FRANK E. HARRISAND STUART A. RICE
1360
the chain molecule. When random flight statistics of chain configuration are applicable, Kuhn has shown1g that the probability of ring closure is inversely proportional to the 2/2 power of the chain length. This leads to the estimate that half of the dichelates formed in PMAL involve amino acid groups separated by less than 30 monomer units. The insensitivity of the chelate distribution t o variations in the copper concentration when PMAL is in large excess, indicates that there is little interference between regions of the polymer chain participating in the complexing of different Cu(I1) ions. At a lower excess of polymer such interference would undoubtedly work against the formation of the dichelate, but such conditions could not be studied since they led to polymer precipitation. The experiments in which the degree of neutralization of PMAL was varied, corresponded to a net charge per monomer unit from 2 = $0.01 to 2 = +0.22.20 Since the charged groups are placed at the end of long side chains attached to the polymer
-
(19) W. Kuhn, KollOid-Z., 68, 2 (1934). (20) 2 a [2Cw (H+)]/Cp where Cub ia the concentration
+
-
of bound copper ion.
Vol. 61
backbone the electrical field due to the polymer is relatively weak. It has, therefore, a correspondingly small effect on the equilibria involving the polyion and its counterions, with the apparent first ionization constant of the amino acid groups being increased only by a factor of 2 at the highest positive charge of PMAL which was investigated. Nevertheless, the electrostatic free energyz1 associated with the displacement by a Cu(I1) ion of the proton of a zwitterionic amino acid group attached to a polymer bearing a net positive charge should hinder the formation of the monochelate. It is hard to see why PMAL seems to form this complex more readily than valine (Fig. 1). On the other hand, in the coordination of a second amino acid group to a monochelate carried by the polymer, protons are lost by the PMAL and a net positive charge of the polymer should favor, on electrostatic grounds, the dichelate formation. A slight tendency for C,B to increase with 2 was observed, but the uncertainties in the interpretation of the spectral data preclude a quantitative evaluation of thiti effect. (21) A. Katchalsky and J. Gillis, Reo. ~ W U . chim., 88, 879 (1949).
A MODEL FOR ION BINDING AND EXCHANGE I N POLYELECTROLYTE SOLUTIONS AND GELS BYFRANK E. HARRIS AND STUART A. RICE^ Department of Chemistry, University of California, Berkeley, California, and Department of Chemistry, Harvard University, Cambridge, Massachusetts Received February $6, 1967
A molecular model for linear and cross-linked polyelectrolytes is described. The model emphasizes the effect of interactions between neighboring charged groups upon both configurational and thermodynamic properties of the polymeric systems. Ion binding is introduced in a ,phenomenological manner, and it is shown that the model predicts far larger amounts of binding to polymers than to small molecules containing similar functional groups. It is found that ion binding is necessary to explain the configurational properties and titration curves of linear polyelectrolytes. Moreover, equilibria among ion pairs and unbound ions are shown to provide a means for understanding of the variation of ion-exchange resin selectivity with cross-linking, exchange capacity and the composition of the solution in contact with the resin.
Recent years have seen much progress toward an understanding of the configurational and titration properties of linear polyelectrolytes in solution. This progress has evolved from the synthesis of several concepts, each of which was originally developed for use elsewhere. A current picture of a polyelectrolyte solution includes consideration of the statistics of coiled chains, of electrostatic interactions between ions in solution, and, of particular interest a t this time, of binding of counter ions t o the polymer. Concurrently, many investigators were studying ion exchange in polyelectrolyte gels. The more comprehensive efforts toward elucidation of the exchange phenomena were not from a microscopic structural point of view, but instead treated the gel as a macroscopic phase of suitable thermodynamic properties. The authors have recently shown, however, how the theory of linear polyelectrolytes can be extended to cross-linked polyelectrolyte gels. They found that a consideration of the binding phenomena in the gels leads naturally to a description of their ion-exchange (1) Society of Fellows, Harvard University.
