**2. Mechanochemical synthesis of Fe3O4 nanoparticles via coprecipitation**

In this method, a tumbling ball mill is used as a reactor, and a suspension of Fe(OH)2 and α-FeOOH as a precursor, which is prepared via coprecipitation, is ball-milled in an organic solvent-free water system. The ball-milling treatment is performed under a cooling condition. Thus, high mechanical energy generated by collision of ball media is applied to the precursor instead of the heat energy, which promotes the solid phase reaction between Fe(OH)2 and α-FeOOH forming Fe3O4 and the crystallization process without the crystal growth caused by the heat energy. This corresponds to the mechanochemical effect. Accordingly, this method does not need any additional heating treatment to improve the crystallinity of the product. In addition, any additives such as surfactants and oxidizing and reducing agents are not required. Consequently, this method provides successfully superparamagnetic Fe3O4 nanocrystals with a size of less than 15 nm. The details of this method are described below.

Typically, 1.5 mmol of ferrous sulfate heptahydrate (FeSO4· 7H2O) and 3.0 mmol of ferric chloride hexahydrate (FeCl3· 6H2O) are dissolved in 60 ml of deionized and deoxygenated water in a beaker. The molar ratio of ferrous ion to ferric ion is 0.5, corresponding to the chemical stoichiometric ratio of the Fe3O4 formation reaction. 30 ml of 1.0 kmol/m3 sodium hydroxide (NaOH) solution is added into the acid solution at a constant addition rate of 3 ml/min under vigorous stirring using a magnetic stirrer in an argon atmosphere. As the pH of the solution increased, Fe(OH)2 and α-FeOOH coprecipitate, according to Eqs. (1) to (3).

$$\text{Fe}^{2+} + 2\text{OH} \text{-} \rightarrow \text{Fe(OH)}\_{2} \tag{1}$$

Fe3+ + 3OH– → Fe(OH)3 (2)

$$\text{Fe(OH)}\_{3} \rightarrow \text{a-FeCOH} + \text{H}\_{2}\text{O} \tag{3}$$

When adding the NaOH solution, the solution temperature is kept below 5°C by ice-cooling in order to avoid the solid phase reaction forming Fe3O4 from Fe(OH)2 and α-FeOOH according to Eq. (4) (Lian et al., 2004).

$$\text{Fe(OH)}\_{2} + 2\text{a-FeCOH} \rightarrow \text{Fe}\_{3}\text{O}\_{4} + 2\text{H}\_{2}\text{O} \tag{4}$$

However, even under cooling, Fe(OH)2 and α-FeOOH partially take place the Fe3O4 formation reaction. This results in a dark brown suspension with a pH of higher than 12 containing Fe(OH)2, α-FeOOH, and a tiny amount of Fe3O4. The suspension thus prepared is subjected to the following ball-milling treatment.

The starting suspension is poured into a milling pot with an inner diameter of 90 mm and a capacity of 500 ml, made of stainless steel (18%Cr-8%Ni). Stainless steel balls with a diameter of 3.2 mm are used as the milling media. The charged volume of balls containing the void formed among them is 40% of the pot capacity, as illustrated in Fig. 1.

high crystallinity without using any environmental-unfriendly additives (Iwasaki et al., 2008, 2009, 2010, 2011b). This chapter describes the outline of this method and the kinetic

**2. Mechanochemical synthesis of Fe3O4 nanoparticles via coprecipitation** 

In this method, a tumbling ball mill is used as a reactor, and a suspension of Fe(OH)2 and α-FeOOH as a precursor, which is prepared via coprecipitation, is ball-milled in an organic solvent-free water system. The ball-milling treatment is performed under a cooling condition. Thus, high mechanical energy generated by collision of ball media is applied to the precursor instead of the heat energy, which promotes the solid phase reaction between Fe(OH)2 and α-FeOOH forming Fe3O4 and the crystallization process without the crystal growth caused by the heat energy. This corresponds to the mechanochemical effect. Accordingly, this method does not need any additional heating treatment to improve the crystallinity of the product. In addition, any additives such as surfactants and oxidizing and reducing agents are not required. Consequently, this method provides successfully superparamagnetic Fe3O4 nanocrystals with a size of less than 15 nm. The details of this

Typically, 1.5 mmol of ferrous sulfate heptahydrate (FeSO4· 7H2O) and 3.0 mmol of ferric chloride hexahydrate (FeCl3· 6H2O) are dissolved in 60 ml of deionized and deoxygenated water in a beaker. The molar ratio of ferrous ion to ferric ion is 0.5, corresponding to the chemical stoichiometric ratio of the Fe3O4 formation reaction. 30 ml of 1.0 kmol/m3 sodium hydroxide (NaOH) solution is added into the acid solution at a constant addition rate of 3 ml/min under vigorous stirring using a magnetic stirrer in an argon atmosphere. As the pH of the solution increased, Fe(OH)2 and α-FeOOH coprecipitate, according to

Fe2+ + 2OH– → Fe(OH)2 (1)

Fe3+ + 3OH– → Fe(OH)3 (2)

 Fe(OH)3 → α-FeOOH + H2O (3) When adding the NaOH solution, the solution temperature is kept below 5°C by ice-cooling in order to avoid the solid phase reaction forming Fe3O4 from Fe(OH)2 and α-FeOOH

 Fe(OH)2 + 2α-FeOOH → Fe3O4 + 2H2O (4) However, even under cooling, Fe(OH)2 and α-FeOOH partially take place the Fe3O4 formation reaction. This results in a dark brown suspension with a pH of higher than 12 containing Fe(OH)2, α-FeOOH, and a tiny amount of Fe3O4. The suspension thus prepared is

The starting suspension is poured into a milling pot with an inner diameter of 90 mm and a capacity of 500 ml, made of stainless steel (18%Cr-8%Ni). Stainless steel balls with a diameter of 3.2 mm are used as the milling media. The charged volume of balls containing

the void formed among them is 40% of the pot capacity, as illustrated in Fig. 1.

analysis of the mechanochemical process.

method are described below.

according to Eq. (4) (Lian et al., 2004).

subjected to the following ball-milling treatment.

Eqs. (1) to (3).

Fig. 1. Schematic illustration of tumbling ball mill used in this work.

After replacement of air in the milling pot with argon, the milling pot is sealed. In order to promote the reaction between Fe(OH)2 and α-FeOOH, the ball-milling treatment is then carried out by rotating the milling pot at rotational speeds of 35 to 140 rpm (corresponding to 16% to 64% of the critical rotational speed (= 220 rpm) determined experimentally based on the behavior of balls containing the suspension) for a given time. During the ball-milling treatment, the milling pot is cooled from its outside in a water bath. Temperature of the water bath is kept at 1.0±0.1°C, and temperature of the suspension is between 1.6°C and 1.7°C within the rotational speed range; this means that the milling pot is cooled enough. After the ball-milling treatment, the obtained precipitate is washed and then dried at 30°C under vacuum overnight.

