517
J. Phys. Chem. 1981, 85, 517-525
However, the same radical can also be formed as a result of hydrogen-atom abstraction by neutral radicals such as H atoms. A relatively low yield of methyl radicals was observed immediately after y irradiation at 77 K and may be due to some dissociation of excited dimethyl carbonate anions formed by electron capture: H3CO\ /C-0” HJCO
-
0 H3C.
t
“C-0/ H3C0
As mentioned in presenting the results of the 13C dimethyl carbonate experiments, a 145-G doublet of 30-G doublets was detected and is most easily seen in the wings of the middle spectrum of Figure 2. These coupling constants are reasonable for 13C (145 G) and P-IH (30 G) nuclei. The four lines are tentatively assigned to radical VI. Support for this assignment comes from consideration ,OW3
H3CQ
CH3
of the P-IH coupling constant. If one assumes that the 30-G coupling in the solid is for VI with one of the C,-H bonds coplanar with the orbital of the unpaired electron on the central carbon and that the cos2 0 relationship holds,36a coupling of (cos20)30 = (1/2)30 = 15 G is predicted for the same radical in solution with the methyl group freely rotating. Norman, Gilbert, and Dobbs have measured a coupling of 14.6 G for this radical in the liquid phase,37but the 13C coupling constant was not reported. Acknowledgment. We are indebted to Dr. C. M. L. Kerr for preliminary work on these systems. Also, we thank Dr. A. Hasegawa for experimental advice in the initial stages of this study. This work was carried out at the University of Tennessee and was supported by the Division of Chemical Sciences, Office of Basic Energy Sciences, US. Department of Energy (Document No. DOE/ER/02968126). Supplementary Material Available: Five figures showing the effects of photobleaching and annealing on the ESR spectra of y-irradiated samples of diethyl and di-n-propyl carbonates (5 pages). Ordering information is given on any current masthead page.
VI (35) B. Eda and M. Iwasaki, J. Chem. Phys., 55, 3442 (1971).
(36) Reference 33, p 170. (37) A. J. Dobbs, B. C. Gilbert, and R. 0. C. Norman, J . Chen. SOC. A, 124 (1971).
Polyelectrolyte Theory. 3. The Surface Potential in Mixed-Salt
Solutions*
G. Welsbuch Dkpartement de Physique, Facult.4 des Sciences de Lumlny, 13288 Marseille C6dex. France
and M. GuQron” Groupe de Biophysique du Laboratoire de Physique de la Matisre Conden&e, 4 Ecole Polytechnique, 9 1 128 Palaiseau, France (Received: October 17, 1980)
The problem of a polyelectrolyte in mixed-salt solutions is studied in Poisson-Boltzmann theory. Simple properties are derived for the counterion distribution near the polyelectrolyte surface. (a) For a planar polyelectrolyte, Grahame’s equation shows that the sum of the ionic concentrations at the surface minus concentrations at large is independent of salt composition and concentration. It is equal to 2 ~ l ~ ( a /where e)~ 1~ is the Bjerrum length and u is the surface charge density. This exact sum rule yields the surface potential and hence the various ionic concentrations at the surface via an algebraic equation. (b) In a large range of conditions,the surface potential of highly charged (6 > 1)cylinders or spheres is close to that of the plane having the same surface charge density. (The parameter 4 is equal to ?rl&,u/e where R, is the radius of curvature). A formula for the surface potential is derived by a perturbation treatment. It is valid even for spheres and cylinders having a large curvature, i.e., R,/X (2?r&X)-’
(1)
where u is the surface charge density, e is the electronic charge, 1~ is the Bjerrum length e2/(4a@kT), which is 0.72 nm in water, and X is the Debye length whose general definition is = 4nlBCniz?
(2)
for a solution containing ions of valency zi and concentration ni. The other symbols have their usual meaning. One finds that for the counterion concentration in the immediate vicinity of the surface of the highly charged plane CIV
= 2?r(u/e)21B
(3)
It is independent of salt valency and concentration. The counterion concentration decays as one moves away from the plane. It is reduced to 0.25(CIV) at a distance T of
T = 12?rzlBu/e1-1 (4) and the integrated charge, up to T,is equal to 0.5 per unit charge on the plane. Thus the distribution of counterions close to the plane is completely independent of the salt concentration at large. We note for future use the value of the potential of the plane in a pure-salt solution, valid for any surface charge density:
where the potential at the surface of the plane, w = 4(0), is given by sinh (zw/2) = 2?rz(u/e)XlB
(6)
and 4 is related to the potential $ by 4 = e$/(kT). An analytical solution for two parallel plates bearing opposite charges, with mono- and divalent counterions and added salt has been published recentlya8 It was shown that, for monovalent salt solutions, the added counterions are evenly distributed over the entire volume. 2.2. Other Shapes. From numerical solutions of the Poisson-Boltzmann equations, one finds that highly charged cylinders in monovalent salt behave similarly to the lane.^^^ The CIV remains finite even in low salt. In paper 1,an empirical formula was proposed: CIV = CIVP1,, exp(-2.3/t)
(7)
where CIVpl,, refers to a plane with the same surface charge density u as the cylinder and 4 is the linear charge density in units of le/lBl. It is related to u by 5 = alBR,U/e (8) (7) Prock, A.; McConkey, G. “Topics in Chemical Physics Based on the Haward Lectures of Peter J. W. Debye”;Amsterdam, 1962; pp 221-5. (8) Engstrom, S.; Wennerstrom, H. J. Phys. Chem. 1978,82, 2711-4 and private communication.
