[Show abstract][Hide abstract] ABSTRACT: Shear-strain and shear-stress correlations in isotropic elastic bodies are
investigated both theoretically and numerically at either imposed mean
shear-stress $\tau$ ($\lambda=0$) or shear-strain $\gamma$ ($\lambda=1$) and
for more general values of a dimensionless parameter $\lambda$ characterizing
the generalized Gaussian ensemble. It allows to tune the strain fluctuations
$\mu_{\gamma\gamma} \equiv \beta V \la \delta \gamma^2 \ra =
(1-\lambda)/G_{eq}$ with $\beta$ being the inverse temperature, $V$ the volume,
$\gamma$ 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 $\mu_{\tau\tau} \equiv \beta V \la \delta\tau^2 \ra$ ($\tau$ being
the instantaneous stress) that $\mu_{\tau\tau} = \mu_{A} - \lambda G_{eq}$ with
$\mu_{A} = \mu_{\tau\tau}|_{\lambda=0}$ being the affine shear-elasticity. For
the stress autocorrelation function $c_{\tau\tau}(t) \equiv \beta V \la \delta
\tau(t) \delta \tau(0) \ra$ this result is then seen (assuming a sufficiently
slow shear-stress barostat) to generalize to $c_{\tau\tau}(t) = G(t) - \lambda
\Geq$ with $G(t)$ being the shear-stress relaxation modulus.
[Show abstract][Hide abstract] ABSTRACT: The shear stress relaxation modulus G(t) may be determined from the shear stress after switching on a tiny step strain γ or by inverse Fourier transformation of the storage modulus G′(ω) or the loss modulus G′′(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 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) = Geq + C(t) for t > 0 with Geq being the static equilibrium shear modulus. A similar relation holds for G′(ω). G(t) and C(t) must thus become different for a solid body and it is impossible to obtain Geq directly from C(t).
[Show abstract][Hide abstract] ABSTRACT: Glasses are inherently out-of-equilibrium systems evolving slowly toward their equilibrium state in a process called physical aging. During aging, dynamic observables depend on the history of the system, hampering comparative studies of dynamics in different glass formers. Here, we demonstrate how glass formers can be directly compared on the level of single-particle jumps, i.e. the structural relaxation events underlying the α-process. Describing the dynamics in terms of a continuous-time random walk, an analytic prediction for the jump rate is derived. The result is subsequently compared to molecular-dynamics simulations of amorphous silica and a polymer melt as two generic representatives of strong and fragile glass formers, and good agreement is found.
[Show abstract][Hide abstract] ABSTRACT: 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.
Physical Review E 02/2015; 91(2). DOI:10.1103/PhysRevE.91.022107 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We present molecular-dynamics simulations for a fully flexible model of polymer melts with different chain length N ranging from short oligomers (N = 4) to values near the entanglement length (N = 64). For these systems we explore the structural relaxation of the supercooled melt near the critical temperature T c of mode-coupling theory (MCT). Coherent and incoherent scattering functions are analyzed in terms of the idealized MCT. For temperatures T > T c we provide evidence for the space-time factorization property of the β relaxation and for the time-temperature superposition principle (TTSP) of the α relaxation, and we also discuss deviations from these predictions for T ≈ T c. For T larger than the smallest temperature where the TTSP holds we perform a quantitative analysis of the dynamics with the asymptotic MCT predictions for the late β regime. Within MCT a key quantity, in addition to T c, is the exponent parameter λ. For the fully flexible polymer models studied we find that λ is independent of N and has a value (λ = 0.735 ) typical of simple glass-forming liquids. On the other hand, the critical temperature increases with chain length toward an asymptotic value T c (∞) . This increase can be described by T c (∞) - T c(N) ∼ 1/N and may be interpreted in terms of the N dependence of the monomer density ρ, if we assume that the MCT glass transition is ruled by a soft-sphere-like constant coupling parameter Γ c = ρ c T c (-1/4), where ρ c is the monomer density at T c. In addition, we also estimate T c from a Hansen-Verlet-like criterion and MCT calculations based on structural input from the simulation. For our polymer model both the Hansen-Verlet criterion and the MCT calculations suggest T c to decrease with increasing chain length, in contrast to the direct analysis of the simulation data.
The European Physical Journal E 02/2015; 38(2):97. DOI:10.1140/epje/i2015-15011-x · 1.76 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We discuss systems for which two carefully derived, yet conflicting, theories coexisted. Dense polymers in two dimensions and star-shaped polymers in the ?-regime are considered. In both cases the two proposed theories are in a sense exact, but turn out to satisfy different crossing rules (for the 2d polymer) or to correspond to different orders of limits. Finally, both theories prove very useful, albeit for different subclasses of physical systems.
