APS March Meeting Dallas, March 22, 2011

FACETING OF MULTICOMPONENT CHARGED ELASTIC SHELLS

Rastko Sknepnek, Cheuk-Yui Leung, Liam C. PalmerGraziano Vernizzi, Samuel I. Stupp, Michael J. Bedzyk, Monica Olvera de la Cruz

APS March Meeting Dallas, March 22, 2011

Motivation

Can electrostatic interactions lead to faceting?

100 nm

Greenfield, M., et al., JACS (2009)

Vernizzi & Olvera de la Cruz, PNAS (2007)

PCDA-OH

PCDA-KKK

-1

+3

Experiments find faceted structures in the 100nm size range.

Minimization of electrostatic energy on fixed geometry reveals that in certain cases faceted structures are energetically favorable.

APS March Meeting Dallas, March 22, 2011

Coarse-graining

Cooke, et al., PRE, 2005tail-tail interaction potential

wc

Cooke, et al., PRE, 2005

+1,+2,+3

-1

gel

liquid

unstable

e

exp

0

ij

i j DH c cDH ij ij

c

rq q V r for r r

V r r

for r r

Electrostatic effects treated within linearized Debye-Hueckel theory:

APS March Meeting Dallas, March 22, 2011

Molecular dynamics of a bilayer patch with 4000 lipids.

T=0.6

Focus on a small region of phase diagram T=0.6, 0.7wc=1.15

1:1 (liquid) 1:2 (ordered) 1:3 (ordered)T=0.7

1:1 (liquid) 1:2 (liquid) 1:3 (“almost” ordered)

APS March Meeting Dallas, March 22, 2011

224 2B

q

k TL h

q q

Use results of linearized Helfrich theory:

,,

, x y

x yx y

i q x q y

q qq qh x y h e

h(x,y) vertical position at (x,y)

s – lateral tension

T=0.7, wc=1.15

Electrostatic interactions significantly increase k.

Estimate of the bending rigidity k

APS March Meeting Dallas, March 22, 2011

Estimate of the Young’s modulus Y

Regular two-dimensional ionic crystals:

square triangular triangular triangular

1:1 1:2 1:3

12 6

'

,

14

2i j

cell celli j ij ij ij

q qE N E

r r r

Total energy:

(0) 31( )

2cell ijkl ij kl ijE c c u u O u extract Y

Estimate: Y3:1/Y2:1»1.8

APS March Meeting Dallas, March 22, 2011

In addition, different valence charges are expected to segregate.

MD simulation of a three component system (1:2 and 1:3) in liquid phase (T=0.9)

+3

+2

-1

Segregation leads to an onset of effective line tension between differently charged regions.

In continuum representation:

t CF dl

APS March Meeting Dallas, March 22, 2011

We find shaped using a discretized version of the continuum theory of elasticity.

Regions with different charge ratios have different elastic properties.

(Seung and Nelson, PRA 1988)

, , 1,

1 .discreteb i i t i t

i t

E n n

bending energy:

All effects of charge are encoded in the elastic properties.

stretching energy:

20,

1

2discretes ij

i j

E k l l

line tension: ,,

1 .2 i j

discretet t t

i j

F We used simulated annealing

Metropolis Monte Carlo simulations to find optimal shapes.

APS March Meeting Dallas, March 22, 2011

Optimal faceted structures khard/ksoft=10 Yhard/Ysoft=5

g=0.1

g=0.3

g=0.6

linetension

hard component

fraction 20% 40% 60% 80%

APS March Meeting Dallas, March 22, 2011

Optimal faceted structures khard/ksoft=30 Yhard/Ysoft=10

g=0.1

g=0.3

g=0.6

linetension

hard component

fraction 20% 40% 60% 80%

100 nm

APS March Meeting Dallas, March 22, 2011

Summary

Funding provided bythe U.S. Department of Energy

Experimental collaborators:

Dr. Megan GreenfieldCheuk LeungProf. Michael BedzykProf. Samuel Stupp

Northwestern High PerformanceComputing System - Quest

• 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