Introduction

Since the 1990s, magnetic Fe3O4 nanoparticles (Fe3O4 NPs) have emerged as an important material in biomedical and magnetic fields. In biomedical fields, Fe3O4 NPs have been used as drug carriers to enhance the magnetic resonance imaging MRI (Gupta et al. 2020; Khmara et al. 2019; Teymourian et al. 2013). The magnetic properties of Fe3O4 NPs enable their usage as a magnetic material in electronics as well as magnetic fluids such as ferrofluids, battery, catalysis and magnetic storage media (Nkutha et al. 2020; Wu and Huang 2017; Yao et al. 2015). Recently Fe3O4 NPs have become one of the outstanding magnetic materials (as an adsorbent) in wastewater treatment applications. They can be used to treat wastewater contaminated by heavy metals such as Cadmium (II), Chromium (VI) and Copper (II) (Chen et al. 2017; Ewis et al. 2020; Hajizadeh et al. 2020; Matome et al. 2020). The advance in terms of combining nanotechnology and membrane technology has greatly expanded the use of Fe3O4 NPs by introducing them as nanofillers in membrane modification and concurrently enhancing the membrane application for wastewater treatment (Ba-Abbad et al. 2017; Daraei et al. 2013; Ewis et al. 2021). However, it is crucial to obtain high reactivity ultrafine Fe3O4 NPs with higher ratio of surface area to volume (Tang and Lo 2013). Therefore, techniques for preparing smaller and higher surface area to volume ratio of Fe3O4 nanoparticles are becoming more important. Various methods have been used to prepare Fe3O4 nanoparticles such as co-precipitation, polyol processes, thermal decomposition and micro-emulsion (Chen et al. 2011; Lu et al. 2013; Zhang et al. 2013). Among them, co-precipitation method has gained popularity because of its simplicity (Hariani et al. 2013), requirement of less expensive equipment and ability to consistently produce smaller Fe3O4 particles. During the co-precipitation method, the effect of process factors such as pH, Fe2+ to Fe3+ ion molar ratio as well as the interaction between process factors have to be considered in order to produce the desired Fe3O4 size for specific applications. The conventional technique for optimizing the process factors needs more and more experimental efforts, which consume more time- and yet ignore the interaction effects between the factors of the process. Such limitations may be overcome by a three-level statistical design as it can minimize the total number of experiments, provide appropriate models for process optimization and give better illustration of the inter-variable interactions. The most well-known statistical approach is response surface methodology (RSM), which consists of a central composite design (CCD) (Lingamdinne et al. 2020), Box–Behnken design (BBD) (Bahari et al. 2019) and D-optimal design (Ba-Abbad et al. 2013; Hariz et al. 2018). RSM has been proven to provide significantly improved results using few experimental runs.

Several studies applied RSM to control the size and properties of metal oxides nanoparticle during the preparation stage. In order to minimize the size and enhance the photocatalyst properties of ZnO nanoparticles, the D-optimal design as one of RSM designs is applied (Ba-Abbad et al. 2013). Three main factors have a significant influence on the particle size, which are molar ratio between starting materials (zinc acetate and oxalic acid), pH of the solution and the calcination temperature. The minimum size produced experimentally was 18 ± 2 nm compared to 22.9 nm which is predicted by the model (Ba-Abbad et al. 2013). Kuldeep Singh et al.(2020) prepared iron oxide nanoparticles using Coriandrum sativum L. Leaf extract (Singh et al. 2020). The study showed that volume of leaf extract, agitation speed, and temperature has a significant influence on the particle size. The developed model using Box–Behnken design produced smallest size of iron oxide nanoparticles (mean 161.5 nm) as the main target of applying the RSM and totally closed with predict under optimum condition (mean 157.3 nm). On the other hand, nickel oxide nanoparticle was synthesized and optimized through the sol–gel method using Box–Behnken design (Ba-Abbad et al. 2015). To control nickel oxide nanoparticle during synthesis to produce the smallest possible particle size, the molar ratio between starting materials (nickel nitrate and citric acid), pH of solution and calcination temperature were chosen as the most significant factors. Nickel oxide nanoparticles were synthesized under predicted condition by Box–Behnken design for the minimum particle size of 13.7 nm. The particle size produced experimentally as 14.3 nm was in a good agreement with the predicted size by Box–Behnken design.

