1228
The Journal of Physical Chemistv, Vol. 83, No. 9, 1979
R. F. Probstein, W.-F. Leung, and Y. Alliance
Determination of Diffusivity and Gel Concentration in Macromolecular Solutions by Ultrafiltration Ronald F. Probsteln, * Woon-Fong Leung, and Yaghoub Alliance Department of Mechanical Engineering. Massachusetts Institute of Technology, Cambridge, Massachusetts 02 139 (Received October 10, 1978) Publication costs assisted by the Office of Water Research and Technology
A method is presented for determining the gel concentration and diffusivity at gelling of macromolecular solutions by comparing measured ultrafiltration limiting fluxes in plane, laminar, and turbulent channel flow with theoretical fluxes obtained from analytical mass transfer solutions. Also given is a method for determining macromolecular solution diffusivity as a function of concentration by comparing ultrafiltrate flux-pressure curves in laminar channel flow with a theoretical, closed form mass transfer solution for the flux-pressure behavior. The diffusivity of bovine serum albumin solution as a function of concentration is found at pH 4.7 (the isoelectric point) and discrepancies among existing literature values are analyzed. Gel concentration and diffusivity at pH 4.7 and 7.4 are also determined. The indirect ultrafiltration approach is shown to provide an accurate and simple means of evaluating macromolecular solution diffusivity as a function of pH, concentration, and temperature, in contrast to present sophisticated techniques that have often proved unreliable.
Introduction Measurements of the diffusivity of macromolecular solutions, particularly at high concentration, are difficult to carry out and important differences exist among reported values. Another and related measurement of importance, for which little data exists, is the concentration a t which gelation takes place and the diffusivity at this Concentration. In the present study we examine bovine serum albumin (BSA), a protein solution of molecular weight 69000. However, the methods described are general and apply to other macromolecular solutions. Various measurements of BSA diffusivity at dilute concentrations (less than 1 g %) and 25 “C show values to 7.1 X of from 6.6 X cm2/s (e.g., Creethl and Charlwood2). The data show little effect of buffer type, pH value, and ionic strength of the solution. A t higher concentration these factors cannot be ignored as the interactions among the charged macroions become more pronounced. Data by Doherty and Benedek3show a strong dependence of BSA diffusivity on the solution ionic strength and the average protein surface charge. Keller et aL4carried out measurements in a diaphragm diffusion cell for solution concentrations up to 31 g % in 0.1 M acetate buffer at pH 4.7, the isoelectric point of BSA. The correlation they gave for diffusivity is 7.1 x 10-7 D = 0.159~ tanh (0.159~) where c is in g of BSA/100 cm3 of solution (g % ) and D is in cm2/s. More recently Phillies et al.5 have measured the diffusivity of BSA in 0.05 M phosphate buffer, 0.05 M acetate buffer (both in 0.2 M saline solution), and 0.15 M pure saline water over the pH range 4.3-7.6. They used a laser scattering spectroscopic technique. Their data showed a significant dependence of the diffusivity on BSA concentration, pH, and buffer used. For pH 5.0, 0.05 M acetate buffer, 0.2 M NaCl solution their measurements are closely correlated by the linear relation D = (5.9 - 0.016~)X lo-’ (2) where, as above, D is in cmz/s and c in g %. Their data around p H 7 scattered appreciably. The important differences existing among the various sets of data, partic0022-3654/79/2083-1228$01 .OO/O
