Journal of Engineering Science and Technology

Vol. 18, No. 3 (2023) 1713 - 1734

© School of Engineering, Taylor’s University

THE ALTERATION OF MECHANICAL BEHAVIOR FOR

Ba0.5Sr0.5Fe12O19/(Ni0.5Zn0.5)Fe2O4 NANOCOMPOSITES

MALAK YASSINE1, NOUR EL GHOUCH2,*,

A. M. ABDALLAH1, KHULUD HABANJAR1, R. AWAD1,3
1Physics Department, Faculty of Science, Beirut Arab University, 11-5020, Beirut, Lebanon
2Chemistry Department, Faculty of Science, Beirut Arab University, 11-5020, Beirut, Lebanon
3Physics Department, Faculty of Science, Alexandria University, 21511, Alexandria, Egypt

*Corresponding Author: n.ghosh@bau.edu.lb

Abstract
Ba0.5Sr0.5Fe12O19/x(Ni0.5Zn0.5)Fe2O4 nanocomposites with x = 10, 20, 30, and 50
wt.% derived from nickel zinc ferrite added to barium strontium hexaferrite were
synthesized and characterized in this work. The prepared nanoparticles and
nanocomposites were manufactured by co-precipitation and high-speed ball
milling approaches. FTIR spectroscopic analysis revealed the absorption peaks
that are characterized of both soft spinel and hard hexagonal ferrites in the
nanocomposites. Additionally, the mechanical parameters have been computed
from the FTIR data. Besides, force constant on A-site and B-site, Debye
temperature, longitudinal, transverse and mean elastic wave velocities of the
prepared samples, have been calculated. Furthermore, Vickers microhardness
( 𝐻𝑉 ) tests were made to analyse the mechanical properties of the prepared
samples under a range of applied loads (0.49-9.8 N) and temperatures (30−80
C). The 𝐻𝑉 measurements revealed that the prepared samples exhibited the
behaviour of normal indentation size effect. Diverse models were theoretically
used to study the obtained 𝐻𝑉 values. The most appropriate theoretical model to
depict the actual 𝐻𝑉 values is the modified proportional sample resistance model.
Besides, with increasing temperature, the BaSr/10%NiZn sample had the highest
𝐻𝑉 values among the nanocomposites.
Keywords: Co-precipitation method, Elastic modulus, Nanocomposites,
Temperature, Vickers microhardness.

1713
1714 M. Yassine et al.

1. Introduction
The wide application of soft spinel and hard hexagonal ferrites in the permanent
magnets and recording media industry [1], makes the nanocomposites one of the
most promising candidate for future industrial investigations. Many literature
reporting the optical, magnetic, and electrical properties of ferrite nanocomposites
[2-4]. However, the mechanical properties for such type of these nanocomposites,
which are related to the strength of these materials, have not yet been studied. The
mechanical parameters of hard/soft ferrite nanocomposites must be determined to
design and fabricate practical devices for applications such as magnetic media disk
application [4].
Knowing the mechanical properties of ferrites is critical because these materials
must be mechanically resistant to avoid the damage during component assembly
and machining [5]. In addition, the mechanical properties of hexaferrites are
significant and are affected by many manufacturing conditions including pressure,
temperature and duration of sintering [6]. Hardness tests are often employed in
industry to evaluate the usability of fabrication materials and, in particular, the
control of heat treatment performance. In this regard, the Vickers microhardness
test is a preferred method for estimating the mechanical properties of hard/soft
ferrite nanocomposites. Besides, nanoparticle inclusion has demonstrated their
efficiency in improving the mechanical characteristics and mechanical parameters
of composites [7]. The wet chemical co-precipitation approach has surpassed other
preparation methods for the synthesis of nanocomposites since it is easy,
affordable, and practical for large-scale production [8].
Vickers microhardness test was performed on many composites such as
Tokunaga et al. [9] reported that the carbon nanotubes and fullerenes, were created
via a procedure of extreme plastic deformation by high pressure torsion. Vickers
microhardness was over 120 𝐻𝑉 and grain size with 5 mass% of fullerene was
decreased to ~ 80 nm, while the grain size without fullerenes is ~500 nm. Also, it
is expected that the fullerenes would be distributed unevenly throughout the
aluminum composites, which is consistent with their mechanical characteristics.
Moreover, Farhat et al. [10] studied the mechanical characteristics of zinc erbium
oxide at different applied loads and dwell times. The load independent
microhardness data of erbium doped zinc oxide nanoparticle samples were revealed
to be best considered by the modified proportional specimen resistance model.
Furthermore, Joshua et al. [11] fabricated AA7068/TiO2 metal matrix composite
via powder metallurgy technique. They found that the inclusion of TiO2 particles
increased the microhardness of the composites.
To the best of our knowledge, no prior literature studied experimentally the
effect of the addition of (Ni0.5Zn0.5)Fe2O4 (C0) to the Ba0.5Sr0.5Fe12O19 (C1)
nanoparticles on the mechanical characteristics of the prepared nanocomposites. As
previously reported in our work [12], C1, C0 nanoparticles, and C1/xC0
nanocomposites, with 4 different ratios of 100:10, 100:20, 100:30, and 100:50,
were prepared by the co-precipitation and high-speed ball milling methods,
respectively. C1 and C0 were found to be the two primary phases of
nanocomposites, according to XRD studies. Also, C1 and C0 had hexagonal and
spherical morphologies, respectively, according to the TEM investigation. The
addition of C0 reduces the nanocomposites’ size. In addition, the magnetic
investigation revealed a majority of dipolar interactions with the C0 addition. When

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 The Alteration of Mechanical Behavior for Ba0.5Sr0.5Fe12O19/ . . . . 1715

compared to C1, the (BH)max of the nanocomposite with 10% of NiZn improved by
10%, reaching a maximum value of 22.89 kJ/m3. These prepared samples were
recognized as promising permanent magnet candidates.
In the current work, (C1), (C0) nanoparticles, and C1/xC0 nanocomposites, with
x = 10, 20, 30 and 50 wt.%, which are denoted as C1.1, C1.2, C1.3 and C1.5,
respectively, were prepared via the co-precipitation and high-speed ball milling
methods. C1 and C0 were considered as hard and soft magnetic materials. The
elastic properties were estimated by Fourier transform infrared spectroscopy. Also,
the Vickers microhardness analysis has been carried out using several empirical
and theoretical models with a dwell period of 30 s. The Vickers microhardness (𝐻𝑉 )
results are analyzed using Meyer’s law, Hays-Kendall approach, elastic/plastic
deformation model, proportional specimen resistance model, and modified
proportional specimen resistance model. The best model for characterizing the
Vickers microhardness of the prepared samples is also discussed. Additionally, the
microhardness curves are used to estimate Vickers microhardness, Young's
modulus, yield strength, and fracture toughness values. Besides, studying the effect
of temperature on the mechanical properties of the prepared samples is another aim
of this work.

