5790
J. Phys. Chem. B 2000, 104, 5790-5796
Mechanism of Reductive Desorption of Self-Assembled Monolayers on the Basis of Avrami Theorem and Diffusion Igor A. Vinokurov,† Mario Morin,‡ and Jouko Kankare*,† Department of Chemistry, UniVersity of Turku, FIN-20014 Turku, Finland, and Ottawa-Carleton Chemistry Institute and Department of Chemistry, UniVersity of Ottawa, 10 Marie-Curie, Ottawa, Canada K1N 6N5 ReceiVed: January 20, 2000; In Final Form: April 5, 2000
A mathematical model for the reductive desorption of self-assembled monolayers has been developed. The model is based on the previously presented model in which desorption occurs as growing empty patches. The overlap of patches is described by using the Avrami theorem, and the diffusion rate or the desorbing compound is taken into account. The resulting mathematical expression is a nonlinear integral equation with two adjustable parameters. The model is fitted to the experimental results on the desorption of nonanethiol from a selfassembled layer on Au(111). Reasonably good fits are obtained at four desorption potentials.
Introduction The process of self-assembly resulting in the well-ordered structures at the molecular level has received considerable attention in recent years.1-19 It is already realized that selfassembly provides powerful means for controlling the chemical nature of the electrode-solution interface. Among others, the thiol-based monolayers on gold have promising applications relevant to various electrochemical sensors.3,6,8-11 The two advantageous analytical peculiarities associated with coating an electrode with a self-assembled monolayer are connected with the property of the monolayer to prevent close approach of solvent and ions to the electrode surface and therefore to decrease the double-layer capacitance. It results in the reduction of non-faradaic background currents and selective reduction of faradaic background currents, because the interfacial electrontransfer reactions are forced to occur either by the long-range electron transfer across the monolayer or at microscopic defect sites in the monolayer.8 This has raised interest in further studies of adsorption/desorption mechanisms of thiols on gold for such knowledge is important to allow better control of their structure and stability. Reductive desorption of self-assembled alkanethiol monolayers has been the subject of several recent works13,17-19 and it appears to be a one-electron process:2,7
CH3(CH2)nS - Au + e- f CH3(CH2)nS- + Au On the basis of chronoamperometric measurements, it was suggested that the reductive removal of chemisorbed thiols starts at etching centers which expand in a circular fashion.17,18 The model used to analyze the chronoamperometric results involves a nucleation/growth mechanism for the holes that are being formed during the desorption process. However, as it was recognized in these works, the nucleation and growth models do not give satisfactory description of chronoamperometric signal to a full extent. At longer times and at higher values of * To whom correspondence should be addressed. E-mail: Jouko.
[email protected]. † University of Turku. ‡ University of Ottawa.
overpotential the experimental current decays more slowly than predicted by these models. The failure of the nucleation and growth models was assigned to their neglect of the interaction between adsorbed molecules.13 It was previously suggested18,19 that chain-chain interactions between adsorbed thiols limit the rate of growth of the etching centers. It also was discussed that at very large overpotentials the rate of growth of the etching centers becomes limited by the diffusion of the reduced thiols away from the edge of the etching centers.18 However, previously existing hypotheses being only on a speculative level this has not resulted in corresponding theoretical description of chronoamperometric signals. We show, in this paper, that a satisfactory analysis of chronoamperograms can be obtained by taking into account the diffusion of released species from electrode surface into solution. Theory The model of etching centers was recently used,17,18 to analyze the reductive removal of an alkanethiol monolayer. In this model, the reductive desorption starts at etching centers that expand in a circular fashion via the reduction of the chemisorbed thiols at their edges. Below we present a modified version of this model that takes into account the diffusion process of the desorbed species. Our model is based on four main approximations and assumptions, which we describe below. 1. The desorption process starts from a dense layer of adsorbed molecules. During the desorption process, new holes are constantly forming on the self-assembled layer. The rate of hole formation is directly proportional to the number of those remaining molecules that haVe not participated in the hole initiation:
dNc ) kc(N0 - Nc) dt
(1)
Here N0 is the total number of adsorbed and detachable molecules on the surface and Nc the number of holes becoming centers of growing empty patches. This assumption is the same as made in the progressive nucleation model. Essentially, it is similar to the rate law of radioactive decay. The rate constant kc may not be directly dependent on the potential used in the desorption, although indirect dependence is probable, e.g., via
10.1021/jp0002502 CCC: $19.00 © 2000 American Chemical Society Published on Web 06/15/2000
