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An Anderson model for electrosorption
J. Electroanal. Chem., 185 (1985)253-261
253
Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
AN ANDERSON
MODEL
FOR ELECTROSORPTION
A.A. KORNYSHEV A. iV. Frumkin Institute o f Electrochemistry o f the Academy o f Sciences of the U.S.S.R., Leninskii Prospekt 31, Moscow (U.S.S.IL)
W. SCHMICKLER lnstitut f-ur Ph),sikalische Chemic, Unioersitftt Bonn, Wegelerstr. 12, D - 5 3 0 0 Bolm (F.R.G.)
(Received 21st June 1984; in revised form 19th November 1984)
ABSTRACT The Anderson model is applied to adsorption on a metal electrode from an electrolyte solution. Self-consistent equations are derived for the partial charge on an adsorbate. Numerical model calculations are performed for a series of s- and p.elech-on systems, ta~dng into account the interaction with the solvent. The r~uits are in line with experimentally dete .rTnined electrosorption valencies.
INTRODUCTION T h e A n d e r s o n m o d e l h a s b e e n w i d e l y a p p l i e d to t h e a d s o r p t i o n o f a t o m s o n m e t a l s u r f a c e s in a v a c u u m [1]. I t s b a s i c i d e a is t h a t t h e m e t a I - a d s o r b a t e i n t e r a c t i o n b r o a d e n s a n d s h i f t s t h e e l e c t r o n i c levels o f t h e a d s o r b a t e s ; t h e s e b r o a d e n e d l e v e l s a r e filled u p to t h e F e r m i level b y e l e c t r o n s s h a r e d w i t h the m e t a l , s o t h a t t h e a d s o r b a t e g e n e r a l l y c a r r i e s a p a r t i a l c h a r g e . W i t h i n t h e f r a m e w o r k o f this t h e o r y , w o r k f u n c t i o n c h a n g e s o n a d s o r p t i o n , t h e i r c o v e r a g e d e p e n d e n c e , a n d t r e n d s in adsorption energies can be rationalized. The.par~lelism between processes at the metal/vacuum and the metal/electrol y t e i n t e r f a c e h a s b e e n r e p e a t e d l y n o t i c e d in r e c e n t t i m e s . W e s h o u l d e x p e c t s i m i l a r m e c h a n i s m s to o p e r a t e ha t h e a d s o r p t i o n p r o c e s s e s in t h e s e s y s t e m s . So, in this w o r k we s h a l l a p p l y t h e A n d e r s o n m o d e l to t h e a d s o r p t i o n at t h e m e t a l / e l e c t r o l y t e i n t e r f a c e . H e r e , t h e s i t u a t i o n is c o m p l i c a t e d b y t h e i n t e r a c t i o n o f t h e a d s o r b a t e wi_.th t h e s o l v e n t , w h i c h is s t r o n g a n d o f l o n g r a n g e d u e t o i t s p r e d o m i n a n t l y c 0 u l o m b i c nature. : -.. W e shall thus e x t e n d the A n d e r s o n m o d e l a n d include terms for the interaction of t h e a d a t o m l w i t h t h e s o l v e n t m o d e s in t h e e f f e c t i v e H a m i l t o n i a n . U s i n g s i m p l e approximations, we shall derive an expression for the charge on the adatom, which d e p e n d s o n - t h e characteristics of the interaction. T o test the validity of the model, we shall g i v e estimates for these quantities, a n d c a l c u l a t e the c h a r g e - t r a n s / e r
0022-072g/85/$03.30
© 1985 Elsevier Sequoia S.A_
254
c o e f f i c i e n t s for a n u m b e r o f s y s t e m s for w h i c h e x p e r i m e n t a l d a t a exist. H e r e we shall limit o u r s e l v e s t~ the c a s e o f s i n g l e - a d a t o m a d s o r p t i o n ; a d a t o m - a d a t o m i n t e r a c tions, w h i c h b e c o m e i m p o r t a n t at h i g h e r c o v e r a g e s , will b e c o n s i d e r e d in a s u b s e q u e n t p u b l i c a t i o n [2]. T h e A n d e r s o n m o d e l h a s p r e v i o u s l y b e e n a p p l i e d to the o l e c t r o s o r p t i o n o f alkali i o n s b y o n e o f us [3]. T h e m a i n d i f f e r e n c e b e t w e e n this a n d the earlier w o r k is t h a t w e shall t r e a t a m u c h w i d e r class o f s y s t e m s a n d t h a t we shall a c c o u n t f o r the i n t e r a c t i o n o f the a d a t o r n s w i t h the fast s o l v e n t m o d e s , w h i c h w a s . p r e v i o u s l y neglected. THE MODEL HAMILTONIAN
W e c o n s i d e r a single a t o m a d s o r b e d at a m e t a l / e l e c t r o l y t e i n t e r f a c e . W e restrict o u r s e l v e s to the c a s e w h e r e o n e a d a t o m o r b i t a l i n t e r a c t s with the m e t a l , a n d w e d e n o t e b y c a the c o r r e s p o n d i n g a d a t o m e l e c t r o n i c energy. F o l l o w i n g N c w n s [4], we w r i t e the t e r m s for the v a l e n c e e l e c t r o n s o n the a d a t o m as:
(1)
~a ~ ' ~ ° + U.~ona_° o
w h e r e U is the c o u l o m b r e p u l s i o n e n e r g y o f t w o v a l e n c e electrons, o is the spin index, a n d the n ' s d e n o t e the usual o c c u p a t i o n o p e r a t o r s . T h e t e r m s f o r the m e t a l e l e c t r o n s a n d for e l e c t r o n e x c h a n g e b e t w e e n the m e t a l a n d the a d a t o m h a v e the usual f o r m :
ko
"ka
w h e r e k d e n o t e s a set o f q u a n t u m n u m b e r s f o r the m e t a l electrons, Izk is the m a t r i x e l e m e n t for e l e c t r o n t r a n s f e r f r o m the a d a t o m to the m e t a l , a n d c + a n d c d e n o t e the c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s for the s t a t e s i n d i c a t e d . I n c o n s i d e r i n g the a d a t o m - s o l v e n t i n t e r a c t i o n , we d i s t i n g u i s h b e t w e e n the librational a n d v i b r a t i o n a l m o d e s o f the solvent, w h i c h are slow c o m p a r e d to the f r e q u e n c i e s o f the e l e c t r o n i c t r a n s i t i o n s , a n d the fast m o d e s d u e to the e l e c t r o n i c p o l a r i z a b i l i t y o f the s o l v e n t molecules. I n the h a r m o n i c a p p r o x i m a t i o n , we c a n w r i t e the H a m i l t o n i a n f o r the slow s o l v e n t m o d e s a n d their i n t e r a c t i o n w i t h the a d s o r b a t e in the f o r m f a m i l i a r f r o m e l e c t r o n t r a n s f e r t h e o r y [5]:
)(
)
z--~.,nao ~_,t~to~ga~.q~ ~"
a
r.,
(3)
H e r e , ~ labels the slow s o l v e n t m o d e s o f f r e q u e n c y t%, m o m e n t u m pp a n d c o o r d i n a t e q~; z is the c h a - g e n u m b e r o f the a t o m ' s core, a n d g~. a r e the c o u p l i n g c o n s t a n t s f o r the i n t e r a c t i o n . T h r o u g h t h e s e t e r m s , the a d s o r b a t e e l e c t r o n i c e n e r g y is a f u n c t i o n o f the s o l v e n t c o o r d i n a t e s q.. T o c a l c u l a t e t h e e q u i l i b r i u m p r o p e r t i e s o f the s y s t e m , o n e c a n follow K r a t s o v a n d M a l s h u k o v [6], c o n s i d e r q~ as the e x t e r n a l p a r a m e t e r s o f the e l e c t r o n i c H a m i l t o n i a n , a n d c o n s i d e r t h e s o l v e n t c o n f i g u r a t i o n f o r w h i c h the e l e c t r o n i c e n e r g y o f the s y s t e m is a m i n i m u m . T h i s l e a d s to t h e c o n d i t i o n :
255
all(q.___..~)) = 0
(4)
( Oq. w h e r e the H e l l m a n - F e y n m a n t h e o r e m has been used. F r o m this, we o b t a i n for the e x p e c t a t i o n values o f the solvent c o o r d i n a t e s in tii-~ m i n i m a l e n e r g y c o n f i g u r a t i o n :
Alternatively, we can follow ref. 3, i n v o k e the H a r t r e e - F o c k a p p r o x i m a t i o n , a n d calculate the e x p e c t a t i o n values. b y replacing the e l e c t r o n i c o p e r a t o r s n , ° in eqn. (3) by their e x p e c t a t i o n values. W i t h i n the h a r m o n i c a p p r o x i m a t i o n , b o t h p r o c e d u r e s give the s a m e results. T h e slow solvent m o d e s thus c o n t r i b u t e a term:
))
,6,