Journal of Statistical Mechanics Theory and Experiment 04/2014; 2014(4):P04024. DOI:10.1088/1742-5468/2014/04/P04024 · 2.40 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The continuous-time random walk (CTRW) describes the single-particle dynamics as a series of jumps separated by random waiting times. This description is applied to analyze trajectories from molecular dynamics (MD) simulations of a supercooled polymer melt. Based on the algorithm presented by Helfferich et al. [Phys. Rev. E 89, 042603 (2014)], we detect jump events of the monomers. As a function of temperature and chain length, we examine key distributions of the CTRW: the jump-length distribution (JLD), the waiting-time distribution (WTD), and the persistence-time distribution (PTD), i.e., the distribution of waiting times for the first jump. For the equilibrium (polymer) liquid under consideration, we verify that the PTD is determined by the WTD. For the mean-square displacement (MSD) of a monomer, the results for the CTRW model are compared with the underlying MD data. The MD data exhibit two regimes of subdiffusive behavior, one for the early α process and another at later times due to chain connectivity. By contrast, the analytical solution of the CTRW yields diffusive behavior for the MSD at all times. Empirically, we can account for the effect of chain connectivity in Monte Carlo simulations of the CTRW. The results of these simulations are then in good agreement with the MD data in the connectivity-dominated regime, but not in the early α regime where they systematically underestimate the MSD from the MD.
Physical Review E 04/2014; 89(4-1):042604. · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Single-particle trajectories in supercooled liquids display long periods of localization interrupted by "fast moves." This observation suggests a modeling by a continuous-time random walk (CTRW). We perform molecular dynamics simulations of equilibrated short-chain polymer melts near the critical temperature of mode-coupling theory Tc and extract "moves" from the monomer trajectories. We show that not all moves comply with the conditions of a CTRW. Strong forward-backward correlations are found in the supercooled state. A refinement procedure is suggested to exclude these moves from the analysis. We discuss the repercussions of the refinement on the jump-length and waiting-time distributions as well as on characteristic time scales, such as the average waiting time ("exchange time") and the average time for the first move ("persistence time"). The refinement modifies the temperature (T) dependence of these time scales. For instance, the average waiting time changes from an Arrhenius-type to a Vogel-Fulcher-type T dependence. We discuss this observation in the context of the bifurcation of the α process and (Johari) β process found in many glass-forming materials to occur near Tc. Our analysis lays the foundation for a study of the jump-length and waiting-time distributions, their temperature and chain-length dependencies, and the modeling of the monomer dynamics by a CTRW approach in the companion paper [J. Helfferich et al., Phys. Rev. E 89, 042604 (2014)].
Physical Review E 04/2014; 89(4-1):042603. · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The continuous-time random walk (CTRW) describes the single-particle dynamics as a series of jumps separated by random waiting times. This description is applied to analyze trajectories from molecular dynamics (MD) simulations of a supercooled polymer melt. Based on the algorithm presented by Helfferich et al. [Phys. Rev. E 89, 042603 (2014), 10.1103/PhysRevE.89.042603], we detect jump events of the monomers. As a function of temperature and chain length, we examine key distributions of the CTRW: the jump-length distribution (JLD), the waiting-time distribution (WTD), and the persistence-time distribution (PTD), i.e., the distribution of waiting times for the first jump. For the equilibrium (polymer) liquid under consideration, we verify that the PTD is determined by the WTD. For the mean-square displacement (MSD) of a monomer, the results for the CTRW model are compared with the underlying MD data. The MD data exhibit two regimes of subdiffusive behavior, one for the early α process and another at later times due to chain connectivity. By contrast, the analytical solution of the CTRW yields diffusive behavior for the MSD at all times. Empirically, we can account for the effect of chain connectivity in Monte Carlo simulations of the CTRW. The results of these simulations are then in good agreement with the MD data in the connectivity-dominated regime, but not in the early α regime where they systematically underestimate the MSD from the MD.