Reductive Desorption of Self-Assembled Monolayers
J. Phys. Chem. B, Vol. 104, No. 24, 2000 5791
strong local fields at the defect sites or concurrent electrolytic processes such as gas formation. 2. The size of the patches is increasing in a conVex fashion. The molecules are desorbed only from the perimeters of the patches, and the rate is proportional to the number of molecules on the perimeter. This assumption follows from the lateral attraction of long-chain adsorbate molecules. This interaction tries to maintain the perimeter as short as possible, keeping its convex or even circular shape. On the other hand, the molecules at the circumference are more loosely held by the interactive forces and consequently more easily desorbed. Let the length of the perimeter of a patch be P and the area S. As the patch grows maintaining its shape, we have
S ) RP2
(2)
where R is more or less constant dependent on the shape taken by the patch. For instance, for a circle it is equal to 1/4π, which can be shown to be the maximum value of R. Let a patch be formed by removing Nm(τ,t) molecules, the process being initiated at the moment τ. If the area of the patch is S and the cross-sectional area of adsorbed molecules a, this number can be estimated to be
S R Nm(τ,t) ) ) P2 a a
(3)
from which we have
P)
xRaN (τ,t) m
(4)
If the diameter of the molecular cross section is d , the number of molecules at the perimeter is
Np )
P ) d
x
a xNm(τ,t) ) γxNm(τ,t) Rd2
(5)
Here we defined a dimensionless shape factor γ which depends only on the shapes of the patches and molecules. If both the shape of the patch and cross section of molecules are circular, γ equals to π. Consequently, γ is a function of time only if the shape of the patches changes during the growth. Except for the random formation of patch centers, the molecules desorbed are assumed to become detached only from the patch perimeters. They follow the first-order kinetics where the rate constant A depends on potential:
dNm ) ANp ) AγxNm dt
(6)
This can be solved to give
xNm(τ,t) ) 1/2∫τ γA dt t
(7)
The nature of A is now clarified by the following assumption: 3. The rate of desorption is proportional to the difference between the saturation concentration and surface concentration of desorbing molecules. Surface concentration is defined here as the concentration in the immediate vicinity of the surface. We divide the interface in two regions. We call the plane of surface-contacting atoms of the adsorbed molecules the inner plane (IP), and the plane of molecules just outside the layer of adsorbed molecules the outer plane (OP). The desorption process is now separated into two steps. The first step is the charge transfer between the molecule at IP and
electrode. In this step it is assumed that the molecule does not appreciably move from IP due to the strong lateral attractive forces of the long-chain self-assembled molecules. The second step is the diffusion/migration of the molecules from IP to OP. The driving force for this step is proportional to the difference between the electrochemical potentials of molecules at OP and IP. The flux of molecules through OP from a single patch is then
J ) -βcOP(µ˜ OP - µ˜ IP)
(8)
where cOP is the concentration at OP and β is the factor including any other variables having an effect on desorption rate. The molecules at OP can be considered free and their electrochemical potential is obtained in the conventional way:
µ˜ OP ) µ°OP + RT ln aOP + zFφOP
(9)
The calculation of the electrochemical potential of the adsorbed molecules at IP poses a less straightforward problem. However, the electrochemical potential at IP can be evaluated by assuming a quasi-equilibrium. Let us assume that we have a surface with adsorbed molecules in equilibrium with a solution and that we gradually increase the concentration in the solution. When the saturation concentration is reached, further attempts to increase the concentration result in the formation of multilayers on the surface. At this point of equilibrium, as the electrochemical potentials at IP and in the solution are equal, we may write sat sat µ˜ IP ) µ˜ sat OP ) µ°OP + RT ln aOP + zFφOP
(10)
In the present case, there are isolated islands of molecules on the surface and we assume that eq 10 is locally valid in the immediate vicinity of the islands. Substitution of eqs 10 and 9 into eq 8 gives
[
J ) βcOPRT ln
]
asat OP zF - (φ - φsat OP) aOP RT OP
(11)
We now make the following two approximations:
asat OP - aOP .1 aOP
(12)
RT | zF
|φOP - φsat OP| , |
(13)
The approximation in eq 12 can be rationalized by the fact that the thiolates are weakly soluble and that the saturation concentration is low. Hence, we do not expect a large difference between the activities in a small region defined by the OP. The other approximation described in eq 13 is based on the fact that the organic monolayer blocks efficiently the applied electric field. The electric fields at OP are thus small. We assume that the screenings at saturation and at the edge of a patch are very similar. It is reasonable to assume that they differ by less than RT/zF. In addition, we assume that the activity coefficients are nearly equal so that the ratio of activities equals the ratio of concentrations. Then we may rewrite eq 11 as
(
J ≈ βNAcOPRT ln 1 +
)
csat OP - cOP ≈ βNART(csat OP - cOP) ) cOP βNART(csat - cs) (14)
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Vinokurov et al.
