We introduce a coarse-grained stochastic network dynamics (CGSND) framework for modeling deformation and rupture in polymer networks. The method replaces explicit molecular dynamics (MD) or coarse-grained molecular dynamics (CGMD) with network-level evolution rules while retaining chain entropic elasticity and force-controlled bond failure. Under uniaxial loading, CGSND reproduces the characteristic nonlinear stress--stretch response of elastomeric networks, including a well-defined ultimate tensile strength and post-peak softening due to progressive bond rupture. Comparison with coarse-grained molecular dynamics (CGMD) simulations shows that CGSND captures the qualitative form of the stress response and the onset of catastrophic damage despite its rate-independent formulation. Analysis of rupture kinetics reveals a pronounced peak in the bond-breaking hazard rate near the ultimate tensile strength in both approaches. In addition, the distribution of broken segment lengths remains statistically indistinguishable from the initial network, indicating that rupture is not biased toward short or long chains. Finally, the evolution of the Gini coefficient of bond force magnitudes reveals strong force localization preceding failure. These results demonstrate that CGSND provides a computationally efficient and physically interpretable framework for connecting force localization and rupture kinetics to macroscopic failure in polymer networks.
We present the first experimental study on the simultaneous capillary instability amongst viscous concentric rings suspended atop an immiscible medium. The rings ruptured upon annealing, with three types of phase correlation between neighboring rings. In the case of weak substrate confinement, the rings ruptured independently when they were sparsely distanced, but via an out-of-phase mode when packed closer. If the substrate confinement was strong, the rings would rupture via an in-phase mode, resulting in radially aligned droplets. The concentric ring geometry caused a competition between the…
It is shown that the intrinsic stress in solvent cast polymer coatings plays a key role in the nucleation of holes in the film. Nucleation is important because it is meanwhile clear that heterogeneous nucleation is the only relevant rupture mechanism for the technologically relevant thickness regime well above 100 nm. The most striking feature is that in contrast to what has been widely believed, the number density of holes scales not algebraically, but exponentially with the film thickness.
We consider the rupture dynamics of a homopolymer chain pulled at one end at a constant loading rate. Our model of the breakable polymer is related to the Rouse chain, with the only difference that the interaction between the monomers is described by the Morse potential instead of the harmonic one, and thus allows for mechanical failure. We show that in the experimentally relevant domain of parameters the dependence of the most probable rupture force on the chain length may be non-monotonic, so that the medium-length chains break easier than the short and the long ones. The qualitative theory …
Progressive damage, which eventually leads to failure, is ubiquitous in biological and synthetic polymers. The simplest case to consider is that of elastomeric materials, which can undergo large reversible deformations with negligible rate dependence. In this paper, we develop a theory for modeling progressive damage and rupture of such materials. We extend the phase-field method, which is widely used to describe the damage and fracture of brittle materials, to elastomeric materials undergoing large deformations. A central feature of our theory is the recognition that the free energy of elasto…
We consider the rupture dynamics of a homopolymer chain pulled at one end at a constant loading rate r. Compared to single bond breaking, the existence of the chain introduces two new aspects into rupture dynamics: the non-Markovian aspect in the barrier crossing and the slow-down of the force propagation to the breakable bond. The relative impact of both these processes is investigated, and the second one was found to be the most important at moderate loading rates. The most probable rupture force is found to decrease with the number of bonds as f_{max} const-(ln(N/r))^(2/3) and finally to ap…
Common experience tells us that a knot significantly weakens the polymer strand in which it is tied, which in turn leads to more facile chain rupture under tensile loading. Using first-principles molecular dynamics calculations we describe the dynamical evolution of the radicals that form after chain rupture of a single knotted alkane molecule in their very early stages of life. They are able to recombine, to form cyclic alkanes and to undergo disproportionation phenomena with nearby chain segments. The breaking of a single knotted polymer chain under mechanical loading is thus predicted to re…
In capillary-driven fluid dynamics, simple departures from equilibrium offer the chance to quantitatively model the resulting relaxations. These dynamics in turn provide insight on both practical and fundamental aspects of thin-film hydrodynamics. In this work, we describe a model trilayer dewetting experiment elucidating the effect of solid, no-slip confining boundaries on the bursting of a liquid film in a viscous environment. This experiment was inspired by an industrial polymer processing technique, multilayer coextrusion, in which thousands of alternating layers are stacked atop one anoth…
