OLC
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- しおり てらわ
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1 OLC
2 Master thesis Molecular Dnamics Studies on Friction Mechanism of OLC Hideaki NISHIMURA Feburar 2013 Department of Mechanical Engineering, Graduate School of Engineering, Kobe Universit, Kobe, Japan
3 OLC OLC Stone-Wales 60n 2 (n=1 6) OLC OLC OLC OLC OLC OLC C 60 OLC 10 2 C 960 C 1500 OLC OLC 10 1 OLC OLC
4 Summar OLC is epected as new solid lubricant because of its spherical structure. For a new insight on the friction properties of OLC thin film, various molecular dnamics (MD) simulations are performed on the compression and friction of isolated fullerene/olc and thin films composed of the arra of fullerenes/olcs. First, we have made spherical fullerenes based on the polhedral rule of 60n 2 (n=1 6) atoms with Stone-Wales defects. OLCs were also made b nesting these fullerenes. Then we performed various compression simulations on isolated fullerene/olc. In the point compression b holding the five-membered ring at the top and bottom of fullerene/olc, the OLCs showed higher strength than the fullerenes because of its multi-laer structure. Detail observation revealed that the internal fullerene in the OLC receives higher force than the isolated fullerene. According to the compression b a flat diamond wall, we also found the following facts; (1) the OLCs show higher stress than the fullerenes at the earl stage of compression, (2) the smaller fullerene/olc shows higher stress than the larger ones, and (3) the OLC shows drastic stress increase when the center C 60 was collapsed. Then we performed various scratch simulations on isolated fullerene/olc b a flat diamond wall changing the indentation depth. The friction coefficient showed the order of 10 2 regardless of the indentation depth or collapse morphologies of the target (fullerene/olc). Here, the target carbons glide without rotation under scratch, ecept the OLCs of half crushed C 960 and C We cannot find out the relationship between the friction coefficient and the sie of fullerenes; however, the large OLC showed lower friction coefficient than the small OLC. Finall, we performed scratch simulations on fullerene/olc thin film. When we introduced the surface roughness of a indenter/substrate, the friction coefficient increased to the order of This magnitude agrees with eperimental result. All the fullerene/olc rotate under the scratch, because the were held b the serrated surface. We found that the fullerene shows much higher friction coefficient than the OLC since the can deform to fit their shape to the surface roughness.
5 Brenner Lennard-Jones OLC i
6 ii
7 1 C [1] 60 (CNT) [2] 1992 Nature Ugarte (Onion-Like Carbon:OLC) [3] ( 1.1) OLC (Transmission Electron Microscope:TEM) nm nm C [nm] [4] [5] Kunetsov OLC [4],[6] Hirata Igarashi OLC 0.1 [7] Jol-Pottu Ohmae OLC [8] [10] [11] [13] C 60 CNT 1
8 1 2 C 60 [14] CNT [15] CNT CNT [16] Hirai, Nishimaki, CNT [17] Deguchi Yamaguchi CNT Stone-Wales [18] CNT CNT [19] Tight Binding [20],[21] OLC TEM OLC C 60 (0.334[nm]) n 60n 2 [22] [23] OLC TEM OLC OLC Saito [24] Terrones 60n 2 Stone-Wales [25] Wang Chang Stone-Wales [26] [27],[28] OLC C 60 C 240 C 540 OLC [29],[30]
9 1 3 OLC OLC OLC OLC 2, OLC OLC 3 5 OLC ( OLC ), nm Fig.1.1 TEM image of OLC. [3]
10 2 2.1 (molecular dnamics method MD ) m i d 2 r i dt 2 = F i (2.1) m i r i i i F i Φ tot F i = Φ tot r i (2.2) (2.1) Verlet Verlet t + t t t i r i (t ± t) Talor r i (t + t) = r i (t) + t dr i (t) dt r i (t t) = r i (t) t dr i (t) dt + ( t)2 2 + ( t)2 2 v i t i d 2 r i (t) + O ( ( t) 3) dt 2 (2.3) d 2 r i (t) + O ( ( t) 3) dt 2 (2.4) dr i dt = v i (t) (2.5) 4
11 2 5 (2.1) (2.5) (2.3) (2.4) r i (t + t) = r i (t) + tv i (t) + ( t)2 2 r i (t t) = r i (t) tv i (t) + ( t)2 2 F i (t) + O ( ( t) 3) m i (2.6) F i (t) + O ( ( t) 3) m i (2.7) r i (t + t) + r i (t t) = 2r i (t) + ( t) 2 F i (t) m i + O ( ( t) 4) (2.8) r i (t + t) r i (t t) = 2 tv i (t) + O ( ( t) 3) (2.9) t + t t r i (t + t) = 2r i (t) r i (t t) + ( t) 2 F i (t) m i + O ( ( t) 4) (2.10) v i (t) = 1 2 t {r i (t + t) r i (t t)} + O ( ( t) 2) (2.11) t + t 2 t t t (t = 0) t = t r i ( t) (2.6) Brenner Tersoff Brenner [31] 2 sp 2 sp 3
12 2 6 Φ tot = i j(>i) f c (r ij ) [ V R (r ij ) B ij V A (r ij ) ] (2.12). r ij i, j V R (r ij ) B ij V A (r ij ) f c (r ij ). D e V R (r ij ) = S 1 ep [ β 2S (r ij R e ) ] (2.13) V A (r ij ) = S D e 2 S 1 ep β S (r ij R e ) (2.14) 1, r ij < R [ ( )] 1 π (rij R f c (r ij ) = 1 + cos 1 ) R 2 R /2, R 1 < r ij < R 2 (2.15) 1 0, r ij > R 2 Bij i j k 3, B ij = 1 2 (B ij + B ji ) (2.16) B ij, B ji i, j B ij B ji. B ij = [1 + G(θ i )f c (r ik )] δ (2.17) k i,j B ji = [1 + G(θ j )f c (r jk )] δ (2.18) k i,j r ik i, k, r jk j, k. θ i i j i k, θ j j i j k., G(θ i )f c (r ik ) ζ ij, G(θ j )f c (r jk ) ζ ji. G(θ) G(θ) = a 0 [ 1 + c2 0 c 2 0 d 2 0 d (h + cos θ) 2 ] (2.19). (2.12) (2.19) C 2
13 2 7 fcc 2.1 Table 2.1 Potential parameters for Brenner potential. D e [ev] 6.0 R e [nm] β [nm 1 ] 0.21 S 1.22 h 1.0 a c d R 1 [nm] 0.17 R 2 [nm] 0.20 h 1 h 0 h [deg.] Brenner, 1, , [deg.],, j, k.