The dried samples thus obtained were characterized according to standard methods. The powder X-ray diffraction (XRD) pattern of samples was measured with CuKα radiation ranging from 2θ = 10 to 80° at a scanning rate of 1.0°/min using a Rigaku RINT-1500 powder X-ray diffractometer. Fig. 2 shows the XRD pattern of samples obtained at various rotational speeds of the milling pot. In all the XRD patterns, clear diffractions indicating Fe(OH)2 phase were not observed because it tends to form amorphous phase. Before the ball-milling treatment, the sample contained amorphous α-FeOOH and Fe3O4 phases. As the milling time elapsed, α-FeOOH gradually disappeared and finally the single-phase of Fe3O4 formed. This reaction can be attributed to the application of the mechanical energy generated by collision of the balls to α-FeOOH and Fe(OH)2. The reaction rate depended strongly on the rotational speed. The time required for completing the Fe3O4 formation reaction was reduced with increasing in the rotational speed; the reaction almost completed in 12 h at 35 rpm, in 9 h at 70 rpm, in 7.5 h at 105 rpm, and in 6 h at 140 rpm. On the other hand, when the ball-milling treatment was not conducted, i.e., the suspension was kept cooling statically at below 2°C in the water bath, the XRD pattern was almost the same as that of the staring precipitate even after 12 h. It was confirmed that the ball-milling treatment promoted the solid phase reaction by the mechanochemical effect. At early stages of the ball-milling treatment, α-FeOOH seemed to increase from the initial. Actually,

Novel Mechanochemical Process

crystallinity of samples was high.

for Aqueous-Phase Synthesis of Superparamagnetic Magnetite Nanoparticles 239

median diameter (number basis) was determined from the obtained size distribution. Fig. 4 indicates the particle size distribution of samples. The samples had a narrow size distribution with a median diameter of about 11 nm. The median size almost agreed with the average crystallite size and the particle size observed by SEM; this means that the

 Fig. 3. SEM images of samples obtained (a) at 35 rpm in 12 h and (b) at 140 rpm in 6 h.

 Fig. 4. Particle size distribution of samples obtained (a) at 35 rpm in 12 h, (b) at 70 rpm in 9

h, (c) at 105 rpm in 7.5 h, and (d) at 140 rpm in 6 h.

suspensions at early stages had light brown colors as compared to the staring suspension. This implies that Fe(OH)2 is oxidized and Fe3O4 is decomposed (hydrolyzed) because Fe(OH)2 and Fe3O4 in the starting solution are relatively unstable. As the milling time elapsed, the color of the suspension became darker brown, and finally black precipitates were obtained. This result also implies the reduction of α-FeOOH. Fe3O4 formed by the mechanical energy is hardly decomposed during ball-milling because it is well crystallized and becomes stable.

The samples had the typical diffraction angles showing relatively high peaks, agreed well with those of Fe3O4 phase. The average crystallite size was calculated from the full-width at half-maximum (FWHM) of the Fe3O4 (311) diffraction peak at 2θ ≈ 35.5° using the Scherrer's formula. The lattice constant was also determined from several diffraction angles showing high intensity peaks. The lattice constant was determined to be between 8.374 and 8.395 Å and close to the standard value of Fe3O4 (= 8.396 Å) rather than γ-Fe2O3 (= 8.345 Å). These results support that the samples had a single Fe3O4 phase rather than γ-Fe2O3 regardless of the milling conditions.

Fig. 2. XRD pattern of samples obtained at (a) 35, (b) 70, (c) 105, and (d) 140 rpm.

The morphology of samples was observed with a field emission scanning electron microscope (FE-SEM; JSM-6700F, JEOL). Fig. 3 shows the SEM image of samples. The samples were spherical nanoparticles with a size of about 10–20 nm.

The hydrodynamic particle size distribution of samples was measured by dynamic light scattering (DLS-700, Otsuka Electronics) for the sample-redispersed aqueous suspension containing a small amount of sodium dodecyl sulphate as a dispersion stabilizer. The

suspensions at early stages had light brown colors as compared to the staring suspension. This implies that Fe(OH)2 is oxidized and Fe3O4 is decomposed (hydrolyzed) because Fe(OH)2 and Fe3O4 in the starting solution are relatively unstable. As the milling time elapsed, the color of the suspension became darker brown, and finally black precipitates were obtained. This result also implies the reduction of α-FeOOH. Fe3O4 formed by the mechanical energy is hardly decomposed during ball-milling because it is well crystallized

The samples had the typical diffraction angles showing relatively high peaks, agreed well with those of Fe3O4 phase. The average crystallite size was calculated from the full-width at half-maximum (FWHM) of the Fe3O4 (311) diffraction peak at 2θ ≈ 35.5° using the Scherrer's formula. The lattice constant was also determined from several diffraction angles showing high intensity peaks. The lattice constant was determined to be between 8.374 and 8.395 Å and close to the standard value of Fe3O4 (= 8.396 Å) rather than γ-Fe2O3 (= 8.345 Å). These results support that the samples had a single Fe3O4 phase rather than γ-Fe2O3 regardless of

Fig. 2. XRD pattern of samples obtained at (a) 35, (b) 70, (c) 105, and (d) 140 rpm.

samples were spherical nanoparticles with a size of about 10–20 nm.

The morphology of samples was observed with a field emission scanning electron microscope (FE-SEM; JSM-6700F, JEOL). Fig. 3 shows the SEM image of samples. The

The hydrodynamic particle size distribution of samples was measured by dynamic light scattering (DLS-700, Otsuka Electronics) for the sample-redispersed aqueous suspension containing a small amount of sodium dodecyl sulphate as a dispersion stabilizer. The

and becomes stable.

the milling conditions.

median diameter (number basis) was determined from the obtained size distribution. Fig. 4 indicates the particle size distribution of samples. The samples had a narrow size distribution with a median diameter of about 11 nm. The median size almost agreed with the average crystallite size and the particle size observed by SEM; this means that the crystallinity of samples was high.

Fig. 3. SEM images of samples obtained (a) at 35 rpm in 12 h and (b) at 140 rpm in 6 h.

Fig. 4. Particle size distribution of samples obtained (a) at 35 rpm in 12 h, (b) at 70 rpm in 9 h, (c) at 105 rpm in 7.5 h, and (d) at 140 rpm in 6 h.

Novel Mechanochemical Process

even at higher rotational speeds.

Average crystallite size [nm]

Milling conditions Rotational speed [rpm] Milling time [h]

Table 2. Impurity content in samples.

**3. Analysis of mechanochemical process** 

**3.1 Reaction rate equation in mechanochemical process** 

Saturation magnetization [emu/g]

Table 1. Properties of samples obtained under various conditions.

35 12

0.39 0.10 0.53 0.08 0.15

Generally, preparation of nanoparticles with a size of less than 20 nm is very difficult by means of grinding techniques using ball mills and bead mills (i.e., break-down methods) even under wet conditions. In particular, tumbling mills are impossible to provide such nanoparticles by grinding because the mechanical energy generated in tumbling mills is too low to reach a particle size less than several micrometers. Accordingly, the mechanical energy is used not for the grinding of coarse (grown) particles but for the Fe3O4 formation reaction; the Fe3O4 nanoparticles are prepared by means of a build-up method. Consequently, the obtained results shown above demonstrate that this synthesis method is effective for the production of superparamagnetic Fe3O4 nanoparticles with good

For kinetically analyzing this mechanochemical process, the reaction rate equation must be derived. Therefore, the change in the concentration of the starting materials with the reaction time is required. In this investigation, the concentration of the starting materials was approximately estimated from the XRD data. As mentioned earlier, in this mechanochemical process, the formation and conversion of α-FeOOH plays an important

Milling conditions Rotational speed [rpm] Milling time [h]

Lattice constant [Å] Median size [nm]

Coercivity [Oe] Zeta potential [mV]

> Cr [wt.%] Ni [wt.%] Na [wt.%] S [wt.%] Cl [wt.%]

dispersibility.

for Aqueous-Phase Synthesis of Superparamagnetic Magnetite Nanoparticles 241

Table 2 gives the impurity content in the samples. The sample contamination had a tendency to increase at higher rotational speeds. However, the use of fluorocarbon resinlined pot and carbon steel balls can decrease the incorporation of Cr and Ni into the product

> 35 12

11.2 8.395 9.5 68.7 9 –16.3 70 9

11.3 8.385 10.9 68.6 7 –15.6

70 9

0.62 0.37 0.19 0.03 0.02 105 7.5

10.8 8.392 10.3 74.6 2 –20.6

105 7.5

1.26 0.61 0.25 0.03 0.02 140 6

11.9 8.374 8.8 75.2 1 –18.1

140 6

1.21 0.69 0.32 0.03 0.04

The magnetic property (magnetization-magnetic field hysteretic cycle) was analyzed using a superconducting quantum interference device (SQUID) magnetometer (Quantum Design model MPMS) at room temperature in the rage of magnetic field between –10 kOe and 10 kOe. Fig. 5 shows the magnetization-magnetic field curve of samples. The samples had a low coercivity, showing superparamagnetism. The saturation magnetization was a little lower than that of the corresponding bulk (= 92 emu/g) because of the smaller size (Lee et al., 1996).

