OCt. 20, 1959
STABILITY CONSTANTS FOR
BINDINGO F UNIVALENT
CATIONS BY
POI- -GROUPS 5295
By means of a first-order power series expansion, we obtain A,
-L
=
(e) x
(m.
- m,’)
(3e)
where m,‘ has been defined in the text as the effective electrolyte concentration. We then have
By combining equations 3a and 3f, we obtain equaas given in the text* NEW BRUNSWICK, XEW
JERSEY
RUTGERS, THESTATE UNIVERSITY]
[CONTRIBUTION FROM THE SCHOOL O F CHEMISTRY OF
Counterion Binding by Polyelectrolytes. 111. Stability Constants for the Binding of Univalent Cations by PO,- -Groups of Polyphosphates from Electrophoresis Measurements1 BY ULRICHP. STRAWS AND PHILIP D. Ross RECEIVED APRIL 23, 1959 The degree of binding of alkali metal ions to long-chain polyphosphates has been determined by electrophoresis measurements a t 0” in aqueous solutions maintained a t 0.2 ionic strength with tetramethylammonium bromide. The method depends on a n empirical calibration with Ca++, Mg++ and M n + + , which are essentially completely bound a t divalent ion to polyphosphate normality ratios up to 0.5. If the alkali metal ions and the tetramethylammonium ions are considered as site-bound, the association of individual POa--groups with the univalent cations follows the Law of Mass Action modified for the effects of the high potential near the polyelectrolyte chain, of the nearest neighbor interactions of adjacent Pos-groups and of the changes in the molecular dimensions. The binding constants decreased in the order of increasing crystal radii of the cations, namely, Li+ > N a + > K + > CS+= (CHa),N+.
I n the first paper of this series, the effects of lithium, sodium, potassium and tetramethylammonium (TMA+) ions on the electrophoretic mobility of long-chain polyphosphates were compared.2 The results indicated that the alkali metal ions were more strongly bound to the polyphosphate than was the TMA+. However, a meaningful quantitative comparison of the data was complicated by many factors, such as variations in the ionic environment and in the molecular dimensions of the polymer chain. I n the work described in this paper, considerable simplification was achieved by comparing the effects of small amounts of various alkali metal ions on the electrophoretic mobility of the polyphosphate chain in the presence of a large excess of TMABr. By keeping the total ionic strength of simple electrolyte a t 0.2 N , the comparison could be made while the polymer chain was maintained in a relatively constant condition. With certain assumptions i t also was found possible to determine the extent of binding quantitatively and to calculate binding constants. Such information contributes to an understanding of polyelectrolyte behavior and may also help to elucidate certain biological phenomena.
Experimental The potassium polyphosphate employed was a Kurrol’s salt (our sample No. H-2170W) with weight-average degree of polymerization P, equal to 9,400.s (1) T h e contents of this paper are contained in a thesis to he submitted b y P. D. Ross t o t h e Graduate School of Rutgers, T h e State University, in partial fufillment of t h e requirements for the degree of Doctor of Philosophy. This investigation was supported by a grant from t h e United States Atomic Energy Commission under Contract AT(30-1)1018. (2) U.P. Strauss, D. Woodside and P. Wineman, J. Phys. Chcrn., 61, 1353 (1957). (3) This sample was prepared by R. Hubbard who also determined its P, by viscometry. T h e sample was washed with water t o remove any low molecular weight species as described by R. Pfanstiel and R. K. Iler, TKISJOURNAL, 74, 6059 (1952).
