Computational Materials Science X

The aggregative growth and oriented attachment of nanocrystals and nanoparticles are reviewed, and they are contrasted to classical LaMer nucleation and growth, and to Ostwald ripening. Kinetic and mechanistic models are presented, and experiments directly observing aggregative growth and oriented attachment are summarized. Aggregative growth is described as a nonclassical nucleation and growth process. The concept of a nucleation function is introduced, and approximated with a Gaussian form. The height (Γmax) and width (Δtn) of the nucleation function are systematically varied by conditions that influence the colloidal stability of the small, primary nanocrystals participating in aggregative growth. The nucleation parameters Γmax and Δtn correlate with the final nanocrystal mean size and size distribution, affording a potential means of achieving nucleation control in nanocrystal synthesis.

Kinetics and Mechanisms of Aggregative Nanocrystal Growth

Abstract A new approach to the modeling of the equiaxed solidification of dendritic alloys is proposed. It is assumed that, in metallic alloys, microstructure formation is primarily controlled by solute diffusion (i.e. there is complete “thermal mixing” at the scale of one grain), and that the dendrite interface is an iso-concentrate at all times. The evolution of one dendritic grain is therefore modelled as follows: (i) complete mixing of solute within the interdendritic liquid; (ii) no back-diffusion in the solid; (iii) spherical solute diffusion in the liquid around the grain envelope; (iv) overall solute balance; (v) overall thermal balance; (vi) growth velocity, υg, of dendrite tips governed by the kinetic equation derived for the isolated dendrite case. By using an explicit finite difference scheme to solve these coupled equations, the concentration profiles, cooling curve, fraction of solid and evolution of dendritic grain envelope can be calculated. The intitial conditions used to start the calculation are provided by two parameters related to nucleation: the initial undercooling and the density of gains. The effect of nucleation and thermal conditions on equiaxed growth are studied. The theoretical predictions of recalescence and of the distribution of an interdendritic eutectic phase are in good agreement with experimental observations.

Solute diffusion model for equiaxed dendritic growth

In an ideal scenario, a phase field model is able to compute quantitative aspects of the evolution of microstructure without explicit intervention. The method is particularly appealing because it provides a visual impression of the development of structure, one which often matches observations. The essence of the technique is that phases and the interfaces between the phases are all incorporated into a grand functional for the free energy of a heterogeneous system, using an order parameter which can be translated into what is perceived as a phase or an interface in ordinary jargon. There are, however, assumptions which are inconsistent with practical experience and it is important to realise the limitations of the method. The purpose of this review is to introduce the essence of the method, and to describe, in the context of materials science, the advantages and pitfalls associated with the technique.

Phase field method

The thermal coarsening (180 °C) of decanethiolate-capped Au nanocrystals is studied at various tetraoctylammonium bromide concentrations. The coarsening kinetics are determined by measuring nanocrystal size distributions (CSDs) as a function of time. The results are shown to be consistent with aggregative nucleation and growth. For each kinetic trial, the time dependence of the aggregative nucleation rate is extracted from the early time CSDs and fitted by a Gaussian profile. The height of the profile is the maximum nucleation rate, Γmax, and the 2σ width is the time window for nucleation, Δtn. These nucleation parameters are shown to control the final mean size and size distribution of the coarsened nanocrystals. The coarsening kinetics are influenced by tetraoctylammonium bromide concentration because the nanocrystals are partially electrostatically stabilized.

Nucleation Control of Size and Dispersity in Aggregative Nanoparticle Growth. A Study of the Coarsening Kinetics of Thiolate-Capped Gold Nanocrystals

