Shear-stress relaxation and ensemble transformation of shear-stress autocorrelation functions

A simulation method for highly entangled polymer nanocomposites: scaling exponents of slip-spring age among free and grafted chains, grafting density and nanoparticle/polymer interaction dependence on particle dispersion

We present an extension of the SLIPLINK technology introduced by A. Likhtman to polymer nanocomposites in order to model explicitly free and grafted chains. Entanglements are explicitly modeled by slip-springs (SS) and follow the constraint release algorithm of destruction/recreation when reaching the chain end. Following the birth/death process, one can compute the age of slip-springs and the entire population age pyramid. We varied nanoparticle volume fraction, grafting density, and polymer/particle interactions to determine structural and dynamic properties of the nanocomposite materials. Scaling laws for slip-springs average age versus chain length have been obtained. While the dynamics of slip-springs between free chains in the nanocomposite is almost identical to that of a pure polymer melt, a characteristic exponent close to 3.7 has emerged governing the average age of slip-springs between grafted chains. The number of inter-particle graft-graft entanglements and their increased average lifetimes have a strong impact on the viscoelastic response of the material and the nanoparticle cluster formation. The emergence of polymer network elasticity will be discussed for high grafting density and high-volume fraction.

General Relations between Stress Fluctuations and Viscoelasticity in Amorphous Polymer and Glass-Forming Systems

Mechanical stress governs the dynamics of viscoelastic polymer systems and supercooled glass-forming fluids. It was recently established that liquids with long terminal relaxation times are characterized by transiently frozen stress fields, which, moreover, exhibit long-range correlations contributing to the dynamically heterogeneous nature of such systems. Recent studies show that stress correlations and relaxation elastic moduli are intimately related in isotropic viscoelastic systems. However, the origin of these relations (involving spatially resolved material relaxation functions) is non-trivial: some relations are based on the fluctuation-dissipation theorem (FDT), while others involve approximations. Generalizing our recent results on 2D systems, we here rigorously derive three exact FDT relations (already established in our recent investigations and, partially, in classical studies) between spatio-temporal stress correlations and generalized relaxation moduli, and a couple of new exact relations. We also derive several new approximate relations valid in the hydrodynamic regime, taking into account the effects of thermal conductivity and composition fluctuations for arbitrary space dimension. One approximate relation was heuristically obtained in our previous studies and verified using our extended simulation data on two-dimensional (2D) glass-forming systems. As a result, we provide the means to obtain, in any spatial dimension, all stress-correlation functions in terms of relaxation moduli and vice versa. The new approximate relations are tested using simulation data on 2D systems of polydisperse Lennard-Jones particles.

Strain correlation functions in isotropic elastic bodies: large wavelength limit for two-dimensional systems

Strain correlation functions in two-dimensional isotropic elastic bodies are shown both theoretically (using the general structure of isotropic tensor fields) and numerically (using a glass-forming model system) to depend on the coordinates of the field variable (position vector r in real space or wavevector q in reciprocal space) and thus on the direction of the field vector and the orientation of the coordinate system. Since the fluctuations of the longitudinal and transverse components of the strain field in reciprocal space are known in the long-wavelength limit from the equipartition theorem, all components of the correlation function tensor field are imposed and no additional physical assumptions are needed. An observed dependence on the field vector direction thus cannot be used as an indication for anisotropy or for a plastic rearrangement. This dependence is different for the associated strain response field containing also information on the localized stress perturbation.

