Monte Carlo Simulation of Interfaces
David L. Cheung
Department of Chemistry & Centre for Scientific Computing
University of Warwick
MIR@W day Monte Carlo: 07/03/2011
What is...?
Soft matter
liquids, liquid crystals, polymers,
colloids, surfactants
Interfaces
liquid-liquid, liquid-gas
solid-liquid
membranes
soft (energies≈ kB T )
fluidity
order
self-organization
DL Cheung (University of Warwick)
MC simulation of Interfaces
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Outline
1
Introduction
2
Nanoparticles at soft interfaces
3
Adsorption on polymer vesicles
4
Conclusions
DL Cheung (University of Warwick)
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Nanoparticles at soft interfaces
Nanoparticles
◮
quantum dots, fullerenes, macromolecules, proteins...
Soft interfaces
◮
liquid-liquid (oil-water), polymer blends, membranes....
Nanoparticle monolayer
Nanoparticle stabilised droplet
Adhesion of nanoparticles onto soft interfaces
◮
for ≈nm particles detachment energy 10 kB T
A Böker et al Soft Matter 2007; S Bon et al Langmuir 2007
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Nanoparticle-interface interaction
Nanoparticle self-assembly
◮
◮
dynamics/kinetics
capillary forces between nanoparticles: structure formation
Adhesion strength not measured experimentally
◮
often estimated from continuum models
Many contributions to interaction
◮
difficult to isolate different effects experimentally
RP Sear et al PRE 1999
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Model
Solvent
◮
non-additive hard sphere (Widom-Rowlinson) mixture
2 components A and B
Vαβ (r ) = (1 − δαβ )Vhs (r , σ)
◮
(diameter σ ≈ 0.3 nm)
demixes above critical density
Nanoparticle
◮
hard-sphere (diameter 3-6 σ)
Grand-canonical Monte Carlo simulations
◮
Free-energy profile (effective NP-interface interaction)
βF (z) = − log P(z)
◮
Wang-Landau sampling
B Widom & JS Rowlinson JCP 1970; FG Wang & DP Landau PRL 2001
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Why do we need Wang-Landau?
P(z)
Detachment energy≈ 10 kB T
To accurately determine free
energy profile need to sample
uniformly across interface
z
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Interface structure
Interface broadened by capillary waves
◮
◮
interface width comparable to nanoparticle size (1-2 nm)
bridging between nanoparticle and interface
DLC & SAF Bon PRL 2009
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Nanoparticle-interface interaction
Free energy profile
7
6
βF(zc )
5
4
3
2
1
0
0
◮
◮
2
4
zc / σ
6
8
10
long-range interaction (> Rc )
detachment energy: 1-10 kB T
DLC & SAF Bon PRL 2009
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Nanoparticle-interface interaction
Free energy profile
7
6
βF(zc )
5
Rc = 1.5σ
Rc = 2.0σ
Rc = 2.5σ
Rc = 3.0σ
4
3
2
1
0
0
◮
◮
◮
2
4
zc / σ
6
8
10
long-range interaction (> Rc )
detachment energy: 1-10 kB T
detachment energy: ∝ Rc2
DLC & SAF Bon PRL 2009
DL Cheung (University of Warwick)
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Can we use continuum theory?
