X. ZHANG et al.: STATISTICAL STUDY OF THE PRIMARY-DENDRITE SPACINGS IN SINGLE-CRYSTAL SUPERALLOYS
149–153
STATISTICAL STUDY OF THE PRIMARY-DENDRITE SPACINGS
IN SINGLE-CRYSTAL SUPERALLOYS
STATISTI^NA ANALIZA PRIMARNIH MEDDENDRITNIH
PROSTOROV V MONOKRISTALNI SUPERZLITINI
Xiaoli Zhang
1
, Tingzhen Xin
1
, Shuaishuai Yue
1
, Xiangbin Meng
2
, Guiqun Liu
1*
1
School of Material Science and Engineering, North Minzu University, Yinchuan 750021, China
2
School of Mechanical and Automotive Engineering, Qilu University of Technology , Shandong Academy of Sciences, Jinan 250353, China
Prejem rokopisa – received: 2020-08-13; sprejem za objavo – accepted for publication: 2020-10-05
doi:10.17222/mit.2020.161
Three generations of single-crystal superalloys (AM3, DD5 and DD90) were produced with directional solidification and seed-
ing. The square-calculation method and direct-measurement method were used to study the primary-dendrite spacings of the
three single-crystal superalloys at different solidification distances (50, 150 and 250) mm. Statistical results showed that, firstly,
as the solidification heights increased, the average primary-dendrite spacings of the three alloys were gradually enlarged; sec-
ondly, the average primary-dendrite spacings calculated with the square-calculation method and the direct-measurement method
were similar, which indicated the high precision of the two methods; thirdly, the primary-dendrite spacings of the three alloys
were distributed randomly in a relatively large range, following a normal distribution. The maximum values of the primary-den-
drite spacings were 4.7–8.5 times larger than the minimum ones at different solidification distances and were different from the
values from previous studies.
Keywords: directional solidification, seeding, single-crystal superalloy, primary-dendrite spacing
Avtorji so izdelali tri generacije monokristalnih superzlitin (AM3, DD5 in DD90) s postopkom direktnega strjevanja in sejanja.
Uporabili so ra~unsko metodo kvadratov in direktne meritve za {tudij razdalje med primarnimi dendritnimi vejami treh
monokristalnih superzlitin na razli~nih razdaljah strjevanja na (50, 150 in 250) mm. Statisti~ni rezultati {tudije so pokazali: 1)
da solidifikacijska vi{ina nara{~a in da povpre~na primarna velikost meddendritnih prostorov postopno nara{~a za vse tri
preiskovane zlitine; 2)drugi~: da sta si povpre~ni velikosti primarnih dendritnih medprostorov, izra~unanih z ra~unsko metodo
kvadratov in metodo direktnega merjenja, podobni kar ka`e na visoko natan~nost obeh metod ter 3) primarni dendritni
medprostori vseh treh preiskovanih zlitin so porazdeljeni naklju~no in v relativno {irokem obmo~ju sledijo normalni velikostni
porazdelitvi. Maksimalne vrednosti velikosti primarnih meddendritnih prostorov so 4,7-krat do 8,5-krat ve~je od najmanj{ih pri
razli~nih razdaljah strjevanja, kar se razlikuje od predhodnih raziskav.
Klju~ne besede: direktno strjevanje, sejanje, monokristalna superzlitina, primarni meddendritni prostori
1 INTRODUCTION
The directional-solidification technology is one kind
of casting technique. According to this technique, molten
materials solidify along the opposite direction of the heat
flow when the unidirectional temperature gradient be-
tween the solid and melt is built using a special method,
which is to ensure the growth direction of crystals in par-
allel and contrary to the direction of the heat flow.
1
Dur-
ing solidification, a negative temperature gradient ap-
pears when the temperature at the solid-liquid interface
goes up due to the latent heat of crystallization. Thus, the
liquid phase is in the supercooled condition. Under this
circumstance, the solid-liquid interface is unlikely to be
flat-shaped. Instead, a lot of of dendrites (the so-called
primary dendrites) start growing into the liquid with
some secondary dendrites formed on the crystal axis.
2
By
studying the primary-dendrite spacing, we can evaluate
the dendritic structure of materials, which is decisive for
the material performance.
1,3
Due to the disappearing of crystal boundaries,
Ni-based single-crystal superalloys exhibit excellent
high-temperature properties and are preferred materials
for aero-engine turbine blades.
4–9
Due to the use of the
seeding method or selecting method, the structure of a
single-crystal superalloy includes cross-shaped den-
drites, making a significant effect on the thermo-mechan-
ical properties and heat treatment of the single-crystal
superalloy.
