Langmuir 1996, 12, 3295-3304
3295
Use of High-Frequency AC Voltammetry To Study Kinetics of Physical Adsorption at Mercury Ultramicroelectrodes Andrzej S. Baranski* and Agata Moyana Department of Chemistry, University of Saskatchewan, 110 Science Place, Saskatoon, Saskatchewan S7N 5C9, Canada Received July 26, 1995. In Final Form: January 16, 1996X Adsorption kinetics of small organic molecules was studied by ac voltammetry with mercury ultramicroelectrodes in the frequency range 0.22-6.25 MHz. It was found that the activation energy of the adsorption process can be expressed as a linear combination of the electrical component of the standard free energy of adsorption (a major contribution) and the energy of lateral interactions (a minor contribution). The results suggest that, during the adsorption process, work against an electrical field must be performed before excess energy (due to the change in the hydrophobic interactions) is released. The final state is reached by a rotational and translational motion of the surfactant molecule. At the zero charge potential the rate constant reaches the maximum value of (4.6 ( 0.3) × 109 s-1. This value indicates that the viscosity of interfacial water at mercury near the zero charge potential is about 1.7 times higher than the bulk viscosity.
Introduction In the last decade remarkable progress in measurements of fast interfacial kinetics has been achieved due to the employment of ultramicroelectrodes. A variety of different techniques suitable for studying rates of interfacial reactions have been developed. The most important approaches include techniques based on the short response time of ultramicroelectrodes (e.g. fast cyclic voltammetry,1-3 submicrosecond chronoamperometry,4 and ac voltammetry5-7) or an enhanced mass transport to ultramicroelectrodes (e.g. steady-state voltammetry8 and electrochemical microscopy).9 The last two techniques are useful in determining rates of some reduction-oxidation (redox) reactions, but they cannot be used in the studies of adsorption kinetics. On the other hand, fast cyclic voltammetry and chronoamperometry have been used to study kinetics of red-ox processes of adsorbed species3,4 and kinetics of chemisorption.10 The method of the determination of kinetics of adsorption based on cyclic voltammetric data was presented in our previous papers.10,11 The method was adequate for studies of relatively slow chemisorption processes but was found unreliable in the determination of faster processes involving physical adsorption of organic surfactants. In general, frequency domain techniques such as ac voltammetry or impedance spectroscopy are better suited for studies of fast interfacial processes compared to time domain electrochemical techniques.7 AC impedance tech* To whom correspondence should be directed. E-mail address:
[email protected]. Fax: (306) 966-4730. X Abstract published in Advance ACS Abstracts, June 1, 1996. (1) Andrieux, C. P.; Garreau, D.; Hapiot, P.; Saveant, J.-M. J. Electroanal. Chem. 1988, 248, 447. (2) Wipf, D. O.; Kristensen, E. W.; Deakin, R. M.; Wightman, R. M. Anal. Chem. 1988, 60, 306. (3) Chen, X.; Zhuang, J.; He, P. J. Electroanal. Chem. 1989, 271, 257. (4) Forster, R. J.; Faulkner, L. R. Anal. Chem. 1995, 67, 1232. (5) Baranski, A. S. J. Electroanal. Chem. 1991, 300, 309. (6) Baranski, A. S.; Winkler, K.; Fawcett, W. R. J. Electroanal. Chem. 1991, 313, 367. (7) Winkler, K.; Baranski, A. J. Electroanal. Chem. 1993, 348, 197. (8) Oldham, K. B.; Zoski, C. G.; Bond, A. M.; Sweigart, D. A. J. Electroanal. Chem. 1988, 248, 467. (9) Mirkin, M. V.; Richards, T. C.; Bard, A. J. J. Phys. Chem. 1993, 97, 7672. (10) Szulborska, A.; Baranski, A. S. J. Electroanal. Chem. 1994, 377, 269. (11) Szulborska, A.; Baranski, A. S. J. Electroanal. Chem. 1994, 377, 23.
S0743-7463(95)00623-8 CCC: $12.00
niques have been applied in the past to studies of adsorption dynamics. In the pioneering work, Lorenz and Mo¨ckel12 in 1956 measured the kinetics of adsorption of various organic compounds at a dropping mercury electrode. The authors estimated the rate constant of the physical adsorption to be in the range 105 s-1. In 1968 Armstrong et al.13 measured the kinetics of the adsorption process of n-butyric acid and cyclohexanol from aqueous solutions at a mercury electrode. The authors concluded that up to 150 kHz the rate of the process was controlled by diffusion. On this basis they estimated the rate constant of the adsorption process for both studied substances to be higher than 106 s-1 and lower than the rotational relaxation of the molecules (ca. 1010 s-1). The discrepancy between the results of these two works is understandable, since at that time researchers worked at the limits of existing experimental techniques. It is surprising, however, that since then there were no subsequent attempts to determine the rates of the interfacial steps of physical adsorption at the metal/ solution interface. We have shown recently that reliable ac voltammetric measurements at ultramicroelectrodes can be carried out at frequencies up to 10 MHz.14 This should be sufficient to obtain a credible determination of the kinetics of molecular exchange in the double layer and hopefully should provide a new insight into interfacial molecular dynamics. Experimental Section Reagents. All solutions were prepared from double-distilled deionized water (Corning Mega-Pure system MP-6A and D-2). Analytical grade H2SO4, Na2SO4, n-propanol, 2-propanol, nbutanol, n-pentanol, n-butyric acid, and methyl ethyl ketone were used as received. Before the experiments all solutions were deaerated for about 10 min with oxygen-free argon. Electrochemical Cell. A standard three-electrode electrochemical cell was used. The auxiliary electrode was made of platinum foil ca. 1 cm2 in surface area. The reference electrode was Hg|Hg2SO4(s)|1 M K2SO4. Preparation of Working Electrodes. The electrodes were prepared by sealing gold wires 5 or 6.25 µm in radius (from Goodfellow Metals Ltd. or Alfa Æsler) into low-melting glass tubing. Then the electrode tip was cut and the electrode polished (12) Lorenz, W.; Mo¨ckel, F. Z. Elektrochem. 1956, 60, 507. (13) Armstrong, R. D.; Race, W. P.; Thirsk, H. R. J. Electroanal. Chem. Interfacial Electrochem. 1968, 16, 517. (14) Baranski, A. S.; Szulborska, A. Electrochim. Acta 1996, 41, 985.
