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.