Micro-structural evolution of polyethylene (PE) across the glass transition temperature (*T*_{g}) is investigated by full atom molecular dynamic (MD) simulation. The specific volume as a function of temperature for PE is obtained, through which the volumetric glass transition temperature was determined. The dihedral distribution of the overall bonds of the amorphous polyethylene chain system keeps consistent with the rotational isomeric state scheme. The dependence of isomer probability of skeletal bonds and average radius of gyration on temperature also can be used to estimate the glass temperature. The micro-structural information obtained from MD simulation should be helpful to understand glass transition mechanism of polymer system.

**Micro-structural evolution of polyethylene across the glass transition temperature is investigated by full atom molecular dynamic simulation.** The relationship between the specific volume and the temperature can be used to determine glass temperature of amorphous polyethylene system. The dihedral distribution of the overall bonds of the amorphous polyethylene chain system keeps consistent with the rotational isomeric state scheme. The dependence of isomer probability of skeletal bonds and average radius of gyration on temperature also can be used to estimate the glass temperature.

This work presents a mathematical model for the cross-linking copolymerization of styrene-divinylbenzene comprising the phase separation and cyclization kinetics. The gelation dynamics was modeled through the Numerical Fractionation technique, and the phase separation was taken into account through modified Flory's equations. A balance of sequences was used in order to ensure cyclization reactions in the model. The present approach tends to predict the copolymerization's main variables and it was compared with porosity data from literature. The cyclization reactions effect on the phase separation behavior could also be assessed. The cyclization rate coefficient was found to be a function of the solvents system used in the reaction. Further details can be provided by the model through the Numerical Fractionation method which allows one to calculate the concentration of chains from different generations, and their effect upon the polymer particle porosity.

**The gelation dynamics is modeled through the Numerical Fractionation technique, which is applied for gel and sol phases (the numbers represent the chain generations).** The cyclization reactions' rate depends on the degree of polymerization and on the good/poor solvent mixture's composition. These reactions are responsible for delaying the gelation process, affecting the porosity of the polymer particle, as shown in simulation results.

The present study concerns analysis of advective–diffusive transport in prototypical industrial mixing devices by the diffusive mapping method. The latter is a recent extension of the conventional mapping method for distributive mixing and enables application of this efficient technique to mixing flows, including molecular diffusion. Primary challenge is detailed numerical analysis of flow and mixing properties inside realistic systems with complex geometries. This is performed by combining the diffusive mapping method with a sophisticated computational scheme for flow simulations in domains with complex (moving) boundaries: the eXtended Finite Element Method (XFEM). The mixing properties are investigated through spectral analysis of the advective–diffusive transport by way of eigenmode decomposition of the mapping matrix. Key topic is the correlation between the evolution of concentration fields and invariant Lagrangian structures in the flow field.

**Mixing in realistic industrial devices is of great relevance.** The diffusive mapping method is a flexible technique for mixing simulations inside complex geometries. This study applies this method to advective–diffusive transport in two prototypical 3D industrial mixers. It is combined with the extended finite element method (XFEM), enabling reliable and efficient computation of evolving concentration fields.

Structural properties of polyaniline (PANI) protonated with camphorsulfonic acid (CSA) have been investigated by means of molecular dynamics simulations. New OPLS-AA based force field has been applied to calculations done within the Gromacs package. All partial charges were derived from DFT quantum calculations using NWChem package. After testing the force field applied to the simulated macromolecules and ions, we have performed a series of long calculations for the system containing (at least) 42 PANI chains of 7 monomers each, doped with 294 CSA ions (equal amounts of both enantiomers). Our calculations show that the only stable and highly ordered structure of the system studied is characterized by double layers of PANI separated by double layers of CSA. The most important conclusion is that identical final result is reached while starting from several quite different initial arrangements. Here the new model is discussed and justified.

**The structure of polyaniline protonated with camphorsulfonic acid has been modeled by molecular dynamics simulations.** All partial charges have been derived from DFT quantum calculations and a new force field has been applied. The resulting stable structure is characterized by double layers of PANI separated by double layers of CSA. The new model is in good agreement with several experimental data.

