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![Page 1: On the use of spatial eigenvalue spectra in transient polymeric networks Qualifying exam Joris Billen December 4 th 2009.](https://reader036.fdocuments.net/reader036/viewer/2022062422/56649ef45503460f94c07853/html5/thumbnails/1.jpg)
On the use of spatial eigenvalue spectra in transient polymeric
networks
Qualifying exam
Joris Billen
December 4th 2009
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Overview
• Transient polymer networks
• Eigenvalue spectra for network reconstruction
• Spatial eigenvalue spectra
• Current work
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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.
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TemperatureSol Gel
Transient polymeric networks• Reversible polymeric gels• Telechelic polymers
Concentration
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• 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
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• 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]
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Transient polymeric network• Study of polymeric network
T=1.0only endgroupsshown
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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
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Degree distribution gel• Bimodal network:
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Degree distribution gel (II)
• 2 sorts of nodes:– Peers– Superpeers
!!)(
k
ekN
k
ekNkP
PSkk
PP
kk
SS
Master thesis M. Wilson
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probabilities to form links?pSS
PPPSPSP
PPSSSSS
NpNpk
NpNpk
adjust :
pPP pPS
One degree of freedom!
Mimicking network
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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.
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Mimicking network (III)
[drawings courtesyof Mark Wilson]
lP
lS
lps
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• 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
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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).
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Spatial dependent networks: construction (I)
• Erdös-Rényi (ER)
Regular ER random network Spatial dependent ER
qconnect
constant qconnect
~ distance
ijij dq ~
measure forspatial dependence
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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
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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(~
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Spatial dependent networks: spectra
Observed effects for high :– fat tail to the right– peak shifts to left– peak at -1
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• Quantification tools:– mth central moment about mean:
– Skewness:
– Number of directed paths that return to starting vertex after s steps:
Analysis of spectra
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Skewness
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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
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Directed paths (II)
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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
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System size dependence
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Relation skewness and clustering coefficient (I)
• Clustering coefficient = # connected neighbors
# possible connections
• Average clustering coefficient
Spatial ER
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Anti-spatial network• Reduction of triangles
• More negative eigenvalues
• Skewness goes to zero for high negative
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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
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Current work
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Current work (I)
• Polymer system under shear
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Current work (II)
stress versusshear:plateau
velocityprofile:shear banding
Sprakel et al.,Phys Rev. E, 79,056306(2009).
preliminary results
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Current work (III)• Changes in topology?
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Acknowledgements
• Prof. Baljon
• Mark Wilson
• Prof. Avinoam Rabinovitch
• Committee members
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Emergency slide I
• Spatial smallworld
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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
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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
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• 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
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• 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)
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Shear banding
S. Fielding, Soft Matter 2007,3, 1262.