Modelling the growth of a single centre

Accepted Manuscript Modelling the growth of a single centre

M.Y. Abyaneh, M. Fleischmann, M.H. Mehrabi PII: DOI: Reference:

S1572-6657(18)30781-1 https://doi.org/10.1016/j.jelechem.2018.11.030 JEAC 12739

To appear in:

Journal of Electroanalytical Chemistry

Received date: Revised date: Accepted date:

26 September 2018 17 November 2018 18 November 2018

Please cite this article as: M.Y. Abyaneh, M. Fleischmann, M.H. Mehrabi , Modelling the growth of a single centre. Jeac (2018), https://doi.org/10.1016/j.jelechem.2018.11.030

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ACCEPTED MANUSCRIPT

Modelling the Growth of a Single Centre M. Y. Abyaneh*, M. Fleischmann(a) and M. H. Mehrabi

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Astrakangatan 62, Stockholm 16552 Sweden

Abstract

In this paper a novel approach for deriving the current-time profile of the growth of a single

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hemispherical centre under a mixed kinetic-diffusion controlled mechanism is conceived. The novelty of the approach lies, first and foremost, on the indisputable assumptions made

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regarding the boundary conditions imposed on the partial differential equation associated with the spherical diffusion. As a result, the variation of the surface concentration of the

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depositing ions with time is formulated. Based on this formulation, we have been able to

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properly characterize the pattern of the growth of a single centre which has been, hitherto, a longstanding problem. The precise portrayal of the results necessitated the exact solutions

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to be obtained numerically.

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Approximate closed-form solutions are derived for the rate of growth of a single centre, as well as for the flow of current to the growth centre, provided that the reactions are assumed to be in a steady-state. It is shown that none of the previous works are capable of providing correct solutions even for the steady-state approximation. Solutions based on the steady-state approximation are shown to closely follow the characteristics of the growth of a single centre provided that reaction rates are small, 𝑘 < 10−2 cm s−1 . In experimental works associated with higher rates of reaction, 𝑘 > 10−2 cm s −1 , it is necessary to involve

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ACCEPTED MANUSCRIPT the exact solutions obtained numerically. The clarification of the range of rate constants for which the growth process is controlled (a) by the rate of charge transfer across the interface, (b) by a mix of charge transfer and diffusion, and (c) by only the rate of diffusion of the depositing ions to the interface, remains one of the most valuable contribution of this

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article. Keywords: nucleation, electrocrystallization, mixed-control, microelectrodes, nanoparticles

List of symbols

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*To whom correspondence should be addressed. Email [email protected] (a) deceased on the 3rd of August 2012

the radius of the growth centre (cm)

𝑐0

bulk concentration of the depositing species (mol cm−3 )

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𝑎

ED

c(a, t) concentration at the surface of the growth centre (mol cm−3 ) diffusion coefficient (cm2 s−1 )

F

the Faraday constant (C mol−1 )

𝑖

current (A)

𝑗

flux density (mol cm−2 s −1 )

𝑘

the net rate constant for crystallization (cm s −1 )

𝑘1

𝑘𝑐0 (mol cm−2 s −1 )

𝑀

molar mass of the depositing species (g mol−1 )

𝑟

radius of the diffusion zone (cm)

𝑡

deposition time (s)

𝜌

density of the deposit (g cm−3 )

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CE

PT

𝐷

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ACCEPTED MANUSCRIPT 1. Introduction There generally exist two distinct methods for the in-situ studies of the kinetics of nucleation and crystal growth during the electrodeposition processes. Both methods employ an applied potential step to an inert electrode and record the resulting variation in

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the flow of current. In one method the applied potential is specifically chosen to be so low as to ensure the rate of growth is controlled by the processes at the electrode/electrolyte interface. Precise equations for the flow of the transient current with the passage of time, current-time transients (CTTs), for a wide range of models have been derived [2-17]. The

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nonlinear regression fit of the recorded CTTs to suitably chosen derived equations has been shown to provide information regarding the kinetics of nucleation and growth [17,18] as

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well as the morphology of the growth centres (see e.g. [12,19]).

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The other method of studying electrocrystallisation processes is to analyze the recorded CTTs at higher potentials for which growth is assumed to be exclusively controlled by the

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diffusion of the depositing species through the solution (see e.g. [20]). A major problem in

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the development of adequate models for the recorded CTTs, for this scenario, is the correct formulation and inclusion of the effects of the overlap of the diffusion zones (see e.g. [20-

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29]). These effects are avoided if attention is restricted to the initial stages of deposition such that the diffusion zones are still independent of each other [30-31]. However, the advent of the application of microelectrodes [32] makes it possible to achieve considerable simplifications in the experimentation and in the modelling. By using inert electrodes of sufficiently small dimensions, it is possible to restrict nucleation and growth to that of a single centre [33-36]. Then the question of the overlap of growth centres and of diffusion zones does not arise.

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ACCEPTED MANUSCRIPT Given that the diffusional flux to a point sink is huge, the growth of a minute centre is invariably controlled by the rate of charge-transfer in the early stages. Thus, the derived current-time (CT) profile for a single potential step experiment should reduce to [1] 𝑀 2

𝑖𝑐𝑡 = 2𝜋𝑧𝐹 ( 𝜌 ) 𝑘1 3 𝑡 2

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(1.1)

where, the subscript 𝑐𝑡 stands for charge transfer controlled growth, z𝐹(C mol−1 ) is the charge transferred per ion, 𝑀(g mol−1 ) and 𝜌(g cm−3 ) are the molar mass and the density of the deposit and 𝑘1 (mol cm−2 𝑠 −1 ) is the flux of ions assimilated by the growth centre.

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However, under appropriate conditions, the flow of current is expected to follow that associated with growth centres the rate of which is controlled solely by the rate of diffusion.

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Alas, the reported equations for the CT profiles resulting from the diffusion-controlled growth of centres are, at best, inconsistent [30,37-43]. Therefore, it is essential to establish

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the exact dependency of current on time for the diffusion-controlled growth prior to getting

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involved with the derivation of the flow of current resulting from a mixed kinetic-diffusion

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controlled mechanism. This will be done in Section 2 of this paper. Attempts have been made to derive equations [30,37,38] for the mixed kinetic-diffusion

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controlled growth of a single hemispherical centre. However, the equations given are incorrect. Due to the importance of the subject, and because of the ambiguity of some of the previous works, it has become necessary to devote Section 5 and most of Section 6 of this paper to their detailed analysis and point out their inadequacy in providing a sound equation for the work in hand. In this paper we derive the characteristics of the growth of a single centre, while making sure that each mathematical step taken during the derivation is based on unambiguous

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ACCEPTED MANUSCRIPT argument and sound foundation. The detailed analysis of the previous works [30,37,38], and comparison of their results with the well-established facts and with the results of the present work, will show that the equations and the analysis offered in this paper should be considered as the benchmark for the proper study of nucleation and the early stages of

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growth of electrodeposits.

2. Current flow resulting purely from the diffusion-controlled growth

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Pure diffusion-controlled growth takes place when the flux of ions reaching the growth centre, 𝐽(mol cm−2 𝑠 −1 ), is sufficiently small in comparison to the flux of ions, 𝑘1 , which

𝑑𝑡

𝑀

𝑀

𝜕𝑐

= ( 𝜌 ) 𝐽 = ( 𝜌 ) 𝐷 𝜕𝑟 (𝑎, 𝑡)

(2.1)

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𝑑𝑎

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of a centre of radius 𝑎 is given by

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could have been assimilated by the growth centre. Under this condition the rate of growth

For a spherical diffusion, the concentration gradient at the surface of the growth centre is

𝜕𝑟

(𝑎, 𝑡) = 𝑐0 (

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𝜕𝑐

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given by [44]

1

√𝜋𝐷𝑡

1

+ 𝑎)

(2.2)

where 𝑐0 (mol cm−3 ) is the bulk concentration of the depositing species. Substituting Equation (2.2) into (2.1) leads to the differential equation 𝑑𝑎 𝑑𝑡

=

𝑀𝑐0 𝜌

𝐷 1/2

𝐷

(𝜋1/2 𝑡 1/2 + 𝑎 )

(2.3)