2. Experimental
C1 and C0 nanoparticles and C1/xC0 nanocomposites were fabricated via the wet
chemical co-precipitation and high-speed ball milling methods, as reported in our
previous work [12].
The functional groups presented in the samples were studied using FTIR, via
Nicolet iS5 spectrometer. Around 2 mg of each sample were finely combined with
approximately 200 mg of KBr powder. The resulting powders were then made into a
disc by pressing it in a mechanical pellet presser, where the pressure was maintained
for several minutes before removing the formed KBr disk. The disc is then ready for
scanning, while 100 % KBr disc was also prepared for use as the background.
The mechanical properties were performed at room temperature with a digital
microhardness tester (MHVD-1000IS). Around 2 g of each sample was pressed into
a disc sintered at 1000 °C for 4 h. The applied static loads were 0.49 - 9.8 N
throughout the measurements, and a dwell period of 30 s. In addition, the prepared
samples are exposed to heat-treatment at different temperatures (30 - 80 °C) with
applied static load of 500 N and a dwell period of 30 s.

3. Theoretical Approach
3.1. FTIR
An FTIR spectrometer continuously records high spectral resolution data over a
wide spectral range. It offers a considerable advantage over dispersed
spectrometers, which can only determine intensity over a small number of
wavelengths at a time. The chemical bonding, metal-oxygen vibration positioning
modes, and elastic parameters of ferrites are all determined from the FTIR data
analysis. The FTIR absorption spectra of ferrites are intensely reliant on the
distribution of various cations in tetrahedral (A sites) and octahedral (B sites) of the
spinel and hexagonal lattices [13]. According to Waldron [14], the higher frequency
absorption band (𝑣1 ) is affected by stretching vibrations of the bond between the

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1716 M. Yassine et al.

cation in the tetrahedral site and oxygen, while the lower frequency absorption band
(𝑣2 ) is resulted by vibrations of the bond between the cation in the octahedral site
and oxygen. The variation in band position ( 𝑣1 and 𝑣2 ) is due to the distance
between the O2- and Fe3+ ions in both octahedral and tetrahedral sites. In tetrahedral
coordination, Fe3+ and O2- are closer together, resulting in a stronger covalent bond.
The following equations are used to compute the tetrahedral (corresponds to A
site) force constant, denoted as 𝑘𝑡 , and the octahedral (corresponds to B site) force
constant, denoted as 𝑘𝑜 , by considering the positions of high (𝑣1 ) and low
frequency (𝑣2 ) absorption band in the infrared spectrum [15].
𝑘𝑡 = 4𝜋 2 𝑣1 2 𝑐 2 𝜇𝑚 , (1)
𝑘𝑜 = 4𝜋 2 𝑣2 2 𝑐 2 𝜇𝑚 , (2)
8 -1
here the force constant and speed of light (2.99 × 10 ms ) are 𝑘 and 𝑐, respectively.
The band frequency of A/B sites in cm-1 is represented by 𝑣. The reduced mass for
the Fe3+ ions and the O2- ions (~2.061 × 10-23 g) is denoted by 𝜇𝑚 [15].
The Debye temperature (𝜃𝐷 ) is a basic property of solids that relates elastic and
thermodynamic characteristics such as specific heat, which is calculated from this
relationship proposed by Anderson [16]:
ℎ𝑐𝑣𝑎𝑣
𝜃𝐷 = = 1.438𝑣𝑎𝑣 , (3)
2𝜋𝑘𝐵

where Boltzmann's and Planck's constants are 𝑘𝐵 and ℎ, respectively, the speed of
ℎ𝐶
light is 𝑐. As a result, in ferrite materials the value of = 1.438 [16]. Also, 𝑣𝑎𝑣 is
2𝜋𝑘𝐵
the average value of the wavenumbers which is equal to (𝑣1 + 𝑣2 )/2 for the ferrites.
The average force constant is obtained using the expression:
𝑘𝑡 +𝑘𝑜
𝑘𝑎𝑣 =
2
, (4)

The study of elastic properties is significant because it defines the binding force
strength of a ferrite material and also helps to reduce the mechanical straining of
these materials during manufacturing and industrial use [17]. FTIR and XRD
analyses are employed to determine the elastic property of ferrite materials [18].
The stiffness constant and other elastic parameters are calculated using force
constants derived from infrared spectra and crystallographic parameters. According
to Waldron [14], stiffness constant 𝐶11 = 𝐶12 for the materials possessing the
cubic spinel crystal structure. Moreover, the materials possessing the hexagonal
crystal structure contain five stiffness constants i.e. 𝐶11 , 𝐶12 , 𝐶13 , 𝐶33 , and 𝐶44 .
Because it is assumed that hexagonal crystals are elastically isotropic, thus, 𝐶11 =
𝐶13 , 𝐶12 = 𝐶13 , and 𝐶44 = 1/2 (𝐶11 − 𝐶12 ) [19]. Stiffness constants 𝐶11 and 𝐶12 are
given by the equations [20]:
𝑘𝑎𝑣
𝐶11 =
𝑎
, (5)
𝐶11 𝜎
𝐶12 = , (6)
1−𝜎

where 𝑎 denotes the lattice constant and 𝜎 represents poison ratio respectively.
Poison’s ratio is calculated by the equation 𝜎 = 0.324 (1 − 1.043𝑃) in which 𝑃