Free energy profile: F (z) = πγz 2 (z ≤ Rc )
7
6
βF(zc )
5
Rc = 1.5σ
Rc = 2.0σ
Rc = 2.5σ
Rc = 3.0σ
4
3
2
1
0
0
2
4
zc / σ
6
8
10
Continuum theory in poor agreement with simulation
◮
underestimates interaction range → flat interface approximation
DLC & SAF Bon PRL 2009
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More complex particles: Janus spheres
Hemispheres of different (A-philic/B-philic) functionality
8
βF(zc )
6
10
4
∆=0
∆ = 0.05
∆ = 0.10
∆ = 0.15
∆ = 0.25
2
0
0
Detachment energy / kBT
12
2
4
zc / σ
6
8
10
8
6
4
2
0
0.05
0.1
2
0.15
0.2
βσ ∆γ
Stability increases with ∆γ
DLC & SAF Bon Soft Matter 2009
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More complex particles: Janus spheres
Hemispheres of different (A-philic/B-philic) functionality
8
βF(zc )
6
10
4
∆=0
∆ = 0.05
∆ = 0.10
∆ = 0.15
∆ = 0.25
2
0
0
Detachment energy / kBT
12
2
4
zc / σ
6
8
10
8
6
4
2
0
0.05
0.1
2
0.15
0.2
βσ ∆γ
Stability increases with ∆γ
Continuum theory: overestimates detachment energy
DLC & SAF Bon Soft Matter 2009; BP Binks et al Langmuir 2000
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Nanoparticles rotate
Theory assumes fixed particle orientation
nm-sized particles have significant orientation freedom
Angle between particle orientation and z
0.8
<cosφ>
0.6
∆γ
0.4
0.2
0
-10
-5
0
z/σ
5
10
DLC & SAF Bon Soft Matter 2009
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Fixing the orientation
Stability increased ≈50% for fixed orientation
Detachment energy / kBT
12
βF(zc )
9
10
6
3
0
0
2
◮
4
zc / σ
6
8
10
8
6
4
2
0
0.05
0.1
2
0.15
0.2
βσ ∆γ
fix orientation experimentally (E or B field?)
DLC & SAF Bon Soft Matter 2009
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Nanoparticle-interface interactions
Have used molecular simulation to study interaction of nanoparticle with
liquid interfaces
◮
◮
◮
interaction is longer ranged than expected - capillary waves
detachment energy is 1 − 10 kB T
detachment energy ∝ Rc2
Comparison with continuum theory (Pieranski approximation)
◮
◮
underestimates detachment energy
underestimates range of interaction - neglect of capillary waves
Janus particles
◮
more stable on interfaces, although orientational freedom important
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Outline
1
Introduction
2
Nanoparticles at soft interfaces
Nanoparticle-interface interactions
Janus particles
3
Adsorption on polymer vesicles
4
Conclusions
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Polymer vesicles
Polymer bilayers forming a fluid-filled
sac
polymersomes - analogous to
liposomes
synthetic minimal cells
Use of polymers enhances the
mechanical stability compared to
liposomes
reaction vessels
drug delivery vehicles
V Ortiz et al, JPCB, 2005
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Armoured vesicles
The stability of polymer vesicles may
be further enhanced by a coating of
nanoparticles
This occurs in many naturally occuring
systems
bacterial S-layers
nanopatterned calcium carbonate
coating on coccolithophorids
Can this be mimicked in synthetic,
polymeric systems?
Y. Shiraiwa, University of Tsukuba
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Experimental system (Bon group, UoW)
Vesicle: poly(n-butyl methacrylate)-b-(N,N-dimethylaminoethyl methacrylate)
block co-polymer (1 µm radius)
Nanoparticles: 120 nm and 200 nm diameter anionic polystyene latex
particles
Distinct packing patterns of nanoparticles on vesicle surface
can we reproduce/predict these packings from simulations?
