Shelini Menezes-Affonso and Paul Ander
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Self-Diffusion Coefficients for Chloride Ion in Aqueous Solutions of Sodium Polyacrylate odium Chloride helini Menezes-Affonso and Paul Anderr Department of Chemistry, Seton Hall Un/versity, South Orange, New Jersey 07079 (Received January 2 1, 1974) Publication costs assisted by Seton Hall University
The concentration dependence of the self-diffusion coefficients of the coion in polyelectrolyte solutions containing simple salts was studied at two concentrations of sodium polyacrylate at several concentrations of sodium chloride in water at 25'. At constant polyelectrolyte concentration, the self-diffusion coefficients of tine coion increase with the added salt concentration, whereas at a fixed salt concentration the cqion selfdiffusion coefficients decrease with increasing polymer concentration. Good correlation has been obtained between the predictions for the self-diffusion coefficient of the coion obtained from the Manning line charge polyion model and the data presented in this study.
It is well recognized that the interactions of small ions with polyelectrolytes in aqueous solution govern many of the solution properties of polyelectrolytes. Attention has been focused on the long-range electrostatic and shortrange specific interactions of the counterions with the polyion.' -5 However, no systematic experimental investigations have been reported concerning the interactions of the coion with the polyiton. This interaction has been dismissed by many by their insistence that the coion, being of the same charge as the polyion, would hardly be affected by the polyion. Yet, the literature contains some results6-14 that indicate that the coion has a noticeable effect on the solution properties of polyelectrolytes. A model for polyelectrolyte solutions developed by Man,ing15-17 presents sXmple limiting laws for both thermodynarnic and transport properties for both the counterion and the coion. According to this theory, a fraction of the counterions coildense onto the polyion above a critical charge density for {,he polyion. The coions interact with the polyion by Ciebye-Hfickel interactions, as do the uncondensed counterions. While the Manning theory has not been tested extensively, a few good correlations between the theory and experimental results have been d i ~ c u s s e d Recently, .~ Dixler and Anderls reported excellent agreement between the theoretical and cxperimental values of the self-diffusion coefficient of sodium ions, in aqueous solutions of sodium polyacrylate in sodium chloride. Here, we discuss the results of the self-diffusion coefficients of chloride ions in aqueous solutions of sodium polyacrylate containing sodium chloride. A capillary methodl9J0 was used along with an equation developed by Ril[cMay21 LB=-
c
212
"(1 - - c , >-t 4
where D is the self-diffusion coefficient, C/Co is the ratio of final to the initial clhloride activities in the capillary, 1 is the length of the capillary, and t is the time of diffusion. The experimental conditions employed in this work were such that 0.5 < G!Co < 0.7, for which range McKay's equation has been shown to be valid. The Journal of Physical Chemistry, Voi. 78, No. 17. 1974
Experimental Section
Material. The sodium polyacrylate used in this investigation was the same used in a previous study.18 The radioactive "C1 (half-life 3 X lo5 years) was obtained from New England Nuclear as a 2.68 N H 36Claqueous solution, which was neutralized with aqueous NaQH. Diffusion Measurements. All diffusion measurements were carried out in a bath thermostated at 25.00 f 0.01'. The diffusion assembly was the same as used previously.18 Precision bore capillaries of 1.60 rnm i.d. and 3.00 em length were used. After time was allowed for diffusion, usually 1 day, each capillary solution was quantitatively transferred to a planchet. All planchets were dried under an infrared lamp while rotating slowly on a sample spinner. After being emptied of the diffused contents, the capillary was rinsed with the original tracer solution and refilled. This was again quantitatively transferred to another planchet and dried. The two planchets were subsequently counted in a Geiger counter to give C and Co. Sample Counting and Validation of Technique. Since 36Cl emits high energy p particles, a Geiger-Muller Counter was selected as a detector. The counting assembly consisted of a Nuclear Chicago 6 - M counler, timer, and a scaler Model 181B. The counting parameter used to calculate the diffusion coefficient according to eq 1 is the square of the ratio C/Co and consequently the error is magnified substantially. In order to minimize the counting errors, each sample was counted at least six times, each count for a duration of 15 min. The diffusion coefficients reported were calculated from the total count obtained from the six measurements for each sample. Each 0 2 value reported i s the average value and the standard deviation obtained for three to six samples. Prior to the self-diffusion measurements in NaPA solutions, the validity of the transfer of capillary contents, the counting techniques, and the diffusion procedure were tested using aqueous NaCl solutions. The diffusion technique described above and eq 1were used to determine the diffusion coefficients of C1- in 0.050 and 0.030 N NaC1 soluand tions. The results obtained were 1.93 f 0.01 X 1.97 f 0.01 X 10-5 cm2/sec for the 0.050 and 0.030 N NaCl,
Solution Propertios of Paslyelectrolytes
1757
respectively. Since these are in good agreement with the corresponding literature values of 1.94 X and 1.96 X cm2/sec,22 the technique was concluded to be adequate for use in this study.
