defects in hard-sphere colloidal crystals
TRANSCRIPT
Defects in Hard-Sphere Colloidal Crystals
CitationPersson Gulda, Maria Christina Margareta. 2012. Defects in Hard-Sphere Colloidal Crystals. Doctoral dissertation, Harvard University.
Permanent linkhttp://nrs.harvard.edu/urn-3:HUL.InstRepos:10406377
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Accessibility
µ
∆Vm = 0.19Vo Vo
∆Sm = 0.49kBT
φ
φ φ
φ
φ
×
µ
µ
µ
µ
µ
µm3
µ 3
µm
µm
µ 3
µ 3
µ 3
µ 3
µ 3
µ 3
µ 2 µ 2
∑∑
16a < 112 > → →
µ
a µ
µ
α
FPK
FPK Fl
Fd
1
Σ
2
µ σ σ
3 3.9 × 10−15
3
fRawStock
fwater
fwater = 0.369%− fRawStock
fRawStock
σd
vsettling g
Fd Fg
vsettling
Fg = mg =4
3πa3∆ρg
a d
∆ρ
η v
a
Fd = 6πηav
vsettling
vsettling =4a2∆ρg
18η
η ∼
1.6×10−3 7×10−7
a2
Uthermal =3kBT
2
Ukinetic =mv2
2
vb
< vb >=
√3kBT
m
kB
2× 10−3 104
φ
F = U − TS
Uthermal =32kBT
F = (3
2kB − S)T
φ
φ
φ
φ
φ ∼
φ
P ρ kB
P = ρkBTZ(φ)
Z φ
Z(β) = 2.557696 + 0.1253077β + 0.1762393β2 − 1.053308β3
+2.818621β4 − 2.921934β5 + 1.118413β6 +12− 3β
β
β
β(φ) = 4(1− φ
φo)
φo
ρSilicaRawStock3
fRawStock
VRawStock = VCellfRawStock
VCell
ACell hCell
VRawStock = ACellhCellfRawStock
mSilica VRawStock
ρSilicaRawStock
mSilica = VRawStockρSilicaRawStock = ACellhCellfRawStockρSilicaRawStock
VSilica mSilica
ρSilica 3
VSilica =mSilica
ρSilica=
ACellhCellfRawStockρSilicaRawStock
ρSilica
VCrystal VSilica
ΦCrystal
VCrystal =VSilica
ΦCrystal=
ACellhCellfRawStockρSilicaRawStock
ρSilicaΦCrystal
VCrystal
ACell hCrystal
hCrystal =hCellfRawStockρSilicaRawStock
ρSilicaΦCrystal
fRawStock ΦCrystal
hCell hCrystal
hCrystal =hCellfRawStock50mg/cm3
ΦCrystal2 g/cm3= 0.025
hCellfRawStock
ΦCrystal
φφ φ
φφ
dc
Ukinetic =mv2b2
mgd d
dc =v2b2g
vb dc ∼ 100
∼ 700
fRawStock
hCell
3
G
G = H − TS
∆S
∆H
∆H ∆h
∆H = n∆h
∆H
p ∆Vv
∆Vv = Vp +∆Vrelax
Vp ∆Vrelax
∆H = n p∆Vv
∆S
∆sc ∆sv ∆sc
∆Sc = kBln((N + n)!
N !n!) = kB[(N + n)ln(N + n)−Nln(N)− nln(n)]
∆G = n∆h− nT∆sv − kBT [(N + n)ln(N + n)−Nln(N)− nln(n)]
δ∆G
δn= ∆h− T∆sv − kBT [ln(N + n) +
N + n
N + n− ln(n)− n
n]
= ∆h− T∆sv + kBT ln(n
N + n) = 0
xv
xv =n
N + n= e
−∆h−T∆svkBT
10−6−10−3
10−4
10−7− 10−4
10−3 10−2 − 10−3
Γ
∆Gm = ∆Hm − T∆Sm = p∆Vm − T∆Sm
Γ = ν × exp(∆Gm
kBT) = ν × exp(
p∆Vm − T∆Sm
kBT)
kB ∆Vm
∆Sm
ν
D λ
ν =6D
λ2
D η
kB a
D =kBT
6πηa
λ 2a
ν
ν =kBT
4πηa3
kB = 1.38× 10−23 T = 300 η = 1.6× 10−3
a = 1.55× 10−6/2 µ ν ∼ 0.6 −1.
