Dendritic Solidification

H. K. D. H. Bhadeshia


A liquid when cooled solidifies. Alternatively, it may solidify when the pressure is decreased or increased, depending on the sign of the density change. Once nucleation has occurred, solidification proceeds by the movement of an interface. The process may generate heat if the enthalpy of the solid is less than that of the liquid. Similarly, solute may partition into the liquid if its solubility in the solid is less than that in the liquid.

Simulated dendrite

Computer simulated image of dendritic growth using a cellular automata technique. Notice the branching on the dendrites. Photograph courtesy of the Institute of Materials, based on the work of U. Dilthey, V. Pavlik and T. Reichel, Mathematical Modelling of Weld Phenomena III, eds H. Cerjak and H. Bhadeshia, Institute of Materials, 1997.

The accumulation of solute and heat ahead of the interface can lead to circumstances in which the liquid in front of the solidification front is supercooled. The interface thus becomes unstable and in appropriate circumstances solidification becomes dendritic. The mechanism of this instability is discussed elsewhere.

A dendrite tends to branch because the interface instability applies at all points along its growth front. The branching gives it a tree-like character which is the orgin of the term dendrite.

Computer simulated image of the dendritic solidification of pure nickel. The simulation is of "free growth", i.e., the solid is growing without contact with anything but the liquid. The degree of undercooling of the liquid in front of the interface is indicated by the adjacent scale. Photograph courtesy of the Institute of Materials, based on the work of U. Dilthey, V. Pavlik and T. Reichel, Mathematical Modelling of Weld Phenomena III, Institute of Materials, 1997.

Growth tends to occur along fast growth directions which are generally <100> for cubic metals.

Nickel dendrite

Solidification of Al-2Cu wt% Liquid

The following video shows how the concentration of copper (in an aluminium 2 wt% copper alloy) in the liquid phase changes as the dendrite solidifies. It is provided by courtesy of Andreas Schäfer, Chair of Computer Science, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany. It is produced in collaboration with colleagues at the Friedrich-Schiller-University Jena

Technological Consequences

Dendrites of Zinc

The following photographs show dendrites of zinc obtained by withdrawing the solid from a melt of impure zinc. The photographs are of samples collected by Professor Paul Howell, Pennsylvania State University. Dendrites
Dendrites of zinc rescued from partially solidified melt. Dendrites
Dendrites of zinc rescued from partially solidified melt.

Dendrites of Ice

When the weather outside is cold, moisture in a warm room can condense on the inner surface to form a thin film of moisture. If the temperature outside is sufficiently low, ice nucleates and grows. The region around the ice crystal becomes depleted in moisture. Moisture then has to arrive to the ice crystal by diffusion through the depleted zone, from the remaining moisture far from the interface. Suppose a small part of the ice crystal accidentally advances further then the rest of the interface. The diffusion distance for that perturbation decreases, and hence the perturbation grows faster. This leads to the formation of a branch, and a branching instability is said to have formed. This leads to the formation of ice dendrites as illustrated below. These pictures were taken at the Harbin Institute of Technology - the temperature outside can be below -20oC. The mechanism described here is essentially how snow-flakes are supposed to form, by the diffusion of water molecules through air on to the ice crystals. Snow-flakes have the dendritic morphology in three dimensions.

Ice dendrites on inner surface of cold window.
Ice dendrites.
Ice dendrites.
Ice dendrites.

Negative Dendrites

When a sheet of ice undergoes internal melting, dendrites of water form inside the ice. It is now the liquid which advances into the solid with an unstable interface. Furthermore, since ice has a lower density than water, a bubble forms inside each dendrite of the water.

Dendrites in Metallic Glass

The following transmission electron micrographs have kindly been provided by Andrew Fairbank with copyright clearance from the University of Wollongong. They show the early stages of dendrites of α-(Fe,Si) growing in the solid-state, from the amorphous Fe82Si4B14 metallic glass during annealing at 433 °C for 60 min.

Dendrites forming in Fe82Si4B14 metallic glass.
Dendrites forming in Fe82Si4B14 metallic glass.
Dendrites forming in Fe82Si4B14 metallic glass.
Dendrites forming in Fe82Si4B14 metallic glass.
Dendrites forming in Fe82Si4B14 metallic glass.

