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The addition of fillers can significantly improve the mechanical behavior of polymers. The responsible mechanisms at the molecular level can be well assessed by particle-based simulation techniques, such as molecular dynamics. However, the high computational cost of these simulations prevents the study of macroscopic samples. Continuum-based approaches, particularly micromechanics, offer a more efficient alternative but require precise constitutive models for all constituents, which are usually unavailable at these small length scales. In this contribution, we derive a molecular-dynamics-informed constitutive law by employing a characterization strategy introduced in a previous publication. We choose silicon dioxide (silica) as an exemplary filler material used in polymer composites and perform uniaxial and shear deformation tests with molecular dynamics. The material exhibits elastoplastic behavior with a pronounced anisotropy. Based on the pseudo-experimental data, we calibrate an anisotropic elastic constitutive law and reproduce the material response for small strains accurately. The study validates the characterization strategy that facilitates the calibration of constitutive laws from molecular dynamics simulations. Furthermore, the obtained material model for coarse-grained silica forms the basis for future continuum-based investigations of polymer nanocomposites. In general, the presented transition from a fine-scale particle model to a coarse and computationally efficient continuum description adds to the body of knowledge of molecular science as well as the engineering community.

In this contribution, molecular dynamics (MD) simulations in combination with continuum mechanical (CM) approaches are performed to investigate particle movements under uniaxial deformations of an amorphous polymer at particle resolution. A coarse-grained (CG) model of atactic polystyrene is used as an exemplary model system. We propose a hybrid molecular dynamics-continuum mechanical (MD-CM) approach to simulate the deformation behavior of the polymer. As a reference, purely molecular dynamics systems are used. The methods are compared with regard to the local displacement of the particles and the global stress-strain behavior of the overall system. The good reproducibility of the system’s mechanical behavior with the hybrid molecular dynamics-continuum mechanical method is shown. Furthermore, it is demonstrated that CPU time can be significantly reduced with the hybrid calculation model.

This contribution presents a partitioned‐domain particle‐continuum coupling method for amorphous polymers with multiple particle‐based domains. The coupling method treats the particle‐based domains with molecular dynamics (MD) simulations and the continuum domain discretized by the Finite Element (FE) method. In the continuum domain, a viscoelastic‐viscoplastic (VE‐VP) constitutive model derived from MD simulation results of the polymer at molecular resolution is employed. The effects of the minimum distances between the domains, the distribution and the number of the MD domains as well as the strain rates are studied under uniaxial tension. This method is a precursor for multiscale simulations of polymer‐based nanocomposites (PNC).

In this contribution, we present a partitioned‐domain method coupling a particle domain and a continuum domain for multiscale simulations of inelastic amorphous polymers under isothermal conditions. In the continuum domain, a viscoelastic–viscoplastic constitutive model calibrated from previous molecular dynamics (MD) simulations is employed to capture the inelastic properties of the polymer. Due to the material's rate‐dependence, a temporal coupling scheme is introduced. The influence of the time‐related parameters on the computational cost and accuracy is discussed. With appropriate parameters, multiscale simulations of glassy polystyrene under various loading conditions are implemented to showcase the method's capabilities to capture the mechanical behavior of polymers with different strain rates and with non‐affine deformations of the MD domain.

Atomistic-to-continuum coupling methods are used to unravel molecular mechanisms of polymers and polymer composites. These multiscale techniques advantageously combine the computational efficiency of continuum approaches while keeping the accuracy of particle-based methods. The Capriccio method [Pfaller et al.

In the context of fracture simulations of polymers, the molecular mechanisms in the vicinity of the crack tip are of particular interest. Nevertheless, to keep the computational cost to a minimum, a coarser resolution must be used in the remaining regions of the numerical sample. For the specific case of amorphous polymers, the Capriccio method bridges the gap between the length and time scales involved at the different levels of resolution by concurrently coupling molecular dynamics (MD) with the finite element method (FEM). Within the scope of the Capriccio approach, the coupling to the molecular MD region introduces non-periodic, so-called stochastic boundary conditions (SBC). In similarity to typical simulations under periodic boundary conditions (PBC), the SBC MD simulations must reach an equilibrium state before mechanical loads are exerted on the coupled systems. In this contribution, we hence extensively study the equilibration properties of non-periodic MD samples using the Capriccio method. From a thorough parameter study, which in general reveals only minor effects of the considered quantities on the equilibrium state, a best set of parameters follows. The behavior of an exemplary system equilibrated with this parameter set is further studied under uniaxial tension and we observe some peculiarities in view of creep and relaxation phenomena. This raises important questions to be addressed in the further development of the Capriccio method.