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 The Alteration of Mechanical Behavior for Ba0.5Sr0.5Fe12O19/ . . . . 1717

is the porosity of the prepared samples. The x-ray density (𝜌𝑥 ) for the prepared
samples has been computed using the following equation [20]:
𝑍×𝑀
𝜌𝑥 = 3 , (7)
𝑁𝑉𝑐𝑒𝑙𝑙

where the Avogadro’s number (6.023×1023 atom/mole) is 𝑁, the molar mass of the
sample is 𝑀, number of molecules per unit cell is 𝑍 (for spinel ferrites 𝑍 = 8 and
for hexaferrites 𝑍 = 2), and the unit cell volume is 𝑉𝑐𝑒𝑙𝑙 .
The bulk density 𝜌𝐵 for the prepared samples has been calculated by the
following equation [21]:
𝑚
𝜌𝐵 = , (8)
𝜋𝑟 2 ℎ

where the mass, radius and thickness are 𝑚, 𝑟 and ℎ, respectively, of each pellet of
the prepared samples.
The porosity (𝑃) of the prepared samples attained from the values of the 𝜌𝑥 and
𝜌𝐵 by the following equation [21]:
𝜌𝐵
𝑃 =1− , (9)
𝜌𝑥

Other elastic moduli for the prepared samples are computed using the following
relations [20]:
(𝐶11 −𝐶12 )(𝐶11 +2𝐶12 )
Elastic’s modulus (GPa): 𝐸 = , (10)
(𝐶11 +𝐶12 )
𝐸
Rigidity modulus (GPa): 𝐺 = , (11)
2(𝜎+1)
1
Bulk modulus (GPa): 𝐾 = (𝐶11 + 2𝐶12 ), (12)
3

The X-ray density 𝜌𝑥 and stiffness constant (𝐶11 ) are used to calculate the
longitudinal elastic wave velocity (𝑉𝑙 ) and transverse wave velocity (𝑉𝑡 ) by the
following equations [22]:
𝐶11
𝑉𝑙 = √ , (13)
𝑑𝑋𝑅𝐷

𝑉𝐿
𝑉𝑡 = , (14)
√3

The mean elastic wave velocity (𝑉𝑚 ) can be estimated using the longitudinal
wave velocity (𝑉𝑙 ) and transverse wave velocity (𝑉𝑡 ) in the following equation [23]:
1/3
𝑉𝑙 3 ×𝑉𝑡 3
𝑉𝑚 = [3 ( )] (15)
𝑉𝑡 3 +2𝑉𝑙 3

3.2. Vickers microhardness

The hardness of the prepared samples is determined by measuring the resistance
that appears when a stiff (diamond) indenter is immersed in the sample’s surface.
The indentation process is divided into two steps: loading by exerting an applied
load for a few seconds and unloading. After that, the indenture's permanent
impression is obtained, and the diagonals (𝑑1 and 𝑑2 ) of the indentation were

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1718 M. Yassine et al.

measured using a calibrated microscope. Then, the average diagonal length of the
indentation (𝑑) is calculated in micrometers according to this equation:
𝑑1 +𝑑2
𝑑= , (16)
2

Hence, the load-dependent microhardness was estimated using the relationship

[24]:
𝛼
2𝐹𝑠𝑖𝑛 𝐹
2
𝐻𝑉 = ≈ 1854.4 × (GPa), (17)
𝑑2 𝑑2

where the applied static load is symbolized by 𝐹 in Newton and the face angle of
136° is signified by 𝛼 for the indenter.

3.2.1. Modelling
3.2.1.1. Meyer’s law
Meyer's law, which is used to determine the behaviour of the material, it is the
simplest approach to explain the trend of the indentation size effect (𝐼𝑆𝐸) or reverse
𝐼𝑆𝐸 ( 𝑅𝐼𝑆𝐸 ) behaviour of materials. It relates the static applied load ( 𝐹 ) and
diagonal length (𝑑) of the indentation [25]:
𝐹 = 𝐴𝑑 𝑛 , (18)
where the typical microhardness constant is 𝐴 and a Meyer's index is the
exponent 𝑛 which describes the 𝐼𝑆𝐸 . The material follows the 𝑅𝐼𝑆𝐸 behaviour
when the 𝑛 value is greater than 2. In contrast, the material displays the 𝐼𝑆𝐸
behaviour if the 𝑛 value is lower than 2. Furthermore, it is widely known that the
Kick's law is effective when the 𝑛 value equals 2, which means that the hardness is
independent to the applied load and given as follows: 𝐹 = 𝐴1𝑘 𝑑 𝑛 .

3.2.1.2. Hays-Kendall (HK) model

The 𝐼𝑆𝐸 behaviour observed in the microhardness tests is achieved by Hays-
Kendall (HK) model for a variety of materials [26]. This model claims that the
minimum load (𝑊) is necessary to produce plastic deformation and below in which
only an elastic deformation is visible at 𝐻𝑉 levels, which is a sample resistance
pressure. The indenter can enter the material beyond a particular load value known
as the critical applied load value. As a result, the indentation size begins to rise after
the material's critical applied load value that is proportional to an effective load
𝐹𝑒𝑓𝑓 = 𝐹 − 𝑊 rather than the applied load, as shown by the following relation:
𝐹 − 𝑊 = 𝐴1 𝑑 2 , (19)
where the load-independent microhardness constant is denoted by 𝐴1 . The
values of 𝑊 and 𝐴1 are obtained from 𝐹 to 𝑑 2 graph for all the prepared samples.
In addition, due to HK approximation, the HK microhardness (𝐻𝐻𝐾 ) and the HK
load-independent microhardness (𝐻𝐻𝐾𝑖𝑛 ) can be expressed using the following
equations, respectively:
𝐹−𝑊
𝐻𝐻𝐾 = 1854.4 × , (20)
𝑑2