Many mathematical models of solidification use the following equation for the estimation of solid volume f S = 1 − exp ( − n 1 τ n 2 ) , where τ is time, n1 and n2 are constants. The fundamentals of the statistical theory of metal solidification were developed in classic papers written by Kolmogorov [A.N. Kolmogorov, Bull. Acad. Sci. USSR. Ser. Math., 3 (1937) 355–359 (in Russian)], Johnson and Mehl [W.A. Johnson, R.F. Mehl, Trans. Metall. Soc. AIME 135 (1939) 416–442], Avrami [M. Avrami, J. Chem. Phys. 7 (1939) 1103–1112; M. Avrami, J. Chem. Phys. 8 (1940) 212–224; M. Avrami, J. Chem. Phys. 9 (1941) 177–184]. The above equation in contemporary literature is named after these authors (K-J-M-A), but this equation is only a particular solution of Kolmogorov's general solution fS = 1−exp(−Ω), where the function Ω is the so-called total extended volume (per unit volume) of the growing grains, if their overlapping is neglected. Kolmogorov's general solution for the Ω-function is an accurate development only in the case of geometric similarity of the growing grains, their uniform distribution and equal orientation in space. Moreover, for an arbitrary moment and direction all grains must have the same absolute value of the growth velocity vector. If the conditions stated above are neglected, the above equations can give overestimated results. The source of this overestimation is the, so-called, screening effect. The presented solution takes into account the screening effect for the calculation of Ω-function. This solution expands the scope of the Kolmogorov's statistical theory of transformation. The proposed solution takes into consideration the growth of the grains of any shape, assuming their uniform distribution in space.

About Kolmogorov's statistical theory of phase transformation

Modelling was carried out to investigate the internal dendrite grains structure formation from a liquid two-component solution in the area adjacent to a mould wall. For the simulation, our own model and computer program based on CAFD (Cellular Automata Finite Differences) were used. In modelling, the effect of process conditions and material-related parameters, e.g. nucleation temperature, heat exchange rate, interfacial energy, crystal orientation with respect to the casting wall, etc. on the nature of the dendritic grain growth was examined. It was demonstrated that the profile of concentration field in a near-mould-wall zone impedes the growth of the solid phase in the direct vicinity of the wall. A local melting down of the grains of a solid phase due to the segregation of admixtures reducing the alloy point of liquidus is also possible.

Modelling of Dendritic Growth during Unidirectional Solidification by the Method of Cellular Automata

Austenitizing as a first stage of the heat treatment of castings to produce the cast iron grades like ADI, ACI, or AGI consists in holding of castings at a temperature co mprised within the range of 800-950 0 C to obtain an austenitic structure of the matrix as a point of departure for ausferritic structure. The temperature and time of austenitizing exert an important effect on the structure parameters and mechanical behaviour of mat erial obtained after the final treatment, which is austempering. A mathematical model of the process has been caharcterized, allowing for the temperature field in cas ting, the field of diffusion in the area of pearlite lamellae, and changes of diffusion coe fficient in function of temperature. A numerical program was developed by means of which the kinetics of austenite growth and cementite and ferrite fading in lamellar pearlite during austenitizing were determined, and a non-stationary field of carbon concentration in the examined system was computed. The temperature field in casting was also verified e xperimentally (a sample of 6x10x15 mm held in salt bath).

Computer simulation of the austenitizing process in cast iron with pearlitic matrix

Computer modeling of a temperature field and a solid phase fraction in casted billets is the base of any numerical simulation of the continuous casting technology. Temperature distribution in an ingot longitudinal and cross section for the same technological parameters is a function of solidification rate and rate of the solidification heat release. Nucleation rate and solid grain growth velocity depend on a melt undercooling below the liquidus temperature, and consequently depend on a temperature value. The results of the primary grain growth and temperature distribution modeling are presented for the square steel continuous casting 160×160 mm produced by CELSA Steel Works in Ostrowiec. For the modeling the ProCAST R ©

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Modeling of the Grain Structure Formation in the Steel Continuous Ingot by Cafe Method

A model of ausferrite growth in ADI was developed together with a simulation program. Their application enables tracing of the process of ausferrite growth, i.e. of the high-carbon ferrite in auste...

Wojciech Kapturkiewicz

Papers

About Kolmogorov's statistical theory of phase transformation

Modelling of Dendritic Growth during Unidirectional Solidification by the Method of Cellular Automata

Computer simulation of the austenitizing process in cast iron with pearlitic matrix

Modeling of the Grain Structure Formation in the Steel Continuous Ingot by Cafe Method

Modelling of ausferrite growth in ADI