To design increasingly tough, resilient, and fatigue-resistant elastomers and hydrogels, the relationship between controllable network parameters at the molecular level to macroscopic quantities that govern damage and failure must be established. Constitutive models based upon statistical mechanics have used variants of the freely jointed chain (FJC) model with rigid links. However, since the free energy state of a polymer chain is dominated by enthalpic bond distortion effects as the chain approaches its rupture point, bond extensibility ought to be accounted for if the model is intended to c…
We examine the thermally-induced fracture of an unstrained polymer chain of discrete segments coupled by an anharmonic potential by means of Molecular Dynamics simulation with a Langevin thermostat. Cases of both under- and over-damped dynamics are investigated, and a comparison with recent studies of bond scission in model polymers with harmonic interactions is performed. We find that the polymer degradation changes qualitatively between the inertial regime and that of heavily damped dynamics. The role of bond healing (recombination) is also studied and probability distributions for the recom…
A simple expression for the composition dependence of the Flory-Huggins interaction parameter of polymer/solvent systems reported earlier is used to model the demixing of polymer solutions into two liquid phases. To this end the system specific parameters zeta and ny of that approach are calculated as a function of temperature using the thermodynamic expressions resulting for the critical conditions on one side and from experimentally determined critical data for polymers of different molar mass on the other side. By means of data reported for the system cyclohexane/polystyrene it is demonstra…
We consider the thermal breakage of a tethered polymer chain of discrete segments coupled by Morse potentials under constant tensile stress. The chain dynamics at the onset of fracture is studied analytically by Kramers-Langer multidimensional theory and by extensive Molecular Dynamics simulations in 1D- and 3D-space. Comparison with simulation data in one- and three dimensions demonstrates that the Kramers-Langer theory provides good qualitative description of the process of bond-scission as caused by a {\em collective} unstable mode. We derive distributions of the probability for scission ov…
The stress response of polymer double networks depends not only on the properties of the constituent networks, but also on the interactions arising between them. Here we demonstrate, via coarse-grained simulations, that both their global stress response and their microscopic fracture mechanics are governed by load sharing through these inter-network interactions. By comparing our results with affine predictions, where stress redistribution is by definition homogeneous, we show that stress redistribution is highly inhomogeneous. In particular, the affine prediction overestimates the fraction of…
The breakage of a polymer chain of segments, coupled by anharmonic bonds with applied constant external tensile force is studied by means of Molecular Dynamics simulation. We show that the mean life time of the chain becomes progressively independent of the number of bonds as the pulling force grows. The latter affects also the rupture rates of individual bonds along the polymer backbone manifesting the essential role of inertial effects in the fragmentation process. The role of local defects, temperature and friction in the scission kinetics is also examined.
In part I of this contribution we have reported how the Flory-Huggins interaction parameter chi can be modeled as a function of chain length within the composition range of pair interaction between the macromolecules by means of the three parameters alpha, zeta and lambda. This contribution presents the extension of the approach to arbitrary volume fractions of the polymer and its application to published data. The resulting equation is simple and requires only the additional parameter ny to incorporate the composition dependence. Its employment to experimental data is very much facilitated by…
This is the first of two parts investigating the Flory-Huggins interaction parameter chi as a function of composition and chain length. Part I encompasses experimental and theoretical work. The former comprises the synthesis of poly(dimethylsiloxane)s with different molar mass and the measurements of their second osmotic virial coefficients in solvents of diverse quality as a function of M via light scattering and osmotic pressures. The theoretical analysis is performed by subdividing the dilution process into two clearly separable states. It yields a simple expression for the range of pair in…
We re-examine the physical origin of the polymer glass transition from the point of view of marginal rigidity, which is achieved above a certain number of intermolecular contacts. In the case of polymer chains in a melt / poor solvent, each monomer has two neighbors bound by covalent bonds and also a number of central-force contacts modelled by the Lennard-Jones (LJ) potential. We find that when the average number of contacts per monomer (covalent and non-covalent) exceeds the critical value z*=4, the system becomes solid and the dynamics arrested - a state that we declare the glass. Coarse-gr…
