Mass transfer kinetics of graphene oxide prepared by chemical oxidation intercalationg assisted ultrasonic field

3.1 Theoretical basis for establishing the model

The graphite structure is consists of three covalent bonds formed by the sp² hybrid orbital electrons and 2Pz orbital electrons of one carbon atom and three adjacent carbon atoms, and arranged into a planar hexagonal network structure. These network structures are linked into parallel planes by van der Waals forces in the form of dispersion forces, constituting the lamellar structure. The spacing between carbon atoms in the lamellar structure is 0.142 nm, and the spacing between the lamellar structure is 0.34 nm (Chuan, 2000). However, the molecular diameter of concentrated H₂SO₄ is about 0.39 nm, which is similar to the distance between graphite layer and exists in the form of non-ion, so it is difficult to intercalate into graphite sheet layer. This intercalation process was confirmed through thermodynamic calculation to be a negative Gibbs free energy process, which is difficult to achieve under conventional conditions (Aronson et al., 1971). Then auxiliary oxidants, such as NaNO₃ and (NH₄)₂SO₄, are often added to generate HSO₄^-, increase the solution's electronegativity, improve the intercalation power, and form H₂SO₄-GIC.

The first step of preparing GO is conversion of graphite into a stage-1 graphite intercalation compound (GIC), and the distance between these layers has been changed to 0.798 nm. The second step is conversion of the stage-1 GIC into the pristine graphite oxide (PGO). This step involves the diffusion of the oxidizing agent MnO₃⁺ into the preoccupied graphite galleries. This rate-determining step makes the entire process diffusive-controlled (Dimiev et al., 2013). A different chemical reaction mechanism is that the oxidant Mn₂O₇ and graphite take place in the main oxidation reaction (Dreyer et al., 2009). This theory is mainly based on the experiments which the oxidant Mn₂O₇ has the strongest oxidability in other oxidation reactions, and thus speculated that the same is true in the oxidation reaction of graphite (Trömel et al., 1987).

We have confirmed the whole preparation process of GO constitutes four distinct independent steps by experimental studies (Chen et al., 2022). In the second step, only Mn₂O₇ is generated in the reaction of KMnO₄ and concentrated H₂SO₄ at 0–4 ℃, and the intercalation has been completed at this temperature (Fig. 1). Then, it is heated to decompose and generate oxygen atoms to oxidize with the carbon atoms at the delocalized and defective positions on the graphite, and further hydrate to generate oxygen-containing groups, which are then stripped into GO. Therefore, it can be considered that the reaction process is intercalation and then oxidation, and the completeness of intercalation diffusion is the most important factor affecting the oxidation quality of graphite sheets.The molecular diameter of Mn₂O₇ is about 0.671 nm, and the single molecular diameter is slightly smaller than the layer spacing of HNO₃-H₂SO₄-GIC. Therefore, Mn₂O₇ molecules can only intercalate from the external high-concentration liquid to the low-concentration liquid between the graphite layers in the form of single molecular layer in the vertical direction and molecular convection diffusion in the horizontal direction, and replace part of the HNO₃-H₂SO₄ mixture to generate new Mn₂O₇-H₂SO₄-GIC compounds.

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Fig. 1

The reaction mechanism of conversion of flake graphite into GO by chemical oxidation process (Chen et al., 2022).

