Thomas A. Witten

Force fluctuations in bead packs

Experimental observations and numerical simulations of the large force inhomogeneities present in stationary bead packs are presented. Forces much larger than the mean occurred but were exponentially rare. An exactly soluble model reproduced many aspects of the experiments and simulations. In this model, the fluctuations in the force distribution arise because of variations in the contact angles and the constraints imposed by the force balance on each bead in the pile.

A Parabolic Density Profile for Grafted Polymers

We study the statistics of a grafted polymer brush, consisting of a set of monodisperse chains in solution, each attached irreversibly by one end to a flat surface. We use a self-consistent field method, valid in the limit of weak excluded volume and at moderately high surface coverage. Exploiting the fact that the chains are highly stretched, we map the problem (in the long-chain limit) onto one involving the motion of classical particles in an equal-time potential, which we can solve exactly. The resulting density profile for the brush takes a parabolic form.

Model for force fluctuations in bead packs

We study theoretically the complex network of forces that is responsible for the static structure and properties of granular materials. We present detailed calculations for a model in which the fluctuations in the force distribution arise because of variations in the contact angles and the constraints imposed by the force balance on each bead of the pile. We compare our results for force distribution function for this model, including exact results for certain contact angle probability distributions, with numerical simulations of force distributions in random sphere packings. This model reproduces many aspects of the force distribution observed both in experiment and in numerical simulations of sphere packings.

Scaling Properties of Stretching Ridges in a Crumpled Elastic Sheet

Strong deformation of a sheet of solid material often leads to a crumpled state having sharp points of high curvature. A scaling property governing the crumpled state has been numerically demonstrated by an examination of the ridges joining pairs of sharp points in a range of simple geometries of variable size. As the linear size X increases sufficiently, the deformation energy grows as X1/3 and consists of similar amounts of bending and stretching energy. The deformation energy becomes concentrated in a fraction of the sheet that decreases as X1/3. Despite this concentration, the local strain in the ridge decreases as X2/3. Nearly all the deformation energy in thin, crumpled elastic sheets was found to be concentrated in ridges that obey these scaling laws.

Mesoscopic self‐assembly of gold islands on diblock‐copolymer films

We describe the fabrication and characterization of self‐assembled gold island arrays on diblock‐copolymer thin films. The natural tendency of these polymers to form ordered phases is used to induce selective aggregation of evaporated gold metal during an annealing process. We obtain well‐defined, nanoscale island arrays aligned with one of the copolymer blocks. Near perfect segregation is achieved between the two domains. Two types of diblock‐copolymer systems are discussed, together with the resulting island patterns.

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(omega) of vibrational modes of angular frequency omega 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(omega) toward a nonzero constant as omega-->0, (b) appearance of a low-frequency cutoff omega*, and (c) power-law increase of omega* with compression. Differences between these two systems occur at a lower frequency. The density of states of the modified system displays an abrupt plateau that appears at omega*, 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 deltaz:deltaz> or =p1/2, 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.

Macrocrystal Ordering in Star Polymer Solutions

We show that a solution of f-armed star polymers develops a peak in its scattering structure function S(q), whose height scales as f3/2. The peak is largest when the separation between neighboring stars is about equal to the radius of a star. Our results follow from general scaling properties of polymers in a good solvent. We predict crystalline ordering of the polymers when the functionality f exceeds a universal threshold fc.

Stress propagation through frictionless granular material

We examine the network of forces to be expected in a static assembly of hard, frictionless spherical beads of random sizes, such as a colloidal glass. Such an assembly is minimally connected: the ratio of constraint equations to contact forces approaches unity for a large assembly. However, the bead positions in a finite subregion of the assembly are underdetermined. Thus to maintain equilibrium, half of the exterior contact forces are determined by the other half. We argue that the transmission of force may be regarded as unidirectional, in contrast to the transmission of force in an elastic material. Specializing to sequentially deposited beads, we show that forces on a given buried bead can be uniquely specified in terms of forces involving more recently added beads. We derive equations for the transmission of stress averaged over scales much larger than a single bead. This derivation requires the ansatz that statistical fluctuations of the forces are independent of fluctuations of the contact geometry. Under this ansatz, the d(d+1)/2-component stress field can be expressed in terms of a d-component vector field. The procedure may be generalized to nonsequential packings. In two dimensions, the stress propagates according to a wave equation, as postulated in recent work elsewhere. We demonstrate similar wave-like propagation in higher dimensions, assuming that the packing geometry has uniaxial symmetry. In macroscopic granular materials we argue that our approach may be useful even though grains have friction and are not packed sequentially.

Glass transition of dense fluids of hard and compressible spheres

We use computer simulations to study the glass transition of dense fluids made of polydisperse repulsive spheres. For hard particles, we vary the volume fraction, phi , and use compressible particles to explore finite temperatures, T>0 . In the hard sphere limit, our dynamic data show evidence of an avoided mode-coupling singularity near phi(MCT) is approximately 0.592; they are consistent with a divergence of equilibrium relaxation times occurring at phi(0) is approximately 0.635, but they leave open the existence of a finite temperature singularity for compressible spheres at volume fraction phi>phi(0). Using direct measurements and a scaling procedure, we estimate the equilibrium equation of state for the hard sphere metastable fluid up to phi(0), where pressure remains finite, suggesting that phi(0) corresponds to an ideal glass transition. We use nonequilibrium protocols to explore glassy states above phi(0) and establish the existence of multiple equations of state for the unequilibrated glass of hard spheres, all diverging at different densities in the range phi in [0.642, 0.664]. Glassiness thus results in the existence of a continuum of densities where jamming transitions can occur.

Crumpling a Thin Sheet

Crumpled sheets have a surprisingly large resistance to further compression. We have studied the crumpling of thin sheets of Mylar under different loading conditions. When placed under a fixed compressive force, the size of a crumpled material decreases logarithmically in time for periods up to three weeks. We also find hysteretic behavior when measuring the compression as a function of applied force. By using a pretreating protocol, we control this hysteresis and find reproducible scaling behavior for the size of the crumpled material as a function of the applied force.