© 1996 American Chemical Society
3296 Langmuir, Vol. 12, No. 13, 1996
Figure 1. Diagram representing the components of the total cell impedance. Dotted lines correspond to elements that are negligible. Symbols used: Rs, solution resistance; Cs, stray capacitance; Cdl, double-layer capacitance; Rdl, resistive element resulting from a finite dielectric relaxation in the double layer; Zc, pseudocapactivie impedance; Ra, activation resistance; ZW, Warburg impedance; ZN, impedance resulting from a nonlinear mass transport. with carborundum paper and, finally, with 0.3 µm alumina. In order to minimize the stray capacitance, the ultramicroelectrode was shielded with copper tubing. Details of the electrode construction are provided in a previous paper.14 Before electrochemical measurements a mercury hemispherical ultramicroelectrode was prepared by electrodeposition of mercury from 0.1 M Hg2(ClO4)2 in 1 M HClO4 solution onto the gold disk ultramicroelectrode at -800 mV versus Hg|Hg22+. The amount of deposited Hg was controlled by monitoring the charge passing during the electrolysis. The mercury electrode was cleaned by washing it in concentrated H2SO4 after deposition. Usually the electrode prepared in this manner gave reproducible results for about 5-20 experiments. After that the electrode was repolished, and a new mercury hemisphere was deposited. Electronic Circuit. The electronic circuit used in measurements included an EG&G Lock-in amplifier model 5202 with a frequency range 0.1-50 MHz, a Tektronix model SG503 highfrequency generator, an IBM compatible personal computer (80486, 66 MHz CPU), a custom built potentiostat, and an ac current transducer. A detailed description of the electronic setup, calibration procedures, and optimum conditions for carrying out accurate measurements was discussed in our previous paper.14
Results and Discussion Data Processing Method. In the Appendix a derivation of all fundamental relations between experimental data and calculated parameters is provided. It is based on a very general description of the process not referring to any specific adsorption isotherm. Only two assumptions were made during the derivation of these equations. The first one is that the activity coefficient of a desorbed surfactant is constant throughout the experiment, and the second one is that the perturbation of the system from equilibrium is very small. This derivation provides a good starting point for discussing the data processing method. The electronic circuit equivalent to the electrode/solution interface is shown in Figure 1. AC experimentation allows the determination of the real and imaginary parts of the total impedance as a function of potential. In addition, if curves are recorded for several different frequencies, these values are also known as functions of frequency. The total impedance, Zt, composed of all components presented in Figure 1, may be successfully used in the determination of the activation resistance, Ra, and hence the kinetics of the adsorption process, only if other components present in the equivalent circuit can be eliminated. The elements shown in Figure 1 with dotted lines are negligible. The stray capacitance, Cs, was practically
Baranski and Moyana
Figure 2. Dependence of the serial capacitance on potential for 1 M Na2SO4 containing 0.4 M methyl ethyl ketone obtained at a Hg hemispherical electrode, 5 µm in radius at 23 °C. Frequencies used in measurements: 0.22 (top curve), 0.39, 0.78, 1.56, 3.13, 6.25 MHz (bottom curve).
eliminated by a special construction of the working electrode and a specially designed calibration procedure. Details of both are given in a previous paper.14 The nonlinear diffusion impedance, ZN, is responsible for the diffusion enhancement effect often observed at ultramicroelectrodes; however, this effect is negligible at the frequencies used. From eqs A27 and A28 it can be seen that for ultramicroelectrodes used in this work this parameter is at least 100 times larger than the Warburg impedance, ZW, if the frequency exceeds 30 kHz. All the ac experiments in this work were performed at frequencies significantly higher than this value. In Figure 1 Rs represents the solution resistance. Its value depends on the electrode size and the conductivity of the electrolyte. Under the experimental conditions of this work (i.e. concentrated electrolyte solutions and frequencies lower than 10 MHz) the solution resistance is expected to be independent of frequency. At higher frequencies this assumption may be invalid due to the Debye-Falkenhagen effect. It should be noted, however, that the real impedance of the electrode, as presented in Figure 3, shows a significant frequency dispersion. This can be explained by the presence of a resistive element, Rdl, resulting from a finite dielectric relaxation in the double layer.14,15 The double-layer capacitance is represented in this diagram as Cdl; the value of this element depends slightly on frequency. The remaining elements in Figure 1 (Zc, Ra, and RW) constitute the adsorption process impedance, ZA. The origin and relation of these elements to the thermodynamic and kinetic parameters of adsorption are given in the Appendix. The double-layer capacitance of the mercury ultramicroelectrode in 0.4 M methyl ethyl ketone in 1 M Na2SO4 solution is presented in Figure 2 as a function of potential at different frequencies. It can be seen that at more positive potentials capacitance values correspond to a surface free of any adsorbed substances. The electrode in this potential range is covered by a layer of water molecules. As the potential becomes more negative and approaches the zero charge potential, adsorption of water is no longer favored (smaller electrostatic interactions) and the electrode becomes covered with a layer of neutral organic molecules, repelled from the bulk of the solution due to a hydrophobic effect. A resulting pseudocapacitance peak due to a change in the electrode coverage is observed. After that capacitance value decreases due to an increase (15) Bockris, J. O’M.; Gileadi, E.; Mu¨ller, K. J. Chem. Phys. 1966, 44, 1445.
