PDAS across a single-crystal casting – why dendrites are not finest next to the chill ring

Primary dendrite arm spacing (PDAS) is one of the key characteristics of a single-crystal casting. It governs segregation, the size of interdendritic pores and eutectic, the homogenisation time in heat treatment and, in the end, the properties of the blade. It is usually treated as a function of two solidification parameters, the thermal gradient G and the velocity v: PDAS = A·v^–0.25·G^–0.5. The higher the gradient, the finer the dendrites.

Hence the familiar picture for a blade in a Bridgman furnace. The side facing the heater and the chill ring cools harder, its gradient is higher and its structure should be finest. The shadowed side facing the centre of the cluster sees less of the cold zone, and its structure should be coarsest. This is how the non-uniformity across an airfoil – and the defects tied to it – is usually explained.

Work by Dariusz Szeliga of Rzeszów University of Technology, Poland, on an industrial furnace shows that this logic does not always hold, and offers another explanation – through the shape and tilt of the solidification front.

How the trial was run

The furnace is an industrial Bridgman unit at Rzeszów University of Technology: two 300 mm heaters, 250 mm chill rings and a 200 mm chill plate. Melting and solidification were done under vacuum.

The wax cluster had a central downsprue with seven rods around it, 12 mm in diameter and 250 mm tall, each with a spiral grain selector and starter. The shell had nine coats: the first an alumina slurry with alumina stucco from a fluidised bed, the rest mullite of a coarser grade. The mould was heated to 1520 °C, filled with CMSX-4 at the same temperature and withdrawn through a ring baffle into the cooling zone at 3 or 6 mm/min.

From each trial a 180 mm rod was taken and cut across at nine levels from the base – 22, 41, 60, 79 mm and so on to the top. On every section PDAS was measured in four areas from the outer side (facing the heater) to the inner side (facing the downsprue). The mushy-zone shape, gradient and solidification velocity were computed in ProCAST; the tilt of the liquidus isotherm was found from the ratio of radial to axial gradient at the 1380 °C liquidus.

The trial

12 mmrod diameter, seven per cluster

1520 °Cmould and pouring temperature

3 and 6 mm/minwithdrawal rates

9 × 4sections along the height and areas across the width

What the sections showed

At 3 mm/min the structure changed monotonically across almost every section: PDAS was finest at the outer side and coarsest at the inner side, on sections two to nine. In the top, ninth section it was 300, 358, 433 and 443 µm from the outer side to the inner. That is the expected picture: the gradient is higher outside.

At 6 mm/min it was different. PDAS was usually coarsest at the inner or outer side and decreased toward the centre: in the ninth section, 390 µm outside, 355 and 350 µm in the middle and 429 µm at the inner side. The lowest section at 3 mm/min behaved the same way. Non-uniformity across the width was greatest in the upper part of the rod and at the transition from the selector to the casting; at 6 mm/min the largest PDAS difference across a section was smaller than at 3 mm/min.

What the simulation showed

The model explained the difference through the front shape. At 3 mm/min the liquidus isotherm is flat over most of the height and tilted toward the outer side of the casting. At the bottom, up to about 50 mm, it is still concave and then turns flat; the tilt is smallest in the middle of the rod, at 60–80 mm, and grows above that. At 6 mm/min the isotherm is concave over the whole height: symmetric in the middle, more curved and tilted at the top and bottom. The asymmetry comes from the uneven effect of the chill rings on the outer side of the mould and from the shadow on the downsprue side.

The axial gradient varies strongly across the width: always highest at the outer side, by the chill ring, lowest at the inner side, where shading is stronger. The solidification velocity is almost uniform across the width: the difference between outer and inner areas in the upper part is 0.078 mm/min at 3 mm/min and 0.84 mm/min at 6 mm/min. Both gradient and velocity change monotonically from the outer side to the inner – the model shows no surge in the centre that could explain the fine dendrites there at 6 mm/min.

Why the model missed it

PDAS computed in ProCAST from local G and v (with the constant A = 487.2 for CMSX-4) barely changes across the width: at 3 mm/min in the ninth section the model gave a difference of only 22 µm between the outer and inner areas, while the sections show about 140 µm. At 6 mm/min the model did not even reproduce the pattern – the minimum in the centre.

The author's explanation lies in the front tilt. As an inclined front moves, the part of the section where the front is lower solidifies last, and that part gets the finest PDAS, whichever way the front tilts; the part that solidifies first gets the coarsest. So with a flat inclined front the minimum and maximum end up on opposite sides, and with a concave front the centre solidifies last.

Quantitatively, the author ties this to the temperature difference across the front. The maximum temperature difference across the rod is about 7 °C for a flat inclined front and about 4.5 °C for a concave one, and the PDAS difference across the width grows linearly with it. The mechanism is convection in the melt ahead of the front: a flat inclined front sets up one large flow loop across the whole section, a concave front several small ones, with a smaller density difference and weaker flow.

Limits of the study

The author is candid about the weak point: the temperature profile shape comes from simulation. In an industrial furnace it can only be confirmed with thermocouples at several points on one level across the casting, and the paper has no such measurement.

A 12 mm rod is not a blade: an airfoil has a different thickness and shape, platforms and a core, and the flow pattern will differ. The trials cover one furnace, one alloy and two rates. But the conclusion itself – that structural non-uniformity across the width is set by the shape and tilt of the front, not only by local G and v – carries over directly to any casting in a Bridgman furnace.