J. M. Deutch

Molecular Theory of Brownian Motion for Several Particles

A molecular derivation is presented for the coupled Langevin equations that describe the motion of heavy particles in a fluid. In contrast to the case of a single heavy particle, the friction tensors which appear depend upon the instantaneous separations between the particles. A Fokker–Planck equation describing the reduced distribution function for the heavy particles is also obtained. Calculations with these equations require evaluation of the friction tensors. The friction tensors are evaluated in two ways, by an approximate macroscopic hydrodynamic calculation and an approximate hydrodynamic fluctuation calculation. Both calculations lead to identical expressions for the friction tensors which are shown to have a long‐range character at large interparticle separations. Finally, it is shown that these long‐range effects cancel in the calculation of the diffusion constant for each particle but remain in the calculation of the relative diffusion constant.

Concentration dependence of the rate of diffusion‐controlled reactions

A theory for the concentration dependence of the rate of diffusion‐controlled reactions is formulated. One of the reacting partners is taken to be a collection of static sinks. The steady state situation for a random distribution of these sinks is studied. The rate coefficient is predicted to increase with concentration of sinks and the dependence on concentration is shown to be nonanalytic.

Fluctuations and transitions at chemical instabilities: The analogy to phase transitions

The properties of a reacting system near an instability are investigated and the analogy between transitions in unstable systems and equilibrium phase transitions is developed in detail. The set of macroscopic steady state rate equations plays the role of an equation of state. The bifurcation points of this set are analogous to transition and critical points of equilibrium phase transitions. Hard transitions of unstable systems correspond to first order and soft transitions to second and higher order phase transitions. Critical exponents are defined for those properties of the unstable systems which are singular at the transition points, and relations between these critical exponents are investigated. Critical fluctuations are studied with stochastic analogs of the macroscopic rate equations. Both master and Langevin equations are considered and lead to the following conclusions: When a transition or a critical point is approached (a) the amplitude of fluctuations grows; (b) the lifetime of these fluctuat...

Analysis of Monte Carlo results on the kinetics of lattice polymer chains with excluded volume

Monte Carlo calculations by Verdier et al. on the kinetics of polymer chains on a lattice have shown a large increase of relaxation times in the presence of excluded volume restrictions, i.e., when two beads of the chain cannot occupy the same lattice site. We show that these long relaxation times must be attributed to the specific choice of the kinetics rather than to the intrinsic nature of the excluded volume interaction. A simplified analytic model which preserves the essential characteristics of the kinetics of Verdier’s model reproduces qualitatively the Monte Carlo results for the realxation of the squares of the Rouse coordinates.

Exact solution of the mean spherical model for strong electrolytes in polar solvents

The mean spherical model (MSM) for dense fluids is solved for an arbitrary mixture of equal radii charged hard spheres with permanent embedded dipole moments. The model provides a treatment of ionic solutions that includes the feature of a molecular solvent. Thus, it gives a basis for investigating deviations from the familiar continuum dielectric model in ionic solution theory. The arbitrary polar‐ionic mixture is first reduced to an effective two component problem. One component is an effective charged species while the other is an effective polar species. This two component problem is solved in terms of three parameters closely related to the thermodynamic functions of the fluid. Nonlinear algebraic equations for these parameters are obtained. Although these equations appear to be analytically intractable for arbitrary ionic and dipolar strengths, explicit results are obtained for low ionic strength. In this limit, the ion‐ion contribution to the Helmholtz free energy is given by the classical Debye‐Hu...

Light Scattering from Binary Solutions

The spectrum of the light scattered by a binary solution is calculated from thermodynamic fluctuation theory and the linearized hydrodynamic equations appropriate to a two‐component fluid. The spectrum consists of three peaks. Expressions are obtained for the positions and widths of the two‐side, Brillouin peaks. In general the central, unshifted Rayleigh peak is found to consist of a superposition of two Lorentzians that involve the combined dynamical effects of heat conduction and diffusion. The condition is stated under which it is possible to separate the central peak simply into two contributions, one arising from diffusion and one from thermal conduction. For many binary systems this separation is justified. In these cases measurement of the spectrum of the scattered light should prove to be an attractive alternative means of measuring the diffusion coefficient of binary solutions.

Structure of Dielectric Fluids. I. The Two‐Particle Distribution Function of Polar Fluids

A molecular fluid of identical molecules with a rigid dipole moment in an arbitrarily shaped volume is considered. The volume may or may not be embedded in a dielectric continuum. It is shown that when an arbitrary external field is applied, the constitutive relation P=(e−1)/4 πE between the local polarization and the local macroscopic electric field is valid under some completely acceptable restrictions. In the establishment of this relation a crucial role is played by a long‐range part of the two‐particle correlation function, for which an explicit expression is obtained. One term in this long‐range part is explicitly dependent on the shape of the sample volume and on the surroundings. The resulting dielectric constant of the molecular fluid can be formally expressed in only the local interactions of the molecules and is thus independent of the surroundings and the shape of the sample.

Light scattering from dilute macromolecular solutions

The polarized light scattering spectrum from a dilute solution of spherical macromolecules is customarily interpreted on the basis of independent particle diffusion. However, it is known that diffusion in such a system is governed by a many particle diffusion equation with cross‐diffusion coefficients Dij that depend on the inverse distance between all pairs of particles (i, j). Here we prove that the spectrum from the system described by the N‐particle diffusion equation is identical to the spectrum obtained from the simple, but incorrect, independent particle diffusion model. The physical reason for this suprising simplication is that the Dij are proportional to Oseens tensor which holds for an incompressible fluid and hence has no longitudinal part. When short‐range forces are taken into account as well as the long‐range hydrodynamic interaction present in Dij, it is possible to obtain simple approximate expression for a k‐dependent effective diffusion constant Deff(k). This expression is evaluated fo...

Frictional properties of dilute polymer solutions. I. Rotational friction coefficient

The theory of Debye and Bueche for the frictional properties of dilute polymer solutions is placed on a microscopic basis. It is shown that the microscopic foundations for the theories of Debye−Bueche and Kirkwood−Riseman are identical, but that the theories differ in their statistical analysis. The Debye−Bueche equations are applied to the rotational friction coefficient of a spherically symmetric polymer with arbitrary radial density distribution. An exact result is derived for a Gaussian distribution of low density. A variational principle of minimum energy dissipation is formulated which is suitable for numerical work.

A comparison of generalized cumulant and projection operator methods in spin‐relaxation theory

The general spin‐relaxation theories of Albers and Deutch and of Argyres and Kelley based on different projection operator methods, and the theory of Freed based on generalized cumulant expansions are compared. It is shown that the first two yield equivalent expressions for the time evolution of the spin density matrix. They are also found to be equivalent to a cumulant expansion based on total ordering of the cumulant operators (TTOC), which is different from the partial time ordering method (PTOC) used by Freed. The TTOC method is found to be the more convenient for the frequency domain (i.e., for calculating spectra), while the PTOC method is for time domain analyses. Examples of the use of the TTOC method are given. Useful expressions are given for the case where the lattice may be treated in terms of classical Markov processes, but, in general, it is found that for such cases the stochastic Liouville method is the more useful for computations.