1. On the use of spatial eigenvalue spectra in transient polymeric networks Qualifying exam Joris Billen December 4 th 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. Temperature Sol Gel Transient polymeric networks Reversible polymeric gels Telechelic polymers Concentration

5. 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

6. 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 courtesy of Mark Wilson]

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

8. Network notations Network definitions and notation –Degree (e.g. k 4 =3) –Average degree: –Degree distribution P(k) –Adjacency matrix –Spectral density: kP(k) 10 20.5 3 40 1 2 3 4 0 0 1 1 1 1 0 1 1 1 1 0 1 2 3 4 node1234

9. Degree distribution gel Bimodal network:

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

11. probabilities to form links? p SS adjust : p PP p PS One degree of freedom! Mimicking network

12. Mimicking network (II) Simulated Gel Model 2 separated networks p ps =0 Model no links between peers p pp =0 Model p pp =0.002 p ps =0.009 p ss =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] lPlP lSlS l ps

14. 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

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. Spatial dependent networks: construction (I) Erdös-Rényi (ER) Regular ER random network Spatial dependent ER q connect constant q connect ~ distance  measure for spatial dependence

17. Spatial dependent networks: construction (II) 1.Create lowest cost network 2.Rewire each link with p >p<p Rewiring probability p 01  Lowest cost ER SD ER if rewired connection probability q ij ~d ij -  Small-world network

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

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

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

21. Skewness

22. Directed paths Spectrum contains info on graph’s topology: Tree: D 2 =4 (1-2-1) (2-1-2) (1-3-1) (3-1-3) D 3 =0 1 2 3 Triangle D 2 =6 D 3 =6 3 2 1 # 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 ER2D triangular lattice T and S increase for spatial network

25. System size dependence

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

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

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

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. 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

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. 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 Number of triangles

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

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