∆Va
∆Sa ∆Ga =
p∆Va − T∆Sa xv
xv2
xv2x2v
= exp(−∆Ga
kBT) = exp(
T∆Sa − p∆Va
kBT)
µ
×
xtemplate ytemplate
µ
ax ay
az xtemplate bx by
bz ytemplate
(ax× δx)2 + (ay × δy)2 + (az × δz)2 = x2template
(bx× δx)2 + (by × δy)2 + (bz × δz)2 = y2template
δx δy δz µ
ztemplate
µ µ
DBody
µ
D2Body = x2 + y2 + z2
z =√
D2Body − x2 − y2
µ
z =√
D2Body − x2 − y2 =
√(2d)2 − 2d2 =
√2d =
√2 ∗ 1.56 = 2.21µ
εz
εx εy ν
εz + εxν + εyν = 0
εx,y =1.63− 1.56
1.56= 4.5%
εz = −2εx,yν = −2× 0.045× 0.37 = −3.3%
ztemplate
ztemplate = z(1 + εz) = 2.21× (1− 0.033) = 2.14µ
(cx× δx)2 + (cy × δy)2 + (cz × δz)2 = z2template
δx δy δz
δx = 0.1538µ
δy = 0.1553µ
δz = 0.1099µ
µ
µ
Nparticles(111) 36× 3 = 108
VBox(111)
VBox(111) µ 3
µ 3
NParticles(100)
VBox(100) µm3
µ 3
µ 3
µ 3
∼
µ
µ
µ µ
u
u =cr
r3= −c∇1
r
∆V =
∫udS = u4πr2
u =∆V
4πr2
u1u2
=∆R1
∆R2=
R22
R21
∆R1 =2.292
1.622(2.29− 2.26)µm = 0.06µm
∆R1 µ
R1 µ
µ
µ
∆R1 µ
µ 3 µ 3
µ
µ
µ µ
µ µ
µm3
µ 3
µ
µ
µmµm
µ µ 3
µ 3
µ 3
P (vf ) =γ
< vf >exp
−γvf< vf >
vf < vf >
γ
µ −3 −γ/< vf > < vf >
µ 3
< vparticle >=< vo > + < vf >=4
3(1.55
2)3π/0.74 + 0.052 = 2.69µm3
vo
vparticle
< vf >
µ 3
µ 3
µ 3
µ 3
µ 3
µ 3
µ 3
µ 3
µ 3
µ 3
µ 3
µm2
µ 2
∆Vv
∆Vv = ∆Vp +∆Vrelax
PVoNkBT
µ 3
µ 3
µ 2
µ 2
∆Vrelax ∆Vv
µ µ µ 3
µ 3 µ 3 µ 3
µ 3
µ 3 µ 3 µ 3
µ 3
∆Vrelax ∆Vv
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 pkBT = ZN
V
PVo
NkBT=
ZVo
V
Γ =1
1115× 1
14× 3600= 1.8× 10−8sec−1
ν = 0.6 −1
∆GmkBT
Γν
PVoNkBT
PVoNkBT
PVoNkBT
Γν 1021
µ 3
rv1 rv2
1
12(11∑
i=1
ri + rv2) = rv1
1
12(11∑
j=1
rj + rv1) = rv2
ri rj
µm3 µm3
µm3
∆Va
∆Ha
∆Sa
∆Ga = ∆Ha − T∆Sa
∆Ga
µ 3
µ 3 ∆Vv
∆Vv = 18× (3.13− 2.75)µm3 = 6.84µm3
µ 3
∆Vv Vrelax µ 3
Γ =4
18× 1
13.5× 3600= 4.6× 10−6sec−1
√4 ±2.3 × 10−6
ν = 0.6 −1
∆GmkBT ±0.7
logΓ
ν= − p∆Vm
kBT+
∆Sm
kB
Vo
logΓ
ν= − pV0
kBT
∆Vm
V0+
∆Sm
kB
∆VmV0
∆SmkB
Γ =3
5× 1
13.5× 3600= 1.2× 10−5sec−1
√4 ±0.7 × 10−5
ν −1 ∆GmkBT
±0.8
∆GmkBT
Γ −1 ∆GmkBT
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
∑
∑
∑
16a < 112 > → → →
√22
∑∑
σy
α
σy = σo + kd−12
110 111
a
±12a[011] ±1
2a[110]
π ±12a[101]
π
T
1
2a[011] → 1
2a[011]T
±12a[110],±
12a[101]
12
1
2a[110] → 1
2a[110]T + 2× 1
6a[112]
1
2a[110] → 1
2a[101]T +
1
6a[211]
12
Σ
16 [112]
16a[112]
pV = Z(V )kBT
K = −V dp
dV
K =ZkBT
V(1− V
