Lecture  II

Classical  nuclea/on  theory

Outline

1. Kramers  theory  

2. Classical  Nuclea2on  Theory  (CNT)  

3. Seeding  method:  crystalliza2on  of  water  

4. Detec2ng  crystalline  regions  

5. Hard  sphere  freezing  

6. Free  energies  are  not  unique  

7. Dynamical  correc2ons:  BenneD-­‐Chandler

Homogeneous  vs.  heterogeneous  nuclea/on

nucleus,  crystallite,  embryo,  cluster,  nanopar2cle

heterogeneous  nuclea2on

Cacciuto, Auer and Frenkel, Nature (2004)

Filion, Hermes, Ni, Dijkstra, JCP (2010)

homogeneous  nuclea2on

Assump/ons  of  CNT

R. P. Sear, Int. Mat. Rev. (2012)

1. One-­‐step  process:  only  one  barrier  significant;  nucleus  consists  of  a  piece  of  new  bulk  phase  

2. Nucleus  grows  one  monomer  at  a  2me  

3. Crystal  laLce  can  be  neglected,  nucleus  is  spherical  

4. Only  long  2me  scale  is  that  of  barrier  crossing  (microscopic  kine2cs  are  fast)  

5. Nuclea2on  rate  does  not  depend  on  history    

6. Nuclea2on  occurs  over  saddle  point  in  free  energy  (nucleus  size  is  reac2on  coordinate)

Escape  over  barrier  -­‐  the  Kramers  problem

@⇢(q, t)

@t=

@

@q

D(q)

@�U(q)

@q⇢(q, t) +D(q)

@⇢(q, t)

@q

�

Smoluchowski  equa2on

U(q)

Time  evolu2on  of  q

q̇ = �D(q)�@U(q)

@q+p

2D(q)⇠(t)

R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford (2001).

D(q)

q0a b

reflecting

absorbing

Mean  first  passage  2me

⌧(q0) =

Z b

q0

dye�U(y)

D(y)

Z y

adz e��U(y)

� = 1/kBT

Escape  over  barrier  -­‐  the  Kramers  problem

U(q)

D(q)

q0a b

reflecting

absorbing

Mean  first  passage  2me

⌧(q0) =

Z b

q0

dye�U(y)

D(y)

Z y

adz e��U(y)

q*

k�1 =

Z

\dq

e�U(q)

D(q)

Z

[dq e��U(q)

Escape  rate

\[

Expand  U(q)  around  q* Kramers’  result

P (q⇤)

U(q) ⇡ U(q⇤) +1

2!2(q � q⇤)2

≫ kBT

✴ Barrier high compared to kBT✴ Diffusions constant D(q) constant on barrier✴ Top of barrier parabolic

k =!D(q⇤)p2⇡kBT

e��U(q⇤)

R[ dq e��U(q)

Crystal  growth  as  diffusion  process

n  =  size  of  the  crystalline  cluster

n*D(n)  =  diffusion  constant  of  cluster  size  n  P(n)  =  equilibrium  cluster  size  distribu2on

k =!D(n⇤)p2⇡kBT

P (n⇤)

ΔF*

Nuclea2on  rate J =k

VEscape  rate

Classical  nuclea/on  theory

n*

ΔF*

Nuclea2on  rate

Diffusion  constant

Nuclea2on  free  energy

Probability  of  having  cri2cal  nucleus

Curvature  of  barrier

Cri2cal  cluster  size  and  barrier

�F (n) = (36⇡)1/3(nv)2/3� � n|�µ|

n⇤ =32⇡

3

�3v2

|�µ|3 �F ⇤ =16⇡

3

�3v2

|�µ|2

P (n⇤) = Ne���F⇤

!2 =|�µ|3n⇤ D(n⇤) = 24

DS

�2n⇤2/3

J =

r�|�µ|6⇡n⇤ 24

DS

�2n⇤2/3⇢ exp

⇢� 16⇡

3kBT

�3

|�µ|2⇢2

�

Zeldovich  factor

kine2c  pre-­‐factor

Nuclea/on  theorem

Macroscopic:  from  temperature   dependence  of  nuclea2on  rate

Microscopic:  sizecri2cal  nucleus

Nuclea2on  theorem  does  not  depend  on  CNT

Moore  and  Molinero,  Nature  (2011)  

