363
Langmuir 1989, 5 , 363-369
E. Caponetti as a NATO Senior Fellow (Bando n. 217.18/03).
Appendix Although the intraparticle form factor given by ( F ( K ) ) for ellipsoidal particles has been expressed in integral form,12J7 no explicit analytical expression permitting analysis and easy computation is available. We have obtained a power series for this expression which permits us to make a direct comparison with the Guinier expansion in eq 6. Although we intend to publish this result sepa(17)Kotlarchyk, M.;Chen, S.-H. J. Chem. Phys. 1983, 79,2461.
rately, the series is of the form 2 A 2 ~4-2 A 2 e 2 ~ 2 3 A 4 ~ 4 ~44A 4 ~ 2+~84A 4 ~-4 115 875 20A6e6~' 2 4 A 6 ~ 4 ~ 36 2 A 6 ~ 24-~ 6 4 A ' ~ ~ ... 165375 As shown in Figure 8 and suggested by Guinier,12differences between intraparticle form factors and the Guinier exponential expression given in eq 6 are minimal for axial ratios, e z 0.4. This allows us to extend the use of eq 6 for our samples to K values several times those appropriate for spheres with acceptable error. Registry No. n-Hexadecane, 544-76-3; 1-pentanol,71-41-0; 1-butanol, 71-36-3; oleic acid, 112-80-1;ethanolamine, 141-43-5.
+ +
+
+
+
Polyelectrolyte Ultrafiltration of Multivalent Ions. Removal of Cu2+by Sodium Poly(styrenesulfonate) K. James Sasaki,t Susan L. Burnett,? Sherril D. Christian,*?+Edwin E. Tucker,+ and John F. ScamehornS Institute for Applied Surfactant Research, The University of Oklahoma, Norman, Oklahoma 73019, Department of Chemistry, The University of Oklahoma, Norman, Oklahoma 73019, and School of Chemical Engineering and Materials Science, The University of Oklahoma, Norman, Oklahoma 73019 Received August 29, 1988. In Final Form: November 8, 1988 Ultrafiltration and equilibrium dialysis methods, using the polyelectrolyte poly(styrenesulfonate) (PSS), have been used to investigatethe removal of Cu(I1) from aqueous streams. Separations have been measured at mole ratios of styrenesulfonate to total copper varying from 101 to 3:1, at NaCl concentrations varying from 0 to 80 mM. Retention ratios (i.e., ratios of the total concentration of copper in the retentate to that in the permeate) as large as lo3 have been measured for solutions containing large PSS:Cu(II) ratios. Retention ratios are shown to increase, at fixed PSS:Cu(II) ratios, as the total concentration of copper decreases. The ultrafiltration and dialysis results are well correlated by an ion-binding model proposed previously (ref 1).
Introduction Previou~lyl-~ we showed that micellar-enhanced ultrafiltration (MEUF), utilizing ionic surfactant micelles, can be an effective method for concentrating and/or removing divalent ions from aqueous streams, in the presence and in the absence of added 1 : l electrolyte. Experimental techniques have been described for using MEUF14 and equilibrium dialysis or semiequilibrium dialy~isl"-'~ to study the removal of both organic and ionic species. We have developed an ion-binding model to predict the concentrations of ions that will pass through an ultrafiltration or dialysis membrane when known concentrations of an ionic surfactant are present in micellar form in an aqueous stream, together with known concentrations of the oppositely charged multivalent and monovalent ions.' The model utilizes a two-phase theory developed by Oosawa" to determine the fraction of each counterion that will be either "bound" to the polyelectrolyte or "free" in the bulk aqueous solution. An important feature of the model is the assumption that the activity of a neutral electrolyte passing through the ultrafiltration or dialysis membrane
* Author
*
to whom correspondence should be addressed. Department of Chemistry. School of Chemical Engineering and Materials Science.
