Contents

## Network Properties of a Simulated Polymeric Gel Objective Real-World Networks MD Simulation of Polymer System Molecular Dynamic Simulation

Cornelius, Keridwen, Managing Editor has reference to this Academic Journal, PHwiki organized this Journal Network Properties of a Simulated Polymeric Gel Special Thanks: Dr. Arlette Baljon Department of Physics, San Diego State University Dr. Avinoam Rabinovitch Department of Physics, Ben Gurion University of the Negev, Israel Joris Billen PhD. C in addition to idate, Department of Computational Science, SDSU Mark Wilson Fall 2008 Objective Real-World Networks MD Simulated Polymer Network Network Basics – Toolbox Degree distribution Clustering coefficient Model Erds-Rényi r in addition to om model Compare Polymer Network in addition to Model Further Development of Model To describe a MD simulation of a polymer system in terms of complex network structure. Investigate to what degree r in addition to om network models describe a polymer network. M.E.J. Newman, SIAM Review, Vol. 45,No . 2,pp . 167256 Real-World Networks Barabasi et. al. Rev. Mod. Phys., (74), No. 1, 2002 In as long as mation Networks Internet World Wide Web Transportation Networks Airports Highways Social Networks Friendships Actors in common movies M.E.J. Newman SAIM Review (45) No.2 pp167-256

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MD Simulation of Polymer System Molecular Dynamic Simulation Bead-spring model of a molecule comprised of repeating units 1000 polymeric chains Each chain consists of eight beads joined with anharmonic potential Functionalized end groups telechelic polymer Each end group can be involve with multiple junctions Truncated Lennard-Jones potential Excluded volume interactions Molecular Dynamic Simulation Reversible junctions between end groups are based upon Monte Carlo process Junctions are updated every ~20th time step Probability of breaking or as long as ming junctions is based upon the energy difference of new in addition to old states as compared to kT Forces calculated in addition to spatial locations are updated Analyzed as long as a variety of temperatures Thermal equalization Spatial in addition to junction data is gathered as long as desired number of time steps

Previous Studies Fluid-like dynamics at high temperature Deviations from liquid behavior at T = 0.75 Aggregate as long as ming transition at T = 0.51 Gelation transition below T = 0.3 Structure exhibits limited movement in addition to flow. End groups are confined to specific aggregates as long as long periods of time Polymer Network Polymer system described in terms of a network Low temperature aggregate as long as ming system Terminology Aggregates Single Bridges Multiple Bridges Loops Network Basics

Degree Distribution Dynamic system statistical averages of many realizations 4 Nodes Links Multiple in addition to Single Arrangement of links defines topology Degree: number of links incident upon a node Probability distribution Functional as long as m gives insight into class of network R in addition to om Poisson distributed Scale free power law distributed Clustering Coefficient Fractional measure of a nodes connectivity Local property describing an individual node Can be extended to a global network property Only average over contributing nodes Degree correlations Average C as a function of degree Functional dependence indicates connections between nodes are dependent upon their degree Model

Erds-Rényi R in addition to om Model Models are created to mimic real-world networks Begin with disconnected nodes, as long as m links between r in addition to omly selected pairs Alternative method: Probabilistically as long as m links through the total number of possible connections Polymer Network Properties Degree Distributions Decreasing initial drop-off at low degree Low temperature onset of secondary peak with higher degree Two communities

Curve Fitting The Distributions Maintain the idea of two independent communities Two Poisson fitting function Iterative minimization of Chi-squared Curve Fitting The Distributions Define the A in addition to B-Community Magnitude in addition to peak location Fraction within the A-Community A-Community Fraction What percentage of the total distribution is attributed to the A-Community Agreement with the onset of the secondary peak Agreement with previous studies T = 0.5

Clustering Coefficient High values of C as long as the polymer network Fall short in describing the high interconnectivity Why the differences Bimodal p(k), proximity Proximity A Fundamental Difference Rewiring the Polymer Network Remove proximity through a r in addition to om rewiring Process: R in addition to omly choose two polymer chains Assure they do not share an aggregate Break chain in addition to connect opposing end groups Maintains the degree distribution of aggregates involved Removes proximity by not accounting as long as the distance between opposing end groups

Clustering Coefficient Dramatic changes in C with first few hundred rewires Convergent value taken as an average of the last 10 values Clustering Coefficient Removal of proximity constraint results in expected values as long as C Clustering Coefficient Functional dependence indicates the polymer is degree correlated Connections between nodes are dependent upon their degree Rewiring process removes this dependence uncorrelated network

Development of the Model Bimodal ER R in addition to om Model Model consists of two communities In order to match the polymer degree distribution Input parameters to the model: Connections are as long as med between r in addition to om pairs in each community Degree distribution comprised of two Poissons Bimodal R in addition to om Model with Proximity Assign r in addition to om spatial coordinates to nodes within 1 X 1 X 1 cell Limit allowed connections between nodes based upon a cut-off distance Degree distribution is unaffected by the addition of the proximity constraint How do we add proximity to the model

Bimodal R in addition to om Model with Proximity p(k) is the same independent of lAB Stronger cut-off constraint C increases Bimodal model with proximity is uncorrelated Polymer Network: T = 0.5 C = 0.215 Bimodal Model: Cut-off = 0.35 C = 0.223 Eigenvalue spectrum Match C with cut-off Match spectrum peak at -1 Find ideal number of LAB How do we determine the type of bimodal model Eigenvalues of the adjacency matrix Eigenvalue spectrum

Formula Chi-squared Correlation coefficient

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