Topological Obstacles Research Articles

Geodesic equivalence via integrability

We suggest a construction that, given an orbital diffeomorphism between two Hamiltonian systems, produces integrals of them. We treat geodesic equivalence of metrics as the main example of it. In this case, the integrals commute; they are functionally independent if the eigenvalues of the tensor giαg¯αj are all different; if the eigenvalues are all different at least at one point then they are all different at almost each point and the geodesic flows of the metrics are Liouville integrable. This gives us topological obstacles to geodesic equivalence.

DNA separation at a liquid-solid interface.

We demonstrate that it is possible to separate a broad band of DNA on a solid substrate without topological obstacles. The mobility was found to scale with molecular size (N) as N(-0.25), while the resolution scaled as N(0.75) indicating that diffusivity on this substrate was minimal. By varying the buffer concentration we were able to show that the mobility for a given chain length scaled with the persistent length (p) as p(1/2). This could be shown to be related to the Gaussian conformation of the chains adsorbed on the surface. A two-dimensional corrugated surface of nonporous silica beads was produced using a self-assembling process at the air/water interface. Even though the surface corrugations were comparable to persistence length we show that they do not affect the mobility, indicating that surface friction rather than topological constraints are the predominant mechanism of separation on a surface.

Coherent scattering function in the reptation model: analysis beyond asymptotic limits.

We calculate the coherent dynamical scattering function S(c)(q,t;N) of a flexible chain of length N, diffusing through an ordered background of topological obstacles. As an instructive generalization, we also calculate the scattering function S(c)(q,t;M,N) for the central piece of length M<or=N of the chain. Using the full reptation model, we treat global creep, tube length fluctuations, and internal relaxation within a consistent and unified approach. Our theory concentrates on the universal aspects of reptational motion, and our results in all details show excellent agreement with our simulations of the Evans-Edwards model, provided we allow for a phenomenological prefactor which accounts for nonuniversal effects of the microstructure of the Monte Carlo chain, present for short times. Previous approaches to the coherent structure function can be analyzed as special limits of our theory. First, the effects of internal relaxation can be isolated by studying the limit N--> infinity, M fixed. The results do not support the model of a "Rouse chain in a tube." We trace this back to the nonequilibrium initial conditions of the latter model. Second, in the limit of long chains (M=N--> infinity ) and times large compared to the internal relaxation time (t/N(2)--> infinity ), our theory reproduces the results of the primitive chain model. This limiting form applies only to extremely long chains, and for chain lengths accessible in practice, effects of, e.g., tube length fluctuations are not negligible.

Brownian dynamics of concentrated polymers in two dimensions

Two-dimensional simulations of the Brownian dynamics of polymers in a grid of topological obstacles were carried out in this study. A first model made use of the exact expression for the free energy of a chain with fixed ends interacting with an obstacle and can be used to derive the elastic force acting on the subchains of a Rouse-like model. With this model we calculated the diffusion coefficient of chains of different lengths constrained by the fixed obstacles. A second model, less rigorous but more efficient computationally, was used to simulate the chain dynamics in fast flows. In such a case, the obstacles were convected according to the shear gradient. During the simulation, the configurational changes of the chain constrained by the obstacles could be monitored and the stresses which developed were calculated. These simulations reveal important effects of the conformational fluctuations at high shear rates.

Dynamics of entanglements: A nonlinear model consistent with the Cox-Merz rule

It is argued that the flow directly contributes a mechanism by which the topological obstacles are renewed. As a consequence, the chain diffusivity is increased (or, equivalently, the friction coefficient is decreased) in fast flows. Such a nonlinear effect is incorporated in a simple dumbbell model so as to produce a constitutive equation in closed form. An extension to the case of multiple relaxation times is also briefly considered. The predictions compare favourably with the Cox-Merz rule as well as with the behaviour of polymer melts in elongational flows.

Polymer chain in a random array of topological obstacles: Classification and statistics of complex loops.

An arbitrary complex polymer loop embedded in three-dimensional space and topologically entangled with a quenched array of randomly distributed parallel rods is considered. It is shown that an ensemble of the complex loops, prepared in different possible ways, is equivalent to the ensemble of cactuslike graphs where the leaves of these cacti are simple loops. The free energy of the cactuslike graphs is calculated in the mean-field approximation. The critical condition for the collapse transition is obtained. The collapsed state of the complex loop with the density distributed over different leaves is shown to be preferable

Curve matching and stereo calibration

The topological obstacles to the matching of smooth curves in stereo images are shown to occur at epipolar tangencies. Matching is possible when these tangencies satisfy certain projective constraints (the tangent lines form corresponding pencils) and metric contraints dependent on the camera geometry. Such points are good matching primitives, even when the image curves correspond to smooth surface profiles. An iterative scheme for improving camera calibration based on these results is derived, and performance demonstrated on real data.

Stress Relaxation and Differential Dynamic Modulus of Polyisobutylene in Large Shearing Deformations

Stress relaxation in shear has been measured at 25°C for a polyisobutylene sample with viscosity average molecular weight 7.7×105, at shear strains γ from 0.2 to 0.5, over a time scale from 102 to 104.5 s. In this range, the relaxation modulus G(γ;t) could be expressed by the product G(0;t)h(γ); at γ=0.5, h=0.76. Simultaneous measurements of the differential storage and loss shear moduli, G′(ω,γ;t) and G″(ω,γ;t) were made throughout the relaxation process by intermittently superposing small oscillating deformations with a maximum additional strain of 0.01, at a frequency of about 0.33 Hz, which falls in the middle of the plateau zone of the polymer. For γ=0.2, G′ and G″ remained unchanged from their values at zero static strain, G′(ω,0) and G″(ω,0) respectively, confirming that the density of entanglements (or topological obstacles, or temporary network junctions), remains constant throughout the relaxation process at small strains. At higher strains, G″ was nearly constant, but G′ was somewhat smaller at the first measurement after imposition of static strain and slowly recovered to its original zero-strain value. The behavior could be qualitatively interpreted by the Doi theory which includes both an equilibration process characterized by a time τe and a disengagement process characterized by a time τd; the decrease in G′ would correspond to loss of entanglements by the contraction of stretched molecules during equilibration and the subsequent increase to reestablishment of new entanglements by reptation during disengagement.