0743-7463/S9/2405-0363$01.50/0
will equal the equilibrium activity of that electrolyte in the retentate solution. By use of this assumption, together (1)Christian, S. D.; Bhat, S. N.; Tucker, E. E.; Scamehorn, J. F.; El-Sayed, D. A. AZChE J. 1988,34, 189. (2)Scamehom, J. F.;E l l i i n , R. T.; Christian, S. D.; Penney, B. W.; Dunn, R. 0.; Bhat, S. N. AICHE Symp. Ser. 1986,250,48. (3)Scamehom, J. F.;Christian, S. D. In Surfactant-Based Separation Processes; Scamehorn, J. F., Harwell, J. H., Eds.; Marcel Dekker: New York; 1989;Chapter 2. (4)El-Sayed, D. A.; Scamehom,J. F.; Christian, S. D., in preparation. (5)Leung, P. S. In Ultrafiltration Membranes and Applications; Cooper, A. R., Ed.; Plenum: New York, 1979; p 415. (6)Dunn, R.0.; Scamehom, J. F.; Christian, S. D. Sep. Sci. Technol. 1985, 20, 257. (7)Dunn, R.0.; Scamehom, J. F.; Christian, S. D. Sep. Sci. Technol. 1987, 22, 763. (8) Bhat, S. N.; Smith, G. A.; Tucker, E. E.; Christian, S. D.; Smith, W.; Scamehorn, J. F. Znd. Eng. Chem. Res. 1987,26, 217. (9) Scamehorn, J. F.; Harwell, J. H. In Surfactants in Chemical/ Process Engineering; Wasan, D. T., Shah, D. O., Ginn, M. E., Ede.; Marcel Dekker: New York, 1988; p 77. (10)Christian, S. D.; Smith, G. A.; Tucker, E. E.; Scamehorn, J. F. Langmuir 1985, 1, 564. (11)Smith, G.A.;Christian, S.D.; Tucker, E. E.; Scamehorn,J. F. J. SolutLon Chem. 1986, 15, 519. (12)Smith, G.A.;Christian, S. D.; Tucker, E. E.; Scamehorn, J. F. Langmuir 1987,3, 598. (13)Smith, G.A. Ph.D. Dissertation, University of Oklahoma, 1986. (14)Higazy, W.; Mahmoud. F. 2.:Taha, A. A,; Christian, S. D. J. Solution Chem. 1988,17, 191.
0 1989 American Chemical Society
364 Langmuir, Vol. 5, No. 2, 1989
Sasaki et al. POLYELECTROLYTE ANION
-
RETENTATE
uLTRirILTRiTIo\J/ IIE!dBRANC
+
PCKXIEATF.
Figure 1. Schematic diagram of polyelectrolyte ultrafiltration
to remove multivalent ions from water.
with material balance equations and the condition of electrical neutrality, excellent predictions can be made of the concentration of components in the permeate solution. An empirical factor is included in the model to account for the effect of an added 1:l electrolyte (usually NaC1) on the removal of the polyvalent counterion. Given the similarities between charged micelles of ionic surfactants and polyelectrolytes, we thought it should be possible to use polyelectrolytes instead of micelles to remove metal ions by ultrafiltration. One potential advantage of substituting polymer macroions for ionic micelles is that it should be possible to use polyelectrolytes a t extremely low concentrations in water to remove trace quantities of highly poisonous or valuable multivalent counterions. Polyelectrolytes do not dissociate into smaller (monomeric) species a t low concentrations, whereas ionic micellar solutions contain increasing fractions of monomers as the total surfactant concentration is diminished toward the critical micelle concentration. Moreover, observations and predictions regarding the removal of ions by MEUF have indicated that the separation efficiency increases to a large limiting value as the total concentration of ions and micelles is reduced, a t a constant mole ratio of micellar surfactant to multivalent ion. Thus, the removal of multivalent ions, such as those found in radioactive or metal-plating industry wastes, could potentially be achieved by using very small concentrations of a polyelectrolyte. Figure 1is a schematic diagram illustrating the proposed treatment of an aqueous stream to remove dissolved multivalent ions by ultrafiltration. A polyelectrolyte having a charge opposite that of the multivalent ions is added to the stream; these ions become highly concentrated in the immediate vicinity of the polyelectrolyte, as compared with the "bulk" aqueous solution.16 When the stream is forced through an ultrafiltration membrane, most of the polyvalent ions remain in the retentate solution, and the permeate solution (under favorable conditions) will contain only very small concentrations of salts other than 1:l electrolytes. The present paper describes the use of sodium poly(styrenesulfonate) (PSS) to remove divalent copper ion from aqueous streams containing Cu2+,with and without added sodium chloride. Excellent retention ratios (i.e., ratios of the total concentration of copper in the retentate (15) Michaels, A. S.; Bixler, H. J. In B o g . Sep. Purif. 1968, I, 143. Strathmann, H. Sep. Sci. Technol. 1980, 15, 1153. (16)Oosawa, F. Polyelectrolytes; Marcel Dekker: New York, 1971.
