The runtime of efficient polynomially bounded class P algorithms will be shifted by a factor while exponential class NP algorithms are only improved by an additive constant.

## Polymer Science: A Comprehensive Reference

Algorithms A 1 , A 2 and A 3 have polynomial run times. An algorithm is said to be efficient if its runtime—which depends on some input N —has a polynomial upper bound. Algorithms A 4 and A 5 on the other hand have no polynomial upper limit. Thus, they are called inefficient. In computer science, the class of problems that can be solved with efficient algorithms i.

Figure 14a. As the set of polynomials is closed under addition, multiplication and composition, P is a very robust class of problems: Combining several polynomial algorithms results into an algorithm which again exhibits a polynomial runtime. Today, it is generally assumed that all problems in P are contained in the class NP , cf. Due to the robustness of the definition of the class P of efficient algorithms, an inefficient algorithm can have a shorter runtime than its efficient counterpart, up to a certain system size N 0. Figure 14b.

Hence, the actual runtime of an algorithm lies somewhere between the worst-case and the average-case runtime behavior, cf. Figure 14c. In particle based simulations with assumed pairwise additive interactions the total force on the particles or atoms in a system depends on the current position of two particles only. If this assumption breaks down and has to abandoned in a simulation model, contributions of more complex non-additive interactions to the total potential have to be considered.

For example, a suitable form of three-body interactions was introduced for the first time by Axilrod and Teller [ ]. Such a potential depends on the position of at least three different particles. This is the main reason why ab initio methods are restricted to very small system sizes. This is also generally true in finite element codes where special care has to be taken when elements start to penetrate each other. Usually one uses so-called contact-algorithms which use a simple spring model between penetrating elements.

The spring forces try to separate the penetrating elements again and the core of the contact algorithm is a lookup-table of element knots which is used to decide whether two elements penetrate each other or not. In principle, one can achieve a further speedup in the execution of a MD program by parallelizing it.

The maximum speedup S f, M of a parallelized code is thus given by:. Analyzing this equation for different pairs of values f, M shows that the actual speedup of a parallelized program is always smaller than the theoretical value as the parallelization itself is expensive. Also, for fixed f , the speedup does not grow linearly with M but approaches a limiting value.

This is particularly important for massive-parallel program implementations with thousands of processors. In this section we review recent numerical applications in the field of shock wave physics based on the numerical methods that have been introduced and discussed in the previous sections. We start our discussion with a succinct introduction into shock wave physics and then focus on modeling polycrystalline solids such as high-performance ceramics.

Results of simulating shock wave propagation in such materials using both, FEM, and a concurrent multiscale particle-based model are presented. Nowadays, shock wave processes and their numerical simulation cover a spectrum that ranges from gas-dynamics of super-sonic objects, over air-blast-waves originating from detonations including their interaction with deformable structures, to the effects of shocks in structures, e.

Shock waves in soft matter have increasingly attracted interest in the field of medical treatment of inflammations or of nephroliths. The specific characteristics distinguishing shock waves from ordinary acoustic waves are the extremely short rise times in the range of nanoseconds, in contrast to microseconds with acoustic waves and their dissipative nature. Reason for the formation of a shock wave is either the super-sonic motion of an object and the related wave-superposition or the pressure-dependency of sound speed which again leads to wave superposition and steepening of the wave front.

An example of the tremendous effect shock waves may have in solids is exhibited in Figure In the vicinity of the impact location the high pressure amplitudes lead to the formation of crater lips under hydrodynamic pressure conditions. As a result, no phase transition of the material occurs, but rather only the high-pressure shock waves are responsible for the lip formation. As the initiated shock travels further into the material, it is reflected at the free surfaces shaping release waves.

At locations where several release waves are super-imposed, a tensile pressure state is established which can lead to instantaneous failure, called spallation. Until the early s, investigations of shock wave formation and propagation were restricted almost completely to gaseous media. Nevertheless, the achievements in gas dynamics set the basis for fundamental work on shock waves in solids.

During the course of his studies on finite-amplitude waves in solids, Riemann [ ] invented the method of characteristics which became the tool of choice for the investigation of wave propagation, until almost a century later von Neumann and Richtmyer [ ] introduced the idea of artificial viscosity AV which refers to the transformation of kinetic energy into heat through the narrow shock transition zone. Although AV was introduced for numerical reasons, it is an elective addition in hydrocodes used to modify a physical process so that it can be more easily computed.