properties. It is the purpose of this paper to survey the methods by which a simple molecular model has been applied to both linear and cross-linked systems, and t o indicate, without mathematical detail, how the assumed model leads to a relatively satisfactory reproduction of many experimental observations. I. Linear Polyelectrolytes.-A basic step in the study of linear polyelectrolytes was taken in 1948 by Kuhn, Kunzle and Katchalsky,a who described the configuration of a polyion as that of a chain which was randomly coiled with the restriction that its mean end-to-end distance be such as to minimize the sum of the configurational and electrostatic free energies. This simple model was able t o predict qualitatively the size changes accompanying changes in charge on a polymer. How(2) Summaries of experimental data and comparison with this model appear in ref. 8 and 11. Much general information is given in: P. Doty and 0. Ehrlich. Ann. Rev. Phus. C h e n . , 8, 81 (1952), and P. J. Flory, “Prinoiples of Polymer Chemistry,” Cornell University Press, Ithaca, N. Y.,1953. (3) W. Kuhn. 0. Kunzle and A. Katchalsky, Hslv. Chim. Acta, 81, 9419 (1948).
Oct., 1957
ION BINDINGAND EXCHANGE IN POLYELECTROLYTE SOLUTIONS AND GELS
ever, this treatment was approximate in that it assumed the charges on the polymer t o have an interaction energy determined only by the end-toend distance, rather than by all the distances between charges on the chain.4 Moreover, it was not shown that a screened coulomb potential was the proper expression to use in deriving the interaction energy between a pair of fixed charges in a solution containing mobile ions. In the same year, Hermans and Overbeek6 proposed a model in which the polymer was regarded as a porous spherical charge distribution, whose radius was to be varied to minimize the total free energy. By using a linearized form of the PoissonBoltzmann equation, they computed the electrostatic free energy without using the screened coulomb potential. Although the Hermans-Overbeek model avoids explicit consideration of the chain configuration, it predicts the size changes of polyions quite well. But neither of the models discussed thus far was conspicuously successful in interpreting the titration curves of polyelectrolytes. Two years later, Huizenga, Grieger and Walls reported experiments which for the first time fully showed the importance of ion binding in polyelectrolyte solutions. They found that up t o 60% of the sodium ions counter to a carboxylate polymer moved with it in electrophoresis. Since these polymers are thought to be free draining under the conditions of this experiment, one concludes that the counter ions are not merely within the volume of the polymer, but are relatively tightly bound to it. The foregoing work, and contributions’ by Flory, Kimball, Cutler and Samelson, Osawa, Imai and Kagawa, and others, led the authors to propose a model8 which was new in that (a) the interactions between charges of the polymer influenced its local configurational properties as well as its end-to-end distance, (b) these interactions affected the tendency toward ionization a t each ionizable group, making the various ionizable groups of a polymer interdependent, rather than independent, as heretofore postulated, and (c) the interactions were large enough to cause important amounts of binding of counter ions, even though negligible amounts of binding would occur if the ionizable groups did not interact. I n implementing this model, it is necessary to allow for the interrelations among its several features. I n particular, to determine the mean size, degree of ionization and binding of a polyion in a solution of given pH, counter ion concentrations and ionic strength, one would first determine the free energy as a function of all six of these variables, and then minimize it with respect to the first three, which are not externally controllable. Moreover, the expression for (4) F. E. Harris and 9. A. Rice, J . Polvms? Sei.. 15, 151 (1955). (5) J. J. Hermans and J. T. G. Overbeek, Rae. 1 m v . chim., 87, 761 (1948). (6) J. R. Huizenga, P. F. Grieger and F. T. Wall, J . Am. Chsm. Soc., 72, 2636, 4228 (1950). ( 7 ) P. J. Flow, J . Chem. Phys., 21, 162 (1953); G. Rimball, M. Cutler and H. Samelson, THISJOURNAL, 56, 57 (1952); F. Osawa. N. Imai, and I. Kagawa, J . Polvmm Sei., 18, 93 (1954). (8) F. E. Harris and 8. A. Rice, THISJOURNAL, 68, 725. 733 (1954) J . Chsm. P h w . , 24, 326, 336 (1956).