ularly at high concentrations, are illustrated in Figure 1. In this study a general procedure is presented for determining indirectly the diffusivity and gel concentration (solubility) of large molecules dissolved in solution from data on ultrafiltration permeate rates of the solvent as a function of applied pressure. Ultrafiltration is a pressure driven membrane separation process in which the hydrostatic pressures applied across the membrane are about 0.7 X lo5 to 7 X lo5 Pa. The solvent passes through the membrane and the macromolecules do not. Other material which does not pass through includes particulate matter, colloids, and suspensions. Rejection is usually close to complete. The rejected material remains behind in a concentrated solution and could provide a high resistance to flow across the membrane. For this reason the solution is usually circulated past the membrane and is removed from the ultrafilter in a concentrated form. In the procedure described here we consider ultrafiltration in an unobstructed plane parallel channel, under conditions of both laminar and turbulent flow. No matter what the flow regime or how high the flow rate past the membrane, there is always some accumulation of the macromolecules at the membrane surface. This leads to the phenomena of concentration polarization, in which Michaels6 has postulated that the high concentration solution causes a local increase in osmotic pressure which lowers the permeate rate through the membrane by reducing the effective driving pressure. The consequence is that an increase in applied transmembrane pressure results in less than a proportional increase in permeate flux. Concentration polarization increases with pressure and the flux curve becomes more nonlinear. At sufficiently high pressure there is a “saturation” effect and the flux approaches a limiting value which is independent of the membrane permeability and pressure, but which depends on the flow speed, geometry, and solution properties. In Michaels’ picture, under conditions of high concentration polarization the concentration of the macromolecular solution at the membrane reaches a condition where gelation takes place and a gel layer forms on the membrane. The gel layer then offers the limiting resistance to the permeate flow. Of importance is that the “polarized’ layer, in which the solution concentration is higher at the membrane surface C9 1979 American Chemical Society
The Journal of Physical Chemistry, Vol. 83, No. 9, 1979
Determination o’f Diffusivity and Gel Concentration
limiting flux data to zero flux enables the gelation concentration to be determined. We have obtained the analogous turbulent flow result for limiting mass flux by a derivation that exactly parallels the derivation of the turbulent flow limiting current flux in electrodialysis under the same flow conditions.8 The results for the two cases can be shown to be completely analogous, including the constants, and the ultrafiltration flux may be written (turbulent flow; Re 2 1100)
pH 7.1 - 7.7 (Phillies et al.)
P
!.
0 x
-1’
Dg
\
L
2
L. I 5
C
V,,im = O . O ~ ~ - S CRen.9 ~ . ~In~ h C,
L
0
1229
I I 1 I I A I 10 15 20 25 30 35‘ 55 Albumin Concentration(g/lOO c m 3 )
I 60
65
Figure 1. Diffueivity of bovine serum albumin solution.
than in the bulk of the fluid, offers a diffusive resistance to the flux. If the flux-pressure curve is known theoretically as a function of the diffusivity, which is concentration dependent, then for a known solution diffusivity behavior the curve could be traced out. Conversely, if the curve could be determined experimentally, then by inverting the solution and assuming uniqueness in the functional behavior of the diffusivity, it should be possible to determine the diffusivity dependence on concentration. It is this latter approach which we adopt here. Gel Concentration Based on Michaels’ gel model,6 the limiting flux for the ultrafiltration of a macromolecular solute has been obtained in a laminar channel flow with a concentration dependent di~ffusivity,~ for the mass transfer problem described. A boundary layer integral method was used and the following simple analytic expression was obtained for the average limiting flux V, over the membrane (channel) length L (laminar flow; Re S 700)
Here, h is the channel half-height, U the mean flow velocity, D the diffusivity at the gel concentration, and cg and c, t i e gelling and bulk solution concentrations, respectively. In the derivation of eq 3 the fraction of solvent ultrafiltered over the channel length was assumed to be small and cg/c, large. Whether the flow is laminar or not is determined by the Reynolds number, Re, where Re = Uh/u, with ua>the bulk kinematic viscosity. This relation closely parallels the widely used formula of Michaels,6 but with the diffusivity evaluated a t the gel concentration rather than the bulk concentration. The ge1atio:n concentration at the membrane surface is evaluated by observing from eq 3 that