If the AV is too small, velocity oscillations about the correct mean value are observed to develop behind a shock. The proper formulation and magnitude needed for an AV has undergone many refinements over the years and culminated in the method pioneered by Godunov in [ ], in which a local elementary wave solution is used to capture the existence and propagation characteristics of shock and rarefaction waves.

Rankine [ ] and Hugoniot [ , ] set the basics for the thermodynamics of shock waves. A fourth equation is needed to find a solution for the Riemann problem described by Equations 38 — On the other hand, a relation between any other pair of the involved variables can be employed to identify the EOS. It involves an experiment, e. Thus, the measured relation between shock velocity and particle velocity. For most crystalline materials, specifically for metals, relation 41 is linear.

Porosity of materials however, leads to significant non-linearities in the shock-particle velocity relation. Experimental investigations of highly porous and inhomogeneous materials face specific complexity concerning a precise representative velocity measurement. Therefore, meso-mechanical simulation of the shock propagation in composite materials on the basis of known component EOS data have become a useful characterization tool see for example [ 92 ].

Performing a shock experiment leads to the so called principal Hugoniot curve representing all possible thermodynamic states available to a material when loaded by shock waves of various amplitudes. In order to find a mathematical description of high pressure states in its vicinity, the Hugoniot curve is utilized as reference curve. The predictive capability of numerical simulation of shock processes such as high- and hypervelocity impact scenarios strongly depends on the quality of the employed EOS.

A wide spectrum of materials has been characterized experimentally in terms of their high-pressure EOS over the last decades. Shock compression experiments [ , ] as well as, more recently, isentropic compression tests [ ] are well established for the identification of reference curves. A fundamental requirement for the shock wave initiation and its stable propagation is the convexity of the related EOS. Bethe [ ] formulated the following two conditions for the existence of shock waves:.

Understanding the micro-structural features of polycrystalline materials such as high-performance ceramics HPCs or metals is a prerequisite for the design of new materials with desired superior properties, such as high toughness or strength.

### Polymer Viscoelasticity: Basics, Molecular Theories, Experiments and Simulations (2nd Edition)

Thus, for enhancing simulation models used for the prediction of material properties on multiscales, there exists a simultaneous need for characterization and ever more realistic representations of micro structures. On the length scale of a few microns to a few hundred microns, many materials exhibit a polyhedral granular structure which is known to crucially influence their macroscopic mechanical properties. In order to overcome this situation by numerical simulations, a detailed and realistic modeling of the available experimental structures is a basic requirement.

With numerical investigations taking explicitly into account the micro structural details, one can expect to achieve a considerably enhanced understanding of the structure-property relationships of such materials [ , ].

With ceramics, the specific shape and size of these polycrystalline grain structures is formed during a sintering process where atomic diffusion on the nanometer scale plays a dominant role. Usually, the sintering process results in a dense micro structure with grain sizes of up to several hundred micrometers.

It is known that both leads to a dramatic increase in hardness which outperforms most metal alloys at considerably lower weight. Producing very small grain sizes in the making of HPCs below nm results again in decreasing hardness [ ]. Hence, there is no simple connection between grain size and macroscopic hardness of a polycrystalline material. The micro structure of densely sintered ceramics can be considered in very good approximation as a tessellation of R 2 with convex polyhedra, i. Figure 16a. A direct, primitive discretization of the micro-photograph into equal-spaced squares in a 2D mesh can be used for a direct simulation of material properties, cf.

Figure 16b. However, with this modeling approach, the grain boundaries on the micrometer scale have to be modeled explicitly with very small elements of finite thickness. Thus, the influence of the area of the interface is unrealistically overestimated in light of the known fact that grain boundaries, which constitute an area of local disorder, often exhibit only a thickness of a few layers of atoms [ ]. Moreover, a photomicrograph is just one 2D sample of the real micro structure in 3D, hence the value of its explicit rendering is very questionable.

Finally, with this approach there is no 3D information available at all. While experimentally measured micro structures in 3D are generally not available for ceramic materials, only recently first reports about measured micro structures of steel have been published [ , ]. Nevertheless, these experiments are expensive and their resolution as well as the number of measured grains still seem to be poor [ ].