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ACCEPTED MANUSCRIPT To solve the above equation, we assume that 𝑎 = 𝑚𝑡1/2 leading to the rate of growth given by 𝑚𝑡 −1/2 /2, which, when substituted into Equation (2.3), yields, 𝑚 2

=

𝑀𝑐0 𝐷 1/2

𝐷

(𝜋1/2 + 𝑚)

𝜌

(2.4)

𝑚=

𝑀𝑐0 𝜌

𝐷

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Solving Equation (2.4) for 𝑚 and taking the positive root leads to

2𝜋𝜌

√ [1 + √1 + ] 𝜋 𝑀𝑐 0

(2.5)

𝑀𝑐0 𝜌

2𝜋𝜌

𝐷𝑡

[1 + √1 + 𝑀𝑐 ] √ 𝜋 0

(2.6)

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𝑎=

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Substitution of 𝑚, given by Equation (2.5), in 𝑎 = 𝑚𝑡1/2 results in [43]

However, the flow of current into a hemispherical growth centre of radius 𝑎 is given by 𝑧𝐹𝜌 𝑀

𝑑𝑎

2𝜋𝑎2 𝑑𝑡

ED

𝑖=

(2.7)

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Differentiating Equation (2.6) with respect to 𝑡 and substituting both the differential as well

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as the value of 𝑎 given by (2.6) into Equation (2.7) leads to

𝑀 2

2𝜋𝜌

3

AC

𝑖𝑑𝑐 = 𝑧𝐹 ( 𝜌 ) (1 + √1 + 𝑀𝑐 ) 0

𝑐0 3 𝐷3/2 1/2 𝑡 𝜋 1/2

(2.8)

where, the subscript 𝑑𝑐 stands for diffusion-controlled. We can now proceed to formulate the flow of current as a function of time when the rate of growth is controlled by a mixed kinetic-diffusion mechanism.

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ACCEPTED MANUSCRIPT 3. The flow of current under a mixed kinetic-diffusion control In general, the rate of growth of a centre is related to the flux of ions, 𝑘1 , assimilated by the 𝑑𝑎

centre via the relationship 𝑑𝑡 =

𝑀𝑘1 𝜌

, provided that 𝑘1 is sufficiently smaller than the flux of

ions, 𝐽, diffusing to the surface. In this scenario, the concentration of the depositing ions at

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the surface of the growth centre is approximately the same as the bulk concentration, 𝑐0 . However, the surface concentration of the depositing ions, 𝑐(𝑎, 𝑡), will start to gradually decrease from its initial magnitude, 𝑐0 , to its final magnitude extremely close to zero, at 𝑑𝑎

𝑀𝐽 𝜌

, where 𝐽 is

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which stage the rate of growth of the centre is determined by 𝑑𝑡 =

determined by the Fick’s laws of diffusion. Therefore, in modelling the growth of a single

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centre under a mixed kinetic-diffusion control, we must relate the flux of ions assimilated by the growth centre, which would neither be 𝑘1 nor 𝐽, to a realistic time-dependent

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parameter.

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We note that the flux of ions assimilated by the growth centre is initially given by 𝑘1 = 𝑘𝑐0, where 𝑘(cm s−1 ) is the net rate constant for crystallization. However, with the passage of

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time, the flux consumed will gradually decrease as a result of a gradual fall in the rate of

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supply of ions. But the decrease in the rate of supply is a direct consequence of the 𝜕𝑐

decrease in the concentration gradient of ions at the surface of the growth centre, 𝜕𝑟 (𝑎, 𝑡), and this concentration gradient is, in turn, related to the concentration of the ions at the surface, 𝑐(𝑎, 𝑡). It then follows that a realistic approach would be to assume 𝜕𝑐

𝐷 𝜕𝑟 (𝑎, 𝑡) = 𝑘𝑐(𝑎, 𝑡),

𝑡≥0

(3.1)

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ACCEPTED MANUSCRIPT Therefore, developing an expression for the flow of current at any given 𝑡 requires, ab initio, the formulation of the variation of the surface concentration of the depositing species with time. 3.1 Variation of the surface concentration of the depositing ions with time

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In a previous work [38] the dependence of the surface concentration of the depositing ions on time was formulated by setting the steady-state approximation of the diffusive flux equal to the flux defined by a simple Butler-Volmer type of equation. However, it is possible to achieve an exact solution for the variation of the surface concentration of the depositing

𝜕𝑐

2𝐷 𝜕𝑐 𝑟 𝜕𝑟

,

𝑎𝑐 ≤ 𝑎 < 𝑟 < ∞ , 𝑡 ≥ 0

(3.2)

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𝜕𝑡

𝜕2 𝑐

= 𝐷 𝜕𝑟 2 +

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species with time by solving directly

𝑡≥0

(3.3)

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lim𝑟→∞ 𝑐(𝑟, 𝑡) = 𝑐0,

ED

with 𝑐(𝑟, 0) = 𝑐0 at 𝑟 ≥ 𝑎 and the boundary conditions given by Equation (3.1) and

In equation (3.2), 𝑎 and 𝑎𝑐 (cm) represent the radius of the hemispherical growth centre

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and of the critical nucleus, respectively, 𝑟(cm) the distance from the origin (located at the

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centre of the hemispherical growth centre), 𝑡(s) the time and 𝑐(𝑟, 𝑡) in (mol cm−3 ) the concentration of the depositing species at any given 𝑟 and 𝑡. Assuming that 𝑢(𝑟, 𝑡) = 𝑟𝑐(𝑟, 𝑡), then Equations (3.1), (3.2) and (3.3) lead, respectively, to 𝜕𝑢 𝜕𝑟

𝜕𝑢 𝜕𝑡

1

𝑘

(𝑎, 𝑡) = ( + ) 𝑢(𝑎, 𝑡), 𝑎 𝐷 𝜕2 𝑢

= 𝐷 𝜕𝑟 2 ,

𝑡≥0

𝑎𝑐 ≤ 𝑎 < 𝑟 < ∞,

(3.4)

𝑡≥0

(3.5)

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ACCEPTED MANUSCRIPT and

lim𝑟→∞

𝑢(𝑟,𝑡)

= 𝑐0 ,

𝑟

𝑡≥0

(3.6)

Given that the Laplace Transform of 𝑢(𝑟, 𝑡) with respect to 𝑡, is denoted by 𝑈(𝑟, 𝑠), i.e. ℒ{𝑢(𝑟, 𝑡)} = 𝑈(𝑟, 𝑠), then Equations (3.4) and (3.5) can be written, respectively, as

𝜕𝑟

1

𝑘

(𝑎, 𝑠) = ( + ) 𝑈(𝑎, 𝑠) 𝑎 𝐷

(3.7)

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𝜕𝑈

𝜕2 𝑈

𝑠𝑈(𝑟, 𝑠) − 𝑟𝑐0 = 𝐷 𝜕𝑟 2 (𝑟, 𝑠) Equation (3.8) has a standard solution

𝑟𝑐0 𝑠

+ 𝐴𝑒

𝑠 𝐷

−√ 𝑟

+ 𝐵𝑒

𝑠 𝐷

√ 𝑟

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𝑈(𝑟, 𝑠) =

(3.8)

(3.9)

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The constant 𝐵 is found to be equal to naught (see Appendix A). Differentiating Equation

𝜕𝑈

(𝑎, 𝑠) =

𝑐0 𝑠

𝑠

𝑠

− 𝐴√𝐷 𝑒𝑥𝑝 (−√𝐷 𝑎)

(3.10)

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𝜕𝑟

ED

(3.9) with respect to 𝑟, with 𝐵 = 0, and then substituting for 𝑟 = 𝑎 leads to

𝑎𝑐0 𝑠

𝑠

+ 𝐴𝑒𝑥𝑝 (−√𝐷 𝑎)

(3.11)

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𝑈(𝑎, 𝑠) =

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Likewise, Equation (3.9) for 𝐵 = 0 and 𝑟 = 𝑎 becomes

Combination of Equation (3.7), (3.10) and (3.11) results in

𝑐0 𝑠

𝑠

𝑠

1

𝑘

− 𝐴√𝐷 𝑒𝑥𝑝 (−𝑎√𝐷 ) = (𝑎 + 𝐷) [

𝑎𝑐0 𝑠

𝑠

+ 𝐴𝑒𝑥𝑝 (−𝑎√𝐷)]

