Chapter 13
Sodium-Ion Interactions with Polyions in Aqueous Salt-Free Solutions by Diffusion Paul Ander Chemistry Department, Seton Hall University, South Orange, NJ 07079
An unambiguous way to study the interactions of counterions with polyelectrolytes in solution is by determining the counterion and coion radioactive tracer diffusion coefficients in the presence of a polyelectrolyte with a common counterion. This has been done principally by FernandezPrini, et al. , Magdelenat, et al. and Ander, et al. . A capillary diffusion technique was employed. Aqueous solutions have been used in almost all studies. This brief review will focus on the experimental findings of the author. Equations have been developed by Manning for counterion and coion diffusion coefficients in polyelectrolyte solutions in the absence and presence of simple salts for a line charge model for the polyelectrolyte. 1,2
3,4
5-13
14-20
DIFFUSION MEASUREMENTS To determine the self-diffusion coefficients of ions, the open-end capillary method originally introduced by Anderson and Saddington was employed without stirring. Due to the high viscosity of polyelectrolyte solutions, a non-stirring technique was found to be more reliable than one employing stirring because of the considerable change of viscosity of the solution as concentration varies. Fawcett and Caton analyzed the source of error in the capillary method of measuring diffusion coefficients. They concluded that the serious source of systematic error results from convective loss of diffusant from the mouth of the capillary, and that accurate results require the use of fine bore capillaries from which convective loss is minimal. In our laboratory all diffusion measurements were carried out in a constant temperature bath at 25.00 + 0.01°C. Precision bore capillaries of 1.00 + 0.005 mm diameter and 2.82 + 0.005 cm length were filled with the desired polyelectrolyte solution containing tracer amount of radioactively labeled ion under investigation. The outside of the filled capillaries were then wiped dry and each capillary was placed into an individual test tube (100 mm χ 13 mm), which was subsequently filled with a radioactively inert solution of identical composition to the solution inside the capillary. Since the solutions inside and outside the capillary were the same polyelectrolyte concentration and the same simple salt concentration, no concentration 21
22
0097-6156/91/0467-0202$06.00/0 © 1991 American Chemical Society
13. ANDER
203
Sodium-ion Interactions with Polyions
gradient existed. The only gradient which existed was that for radioactively-labeled cation, which caused it to diffuse out of the capillary. After sufficient time was allowed for diffusion, usually twenty-four hours, the outer solution was carefully withdrawn by means of an aspirator. Upon removal of the capillary from the test tube, the outside of capillary was dried and place with its open-end down into a vial containing 5 ml of scintillation fluid (Ready-Solve HP, Beckman). Each vial was then centrifuged for five minutes, followed by shaking and mixing for thirty seconds on a Vortex Geni (Scientific Industrial Inc.) to insure through mixing of the capillary contents with the scintillation liquid. The radioactive content of each capillary, C , was determined using a Model LS 7500 Beckman liquid scintillation system. Each sample was counted to a minimum of 20,000 counts, which corresponds to about 0.7% counting error. The initial radioactive content of capillary, C , was determined by the same procedure, using a capillary of identical bore and length with no diffusion time allowed. A minimum of six capillaries was used to determine each value of C and C . Diffusion coefficients, D, were calculated using the expression23,24 which is obtained from integrating of Fick's equation Q
0
C/C
2
0
2
2
= Σ 8 Ti- (2n+l)-2 p ( - π 2 ( 2 η + 1 ) 0 ΐ / ^ ) η
ο
(1)
eX
where L is the length of the capillary in centimeters and t is the time allowed for diffusion in seconds. The McKay's approximation solution of Fick's equation25 D=(^^)(l-C/C )2(L2/t)
(2)
0