to that in the permeate) are determined for the solutions containing little or no NaC1; reasonably good separations are achieved even in the presence of 4C-80 mM NaCl. All of the results are correlated with the polyelectrolyte binding model described previously1 but assuming that the PSS anions are rodlike, rather than spherical, as was assumed for micelles of cationic and anionic surfactants. Experimental Section Sodium poly(styrenesu1fonate) (PSS),having an average molecular weight of approximately85000, was obtained from Aldrich Chemical and utilized without further purification. Elemental analysis of the compound and the observed change in weight on drying were consistent with the empirical formula NaC8H7S03.2H20.CuC1, (MallinckrcdtChemical Works, analytical reagent grade) and NaCl (Fisher Scientific, reagent grade) were used without furhter purification. Ultrafiltration and equilibrium dialysis experiments were performed as described previ~usly.'-~@'~ In the ultrafiltration studies, 300-mL solutions containing known concentrations of PSS, CuC12, and NaCl were introduced into the batch ultrafiltration cell. A pressure of 414 kPa (nitrogen gas) was applied, and samples (approximately30 mL each) were collected until nearly two-thirds of the liquid had passed through the ultrafiltration membrane (having a molecular weight cutoff of 5000 daltons). The experimental values of permeate concentrations in Table I correspond to the midpoint of each ultrafiltration run, at which point approximately 100 mL of the liquid had passed through the membrane. Included in Table I are values of the retentate concentrations of PSS, Cu2+,and NaC1, also at the midpoint of the run. Table I1 includes similar results for the equilibrium dialysis runs, for 5-mL retentate and permeate solutions,using a membrane having a molecular weight cutoff of 6000 daltons. The analytical concentrations of species reported in Table I1 were determined approximately 16-20 h after the start of each experiment. Atomic absorption spectrometryand visible spectrophotometric analysis using the reagent PAR17J8were employed to determine concentrations of copper in both the ultrafiltration and equilibrium dialysis experiments. In none of the experiments was PSS detected in the permeate solutions. To avoid problems with the adsorption of Cu2+on ultrafiltration membranes, each membrane was soaked overnight in the retentate solution prior to an ultrafiltration run. The first 20 or 30 mL of permeate solution was discarded, because this sample ordinarily contained anomalously large concentrations of copper. Dialysis membranes were first soaked for approximately 12 h with distilled water and then treated overnight with an aqueous Cu2+solution (approximately 1 mM). Finally, they were allowed to stand overnight in distilled water before use. Dialysis and ultrafiltration experiments were performed at 30 "C. Discussion Ion-Binding Model. Consistent with our previous modeling of ultrafiltration and semiequilibrium dialysis results for the removal of organic and ionic solutes, we assume that in either the ultrafiltration or dialysis experiments the species passing through the membrane are at thermodynamic equilibrium with the retentate solution. That is, CuClz and NaCl will have the same thermodynamic activities in the permeate solution as in the retentate solution, although of course the PSS is not present in the permeate. Because the ionic strengths of the solutions containing no added NaCl are quite small, we neglect activity coefficient effects and equate the corresponding ion products for the two electrolytes:
(17) Anderson, R. G.; Nickless, G. Analyst 1969, 92, 207. (18) Iwamoto, T. Bull. Chem. Soc. Jpn. 1969, 34, 605.
Langmuir, Vol. 5, No. 2, 1989 365
Polyelectrolyte Ultrafiltration of Multivalent Ions Table I. Polyelectrolyte Ultrafiltration of Cu2+
0.003 11 0.003 02 0.001 67 0.001 74 0.001 18 0.001 20 0.001 23 0.001 20 0.001 19 O.OO? 10 0.001 09 0.000 990 0.000 820 0.001 97 0.003 16 0.002 04 0.001 97 0.003 78 0.003 10 0.004 62 0.006 22 0.001 02 0.003 41 0.002 52 0.006 55 0.002 14 0.001 99 0.006 78 0.008 62 0.011 5 0.001 83 0.002 98 0.005 58 0.003 66 0.002 48 0.000 997 0.000 904 0.000 951 0.000 999 0.000 999 0.009 84 0.000 962
0.010 7 0.002 50 0.038 5 0.020 1 0.002 54 0.0100 0.008 48 0.012 5
