J. Phys. Chem. 1992, 96, 7816-7820
7816
of acceptor molecules. However, as the fluorescence decay of HHC in the absence of acceptor molecules deviates from single-exponential behavior, it is not fruitful in the present case to analyze the results from timeresolved fluorescence measurements in this way.
concl~ions
This study has shown that FET studies can be used to obtain information about the homogeneity of fluorescent components in air-water monolayers when only a few mole percent of a particular component is present and the inhomogeneities present are only of molecular scale. Thus, FET studies can be used to probe the homogeneity of components in air-water monolayers that are not accessible by other methods. Acknowledgment. F.G.and P.J.T. gratefully acknowledge financial support from the Australian Research Council. NO. HHC, 26038-83-5; RhB-DPPE, 1261 11-99-7; DPPC, 63-89-8; DOPC, 4235-95-4.
References and Notes (1) Vanderlick, T. K.; MBhwald, H. J. Phys. Chem. 1990, 94, 886. Mahwald, H. Angnv. Chem., Int. Ed. Engl. 1988, 27, 728. (2) McConnell, H. M.; Rice, P. A.; Benvegnu, D. J. J. Phys. Chem. 1990, 94, 8965. ( 3 ) Gaines. G.L., Jr. Insoluble Monolayers at Liquid-gas Interfaces; Interscience: New York, 1966. (4) Urquhart, R.S.;Hall, R. A.; Thistlethwaite,P. J.; Grieser, F. J. Phys. Chem. 1990,94,4173. 15) MBbius. D.: M6hwald. H.Adu. Mater. 1991. 3. 19. (6j M8bius; D.f Bucher, H.;Kuhn, H.; Sonderman, J. Ber. Bunsen-Ges. Phys. Chem. 1%9,73, 845. (7) Hall, R.A.; Haya, D.; Thistlethwaite,P. J.; Griaer, F. Colloids Surf. 1991, 56, 339.
(8) Pecls-II Luminescence Software Condensed Operating Inrtructionrf w Models LS-3. 43-4, LS-5;Perkin-Elmer: Oak Brook, IL, 1983; p 5-29. (9) Parker, C. A.; Ress, W. T. Analyst (London) 1960, 85, 587. (10) Hauser, M.; Klein, U. K. A.; Ghele. U. 2.Phys. Chem. N.F. 1976, 101, 255. ( 1 1) Kellerer, H.; Blumen, A. Biophys. J. 1984, 46, 1 . (12) Baumann, J.; Fayer, M. D. J. Chem. Phys. 1986, 85, 4087. (13) Wolber, P. K.; Hudson, B. S. Biophys. J. 1979, 28, 197. (14) F6rster, Th. Discuss. Faraday Soc. 1959, 27, 7. (15) Tweet, A. G.; Bellamy, W. D.; Gaina, G.L., Jr. J. Chem. Phys. 1964, 41, 2068. (16) Grieser, F.; Thistlethwaite, P. J.; Urquhart, R. S.;Patterson, L. K. J . Phys. Chem. 1987, 91, 5286. (17) Villallonga, F. Biochim. Biophys. Acta 1968, 163, 290. (18) Phillips, M. C.; Chapman, D. Biochim. Biophys. Acta 1968,163,301. (19) Drummond, C. J.; Grieser, F. Photochem. Phorobiol. 1987, 45, 19. (20) Mikes, V. Collect. Czech. Chem. Commun. 1979, 44, 508. (21) Moriya, T. Bull. Chem. Soc. Jpn. 1983, 56, 6 . (22) Moriya, T. Bull. Chem. Soc. Jpn. 1988,61, 1873. (23) Zinsli, P. E. J. Photochem. 1974/1975, 3, 55. (24) OConnor, D. V.; Phillip, D. Time-correlated single photon countins Academic: London, 1984. (25) Anfinrud, P. A.; Hart, D. E.; Struve, W. S.J. Phys. Chem. 1988,92, 4067. (26) Carum, F.; Grieser, F.; Murphy, A.; Thistlethwaite,P. J.; Urquhart, R.; Almgren, M.; Wistus, E. J. Am. Chem. Soc. 1991, 113,4838. (27) Bauer, R. K.; Kowalczyk, A. 2.Naturforsch., A: Phys., Phys. Chem., Kosmophys. 1980, 35,946. (28) Birks, J. B. Phorophysics of aromatic molecules; Wiley-Interscience: London, 1970; p 9 1 . (29) Tamai, N.; Yamazaki, T.; Yamazaki, I. Chem. Phys. Lett. 1988,147, 25. (30) MacDonald, R. I. J. Biol. Chem. 1990, 265, 13533. (31) Lbche, M.; Sackmann, E.; M6hwald, H. Ber. Bunsen-Ges. Phys. Chem. 1983, 87, 848. (32) Cadenhead, D. A. In Structure and properties of cell membranes; Benga, G.,Ed.;CRC: Boca Raton, FL, 1985; Vol. 3, Chapter 2. (33) DBrfler, H.-D. Adv. Colloid Interface Sci. 1990, 31, 1. (34) Yamazaki, I.; Tamai, N.; Yamazaki, T. J. Phys. Chem. 1990,94,516.
