H. Magdelenat Service de Radiopathologie Fondallon Curie 26. Rue d,Ulm 75231 Paris Cedex 05. France P. ~ u r q , 'P. Tivant,' M. Chemla,' R. Menez2 and M. Drifford2
I1
Apparent Ionic Charge in Polyelectrolyte Solutions
I
The concept o f "apparent charge"is a good approximation o f the actual structural charge on particles at finite concentrations in solutions of electrolytes; thus it reflects the complexity o f thestructure o f charged particles in solution. Most of the physical chemistry of ionic solutions is governed by the strength of coulombic interactions between charged particles. T h e knowledge of t h e electrical charge of these in;rracrlnC: particlrs i? rh&ef$,rr fundamental t o any rheorerical o r ilumritative intrrprctnt~ono i experimental dara on electroiyte solutions. Our purpose, in this paper, is t o show how t h e comparison of the kxierimental values of the average displacement of charged particles obtained under t h e action of thermal motion alone (self ,~~ diffusion) with those obtained bv t h e action of a n external electric field (electrical mobility), leads t o the concept of "apparent charge," which is a good approximation of their actual structural charge in a given electrolyte solution a t finite concentration a n d t h u s reflects t h e complexitv of their structure. At infinite dilution, the Nernst-Einstein relationship correlates the electrical mobility u o and t h e self-diffusion coefficient D o t o the structural charge of t h e particle according to ~~~~~~~
~
uQIDo= Z,elkT
(1)
where kT is t h e Boltzmann factor and e t h e elementary rharee. For finite values of t h e concentration, t h e electrical mohilitv. u . is different from u o and the self-diffusion coefficient. D, k'different from Do.Thus, if we define the apparent charge, Zap,by Z,. = (ulD). (kTle) (2) its value is generally different from Z,?.However, in most cases, t h e difference is so small t h a t the structural charge can be derived without ambiguity, and even, i n some cases, theoretical corrections are available which . give t h e actual charge of the ionic species. T h e methodologv for t h e measurements of self-diffusion coefficients ( D ) aLd electrical mobilities ( u ) is presented, which consists, in the case of small ions, i n radio-tracer methods, or, in t h e case of macromolecular ions, in photon correlation methods (dynamic light scattering, electrophoretic light scattering). T h e relevance and usefulness of the concept of apparent charge in various fields of solution electrochemistry are illustrated by recent or original examples, including 1) symmetrical (1-1)electrolyte aqueous solutions where the apparent charges of the ions remain close to their nominal value in a large
range of concentration. 'Lahoratnire d'Electrnchimie. ERA310, Universite Pierre et Marie Curie, 4, Place Jussieu, 15320 Paris Cedex 05, France. ZServicede Chimie Physique, Division de Chimie, C.E.N., Saclay, B. P. N2,91190 G1F Sur Yvette, France. 12 1 Journal of Chemical Education
2) symmetrical electrolyte solutions (2-Z), in water, or (1-1) in mixtures of water and sblvent of low dielectric constant where Z ,. evidences pair formation. 3) complexing electrolyte solutions, such as silver salts solutions where Zaqreflects the formation of various proportions of such ionic soecles . as AZ(XK("-'). .. . 4 , p ~ ~ l ~ r l r r t r t ~ l ~ l ( where . a ~ ~ l%~. , ,~L ~t u~ ~he~ wrd n s 39 B meniuremml UI c ~ , u n t ~ r t ~ ~ n - e c ~ n d c nun i i ~rhe t i c ~psdv~cm. n
Experimental Method Tracer Methods Determination of Self-Diffusion Coefficients: As in any experimental device using radioactive isotopes in the determination of self-diffusion coefficients, one must create an heterogeneity of the isotropic concentration and then follow the homogenization of the whole-medium. Since its introduction by Anderson and Saddington ( I ) , the open end capillary method has been widely used for ionic solutions, giving values within 1% accuracy, when adsorption is negligible or can he accounted far. Asilica capillary tube (1 # 3 cm) with constant section ( I mm 6 )and plane bottom is filled with the electrolytic solution labeled with the investigated radioactive ion or molecule of initial volumic activity Ci. The capillary tube is then immersed in a large excess of a thermostated (25 0.l0C) inactive solution of identical chemical composition. The capillary tube is placed an a Teflon basket which is slowly rotated ( u = 2 cmlmn) in order to avoid radiotracer accumulation at the open end of the capillary, without convection. After a sufficient amount of diffusion time t, the final isotopic concentration of each capillary is measured. The ratio y = C,/C, leads to the self diffusion coefficient of the radiolabeled species by solving Fick's equation with the proper limit conditions ( 2 , 3 ) Thus. when r < 0.6
*
while when y
> 0.6
s12 12 D =-(I - y)2 = 0.7854-(1 - y)" (4) 4t t A Fortran program gives the mean value and the statistical accuracy of the determination from eight capillary tubes run simultaneously. When the isotooe is a eamma emitter, the capillary tubes can be counted directl; in a y counter with ordinary care for spectrometry ~
~
and eeornetrv. " . . ~ ~ ~ ~ ~ ~ ,
When it is a 0 emitter, C, and C, are activities per mg of solution and the content of each tube has to be weighed and counted by liquid scintillation. Determination of Electrical Mobilities: Basically, an initially labeled zone migrates in a chemically homogeneous solution under the action of an applied electric field (electrophoresis). The distance of mieration after a riven time is ormortional to the electrical mobility . . of 'the labeled species. The present study has been carried out by electrophoresis on cellulose acetate strips (Cellogel 30 X 2.5 cm) whenever possible or on cellulose paper strips (Schleicher-Schull n" 20434 at low pH where cellulose acetate is improper. The strips are soaked with the inactive electrolyte solution and placed on a thermostated electrophoresis apparatus (Pherograph
..