Therefore, the aim of this study is to produce the smallest size of Fe3O4 NPs using co-precipitation method without adding any additive such as capping agents or surfactants. To investigate this aim, the preparation conditions such as molar ratio between Fe2+ and Fe3+ ions as well as pH were controlled, and the optimum process parameters were determined using Response Surface Methodology (RSM). These conditions of experiments were optimized using Box–Behnken design with particle size of Fe3O4 NPs as the response. Nevertheless, the literature shows no comprehensive optimization of these factors (Fe2+ and Fe3+ ions as well as pH) to produce the smallest possible size of Fe3O4 NPs. To the best of the authors’ knowledge, there are no studies on optimizing the synthesis condition of Fe3O4 NPs using co-precipitation method with wide range of Fe2+ and Fe3+ ions molar ratio (1 to 3) and short range of pH (10–12). The previous study reported the ratio between Fe2+ and Fe3+ is 1:2, but ignored the effect of other ratios that can possibly produce smaller size of Fe3O4 NPs considering other factor such as pH. On the other hand, desired pH of solution was selected as factor in this study because to avoid any excess of ammonium hydroxide (ammonia) that used as volume (ml) in previous studies, which have been influenced on particle growth (Liu et al. 2022). This study shows complex interactions between factors that can be controlled to produce the minimum possible size of Fe3O4 NPs, which is not reported elsewhere. The results are expected to be the base for future studies aiming at developing Fe3O4 NPs using the co-precipitation method.

Materials and methods

Materials and methods

In order to synthesize Fe3O4 NPs, ferric chloride hexahydrate (FeCl3·6H2O) which was supplied by Merck, Germany and ferrous sulfate heptahydrate (FeSO4·7H2O) was ordered from Bendosen, Malaysia. The Ammonium hydroxide (NH4OH) with 30% of ammonia in water was obtained from R&M Chemicals, Malaysia. Fe3O4 nanoparticles were synthesized by the co-precipitation method. Initially, FeCl3 ·6H2O and FeSO4·7H2O were dissolved in ethanol and pure water, respectively. After both solutions had completely dissolved, solution of FeSO4·7H2O was added slowly to the other solution of FeCl3·6H2O. The final mixture was heated to 35 °C followed by the addition of NH4OH to reach the desired pH. The solution mixture was stirred vigorously under heating at 60 °C for 2 h. The resulting solution was then left to cool followed by washing. Repeated washing was carried out using deionized water until the pH reached 7 ± 0.2 and the washing process was performed once with ethanol to remove remaining water. The final powder was kept overnight at 75 °C in an oven to remove remaining water and ethanol. Figure 1 shows the flow diagram of experimental procedures to synthesize Fe3O4 NPs.

Fig. 1
figure 1

Flow diagram for Fe3O4 NPs synthesis

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The chemical reaction between the reactants to produce Fe3O4 NPs can be expressed by Eq. (1):

$${\text{Fe}}^{2 + } + 2{\text{Fe}}^{3 + } + 8{\text{OH}}^{ - } \mathop{\longrightarrow}\limits^{{60^{ \circ } {\text{C}}}}{\text{Fe}}_{3} {\text{O}}_{4} + 4{\text{H}}_{2} {\text{O}}$$
(1)

To test of Fe3O4 NPs performance for removal of Congo red (CR) dye as model of pollutant, the experiments were carried out in conical flask 50 ml. About 30 ml of CR solution with 20 ppm as initial concentration was shacked with certain dosage of Fe3O4 NPs as 0.05 and 0.1 g for 1 h at room temperature. After separation of Fe3O4 NPs using external magnet, final concentration of CR was measured by UV–Vis spectrophotometer (Perkin Elmer, lambda 35). The removal (R %) of CR was calculated using Eq. (2):

$$R \%=\frac{{C}_{0}-C}{{C}_{0}}\times 100$$
(2)

where C0 and C are the initial and final concentration of CR, which is determined by UV–Vis absorbance before and after adsorption process using Fe3O4 NPs.