Polyelectrolyte Theory. 4. Algebraic Approximation for the Poiseon-Boltzmann Free Energy of a Cylinder M.Gueron*vt and J.-Pb. Demaret' Groupe de Biophysique and Laboratoire de Biochimie, Ecole Polytechnique. 91 128 Palaiseau, France (Received: March 5, 1992; In Final Form: June 16, 1992)
Starting from the case of the plane as a zero-order approximation, an algebraic expression was developed previously for the Poisson-Boltzmann value of the surface potential of a highly charged cylinder in a wide range of salt concentrations. We extend this work to the case of weakly charged cylinders. This enables us to compute the free energy of a charged cylinder. The free energy is the sum of three definite integrals which can be expressed algebraically. The method applies to the case of solutions of ions of finite size. In all cases, the free energy may be computed quickly and simply.
1. Introduction It was previously shown that the Poisson-Boltzmann surface potential of a highly charged polyelectrolyte is dependent on its shape. This statement was first supported by numerical integration of the equations' and later by analysis based on the concept of a scaling length I, for the electric field2
4
= -(dp/Wo/(d2~/~2)o
(1)
where cp is the reduced potential, i.e., the electrostatic potential divided by kT/e; T is the temperature and e is the electronic charge. The zero index indicates that the derivativa are computed at the polyelectrolyte surface. 'Group de Biophysique. f Laboratoire de Biochimie. Current address: L.P.C.B., Institut Curie et Universitd Paris VI, 1 1 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France.
0022-3654/92/2096-7816$03.00/0
Near a charged plane, the electric field and the charge concentration can be described with the help of the scaling length I,, providing a unified description of the linear and nonlinear by the charge re*a* The plane is Or by the area A = e/a*Or by the thickness Th Th = A / ( ~ T I B Z ) (2) where IB is the Bjerrum length, e 2 / ( 4 ~ @ k T ) and , -ze is the counterionic charge. The scaling length I, is found to be approximately equal to the smallest of Th and of the Debye length A. The case X < Th (and hence I, A) corresponds to the linear regime. The Poisson-Boltzmann solution for the plane is expected to be a good approximation for other shapes than the plane if the radius of curvature is larger than the scaling length. For a cylinder of radius a (curvature radius 2a), eq 2 becomes Th = a / ( [ z ) (3) 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 19, 1992 7817
Polyelectrolyte Theory where fe is, as usual, the linear charge along an axial length lg. With these conventions, le, Th, Az, and fz are always positive; z, f , A, and Cp(0) are positive for a polyanion. If the ionic strength is not too high, Th is smaller than A and determines the scaling length le. The condition that 1, be smaller than the radius of curvature is then simply 2fz
>1
same surface charge density is a good approximation for that of the cylinder. Based on such considerations and on a sum-rule equivalent to Grahame’s e q ~ a t i o n we , ~ derived, by a perturbation approach, a formula for the reduced potential 40)at the surface of a strongly charged sphere or cylinder4
Y
fz’- 0.45
+ 1.8(t/[~’)I/~+ 3t2/([z’)
(5)
(6)