14 θ = 30 θ = 60 θ = 120 θ = 180 E [ev] r [nm] Fig.2.1 Relationship between potential energ and bond length r[nm] θ[deg] Fig.2.2 Relationship between stable bond length and bending angle.
15 Lennard-Jones Van der Waals Lennard-Jones [32] Φ tot = i j( i) ( ) 12 ( ) 6 4ϵ σ σ (2.20) r ij r ij 1 2 (2.20) 2.2, 2.3 Table 2.2 Potential parameters for Van der Waals. ϵ σ [ev] [nm] E [ev] r [nm] Fig.2.3 Relationship between Van der Waals potential and atomic distance.
16 (2.2) Brenner ij. r i, r j, r k Φ ij = V R (r ij ) B ij + B ji V A (r ij ) (2.21) 2 F i = Φ ij r i, F j = Φ ij r j, F k = Φ ij r k, j > i.. i F i = [ Bij V A(r ij ) V R(r ij ) ] r ij + 1 [ r ij 2 V Bij A(r ij ) r i + B ] ji ri (2.22). r. V R (r ij ), V A (r ij ) r ij D e V R(r ij ) = β 2S S 1 ep [ β 2S (r ij R e ) ] r ij (2.23) r ij 2 V A(r S D e 2 ij ) = β S S 1 ep β S (r ij R e ) r ij (2.24) r ij. f c (r ij ) r ij f c (r ij ) 1 ( ) π π (rij R 1 ) sin : R 2 > r ij > R 1 = 2 R 2 R 1 R 2 R 1 (2.25) r ij 0 : r ij < R 1, R 2 < r ij. B ij, B ji r i. B ij r i ζ ij r i = = δ(1 + ζ ij ) δ 1 ζ ij (2.26) r i ( i f c(r ik )G(θ i ) r ik + f c (r ik )G (θ i ) cos θ ) i (2.27) r ik r i k i,j G (θ) G(θ) cos θ. G (θ) = G(θ) [ cos θ = a 2c 2 ] 0(1 + cos θ) 0 [d (1 + cos θ) 2 ] 2 (2.28)
17 2 11 cos θ. j, i, k B ji. cos θ i = r ij r ik (2.29) r ij r ik ( cos θ i 1 = cos θ ) ( i rij 1 + cos θ ) i rik (2.30) r i r ik r ij r ij r ij r ik r ik B ji = (1 + ζ ji ) δ B ji r i = δ (1 + ζ ji ) δ 1 ζ ji r i (2.31) j ζ ji = f c (r jk )G (θ j ) cos θ j (2.32) r i r i k i,j cos θ j = r ji r jk r ji r jk (2.33) cos θ j = 1 r jk + cos θ j r ji r i r ji r jk r ji r ji (2.34) F i = [ Bij V A(r ij ) V R(r ij ) ] r ij r ij V A(r ij ) ( δ (1 + ζ ij ) δ 1) i V A(r ij ) ( δ (1 + ζ ji ) j k i,j j δ 1) k i,j f c (r ik )G (θ i ) ( ) 1 r ik cos θ i r ij ) cos θ i r ik f c(r ik )G(θ i ) + f c (r ik )G (θ i ) ( 1 f c (r jk )G (θ j ) cos θ j r ji f c (r jk )G (θ j ) 1 r ji F j = [ Bij V A(r ij ) V R(r ij ) ] ( r ) ij + 1 [ r ij 2 V Bij A(r ij ) r i r ji r ji rjk r jk + B ] ji r i r ij (2.35) r ij r ij r ik r ik. B ij ζ ij ζ ij r i = i k i,j ( f c (r ik )G (θ i ) cos θ i r j ) (2.36) ζ ji r j = j k i,j cos θ i = 1 r ik + cos θ i r j r ij r ik r ij ( f c(r jk )G(θ j ) r jk + f c (r jk )G (θ j ) cos θ j r jk r j r ij r ij (2.37) ) (2.38)
18 2 12 ( cos θ j 1 = cos θ ) ( j rji 1 + cos θ ) j rjk (2.39) r j r jk r ji r ji r ji r jk r jk F j = [ Bij V A(r ij ) V R(r ij ) ] ( r ) ij r ij V A(r ij ) ( δ (1 + ζ ij ) δ 1) i V A(r ij ) ( δ (1 + ζ ji ) k k i,j j δ 1) k i,j F k = 1 2 V A(r ij ) f c (r ik )G (θ i ) cos θ i r ij f c (r ik )G (θ i ) 1 r ij r ij ij r ik r ik f c (r jk )G (θ j ) ( 1 r jk cos θ ) j r ji f c(r jk )G(θ j ) + f c (r jk )G (θ j ) ( 1 r ji cos θ ) j r jk [ Bij r k + B ] ji r k (2.40) r ji r ji rjk r jk. B ij, B ji ζ ij, ζ ji ζ ij r k = ζ ji r k = i k i,j j ( k i,j ( ( f c(r ik )G(θ i ) r ) ik + f c (r ik )G (θ i ) cos θ ) i r ik r k ( f c(r jk )G(θ j ) r ) jk + f c (r jk )G (θ j ) cos θ ) j r jk r k (2.41) (2.42). F k = 1 2 V A(r ij ) ( δ (1 + ζ ij ) δ 1) i V A(r ij ) ( δ (1 + ζ ji ) k i,j j δ 1) k i,j Van der Waals f c (r ik )G (θ i ) 1 r ik f c(r ik )G(θ i ) + f c (r ik )G (θ i ) ( cos θ i r ik ) f c (r jk )G (θ j ) 1 r jk f c(r jk )G(θ j ) + f c (r jk )G (θ j ) cos θ j r jk r ij r ij r ik ik r ji r ji rjk r jk F i = Φ(r ij) r ij = 4ϵ 12 ( ) 12 σ + 6 ( ) 6 σ r ij (2.43) r j( i) ij r ij r ij r ij r ij F j = F i (2.44) i j k r ij, r ik i, j i, k θ i r ij, r ik
19 N 1step N (N 1) N ( r c ) 2.4 r c r fc r c N (r c N 1) r fc r c (N (N 1) ) rfc rc Fig.2.4 Schematic of bookkeeping method.