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Figure 3. Dependence of the real impedance on potential. Solution and experimental conditions as in Figure 2.
in the Helmholtz layer thickness. As the potential is changed in the negative direction, pass the zero charge potential, the charge on the electrode causes the attraction of water dipoles, and desorption of organic molecules is observed. Once again, due to changes in the electrode coverage, a pseudocapacitance peak is observed at negative potentials. At even more negative potentials the capacitance returns to the value characteristic for an adsorbatefree surface. The potential dependence of the real impedance at different frequencies is shown in Figure 3 for the same system as above. The observed peaks reflect changes of the real component of the Warburg impedance and of the activation resistance due to the adsorption process. Frequency dispersion observed outside these peaks can be explained as mentioned earlier by the dielectric relaxation in the double layer. As the frequency increases, the values of the real impedance approach a constant value, which provides a good approximation of the solution resistance. The value of the real impedance measured at 6.25 MHz at a potential ca. -250 mV was treated as the solution resistance, Rs. It can be seen that the curves obtained at lower frequencies are very noisy. This results from the fact that at these frequencies the face angle approaches 90°, making an accurate determination of the real impedance impossible. In the first step of data processing Rs was subtracted from the total cell impedance:
Z1 ) Zt - Rs
(1)
The impedance obtained after this subtraction is composed of three major elements connected in parallel: the double-layer capacitance, the dielectric relaxation resistance, and the adsorption impedance. In order to calculate the adsorption impedance, Z1 should be converted into an admittance:
Y1 ) 1/Z1
(2)
This allows a subtraction of admittances associated with the Cdl and Rdl elements. The subtraction is possible because the potential dependence of Cdl and Rdl is small compared to the potential dependence of the adsorption admittance (see Figure 4). At potentials far from the peak the admittance is determined only by Cdl and Rdl. To take advantage of this property, the curves before and after the peak were fitted into a fifth- or sixth-order polynomial expression via the least-square method. The obtained regression coefficients were used to estimate the admittance due to the double-layer capacitance and the dielectric
Figure 4. Subtraction of the admittance corresponding to Cdl and Rdl from the real and imaginary part of the total cell admittance corrected for solution resistance. Dashed line represents an estimation of the background admittance by a sixth-order polynomial curve fitting. Experimental points (markers) were obtained for 0.5 M propan-2-ol in 1 M Na2SO4 with a 5 µm Hg hemispherical electrode at a 6.25 MHz frequency.
relaxation resistance in the potential region where the adsorption/desorption process takes place, and consequently, these estimated values were subtracted from Y1. The procedure is graphically illustrated in Figure 4. Two separate operations were performed on the real and the imaginary components of Y1, and as a result the real, YA,re, and imaginary, YA,im, components of the adsorption admittance were obtained. Subsequently the admittance was converted into impedance:
ZA,re )
YA,re 2 YA,re
+
2 YA,im
, ZA,im ) -
YA,im 2 YA,re
2 + YA,im
(3)
with
ZA ) ZA,re + jZA,im ) Ra + Zc + ZW
(4)
where j ) x-1. A general equation describing the adsorption impedance is given in the Appendix (eq A25). This equation can be simplified after taking into account the specific properties of the studied system. Literature data indicate that most organic surfactants follow the Frumkin isotherm. Consequently we can assume that the interaction parameter, g, defined by eq A11 is practically independent of the electrode coverage and potential and that the number of solvent molecules replaced by one adsorbate molecule, κ, is equal to 1. The last condition requires some explanation. Since organic molecules are larger than water, κ in eq A1 should be larger than one. However, it was postulated by Damaskin et al.16 and Parsons17 that the adsorbate molecule displaces small clusters of water molecules (formed due to hydrogen bonds) of the same size as the adsorbate rather than single water molecules. As a consequence the probability of the adsorbate to displace κ water molecules from the electrode surface is greater than (1 - θi)κ, and it can be well approximated by (1 - θi). When these simplifications are taken into account, the three terms in eq A25 can be described by
Ra )
RT γ F AΓmθikd 2
2
(5)
(16) Damaskin, B. B.; Petri, O. A.; Batrakov, V. V. Adsorption of Organic Compounds on Electrodes; Plenum Press: New York, 1971. (17) Parsons, R. J. Electroanal. Chem. 1964, 7, 136; 1964, 8, 93.
3298 Langmuir, Vol. 12, No. 13, 1996
Zc ) -
[1 + gθi(1 - θi)]RT γ2F2AΓmωθi(1 - θi)
ZW )
Baranski and Moyana
j
RT(1 - j) γ F2ACbx2Dω 2
(6)
(7)
where kd is the desorption rate constant, A is the electrode area, γ is the electrosorption valency (defined by eq A10), θi is the surface coverage with the adsorbate at the electrode surface, Γm is the surface concentration of the adsorbate corresponding to monolayer coverage, expressed in mol/cm2, ω is the angular frequency of the excitation potential, D is the diffusion coefficient, Cb is the concentration of the adsorbate near the electrode surface, and all other symbols have their usual meaning. According to eqs 5-7 both real and imaginary parts of the Warburg impedance are equal in value but opposite in sign, Zc is purely imaginary and depends on frequency, and the activation resistance is independent of frequency and is purely real. Therefore, if the values of the real and the imaginary components of the adsorption impedance are added, the effect of the Warburg impedance is eliminated:
ZA,re + ZA,im ) |Zc| + Ra ) [1 + gθi(1 - θi)]RT - 2 2 + Ra ) βτ + Ra (8) γ F AΓmωθi(1 - θi) where τ ) 1/ω is the time parameter. Figure 5 shows a 3D plot of ZA,re + ZA,im versus the time parameter and the potential; the plot was obtained from data presented in Figures 2 and 3. Ra and β were obtained by the least-square linear regression of ZA,re + ZA,im versus the time parameter; the potential dependencies of these parameters are given in Figure 6. Ra contains information about kinetics, and β, information about the thermodynamics of the process. In order to extract this information, the electrosorption valency must be determined. The electrosorption valency was calculated from the differential capacity data obtained at relatively low frequencies (750 Hz) to ensure that the results are not affected by the activation resistance and the Warburg impedance. AC measurements at such low frequencies are difficult to carry out at ultramicroelectrodes because of inherently low currents and a strong edge effect, likely caused by an imperfect sealing of the electrode.18,19 To avoid potential problems, these measurements were performed at the static mercury drop electrode (SMDE) 0.0106 cm2 in surface area. A change in the charge density at the electrode, ∆σads caused by the adsorption process can be calculated in the following way:
∆σads )
∫EE[Cθeq(E) - C0(E)] dE 0
i
(9)
where Cθeqi (E) is the differential capacitance of the electrode at equilibrium with a solution containing surfactant and C0(E) is the capacitance of the electrode in the same supporting electrolyte in the absence of surfactant. (18) Baranski, A. S. J. Electrochem. Soc. 1986, 133, 93. (19) Harman, A. R.; Baranski, A. S. Can. J. Chem. 1988, 66, 1036. (20) Karolczak, M. Langmuir 1990, 6, 863. (21) Pezzatini, G.; Foresti, M. L.; Moncelli, M. R.; Guidelli, R. J. Colloid Interface Sci. 1991, 146, 453. (22) Moncelli, M. R.; Foresti, M. L.; Guidelli, R. J. Electroanal. Chem. 1990, 295, 225. (23) Baikerikar, K. G.; Hansen, R. S. Langmuir 1991, 7, 1963.
