J . Phys. Chem. 1990, 94, 2175-2180 TABLE 11: Molecular Orbital Overlap Population, POPtj, and the Combination Coefficients of Atomic Orbitals (a,,aj) of the Concerned Carbon-Carbon Bonds, Obtained by an EHMO Calculation of HOMO for a Model Chain (*PPy) at Various Oxidation States
oxidn state *PPO *Pp'+ *PP2+
c,-c,
'8-'8
0.909 0.943 0.916
0.990 1.038 1.085
-0.358, 0.219
-0.318, 0.318
ca-c8
1.161 1.102 1.043
ai@j
* P p , *PP+
-
-0.358, -0.318
PP2+ lead to an increase of C,-C, as well as C,-C, bond orders, but to the decrease of C,-C, bond order. This expectation is clearly demonstrated by the increase of POP, from * P p to *PP'+, and to *PP2+ (see Table 11). The above-calculated changes of carbon-carbon bond order of PPy as its oxidation degree increases is qualitatively quite compatible with the Raman observation of frequency shift of the double bonds. In fact, as regards the P p and PP2+ species, the calculated results are in agreement with those reported by BrEdas et al. from a b initio calculation, in which aromatic-like and quinoid-like structures have been concluded respectively for the neutral and bipolaron statesS4 Thus, the Raman observation provides an evidence for the predicated structural changes associated with the formation and transformation of the proposed three PP2+process, the double bond species. For the P P PP'+ component of the inter-ring Cwd and the intraring C,, increases, whereas that of the intraring C,, decreases. Similar Raman spectroscopic behavior of compounds with structure similar to PPy was reported. Consider, for example, the methylviologen reduction (MV2+ MV'+) in which a change from intraring double bond to inter-ring double bond was suggested. Theoretical calculation and Raman spectroscopic determination indicated that it led to an increase (10-20 cm-') of uCx (at ca. 1660 cm-').I9 This situation is quite similar to our
- -
-
2175
discussion on the PPy case and can be considered as a support for our present interpretation. In fact, our interpretation of the Raman shift also agrees with the Raman studies recently reported by Hester et aLI3 Their explanation for possible configuration changes of C=C bonds of PPy chain was based on a comparison of the vc=c bands between polyacetylene and polypyrrole. Summary
The electrochemical redox process of polypyrrole films coated on electrodes has been investigated by in situ ESR and Raman spectroscopic methods. The good agreement between the in situ ESR data and the theoretical kinetic predictions on the basis of the polaron-bipolaron model of PPy confirms, straightforwardly, that the electrochemical formation of a polaron and its transformation to a bipolaron are both kinetically and thermodynamically dependent on the applied potential. It supports the conclusion that both the polaron and bipolaron are involved in the electrochemical charge transport of PPy.'+ The in situ Raman data, in correlation with the ESR conclusion, provide evidences about the structural change of PPy's carbon-carbon bonds associated with the polaron and bipolaron formation, suggestive of an increase of the inter-ring double bond component for the process from neutral PPy to polaron state and further to bipolaron state. We emphasize here the significance of the combination of the in situ ESR and in situ Raman studies which, with further precise evaluation of the various parameters, may lead to insight into the real picture of the polarons and bipolarons involved in the electrochemical redox process of PPy films. Acknowledgment. We thank F. L. Wang, T. J. Zhou, Y. Zhang, and A. Preusser for discussions about computations, and W. X. Huang for ESR experimental assistance. Registry No. PPy, 30604-81-0; LiCIO,, 7791-03-9; Na2S04,775782-6; KNO,, 7757-19-1. (19) Hester, R. E.; Suzuki, S. J . Phys. Chem. 1982, 86, 4626.
