arXiv subjects

Annette Zippelius

Publications and source records attributed to Annette Zippelius.

87 records · Page 5Linked to original sources

arXiv2001

Energy landscape of a Lennard-Jones liquid: Statistics of stationary points

Molecular dynamics simulations are used to generate an ensemble of saddles of the potential energy of a Lennard-Jones liquid. Classifying all extrema by their potential energy u and number of unstable directions k, a well defined relation k(u) is revealed. The degree of instability of typical stationary points vanishes at a threshold potential energy, which lies above the energy of the lowest glassy minima of the system. The energies of the inherent states, as obtained by the Stillinger-Weber method, approach the threshold energy at a temperature close to the mode-coupling transition temperature Tc.

cond-mat.stat-mech↗

arXiv2000

Stress Relaxation of Near-Critical Gels

The time-dependent stress relaxation for a Rouse model of a crosslinked polymer melt is completely determined by the spectrum of eigenvalues of the connectivity matrix. The latter has been computed analytically for a mean-field distribution of crosslinks. It shows a Lifshitz tail for small eigenvalues and all concentrations below the percolation threshold, giving rise to a stretched exponential decay of the stress relaxation function in the sol phase. At the critical point the density of states is finite for small eigenvalues, resulting in a logarithmic divergence of the viscosity and an algebraic decay of the stress relaxation function. Numerical diagonalization of the connectivity matrix supports the analytical findings and has furthermore been applied to cluster statistics corresponding to random bond percolation in two and three dimensions.

cond-mat.soft↗

arXiv2000

Free cooling of particles with rotational degrees of freedom

Free cooling of granular materials is analyzed on the basis of a pseudo-Liouville operator. Exchange of translational and rotational energy requires surface roughness for spherical grains, but occurs for non-spherical grains, like needles, even if they are perfectly smooth. Based on the assumption of a homogeneous cooling state, we derive an approximate analytical theory. It predicts that cooling of both rough spheres and smooth needles proceeds in two stages: An exponentially fast decay to a state with stationary ratio of translational and rotational energy and a subsequent algebraic decay of the total energy. These results are confirmed by simulations for large systems of moderate density. For higher densities, we observe deviations from the homogeneous state as well as large-scale structures in the velocity field. We study non-Gaussian distributions of the momenta perturbatively and observe a breakdown of the expansion for particular values of surface roughness and normal restitution.

cond-mat.stat-mech↗

arXiv2000

Critical Dynamics of Gelation

Shear relaxation and dynamic density fluctuations are studied within a Rouse model, generalized to include the effects of permanent random crosslinks. We derive an exact correspondence between the static shear viscosity and the resistance of a random resistor network. This relation allows us to compute the static shear viscosity exactly for uncorrelated crosslinks. For more general percolation models, which are amenable to a scaling description, it yields the scaling relation $ k=ϕ-β$ for the critical exponent of the shear viscosity. Here $β$ is the thermal exponent for the gel fraction and $ϕ$ is the crossover exponent of the resistor network. The results on the shear viscosity are also used in deriving upper and lower bounds on the incoherent scattering function in the long-time limit, thereby corroborating previous results.

cond-mat.stat-mech↗

arXiv2000

Saddles on the potential energy landscape of a Lennard-Jones liquid

By means of molecular dynamics simulations, we study the stationary points of the potential energy in a Lennard-Jones liquid, giving a purely geometric characterization of the energy landscape of the system. We find a linear relation between the degree of instability of the stationary points and their potential energy, and we locate the energy where the instability vanishes. This threshold energy marks the border between saddle-dominated and minima-dominated regions of the energy landscape. The temperature where the potential energy of the Stillinger-Weber minima becomes equal to the threshold energy turns out to be very close to the mode-coupling transition temperature.

cond-mat.stat-mech↗

arXiv1999

Dynamics of inelastically colliding spheres with Coulomb friction: Relaxation of translational and rotational energy