The integration must start at potential E0, where no change in the charge density takes place. This can be any potential at which adsorption is thermodynamically impossible or the zero charge potential. The electrosorption valency is given by the ratio of the number of electrons leaving the interface to the number of adsorbate molecules entering the interface; consequently, in the potential region where θeq i approaches 1 this parameter can be calculated as follows:
γ)
∆σads FΓm
(10)
Since the change in an electrical charge is also equal to the derivative of the free energy of the interface with respect to potential, this calculation is in agreement with a definition of this parameter provided by eq A10. Values of Γm used in these calculations were taken from the literature, and they are listed in Table 1. Figure 7 presents ∆σads and γ calculated from experimental data obtained for 0.2 M 1-butanol in 1 M Na2SO4. The relationship of γ versus E, numerically defined in the potential region where θeq i ≈ 1, was fitted to a third-order polynomial, and the resulting regression coefficients were used to extrapolate this relationship to smaller electrode coverages. The result of this operation is shown in Figure 7 by a dotted line. Figure 7 also shows the electrosorption valency calculated from the data given by Koppitz et al.24 for the same surfactant. The authors reported only two parameters: the values of γ and dγ/dE at the zero charge potential. As shown in Figure 7, these parameters are in accord with our results. When γ(E) is known the normalized β parameter can be calculated as a function of potential:
[γ(E)F]2AΓm β˜ (E) ) β RT
(11)
According to eq 6:
β˜ (E) )
1 +g θi(E)[1 - θi(E)]
(12)
If change of g with potential is negligible, then this function reaches a minimum, β˜ min, at E ) E1/2, where θi(E1/2) ) 0.5; therefore, the electrode coverage can be calculated as
x
θi(E) ) 0.5 ( 0.5
β˜ - β˜ min
β˜ - β˜ min + 4
(13)
The second term in this equation describes a difference in coverage due to a change of potential from E1/2 to E. The nature of the process requires that θi(E) decreases monotonically when potential moves away from the zero charge potential (Epzc). Therefore, the second term in eq 13 must have a positive sign when E1/2 < Epzc and E > E1/2 or when E1/2 > Epzc and E < E1/2; the negative sign must be used when E1/2 < Epzc and E < E1/2 or when E1/2 > Epzc and E > E1/2. When θi(E) is known, the apparent equilibrium constant of adsorption, Kads, and the rate constant of the desorption process, kd, can be calculated from eqs A7 and 5, respectively. The rate of the adsorption process is calculated as kdKads. (24) Koppitz, F. D.; Schultze, J. W.; Rolle, D. J. Electroanal. Chem. 1984, 170, 5.
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Figure 5. Sum of real and imaginary components of the adsorption process impedance plotted as a function of potential and time parameter. On the basis of data presented in Figures 2 and 3.
Figure 6. Dependence of the activation resistance and the slope, β, of the sum of real and imaginary components of the adsorption process impedance versus τ on potential. Values determined for the frequency range 0.22-6.25 MHz. Conditions as in Figure 2.
The standard free energy of adsorption, ∆G°ads ) -RT ln Kads, can be represented as a sum of electrical, ∆G°elc, and nonelectrical, ∆G°nel, components. The electrical component of the standard free energy can be obtained by an integration of the electrosorption valency from the potential of maximum adsorption, Em (i.e. the potential at which the electrosorption valency is equal to zero):
∫EE γ(E) dE
∆G°elc(E) ) F
m
(14)
Consequently, the nonelectrical component is given by
∆G°nel(θi) ) -RT ln[Kads(E,θi)] - ∆G°elc(E)
(15)
The dependence of ∆G°nel on θi can be approximated by a linear function:
∆G°nel ) ∆G°m + gRTθi
(16)
where ∆G°m is the standard free energy of adsorption at the potential of maximum adsorption and θi f 0. Parameters describing the equilibrium of adsorption for all studied systems are given in Table 1. The parameters ∆G°m and g were obtained by the least-square regression
of ∆G°nel versus θi. In order to compare our values of ∆G°m with the literature data, a correction for a different definition of the standard state must be made. In the literature the equilibrium constant expression (eq A7) is usually written with the concentration of surfactant at the initial state expressed in terms of the mole fraction and at the final state in terms of the surface coverage. In adsorption kinetics such a formulation is unacceptable (definition of the standard state changes as a molecule passes through the activation barrier). Therefore, we have to choose one way of expressing the concentration of the adsorbate before and after the adsorption event. We have decided to use cross-section coverages which describe the fraction of the surface occupied by the molecules of interest at selected cross sections of the solution. Both the crosssection coverage, θ0, and the mole fraction, x0, are dimensionless, and for θ0 , 1 they are proportional to each other: x0 ) (aw/a0)θ0 (where aw and a0 are the surface areas occupied by one water and one surfactant molecule, respectively). This allows a simple recalculation of equilibrium constants and standard free energies defined in different systems. The adjusted values are given in the third column of Table 1 in parentheses, and they are followed by literature values. A discrepancy in the case of n-butanol is probably due to the fact that in our case ∆G°m was obtained by a linear extrapolation (i.e. assuming that g is independent of potential). However, Moncelli et al.25 have shown that in the case of n-butanol g decreases significantly when the electrode potential approaches the zero charge potential. Kinetics of Adsorption. AC voltammograms at mercury ultramicroelectrodes in the frequency range 0.22-6.25 MHz were obtained for aqueous 1 M Na2SO4 containing one of the following adsorbates: 0.25 M 2-propanol, 0.5 M 2-propanol, 0.5 M n-propanol, 0.5 M n-butanol, 0.4 M methyl ethyl ketone, or 0.2 M n-butyric acid. In general the quality of the results was satisfactory. The kinetic data obtained around θi ) 0.5 were the most accurate. The results obtained for negative adsorption peaks were usually more scattered but showed a more regular behavior than the results obtained at positive potentials. The major source of errors is probably associated with the subtraction of Cdl and Rdl components (see Figure 4). We also noted some problems with systems for which g < -3 (e.g. n-butanol). These problems may (25) Moncelli, M. R.; Foresti, M. L.; Guidelli, R. J. Electroanal. Chem. 1990, 295, 225.
