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

https://github.com/suchow/