. two cases must be distinguished:solutions without ionic association and those with ionic association or complexation. Francfort). The temperature of the strip is 5'C + 0.1. Radiolabeled solutions are then applied on the strips by means of a special applicator giving well defined thin zones. A constant electric field is then applied (8 Vlcm). The migration of a reference ion of known mobility in the experimental conditions is determined simultaneously, on the same strip, since only relative electrical mobilities can be measured by this technique. Eleetroosmase is estimated by measuring the migration of a non-electrolyte molecule II4C urea or glucose). Analysis of the radioactivity distribution is carried out either, in the case o f a 7 emitter, by cutting the strip transversally into 2-mm strips which are counted individually, or, in the case uf a fl emitter, by means of a linear scanner, equipped with adetection slot ( 1 X 16 mm), which records the radioactive distribution curve and its integral which is used to determine the barycenter of the distribution, defining the migration distance from the origin (deposit). Sample Preparation: Electrolyte solutions were prepared from Merck Suprapur salts with twiee distilled water. For polyelectrolyte sdutims, we used Chondroitin Sulfate, isomer 4, as Nasalt from Serva (puriss) a t constant concentration (0.5 gh). The equivalent imic concentration was determined by NaOH titration of the corresponding acid form of ehondmitin sulfate obtained by ion exchange of the sodium salt on a 10 fold excess Dawex 50 X8 (Ht). A 0.5 gh chondroitin sulfate solution is 1.7 X 10-"equivalent. To this solution simple electrolyte solutions were added with negligible total volume change. Radioactive tracers were obtained as high specific activity aqueous chlorides from Amersham (England) or C.I.S. (France): 22Na,'"a, K a n b e interpreted either in terms of contact ion pairs (Fnoss (9)or of Bjerrum t y p e ion pairs (Justice (12)). S u c h association phenomena have heen extensively studied in t h e case of 2-2 electrolytes in water and of 1-1electrolytes in low dielectric constant solutions (9-12).
+
Volume 55, Number 1. January 1978 I 13
Table 1. Concentration Dependence of the Apparent Charge of Aqueous 1-1 Electrolytes C molar
0.01
0.02
0.03
ex0
0.0972
0.951
0.943
L.L
ew
0.960 0.943
L.L exp
0.943 0.956
L.L
0.950
0.04
0.05
0.1
0.25
05
0.84
0.789
0.660
KC! Zap NaF 2,
0.932
LiCl Z, 0.910
Calculated from theoretical limiting taws (Ref 19). Table 2. Apparent Charge for CsBr In Water-Dioxan Mixtures CsBr (mole/l)
0.0005
0,001
0.002
0.003
0.004
0.859
0.991 0.561
0.974 0.472
0.964 0.429
0.951 0.404
280
zap60%
z.,
dioxan 80% dioxan
Owing tu t h e i w t that c m d u r t a n c r theory a n d meniuremrnts h a w rlrarlv sctrlerl t h e concept of ion pairme in simple clectrolyte s o ~ ~ t i o nsuch s , systems will serve as riferencei to give t h e most r i m r o u s foundation of t h e conceut of a u p a r e n t charge that we i n t e n d t o develop hereafter a s a n ori&ml description of ionic association process
0.005
0.006
0.008
0.01
0.02
0.984 0.935
0.925
0.972 0909
0.951
0.933
~
D+ = (I - n)D$
0.05
0.943 0.837
0.760
D- = (1 - a)D!
