Macroscopic counterion diffusion in solutions of cylindrical

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J . Phys. Chem. 1985,89, 3385-3391

3385

Macroscopic Counterion Dlffusion in Solutions of Cylindrical Polyelectrolytes Lars G. Nilsson,* Lars Nordenskiold, Division of Physical Chemistry, Arrhenius Laboratory, University of Stockholm, S- 106 91 Stockholm, Sweden

Peter Stilbs, Institute of Physical Chemistry, Uppsala University, S - 751 21 Uppsala, Sweden

and William H. Braunlin Physical Chemistry 2, Chemical Center, University of Lund, S-221 00 Lund, Sweden (Received: August 27, 1984; In Final Form: February 28, 1985)

Within the Poisson-Boltzmann approximation and the cylindrical cell model of polyelectrolytes the ratio of the effective and the "free" counterion diffusion constants, DIDo,has been derived. If the surface charge of the macroion is regarded as being uniform, the diffusion quotient of an isotropic solution is DIDo = + '/3DL/Do, where D, is the diffusion constant perpendicular to the macroion. An expression for D, is obtained by using the method of Bell to solve the stationary-state Smoluchowski diffusion equation. 'Li FT NMR self-diffusion experiments have been performed on double helical calf thymus LiDNA solutions of different salt concentrations (LiCl and MgC12). The observed diffusion quotients are smaller than their values in DNA-free solution but approach unity upon titration with salt. Numerical calculations, based on the PoissonBoltzmann-Smoluchowski model, of lithium diffusion in these DNA systems have been performed. Calculations have also been performed of sodium and calcium ioh diffusion pertinent to previous tracer diffusion measurements in solutions of DNA, chondroitine sulfate, and polyacrylate. The calculated monovalent diffusion quotients are found to agree satisfactorily with experimental values while the experimentally observed small diffusion quotients for the divalent calcium ions in DNA and chondroitine sulfate solutions cannot be reproduced with this model.

Introduction It is well established that long-range electrostatic interactions are of considerable importance for many properties of Solutions containing charged polyions,' micelles? biological membra ne^,^ and charged aggregates in general. Nucleic acids, for example, double helical DNA, are highly charged polyelectrolytes. As such, their flexibility, conformation, stability of secondary and tertiary structure, and binding with charged ligands (e.g., proteins) are largely governed by these long-range electrostatic force^.^-^ A consequence of the highly charged nature of DNA is that the small-ion distribution near and between DNA molecules to a large extent determines the physical and chemical properties, thermodynamic as well as dynamic, of DNA systems. In order to theoretically describe the ion distribution it is most convenient to employ the cylindrical cell model,' which results in well-defined, electroneutral subsystems containing one macroion, their shapes being time independent. The accumulation of counterions near the polyion is then usually described within the Poisson-Boltzmann (PB) theory! Alternatively, the accumulation of counterions in the vicinity of DNA and other cylindrical polyelectrolytes can be described by Mannings counterion condensation (CC) theory? Much experimental work has been directed at quantifying the extent of counterion accumulation near double helical and the results have been interpreted within (1) Oosawa, F. 'Polyelectrolytes"; Marcel Dekker: New York, 1971. (2) Wennerstrbm, H.; Lindman, B. Phys. Rep. 1979, 52, 1. (3) McLaughlin, S.A. "Current Topics in Membrane Transport"; Bronner and Kleinzeller, Eds.; Academic Press: New York, 1977; Vol. 9. (4) Record, M. T., Jr.; Mazur, S. J.; Melancon, P.; Roe, J.-H.; Shaner, S. L.; Unger, 1. Annu. Reu. Biochem. 1981, 50, 997. (5) Anderson, C. F.; Record, M. T., Jr. Annu. Reu. Phys. Chem. 1982,33, 191. (6) Anderson, C. F.; Record, M. T., Jr. "Structure and Dynamics: Nucleic Acids and Proteins"; Clementi, E., Sarma, R. H., Eds.; Adenine Press: New York, 1983; pp 301-319. (7) Hill, T. L. 'Statistical Thermodynamics"; Addison-Wesley: Reading, MA, 1960. (8) Katchalsky, A. Pure Appl. Chem. 1971, 26, 327. (9) Manning, G. S. Acc. Chem. Res. 1979, 12, 443. (10) Manning, G. S.;Q. Reo. Biophys. 1978, Z I , 179. (11) Anderson, C. F.; Record, M. T., Jr.; Hart, P. A. Biophys. Chem. 1978, 7, 301. (12) Bleam, M. L.; Anderson, C. F.; Record, M. T., Jr. Proc. Narl. Acad. Sci. U.S.A. 1980, 77, 3085.

either the PB or the CC model. In order to describe equilibrium properties of polyelectrolyte systems with different geometries the Poisson-Boltzmann equation has been extensively used:

In this equation # denotes the average electrostatic potential, while ni and zi are respectively the number concentration at the cell boundary and the valence of ion species i. elto is the permittivity which is taken to be uniform throughout the entire system. T is the absolute temperature, k is the Boltzmann constant, and e is the elementary charge. PB theory has been found to give a qualitatively very good description of the ion-binding properties of a large number of systems."J6-'* For monovalent ions the PB ion distribution usually is in very good agreement with results from Monte Carlo simulations.'+22 The agreement is less satisfactory for multivalent ions due to ion correlation effect^.^,'^ Thermodynamic properties which are determined by the ion concentration at the cell boundary, such as ion activities and osmotic pressure, seem to be less well described than the ion d i s t r i b u t i ~ n . ~ ~ ~ ~ J ~ How the simple PB description suffices to describe dynamic observables in polyelectrolyte systems, e.g., counterion diffusion over macroscopic distances, has been less well studied. In 1962 Lifson and Jacksonz5 derived an expression for the macroscopic (1 3) Murk Rose, D.; Bleam, M. L.; Record, M. T., Jr.; Bryant, R. G. Proc. Natl. Acad. Sei. U.S.A 1980, 77, 6289. (14) Bleam, M. L.; Anderson, C. F.; Record, M. T., Jr. Biochemistry 1983, 22, 5418. (15) Braunlin, W. H.; Nordenskibld, L. Eur. J . Biochem. 1984, 142, 133. (16) Manning, G. S. J . Chem. Phys. 1975,62, 748. (17) McLaughlin, S. A.; Szabo, G.; Eisenman, G. J . Gen. Physiol. 1971, 58, 667. (18) Wennerstrbm, H.;Lindman, B.; Lindblom, G.; Tiddy, G. J. T. J . Chem. SOC.,Faraday Trans. I 1979, 75, 663. (19) Torrie, G. M.; Valleau, J. P. Chem. Phys. Leu. 1979, 65, 343. (20) Jonsson. B.; Wennerstrbm, H.; Halle, B. J . Phys. Chem. 1980, 84, 2179. (21) Bratko, D.; Vlachy, V. Chem. Phys. Leu. 1982, 90, 434. (22) Le Bret, M.; Zimm, B. H. Biopolymers 1984, 23, 271. (23) Fixman, M. J . Chem. Phys. 1979, 70, 4995. (24) Guldbrand, L.; Jonsson, B.; Wennerstrom, H.; Linse, P. J . Chem. Phys. 1984, 80, 2221.

0022-3654/85/2089-3385$01.50/0 0 1985 American Chemical Society

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The Journal of Physical Chemistry, Vol. 89, No. 15, 1985

counterion self-diffusion quotient in solutions consisting of parallel plane polyelectrolytes by the mean first passage time method. They considered the plane polyelectrolytes to be infinitely large, thus reducing the dimensionality of the problem to one. In effect their method amounts to a direct integration of the Smoluchowski mean field equation26

Heref(q,t)dq is the probability of finding a tagged ion within the volume element dq at q at time t, and Do is the diffusion constant in the absence of the external field produced by the macroions. When considering two- or three-dimensional cases the Smoluchowski equation is not separable, thus making the method of mean first passage times less tractable. Instead Jackson and C0rie11~~ considered a system of parallelepipedic cells with an imposed constant macroscopic gradient of tagged counterions, thus turning the problem into one of stationary-state diffusion. and Bell and Dunning29 applied this stationary-state diffusion method to particles and cells of spherical symmetry. A theory of counterion self-diffusion in solutions of cylindrical polyelectrolytes based on the CC theory has been presented by Manning.3o Yoshida31 adopted the cylindrical cell model and applied Mannings method to derive an expression for the counterion self-diffusion quotient. The result was then applied to salt-free polyelectrolyte solutions together with the analytical Poisson-Boltzmann result to obtain a limiting law expression for the counterion self-diffusion coefficient. Recently Belloni et al.32extended the treatment of Bell and D ~ n n i n g ~to* the , ~ ~cylindrical case, but made the erroneous assumption that the cell-averaged diffusion constant is independent of the direction of the cell relative to the gradient of tagged counterions. In this paper we present the correct result for macroscopic counterion self-diffusion in solutions of cylindrical polyelectrolytes, in terms of the PB cell and stationary-state Smoluchowski models. Macroscopic counterion diffusion in cylindrical polyelectrolyte solutions has with one exception33been measured by radioactive tracer methods.3443 Huizenga et al.34 studied the sodium counterion self-diffusion in sodium polyelectrolyte solutions at different degrees of neutralization. In a series of important experimental papers Ander and c o - w o r k e r ~have ~ ~ ~studied ~ the self-diffusion of counter- and co-ions in different polyelectrolyte solutions in the presence of added salt. In one of these papers sodium and calcium diffusion was measured in NaDNA solutions.j* In all these papers the experimental results were compared with the predictions of Mannings diffusion theory.30 Turq and c o - ~ o r k e r shave ~ ~ , reported ~~ results on counterion self-diffusion of Na+, Ca2+,Sr2+,and La3+ in solutions of chondroitine sulfate. The tracer method is a time consuming procedure which requires a suitable isotope labeling and also the system rheology may cause problems. In comparison, the FT NMR pulsed gradient

