# On the influence of alloy composition on the additive manufacturability of
Ni-based superalloys

Joseph N. Ghoussoub, Yuanbo T. Tang  William J B. Dick-Cleland  André A.N.
Németh  Yilun Gong  D. Graham McCartney  Alan C.F. Cocks  Roger C. Reed
Department of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH, UK
Alloyed Ltd. Yarnton, Kidlington OX5 1QU, UK  Department of Engineering
Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK

###### Abstract

The susceptibility of nickel-based superalloys to processing-induced crack
formation during laser powder-bed additive manufacturing is studied. Twelve
different alloys – some of existing (heritage) type but also other newly-
designed ones – are considered. A strong inter-dependence of alloy composition
and processability is demonstrated. Stereological procedures are developed to
enable the two dominant defect types found – solidification cracks and solid-
state ductility dip cracks – to be distinguished and quantified. Differential
scanning calorimetry, creep stress relaxation tests at 1000 ∘C and
measurements of tensile ductility at 800 ∘C are used to interpret the effects
of alloy composition. A model for solid-state cracking is proposed, based on
an incapacity to relax the thermal stress arising from constrained
differential thermal contraction; its development is supported by experimental
measurements using a constrained bar cooling test. A modified solidification
cracking criterion is proposed based upon solidification range but including
also a contribution from the stress relaxation effect. This work provides
fundamental insights into the role of composition on the additive
manufacturability of these materials.

###### keywords:

Additive Manufacturing , Ni-based superalloys , Cracking mechanisms , Laser-
Powder Bed Fusion , Alloy design

††journal: Springer

##  1 Introduction

Additive manufacturing (AM) is a new process, which is particularly promising
for metals and alloys [1]. It opens up the possibility of multifunctional
component design at significant geometrical precision, with close integration
to CAD-based systems to enable true digital manufacturing. But like any
process, it must rely upon the material feedstock for its success, and therein
lies a dilemma. Can existing metallic alloys be made to process well under the
extreme cooling rates and stress-states induced [2, 3]? Indeed, much current
research emphasizing process optimization assumes that this will be the
case.Or alternatively, will new grades of alloy be necessary with novel
compositions and behavioral characteristics designed specifically for the
process [4]?

Here, the nickel-based superalloys are considered. They are best known for
their applications in aeronautics, hypersonics, rocketry and high temperature
technologies more broadly [5]. They are an excellent test-case for the
additive manufacturing process, for several reasons. First, their
compositional complexity requires as many as 15 different alloying elements to
be present; this broadens the freezing range and thus exacerbates
solidification-related cracking whereby the remaining liquid fails to fill the
partially solidified microstructure which is then pulled apart by shrinkage
effects [6], see Figure 1. Recent work on solidification cracking in AM has
revealed that grain boundary misorientation is critical to susceptibility; hot
cracks form only on high angle grain boundaries (HAGB) [7] and that grain
boundary segregation and ensuing liquid film stability influence crack
formation [8, 9]. Second, the superalloys are prone to solid-state cracking
due to a lack of ductility particularly in the so-called ductility dip regime
of 700 to 900 ∘C. In the literature, the terminology used for this phenomenon
is confusing, being referred to variously as ductility dip cracking, strain-
age cracking and post weld heat treatment cracking. Here, we have chosen to
use the all-encompassing term solid-state cracking to reflect the fact that –
in the AM process – such cracks propagate when the material is in the solid-
state [10]. Solid-state cracks are characterized by their morphologies which
are long and sharp, in contrast to the more meandering nature of
solidification cracks.Work in recent years has successfully mitigated such
crack formation by increased solid solution strengthening [11], or by control
of gamma prime ($\gamma^{\prime}$) content to limit the strength induced and
improve ductility [12]. And third, the multifunctional nature of their
applications necessitates thin-walled and porous structures, to promote for
example lightweighting and heat transfer.

The work reported in this paper was motivated with the above in mind. Here, we
have avoided the temptation to fix on one alloy system, and then conduct
processing trials exhaustively in an effort to work out the best combination
of key process variables (KPVs) to optimise manufacturing. Instead, consistent
with the ushering in of new processing methods for the superalloys — vacuum
melting, powder metallurgy, Bridgeman single crystal processing for example —
which have always led to new compositional grades, we have taken an
alternative approach: work out which alloys perform best for the process, and
why. This has necessitated experimentation on a range of different alloys,
including some designed specifically for additive manufacturing.

##  2 Experimental Methods

###  2.1 Choice of alloy compositions

Twelve different superalloy compositions have been studied in this work – five
novel compositions and seven better-known so-called heritage grades, including
the cast/wrought alloys IN625, IN718, and Waspaloy of moderate
$\gamma^{\prime}$ content but then also IN713, IN738LC, IN939 and CM247LC
which are widely employed for investment casting applications – these display
excellent creep properties on account of their high $\gamma^{\prime}$
fraction. The new alloys termed ABD850AM, ABD900AM, and AM-Dev are new alloys
designed specifically for the AM process, with the objective of maintaining
both performance and processability. Lastly, ExpAM and ABD850AM+CB are
experimental compositions, with low and high carbon and boron contents
respectively, aimed at assessing the influence of C and B, as well as a very
high $\gamma^{\prime}$ volume fraction in the case of ExpAM. Feedstock powders
were produced via argon gas atomization; their particle size distributions
each had D10 and D90 values ranging from 19.2 to 56.1 $\upmu$m respectively
and D50 between 32.3 and 36.5 $\upmu$m. The measured alloy compositions as
determined by inductively coupled plasma-optical emission spectroscopy (ICP-
OES) and ICP-combustion analysis are given in Table 1.

Table 1: Measured alloy powder compositions (wt. %). Alloy | Ni | Al | Co | Cr | Fe | Mo | Nb | Ta | Ti | W | C | B | Si | Hf
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
ABD850AM | Bal | 1.3 | 18.0 | 19.9 | - | 2.02 | 0.64 | 0.48 | 2.25 | 4.85 | 0.010 | 0.003 | - | -
ABD850AM+CB | Bal | 1.5 | 18.6 | 19.7 | - | 2.03 | 0.37 | 0.57 | 2.42 | 5.05 | 0.133 | 0.007 | - | -
ABD900AM | Bal | 2.1 | 20.3 | 17.1 | - | 2.09 | 1.85 | 1.21 | 2.39 | 3.06 | 0.047 | 0.003 | - | -
AM-Dev | Bal | 4.2 | 19.1 | 8.7 | - | 1.20 | 3.60 | 5.50 | 0.12 | 7.10 | 0.030 | 0.005 | - | -
CM247LC | Bal | 5.5 | 9.3 | 8.3 | - | 0.55 | - | 3.20 | 0.73 | 9.66 | 0.070 | 0.020 | - | 1.4
ExpAM | Bal | 5.6 | 7.6 | 11.7 | - | 2.00 | 1.10 | - | 2.00 | 3.70 | 0.004 | <0.001 | - | -
IN625 | Bal | 0.1 | - | 19.9 | 3.80 | 8.81 | 3.65 | - | 0.17 | - | 0.049 | <0.001 | - | -
IN713 | Bal | 6.0 | - | 12.0 | 0.20 | 4.30 | 2.10 | - | 0.70 | - | 0.060 | 0.010 | - | -
IN718 | Bal | 0.6 | 1.1 | 19.2 | 17.2 | 3.20 | 4.90 | - | 0.80 | - | 0.080 | 0.001 | 0.35 | -
IN738LC | Bal | 3.3 | 8.5 | 15.7 | 0.02 | 1.73 | 0.87 | 1.69 | 3.29 | 2.71 | 0.110 | 0.011 | 0.03 | -
IN939 | Bal | 1.8 | 18.8 | 22.5 | - | 0.00 | 1.00 | 1.30 | 3.60 | 1.60 | 0.160 | 0.011 | - | -
Waspaloy | Bal | 1.3 | 12.0 | 20.0 | 0.04 | 4.37 | 0.00 | - | 3.00 | - | 0.050 | 0.008 | 0.07 | -

###  2.2 Additive manufacturing

Processing was carried out by the laser powder bed fusion (L-PBF) method in an
argon atmosphere using a Renishaw AM 400 pulsed fiber laser system of
wavelength 1075 nm and build plate size of 250$\times$250 mm2. The processing
parameters employed were: laser power 200 W, laser spot size diameter 70
$\upmu$m, powder layer thickness 30 $\upmu$m, and pulse exposure time 60
$\upmu$s. A ‘meander’ laser scan path pattern was used with hatch spacing of
70 $\upmu$m and laser speed of $0.875$ m/s, the path frame of reference was
rotated by 67 ∘ with each layer [13]. Consistent with Renishaw’s recommended
practice, the speed on the borders was reduced to $0.5$ m/s. To assess the
composition-dependence of the severity of cracking, cubes of dimensions
10$\times$10$\times$10 mm3 were printed. Vertical bars of dimensions
10$\times$10$\times$52 mm3 were also printed to assess the mechanical
properties of the alloy variants. These were manufactured with 16 inverted
pyramid legs to allow for easy removal from the baseplate. Cylinders of
diameter 3 mm and height 1 mm were produced for differential scanning
calorimetry (DSC). Each alloy and geometry was processed with identical
parameters, as determined to be optimal for minimizing cracking in this
manufacturing system [14].