(3.12)

Solving Equation (3.12) for 𝐴 leads to

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ACCEPTED MANUSCRIPT 𝑠 𝐷

𝑎2 𝑐0 𝑘𝑒𝑥𝑝(𝑎√ )

𝐴=−

(3.13)

𝑠 𝐷

𝑠(𝐷+𝑎𝑘+𝑎𝐷√ )

Substituting 𝐴, from Equation (3.13), and 𝐵 = 0 into Equation (3.9) and taking the inverse

𝑢(𝑟, 𝑡) = 𝑟𝑐0 −

𝑘𝑎𝑐0 √𝐷

ℒ

−1

{

𝑒

−√

𝑠 (𝑟−𝑎) 𝐷 √𝐷

𝑘

}

𝑠(√𝑠+ + ) 𝑎 √𝐷

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Laplace Transform will lead to

(3.14)

𝑘𝑐0 √𝐷

ℒ −1 {

1

}

√𝐷 𝑘 𝑠(√𝑠+ + ) 𝑎 √𝐷

(3.15)

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𝑐(𝑎, 𝑡) = 𝑐0 −

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Given that 𝑢(𝑟, 𝑡) = 𝑟𝑐(𝑟, 𝑡), then Equation (3.14) will take the form

In order to perform the above inverse transform, we define

√𝐷 𝑎

+

𝑘 √𝐷

=

𝐷+𝑘𝑎 𝑎√𝐷

(3.16)

ED

𝛽=

𝐴̃

𝐵̃

√

+𝑠+ 𝑠

𝐶̃

(3.17)

√𝑠+𝛽

AC

𝑠(√

= 𝑠+𝛽)

CE

1

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Then, using partial fraction we get

1 1 Obviously, the equalities 𝐴̃ = −𝐶̃ = − 𝛽2 𝑎𝑛𝑑 𝐵̃ = 𝛽 must hold for the above equation to

be true. Thus,

ℒ −1 {

1

1

} = ℒ −1 {− 𝛽2

√𝐷 𝑘 𝑠(√𝑠+ + ) 𝑎 √𝐷

√𝑠

1

+ 𝛽𝑠 + 𝛽2 (

1 √𝑠+𝛽)

}

(3.18)

which for 𝛽 > 0 leads to

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ACCEPTED MANUSCRIPT ℒ −1 {

1

1

2

} = 𝛽 [1 − 𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽 √𝑡)]

(3.19)

√𝐷 𝑘 𝑠(√𝑠+ + ) 𝑎 √𝐷

Inserting the result of the inverse Laplace transform from Equation (3.19) into (3.15) and combining the resulting equation with Equation (3.16) leads to the intended goal of this

with time. Thus,

𝐷+𝑘𝑎

2

[1 − 𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽 √𝑡)]

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3.2. Testing the validity of Equation (3.20)

(3.20)

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𝑘𝑐0 𝑎

𝑐(𝑎, 𝑡) = 𝑐0 −

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section, that is, to the variation of the surface concentration of the depositing ions, 𝑐(𝑎, 𝑡),

We note that Equation (3.1) and (3.20) results in

(𝑎, 𝑡) =

𝑘𝑐0 𝐷+𝑘𝑎

[1 +

𝑘𝑎

2

𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽√𝑡)]

ED

𝜕𝑐 𝜕𝑟

𝐷

(3.21)

2

PT

Furthermore, given that 𝛽 is always positive

CE

lim𝑡→∞ 𝛽√𝑡𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽√𝑡) =

1 √𝜋

(3.22)

2

AC

Equation (3.22) suggests that for relatively large 𝑡’s 𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽√𝑡) ≈ 𝛽

1 √𝜋𝑡

(3.23)

The truth of Equation (3.23), at relatively large 𝑡’s, can be demonstrated graphically by choosing any realistic figure for 𝛽, Equation (3.16). The diffusion coefficient can be assumed to be of the order of 𝐷 = 10−5 cm2 s −1. Then, for a small centre of 50Å radius, the magnitudes of 𝛽 range roughly from 6300 𝑠 −1/2 for a very slow reaction rate, 𝑘 = 11

ACCEPTED MANUSCRIPT 10−4 cm s−1 , to about 38000 𝑠 −1/2 for a very fast reaction rate, 𝑘 = 102 cm s−1. However, for the same above-mentioned 𝑘-values and for the largest possible centre, such as 𝑎 = 104 Å, the size of an electrode of radius 1 𝜇𝑚, 𝛽 ranges from about 31.6 to 31600 𝑠 −1/2. 1 √𝜋𝑡

versus 𝑡 for 𝛽 = 3000 𝑠 −1/2 , a relatively

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2

Fig. 1 is a comparison of 𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽 √𝑡) with 𝛽

2

fast reaction. It can be seen that for all values of 𝑡 > 1μs, the two functions, 𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽 √𝑡) and 𝛽

1 √𝜋𝑡

, would become, for all practical purposes, one and the same. Now that we have

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verified the validity of Equation (3.23), we substitute this equation into (3.21) after inserting the definition of 𝛽 from Equation (3.16). The resulting equation is then given by

𝜕𝑟

(𝑎, 𝑡) ≈

𝑘𝑐0

[1 + 𝐷+𝑘𝑎

𝑘𝑎2 ] √𝜋𝐷𝑡(𝐷+𝑘𝑎)

MA

𝜕𝑐

(3.24)

ED

We can now proceed to examine Equation (3.24), and by implication Equation (3.20) and

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(3.21), to ascertain whether Equation (3.24) is qualified to be used as a base for the exact formulation of the current flowing through a single centre at any given time 𝑡. For this

CE

reason we first test whether the flow of current to be derived from (3.24) coincides at

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sufficiently long times with Equation (2.8), an equation derived when the rate of growth is controlled by the rate of diffusion. Given that large values of 𝑎 corresponds to the growth of a single centre at long times, we now need to, only, demonstrate that at sufficiently large values of 𝑎, Equation (3.24) can converge to Equation (2.2), and by implication to Equation (2.8). We note that Equation (3.24) is valid for all 𝑎’s and, furthermore, for a sufficiently large 𝑎, 𝑘𝑎 ≫ 𝐷, which in turn leads to the term (𝐷 + 𝑘𝑎) ≈ 𝑘𝑎. Consequently, Equation (3.24) under such conditions can

12

ACCEPTED MANUSCRIPT be seen to reduce to Equation (2.2), thereby demonstrating the aptness of Equation (3.20) and (3.21). The next step is to demonstrate that Equation (3.21) can also lead to Equation (1.1) for sufficiently small 𝑎. It can, graphically, be demonstrated that the term

𝑘𝑎 𝐷

2

𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽√𝑡) in

SC RI PT

Equation (3.20) or (3.21) is negligible compared to 1, for all values of 0 < 𝑎 < 1 μm and for both slow (𝑘 = 10−4 cm/s) and fast reaction rates (𝑘 = 102 cm/s). Furthermore, for small values of 𝑎, corresponding to sufficiently small growth times, 𝐷 ≫ 𝑘𝑎 and Equation (3.21) is thereby reduced to

≈

𝑘𝑐0

NU

𝑟=𝑎

𝐷

But the rate of growth is given by

𝑑𝑡

=

𝑀

𝑑𝑐

𝐷 (𝑑𝑟) 𝜌

𝑟=𝑎

=

𝑀 𝜌

𝑘𝑐0

ED

𝑑𝑎

MA

𝑑𝑐

(𝑑𝑟)

(3.25)

(3.26)

𝑀𝑘𝑐0 𝜌

𝑡

CE

𝑎=

PT

The radius of the growth centre at sufficiently short time is, therefore, given by

(3.27)

AC

The growth current can then be obtained by substituting Equations (3.27) and (3.26) into (2.7). Thus,

𝑀 2

(𝑖𝑠ℎ𝑜𝑟𝑡 𝑡𝑖𝑚𝑒 )𝑐𝑡 ≈ 2𝜋𝑧𝐹 ( ) (𝑘𝑐0 )3 𝑡 2 𝜌

(3.28)

Equation (3.28) is the same as Equation (1.1), considering that 𝑘1 = 𝑘𝑐0 . It then follows that Equation (3.20), at sufficiently short times, has predicted the expected form given by Equation (1.1).

