Materials Transactions, Vol. 60, No. 2 (2019) pp. 263 to 268© 2019 Japan Society of Powder and Powder Metallurgy

Quantitative Analysis on Light Elements Solution Strengthening in Pure Titanium
Sintered Materials by Labusch Model Using Experimental Data+1
Shota Kariya1,2,+2, Mizuki Fukuo1,+2, Junko Umeda2 and Katsuyoshi Kondoh2,+3
1

Department of Mechanical Engineering, Graduate School of Osaka University, Suita 565-0871, Japan
Joining and Welding Research Institute, Osaka University, Ibaraki 567-0047, Japan

2

Solid solution strengthening eﬀect by oxygen (O) and nitrogen (N) atoms of ¡-titanium (Ti) materials was quantitatively evaluated using
Labusch model by consideration of the experimental data. When using Labusch model to predict solid solution strengthening improvement, anapplication of the isotropic strains by solute elements is generally assumed to estimate Fm value. It is, however, diﬃcult to exactly calculate Fmvalue for ¡-Ti materials with O and N solute atoms because the anisotropic strains are induced in ¡-Ti crystal with hcp structure by theseelements. In this study, Fm value was experimentally derived from the relationship between 0.2% yield stress and solute elements (O and N
atoms) content of powder metallurgy Ti materials. As a result, the strengthening improvement was proportional to c2/3/Sf (c: soluted atom
content, Sf: Schmid factor), and its factor of proportionality of Ti­O and Ti­N materials was 4.17 © 103 and 3.29 © 103, respectively. According
to this analysis, it was clariﬁed that Fm value of Ti­O and Ti­N materials was 6.22 © 10¹10 and 5.21 © 10¹10, respectively, and then theestimated strengthening improvement by using these values was signiﬁcantly agreed with the experimental results of PM Ti materials with O andN solution atoms. [doi:10.2320/matertrans.Y-M2018849](Received September 11, 2018; Accepted October 19, 2018; Published January 25, 2019)
Keywords: solid solution, titanium, oxygen, nitrogen, Labusch model

1.

Introduction

Titanium (Ti) has excellent properties such as high speciﬁcstrength, excellent corrosion resistance and high biocompatibility due to the osseointegration.1,2) These performances
make it possible to apply Ti materials to the medical devices
and transport instruments.3­7) From a viewpoint of thestrength improvement, there is a serious problem of theconventional Ti­6 Al­4 V alloy (ASTM B348 Gr5) because
it contains vanadium element which is one of toxic andexpensive rare metals.8) Regarding this issue, recently theubiquitous element such as oxygen, nitrogen, carbon, silicon,
iron etc., are employed instead of rare metals used as thealloying elements to improve the mechanical strength of Ti
materials. For instance, there are Ti materials strengthened
by TiC particles produced via a reaction between additional
carbon particles and titanium matrix,9,10) grains reﬁnement
using silicon addition,11,12) and solid solution using oxygen or
nitrogen atoms.13­18) In these materials, Orowan mechanismand Hall-Petch relationship are often used to quantitatively
estimate the eﬀect of ubiquitous element addition on theincrement of their yield strength.19­21) Furthermore, in regard
to estimation of solid solution strengthening in metals,
Cottrell proposed the model regarding the pinning eﬀect on
dislocations movement by solute atoms,22) and Fleischer
and Friedel explained the role of obstacles touched bydislocations under the stress applied.23,24) However, noconsideration on the interaction between the solute atoms
was taken in these models. Mott, Nabarro and others
established many theories which overcome this problem,25)
and many researchers previously employed these models toquantitatively understand and calculate the solid solution+1

This Paper was Originally Published in Japanese in J. Jpn. Soc. Powder
Powder Metallurgy 65 (2018) 407­413.
+2
Graduate Student, Osaka University+3
Corresponding author, E-mail: kondoh@jwri.osaka-u.ac.jp

strengthening eﬀect. Labusch also established the similar
model regarding the solid solution strengthening behavior.26)
In this model, the strengthening increment was predicted byusing the premises that solute atoms apply isotropic strain.
Therefore, it is diﬃcult to apply Labusch model on ¡-Ti
material having an anisotropic strain due to its hcp
structure.27) At this moment, there is no report to show agood agreement between experimental data and calculated
results by using above-mentioned models regarding theinterstitial solid solution strengthening of ¡-Ti materials.
This study discussed the eﬀect of the interstitial solid
solution elements such as oxygen and nitrogen on the
microstructures and mechanical properties of powder metallurgy (PM) Ti materials with oxygen or nitrogen atoms. In
particular, the parameters used in Labusch model was
obtained from the experimental data, and the solid solution
strengthening increment of PM Ti­O and Ti­N materials
was quantitatively calculated. As a result, a good agreement
between experimental data and calculated results was
obtained.
2.