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Simulations
Monte Carlo simulations
nanoparticles interact through modified
LJ+Yukawa potential
24
r 12 req
exp(−r /ξ)
eq
V (r ) = ǫ
+A
−2
r
r
r /ξ
A is related to the charge density on the
particle surface
model vesicle as large sphere
◮
◮
gradually add particles onto vesicle - NPs
irreversibly adsorbed
NPs move on the surface (modify normal
move acceptance to ensure uniform
sampling on spherical surface)
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Comparison
Packing patterns on polymer vesicles (after 14 hours annealing) and from
simulations (insets)
56%, 78%, and 91% encounter probabilities (relative likelihood of smaller
particle attaching to vesicle)
Good agreement between experimental and simulation packing patterns
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Controlling the pattern
Study the effect of changing the the surface charge density
56 % small particles
Increasing charge density→
79 % small particles
91 % small particles
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Adsorption on polymer vesicles
Experimentally it has been shown that it is possible to coat polymer
vesicles in ’nanoparticle-armour’
Using simple MC simulations can reproduce experimentally observed
packing patterns on polymer vesicles
Find changes in packing patterns with increasing surface charge density
Further work is to quantify the simulation and experimental packing
patterns
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Conclusions
Molecular simulation provides a powerful tool for the investigation of
interfacial systems
Gives molecular insight that is complementary to experimental studies
◮
◮
◮
test/refine theories
study molecular details difficult/impossible to resolve experimentally
gives precise control over the system parameters
References:
Nanoparticle-interface interactions
DLC & S. Bon, Phys. Rev. Lett., 102, 066103 (2009)
DLC & S. Bon, Soft Matter, 5, 3969 (2009)
Armoured vesicles
R Chen et al, J. Am. Chem. Soc., 133, 2151 (2011)
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Acknowledgements
Collaborators
Warwick
Ale Troisi, Sara Fortuna, Dan Jones, Natalie Martsinovich, David
McMahon, Jack Sleigh
Stefan Bon, Rong Chen, Daniel Pearce
Rebecca Notman
Mike Allen, Paul O’Brien
Paola Carbone, Andrew Masters (Manchester), Mark Wilson (Durham),
Phil Cox (B’ham)
Martin Oettel, Friederike Schmid (Mainz), Matthias Schmidt
(Bayreuth/Bristol)
Computers
Centre for Scientific Computing, UoW
Funding
EPSRC, ERC, Leverhulme Trust, University of Warwick
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Model & method
Study system using grand-canonical MC simulations
◮
◮
simulations at different (solvent) chemical potentials βµ = 0.15...0.35
(βµc ≈ 0.04)
confine system in z-direction: localises the interface near cell centre
Calculate free energy profile
◮
βF (zc ) = − log P(zc )
◮
find probability distribution using Wang-Landau sampling
divide separation into overlapping windows
◮
Calculate interfacial tension
FG Wang & DP Landau, PRL, 2001
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System-size dependence
1.8
1.5
1.5
1.75
1.45
1.7
1.4
1.65
1.35
Lx
1
0.5
0
0
2
4
6
zc / σ
1.6
10
12
14
16
Lx / σ
18
∆F/∆FPier
2
β∆F
βF (zc )
System-size dependence (lateral box length Lx )
1.3
20
Consistent with decrease in γ
◮
◮
interaction range increases
barrier decreases
DLC & SAF Bon PRL 2009
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Varying wettabilities
Stability controlled by changing surface
tensions
◮
◮
Experimental systems:
different γ for two components
polystyrene-water:
γ=32 mN m−1
polystyrene-hexadecane: γ=14.6 mN
m−1
Simulate non-symmetric case
◮
◮
◮
use different radii for the two
components
RA/B = Rc ∓ ∆
Rc = 2.5σ and ∆ ≤ 0.1Rc
βµ = 0.15
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Diversion: calculating γiP
To make contact with continuum theories need to calculate γAP and γBP
Bresme & Quirke
◮
free energy change from change in nanoparticle radius
∆F = (4πRc2 ∆R)P + (8πRc ∆R)γ
0.05
2
βσ γA/B
0
-0.05
-0.1
-0.15
0
0.05
0.1
0.15
∆/σ
0.2
0.25
∆γ < 8 mN m−1
F Bresme & N Quirke, PRL 1998
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Towards the continuum limit
As particle size increases continuum theory becomes more accurate
1.5
Rc
1
∆F
F
1.4
1.35
0.5
1.3
0
0
1.5
2
1
2
2.5
Rc / σ
3
3
4
z/Rc
DLC & SAF Bon PRL 2009
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Not just hard spheres
Binary Lennard-Jones fluid
40
35
ρ=0.50
ρ=0.60
ρ=0.69
Free energy / kBT
Free energy / kBT
50
30
20
10
0
0
30
25
ε=0.00
ε=0.25
ε=0.50
ε=0.75
ε=1.00
20
15
10
5
2
4
zc / σ
6
8
10
0
0
2
4
zc / σ
6
8
10
Qualitative results same as for WR mixture
Attractive NP-fluid interactions decrease NP stability
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