13

ACCEPTED MANUSCRIPT We have therefore verified the validity of Equation (3.20) by testing its predictive characteristics at both limits, sufficiently short and sufficiently long times. It then follows that the exact equation representing the flux of ions incorporated into a single hemispherical growth centre is given by, see Equation (3.21) 𝑘𝑐 𝐷

𝑘𝑎 𝐷

2

𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽 √𝑡)]

(3.29)

SC RI PT

𝜕𝑐

0 D𝜕𝑟 (𝑎, 𝑡) = 𝐷+𝑘𝑎 [1 +

Combining Equation (3.28) with Equation (2.1) leads to the rate of growth 𝑑𝑎 𝑑𝑡

𝑀𝑘𝑐 𝐷

0 = 𝜌(𝐷+𝑘𝑎) [1 +

𝑘𝑎 𝐷

2

𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽 √𝑡)]

(3.30)

NU

Unfortunately, there seems to be no closed-form solution to Equation (3.30) associating 𝑎 with time 𝑡. However, it is possible to numerically deduce from Equation (3.30) the

MA

characteristics of 𝑎 as a function of 𝑡. This task will be fulfilled in Section 4.

ED

3.3. Solution based on the steady-state approximation In this section we endeavor to achieve an approximate closed-form solution to Equation

PT

(3.30). As mentioned in the previous section, it can graphically be demonstrated that, for

CE

both slow and fast reaction rates, the transient term,

𝑘𝑎 𝐷

2

𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽 √𝑡), is negligible

AC

compared to 1, for all values of 0 < 𝑎 < 1 𝜇𝑚. Henceforth in this section we seek a closed 𝑀𝑘𝑐 𝐷

0 form solution for Equation (3.30) based only on the steady-state term, 𝜌(𝐷+𝑘𝑎) .

To show that this term is indeed representing a steady-state scenario, we set

𝜕𝑐 𝜕𝑡

in

equation (3.2) to naught. Thus, 𝑑2 𝑐 𝑑𝑟 2

2 𝑑𝑐

+ 𝑟 𝑑𝑟 = 0

(3.31)

We can rewrite (3.31) in the form 14

ACCEPTED MANUSCRIPT 𝑑

𝑑𝑐

(𝑟 2 𝑑𝑟) = 0 𝑑𝑟

giving 𝑐 =

𝐴1

(3.32)

+ 𝐵1

𝑟

(3.33)

Applying the boundary condition, Equation (3.3), to Equation (3.33) gives 𝐵1 = 𝑐0 .

combining the result with Equation (3.1) leads to 𝑑𝑐

𝐷( )

𝑑𝑟 𝑟=𝑎

𝐴1 𝑎

)=−

𝐴1 𝐷 𝑎2

−𝑘𝑎2 𝑐0 𝑘𝑎+𝐷

(3.34)

(3.35)

NU

giving 𝐴1 =

= 𝑘 (𝑐0 +

SC RI PT

Differentiation of Equation (3.33) with respect to 𝑟, after the substituting 𝑐0 for 𝐵1, and then

𝑑𝑐

(𝑑𝑟)

MA

Substitution of 𝐴1 from (3.35) into (3.34) leads to 𝑘𝑐

𝑟=𝑎

0 = 𝑘𝑎+𝐷

(3.36)

𝑑𝑡

𝑀𝑘𝑐 𝐷

0 = 𝜌(𝐷+𝑘𝑎)

PT

𝑑𝑎

ED

We note that combination of Equation (3.36) with (2.1) leads to

(3.37)

AC

Equation (3.30).

CE

which is indeed the steady-state term, that is the term in front of the square bracket in

Integration of Equation (3.37) leads to the quadratic equation 𝑘𝑎2 2

+ 𝐷𝑎 =

𝑀𝑘𝑐0 𝐷 𝜌

𝑡

(3.38)

with the positive root given by 1

𝐷

𝑎 = 𝑘 (1 +

2𝑘 2 𝑀𝑐0 𝑡 2 𝐷𝜌

𝐷

) −𝑘

(3.39)

15

ACCEPTED MANUSCRIPT Replacing 𝑎 and d𝑎/𝑑𝑡, given by Equation (3.39) and (3.37), respectively, into Equation (2.7) 𝑖=

leads to

2𝜋𝑧𝐹𝐷 2 𝑐0 𝑘

1

[(1 +

2𝑘 2 𝑀𝑐0 𝑡 2 𝐷𝜌

) − 2 + (1 +

2𝑘 2 𝑀𝑐0 𝑡 𝐷𝜌

−1/2

)

]

(3.40)

The short-time characteristics of Equation (3.40) is derived from the leading term of the

𝑖𝑡→0 ≈

2𝜋𝑧𝐹𝐷 2 𝑐0 1 2𝑘 2 𝑀𝑐0 𝑡 𝑘

[4 (

𝐷𝜌

2

𝑀 2

SC RI PT

binomial expansions within the square bracket. Thus,

) ] = 2𝜋𝑧𝐹 ( 𝜌 ) (𝑘𝑐0 )3 𝑡 2

which is the expected form given by Equation (1.1).

(3.41)

NU

Having shown that Equation (3.41) does, indeed, provide the correct expression for the

MA

short-time behaviour of the CT profile, we now test the long-time characteristics of Equation (3.40). We note that Equation (3.40), at sufficiently long time, reduces to

ED

𝑘

[

]≈

1

2𝜋𝑧𝐹𝐷 2 𝑐0 2𝑘 2 𝑀𝑐0 𝑡 2 𝑘

(

𝐷𝜌

)

(3.42)

PT

𝑖𝑡→∞ ≈

2𝜋𝑧𝐹𝐷 2 𝑐0

1 2𝑘2 𝑀𝑐0 𝑡 2 ) +1 −2( 𝐷𝜌 1 2𝑘2 𝑀𝑐0 𝑡 2 ( ) 𝐷𝜌

2𝑘2 𝑀𝑐0 𝑡 𝐷𝜌

CE

which, in turn, is reduced to

𝑀 1/2

(𝐷𝑐0 )3/2 𝑡1/2

(3.43)

AC

𝑖𝑡→∞ ≈ 2√2𝜋𝑧𝐹 ( 𝜌 )

It then follows that, should Equation (3.40) correctly represents the steady-state approximation scenario, then its long-time characteristic, Equation (3.43), should also be deduced from the steady-state approximation to Equation (2.2). The steady-state term of Equation (2.2) is given by 𝜕𝑐 𝜕𝑟

(𝑎, 𝑡) =

𝑐0 𝑎

(3.44)

16

ACCEPTED MANUSCRIPT Substitution of the concentration gradient from Equation (3.44) into Equation (2.1) and integration of the result leads to 2𝐷𝑐0 𝑀𝑡 1/2

𝑎≈(

)

𝜌

(3.45)

𝑑𝑎 𝑑𝑡

1

≈2(

SC RI PT

The rate of growth is, therefore, given by 2𝐷𝑐0 𝑀 1/2 −1/2 ) 𝑡 𝜌

(3.46)

The steady-state approximation to a purely diffusion controlled growth is then obtained by

NU

Substituting (3.45) and (3.46) into Equation (2.7). The result of this operation leads to Equation (3.43), which represents the long-time characteristic of the CT profile derived for

MA

the steady-state approximation, Equation (3.40).

ED

Equation (3.43) is also given elsewhere [39] and subsequently used by many researchers (see e.g.[41]). But it must be borne in mind that (3.43) represents, only, a steady-state

PT

approximation to the exact equation, Equation (2.8) [43]. Equation (3.40) can be written in 𝑡

CE

the form of a dimensionless current 𝐼 versus a dimensionless time 𝜏. Thus1, 𝑖𝑘

𝑡 1/2

𝐼 = 2𝜋𝑧𝐹𝐷2 𝑐 = [(1 + 𝜏)

AC

0

𝑡 −1/2

− 2 + (1 + 𝜏)

]

(3.47)

where, 𝜏 is defined by 𝐷𝜌

𝜏 = 2𝑘 2 𝑀𝑐

0

(3.48)

4. Results 1

Equation (3.47) had been derived by the late Professor Martin Fleischmann FRS in a private correspondence nd sent to the leading author, Professor Morteza Abyaneh, on the 2 of September 2004. An equation deceptively similar in structure to Equation (3.47), but erroneous and highly tortuous in its definition of 𝐼 and 𝜏, has been provided in Ref. [45].