g3 -
vw,b(
In cg - In c,
(4)
a form appropriate for varying flow rate Q. Assuming the functional relationship of eq 4 to be correct, then the limiting flux when plotted against bulk concentration should be linear on a semilog plot. Extrapolation of the
where S c = u,/D, is the Schmidt number based on the bulk kinematic viscosity and diffusivity. Note here that the average flux is independent of the channel length and hence is the same as the local flux at any point. This is a consequence of the fact that the turbulent concentration polarization layer is modeled as a one-dimensional thin film. For varying flow rate, expressed by varying Reynolds number Re = Uh/u,, the turbulent limiting flux is
VwhmRe-n.9
-
In cg - In c,
(6)
Here, the weak dependence of Sc1I4on c, through the dependeme of D , on c, has been neglected, so that as for laminar flow the limiting flux varies linearly with the logarithm of the bulk concentration. We may therefore use eq 4 or 6 to determine cgand we show below that either approach gives the same result. Ultrafiltration experiments in a parallel plate channel have been carried out in both laminar and turbulent flow with the system pressurized typically in the range of 0.7 X l o 5 to 4 X lo5 Pa. The channel length was 48 cm, the width 7.62 cm, and the height about 0.38 cm. The working solution was BSA a t pH 4.7 (the isoelectric point) in 0.1 M acetate buffer. Amicon UM-10 membranes with a macromolecular rejection above 99.3% were used during the course of experiments. BSA concentrations were measured by UV light absorption with a spectrophotometer at the absorption peak of 280 my. Additional experimental details may be found in ref 7 and 9. Typical experimental ultrafiltrate flux-pressure curves for laminar and turbulent flow are shown in Figures 2 and 3, respectively. It can be seen that gel polarization in turbulent flow requires considerably higher bulk concentrations than in laminar flow. Figure 4 is a semilog plot of the pressure independent limiting flux as a function of bulk concentration in laminar and turbulent flow, as suggested by eq 4 and 6. The gel concentration from extrapolation to zero flux is determined to be 34 g/ 100 cm3 from both the laminar and turbulent flow data.lg The striking agreement between the results from the two approaches lends much support to the procedure. In ref 9, using eq 4 and the laminar flow data of Blatt et al.1° the gel concentration at pH 7 was found to be 58 g/100 cm3 (see footnote 11). Kozinski and Lightfoot12 measured the solubility of BSA at pH 6.8 and found it to be 58.5 g/ 100 cm3. At that concentration they observed the solution is like an “immobile gel”. There are no solubility data known to us in the literature at pH 4.7 to compare our result with but it is known that the solubility is at a minimum at pH 4.7, the isoelectric point of BSA, so that the lower value of cg at pH 4.7 in comparison with pH 7 is consistent. Diffusivity at Gel Concentration From the experimental limiting flux data in laminar flow, and using the value of cg = 34 g/ 100 cm3 determined
1230
The Journal of Physical Chemistry, Vo/. 83, No. 9, 1979
R. F. Probstein, W.-F. Leung, and Y. Alliance
I _ _ _ _ ?
r
003
c!g/100cm3)
P 0CI
1, ;: 1
Figure 4. Determination of gelation concentration for bovine serum albumin solution at pH 4.7 from limiting flux data in laminar and turbulent flow.
--I
? 4
0
,
0
3
A
2
2 44
I
2
1
3
,
4
~p x I O " ( P ~ )
Figure 2. Flux-pressure data for ultrafiltration in plane laminar channel flow. 0 04
003
IC 100
I
1
I
I
Re
I
I
I l l 1
uh
1000
1
6 30
:
v03
002 \
-5
Flgure 5. Comparison between theory and experiment for limiting flux in laminar and turbulent channel ultrafiltration of bovine serum albumin solution at pH 4.7.
9 Q(t/min)
00
v
20
x
18
0
16 14 12
A 0
e v
A 0
1
,
cmi4/100cm3~
10 9
,
I
7 2
5.40 5.51 5.21 4 97 4.71 4.77 4.75
,
,
4.68
3
Ap x 10-5(Po) Figure 3. Flux-pressure data for ultrafiltration in plane turbulent channel flow.