1361
the free energy must reflect the fact that not all chain configurations of the same end-to-end distance will have the same electrostatic interaction, nor will all arrangements of the same total number of charges be equi-energetic for a given chain configuration. A further discussion of these and related points will be in order after a more detailed definition of our molecular model. To specify the model more completely, consider for definiteness a weak polyacid, each functional group of which can be un-ionized (zero charge), ionized (- 1 charge), or ion paired (zero charge). It is assumed that an intrinsic “chemical” free energy change can be assigned to the conversion of an ionizable group from one to another of these charge states, so that, ignoring interactions among the charges, the ionization and binding would be describable by equilibrium constants. The binding is thus treated phenomenologically. The ion pairs introduced here may be described as “sitebound,” to distinguish them from ions merely required to be near the polymer to maintain electrical neutrality. The chain configurations available to the polymer are assumed to be just those accessible to an otherwise similar uncharged polymer, so that the effect of the electrostatic interactions is to alter the distribution among these configurations. The chain configurations will be described in the manner first proposed by K ~ h nby , ~regarding the polymer as a series of rigid links connected by universal joints. The lengths of the links are chosen so as to allow as well as possible for all charge-independent forces restricting the short range flexibility of the chain. The Kuhn model makes all the accessible configurations of equal non-electrostatic energy, so that the charge interactions completely determine the manner in which various configurations are weighted. Thus the charge interactions have two effects: they influence the equilibria among un-ionized, ionized and bound functional groups, and they affect the configurational distribution. These interactions are, of course, t o be calculated keeping in mind that the space between the charges is occupied by an ionic solution, and hence the calculation depends upon the ionic strength and other properties of the solution. In accordance with the over-all approach sketched in the third preceding paragraph, we first turn to the formulation of the free energy when the mean size and degrees of ionization and binding are specified. It is convenient to begin by assigning t o the polymer a specific chain configuration, and to specify which particular functional groups are to be un-ionized, ionized and ion paired. One may then consider the free energy of this visualizable, but unattainable “state” of the polymer. On a compietely microscopic scale, this “state” is really a large number of states, because we do not specify the distributions of the small mobile ions of the system, the energy levels of the various species, etc. It is thus indeed relevant to consider the free energy of this “state.” Each change in the configuration or in the status of the functional groups results in a new “state,” with its associated (9) W.Kuhn, Kolloid. Z.,75, 258 (1936): 87,3 (1939).
1362
FRANKE. HARRISAND STUARTA. RICE
free energy. These “states,” together with their free energies, may be taken as the starting point for a statistical mechanical calculation of the over-all free energy when only the numbers of ionized, unionized, and bound groups, and the average configuration are specified. The free energy of a “state” of the type outlined above will consist only of the intrinsic free energies of the functional groups plus the electrostatic interaction free energy. The configuration of the chain will not contribute by virtue of the Kuhn chain model. The only problem is in the calculation of the interaction free energy for the charge distribution of each “state.” The authors have shown1° that in spite of the presence of fixed charges, the Debye-Huckel potential may be used for this purpose. The demonstration involves approximations not much more restrictive than those of the ordinary Debye-Huckel theory. The methods outlined above provide, a t least in principle, a way of calculating the free energy of a ‘(state” of specified configuration and charge distribution. At this point, mathematical complexity forces an approximation. We choose to calculate exactly for each individual “state” only the electrostatic interaction between nearest neighboring functional groups along the chain, and to approximate the remainder of the interaction under the assumption that the charge be distributed evenly along the chain. This approximation has the effect of preserving the strong influence a charged group has upon the ionization and binding a t nearest neighboring groups, while making tractable the much less sensitive calculation of the interactions which promote configurational expansion. To be more explicit, we have just assumed that if the net charge is specified, the distribution of that charge along the chain is independent of the chain configuration, and that the free energy difference among configurations is to be calculated assuming the charge to be uniformly distributed along the chain. The remainder of the over-all free energy now separates into two parts: (a) the configurational free energy of a uniformly charged chain, and (b) the distributional free energy of a chain of charged and uncharged sites with near neighbor interaction. The computation of (a) is essentially a biased random walk problem, while (b) is formally equivalent to the calculation of the spin distribution in B onedimensional Ising lattice. The details of both these computations are described elsewhere.* Qualitatively, it is apparent that increasing the charge on a chain will tend to restrict its configurational freedom, or increase its configurational free energy. One would likewise expect that computation (b) would result in weighting heavily only those charge distributions in which not very many more charges than necessary are nearest neighbors. When the free energy, as computed by the methods just outlined, is minimized with respect to the number of ionized and ion paired groups, it is found that the electrostatic interactions cause the net charge of the polymer to be much less than if no such interactions existed. This means that the (10)