17

ACCEPTED MANUSCRIPT Given that Equation (3.30) has no closed form solution, we are forced to solve the equation numerically. To do so we need to first provide data for all of the parameters in Equation (3.30). The following realistic data are assumed: 𝐷 = 10−5 cm2 s −1 , 𝑐0 = 10−3 mol cm−3 , 𝑀 = 60 g mol−1 , 𝜌 = 10 g cm−3 , with 𝑘 ranging from 10−4 to 102 cm s−1 and 𝛽 defined

SC RI PT

by Equation (3.16). Mathematica software, version 11, gives the solution as an Interpolating function by using Hermite interpolation, a method of interpolating data as a polynomial function. Figures 2 compares the plots of 𝑎 as a function of 𝑡 for (a) the numerical solution to Equation (3.30), (b) the steady-state approximation, Equation (3.38), (c) the kinetically controlled growth, Equation (3.27) and (d) the diffusion-controlled growth of a single centre

NU

(Equation (2.6)) for a range of 𝑘-values. Figures 2A, 2B, 2C and 2D are drawn, respectively,

MA

for 𝑘 = 10−3 , 𝑘 = 10−2 , 𝑘 = 10−1 and 𝑘 = 1 cm s −1. The range of time is chosen such that

ED

the final size of the centre is approximately 10 μm.

It can be seen from the above-mentioned figures that the deposition time, 𝑡, for which the

PT

growth is predominantly controlled by the rate of charge-transfer, kc, across the interface is

CE

progressively reduced, the grater the net crystallization rate constant, 𝑘. Furthermore, it is observed that with increase in 𝑘 the rate-controlling mechanism becomes progressively

AC

dominated by the rate of diffusion. It is also noted that the result predicted from the steadystate approximation is in complete agreement with that estimated from the numerical solution at low rate constants, Fig. 2A and 2B. However, at higher rates of deposition, Fig. 2C and 2D, the steady-state result, Equation (3.40), progressively diverges from the results estimated by the numerical solution to Equation (3.30). At rate constants 𝑘 ≥ 0.1 cm s −1, the rate-determining step during the entire deposition time, 𝑡, practically ceases to be the rate at which charge is transferred at the interface. At these higher rates, 𝑘 > 0.1 cm s −1 ,

18

ACCEPTED MANUSCRIPT the growth profile would progressively follow the same pattern as that dictated by diffusion, Equation (2.6). In contrast, for 𝑘 < 10−3 cm s−1, the charge-transfer controlled growth has been observed to be the sole rate-controlling mechanism of growth.

SC RI PT

The flow of current as a function of 𝑎 is given by Equation (2.7). Therefore, combining the value of 𝑎, obtained from the numerical solution of Equation (3.30) as well as its differential with respect to 𝑡, with Equation (2.7) results in the exact form of current-time (CT) profile. Fig. 3 outlines the CT profiles predicted (a) by using the above-mentioned procedure, (b) from the steady-state approximation, Equation (3.40), (c) from the kinetically controlled

NU

growth of a single centre, Equation (3.28) and, finally, (d) from the diffusion-controlled

MA

growth, Equation (2.8). Fig. 3A and 3B represent the profiles obtained, respectively, for 𝑘 = 10−3 and 𝑘 = 10−1 cm s −1 . The results predicted by these figures reinforce the

ED

faultlessness of the results displayed in Fig. 2. For example, In line with the findings from Fig. 2, Fig. 3B shows that, at higher rate constants, the results obtained from the steady-state

CE

PT

approximation estimates lower values of the flow of current with the passage of time.

AC

5. Comparison with previous theoretical work To expose the novelty of this work and the advances made over the previously reported work, it becomes necessary to analyse the results of the previous attempts in formulating the characteristics of the growth of a single centre. There are three major theoretical work [30, 37, 38] which deal with this topic. The first work was reported in 1975 [30], while the second and the third work were reported in 1983 [37] and 1994 [38], respectively. In this section the results of each of these works will be analysed under the headings which relate

19

ACCEPTED MANUSCRIPT to the year they have been published. However, we need to first establish whether any of these works have been able to provide the correct solution to the steady-state approximation, Equation (3.47), before moving further. 5.1 The 1975 publication [30]

SC RI PT

This article is written for the purpose of formulating the growth of mercury droplets on an inert substrate to study the 𝐻𝑔22+ /𝐻𝑔 electrode reaction. The method used for the derivation of the required equation is based on the classical point-plane problem in electrostatics [46] and its application in electrochemistry [47]. The dimensionless current-

NU

time profile, 𝐼, resulting from the application of a potential step is given by Equation (15) of

𝑖𝑀𝛽 2

2

]

(5.1)

ED

𝐼 = 𝜋𝜌𝐹𝛾 = [

𝑡 1/2 −1 𝜏 1/4 𝑡 (1+ ) 𝜏

(1+ )

MA

Ref. [30], that is by

1

𝑡 𝑡 (1+ )−2(1+ )2 +1 𝜏 𝑡 1/2 (1+ ) 𝜏

𝑡 1/2

= [(1 + 𝜏)

CE

𝜏

𝑡 −1/2

− 2 + (1 + 𝜏)

AC

𝐼=

PT

Equation (5.1) can be written in the form

]

(5.2)

The dimensionless current given by Equation (5.2) has the same structural form as Equation 1

(3.47) derived for the steady-state scenario. However, using the definitions 𝛽, 𝛾 and 𝜏

provided by Equations (10), (11) and (16) of the Ref. [30] together with the Butler-Volmer equation −𝑧𝛼𝐹𝜂

𝑘 = 𝑘0 𝑒𝑥𝑝 (

𝑅𝑇

𝑧𝐹𝜂

) [1 − 𝑒𝑥𝑝 ( 𝑅𝑇 )]

(5.3)

20

ACCEPTED MANUSCRIPT lead to the definition of 𝜏 in terms of 𝑘, that is to 𝑧𝐹𝜂 )] 𝑅𝑇 𝑀𝑘 2 𝑐0 (1−2𝛼) 𝑐1 2𝛼

𝜌𝐷[1−𝑒𝑥𝑝(

𝜏=

(5.4)

𝑖𝑘𝑐1𝛼

𝐼=

(1+𝛼)

8 𝜋 𝑧𝐹𝐷 2 𝑐0

[1−𝑒𝑥𝑝(

SC RI PT

Substituting the definitions of 𝛽 and 𝛾 provided by Ref. [30] into Equation (5.1) leads to

𝑧𝐹𝜂 2 )] 𝑅𝑇

(5.5)

It is noted that, despite the similarity between Equation (5.2) and our equation derived for the steady-state approximation, Equation (3.47), neither 𝜏 nor 𝐼 defined in the above

NU

equations are the same as the definitions of 𝜏 and 𝐼 provided by us, Equation (3.48) and

5.2 The 1983 publication [37]

ED

MA

(3.47).

