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.

Different types of spatial correlation functions for non-ergodic stochastic processes of macroscopic systems.

Focusing on non-ergodic macroscopic systems, we reconsider the variances [Formula: see text] of time averages [Formula: see text] of time-series [Formula: see text]. The total variance [Formula: see text] (direct average over all time series) is known to be the sum of an internal variance [Formula: see text] (fluctuations within the meta-basins) and an external variance [Formula: see text] (fluctuations between meta-basins). It is shown that whenever [Formula: see text] can be expressed as a volume average of a local field [Formula: see text] the three variances can be written as volume averages of correlation functions [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text]. The dependences of the [Formula: see text] on the sampling time [Formula: see text] and the system volume V can thus be traced back to [Formula: see text] and [Formula: see text]. Various relations are illustrated using lattice spring models with spatially correlated spring constants. .

Role of torsional potential in chain conformation, thermodynamics, and glass formation of simulated polybutadiene melts.

For polymer chains, the torsional potential is an important intramolecular energy influencing chain flexibility and segmental dynamics. Through molecular dynamics simulations of an atomistic model for melts of cis-trans-1,4-polybutadiene (PBD), we explore the effect of the torsions on conformational properties (bond vector correlations and mean-square internal distances), fundamental thermodynamic quantities (density, compressibility, internal energy, and specific heat), and glass transition temperature Tg. This is achieved by systematically reducing the strength of the torsional potential, starting from the chemically realistic chain (CRC) model with the full potential toward the freely rotating chain (FRC) model without the torsional potential. For the equilibrium liquid, we find that the effect of the torsions on polymer conformations is very weak. Still weaker is the influence on the monomer density ρ and isothermal compressibility κT of the polymer liquid, both of which can be considered as independent of the torsional potential. We show that a van der Waals-like model proposed by Long and Lequeux [Eur. Phys. J. E 4, 371 (2001)] allows us to describe very well the temperature (T) dependence of ρ and κT. We also find that our data obey the linear relation between 1/kBTρκT and 1/T (with the Boltzmann constant kB) that has recently been predicted and verified on the experiment by Mirigian and Schweizer [J. Chem. Phys. 140, 194507 (2014)]. For the equilibrium liquid, simulations result in a specific heat, at constant pressure and at constant volume, which increases on cooling. This T dependence is opposite to the one found experimentally for many polymer liquids, including PBD. We suggest that this difference between simulation and experiment may be attributed to quantum effects due to hydrogen atoms and backbone vibrations, which, by construction, are not included in the classical united-atom model employed here. Finally, we also determine Tg from the density-temperature curve monitored in a finite-rate cooling process. While the influence of the torsional potential on ρ(T) is vanishingly small in the equilibrium liquid, the effect of the torsions on Tg is large. We find that Tg decreases by about 150 K when going from the CRC to the FRC model.

Theory of length-scale dependent relaxation moduli and stress fluctuations in glass-forming and viscoelastic liquids.

The spatiotemporal correlations of the local stress tensor in supercooled liquids are studied both theoretically and by molecular dynamics simulations of a two-dimensional (2D) polydisperse Lennard-Jones system. Asymptotically exact theoretical equations defining the dynamical structure factor and all components of the stress correlation tensor for low wave-vector q are presented in terms of the generalized (q-dependent) shear and longitudinal relaxation moduli, G(q, t) and K(q, t). We developed a rigorous approach (valid for low q) to calculate K(q, t) in terms of certain bulk correlation functions (for q = 0), the static structure factor S(q), and thermal conductivity κ. The proposed approach takes into account both the thermostatting effect and the effect of polydispersity. The theoretical results for the (q, t)-dependent stress correlation functions are compared with our simulation data, and an excellent agreement is found for qb̄≲0.5 (with b̄ being the mean particle diameter) both above and below the glass transition without any fitting parameters. Our data are consistent with recently predicted (both theoretically and by simulations) long-range correlations of the shear stress quenched in heterogeneous glassy structures.

Simple models for strictly non-ergodic stochastic processes of macroscopic systems.

We investigate simple models for strictly non-ergodic stochastic processes [Formula: see text] (t being the discrete time step) focusing on the expectation value v and the standard deviation [Formula: see text] of the empirical variance [Formula: see text] of finite time series [Formula: see text]. [Formula: see text] is averaged over a fluctuating field [Formula: see text] ([Formula: see text] being the microcell position) characterized by a quenched spatially correlated Gaussian field [Formula: see text]. Due to the quenched [Formula: see text]-field [Formula: see text] becomes a finite constant, [Formula: see text], for large sampling times [Formula: see text]. The volume dependence of the non-ergodicity parameter [Formula: see text] is investigated for different spatial correlations. Models with marginally long-ranged [Formula: see text]-correlations are successfully mapped on shear stress data from simulated amorphous glasses of polydisperse beads.