0.012 4 0.012 5 0.007 53 0.0124 0.005 08 0.004 46 0.003 85 0.003 31 0.002 93 0.002 58 0.002 25 0.001 28 0.002 06 0.005 13 0.008 24 0.005 36 0.005 13 0.009 96 0.011 3 0.019 3 0.025 8 0.004 36 0.021 8 0.010 8 0.020 5 0.005 55 0.005 23 0.029 1 0.023 0 0.030 9 0.015 6 0.012 4 0.012 4 0.031 2 0.010 8 0.008 52 0.008 52 0.008 52 0.008 52 0.008 52 0.008 52 0.004 53
0.000 047 2 0.000041 6 0.000011 2 0.000 004 60 0.000 006 60 0.000 008 80 0.000 022 2 0.000 035 7 0.000 073 7 0.000 086 1 0.000 126 0.000 370 0.000 040 0 0.000 104 0.000 228 0.000 118 0.000 109 0.000 263 0.000 062 0 0.000 084 0 0.000 135 0.000 004 00 0.000 020 0 0.000 028 0 0.000 554 0.000 116 0.000 104 0.000 128 0.000 784 0.001 12 0.000 000 990 0.000 041 0 0.000 740 0.000 015 0 0.000 134 0.000 006 62 0.000 192 0.000 098 6 0.000 003 09 0.000 040 0 0.000 032 5 0.000 193
0.000 030 0 0.000 025 2 0.000 006 25 0.000 002 71 0.000 003 63 0.000 006 24 0.000 014 9 0.000 032 0 0.000 069 5 0.000 085 0 0.000 159 0.000414 0.000 035 6 0.000 101 0.000 200 0.000 101 0.000 101 0.000 246 0.000040 7 0.000055 2 0.000 098 5 0.000 002 79 0.000012 2 0.000 016 1 0.000 524 0.000 116 0.000 096 7 0.000 106 0.000 770 0.001 13 0.000 002 37 0.000 024 2 0.000 817 0.000 009 44 0.000 198 0.000 003 95 0.000 184 0.000 073 7 0.000 004 03 0.000 026 1 0.000 019 5 0.000 101
PSS: sodium poly(styrenesu1fonate) (monomolarity). ret, retentate solution; per, permeate solution. = 4.15, (Y = 10.4. CRejection(%) = (1 - [total copper],,/[total copperIret) X 100%.
where the subscripts ret and per denote the retentate and permeate solutions. The concentrations in eq 1 and 2 pertain to the unbound ions, i.e., the ions not adsorbed on or associated with the polyelectrolyte. We also require that the solutions passing through the membrane be electrically neutral, so that the concentration of chloride ion must equal that of Na+ plus twice that of Cu2+. We assume here that the PSS macroion exists in an extended, rodlike form’6J+21rather than in the spherical form assumed previously’ for ionic surfactant micelles a t relatively low concentrations in aqueous solution. In order to calculate the fractions of each type of ion that are bound and free, we employ the two-phase approximation of Oosawa,16 which can be used to relate the extent of counterion binding of the monovalent and divalent ions to the intensity of the surface potential of the polyelectrolyte. A basic assumption of the Oosawa model is that the ratio of the concentration of a given counterion in the immediate vicinity of the polymer to its concentration in the bulk solution is given by the Boltzmann equation, using (19) Manning, G. S. In Polyelectrolytes; Selegny, E., Ed.; D. Reidel: Boston, 1974; p 9. (20) Nagasawa, M. In Polyelectrolytes; Selegny, E., Ed.; D. Reidel: Boston, 1974; p 57. (21) Manning, G. S. Annu. Reu. Phys. Chem. 1972, 23, 117.
98.48 98.62 99.33 99.74 99.44 99.27 98.19 97.03 93.81 92.17 88.44 62.67 95.12 94.72 92.78 94.22 94.47 93.04 98.00 98.18 97.83 99.61 99.41 98.89 91.54 94.58 94.76 98.11 90.90 90.26 99.95 98.62 86.74 99.59 94.60 99.34 78.75 89.64 99.69 96.00 99.67 79.93
Calculated by text model using
8“
a mean difference of potential between the two regions. For the assumed rodlike macroion, the Oosawa equations are In [(I - P)PI = In [$/(I - 4)l
+ (Pq + P’q?zQ In (I/$) (3)
In
- P’)/P’I =
In [4/(1 - d
l + (Pq + Pq?z’Q In ( 1 / d (4)
where P and p’ are the degrees of dissociation of the monovalent and divalent ions, respectively, 4 is the fraction of the total solution volume in which the bound ions are located, q and q’are the fractions of the free ion charge carried by the two types of counterion, z and z’ are the magnitudes of the counterion charges (here, 1for Na+ and 2 for Cu2+),and Q is the dimensionless potential parameter reflecting the charge density on the macroion. In using Oosawa’s equation to model micellar-enhanced ultrafiltration results, we have found that the choice of the variable 4 does not strongly affect the calculated results. Therefore, we have taken 4 to be equal to the apparent molar volume of the styrenesulfonate (approximately 0.20 L mol-’) multiplied by the total molarity of styrenesulfonate units in solution. As in our previous studies of the removal of divalent copper using anionic surfactant2 and the removal of
Sasaki et al.