Physical Review E 03/2014; 89(4). DOI:10.1103/PhysRevE.89.042604 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Single-particle trajectories in supercooled liquids display long periods of localization interrupted by "fast moves." This observation suggests a modeling by a continuous-time random walk (CTRW). We perform molecular dynamics simulations of equilibrated short-chain polymer melts near the critical temperature of mode-coupling theory Tc and extract "moves" from the monomer trajectories. We show that not all moves comply with the conditions of a CTRW. Strong forward-backward correlations are found in the supercooled state. A refinement procedure is suggested to exclude these moves from the analysis. We discuss the repercussions of the refinement on the jump-length and waiting-time distributions as well as on characteristic time scales, such as the average waiting time ("exchange time") and the average time for the first move ("persistence time"). The refinement modifies the temperature (T) dependence of these time scales. For instance, the average waiting time changes from an Arrhenius-type to a Vogel-Fulcher-type T dependence. We discuss this observation in the context of the bifurcation of the α process and (Johari) β process found in many glass-forming materials to occur near Tc. Our analysis lays the foundation for a study of the jump-length and waiting-time distributions, their temperature and chain-length dependencies, and the modeling of the monomer dynamics by a CTRW approach in the companion paper [J. Helfferich et al., Phys. Rev. E 89, 042604 (2014), 10.1103/PhysRevE.89.042604].
Physical Review E 03/2014; 89(4). DOI:10.1103/PhysRevE.89.042603 · 2.29 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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.
The European Physical Journal E 02/2014; 37(2):9968. DOI:10.1140/epje/i2014-14012-7 · 1.76 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Melts of unconcatenated and unknotted polymer rings are a paradigm for soft matter ruled by topological interactions. We propose a description of a system of rings of length N as a collection of smaller polydisperse Gaussian loops, ranging from the entanglement length to the skeleton ring length \sim N^{2/3} , assembled in random trees. Individual rings in the melt are predicted to be marginally compact with a mean square radius of gyration R_g^2 \sim N^{2/3}(1-\text{const} \cdot N^{-1/3}) . As a rule, simple power laws for asymptotically long rings come with sluggish crossovers. Experiments and computer simulations merely deal with crossover regimes typically extending to N\sim 10^{3\text{-}4} . The estimated crossover functions allow for a satisfactory fit of simulation data.
[Show abstract][Hide abstract] ABSTRACT: By considering Voronoi tessellations of the configurations of a fluid, we
propose two new conserved fields, which provide structural information not
fully accounted for by the usual 2-point density field fluctuations (structure
factor). One of these fields is scalar and associated to the Voronoi cell
volumes, whereas the other one, termed the "geometrical polarisation", is
vectorial, related to the very local anisotropy of the configurations. We study
the static and dynamical properties of these fields in the supercooled regime
of a model glass-forming liquid. We show in particular that the geometrical
polarisation is both statically correlated to the force field and contrary to
it develops a plateau regime when the temperature is lowered. We attribute this
behaviour to the microsopic disorder of the underlying inherent structures (IS)
which dictate the dynamics on time scales larger than the true microscopic
time, in the strong supercooled regime. In this respect, this work raises the
issue of to what extent the inter IS dynamics, intrinsically anisotropic and
collective (cf. T.B. Schr{\o}der et al. {\it J. of Chem. Phys.}, {\bf 112},
9834 (2000)), could be related to their polarisation field.
The European Physical Journal E 12/2013; 37(6). DOI:10.1140/epje/i2014-14046-9 · 1.76 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Recent computational studies on melts of nonconcatenated rings suggest compact configurations of fractal dimension df = 3. This begs the question of whether the irregular surfaces of these compact rings may be characterized by a fractal surface dimension ds < 3. We revisit the scaling analysis of the form factor by Halverson et al. [J. Chem. Phys. 134, 204904 (2011)] implying ds ≈ 2.8. Our analysis suggests that this conclusion might be due to the application of the Generalized Porod Law at large wavevectors where length scales other than the total chain size do matter. We present an alternative "decorated Gaussian loop" model which does not require ds < 3.
The Journal of Chemical Physics 12/2013; 139(21):217101. DOI:10.1063/1.4833140 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: 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. The stress fluctuation representation F Row|λ=1 of the compression modulus K in the NVT-ensemble is derived directly (without a microscopic displacement field) using the well-known thermodynamic transformation rules between conjugated ensembles. The transform is made manifest by computing the Rowlinson functional FRow also in the NPT-ensemble where FRow|λ=0 = Kf0(x) with x = Pid/K being a scaling variable, Pid the ideal pressure and f0(x) = x(2-x) a universal function. By gradually increasing λ by means of an external spring potential, the crossover between both classical ensemble limits is monitored. This demonstrates, e.g., the lever rule FRow|λ = K[λ + (1 - λ)f0(x)].