Olefin block copolymers (OBCs) are new class of thermoplastic elastomers having low glass transition temperature soft blocks and highly crystalline hard blocks synthesized by reversible chain shuttling between two catalysts with considerably different comonomer responses through a chain transfer agent. Theoretical representation of kinetics and microstructure evolution in chain shuttling polymerization (CSP) is of vital importance especially since the existing characterization tools have severe limitations in retrieving the blocky nature of OBCs. In this work, we correspondingly develop an effective model to represent CSP in practical conditions and compare our predictions to the existing experimental data. We illuminate kinetics and microstructure development in both OBC chains and individual blocks. We specifically clarify the effect of varying the reversibility of transfer reactions through tuning chain shuttling agent and hydrogen concentrations on OBC chains and blocks in terms of their corresponding molecular weight and chemical composition distribution and draw guidelines for achieving OBCs with desired properties.

**An effective model is developed for representation of kinetics and microstructure evolution in the** course of chain shuttling polymerization in the view of both OBC chains and individual hard and soft blocks.

Monte Carlo (MC) methods were applied to the complex kinetic model of butyl acrylate polymerizations. The MC simulator mcPolymer developed in house allows for handling chain-length-dependent termination kinetics. The simulations provide detailed information on the microstructure of each individual polymer chain. For example, the number of short chain branches (SCBs) on each polymer chain and the length of the monomer sequence between two short chain branches are captured. It is shown that the maximum of the branching density distribution is shifted to shorter chain lengths with reaction time. Variation of initiator concentration does not lead to significant changes in the sequence length distributions and branching distributions as long as the same monomer conversion is reached.

The Monte Carlo (MC) simulator mcPolymer is capable of handling of a large number of molecules, allowing for the implementation of a chain-length-dependent termination model. It was applied to the complex kinetic model of butyl acrylate polymerizations, being able to reproduce experimental findings. Furthermore, detailed information on the microstructure of each individual polymer chain was extracted.

The kinetic model of the amphiphilic copolymers with hyperbranched core and linear arms was developed for both semibatch and one-pot polymerization. The molecular size distribution functions of the species obtained and the various molecular parameters were analytically derived. The topological structures and molecular parameters of the products depend on initial mole feed ratios of the reactants. Under the same reaction conditions, the polydispersity index of the products resulted from semibatch is smaller than that of the one-pot process and there are more terminal units on the surface of hyperbranched macroinitiators from semibatch process. Accordingly, the topological structures of the amphiphilic hyperbranched copolymers and the successive self-assembly shapes can be designed by the suitable initial conditions of the polymerization.

**The kinetic model of the amphiphilic copolymers with hyperbranched core and linear arms was developed.** The molecular size distribution functions of the species obtained were analytically derived. Accordingly, the topological structures of the amphiphilic hyperbranched copolymers and the successive self-assembly shapes can be designed.

The generation of fully cross-linked structure is a central task in molecular dynamics simulations of thermosetting polymers. Several algorithms for generating such structures exist, but it is so far not clear what impact the choice of algorithm has on thermal and material properties one typically wants to use simulations to predict. We generated cross-linked systems comprised of stoichiometric amounts of diglycidyl ether of bisphenol A and poly(oxyproplylene) diamine using two methods: (1) a single-step approach, in which cross-link bonds are assigned based on a Monte-Carlo algorithm that minimizes aggregate bond lengths, and (2) a multi-step approach, which uses an incrementally increasing capture radius to identify bonding partners. The choice of cross-linking method has only a minimal impact on thermal and mechanical properties. The minimum nitrogen-to-nitrogen contour-length distributions are also insensitive to the method. However, significant differences were found in the molecular weight distribution of fragments formed by cutting each POP cross-linker between the amines: the single-step method results in fewer, larger fragments compared to the multi-step method. This indicates that the networks formed by the two methods are qualitatively different, and underscores the need for further studies to characterize the influence of polymer network connectivity both in simulations and experiments.

MD simulations with ad-hoc cross-linking algorithms have been used to generate molecular models of fully cross-linked epoxy materials. Two such algorithms are compared, and it is shown that glassy-state thermal and mechanical properties were not significantly influenced by the algorithm choice. This result notwithstanding, it is shown that the two algorithms result in very different network isomers, pointing toward a possible experimental method of structure validation in future work.

General step-growth polymerization systems of order 2 are considered, i.e. systems of type “AfiBgi”. We describe an algorithmic method to calculate the molecular size distribution (MSD). Input to the algorithm is the “recipe”: a list of the monomers involved stating their A and B functionalities and their molar amounts, and the degree of conversion. Output is the MSD and its moments. Three main steps lead from input to output: (i) setting up a polynomial equation for the generating function that generates the MSD, (ii) transforming this polynomial equation to a differential equation, and (iii) transforming the latter one further to a recurrence equation. The recurrence yields the MSD and is of constant order.

