Correction of Flotation Coefficients, Derived from Ultracentrifugation, for Pressure a N D Concentration

-Flotation coefficients are usually determined from data obtained under high pressure (cell bottom) conditions. A power series expansion is proposed for correction to atmospheric pressure (meniscus) conditions. The same expansion may be applied to the concentration correction of flotation coefficients. From sedimentation experiments with polymer solutions in the analytical ultracentrifuge, we can derive sedimentation coefficients of the polymeric component. A widely applied procedure is to identify the sedimentation coefficient with the slope of the best straight line through the data points In x* and 26o2t (where co = angular speed in radians per second, t = time in sec, x* = (r*/r,.) 2, r* = distance of the top of the Schlieren peak to the centre of rotation, r,, = distance of the meniscus to the centre of rotation). This procedure leads to sedimentation coefficients which depend on the concentration, and are not corrected to the pressure at the meniscus (where p = 1 arm.). We will denote such an uncorrected sedimentation coefficient by s. Several procedures have been developed to obtain "true" sedimentation coefficients, corrected for pressure and concentration [1-4]. These procedures apply to dilute solutions. When the solvent is heavier than the polymer, flotation occurs. Now the boundary layer travels from the high pressure region (where the pressure is of the order of 100 atm.) at the cell's bottom towards the low pressure region near the meniscus. The procedures for correction for pressure which apply to sedimentation do not hold for flotation. The reason is that corrections for pressure have to be carried out for a region in the cell where the boundary layer has not proceeded "too far" away from its starting point (the meniscus, for sedimentation; the bottom for flotation). This is connected with the requirement that the time parameter r 2So6o2t (where So is the "true" sedimentation coefficient) should not be "too far" from zero [1]. As for r --~ 0 in the flotation case, the boundary is still in the high pressure region; in contrast with the sedimentation case, we cannot use the correction procedures of sedimentation for flotation without appropriate modification. As far as we are aware, no proposals for the correction of flotation coefficients (needed for a study on permeability [6]) have been published. Pouyet and Dayantis [5] suppose for their semi-dilute solutions (concentration > say 2~o) that the pressure effect affects all flotation coefficients to the same extent, i.e. as if all the concentrations were multiplied by the 51 same (compression) factor. In a plot of log [s f vs log c this would result in a shift to the right of the whole plot, without any influence on the slope of the straight line which is characteristic for the dependence of log fsJ on log c in the semi-dilute region. In this paper we consider in more detail if the latter conclusions can be justified by a less qualitative reasoning. We start from the correction procedure of Dishon et at. [1] for sedimentation coefficients obtained in dilute solution. Dishon et al. [1] wrote the solution of the (diffusion free) differential equation describing the sedimentation process (the so-called Lamm equation) as a series expansion in powers of the time parameter ~. It holds only for 3 ,0 : lnx* = r/(~ + i) (2~ + 1)m ~r2 + 0( .c3} (1) 2(~ + 1) 3