First-order phase transitions on electrodes: the role of surface diffusion

f ii . I I I H I nl

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s u r f a c e s c i e n c e

ELSEVIER Surface Science 359 (1996) 280 290

First-order phase transitions on electrodes: the role of surface diffusion

L. Pohlmann a, C. Donner h,., H. Baumg/irtel b a Hahn Meitner lnstitut, Glienicker Strasse I00, 14109 Berlin, Germany

b lnstitutffir Physikalische und Theoretische Chemie, Freie Universitiit Berlin, Takustrasse 3, 14195 Berlin, Germany

Received 4 September 1995; accepted for publication 16 January 1996

Abstract

Two-dimensional first-order phase transitions of molecules adsorbed on homogeneous surfaces often proceed via nucleation and growth processes. In the present paper, a mathematical model is developed which takes into account the simultaneous processes of adsorption of the molecules from the bulk, surface diffusion of the adsorbed molecules and their consumption at the edge of the growing nuclei. From the numerical solutions of the model equations, several regimes of the process of nucleation and growth can be obtained depending on the ratios of the system parameters. The existence of these different growth regimes is provided experimentally in the 5-bromocytosine-mercury system.

Keywords: Adsorption kinetics; Biological molecules DNA bases; Computer simulations; Growth; Models of surface kinetics; Nucleation; Surface diffusion; Surface thermodynamics

I. Introduction

Lorenz was the first to discover that the potential-dependent adsorption behaviour of certain organic substances dissolved in electrolytes is characterized by sharp discontinuities in the relation- ship between double-layer capacity and potential [1]. Armstrong interpreted these phenomena for the first time as first-order phase transitions between a two-dimensional "gas-like" phase and a condensed phase [2]. In the potential range between these discontinuities, capacity is low and approximately independent of the potential. In the literature this region is generally called the "pit region".

* Corresponding author.

Several classes of organic neutral molecules can undergo a first-order phase transition at electrodes forming monolayer films [3]. The kinetics of this film formation generally include nucleation and growth processes and other first-order transitions on surfaces observed in electrochemistry, for exam- ple underpotential deposition, anodic film formation and reconstruction of solid electrodes [4-6].

The main difference between the first two examples and film formation of neutral adsorbed molecules is the lack of reduction-oxidation (redox) reactions in the latter case. This means that the electrode is ideally polarized in the investigated potential range (in this range no faradaic current occurs). The equilibrium concentration of adsorbed molecules at the surface is determined by a potential-dependent adsorption isotherm, whereas the concentration of molecules undergoing redox reac-

0039-6028/96/$15.00 Copyright © 1996 Elsevier Science B.V. All rights reserved P H S0039-6028 (96) 00362-7

L. Pohlmann et al./Surface Science 350 f 1996) 280-290 281

tion inthe former cases is determined by the potential based on the Nernst equation.

There is another important difference between first-order phase transition during surface reconstruction and the condensation of neutral molecules. During a reconstruction process the number of metal atoms on a surface remains constant, whereas during a condensation process of neutral molecules on surfaces, the total number of molecules is not constant. Film formation includes the exchange of molecules between the bulk and the surface by adsorption and desorption. Therefore, it resembles a thermodynamic open system. The highest possible concentration (equilibrium concentration Ceq) of adsorbed molecules (often called gas-like molecules) on the surface without condensation is defined by the potential-dependent adsorption isotherm [7].

As a consequence of the fixed number of surface atoms in the reconstruction process, the two phases exist simultaneously in a narrow potential range [8]. However, in the case of film formation the coexistence of condensed islands and gas-like molecules is only possible at fixed potential values. The so-called "pit edge" potentials are thermodynami- cally defined by the intersection of the adsorption isotherm Ceq(E ) with the saturation concentration csat. Only a differential deviation from the pit edge potential leads to the complete conversion of the thermodynamic unstable phase to the stable phase, and as a result, the surface is homogeneously covered with one phase, which is either "film-like" or "gas-like".

