Normalised Enthalpy and the Keyhole Threshold

In 2014 a group at Lawrence Livermore — King, Barth, Castillo, Gallegos, Gibbs, Hahn, Kamath and Rubenchik — published in the Journal of Materials Processing Technology an observation of keyhole-mode melting in laser powder-bed fusion, together with a dimensionless group that predicts when it starts. That group is the normalised enthalpy, and it is what a reader who came to this site for volumetric energy density actually needed.

It is written $\Delta H/h_s = A P / \left(h_s \sqrt{\pi D v \sigma^{3}}\right)$. The numerator, $A P$, is the power actually absorbed. The denominator is the enthalpy the material needs to reach melting, $h_s = \rho c_p T_m$, multiplied by the volume rate that a moving heat source of radius $\sigma$ travelling at $v$ through a material of thermal diffusivity $D$ heats up. Divide one by the other and the units cancel exactly: a dimensionless statement of how many times over the beam is heating its own interaction volume to melting. The grouping comes from moving-heat-source conduction theory, not from an accountant's division, which is why it behaves like physics and $E_v$ does not.

The number to remember is roughly 30. Below about 6 there is often not enough energy to sustain a continuous track and the bead balls up. Between about 6 and 30 the melt pool is in conduction mode: shallow, roughly hemispherical, wider than it is deep, stable, and this is where good parameter sets live. Above about 30 the surface is pushed down by recoil pressure into a deep narrow vapour cavity — keyhole mode — which is much deeper than it is wide and which oscillates and collapses. Each collapse can pinch off a bubble of metal vapour that freezes in as a round pore, and keyhole porosity is invisible from the surface. King's threshold was established on 316L and matched against sectioned melt pools; treat it as a landmark rather than a constant of nature, because it moves with the material and with the powder layer.

The most instructive thing about this expression is where the beam radius sits. Power enters linearly. Speed enters under a square root. But the radius enters as $\sigma^{3}$ under that square root, which is $\sigma^{3/2}$ — so it is the strongest single lever in the whole group. Focus a beam from 50 µm radius down to 25 µm at unchanged power and the normalised enthalpy rises by a factor of $2^{3/2}$, nearly three, which can walk a comfortable process straight into keyholing without a single setting having been changed on the machine. This is exactly the kind of thing volumetric energy density cannot see, because it contains no spot size at all, and it is why spot size drift — a contaminated window, a defocused plane, a beam that grows toward the corners of a large plate — shows up as defects that nobody can trace to a parameter.

The awkward input is $A$. Absorptivity is not a material constant: it depends on alloy, on powder oxidation and roughness, on the angle of incidence, and on whether a keyhole has opened, which makes the relation mildly circular right at the threshold it is trying to predict. This is another reason to read 30 as a boundary region. The practical way to handle it is to calibrate: find the power at which your own sectioned tracks start showing keyhole depths, set $\Delta H/h_s = 30$, and read off the absorptivity that implies. That number belongs to your powder in your machine's atmosphere, which beats any handbook figure, and it is worth recording beside the powder lot number — a reused, oxidised powder absorbs quite differently from a virgin one.

One last note on $h_s$. It is the enthalpy per unit volume needed to bring the material to its melting point, $\rho c_p T_m$, and it is roughly 9.4 J/mm³ for 316L stainless, 6.0 for Ti-6Al-4V and 2.9 for AlSi10Mg. Note that it does not include the latent heat of fusion, which is a deliberate simplification in the original formulation and part of why the threshold is calibrated rather than derived.