366 Langmuir, Vol. 5, No. 2, 1989 Table 11: Equilibrium Dialysis of Cuz+ [CUP+lr.tl M 0.004 53 0.004 51 0.004 13 0.004 15 0.003 71 0.003 67 0.002 68 0.002 81 0.002 26 0.002 31 0.001 84 0.001 83 0.000 908 0.000 438 0.000448 0.008 59 0.007 78 0.007 78 0.006 71 0.007 67 0.006 24 0.006 89 0.004 93 0.004 96 0.003 97 0.003 94 0.003 02 0.003 11 0.002 04 0.002 08 0.010 6 0.010 4 0.009 42 0.009 68 0.009 71 0.008 95 0.006 80 0.006 75 0.005 35 0.005 86 0.003 83 0.003 87 0.002 06 0.002 06 0.001 17 0.001 15 0.002 63 0.002 68 0.002 56 0.002 56 0.002 34 0.002 45 0.002 22 0.002 23 0.001 95 0.001 95 0.001 74 0.001 68 0.001 04 0.001 11 0.000991 0.001 01 0.000 897 0.000 874 0.000 756 0.000 744 0.000 584 0.000 527
[NaClIret, M
[PSSlret," M
0.006 20 0.006 20 0.012 4 0.012 4 0.024 8 0.024 8 0.025 8 0.025 8 0.049 1 0.049 1 0.075 0 0.075 0 0.004 99 0.004 99 0.009 a4 0.009 84 0.019 9 0.019 9 0.040 0 0.040 0 0.080 0 0.080 0
0.012 8 0.012 8 0.011 5 0.011 5 0.010 2 0.010 2 0.007 66 0.007 66 0.006 38 0.006 38 0.005 10 0.005 10 0.002 55 0.001 27 0.001 27 0.038 3 0.034 1 0.034 1 0.029 8 0.029 8 0.025 5 0.025 5 0.021 3 0.021 3 0.017 0 0.017 0 0.012 8 0.012 8 0.008 50 0.008 50 0.119 0.119 0.102 0.102 0.085 2 0.085 2 0.068 2 0.068 2 0.051 1 0.051 1 0.034 0 0.034 0 0.017 0 0.017 0 0.008 53 0.008 53 0.012 5 0.012 5 0.012 5 0.012 5 0.012 5 0.012 5 0.012 5 0.012 5 0.012 3 0.012 3 0.012 5 0.012 5 0.008 51 0.008 51 0.008 38 0.008 38 0.008 39 0.008 39 0.008 51 0.00851 0.00854 0.008 54
exptl 0.000 223 0.000 225 0.000 214 0.000 229 0.000 170 0.000 181 0.000 126 0.000 122 0.000 070 9 0.000 103 0.000 080 9 0.000 078 6 0.000 037 4 0.000 019 8 0.000 009 09 0.000 147 0.000 107 0.000 103 0.000091 7 0.000 102 0.000 064 5 0.000 081 7 0.000 048 0 0.000055 1 0.000 033 3 0.000 029 9 0.000018 5 0.000021 0 0.000007 22 0.000 008 55 0.000 126 0.000 117 0.000 095 2 0.000 102 0.000 091 6 0.000 082 0 0.000 039 2 0.000 044 6 0.000 024 4 0.000 022 4 0.000 005 74 0.00000651 0.000002 16 0.000 001 52 0.000 000 783 0.000 000 758 0.000 080 3 0.000 081 9 0.000 170 0.000 181 0.000 357 0.000 331 0.000 403 0.000 403 0.000671 0.000 669 0.000 836 0.000 876 0.000 004 47 0.000 009 48 0.000 023 7 0.000 021 9 0.000 083 5 0.000 065 5 0.000 219 0.000 193 0.000 308 0.000 250
calcdb 0.000 216 0.000 211 0.000 196 0.000 204 0.000 177 0.000 165 0.000 086 0 0.000 120 0.000 068 1 0.000081 5 0.000 055 4 0.000051 7 0.000015 1 0.000 003 13 0.000 003 88 0.000 136 0.000 118 0.000 118 0.000 086 0 0.000 160 0.000 093 5 0.000 151 0.000 051 3 0.000 052 7 0.000 034 8 0.000 033 2 0.000021 3 0.000 024 6 0.000 010 4 0.000011 5 0.000 055 1 0.000 052 5 0.000 044 9 0.000 048 6 0.000 057 9 0.000 045 5 0.000 025 0 0.000 024 5 0.000016 2 0.000021 3 0.000 008 91 0.000 009 25 0.000 002 80 0.000 002 79 0.000001 08 0.000001 01 0.000 105 0.000 113 0.000 220 0.000218 0.000411 0.000 452 0.000 388 0.000 391 0.000 653 0.000 650 0.000 821 0.000 781 0.000 011 2 0.000 013 1 0.000 029 4 0.000 030 8 0.000 079 0 0.000 075 4 0.000 177 0.000 173 0.000 291 0.000 260
rejection,' % 95.24 95.33 95.24 95.08 95.23 95.51 96.79 95.74 96.98 96.47 96.99 97.17 98.34 99.29 99.13 98.41 98.48 98.48 98.72 97.92 98.50 97.80 98.96 98.94 99.13 99.16 99.30 99.21 99.49 99.45 99.48 99.50 99.52 99.50 99.40 99.49 99.63 99.64 99.70 99.64 99.77 99.76 99.86 99.86 99.91 99.91 96.00 95.79 91.42 91.46 82.44 81.57 82.57 82.50 66.54 66.60 52.93 53.35 98.93 98.82 97.04 96.96 91.20 91.37 76.59 76.72 50.13 50.63
PSS: sodium poly(styrenesu1fonate) (monomolarity). ret, retentate solution; per, permeate solution. Calculated by text model using Qo = 4.52, a = 14.2. CRejection( % ) = (1 - [total copper],,/[total
copper],,,)
chromate anions using cationic surfactant,' we observe here the deleterious effect of added NaCl on the separation of divalent copper by PSS. Added sodium chloride acts to
X
100%.
