Defects in Hard-Sphere Colloidal Crystals
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- Αλθαία Γεννάδιος
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1 Defects in Hard-Sphere Colloidal Crystals The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. Citation Accessed Citable Link Terms of Use Persson Gulda, Maria Christina Margareta Defects in Hard- Sphere Colloidal Crystals. Doctoral dissertation, Harvard University. November 13, :05:51 PM EST This article was downloaded from Harvard University's DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at (Article begins on next page)
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3 µ V m =0.19V o V o S m =0.49k B T
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12 φ φ φ φ φ
13 µ
14 µ µ µ µ µm 3 µ 3 µm µm µ 3 µ 3 µ 3 µ 3
15 µ 3 µ 3 µ 2 µ 2
16 1 a<11 2 > 6 µ a µ
17 µ α
18 F PK F PK F l F d
19
20 1
21
22 Σ
23 2
24 µ σ σ f RawStock f water
25 f water =0.369% f RawStock f RawStock σ d v settling g F d F g v settling F g = mg = 4 3 πa3 ρg a d ρ η v a F d =6πηav v settling v settling = 4a2 ρg 18η
26 η a 2 U thermal = 3k BT 2 U kinetic = mv2 2 v b <v b >= 3kB T m k B
27 φ F = U TS U thermal = 3 2 k BT F =( 3 2 k B S)T φ φ φ φ φ φ
28 P ρ k B P = ρk B TZ(φ) Z φ Z(β) = β β β β β β β β β β(φ) = 4(1 φ φ o ) φ o ρ SilicaRawStock 3 f RawStock
29 V RawStock = V Cell f RawStock V Cell A Cell h Cell V RawStock = A Cell h Cell f RawStock m Silica V RawStock ρ SilicaRawStock m Silica = V RawStock ρ SilicaRawStock = A Cell h Cell f RawStock ρ SilicaRawStock V Silica m Silica ρ Silica 3 V Silica = m Silica ρ Silica = A Cellh Cell f RawStock ρ SilicaRawStock ρ Silica V Crystal V Silica Φ Crystal V Crystal = V Silica Φ Crystal = A Cellh Cell f RawStock ρ SilicaRawStock ρ Silica Φ Crystal V Crystal A Cell h Crystal
30 h Crystal = h Cellf RawStock ρ SilicaRawStock ρ Silica Φ Crystal f RawStock Φ Crystal h Cell h Crystal h Crystal = h Cellf RawStock 50 mg/cm 3 Φ Crystal 2 g/cm 3 =0.025 h Cellf RawStock Φ Crystal
31 φ φ φ φ φ
32
33
34 d c U kinetic = mv2 b 2 mgd d d c = v2 b 2g
35 v b d c f RawStock h Cell
36
37 3
38 G G = H TS S H H h H = n h
39 H p V v V v = V p + V relax V p V relax H = np V v S s c s v s c S c = k B ln( (N + n)! )=k B [(N + n)ln(n + n) Nln(N) nln(n)] N! n! G = n h nt s v k B T [(N + n)ln(n + n) Nln(N) nln(n)]
40 δ G δn = h T s v k B T [ln(n + n)+ N + n N + n ln(n) n n ] n = h T s v + k B T ln( N + n ) = 0 x v x v = n N + n = h T sv e k B T Γ
41 G m = H m T S m = p V m T S m Γ= ν exp( G m k B T )=ν exp(p V m T S m ) k B T k B V m S m ν D λ ν = 6D λ 2 D η k B a D = k BT 6πηa λ 2a ν ν = k BT 4πηa 3 k B = T = 300 η = a = /2 µ ν