In practice the experimentally determined pit edge (determined by the single-potential step method from outside the pit region into the pit region) will be slightly shifted in direction of the middle of the pit (to a concentration of gas-like molecules higher than the saturation concentration) due to the kinetic inhibition of nucleation. Therefore, a so-called "critical nucleation concentration" ccrit exists. Belows this concentration, nucleation cannot be observed on a reasonable time scale (Fig. 1), whereas the growth of already formed nuclei can still proceed [9]. The overall kinetics of phase formation involve the partial processes of (i) adsorption and desorption of mon- omers, and (ii) nucleation and growth, both with

( ,

(} , _ / /

/ --4 ~ - /

~, , s @ t h . . . . . . . . . . . . / J . . . . . . . . . . . . . . .

! experi 'T'e~tol Plt "egi9n I , i klne is ~h6se irons ~ [

S :Ber4qodynor8 c p~ese t~cnsiti&r,

, I n . . . . .

: : t 0 r i : [,3 '

Fig. 1. Schematic representation of the potential dependence of the concentration of gas-like molecules and the conditions for first-order phase transitions.

different kinetics. This is discussed in detail in Refs. [ 10,11]. The processes (i) are functions of the potential, and the processes (ii) are functions of the actual concentration of gas-like adsorbed molecules at the surfaces.

It is not, therefore, surprising that during a condensation process at a fixed potential, the rate of nucleation as well as the rate of growth is time- dependent. It is impossible to obtain exact mathematical solutions for the kinetics of film formation because necessary information about the adsorption of gas-like molecules is not available. Therefore, in Refs. [ 10,11] a new formalism based on splitting the entire process into several time intervals with quasi-stationary conditions in each has been introduced. Following this approach it was possible for the first time to interpret capacity- -time plots in wide temperature and potential ranges using a model which accounts for the simultaneous processes of adsorption and condensation. In the present paper the limits of the applicability of this model with regard to growth law will be analyzed.

2. Theory of surface diffusion during island growth

It is well known that the growth rate of nuclei depends on the supersaturation of gas-like molecules, the rate constant of growth, and the surface diffusion coefficient. Supersaturation is a consequence of the adsorption kinetics and is therefore time-dependent. Depending on the ratio of the

282 L. Pohlmann et al./Surface Science 359 (1996) 280-290

growth-rate constant to the surface diffusion coefficient, the growth process itself will be either diffusion-limited or kinetically-limited or, obviously, may be controlled by mixed kinetics. In the case of kinetic limitation, the growth rate depends only on the homogeneous surface concentration of gas-like molecules. In all other cases, it is necessary to solve the diffusion problem with a number of randomly distributed nuclei with different sizes and different growth laws individually. This requires consideration of the actual concentrations at the periphery of the different nuclei. This general problem cannot be solved. Therefore, simplified models were used by several authors to obtain approximate functions describing the time evolution of growth rates. The model most often used describes the completely diffusion-limited growth of one nucleus in an infinite medium, which leads

to the well known 1/V~ law [12-14]. However, this model is inadequate to account for the real processes during growth of nuclei for the following reasons.

(1) The assumption of completely diffusion-limited growth, i.e. of a constant zero concentration Co at the periphery of the nucleus from the very beginning of the process, requires a fast concentration change from c o > 0 at t = 0 to c = 0 at t > 0, which is not very realistic.

(2) The semi-infinite geometry of the model becomes obsolete if the diffusion zones of the individual nuclei begin to overlap.

(3) The model requires a dosed system in which the molecules consumed at the periphery of the nucleus cannot be replaced from the bulk phase. This is obviously in contradiction with the nucleation problem discussed here, which is an open system due to the adsorption of molecules from the bulk solution.

Consequently the model developed here has to avoid the shortcomings above. It consists of a diffusion equation (Eq. (1)) completed by a linear source term describing the adsorption kinetics according to Langmuir behaviour. Furthermore, mixed boundary conditions (third-order boundary conditions, Eq. (3)) are introduced, connecting the diffusional flux of molecules of the periphery of the nuclei with concentration-dependent growth rate.

