Download or read book Investigating Polymer Physics with Single Molecule Experiment and Brownian Dynamics Simulation written by Charles Martin Schroeder and published by . This book was released on 2004 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Studying Topologically Complex DNA at the Single molecule Level written by Wan Yuan Beatrice Soh and published by . This book was released on 2020 with total page 222 pages. Available in PDF, EPUB and Kindle. Book excerpt: Over two decades ago, with advances in microfabrication techniques and fluorescence microscopy, single-molecule studies emerged as a powerful approach to investigate polymer dynamics at the molecular level. By providing a platform for the direct observation and precise manipulation of individual polymer molecules, single-molecule studies allow for the probing of microscopic interactions that give rise to the macroscopic properties of the polymer system. Single molecule studies have been widely used to investigate the static and dynamic properties of double-stranded deoxyribonucleic acid (DNA) as a model polymer. Such studies not only help to develop a fundamental understanding of key topics in polymer physics that cannot be easily accessed via traditional bulk experimental methods, but also facilitate the development of emerging DNA mapping and sequencing techniques. The majority of single-molecule studies to date have involved linear DNA molecules. It is known that topological constraints on the molecular level have a signicant influence on polymer dynamics. A nascent area in the field of polymer physics is the study of polymers with complex topologies. In this thesis, we present a series of single-molecule experiments and Brownian dynamics simulations used to investigate the polymer physics of topologically complex DNA. Specically, we focus on knotted polymers, ring polymers and catenated polymer networks. To investigate the impact of a knot on polymer dynamics, we employ a combined approach of single-molecule experiments and Brownian dynamics simulations. We study experimentally the steady-state behavior of knotted polymers in planar elongational fields and nd that the presence of a knot leads to a faster relaxation time and, accordingly, a shift in the coilstretch transition for the molecule. In consequence, the untying of a knot near the coilstretch transition can give rise to dramatic changes in chain conformation. We use Brownian dynamics simulations to study in detail the impact of the knot untying process on polymer dynamics in planar elongational fields and complement the simulations with experimental results. As a knot moves o the chain in an elongational field, the knot size changes due to the non-uniform tension prole along the chain and causes a change in the eective Weissenberg number, which in turn leads to a change in chain extension. With the use of simulations, we further investigate the knot untying process by probing the topological pathway of an untying knot. We study the distributions of knot conformational states and knot untying pathways on uniformly tensioned chains and chains subjected to elongational fields, and demonstrate that external fields can be used to influence how a knot unties from a chain. Next, we shift focus to ring polymers. We use single-molecule experiments to study the dynamics of self-entangled circular DNA. Our results demonstrate that ring polymers can self-entangle by forming self-threadings, and that such threadings can lead to a signicant slowdown in polymer dynamics. It seems counterintuitive that self-entanglements can arise in ring polymers, which lack chain ends. To delve into the physics of self-entanglements on circular chains, we implement a macroscopic system that allows for the direct visualization of chain conformation. We investigate the formation of self-entanglements on granular chains subjected to a tumbling motion, and use the well-studied self-entanglements on linear chains as a framework for interpreting self-entanglements on circular chains. We develop a method to characterize the self-entanglements on circular chains with known topological descriptors from knot theory and propose a general mechanism for the self-entanglement of circular chains. Finally, we consider the deformation dynamics of catenated DNA networks. A kinetoplast is a complex network of catenated DNA rings that resembles a two-dimensional polymeric system. We perform single-molecule experiments to study the deformation response of kinetoplasts in a planar elongational field. Our results demonstrate that kinetoplasts deform in a stagewise fashion and undergo transient deformation at large strains, as a result of conformational rearrangements from a metastable state. In contrast to linear polymers that display a coil-stretch transition, kinetoplasts do not exhibit an abrupt transition between the non-deformed and deformed states.

