Elastic effects on heterogeneous nucleation

Heterogeneous nucleation and the growth of microstructures are often accompanied by elastic deformations which can significantly influence the thermodynamical properties and kinetics of these processes. Despite the obvious relevance of these phenomena, generally accepted theories which take into account the elastic long-range interactions in a consistent way are not yet available, especially under consideration of the atomic structures and ordering. In this project, we investigate the influence of these new effects on the dynamical processes of heterogeneous nucleation and microstructure development.

Motivation

Heterogeneous nucleation and the growth of microstructures are often accompanied by elastic deformations which can significantly influence the thermodynamical properties and kinetics of these processes. Lattice mismatches, density differences or concentration gradients are primary origins for these effects, which lead to the appearance of new lengthscales and noticeable consequences for the material properties. Despite the obvious relevance of these phenomena, generally accepted theories which take into account the elastic long-range interactions in a consistent way are not yet available, especially under consideration of the atomic structures and ordering. In this project, we investigate the influence of these new effects on the dynamical processes of heterogeneous nucleation and microstructure development. The general aim is to obtain generalizations of classical scaling relations by analytical and numerical methods, and to compare them to experimental results.

State of the art

Fig. 1: Sketch of the influence of elastic effects on heterogeneous nucleation.

Heterogeneous nucleation is a fascinating and challenging topic already for a long time. Experimental investigations are known to be difficult in metallic systems, and therefore the investigation of colloidal systems is a promising approach to observe these processes for obtaining a deeper understanding. In the framework of classical descriptions nucleation is understood as an energetic competition between a surface and a bulk term. Depending on interfacial energies, heterogeneous nucleation at walls or impurities is often favored in comparison to homogeneous nucleation. This can be understood from a macroscopic relation by Turnbull which links the energy barriers for homogeneous and heterogeneous nucleation by a catalytic potency factor which depends only on the contact angle between the substrate and the adsorbate. This itself is determined by the interfacial energies present at the triple junction. This relation is based on the assumption that the nucleus forms as a spherical cap on the surface, which neglects surface anisotropies. It has recently been verified for a nanoscopic model using an Ising lattice gas model, and it has been pointed out that line tension effects are important and lead to a modification of the Turnbull formula, and in particular negative line tensions have been found.

Newkirk and Turnbull generalized the expression by taking into account lattice mismatch effects, which lead to the appearance of an additional bulk energy term, which is quadratic in the strain mismatch (disregistry). The comparison to experimental results for nucleation of ammonium iodide from aqueous solutions on mica however does not lead to quantitative agreement and the elastic constants have to be assumed to be 1 or 2 orders of magnitude smaller then the theoretical expectation to obtain agreement between theory and experiment. This unsatisfactory finding might be attributed to the fact that the actual elastic problem is not solved but a homogeneous strain distribution is assumed. In reality, however, strong strain relaxation takes place by full elastic equilibration, since the cap of the nucleus is subjected to a hydrostatic stress only.

Furthermore, the central condition, the spherical shape of the cap, will most likely be violated by the presence of elastic effects, since they typically induce nonhydrostatic stress states inside the nucleus, and therefore the chemical potential at the top surface, which determines the shape of the cap, is no longer given by curvature effects only. This is also known from heteroepitaxy, where island formation (apart from faceting effects) happens in more complicated structures than spherical caps. This example already shows that the influence of elastic effects on heterogeneous nucleation is - despite their importance - not yet sufficiently understood.

In general, major reasons for the appearance of elastic effects are:

  • Lattice mismatches or structural dissimilarities between substrate and nucleus,
  • Heterogeneous nucleation on curved substrates, which leads to deformations of the growing phase,
  • Defects and grain boundaries,
  • Concentration and thermal gradients.

Recent results

Fig. 2: Steady-state pattern of the two-phase finger in eutectic alloys from a phase-field calculation.
  • Phase field models are essential for studying heterogeneous nucleation and microstructure evolution. Due to the presence of a substrate, at least three phases are relevant in the system, and typically multi-order parameter phase field models are therefore used for mesoscale simulations. A widely used one is based on the model by Steinbach and Pezzolla, and before in this model the formation of triple junctions was not clearly understood. In particular, in this model so called dynamical equilibria at triple junctions seemed to be possible, which would lead to a violation of of Young's law. This condition, however, is extremely important for the correct modeling of heterogeneous nucleation, since the contact angles crucially determine the energetic barriers. Here we have confirmed analytically that such dynamical states do not exist, and that an equilibrium solution therefore does indeed correspond to a minimum of the free energy; this implies that Young's law must be satisfied in the framework of the model. We have also shown that Young's law is a consequence of the interface kinetic equilibrium and not due to a mechanical force balance.
  • We compared different scenarios for dendritic melting of alloys with respect to the front propagation velocity. In contrast to conventional dendritic growth, selection can here be also due to the presence of a grain boundary or coherence strains, and the propagation speed is higher. The most favorable situation is partial melting, where two parabolic fronts, one melting and one solidifying interface, are moving together, since the process is then determined by diffusion in the thin liquid layer. There, and also in phase field simulations of melting in peritectic and eutectic systems, we observed a rotation of the triple junction relative to the growth direction. Finally, we discussed the role of elastic effects due to density and structural differences on solid-state phase transformations, and we found that they significantly alter the selection principles. In particular, we obtained free dendritic growth even with isotropic surface tension. This is investigated by Greens function methods and a phase field approach for growth in a channel and illustrated for the formation of a twin phase.
  • We have investigated the steady-state solidification in eutectic alloys, with a three-phase equilibrium. The scenario under consideration consisted of finger growth where the asymptotic shape of the solidification front is parabolic far behind the tip. Here, a two-phase finger situation was studied where one solid phase is present only in a central lamella, while the exterior part consists of another solid phase with Ivantsov asymptotics. This is due to the thermodynamic solid-solid equilibrium below the eutectic temperature. This pattern has been observed experimentally after the destabilization of a periodic planar front. We have developed a model which captures the essential physics; a central aspect is the relevance of two control parameters, i.e. the deviation from the eutectic temperature and the deviation from the eutectic concentration, which can lead to qualitatively very different patterns. The underlying numerical equations were solved with boundary integral techniques and phase field methods.

This project is part of the DFG Priority Program "Heterogeneous Nucleation and Microstructure Formation"

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