F. E. Hwris and S. A. Rice, J . Chem. Phyr.. !M, 955 (1956).
Vol. 61
free energy increase associated with forming unionized groups or ion pairs is more than compensated for by the reduction in electrostatic free energy, until many more than the “normal” number of ion pairs and un-ionized groups have been created. The effect is much larger than one might a t first suppose, because the charges are frequently close enough together that the effective dielectric constant of the region between them is far lower than that of normal water. Moreover, as the number of charged groups gets fairly large, the vast majority of the possible distributions of the charges along the chain are extremely unfavorable energetically, so that the free energy contains a large negative entropy term. The interaction free energy thus makes it progressively more difficult to remove successive protons from a weak polyacid, spreading the titration curve in the manner experimentally observed. When the protons finally are removed, many of them are replaced by counter ions having little intrinsic affinity for ion pairing. The configurational expansion accompanying neutralization of a polyacid is calculable by minimizing the free energy with respect to its average end-to-end distance. It is found that the binding and reduced ionization cause the expansion predicted for the polymer t o be in much better agreement with experiment than if the binding or reduced ionization were ignored. Moreover, the amount of binding predicted to satisfy the titration and configuration properties is consistent with the amount of site binding observed in electrophoresis.6 11. Polyelectrolyte Gels.-The model set forth in the preceding section may be applied to polyelectrolyte gels.”J2 The basic ideas are the same as those already described, but the details are significantly affected by the cross-linking of the gels. The specific systems of most interest here are strong electrolyte ion exchangers in equilibrium with various kinds of monatomic counter ions. Examples of such systems are the sulfonated crosslinked polystyrene exchangers in equilibrium with mixtures of alkali metal ions. Consider, therefore, a cross-linked gel containing a number of functional groups which can be ionized (- 1 charge) or paired with either of two types of counter ion to give an electrically neutral ion pair. As before, we assume that the intrinsic free energy associated with formation of each type of ion pair has a value which depends upon the kind of counter ion and upon its concentration in the external solution. The ion-exchange selectivity is introduced by assuming different intrinsic free energies for formation of different kinds of ion pairs. Exchange selectivity only results, however, when the conditions are such that significant numbers of ion pairs are formed. Just as for linear polyelectrolytes, it is desired to compute the free energy for a given configuration and charge distribution of the gel. As pointed out before, the Debye-Huckel screened coulomb potential may be used to calculate the electrostatic interactions. However, the cross-linked systems present problems not encountered in dealing with (11) 8. A. Rice and F. E. Harris, 2. physik. Chem., 8, 207 (1958) (12) F. E,Harris and S. A. Rice, J . Chsm. Phys., 14, 1258 (1956)
.
Oct., 1957
IONBINDING AND EXCHANGE IN POLYELECTROLYTE SOLUTIONS AND GELS
1363
linear polyelectrolytes. Many exchange resins are the gel will be assumed to be on a lattice, whose cross-linked sufficiently that there are reasonably scale will be a function of the gel volume. The diswell defined regions inside and outside the ex- tributional free energy of the gel can then be calcuchanger, respectively. The interior region is charac- lated subject to the same sort of approximations as terized by different ion concentrations and by the enter into the simple cell theories of liquids. The calculation of the gel free energy will be presence of enough organic matter to cause the solution to be quite different from that outside. The complete when the configurational free energy and authors’ analysislO confirming the use of the the electrostatic free energy are each characterized screened coulomb potential shows that the screen- as a function of gel volume. As pointed out in the ing constant to be used must involve the proper di- ast paragraph, the electrostatic interactions may be electric constant for the interior region, and must handled by the methods of the cell liquid theory. recognize that the space occupied by organic matter The analogy is exact if ion paired groups are identiis not accessible to the mobile ions. However, the fied as the holes of the liquid theory, and the bulk concentrations should be used, as the different charges as occupied cells. The details have been ion concentrations within the resin are allowed for given elsewhere.11 The configurational free energy in the derivation of the screened coulomb potential. is just that of an uncharged gel, so that the theory The large concentration differences between re- of rubber elasticity may be applied. We have gions internal and external to an ion-exchange found it convenient to use the theory in the form resin have led many investigators t o include an given by Flory. l 4 From the free energy function determined in the “osmotic” contribution to the free energy when describing an ion exchange resin on a phenomenolog- manner just described, one can relate the external ical basis.ls The present model, however, is en- solution composition to the volume and ion pairing tirely molecular in character, and i t would be in- in a resin. Our model differs from the traditional correct for us to include such a contribution in ad- approaches13 in that our only source of selectivity dition t o the direct calculation