PT

This article [37] attempts to provide some useful and straightforward results using

CE

reasonable approximations to the task of describing the growth rates of a single hemispherical nucleus. A task which, in the author’s own word “has eluded researchers”,

AC

because “mass transport occurs across an expanding nucleus/electrolyte interface, and so the mathematical solution is non-trivial for many acceptable initial and boundary conditions.” However, the boundary condition which the author has imposed on the differential equation is 𝜕𝑐

𝐷 𝜕𝑟 (𝑎, 𝑡) = 𝑘𝑓 𝑐(𝑎, 𝑡) − 𝑘𝑏

𝑡>0

(5.6)

21

ACCEPTED MANUSCRIPT where 𝑘𝑓 (cm s−1 ) seems to represent the concentration-independent rate constant for the metal deposition and 𝑘𝑏 (mol cm−2 s −1 ) the concentration-dependent rate constant for the metal dissolution. This imposed boundary condition is, obviously, not the same as that assumed by us, Equation (3.1), written below for the ease of comparison 𝜕𝑐

𝐷 𝜕𝑟 (𝑎, 𝑡) = 𝑘𝑐(𝑎, 𝑡)

SC RI PT

𝑡≥0

(5.7)

Let us now proceed with probing further into the results obtained by Fletcher [37]. We note that the CT profile is given in an implicit form by Equation (52) and (53) of Ref. [37], which,

𝑧𝐹𝜂

2𝑀𝐷𝑐0

𝐷 𝑘𝑓

𝑎 = {(𝑘 ) +

𝜌

𝑓

𝑧𝐹𝜂

)

𝑡 [1 − 𝑒𝑥𝑝 ( 𝑅𝑇 )]}

ED

with

𝑎+

(5.8)

1/2

2

𝐷

𝑎2

MA

𝑖 = 2𝜋𝑧𝐹𝐷𝑐0 [1 − 𝑒𝑥𝑝 ( 𝑅𝑇 )] (

NU

when written in the form of our notations (𝜂 < 0), are, respectively, presented as

𝐷

−𝑘

𝑓

(5.9)

PT

assuming that 𝑎 = 0 when 𝑡 = 0. Equation (5.9) can be written as 1/2

2

− 𝐴′

CE

𝑎 = {𝐴′ + 𝐵 ′ 𝑡}

AC

where,

𝐷

{

𝐴′ = 𝑘 = 𝑓

𝐵′ =

2𝑀𝐷𝑐0 𝜌

Then, the term (

𝐷 𝑘0 𝑒𝑥𝑝(

−𝛼𝑧𝐹𝜂 ) 𝑅𝑇

𝑧𝐹𝜂

(5.11)

[1 − 𝑒𝑥𝑝 ( 𝑅𝑇 )]

𝑎2 𝑎+

(5.10)

𝐷 𝑘𝑓

) in Equation (5.8) can be written as

22

ACCEPTED MANUSCRIPT

(

2

𝑎2 𝐷 𝑎+ 𝑘𝑓

)=

[(𝐴′ +𝐵′ 𝑡)

1/2

2

(𝐴′ +𝐵′ 𝑡)

2

−𝐴′ ]

= 𝐴′ [

1/2

𝐵′ 𝑡

𝐵′ 𝑡

𝐴

𝐴 1/2

1/2

(1+ 2 )−2(1+ 2 ) ′ ′ 𝐵′ 𝑡

+1

]

(5.12)

(1+ 2 ) ′ 𝐴

Now the CT profile given by Equation (5.8) can be written in an explicit form provided that 𝐴′

2

𝐵′

. Thus,

𝑡 1/2

𝑧𝐹𝜂

𝑖 = 2𝜋𝑧𝐹𝐷𝑐0 𝐴′ [1 − 𝑒𝑥𝑝 ( 𝑅𝑇 )] [(1 + 𝜏′ )

SC RI PT

𝜏′ =

𝑡 −1/2

− 2 + (1 + 𝜏′ )

]

(5.13)

By substituting for 𝐴′ the definition provided by Equation (5.11), the CT profile described by

−𝛼𝑧𝐹𝜂 ) 𝑅𝑇 𝑧𝐹𝜂 2𝜋𝑧𝐹𝐷 2 𝑐0 [1−𝑒𝑥𝑝( )] 𝑅𝑇

𝑖𝑘0 𝑒𝑥𝑝(

𝑡 1/2

= [(1 + 𝜏′ )

𝑡 −1/2

− 2 + (1 + 𝜏′ )

]

(5.14)

MA

𝐼=

NU

Equation (5.13) can be written in an explicit dimensionless form

𝑖𝑘 2𝜋𝑧𝐹𝐷 2 𝑐0 [1−𝑒𝑥𝑝(

𝑧𝐹𝜂 )] 𝑅𝑇

2

𝑡 1/2

= [(1 + ′ ) 𝜏

𝑡 −1/2

− 2 + (1 + ′ ) 𝜏

]

(5.15)

PT

𝐼=

ED

Equation (5.14) together with Equation (5.3) leads to

CE

The dimensionless current, as defined by Equation (5.15), has the same structural form as Equation (15) of Ref. [30], which is Equation (5.2) of this article, as well as Equation (3.47)

AC

derived for the steady-state scenario. The mute question is whether 𝜏 ′ in Equation (5.14) is the same as 𝜏 defined by Equation (3.48). Substituting the parameters 𝐴′ , 𝐵 ′ and 𝑘𝑓 defined in (5.11), in 𝜏 ′ =

𝜏′ =

′2

𝐴

𝐵′

𝐴′

2

𝐵′

leads us to the answer to this question.

(

= 2𝑀𝐷𝑐0 𝜌

𝐷 ) 𝑘𝑓

2

[1−𝑒𝑥𝑝(

𝑧𝐹𝜂 )] 𝑅𝑇

=

𝑧𝐹𝜂 )] 𝑅𝑇 2 𝑘

𝜌𝐷[1−𝑒𝑥𝑝( 2𝑀𝑐0

(5.16)

23

ACCEPTED MANUSCRIPT However, we note that although Equation (5.16) is not the same as Equation (3.48) but it would be exactly the same as Equation (5.4), derived eight years earlier [30], provided that 𝛼 = 1/2 and 𝑐1 = 2𝑐0 . Using the same assumption, we note that Equation (5.15) turns out to be structurally the same as Equation (5.5), resulting from the work carried out previously

SC RI PT

in 1975 [30]. It is intriguing that despite these profound similarities, there are no references to the work carried out in 1975 [30] in Ref. [37]. Nonetheless, it is clearly demonstrated that the work carried out in 1983 [30] is not compatible with the present work, even for the

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steady-state approximation.

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5.3 The 1994 publication [38]

This article was written to derive equations for the deposition/dissolution of a single growth

ED

centre in both single- and double-potential step experiments. The flow of current resulting

PT

from the application of a single potential step was given in a dimensionless form by Equation (14) of Ref. [38], that is by

CE

1+𝑇1 −(1+2𝑇1 )1/2 ] (1+2𝑇1 )1/2

(5.17)

AC

𝐼=[

𝑡

where, 𝑇1 = 𝜏′ and 𝜏1′ , written in our notation, is defined by 1

𝜏1′ =

𝜌𝐷[1−𝑒𝑥𝑝(

𝑧𝐹𝜂 )] 𝑅𝑇

𝑀𝑘 2 𝑐0

(5.18)

Equation (5.17) and (5.18) is obviously different from our Equation (3.47) and (3.48).

24

ACCEPTED MANUSCRIPT 6. General discussion The novelty of this work lies primarily on the choice we had to make regarding the boundary condition, Equation (3.1). The significant role played by the boundary condition in acquiring a sensible result from a given partial differential equation is well known by those working in

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this field. Though this important notion has also been mentioned in a previous theoretical work [37] on the topic in hand, the author of the article had chosen a dubious boundary condition, Equation (42) of reference [37] or Equation (5.6) of this article, without any given justification whatsoever. It is because of the significance of the choice of this boundary

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condition that we had devoted almost the whole of the introduction to Section 3 to a discussion which ultimately lead to a sound boundary condition, Equation (3.1). It is

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therefore of no surprise that equations derived elsewhere [37] have been found to be

ED

erroneous in expressing the growth of a single centre. It is helpful to note that assumptions made in Equation (42) of Ref. [37] consist of two parts,

PT

the second part of which was presented by Equation (5.6) and was shown to be false. The

𝑖 𝑛𝐹𝐴

AC

Ref. [37].

