A computational approach to
understanding crystal nucleation
and growth
John Harding and Colin Freeman
Department of Engineering Materials
University of Sheffield
M k Rodger
Mark
R d
and
dD
David
id Q
Quigley
i l
Department of Chemistry
University of Warwick
1
Biomaterials at the micro and nanoscale
F 3O4 nanoparticles
Fe
ti l
nanocrystalline
y
hydroxyapatite paste
(a) Fe3O4 crystals in magnetotactic bacteria (courtesy Patrick James, Univ. of Southampton, UK), (b) Ostim®
nanocrystalline hydroxyapatite paste (courtesy, Franz-Xavier Huber), Huber, F. X.; Belyaev, O.; Hillmeier, J.; Kock, H. J.;
H b C
Huber,
C.; M
Meeder,
d P
P. JJ.; Berger,
B
I B.
I.
B Musculoskeletal
M
l k l t l Di
Disorders
d
2006 7,
2006,
7 14
SiO2 diatom frustrule
red abalone plates
(c) amorphous SiO2 diatom frustule (courtesy Kenneth Bart, Hamilton College,
Clinton NY), and (d) Nacreous plate structure in red abalone; Tian, Z. R. R.; Voigt,
J. A.; Liu, J.; McKenzie, B.; McDermott, M. J.; Rodriguez, M. A.; Konishi, H.; Xu, H.
F. Nat. Mater. 2003,, 2,, 821.
2
Hierarchy in biomaterial systems
(A) Consolidated silica nanoparticles deposited around a preformed organic axial filament (shown on the right). (B) Lamellar structure
of spicule made of alternating organic and silica layers. Inset depicts the organically glued interlayer region. (C) Bundling of spicules.
(D) (Right) Vertical and horizontal ordering of bundled spicules forming a square-lattice
square lattice cylindrical cage with every second cell
reinforced by diagonal elements. (Left) The node structure. (E) Cementation of nodes and spicules in the skeletal lattice with layered
silica matrix. (Inset) Fiber-reinforced composite of an individual beam in the strut. (F) Surface ridges protect against ovalisation of the
skeleton tube (G) Flexural anchoring of the rigid cage into the soft sediments of the sea floor.. (J.C. Weaver, J. Aizenberg, G.E.
3
Fantner et al J. Struc. Bio. 158, (2007) 93.)
The seven-fold
hierarchy of bone
Level 1: Isolated crystals from human
bone (left side) and part of an
unmineralized and unstained collagen
fibril from turkey tendon
Level 2: TEM micrograph of a
mineralized collagen fibril .
Level 3: TEM micrograph of a thin
section of mineralized turkey tendon.
Level 4: Four fibril array patterns of
organization found in bone
bone.
Level 5: SEM micrograph of a single
osteon from human bone.
Level 6: Light micrograph of a
f t d section
fractured
ti through
th
h a ffossilized
ili d
(about 5500 years old) human femur.
Level 7: Whole bovine bone
(scale: 10 cm).
Wiener & Wagner Annu. Rev. Mater.
Sci. 1998.28:271-298.
4
A hierarchy of methods
Atomistic Simulations
Electronic
Structure
Many
processes
occur
Based on Born Model
of
Calculations
To
Model
the
on
a time scale too long
Solids.
Highly
accurate
tobiological
be studied
via
METADISE (Watson et al)
atomistic
calculations
Energy
Minimisation
interface
with
Results
from
relatively
DL_POLY
(Smith)
minerals
we
small
are
used
Dynamics
RMolecular
Results
ltsystems
are
th
therefore
f will
to
fit and
verify
potential
need
to
make
used
to develop
mesoPolarisability
described
models.
models
(based
on DPD
use
of
simulation
using
g Shell
Model
for example in this case)
techniques
Such
models
canused
also to:
Atomistic
Calcs
Systems
typically
contain
based
onatoms
aatomistic
suggest
further
Scan
configuration
10 – 100000
and of
calculations
space
the
order of
in size
range
ofnm
length
(identify
important
scales
andGood
…but
Requires
Results
from mesoscale
systems)
accuracy
Quality
Potential
calculations
couldetc
also
Study Interfaces
Parameters
be
fed into
continuum
(Where
system
to large)
composite models
5
Coccoliths – crystal control
J.R. Young, S.A. Davis, P.R. Bown, and S. Mann
6
Journal of Structural Biology 126, 195–215 (1999)
Structure of an eggshell.
