Marshall Fixman

Bio: Marshall Fixman is an academic researcher from Yale University. The author has contributed to research in topics: Radial distribution function & Intrinsic viscosity. The author has an hindex of 34, co-authored 69 publications receiving 4434 citations.

General theory of diffusion‐controlled reactions

Abstract: A formal mathematical description of diffusion‐controlled bimolecular reactions is presented. The theory is completely general with respect to the kinds of reaction processes that may be considered. Besides the presentation of the general formalism, the paper also contains several examples illustrating the application of the theory to a simple many‐particule system for simple catalytic bimolecular reactions, including fluorescence quenching. The manner in which approximate solutions may be obtained is also outlined.

Simulation of polymer dynamics. I. General theory

Abstract: Methods are presented for the simulation of conformational relaxation processes in polymer chains. The chains interact with the solvent through frictional and Langevin forces, have constrained bond lengths and angles, and smooth rotational energy functions. Relationships are obtained between the stochastic difference equations that form the basis for computer simulation, and equivalent Langevin differential equations and Smoluchowski diffusion equations. The presence of constraints and the associated action of potential forces and Langevin forces on the same time scale leads to somewhat intricate algorithms and correction terms. However, the time increment during a simulation step can be increased by a factor of 1000 or more over what is required if vibrational potentials are used to preserve the primary chain structure. Computation time is proportional to chain length in the absence of hydrodynamic interaction and excluded volume forces.

Diffusion‐controlled intrachain reactions of polymers. I Theory

Abstract: A theory of diffusion‐controlled intramolecularly catalyzed reactions of polymer chains in dilute solution is formulated. A closure approximation yields a soluble integral equation for (essentially) the transient rate of the reaction. The time dependent solution consists of a sum of simple exponential terms whose coefficients and decay constants can be readily calculated. The theory is also applied to luminescence quenching and ring‐chain interconversion.

The Poisson–Boltzmann equation and its application to polyelectrolytes

Abstract: The validity of the Poisson–Boltzmann (PB) equation is reconsidered on the basis of functional expansion techniques supplemented by the mean spherical approximation. In the application of greatest interest a strong Coulomb potential originating in an external source, such as a polyelectrolyte molecule, acts on a salt solution of small mobile ions. Where the local charge density of mobile ions is high, substantial errors may occur in the PB approximation that relates charge density to mean potential. However, the solution to the PB equation is nevertheless a good approximation in the indicated application because a quite small percentage change in the electrostatic potential can compensate large errors in the Boltzmann distribution. An application to DNA illustrates this compensation, and also its impending failure at bulk salt concentrations in excess of 0.1M. A two phase (or condensation) model is derived as an approximation to the PB equation and retains fair accuracy even at substantial salt concentrations, where the limiting laws lose theoretical validity.

Classical statistical mechanics of constraints: a theorem and application to polymers.

Abstract: A classical system of mass points subject to holonomic constraints has a kinetic energy dependent on the coordinates as well as the moments of the remaining degrees of freedom. Coordinate averages formed in the reduced space of unconstrained coordinates and their conjugate momenta then involve a metric determinant that may be difficult to evaluate. A theorem is derived that permits a relatively easy evaluation when the constraints are distances between particles, and an application is made to a Kramers type freely jointed chain.

Reaction-rate theory: fifty years after Kramers

Abstract: The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Theory of Dynamic Critical Phenomena

Abstract: An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.

Dynamics of entangled linear polymer melts: A molecular‐dynamics simulation

Abstract: We present an extensive molecular‐dynamics simulation for a bead spring model of a melt of linear polymers. The number of monomers N covers the range from N=5 to N=400. Since the entanglement length Ne is found to be approximately 35, our chains cover the crossover from the nonentangled to the entangled regime. The Rouse model provides an excellent description for short chains N

A unified view of polymer, dumbbell, and oligonucleotide DNA nearest-neighbor thermodynamics

Abstract: A unified view of polymer, dumbbell, and oligonucleotide nearest-neighbor (NN) thermodynamics is presented DNA NN DG° 37 parameters from seven laboratories are presented in the same format so that careful comparisons can be made The seven studies used data from natural polymers, synthetic polymers, oligonucleotide dumbbells, and oligonucleotide duplexes to derive NN parameters; used dif- ferent methods of data analysis; used different salt concen- trations; and presented the NN thermodynamics in different formats As a result of these differences, there has been much confusion regarding the NN thermodynamics of DNA poly- mers and oligomers Herein I show that six of the studies are actually in remarkable agreement with one another and explanations are provided in cases where discrepancies re- main Further, a single set of parameters, derived from 108 oligonucleotide duplexes, adequately describes polymer and oligomer thermodynamics Empirical salt dependencies are also derived for oligonucleotides and polymers

Diffuse-interface methods in fluid mechanics

Abstract: We review the development of diffuse-interface models of hydrodynamics and their application to a wide variety of interfacial phenomena. These models have been applied successfully to situations in which the physical phenomena of interest have a length scale commensurate with the thickness of the interfacial region (e.g. near-critical interfacial phenomena or small-scale flows such as those occurring near contact lines) and fluid flows involving large interface deformations and/or topological changes (e.g. breakup and coalescence events associated with fluid jets, droplets, and large-deformation waves). We discuss the issues involved in formulating diffuse-interface models for single-component and binary fluids. Recent applications and computations using these models are discussed in each case. Further, we address issues including sharp-interface analyses that relate these models to the classical free-boundary problem, computational approaches to describe interfacial phenomena, and models of fully miscible fluids.