above for pH 4.7, the value of D, is found from eq 3 to be 5.6 X cm2/s. This same value of diffusivity was found over a range of limiting fluxes corresponding to different flow speeds. The physiochemical condition of the BSA solution is the same as in the work of Keller et al.,4 however, the value of D, obtained indirectly from the ultrafiltration measurements is four times higher than their value of 1.4 X cm2/s (extrapolated from 31 to 34 g/100 cm3) as measured by a diaphragm diffusion cell method. This is shown in Figure 1. Our value of D, is close to but a little higher than the Phillies et al.5 value of 5.3 X cmz/s (extrapolated from 20 to 34 g/100 cm3) at pH 5.0 in a similar acetate buffer. Their data were obtained by a laser spectroscopic technique. Using the ultrafiltration data of ref 9, the value of D, at pH 7.4 had previously been
found7 to be 6.7 X cmz/s by the method described here. This value is also in general agreement with the scattered data reported by Phillies et aL5 for pH 7.1-7.7. Using the values of c, and D, found above for pH 4.7, we have calculated the limiting flux values in laminar and turbulent flow given by eq 3 and 5 and compared them in Figure 5 with the measured values (drawn from Figures 2 and 3). The agreement is excellent. It may be noted that the transition between laminar and fully developed turbulent flow spans a Reynolds number range between 700 and 1100, in good agreement with the early channel flow measurements of Davies and White13 and the more recent data of Isaacson and Sonin.14 It is important to recognize that the functional dependence of the limiting flux on D, is different for the two flow conditions: Vwiim Dg213 for laminar flow but V,,, D, for turbulent flow. We conclude, therefore, that the value of D, determined from the laminar result must in fact be more than a matched constant in order to fit the two theories, each with a different power law dependence on D,. Moreover, the excellent agreement between theory and experiment lends weight to the correctness of the value determined. The procedure outlined could be applied to the determination of the diffusivities at gelation for different pH ranges and ionic and buffer strengths than those considered here, as well as to other macromolecular solutions.
-
-
The Journal of Physical Chemisfry, Vol. 83, No. 9, 1979
Determination of Diffusivity and Gel Concentration
Diffusivity-Concentration Relation As described earlier, a t low degrees of concentration polarization ithe departure from a linear flux-pressure relationship is assumed to be a consequence of an increased osmotic pressure a t the membrane because of the higher solution concentration there. The amount of flux transmitted through the membrane at any point along the channel may be expressed by the relation15 (7) u, = N A P - dcw)l Here, uw is the local flux corresponding to the local osmotic pressure, T , ad the solution at the wall concentration, A is the membrane permeability coefficient, and Ap the applied transmembrane pressure difference. The determination of the concentration a t the membrane surface (the wall) is a mass transfer problem. In our discussion we assume the flow to be a laminar, fully developed channel flow, with perfectly rejecting membranes constituting the parallel channel walls. The concentration diffusion layer next to the membrane is assumed to be thin compared to the channel width but growing along the channel. The plane laminar, convective diffusion equation with a concentration dependent diffusivity together with the appropriate boundary conditions defines the permeate flux through the membrane by defining the concentration a t the membrane. The concentration at the membrane is coupled to the osmotic pressure through the boundary condition of complete solute rejection U W C W = [D(ac/ay)l, (8) with y distance measured normal to the membrane. The problem as described has been solved in closed form by a boundary layer integral method.16 The solution depends, of course, on the behavior of diffusivity on concentration, D ( c ) , as well as on the osmotic pressure dependence 011 concentration ~ ( c ) .Reliable data exist for the osmotic pressure behavior. What we propose here is that the full functional diffusivity-concentration behavior can be obtained by matching the low polarization integral solution with experirnental ultrafiltration data by varying the functional form of the diffusivity dependence until agreement is achieved between theory and experiment. In doing this, however, we note that reliable values of the diffusivity are available at low concentration, and from the results described in the preceding section, at the gelling concentration. Such an indirect “bootstrap” determination of diffusivity is considerably simpler than the direct experimental methods now employed and, as we shall show, is probably more accurate in many cases. The details of the integral mass transfer analyses may be found in ref 16. The results required to compute the flux-pressure curve are given in the Appendix. For the calculations presented here the osmotic pressure data for BSA a t pH 4.;‘ were obtained from the measurements of Vilker.17 These data are in reasonable agreement with earlier data by Scatchard et a1.,18 although at higher pH they are consistently higher. The diffusivity-concentration relations used in the calculations were those of Keller et al. and Phillies et al. given by eq 1and 2, respectively, and the relation proposed here
D
= (7 - 0 . 0 4 1 ~ x )
(9)
with D in cm2/8 and c in g %. Equation 9 has been defined to agree with the dilute concentration data in the literature and the high concentration value determined in the last section. The comparison of the different diffusivity behaviors is shown in Figure 1. Calculations using the formulas given in the Appendix have been carried out a t pH 4.7 for c, = 2.44 g/100 cm3
O o 3 I
;I
o
Experiment lcm: 2 4 4 g / 1 0 0 c m 3 , Re.21:
Theory
D(Keller e l 0 1
0
1
1231
2
1
3
Ap~lo-~(Po)
Figure 6. Comparison of ultrafiltrate flux-pressure curves for bovine serum albumin solution at pH 4.7 with different assumed diffusivity behavior in laminar channel flow at fixed bulk flow rate.
