Phase equilibria in solutions of rod-like particles

Abstract

A partition function for a system of rigid rod-like particles with partial orientation about an axis is derived through the use of a modified lattice model. In the limit of perfect orientation the partition function reduces to the ideal mixing law ; for complete disorientation it corresponds to the polymer mixing law for rigid chains. A general expression is given for the free energy of mixing as a function of the mole numbers, the axis ratio of the solute particles, and a disorientation parameter. This function passes through a minimum followed by a maximum with increase in the disorientation parameter, provided the latter exceeds a critical value which is 2e for the pure solute and which increases with dilution. Assigning this parameter the value which minimizes the free energy, the chemical potentials display discontinuities a t the concentration a t which the minimum first appears. Separation into an isotropic phase and a some what more concentrated anisotropic phase arises because of the discontinuity, in confirmation of the theories of Onsager and Isihara, which treat only the second virial coefficient. Phase separation thus arises as a consequence of particle asymmetry, unassisted by an energy term . Whereas for a large-particle asymmetry both phases in equilibrium are predicted to be fairly dilute when mixing is athermal, a comparatively small positive energy of interaction causes the concentration in the anisotropic phase to increase sharply, while the concentration in the isotropic phase becomes vanishingly small. The theory offers a statistical mechanical basis for interpreting precipitation of rod-like colloidal particles with the formation of fibrillar structures such as are prominent in the fibrous proteins. The asymmetry of tobacco mosaic virus particles (with or without inclusion of their electric double layers) is insufficient alone to explain the well-known phase separation which occurs from their dilute solutions at very low ionic strengths. Higher-order interaction between electric double layers appears to be a major factor in bringing about dilute phase separation for these and other asymmetric colloidal particles bearing large charges, as was pointed out previously by Oster.

3-D phase diagram of HPC/H2O/H3PO4 tertiary system

“…HPC is known to be one of the few cellulose derivatives able to form LC phases with both thermotropic and lyotropic characteristics (Flory 1956(Flory , 1978Budgell 1989). This characteristic originates from the chemical structure of HPC, which contains primary and secondary hydroxypropyl groups and residual hydroxyl moieties.…”

Understanding and Describing the Liquid-Crystalline States of Polypeptide Solutions: A Coarse-Grained Model of PBLG in DMF

“…The lattice theory of Flory 117,120 has been employed to explain the phase behaviour of the PBLG-DMF system and several qualitative similarities between the theoretical description and experiment were highlighted. 22,23 Because of the discrete treatment of the medium in this type of lattice treatment and the artificial sectioning of the rod-like particles, it becomes difficult to make a direct comparison between the lattice model and the real macromolecule.…”

An analytical equation of state for describing isotropic-nematic phase equilibria of Lennard-Jones chain fluids with variable degree of molecular flexibility

“…This type of behaviour, referred to as Type III behaviour, 39 was initially predicted from a lattice theory for solutions of rigid polymers by Flory and co-workers. 47,48 Later, the same behaviour was predicted from several Onsager-based approaches. [39][40][41][42]45 Experimentally, the existence of Type III behaviour is observed for solutions of poly(γ-benzyl-L-glutamate) (PBLG), 46,49,50 polysaccharide Schizophyllan, 51,52 and -hexa-alkylbenzene FIG.…”