# Heterogeneous Substrates Modify Non-Classical Nucleation Pathways: Reanalysis of Kinetic Data from the Electrodeposition of Mercury on Platinum Using Hierarchy of Sigmoid Growth Models

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## Abstract

**:**

_{max}and τ. The universal character of the studied phenomenon was revealed when replotting the original data as α ≡ N(t)/N

_{max}vs. t/τ. Yet the simplest model, the recently introduced α

_{21}model which is aimed to describe diffusion-limited growth in 2D, α

_{21}= tanh

^{2}(2t/τ

_{21}), fits all datasets with an R

^{2}≥ 0.989. This can be rationalized by attracting the non-classical notion of two-step nucleation—the nuclei form in a metastable phase which, in this case, grows on the electrode surface. Beyond the universality, we find the dependence of the two obtained scales on the overvoltage, which is increased systematically from 83 to 88 mV to generate the six N(t) datasets for each of the two electrode types—planar and hemispherical. Surprisingly, for one of them, the planar electrode, there is a discontinuity in the dependence—an almost horizontal jump from 85 to 86 mV, while for the hemispherical electrode, τ decreases smoothly.

## 1. Introduction

_{21}model, which is a particular solution of the general model of crystal growth α

_{Dg}in decaying supersaturation [6], shows a fit with high numerical precision of all 12 datasets from [1]. Further, we will show that the additional parameters from the more complex models do not show a clear dependence on the overpotential, the control parameter that is varied systematically in the experiment to produce the different datasets. Thus, more parameters do not bring more understanding of the phenomenon of heterogeneous nucleation in the electrodeposition of mercury on platinum and could even hide the true understanding of the phenomena whilst lacking a physical insight. Eventually, we conclude with the hypothesis that the numerically obtained behavior from the α

_{21}model is a clear manifestation of the non-classical two-step nucleation process [7]—the nuclei form uniformly in the growing two-dimensional metastable phase and its growth is described by the α

_{21}model.

## 2. Hierarchy of Models (HoM)

_{max}—the maximal number of the observed nuclei at the end of the process and the timescale of the phenomenon generally denoted by τ. The α

_{21}-model contains only these two parameters, which means that after using these two scales to re-scale the original data they should collapse onto a single or master curve defined by the mathematical expression of the model without any parameters in it. In the model of Johnson–Mehl–Avrami–Kolmogorov (JMAKn), an additional “tweaking” parameter is used—the so called Avrami exponent n to which the ratio of the time and the timescale is raised, denoted by τ

_{J}. The dimensionless equation of the Richards model contains two more parameters in addition to N

_{max}and τ

_{R}(which itself is composed by the fit parameters)—the additional power q and the dimensionless constant K, a combination of three of the model parameters but N

_{max}—the power q and the two particular timescales, the time to the inflection point t

_{i}

_{,}and the inverse of the kinetic coefficient τ

_{k}. Anticipating a main result from our study, it is not the number of parameters that improves the fit quality! In opposite, the use of the two more complex models, JMAKn and Richards, does not increase the numerical precision but results in values for the exponents n and q, respectively, that do not allow a consistent treatment. Thus, we can conclude that the two more complex models are “artificially good” [6] in describing the concrete experimental data and, even worse, the high numerical precision of their output can hide the physics behind the phenomenon observed.

#### 2.1. The α_{21} Model [6]

_{21}model is a particular case of the α

_{Dg}model, introduced recently in [6] to describe two-dimensional, D = 2, diffusion-limited, g = 1, crystal growth with exhaustion of the initial supersaturation. It is suited for the purposes of our present study as the following:

_{max}tanh

^{2}(2t/τ

_{21})

^{2}(2t/τ

_{21}), results from the integration of the following differential equation:

_{21}= (l/l

_{max})

^{2}where l is the side length of the growing in 2D square(s) and l

_{max}is the maximal size achievable with the initial supersaturation. Thus, in the right hand side of Equation (2) the positive feedback becomes 4l/l

_{max}, which is the perimeter of the growing square(s).

_{Dg}model is a further development of the idea behind the model of crystallization in three dimensions, describing the evolution of the rescaled size of the growing cube(s) [15] to obtain a differential equation for the size or, as in [6], for the transformation ratio α. In the context of this study, α could be defined as α ≡ N/N

_{max}.