###  2.3 Assessment of additive manufacturability

The printability was assessed by measurement of cracking severity and
morphology. First, extensive quantitative stereological assessment was made
with optical micrographs taken using a Zeiss Axio A1 optical microscope at
100$\times$ magnification. For this purpose, the cube samples were sectioned
on (i) planes with normal corresponding to the build direction (termed XY
plane) at mid-height and (ii) planes with normal perpendicular to the build
direction (termed XZ). The severity of cracking was determined by imaging the
XY plane at the sample height midpoint and on the XZ plane through the center;
5 images were considered per alloy. In this analysis the bulk of the sample –
400 $\upmu$m from the sample edge – was considered. The ImageJ software was
used to compare the alloys by determining the crack count density (cracks/mm2)
and crack length density (mm/mm2) – note that the crack length was defined as
the caliper diameter corresponding to the largest line length across each
crack.

In the case of a pulsed laser, solidification occurs in discrete melting
events; it follows that a solidification crack must be of length equal to or
smaller than a single melt pool radius. Conversely, any crack of length
greater than one melt pool radius must have propagated as a solid-state crack,
although of course one cannot eliminate the possibility that the solid-state
crack propagated from a solidification crack. In order to provide quantitative
information, the perimeter to area ratio (units $\upmu$m-1) of cracks was
assessed. Images were taken at a magnification of 100$\times$, giving a
resolution high enough to identify the jagged features of solidification
cracks. A further 70 images were taken of each alloy to compare the
contributions of each mechanism on a representative scale; where cracking
occurred, many thousands of cracks were analyzed.

To garner further information concerning cracking mechanisms at higher spatial
resolution, a Zeiss Merlin Gemini 2 field emission gun scanning electron
microscope (FEG-SEM) was used equipped with an Oxford Instruments XMax
150mm/mm2 energy-dispersive X-ray spectroscopy (EDX) detector. Line scans were
performed to determine qualitatively the composition of carbides observed in
the microstructure. Scans comprised 500 points at an accelerating voltage of 5
kV and probe current 500 pA resulting in an estimated interaction width and
depth of 100 nm. The microstructure of the alloys after manufacturing always
comprised $\gamma$ and MC carbide with $\gamma^{\prime}$ absent; an example of
the CM247LC as-printed microstructure compared to the heat-treated
microstructure following electrolytic etching with 10% phosphoric acid at 3 V
direct current is given in Figure 1. In what follows, it is emphasized that
only material in the as-printed condition is considered.

Figure 1: (a) Schematic diagram showing the L-PBF build process, whereby a
laser melts powder in consecutive layers, (b) the material state and length
scale at which solidification and solid-state cracking occur with
corresponding SEM image examples of a (c) solidification crack and (d) a
solid-state crack in IN939 as well as (e) SEM micrographs of etched CM247LC in
the as-printed condition showing the $\gamma$ matrix and MC carbide and
CM247LC in the heat treated condition (reproduced from [4]) with
$\gamma^{\prime}$ precipitates.

###  2.4 Assessment of mechanical and stress relaxation behaviour

Uniaxial tensile, isothermal stress relaxation and constrained cooling tests
were performed with an Instron electro-thermal mechanical testing (ETMT)
machine. Specimens were machined with axis along the build direction using
electrical discharge machining (EDM) with dimensions as given in Figure 2. To
negate the influence of any machining-induced surface roughness, all surfaces
were polished to a 4000 grit finish by hand. For each of the three types of
test, joule heating under free expansion conditions ($\sigma=0$ MPa) was
employed to reach the test temperature using a heating rate of 200 K/s, thus
avoiding any second phase precipitation. Since it was needed for the
interpretation of our results, an estimate of the average linear coefficient
of thermal expansion between 300 and 800 ∘C was made during this pre-test
sample heating as shown in Figure 3. The coefficient of thermal expansion is
found to have a dependence on the Al content, as previously reported in [15].

Figure 2: A schematic illustration of temperature, strain and stress profile
as a function of time during the stress relaxation and constrained bar cooling
test and the sample geometry used.

Isothermal uniaxial tensile tests were performed for all the alloys at room
temperature and at 800 ∘C – this temperature being taken to be representative
of where the ductility dip is likely to be severe [4]. Specimens were strained
rapidly at a rate of $10^{-2}$ s-1 to reduce the influence of any dynamic
precipitation during testing. An iMetrum non-contact digital image correlation
system was employed for measuring the strain; 3 repeats were performed for
each alloy. The elastic modulus of each alloy was determined between 150-300
MPa. The modulus determined at 800 ∘C correlated with increasing equilibrium
$\gamma^{\prime}$ volume fraction, see Figure 3, though was likely due to the
contributions of $\gamma^{\prime}$ formers in solid solution of $\gamma$.

Stress relaxation tests were performed to quantify the capacity of the alloys
to relieve the stress accumulated by processing. Tensile specimens were heated
to 1000 ∘C allowing for free expansion to occur as shown in Figure 2, followed
by a loading to 100 MPa at 25 MPa/s after which the sample grips were
instantaneously locked into position, in order to observe the relaxation of
the stress during cooling [16]. The ability to stress relax was benchmarked by
comparing the stress remaining after 60 s. Further stress relaxation tests at
various supersolvus temperatures (1100-1200 ∘C) were performed on the
ABD850AM+CB, Waspaloy, and IN939 alloys by the same procedure.

A constrained bar cooling test was performed to simulate the stress build up
expected of additive manufacturing. Tensile specimens were heated under free
expansion conditions to 1100 ∘C then left for 20 s to ensure thermal-
mechanical equilibrium. The displacement of the ETMT grips was then fixed, and
the sample was cooled to 300 ∘C. The accumulated stress during cooling was
measured for the various alloys at cooling rates of 100 K/s, 80 K/s, 50 K/s,
and 25 K/s.

Figure 3: (a) The average coefficient of thermal expansion $\alpha$ measured
during pre-test heating under free expansion conditions on the ETMT vs the Al
content (wt.%) and (b) the average elastic modulus determined at 800 ∘C vs the
equilibrium $\gamma^{\prime}$ volume fraction at 800 ∘C fraction acquired by
Thermo-Calc (TTNi8 database). Error bars represent the minimum and maximum
values measured.

###  2.5 Assessment of solidification range and phase transformation
temperatures

A NETZSCH 404 F1 Pegasus DSC was used to compare solidification behaviour of
the as-processed microstructure, albeit at cooling rates far lower than those
experienced during AM processing. The alloys were subjected to a thermal
history consisting of (i) heating to $700\,$∘C at 20 ∘C/min, (ii) heating from
700 ∘C to 1450 ∘C at 10 ∘C/min (iii) cooling to 700 ∘C at the same rate, and
(iv) cooling to room temperature at 20 ∘C/min. The DSC signal from the first
heat causes the transformation from a non-equilibrium additively manufactured
microstructure to a homogeneous liquid. The first cool was indicative not of
AM conditions, but of the solidification path from homogeneous liquid to solid
at 10 ∘C/min. The thermophysical properties of interest derived from the DSC
signal upon heating were the onset of incipient melting
$T_{\mathrm{S^{\prime}}}$, and the liquidus $T_{\mathrm{L}}$, as illustrated
in Figure 16a. From cooling the onset of solidification
$T_{\mathrm{L^{\prime}}}$ and the solidus temperature $T_{\mathrm{S}}$ were
extracted as shown in Figure 16b. The solidus and onset of incipient melting
as well as liquidus and onset of solid nucleation temperatures were estimated
following the procedures of Boettinger, Chapman, and Quested [17, 18, 19]. In
order to evaluate legacy crack criteria on the basis of an experimentally
determined solidification path, the solid fraction as a function of
temperature was estimated by considering it proportional to the ratio of the
enthalpy of the transformation at $T$ to the total enthalpy of the phase
transformation. This estimate is subject to possible error because other phase
transformations may have occurred throughout the freezing range.

Modeling of solidification under equilibrium and Scheil-Gulliver assumptions
was implemented using Thermo-Calc employing the TCNI8 and TTNi8 databases to
simulate the alloy solidification path [20]. In the case of the Scheil models,
solidification was considered to have ended at a liquid fraction remaining of
0.01. The propensity for backdiffusion of C to occur was deemed negligible in
accordance with the dimensionless back diffusion parameter $\alpha$ [21].

$\alpha=\frac{4D_{\mathrm{C}}}{\lambda_{\mathrm{cell}}^{2}}\frac{\Delta
T_{\mathrm{freezing}}}{\mathrm{d}T/\mathrm{d}t}$ (1)

where $D$ is the chemical diffusion coefficient of carbon in $\gamma$ at
temperature [22], $\lambda_{\mathrm{cell}}$ is the cell spacing, and $\Delta
T_{\mathrm{freezing}}$ is the magnitude of the freezing range. Given an
experimentally determined $\lambda_{\mathrm{cell}}$ of 1 $\upmu$m [4], and
typical cooling rates reported in the literature [23, 24, 25, 26] the value of
$\alpha$ is substantially less than unity. Hence, backdiffusion of C was
deemed negligible and modeling of solidification under Scheil assumptions was
considered a reasonable first approximation.