could be used. From the approximation used in deriving Equation 2, when the ratio of C / C is equal to 0.45, an error of 0.5% results in the value of D. The higher the ratio, the smaller is the error. However, the higher the ratio, the larger is the error due to experimental manipulation. To avoid these problems, Equation 1 was used to determine the experimental values of D. This expression was inverted and solved for D with the aid of computer. Q
RESULTS The most important characteristic of a polyelectrolyte is the reduced charge density or the charge density parameter ξ, 2
ζ = e / ekTb
(3)
where e is the fundamental charge, ε is the dielectric constant of the medium, k is the Boltzmann constant, Τ is the absolute temperature and b is the average axial distance between charges. The dominant polyelectrolyte theory is that of Manning, 1^15,16 which developed equations that can be easily correlated with several experimental thermodynamic and transport properties. The polyelectrolyte is modeled as an infinite line charge whose counterions can be totally dissociated from the polyelectrolyte or partially dissociated from (and partially condensed onto) the polyion. The theory has the polyelectrolyte dissociating its counterions completely if ξ < ξ where ξ = IZ with Z is the counterion charge, and 0
c
c
c
204
WATER-SOLUBLE POLYMERS
D i / D ^ f [ l - p ( ? c , ξ-1χ)]
(4)
A
where Dj/D^ are the tracer diffusion coefficients of the i th ion in the condensation term
f =U c
ξ " * + D/(X + 1
( 5 )
D
where X = N / N , the equivalent concentration of polyelectrolyte to that of simple salt. The value of f is unity for coions and for counterions if ξ < ξ . The value of Α(ξ > ξ Χ ) or simply A is 0.87 for salt-free solutions of polyelectrolytes with monovalent counterions, i.e., in the limit of X = or practically at low simple salt concentration. This is an electrostatic interaction term between small ions and the polyion. For polyelectrolytes with ξ > ξ some counterion condensation occurs to reduce ξ to its effective value and f ^ 1. From Equation 5, f is the fraction of total uncondensed counterions. Salt-free diffusion results were usually obtained over the concentration range of polyelectrolyte from 10*^ < Np < 10-*. If one examines the salt-free counterion diffusion coefficients in aqueous polyelectrolyte solutions in Figures 1 and 2, it is noted generally that a minimum in the curve of Dj/Di vs. Np occurs when the tracer ion and the counterion are the same. Sometimes the minimum is gentle, sometimes more pronounced and a few times Dj/D°i is found to be fairly constant as Np is varied between 10-^ and 10-lN. The minimum Dj/dJ value is, of course, a measure of maximum counterîôn-polyion interaction and usually occurs between 10-2 and 10-3 N . One can rationalize the minimum by understanding that as the polyion in a dilute solution is further diluted, the concomittant elongation of the polyion and its Debye Huckel atmosphere expansion causes some counterions in the atmosphere to become more free. On the other side of the minimum, increasing concentration causes greater overlap of the polyions and their ionic atmospheres and counterions become more free. A perusal of the salt-free polyelectrolyte solutions literature will show the same concentration trend when the counterion activity coefficients are determined. The Disja/D^a trend for vinylic type polyelectrolytes is illustrated in Figure 1 for fully or almost fully charged polyacrylate (PA) and polystyrenesulfonate (PSS) polyelectrolytes. Table I p
s
α
_ 1
0
00
0
Table I. Comparison of the Minimum Df^ /DNa Experimental Values with Those Predicted Theoretically for Several Polyelectrolytes in Salt-Free Solutions a
polyelectrolyte
ionic group on polyelectrolyte
i-carrageenan
OS05
1.66
0.60
0.65
0.52
alginate
COO"
1.43
0.70
0.65
0.61 0.35
ξ
ξ-ι
exptl eq 4
polyvinylsulfonate
S05
2.5
0.40
0.63
polystyrenesulfonate
S05
2.6
0.38
0.62
0.34
polystyrenecarboxylate
COO'
2.6
0.38
0.45
0.32
polyacrylate
COO"
2.7
0.37
0.38
0.32
polyacrylate
COO'
2.7
0.37
0.40
0.32
OSO3
2.85
0.35
0.27
0.30
0.33
0.30
0.29
dextran sulfate heparin
COO-,OSO5,NSO53.0
13. ANDER
205
Sodium-Ion Interactions with Polyions LOI
0.8
-A
Δ
0.6
0.4 ANaPVS ONaPSS • NaPSC QNaPA
0.2
0.0 1.0
2.0 -LOG N
4.0
3.0 p
Figure 1. E > / D dependence on the equivalent concentration and nature of polyelectrolyte in aqueous salt-free solutions. N a
N a
1.00
2.00 3.00 -LOG N
4.0 U
p
Figure 2.
