COMPUTER SIMULATION OF TRACER DIFFUSION IN GEL NETWORK ZHOU HUAI A THESIS SUBMITED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CHEMICAL AND BIOMOLECULAR ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2010 Computer simulation of tracer diffusion in gel network ACKNOWLEDGEMENT I would like to express my sincere appreciation to my supervisor, Prof. Chen Shing Bor for his patient guidance, effective support and wise encouragement throughout this research. His rigorous attitude towards research and serious working manner give me a very deep impression and benefit me a lot for my future career. Many thanks also go to all my labmates and friends: Mr. Kok Hong, Mr. Zhou Tong, Mr. Zhao Guangqiang, Miss. Shen Yiran, Miss Chieng Yu Yuan, Miss. Moe Sande, Mr. Zheng Zhangfeng, Miss Xue Changying, Mr. Li Jianguo, and Dr. Lim Wee Chuan, for their support and assistant through the project. They made my stay in NUS much enjoyable. My family members are also thanked for their support in my whole period of study. Finally, I wish to express my thanks to the National University of Singapore for providing the financial support for this project and the research scholarship throughout my whole period of candidature. i Computer simulation of tracer diffusion in gel network TABLE OF CONTENT ACKNOWLEDGEMENT i TABLE OF CONTENT ii SUMMARY v NOMENCLATURE vii LIST OF FIGURES xi LIST OF TABLES Chapter Introduction xviii 1.1 The need to understand the tracer diffusion in gel 1.2 Significance of computer simulations 1.3 Theoretical prediction of tracer diffusion in gel: computer simulation 1.4 Research objectives 1.5 Outline of the thesis Chapter Literature review 2.1 Theoretical background of computer simulation 10 11 2.1.1 Simulation methods 11 2.1.2 Interaction potentials 13 2.1.3 Coarse-graining and hybrid model 22 2.1.4 Periodic boundary condition (PBC) 26 2.2 Computer simulation of tracer diffusion in gel 27 2.2.1 Models of gel network 27 2.2.2 Diffusion behavior in polyelectrolyte gel 30 2.3 Physical models of diffusion in gel 35 2.4 Experimental study 37 ii Computer simulation of tracer diffusion in gel network Chapter Brownian dynamics simulation of tracer diffusion in a cross-linked network 39 3.1 Introduction 39 3.2 Description of the methods 43 3.3 Results and discussions 50 3.3.1 Uncharged network 52 3.3.2 Charged network 66 3.4 Conclusion 90 Chapter Brownian dynamics simulation of chain diffusion in the polyelectrolyte gel network 92 4.1 Introduction 92 4.2 Description of the methods 97 4.3 Results and discussions 102 4.3.1 Uncharged chain and gel 103 4.3.2 Charged chain and gel 114 4.4 Conclusion 126 Chapter Dissipative particle dynamics simulation of tracer diffusion in gel network 127 5.1 Introduction 127 5.2 Description of the methods 133 5.3 Results and discussions 138 5.3.1 Solvent behavior 138 5.3.2 Diffusion behavior of network beads 141 5.3.3 Tracer diffusion in gel network 145 5.4 Conclusion 153 iii Computer simulation of tracer diffusion in gel network Chapter Conclusion and recommendations 154 6.1 Concluding remarks 154 6.2 Recommendations 157 REFERENCES 158 iv Computer simulation of tracer diffusion in gel network SUMMARY Computer simulation is used to study tracer or chain diffusion in polyelectrolyte gels. Owing to the limitation of present computer power, a mesoscopic approach is adopted to handle long time dynamics in this thesis. Brownian Dynamics (BD) simulation is mainly employed to study the self-diffusion of tracer particles and polymer chain in a cross-linked gel network based on a coarse-grained bead-spring lattice model with a truncated Lennard-Jones potential representing the excluded volume effect and a screened electrostatic interaction accounting for charge effect. Several effects are investigated including the network porosity, flexibility, degree of cross-linking (for tracer particle diffusion only), and electrostatic interaction. In addition, Dissipative Particle Dynamics (DPD) method is implemented to examine hydrodynamic interaction for tracer diffusion in gel network that is ignored by BD simulation. For tracer particle diffusion, the long-time diffusivity of tracer particle is studied in both uncharged and charged system. It is interesting to find that for charged system the diffusion is further hindered by the electrostatic interaction, regardless of whether the tracer particle and the network are oppositely or similarly charged. However, there exists a difference in the hindrance mechanism between the two cases. For the polymer chain diffusion, the conformation and dynamic properties of polymer chains are examined. For uncharged system, a decrease in diffusivity of chain is observed with the decrease of the v Computer simulation of tracer diffusion in gel network porosity of the network. The difference in diffusion behavior of an oppositely and similarly charged chain in gel network is discussed for varied charge amount or Debye length. The static properties of the chain are used to explain the difference between the two cases, such as the average bond angle, the mean-square end-to-end distance, the mean-square radius of gyration, and the three average eigenvalues of the moment of inertia tensor. Finally, the applicability of the DPD method to study the hydrodynamic interaction for the tracer diffusion in gel network is demonstrated, and the advantages or disadvantages of the DPD and BD method are also addressed. These computer simulation results based on the simplified coarse-grained model