Correlations of tensor field components in isotropic systems with an application to stress correlations in elastic bodies

Correlation functions of components of second-order tensor fields in isotropic systems can be reduced to an isotropic fourth-order tensor field characterized by a few invariant correlation functions (ICFs). It is emphasized that components of this field depend in general on the coordinates of the field vector variable and thus on the orientation of the coordinate system. These angular dependencies are distinct from those of ordinary anisotropic systems. As a simple example of the procedure to obtain the ICFs we discuss correlations of time-averaged stresses in isotropic glasses where only one ICF in reciprocal space becomes a finite constant e for large sampling times and small wave vectors. It is shown that e is set by the typical size of the frozen-in stress components normal to the wave vectors, i.e., it is caused by the symmetry breaking of the stress for each independent configuration. Using the presented general mathematical formalism for isotropic tensor fields this finding explains in turn the observed long-range stress correlations in real space. Under additional but rather general assumptions e is shown to be given by a thermodynamic quantity, the equilibrium Young modulus E. We thus relate for certain isotropic amorphous bodies the existence of finite Young or shear moduli to the symmetry breaking of a stress component in reciprocal space.

Structural Features and Nonlinear Rheology of Self-Assembled Networks of Cross-Linked Semiflexible Polymers

Disordered networks of semiflexible filaments are common support structures in biology. Familiar examples include fibrous matrices in blood clots, bacterial biofilms, and essential components of cells and tissues of plants, animals, and fungi. Despite the ubiquity of these networks in biomaterials, we have only a limited understanding of the relationship between their structural features and their highly strain-sensitive mechanical properties. In this work, we perform simulations of three-dimensional networks produced by the irreversible formation of cross-links between linker-decorated semiflexible filaments. We characterize the structure of networks formed by a simple diffusion-dependent assembly process and measure their associated steady-state rheological features at finite temperature over a range of applied prestrains that encompass the strain-stiffening transition. We quantify the dependence of network connectivity on cross-linker availability and detail the associated connectivity dependence of both linear elasticity and nonlinear strain-stiffening behavior, drawing comparisons with prior experimental measurements of the cross-linker concentration-dependent elasticity of actin gels.

Stress relaxation in tunable gels

Hydrogels are a staple of biomaterials development. Optimizing their use in e.g. drug delivery or tissue engineering requires a solid understanding of how to adjust their mechanical properties. Here, we present a numerical study of a class of hydrogels made of 4-arm star polymers with a combination of covalent and reversible crosslinks. This design principle combines the flexibility and responsivity associated with reversible linkers with stability provided by chemical crosslinks. In molecular dynamics simulations of such hybrid gel networks, we observe that the strength of the reversible bonds can tune the material from solid to fluid. We identify at what fraction of reversible bonds this tunability is most pronounced, and find that the stress relaxation time of the gels in this tunable regime is set directly by the average lifetime of the reversible bonds. As our design is easy to realize in the already widely-used tetraPEG gel setting, our work will provide guidelines to improve the mechanical performance of biomedical gels.

Fluctuations of non-ergodic stochastic processes

We investigate the standard deviation [Formula: see text] of the variance [Formula: see text] of time series [Formula: see text] measured over a finite sampling time [Formula: see text] focusing on non-ergodic systems where independent "configurations" c get trapped in meta-basins of a generalized phase space. It is thus relevant in which order averages over the configurations c and over time series k of a configuration c are performed. Three variances of [Formula: see text] must be distinguished: the total variance [Formula: see text] and its contributions [Formula: see text], the typical internal variance within the meta-basins, and [Formula: see text], characterizing the dispersion between the different basins. We discuss simplifications for physical systems where the stochastic variable x(t) is due to a density field averaged over a large system volume V. The relations are illustrated for the shear-stress fluctuations in quenched elastic networks and low-temperature glasses formed by polydisperse particles and free-standing polymer films. The different statistics of [Formula: see text] and [Formula: see text] are manifested by their different system-size dependences.

Ensemble fluctuations matter for variances of macroscopic variables

Extending recent work on stress fluctuations in complex fluids and amorphous solids we describe in general terms the ensemble average [Formula: see text] and the standard deviation [Formula: see text] of the variance [Formula: see text] of time series [Formula: see text] of a stochastic process x(t) measured over a finite sampling time [Formula: see text]. Assuming a stationary, Gaussian and ergodic process, [Formula: see text] is given by a functional [Formula: see text] of the autocorrelation function h(t). [Formula: see text] is shown to become large and similar to [Formula: see text] if [Formula: see text] corresponds to a fast relaxation process. Albeit [Formula: see text] does not hold in general for non-ergodic systems, the deviations for common systems with many microstates are merely finite-size corrections. Various issues are illustrated for shear-stress fluctuations in simple coarse-grained model systems.