Box–Behnken experimental design (BBD)

The statistical technique that uses a sequence of designed experiments to get optimal responses with given operating conditions is call response surface methodology. The BBD which is one of the RSM standard designs was selected to investigate the effect of the process factors according to the finally established empirical model. The BBD (Minitab 19) was selected in this study because of its ability to create designs with desirable statistical properties. BBD requires only a small number of experimental runs to determine the complex response function (Krishna and Padma Sree 2013). The derived complex response function can be fitted into a second-order polynomial model as shown in Eq. (3)

$$Y = C_{0} + a = 1\sum\limits_{a = 1}^{A} {C_{a} x_{a} } + \sum\limits_{i = 1}^{A} {C_{aa} x_{a}^{2} } + \sum\limits_{a = }^{A - 1} {\sum\limits_{b = a + 1}^{A} {C_{ab} x_{a} x_{b} } }$$
(3)

where Y is the predicted response associated with each factor level combination, and C0, Ca,Caa and Cab are the intercept term coefficient, linear term coefficient, quadratic term coefficient, and interaction coefficient, respectively. A xa and xb indicate the independent variables. Additionally, k is the number of the independent variables. In this study, three factors are considered which are Fe2+ ion molar ratio, Fe3+ ion molar ratio, pH, to determine their effect on the particle size as well as understand the interaction between them. The molar ratio of Fe2+ and Fe3+ ions was varied between 1 and 3 and the pH of the solution was ranged from 10 to 12 using ammonium hydroxide. The variations of the process parameters are summarized in Table 1. A combination of these process parameters was used to determine the response of the particle size, as given in Table 2. However, the BBD contained a total of experiments as (17) with (12) experiments organized in a factorial design at the midpoint of the edges of the process space and (5) replicate experiments at the center point to evaluate of experiments error (Gidwani and Vyas 2016).

Table 1 Process parameter used for Box–Behnken design
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Table 2 Design layout of Box–Behnken and response for synthesized Fe3O4 NPs
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Characterization

X-ray diffraction (XRD)

The crystal size and structure of synthesized Fe3O4 NPs were determined using XRD technique. The analysis was achieved using a diffraction meter (Bruker AXS D8 Advance, Germany). The Fe3O4 NPs samples were scanned of a 2θ angle from 55 to 80°. The Scherrer equation Eq. (4) was applied to calculate the crystal size average.

$$D = \frac{K\lambda }{{\beta \cos \theta }}$$
(4)

where β is the peak width at half-maximum (full width at half maximum (FWHM),), K is the constant of Scherer equation which K = 0.89, λ is X-ray wavelength which λ = 1.5406 Å and θ is the Bragg diffraction angle.

Morphology analysis using Transmission electron microscope (TEM)

The particle shape and size of synthesized Fe3O4 NPs were determined by a transmission electron microscope with model of JEOL, JEM-2010, Netherland. A small amount of Fe3O4 NPs was dispersed in ethanol using sonication. After that, one drop of the suspended solution was dropped on a carbon-coated grid with 400 mesh. After a certain time, the sample was dried and introduced to the TEM machine.

Results and discussion

Box–Behnken design model for Fe3O4 particle size

The effect of the independent variables such as molar ratio of Fe2+ to Fe3+ and pH on the response of Fe3O4 particle size were determined with a quadratic polynomial model created by analysis of variance (ANOVA) in Eq. (5).

$$\begin{gathered} {\text{Y}} = - {633}.{5} + {15}.0{7}\left( {{\text{Fe}}^{{{2} + }} } \right) + {51}.{92}\left( {{\text{Fe}}^{{{3} + }} } \right) + 108.7 \left( {{\text{pH}}} \right) - {6}.{92}0\left( {{\text{Fe}}^{{{2} + }} } \right)^{{2}} - {6}.{263}\left( {{\text{Fe}}^{{{3} + }} } \right)^{{2}} - {4}.{91}0\left( {{\text{pH}}} \right)^{{2}} \hfill \\ - 0.{125}\left( {{\text{Fe}}^{{{2} + }} *{\text{ Fe}}^{{{3} + }} } \right) + {1}.{497}\left( {{\text{Fe}}^{{{2} + *}} {\text{pH}}} \right) - {2}.{52}0\left( {{\text{Fe}}^{{{3} + }} *{\text{pH}}} \right) \hfill \\ \end{gathered}$$
(5)

where Y is the predicted particle size included all linear term, quadratic term, and interaction coefficients. This model can be applied to predict the particle size of Fe3O4. The developed Eq. (4) also shows that the main process parameter influencing the Fe3O4 particle size is the Fe2+ ion molar ratio followed by pH, as shown in Table 3.