We investigate the free cooling of inelastic rough spheres in the presence of Coulomb friction. Depending on the coefficients of normal restitution $ε$ and Coulomb friction $μ$, we find qualitatively different asymptotic states. For nearly complete normal restitution ($ε$ close to 1) and large $μ$, friction does not change the cooling properties qualitatively compared to a constant coefficient of tangential restitution. In particular, the asymptotic state is characterized by a constant ratio of rotational and translational energies, both decaying according to Haff's law. However, for small $ε$ and small $μ$, the dissipation of rotational energy is suppressed, so that the asymptotic state is characterized by constant rotational energy while the translational energy continues to decay as predicted by Haff's law. Introducing either surface roughness for grazing collisions or cohesion forces for collisions with vanishing normal load, causes the rotational energy to decay according to Haff's law again in the asymptotic long-time limit with, however, an intermediate regime of approximately constant rotational energy.

cond-mat.stat-mech↗

arXiv1999

Shear viscosity of a crosslinked polymer melt

We investigate the static shear viscosity on the sol side of the vulcanization transition within a minimal mesoscopic model for the Rouse-dynamics of a randomly crosslinked melt of phantom polymers. We derive an exact relation between the viscosity and the resistances measured in a corresponding random resistor network. This enables us to calculate the viscosity exactly for an ensemble of crosslinks without correlations. The viscosity diverges logarithmically as the critical point is approached. For a more realistic ensemble of crosslinks amenable to the scaling description of percolation, we prove the scaling relation $k=ϕ-β$ between the critical exponent $k$ of the viscosity, the thermal exponent $β$ associated with the gel fraction and the crossover exponent $ϕ$ of a random resistor network.

cond-mat.stat-mech↗

arXiv1999

The amorphous solid state: a locally stable thermodynamic phase of randomly constrained systems

The question of the local stability of the (replica-symmetric) amorphous solid state is addressed for a class of systems undergoing a continuous liquid to amorphous-solid phase transition driven by the effect of random constraints. The Hessian matrix, associated with infinitesimal fluctuations around the stationary point corresponding to the amorphous solid state, is obtained. The eigenvalues of this Hessian matrix are all shown to be strictly positive near the transition, except for one--the zero mode associated with the spontaneously broken continuous translational symmetry of the system. Thus the local stability of the amorphous solid state is established.

cond-mat.dis-nn↗

arXiv1999

Granular cooling of hard needles

We have developed a kinetic theory of hard needles undergoing binary collisions with loss of energy due to normal and tangential restitution. In addition, we have simulated many particle systems of granular hard needles. The theory, based on the assumption of a homogeneous cooling state, predicts that granular cooling of the needles proceeds in two stages: An exponential decay of the initial configuration to a state where translational and rotational energies take on a time independent ratio (not necessarily unity), followed by an algebraic decay of the total kinetic energy $\sim t^{-2}$. The simulations support the theory very well for low and moderate densities. For higher densities, we have observed the onset of the formation of clusters and shear bands.

cond-mat.stat-mech↗

arXiv1998

Coefficient of restitution for elastic disks

We calculate the coefficient of restitution, $ε$, starting from a microscopic model of elastic disks. The theory is shown to agree with the approach of Hertz in the quasistatic limit, but predicts inelastic collisions for finite relative velocities of two approaching disks. The velocity dependence of $ε$ is calculated numerically for a wide range of velocities. The coefficient of restitution furthermore depends on the elastic constants of the material via Poisson's number. The elastic vibrations absorb kinetic energy more effectively for materials with low values of the shear modulus.

cond-mat↗

arXiv1997

Dynamics of inelastically colliding rough spheres: Relaxation of translational and rotational energy

We study the exchange of kinetic energy between translational and rotational degrees of freedom for inelastic collisions of rough spheres. Even if equipartition holds in the initial state it is immediately destroyed by collisions. The simplest generalisation of the homogeneous cooling state allows for two temperatures, characterizing translational and rotational degrees of freedom separately. For times larger than a crossover frequency, which is determined by the Enskog frequency and the initial temperature, both energies decay algebraically like $t^{-2}$ with a fixed ratio of amplitudes, different from one.

