4898
Langmuir 1999, 15, 4898-4906
Interference of Adsorption, Condensation, and Double-Layer Charging in Kinetic Studies of Film Formation. 1. Constant Double-Layer Potential C. Donner* and L. Pohlmann Free University of Berlin, Institute of Physical and Theoretical Chemistry, Takustrasse 3, 14157 Berlin, Germany Received December 30, 1998. In Final Form: April 7, 1999
The current response after a single potential step experiment reflects the kinetics of nonfaradaic phase transitions. In the present paper, a classification of possible shapes of current transients under potentiostatic control is proposed. The possible shapes of current transients were simulated by a model, which describes the condensation process as a coupled process of adsorption, nucleation, and growth. Due to the coupling of these three processes, the surface concentration of the expanded phase and therefore the supersaturation as the driving force for the condensation become time dependent. The classification of the transients is carried out according to the location of the final potential in respect to the potential of maximum adsorption on one hand and according to an additional possible dipole contribution due to a reorientation of molecules during the phase transition on the other hand. Apart from the well-known nonmonotonic current transients, which stand for a phase transition process independent of whether the phase transition is of faradaic or nonfaradaic nature, new shapes of current transients were additionally simulated. From the appearance and the location of such new shapes in a system, one can draw conclusions about the direction of reorientation during the phase transitions as well as about the location of the potential of maximum adsorption.
1. Introduction In electrochemical context two-dimensional first-order phase transition processes involve both faradaic electrocrystallization processes including UPD phenomena1 and nonfaradaic film formation processes of organic neutral molecules in adsorbate layers.2 A common characteristic for the kinetics of two-dimensional faradaic and nonfaradaic first-order phase transitions consists of the nucleation and growth mechanism of the condensed phase out of the supersaturated expanded phase.3 In most experiments the kinetics is investigated by measurements of the current and, for nonfaradaic transitions, also of the capacity response after a potential jump into the condensation region. If the following comments are restricted to the measurements of the current response, that is to say, that for both faradaic and nonfaradaic phase transitions the shapes of the nonmonotonic transients as shown in Figure 2a are principally the same, even this characteristic shape is often the only electrochemical feature to identify the current response as a nucleation and growth mechanism. However, this transient behavior is neither a sufficient3 nor a necessary4 criterion for first-order phase transitions. There are principal differences between the two kinds of phase transitions on electrodes. In nonfaradaic processes, apart from the transient shape as shown in Figure 2, other shapes, the so-called inverted current transients, are found experimentally5 (see Figures 3, 4, and 7). (1) Fleischmann, M.; Thirsk, H. R. In Advances in Electrochemistry and Electrochemical Engineering; Delahay, P., Ed.; Wiley: New York, 1963; p 3. (2) de Levie, R. Chem. Rev. 1988, 88, 599. (3) Budevski, E.; Staikov, G.; Lorenz, W. J. Electrochemical Phase Formation and Growth, VCH Verlagsgesellschaft mbH: Weinheim, Germany, 1996. (4) Bosco, E.; Rangarajan, S. K. J. Chem. Soc., Faraday Trans. 1 1981, 77, 1673. (5) Donner, C.; Kirste, St.; Pohlmann, L.; Baumga¨rtel, H. Langmuir 1998, 14, 6999.
Figure 1. Schematic representation of charge-potential curves in phase transition experiments for a nonfaradaic film formation experiment. The charge potential curves represents (a) the adsorbate free surface (q0 vs E), (b) the surface covered with the equilibrium concentration of the expanded adsorbate phase (qequ vs E), (c) surface covered with the condensed phase without reorientation (qa vs E), and (d) surface covered with the condensed phase with reorientation (qfilm vs E). The final potentials Ei are chosen to be characteristic for the different charge ratios of the plot: E1, q0 < qequ < qfilm < qa; E2, q0 > qequ > qa > qfilm; E3, q0 < qfilm < qequ < qa; E4, qfilm < q0 < qequ < qa.
For the modeling of the kinetics of nucleation and growth processes in two-dimensional systems, several proposals exist. The best known model equally applied for the interpretation of electrocrystallization and film formation processes was developed by Bewick, Fleischmann, and Thirsk6 (BFT model), which distinguishes between a progressive/instantaneous nucleation and a constant/ surface diffusion controlled growth. In later decades, this model received some extensions regarding both the nucleation process by the establishment (6) Southhampton Electrochemistry Group. Instrumental Methods in Electrochemistry; Ellis Horwood Series in Physical Chemistry; Ellis Horwood: London, 1993; Chapter 9.
10.1021/la981763r CCC: $18.00 © 1999 American Chemical Society Published on Web 06/05/1999
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Figure 2. Current transients obtained for a final potential E1 (Figure 1): (a) without reorientation; (b) with reorientation. Key: (‚‚‚) adsorption current; (s and full circles) total current response; (- - -) reorientation current.