6
According to the earlier theoretical and experimental
studies, there is a relationship between primary-dendrite
spacing and solidification parameters (c
0
, G
L
, v), or
superalloy properties, and a theoretical model of the den-
dritic-structure characteristic parameter was also estab-
lished.
10–16
However, a lot of recent research.
13,17–20
proved that there is a wide range of tolerance for stable
primary-dendrite spacing under a given solidification
condition. The tolerance of dendrite spacing is related to
the solidification history. J. D. Hunt et al.
13,21
indicated
that the biggest spacing between dendrites or crystal
cells is twice the size of the lowest one. Furthermore,
G. L. Ding et al.
17
revealed that the biggest spacing be-
tween dendrites or crystal cells is more than twice the
Materiali in tehnologije / Materials and technology 55 (2021) 1, 149–153 149
UDK 519.23:551.312.1:669-172 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 55(1)149(2021)
*Corresponding author's e-mail:
gqliu10b@alum.imr.ac.cn (Guiqun Liu)

size of the lowest one, but they did not specify how many
times it occurs. In fact, a lot of errors cannot be avoided
during a dendrite-spacing measurement because of the
diversity of the crystallographic orientations and crystal
boundaries in polycrystalline materials. For this reason,
it is necessary to examine the dimensions of a pri-
mary-dendrite spacing without the diversity in the crys-
tallographic orientations and crystal boundaries.
A primary-dendrite spacing refers to the spacing be-
tween primary-dendrite axes of two neighboring den-
drites. As for now, the most accurate measurement of a
primary-dendrite spacing is the direct-measurement
method. Although the direct-measurement method is
more accurate, it requires massive work. However, in
metallic materials, cross-shaped dendrites grow in a
square pattern, indicating axes of primary dendrites ar-
ranged in the form of a square lattice, which means that
the primary-dendritic spacing can be determined with the
number of the lattice squares. This method is called the
square-calculation method, which is simple and requires
less work. The materials used in this study were three
kinds of a Ni-based superalloy (the first-generation
AM3, the second-generation DD5 and the third-genera-
tion DD90). The three superalloys were [001] single-
crystal superalloys made with the seeding method.
2. EXPERIMENTAL PART
The chemical compositions of the three single-crystal
superalloys (AM3, DD5 and DD90) used in the present
work are listed in Table 1. In order to assure the same
compositions of the samples and seeds, the chemical
compositions of the cast superalloys and seeds are the
same as in Table 1.
The three Ni-based single-crystal superalloys were
produced with directional solidification and seeding in
order to eliminate the effects of the grain boundaries and
crystallographic orientation on the primary-dendrite
spacings. To ensure the directions of the dendritic growth
and thermal gradient are the same, the [001] direction of
every seed was aligned to the growth direction and the
[010]/[100] direction was parallel or vertical to the plat-
form edges. The length, width and height of every seed
were8×8×18mm.Thelength, width and height of ev-
ery specimen were 10 × 10 × 300 mm.
Directional experiments were performed in a
Bridgman high-rate solidification (HRS) furnace with a
twin-zone resistance heater. In order to obtain a partial
melt of the top of every seed, the mold was preheated to
1600 °C. At the same time, the superalloy was super-
heated to 1600 °C and kept at the temperature for 5 min
to homogenize the melt. The melt was then poured into
an empty mold. Its dwell time was 10 min to ensure the
system reached thermal equilibrium. Then, the mold was
at a constant speed of 6 mm/min. The samples were
cleaned when the mold shells were cooled to room tem-
perature.
Cross-section samples at different solidification dis-
tances (50, 150 and 250) mm were cut with wire cutting
and used to investigate the dendritic structure with a
LEICA optical microscope. All the samples were mi-
cro-etched in a mixture of HCl (80 mL) + H
2
SO
4
(5 mL)
+ CuSO
4
(20 g) + H
2
O (100 mL). Finally, primary-den-
drite spacings were calculated using metallographic im-
ages and image-analysis software Image-Pro Plus 6.0. In
order to reduce experimental errors, the same
metallographic images were used for direct measurement
and square calculation.
Primary-dendrite spacing is the distance between the
dendritic trunks of two adjacent dendrites, so the most
accurate statistical method of primary-dendrite spacing is
direct measurement. This method is carried out as fol-
lows: the distance between the dendritic trucks of two
adjacent dendrites is measured directly. After many mea-
surements, the average values are taken. In this study, the
metallographic images were opened in Image-Pro Plus
6.0. The primary-dendritic spacings of the adjacent den-
drites were measured directly and the average pri-
mary-dendritic spacings were calculated after multiple
measurements.