Fig. 5. Magnetization-magnetic field curve of samples obtained (a) at 35 rpm in 12 h, (b) at 70 rpm in 9 h, (c) at 105 rpm in 7.5 h, and (d) at 140 rpm in 6 h.

Table 1 summarizes the properties. The zeta potential was measured with a zeta potential analyzer (Zetasizer Nano ZS, Malvern Instruments). As can be seen in Table 1, the Fe3O4 nanoparticles with similar properties were obtained regardless of the rotational speed. The zeta potential of samples was relatively high with adding neither anti-aggregation agents nor organic solvents in the synthesis and almost the same even when the mechanical energy applied to the suspension per unit time was varied. This reveals that the applied mechanical energy hardly affected the dispersibility of Fe3O4 nanoparticles.

In this method, the sample contamination caused by the wear of milling pot and balls is concerned. Therefore, the chemical component of samples was determined by means of an energy dispersive X-ray spectrometer (EDS; JED-2300F, JEOL) equipped with the FE-SEM.

Table 2 gives the impurity content in the samples. The sample contamination had a tendency to increase at higher rotational speeds. However, the use of fluorocarbon resinlined pot and carbon steel balls can decrease the incorporation of Cr and Ni into the product even at higher rotational speeds.


Table 1. Properties of samples obtained under various conditions.


Table 2. Impurity content in samples.

240 Materials Science and Technology

The magnetic property (magnetization-magnetic field hysteretic cycle) was analyzed using a superconducting quantum interference device (SQUID) magnetometer (Quantum Design model MPMS) at room temperature in the rage of magnetic field between –10 kOe and 10 kOe. Fig. 5 shows the magnetization-magnetic field curve of samples. The samples had a low coercivity, showing superparamagnetism. The saturation magnetization was a little lower than that of the corresponding bulk (= 92 emu/g) because of the smaller size (Lee et

 Fig. 5. Magnetization-magnetic field curve of samples obtained (a) at 35 rpm in 12 h, (b) at

Table 1 summarizes the properties. The zeta potential was measured with a zeta potential analyzer (Zetasizer Nano ZS, Malvern Instruments). As can be seen in Table 1, the Fe3O4 nanoparticles with similar properties were obtained regardless of the rotational speed. The zeta potential of samples was relatively high with adding neither anti-aggregation agents nor organic solvents in the synthesis and almost the same even when the mechanical energy applied to the suspension per unit time was varied. This reveals that the applied mechanical

In this method, the sample contamination caused by the wear of milling pot and balls is concerned. Therefore, the chemical component of samples was determined by means of an energy dispersive X-ray spectrometer (EDS; JED-2300F, JEOL) equipped with the FE-SEM.

70 rpm in 9 h, (c) at 105 rpm in 7.5 h, and (d) at 140 rpm in 6 h.

energy hardly affected the dispersibility of Fe3O4 nanoparticles.

al., 1996).

Generally, preparation of nanoparticles with a size of less than 20 nm is very difficult by means of grinding techniques using ball mills and bead mills (i.e., break-down methods) even under wet conditions. In particular, tumbling mills are impossible to provide such nanoparticles by grinding because the mechanical energy generated in tumbling mills is too low to reach a particle size less than several micrometers. Accordingly, the mechanical energy is used not for the grinding of coarse (grown) particles but for the Fe3O4 formation reaction; the Fe3O4 nanoparticles are prepared by means of a build-up method. Consequently, the obtained results shown above demonstrate that this synthesis method is effective for the production of superparamagnetic Fe3O4 nanoparticles with good dispersibility.

### **3. Analysis of mechanochemical process**

#### **3.1 Reaction rate equation in mechanochemical process**

For kinetically analyzing this mechanochemical process, the reaction rate equation must be derived. Therefore, the change in the concentration of the starting materials with the reaction time is required. In this investigation, the concentration of the starting materials was approximately estimated from the XRD data. As mentioned earlier, in this mechanochemical process, the formation and conversion of α-FeOOH plays an important

Novel Mechanochemical Process

of single α-FeOOH phase.

140 rpm.

be expressed by

reaction time t was newly defined as

for Aqueous-Phase Synthesis of Superparamagnetic Magnetite Nanoparticles 243

mechanochemical effect according to Eqs. (6) and (7), respectively, resulting to the formation

Fe(OH)2 → α-FeOOH + ½H2 (6)

 Fe3O4 + 2H2O → 2α-FeOOH + Fe(OH)2 (7) After that, α-FeOOH is partially reduced to Fe(OH)2 by the generated H2 gas, and the formed Fe(OH)2 reacts immediately with α-FeOOH, resulting to the formation of Fe3O4.

Fig. 7. Change in concentration Cg of α-FeOOH with milling time tm at rotational speed of

 2α-FeOOH + Fe(OH)2 → Fe3O4 + 2H2O (4) From these reaction equations, the overall apparent equation for the formation of Fe3O4 can

 3α-FeOOH + ½H2 → Fe3O4 + 2H2O (9) Consequently, in this analysis, it can be assumed that the single α-FeOOH phase is gradually converted to Fe3O4 after the milling time of 1800 s according to Eq.(9). Here, the

Fig. 8 shows the change in the concentration ratio Cg/C0 with the reaction time t, where C0 indicates the initial concentration of α-FeOOH at t = 0. Cg/C0 decreased exponentially with t, suggesting that the conversion reaction from α-FeOOH to Fe3O4 may be described by the

dCg/dt = –kCgn (11)

n-th order rate equation. In this case, the reaction rate equation is expressed by

α-FeOOH + ½H2 → Fe(OH)2 (8)

t = tm – 1800 (10)

role rather than Fe(OH)2. Accordingly, the temporal change in the content of α-FeOOH in the product was focused on. In order to estimate the content of α-FeOOH from the XRD data, the relationship between the content of α-FeOOH and the diffraction intensity is needed as an analytical curve. Therefore, virtual products with various compositions of α-FeOOH and Fe3O4, expressed by the molar ratio γ of α-FeOOH to the α-FeOOH–Fe3O4 mixture, were artificially prepared by mixing appropriate amounts of α-FeOOH and Fe3O4. The XRD analysis for the virtual products was conducted, and the diffraction intensity I at 2θ = 21.2° corresponding to the (011) plane of α-FeOOH was measured. The intensity ratio ξ was calculated, defined by

$$
\xi = (\mathbf{I} - \mathbf{I}\_m) / (\mathbf{I}\_\% - \mathbf{I}\_m) \tag{5}
$$

where Ig and Im indicate the peak intensity at 2θ = 21.2° for α-FeOOH and Fe3O4, respectively. Fig. 6 shows the intensity ratio ξ as a function of the molar ratio γ. Using this analytical curve, the concentration Cg of α-FeOOH in the suspension was estimated.