+ aDp + aDp
(11) (12)
~~~
A=(l-a)hi
(13)
Zap = -AFI(DtF+ D-F+ 2nDpl(l - n))
(14)
RT FZ
and we define for any value of the concentration
where F is the Faraday, D+ and D- the selfdiffusion coefficients of cation and anion, respectively, and h the electrical conductance of the electrolytic solution. In such solutions, the apparent eharge of an ionic species should he simply identical to its ordinary ionic charge, a t least a t high dilutions. In Table 1 are represented the values of Z,. with concentration for various aqueous 1-1 electrolytes a t 298OK. The values uf the apparent eharge tend toward the ideal value, 2 = 1, when concentration decreases, and decrease with increasing concentrations. For concentrations smaller than 0.1 M the relative diminution in apparent eharge is always smaller than 20%. The experimental variation of Zapis, a t least at high dilution, in agreement with the limiting law. Physically, the decrease in apparent eharge with increasing concentration derives from the fact that the ionic mobilities (or conductances) are not only lowered by the so called relaxation effect (long range coulomhic interactions) as are self-diffusion coefficients, hut also by the hydrodynamic countercurrent due to the flows of cations and anions in opposite directions under the action of the electrical field (electrophoretic effect). Experimental results and their interpretation in terms of electrostatic and hydmdynamic interactions are a verification of the Nerst-Einstein relationship a t high dilution and give the domain of validity and the accuracy for the concept of apparent eharge in aqueous solutions of simple 1-1 electrolytes. h) For 2-2 electrolytes in water or for 1-1 electrolytes in low dielectric omstant media, the ion-pairing phenomenon lowers the apparent charge hy the formation of neutral entities of zero charge. In these cases. the comoarison between ionic self-diffusion and
14 / Journal of Chemical Mucation
0.03
CsBr in water-dioxan mixtures of varying proportions illustrates the situation (Table 2). The decrease of the apparent charge with increasing content of dioxan can be interpreted in a chemical model involving the formation uf neutral oairs of zero charee which do not oartieioate in the electrical transport. Taking n as the pair proportion and D, as the self-diffusion coefficient of the ion pair we have
a ) For svmmetrical 1-1 electrolvte solutions. where no " aoueous . ionic association is present, it is more simple to define thk apparent charee from both ionic self-diffusion coefficient a n d eleet.ricnl .~~ ~.~~~~~ - ~ - ~eon~... ..~..~ ductance because of a better precision in the determination of eleetrical conductance than of electrical mobility. At infinite dilution we and have ~~
0.025
where ions.
DL, DL, h f are the transport coefficients of the unassaciated
stant of the medium (12). C) For complex electrolyte solutions, the relevance of the concept of ionic apparent eharge can he illustrated hy an example: the behavior of silver ions in alkali halides and cyanides ( X - ) . It is well known that in such media, silver ion is precipitated as AgX and that redissolution occurs either in presence of an excess of X-, by formation of AgX,-("-')-type complexes, or upon addition of other soluble silver salts, hy formation of Ag,Xm-'-type complexes. The accurate measurement of the solubility ofsilver halides has been a problem for a long while, due to the bulky character of usual thermodynamical methods (16), until Lieser (17),using radiotracer techniques, gave satisfactory values. The experimental proof for such associations as AgX, -I"-" or Ag,Xm-' has been given by King et al. (18)who have shown that in sodium iodide solutions, radioactive silver moved as an anionic species. Moreover, the accurate solubility determinations of Lieser led to the determination of the stability constants of these various com~lexes,when farmer determinations (ootentiometrv. conducnon 1) In the presence of chlorides the soluhility of silver is always small but increases with increasing concentration of chloride beyond the stoichiometric precipitation of AgCI. 2) In the presence of bromides and iodides the solubility of silver beyond the stoichiometrie precipitatiun of AgBr and Agl is much higher than in the case of chloride ion.
