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Network Properties of a Simulated Polymeric Gel Mark Wilson Fall 2008 Special Thanks: Dr. Network Properties of a Simulated Polymeric Gel Mark Wilson Fall 2008 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 Ph. D. Candidate, Department of Computational Science, SDSU

Objective 1) To describe a MD simulation of a polymer system in terms of Objective 1) To describe a MD simulation of a polymer system in terms of complex network structure. 2) Investigate to what degree random network models describe a polymer network. • Real-World Networks • MD Simulated Polymer Network • Network Basics - Toolbox • Degree distribution • Clustering coefficient • Model M. E. J. Newman, SIAM Review, Vol. 45, No. 2, pp. 167– 256 • Erdős-Rényi random model • Compare Polymer Network and Model • Further Development of Model

Real-World Networks • Information Networks • Internet • World Wide Web • Transportation Networks Real-World Networks • Information Networks • Internet • World Wide Web • Transportation Networks • Airports • Highways • Social Networks • Friendships • Actors in common movies M. E. J. Newman SAIM Mod. Phys. , (74), No. 1, 2002 Barabasi et. al. Review (45) No. 2 pp 167 -256

MD Simulation of Polymer System MD Simulation of Polymer System

Molecular Dynamic Simulation • Bead-spring model of a molecule comprised of repeating units • 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 Molecular Dynamic Simulation • Reversible junctions between end groups are based upon Monte Carlo process • Junctions are updated every ~20 th time step • Probability of breaking or forming junctions is based upon the energy difference of new and old states as compared to k. T • Forces calculated and spatial locations are updated • Analyzed for a variety of temperatures • Thermal equalization • Spatial and junction data is gathered for desired number of time steps

Previous Studies • Fluid-like dynamics at high temperature • Deviations from liquid behavior at Previous Studies • Fluid-like dynamics at high temperature • Deviations from liquid behavior at T = 0. 75 • Aggregate forming transition at T = 0. 51 • Gelation transition below T = 0. 3 • Structure exhibits limited movement and flow. End groups are confined to specific aggregates for long periods of time

Polymer Network • Polymer system described in terms of a network • Low temperature Polymer Network • Polymer system described in terms of a network • Low temperature – aggregate forming system • Terminology • Aggregates • Single Bridges • Multiple Bridges • Loops

Network Basics Network Basics

Degree Distribution • Dynamic system – statistical averages of many realizations • 4 Nodes Degree Distribution • Dynamic system – statistical averages of many realizations • 4 Nodes • Links – Multiple and Single • Arrangement of links defines topology • Degree: number of links incident upon a node • Probability distribution • Functional form gives insight into class of network • Random – Poisson distributed • Scale free – power law distributed k 1 2 3 4 5 n(k) 1 1 0 p(k) 0. 25 0

Clustering Coefficient • Fractional measure of a node’s connectivity 0 £ ci £ 1 Clustering Coefficient • Fractional measure of a node’s connectivity 0 £ ci £ 1 • Local property describing an individual node ci = Ei zi ( zi - 1) 2 Number of existing links between neighbors Total number of possible links • Can be extended to a global network property • Only average over contributing nodes • Degree correlations • Average C as a function of degree c (k ) = 1 Nk å ci k é k - k ù ê ú C= 2 N ê ú k ë û 2 Y (k ) • Functional dependence indicates connections between nodes are dependent upon their degree 2

Model Model

Erdős-Rényi Random Models are created to mimic real-world networks • Begin with N disconnected Erdős-Rényi Random Models are created to mimic real-world networks • Begin with N disconnected nodes, form l links between randomly selected pairs Alternative method: • Probabilistically form links through the total number of possible connections p (k ) = lk e - l ¥ k! k = å k p(k ) k =0 l= k

Polymer Network Properties Polymer Network Properties

Degree Distributions • Decreasing initial drop-off at low degree • Low temperature onset of 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 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 and B-Community • Magnitude and peak Curve Fitting The Distributions • Define the A and B-Community • Magnitude and peak location • Fraction within the A-Community

A-Community Fraction • What percentage of the total distribution is attributed to 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 for the polymer network • Fall short Clustering Coefficient • High values of C for the polymer network • Fall short in describing the high interconnectivity • Why the differences? Bimodal p(k), proximity é k 2 - k ù k ê ú = C 2 N ê ú k ë û 2

Proximity – A Fundamental Difference Proximity – A Fundamental Difference

Rewiring the Polymer Network • Remove proximity through a random rewiring • Process: • Rewiring the Polymer Network • Remove proximity through a random rewiring • Process: • Randomly choose two polymer chains • Assure they do not share an aggregate • Break chain and connect opposing end groups • Maintains the degree distribution of aggregates involved • Removes proximity by not accounting for the distance between opposing end groups

Clustering Coefficient • Dramatic changes in C with first few hundred rewires • Convergent 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 for C Clustering Coefficient • Removal of proximity constraint results in expected values for C

Clustering Coefficient • Functional dependence indicates the polymer is degree correlated • Connections between 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 Development of the Model

Bimodal ER Random Model • Model consists of two communities • In order to Bimodal ER Random Model • Model consists of two communities • In order to match the polymer degree distribution • Input parameters to the model: k A ¥ N A = N å p A (k ) k =0 2 l A + l AB = N A k. B l AB ¥ N B = N å p. B (k ) k =0 2 l B + l AB = N B k B • Connections are formed between random pairs in each community • Degree distribution comprised of two Poissons

Bimodal Random Model with Proximity • How do we add proximity to the model? Bimodal Random Model with Proximity • How do we add proximity to the model? • Assign random spatial coordinates to nodes within 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

Bimodal Random Model with Proximity • p(k) is the same independent of l. AB Bimodal Random Model with Proximity • p(k) is the same independent of l. AB • 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 • How do we determine the type of bimodal model? • Eigenvalues Eigenvalue spectrum • How do we determine the type of bimodal model? • Eigenvalues of the adjacency matrix • Match C with cutoff • Match spectrum peak at -1 • Find ideal number of LAB

Eigenvalue spectrum Eigenvalue spectrum

Conclusion • Described a polymer system in terms of a complex network • Investigated Conclusion • Described a polymer system in terms of a complex network • Investigated two defining network properties of the polymer • Degree distribution – bimodal nature – two independent communities • A-Community becomes prevalent at T = 0. 5 • Clustering coefficient – highly interconnected at low temperatures • Rewiring – found that proximity plays a role in C • Used random model to describe the polymer network • Bimodal degree distribution – predicts polymer distribution well • C was matched by adding proximity constraint to the model • Eigenvalue spectrum - Topology as a function of temperature

Tails of the Distributions • Where does the random model fall short? • Log-log Tails of the Distributions • Where does the random model fall short? • Log-log scale • Power law distributed at T = 0. 5?

Adjacency Matrices • 4 Nodes • Single and Multiple links • Loops • Arrangement Adjacency Matrices • 4 Nodes • Single and Multiple links • Loops • Arrangement of links defines topology • Described by an adjacency matrix • matrix describes the connectivity • Entries contain the number of links between node and node • Symmetric matrix

Equation of Motion K. Kremer and G. S. Grest. Dynamics of entangled linear polymer 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

Formula Chi-squared Correlation coefficient Formula Chi-squared Correlation coefficient