J. H. Weiner

Contribution of covalent bond force to pressure in polymer melts

Molecular dynamics simulations of the freely jointed effective hard sphere model of a polymer melt are performed, with continuous covalent and noncovalent potentials employed for computational convenience. The virial theorem is used to determine the melt pressure from the simulations. Excellent agreement is found over a wide range of density and degree of polymerization with the recently introduced equation of state of Honnell and Hall. The atomic pressure is defined as the contribution made by each atom to the compressibility factor of the system; it is found to be uniform along the interior atoms of the polymer chain, with slightly different values for the end atoms. This uniform value is almost independent of N, the number of bonds in the chain, and explains the small dependence on N of the pressure that is observed. Particular attention is paid to the contribution to the pressure made by the force in the covalent bonds of the system. For packing fractions greater than 0.3, approximately 30% of the pressure in the melt arises from this source.

Brownian dynamics study of a polymer chain of linked rigid bodies

A Brownian dynamics model for the backbone chain of a macromolecule is developed as a system of linked rigid bodies so that constraints on valence angles and bond lengths are satisfied exactly. For comparison, a corresponding flexible model is developed in which bond lengths and valence angles are held nearly constant by strong harmonic potentials. Equilibrium properties and barrier crossing rates are examined theoretically and by computer simulation of both models, with differences arising due to the presence of constraints in the rigid case. A compensating potential based on the metric determinant of unconstrained coordinates in the rigid model is found to eliminate the effect of constraints. Barrier crossing rates in the transition state approximation are studied when a force fixed in space is applied to the end atoms of the three‐bond chain. An exact transition state rate formula developed for this case predicts curved Arrhenius plots of barrier crossing rates; this result is confirmed by computer sim...

Particle Method for the Numerical Solution of the Time‐Dependent Schrödinger Equation

A method for the solution of the time‐dependent Schrodinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two‐dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed. This particular check problem was chosen because of its close relationship to the wave‐packet dynamics of chemical reaction rates.

Transmission function vs energy splitting in tunneling calculations

Two methods have been widely employed for computing tunneling rates in a double‐well potential, one based on a transmission function, the second on energy splitting. Semiclassical calculations (Brickmann and Zimmermann) show that the transmission method leads to lower rates than the splitting method. It is shown here without employing the semiclassical approximation that a more accurate relation between the energy splitting and the transmittion function may be obtained by using a decomposition of the stationary states into scattering states. This relation is then used to provide an analytical basis for the usual heuristic picture in which the particle oscillates with classical frequency in one well and has a probability of tunneling to the other well equal to the transmission function in each classical period of oscillation. It is concluded that the transmission method should give more meaningful results, particularly in situations where interactions of the system with a heat bath have significant effects...

Nature of Stress on the Atomic Level in Dense Polymer Systems

Stress in dense polymer systems is classically viewed as being molecular in character and is based on the entropic spring concept. A description on the atomic level has been developed on the basis of extensive computer simulations. An important new concept is the intrinsic monomer stress (IMS), the individual monomer contribution to the macroscopic stress referred to a local moving coordinate system in which the backbone bonds attached to that monomer are fixed. The IMS is time-independent and, for a given polymer system at fixed density, has the same value in the equilibrium melt, with the melt undergoing stress relaxation, and in the deformed cross-linked system.

Time‐Dependent Perturbation Calculations Based on the Hydrodynamic Analogy to Quantum Mechanics

A new formulation is presented for the problem of the time‐dependent perturbation of bound states. It is based on the hydrodynamic analogy to quantum mechanics and leads to the physical picture of a continuum of particles which oscillate in the vicinity of their equilibrium positions under the influence of the perturbing field. The corresponding mathematical formulation involves a single partial differential equation which is second order in time and fourth order in the spatial coordinates. The procedure is applied to the time‐dependent polarizability of the hydrogen atom in its ground state. It was possible to obtain an exact series solution which yields results in full agreement with those previously obtained by other methods.

Quantum rate theory for a symmetric double‐well potential

A quantum rate theory is presented for a symmetric double‐well potential which is defined by a piecewise quadratic function. The theory is based on stationary states which are decomposed into‐ and left‐moving states. The flux and transmission coefficients for the latter are found in terms of parabolic cylinder functions and are thermally averaged. Close analogies between the quantum and classical formulations are found when an appropriate phase space representation is used. The theory shows good agreement with experimental results for the diffusion of hydrogen and deuterium in niobium, but the agreement is poorer for the same process with the host metal vanadium and the theory does not predict the observed anomalous isotope effect for this process in palladium.

SIMULATED POLYMER MELT STRESS RELAXATION. I: PLATEAU BEHAVIOR

The computer simulation of stress relaxation in a model polymer melt is performed with a nonequilibrium molecular dynamics algorithm. The chains are freely jointed with N=30, 100, and 200 bonds and with both intra‐ and interchain excluded volume interactions. For N=200, the model exhibits incipient plateau behavior as evidenced by an inflection point in the stress relaxation history. Comparison is made between the stress history as computed on the atomic level by the virial stress formula and on the molecular level using the entropic spring formulation. The two histories are in agreement only for the case of N=200, with entanglement length Ne=40, and only for the time period following the inflection point.

A generalization of Kramers’ rate formula to include some anharmonic effects

Consideration from the Langevin approach of Brownian‐motion effects on a particle in a parabolic barrier potential leads to a transmission function which gives the probability that the particle will surmount the barrier. When used in conjunction with an approximate low‐temperature normalization condition, the Kramers rate formula, originally derived using the Fokker–Planck approach, is reproduced. The rate formula is then generalized by including anharmonic effects due to the presence of the barrier as they enter in an exact normalization condition. The generalized Kramers formula has a temperature dependence of the frequency factor which is verified by computer simulation for a periodic and double‐well potential. Data from computer experiments are fitted using both the original and generalized formulas. The generalized formula is found to be useful in extracting information on the barrier height and friction coefficient from the experimental data.

Brownian dynamics study of a polymer chain of linked rigid bodies. II. Results for longer chains

As an extension of a previous paper, computer simulation results for a polymer chain of linked rigid bodies are presented for chains with between four and fifteen bonds. Previous theoretical results for a three‐bond chain are found by computer simulation to apply to longer chains. In particular, a Fixman potential for an N‐bond chain is developed as the sum of three‐bond Fixman potentials, and transition state rates measured in computer simulation for a four‐bond chain compare reasonably well with the theory for a three‐bond chain. A further study of the motion of the free end of a chain with one end fixed indicates correlation effects diminish the possibility of a ’’whipping motion’’ of the chain end.