20 b b N b 0 b Fig.2.5 Schematic of domain decomposition method.
21 (2.45) 1 2 mα v α i v α i = 3 2 k BT (2.45) m α : α v α i k B : T α : Boltmann = [J/K] T 0 α v α i 0 v α i 0 (2.46) ( ) 0.5 vi α 3kB T 0 0 = (2.46) T α (2.47) v α i = m α ( ) 0.5 3kB T (2.47) m α (2.46) (2.47) vi α 0 vi α = ( T0 T ) 0.5 (2.48) T T 0 (2.48) Verlet r α i (t+ t) 2.49 T 0 /T r α i (t + t) r α i (t + t) = r α i (t + t) r α i (t) = r α i (t) r α i (t t) + ( t) 2 F α i (t) m α (2.49) [33]
22 [23] Euler 12 (Isolated Pentagon Rule) C 60 [23] 60n 2 [34] (Stone-Wales ) C 240 C 60 C C C 960 (a) Before inserting (b) After inserting Fig.2.6 Snapshots of Stone-Wales defect.
23 2 17 (a) Before inserting (b) After inserting Fig.2.7 Snapshots of epanded C 540. (a) Before inserting (b) After inserting Fig.2.8 Snapshots of epanded C 960.
24 n 2 n=1 6 ( ) 20000[fs], 0.145[nm]. Mawell Boltmann 10[K] 1.0[fs] n r OLC n r 3.1 C nm ( ) (0.336nm) 3.1 C
25 3 19 Table 3.1 Fullerene parameter n, radius and radius difference after relaation. fullerene n radius [nm] radius difference [nm] C C C C C C (a) top view (b) side view ( 1 ) (c) side view ( 2 ) Fig.3.1 Snapshots of C 1500 after relaation.
26 n= [/fs] Mawell Boltmann 10[K] 1.0[fs] 3.2 C (ii) 3.2(i) 3.2(i) ε < 0.05 (a)ε = (b)ε = 0.03 (b)ε = 0.03 (c)ε = (ii) (b) (b) (c) 3.2(i) 0.05 < ε < 0.22 (c)ε = 0.05 (d)ε = (ii) (e) 3.3 C (i) (a) (b) (b) (c) 3.3(ii) C 540 (b) 3.3(i) < ε < 0.3 (c) (d) C 540 (d)
27 (ii) (e) 3.4 C (i) (a) (c) 3.4(ii) (d) (f) C C (i) (a) (c) (d) ( 3.5(ii) (e) ) 3.6 C 540 ε = 0.17 σ = 2.21[GPa] C 960 ε = 0.23 σ = 1.14[GPa] C 1500 ε = σ = 1.12[GPa] C 2160 ε = σ = 0.52[GPa] bond 3.7 ( 2.7 ) 3.7 C 540 C 1500 C 960 C 2160 C 540 C 1500 C 960 C 2160
28 3 22 Compressive stress, -σ, GPa (a) (b) (c) (d) (e) C 540 (f) Compressive strain, ε (i) Stress - strain curve of C 540 under compression. close-up bottom view (a) ε =0.005 (b) ε =0.03 (c) ε=0.05 close-up bottom view (d) ε =0.17 (e) ε =0.19 (f) ε=0.215 (ii) Snapshots of C 540 under compression. Fig.3.2 Compression of C 540 b holding the five-membered ring at the top and bottom.
29 3 23 C 960 Compressive stress, -σ, GPa (a) (b) (c) (d) (e) (f) Compressive strain, ε (i) Stress - strain curve of C 960 under compression. (a) ε =0.045 (b) ε =0.055 close-up top view (c) ε=0.095 (d) ε =0.23 (e) ε =0.245 close-up top view (f) ε=0.27 (ii) Snapshots of C 960 under compression. Fig.3.3 Compression of C 960 b holding the five-membered ring at the top and bottom.
30 3 24 C 1500 Compressive stress, -σ, GPa (b) (a) (c) (d) (e) Compressive strain, ε (f) (i) Stress - strain curve of C 1500 under compression. (a) ε =0.015 (b) ε =0.025 close-up bottom view (c) ε =0.06 (d) ε =0.175 (e) ε =0.18 close-up bottom view (f) ε=0.295 (ii) Snapshots of C 1500 under compression. Fig.3.4 Compression of C 1500 b holding the five-membered ring at the top and bottom.
31 3 25 C 2160 Compressive stress, -σ, GPa (a) (d) (e) (c) (f) (b) Compressive strain, ε (i) Stress - strain curve of C 2160 under compression. (a) ε =0 (b) ε =0.01 close-up top view (c) ε =0.125 (d) ε =0.245 (e) ε =0.255 close-up top view (f) ε=0.3 (ii) Snapshots of C 2160 under compression. Fig.3.5 Compression of C 2160 b holding the five-membered ring at the top and bottom.
