arXiv subjects

J. Baschnagel

Publications and source records attributed to J. Baschnagel.

At least 19 recordsLinked to original sources

arXiv2023

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

cond-mat.stat-mech↗

arXiv2023

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 forth-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 wavevectors. It is shown that e is set by the typical size of the frozen-in stress components normal to the wavevectors, 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.

cond-mat.stat-mech↗

arXiv2022

Spatial correlation functions for non-ergodic stochastic processes of macroscopic system

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

cond-mat.stat-mech↗

arXiv2022

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 or the shear-stress relaxation modulus, 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 $τ_p$ of mode $p$ must scale as $τ_p \sim (N/p)^{5/3}$ rather than the usual square power law for linear chains.

cond-mat.soft↗

arXiv2022

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 $G(t) = μ_A - h(t)$ for the computational determination of the shear stress relaxation modulus $G(t)$. Here, $μ_A = G(0)$ 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 $G(t)=c(t)$ using the shear stress auto-correlation function $c(t)$ becomes incorrect in this limit.

cond-mat.soft↗

arXiv2021

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

We investigate simple models for strictly non-ergodic stochastic processes $x_t$ ($t$ being the discrete time step) focusing on the expectation value $v$ and the standard deviation $δv$ of the empirical variance $v[x]$ of finite time series $x$. $x_t$ is averaged over a fluctuating field $σ_{r}$ ($r$ being the microcell position) characterized by a quenched spatially correlated Gaussian field. Due to the quenched field $δv(Δt)$ becomes a finite constant, $Δ_{ne} > 0$, for large sampling times $Δt$. The volume dependence of the non-ergodicity parameter $Δ_{ne}$ is investigated for different spatial correlations. Models with marginally long-ranged $\fr$-correlations are successfully mapped on shear-stress data from simulated amorphous glasses of polydisperse beads.

cond-mat.dis-nn↗

arXiv2021

Fluctuations of non-ergodic stochastic processes

We investigate the standard deviation $δv(\tsamp)$ of the variance $v[\xbf]$ of time series $\xbf$ measured over a finite sampling time $\tsamp$ 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 $v[\xbf_{ck}]$ must be distinguished: the total variance $\dvtot = \dvint + \dvext$ and its contributions $\dvint$, the typical internal variance within the meta-basins, and $\dvext$, 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 $\svint$ and $\svext$ are manifested by their different system-size dependence

cond-mat.stat-mech↗

arXiv2020

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 $v(Δt)$ and the standard deviation $δv(Δt)$ of the variance $v[\mathbf{x}]$ of time series $\mathbf{x}$ of a stochastic process $x(t)$ measured over a finite sampling time $Δt$. Assuming a stationary, Gaussian and ergodic process, $δv$ is given by a functional $δv_G[h]$ of the autocorrelation function $h(t)$. $δv(Δt)$ is shown to become large and similar to $v(Δt)$ if $Δt$ corresponds to a fast relaxation process. Albeit $δv = δv_G[h]$ 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.

cond-mat.stat-mech↗

arXiv2018

Shear-stress fluctuations in free-standing polymer films

Using molecular dynamics simulation of a polymer glass model we investigate free-standing polymer films focusing on the in-plane shear modulus $μ$ and the corresponding shear-stress relaxation modulus $G(t)$ as functions of temperature $T$, film thickness $H$ (tuned by means of the lateral box size $L$) and sampling time $Δt$. Various observables are seen to vary linearly with $1/H$ demonstrating thus the (to leading order) linear superposition of bulk and surface properties. In agreement with recent studies on three-dimensional polymer glass-formers, $μ$ and $G(t)$ are found to decrease continuously with $T$. A jump-singularity is not observed. Confirming the time-translational invariance of our systems, the $Δt$-dependence of $μ$ is traced back to $G(t)$.

cond-mat.soft↗

arXiv2017

Shear-stress fluctuations and relaxation in polymer glasses

We investigate by means of molecular dynamics simulation a coarse-grained polymer glass model focusing on (quasi-static 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 non-monotonous 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 $G(t)$. This implies that the shear viscosity $η(T)$ may be readily obtained from the $1/Δt$-decay of $μ_{sf}$ above the glass transition.

cond-mat.soft↗

arXiv2017

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 non-monotonous 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 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.

cond-mat.soft↗

arXiv2016

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 $Δdt$ being the sampling time and $t_*(f) \sim 1/f$ the Maxwell relaxation time (set by the spring recombination frequency $f$) the dimensionless parameter $Δx = dt/t_*(f)$ is systematically scanned from the liquid limit ($Δdx \gg 1)$ to the solid limit ($Δx \ll 1$) 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 $G(t) = C(t)$ using the (rescaled) shear-stress autocorrelation function $C(t)$. Lower bounds for the $m$ configurations required by both relations are given.