In this way, the diffusion and adsorption between two simultaneously created nuclei will be described. For the sake of simplicity, the analysis will here be restricted to the one-dimensional case

(~C ~2C 8t = D ~x 2 + ka(ceq - c) c(O,x) = Co, ( 1 )

8c x = 0 D ~ x ~- kw(C - tsar) (2)

OC x = L t3~ = - k w ( c - c sa t )

where c is the concentration of gas-like adsorbed molecules, Ceq is the equilibrium concentration of gas-like molecules according to the Langmuir isotherm, csat is the saturation concentration, ka is the rate constant of adsorption, kw is the rate constant of growth of nuclei of the condensed phase, D is the surface diffusion coefficient, x is the distance from one nucleus, L is the location of the other nucleus, and Co is the uniform concentration at t = 0. In the following simulations, Co is set equal to Cerit.

Although this model goes far beyond the diffusion models commonly used in two-dimensional nucleation theory, it does not consider the linear growth of the nuclei. In general, the distance L between the nuclei decreases, leading to a moving boundary problem. Furthermore, it seems hopeless to tackle the full problem of randomly distributed cylindrical nuclei of different ages on the surface in the approach used here.

The problem (Eqs.(1) and (2)) is linear and therefore analytically solvable in terms of series expansions. However, due to the poor convergence of these series (which requires their numerical evaluation) it seems reasonable to solve the differential equation directly by a finite difference method. Thus on the basis of the Cranck- Nicholson scheme, Eqs. (1) and (2) were solved for different parameter sets. It transpires that two qualitatively different cases of time-dependent concentration profiles can be distinguished.

2.1. Model A

If the critical nucleation concentration is significantly lower than the equilibrium concentration

L. Pohlmann et al./Surface Science 359 (1996) 280-290 283

(co,it < C,q) and the effective surface diffusion rate is smaller than the adsorption rate (D/L 2 << ka) one obtains two different types of the following concentration time profiles.

Fig. 2 presents the concentration profiles at different times if the surface diffusion rate is also smaller than the growth rate. Here the concentration profiles immediately after time t = 0 begin to intersect the line characterized by the critical nucleation concentration. After a certain time interval, the concentration profiles then exceed the critical nucleation concentration in the whole region o < x < L. The corresponding diffusional flux decreases quickly at the beginning of the process and then increases until reaching a stationary value (Fig. 3).

5 0

o c O o

I t = 0 . 4 1 I

/ %-- - - . ,x

Ft 0

i i

d i s t a n c e x

Uc rit

Fig. 2. Concentration profiles of gas-like molecules at different times with surface diffusion rate smaller than growth rate and adsorption rate. Dimensionless parameters of the simulations: ~ = ka(L2/D) -- 10, 7 = (1 - ( c s a t / C e q ) ) = 0.9, Ucrl t = (%it -- Cs~t/C~q) = 0.3, Ct = kw(L/D) = 5, time unit = L2/D, length unit = L.

In all of these simulations, normalized concentrations U=(c-c~t)/C~q are used, so that in Figs. 1-8 c~t corresponds to the zero line.

If, on the other hand, the diffusion rate is greater than the growth rate, the concentration profiles are all the time above the critical nucleation concentration, as shown in Fig. 4. The diffusional flux increases from the very beginning (Fig. 5). Experimentally, the conditions of model A are fulfilled, apart from the pit edge in the direction of the middle of the pit.

2.2. Model B

If the concentration Cerit when nucleation starts is very close to the equilibrium concentration (Ccrit- "~ Ceq; however ccrit is always less than Qq), and the effective surface diffusion rate is smaller than the growth rate constant (D/L<< kw), which is again

g

2

o u

I t ,-4.7 ~, 1 - i

I ; t = O

d i s t a n c e x

be f i t

Fig. 4. Concentration profiles of gas-like molecules at different times with surface diffusion rate greater than growth rate but smaller than adsorption rate. ct = 0.5, fl = 4, y = 0.9, Ucrit = 0.3.