CE

assumption made in the first part, is shown below to correct the probable typing error in

𝜕𝐶

= 𝐷 (𝜕𝑟 )

(6.1)

𝑟=𝑅

where, 𝑅 = 𝑎 and 𝑛 = 𝑧 in our notations. 𝐴, which is not defined in [37], is most probably the area of the electrode. The relation shown in Equation (6.1) is utterly wrong because it 𝜕𝐶

implies that the flux, the term 𝐷 ( 𝜕𝑟 )

𝑟=𝑅

, is directly related to the flow of current. The flux

25

ACCEPTED MANUSCRIPT is actually related to the rate of growth through Equation (2.1), while the flow of current is given by Equation (2.7). It may be assumed by the readers of this article that we may not have taken account of the back reaction (the rate of dissolution), because the assumption made by Fletcher, Equation

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(42) of Ref. [37], Equation (5.6) of this article, has both terms, 𝑘𝑓 and 𝑘𝑏 . It would therefore be useful to mention that the rate constant 𝑘 in our article indirectly takes care of both forward and back reaction through Butler-Volmer equation, Equation (5.3). It is instructive to bear in mind that processes that are solely controlled by diffusion cannot, rationally, be

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dependent on the overpotential. However, the flow of current at sufficiently long times,

𝑀 1/2

𝑧𝐹𝜂

3/2

{2𝐷𝑐0 [1 − 𝑒𝑥𝑝 ( 𝑅𝑇 )]}

𝑡1/2

(6.2)

ED

𝑖𝑡→∞ ≈ 𝜋𝑧𝐹 ( 𝜌 )

MA

according to 1983 [37] and 1994 paper [38], is predicted to be

We note that our exact formulation as well as our steady-state approximation, Equation

PT

(3.40), both lead for the diffusion-controlled growth of a single centre to potential-

CE

independent current profiles, Equation (2.8) and Equation (3.43), respectively. One of the most valuable contribution of this paper is the clarification of the range of rate

AC

constants for which the growth process will be controlled either by the rate of charge transfer across the interface or the rate of diffusion of depositing ions to the interface. Fig. 4 separates the exact region that the rate of growth is controlled only by the charge transfer across the interface, 𝐾𝐶, with the region 𝑀𝐶, in which a mixed kinetic-diffusion controlled growth is operative. Fortuitously, the curve (the full line in Fig. 5) which separates the two regions 𝐾𝐶 and 𝑀𝐶, follows an exponential curve defined roughly by 𝑘 = 0.01𝐸𝑥𝑝(−0.048𝑡). For rate constants, 𝑘 < 10−4 cm s−1 , the process is controlled

26

ACCEPTED MANUSCRIPT solely by the rate of charge transfer, independent of the size of the growth centre, whilst for 𝑘 > 10−4 cm s −1 but less that 10−2 cm s−1 the demarcation between 𝐾𝐶 and 𝑀𝐶 is dependent on the size of the growth centre or in other words by the passage of time. Fig.5 shows the three regions in which 𝑀𝐶, 𝐾𝐶 and 𝐷𝐶 (the region in which diffusion is the sole

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rate-determining step. The upper line in Fig. 5 is drawn for showing that with increasing rate constant the growth would be faster and the size limit which is imposed by the size of microelectrode is reached sooner. Therefore, the region above the upper line is the region where it is inconceivable to perform experiments on microelectrode. We have assumed this limit to be less than 10 μm. In fact, the upper line should be a curve with an inward

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curvature; the curve is represented in the form of a line for the sake of the neatness of the

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graph.

Recently a paper was published which had evaluated the growth of a single centre using

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finite element simulation [48]. It must be borne in mind that simulations are necessary

PT

when a problem cannot be tackled mathematically. Having derived exact formulation for the size of the growth centres, as well as for the flow of current to a single growth centre, all

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the information in that paper [48], and much more, can be retrieved, very effectively and

AC

with great precision, from the present article. For example, it is so easy to plot the behaviour of current as a function of concentration of the electrolyte and time by keeping all other variables constant and change the values of concentration only.

27

ACCEPTED MANUSCRIPT 7. Conclusions The dependence of the concentration of the depositing ions as a function of the radius of a single hemispherical centre, 𝑎, and as a function of time, 𝑡, has been formulated, Equation (3.21). This derived equation has been verified to form a solid base upon which the growth

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profile of the centre can be characterized. An equation for the rate of growth as a function of both 𝑎 and 𝑡, in the form of a first-order differential equation, Equation (3.30), for which a closed-form solution was not found, was provided. To obtain a closed-form result without implicating a numerical solution, the transient term of Equation (3.30) was ignored in favour

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of the steady-state term. Henceforth, a closed form solution was found for the steady-state approximation, Equation (3.40).

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The results of the previous theoretical papers [30,37,38] on the topic in hand were analysed in detail and it was shown that they were all inconsistent with our results, even when the

ED

results of the steady-state scenario have been considered. The demarcation between the

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three modes of rate-controlling mechanism has been delineated, Fig. 5. The exact formulation of the size of growth centre, as well as the flow of current to the growth centre,

CE

at any given time 𝑡, is given in the form of an interpolating function and compared with the

AC

results obtained from the steady-state approximation. It is found that the steady-state approximation offers a superb estimate when the rate constants are even as high as 𝑘 = 10−2 cm s −1 . At higher values of 𝑘 ≥ 10−1 cm s−1 , Fig. 2B and 2C, it is best to use the numerically obtained interpolating function.

28

ACCEPTED MANUSCRIPT

References 1. M. Fleischmann and H.R. Thirsk, in Advances in Pure and Applied Electrochemimistry, Vol. 3, p. 123, P. Delahay, Editor, Interscience, 1963. 2. M. Y. Abyaneh, Electrochimica Acta, 36 (1991) 727 (1991).

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3. A. Bewick, M. Fleischmann and H.R. Thirsk, Trans. Faraday Soc., 58, 2200 (1962). 4. R. D. Armstrong, Fleischmann and H.R. Thirsk, J. Electroanal. Chem., 11, 208 (1966). 5. M Y Abyaneh and M Fleischmann, J. Electroanal. Chem., 119, 187 (1981).

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6. M. Y. Abyaneh, Electrochimica Acta, 27, 1329 (1982). 7. M Y Abyaneh, J. Electroanal. Chem., 209, 1 (1986).

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8. M. Y. Abyaneh, Electrochimica Acta, 27, 1329 (1982). 9. M. Y. Abyaneh, J. Electroanal. Chem., 366, 191 (1994).

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10. M. Y. Abyaneh, J. Electroanal. Chem., 387, 29 (1995).

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11. M. Y. Abyaneh, J. Electroanal. Chem., 530, 108 (2002). 12. M. Y. Abyaneh, J. Electrochem. Soc., 151, C737 (2004).

CE

13. M. Y. Abyaneh, J. Electrochem. Soc., 151, C743 (2004).

AC

14. M. Y. Abyaneh, J. Electrochem. Soc., 151, C194 (2004). 15. M. Y. Abyaneh, J. Electrochem. Soc., 152, C776 (2005). 16. M. Y. Abyaneh, J. Electrochem. Soc., 154, D5 (2007). 17. M. Y. Abyaneh and M. Fleischmann, J. Electroanal. Chem., 119, 197 (1981). 18. M. Y. Abyaneh and M. Fleischmann, Trans. Inst. Met. Fin., 58, 9 (1980). 19. M. Y. Abyaneh, V. Saez, J. Gonzales-Garcia and T. J. Mason, Electrochimica Acta, 55, 3572 (2010). 20. B.R. Scharifker and J. Motsany, J. Electroanal. Chem., 177, 13 (1984).

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ACCEPTED MANUSCRIPT 21. M. Sluyters-Rehbach, J.H.O.J. Wijenberg E. Bosco and J.H. Sluyters, J. Electroanal. Chem., 236, 1 (1987). 22. M.V. Mirkin and A.P. Nilov, J. Electroanal. Chem., 283, 35 (1990). 23. V.A. Isaev and A.N. Baraboshkin, J. Electroanal. Chem., 377, 33 (1994). 24. P.C.T. D'Ajello, M.L. Munford and A.A. Pasa, J. Chem. Phys., 111, 4267 (1999).

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25. L. Heerman and A. Tarallo, J. Electroanal. Chem., 377, 33 (1994). 26. M. Y. Abyaneh and M. Fleischmann, J. Electrochem. Soc., 138, 2491 (1991). 27. Alexander Milchev and Luc Heerman, Electrochim. Acta, 48, 2903 (2003). 28. M. Y. Abyaneh, ‘Developments in Electrochemistry: Science Inspired by Martin Fleischmann’, Derek Pletcher et.al, Eds, Wiley, 59-64 (2014).

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29. D. Branco P., K. Saavendra, M. Palmer-Pardavé, C.Bonras, J. Mostany and B. R. Scharifker, J. Electroanal. Chem. 765, 140 (2016).

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30. P. Bindra, A.P. Brown, M. Fleischmann and D. Pletcher, Electroanalytical Chemistry and Interfacial Electrohemistry, 58, 31 (1975).

