Transcript Polymer Wrapping of Carbon Nanotubes

Spontaneous ordering of semiflexible
polymers on nanotubes and nanospheres
Simcha Srebnik
Chemical Engineering
Technion
Why study semiflexible polymers?
• Biopolymers
– double-stranded DNA
– unstructured RNA
– unstructured polypeptides (proteins).
• Semiflexible Polymers
– aromatics
– bulky side groups
Unlike the ideal chain, there is no consistent model that
describes their behavior
Polymer statistics
• The semiflexible chain
– N=104, lp = 1 (ideal), 6.5 (e.g., polyacrylamide),
500 (α-helix)
200
20
150
-20
50
0
50
R
-60
2 12
 6102 l
0
where
100
2
 nl 
lp 2


 n l  np

R0
n 
l
 p
2
p p
0
R
2 12
200
300
0
-500
 104 l
0
R02  nl 2
0 - 100
0
R
2
-20
-50
00
- 30
-100
-100
For flexible
chains,
40
20
500
60
500
00
- 50
-80
80
00
- 60
-50
0
00
- 20
 102 l
00
- 40
12
500
0
100
R2
0
-40
00
- 70
100
00
- 40
0
0
0
0
-2
0
The wormlike chain model
• Kratky-Porod chains
– the orientation correlation function for a worm-like
chain follows an exponential decay
i–1
i

si  si  x
 cosq x ~ exp xl l p
si si  x
qi
si
R
2
 RR  l
2

L
0
sdn   sdn
L
L lp 
 2l   1  e 
l

 p

2
p
Kratky and Porod, Recl. Trav. Chim. Pays-Bas 68 (1949) 1106
L
0

Scaling of semiflexible chains
• The KP model accurately
predicts end-to-end
distance for the entire
range of chain flexibility
– Drawback
• Cannot obtain end-to-end
distance distribution for
comparison with
experiments (S(k))
• Other exact theories exist,
but solution is numerical
and extension to other
related problems (e.g.,
external forces, geometrical
constraints) is difficult.
flexible
Coarse-grained simulation
• Use simplified models of ‘pearl
necklace’ polymer chains
– Ideal (ghost particles)
– excluded volume (hard sphere)
– Lennard-Jones (soft sphere)
N 2
U
   cos i  12
k BT
i 1
|
5
6
8
10
10
15
23
230
7000
13000
30000
4
3
U/kbT
Polymer
lp/l0
Poly(ethylene oxide)
2.5
Poly(propylene)
3
Poly(ethylene)
3.5
Poly(methyl methacrylate)
4
Poly(vinyl chloride)
4
Poly(styrene)
5
Poly(acrylamide)
6.5
Cellulose diacetate
26
Poly(para-benzamide)
200
DNA (in double helix)
300
Poly(benzyl-l-glutamate) (α-helix) 500

2
1
0
-3
-2
-1
0

lp ~  0.6
1
2
3
Modeling ‘ideal’ semiflexible chains
• Current computer resources limit our simluations
to chains with ~102 monomers.
– Develop model for analyzing conformational behavior
of very long chains.
– Limited to non-interacting systems.
e
s
i+1
ri 1  ri 
i
i–1
l
u s , i  au e , i
si  e i
l  ri 
l
si  e i
u s , i  au e , i
Polymer adsorption on curved manifolds
• Noncovalent functionalization of nanotubes using
polymer wrapping
–
–
–
–
Dispersion of CNTs in aqueous or organic media
Mechanical reinforcement
Fluorescent labeling
Sensors and biosensors (conjugated
polymers/biopolymers)
• Polymer in or on spheres
– DNA packaging in viruses, vesicles, or cells
– Protein encapsulation
– Colloidal and micellar suspensions
Carbon Nanotubes
• First reported by IIjima in 1991 (“microtubules”)
– Nature 354 (1991) 56-58.
– Over 5000 citations!
11
Examples of helical wrapping
HupR protein
on MWNTs
Balavoine and Shultz.
Angew. Chem., 1999, 1912
PmPV coating
B. McCarthy, J. N. Coleman. J.
Phys. Chem. B, 2002, 2210
DNA
Zheng et al., Nature
materials, 2 (2003)338.
12
Forces leading to helical wrapping
• Molecular modeling suggests that ssDNA can bind to
carbon nanotubes through -stacking, resulting in helical
wrapping. (Zheng et al., Nature Materials 2 (2003) 338).
• Alignment of backbone aromatic rings was also thought
to determine interactions between CNTs and polymers
(Zaiser and coworkers, J Phys Chem B 109 (2005) 10009; Coleman and coworkers, J Phys Chem B 106
(2002) 2210-2216).
– Note: all molecular modeling studies based their conclusions
regarding polymers on short oligomers
• Shinkai and coworkers used TEM and AFM to confirm
periodic helical structure of polysaccharides adsorbed on
CNTs. Argue that helical pattern is observed because of
their strong helix-forming nature. (JACS 127 (2005) 5875-5884)
• ‘General phenomenon’ argued by Baskaran et al. from
studies on various polymers. (Chem Mater 17(2005)3389)
Smalley’s postulate
• Monolayer wrapping results
from a thermodynamic drive
to eliminate the hydrophobic
interface between the tubes
and their aqueous medium.
• Random adsorption is not likely
to result in sufficient coverage;
single tight coil would introduce
significant bond-angle strain in
the polymer backbone;
• multiple helices are the likely
configuration.
Smalley and coworkers, Chem Phys Lett 342 (2001) 265
Simplest MC simulation
Recipe: adsorption and frustration.
•
•
•
•
•
Dilute semiflexible polymer solution
Impenetrable infinite cylinder
Periodic boundaries
100
LJ interactions
80
MC moves
– Reptation
– Kink-jump
– Pivot
• Metropolis acceptance
p  minexp U new  U old  kT ,1
106
–
equilibration moves
– Averages every 103 for
additional 107 iterations
60
40
20
0
40
20
0
-20
-40
-40
-20
0
20
40
Potential of nanotube
• Surface-averaged Lennard-Jones potential
between the CNT and monomers:
U cyl 
Router 2  z
   rdr  d  dz  U
LJ
Rinner 0 0
2
Router
0
Rinner
U cyl  8  d