42 V a S a G a = p V a T S a x v x v2 x v2 x 2 v = exp( G a k B T )=exp(t S a p V a ) k B T
43 µ
44
45
46 x template y template µ
47 ax ay az x template bx by bz y template (ax δx) 2 +(ay δy) 2 +(az δz) 2 = x 2 template (bx δx) 2 +(by δy) 2 +(bz δz) 2 = y 2 template δx δy δz µ z template µ µ D Body µ D 2 Body = x2 + y 2 + z 2 z = DBody 2 x2 y 2 µ z = DBody 2 x2 y 2 = (2d) 2 2d 2 = 2d = = 2.21µ
48 ε z ε x ε y ν ε z + ε x ν + ε y ν =0 ε x,y = =4.5% ε z = 2ε x,y ν = = 3.3% z template z template = z(1 + ε z ) = 2.21 ( ) = 2.14µ (cx δx) 2 +(cy δy) 2 +(cz δz) 2 = z 2 template δx δy δz δx =0.1538µ δy =0.1553µ δz =0.1099µ
49 µ
50
51
52 µ
53
54 N particles(111) 36 3 = 108 V Box(111) V Box(111) µ 3
55
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57 µ 3 N P articles(100) V Box(100) µm 3 µ 3 µ 3 µ 3
58 µ
59
60
61 µ µ µ ū ū = c r r 3 = c 1 r V = u ds = u4πr 2 u = V 4πr 2 u 1 u 2 = R 1 R 2 = R2 2 R 2 1 R 1 = ( )µm =0.06µm R 1 µ R 1 µ
62 µ µ R 1 µ µ 3 µ 3 µ µ
63
64 µ µ µ µ
65 µm 3 µ 3
66
67 µ µ
68 µm µm
69 µ µ 3 µ 3 µ 3
70
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72 P (v f )= γ <v f > exp γv f <v f > v f <v f > γ µ 3 γ/< v f > <v f > µ 3 <v particle >=< v o > + <v f >= 4 3 ( )3 π/ = 2.69µm 3 v o v particle <v f >
73
74 µ 3
75 µ 3
76 µ 3 µ 3 µ 3 µ 3 µ 3 µ 3
77 µ 3 µ 3
78 µ 3 µm 2 µ 2 V v V v = V p + V relax PVo Nk B T
79 µ 3 µ 3
80 µ 2 µ 2
81 V relax V v µ µ µ 3 µ 3 µ 3 µ 3 µ 3 µ 3 µ 3 µ 3 µ 3 V relax V v µ µ µ 3 µ 3 µ 3 µ 3 µ 3 µ 3 µ 3 µ 3 µ 3 µ 3 µ 3 ρ N/V p k B T = ZN V PV o Nk B T = ZV o V
82
83 G m k B T Γ= = sec 1 ν =0.6 1 Γ ν PV o Nk B T PV o Nk B T PV o Nk B T Γ ν 1021
84
85
86
87 µ 3 r v1 r v ( r i + r v2 ) = r v1 i= ( r j + r v1 ) = r v2 j=1 r i r j µm 3 µm 3
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89
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91 µm 3 V a H a S a G a = H a T S a G a µ 3 µ 3 V v V v = 18 ( )µm 3 =6.84µm 3 µ 3 V v V relax µ 3 Γ= = sec 1
92
93
94 4 ± ν =0.6 1 Gm k B T ±0.7
95
96 log Γ ν = p V m k B T + S m k B V o log Γ ν = pv 0 V m + S m k B T V 0 k B V m V 0 S m k B
97
98
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100 Γ= = sec 1 4 ± ν 1 Gm k B T ±0.8 G m k B T Γ 1 Gm k B T (4.6 ± 2.3) ± 0.7 (1.2 ± 0.7) ± 0.8
101
102
103 4
104 1 6 a<11 2 > 2 2
105
106 σ y α σ y = σ o + kd 1 2
107 a ± 1 2 a[01 1] ± 1 2a[110] π ± 1 2a[101] π T 1 2 a[01 1] 1 2 a[0 11] T ± 1 2 a[110], ± 1 2a[101] a[110] 1 2 a[110] T a[11 2]
108 1 2 a[110] 1 2 a[101] T a[2 1 1] 1 2 Σ 1 6 [ 1 1 2] 1 6 a[ 1 12]
109 pv = Z(V )k B T K = V dp dv K = Zk BT V (1 V Z δz δv ) µ
110 γ µ = 1 V δ 2 F δγ 2 γ U thermal µ = T V δ 2 S δγ 2 µ o a 6 (α) α ε = bcos(α) L