The color code exhibits the pressure profile. A different way of generating micro structures, is based on classical Voronoi diagrams in d -dimensional Euclidean space E d and their duals—the Delaunay triangulations—which both constitute important models in stochastic geometry and have been used in various scientific fields for describing space-filling, mosaic-like structures resulting from growth processes.

Voronoi diagrams are geometric structures that deal with proximity of a set of points or more general objects. Often one wants to know details about proximity: Who is closest to whom? The origin of this concept dates back to the 17th century. In his book on the principles of philosophy [ ], R. Descartes claims that the solar system consists of vortices.

His illustrations show a decomposition of space into convex regions, each consisting of matter revolving round one of the fixed stars.

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Even though Descartes has not explicitly defined the extension of these regions, the underlying idea seems to be the following: Let a space T and a set S of sites p in T be given, together with the notion of the influence a site p exerts on a point x of T. This concept has independently emerged, and proven useful, in various fields of science. Different names particular to the respective field have been used, such as medial axis transform in biology or physiology, Wiegner-Seitz zones in chemistry and physics, domains of action in crystallography, and Thiessen polygons in meteorology.

The mathematicians Dirichlet [ ], and Voronoi [ ] were the first to formally introduce this concept. They used it for the study of quadratic forms; here the sites are integer lattice points, and influence is measured by the Euclidean distance. The resulting structure has been called Dirichlet tesselation or Voronoi diagram , cf. Figure 17a , which has become its standard name today. Voronoi was the first to consider the dual of this structure, where any two point sites are connected whose regions have a boundary in common, cf.

Figure 17b. Later, Delauney [ ] obtained the same by defining that two point sites are connected if and only if they lie on a circle whose interior contains no point of T. After him, the dual of the Voronoi diagram has been denoted Delaunay tesselation or Delaunay triangulation.

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Voronoi tessellations in R 2 have been used in many fields of materials science, e. Ghosh et al. However, these models have major drawbacks such as limitations to two dimensions and a generic nature of the structures as they are usually not validated with actual experimental data. Besides its applications in other fields of science, the Voronoi diagram and its dual can be used for solving numerous, and surprisingly different, geometric problems.

Moreover, these structures are very appealing, and a lot of research has been devoted to their study about one in every 16 papers in computational geometry , ever since Shamos and Hoey [ ] introduced them to this field. The reader interested in a complete overview over the existing literature should consult the book by Okabe et al. In a recent approach to micro structural modeling of polycrystalline solids it was suggested to use power diagrams PDs along with a new optimization scheme for the generation of realistic 3D structures [ ].

PDs are a well studied generalization of Voronoi diagrams for arbitrary dimensions [ ] and have some major advantages over Voronoi diagrams as outlined in [ ]. The suggested optimization is based on the statistical characterization of the grains in terms of the distribution of the grain areas A and the grain perimeters P obtained from cross-section micro-photographs, cf. An important result obtained using this method is that neither the experimental area nor the perimeter distribution obey a Gaussian statistics which is contrary to what was claimed e.

The right picture at the bottom of a and b exhibits a corresponding 2D virtual cut through the 3D PD, i. Clearly, the histograms both show no Gaussian distribution as was claimed, e. The optimization scheme for the generation of realistic polycrystalline 3D structures is based on comparing all polyhedral cells typically at least This comparison is performed for each coordinate axis by generating a set of parallel, equidistant 2D slices typically slices for each of the three coordinate directions through the cube and perpendicular to the respective axis, see Figure For each 2D slice the grain sizes A are calculated and combined into one histogram.

The same is done for the perimeter P. Then, the calculated histograms are compared with the experimental histograms A i exp and P i exp by calculating the first k central moments of the area and perimeter distributions A i and P i , respectively. A figure of merit m of conformity is defined according to which the PDs are optimized [ ]:. The figure of merit m in Equation 46 is first calculated from the initial PD generated by a Poisson distribution of generator points. Using a reverse Monte-Carlo scheme, one generator point is chosen at random, its position modified and m is checked again. If m has decreased, the MC move is accepted, otherwise it is rejected.

In Figure 20 we present the resulting histogram of an optimized PD for Al 2 O 3 and show the time development of the figure of merit m for this sample, following the proposed optimization scheme described above. Optimization scheme as suggested in [ ]. The bar graphs show the respective histograms of experimental data. After and optimization steps, the maximum step size of the reverse MC algorithm changing the position of a generator point was increased, which shows a direct influence on the speed of optimization.