where w is the reduced surface potential of the plane with the same surface charge density, under the same salt conditions; -z’e is equal to the counterionic charge if all ions have the same (absolute) charge; it is an appropriate equivalent charge in the case of a mixed solution; c is equal to a/A. This formula has the proper behavior for the limiting case of 03. t 0 3 , c/([z’) = Th/A) and the proper dethe plane ( f pendence on z’. It results in small errors on the reduced potential (smaller than 0.1) for a wide range of values of E and i, covering biopolymers in all practical cases. To evaluate Cp(0) by (S), the only requirement is the computation of the surface potential w of the plane with the same surface charge density, u. In a pure solution, w is equal to (2a/z) arcsinh (A/Th). In a mixed solution, w is obtained as the solution of an algebraic equation by using the sum rule for ionic concentrations (Grahame’s equation). The above analysis has been applied to the description of the ionic distribution near a polyelectrolyte,4 to the computation of the distribution of counterions in the case of mixed solutions, and to the investigation of counterion bindingn5The knowledge of 40) also enables one to compute easily a complete numerical solution of the Poisson-Boltmann equation, and thus obtain the potential at all points. In o@er applications, an approximation for the electostaticfree energy F is more convenient than one for the surface potential. For instance, thermodynamic properties such as osmotic pressure or counterion activity are directly related to the free energy. The free energy may also be used in the study of ion binding and in the comparison of the stabilities of different structures, such as B-DNA and Z-DNA.6 In the present paper we shall again make reference to the planar geometry, and use the value of the surface potential (5) as a starting point to obtain an approximate expression in closed form for the electrostatic free energy F of a charged cylinder. The free energy per phosphate is given by7,*
- -
F = k T S 0’ d O , s ) ds
+
S
F1 = k T S (P(0,s) ds 0
I
F2 = k T J 4 0 , s ) ds
(4)
When this condition, which could be taken to define the strongly charged cylinder, is satisfied, the potential of a plane with the
40) = 0 - 2/(z‘y)
energy integral into two, for S = l / &rather than S = 1/(2f). We therefore rewrite (7) as F = FI F2, with
(7)
where Tis the temperature and 40,s)is the surface potential of a polyelectrolyte whose charges are s times the true phosphate charges, immersed in a solution whose ionic charges are invariant. The evaluation of F requires expressions for the potential of both weakly charged and strongly charged cylinders, depending on the value of s. As a result, the free energy is the sum of three definite integrals, each of which is expressed algebraically. We consider henceforth a polyelectrolyte in a solution of monovalent salt. The extension to z-valent salt is given in the Appendix. 2. Principle of the Calculation According to q 4 the transition between weakly and strongly charged cylinder corresponds to 2sI = 1. We shall split the free
(7’)
In the first integral, F,, we use an expression of 40)computed for a weakly charged cylinder. In the second integral, F2, we start from eq 5 and write F2 = Fp + F,,,, with
Fp = k T J 1 w ( s ) ds F,,, =
~TL I
corr(s) d~
(8)
where corr(s) = -2/Y([s). F is the sum of the three definite integrals, F I , Fp, and F,,,. As a fmt step we compute the corresponding indefinite integrals, 11,Zp,and I,,,. From these intermediary results we also obtain expressions for the free energy of the weakly charged cylinder and of the plane. 21. The Weakly Charged Cylinder. The Debye-Huckel surface potential is
d o ) = 2 EsKo(4/(4(4)
(9)
where KOand K,are the modified Bessel functions. For all values of t, we use the approximation cpD(0) = 2[s log (1 + l / t ) (9’) The Poisson-Boltzmann potential is smaller by a factor B, which has been studied by Stigter? For 0 < ,$s < 1.5 and 0.1 < e < 10, we approximate B as @=1
+ ([s)2/(5 + 202)
In the same range, we obtain therefore
40) = 2[s[log (1
+ l / € ) ] / ( l + ([s)2/(5 + 202))
This has the form p s / ( 1 qs2). Hence
+
Z,/kT = ( 1 / f ) ( 5
+ qs2) whose integral is @/2q) log (1
+ 202) [log (1 + 1/€)] x [log (1
+ ( S 0 2 / ( 5 + 20c2))1 (10)
where Sf is the linear charge of the cylinder. For our application, the final charge is
sg = 1 Hence Fl/kT (l/f)(5