Relaxation moduli of glass-forming systems: temperature effects and fluctuations.

Equilibrium and dynamical properties of a two-dimensional polydisperse colloidal model system are characterized by means of molecular dynamics (MD) and Monte Carlo (MC) simulations. We employed several methods to prepare quasi-equilibrated systems: in particular, by slow cooling and tempering with MD (method SC-MD), and by tempering with MC dynamics involving swaps of particle diameters (methods Sw-MD, Sw-MC). It is revealed that the Sw-methods are much more efficient for equilibration below the glass transition temperature Tg leading to denser and more rigid systems which show much slower self-diffusion and shear-stress relaxation than their counterparts prepared with the SC-MD method. The shear-stress relaxation modulus G(t) is obtained based on the classical stress-fluctuation relation. We demonstrate that the α-relaxation time τα obtained using a time-temperature superposition of G(t) shows a super-Arrhenius behavior with the VFT temperature T0 well below Tg. We also derive novel rigorous fluctuation relations providing isothermic and adiabatic compression relaxation moduli in the whole time range (including the short-time inertial regime) based on correlation data for thermostatted systems. It is also shown that: (i) the assumption of Gaussian statistics for stress fluctuations leads to accurate predictions of the variances of the fluctuation moduli for both shear (µF) and compression (ηF) at T⪆Tg. (ii) The long-time (quasi-static) isothermic and adiabatic moduli increase on cooling faster than the affine compression modulus ηA, and this leads to a monotonic temperature dependence of ηF which is qualitatively different from µF(T) showing a maximum near Tg.

General relations to obtain the time-dependent heat capacity from isothermal simulations.

It is well-known that time-dependent correlation functions related to temperature and energy can crucially depend on the thermostatting mechanism used in computer simulations of molecular systems. We argue, however, that linear response functions must be considered as universal properties of physical systems. This implies that the classical fluctuation equation for the transient heat capacity, cv(t), is not applicable to the thermostatted molecular dynamics (apart from long enough times). To improve on this point, we derive a number of exact general expressions for the frequency-dependent heat capacity in terms of energy correlation functions, valid for the Nosé-Hoover and some other thermostats. We also establish a general relation between auto- and cross correlation functions of energy and temperature. Recommendations on how to use these relations to maximize the numerical precision are provided. It is demonstrated that our approach allows us to obtain cv(t) for a supercooled liquid system with high precision and over many decades in time reflecting all pertinent relaxation processes.

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.

Glassy dynamics of a binary Voronoi fluid: a mode-coupling analysis.

The binary Voronoi mixture is a fluid model whose interactions are derived from the Voronoi-Laguerre tessellation of the configurations of the system. The resulting interactions are local and many-body. Here we perform molecular-dynamics (MD) simulations of an equimolar mixture that is weakly polydisperse and additive. For the first time we study the structural relaxation of this mixture in the supercooled-liquid regime. From the simulations we determine the time- and temperature-dependent coherent and incoherent scattering functions for a large range of wave vectors, as well as the mean-square displacements of both particle species. We perform a detailed analysis of the dynamics by comparing the MD results with the first-principles-based idealized mode-coupling theory (MCT). To this end, we employ two approaches: fits to the asymptotic predictions of the theory, and fit-parameter-free binary MCT calculations based on static-structure-factor input from the simulations. We find that many-body interactions of the Voronoi mixture do not lead to strong qualitative differences relative to similar analyses carried out for simple liquids with pair-wise interactions. For instance, the fits give an exponent parameter λ ≈ 0.746 comparable to typical values found for simple liquids, the wavevector dependence of the Kohlrausch relaxation time is in good qualitative agreement with literature results for polydisperse hard spheres, and the MCT calculations based on static input overestimate the critical temperature, albeit only by a factor of about 1.2. This overestimation appears to be weak relative to other well-studied supercooled-liquid models such as the binary Kob-Andersen Lennard-Jones mixture. Overall, the agreement between MCT and simulation suggests that it is possible to predict several microscopic dynamic properties with qualitative, and in some cases near-quantitative, accuracy based solely on static two-point structural correlations, even though the system itself is inherently governed by many-body interactions.

Composition fluctuations in polydisperse liquids: Glasslike effects well above the glass transition.

