Use of the Poisson-Boltzmann Equation To Predict Ion Condensation

17 Use of the Poisson-Boltzmann Equation To Predict Ion Condensation Around Polyelectrolytes

Downloaded by MICHIGAN STATE UNIV on February 18, 2015 | http://pubs.acs.org Publication Date: March 28, 1986 | doi: 10.1021/bk-1986-0302.ch017

Bruno H . Zimm Department of Chemistry, B-017, University of California, San Diego, La Jolla, CA 92093

We summarize recent work showing that condensation can be derived as a natural consequence of the PoissonBoltzmann equation applied to an infinitely long cylindrical polyelectrolyte in the following sense: Nearly all of the condensed population of counter-ions is trapped within a finite distance of the polyelectro lyte even when the system is infinitely diluted. Such behavior is familiar in the case of charged plane sur faces where the trapped ions form the Gouy double layer. The difference between the plane and the cylinder is that with the former all of the charge of the double layer is trapped, while with the latter only the condensed population is trapped. Manning's c o n d e n s a t i o n t h e o r y (J_) d e s c r i b e s the b e h a v i o r o f s m a l l i o n s around a l o n g , h i g h l y charged p o l y e l e c t r o l y t e by p o s t u l a t i n g t h a t t h e r e a r e two p o p u l a t i o n s o f c o u n t e r - i o n s , one normal, and one "condensed". I f we have c o u n t e r - i o n s w i t h v a l e n c e z, t h e f r a c t i o n , F^, o f these i o n s i n t h e condensed p o p u l a t i o n i s F

M

- 1 - 1/ζξ

ξ, c a l l e d t h e l i n e a g c h a r g e - d e n s i t y parameter o f the p o l y e l e c t r o l y t e , i s e q u a l t o e /4|?e DkTb, where e i s the e l e c t r o n charge, e the c a p a c i t i v i t y o f the vacuum, D t h e d i e l e c t r i c c o n s t a n t o f t h e s o l v e n t , kT a s u s u a l , and b i s the a x i a l s p a c i n g o f the ( u n i v a l e n t ) charges on the p o l y e l e c t r o l y t e . The above f o r m u l a h o l d s i f t h e charge d e n s i t y on the p o l y e l e c t r o l y t e i s h i g h enough so t h a t ξ i s g r e a t e r than 1/z; o t h e r w i s e t h e r e i s no condensed p o p u l a t i o n . T h i s t h e o r y has s c o r e d a number o f remarkable s u c c e s s e s , p a r t i c u l a r l y i n r e g a r d t o DNA a s the p o l y e l e c t r o l y t e . (For r e v i e w s , see Manning (2) and Anderson and Record (3^). ) For a l o n g time, however, t h e l i t e r a t u r e d i d n o t g i v e a c l e a r p i c t u r e o f how these condensed i o n s were d i s t r i b u t e d i n space; t h e r e was some disagreement between Manning's own p i c t u r e , i n which t h e 0

0097-6156/86/ 0302-0212$06.00/ 0 © 1986 American Chemical Society

In Coulombic Interactions in Macromolecular Systems; Eisenberg, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.

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Ion Condensation Around Polyelectrolytes