3300 Langmuir, Vol. 12, No. 13, 1996
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Table 1. Thermodynamic Parameters of Adsorption of Selected Aliphatic Compounds at Mercury in 1 M Na2SO4 regression coefficients of electrosorption valencyc 1010Γm/mol cm-2
substance 0.5 M propan-1-ol
5.41d
0.5 M propan-2-ol
4.8e
0.2 M butan-1-ol
5.1f
0.2 M butyric acid
5.1h
0.4 M methyl ethyl ketone
4.9i
a
∆G°m /kJ mol
-1
-10.7
gb -2.42
-8.94 (-12.5,e -12.1) -12.1 (-17.6,f -15.1) -12.2 -11.8
-2.34 (-2.2e) -3.39 (-3.0g) -3.06 -3.05
a3 0.0 0.0269 0.1063 0.0195 0.0766
a2 0.3286 (γ+ ) 0.22) 0.3302 (γ+ ) 0.21) 0.6406 (γ+ ) 0.32) 0.3007 (γ+ ) 0.24) 0.4563 (γ+ ) 0.26)
a1 0.9353 (γ- ) -0.085) 0.9397 (γ- ) -0.13) 1.2978 (γ- ) -0.14) 0.8968 (γ- ) -0.12) 0.9871 (γ- ) -0.14)
a0 0.5722 0.6472 0.7352 0.6294 0.6093
a The standard free energy of adsorption at the maximum adsorption potential extrapolated to the zero electrode coverage for a standard state defined by eq A7. Literature values are given in parentheses, followed by our results recalculated to the same standard state. b Interaction parameter determined from ac voltammetric measurements at ultramicroelectrodes; an average value for the positive and the negative adsorption peaks. Literature values given in parentheses. c Based on the equation γ(E) ) a3E3 + a2E2 + a1E + a0, where E is in volts versus Hg|Hg2SO4(s)|1 M K2SO4. d Data from ref 20. e Data from ref 21. f Data from ref 22. g Data for saturated Na2SO4 from ref 23. h Assumed the same value as for butan-1-ol.22 i Assumed the same value as for butan-2-ol.24
Figure 7. Change of charge at the electrode surface caused by the adsorption process (markers) and the electrosorption valency as functions of potential. The dotted line represents the electrosorption valency calculated from experimental data according to equation 10. The equation gives the regression parameters allowing determination of γ as a function of potential. The dashed line shows the values reported by Koppitz et al.24 for n-butanol. Experimental data obtained in 0.2 M n-butanol solution in 1 M Na2SO4 supporting electrolyte with an SMDE 0.0106 cm2 in surface area at the frequency 750 Hz.
be associated with a too simplistic treatment of the interaction parameter in this work. Two general trends were noted: (i) adsorption on the negative side of the zero charge potential is usually faster than that on the positive side, and (ii) surfactants adsorbing farther from the zero charge potential show lower rate constants. It was found that for all studied systems the rate of the adsorption process shows a good correlation with the electrical component of the standard free energy of adsorption. In order to evaluate more exactly the relative contribution of ∆G°elc and ∆G°nel to the activation energy, the multiple linear regression method was used (ln(ka) against ∆G°elc and ∆G°nel.) On this basis the following empirical relation was found:
(
ka ) k0 exp -
)
R∆G°elc + χ(∆G°nel - ∆G°m) RT
(17)
where k0 ) (4.6 ( 0.3) × 109 s-1, R ) 1.09 ( 0.02, and χ ) 0.36 ( 0.02. Errors are reported as the standard deviations obtained by the least-square method. The kinetic data for all studied systems were included in the analysis. The agreement between the experimental data and eq 17 is graphically shown in Figure 8. A large value of R indicates that in the studied adsorption processes ∆G°elc rises before ∆G°nel drops; in
Figure 8. Dependence of the adsorption rate constant on the linear combination of electrical and nonelectrical components of the standard free energy of adsorption (R ) 1.09 and χ ) 0.36, details in text). Data points were obtained in 1 M Na2SO4 containing 0.4 M methyl ethyl ketone ([), 0.5 M butan-1-ol (4), 0.25 M propan-2-ol (9), 0.5 M propan-2-ol (]), 0.5 M propan1-ol (O), or 0.2 M butyric acid (b). Results for both negative and positive (with respect to Epzc) adsorption peaks are included.