Results and diwussions Self-diffumon coefficients for C1- Dp were determined in dilute aqueous Rolutions of NaPA containing NaCl at 2 5 O . Completely rieutraiized NaPA at concentrations of 0.0100 and 0.0300 Ar were used. The NaCl concentrations ranged from 0.0011 to 0.0300 iY. Thus the value of X, defined by
0 95
-
D~/D;’
0.90
where ne and n, are equivalent concentrations of the polyelectrolyte and sodium chloride, respectively, ranged from 1to 9 for both polyelectrolyte concentrations. From the experimental results depicted in Table I it is evident that the D Qvalues decrease with increasing values of X, leveling off to a (constant value a t high X values. Also the Dz values decrease with a decrease in concentration of the added salt, ab constant polymer concentration. (It should be noled that this tendency is in contrast with the behavior in simple sa It s o l ~ t i o n s ~where ~ , ~ * the opposite trend is observed.) Inspection of the data in Table I further reveals that at a fixed concentration of simple salt, the diffusion coefficients decrease with increasing polymer concentration. Most important is that 0 2 is found to be a function of X only under the conditions studied. A test of the Manning theory is to compare the experimental self-diffusion coefficients of the counterion and the coion with those predicted. Very good agreement has been observed in the case of the counterion.ls The experimental data obtained for tlne self-diffusion coefficients of the C1ion were correlated wii h Manning’s theoretical predictions. Based on the assumption that the condensed counterions have negligible mobility while all the uncondensed small ions are subject to Clebye -Huckel interactions with the polyions, Manning dcmved expressions to calculate the self-diffusion coefficients of the mobile ions. The diffusion coefficient on an rth ion is given by
-
-
0.85 1
2
3
4
5
6
7
8
9
X @/D20 vs. X. Comparison of theoretical curve (solid line) with experimental 4 values for ne = O.01QQ N, 0, and 0.0300
Figure 1. Plot of N, 0 , with
= 2.033 X
cm*/sec.
TABLE I: Self-Diffusion Coefficients for 61- in NaPA-NaCI-H20 Solutions at 25 O D S X 106, cm2/sec X
1 2 3 4 5 6 7 8 9
ne = 0,0100N
1.92 1.85 1.82 1.81 1.80 1.78 1.79 1.78 1.78
TABLE 11: Values for X
&
ne = 0.0300
0.02
1 . 9 1 f 0.03 1.88 c 0.01 1.84 I:0.03 1.81 -f. 0 . 0 1 1.713 & 0 . 0 3 1.80 & 0.02 1.81 I O . 0 1 1.79 f 0.01 1 . 7 7 & 0.02
=t0.03 jl 0.01 i 0.00 I :0.01 I :0.03 & 0.01 f 0.00 f 0.01
=
N
1 to 9
a: -1x
A
E -1x
A
0.351 0.702 1.053 1.403 1.754
0.125 0.194 0.235 0.263 0.283
2.105 2.456 2 307 3.150
0.298 0 310 0,319 0.326
~
(3 where i = 1 and 2 represents the uncondensed counterions and the coions, i>espectively;Dl is the self-diffusion coefficients of small ions in the polyelectrolyte solution and D,O i s its value a t infinite dilution in the absence of polyelectrolyte; and the quantity il is given by the series
A = m*=-w
m2 =
s’
where nl,~n2# 0 , 0; > 1; ml and m2 are integers and .$ is a linear charge density parameter given by
and 2, is the valence of a charged group on the polyion, e is the unit of electrical charge, h is the Boltzmann constant, T is the absolute temperature, and b the distance between for a fully charged vinylic charges on the poiyian, 2.5 polyelectrolyte making ( --- 2.85 for the present study. Since the series A wa!; rapidly convergent, the summation for values of ( m l ,m2) was (evaluated only from -10 to +lo, ex-
cluding the point (ml, mn) = (0,O). The values obtained for A for X values from 1to 9 are given in Table 11. The limiting laws are specifically formulated to be strictly valid a t infinite dilution. However, tlne properties of polyelectrolyte solutions a t the lowest measurable concentrations are taken to be very close to those a t infinite dilution. The range of X values was chosen because according to Manning theory the “polyelectrolyte effect” on the diffusion coefficients of the coion is especially evident a t high values of X. Using the limiting value for DzO = 2.033 X cm2/secQ2 and the values obtained as described above for X and A, theoretical values were computed from eq 3 for D2. It can be seen from Figure 1that the D2/Dz0 values obtained from self-diffusion experiments are in close agreement with the values calculated from eq 3, within range of concentrations investigated. This lends confirmation to LManning’s theory for self-diffusion, since both the self-diffusion coefficients The Journai of Physical Chemistry. Vol. 78, No. 7 7. 1974