Polyelectrolytes in Mixed-Salt Solutions
The Journal of Physical Chemistry. Vol. 85, No. 5, 1981 519
Equation 8 provides a definition of E for polyelectrolytes of any shape, as a function of R,, the radius of curvature, equal to twice the radius a for a cylinder. In the present work, a generalized and corrected form of eq 7 is derived on the basis of a perturbation analysis (see sections 5.1 and 5.2). The counterion concentration is reduced to 0.25tCIV) in approximately the distance T given by eq 4, which can also be written in the form of eq 9. Thus the highly T = RC/(20 t 9)
r is the distance from the plane surface, from the cylinder axis, or from the center of the sphere, as is appropriate. When one uses eq 8 for the relation between a and s, the Poisson-Boltzmann equations are as follows:
charged cylinder exhibits like the plane a counterion distribution in its vicinity which is remarkably independent of the salt concentration. For spheres, one finds approximately the same results as for cylinders having the same surface charge density and the same curvature, provided the salt concentration is large enough. One can then use eq 7-9, E being defined by eq 8 for shapes different from the cylinder. The high-salt requirements amount to e 2 0.1 where e = R,/(2X) (10)
where j = 0, 1, and 2 for the plane, the cylinder, and the sphere, respectively, and j c is the coordinate of the polyelectrolyte surface. We have defined the relative ionic concentrations aiby
sb)= Cai[exp(zi$) - 11
(14)
For Rc/2 = 1 nm, this corresponds to 1 mM salt, and one then concludes that the counterionic distribution is shape independent in all biological experiments.‘j At lower salt concentrations, the CIV of the sphere decreases slowly and goes to zero with the salt concentration. 2.3. Comparison with Condensation Theory. The constancy of the counterion distribution vs. salt concentration is reminiscent of the “condensation” of condensation theory.“” We have, however? pointed to a number of contradictions between condensation theory and Poisson-Boltzmann. Among the most important, one notes the following. (a) The number of “condensed ions” 1 - 4-l plays no role in Poisson-Boltzmann theory, and the corresponding “condensation radius” becomes infinite in zero salt whereas condensation theory claims it is small and salt independent. (b) The counterion concentration near a cylinder varies as a-2 in Poisson-Boltzmann and goes to zero for a cylinder of large radius at constant 6. This is expected since the surface charge density then goes to zero. (A plane of finite charge density has 5 = according to eq 8). In contrast, the counterion concentration of condensed counterions is independent of a in condensation theory (eq 16 of ref 11) and therefore remains high even in the limit of zero surface charge density! We have showns where the mathematical demonstration of these results is in error. (c) In condensation theory the counterion condensation is specifically related to the cylindrical geometry whereas the Poisson-Boltzmann distribution is largely independent of the polyelectrolyte shape in “high salt”, corresponding to the conditions of many experiments which condensation theory seeks to interpret.
S = s(je)/(4*lBX2)
(15)
ddJ dyl.