The European Physical Journal E 11/2013; 36(11):9945. DOI:10.1140/epje/i2013-13131-y · 1.76 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The density crossover scaling of thermodynamic and conformational properties of solutions and melts of self-avoiding and highly flexible polymer chains without chain intersections confined to strictly two dimensions (d = 2) is investigated by means of molecular dynamics and Monte Carlo simulations of a standard coarse grained bead-spring model. We focus on properties related to the contact exponent set by the intrachain subchain size distribution. With R ∼ N
ν being the size of chains of length N and ρ the monomer density, the interaction energy e
int between monomers from different chains and the corresponding number n
int of interchain contacts per monomer are found to scale as
with ν = 3/4 and θ2 = 19/12 for dilute solutions and ν = 1/d and θ2 = 3/4 for N≫ g(ρ) ≈ 1/ρ2. Irrespective of ρ, long chains thus become compact packings of blobs of contour length
with d
p = d − θ2 = 5/4 being the fractal line dimension. Due to the generalized Porod scattering of the compact chains, the Kratky representation of the intramolecular form factor F(q) reveals a non-monotonous behavior approaching with increasing chain length and density a power-law slope
$F(q)q^d /\rho \approx 1/(qR)^{\theta _2 } $
in the intermediate regime of the wavevector q. The specific intermolecular contact probability is argued to imply an enhanced compatibility for polymer blends confined to ultrathin films. We comment briefly on finite persistence length effects.
Polymer Science Series C 09/2013; 55(1). DOI:10.1134/S1811238213070072 · 1.04 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Comparing isotropic solids and fluids at either imposed volume or pressure, we investigate various correlations of the instantaneous pressure and its ideal and excess contributions. Focusing on the compression modulus K, it is emphasized that the stress fluctuation representation of the elastic moduli may be obtained directly (without a microscopic displacement field) by comparing the stress fluctuations in conjugated ensembles. This is made manifest by computing the Rowlinson stress fluctuation expression Krow of the compression modulus for NPT-ensembles. It is shown theoretically and numerically that Krow∣P = Pid(2 - Pid∕K) with Pid being the ideal pressure contribution.
The Journal of Chemical Physics 05/2013; 138(19):191101. DOI:10.1063/1.4807305 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The shear modulus G of two glass-forming colloidal model systems in d = 3 and d = 2 dimensions is investigated by means of, respectively, molecular dynamics and Monte Carlo simulations. Comparing ensembles where either the shear strain γ or the conjugated (mean) shear stress τ are imposed, we compute G from the respective stress and strain fluctuations as a function of temperature T while keeping a constant normal pressure P. The choice of the ensemble is seen to be highly relevant for the shear stress fluctuations μF(T) which at constant τ decay monotonously with T following the affine shear elasticity μA(T), i.e., a simple two-point correlation function. At variance, non-monotonous behavior with a maximum at the glass transition temperature Tg is demonstrated for μF(T) at constant γ. The increase of G below Tg is reasonably fitted for both models by a continuous cusp singularity, G(T)∝(1 - T∕Tg)(1∕2), in qualitative agreement with recent theoretical predictions. It is argued, however, that longer sampling times may lead to a sharper transition.
The Journal of Chemical Physics 03/2013; 138(12):12A533. DOI:10.1063/1.4790137 · 2.95 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this chapter we return to mathematical foundations and discuss the theory of stable distributions as an extension to the central limit theorem and the canonical representation of stable distributions. The solution of the one-dimensional Weierstrass random walk is presented in detail, including its fractal properties and super-diffusive behavior. Fractal time random walks and their relation to subdiffusive behavior are introduced next. Finally, we discuss the truncated Lévy flight which will be employed in Chap. 5 for the description of financial fluctuations.
[Show abstract][Hide abstract] ABSTRACT: The truncation of a pair potential at a distance r_cut is well-known to imply
in general an impulsive correction to the pressure and other moments of the
first derivatives of the potential. That depending on r_cut the truncation may
also be of relevance to higher derivatives is shown theoretically for the Born
contributions to the elastic moduli obtained using the stress-fluctuation
formalism in d dimensions. Focusing on isotropic liquids for which the shear
modulus G must vanish by construction, the predicted corrections are tested
numerically for binary mixtures and polydisperse Lennard-Jones beads in,
respectively, d=3 and d=2 dimensions.
Physical Review E 10/2012; 86(4-2):046705. DOI:10.1103/PhysRevE.86.046705 · 2.29 Impact Factor