of the detailed inter- enters through the intrinsic ion pairing constants. actions. The concentration differences are merely We have assumed that the size of the ions enters in a consequence of the forces we have already con- no other way. Another difference is our assumpsidered. Our present point of view incidentally tion that there is an equilibrium between paired removes a conceptual difficulty in treating resins of and free charged groups, with only the paired low cross-linking. For if an osmotic description groups contributing to a selective effect. This is used in preference to a detailed model, ambiguity concept leads to the conclusion that exchange arises when the degree of cross-linking is diminished selectivity is increased when conditions are such to the point where the regions inside and outside that ion pairing increases. Since the ion pairing the resin cease to be well defined. only occurs when necessary to reduce strong elecAlthough we have in principle a means of deter- trostatic fields, we may understand why resins of mining the free energy a t every specified configura- high exchange capacity should be more selective tion and charge distribution, we must again make than resins of lower capacity. Likewise, resins approximations to simplify the calculation. The which, because of high cross-linking, cannot reduce difference in configurational properties of linear their electrostatic interactions through expansjon, systems and gels suggests that the approxima- will be more selective than 1oosely cross-linked tions used here be somewhat different from those of resins which can easily swell. Finally, let us examine more closely the form of the preceding section. In a gel, many of the spatially near neighboring functional groups will not be the ion-exchange selectivity quotient, to see what near neighbors along the polymer chains, so that it behavior is to be expected when the ion composidefinitely will be necessary to consider interactions tion is altered. The selectivity quotient, which between groups which are not closely connected. we shall call S$, is the equilibrium quotient for the In fairly concentrated gels, near neighbors to a reaction given group will in general be distributed so that (free ion)l (ion in resin)^ = (free ion)^ (ion in resin)] most motions a t constant gel volume will not seriously affect t,he electrostatic energy, with some That is pairs of charges approaching more closely as others SIP = c2nt/clnz (1) are carried further apart. This is a radically djfferent state of affairs than that of a linear polymer, where ci is the external concentration of ion i and where a t a constant end-to-end distance the elec- ni is the total number of ions of type i in the resin. trostatic energy depends critically upon the local Now the nil which are measured, differ from the coiling of the chain. We therefore assume, with- numbers of ion pairs because the ni include in out gross inconsistency, that all gel configurations addition the free ions of type i within the resin of the same volume are electrostatically equivalent, volume. Since the free ions are not selectively and that the interaction between the charge state influenced, their concentrations will be proporof the gel and its configuration depends only upon tional t o those of the external solution. The total the gel volume. For the purpose of calculating the number of ion pairs will, as pointed out already, deelectrostatic interactions, the functional groups of pend upon the electrostatic interactions. However, the relative numbers of bound ions of each (13) Among the more well known phenomenological theories are kind are independent of these interactions. This those of H. P. Gregor. J . Am. Chem. Soc., 70, 1293 (1948); 75, 642, 3537 (1951); C. W. Davies and G . D. Yeoman, Trans. Faraday Soe., is because the replacement of one kind of bound
+
49, 968, 975 (1953); E. Glueckauf, PTOC.Roy. SOC. (London), A214, 207 (1952).
(14) P.
J. Flory. ref. 2.
+
H. A. SAROFF
1364
ion by another does not affect the electrostatic energy. I n other words, whether or not the electrostatic interactions are ignored, the reaction (free ion), .j- (ion pair)z = (free ion)a
+ (ion pair),
will have the same equilibrium constant] K K =
CZXI/CIXZ
(2)
where xi is the number of ion pairs involving ions of type i. Since K is in general not unity, we see that ni contains one part which is proportional to ci and one part which is not. Consequently, we may predict that S'z should vary with the ionic composition of the solution, even if the number of ion pairs remains constant. A more quantitative investigation of t,he behavior of S'Z may be made if we know the total numbers of unbound ions in the resin. At normal resin capacities and external solution concentrations, there will be negligible amounts of extra electrolyte in the resin, so that
Vol. 61
the total number of unbound counter ions will be essentially the same as the number of charged functional groups. This last stipulation enables SlZto be expressed in terms of c2/c1, K and the total number of ion pairs. It may then be shown that the variation of S'Zwith c2/cl is in the direction ordinarily observed experimentally, that is, S12is increased by an increase in cz/q, whether K be smaller or larger than unity. A detailed application of the equations of the model indicates that the total amount of binding does not change very much with changes in the exchange state at constan t total ionic strength, so that the conclusions sketched above are confirmed by a more thorough analysis. Acknowledgment.-It is a pleasure t o acknowledge our great indebtedness t o Professor Paul Doty, who originally introduced us t o this problem and who carefully examined much of our earlier work. His advice and enthusiasm have been invaluable.
