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  • TITLE
  • CERTIFICATE
  • DECLARATION
  • ACKNOWLEDGEMENT
  • CONTENTS
  • 1. INTRODUCTION
  • 2. REVIEW OF LITERATURE
  • REFERENCES
  • 3. CONFIGURATIONAL STATISTICS OF CHAIN MOLECULES
  • 3.1 Random Walk in One, Two and Three Dimensions
  • Fig.3.1. The twelve possible SAWs of a rook over a 3 by 3 chessboard, moving from one corner to the opposite corner
  • 3.2 Spatial Configuration of Chain Molecules-Simplified Model Chains
  • Fig.3.2. Diagrammatic representation of a configuration of skeletal atoms of a hypothetical chain, consisting of n bonds
  • 3.2.1 The Freely Jointed Chain
  • Fig.3.3. A portion of the polyethylene chain in all transform, Including bonds1-1 to 1+4. 01 i: r the supplement of the llh bond angle and $1 Is therotation angle about bond i
  • 3.2.2 The Freely Rotating Chain
  • 3.2.3 Neighbour Correlations in Real Chains
  • 3.2.4 Chains with Separable Configurational Energies
  • 3.2.5 Chain Molecule with Interdependent Rotational Potentials
  • 3.2.6 Bond Rotational Potentials for Simple Molecules
  • Table 3.1. Rotational Barrier heights in a few simple molecules
  • Fig.3.5. The conformational energy of n-butane as a function of rotation angle, 4, about the central C-C bond, with terminal groups in the staggered conformation.
  • 3.2.7 Statistical Weight Matrices for Interdependent Bonds
  • 3.2.8Chain Model in the Rotational Isomeric State (RIS) Approximation
  • Fig.3.6. Ramachandran diagram for internal rotations in n-pentane for Φ1 = Φ4 =0, with contours shown at intervals of 1 Kcal mol-1. x marks indicate minima.
  • 3.2.9 Calculation of Mean Square Moments
  • 3.3 Configuration of Chain Molecules in Dilute Solution
  • Fig.3.8. Short-range (s) and long-range (I) forces in chainlike macromolecules
  • 3.3.1 The Theta Temperature
  • References
  • 4. COMPUTATION METHODS
  • 4.1 Statistical Mechanical Preliminaries
  • 4.2 Common Statistical Ensembles
  • computation Methods
  • 4.3 Florys Matrix Multiplication Method
  • 4.3.1 Chains with Identical Bonds
  • Table 4.1. Bond lengths, bond angles and interaction energies for RIS states ofpolyethylene (PE) and polytetrafluoroethylene (PTFE)
  • Table 4.2. Bond length and bond angle and interaction energies for RIS states of polymeric sulphur
  • 4.3.2 Regularly Repeating Sequence
  • 4.4 Monte Carlo Method
  • 4.4.1 A Monte Carlo Trajectory
  • Fig.4.1. The geometry for the hit and miss integration to find the value of x
  • Fig.4.2. Accepting moves to higher energy configurations in MC simulation
  • 4.4.2 The Metropolis Procedure (Simulation of chains with excluded volume)
  • 4.4.3 Initial Configuration
  • 4.4.4 Equilibration
  • 4.4.5 Random Number Generators
  • Testing the random number generator
  • 4.4.6 Potential Functions
  • Fig. 4.3. Lennard-Jones type potential U (r)
  • Potential functions used in the MC simulation of polyethylene (PE)
  • Potential function used in MC simulation of polytetrafluoroethylene (PTFE)
  • Calculation of parameters in the LJ-CP potential function
  • 4.4.7Monte Carlo Simulation of Random Walk
  • 4.5 Exact Enumeration Method
  • 4.5.1 Exact Enumeration of Short Random Walks
  • References
  • 5. RESULTS AND DISCUSSION
  • 5.1 Random Walk in One, Two and Three Dimensions
  • Table 5.1. Monte Corlo simulation data for one- and two dimensional random walk - Comparison with formula (5.4)
  • Table 5.2. Arc Sine Law for last visits in one-dimensional random walk of 20 stepsand its analogue in two-dimensional random walk
  • Table 5.3 Exact enumeration data for two-and three -dimensional random walk -comparison with formula (5.40) and (5.8) respectively
  • 5.2 Polyethylene
  • Fig. 5.1. Characteristic ratios ln as a function of number of monomer units, n
  • Fig.5.2. Characteristic ratios /n as a function of number of monomer units, polyethylene under Kihara Convex Core Potential
  • 5.2.1 Results for PE under the Kihara Convex Core Potential
  • Fig.5.3. Expansion coefficient a% as a function of number of monomer units, n, Polyethylene under Kihara Convex Core Potential
  • Fig.5.4. Characteristic ratios /n as a function of in n, polyethylene under Kihara Convex Core Potential
