Theory of polymer breaking under tension

We consider the breaking of a polymer molecule which is fixed at one end and is acted upon by a force at the other. The polymer is assumed to be a linear chain joined together by bonds which satisfy the Morse potential. The applied force is found to modify the Morse potential so that the minimum becomes metastable. Breaking is just the decay of this metastable bond, by causing it to go over the barrier. Increasing the force causes the potential to become more and more distorted and eventually leads to the disappearance of the barrier. The limiting force at which the barrier disappears is D(e)a/2,D-e with a the parameters characterizing the Morse potential. The rate of breaking is first calculated using multidimensional quantum transition state theory. We use the harmonic approximation to account for vibrations of all the units. It includes tunneling contributions to the rate, but is valid only above a certain critical temperature. It is possible to get an analytical expression for the rate of breaking. We have calculated the rate of breaking for a model, which mimics polyethylene. First we calculate the rate of breaking of a single bond, without worrying about the other bonds. Inclusion of other bonds under the harmonic approximation is found to lower this rate by at the most one order of magnitude. Quantum effects are found to increase the rate of breaking and are significant only at temperatures less than 150 K. At 300 K, the calculations predict a bond in polyethylene to have a lifetime of only seconds at a force which is only half the limiting force. Calculations were also done using the Lennard-Jones potential. The results for Lennard-Jones and Morse potentials were rather different, due to the different long-range behaviors of the two potentials. A calculation including friction was carried out, at the classical level, by assuming that each atom of the chain is coupled to its own collection of harmonic oscillators. Comparison of the results with the simulations of Oliveira and Taylor [J. Chem. Phys. 101, 10 118 (1994)] showed the rate to be two to three orders of magnitude higher. As a possible explanation of discrepancy, we consider the translational motion of the ends of the broken chains. Using a continuum approximation for the chain, we find that in the absence of friction, the rate of the process can be limited by the rate at which the two broken ends separate from one another and the lowering of the rate is at the most a factor of 2, for the parameters used in the simulation (for polyethylene). In the presence of friction, we find that the rate can be lowered by one to two orders of magnitude, making our results to be in reasonable agreement with the simulations.

Ab initio, DFT and transition state theory calculations on 1,2-HF, HCI and CIF elimination reactions from CH2F-CH2Cl

This paper reports ab intio, DFT and transition state theory (TST) calculations on HF, HCI and CIF elimination reactions from CH2Cl-CH2F molecule. Both the ground state and the transition state for HX elimination reactions have been optimized at HF, MP2 and DFT calculations with 6-31G*, 6-31G** and 6-311++G** basis sets. In addition, CCSD(T) single point calculations were carried out with MP2/6-311++G** optimized geometry for more accurate determination of the energies of the minima and transition state, compared to the other methods employed here. Classical barriers are converted to Arrhenius activation energy by TST calculations for comparisons with experimental results. The pre-exponential factors, A, calculated at all levels of theory are significantly larger than the experimental values. For activation energy, E-a DFT gives good results for HF elimination, within 4-8 W mol(-1) from experimental values. None of the methods employed, including CCSD(T), give comparable results for HCI elimination reactions. However, rate constants calculated by CCSD(T) method are in very good agreement with experiment for HCI elimination and they are in reasonable agreement for HF elimination reactions. Due to the strong correlation between A and E., the rate constants could be fit to a lower A and E-a (as given by experimental fitting, corresponding to a tight TS) or to larger A and E-a (as given by high level ab initio calculations, corresponding to a loose TS). The barrier for CIF elimination is determined to be 607 U mol(-1) at HF level and it is unlikely to be important for CH2FCH2Cl. Results for other CH2X-CH2Y (X,Y = F/Cl) are included for comparison.

A waveguide problem involving a thick iris in the theory of electromagnetism

The problem of electromagnetic wave propagation in a rectangular waveguide containing a thick iris is considered for its complete solution by reducing it to two suitable integral equations, one of which is of the first kind and the other is of the second kind. These integral equations are solved approximately, by using truncated Fourier series for the unknown functions. The reflection coefficient is computed numerically from the two integral equation approaches, and almost the same numerical results are obtained. This is also depicted graphically against the wave number and compared with thin iris results, which are computed by using complementary formulations coupled with Galerkin approximations. While the reflection coefficient for a thin iris steadily increases with the wave number, for a thick iris it fluctuates and zero reflection occurs. The number of zeros of the reflection coefficient for a thick iris increases with the thickness. Thus a thick iris becomes completely transparent for some discrete wave numbers. This phenomenon may be significant in the modelling of rectangular waveguides.

Theory of Gas Hydrates: Effect of the Approximation of Rigid Water Lattice

One of the assumptions of the van der Waals and Platteeuw theory for gas hydrates is that the host water lattice is rigid and not distorted by the presence of guest molecules. In this work, we study the effect of this approximation on the triple-point lines of the gas hydrates. We calculate the triple-point lines of methane and ethane hydrates via Monte Carlo molecular simulations and compare the simulation results with the predictions of van der Waals and Platteeuw theory. Our study shows that even if the exact intermolecular potential between the guest molecules and water is known, the dissociation temperatures predicted by the theory are significantly higher. This has serious implications to the modeling of gas hydrate thermodynamics, and in spite of the several impressive efforts made toward obtaining an accurate description of intermolecular interactions in gas hydrates, the theory will suffer from the problem of robustness if the issue of movement of water molecules is not adequately addressed.

Novel Algorithm for estimation of Swell Pressure of Fine Grained Soils Based on Diffuse Double Layer (DDL) Theory

Use of engineered landfills for the disposal of industrial wastes is currently a common practice. Bentonite is attracting a greater attention not only as capping and lining materials in landfills but also as buffer and backfill materials for repositories of high-level nuclear waste around the world. In the design of buffer and backfill materials, it is important to know the swelling pressures of compacted bentonite with different electrolyte solutions. The theoretical studies on swell pressure behaviour are all based on Diffuse Double Layer (DDL) theory. To establish a relation between the swell pressure and void ratio of the soil, it is necessary to calculate the mid-plane potential in the diffuse part of the interacting ionic double layers. The difficulty in these calculations is the elliptic integral involved in the relation between half space distance and mid plane potential. Several investigators circumvented this problem using indirect methods or by using cumbersome numerical techniques. In this work, a novel approach is proposed for theoretical estimations of swell pressures of fine-grained soil from the DDL theory. The proposed approach circumvents the complex computations in establishing the relationship between mid-plane potential and diffused plates’ distances in other words, between swell pressure and void ratio.

Classical screw theory: A fresh look using dual eigenvalue approach

This paper presents a novel algebraic formulation of the central problem of screw theory, namely the determination of the principal screws of a given system. Using the algebra of dual numbers, it shows that the principal screws can be determined via the solution of a generalised eigenproblem of two real, symmetric matrices. This approach allows the study of the principal screws of the general screw systems associated with a manipulator of arbitrary geometry in terms of closed-form expressions of its architecture and configuration parameters. The formulation is illustrated with examples of practical manipulators.

A relook at Reissner's theory of plates in bending

Shear deformation and higher order theories of plates in bending are (generally) based on plate element equilibrium equations derived either through variational principles or other methods. They involve coupling of flexure with torsion (torsion-type) problem and if applied vertical load is along one face of the plate, coupling even with extension problem. These coupled problems with reference to vertical deflection of plate in flexure result in artificial deflection due to torsion and increased deflection of faces of the plate due to extension. Coupling in the former case is eliminated earlier using an iterative method for analysis of thick plates in bending. The method is extended here for the analysis of associated stretching problem in flexure.