Fig. 6. Relationship between intensity ratio ξ and molar ratio γ of α-FeOOH.

Fig. 7 shows the change in Cg with the milling time tm at the rotational speed of 140 rpm as an example. Cg increased immediately after the ball-milling treatment began and then decreased with the milling time. The similar tendency was observed at other rotational speeds. The highest Cg observed in 1800 s almost agreed with the concentration of total iron in the starting solution (≈ 4.5 mmol/90 ml = 50 mol/m3). This implies that the total iron in the starting solution may be converted to α-FeOOH at an initial stage of the ball-milling treatment, and after that, Fe3O4 may form from α-FeOOH by the mechanochemical effect. Thus, the formation path of Fe3O4 in this mechanochemical process can differ from that in conventional coprecipitation processes.

From these results shown above, a possible reaction mechanism of Fe3O4 formation can be constructed as follows. At an initial stage of the ball-milling treatment, amorphous phases of Fe(OH)2 and Fe3O4 in the staring suspension are rapidly oxidized and hydrolyzed by the

role rather than Fe(OH)2. Accordingly, the temporal change in the content of α-FeOOH in the product was focused on. In order to estimate the content of α-FeOOH from the XRD data, the relationship between the content of α-FeOOH and the diffraction intensity is needed as an analytical curve. Therefore, virtual products with various compositions of α-FeOOH and Fe3O4, expressed by the molar ratio γ of α-FeOOH to the α-FeOOH–Fe3O4 mixture, were artificially prepared by mixing appropriate amounts of α-FeOOH and Fe3O4. The XRD analysis for the virtual products was conducted, and the diffraction intensity I at 2θ = 21.2° corresponding to the (011) plane of α-FeOOH was measured. The intensity ratio ξ

where Ig and Im indicate the peak intensity at 2θ = 21.2° for α-FeOOH and Fe3O4, respectively. Fig. 6 shows the intensity ratio ξ as a function of the molar ratio γ. Using this analytical curve, the concentration Cg of α-FeOOH in the suspension was estimated.

Fig. 6. Relationship between intensity ratio ξ and molar ratio γ of α-FeOOH.

conventional coprecipitation processes.

Fig. 7 shows the change in Cg with the milling time tm at the rotational speed of 140 rpm as an example. Cg increased immediately after the ball-milling treatment began and then decreased with the milling time. The similar tendency was observed at other rotational speeds. The highest Cg observed in 1800 s almost agreed with the concentration of total iron in the starting solution (≈ 4.5 mmol/90 ml = 50 mol/m3). This implies that the total iron in the starting solution may be converted to α-FeOOH at an initial stage of the ball-milling treatment, and after that, Fe3O4 may form from α-FeOOH by the mechanochemical effect. Thus, the formation path of Fe3O4 in this mechanochemical process can differ from that in

From these results shown above, a possible reaction mechanism of Fe3O4 formation can be constructed as follows. At an initial stage of the ball-milling treatment, amorphous phases of Fe(OH)2 and Fe3O4 in the staring suspension are rapidly oxidized and hydrolyzed by the

ξ = (I – Im)/(Ig – Im) (5)

was calculated, defined by

mechanochemical effect according to Eqs. (6) and (7), respectively, resulting to the formation of single α-FeOOH phase.

$$\text{Fe(OH)}\_{2} \rightarrow \text{a-FeCOH} + \text{l/} \text{H}\_{2} \tag{6}$$

$$\text{Fe}\_3\text{O}\_4 + 2\text{H}\_2\text{O} \rightarrow 2\text{a-FeCOOH} + \text{Fe(OH)}\_2\tag{7}$$

After that, α-FeOOH is partially reduced to Fe(OH)2 by the generated H2 gas, and the formed Fe(OH)2 reacts immediately with α-FeOOH, resulting to the formation of Fe3O4.

Fig. 7. Change in concentration Cg of α-FeOOH with milling time tm at rotational speed of 140 rpm.

$$\text{g-FeOOH} + \text{\color{red}{\cdot $}} \text{\color{red}{\cdot$ }} \text{H}\_2 \rightarrow \text{Fe(OH)}\_2 \tag{8}$$

$$2\text{a-FeCOH} + \text{Fe(OH)}\_2 \rightarrow \text{Fe}\_3\text{O}\_4 + 2\text{H}\_2\text{O} \tag{4}$$

From these reaction equations, the overall apparent equation for the formation of Fe3O4 can be expressed by

$$\text{Эг-FeCOOH} + \text{үн}\_2 \rightarrow \text{Fe}\_2\text{O}\_4 + 2\text{H}\_2\text{O} \tag{9}$$

Consequently, in this analysis, it can be assumed that the single α-FeOOH phase is gradually converted to Fe3O4 after the milling time of 1800 s according to Eq.(9). Here, the reaction time t was newly defined as

$$\mathbf{t} = \mathbf{t}\_{\mathrm{m}} - 1800 \,\tag{10}$$

Fig. 8 shows the change in the concentration ratio Cg/C0 with the reaction time t, where C0 indicates the initial concentration of α-FeOOH at t = 0. Cg/C0 decreased exponentially with t, suggesting that the conversion reaction from α-FeOOH to Fe3O4 may be described by the n-th order rate equation. In this case, the reaction rate equation is expressed by

$$\text{dC}\_{\mathcal{R}} / \text{dt} = -\text{kC}\_{\mathcal{R}} n \tag{11}$$

Novel Mechanochemical Process

initial concentration on the reaction rate small.

(b) C0 = 12.5 mol/m3.

for Aqueous-Phase Synthesis of Superparamagnetic Magnetite Nanoparticles 245

 Fig. 9. Changes in (a) order of reaction n and (b) rate constant k with rotational speed.

Fig. 10 shows the effect of the initial iron concentration C0 on the reaction rate. There was no noticeable difference among the results. Fig. 11 shows the change in n and k with C0. n and k were almost constant against the variation in C0. In this mechanochemical process, the order of reaction was relatively small. This means that the concentration dependence of the reaction rate is low in this system where the reaction proceeds using the mechanical energy, unlike those using the heat energy. Generally, in liquid-phase reaction processes using heating treatments, the reaction solution is heated overall and the reaction proceeds everywhere in the solution. Thus, higher concentration of staring materials tends to lead to faster reaction rate. On the other hand, in this mechanochemical process, the mechanical energy promoting the reaction is applied to the suspension on the impact points of the balls, which distribute discretely in the milling pot. Thus, even when the concentration is high, the suspension which can receive the mechanical energy is limited. This makes the effect of the

Fig. 10. Effect of initial concentration C0 on reaction rate at 140 rpm: (a) C0 = 25 mol/m3 and

where k and n are the rate constant and the order of reaction, respectively. By solving this differential equation using the boundary condition, Cg = C0 at t = 0, the concentration ratio Cg/C0 is expressed by

$$\mathbf{C\_{p}}/\mathbf{C\_{0}} = \{1 + (\mathbf{n-1})\mathbf{k}\mathbf{C\_{0}}^{n-1}\mathbf{t}\}^{1/(1-n)}\tag{12}$$

As shown in Fig. 8, the data of the concentration of α-FeOOH against the reaction time were fitted to Eq. (12), and the values of n and k were determined. Fig. 9 shows n and k as a function of the rotational speed, respectively. n was kept almost constant, about 0.6, regardless of the rotational speed. Accordingly, the formation reaction of Fe3O4 expressed by Eq. (9) may be described by the 0.6th-order rate equation. On the other hand, k increased with increasing in the rotational speed. At higher rotational speeds, higher mechanical energy is generated per unit time in the ball-milling treatment. Thus, this phenomenon relating to the increase in k in this mechanochemical process is analogous to an increase in the reaction rate at higher temperatures, at which greater amounts of heat energy are given to the system.

