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. 2012. Defects in Hard- Sphere Colloidal Crystals. Doctoral dissertation, Harvard University. November 13, 2017 8:05:51 PM EST http://nrs.harvard.edu/urn-3:hul.instrepos:10406377 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 http://nrs.harvard.edu/urn-3:hul.instrepos:dash.current.terms-ofuse#laa (Article begins on next page)
µ V m =0.19V o V o S m =0.49k B T
φ φ φ φ φ
µ
µ µ µ µ µm 3 µ 3 µm µm µ 3 µ 3 µ 3 µ 3
µ 3 µ 3 µ 2 µ 2
1 a<11 2 > 6 µ a µ
µ α
F PK F PK F l F d
1
Σ
2
µ σ σ 3 3.9 10 15 3 f RawStock f water
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
η 1.6 10 3 7 10 7 a 2 U thermal = 3k BT 2 U kinetic = mv2 2 v b <v b >= 3kB T m k B 2 10 3 10 4
φ F = U TS U thermal = 3 2 k BT F =( 3 2 k B S)T φ φ φ φ φ φ
P ρ k B P = ρk B TZ(φ) Z φ Z(β) = 2.557696 + 0.1253077β +0.1762393β 2 1.053308β 3 +2.818621β 4 2.921934β 5 +1.118413β 6 + 12 3β β β β(φ) = 4(1 φ φ o ) φ o ρ SilicaRawStock 3 f RawStock
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
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
φ φ φ φ φ
d c U kinetic = mv2 b 2 mgd d d c = v2 b 2g
v b d c 100 700 f RawStock h Cell
3
G G = H TS S H H h H = n h
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)]
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 10 6 10 3 10 4 10 7 10 4 10 3 10 2 10 3 Γ
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 =1.38 10 23 T = 300 η =1.6 10 3 a =1.55 10 6 /2 µ ν 0.6 1.
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
µ
x template y template µ
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 1.56 = 2.21µ
ε z ε x ε y ν ε z + ε x ν + ε y ν =0 ε x,y = 1.63 1.56 1.56 =4.5% ε z = 2ε x,y ν = 2 0.045 0.37 = 3.3% z template z template = z(1 + ε z ) = 2.21 (1 0.033) = 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
µ
µ
N particles(111) 36 3 = 108 V Box(111) V Box(111) µ 3
µ 3 N P articles(100) V Box(100) µm 3 µ 3 µ 3 µ 3
µ
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 = 2.292 (2.29 2.26)µm =0.06µm 1.622 R 1 µ R 1 µ
µ µ R 1 µ µ 3 µ 3 µ µ
µ µ µ µ
µm 3 µ 3
µ µ
µm µm
µ µ 3 µ 3 µ 3
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 (1.55 2 )3 π/0.74 + 0.052 = 2.69µm 3 v o v particle <v f >
µ 3
µ 3
µ 3 µ 3 µ 3 µ 3 µ 3 µ 3
µ 3 µ 3
µ 3 µm 2 µ 2 V v V v = V p + V relax PVo Nk B T
µ 3 µ 3
µ 2 µ 2
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
G m k B T Γ= 1 1115 1 14 3600 =1.8 10 8 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
µ 3 r v1 r v2 1 11 12 ( r i + r v2 ) = r v1 i=1 1 11 12 ( r j + r v1 ) = r v2 j=1 r i r j µm 3 µm 3
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 (3.13 2.75)µm 3 =6.84µm 3 µ 3 V v V relax µ 3 Γ= 4 18 1 13.5 3600 =4.6 10 6 sec 1
4 ±2.3 10 6 ν =0.6 1 Gm k B T ±0.7
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
Γ= 3 5 1 13.5 3600 =1.2 10 5 sec 1 4 ±0.7 10 5 ν 1 Gm k B T ±0.8 G m k B T Γ 1 Gm k B T 1.8 10 8 (4.6 ± 2.3) 10 6 11.8 ± 0.7 (1.2 ± 0.7) 10 5 10.8 ± 0.8
4
1 6 a<11 2 > 2 2
σ y α σ y = σ o + kd 1 2
1 10 111 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] 1 2 1 2 a[110] 1 2 a[110] T +2 1 6 a[11 2]
1 2 a[110] 1 2 a[101] T + 1 6 a[2 1 1] 1 2 Σ 1 6 [ 1 1 2] 1 6 a[ 1 12]
pv = Z(V )k B T K = V dp dv K = Zk BT V (1 V Z δz δv ) µ
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
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
E µ = 2(1 + ν) Z z o U elastic = 1 2 Eε2 0z U elastic F elastic = 1 2 LEε2 0
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
1 6 a<11 2 >
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
µ a µ
h(number of layers) = 0.022t + 28.38 (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
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
(ˆx, ŷ, ẑ)
µ
[ˆx00] 1 6 [ 12 1] [0ŷ0] 1 3 [ 1 1 1] [00ẑ] 1 2 [ 101] ˆx 1 6 2 6 1 6 ŷ 1 3 1 3 1 3 ẑ 1 2 1 2 [ˆx00] 1 6 [121] [0ŷ0] 1 3 [ 11 1] [00ẑ] 1 2 [ 101]
ˆx 1 6 2 6 1 6 ŷ 1 3 1 3 1 3 ẑ 1 2 1 2 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, 0.8944, 0.1278]
bt,exp = [0.2756, 0.7203, 0.2759] µ µ
a A perp article = 3a 2 4 µ
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
α
ε = bcosαasinα V 8.3 10 4 9.9 10 4 4.0 10 4 6 10 3 2 10 3
F PK F PK b σ d s F PK =( σ b) d s F PK = F d +2F l
F PK
F PK F l F d
F PK = F d + F l F d = 4η π Sv S µ η 1.6 10 3 µ 1.5 10 15 F l = 1 2 µb2 µ =2 b =0.81µ 6.5 10 13 F PK = σb S σ = µ ε = µ(ε o ε) ε = 1 2 µb2 µb S =8 10 3 ε ε o ε
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 8 10 3 µb ε b ε b =8 10 3 (1 + 2 S 3 r ) r µ ε o ε
5
V m =0.19V o V o S m =0.49k B T
6
10 4
µ µ
2
o
7
8 µ