2 1 0 .8 [

Umax

f lux

0 t ime 0 .4 C' t ime 5 .75

Fig. 3. Diffusional flux at the periphery of the nucleus and the maximum concentration of gas-like molecules at the surface corresponding to Fig. 2. Simulation time interval = 0.4L2/D.

x E

Ucr i t

Fig. 5. Diffusional flux and maximum concentration of gas- like molecules at the surface corresponding to Fig. 4. Simulation time interval = 3.75LZ/D.

284 L. Pohlmann et at~Surface Science 359 (1996) 280 290

greater than the adsorption rate constant (kw/L> ka), one obtains concentration-time profiles as shown in Fig. 6. In this case the linear growth rate of the nucleus (proportional to the diffusional flux at the periphery of the nucleus) can be very well approximated by an equation describing mixed kinetics of surface reactions and diffusion without adsorption [ 15] (Fig. 7)

dr kwco dt - 1 + k~/-~t~t/D' (3)

where r is the radius of the growing nucleus. This result is not surprising, because under the

conditions of slow adsorption and fast consumption of the adsorbed molecules in the condensation process, the two-dimensional system can be consid-

t - 0

o "- . . . . . . . . - - : ~ t < t -

o ' ' ' " " ' ' ' '

q) f' w'l " " 'x '~

LIs rit

, - . , ,

d i s t o n c e x

Fig. 6. Concentration profiles of gas-like molecules at different times if (i) ccrit ~ C,q, Ucrit = 0.9, (ii) surface diffusion rate smaller than the growth rate (~ = 10), and (iii) surface diffusion rate greater than the adsorption rate (fl = 1, t' = 0.9).

t l

-,4

"" . Umax

\\ f l u x

t ime 0.9~

1' Ucr i t

E D

Fig. 7. Diffusional flux and average concentration of gas-like molecules at the surface corresponding to Fig. 6. Simulation time intervals = 0.96LZ/D; solid line represents the approxima- tive formula (Eq. (3)).

ered as a quasi-closed system. With respect to the film formation these conditions will be fulfilled near the edge of the condensation region (pit region). This seems reasonable, because near the pit edge the mean distance between neighbouring nuclei L increases, whereas the growth rate constant kw does not significantly depend on the potential.

The opposite subcase, D/L>> kw, means a complete absence of diffusional limitation and is always contained in model A for the limit K3 = t (this limit case has already been dealt with elsewhere [11]).

The profiles of diffusional fluxes (Figs. 3, 5 and 7) obtained so far are valid if no further nucleation takes place. Therefore, the experimentally investigated cases of mononucleation [16] could be described by this model.

However from the concentration profiles (Figs. 2 and 4) it is clearly seen that if the distance between the nuclei is greater than the diffusional length

~/D/ka, the actual concentration between two nuclei exceeds the critical nucleation concentration. Thus the nucleation process must still proceed. This is in accordance with de Levie [17], who stated that instantaneous homogeneous nucleation is not possible.

Nucleation and growth will proceed until the distance between the nuclei becomes smaller than the diffusional length (i.e. the depletion zones begin to overlap). Then, nucleation stops and further increase of the condensed phase is due only to the growth of already existing islands. This can be illustrated using Eqs.(1) and (2) and assuming that after a certain time interval, another nucleus will appear between the two existing nuclei, effectively bisecting the internuclear distance. This process of subsequent nucleation repeats until the maximum concentration between the nuclei falls below the critical nucleation concentration. One typical simulation of this process is shown in Fig. 8.

In a real system there will be many randomly distributed nuclei and a further stochastic nucleation process proceeding until the concentration decreases below the critical nucleation concentration at each area on the surface. Therefore it is reasonable to use a smooth concentration-time


x ~ " ~ ',U.r'qax

0 t ime

E Uc tit

0 5 7 ~

Fig. 8. Diffusional flux and average concentration of gas-like molecules at the surface assuming that after a certain time interval another nucleus will appear between both nuclei, effectively bisecting the internuclear distance (parameters as in Fig. 4).

curve (obtained by averaging over many simulations analogous to Fig. 8) to describe overall nucleation and growth processes.