Experimental Procedure

2.1 Materials fabrication
Commercial pure Ti powder (TC-450, TOHO TECHNICAL SERVICE Co., Ltd) and TiO2 particles (KOJUNDOCHEMICAl LABORATORY Co., Ltd) were used as starting
materials. Pure Ti powder prepared by the hydride anddehydride (HDH) process, having a mean particle size of
27.8 µm, shows a typical irregular morphology. TiO2
particles with around 3.6 µm mean particle size were
employed as oxygen providers to pure Ti materials. In thisexperiment, each elemental mixture (200 g) of pure Ti andTiO2 powder such as Ti­0.5 mass% TiO2, Ti­1.0 mass%
TiO2, Ti­1.5 mass% TiO2 and Ti­2.0 mass% TiO2, was put
into the jar, and uniformly mixed by rock milling process(60 Hz, 3.6 ks).

264

S. Kariya, M. Fukuo, J. Umeda and K. Kondoh

Elemental mixture of Ti-TiO2 powder
Spark Plasma Sintering (SPS)
1000Υ for 1.8 ks

1100Υ for 1.8 ks
Homogenization
1000Υ for 86.4 ks

Hot extrusion (extrusion ratio; 36)
1000Υ

1100Υ

Ti1000-O

Ti1100HT-O

2.2 Materials characterization
Oxygen (O) and nitrogen (N) contents of the extruded rods
were measured using a highly accurate O/N/H analyzer(EMGA-930, HORIBA, Ltd.). The microstructures of ﬁnal
products were characterized using X-ray diﬀraction (XRD,
XRD-6100, SHIMADZU Corp.) and electron back-scatter
diﬀraction (EBSD) that attached to a ﬁeld emission scanningelectron microscopy (FE-SEM, JSM-6500F, JEOL Ltd.).
EBSD samples were polished by electro-chemical etching
using 95 vol% CH3COOH + 5 vol% HClO4 solution with
20 V for 60³90 s. The tensile properties of these samples were
evaluated at room temperature under a nominal strain rate of
5.0 © 10¹4/s using a mechanical tensting machine (Autograph AG-X 50 kN, SHIMADZU CORPORATION). At least
two tensile specimens with 15 mm in gauge length and 3 mm
in diameter were machined from each extruded rod.

Fig. 1 Flowchart of specimen preparation of Ti1000-O and Ti1100HT-O
materials by powder metallurgy route.

3.

The powder mixture was consolidated using a spark
plasma sintering (SPS, SPS-1030, SPS Syntex Inc.). The SPS
conditions at 1000 and 1100°C for 1.8 ks in vacuum (6 Pa)
were selected, and 30 MPa pressure was applied to thepowder in the carbon die. The diameter of each SPSed
columned billet was 42 mm.
The homogenization heat treatment at 1000°C for 86.4 ks
was applied to the latter billet by using the vacuum furnace(FT-1200R-120, FULL-TECH FURNACE Co., Ltd). The
SPSed billets were extruded by a hydraulic direct pressmachine (SHP-200-450, Shibayama kikai Co., Ltd), where
the extrusion speed was 3 mm s¹1, the extrusion die diameterwas 7 mm and the extrusion ratio was 36. Before hot
extrusion, the billet was pre-heated at 1000 and 1100°C for
300 s in Ar gas atmosphere using the image furnace, andimmediately supplied to the extrusion process. Each extruded
rod was named as Ti1000-O and Ti1100HT-O, respectively
as shown in Fig. 1.