𝐻𝐻𝐾𝑖𝑛 = 1854.4 × 𝐴1 (21)

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 The Alteration of Mechanical Behavior for Ba0.5Sr0.5Fe12O19/ . . . . 1719

3.2.1.3. Elastic-plastic deformation (EPD) model

Indentation load dependence was provided by Bull et al. [27] and Upit et al. [28]
by including an elastic component ( 𝑑0 ) into the observed plastic indentation
semidiagonal (𝑑) by the equation [29]:
𝐹 = 𝐴2 (𝑑 + 𝑑0 )2 , (22)
where the real microhardness constant is 𝐴2 , from which the EPD
microhardness (𝐻𝐸𝑃𝐷 ) and the EPD load-independent microhardness (𝐻𝐸𝑃𝐷𝑖𝑛 ) are
calculated according to the following equations, respectively:
𝐹
𝐻𝐸𝑃𝐷 = 1854.4 × (𝑑+𝑑 2 , (23)
0)

𝐻𝐸𝑃𝐷𝑖𝑛 = 1854.4 × 𝐴2 (24)

3.2.1.4. Proportional sample resistance (PSR) model

PSR model can characterize the ISE behaviour for various materials, as reported by
Li and Bradt [30]. Also, PSR model is considered as the alteration of HK model by
replacing 𝑊 in Eq. (19) by the term 𝛼1 𝑑 as the following relation:
𝐹 = 𝛼1 𝑑 + 𝛽𝑑 2 , (25)
where the surface energy and the true microhardness coefficient are represented by
𝛼1 and 𝛽 [30]. These two terms can be used to calculate the PSR microhardness
(𝐻𝑃𝑆𝑅 ) and the PSR load-independent microhardness (𝐻𝑃𝑆𝑅𝑖𝑛 ) and they are given
by the following equations, respectively:
𝛼1 𝑑+𝛽𝑑 2
𝐻𝑃𝑆𝑅 = 1854.4 × , (26)
𝑑2

𝐻𝑃𝑆𝑅𝑖𝑛 = 1854.4 × 𝛽, (27)

3.2.1.5. Modified proportional sample resistance (MPSR) model

When the impact of machining-induced plastically deformed surface on
microhardness testing is considered, the PSR model is modified using Eq. (28) [31]:
𝐹 = 𝛼2 + 𝛼3 𝑑 + 𝛼4 𝑑 2 , (28)
where the constant 𝛼2 signifies the minimum applied load for the impression length,
and it is the same to the load independent constant (𝑊) in the HK model. Moreover,
the constants 𝛼3 and 𝛼4 are equivalent to 𝛼1 and 𝛽 in the PSR model. The MPSR
microhardness (𝐻𝑀𝑃𝑆𝑅 ) and the MPSR load-independent microhardness (𝐻𝑀𝑃𝑆𝑅𝑖𝑛 )
can be estimated according to the following equations, respectively:
𝛼2 +𝛼3 𝑑+𝛼4 𝑑 2
𝐻𝑀𝑃𝑆𝑅 = 1854.4 × , (29)
𝑑2

𝐻𝑀𝑃𝑆𝑅𝑖𝑛 = 1854.4 × 𝛼4 (30)

The following equations can be used to calculate the elastic modulus (𝐸𝑚 ), yield strength
(𝑌), fracture toughness (𝐾𝑓 ), and the brittle index (𝐵𝑖 ) values based on the microhardness:
1
𝐻𝑉 = 𝐸𝑚 (1 − 1.9𝑃 + 0.9𝑃2 ), (31)
20

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1720 M. Yassine et al.

𝐻𝑉
𝑌≈ , (32)
3

𝐾𝑓 = √2𝐸𝛼1 , (33)
𝐻𝑉
𝐵𝑖 = . (34)
𝐾𝑓

4. Results and Discussion

4.1. FTIR
The prepared materials' FTIR spectra are recorded in the range 4000 to 400 cm -1,
as shown in Fig. 1(a). The absorption band around 3426-3445 cm-1 is ascribed to
the intermolecular hydrogen bond of 𝑣(O-H, stretch) [32]. The adsorption band at
1624-1642 cm-1 correspond to the bending mode of water molecules absorbed
during sample preparation process 𝛿 (H-O-H, bend) [33]. The other absorption
bands at 2340-2367 cm-1 originate from axial symmetrical deformation of CO 2 of
𝑣𝑠 (C=O, stretch) [33]. Besides, the FTIR spectra in the range of 400-700 cm-1 of
the prepared samples deconvoluted, to determine their absorption bands in Fig.
1(b). There are clear three principle absorption bands for C 1 and the
nanocomposites and two principle absorption band for C0 recorded in the range of
600- 400 cm-1 in the infrared spectra. A high vibration band between 𝑣1 (539- 602
cm-1) corresponds to the tetrahedral A site metal-oxygen bonding of Fe-O bending
by Fe-O4 and a low vibration band in the range of 𝑣2 (400- 443 cm-1) attributes to
the octahedral B site metal-oxygen vibrations of Fe-O stretching of Fe-O6 [34, 35].
The absorption frequency band for A and B sites are presented in Table 1. Because
A sites have shorter bond lengths than B sites, the 𝑣1 (values are much greater than
those of 𝑣2 as seen in Table 1).
These bands exhibit a slight shift in frequency to high frequency at x = 20 wt.%.
This is due to cation rearrangement between different sites of the C1 and C0 structures
[36]. The band shift of the 𝑣1 and 𝑣2 frequencies shows a change in the force
constant, because the vibration frequency is related to the force constant (𝐾). Table
1 lists the computed values of the force constants 𝑘𝑡 and 𝑘𝑜 for A and B sites,
respectively. It is shown that the force constants of the 𝑘𝑡 and 𝑘𝑜 increase with the
addition of C0 up to x = 20 wt.%. This rise in the force constants is due to an increase
in the vibrating frequency of the tetrahedral and octahedral sites, as a result of
shortening the bond length [37]. Conversely, the force constants 𝑘𝑡 and 𝑘𝑜 decrease
with further additions of C0, exhibiting the same pattern as the absorption frequencies
(𝑣1 and 𝑣2 ) [37]. Due to this decrease in force constants, as interatomic separation
rises, the bonds at the octahedral and tetrahedral sites become weaker [38].
The variation in Debye temperature 𝜃𝐷 for the prepared samples can be noticed
in Table 1. It is shown that 𝜃𝐷 decreases from pure BaSr to x = 10 wt.%, then
increases up to a maximum value at x = 20 wt.% and slowly decreases thereafter.
The theory of specific heat can be explained this shift in Debye temperature [39].
This theory states that when 𝜃𝐷 drops, the electrons absorb some of the heat,
proving that electrons are responsible for the conduction for these samples to
become n-type. On the other hand, the abrupt increase in 𝜃𝐷 may indicate a
transformation in sample conductivity from conduction electrons (n-type) to holes
(p-type) with the addition of 𝑁𝑖𝑍𝑛 up to 20 wt.%. Additionally, the elevated Debye