(25) Lifson, S.; Jackson, J. L. J. Chem. Phys. 1962, 36, 2410. (26) Chandrasekhar, S. Rev. Mod. Phys. 1943, 15, 1. (27) Jackson, J. L.; Coriell, S. R. J. Chem. Phys. 1963, 38, 959. (28) Bell, G. M. Trans. Faraday SOC.1964,60,1752. (29) Bell, G. M.; Dunning, A. J. Trans. Faraday SOC.1970. 66, 500. (30) Manning, G. S. J. Chem. Phys. 1969, 51, 934. (31) Yoshida, N. J. Chem. Phys. 1978, 69,4867. (32) Belloni, L.; Drifford, M.; Turq, P. Chem. Phys. 1984, 83, 147. (33) Stilbs, P.; Lindman, B. J. Magn. Reson. 1982, 48, 132. (34) Huizenga, J. R.; Grieger, P. F.; Wall, F. T. J. Am. Chem. SOC.1950, 72, 4228. (35) Dixler, D. S.; Ander, P. J. Phys. Chem. 1973, 77, 2684. (36) Menezes-Alfonso, S.; Ander, P. J. Phys. Chem. 1974, 78, 1756. (37) Kowblansky, M.; Ander, P. J. Phys. Chem. 1976,80, 297. (38) Trifiletti, R.; Ander, P. Macromolecules 1979, 12, 1197. (39) Ander, P.; Casiero, M.; Geer, T.; Leung-Louie, L.; Sapjeta, J.; Savner, S . Marcomolecules 1982, 15, 1333. (40) Ander, P.; Kardan, M. Macromolecules 1984, 17, 243 1. (41) Ander, P.; Kardan, M. Macromolecules 1984, 17, 2436. (42) Magdelenat, H.; Turq, P.; Chemla, M. Biopolymers 1974, 13, 1535. (43) Magdelenat, H.; Turq, P.; Tivant, P.; Chemla, M.; Menez, R.; Drifford, M. Biopolymers 1979, 18, 187.

Nilsson et al. TABLE I: Lithium Self-Diffusion Coefficients in Calf Thymus DNA at Various LiCl Contents [PI, mM 15.2 15.2 15.2 15.1 15.1 15.0 14.9 14.8 14.5

0

[LiCl], mM 8.65 13.4 20.4 29.8 41.5 57.6 80.3 102.7 146.5 100.0

[LiCl]/[P]"

D,lo9 m2 s-l

0.567 0.879 1.346 1.970 2.749 3.840 5.399 6.960 10.075

0.713 f 0.013 0.759 f 0.019 0.810 f 0.005 0.864 f 0.013 0.882 f 0.005 0.968 f 0.01 1 1.028 f 0.029 1.027 f 0.014 1.044 f 0.014 130 = 1.065 f 0.01 1

"Salt equivalents per phosphate.

TABLE II: Lithium Self-Diffusion Coefficients in Calf Thymus DNA and LiCl at Various M E l , Contents

[PI, mM

[LiCll, mM

15.2 15.2 15.2 15.1 15.0 14.8 14.6 14.4 14.2 0

8.65 8.64 8.61 8.56 8.50 8.41 8.30 8.18 8.08 100.0

[MgC121, (2[MgCl,l + [LiCI])/[P]' mM 0.567 0 0.142 0.586 0.623 0.426 0.989 0.698 1.684 0.792 2.707 0.932 4.039 1.119 1.306 5.335 6.600 1.494

0

D,lo9 m2sY1 0.713 f 0.013 0.727 f 0.026 0.751 f 0.033 0.809 f 0.013 0.867 f 0.044 0.966 f 0.014 0.980 f 0.018 1.024 f 0.015 1.068 f 0.032 Do = 1.065 f 0.011

"Salt equivalents per phosphate.