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of the counterion and the coion are found to be in good agreement with the predicted values. The genera' trend predicted is for the D2 of the coion to rapidly decrease with increasing X values and then to level off to a constant value a t higher values of X over a wide range of concentrations. To investigate the validity of this conclusion two sets of data were obtained keeping X constant and varying the values of ne and n,. From Figure 1 it can be seen that the Dz values for the two sets of points are in good agreement and also display the expected trend. It should be nlotod frorn Figure 1 that the experimental points fall about 3% below the theoretical curve. This small discrepancy may be thought to be due to the neglect of small ion-small ion interactions in the theory, which was discussed by Wells.25 However, Dixler and Anderls found no need to consider small ion-small ion interactions since this small correction would be hard to detect in the larger scatter of the points. Theoretically the limiting value of D2 a t n, = 0, Le., a polyelectrolyte with no simple salt present, could be obtained by evaluating; A at X = m , which is 0.40, and inserting this value into eq 3. This gives D2/Dzo = 0.87 and hence Bz = 1.78 >< lom5cm2/sec for the limiting value. It is interesting thal for both polyelectrolyte concentrations employed in this study a constant value of 1.79 f 0.03 X cm2/sec was obtained for D from X = 4 to X = 9. At X = 4, Da/D$ has a theoretical value of 0.91, which is approximately 5% grciater than 0.87, the limiting value a t X = m which is predicted from the theory. One additicmal prediction to be made from Manning relationship is t i a t
that is, for a maximally charged polyelectrolyte the relative decrease in the self-diffusion coefficient of the counterion is larger than that for the coion. For this case the diffusion coefficieinte 01' the condensed counterions are assumed to
The Journai oil Physicai Chemistry, Vol. 78, No. 7 7, 1974
Shelini Menezes-Atfonso and Paul Ander
be zero and those of the uncondensed counterions are given by the equation
The experimental data for DNa+18 and Del- confirm the above result. Furthermore, these self-diiffusion results justify an assumption of the Manning model that the mobility of the condensed counterions is negligible.
Acknowledgment. We are grateful to Dr. G. S. Manning for his stimulating discussions.
References and Notes (1) S. A. Rice and M. Nagasawa, "Polyelectrolyte Solutions," Academic Press, New York, N. Y., 1961,,Chapter 9. (2) F. Oosawa, Polyelectrolytes, Marcel Dekker, New York, N. Y., 1971, p 160. (3) R. W. Armstrong and U. P. Strauss, "Polyeiectrolytes," in "Encyclopedia of Polymer Science and Technology," Vol. X, Wiley, New York. N. Y., 1969. (4) G. S. Manning, Annu. Rev. Phys. Chem., 23, 117 (1972). (5) A. Katchalsky, Pure Appl. Chem.,26, 327 (1971). (6) P. H. von Hippel and T. Schleigh, Accounts Chem. Res., 2, 257 (1969). (7) H. Eisenberg and G. R. Mohan, J. Phys. Chem., 63, 671 (1959). (8) H. Eisenbergand D. Woodside, J. Chem. Phys., 36, 1844 (1962). (9) A. Takahashi, S. Yamori, and T. Kagawa, Nippon Kagaku Zasshi, 83, 11, 14 (1962). ( I O ) J. T. G. Overbeek, A. Vrij, and H. F. Huisman, h "Electromagnetic Scattering," M. Kerker, Ed., Macmillan, New York, N. Y., 1962, p 321. (1 1) R. L. Gustafson, J. Phys. Chem., 67, 2549 (1963). (12) M. Nagasawa, I. Kagawa, and M. I. Izumi, J. Polym. So.,37,375 (1959). (13) T. J. Podlas and P. Ander, Macromolecules,2, 432 (1969). (14) T. J. Podlas and P. Ander, Macromolecules,3, 154 (1970). (15) G. S. Manning, J. Chem. Phys., 51,924 (1969). (16) G. S. Manning, J. Chem. Phys.,51,934(1969). (17) G. S. Manning, J. Chem. Phys., 51, 3249 (1969). (18) D.S. Dixler and P. Ander, J. Phys. Chem.,77, 2684 (1973). (19) J. A. Anderson and D. J. Saddlngton, J. Chem. SOC., S381 (1949). (20) R . Fernandez-Prini, E. Baumgarter, S. Liberrnan, and A. E. Lagos, J. Phys. Chem., 73, 1420 (1969). (21) R . McKay, Proc. Phys. SOC.,42, 547 (1930). (22) J. H. Wang and S. Miller, J. Amer. Chem. SOC.,74, 1611 (1952). (23) R. Mills, J. Phys. Chem., 61, 1631 (1957). (24) R. S. Robinson and R . H. Stokes, "Electrolyte Solutions," Butterworths, London, 1959. (25) J. D. Wells, Biopolymers, 12, 223 (1973).