Collective Ionic Diffusion in Polysaccharide Solutions Jean-Pierre Simonin,* Patricia Tivant, Pierre Turq, and Embarek Soualhia Laboratoire d'Electrochimie, BBt. F, Tour 74, UniuersitP Pierre et Marie Curie, 8 rue Cuuier. 75005 Paris, France (Received: August 14, 1989)
For two different polyelectrolytes (chondroitin sulfate and heparin), the influence of added salt concentration (NaCl) on the diffusive modes of the macroion and of its counterion is studied in coupling diffusion experiments by the tracer technique using the open-end capillary setup. The slow transition between the purely coupled diffusion process, called chemical diffusion (no salt added), and the individual motion, or self-diffusion (with excess added salt), can be observed in polyelectrolytesolutions. The experimental results are qualitativelydescribed by normal-mode and finite difference calculations,starting from an estimation of the effective charge of the macroion. This charge is obtained from chemical diffusion data, with a correction by activity coefficients as described by the mean spherical approximation (MSA). It is compared to theoretical results from Manning's theory and a Poisson-Boltzmann cell calculation. It is found that experimental and theoretical results on the charge of the macroion are in good agreement in the case of heparin. On the other hand, a significant discrepancy is found for chondroitin sulfate, which can be interpreted as originating from association in the solution of this macroion.
The study of coupled-diffusion processes in electrolytes has shown3 that the diffusive behavior of a tracer can exhibit a con*To whom correspondence should be addressed.
0022-3654/90/2094-2175$02.50/0
(1) Chu, B. Laser Light Scattering, Academic Press: New York, 1974. (2) Anderson, J. S.; Saddington, K. J. J . Chem. SOC.1949, 2, 381. (3) Turq, P.; Chemla, M.; Latrous, H.; M'Halla, J. J . Phys. Chem. 1977, 81, 485.
0 1990 American Chemical Society
2176
The Journal of Physical Chemistry, Vol. 94, No. 5, 1990
serum albumin5q6 (BSA), poly(~-lysine),~ and short DNA rods8 The purpose of the present work is to show that corresponding results can be obtained for polyelectrolyte solutions, using a tracer technique. The basic advantage of such a technique is that, unlike QELS, it does not require the presence of a salt in the solution of polyion, which permits one, as will be seen below, to obtain information on the effective charges of polyions. We now recall briefly the basic ideas of coupled-diffusion phenomena in electrolytes and examine in which way they can be applied to solutions containing polyions. First, for all types of diffusive processes in electrolytes, the local dynamic electroneutrality condition is always satisfied (on a semimacroscopic scale); the ions move and, at the same time, the system stays electrically neutral locally. As a consequence, in the absence of any support electrolyte, both ions of a binary electrolyte must move at the same velocity, whatever their "self-velocity" may be. This phenomenon, known as the "chemical diffusion", can be described by one diffusion coefficient, the so-called Nernst-Hartley diffusion coefficient Dnh. Applied to a pure polyelectrolyte solution, this condition implies that the diffusion coefficient measured, when either the polyions or the counterions are labeled, should have the same value Dnh. Note that the QELS technique does not allow one to get this limit, because the scattered light is equal to zero when there is no salt added to the solution. However, making measurements for different salt concentrations, and using an extrapolation method, it is possible to determine this collective diffusion coefficient 0,. On the contrary, with the tracer techniques, it is possible to explore the largest range of diffusion coefficients; thus, one can couple, in a wide range of concentration, salt and tracer concentration gradients. Two anionic polysaccharides have been studied: chondroitin sulfate and heparin. Their charge densities are very different, which makes it interesting to observe whether the behavior of these macromolecules is related only to their charge density or, on the contrary, if the results must be interpreted in a more general framework. Moreover, the low-concentration regime was chosen, in order to compare the behavior of the polyelectrolytes to that of simple electrolytes. The experiments are described in the first section. Next, the theoretical models applicable to the present study are studied in the second section. Lastly, the experimental results are interpreted by use of these models. 2. Experimental Section 2.1. Materials. The two types of biological macromolecules studied, chondroitin sulfate and heparin, were obtained in the following forms: chondroitin sulfate (isomer 4) was purchased from Sigma (Type 1, Ref. C-4134), as a sodium salt; the commercial sodium heparinate, whose anticoagulant activity was roughly equal to 160 UI/mg, was provided by Choay Institute. Without any data furnished by the supplier, one of our purposes was to characterize these two polysaccharides; thus, the weight-average molecular weight M,, and the self-diffusion coefficient of each polyion, Dpo, were measured by QELS.Iq8 The equivalent concentration was also determined in the following way: the macromolecule, as a sodium salt is passed through (4) Tivant, P.; Turq, P.; Drifford, M.; Magdelenat, H.; Menez, R. Bio. polymers 1983, 22, 643. (5) Raj, T.; Flygare, W. H. Biochemistry 1974, 13, 3336. (6) Doherty, P.; Benedek, G. B. J . Chem. Phys. 1974, 61, 5426. (7) Lin, S . C.; Lee, W. I . ; Schurr, J. M. Biopolymers 1978, 17, 1041. (8) Tanford, C. Physical Chemistry of Macromolecules; Wiley: New York, 1961. (9) (a) Manning, G. S. J . Chem. Phys. 1969, 51, 924. (b) Manning, G. S.1969, SI, 934. (10) Tivant, P.; Perera, A.; Turq, P.; Belloni, L. Biopolymers 1989, 28, 1179. ( 1 1 ) Tirado, M. M.; Garcia de la Torre, J. J . Chem. Phys. 1979, 71, 2581. (12) Bell, G. A.: Dunning, A. J. Trans. Furaduy SOC. 1970, 66, 500. (13) Katchalsky, A.; Alexandrowicz, Z.; Kedem, 0. Chemical Physics of Ionic Solurions; Conway, B. E., Barradas, R. G., Eds.; Wiley: New York, 1966. (14) Belloni, L.;Drifford, M.; Turq, P. Chem. Phys. 1983, 83, 147-154.