shed light on the diffusion behaviour of a tracer particle or chain at mesoscopic level. The unusual behaviour of tracer or chain caused by the attractive electrostatic force is intriguing, which can be explained by electrostatic entrapment effect. This effect is dependent on the charge, double layer thickness, and porosity. vi Computer simulation of tracer diffusion in gel network NOMENCLATURE Abbreviations Description BD Brownian dynamics COM center of mass DNA deoxyribonucleic acid DPD Dissipative particle dynamics EVE Excluded volume effect MC Monte Carlo MD Molecular dynamics MSD mean square displacement RDF radial distribution function SDE stochastic differential equation VV velocity Verlet integrator Symbols a Radius of the particle aij The maximum repulsion force between particle i and j D Long-time diffusivity of polyelectrolyte D0 Diffusivity of a particle at infinite dilute solution e Unit vector vii Computer simulation of tracer diffusion in gel network E1, E2, E3 Eigenvalues of the moment of inertia tensor E Moment of inertia tensor F Force vector of all the particles Fi Force on particle i FCi Conservative force on particle i FDi Dissipative force on particle i FRi Random force on particle i g(r) Radial distribution function gbb(r) Radial distribution function for bead to bead gbc(r) Radial distribution function for bead to COM gtb(r) Radial distribution function for tracer to bead H The minimum separation distance between the surfaces of particles K Jump frequency, which depends on temperature and diffusant size k Spring constant for network kB Boltzmann constant ks Spring constant for chain l0 Equilibrium bond length for network l0s Equilibrium bond length for chain lB Bjerrum length L Length of simulation box m Bead mass viii Computer simulation of tracer diffusion in gel network M The number of beads on one chain MSD Mean square displacement N Number of beads of network per dimension Q Total effective charge on beads r Distance between two particles rc Cut-off length rcm Center of mass of a chain ri Position of bead i Rg Radius of gyration of a chain Rn end-to-end distance of a chain Rn end-to-end vector R random force t Time T Temperature Uij Total interaction energy Uel Electrostatic interaction energy Uex Lennard-Jones potential Usp Elastic bond energy v Exponent of scaling laws v velocity ix Chapter tracer diffusivity owing to a high possibility for the formation of larger openings for the tracer to escape from a unit cell to an adjacent one. The electrostatic interaction gives rise to distinct behaviors between similarly- and oppositely-charged cases. The tracer diffusivity for the latter case is not a monotonic function of the network porosity and double layer thickness. This interesting behavior is ascribed to the tracer entrapment by the network beads due to the electrostatic attraction. The entrapment effect is strong at high porosities, because the tracer-bead pair can hardly be interfered by remaining beads, which are far away. At low enough porosity, the tracer diffusivity may also increase with increasing cross-linking. For chain in network, Brownian dynamic simulation is also used to investigate the diffusion and conformation of chain molecules in a gel network based on a coarse-grained bead-spring model. The chain and network are free-draining, and prohibition of connector crossing is not taken into account. Several effects on the chain diffusion and conformation have been examined including the chain flexibility, chain length, network flexibility, network porosity, and charge. For uncharged systems, a decrease in the chain or network flexibility can slow down the chain diffusion. However, the decrease in diffusivity is not noticeable unless the flexibility is high enough. The chain diffusivity declines with decreasing network porosity, while the end-to-end distance and radius of gyration not show significant changes. At low enough porosity, the chain diffusion takes the mode like reptation, evidenced by the change in bond angle distribution. For charged systems, the 155 Chapter chain diffusion exhibits different behaviors between the oppositely and similarly charged cases. For high network porosity, the chain diffusion could be strongly hindered for the oppositely charged cases due to electrostatic entrapment. The chain diffusivity can therefore become a non-monotonic function of porosity, in contrast to the behavior of its similarly charged counterpart. To address the hydrodynamic interaction, Dissipative Particle Dynamics has been used to study the diffusion behavior of a tracer particle in gel network. This method can produce results to show some tendency of tracer diffusion in gel network or dynamics of network itself. The long-time diffusion coefficient of the tracer particle is found to decrease with decreasing network porosity and increasing spring constant. However, the diffusion behavior of tracer particle in uncharged gel network shows a discrepancy between BD and DPD, when the network porosity is low enough. As a relatively ‘soft’ potential implemented in DPD method in contrast to the truncated Lennard-Jones potential representing the excluded volume effect in BD method, the steric effect may not be appropriately modeled in DPD method. However, for the charged gel network where the hydrodynamic interaction appears important, the mesoscopic approach -DPD is promising if a bettermodel for steric effect can be adopted without compromising the advantage of using a larger time step. 156 Chapter 6.2 Recommendations It is important to address two points: the negligence of hydrodynamic interaction in the study of BD simulation, and an improvement for the simulation model in BD or DPD. Although DPD can include the hydrodynamics interaction, it is still a developing method with debate, which could be further improved in the future. Hydrodynamic interaction is long-ranged in nature, and can in principle be examined in BD using available hydrodynamic models, such as the Rotne-Prager tensor, together with the Ewald-sum technique. 