Unravelling the dynamics of semidilute polymer solutions using Brownian dynamics

A polymer solution has three concentration regimes: (i) dilute (ii) semidilute and (iii) concentrated. There are a number of contexts involving polymer solutions, such as in the spinning of nanofi bers or in ink jet printing, where in order to achieve the most optimal outcome the concentration of polymers must be in the semidilute regime. In many biological contexts as well, such as the diffusion of protein and other biomolecules, the essential physics occur in the semidilute regime. Therefore, it is extremely important to understand the behavior of semidilute polymer solutions from the fundamental and also from the technological point of view. A significant amount of research has been carried out in the dilute and concentrated regimes in the past by means of experiments, theories and computer simulations. These two regimes have been explored successfully because the behavior of polymer solutions in the dilute and concentrated regimes can be understood by studying the behavior of single molecules. In the dilute case the motivation for this is obvious, while in the concentrated case, by treating all the molecules that surround a particular molecule as obstacles that constrain its motion, the entire problem is reduced to understanding the motion of a polymer in a tube. This approximation, however, is not valid in the semidilute regime, which lies between the dilute and concentrated regimes, because of all the many-body interactions, that arise in this regime. The main focus of this thesis is to develop an optimized Brownian dynamics (BD) simulation algorithm for semidilute polymer solutions at and far from equilibrium, that is capable of accounting for the many-body interactions. The goal is to use this algorithm to predict various physical properties for a range of concentrations and temperatures and to interpret the results in terms of the blob scaling theory. The development of a BD simulation algorithm for multi-chain systems requires the consideration of a large system of polymer chains coupled to one another through excluded volume interactions (which are short-range in space) and hydrodynamic interactions (which are long-range in space). In the presence of periodic boundary conditions, long-ranged hydrodynamic interactions are frequently summed with the Ewald summation technique (Beenakker, 1986; Stoltz et al., 2006). By performing detailed simulations that shed light on the influence of several tuning parameters involved both in the Ewald summation method, and in the efficient treatment of Brownian forces, we describe the development of a BD algorithm in this thesis, in which the computational cost scales as O(N^{1.8}), where N is the number of monomers in the simulation box. It is also shown that Beenakker's original implementation of the Ewald sum, which is only valid for systems without bead overlap, can be modified so that _ solutions can be simulated by switching off excluded volume interactions. Comparison of the predictions by the BD algorithm of the gyration radius, the end-to-end vector, and the self-diffusion coefficient with the hybrid lattice Boltzmann-Molecular dynamics (LB-MD) method (Ahlrichs and Dunweg, 1999) shows excellent agreement between the two methods. This study has been published in the paper Jain et al. (2012). The behavior of semidilute polymer solutions at equilibrium varies significantly with concentration and solvent quality. These effects are reflected in the concentration driven crossover from the dilute to the concentrated regime, and in the solvent quality driven crossover from theta solvents to good solvents in the phase diagram of polymer solutions. This double crossover region for concentration above the overlap concentration, is explored by Brownian dynamics simulations to map out the universal crossover scaling functions for the gyration radius and the single-chain diffusion constant. Scaling considerations (Rubinstein and Colby, 2003), our simulation results, and recently reported experimental data (Pan, Nguyen, Sunthar, Sridhar & Prakash, Pan et al.) on the polymer contribution to the zero-shear rate viscosity obtained from rheological measurements on DNA systems support the assumption that there are simple relations between these functions, such that they can be inferred from one another. This study has been published in the paper Jain et al. (2012). Unlike the simulation of equilibrium systems where periodic boundary conditions (PBCs) are used in an orthogonal cell to get rid of wall effects, for the simulation of far from equilibrium systems, appropriate PBCs need to be used such that they are compatible with any particular imposed flow. One should also be able to carry out the simulation for an arbitrary amount of time. Commonly, the Lees Edwards PBC (Lees and Edwards, 1972) is used for planar shear flow and the Kraynik-Reinelt PBC (Kraynik and Reinelt, 1992) is used for planar elongational flow. These PBCs have been used and tested in molecular dynamics simulations (Bhupathiraju et al., 1996; Todd and Daivis, 1998) and multi-chain BD simulations (Stoltz et al., 2006). In this thesis PBCs that can handle a planar mixed flow (which is a linear combination of planar elongational flow and planar shear flow) (Hunt et al., 2010) is implemented in a multi-chain BD simulation algorithm for semidilute polymer solutions. Preliminary results on the validation of the planar mixed flow algorithm are presented. References: 1. Beenakker, C. W. J., 1986: Ewald sum of the Rotne-Prager tensor. J.Chem.Phys., 85, 1581-1582. 2. Stoltz, C., J. J. de Pablo, and M. D. Graham, 2006: Concentration dependence of shear and extensional rheology of polymer simulations: Brownian dynamics simulations. J.Rheol., 502, 137. 3. Ahlrichs, P. and B. Dunweg, 1999: Simulation of a single polymer chain in solution by combining Lattice Boltzmann and molecular dynamics. J.Chem.Phys., 111, 8225. 4. Jain, A., P. Sunthar, B. Dunweg, and J. R. Prakash, 2012: Optimization of a Brownian-dynamics algorithm for semidilute polymer solutions. Phys. Rev. E, 85, 066703. 5. Rubinstein, M. and R. H. Colby, 2003: Polymer Physics. Oxford University Press 6. Pan, S., D. A. Nguyen, P. Sunthar, T. Sridhar, and J. R. Prakash Universal solvent quality crossover of the zero shear rate viscosity of semidilute DNA solutions. 2011arXiv1112.3720P. 7. Jain, A., B. Dunweg, and J. R. Prakash, 2012: Dynamic crossover scaling in polymer solutions. Phys. Rev. Lett., 109, 088302. 8. Lees, A. W. and S. F. Edwards, 1972: The computer studies of transport processes under extreme conditions. J. Phys. C: Solid State Phys., 5, 1921-1929. 9. Kraynik, A. M. and D. A. Reinelt, 1992: Extensional motions of spatially periodic lattices. Int. J. 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