Effect of hard magnetic ferrite (Ba_0.5Sr_0.5Fe₁₂O₁₉) nanoparticles on the mechanical properties of the (Bi, Pb)-2223 phase

Abstract

To manifest the effect of hard magnetic Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles on the mechanical performance of the (Bi,Pb)-2223 superconducting phase, nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ, with x = 0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.10, and 0.20 wt%, were synthesized using a conventional solid-state reaction method. The X-ray diffraction (XRD) data revealed that adding nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) to the host (Bi,Pb)-2223 phase preserved the orthorhombic structure. The porosity (P%) calculations revealed a decrease until x = 0.04 wt%, which suggests that the addition of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x reduces the number of voids and improves inter-grain connections, as confirmed by SEM micrographs. The superconducting transition temperature (T_c) increased to 112 K with the inclusion of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) up to x = 0.04 wt%. Vickers microhardness (H_V) measurements were conducted at various applied loads (0.245–9.800 N) and a duration time of 45 s. The H_V number increased with the addition of x up to x = 0.04 wt% but then decreased with further addition. Various models were employed for analysis and modelling of Vickers hardness (H_V) versus test load (F), including Meyer’s law, Hays–Kendall (H–K) model, the elastic/plastic deformation (EPD) model, the proportional sample resistance (PSR) model, the modified proportional sample resistance (MPSR), and indentation-induced cracking (IIC) model. It was found that the PSR model was the most appropriate theoretical model for describing the microhardness of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composites. Moreover, the elastic modulus (E), yield strength (Y), fracture toughness (K), brittleness index (B), and elastic stiffness coefficient (C11) were estimated as a function of the inclusion of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x. Furthermore, the indentation creep test (time-dependent Vickers microhardness) revealed that the dislocation creep mechanism exists in composite samples with low concentrations (x < 0.05 wt%), whereas the dislocation climbs creep mechanism was observed for x ≥ 0.05 wt%.

1 Introduction

From the application point of view, the Bi–Sr–Ca–Cu–O (BSCCO) high-temperature superconductors (HTSCs) have attracted considerable interest from scientists all over the world because of their extraordinary qualities and numerous applications in various fields of technology [1]. Bi₂Sr₂Ca_n–1Cu_nO_2n+4+δ, is the general formula of BSCCO superconductors, where n = 1, 2, and 3 denote the number of CuO₂ layers along the c-axis, which ultimately determines the number of unit cells in the superconductor [2]. The superconducting transition temperature T_c at which these materials transition to superconductivity rises as the number of CuO₂ layers increases. There are three main phases that define the family of BSCCO superconductors, including the very low Bi-2201 phase (n = 1, T_c ~ 20 K), the low Bi-2212 phase (n = 2, T_c ~ 85 K), and the high Bi-2223 phase (n = 3, T_c ~ 110 K), respectively. The high Bi-2223 phase has many characteristics, that make it the most desirable of these phases, including high T_c (~ 110 K), critical current density, J_c (~ 106 A/cm²), the extremely high value of upper critical magnetic field at liquid helium temperatures (H_{c2 (4.2 K)} > 100 T), atmospheric stability, and especially an excellent mechanical performance [3,4,5,6,7,8,9]. As a result, it was used in numerous industrial setups, including power cables [10], magnetic sensors [11], high-field magnets [12, 13], and magnetic resonance imaging systems [14, 15]. However, it still has many manufacturing issues, such as very low ductility and high brittleness due to their ceramic nature, resulting in poor mechanical properties for practical applications [16,17,18]. Therefore, to improve these restrictions, the microstructure and mechanical parameters of ceramic superconducting materials should be improved. Therefore, many researchers directed their efforts towards overcoming this limitation in diverse ways. The most efficient way is by adding nanoparticles into the pores between the grains in the structure of superconductor composites [19, 20]. The Vickers microhardness (H_V) technique is a favorite for calculating the mechanical parameters of superconducting materials, because it is easy, efficient, simple, accurate, and nondestructive. Additionally, crucial structural factors like strength and deformation characteristics can be identified [19, 21]. With the addition of various nanoparticles with different percentages of the (Bi,Pb)-2223 superconducting phase, many researchers have been working to examine and enhance the phase formation, microstructure, electrical transport, and mechanical properties. Rahal et al. [22] investigated the superconducting and mechanical properties of the composite (SnO₂)_x(Bi,Pb)-2223 with different sintering temperatures, and found that the addition of nano-SnO₂ particles enhanced the mechanical properties of superconductors. Habanjar et al. [23] reported the impact of BaFe₁₂O₁₉ nanoparticles on the (Bi,Pb)-2223 phase. Their results showed a rise in the values of microhardness, H_v, in the (Bi, Pb)-2223 phase with the addition of nanoparticles. Awad et al. [24] discovered that adding SnO₂ nanoparticles improved the mechanical properties of the (Bi,Pb)-2223 phase until x = 0.4 wt%. Also, El Ghouch et al. [19] found that the addition of two transition metals (FeSiW₁₁ and MnSiW₁₁) nanoparticles into the (Bi,Pb)-2223 phase increased the mechanical parameters up to 0.04 wt% and 0.08 wt%, respectively.