##  3 Results

###  3.1 On the assessment of cracking severity, morphology, and mechanism

Our experimentation involving extensive characterization of cracking provides
proof of the significant effect of composition on processability. This is
apparent in the optical micrographs of the XY transverse section – normal
corresponding to build direction – in Figure 4. Cracking is prevalent in the
legacy alloys IN713, IN738LC, IN939, and CM247LC as well as the two
experimental compositions – ExpAM and ABD850AM+CB. The other alloys examined
here process largely crack-free. The distribution of cracks across the XY
plane is uniform in IN713 and ExpAM, whereas the others cracked to a greater
extent at the edges, where the effective laser scan speed was reduced. If one
puts aside the prevalence for surface cracking for one moment and considers
the cracking information for the bulk, the data for different measures of
crack severity developed here – count density and length density – are
consistent with each other, see Figure 6. When examined on the longitudinal XZ
plane, there is a clear disposition for the cracks to be aligned along the
build direction, see Figure 5, with SEM images indicating that cracks of both
solid-state type and solidification type are present in all cracked alloys.
Solid-state cracks tend to exhibit a long and straight morphology, whereas
solidification cracks appear more jagged.

Figure 4: Optical images of the XY plane for each alloy with a binary
threshold applied exemplifying cracking or lack thereof in each alloy. It is
evident the ExpAM, CM247LC, IN939, IN713, IN738LC and ABD850+CB demonstrated
extensive cracking and hence are not readily processable. Figure 5: Optical
micrographs taken of the XZ plane exemplifying cracks observed at the edge and
within the bulk of CM247LC, IN939, ExpAM, and IN713, as well as SEM
micrographs of solidification and solid-state cracks observed in respective
alloys on the XZ plane. Figure 6: Bar chart showing crack length density and
count count density for the six unprocessable alloys.

In order to strengthen our findings further, detailed quantitative
stereological analysis has been employed to quantify the extent to which
solidification and solid-state cracks occur in each alloy, and to allow for
distinguishing between them. We emphasize first the quantification of cracks
at depth in the bulk, since these are the greatest in number and since it may
be possible to improve the surface cracking susceptibility by tailoring the
processing conditions at the surface still further. The morphology of cracks
in different alloys is summarized by the kernel density estimate of the
probability distribution of the perimeter to area ratio $\upmu$m-1 of cracks
in each of the 6 cracked alloys, see Figure 7. Solidification cracks,
distinguished by their characteristic jagged morphology and lower perimeter to
area ratio, arise to greater extent in the IN738LC, IN939, CM247LC and
ABD850AM+CB. This is proven by the deviation of their perimeter to area ratio
distributions, which possess a smaller median, interquartile range, and
lower/upper adjacent values relative to IN713 and ExpAM. Solid-state cracking
is the dominant mechanism in IN713 and ExpAM; in these alloys, cracks are
longer and straighter, exemplified by the shift of their perimeter to area
distributions towards greater values. Nevertheless, overlap between the
distributions for some alloys is apparent for approximately 0.5 $<$
perimeter/area $<$ 0.7, suggesting that the two mechanisms are both operating
simultaneously and competing against each other. Cracks in this range are of
mixed mode, indicating they began as solidification cracks and then propagated
further in the solid-state. The probability distribution of ABD850AM+CB is cut
off at 0.19 and 0.67 as the distribution is comprised of only 24 cracks
observed.

Finally, consider again the propensity for some alloys – IN738LC, IN939,
CM247LC and ABD850AM+CB – to exhibit increased cracking at sample edges. In
these alloys, cracks near edges observed on the XY plane have uniform spacing
of $\approx$ 80 $\upmu$m - this periodicity is attributed to the preferential
nucleation of cracks to minimize strain energy [27]. The slower scan speed
used at the sample edge resulted in coarser microstructure less able to
accommodate strain, with larger grains with increased texture component as
compared to the bulk [28]. Furthermore, the sample edge was processed with
greater energy density and larger melt pools, both of which have been observed
in the literature to exacerbate solidification cracking in particular [29].
ExpAM and IN713 crack at the edges, though not comparatively more than in the
bulk. However, the 4 alloys mentioned above which are prone to more
solidification cracking in the bulk exhibit more cracking at the edge
(relative to their bulk) as they are more sensitive to the local
microstructure and processing conditions.

Figure 7: Violin plots showing the median, interquartile range, lower/upper
adjacent values, and probability distribution of the perimeter/area ratio. As
well as examples of solidification and solid-state cracks and their
corresponding perimeter/area values.

###  3.2 On the influence of mechanical properties on additive
manufacturability

####  3.2.1 Tensile response of the alloys

The temperature-dependence of the tensile response at $\rm 10^{-2}s^{-1}$ is
illustrated in Figure 8, for the alloys IN625, AM-Dev, and CM247LC. Note that
the starting microstructure corresponds to the as-processed condition. The
heritage alloys IN625 and CM247LC serve as lower and upper bounds respectively
for flow stress across the full temperature range due to the appreciably
smaller or larger extent of alloying. A significant degree of ductility is
evident between 25 ∘C and 600 ∘C; however, a significant drop in ductility is
found between 800 ∘C and 1000 ∘C.

The above findings have justified a thorough investigation of the composition-
dependence of the strain-to-failure in the ductility-dip regime. The
temperature of 800 ∘C was chosen for this study. The ductility at 800 ∘C has
been found to vary within the range 1% to 40% for the alloys studied, see
Figure 9. Alloys which cracked during processing were found to exhibit
significantly reduced ductility. In particular, the alloys IN713 and ExpAM –
which are prone to solid-state cracking – failed by brittle fracture at a
strain of $<$ 1.5% with little necking. In this case, their engineering strain
to failure has been defined as the strain at which brittle fracture takes
place. For the remaining alloys, the engineering strain to failure is taken as
the strain at which the sample comes apart, occurring with little/no necking,
and others such as IN625 and IN718 occurring after significant necking.
Serrated flow was observed in all the alloys with the exception of ExpAM,
IN713, and CM247LC. This is attributed to the interaction of solute atoms and
moving dislocations [30, 31].

To summarize, it is clear that the processable alloys maintain some ductility
at 800 ∘C whereas the unprocessable alloys do not. The engineering strain to
failure and 0.2% offset engineering stress (referred to as the flow stress)
measured for each alloy at 800 ∘C are given in Figure 9 – one can see that
alloys that developed cracks during processing each fail at engineering strain
to failure less than 7% at 800 ∘C. Conversely, all the readily processable
alloys were able to accommodate greater than 7% engineering strain. A
strength/ductility trade-off is apparent. In all 12 alloys, good ductility
($>$15% engineering strain to failure) is observed during testing at room
temperature. This confirms a negligible influence of pre-existing defects at
high temperature and the relevance of a mechanism of high-temperature
embrittlement.

Figure 8: The isothermal tensile response of (a) IN625, (b) AM-Dev, and (c)
CM247LC at 25, 600, 800, 1000, and 1100 ∘C in the as-printed state. Summary of
the temperature dependence of (d) the elastic modulus, (e) the flow stress,
and (f) the strain to failure. Figure 9: The tensile response of (a) the
processable heritage alloys (b) unprocessable heritage alloys (c) processable
novel alloys (d) unprocessable novel alloys at 800 ∘C. (e) the mean flow
stress vs the mean engineering strain to failure at 800 ∘C after 3 tests,
error bars show the minimum and maximum values of the 3 repeats. (f) the flow
stress vs the engineering strain to failure at 25 ∘C.

####  3.2.2 Stress relaxation behaviour

The isothermal stress relaxation responses of the different alloys at 1000 ∘C
are compared in Figure 10, for an initially imposed stress of 100 MPa. The
curves possess some common features – but also differences. A significant
transient is observed within the first 10 s which is characterized by a
rapidly decreasing stress, which develops into a more gradual decay at longer
times.

The most creep resistant alloys are those of higher Al, Ti, Ta and Nb contents
such as CM247LC, IN939, IN738LC and AM-Dev. However, also apparent is that
some alloys such as ABD850AM, ABD900AM and AM-Dev – which print well – do not
relieve stress very much more slowly than those – for example ABD850AM +CB and
ExpAM – which print poorly. It follows that the capacity for stress relaxation
cannot alone be responsible for conferring additive manufacturability.

The above findings warrant further detailed quantitative analysis, and this is
developed in Section 4. But the modelling presented there requires an estimate
of the temperature-dependence of the stress relaxation phenomenon, and thus a
first estimate of a constitutive law for this effect. For some of the alloys –
specifically IN939, Waspaloy, AM-Dev and ABD850AM+CB – further testing has
been carried out for a number of temperatures, see Figure 11. As expected, a
strong dependence upon temperature is displayed.

Figure 10: The stress relaxation behavior of the (a) processable heritage
alloys (b) unprocessable heritage alloys (c) processable novel alloys (d)
unprocessable novel alloys at 1000 ∘C. Figure 11: The stress relaxation
behavior of (a) ABD850AM+CB (b) AM-Dev (c) Waspaloy and (d) IN939 at various
temperatures.

####  3.2.3 Stress development during constrained cooling

The results presented thus far indicate that lack of printability may in some
way be associated with the build of stress during constrained cooling coupled
with a lack of ductility in the mid-temperature regime. This motivated the
assessment of the non-isothermal stress build-up during constrained cooling
using the fixed-grips test involving cooling from 1100 ∘C at 50 ∘C/s, see
Figure 12.