7
What are the issues?
• Nucleation takes place in a container (vesicle) that
permits control of the solution chemistry. Organic
molecules are present both in solution and in an
insoluble matrix – both can control the growth
• The
Th systems
t
grow in
i the
th presence off water.
t A proper
treatment of the water structure is essential
• Classical
Cl
i l ((unit
it b
by unit)
it) nucleation
l ti models
d l are often
ft
inappropriate. In many cases, the system first
nucleates nanoparticles and then assembles them
them.
• The hierarchy both invites the use of a set of models
linked together and necessitates this approach.
8
Non-classical crystallisation
9
Crystal nucleation and growth
H. Cölfen and S. Mann, Angewandte Chemie International Edition,
2003, 42, 2350-65 (adapted).
10
Mesocrystals
From mesocrystals to single
crystals by changing the solution
Calcium carbonate crystals
precipitated from a solution containing
Ca and PEO22-PNaStS49 at a fixed
[Ca]:[S] molar ratio of 1.25:1 and Ca
concentrations of (a) 10, (b) 5, (c) 2.5,
(d) 1, (e) 0.5, and (f) 0.1 mM. These
crystals are representative of the
entire sample population.
Alex N. Kulak, Peter Iddon, Yuting Li,
Steven P. Armes, Helmut Co¨lfen,
Oskar Paris, Rory M. Wilson, and
Fiona C. Meldrum, J. AM. CHEM.
SOC 2007,
SOC.
2007 129,
129 3729-3736
11
The problem of force-fields
Slab of Mineral
All atoms
freeMolecule
to relax
Organic
/ Bio
H b
Has
been
extensively
ti di l
M
Many
wellll d
derived,
studied force
and for
many
generic
fields
exist
minerals
potential
eg
g CVFF, reliable
CHARMM,
parameters
exist.
AMBER,
COMPASS
For CaCO3etc
these include
DREIDING
models byupdated
Pavese have
et al
Regularly
updated,
and several
and
been
appliedby
to Gale
a wide
collaborators.
variety
of situations.
B th have
Both
hmoreb
been
Th
These
often
ft include
i l d
successfully
applied to all
aqueous
conditions.
polymorphs, flat and
stepped surfaces.
12
Water D
Density Relative to Bulk
The Effect Of Water
3
2.5
2
1.5
1
0.5
0
0
2
4
6
Distance from Surface
8
Clear Layering of Water upto
10A from surface
I broad
In
b d agreementt with
ith
earlier experimental data
13
Kerisit and Parker JACS 126:10152 (2004)
Simulating nucleation
• There are a few direct simulations of nucleation
– usually need high subcooling or supersaturation
• Can bias the simulation in various ways
– most methods involve adding a bias potential to the
simulation but vary in how this is done
simulation,
done.
– all methods face the same issue – how to choose
the direction to bias it in
– metadynamics (Laio and Parrinello) is one of the
most successful of recent methods.
14
Metadynamics Illustrated
Small Gaussian bias potentials are
added to current location in order
parameter space at intervals Taug.
Pushed over free energy barriers
into unexplored regions.
Provided motion of order parameters
is adiabatically slow, the free energy
is recovered as the negative of the
total bias.
15
Metadynamics Approach
[Laio & Parrinello P.N.A.S. 99 12562 (2002)]
• Describe system in terms of order parameters
λ = ((λ1,λ2….λM) which describe nucleation/growth
g
p
pathway.
y
The derivative of the Gibbs free-energy w.r.t. λ is evaluated
via MD or MC.
• “Thermodynamic
“Th
d
i fforce”” acting
ti on th
the order
d parameters
t
iis :
• Augment Ftherm
with Vaug(λ1,λ
λ2….λ
λM) – history dependent.
dependent
th
– Pushes the system away from previously visited λ.