to the e l e c t r o n i c H a m i l t o n i a n . Finally, we c o n s i d e r the i n t e r a c t i o n with the fast p o l a r i z a t i o n m o d e s o f the interface. N o r m a l l y , b o t h the s u r f a c e p l a s m a m o d e s o f the metal a n d the e l e c t r o n i c solvent m o d e s are f a s t e r t h a n the e l e c t r o n e x c h a n g e b e t w e e n the metal a n d the a d s o r b a t e s [12]. W e shNl assume, in the following, that this c o n d i t i o n holds. T h e n these i n t e r a c t i o n s shift the a d s o r b a t e level e= a n d the c o r r e l a t i o n e n e r g y U, as was s h o w n b y I-tewson a n d N e w n s [7]. E s t i m a t e s for the shifted values ga a n d 0 w411 be given in the next section. T o e s t i m a t e the o c c u p a t i o n p r o b a b i l i t y o f the a d s o r b a t e level, we m a k e the simplifying a s s u m p t i o n that the level width:
:, ( , 0 ) = ~Y'.I Vk I"-6( ~' - ~ )
(7)
k
is i n d e p e n d e n t o f the e l e c t r o n i c e n e r g y ~o. C o n s i d e r i n g o n l y n o n - m a g n e t i c s o l u t i o n s in a restricted H a r t r e e - F o c k a p p r o x i m a t i o n so that b o t h spin states in the orbital are o c c u p i e d to the s a m e extent, the effective e n e r g y level E~ of a n orbital is:
E~=~
+ 0 ( , , > + 2(~ - 2 ( , , > ) E s
(8)
where t,
is the e n e r g y o f s o l v a t i o n o f an a d s o r b a t e with unit charge; it can also be i n t e r p r e t e d as the e n e r g y o f solvent r e o r g a n i z a t i o n o f the a d s o r b a t e , a c o n c e p t familiar f r o m e l e c t r o n t r a n s f e r theory. ( n > = ( n ~ o ) is the o c c u p a t i o n p r o b a b i l i t y o f an e l e c t r o n i c a d s o r b a t e state, which is given b y the farr'Aliar relation [1]:
(,,) = L cot - ' E~ - EF ~r
A
(9)
256
w h e r e E F is the F e r m i e n e r g y o f the metal. W h e n the r e l e v a n t s y s t e m p a r a m e t e r s a r e k n o w n , the o c c u p a t i o n p r o b a b i l i t y ( n ) c a n be c a l c u l a t e d f r o m eqns. (8) a n d (9) in a self-consistent manner. It h a s b e e n p o i n t e d o u t b y K r a t s o v a n d M a l s h u k o v [6] t h a t eqn. (9) m a y h a v e several solutions. In o u r case, eqn. (9) m a y h a v e m o r e t h a n o n e s o l u t i o n if the relation: 4E,- /.7/ > 1 (10) A~ is fulfilled; this will be the case if the i n t e r a c t i o n with the s o l v e n t is s t r o n g a n d t h a t with the m e t a l is w e a k . In the following, we shall a l w a y s give the s o l u t i o n w i t h the lowest energy, if m o r e t h a n o n e s o l u t i o n exists. A n e x p l o r a t i o n o f the c o n s e q u e n c e s o f the e x i s t e n c e o f several s o l u t i o n s will b e g i v e n e l s e w h e r e [8]. ESTIMATES OF THE SYSTEM PARAMETERS
In o r d e r to c a l c u l a t e the c h a r g e d i s t r i b u t i o n in a p a r t i c u l a r a d a t o m / s u b s t r a t e . s y s t e m , we m u s t derive e x p r e s s i o n s for the v a r i o u s q u a n t i t i e s t h a t d e t e r m i n e the effective a d a t o m e l e c t r o n i c level a c c o r d i n g to eqn. (8). T h e m o d i f i e d a d s o r b a t e level ~ a n d the C o u l o m b r e p u l s i o n 0 c a n b e e s t i m a t e d in the f o l l o w i n g w a y : Let us c o n s i d e r the case where, o n the n e u t r a l a t o m , the a d s o r b a t e o r b i t a l is h a l f - f i l l e d - - t h i s will b e so in all the s y s t e m s c o n s i d e r e d b e l o w e x c e p t for the a d s o r p t i o n o f lead. T h e h y p o t h e t i c a l e n e r g y r e q u i r e d to ionize the n e u t r a l a d s o r b e d a t o m while k e e p i n g the slow solvent m o d e s fixed is: ~ = - - I I + tim + Er (11) w h e r e I~ is the e n e r g y o f i o n i z a t i o n o f the a t o m in v a c u o , ~im is the s e l f - e n e r g y d u e to the fast p o l a r i z a t i o n m o d e s o f the interface, a n d ef is the i n t e r a c t i o n e n e r g y w i t h the fast b u l k s o l v e n t m o d e s , b o t h t a k e n for a singly c h a r g e d ion. Similarly, the e n e r g y r e q u i r e d to a d d a n e l e c t r o n is: (12)
- - ~ a - - 0 = A -at- ~im "~ ~f
w h e r e A is the e l e c t r o n a f f i n i t y o f the a t o m . F r o m these r e l a t i o n s we derive:
--~'| q'- (ira "1- ( f 0=~il--A - - 2 ( e l m + e l ) (13) F o r the a d s o r p t i o n o f lead, w h e r e the v a l e n c e p - o r b i t a l is d o u b l y o c c u p i e d in the n e u t r a l state, we o b t a i n instead: ~a = - l n + 3or + 3 E l m , a n d 0---- I 2 -- 1 i -- 2~ r - - ~-~irn; 12 is the s e c o n d i o n i z a t i o n energy. F o r the c a l c u l a t i o n o f the i n t e r a c t i o n with the s o l v e n t m o d e s , we n e e d a n explicit m o d e l for the s o l v a t i o n o f a d a t o m s . W e shall use the n o n - l o c a l dielectric m o d e l o f D o g o n a d z e a n d K o r n y s h e v [9]; here, o n e d i s t i n g u i s h e s b e t w e e n the electronic, the v i b r a t i o n a l , a n d the l i b r a t i o n a l solvent m o d e s , which h a v e d i f f e r e n t c o r r e l a t i o n l e n g t h s X t, X 2 a n d X 3, respectively. T h e e l e c t r o n i c m o d e s are fast, the o t h e r t w o are slow. T h e i n t e r a c t i o n e n e r g y o f a fully s o l v a t e d ion with the slow b u l k m o d e s is:
(a =
A ~tG~.... =
( z1e ° )i22 {r ( coP,
1)F(A2/ri)+(-~i~ - l)F(A3/ri))
(14a)
257
w h e r e ¢~ is the i n f r a r e d , cop, the optical, a n d c the static dielectric c o n s t a n t (cir = 4.9 for water); r i is the ionic radius. F o r the fast m o d e s , this e n e r g y is: AGhast =
(2e°)a2ri (I-
",,ptl
)F(Xl/r,)
F(x) = "l--(1-
e-X)/x
(141o)
F o r n u m e r i c a l calculations, we h a v e taken: AI = 0.I rim, A2 = 0.12 rim, X3 = 0.68 n m in a c c o r d with ref. 9. E q u a t i o n s (14a) a n d (14b) t o g e t h e r give quite g o o d values f o r the energies o f h y d r a t i o n o f alkali a n d halide ions, b u t they d o not w o r k q u i t e so well for o t h e r m e t a l ions. W e r e q u i r e the energies o f i n t e r a c t i o n o f a singly c h a r g e d a d s o r b a t e with the slow a n d the fast solvent m o d e s . T o a c c o u n t for the fact that the a d s o r b a t e s are o n l y p a r t i a l l y solvated a n d for the deficiencies o f the s o l v a t i o n m o d e l , we have set:
Es:aX{( 1EopL
~i~1) F ( ~ t ~ d / r ) + (
ef=etX(1--Coptl )F(X1/r )
1E.~
1) (15)
w h e r e a is the d e g r e e o f s o l v a t i o n o f the a d a t o m a n d the c o n s t a n t X has b e e n fixed in such a way that - - z 2 ( E , + ¢ r ) / a e q u a l s the e x p e r i m e n t a l e n e r g y o f solvation, AGS, o f the ion; f o r the alkali a n d halide ions, X is close to the theoretical value o f W e have p e r f o r m e d n u m e r i c a l c a l c u l a t i o n s for a = 1 / 3 a n d a = 2 / 3 ; the f o r m e r value r o u g h l y c o r r e s p o n d s to a d s o r p t i o n o n a step site, the latter to a d s o r p t i o n o n a flat surface (terrace). By setting t x = 0 , we get the case for a d s o r p t i o n f r o m the v a c u u m . Finally, we h a v e to specify the i n t e r a c t i o n with the fast p o l a r i z a t i o n m o d e s o f the interface. F o r the m e t a l / s o l v e n t interface, n o e x p r e s s i o n b a s e d o n q u a n t u m - m e c h a nical c a l c u l a t i o n s exists. W e use a p h e n o m e n o l o g i c a l e x p r e s s i o n d e r i v e d f r o m a s p a c e dispersion m o d e l [10]:
eo/2r.
eo
(Cop t -- 1) -- ~:2r 2
¢im= 4¢op ' (cop , + 1) + K2r 2
(16)
w h e r e .~ is the inverse T h o m a s - F e r m i s c r e e n i n g length; we h a v e taken 1 / ~ = 0.05 n m for n u m e r i c a l calculations, which is the typical value. T h e radius r o f the a d a t o m d e p e n d s o n its charge. W e have used the f o l l o w i n g i n t e r p o l a t i o n f o r m u l a d u e to B r o w n [11]:
r=[r~(z-q)+r'3q]'/3
(17)
w h e r e r, a n d r; are the radii o f the a t o m a n d ion, respectively; z is the c h a r g e n u m b e r o f the ion arid q that o f the a d s o r b a t e .
