Interface evolution in phase transformations ruled by nucleation and growth

https://doi.org/10.1016/j.physa.2020.124981Get rights and content

Highlights

  • The interface evolution in transformations by nucleation and growth is modeled.

  • The solution for the general case of spatially correlated nuclei is derived.

  • Closed form kinetics are given for homogeneous and heterogeneous nucleation.

  • The approach is suitable for describing experimental data on recrystallization and film deposition.

Abstract

An analytical model for the evolution of the boundary of the new phase in transformations ruled by nucleation and growth is presented. Both homogeneous and heterogeneous nucleation have been considered: The former includes transformations in 2D and 3D space and the latter nucleation and growth on flat solid substrate. The theory is formulated for the general case of spatially correlated nuclei, arbitrary nucleation rate and power growth law of nuclei. In the case of heterogeneous nucleation, spheroidal nuclei have been assumed and the dependence of the kinetics on contact angle investigated. The validity of the present approach is deemed through comparison with experimental data from literature which also comprise oxide growth by ALD (Atomic Layer Deposition) metal electrodeposition at solid substrate and alloy recrystallization.

Introduction

The synthesis and the design of solid systems with the desired interface morphology is a topic of great moment in Materials Science. This is due to the important role played by the interface whenever it is the locus, in a device, where chemical and physical processes take place. The term interfacial phenomena is intended here to include processes occurring at the solid–gas, solid–solid and solid–liquid interfaces. As a few examples, it is worth mentioning the adsorption in porous systems at the gas–solid interface, the 2D-interfaces in chemical sensors and the catalytic activity of supported metal catalysts where the chemical reaction occurs at the border between metal or oxide surfaces and gas phase [1], [2], [3], [4]. In gas sensors the interaction between gas molecules and materials mainly takes place on the surface, hence the number of surface atoms is crucial for determining the performance of the sensor [2].

In electrochemistry, the morphology of the liquid–solid interface is important for the electrodeposition of metals at electrodes via nucleation and growth. In fact, under potentiostatic conditions and in case of either interface or diffusion-controlled growths, the kinetics of metal deposition depends upon the evolution of the extension of the metal/liquid interface. It is the evolution of this interface that affects the shape of the cronoamperometric curve [5], [6], [7].

For the functionality of nano-composites, interfaces may play an important role for controlling the transport of ions, electrons and phonons, either in matrix or through the filler. For instance, in dye-synthesized solar cells, photogenerated positive charges are normally considered to be carried away from the dyes by a separate phase of hole transporting material (HTM). It has been shown through experiments that the regeneration yield, ascribed to the hole diffusion, is linked to the extension of the interface between the TiO2 and HTM [8]. Besides, during the growth of tin-based alloy films the nucleation and growth of voids leads to the formation of interfaces whose extension is significant in controlling the properties of solder joints of microelectronic devices [9]. Another field that is currently attracting attention from researchers is that of thin oxide layer growth by thermal Atomic Layer Deposition (ALD). This highly scalable method is promising for using III–V semiconductors as replacement of Si in MOSFET transistors. The oxide growth rate in ALD is linked to the extension of the surface area of the oxide, which exhibits a maximum before impingement among islands becomes significant [10].

The present article is devoted to study the evolution of the interface extension of the new phase in transformations occurring by nucleation and growth. An analytical model has been developed which comprises both cases of homogeneous and heterogeneous nucleation. With respect to previously developed modeling, which are limited to random nucleation and constant rate of nucleus growth [7], [11], [12] (and references therein), the present theory is formulated for the general case of spatially correlated nucleation, power growth law of nuclei and arbitrary nucleation rate.

The article is divided as follows: Section 2.1 is devoted to the definition of the probability functions that are needed in the stochastic approach developed in sect.2.2 for computing the kinetics of interface extension. For correlated nuclei these probabilities are given in terms of n-dots correlation functions. In Section 2.3 the approach is applied to the model case of random nucleation with spherical and ellipsoidal-cap nuclei, for homogeneous and heterogeneous nucleation, respectively. The last two sections are devoted to numerical results and application of the model to describe experimental data.

Section snippets

Definition of the probabilities

We consider phase transformations ruled by nucleation and growth of spherical or circular nuclei in a homogeneous system. The growth law is given by the function r(t,t), that is the radius of the nucleus,1 at time t, with t<t being the birth time of the nucleus. The nucleation rate of actual nuclei is indicated as Ia(t) and the “phantom-included” nucleation rate (compatible with correlation constraints) as I(t).

Conclusions

We developed a theoretical approach for the kinetics of interface extension in transformations ruled by nucleation and growth. The approach applies to the general case of correlated nucleation and requires the knowledge of the Pct|t conditional probability in terms of correlation functions. The random case was discussed in detail for both homogeneous and heterogeneous nucleation. In the former case an analytical solution is determined which is manageable for describing experimental data. The

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (33)

  • YamazoeN.

    Sensors Actuators B

    (1991)
  • SáezV. et al.

    Mater. Chem. Phys.

    (2011)
  • IsaevV.A. et al.

    Journal of Electroanalytical Chemistry

    (2018)
  • BoscoE. et al.

    J. Electroanal. Chem.

    (1982)
  • TomelliniM. et al.

    Physica A

    (2019)
  • Palomar-PardavéM. et al.

    Electrochem. Acta

    (2005)
  • TomelliniM. et al.

    Surf. Sci. Lett.

    (1996)
    TongW.S. et al.

    Acta Mater.

    (2000)
  • RiosP.R. et al.

    Mater. Res.

    (2006)
  • FarjasJ. et al.

    Acta Mater.

    (2006)
  • MehtaD.A. et al.

    J. Heat Treat.

    (1981)
  • Mendoza-HuizarL.H. et al.

    J. Mex. Chem. Soc.

    (2009)
  • DelgadoaD. et al.

    Catal. Today

    (2019)
  • ZhangJ. et al.

    Adv. Mater.

    (2016)
  • FujitaniT. et al.

    Catal. Lett.

    (1994)
  • KongD. et al.

    J. Electrochem. Soc.

    (2018)
  • MoiaD. et al.

    J. Phys. Chem. C

    (2015)
  • Cited by (0)

    View full text