Robert J. Rubin

Adsorption of a Chain Polymer between Two Plates

A lattice model of adsorption of an isolated chain polymer between two plates is investigated using a matrix formalism and a grand canonical ensemble (GCE) formalism. The matrix formalism is particularly convenient for calculating the polymer segment density as a function of the distance from one of the plates for different fixed plate separations. The GCE formalism can be used to calculate the fraction of loops (sequences of polymer segments whose ends are in contact with one plate and whose intermediate segments lie between the two plates), bridges (sequences of polymer segments whose ends are in contact with different plates and whose intermediate segments lie between the two plates), and trains (sequences of polymer segments which are wholly in contact with one plate or the other). All of the foregoing quantities have been calculated in the limit of infinite molecular weight as a function of the distance of separation between the plates and the energy of adsorption of a polymer segment on a plate. The...

Random‐Walk Model of Chain‐Polymer Adsorption at a Surface

A random‐walk lattice model of adsorption of an isolated polymer chain at a solution surface is investigated. One‐dimensional characteristics of the monomer‐unit distribution are determined analytically in the limit of long polymer chains, neglecting the self‐excluded volume. The mean number of monomer units adsorbed in the surface layer ν(θ, N) is determined, assuming that one end of the polymer chain lies in the surface layer. The parameter N is the number of monomer units in the chain, and θ is the adsorption energy of each monomer unit in the surface layer measured in units of kT. In addition, the mean distance of the free end of the chain from the surface z(θ, N) is determined. The lattice models considered include the simple‐cubic, hexagonal‐closepacked, face‐centered‐cubic, and body‐centered‐cubic lattices. In the limit in which N→∞, both ν(θ, N) and z(θ, N) exhibit a very interesting discontinuity at a lattice‐dependent adsorption energy θc. For example for θ>θc, ν(θ, N) (which is also proportiona...

Statistical Dynamics of Simple Cubic Lattices. Model for the Study of Brownian Motion. II

The system considered is an n‐dimensional cubic crystal with nearest‐neighbor central and noncentral harmonic forces in which the mass M of one of the lattice particles is relatively large. It is assumed that the velocities and positions of the light particles in the system (mass m) are normally distributed, at time t=0, as in thermal equilibrium. The conditional velocity distribution for the heavy particle at time t is then a normal distribution with a time‐dependent mean value. This mean value is the velocity autocorrelation function. The dispersion of the distribution is shown to be a simple function of the autocorrelation. In the limit M/m≫1 in the one‐ and two‐dimensional lattices, the autocorrelation function is, respectively, a damped exponential and a damped oscillating exponential. These different types of statistical behavior are related to the different dynamic properties of the medium with which the heavy particle interacts.

Abnormal Lattice Thermal Conductivity of a One‐Dimensional, Harmonic, Isotopically Disordered Crystal

Energy transport is investigated in a model system for which exact analytic results can be obtained. The system is an infinite, one‐dimensional harmonic crystal which is perfect everywhere except in a finite segment which contains N isotopic defects. Initially, the momenta and displacements of all atoms to the left of the defect region are canonically distributed at a temperature T, and the right half of the crystal is at a lower temperature. This initial nonequilibrium state evolves according to the equations of motion, and ultimately a steady state is established in the vicinity of the region containing the defects. The thermal conductivity is calculated from exact expressions for the steady state energy flux and thermal gradient. For a crystal in which the N isotopic defects are distributed at random but in which the overall defect concentration is fixed, we demonstrate that the thermal conductivity approaches infinity as least as fast as N1/2. A Monte Carlo evaluation of the thermal conductivity for a...

Quantum‐Mechanical Calculation of the Probability of an Exchange Reaction for Constrained Linear Encounters

A numerical procedure suitable for use with a high‐speed computing machine is developed for calculating the average quantum‐mechanical probability of the exchange reaction BC+A→B+CA for constrained linear encounters at temperature T when BC is initially in its ground or first excited vibrational state. The average refers to the average over the relative momentum frequency distribution of collisions between BC and A at temperature T. The procedure, which involves the numerical solution of the time‐dependent Schrodinger equation, is sufficiently general so that any three‐atom potential energy surface may be used. Two sample calculations have been performed using a simple potential energy surface. The results of these quantum‐mechanical calculations for the average probability of reaction are compared with the corresponding classical quantities which are obtained in an elementary fashion.