At the cooling rate of 50 ∘C/s – close to the fastest possible using the ETMT
machine’s configuration – the alloys develop stress to different extent, see
Figure 12. Of the alloys assessed, stress develops in CM247LC and IN625 to the
greatest and least extents respectively. Interestingly, we have never been
able to replicate cracking in this constrained-bar test, at least without a
further imposition of a further superimposed mechanical strain. When cooling
is carried out at the slower rate of 25 ∘C/s the stress accumulated is
concomitantly less, as discussed in the later Section 4.

Figure 12: Comparison of the stress developed during constrained bar cooling
of the CM247LC, IN939, Waspaloy, ABD850AM+CB, and IN625 alloys at a cooling
rate of 50 ∘C/s.

###  3.3 Characterisation of as-printed microstructure and influence of
freezing range

####  3.3.1 As-printed microstructure

The as-printed microstructure in all cases comprises long columnar grains
preferentially aligned in the build direction. There are three distinct
characteristics: (i) cells of the matrix $\gamma$ with very small or no
secondary dendrite arms; (ii) a narrow intercellular region of differing
etching characteristics and therefore likely composition, and (iii) carbides
residing largely in the intercellular region. It is emphasized that no
$\gamma^{\prime}$ precipitate phase has been detected in any of the alloys in
the as-printed state or following tensile testing at 800 ∘C. Recent data
acquired via synchrotron x-ray diffraction has provided further corroboration
that superalloys processed in this manner are devoid of $\gamma^{\prime}$ in
the as-printed state [28]. The rapid cooling rate results in very fine
cellular morphologies of spacing in the range 300 nm to 1.5 $\upmu$m depending
on the proximity to the sample edge.

Figure 13 shows the enrichment of the intercellular carbides in CM247LC,
IN625, and ExpAM as detected by EDX linescans. A table of the enrichment
observed in each alloy and confirmation from the literature is made in Table
2. The literature indicates that the carbides are likely of MC type of face-
centered cubic structure. The presence of Cr enrichment in only ExpAM – the
most severely cracked alloy – suggests a deleterious effect of chromium-rich
carbides on processability.

In CM247LC, EDX analysis at high magnification reveals the presence of a
continuous film of solute enrichment at the solidification crack tips. The
enrichment of Ta and Hf separate from the carbide, shown in Figure 14,
confirms the partitioning of these solutes to the final solidifying liquid. In
IN713 – which cracks predominantly by the solid-state mechanism – no such
continuous solute enrichment was observed, only discontinuous phases as shown
in Figure 14. This microstructural difference between solidification and
solid-state cracking confirms that CM247LC’s susceptibility to solidification
cracking stems from the solute enrichment of the solidifying liquid and
subsequent widening of the freezing range.

Figure 13: Backscattered electron (BSE) micrographs of the as-printed microstructure after tensile testing at 800 ∘C showing the $\gamma$ matrix and the intercellular carbides as well as EDX linescans across the carbides showing the carbide enrichment in (a) CM247LC, (b) IN625, and (c) ExpAM. Figure 14: SEM and EDX analysis at the crack tip for a typical solid-state crack and solidification crack observed in (a) IN713 and (b) CM247LC respectively. Table 2: Summary of phase transformation temperatures and carbide enrichment determined experimentally. Alloy |  $T_{\mathrm{S^{\prime}}}$ (∘C) |  $T_{\mathrm{L}}$ (∘C) |  $T_{\mathrm{L^{\prime}}}$ (∘C) |  $T_{\mathrm{S}}$ (∘C) | Carbide Enrichment | Corroboration of Carbide Enrichment
---|---|---|---|---|---|---
ABD850AM | 1329 | 1384 | 1368 | 1264 | - | -
ABD850AM+CB | 1276 | 1351 | 1340 | 1209 | Nb,Ta | -
ABD900AM | 1270 | 1363 | 1347 | 1233 | Nb, Ta | -
AM-Dev | 1279 | 1366 | 1352 | 1229 | Nb,Ta | -
CM247LC | 1258 | 1375 | 1366 | 1266 | Ta,Hf,W | [32, 33]
ExpAM | 1309 | 1361 | 1344 | 1254 | Cr,Co,Mo | -
IN625 | 1285 | 1357 | 1308 | 1234 | Nb | [34, 35]
IN713 | 1240 | 1358 | 1341 | 1238 | Nb,Mo | [36, 37]
IN718 | 1242 | 1342 | 1296 | 1182 | Nb,Mo | [38]
IN738LC | 1279 | 1351 | 1340 | 1200 | Ta,W,Nb,Mo | [39, 40]
IN939 | 1203 | 1326 | 1319 | 1174 | Ta,Nb | [41]
Waspaloy | 1292 | 1371 | 1355 | 1265 | Mo,Ti | [42]

####  3.3.2 Differential scanning calorimetry and thermodynamic modelling

The magnitude of the freezing range may well be a good indicator of the
solidification cracking susceptibility – the alloys prone to predominately
solidification type cracks freeze over wide solidification intervals. Figure
15 shows the magnitude of the freezing range (i) as measured by differential
scanning calorimetry and (ii) and (iii) from modeling of the solidification
path by thermodynamic simulation under the Scheil conditions with the TCNI8
and TTNi8 thermodynamic databases, respectively. Estimates from DSC and the
TCNI8 database show the solidification cracking prone alloys ABD850AM+CB,
IN738LC, and IN939 have the widest freezing range of the 12 investigated. On
the other hand, TTNi8 predicts the solidification prone cracking alloys to
have the 4 widest freezing ranges.

The DSC signals and labeled phase transformation temperatures of ExpAM are
shown in Figure 16. The simulations predict significantly lower solidus
temperatures than the DSC tests and the full suppression of $\gamma^{\prime}$
precipitation even in the alloys with highest $\gamma^{\prime}$ former
content; this suppression is confirmed by characterization in Figure 1.
Liquidus and solidus temperatures determined by DSC during heating and cooling
are summarized in Table 2. One needs to bear in mind of course that it is not
possible to replicate the extreme heating/cooling rates of the AM process in
the DSC.

Figure 15: Summary of the magnitude of the freezing range determined (a)
experimentally by DSC and by scheil model using (b) the TCNI8 and (c) the
TTNi8 thermodynamic databases. Figure 16: DSC signal from ExpAM and the
respective phase transformations determined by interpretation of (a) the
heating cycle and (b) the cooling cycle.

##  4 Analysis and Discussion

The results presented above are insightful and confirm unambiguously that a
strong inter-dependence exists between processability and alloy composition.
Nevertheless, further understanding can be gained via more detailed analysis.
We consider each of the solid-state and solidification cracking phenomena in
turn.

###  4.1 Modelling of stress development during cooling leading to solid-state
cracking

Figure 12 confirms that the stress experienced during cooling in the
constrained-bar test depends upon alloy composition. But various factors will
contribute to this to varying extents – the composition-dependence of elastic
modulus and thermal expansion coefficient of Figure 3, but also a possible
stress relaxation effect – and it is helpful to deconvolute these in order to
determine which is the more important. Further experimentation has indicated
that cooling rate does indeed influence somewhat the build up of stress, see
Figure 17 which includes data for ABD850AM+CB, Waspaloy and IN939. Clearly,
slower cooling provides more time for stress relaxation. Particularly for the
stronger IN939 alloy, the initial rate of change of stress with temperature
shows no dependence upon cooling rate, implying the need for interpretation
based upon thermal-elastic-plastic behavior with no creep relaxation playing
any role in that initial regime. Figure 18 provides further insight into what
is happening in these tests – consider the case of the least strong and most
strong alloys considered here, IN625 and CM247LC respectfully. The variation
of the measured stress with temperature for the two cases matches closely the
product of elastic modulus, thermal expansion efficient and temperature
difference $E\alpha\Delta T$ for several hundred degrees below the initial
temperature of 1100 ∘C. Particularly for the stronger alloy CM247LC, the
product $E\alpha\Delta T$ lies below the measured uniaxial flow stress
initially but surpasses it in the ductility-dip regime. For the weaker alloy
IN625, the measured stress in the constrained bar test determined at low
temperature matches well the measured flow stress consistent with only a weak
work-hardening effect.

Figure 17: The stress evolution during cooling at different rates in (a)
ABD850AM+CB, (b) Waspaloy, and (c) IN939. Figure 18: The stress evolution
during cooling at 50 ∘C/s superimposed with the temperature dependent flow
stress and linear thermal-elastic stress development for (a) IN625 and (b)
CM247LC.

