Kinetics and Thermodynamics of Multistep Nucleation and
Self-assembly in Nanosize Materials
Brussels, March 25-26, 2010
The Two-step Mechanism of Nucleation
of Crystals from Solution
Peter G. Vekilov, Oleg Galkin, Luis Filobelo, Weichun Pan,
Anatoly Kolomeisky (Rice), Dimo Kashchiev (IPC), Vas Lubchenko
Department of Chemical and Biomolecular Engineering,
Department of Chemistry, University of Houston
NIH, NASA, NSF, Welch Foundation
The Goal: Crystals with “Just-right”…
 Number
 Polymorph
 Morphology
 Habit
 Size
 Size distribution
Requires data on:
 Solution PChem
 Phase diagrams
 Metastable
states
 Nucleation
mechanisms
 Growth mechanisms
 Agglomeration
…
Crystallization and Nucleation
Crystallization
Nucleation
Crystallization and Nucleation
Crystallization
Nucleation
Concentration
Crystallization as
Sequential Transition
along Two
Order Parameters
Structure
 Two-step mechanism:
suggested by t W & F, T & O
for critical point for L-L phase
separation for proteins
 Everywhere else in phase
diagram—classical crystal
nucleation predicted
P.R. ten Wolde, D. Frenkel,Science 277 (1997) 1975
V. Talanquer, D.W. Oxtoby, J. Chem. Phys. 109 (1998) 223
Solubility
Temperature
 Classical viewpoint: direct
nucleation along a “diagonal
line” envisioned;
…
Gelation
Binodal
Spinodal
Protein Concentration
Concentration
The Two-step Mechanism
 It may apply to all crystals
(and other ordered solids)
forming in solution
Galkin, O. & Vekilov, P. G. (2000)
Proc. Natl. Acad. Sci. USA 97, 6277
Vekilov, P. G. (2004)
Crystal Growth and Design 4, 671
Solubility
Temperature
 It operates in all areas of the
phase diagram
Structure
…
Gelation
Binodal
Spinodal
Protein Concentration
TL-L
Clys = 50 mg/ml
Clys = 80 mg/ml
0.4
Solubility
Temperature
Homogeneous Nucleation Rate J [cm-3 s-1]
The Nucleation Rate
0.2
Gelation
Binodal
T
L-L
Spinodal
Protein Concentration
0
5
10
15
20
Temperature T [°C]
25
30
 Maximum in J(T)
 Exponential increase at intermediate DT’s; by weak decrease at higher DT’s
 T of maximum shifts with concentration
Two steps:
Which One is Rate
Determining
Concentration
Structure
…
Critical
level of
ordering
Critical size
of ordered
nucleus
Rate of cluster formation
J1 ~ J01exp(– DG1*/kBT)
DG2*
crystals
 Is J02 more important than DG2*?
DG1*
DGL-L
DG2*
dense liquid
 Is J01 more important than
DG1*?
solution
*
*
 Is DG1 > DG2 ?
Free Energy G
Rate of nucleation within clusters
J2 ~ J02exp(–DG2* / kBT)
DG1*
DGL-L
Nucleation Reaction Coordinate
Nucleation of Dense Liquid Droplets
T – TL-L = 0.7 oC
3.53 s
6.09 s
10.58 s
1.92 s
7.37 s
T – TL-L = 1.3 oC
0.96 s
• Number of droplets increases with time
• Faster nucleation at higher DT’s
 Characteristics of nucleation regime of droplet generation
Nucleation Rate of Dense Phase Droplets
Droplets in Viewfield
400
M. Shah, et al.,
J. Chem. Phys. 121 (2004) 7505
300
200
• Number of droplets
increases in time—
nucleation regime
Nucleation Rate = 4.3 x 109 cm-3s-1
• Nucleation rate ~109 cm-3s-1
significantly higher
than rates of crystal
nucleation ~ 0.1 – 1 cm-3s-1
100
0
0
2
4
6
Time [s]
8
10
 Structuring of dense liquid quasi-droplet is the rate determining stage
 Equilibrium between solution and clusters: msolution = mclusters
 Dm(solution,crystal) = Dm(clusters,crystal)
Other proteins:
• Ferritin crystals grown at
s = 4.2, where n*  1
• Are protein crystals always
grown in spinodal regimes?
-3 -1
• Spinodal – boundary between
metastable and unstable two –
phase areas
Nucleation Rate J [cm s ]
Why is the maximum in J(T) sharp?
