lecture ii cnt - società italiana di fisica · r. zwanzig, nonequilibrium statistical mechanics,...
TRANSCRIPT
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
∑−=+
=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