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 The Alteration of Mechanical Behavior for Ba0.5Sr0.5Fe12O19/ . . . . 1721

temperature may influence by the elevated rigidity at x = 20 wt.% and then

decreases with further additions [40].

(i) (ii)

1642
2356
C0

3428
C0

1630
C1.5 3445

2357
C1.5
Transmittance (a.u.)

Absorbance (a.u.)
2356

1631
C1.3
3431

C1.3

1630
2340
C1.2
3430

C1.2

1624
2356

C1.1
3426

C1.1
2360

1630

C1
3431

C1

4000 3500 3000 2500 2000 1500 1000 500 700 650 600 550 500 450 400

Wavenumber (cm-1) Wavenumber (cm-1)

Fig. 1. (a) Full range FTIR spectra and (b) the deconvoluted
spectra in the fingerprint region for the prepared samples.

Table 1. Vibrational band frequencies (𝛎𝟏 and 𝛎𝟐 ),

force constants (𝒌𝒕 , 𝒌𝒐 , and 𝒌𝒂𝒗 ), X ray densities (𝝆𝒙 ), bulk densities (𝝆𝑩 ),
Porosities (𝑷), Debye temperature (𝜽𝑫 ) and Poisson's ratio (𝝈).
x (wt.%) C1 C0 C1.1 C1.2 C1.3 C1.5
𝛎𝟏
599 583 598 602 600 594
(GPa)
𝛎𝟐
443 394 442 443 442 439
(GPa)
𝒌𝒕
262.99 249.13 262.11 265.63 263.87 258.62
(GPa)
𝒌𝒐
143.84 113.78 143.19 143.84 143.19 141.26
(GPa)
𝒌𝒂𝒗
203.42 181.45 202.65 204.74 203.53 199.94
(GPa)
𝝆𝒙
5.22 5.38 5.34 5.44 5.56 5.83
(g/cm3)
𝝆𝑩
2.87 4.38 3.65 2.96 1.17 1.97
(g/cm3)
𝑷
0.45 0.19 0.32 0.46 0.79 0.66
(g/cm3)
𝜽𝑫
749.198 702.463 747.760 751.355 749.198 742.727
(K)
𝝈
0.17 0.26 0.22 0.17 0.06 0.10
(GPa)

The variation of 𝜌𝑥 , with C0 concentration has been adduced also in Table 1.

The value of 𝜌𝑥 rises with the addition of C0. This behavior is explained by the

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1722 M. Yassine et al.

decrease in the unit cell volume of the prepared samples as the unit cell volume is
inversely proportional to X-ray density. Moreover, the values of 𝜌𝐵 are less than
that of 𝜌𝑥 as seen in Table 1, owing to the presence of inter and intra granular
porosity created during sintering process [41]. The porosity is found to increase
from pure BaSr to 𝑥 = 20 wt.%, then increases up to a maximum value at 𝑥 = 30
wt.% and decreases with further addition. This behavior is recognized to the
irregular shape of grain size in the prepared samples. In addition, the decreasing
trend in porosity indicates that some densification existed in case of C 1.1 and C1.5,
that is owing to increase in bulk density in these samples [42].
The obtained values of Poisson's ratio 𝜎 exhibit nonlinear behavior with the
addition of C0 (in the range of 0.06-0.26) as illustrated in Table 1. The obtained
values of 𝜎 lie in the range from -1 to 0.5 which is in agreement with the theory of
isotropic elasticity [43].
Figure 2(a) shows the variation of stiffness constants, 𝐶11 and 𝐶12 , as function
of concentration of C0. The rise in 𝐶11 and 𝐶12 is attributed to the loss of retained
water from the samples with the addition of C0, but the abrupt decline can be
attributed to the transformation process, in which the unit cell volume rises rapidly.
However, stiffness constant, 𝐶11 and 𝐶12 are determined by atomic bonding
tightness and force constant, therefore their decline may be related to weak atomic
bonding between Ba2+, Sr2+, Ni2+, Ni3+, Zn2+, Fe2+, and Fe3+ cations as the addition
of C0 increases.
The variation of elastic moduli: Young's modulus (𝐸), rigidity modulus (𝐺) and
bulk modulus (𝐾) against the concentration of C0 is presented in Fig. 2(b). It is seen
that the elastic moduli drop, indicating that the inter-atomic bonding between
distinct atoms in the crystal lattice weakens with the addition of C0 [43]. The
increase in cation repulsion between electrons with the addition of C0, resulting in
a drop in elastic moduli. In contrast, the increase in the elastic moduli is due to the
cations being rearranged within the crystal lattice, which will enhance
crystallization process [44]. Consequently, the strength of the bonding between the
corresponding ions increases.

Fig. 2. Variation of , 𝑪𝟏𝟏 (a) and 𝑪𝟏𝟐 and (b) elastic

moduli constants as a function of composition 𝒙.

The variation of longitudinal wave velocity (𝑉𝑙 ), transverse wave velocity (𝑉𝑡 )
and the mean elastic wave velocity (𝑉𝑚 ) of the prepared samples are shown in Fig.