where we have multiplied the flux by the Avogadro number NA to give the flux in number of molecules crossing a unit area in a unit time. For convenience, subscript OP has been dropped and subscript s added to denote “surface”. When multiplied by the area through which desorbed molecules are passing, Npa, we obtain the growth rate of a patch:
dNm ) JNpa dt ) βaNANpRT(csat - cs) ) βγaNART(csat - cs)xNm (15) Comparison with eq 6 shows that eq 15 has the same form. In order to conform with the potential dependency of A in eq 6, and regarding the progressive electron transfer from the metal to thiol as the driving force for the growth of the holes, we assume that parameter β depends on potential. Equation 15 can be solved, giving
xNm(τ,t) ) k∫τ γ(τ)(csat - cs(u)) du
(16)
k ) 1/2aNAβRT
(17)
t
where
and assume that the shape factor remains constant during the patch growth. According to assumption 1, the rate of appearance of holes follows the first-order kinetics, giving from eq 1
Nc ) N0(1 - e-kcτ)
(19)
B)
∫τt(csat - cs) du]2 dτ
∫0t dNh m ) N0kck2
∫0t e-k τγ2(τ){∫τt[csat - cs(u)] du}2 dτ c
N0 1 N0k2γ j 2 ) 1/4 (Rβγ j NART)2 Ntot Ntot
Combining (22) with (21) and (20) gives
(26)
with the following initial and boundary conditions:
c(0,x) ) 0; c(t,∞) ) 0
(27)
Here D is diffusion coefficient of desorbed species, c their concentration, t time from the beginning of the desorption process, and x distance from the electrode surface. The relevant solution of eq 26 in Laplace plane is
(x ) s x D
cj(s,x) ) cj(s,0) exp -
(28)
xDs(∂x∂cj)
(20)
(21)
cj(s,0) ) -
dθ ∂c J ) Γs ) -D dt ∂x x)0
( )
(30)
Here Γs is the limiting value of adsorption at full surface coverage. Taking Laplace transform of eq 30, and substituting into (29) we have
(22)
(23)
(29)
x)0
From Fick’s first law:
cj(s,0) )
Here Ntot is the total number of adsorbed species at full surface coverage. Apparently we have Ntot g N0, the equality being valid only for the perfectly homogeneous monolayers. From Avrami’s theorem20-22 we get for the coverage by the patches:
θ ) 1 - e-θext
(25)
∂2c ∂c )D 2 ∂t ∂x
The extended coverage by patches is then
Next θext ) Ntot
(24)
which gives
The overlap of patches is now taken into account by using the Avrami theorem.20-22 It is based on the concepts of the extended number of released molecules, Next, and the extended coverage by patches, θext. For the extended number of released molecules at the time t we can write
Next )
c
Here we have also made an approximation by taking an average value γ j for the shape factor γ(τ). In order to estimate cs, we have to make a further assumption: 4. The system obeys laterally homogeneous planar diffusion. Although the surface of the electrode is partly occupied by adsorbed molecules, and partly by empty patches of various sizes, we assume that the concentration in contact with the electrode surface is laterally homogeneous. This is an approximation commonly used in various applications dealing with, e.g., rough electrodes. At least in the case of single-crystal electrodes as used in this work, the reductive desorption occurs homogeneously over all the surface, and we may assume that this assumption is very well warranted. To take into account the diffusion of released species from electrode surface into solution we consider the diffusion equation
At the time moment t we have the number of released molecules, from the patches initiated within time interval dτ:
dN h m ) Nm(τ,t) dNc ) N0kck2e-kcτγ2[
∫0t e-k τ{∫τt[csat - cs(u)] du}2 dτ}
where
(18)
The number of patches initiated at time τ within the time interval dτ is then
dNc ) N0kce-kcτ dτ
θ(t) ) 1 - exp{-Bkc
Γs s θ h xD xs
(31)
which gives finally the surface concentration in terms of the semiderivative of the coverage of patches:
cs )
Γs
xπD
Γs d1/2θ ∂θ dσ ) xD dt1/2 t - σ ∂σ
∫0tx 1
Substitution into (24) gives
{
θ(t) ) 1 - exp -Bkc
{[
]} }
Γ d1/2θ 1/2 D dt
∫0te-k τ ∫τt csat - x s c
(32)
2
dτ
Reductive Desorption of Self-Assembled Monolayers
{
) 1 - exp -Bkc(csat)2