3. Experimental

The kinetics of the film formation of 5 mmol dm -a 5-bromocytosine (Janssen Chimica, 99% purity) in 0.5 moldm -3 Na2SO4 solution (adjusted to pH 7 by an NaH2POa/Na2HPO4 buffer) on a slowly dropping mercury electrode were investigated by measuring the time dependence of the double-layer capacity (C-t curves) after a potential step from a value outside the film formation region ( - 100 mV at the anodic edge) to potentials inside the pit region. The potential step was carried out 17.7 s after the beginning of drop formation. Therefore, the increase in drop area within the measuring time does not exceed 5%, so experimental conditions approximately equal to measurements at stationary mercury electrodes. The frequency of the alternating current was 10 kHz and the amplitude was 3 mV. A P t wire was employed as an auxiliary electrode and a SCE separated from the electrochemical cell by a current bridge was used as a reference electrode.

All electrolyte and buffer components used were of suprapure quality (Merck, Darmstadt). The water was thrice-distilled, and solutions were deaerated with nitrogen before measurements were started.

The calculation of the film coverage 0 is the result of the equation

Co - C~ 0 = , (4)

C 0 - - Cfilm

where Cocgas) is the capacity of the film-free surface at t = 0, Ct is the capacity at time t, and Cnlm is the capacity of the completely film-covered surface.

It should be noted that at t = 0 the surface is always covered with gas-like adsorbed molecules, and the concentration of this molecule changes during the condensation process as described in Section 2. Nevertheless, Eq. (4) can be used for determination of the film coverage, when the differences between the capacities (Co~g~s~- Cfilm)

are sufficiently large to allow us to neglect the changes of C~g~s) during the condensation in comparison to the time-dependent decrease of Ct. This condition is not fulfilled near the critical temperature of phase transition where the difference between the densities of both phases is small [ 11 ]. Consequently, the investigation of the temperature dependence of film kinetics by alternating current polarography is generally limited.

In the present case the critical temperature of the system is 36°C and the experimental data were analysed up to a temperature of 24°C, which is sufficiently below the critical point of condensation.

The surface coverage curves were fitted by a non-linear regression program according to the Newton method [ 18].

4. Derivation of model equations and interpretation of experimental results

In general, the mathematical expression for the time dependence of the surface coverage 0 is the convolution integral of crystallization combined with the well-known Avrami theorem [ 19-21]

0 = 1 - exp(-O~xt), (5)

= 1 dN 0¢xt ~ fo (~-~)~=~ u( f t G(U)dU)Z dz, (6,

where 0ex t is the sum of cylindrical film islands without overlap, S E is the electrode area, dN/dt is

286 L. Pohlmann et aL /Surface Science 359 (1996) 280 290

the nucleation rate, and G(c) is the concentration- dependent linear growth rate.

Eqs. (5) and (6) are valid independently from the specific laws of nucleation and growth. Since the nucleation and growth rate in general are dependent on the surface concentration, it is necessary to know the average concentration-time profile in order to obtain an explicit formulation for the law of coverage increase according to Eqs. (5) and (6). However, the concentration-time profile is determined by the balance of the adsorption and condensation kinetics, and therefore knowledge of the law of coverage increase only is required. Nevertheless, on the basis of simulated curves (Section 2) one can deduce different scenarios of concentration evolution in dependence on the ratios between adsorption and condensation parameters (models A and B). This qualitative knowledge allows us to modify the convolution integral (Eq. 6) to obtain approximate formulae for several cases.

4.1. Mathematical expression to describe the time dependence of film formation according to model A

In this case, there will be increasing concentration until a maximum is reached. After that, concentration decreases towards the saturation concentration until complete overlapping of islands terminates the process. Generally speaking, in the second stage of this process the concentration falls below the critical nucleation concentration, leading to the truncation of nucleation (Fig. 8). As described in a previous paper [ 10] the nucleation process can be split approximately into two time intervals. In the first, the nucleation rate is constant.