Electrophoretic mobilities were determined at 0’ in a Perkin-Elmer Model 38 Tiselius apparatus by the method previously described. Single sharp boundaries were obtained by slowly adding solvent from a glass tube with a fine capillary tip when bringing the boundaries into view. This improvement in the technique is due to M. Levy. In general, the ascending boundaries moved 1-5% faster than the descending boundaries. Average values of the mobilities are used throughout this paper. There was no effect of polyphosphate concentration on the mobility in the concentration range employed (0.01-0.02 N ) . The pH of all solutions was maintained between 6.2 and 7.5, in which range it did not affect the mobility.2 Each solution of the potassium polyphosphate in 0.2 N TMABr was introduced into a Visking cellulose bag and dialyzed against several portions of this TMABr solution to eliminate the potassium ion. Then the dialysis was continued against several portions of the alkali metal bromide-TMABr solution against which the boundaries were to be formed.6 Tumbling a t 5’ was employed to ensure the attainment of equilibrium. Dialysis for electrophoresis of the divalent counterions studied (Ca++, M g + + and M n + + )was conducted by a slightly different technique. After the removal of the IC+,carried out as described above, a portion of the polyphosphate solution was pipetted into a new dialysis bag and an amount of divalent cation ( M + + ) bromide solution, calculated to give the desired degree of binding, was added from a micro-buret. The dialysis bag was sealed and immersed in a twenty-fold excess of 0.20 M TMABr. Equilibrium was obtained by tumbling for 24-36 hr. a t 5 O . S Conductivity water prepared by passing distilled water through a mixed-bed ion-exchange resin and whose specific conductance ranged from 0.3 to 0.8 X reciprocal ohms per cm. a t 0’ was used to make up all solutions employed in this investigation. ~~
(4) U. P. Strauss, N. L. Gershfeld and H. Spiera, i b i d . , 76, 5909 (1954). (5) I t was previously established that after three t o four changes, the outside solution maintains its original composition. (6) As a check in a few cases, both the inside and outside solutions were analyzed for C a + + after completion of the dialysis. These analyses were carried out by flame photometry using the internal standard procedure. (L.L. Merritt, H. H. Willard and J. A. Dean, “Instrumental Methods of Analysis,” D. Van Nostrand Co., New York, N. Y.,1951, pp. 79-83). All of the added C a + + was found in t h e inside solutions containing t h e polyphosphate, while no C a t + could be detected in t h e outside solutions. These analyses, which were carried o u t u p t o C a + + t o polyphosphate normality ratios of 0.3 and were accurate t o rt0.02 in this quantity, lend support t o the assumption of virtually complete binding of Ca in this range. + +
ULRICHP. STRAUSS AND PHILIPD. Ross
5296
TABLE
ELBCTROPHORETIC MORILI rms +
None" Li +
Na
+
(hl+)''
X 104
t
BY
~ T M A
2 07 ii 397 i, 603 0 0025 1 99 382 U 09 53 005 1 91 ,367 17 46 01 1 78 342 29 17 03 1 62 311 45 24 05 1 5.3 294 h4 17 10' 1 41 271 20 01
03 05 10'
1 23 1 93 1 70
236 370
1 55
29X Zi.1 21'1 3XX 3% 365 '322 271 384 372 344 305
1 42
I
DERIVEDBISDINGPARAMETBRB
-e+/
1'
hl
AND
Vol. 81
326
i6
15 37 52
48 30 18
kT
3 04 2 92 2 8!1 2 61 2 38 2 24 2 07 180 2 83 2 49 2 27 2 ox 167 2 96 2 87 2 79 2 46 2 07 2 93 2 84 2 62
KTMA~
KhiC
Ku'C
ii
5 1 i1 4 5 6 f I1 9 6 2 i 7
13 i 2 9 i
2 7 i 2 4 f 2 I i .3 6 i
f !I i i 4 17 i .i 1 6 2 4 3 3
1J 16 2 0
I 4
+ +
0 7 i I 1 i 1 (5
+
4 4 3
2 2
083 i 0 I196 f 110 i 136 i 162 i 155 f
Ku/ KThia
(104
..
004
OOt
13 14 1,i 11
00.1
11
004 (104
Khl
(cor.)d . ,
1.1 1.2
I .2 0 R $1
. .
1 2
1 2
, .
110 i I60 172 f
+
004
004 NIS
5.9 7.0 8.7
0 .5
.6
-,
..
