FACETING OF MULTICOMPONENT
CHARGED ELASTIC SHELLS
Rastko Sknepnek, Cheuk-Yui Leung, Liam C. Palmer
Graziano Vernizzi, Samuel I. Stupp, Michael J. Bedzyk,
Monica Olvera de la Cruz
APS March Meeting
Dallas, March 22, 2011
Motivation
Experiments find faceted structures in the
100nm size range.
-1
PCDA-OH
PCDA-KKK
+3
Greenfield, M., et al., JACS (2009)
Minimization of electrostatic energy on fixed
geometry reveals that in certain cases
faceted structures are energetically
favorable.
Vernizzi & Olvera de la Cruz, PNAS (2007)
Can electrostatic interactions lead to faceting?
APS March Meeting
Dallas, March 22, 2011
Coarse-graining
+1,+2,+3
wc
e
-1
tail-tail interaction potential
Cooke, et al., PRE, 2005
unstable
liquid
gel
Electrostatic effects treated within
linearized Debye-Hueckel theory:
V D H  rij 

exp    rij 
 qi q j
 V D H  rc 

e rij

0

for
r  rc
for
r  rc
Cooke, et al., PRE, 2005
APS March Meeting
Dallas, March 22, 2011
Molecular dynamics of a bilayer patch with 4000 lipids.
Focus on a small region of
phase diagram T=0.6, 0.7
wc=1.15
T=0.6
1:1 (liquid)
1:2 (ordered)
1:3 (ordered)
T=0.7
1:1 (liquid)
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1:2 (liquid)
1:3 (“almost” ordered)
Dallas, March 22, 2011
Estimate of the bending rigidity 
Use results of linearized Helfrich theory:
2
L
2
hq

h  x, y  
k BT
q q
4
h(x,y) vertical position at (x,y)

qx ,q y
hq x , q y e

i qx x  q y y

2
 – lateral tension
Electrostatic interactions significantly
increase .
T=0.7, wc=1.15
APS March Meeting
Dallas, March 22, 2011
Estimate of the Young’s modulus Y
Regular two-dimensional ionic crystals:
square
triangular
1:1
triangular
triangular
1:2
1:3
Total energy:
E  N cell E cell 
1

2
i, j


qi q j

'

 4e  

  rij
rij






12


r
 ij




6




extract Y
E cell  c
(0)

1
2
c ijkl u ij u kl  O ( u ij )
3
Estimate: Y3:1/Y2:11.8
APS March Meeting
Dallas, March 22, 2011
In addition, different valence charges are
expected to segregate.
+3
+2
-1
MD simulation of a three component system
(1:2 and 1:3) in liquid phase (T=0.9)
Segregation leads to an onset of effective
line tension between differently charged
regions.
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In continuum representation:
Ft  

dl
C
Dallas, March 22, 2011
Regions with different charge ratios have different elastic properties.
All effects of charge are encoded in the elastic properties.
We find shaped using a discretized version of the continuum theory of elasticity.
(Seung and Nelson, PRA 1988)
stretching energy:
bending energy:
discrete
Eb

  i  1  n i , t  n i , t  1 .
Es
i ,t
line tension:
Ft
discrete


1


2
i, j
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ti ,t j
.
discrete

1
2

k ij  l  l 0 
2
i, j
We used simulated annealing
Metropolis Monte Carlo simulations
to find optimal shapes.
Dallas, March 22, 2011
Optimal faceted structures
hard/soft=10
Yhard/Ysoft=5
line
tension
=0.1
=0.3
=0.6
hard
component
fraction
20%
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40%
60%
80%
Dallas, March 22, 2011
Optimal faceted structures
hard/soft=30
Yhard/Ysoft=10
line
tension
=0.1
=0.3
=0.6
hard
component
fraction
20%
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40%
60%
80%
Dallas, March 22, 2011
Summary
•
•
•
•
We show that electrostatic interaction can lead to lipid crystallization
Charge significantly renormalizes elastic properties
Different regions segregate – effective line tension
Resulting shapes are faceted
Experimental collaborators:
Dr. Megan Greenfield
Cheuk Leung
Prof. Michael Bedzyk
Prof. Samuel Stupp
Funding provided by
the U.S. Department of Energy
APS March Meeting
Northwestern High
Performance
Computing System Quest
Dallas, March 22, 2011