In order to investigate the solid solution strengthening
eﬀect by oxygen and nitrogen elements on the mechanicalproperties of PM titanium materials, each content of Ti1000O and Ti1100HT-O samples was analyzed. Oxygen andnitrogen contents of raw pure Ti powder, Ti1000-O andTi1100HT-O materials were shown in Table 1, where thecalculated oxygen content from TiO2 addition was also
shown. Nitrogen content of each material was 0.07 at%, thatis, there was no signiﬁcant diﬀerence in all samples. On theother hand, the oxygen content of each material gradually
increased with increase in the TiO2 addition, and agreed wellwith the calculated value as shown in Table 1. It means thatthe oxygen content of the extruded materials can bequantitatively controlled by TiO2 particle addition. Ti
material with 2 mass% TiO2 addition (Ti1100HT-2.76O) has
an extremely high oxygen content of 2.76 at% compared with
ASTM/ASME Grade 1 (0.54 at%) and Grade 4 (1.19 at%)pure Ti.
XRD analysis on Ti1000-O and Ti1100HT-O wasconducted in order to evaluate the solid solution behavior

Results and Discussion

Table 1 Oxygen and nitrogen contents of raw pure Ti powder and PM Ti­O materials in this study, and calculated oxygen content from
TiO2 addition of each material.

Quantitative Analysis on Light Elements Solution Strengthening in Pure Titanium Sintered Materials by Labusch Model

(200)

2.76O
1.68O
0.60O
68

70

72

74

76

2.965
2.960
2.955
2.950
䖃 Ti1000
䚽 Ti1100HT

2.945
2.940

1

2

Fig. 2 XRD proﬁles of Ti1100HT-O materials with diﬀerent oxygen
contents.

IPF maps and inverse pole ﬁgures of Ti1100HT-O
materials with oxygen content of 0.60 at%, 1.68 at% and2.76 at% are shown in Fig. 4. The matrix of all samples
consisted of equiaxial ¡-Ti grains, not acicular grains. The
cooling rate after extrusion was about 6°C/s, and was enoughto prevent the formation of martensite phases due to thetransformation from ¢ to ¡-Ti,29) and resulted in theformation of equiaxial ¡-Ti grains in the extruded materials.

4.700
4.695
4.690
4.685
䖃 Ti1000
䚽 Ti1100HT

4.680
4.675
0

3

Oxygen content (at.%)

1

2

3

Oxygen content (at.%)Dependence of lattice parameters in (a) a-axis and (b) c-axis on oxygen content of Ti1000-O and Ti1100HT-O materials.

(a) Ti1100HT-0.60O

50 μm

Fig. 478

Diffraction angle, 2θ / deg.

Lattice constant in c-axis (ύ)

2.970

0

Fig. 3

(112)

(b)

(a)

Lattice constant in a-axis (ύ)(201)
Ti(103)

Intensity (a.u.)in ¡-Ti crystal with a hexagonal close packing (hcp)structure. It is well known that ¡-Ti crystal lattice isexpanded in c-axis direction by the interstitial solid solution
elements such as oxygen and nitrogen atoms.27) Figure 2
indicates XRD analysis results of Ti1100HT-O with various
oxygen contents. In this study, each XRD proﬁle shows nopeak shift at 74.04° corresponding to (200) cylindrical plane
which is not aﬀected by the lattice expansion behavior in
c-axis direction. On the other hand, ¡-Ti peaks at 70.66°,
76.22° and 77.55° related to basal and prismatic planes
obviously shifted to a lower angle due to the interstitial solidsolution phenomenon. By using the XRD peak shifts shownin Fig. 2 and that of Ti1000-O, the dependence of the lattice
parameters of ¡-Ti hcp structure in a-axis and c-axis
directions on the oxygen content were calculated. As shownin Fig. 3, the lattice constant in c-axis direction (b)proportionally increased with increase in oxygen contents
while that in a-axis (a) revealed no change in increasing theoxygen content. Previous studies reported that the TiO2
particles were thermally decomposed during sintering with
Ti powder.13,14,28) In this study, too, oxygen atoms should bein solid solution, because we applied higher temperature andlonger time in the heat treatment to the SPSed billets.
Therefore, the additive TiO2 particles were completely
decomposed and all oxygen atoms originated from TiO2
particles were soluted into ¡-Ti crystals of Ti1100HT-O andTi1000-O materials. It resulted in the lattice expansion in
c-axis direction.