Z
δZ
δV)
µ
γ
µ =1
V
δ2F
δγ2
γ
Uthermal
µ = −T
V
δ2S
δγ2
µ
o
a6
(α)
α
ε =bcos(α)
L
Uelastic =1
2Eε2elastich
εelastic
ε0 ε
εelastic = ε0 − ε
Uelastic
Uelastic =1
2E(ε0 −
bcos(α)
L)2h
Udislocation =1
L
µb2
4π(1− ν)ln
R
ro
ν
ro
1
L=
ε0bcos(α)
− 1
4π(1− ν)
µ
E
1
cos2(α)
1
hln
4h
b
hc
hc =1
4π(1− ν)
µ
E
1
ε0
b
cos(α)ln
4hcb
E
µ= 2(1 + ν)
Z
zo
Uelastic =1
2Eε20z
Uelastic
Felastic =1
2LEε20
Frepel =µ(bcos(α))2
4π(1− ν)z
zo
zo =(bcos(α))2
2π(1− ν)Lε20
µ
E
µm
µ
1.025a
16a < 112 > → →
a µ
a
Aparticle
Aparticle =a2√2
µ
Aparticle
Nlayer =Atotal
Aparticle=
√2L2
a2
µ Nlayer
µ
µ aµ
h(Number of layers) = 0.022t+ 28.38 (0 ≤ t ≤ 900min)
tlayera
2√2
µ
h(µm) = 0.018t+ 23 (0 ≤ t ≤ 900min)
J
vsettling
nColloidsInSolution
J = vsettlingnColloidsInSolution
Γ
ACrossSection
Γ = J A = vsettlingnColloidsInSolutionACrossSection
nColloidsInSolution
ρSilicaInRawStock
mcolloid
fRawStock
nColloidsInSolution =ρSilicaInRawStock
mcolloidfRawStock
Γ Nlayer =ACrossSection
Aparticle=
ACrossSection√2a2
tlayer
hTheory
hTheory =Γ
Nlayertlayer =
vsettlingρSilicaInRawStockfRawStockAparticle tlayermcolloid
ρSilicaInRawStock vsettling mcolloid Aparticle
tlayer fRawStock
µ
ρColloidsInSolution
o
o o
o
(x, y, z)
µ
[x00] → 1√6[121]
[0y0] → 1√3[111]
[00z] → 1√2[101]
x 1√6
2√6
1√6
y 1√3
1√3
1√3
z 1√2
1√2
[x00] → 1√6[121]
[0y0] → 1√3[111]
[00z] → 1√2[101]
x 1√6
2√6
1√6
y 1√3
1√3
1√3
z 1√2
1√2
a6 [112]
a6 [112]
a6 [112]
bt
a
6[112]LeftCrystalSystem =
a
6[112]LeftCrystalSystem+
a
6[112]RightCrystalSystem+bt
bt = −a
6[112]RightCrystalSystem
bt
bt = [0.2556,−0.8944,−0.1278]
bt,exp = [0.2756,−0.7203,−0.2759]
µ
µ
AperParticle =
√3a2
4
a
µ
L2
L1
∆ε =b
L1
∆ε =b
L1(A
hL2) =
bA
V
Aeffective
L1
Aeffective = Asinα
α o
L1
beffective = bcosα
α
∆ε =bcosαAsinα
V
∼ 8.3×10−4
∼ 9.9× 10−4
∼ 4.0 × 10−4
∼ 6× 10−3
∼ 2 × 10−3
FPK
FPK b ¯σ
ds
FPK = (¯σ · b)× ds
FPK = Fd + 2Fl
FPK
FPK Fl
Fd
FPK = Fd + Fl
Fd =4η
π∆Sv
∆S
µ η 1.6×10−3
µ
1.5× 10−15
Fl =1
2µb2
µ = 2 b = 0.81µ 6.5×10−13
FPK = σ b∆S
σ = µ∆ε = µ(εo − ε)
∆ε =12µb
2
µb∆S= 8× 10−3
ε εo−ε
∆S
∆r FPK FPK∆r
FPK2∆r
FPK(3∆r) = T (3∆r) + 2T∆S
FPK
FPK
∆S=
T
∆S+
2T
3∆r
T∆S 8 × 10−3µb
∆εb
∆εb = 8× 10−3(1 +2∆S
3∆r)
∆r µ
εo − ε
5
∆Vm = 0.19Vo Vo
∆Sm = 0.49kBT
6
×10−4
µ × µ
× 2
o
7
8
µ