3 
 63
d  

11 / 2
5/ 2 
16x 
 512x
where
x  ( D  R) 2   2  2 ( D  R)  cos
• The total potential energy of a given polymer configuration
is given by:
U tot  U cyl ( ri )  U B (q )  U LJ ( rij )
i 
i j i

 q




adsorption
multiple helix




frustratio n
00
100
80
80
60
60
40
40
20
20
0
50
0
50
0
0
50
0
R=2, N=100
1
lp
0.8
0.6
fads
1
0.4
0.8
0.6
fads
0.4
0.2
0.2
0
0
50
10
20

30
40
50
5
0
2
4
lp
44
1.6
6
R
3
Nt
8
2
1.2
3
4
2
5
1
3
1
0
0
Nt
10
20

30
40
50
2
1
k=0
0
0
2
4
R
6
8
k=50
Effect of concentration
N=100, R=3, k=50
120
80
k=0
k=5
k=10
k=50
k=100
60
40
20
0
0
2
4
6
8
0
2
4
Nc
6
8
Nc=2
Nc=3
4
3
Nt
% ads
100
2
1
0
Nc=5
Nc=8
Transitions
G ( m) 
1
N 2
g (m, i )

N  3 i 2
1/( N  m  1) j 1
N  m1
g (m, i) 
(cos qi , j  cos qi , j )(cos qi , j m  cos qi , j )
1/( N  1) j 1 (cos qi , j  cos qi , j )2
N 1
G(m)  exp(m /  ) cos(2 m / P)
80
helix
60
 40
20
adsorption
0
0
20

40
60
Helical pitch
• Helical pitch depends on NT radius and chain
flexibility
50
q av, degrees
40
q
30
20
lp
10
0
0.1
1
10
R/l
100
What drives helical polymer wrapping?
• Hydrophobic drive?
– Monolayer adsorption also achieved with weak
interactions between monomers and tube for
semiflexible chains
– Not sufficient to induce helicity
• Helical polymers?
– Too stringent, semiflexible polymers sufficient
• Helicity of nanotube (-stacking)
– Geometry (tube radius) and chain flexibility provide
strong drive for helical wrapping
VIM on sphere
104
103
102
e
s
B
i+1
<R2 >/lp 2
A
101
100
10-1
i
10-2
10-3
i–1
10-4
10-2
O
10-1
100
101
102
103
104
L/lp
1.0
5s
0.9
10s
0.8
2s
Position of bead i+1 is determined
from a point along the path of a great
circle connecting monomer i and the
intersection of line OA with the
surface of the sphere.
<R2>/lp2
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0
2
4
L/lp
6
8
10
Polymer wrapping of a sphere
N=1000 monomers confined to a sphere with radius =10s
40
35
Ref. 10
VIM
30
lp,min
25
20
15
10
5
0
0
5
10
15

20
25
Conclusions
• weak surface interactions are sufficient to overcome low
entropy barrier of semiflexible chains and lead to
monolayer adsorption
• helix is a stable ‘universal’ state for polymers determined
solely by surface curvature (NT and sphere) and polymer
bending energy.
• geometry determines helical pitch at intermediate radii for
semiflexible chains
• multiple helices form due to vdW interactions between
monomers which are sufficient to overcome (small)
translational entropy of adsorbed chains
Conclusions (2)
• Available computational resources limit our simulations to
relatively short chains
– The semiflexible chain can be effectively modeled through a
summation of energy and entropy ‘vectors’ that determine the
growth or position of a monomer based solely on the two previous
monomers
Acknowledgement
•
•
•
•
•
Liora Levi
Yevgeny moskovitz
Hely Oizerovich
Inna Gorevitz
Iliya Kusner
• ISF
• Rubin Scientific and Medical Research Fund