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Accordingly, in consideration of the convenience of theoretical calculation, this paper proposes the following hypothesis about mass transfer kinetics of graphite oxidation intercalation: (1) It is assumed that the graphite sheets are circular layered with radius r. In the first step, the mixed solution of concentrated sulfuric acid, nitric acid and HSO₄^- diffuses and intercalates between graphite layers to form HNO₃-H₂SO₄-GIC graphite intercalation compound with uniform concentration in the whole region. (2) In the second step, a certain proportion of potassium permanganate is added to the above solution, and dark green Mn₂O₇, HSO₄^- and H₂SO₄ liquid mixtures are generated after the reaction. The main oxidizing substance is Mn₂O₇, and other substances, such as HNO₃, HSO₄^- and H₂SO₄, can be regarded as mass transfer carriers because the first intercalation process has reached dynamic mass transfer equilibrium.When other substances in the mixture except Mn₂O₇ are in contact with the intercalated substance in the first step, they act as a mass transfer bridge, which is equivalent to the molecular convection–diffusion mass transfer of Mn₂O₇ in the HNO₃/H₂SO₄ stagnant liquid in the nanosized flat channel. (3) the main concentration of oxidizer solution around the graphite sheet layer is uniform. Since the spacing between the graphite layers is very small, only single molecules can migrate into the graphite layer by radial diffusion, and the flow rate of the fluid is small, so it is assumed that the flow form of the intercalation solution is laminar flow. Moreover, the mass of concentrated sulfuric acid actually used is much greater than that of potassium permanganate, so the mixture can be considered as dilute solution of Mn₂O₇. After the diffusion mass transfer is completed, Oxygen atoms generated by Mn₂O₇ in the graphite interlayer oxidize with C⚌C on the graphite layer to generate PGO and consume Mn₂O₇ at the same time. Oxidation reaction is a first-order irreversible reaction with a fast reaction rate, far greater than the diffusion rate. Therefore, mass transfer process of oxidizer intercalation is the key control process for the preparation of GO. The equation is as following: $$3\text{M}{\text{n}}_2{O}_7\stackrel{35-{45}^{°}C}{\to }6Mn{O}_2+3O_2+O_3$$

3.2 The mass transfer kinetics model

Based on the above assumptions, it is determined that the graphite oxidation intercalation process conforms to the laminar flow two-phase molecular convection–diffusion mass transfer theory on a flat circular wall. That is, component A (Mn₂O₇) diffuses steady-state molecular convection to stagnant component B (HNO₃/H₂SO₄), and the solution system is dilute solution (Fig. 2). As the liquid concentration around the graphite sheet is the same and the diffusion driving potential is the same, the change of the concentration of component A along the angular variable is zero, that is, dC/dε = 0. There are monomolecular channels between layers, so the concentration change of component A along the direction perpendicular to the graphite sheet y is also zero, dC/dy = 0. In this intercalation process, only the concentration of component A changes along the radial direction r, namely −dC/dr.

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Fig. 2

Schematic diagram of Mn₂O₇ intercalation mass transfer model.

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The mass transfer coefficient k_m can be obtained by solving the mass transfer equation (Chen et al., 2009):

In the formula, D_AB is the diffusion coefficient of component A through stagnant component B (m²/s), ΔZ is the radial diffusion distance of component A (m), C_av is the total average concentration of the mixture (Kmol m⁻³), Cr_A1 and Cr_A2 are the molar concentration of component A at points 1 and 2 (mol/L), T_A is the number of moles of solute component A leaving unit interface in unit time (kmol/m⁻² s⁻¹). C_A is the concentration of solute component A in the interfacial fluid (mol/L), that is, the concentration of the fluid in equilibrium with the solid at the temperature and pressure of the system. C_∞ is the concentration at a certain point in the graphite lamellar outflow field (mol/L).

The diffusion process can be described by Fick's Law. Component B is static relative to component A, so its diffusion coefficient D_AB formula is (Dai et al., 2005):

Where, J_A is the mass of A diffused per unit area in unit time (mol/m⁻² s⁻¹), The diffusivity varies from system to system and also with temperature, pressure, and composition.

According to the relationship between the driving force and the viscous force that molecules are subjected to in the diffusion process, D_AB of molecular diffusion coefficient can be derived according to Fick's law:

Where R is the molar gas constant (J/mol K⁻¹), T is the system temperature (K), N_A is Avogadro's constant (mol⁻¹), η is the solution viscosity (P_a s), and r is the effective flow radius of molecules (m).