A THEORY FOR THE BINDING OF CHLORIDE IONS TO SERUM ALBUMIN BASED ON A HYDROGEN BONDED MODEL BY H. A. SAROFF National Institute of Arthritis and Metabolic Diseases, National Institutes of Health, Public Health Service, U . S. Department of Health, Education, and Welfare, Bethesda, Margland Received February 86, IS67
The binding of chloride ion to serum albumin is treated by considering hydrogen bonding between the carboxylate ion and cationic nitrogen groups as well as the sim le electrostatic effects. Equations are derived for this competitive model, and satisfactory fits to the chloride binding and Eydrogen bindingdata are found in the pH region from 2 to 5. The number of sets of sites required to fit the chloride binding data are three. The first set has four sites with an association constant, k A = 2500; the second set has 12 sites with k~ = 50; the third set has 84 sites with k~ = 1. The intrinsic p K of the carboxyl groups required to fit the data is 5.0 for all the assumed 100 groups taking part in hydrogen bonding. The intrinsic pK of the basic nitroqen groups is taken to be 7.0 for the first 16 and greater than 9.0 for the remaining 84.
Introduction bonding model follows the ideas of Laskowski The sites to which anions bind in serum albumin and Scheraga,' and a similar competitive reaction are undoubtedly the hundred positively charged has been applied to explain the binding of Na+ nitrogen centers of arginine, lysine and histidine. l s 2 and K + to m y o ~ i n . ~ , ~ Binding Equations.-The competitive reactions The pH dependence of the binding indicates that the groups controlling the binding (in addition t o assumed to occur are the binding sites themselves) have a pK associated (A) !-COOHZN-J + H + = (-COOH HZN-j. with the carboxyl groups. Electrostatic theory p ( C 0 0 H . HpN) ka, = has been applied to explain the binding by Scatchp(COO-. HsN)cH+ ard and co-workers.a-6 (B) (-COO- H3+N-J Hf J-COOH Hg+N-J. Some inconsistencies occur with the application p(C00H. H 3 ' N L of the simple electrostatic theory. Data in the = coo-, H,+N)CH+ electro~ acid region do not fit the t h e ~ r y . The static correction term, w,for the binding of anions (C) /-COO- AHaN--J + H + = J-COOH AHBN-1. = p ( C 0 0 H . AHIN) and hydrogen ion differ.5~6 P(coo-. A H ~ N ) C E + This communication extends the binding theory J-COOHzN--J + H + = /-COOHa+N-J. (D) by including competitive hydrogen bonding as well as the simple electrostatic theory. The hydrogen Pmoo-. H:+N) kHl = coo-, H,N)CH+ (1) G. E. Perlmann, J . B i d . Chem., 137, 707 (1941).
+
kH1
(2) G. Scatchard and E. S. Black, THISJOURNAL, 53, 88 (1949). (3) G. Scatchard, Ann. N . Y . Acad. Sei., 61, 660 (1949). (4) G. Scatchard, I. H. Scheinberg and S. H. Armstrong, Jr., J . A m . Ckem. Soc., 71, 5 3 5 , 641 (1950). ( 5 ) G. Scatchard, J. S. Coleman and A. L. Shen, ibid., 79, 12
(1957). (G)
C. Tanford. S. A. Swanson and W. S. Shore, ibid., 77, 6414
(1955).
(E) J-COOH
HzN-J
+ Hf
= I-COOH
Ha+N-/. p ( C O O H , Ha'N)
kHz
= p(CooH, H z N ) C H +
(7) M. Laskowski, Jr., and H. A. Scheraga, ibid., 76, 6305 (1954). (8) M. 8. Lewis and H. A. Saroff, ibid.. 79, 2112 (1957). (9) H.A. Saroff, Arch. Biochem. Biophys., in press.