Parameter-free predictions of the viscoelastic response of glassy polymers from non-affine lattice dynamics

We study the viscoelastic response of amorphous polymers using theory and simulations. By accounting for internal stresses and considering instantaneous normal modes (INMs) within athermal non-affine theory, we make parameter-free predictions of the dynamic viscoelastic moduli obtained in coarse-grained simulations of polymer glasses at non-zero temperatures. The theoretical results show very good correspondence with rheology data collected from molecular dynamics simulations over five orders of magnitude in frequency, with some instabilities that accumulate in the low-frequency part on approach to the glass transition. These results provide evidence that the mechanical glass transition itself is continuous and thus represents a crossover rather than a true phase transition. The relatively sharp drop of the low-frequency storage modulus across the glass transition temperature can be explained mechanistically within the proposed theory: the proliferation of low-eigenfrequency vibrational excitations (boson peak and nearly-zero energy excitations) is directly responsible for the rapid growth of a negative non-affine contribution to the storage modulus.

Long-range stress correlations in viscoelastic and glass-forming fluids

A simple and rigorous approach to obtain stress correlations in viscoelastic liquids (including supercooled liquid and equilibrium amorphous systems) is proposed. The long-range dynamical correlations of local shear stress are calculated and analyzed in 2-dimensional space. It is established how the long-range character of the stress correlations gradually emerges as the relevant dynamical correlation length l grows in time. The correlation range l is defined by momentum propagation due to acoustic waves and vorticity diffusion which are the basic mechanisms for transmission of shear stress perturbations. We obtain the general expression defining the time- and distance-dependent stress correlation tensor in terms of material functions (generalized relaxation moduli). The effect of liquid compressibility is quantitatively analyzed; it is shown to be important at large distances and/or short times. The revealed long-range stress correlation effect is shown to be dynamical in nature and unconnected with static structural correlations in liquids (correlation length ξ_s). Our approach is based on the assumption that ξ_s is small enough as reflected in weak wave-number dependencies of the generalized relaxation moduli. We provide a simple physical picture connecting the elucidated long-range fluctuation effect with anisotropic correlations of the (transient) inherent stress field, and discuss its implications.

Stress auto-correlation tensor in glass-forming isothermal fluids: From viscous to elastic response

We develop a generalized hydrodynamic theory, which can account for the build-up of long-ranged and long-lived shear stress correlations in supercooled liquids as the glass transition is approached. Our theory is based on the decomposition of tensorial stress relaxation into fast microscopic processes and slow dynamics due to conservation laws. In the fluid, anisotropic shear stress correlations arise from the tensorial nature of stress. By approximating the fast microscopic processes by a single relaxation time in the spirit of Maxwell, we find viscoelastic precursors of the Eshelby-type correlations familiar in an elastic medium. The spatial extent of shear stress fluctuations is characterized by a correlation length ξ which grows like the viscosity η or time scale τ ∼ η, whose divergence signals the glass transition. In the solid, the correlation length is infinite and stress correlations decay algebraically as r-d in d dimensions.

Dynamics of Vitrimers: Defects as a Highway to Stress Relaxation

We propose a coarse-grained model to investigate stress relaxation in star-polymer networks induced by dynamic bond-exchange processes. We show how the swapping mechanism, once activated, allows the network to reconfigure, exploring distinct topological configurations, all of them characterized by complete extent of reaction. Our results reveal the important role played by topological defects in mediating the exchange reaction and speeding up stress relaxation. The model provides a representation of the dynamics in vitrimers, a new class of polymers characterized by bond-swap mechanisms which preserve the total number of bonds, as well as in other bond-exchange materials.

