-1E-09 -5E-10 0 5E-10 1E-09
-50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0
x-z y-z axis-z
(c)
x, y (Å)
z(Å)
-1E-09 -5E-10 0 5E-10 1E-09
-50.0 -40.0 -30.0 -20.0 -10.0 0.0 10.0 20.0 30.0 40.0
x-z y-z axis-z