self-diffusion method,33,44,45 briefly described below, has great advantages. It is a considerably faster method applkable to systems of different rheology. It is also possible to monitor the self-diffusion of several different species in the same system in one experiment. Undoubtedly, the greatest advantage is the direct correspondence between the experimental observable and the macroscopic self-diffusion constant as it is defined. In this paper we present the results of 7Li FT N M k self-diffusion measurements on calf thymus LiDNA solutions with different amounts of added LiCl and &lgCl2. The experimental results are compared to diffusion quotients calculated from the steady-state Poisson-Boltzmann-Smoluchowski (PBS) cell model. We also present calculations on other polyelectrolyte systems for which tracer counterion self-diffusion data are available in the literature. Materials and Methods Calf thymus LiDNA was purchased from Worthington Co. and prepared as described e l s e ~ h e r e . ' IBriefly ~ ~ ~ the procedure involved room temperature phenol and ether extractions to remove protein contaminants, an ethanol precipitation step, followed by sonication and extensive dialysis, first against EDTA to remove divalent cations and then *against solutions of progressively lower LiCl concentrations. The DNA concentration was determined from the absorbance at 260 nm, and the concentration of lithium ions was determined by atomic absorption spectrometry, the results being 15.2 mM DNA phosphate and 23.8 mM lithium. Titrations were carried out by adding microliter amounts of LiCl and MgC12 to final concentrations as given in Tables I and 11. 7Li FT N M R self-diffusion measurements were carried out at 25 f 0.5 OC on a JEOL FX-100 spectrometer at 38.7 MHz 7Li resonance frequency. The samples were contained in 8-mm sample tubes which were placed in IO-" tubes with D 2 0 in the annulus as an external deuterium lock. This method of spin diffusion measurements by FT pulsed gradient spin-echo experiments is based on James and McDon a l d ' ~FT~ ~N M R implementation of Stejskal's and Tanner's47 (44) Stilbs, P.;Moseley, M. E. Chem. Scr. 1980, 15, 176. (45) Stilbs, P. J. Colloid Interface Sci. 1982, 87, 385. (46) James, T. L.; McDonald, G. G.J. Magn. Reson. 1973, 11, 58.

The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 3387

Cylindrical Polyelectrolytes pulsed field-gradient method, and further developed by Stilbs and M ~ s e l e y . ~ It. ~utilizes ~ the ordinary homospoil coils of the spectrometer for generating the softwarecontrolled magnetic field gradient pulses which in our experiments were around 1 G cm-’ and did not vary more than *OS% in succesive experiments. The calibration of G was carried out as described in ref 44. A series of absorption spectra with different gradient pulse lengths is generated, and the amplitudes follow the relation A

a

exp(-y2G2b2D(A- 6/3))

Y

f

(3)

where y is the magnetogyric ratio of the nucleus, G is the magnetic field gradient, 6 is the gradient duration, D is the self-diffusion constant, and A is the common 90-180° rf and gradient pulse interval. The yalue of A was 200 ms for all durations, 6, of the symmetrically placed gradient pulses which varied from 40 to 180 ms. Under cdhstant rf pulse interval conditions, T2 attenuation effects on the echo are constant in this experiment. The diffusion constant, D, was calculated from eq 3 off-line with an iterative nonlinear least-squares method. For more details about the pulsed field gradient diffusion experiment in the present low-gradient high-resolution form, and about the evaluation of the diffusion coefficient, thb reader is referred to the work of Stilbs and M ~ s e l e y . ” . ~The ~ “free” diffusion constant Do was measured in a 0.1 M LiCl solution and was found to be 1.07 X 10” m2 6’ at 25 OC.

Theory an8 Computing Methods Consider a cell in a laboratory fixed coordinate system, as indicated by the superscript L. According to Fick’s first law, we have for the cell-averaged flow of tracer ions J-

= -GLDL

(4)

GL is the macroscopic gradient of tracer ions, i.e., the gradient at the cell surface in a laboratory direction, and DL is the cellaveraged, 3 X 3 (nondiagonttl) representation of the diffusion tensor. The cell-averaged flow expressed in molecular coordinates, p,is trdnsformed to laboratory coordinates by

J- = J ~ D ( ~ ) ( Q )

(5)

t’

where D(’)(Q)is the W i p e r rotation matrix. Q denotes the Euler angles for rotating the molecular coordinate system into the laboratory system. Usihg Fick’s law for and transforming GM into GL gives

J- = -GMDMD(”(Q)= -GLD(’)(-Q)DMD(’)(Q)

(7)

For a cylindrical cell DMhas the diagonal elements D,, D and DL,and zero-valued off-diagonal elements. 11 and I re& to directions parallel and perpendicular to the cylinder *is, respectively. Averaging over all Euler angles we find that (Dki) = (Y3Df: + 73DWkI

(8)

and thus, for an isotropic solution DL =

Y3DY

+ 73DY

(9)

When end effects are neglected, i.e., the cylinders are treated as infinitely long, and the macroion charge is regarded as uniformly smeared out Over the macroion surface, there will be no periodic axial potential gradient. Assuming the diffusion constant to be independent of concentration, the axial diffusion constant, D,,,will equal the “free” diffusion constant, Do. This means that the diffusion constant in the region near the polyion, where one also has a large deviation from electroneutrality,is assumed identical with the value of Do in a dilute electroneutral solution. Note that since the cylinders are infinitely long they will not affect the axial diffusion and therefore Do is the diffusion constant in the corre(47) Stejskal, E. 0.; Tanner, J. E. J. Chem. Phys. 1965, 42, 288.