The random walk of a bonded monomer in a polymer melt is anisotropic due to local order and bond connectivity. We investigate both effects by molecular-dynamics simulations on melts of fully-flexible linear chains ranging from dimers ($M=2$) up to entangled polymers ($M=200$). The corresponding atomic liquid is also considered as reference system. To disentangle the influence of the local geometry and the bond arrangements, and reveal their interplay, we define suitable measures of the anisotropy emphasising either the former or the latter aspect. Connectivity anisotropy, as measured by the co…
Hybrid molecular dynamics/Monte Carlo simulations used to study melts of unentangled, thermoreversibly associating supramolecular polymers. In this first of a series of papers, we describe and validate a model that is effective in separating the effects of thermodynamics and chemical kinetics on the dynamics and mechanics of these systems, and is extensible to arbitrarily nonequilibrium situations and nonlinear mechanical properties. We examine the model's quiescent (and heterogeneous) dynamics, nonequilibrium chemical dynamics, and mechanical properties. Many of our results may be understood …
We introduce a kinetic model of coarsening of transient polymer networks during the intermediate stages of phase separation in polymer systems. The model explicitly takes into account the effect of the connectivity of polymer chains. We show that during the intermediate stages the size of the droplet grows according to an unusual coarsening law $R \sim t^{1/(d + 3)}$. When the network structure breaks up, the coarsening law may cross over to $R \sim \ln t^{-1}$, until only at very late stages the Lifshitz-Slyozov coarsening law $R \sim t^{1/3}$ is recovered.
We consider a microscopic model for the failure of soft adhesives in tension based on ideas of bond rupture under dynamic loading. Focusing on adhesive failure under loading at constant velocity, we demonstrate that bi-modal curves of stress against strain may occur due to effects of finite polymer chain or bond length and characterise the loading conditions under which such bi-modal behaviour is observed. The results of this analysis are in qualitative agreement with experiments performed on unconfined adhesives in which failure does not occur by cavitation.
We study the influence of thermal fluctuations on the fracture of elastic networks, via simulations of the uniaxial extension of central-force spring networks with varying rigidity, i.e. connectivity. Studying their failure response, both at the macroscopic and microscopic level, we find that an increase in temperature corresponds to a more homogeneous stress (re)distribution and induces thermally activated failure of springs. As a consequence, the material strength decreases upon increasing temperature, the damage is spread over larger lengthscales and a more ductile fracture process is obser…
In this paper, a micro-mechanical model is introduce to characterize the stress softening phenomenon in double network hydrogels. Damage in polymer matrices under deformation, which appears as the rupture of cross-linking, is assumed to result in stress softening. The evolution of a polymer matrix is described with the network evolution method by Dargazany and Itskov [Dargazany and Itskov, International Journal of Solids and Structures, 2009, 46, 2967]. The model includes a number of physically-motivated material parameters and obtains good agreement against different sets of experimental data…
A coarse-grained simulation model eliminates microscopic degrees of freedom and represents a polymer by a simplified structure. A priori, two classes of coarse-grained models may be distinguished: those which are designed for a specific polymer and reflect the underlying atomistic details to some extent, and those which retain only the most basic features of a polymer chain (chain connectivity, short-range excluded-volume interactions, etc.). In this review we mainly focus on the second class of generic polymer models, while the first class of specific coarse-grained models is only touched upo…
The concept of network immunity, i.e., the robustness of the network connectivity after a random deletion of edges or vertices, has been investigated in biological or communication networks. We apply this concept to a self-assembling, physical network of microemulsion droplets connected by telechelic polymers, where more than one polymer can connect a pair of droplets. The gel phase of this system has higher immunity if it is more likely to survive (i.e., maintain a macroscopic, connected component) when some of the polymers are randomly degraded. We consider the distribution $p(σ)$ of the num…
By performing extensive simulations with unprecedentedly large system sizes, we unveil how rigidity influences the fracture of disordered materials. We observe the largest damage in networks with connectivity close to the isostatic point and when the rupture thresholds are small. However, regardless of network and spring properties, a more brittle fracture is observed upon increasing system size. Differently from most of the fracture descriptors, the maximum stress drop, a proxy for brittleness, displays a universal non-monotonic dependence on system size. Based on this uncommon trend it is po…