10
0
10
-1
10
-2
•
10.0oC
12.6oC
15.0oC
n* = 1
2.8
3.2
3.6
Supersaturation Dm/ kBT
Spinodal can be defined from n*  1
L.F. Filobelo, et al.,
J. Chem. Phys. 123, 014904 (2005)
Pre-exponential Factors and Barriers for
Structuring
50
Liquidus or solubility of crystals
Gelation line
o
Temperature [ C]
40
Solution-crystal
spinodal
30
L-L coexistence
20
10
0
D.N. Petsev, et al.,
J. Phys. Chem. B 107 (2003) 3921
L.F. Filobelo, et al.,
J. Chem. Phys. 123 (2005) 014904
L-L spinodal
-10
0
20 mm
100
200
Concentration [mg/ml]
300
Why is the maximum in J(T) sharp?
50
Liquidus
Gelation line
o
Temperature [ C]
40
 J(T) reaches sharp max
at solution-crystal spinodal
30
L-L coexistence
20
10
0
Liquid-liquid (L-L) spinodal
-10
0
100
200
Concentration [mg/ml]
300
Phenomenological Theory of Two-step Nucleation
-1
-3 -1
Nucleation Rate J [cm s ]
012
0.5
u1 (T )
1
1



u0 (T ) u0 (T )u 2 (T ) u 2 (T )
– mean first-passage time
J = -1 , 2 – rate-limiting
E
U 2 exp( 2 )
k BT
J
U
DG
1  1 exp(
)
U0
k BT
U 2  k2
C1 T
 (C1 , T )
0.4
0.3
0.2
0.1
0.0
Viscosity inside dense liquid
 0 1 C1 expk C1 exp E / k BT 
E 2 (T ) 
E*
Te  T  2

T  T  2
1  e
 Te  Tsp 2







 Single adjustable k2 reproduces
3 complex kinetic curves
W. Pan, et al., J. Chem. Phys. 122, 174905 (2005)
276
280
284 288 292
Temperature T [K]
296
300
T = 12.6 oC
-3 -1
Nucleation Rate J [cm s ]
Nucleation barrier on approach to spinodal
80 mg ml
-1
50 mg ml
0.2
0.1
0.0
20
30
40
50
-1
Concentration C [mg ml ]
60
The Pre-exponential Factor in the Nucleation Rate Law
DG *
J  J0 exp(
)
kBT
From experiments:
J0 ~ 1010 cm-3s-1
From classical theory
J0 ~ 1020 cm-3s-1
???
R.P. Sear,
J. Phys. Chem. B 110 (2006) 21944
E2
)
k BT
J
U
DG
1  1 exp(
)
U0
k BT
U1
DG 1
exp(
)  ,   cluster volume fraction
U0
kBT 
U 2 exp(
Low J0—due to nucleation within clusters
Volume Fraction
From phenomenological theory:
10
-7
0
50
100
150
200
Time of Monitoring [min]
250
The Two-step Mechanism for Other Crystals
 Glycine, urea
B. Garetz, et al., Phys. Rev. Lett. 89, 175501 (2002)
J.E. Aber, et al., Phys. Rev. Lett. 94, 145503 (2005)
D.W. Oxtoby, Nature 420, 277 (2002)
 Charged colloid crystals
M. E. Leunissen, et al., Nature 437, 235 (2005)
 NaClO3
R.Y. Qian, G.D. Botsaris,
Chem. Eng. Sci. 59, 2841 (2004)
 NaCl nucleation from solution (MD simulation)
D. Zahn,
Phys. Rev. Lett. 92, 040801 (2004)
 Calcite nucleation
L. Gower, Chem. Rev. 108, 4551 (2008)
D. Gebauer, et al., Science 322, 1819 (2008)
Theoretical Justification of Generality of 2step Mechanism
J. Lutsko, G. Nicolis, Phys. Rev. Lett., 96 (2006) 046102
 For protein molecules
b
c
DG/kBT
a
c
b
a
 For small molecules
e
f
DG/kBT
d
Reaction Coordinate
Two-step barrier always lower than direct barrier
f
e
d
q(T) much stronger than R(T)
contradicts 1-step nucleation and agrees with 2-step
0.6
1/R [s mm-1]
Clusters and HbS Polymer Nucleation
0.4
0.2
0.0
D. Kashchiev, et al.,
J. Chem. Phys. 122 (2005) 244706
0
Polymers are perpendicular to plane of polarization of
polarized light
t=12 s n=1
t=15 s n=2
15
20
q [s]
90
25
60
30
150
180
0
330
210
240
300
800
270
Radius [nm]
600
O. Galkin, P.G. Vekilov, J. Mol. Biol. 336, 43 (2004)
O. Galkin, et al., J. Mol. Biol. 365, 425 (2007)