258
ESTIMATES FOR THE CHARGE TRANSFER
IN PARTICULAR
SYSTEMS
Using the formulae presented above, we have estimated the charge o n a number of adsorbates for which experimental data are available. Since the version of the Anderson model that we have used here works best for w e a k adsorbateLsubstrate interaction, we have used here works best for weak adsorbate-substrate interaction , we have generally performed calculations for the adsorption o n mercury electrodes. For several adsorbates, experimental data on mercury are not available. The data employed in the estimates are given in Table 1, and in Table 2 o n e finds the values calculated for the charge transfer h o n adsorption, i.e. the charge number z of the ion minus the charge number q of the adsorbate. We have performed calculations for level widths A = 0.5 and 1 eV, which comprises the range usually observed in field emission experiments [1,12]. We have not studied the potential dependence of the charge transfer. Our calculations refer to the situation where the outer potential drop between the metal and the adatom vanishes; the corresponding electrode potential is generally not too far from the potential of zero charge (pze). Unfortunately, the charge transfer h on adsorpfion is not a measurable quantity. It is, however, closely ,,elated to the so-called electrosorption valency 3' introduced by Vetter and Schultze [13]. Neglecting comparatively small dipole terms, we have: 3"~q= gz + X(1 - - g ) (18) where "rN is the electrosorption valency at the pzc and g is a small geometrical TABLE 1 D a t a for the e s t i m a t i o n o f c h a r g e - t r a n s f e r c o e f f i c i e n t s " ion
Ii b / e V
A/eV
10 r i ¢ / n m
10
K +
4.34 4.18 3.89
0.47 d 0.42 c 0.39 d
1.33 1.48 1.68
2.26 • 2.42 = 2.61 c
3.51 3.33 3.09
12.96 11.81 10.45
3.61 b 3.36 b 3.06 b
1.81 1.95 2.16
0.99 c 1.14 ~ 1.33
3.29 3.15 2.67
7.72 7.57 6.11 7.41
1.80 t 0.88 g
0.96 1.26 1.40 1.20
1.27 1.44 1.73 1.75
5.92 5.02 3.57 15.6
Rb ÷ Cs ~ CI-
Br1C u 2+ Ag*
Ti + P b 2+
h
r~/nm
~ ~ ~ ~
AG~/eV
a 11 a n d A are the first i o n i z a t i o n e n e r g y a n d the electron a f f i n i t y o f the neutral a t o m ; A w a s set to z e r o w h e r e it w a s n o t available. W o r k f u n c t i o n s : H g 4.5 e V , P t 5 . 0 3 e V , A u 4 . 8 0 e V , A g 4 . 3 0 t:V ( e l e c t r o c h e m i c a l values after Trasatti [15l). b F r o m C o t t o n a n d W i l k i n s o n [16]. ¢ A f t e r P a u l i n g [17]. d A f t e r Weiss [181.
E s t i m a t e d f r o m the lattice c o n s t a n t o f the e l e m e n t , taken f r o m A s h e r o f t a n d M e r m i n [19]. t A f t e r C l e m e n t i [20]. s A f t e r M o i s e i w i t c h [21]. h 12 = 15.03 e V .
259 TABLE
2
Calculated charge-transfer coefficients System
2k
"YN a
A=0.5eV vae.
A=I et = 1 / 3
a = 2/3
eV ~t = 1 , / 3
~t = 2 , / 3
K/Hg Rb/Hg Cs/Hg
0.15 0.15 0.14
0.12 0.12 0.11
0.08 0.08 0.07
0.30 0.29 0.27
0.25 0.24 0.23
0.16 0.16 0.16
0.12 0.15 0.18
Cl/Hg Br/Hg I/Hg
- 0.21 -- 0 . 2 6 --0.34
- 0.17 -- 0 . 2 0 -0.38
- 0.10 - 0.12 -0.17
-- 0 . 3 2 - 0.38 --0.47
- 0.27 -- 0 . 3 2 -0.41
- 0.19 - 0.22 --0.30
- 0.2 -- 0 . 3 4 -0.45
Cu/Pt Ag/Au Tl/Pt Pb/Ag
1.85 0.73 0.26 1.81
1.42 0.47 0.19 1.83
1.90 0.80 0.42 1.69
1.77 0.68 0.35 1.71
1.16 0.26 0.21 1.45
1.09 b 0.12 0.10 b 1.70 b
vac.
1.8 0.6 c 0.85 2.0
a E l e e t r o s o r p t i o n v a l e n c S e s w e r e t a k e n f r o m K o p p i t z a n d S c h u l t z e [22] u n l e s s o t h e r . v i s e i n d i c a t e d . b More than one solution exists; the one with the It:vest energy has been chosen. c T a k e n f r o m M e l n i e k e e t al. [23].
f a c t o r that describes the p e n e t r a t i o n o f the a d s o l b a t e i n t o the d o u b l e - l a y e r field. F o r cations, the c h a r g e t r a n s f e r ~k is, in general, positive, for a n i o n s negative, i.e. the ions are partially d i s c h a r g e d o n a d s o r p t i o n . C o n s e q u e n t l y we h a v e I vNI > IX I- T h e e x p e r i m e n t a l values for YN in T a b l e 2 are for sma21 coverages, since o u r m o d e l , in which a d a t o m - a d a t o m i n t e r a c t i o n s are neglected, refers to this situation. T h e a d a t o m s investigated fall into three classes, w h i c h we shall discuss s e p a r a t e l y . A l k a l i ions
T h e s e are c h a r a c t e r i z e d b y a small i o n i z a t i o n energy, so that the e n e r g y o f the v a l e n c e e l e c t r o n lies a b o v e the F e r m i level o f m e r c u r y . H e n c e the o c c u p a t i o n p r o b a b i l i t y o f this level is small, a n d the alkali ions k e e p their positive c h a r g e a l m o s t c o m p l e t e l y o n a d s o r p t i o n . T h e ionic state is also f a v o u r e d b y the i n f l u e n c e o f the solvation a n d b y i m a g e forces. T h e r e f o r e the a d a t o m s in e l e c t r o c h e m i c a l s y s t e m s always c a r r y a g r e a t e r c h a r g e than the s a m e a d a t o m s a d s o r b e d f r o m the v a c u u m , a n d the c h a r g e increases with the d e g r e e o f s o l v a t i o n ; this is t r u e for ~11 systems. F r o m p h o t o e m i s s i o n data, the level w i d t h A o f alkali a d s o r b a t e is e s t i m a t e d to b e a b o u t 0.5 eV [12]; o n liquid m e r c u r y , we w o u l d e x p e c t the a d a t o m s to i n t e r a c t still relatively strongly, since the s u r f a c e s h o u l d n o t c o n t a i n a n y steps, so that ot = 2 / 3 w o u l d b e a b e t t e r estimate. T h e c o r r e s p o n d i n g c h a r g e - t r a n s f e r c o e f f i c i e n t s are in line with the o b s e r v e d e l e c t r o s o r p t i o n valencies. T h e y are also o f the s a m e o r d e r o f m a g n i t u d e as those e s t i m a t e d in a p r e v i o u s p u b l i c a t i o n [3]. H a l i d e ions
F o r these systems, we h a v e a s s u m e d that the p - o r b i t a l p e r p e n d i c u l a r to the m e t a l surJ'ace, which f o r m s the o - b o n d , gives the d o m i n a n t c o n t r i b u t i o n to the b o n d , a n d
260
we h a v e n e g l e c t e d the i n t e r a c t i o n with the o t h e r o r b i t a l s . T h e h a l i d e a t o m s h a v e high i o n i z a t i o n energies a n d e l e c t r o n affinities. H e n c e the o u t e r p - o r b i t a l s a r e a l m o s t c o m p l e t e l y filled, a n d the a d s o r b e d ions still c a r r y m o s t o f their n e g a t i v e charge. Since the i o n i z a t i o n energies, the e l e c t r o n affinities, a n d the energies Of s o l v a t i o n d e c r e a s e in the series f r o m F to I, there is a n a c c o m p a n y i n g d e c r e a s e o f the n e g a t i v e c h a r g e on the a d s o r b e d ions. T h e e l e c t r o s o r p t i o n v a l e n c i e s follow the s a m e trend, a n d a r e in line w i t h values o f A o f the o r d e r o f 0.5 - 1 eV. O t h e r m e t a l ions