Simonin et al. TABLE I characteristics mol wt, g/mol self-diffusion coeff, cm2/s equiv concn, equiv/L dist between two charged groups, b, A chain length, k, radius, k, struct charge, 2,
TABLE I1 CE(in) - CE(0Ut)' CT(in) - CT(0Ut)' nature of the diffusion process
chondroitin 20000 4.5 x 10-7 3.5 x 10-3 5 . 8 f 0.2 435 f 15 6fl -7 5
heparin 9000 8.0 x 10-7 5.5 x 10-3 2.9 f 0.3 I60 f 15 10 f 2 -55
0
#O
#O
#O
#O
0
#O
selfdiffusion
coupled diffusion (general case)
coupled diffusion (transient regime)
collectiveb diffusion
#O
C, and CT are respectively the electrolyte (or polyelectrolyte) and the tracer concentrations, and the concentrations C(in) and C(out) respectively refer to concentrations within and outside the capillary. *In this case, (CT/CE)~",= (CT/CE),,,; the diffusion process observed is also that of the unlabeled species of the electrolyte, because the labeled species concentration is proportional to the electrolyte concentration everywhere.
a column of resin which exchanges the cations, and which is initially in its acid form (Dowex SOW-XS). The polyelectrolyte is then completely protonated and, next, it is titrated with NaOH. The distance between ionized groups of the macromolecule, denoted by 6, related9 to the charge density parameter of Manning tS,has been determined from the sulfate content, along with the titration data. The structural charge has been obtained either from the data provided by the supplier (weight percentage of atoms or groups of the macromolecule) or, more simply, by multiplying the equivalent concentration (in equiv/g) by the molecular weight. These data yield an estimation of the total chain length of the particle. The average radius of a macroion has been estimated from a fitting of the diffusion coefficients of the counterions by using a Poisson-Boltzmann cell model (see section 3.1.2); it was shown in ref 10 that the use of this model can lead to a good agreement with the experimental data. Lastly, the values of the ratios of the length of a chain to its radius are consistent with those that can be calculated from self-diffusion data on the macroions, using analytical expressions for the diffusion coefficients of rigid rods." The physical characteristics of the macroions are summarized in Table I. 2.2. Description of the Experiments. The experimental method used here was the open-end capillary technique, suitable for the determination of self-diffusion coefficients:2 the ionic species of interest is labeled with a radioactive isotope and introduced into the capillary; next, the latter is immersed into a container, whose volume is much larger than that of the capillary. In the following, the polyion will be denoted by Pz-. Both ionic species, Na+ and the polyion, are labeled with a radioactive tracer. Na+ is easily labeled with "Na. The polyion is not directly labeled. We use the following property of a charged polyion: if a small quantity of bivalent cations like Sr2+,Co2+,etc. is added to the polyelectrolyte solution, these bivalent ions are completely boundlo or "condensed" on the polyion, because of the high charge density of the latter. Consequently, they move exactly at the same velocity as the polyion. The isotope 85Sr was used to label the polyion. Each of the active elements have been purchased from Amersham (England) as carrier free chlorides in aqueous solutions. After a known lapse of time, the capillary is removed and the remaining activity is measured with a gamma counting system. This activity is compared to the initial activity of the capillary. A particular experiment which can be performed consists in replacing the solution in the container by pure water; this is the case of the chemical diffusion mentioned above, in which it is expected that both species (the polyion and the counterion) diffuse at the same velocity. The various experiments differ in the amount of salt (sodium chloride was chosen) added to a polyelectrolyte solution.