O. Galkin, et al., Biophys. J. 92, 902 (2007)
P.G. Vekilov, Brit. J. Haematol. 139, 173 (2007)
t=0 n=0
10
120
Dependencies of r, Vl and Nl of mesoscopic metastable
clusters
on C and T
follow those of nucleated polymers
 Clusters are precursors for polymer nuclei
5
400
200
0
t=18 s n=3
o
o
10 C
o
25 C
15 C
o
30 C
20
o
20 C
40
60
Time of Monitoring [min]
t=21 s n=5
Aggregation Precedes Ordering in Biological Self-assembly
 Hemoglobin assembly—from 2 a-chains, 2 b-chains and 4 heme-moieties
after translocation
a- and b-chains associate
prior to folding
K. Adachi, et al.,
J Biol Chem 277, 13415 (2002)
Hemes attach to a2b2 complex
and then enter assigned slots
G. Vasudevan, M. J. McDonald,
Curr Protein Pept Sci 3, 461 (2002)
 Nucleation of prion-protein fibers—via a disordered toxic fluid-like cluster
Molten Oligomer
Nucleus
Fibril
R. Krishnan, S. L. Lindquist, Nature 435, 765 (2005)
A. Lomakin, et al., Proc. Natl. Acad. Sci. USA 93, 1125 (1996)
E. H.Koo, et al., Proc. Natl. Acad. Sci. USA 96, 9989 (1999)
Density
The Two-step Mechanism
…
 What are the consequences
for the nucleation kinetics?
Structure
 What are the precursors above
the L-L coexistence line?
 Does it offer new “handles” for
control?
Temperature
Solubility
Gelation
Binodal
Spinodal
Protein Concentration
100 µm
Crystals Do Not Nucleate Within Liquid Droplets
t=4h
t=6h
t=8h
t = 10 h
 What else may
be precursor?
t = 12 h
t = 14 h
t = 16 h
Direct Observation of Clusters
lumazine synthase 3 mg/ml 1.3 M phosphate 24 C
 Clusters shrink in height:
Height [nm]
 Steps generated from
clusters merge
continuously with
underlying lattice
 Clusters represent
hidden dense liquid
200
200
100
100
0
0
0
2.5
5
7.5 10
Surface Coordinate [mm]
0
2.5
5
7.5 10
Surface Coordinate [mm]
O. Gliko, et al., J. Amer. Chem. Soc. 127 (2005) 3433
W. Pan, et al., Biophys. J. 92 (2007) 267
O. Gliko, et al., J. Phys. Chem. B 111 (2007) 3106
Evidence for Mesoscopic Clusters in Protein Solutions
Dynamic light scattering determinations
oxy HbS
C=169.6 mg/ ml
deoxy HbS
C=131.2 mg/ml
0.6
100
Lysozyme78 mg ml-1, 20 mM HEPES, pH = 7.8
10
0
Clusters
0.3
HbS
molecules
0.0
1E-4
0.01
1
100
Delay Time  [ms]
 Cluster size n100 nm
 Cluster lifetime >15 ms to 10 s
100
150
200
250
400
0
50
100
150
deoxy-HbS
1000
 Steady volume 10-8—10-3 V:
not crystals
 Fast decay rate indicates that
clusters are hidden liquid
50
LuSy 8.1 mg/mL 1.3 M phosphate pH = 8.7
800
Radius [nm]
g2() – 1
0.9
67.2 mg/ml
131.2
0.15 M phosphate pH = 7.2
100
0
50
100
150
Time of Monitoring [min]
O. Gliko, et al., J. Amer. Chem. Soc. 127 (2005) 3433
The Angular Dependence
Decay rate G2 [ms]
2.5
q = 4pn/l sin(q/2)
2.0
q– scattering angle
Cluster peak
1.5
For freely diffusing clusters
G2 = Dq2
1.0
0.5
0.0
0
200
400
q2
600
Loose network is
anisotropic environment
G2 ≠ Dq2
[mm-2]
Freely-diffusing clusters G2 = Dq2
Their size – from Einstein-Stokes law
D = kT / 6pR2
The Free Energy Cost of Higher Concentration
15
K /Rq  ( /) /RT
0
H

DG ~ 10 kBT
-2
10
-4
-6
5
 Fraction of protein in clusters:
exp(-10) = 4 × 10-5 an overestimate
 Clusters must contain only a few
molecules
 Cluster lifetime must be
O(diff ) = 10 ms

DG/NkBT
L
MwK/Rq
DG     dV  D ( V )
-8
0
0
100
200
300
400
-10
-1