The main purpose of this work is to investigate the effect of hard magnetic ferrites’ Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles on the mechanical characteristics of the (Bi_1.8Pb_0.4)Sr₂Ca_2.1Cu_3.2O_10+δ superconductor phase, besides phase formation, microstructure, and electrical resistivity measurements. In addition, the microhardness values (H_V) have been examined according to several theoretical models in the load-dependent and independent (plateau region) regions. Moreover, the variation of microhardness values (H_V) with time in the range of duration (15–60 s) using the indentation creep mechanism was investigated.

2 Experimental procedure

Conventional solid-state reaction technique was used to prepare the superconducting phase nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi_1.8Pb_0.4)Sr₂Ca_2.1Cu_3.2O_10+δ with x = 0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.10, and 0.20 wt%. Appropriate stoichiometric amounts of Bi₂O₃, PbO, SrCO₃, CaO, and CuO (Sigma-Aldrich chemicals, > 99% purity) were mixed, ground in an agate mortar, and sifted using a 125 μm sieve. The resulting powder of the homogeneous mixture was calcinated twice in air at 1093 K for 24 h, with grinding and sifting between the two stages of the calcination process. The resulting powder was ground, sifted, and then pressed into a disc (1.3 cm in diameter and 0.3 cm in thickness) under a pressure of 0.98 GPa, using a Pye Unicam hydraulic press. The pellets were then sintered in air for 96 h at 1118 K at heating and cooling rates of 4 K/min and 2 K/min, respectively. At the end of the process, the final pellets were ground and sieved to become a homogenous powder, and the Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles of average grain size (48.987 nm) synthesized via the wet chemical coprecipitation technique, as previously reported [25], were added with different weight percentages (x = 0.00, 0.01, 0.02, 0.03, 0.04, 0.05, 0.10, and 0.20 wt%) to the resultant powders. The resulting mixture was pressed again at 0.98 GPa and sintered in air at 1118 K with a heating rate of 2 K/min and a cooling rate of 1 K/min to afford the desired nanoparticle-superconductor composite.

X-ray diffraction (XRD) was used to examine the crystal structure of the prepared nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples using the X'Pert Pro (PANalytical) device with a Cu target (λ = 1.5406 Å) (at 40 kV and 40 mA) in the range 3° ≤ 2θ ≤ 65°. The surface morphologies of the samples were studied via scanning electron microscopy (SEM) using a JSM-IT200 device. The electrical resistivity measurements were carried out using the four-probe technique from room temperature down to zero-resistivity temperature (T₀) via a closed cryogenic refrigeration system (Displex) employing helium gas as a working medium. The Vickers microhardness device, model IN-412A, was used to measure the microhardness of composite samples at the various indentation test loads from 0.245 to 9.8 N at a dwell time of 45 s, at room temperature. The same process was performed five times at the various locations on the sample surface to find a more accurate average value. In addition, an indentation creep test was performed on the prepared samples at various dwell times (t = 15, 30, 45, and 60 s) and for fixed loads of 2.94, 4.90, and 9.80 N.

3 Theoretical approach

The Vickers microhardness tester is the most commonly used for hardness measurement, which involves applying a fixed load to a diamond indenter and measuring the dimension of the resultant indentation on the surface of the specimen [26]. The Vickers microhardness (H_V) values are calculated according to the following equation:

$${H}_{\mathrm{V}} = 1854.4 \left(\frac{F}{{d}^{2}}\right) \left(\mathrm{GPa}\right),$$

(1)

where F and d are the applied test load and diagonal length of the indentation in newton (N) and micrometer (μm), respectively. To studying the mechanical properties of ceramic materials, several models have been proposed to interpret the Vickers microhardness data (H_V) behavior and estimate the true microhardness value.

3.1 Meyer’s law

Meyer’s law is used to explain the indentation size effect (ISE) behavior of materials. The relation between the applied load F(N) and the indentation diagonal length d (μm) is given by:

$$F= A{d}^{n},$$

(2)

where A is the standard hardness constant, and n is Meyer’s number. The Meyer number is a leading indicator for defining a compound's mechanical behavior as defined by a stress applied to the surface. In this context, if the value of n > 2, the material shows normal indentation size effect (ISE) behavior, whereas if n < 2, the material will show reverse indentation size effect (RISE) behavior, and if n = 2, the microhardness is independent of the applied load [27, 28].

3.2 Hays–Kendall approach (H-K)

H–K suggested that there is a threshold load (W) to begin the noticeable indentation in the sample surface [29]. As a result, they suggested an effective indentation load ${F}_{\mathrm{eff}} = F - {W}_{HK}$, which produces the measured indentation as follows:

$$F - {W}_{HK} = {A}_{1}{d}^{2},$$

(3)

where A₁ is an independent constant from the applied load.

The true hardness of the plateau region according to the Hays–Kendall model, H_HK, could be calculated according to Eq. (4)

$${H}_{HK} = 1854.4 \times {A}_{1}.$$

(4)

3.3 Elastic/plastic deformation model (EPD)

The elastic/plastic deformation (EPD) model focuses on the elastic recovery that is created by the closeness of the indentation impression. Bull et al. [30] suppose that the indentation includes a portion of elastic distortion, which relaxed when the applied load was removed. Therefore, the measured semi-diagonal plastic indentation (d) must be combined with an elastic component (d₀) to determine the microhardness [31]. Hence, the load force must be corrected according to the following relationship:

$$F ={ A}_{2} {(d + {d}_{0})}^{2},$$

(5)

where A₂ is constant.