From the recipe straight to the MSD. For step-growth polymerized systems of general type “AfiBgi”, a computer algebra method is presented that leads via a few transformation steps from the recipe to the MSD.

A kinetic scheme describing Catalytic Chain Growth (CCG) polymerization was developed and implemented into the computer program PREDICI, by which experimental concentration versus time traces of the participating individual species obtained from online NMR spectroscopy as well as full molecular weight distributions could successfully be modeled. The obtained kinetic coefficients are of high statistical significance. The method was demonstrated on the CCG of styrene-*d*8 in toluene-*d*8 using Cp*_{2}ZrCl_{2} as the catalyst precursor and dibenzyl magnesium as the transfer agent. The quality of the modeling increased dramatically when assuming a chain-length dependent propagation. The extraordinarily good quality of the simulation for all participating species proves that the underlying kinetic model is very likely to be correct. The simulations also demonstrate that this CCG system is a controlled process and that increasing the concentration of the catalyst precursor may lead to a reduced overall polymerization rate.

A kinetic scheme describing Catalytic Chain Growth (CCG) polymerization was developed and implemented into the computer program PREDICI, by which experimental concentration versus time traces of the participating individual species obtained from online NMR spectroscopy as well as full molecular weight distributions could successfully be modeled. The method was demonstrated on the CCG of styrene-d8 in toluene-d8 using Cp*_{2}ZrCl_{2} as the catalyst precursor and dibenzyl magnesium as the transfer agent.

In this study propagation kinetics of free radical polymerization of N-methylacrylamide (NMAAm) is studied with density functional theory calculations. The propagation rate constant ratio of N,N-dimethylacrylamide (DMAAm) and NMAAm (k_{NMAAm}/k_{DMAAm}) is evaluated via model reactions at dimeric stage. The most favorable modes of addition is shown to be determined by the steric effects and hydrogen bonding interactions between the reactive fragments. Gauche and trans orientations are preferred as the least energetic additions in pro-meso and pro-racemo attacks, respectively. Benchmark studies with various density functionals (M05-2X, M06-2X, MPWB1K, BMK) combined with 6-311 + G(3df,2p) basis set assess the calculations. The k_{NMAAm}/k_{DMAAm} ratio obtained in this study is in line with the experimental value. The addition reaction barrier via dimeric associates in case of NMAAm does not yield significant difference than the barrier via monomeric species.

Propagation kinetics in free radical polymerization of N,N-dimethylacrylamide and N-methylacrylamide is modeled with quantum chemical calculations at dimeric model stage. The propagation rate constant ratio of these monomers is calculated with various density functionals. Calculations shed light on the electronic and steric effects and hydrogen bonding interactions within the reactive species that co-play a role in determining the favorable modes of additions.

The linear rheology of unentangled Maxwellian transient networks formed from reversibly associating telechelic polymers can be characterized by a plateau modulus and a shear relaxation time. The concentration dependences of both these properties have been modeled using a variety of theories. An expression is introduced here that allows the concentration dependence of shear relaxation time to be determined (to within a constant, the microscopic chain lifetime) directly from the concentration dependence of the plateau modulus, subject to some approximations. This prediction is tested against experimental results for hydrophobic ethoxylated urethane (HEUR) and against results of simple-model simulations. The nature of the approximations needed to derive this relationship and the scope of its applicability are discussed.

**A simple equation relating the microscopic lifetime of associating chains in a transient network to the shear stress relaxation time,** in terms of the concentration dependence of the plateau modulus, is proposed and tested against published experimental data on associating polymers near the gelation transition and against simulated networks well above the gelation transition.

We examine the simplest relevant molecular model for large-amplitude oscillatory shear flow of a polymeric liquid: The dilute suspension of rigid dumbbells in a Newtonian solvent. We find explicit analytical expressions for the orientation distribution, and specifically for test conditions of frequency and shear rate amplitude that generate higher harmonics in the shear stress and normal stress difference responses. We solve the diffusion equation analytically to explore molecular orientation induced by oscillatory shear flow. We see that the orientation distribution is neither even nor odd. We find zeroth, first, second, third, and fourth harmonics of the orientation distribution, and we have derived explicit analytical expressions for these. We provide a clear visualization of the orientation distribution in large-amplitude oscillatory shear flow in spherical coordinates all the way around one full alternant cycle. The Newtonian distribution is nearly isotropic, the linearly viscoelastic, only slightly anisotropic, and the nonlinear viscoelastic, highly anisotropic.