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 The Alteration of Mechanical Behavior for Ba0.5Sr0.5Fe12O19/ . . . . 1723

3. The decrease in velocities may be related to decrease in force constant. The

consistent finding for all ferrites is that transverse wave velocities (𝑉𝑡 ) has lesser
values than longitudinal wave velocity (𝑉𝑙 ) [45].

Fig. 3. Variation of (𝐕𝐥 ), (𝐕𝐭 ) and (𝐕𝐦 ) as a function of composition 𝒙.

4.2. Vickers Microhardness

The Vickers microhardness 𝐻𝑉 was calculated according to eq. (17) and plotted as
a function of the applied load for the prepared samples in Fig. 4. As the applied
load rises up to 2.94 N, 𝐻𝑉 drops quickly (a load dependent segment), after that 𝐻𝑉
achieves saturation, which is nearly the plateau region, for higher loads 𝐹 >2.94 N
(a load independent segment). This decrease in 𝐻𝑉 values is known as the
indentation size effect (𝐼𝑆𝐸). Because the material has weak grain boundaries and
the indenter mainly touches the surface layers, 𝐻𝑉 is large at low indentation loads
[7]. Whereas the impact of the inner layers becomes more relevant as the indenter's
penetration depth rises at higher applied loads. For that reason, 𝐻𝑉 values remain
almost constant as the applied load increases [10]. Additionally, the inset of Fig. 4
shows that 𝐻𝑉 increases by adding 𝑁𝑖𝑍𝑛 up to 𝑥 = 20 wt.% into BaSr for all
applied loads (𝐹). This enhancement is due to a decrease in porosity or resistance
to fracture propagation inside the grains, as well as an increase in grain connectivity
[46]. However, 𝐻𝑉 decreases with further additions due to a growth in the grain
boundary weak links, specimen cracking/porosity, and impurity phases [46].
Table 2 includes the fitting equations for all the prepared samples that describe
the change of 𝐻𝑉 with applied load using quadratic formulas. The variation of the
𝐹 2 coefficients increases from 0.00171 to 0.0189 with the addition of C0 from 10
to 20 wt.%. This result is explained by the fact that there are less fractures and
dislocations in these samples associated with rise in 𝐻𝑉 values [46]. On the other
hand, the variation of the 𝐹 2 coefficients decreases with further additions of C0 up
to 50 wt. %. This is attributed to the quick growth in fractures and dislocations in
these samples [46].

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 1724 M. Yassine et al.

Fig. 4. Variation of experimental 𝑯𝑽 as a function of the applied static load 𝐅

for the prepared samples at a dwell time of 30 s. The inset shows the
variation of , 𝑯𝑽 with 𝒙 for 𝑭 = 0.49, 0.98, 1.96, 2.94, 4.90, and 9.80 N.

Table 2 Fitting parameters obtained from

the theoretical models for the prepared samples.
x (wt.%) C1 C0 C1.1 C1.2 C1.3 C1.5
𝐻𝑉 = 𝐻𝑉 = 𝐻𝑉 = 𝐻𝑉 = 𝐻𝑉 = 𝐻𝑉 =
Fitting 0.0144F2 - 0.0344F2- 0.0171F2- 0.0189F2 - 0.0131F2 - 0.0101F2 -
relations 0.2106F + 0.4669F + 0.2465F + 0.2696F + 0.1906F + 0.1534F +
1.262 2.3904 1.3863 1.5034 1.1794 1.0702
𝑹𝟐 1 0.863 0.982 0.969 0.994 0.999
𝑯𝑽
0.586 1.103 0.618 0.684 0.573 0.552
(GPa)
𝑯𝑯𝑲𝒊𝒏
0.549 1.037 0.571 0.636 0.541 0.512
(GPa)
𝑯𝑬𝑷𝑹𝒊𝒏
0.463 0.890 0.482 0.538 0.464 0.445
(GPa)
𝑯𝑷𝑺𝑹𝒊𝒏
0.433 0.831 0.447 0.501 0.434 0.419
(GPa)
𝑯𝑴𝑷𝑺𝑹𝒊𝒏
0.602 1.113 0.630 0.697 0.588 0.565
(GPa)

Several theoretical models are applied to determine the true microhardness or

load-independent microhardness, which is the change of 𝐻𝑉 with the applied static
load 𝐹 in the plateau limit regions. So, the relationships between the applied load
and the indentation diagonal length are given below to characterize the 𝐼𝑆𝐸
behavior of the prepared samples.
The 𝑙𝑛𝐹 − 𝑙𝑛𝑑 graphs of the prepared samples are shown in Fig. 5, in which 𝑛
values, that are obtained from the slope of the linear plot, are found to be less than 2

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 The Alteration of Mechanical Behavior for Ba0.5Sr0.5Fe12O19/ . . . . 1725

for each sample, verifying normal 𝐼𝑆𝐸 behavior. The vertical intercept of the graph
indicates 𝐴 value, and the 𝑛 and 𝐴 values are summarized in Table 3. Furthermore,
normal 𝐼𝑆𝐸 behavior is considered to be correlated to geometrically necessary
dislocations (GNDs). Consequently, this behavior can be explained as the prepared
samples have high hardness under low loading conditions. As a result, the
dislocations are assumed to regulate the deformation of materials with lower 𝑛 values
more than those with higher 𝑛 values [47]. As well, upon adding 10 wt.% of C0 to C1,
the value of 𝑛 decreases from 1.577 to 1.553, suggesting less lattice damage owing
to the migration of Ni2+ and Zn2+ ions in C0 into Ba2+and Sr2+ ions in C1 [10]. In
contrast, the value of 𝑛 increases from 1.553 to 1.623 with the addition of C0 from 10
to 50 wt.%, indicating larger lattice damage. This can be due to the inclusion of larger
amount of C0 to C1. It is also important to classify the hardness of these materials, if
the value of 𝑛 is between 1 and 1.6, the material is hard and if it is greater than 1.6,
the material is soft [10]. Hence, all the prepared samples are classified as soft
materials, except for C1.5 is considered as hard material.
Figure 6 illustrates the linear fitting of 𝐹 against 𝑑 2 for the prepared samples.
These obtained values of load independent microhardness for all the samples are
tabulated in Table 2. As seen from Table 3 the values of 𝑊 are positive for all the
prepared samples (exhibiting the normal 𝐼𝑆𝐸 behavior) and are almost greater than
the lowest applied load employed in this research (0.49 N). This means that the
applied loads are strong enough to create elastic and plastic deformations [7], and that
once the indenter is removed, an elastic relaxation occurs on the sample's surface.
Further, the values of 𝐴1 change in the same way as the variation of 𝐻𝑉 with 𝐹 with
the addition of C0, verifying the 𝐼𝑆𝐸 trend [7]. The observed microhardness values in
the plateau region (𝐻𝐻𝐾𝑖𝑛 ) are found to be nearly close to the theoretical 𝐻𝑉 data
given in Table 2. The deviation between the true hardness and the calculated
microhardness from HK model varies from 5.5 % to 7.6 %. As a result, the HK model
is inapplicable to studying true microhardness behavior for the prepared samples.