x -L

/

Figure 1. The cylindrical cell. u is the cell radius, b the macroion radius, and xo the center-of-cell x coordinate. r, a,and z are the cylindrical coordinates.

sponding symmetrical electrolyte solution. Dropping the superscripts, eq 9 then reduces to D/DO = 73 + 7 3 ( D L / D O )

(10)

Thus, under these assumptions, it is necessary only to calculate the anisotropic diffusion quotient, DJDo, in order to find the isotropic diffusion quotient, DIDo. In order to do this we now employ the stationary-state method as originally done by Bell,”J9 but in our case for cylindrical symmetry. For details on the method of derivation the reader is referred to Bell’s original work. The stationary-state Smoluchowski equation can be written 0.j = 0

(6)

Then it follows from eq 4 that

W = D(~)(-Q)~D(~)(Q)

4

j = -Do(Vn

(11)

+ znV4)

(12)

in which the Einstein relation, Do = k T / q , has been used. j is the (microscopic) flow, z the valence, n the number concentration, and q the hydrodynamic friction coefficient of the tracer ions. 4 = e$/kT is the reduced potential. In Figure 1 is shown the cylindrical cell with radius a containing the macroion with radius b. Impose on this system a macroscopic gradient, G,of tracer ions and an opposite gradient, -G, of nontracer ions of species i. Let this gradient be macroscopically constant in the x direction, i.e., let n(a,a) = G(xo a cos a),where xo is the center-of-cell x coordinate and a is the azimuthal angle as defined in Figure 1. It is then reasonable to try to satisfy the stationary-state Smoluchowski equation with a solution of the form n(r,a) = f o ( r ) + f ( r ) cos a (1 3)

+

where = Gxo f l u ) = Ga

(14a) ( 14b)

We regard the macroions as being nonpenetrable, and therefore the radial flow at the macroion surface must be zero, j,(b) = 0. From eq 12 and 13 we obtain the boundary conditions at the macroion surface: fo’(b) + zf,(b)4’(b) = 0 f(b)

+ zflb)4’(b) = 0

(154 (15b)

3388 The Journal of Physical Chemistry, Vol. 89, No. 15, 1985

Nilsson et al.

where x(a) is determined from the fit-order nonlinear differential equation x' + X(X/r - zr&) - 1 / r = 0 (22) under the boundary condition x(b) = 0

Figure 2. The flow through the surface S2perpendicular to the x axis is obtained when integrating the radial flow over the remaining cylindrical cell surface SIaccording to eq 19. The cell averaged flow, cq 20, then follows when integrating over x from -a to u and dividing by the volume.

Here prime indicates derivative with respect to r, as in the following. Taking the divergence of ( 12), transforming to cylindrical coordinates, and separating variables gives two coupled ordinary differential equations, one determining the radial and one determining the azimuthal factors in the two terms describing the ion distribution within the cell, eq 13. One of these radial factors, fo,is easily solved for, giving the tracer ion distribution as n(r,a) = Gxo exp(-zc#J) + A r ) cos a (16) where f is determined by (rf'+ zrj"')' - f / r = 0

(17)

under the boundary cdhditions (14b) and (15b). The first term in (16) gives the equilibrium distribution and the last term is the perturbation caused by the macroscopic gradient. The macroscopic diffusion constant, D,, is defined as J, = -D,G, where J, = (j,)and G = a(n)/dx0. Only the equilibrium term in eq 16 contributes to the average concentration: (i,) = -DLG(exp(-zd))

(23)

In eq 21 c denotes the total concentration of species i, i.e., both tracer ions and nontraser ions. The diffusion quotient is thus directly related to the ratio between the concentration at the cell boundary and the mean concentration in the cell. This quantity is small a t low salt content but, if the macroion radius is small compared to the cell radius, approaches unity at large quantities of added salt. Under these conditions the quantity x(a) also approaches unity with added salt and is of the order of 1.5 in the low salt cases. The result given above in eq 21-23 for DJD0 (the second term in eq 21) is, if z/3 is substituted by unity, identical with the final result for DIDo obtained by Belloni et al.32(their eq 3.17). Thus the contribution from DI,/Dbwhich in our model is 1/3, is neglected by these authors. This is due to an erroneous result that the cell-averaged diffusion constant is independent of the angle between the gradient of tagged ions and the orientation of the cell. It is not clear to us how the authors of ref 32 arrived a t this obviously faulty conclusion. The consequence of this error is a systematic underestimation of the diffusion quotients. If we compare our result, eq 21, with that obtained by Yoshida (eq 24 in ref 31), it can be shown by making a suitable variable transformation that the two results are identical. Yoshida uses the PB cell model, but his calculation of the diffusion quotient, which follows Manning,'O is different from ours but yields the same result. It should however be pointed out that our general and perhaps intuitively obvious result, eq 9, does not follow from Yoshida's derivation. Within the cylindrical cell model eq 9 is valid for any potential. It can thus be used as a starting point for a generalization of eq 21 to include the effects of a discrete charge distribution on the polyion. Furthermore, in his calculations of diffusion quotients Yoshida uses his salt-free PB limiting law expression of his eq 24. Turning again to eq 21 it is of interest to note that due to the uniform charge approximation there is a lower limiting value of the diffusion quotient of 1/3. It should also be noted that the diffusion quotient can never reach unity, not only as a consequence of the electric field, but also as a result of the physical obstacles the macroions constitute. Consider the limiting cases of uncharged macromolecules or very high salt content, i.e., when $ is zero everywhere ih the free volume. Equation 17 is then easily solvable, giving x ( a ) = (1 - ( b / ~ ) ~1)+/ ( b / ~ ) ~In) this . limiting case we also get c ( a ) / ( c ) = 1/(1 - ( b / a ) z )and thus