+ 204[log (1 + l/c)][log
(1 1)
(1
+ 1/(5 + 20€2))]
(12) Note: An approximation to the free energy of a weakly charged cylinder (WCC)is obtained by setting S = 1 in eq 10: F,,/kT = (1/[)(5 + 2Ot2)[log (1 + l/t)][log (1 + [2/(5 + 20€2))] (13) 2.2. Free Eaergy of the Plane (Exact Expdom). The surface potential is w = 2 arcsinh (sA/Th) (14) whose integral is
I , / k T = 2s arcsinh (sA/Th) - 2(s2 + (Th/A)2)1/2 (15) We choose a plane having the same surface charge density as the cylinder under consideration; hence Th/A=e/[, by eq 3. This gives
Z p / k T = 2s arcsinh (sE/e) - 2(s2 + ( c / [ ) ~ ) I / ~
(16)
7818 The Journal of Physical Chemistry, Vol. 96, No. 19, 1992
Since Fp = Zp(l)
- Zp(l/[),
;; ;;
we have
F p / k T = 2 arcsinh ([/e) - 2( 1 + ( c / € ) ~ ) ' -/ ~ (2/5) arcsinh ( I / € ) 2(1
+
+ e2)1/2/[
(17)
For computations, it is convenient to use the relation arcsinh X = log (X+ (x2 + 1)1/2)
(18)
- (1 + (Th/X)2)1/2+ Th/X]
52 55
where Th is given by (2), with z = 1. This result is well-known (see, for instance, ref 9, and references quoted therein).
Term for tbe Strongly chuged Cylinder (scc).
To integrate the correction term -2/Y in (9,we approximate ([s)-I/~ and (&)-I by two quadratic polynomials in [s, adjusted to give the correct values for €8 = 1, 2, and 5. Then, 1/(A
and linear char:. deniity para..t.r 'T is lo: of concentration lrrle/ll 'Lambda 1s the Debya length Inml
+ B([s) + C([s)')
with
: f is Ifrae energy per phosphat.l/hT 'multiply b y t w o for frem energy per base pair
60 'PI ir the f r w anergy of a 'weakly charged" cylinder, with linear charge 6 1 'density S*XI=I, hence S=I/XI laquation6 7 and 7'1. 6 2 'The char:in: oparatien continues, from 1 t o XI, requirini .rf sner:y F2. 6 4 ' I P ia the free energy increment far the plan.. 66 'FCORR is the corresponding correction f o r the cylinder. 68
100 ' l l 0 'FREE ENERGY OF WEAKLY CHARGED CYLINDER , , E
11s
-2/Y
a++*++~++*+****+~*x**+aaa++*+*++a+a**~a~*++*a+***x*+a+aa*a+**a**~a~
5 A=I:Xl=k.Z 'radius In.) 30 CONC*lO"T 32 LAMBDA=I/SPRl1O.1xCONCI 3 4 !PS=A/LAIIBDA 50 'F=Fl+FZ : FZ=FP+FCORR
(19)
2.3. C o r "
++**a++++**++++*aanaa*~+***+t+++++**+*++s*n+++~***+a++*ax*+a~a++na~ ELECTROSTATIC FREE ENERGY of charled cylinder I w n o v a l m t sa111
40
Note: The free energy of a plane is ZP(1) - I&O) Fplsne/kT= 2[arcsinh (h/Th)
Gueron and Demaret
--> PI
(equation 121
I
I20 F11=5kll+4+EPS*EPSl
:
F12=LOClIrl/EPS~:Fl3=L~ll+l/Flll
125 F1=FIIfFIZaF13/XI 130 ' 132 ' 140 'INTEGRATION Of POTENTIAL OF PLANE --> FP
150 ' 160 U:XI/EPS 175 FOR J=O TO I :S=1*JfiI/Xl-lI:Y=SPRIiStUl^2+1r 180 FPPIJI = S*LCGIS*U+WI-W/U:NEXT J 185 FP=Zri PPPIOI-FPPIII 1 300 ' 302 310 'CORRECTION TERM FOR CYLINDER - - > FCORR 320 370 E S = S P R I E P S I : L ~ = E P S I E P S 380 ' 385 A* -.45 + 2.34rES 3.79fEZ - 0.72fES - 1.59112 390 8; I 395 C: + 0.09.ES + 0.20IEZ
Isquation 171
(equation 201
f
A = -0.45
B
+ 2.346 + 3.79~~
1 -0.726 - 1.59~~
c = 0.096 + 0.20e2
397 ' 400 Llm(I=XI: LIMLO=l
402 410 430 440 445
'
' integration limits
M S U B 470: FCORR=INTEGRALtl-ZI/XI FZ=FP+FCORR : F=Fl+FZ :PRINT F ' I + I I I + + i f + +R E S U L T w a + x x I + s + E a I * ~ RETURN
(50 STOP 460 ' 470 'Subroutine for i n t q r a t i o n of I/iA*BX+CX+Xl 4 8 0 'Different integration f o r w l a s 1Iin.r 520 and 5361 are used, 485 'depending on the sign of the discr!.inrnt P 487 ' 490 ~ I = ~ ~ C + L I ~ H I : L O . Z + C ~ L I ~ L O 500 P= 4 I A x C - B I B : IF l P < = O i GOT0 530 510 as=saRccx 5 2 0 INTEG=ATNliHi+8~/~SI-ATNl~LO*8I~4S~:INTEGRAL=2IINTEG/PS:GOTO 560 525 530 ' I f N = O , the r o o t s of A 0.X + CfXrX = 0 are real. In our conditions 531 'they ar* smaller than a11 values in the integration range. The 532 'appropriate f o r m u l a i~ g i v e n below. 534 ' 536 PS = sPRi-ai 538 INTEG~lHI+B-PS~fILO~8~PSl/I~l+8*PSl/ILO+8-PSl 5 4 0 INTEGRAL = LOCIINTEGIIPS 560 RETURN ................