**We examine the simplest relevant molecular model for large-amplitude oscillatory shear flow of a polymeric liquid: the dilute suspension of rigid dumbbells in a Newtonian solvent.** We find explicit analytical expressions for the orientation distribution, and use these expressions to examine the detailed shape of the orientation distribution with detailed visualizations all the way around one full alternant cycle.

**Front Cover:** The assumption of proportionality between the viscosity and radius-of-gyration branching indices, g' and g, is suggested on the basis of the proposal of Stockmayer and Fixmann and on the data of Kurata and Fukatsu as well as our own, rather than the exponential relation assuming the value of the branching exponent ε = 0.5, based on data of Zimm and Kilb. Further details can be found in the article by M. Netopilík on page 80.

**Back Cover:** Star polymers with arms of two different homopolymers with a weak mutual attraction are investigated using Monte Carlo simulations, and the effect of the polymer architecture on the complexation behavior of the polymers is studied. The star architecture promotes an internal complexation, which is indicated by a reduction of the end-to-end distance of the polymer chains. Further details can be found in the article by Pascal Hebbeker, Felix A. Plamper, and Stefanie Schneider* on page 110.

Randomly branched Gaussian chains, including regular and random stars, were modeled. The chains were generated by repeated random movement of the end-point in a cubic grid. The radius-of-gyration- and viscosity-volume branching indexes, *g* and *g*′, were calculated for randomly branched molecules and compared with the data taken from the literature. The dependence of *g*′ on *g*, constructed from our data as well as from those by Kurata and Fukatsu, appears linear. The experimental data taken from the literature do not contradict this finding. On the other hand, the values of the branching exponent *ϵ* are highly scattered and a reliable average cannot be found. This findings support the value of the branching exponent of *ϵ* = 1, relating *g* and *g*′, proposed also by other authors.

**Numerically calculated values of hydrodynamic branching index in dependence of the radius-of-gyration branching index, g**

For the simple ABC linear terpolymer composed of a solvophilic A block and two solvophobic B and C blocks, the solution-state self-assembly is systematically investigated by dissipative particle dynamics (DPD) simulations, and several complex multicompartment micelles beyond the conventional wisdom, such as helix-on-sphere, cage, ring, bowl, raspberry-onion and so on, are predicted from the simulations. The detailed phase diagrams clearly point out the regions of building these complex multicompartment micelles. Thus, the complex multicompartment micelles can be obtained from the simple linear ABC terpolymers. This work advances the molecular-level understanding of multicompartment micelles from the simple linear ABC terpolymers, which will be useful for the future application of novel micelles.

**Rich multicompartment micelles,** such as raspberry-onion, helix-on-sphere, cage, ring, worm, bowl, can be formed by the self-assembly of the simplest linear ABC terpolymers in solutions, which is beyond the traditional understanding.

Surface-initiated controlled radical polymerization, such as ATRP, has been proven to be a powerful method for preparation of well-controlled functional grafted polymers. There are thousands of experimental works published, by varying polymers, surfaces, and applications. In comparison, theoretical developments are very lacking. Many fundamental questions still remain to be answered. These questions include, but not limited to: What determine the surface initiator efficiency? How many chains per square nanometer can be fully grown? What is the maximum thickness can polymer grow to? If chains are simultaneously grown from surface and solution, which has higher molecular weight? Answers to these questions are most helpful for innovation and further development in this important area. In this work, we employed recently developed theories to explain experimentally observed kinetic profiles of polymer layer thickness growth. What limits the growth of surface chains? Is it the monomer starvation in grafting layer that slow down propagation? Or is it the termination of radicals that stop the chain growth? It was found that no existing models could fully explain the often contradictory experimental observations. It is our hope that this work will provoke further discussions and inspire more effort in resolving the fundamental issues of this area.

This paper discusses a fundamental question: how thick can a polymer layer grow to in surface-initiated ATRP (SI-ATRP)? or equivalently, what limits the growing of chains from surface? Two models developed from existing theories are compared to experiment trends from literature. It is shown that the current understanding of the mechanism behind SI-ATRP is still incomplete and requires further investigations.

Within the broad class of hyperbranched polymers, highly symmetrical objects (such as dendrimers and Vicsek fractals) are of special theoretical interest. Here we study, using the MARTINI force-field, polyamidoamine Vicsek fractals (PVF) in silico, focusing on their structure and dynamics in dilute solution. Our extensive microsecond-long simulations show that the radius of gyration of PVF scales with the molecular weight as *N*^{0.54}, behavior rather close to that of stars and considerably distinct from that of dendrimers. The study of the radial density profiles indicates that different parts of the PVF interpenetrate significantly, fact which stresses the soft and sparse character of PVF. These results are also supported by our findings for the rotational autocorrelation functions.

