Equilibrium Between Sedimentation and Diffusion

15 The relationship between the flux due to sedimentation and that due to diffusion. At equilibrium, the two are equal. 

Back-diffusion occurs at a rate that increases with the buildup of a concentration gradient. When equilibrium is finally reached, we may write 

If we substitute the value for the rate of sedimentation under gravity, Equation (4), we obtain 

If the sedimentation occurs in a centrifugal field, on the other hand, g must be replaced by u2x in Equation (4)  

Equations (82) and (83) are easily integrated to produce expressions that give c as a function of x at equilibrium. Defining c, and c2 to be the equilibrium concentrations at xt and jc2, respectively, and then integrating, we obtain for Equation (82) 

---

Alternatively, in sedimentation equilibrium experiments, the ultracentrifuge is operated at slow speeds for longer time to permit the solute molecules to attain equilibrium between sedimentation and diffusion. If the sample is monodisperse, its concentrations cb c2 can be measured at 2 positions xh x2 in the cell. 

The sedimentation velocity determination is dynamic and can be completed in a short period of time. The sedimentation equilibrium method gives quantitative results, but long periods of time are required for centrifugation at relatively low velocities to establish equilibrium between sedimentation and diffusion. 

Calculate the equilibrium between sedimentation and diffusion for the particle given in problem 9 assuming hindered settling at = 50%. 

Equations (V.7) and (V.l 1) reflect the increase in the sedimentation rate with increasing particle size, and the increase in the rate of diffusion with a decrease in particle size. Thus, one may expect that for particles of intermediate size the sedimentation and diffusion fluxes can balance each other, i.e.Jd +js = 0, which leads to an equilibrium between sedimentation and diffusion ... 

One can use a thermodynamic approach to derive the equation describing the condition of equilibrium between sedimentation and diffusion. The derivation involves the assumption of a constant gravity-chemical potential (i.e. the generalized chemical potential which includes the action of the external gravity field) and the assumption that the laws established for ideal systems can be applied to dilute dispersions, i.e. 

For disperse systems consisting of particles with radii less than 0.1 pm direct observation of the equilibrium between sedimentation and diffusion is difficult or even impossible, as the equilibrium is reached extremely slowly. 

Smaller centrifugal forces are required for this method, but several days of continuous running are necessary to reach an equilibrium between sedimentation and diffusion. The molecular weight is calculated from the equation ... 

If a sedimentation experiment is carried out long enough, a state of equilibrium is eventually reached between sedimentation and diffusion. Under these conditions material will pass through a cross section perpendicular to the radius in both directions at equal rates downward owing to the centrifugal field, and upward owing to the concentration gradient. It is easy to write expressions for the two fluxes which describe this situation ... 

At low rotor revolution numbers an equilibrium state can be reached between sedimentation and diffusion. Now, a time-independent concentration gradient is established, i.e., (dddt) = 0. Under these conditions, the Svedberg equation becomes ... 

It is our objective in this chapter to outline the basic concepts that are behind sedimentation and diffusion. As we see in this chapter, gravitational and centrifugal sedimentation are frequently used for particle-size analysis as well as for obtaining measures of solvation and shapes of particles. Diffusion plays a much more prevalent role in numerous aspects of colloid science and is also used in particle-size analysis, as we see in Chapter 5 when we discuss dynamic light scattering. The equilibrium between centrifugation and diffusion is particularly important in analytical and preparative ultracentrifuges. 

In this method, usually a density profile is formed in the ultracentrifugation cell by centrifuging solutions for instance of sucrose or Metrizamide in water or methanol until the equilibrium state between sedimentation and diffusion of the dissolved molecules has been established. Figure 1 shows schematically two of such equilibrium density profiles (dotted lines). 

The equilibrium between gravitational or centrifugal force and diffusion is routinely taken advantage of in colloid science, as illustrated in Vignette II. Our objective in this chapter is to examine the effects of sedimentation and diffusion, first taken separately and then combined, on particles in the colloidal size range. 

This chapter contains one of the more diverse assortments of topics of any chapter in the volume. In it we discuss the viscosity of polymer solutions, especially the intrinsic viscosity the diffusion and sedimentation behavior of polymers, including the equilibrium between the two and the analysis of polymers by gel permeation chromatography (GPC). At first glance these seem to be rather unrelated topics, but features they all share are a dependence on the spatial extension of the molecules in solution and applicability to molecular weight determination. 

Because of the similarity of transport in biotilms and in stagnant sediments, information on the parameters that control the conductivity of the biofilm can be obtained from diagenetic models for contaminant diffusion in pore waters. Assuming that molecular diffusion is the dominant transport mechanism, and that instantaneous sorption equilibrium exists between dissolved and particle-bound solutes, the vertical flux ( ) through a stagnant sediment is given by (Berner, 1980)... 

Describe the relation between sedimentation/diffusion equilibrium and the barometric equation. 

The sedimentation equilibrium procedure, on the other hand, depends upon maintaining a centrifugal force just sufficient to give a measurable concentration change in relation to the height of the cell. The establishment of this equilibrium between the settling and diffusion rates of the dispersed phase enables the degree of polymerization of the material to be calculated from the equation ... 

In solid-liquid mixing design problems, the main features to be determined are the flow patterns in the vessel, the impeller power draw, and the solid concentration profile versus the solid concentration. In principle, they could be readily obtained by resorting to the CFD (computational fluid dynamics) resolution of the appropriate multiphase fluid mechanics equations. Historically, simplified methods have first been proposed in the literature, which do not use numerical intensive computation. The most common approach is the dispersion-sedimentation phenomenological model. It postulates equilibrium between the particle flux due to sedimentation and the particle flux resuspended by the turbulent diffusion created by the rotating impeller. 

Fig. 15.7 Calculation of the equilibrium between oxygen consumption occurring during the oxidation of sedimentary organic matter near the sediment surface and resupply of oxygen by means of diffusion. Here, oxygen consumption is assumed to follow first-order kinetics. Application of Pick s Second Law in an explicit numeric solution permits a reliable calculation of the times required for the adjustment of a stationary condition.

---