32 3 26 Compressive stress, -σ, GPa C 540 C 960 C 1500 C Compressive strain, ε Fig.3.6 Stress - strain curves of fullerenes under compression. (a) C540 (b) C960 (c) C1500 (d) C2160 Fig.3.7 Top view of fullerenes after stress peak.
![3 27 3.2.2 n=3 6 0.44[nm] 20000[fs] 3.8 1[nm] D 3/4D 0.7[nm] 1.0 10 4 [nm/fs]. 7.](/docs-images/109/186975928/images/33-0.jpg "1[nm] Van der Waals Mawell Boltmann 10[K] 1.0[fs] 1nm 7.1nm Fig.3.")
33 n= [nm] 20000[fs] 3.8 1[nm] D 3/4D 0.7[nm] [nm/fs]. 7.1[nm] Van der Waals Mawell Boltmann 10[K] 1.0[fs] 1nm 7.1nm Fig.3.8 Schematic of wall compression (C 1500 ).
34 C (ii) 3.9(i) 3.9(i) (i) (a) 0.6[nm] Van der Waals Van der Waals 0.37[nm] 0.6[nm] (1[nm]-0.37[nm]) (a) (d) 1.15[nm] 3.9(ii) (d) (e) 3.2(ii) (f) 1.7[nm] 3.9(ii) (f) 3.10 C (i) (a) (b) (c) (c) (e) (a) (b) 3.10(ii) 3.3(ii) (b) (e) (f) (h) (e) C 540 (f) 3.10(ii) (f) (g) (h) 3.11 C 1500 C (ii) (f)
35 C 2160 (h) 3.12(ii) (c) (d) 3.5(ii) 3.12(ii) (f) (g) (h) 3.12(ii) (h) (i) (h) Van der Waals 3.13 C C [nm] C [nm] C [nm] C [nm] (C (ii) (f) C (ii) (h) C (ii) (f) C (ii) (i)) 3.14 C 540 C 1500 C 2160
36 3 30 C 540 Compressive stress, -σ, GPa 4 (d) (f) 2 (e) (b) (c) 0 (a) Displacement of upper wall, nm (i) Stress - displacement curve of C 540 under compression. (a) 0.6nm indentation (b) 0.85nm indentation (c) 0.9nm indentation (d) 1.15nm indentation (e) 1.4nm indentation (f) 1.85nm indentation (ii) Snapshots of C 540 under compression. Fig.3.9 Compression of C 540 b diamond wall.
37 3 31 C 960 Compressive stress, -σ, GPa (a) (c) (b) (f) (g) (e) (d) (h) Displacement of upper wall, nm (i) Stress - displacement curve of C 960 under compression. (a) 0.85nm indentation (b) 0.95nm indentation (c) 1.3nm indentation (d) 1.8nm indentation (e) 2.0nm indentation (f) 2.3nm indentation (g) 2.4nm indentation (h) 2.5nm indentation (ii) Snapshots of C 960 under compression. Fig.3.10 Compression of C 960 b diamond wall.
38 3 32 C 1500 Compressive stress, -σ, GPa (a) (c) (b) (d) (e) (f) Displacement of upper wall, nm (i) Stress - displacement curve of C 1500 under compression. (a) 0.8nm indentation (b) 0.85nm indentation (c) 1.4nm indentation (d) 1.7nm indentation (e) 2.2nm indentation (f) 2.9nm indentation (ii) Snapshots of C 1500 under compression. Fig.3.11 Compression of C 1500 b diamond wall.
39 3 33 C 2160 Compressive stress, -σ, GPa 4 2 (f) (c) (i) (a) (g) (h) 0 (b) (d) (e) Displacement of upper wall, nm (i) Stress - displacement curve of C 2160 under compression. (a) 0.8nm indentation (b) 0.85nm indentation (c) 1.9nm indentation (d) 2.2nm indentation (e) 2.6nm indentation (f) 2.9nm indentation (g) 3.0nm indentation (h) 3.3nm indentation (i) 3.7nm indentation (ii) Snapshots of C 2160 under compression. Fig.3.12 Compression of C 2160 b diamond wall.
40 3 34 Compressive stress, -σ, GPa C 540 C 960 C 1500 C Displacement of upper wall, nm Fig.3.13 Stress - displacement curves of fullerenes under compression. (a) C540 (b) C960 (c) C1500 (d) C2160 Fig.3.14 Snapshots of fullerenes at the point just before the drastic stress increase.
![3 35 3.3 3.3.1 n=3 6 3.15 1[nm] 4 C 540 10.7[nm] C 960 12.8[nm] C 1500 14.9[nm] C 2160 17.1[nm] 1nm+0.1D 1nm+0.](/docs-images/109/186975928/images/41-0.jpg "3D 1nm+0.5D 1nm+0.7D 4 (D ) 3D 1.0 10 4 [nm/fs]. 1nm 14.9nm Fig.3.15 Schematic of scratch simulation (C 1500 ).")
41 n= [nm] 4 C [nm] C [nm] C [nm] C [nm] 1nm+0.1D 1nm+0.3D 1nm+0.5D 1nm+0.7D 4 (D ) 3D [nm/fs]. 1nm 14.9nm Fig.3.15 Schematic of scratch simulation (C 1500 ).