N a ion diffusion coefficient dependence on the equivalent concentration for (•K-carrageenan, (O)alginate, 0)heparin and (e)dextran sulfate. +
206
WATER-SOLUBLE POLYMERS
lists the minimum values obtained for these polyelectrolytes, where it is noted that the vinylic carboxylate polymers have values close to their theoretical values of 0.87 ξ - S while the sulfonates appear to dissociate more counterions that is predicted theoretically. To test the Manning theory it is best to use polyelectrolytes close to the theoretical model. Thus small ion tracer diffusion studies were performed with long, stiff ionic polysaccharides. Figure 2 shows D N / D N vs log Np for sodium dextran sulfate ( ξ = 2.85), sodium iota-carrageenan (ξ = 1.66), sodium alginate ( ξ = 1.43) and sodium heparin ( ξ = 3.0), respectively. 10,11 The first two polysaccharides have pendant sulfate groups, the third pendant carboxyl groups and the fourth pendant carboxyl, sulfate and N-sulfonate groups. Each curve has a pronounced minimum at approximately 10-2 to 10-^N. Table I indicates that the experimental minimum values for ionic polysaccharides are close in accord with the theoretical value of 0.87 ξ - A l s o , the counterion diffusion ratios seem to be independent of the nature of the ionic group, as is predicted from the theory. However, the theory appears to be incorrect in predicting that the counterion diffusion ratio is independent of concentration for salt-free solutions. An interesting study was the evaluation of the Djsj /DN * aqueous salt-free solutions of polyelectrolytes with similar structures whose charge density could be varied. To this end measurements sodium ion diffusion measurements were performed using the sodium salts of aerylate/acrylie acid(NaPA/HPA), acrylate/acrylamide(NaPA/PAM) and acrylate/N,N-dimethacrylamide(NaPA/PDAM) over the ranges 5 χ 10"^ < Np < Ι Ο " and 0.20 < ξ ξ > 0.95 with a slope of 0.79 + 0.04 and an intercept of 0.11 + 0.03. The diffusion ratio falls precipitously Tabout 22%) at ξ = 1 to a constant diffusion ratio of 0.631 + 0.017 for ξ < 1. The few Na+ ion diffusion coefficients in NaPA/HPA solutions obtained by Wall et a l . , shown in Figure 3, agree with those presented here for ξ > 1. For NaPA/PAM and N a P A / P D A M , Figure 3 shows that DNa/DNa linear in ξ" for 2.7 < ξ 1, then almost exact accord with the theoretical slope of Equation 6 is achieved, i.e., 0.88, 0.87, and 0.87 for NaPA/HPA, N a P A / P A M , and N a P A / P D A M , respectively. The similar results obtained for the three polyelectrolytes indicate that the polyelectrolytes have their carboxyl groups distributed along the chain in the same manner, most probably random. Other facts support this. This discontinuity at ξ = ξ» for NaPA/HPA and the change in slope at ξ = Ê£ for N a P A / P A M and NaPA/PDAM are evident from Figure 3. With counterion condensation occurring at ξ^, the reduction of charge along the chain caused it to coil, thereby facilitating intramolecular hydrogen bond formation at lower charge densities, i.e., ξ < 1, for NaPA/HPA. Cooperative hydrogen bonding by two adjacent carboxylate groups in D
208
WATER-SOLUBLE POLYMERS