T h e third g r o u p c o m p r i s e s m e t a l ions w h i c h p l a y a role in u n d e r p o t e n t i a l d e p o s i t i o n : A g +, C u 2+, P b 2+, Tl +. In e a c h case, we h a v e o n l y c o n s i d e r e d t h e i n t e r a c t i o n o f the e n e r g e t i c a l l y highest o r b i t a l with the s u b s t r a t e . T h e energies o f i o n i z a t i o n o f the a t o m s lie in b e t w e e n those for the alkali a n d t h o s e for the h a l i d e ions, a n d so we o b t a i n sizable c h a r g e - t r a n s f e r coefficients. T h e e x p e r i m e n t a l values f o r the e l e c t r o s o r p t i o n valencies are for p o l y c r y s t a l l i n e e l e c t r o d e s a n d thus p r o b a b l y c o r r e s p o n d to a d s o r p t i o n o n s t e p sites, so t h a t the d e g r e e o f stflvation s h o u l d b e small. Also, m o s t e x p e r i m e n t a l d a t a for u n d e r p o t e n t i a l d e p o s i t i o n a r e for high c o v e r a g e s , w h e r e the a d a t o m s are a l m o s t c o m p l e t e l y d i s c h a r g e d d u e to C o u l o m b r e p u l s i o n a n d lateral i n t e r a c t i o n s . O n l y in a few cases h a s the e l e c t r o s o r p t i o n v a l e n c y b e e n s t u d i e d as a f u n c t i o n o f the c o v e r a g e ; these s t u d i e s i n d i c a t e t h a t at small c o v e r a g e s X is s m a l l e r t h a n the c h a r g e n u m b e r z o f the ion, b u t t h a t h t e n d s to z as the c o v e r a g e a p p r o a c h e s unity; i.e. the ions a r e d i s c h a r g e d with i n c r e a s i n g c o v e r a g e . O u r c a l c u l a t e d c h a r g e - t r a n s f e r c o e f f i c i e n t s fit r a t h e r well i n t o this picture. DISCUSSION C o n s i d e r i n g the s i m p l i c i t y o f the m o d e l , the c a l c u l a t e d c h a r g e - t r a n s f e r coefficients fit the e x p e r i m e n t a l d a t a s u r p r i s i n g l y well. T h e e l e c t r o s o r p t i o n valencies o b s e r v e d are g e n e r a l l y in line with the partial c h a r g e - t r a n s f e r c o e f f i c i e n t s c a l c u l a t e d for r e s o n a n c e w i d t h s o f 0 . 5 - 1 . 0 eV, which is also the r a n g e o f line w i d t h s o b s e r v e d in field e m i s s i o n e x p e r i m e n t s . P e r h a p s e v e n m o r e i m p o r t a n t t h a n this overall a g r e e m e n t is the fact t h a t the o b s e r v e d t r e n d s are c o r r e c t l y r e p r o d u c e d . In p a r t i c u lar, the d i f f e r e n t b e h a v i o u r o f the alkali ions, w h i c h r e m a i n highly p o s i t i v e l y c h a r g e d , the halide ions, w h i c h c a r r y a high n e g a t i v e charge, a n d the o t h e r m e t a l ions, w h i c h h a v e a n i n t e r m e d i a t e p o s i t i v e charge, is b o r n e o u t in the c a l c u l a t i o n s . Also, t r e n d s w i t h i n these g r o u p s a r e c o r r e c t l y p r e d i c t e d ; thus, the h a l i d e ions b e c o m e less c h a r g e d as we g o f r o m F - to I - . T h i s success o f the m o d e l is p r o b a b l y m a i n l y d u e to the fact t h a t the m o s t i m p o r t a n t p a r a m e t e r s in the m o d e l are the e n e r g i e s o f i o n i z a t i o n o f the a t o m s , the w o r k f u n c t i o n o f the m e t a l s u b s t r a t e , a n d f o r the h a l i d e a n d alkali ions, also the e l e c t r o n affinities, for which reliable d a t a exist. W h i l e the s o l v a t i o n p a r a m e t e r s E s arid h r are i m p o r t a n t , in t h a t t h e y f a v o u r the c h a r g e d s t a t e o f the a d a t o m , they d o n o t v a r y s o m u c h a n d thus d o not d i s t u r b the p a t t e r n g e n e r a t e d b y the o t h e r parameters.
261
It is of interest to c o m p a r e our work with the results obt ai ned by ot her methods. Recently, Weissmann and C o h a n [14] have studied the adsorpt i on of F - , C I - and K + on small clusters of silver by the iterative e x t e n d e d H~ekel m et hod. Qualitatively, they observed the same trends as we did: a d a t o m s of F and CI carry negative charges, F being m o r e highly charged than CI, and K carries a large positive charge. As we have pointed out, our present work contains a n u m b e r of simplifying assumptions, which should be studied and improved upon in future work. We think that our results are encouraging and that they demonstrate the applicability of the
A n d e r s o n model to electrochemical interfaces. ACKNOWLEDGEMENTS
We would like to thank Professor Dr. J.W. Schultze, D~sseldorf, for useful discussions on electrosorption valencies. Financial support by the D eut sche Forschungsgemeinschaft and by the Verband der Chemischen Industrie is gratefully acknowledged. A.A.K. aknowledges the hospitality of the University of Diasseldorf, where the m aj or part of this work was perform ed. REFERENCES 1 F o r a review of this model see: J.P. Muscat and D.M. Newns, Progr. Surf. Sci., 9 (1978) 1; J.P. G a d z u k in J.M. Blakely (Ed.), Surface Physics of Materials, Vol. 2, Academic Press, New York, 1975. 2 A. Kornyshev and W. Schmiclder, manuscript in preparation. 3 W. Sehmiekler, J. Eleetroanal. Chem., 100 (1979) 533. 4 D.M. Newns, Phys. Lett., 33A (1970) 43; Phys. Rev. Lett., 25 (1970) 1575. 5 For a recent review see: J. Ulstrup, Charge Transfer Processes in C o n d e n s e d Media, Springer Verlag, Berlin, Heidelberg, New York, 1979. 6 V . ~ Kratsov and A.G. Malshukov, Soy. Phys. JETP, 48 (1978) 248 (English translation). 7 A.C. Hewson a n d D.M. Ne-..-'as, J. Appl. Phys. Suppl., 2 (1974) 2121. 8 W. Schmickler, Chem. Phys. Lett., in press. 9 R.R. Dogonadze a n d A.A. Kornyshev, J. Chem. Soc. F a r a d a y Trans. 2, 70 (!0,74) !121. 10 M.A. Vorotyntsev, A.A. Kornyshev and A.I. Rubinstein, Elektrokhim:,ya, 13 (19"77) 1767; A..A. Kornyshev, Electrochim. Acta, 26 (1981) 1. 11 O.M. Brown, private communication. 12 J.W. Gadzuk, J.K. H a r t m a n n a n d T.N. Rhodin, Phys. Rev., B4 (1971) 241. 13 K d . Vetter and J.W. Schultze, Ber. Bunsenges. Phys. Chem., 76 (1972) 920, 927. 14 M. Weissmann and N.V. Cohan, J. E!ectroanal. Chem., 163 (!984) 381. 15 S. Trasatti in H. Gerischer a n d C.W. Tobias (Eds.), A d v a n c e s in Electrochemistry a n d Electrochemical Engineering, Vol. 10, Wiley-lnterscience, New York, 1977. 16 F.A. Cotton a n d G. Wilkinson, A d v a n c e d Inorganic Chemistry, 3rd ed., lnterseience, New York, 1977. 17 L. Pauling, The N a t u r e of the Chemical Bond, 3rd ed., Cornell University Press, Ithaca, NY, 1960. 18 A.W. Weiss, Phys. Rev., 166 (1968) 76. 19 N.W. Ashcroft a n d N.D. Mermin, Solid State Physics, ltolt, Rinehart and Winston, Philadelphia, 1976. 20 E. Clementi, Phys. Rev., 135 (1964) A980. 21 B.L. Moiseiwitch, Adv. Atomic Mol. Phys., 1 (1965) 327. 22 F.D. Koppitz a n d J.W. Schultze, Eleetrochim Aeta, 21 (1976) 327. 23 L.S. Melnieke, T.M. R i e d h a m m e r a n d S. Bruckenstein, Proceedings of the 3rd Symposium o n Electrode Processes 1979, T h e Electrochemical Society, Princeton, N J, 1980, p. 306.
