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.

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.

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. .

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.

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.

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.

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.

Marginally compact hyperbranched polymer trees.

Assuming Gaussian chain statistics along the chain contour, we generate by means of a proper fractal generator hyperbranched polymer trees which are marginally compact. Static and dynamical properties, such as the radial intrachain pair density distribution ρpair(r) or the shear-stress relaxation modulus G(t), are investigated theoretically and by means of computer simulations. We emphasize that albeit the self-contact density diverges logarithmically with the total mass N, this effect becomes rapidly irrelevant with increasing spacer length S. In addition to this it is seen that the standard Rouse analysis must necessarily become inappropriate for compact objects for which the relaxation time τp of mode p must scale as τp ∼ (N/p)5/3 rather than the usual square power law for linear chains.

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.

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.

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.

Combustion Processes as a Source of High Levels of Indoor Hydroxyl Radicals through the Photolysis of Nitrous Acid.

Hydroxyl radicals (OH) are known to control the oxidative capacity of the atmosphere but their influence on reactivity within indoor environments is believed to be of little importance. Atmospheric direct sources of OH include the photolysis of ozone and nitrous acid (HONO) and the ozonolysis of alkenes. It has been argued that the ultraviolet light fraction of the solar spectrum is largely attenuated within indoor environments, thus, limiting the extent of photolytic OH sources. Conversely, the ozonolysis of alkenes has been suggested as the main pathway of OH formation within indoor settings. According to this hypothesis the indoor OH radical concentrations span in the range of only 10(4) to 10(5) cm(-3). However, recent direct OH radical measurements within a school classroom yielded OH radical peak values at moderate light intensity measured at evenings of 1.8 × 10(6) cm(-3) that were attributed to the photolysis of HONO. In this work, we report results from chamber experiments irradiated with varying light intensities in order to mimic realistic indoor lighting conditions. The exhaust of a burning candle was introduced in the chamber as a typical indoor source causing a sharp peak of HONO, but also of nitrogen oxides (NOx). The photolysis of HONO yields peak OH concentration values, that for the range of indoors lightning conditions were estimated in the range 5.7 ×· 10(6) to 1.6 × 10(7) cm(-3). Excellent agreement exists between OH levels determined by a chemical clock and those calculated by a simple PSS model. These findings suggest that significant OH reactivity takes place at our dwellings and the consequences of this reactivity-that is, formation of secondary oxidants-ought to be studied hereafter.

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

We revisit the relation between the shear-stress relaxation modulus G(t), computed at finite shear strain 0<γ≪1, and the shear-stress autocorrelation functions C(t)|(γ) and C(t)|(τ) computed, respectively, at imposed strain γ and mean stress τ. Focusing on permanent isotropic spring networks it is shown theoretically and computationally that in general G(t)=C(t)|(τ)=C(t)|(γ)+G(eq) for t>0 with G(eq) being the static equilibrium shear modulus. G(t) and C(t)|(γ) thus must become different for solids and it is impossible to obtain G(eq) alone from C(t)|(γ) as often assumed. We comment briefly on self-assembled transient networks where G(eq)(f) must vanish for a finite scission-recombination frequency f. We argue that G(t)=C(t)|(τ)=C(t)|(γ) should reveal an intermediate plateau set by the shear modulus G(eq)(f=0) of the quenched network.