Stress-stress fluctuation formula for elastic constants in the NPT ensemble

Several fluctuation formulas are available for calculating elastic constants from equilibrium correlation functions in computer simulations, but the ones available for simulations at constant pressure exhibit slow convergence properties and cannot be used for the determination of local elastic constants. To overcome these drawbacks, we derive a stress-stress fluctuation formula in the NPT ensemble based on known expressions in the NVT ensemble. We validate the formula in the NPT ensemble by calculating elastic constants for the simple nearest-neighbor Lennard-Jones crystal and by comparing the results with those obtained in the NVT ensemble. For both local and bulk elastic constants we find an excellent agreement between the simulated data in the two ensembles. To demonstrate the usefulness of the formula, we apply it to determine the elastic constants of a simulated lipid bilayer.

Elastic moduli of a Brownian colloidal glass former

The static, dynamic and flow-dependent shear moduli of a binary mixture of Brownian hard disks are studied by an event-driven molecular dynamics simulation. Thereby, the emergence of rigidity close to the glass transition encoded in the static shear modulus [Formula: see text] is accessed by three methods. Results from shear stress auto-correlation functions, elastic dispersion relations, and the elastic response to strain deformations upon the start-up of shear flow are compared. This enables one to sample the time-dependent shear modulus [Formula: see text] consistently over several decades in time. By that a very precise specification of the glass transition point and of [Formula: see text] is feasible. Predictions by mode coupling theory of a finite shear modulus at the glass transition, of α-scaling in fluid states close to the transition, and of shear induced decay in yielding glass states are tested and broadly verified.

Shear-stress fluctuations and relaxation in polymer glasses

We investigate by means of molecular dynamics simulation a coarse-grained polymer glass model focusing on (quasistatic and dynamical) shear-stress fluctuations as a function of temperature T and sampling time Δt. The linear response is characterized using (ensemble-averaged) expectation values of the contributions (time averaged for each shear plane) to the stress-fluctuation relation μ_{sf} for the shear modulus and the shear-stress relaxation modulus G(t). Using 100 independent configurations, we pay attention to the respective standard deviations. While the ensemble-averaged modulus μ_{sf}(T) decreases continuously with increasing T for all Δt sampled, its standard deviation δμ_{sf}(T) is nonmonotonic with a striking peak at the glass transition. The question of whether the shear modulus is continuous or has a jump singularity at the glass transition is thus ill posed. Confirming the effective time-translational invariance of our systems, the Δt dependence of μ_{sf} and related quantities can be understood using a weighted integral over G(t).

Emergence of Long-Ranged Stress Correlations at the Liquid to Glass Transition

A theory for the nonlocal shear stress correlations in supercooled liquids is derived from first principles. It captures the crossover from viscous to elastic dynamics at an idealized liquid to glass transition and explains the emergence of long-ranged stress correlations in glass, as expected from classical continuum elasticity. The long-ranged stress correlations can be traced to the coupling of shear stress to transverse momentum, which is ignored in the classic Maxwell model. To rescue this widely used model, we suggest a generalization in terms of a single relaxation time τ for the fast degrees of freedom only. This generalized Maxwell model implies a divergent correlation length ξ∝τ as well as dynamic critical scaling and correctly accounts for the far-field stress correlations. It can be rephrased in terms of generalized hydrodynamic equations, which naturally couple stress and momentum and furthermore allow us to connect to fluidity and elastoplastic models.

Shear Modulus and Shear-Stress Fluctuations in Polymer Glasses

Using molecular dynamics simulation of a standard coarse-grained polymer glass model, we investigate by means of the stress-fluctuation formalism the shear modulus μ as a function of temperature T and sampling time Δt. While the ensemble-averaged modulus μ(T) is found to decrease continuously for all Δt sampled, its standard deviation δμ(T) is nonmonotonic, with a striking peak at the glass transition. Confirming the effective time-translational invariance of our systems, μ(Δt) can be understood using a weighted integral over the shear-stress relaxation modulus G(t). While the crossover of μ(T) gets sharper with an increasing Δt, the peak of δμ(T) becomes more singular. It is thus elusive to predict the modulus of a single configuration at the glass transition.