Thus, the true hardness according to the EPD model, H_EPD, could be calculated using Eq. (6):

$${H}_{EPD} = 1854.4 \times {A}_{2}.$$

(6)

3.4 Proportional specimen resistance model (PSR)

According to Li and Bradt [32], the Hays–Kandel approach was modified by replacing the constant threshold load W by a load-dependent term ${\alpha }_{1}d$, where α₁ refers to the test sample’s elastic resistance as well as the fractional resistance of the surface cracks [33]. The ISE behavior is governed by Eq. (7), as follows:

$$F = {\alpha }_{1}d + {\alpha }_{2}{d}^{2},$$

(7)

where ${\alpha }_{2}$ is constant related to the true microhardness.

Accordingly, the true hardness, ${H}_{PSR}$, can be calculated using the following relation:

$${H}_{PSR} = 1854.4 \times {\alpha }_{2}.$$

(8)

3.5 Modified proportional specimen resistance (MPSR) model

Gong et al. [34] suggested a modified proportional specimen resistance (PSR) model according to the following equation:

$$F = {\alpha }_{0}+ {\alpha }_{1}d + {\alpha }_{2}{d}^{2}.$$

(9)

Here, the parameter ${\alpha }_{0}$ is related to the residual surface stresses associated with the surface grinding and polishing processes during sample preparation. The parameters ${\alpha }_{1}$ and ${\alpha }_{2}$ are constants as defined in the PSR model and have the same physical meaning.

Using the following equation, one can find the values of load independence with respect to the MPSR model:

$${H}_{MPSR} = 1854.4 \times {\alpha }_{2}.$$

(10)

3.6 Indentation-induced cracking (IIC) model

Finally, the indentation-induced cracking (IIC) model is a useful approach to deciding what behavior (ISE or RISE) of the material is present and calculating the microhardness parameters [35], according to the following equation:

$${H}_{\mathrm{V}}={\lambda }_{1}{K}_{1}\left(\frac{F}{{d}^{2}}\right)+{K}_{2}\left(\frac{{F}^{5/3}}{{d}^{3}}\right).$$

(11)

In this relation, λ₁ is constant, K₁ is constant depending on the indenter’s geometry, and K₂ is constant depending on the applied load. For brittle materials (ceramic), the constant of λ₁ ≅ 0, so Eq. (11) becomes

$${H}_{\mathrm{V}}=K{\left(\frac{{F}^{5/3}}{{d}^{3}}\right)}^{m}.$$

(12)

Herein, the parameters of K and m are ascribed to the load-independent constants.

Furthermore, the value of m is like the n value in Mayer’s law, where both can be used to identify the indentation size effect (ISE) behavior of materials. The m value has a crucial value of 0.6; if m > 0.6, the material shows normal ISE behavior, whereas m < 0.6 exhibits RISE. Additionally, the values of load-independent parameters in the plateau region are calculated such as elastic modulus (E), yield strength (Y), fracture toughness (K_f), brittleness index (B_i), and elastic stiffness coefficient (C₁₁), according to the following formulizations [36]:

$$E = 81.9635\times {H}_{\mathrm{V}},$$

(13)

$${B}_{i}={H}_{\mathrm{V}}/{K}_{f},$$

(14)

$$Y = \frac{{H}_{\mathrm{V}}}{3},$$

(15)

$${C}_{II}= {{H}_{\mathrm{V}}}^{7/3},$$

(16)

$${K}_{f} = \sqrt{2E{\alpha }_{1},\;}\quad \left({\alpha }_{1}:\mathrm{ surface energy}\right).$$

(17)

3.7 Indentation creep

Indentation creep is an important indentation in many applications of ceramic materials. It is defined as the relative change in the indentation depth, while the applied load is kept constant [37]. Using the Sargent–Ashby model [38], you can investigate indentation creep behavior in terms of the following relationship:

$${H}_{\mathrm{V}}\left(t\right)= \frac{{\varepsilon }_{0}}{{(nc{\varepsilon }_{0}t)}^{1/\mu }}.$$

(18)

H_V(t) denotes the time-dependent Vickers microhardness. ε₀ and c are constants, and $\mu$ is the stress exponent. The values of stress exponent μ illustrated the details of the deformation mechanism. It was reported that: if the value of μ is close to 1, the sample has diffusion creep [39], and if it is close to 2, the grain boundary will slide [40]. Dislocations climbs exist for values of μ ranging from between 4 and 6 [41]. When the values of μ are between 3 and 10, dislocation creep dominates [42, 43].

4 Results and discussion

4.1 Ba_0.5Sr_0.5Fe₁₂O₁₉ characterization

The characteristics and analyses of Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles were previously reported by Yassine et al. [25], and the findings are summarized in Table 1.

Table 1 Characteristic parameters of Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles

4.2 X-ray power diffraction (XRD)

The XRD patterns of the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composite are displayed in Fig. 1a. The main peaks in all the samples have been well indexed as the main phase of (Bi, Pb)-2223 superconducting phase, with orthorhombic crystal structure and space group A2aa which matches with (JCPDS card no. 48–0690: Calcium Copper Lead Strontium Bismuth Oxide: (Bi,Pb)₂Sr₂Ca₂Cu₃O₁₀) [8, 44]. Aside from some minor phases like the (Bi,Pb)-2212 superconducting phase, matching with (JCPDS card no. 41–0317: Bismuth Strontium Calcium Copper Oxide: Bi₂Sr₂CaCu₂O₈), and the non-superconducting phase of Ca₂PbO₄, matching with (JCPDS card no. 48–1533: Calcium Lead Oxide: Ca₂O₄Pb), which are commonly formed during the preparation of the high-T_c (Bi,Pb)-2223 phase [45, 46]. The superconducting phase of (Bi,Pb)-2212 frequently forms as a minority phase in the (Bi,Pb)-2223 high phase and cannot be categorized as an impurity phase [47, 48]. While the partially melted liquid Ca₂PbO₄ phase formed during the heat treatment of Pb-doped Bi-based superconductors (Pb-rich compound) is essential for the development of the (Bi,Pb)-2223 phase [49]. Pb lead reacts with CaCO₃ to produce the highly reactive intermediate phase Ca₂PbO₄, which greatly aids in the phase formation of Bi-2223. This intermediate phase readily combines with other substances to form a liquid phase, acting as a reservoir and a fast cation diffusion pathway, and assisting in the formation of the Bi-2223 phase [50, 51]. The mixture of the Bi-2212 phase, which formed during the sintering process at temperatures between 750 and 800 °C, and the partially melted liquid Ca₂PbO₄ phase, which formed above at a temperature of 820 °C, provides the best approximation of the formation of the Bi-2223 phase among several different methods for formation of the Bi-(2223) phase [52,53,54,55]. To stop the growth of secondary phases, which become less reactive as they grow larger, it is crucial to quickly synthesize the Bi-2223 phase during the sintering step. Therefore, this reaction must proceed quickly to prevent segregation and the growth of unnecessary secondary phases, to maximize the conversion of the calcined material to the Bi-2223 phase. A suitable phase composition (Bi-2212, Ca₂PbO₄, Ca₂CuO₃, CuO) must be obtained during calcination to enable rapid Bi-2223 phase formation [52, 56,57,58]. However, the presence of the minor phases (Bi-2212 and Ca₂PbO₄) after the sintering process may be due to their not having completely transformed into the Bi-2223 phase. Although the secondary phases (Bi, Pb)-2212 and Ca₂PbO₄ have increased as x increases up to 0.20 wt%, the major phase (Bi, Pb)-2223 remained. This majority could be due to the substitution of Pb²⁺ at Bi³⁺ sites, which makes the (Bi,Pb)-2223 phase more stable in the Pb-rich environment [52]. Moreover, the patterns do not show any peaks for barium, strontium and their oxides or iron-based compounds, such as SrFeO_3+x (JCPDS No. 34–0641, 40–0906)[59] and Fe₂O₃ (JCPDS No. 24–0072, 33–0664) phases [60, 61]. The addition of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) causes no change in peak positions. As a result, the Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles have no effect on the (Bi, Pb)-2223 phase’s structure. Similar findings of unchanged peak positions and undetected nanoparticle-rich or oxide phases were obtained by adding BaSnO₃ [62], ZrO₂ [63], Al₂O₃ [64], SnO₂ [22, 65], and ZnO [66] nanoparticles into the (Bi,Pb)-2223 phase.

Fig. 1

a XRD pattern for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples, b Rietveld refinement of 0.00, 0.04, and 0.20 wt% patterns, and c crystal structures of pure sample

The X-ray diffraction patterns for all samples were refined using the Fullprof program [67]. The details of the refinement procedure were done as follows: In the first step, the intensities of the experimental data were fitted using the pseudo-Voight function to refine the peak shape parameters, the linear interpolation function was used to describe the background between 38 points, and the Berar–Baldinozzi method was applied for the asymmetry correction. The parameters, such as scale factors, lattice parameters, instrumental profile, peak shape parameters, and structure parameters, were then refined. At the end of the refinement, the agreement factors (R_P, R_wp) of (7.92, 10.20), (8.20, 10.50), and (8.29, 10.50), and the goodness-of-fit parameters (χ²) of 1.58, 1.67, and 1.84, respectively, were obtained for samples with x = 0.00, 0.04, and 0.20 wt%, respectively, as shown in the graphs of Fig. 1b. Furthermore, Fig. 1c visualizes the crystal structures of a pure (Bi,Pb)-2223 sample, drawn with VESTA software [68] through Fullprof output data. As shown, the unit-cell structure consists of two blocks of BiO–SrO–CuO₂–Ca-CuO₂–Ca-CuO₂–SrO–BiO sequences along the c-axis. Moreover, the estimated average bond length of Cu–O in the CuO₂ plane is 1.89 Å, which is within the expected equilibrium distance range (1.80–2.00 Å), whereas the average Bi–O-bond length is 2.72 Å, which is within the expected equilibrium distance range (2.30–2.70 Å) [69,70,71].

The lattice parameters of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples were obtained from the Rietveld refinement and are listed in Table 2. It is clear that the lattice parameters (a and b) are almost unchanged, while there is a slight fluctuation without systematic trend in c-lattice parameter values that may be caused by a change in the distance between CuO₂ layers, which could affect how charges are transferred to these layers after the addition of some nanoparticles and the occurrence of a disordered rearrangement of oxygen vacancies. Similar behavior was mentioned early on for (Bi,Pb)-2223 phase inclusion with different nanoparticles, such as (K₅[SiFe(OH₂)W₁₁O₃₉]·14H₂O,[FeSiW₁₁]) [72] and TiO₂ [73]. Moreover, the orthorhombic distortion OD (b-a/a) is calculated and presented in Table 2. It is obvious that all values of OD are less than 1% for all composite samples, approving the stability of the (Bi, Pb)-2223 phase, and there is no transition to a tetragonal structure [74]. This suggests that the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x addition does not enter the crystal structure of the (Bi, Pb)-2223 phase but rather settles between the grains at low concentrations while sticking above the grains at higher concentrations [23, 62, 75, 76], as discussed in detail in the SEM section.