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31. P. Bindra, A.P. Brown, M. Fleischmann and D. Pletcher, Electroanalytical Chemistry and Interfacial Electrohemistry, 58, 39 (1975).

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32. M. Fleischmann, S. Pons, D.R. Rolison and P.P. Schmidt, Eds., Ultramicroelectrodes, Datatech Systems, Morganton, NC, 1987.

CE

33. E. Budevski, W. Bostanoff, W. Witanoff, Z. Stoinoff, Z. Kotzewa and R. Kaischev, Electrochim. Acta, 11, 1697 (1966).

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34. B. Scharifker and G.J. Hills, J. Electroanal. Chem., 130, 81 (1981). 35. G. Gunawardena G.J. Hills and B. Scharifker, J. Electroanal. Chem., 130, 99 (1981). 36. M. Fleischmann, L.J. Li and L.M. Peter, Electrochim. Acta, 34, 475 (1989). 37. Stephen Fletcher, J. Chem. Soc., Faraday Trans. I, 79, 467 (1983). 38. Martin Fleischmann, Stanley Pons, Joao Sousa and Jamal Ghoroghchian, J. Electroanal. Chem., 366, 171 (1994). 39. G. J. Hills, D. J. Schiffrin and J. Thompson, Electrochim. Acta, 19, 657 (1974).

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ACCEPTED MANUSCRIPT 40. Gamini Gunawardena, Graham Hills, Irene Montenegro and Benjamin Scharifker, J. Electroanal. Chem., 138, 225 (1982). 41. Benjamin Scharifker and Graham Hills, Electrochim. Acta, 28, 879 (1983). 42. S. Fletcher and D. B. Matheus, J. appl. Electrocchem. 11, 1 (1981). 43. M. Y. Abyaneh, J. Electroanal. Chem., 586, 196 (2006).

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44. J. Koryta, J. Dvorak and L. Kavan, `Principles of Electrochemistry`, John Wiley and Sons Ltd, p. 110. 45. Yuliy D. Gamburg, Giovanni Zangari, p. 116 –117, Theory and Practice of Metal Electrodeposition, Springer (2011). 46. W. Thompson, Mathematical and Physical papers, I, Cambridge University Press, London (1882).

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47. C. Kasper, Trans. Electrochemical Soc., 77, 353 (1940).

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48. M. H. Mamme, J. Deconinck and J. Ustarroz, Electrochimica Acta, 258, 662 (2017).

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Appendix

PT

Let us consider Equation (3.6) and remind ourselves of the definition of limit. The definition

(A1)

𝑢(𝑟,𝑡) 𝑟

= 𝑐0

AC

lim𝑟→∞

CE

states that

If, for every 𝜀 > 0, there exists a corresponding number 𝑀𝜀 > 0, such that for all 𝜀

−𝜀 <

𝑢(𝑟,𝑡) 𝑟

− 𝑐0 < 𝜀,

for every 𝑟 > 𝑀𝜀

(A2)

Multiplying the above inequalities by 𝑒 −𝑠𝑡 results in

31

ACCEPTED MANUSCRIPT −𝜀𝑒 −𝑠𝑡 <

𝑢(𝑟,𝑡) 𝑟

𝑒 −𝑠𝑡 − 𝑐0 𝑒 −𝑠𝑡 < 𝜀𝑒 −𝑠𝑡

for 𝑠 > 0

(A3)

It then follows that ∞ 𝑢(𝑟,𝑡)

∞

Thus, for 𝑠 > 1 and 𝑡 > 0 𝑈(𝑟,𝑠)

−𝜀 <

𝑟

−

𝑐0 𝑠

<𝜀

for every 𝑟 > 𝑀𝜀

𝑈(𝑟,𝑠)

lim

=

𝑟

𝑢(𝑟,𝑡) 𝑟

we have

𝑈(𝑟,𝑠) 𝑟

𝑐0

(A5)

(A6)

MA

𝑠

(A4)

. It then follows that

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Equation (A5) is similar to (A2) except for

𝑟→∞

∞

𝑒 −𝑠𝑡 𝑑𝑡 − 𝑐0 ∫0 𝑒 −𝑠𝑡 𝑑𝑡 < ∫0 𝜀𝑒 −𝑠𝑡 𝑑𝑡

𝑟

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∞

− ∫0 𝜀𝑒 −𝑠𝑡 𝑑𝑡 < ∫0

If we now divide Equation (3.9) by 𝑟 we get

=

𝑠

𝑈(𝑟,𝑠)

Thus, lim

𝑟

−√

𝑠 𝑟 𝐷

𝑟

𝑠

+

√ 𝑟 𝐵𝑒 𝐷

𝑐0

= lim ( 𝑠 + 𝑟→∞

(A7)

𝑟

𝐴𝑒

−√

𝑟

𝑠 𝑟 𝐷

𝑠

+

√ 𝑟 𝐵𝑒 𝐷

𝑟

𝑠

)=

𝑐0 𝑠

√ 𝑟 𝐵𝑒 𝐷

+ lim ( 𝑟→∞

𝑟

)

(A8)

CE

𝑟→∞

+

𝐴𝑒

ED

𝑟

𝑐0

PT

𝑈(𝑟,𝑠)

𝑐0 𝑠

AC

Equation (A6) and (A8) result in

=

𝑐0 𝑠

+ lim ( 𝑟→∞

𝑠

√ 𝑟 𝐵𝑒 𝐷

𝑟

)

(A9)

which can only be true if 𝐵 = 0.

32

ACCEPTED MANUSCRIPT 2

Fig. 1 Comparison of the plots of 𝑒 𝛽 𝑡 𝑒𝑟𝑓𝑐(𝛽√𝑡) and 𝛽

1 √𝜋𝑡

versus 𝑡 for 𝛽 = 3000 𝑠 −1/2 .

Fig. 2 Plots of 𝑎 versus 𝑡 for 4 different crystallisation rate constant, 𝑘, for values obtained from (a) the numerical solution to Equation (3.30), (b) the steady-state approximation, Equation (3.39), (c) the kinetically controlled growth, Equation (3.27) and (d) the diffusioncontrolled growth of a single centre, Equation (2.6).

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Fig. 3 Plots of 𝑖 versus 𝑡, for 2 different crystallisation rate constants, 𝑘, for values obtained from (a) the numerical solution, (b) the steady-state approximation, Equation (3.40), (c) the kinetically controlled growth, Equation (3.28) and (d) the diffusion-controlled growth of a single centre, Equation (2.8).

Fig. 4 Plot of the rate constant, 𝑘, against time, 𝑡, to establish the conditions at which the growth process changes from being controlled by charge-transfer to that controlled by a

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mixed kinetic-diffusion.

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Fig. 5 Plot of 𝑙𝑜𝑔𝑘 versus 𝑡, defining the regions which are being controlled by the pure charge transfer process, 𝐾𝐶, by a mixture of charge-transfer and diffusional processes, 𝑀𝐶,

AC

CE

PT

ED

and by diffusional processes alone, 𝐷𝐶.

33

ACCEPTED MANUSCRIPT Highlights In this paper a novel approach for deriving the current-time profile of the growth of a single hemispherical centre under a mixed kinetic-diffusion controlled mechanism is conceived. The novelty of the approach lies, first and foremost, on the indisputable assumptions made

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regarding the boundary conditions imposed on the partial differential equation associated with the spherical diffusion. As a result, the dependence of the concentration of the depositing ions at the surface of the microelectrode is formulated. Based on this formulation, we have been able to properly characterize the growth of a single centre which

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has been, hitherto, a longstanding problem. The precise portrayal of the results necessitated

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the exact solutions to be obtained numerically.

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Approximate closed-form solutions are derived for the rate of growth of a single centre, as

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well as for the flow of current to the growth centre, provided that the reactions are assumed to be in a steady-state. It is shown that none of the previous works are capable of

CE

providing correct solutions even for the steady-state approximation. The clarification of the

AC

range of rate constants for which the growth process is controlled (a) by the rate of charge transfer across the interface, (b) by a mix of charge transfer and diffusion, and (c) by only the rate of diffusion of the depositing ions to the interface, remains one of the most valuable contribution of this article.

34

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5