3 1 i 3 1 9 i 2 1.7 0.14 1194 i O l i l 0 7 f 1 0 16 i 08 1 4 11 1 96 12 6 i 1 I6 i 04 118 i 005 10 1.2 17 f 03 145 i 005 05 1 90 18 46 6 =t 1 .11 1 3 10 1 68 10 f 1 38 f 0.1 39 29 284 i 006 .. 20 1 41 73 79 i OX 1i f 1 il,117 03 2 00 0 ?,7 zt 11 09 i 03 111 f 005 08 08 54 cs .(I7 12 f 0,3 140 i 006 0S 1 93 14 49 44 f !J8 9 ,116 . I 19 i 03 258 i 1JO8 10 1 79 39 f !)6 28 38 20 1 59 70 2 33 1 11 f ox 11 i 113 The solutions against which the polyphosphate solutions were dialyzed contained ( M + )equivalents of alkali metal io11 These results were outside the limits of the calibration method for ob:ind 0.2 -(M+) equivalents of T M A + per liter. taining values of @ and K . The precision estimates are based on a n experimental uncertainty of 0.02 in the values of u . These values were obtained by multiplying K M by ( 0 . 0 8 3 / K ~ ~ See ~). .ill of the K's have units of reciprocal normality. text.
K+
+
20
1 14
01 03
2 02
78 05
50 51)
Viscosities were measured in a Bingham viscometer as previously described .z
most of the cation content is TMA+ and the polymer chain therefore is roughly in the same condition, i is decreased by the different cations in the Results and Discussion Electrophoretic Mobility and Degree of Ioniza- order Li+ > E a + > K + > Cs+. Thus, the binding tion.-The effect of the alkali metal ions on the of the alkali metal ions increases with diminishing sizes of the anhydrous cations, a result which electrophoretic mobility ZL, expressed in cm.2/ volt-sec., of the polyphosphate is shown in the indicates that the P03--group penetrates through first three columns of Table I, where XI+ represents the hydration layer of the cations. lo Degree of Binding of Alkali Metal Ions.-It is the alkali nietal ion, and (M+) its normality in the solution against which the polyphosphate solution clear that with increased binding of alkali metal ions was dialyzed. The ThI.I+ normality of each the electrical potential of the polyelectrolyte is solution was 0.2 - ( A I + ) , so that the total normality reduced, and therefore some previously bound was constant a t 0.2. In all cases bromide was the TMA+ ions are released. Consequently the obby-ion. The mobility is seen to decrease with in- served decrease in i must be less than the degree of creasing (M+),to a different degree with each alkali binding of the alkali metal ions and cannot be used metal ion. I t has been established previously that as a quantitative measure of this binding. One at the total electrolyte normality employed here, way to obtain such a measure is by means of an the polyphosphate chains are free draining.'>.l empirical calibration with cations which are known Under these conditions, according to several recent to be, for all practical purposes, completely bound. theoretical treatment^,^,^ the electrophoretic mo- The ions chosen for this purpose were C a + + , bility is proportional to the degree of ionization X g + + and A h + + . These ions are very strongly i of the polyelectrolyte. By a combined study bound bypolyphosphatesll, l 2 buthave little tendency of conductivity and electrophoresis of long-chain to form complexes with other species occurring in our solutions. The electrophoresis results for polyphosphates a t O 0 , the relationship these ions are shown in Fig. 1, where u is plotted u = 5 . 2 x 10-4i (1) against the ratio of the divalent cation normality was found to hold in 0.2 N TMABr, KBr, NaBr to the polyphosphate normality. I t is seen that and LiBr ~ o l u t i o n s . ~We shall therefore assume as long as this ratio is 0.5 or less, the results with equation 1 to be valid in all of our systems. The the three cations fall on the same straight line, and resulting values of i are shown in the fourth column we may therefore assume essentially complete of Table 1. It is seen from these results that for a binding.13 The above ratio is then equal to BM, given low value of (M+), e.g., 0.01 or 0.03, where (10) See footnote 28 of ref. 2. (7) U. P. Strauss and S. Bluestone, THISJOURNAL, 81, 5292 (1959). (8) J. T h . G. Overbeek and D. Stigter, Rec. t v ~ u ' .chim., 76, 543 (1956). (9) J . J . Hermans and H . Fujita, Proc. A k o d . Amsterdam. B68, 182 (1955)
(11) J. R. Van Wazer and D. A. Campanella, THIS JOURNAL, 72, 655 (1950). (12) R. M . Smith and R. A. Mberty, i b i d . , 78, 2376 (1956). (13) Since t h e binding constants are known t o be different for these cations.52 incomplete binding would result in different ciirvcs i n Pig. I .