253
Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
AN ANDERSON
MODEL
FOR ELECTROSORPTION
A.A. KORNYSHEV A. iV. Frumkin Institute o f Electrochemistry o f the Academy o f Sciences of the U.S.S.R., Leninskii Prospekt 31, Moscow (U.S.S.IL)
W. SCHMICKLER lnstitut f-ur Ph),sikalische Chemic, Unioersitftt Bonn, Wegelerstr. 12, D - 5 3 0 0 Bolm (F.R.G.)
(Received 21st June 1984; in revised form 19th November 1984)
ABSTRACT The Anderson model is applied to adsorption on a metal electrode from an electrolyte solution. Self-consistent equations are derived for the partial charge on an adsorbate. Numerical model calculations are performed for a series of s- and p.elech-on systems, ta~dng into account the interaction with the solvent. The r~uits are in line with experimentally dete .rTnined electrosorption valencies.
INTRODUCTION T h e A n d e r s o n m o d e l h a s b e e n w i d e l y a p p l i e d to t h e a d s o r p t i o n o f a t o m s o n m e t a l s u r f a c e s in a v a c u u m [1]. I t s b a s i c i d e a is t h a t t h e m e t a I - a d s o r b a t e i n t e r a c t i o n b r o a d e n s a n d s h i f t s t h e e l e c t r o n i c levels o f t h e a d s o r b a t e s ; t h e s e b r o a d e n e d l e v e l s a r e filled u p to t h e F e r m i level b y e l e c t r o n s s h a r e d w i t h the m e t a l , s o t h a t t h e a d s o r b a t e g e n e r a l l y c a r r i e s a p a r t i a l c h a r g e . W i t h i n t h e f r a m e w o r k o f this t h e o r y , w o r k f u n c t i o n c h a n g e s o n a d s o r p t i o n , t h e i r c o v e r a g e d e p e n d e n c e , a n d t r e n d s in adsorption energies can be rationalized. The.par~lelism between processes at the metal/vacuum and the metal/electrol y t e i n t e r f a c e h a s b e e n r e p e a t e d l y n o t i c e d in r e c e n t t i m e s . W e s h o u l d e x p e c t s i m i l a r m e c h a n i s m s to o p e r a t e ha t h e a d s o r p t i o n p r o c e s s e s in t h e s e s y s t e m s . So, in this w o r k we s h a l l a p p l y t h e A n d e r s o n m o d e l to t h e a d s o r p t i o n at t h e m e t a l / e l e c t r o l y t e i n t e r f a c e . H e r e , t h e s i t u a t i o n is c o m p l i c a t e d b y t h e i n t e r a c t i o n o f t h e a d s o r b a t e wi_.th t h e s o l v e n t , w h i c h is s t r o n g a n d o f l o n g r a n g e d u e t o i t s p r e d o m i n a n t l y c 0 u l o m b i c nature. : -.. W e shall thus e x t e n d the A n d e r s o n m o d e l a n d include terms for the interaction of t h e a d a t o m l w i t h t h e s o l v e n t m o d e s in t h e e f f e c t i v e H a m i l t o n i a n . U s i n g s i m p l e approximations, we shall derive an expression for the charge on the adatom, which d e p e n d s o n - t h e characteristics of the interaction. T o test the validity of the model, we shall g i v e estimates for these quantities, a n d c a l c u l a t e the c h a r g e - t r a n s / e r
0022-072g/85/$03.30
© 1985 Elsevier Sequoia S.A_
254
c o e f f i c i e n t s for a n u m b e r o f s y s t e m s for w h i c h e x p e r i m e n t a l d a t a exist. H e r e we shall limit o u r s e l v e s t~ the c a s e o f s i n g l e - a d a t o m a d s o r p t i o n ; a d a t o m - a d a t o m i n t e r a c tions, w h i c h b e c o m e i m p o r t a n t at h i g h e r c o v e r a g e s , will b e c o n s i d e r e d in a s u b s e q u e n t p u b l i c a t i o n [2]. T h e A n d e r s o n m o d e l h a s p r e v i o u s l y b e e n a p p l i e d to the o l e c t r o s o r p t i o n o f alkali i o n s b y o n e o f us [3]. T h e m a i n d i f f e r e n c e b e t w e e n this a n d the earlier w o r k is t h a t w e shall t r e a t a m u c h w i d e r class o f s y s t e m s a n d t h a t we shall a c c o u n t f o r the i n t e r a c t i o n o f the a d a t o r n s w i t h the fast s o l v e n t m o d e s , w h i c h w a s . p r e v i o u s l y neglected. THE MODEL HAMILTONIAN
W e c o n s i d e r a single a t o m a d s o r b e d at a m e t a l / e l e c t r o l y t e i n t e r f a c e . W e restrict o u r s e l v e s to the c a s e w h e r e o n e a d a t o m o r b i t a l i n t e r a c t s with the m e t a l , a n d w e d e n o t e b y c a the c o r r e s p o n d i n g a d a t o m e l e c t r o n i c energy. F o l l o w i n g N c w n s [4], we w r i t e the t e r m s for the v a l e n c e e l e c t r o n s o n the a d a t o m as:
(1)
~a ~ ' ~ ° + U.~ona_° o
w h e r e U is the c o u l o m b r e p u l s i o n e n e r g y o f t w o v a l e n c e electrons, o is the spin index, a n d the n ' s d e n o t e the usual o c c u p a t i o n o p e r a t o r s . T h e t e r m s f o r the m e t a l e l e c t r o n s a n d for e l e c t r o n e x c h a n g e b e t w e e n the m e t a l a n d the a d a t o m h a v e the usual f o r m :
ko
"ka
w h e r e k d e n o t e s a set o f q u a n t u m n u m b e r s f o r the m e t a l electrons, Izk is the m a t r i x e l e m e n t for e l e c t r o n t r a n s f e r f r o m the a d a t o m to the m e t a l , a n d c + a n d c d e n o t e the c r e a t i o n a n d a n n i h i l a t i o n o p e r a t o r s for the s t a t e s i n d i c a t e d . I n c o n s i d e r i n g the a d a t o m - s o l v e n t i n t e r a c t i o n , we d i s t i n g u i s h b e t w e e n the librational a n d v i b r a t i o n a l m o d e s o f the solvent, w h i c h are slow c o m p a r e d to the f r e q u e n c i e s o f the e l e c t r o n i c t r a n s i t i o n s , a n d the fast m o d e s d u e to the e l e c t r o n i c p o l a r i z a b i l i t y o f the s o l v e n t molecules. I n the h a r m o n i c a p p r o x i m a t i o n , we c a n w r i t e the H a m i l t o n i a n f o r the slow s o l v e n t m o d e s a n d their i n t e r a c t i o n w i t h the a d s o r b a t e in the f o r m f a m i l i a r f r o m e l e c t r o n t r a n s f e r t h e o r y [5]:
)(
)
z--~.,nao ~_,t~to~ga~.q~ ~"
a
r.,
(3)
H e r e , ~ labels the slow s o l v e n t m o d e s o f f r e q u e n c y t%, m o m e n t u m pp a n d c o o r d i n a t e q~; z is the c h a - g e n u m b e r o f the a t o m ' s core, a n d g~. a r e the c o u p l i n g c o n s t a n t s f o r the i n t e r a c t i o n . T h r o u g h t h e s e t e r m s , the a d s o r b a t e e l e c t r o n i c e n e r g y is a f u n c t i o n o f the s o l v e n t c o o r d i n a t e s q.. T o c a l c u l a t e t h e e q u i l i b r i u m p r o p e r t i e s o f the s y s t e m , o n e c a n follow K r a t s o v a n d M a l s h u k o v [6], c o n s i d e r q~ as the e x t e r n a l p a r a m e t e r s o f the e l e c t r o n i c H a m i l t o n i a n , a n d c o n s i d e r t h e s o l v e n t c o n f i g u r a t i o n f o r w h i c h the e l e c t r o n i c e n e r g y o f the s y s t e m is a m i n i m u m . T h i s l e a d s to t h e c o n d i t i o n :
255
all(q.___..~)) = 0
(4)
( Oq. w h e r e the H e l l m a n - F e y n m a n t h e o r e m has been used. F r o m this, we o b t a i n for the e x p e c t a t i o n values o f the solvent c o o r d i n a t e s in tii-~ m i n i m a l e n e r g y c o n f i g u r a t i o n :
Alternatively, we can follow ref. 3, i n v o k e the H a r t r e e - F o c k a p p r o x i m a t i o n , a n d calculate the e x p e c t a t i o n values.