Interface Stability and Diffusion Bonding

Some materials cannot be welded by conventional techniques because the high temperatures involved would destroy their properties. For such materials, diffusion bonding is an attractive solution because it is a solid state joining technique, which is normally carried out at a temperature much lower than the melting point of the material.

Diffusion bonding is a candidate process for joining many aluminium based materials including a variety of artificial composites. Unfortunately, the method has been beset by difficulties, particularly that the bond line remains a plane of weakness. This is because the bond plane is a site for impurity segregation, where oxide particles may also be trapped. In addition, there can be problems in ensuring the continuity of the metallic bond.

Equipment used for temperature gradient diffusion bonding

Shirzadi and Wallach (Materials Science and Metallurgy, University of Cambridge) invented a disarmingly simple method of breaking up the planar bond into an unstable interface which develops into a three-dimensionally 'sinusoidal' or cellular surface. A small temperature gradient was applied at the bond, causing the interface instability. This concept is taught in many undergraduate courses but it took imagination and foresight on the part of Shirzadi and Wallach to apply it to transient liquid phase bonding. The method is incredibly successful, leading to a vast increase in bond strength, and has been granted a UK patent, No. 9709167.2, the Granjon Prize of the International Institute of Welding and the Cook-Ablett Award of the Institute of Materials.

Computer-generated Movies Showing the Evolution of Microstructure

Some discussion of the transformations illustrated in these movies can be found in Metals and Alloys lectures.

The following computer simulations of grain growth in two-dimensions have been provided by courtesy of V. Pavlik and U. Dilthey of the ISF-Welding Institute of Aachen University in Germany. The solidification simulations use a technique called "cellular automata" in combination with finite difference methods. Cellular automata allow non-trivial processes and patterns to be computed starting with simple deterministic rules. Solute concentration contours in the parent phase are represented by colours. Look out for the development of secondary dendrite arms, coarsening of these arms, the development of solute segregation due to non-equilibrium solidification. Note how the microstructure changes radically as a function of the solidification parameters (velocity and temperature gradient)

Dendritic solidification in Fe-0.11 wt% C velocity 10 mm/s, temperature gradient 100 K/mm. Have a careful look at the development of the secondary dendrite arms. The initial spacing between the secondary dendrite arms is much finer than in the final microstructure. This is because of coarsening - some of the finer arms dissolve as the coarser ones grow. The later stages also show the coalescence of the dedrite arms (both primary and secondary).

"Seaweed" solidification structure in Fe-0.11 wt% C. velocity 10 mm/s, temperature gradient 300 K/mm

Solidification to a dendritic microstructure where the primary phase to solidify is delta-ferrite, in Fe-0.15 wt% C. velocity 0.1 mm/s, temperature gradient 15 K/mm. The dendrites in this simulation are constrained to grow along particular crystallographic directions determined at the nucleation stage. This is why they grow at an angle even though the temperature gradient is vertical. The simulation illustrates selection during growth, i.e., dendrites which are better oriented to the temperature gradient overgrow those which are not well oriented. This movie has been provided by courtesy of Janin Taiden of Access in Germany.

The following movie is a simulation of eutectic solidification in an Al--Si system. It has been provided by courtesy of Britta Nestler of RWTH in Aachen, Germany. The simulation is based on a technique known as "phase field" modelling. In this method, the boundary is treated as a continuous transition between adjacent grains across a thin layer of finite thickness. The value of a phase-field variable then identifies the location of the boundary and of each grain. The advantage of this method is that the boundary becomes a part of the system so that it does not have to be determined explicitly in the solution. Notice how the eutectic spacing changes as solidification proceeds, and the nature of the solute diffusion field at the solidification front. The diffusion distance is not very large, about equal to the spacing of the lamellae.

The simulations presented here have been provided for teaching purposes via the good offices of Dr Vitali Pavlik of Aachen University in Germany.


Battery Fires

Professor Clare Grey and her team at Cambridge University have concluded that metal fibres in the form of dendrites grow within lithium batteries which are charged rapidly. These cause short circuits, overheating and in some cases, fires.

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