Fig. 8. Change in concentration ratio Cg/C0 with reaction time t at rotational speeds of (a) 35 rpm, (b) 70 rpm, (c) 105 rpm, and (d) 140 rpm.

where k and n are the rate constant and the order of reaction, respectively. By solving this differential equation using the boundary condition, Cg = C0 at t = 0, the concentration ratio

 Cg/C0 = {1+(n–1)kC0n–1t}1/(1–n) (12) As shown in Fig. 8, the data of the concentration of α-FeOOH against the reaction time were fitted to Eq. (12), and the values of n and k were determined. Fig. 9 shows n and k as a function of the rotational speed, respectively. n was kept almost constant, about 0.6, regardless of the rotational speed. Accordingly, the formation reaction of Fe3O4 expressed by Eq. (9) may be described by the 0.6th-order rate equation. On the other hand, k increased with increasing in the rotational speed. At higher rotational speeds, higher mechanical energy is generated per unit time in the ball-milling treatment. Thus, this phenomenon relating to the increase in k in this mechanochemical process is analogous to an increase in the reaction rate at higher temperatures, at which greater amounts of heat energy are given

Fig. 8. Change in concentration ratio Cg/C0 with reaction time t at rotational speeds of (a) 35

rpm, (b) 70 rpm, (c) 105 rpm, and (d) 140 rpm.

Cg/C0 is expressed by

to the system.

Fig. 9. Changes in (a) order of reaction n and (b) rate constant k with rotational speed.

Fig. 10 shows the effect of the initial iron concentration C0 on the reaction rate. There was no noticeable difference among the results. Fig. 11 shows the change in n and k with C0. n and k were almost constant against the variation in C0. In this mechanochemical process, the order of reaction was relatively small. This means that the concentration dependence of the reaction rate is low in this system where the reaction proceeds using the mechanical energy, unlike those using the heat energy. Generally, in liquid-phase reaction processes using heating treatments, the reaction solution is heated overall and the reaction proceeds everywhere in the solution. Thus, higher concentration of staring materials tends to lead to faster reaction rate. On the other hand, in this mechanochemical process, the mechanical energy promoting the reaction is applied to the suspension on the impact points of the balls, which distribute discretely in the milling pot. Thus, even when the concentration is high, the suspension which can receive the mechanical energy is limited. This makes the effect of the initial concentration on the reaction rate small.

Fig. 10. Effect of initial concentration C0 on reaction rate at 140 rpm: (a) C0 = 25 mol/m3 and (b) C0 = 12.5 mol/m3.

Novel Mechanochemical Process

Stiffness for the ball-to-ball collision:

Stiffness for the ball-to-pot wall collision:

Damping coefficient:

condition (Gudin et al., 2007).

contacting balls.

for Aqueous-Phase Synthesis of Superparamagnetic Magnetite Nanoparticles 247

where **δ**, **v**, κ, η, μ, r, and **n** are overlap displacement between contacting balls, relative velocity of contacting balls, stiffness, damping coefficient, sliding friction coefficient, radius of a ball, and unit vector of normal direction at a contact point, respectively. The subscripts n and t mean the components of normal and tangential directions at a contact point, respectively. The subscripts i and j indicate the number of contacting balls. κ and η are

where Y and σ are Young's modulus and Poisson's ratio, respectively. The subscripts b and w indicate ball and pot wall, respectively. α is the constant depending on the restitution coefficient and was determined to be 0.20 based on the experimental value of restitution coefficient (= 0.75) according to the method proposed by Tsuji et al. (1992). The simulation parameters are summarized in Table 3. The sliding friction coefficient used in the calculation was determined based on the critical rotational speed measured experimentally under wet

The impact energy of each ball in a single collision, Ei, was defined as the kinetic energy of a

 Ei = (1/2)M|**v**|2 (24) Here, **v** is the relative velocity of a ball at the moment when contacting. The ball behavior was simulated using the calculation parameters presented in Table 3, and the impact energy of balls was determined. The balls collide with each other at various relative velocities during ball-milling, resulting in the generation of various amounts of the impact energy. Thus, the impact energy of balls is distributed, as shown in Fig. 12. The distribution of the impact energy shifted to a high energy range with increasing in the rotational speed because

Next, the impact energy of balls per unit time, E, was defined as the total kinetic energy of

ball contacting with another ball or the pot wall (Kano et al., 2000).

of vigorous motion of the balls at higher rotational speeds.

determined from the following equations (Tsuji et al., 1992).

**F**t = –κt**δ**t–ηt**v**t (when |**F**t|≤ μ|**F**n|) (16)

**v**t = **v**–(**v**· **n**)**n**+r(**ω**i+**ω**j)×**n** (18)

κn = (2r)1/2Yb/[3(1–σb2)] (19)

κt = 2(2r)1/2Yb**δ**n1/2/[2(1+σb)(2–σb)] (20)

κn = (4/3)r1/2/[(1–σb2)/Yb+(1–σw2)/Yw] (21)

κt = 8r1/2Yb**δ**n1/2/[2(1+σb)(2–σb)] (22)

ηn = ηt = α(Mκn)1/2**δ**n1/4 (23)

**F**t = –μ|**F**n|(**v**t/|**v**t|) (when |**F**t|> μ|**F**n|) (17)

Fig. 11. Changes in n and k with initial concentration C0.

#### **3.2 Mechanical energy generated in mechanochemical process**

In order to analyze in more details this mechanochemical process, the mechanical energy generated by collision of the balls, i.e., the impact energy of balls, was numerically analyzed by simulating the behavior of balls in the milling pot by means of the discrete element method (DEM). Based on the analysis results, the contribution of the mechanical energy to the Fe3O4 formation reaction was investigated, and the reaction mechanism in this system was analyzed.

For calculating the impact energy of balls, the behavior of balls in the milling pot under wet condition was simulated using the three-dimensional DEM. This simulation model describes the motion of each ball based on Newton's second law for individual ball, allowing for the external forces acting on the ball (Cundall and Strack, 1979). In this model, the interaction between ball and atmosphere gas (argon) was neglected because the contact force acting on the colliding balls is much stronger than the drag force acting on the balls in the translational motion. The fundamental equations of translational and rotational motions of a ball are expressed as follows:

$$\mathbf{d}^2 \mathbf{X} / \text{d}t^2 = (\mathbf{F} / \mathbf{M}) \mathbf{+g} \tag{13}$$

$$\mathbf{d}\mathbf{a} / \mathbf{d}\mathbf{t} = \mathbf{T} / \mathbf{I}\_{\mathbb{b}} \tag{14}$$

where **X**, M, **ω**, and Ib are mass, position, inertia moment, and angular velocity of a ball, respectively. t, **g**, **F**, and **T** are time, gravity acceleration, contact force, and torque caused by the tangential contact force, respectively. **X** and **ω** were calculated by integrating Eqs. (13) and (14) with respect to time between t and t+∆t.