Span of a Polymer Chain

The x span of a polymer chain is defined as the difference between the largest x coordinate and the smallest x coordinate of any segments in the chain. The polymer chain is represented by a nearest‐neighbor lattice‐model random walk in which the mean square displacement of each component of a single step is 13. The distribution function of the x span of an N step random walk is obtained; and the following asymptotic formulas are obtained for its first and second moments, respectively, 2(2N/3π)1/2 and 4 ln2 N/3. The corresponding moments of the magnitude of the x component of the end‐to‐end distance are (2N/3π)1/2 and N/3. Excluded volume effects are not considered. It is noted that the problem of calculating the first moment of the x span is identical with the one‐dimensional case of the Dvoretzky‐Erdos problem, namely, the calculation of the average number of different lattice sites visited in an N‐step random walk on a d‐dimensional lattice.

Random walks on lattices. The problem of visits to a set of points revisited

A general method is outlined for calculating the statistical properties of the number of visits to a set of points in a random walk. In illustrative examples, known results and new results are easily derived.

The theory of ordered spans of unrestricted random walks

The spans of ann-step random walk on a simple cubic lattice are the sides of the smallest rectangular box, with sides parallel to the coordinate axes, that contains the random walk. Daniels first developed the theory in outline and derived results for the simple random walk on a line. We show that the development of a more general asymptotic theory is facilitated by introducing the spectral representation of step probabilities. This allows us to consider the probability density for spans of random walks in which all moments of single steps may be infinite. The theory can also be extended to continuous-time random walks. We also show that the use of Abelian summation simplifies calculation of the moments. In particular we derive expressions for the span distributions of random walks (in one dimension) with single step transition probabilities of the formP(j) ∼ 1/j1+α, where 0<α<2. We also derive results for continuous-time random walks in which the expected time between steps may be infinite.

Ordered spans of unrestricted and self‐avoiding random‐walk models of polymer chains. I. Space‐fixed axes

An N‐step random walk on a cubic lattice is adopted as a model of a random polymer chain. The spans, or extents, of each random walk configuration in the principal lattice directions are arranged in order of magnitude, ξ3?ξ2⩾ξ1. In the case of the unrestricted random walk, the average values of the ordered spans 〈ξi,u〉 and 〈ξ2i,u〉, i=1, 2, and 3, are calculated analytically in the limit of large N. The limiting relative values of the first moments, 〈ξi,u〉 are 1.637 : 1.267 : 1; and the limiting values of the second moments 〈ξ2i,u〉 are 2.710 : 1,600 : 1. In the case of the restricted or self‐avoiding walk, the corresponding average spans 〈ξi,r〉 and 〈ξ2i,r〉 are estimated for N?150 by using a Monte Carlo procedure. The same Monte Carlo procedure is used to estimate the values of 〈ξi,u〉 and 〈ξ2i,u〉 for N?1000. On the assumption that the rate of approach of the average ordered spans of the self‐avoiding walks to their asymptotic forms is similar to the rate of approach of the average ordered spans of the unres...

Random‐Walk Model of Adsorption of a Chain‐Polymer Molecule on a Long Rigid‐Rod Molecule

A lattice model of adsorption of a flexible chain molecule on a rodlike molecule is investigated. The rodlike molecule is represented by the lattice sites on the z axis of a simple‐cubic lattice; sites which are nearest neighbors to the z axis are adsorbing sites. The dimensionless adsorption energy per monomer unit is θ = e/kT. The problem of enumerating polymer‐chain configurations taking into account the increased probability of occupying adsorbing sites and the zero probability of occupying z‐axis sites is formulated and solved as a random‐walk problem. This model is a natural generalization of a random‐walk model of adsorption on a plane solution surface. The average fraction of monomer units in adsorbing sites fR(θ) is computed in the limit in which the number of monomer units in the polymer chain approaches infinity. There is a critical value of the adsorption energy θc = ln (6/5) such that for θ θc, fR(θ) is an increasing function of θ with all derivatives equal to zero at θc ...