After 1. The inset shows the corresponding time development of m of the perimeter red and area distribution of the third central moment. Having an efficient means to generate realistic polycrystalline structures, they can be meshed and be used for a numerical FEM analysis, cf. For simulations of macroscopic material behavior, techniques based on a continuum approximation, such as FEM or SPH are almost exclusively used.

Figure 21 shows a 3D tile of a meshed PD. In a continuum approach the considered grain structure of the material is typically subdivided into smaller finite elements, e. Tetrahedral elements at the surface can either be cut, thus obtaining a smooth surface, or they can represent a more realistic surface coarseness. Also displayed is an enlarged section of the 3D tetrahedral mesh at the surface of the virtual specimen. Upon failure, the elements are separated according to some predefined failure modes, often including a heuristic Weibull distribution [ , ] which is artificially imposed upon the system.

In a the granular surface structure, its mesh and a detailed augmented section of the mesh at the surface are displayed. Illustration of the multiscale problem. With concurrent FEM methods which include micro structural details, only a very small part of a real system can actually be simulated due to the necessary large number of elements.

Figure taken from [ ]. Figure 22 illustrates the disadvantages and the multiscale problem associated with FEM simulations in which micro structural details are included. On the left, a high-speed camera snapshot of an edge-on impact experiment The enlargements in the middle and on the right show the small size of the region that is actually accessible to FEM analysis in a concurrent multiscale simulation approach. With FEM only a very small part of a macroscopic system can actually be simulated due to the necessary large number of elements.

This is why in FEM simulations of polycrystalline materials, in order to be able to simulate a sufficient number of grains, often only two dimensions are considered in the first place. For most codes, an element number exceeding a few dozen millions is the upper limit which is still feasible in FEM simulations on the largest super computer systems.

More severe, the constitutive equations for the material description which are needed in a phenomenological description, are derived from experiments with idealized load conditions. This often leads to many fit parameters in models, which diminishes their physical value. In addition, FEM generally has many computational problems numerical instabilities when it comes to very large element distortions in the vicinity of the impact region where the stresses, strain rates, and deformations are very large.

The time scale of a multiscale FEM simulation does not a priori fit to the timescale of the experiment; thus, parameter adaptations of the included damage model are necessary but are often unphysical. The multiscale problem associated with FEM simulations described in Figure 22 is further worsened by the fact that the results of FEM analyses of highly dynamic processes are often strongly influenced by mesh resolution and mesh quality [ , ], which, from a physical point of view, is not acceptable, since the physical properties of a system should be invariant to the arbitrarily chosen spatial resolution of the problem.

Snapshots of simulations of the edge-on impact system of Figure 22 using a primitive discretization of the geometry of the system in terms of hexahedral elements. Investigations of materials which involve multiple structure levels, such as nano- and polycrystalline solids, require large ensembles of atoms to accurately reflect the structures on the atomic and microscopic levels. For systems of reasonable size, atomistic simulations are still limited to following the dynamics of the considered systems only on time scales of nanoseconds. Such scales are much shorter than what is needed to follow many dynamic phenomena that are of experimental interest [ 80 , ].

Whether a material under load displays a ductile, metal-like behavior or ultimately breaks irreversibly, depends on the atomic crystal structure and on the propagation of defects in the material. Broken atomic bonds cracks and dislocations are the two major defects determining mechanical properties on the atomic scale.

Molecular dynamics investigations of this type using generic models of the solid state have lead to a basic understanding of the processes that govern failure and crack behavior, such as the instability of crack propagation [ 24 , — ], the dynamics of dislocations [ 33 , 80 , , ], the limiting speed of crack propagation [ 35 , , ], the brittle-to-ductile transition [ 35 , , , ], or the universal features of energy dissipation in fracture [ ]. Most metals are crystalline in nature, i.

When crystals form, they may solidify into either a polycrystalline solid or a single crystal. In a single crystal, all atoms are arranged into one lattice or a crystal structure. The structure of single crystals makes them ideal for studies of material response to shock loading. When a highly ordered material, such as a metal crystal, is put under a planar shock, the crystal is compressed along the direction of the shock propagation, see Figure 24a. This uniaxial response can remain elastic so that, once the disturbance is removed, the lattice will relax back to its original configuration.