Figure 3. Current transients obtained for a final potential E2: (a) without reorientation; (b) with reorientation. Key: (‚‚‚) adsorption current; (s and full circles) total current response; (- - -) reorientation current.
of the more general exponential law of nucleation7 and the growth process by variation of the growth rate with the nucleus size (see for example ref 8). The above cited models consider only the nucleation and growth process itself. Truly, the current transients are the result of competing adsorption and condensation processes. For homogeneous and heterogeneous surfaces at first a critical nucleation concentration must be reached before nucleation can start. After the nucleation has started, the supersaturation on the electrode is determined by the consumption of molecules of the expanded phase due to the growing islands on one hand and by the adsorption of molecules from the bulk to the expanded phase on the other hand. The problem of the coupling of these adsorption and condensation processes is treated in several papers under different experimental conditions. Without claiming completness at this point, one should mention the model of Staikov, Ju¨ttner, Lorenz, and Schmidt9 for the description of phase transitions in faradaic systems and the model of Guidellli10 for the description of phase transitions in chemisorbed adsorbate systems. The first model assumes a constant supersaturation during the phase transition process, whereas the second one is based on a kinetics limited by bulk diffusion. Another model for describing explicitly the coupling in nonfaradaic phase transitions is proposed in ref 11, in which a time dependent supersaturation is the consequence of the mass balance equation, containing a source
term, the adsorption of organic molecules from the bulk onto the electrode, and a sink term, the consumption of exactly these molecules due to the growing condensed islands. Other models containing adsorption and condensation current parts are based on a proposal of Rangarajan et al.4 This approach was elaborated in several papers, e.g. in ref 12. The common feature of these models is the fact that adsorption and condensation current parts, often with different rate constants, are simply added. Physically, this means that the adsorption and condensation processes are taking place independent of each other on different fixed parts of the electrode surface. This scenario is only conceivable on heterogeneous surfaces in a kinetic connotation under very restricted experimental conditions, such as direct impingement of adatoms in condensed islands. It is also possible that both processes are competing for the occupation of the electrode surface. Unfortunately, the consideration of this competition was inconsistently implemented in the above cited papers, because, as a consequence of these models, the adsorption equilibrium concentration would depend on the area of the free surface (see ref 13). As a rule in experimental papers the above-described models are applied to fit or to interpret the commonly known current transients (see Figure 2), regardless of whether the current transients originate from faradaic or from nonfaradaic experiments. However, the validity of a model for the interpretation of current transients in nonfaradaic experiments must be proved by its ability to reproduce inverted current transients. The aim of this paper is to simulate a complete set of possible shapes of current transients in nonfaradaic film formation experiments under potentiostatic conditions using the model proposed in ref 11. The influence of the
(7) Retter, U. J. Electroanal. Chem. 1982, 136, 167. (8) Philipp, R. J. Electroanal. Chem. 1990, 290, 67. (9) Staikov, G.; Ju¨ttner, K.; Lorenz, W. J.; Schmidt, E. Electrochim. Acta 1978, 23, 305. (10) Guidelli, R.; Foresti, M. L.; Innocenti, M. J. Phys. Chem. 1996, 100, 47. (11) Pohlmann, L.; Donner, C.; Baumga¨rtel, H. J. Phys. Chem. B 1997, 101, 10198.
(12) Arulraj, A.; Noel, M. Electrochim. Acta 1988, 979.
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reorientation during the phase transition is analyzed. The simulated current curves are classified according to their appearance regarding the potential of maximum adsorption and their different charge contributions of adsorption and reorientation depending on the final potential. The coupling of the double-layer charging with the whole adsorption process leads to a nonconstant double-layer potential in the interface during condensation. This condition, under which the double-layer charging is located in the same time region as the adsorption process, is often fulfilled during nonfaradaic phase transitions on solid electrodes. The simulation and classification of nonpotentiostatic transients require an extension of the model11 used in this paper and will be the aim of a following paper (part 2). 2. Theory To make the simulations and the subsequent classification of potentiostatic current transients more transparent, this section should begin with reflections about the differences between faradaic and nonfaradaic transients. 2.1. Differences between Nonfaradaic and Faradaic Monolayers. In faradaic phase transition experiments such as electrocrystallization, the current response contains the following charge contributions. The main part is caused by the discharge of metal adatoms on the surface; hence, every adatom provides the same whole electron charge. For a rebuilding of a complete monolayer, the same charge must flow independently from the final potential, where the process takes place, and from the mechanism of the process. This means that for integration of current transients, measured at different potentials, the same charges must always be obtained. If additionally the capacity properties of the two metal monolayers are different, furthermore, a charge contribution is caused due to the charging or discharging of the one metal capacity in relation to the other one. However, as a rule, this capacity charge contribution is very small compared to the faradaic charge contribution, so that in experiments this charge contribution is often neglected. From this it can be concluded that by neglecting the double-layer effects the charge, flowing in single potential step experiments, must be identical with the charge under the peak, which belongs to the phase transition in potential scan experiments. Indeed, this agreement is found in many electrocrystallization experiments. In nonfaradaic phase transition experiments, two fundamental differences exist in comparison to faradaic transitions (Figure 1). If one considers phase transitions without reorientation of dipoles in the condensed phase, every adsorbed molecule contributes the same charge to the capacity, independent of whether the molecule is located in the condensed or the expanded adsorption state. The charge-potential curves for the adsorbate free surface, the surface covered with the complete monolayer of the expanded phase, and the surface covered with the complete monolayer of the condensed phase intersect in only one potential, the potential of maximum adsorption Em. This is experimentally confirmed in various systems.14,15 For systems, however, where the potential of (13) Retter, U.; Pohlmann, L.; Donner, C. Manuscript in preparation. (14) Guidelli, R. In Adsorption of Molecules at Metal Electrodes; Lipkowski, J., Ross, P. N., Eds.; VCH Verlagsgesellschaft: Weinheim, Germany, 1992; Chapter 1. (15) Lipkowski, J.; Stolberg, L. In Adsorption of Molecules at Metal Electrodes; Lipkowski, J., Ross, P. N., Eds.; VCH Verlagsgesellschaft: Weinheim, Germany, 1992; Chapter 4.