_{i}, N

_{i}) closer to, but not on the line, N(t)/N

_{max}= t/τ

_{21}:

_{i}/N

_{max}= 1/3

_{i}= 0.329τ21

_{21}model is the model with the lowest number of parameters from the HoM, only N

_{max}and τ

_{21}, and we will use these scales first to rescale the number of nuclei N and the time t, a program that is already constructed in [8] to study protein nucleation. The rescaling operation of the N-axis can also be considered as another formulation of the so-called transformation ratio, defined for the purpose of the present study as α ≡ N/N

_{max}. Ideally, all the rescaled data should collapse onto the same master curve and a failure to conform to this expectation for some of the datasets should be considered a failure of the model to adequately describe this particular dataset. An important aspect in favor of the α

_{21}model is that it is lacking a “tuning exponent” as in the other two models from our hierarchy. Therefore, only the α

_{21}model naturally provides a master curve N/N

_{max}= tanh

^{2}(2t/τ

_{21}), while for the other two more complex models, JMAKn and Richards, the “master curves” are as many as there are different values obtained for the “tuning exponents”—n and q, correspondingly. In such situations, one could force the collapse by fixing the excessive parameter(s) to certain presumed value(s), but this has consequences. Conceptually, this means judgement that is outside the numerical procedure. Numerically, this would eventually lead to a decrease in the numerical precision, and we will see an example below.

_{max}found along the two axes t and N, respectively. A similar strategy was adopted in [16] to study the cloud condensation nuclei (CCN) as counted in 20 boxes by size at 6 different supersaturations. Then, to map them onto universal distributions after rescaling each of the 20 counts by the total number of CCN, in order to study further the peculiarities of the scale, using one for each value of the supersaturation the “total number of CCN” (corrected to concentration in this particular case [16]) and how it correlates with the meteo-elements.

_{21}is substituted with tanh

^{2}(2t/τ

_{21}).

_{Dg}model(s) could reveal the links between the three different parts of the model(s) for different combinations of the concrete, presumably integer values of D and g. Here, we should point out only that the value of the fully developed chaos r = 2.6, see Figure 1, is less than the coefficient of 4 in (2). Thus, one could integrate (2) with a general coefficient of r instead of 4 to obtain a family of curves, one for each value of r, in order to follow how the generalization reflects on the integral behavior of the model.

#### 2.2. The Johnson–Mehl–Avrami–Kolmogorov Model (JMAKn)

_{21}is also studied in detail in [6] and, in particular, it is found that the conversion factor between the two timescales is as follows:

_{21}

#### 2.3. Richards Model

_{k}, the inverse the kinetic coefficient k, is used. Thus, the positive feedback in Equation (13), when q > 1, is represented by kN/(q − 1), while the negative one is −k${N}^{q}$/[(q − 1)${N}_{\mathrm{max}}^{q-1}$]. When q < 1, the role of the two feedbacks is reverted. This is the value of q that fine-tunes the position of the inflection point of the model and, in [10], the exact dependence is found: ${N}_{i}\left({t}_{i}\right)={N}_{\mathrm{max}}{q}^{\frac{1}{1-q}}$. This points at the way of finding the model’s timescale τ

_{R}when using the second timescale in (12) and the time to the inflection point t

_{i}, which comes into the model only after finding the integration constant from the integration of Equation (13):

^{1/(q−1)}t

_{i}/t

_{k}and remains the only parameter in addition to q. This non-dimensionalization procedure brings the inflection point of the model to the line α ≡ N/N

_{max}= t/τ

_{R}, independent of K. Note that the inflection point is the same for a fixed value of q for any value of K, and that this provides the basis of a further study by considering the subtle connection between the integral curve and the corresponding chaotic map. In the “map-language”, K is usually denoted by r as in the two chaotic maps shown above. It is also worth mentioning here that the inflection points of the two previous models from the HoM are only close to the line α ≡ N/N

_{max}=t/τ

_{R}[6]. In addition, it is worth stressing that for the non-dimensionalization of the differential Equation (13), namely, the time-scale τ

_{R}, Equation (14), is to be used.

_{21}model. Thus, one can re-visit concrete situations where the Gomperz model was used but the α

_{21}model was possibly applicable as well, not only in the technical sense but also in terms of the physics behind [6].

_{c}= q

^{q}

^{/(q−1)}, for which the model curve still crosses the line N/N

_{max}= t/τ

_{R}once the value of K or r is the same as in the corresponding chaotic map showing the fully developed chaos. For K > K

_{c}, the integral curve of the model with a fixed value of q crosses the line N/N

_{max}= t/τ

_{R}three times already, with the midpoint of these three points being the inflection point.