))
,6,
to the e l e c t r o n i c H a m i l t o n i a n . Finally, we c o n s i d e r the i n t e r a c t i o n with the fast p o l a r i z a t i o n m o d e s o f the interface. N o r m a l l y , b o t h the s u r f a c e p l a s m a m o d e s o f the metal a n d the e l e c t r o n i c solvent m o d e s are f a s t e r t h a n the e l e c t r o n e x c h a n g e b e t w e e n the metal a n d the a d s o r b a t e s [12]. W e shNl assume, in the following, that this c o n d i t i o n holds. T h e n these i n t e r a c t i o n s shift the a d s o r b a t e level e= a n d the c o r r e l a t i o n e n e r g y U, as was s h o w n b y I-tewson a n d N e w n s [7]. E s t i m a t e s for the shifted values ga a n d 0 w411 be given in the next section. T o e s t i m a t e the o c c u p a t i o n p r o b a b i l i t y o f the a d s o r b a t e level, we m a k e the simplifying a s s u m p t i o n that the level width:
:, ( , 0 ) = ~Y'.I Vk I"-6( ~' - ~ )
(7)
k
is i n d e p e n d e n t o f the e l e c t r o n i c e n e r g y ~o. C o n s i d e r i n g o n l y n o n - m a g n e t i c s o l u t i o n s in a restricted H a r t r e e - F o c k a p p r o x i m a t i o n so that b o t h spin states in the orbital are o c c u p i e d to the s a m e extent, the effective e n e r g y level E~ of a n orbital is:
E~=~
+ 0 ( , , > + 2(~ - 2 ( , , > ) E s
(8)
where t,
is the e n e r g y o f s o l v a t i o n o f an a d s o r b a t e with unit charge; it can also be i n t e r p r e t e d as the e n e r g y o f solvent r e o r g a n i z a t i o n o f the a d s o r b a t e , a c o n c e p t familiar f r o m e l e c t r o n t r a n s f e r theory. ( n > = ( n ~ o ) is the o c c u p a t i o n p r o b a b i l i t y o f an e l e c t r o n i c a d s o r b a t e state, which is given b y the farr'Aliar relation [1]:
(,,) = L cot - ' E~ - EF ~r
A
(9)
256
w h e r e E F is the F e r m i e n e r g y o f the metal. W h e n the r e l e v a n t s y s t e m p a r a m e t e r s a r e k n o w n , the o c c u p a t i o n p r o b a b i l i t y ( n ) c a n be c a l c u l a t e d f r o m eqns. (8) a n d (9) in a self-consistent manner. It h a s b e e n p o i n t e d o u t b y K r a t s o v a n d M a l s h u k o v [6] t h a t eqn. (9) m a y h a v e several solutions. In o u r case, eqn. (9) m a y h a v e m o r e t h a n o n e s o l u t i o n if the relation: 4E,- /.7/ > 1 (10) A~ is fulfilled; this will be the case if the i n t e r a c t i o n with the s o l v e n t is s t r o n g a n d t h a t with the m e t a l is w e a k . In the following, we shall a l w a y s give the s o l u t i o n w i t h the lowest energy, if m o r e t h a n o n e s o l u t i o n exists. A n e x p l o r a t i o n o f the c o n s e q u e n c e s o f the e x i s t e n c e o f several s o l u t i o n s will b e g i v e n e l s e w h e r e [8]. ESTIMATES OF THE SYSTEM PARAMETERS
In o r d e r to c a l c u l a t e the c h a r g e d i s t r i b u t i o n in a p a r t i c u l a r a d a t o m / s u b s t r a t e . s y s t e m , we m u s t derive e x p r e s s i o n s for the v a r i o u s q u a n t i t i e s t h a t d e t e r m i n e the effective a d a t o m e l e c t r o n i c level a c c o r d i n g to eqn. (8). T h e m o d i f i e d a d s o r b a t e level ~ a n d the C o u l o m b r e p u l s i o n 0 c a n b e e s t i m a t e d in the f o l l o w i n g w a y : Let us c o n s i d e r the case where, o n the n e u t r a l a t o m , the a d s o r b a t e o r b i t a l is h a l f - f i l l e d - - t h i s will b e so in all the s y s t e m s c o n s i d e r e d b e l o w e x c e p t for the a d s o r p t i o n o f lead. T h e h y p o t h e t i c a l e n e r g y r e q u i r e d to ionize the n e u t r a l a d s o r b e d a t o m while k e e p i n g the slow solvent m o d e s fixed is: ~ = - - I I + tim + Er (11) w h e r e I~ is the e n e r g y o f i o n i z a t i o n o f the a t o m in v a c u o , ~im is the s e l f - e n e r g y d u e to the fast p o l a r i z a t i o n m o d e s o f the interface, a n d ef is the i n t e r a c t i o n e n e r g y w i t h the fast b u l k s o l v e n t m o d e s , b o t h t a k e n for a singly c h a r g e d ion. Similarly, the e n e r g y r e q u i r e d to a d d a n e l e c t r o n is: (12)
- - ~ a - - 0 = A -at- ~im "~ ~f
w h e r e A is the e l e c t r o n a f f i n i t y o f the a t o m . F r o m these r e l a t i o n s we derive:
--~'| q'- (ira "1- ( f 0=~il--A - - 2 ( e l m + e l ) (13) F o r the a d s o r p t i o n o f lead, w h e r e the v a l e n c e p - o r b i t a l is d o u b l y o c c u p i e d in the n e u t r a l state, we o b t a i n instead: ~a = - l n + 3or + 3 E l m , a n d 0---- I 2 -- 1 i -- 2~ r - - ~-~irn; 12 is the s e c o n d i o n i z a t i o n energy. F o r the c a l c u l a t i o n o f the i n t e r a c t i o n with the s o l v e n t m o d e s , we n e e d a n explicit m o d e l for the s o l v a t i o n o f a d a t o m s . W e shall use the n o n - l o c a l dielectric m o d e l o f D o g o n a d z e a n d K o r n y s h e v [9]; here, o n e d i s t i n g u i s h e s b e t w e e n the electronic, the v i b r a t i o n a l , a n d the l i b r a t i o n a l solvent m o d e s , which h a v e d i f f e r e n t c o r r e l a t i o n l e n g t h s X t, X 2 a n d X 3, respectively. T h e e l e c t r o n i c m o d e s are fast, the o t h e r t w o are slow. T h e i n t e r a c t i o n e n e r g y o f a fully s o l v a t e d ion with the slow b u l k m o d e s is:
(a =
A ~tG~.... =
( z1e ° )i22 {r ( coP,
1)F(A2/ri)+(-~i~ - l)F(A3/ri))
(14a)
257
w h e r e ¢~ is the i n f r a r e d , cop, the optical, a n d c the static dielectric c o n s t a n t (cir = 4.9 for water); r i is the ionic radius. F o r the fast m o d e s , this e n e r g y is: AGhast =
(2e°)a2ri (I-
",,ptl
)F(Xl/r,)
F(x) = "l--(1-
e-X)/x
(141o)
F o r n u m e r i c a l calculations, we h a v e taken: AI = 0.I rim, A2 = 0.12 rim, X3 = 0.68 n m in a c c o r d with ref. 9. E q u a t i o n s (14a) a n d (14b) t o g e t h e r give quite g o o d values f o r the energies o f h y d r a t i o n o f alkali a n d halide ions, b u t they d o not w o r k q u i t e so well for o t h e r m e t a l ions. W e r e q u i r e the energies o f i n t e r a c t i o n o f a singly c h a r g e d a d s o r b a t e with the slow a n d the fast solvent m o d e s . T o a c c o u n t for the fact that the a d s o r b a t e s are o n l y p a r t i a l l y solvated a n d for the deficiencies o f the s o l v a t i o n m o d e l , we have set:
Es:aX{( 1EopL
~i~1) F ( ~ t ~ d / r ) + (
ef=etX(1--Coptl )F(X1/r )
1E.~
1) (15)
w h e r e a is the d e g r e e o f s o l v a t i o n o f the a d a t o m a n d the c o n s t a n t X has b e e n fixed in such a way that - - z 2 ( E , + ¢ r ) / a e q u a l s the e x p e r i m e n t a l e n e r g y o f solvation, AGS, o f the ion; f o r the alkali a n d halide ions, X is close to the theoretical value o f W e have p e r f o r m e d n u m e r i c a l c a l c u l a t i o n s for a = 1 / 3 a n d a = 2 / 3 ; the f o r m e r value r o u g h l y c o r r e s p o n d s to a d s o r p t i o n o n a step site, the latter to a d s o r p t i o n o n a flat surface (terrace). By setting t x = 0 , we get the case for a d s o r p t i o n f r o m the v a c u u m . Finally, we h a v e to specify the i n t e r a c t i o n with the fast p o l a r i z a t i o n m o d e s o f the interface. F o r the m e t a l / s o l v e n t interface, n o e x p r e s s i o n b a s e d o n q u a n t u m - m e c h a nical c a l c u l a t i o n s exists. W e use a p h e n o m e n o l o g i c a l e x p r e s s i o n d e r i v e d f r o m a s p a c e dispersion m o d e l [10]:
eo/2r.
eo
(Cop t -- 1) -- ~:2r 2
¢im= 4¢op ' (cop , + 1) + K2r 2
(16)
w h e r e .~ is the inverse T h o m a s - F e r m i s c r e e n i n g length; we h a v e taken 1 / ~ = 0.05 n m for n u m e r i c a l calculations, which is the typical value. T h e radius r o f the a d a t o m d e p e n d s o n its charge. W e have used the f o l l o w i n g i n t e r p o l a t i o n f o r m u l a d u e to B r o w n [11]:
r=[r~(z-q)+r'3q]'/3
(17)
w h e r e r, a n d r; are the radii o f the a t o m a n d ion, respectively; z is the c h a r g e n u m b e r o f the ion arid q that o f the a d s o r b a t e .
