Osmotic Flow Effects on the Dialysis Escape Curves of Solutes Dien-Feng Shieh, Jan Feijen,’ and Donald J. Lyman Department of Materials Science and Engineering, University of Utah, Salt Lake City, UT 84 7 12
While dialysis has been widely used as a separation technique for a long time, Craig and his coworkers ( I , 2 ) recently developed it as a simple and convenient method to characterize a variety of biochemical solutes. This characterization is based on the shape of the escape curves of the solutes during aqueous thin-film dialysis. With this thin-film technique, the dialyzing solution (about 0.6 ml) can be uniformly distributed over 40-50 cm2 of membrane area. The outside solution (10to 20 times the volume of the dialyzing solution) is replaced periodically with fresh solvent and the concentration of the dialyzed solute measured a t that time. A plot of percentage of solute remaining vs. time on a semilog scale is called the escape curve. Different shapes can be observed for different solute, solvent, and membrane combinations. The systems with straight escape curves were described as “ideal”; nonlinearity in the escape curves was explained in terms of impurity, association, or aggregation of the solutes studied (3). Recently, Stewart et al. ( 4 ) carried out computer calculations, based on first-order kinetic models, to show how solute impurity, association, or aggregation could cause the escape curves to deviate from linearity. While no proof was mentioned about the kinetic models used in their calculations, deviation from linearity can still be observed even with pure homogeneous solutes that do not associate ( I ) . Though many factors affect the observed shape of the escape curves, one phenomenon, Le., osmotic flow, although well known itself, has been neglected. In this paper, we would like to stress its importance in affecting the shape of the escape curves. Computer calculations in terms of three phenomenal parameters-permeability coefficient, filtration coefficient, and reflection coefficient-were used to illustrate the effects. The calculations are qualitatively confirmed by experimental results reported for the thin film dialysis measurements.
THEORY A N D CALCULATIONS A general dialysis system can be described as consisting of two compartments with volumes V I and Vz separated by a membrane with effective area S. The concentrations of the solutes in compartments 1 and 2 are designated as Csl and Cs2.The mass flux J s through the membrane is given by
where U is defined as the overall permeability. From the mass balance, we obtain d ( ViCs,)
____ df
-
-JsS
(2)
where Cslo = Csl ( t = 0). Plotting In CsI/CsIoas a function of time leads to a linear plot only when ( U S ) / V 1is constant. Depending on the dialysis system used, U and S are approximately constant, but V1 can change significantly as a result of osmotic flow. Furthermore, although the outside concentration CsP can be kept nearly zero by frequent replacement of the solvent in compartment 2, a more accurate description of the transport phenomena in the case of longer time intervals between the change of solvent ( t , - t,-1) is given by
where (Csl), is the concentration in compartment 1 a t time t , and (Csl),-l is the concentration a t time t,-1. If ( t , t , - l ) is small, Equation 3 is the same as Equation 4. When ( t , - t , - l ) is large, Equation 4 should be used to describe the experimental system. I t should be noted here that the derivation of Equation 4 is based on the assumption of constant V1 and V2 during the interval ( t , - t,-1). However, as a result of osmotic flow, V1 will increase as given by
where JV is the volumetric flow rate. When t = 0, V I = v10.
In the case where Vs is much larger than V I , V2 can be considered reasonably constant. In order to describe this osmotic flow, we used general phenomenal equations derived from the irreversible thermodynamic theory. Three experimentally measurable parameters, the filtration coefficient L,, the reflection coefficient u, and the permeability coefficient w can be used to describe the phenomena ( 5 ) . The volumetric flow rate Jv is given by Jv = L , ( A p
- OAT)
(6)
The parameter cr can vary from zero for solutes that have the same permeability as water, to one for solutes that are completely rejected by the membrane. The hydrostatic pressure difference is given by &I whereas the theoretical osmotic pressure difference AT equals RTACs/M,. ( h C s is the mean concentration difference across the membrane.) The net solute flux across the membrane J s is given by:
If V2 >> V1 and Cs2