Fig. 5. 𝒍𝒏𝑭 versus 𝒍𝒏𝒅 for the prepared samples at a dwell time of 30 s.

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 1726 M. Yassine et al.

Table 3 Fitting relationship for 𝑯𝑽 values with regard

to the applied test load; experimental and theoretical microhardness
analysis results for the prepared samples.
x (wt.%) C1 C0 C1.1 C1.2 C1.3 C1.5
Meyer’s 𝒏 1.577 1.557 1.553 1.561 1.598 1.623
parameters 𝑨 ×10-3 (N/m2) 2.681 4.575 3.106 3.206 2.361 2.03
𝑾 (N) 0.53 0.41 0.55 0.52 0.49 0.52
HK’s
𝑨𝟏 ×
parameters (N/m2) 2.96 5.59 3.08 3.43 2.92 2.76
10-4
𝑨𝟐 ×
EPD’s (N/m2) 0.25 0.48 0.26 0.29 0.25 0.24
10-3
parameters
𝒅𝟎 (m) 20.61 12.91 20.85 18.89 19.45 19.64
𝜶𝟏
PSR’s (N/m) 1.341 1.596 1.426 1.429 1.235 1.18
×10-2
parameters
𝜷 ×10-4 (N/m2) 2.34 4.48 2.41 2.70 2.34 2.26
𝜶𝟐 (N) 0.15 0.23 0.1 0.13 0.17 -0.01
MPSR’s
𝜶𝟑 ×10-2 (N/m) 0.97 0.64 1.17 1.07 0.81 1.29
parameters
𝜶𝟒 ×10-4 (N/m2) 2.51 5.19 2.52 2.89 2.54 2.18

Fig. 6. Effective load 𝑭 as a function of 𝒅𝟐

for the prepared samples at a dwell time of 30 s.

Figure 7 demonstrates the linear plot of 𝐹 0.5 versus 𝑑 for the prepared samples.
𝐴2 and 𝑑0 are identified and Table 2 illustrates their findings. The results from the
Hays-Kendall model are confirmed by the positive values of 𝐴2 and 𝑑0 , which
demonstrate that there is elastic deformation in the prepared samples along with
plastic deformation, verifying the 𝐼𝑆𝐸 behavior [10].
Table 3 clearly shows that the (𝐻𝐸𝑃𝐷𝑖𝑛 ) values are relatively away from the
original microhardness values in the plateau region; hence, this model is
insufficient in determining the actual hardness values for all the prepared samples.
The deviation between the true microhardness and the calculated microhardness
from EPD model varies from 19.0 % to 22.0 %.

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 The Alteration of Mechanical Behavior for Ba0.5Sr0.5Fe12O19/ . . . . 1727

Fig. 7. Linear plot of 𝑭𝟎.𝟓 versus 𝒅 for

the prepared samples at a dwell time of 30 s.
Figure 8 depicts a plot of (𝐹/𝑑) versus 𝑑 for the prepared samples, following
PSR model. 𝛼1 denotes the y-intercept and 𝛽 denotes the slope, and their values are
listed in Table 3. The tabulated data demonstrate that all the prepared samples
exhibit positive values of 𝛼1 and 𝛽 , supporting the HK model results. These
findings show the presence of elastic as well as plastic deformation, demonstrating
𝐼𝑆𝐸 behavior in the prepared samples. Moreover, 𝛼1 increases with the addition of
C0 up to 20 wt.% due to energy dissipation via fractures at the interfaces [7].
Furthermore, the sample transition values to the plateau differ from the load
independent hardness values calculated using the PSR model. This demonstrates
that this model is inadequate for identifying the true hardness levels in this
investigation. The deviation between the true hardness and the calculated
microhardness from PSR model varies from 24.1 % to 27.7 %.

Fig. 8 . 𝐅/𝐝 versus 𝐝 for the prepared samples at a dwell time of 30 s.

Figure 9 illustrates the change of 𝐹 with 𝑑 for the prepared samples. Table 3
provides the MPSR parameters 𝛼2 , 𝛼3 and 𝛼4 , which are derived using a standard
polynomial fit to the data. The positive values of 𝛼2 , describe that the samples suffer

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1728 M. Yassine et al.

from both elastic and plastic deformation. Furthermore, the experimental 𝐻𝑉 values
were quite close to the true microhardness values measured. The error between the
true microhardness and calculated microhardness from MPSR model varies from
0.9% to 2.8%. As a result, the MPSR model is the most credible model for discussing
the 𝐻𝑉 findings and the mechanical properties of all the prepared samples in the
plateau region. Moreover, the residual surface stress contribution of the MPSR model
to the observed 𝐼𝑆𝐸 behavior is negligible compared to the other models.

Fig. 9. 𝑭 versus 𝒅 for the prepared samples at a dwell time of 30 s.