3.3 Mass transfer kinetics model enhanced by ultrasonic field

Ultrasound refers to sound waves whose frequency ranges from 2 × 10⁴ to 2 × 10⁶ Hz. In the process of graphite oxidation preparation of GO, the intervention of ultrasonic field can effectively improve the oxidation rate, increase the chemical yield, and achieve the purpose of process strengthening (Li, 1994). Therefore, it is extremely important to evaluate the degree of enhanced mass transfer by ultrasonic field as the basis for optimizing ultrasonic chemical reactor. As an essential parameter of mass transfer in chemical reaction process, mass transfer coefficient can effectively evaluate the enhancement degree of ultrasonic field on liquid–solid reaction process. In addition, it can directly establish mathematical relations with measurable parameters such as chemical reaction rate, conversion rate and mass transfer rate, which is conducive to further experimental verification of ultrasonic field strengthening, so it is of great significance to quantitatively solve the mass transfer coefficient (Ma et al., 2011; Kanaziri et al., 1955).

Because the graphite layer is a two-dimensional layered structure, a two-dimensional diffusion mass transfer model was established to describe the kinetic characteristics of oxidation intercalation. However, the liquid around the graphite sheet is homogeneous, the concentration of oxidizer is the same, the driving force is the same along the circumference of the circular graphite sheet, and the concentration gradient of Angle vector is zero, the model can be simplified to a one-dimensional diffusion model along the radial direction. This model is based on the physical idea that ultrasonic field changes the fluid field, thus changing the mass transfer field, and then changing the mass transfer coefficient. A dynamic model is proposed to quantitatively solve the mass transfer coefficient.The steps of the model building process are as follows:

The sound pressure P in the liquid is calculated by using the harmonic dynamic equation of ultrasonic propagationin the medium of oxidation system (Jiao, 2014).

In the formula, ρ_c is the densityof the mixed liquid (Kg m⁻³), C_c sound propagation velocity of the mixed liquid (m/s), ▽P is the intensity gradient of sound pressure propagating along the radial direction (Pa), ω is the circular frequency of the ultrasonic wave source (MHz), ω = 2πf.

By substituting P into the momentum transfer equation of the fluid field, the force F causing the liquid flow in unit volume can be obtained:

ρ₀ and c₀ correspond to the density and sound velocity of the solution respectively, and α is the sound absorption coefficient of the solution.

By substituting F into momentum transfer Eq. (8) and continuity Eq. (9) for simultaneous solution, liquid flow rate U can be obtained:

Where I is the identity matrix, u, μ and P are liquid flow rate (m/s), liquid viscosity (cp) and pressure in liquid respectively (Pa). F is the force causing the fluid to flow per unit volume (kg m⁻² s⁻²). The superscript T represents the transformation matrix. The equation assumes that the density of the liquid in the reaction vessel is constant.

In the actual reaction, the substance A (Mn₂O₇) has been completely formed at 0–4 ℃, and no new substance A is formed during the intercalation process, and its molar rate r = 0. The oxidation reaction between Mn₂O₇ and graphite sheet takes place at 35–45 ℃. This reaction is an irreversible first-order reaction, and there is no further reaction.Therefore, the surface concentration C_AS of Mn₂O₇ solution on the surface of graphite sheet and the radial dC_A/ dr concentration gradient relative to the graphite layer can be simultaneously solved by using the mass transfer differential Eq. (10), mass transfer flux Eq. (11), Eq. (12) of mass transfer flux and reaction rate, and the reaction rate Eq. (13) of Mn₂O₇ solution on the surface of graphite sheet.

k in Eq. (13) can be obtained by Eq. (14):

Where D is the diffusion coefficient of substance A (m²/s), C_A is the molar concentration of substance A (mol/L), C_AS is the surface molar concentration of substance A solution on the surface of graphite sheet (mol/L), r * is the molar formation rate of substance A (kmol/m⁻² s⁻¹), N is the total mass transfer flux of substance A (kmol/m⁻² s⁻¹), N₀ is the flux of substance A on the surface of graphite sheet (kmol/m⁻² s⁻¹), q is the surface reaction rate (kmol/m⁻² s⁻¹), n is unit normal vector, u is the liquid flow rate in the system (m/s), k is the surface reaction rate constant (s⁻¹), Ea and K₀ are the reaction activation energy (kmol/m⁻² s⁻¹) and Arrhenius parameters under ultrasonic participation (kmol/m⁻² s⁻¹).