42 C nm+0.1D 1nm+0.3D ±0.1 1nm+0.5D ±0.05 1nm+0.7D ±0.02 1nm+0.1D (a) C 540 ( ) (b) (d) ( 3.2) 10 2 C 960 C 2160 C 540 C C nm+0.3D C 540 C C 1500
43 3 37 1nm+0.5D C nm+0.7D Van der Waals ( 3.5 ) nm+0.7D
44 3 38 average average Friction coefficient, µ Friction coefficient, µ Displacement of upper wall, nm Displacement of upper wall, nm (a) 1nm+0.1D indentation (b) 1nm+0.3D indentation average average Friction coefficient, µ Friction coefficient, µ Displacement of upper wall, nm Displacement of upper wall, nm (c) 1nm+0.5D indentation (d) 1nm+0.7D indentation Fig.3.16 Friction coefficient - displacement curves of C 1500 under scratch.
45 3 39 Table 3.2 Friction coefficient of fullerenes (1nm+0.1D indentation). fullerene C 540 C 960 C 1500 C 2160 friction coefficient (a) C540 (b) C960 (c) C1500 (d) C2160 Fig.3.17 Snapshots of fullerenes after indentation (1nm+0.1D indentation). 14.9nm (a) 0nm scratch (b) D (3.61nm) scratch (c) 2D (7.22nm) scratch (d) 3D (10.83nm) scratch Fig.3.18 Snapshots of C 1500 under scratch (1nm+0.1D indentation).
46 3 40 Table 3.3 Friction coefficient of fullerenes (1nm+0.3D indentation). fullerene C 540 C 960 C 1500 C 2160 friction coefficient (a) C540 (b) C960 (c) C1500 (d) C2160 Fig.3.19 Snapshots of fullerenes after indentation (1nm+0.3D indentation). 14.9nm (a) 0nm scratch (b) D (3.61nm) scratch (c) 2D (7.22nm) scratch (d) 3D (10.83nm) scratch Fig.3.20 Snapshots of C 1500 under scratch (1nm+0.3D indentation).
47 3 41 Table 3.4 Friction coefficient of fullerenes (1nm+0.5D indentation). fullerene C 540 C 960 C 1500 C 2160 friction coefficient (a) C540 (b) C960 (c) C1500 (d) C2160 Fig.3.21 Snapshots of fullerenes after indentation (1nm+0.5D indentation). 14.9nm (a) 0nm scratch (b) D (3.61nm) scratch (c) 2D (7.22nm) scratch (d) 3D (10.83nm) scratch Fig.3.22 Snapshots of C 1500 under scratch (1nm+0.5D indentation).
48 3 42 Table 3.5 Friction coefficient of fullerenes (1nm+0.7D indentation). fullerene C 540 C 960 C 1500 C 2160 friction coefficient (a) C540 (b) C960 (c) C1500 (d) C2160 Fig.3.23 Snapshots of fullerenes after indentation (1nm+0.7D indentation). 14.9nm (a) 0nm scratch (b) D (3.61nm) scratch (c) 2D (7.22nm) scratch (d) 3D (10.83nm) scratch Fig.3.24 Snapshots of C 1500 under scratch (1nm+0.7D indentation).
49 4 OLC OLC 30000[fs] OLC C 1500 OLC C 1500 C 540 C 960 C 1500 C C OLC 43
50 4 OLC (a) top view (b) side view ( 1 ) (c) side view ( 2 ) Fig.4.1 Snapshots of C 1500 after relaation.
51 4 OLC OLC [/fs] 4.2 C (i) ε < 0.05 (a)ε = 0.01 (b)ε = 0.02 (c)ε = (ii) (b) 4.2(i) (c)ε = 0.05 (d)ε = (ii) (e) C 540 ( 3.7(a) ) C C (i) 4.3(ii) (a) C 540 (b) (e) (b) C 540 (e) C 960 C (a) C (b) 4.4 C (i) ε < 0.05
52 4 OLC (ii) (b) C 540 (c) (e) 4.4(ii) (c) C 540 (d) C 1500 (e) C C (ii) (b) C 540 (c) C 1500 (d) C 960 (e) C OLC OLC OLC 4.5 C 2160 C 540 C 1500 C 960 C C 960 C 1500 C C 540 ( ) OLC
53 4 OLC 540 Compressive stress, -σ, GPa (a) (d) (e)(f) (c) (b) Compressive strain, ε (i) Stress - strain curve of C 540 under compression. (a) ε =0.01 (b) ε =0.02 (c) ε=0.05 (d) ε =0.215 (e) ε =0.225 (f) ε=0.245 (ii) Snapshots of C 540 under compression. Fig.4.2 Compression of C 540 b holding the five-membered ring at the top and bottom.
54 4 OLC 960 Compressive stress, -σ, GPa (a) (c) (d) (b) (e) Compressive strain, ε (i) Stress - strain curve of C 960 under compression. (a) ε =0.065 (b) ε =0.245 close-up C540 (c) ε =0.255 (d) ε =0.265 (e) ε=0.275 (ii) Snapshots of C 960 under compression. Fig.4.3 Compression of C 960 b holding the five-membered ring at the top and bottom.
55 4 OLC 1500 Compressive stress, -σ, GPa (a) (c) (e) (d) (b) Compressive strain, ε (i) Stress - strain curve of C 1500 under compression. (a) ε =0.025 (b) ε =0.04 (c) ε=0.2 close-up C540 (d) ε =0.22 (e) ε =0.245 close-up C960 (ii) Snapshots of C 1500 under compression. Fig.4.4 Compression of C 1500 b holding the five-membered ring at the top and bottom.
56 4 OLC 2160 Compressive stress, -σ, GPa (a) (b)(c) (d) (e) Compressive strain, ε (i) Stress - strain curve of C 2160 under compression. (a) ε =0.06 (b) ε =0.16 close-up C540 (c) ε=0.185 close-up C1500 (d) ε =0.23 close-up C960 (e) ε =0.26 (ii) Snapshots of C 2160 under compression. Fig.4.5 Compression of C 2160 b holding the five-membered ring at the top and bottom.
