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.Comment: 13 pages, 14 figure

Formation and Equilibrium Properties of Living Polymer Brushes

Polydisperse brushes obtained by reversible radical chain polymerization reaction onto a solid substrate with surface-attached initiators, are studied by means of an off-lattice Monte Carlo algorithm of living polymers (LP). Various properties of such brushes, like the average chain length and the conformational orientation of the polymers, or the force exerted by the brush on the opposite container wall, reveal power-law dependence on the relevant parameters. The observed molecular weight distribution (MWD) of the grafted LP decays much more slowly than the corresponding LP bulk system due to the gradient of the monomer density within the dense pseudo-brush which favors longer chains. Both MWD and the density profiles of grafted polymers and chain ends are well fitted by effective power laws whereby the different exponents turn out to be mutually self-consistent for a pseudo-brush in the strong-stretching regime.Comment: 33 pages, 11 figues, J.Chem. Phys. accepted Oct. 199

Dynamical Monte Carlo Study of Equilibrium Polymers : Static Properties

We report results of extensive Dynamical Monte Carlo investigations on self-assembled Equilibrium Polymers (EP) without loops in good solvent. (This is thought to provide a good model of giant surfactant micelles.) Using a novel algorithm we are able to describe efficiently both static and dynamic properties of systems in which the mean chain length \Lav is effectively comparable to that of laboratory experiments (up to 5000 monomers, even at high polymer densities). We sample up to scission energies of $E/k_BT=15$ over nearly three orders of magnitude in monomer density $\phi$, and present a detailed crossover study ranging from swollen EP chains in the dilute regime up to dense molten systems. Confirming recent theoretical predictions, the mean-chain length is found to scale as \Lav \propto \phi^\alpha \exp(\delta E) where the exponents approach $\alpha_d=\delta_d=1/(1+\gamma) \approx 0.46$ and $\alpha_s = 1/2 [1+(\gamma-1)/(\nu d -1)] \approx 0.6, \delta_s=1/2$ in the dilute and semidilute limits respectively. The chain length distribution is qualitatively well described in the dilute limit by the Schulz-Zimm distribution \cN(s)\approx s^{\gamma-1} \exp(-s) where the scaling variable is s=\gamma L/\Lav. The very large size of these simulations allows also an accurate determination of the self-avoiding walk susceptibility exponent $\gamma \approx 1.165 \pm 0.01$. ....... Finite-size effects are discussed in detail.Comment: 15 pages, 14 figures, LATE

Effects of compression on the vibrational modes of marginally jammed solids

Glasses have a large excess of low-frequency vibrational modes in comparison with most crystalline solids. We show that such a feature is a necessary consequence of the weak connectivity of the solid, and that the frequency of modes in excess is very sensitive to the pressure. We analyze in particular two systems whose density D(w) of vibrational modes of angular frequency w display scaling behaviors with the packing fraction: (i) simulations of jammed packings of particles interacting through finite-range, purely repulsive potentials, comprised of weakly compressed spheres at zero temperature and (ii) a system with the same network of contacts, but where the force between any particles in contact (and therefore the total pressure) is set to zero. We account in the two cases for the observed a) convergence of D(w) toward a non-zero constant as w goes to 0, b) appearance of a low-frequency cutoff w*, and c) power-law increase of w* with compression. Differences between these two systems occur at lower frequency. The density of states of the modified system displays an abrupt plateau that appears at w*, below which we expect the system to behave as a normal, continuous, elastic body. In the unmodified system, the pressure lowers the frequency of the modes in excess. The requirement of stability despite the destabilizing effect of pressure yields a lower bound on the number of extra contact per particle dz: dz > p^(1/2), which generalizes the Maxwell criterion for rigidity when pressure is present. This scaling behavior is observed in the simulations. We finally discuss how the cooling procedure can affect the microscopic structure and the density of normal modes.Comment: 13 pages, 8 figure

Models of stress fluctuations in granular media

We investigate in detail two models describing how stresses propagate and fluctuate in granular media. The first one is a scalar model where only the vertical component of the stress tensor is considered. In the continuum limit, this model is equivalent to a diffusion equation (where the r\^ole of time is played by the vertical coordinate) plus a randomly varying convection term. We calculate the response and correlation function of this model, and discuss several properties, in particular related to the stress distribution function. We then turn to the tensorial model, where the basic starting point is a wave equation which, in the absence of disorder, leads to a ray-like propagation of stress. In the presence of disorder, the rays acquire a diffusive width and the angle of propagation is shifted. A striking feature is that the response function becomes negative, which suggests that the contact network is mechanically unstable to very weak perturbations. The stress correlation function reveals characteristic features related to the ray-like propagation, which are absent in the scalar description. Our analytical calculations are confirmed and extended by a numerical analysis of the stochastic wave equation.Comment: 32 pages, latex, 18 figures and 6 diagram

Vibrations of amorphous, nanometric structures: When does continuum theory apply?

Structures involving solid particles of nanometric dimensions play an increasingly important role in material sciences. These structures are often characterized through the vibrational properties of their constituent particles, which can be probed by spectroscopic methods. Interpretation of such experimental data requires an extension of continuum elasticity theory down to increasingly small scales. Using numerical simulation and exact diagonalization for simple models, we show that continuum elasticity, applied to disordered system, actually breaks down below a length scale of typically 30 to 50 molecular sizes. This length scale is likely related to the one which is generally invoked to explain the peculiar vibrational properties of glassy systems.Comment: 4 pages, 5 figures, LATEX, Europhysics Letters accepte

Inhomogeneous elastic response of silica glass

Using large scale molecular dynamics simulations we investigate the properties of the {\em non-affine} displacement field induced by macroscopic uniaxial deformation of amorphous silica,a strong glass according to Angell's classification. We demonstrate the existence of a length scale $\xi$ characterizing the correlations of this field (corresponding to a volume of about 1000 atoms), and compare its structure to the one observed in a standard fragile model glass. The "Boson-peak'' anomaly of the density of states can be traced back in both cases to elastic inhomogeneities on wavelengths smaller than $\xi$, where classical continuum elasticity becomes simply unapplicable