reduce the potential of the polyelectrolyte anion, presumably by compressing the electric double layer. Moreover, the Na+ ions present in the added electrolyte (as well as
Polyelectrolyte Ultrafiltration of Multivalent Ions
Langmuir, Vol. 5, No. 2, 1989 367
the cations already present in sodium PSS) compete with the Cu2+ ions for binding sites near the surface of the macroion. Previously,l we accounted for the reduction in potential at the surface of the (assumed) spherical micelles by using the empirical equation
P = P/(l+ cy[NaCl]1/2)
(5)
where is the potential parameter for spheres in the absence of added NaCl and P is the value of this parameter when sodium chloride is present at the concentration [NaCl]. a is an adjustable parameter used to account for the effect of added 1:l electrolyte in modeling all the results. We assume here that an equation analogous to eq 5 may be useful in correlating the variation of Q with [NaCl]. Thus, we write
Q = QO/(l + ~t[NaCl]'/~)
(6)
where Qo is the potential parameter for rodlike macroions in the absence of added NaCl and Q is the value of the parameter a t a given molar concentration of NaC1. Equations 1-4 and 6, together with material balance equations and the condition of electrical neutrality, constitute a mathematical model for predicting ultrafiltration and equilibrium dialysis results. The model involves only the two parameters Qo and a; by solving the equations simultaneously, one can predict the concentration of Cu2+ in the permeate solution for any given total equilibrium concentrations of PSS, Cu2+,and NaCl in the retentate solution. In the following section, we describe the use of nonlinear least-squares analysis to fit all of the data to the model. However, it is of interest first to show that the equilibrium dialysis data, for retentate solutions containing [ P s s ] to [total copper] ratios of approximately lO:l, 5:1, and 3:l in the absence of added NaC1, can be predicted reasonably well by using the theoretical value of Qo for a linear polymer having the structure of sodium polystyrenesulfate. In Oosawa's development,16Qo is shown to vary inversely with the distance between adjacent charges, projected onto the axis of the rodlike polymer. Thus, assuming that the charges on the sulfonate groups are separated by a linear distance of 2.53 A along an extended, saturated hydrocarbon chain,16p22 one can predict that Qo will equal 2.83. Figure 2 shows how well the model, utilizing this value of Qo, predicts the results for solutions containing no added salt. There are, to be sure, systematic deviations of the experimental results from the theory: the results for retentate solutions having approximately a 5:l ratio of [PSS] to [total copper] show that copper concentrations in the permeate are significantly smaller than predicted, and the observed values of [Cu2+Ipe,for the 3:l solutions are also somewhat smaller than theoretical a t the higher concentrations of copper. These deviations might be attributed partly to the specific binding of Cu2+to the sulfonate ions or to the bridging of Cu2+between sulfonate groups of a given chain or in two chains. It is also possible that bulky arenesulfonate groups might prevent the carbon atoms in the polymer chain from being fully extended, in exclusively trans configuration^;^^ this would increase the theoretical value of Qo. Moreover, our neglect of activity coefficient effects for the free Na+ and Cu2+ions may result in calculating permeate concentrations of these counterions that are too large for a given value of Qo. A more detailed treatment of counterion condensation in polyelectrolyte solutions (see ref 24) might improve the prediction of (22) Oman, S.; Dolar, D. 2. Phys. Chem. Neue Folge 1967, 56, 1. (23) van der Helm, D., private communication.