Similar to nucleation, the growth rate will also go through a maximum, which allows us to replace it by an averaged constant growth rate in a certain time interval.

0¢xt = SE rkb~Z kwcav dU dz 0 < t < t',

(7)

is stopped

1 t' dO) 2 dz 0ex t = ~E fO rkblZ(fzt kwcav l>_~t', (8)

where t' is the time of nucleation truncation. From a comparison of Figs. 2 and 4, it can be

seen that in the case of slow diffusion and fast growth rate (Fig. 2), the concentration gradient at the periphery of nuclei is very steep. This indicates that after the time interval where constant growth takes place a time interval follows with diffusion-

limited growth, leading to a 1/x~ law [ 11 ], com- pleting Eqs. (7) and (8) by

= 1

+ Cav/X/Tz(U-K3)/D dU dz (9) K3

In the following, this model is called model A. By solving Eqs. (7) (9) and fitting the resulting formula to the experimental data, four constants can be identified which are connected in a certain way with the model parameters [11] K I = (kwC~v)2rkbTZ/SE, 2 K 2 = 4CavDrkb/SE, and K 3 = truncation time of constant growth.

4.1. Mathematical expression to describe the time dependence of film formation according to model B

In this case, the concentration at the periphery of the nuclei decreases towards the saturation concentration from the very beginning of the process (Fig. 7). The concentration away from the nuclei is approximately constant for the opening time interval and then decreases below the critical nucleation concentration. This allows us to adopt the truncated nucleation model as described above. The growth rate can be described by Eq. (3) for the whole process. Model B then has the form

Oox,=¼f/rk .(f'kwcav/ (1 + k w x / ~ ( U - z)/D) dU dz,

whereas in the second time interval the nucleation 0 < t < t' (10)


Oext= ~ fot' rkbrt( fzt kwCav/

(1 +k~/~(U-z)/m)dU dz,

t>_t' (11)

The constants obtained from the curve fit here are g l = 2 4c,vrkbD/SE, K2 kwx/n/D, and t '= t ime of nucleation truncation.

It should be noted that from models A and B some special cases (e.g. instantaneous or pro- gressive nucleation with constant growth) can be derived in dependence from the value of the constants t' and K 3 (see Ref. [ 11 ]).

These models are applicable in different potential regions according to the conditions mentioned in Section 2. Because the equilibrium concentration of gas-like molecules depends on the potential in a Gaussian-like way, whereas the saturation and critical nucleation concentration are independent from the potential (Fig. 1), model A will apply in direction of the middle of the pit, and model B will be valid near the pit edge.

Fig. 9 shows some examples for short coverage-- time transients measured at 24°C at the anodic part far from the pit edge (305, -314 and -320 mV). The curves in Fig. 9a represent the non-linear regression fit of model A following Eqs. (7)-(9). The curves in Fig. 9b represent the non-linear regression fit of model B according to Eqs. (10) and (11). It is clearly shown that model A describes the transients in good approximation, whereas the fitted curves of model B differ from the experimental curves, especially at surface cover- ages above 0.8. The fitting parameters of models A and B are listed in Table 1. The lack of fit for model B is clearly greater than the lack of fit for model A. The good fit of model A is in accordance with the results described in Ref. [11] for short transients in the same electrolyte system and with the same concentration of 5-bromocytosine, but measured at a temperature of 8°C. Another result was obtained for longer coverage-time transients (for potentials near the pit edge) and is shown in Fig. 10. As in Fig. 9, models A and B were fitted to the experimental data (see Table 1). Whereas

,3 @

@ > ,},

,©

,Q

1 0

0 6

/ I / I / ¢' 0 4 / f , , f

O C 0.00 0 08 O! 6 0 2 4

t i c s @ / ' s

0 31! C 40

.:.b ,33 ,'St,, L}

0 > C)

©

O8

('j ..},

0 6 -7 . /

:S ' "

0 2

o o ~ - ~ , 0 03 0 08 01 (- 0 24 0 32

] R-',6 "