– Grow a bias potential to overcome energy barriers.
– Pushed
P h d iinto
t rare configurations
fi
ti
tto overcome entropic
t i
barriers.
16
Order parameters
Q6 Steinhardt order
p
parameter
[ Steinhardt et al. Phys.Rev.B.
28 784 ((1983)) ]
k runs over all nearest neighbour separation vectors.
Contributions reinforce for an ordered system, and
cancel for a disordered system.
17
Phase assignment
Ca2+
CO32-
• Based on computation of local per-ion order parameters
and comparison to bulk reference values.
18
Changing the size of the cluster
75
calcite
192
75 units
minimum “B”
B
G/kBT
vaterite
DQ
Quigley
g y and PM Rodger,
g
J. Chem. Phys. 128 (2008)
221101
19
The effect of adding a protein
G/kBT
G/kBT
Quigley,
Q
i l
F
Freeman,
Rodger, Harding in
preparation
G/kBT
G/kBT
•
•
•
•
Freeze protein+nanoparticle, equilibrate water for 300ps.
Heat in 50k increments of 300ps up to 310k.
Perform ~40
~40,000
000 metadynamics steps as in pure water
water.
Depending on orientation can stabilise any of the three 20
accessible phases (about 22000 water molecules not shown).
Adsorption: polysaccharides
M. Yang, S.L. Stipp, J.H. Harding. Crys. Eng.
Des. (2008) in press
surface
galactose
mannose
(00.1,CO3
2-)
-93.7
-40.7
-60.4
-40.8 -58.9
-91.5
-81.6
23.2
-51.6 -50.4
-41 2
-41.2
-43 7
-43.7
-73 4
-73.4
-36 4 -48.7
-36.4
-48 7
-50.8
-19.0
-81.8
-31.7 -45.8
-22.9
-29.6
-60.6
-55.5 -42.1
-11.1
28.9
0.6
-45.2
-6.7
2.0
15.7
1.0
-24.0
-1.3
-10.6
10 6
21 9
21.9
12 5
12.5
96
9.6
84
8.4
xylose
rhamnose
average
polar surface
(1-2.0)
acute stepped
surface
(31 8)
(31.8)
acute stepped
surface
(10.0)
planar surface
(10.4)
planar surface
((21.4))
obtuse surface
(31.16)
obtuse surface
(00.1,C
(00
1C
a2+)
polar surface
21
Absorption – peptides
• Large number of cases to consider
• Need free energies not energies –
experimental evidence is clear that
many cases are entropically driven
although some are enthalpically
driven.
• Experimental evidence that in
some cases the presence of organic
molecules speeds up the growth, in
others it slows it down
down.
M. Yang, unpublished work
22
Surface – Protein
Build 8-layer calcite slabs
~10000 sqÅ area.
Highest and lowest energy configurations
selected for each position and the system is
solvated with ~20,000 water molecules.
Protein placed in 6 different positions on the
surface and in 64 different orientations to
explore various configurations.
Run for 200 ps in vacuum.
C.L. Freeman, unpublished work
R ffor 2 ns.
Run
Highest and lowest energy positions selected and run
until the system shows stability.
23
Residues and the Surface
(2)
(1)
3x
ARG
5 ARG
5x
3 SER
3x
3 ALA/GLY
3x
ARG – N-H hydrogen bonds to carbonate oxygen and N-Ca.
SER – O-H
O H hhydrogen
d
bbonds
d tto carbonate
b t oxygen and
d O-Ca.
OC
ALA/GLY – N-Ca/C=O-Ca of residue backbone.
24
25
What needs to be done
• Investigate the effect of internal water on
the transformation of nanoparticles
• Investigate effects of several nanoparticles
and proteins
proteins, polysaccharides,
polysaccharides peptides
• Devise mesoscale models to use the
information from atomistic simulations and
build models for mesocrystal formation.
• Do this all for calcium phosphates (including
apatites) where there is as least as much
data as for carbonates.
26