The Journal of Physical Chemistry, Vol. 94, No. 5, 1990 2177
Ionic Diffusion in Polysaccharide Solutions Moreover, two types of experiments were carried out: (i) Sodium chloride is homogeneous throughout the system and the polyelectrolyte is present only inside the capillary. (ii) The capillary is filled with the polyelectrolyte and NaC1, but the container is filled with water. So, there are gradients of concentrations of both polyelectrolyte and NaCl solutions. Using the tracer techniques, it is possible to explore the largest range of diffusion coefficients. Thus, one can couple, in a wide range of concentration, salt and tracer concentration gradients. These different possibilities are summarized in Table 11. We chose to describe an experiment by the apparent diffusion coefficients of the polyion and of the counterion. By apparent diffusion coefficient we mean the diffusion coefficient which would correspond to a self-diffusion measurement. It is calculated from the ratio y of the final to the initial activity of the capillary Y = AfinadAinitiat
(1)
is the mean distance between monovalent charged groups. The hypotheses of Manning’s theory are as follows: (i) the real polyion is schematized by a uniformly charged thread (of linear charge density X) of infinite length; (ii) the interactions between polyions are neglected; (iii) the interactions between the counterions occurring in the vicinity of the polyion are neglected; (iv) the dielectric constant of the medium is that of the pure solvent. Clearly, assumption (i) should apply mostly to the case of a pure solution of polyelectrolyte, without added salt, so that the inner repulsions between charged groups ensure a good rigidity of the polyion, and assumption (ii) should be verified for dilute solutions of polyelectrolytes (concentration lower than the critical value C*). An important parameter appearing in the model is the parameter f , defined by
using the equation Dapp= ( 4 L 2 / t ) ( ( l / r 2 )In (8/7r2y)
+ ~ ‘ ~ y ~ / ( 9 . 8(2)~ ) )
in which L is the length of the capillary and t is the time of immersion of the capillary. Equation 2 is valid when y < 0.6 with a precision better than 0.01%. When y > 0.6 the formula Dapp= rL2(1 - ~ ) ~ / ( 4 t )
(3)
can be applied.
3. Theory In electrolyte solutions containing ions of charge greater than unity, the association phenomenon (in the Bjerrum sense) constitutes the first departure from ideality. It originates from the strong Coulombic interaction at short distance between ions of opposite charges. It is negligible only in the case of uni-univalent electrolytes. The very high charge of the macroions causes a high level of association between the latters and their counterions, which must be described by theoretical tools adapted to these particular objects. In the present work, we give first a simple description of the experimental diffusion data, neglecting all departures from ideality, except that resulting in the condensation of counterions on the polyions. A complete and precise description of the nonideality effects in polyelectrolyte solutions, with added salt, is a difficult matter: it includes the calculation of activity coefficients as well as the description of hydrodynamical interactions and relaxation effects. This work is presently in progress in our group and will be the subject of a subsequent article. Thus, in sections 3.2 and 4.2, the system we will consider is the “renormalized” system in which a fixed part of the counterions is bound to the polyions while the rest of the counterions is free. This system constitutes the new reference state. In section 4.1, for the more simple system of pure solutions of polyions, an estimation of mean sodium ion/polyion activity coefficients is used to get a more refined calculation of the effective charges of heparin and chondroitin sulfate in our samples. The second basic interaction phenomenon occurring between the ions arises from electrostatic forces at a larger scale. These forces will be considered to act on the species of the new reference state (free counterions and modified polyions). A consequence of this long-range interaction is that the observation of the system on a semimacroscopic scale, on lapses of time of the order of 1 s, leads to the local dynamical electroneutrality condition. 3.1. Study of Condensation. 3.1 .I. Manning’s Theory. The description of the condensation of small counterions on polyions by Manning’s theoryg is a celebrated model. We first recall its principles and results and next apply them to our systems. Let us consider that the polyion is in a linear conformation. Its total length is Lo and its structural charge is Zs < 0. If all the charged sites have a charge Z = -1, then the mean linear charge density on the polyion is X = -e/b (4) in which e is the elementary charge and
where LB is the Bjerrum length