Donner and Pohlmann
maximum adsorption is not constant, one can take the invariance of Em as a reasonable first approximation. For a constant final potential the entire charge, which must flow for a rebuilding of a complete condensed organic monolayer is always the same. But for different final potentials the total charge difference ∆q ) q0 - qfilm for a transition between the adsorbate free surface and a surface covered with a condensed film is also different as is shown in Figure 1. The nearer the final potential lies at Em, the smaller is the absolute charge, which must flow during phase transition and, consequently, at Em no charge flow can be detected.16 Exactly at Em, the current response in potential step experiments becomes inverted. The sign of the current, flowing to build up the adsorption equilibrium of the expanded monolayer (curve b in Figure 1) and to build up the condensed layer (curve c in Figure 1) is always the same and depends on the position of the final potential relative to Em. If reorientation occurs during the phase transition, an additional charge contribution is obtained due to the additional negative or positive dipole contribution of the organic molecules in the condensed phase (see, for example, curve d in Figure 1). As a consequence the charge potential curve of the condensed monolayer is shifted to more positive or more negative charge values, respectively. In the same manner the potential of maximum adsorption Em is shifted to E′m, and an additional point of intersection between the charge potential curves of the condensed and noncondensed phase, at the crossing point potential Ecp, appears. Now, between the potential region Ecp and Em the pure adsorption current to build up the adsorption equilibrium of the expanded phase and the condensation current have the opposite sign. The three potentials Em, E′m, and Ecp differ from each other by the different charge contributions according to the adsorption and condensation processes. At the potential Em, only the charge flowing due to reorientation can be detected. On the other hand, at E′m, the charge contribution due to the entire adsorption process and due to reorientation is equal but opposite in sign. Finally, at Ecp, only the adsorption current can be detected, while the reorientation current vanishes. Indeed, in such a more complicated case, the inversion of current transients takes place at final potentials somewhere between Ecp and Em depending on the kinetic parameters. Generally, the charge difference between the complete monolayers with and without reorientation is also a function of the applied potential. For sake of simplicity in the following simulations, we assume a potential independent charge difference between the two complete monolayers. This simplification does not influence the principal classification by any means. Contrary to faradaic experiments, the charge belonging to the condensation peak in scan experiments and the charge obtained in single potential step experiments must be different just as the charges belonging to the condensation and the dissolution peak are different due to their different potential positions. It seems to be possible that also in chemisorption systems or in systems with very slow faradaic reactions of organic molecules, in which the faradaic and the nonfaradaic current parts are of comparable magnitude, inverted current transients may be obtained.17 2.2. Modeling the Current Transients. The shape of the different current transients changing with the final potential was briefly explained in ref 16 and at first (16) Buess-Herman, Cl. In Trends in Interfacial Electrochemistry; Silva, A. F., Ed.; D. Reidel: Dordrecht, The Netherlands, 1986; p 205. (17) Philipp, R. Private communication and unpublished results.