We study a two-dimensional glass-forming system of slightly polydisperse (LJ) particles using molecular dynamics simulations and demonstrate that in the liquid regime (well above the vitrification temperature) this model shows a number of features typical of the glass transition: (i) the relation between compressibility and structure factor S(q) is strongly violated; (ii) the dynamical structure factor S(q,t) at low q shows a two-step relaxation; (iii) the time-dependent heat capacity c_{v}(t) shows a long-time power-law tail. We show that these phenomena can be rationalized with the idea of composition fluctuations and provide a quantitative theory for the effects (i) and (ii). It implies that such effects must be inherent in all polydisperse colloidal models, including binary LJ mixtures.

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 ξs). Our approach is based on the assumption that ξs 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.

Continuous-time random-walk approach to supercooled liquids: Self-part of the van Hove function and related quantities.

From equilibrium molecular dynamics (MD) simulations of a bead-spring model for short-chain glass-forming polymer melts we calculate several quantities characterizing the single-monomer dynamics near the (extrapolated) critical temperature [Formula: see text] of mode-coupling theory: the mean-square displacement g0(t), the non-Gaussian parameter [Formula: see text] and the self-part of the van Hove function [Formula: see text] which measures the distribution of monomer displacements r in time t. We also determine these quantities from a continuous-time random walk (CTRW) approach. The CTRW is defined in terms of various probability distributions which we know from previous analysis. Utilizing these distributions the CTRW can be solved numerically and compared to the MD data with no adjustable parameter. The MD results reveal the heterogeneous and non-Gaussian single-particle dynamics of the supercooled melt near [Formula: see text]. In the time window of the early [Formula: see text] relaxation [Formula: see text] is large and [Formula: see text] is broad, reflecting the coexistence of monomer displacements that are much smaller ("slow particles") and much larger ("fast particles") than the average at time t, i.e. than [Formula: see text]. For large r the tail of [Formula: see text] is compatible with an exponential decay, as found for many glassy systems. The CTRW can reproduce the spatiotemporal dependence of [Formula: see text] at a qualitative to semiquantitative level. However, it is not quantitatively accurate in the studied temperature regime, although the agreement with the MD data improves upon cooling. In the early [Formula: see text] regime we also analyze the MD results for [Formula: see text] via the space-time factorization theorem predicted by ideal mode-coupling theory. While we find the factorization to be well satisfied for small r, both above and below [Formula: see text] , deviations occur for larger r comprising the tail of [Formula: see text]. The CTRW analysis suggests that single-particle "hops" are a contributing factor for these deviations.

Voronoi glass-forming liquids: A structural study.

We introduce a theoretical model of simple fluid, whose interactions, defined in terms of the Voronoi cells of the configurations, are local and many-body. The resulting system is studied both theoretically and numerically. We show that the fluid, though sharing the global features of other models of fluids with soft interactions, has several unusual characteristics, which are investigated and discussed.

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).

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.

A novel interferometric method for the study of the viscoelastic properties of ultra-thin polymer films determined from nanobubble inflation.

Glass formation and glassy behavior remain as the important areas of investigation in soft matter physics with many aspects which are still not completely understood, especially at the nanometer size-scale. In the present work, we show an extension of the "nanobubble inflation" method developed by O'Connell and McKenna [Rev. Sci. Instrum. 78, 013901 (2007)] which uses an interferometric method to measure the topography of a large array of 5 µm sized nanometer thick films subjected to constant inflation pressures during which the bubbles grow or creep with time. The interferometric method offers the possibility of making measurements on multiple bubbles at once as well as having the advantage over the AFM methods of O'Connell and McKenna of being a true non-contact method. Here we demonstrate the method using ultra-thin films of both poly(vinyl acetate) (PVAc) and polystyrene (PS) and discuss the capabilities of the method relative to the AFM method, its advantages and disadvantages. Furthermore we show that the results from experiments on PVAc are consistent with the prior work on PVAc, while high stress results with PS show signs of a new non-linear response regime that may be related to the plasticity of the ultra-thin film.

Molecular dynamics simulation of the capillary leveling of viscoelastic polymer films.

Surface tension-driven flow techniques have recently emerged as an efficient means of shedding light into the rheology of thin polymer films. Motivated by experimental and theoretical approaches in films bearing a varying surface topography, we present results on the capillary relaxation of a square pattern at the free surface of a viscoelastic polymer film, using molecular dynamics simulations of a coarse-grained polymer model. Height profiles are monitored as a function of time after heating the system above its glass-transition temperature and their time dependence is fitted to the theory of capillary leveling. Results show that the viscosity is not constant, but time dependent. In addition to providing a complementary insight about the local inner mechanisms, our simulations of the capillary-leveling process therefore probe the viscoelasticity of the polymer and not only its viscosity, in contrast to most experimental approaches.

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.