We consider the fracture of a free-standing two-dimensional (2D) elastic-brittle network to be used as protective coating subject to constant tensile stress applied on its rim. Using a Molecular Dynamics simulation with Langevin thermostat, we investigate the scission and recombination of bonds, and the formation of cracks in the 2D graphene-like hexagonal sheet for different pulling force $f$ and temperature $T$. We find that bond rupture occurs almost always at the sheet periphery and the First Mean Breakage Time $<τ>$ of bonds decays with membrane size as $<τ> \propto N^{-β}$ where $β\appro…
We present simulations on a binary blend of bead-spring polymer chains. The introduction of monomer size disparity yields very different relaxation times for each component of the blend. Competition between two different arrest mechanisms, namely bulk-like dynamics and confinement, leads to an anomalous relaxation scenario for the fast component, characterized by sublinear time dependence for mean squared displacements, or logarithmic decay and convex-to-concave crossover for density-density correlators. These anomalous dynamic features, which are observed over time intervals extending up to f…
We report on quantitative comparisons between simulation results of a bead-spring model and mode-coupling theory calculations for the structural and conformational dynamics of a supercooled, unentangled polymer melt. We find semiquantitative agreement between simulation and theory, except for processes that occur on intermediate length scales between the compressibility plateau and the amorphous halo of the static structure factor. Our results suggest that the onset of slow relaxation in a glass-forming melt can be described in terms of monomer-caging supplemented by chain connectivity. Furthe…
It is commonly accepted that in concentrated solutions or melts high-molecular weight polymers display random-walk conformational properties without long-range correlations between subsequent bonds. This absence of memory means, for instance, that the bond-bond correlation function, $P(s)$, of two bonds separated by $s$ monomers along the chain should exponentially decay with $s$. Presenting numerical results and theoretical arguments for both monodisperse chains and self-assembled (essentially Flory size-distributed) equilibrium polymers we demonstrate that some long-range correlations remain…
The elasticity of disordered and polydisperse polymer networks is a fundamental problem of soft matter physics that is still open. Here, we self-assemble polymer networks via simulations of a mixture of bivalent and tri- or tetravalent patchy particles, which result in an exponential strand length distribution analogous to that of experimental randomly crosslinked systems. After assembly, the network connectivity and topology are frozen and the resulting system is characterized. We find that the fractal structure of the network depends on the number density at which the assembly has been carri…
We study the dynamics of a single chain polymer confined to a two dimensional cell. We introduce a kinetically constrained lattice gas model that preserves the connectivity of the chain, and we use this kinetically constrained model to study the dynamics of the polymer at varying densities through Monte Carlo simulations. Even at densities close to the fully-packed configuration, we find that the monomers comprising the chain manage to diffuse around the box with a root mean square displacement of the order of the box dimensions over time scales for which the overall geometry of the polymer is…
We develop a segment-scale, force-based theory for the breakdown of the unentangled Rouse model and subsequent emergence of isotropic mesoscopic localization and entropic elasticity in chain polymer liquids in the absence of ergodicity-restoring anisotropic reptation motion. The theory is formulated in terms of a conformational N-dynamic-order-parameter Generalized Langevin Equation approach. It is implemented using a field-theoretic Gaussian thread model of polymer structure and closed in a universal manner at the level of the chain dynamic second moment matrix. The physical idea is that the …
We present a novel simulation technique derived from Brownian cluster dynamics used so far to study the isotropic colloidal aggregation. It now implements the classical Kern-Frenkel potential to describe patchy interactions between particles. This technique gives access to static properties, dynamics and kinetics of the system, even far from the equilibrium. Particle thermal motions are modeled using billions of independent small random translations and rotations, constrained by the excluded volume and the connectivity. This algorithm, applied to a single polymer chain leads to correct static …
The glass transition is a long-standing unsolved problem in materials science. For polymers, our understanding of glass-formation is particularly poor due to the added complexity of chain connectivity and flexibility; structural relaxation of polymers thus involves a complex interplay between intra- and inter-molecular cooperativity. Here we study how the glass transition temperature Tg varies with molecular weight M for different polymer chemistries and chain flexibilities. We find that Tg(M) is controlled by the average mass (or volume) per conformational degree of freedom, and that a `local…