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described theoretically and experimentally in ref 5. The direction of the adsorption and condensation current depends sensitively on the position of the final potential regarding the potential of maximum adsorption Em on one hand and the crossing point potential Ecp in reorientation systems on the other hand. In ref 5, this change in the shape of transients was exploited to determine the otherwise unaccessible potential of maximum adsorption Em in condensed layers. To simulate typical current-time transients, the model of coupled adsorption and condensation11 was used. In this model it was assumed that the adsorption kinetics of the expanded phase can be decomposed into a Langmuir kinetics on one side, which would build up the equilibrium surface concentration of the expanded phase, and on the other side into a condensation kinetics via nucleation and growth, if the critical surface concentration is exceeded. To be more explicitly, the lower branch of the more adequate lattice gas isotherm (or of the Frumkin isotherm, as an intermediate approximation) is replaced by a Langmuir isotherm, whereas above a critical surface concentration the steplike isotherm according to the lattice gas model takes place. This is obviously an approximation, but it is much easier to handle in comparison with the lattice gas model. This approximation is justified, because usually the equilibrium concentration of the expanded phase is much smaller than the surface concentration inside the condensed film. The coupling between the adsorption, the nucleation, and the growth processes is mediated by the surface concentration of the expanded phase, which is assumed to be uniformly distributed at the film free surface, i.e the role of the surface diffusion was neglected for sake of simplicity. For the simulations the following coupled differential equations in their dimensionless form must be solved numerically (see, for detail, in ref 11). (1) This first equation is the mass balance equation for the normalized surface concentration of the adsorbate in the expanded phase Γa:
dpdl ) 2π(Γa - δ)ndl dtdl
dΓa ) R(γ - Γa) - (Γa - δ)(1 - Γa)pdl dtdl
(1)
It should be noted here that in ref 11 the (1 - Γa) part in the last term was absent. This additional term denotes the normalized difference between the density of the condensed phase and the actual surface concentration of the expanded phase. However, the error which occurs by omitting this factor is not significant since the concentration of the expanded phase is usually much smaller than the film density. (2) The second equation gives the time dependence of the surface coverage Θ of the condensed phase (defined as the fraction of the film covered surface area in respect to the whole surface area):
dΘ ) (1 - Θ)(Γa - δ)pdl dtdl
(2)
For the extended surface coverage Θext then, according to the Avrami theorem, the following equation is valid:
dΘext ) (Γa - δ)pdl dtdl
(2a)
(3) Then we have the time dependence of the total periphery length pdl of all islands of the condensed phase:
(3)
(4) Finally the nucleation rate ndl according to the Volmer-Zeldovich law (Z here is the Zeldovich preexponential factor) is given:
(
( ))
dndl Γa ) Γa2Z exp -K/ln dtdl δ
(4)
The units used are defined as follows: concentration unit, Γfilm; time unit, tunit ) 1/(knkw2Γfilm4)1/3; length unit, lunit ) (kw/knΓfilm)1/3. Here Γfilm is the surface concentration of the condensed monolayer, and kn and kw are the rate constants of nucleation and growth. From these units the new dimensionless variables are as follows:
tdl ) t/tunit; Γa ) Γ/Γfilm; pdl ) plunit; ndl ) n(lunit)2 The dimensionless parameters are then defined as follows: K, unchanged constant of the Volmer-Zeldovich law; R ) ka/(knkw2Γfilm4)1/3; ka ) kadscbulk + kdes, effective rate constant of adsorption; δ ) Γsat/Γfilm; Γsat, saturation concentration of the expanded adsorbate phase; γ ) Γeq/ Γfilm; Γeq ) kadscbulkΓmax/(kadscbulk + kdes), equilibrium concentration of noncondensed molecules. Γmax is the maximum possible surface concentration of adsorbed molecules in the expanded phase (according to the formal limit of infinite bulk concentration cbulk f ∞). The obtained current response itself reflects the increase of the number of adsorbed molecules up to a rebuilding of a complete condensed monolayer plus the contribution of the dipole reorientation during the phase transition. To derive the expression for the time dependent current density according to the surface charging (or discharging), one can start with the mean charge density of an electrode surface, which is partially covered by the condensed phase of the adsorbate (for the derivation, see Appendix, eq A9)
q(Γa, Θ, E) ) qexp(Γa, E)(1 - Θ) + qfilm(E)Θ (5a) with the surface charge (per unit area) of the expanded adsorbate phase
qexp(Γa, E) ) q0(E)(1 - κΓa) + qa(E)κΓa
(5b)
and κ ) Γfilm/Γmax. For simplicity, in the following it will be assumed that κ ) 1; i.e., Γmax ) Γfilm. This is valid, if the occupation area of the adsorbed molecules does not change significantly during the condensation process. The electrical variables can be nondimensionalized formally by dividing by some convenient SI units: Uunit ) 1 V, Qunit ) 1 µC, Cunit ) 1 µF. Then the time derivation of q(t), eq 5, gives the mean current density i(t) reflecting the changes in the surface charge due to the adsorption and condensation of the neutral molecules:
i(t) ) [qfilm(E) - qexp(Γa, E)]
dΘ + dt
(1 - θ)[qa(E) - q0(E)]κ
dΓa (6) dt
This equation attributes the total current density i(t) to two partial current densities, where the first one describes the charging due to the growing condensed film (dΘ/dt),