The integral of (20) is given in textbooks: there are three different expressions, depending on the relative values of A, B, and C. 2.4. Free Energy of the Strongly Charged Cylinder. This is the sum of F1(12), Fp (17), and the correction term , F Rather than write this out in detail, we give in Figure 1 a BASIC program which computes FI,F,,,and F , and produces their sum as output. 3. Resulh
The free energy is the sum of three terms. The first, F,,is the free energy required to charge the cylinder up to a linear charge of 1 (weakly charged cylinder). The second is Fp'the free energy for charging a plane from a surface charge density corresponding to unity linear charge of the cylinder to f . The third is F , which, added to Fp,corresponds to the charge of the cylinder rather than the plane. The latter term is smaller than kT per phosphate and it is less than 20% of the free energy in the range considered. The free energy is plotted directly in Figure 2A, and with reference to that of the plane with the same surface charge density in Figure 2B. For DNA (4 = 4.2, a = 1 nm) in the conditions of most biochemical measurements (e = 1 corresponds to a monovalent salt concentration of 100 mM), the free energy of the plane is a good approximation. Hence shape effects on electrostatic properties should be modest in such conditions. In order to evaluate the quality of the algebraic approximation, we have compared our results with those obtained by numerical integration of the Poisson-Boltzmann equation for a series of values of 4 and e. Table I shows the excellent quality of the algebraic approximation over a large range of conditions. 4. Generalization to Ions of Finite Size
The Poisson-Boltzmann model analyzed above assumes that the ions in the solution are point charges and that the surface charge of the polyelectrolyte has zero thickness. The model may be generalized by introducing a minimum approach distance, d, between the cylinder and the ions, equal to the sum of the ionic radii.IO The effect of finite ionic size on the distribution of distances between solute ions is ignored. In the model, the cylinder of radius a + d is inaccessible to the center point of the ions. The effect of each (nonpolarizable) ion is the same as if the charge is localized at its center. As for the polyelectrolyte charge at radius a, it has, by Gauss's theorem, the same effect as if it were spread out at radius a d. As a result, the potential beyond radius a d is the same as that of a cylinder of radius a + d, with a superficial charge whose
+
+
Figure 1. BASIC program for computing the free energy of a strongly charged cylinder. F denotes the free energy per phosphate divided by kT. The ions are point charges; for ions of finite radius, the supplementary contribution is given by eq 23. TABLE I: Precision of the Algebraic Expression for the Free Energy of Cvlinders" € 1 2 4 = 4.2 t [=I 0.01 4.207 6.907 9.600 0.000 0.003 0.002 0.1 2.193 3.815 5.752 O.Oo0 0.005 0.004 1 0.680 1.307 2.317 O.Oo0 0.011 0.010 10 0.095 0.192 0.400 0.000 0.000 0.002
"For each value of c and [, we give the "exact" value of the free energy per electronic charge in units of kT, and, in italics, the error of the algebraic expression (exact value minus algebraic). The Yexact" value was computed by numerical integration of the potential, as given by the algebraic expression in ref 4. The precision of the expression is better than 10-2kT, as checked against values of the potential computed by numerical solution of the Poisson-Boltzmann equation using a fourth-order Runge-Kutta method. The integration was carried out by Simpson's method. The number of integration steps was increased until the free energy value stabilized up to the fifth significant figure. The algebraic expression is F I+ Fp F,,, as given by eqs 12, 17, and 20, combined as is Figure 1.