The enhanced mechanical behavior of polymer nanocomposites with spherical filler particles is attributed to the formation of matrix-filler interphases. The nano-scale leads to particularly high interphase volume fractions while rendering experimental investigations extremely difficult. Previously, we introduced a molecular dynamics-based interphase model capturing the crucial spatial profiles of elastic and inelastic properties inside the interphase. This contribution demonstrates that our model captures polymer nanocomposites’ essential characteristics reported from experiments. To this end, we thoroughly verify and validate the model before discussing the resulting local plastic strain distribution. Furthermore, we obtain a reinforcement in terms of the overall stiffness for smaller particles and higher filler contents, while the influence of particle spacing seems negligible, matching experimental observations in the literature. This paper proposes a methodology to unravel the underlying complex mechanical behavior of polymer nanocomposites and to translate the findings into engineering quantities accessible to a broader audience and technical applications.

Epoxy resin plasticity and damage was studied from molecular dynamic simulations and interpreted by the help of constitutive modelling. For the latter, we suggested a physically motivated approach that aims at interpolating two well-defined limiting cases; namely, pulling at the vanishing strain rate and very rapid deformation; here, taken as 50% of the speed of sound of the material. In turn, to consider 0.1–10-m/s-scale deformation rates, we employed a simple relaxation model featuring exponential stress decay with a relaxation time of 1.5 ns. As benchmarks, deformation and strain reversal runs were performed by molecular dynamic simulations using two different strain rates. Our analyses show the importance of molecular rearrangements within the epoxy network loops for rationalizing the strain-rate dependence of plasticity and residual stress upon strain reversal. To this end, our constitutive model reasonably reproduced experimental data of elastic and visco-elastic epoxy deformation, along with the maximum stress experienced before fracturing. Moreover, we show the importance of introducing damage elements for mimicking the mechanical behavior of epoxy resins.

This contribution introduces an unconventional procedure to characterize spatial profiles of elastic and inelastic properties inside polymer interphases around nanoparticles. Interphases denote those regions in the polymer matrix whose mechanical properties are influenced by the filler surfaces and thus deviate from the bulk properties. They are of particular relevance in case of nano-sized filler particles with a comparatively large surface-to-volume ratio and hence can explain the frequent observation that the overall properties of polymer nanocomposites cannot be determined by classical mixing rules, which only consider the behavior of the individual constituents. Interphase characterization for nanocomposites poses hardly solvable challenges to the experimenter and is still an unsolved problem in many cases. Instead of real experiments, we perform pseudo experiments using our recently developed Capriccio method, which is an MD-FE domain-decomposition tool specifically designed for amorphous polymers. These pseudo-experimental data then serve as input for a typical inverse parameter identification. With this procedure, spatially varying mechanical properties inside the polymer are, for the first time, translated into intuitively understandable profiles of continuum mechanical parameters. As a model material, we employ silica-enforced polystyrene, for which our procedure reveals exponential saturation profiles for Young’s modulus and the yield stress inside the interphase, where the former takes about seven times the bulk value at the particle surface and the latter roughly triples. Interestingly, hardening coefficient and Poisson’s ratio of the polymer remain nearly constant inside the interphase. Besides gaining insight into the constitutive influence of filler particles, these unexpected and intriguing results also offer interesting explanatory options for the failure behavior of polymer nanocomposites.