Protein Mass Concentration [mg ml ]
Clusters must contain a chemical species with slow decay rate
Open Question about the Clusters
(a)
-1
10
(b)
-1
2
-3
10
-5
10
80 mg ml
o
57
o
77
o
96
o
116
o
140
150 mg ml
-7
10
(c)
-1
2
-3
10
(d)
334 mg
-1
228 mg ml
-5
-7
-3
10
-1
10
1
10
 sin2(q/2) [ms]
3
10
-5
10
-3
10
Cluster volume fraction
less than suggested by
equilibrium free energy
Clusters are fluid
internally
Complicated
dependence of cluster
volume fraction on
protein concentration
ml-1
10
10 -5
10
Macroscopic lifetimes
Cluster size independent
of protein concentration
-1
10
Mesoscopic size
-1
10
1
10
 sin2(q/2) [ms]
3
10
Cluster shoulder--nonexponential at high
concentrations
Microscopic Scenario and the Cluster Radius
Clusters consist of offequilibrium mixture of
monomers and oligomers,
kinetically stabilized
Cluster radius R is determined
by the decay rate and
diffusivity of the oligomers:
R  (D2/k2)1/2
Since R ≃ 100 nm, oligomer
lifetime
k2-1  10 ms

Cluster radius does not depend on concentration
The Cluster Lifetime
-1
2
10
-3
10
-1
228 mg ml
-5
10
-7
10 -5
10
-3
-1
10
10
1
10
3
10
-1
2
10
R2
2
D1
-3
10
Cluster size fluctuates with
characteristic time
 The imprint of these fluctuations on the
correlation functions scales with q2
334 mg ml-1
-5
10
  ≈ 10 ms
is the lower bond of cluster lifetime
-7
10 -5
10
-3
10
-1
1
10
10
2
*sin (q2) [ms]
3
10
The Oligomer Mechanism: Hydration
80
40
-3
10
2()
Free Energy [kJ/mol]
-1
10
0
-5
-40
10
2
4
6
8
10
12
14
Separation [Å]
-7
10
-1
150 mg ml
20 mM Hepes, pH 7.8
0.1M acetate, pH 4.5
20 mM phosphate, pH 7.8
-5
10
-3
10
-1
1
10
10
2
*sin (q2)[ms]
Water structuring leads to secondary minima in interaction
between nanoscopic solutes
Small ions, i.g., HPO42- , are known to disturb the hydration
shell
3
10
The Oligomer Mechanism: Domain Swapping
-1
80 mg ml
No urea
0.2 M urea
0.5 M
1M
-1
10
-3
 2( )
10
-1
150 mg ml
No urea
0.2 M urea
0.5 M
1M
-5
10
-7
-3
10
-1
1
10
10
2
*sin (q/2) [ms]
3
10
Attraction between solvent
exposed hydrophobic residues,
“domain swapping”
Urea can be used to control
degree of unfolding
-5
-3
10
10
-1
10
1
10
3
10
*sin2(q/2) [ms]
0.5 M urea
0.5 M urea
no urea
10
0
-2
-4
-6
-8
5
DG/NkBT
-5
10
MwK/Rq
10
-10
0
100 200 300 400 500
-1
Lysozyme Concentration [mg ml ]
Summary and Conclusions
Assembly of ordered arrays
crystals, oligomers, fibers, etc.
is preceded by association into disordered clusters
The precursor is a metastable mesoscopic liquid cluster
Rate of crystal nucleation is determined by structuring of dense quasidroplet
Polymorph selection is determined by kinetics factors rather than by high
barriers
The low volume fraction of the nucleation precursors delays nucleation by
~ 1010
Understanding and control of nucleation in solution requires insights into
the solution physicochemical mechanism nano- and mesostuctures
So What?
Clusters are needed for nucleation of crystals.
To enhance clusters:
moderate intermolecular attraction or repulsion
proper water structure around the protein molecules
Crystal nucleation occurs in a spinodal regime
g is not important
Simpler picture of nucleation and role of additives
Heterogeneous particles may affect polymorph selection via structural
similarity