J = exp

⇢� 16⇡

3kBT

�3

|�µ|2⇢2

� n⇤ =@kBT ln J

@�µ� @kBT ln

@�µ

water hard  spheres

Harland  van  Megen,  PRE  (1997)

n⇤ =32⇡

3

�3v2

|�µ|3 �F ⇤ =16⇡

3

�3v2

|�µ|2

Seeding  method

Sanz, Vega, Espinosa, Caballero-Bernal, Abascal, and Valeriani, JACS 135, 15008 (2013)

TIP4P/ice  &  TIP4P/2005  15K-­‐35K  below  mel2ng  (moderate  undercooling)  γ=29  mN/m  (by  fiLng  CNT)  Δμ  from  thermodynamic  integra2on  N=2  x  105  Tm  =  252  K  for  TIP4P/2005    Tm  =  272  K  for  TIP4P/ice  

TIP4P/2005

✴ Size of critical cluster✴ Surface free energy✴ Nucleation rate

n⇤ =32⇡

3

�3v2

|�µ|3 �F ⇤ =16⇡

3

�3v2

|�µ|2

Nuclea/on  rate  from  seeding  method

Sanz,  Vega,  Espinosa,  Caballero-­‐Bernal,  Abascal,  and  Valeriani,  JACS    135,  15008  (2013)    

J =

r�|�µ|6⇡n⇤ D(n⇤

)⇢ exp

⇢�16⇡

3

��3

|�µ|2⇢2

�

h[n(t)� n⇤]2i ⇡ 2D(n⇤)t

h[n(t)�

n⇤ ]

2i⇡

2D(n

⇤ )t

It  is  impossible  that  ice  nucleates  homogeneously  at  temperatures  above  -­‐20  C  

�

qlm (i) =1

Nb (i)Ylm (rij )

j=1

Nb (i)

∑

�

Ylm (ϑ,ϕ)

spherical  harmonics

P. Steinhardt, D. R. Nelson, and M. Ronchetti, PRB 28, 784 (1983)

rotationally  invariant

�

ql (i) =4π2l +1

qlm (i)2

m=−l

l

∑

Detec/ng  local  order  Steinhardt  bond  order  parameter

l=0

l=1

l=2

l=3

Dis/nguishing  solid  structures

∑−=+

=l

lmlml iq

liq 2)(

124)( π

W. Lechner and C. Dellago, JCP 129, 114707 (2008)

q4_

q6_

FCCHCPBCCliquid

Averaged  local  order  parameter

q̄lm(i) =1

Nb + 1

"qlm(i) +

NbX

k=1

qlm(nk)

#

�

Nbonds = Θ(sij − 0.5)Nb

∑

�

sij =4π2l +1

qlm (i)m=−l

l

∑ qlm* ( j)

ql (i)ql ( j)

structural  correlations  

Dis/nguishing  liquid  from  solid

sij  =  1        perfect  crystal  sij  =  0        liquid

if  sij  >  0.5:    i  and  j  connected    

n  =  size  of  largest  cluster  of  crystalline  par2cles  

Freezing  of  hard  spheres

p

φ0.494 0.545 0.74

Hard  sphere  freezing  Kirkwood  (1941)

Alder  and  Wainwright  (1957)  Wood  and  Jacobsen  (1957)

φ  =  packing  frac2on

Nuclea/on  of  hard  sphere  crystals

Auer  and  Frenkel,  Nature  (2001)  Gasser,  Weeks,  Schofield,  Pusey,  Weitz,  Science  (2001)

k =!D(n⇤)p2⇡kBT

P (n⇤)

n(x)  =  size  of  the  largest  crystalline  cluster

Compute  P(n)  in  computer  simula2on

P (n) =

Zdx⇢(x)�[n� n(x)] = h�[n� n(x)]i

Free  energy  F(n)