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This contribution aims at a systematic investigation of staggered solution schemes for the computation of coupled domains having different resolutions in space, a problem frequently arising in multi-scale modeling of materials. To couple a standard finite element domain with a high resolution atomistic or coarse-grained, i.e. particle-based domain, a so-called bridging domain is considered. In this handshake region a total energy, which is the sum of the weighted energies of both domains, needs to be formulated. Interactions in the particle domain are modeled by potential functions, e.g. a harmonic potential in the simplest case or the Lennard-Jones potential to consider also anharmonic interactions between the particles. The main goal is to separate the computation of finite element and particle domains as much as possible, amongst others to calculate the different domains on several CPUs. In the present work, the governing equations of the coupling method are presented. The energy functions of continuum, particle domain and bridging domain are recapitulated and the coupling constraint is set up. For the sake of simplicity, these relations are reformulated for the case of a one dimensional system. On the one hand, this system is computed monolithically without any separation of domains. On the other hand, various staggered solution schemes are derived systematically. The relevant equations of each scheme are given in detail together with the sequent iteration steps. All staggered schemes are investigated qualitatively, e.g. by their convergence behavior, as well as quantitatively by comparing the staggered solutions with the monolithic solution.

A new simulation technique is introduced to couple a flexible particle domain as encountered in soft-matter systems and a continuum which is solved by the Finite Element (FE) method. The particle domain is simulated by a molecular dynamics (MD) method in coarse grained (CG) representation. On the basis of computational experiences from a previous study, a staggered coupling procedure has been chosen. The proposed MD-FE coupling approximates the continuum as a static region while the MD particle space is treated as a dynamical ensemble. The information transfer between MD and FE domains is realized by a coupling region which contains, in particular, additional auxiliary particles, so-called anchor points. Each anchor point is harmonically bonded to a standard MD particle in the coupling region. This type of interaction offers a straightforward access to force gradients at the anchor points that are required in the developed hybrid approach. Time-averaged forces and force gradients from the MD domain are transmitted to the continuum. A static coupling procedure, based on the Arlequin framework, between the FE domain and the anchor points provides new anchor point positions in the MD-FE coupling region. The capability of the new simulation procedure has been quantified for an atactic polystyrene (PS) sample and for a PS���silica nanocomposite, both simulated in CG representation. Numerical data are given in the linear elastic regime which is conserved up to 3% strain. The convergence of the MD-FE coupling procedure has been demonstrated for quantities such as reaction forces or the Cauchy stress which have been determined both in the bare FE domain and in the coupled system. Possible applications of the hybrid method are shortly mentioned.

Polymers and, in particular, polymer composites are known for the enormous adjustability of their mechanical, chemical, and thermal behavior. Multiscale methods are increasingly employed to unravel the polymer microstructure's impact on the material properties. These methods combine the accuracy of particle-based techniques with the efficiency of continuum mechanical approaches. Amorphous polymers pose a special challenge since their microstructure does not continue periodically, and therefore special attention needs to be paid to the particle domain boundary. In this study, we introduce a coupling via an interface between the continuum and the particle domain. Padding atoms as particle representations of the continuum, which serve as interaction partners for the atoms in the particle region, allow for the transfer of displacements and forces between the domains. We present a straightforward 1D example with simple interactions, evaluate the scheme's performance, discuss the resulting energy contributions, and identify an optimal set of coupling parameters. Eventually, this forms the basis for future 3D implementations.

This contribution presents a phenomenological viscoelastic-viscoplastic constitutive model informed by coarse-grained (CG) molecular dynamics (MD) simulations of pure glassy polystyrene (PS). In contrast to experiments, where viscoplasticity is caused by various effects simultaneously, these effects can be decomposed in MD simulations by adjusting the MD system. In the MD simulations considered here, neither bond breakage nor cross-links are introduced; instead, we focus on the intermolecular interaction of polymer chains. We employ a thermo-dynamically consistent generalized Maxwell framework parallelly comprising an elastic, a viscoelastic, and several elasto-viscoplastic modules with different yield stress to capture the viscoelastic and the viscoplastic mechanical behavior simultaneously. The yield stresses decrease with the maximum deformation the MD system has experienced. The constitutive model presented here is based on 10 material parameters, which can be identified by a few data sets and fits well the CG MD simulations of PS under uniaxial and biaxial deformation with time-proportional and cyclic loading conditions in a wide range of strain rates (0.1%/ns–20%/ns).

This contribution investigates, based on molecular dynamics (MD), the mechanical deformation behavior of polystyrene-silica nanocomposites and focuses on the influence of micromechanical properties as e.g. filler particle size and filler mass fraction. With regard to simulations of macroscopic problems with system sizes not capable by pure MD approaches, our investigations are complemented by hybrid molecular dynamics – finite element (MD-FE) simulations. Our computational approach shows that an increasing total interfacial area between the nanoparticles and the polymer matrix stiffens the nanocomposite. As expected, small nanoparticles have more significant impact on the macroscopic mechanical properties of nanocomposites than large ones. We show that, for the same mass fraction of nanoparticles, the Young’s modulus increases by about 4–5% when the nanoparticle diameter is decreased from 5 to 2 nm. Furthermore, we find that: (i) the end-to-end distances of free polymer chains in the vicinity of nanoparticles are larger than in the bulk; (ii) the addition of nanoparticles slows down the global dynamics of free polymer chains; and (iii) the interphase thickness of nanocomposites is about 1–1.5 nm. Beyond that, we study structural properties at the microscale under uniaxial tension and find that the presence of nanoparticles hinders the orientation of free polymer chains under deformation. This hindrance is more pronounced for small nanoparticles and high mass fractions. Polymer structural descriptors as the chain end-to-end vector and a molecular anisotropy parameter largely change in line with the geometrical transformation of the whole sample.

Nano-filled polymers are becoming more and more important to meet the continuously growing requirements of modern engineering problems. The investigation of these composite materials at the molecular level, however, is either prohibitively expensive or just impossible. Multiscale approaches offer an elegant way to analyze such nanocomposites by significantly reducing computational costs compared to fully molecular simulations. When coupling different time and length scales, however, it is in particular important to ensure that the same material description is applied at each level of resolution. The Capriccio method (Pfaller et al, 2012, 2013), for instance, couples a particle domain modeled with molecular dynamics (MD) with a finite element based continuum description and has been used i.a. to investigate the effects of nano-sized silica additives embedded in atactic polystyrene (PS), cf. Pfaller et al (2016); Liu et al (2017). However, a simple hyperelastic constitutive law is used so far for the continuum description which is not capable to fully match the behavior of the particle domain. To overcome this issue and to enable further optimization of the coupling scheme, the material model used for the continuum should be derived directly from pure MD simulations under thermodynamic conditions identical to those used by the Capriccio method. To this end, we analyze the material response of pure PS under uniaxial deformation using strain-controlled MD simulations (Ries et al, 2019). Analogously, we perform simulations under pure shear deformation to obtain a comprehensive understanding of the material behavior. As a result, the present PS shows viscoelastic characteristics for small strains, whereas viscoplasticity is observed for larger deformations. The insights gained and data generated are used to select a suitable material model whose parameters have to be identified in a subsequent parameter optimization.

We present a web-based software tool for the interactive visualization of stresses around an elliptical hole in an infinite plate under simple tension ( http://www.ltm.fau.de/plate ). The analytical solution of this fundamental problem dates back more than 100 years and has served as a vital contribution in many disciplines of solid mechanics ever since. The presented software provides insight into this important problem by instantly rendering the analytical solution for immediate changes of the dimensions and orientation of the hole, and by allowing the user to zoom closer into regions of interest. The application is openly accessible from modern mobile and desktop devices for the use in education and research and may provide benchmark solutions for numerical methods.

In this contribution, we present a characterization methodology to obtain pseudo experimental deformation data from CG MD simulations of polymers as an inevitable prerequisite to choose and calibrate continuum mechanical constitutive laws. Without restriction of generality, we employ a well established CG model of atactic polystyrene as exemplary model system and simulate its mechanical behavior under various uniaxial tension and compression load cases. To demonstrate the applicability of the obtained data, we exemplarily calibrate a viscoelastic continuum mechanical constitutive law. We conclude our contribution by a thorough discussion of the findings obtained in the numerical pseudo experiments and give an outline of subsequent research activities. Thus, this work contributes to the field of multiscale simulation methods and adds a specific application to the body of knowledge of CG MD simulations.

A recently developed hybrid method is employed to study the mechanical behavior of silica-polystyrene nanocomposites (NCs) under uniaxial elongation. The hybrid method couples a particle domain to a continuum domain. The region of physical interest, i.e., the interphase around a nanoparticle (NP), is treated at molecular resolution, while the surrounding elastic continuum is handled with a finite-element approach. In the present paper we analyze the polymer behavior in the neighborhood of one or two nanoparticle(s) at molecular resolution. The coarse-grained hybrid method allows us to simulate a large polymer matrix region surrounding the nanoparticles. We consider NCs with dilute concentration of NPs embedded in an atactic polystyrene matrix formed by 300 chains with 200 monomer beads. The overall orientation of polymer segments relative to the deformation direction is determined in the neighborhood of the nanoparticle to investigate the polymer response to this perturbation. Calculations of strainlike quantities give insight into the deformation behavior of a system with two NPs and show that the applied strain and the nanoparticle distance have significant influence on the deformation behavior. Finally, we investigate to what extent a continuum-based description may account for the specific effects occurring in the interphase between the polymer matrix and the NPs.

We report on a systematic study of the Coulomb-blockade effects in nanofabricated narrow constrictions in thin (Ga,Mn)As films. Different low-temperature transport regimes have been observed for decreasing constriction sizes: the Ohmic, the single-electron tunneling (SET), and a completely insulating regime. In the SET, complex stability diagrams with nested Coulomb diamonds and anomalous conductance suppression in the vicinity of charge degeneracy points have been observed. We rationalize these observations in the SET with a double ferromagnetic island model coupled to ferromagnetic leads. Its transport characteristics are analyzed in terms of a modified orthodox theory of Coulomb blockade which takes into account the energy dependence of the density of states in the metallic islands.

We present electronic transport measurements of a single wall carbon nanotube quantum dot coupled to Nb superconducting contacts. For temperatures comparable to the superconducting gap peculiar transport features are observed inside the Coulomb blockade and superconducting energy gap regions. The observed temperature dependence can be explained in terms of sequential tunneling processes involving thermally excited quasiparticles. In particular, these new channels give rise to two unusual conductance peaks at zero bias in the vicinity of the charge degeneracy point and allow to determine the degeneracy of the ground states involved in transport. The measurements are in good agreement with model calculations., Comment: 5 pages, 4 figures

We present a microscopic theory of transport through quantum dot set-ups coupled to superconducting leads. We derive a master equation for the reduced density matrix to lowest order in the tunneling Hamiltonian and focus on quasiparticle tunneling. For high enough temperatures transport occurs in the subgap region due to thermally excited quasiparticles, which can be used to observe excited states of the system for low bias voltages. On the example of a double quantum dot we show how subgap transport spectroscopy can be done. Moreover, we use the single level quantum dot coupled to a normal and a superconducting lead to give a possible explanation for the subgap features observed in the experiments published in Appl. Phys. Lett. 95, 192103 (2009)., 18 pages, 20 figures, revised according to published version

The procedure proposed here couples a continuum with a particle region by introducing an overlapping bridging domain where both regions exchange information. Beyond existing methods, the particle region is computed by Molecular Dynamics (MD) at finite temperature, using a code that has been developed by our co-workers of the Theoretical Physical Chemistry Group at TU Darmstadt. It is also used in the wide field of physical chemistry for various approaches. In contrast to MD procedures employing periodic boundary conditions (PBC), stochastic boundary conditions (SBC) allow for the definition of static anchor points. These additional artificial particles are coupled to the continuum part and serve as information transmitter. The continuum is treated classically by the Finite Element Method (FEM) which has been adapted based on the Arlequin and the bridging domain method. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

This contribution investigates schemes to couple domains of different resolutions in space, a problem frequently arising in multi-scale modelling of materials. To couple a standard finite element domain with a high resolution atomistic or molecular, i.e. particle-based domain, a so-called bridging domain is considered, e.g. Xu and Belytschko [1], Zhang, Khare, Lu and Belytschko [2] and Bauman et al. [3]. The main goal is to separate the computation of finite element and atomistic domains as much as possible, amongst others, to calculate the different domains on several CPUs. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)

A scheme is described for performing molecular dynamics simulations on polymers under nonperiodic, stochastic boundary conditions. It has been designed to allow later the embedding of a particle domain treated by molecular dynamics into a continuum environment treated by finite elements. It combines, in the boundary region, harmonically restrained particles to confine the system with dissipative particle dynamics to dissipate energy and to thermostat the simulation. The equilibrium position of the tethered particles, the so-called anchor points, are well suited for transmitting deformations, forces and force derivatives between the particle and continuum domains. In the present work the particle scheme is tested by comparing results for coarse-grained polystyrene melts under nonperiodic and regular periodic boundary conditions. Excellent agreement is found for thermodynamic, structural, and dynamic properties.