0.000
0.002
I
I
I
I
I
0.004
0006
0.008
0.010
0.012
[Culrct I M
Figure 2. Equilibrium dialysis results for retentate solutions containing approximately 3:l (m), 5:l (X), and 101 (0) mole ratios of styrenesulfonate to total copper in the absence of added NaCl. Solid curves represent predictions of the polyelectrolyte model described in the text, obtained by using the theoretical value of the charge density parameter, Qo = 2.83; dashed lines represent predictions made with Q" = 4.25, fitted to all of the equilibrium dialysis and ultrafiltration data. [CUI,, and [CUI,, denote total concentrations of copper in the permeate and retentate com-
partments. ultrafiltration results, although at the sacrifice of the simplicity of the present model. But for many purposes, satisfactory estimates of the separation of copper by equilibrium dialysis or ultrafiltration can be made with the theoretical value of Qo, calculated for the geometry of the extended chain without using any adjustable parameters. Nonlinear Least-SquaresAnalysis of Results. Details of a nonlinear least-squares method, capable of correlating all of the data of each type with the mathematical model described above, are given in ref 1. The penultimate column in Tables I and I1 lists the computed values of the concentration of Cu2+ in the permeate for all of the experiments; these concentrations were calculated by using separate values of the adjustable parameters cy and Qo to fit the ultrafiltration and dialysis results. Constants used in correlating the results are as follows: for the equilibrium dialysis runs, cy = 14.2 f 1.0 and Qo = 4.52 f 0.21; for the ultrafiltration experiments, cy = 10.0 f 0.4 and Qo = 4.15 f 0.07. The root mean square deviation in the permeate copper concentration for the two types of data is 3.26 X M (corresponding to a mean relative error of approximately 35%) for the dialysis data and 2.77 X M (corresponding to a mean relative error of approximately 36%) for the ultrafiltration results. When all of the ultrafiltration and dialysis results are treated together as a single collection of data, the values a = 12.3 f 0.5 and Qo = 4.25 f 0.09 are obtained, and the root mean square deviation in the permeate Cu2+concentration is 3.12 X lo+ M (corresponding to a mean relative error of 37%). The derived value of Qo is approximately 50% greater than the theoretical value for poly(styrenesulfonate), assuming an extended chain structure with complete sulfonation of the styrene (vide supra). D 0 1 a r ~has ~ also noted that Qo is (24) Manning, G . S. J. Phys. Chem. 1984, 88,6654.
Sasaki et al.
368 Langmuir, Vol. 5, No. 2, 1989 0 OW3
0 wo2
B . -b .I
a
0
\
--rrrrrr
0.0000
[NaCl] / M
Figure 3. Effect of added NaCl on separations: dialysis results (0) and ultrafiltration results (m)for retentate solutions containing approximately a 9:l ratio of styrenesulfonate to total copper at [total copper] = 0.92 mM. Dashed curve predicted by using Qo = 4.25 and CY = 12.3 (see text).
greater than predicted for the extended chain; thus, the best fit of osmotic coefficient data for polystyrenesulfonates requires values of Qo 30% greater than theoretical for Mg2+and 60% greater for Ca2+. Given the wide ranges of concentrations of PSS, NaC1, and CuC1, used in the present experiments, we consider that the results obtained by using the model provide an excellent demonstration of its utility in predicting the degree of separation that can be achieved by polyelectrolyte ultrafiltration. The final column in Tables I and I1 includes values of the rejection, defined as R = 1 - [total copper],,,/[total copperIret Values of R vary from 0.999 (99.9%) in the case of solutions at large [PSS]:[totd copper] ratios to values as low as 47% in solutions containing 80 mM NaC1. The dashed lines in Figure 2 represent the least-squares fitted curves for dialysis results, for retentate solutions containing approximately 3:1, 5:1, and 1 O : l ratios of PSS to [total copper]. Some of the scatter in the plots and the deviation of points from the theoretical lines arise because the ratios are not exactly equal to the nominal values. However, the functional dependence of [CuP+Ipe,on [total copper],,, for any given ratio of styrenesulfonate to copper, is well established. Both the experimental study and the analysis of data show that the rejection of copper(I1) in an ultrafiltration process will continue to increase as the aqueous stream contains less and less PSS and copper, provided that the concentrations of these species are maintained at fixed ratios. This favorable result is a consequence of the fact that the relative extent of binding of divalent ions to a polyelectrolyte containing mostly monovalent counterions increases monotonically with in~
~~
(25) D o h , D. In Polyelectrolytes; Selegny,E., Ed.;D. Reidel: Boston, 1974; p 102.