0 48

Fig. 9. Surface coverage-time curves and the fitted model curves for the system 5-bromocytosine in 0.5 tool din-3 Na2 SO4 solution at 24°C after a potential step from - 1 0 0 m V to a, -320; b, -314; and c, -305mV. The points represent experimental data, the solid lines were calculated using Eqs. 7-9 and model A in Fig. 9a, and with Eqs.(10) and (11) and model B in Fig. 9b.

model A describes the measured transients fairly well with a nearly constant lack of fit, model B shows a tendency for better approximations towards potentials close to the pit edge (the lack of fit of model B decreases in the direction of the pit edge and becomes smaller than that of model A for the potentials -269, -275 and -281 mV).

The results of curve fitting show that the model considerations described in Sections 2 and 4 lead to a consistent picture at the electrode surface. In regions where the difference between the equilibrium concentration Ceq and the critical nucleation concentration Ccrit is high, model A is a good tool to describe the kinetics. On the other hand, only in regions where the difference between the two concentrations is very small (near the pit edge) is the condition for application of model B (i.e. a thermodynamic quasi-closed system) satisfied. This

288 L. Pohlmann et al./Surface Science 359 (1996) 280 290

Table 1 Data for regression of the experimental O-t curves at 24°C and different potentials according to models A and B

Regression according to model A Regression according to model B

PotentiaP K1 K 2 t' K 3 Lack of K~ K 2 t' Lack of (mV) (s -3) (s 2) (ms) (ms) fit b (s 2) (s-V2) (ms} fit b

-269 30.71+0.016 5.28_+0.011 79_+0.9 899_+1.0 0.019 29.88+0.150 6.40_+0.042 833_+22.0 0.014 -275 59.60+0.015 2.19_+0.011 73+0.6 760-+1.0 0.007 26.46+0.107 12.54-+0.051 519-+10.0 0.012 -281 198.48 + 0.056 12.72 _+ 0.023 70 _+ 1.0 620_ 0.8 0.014 24.38 + 0.091 21.32_+ 0.066 395 + 9.0 0.012 -287 169.07 _+ 0.020 6.53 ± 0.009 63 _+ 0.5 450 + 0.5 0.015 22.71 + 0.085 35.91 -+ 0.093 276-+ 7.0 0,013 -299 552.66_+0.171 10.50-+0.268 39-+2.0 330-+1.3 0.013 30.97-+0.116 68.84+0.138 201___7.0 0.013 -305 1089.40 + 0.034 9.69 _+ 0.016 42 + 0.3 236 + 0.5 0.013 27.64__+ 0.046 157.82 ± 0.104 145 -+ 3.0 0,013 - 314 2852.70 -+ 0.064 57.13 ___ 0.027 38 _+ 0.3 129 _+ 0.9 0.014 26.80 _+ 0.035 428.74 _ 0.153 70 ___ 2.0 0.031 -320 7349.67+0.046 94.15_+0.013 31+0.1 90_+0.1 0.012 31.18+0.122 873.33_+0.777 47+5.0 0.039

a Versus SCE.

b N / ] ~ ( ( 0 e x _ _ 0 t h e o ) 2 ) / ( n - - 1)/(Z Oex/n), where 0~x and 0 t h e o a r e experimental and theoretical value of surface coverage, respectively, and n is the number of measuring points.

° I 0 0

08

a b L @

0.6

8 0 2

J~

, i i i , ' 0 0 0 0.32 0 .64 0 9 6 1 2 8 1 60

time / s

l O Q o o 0 8

@ > 0 6 0 ( J

@ 0 4 0 © "t- • 0.2 CO

O 0 , , , ,

0 O0 0 3 2 0 .64 0 96 1 28 1 6 0

t i m e / s

Fig. 10. Surface coverage-time curves and the fitted model curves for the system 5-bromocytosine in 0.5 tool dm-3 Na2 SO4 solution at 24°C after a potential step from -100mV to a, -299; b, -287; c, -281; d, -275; and e, -269 inV. Points represent experimental data, and solid lines were calculated with Eqs. (7)-(9) and model A in (a), and with Eqs. (10) and (11) and model B in (b).

is in accordance to the evolution of the lack of fit of model B towards the pit edge.