Download or read book Polymer Dynamics in Dilute Media written by Shikha Somani and published by Stanford University. This book was released on 2011 with total page 135 pages. Available in PDF, EPUB and Kindle. Book excerpt: Polymers undergo a sharp coil to stretch conformational transition in extension dominated flows when the strain rate exceeds a critical value. Dramatic change in flow behavior is known to occur at the coil-stretch transition, making it useful for several commercial applications. Despite decades of study, this phenomenon remains surrounded with controversy as the effect of solvent properties and fluid flow elements on this transition is not fully understood. In this work, we present a study of the coil-stretch transition and related hysteresis phenomenon using stochastic computer simulations. We first investigate the effect of solvent quality on the coil-stretch transition using Brownian dynamics simulations. Unlike experiments, which are plagued with problems related to polydispersity of polymers and inaccurate control over flow profiles, simulations offer a powerful platform to systematically study the effect of solvent quality while keeping all other parameters in the system constant. The system consists of a polymer subjected to planar elongational flow in both theta solvents and good solvents. The polymer is represented by a bead-spring chain model undergoing elongational flow. Solvent-mediated effects such as fluctuating hydrodynamic interactions (HI) and excluded volume (EV) are included rigorously. Conformational hysteresis is understood in terms of a 1-D energy landscape theory with an activation energy barrier for transition. At steady state, depending upon the flow rate, the energy landscape can either have one or two energy wells. An energy landscape with one well corresponds to the coiled state at low flow rate and stretched state at high flowrate. The double welled landscape corresponds to the hysteretic regime where both coiled and stretched conformational states coexist across the ensemble population. A key factor in determining the effect of solvent quality is the use of a proper measure of solvent quality. In almost all earlier studies, the effect of molecular weight on solvent quality has been neglected, producing inconsistent results. Here, the solvent quality is quantified carefully such that the effect of molecular weight and temperature is taken into account. Contrary to earlier findings, it is observed that with improvement in solvent quality, the chains unravel faster and the critical strain rate at which the coil to stretch transition takes place decreases. Furthermore, the solvent quality has a profound effect on the scaling of the critical strain rate with molecular weight and on both the transient and steady state properties of the system. Universal functions are shown to exist for the observed dynamic and static properties, which will prove useful in determining the operating parameters for experiments. In particular, the ratio of the two different relaxation times (longest relaxation time and zero shear rate viscosity) is found to be a universal function of solvent quality independent of molecular weight. The relaxation times (both the longest relaxation time and the zero shear rate viscosity) increase while the critical strain rate is found to decrease with solvent quality. Next, the study of conformational hysteresis is extended to more complicated 3-D flows to understand the effect of flow vorticity on this phenomenon. Heretofore, there has been no systematic methodology for studying the dynamical interactions between polymer molecules and elementary flow patterns in three-dimensional flows. Such a framework is essential not just for gaining valuable insights into the physics of complex fluids at a fundamental level, but it is also crucial for various important applications like turbulent drag reduction where the underlying physical mechanisms involve dynamical interactions between polymers and turbulence fine scale flow features. Such a study is presented here to provide a framework to interpret complex fluid phenomenon in terms of elementary flow patterns. We investigate the conformational hysteresis using rigorous Brownian dynamics simulations and specifically explore the effect of flow vorticity on the lifetime and width of the hysteresis window in 3-D flows. A systematic procedure is developed with careful eigenvalue analysis to explore the sole effect of vorticity on polymer dynamics keeping the principal strain rate fixed. It is observed that the hysteresis width shrinks due to increase in flow vorticity irrespective of the flow type (bi-extensional, bi-compressional, spiral-inwards, spiral-outwards etc). This is further traced to the alignment of eigenvectors with the principal eigenvector direction leading to enhanced fluctuations. Vorticity is found to have a significant effect on both the transient and the steady state properties. Understanding the effect of vorticity on polymer conformational hysteresis can further help in understanding the fundamental processes in complex flows.

Download or read book On the Simulation and Theory of Polymer Dynamics in Sieving Media written by Martin Kenward and published by . This book was released on 2007 with total page 372 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Handbook of Materials Modeling written by Sidney Yip and published by Springer Science & Business Media. This book was released on 2007-11-17 with total page 2903 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first reference of its kind in the rapidly emerging field of computational approachs to materials research, this is a compendium of perspective-providing and topical articles written to inform students and non-specialists of the current status and capabilities of modelling and simulation. From the standpoint of methodology, the development follows a multiscale approach with emphasis on electronic-structure, atomistic, and mesoscale methods, as well as mathematical analysis and rate processes. Basic models are treated across traditional disciplines, not only in the discussion of methods but also in chapters on crystal defects, microstructure, fluids, polymers and soft matter. Written by authors who are actively participating in the current development, this collection of 150 articles has the breadth and depth to be a major contributor toward defining the field of computational materials. In addition, there are 40 commentaries by highly respected researchers, presenting various views that should interest the future generations of the community. Subject Editors: Martin Bazant, MIT; Bruce Boghosian, Tufts University; Richard Catlow, Royal Institution; Long-Qing Chen, Pennsylvania State University; William Curtin, Brown University; Tomas Diaz de la Rubia, Lawrence Livermore National Laboratory; Nicolas Hadjiconstantinou, MIT; Mark F. Horstemeyer, Mississippi State University; Efthimios Kaxiras, Harvard University; L. Mahadevan, Harvard University; Dimitrios Maroudas, University of Massachusetts; Nicola Marzari, MIT; Horia Metiu, University of California Santa Barbara; Gregory C. Rutledge, MIT; David J. Srolovitz, Princeton University; Bernhardt L. Trout, MIT; Dieter Wolf, Argonne National Laboratory.