Table 3 ANOVA results for quadratic model of synthesized Fe3O4 NPs
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Statistical model for Fe3O4 particle size analysis

Effect of Fe.2+ ion molar ratio

Figure 2 illustrates the effect of Fe2+ ion molar ratio on Fe3O4 NPs size. The relationship between Fe2+ ion molar ratio and particle size appeared as a parabola shape in a second-order equation. Figure 2 shows that Fe3O4 particle size increases as the Fe2+ ion molar ratio increases from 1.0 to 2.5 and then decreases slightly when the molar ratio is more than 2.5. During the reaction, Fe2+ ion reacts with the OH− to form Fe(OH)2 by following Eq. (6). It is worth mentioning that as the amount of Fe2+ ion increases, more Fe(OH)2 is generated. Fe(OH)2 generated from Eq. (6) acts as an intermediate in Eq. (7), which reacts with O2 to produce FeOOH and water. More Fe(OH)2 and FeOOH compounds are produced at higher Fe2+ ion molar ratio, which causes Fe3O4 particle size to increase accordingly (Coppola et al. 2016). As shown in Eq. (8), the reaction of Fe(OH)2 with FeOOH ultimately produces Fe3O4 NPs. However, as the Fe2+ ion molar ratio increases to more than 2.5, Fe3O4 particle size starts to decrease. This was because the saturated condition has been reached (Yang et al. 2008).

$${\text{Fe}}^{2 + } + 2{\text{OH}}^{ - } \to {\text{Fe}}\left( {{\text{OH}}} \right)_{2}$$
(6)
$$4{\text{Fe}}\left( {{\text{OH}}} \right)_{2} + {\text{O}}_{2} \to 4{\text{FeOOH}} + 2{\text{H}}_{2} {\text{O}}$$
(7)
$${\text{Fe}}\left( {{\text{OH}}} \right)_{2} + 2{\text{FeOOH}} \to {\text{Fe}}_{3} {\text{O}}_{4} + 2{\text{H}}_{2} {\text{O}}$$
(8)
Fig. 2
figure 2

Effects of Fe2+ ion molar ratio on Fe3O4 NPs size

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Effect of pH

The effect of pH on the Fe3O4 NPs particle size also exhibited a quadratic shape of second-order relationship, as shown in Fig. 3. It can be observed that Fe3O4 particle size increased as the pH increased from 10 up to 11. Further increase in the pH has a divergent effect on Fe3O4 particle size reduction. However, within the reaction, the solution pH was adjusted using the NH4OH which formed hydroxide (OH−) ions and these ions reacted with Fe2+ and Fe3+ ions to form an intermediate compound of FeOOH. The amount of Fe2+ ion, Fe3+ ion and OH−(adjusted by pH) in the reaction solution exhibited a great effect on the final size of the Fe3O4 NPs produced. As long as the amount of OH− is sufficient, small Fe3O4 particle size is formed due to low concentration of Fe(OH)2 and FeOOH (Eq. (8)), which caused control of particle growth. However, at pH 11 where the amount of OH− was in excess and concentration of Fe(OH)2 and FeOOH was higher, there was a fast formation of Fe3O4 particles. This growth eventually resulted in a bigger Fe3O4 particle size. The particle size decreased at pH more than 11 (More OH− ions, available) because of the rapid formation of ferric hydroxide, as in Eq. (9 and 10) as follows (Suppiah and Abd Hamid 2016).

$${\text{Fe}}^{3 + } + 3{\text{OH}}^{ - } \to {\text{Fe}}\left( {{\text{OH}}} \right)_{3}$$
(9)
$${\text{Fe}}\left( {{\text{OH}}} \right)_{2} + 2{\text{Fe}}\left( {{\text{OH}}} \right)_{3} \to {\text{Fe}}_{3} {\text{O}}_{4} + 4{\text{H}}_{2} {\text{O}}$$
(10)
Fig. 3
figure 3

Effects of pH on Fe3O4 NPs size

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Response surface diagram and optimization