t is the dielectric constant of the solvent, kB is the Boltzmann constant, and T is the temperature. The value of LB is 7 . 1 3 A in water at 25 O C . The results of the theory then are as follows: (i) no condensation occurs if 6, < l/lZcl, where Z, is the charge of the counterions; (ii) in the opposite case, condensation occurs which lowers the effective charge of the polyion to the value Zeffsuch that
teff
= 1 /Vel
(7)
with, as in eq 5 Considering positive counterions and a negative polyion, eq 7 and 8 lead to
Zeff = -LO/(LBZC)
(9)
The fraction a of the counterions condensed on the polyion can then be expressed simply. It results from its definition that a = 1
- Zeff/Z,
(10)
which, from eq 4a, 5, 7, 8, reads a =1
- l/(ZCfS)
( 1 1)
3.1.2. Poisson-Boltzmann Treatment. A Poisson-Boltzmann cell model12 (PBC) was used. It is now well-known and it has been used by numerous authors, in particular in the cylindrically symmetrical case.13 In this model the polyions are uniformly distributed in the form of a regular lattice. In our case of linear polyions, the solution is separated into elementary cylindrical cells, each of them similar and neutrally charged. The radius of the cells is related to the concentration of the solution. The precise ingredients of the calculation can be found e1~ewhere.l~ 3.2. Calculation of the Apparent Diffusion Coefficients. 3.2.1. Finite Difference Treatment. The first method used in the well-known finite-difference technique. The principle of the calculation is to replace, in the equations governing the evolution of the system, all infinitesimal increments, symbolically denoted by d’s, as for instance dC which stands for a concentration variation (in time or space), by finite increments denoted by A’s. Thus, both time and space are discretized in finite cells of respective length Ax and At. It is known that, when dealing with the heat conduction Fourier equation in its simplest form, there must be a simple relation between those two quantities15
(15) For example: Jenson, V. G . ;Jeffreys, G . V. Mathematical Methods in Chemical Engineering, Academic Press: London, 1911.
2178
The Journal of Physical Chemistry, Vol. 94, No. 5, 1990
where a is the heat conduction constant. In the case of diffusion equations involving several species which are coupled in their diffusion and for which the diffusion coefficients are written Di, we have foundI6 that it suffices to choose At in such a way that it is related to Ax in the following way
and to take M to be at least equal to 5 to achieve the convergence (in the computational sense) of the procedure. When M = 1, At is of the order of the mean diffusion time of the faster species between contiguous cells. In the present case we wanted to improve a particular point of the algorithm, dealing with the fact that the initial concentration profiles are varying abruptly in space, at least for some species, at the opening of the capillary; for example, the tracer is present inside the capillary but not outside. Therefore, one may expect a much too rapid variation of these concentrations in the first time steps of the simulation, leading to large errors for these profiles at short times; since coupling processes are also involved here, even negative transient concentrations can be obtained. In order to avoid this problem we have chosen to take a varying time step, that is, a varying M (defined by eq 13), in the following way. The diffusion equations of the system consist in a set of partial differential equations
aci/at = Di(d2Ci/dx2- pZie d ( C , ~ ) / d x )
+ At A j j ( t )
where AiJ(t)can be expressed with the charges, diffusion coefficients, and concentrations of all species. Then, in the first time steps of the calculation, one can impose in each cell the relative variation rate of all concentrations to be smaller (or equal) than a definite limit, which we call A. Expressing this condition for all species and all cells yields the sufficient condition At = min [AIAij(t)l/CiJ(t)]
ij
This expression corresponds to a varying value of M , which is at the first time step; typically of the order of 100 with X = it then decreases and we have chosen M not to be less than 5. This way of proceeding ensures good behavior of the simulation at short and long times. The charges of the polyions were taken from experimental data of chemical diffusion experiments (see section 4.1). The values of the diffusion coefficients of sodium and chloride ions were taken at infinite dilution. Let us mention lastly that the system must be described with the use of four species: the polyion, its counterion Na+, chloride ion (when salt is added), and the tracer, labeled sodium, which must be considered as an independent species. 3.2.2. Normal-Mode Treatment. The second method, known as the normal-mode analysis, is the solution of the near-toequilibrium linearized diffusion equations of the system; it has been widely used, up till now, in the work of our group,I6-l9 for (16) Simonin, J. P.; Gaillard, J. F.; Turq, P.; Soualhia, E. J. Phys. Chem. 1988. 92. 1696. (17)-Turq, P.; Orcil, L.;Chemla, M.; Mills, R. J . Phys. Chem. 1982, 86, 4062
