on the use of spatial eigenvalue spectra in transient polymeric networks qualifying exam joris...
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On the use of spatial eigenvalue spectra in transient polymeric
networks
Qualifying exam
Joris Billen
December 4th 2009
Overview
• Transient polymer networks
• Eigenvalue spectra for network reconstruction
• Spatial eigenvalue spectra
• Current work
Transient polymeric networks*
*’Numerical study of the gel transition in reversible associating polymers’, Arlette R. C. Baljon, Danny Flynn, and David Krawzsenek, J. Chem. Phys. 126, 044907 2007.
TemperatureSol Gel
Transient polymeric networks• Reversible polymeric gels• Telechelic polymers
Concentration
• Examples– PEG (polyethylene glycol) chains terminated by
hydrophobic moieties
– Poly-(N-isopropylacrylamide) (PNIPAM)
• Use:– laxatives, skin creams, tooth paste, Paintball fill,
preservative for objects salvaged from underwater, eye drops, print heads, spandex, foam cushions,…
– cytoskeleton
Telechelic polymers
• Bead-spring model
• 1000 polymeric chains, 8 beads
• Reversible junctions between end groups
• Molecular Dynamics simulations
with Lennard-Jones interaction between beads and
FENE bonds model chain structure and junctions
• Monte Carlo moves to form and destroy junctions
• Temperature control (coupled to heat bath)
Hybrid MD / MC simulation
[drawing courtesyof Mark Wilson]
Transient polymeric network• Study of polymeric network
T=1.0only endgroupsshown
Network notations• Network definitions and notation
– Degree (e.g. k4=3)
– Average degree:– Degree distribution P(k)– Adjacency matrix– Spectral density:
k P(k)1 0
2 0.5
3 0.5
4 0
1
2
3
4
0 0 1 1
0 0 1 1
1 1 0 1
1 1 1 0
1
2
3
4
node 1 2 3 4
5.22
1
N
lkPkk i
N
ii
N
j=jλλδ
N=ρ(λ)
1
1
Degree distribution gel• Bimodal network:
Degree distribution gel (II)
• 2 sorts of nodes:– Peers– Superpeers
!!)(
k
ekN
k
ekNkP
PSkk
PP
kk
SS
Master thesis M. Wilson
probabilities to form links?pSS
PPPSPSP
PPSSSSS
NpNpk
NpNpk
adjust :
pPP pPS
One degree of freedom!
Mimicking network
Mimicking network (II)
SimulatedGel
Model2 separatednetworkspps=0
Modelno linksbetween peersppp=0
Modelppp=0.002pps=0.009pss=0.04
‘Topological changes at the gel transition of a reversible polymeric network’, J. Billen, M. Wilson, A. Rabinovitch and A. R. C. Baljon, Europhys. Lett. 87 (2009) 68003.
Mimicking network (III)
[drawings courtesyof Mark Wilson]
lP
lS
lps
• Proximity included
in mimicking gel
• Asymmetric spectrum
• Spectrum to estimate maximum connection length• Many real-life networks are spatial
– Internet, neural networks, airport networks, social networks, disease spreading, polymeric gel, …
Spatial networks
Eigenvalue spectra of spatial dependent networks*
* ’Eigenvalue spectra of spatial-dependent networks’, J. Billen, M. Wilson, A.R.C. Baljon, A. Rabinovitch, Phys. Rev. E 80, 046116 (2009).
Spatial dependent networks: construction (I)
• Erdös-Rényi (ER)
Regular ER random network Spatial dependent ER
qconnect
constant qconnect
~ distance
ijij dq ~
measure forspatial dependence
Spatial dependent networks: construction (II)
1.Create lowest cost network
2.Rewire each link with p
>p
<p
Rewiring probability p
0 1
Lo
wes
t co
st
ER
SD
ER
if rewired connection probability qij~dij
-
• Small-world network
4
Spatial dependent networks: construction (III)
• Scale-free network
Regular scalefreeRich get richer
Spatial dependent scalefree:Rich get richer... when they are close
qconnect
~degree k qconnect
~(degree k,distance dij)
1
5
1
1
1
1
2
1
4
1
1
11
22
ijjji dkq )1(~
Spatial dependent networks: spectra
Observed effects for high :– fat tail to the right– peak shifts to left– peak at -1
• Quantification tools:– mth central moment about mean:
– Skewness:
– Number of directed paths that return to starting vertex after s steps:
Analysis of spectra
Skewness
Directed paths
N
j=
kjk λ=D
1
• Spectrum contains info on graph’s topology:
Tree:D2=4(1-2-1)(2-1-2)(1-3-1)(3-1-3)
D3=0
1
2 3
TriangleD2=6D3=6
32
1
# of directed paths of k steps returning to the same node in the graph
Directed paths (II)
Number of triangles
• Skewness S related to number of triangles T
ER spatial ER 2Dtriangular lattice
• T and S increase for spatial network15
1
90
1
2
1
2
kkS
kkNT
N
kkS
kkT
1
6
1
2
1
2
3
6
NT
S
System size dependence
Relation skewness and clustering coefficient (I)
• Clustering coefficient = # connected neighbors
# possible connections
• Average clustering coefficient
Spatial ER
Anti-spatial network• Reduction of triangles
• More negative eigenvalues
• Skewness goes to zero for high negative
Conclusions
• Contribution 1: Spectral density of polymer simulation– Spectrum tool for network reconstruction– Spectral density can be used to quantify spatial
dependence in polymer
• Contribution 2: Insight in spectral density of spatial networks– Asymmetry caused by increase in triangles– Clustering and skewed spectrum related
Current work
Current work (I)
• Polymer system under shear
Current work (II)
stress versusshear:plateau
velocityprofile:shear banding
Sprakel et al.,Phys Rev. E, 79,056306(2009).
preliminary results
Current work (III)• Changes in topology?
Acknowledgements
• Prof. Baljon
• Mark Wilson
• Prof. Avinoam Rabinovitch
• Committee members
Emergency slide I
• Spatial smallworld
Emergency slide II
• How does the mimicking work?– Get N=Ns+Np from simulation– Determine Ns and Np from fits of bimodal– Determine ls / lp / lps so that
0
)(k
AA kpNN
0
)(k
BB kpNN
Equation of Motion
)(tWrUr iiij
iji
FENEij
LJijij UUU
K. Kremer and G. S. Grest. Dynamics of entangled linear polymer melts: Amolecular-dynamics simulation. Journal of Chemical Physics, 92:5057, 1990.
W
•Interaction energy
•Friction constant
•Heat bath coupling – all complicated interactions
•Gaussian white noise
• Skewness related to number of triangles T
• P (node and 2 neighbours form a triangle) =
possible combinations to pick 2 neighbours X
total number of links / all possible links
ER spatial ER
Number of triangles
• Relation skewness and clustering:
however only valid for high <k> when <ki(ki-1)> ~ ki(ki-1)
can be approximated
by
Spatial dependent networks: discussion (IV)
Shear banding
S. Fielding, Soft Matter 2007,3, 1262.