[Culret / M
Figure 4. Predicted dependence of retention ratio (definedas total concentration of copper in the retentate solution divided by total concentration of copper in permeate) on total concentration of copper in the retentate, for an assumed mole ratio of styrenesulfonateto total copper of 101. The curve was calculated by using the theoretical value Qo = 2.83.
creasing dilution. In contrast, most separation methods become markedly less efficient a t higher dilutions. Some of the results for separations in which NaCl is added to retentate solutions containing nearly constant concentrations of PSS and total copper are shown in Figure 3, for both dialysis and ultrafiltration experiments. The solid curve represents permeate concentrations calculated by the nonlinear least-squares fit of all the data, obtained by using the parameter values a = 12.3 and @ = 4.25. The effect of NaCl in decreasing rejections, or increasing the concentration of copper that penetrates the membrane, is similar to that observed in the removal of divalent ions by ionic surfactant However, the separation efficiencies are still reasonably large in the presence of NaCl at concentrations several times that of the styrenesulfonate groups; moreover, as a solution containing a given ratio of NaC1:PSS:total copper is diluted with water, the rejections rapidly increase toward unity. Potential Advantages of Polyelectrolyte Ultrafiltration in Removing Trace Quantities of Multivalent Ions. The remarkable fact that separations become better and better as the total concentration of polyelectrolyte and multivalent ion is decreased, at a constant ratio of the two species, makes it seem likely that polyelectrolyte ultrafiltration may become a successful industrial method for concentrating ions that are either highly toxic at very low solution concentrations or very valuable. Figure 4 is a log-log plot of the retention ratio (equal to the total concentration of the divalent ion in the retentate divided by that in the permeate) vs the total concentration of the divalent ion in the retentate, for an assumed ratio of styrenesulfonate units to total metal ion equal to 101. The solid curve represents predictions of the polyelectrolyte model with @ = 2.83, calculated by assuming PSS is a fully extended chain. (Values of Qo obtained by the nonlinear
Langmuir, Vol. 5, No. 2, 1989 369
Polyelectrolyte Ultrafiltration of Multivalent Ions least-squares method described above are somewhat larger than this, so separations might be even better than indicated by Figure 4.) Nonetheless, the model predicts that only small amounts of PSS would be required for virtually quantitative removal of divalent or trivalent cations, present a t small concentrations. With each 10-fold reduction in [total copperIret, and a corresponding 10-fold decrease in [PSS], the retention ratio increases by nearly a factor of 10. The important conclusion reached in the preceding paragraph-that separations become better on increasing dilution-warrants further discussion. Equations 3 and 4,which form the basis of the theory for predicting ultrafiltration rejections, can be combined to give
[(I - p')/p'1/[(1 - @)/PI2 = (1 - 4)/4
(7)
which may be rewritten as [cu'+]b,ret/ [Cu2+If,ret= ([Na+Ib,ret/[Na+I,,,et)2(1 - 4)/4 (8) where the subscripts b,ret and f,ret refer to ion concentrations bound and free, respectively, in the retentate compartment. Independent of the value of Q (the potential parameter in the Oosawa theory), the ratio of bound to free Cu2+will vary quadratically as the ratio of bound to free Na+, with a proportionality constant depending only on 4 (the volume fraction of polyelectrolyte). At sufficiently small concentrations of the polyelectrolyte, with Na+ ions in large excess to Cu2+ ions, the ratio [Na+]b,ret/[Na+]f,retbecomes nearly constant and approximately equal to (Q - l)/Q, provided that Q > 1. In this same limit, 4 in the numerator of the factor (1- +)/4 can be neglected, so that the term becomes proportional to the reciprocal of the PSS concentration in the retentate. Thus eq 8 predicts that when 4