other words, before excess energy (due to the change in the hydrophobic hydration) is released work must be performed against the electrical field. A mechanism suggested in Figure 9 is in agreement with this conclusion. A surfactant most likely approaches the electrode with a random orientation of the hydrophobic group with respect to the metal. When this approach takes place, ions from the double layer are pushed away. Since the centers of the charges at the surface site and in solution are farther apart, the local double-layer capacitance decreases. In order to explain R close to one (or higher), we have to assume that the double-layer capacitance at the transition state is almost the same (or lower) as that at the final state. After reaching the maximum energy state, the reaction proceeds by translation through the last layer of water molecules and rotation (from a random to a fixed orientation with the hydrophobic group pointing toward the surface, an average change of the angle by π/2). When this process occurs, a part of the hydrophobic group is removed from solution and the excess energy is dispersed in the system. It should be noted that at larger coverages the transition state is stabilized by attractive lateral interactions with adsorbed surfactant molecules. This explains a 36% contribution of (∆G°nel - ∆G°m) to the activation energy (since ∆G°nel - ∆G°m < 0, this contribution lowers the activation barrier). The parameter k0
Physical Adsorption at Mercury Ultramicroelectrodes
Langmuir, Vol. 12, No. 13, 1996 3301
Figure 9. Simplified diagram of the proposed mechanism of the physical adsorption process (see details in text). Circles with a negative sign indicate centers of the counter charge for given surface sites rather than specific ions in solution. The orientation of the surfactant molecule is random in the initial state and fixed in the final state. Lines represent changes in the electrical (∆G°elc) and nonelectrical (∆G°nel) components of the standard free energy of adsorption as well as overall changes in the standard free energy of the reacting system (∆G°elc + ∆G°nel).
represents the rate constant of the adsorption process at the zero charge potential and for θi f 0. This parameter must account for the translation and rotation of surfactant in the double layer. We will express this parameter in terms of bulk values of rotational (Dr) and translational (Dt ) diffusion coefficients, and we will include a new activation energy term (the interfacial firction, ∆G°if) to account for the extra friction associated with molecular motions in the double layer as compared with the bulk solution. On the basis of Einstein’s equation26 the halflife of the translational step is
d2w 〈d〉2 τt ) ) 2Dt 4Dt
(18)
where 〈d〉 is the root mean square (rms) displacement of surfactant molecules in time τt and dw is the diameter of the water molecules. By taking dw ) 3.1 × 10-8 cm and Dt ) 10-5 cm2/s,27 we obtain τt ) 2.4 × 10-11 s. The half-life of the rotational motion is
τr )
〈Θ〉2 π2 ) 4Dr 32Dr
(19)
where Θ is the rms displacement angle in radians. The rotational diffusion coefficient of the water molecules in liquid water is 7 × 1010 s-1 at 25 °C (ref 22, p 5.44). The value of Dr is inversely proportional to the molecular volume (ref 22, Chapter 6.5g) (a fivefold decrease for the studied compounds with respect to water is expected on this basis). In addition, the rotational diffusion coefficient is smaller if rotation of a nonspherical molecule about the axis perpendicular to its larger dimension is considered. Consequently, Dr ) 5 × 109 s-1 can be expected (26) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: London, 1991; Vol. 1, Chapter 6.3. (27) Lide, D. R., Ed. CRC Handbook of Chemistry and Physics, 73rd ed.; CRC Press: Boca Raton, FL, 1993; pp 6-175.
for the studied compounds in bulk solution. If this value is entered into eq 19, τr ) 6.2 × 10-11 is obtained. On the basis of the experimental results the average half-life for the studied process, τ1/2, at the zero charge potential is
τ1/2 )
ln 2 ) 1.5 × 10-10 s-1 k0
(20)
Therefore we can evaluate the interfacial friction term as
( )
∆G°if ) RT ln
τ1/2 τt + τr
(21)
For the studied systems this term is very small (about 0.56RT). This result is interesting because it indicates that at mercury electrodes the viscosity of interfacial water near the zero charge potential is only slightly larger (say, 1.7 times) than the viscosity of bulk water. Recently, the adsorption kinetics of pentanoic acid at the solution/air interface was studied by ripplon spectroscopy.28 The authors concluded that the rate of the process is controlled by diffusion and that the diffusion process in close vicinity to the surface (1000-10 nm) is the same as that in the bulk phase. This is in accord with our results, although it should be noted that in the case of our research the diffusion processes in the double-layer region, i.e. less than 1 nm from the surface, are examined. The results presented in Figure 8 indicate that adsorption rate constants in the range of 109 s-1 can be measured with high-frequency ac voltammetry at ultramicroelectrodes. It should be stressed that these results are within the limits of the applied method. Analysis of eq A6 indicates that the rate at which the studied system relaxes is determined by kd + kaθ0. In performed mesaurements this term was lower than 108 s-1 and it was comparable with (or smaller than) the maximum angular frequency (28) Sakai, K.; Takagi, K. Langmuir 1994, 10, 257.
3302 Langmuir, Vol. 12, No. 13, 1996
Baranski and Moyana
Canada (NSERC), through an operating grant, is gratefully acknowledged. Appendix Electrical Current of Adsorption. Let us consider an adsorption process occurring at a metal/solution interface under controlled potential conditions. The interfacial step of this process can be represented by the following reaction:
A0 + κSi a Ai + κS0
Figure 10. Modulus and phase angle of cell impedance as a function of angular frequency (the Bode plot). Markers represent experimental data obtained with a Hg hemispherical electrode, 5 µm in radius for 1 M Na2SO4 containing 0.4 M methyl ethyl ketone at 23 °C. The electrode potential was -1.861 V versus Hg|Hg2SO4(s)|1 M K2SO4. The solid line represents the cell impedance calculated according to kinetic, thermodynamic, and equivalent circuit parameters determined in this work (details in the text). The dashed line represents the cell impedance expected for the same process controlled only by the rate of diffusion.