Table 2 XRD characteristics parameters and average grain sizes for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples

The relative intensity method was used to compute the values of the relative volume fraction of the main phase and the impurity phases [77], and their values are presented in Table 2. The volume fraction of the (Bi,Pb)-2223 phase increased to 87.87% for x = 0.04 wt% by enhancement percentage of 7.588% and then decreased for further addition. Conversely, the volume fraction of (Bi,Pb)-2212 and Ca₂PbO₄ impurity phases show opposite trends. According to these findings, a small addition of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) up to 0.04 wt% improved the phase formation of (Bi, Pb)-2223, whereas a higher concentration degraded the main phase. This improvement could be explained by the addition of a low concentration of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉), which improved the viscosity, homogeneity, and rate of formation of the (Bi, Pb)-2223 phase at the reaction temperature [64]. While the degradation of the volume fraction of the (Bi,Pb)-2223 phase at high concentrations of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) occurs due to an alteration in the reaction rate and promotes the development of other phases, this causes the formation of the (Bi, Pb)-2223 phase to be slightly delayed. These results coincide with those mentioned in earlier reports when BaSnO₃ [62], Al₂O₃ [64], MgO [11], and Eu₂O₃ [78] nanoparticles were added to the (Bi,Pb)-2223 superconducting phase. Moreover, the ferrimagnetic nature of Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles, with the saturation magnetization (M_s = 56.619 emu/g) and the coercivity field (H_C = 5813.9 G), which encourages Cooper pair breaking and retards the formation of the (Bi,Pb)-2223 phase at high concentrations, may also be responsible for this delay [79].

The porosity percentage (P%) of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi, Pb)-2223 composites was calculated using Eq. (19) and is plotted in Fig. 2

$$P\left(\%\right)=\left(1- \frac{{\rho }_{\mathrm{exp}}}{{\rho }_{\mathrm{th}}}\right)\times 100,$$

(19)

where ρ_exp and ρ_th are the experimental and theoretical densities, respectively.

Fig. 2

Variation the porosity% with x(wt%) of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composite samples

It was discovered that the value of (P %) for our samples ranged from 20 to 40%, which is acceptable for ceramic superconductor materials [80]. Furthermore, as the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) concentration increased up to x = 0.04 wt%, the porosity decreased by 5.688%, indicating that the addition of a small amount of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) improved the microstructure of these samples. Then, for x = 0.05 wt%, a slight increase and a noticeable increase for x = 0.10 and 0.02 wt% resulted in a significantly increased porosity, increasing the chance of the formation of voids and pores, which weakens the links between the grains for these samples, as will be discussed in detail in the SEM section.

4.3 SEM

SEM micrographs of selected nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composites, with x = 0.00, 0.01, 0.04, 0.05, and 0.20 wt%, are shown in Fig. 3. All samples shared flaky layers of random alignment and large platelet-like structures, which are hallmarks of the main phase (Bi,Pb)-2223 [64]. Besides, the appearances of some grains with irregular shapes were designated for the impurity phase Ca₂PbO₄ with different percentages in all micrographs of composite samples, as in the composite sample with x = 0.05, and 0.10 wt%, respectively. Moreover, there are still tiny spherical grains that might be signs of Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles in the (Bi,Pb)-2223 phase, which agglomerate and stack on the grain’s surface as the concentration of nanoparticles increases, as seen in x = 0.20 wt%. The average grain size was calculated using the ImageJ program [81]. The distribution of grain sizes can be clearly seen from the histogram in Fig. 3. It was noted that the average grain size increased as the concentration of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) increased up to x = 0.04 wt% and then decreased for further concentrations, as shown in Table 2. Meaning that the addition of a small amount of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) improved the grain’s growth, and this indicates an enhancement in the grain’s conductivity. The nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) added to fill voids and cracks reduced the porosity of these samples [82,83,84]. While the voids in the composite samples with x = 0.05 wt% increased slightly and the grain size began to degrade, they were still better than the pure sample. The voids increased significantly for samples as the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) concentration increased, as seen in the micrograph for x = 0.20 wt%, and their grain size decreased. This is due to the agglomeration of the Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles at high concentration, which produces the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) screening effect. As a result, nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) acted as a barrier to grain growth and demonstrated weak links between them in these samples [45, 46, 85,86,87].

Fig. 3

Some selected SEM micrographs with grain size distributions for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composite. *The orange circle for (Ca₂PbO₄). *The yellow circle for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉). *The red color represents the voids between the grains