In metal materials, cross-shaped dendrites are gener-
ally arranged in a square pattern, that is to say, four den-
drites around a dendrite form a square. Therefore, the
square-calculation method is carried out as follows:
firstly, a large number of transverse metallographic im-
ages at different locations are taken with a metallo-
graphic microscope. Secondly, the number of dendrites
in each image is counted as the N. The dendrite trunk of
each dendrite is taken as the counting point in statistics.
A dendrite trunk is neglected when it is outside the im-
age boundary. A dendrite trunk is 0.5 of a counting point
at the image boundary, and it is one counting point when
it is inside the image. Then, the S area of the image is
calculated in accordance with the magnification of the
image. Finally, the average primary-dendrite spacing is
defined as: 1
=( S/N)
0.5
.
3. RESULTS AND DISCUSSION
3.1 Dendrite morphology
Figure 1 shows dendritic morphologies of the trans-
verse section of the AM3 single-crystal superalloy at dif-
X. ZHANG et al.: STATISTICAL STUDY OF THE PRIMARY-DENDRITE SPACINGS IN SINGLE-CRYSTAL SUPERALLOYS
150 Materiali in tehnologije / Materials and technology 55 (2021) 1, 149–153
Table 1: Chemical compositions of the three alloys (w/%)
Alloys Cr Co Mo Re W Al Ti Ta Hf C Ni
A M 385 . 52 . 2–5623 . 5–– Bal.
DD5 6.9 7.6 1.6 3.1 5.3 6.2 – 6.6 0.14 0.038 Bal.
DD90 5 12 1566–8–– Bal.

ferent solidification distances after directional solidifica-
tion. The dendritic morphologies are cross-shaped and
the direction of the secondary-dendrite arm is the same.
However, the adjacent dendrites are not arranged in a
square. By comparing the transverse microstructures at
different solidification heights, it was found that with an
increase in the solidification height, the tertiary-dendrite
arms become more and more developed. As shown in
Figure 1c, a large number of tertiary-dendrite arms ap-
pear at a solidification height of 250 mm. In addition, the
dendrite morphologies of the three alloys (AM3, DD5
and DD90) were found to be identical when the trans-
verse microstructures of the three alloys were compared.
3.2 Statistics of primary-dendrite spacing
According to the square-calculation method, the
more dendrites are counted, the fewer errors are gener-
ated during the statistical treatment of the average pri-
mary-dendrite spacing. In order to reduce the errors, we
selected metallographic pictures at a 15× magnification.
It was found that the number of dendrites in the
metallographic pictures of the three superalloys with dif-
ferent solidification heights (50, 150 and 250) mm was
about 400–600. Then, the average primary-dendrite
spacing was calculated with the following formula:  1
=
(S/N)
0.5
. The average primary-dendrite spacings are
shown with the black histograms in Figure 2. When the
solidification heights are the same, the average pri-
mary-dendrite spacing of the first-generation AM3 alloy
is slightly larger than those of the second-generation
DD5 and third-generation DD90. Also, the average pri-
mary-dendrite spacings of the second-generation DD5
and third-generation DD90 are approximately the same.
In order to reduce the errors of direct measurement,
the primary-dendrite spacing of every alloy was mea-
sured 500–1000 times at the same solidification height.
The average primary-dendrite spacings are shown with
the histograms in Figure 2. For the square-calculation
method and direct-measurement method, the change rule
for the average primary-dendrite spacing was identical.
When the solidification height was the same, the average
primary-dendrite spacing of the first-generation AM3 al-
loy was slightly larger than those of the second-genera-
tion DD5 and third-generation DD90. In addition, the
average primary-dendrite spacings of the second-genera-
tion DD5 and third-generation DD90 were similar.
When comparing the average primary-dendrite spac-
ing calculated with the square-calculation method and
direct-measurement method, it was found that the differ-
ence between the two methods was small, showing that
the accuracy of the square-calculation method is high.
Although the accuracy of direct measurement is high, it
is time-consuming and laborious. In order to improve the
work efficiency, the square-calculation method can re-
place the direct-measurement method for calculating the
primary-dendrite spacing.
The average primary-dendrite spacings of the three
single-crystal superalloys are shown in Figure 3. With
the increase in the solidification height, the average pri-
X. ZHANG et al.: STATISTICAL STUDY OF THE PRIMARY-DENDRITE SPACINGS IN SINGLE-CRYSTAL SUPERALLOYS
Materiali in tehnologije / Materials and technology 55 (2021) 1, 149–153 151
Figure 2: Average primary-dendrite spacings of the three alloys: a) 50 mm solidification height, b) 150 mm solidification height, c) 250 mm so-
lidification height
Figure 1: Optical micrographs showing the transverse microstructures of AM3 alloy at different solidification heights and a 50× magnification:
a) 50 mm; b) 150 mm; c) 250 mm

mary-dendrite spacings of the three alloys increased sig-
nificantly. Especially in the later stage of solidification,
the average primary-dendrite spacing increased faster
than in the earlier stage. In addition, the average pri-
mary-dendrite spacing of the first-generation single-crys-
tal superalloy AM3 was slightly larger than those of the
second-generation DD5 and third-generation DD90.