TABLE 111 heparin (Z, = -55) chondroitin sulfate (Z, = -75)
ZefAManning) -22.5 f 2.0 -61.0 f 2.0
Z,,I(PBC) -23.0 f 0.1 -61.7 f 0.2
Z,&exp) -21 -35
it gives analytical expressions for the observables of the system, and it always provides with a good estimation (in reasonable agreement with both experimental and finite difference results) for these quantities, even though the initial situation is far from equilibrium. The technique can be used to describe only the case where the outside solution is pure water. In the presence of salt, the equilibrium concentration of the tracer (labeled sodium ion, or labeled polyion) is zero, and therefore the result (which involves the equilibrium concentrations of the species) would be independent of the initial excess of concentration of the labeled species. This would yield a horizontal line when plotting the apparent diffusion coefficient of the species versus added salt, with a discontinuity at the point of origin. This is obviously unphysical. In the case where the outside solution is pure water, the equilibrium concentrations are vanishingly small, but are proportional to their initial values inside the capillary; it is found that the result is expressed as a function of the respective concentration ratios. The principle of the method has been exposed in previous article^.'^-^^ The concentrations are written in the form of an equilibrium part Ciq and an excess part SCi
(14)
for each species i (p = l / k B T with kB the Boltzmann constant); Di is the self-diffusion coefficient, Zie the charge, and Ci the local concentration at time t of that species; E is the electrical diffusion field originating from the local heterogeneity of charge. The discretized diffusion equation for a species i in cell j can then be written as Cij(t+At) = Cjj(t)
Simonin et al.
and the set of equations (14) is linearized with respect to equilibrium. After a Fourier transform on space and a Laplace transform on time, a linear system is obtained, which can be written in a compact matrix form and solved with standard linear algebra. The only original point here is the way the concentration profiles are repeated in space” in order to express the concentration profiles as cosine Fourier series. To achieve this, a virtual periodic system is imagined, respecting the boundary conditions of the real system. Here, the latter are as follows: no flow of matter at the bottom of the capillary ( x = 0) and a zero concentration of each species at the opening of the tube. Let us call L the length of the capillary. The strategy is to proceed as follows: first, at time t = 0, if C, is the initial concentration of species 1 (between x = 0 and x = L ) , C = -Cl is taken for the concentration of this species between x = L and x = 2L; the use of this negative concentration is of course only a mathematical trick to solve the problem. Lastly the system is symmetrized with respect to x = 0 in order to expand the concentrations in cosine Fourier series after having repeated periodically the same pattern with a spatial period of 4L. The explicit calculus, though simple in principle, is long and will not be detailed here. It was performed with the use of a computer programming system, MACSYMA, a symbolic manipulation system. The final result is given in the Appendix. 4. Results
When the variation of some parameter (effective charge, apparent diffusion coefficient) is studied as a function of the amount of salt added to a solution of polyion, the ratio r = CSalt/Ce, (18) in which C,., and C, are the concentration of salt added and the equivalent concentration of the polyion, varies between 0 and 2 for heparin (C, = 5.4 X equiv/L), and between 0 and 4 for chondroitin sulfate (C, = 3.5 X equiv/L). 4.1. Effective Charges of the Polyions. The results from Manning’s theory, the PBC procedure (section 3.1.2), and the experimental results are summarized in Table 111. The physical characteristics of the macroions were taken from Table I. The experimental charges were deduced from the measurement of the observed Nernst-Hartley diffusion coefficient Dnhabsof the system Na+/polyion at high dilution (concentration C = 1 g/L; mol/L); we took the polyion concentration of the order of
The Journal of Physical Chemistry, Vol. 94, No. 5, 1990 2179
Ionic Diffusion in Polysaccharide Solutions
15 -
r Figure 1. Results for chondroitin sulfate. Apparent diffusion coefficients versus the ratio of added salt concentration to the equivalent concentration of polyion. Upper curves: diffusive modes of the counterion (sodium ion). Lower curves: polyion. Experimental points: (W) no salt in the outside vessel; (A)with salt. Curves 1: finite-difference calculations (with salt in the outside vessel). Curves 2: normal-mode calculations (no salt). Curves 3: finite-difference calculations (no salt). DNaand Dp indicate the values of the self-diffusion coefficients of the sodium ion and of the polyion.