The statistical analysis of variance (ANOVA) is essential to investigate the significance of the predicted model. Table 3 shows all levels of Fe2+ ion molar ratio and pH are significant since the P value < 0.0001. The parameter is statistically significant when the P value < 0.05. The ‘lack of Fit is 0.46 which implies that the parameter itself is not significant. Contour diagrams of interaction between process parameters that show the effect on Fe3O4 nanoparticle size response are given in Fig. 4. These diagrams can be used to recognize the effect of these factors on Fe3O4 particle size. Figure 4(A1 and A2) shows the interaction between Fe2+ and Fe3+ ion molar ratios on Fe3O4 particle size. From this interaction, the contour diagrams suggest that 1.00 mol of Fe2+ ion molar ratio should be used with 3.00 mol of Fe3+ ion molar ratio to produce minimal Fe3O4 particle size. The effect of Fe2+ ion molar ratio and pH on Fe3O4 nanoparticle size response are shown in Fig. 4(B1 and B2). The interaction effect of Fe2+ ion molar ratio and pH also showed a quadratic relation to the response of Fe3O4 particle size. The size increases as the pH of solution is more than 10.0 (with constant of Fe2+ ion) and then decreases observing the smallest size at pH 12. This is attributed to excess OH− ions that promotes the production of Fe(OH)2 and FeOOH (Eq. (7)) and fasten the formation of Fe3O4 particles. At the same time, a similar effect was observed on the Fe3O4 particle size as the Fe2+ ion (with constant pH) ratio increased more than 1.0 followed by particle size reduction when the ratio of Fe2+ ion is more than 1.0. The interaction between Fe2+ ion molar ratio and pH have the strongest effects on Fe3O4 particle size. These factors also showed the most significant effects on Fe3O4 particle size based on the p value obtained from ANOVA analysis. Figure 4(C1 and C2) indicates the effect of Fe3+ ion molar ratio and pH of the medium on Fe3O4 nanoparticle size. These factors have less effect on particle size when they were working together than Fe2+ ion molar ratio and pH in Fig. 4(B1, B2). A similar pattern in particle size is observed when pH value ranged from 10 to 12 and Fe3+ ion molar ratio ranged from one to three. This phenomenon was also observed in other works which confirmed a weaker influence of Fe3+ ion molar ratio on the size of Fe3O4 (Chen et al. 2010; Rajput et al. 2016).

Fig. 4
figure 4

Interactions between (A1, A2) Fe2+ ion to Fe3+ ion molar ratio, (B1, B2) Fe2+ ion molar ratio and pH, (C1, C2) Fe3+ ion molar ratio and pH on Fe3O4 NPs size

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Pareto chart for the ANOVA table is given in Fig. 5 and important to show each independent variables with their effects as percentage on the response. The most effects factors from chart are Fe2+ ion molar ratio and pH, which are listed in their higher order of significance to control of Fe3O4 NPs size. These results showed in good agreement with analysis data from RSM diagrams and ANOVA table.

Fig. 5
figure 5

Pareto chart for standardized effects of factors on Fe3O4 NPs size

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Model regression verification

The empirical model of Fe3O4 particle size which indicates the validity and the reliability of the model as the value of R-squared (0.9752) and Adj R-squared (0.9736), are relatively high. Figure 6a shows the plot of the normality % probability versus studentized residuals, which confirms that errors are under normal distribution. As can be seen from Fig. 6a, the tabulation of the residuals are close to the straight line, which confirms that the distribution is normal. If the tabulation of residuals is far from the straight line, the error will be significant and this will affect the significance of the predicted model. Figure 6b shows the balanced scattering of points above and below the x-axis in which no unusual structure was observed. Figure 6cshows the plot of predicted versus actual value of Fe3O4 particle size. These results confirmed the higher adequacy of the model with the correlation coefficient, R2. At the same time, the empirical model showed that experimental results are in a good fit with model. ANOVA analyses suggest that the model derived by the Box–Behnken design is reliable in terms of predicting the Fe3O4 NPs size.

Fig. 6
figure 6

Diagnostic plots of Fe3O4 synthesis a normal % probability versus studentized residuals, b studentized residuals versus predicted, c predicted plot versus actual

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To achieve the optimum condition for Fe3O4 minimal size, four optimization criteria are required, (as minimum, maximum, within the range and target) and the values of these criteria were applied as shown in Table 4.

Table 4 Optimization of all factors to obtain minimum size for Fe3O4 NPs
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Accordingly, the model equation predicted that the minimal size of particles is 6.80 nm, which is produced by 1.00 molar ratio of Fe2+ ion and 3.00 molar ratio of Fe3+ ion with pH at 12.00, as given in Fig. 7. These results are in a good agreement with the interpretation of the contour diagram. To verify the validity of the result, experiments were carried out with three replicates to prepare Fe3O4 nanoparticles in optimal conditions. The average result obtained from the XRD was 10 ± 3 nm and that from TEM was 10 ± 2 nm. This confirmed the effectiveness and reliability of the predicted values from the prediction model.