However, under high-stress conditions, the configuration of atoms in the lattice may be changed irreversibly. Irreversible changes in phase and the development of defects at the atomic level lead to macroscopic changes, such as plasticity, melting, or solid-to-solid phase transformations. When the dynamic compression is removed, the shock-modified micro structure may influence the formation and growth of voids, cracks, and other processes that may cause the material to fail, see Figure 24b. These atomistic changes can dramatically affect a materials behavior, such as its thermodynamic state, strength, and fracture toughness.

Few data are available on the phase transformations that occur under highly dynamic stress conditions or on the defects and voids that may form and grow as a result. MD methods for typical engineering applications on dislocation dynamics, ductility and plasticity, failure, cracks and fracture under shock loading in solids were extended to large-scale simulations of more than 10 8 particles during the late s by Abraham and Coworkers [ 33 , 34 ], Holian and Lomdahl [ ], Zhou [ ] and others [ , ].

Today, many-particle MD simulations taking into account the degrees of freedom of several billion particles have been simulated in atomistic shock wave and brittle to ductile failure simulations [ — ]. Initially it will compress uniaxially and then relax plastically through defects on the nanoscale, a process known as the one-dimensional to three-dimensional transition. The material may also undergo a structural transformation, represented here as a cubic to hexagonal change.

The transformation occurs over a characteristic time scale. The new phase may be polycrystalline solid or melt. Once pressure is released, the microvoids that formed may grow, leading to macroscopic damage that causes the solid to fail. As the shock wave releases, the voids grow and may coalesce, resulting in material failure. In the following we discuss a recently proposed concurrent multiscale approach for the simulation of failure and cracks in brittle materials which is based on mesoscopic particle dynamics, the Discrete Element Method DEM , but which allows for simulating macroscopic properties of solids by fitting only a few model parameters [ ].

## ISBN 13: 9789814313032

Instead of trying to reproduce the geometrical shape of grains on the microscale as seen in two-dimensional 2D micrographs, in the proposed approach one models the macroscopic solid state with soft particles, which, in the initial configuration, are allowed to overlap, cf. Figure 25a. The overall system configuration, see Figure 25b , can be visualized as a network of links that connect the centers of overlapping particles, cf. Figure 25c. The degree of particle overlap in the model is a measure of the force that is needed to detach particles from each other.

The force is imposed on the particles by elastic springs. This simple model can easily be extended to incorporate irreversible changes of state such as plastic flow in metals on the macro scale. However, for brittle materials, where catastrophic failure occurs after a short elastic strain, in general, plastic flow behavior can be completely neglected. Additionally, a failure threshold is introduced for both, extension and compression of the springs that connect the initial particle network.

By adjusting only two model parameters for the strain part of the potential, the correct stress-strain relationship of a specific brittle material as observed in macroscopic experiments can be obtained. The model is then applied to other types of external loading, e. The particle Model as suggested in [ ]. Here, only two particles are displayed. In the model the number of overlapping particles is unlimited and each individual particle pair contributes to the overall pressure and tensile strength of the solid. Here, only the 2D case is shown for simplicity. In essence, E is indeed independent of N.

When the material is put under a low tension the small deviations of particle positions from equilibrium will vanish as soon as the external force is released. Each individual pair of overlapping particles can thus be visualized as being connected by a spring, the equilibrium length of which equals the initial distance d 0 ij. This property is expressed in the cohesive potential by the following equation:.

The total potential is the following sum:. Failure is included in the model by introducing two breaking thresholds for the springs with respect to compressive and to tensile failure, respectively. If either of these thresholds is exceeded, the respective spring is considered to be broken and is removed from the system. A tensile failure criterium is reached when the overlap between two particles vanishes, i.

Particle pairs without a spring after pressure or tensile failure still interact via the repulsive potential and cannot move through each other. An appealing feature of this model, as opposed to many other material models used for the description of brittle materials, see e. These model parameters can be adjusted to mimic the behavior of specific materials.