258
ESTIMATES FOR THE CHARGE TRANSFER
IN PARTICULAR
SYSTEMS
Using the formulae presented above, we have estimated the charge o n a number of adsorbates for which experimental data are available. Since the version of the Anderson model that we have used here works best for w e a k adsorbateLsubstrate interaction, we have used here works best for weak adsorbate-substrate interaction , we have generally performed calculations for the adsorption o n mercury electrodes. For several adsorbates, experimental data on mercury are not available. The data employed in the estimates are given in Table 1, and in Table 2 o n e finds the values calculated for the charge transfer h o n adsorption, i.e. the charge number z of the ion minus the charge number q of the adsorbate. We have performed calculations for level widths A = 0.5 and 1 eV, which comprises the range usually observed in field emission experiments [1,12]. We have not studied the potential dependence of the charge transfer. Our calculations refer to the situation where the outer potential drop between the metal and the adatom vanishes; the corresponding electrode potential is generally not too far from the potential of zero charge (pze). Unfortunately, the charge transfer h on adsorpfion is not a measurable quantity. It is, however, closely ,,elated to the so-called electrosorption valency 3' introduced by Vetter and Schultze [13]. Neglecting comparatively small dipole terms, we have: 3"~q= gz + X(1 - - g ) (18) where "rN is the electrosorption valency at the pzc and g is a small geometrical TABLE 1 D a t a for the e s t i m a t i o n o f c h a r g e - t r a n s f e r c o e f f i c i e n t s " ion
Ii b / e V
A/eV
10 r i ¢ / n m
10
K +
4.34 4.18 3.89
0.47 d 0.42 c 0.39 d
1.33 1.48 1.68
2.26 • 2.42 = 2.61 c
3.51 3.33 3.09
12.96 11.81 10.45
3.61 b 3.36 b 3.06 b
1.81 1.95 2.16
0.99 c 1.14 ~ 1.33
3.29 3.15 2.67
7.72 7.57 6.11 7.41
1.80 t 0.88 g
0.96 1.26 1.40 1.20
1.27 1.44 1.73 1.75
5.92 5.02 3.57 15.6
Rb ÷ Cs ~ CI-
Br1C u 2+ Ag*
Ti + P b 2+
h
r~/nm
~ ~ ~ ~
AG~/eV
a 11 a n d A are the first i o n i z a t i o n e n e r g y a n d the electron a f f i n i t y o f the neutral a t o m ; A w a s set to z e r o w h e r e it w a s n o t available. W o r k f u n c t i o n s : H g 4.5 e V , P t 5 . 0 3 e V , A u 4 . 8 0 e V , A g 4 . 3 0 t:V ( e l e c t r o c h e m i c a l values after Trasatti [15l). b F r o m C o t t o n a n d W i l k i n s o n [16]. ¢ A f t e r P a u l i n g [17]. d A f t e r Weiss [181.
E s t i m a t e d f r o m the lattice c o n s t a n t o f the e l e m e n t , taken f r o m A s h e r o f t a n d M e r m i n [19]. t A f t e r C l e m e n t i [20]. s A f t e r M o i s e i w i t c h [21]. h 12 = 15.03 e V .
259 TABLE
2
Calculated charge-transfer coefficients System
2k
"YN a
A=0.5eV vae.
A=I et = 1 / 3
a = 2/3
eV ~t = 1 , / 3
~t = 2 , / 3
K/Hg Rb/Hg Cs/Hg
0.15 0.15 0.14
0.12 0.12 0.11
0.08 0.08 0.07
0.30 0.29 0.27
0.25 0.24 0.23
0.16 0.16 0.16
0.12 0.15 0.18
Cl/Hg Br/Hg I/Hg
- 0.21 -- 0 . 2 6 --0.34
- 0.17 -- 0 . 2 0 -0.38
- 0.10 - 0.12 -0.17
-- 0 . 3 2 - 0.38 --0.47
- 0.27 -- 0 . 3 2 -0.41
- 0.19 - 0.22 --0.30
- 0.2 -- 0 . 3 4 -0.45
Cu/Pt Ag/Au Tl/Pt Pb/Ag
1.85 0.73 0.26 1.81
1.42 0.47 0.19 1.83
1.90 0.80 0.42 1.69
1.77 0.68 0.35 1.71
1.16 0.26 0.21 1.45
1.09 b 0.12 0.10 b 1.70 b
vac.
1.8 0.6 c 0.85 2.0
a E l e e t r o s o r p t i o n v a l e n c S e s w e r e t a k e n f r o m K o p p i t z a n d S c h u l t z e [22] u n l e s s o t h e r . v i s e i n d i c a t e d . b More than one solution exists; the one with the It:vest energy has been chosen. c T a k e n f r o m M e l n i e k e e t al. [23].
f a c t o r that describes the p e n e t r a t i o n o f the a d s o l b a t e i n t o the d o u b l e - l a y e r field. F o r cations, the c h a r g e t r a n s f e r ~k is, in general, positive, for a n i o n s negative, i.e. the ions are partially d i s c h a r g e d o n a d s o r p t i o n . C o n s e q u e n t l y we h a v e I vNI > IX I- T h e e x p e r i m e n t a l values for YN in T a b l e 2 are for sma21 coverages, since o u r m o d e l , in which a d a t o m - a d a t o m i n t e r a c t i o n s are neglected, refers to this situation. T h e a d a t o m s investigated fall into three classes, w h i c h we shall discuss s e p a r a t e l y . A l k a l i ions
T h e s e are c h a r a c t e r i z e d b y a small i o n i z a t i o n energy, so that the e n e r g y o f the v a l e n c e e l e c t r o n lies a b o v e the F e r m i level o f m e r c u r y . H e n c e the o c c u p a t i o n p r o b a b i l i t y o f this level is small, a n d the alkali ions k e e p their positive c h a r g e a l m o s t c o m p l e t e l y o n a d s o r p t i o n . T h e ionic state is also f a v o u r e d b y the i n f l u e n c e o f the solvation a n d b y i m a g e forces. T h e r e f o r e the a d a t o m s in e l e c t r o c h e m i c a l s y s t e m s always c a r r y a g r e a t e r c h a r g e than the s a m e a d a t o m s a d s o r b e d f r o m the v a c u u m , a n d the c h a r g e increases with the d e g r e e o f s o l v a t i o n ; this is t r u e for ~11 systems. F r o m p h o t o e m i s s i o n data, the level w i d t h A o f alkali a d s o r b a t e is e s t i m a t e d to b e a b o u t 0.5 eV [12]; o n liquid m e r c u r y , we w o u l d e x p e c t the a d a t o m s to i n t e r a c t still relatively strongly, since the s u r f a c e s h o u l d n o t c o n t a i n a n y steps, so that ot = 2 / 3 w o u l d b e a b e t t e r estimate. T h e c o r r e s p o n d i n g c h a r g e - t r a n s f e r c o e f f i c i e n t s are in line with the o b s e r v e d e l e c t r o s o r p t i o n valencies. T h e y are also o f the s a m e o r d e r o f m a g n i t u d e as those e s t i m a t e d in a p r e v i o u s p u b l i c a t i o n [3]. H a l i d e ions
F o r these systems, we h a v e a s s u m e d that the p - o r b i t a l p e r p e n d i c u l a r to the m e t a l surJ'ace, which f o r m s the o - b o n d , gives the d o m i n a n t c o n t r i b u t i o n to the b o n d , a n d
260
we h a v e n e g l e c t e d the i n t e r a c t i o n with the o t h e r o r b i t a l s . T h e h a l i d e a t o m s h a v e high i o n i z a t i o n energies a n d e l e c t r o n affinities. H e n c e the o u t e r p - o r b i t a l s a r e a l m o s t c o m p l e t e l y filled, a n d the a d s o r b e d ions still c a r r y m o s t o f their n e g a t i v e charge. Since the i o n i z a t i o n energies, the e l e c t r o n affinities, a n d the energies Of s o l v a t i o n d e c r e a s e in the series f r o m F to I, there is a n a c c o m p a n y i n g d e c r e a s e o f the n e g a t i v e c h a r g e on the a d s o r b e d ions. T h e e l e c t r o s o r p t i o n v a l e n c i e s follow the s a m e trend, a n d a r e in line w i t h values o f A o f the o r d e r o f 0.5 - 1 eV. O t h e r m e t a l ions