Download or read book Polymer Physics Experiments with Single DNA Molecules written by Douglas E. Smith and published by . This book was released on 1999 with total page 462 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Unravelling the Dynamics of Semidilute Polymer Solutions Using Brownian Dynamics written by Aashish Jain and published by . This book was released on 2013 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. Multiphase Flow, 18, 1045. 10. Bhupathiraju, R., P. T. Cummings, and H. D. Cochran, 1996: An efficient parallel algorithm for non-equilibrium molecular dynamics simulations of very large systems in planar Couette flow. Mol.Phys., 88(6), 1665-1670. 11.Todd, B. D. and P. J. Daivis, 1998: Non-equilibrium molecular dynamics simulations of planar elongational flow with spatially and temporally periodic boundary conditions. Phys. Rev. Lett., 81, 1118. 12. Hunt, T. A., S. Bernardi, and B. D. Todd, 2010: A new algorithm for extended nonequilibrium molecular dynamics simulations of mixed flow. J.Chem.Phys., 133(15), 154116.

Download or read book Introduction to Polymer Physics written by Masao Doi and published by Oxford University Press. This book was released on 1996 with total page 148 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a concise and clearly written introduction to the modern theory of polymer physics. The book describes basic concepts and methods of investigating the statistical properties of the assembly of chain-like molecules. The topics discussed include scaling theory, concentration fluctuation, gels, and reptation. Both graduate students and researchers in physics, physical chemistry, chemical engineering, and materials science will find this an extremely useful textbook and reference work.

Download or read book Polymer Physics and Engineering written by and published by Springer. This book was released on 2003-07-01 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Particle Methods for Multi Scale and Multi physics written by Moubin E. T. Al LIU and published by World Scientific. This book was released on 2015-12-28 with total page 400 pages. Available in PDF, EPUB and Kindle. Book excerpt: Multi-scale and multi-physics modeling is useful and important for all areas in engineering and sciences. Particle Methods for Multi-Scale and Multi-Physics systematically addresses some major particle methods for modeling multi-scale and multi-physical problems in engineering and sciences. It contains different particle methods from atomistic scales to continuum scales, with emphasis on molecular dynamics (MD), dissipative particle dynamics (DPD) and smoothed particle hydrodynamics (SPH). This book covers the theoretical background, numerical techniques and many interesting applications of the particle methods discussed in this text, especially in: micro-fluidics and bio-fluidics (e.g., micro drop dynamics, movement and suspension of macro-molecules, cell deformation and migration); environmental and geophysical flows (e.g., saturated and unsaturated flows in porous media and fractures); and free surface flows with possible interacting solid objects (e.g., wave impact, liquid sloshing, water entry and exit, oil spill and boom movement). The presented methodologies, techniques and example applications will benefit students, researchers and professionals in computational engineering and sciences --