(18)

From the Smoluchowski eq 12 and from the expression for the tracer ion distribution (16) we can also obtain this cell-averaged flow by applying Gauss' divergence theorem, which simply states that there will be no accumulation or depletion of tracer ions in the cell under stationary-state conditions: JVj-dV = l j r dS1 + Sj, dS2 = 0

(19)

S1and Szdenote the surfaces as shown in Figure 2. We integrate

the radial flow at the cell surface, given from (12) and (16 ) , over

SIand then integrate (19) over x and divide by the volume, resulting in:

ci,) = -D$'(a) (20) The anisotropic diffusion quotient, Dl/Do, is now obtained froin (18) and (20). In order to eliminate the dependence on the gradient we make the substitution X(r) = rf'(r)/Ar) zr$'(r), and according to (10) we finally conclude for the isotropic case:

+

We note that the limiting behavior as the macromolecule radius approaches zero, or as the dilution approaches infinity, is correct, the diffusion quotient becoming the free value of unity. On the other hand, when the quotient b / a approaches its maximum value of unity, the diffusion quotient eventually equals 2/3. This is a reflection of the fact that while radial motion becomes increasingly restricted, the tangential and of course also the axial motions are unaffected. In Figure 3 this obstacle effect on the diffusin quotient is shown as a function of DNA concentration. The effect is negligible at monomer concentrations below 50 mM but is considerable at 500 mM and above. The potentials necessary to calculate the theoretical diffusion quotients were obtained froin a fourth-order RungeKutta solution of the Poisson-Boltzmann eq 1. The input values for the cell boundary concentrations of the different ions were obtained interatively by the Newton-Raphson method. This procedure works in general very well, as long as the salt content is not too high. In these cases it was more convenient to manually control the input

The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 3389

Cylindrical Polyelectrolytes

4 .0

0 .0

\

0.6 0" \

0

u o0 . 0

0 .OO 0

200 400 600 000 DNA concentrotcon 1 m M I

2

0

s

6

4

2

40s

0

4000

Figure 3. Counterion self-diffusionquotient in an isotropic DNA solu-

tion, hypothetically consisting of uncharged DNA molecules, vs. DNA concentration, according to eq 24. The DNA radius is 10.0 A. concentrations until sufficient precision was obtained. The solution of the PB equation gives both the concentration at the cell boundary, c(a), appearing in (21), as well as the derivative of the potential, which was used in a fourth-order RungeKutta solution of the first-order nonlinear differential eq 22 to get the dimensionless quantity ~ ( a ) The . diffusion quotients were all calculated to a precision of better than 0.2%, which normally required 500 step or more in the solution of the PB equation. In all calculations the relative dielectric constant E , was assumed to .be 79.9. For DNA we used the temperature 298.2 K, the radius of DNA was set to 10.0 A, and the surface charge density to -0.150 C m-*. For chondroitine sulfate the corresponding quantities were set to 293.6 K,6.00 A, and -0.0680 C m-2, respectively, and for poly(acrylic acid) to 298.2 K, 3.00 A, and -0.340 C m-z (at 100% neutralization), respectively. RWultS

The experimental 'Li self-diffusion constants in LiDNA solutions titrated with LiCl and MgCl2 as measured by the FT NMR technique briefly outlined above are shown in Tables I and 11. As is to be expected, the lower the salt content the more reduced is the d h i o n constant, a clear reflection of the decreasing surface potential. Upon titration with salt the diffusion constant increases toward D,,, the diffusion constant in a DNA-free solution (obstacle effects neglected). The lithium diffusion constant increases much more rapidly (on a per charge as well as per molar basis) upon titration with the divalent counterion Mg2+ than is the case for the LiCl titration. The calculated self-diffusion quotients for the LiDNA solutions with different amounts of added LiCl and MgClz are shown in Figures 4 and 5 together with the experimental results. In Figure 4 are also shown the diffusion quotients calculated from Manning's theory of counterion diffusion in cylindrical polyelectrolyte solutions, based on the CC model (eq 32 and 39 of ref 30). It is seen that the theoretical PBS results agree quite satisfactorily with the experimental ones over a wide range of added salt. The quotients calculated from Manning's model agree less satisfactory, the diffusion quotients being consistently too small compared to those from experiment. Trifletti and Ander3*measured the sodium tracer diffusion in NaDNA/NaCl solutions as a function of the ratio of phosphate to salt concentration, but at constant salt concentration. Curves similar to our Figure 4, but over a wider composition range, were then obtained. These results cannot be exactly compared to ours because the composition of the solutions at a given ratio was different. However, if the resulting diffusion quotients are compared only as a function of the ratio of phosphate to added salt, our curve in Figure 4 is very similar to the corresponding curves obtained by these authors.