The constants obtained from the models A and B cannot be compared with each other, although in a formal comparison the following relations could be obtained

Kl(model B)= Ka(model A),

K2(model B)= 2(K1/K2)°'S(model A).

However, one must not forget that the average concentrations of gas-like molecules in models A and B are defined in a different way.

Nevertheless it is evident that the phase transition on homogeneous electrode surfaces in thermodynamic open systems cannot be examined without considering the role of the simultaneous adsorption process. This can be shown if one attempts to fit the simpler well-known models of the exponential law of nucleation combined with constants [22] or surface-diffusion controlled growth [ 14]. The fitting parameters of the former model (exponential law of nucleation with constant growth) are listed in Table 2. It can be clearly seen that for all potentials the lack of fit is greater than in Table 1 for models A and B. Moreover, the relative errors of the fitting parameters far from the pit edge are significantly higher than the relative errors listed in Table 1. It was very difficult to


Table 2 Data for regression of the experimental O-t curves at 24°C and different potentials according to exponential law and constant growth

Potential a K1 b K2 c Lack of (mV) (s -2) (s 1) fit d

- 269 22.26 _ 0.148 2.44 _+ 0.019 0.018 - 2 7 5 29.51 _+ 0.289 4.26 _+ 0.043 0.017 -281 31.35 _ 0.491 6.12 _+ 0.094 0.020 - 2 8 7 43.54 +_ 1.061 9.82 _+ 0.198 0.027 - 299 73.76 _ 2.458 19.98 _+ 0.466 0.026 - 305 48.36 _+ 1.606 44.44 _+ 0.786 0.023 -- 314 110.00 +_ 10.782 84.04 +_ 3.922 0.050 - 3 2 0 151.65 _+ 24.858 165.66 _+ 10.661 0.057

a Versus SCE. b K1 = lzNok2/SE"

c K2 = rate constant nucleation.

d X/Y~((Oex--Otheo)Z)/(n--1)/(ZOex/n), where 0ex and 0theo are experimental and theoretical value of surface coverage, respectively, and n is the number of measuring points.

molecules an exact modelling of the phase transition is impossible. Nevertheless, with the models proposed above and their special cases, it was possible to find a logical explanation for the transients in wide potential and temperature ranges. One can expect the models to be also valid for phase transitions on solid electrodes, because in heterogeneous nucleation the number of latent nuclei is not a constant No, but a function of time- dependent supersaturation. Consideration of the simultaneous processes of adsorption and phase transitions, including nucleation and growth, and their influence upon each other, are generally indi- cated in thermodynamic open systems.

The measured induction periods were very short and therefore could not be separated by non-linear curve fitting. The dependence of the phase transition mechanism on the initial conditions will be the subject of a forthcoming paper.

fit the second model (exponential law of nucleation with surface-diffusion controlled growth) to the experimental data. The fitting procedure was con- vergent for only one potential ( -314mV). The relative errors exceed the fit parameters by many times: Kx = rcNokdD/SE = 0.14 + 91.997 s-l; K 2 =

kkb = 1000 _ 7751.9 S-% lack of fit = 0.053. (No is the number of latent nuclei, kD is the rate of constant of surface diffusion controlled growth, and kkb is the rate constant of nucleation). Therefore this model is not appropriate to describe the data presented here.

5. Conclusions

In this paper it was theoretically shown on the basis of the diffusion equation in addition to the adsorption source term that the models proposed in Refs. [-10,11] are based on realistic physical assumptions. An additional formalism (model B) was derived from the simulated curves, which was valid under defined conditions (slow adsorption kinetics, co,it ~ Ceq) at the pit edge.

Without information about the temperature and potential dependence of the adsorption of gas-like

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First-order phase transitions on electrodes: the role of surface diffusion

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