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The Journal of Chemical Physics, 122(12), 124905. Zhou, T., and Chen, S.B., 2006. Computer simulations of diffusion and dynamics of shortchain polyelectrolytes. Journal of Chemical Physics, 124, 034904. 170 [...]... dot line is the linear fit of points Figure 3.5 Mean square displacement of the tracer in a 100% cross-linking 54 uncharged network Figure 3.6 Normalized tracer diffusivity versus time for a 100% 56 cross-linking uncharged network with varying spring constant at β= 0.818 (a) and 0.728 (b) xi Computer simulation of tracer diffusion in gel network Figure 3.7 Radial distribution function for uncharged network. .. The normalized diffusivity of tracer in gel network with various 145 xvi Computer simulation of tracer diffusion in gel network porosities Figure 5.6 the normalized diffusivity of tracer in gel network with various 147 flexibilities, porosity of network is 0.886; (a) by DPD method (b) comparison between DPD and BD Figure 5.7 Comparison of tracer diffusivity in uncharged gel network with 149 various porosities... presented in Chapter 2, in which an short overview of computer simulation in polymer gel system with particular emphasis on the previous research works in this field are presented In Chapter 3, we study the tracer diffusion in cross-linked polyelectrolyte gel network Four effects on the diffusion of tracer are investigated by Brownian Dynamics simulation, including the flexibility, cross-linking degree of. .. construct according to the change of surrounding conditions It also can swing due to conflictions by the surrounding molecules (e.g., solvent or the polymer molecules) Consequently, the continual changes of a gel network affect the tracer diffusion in the gel network Second, the diffusion of a tracer in gel network is sensitive to the tracer- gel interaction potential which depends on various surrounding conditions,... prediction of tracer diffusion in gel network: computer simulation Some scientists have successfully obtained the tracer diffusivity in gel network using computer simulation However, computer simulation of a gel system is time-consuming in contrast to the computation for the dilute polymer solution This is because if an atomistic level model is applied to each molecule, there will be millions of atoms... Also, since there is no study about tracers or polymer chains diffusing in gel network 7 Chapter 1 taking into accounts both flexibility and hydrodynamic interaction, we are motivated to investigate the diffusion of tracer particles in a flexible cross-linked network by DPD methods It aims to compare the results of DPD simulation with those of BD simulation, and address the suitability of DPD simulation. .. tracer diffusion in gel network A polymer gel is an elastic cross-linked polymer network with a fluid filling the interstitial space of the network Polymer gels are wet and soft and look like a solid material, but are capable of undergoing large deformations Living organisms are largely made of gels Except for bones, teeth, nails, and the outer layers of skin, mammalian tissues are highly aqueous gel. .. radius of gyration Figure 4.3 Average eigenvalues for the moment of inertia tensor of a 106 diffusing chain in a rigid network with β=0.934 Figure 4.4 Variation of chain diffusivity with the spring constant of the 107 network for β= 0.934 and ks=10 Figure 4.5 Diffusivity of the chain with ks=10 versus the network porosity 108 for a rigid network or a flexible network (k=80) xiv Computer simulation of tracer. .. degree of the network, excluded volume effect, and charge effect To examine the diffusion of polymer chains in gel network, we investigate the effects of chain length, porosity of network and charge effect on the diffusivity and structural properties of the chain in Chapter 4 Chapter 5 presents the DPD simulation of tracer diffusion, and addresses the comparison between BD simulation and DPD simulation, ... and review the recent progress in the study of tracer diffusion in gel network by computer simulation Firstly, we briefly introduce the fundamental roles of computer simulation method and model Secondly, we describe the previous computer simulation works on the network of polymer gel Thirdly, we review several relevant research works about the tracer diffusion in polymer gel We then discuss the previous . diffusivity of tracer in gel network with various 145 Computer simulation of tracer diffusion in gel network xvii porosities Figure 5.6 the normalized diffusivity of tracer in gel network. Computer simulation of tracer diffusion in gel network v SUMMARY Computer simulation is used to study tracer or chain diffusion in polyelectrolyte gels. Owing to the limitation of present. the Computer simulation of tracer diffusion in gel network vi porosity of the network. The difference in diffusion behavior of an oppositely and similarly charged chain in gel network is discussed