**Highly symmetrical, deterministic structures are important representatives for hyperbranched macromolecules; in particular, fractal structures cover a broad class of model polymeric systems.** Here, polyamidoamine Vicsek fractals are studied by employing extensive molecular dynamics simulations along with the coarse-grained MARTINI force-field to unravel their structural and dynamic characteristics in dilute solution.

Star polymers with arms of two different homopolymers with a weak mutual attraction have been investigated using a coarse grained polymer model and Monte Carlo (MC) simulations. The effect of the polymer architecture on the complexation behavior has been studied for various numbers of arms (up to 8) starting from a diblock copolymer and compared to diblock copolymers as well as alternating linear copolymers. It was found that the star architecture promotes an internal complexation, which is indicated by a reduction of the end-to-end distance of the polymer chains. The local concentration of the weakly attractive partners is important to promote their interaction and to harvest the attraction. The results were compared to experimental data.

**Polymeric systems consisting of two different weakly attractive components are investigated using Monte Carlo simulations, comparing various architectures.** When arranged in a miktoarm star polymer, complexation is promoted compared to a linear block copolymer and a strictly alternating copolymer.

A Markovian model is proposed to investigate the complex molecular buildup processes during the self-condensing vinyl polymerization. The weight-average chain length is represented by the analytic formula, . By application of the Markovian model to the Monte Carlo method, one can investigate the structure of the polymer formed one by one, and the full distributions of chain lengths, degrees of branching, and molecular dimensions are investigated. When the polymer molecules are fractionated by the chain length, the universal curves for the degree of branching, and the molecular dimension, which do not change during the course of polymerization were discovered.

**A Markovian model is proposed for the self-condensing vinyl polymerization (SCVP), to investigate the full distributions of chain lengths, degrees of branching, and molecular dimensions.** When the polymer molecules are fractionated by the chain length, the universal curves for the degree of branching, and the molecular dimension, which are invariant during the course of polymerization were discovered.

The statistical description of a polymer chain within the nematic system of many chains is considered. The chain entropy and free energy are numerically calculated on the basis of fundamental definitions. No constraints are imposed on the elongation of the chain end-to-end distance, which can change from zero to a value for maximally stretched chains, or on the strength of the internal nematic interactions within the system. Obtained results are compared with those calculated by approximate formulas for varying strength of interactions and length of chains.

Numerical exact calculations show that for long chains the chain end-to-end probability can be Gaussian or non-Gaussian. It depends on the intensity of the orienting interactions inside the system of chains. That result verifies often accepted assumption that long chains are always Gaussian.

Influence of size and shape of simulation box on lamellar morphology in DPD simulation is systematically studied. The 3D Voronoi Tessellation and cluster analysis is employed. It is shown that orientation and number of lamellar planes determines the value of lamellar spacing in box with periodic boundary conditions. The role of initial conditions and relaxation scenario on lamellar morphology is also studied. Scaling the box dimensions and equalizing the diagonal pressure tensor components is used to obtain natural lamellar spacing. Where the estimation of natural lamellar spacing is unclear, the intersection of the off-diagonal components of the pressure tensor is used. Finally, examples where maximum lamellar spacing achieved by scaling of the box is smaller than the natural lamellar spacing are presented.

**Commensuration of lamellar morphology and simulation box** is studied by means of dissipative particle dynamics. The lamellar spacing is found to be influenced by incommensurability effect even for large cubic simulation boxes. The adjusted box dimensions and components of pressure tensor are used for estimating the natural lamellar spacing for generally oriented lamellar planes.

Reactive blending of PET and PEN has been simulated by numerical integration of kinetic differential equations and Monte Carlo (MC) approaches. Accurate modeling of transesterification reaction at different temperatures, times, and PET/PEN feed compositions assured precise control of architecture of PET–PEN copolymer chains. Microstructural features like the number and weight distribution of ethylene terephthalate and ethylene naphthalate segments, molecular weight distribution, and polydispersity were monitored. The developed MC code can satisfactorily capture the extent of reaction, the randomness coefficient of PET–PEN copolymers, and compositional variations as found by ^{1}H-NMR.

**A detailed information on microstructural evolutions during reactive blending of PET and PEN based on fitting of results of numerical integration of differential equations and Kinetic Monte Carlo simulations** to experimental data is reported. A broad understanding of transesterification kinetics which enables precise control of architecture of PET-PEN copolymer chains during reactive blending is provided.