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condensed i o n s o c c u p i e d a f i x e d and r a t h e r s m a l l volume around the p o l y e l e c t r o l y t e ( £ ) , and s o l u t i o n s o f the Poisson-Boltzmann (GouyChapman) e q u a t i o n (J5,6^, i n which the condensed i o n c l o u d was shown t o be d i f f u s e and to extend out t o d i s t a n c e s t h a t approached i n f i n i t y at i n f i n i t e d i l u t i o n . R e c e n t l y Marc Le B r e t and I (7_ Q) have been a b l e to remove most o f the s o u r c e s o f d i s a g r e e m e n t by examining i n d e t a i l the s o l u t i o n s o f the Poisson-Boltzmann e q u a t i o n w i t h the p o l y e l e c t r o l y t e modeled as an i n f i n i t e l y l o n g charged c y l i n d e r . In the s o l u t i o n s o f t h i s e q u a t i o n a t i n f i n i t e d i l u t i o n we f i n d a "bound", or "condensed", c l o u d o f c o u n t e r - i o n s , a c l o u d d e f i n e d by t h e f a c t t h a t e s s e n t i a l l y a l l o f i t remains w i t h i n a f i n i t e d i s t a n c e o f the p o l y - i o n even a t i n f i n i t e d i l u t i o n . The number o f i o n s i n the c l o u d i s e x a c t l y t h a t g i v e n by Manning's c o n d e n s a t i o n formula. (We put the words "bound" and "condensed" i n q u o t a t i o n marks because they have been used i n many d i f f e r e n t senses i n the p a s t ; here they a r e i n t e n d e d o n l y i n the sense j u s t d e f i n e d . We a r e not p r e p a r e d t o argue whether they are the best words f o r the purpose.) The d e t a i l s o f t h i s c a l c u l a t i o n have a l r e a d y been p u b l i s h e d (7_,8) so we p r e s e n t here o n l y a b r i e f d e s c r i p t i o n and the main c o n c l u s i o n s . The system t h a t we c o n s i d e r i s an i n f i n i t e l y l o n g c y l i n d r i c a l p o l y - i o n e n c l o s e d i n an o u t e r c o n c e n t r i c c y l i n d r i c a l c o n t a i n e r f i l l e d w i t h s o l v e n t and c o u n t e r - i o n s o f v a l e n c e ζ but w i t h no added s a l t ; t h i s i s one case i n which a n a l y t i c , as opposed t o n u m e r i c a l , s o l u t i o n s of the Poisson-Boltzmann e q u a t i o n are a v a i l a b l e . Numeri c a l s o l u t i o n s f o r the case where added s a l t i s p r e s e n t show much the same p i c t u r e , however, so t h i s l i m i t i n g case w i t h c o u n t e r - i o n s o n l y i s s t i l l of general i n t e r e s t . The Poisson-Boltzmann e q u a t i o n f o r t h i s system was s o l v e d l o n g ago (9>10). With t h i s s o l u t i o n , which g i v e s the c o n c e n t r a t i o n o f c o u n t e r i o n s a t any p o s i t i o n , we can proceed i n t h e f o l l o w i n g way t o f i n d t h e condensed f r a c t i o n o f c o u n t e r - i o n s . We c o n s i d e r the case where the l i n e a r charge d e n s i t y on the p o l y - i o n i s h i g h , so t h a t ξ i s g r e a t e r than 1/z. From the s o l u t i o n we can c a l c u l a t e the r a d i u s , r ( F ) , measured from the c y l i n d e r a x i s , t h a t c o n t a i n s a g i v e n f r a c t i o n , F, o f t h e c o u n t e r - i o n s , and we can f i n d the l i m i t i n g v a l u e o f t h i s r a d i u s as the average c o n c e n t r a t i o n o f c o u n t e r - i o n s goes t o z e r o , which happens when the o u t e r c y l i n d e r i s made i n f i n i t e l y large. When we do t h i s we f i n d t h a t the b e h a v i o r o f r ( F ) i s v e r y d i f f e r e n t depending on whether F i s g r e a t e r or l e s s than the F of Eq. (1). When F > F^ then


9

9

M

-1 /2 r ( F ) QC c which goes to i n f i n i t y as c, the average c o n c e n t r a t i o n , goes t o z e r o , but when F < F then M w

r(F) = a e x p [ F / ( ^ - 1 ) ( F

-F))]

which remains f i n i t e . Thus i f we take any f r a c t i o n l e s s than F of c o u n t e r - i o n s , t h i s f r a c t i o n remains w i t h i n a f i n i t e r a d i u s o f the p o l y - i o n a t i n f i n i t e d i l u t i o n ; t h e s e are the s o - c a l l e d condensed counter-ions. These c o u n t e r - i o n s form a c l o u d around the p o l y - i o n , a l l o f the c l o u d l y i n g w i t h i n a f i n i t e r a d i u s o f the p o l y - i o n . In c o n t r a s t , i o n s i n e x c e s s o f the f r a c t i o n F^ d i l u t e away to i n f i n i t y at I n f i n i t e d i l u t i o n . M


214

COULOMBIC INTERACTIONS IN MACROMOLECULAR SYSTEMS


Away f r o m i n f i n i t e d i l u t i o n , and even when some added s a l t i s p r e s e n t , t h e s i t u a t i o n o f t h e condensed c l o u d i s not much d i f f e r e n t . At f i n i t e average c o n c e n t r a t i o n s t h e c l o u d moves i n somewhat c l o s e r t o t h e p o l y - i o n , and o f c o u r s e t h e non-condensed i o n s a r e p r e s e n t also. The c o n c e n t r a t i o n o f c o u n t e r - i o n s a t t h e s u r f a c e o f t h e p o l y - i o n i s remarkably h i g h . F o r example, w i t h a p o l y - i o n w i t h t h e approximate c h a r a c t e r i s t i c s o f DNA (ξ-4 and a r a d i u s o f 1.25nm), t h e c o n c e n t r a t i o n o f u n i v a l e n t c o u n t e r - i o n s a t t h e s u r f a c e s t a y s near 3 m o l a r even a t i n f i n i t e d i l u t i o n , and t h i s c o n c e n t r a t i o n i s remark a b l y i n s e n s i t i v e to the o v e r a l l average c o n c e n t r a t i o n unless the l a t t e r exceeds 1 molar. The

Plane a l s o Shows " C o n d e n s a t i o n "