Numerical determination of shear stress relaxation modulus of polymer glasses

Focusing on simulated polymer glasses well below the glass transition, we confirm the validity and the efficiency of the recently proposed simple-average expression [Formula: see text] for the computational determination of the shear stress relaxation modulus G(t). Here, [Formula: see text] characterizes the affine shear transformation of the system at t = 0 and h(t) the mean-square displacement of the instantaneous shear stress as a function of time t. This relation is seen to be particulary useful for systems with quenched or sluggish transient shear stresses which necessarily arise below the glass transition. The commonly accepted relation [Formula: see text] using the shear stress auto-correlation function c(t) becomes incorrect in this limit.

Three-body potential for simulating bond swaps in molecular dynamics

Novel soft matter materials join the resistance of a permanent mesh of strong inter-particle bonds with the self-healing and restructuring properties allowed by bond-swapping processes. Theoretical and numerical studies of the dynamics of coarse-grained models of covalent adaptable networks and vitrimers require effective algorithms for modelling the corresponding evolution of the network topology. Here I propose a simple trick for performing molecular dynamics simulations of bond-swapping network systems with particle-level description. The method is based on the addition of a computationally non-expensive three-body repulsive potential that encodes for the single-bond per particle condition and establishes a flat potential energy surface for the bond swap.

Strain Pattern in Supercooled Liquids

Investigations of strain correlations at the glass transition reveal unexpected phenomena. The shear strain fluctuations show an Eshelby-strain pattern [∼cos(4θ)/r^{2}], characteristic of elastic response, even in liquids, at long times. We address this using a mode-coupling theory for the strain fluctuations in supercooled liquids and data from both video microscopy of a two-dimensional colloidal glass former and simulations of Brownian hard disks. We show that the long-ranged and long-lived strain signatures follow a scaling law valid close to the glass transition. For large enough viscosities, the Eshelby-strain pattern is visible even on time scales longer than the structural relaxation time τ and after the shear modulus has relaxed to zero.

Shear-stress fluctuations in self-assembled transient elastic networks

Focusing on shear-stress fluctuations, we investigate numerically a simple generic model for self-assembled transient networks formed by repulsive beads reversibly bridged by ideal springs. With Δt being the sampling time and t_{☆}(f)∼1/f the Maxwell relaxation time (set by the spring recombination frequency f), the dimensionless parameter Δx=Δt/t_{☆}(f) is systematically scanned from the liquid limit (Δx≫1) to the solid limit (Δx≪1) where the network topology is quenched and an ensemble average over m-independent configurations is required. Generalizing previous work on permanent networks, it is shown that the shear-stress relaxation modulus G(t) may be efficiently determined for all Δx using the simple-average expression G(t)=μ_{A}-h(t) with μ_{A}=G(0) characterizing the canonical-affine shear transformation of the system at t=0 and h(t) the (rescaled) mean-square displacement of the instantaneous shear stress as a function of time t. This relation is compared to the standard expression G(t)=c[over ̃](t) using the (rescaled) shear-stress autocorrelation function c[over ̃](t). Lower bounds for the m configurations required by both relations are given.