SoCt equcvoCmnts pmr phoophote

Figure 4. Lithium self-diffusion quotients vs. salt equivalents per phosphate, [LiCl]/[P], during titration of 15.25 mM calf thymus LiDNA with LiCl. Solid curve is our theoretical PBS results, dashed curve is the Manning CC diffusion quotients, and bars are our experimental results (Table I).

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3390 The Journal of Physical Chemistry, Vol. 89, No. 15, 1985

Nilsson et al. 4

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Figure 8. Sodium self-diffusion quotients vs. percent NaOH neutralization of poly(acry1ic acid), PAA. Solid curve is the calculated PBS result and bars are the experimental tracer ion results of Huizenga (ref 34).

the “condensed” counterions have no mobility at all, i.e., that there is no axial diffusion of those ions along the polyion. On the other hand, the contribution to the total diffusion quotient from the uncondensed ions is calculated in principally the same way as we do here, but the Debye-Hiickel potential is used instead of the 0 .e full PB potential. This should therefore give an overestimated contribution. The deviation of the total diffusion quotient, as compared to experimental values, is however dominated by the 0.6 immobility of the condensed ions. / Dixler and Ander3$have studied the salt dependence of sodium 0” \ diffusion in sodium polyacrylate/NaCl solutions by the tracer c1 method. The salt dependence observed in this system showed the 0.4 same qualitative behavior as our data for DNA. In this case Mannings modePo was found to be in good agreement with the experiments. I 0.2 For the titration of LiDNA with MgC12 (Figure 5) the i I agreement between calculated and experimental Li+ diffusion quotients is likewise satisfactory. As already mentioned, the 0.0 . .1 . .1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lithium diffusion constant increases much more rapidly than is the case for the LiCl titration. This is a consequence of the fact 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 i . 0 1.2 4.4 that preferentially multivalent counterions accumulate near the SoCt equcvoCsnts per suCphate charged DNA, thereby increasing the entropy of the system. The Figure 7. Sodium and calcium diffusion quotients vs. salt equivalents per monovalent counterions are in this way to a large extent forced sulphate, 2[CaCl,]/[SO,], during titration of 3.40 mM sodium choninto the bulk, and a smaller fraction of these counterions will be droitine sulfate with CaCll. The upper curve and bars are the theoretical exposed to the large potential gradient near the DNA surface. PES sodium and the experimental (ref 42 and 43) tracer sodium diffusion The magnesium titration data show small but qualitatively quotients, respectively. The lower curve and bars are the corresponding calcium values. different deviations from the theoretical predictions when compared to the lithium titration. The calculated PBS diffusion model are both expected to result in overestimations of the quotients fall above the experimental ones in the beginning of the counterion diffusion quotients. The PB mean field approximation titration but below after the addition of some magnesium. This which amounts to neglecting the ion correlation e f f e c t ~will ~ ~ , ~ ~ might possibly be a reflection of the fact that correlation effects, result in an underestimation of the counterion concentration in neglected in the PB mean field approximation, are much more the vicinity of the charged surface and thus an overestimation of important for divalent than for monovalent ions. the diffusion quotient. Furthermore, the assumption of a uniform For the self-diffusion of divalent counterions (Ca2’) in the macroion charge distribution of course also results in an overNaDNA and sodium chondroitine sulfate systems the situation estimated diffusion quotient. Both of these approximations are is very different (Figures 6 and 7). Here it is clear that the model fails. At low amounts of added calcium it can be seen that the expected to be of minor importance at high salt concentrations, a reflection of the fact that added salt mainly goes into the bulk experimental calcium diffusion quotients are very small and in solution as well as the fact that the shielding of the DNA phosfact approach zero at very low ratios of salt equivalents to polyion phate charges increases. In the hypothetical case of no, or very monomer concentrations. As previously mentioned, a consequence little, added salt and a still existing double helical DNA structure of the uniform-charge assumption is that diffusion quotients of there is reason to assume a considerable reduction of axial difless than 1/3 are impossible within the present model. This of fusion. course holds regardless of correlation effects. Therefore, in order The diffusion quotients calculated from Manning’s modePo to describe self-diffusion of multivalent counterions in solutions (dashed curve in Figure 4) are considerably smaller than the of cylindrical polyelectrolytes it seems necessary to properly account for the axial diffusion. Only at high concentrations of added experimental results. The same trend was also found by Trifiletti and Ander for their sodium diffusion quotients in NaDNA/NaCl salt can this uniform-charge approximation be realistic. In the solutions.38 This effect is probably due to the approximation that low and intermediate concentration range correlation effects should 4.0