C o n d e n s a t i o n i n t h e above sense, a c l o u d o f i o n s r e m a i n i n g w i t h i n a f i n i t e d i s t a n c e even a t i n f i n i t e d i l u t i o n , i s not u n i q u e t o t h e i n f i n i t e l y l o n g charged c y l i n d e r , a l t h o u g h t h e phenomenon i s not u s u a l l y known by t h a t name. The mobile i o n s o f t h e double l a y e r n e x t t o a charged i n f i n i t e p l a n e s u r f a c e behave i n t h e same way. The s o l u t i o n o f t h e P o i s s o n - B o l t z m a n n e q u a t i o n f o r t h i s case has been known even l o n g e r than f o r t h e c y l i n d e r (_11_) ; f r o m i t we can c a l c u l a t e t h e b e h a v i o r of t h e mobile i o n s as t h e i r average c o n c e n t r a t i o n approaches z e r o . L e t Q be t h e amount o f f i x e d charge p e r u n i t a r e a on t h e s o l i d p l a n a r s u r f a c e , and l e t 1 be t h e s o - c a l l e d B j e r r u m l e n g t h d e f i n e d by β

1

2

B

= e /(4*e DkT) 0

and l e t X be a d i s t a n c e measured from t h e s o l i d s u r f a c e t o a p l a n e i n the l i q u i d . T h i s p l a n e and t h e s o l i d s u r f a c e e n c l o s e between them a c e r t a i n f r a c t i o n , F, o f t h e t o t a l m o b i l e - i o n charge, which t o t a l c h a r g e p e r u n i t a r e a o f s u r f a c e i s e q u a l t o and o f o p p o s i t e s i g n t o Q. As w i t h t h e c y l i n d e r , we can c a l c u l a t e X ( F ) , t h e d i s t a n c e c o r r e s p o n d i n g t o a g i v e n F, and take t h e l i m i t a t i n f i n i t e dilution. T h i s l i m i t i s (8) ( f o r a s y m m e t r i c a l e l e c t r o l y t e )

which has a f i n i t e v a l u e f o r any F l e s s than u n i t y . Thus almost a l l o f t h e m o b i l e - i o n charge s t a y s w i t h i n a f i n i t e d i s t a n c e X o f t h e s o l i d p l a n e s u r f a c e a t i n f i n i t e d i l u t i o n , and t h i s c h a r g e s a t i s f i e s t h e same d e f i n i t i o n o f "condensed" a s t h e f r a c t i o n F^ o f t h e c y l i n d e r case. The o n l y d i f f e r e n c e i s t h a t i n t h e case o f t h e p l a n e e f f e c t i v e l y a l l o f t h e charge i s "condensed", w h i l e i n t h e case o f the c y l i n d e r o n l y the f r a c t i o n F^, which has a v a l u e anywhere between 0 and 1 depending on t h e l i n e a r charge d e n s i t y on t h e c y l i n d e r , i s "condensed". I n c o n c l u s i o n , we can say t h a t t h e P o i s s o n - B o l t z m a n n d e s c r i p t i o n o f t h e condensed p o p u l a t i o n o f mobile i o n s near a charged c y l i n d e r i s s i m i l a r to the Poisson-Boltzmann d e s c r i p t i o n o f the m o b i l e i o n s i n t h e Gouy d i f f u s e double l a y e r a t a charged p l a n a r s u r f a c e , a d e s c r i p t i o n t h a t has been w e l l known f o r a l o n g time. I n b o t h c a s e s t h e i o n s a r e "bound", o r "condensed", i n t h e sense t h a t


17. ZIMM

Ion Condensation Around Polyelectrolytes

they cannot d i l u t e indefinitely.

215

away as t h e volume o f t h e system i s expanded

Literature Cited


1. 2. 3.

Manning, G.S. J. Chem. Phys. 1969, 51, 924-933, 3249-3252. Manning, G.S. Q. Rev. Biophys. 1978, 11, 179-246. Anderson, C.F.; Record, M.T. Ann. Rev. Phys. Chem. 1982, 33, 191-222. 4. Manning, G.S. Biophys. Chem. 1977, 7, 95-102. 5. MacGillivray, A.D. J. Chem. Phys. 1972, 56, 81-83, 83-85. 6. Gueron, M.; Weisbuch, M. Biopolymers 1980, 19, 353-382. 7. Le Bret, M.; Zimm, B.H. Biopolymers 1984, 23, 287-312. 8. Zimm, B.H.; Le Bret, M. J . Biomolec. Structure and Dynamics 1983, 1, 461-471. 9. Alfrey, T.; Berg, P.W.; Morawetz, H. J. Polymer Sci. 1951, 7, 543-547. 10. Fuoss, R.M.; Katchalsky, Α.; Lifson, S. Proc. Nat. Acad. Sci., U.S. 1951, 37, 579-589. 11. Verwey, E.J.W.; Nissen, K.F. Phil. Mag. 1939, (7)281, 435-446. RECEIVED

June 5, 1985


Use of the Poisson-Boltzmann Equation To Predict Ion Condensation

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