Table 3. Apparent Charge and Solubility of Sodium Chloride C NaCl in moleli Solubility of Agt in 1 0 V mole11
0.4 14.2 -1.51
zapAst
1 70.3 -1.92
Table 4. Apparent Charge and Solubility of Sodium Bromide C NaBr in
m&/l
0.05
0.1
0.4
1
Table 5. Apparent Charge and Solublllty of Sodium Iodide CNal in male11 Sol~bililyof Ag+ in 10-%male/i
2,
Ast
0.1 0.02 0.05 1.14 6.41 24.9 -1.52 -1.55 -1.60
0.4 84.20 -1.95
1 84.10 -2.58
Table 6. Apparent Charge and Concentrations of Silver and Cyanlde C CN molell C Ag mole11 ZapAgt
0.201 0.1 -0.831
0.3 0.1 -0.961
1 0.1 -1.609
2 0.1 -1.913
3) For cyanides, after the precipitation of AgCN, the reaction of formation of A g i C N k is a complete one and the solubility of
A g ( C N k is noticeable. We have therefore reported (Tables 3-61, in the case of chloride. hromide, and iodide, not only the value of the apparent charge as obtained from the comparison of the self-diffusion coefficient and the electrical mobility of silver, hut thevalue of the solubility in water for the halide cmcentration. Fur cyanides, both silver and cyanide concentrations are reported in addition to the value of the apparent charge. The values of apparent charge exhibit the presence of Ag(X2)and Ag(X4- only far CI-, Br-, and CN-. For the highest concentrations of I- the oresence of A d I P must also be taken into account. The above values of the apparent charge of silver can be more q u a n t i t a t d y analyzed in terms of charges and properties of the different entities present in the solution. Even if the electrostat~c corrections (relaxation and electrophoretic effects, activity coefficients) are not introduced, the proportions of the different entities are in agreement with the available data for the thermodynamical constants of formation of the different species (23)
Polyelectrolyte Solutions A polyelectrolyte solution is characterized by the presence of polgelectrolyte macromolecules, which hear a high charge density (linear and ncgati\e in our e w m p l e ) , by counterions and yenero.1~In ions d . i n added supporting electrdytr. In such solution; it w a c lntercatinr to f8~llou, distinctively the transport propertit,.; of the pulyim itselirl~yelectrophuretic light sratterinr ,IS devel,rv~din the next stxtionl and thc~st!id the counterions by use of radiotracers as i n ordinary electrolyte solutions.
Apparent Charge of Counterions T h e association properties, and t h u s t h e transport properties of such counterions are mainly governed by the charge density of t h e polyelectrolyte chain (24, 2.5). If this charge density is sufficiently high, some new features will occur, such as ionic condensation along the polyelectrolyte chain, and, as a consequence, considerable changes in t h e apparent charge of t h e counterions will occur. However, in most cases, i t will be possible t o distinguish between condensed and uncondensed counterions: t h e free fraction of t h e counterions hehaves almost a s free ions in simple electrolyte solutions, t h e slackening of t h e ions in the coulombic field of the highly charaed ~. o l g i o nbeinp always small. With regard to condensed . ro11n16&1i*,their (1rgn.e of reedo om a h : : the ~x,lyelectrulytt~ r h . m i. nlmc~rzc!ru . ~ n dtheir transport pnriimeters are nearly t:qual t o rhar of the marromolerulnr ion itieli. Figurr I lllui-
Figure 1. Variation of the apparent charge with the nature and concentration of caunterions in polyalectroiyte solutions (chondroitinsulfate,Na salt).
trates the variations of t h e apparent charge of different counterions in t h e following systems 1) The polyelectrolyte macromolecule was chondroitin sulfate (Na salt, puriss), isomer 4, from Serva. In dilute neutral solutions it can be considered as a long line charge bearing appnximately 50 negative charge carhoxyl and sulfate groups, according to structural studies (26). 2) Different simple salts ichlurides) were added to the polyelectrolyte solution since the apparent charge of caunterions varies theoretically (27) with: a) the nature of the added counterion, essentially with its nominal valence, and b) the concentration ratio of the added counterion to the polyelectrolyte.
Full theoretical treatment for multivalent counterions i n this system, according t o Manning's electrostatic theory (27-29) of polyelectrolyte solutions has been developed in previous works (30). We observe on Figure 1,that, when the condensed fraction of counterions is small (low charge density of t h e polyion or/ and large excess of added counterions) their apparent charge is almost t h a t of the ion in polyelectrolyte-free solution (right part of Fig. 11, whereas in the case of a n important condensation (high charge density of the polyion orland large excess of polyion concentration), the sign of the apparent charge may be changed and its value may even he of t h e order of magnitude of t h a t of the polyion (left part of curves b, c, on Fig. 1). T h e apparent charge of t h e initial counterion (here Na+) is never so drastically changed since its minimal concentration is t h e equivalent concentration of t h e polyion and a fraction of free ions of such nature is always present (curve a on Fig. 1).