_{Dg}model with concrete values of D = 1, 2, 3 and g = 1, 2 in order to bracket the studies in which the Richards model was used. However, one could also try the α

_{Dg}model to reanalyze the datasets used there. Anticipating such a study, when fitting α

_{21}with the Richards model one obtains q = 0.823, far below the logistic model with q = 2, and K = 3.842, well beyond the value of K

_{c}= 2.474, see Figure 3 (also for the slow decrease of R

^{2}apart from q = 0.823).

## 3. Results

_{21}model, while the other two models serve more to verify the results from the simplest one in the HoM. For them we give only specific details of general interest in the Appendix A such as the rescaled data according to the scales found and the values of n and q, respectively.

#### 3.1. Fitting with the α_{21} Model

#### 3.2. Summary of the Fitting Procedure

^{2}obtained do not allow for discrimination between the models in our hierarchy and, thus, physical considerations must come into the play.

## 4. Discussion and Conclusions

_{21}model of crystallization, which has an autocatalytic loop by definition.

_{max}and the time-scale τ—for a given (and fixed) driving force (overvoltage) of the phenomenon. Thus, it is tempting to study the obtained data in a quantitative manner. A major result of this study is the dependence of the obtained time-scales on the overvoltage, smoothly decreasing for one of the cathodes—the hemispherical one, and discontinuous for the planar electrode. The combination of the two behaviors resembles the so-called “cusp catastrophe” known to the general public from the isotherms drawn from the Van der Waals equation above and below T

_{c}.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{21}model. Hence, they are eliminated by the action of a thinking experiment, the so called “Ockham’s razor”. The referenced sources and equations are presented in the main text.

#### Appendix A.1. Fitting with the JMAKn Model

**Figure A1.**Rescaling the data for the planar electrode (Figure 5 in [1]) with the fit parameters from JMAKn, see also Table 1, Equation (7), the dotted line is $\alpha \equiv N/{N}_{\mathrm{max}}=t/{\tau}_{\mathrm{JMAK}}$. There is no master curve since n is different for each value of the overpotential.

**Figure A3.**Values of n from (7) used to fit data for the planar electrode (from Figure 5) in [1] with the JMAKn model.

**Figure A4.**Values of n from (7) used to fit data for the hemispherical (from Figure 6) [1] with the JMAKn model.

#### Appendix A.2. Fitting with the Richards Model

**Figure A5.**Rescaling the data for the planar electrode (Figure 5 in [1]) with the fit parameters from the Richards model, Equation (12). Since there is not a fixed single value of q (see the figure below), there is no master curve in the usual sense.

**Figure A6.**Rescaling the data for the hemispherical electrode (Figure 6 in [1]) with the fit parameters from the Richards model, Equation (12). Here, the failure of the data collapse is clearly seen, especially when compared to the fits with the previous models from the HoM.

**Figure A7.**Values of the “tuning” exponent q in the Richards model, Equation (12), as obtained from fitting Figure 5 in [1].

**Figure A8.**Values of the “tuning” exponent q in the Richards model, Equation (12), as obtained from fitting Figure 6 in [1].

_{21}, while in order to plot curves from the Richards model, one should chose values not only for q but also for K!

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**Figure 3.**Fitting a dataset of 30 points drawn from the α

_{21}-model with the Richards model. It is evident that R

^{2}cannot distinguish between the values of q taken in a large interval.

**Figure 4.**Fitting the experimental data for the planar electrode, digitized from Figure 5 from [1], with α

_{21}.

**Figure 5.**Fitting the data for hemispherical electrode, digitized from Figure 6 from [1], with α

_{21}.

**Figure 6.**Rescaling the data for the planar electrode in [1] with the fit parameters obtained from α

_{21}. The dotted line is N/N

_{max}= t/τ

_{21}and the solid curve is the “master curve” N/N

_{max}= tanh

^{2}(2t/τ

_{21}).

**Figure 7.**Rescaling the data for the hemispherical electrode in [1] with the fit parameters from α

_{21}. The dotted line is N/N

_{max}= t/τ

_{21}and the solid curve is N/N

_{max}= tanh

^{2}(2t/τ

_{21}).

**Figure 8.**Parameters from fitting the data for the planar electrode (from Figure 5 [1]) with α

_{21}. Circles are for N

_{max}and squares—for τ

_{21}. The slope of the solid line is ~−1, while the slope of the dotted line is ~−0.5.

**Figure 9.**Parameters from fitting the data for hemispherical electrode (from Figure 6 in [1]) with α

_{21}. Circles are for N

_{max}and squares—for τ

_{21}.

