Coil Radius Research Articles

Theory of Dilute Polymer Solution. II. Osmotic Pressure and Frictional Properties

The osmotic second virial coefficient of flexible polymer solutions is calculated on the basis of the pearl-necklace model with continuous medium approximation. Proper account is taken not only of the intermolecular interaction but also of the intramolecular interaction of segments or the so-called ``excluded volume effect.'' Evaluation is carried out to the triple contact approximation. The results show that the intramolecular interaction has a marked influence on the molecular weight dependence of the coefficient. Then, theories of the intrinsic viscosity and frictional coefficient are developed on the same model as above, following the Kirkwood and Riseman scheme. The results show that, if α is the linear expansion factor of the polymer coil, the intrinsic viscosity is asymptotically proportional to α2 for the free-draining case and α2.43 for the impermeable case. This implies that even in the limit for impermeable molecule, the hydrodynamic radius of the polymer coil is not proportional to its statistical radius. Finally a thorough comparison of the present theories with experimental data is made, taking solutions of polystyrene in cyclohexane as an example. The results prove that the theories are very fitted for structural interpretation of experimental data.

Theory of Dilute Polymer Solutions (III) Intrinsic Viscosity

A theory of the intrinsic viscosity [η] of chain polymer solutions is developed on the basis of the Kirkwood and Riseman scheme, taking some modifications concerning the excluded volume effect into account. If the volume effect is represented by the linear expansion factor α of the root mean square radius of gyration, it is concluded that [η] is proportional to α 2 for the free-draining molecule and α 2.43 for the non-free-draining molecule. This implies that the effective hydrodynamic radius of the polymer coil is not proportional to the statistical one in the limit, where the rigid sphere model is asymptotically valid; in other words, the more the polymer chain is expanded, the deeper the solvent molecules penetrate into the polymer coil.

An Approximate Solution of the Natural Vibrations of Conical Coil Springs

The author treats through an approximate method the natural vibrations of the conical coil springs which have the Archimedes' spiral in their planes. The conditions of approximation are shown by two items neglecting the effect of the helical angle and supposing the wire of spring to be fine enough as compared with the radius of the coil, just as in the equation of motion in the natural vibrations of cylindrical coil springs. So a partial differential equation is to be made by the author for the natural vibrations in the direction of coil axis which are produced only by the torsion of a fine spring wire, and to be treated in connection with its solution when the boundary conditions are as follows : (1) both ends are fixed, (2) one end is fixed and the other end free. The results are obtained that the natural frequencies of conical coil springs have been calculated in proportion to the ratio of the small end radius of coil to the large end one, and several forms of the normal mode of vibrations have been shown in both cases respectively.

The dissociation and specific viscosity of polymethacrylic acid

AbstractSolutions of polymethacrylic acid have been partially neutralized to different extents, and the pH values and viscosities of the resulting solutions determined. These measurements were carried out at several different potassium chloride concentrations, and on several polymer samples. The calculated dissociation constants of the polymer acid were substituted in the equation developed by J. Th. G. Overbeek, which expresses the change in dissociation constant of a polybasic acid as a function of the radius of the molecular coil and of the charge on the molecule. The derivation of this equation treats the polymer molecule as a sphere containing a continuous charge. The radii found are of the right order and change in the expected way with changing ionic strength and ionisation. Substitution of these radii in the Debije viscosity equation gives satisfactory agreement between observed and experimental viscosities at low degrees of ionisation. At higher degrees of dissociation it would seem that the treatment of the polymer coil as a sphere is no longer satisfactory, although a better approximation is obtained if the charge is considered to be entirely on the surface of the sphere.

Extension and Period of a Conically Wound Spring

By imposing certain restrictions on the shape of a conically wound spring, it is valid to assume that the extension under a load is due entirely to torsion. It is found that the extension is then given to an approximation by the expression: $Z=\frac{2Mg{R}^{2}L}{3\ensuremath{\pi}n{{r}_{1}}^{4}}$ $\mathrm{Mg}$ is the load, $R$ is the radius of the end coil, $L$ and ${r}_{1}$ are respectively the length and radius of the wire of which the spring is made, and $n$ is its rigidity modulus. For a truncated conically wound spring, this expression becomes: $Z=\frac{2MgN(R+r)({R}^{2}+Rr+{r}^{2})}{3n{{r}_{1}}^{4}}$ $N$ is the number of coils of the spring, and $r$ is the radius of the small end coil. By making certain assumptions the period is shown to be: $T=2\ensuremath{\pi}{\left(\left(1+\frac{\mathrm{xm}}{M}\right)\frac{z}{g}\right)}^{\frac{1}{2}}$ $m$ is the mass of the spring, and $X$ is a function of the ratio of the radii of the two end coils.

Period of A Loaded Conical Jolly Balance Spring

An expression for the period of a conical Jolly balance spring is deduced theoretically and the result is verified by experimental data. By some assumptions which simplify the equation of motion it is reduced to that for simple harmonic motion from which the period is obtained. The period is: T=2π(M+m03k)hmgMo is the mass of the load, m0 is the mass of the spring, h is the extension of the spring for a load m and C = (r2/r1 − 1) K=5[(c+1)2+1]2(c+2)3(160+480c+696c2+620c3+360c4+135c5+30c6+3c7)r2 is the radius of the large end coil, r1 is the radius of the small end coil. Very approximately, K = 1+C/3 so that m03k=m02+r2/r1