HOME
Search & Results
Full Text
Thesis Details
Page:
218
Full Screen
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