General dynamic theory of macromolecular networks. I. Definitions and classification

The problem of the structural theory of macromolecular networks is formulated and discussed in general terms. The conditions required for a system to become a homogeneous macromolecular network are defined and discussed. Networks are divided according to the nature of their junctions into three classes: energetic (with chemical or quasi-chemical crosslinks), topological (with entangled chains), and contact (with frictional interactions). The main features of these three classes are discussed. A distribution density function ψ describing the configurations of macromolecules in network systems is introduced. The phase space of variables is 4(N + 1)-dimensional and includes the coordinates of (N + 1) vectors h _i joining the adjacent network junctions and (N + 1) contour lengths l_i of the network chains. The system of simultaneous equations required for the determination of the function ψ includes the equation of continuity, kinematic equations for the deformation velocity of the individual junctions, the force balance equation needed for the determination of sliding rates l_i, kinetic equations for the processes of junction breakage and reformation, and the equilibrium distribution of network junctions defining the initial conditions for the distribution function ψ.     

Irreversible deformation of macromolecular networks

Based upon a thermodynamical approach, the generalized Onsager type of relaxation of van der Waals networks is presented. By linearly and identically coupling the set hidden variables to the network, the memory function of the system can be related to the equilibrium strain-energy function. The relaxation behavior of real networks on stretching can quantitatively be described by means of a distribution of relaxation times known from small strain experiments. Some new and interesting conclusions are discussed as to how the macroscopically non-linear visco-elastic response might be interpreted.     

General theory of deformation of viscoelastic substances

The general theory of the deformation of viscoelastic substances developed in two previous papers for the one-dimensional case is now extended to three dimensions. The integro-differential equations obtained are generalizations of the well-known Navier-Cauchy equations and, like these, only apply to small deformations. Some special cases are considered in detail and lead to interesting conclusions. In particular, the theory predicts that for incompressible high polymers, one should obtain the same creep and relaxation functions when the material is subjected to extension, torsion, or bending.     

Dynamics of macromolecular chains. II. Orientation relaxation generated by elementary three-bond motions and notion of an independent kinetic segment

In the preceding paper, general equations were established for the motions of chains confined to a tetrahedral lattice. In the present paper, bond orientation correlation and autocorrelation functions are explicitly calculated for the case where only three-bond elementary motions are considered. Effects due to the chain end are analyzed and the relaxation time distribution function is established. The expressions obtained reflect the influence of the chain structure. Finally, to characterize the dynamic behavior of chains in orientation relaxation experiments, the notion of an independent kinetic segment is proposed.     

Dynamics of macromolecular chains. V. Interpretation of the dielectric relaxation data

The autocorrelation function of orientations, derived for the model of conformational jumps in a chain described in a tetrahedral lattice, has been tentatively applied to dielectric relaxation data. The complex dielectric relaxation of polymers can be interpreted by the aid of only two relaxation processes. In the case of poly(p-chlorostyrene) in solution, the characteristic times that we have calculated agree favorably with the fluorescence-depolarization results. The model has been shown to be consistent with the empirical decay function exp[? (t/τ)^β] proposed by Williams.     

Dynamics of macromolecular chains. I. Theory of motions on a tetrahedral lattice

The bond correlation function for a macromolecular model chain on a tetrahedral lattice is derived by considering elementary three-bond and four-bond motions. The method of calculation allows the conformational structure of the chain to be involved in the final equations. Moreover, when the three-bond motions are considered alone, no linearization assumption is required; hence, the theory is valid at short times.     

Dynamics of macromolecular chains. VII. Nuclear magnetic relaxation of monosubstituted polystyrenes in solution

The ¹H spin-echo and ¹³C spin–lattice relaxation times have been measured for solutions of polystyrene derivatives: ortho, meta, and para-halo (F, Cl, Br) and ortho, meta, para, and α-methyl. Results obtained from these two techniques permit comparison of the intramolecular mobility of these polymers with that of polystyrene. Poly(α-methylstyrene) does not differ from polystyrene except for a slight slowing of both segmental reorientation and internal phenyl-group motions and apparent hindrance of the methyl-group rotation. Segmental reorientation of poly(m-methylstyrene) is similar to that of polystyrene; rotation of the methyl group is free, while the internal phenyl-ring process is slower. Poly(p-methylstyrene) and poly(o-methylstyrene) also contain freely rotating methyl groups; the intramolecular mobility decreases from the para to the ortho position of the substituent. Finally, in poly(o-bromostyrene) and poly(o-chlorostyrene), the internal motion of the phenyl ring is completely overshadowed by the segemental reorientation, which is itself quite reduced.     