**Table 1.**Parameter values found from fitting Figure 5 from [1], that was obtained from the planar electrode, with the models from the HoM. Note that the timescale from the Richards model is not obtained directly from the fit, see Equation (14), and since no conclusions are drawn based on the values of t

_{i}, t

_{k}and K = q

^{1/(q−1)}t

_{i}/t

_{k}, they are not shown in the tables.

Over-Voltage, mV | α_{21} | JMAKn | Richards | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

N_{max} | τ_{21} | R^{2} | N_{max} | τ_{JMAK} | n | R^{2} | N_{max} | τ_{R} | q | R^{2} | |

83 | 30.43 | 5.22 | 0.9974 | 29.9 | 5.6 | 1.89 | 0.9977 | 30.29 | 5.20 | 0.83 | 0.9967 |

84 | 44.61 | 4.11 | 0.9937 | 45.78 | 4.75 | 1.39 | 0.9986 | 46.28 | 3.93 | 0.53 | 0.9981 |

85 | 56.94 | 3.19 | 0.9986 | 56.86 | 3.51 | 1.67 | 0.9980 | 57.36 | 3.28 | 0.64 | 0.9984 |

86 | 74.51 | 3.20 | 0.9976 | 75.06 | 3.61 | 1.58 | 0.9992 | 75.60 | 3.09 | 0.79 | 0.9988 |

87 | 92.7 | 2.62 | 0.9988 | 92.37 | 2.89 | 1.70 | 0.9995 | 93.17 | 2.52 | 0.96 | 0.9989 |

88 | 111.01 | 2.23 | 0.9974 | 109.19 | 2.40 | 1.86 | 0.9990 | 109.57 | 2.09 | 1.35 | 0.9983 |

**Table 2.**Parameter values found from fitting Figure 6 from [1] from the models of the hierarchy. Note that the timescale from the Richards model is not obtained directly from the fit, see Equation (14).

Over-Voltage, mV | α_{21} | JMAKn | Richards | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

n_{max} | τ_{21} | R^{2} | N_{max} | τ_{JMAK} | n | R^{2} | n_{max} | τ_{R} | q | R^{2} | |

83 | 24.41 | 7.27 | 0.9966 | 23.92 | 7.79 | 1.91 | 0.9978 | 24.12 | 6.95 | 1.18 | 0.9963 |

84 | 33.84 | 5.17 | 0.9977 | 33.68 | 5.69 | 1.72 | 0.9978 | 33.93 | 4.98 | 0.93 | 0.9971 |

85 | 48.69 | 4.38 | 0.9969 | 48.97 | 4.92 | 1.55 | 0.9985 | 48.19 | 4.62 | 0.43 | 0.9987 |

86 | 72.78 | 3.57 | 0.9890 | 74.50 | 4.13 | 1.38 | 0.9967 | 73.07 | 3.36 | 0.51 | 0.9948 |

87 | 103.77 | 2.82 | 0.9923 | 106.51 | 3.30 | 1.38 | 0.9993 | 103.81 | 2.68 | 0.54 | 0.9990 |

88 | 151.22 | 2.60 | 0.9891 | 156.27 | 3.03 | 1.35 | 0.9971 | 151.30 | 2.43 | 0.52 | 0.9958 |

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**MDPI and ACS Style**

Kleshtanova, V.; Ivanov, V.V.; Hodzhaoglu, F.; Prieto, J.E.; Tonchev, V.
Heterogeneous Substrates Modify Non-Classical Nucleation Pathways: Reanalysis of Kinetic Data from the Electrodeposition of Mercury on Platinum Using Hierarchy of Sigmoid Growth Models. *Crystals* **2023**, *13*, 1690.
https://doi.org/10.3390/cryst13121690

**AMA Style**

Kleshtanova V, Ivanov VV, Hodzhaoglu F, Prieto JE, Tonchev V.
Heterogeneous Substrates Modify Non-Classical Nucleation Pathways: Reanalysis of Kinetic Data from the Electrodeposition of Mercury on Platinum Using Hierarchy of Sigmoid Growth Models. *Crystals*. 2023; 13(12):1690.
https://doi.org/10.3390/cryst13121690

**Chicago/Turabian Style**

Kleshtanova, Viktoria, Vassil V. Ivanov, Feyzim Hodzhaoglu, Jose Emilio Prieto, and Vesselin Tonchev.
2023. "Heterogeneous Substrates Modify Non-Classical Nucleation Pathways: Reanalysis of Kinetic Data from the Electrodeposition of Mercury on Platinum Using Hierarchy of Sigmoid Growth Models" *Crystals* 13, no. 12: 1690.
https://doi.org/10.3390/cryst13121690