For estimating the contact force acting on a ball, the Hertz-Mindlin contact model was used. The contact forces of normal and tangential directions, **F**n and **F**t, were estimated using the following equations:

$$\mathbf{F}\_{\mathbf{n}} = (-\mathbf{x}\_{\mathbf{n}} \mathbf{\hat{o}}\_{\mathbf{n}}^{3/2} \mathbf{-} \mathbf{r}\_{\mathbf{n}} \mathbf{v}\_{\mathbf{n}} \cdot \mathbf{n}) \mathbf{n} \tag{15}$$

$$\mathbf{F}\_{\mathbf{t}} = \mathbf{-}\mathbf{x}\_{\mathbf{t}} \mathbf{\hat{s}}\_{\mathbf{t}} - \mathbf{r}\_{\mathbf{t}} \mathbf{v}\_{\mathbf{t}} \text{ (where } \|\mathbf{F}\_{\mathbf{t}}\| \le \boldsymbol{\mu} \, |\, \mathbf{F}\_{\mathbf{n}}\, | \, \mathbf{y} \tag{16}$$

$$\mathbf{F}\_{\mathbf{t}} = \text{--}\boldsymbol{\mu} \left| \mathbf{F}\_{\mathbf{n}} \right| \left( \mathbf{v}\_{\mathbf{t}} / \left| \mathbf{v}\_{\mathbf{t}} \right| \right) \text{ (when } \left| \mathbf{F}\_{\mathbf{t}} \right| \succeq \boldsymbol{\mu} \left| \mathbf{F}\_{\mathbf{n}} \right| \text{)}\tag{17}$$

$$\mathbf{v}\_t = \mathbf{v} - (\mathbf{v} \cdot \mathbf{n})\mathbf{n} + \mathbf{r}(\mathbf{a}\_l + \mathbf{a}\_l) \times \mathbf{n} \tag{18}$$

where **δ**, **v**, κ, η, μ, r, and **n** are overlap displacement between contacting balls, relative velocity of contacting balls, stiffness, damping coefficient, sliding friction coefficient, radius of a ball, and unit vector of normal direction at a contact point, respectively. The subscripts n and t mean the components of normal and tangential directions at a contact point, respectively. The subscripts i and j indicate the number of contacting balls. κ and η are determined from the following equations (Tsuji et al., 1992).

Stiffness for the ball-to-ball collision:

$$\mathbf{x}\_{\mathbf{n}} = (\mathbf{2}\mathbf{r})^{1/2}\mathbf{Y}\_{\mathbf{b}} / \left[\Im(\mathbf{1} - \mathbf{o}\_{\mathbf{b}}\mathbf{2})\right] \tag{19}$$

$$\mathbf{x}\_{\mathsf{t}} = \mathsf{Z}(2\mathbf{r})^{1/2} \mathbf{Y}\_{\mathsf{b}} \mathsf{S}\_{\mathsf{b}} 1/2 / \left[ \mathbf{2}(1+\mathsf{o}\_{\mathsf{b}})(2-\mathsf{o}\_{\mathsf{b}}) \right] \tag{20}$$

Stiffness for the ball-to-pot wall collision:

$$\mathbf{x}\_{\rm n} = (\mathbf{4}/3)\mathbf{r}^{1/2}/[(1-\mathbf{o}\,\mathbf{e}^2)/\mathbf{Y}\_{\rm b} + (1-\mathbf{o}\,\mathbf{e}^2)/\mathbf{Y}\_{\rm w}] \tag{21}$$

$$\mathbf{x}\_{\mathsf{k}} = 8\mathbf{r}^{1/2}\mathbf{Y}\_{\mathsf{b}}\mathbf{\mathsf{G}}\_{\mathsf{n}}{}^{1/2} / \left[2(\mathbf{1}\mathsf{+}\mathsf{o}\_{\mathsf{b}})(\mathbf{2}\mathsf{-}\mathsf{o}\_{\mathsf{b}})\right] \tag{22}$$

Damping coefficient:

246 Materials Science and Technology

In order to analyze in more details this mechanochemical process, the mechanical energy generated by collision of the balls, i.e., the impact energy of balls, was numerically analyzed by simulating the behavior of balls in the milling pot by means of the discrete element method (DEM). Based on the analysis results, the contribution of the mechanical energy to the Fe3O4 formation reaction was investigated, and the reaction mechanism in this system

For calculating the impact energy of balls, the behavior of balls in the milling pot under wet condition was simulated using the three-dimensional DEM. This simulation model describes the motion of each ball based on Newton's second law for individual ball, allowing for the external forces acting on the ball (Cundall and Strack, 1979). In this model, the interaction between ball and atmosphere gas (argon) was neglected because the contact force acting on the colliding balls is much stronger than the drag force acting on the balls in the translational motion. The fundamental equations of translational and rotational motions of a

d2**X**/dt2 = (**F**/M)+**g** (13)

 d**ω**/dt = **T**/Ib (14) where **X**, M, **ω**, and Ib are mass, position, inertia moment, and angular velocity of a ball, respectively. t, **g**, **F**, and **T** are time, gravity acceleration, contact force, and torque caused by the tangential contact force, respectively. **X** and **ω** were calculated by integrating Eqs. (13)

For estimating the contact force acting on a ball, the Hertz-Mindlin contact model was used. The contact forces of normal and tangential directions, **F**n and **F**t, were estimated using the

**F**n = (–κn**δ**n3/2–ηn**v**n· **n**)**n** (15)

Fig. 11. Changes in n and k with initial concentration C0.

was analyzed.

ball are expressed as follows:

following equations:

and (14) with respect to time between t and t+∆t.

**3.2 Mechanical energy generated in mechanochemical process** 

$$\mathbf{r}\_{\rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \rm \$$

where Y and σ are Young's modulus and Poisson's ratio, respectively. The subscripts b and w indicate ball and pot wall, respectively. α is the constant depending on the restitution coefficient and was determined to be 0.20 based on the experimental value of restitution coefficient (= 0.75) according to the method proposed by Tsuji et al. (1992). The simulation parameters are summarized in Table 3. The sliding friction coefficient used in the calculation was determined based on the critical rotational speed measured experimentally under wet condition (Gudin et al., 2007).

The impact energy of each ball in a single collision, Ei, was defined as the kinetic energy of a ball contacting with another ball or the pot wall (Kano et al., 2000).

$$\mathbf{E}\_i = (1/2)\mathbf{M} \, |\, \mathbf{v} \, |^2 \tag{24}$$

Here, **v** is the relative velocity of a ball at the moment when contacting. The ball behavior was simulated using the calculation parameters presented in Table 3, and the impact energy of balls was determined. The balls collide with each other at various relative velocities during ball-milling, resulting in the generation of various amounts of the impact energy. Thus, the impact energy of balls is distributed, as shown in Fig. 12. The distribution of the impact energy shifted to a high energy range with increasing in the rotational speed because of vigorous motion of the balls at higher rotational speeds.

Next, the impact energy of balls per unit time, E, was defined as the total kinetic energy of contacting balls.

Novel Mechanochemical Process

rotational speeds.

for Aqueous-Phase Synthesis of Superparamagnetic Magnetite Nanoparticles 249

Fig. 14 shows the relationship between the rate constant k and the impact energy E. k increased with increasing in E. This result reveals that the reaction rate increases under high mechanical energy fields. As mentioned above, this is analogous to increase of the reaction rate caused by temperature rise of the system, i.e., increase of the heat energy given to the system. As can be seen in Fig. 14, the value of k at the rotational speed of 140 rpm was not so large while relatively great amount of mechanical energy applied to the suspension. This suggests that the impact energy was not effectively used for progress of the reaction at high

Fig. 14. Relationship between rate constant k and impact energy E of balls per unit time.