+
density is reduced from the original by the factor a / ( a + 6). The linear charge parameter [ is unchanged and t becomes (a d)/h. The expression for the surface potential as a function of [ and e (eqs 2, 3, 5 , 6, and 14) is unchanged. On the other hand, the potential of the polyelectrolyte charges is changed, since they are at radius a, rather than a d.
+
+
The Journal of Physical Chemistry, Vol. 96, No. 19, 1992 7819
Polyelectrolyte Theory
study of the B-Z transition of DNA (manuscript in preparation).
F/k
5. Discussion In principle, all equilibrium properties can be derived from the free energy. For instance, the surface potential is
-2
0
-1
log E
1
10
Figwe 2. The free energy Fof strongly charged cylinders (E = 1, 2,4.2), as a function of t cylinder radius/Debye length. For DNA (1 nm radius), log t = 0 corresponds to a concentration of 100 mM monovalent salt. (A, top) The free energy. (B, bottom) The ratio of the free energy to that of a plane sharing the same surface charge density. The linear charge parameter of 4.2 is that of B-DNA.
Therefore, the energy required for charging the cylinder will also change and the free energy will be altered. To compute the potential difference between the cylinder charges and the cylinder surface, consider the coaxial capacitor of length le, whose inner and outer radii are a and a + d. The capacity is
c = 2*coDplB/10g (1 + d / a )
(21)
where Dp is the dielectric constant due to ionic polarizability, for which we may take the traditional value D = 2. The charge along le is €e; hence the potential difference p[kT/e) is equal to te/C, where the index f stands for finite ion size. We obtain cpr = 2(€D/Dp) log (1
+ d/a)
(22)
where D is the solvent dielectric constant; D = 80. The charging process gives rise to a supplementary term in the free energy Fr:
F f / k T = (€D/Dp) log (1
-
+ d/a)
(23)
If we take for d the sum of ionic radii of oxygen (0.146) and sodium (0.098),and consider the case of DNA, with a = 1 nm and 4 = 4.2, we find F f / k T 37. This large value suggests that small counterions will displace larger ones in the vicinity of the polyelectrolyte, as has been pointed out previously.1° The free energy computed here is independent of ionic concentration. However, a concentration dependence would appear on incorporating a variation of the dielectric constant near the polyelectrolyte surface. The present treatment of ffite ions should be used with caution. The coaxial capacitor created around the polyelectrolyte may be adequate in the case of ions interacting with a uniformly charged cylinder, but the uniform cylinder is itself a model. A good model of ions interacting with a cylinder may be a poor model of ions interacting with the true polyelectrolyte. Such is indeed the case when the latter includes solution up to and within its radius of charge (e.g., in the grooves of B-DNA). This short-circuits the capacitor of the model! The region forbidden to the center of ions is better modeled as discontinuous or porous, as described in our
k T d O ) = d@/dq) (24) where q is the phosphate charge. This formula is equivalent to (7). Other properties are also obtained by derivation of the free energy with respect to the appropriate variable. It should be noted that the free energy computed in the present work is that of an isolated polyelectrolyte and is therefore not convenient for the computation of colligative properties such as counterion activity or osmotic pressure. The derivatives may be computed analytically starting from the equations obtained here, where t and c are the dependent variables. But it is simpler to compute an increment of F directly, by using a program such as the one in Figure 1. Our method may be compared with the alternatives based on numerical integration of the Poisson-Boltzmann equation, whose results are in principle as precise as desired. The comparison is required for the evaluation of the precision of the algebraic expression, as carried out above. In our opinion, the numerical method is less illuminating and less convenient for applications. Furthermore, it has difficulties: convergence must be obtained and successive integrations may lead to numerical errors. In the course of our study of the B-Z transition of DNA, we detected such an error in the literature by comparison with the algebraic expression. Stigter has tabulated the ratio of the Poisson-Boltzmann free energy of a cylinder to the value computed in the linear (Debye) theory as a function o f t and of the surface potential. For strongly charged cylinders, the ratio may be as small as 114. This displays the importance of nonlinear effects and shows that the weakly charged cylinder is not a very good starting point for the description of the strongly charged cylinder. This may be compared with our approach in which we use the plane as the reference polyelectrolyte, we compute an algebraic approximation to the free energy, and we display the latter as a function of c and the linear charge density [. 6. Conclusion
We have obtained a good approximation for the PoissonBoltzmann free energy of charged cylinders, using only algebraic expressions in closed form and requiring therefore no integration of a differential equation. It is implemented in a simple BASIC program (Figure 1). The results can be used for any cylinder, weakly or strongly charged. The finite size of ions may be taken into account. The approximations are satisfactory for < e < 10. The corresponding range of salt concentrations is lO-’-lO M for a cylinder of 1-nm radius. The results are for solutions of monovalent salt. The straightforward extension to z-valent salt is given in the Appendix. Our method makes it simple to evaluate Poisson-Boltzmann free energies and to grasp how the effects of polyelectrolyte shape and of ion concentration interplay. The emphasis on the planar approximation proves useful for investigations of polyelectrolyte properties, such as the B-Z transition6 of DNA. APWX
The Poteatial dFreeEnergyinsdutionscon~z-v~ Ions. The formulas in the Introduction (eq 1-7) are valid for solutions containing ions of whatever valency z. This is also the case for eqs 21-24. On the other hand, the computations of sections 2 4 , including Figure 1, are for monovalent salt only. We now show how these same equations may be used in the case of a solution of z-valent cations and anions. The Poisson-Boltzmann equations are d2p/d9 + ( l / r ) dp/dr = (A%)-’ sinh (zcp) (Al) dcp/r = -25/b
(A21
J. Phys. Chem. 1992,96. 7820-7823
7820
Hence, 2 9 is a function of z l :
J5,4
dZ,E,d = (1
(A31
where the fmt symbol in the parentheses refers to the ionic valency, the second to the linear charge density of the cylinder, and the third is a/A, as defined in the Introduction. (a) As a first application, consider the form of the surface potential of a cylinder used by Stigter?
dl,€,d= cpD(l,t,e)/@(1,5,4
(A4)
The Debye-HILckel potential, c p ~ ,is independent of z and proportional to 5 (eq 9). Hence it satisfies eq A l , as should be expected. Since must also satisfy eq Al, Stigter's factor @ must obey the following relation:
@(z,64 = @(1. J5,4
(AS)
In other words, the correction term @ for the bharged cylinder in z-valent salt is equal to the correction term for a (z[)-charged cylinder in monovalent salt (with the same value of e). (b) Consider now the free energies. We have I
F(z,b4/kT =
1
dz,sF,4 ds =
( l / z ) d L z s t , d ds (A4)
= (l/z)F(l,zE,e)/kT
(A6)
Hence, the free energy of a &charged cylinder in a solution of z-valent ions is equal to 1/z times the free energy of a (I[)charged cylinder in monovalent salt (with the same value of e). (c) This leads to the following practical rules: (i) To obtain the surface potential or the free energy of a cylinder (4)in z-valent salt, apply the procedures of the present paper to a cylinder with linear charge density ( z l ) in monovalent salt. Divide the result by z. This rule is to be used with all procedures described from section 2 onwards, including Figure 1. (ii) To obtain the surface potential or the free energy of a plane ( u ) in z-valent salt, apply eqs 14 and 19 to a plane with surface charge density zu (hence Th = A/(27rlBz)) in monovalent salt. Divide the result by z.