OCt. 20, 1959
STABILITY
CONSTANTS FOR BINDINGOF UNIVALENT CATIONS BY PO3-
the fraction of phosphate groups which are neutralized by bound metallic cations.14 The line in Fig. 1 can be represented by the equation j 3 ~= 2.08
-uX
lo';
5 0.5
+
(3)
The quantity ( h l f ) e 5 is the effective concentration of M + near the polymer chain and may be given by the Boltzmann relation (M+),ir = (M+) exp( -e+/kT)
(4)
where e is the charge on the cation, k the Boltzrnann constant and T the absolute temperature. The quantity $is the electrostatic potential a t the surface of the polymer chain and is here assumed to be equal to the zeta potential, The validity of this assumption has been established both on theoretical* and experimentalz0,21grounds. The zeta potential is calculated from the electrophoretic mobility by the relation
r.
1j-l =
Gvu/D
p.
(2)
As the abscissa in Fig. 1 exceeds 0.5, the mobility values begin to diverge, which means that at least the Ca++ and Mg++ ions are not completely bound in this region. The relative position of the curves suggests the binding order Mn++>>Mg++>Ca++, which is in agreement with the findings of others. l 2 One more requirement is necessary for the validity of our calibration scheme, namely, that there is also free draining in the presence of the divalent cations. The intrinsic viscosities were therefore determined a t 50% binding of Mn++, Mg++ and Ca++ and found to be 3.3, 3.7 and 2.8, respectively, a t Oo. Since these values are much larger than 0.507, the value determined by Hubbardl6 in the theta solvent 0.415 N NaBr16 a t 25', the polymer chain is quite expanded and therefore free draining.17 The values of phl for the alkali metal ions are given in the fifth column of Table I. For all the solutions containing both M + and T h l A + , PM was calculated by eq. 2.18 For the 0.2 iV alkali metal bromide solutions, / 3 was ~ taken to be equal to (1 - i).' Binding Constants.-Writing the equation for the binding phenomenon, M.+ POS- = MP03,we may ~ the ~ define an apparent binding constant K M ' ,by relation
(5)
where 7 and D are the viscosity and dielectric constant of the solvent, respectively.s Since the polyphosphate always moved toward the anode, {. and hence +, are negative. (14) I n defining p ~ i t, is understood t h a t one equivalent of POagroups is bound by one equivalent of cation. If more than one equivalent of POa--groups were bound, i t would not be detected by our experimental method, nor would i t a5ert any of our conclusions. (15) R. Hubbard, unpublished results. (16) U. P. Strauss and P. L. Wineman, THISJ O U R N A L , 80, 2366 (1958). (17) T h e different values of t h e intrinsic viscosity obtained furnish support for t h e previously reported conclusions t h a t , at equal degrees of binding, different cations bave di5erent effects on t h e solvent affinity of t h e polymer chain. (18) T h e values of OM calculated in this manner are independent of t h s values of i as obtained from eq. 1. (19) I n t h e calculation of Kr',the POa--groups are assumed t o be independent. T h e effect of nearest neighbor interactinns is considered below. (20) A. Katchalsky, J . Polynter Sci., 12, 159 (1954). (21) A. Katchalsky, N. Shavit and H. Eisenberg, ibid.. 13, 69 (19S41
-GROUPS 5297
2.0
-
1.8 t .
- -
0 X
1
1.6 -
1.2
0.0
0.2 Normality
0.6
0.4 Ratio
of
0.8
M++ t o
PO;.