The Fe3O4 formation reaction can occur when the impact energy exceeding a threshold value (corresponding to the activation energy) applies to α-FeOOH at the contact points. Smaller impact energy than the threshold value cannot promote the reaction. Assuming

**3.3 Mechanical energy required for Fe3O4 formation reaction** 

Fig. 13. Change in impact energy of balls with rotational speed.

$$\mathbf{E} = \sum^{\notin} \quad \text{(1/2)}\\\mathbf{M} \|\mathbf{v}\|^2 \tag{25}$$


Table 3. Simulation parameters used in calculations.

Fig. 12. Effect of rotational speed on impact energy distribution.

f is the average number of contact points per unit time. Eq.(25) means that the kinetic energy of balls just before contacting (i.e., at |**δ**| = 0) is summed up for all the contact points within unit time but the kinetic energy during contacting (i.e., within |**δ**| > 0) is not calculated. Therefore, the impact energy thus defined corresponds to the maximum kinetic energy of balls when colliding. Fig. 13 shows the calculation result of the impact energy per unit time. The impact energy was approximately proportional to the rotational speed. This implies that the impact energy depends on the number of revolutions of the pot per unit time rather than the kinetic energy of the rotational motion of the pot. The impact energy considerably varied depending on the rotational speed, resulting in variation of the milling time required for completing the Fe3O4 formation reaction. However, the properties of obtained Fe3O4 nanoparticles were almost the same even when the rotational speed was varied as shown above. This reveals that the grinding of Fe3O4 nanoparticles do not occur in this process and that the impact energy greatly influences the formation process of Fe3O4 nanoparticles (in particular, the reaction rate) rather than the properties of products.

(1/2)M|**v**|2 (25)

7218 3.2 mm 90 mm 80 mm 40% 7930 kg/m3 197 GPa 0.30 0.23 1.0 µs

E = f

Ball filling ratio to pot capacity

Young's modulus of ball and pot Poisson's ratio of ball and pot Sliding friction coefficient

Density of ball and pot

Fig. 12. Effect of rotational speed on impact energy distribution.

particular, the reaction rate) rather than the properties of products.

f is the average number of contact points per unit time. Eq.(25) means that the kinetic energy of balls just before contacting (i.e., at |**δ**| = 0) is summed up for all the contact points within unit time but the kinetic energy during contacting (i.e., within |**δ**| > 0) is not calculated. Therefore, the impact energy thus defined corresponds to the maximum kinetic energy of balls when colliding. Fig. 13 shows the calculation result of the impact energy per unit time. The impact energy was approximately proportional to the rotational speed. This implies that the impact energy depends on the number of revolutions of the pot per unit time rather than the kinetic energy of the rotational motion of the pot. The impact energy considerably varied depending on the rotational speed, resulting in variation of the milling time required for completing the Fe3O4 formation reaction. However, the properties of obtained Fe3O4 nanoparticles were almost the same even when the rotational speed was varied as shown above. This reveals that the grinding of Fe3O4 nanoparticles do not occur in this process and that the impact energy greatly influences the formation process of Fe3O4 nanoparticles (in

Number of balls Ball diameter Pot diameter Pot depth

Time step

Table 3. Simulation parameters used in calculations.

Fig. 13. Change in impact energy of balls with rotational speed.

Fig. 14 shows the relationship between the rate constant k and the impact energy E. k increased with increasing in E. This result reveals that the reaction rate increases under high mechanical energy fields. As mentioned above, this is analogous to increase of the reaction rate caused by temperature rise of the system, i.e., increase of the heat energy given to the system. As can be seen in Fig. 14, the value of k at the rotational speed of 140 rpm was not so large while relatively great amount of mechanical energy applied to the suspension. This suggests that the impact energy was not effectively used for progress of the reaction at high rotational speeds.

Fig. 14. Relationship between rate constant k and impact energy E of balls per unit time.

#### **3.3 Mechanical energy required for Fe3O4 formation reaction**

The Fe3O4 formation reaction can occur when the impact energy exceeding a threshold value (corresponding to the activation energy) applies to α-FeOOH at the contact points. Smaller impact energy than the threshold value cannot promote the reaction. Assuming

Novel Mechanochemical Process

rotational speed.

for Aqueous-Phase Synthesis of Superparamagnetic Magnetite Nanoparticles 251

Fig. 16. Relationship between threshold value of impact energy and coefficient b.

Fig. 17. Change in number of collisions generating impact energy more than 93.7 nJ with

This threshold value of the impact energy in a single collision, the number of collisions shown in Fig. 17, and the completion time of the Fe3O4 formation reaction shown in Fig. 8 were used to estimate the accumulative impact energy required for completing the Fe3O4 formation reaction. Fig. 18 shows the variation in the accumulative impact energy with the rotational speed. It was found that the accumulative impact energy was almost constant regardless of the rotational speed. Because the accumulative impact energy is the mechanical energy required for synthesizing 1.5 mmol of Fe3O4, the accumulative impact energy per 1 mol of Fe3O4 is defined as the apparent activation energy. Using the average of the accumulative impact energy, the apparent activation energy was determined to be 15.6 MJ/mol, which was independently of the rotational speed. This result is also analogous to the fact that generally the activation energy is independently of the reaction temperature in reaction systems using heat energy. However, the apparent activation energy thus

that the region where the impact energy applies is extremely small, even though the impact energy larger than the threshold value generates at the contact point, α-FeOOH far away from there may not react using the surplus energy. Thus, it is considered that progress of the reaction at the contact points depends on whether the impact energy exceeding the threshold value is given to α-FeOOH or not, and that the surplus energy cannot promote the reaction. Accordingly, the reaction rate can be proportional to the number of collisions that the reaction occurred. The relationship between the rate constant k and the number nt of collisions per unit time with the energy exceeding predetermined threshold values is shown in Fig. 15. As can be seen in Fig. 15, the correlation between k and nt was expressed by

$$\mathbf{k} = \mathbf{a}\mathbf{n}\_t\mathbf{b} \tag{26}$$

where a and b are the coefficients depending on the threshold value. Fig. 16 shows the relationship between the predetermined threshold value and the coefficient b. As indicated in Eq. (26), when b = 1, k is proportional to nt. From Fig. 16, the threshold value giving b = 1 was determined to be 93.7 nJ. The analysis result reveals that the Fe3O4 formation reaction occurs at the contact points where the impact energy exceeding 93.7 nJ generates, regardless of the rotational speed. Consequently, this threshold value may be closely related to the activation energy in this reaction system. Fig. 17 shows the number of collisions per unit time that the impact energy more than 93.7 nJ generates as a function of the rotational speed. The number of collisions increased with increasing in the rotational speed.

Fig. 15. Relationship between rate constant k and number nt of collisions of balls per unit time for various threshold values of impact energy.

that the region where the impact energy applies is extremely small, even though the impact energy larger than the threshold value generates at the contact point, α-FeOOH far away from there may not react using the surplus energy. Thus, it is considered that progress of the reaction at the contact points depends on whether the impact energy exceeding the threshold value is given to α-FeOOH or not, and that the surplus energy cannot promote the reaction. Accordingly, the reaction rate can be proportional to the number of collisions that the reaction occurred. The relationship between the rate constant k and the number nt of collisions per unit time with the energy exceeding predetermined threshold values is shown in Fig. 15. As can be seen in Fig. 15, the correlation between k

k = ant

where a and b are the coefficients depending on the threshold value. Fig. 16 shows the relationship between the predetermined threshold value and the coefficient b. As indicated in Eq. (26), when b = 1, k is proportional to nt. From Fig. 16, the threshold value giving b = 1 was determined to be 93.7 nJ. The analysis result reveals that the Fe3O4 formation reaction occurs at the contact points where the impact energy exceeding 93.7 nJ generates, regardless of the rotational speed. Consequently, this threshold value may be closely related to the activation energy in this reaction system. Fig. 17 shows the number of collisions per unit time that the impact energy more than 93.7 nJ generates as a function of the rotational speed. The number of collisions increased with increasing in the

Fig. 15. Relationship between rate constant k and number nt of collisions of balls per unit

time for various threshold values of impact energy.

b (26)

and nt was expressed by

rotational speed.