Dapp x l o 6 ( c m 2 I
0
.5
1.5
1
2
2.5
r Figure 2. Results for heparin. Apparent diffusion coefficients versus the ratio of added salt concentration to the equivalent concentration of pc ,ion. Upper curves: diffusive modes of the counterion (sodium ion). Lower curves: polyion. Experimental points: (W) no salt in the outside vessel; (A) with salt. Curves 1: finite-difference calculations (with salt in the outside vessel). Curves 2: normal-mode calculations (no salt). Curves 3: finite-difference calculations (no salt). DNaand Dp indicate the values of the self-diffusion of the sodium ion and of the polyion.
mean value of the two Nernst-Hartley diffusion coefficients obtained with the sodium tracer and the labeled polyion, successively yielding
&''= 5.65 X cm2/s for chondroitin sulfate &'' = 5.8 x IOd cm2/s for heparin Owing to the very high charge of the polyions, the influence ofthe departures from ideality must be evaluated. For this P u W e we have used the expressions from the mean Spherical approximation (MSA) theory for activity coefficients.20 Due to the low (20) Hard-sphere contribution: Humffray, A. A. J . Phys. Chem. 1983, 87, 5522. Electrostatic contribution: Vericat, F.; Grigera, J. R. J . Phys. Chem. 1982, 86, 1030.
concentration of the solutions we suppose that the electrophoretic effect can be neglected. Relaxation effects are also very small in the chemical diffusion experiments and are therefore also neglected. A corrected Nernst-Hartley diffusion coefficient is then deduced from the relation21 Dnh = onhob/( 1 + d log y+/d log c) (19) in which y+ is the mean activity coefficient of the sodium ion and the polyion, and C is the concentration of the solution. Also Dnh =
(lzl +
~ ) D N $ P / ( D N+~ IZIDp)
(19a)
(21) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions, 2nd ed.; Butterworths: London, 1959.
2180 The Journal of Physical Chemistry, Vol. 94, No. 5, 1990
where Z is the charge of the polyion and DNais the self-diffusion coefficient of the sodium ion. We have calculated the activity coefficients at a mean concentration, equal to half of the initial concentration. The diameters of the species were estimated from their hydrodynamical radii (Einstein-Stokes formula). The experimental effective value of the charge of the macroion, whose definition comes from (19a) Zeff
= -(DNa/DP)(Dnh - & ’ ) / ( h a- Dnh)
(20)
is calculated iteratively as to be consistent with the effect of activity coefficients in eq 19. The direct application of eq 20, taking for Dnhthe observed on),*, yields values of the charge which are much lower in absolute value: for heparin the result is 2 = -1 1, and for chondroitin Z = -20. These values were used for the finite-difference and normal-mode calculations. We have found that the PBC calculation leads to a constant value, as a function of r, for the effective charge of the polyions. This result is consistent with Manning’s theory. Moreover, the values derived from these procedures are nearly identical, as was found previously by various worker^.^^^^^ These two results are fairly consistent with the experimental corrected value for heparin, but a large discrepancy is observed in the case of chondroitin sulfate. The case of chondroitin seems difficult to interpret. At least in one previous such a difficulty was found experimentally for chondroitin sulfate: the results for the measurement of the apparent molecular weight, from light scattering experiments and analytical centrifugation (Archibald’s method, sedimentation equilibrium, and sedimentation velocity), were found to be strongly concentration dependent, with a maximum at about 1 g/L. The authors concluded the existence of association of molecules of chondroitin. This phenomenon may be at the origin of the discrepancy observed in our results. The precise explanation would require further studies of the associative behavior of chondroitin solutions, and its influence on the physical parameters measured by different techniques. 