polycarboxylates has been discussed by Begala and Strauss.32 This would increase the charge density of the coil and result in lower DNa l u e s , as is observed. Such an amount of coiling is not as pronounced for the other two polyelectrolytes at = * because intramolecular hydrogen bond formation is not probable and hence, only a change in slope occurs at KQ. The constant DN /D[s| values for ξ < ξ which were obtained for all three polyelectrolytes indicate that the effective charge density of the coils is fairly constant in this charge density region and that rod-like behavior does not occur since Equation 7 is not obeyed. It is interesting that the electrophoretic mobilities of 6-6 ionene bromide in 4.0 m M KBr remained fairly constant for ξ < 1. followed by a precipitous drop at ξ = 1 to a constant value for ξ_> 1.31 Ware explains these results by noting that near ξ = 1, the spacing of the condensed ions predicted by condensation theory is greater than the Debye screening length, resulting in further counterion condensation. Perhaps KQ is the effective value for ξ > ξ as well as for ξ >Çc to give stability to the solution. For ξ > ξ this stability is achieved by counterion condensation reducing the charge on the chain. For ξ < ξ this stability is achieved by a slight conformation change to increase the charge density of the chain. (It would be interesting to examine this idea by noting the change in the radius of gyration of the chain as ξ decreases below ξ . ) For each of these polyelectrolytes below ξ ο the N a ion interacts with a polyion of constant effective charge density no matter what the value of the stoichiometric charge density. The only way this can happen is that the rodlike polyelectrolyte folds when ξ < ξ to a constant effective ξ value. Dr. Marie Kowblansky, as a graduate student in my laboratory, first suggested that an explanation of these observations can be that unstable localized "loops" formed when ξ < ^ and that the critical charge density parameter ξ is a stability point for the conformation of the polyelectrolyte. It is tempting to speculate that this constant effective ξ is ξ , i.e., at ξ£ the free energy of the solution is minimized. Recently, Manning used his condensation theory to account for these global polyelectrolyte transitions quantitatively by correlating them with the linear critical charge density for counterion condensation.33 He showed that a mechanical instability of the locally folded structures can arise at the condensation critical point. It was then of interest to see if the behavior obtained for sodium polyacrylates of varying charge density would be similar for sodium carboxymethylcellulose (NaCMC). Aqueous salt-free solution of NaCMC with ξ values of 0.53, 0.67, 0.89, 1.00, 1.03, 1.32 and 2.04 were prepared with a concentration range of 1 χ 10"^ to 5 χ 10~ N, the same concentration range used for the sodium polyacrylates. The Na+ ion diffusion results for NaCMC are shown in Figure 4, which have the characteristics of the previous^ figures for D N / D f t a 8 p> i«e., shallow minima with ( D N a / D y ) i decreasing as ξ increases. Also, ( D N a / ^ N a W n (D[s|a/ Na)x=10 close in value at each charge density. Table III lists these values, along with the values predicted by Equation 6. A plot for NaCMC according to Equation 6 is in Figure 3 is linear, where the experimental slope 0.32 + 0.04 and the intercept 0.41 + 0.03. These values are quite different from those predicted by Equation 6. This might be due to both the hydrophilic nature of the NaCMC surface and the rigid backbone of this polymer as compared to the hydrophobic surface and flexible backbone of the vinylic polymers. In the region of ξ > 1.32, for which the NaCMC polymers have more than one carbonyl group per glucose ring (DS > 1), the charge density is more likely high enough to make the NaCMC and va
a
a
0
α
0
€
+
0
0
α
0
2
a
a
D
r n
v s
l o
N
a n c i
n
a r e
13. ANDER
Figure 3.
