The statistics of stiff chains, with applications to light scattering

Synopsis When the directions of successive steps of a path in space are correlated, the position of the endpoint of a path of given length will show correlation with the orientation at this endpoint. The present paper deals with the limiting case in which the length of the steps approaches zero. Using the method introduced by Kramers6) in the theory of Brownian movement, a differential equation is set up for the distribution of positions and orientations. It is shown that the result is in conformity with that derived by Daniels5) who used a different method and eliminated the orientation from the very beginning. The Laplace transform of the intensity of the light scattered in a given direction is derived, and it is shown that the asymptotic result for very long chains is identical with that derived by previous authors for Gaussian chains, while that for very short chains is the same as the result for stiff rods. Formulae are also derived for the intensity of the light scattered by optically anisotropic chains, where the correlation between position and orientation is a decisive factor. The relation between the intensity of the light and the angle of scattering is shown to depend on the optical anisotropy of the chains, except at the limit of very long chains, as was to be expected. Attention is drawn to the fact that the model should be applicable also to certain problems concerned with the scattering of nuclear particles.

[1]  W. Kuhn,et al.  Beziehungen zwischen Molekülgröße, statistischer Molekülgestalt und elastischen Eigenschaften hochpolymerer Stoffe , 1936 .

[2]  A. Peterlin.  Statistics of linear macromolecules with a relatively short chain. IV. Molecular dimensions from viscosity and sedimentation , 1952 .

[3]  W. Kuhn,et al.  Über die Gestalt fadenförmiger Moleküle in Lösungen , 1934 .

[4]  H. Benoǐt.  Calcul de l'écart quadratique moyen entre les extrémités de diverses chaînes moléculaires de type usuel , 1948 .

[5]  P. Debye.  Light Scattering in Solutions , 1944 .

[6]  S. Chandrasekhar.  Stochastic problems in Physics and Astronomy , 1943 .

[7]  W. J. Taylor,et al.  Average Length and Radius of Normal Paraffln Hydrocarbon Molecules , 1948 .

[8]  B. Rossi,et al.  Cosmic-Ray Theory , 1941 .

[9]  W. Kuhn,et al.  Beziehungen zwischen elastischen Konstanten und Dehnungsdoppelbrechung hochelastischer Stoffe , 1942 .

[10]  C. Tchen.  Random Flight with Multiple Partial Correlations , 1952 .

[11]  R. Kubo.  Statistical Theory of Linear Polymers. IV. Effects of Hinderance for Internal Rotation , 1948 .

[12]  J. E. Moyal.  XCV. The momentum and sign of fast cosmic ray particles , 1950 .

[13]  Paul J. Flory,et al.  The Configuration of Real Polymer Chains , 1949 .

[14]  J. Hermans.  An estimate of the volume effect in coiling long chain molecules , 1950 .

[15]  H. Kramers.  Brownian motion in a field of force and the diffusion model of chemical reactions , 1940 .

[16]  Elliott W. Montroll,et al.  Markoff Chains and Excluded Volume Effect in Polymer Chains , 1950 .

[17]  Henry Eyring,et al.  The Resultant Electric Moment of Complex Molecules , 1932 .

[18]  M. Klamkin,et al.  The Excluded Volume of Polymer Chains , 1952 .

[19]  Bern,et al.  Zur mathematisch‐statistischen Theorie der Kettenmoleküle , 1950 .

[20]  P. Debye,et al.  Molecular-weight determination by light scattering. , 1947, The Journal of physical and colloid chemistry.

[21]  F. C. Collins,et al.  Excluded Volume Effect in Polymer Chains. I , 1951 .

[22]  E. Guth,et al.  Zur innermolekularen, Statistik, insbesondere bei Kettenmolekiilen I , 1934 .

[23]  J. Kirkwood,et al.  Errata: The Intrinsic Viscosities and Diffusion Constants of Flexible Macromolecules in Solution , 1948 .