F (n) = �kBT lnP (n)

k =!D(n⇤)p2⇡kBT

e��F (n⇤)

Rate

cri2cal  cluster

•  Generate  test  configura2on    •  Calculate  energy    •  Accept  or  reject

Markov  chain  Monte  Carlo  simula/on

Compu/ng  the  nuclea/on  free  energyUmbrella  sampling  with  bias  UB(x)

Parallel  replica  

Auer  and  Frenkel,  J.  Chem.  Phys.  (2004)

UB[n(x)] =1

2Kj [n(x)� nj ]

2

Nuclea2on  free  energy  

Compu/ng  the  kine/c  prefactor

Auer  and  Frenkel,  J.  Chem.  Phys.  (2004)

k =!D(n⇤)p2⇡kBT

P (n⇤)

Kine2c  prefactor

h[n(t)� n⇤]2i ⇡ 2D(n⇤)t

φ  =  0.5277 ✴ Does not rely on shape of barrier✴ Only assumption: diffusive in n

Nuclea/on  free  energy

S. Jungblut and C. Dellago, JCP 134 10450 (2011)

Lennard-Jones, T=0.5 (28% undercooling)

R. P. Sear, Int. Mat. Rev. (2012)

1. One-­‐step  process:  only  one  barrier  significant;  nucleus  consists  of  a  piece  of  new  bulk  phase  

2. Nucleus  grows  one  monomer  at  a  2me  

3. Crystal  laLce  can  be  neglected,  nucleus  is  spherical  

4. Only  long  2me  scale  is  that  of  barrier  crossing  (microscopic  kine2cs  are  fast)  

5. Nuclea2on  rate  does  not  depend  on  history    

6. Nuclea2on  occurs  over  saddle  point  in  free  energy  (nucleus  size  is  reac2on  coordinate)

How  classical  is  nuclea/on?Ostwald’s  step  rule(1897)  (ice  from  cold  vapor)

Nuclea2on  near  curved  objects  Role  of  defects?

Other  slow  processes    (charge  ordering,  glassy  dynamics)

Sanz, Valeriani, Frenkel, Dijkstra, PRL (2012)

Thermal  history  in  porous  media

Shape,  structure  maDer

non-classical nucleation

Free  energy  landscapes  are  not  unique

PZ(z) = PQ['�1(z)]

����d'�1

dz

����

PQ(q) =

Zdx⇢(x)�[q � q(x)]

FQ(q) = �kBT lnPQ(q)

FZ(z) = FQ['�1(z)]� kBT ln

����d'�1

dz

����

FQ(q) = (1� q2)2 + C

FZ(z) = z2 + C 0

z = '(q) q = '�1(z)z

Transi/on  rates  are  unique

C(t) =hhA [q(0)]hB [q(t)]i

hhAi

q  evolves  according  to  Langevin  equa2on

z

hB(x) = ✓[q(x)]

hA(x) = 1� ✓[q(x)]

BenneW-­‐Chandler  approach  

Time  correla2on  func2on

C(t) ⇡ kABt

Phenomenological  kine2cs

hhA(x0)hB(xt)ihhAi

= hhBi⇣1� e

�t/⌧rxn

⌘

⌧�1

rxn

= kAB + kBAC(t) =

hhA(x0)hB(xt)ihhAi

BenneW-­‐Chandler  approach  

dynamical  correc2on   from  free  energy  

transmission  coefficient   = kAB/kTST

k(t) =dC(t)

dt

D. Chandler, JCP (1978)

Transi/on  state  theory  (TST)

Assump2on  of  TST:  no  recrossings  of  dividing  surface

r1

r2

A

TS

Harmonic  TST

kTST =hq̇�[q � q⇤]✓[q̇]i

h�[q⇤ � q]i ⇥ h�[q⇤ � q]ih✓[q⇤ � q]i

dividing  surface

B

Kramers  turnover

D. Chandler, Lerici Lectures (1997)