111 U elastic = 1 2 Eε2 elastic h ε elastic ε 0 ε ε elastic = ε 0 ε U elastic U elastic = 1 2 E(ε 0 bcos(α) ) 2 h L U dislocation = 1 µb 2 L 4π(1 ν) ln R r o ν r o 1 L = ε 0 bcos(α) 1 4π(1 ν) µ 1 1 E cos 2 (α) h ln4h b h c h c = 1 µ 1 4π(1 ν) E ε 0 b cos(α) ln4h c b
112 E µ = 2(1 + ν) Z z o U elastic = 1 2 Eε2 0z U elastic F elastic = 1 2 LEε2 0
113 F repel = µ(bcos(α))2 4π(1 ν)z z o z o = (bcos(α))2 µ 2π(1 ν)lε 2 0 E µm µ 1.025a
114 1 6 a<11 2 >
115 a µ a A particle A particle = a2 2 µ A particle N layer = A total 2L 2 = A particle a 2 µ N layer µ
116 µ a µ
117
118 h(number of layers) = 0.022t (0 t 900min) t layer a 2 µ 2 h(µm) = 0.018t + 23 (0 t 900min) J v settling n ColloidsInSolution J = v settling n ColloidsInSolution Γ A CrossSection Γ= JA= v settling n ColloidsInSolution A CrossSection n ColloidsInSolution ρ SilicaInRawStock m colloid f RawStock n ColloidsInSolution = ρ SilicaInRawStock m colloid f RawStock Γ N layer = A CrossSection A particle = A CrossSection 2a 2 t layer
119 h T heory h T heory = Γ N layer t layer = v settlingρ SilicaInRawStock f RawStock A particle t layer m colloid ρ SilicaInRawStock v settling m colloid A particle t layer f RawStock µ ρ ColloidsInSolution o o o o
120 (ˆx, ŷ, ẑ)
121
122
123 µ
124 [ˆx00] 1 6 [ 12 1] [0ŷ0] 1 3 [ 1 1 1] [00ẑ] 1 2 [ 101] ˆx ŷ ẑ [ˆx00] 1 6 [121] [0ŷ0] 1 3 [ 11 1] [00ẑ] 1 2 [ 101]
125
126
127 ˆx ŷ ẑ a 6 [112] a 6 [112] a 6 [112] b t a 6 [112] LeftCrystalSystem = a 6 [112] LeftCrystalSystem+ a 6 [112] RightCrystalSystem+ b t bt = a 6 [112] RightCrystalSystem b t bt = [0.2556, , ]
128
129
130 bt,exp = [0.2756, , ] µ µ
131
132
133 a A perp article = 3a 2 4 µ
134
135
136 L 2 L 1 ε = b L 1 ε = b L 1 ( A hl 2 )= ba V A effective L 1 A effective = Asinα α o L 1 beffective = bcosα
137
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139 α
140 ε = bcosαasinα V
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142
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144 F PK F PK b σ d s F PK =( σ b) d s F PK = F d +2F l
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146 F PK
147 F PK F l F d
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149 F PK = F d + F l F d = 4η π Sv S µ η µ F l = 1 2 µb2 µ =2 b =0.81µ F PK = σb S σ = µ ε = µ(ε o ε) ε = 1 2 µb2 µb S = ε ε o ε
150 S r F PK F PK r F PK 2 r F PK (3 r) =T (3 r)+2t S F PK F PK S = T S + 2T 3 r T S µb ε b ε b = (1 + 2 S 3 r ) r µ ε o ε
151
152
153 5
154 V m =0.19V o V o S m =0.49k B T
155 6
156 10 4
157 µ µ
158 2
159 o
160
161
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163 7
164 8 µ
165
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