On the use of spatial eigenvalue spectra in transient polymeric networks

1. On the use of spatial eigenvalue spectra in transient polymeric networks Qualifying exam Joris Billen December 4th 2009

2. Overview • Transient polymer networks • Eigenvalue spectra for network reconstruction • Spatial eigenvalue spectra • Current work

3. 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.

4. Transient polymeric networks • Reversible polymeric gels • Telechelic polymers Concentration Gel Sol Temperature

5. Telechelic polymers • 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

6. Hybrid MD / MC simulation • 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) [drawing courtesy of Mark Wilson]

7. Transient polymeric network • Study of polymeric network T=1.0 only endgroups shown

8. 3 2 4 node 1 2 3 4 1 0 0 1 1 0 0 1 1 1 1 0 1 1 1 1 0 1 2 3 4 Network notations • Network definitions and notation • Degree (e.g. k4=3) • Average degree: • Degree distribution P(k) • Adjacency matrix • Spectral density:

9. Degree distribution gel • Bimodal network:

10. Degree distribution gel (II) • 2 sorts of nodes: • Peers • Superpeers Master thesis M. Wilson

11. adjust : Mimicking network probabilities to form links? pSS pPP pPS One degree of freedom!

12. Mimicking network (II) Simulated Gel Model 2 separated networks pps=0 Model no links between peers ppp=0 Model ppp=0.002 pps=0.009 pss=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.

13. Mimicking network (III) [drawings courtesy of Mark Wilson] lP lS lps

14. Spatial networks • 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, …

15. 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).

16. a measure for spatial dependence Spatial dependent networks: construction (I) • Erdös-Rényi (ER) Spatial dependent ER Regular ER random network qconnect~ distance qconnect constant

17. >p <p SD ER Lowest cost a ER Rewiring probability p 0 1 Spatial dependent networks: construction (II) • Small-world network 1.Create lowest cost network 2.Rewire each link with p if rewired connection probability qij~dij-a

18. Spatial dependent networks: construction (III) • Scale-free network Spatial dependent scalefree: Rich get richer... when they are close Regular scalefree Rich get richer qconnect~degree k qconnect~(degree k,distance dij) 1 1 1 1 4 5 4 1 2 1 2 2 1 1 1 1

19. Spatial dependent networks: spectra Observed effects for high a: • fat tail to the right • peak shifts to left • peak at -1

20. Analysis of spectra • Quantification tools: • mth central moment about mean: • Skewness: • Number of directed paths that return to starting vertex after s steps:

21. Skewness

22. Tree: D2=4 (1-2-1) (2-1-2) (1-3-1) (3-1-3) D3=0 Triangle D2=6 D3=6 1 1 3 3 2 2 Directed paths • Spectrum contains info on graph’s topology: # of directed paths of k steps returning to the same node in the graph

23. Directed paths (II)

24. Number of triangles • Skewness S related to number of triangles T ERspatial ER 2D triangular lattice • T and S increase for spatial network

25. System size dependence

26. Spatial ER Relation skewness and clustering coefficient (I) • Clustering coefficient = # connected neighbors # possible connections • Average clustering coefficient

27. Anti-spatial network • Reduction of triangles • More negative eigenvalues • Skewness goes to zero for high negative a

28. 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

29. Current work

30. Current work (I) • Polymer system under shear

31. Current work (II) Sprakel et al., Phys Rev. E, 79,056306(2009). preliminary results stress versus shear: plateau velocity profile: shear banding

32. Current work (III) • Changes in topology?

33. Acknowledgements • Prof. Baljon • Mark Wilson • Prof. Avinoam Rabinovitch • Committee members

34. Emergency slide I • Spatial smallworld

35. ¥ ¥ å å = = N N p ( k ) N N p ( k ) A A B B = = 0 0 k k 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

36. Equation of Motion K. Kremer and G. S. Grest. Dynamics of entangled linear polymer melts: A molecular-dynamics simulation. Journal of Chemical Physics, 92:5057, 1990. • Interaction energy • Friction constant • Heat bath coupling – all complicated interactions • Gaussian white noise

37. Number of triangles • 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 ERspatial ER

38. Spatial dependent networks: discussion (IV) • Relation skewness and clustering: however only valid for high <k> when <ki(ki-1)> ~ ki(ki-1) can be approximated by

39. Shear banding S. Fielding, Soft Matter 2007,3, 1262.