Feb., 1959
TRACER DIFFUSION COEFFICIENTS OF STRONTIUM-90
269
TRACER DIFFUSION COEFFICIEKTS OF STROKTIURI: 10s .IN AQUEOUS POLYSTYRENE SUEFOKIC ACID SOLUTIOXS' BY J. P. Dux AND J. STEIGMAN~ Department of Chettaistry, Polytechnic Insfitute of Brooklyn, Brooklyn, N e w York Received August $6,1968
Tracer diffusion coefficients of strontium-00 in dilute solution of polystyrenesulfonic acid were measured a t 25". The coefficients were low and decreased markedly with increasing time of diftusion. This was explained on the basis of irreversible binding of strontium ions to the polyions. Theory showed that the number average diffusion coefficient of the polyions could be calculated from the strontium coefficient,s It WYRSfound experimentally that the tracer diffusion coefficients of sodium and cesium ions were practicnlly unaffected by the polyacid, in coiitrsst t o strontium. Preparation of polystyrene sulfonic acid with radioactive sulfur permitted the simultaneous measurement of the self-diffusion coefficient of the polymer and the tracer diffusion of strontium-90, 111 the h i t the two diffusion coefficients approached the same value, showing that the strontium moved with the polyioii.
I. Introduction Among the outstanding achievements of radiochemical techniques is the measuremelit of selfdiff usioii coefficieiits and tracer diffusion coefficients. A previous communicatioii by the authors3 described a series of experiments in which the selfdiffusion Coefficient of strontium as counterion to polystyreiiesulfoiinte ions in acid solution mas studied. This paper is concerned with the measurement of tracer diffusion coefficients of strontium in solutions of polystyreiiesulfoiiic acid. The tracer diffusion coefficient of an ion is one which is measured for a solution in which the concentratioii of the diffusing ion is negligibly small compared to the conceiitratioiis of the other electrolytes which are present. For this situatioii Onsager4 has shown that both the diff usion potential and the electrophoretic effect may be neglected. The activity coefficient of the diffusing ion is essentially constant over the diffusion path, in spite of the fact that its conceiitratioii varies from zero t.0 a finite (but extremely small) value. Because of the very low concentration of the ion in question, these diffusion coefficients nre most accurately measured by radiocheiiiical methods. Other electrolytes must be present in more coiivent,ional concentrations i n order for the activity coefficient of the tracer to reinnin constant as well as for the purpose of minimizing side effects like adsorption on container walls niid interaction with impurities which are foulid even in the most carefully purified water. In this research the polyelectrolyte polystyreiiesulfonic acid was present in concentrations of approximately molar i n sulfonate ions, while the strontium-ion concentration (us the radioactive isotope strontium-90) was of the order of niolttr or less. Under these coiiditions of concentration it was found that the value of the tracer diffusion coefficient was unexpectedly low and decreased with increasing time of diffusion, coiitrary t o the behavior of such solutions at higher strontium concentrat i o n ~ . The ~ effect can be interpreted as an irreversible binding of stroiitium to the polyelectrolyte, or else an exchange between bound and free strontium which is slower than the diffusion proc(1) From a thesis submitted b y James P. D u x t o t h e Graduate School of T h e Polytechnic Institute of Brooklyn in partial fulfillment of t h e requirements for t h e degree of Doctor of Philosophy. (2) T o whom inquiries should be addressed. (3) J. P. Dux and J. Steigman, THISJOURNAL, 62, 288 (10.58). (1) L. Onsnger, Ann. N . Y.Acad. Sci., 4 6 , 2 7 8 (1945).