Fig. 1.-Calibration of electrophoresis method with divalent cations: 0, C a + + ; 0 , M g + + ; (3, Mn++.
The quantity (POa-)r is the normality of free phosphate groups. Its value depends on how we consider the ThlA i- binding. Calculations show that if we ignore the bound T A L A + groups, K M ' and especially K M ' ~are not constant but change considerably with increasing (ill+ Consequently, ). we shall consider the T h M + also as site-bound. The fraction of POa--groups t o which T M d + is bound, PTMA,is given by the relation
PTMA = 1
- i - PM
(6)
The values of @TMA and (- e$/kT) are given in the sixth and seventh columns of Table I, respectively. Equation 3 can now be expressed in the form PY KM' = exp(e+/kT) (7) (M +)i The values of KM',calculated by eq. 7 , are given in the eighth column of Table I. They are seen to vary by a factor of about two to three for each cation. More relevant values for the binding constants are obtained when the effect of nearest neighbor interactions of POa--groups is taken into account.2a-28 A theoretical treatment based on the one-dimensional k i n g lattice involving the binding of two univalent species gives the equationsz6
+
~ K = M log ( M + ) - 0.43e$/kT - A ~ K
t) + i)
lOg[(l - ~ ) / P M ]- 0.43 cosh-' (1 f
~ K T M=Alog (TMA+) - 0.43 e$/kT ApK log[(l - ~)/PTMA] - 0.43 cash-1 (1
+
(8) (9)
(22) K M is defined as t h e binding constant for which nearest neighbor interactions of POa--groups are taken into account. The details of the calculation will be given below. (23) R . A. Marcus, J . P h y s . Chem., 6 8 , 621 (1964). (24) F. E. Harris and S. A. Rice, ibid., 68, 725 (1954) (25) T. L. Hill, J . Polymer S c i . , 93, 549 (1957). (26) S Lifson. J . Chenz. Phvs.. 26. 727 (1957).
ULRICHP. STRAUSS AND PHILIPD. Ross
5298
Vol. 81
where6 = [(l- 2i)2/2i(l - 41 and ApK = logv = seen that the first and second binding constants may -0.43e2/D~rkt, D E and r being the effective di- go in opposite directions as the size of the cation electric constant and the distance between neigh- changes. This result explains to some extent the boring POo--groups, respectively. In the ninth and frequently observed complexity of the binding tenth columns of Table I, values of K M and KTMA behavior of polyelectrolytes and ion-exchange resins. are given, calculated by eq. 8 and 9, respectively, If the values of KTMA are affected by changing with the assumptions that r = 3 A. and D E = 89. molecular dimensions in the same way as those of The effect of changing the assumed value of D E is K M , the ratio K M I K T M should A be constant for a not very great. Thus, if one were to set D E equal given alkali metal ion. This ratio is also indeto 28, the values of the binding constants would pendent of the electrical potential, of whether the be about one-half those in Table I nearest neighbor interactions are considered or not, It is seen that the values of K M and K T M Aaxe and hence of the effective dielectric constant and reasonably constant for a given cation a t low values the distance between charged groups. Thus, it is of (M+). I n the case of Na+, K f and Csf, how- not surprising that, on the whole, the values of ever, there seems to be a tendency for K M to in- KMIKTMAshow less variation for each alkali crease a t the higher alkali metal ion concentrations. metal ion than do the values of K M . ~ ' If one plots K M against (M+) on a double logarithThis result suggests a way of extrapolating the mic graph, one finds that for each of these ions values of KM, a t constant ionic strength of 0.2, the curve rises smoothly to the value of K M a t to infinite dilution of M f . The extrapolation, (Mf) = 0.2. The effect is especially noticeable which because of the scattering in the KM-values with K T M Awhich increases with increasing (M+) would be difficult to perform directly, is achieved for all four alkali metal ions. We believe that this by multiplying each K M by the quantity (0.083/ effect is caused by the contraction of the polymer K T M A ) .The ~ ~ resulting corrected values of K M coil which occurs as (AI+) increases. As the denoted by KM(COr.), are given in the last column polymer coil contracts there will be more points of of Table I and are seen to be quite constant. contact between