209
Sodium-Ion Interactions with Polyions
D / D N a dependence on ξ" for N a P A / H P A , N a P A / P A M , N a P A / P D A M , and NaPA/HPA26. T solid line is the Manning theory prediction. 1
N a
n e
Figure 4.
D /rj dependence on the normality of NaCMC of varying charge density in aqueous salt-free solutions. N a
N a
210
WATER-SOLUBLE POLYMERS
vinylic polymers seem equally rigid. The hydrophilic nature of the NaCMC polymers, due to the hydroxyl groups, becomes more significant in this region. The attraction of water molecules to the NaCMC backbone screens the charges, thereby weakening the long-range polyion-counterion interactions and leads to the observed higher diffusion coefficients for the NaCMC polymers as compared to the vinylic polymers in Figure 3. For intermediate ζ values, 1.32 > ζ > 1, the weaker electrostatic repulsive forces between the charged groups on the chains permit the polyions to coil to a greater extent and the diffusion coefficient rises. Consequently the sodium ions that interact with the more flexible vinylic polymers, which can coil more easily than the polysaccharides, have greater diffusion coefficients in this region. This results in the smaller slope in the region of ξ > 1 for the NaCMC polymers as compared to the vinylic polymers. The rodlike model of Manning predicts that condensation does not occur for ξ < 1 for sodium polyelectrolytes. Since NaCMC was used in this study because ionic polysaccharides are relatively stiffer than vinylic polyelectrolytes, for ξ < ^ one would expect the experimental diffusion curve to approach unity as ξ approaches zero. It is obvious from Figure 3 that for NaCMC for ζ < 1, a constant value of D N / D N = 0.72 has been reached. When this line of zero slope for ξ < 1 is extrapolated to the experimental NaCMC line for ξ > 1, they intersect close to ξ = 1, the ξ value. This again, as for NaPA/PAM and NaPA/PDAM, indicates that the critical charge density parameter is correctly predicted from theory. (The discontinuity observed for NaPA/HPA at ξ = ζ also shows this!) Figure 3 also shows that for ξ < the Djsj /DN points for each polymer are all below those predicted by the rodlike theories, indicating that these models are inappropriate in this range. Similar results were obtained for sodium ion activity coefficients for aqueous solutions of sodium pectinate, ** N a C M C , ^ and N a P A / H P A . The zero slope line for NaCMC for ξ < ξ lies close to the zero slope lines for NaPA/PAM and NaPA/PDAM. Also, for ξ < ^ the D / D N s for NaPA/HPA are constant. For each of these polyelectrolytes below the N a ion interacts with a polyion of constant effective charge density no matter what the value of the stoichiometric charge density. From Tables I, II and III, it is noted that the quantities (Djsj / Na)min a
a
0
0
a
3
3
a
3 6
0
v a l u e
N a
a
+
D
a
Table III.