57 4 OLC 51 Compressive stress, -σ, GPa Compressive strain, ε Fig.4.6 Stress - strain curves of OLCs under compression. Compressive stress, -σ, GPa C 540 C 960 C 1500 C Compressive strain, ε Fig.4.7 Stress - strain curves of fullerenes under compression.
58 4 OLC OLC 7.1[nm] 7.1[nm] 0.7[nm] [nm/fs]. 4.8 C (i) (i) Van der Waals (a) 0.7[nm] (c) 1.6[nm] (e) 1.9[nm] 4.8(ii) ( 4.2(ii) ) (c) (e) (c) (d) C 540 (e) C OLC 1.0[nm] OLC C 540 Van der Waals 0.7[nm] C (ii) (b) C 1500 OLC C (ii) OLC
59 4 OLC 53 C 1500 C OLC ( 0.9[nm] ) OLC
60 4 OLC 540 Compressive stress, -σ, GPa 10 (c) (e) (d) 5 (b) 0 (a) Displacement of upper wall, nm (i) Stress - displacement curve of C 540 under compression. (a) 0.7nm indentation (b) 0.85nm indentation (c) 1.6nm indentation (d) 1.7nm indentation (e) 1.9nm indentation close-up C60 (ii) Snapshots of C 540 under compression. Fig.4.8 Compression of C 540 b diamond wall.
61 4 OLC 55 Compressive stress, -σ, GPa Displacement of upper wall, Fig.4.9 Stress - displacement curves of OLCs under compression. Compressive stress, -σ, GPa C 540 C 960 C 1500 C Displacement of upper wall, nm Fig.4.10 Stress - displacement curves of fullerenes under compression.
62 4 OLC OLC OLC 4 C [nm] C [nm] C [nm] C [nm] 1nm+0.1D 1nm+0.3D 1nm+0.5D 1nm+0.7D nm+0.1D C OLC (a) 5 1 OLC OLC 4.12 OLC C OLC C nm+0.3D 4.14 OLC nm+0.1D
63 4 OLC 57 ± OLC 1nm+0.1D 4.16 C nm+0.5D C nm+0.3D (b) (c) (d) 3.5[nm] 4.18 OLC C 60 OLC ( 4.3) 1nm+0.1D 1nm+0.3D C OLC 4.19 (b) (d) 4.17 (a) (b) C 60 C 240 (b) (c) (c) (d) 4.17 (c) (e) (f) nm+0.5D C 960 C 540 C 2160 C 960 C nm+0.7D
64 4 OLC C OLC 1nm+0.5D OLC 1nm+0.5D 4.21 OLC C nm+0.5D bond
65 4 OLC 59 average Friction coefficient, µ Displacement of upper wall, nm Fig.4.11 Friction coefficient - displacement curve of C 1500 under scratch (1nm+0.1D indentation). Table 4.1 Radius of contact area and friction coefficient of OLCs (1nm+0.1D indentation). OLC R c [nm] friction coefficient C C C C Fig.4.12 Snapshots of OLCs after indentation (1nm+0.1D indentation). 14.9nm (a) 0nm scratch (b) D (3.61nm) scratch (c) 2D (7.22nm) scratch (d) 3D (10.83nm) scratch Fig.4.13 Snapshots of C 1500 under scratch (1nm+0.1D indentation).
66 4 OLC 60 average Friction coefficient, µ Displacement of upper wall, nm Fig.4.14 Friction coefficient - displacement curve of C 1500 under scratch (1nm+0.3D indentation). Table 4.2 Radius of contact area and friction coefficient of OLCs (1nm+0.3D indentation). OLC R c [nm] friction coefficient C C C C Fig.4.15 Snapshots of OLCs after indentation (1nm+0.3D indentation). 14.9nm (a) 0nm scratch (b) D (3.61nm) scratch (c) 2D (7.22nm) scratch (d) 3D (10.83nm) scratch Fig.4.16 Snapshots of C 1500 under scratch (1nm+0.3D indentation).
67 4 OLC 61 average Friction coefficient, µ (d) 0.02 (b) (c) Displacement of upper wall, nm Fig.4.17 Friction coefficient - displacement curve of C 1500 under scratch (1nm+0.5D indentation). Table 4.3 Radius of contact area and friction coefficient of OLCs (1nm+0.5D indentation). OLC R c [nm] friction coefficient C C C C Fig.4.18 Snapshots of OLCs after indentation (1nm+0.5D indentation). 14.9nm (a) 0nm scratch (b) 2.8nm scratch (c) 3.5nm scratch surface closeup (d) 5.0nm scratch (e) 7.0nm scratch (f) 3D (10.83nm) scratch Fig.4.19 Snapshots of C 1500 under scratch (1nm+0.5D indentation).
68 4 OLC 62 average Friction coefficient, µ Displacement of upper wall, nm Fig.4.20 Friction coefficient - displacement curve of C 1500 under scratch (1nm+0.7D indentation). Table 4.4 Radius of contact area and friction coefficient of OLCs (1nm+0.7D indentation). OLC R c [nm] friction coefficient C C C C Fig.4.21 Snapshots of OLCs after indentation (1nm+0.7D indentation). 14.9nm (a) 0nm scratch (b) D (3.61nm) scratch (c) 2D (7.22nm) scratch (d) 3D (10.83nm) scratch Fig.4.22 Snapshots of C 1500 under scratch (1nm+0.7D indentation).
69 OLC C 540 C 2160 OLC C 540 C small serrate (S) large serrate (L) (C 540 C [nm] C 2160 C [nm] ) 30000[fs] 1000[nN] 7.5[nm] [nm/fs]. OLC ( [nm 2 ])
70 5 64 Table 5.1 Diameter, number of atoms and coverage rate in simulation models. model name diameter [nm] No. of atoms coverage rate [%] C C C C nm 14.9nm (a) Small serrate indenter 1.06nm 0.55nm 1.06nm 14.9nm 14.9nm 0.55nm (b) Large serrate indenter 1.24nm 2.49nm Fig.5.1 Dimensions of diamond wall indenter.