[
∫0te-k τ (t - τ) c
|] }
d-1/2θ -1/2 csatxD dt Γs
t τ
J. Phys. Chem. B, Vol. 104, No. 24, 2000 5793
2
dτ
(33)
Hence we have two relationships, eqs 35 and 33, between current I and coverage θ, and the numerical comparison of these relationships gives a method to estimate parameters kc and B. Let the current I(t) as the function of time be known from a chronoamperometric experiment. This function can be represented by an interpolation polynomial:
Here the semiintegral is defined as the Riemann-Liouville integral23 -1/2
d 1 f ) -1/2 dt xπ
f(u)
∫0t x
t-u
du
(34)
Equation 33 describes the coverage by patches as a function of time, although the functional dependence is given by a rather awkward nonlinear integral equation, which can be solved only numerically. It is interesting to note that eq 33 contains only two adjustable parameters, kc and B, since csat, Γs, and D can be determined independently. To check the validity of the theory, we should fit the results of chronoamperometric experiments to eq 33. The current, I(t), flowing during the chronoamperometric experiment, is approximately related to coverage by
I(t) ) Qtot
dθ dt
(35)
where Qtot is the integrated charge of the chronoamperograms, known from experiment. In fact, the charge can be separated into two parts, capacitive, Qc, and faradaic, Qf. For the capacitive charge we may write
dQc ) C dφM + φM dC
(37)
The faradaic charge is obtained from
dQf ) -FΓs dθ
(38)
Hence the total change of charge is
dQ ) dQc + dQf ) -FΓs + φM(Cb - Cf) dθ
(39)
(41)
The coverage θ is then obtained from eq 35:
θ(t) )
1
N
∑
tk+1
ak Q k)0 k + 1
(42)
For eq 33 we need a semiintegral:23 2 k+ 1/2 d-1/2tk (k!) (4t) ) dt-1/2 (2k + 1)!xπ
(43)
Substitution of eq 43 and (42) into (33) gives
{
θ(t) ) 1 - exp -B′
[
∫0te-k τ (t - τ) c
] }
2 4 k!(k + 1)! k+ 3/2 k+ 3/2 -τ ) dτ C ak (t (2k + 3)! k)0 N
∑
k
(44)
where
B′ ) Bkc(csat)2; C )
8Γs Qc xπD sat
This equation can now be used for the estimation of parameters kc and B′ by a nonlinear fitting procedure. A program compiled in C language was written for this purpose. Experimental Section Solubility of Potassium 1-Nonanethiolate. A portion of 600 µL of 1-nonanethiol (Aldrich, 95%) was injected into 40.0 mL of 0.18 M solution of potassium hydroxide in water which was deaerated by bubbling 99.999% nitrogen for 2 h. The solution was kept in a bottle in a glovebox where oxygen content was under 1 ppm with occasional shaking for several days. Ten milliliter samples were taken, mixed with the excess of silver nitrate solution to precipitate mercaptan as silver mercaptide, and finally the excess silver ion was titrated with ammonium thiocyanate with ferric ammonium sulfate as indicator. The main difficulty in the determination was strong emulsion formation during the dissolution process. This apparently diminishes the accuracy of the result which after 4 determinations is (4 ( 1) × 10-3 mol dm-3 at about 22 °C. Results and Discussion
and current
dθ dQ dθ ) [-FΓs + φM(Cb - Cf)] ) Qtot I) dt dt dt
aktk ∑ k)0
(36)
where C is the differential capacitance of the double layer and φM is the potential difference between the electrode and solution. In the present experiment, the applied potential is changed instantaneously from one potential to another and kept constant thereafter. During the potential step and immediately after it, the capacitive current is caused by charging the double-layer capacitance which itself is dependent on the potential. This process can be seen as a sharp peak in the beginning of the chronoamperogram. After the step, potential is constant and the charging current is entirely caused by the capacitance of the bare electrode Cb and the thiol-coated electrode Ct and the change of coverage:
dQc ) φM dC ) φM (Cb - Ct) dθ
N
I(t) )
(40)
Here Qtot does not include the initial sharp peak of charging current. Consequently, although the measured current includes a capacitive contribution, it is still directly proportional to the derivative of coverage.