Simulation of the brownian motion of macromolecular chains. II. Chain dynamics

A simulation of the Brownian motion of polyethers and polyethylene is reported. The chains under consideration are confined to a tetrahedral lattice and subjected to conformational energy conditions. The dynamic behavior of the whole chain is studied. The diffusion of individual chain atoms reflects the chemical nature of the chain, and the diffusion of the center of mass, in connection with experimental results, affords a time scale for the simulation. The relaxation of bond orientation, which is connected with fluorescence depolarization, is found to be consistent with the theory of Valeur and co-workers. The relaxation of the total dipole is interpreted in terms of conformational features of the motions and correlations between neighboring dipoles. Finally, relations between chemical structure and dynamic behavior are established. Three classes of polyether chains are to be distinguished: the rigid chain poly(methylene oxide), the highly flexible poly(ethylene oxide) and an intermediate class typified by polyethylene.     

Dynamics of macromolecular chains. IV. Quenching and fluorescence polarization studies of polystyrene in solution

Combined measurements of quenching and fluorescence polarization corroborate the results previously obtained by the polarized fluorescence anisotropy decay technique and, more especially, confirm the form of the proposed orientation autocorrelation function. The use of this method to determine the mean relaxation time of the relaxation time distribution is also explained.     

Non-relativistic self-consistent-field theory. II

The condition to be satisfied by the orbitals which minimize the orbital energies is determined. This condition provides a link with the Thomas-Fermi approximation.

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Zusammenfassung Es wird eine Bedingung bestimmt, die von den Orbitalen, die die Orbital-Energien auf ein Minimum herabsetzen, erfüllt wird. Diese Bedingung bringt eine Verbindung zu der Thomas-Fermi-Methode.

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Résumé On a déterminé la condition qui doit Être satisfaite par les orbitales qui produisent un minimum pour les énergies des orbitales. Cette condition permet d'établir un rapport avec la théorie de Thomas-Fermi.

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This work has been supported in part by the National Research Council of Canada and has been presented in part at the 47th Annual Canadian Chemical Conference, Kingston, Ontario, Canada, June 1–3, 1964, and at the 2nd Gordon Research Conference on Theoretical Chemistry, New Hampton, N. H., U. S. A., June 29–July 3, 1964.     

On the theory of electron correlation in atoms and molecules. II. General cluster expansion theory and the general correlated wave functions method

An exact cluster expansion of many electron wave functions is derived, beginning with a finite linear combination of Slater determinants rather than the more usual single determinant. This general cluster expansion is found to apply both in the case where all possible Slater determinants from a finite set of spin orbitals are included in the linear combination, and in the case where the number of determinants is restricted. The special properties of that finite linear combination of determinants closest to the exact wave function in the least squares sense are studied. These properties lead to the derivation of a general correlated wave functions method, illustrating again the close relationship between methods of this type and cluster expansion theory. Additional approximations, necessary for practical calculations, are set out.     

Dynamics of macromolecular chains. III. Time-dependent fluorescence polarization studies of local motions of polystyrene in solution

Fluorescence anisotropy decay experiments are described for polystyrene in various ethylacetate-tripropionin mixtures. Decay curve trends agree with the proposed theoretical autocorrelation function. Study of the effects of viscosity shows that the mean relaxation time varies according to a nonlinear law for low viscosities and that the relaxation time θ, reflecting the effects of the possible departures from the motions permitted by an ideal tetrahedral lattice, obeys a law of the type: θ = α + bη. Furthermore, the effects of the direction of the fluorophore transition moment are examined.     

Dynamics of macromolecular chains. VI. Carbon 13 and proton nuclear magnetic relaxation of polystyrene in solution

Spin-lattice ¹H and ¹³C nuclear magnetic relaxation (NMR) times T₁ have been measured for solutions of polystyrene in hexachlorobutadiene at two different frequencies. Some nuclear Overhauser enhancements and linewidths have also been determined. At 15 and 25 MHz the relaxation times T₁ of the ortho and meta carbons show two different dependences on temperature. These measurements indicate internal motion of phenyl groups around the C_α—C_para axis. A single isotropic correlation time is inadequate to explain the relaxation data for the para carbon. Use of a diamond-lattice motional model reveals that segmental reorientation of the chain backbone of polystyrene can be described in terms of two correlation times, ρ characterizing the three-bond motion process, and θ reflecting either isotropic motions of subchains or departure from an ideal lattice. Data on low-molecular-weight polystyrene indicate the participation of overall rotatory diffusion in the relaxation process. This motion is no longer efficient in high-molecular-weight polymers, where relaxation is due to segmental reorientation.