and Q = 2 L/min (Re = 220) using Vilker’s osmotic pressure datal7 and the three diffusivity relations of eq 1, 2, and 9. The results of these calculations and a comparison with experimental data taken from Figure 2 are shown in Figure 6. The comparison would seem to justify eq 9. It shows that the data of ref 5 are likely to be more nearly correct, and it casts serious doubts on the measurements of ref 4. It can be seen in Figure 6 that the diffusivity function of Keller et aL4 gives rise to a maximum in the flux at low pressure, which is in qualitative disagreement with our measurements and those of others. The maximum in the flux is a consequence of the fact that the diffusivity of eq 1 decreases rapidly with concentration, resulting in extremely large polarization resistance. In order to show that the diffusivity behavior of eq 9 provides the same level of agreement between theory and experiment under different conditions, we have shown in Figure 7 the comparison using eq 9 at different flow rates and bulk concentrations. We have also shown in Figure 7 the limiting flux values given by eq 3. The comparison of Figure 7 provides a strong confirmation of the results, and the validity of the osmotic pressure and gel model approaches. Conclusions From a comparison of experimental ultrafiltration flux-pressure excursion curves in laminar channel flow with a closed form theoretical mass transfer solution it is possible in a relatively simple manner to determine the diffusivity characteristics of a macromolecular solution. Diffusivity as a function of concentration for bovine serum albumin solutions at pH 4.7 as determined by this method casts doubts on the data of Keller et aL4 obtained with a diaphragm diffusion cell, but shows that the measurements of Phillies et aln5obtained using a laser scattering technique are much more nearly correct. Similarly, the comparison of measured limiting ultrafiltrate fluxes with analytic expressions obtained for this quantity in laminar and turbulent channel flow on the basis of a gel resistance model enables the gelation concentration and gel diffusivity to be determined for macromolecular solutions. Values of these properties are obtained for bovine serum albumin solutions at pH 4.7 and 7.4. The methods developed enable solution diffusivities and gel concentrations
1232
The Journal of Physical Chemistry, Vol. 83,
No. 9, 1979
R. F. Probstein, W.-F. Leung, and Y. Alliance
-
= ApAh/D,
Osmotic pressure model Gel model
-_-I
D~ ; 5 5 5 x ~ o ' ~ c m ~ / s e c
I
I
Cq .34g/100cm3
T =
T/P,
b = n,Ah/D,
(A4)
The boundary layer integral method solution16yields the following for dSldY:
I
i
-4
*I
[/&
502
where f = y6*2(1 - 2/6*) 0
--- - - - -
,-"-
-- -- --
---
3 E
C-O-~-------
- -.
a(t/min~ 001
c,(g/100
A
6
165
0
4
2 74
3
3
2 57
7
2
2 44
and where
cn3j
d6* -_ -6* _ dr
Y
o I I
1
I
1
4
2
~pxlO-s(Pai
Figure 7. Comparison of theory and experiment for ultrafiltrate fluxpressure curves for bovine serum albumin solution at pH 4.7 in laminar channel flow with different bulk flow rates and an assumed diffusivity given by eq 9.
to be obtained relatively simply as a function of concentration, pH, and temperature.