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and the second one is the current flowing due to the changes in the surface concentration of the expanded adsorbate phase (dΓa/dt) at the film free surface parts. But this equation is somewhat complicated, because the difference in the first square brackets is still explicitly time dependent due to time dependence of Γa(t). Equation 6 can be transformed into an equivalent form. For this, eq 1 will be used, which can be written in the following symbolic form
[ ] [ ]
dΓa dΓa ) dtdl dt
+
-
dΓa dt
(1a)
-
denoting explicitly the source and the sink term
[ ] [ ] dΓa dt
+
) R(γ - Γa);
dΓa dt
-
) (Γa - δ)(1 - Γa)pdl ) (1 - Γa)
dΘext (1b) dtdl
This leads with eq 6 to
dΘ + dt dΓa (1 - Θ)[qa(E) - q0(E)]κ dt
i(t) ) [qfilm(E) - qexp(Γa, E)]
[[ ]
- (1 - Γa)
[ ]
-
+
]
dΘext dtdl
or
i(t) ) (1 - Θ)[qa(E) - q0(E)]κ
dΓa dt
+
[(qa(E) - qfilm(E)) + (κ - 1)(qa(E) - q0(E))Γa]
dΘ (7) dt
This equation becomes much simpler, if we follow the above-mentioned assumption of κ ) 1:
i(t) ) (1 - Θ)[qa(E) - q0(E)]
[ ] dΓa dt
+
-
[(qa(E) - qfilm(E))]
dΘ (7a) dt
In this equation now the differences in the square brackets both are independent of the surface concentration Γa(t) and are dependent only on the applied potential E. Here the square brackets with the “+”-subscript denote only the source process in the balance, eq 1. This restriction to the source term is essential, because the molecules, which are leaving the expanded phase (and which are described by the sink term), nevertheless, remain on the surface, becoming incorporated in the condensed phase. This means that the charge flowing due to the adsorption of one molecule occuring in that moment when the molecule arrives at the surface. If then later this adsorbed molecule is incorporated in the growing condensed phase, no additional charge will flow, so long as the molecule does not change its orientation (this is at least valid in the scope of the above-mentioned Frumkin assumption). Therefore, the true current reflecting the adsorption process is expressed by the first term in eq 7a (using here also the definition in eq 1b):
iads(t) ) (1 - Θ) (qa(E) - q0(E)) R(γ - Γa)
(8)
However, if during condensation of an adsorbed molecule its orientation and its electronic properties are changed (i.e. if per definition qa * qfilm), an additional current occurs
at the moment of the condensation. This reorientational current is described by the second term of eq 7a (using also eq 2):
irear(t) ) -(qa(E) - qfilm(E)) (1 - Θ)(Γa - δ)pdl (9) Equation 9 becomes zero, if during the phase transition no reorientation occurs. In the following section, on the basis of eqs 1-4, all possible types of current transients (eqs 7-9) are simulated (for the case of κ ) 1). The transients are classified according to their position of the final potential in respect to the potential of intersection between the noncondensed and the condensed state Ecp on one hand and the potential of maximum adsorption Em on the other hand.18 The effect of the reorientation on the current transients is also discussed. 3. Numerical Simulations If no reorientation occurs during the phase transition, then the equilibrium potential-charge curves of the pure electrolyte, the noncondensed state, and the condensed state intersect at only one potential Em (curves a-c, Figure 1). Otherwise, if reorientation takes place during phase transition, then two additional intersection points occur, one at the potential E′m between curves a and d in Figure 1 and the other one at the potential Ecp between curves b and d. The charge-potential curve d of the condensed state represents the case where this condensed phase yields a more positive dipole contribution than the noncondensed phase. The PZC is therefore shifted into the positive and the Em into the negative direction. The simulations presented in the following are dealing with this according to the experimental results in the system thymine/NaClO4/H2O on mercury.5 In the opposite case, where during the phase transition a negative reorientational charge contribution is obtained, the corresponding charge-potential curve d would be located at more positive charges in comparison to curve c. This case was found in the system adenine/NaClO4/ H2O on mercury.15 Principally, the expected currenttransient shapes are the same in both cases. However, the strengthening or leveling effect of the additionally obtained charge caused by the reorientation is opposite in its effects on the shape of current transients for both cases. If one assumes that immediately after the potential jump is finished the electrode surface is free from adsorbate (Figure 1 curve a), then at first the critical nucleation concentration must be reached before the condensation can start. The critical nucleation concentration vs potential curve is located between curves a and b in Figure 1 and is determined by the dimensionless parameters δ and K. The following simulations of possible current transients shown in Figures 2-5 and 7 are carried out with a constant set of parameters: R ) 0.1; γ ) 0.233; δ ) 0.166, κ ) 1; and K ) 0.076. This parameter set is borrowed from ref 11. The entire adsorption kinetics is then the same in all cases. Solely the charge differences ∆q ) q0 - qa (for the no reorientation case) and ∆q ) q0 - qfilm (for the reorientation case), respectively, are functions of the final potential. Starting from charge differences ∆q ) q0 - qa/film, experimentally found in ref 5, the values for the charge differences used in the presented simulations are listed in Table 1. Without reorientation effects the charge densities qa and qfilm have the same value and row 3 in Table 1 can be neglected. Parts a and b of Figure 2 show the well-known current transients, which are commonly measured and used to
Kinetic Studies of Film Formation
Figure 4. Current transients for different charge differences listed in Table 1. The curves represent the transition from “normal” to “inverted” current transients between the final potentials E1 and E2.
