Scaling Laws for Ideal and Real Macromolecular Chains from Relativistic Theory of Self-Diffusion in a Liquid Medium

Abstract

Scaling laws concerning a solution of linear macromolecules have been investigated by means of a relativistic theory of self-diffusion in a simple liquid. A pure liquid medium has been modeled as a four-dimensional continuum characterized by a spacetime metric, where the equation for the Brownian movement, ∝ Dt, is interpreted as a covariant law provided by an invariant self-diffusion coefficient. The local diffusivity change due to the presence of a chain molecule has then been associated with a geometrical coordinate transformation that can be studied by using classical relativistic formalisms. Special and general theories of relativity (SR and GR) have been used to study the behavior of ideal and real chains, i.e., single coils and concentrated polymer solutions, respectively. Application of SR gives scaling laws for the average end-to-end distance and global relaxation time of an ideal macromolecule (i.e., with no excluded volume effects). Einstein equations of GR for a curved and empty space return the Stokes law and the Flory formula for a real coil in a four-dimensional space. In a concentrated dispersion, they predict a new scaling behavior (involving mean size, time, viscosity, and diffusion coefficients) of which the current values of universal exponents represent a solution in both low and high molecular weight regimes and for ideal and real chains. In the end, it is suggested a correspondence between Brownian motion and field theory, and the interpretation of a polymer chain-in-a-tube as a geodesic path.