4.4 Electrical resistivity measurements

The electrical resistivity dependence on temperature of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 samples with 0.0 ≤ x ≤ 0.2 wt% is represented in Fig. 4. All samples show metallic-like behavior from room temperature down to the onset temperature (${T}_{c}^{\mathrm{onset}})$, followed by a superconducting transition at T_c. A small curvature in the electrical resistivity data is observed above T_c, indicating superconducting thermodynamic fluctuations [33]. These fluctuations are due to the appearance of Cooper pairs in a finite range above T_c [88]. The values of T_c are determined from dρ/dT versus T plot as the temperature corresponds to the maximum dρ/dT as shown in the inset of Fig. 4, and their values are displayed in Table 3. The zero-resistivity temperature, T₀, also exhibits the same patterns as T_c, rising from 93.250 to 103.500 K when x increased from 0.00 to 0.04 wt%, and then falling to 80.00 K when x = 0.20 wt%. As seen from Table 3, as x increases from 0.00 to 0.04 wt%, the T_c increases from 107.831 to 112.440 K, and then, it is reduced for further addition. The increases in T_c can be attributed to the increase in the (Bi, Pb)-2223 volume fraction phase, the improvement in grain connectivity, and the increasing grain size in the low concentrations, as confirmed by the SEM result. The decrease in T_c values, on the other hand, is attributed to an increase in the impurity phases, and a weak link in grains conductivity due to nanoparticles’ agglomeration, which results in scattering and the breaking of the Cooper pairs due to the ferrimagnetic nature of the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉), with saturation magnetization (M_s = 56.619 emu/g) and the coercivity field (H_C = 5813.9 G) [89]. The values of the superconducting transition width, ∆T = T_c–T₀, which determine the quality of the superconducting transition and the purity of the sample, are shown in Table 3. It is noted that the transition to T₀ for the samples with x = 0.00 to 0.05 wt% seems to be nearly sharp, which indicates that the phase formations of these samples are nearly identical to single-phase formations [90]. On the other hand, the samples with x = 1.0 and 0.2 wt% showed a broadening in the transition to T₀; this broadening being attributable to the dominance of impurities on the phase formation of these samples, which resulted in a decrease in the T_c value. As a result, for x > 0.05 wt%, high values of the transition width, ∆T, may explain the appearance of macroscopic inhomogeneity and impurity phases.

Fig. 4

Electrical resistivity of (Bi,Pb)-2223 with various amount of Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles

Table 3 Variation of T_c, ΔT, T₀, ρ_room, ρ₀, and α for all (Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 samples

The normal state resistivity (ρ_room) at 300 K, residual resistivity (${\rho }_{0}$) at 0 K, and the resistivity temperature coefficient ($\alpha$) were obtained by the fitting of the resistivity data in range of temperature 150 ≤ T ≤ 300 K according to Matthiessen’s rule [91], and their values are presented in Table 3

$$\uprho \left(\mathrm{T}\right)={\uprho }_{0}+\mathrm{\alpha T}.$$

(20)

As seen in Table 3, the lower values of the resistivity parameters ρ_room, ρ₀, and $\alpha$ of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 samples at x = 0.04 wt%, indicate a high purity of this sample, as obtained from XRD data, and good links between superconducting grains, as well as obtained from the SEM microstructural, indicating the addition of the Ba_0.5Sr_0.5Fe₁₂O₁₉ nanoparticles had a positive effect in the (Bi, Pb)-2223 superconductors’ phase.

4.5 Vickers microhardness measurement

Figure 5 displays Vickers microhardness values (H_V) that are obtained with precision from Eq. (1). Each sample clearly exhibits non-linear microhardness behavior (load-dependent region) and then progresses to sustain saturation as the applied load exceeds 3 N (load-independent region). This behavior can be explained based on the penetration depth of the indenter [92]. The indentation size effect (ISE), which describes non-linear behavior in a load-independent region, has been widely reported in the literature [27, 28, 93, 94]. Furthermore, H_v values increased with nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) concentration up to x = 0.04 wt% by 11.995% of the pure sample, indicating that the small amount of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) has a noticeable influence on increasing microhardness, indicating an improvement in the mechanical properties of these samples. This improvement is attributed to the SEM results, which revealed an increase in grain connectivity due to porosity reduction and an increase in specimen resistance to crack propagation among grains due to the addition of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉). While it is noted that there was a slight decrease in H_v values for x = 0.05 wt%, it is still higher than the composite sample with x = 0.03 wt%. The values of H_v then decreased significantly with an additional increase up to x = 0.2 wt%. This decrease is attributed to the unsatisfactory results for the porosity and SEM micrograph analysis of these samples.

Fig. 5

Variation of H_v versus F for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composite with (0.00 ≤ x ≤ 0.20 wt%)

4.5.1 Analyses and modelling of microhardness

The (ISE) behavior for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples was investigated, using Meyer’s law according to Eq. (2) and the graph of $\mathrm{ln}F$ versus $\mathrm{ln}d$, as shown in Fig. 6 for some selected samples. The values of the constants A and Meyer’s number n were determined from the fitted curves of the experimental data, as shown in Fig. 6.

Fig. 6

$\mathrm{Ln}F$ versus $\mathrm{Ln}d$ according to Meyer’s low for some selected nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)x/(Bi,Pb)-2223 composite samples

It is clear that from Table 4, all samples have Meyer’s numbers in the range of (1.682– 1.694) for all samples, confirming normal ISE behavior. Furthermore, the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)x/Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composite indicates a high-temperature superconductor with a hard material range [95]. Moreover, the values of Meyer’s constant A are not appreciated for high-temperature superconductors due to their small values. These results are similar to those found by Abdeen et al. [33] in their study to investigate the electrical and mechanical properties of (Bi,Pb)-2223 by substitution of Ca²⁺ by Ho³⁺ ions.

The H–K approach was used to analyze the microhardness of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/ Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composite samples. The H_V values were calculated according to Eq. (3). The values of the fitting parameters W_HK and A₁ obtained from Fig. 7 are summarized in Table 4. In addition to that, the load-independent microhardness H_H-Kin values were calculated by Eq. (4) and are listed in Table 5.