Figure 5. Current transients obtained for a final potential E3: (‚‚‚) adsorption current; (s and full circles) total current response; (- - -) reorientation current.
interpret a kinetic process as a nucleation and growth process. These curves are obtained if the final potential is located at E1 in Figure 1. The curves in parts a and b of Figure 2 are distinct due to their reorientation current part. As an example, Figure 2a represents the simulation of a simple current transient without reorientation. The transient total current reflects the pure adsorption kinetics of the noncondensed molecules to build up a condensed monolayer. The total current response is the only one, which can be measured in current-time experiments. In Figure 2b, where reorientation takes place, it is the sum of the adsorption and the reorientation current. The shape of the presented transient in Figure 2b is formally the same as it shown in Figure 2a. But the transient in the latter case is composed of different parts of charges according to eq 7. Generally can be said, that the reorientation effect weakens the total current response. The minimum and maximum of the pure adsorption kinetics (‚‚‚) are slightly shifted to shorter times with regard to the total current transient (s with full circles). This effect is caused by the delayed start of the condensation process and their negative charge contribution. Depending on the ratio between the pure adsorption kinetics and the growth rate of the condensed islands, the “discrepancy” between the adsorption current and the total current response is more or less pronounced. Furthermore, it is not conclusive that an exponential decaying part of the total current transient up to zero is a hint for a fast adsorption kinetics, because the current response is always lower than the true adsorption current at every time during condensation. The pure comparison between the charge flowing before (exponential decaying part of the current transient) and after condensation starts, in
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general, allows no quantitative conclusions about the increase of density in the condensed phase on one hand and the possible number of surface defects19 on the other hand, if the dipole and therefore the charge contribution of the condensed and noncondensed phases are unknown. If the final potential is located at E2, the typical shapes of current transients shown in Figure 3 are obtained. For the transients without reorientation (shown in Figure 3a) is also valid, that the real adsorption current and the total current response are identical. The direction of charge flow is merely reversed in comparison with Figure 2a. If reorientation takes place at the final potential E2, the total current response is strengthened compared with the pure adsorption curve (see Figure 3b). At the same time one must not overlook that also in this case the maxima and minima of the total curve and the pure adsorption curve are slightly shifted due to the different adsorption and condensation kinetics. Also valid for these transients is the point that the ratio between the charge flowing up to nucleation and the charge flowing after nucleation starts reflects by no means the ratio of the density of the noncondensed and the condensed phase and the density of surface defects, respectively. For the final potential E1 as well as for E2 the charge necessary to build up a noncondensed or condensed monolayer has the same sign, independent of reorientation effects. For simulations carried out for constant dimensionless parameters R, δ, γ, and K, the transition between “normal” and “inverted” current transients can be achived by variation of the final potential as is shown in Figure 4 (charge differences used from Table 1). The nearer the final potential is located in the vicinity of Em, the lower is the absolute value of charge flowing during the whole adsorption process. This affects both the exponential decaying part and the following parabola-like part of the current transient, respectively. In real experimental systems, not only is the absolute value of charge changing by the final potential but also the adsorption kinetics itself is strongly potential dependent. Due to these two effects, the sensitivity of current detection in the immediate vicinity of the potential of maximum adsorption Em is limited. A special feature in the charge-potential characteristic is obtained during reorientation processes in the potential region between Ecp and Em due to the opposite sign of the charge flowing by simple adsorption and by reorientation, respectively. A detailed characteristic for the charge ratios consists of that, that between Ecp and E′m the difference ∆q1 ) q0 - qa is greater than the difference ∆q2 ) qa qfilm, whereas in the region between E′m and Em the ratio between these differences is the opposite. In Figure 5 an example for a current transient obtained at the final potential E3 in Figure 1 is shown. Surprisingly, the sign of the simulated current transient located almost in the middle of the region between Ecp and E′m is not clearly negative or positive, but the zero charge line is intersected two times. This current oscillation depends in their size effect and their clearness on the charge difference between q0, qa, and qfilm on one hand and on the ratio between the pure adsorption and the growth kinetics on the other hand. If one decreases the difference between q0 and qa (E3 would then be located closer to E′m), one obtains in the simulations the inverted current transients (18) Donner, C; Kirste, St.; Pohlmann, L.; Baumga¨rtel, H. Presented at the Portucalensis Conference on Electrified Interfaces, Povoa de Varzin, Portugal, 1998. (19) Ho¨lzle, M. H., Wandlowski, Th.; Kolb, D. Surf. Sci. 1995, 335, 281.
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Table 1. Charge Values used for the Simulations of the Current Transients Shown in Figures 2-5 and 7a E1 (Figure 2) q0 qa qf a
-9.96 -6.79
-10.70 -7.18 -8.15
-11.90 -7.65
E3 (Figure 5)
E4 (Figure 7)
-7.90 -6.12 -7.18
-5.95 -5.34 -6.43
E2 (Figure 3) -3.95 -4.62
-3.28 -4.45 -5.54
-4.37 -4.81
For Figure 4, it was assumed that no reorientation effects take place in the system.