The color code displays the coordination numbers: blue 0 , green 4 , yellow 6 , red 8. In the following, some simulation snapshots of applications of the proposed multiscale model adapted from [ ] are displayed. Here, the model is applied to simulate a virtual material specimen of macroscopic dimensions under tensile, shear and impact load. In quasi-static load experiments, the involved physical processes occur on time scales small enough so that the system under investigation is always very close to equilibrium.

During relaxation the particles may move and bonds may break. Equilibrium is reached per definition as soon as no bonds break during consecutive timesteps. After reaching equilibrium, the external strain is increased again and the whole procedure is iterated until ultimate failure occurs. Results of the simulations are displayed in the picture series of Figure 28 , which shows 4 snapshots of the fracture process, in which the main features of crack instability, as pointed out by Sharon and Fineberg [ ] onset of branching at crack tips, followed by crack branching and fragmentation are well captured.

At first, many micro-cracks are initiated all over the material by failing particle pair bonds. These micro-cracks lead to local accumulations of stresses in the material until crack tips occur where local tensions accumulate to form a macroscopic crack. This crack ultimately leads to a macroscopic, catastrophic failure of the model solid, which corresponds very well to the fracture pattern of a brittle material. Similar FEM simulations, see Figure 29 , using about 50 million elements, still exhibit a strong dependence of the number of elements and of element size [ ].

### Monographs, contributions to books

One advantage of the proposed particle model in contrast to FEM models is, that many systems with the same set of parameters and hence the same macroscopic properties but statistically varying micro-structure i. By way of performing many simulations a good statistics for corresponding observables can be achieved. The color code is: Force-free bonds green ; Bonds under tensile load red. Snapshots of a 2D FEM simulation of the fracture process of a PMMA poly-methyl methacrylate plate which is subject to an initial uniform strain rate in vertical direction.

Thus, the model solid has to be artificially pre-notched and it still exhibits a strong dependence on the mesh size and the number of elements. Adapted from [ ]. The results of uni-axial shear simulations are presented in Figure Only the color-coded network of particles is shown: stressfree green , tension red and pressure blue. Starting from the initially unloaded state, the top and bottom layer of particles is shifted in opposite directions. At first, in Figure 30a , the tension increases over the whole system. Then, as can be seen from Figure 30b and c , shear bands form and stresses accumulate, until failure due to a series of broken bonds occurs.

The color code is the same as in Figure 28 except for particle bonds under pressure which are coded in blue. These oxide and non-oxide ceramics represent two major classes of ceramics that have many important applications. The impactor hits the target at the left edge. This leads to a local compression of the particles in the impact area. The top series of snapshots in Figure 31a shows the propagation of a shock wave through the material. The shape of the shock front and also the distance traveled by it correspond very well to the high-speed photographs in the middle of Figure 31a.

These snapshots were taken at comparable times after the impact had occurred in the experiment and in the simulation, respectively. After a reflection of the pressure wave at the free end of the material sample, and its propagation back into the material, the energy stored in the shock wave front finally disperses in the material. One can study in great detail the physics of shock waves traversing the material and easily identify strained or compressed regions by plotting the potential energies of the individual pair bonds.

Also failure in the material can conveniently be visualized by plotting only the failed bonds as a function of time, cf. A simple measure of the degree of damage is the number of broken bonds with respect to the their total initial number. Figure 31b exhibits the results of this analysis.

For all impact speeds the damage in the SiC-model is consistently larger than in the one for Al 2 O 3 which is also seen in the experiments. Results of a simulation of the edge-on-impact EOI geometry, cf. Figure 23 , except this time, the whole macroscopic geometry of the experiment can be simulated while still including a microscopic resolution of the system. The impactor is not modeled explicitly, but rather its energy is transformed into kinetic energy of the particle bonds at the impact site.

The time interval between the high-speed photographs is comparable with the simulation snapshots above. Thermodynamic and Kinetic Polymerizability 4. Equilibrium Copolymerization in Ring-Opening Polymerization 4. Organocatalyzed Ring-Opening Polymerizations 4. Stereoselective Ring-Opening Polymerization of Epoxides 4. Ring-Opening Polymerization of Cyclic Acetals 4.

ROP of Cyclic Esters. Mechanisms of Ionic and Coordination Processes 4. Polymerization of Oxazolines 4. Radical Ring-Opening Polymerization 4. High-Molecular-Weight Poly ethylene oxide 4. Nonlinear Macromolecules by Ring-Opening Polymerization 4. Chain Extension by Ring Opening 4. Ring-Opening Dispersion Polymerization 4. Oligomeric Poly ethylene oxide s. Functionalized Poly ethylene glycol s. PEGylation 4. Polyphosphoesters 4. Ring-Opening Polymerization of Cyclic Esters 4.