cond-mat.stat-mech↗

arXiv1997

Cooling dynamics of a dilute gas of inelastic rods: a many particle simulation

We present results of simulations for a dilute gas of inelastically colliding particles. Collisions are modelled as a stochastic process, which on average decreases the translational energy (cooling), but allows for fluctuations in the transfer of energy to internal vibrations. We show that these fluctuations are strong enough to suppress inelastic collapse. This allows us to study large systems for long times in the truely inelastic regime. During the cooling stage we observe complex cluster dynamics, as large clusters of particles form, collide and merge or dissolve. Typical clusters are found to survive long enough to establish local equilibrium within a cluster, but not among different clusters. We extend the model to include net dissipation of energy by damping of the internal vibrations. Inelatic collapse is avoided also in this case but in contrast to the conservative system the translational energy decays according to the mean field scaling law, E(t)\propto t^{-2}, for asymptotically long times.

cond-mat.stat-mech↗

arXiv1997

Universality and its Origins at the Amorphous Solidification Transition

Systems undergoing an equilibrium phase transition from a liquid state to an amorphous solid state exhibit certain universal characteristics. Chief among these are the fraction of particles that are randomly localized and the scaling functions that describe the order parameter and (equivalently) the statistical distribution of localization lengths for these localized particles. The purpose of this Paper is to discuss the origins and consequences of this universality, and in doing so, three themes are explored. First, a replica-Landau-type approach is formulated for the universality class of systems that are composed of extended objects connected by permanent random constraints and undergo amorphous solidification at a critical density of constraints. This formulation generalizes the cases of randomly cross-linked and end-linked macromolecular systems, discussed previously. The universal replica free energy is constructed, in terms of the replica order parameter appropriate to amorphous solidification, the value of the order parameter is obtained in the liquid and amorphous solid states, and the chief universal characteristics are determined. Second, the theory is reformulated in terms of the distribution of local static density fluctuations rather than the replica order parameter. It is shown that a suitable free energy can be constructed, depending on the distribution of static density fluctuations, and that this formulation yields precisely the same conclusions as the replica approach. Third, the universal predictions of the theory are compared with the results of extensive numerical simulations of randomly cross-linked macromolecular systems, due to Barsky and Plischke, and excellent agreement is found.

cond-mat.dis-nn↗

arXiv1997

Dynamical signatures of the vulcanization transition

Dynamical properties of vulcanized polymer networks are addressed via a Rouse-type model that incorporates the effect of permanent random crosslinks. The incoherent intermediate scattering function is computed in the sol and gel phases, and at the vulcanization transition between them. At any nonzero crosslink density within the sol phase Kohlrausch relaxation is found. The critical point is signalled by divergence of the longest time-scale, and at this point the scattering function decays algebraically, whereas within the gel phase it acquires a time-persistent part identified with the gel fraction.

cond-mat.soft↗

arXiv1996

Randomly Crosslinked Macromolecular Systems: Vulcanisation Transition to and Properties of the Amorphous Solid State

As Charles Goodyear discovered in 1839, when he first vulcanised rubber, a macromolecular liquid is transformed into a solid when a sufficient density of permanent crosslinks is introduced at random. At this continuous equi- librium phase transition, the liquid state, in which all macromolecules are delocalised, is transformed into a solid state, in which a nonzero fraction of macromolecules have spontaneously become localised. This solid state is a most unusual one: localisation occurs about mean positions that are distributed homogeneously and randomly, and to an extent that varies randomly from monomer to monomer. Thus, the solid state emerging at the vulcanisation transition is an equilibrium amorphous solid state: it is properly viewed as a solid state that bears the same relationship to the liquid and crystalline states as the spin glass state of certain magnetic systems bears to the paramagnetic and ferromagnetic states, in the sense that, like the spin glass state, it is diagnosed by a subtle order parameter. In this review we give a detailed exposition of a theoretical approach to the physical properties of systems of randomly, permanently crosslinked macromolecules. Our primary focus is on the equilibrium properties of such systems, especially in the regime of Goodyear's vulcanisation transition.

cond-mat↗