Acknowledgment. The authors thank Dr. Joseph Shen of Abcor Corp. for many helpful suggestions. This research was supported by the Office of Water Research and Technology of the U.S. Department of Interior. Appendix
Formulas from Boundary Layer Integral Solution for Calculating Ultrafiltrate Flux-Pressure Curves i n Plane, Laminar Channel Flow. The local flux through the membrane, v,, can be calculated from eq 7 once the solution concentration at the membrane, c,, is known as a function of distance along the channel, x, from a solution of the convective mass transfer problem, The average flux for a channel of length L is then given by
In ref 16 it is shown that an implicit expression for c,(x) is given by a quadrature, which for numerical computation is most conveniently written in the dimensionless form
Here, S is the fraction of permeate ultrafiltered from the channel inlet to the point X, and y is the dimensionless wall concentration where
The parameter y measures the degree of concentration polarization. The dimensionless pressures and membrane permeability are defined by
(-49) The dimensionless diffusion coefficient is reduced with respect to the bulk value D, = D,/ D,
The permeability of the Amicon UM-10 membranes used in the experiments ranged from 2.67 X to 3.63 X cm3/(cm2)(min)(kPa).Interpolation of the osmotic pressure data of VilkeP a t pH 4.7, with c in g 70 and P in Pascals may be represented by
+
r4,7(Pa)= 2.30 X 103c - 1.34 X 102c2 3 . 0 9 ~ ~ ( A l l ) Osmotic pressure data for BSA at other pH may be found in ref 17. References and Notes (1) J. M. Creeth, J . Biochem., 51, 10 (1952). (2) P. A. Charlwood, J . Phys. Chem., 57, 125 (1953). (3) P. Doherty and G. B. Benedek, J . Chem. Phys., 61, 5426 (1974). (4) K. H. Keller, E. R. Canales, and S. I. Yum, J . Phys. Chem., 75, 379 (1971). (5) G. D. Phillies, G. EL Benedek, and N. A. Mazer, J . Chem. Phys., 65, 1883 (1976). (6) A. S. Michaels, Chem. Eng. Prog., 64 (12), 31 (1968). (7) R. F. Probstein, J. S. Shen, and W. F. Leung, Desalination, 24, 1 (1978). (8) R. F. Probstein, A. A. Sonin, and E. Gur-Arie, Desalination, 11, 165 (1972). (9) J. S. Shen and R. F. Probstein, Ind. Eng. Chem., Fundam., 16, 459 (1977) (IO) W. F. Biatt, A. Dravid, A. S. Michaels, and L. Nelson, "Membrane Science and Technology", J. E. Flinn, Ed., Plenum Press, New York, 1970, p 47. (11) M. R. Doshi has pointed out to us that the flow rates of ref 10 were misinterpreted in ref 9 to be a factor of 4 higher than they were. This constant factor does not affect the determination of c . (12) A. A. Kozinski and E. N. Lightfoot, AIChE J., 18, 1030 (6972). (13) S. J. Davies and C. M. White, Proc. Roy. SOC. London, Ser. A, 119, 92 (1928). (14) M. S. Isaacson and A. A. Sonin, Ind. Eng. Chem., Process Des. Dev., 15, 313 (1976). (15) U. Merten, Ind. Eng. Chem., Fundam., 2, 229 (1963). (16) W. F. Leung and R. F. Probstein, Ind. Eng. Chem., Fundam., in press. (17) V. L. Vilker, Ph.D. Thesis, Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Mass., 1975. (18) . . G. Scatchard, A. C. Batchelder, and A. Brown, J. Am. Chem. SOC., 68, 2320 (1946). IN PROOF: Recent experiments suggest that, for laminar (19) NOTEADDED flow, higher bulk concentrations than those of Figure 4 should be used to reliably extrapolate !he gel concentration.