Modeling of the constrained bar test puts the thermal-mechanics of this
situation on a quantitative basis. The net rate of straining is zero;
therefore the rates of elastic, plastic, creep and thermal straining
$\dot{\varepsilon}_{\mathrm{elastic}}$,
$\dot{\varepsilon}_{\mathrm{plastic}}$, $\dot{\varepsilon}_{\mathrm{creep}}$
and $\dot{\varepsilon}_{\mathrm{thermal}}$ must satisfy

$\dot{\varepsilon}_{\mathrm{elastic}}+\dot{\varepsilon}_{\mathrm{plastic}}+\dot{\varepsilon}_{\mathrm{creep}}+\dot{\varepsilon}_{\mathrm{thermal}}=0$
(2)

In this expression, $\dot{\varepsilon}_{\mathrm{thermal}}=\alpha\dot{T}$ is
determined by the cooling rate, we find that as the specimen is cooled one of
the mechanical strain-rates ($\dot{\varepsilon}_{\mathrm{elastic}}$,
$\dot{\varepsilon}_{\mathrm{plastic}}$ or
$\dot{\varepsilon}_{\mathrm{creep}}$) dominates, with the range of dominance
of each mechanism and therefore the detailed response depending on cooling
rate and the mechanical properties of the material. Initially, the stress
$\sigma$ equals zero. As the sample is cooled the response is elastic and

$\dot{\varepsilon}_{\mathrm{elastic}}=\frac{\dot{\sigma}}{E}=-\alpha\dot{T}$
(3)

Giving, $\sigma=\sigma_{\mathrm{elastic}}=E\alpha\Delta T$, as shown in Figure
18 for IN625 and CM247LC. as the stress increases eventually creep takes over
as the dominant mechanism. For a material that creeps according to Norton’s
Law we find

$\sigma=\sigma_{\mathrm{creep}}\\{T\\}=\left(-\frac{\alpha\dot{T}}{A}\right)^{1/n}\mathrm{exp}\left\\{\frac{Q}{nRT}\right\\}$
(4)

where $R$ is the ideal gas constant and the parameters approximated by fitting
are the creep exponent $n$ (4.8 for both Waspaloy and CM247LC), the activation
energy for creep $Q$ (800 kJ/mol for Waspaloy and 1000 kJ/mol for CM247LC) and
the pre-exponential constant in Norton’s Law $A$ ($6.1\times 10^{-7}$ MPa-n
s-1 for both alloys). The actual stress is the smallest of those given by the
elastic response and equation (4). The initial response is elastic, but as the
sample is cooled the stress increases, creep takes over as the dominant
mechanism and the stress is given by equation (4). For Waspaloy, which has a
low creep strength at elevated temperatures, the initial elastic range is very
short and creep becomes the dominant process at low stresses, as shown in
Figure 19a, As the temperature is decreased further the creep strength and
therefore the stress given by equation (5) increase. In the case of Waspaloy,
we find that eventually elastic deformation dominates again. This transition
occurs when

$\dot{\sigma}_{\mathrm{creep}}=\left(\frac{-Q}{nR{T}^{2}}\right)\left(-\frac{\alpha\dot{T}}{A}\right)^{1/n}\mathrm{exp}\left\\{\frac{Q}{nRT}\right\\}=\dot{\sigma}_{\mathrm{elastic}}=E\alpha\dot{T}$
(5)

Substituting (4) into (5) gives a relationship between the stress and
temperature at the creep-elastic transition

$\sigma_{\mathrm{ce}}=\frac{E\alpha nR{T_{\mathrm{ce}}}^{2}}{Q}$ (6)

which we have designated using the subscript ce. The actual stress and
temperature a the transition is given by the intersection of equations (4) and
(6) on a plot of $\sigma$ against $T$. When $T<T_{\mathrm{ce}}$ the response
is elastic the stress is given by

$E\alpha(T_{\mathrm{ce}}-T)+\sigma_{\mathrm{ce}}$ (7)

until the onset of plastic deformation. For quantitative calculations one
needs a plasticity law, taken here to be the Ramberg-Osgood relationship with
$\sigma_{\mathrm{y}}$ the material’s yield stress and $m$ a material-dependent
constant, fitted to be 3.4 and 3.6 for Waspaloy and CM247LC respectively,
consistent with

$\dot{\varepsilon}_{\mathrm{plastic}}=m\frac{\dot{\sigma}}{E}\left(\frac{\sigma}{\sigma_{\mathrm{y}}}\right)^{m-1}$
(8)

where for simplicity we have assumed that $\sigma_{\mathrm{y}}$ does not vary
with temperature. The transition to plastic deformation occurs when $\sigma$
given by equation (4) and (7) equals $\sigma_{\mathrm{y}}$. We designate the
temperature at which this occurs as $T_{\mathrm{y}}$. It follows that
subsequent response is given by

$\alpha(T_{\mathrm{y}}-T)=\frac{\sigma_{\mathrm{y}}}{E}\left[\left(\frac{\sigma\\{T\\}}{\sigma_{\mathrm{y}}}\right)^{m}-1\right]$
(9)

and we can describe the stress as follows for $T<T_{\mathrm{y}}$

$\sigma_{\mathrm{y}}\left(1+\alpha\frac{E}{\sigma_{\mathrm{y}}}(T_{\mathrm{y}}-T)\right)^{1/m}$
(10)

To summarize, the stress development during cooling can be represented by the
equation set:

$\sigma\\{T\\}=\begin{cases}E\alpha\Delta T&\text{if $T>T_{\mathrm{ce}}$
$\cap$ $\sigma_{\mathrm{elastic}}<\sigma_{\mathrm{creep}}$}\\\
\left(-\frac{\alpha\dot{T}}{A}\right)^{1/n}\mathrm{exp}\left\\{\frac{Q}{nRT}\right\\}&\text{if
$T>T_{\mathrm{ce}}\cap\sigma_{\mathrm{creep}}<\sigma_{\mathrm{elastic}}$}\\\
E\alpha(T_{\mathrm{ce}}-T)+\sigma_{\mathrm{creep}}\\{T_{\mathrm{ce}}\\}&\text{if
$T_{\mathrm{ce}}>T>T_{\mathrm{y}}$}\\\
\sigma_{\mathrm{y}}\left(1+\alpha\frac{E}{\sigma_{\mathrm{y}}}(T_{\mathrm{y}}-T)\right)^{1/m}&\text{if
$T<T_{\mathrm{y}}$}\end{cases}$ (11)

It has been found that the above formulation can largely rationalize the
temperature-dependence of the build of stress with decreasing temperature, see
Figure 19 for Waspaloy and CM247LC. All the above regimes of behavior are not
exhibited by all materials. In the case of Waspaloy, the initial elastic
response is small and creep dominates for the slow cooling rate considered in
Figure 19 until approximately 900 ∘C when the behavior is elastic; and it
becomes thermal-plastic at $\sim$700 ∘C. For CM247LC which displays much
greater creep resistance, the deformation is initially linear thermal-elastic
over a wide temperature regime until the onset of rate-insensitive plasticity
again at $\sim$700 ∘C. In each case the temperature range over which creep
occurs is small and therefore limited creep deformation occurs during
cooldown. In order to compare the plastic strain accumulation to the
determined composition-dependent ductility exhaustion limit, a strain-rate
independent first order approximation can be made of the plastic strain as
follows

$\varepsilon_{\mathrm{plastic}}\\{T\\}\approx E\alpha\Delta
T-\frac{\sigma\\{T\\}}{E}$ (12)

Our calculations indicate a critical minimum ductility in the range 0.5 to 1%
is likely to be needed for the avoidance of ductility-dip cracking in the
critical temperature range of 700 ∘C to 900 ∘C. These findings rationalize the
poor resistance of the likes of IN713 and ExpAM to solid-state cracking.

To summarize, the analysis presented in this section allows a number of
critical points to be confirmed. First, the balance of evidence suggests that
solid-state cracking – in susceptible alloys – will occur in the ductility-dip
regime due to the brittleness which is prevalent there. Second, since the
cooling rates of the AM process are extremely rapid [23, 24, 25, 26], the
development of stress will be thermal-elastic-plastic in origin, such that for
the highest strength alloys there will be little influence of stress
relaxation on the composition-dependence of cracking susceptibility. Third,
nonetheless the alloy composition is important: the greater the extent of
alloying, the greater the stress build up to drive cracking and the greater
the risk of the yield stress being exceeded in the ductility-dip regime. Thus,
the cracking risk is exacerbated by greater alloy strength but for any given
strength, the risk is ameliorated if the low-temperature ductility is greater.
These two factors are strongly correlated via the strength/ductility
relationship. Finally, the critical ductility needed to avoid solid-state
cracking appears to be in the range 0.5 to 1% in the ductility dip regime.
Obviously, the above applies only to solid-state cracking and ignores the
solidification cracking effect which is now considered in the following
section.

Figure 19: Comparison of the experimentally measured stress during cooling and
the fitted thermal-elastic-plastic-viscoplastic model for stress (left axis)
and strain (right axis) development of (a) Waspaloy and (b) CM247LC.

###  4.2 Analysis of solidification cracking susceptibility

So far, the occurrence of solidification cracking has been correlated with the
magnitude of the freezing range and presence of continuous films of solute
enrichment at crack tips. But a number of composition-dependent solidification
cracking criteria should be considered, for example those due to Clyne &
Davies (CD) [43], Rappaz, Drezet, & Gremaud (RDG) [44] and Kou [45]. In each
case, a parameter $\Phi$ can be identified, which is predicted to scale with
cracking susceptibility. The CD approach for instance considers the
dimensionless ratio of the time during solidification during which the alloy
is vulnerable to cracking $\Delta t_{\mathrm{vulnerable}}$ to the time during
which it can relieve the accumulated stress $\Delta
t_{\mathrm{stress\>relief}}$. Assuming a constant cooling rate this ratio is
proportional to each respective temperature range, consistent with

$\Phi_{\mathrm{CD}}\propto\frac{\Delta T_{\mathrm{vulnerable}}}{\Delta
T_{\mathrm{stress\>relief}}}=\frac{T_{f_{\mathrm{s}}=0.99}-T_{f_{\mathrm{s}}=0.9}}{T_{f_{\mathrm{s}}=0.9}-T_{f_{\mathrm{s}}=0.4}}$
(13)