## Linear Viscoelasticity from Molecular Dynamics Simulation of Entangled Polymers | Macromolecules

Template Polymerization 4. Mechanistic Aspects of Solid-State Polycondensation 4. Radical Polymerization at High Pressure 4. Electroinitiated Polymerization 4. Photopolymerization 4. Frontal Polymerization 4. Microwave-Assisted Polymerization 4. Polycondensation 5. Introduction and Overview 5. Sequence Control in One-Step Polycondensation 5. Nonstoichiometric Polycondensation 5. Chain-Growth Condensation Polymerization 5. Oxidative Coupling Polymerization 5. Advances in Acyclic Diene Metathesis Polymerization 5.

Enzymatic Polymerization 5. Nonlinear Polycondensates 5. Post-Polymerization Modification 5. Supramolecular Polymers 5. Chemistry and Technology of Step-Growth Polyesters 5. Biodegradable Polyesters 5. Polycarbonates 5. Aromatic Polyethers, Polyetherketones, Polysulfides, and Polysulfones 5. Chemistry and Technology of Polyamides 5. Lyotropic Polycondensation including Fibers 5. Polyimides 5. High-Performance Heterocyclic Polymers 5. Metal-Containing Macromolecules 5. Phosphorus-Containing Dendritic Architectures 5.

Epoxy Resins and Phenol-Formaldehyde Resins 5. High-Temperature Thermosets 5. Macromolecular Architectures and Soft Nano-Objects 6. Introduction 6. Synthesis and Properties of Macrocyclic Polymers 6. Polymers with Star-Related Structures 6. Dendrimers 6. Hyperbranched Polymers 6. Molecular Brushes 6. Spherical Polymer Brushes 6. Model Networks and Functional Conetworks 6. Polymer Nanogels and Microgels 6. Controlled Composition 6. Well-Defined Block Copolymers 6. Graft Copolymers and Comb-Shaped Homopolymers 6. Synthetic—Biological Hybrid Polymers 6. Dynamic Supramolecular Polymers 6.

Stereocontrolled Chiral Polymers 6. Rigid—Flexible and Rod—Coil Copolymers 6. Nanostructured Polymer Materials and Thin Films 7. Introduction 7. Block Copolymers in the Condensed State 7. Block Copolymer Thin Films 7. Block Copolymers under Confinement 7. Assemblies of Polymer-Based Nanoscopic Objects 7. Hybrid Polymer—Inorganic Nanostructures 7. Nanostructured Electrospun Fibers 7. Soft Lithographic Approaches to Nanofabrication 7. Nanoimprint Lithography of Polymers 7. Modeling Mixtures of Nanorods and Polymers 7. Sterically Stabilized Nanoparticles in Solutions and at Interfaces 7.

Electrical Conductivity of Polymer Nanocomposites 7. Polymer Dynamics in Constrained Geometries 7. Polymer Nanomechanics 7. Polymers for Advanced Functional Materials 8. Introduction — Applications of Polymers 8. Photoresists and Advanced Patterning 8. Rapid Prototyping 8. Polymer-Based Sensors 8. Electroactive Liquid Crystalline Polymers 8. Nanocomposites and Hybrid Materials 8. Polymer Photonics 8. Optical Fibers 8. Adhesives and Sealants 8.

Polymer Membranes 8. Use the link below to share a full-text version of this article with your friends and colleagues. Learn more. If you have previously obtained access with your personal account, Please log in. If you previously purchased this article, Log in to Readcube. Log out of Readcube.

Click on an option below to access. Log out of ReadCube. We indicate how this basic result can be derived from chaotic, interacting Hamiltonian systems which include densely packed polymer molecules. Recent quasielastic neutron scattering experiments and molecular dynamics simulations are discussed and the results are shown to support this result as well.

An application of the coupling model to find how the viscoelasticity of a polymer depends on the chemical structure of the monomer through the coupling parameter of the local segmental motion is given to illustrate the utility of the model. Volume 90 , Issue 1. The full text of this article hosted at iucr. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username.