  • Fig.5.5. Ratio of mean square end to end distance to mean square radius of gyration as a function of number of monomer units, Polyethylene under Kihara Convex Core Potential
  • Fig.5.6. /n as a function of chain length n, Polyethylene under Kihara Convex Core Potential
  • 5.2.2 Evaluation of Cluster Integrals
  • Fig.5.7. Cluster integral per -CH2 group, β, from perturbation theory.Polyethylene under Kihara Convex Core Potential
  • Fig.5.8. Cluster integral per -CH2 group, βl, from Flory-Stockmayer theory.Polyethylene under Kihara Convex Core Potential
  • 5.2.3 Calculation of Theta Temperatures
  • 5.3 Polytetrafluoroethylene
  • 5.3.1 Calculation of Unperturbed Dimensions
  • 5.3.2 Results for PTFE under the Kihara Convex Core Potential
  • Fig.5.14. PTFE under Kihara convex core potential - characteristic ratio as afunction of chain length
  • Fig.5.16. Expansion coefficient a% as a function of chain length n, for PTFE under Kihara convex core potential
  • Fig.5.17. Mean square radius of gyration as a function of number of monomer units
  • 5.3.3 Effective Cluster Integrals
  • Fig.5.19. Cluster integral per -CF2- group from perturbation theory - PTFE under Kihara convex core potential as a function of chain length
  • Fig.5.20 Cluster integral per -CF2- group from Flory-Sfockmayer theory for PIFE under Kihara potential
  • Fig.5.21. Effective cluster integral per freely jointed segment from perturbation theory as a function of number of freely jointed segments. PIFE underKihara convex core potential
  • Fig.5.22. Effective cluster integral per effective freely jointed segment from Flory Stockmayer theory - as a function of number of effective freely jointed segments. PTFE under Kihara convex core potential
  • 5.3.4 PTFE under the Maitland Potential
  • Fig.5.23. PTFE under Maitland potential characteristic ratio as a function of chainlength
  • Fig.5.24. PTFE under Maitland potential. Expansion coefficient α2 x as a function of chainlength, n
  • Fig.5.25. PTFE under Maitland potential. Cluster integral per -CF2 group from perturbation theory
  • Fig.5.26. PTFE under Maitland potential. Cluster integral per -CF2 group as a function of chainlength, n from Flory-Stockmayer theory
  • Fig.5.27. PTFE under Maitland potential. Cluster integral per freely jointed segment from perturbation theory
  • Fig.5.28. PTFE under Maitland potential. Effective cluster integral per freely jointed segrnt from Flory-Stockrnayer theory
  • 5.3.5 PTFE under the Consistent Potential (CP)
  • Fig.5.29. PTFE under Consistent potential. Characteristic ratio /n as a function of chain length, n
  • Fig.5.30. PTFE under Consistent potential. Expansion coefficient α2x as a function of chain length, nl
  • Fig.5.31. PTFE under Consistent potential. Cluster integral per -CF2 group from perturbation theory
  • Fig.5.32. PTFE under Consistent potential. Cluster integral per -CF2- group from Flory Stockrnayer theory
  • Fig.5.33. PTFE under consistent potential. Effective cluster integral per freely jointed segment from perturbation theory
  • Fig.5.34. PTFE under consistent potential. Effective cluster integral per freely jointed segment from Flory Stockmayer theory
  • 5.4 Polymeric Sulphur
  • Fig.5.35. Characteristic ratios /n of polymeric sulphur, in absence of any potential functions
  • Fig.5.36. Characteristic ratios /n, with all the rings excluded. Results from Exact Enumeration Method, d is the cut-off distance
  • Fig.5.37. The plot of unperturbed /n and perturbed /n obtained by Monte Carlo method
  • 5.5 Cis- (l, 4) -polybutadiene and cis- (1, 4) -polyisoprene
  • 5.5.1 Calculation of Unperturbed Dimensions of cis- (1, 4) -polybutadiene
  • Table 5.5. Characteristic ratio (C.) for cis- (1.4) -polybutadiene (σ = 1.4, τ = 0.05, 1 =, 343 K)
  • Table 5.6. Characteristic ratio (C.) for cis- (1.4) -polybufadiene (a = 0.54, r = 0.1, = 343 K)
  • 5.5.2 Calculation of Unperturbed Dimensions of cis- (1, 4) -polysoprene
  • 5.5.3 The Four Bond Unit Model
  • Table 5.7. Characteristic ratio Cn for cis- (1, 4) -polybutadiene by four bond unit model σ = 1.0.τ = 0.101
  • References
  • 6. CONCLUSION
  • Scope for further work
  • Reference
  • APPENDICES
  • APPENDIX 1 The generator matrix for pe
  • APPENDIX 2 The generator matrix for ptfe
  • APPENDIX 3
  • APPENDIX 4
  • APPENDIX 5
  • APPENDIX 6
  • APPENDIX 7