The CD criterion can be criticized on account of its rather arbitrary
definition of the range during which the material is vulnerable; however,
keeping this constant for the 12 alloys allows for comparison. The alternative
RDG criterion is based on a balance of pressure, such that the formation of
hot tears depends upon the maximum strain rate
$\dot{\varepsilon}_{\mathrm{max}}$ before a cavitation pressure $\Delta
p_{\mathrm{c}}$ results in the formation of a void. Both the mechanical
contribution and the ability for liquid to back fill are dictated by integrals
of the solid fraction vs temperature $f_{\mathrm{s}}(T)$, $A$, $B$, and $C$
[44] consistent with

$\Phi_{\mathrm{RDG}}\propto\frac{1}{\dot{\varepsilon}_{\mathrm{max}}}=\frac{(1+\beta)B}{G}\bigg{[}\frac{\Delta
p_{\mathrm{c}}G\lambda_{2}^{2}}{180\mu}-v_{\mathrm{T}}\beta A\bigg{]}^{-1}$
(14)

where

$A=\int_{T_{\mathrm{cg}}}^{T_{\mathrm{L}}}\frac{f_{\mathrm{s}}\\{T\\}^{2}}{(1-f_{\mathrm{s}}\\{T\\})^{2}}\mathrm{d}T$
(15)

and

$B=\int_{T_{\mathrm{cg}}}^{T_{\mathrm{L}}}\frac{C}{(1-f_{\mathrm{s}}\\{T\\})^{3}}\mathrm{d}T$
(16)

where

$C=f_{\mathrm{s}}\\{T\\}^{2}\int_{T_{\mathrm{\mathrm{cg}}}}^{T_{\mathrm{L}}}f_{\mathrm{s}}\\{T\\}\mathrm{d}T$
(17)

For the calculations carried out here, the viscosity $\mu$, the thermal
gradient $G$, the velocity of the isotherms $v_{\mathrm{T}}$, and the
shrinkage factor $\beta$ have been taken from [44], the secondary arm spacing
$\lambda_{2}$ from [4]. The grain coalescence temperature $T_{\mathrm{cg}}$ is
taken to correspond to $T_{f_{s}=0.99}$. The hot cracking susceptibility is
proportional to $\dot{\varepsilon}^{-1}_{\mathrm{max}}$ and values are
normalized between zero and unity. The RDG criterion suffers from a similar
criticism to the CD criterion, as the grain coalescence temperature – critical
to evaluating the integrals $A$, $B$ and $C$ – is not well understood.
Furthermore, approximating liquid filling by the flow of fluid through a
packed bed may not be the most appropriate for the cellular AM
microstructures. Finally, the Kou solidification cracking index is formulated
around the balance of increasing spatial dimensions associated with solid
formation and volumetric liquid flow within a differential control volume
between two growing grains [45]. Here, the cracking susceptibility is taken to
be proportional to the maximal value of the ratio of change in temperature
$\mathrm{d}T$ relative to the change in square root in the solid fraction
$\mathrm{d}(\sqrt{f_{\mathrm{s}}})$ near the final stages of solidification,
consistent with

$\Phi_{\mathrm{Kou}}\propto\bigg{|}\frac{\mathrm{d}T}{\mathrm{d}(\sqrt{f_{\mathrm{s}}})}\bigg{|}_{f_{\mathrm{s}}\rightarrow
1}$ (18)

Figure 20 summarizes the predictions of the different criteria considered. One
can see that, with the exception of the CD index evaluated with the
experimentally-evaluated DSC traces (which probably gives an underestimate of
the solidification range) – the indices are in line with the findings of our
experimentally-determined solidification cracking susceptibility. However, no
single index predicts correctly the composition-dependence of solidification
cracking for all methods of determining $f_{\mathrm{s}}\\{T\\}$. Moreover, no
one method of estimating $f_{\mathrm{s}}\\{T\\}$ correctly predicts
solidification cracking across all the indices. The strength of applying these
legacy criteria to multi-component Ni-superalloys is their sensitivity to the
final stages of solidification whereby the final liquid is solute enriched and
stable at lower temperatures. This in turn presents a problem, as
$f_{\mathrm{s}}\\{T\\}$ is discontinuous when a new phase becomes stable,
making the exact numerical procedure by which the Kou criterion is evaluated
highly sensitive to these discontinuities.

One can argue that a modified criterion is needed to take account of
contributions of mechanical and stress relaxation properties. A very detailed
analysis is beyond the scope of this paper, but approximate calculations can
be made as follows. Assume the contribution of stress relaxation is more
pronounced at near solidus temperatures as is suggested by the strong
temperature dependence of stress relaxation of Figure 11. A modified criterion
could consider the solidification cracking susceptibility to be proportional
to the ratio of strain accumulated during the vulnerable regime $\alpha\Delta
T_{\mathrm{vulnerable}}$ to the strain relieved in the stress relief regime.
We apply this as shown in Equation 19 using experimentally determined values
of $E$, and $\alpha$, and the $\mathrm{d}\sigma/\mathrm{d}t$ derived from the
secant line of the stress relaxation behavior from 0 to 30 s at 1000 ∘C,
according to:

$\Phi_{\mathrm{CD\>modified}}\propto\frac{\varepsilon_{\mathrm{vulnerable}}}{\varepsilon_{\mathrm{stress\>relief}}}=\frac{\alpha\Delta
T_{\mathrm{vulnerable}}}{\left(\frac{\dot{\sigma}}{E}\right)\frac{\Delta
T_{\mathrm{stress\>relief}}}{\partial T/\partial t}}$ (19)

This is in essence a mechanically-weighted CD index. The modified index
increases the relative predicted susceptibility of the solidification cracking
alloys IN738LC, IN939, and CM247LC appreciably, see Figure 20, by accounting
for thermal expansion, stiffness and the capacity of the material to stress
relax.

Figure 20: Solidification cracking susceptibility as predicted by the Clyne-
Davies criterion, the Rappaz-Drezet-Gremaud criterion, the Kou criterion, and
a modified Clyne-Davies criterion, each determined as a function of the
solidification path determined experimentally and via scheil with the TCNI8
and TTNi8 databases.

##  5 Final considerations

An intriguing aspect of the as-processed microstructures produced in the
present work is the absence of the $\gamma^{\prime}$ phase due to the
suppression of the $\gamma\rightarrow\gamma+\gamma^{\prime}$ reaction caused
by the rapid cooling rates achieved. It follows that the varying strength
levels measured arise from different degrees of solid-solution hardening,
since the same processing parameters yield similar in-plane grain size
distributions and thus approximately similar degrees of sub-structure
hardening via the Hall-Petch effect. Thus one has a very supersaturated
solution of hardening elements present in significant concentrations in the
matrix $\gamma$ phase. For this situation, there is a complete paucity of
information in the literature for the relative contributions of the elements
to the strengthening levels achieved in this situation. Although measurements
have been made on just 12 different alloys in this work, it has proven
possible to come to interesting findings in the following way. Assume that the
contributions to the strengthening of each element $i$ via a coefficient
$k_{i}$ can be summed according to a simple model consistent with [46]

$\sigma_{\mathrm{solid\>solution}}=\bigg{(}\sum_{i}k_{i}^{\frac{1}{n}}x_{i}\bigg{)}^{n}$
(20)

where $x_{i}$ the concentration of solute $i$, $k_{i}$ is the respective
strengthening constant and the value of $n$ is taken as 2/3 [46]. A multi-
regression fitting procedure then allows an estimate of the values of $k_{i}$,
see Table 3. The values obtained are compared with those determined by Roth et
al [47] on much more dilute alloys, via a trawling of the literature. Similar
trends are observed, which lends confidence to our strength measurements. When
the coefficients $k_{i}$ are plotted as a function of the location of element
$i$ in the d-block of transition metals, one sees that the elements from the
far-west of the d-block confer a greater degree of hardening, on an atomic
percent basis. Moreover, it would appear that the extent of hardening for each
column in the d-block is consistent with $\rm 5d>4d>3d$, see Figure 21. Our
data are sparse, but our findings are worthy of future investigations along
these lines.

Table 3: Strengthening coefficients for solution strengthening in nickel for AM at 800 ∘C in comparison to [47] (MPa at Fraction1/2). Alloying Element | $k_{\mathrm{i}}^{\mathrm{AM}}$ |  $k_{\mathrm{i}}$ [47]
---|---|---
Al | 1787 | 225
Co | 43 | 39
Cr | 285 | 337
Fe | - | 153
Mo | 795 | 1015
Nb | 1155 | 1183
Ta | 3736 | 1191
Ti | 1438 | 775
W | 1813 | 977
C | 5355 | 1061
Hf | - | 1401
Figure 21: (a) Comparison of the experimentally determined flow stress and the
solid solution strengthening model at 800 ∘C and (b) strengthening
coefficients as a function of atomic number for solutes in the 3d, 4d, and 5d
groups.