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Cylindrical Polyelectrolytes also be important. It should also be mentioned that for the calcium ion diffusion in DNA solutions, Manning’s diffusion model instead underestimates the diffusion quotients considerably at intermediate and low ratios of salt equivalents to polyion monomers.38 Belloni et al.32used their erroneous result (see the theoretical section above) to calculate corresponding curves to those given in our Figure 7. For calcium ion diffusion they obtained a satisfactory agreement with the experimental points but that is purely coincidental and due to the underestimation (within their model) of the diffusion quotients. An improvement of the model for the electrostatic potential to take into account the discrete nature of the polyion charge distribution would certainly improve the calculated diffusion quotients for Ca*+. However, this may probably not be enough to account for the very low diffusion quotients at small amounts of added CaC12. Instead it may be possible that a significant fraction of the calcium ions is “site-bound” to the discrete charges on the polyion, thus being completely imm0bi1ized.l~ The sodium diffusion in the Na-ChS04 system (Figure 7) is, just as for the lithium diffusion in the DNA systems, well accounted for by the present model. The deviations are smaller than in the DNA systems, due to the lower surface charge density on ChS04 compared to (B)DNA (-0.068 and -0.150 cm-2, respectively) which is of course also manifested in the generally larger diffusion quotients for the ChS04 system. The calculated diffusion quotients for sodium in the poly(acry1ic acid) (PAA) system as a function of the degree of neutralization of the PAA (Figure 8) deviate considerably from the experimental values even at low charge densities for PAA. It seems reasonable that in this salt-free case a discrete charge model would account for much of the deviations although one possibly have to look for other explanations as well. It should be mentioned that in this case Manning’s model gave quite satisfactory agreement with the experiments.30 Apart from the specific comments made so far it seems appropriate to briefly discuss some general questions relating to the present model. The cell model itself, which is equivalent to idealizing the macroion-macroion correlation by setting it equal to one, seems rather crude, but does apparently work well and certainly to a very large extent reduces the complexity of many problems. The assumption of cylindrical symmetry should be appropriate as long as the persistence length of the macroion is considerably longer than the Debye screening length. For DNA, which has a very large radius of gyration$* this is certainly true. (48)Kam,Z.; Borochov, N.; Eisenberg, H.Biopolymers 1981,20,2671.

The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 3391 The neglect of end effects should also be of little importance in the DNA systems studied here, the length of the DNA molecules being of the order of 15 OOO A. Neither the Na-ChSO, length42-43 nor the PAA length34 is originally given. It is reasonable that end effects play a minor role as long as the cell radius to cell length quotient is small. The “free” diffusion constant, Do,has been taken uniform throughout the cell. In the bulk solution this is a reasonable assumption, but close to the macroion surface, where the counterion concentration can be several molar, the “free” diffusion constant is less than its bulk value. Nevertheless this should be considered as a minor approximation. Concerning the assumption of uniform dielectric permittivity throughout the system a few points are worth consideration. Two effects, the high counterion concentration close to the macroion and the orientation of the water molecules, result in a reduced dielectric constant and thus increase the macroion-counterion interaction. The dielectric discontinuity at the macroion surface should also enhance this effect. The neglect of these dielectric saturation and image charge effects might possibly be of some importance in the case of low salt content and high surface charge density, when a great part of the counterions are in the immediate vicinity of the macroion surface. Needless to say there are also many aspects of the Smoluchowski equation itself to although this is probably of minor importance in the present context. As a final remark we can conclude that the PB cylindrical cell model and the steady-state Smoluchowski diffusion model do account reasonably well for the monovalent counterion diffusion quotient in solutions of DNA. This also holds for other cylindrical polyelectrolytes, the agreement being better for lower surface charge densities and higher salt concentrations. The model fails however to account properly for the diffusion of divalent counterions. This failure can be traced to a neglect of both correlation effects and reduced axial diffusion due to the discrete nature of the polyion charge distribution. It should however also be recognized that the small ion diffusion quotients for divalent ions can also in part be due to “site-binding” to the discrete charges on the polyion. Acknowledgment. We are indebted to Mark Paulsen for help with the preparation of the LiDNA sample and to Tom Record in whose laboratory the sample was prepared. We are grateful to HBkan Wennerstrlim for valuable discussions. W.B. acknowledges the support of a postdoctoral fellowship from the Swedish Natural Science Research Council (NFR). Registry No. Li, 1439-93-2.