Table 4 demonstrates the values of 𝐸𝑚 , 𝑌, 𝑘𝑓 and 𝐵𝑖 for the prepared samples.
The values of 𝐸𝑚 acquired from FTIR measurements are higher than those obtained
from 𝐻𝑉 measurements, but they follow a similar variation trend as shown in Fig.
10. This implies that the approach used to calculate elasticity parameters from FTIR
measurements is accurate and acceptable. The values of 𝐸𝑚 and 𝑌 change in the
same trend of 𝐻𝑉 owing to their dependency on the hardness. The elastic
components' contribution to the fracture toughness (𝑘𝑓 ) of nanocomposites with 𝑥
= 10 and 20 wt.%, are comparatively larger than for nanocomposites with 𝑥 = 30
and 50 wt.%, which results in a variation of their abilities to resist cracks (fracture
behavior). So, the microstructure is important that can influence the fracture
behavior [48]. Due to the inverse correlation between the fracture toughness and
the ductility, samples with higher addition of C0 show a higher ductility [48]. In
addition, particle distribution have an impact on the ductility of the nanocomposites
[49]. Furthermore, C1.2 has highest value of 𝐵𝑖 , which has the most tendency to
break when deformed. Moreover, because of the 𝐼𝑆𝐸 behavior, the reduction of
𝐸𝑚 , 𝑌, and 𝑘𝑓 values with applied load is also predicted.

Table 4 The calculated load-dependent 𝑯𝑽 , 𝑬, 𝒀, 𝑲𝒇 ,

and 𝑩𝒊 values for the prepared samples.
𝑯𝑽 𝑬𝒎 𝒀 𝑲𝒇 𝑩𝒊
x (wt.%)
(GPa) (GPa) (GPa) (GPa/μm1/2) (μm1/2)
C1 0.586 35.772 0.195 1.135 0.516
C0 1.103 32.607 0.368 1.699 0.649
C1.1 0.618 25.263 0.206 1.202 0.514
C1.2 0.684 42.587 0.228 1.266 0.540
C1.3 0.573 188.373 0.191 1.077 0.532
C1.5 0.552 80.777 0.184 1.033 0.534

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 The Alteration of Mechanical Behavior for Ba0.5Sr0.5Fe12O19/ . . . . 1729

Fig. 10. Variation of 𝑬 and 𝑬𝒎 as a function of composition 𝒙.

Hardness measurements are performed on the prepared samples at 30, 40, 50,
60, 70 and 80 °C while keeping the dwell period constant at 30 s. Fig. 11 indicates
that 𝐻𝑉 values of C1 and C0 nanoparticles decline at higher temperature while 𝐻𝑉
values remain almost similar for C1/xC0 nanocomposites. The increase in
dislocation density (defects) results in a decrease in hardness as temperature
increases [50]. From Fig. 11, the C1.1 sample has the most enhanced 𝐻𝑉 as
temperature increases from 30 to 80 °C. This is due to the relatively uniform
distribution of C0 nanoparticles among the C1 nanoparticles in C1.1 sample [51].
This means that C1.1 has less spaces between its nanoparticles at 80 °C [52]. C1.1
sample shows the highest 𝐻𝑉 values among the nanocomposites with increasing
temperature, which is the most appropriate sample used in magnetic applications
[53] as reported in our previous work [12].

Fig. 11. 𝑯𝑽 as a function of the temperature for the prepared

samples with applied static load of 500 N and a dwell period of 30 s.

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1730 M. Yassine et al.

5. Conclusions
Magnetically hard/soft ferrite C1/xC0 nanocomposites with x = 10, 20, 30, and 50
wt.% have been prepared using the co-precipitation and high-speed ball milling
techniques. FTIR spectroscopic analysis confirmed the purity of both spinel soft
and hexagonal hard ferrites in the nanocomposites. The stiffness constant, elastic
moduli and wave velocity values can be determined from FTIR. The 𝐻𝑉
measurements revealed that the prepared samples exhibited the behaviour of
normal indentation size effect. The 𝐻𝑉 values improved with the addition of C0 up
to x = 20 wt.% for all applied loads (𝐹), suggesting the decrease in porosity or
resistance to crack propagation within the grains, and an increase strength of the
bonds between the grains. Normal 𝐼𝑆𝐸 behaviour was found when 𝐻𝑉 declined as
the applied load increased for all the prepared samples. Experimental results of 𝐻𝑉
were examined using Meyer’s law, HK approach, EPD model, PSR model and
MPSR model. The MPSR model was the best in describing the actual
microhardness values. However, the mechanical parameters 𝐸𝑚 , 𝑌, 𝑘𝑓 and 𝐵𝑖 for
the prepared samples computed from 𝐻𝑉 values performed to follow the same trend
as 𝐻𝑉 values with the addition of C0. Further, the C1.1 sample exhibited the greatest
𝐻𝑉 values among the nanocomposites as temperature increased, confirming the
relatively uniform distribution of C0 nanoparticles into the BaSr nanoparticles.

Acknowledgments
This research was carried out in the Physics Department, Faculty of Science, Beirut
Arab University, Lebanon, at the Specialized Materials Science Laboratory and the
Advanced Materials Preparation Laboratory.

Nomenclatures

Bi Brittle index
E Young’s modulus from FTIR
Em Elastic modulus
G Rigidity modulus from FTIR
HEPDin EPD Load-independent microhardness
HHKin HK Load-independent microhardness
HMPSRin MPSR Load-independent microhardness
HPSRin PSR Load-independent microhardness
HV Load-dependent microhardness
k Force constant
K Bulk modulus from FTIR
Kf Fracture toughness
Vl Longitudinal wave velocity
Vm Mean elastic wave velocity
Vt Transverse wave velocity
Y Yield strength

Greek symbols
α1 Surface energy
α2 minimum applied load for the impression length
α3 Surface energy

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 The Alteration of Mechanical Behavior for Ba0.5Sr0.5Fe12O19/ . . . . 1731

β True microhardness coefficient

θD Debye temperature
ρB Bulk density
ρx X-ray density

Abbreviations
ISE Indentation size effect behaviour
RISE Reverse indentation size effect behaviour

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