Statistical characterization of cobalt–zinc ferrite doped with gadolinium ions

Central composite design

The main purpose of this article is to study the effect of the molarity of sodium hydroxide and the concentration of gadolinium (controllable variables) on: Td, Mmax, Hci, and pyromagnetic coefficient. The ferrite nanocrystals were synthesized by co-precipitation; where NaOH is a precipitating agent that promotes the ferrite’s growth, making its molarity critical. The controllable variable levels chosen were previously studied varying in range [0.24–0.72 (M)] for the sodium hydroxide and [0.01–0.03] for gadolinium, in the formation of Co_0.7Zn_0.3Fe_2-yGd_yO₄ ferrite.

Designing experiments, in general, consists of three phases: planning, execution, and analysis. The use of designing tools, such as the one proposed for this article, provides the smallest number of runs aiming to obtain the most information. Research laboratories can greatly benefit from these characteristics owing to their usual scarcity of resource. Design of experiments are also useful for subsequent system optimization, as shown in Garcia-Loera et al. ⁴ and Hernandez et al. ⁵ The design chosen for this work was the central composite design for various reasons:

The CCD, in short, studies the effect of each of multiple factors and their interactions, usually up to a second order. The design for this work consists of corner, central, and axial points with four replicates in the center, as seen in Figure 1. Statistical characterization of the magnetic and structural properties was performed using a significance of α = 0.05. The coded values were used to perform every analysis throughout the experiment. For the practitioner, it is important to note that the CCD can be manipulated to sample each factor at three levels, which results in the more typical configuration of a 3² factorial experiment.

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Figure 1.

Central composite design with coded values.

Statistical characterization

The transformation of the natural variables (NaOH molarity and Gd-concentration) to the coded variables for each factor was carried out using the coordinates of the central composite design. For each combination of factors, the ferrite nanoparticles were synthesized by the conventional co-precipitation method. The corresponding structural analysis of the powdered samples was carried out on a Siemens D5000 X-ray diffractometer. The magnetic properties Mmax (emu/g) and Hci (Oe) were obtained on a LakeShore 7400 Series vibrating sample magnetometer. Magnetization-Applied magnetic field (M-H) measurements were performed at room temperature in fields up to 22 kOe. Results for the properties measured are displayed in Table 1. All equipment and software used in this work were available at UPRM facilities.

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Table 1.

Results for the structural and magnetic properties of the ferrite.

Gd (coded)	NaOH (coded)	Gd	NaOH (M)	X’s (nm)	a(nm)	Mmax (emu/g)	Hci (Oe)
1	−1	0.02	0.31	17	0.0836	57	40
1	1	0.02	0.65	26	0.0836	58	19
−1	−1	0.01	0.31	34	0.0835	58	31
−1	1	0.01	0.65	35	0.0836	59	73
0	0	0.02	0.48	22	0.0835	57	21
0	0	0.02	0.48	22	0.0835	57	25
0	0	0.02	0.48	24	0.0836	57	32
0	0	0.02	0.48	23	0.0835	61	28
−1.414	0	0.01	0.48	12	—	58	33
1.414	0	0.03	0.48	33	—	52	29
0	1.414	0.02	0.72	11	—	59	39
0	−1.414	0.02	0.24	29	—	47	36

The structural properties of the Co_0.7Zn_0.3Fe_2-yGd_yO₄ ferrite are studied since X (nm) is related to the maximum magnetization and the formation of a ferrofluid suspension. The XRD patterns confirm the formation of the ferrite phases for the corner points as seen in Figure 2.

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Figure 2.

X-ray diffraction analysis for corner points.

The average size of the crystal X(nm) and lattice parameter (structural properties) were analyzed statistically using MINITAB16, as well as for the remaining properties. The analysis of variance for the structural properties, average size of the crystal, and lattice parameter showed no linear effect, interaction, or curvature as functions of the desired variables, thus neither property can be manipulated through variation of gadolinium or sodium hydroxide. The statistical analysis of Mmax (emu/g) revealed that neither sodium hydroxide nor gadolinium had a significant effect on magnetization, which assumes values of between 57 and 61 emu/g. This slight variation is within the experimental error. In turn, the analysis of variance for coercivity (Oe) suggested an interaction between the controllable variables with a p-value of 0.009 and a negative coefficient. The results are displayed in Table 2 and Figure 3.

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Table 2.

Analysis of variance for coercivity (hci).

Term	Coefficient	p-value
Constant	23.250	0.004
Block	3.250	0.284
Gd	−6.332	0.067
NaOH	3.155	0.297
Gd × Gd	5.500	0.158
NaOH × NaOH	8.750	0.046
Gd × NaOH	−15.750	0.009

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Figure 3.

Surface plot for coercivity (Hci) as a function of NaOH molarity and Gd atomic percentage.

Our results suggested that an increase in the atomic concentration of gadolinium-ions in the ferrite lattice would cause coercivity to decrease. This decreasing trend in coercivity can be attributed to the null contribution of Gd³⁺ (4f⁷) ions to the angular momentum, which is conducive to vanish the spin–orbit interaction associated with coercivity. ⁶ This behavior has been previously reported in Urcia-Romero. ⁷

Figure 3 graphically shows the interaction detected by statistical means. This figure shows the variation of coercivity as a function of both, gadolinium-ion atomic percentage and sodium hydroxide molarity. In this case, the response surface suggests an enhancement of coercivity in the sodium hydroxide and gadolinium doping ranges of 0.625–0.675 M and 0.010–0.012%, respectively. The corresponding average crystal size was around 35 nm. This rising trend in coercivity can be explained in terms of the enlargement of the crystal within the single magnetic domain region^8–9 and the eventual variation in the ionic distribution in the ferrite lattice. This hypothesis will be corroborated by Mössbauer analyses and the results will be reported elsewhere in the future.