Despite previous work having suggested that improving the solid solution
strengthening can improve processability [11], the research presented here
indicates that it is the exhaustion of ductility – not strength – which must
be conferred for crack-free printing if the best strength levels are to be
approached. This is summarized succinctly in Figure 23. Plotting the alloys’
ductility vs cracking severity indicates alloys become unprocessable when
their ductility is less than 7%, though it should be noted that it would
appear to be possible to achieve strength levels comparable to those of
CM247LC without embrittlement. In general, a strength/ductility trade-off is
apparent as the macroscopic elongation in the ductility-loss regime ($\sim$800
∘C) is governed by the balance of plasticity of grain interiors and the
cohesion of grain boundaries. For very strong alloys – such as CM247LC and
ExpAM – the failure mechanism is purely intergranular – see Figure 22 –
implying grain boundaries are the ductility-limiting feature of the
unprocessable alloys. Whereas for alloys such as IN718 and IN625, the fracture
surfaces show only transgranular features such as those due to micro-void
coalescence – suggesting grain boundaries are sufficiently cohesive, making
the grain interior the ductility-limiting feature. The fracture surfaces of
the moderate ductility ABD alloys such as ABD850AM and 1005AM are somewhere in
between: they exhibit features characteristic of both intergranular and
transgranular fracture, see Figure 22. This work finds that despite the
presence of such carbides in the processable and ductile Nb/Mo-rich carbide
forming alloys IN625, IN718 and the ABD alloys, their occurrence in the un-
processable IN713 alloy suggests that carbide enrichment alone does not
guarantee grain boundary properties. It seems likely that the ratio of grain
boundary strength to grain strength – in this case controlled by the
contribution of elements in solid solution – dictates the extent to which
fracture is inter- or intragranular. These need to be balanced to maximize
performance whilst maintaining processability. Clearly, further advances in
the field of grain boundary engineering are needed if further improvements in
property levels are to be achieved.

Figure 22: Fracture surfaces of CM247C, AM-Dev, and IN625 after tensile tests
at 800 ∘C illustrating the transition from intra/transgranular to
intergranular fracture. Figure 23: Processing map demonstrating four classes
of alloys based upon measurements of their cracking severity (left axis) and
flow stress at 800 ∘C (right axis) as a function of their the ductility at 800
∘C.

##  6 Summary and Conclusions

In this paper, the additively manufacturability of nickel-based superalloys
has been considered in detail. The laser powder bed fusion technique is used.
Twelve different alloys have been studied, some of the heritage type but also
some new ones designed specifically for this process. Emphasis has been placed
on the influence of composition on cracking susceptibility. The following
specific conclusions can be drawn from this work:

  1. 1.

A dependence of alloy composition on processability has been proven. Of the
twelve alloys studied, quantitative stereology confirmed that half are
susceptible to processing-induced crack defect formation under the
experimental conditions employed. In particular, IN713 and ExpAM compositions
were shown to be prone to solid-state cracking, whilst CM247LC, IN738LC,
ABD850AM+CB and IN939 exhibited a tendency towards solidification cracking.

  2. 2.

Solid-state cracking arises from a significant loss of tensile ductility at
800 ∘C ($<$7%) which was prevalent in all unprocessable alloys – as
exemplified in particular by the IN713 and ExpAM alloys for which it was
$<$1%. Stereological analysis indicates that this brittleness is intergranular
in form implying grain boundary weakness and exacerbated by increased
$\gamma^{\prime}$ former content.

  3. 3.

A thermal-elastic-plastic-viscoplastic analysis of a 1D constrained bar has
shed light on the factors exacerbating solid-state cracking. It seems likely
that this effect is not strongly affected by creep-driven stress-relaxation,
under the cooling rates experienced. Instead, processing is compromised by
brittleness in the ductility-dip regime. Nevertheless, at any given strength
level it appears to be possible to find alloys which are processable, and
others which are not.

  4. 4.

When solidification cracking occurs, it is in alloys that freeze over a wide
temperature interval and relieve stress (by creep processes) at a slower rate.
Increased C content and the presence of of Hf/Ta enriched crack tips is
characteristic of the alloys which cracked by the solidification mechanism.

  5. 5.

Solidification cracking criteria following Clyne-Davies, Rappaz, and Kou have
been considered as a means of rationalizing our findings concerning
solidification cracking; these are shown to have some predictive power.
However, they do not account for the composition-dependent near-solidus
mechanical properties which may be important for the stress relaxation
processes needed for crack-free processing. A modified Clyne-Davies model is
proposed to account for this.

  6. 6.

The processing/performance relationships of existing and newly designed alloys
have been compared. It has been shown that the strength levels of the former
can be approached whilst engendering the AM processability which is apparently
lacking.

  7. 7.

Our results suggest pathways by which the new alloys may be further improved.
Since creep relaxation does not seem to play a role in the solid-state
cracking phenomena, it might prove possible to design alloys of creep
resistance equivalent to the very best conventionally-cast grades which are
amenable to AM processing, provided that they resist solidification cracking.

##  7 Acknowledgments

The financial support of this work by Alloyed Ltd. as well as the the The
Natural Sciences and Engineering Research Council of Canada (NSERC) in the
Chemical, Biomedical and Materials Science Engineering division award number
532410. The authors acknowledge funding from Innovate UK, under project number
104047, specifically the Materials and Manufacturing Division.

##  8 References

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# Jet measurements in polarized p+p collisions at STAR at RHIC

Xuan Li[1] for the STAR Collaboration  1) Physics Department, Temple
University, Philadelphia, 19122, USA<EMAIL_ADDRESS>
###### Abstract

Jet production in polarized $p+p$ collisions at $\sqrt{s}=200$ GeV and
$\sqrt{s}=500$ GeV provides a powerful probe to study gluons inside the
proton. The Solenoidal Tracker at RHIC (STAR) has the capability, with nearly
full azimuthal ($2\pi$) coverage, to reconstruct jets at mid-rapidity
($|\eta|<1$). The latest STAR inclusive jet longitudinal double-spin asymmetry
$A_{LL}$ measured in 200 GeV $p+p$ collisions provides better constraints on
the polarized gluon distribution $\Delta g(x)$ for $0.05<x<0.2$ than previous
measurements. A recent global QCD fit (DSSV 2014) which includes the 2009 RHIC
results provides the first evidence of non-zero gluon contribution to the
proton spin. A new inclusive jet cross section using the anti-$k_{T}$
algorithm provides potential insights into the unpolarized gluon distribution
function, and the new inclusive jet $A_{LL}$ measurement in 510 GeV $p+p$
collisions shows consistent $x_{T}$ scaling with the 200 GeV result. Future
measurements with continuing high energy polarized proton-proton running at
$\sqrt{s}=500$ GeV at RHIC and detector upgrades in the forward direction will
explore the gluonic contribution to the proton spin at low $x$ region.

##  1 Introduction

One of the main objectives for the Relativistic Heavy Ion Collider (RHIC) spin
program is to quantitatively determine the polarized gluon distribution
$\Delta g(x,Q^{2})$ through longitudinally polarized proton-proton collisions.
Since 2009, RHIC has collided longitudinally polarized proton beams at
$\sqrt{s}=200$ GeV and $\sqrt{s}=500/510$ GeV. The Solenoidal Tracker at RHIC
(STAR) has both tracking detectors and electromagnetic calorimeters with
nearly full azimuthal coverage, which allows for final states, such as jets,
to be well reconstructed at mid-rapidity ($|\eta|<1$). In leading order QCD
calculations, mid-rapidity jet production is dominated by quark+gluon and
gluon+gluon interactions. Therefore hard scattered gluons can be probed by jet
measurements at STAR in $p+p$ collisions. The STAR collaboration has submitted
for publication the measurement of the inclusive jet double spin asymmetry
$A_{LL}$ in 200 GeV $p+p$ collisions based on data collected in the 2009
running period [1]. A global QCD fit, which includes the 2009 STAR jet result,
extracts a non-zero gluon spin contribution ($\Delta g(x,Q^{2})$) to the
proton in the $0.05<x<0.2$ region [2]. This contribution reports a new
inclusive jet cross section measurement at STAR from the 2009 data set. The
new measured inclusive jet cross section in data is consistent with the next-
leading order (NLO) perturbative QCD calculations within uncertainties for the
$0.1<x<0.5$ region.

The uncertainty of $\Delta g(x,Q^{2})$ remains large for the low $x$ region
($x<0.05$). STAR can begin to probe some of this lower $x$ region by measuring
asymmetries of final states at different center of mass energies, such as the
inclusive jet $A_{LL}$ in 510 GeV $p+p$ collisions, and by moving to more
forward rapidities.

Figure 1:  2009 inclusive jet cross section preliminary result in 200 GeV
$p+p$ collisions within jet $|\eta|<1.0$ (top left panel), $|\eta|<0.5$ (top
right panel) and $0.5<|\eta|<1.0$ (bottom left panel) pseudorapidity regions.
The results are compared with underlying event and hadronization corrected NLO
pQCD calculations with CT10 (blue) and NNPDF3.0 (red) PDF sets with
factorization and renormalization scaling uncertainties.

##  2 Inclusive jet cross section in 200 GeV p+p collisions

A primary goal of STAR in the 2009 $p+p$ run at $\sqrt{s}=200$ GeV is to study
the dynamics and the spin structure of the proton via jet measurements. STAR
uses Jet Patches (JP) [3], which are defined as groups of adjacent calorimeter
towers inside the electromagnetic calorimeter used to trigger on jet events.
The anti-$k_{T}$ jet algorithm [4] with a cone radius of R=0.6 is applied in
the 2009 inclusive jet analysis, to replace the mid-point cone algorithm used
in earlier analysis [5]. The anti-$k_{T}$ algorithm reduces the underlying
event contribution and pile up backgrounds to the reconstructed jets. Particle
level jets which are based on final state particles and parton level jets,
which are formed from hard scattered partons (excluding those from the
underlying events and beam remnants), can be reconstructed in the simulation.
Jets reconstructed from the detector response (e.g. charged tracks or neutral
energy depositions) in data are defined as detector level jets. Detector level
jets can be reconstructed in simulations as well with the same trigger and
detector conditions as in the data. In simulation, a detector level jet and a
parton or particle level jet are considered to match each other when the
distance between the two in $\eta-\varphi$ space is minimal and less than the
jet cone radius.