71 5 65 (a) C540 (b) C nm 5.0nm Fig.5.2 Top view of fullerene/olc arra.
72 small serrate C 540 C OLC C 2160 C [nN] 5.2 OLC 3 4 OLC C 2160 < C 2160 < C 540 =.. C 540 C 540 C 540 C 2160 C 2160 OLC Van der Waals (VDW) 5.4 OLC VDW VDW OLC C 540 C 540 C 2160 C OLC ( ) OLC (c) (d) OLC (a) (b) C 2160 C 540 C 540 C 2160 C C 540 C C 540 (7.5[nm])
73 C 540 large serrate C 540 C small serrate small serrate C 2160 C C 540 C 540 C 2160 C large serrate OLC VDW 5.4 C 540 C 540 C 2160 C 2160 VDW 5.4 C 540 C 2160 OLC C 540 C 2160 OLC 5.10 OLC 5.10 (c) (d) OLC OLC VDW 5.10 (a) (c) C 540 C 540 C 540 small serrate C (a) (c) C 540 C 540 (b) (d) C 2160 C 2160 small serrate large serrate C 2160 C 2160 C 540 C
74 5 68 large serrate small serrate VDW 5.3 C 540 C C 540 C 540 small serrate
75 5 69 average= average= Friction coefficient, µ Friction coefficient, µ Displacement of indenter, nm Displacement of indenter, nm (a) C540 Fig.5.3 Friction coefficient - displacement curves under scratch (small serrate indenter). Table 5.2 Indentation depth and friction coefficient (small serrate indenter). model name indentation depth [nm] friction coefficient C 540, small serrate indenter (C 540 -S) C 2160, small serrate indenter (C S) C 540, small serrate indenter ( C 540 -S) C 2160, small serrate indenter ( C S)
76 5 70 (10 5 ) C 540 -S C S 2.0 Number of VDW bonds Displacement of indenter, nm Fig.5.4 Number of VDW bonds - displacement curves under scratch (small serrate indenter). (a) C540-S (b) C2160-S element closeup element closeup Fig.5.5 Snapshots after indentation (small serrate indenter).
77 第5章 薄膜構造での摩擦シミュレーション (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch Fig.5.6 Snapshots of C540 -S under scratch. (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch Fig.5.7 Snapshots of C540 -S under scratch. 71
78 5 72 average= average= Friction coefficient, µ 0.1 Friction coefficient, µ Displacement of indenter, nm Displacement of indenter, nm (a) C540 Fig.5.8 Friction coefficient - displacement curves under scratch (large serrate indenter). Table 5.3 Indentation depth and friction coefficient (large serrate indenter). model name indentation depth [nm] friction coefficient C 540, large serrate indenter (C 540 -L) C 2160, large serrate indenter (C L) C 540, large serrate indenter ( C 540 -L) C 2160, large serrate indenter ( C L)
79 5 73 (10 5 ) C 540 -L C L 2.0 Number of VDW bonds Displacement of indenter, nm Fig.5.9 Number of VDW bonds - displacement curves under scratch (large serrate indenter). (a) C540-L (b) C2160-L element closeup element closeup Fig.5.10 Snapshots after indentation (large serrate indenter).
80 第5章 薄膜構造での摩擦シミュレーション (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch Fig.5.11 Snapshots of C540 -L under scratch. (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch Fig.5.12 Snapshots of C540 -L under scratch. 74
![5 75 5.2 5.2.1 OLC 5.13 small serrate 80000[fs] OLC 30000[fs] 1nm 14.9nm Fig.5.13 Schematic of simulation model ( C 540, small serrate substrate, small serrate indenter).](/docs-images/109/186975928/images/81-0.jpg "5.2.2 small serrate 5.14 5.14 (d) C 2160 (a) (c) 1.0[nm] small serrate 5.4 5.2")
81 OLC 5.13 small serrate 80000[fs] OLC 30000[fs] 1nm 14.9nm Fig.5.13 Schematic of simulation model ( C 540, small serrate substrate, small serrate indenter) small serrate (d) C 2160 (a) (c) 1.0[nm] small serrate
82 5 76 C 540 C C C C 2160 < C 540 < C 2160 < C C (d) large serrate (b) C (a) (c) (a) C 540 3[nm] (c) (d) OLC C C C C large serrate small serrate
83 large serrate C 2160 small serrate 5.22 C C C 2160 C 540 C 540 C C C 540 C 2160 C C 2160 C 2160
84 5 78 average= average= Friction coefficient, µ Friction coefficient, µ Displacement of indenter, nm Displacement of indenter, nm (a) C540 (b) C2160 average= average= Friction coefficient, µ Friction coefficient, µ Displacement of indenter, nm Displacement of indenter, nm Fig.5.14 Friction coefficient - displacement curves under scratch (small serrate substrate, small serrate indenter).
85 5 79 Table 5.4 Indentation depth and friction coefficient (small serrate substrate, small serrate indenter). model name indentation depth [nm] friction coefficient C 540, small serrate substrate, small serrate indenter (C 540 -SS) C 2160, small serrate substrate, small serrate indenter (C SS) C 540, small serrate substrate, small serrate indenter ( C 540 -SS) C 2160, small serrate substrate, small serrate indenter ( C SS) (a) C540-SS (b) C2160-SS element closeup element closeup Fig.5.15 Snapshots after indentation (small serrate substrate, small serrate indenter).