Fig. 16. Relationship between threshold value of impact energy and coefficient b.

Fig. 17. Change in number of collisions generating impact energy more than 93.7 nJ with rotational speed.

This threshold value of the impact energy in a single collision, the number of collisions shown in Fig. 17, and the completion time of the Fe3O4 formation reaction shown in Fig. 8 were used to estimate the accumulative impact energy required for completing the Fe3O4 formation reaction. Fig. 18 shows the variation in the accumulative impact energy with the rotational speed. It was found that the accumulative impact energy was almost constant regardless of the rotational speed. Because the accumulative impact energy is the mechanical energy required for synthesizing 1.5 mmol of Fe3O4, the accumulative impact energy per 1 mol of Fe3O4 is defined as the apparent activation energy. Using the average of the accumulative impact energy, the apparent activation energy was determined to be 15.6 MJ/mol, which was independently of the rotational speed. This result is also analogous to the fact that generally the activation energy is independently of the reaction temperature in reaction systems using heat energy. However, the apparent activation energy thus

Novel Mechanochemical Process

**4. Conclusion** 

**5. References** 

6843.

*B*, 71, 154–159.

synthesizing functional nanoparticles.

978-3527303632, Weinheim, Germany.

for Aqueous-Phase Synthesis of Superparamagnetic Magnetite Nanoparticles 253

for synthesizing Fe3O4 nanoparticles in water system, for promoting the formation reaction and increasing the crystallinity, the starting suspension in the vessel is heated from the outside by conductive heat transfer, resulting in temperature rise of the whole system. The heating is continued to keep the reaction temperature. This causes aggregation of the precipitates, leading to the growth of Fe3O4 nanoparticles. On the contrary, in the synthesis process with the ball-milling treatment, the suspension may be heated in the contact points between balls, which are extremely small regions, and then is cooled immediately because the pot and the balls are cooled enough. Therefore, local temperature rise of the suspension occurs instantaneously, and the heat energy is hardly stored in the system. In addition, the contact points exist discretely in the pot; the discrete heating occurs everywhere. Consequently, the heating type in the synthesis process is the internal heating, which is regarded as a non-uniform heating from a microscopic viewpoint but a uniform heating from a macroscopic one. This inhibits aggregation of the precipitates effectively, and the nucleation frequently occurs rather than the particle growth, resulting in the formation of the ultrafine Fe3O4 nanoparticles with high crystallinity and relatively narrow size distribution. Even at low rotational speeds, the Fe3O4 nanoparticles with a size of about 10 nm can be formed; this means that the

A novel process for preparing superparamagnetic Fe3O4 nanoparticles with high crystallinity in water system has been developed, in which a cooled tumbling ball mill is used as the reaction field. It has been confirmed that this method provides successfully the Fe3O4 nanoparticles having a size of less than 15 nm without using any conventional heating techniques. This mechanochemical process was kinetically analyzed, indicating that the Fe3O4 formation reaction obeys the 0.6th-order rate equation. In addition, the mechanical energy (i.e., the impact energy of balls) promoting the Fe3O4 formation reaction was also analyzed using the numerical simulation method. The rate constant of the reaction was investigated based on the mechanical energy. As a result, the apparent activation energy of the reaction was estimated. This mechanochemical process may contribute to the production of superparamagnetic Fe3O4 nanoparticles under environmentally friendly conditions and be applied to another reaction systems

Buxbaum, G. & Pfaff, G. (2005). *Industrial Inorganic Pigments.* Third edition, Wiley-Vch, ISBN

Buyukhatipoglu, K.; Miller, T.A. & Morss Clyne, A. (2009). Flame synthesis and in vitro

Can, K.; Ozmen, M. & Ersoz, M. (2009). Immobilization of albumin on aminosilane modified

biocompatibility assessment of superparamagnetic iron oxide nanoparticles: Cellular uptake, toxicity and proliferation studies. *J. Nanosci. Nanotechnol.*, 9, 6834–

superparamagnetic magnetite nanoparticles and its characterization. *Colloids Surf.* 

aggregation-inhibition effect is confirmed in low energy fields.

determined was considerably larger than the activation energy in conventional liquid-phase reaction systems because the impact energy was the maximum mechanical energy which α-FeOOH is able to receive. When both the net mechanical energy transferred from the balls to α-FeOOH and the amount of α-FeOOH receiving the energy are known, the true activation energy of the mechanochemical reaction can be determined.

Fig. 18. Variation in accumulative impact energy required for completing Fe3O4 formation reaction with rotational speed.

In general mechanochemical processes, when the mechanical energy caused by shear, compression and friction actions of balls applies to particulate materials under dry condition, the surface energy of particles can increase due to the physical change such as distortion of the crystal lattice, increase of the surface area, and appearance of newly formed crystal surface. This causes the mechanochemical activation of particles. Under wet condition, however, the particles are difficult to undergo the mechanochemical activation because the increased surface energy is reduced by the solvent. Furthermore, it is very difficult to apply the mechanical energy effectively to nanoparticles. Accordingly, in this synthesis process, the mechanochemical activation of nanoparticles is difficult to occur, and the solid phase reaction from α-FeOOH to Fe3O4 may hardly proceed by direct contribution of the mechanical energy. However, at the rotational speed of 35 rpm, i.e. in a low mechanical energy field, the Fe3O4 formation reaction surely proceeded while the reaction rate was relatively low. There is no doubt that the applied mechanical energy promotes the reaction; the reaction mechanism in this synthesis process is considered that the reaction may proceed not by the mechanochemical activation of α-FeOOH but by local and rapid heating and/or through a different reaction path. In the conventional methods for synthesizing Fe3O4 nanoparticles in water system, for promoting the formation reaction and increasing the crystallinity, the starting suspension in the vessel is heated from the outside by conductive heat transfer, resulting in temperature rise of the whole system. The heating is continued to keep the reaction temperature. This causes aggregation of the precipitates, leading to the growth of Fe3O4 nanoparticles. On the contrary, in the synthesis process with the ball-milling treatment, the suspension may be heated in the contact points between balls, which are extremely small regions, and then is cooled immediately because the pot and the balls are cooled enough. Therefore, local temperature rise of the suspension occurs instantaneously, and the heat energy is hardly stored in the system. In addition, the contact points exist discretely in the pot; the discrete heating occurs everywhere. Consequently, the heating type in the synthesis process is the internal heating, which is regarded as a non-uniform heating from a microscopic viewpoint but a uniform heating from a macroscopic one. This inhibits aggregation of the precipitates effectively, and the nucleation frequently occurs rather than the particle growth, resulting in the formation of the ultrafine Fe3O4 nanoparticles with high crystallinity and relatively narrow size distribution. Even at low rotational speeds, the Fe3O4 nanoparticles with a size of about 10 nm can be formed; this means that the aggregation-inhibition effect is confirmed in low energy fields.