Figure 2:  Preliminary results of the measured inclusive jet cross section
compared with NLO pQCD theory calculations using CT10 PDFs for jet
$|\eta|<1.0$ (top left panel), $|\eta|<0.5$ (top right panel) and
$0.5<|\eta|<1.0$ (bottom left panel) in 2009 200 GeV $p+p$ collisions. The
yellow band stands for the systematic error, the grey band stands for the
underlying event and hadronization correction on the theory calculations, the
blue band stands for the CT10 scaling uncertainty and the green band stands
for the CT10 PDF uncertainty.

For the inclusive jet cross section study, the detector jet transverse
momentum $p_{T}$ measured in the data is corrected to the particle jet $p_{T}$
via unfolding. The primary sources of the systematic errors include the
tracking resolution, the tracking inefficiency and the electromagnetic
calorimeter resolution. The dominant systematic uncertainty of this
measurement comes from the calibration of the electromagnetic calorimeter.
Figure 1 shows the preliminary result of the 2009 inclusive jet cross section
versus particle jet $p_{T}$ for jets measured in $|\eta|<1.0$ (top left
panel), $|\eta|<0.5$ (top right panel) and $0.5<|\eta|<1.0$ (bottom left
panel) pseudorapidity regions in 200 GeV $p+p$ collisions. The total
luminosity determined by the Beam Beam Counter (BBC) is 19 $\textrm{pb}^{-1}$
with $8\%$ scaling uncertainty. Results in data are compared with NLO pQCD
calculations with the CT10 [6] and the NNPDF3.0 [7] PDF sets after underlying
event and hadronization corrections are applied to the pure NLO calculations.
The latter is determined by PYTHIA 6.426 simulations by studying the
difference between the particle level jet cross section and the hard
scattering parton jet cross section for each jet $p_{T}$ bin. The maximum
value of the hadronization and underlying event correction is around $5\%$. In
addition to this, a ratio is defined as the difference between the measured
and theory calculated inclusive jet cross section normalized by the theory
calculations. Figure 2 shows this ratio using the CT10 PDF set for the theory
calculations. Data are in good agreement with the NLO pQCD calculations when
using the CT10 PDF set. The gluon distribution function $g(x,Q^{2})$ has large
uncertainties in the high x region ($x>0.1$). The new STAR inclusive jet cross
section measurements are divided into three different pseudo-rapidity regions
($|\eta|<1.0$, $0.5<|\eta|<1.0$ and $|\eta|<0.5$) to access the unpolarized
gluon distribution functions in different x regions. This result is expected
to provide new input to constrain $g(x,Q^{2})$ in the RHIC sensitive x region
($0.1<x<0.5$).

##  3 Inclusive jet longitudinal double spin asymmetries $A_{LL}$ in 200/510
GeV p+p collisions

The double spin asymmetry $A_{LL}$ of jets probes the helicity distribution of
involved partons. Therefore, the gluon polarization contribution to the proton
can be directly probed by measuring inclusive jets or di-jets. From the 2009
RHIC run, STAR collected 200 GeV $p+p$ data which contain 20-fold more
statistics than the previous result [8]. Figure 4 shows the 2009 STAR mid-
rapidity inclusive jet double spin asymmetry $A_{LL}$ versus the parton jet
$p_{T}$ in 200 GeV proton-proton collisions [1]. The measured detector jet
$A_{LL}$ values are corrected to the parton jet $A_{LL}$ in each jet $p_{T}$
bin according to calculations done in the simulation with
polarized/unpolarized PDFs. The systematic error on the product of the
polarization of the two beams is $6.5\%$. The jet acceptance is divided into
$|\eta|<0.5$ (top panel) and $0.5<|\eta|<1$ (bottom panel) to access different
$x$ regions. The colored lines represent the NLO pQCD inclusive jet $A_{LL}$
calculations with different polarized parton distributions used as inputs [9,
10, 11, 12, 13]. The data points fall between the LSS10p [9] and the DSSV [11]
model predictions.

Figure 3:  STAR 2009 inclusive jet $A_{LL}$ versus parton jet $p_{T}$ in 200
GeV $p+p$ collisions [1]. The $A_{LL}$ of the inclusive jets with $|\eta|<0.5$
is shown in the top panel, and the results with $0.5<|\eta|<1.0$ are shown in
the bottom panel. The error bars are statistical, while the gray boxes
represent the systematic uncertainties.

Figure 4:  Top panel: STAR 2009/2012 inclusive jet $A_{LL}$ versus
$x_{T}=2p_{T}/\sqrt{s}$ in 200/510 GeV $p+p$ collisions [15]. 2009 inclusive
jet $A_{LL}$ with $|\eta|<1.0$ is shown in blue and the 2012 result with
$|\eta|<0.9$ is shown in red. Bottom panel: Recent global analysis from the
DSSV group on the running integral of $\Delta g$ as a function of $x_{min}$ at
$Q^{2}=10$ GeV2 [15]. Inner and outer bands stands for the 90 $\%$ C.L.
uncertainties with and without including the projection of up to 2015 RHIC
data.

The DSSV group is the first to apply a global QCD fit on the combined data
samples from DIS, Semi-Inclusive DIS (SIDIS) and RHIC $p+p$ experiments.
Recently, the NNPDF group performed a global QCD fit to extract polarized PDFs
based on the polarized DIS and polarized RHIC $p+p$ data. The extracted gluon
polarization contribution from the latest NNPDF fit in the range $0.05<x<0.2$
is $0.17\pm 0.06$ including the 2009 STAR jet data [10]. The latest DSSV
update, which includes the 2009 STAR inclusive jet $A_{LL}$ [1] and 2009
PHENIX inclusive $\pi^{0}$ $A_{LL}$ [14], exhibits a positive polarized gluon
helicity function [2]. The fitted integral of the gluon helicity function in
this $x$ region ($\Delta G=\int_{0.05}^{1}\\!\Delta g\,\mathrm{d}x(Q^{2}=10$
GeV${}^{2})=0.2^{+0.06}_{-0.07}$ within $90\%$ confidence level) indicates a
first non-zero gluon polarization contribution to proton observed at RHIC.

Higher energy (i.e. 500/510 GeV) $p+p$ collisions at RHIC provide a means to
access lower x gluons. In 2012, STAR recorded around 50 $\textrm{pb}^{-1}$
luminosity of longitudinal polarized $p+p$ data with an average beam
polarization of $53\%$. The jets were reconstructed with the anti-$k_{T}$ jet
algorithm. The cone radius R was reduced to 0.5 in order to help suppress the
backgrounds from pile up and enhance the detector-level jet and the parton-
level jet association probabilities. The top panel of figure 4 shows the STAR
preliminary result of the inclusive jet longitudinal double spin asymmetries
$A_{LL}$ measured in 2012 510 GeV $p+p$ collisions in comparison with the 2009
inclusive jet $A_{LL}$ as a function of $x_{T}$. The 2012 510 GeV inclusive
jet $A_{LL}$ result is in agreement with the 2009 200 GeV result in the
overlapping region, and extends to the lower $x_{T}$ region.

RHIC operated another 200 GeV longitudinally polarized proton-proton run in
2015. The inclusive jet $A_{LL}$ measurement from the combined statistics from
2009 to 2015 will provide better constraints with improved statistics on the
gluon polarization in the same $x$ region as the 2009 result. In addition to
the 200 GeV longitudinal proton-proton run, RHIC also performed 510 GeV
longitudinal proton-proton collisions with around $600\ \textrm{pb}^{-1}$
delivered luminosity in 2012 and 2013. The di-jet production can access low x
gluons via asymmetric partonic scattering. Gluon polarization in the lower $x$
region will be probed by the ongoing 2012/2013 di-jet analysis at the higher
center of mass collision energy. Impacts from these data sets together with
the PHENIX forward $\pi^{0}$ $A_{LL}$ projections are shown in the bottom
panel of Figure 4. The inner uncertainty band stands for the DSSV global QCD
fit within a $90\%$ confidence level limit on the running integral of $\Delta
g$ including the projected uncertainties up to the 2015 running period.

##  4 Summary and Outlook

The latest results of the STAR jet measurements improve our understanding of
both the unpolarized and polarized gluon distribution functions. The new
inclusive jet cross section is expected to reduce the uncertainty on the gluon
distribution function in high x region ($x>0.1$). A non-zero gluon spin
contribution in the range of $0.05<x<1$ is indicated from the 2009 inclusive
jet double spin asymmetry $A_{LL}$ result. The total gluon spin contribution
could be $90\%$ confidence level possibility of the same order as the quark
polarization contribution to the proton spin.

The uncertainties of the polarized gluon PDF at low $x$ remain large compared
to the currently probed region. Longer 500 GeV longitudinally polarized
proton-proton operations at RHIC and a possible forward upgrade will provide
opportunities to measure di-jet production at STAR in the forward
pseudorapidity with high precision data. Such observables are under study, and
future analysis will impact the determination of the gluon helicity dependent
PDFs and provide new constraints for the proton spin at low $x$ region.

##  5 References

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