Fig. 7
figure 7

The optimum conditions predicted by the Box–Behnken design for minimal size of Fe3O4 NPs

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Characterization of Fe3O4 nanoparticles

The average particle size of the Fe3O4 NPs for three replicates was evaluated by XRD analysis. The size of Fe3O4 determined by XRD analysis was found to be 10 ± 3 nm. Figure 8 shows the XRD diagram for the Fe3O4 NPs using optimal condition as 1.00 molar ratio of Fe2+ ion and 3.00 molar ratio of Fe3+ ion with pH at 12.00. All peak positions of Fe3O4 matched at 2θ of 30.21°, 35.72°, 43.34°, 53.35°, 57.37° and 63.07o were found. The position of peaks shows exactly for data of diffraction face-centered cubic (FCC) crystalline as reported in previous studies (Petcharoen and Sirivat 2012; Valenzuela et al. 2009).

Fig. 8
figure 8

X-ray patterns of Fe3O4 NPs prepared under optimum conditions

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The TEM analysis was carried out to support XRD analysis results. The TEM image of Fe3O4 NPs which was prepared under optimal conditions showed almost spherical shapes with an average of 10 ± 2 nm, as given in Fig. 9. These results are in a good agreement with the result from the XRD analysis. The agglomeration of Fe3O4 NPs was mainly contributed by the drying procedure during preparation of TEM samples.

Fig. 9
figure 9

Transmission electron microscope (TEM) of Fe3O4 NPs prepared under optimum conditions

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Figure 10 shows the M–H curve measurements of the Fe3O4 NPs prepared under optimum conditions, which saturation magnetization value was found to be 49.729 emu/g. The magnetic properties of Fe3O4 NPs displayed typical ferromagnetic behavior and were close to those reported elsewhere (Beyaz et al. 2009; Rahdar and Arabi 2012).

Fig. 10
figure 10

Room-temperature magnetic hysteresis loops for Fe3O4 NPs prepared under optimum conditions

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The adsorption test on Congo red (CR) removal was done to study the applicability of the synthesized of Fe3O4 NPs. Tests were carried out using the optimal synthesis condition of Fe3O4 NPs only. Two sets of experiments were conducted to determine effect of the certain amount of Fe3O4 NPs from 0.05 to 0.1 g on CR removal efficiency. However, higher removal of CR was achieved as 82.34% using 0.1 g compare 58.79% by 0.05 g under the same experimental conditions. Furthermore, as shown in Fig. 11, Fe3O4 NPs can be easily separated by a magnetic field from the CR aqueous solution. In general, the co-precipitation method in this study managed to produce smaller Fe3O4 NPs because of its high flexibility compared with the other synthesis methods, producing nanoparticles with various properties. In terms of ease of operation, this method was simple and produced consistently small Fe3O4 NPs in the range of 10 nm. This method does not require the usage of any capping agent or surfactant (or any other additives) and thus reduces the cost of chemicals. Furthermore, this method does not require the use of high temperature and pressure and specialized equipment. This study showed an improved BBD optimization to produce of smaller Fe3O4 NPs sizes and control effects of the process parameters and their interaction during preparation stage. The bigger Fe3O4 NPs obtained with the co-precipitation method can be endorsed to their lack conditions out of predicted optimal conditions for selected variables within the preparation process (Hou et al. 2014).

Fig. 11
figure 11

Adsorption of Congo red using different amount of Fe3O4 NPs a 0.0 g b 0.05 g c 0.10 g

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Conclusions

Fe3O4 nanoparticles were successfully optimized via a Box–Behnken design. The effects of process parameters (factors) and their interactions on the Fe3O4 NPs size response were investigated by ANOVA analysis. The design showed that Fe2+ ion molar ratio and pH had the highest effect on the response (particle size). The empirical model shows good validity and reliability as the R-squared (0.9752) and Adj R-squared (0.9736) are relatively high. The optimal conditions determined by the empirical model for minimal Fe3O4 NPs size were observed at 1.00 mol Fe2+ ion with 3.00 mol Fe3+ ion with pH at 12.00. The predicted particle size of Fe3O4 was 6.80 nm and the experimental values were found to be 10 ± 3 nm and 10 ± 2 nm, as evaluated by XRD and TEM analysis, respectively. Fe3O4 particle size from experimental and theoretical values shows that the predicted empirical model is reliable, and could be utilized as a suitable optimization model. Finally, the optimized Fe3O4 NPs showed about 80% removal of Congo red (CR) dye, which confirms their applicability in adsorption process for future applications.