7162
J. Phys. Chem. 199498, 7162-7169
Reorganization Energy for Electron Transfer at Film-Modified Electrode Surfaces: A Dielectric Continuum Model Yi-Ping Liu and Marshall D. Newton' Depariment of Chemisiry. Brookhaven National Laboratory, Upion. New York 11973-5ooO Received: February 14, 1994; In Final Form: May 10. 19948
Electron transfer between an electrode and a redox couple separated from the electrode by an organic film is analyzed using a local dielectric continuum model. The solvent reorganization energy (E1)is calculated as a function of the cavity radius of the redox group, the thickness of the organic film, and the static and optical dielectric constants of the solvent, the organic film, and the electrode. Comparison is made between results for E. based on alternative formulations in terms of electric field vectors, and significant differences are found, both quantitative and qualitative
1. lntroduetion
aqueousphase
The goal of achieving detailed mechanistic understanding of electron-transfer kinetics continuesto stimulate intensiveefforts, both theoretical and experimental, focused on a broad array of homogeneous and heterogeneous processes of physical and biochemical interest.12 Of particular importance are efforts to elucidate the role of intervening spacer groups in controlling electrontransfer between widely separateddonor (D) and acceptor (A) groups. It is now widely understood that variations in D/A separations may have a significanteffect on theactivation energy as well as on the electronic coupling of the D and A sites, and both effects must be taken into account in analyzing the dependence of rate constant (kc,)on D/A separation (rDA)?-J This dependenceis often found to exhibit, at least approximately, an exponential behavior,)-s
-
For saturated hydrocarbon spacers, B values are typically found to havcmagnitudesof I A-1. Thecontributionoftheactivation energy ( E ' ) to the overall 0 value arises primarily from the well-
known dependence of E' on the solvent reorganization energy? E,. and, hence. on rDA?For example, the familiar classial twosphere D/A modelof Marcusbfor homogeneouselectrontransfer yields
E,h"=(I/tOP- l/t")(1/2aD+ 1/2aA- I/roA) (2) where@andt"are, respectively,theopticalandstaticdielectric constants,andaoand aAare,rCSPCCtiveIy,theCBvityradiiforthe D and A sites. Aside from its crucial role in chargetransfer kinetics, we also note that medium reorganization plays a closely related role in controlling the energetics and line shapes of charge-transfer spcctra, e.g., as represented by the Stokes shift, which equals twice the reorganization energy in the case of linear systems. Of particular interest in rcccnt years has been electron transfer mediated byn-alkanespccrgroups."'9 Muchoftheexperimental kinetic data has been provided by n-alkancthiol films,'G2' which facilitate efficient interfacial electron transfer between a ( I I I ) gold surface (to which the thiol group is attached) and various inorganic redox systems which arc situated (either via van der Waals contact or by covalent attachment) at the interfacebetween the film and an aqueous phase."l,'",14 A generic three-zone system of this type is illustrated in Figure 1, where zones I, 11, and 111 correspond, respectively. to the aqueous phase, the film. *Ab"
published in Adunnn ACS Absrroctr. June IS. 1994.
0022-3654/94/2098-7 162604.50/0
t
z
Figwe 1. Schematicreprcaentationofarcdoicouplcina rphcricalcavity of radius a. scparatcd from a film of width 1. by a distancc d. The paint charge Aq is defined in cq 13. I and p are qlindrical coordinates. and e, (i = 1. II. 111) are the dielatric constants for the three zones.
and the metal electrode. and the redox species is in a mherical cavity of radius a. The deoendence of DIA electronic coudine on the leneth of trans-staggeredn-alkylchainsand other saturated organic spacers has been the subject of numerous recent theoretical and computational studies.l5-'9.** However, a detailed quantitative understandingof the analogousvariation of E, is presently lacking for complex systems (e.g., the homogeneous cases dealt with in refs 3-5 and 12b or the heterogeneous cases reported in refs 8-1 1, 12a, and 14). where simple electrostatic models like eq 2 are either not applicable or are of uncertain accuracy. A close analog of eq 2 is commonly used for two-zone heterogeneous sytems,6423 (e.g.. at an aqueous solutionlmetal electrode interface): .