Figure 6. Experimenttally obtained current transient for the system 14 mmol thymine/0.1 M NaClO4 at a temperature of 20 °C. Potential step: -1700 to -820 mV vs Ag/Ag+.
Figure 7. Current transients obtained for a final potential E4: (‚‚‚) adsorption current; (s and full circles) total current response; (- - -) reorientation current.
experimentally found in ref 5. From this point of view, the “true” inverted current transients described in ref 5 are obtained somewhere between Ecp and E′m, but are not exactly at Ecp. Therefore, the oscillating current transients appear only in a very narrow potential region on one hand and in a region with small differences in the charges on the other hand. Due to this, these transients are very difficult to detect. Experimentally the oscillating currents are obtained in the system thymine/H2O/NaClO4 (Figure 6). The same transients were observed during the anodic film formation of adenine on the mercury electrode.17 This is a proof that also in chemisorption systems the general principles of the coupled adsorption and film formation processes and, therefore, a time dependent supersaturation according to the model described above must be valid. At the potential E′m, the charge flow due to the pure adsorption and the charge flow due to the reorientation are exactly the same. Owing to the delayed condensation at this potential, the inversion of the current transients becomes unambiguous. Between the potentials E′m and Em, the charge belonging to the pure adsorption is smaller than the charge belonging to the pure condensation. Therefore, the reorientation part of the current is dominant, and the current transients appear inverted (Figure 7). Comparison of the adsorption current, the simulated current response, and the reorientation curve reveals that the simulated current transients reflect the condensation kinetics more than the adsorption kinetics. The more the differences ∆q ) q0 qa (the final potential must be closer located to Em) become small, the more the total current transients become closer to the reorientation curve. At exactly Em, the total current transient would reflect the pure condensation kinetics contrary to systems where no reorientation occurs, and therefore no charge could be detected at this potential.
In general, the exponential decaying part and the parabola-like part of current transients are not well separated, and therefore, the total current transients can only be described with a coupled adsorption and condensation model, where the adsorption and the condensation depend on each other. Outside the potential region between Ecp and Em the total current signal is strengthened or weakened according to the negative or positive dipole contribution of the adsorbed molecules. Compared with the pure adsorption curve, the minima and maxima in the total current transients are slightly shifted. Between the potential regions Ecp and E′m, the oscillating current response is the result of the opposite sign of the adsorption current on one hand and the reorientation current on the other hand. Between E′m and Em the current transients become closer to the condensation kinetics, and only at the potential Em does the current transient reflect exactly the condensation kinetics. From the presented simulations it becomes clear that, in general, the modeling of experimental data requires, on one hand, a model which describes the whole adsorption and condensation kinetics in a coupled way, and on the other hand, information about the dipole contributions of the adsorbed molecules is required. A further complication is generated when the doublelayer charging and the pure adsorption kinetics possess the same time constant. Then also these two processes are coupled and the model used in this paper must be modified. The discussion of this case is the aim of the following paper (part 2).
4. Summary The simulations presented above demonstrate the complexity of possible current transients in nonfaradaic phase transition experiments.
Appendix For the derivation of the mean charge density of an electrode surface, which is at a given moment covered with growing islands of the condensed phase, the following will be assumed. In general, the charge in the electrical double layer depends on the presence and the concentration of adsorbed neutral organic molecules at the surface (not accounting here for specific adsorption of ions). Then, according to the Frumkin model of parallel plate condensors, the total
Kinetic Studies of Film Formation
charge excess in the double layer can be calculated by summation of the charge portions of all the adsorbed molecules at the interface, organic ones and solvent molecules. This assumption of linear superposition neglects possible nonlinear interaction effects, which can lead to a dependence of the molar partial charge of the adsorbate on the adsorbate concentration. Nevertheless, this assumption in most cases turns out to be realistic and is therefore widely used in the literature. Furthermore, it will be assumed that the adsorbed organic molecules can exist in two different states: the adsorbed molecules in the expanded phase and the molecules in the condensed phase. In this way, the surface is coverd with three types of molecules (the solvent molecules, the molecules of the expanded phase, and the molecules of the condensed phase). In general, each of these types correspond to a different charge in the double layer per mole adsorbate or solvens (at a given potential). Consequently, each of these types has in general a different molar occupation area. If now the surface of the electrode (with an area of S0) is covered with a coverage of 0 < Θ < 1 by the condensed phase (with an area of SF, i.e., Θ ) SF/S0), and at the remaining places between the condensed islands the adsorbed molecules are present in the expanded phase, the total charge (per unit surface) in the double layer is obtained by the sum of all parts:
Langmuir, Vol. 15, No. 14, 1999 4905
q ) q0
Γsol Γmax sol
(1 - Θ) + qaκΓa(1 - Θ) + qfΘ with κ)
Γfilm Γmax a
(A4)
In this equation the (unknown) actual surface concentration of adsorbed water molecules Γsol can be easily obtained, because the maximum number of possible adsorption sites at a given surface is a fixed number. One has only to consider that, generally speaking, one randomly adsorbed molecule replaces ma water molecules, whereas one in the condensed form adsorbed molecule replaces mfilm molecules of water (the ratios ma and mfilm do not have to be integer numbers). Consequently, the following relations between the three possible complete surface coverages must be valid: max ) mfilmΓfilm Γmax sol ) maΓa
(A5)
This way, at any stage of adsorption and film formation the following balance of surface places must hold:
Γsol(1 - Θ) + maΓ(1 - Θ) + mfilmΓfilmΘ ) Γmaxsol ) const (A6) From this one obtains
Γsol(1 - Θ) ) Γmaxsol - maΓ(1 - Θ) - mfilmΓfilmΘ ) (Γmaxsol - maΓ)(1 - Θ) (A7)
mol
Here, the q ’s are the partial double-layer charges per mol adsorbate (in charge per mol) of the three types of molecules, and the Γ’s are the actual surface concentrations (in moles per unit surface), respectively. This equation is inconvenient due to the presence of molar partial surface charges. However, the latter can be easily removed by the introduction of partial surface charges per area, which can be defined in the following way: mol max mol max mol q0 ) Γmax qa , qfilm ) Γfilm qf (A2) sol qsol , qa ) Γa
The superscript “max” at the surface concentration symbols means here that the maximum possible concentrations (i.e., complete monolayers) of the three types of molecules are considered. Then the new quantities q0, qa, and qfilm denote the surface charge densities of the free electrolyte (q0), of an electrode surface completely covered with the expanded adsorbate phase (qa), and of the surface covered with the condensed film (qfilm). With these kinds of partial surface charges, eq A1 becomes
It is reasonable to assume that the film concentration is constant, i.e., that Γfilm ) Γmax film . If furthermore the normalized surface concentration Γa ) Γ/Γfilm of the expanded phase is used, eq A3 becomes
using the property in (A5). If this is inserted in eq A4, after simple transformations using the definition of Γa, this results finally in
q(Γa, Θ) ) q0(1 - kΓa)(1 - Θ) + qaκΓa(1 - Θ) + qfilmΘ (A8) This equation contains only experimentally accessible or theoretically calculable parameters and defines the dependence of the mean surface charge per unit area g(Γa, Θ) on the amount of the adsorbed molecules in the expanded phase and on the area occupied by the condensed phase as well. Equation A8 can also be rewritten to obtain additional insight
q(Γa, Θ) ) qexp(Γa)(1 - Θ) + qfilmΘ
(A9)
qexp(Γa) ) q0(1 - κΓa) + qaκΓa
(A10)
with
Here, qexp(Γa) is the surface charge per unit area of the expanded adsorbate phase. Differently from the quantity qa, which depends only on the applied potential, this quantity also depends on the surface concentration Γa. From eq A9, it becomes clear that the mean surface charge q(Γa, Θ) is nothing else than the mean of the partial surface charges of the expanded and the condensed phases, repectively, weighted by the surface coverage of the condensed phase. List of Symbols Em, E′m ) potentials of maximum adsorption for systems without and with reorientation during the condensation process
4906 Langmuir, Vol. 15, No. 14, 1999 Ecp ) crossing point potential for systems with reorientation during the condensation process cbulk ) bulk concentration of the organic molecules ka ) effective rate constant of adsorption, ka ) kadscbulk + kdes, with kads ) adsorption rate constant and kdes ) desorption rate constant Γmax ) maximum possible surface concentration of the expanded adsorbate phase (corresponding to the hypothetical limit of cbulk f ∞) Γeq ) equilibrium concentration of the expanded adsorbate phase for given bulk concentration and double-layer potential, Γeq ) kadscbulkΓmax/(kadscbulk + kdes) Γfilm ) surface concentration of the condensed adsorbate phase Γa ) surface concentration Γ of the adsorbate in the expanded phase, normalized to the film density: Γa ) Γ/Γfilm κ ) ratio of Γmax/Γfilm δ ) saturation concentration of the expanded adsorbate phase Γsat, normalized to the film density, δ ) Γsat/Γfilm γ ) normalized equilibrium adsorbate concentration, γ ) Γeq/Γfilm pdl ) dimensionless total periphery length of all islands of the condensed phase
Donner and Pohlmann Θ ) surface coverage of the condensed phase (defined as the fraction of the film covered surface area SF in respect to the whole surface area S0) ndl ) dimensionless nucleation rate ndl kn ) preexponential nucleation rate constant K ) constant in the exponent of the nucleation law kw ) rate constant of the linear growth of the condensed phase R ) dimensionless effective rate constant of adsorption mol mol qmol sol , qa , qfilm ) the partial double layer charges per mol adsorbate for the solvent molecules, the molecules in the expanded phase and the molecules in the condensed phase qsol, qa, qfilm ) the partial double-layer charges per unit surface (surface charges) for surfaces completely covered with solvent molecules, molecules in the expanded phase and molecules in the condensed phase, respectively qexp(Γa) ) surface charge of the expanded adsorbate phase
LA981763R