Fig. 7

${\varvec{F}}$ versus ${{\varvec{d}}}^{2}$ according to H–K model for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composite

Table 4 Parameters of different theoretical models for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composite

Table 5 Variation of load-independent microhardness H_vin with the deviation percentage, and average H_v in plateau region according to different models for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples

According to Table 4, the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples positive W_HK values show that the applied load is sufficient to cause both elastic and plastic deformation [96]. Furthermore, the microhardness H_V values of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composite obtained from Eq. (3) are lower than our experimental H_V values as seen in Fig. 8 for samples with x = 0.00, 0.03, 0.04, and 0.20 wt%. Additionally, the true microhardness H_H-Kin values, which were calculated by Eq. (4) as seen in Table 5, are lower compared with the values of H_Vexp that were obtained in the plateau region. Therefore, the H–K approach is not suitable to describe our experimental data for all samples.

The EPD model was used to calculate and analyze the H_V values of prepared nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi, Pb)-2223 composite samples using Eq. (5). The parameter values of d₀ and A₂ were obtained from the fitted curves of F ^0.5 versus d as given in Fig. 9 and are listed in Table 4. Moreover, the load-independent microhardness H_EPDin values were calculated using Eq. (6) and are tabulated in Table 5.

Fig. 8

Variation of experimental and theoretical H_V values according to H–K, EPD, PSR, and MPSR models with the applied load for following composite samples: a pure, b nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_0.03/(Bi,Pb)-2223, c nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_0.04/(Bi,Pb)-2223, and d nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_0.20/(Bi,Pb)-2223

Fig. 9

F ^0.5 versus ${\varvec{d}}$ according to EPD for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples

From Table 4, the values of d₀ are positive, which confirms that all composite samples exhibit the ISE behavior. On the other hand, the smallest value of d₀ (9.584 μm) and the best value for the parameter A₂ for x = 0.04 wt%. It means that adding nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_0.04 improved the mechanical properties of this sample. Furthermore, as shown in Fig. 8, the theoretical H_V values for the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/Bi_1.8Pb_0.4Sr₂Ca₂Cu_3.2O_10+δ composite samples calculated from Eq. (5) are lower than the experimental H_V values for selected samples with x = 0.00, 0.03, 0.04, and 0.20 wt%. In addition, as shown in Table 5, the true microhardness values H_EPDin calculated using Eq. (6) were lower than the H_Vexp values obtained in the plateau region. Thus, the (EPD) model is not suitable to explain our H_V experimental data.

The PSR model was used to analyze the microhardness of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples. According to Eq. (7), the H_V values were also calculated through a plot of F/d against d that yielded a straight line as seen in Fig. 10, where the best-fit parameters of  ${\alpha }_{1}$ and ${\alpha }_{2}$ can easily be obtained by regression analysis and are depicted in Table 4. Moreover, the H_PSRin values are estimated from Eq. (8) and summarized in Table 5.

Fig. 10

F/d versus ${\varvec{d}}$ according to PSR model for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/ (Bi,Pb)-2223 samples

It is noted that in Table 4, the parameter ${\alpha }_{1}$ values are positive for all samples, confirming that all samples present normal (ISE) behavior as well as the production of both elastic and plastic deformations at the same time. By increasing the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) up to x = 0.04 wt%, the values of parameter ${\alpha }_{2}$ increased, indicating an improvement in the mechanical properties of these samples. Furthermore, the calculated theoretical H_V values for all nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi, Pb)-2223 samples using the PSR model from Eq. (7) are nearly equal to the values of our experimental H_V data, as shown in Fig. 8 for samples with x = 0.00, 0.03, 0.04, and 0.20 wt%. For all composite samples, the points are perfectly matched, as shown in the inset figure in Fig. 8c. In addition, it is apparent from Table 5 that the true microhardness values of H_PSRin were calculated from Eq. (8). It was found that their values were the closest to the H_Vexp values that were obtained in the plateau region from the other models. As well, it can be seen in Table 5 that the PSR model has the smallest deviation percentage (D%) in comparison with the other models. It implies that the theoretical PSR model is appropriate for explaining the general mechanical behavior of (Bi, Pb)-2223 superconducting ceramic systems.

The microhardness data for the prepared nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi, Pb)-2223 composite samples were also analyzed using the MPSR model. The H_V values were determined using Eq. (9). The values of ${\alpha }_{0}$, ${\alpha }_{1}$, and ${\alpha }_{2}$ parameters are obtained from the quadratic regressions of a plot of F against d, as seen in Fig. 11, and they are tabulated in Table 4. Furthermore, the H_MPSRin values are obtained from Eq. (10) and summarized in Table 5.

Fig. 11

${\varvec{F}}$ versus ${\varvec{d}}$ according to MPSR model for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 samples

Table 4 shows that the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi, Pb)-2223 composite samples exhibit typical ISE behavior due to positive MPSR parameter α₂ values. The values of parameter α₂ increased with the increment in the nanoaddition concentration up to the value of x = 0.04 wt% and then decreased to x = 0.20 wt%. This means that the sample with the highest nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) concentration has the best mechanical durability, strength, and fracture toughness against the applied test load [97]. Furthermore, the H_V values calculated by Eq. (9) for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)x/(Bi, Pb)-2223 composite samples are nearly identical to the H_V experimental values, as shown in Fig. 8. However, it is noted that the values of H_MPSR conducted in the low load range are slightly increased, which is attributed to the effect of the existence of the indentation load term on hardness testing conducted in the low load range [34, 98]. Based on this, we can conclude that the MPSR model can provide a satisfactory description of our experimental data within a specific range. Moreover, the true microhardness values H_MPSRin which are obtained from Eq. (10), as seen in Table 5, are slightly close to the H_Vexp values that were obtained in the plateau region. In particular, the samples with x = 0.03 and 0.04 wt% are much closer than other samples, as shown in the inset figure in Fig. 8b. However, the MPSR model is considered unsatisfactory, because it does not interpret all our experimental results compared with the PSR model.

