We study experimentally the jamming phenomenon of granular flow of monodisperse disks of D = 5 mm diameter in a two-dimensional hopper with opening R. The jamming probability J(d) is measured where d identical withR/D. We found that J(d) decreases from 1 to zero when d increases from 2 to 5. From observing the disk configurations of the arch in the jamming events, the jamming probability can be explained quantitatively by treating the arch as the trajectory of a restricted random walker.

Grafted polymer layers under variable solvent conditions are studied by Monte Carlo simulations using the bond fluctuation model. Structural information such as monomer density profiles, brush thickness, mean-square displacement of monomers, and positions of the monomers along the chain are obtained for temperatures above, at, and below the Θ point. In particular, the scaling of the brush thickness is formulated and verified by the simulation data. At the Θ point, more extensive simulations are performed to investigate the structural and dynamical properties. While the brush thickness at the Θ point agrees very well with the scaling and self-consistent field predictions, the latter deviate from our results for the monomer density profile and the distribution function of the free chain ends. It is not clear whether these deviations result from our chains being too short or from a basic inadequacy of the theory. For the dynamics at the Θ point, both the relaxation of the chain configurations and the mean-square time displacement of the chains are studied. The relaxation time τ for a polymer brush of chain length N and surface coverage σ are found to behave as τ∼N3σb with an apparent exponent b≂1.6. For poor solvent conditions below the Θ point, we find that the layer (with randomly fixed grafting sites) develops considerable lateral inhomogenity in its density, which has not been predicted by previous theories.

The bond fluctuation model of polymer chains on lattices is used to study layers of polymers anchored with one end at a hard wall, assuming good solvent conditions and repulsive interactions between the monomers and the wall. Chain lengths from N=10 to N=80 and grafting densities σ from 0.025 to 0.20 are considered, both for the ‘‘quenched’’ case, where the anchor points are kept fixed at randomly chosen surface sites, and the ‘‘annealed’’ case, where lateral diffusion of the anchored ends at the wall is considered. Profiles of monomer density and free end density, chain linear dimensions parallel and perpendicular to the wall, as well as corresponding mean square displacements of inner and end monomers are studied and discussed in the light of current theoretical predictions, and it is shown that most of these properties can be understood in terms of appropriate scaling concepts. Both the relaxation of the total chain configurations and the time dependence of monomer mean square displacements are studied. In the annealed case the lateral diffusion constant D is found to behave as D∼σ−qN−p, where q=2/3 and p crosses over from p≊1 at small σ to p≊2 at large σ. The results for the relaxation time τ are consistent with the recent scaling prediction τ∼σbNa with a=3 and b=2/3.

Endgrafted polymers at surfaces exposed to a shear flow are modeled by a nonequilibrium Monte Carlo method where the jump rate of effective monomers to neighboring lattice sites against the flow direction is smaller than in the flow direction, assuming that this difference in jump rates is proportional to the local velocity of the flowing fluid. In the dilute case of isolated chains, the velocity profile is assumed linearly increasing with the distance from the surface, while for the case of polymer brushes the screening of the velocity field is calculated using a parabolic density profile for the brush whose height is determined self-consistently. Linear dimensions of isolated chains are obtained over a wide range of shear rates γ̇, and the deformation of the coil structure by the shear is studied in detail. For brushes it is found that the density profile and the distribution for the perpendicular coordinate zN of the free ends differ only little from the shear-free case, while the distribution of the free end coordinate xN parallel to the wall in the flow direction gets strongly modified. It is shown that the average scaled chain trajectory (〈zi〉/〈zN〉 as a function of 〈xi〉/〈xN〉, i labels the monomers along the chain) is a universal function independent of shear rate, while 〈xN〉 depends on γ̇, chain length N and grafting density σ in scale form, 〈xN〉/(σ1/3N3γ̇) is a function of N2σ5/3 only. Our results are compared with the recent theories of Rabin and Alexander or Barrat, respectively, and both similarities and differences are noted and discussed. The observed increase of the coil radius with the inclination of the chain produced by the flow is somewhat smaller in our model than it was in those theories.

The replica-symmetry-breaking (RSB) solution of the infinite-range Ising spin glass in the presence of a transverse field is obtained. The quenched free energy and the phase boundary of the glass transition temperature versus the transverse field are calculated at first-step RSB without using the static approximation. We demonstrate that replica symmetry (RS) has to be broken in the spin-glass phase by comparing the free energies of the RSB and RS solutions. No evidence is found to support an intermediate spin-glass phase with replica symmetry.