265(b) Ti1100HT-1.68O

50 μm(c) Ti1100HT-2.76O

Extrusion
direction

50 μm

IPF maps of Ti1100HT-O materials with (a) 0.60 at%, (b) 1.68 at% and (c) 2.76 at% oxygen.

S. Kariya, M. Fukuo, J. Umeda and K. Kondoh

0.35

1200
1100
1000
900
800
700
600
500
400
300

0.30
0.25
0.20
0.15
0.10

䚽 䖃 YS䕻 䕺 UTS䕧 䕦 Elong.

012
Oxygen content (at.%)Elongation, δTable 2 Mean grain size and Schmid’s factor measurements of Ti1000-O
and Ti1100HT-O materials.

Tensile strength, σ /MPa

266

0.05
0.00
3

Fig. 5 Tensile properties dependence on oxygen content of Ti1000-O(closed) and Ti1100HT-O materials (opened).

1200
O solid solution

Yield stress, σY S /MPa

All specimens showed strong (10-10) cylindrical textures
along the extrusion direction, that is, these texture formation
was not related to the oxygen content of Ti1100HT-O
materials. Table 2 shows the mean ¡-Ti grain size andSchmid factor of PM Ti materials prepared in this study.
Ti1000-O materials had ﬁner grains compared to Ti1100HTO materials. This result is interpreted that the higher
temperature SPS and heat treatment ¢ phase which wereapplied to Ti1100HT-O materials resulted in the signiﬁcantgrain coarsening. In all specimens, the oxygen content
increment caused ¡-Ti grains reﬁnement because of the solute
drag eﬀect by agglomeration of oxygen soluted atoms at ¡-Ti
grain boundaries.30­32) For example, the grain size decreased
from 11.6 µm to 8.2 µm in case of Ti1000-O and from
26.1 µm to 11.3 µm in Ti1100HT-O materials. It is obviousthat a small amount of oxygen solution was eﬀective toaccelerate the grains reﬁnement. When more than about 1 at%oxygen was soluted in the materials, however, the grain size
was constant and never drastically changed because so muchoxygen atoms existed at the boundaries and then the above
mentioned solute drag eﬀect was saturated. On the other
hand, Schmid factor was 0.443³0.459 and 0.436³0.455 in
Ti1000-O and Ti1100HT-O, respectively. In both materials,
this factor was independent of the oxygen content. In general,
Schmid factor is aﬀected by the crystal structure and plastic
working conditions.33) As shown in Fig. 3(b), however, theoxygen solution in ¡-Ti grains of these specimens caused thelattice constant change from 4.680 ¡ to 4.695 ¡ in c-axis
direction, that is, the expansion ratio was only 0.3%.
Therefore, the lattice expansion by oxygen solution had very
few eﬀects on Schmid factor changes of PM extruded Ti
materials, and the extrusion process strongly inﬂuenced
Schmid factor of each specimen. In general, it is reported thatcrystal grains induced with plastic strain has multi-color ordiﬀerent shades.34­36) However, IPF maps of extruded
Ti1100HT materials with various oxygen contents showed
the grains with unicolor. It means that the introduced strainsin the matrix during hot extrusion was relieved due todynamic recrystallization.
Figure 5 indicates tensile properties of Ti1000-O andTi1100HT-O materials. 0.2% yield stress (0.2% YS) andultimate tensile stress (UTS) increased with increase in theoxygen addition. The 0.2% YS and UTS were 445 and602 MPa of Ti1100HT-0.60O, 765 and 903 MPa of

1000

Grain refinement

800
600
400
200

Base

0
0.60

1.68

2.76

Oxygen content (at.%)Fig. 6 Strengthening factors contribution to yield stress of Ti1100HT-O
materials.