of the excitation wave form (4 × 107 Hz). The rate of 5 × 109 s-1 is equivalent to the heterogeneous rate constant of 150 cm/s (the conversion factor is the thickness of a monolayer of water molecules). AC voltammetry at standard size electrodes was successfully used in studies of electrode processes with standard rate constants up to about 1.5 cm/s.5 In such studies frequencies up to 5 kHz and electrodes about 0.025 cm in radius were usually employed. Here, we measured 100 times faster kinetics by using 1000 times higher frequencies and 50 times smaller electrodes. Furthermore, measurements of physical adsorption kinetics are easier than measurements of redox kinetics. Since the typical value of γ for the physical adsorption is less than 0.3, the activation resistance is at least 10 times higher than it would be for a redox process with the same rate constant (see eq 5). Consequently, errors due to an inaccurate evaluation of the solution resistance are greatly reduced. The analysis of the total cell impedance in Figure 10 clearly shows that the experimental conditions used in this work were appropriate for measurements of physical adsorption kinetics. Markers in this figure represent experimental data obtained at a Hg hemispherical electrode, 5 µm in radius for 1 M Na2SO4 containing 0.4 M methyl ethyl ketone at 23 °C. The electrode potential was -1.861 V versus Hg|Hg2SO4(s)|1 M K2SO4. The solid lines represent the cell impedance calculated according to the equivalent circuit shown in Figure 1 using eqs 5-7 with the following parameters: ka ) 3 × 108 s-1 (kd ) 7.6 × 106 s-1), A ) 1.6 × 10-6 cm2, γ ) -0.14, Γm ) 4.9 × 10-10 mol/cm2, θi ) 0.5, g ) -3.05, Cb ) 0.4 M, D ) 10-5 cm2/s, Cdl ) 15.5 µF/cm2, and Rs ) 4500 Ω. The contribution to the cell impedance from the equivalent circuit elements, Cs, Zdl, and ZN is very small, and it was omitted in the calculation. The dashed lines in Figure 10 represent the cell impedance expected for the same process controlled only by the rate of diffusion (i.e. for ka ) ∞). It is clear that the effect of adsorption kinetics on the cell impedance is much larger than the experimental errors. Consequently, it proves that kinetic parameters reported in this work can be reliably determined by the applied method.
(A1)
where A and S represent adsorbate and solvent molecules, respectively; the subscripts i and 0 refer to the location of molecules at the interface and outside the interfacial region, respectively; and κ represents the number of solvent molecules replaced by one adsorbate molecule. Each elemental step described by reaction A1 is associated with a flow of γ electrons through the external circuit. This flow arises from a movement of dipoles and ions in the interfacial region caused by the adsorption event. In the case of chemisorption a significant contribution to the electrical current also arises from the charge transfer associated with the bond formation between an adsorbate and the electrode.29 Consequently, the electrical current associated with the adsorption process can be given by
[ (
ia ) -γFA kaΓ0 1 -
)
Γi Γm
]
κ
- kdΓi
(A2)
where F is Faraday’s constant, A is the electrode surface area, Γi and Γ0 are the surface concentrations of the adsorbate at the interface and outside of the interfacial region, respectively, Γm is the surface concentration of the adsorbate in a monolayer, and ka and kd are the adsorption and desorption rate constants, respectively. Since the concentrations of both reactant and product are expressed in moles per surface area, both ka and kd are measured in inverse seconds. Γ0 is a bulk property which can be defined for any cross section in the solution. In dilute solutions this quantity is proportional to the volume concentration, Cb, and to the mole fraction, xb:
Γ0 ) Cbds )
xb a s NA
(A3)
where ds is the diameter of the solvent molecules, as is the average cross-sectional area of a solvent molecule, and NA is Avogadro’s constant. More strictly:
ds )
( ) V hs
NA
1/3
and as )
( ) V hs
2/3
NA
(A4)
where V h s is the molar volume of a solvent. The sign of the current in eq A2 follows the generally accepted sign convention (a change of potential in the positive direction from equilibrium causes a positive current). A relation between the γ parameter and the free energy of adsorption will be discussed later. The description of the process can be simplified by introducing cross-sectional coverages, θi and θ0, a normalized current, ı˜, and a normalized potential, E ˜:
θi )
Γi Γ0 ia EF , θ0 ) , ı˜ ) , E ˜ ) Γm Γm FAΓm RT
(A5)
Now eq A2 can be written as Acknowledgment. The financial support of the Natural Sciences and Engineering Research Council,
(29) Schultze, J. W.; Koppitz, F. D. Electrochim. Acta 1976, 21, 337.