Elastic moduli and vibrational modes in jammed particulate packings

When we elastically impose a homogeneous, affine deformation on amorphous solids, they also undergo an inhomogeneous, nonaffine deformation, which can have a crucial impact on the overall elastic response. To correctly understand the elastic modulus M, it is therefore necessary to take into account not only the affine modulus M_{A}, but also the nonaffine modulus M_{N} that arises from the nonaffine deformation. In the present work, we study the bulk (M=K) and shear (M=G) moduli in static jammed particulate packings over a range of packing fractions φ. The affine M_{A} is determined essentially by the static structural arrangement of particles, whereas the nonaffine M_{N} is related to the vibrational eigenmodes. We elucidate the contribution of each vibrational mode to the nonaffine M_{N} through a modal decomposition of the displacement and force fields. In the vicinity of the (un)jamming transition φ_{c}, the vibrational density of states g(ω) shows a plateau in the intermediate-frequency regime above a characteristic frequency ω^{*}. We illustrate that this unusual feature apparent in g(ω) is reflected in the behavior of M_{N}: As φ→φ_{c}, where ω^{*}→0, those modes for ω<ω^{*} contribute less and less, while contributions from those for ω>ω^{*} approach a constant value which results in M_{N} to approach a critical value M_{Nc}, as M_{N}-M_{Nc}∼ω^{*}. At φ_{c} itself, the bulk modulus attains a finite value K_{c}=K_{Ac}-K_{Nc}>0, such that K_{Nc} has a value that remains below K_{Ac}. In contrast, for the critical shear modulus G_{c}, G_{Nc} and G_{Ac} approach the same value so that the total value becomes exactly zero, G_{c}=G_{Ac}-G_{Nc}=0. We explore what features of the configurational and vibrational properties cause such a distinction between K and G, allowing us to validate analytical expressions for their critical values.

Glass transition of two-dimensional 80-20 Kob-Andersen model at constant pressure

We reconsider numerically the two-dimensional version of the Kob-Andersen model (KA2d) with a fraction of 80% of large spheres. A constant moderate pressure is imposed while the temperature T is systematically quenched from the liquid limit through the glass transition at [Formula: see text] down to very low temperatures. Monodisperse Lennard-Jones (mdLJ) bead systems, forming a crystal phase at low temperatures, are used to highlight several features of the KA2d model. As can be seen, e.g. from the elastic shear modulus G(T), determined using the stress-fluctuation formalism, our KA2d model is a good glass-former. A continuous cusp-singularity, [Formula: see text] with [Formula: see text], is observed in qualitative agreement with other recent numerical and theoretical work, however in striking conflict with the additive jump discontinuity predicted by mode-coupling theory.

Simple average expression for shear-stress relaxation modulus

Focusing on isotropic elastic networks we propose a simple-average expression G(t)=μ_{A}-h(t) for the computational determination of the shear-stress relaxation modulus G(t) of a classical elastic solid or fluid. Here, μ_{A}=G(0) characterizes the shear transformation of the system at t=0 and h(t) the (rescaled) mean-square displacement of the instantaneous shear stress τ[over ̂](t) as a function of time t. We discuss sampling time and ensemble effects and emphasize possible pitfalls of alternative expressions using the shear-stress autocorrelation function. We argue finally that our key relation may be readily adapted for more general linear response functions.

Thermal fluctuations, mechanical response, and hyperuniformity in jammed solids

Jamming is a geometric phase transition occurring in dense particle systems in the absence of temperature. We use computer simulations to analyze the effect of thermal fluctuations on several signatures of the transition. We show that scaling laws for bulk and shear moduli only become relevant when thermal fluctuations are extremely small, and propose their relative ratio as a quantitative signature of jamming criticality. Despite the nonequilibrium nature of the transition, we find that thermally induced fluctuations and mechanical responses obey equilibrium fluctuation-dissipation relations near jamming, provided the appropriate fluctuating component of the particle displacements is analyzed. This shows that mechanical moduli can be directly measured from particle positions in mechanically unperturbed packings, and suggests that the definition of a "nonequilibrium index" is unnecessary for amorphous materials. We find that fluctuations of particle displacements are spatially correlated, and define a transverse and a longitudinal correlation length scale which both diverge as the jamming transition is approached. We analyze the frozen component of density fluctuations and find that it displays signatures of nearly hyperuniform behavior at large length scales. This demonstrates that hyperuniformity in jammed packings is unrelated to a vanishing compressibility and explains why it appears remarkably robust against temperature and density variations. Differently from jamming criticality, obstacles preventing the observation of hyperuniformity in colloidal systems do not originate from thermal fluctuations.