- = - 4 a l & S / e = -2€/c
ai
= n i / ( C n i z ; ) 4dBX2ni
(13)
3.2. Sum Rule for the CIVs. For any shape (plane, cylinder, sphere), we define
where j c is the coordinate of the polyelectrolyte surface. By eq 13
S = C(CIVi - ni)
(16)
where CIVi is the concentration of ion i at the surface. Returning to the plane, the Poisson-Boltzmann equation in mixed salt is d2$/dy2 = Caizi exp(zi$)
(17)
We multiply by d$/dy and integrate, using the conditions (d$/dy), = $(a) = 0. This gives %(d$/dy)2 = sb)
(18)
By combination of eq 14, 18, and 12, we find 40)
Cai[exp(ziw) - 11 = 8 (?rlBXa/e)2= 2 f 2 / e 2 (19)
This is Grahame’s equation, known for a long time by electro~hemists.~~J~ Note that, although 5 is infinite for a plane, € / R eand ,$/e are well defined, according to eq 8. Being algebraic in exp w, eq 19 provides the means for easy computation of the surface potential, w, and hence of the CIVs, given by CIVi = ni exp(ziw) (20) Using eq 8 and 10, one can rewrite Grahame’s equation as a sum rule for the CIVs:
s c(cIvi- ni) = 2 * 1 ~ ( a / e =) ~2 f 2 / ( T l ~ R , 2 ) E
(21)
3. The Plane in Mixed Salt This case is important for two reasons. First it corresponds to the problem of charged membranes and electrodes. Second the results and the mathematical methods are the basis for the study of spheres and cylinders in section 5. We therefore analyze in detail the problem of the plane in mixed salt, including an example. An exact algebraic derivation is provided for the potential and CIVs. 3.1. Poisson-Boltzmann Equations. We use a system of variables which will be applicable to the shapes of planes, cylinders and spheres. We define y = r/X, where
This equation shows that for the plane the sum of excess concentrations of all ions close to the plane is independent of salt concentration and composition and proportional to the square of the surface charge density. This result generalizes those obtained earlier.s 3.3. The Plane in Mixed Salt. An Example. As an example, we describe the counterion distribution for a mixture of mono- and divalent counterions near a moderately or highly charged plane ($to) > 1). We neglect the free-ion concentrations as well as the CIV of co-ions, compared to the CIVs of the counterions. (The free-ion concentrations would be taken into account by replacing
(9) Oosawa, F.“Polyelectrolytes”,Marcel Dekkar: New York, 1971. (10)Mmn/ng, G. S. J. Chern. Phys. 1969,51, 924-933. (11) Manning, G. S. Q.Reo. Biophys. 1978,11, 179-246.
(12) Grahame, D.C. Chern. Reo. 1947,41,441-501. (13)I+ A,; , McLaughin, A,; McLaughin,S., submitted for publication in Btochrrn. Biophys. Acta.
520
The Journal of Physical Chemktty, Vol. 85,No. 5, 7981
y=CIV*/S 1
I
I
I
I
I
I
I
I
I
1
1
0
I
2
I
I
I
I
6
4
I
+
S by (S + Cni)in eq 22-24). Equation 16 is approximated by s = CIV2 + CIVl (22) Since CIVi = ni exp(ziw), we have
+ (1 + 4n2S/n12)1/2]
CIV2 3: S(n2S/n12) (25)
Thus CIV2is proportional to n2at constant nl. At constant n2, CIV2varies as nc2, so that selective accumulation of divalent ions is Tore pronounced in low salt. We can also write CIV2 PI: n 3[, the perturbation approach fails.
w.
A.2. An Example. We consider a cylinder of radius a = R,/2 = 1 nm having [ = 3, in a solution containing 10 mM NaCl and 10 r M MgC12. (a) Computation of e. The Debye length given by eq 2 is 3.0 nm. Hence E = R,/(2A) = 0.33. (b)Computation of the Surface Potential for the Plane. The value of S is 2t2/(7rl&,2) = 2.0 X loz7m9, or 3.32 M/L. To obtain w, we may solve eq 21, or we may use eq 30 or Figure 1. Let us use Figure 1. We compute x : x = Sn2/nI2 = 0.332
This gives y = 0.21; hence, CIVz = 0.70 M/L. The potential is given by exp(2w) = CIVz/nz = 7 x lo4 Hence = 5-58
(The number of accumulated divalent ions Nzpfor the plane is given by eq 34. Considering the derivation of the equivalent salt valency (eq 51), this is the same as N2p= 1 - zP1. By eq 51, z' = 1.08; hence, Nz (c) Solution for the Cylinder. The va ue =of z obtained from eq 51, z' = 1-08, is introduced into eq 50, giving 4(e) = 5.04
P 0.074J
Hence CIVl = nl exp[4(t)] = 1.55 M/L CIV2 = n2 exp[24(~)]= 0.24 M/L CIV1+ CIV2 = 1.79 M/L The exact value of the surface potential is @(E) = 5.09. The error is 0.05, resulting in errors on the CIVs below 11%. An evaluation of the integrated number of divalent ions may be obtained with the procedure of section 4, i.e., by using the value Szof CIVzfor the cylinder in pure divalent salt. In thii case, by eq 50, the potential deviates from that of the plane by 0.17. Hence Sz = 2.38 M/L Using the value CIV2 of the cylinder in the mixed-salt solution, we find
NZ
N
CIVZ/S2 = 0.035 3 - CIVZ/S2
Note that this value is only half the value N2p for the plane. This result may be compared with the exact one: Nz = 0.0313 The error on the integrated number of divalent ions is -12%.