86 第5章 薄膜構造での摩擦シミュレーション 80 (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch closeup closeup closeup closeup Fig.5.16 Snapshots of C540 -SS under scratch. (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch closeup closeup closeup Fig.5.17 Snapshots of C2160 -SS under scratch. closeup
87 第5章 薄膜構造での摩擦シミュレーション 81 (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch closeup closeup closeup closeup Fig.5.18 Snapshots of C540 -SS under scratch. (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch closeup closeup closeup Fig.5.19 Snapshots of C2160 -SS under scratch. closeup
88 5 82 average= average= Friction coefficient, µ 0.2 Friction coefficient, µ Displacement of indenter, nm Displacement of indenter, nm (a) C540 (b) C2160 average= average= Friction coefficient, µ 0.2 Friction coefficient, µ Displacement of indenter, nm Displacement of indenter, nm Fig.5.20 Friction coefficient - displacement curves under scratch (small serrate substrate, large serrate indenter).
89 5 83 Table 5.5 Indentation depth and friction coefficient (small serrate substrate, large serrate indenter). model name indentation depth [nm] friction coefficient C 540, small serrate substrate, large serrate indenter (C 540 -SL) C 2160, small serrate substrate, large serrate indenter (C SL) C 540, small serrate substrate, large serrate indenter ( C 540 -SL) C 2160, small serrate substrate, large serrate indenter ( C SL) (a) C540-SL (b) C2160-SL element closeup element closeup Fig.5.21 Snapshots after indentation (small serrate substrate, large serrate indenter).
90 第5章 薄膜構造での摩擦シミュレーション 84 (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch closeup closeup closeup closeup Fig.5.22 Snapshots of C540 -SL under scratch. (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch closeup closeup closeup Fig.5.23 Snapshots of C2160 -SL under scratch. closeup
91 第5章 薄膜構造での摩擦シミュレーション 85 (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch closeup closeup closeup closeup Fig.5.24 Snapshots of C540 -SL under scratch. (a) 0nm scratch (b) 2.5nm scratch (c) 5.0nm scratch (d) 7.5nm scratch closeup closeup closeup Fig.5.25 Snapshots of C2160 -SL under scratch. closeup
92 6 OLC OLC, OLC. 2, Stone-Wales 3 C nm 60n 2 n 1 bond. C 540 C 960 C 1500 C
93 OLC 3 OLC OLC C 60 OLC OLC nm+0.5D(D: ) C 960 C OLC C 540 C 2160 C 540 C 2160 ( 2 ) 10 2 C 2160 < C 2160 < C 540 < C C 2160 < C 540 < C 2160 < C 540 OLC
94 6 88 OLC 3 4 ( OLC )
95 [1] Kroto, H.W., Heath,J.R., O Brien, S.C., Curl, R.F., and Smalle, R.E., Nature, 318, (1985), 162. [2] Iijima, S., Nature, 354, (1991), 56. [3] Ugarte, D., Nature, 359, (1992), 707. [4] Kunetsov, V.L., Butenko, Yu.V., Chuvilin, A.L., Romanenko, A.I., Okotrub, A.V., Chem.Phs.Lett., 336, (2001), [5],,, 67, (2001), [6] Kunetsov, V.L., Chuvilin, A.L., Butenko, Yu.V. et al, Chem.Phs.Lett., 222, (1994), 343. [7] Hirata, A., Igarashi, M., Kaito, T., Tribol.Int., 37, (2004), [8] Matsumoto, N., Jol-Pottu, L., Kinoshita, H., Ohmae, N., Diamond.Relat.Mater, 16, (2007), [9] Jol-Pottu, L., Vacher, B., Ohmae, N. et al, Tribol.Lett., 30, (2008), [10] Jol-Pottu, L., Kinoshita, H. et al, Tribol.Int., 41, (2008), [11] Cabioc h, T., Riviere, J.P., Delafond, J., J.Mater.Sci., 30, (1995), [12] Banhart, F., Fuller, T., Redlich, Ph.D., Ajaan, P.M., Chem.Phs.Lett., 269, (1997), [13] Cabioc h, T., Jaouen, M. et al, Appl.Phs.Lett., 73, (1998), [14] (B ) 63, 611, (1997), [15] (B ) 76, 764, (2010),
96 90 [16],,,, 55, (2006), 754. [17] Hirai, Y., et al., Jpn J Appl Phs., 42 6B, (2003), [18] Deguchi, H., Yamaguchi, Y. et al, Chem.Phs.Lett., 503, (2011), [19] 10, 13, (2010). [20] Chico, L., Crespi, V.H., Benedict, L.X., Louie, S.G., Cohen, M.L., Phs.Rev.Lett., 76, 6, (1996), 971. [21] Lee, Y.H., Kim, S.G., Tomanek, D., Phs.Rev.Lett., 78, 12, (1997), [22], (2007), 531,. [23],, (1997), 79,. [24] Saito, R. et al, Chem.Phs.Lett., 195, (1992), [25] Terrones, M. et al, Structural.Chem., 13, (2002), [26] Wang, B.-C., Wang, H.-W., Chang, J.-C., Tso, H.-C., J.Mol.Struct., 540, (2001), [27] Lu, J.P., Yang, W., Phs.Rev.B., 49, 16, (1994), [28] Bakowies, D. et al, J.Am.Chem.Soc., 117, (1995), [29] Jol-Pottu, L., Buchol, E.W., Matsumoto, N. et al, Tribol.Lett., 37, (2010), [30] Buchol, E.W., Phillpot, S.R. et al, Com.Mater.Sci., 54, (2012), [31] Brenner, D.W., Phs.Rev.B., 42, 15, (1990), [32] Ahmadieh, A.A., Rafiadeh, H.A., Phs.Rev.B., 7, (1973), [33],, (1990),. [34] (2011).
97 OLC 1 ( 16 ),, (2011.5) OLC 24 CMD2011,, ( ) 25 CMD2012,, ( ) 91
98 92
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3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e
80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0
79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t
2000年度『数学展望 I』講義録
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