different portions of the polymer These constants are useful for comparing the relachain. At such points of contact the binding of tive binding order of polyphosphates for the unications should be strengthened. One might further valent cations, which is Li+ > Na+ > K + > Csf = expect that the size of the cation should play a role. TMAf. This order has been partially observed Thus, a very small cation such as Li+ might be by others for long-chain polyphosphates, for buried sufficiently a t its first binding site to make lower phosphate polymer^,^^-^* for phosphoproi t difficult for a second potential binding site in teins33 and phosphonous and phosphonic ion-exanother portion of the polymer chain to come change resins. 3 4 close enough for additional complexing. This would The results show that the association of inexplain the absence of the coiling effect on R L ~dividual POs--groups with univalent cations follows and the reversal of the normal binding order in the Law of Mass Action, modified for the effects 0.2 N Lif and Na+ solutions.27 The situation can of the potential, of nearest neighbor interactions probably be described by two binding constants between adjacent P03--groups and of the changes for each cation. The first binding constant gives in the molecular dimensions of the polymer. the binding to the first binding site and is probably The reported values of KM(cor.), K Mand KTXA also quite well represented by Khf a t low (AIf)where the provide a t present the best estimates for the polyphosphate chain is extended. This constant hypothetical binding constants of the individual is seen to increase with decreasing size of the POa--groups for the various univalent cations in a anhydrous cation. The second binding constant constant ionic medium of T ~ I L I B ~ . ~ ~ gives the binding of the M + ion to an additional XEWJERSEY binding site on the polymer chain. The necessary N E WBRUNSWICK, data concerning molecular dimensions are not yet (28) In t h e case of Li', this result is somewhat misleading It available; therefore these constants cannot be is quite clear t h a t t h e values of KL,. while fluctuating fairly widely, determined at this time. However, from the remain essentially constant while those of KTUA increase consistently decreases as expected information available so far one can say that this with increasing (ht +). Therefore KL~/KTM\IA from the previous discussion. I t is t h e fluctuations in t h e value of second constant is smaller for Lif than for the other K L which ~ obscure this decrease of KLJKTMAin the table. alkali metal ions treated in this paper. It is (29) T h e value 0.083 represents K'rjra in the absence of alkali metal (27) T h i s reversal is enhanced by t h e expected reciprocal effect; namely, t h a t a t a given degree of binding, & f 9 of two cations t h e one whose K%fis affected more strongly by the coiling of t h e polymer chain should also be more effective in tying different parts of t h e chain together and t h u s in causing t h e coiling of t h e chain. I t has been shown, for instance, t h a t t h e molecular dimensions are very much smaller in 0 2 N N a B r t h a n in 0.2 N LiRr, despite the f a c t t h a t the degree of ionization is almost the same.' T h e phenomenon discussed here also furnishes a simple explanation f o r t h e different solvent affinities of site-bound MPOs groups which were reported previously.' T h i s discussion m a y also be applicable t o t h e observation t h a t carrageenin is gelled by K +, R b and Csf h u t not by L i + and N a + (D. B. Smith and W. 1% Cook, A r c h . Biochem., 46, 232 (1953)). +
ion.
See Table I .
(30) R. M. Smith and R . A . Alberty, J . Phys. Chem., 60, 180 (1956). (31) J. I. Watters, S . M. Lambert and E. D. Loughran, THISJ O U R NAL,79, 3661 (1967). ( 3 2 ) S. M . Lambert and J. I. Watters, ibid.,79, 4262 (1457). (33) C. W.Carr and W. P. Engelstadt, Avch. Biochrm. Biophys., 77,
158 (1958). (34) J . I. Bregman and Y .M u r a t a , THISJ O U R N A L , 74, 1867 (1982). (35) I t is of interest t h a t in the case of T M A + and Cs+ t h e binding constants are close t o the value 0.05 obtained for t h e strong electrolyte HXOI by a n optical method (0. Redlich and J. Bigeleisen, ibid., 66, 1883 (1943)).