ess. In either case, with a polydisperse polymeric system, free strontium ions will diffuse out first, followed by strontium bouiid to lower molecular weight polymers, and finally by s1,roiitium bound to the larger, more slo~vlymoving polynnions. Hence the diffusion coefficient will decrease as the diffusion time is increased. 11. Theory It has been shonrn5 that it is convenient to divide small counterions in the vicinity of large polyioiis into two groups: those which are bound to a polyion, and heiice move with it, mid those which are free, and move iudepeiideiitly of it. There will be some finite rate of exchange between bound and free couiiterions. It also has been demonstrated3 that if the exchange is rapid compared to the diffusion, the measured diffusion coefficient, of the counterion will be independent of the time of diffusion, and will be equal to the msan diffusion coefficient of the counterion. Thus, if D is the observed cliff usioii coefficient, then D = f A D A d- f p b p (1) where j.4 is the fraction of counterions which are free, moving with coefficieiit DA, f,is the fraction of counterions which are hound and D, is the weighted cliff usioii coefficieiit of the bouiid coutiterions. That is Dp = unDn (2) n
where D, is the diffusion coefficient of the nth polyion, and g, is the fractioii of the total of the houiid counterioiis which are bound to the iztli polyion. One coiisequeiice of this rapid exchange is that D is independent of time. However, if the exchange rate betweeti bound and free counterions is slow, the situation is niarkedly different. The limiting case will be emmilied, i l l which all counterions are bound, and there is 110 exchange of these ions between the polyioiis and the solution, or among the polyions. Hence the counterioii moves with the mobility of the polyioii to which it is bound, and, if radioactive, will trace the diffusion of the polyelectrolyte. The assumption is made that the polyelectrolyte solution is dilute so that the polyions diffuse iiidependentlg of each other. We may then write Fick's law separately for each polyioii. If there are n of the latter (i.e., ( 5 ) J. Huieenga, P. Grieger and 1'. Wall, J. A m . Chem. Soc.. 72, 26313 (1950).
J. P.Dux
270
AND
if we have a degree of polymerization n) we may describe linear diffusion in this maimer
J. STEIGMAN
Vol. 63
in the solution. If the espoiieiitial in this equatioii is expanded as a power series, we obtain
(3)
in which C, is the concentration of the nth polyion (tagged), 2 represents distance along the diffusion path, t is time and D, is the diffusion coefficient of the radioactive bound tracer, and hence that of the polyion to which it is attached. Since it is assumed that there is no exchange betweeii polyions13we must solve each equation 3 separately according to the boundary conditions of the experiment in which D, is measured. The technique used to measure diffusion coefficients in this research was the opeii-ended cnpillary method originated by Anderson aiid Saddington.6 The rigorous solution of Fick's law for these particular conditions yields
in which y n is the fraction of the initial concentration of n-polyion remaining in the capillary a t time t, C, is the average coiicentration of n polyion remaining in the capillary a t time t, Coli is the initial concentration in the capillary of the same polyion, and I is the capillary length. If Dnt/n2 3 0.2, all terms in the sum except the first may be neglected, so that (5)
where 9 is a constant of the system, and is equal to n2/412. The total conceiitration of tagged polyion (symbol CT) which remains in the capillary a t time t is the sum of the concentrations of the individual polyions, C,,, so that
cyT=
Cn = n
$n
p,, e-BD,d
(0)
Dividing hy the total initinl coiiceiitratioii COT %On, we obtain
=
in which,f",lis the fixtioil of the iiiitial pulyioii roilcentratioii which is preseiit a s the n-mer; t h a t is foil = Co,/Co~. If ive iiow defiiie n difi'usioii voeficient DT by the equatioii
after substitution in equatioii 7 , \ye 1
u1)hiii
DT = - 111 Z SoTi.C - B D ~ ' ~ et
(9)
This equation shows that DT will be a functioii of time, to the extent that the logarithm of the sum is not equal to the sum of the logarithms of the individual terms. It is to be expected, then, that DT will decrease with time as the smaller, faster-moving polyions diffuse out of the capillary first, followed by the larger particles. The sum represented in equation 7 is the average value of the function e-eDut for the particular molecular-weight distribution of the polyions present ( G ) J. S. dndeison and li. Snddington, J. Chem. Soc., S381 (1949).