Download or read book Polymer Dynamics in Confined and Concentrated Media written by Ajey Krishnamurty Dambal and published by . This book was released on 2009 with total page 432 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Polymer Dynamics in Dilute Media written by Shikha Somani and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Polymers undergo a sharp coil to stretch conformational transition in extension dominated flows when the strain rate exceeds a critical value. Dramatic change in flow behavior is known to occur at the coil-stretch transition, making it useful for several commercial applications. Despite decades of study, this phenomenon remains surrounded with controversy as the effect of solvent properties and fluid flow elements on this transition is not fully understood. In this work, we present a study of the coil-stretch transition and related hysteresis phenomenon using stochastic computer simulations. We first investigate the effect of solvent quality on the coil-stretch transition using Brownian dynamics simulations. Unlike experiments, which are plagued with problems related to polydispersity of polymers and inaccurate control over flow profiles, simulations offer a powerful platform to systematically study the effect of solvent quality while keeping all other parameters in the system constant. The system consists of a polymer subjected to planar elongational flow in both theta solvents and good solvents. The polymer is represented by a bead-spring chain model undergoing elongational flow. Solvent-mediated effects such as fluctuating hydrodynamic interactions (HI) and excluded volume (EV) are included rigorously. Conformational hysteresis is understood in terms of a 1-D energy landscape theory with an activation energy barrier for transition. At steady state, depending upon the flow rate, the energy landscape can either have one or two energy wells. An energy landscape with one well corresponds to the coiled state at low flow rate and stretched state at high flowrate. The double welled landscape corresponds to the hysteretic regime where both coiled and stretched conformational states coexist across the ensemble population. A key factor in determining the effect of solvent quality is the use of a proper measure of solvent quality. In almost all earlier studies, the effect of molecular weight on solvent quality has been neglected, producing inconsistent results. Here, the solvent quality is quantified carefully such that the effect of molecular weight and temperature is taken into account. Contrary to earlier findings, it is observed that with improvement in solvent quality, the chains unravel faster and the critical strain rate at which the coil to stretch transition takes place decreases. Furthermore, the solvent quality has a profound effect on the scaling of the critical strain rate with molecular weight and on both the transient and steady state properties of the system. Universal functions are shown to exist for the observed dynamic and static properties, which will prove useful in determining the operating parameters for experiments. In particular, the ratio of the two different relaxation times (longest relaxation time and zero shear rate viscosity) is found to be a universal function of solvent quality independent of molecular weight. The relaxation times (both the longest relaxation time and the zero shear rate viscosity) increase while the critical strain rate is found to decrease with solvent quality. Next, the study of conformational hysteresis is extended to more complicated 3-D flows to understand the effect of flow vorticity on this phenomenon. Heretofore, there has been no systematic methodology for studying the dynamical interactions between polymer molecules and elementary flow patterns in three-dimensional flows. Such a framework is essential not just for gaining valuable insights into the physics of complex fluids at a fundamental level, but it is also crucial for various important applications like turbulent drag reduction where the underlying physical mechanisms involve dynamical interactions between polymers and turbulence fine scale flow features. Such a study is presented here to provide a framework to interpret complex fluid phenomenon in terms of elementary flow patterns. We investigate the conformational hysteresis using rigorous Brownian dynamics simulations and specifically explore the effect of flow vorticity on the lifetime and width of the hysteresis window in 3-D flows. A systematic procedure is developed with careful eigenvalue analysis to explore the sole effect of vorticity on polymer dynamics keeping the principal strain rate fixed. It is observed that the hysteresis width shrinks due to increase in flow vorticity irrespective of the flow type (bi-extensional, bi-compressional, spiral-inwards, spiral-outwards etc). This is further traced to the alignment of eigenvectors with the principal eigenvector direction leading to enhanced fluctuations. Vorticity is found to have a significant effect on both the transient and the steady state properties. Understanding the effect of vorticity on polymer conformational hysteresis can further help in understanding the fundamental processes in complex flows.

Download or read book Topics in Computational Polymer Physics written by Kamyar Modjtahedzadeh (Graduate student) and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Abstract: Polymer physics studies the mechanical properties and kinetics of monomers and polymers. In this research, we study molecular dynamics and growing self-avoiding walks on basic levels. For the molecular dynamics, we wish to learn how equilibrium poly[n]catenanes behave so we can compare experiments to theory. To do so we simulate [2]catenanes, starting with straightforward simulations and then simulations with more complex collisions. However, before we do that, we make sure that Brownian dynamics algorithms can reproduce known physics to understand the scaling exponents relating physical properties of the polymer systems of interest. When we realize that our Brownian dynamics algorithms work on single polymer chains, we attempt to obtain the equilibrium properties of [2]catenanes and see how hydrodynamics affect them. Moreover, the latter part of this letter; i.e., the growing self-avoiding walk section, is a continuation of a study set about by Wyatt Hooper and Alex Klotz. The growing self-avoiding walk model has only been inspected on a lattice and it is obscure how much of the actual effects are due to the lattice. Therefore, we simulate it off any lattice to find more general principles governing the walk.

Download or read book An Experimental and Theoretical Investigation of Polymer Conformation During Flow Through a Dilute Fiber Bed written by Anthony Richard Evans and published by . This book was released on 1995 with total page 516 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Stochastic Processes in Polymeric Fluids written by Hans C. Öttinger and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book consists of two strongly interweaved parts: the mathematical theory of stochastic processes and its applications to molecular theories of polymeric fluids. The comprehensive mathematical background provided in the first section will be equally useful in many other branches of engineering and the natural sciences. The second part provides readers with a more direct understanding of polymer dynamics, allowing them to identify exactly solvable models more easily, and to develop efficient computer simulation algorithms in a straightforward manner. In view of the examples and applications to problems taken from the front line of science, this volume may be used both as a basic textbook or as a reference book. Program examples written in FORTRAN are available via ftp from ftp.springer.de/pub/chemistry/polysim/.

Download or read book 23rd International Colloquium Tribology written by Arshia Fatemi and published by expert verlag. This book was released on 2022-02-14 with total page 526 pages. Available in PDF, EPUB and Kindle. Book excerpt: The conference provides an international exchange forum for the industry and the academia. Leading university researchers present their latest findings, and representatives of the industry inspire scientists to develop new solutions.

Download or read book Annual Commencement written by Stanford University and published by . This book was released on 2004 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: