A numerical implementation of two-phase flows of Newtonian fluids with a non-linear viscoelastic interface is validated and applied to the case of uniaxial extension of a drop in a matrix fluid. Second-order convergence in space and time is obtained with a Lagrangian-based interface tracking finite element method. The flow problem is analysed using dimensionless groups based on the relative magnitudes of the viscoelastic interfacial extra stress, the interfacial tension and the viscous stress of the bulk fluids. After fitting the intrinsic viscoelastic stress-strain behaviour of interfaces in shear to experimental results from the literature, the influence of interfacial rheology on the drop shape and interfacial stress is investigated. The drop shape is not significantly influenced by the viscoelastic properties of the interface if the interfacial viscoelastic stress, interfacial tension and bulk viscous stress are of the same order of magnitude. However, the interface develops distinct stress profiles for varying interfacial viscoelastic properties. For relatively large viscoelastic interface stress compared to interfacial tension and bulk viscous stress, simulations become unstable. Eventually, the interfacial viscoelastic stress exceeds the interfacial tension and bulk viscous stress, and possibly buckling of the interface occurs, caused by compressive stresses at the drop tip.

The elastic properties of a and /i crystals of isotactic polypropylene are obtained by molecular dynamics simulations of a crystalline domain resembling an infinite crystal. The helical polymer chains are modeled with an all-atom force-field. The assessment of the pristine a2 lattice agreed with published results of structure, density, thermal expansion, and stiffness. Monoclinic a systems with particular imperfections are sampled to assess the effect of defects on conformational stability and elastic properties. The sensitivity of elastic moduli with temperature correlates with the helicity disruption: the more chain-conformational defects, the stronger the decrease inelastic modulus. The non-perfect chiral up-down a1 lattice and the arrangement with one vacancy also display a lower stiffness than a 2 , which can be attributed to a less dense crystal and decreased inter-chain cooperative forces due to periodicity disruption. Two variations of the metastable /i modification were sampled to assess the most energetically favorable configurations. Then, both were subjected to the same procedure, validated first for a . A trigonal mono-chiral system, /i2, and astable bi-chiral one with a four-chain frustrated orthorhombic cell, /i1, were found, both presenting novel characteristics. Both /i structures display a less stable conformation than a 2 , observed by a higher specific volume, a lower transition temperature, and amore significant dependence of the elastic moduli with temperature. Remarkably, the mono-chiral /i2 crystal showed higher elastic modulus than any other crystal below room temperature, related to a more efficient global methyl interlocking between chains. The fact that the experimental value for the density of the /i-kind of crystal is in between the values that we obtained from the simulations of /i2 and /i1 implies that the experimental observations consist of both of these /i crystals, where /i1 might work as an interface between monochiral /i2 layers.

The Eulerian distortion field is an essential ingredient for the continuum modeling of finite elastic and inelastic deformations of materials; however, its relation to finer levels of description has not yet been established. This paper provides a definition of the Eulerian distortion field in terms of the arrangement of the constituent microscopic particles, which is beneficial for fundamental studies as well as for the analysis of computer simulations, e.g., molecular dynamics simulations. Using coarse graining and nonequilibrium thermodynamics, the dynamics of the Eulerian distortion field is examined in detail and related to the underlying dynamics of the particles. First, the usual kinematics of the distortion and the known expression for the Cauchy stress tensor are recovered. And second, it is found that the Mandel stress and the plastic deformation-rate tensor in the natural configuration constitute the relevant force–flux pair for the relaxation of the distortion. Finally, the procedure is illustrated on two examples, namely on an amorphous solid and on a crystalline solid with one slip system.

An out-of-equilibrium simulation method for tracking the time evolution of glassy systems (or any other systems that can be described by hopping dynamics over a network of discrete states) is presented.

The effects of temperature, pressure, and imposed strain on the structural transition pathways of glassy atactic polystyrene (aPS) are studied for a wide range of conditions. By employing an atomistic description of the system, we systematically explore its free energy landscape, emphasizing connections between local free energy minima. A triplet of two minima connected to each other via a first-order saddle point provides the full description of each elementary structural relaxation event. The basis of the analysis is the potential energy landscape (PEL), where efficient methods for finding saddle points and exploring transition pathways have been developed. We then translate the stationary points of the PEL to stationary points of the proper free energy landscape that obeys the macroscopically imposed constraints (either stress- or strain-controlled). By changing the temperature under isobaric conditions (i.e., Gibbs energy landscape), we probe the temperature dependence of the transition rates of the subglass relaxations of aPS, thus obtaining their activation energies by fitting to the Arrhenius equation. The imposition of different strain levels under isothermic conditions allows us to estimate the apparent activation volume of every elementary transition. Our findings are in good agreement with experimental observations for the same system, indicating that both length- and time-scales of the structural transitions of glassy aPS can be obtained by proper free energy minimization of atomistically detailed configurations.

A framework is presented for the formulation of a class of continuum constitutive models for sharp interfaces with non-linear viscoelastic behaviour due to a considerable isotropic interfacial microstructure. For the formulation of a thermodynamically consistent elastoviscoplastic interface constitutive model we adapt an approach successful in describing the behaviour of bulk polymer glasses. The model has a clear separation between dilatation and shear, and is used to predict phenomena related to the plasticity of interfaces observed in the experimental literature, which is relevant for many applications. Stress-strain predictions in standard interfacial rheological flows, i.e. shear and dilatation, are investigated numerically. A predominantly elastic response is obtained at small deformations, with a transition to primarily plastic flow at high stress levels. In interfacial shear flow, strain softening and eventually a plastic plateau occur upon further deformation beyond the yield point. The yield stress and strain and (the relative strength of) the stress overshoot in interfacial shear flow are shown to be controlled by two dimensionless groups of parameters in the model. In interfacial dilatation, the model predicts elastoviscoplastic behaviour with a stress maximum and a decreasing stress without a plateau at even larger deformations. These phenomena are studied for various choices for the parameters in the model.

For suspensions with electromagnetic particles exposed to an external field, we examine the effect of the solidification of the suspending medium on the formation of particle structures, representative of the curing of a photo-reactive resin during stereolithography. To that end, Brownian Dynamics (BD) simulations are examined in which the solidification of the suspending medium can be incorporated by increasing its viscosity in the course of time. For illustrative purposes, it is assumed that the viscosity function is known apriori in explicit and parametrized form, however, experimental data can be used as well. It is demonstrated that one can study the effects of the viscosity increase by a transformation of time, akin to the time–temperature superposition principle, but here also in the presence of thermal noise on the suspended particles. Therefore, instead of performing BD simulations with a continuously increasing viscosity (computationally inefficient), we advocate performing simulations at a constant (low) viscosity and subsequently transform time (nonlinearly) for re-interpretation of the simulation results. So doing, one can predict the formation of particle structures during on-going solidification of the suspending medium. In practice, the viscosity increase is so drastic that further evolution of the particle structure can be considered as arrested after the characteristic transition-time of the viscosity is reached. Semi-quantitative rules of thumb are formulated for the 3D-printing practitioner. Graphical Abstract

The effect of confinement on the conformation of N dumbbells in D dimensions close to a non-interacting and rigid flat wall is examined. Using statistical mechanics and numerical calculations, the partition coefficient and the confinement-induced change in the configurational entropy are calculated as a function of the conformation tensor [Formula: see text] and of the distance of the dumbbells from the wall. Analytical predictions and numerical results for [Formula: see text] concerning the behavior close to the limiting cases (onset of and saturation of confinement) agree favorably; in one case where an analytical prediction has not been achieved, a thorough numerical study establishes the limiting behavior nevertheless. Beyond these limiting cases, the overall behavior of the partition coefficient and the configurational entropy has been examined as well in detail, for various choices of the parameters. Furthermore, it is shown that the effect of confinement for [Formula: see text] is captured entirely by the partition coefficient determined for [Formula: see text]. In general, the average extension of the dumbbells in the direction perpendicular to the wall is decreased the closer the dumbbells are to the wall. Also, the decay of the partition coefficient with increasing extension of the dumbbells becomes steeper, i.e., more localized, the higher the number of dumbbells N. Finally, it is discussed under what conditions these results can be used also for the case of slab- (i.e., slit-) confinement.

Transition pathways on the energy landscape of atactic polystyrene (aPS) glassy specimens are probed below its glass-transition temperature. Each of these transitions is considered an elementary structural relaxation event, whose corresponding rate constant is calculated by applying multidimensional transition-state theory. Initially, a wide spectrum of first-order saddle points surrounding local minima on the energy landscape is discovered by a stabilized hybrid eigenmode-following method. Then, (minimal-energy) "reaction paths" to the adjacent minima are constructed by a quadratic descent method. The heights of the free energy, the potential energy, and the entropy barriers are estimated for every connected triplet of transition state and minima. The resulting distribution of free energy barriers is asymmetric and extremely broad, extending to very high barrier heights (over 50 k(B)T); the corresponding distribution of rate constants extends over 30 orders of magnitude, with well-defined peaks at the time scales corresponding to the subglass relaxations of polystyrene. Analysis of the curvature along the reaction paths reveals a multitude of different rearrangement mechanisms; some of them bearing multiple distinct phases. Finally, connections to theoretical models of the glass phenomenology allows for the prediction, based on first-principles, of the "ideal" glass-transition temperature entering the Vogel-Fulcher-Tammann (VFT) equation describing the super-Arrhenius temperature dependence of glassy dynamics. Our predictions of the time scales of the subglass relaxations and the VFT temperature are in favorable agreement with available experimental literature data for systems of similar molecular weight under the same conditions.

The structure formation of particles with induced dipoles dispersed in a viscous fluid, under a spatially and temporarily uniform external electric or magnetic field, is investigated by means of Brownian Dynamics simulations. Dipole–dipole interactions forces, excluded volume forces and thermal fluctuations are accounted for. The resulting structures are characterized in terms of average orientation of their inter-particle vectors (second Legendre polynomial), network structure, size of particle clusters, anisotropy of the gyration tensor of every cluster and existence of (cluster) percolation. The magnitude of the strength of the external field and the volume fraction of particles are varied and the structural evolution of the system is followed in time. The results show that the characteristic timescale calculated from the interaction of only two dipoles is also valid for the collective dynamics of many-particle simulations. In addition, the magnitude of the strength of the external field in the range of values we investigate influences only the magnitude of the deviations around the average behavior. The main characteristics (number density of branch-points and thickness of branches) of the structure are mainly affected by the volume fraction. The possibility of 3D printing these systems is explored. While the paper provides the details about the case of an electric field, all results presented here can be translated directly into the case of a magnetic field and paramagnetic particles.

The effect of time-dependent external fields on the structures formed by particles with induced dipoles dispersed in a viscous fluid is investigated by means of Brownian Dynamics simulations. The physical effects accounted for are thermal fluctuations, dipole-dipole and excluded volume interactions. The emerging structures are characterised in terms of particle clusters (orientation, size, anisotropy and percolation) and network structure. The strength of the external field is increased in one direction and then kept constant for a certain amount of time, with the structure formation being influenced by the slope of the field-strength increase. This effect can be partially rationalized by inhomogeneous time re-scaling with respect to the field strength, however, the presence of thermal fluctuations makes the scaling at low field strength inappropriate. After the re-scaling, one can observe that the lower the slope of the field increase, the more network-like and the thicker the structure is. In the second part of the study the field is also rotated instantaneously by a certain angle, and the effect of this transition on the structure is studied. For small rotation angles ( $$\theta \le 20^{{\circ }}$$ ) the clusters rotate but stay largely intact, while for large rotation angles ( $$\theta \ge 80^{{\circ }}$$ ) the structure disintegrates and then reforms, due to the nature of the interactions (parallel dipoles with perpendicular inter-particle vector repel each other). For intermediate angles ( $$20<\theta <80^{{\circ }}$$ ), it seems that, during rotation, the structure is altered towards a more network-like state, as a result of cluster fusion (larger clusters). The details provided in this paper concern an electric field, however, all results can be projected into the case of a magnetic field and paramagnetic particles.

The relevant parts of the microstructure participating in the plastic deformation of glassy polystyrene are identified. This is achieved by performing quasi-static deformation simulations of an atomistic representation of polystyrene, within a thermodynamic framework, under experimentally realistic boundary conditions (controlled compressive strain, atmospheric pressure in the lateral directions, and room temperature). In particular, it is shown that discontinuities in the free energy in the course of deformation are representative of microscopic plastic events, which are strongly correlated with the heterogeneous non-affine deformation of the phenyl-rings. However, despite the induced anisotropy during deformation, the phenyl-rings do not show any preferred orientation during deformation. The rearrangements of the microstructure during the discontinuities are analyzed further in terms of a local best-fit deformation gradient tensor (a modified version of the procedure introduced by Falk and Langer in 1998) as well as the stretch- and rotation-components of the local deformation of clusters of atoms that show significant non-affine deformation. This analysis shows that the local deformation is highly heterogeneous, and locally the strains and rotations can be orders of magnitude larger than the imposed deformation, giving rise to plastic deformation of the material.

An elasto-viscoplastic constitutive model was previously developed and successfully used to describe the anisotropic time-dependent response of short fiber reinforced thermoplastics. It is demonstrated that when it is applied to systems with a higher level of anisotropy, the apparent anisotropy of the rate-dependent yield stress deviates from the Hill parameters that are used to define it. This becomes more apparent if the level of anisotropy is increased. The origin of this discrepancy is mathematically investigated and shown to be related to the viscoplastic multiplier of the associated flow rule employed. A solution is proposed that completely restores consistency and extends the applicability of the model to systems with high anisotropy.

A methodology for the characterization of particle structures, especially networks, is developed. This scheme combines 3D image analysis techniques with graph theory tools for the simplification of a structure of thick agglomerates to its skeleton. The connectivity graph of the initial structure is compared with the one of the corresponding skeleton, as a measure of simplification. Examples are used to illustrate the effectiveness of our scheme. Particle structures obtained by Brownian Dynamics simulations are characterized qualitatively and quantitatively. Instead of looking at the characteristics of the structure at the level of the individual particles or neighborhoods of particles, our scheme results in quantitative measures of the network, e.g. the number density of branch-points, the degree of branch-points, and the thickness and the orientation of the branches.

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Two alternative routes are taken to derive, on the basis of the dynamics of a finite number of dumbbells, viscoelasticity in terms of a conformation tensor with fluctuations. The first route is a direct approach using stochastic calculus only, and it serves as a benchmark for the second route, which is guided by thermodynamic principles. In the latter, the Helmholtz free energy and a generalized relaxation tensor play a key role. It is shown that the results of the two routes agree only if a finite-size contribution to the Helmholtz free energy of the conformation tensor is taken into account. Using statistical mechanics, this finite-size contribution is derived explicitly in this paper for a large class of models; this contribution is non-zero whenever the number of dumbbells in the volume of observation is finite. It is noted that the generalized relaxation tensor for the conformation tensor does not need any finite-size correction.

Fluctuating viscoelasticity for conformation-tensor-based models is studied at equilibrium, in simple-shear deformation, and in uniaxial extension. The models studied are the upper-convected Maxwell model, the FENE-P model with finite chain-extensibility, and the Giesekus model with anisotropic drag. Using numerical simulations, the models are compared in detail both with each other and with analytical predictions for the Maxwell model. At equilibrium, the models differ only marginally, both in terms of static and dynamic characteristics. When deformed, the average mechanical response of the Maxwell model is unaffected by the strength of thermal fluctuations, while the mechanical response of the FENE-P and Giesekus models show a slight decrease the stronger the fluctuations in simple shear, whereas the decrease in uniaxial extension is marginal. For all models, the standard deviation of the mechanical response increases with increasing strength of fluctuations, and the magnitude of the standard deviation relative to the average for given fluctuation strength generally decreases the stronger the deformation, this effect being stronger for uniaxial extension than for simple-shear deformation.

A rigorous thermodynamic framework is developed for performing free energy calculations of polymer glasses described by classical molecular forcefields. The proper free energy connected to all combinations of imposed external conditions (strain, stress) is derived from a well defined Helmholtz energy calculated from the detailed atomistic configurations. The quasi-harmonic approximation (QHA) is employed around local minima of the system on its potential energy landscape. The ingredients of the methodology, i.e. the derivatives of the free energy with respect to atomic positions (forces, elements of the Hessian) and the relevant derivatives with respect to strain (stresses and elastic constants), are calculated through tractable analytic expressions which are thoroughly presented and discussed. The derivation is general and readily applicable to any system described by classical molecular forcefields. As a proof of concept, we study the mechanical response of long-chain glassy atactic polystyrene, in the elastic and the plastic regime (compressive strain up to 100%). The macroscopic stress–strain curve for polystyrene is obtained and the evolution of the microscopic polymer configurations in the course of deformation is studied.

Capturing the behavior of complex materials requires connecting dynamics on multiple scales.

The ageing kinetics of amorphous atactic (a-PS), isotactic (i-PS), and syndiotactic (s-PS) polystyrene were studied by means of flash-differential scanning calorimetry. The specimens were aged for up to 2 h at six different ageing temperatures: the optimum ageing temperature, that is, the temperature at which the enthalpy overshoot at the glass transition is maximal for the given elapsed time, and five ageing temperatures ranging from 20 to 80 K below the optimum ageing temperature. A logarithmic increase of the enthalpy overshoot with ageing time is observed for specimens at their optimum ageing temperatures. For temperatures significantly lower than the optimum, there is a range where the enthalpy overshoot is constant, but for higher temperatures (still below the optimum), a logarithmic increase is also observed. Moreover, the ageing kinetics appear to depend on tacticity, with s-PS and i-PS exhibiting the slowest and fastest ageing kinetics, respectively, and a-PS exhibiting ageing kinetics between these two extremes.