cond-mat.stat-mech↗

arXiv2015

Simple-average expressions for shear-stress relaxation modulus

Focusing on isotropic elastic networks we propose a novel 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 and its equilibrium modulus $\G_{eq} = \lim_{t \to \infty} G(t)$. 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 $\hatτ(t)$ as a function of time $t$. While investigating sampling time effects we also discuss the related expressions in terms of shear-stress autocorrelation functions. We argue finally that our key relation may be readily adapted for more general linear response functions.

cond-mat.stat-mech↗

arXiv2015

Fluctuation-dissipation relation between shear stress relaxation modulus and shear stress autocorrelation function revisited

The shear stress relaxation modulus $G(t)$ may be determined from the shear stress $τ(t)$ after switching on a tiny step strain $γ$ or by inverse Fourier transformation of the storage modulus $G^{\prime}(ω)$ or the loss modulus $G^{\prime\prime}(ω)$ obtained in a standard oscillatory shear experiment at angular frequency $ω$. It is widely assumed that $G(t)$ is equivalent in general to the equilibrium stress autocorrelation function $C(t) = βV \langle δτ(t) δτ(0)\rangle$ which may be readily computed in computer simulations ($β$ being the inverse temperature and $V$ the volume). Focusing on isotropic solids formed by permanent spring networks we show theoretically by means of the fluctuation-dissipation theorem and computationally by molecular dynamics simulation that in general $G(t) = G_{eq} + C(t)$ for $t > 0$ with $G_{eq}$ being the static equilibrium shear modulus. A similar relation holds for $G^{\prime}(ω)$. $G(t)$ and $C(t)$ must thus become different for a solid body and it is impossible to obtain $G_{eq}$ directly from $C(t)$.

cond-mat.stat-mech↗

arXiv2015

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

We revisit the relation between the shear stress relaxation modulus $G(t)$, computed at finite shear strain $0 < γ\ll 1$, 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.

cond-mat.stat-mech↗

arXiv2015

Shear-strain and shear-stress fluctuations in generalized Gaussian ensemble simulations of isotropic elastic networks

Shear-strain and shear-stress correlations in isotropic elastic bodies are investigated both theoretically and numerically at either imposed mean shear-stress $τ$ ($λ=0$) or shear-strain $γ$ ($λ=1$) and for more general values of a dimensionless parameter $λ$ characterizing the generalized Gaussian ensemble. It allows to tune the strain fluctuations $μ_{γγ} \equiv βV \la δγ^2 \ra = (1-λ)/G_{eq}$ with $β$ being the inverse temperature, $V$ the volume, $γ$ the instantaneous strain and $G_{eq}$ the equilibrium shear modulus. Focusing on spring networks in two dimensions we show, e.g., for the stress fluctuations $μ_{ττ} \equiv βV \la δτ^2 \ra$ ($τ$ being the instantaneous stress) that $μ_{ττ} = μ_{A} - λG_{eq}$ with $μ_{A} = μ_{ττ}|_{λ=0}$ being the affine shear-elasticity. For the stress autocorrelation function $c_{ττ}(t) \equiv βV \la δτ(t) δτ(0) \ra$ this result is then seen (assuming a sufficiently slow shear-stress barostat) to generalize to $c_{ττ}(t) = G(t) - λ\Geq$ with $G(t)$ being the shear-stress relaxation modulus.

cond-mat.stat-mech↗

arXiv2015

Compressibility and pressure correlations in isotropic solids and fluids

Presenting simple coarse-grained models of isotropic solids and fluids in $d=1$, $2$ and $3$ dimensions we investigate the correlations of the instantaneous pressure and its ideal and excess contributions at either imposed pressure (NPT-ensemble, $λ=0$) or volume (NVT-ensemble, $λ=1$) and for more general values of the dimensionless parameter $λ$ characterizing the constant-volume constraint.

cond-mat.stat-mech↗

arXiv2015

Hyperbranched polymer stars with Gaussian chain statistics revisited

Conformational properties of regular dendrimers and more general hyperbranched polymer stars with Gaussian statistics for the spacer chains between branching points are revisited numerically. We investigate the scaling for asymptotically long chains especially for fractal dimensions $d_f = 3$ (marginally compact) and $d_f = 2.5$ (diffusion limited aggregation). Power-law stars obtained by imposing the number of additional arms per generation are compared to truly self-similar stars. We discuss effects of weak excluded volume interactions and sketch the regime where the Gaussian approximation should hold in dense solutions and melts for sufficiently large spacer chains.

cond-mat.soft↗