References and Notes (1) Guiron, M.; Weisbuch, G. Biopolymers 1980, 19, 353-382. (2) Weisbuch, G.; Guiron, M.J . Phys. 1983,41,251-256. (3) Grahame, D. C. Chem. Rev. 1947,41,441-501. (4) Weisbuch, G.; Guiron, M. J . Phys. Chem. 1981, 85, 517-525. (5) Gubron, M.; Weisbuch, G. Biochimie 1981, 63, 821-825. (6) Guiron, M.; Demaret, J. P.Proc. Natl. Acad. Sci. U.S.A. 1992, 89, 5740-5743. (7) Fixman. M. J . Chem. Phvs. 1979. 70. 4995-5005. (8) Lukashin, A.; Beglov, D.'B.; Frank-Kamenetskii, M. D. J . Biomol. Struct. Dyn. 1991, 8, 1 1 13-1 118. (9) Stigter, D. J. Colloid Interface Sei. 1975, 53, 296-306. (10) Gregor, H. P.; Gregor, J. M. J . Chem. Phys. 1977,66, 1934-1939.
A Kinetic Study of the State of the Proton at the Surface of Dodecyl Sulfate Micelles
Luis Garcia-Rio, J. RamC Leis,* M. Elena P e h , Departamento de Quimica fisica, Facultad de Quimica, Universidad de Santiago, 15706 Santiago de Compostela, Spain
and Emilia Iglesias Departamento de Quimica Fundamental, Facultad de Ciencias, Universidad de La Coruiia, La Coruiia, Spain (Received: October 30, 1991)
The fact that kinetic solvent isotope effects are abnormally low for proton-transfer reactions involving H 3 0 +as the acid has been used to test the state of the proton at the surface of dodecyl sulfate micelles. The observed low value of the solvent isotope effect on the kinetics of the acid denitrosation of N-methyl-N-nitroso-ptoluenesulfonamide in the presence of micelles suggests that the proton donor is H,O+and hence that 'micellar" proton is tightly bound to a water molecule and in a state of solvation similar to that in bulk water. Covalent bonding between the surfactant head groups and the protons seem unlikely. Analysis of the activation parameters for the reaction in water and in dodecyl sulfate micelles confirms this hypothesis, because only the entropy of activation differs appreciably and not the enthalpy. These findings, together with the NMR evidence on the location of the substrate, are consistent with the idea that, within the micelle, reaction takes place in the highly hydrated Stern layer.
Introduction In recent years, micellar reaction media have attracted considerable attention as means of controlling the rates of reactions of chemical, industrial, or biological interest. In most cases,simple electrostatic considerations suffice for prediction of whether a reaction will be catalyzed or inhibited by cationic, anionic, or nonionic micelles, but several quantitative theories based on the pseudophase model have also been developed.' According to this model, a micellar "pseudophase" is distributed uniformly through the bulk (generally aqueous) phase of a micellar system, and reactions take place in both phases. The relative importance of these two pathways depends on the reaction rates in each phase and the distribution of reagents between the two. Most studies in this field have concerned reactions between an organic substrate and a reactive anion in the presence of cationic micelles. In general, it has been concluded that under these
circumstances micelles catalyze the reaction primarily by concentrating both reagents in the small volume of the Stern layer around the micelles, because the estimated rate constant for the reaction in the micellar region is generally found to be similar or even slightly less than that for the reaction in water.' This behavior is probably due to the Stern layer having high water content. The activity of water in this layer is almost as great as it is in the aqueous p h a ~ e , although ~.~ the effective dielectric constant of the micellar interphase seems to be significantly less than that of water (generally being close to that of ethanol): One of the most extensively explored pseudophase models is pseudophase ion exchange (PIE). One of its basic assumptions is that in systems containing two ions capable of acting as counterions to the micelle, their competition for micellar charges is governed by an equilibrium similar to that governing the behavior of ionexchange resins. As mentioned, most work in this area has in-
0022-3654f 92f 2096-782OSQ3.00/0 @ 1992 American Chemical Society