Physical Adsorption at Mercury Ultramicroelectrodes
ı˜ ) -γ[kaθ0(1 - θi)κ - kdθi]
Langmuir, Vol. 12, No. 13, 1996 3303
(A6)
If the adsorption current approaches zero, the system approaches equilibrium and then
Kads(E ˜ ,θi) )
[
]
˜ ,θi) ∆G°ads(E ka θeq i ) eq ) exp eq κ kd θ (1 - θ ) RT 0
i
(A7) where Kads is the apparent equilibrium constant of the process. This constant is a function of the electrode potential, and it usually retains some dependence on the electrode coverage, θi, arising from the nonideal behavior of the adsorbate at the electrode surface. It should be noted that because the cross-sectional coverage outside of the interfacial region is usually very small, Kads is expected to be independent of θ0. By combining eqs A6 and A7, we can obtain
[
]
Kads -1 Qads
˜i ) -γkdθi
(A8)
where Qads ) θi/[θ0(1 - θi)κ] represents the reaction quotient. The term in parentheses describes a departure of the system from equilibrium. The rate of the adsorption process can be written as
[
˜ ,θi) ) Z exp ka(E
]
(A9)
() ∂i˜
∆i˜ )
∂E ˜
θi,θ0
(
)
˜ ,θi) 1 ∂∆G°ads(E RT
∂E ˜
1 )-
( ) ∂Kads
Kads ∂E ˜
θi
(
)
˜ ,θi) 1 ∂∆G°ads(E RT ∂θi
)-
E ˜
[
( )
1 ∂Kads Kads ∂θi
R)
(
)
∂∆G°ads(E ˜ ,θi)
θi
( )
Kads ∂ka ) ka ∂Kads
χ)
(
(
) ( ) E ˜
(A15)
-
) x
]
τ-t dt (A16) r2
D
() ∂i˜
∂E ˜
() ∂i˜
∆E ˜ (s) +
θi,θ0
(A11)
∂θi
θ0,E ˜
∆θi(s) ) -
E ˜
(A12)
() ∂i˜
∆θi(s) +
∂θ0
∆i˜(s)
∆θ0(s)
E ˜ ,θi
(A17)
(A18)
sγ
F∆i˜(s)
(A19)
γ(xDs + D/r)
θi
Kads ∂ka
ka ∂Kads
∫0τ∆i˜ dt
τ-t erfc r2
∆θ0(s) )
)
∂∆G°ads(E ˜ ,θi)
(A14)
∆θ0
E ˜ ,θi
where D is the diffusion coefficient of the adsorbate in solution. By applying a Laplace transformation, eqs A14-A16 can be converted from the time domain into the frequency domain:
(A10)
The normalized adsorption process impedance is given ˜ (s)/∆ı˜(s), and it can be calculated from eqs by Z ˜ A ) ∆E A17-A19:
and the constant potential symmetry factor # ∂∆Gads (E ˜ ,θi)
∂θ0
-1/2
the constant coverage symmetry factor # ∂∆Gads (E ˜ ,θi)
1 γ
∫0τ∆i˜(t) (πDτ r-2 t)
ds γr
∆i˜(s) )
g)
θ0,E ˜
A change in θ0 is related to a change in adsorbate concentration near the electrode surface, and that last quantity can be obtained from the convolution of current with an appropriate mass transport function.19 In the case of diffusion to a spherical (or hemispherical) electrode of radius r, we can write
θi
the lateral interaction parameter
() ∂i˜
∆θi +
exp D
is the where Z is the frequency factor and activation energy of the process. Four important partial derivatives can be defined:
γ)
∂θi
∆θi ) -
# ∆Gads (E ˜ ,θi)
the electrosorption valency
() ∂i˜
∆E ˜ +
˜ are functions of time, and they are ∆ı˜, ∆θi, ∆θ0, and ∆E interrelated. A change in θi occurring in a certain time interval, τ, can be obtained by the integration of the changes in current:
∆θ0 )
# (E ˜ ,θi) ∆Gads
RT
derived by using a standard mathematical procedure.30 According to eq A6 the adsorption current is a function ˜ . A change in ı˜ caused by of three variables: θi, θ0, and E small peroidic changes of these variables can be easily calculated:
(A13)
E ˜
[ ()
Z ˜A ) 1 +
1 ∂i˜ γs ∂θi
θ0,E ˜
( ) ]( ) ∂i˜
F -
γ(xDs + D/r) ∂θ0
∂E ˜
E ˜ ,θi
29
It can be shown that γ defined by eq A10 is identical with the one defined by eq A2. In general the parameters γ, g, R, and χ are expected to vary with potential and electrode coverage, but in a narrow range of potentials and coverages they can be treated as constants. Description of the Electrode Impedance. The electrode impedance due to an adsorption process can be
∂ı˜
θi,θ0
(A20)
From eqs A8, A10, and A11 we can obtain (30) Sluyters-Rehbach, M.; Sluyters, J. H. In Comprehensive Treatise of Electrochemistry; Yeager, E., Bockris, J. O’M., Conway, B. E., Sarangapani, S., Eds.; Plenum Press: New York and London, 1984; Vol. 9, Chapter 5.4.1.
3304 Langmuir, Vol. 12, No. 13, 1996
() ∂i˜
∂E ˜
θ0,θi
() ∂i˜
∂θi
(
) γkd
θ0,E ˜
using eqs A5 and A3.
Kads ) γ2θikd Qads θi
Kads
Qads
Baranski and Moyana
)[ ( ) ( ) ]
-1 γ
∂kd
+ kd
∂E ˜
θi
∂γ
∂E ˜
ZA ) (A21)
θi
Kads 1 + θi(κ - 1) + gθi(1 - θi) Qads (1 - θi)
( )[ ( ) ( ) Kads ∂kd - 1 γθi Qads ∂θi ∂i˜
∂θ0
E ˜
) -γkdθi
θi,E ˜
]
( )
+ γkd (A22)
Kads θ0Qads
(A23)
+ kdθi
∂γ ∂θi
E ˜
Since the perturbation applied to the system is very small, the system is always near the equilibrium and Qads ≈ Kads; by taking this into account, we can greatly simplify eqs A21-A23 and obtain an explicit description of the normalized adsorption impedance:
Z ˜A )
1 + θi(κ - 1) + gθi(1 - θi) 1 + + γ θikd γ2sθi(1 - θi) F (A24) γ2θ0(xDs + D/r)
RT γ F AΓmθikd [1 + θi(κ - 1) + gθi(1 - θi)]RT + j γ2F2AΓmωθi(1 - θi) RT (A25) 2 2 γ F ACb(xjωD + D/r) 2
2
The first term in this equation represents the activation resistance, Ra, the second term describes the reactance associated with the adsorption pseudocapacitance, Zc, and the third term describes the mass transport impedance, Ztr. Ztr can be conveniently represented as the Warburg impedance, ZW, connected in parallel with a resistive element, ZN, which describes the contribution to mass transport from a nonlinear diffusion.
Ztr )
(
1 1 + ZW ZN
)
-1
(A26)
where
2
ZW )
RT(1 - j) γ F2ACbx2Dω 2
(A27)
and Since we are only interested in sustained periodic changes, we can set the real parts of the complex variable s to zero, so s ) jω (where ω is the angular frequency and j ) x-1). We can also convert the normalized adsorption impedance into the quantity measured in experiments by
ZN ) LA9506234
rRT γ F2ACbD 2
(A28)