ξ
Sodium Ion Diffusion Ratios Obtained from Minimum Values in Saltfree NaCMC Solutions and from Constant Values in Solutions of High Polyelectrolyte to NaCI Normalities
ξ"
ο ο ο ( N a / D N a W r ii(D a/D a)x=10 ( N a / D ) * X = oo(l-r)
1
D
D
N
0.53 0.67 0.89 1.00 1.03 1.32 2.04
1.89 1.49 1.12 1.00 0.97 0.76 0.49
0.73 0.72 0.72 0.72 0.70 0.65 0.56
N
N a
0.74
0.72
0.72
0.72 0.69
0.71 0.68
0.71 0.68
0.64 0.57
0.63 0.53
0.63 0.53
Obtained from slope of Equation 10 and of ξ "
1
are close in value, especially if
ξ > ξ . 0
This suggests that
211
Sodium-Ion Interactions with Polyions
13. ANDER
(DNa/DNa) S * S fraction of the polyelectrolyte and that the salt-free polyelectrolyte dissociates the same fraction of counterions whether salt is present or not. Counterion additivity rules are based on this idea. For N a ion diffusion the additivity rule could be written as v e s
t n e
c n a r
e
+
(Np+NsXDNa D N a ) = N p ( D a / D N a ) 7
x
N
x = TO
+ N (DN /DNa)x s
(8)
a
If Equation 8 is divided by N on both sides, it can be written in term of concentration parameter X and can be rearanged as s
( D a / D N a ) = (χ-l N
x
1 )~ * ( D N a / D N )
+
a
x= œ
+
2. It seems from Figure 5 that above ξ-* = 1, an additivity rule for diffusion coefficient does not exist. It is worthwhile to test whether this additivity rule is valid for NaPA, NaPSC, NaPVS, and NaPSS of constant reduced charge density of 2.6. If Equation 8 is divided by (X+l)"*l on both sides, it can be written as D
a
(X + l ) ( D a / D N a ) N
x
= X(D a/DN ) N
a
x = œ
a
i n
+ 1
(10)
Since the diffusion coefficient ratio in salt-free solutions, (DNa/Dfsjab(= , is constant for a given ξ value, Equation 9 plots linearly for (X+l)(DNa/DNab( · X with intercept of unity and slope of (DNa/ Na)x= «>, if the additivity law is valid for N a ion diffusion in polyelectrolyte solutions. An illustrative plot of Equation 10 is shown in Figure 6. The ordinate intercept of resulting lines, listed in Table IV, are close to unity within the experimental error, giving some validity to the additivity rule. It was of interest to compare the slopes of these lines, (DNa/DNab(= , with the minimum values of D N a / D N obtained in salt-free polyelectrolyte solutions from Figure 6. Excellent agreement between these two sets of results, listed in Table IV gives some validity to additivity rule. These 00
ν δ
D
+
00
a
Table IV. Slopes and Intercepts Obtained From Plots of Equations 10 and 11 EQUATION 10
NaPA NaPSC NaPVS NaPSS
Intercept 1.15 + 0.03 1.14 + 0.04 1.06 + 0.02 1.04 + 0.02
Slope 0.38 + 0.02 0.43 + 0.02 0.62 + 0.03 0.62 + 0.03
EQUATION 11 Intercept -0.15 + 0.09 -0.14 + 0.09 -0.05 + 0.04 -0.03 + 0.03
Slope 0.62 + 0.03 0.58 + 0.03 0.38 + 0.03 0.39 + 0.02
q-r) 0.38 0.42 0.62 0.62
+ + + +
0.02 0.02 0.05 0.03
212
WATER-SOLUBLE P O L Y M E R S
Figure 6.
The average (Djsj /DN )(X + 1) values vs. X for vinylic polyelectrolytes of approximately the same ξ values in aqueous NaCI solutions. a
a
13. ANDER
213
Sodium-Ion Interactions with Polyions
experimental results, therefore, suggest that the properties based on longrange coulombic interactions can be approximated for polyelectrolytes in salt-containing solutions, by appropriate measurements in salt-free solutions. Equation 10 could be arrived at with different reasoning. In aqueous solutions of polyelectrolytes with N a as with counterion, rNp equivalents of Na+ ions of the total ( N + N ) equivalents are condensed on the polyion and the fraction of condensed N a ions f ^ in solution is *9>37 +
p
s
+
f
a
Na = r N p / ( N + N ) = r X / ( l + X) p
(11)
s
where r is the fraction of condensed or bound N a ions originally on the polyelectrolyte and (1 - r) is the charge fraction of the polyelectrolyte, which is the degree of dissociation of the polyelectrolyte or the charges on the polyelectrolyte uncompensated for by counterions. The fraction of Nat ions in the solution that are "free" is ÎN = l-*Na or +