However, there was no significant difference between the
second-generation DD5 and third-generation DD90.
The primary-dendrite spacings of the AM3 superal-
loy were measured 955 times, 934 times and 578 times,
respectively, at solidification heights of (50, 150 and
250) mm. Specific values of the primary-dendrite spac-
ings are shown in Figure 4. The primary-dendrite spac-
ing was not a certain value, but a relatively large range of
random change. The primary-dendrite spacings were
mainly distributed in a range of 100–600 μm. In addition,
the primary-dendrite spacings were measured 1129
times, 956 times and 1023 times, respectively, at solidifi-
cation heights of 50, 150 and 250 mm of the DD5 super-
alloy. They were 1006 times, 912 times and 695 times
larger than those of the DD90 superalloy. The variation
rule of the primary-dendrite spacings of superalloys DD5
and DD90 was the same as that of superalloy AM3.
The lower and upper limits of the primary-dendrite
spacings calculated with direct measurement are shown
in Table 2. The primary-dendrite spacings of the three
superalloys were distributed in a large range. There were
upper and lower limits, which was consistent with earlier
research conclusions.
13,17,21
At different solidification
heights of the three superalloys, the difference between
the upper and lower limits of the primary-dendrite spac-
ings was large. With the increase in the solidification
height, the ratio of the upper limit to the lower limit was
(6.9, 6.8 and 4.7) for superalloy AM3, respectively, (6.7,
8.5 and 5.0) for superalloy DD5 and (6.9, 8.5 and 6.8)
for superalloy DD90. Therefore, the upper limit of the
X. ZHANG et al.: STATISTICAL STUDY OF THE PRIMARY-DENDRITE SPACINGS IN SINGLE-CRYSTAL SUPERALLOYS
152 Materiali in tehnologije / Materials and technology 55 (2021) 1, 149–153
Figure 4: Primary-dendrite spacings at different solidification heights of the AM3 superalloy calculated with direct measurement: a) 50 mm; b)
150 mm; c) 250 mm
Table 2: Primary-dendrite spacings of the three superalloys calculated with direct measurement
Superalloys
50 mm height 150 mm height 250 mm height
Lower limit Upper limit Ratio Lower limit Upper limit Ratio Lower limit Upper limit Ratio
AM3 82 569 6.9 90 609 6.8 136 645 4.7
DD5 87 584 6.7 70 593 8.5 119 602 5.0
DD90 98 676 6.9 80 679 8.5 102 694 6.8
Lower limit: the minimum value of the primary-dendrite spacings
Upper limit: the maximum value of the primary-dendrite spacings
Ratio: The ratio of the upper limit to the lower limit
Figure 3: Average primary-dendrite spacings of the three superalloys
at different solidification heights

three superalloys was about 4.7–8.5 times higher than
the lower limit. This was not consistent with the conclu-
sions of references,
17,21
as the difference is an almost
doubled value. However, our conclusion is similar to that
from reference
13
where the difference is far more than a
doubled value. Because the difference between the upper
and lower limits of the primary-dendrite spacing is large,
the actual primary-dendrite spacing generally refers to
the average value, as shown in Figures 2 and 3.
4 CONCLUSIONS
(1) With the increase in the solidification height dur-
ing directional solidification, the average primary-den-
drite spacing of the three single-crystal superalloys in-
creases gradually. Additionally, the increasing rate of the
average primary-dendrite spacing in the late solidifica-
tion stage is significantly higher than that in the early so-
lidification stage.
(2) For the three single-crystal superalloys (AM3,
DD5 and DD90), the difference in the average pri-
mary-dendrite spacing determined with the square-calcu-
lation method and direct-measurement method is not ob-
vious, which indicates that the accuracy of both methods
is high.
(3) The primary-dendrite spacings of the three sin-
gle-crystal superalloys are distributed randomly in a rela-
tively large range. The upper limit is 4.7–8.5 times
higher than the lower limit. They are different from the
values from the previous studies.
Acknowledgements
This work was supported by the Key Research and
Development Program of Ningxia (2018BEE03017,
2019BDE03016) and the Natural Science Foundation of
Ningxia (2018AAC03249).
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Materiali in tehnologije / Materials and technology 55 (2021) 1, 149–153 153