Finally, the concentration C_AS and radial concentration gradient dC_A/dr of Mn₂O₇ solution at the edge of graphite sheet were calculated by using the acoustic field, laminar flow field and mass transfer field calculation module of COMSOL Multiphysics software. The value of mass transfer coefficient K_u in the intercalation oxidation reaction between graphite and Mn₂O₇ under the condition of introducing ultrasonic field can be obtained by Eq. (15):

3.4 Model simulation calculation

Establish the simulation module according to the COMSOL Multiphysics 5.5 Software simulation procedure and the model shown in Fig. 2. The model is a two-phase molecular convection–diffusion mass transfer model with an applied ultrasonic field. The scaly graphite used in laboratory synthesis of GO is generally of 2–3 mm diameter with lamellar spacing of about 0.34 nm. After the formation of H₂SO₄‐GIC in the first stage of intercalation reaction, the spacing increases by 0.798 nm due to the presence of sulfuric acid in the middle lamellar interval (Dimiev et al., 2014). The purpose of this model is to reveal the influence of the mass transfer process on the reaction by simulation calculation. For the convenience of calculation in the COMSOL Multiphysics 5.5 Software, the model is simplified as the Mn₂O₇ solution (substance A) generated after the reaction of concentrated sulfuric acid and potassium permanganate is transferred to two graphite laminate slits with length of 20 µm and width of 0.798 nm filled with concentrated sulfuric acid (substance B), which is two-dimensional diffusion model. That is, the upper and lower layers are graphite sheets, the two layers are filled with concentrated sulfuric acid in the middle, and the graphite layer is surrounded by a homogeneous mixed solution of Mn₂O₇ and concentrated sulfuric acid. The model was drawed in COMSOL Multiphysics 5.5 as showing in Fig. 3.

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Fig. 3

Mass transfer kinetics model of Mn₂O₇ intercalated oxidation of H₂SO₄-GIC.

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3.4.1

3.4.1 Sound pressure field

Sound pressure field simulation is calculated by using the pressure acoustics and steady-state interface in the acoustic module of COMSOL Multiphysics 5.5 software. When ultrasonic waves propagate in the medium, acoustic impedance sound pressure boundary(SPB) conditions need to be determined for air and mixed solution, mixed solution and graphite surface, graphite surface and concentrated sulfuric acid (Fig. 4) (Bauer and Aircr, 1977). The calculation formula for boundary conditions of impedance is:

(16)

$$n·\left(-\frac{1}{\rho }\mathrm{\nabla }p\right)=\frac{i\omega p}{Z_e}$$

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Fig. 4

Mass transfer kinetics sound pressure boundary (SPB) model of Mn₂O₇ intercalated oxidation of H₂SO₄-GIC.

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$Z_e={\rho }_e{c}_e$ , the input impedance of the external region (ρ_e and c_e are respectively the density (Kg m⁻³) and sound velocity of the medium in the external region (m/s), i.e. Mn₂O₇ and concentrated sulfuric acid mixed solution, n is the normal vector).

The calculation of impedance requires related variables, such as the sound speed and density of concentrated sulfuric acid and graphite in the model, which can be obtained by consulting the data.The density and sound velocity of Mn₂O₇ mixed solution after the reaction of potassium permanganate and concentrated sulfuric acid was measured experimentally.Specific values are shown in Table 1:

Table 1 Sound velocity and density of medium.

medium	Density/Kg m−3	sound velocity/m/s	dynamic viscosity/cp
M1	1970	340	27,500
M2	2250	1800	200
M3	1840	340	26,700

M1: mixed solution of Mn₂O₇ and concentrated sulfuric acid; M2: graphite; M3: concentrated sulfuric acid.