a
*Na = «1 - r ) N + N ) / ( N + N ) p
s
p
(12)
s
Counterion condensation onto polyelectrolytes has been operationally defined as association such that the total fraction of polyion sites compensated for with counterion remains invariant over a wide range of X values 18. if the interaction of N a ions with polyelectrolytes is properly described as "counterion condensation", then a plot of f^ja (X + 1) vs. X should be linear with slope r. Note that this would indicate that the fraction of sodium ions dissociated from the polyelectrolyte is constant and independent of the concentration of polyelectrolyte and of simple salt. To evaluate r from the diffusion measurements presented here, an assumption must be made. It's plausable to assume that fNa+ given by DNa/DNa small coion-p^lyion interactions, if at all present, it does not affect the assumption. So, f Na given by +
i s
t n a t
i S
*Na = 1 -DNa/D&a
( 1 3 )
and plots were made of i^ail + X) vs. X for each NaPA/HPA polyelectrolyte with ξ>1. The results from such plots indicated linearity was obtained for 0.1 < X < 10, as shown in Figure 7 and Table V. The charge fractions (1 - r) Table V. Parameters of Equation 11 Obtained for NaPA/HPA Copolymers* ξ 2.73 2.22 1.82 1.54 1.33 1.18 1.05
Slope 0.57 0.50 0.43 0.34 0.28 0.24 0.19
+ + + + + + +
0.08 0.06 0.06 0.04 0.05 0.03 0.03
Intercept -0.13 + 0.10 -0.08 + 0.05 -0.08 + 0.06 -0.06 + 0.06 0.08 + 0.04 0.06 + 0.04 0.00 + 0.00
(1 - r) 0.43 0.50 0.58 0.66 0.72 0.72 0.81
+ + + + + + +
0.06 0.06 0.08 0.08 0.13 0.10 0.13
(DNa/ Na)x= D
0.41 0.46 0.53 0.64 0.70 0.79 0.85
+ + + + + + +
0.01 0.01 0.01 0.02 0.01 0.02 0.03
* f ( X + D vs. X . N a
is plotted against ξ-* in Figure 8. It should first be noted from Table V and
214
WATER-SOLUBLE POLYMERS
Figure 8 that within the indicated experimental error, the calculated charge fraction (1 - r) have values close to the theoretical value (|Z ι | ξ * ) , especially for the range 1.3 and ( 1 - r ) independent of N and both parameters are close in value for each polyelectrolyte studied here. -
D
D
a r e
=
a
D
+
χ
a
+
3
2 3
38
n
+
D
a
a
a
+
χ
+
vs
x
s
00
b o t n
ANDER
Figure 8.
Sodium-Ion Interactions with Polyions
215
Experimental values of (1 -r) and D ^ a / ^ N a dependence on ξ- for N a P A / H P A . The line is from the Manning theory. 1
216
WATER-SOLUBLE POLYMERS
Equations 9 and 11 stem from the same concept that Na+ ions are either condensed onto the polyion and do not contribute to the measured DNa that the Na+ ions are free so as to "not" interact with the polyion and contribute to the measured DNa+interpretation of Wall for (DN /Drsj )x = oo is the degree of dissociation of the polyelectrolyte is consistent with the value obtained for (1 - r) from Equation 11. The theoretical equation of Manning, Equation 4, can be cast in the same form as Equations 17 and 19 if the Debye-Huckel interaction term vanishes. Then, the theoretical value for the fraction of Na+ ions dissociated from the polyion (|Z [ |ξ)-1 is identified with (DN /Drsja)x = œ of Eq 10 and (1-r) of Equation 9. It becomes clear then as to why Equations 9 and 11 should appear to be valid. For high charge density polyelectrolytes, i.e., those used in the present study, the contribution of the condensation term to Na/°Na Equation 4 is much greater than that of the interaction term A. +
T
a
h
e
o r
26
a
a
D
i n
ACKNOWLEDGMENTS The author is gratefully indebted to his students for their contributions. LITERATURE CITED ( 1) ( 2) ( 3) ( 4) ( 5) ( 6) ( 7) ( 8) ( 9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27)
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