Ti1100HT-1.68O and 1007 and 1094 MPa of Ti1100HT2.76O, respectively. On the other hand, the elongation to
failure was more than 20% when oxygen content was lessthan 2.34 at%. while it was 7.5% in case of oxygen content
over 2.76 at%. The principal slip plane of hcp metals strongly
depends on c/a ratio, where a basal plane slipping is
dominant in long hcp structure (c/a > 1.61) and a prismatic
plane slipping occurs in short hcp one (c/a < 1.61).37) In
addition, it is also reported that ¡-Ti materials with oxygen
solid solution show poor ductility because the principal slip
system changes from a prismatic plane to basal plane slipping
due to the lattice expansion in c-axis direction by oxygen
atoms solution. The strengthening contribution factors on
0.2% YS of Ti1100HT-O materials with diﬀerent oxygen
contents were shown in Fig. 6. As mentioned above, since
TiO2 particles were completely dissolved and there was noresidual strain in EBSD analysis results, Therefore, it was
deduced that there are neither eﬀects of TiO2 particles
dispersion nor eﬀects of residual strain on 0.2% YS
increment. The major strengthening factors in Ti­O materials
were both the grain reﬁnement and the oxygen solid solution.

Quantitative Analysis on Light Elements Solution Strengthening in Pure Titanium Sintered Materials by Labusch Model

267

Fig. 7 Relationship between increment of yield stress by (a) oxygen and (b) nitrogen solid solution strengthening eﬀects and material’s
factor in Labusch model.

Therefore, the oxygen solid solution strengthening factor
could be considered quantitatively after modifying the grain
reﬁnement eﬀect by Hall-Petch relationship. Hall-Petch
constant of 15.7 MPa/mm2 was used.20) 0.2% YS increment
by grain reﬁnement from Ti1100HT-0.60O to Ti1100HT1.68O and Ti1100HT-2.76O was 50 MPa and 37 MPa,
respectively. These values were signiﬁcantly small comparedto oxygen solution strengthening of 269 MPa (Ti1100HT1.68O) and 525 MPa (Ti1100HT-O-2.76O). It was concluded
that the solid solution strengthening by oxygen atoms in ¡-Ti
was the main factor to improve the mechanical strength ofTi­O materials.
In order to theoretically study the oxygen solution
strengthening behavior of PM Ti­O materials, the quantitative evaluation of its strengthening eﬀect by using Labusch
model26) dealing with the interaction between edge dislocations and solid solution atoms. In this model, the elastic
interaction energy E is given by eq. (1), where " is a volume
of solid solution atoms and the stress ﬁeld · in solute atom is
considered to be a constant on the basis of hypothesis that
distance between dislocation and solid solute atom isnegligibly small compared with radius of solid solute atoms.
E ¼ · ij ¾ij :ð1ÞWhen the solution atoms give isotropic stress, ¾ij is equal
to ¾ ¤ij. ¤ij corresponds to Kronecker’s delta,38) and then ¤11,
¤22 and ¤33 should be 1 and other components are 0.
Therefore, eq. (1) could be expressed as the below eq. (2);E ¼ ð· 11 ¾11 þ · 22 ¾22 þ · 33 ¾33 Þ¼ ð· 11 þ · 22 þ · 33 Þ¾:

ð2Þ

Expanding eq. (2) into eq. (3) under the conditions of edge
dislocation, the elastic interaction energy about edge
dislocation is shown as follows;
E ¼ Gb