We used our model to describe the chronoamperograms of the reductive desorption of a self-assembled monolayer of 1-nonanethiol from a Au(111) electrode18 in a 0.1 M KOH solution. This system was chosen because of its simplicity and previous electrochemical characterization. Also, the nonanethiolates are only weakly soluble and their alkane chain is long enough for the interadsorbate interactions to be significant. Hence, it should be possible to access the validity of our
5794 J. Phys. Chem. B, Vol. 104, No. 24, 2000
Vinokurov et al.
approximations. The use of a single-crystal substrate also ensures better homogeneity of the formation and growth of patches. In a previous study, it was shown that for large overpotentials (e.g., final potential 0.25 V more negative than the potential of peak reductive current of -0.99 V vs SCE) the experimental current decays more slowly than predicted by the models of progressive and instantaneous etching center creation. One of the possible explanations discussed in ref 18 considers that the rate of growth of the etching centers becomes limited by the diffusion rate of the reduced thiols away from the perimeter of the patches. However, this proposition was not tested mathematically. We now present fits of four chronoamperograms for reduction of the nonanethiol monolayer with final potentials -0.10, -0.15, -0.2, and -0.25 V (relative to the potential of peak reductive current) with our model. The measured chronoamperograms were first fitted with an 11th degree polynomial in order to get coefficients ai. Equation 44 was then used for the fitting procedure. Parameters kc, B′, and (in the case of -0.25 V) C were varied, coverage θ was calculated, and the current was calculated from θ by numerical differentiation. The current from this procedure and the polynomially fitted observed current (eq 37) were subtracted from each other, the difference was squared and summed at ca. 1000 points, and the sum was minimized. This was first done in the case of final voltage -0.25 V where the effect of diffusion was strongest. Parameter C contains the saturation concentration and diffusion coefficient and is assumed to be the same for all the chronoamperograms. Hence, C was subsequently kept constant in the fits at -0.2, -0.15, and -0.10 V, and only kc and B′ were varied. The results of the fitting procedure are shown in Figure 1 and Table 1. As one can see, the fit is nearly perfect in all four cases. The saturation concentration obtained from experimental solubility measurement was combined with parameter C, total charge Q ) -114 µC cm-2, and the coverage18 Γs ) 7.2 × 10-10 mol cm-2 to obtain the diffusion coefficient. The value obtained is between 7.4 × 10-6 and 2.0 × 10-5 cm2 s-1, not far from the estimated values for compounds of this molecular size. This can be compared with the value 5.9 × 10-6 cm2 s-1 obtained for 2-mercaptoethanesulfonate by Calvente et al.24 using voltammetric methods. The mechanisms called progressive and instantaneous etching center formation and growth in the previous publications17,18 have been originally derived for the nucleation and growth processes of metal deposition25 using the Avrami theorem as the basis. These mechanisms have been generally used as mutually exclusive, perhaps not always realizing that they represent extreme limits of the same model. In fact, if we allow in our more general eq 33 the diffusion limitation to vanish by D f ∞, we have Df∞
θ(t) 98 1 - exp[-Bkc(csat)2
[
) 1 - exp -
∫0t e-k τ(t - τ)2 dτ] c
2B(csat)2 (1 - kct + 1/2kc2t2 - e-kct) 2 kc
]
(45)
Current is obtained by applying eq 35 to eq 45:
[
sat 2 2B(csat)2 dθ 2BQ(c ) -kct (e - 1 + kct) exp (1 I)Q ) dt kc k2 c
]
kct + 1/2kc2t2 - e-kct) (46)
Figure 1. Fitting of eq 44 to the experimental results18 of the desorption of 1-nonanethiol from Au(111) at -250, -200, -150, and -100 mV (relative to the potential of peak reductive current at -0.99 V vs SCE).
If we now assume that the formation of etching centers is a very fast process so that they are effectively formed instanta-
Reductive Desorption of Self-Assembled Monolayers
J. Phys. Chem. B, Vol. 104, No. 24, 2000 5795
TABLE 1: Parameters Obtained by Fitting Eq 44 to the Experimental Chronoamperograms18 of the Desorption of 1-Nonanethiol from Au(111) E vs (V)
kc (s-1)
B′ (s-3)
C (cm2 s1/2 C1-)
-0.10 -0.15 -0.20 -0.25
17.55 124.4 2044 7182
457.4 2.97 × 104 3.13 × 106 3.16 × 107
2100 (fixed) 2100 (fixed) 2100 (fixed) 2100
Figure 2. Linear relationship between the desorption potential (relative to the potential of peak reductive current at -0.99 V vs SCE) and logarithm of the ratio of parameters B′ and kc (cf. eqs 49 and 50).
