C. F. Curtiss

A kinetic theory for polymer melts. I. The equation for the single‐link orientational distribution function

In this series of papers we show how a kinetic theory of undiluted polymers can be developed using the Curtiss–Bird–Hassager phase‐space formulation. The polymer molecule is modeled as a Kramers freely jointed bead–rod chain. The objective is to obtain a molecular‐theory expression for the stress tensor from which the rheological properties of polymer melts can be obtained. This development is put forth as an alternative to the Doi–Edwards theory; using a very different approach, we have rederived some of their results and generalized or extended others. In this first paper we develop the partial differential equation for the chain configurational distribution function, and then proceed to get the equation for the orientational distribution function for a single link in the chain. A modification of Stokes’ law is introduced that includes a tensor drag coefficient, characterized by two scalar parameters ζ (the friction coefficient) and e (the link tension coefficient). In addition, to describe the increase of the drag force on a bead with chain length, at constant bead density, a ’’chain constraint exponent’’ β is used, which can vary from zero (the Doi–Edwards limit) to about 0.5. Solutions to the partial differential equation for the single‐link distribution function are given in several forms, including an explicit series solution to terms of third order in the velocity gradients.

Resonance Theory of Termolecular Recombination Kinetics: H+H+M→H2M

A theory is formulated for atomic recombination reactions which is based upon the identification of the set of transition complexes, X2i, as specific quasibound states or orbiting resonances. The conventional “energy‐transfer mechanism” is assumed, since it has been justified under many experimental situations. Calculations, based on a modified distorted‐wave approximation, demonstrate that the main contribution to the rate is that arising from rotational (rather than vibrational) transitions downwards from the quasibound to the bound states. Computations were carried out for the reaction H+H+M→H2+M for M = He, H2, and Ar making use of detailed ab initio knowledge of the spectrum of quasibound states and their wave‐functions. Good agreement was found between the experimental rate constant and that calculated by the present resonance theory. The theory predicts a maximum in the rate in the temperature range between 65° and 100°K, attributed mainly to one particular quasibound state: υ = 14, j = 5. This sug...

Molecular Collisions. VIII

The set of integral equations describing the molecular scattering process developed in the previous papers of this series is reformulated as a set of differential equations, in which the coupling is associated with a Hermitian matrix. Exact eigenvectors and eigenvalues of this matrix are developed. These eigenvectors may be used as the basis of a numerical solution of the equations. They also lead to an approximate solution which is valid when the difference of the wavenumbers in the entrance and exit channels is small. The approximate solution may be considered as the lowest‐order term in a series development.

The Free Volume for Rigid Sphere Molecules

The free volume is calculated for non‐interacting rigid sphere molecules taking into account the exact geometry imposed by the face‐centered cubic packing. The size and shape are quite different from that of the inscribed spheres which correspond to the Lennard‐Jones and Devonshire approximation. At high densities the equation of state obtained from the exact treatment agrees well with the Eyring or Lennard‐Jones and Devonshire equation. When the specific volume is greater than twice the cube of the collision diameters, the molecules are no longer confined to cages formed by neighboring molecules. At these low densities the free volume concept is ambiguous. The equation of state depends upon the shape and orientation of the cells with respect to the lattice positions of the molecules. A particular choice is considered which leads to an equation of state that at low densities is accurate through the second virial coefficient. There are other shapes and orientations of the cells which would lead to other eq...

A kinetic theory for polymer melts. II. The stress tensor and the rheological equation of state

An expression for the stress tensor of an undiluted polymer is derived by starting with the phase‐space kinetic theory of Curtiss, Bird, and Hassager. The final formula contains two integrals, the first of which is the Doi–Edwards expression; the second integral, multiplied by the ’’link tension coefficient’’ e, is new, and consequently a rheological equation of state is obtained whose properties are different from those of the Doi–Edwards theory. As illustrations, the constants in the retarded‐motion expansion are determined to third order, and the relaxation modulus of linear viscoelasticity is obtained.

Kinetic Theory of Nonspherical Molecules. V

In the kinetic theory of rigid nonspherical molecules, discussed previously, the transport coefficients were expressed in terms of certain multi‐dimensional integrals, the brace expressions. In the present discussion these integrals are considered in detail. The integrations over the linear and angular velocities are carried out explicitly and the problem is further reduced to an integral of geometrical parameters over the surfaces of two bodies. The resulting expressions for the transport coefficients apply to any rigid nonspherical convex body in which the center of mass is a center of symmetry. As an example, the final integrations are carried out explicitly for the spherocylindrical model. The shear viscosity, the bulk viscosity, and the thermal conductivity, for this model are presented as functions of two parameters characteristic of the shape and mass distribution of the molecule.

The Theory of Flame Propagation

The characteristics of steady‐state one‐dimensional flames are expressed in terms of a set of first‐order ordinary differential equations suitable for solution by differential analyzers or high speed digital computing devices. Arbitrary systems of chemical kinetics and reaction rates can be investigated. The effect of ambient temperature, pressure, heat transfer from the flame to the flame holder, diffusion of free radicals, thermal conductivity, etc., are easily estimated. The equations which we use are the ordinary hydrodynamic equations of change generalized to include the effect of the chemical reactions. In these the usual expressions for reaction rates are introduced, that is, the rate at which the composition would change in a closed vessel under the local conditions of temperature and density. Equations expressing the diffusion velocities in terms of the composition gradients are given. The flame holder has been idealized in the form of a porous plug through which the fuel can pass freely from lef...

Symmetric Gaseous Diffusion Coefficients

The kinetic theory development of expressions for the diffusion velocities in multicomponent gaseous mixtures is reconsidered. A set of equations for multicomponent diffusion coefficients which are symmetric in the indices is developed. The special cases of two‐and three‐component systems are considered explicitly.

A kinetic theory for polymer melts. IV. Rheological properties for shear flows

Curtiss and Bird recently developed a kinetic theory for undiluted polymers, modeling the fluid as a collection of interacting Kramers freely jointed bead–rod chains. The theory yielded a constitutive equation in which the stress tensor is given as the sum of two integrals over the strain history. Here, the constitutive equation is used to calculate the steady‐state shear flow rheological properties (shear‐rate dependent viscosity and normal‐stress functions) and also unsteady‐state shear flow responses (shear and normal stress growth at the inception of shear flow, shear and normal stress relaxation after the cessation of shear flow, and the stress response in small‐amplitude oscillatory motion). The rheological functions for an earlier slip‐link network theory by Doi and Edwards are also obtained.

Kinetic Theory of Dense Gases

The Boltzmann equation is found to be determined to arbitrary order in the density as a consequence of the molecular chaos assumption. A three‐body collision approximation is given explicitly and shown to be identical to that implied by the Bogolyubov assumptions; the relation between the molecular chaos assumption and the Bogolyubov assumptions is discussed. It is shown that the Enskog modification of the Boltzmann equation for dense gases composed of rigid spheres is correct, subject to the molecular chaos assumption. The current disagreement over the correct form for this equation is discussed.