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)

3 µ V m =0.19V o V o S m =0.49k B T

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

20 1

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

34 d c U kinetic = mv2 b 2 mgd d d c = v2 b 2g

35 v b d c f RawStock h Cell

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

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

52 µ

54 N particles(111) 36 3 = 108 V Box(111) V Box(111) µ 3

57 µ 3 N P articles(100) V Box(100) µm 3 µ 3 µ 3 µ 3

58 µ

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

64 µ µ µ µ

65 µm 3 µ 3

67 µ µ

68 µm µm

69 µ µ 3 µ 3 µ 3

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 >

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

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

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

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

94 4 ± ν =0.6 1 Gm k B T ±0.7

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

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

138

139 α

140 ε = bcosαasinα V

141

142

143

144 F PK F PK b σ d s F PK =( σ b) d s F PK = F d +2F l

145

146 F PK

147 F PK F l F d

148

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

162

163 7

164 8 µ

165

166

167

168

169

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