Finally, IIC model was used to examine the crucial changes in the general mechanical characteristic features in addition to the load-independent microhardness parameters for the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples. The H_V values are calculated from Eq. (12). The parameter values of m and k are extracted from the plotting curves in Fig. 12 and listed in Table 4.

Fig. 12

Ln H_V versus ln(F^5/3/d³) according to IIC model for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 samples

From Table 4, the values of parameter m for all nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 samples are smaller than 0.6, which indicates normal ISE behavior. These results confirmed the results that were acquired through Meyer’s law. Besides, it is evident in Table 5 that the true microhardness H_V values, which are calculated according to Eq. (12) of the IIC model, are very far from the H_V values that were obtained in the plateau region. Thus, this model is not appropriate for the examination of our experimental data.

Table 6 clarifies the calculated values of important mechanical parameters in the plateau region E, Y, K_f, Bi, and CII for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi, Pb)-2223 composite samples using Eqs. (13), (14), (15), (16), and (17), respectively. All parameters follow the same trend, increasing systematically with increasing the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) concentration up to x = 0.04 wt%, and then decreasing to x = 0.20 wt%, as shown in Table 6. The toughness (K_f) of high-temperature superconductors is the most important parameter, because it affects the material's mechanical behavior, which is the material's capacity to withstand cracks [99]. The toughness increased at the optimum of 0.04 wt% by 6.529% of the pure sample due to an increase in the surface energy α₁. That is, a small amount of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) enhanced the mechanical properties of the superconducting ceramic systems of the (Bi, Pb)-2223 phase. On the contrary, they decreased for high doping x > 0.04 wt%. which was attributed to the poor results that were obtained from the microstructure analysis for these samples. These observations were reported in the literature [22, 23, 79, 100].

Table 6 Calculated parameters in plateau region for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 samples

4.6 Indentation creep analysis

Figure 13a–c shows the fluctuation of microhardness H_V with time at fixed loads of 2.94, 4.90, and 9.80 N, respectively, for our nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composite samples. As seen in these figures, the values of Vickers microhardness H_V decrease slightly and linearly with increasing dwell time for all samples. This behavior supports the indenter's basis for penetration depth and appears to represent the typical relationship between microhardness and time [101]. This fall in the hardness values of the (Bi,Pb)-2223 phase with dwell time indicates material-bearing creep deformation [19].

Fig. 13

Variation of H_v versus time at different applied loads a F = 2.96 N, b F = 4.9 N, and c F = 9.80 N for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 samples

Furthermore, the power-law indentation creep behavior of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)x/(Bi, Pb)-2223 composite samples was investigated using Eq. (18) by plotting Ln H_v versus Ln t, as shown in Fig. 14a–c, with loads (a) F = 2.96 N, (b) F = 4.9 N, and (c) F = 9.80 N. From the slop of the fitted lines, the stress exponent μ values were obtained and are shown in Table 7. It is clear that the values of the stress exponent ranged between 10 and 12.2 for samples with low nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) concentrations up to 0.04 wt%, which indicates that the operative creep mechanism of these samples is dislocation creep [22]. The stress exponent for high-doped x > 0.05 wt%, on the other hand, ranges between 4.2 and 7.6. Thus, the mechanism of these samples is dislocation climbs creep.

Fig. 14

Fluctuation of Ln H_v vs Ln t at different applied loads, F, a 2.96 N, b 4.9 N, and c 9.80 N for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 composites

Table 7 Variation of μ with different applied load F for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi,Pb)-2223 samples

5 Conclusion

Polycrystalline nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉)_x/(Bi, Pb)-2223 superconducting composites with x = 0.0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.1, and 0.2 wt% have been successfully prepared. The XRD results showed the hard magnetic ferrite nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) does not enter the crystal structure of the prepared phase and instead is distributed and settled between the superconducting grains. The SEM analysis showed that the average grain sizes increased up to x = 0.04 wt% and then decreased for a high concentration of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) accompanied by an enhancement in the grain connectivity and a drop in porosity percentage (P%) from 30.139 to 24.409%. According to measurements of Dc-electrical resistivity, the addition of nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) up to x = 0.04 wt% induced an improvement in the (Bi.Pb)-2223 superconductor’s (T_c), The Vickers microhardness measurements showed that all samples have ISE behavior. Furthermore, the microhardness was increased by increasing the nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) concentration up to 0.04 wt%, while it was decreased for x ≥ 0.05 wt%. Furthermore, different theoretical models were utilized for investigating true microhardness.  The PSR model was the best theoretical model that described the obtained experimental data, which produced significantly closer results to the experimental true microhardness. Furthermore, the indentation creep mechanism demonstrated in our samples is dislocation creep for nano-(Ba_0.5Sr_0.5Fe₁₂O₁₉) concentrations of x < 0.05 wt% and dislocation climbs for x ≥ 0.05 wt%.