or
in which is the mean diffusion coefficient (defined as ZlnfmDn),Dn2 is the mean square diffusion coefficient, etc. Thus, if we were to plot y a,gaiiist et and fit a power series in et to the resulting curve, it should be possible theoretically to calculate not only 5, but also the various moments of the distribution curve for D over n from the successive coefficients. However, there are two restrictions on this procedure, First, in order for equation 5 to hold, it is necessary that D,t/n2 > 0.2. Secondly, the power series in equation 10 must be convergent, and this requires that the argument D,Ot must be less than one for all values of D,. The work which is reported here includes the measurement of the tracer diffusion coefficient of strontium ions for different times of diffusion in solutions of polystyrenesnlfoliic acid derived from different polymers, the measurement of the tracer diffusion coefficients of sodium and cesium in one of these solutioiw, and the simultaneous measurement of the self-diffusioncoefficient of a sulfonated polystyrene aiid the tracer diff usioii coefficient of strontium in the same solution. 111. Experimental Materials .-Unless otherwise noted, reagent grade materials were used. Polystyrene Sulfonic Acids.-Three different samples of polystyi,ene were su1fonat)ed. Product A was prepared from n polymer of weight average molecular weight 1230. Product8s(:-1 and C-2 were made f r o m commercial polystyrene ol' ivcight :Lvrrage molecular weight 32,500. A and C-1 \v(~t'e c i ~ ~ i ~ i binr da prevlous paper.3 Products B-1 and 13-2, like Product A , \wie n i d e from polystyrene prepared 1iy rlinin transfer in c:ni-l)on tet,rachloride. However, it was sthsequently fractionated by precipitation from dioxane\v:tt,er solution. 1t.p number aver:rge molecular weight was 053. It, rtyrese~it~ed a much more sharply defined poly111er than t,hc polystyrene of Product A . P d u c t s A anti C-1 were nlnde by sulfonntsioii nit'li ]lot, concciitrnted sulfuric acid.3 Pyoducts (2-2, B-1 alld €3-2 ivere prepared by sulfonat.ion with anhydrouR sulfur trioxide nccot,ding to the method of Baer.7 Redistilled dichloipdicthyl ct8liri 0.2 for all species of ions, we may write from equation 7
and
where
tilid hence ,I-,, = ( I
- j.i.rn
By substitution in equation 13, we get
If we now have two observations of time tl and L2, we may write
y
and
v‘ at
aiid yz
-
y2’ =
8
- f,e-ODstz 7r2
.- j
yz’
(17)
These two expressions m&y be solved simultaneously for fs and D,. When this analysis is applied to the data of Table 111, the following values are obtained: (a) for product B-1, the fraction of sulfur-35 present as Sod= is 0.298, and the diffusion coefficient of Sodis 0.98 X lo-” cm.2/sec.; (b) for Product B-2, the fractioii of sulfur-35 present as Sod- is 0.111, and the diffusion coefficient of Sod=is 0.95 X cm.z/ sec. The limiting diffusioii coefficient from the Neriist equation for sulfate ion a t 25” is 1.08 X cm.2/ sec. The agreement is much better than expected, i n view of the experiniental errors involved. It will be noted that although treatment of product B-2 with o small quantity of ion-exchange resin removed a very large fraction of the sulfate ion, the latter was by no mems reduced to a negligible value. Possibly percolation of the radioactive solution through a column of resin would have been a more efficient procedure for removal of sulfuric acid than the shake-out batch technique which mas nctually used. The batch technique mas adopted because of the small quantities of material involved in the sulfonation, and the possibility of greater loss of radioactive polymer by adsorption in a column. It is concluded that strontiuni ion, present in very lorn levels of concentration, is adsorbed on the polystyrene sulfonate ions and diffuses with the coefficient of the polyion to which it is attached. This suggests the possibility of measurement of the diffusion coefficients of strongly dissociated polyelectrolytes by adding traces of suitable radioactive counterions. Acknowledgment.-This research was supported by the U. S. Army Signal Corps under contraet number Dh36-030, Sc55DG. We are grateful to Dr. L. hrond, who performed the sulfonations with sulfur trioxide reported in this paper. We thank Professor F. C. Collins for many helpful discussions.