Apparent Charge of Macromolecular Polyions From Dynamic Light Scattering I n this preliminary work we have studied both thermal (measurement of self-diffusion coefficients) and electrophoresis coupled diffusion (measurement of electrical mobilities) of chondroitin sulfate in various electrolyte solutions so t h a t we could derive the apparent charge of the macromolecule in the limits of the Nerst-Einstein relationship (eqn. (2)),as for simple ions. We were, in particular, able t o follow the variations of t h e polyion charge when increasing concentrations of calcium chloride were added to a solution of the sodium salt of chondroitin sulfate. Self-Diffusion Coefficient of Chondroitin Sulfate (Na S a l t ) i n P u r e Water: Power spectra weie obtained a t room Volume 55, Number 1. January 1978 1 15
Figure 3. Doppler shin of lhe photocunent Spectrum as a hrnction of the external electric field (chondroitin sulfate.Na salt).
Figure 2. Variation of the Lorentria" linewidth I'(Hr)of me photocurrent specbum with the scattering angle ilisin21112). tor chondroitin sulfate (Na salt).
temperature (293'K) for scattering angles varying between 5 and 10'. For each scatterine snec,. angle. . . the nhotocnrrent .
trum wns fitted 1 0 the rlr,iv.;t thnm,t~sall.orcnt;.ian 11) an 4I'l. cumpuler 1,rogrnm. F~yure2 ihrws rhr wriation of the I.ort:nt~ianIinwidrh with sin- I, 2 'l'ne ilnpc 111 rh~!l~nearl ~ l d lead; r($ I hv translational wli-difruwn n,efficicnr I ) ,
D,= 13.42 + 0.06)lo-' ern' scl This value is in good agreement with one determined previously (31) by sedimentation equilibrium. Electrical Mobility of Chondroitin Sulfate (Na Salt) in Pure Water in the Presence of a n External Electric F~eld: Measures were carried out a t 5" scattering angle for increasing values of the applied field. The electrical mobility derived from the slope of the linear plot Au = f(E(V/cm)), (Fig. 3, is Iutroducinr the exnerimental values of Di and u in the Neril-Ilinarpin relationshrp,a~hpr,;n.,li found a \.due i~itht.;ipparrnt charfie ,,fchondnritin sulfate dissolvprl as svdium salt in wrk. (26) (14000 Dalton) and its specific equivalent concentration (34 X lo-' eq/g). The structural number for the hare polyion is 48 hut it is reduced to 39 in solution by the neutralizing effect of Na condensation, as derived from Manning's treatment with the involved charge density parameter (3) [, = 1.24. The agreement with the experimental value is quite good. Effect of the Addition of Bivalent Counterions to the Sodium Salt of Chondroitin Sulfate: We have studied the influence of adding increasing concentration of Ca2+ (as CaCI2) on the electrical mobility and apparent charge of chondroitin sulfate initially introduced as sodium salt. For each Ca2+ concentration, the Doppler shift Au was measured and the corresnondine mobilitv and charee derived as above, assuminn
neutralize an increasing part of its charge, then remains constant and equal to -16, since no more Ca2+ions can condense. The theoretical corresponding value should he -19, the charge density parameter being reduced to its critical value, % (30). 16 / Journal of Chemical Education
Figure 4. Variation of the apparent charge of chondroitin sulfate with the concentration of added calcium chloride.
Thus, in the case of polyelectrolyte solutions, the comparison of the apparent charges of counterions and polyions gives quantitative and qualitative information on the nature of the binding and the remaining degrees of freedom of the counterions. In particular, if, even after a change in sign, the apparent charge of the condensed counterion does not reach the value corresponding to that of the polyion itself, it means that there remain a t least some deerees of freedom for the condensed ions on the polyelectmlyte chain, corresponding to the nossihilitv of site to site or continuous motion of these ions along thechain. I t must however he kent in mind that the derivation of the apparent charge of macromolecular ions is theoretically possible only when the macromolecule is not nreferentiallv oriented in the direction of the electric field. weak electric Fields as those used in our experiments are suitable ~
~
~
Conclusion The preceding study has shown the relevance of the concept of apparent charge in various electrolytic media. In simple 1-1 electrolyte solution, this concept is identical to the ordinary concept of individual ionic charge. In the case of associated and complex electrolyte solution, the apparent charge reflects the presence of neutral ionic nairs and/or chareed comnlexes in an unambiguous manner: In the case of &yelec~rolyte solutions, the concept o f apparent charge appears to he very
powerful to follow the condensation of counterions along the polyelectrolyte chain and permits a direct and consistent
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