neously in the very beginning of the desorption process, this means that kc f ∞. Applying this to eq 46 gives
Iinst ) 2BQ(csat)2t exp-B(csat)2t2
(47)
On the other hand, if the formation of etching centers is slow and thus kc is small, we take a few terms from the series expansion of exponential function and expand (46) as
Iprogr = BkcQ(csat)2t2 exp-1/3Bkc(csat)2t3
(48)
Equations 47 and 48 have the functional forms of the “classical” expressions for the instantaneous and progressive nucleation and growth, respectively. It should be noted that whereas eq 47 represents an exact limit, eq 48 is composed only from the first nonzero terms of the series expansion and consequently the accuracy of the commonly used equation for the current in the progressive nucleation is often questionable, especially at high values of time. Our eq 33 includes both progressive and instantaneous formation and growth of etching centers, their intermediate mechanisms, and in addition, the influence of the diffusion rate of the desorbed species. Parameter B contains potential-dependent β as a factor and hence we should observe potential dependency of B′ . Taking logarithm we have
[
]
N0 1 B′ ( / aγ j NART)2 + 2 ln β (49) ln ) ln(Bcs2) ) ln kc Ntot 2 where the constant terms have been lumped together. Figure 2 shows the linear relationship of this logarithm with potential and we have
B′ ln ) 0.079 - 34.5E kc
(50)
Assuming a one-electron Tafel-like exponential dependence of β on potential, i.e.
[
β ) β0 exp -
]
RF(E - E0) RT
(51)
substituting this into eq 49, and comparing with the experimental result of eq 50, we obtain for the transfer coefficient a value R ) 0.44 ( 0.05. This is within the range commonly observed for a large number of electrochemical processes. Conclusions The model described in this work is an attempt to explain the shape of the current transients observed during the electrochemical desorption of self-assembled monolayers of thiolates. We found that the experimental results can be explained by including the effect of diffusion on the process. Our model involves a number of reasonable assumptions. The first assumption, the first-order formation rate of etching centers, is a common assumption in the theory of nucleation. According to the second assumption, the desorption process produces empty patches which grow keeping their more or less regular shape. This is a direct consequence of the fundamental property of self-assembled films, i.e., the mutual attractive interaction of long-chain molecules. The third assumption that needed more justification is actually intuitively understandable. It simply tells that desorption is strongly suppressed as the concentration of adsorbing molecules approaches saturation. In fact, this assumption is directly related to the analogous semiempirical rate law of dissolution. The fourth assumption deals with the lateral homogeneity of concentration, which is a necessary and common approximation for obtaining a tractable mathematical expression. We showed that the expression derived for the time dependence of coverage and desorption current is rather general. The mathematical form of the expression is, however, complex and the behavior of current-time function is not easily revealed. However, credibility of our model is obtained from the fact that the progressive and instantaneous nucleation models are asymptotical limits of our model. We tested our model by fitting chronoamperograms of the reductive desorption of adsorbed 1-nonanethiol from a gold(111) surface. The excellent fits, obtained for a range of overpotentials, provide support to the approximations made in our model. Although a good fit is a prerequisite for the acceptance of the model, alone it is not sufficient to validate a model. Hence, we used the parameters estimated in the fitting procedure to calculate observable properties. In the present case, we measured the saturation concentration of nonanethiol and used it together with the estimated parameter C to obtain the diffusion coefficient of nonanethiolate. The estimated value is within very reasonable limits for a molecule of this size. Additional support for the model gave the potential dependence of parameter B′, which yields an acceptable value for the transfer coefficient. It is believed that the origin of the potential dependence of B′ is related to the faradaic reduction of the chemisorbed thiol. Acknowledgment. Financial aid from the Academy of Finland is gratefully acknowledged (grant no. 30579). The authors thank Natalia Kotcharova for the experimental work in the solubility measurements. References and Notes (1) Bunding-Lee, K. A.; Mowry, R.; McLennan, S.; Finklea, H. O. J. Electroanal. Chem. 1988, 246, 217. (2) Widrig, C. A.; Chung, C.; Porter, M. D. J. Electroanal. Chem. 1991, 310, 335. (3) Steinberg, S.; Tor, Y.; Sabatani, E.; Rubinstein, I. J. Am. Chem. Soc. 1991, 113, 5176.
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