For this research system, the boundary condition is set as sound pressure. According to different ultrasonic power, it can be calculated by following Eqs. (17) and (18):

(17)

$$I=W/S$$

(18)

$$I=P^2/\left(2\rho c\right)$$

P is sound pressure amplitude (Pa), I is ultrasonic sound intensity (W/m⁻²), W is ultrasonic power (W), S is the area of the interface (m²), ρ is the density of medium (Kg m⁻³), C is the sound velocity of the medium (m/s).

3.4.2

3.4.2 Fluid field

The fluid field simulation is calculated by using the interface of “creep flow” and “steady state” in COMSOL Multiphysics 5.5 software. The “peristaltic flow” interface is used to simulate fluid flows at very low Reynolds numbers where the inertia term in the Navier-Stokes equation is negligible. The interface is mainly used to solve stokes equation related to momentum conservation and continuity equation related to mass conservation. The geometry of boundary conditions in the fluid field is shown in Fig. 5. The range of the simulated fluid field is region 1, where the non-slip boundary condition is described as the stationary solid wall. The four surfaces and four sides of graphite need to be defined as not slip, that is, the velocity is zero. Dynamic viscosity of graphite, concentrated sulfuric acid and mixed solution needed for fluid field simulation calculation can be obtained by consulting the data, as showed in Table 1.

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Fig. 5

The model of mass transfer kinetic fluid field slip of Mn₂O₇ intercalated oxidation of H₂SO₄-GIC.

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3.4.3

3.4.3 Mass transfer field

The “thin matter transfer” and “transient” interfaces in COMSOL Multiphysics 5.5 software are used to simulate the mass transfer field.The “thin matter Transfer” interface is used to calculate the concentration field of thin solutes in solvents. It can calculate the transfer and reaction of substances dissolved in gas, liquids or solids.The driving forces for the transfer can be diffusion described by Fick's law, convection coupled with fluid flow, and migration coupled to an electric field. The “transient” interface is used to calculate chemical composition changes over time. For the mass transfer field of the study system, the geometry of its boundary conditions is shown in Fig. 6. The range of the simulated mass transfer field is region 1. No chemical reaction occurs on the solution surface far from the graphite sheet, and no flux is defined. Boundary conditions can be expressed as (19):

(19)

$$\text{-n}\bullet N=0$$

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Fig. 6

Mass transfer kinetic flux model of Mn₂O₇ intercalated oxidation of H₂SO₄-GIC.

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For the graphite surface, as it reacts with the solution, the boundary condition of flux needs to be defined. The boundary description condition is as follows:

(20)

$$\text{-n}\bullet N=N_0$$

(21)

$$N_0=q$$

Where, N₀ is graphite surface mass flux (kmol/m⁻² s⁻¹), q is surface reaction rate (kmol/m⁻² s⁻¹), and n is normal vector.

The diffusion coefficient D of Mn₂O₇ and concentrated sulfuric acid mixed solution should be determined when the mass transfer field is simulated, and the diffusion coefficient D at 35 ℃ can be calculated by Eq. (5). Using the post-processing function of COMSOL Multiphysics 5.5 software, set the initial pressure of sound pressure and the initial pressure of fluid flow rate to zero, establish a triangular mesh model, calculate the C_AS of solution concentration in graphite interlayer, pressure and flow rate distribution diagram.Its triangular mesh model is shown in Fig. 7.

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Fig. 7

Triangular mesh diagram of mass transfer kinetics model of Mn₂O₇ intercalated oxidation of H₂SO₄-GIC.

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3.5 Calculation results and discussion