ð1 þ ¯Þy2
3³ð1  ¯Þ ðx þ y2 Þð3Þwhere G is stiﬀness, b is size of burger’s vector, ¯ is Poisson
coeﬃcient, and (x, y) is relative coordinates of the solid solute
atom from an edge dislocation. Partial diﬀerentiating eq. (3)by a valuable x, the interaction force F (eq. (4)) between asolid solute atom and edge dislocation is obtainable.
F¼@E
ð1 þ ¯Þ
2xy
¼ Gb
:
@x
3³ð1  ¯Þ ðx2 þ y2 Þ2ð4ÞIn Labusch model, the increment of 0.2% YS "·YS is shownin the below equation with Fm representing the maximum
value of F,1=3¸
c2=3 Fm 4 w
· YS ¼¼
ð5Þ
SFSF 4Gb9
where c is atomic ration of interstitial solid solute atom, SF is
Schmid factor, and w is parameter representing the area
aﬀected by interaction between an edge dislocation and solid
solute atom. (Fm4w/4Gb9)1/3 is constant on Ti­O material,and c2/3/SF is various on oxygen content and fabrication
process. However, solid solute oxygen atom applies anisotropic stress and strain on titanium crystal. Therefore, it is
not possible to determine the parameter, Fm theoretically. In
this study, hence, we derived this parameter experimentallyfrom the tensile-test data as follows. In Fig. 7(a), 0.2% YS
increment by the solid solution oxygen atom was plottedagainst the material’s factor, (c2/3/SF). We can determine thisfactor experimentally. It is obvious that a linear relationshipwas obtained as shown in Fig. 7(a). Here, it is noted that thestraight line should pass the zero point. From the linear
relationship, it was derived that the constant, (Fm4w/4Gb9)1/3
of Ti­O materials is 4.17 © 103 and the parameter, Fm is6.22 © 10¹10.
It is also reported that nitrogen atoms have a solid solution
strengthening eﬀect in ¡-Ti materials in similar to oxygen
atoms.27) In Fig. 7(b), 0.2% YS increment was plottedagainst the factor, c2/3/SF in Ti­N materials. We employed
the experimental data in previous study.16) It is obvious that
0.2% YS increment again increased in proportion to thematerial’s factor, c2/3/SF (R2 = 0.943). It was derived that theconstant, (Fm4w/4Gb9)1/3 of Ti­N materials is 3.29 © 103
and the parameter, Fm is 5.21 © 10¹10.
In order to evaluate the adequacy of the above Fm values ofTi­O and Ti­N materials, the calculated 0.2%YS incrementof three kinds of Ti­O­N materials (Ti1100HT-0.97O-0.78N,
Ti1100HT-1.25O-0.53N, Ti1100HT-1.54O-0.30N) by using
Fm value was compared with the experimental value. The
results are summarized in Table 3. Each Ti­O­N material
was fabricated by using the pre-mixed Ti­TiO2­TiN
powders. Experimental data of 0.2% YS was 800 MPa for
Ti1100HT-0.97O-0.78N, 789 MPa for Ti1100HT-1.25O0.53N and 795 MPa for Ti1100HT-1.54O-0.30N, respectively. Each calculated 0.2%YS was modiﬁed by excluding
the grains reﬁnement strengthening eﬀect. In Labusch model,

268

S. Kariya, M. Fukuo, J. Umeda and K. Kondoh
Table 3Calculated solid solution strengthening eﬀect and 0.2% YS increment modiﬁed from tensile test results.

the total increment of 0.2%YS of Ti­O­N materials was asum of the increment by each solution atom because theinteraction between a solid solute atom and dislocation was
subscribed as an integrated value. As shown in Table 3, thecalculated values of Ti­O­N materials by using Labusch
model showed a good agreement with the experimental data.
It is concluded that the solid solution strengthening of Ti­O
and Ti­N materials can be successfully and quantitatively
estimated by Labusch model with a suitable Fm value
obtained by the experimental method proposed in this study.
4.

Conclusion

It is diﬃcult to calculate 0.2% YS increment by oxygen or
nitrogen solid solution theoretically by Labusch model, sincesolid solution of oxygen applies anisotropic strain on ¡-Ti
materials having a hcp structure. In this study, thereby, 0.2%
YS increment by oxygen or nitrogen solid solution in the ¡-Ti
materials was successfully calculated by the following
procedures. At ﬁrst, material’s factor, c2/3/SF was plottedagainst 0.2% YS increment by assuming the parameter,(Fm4w/4Gb9)1/3 is a constant. It was shown that linear
relationship is obtainable between the 0.2% YS increment andthe material’s parameter both in Ti­O and Ti­N materials.
Then, the proportion coeﬃcient and hence the maximum force(Fm) between the edge dislocation and solid solution atom wasdetermined from such a linear relationship. Fm was 6.22 ©
10¹10 for Ti­O materials and 5.21 © 10¹10 for Ti­N materials.
Finally, the present model was applied to Ti­O­N materials. It
was concluded that the calculated 0.2% YS increment shows agood agreement with the experimental results.
Acknowledgments
This work was partially supported by the Japan Scienceand Technology Agency (JST) under Industry­Academia
Collaborative R&D Program “Heterogeneous Structure
Control: Towards Innovative Development of Metallic
Structural Materials”.
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