T h e third g r o u p c o m p r i s e s m e t a l ions w h i c h p l a y a role in u n d e r p o t e n t i a l d e p o s i t i o n : A g +, C u 2+, P b 2+, Tl +. In e a c h case, we h a v e o n l y c o n s i d e r e d t h e i n t e r a c t i o n o f the e n e r g e t i c a l l y highest o r b i t a l with the s u b s t r a t e . T h e energies o f i o n i z a t i o n o f the a t o m s lie in b e t w e e n those for the alkali a n d t h o s e for the h a l i d e ions, a n d so we o b t a i n sizable c h a r g e - t r a n s f e r coefficients. T h e e x p e r i m e n t a l values f o r the e l e c t r o s o r p t i o n valencies are for p o l y c r y s t a l l i n e e l e c t r o d e s a n d thus p r o b a b l y c o r r e s p o n d to a d s o r p t i o n o n s t e p sites, so t h a t the d e g r e e o f stflvation s h o u l d b e small. Also, m o s t e x p e r i m e n t a l d a t a for u n d e r p o t e n t i a l d e p o s i t i o n a r e for high c o v e r a g e s , w h e r e the a d a t o m s are a l m o s t c o m p l e t e l y d i s c h a r g e d d u e to C o u l o m b r e p u l s i o n a n d lateral i n t e r a c t i o n s . O n l y in a few cases h a s the e l e c t r o s o r p t i o n v a l e n c y b e e n s t u d i e d as a f u n c t i o n o f the c o v e r a g e ; these s t u d i e s i n d i c a t e t h a t at small c o v e r a g e s X is s m a l l e r t h a n the c h a r g e n u m b e r z o f the ion, b u t t h a t h t e n d s to z as the c o v e r a g e a p p r o a c h e s unity; i.e. the ions a r e d i s c h a r g e d with i n c r e a s i n g c o v e r a g e . O u r c a l c u l a t e d c h a r g e - t r a n s f e r c o e f f i c i e n t s fit r a t h e r well i n t o this picture. DISCUSSION C o n s i d e r i n g the s i m p l i c i t y o f the m o d e l , the c a l c u l a t e d c h a r g e - t r a n s f e r coefficients fit the e x p e r i m e n t a l d a t a s u r p r i s i n g l y well. T h e e l e c t r o s o r p t i o n valencies o b s e r v e d are g e n e r a l l y in line with the partial c h a r g e - t r a n s f e r c o e f f i c i e n t s c a l c u l a t e d for r e s o n a n c e w i d t h s o f 0 . 5 - 1 . 0 eV, which is also the r a n g e o f line w i d t h s o b s e r v e d in field e m i s s i o n e x p e r i m e n t s . P e r h a p s e v e n m o r e i m p o r t a n t t h a n this overall a g r e e m e n t is the fact t h a t the o b s e r v e d t r e n d s are c o r r e c t l y r e p r o d u c e d . In p a r t i c u lar, the d i f f e r e n t b e h a v i o u r o f the alkali ions, w h i c h r e m a i n highly p o s i t i v e l y c h a r g e d , the halide ions, w h i c h c a r r y a high n e g a t i v e charge, a n d the o t h e r m e t a l ions, w h i c h h a v e a n i n t e r m e d i a t e p o s i t i v e charge, is b o r n e o u t in the c a l c u l a t i o n s . Also, t r e n d s w i t h i n these g r o u p s a r e c o r r e c t l y p r e d i c t e d ; thus, the h a l i d e ions b e c o m e less c h a r g e d as we g o f r o m F - to I - . T h i s success o f the m o d e l is p r o b a b l y m a i n l y d u e to the fact t h a t the m o s t i m p o r t a n t p a r a m e t e r s in the m o d e l are the e n e r g i e s o f i o n i z a t i o n o f the a t o m s , the w o r k f u n c t i o n o f the m e t a l s u b s t r a t e , a n d f o r the h a l i d e a n d alkali ions, also the e l e c t r o n affinities, for which reliable d a t a exist. W h i l e the s o l v a t i o n p a r a m e t e r s E s arid h r are i m p o r t a n t , in t h a t t h e y f a v o u r the c h a r g e d s t a t e o f the a d a t o m , they d o n o t v a r y s o m u c h a n d thus d o not d i s t u r b the p a t t e r n g e n e r a t e d b y the o t h e r parameters.
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It is of interest to c o m p a r e our work with the results obt ai ned by ot her methods. Recently, Weissmann and C o h a n [14] have studied the adsorpt i on of F - , C I - and K + on small clusters of silver by the iterative e x t e n d e d H~ekel m et hod. Qualitatively, they observed the same trends as we did: a d a t o m s of F and CI carry negative charges, F being m o r e highly charged than CI, and K carries a large positive charge. As we have pointed out, our present work contains a n u m b e r of simplifying assumptions, which should be studied and improved upon in future work. We think that our results are encouraging and that they demonstrate the applicability of the
A n d e r s o n model to electrochemical interfaces. ACKNOWLEDGEMENTS
We would like to thank Professor Dr. J.W. Schultze, D~sseldorf, for useful discussions on electrosorption valencies. Financial support by the D eut sche Forschungsgemeinschaft and by the Verband der Chemischen Industrie is gratefully acknowledged. A.A.K. aknowledges the hospitality of the University of Diasseldorf, where the m aj or part of this work was perform ed. REFERENCES 1 F o r a review of this model see: J.P. Muscat and D.M. Newns, Progr. Surf. Sci., 9 (1978) 1; J.P. G a d z u k in J.M. Blakely (Ed.), Surface Physics of Materials, Vol. 2, Academic Press, New York, 1975. 2 A. Kornyshev and W. Schmiclder, manuscript in preparation. 3 W. Sehmiekler, J. Eleetroanal. Chem., 100 (1979) 533. 4 D.M. Newns, Phys. Lett., 33A (1970) 43; Phys. Rev. Lett., 25 (1970) 1575. 5 For a recent review see: J. Ulstrup, Charge Transfer Processes in C o n d e n s e d Media, Springer Verlag, Berlin, Heidelberg, New York, 1979. 6 V . ~ Kratsov and A.G. Malshukov, Soy. Phys. JETP, 48 (1978) 248 (English translation). 7 A.C. Hewson a n d D.M. Ne-..-'as, J. Appl. Phys. Suppl., 2 (1974) 2121. 8 W. Schmickler, Chem. Phys. Lett., in press. 9 R.R. Dogonadze a n d A.A. Kornyshev, J. Chem. Soc. F a r a d a y Trans. 2, 70 (!0,74) !121. 10 M.A. Vorotyntsev, A.A. Kornyshev and A.I. Rubinstein, Elektrokhim:,ya, 13 (19"77) 1767; A..A. Kornyshev, Electrochim. Acta, 26 (1981) 1. 11 O.M. Brown, private communication. 12 J.W. Gadzuk, J.K. H a r t m a n n a n d T.N. Rhodin, Phys. Rev., B4 (1971) 241. 13 K d . Vetter and J.W. Schultze, Ber. Bunsenges. Phys. Chem., 76 (1972) 920, 927. 14 M. Weissmann and N.V. Cohan, J. E!ectroanal. Chem., 163 (!984) 381. 15 S. Trasatti in H. Gerischer a n d C.W. Tobias (Eds.), A d v a n c e s in Electrochemistry a n d Electrochemical Engineering, Vol. 10, Wiley-lnterscience, New York, 1977. 16 F.A. Cotton a n d G. Wilkinson, A d v a n c e d Inorganic Chemistry, 3rd ed., lnterseience, New York, 1977. 17 L. Pauling, The N a t u r e of the Chemical Bond, 3rd ed., Cornell University Press, Ithaca, NY, 1960. 18 A.W. Weiss, Phys. Rev., 166 (1968) 76. 19 N.W. Ashcroft a n d N.D. Mermin, Solid State Physics, ltolt, Rinehart and Winston, Philadelphia, 1976. 20 E. Clementi, Phys. Rev., 135 (1964) A980. 21 B.L. Moiseiwitch, Adv. Atomic Mol. Phys., 1 (1965) 327. 22 F.D. Koppitz a n d J.W. Schultze, Eleetrochim Aeta, 21 (1976) 327. 23 L.S. Melnieke, T.M. R i e d h a m m e r a n d S. Bruckenstein, Proceedings of the 3rd Symposium o n Electrode Processes 1979, T h e Electrochemical Society, Princeton, N J, 1980, p. 306.