E,"=
I
(l/pP- l/fs')(l/2a-a/2r)
-
(3)
where PF' and tat refer to the solution phase, a is the cavity radius of the redox species, and r is the separation of the redox species and its image in the metal electrode (twice the distance of the redox species from the interface). For ideal metallic screening (metallic c = -),the factor a is unity, whereas a = 0 in the limit where metallic screening is quenched (i.e., no image contribution),6423-26 In the present paper we formulate and evaluate E, for the three-zone system (Figure I ) by solving Poisson's equation (see below) at the dielectric continuum level. Our primary objective is to obtain estimates for the expected variation of E, with film width (Lin Figure 1) and hence to assess the likely importance ofsuchvariation in theanalysisofongoing experimentalwork.9.2' 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 29, 1994 7163
Reorganization Energy for Electron Transfer It is also of interest to understand the relationship between the detailed results for the three-zone system and the simple twozone model represented by eq 3. We shall implement the dielectric continuum model in the limit of spatially local response. Previous work has suggested the possibility of significant consequences from nonlocal effects.2* However, the ability to formulate the details of a nonlocal dielectric function is still at a preliminary stage (with regard to both theaqueous and themetallic zones28-30), and a recent experimental study has suggested that nonlocal effects may be minor when one is dealing with relatively large (on the scale of solvent molecule radii) redox species,3l as in the present study, which specifically addresses the case of a ferrocene ferrocenium (Fc) redox couple in aqueous solution (the subject of the experimental studies reported in refs 9 and 27). Finally, we note that the previous literature offers distinct formulations of Es for a given dielectric continuum system, as described in the next section (e.g.,cf. refs 6,32-34 and refs 35-37). While these distinctions are often minor, especially in applications to homogeneous pr0cesses, we shall find significantquantitativedifferences in the applications reported below.
2. Formulation of Solvent Reorganization The traditional formulation of reorganization energy is based on initial (i) and final (0 states defined, respectively, in terms of charge densities pi and pf.6,32-34 The process of “reorganizing” a state characterized by a given charge density,say state i, involves proceeding from the case where the medium is fully equilibrated to pi (with regard to both inertial and optical modes) to the situation where only the optical medium modes are at equilibrium with pi, while the inertial modes adopt a configuration which would be in equilibrium with pr. For the case of linear coupling to the medium and longitudinal fieldsonly, the reorganization process may be represented in terms of the Maxwell displacement field D as follows:3*
opposed to process 4, which underlies the traditional formulation introduced by Marcus6932 and adopted by others,33,34where the D field changes subject tofixed charge density (Le., V*D, = V-D, = 45~pi;see also ref 41)- Equation 7 leads to the following analog of eq 6:35-37
where the prime on E’, is to distinguish it from E, in eq 6. Although the formal distinctions between eqs 6 and 8 are significant, the quantitative difference may frequently be minor, since it is well-known that the D field often depends only weakly on e and thus may be replaced to a good approximation with the vacuum field, in which case eqs 6 and 8 become equivalent.33J6b However, significant differences may be expected in applications to heterogeneous electron transfer of the type reported here, and some of the results obtained in the present study based on eq 6 are found to differ qualitatioely from results of earlier studies based on eq 8.37b.c It is for this reason that we have gone to some pains to elaborate the origins of these different expressions for
Es In the actual implementation of eq 6 an equivalent form is employed, based on use of Maxwell’s equations and integration by parts:32,33,40b
where the equilibrium potential 4eqand displacement field D, are related by
E, = -V4,
(loa)
D,=eE, with e = cop or
est,
as required in eq 9, and where
V-D,(Apif,e) = 417ApX
(11)
The desired potential, 4, is obtained from the following Poisson’s equation:
V24, = -(4?r/e)Apif Note that in general the dielectric functions may exhibit spatial dependence of the form e = ~(r,r‘).~&~l The limiting case of a spatially local, homogeneous, isotropic medium (or of such zones within a medium) yields the dielectric constants defined in eqs 2 and 3, and this limit will be employed within each of the zones defined in Figure 1 (the treatment of the cavity assigned to the redox couple in zone I is described in the next section).39 In terms of D , fields, E*may now be displayed (see Appendix A) as the difference between the solvation free energy of the (difference) density Apfin a medium with only an opticaldielectric response (cop) and that in a medium with the full (est) response: 32,33,40
In contrast to the foregoing, we now consider an alternative formulation in which, by construction, the D fields of the initial equilibrium state and the reorganized state are e q ~ a l , ’ ~ , ~ ~
Wrwr8= D,
E
Dq(~i,egt)
(7)
where theprime is to distinguish, ’D from DmS in process 4.41 Thus according to this view,35-37 the process of reorganization is supposed to take place subject to fixed displacement field, as
(12)