4.2. Apparent Diffusion Coefficient. The transition of the diffusive modes of the polyion/counterion system, from collective to individual diffusion, as salt is added to the polyelectrolyte solution, is shown in Figures 1 and 2. The agreement of the theoretical predictions with the experiments is qualitative on the whole, but generally not quantitative; as expected, the diffusive coupling between the species is stronger when no salt is present outside the capillary; on the other hand, the finite difference curves accounting for the apparent diffusive mode of the sodium ion and of the polyion are in each case above the experimental points; taking into account properly the departures from ideality, their influence on the transport properties of the system should reduce these deviations. Unfortunately, the complexity of both the system and the process makes it impossible to anticipate the sign and order of magnitude of such corrections. Further careful studies are needed to achieve this treatment. An appreciable increase of the apparent diffusion coefficient of the polyion is noticed, especially with chondroitin sulfate, when the ratio r is increased beyond a critical value. This artifact is due to a partial decondensation of the strontium ions caused by a screening effect created by the ionic atmosphere around the macroions on which the labeling strontium ions were previously (22) Wilson, R. W.; Rau, D. C.; Bloomfield, V. A. Biophys. J . 1980, 30, 317. (23) Klein, B. K.; Anderson, C. F.; Record, Jr., M. T. Biopolymers 1981,
20, 2263. (24) Lemerle, J . ; Magdelenat, H.; Lefebvre, J. Bull. SOC.Chim. Fr. 1974, 5-6, 753.
Simonin et al. condensed. This phenomenon can be qualitatively interpreted by the PBC procedure; taking the concentration of strontium ions equal to lo4 mol/L and the data from Table I yields respectively, for r = 0, 2, 4, rates of free strontium ions of 6 X IO4%, 38%, and 58%. Such an effect was observed previo~sly.~~ It was shown that in conditions of a very large excess of sodium chloride, corresponding to r 2 100, the strontium ions are completely free. It is seen in Figure 1 that no decondensation phenomenon seems to occur in the case where the outside vessel is filled with pure water; we suppose that, in this case, the effective rate of the salt concentration to the equivalent concentration of the macroion, on the length scale of the capillary on which the diffusion process effectively occurs, is much less than the ratio r of the initial homogeneous distribution of matter. It is observed that the diffusive process of the polyions is modified by the effect of the coupled diffusion arising from the counterions. Combining the results provided by the experimental and theoretical curves, it can be said that this phenomenon occurs up to high rates of added salt. A similar result was found in ref 6. Acknowledgment. This work was performed under the technical guidance of N. Pruliere. Appendix The index 1 refers to the polyion, index 2 to the sodium ion, and index 3 to the chloride ion. The quantity y, defined by eq 1, can be written for the polyion and the sodium ion, as YI = g,,,s1+ g,,2s2
(A. 1)
with m
sl =
+ 1)2
k=O
q k = (2k
+ 1)~/(2L)
(A.2)
(‘4.3)
s2is defined by the same expression as (A.2) replacing D+ by D-, D, and D- being the normal diffusive modes of the system 3
D* = ( 1 / 2 ) ( 2 D I ( l - ti)
f
Do)
(‘4.4)
I=I
where t , is the transport number of species i, and
Do = [ C ( ( D ,- D,)’tk2 P
+ 2(D, -D,)(D,- D k ) t , t k ) ] ’ / 2
(AS)
in which p means that the summation is made on the three cyclic permutations on the indices { 1, 2, 31, and i = p ( l), j = p ( 2 ) , k = P(3). Lastly, in (A.l), the g,,’s are gI,l
g1,2= (-@3 g2.l
+ r t , ) - (D3 - D + ) ) / D , - D 2 ) ( t 3+ r t l ) + (D3 - D - ) ) / D o
= ( ( 4 - D2)(t3
= ((O3 - Dl)(t3
+ r’r2) - ( D 3
- D+))/DO
g2.2 = (-(D3 - Dl)(t3 + r’r2) + (O3- D - ) ) / D O r being defined by eq 18, and r’= r / ( r
+ 1)
(-4.6) (A.7) (A.8) (A*9) (A.lO)
is the ratio of the added salt concentration to the total sodium ion concentration. The apparent diffusion coefficients are next calculated from eq 2 and 3. Registry No. Chondroitin sulfate, 9007-28-7; heparin, 9005-49-6. ( 2 5 ) Magdelenat, H.; Turq, P.; Chemla, M. Biopolymers 1974, 13, 1535.