3.5.1

3.5.1 Concentration distribution of Mn₂O₇

In order to facilitate the investigation of the influence of the duration of ultrasonic field intervention on the concentration distribution of Mn₂O₇ inserted into the graphite sheet, the ultrasonic field with a frequency of 28 Hz and a power of 480 W was defined to be emitted from the left side of the model. Considering that the intercalation process of intercalated materials takes time, the concentration distribution will be different with the length of ultrasonic action time, so the duration of action was set at 5 min and 10 min respectively. COMSOL Multiphysics 5.5 software was used to simulate the calculation results, as showed in Fig. 8.It can be seen from Fig. 8 that the addition of ultrasonic field have little influence on the concentration distribution of Mn₂O₇ solution outside the graphite layer, but it can be seen from the flow velocity curve that ultrasonic field can promote the diffusion movement of solution.This is the fact that it is assumed that the solution outside the graphite layer has been evenly mixed before the ultrasonic field intervention. Ultrasonic field intervention only accelerates the directional diffusion rate of the solution molecules, and does not contribute to the solute redistribution. The concentration from the innermost to the edge of the graphite layer is significantly greater than that from the middle. The diffusion distance of Mn₂O₇ promoted by ultrasonic field for 10 min (Fig. 8b) is 0.7 µm larger than that of Mn₂O₇ promoted by ultrasonic field for 5 min (Fig. 8a), indicating that ultrasonic field can promote the diffusion of interlayer oxides to the center of the graphite sheet.

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Fig. 8

The distribution diagram of the effect of ultrasonic time on Mn₂O₇ concentration: (a) 5 min, (b) 10 min.

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3.5.2

3.5.2 Diffusion velocity distribution of Mn₂O₇

It is the similar to the research method of the influence of duration of ultrasonic field intervention on concentration distribution. An ultrasonic field with a frequency of 28 Hz and a power of 480 W was defined to be emitted from the left side of the model, and the duration of action was set at 5 min and 10 min respectively. COMSOL software was used to simulate diffusion velocity distribution of Mn₂O₇, and the results were shown in Fig. 9. As can be observed in Fig. 9a, after 5 min of ultrasonic field, the diffusion rate of Mn₂O₇ solution outside the graphite layer changes little, and the diffusion rate of solution inside the interlayer changes even less.

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Fig. 9

The distribution diagram of the effect of ultrasonic time on Mn₂O₇ diffusion velocity: (a) 5 min, (b) 10 min.

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However, both internal and external diffusion flow patterns are obviously affected by the ultrasonic field, which changes from laminar flow to turbulent flow.The information given in Fig. 9b is that after 10 min in the ultrasonic field, the turbulence intensity of Mn₂O₇ solution is higher than that after 5 min, and the diffusion rate is slightly higher. The diffusion rate inside the graphite layer also increases to some extent, but the change is not really obvious.The results show that the flow pattern of interlaminar mass transfer process in Mn₂O₇ solution intercalated graphite sheet is affected by ultrasonic, but the diffusion rate is not affected. This is consistent with the hypothesis that the spacing between graphite intercalation layers is only 0.798 nm, while the molecular diameter of Mn₂O₇ is 0.69 nm, and the space between upper and lower layers is only enough for monolayer to enter and carry out monolayer diffusion mass transfer.The solution in the layer cannot be affected by ultrasonic cavitation effect like the solution in the outside, and only the change of external pressure forces the flow pattern of the liquid in the inner layer to change, which has much less influence on the diffusion rate.

3.5.3

3.5.3 Pressure distribution of Mn₂O₇ intercalated solution

Fig. 10a shows the pressure distribution contour line of Mn₂O₇ intercalated solution in an ultrasonic field for 5 min. Solution pressure in the graphite layer is significantly higher than that in the outside, the pressure near the ultrasonic occurrence end (left end) is higher than that at the far end (right end). Fig. 10b shows the pressure distribution contour line of the ultrasonic field for 10 min. It can be found that the solution pressure both inside and outside the graphite layer is greater than that of the solution for 5 min, and the pressure in the center of the graphite layer gradually increases from left to right inside the layer. This indicates that the longitudinal mechanical wave formed by ultrasonic cavitation effect can compress and sparge the liquid, and has a great influence on the pressure distribution of graphite sheet solution. This result suggests that the longer the ultrasonic field action time is, the greater solution pressure between graphene sheets will be, which is conducive to the lamellar peeling and the formation of GO. This simulation result is consistent with the experimental results (Zou et al. 2011).

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Fig. 10

The distribution diagram of the effect of ultrasonic time on Mn₂O₇ diffusion pressure field:(a) 5 min, (b) 10 min.

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