Kinetic analysis of spinel formation from powder compaction of magnesia and alumina

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Kinetic analysis of spinel formation from powder compaction of magnesia and alumina Lan Hong∗, Weipeng Chen, Dong Hou∗∗ School of Iron and Steel, Soochow University, Suzhou, 215021, China

ARTICLE INFO

ABSTRACT

Keywords: Spinel formation Magnesia and alumina Powder compaction Kinetics Electrochemical potential Diffusion Expansion

A kinetic investigation into the formation of spinel from alumina (Al2O3) and magnesia (MgO) powder compaction with a stoichiometric mixing molar ratio of 1:1 was conducted in the temperature range of 1573 K to 1773 K over a certain time interval up to 25 h. The samples were pressed at pressures of 125, 375 and 750 MPa. The progress of the reaction was evaluated by monitoring the expansion ratio instead of the thickness of the spinel layer that was generated. The expansion ratio increases with increasing pressing pressure and holding time, and high temperature favored spinel formation. However, densification was observed at temperatures above 1673 K due to the occurrence of sintering between the powders. A kinetic model taking electrochemical potential as the driving force of the reaction was established, and the apparent activation energy was calculated to be 310.6 kJ/mol in the temperature range between 1573 K and 1673 K. The reaction was controlled by the inter-diffusion of Al3+ and Mg2+ ions in the spinel layer that was formed.

1. Introduction Spinel exhibits outstanding corrosion resistance against steelmaking slags because its structural integrity is unaffected by the intrusion of otherwise detrimentally reactive components, such as Fe2+ and Mn2+. As a result of this advantageous property, spinel-containing refractories are widely applied as a durable lining material in steelmaking vessels, such as refining ladles and RH degassers [1–3]. Spinelcontaining refractories are formulated in two ways, depending on the source of the spinel: synthetic, or pre-formed, and in situ, or selfforming. For pre-formed varieties, spinel is first synthesized ex situ by fusion or sintering at high temperatures up to 1750 °C before being added into the refractory [4]. Despite facilitating easy manipulation of the lining volume stability under service, some obvious disadvantages, ranging from high cost to reduced corrosion resistance due to the discontinuous distribution of spinel, are exhibited by these pre-formed varieties. As a result, the preferred means of producing economical spinel-containing refractories is through the in situ generation of spinel from reactive alumina and magnesia species in service [5,6]. Some extensive studies on the performance of in situ spinel containing refractories already exist, including those regarding the influence of various ingredients, such as silica, binder and magnesia, on the refractory properties, particularly corrosion resistance against molten

∗

melts [7–24]. The corrosion resistance against molten slag of the in situ spinel-containing refractories is largely dependent on the spinel formation and the porosity of the formed spinel. The spinel formation can be accelerated, and the density and the porosity of the formed spinel can be adjusted by additives, such as ZrO2 [25], TiO2 [26,27], LiF and CaCO3 [28], MgF2 [29] and ZnO [30], as sintering agents. For in situ spinel-containing refractories, the formation reaction of spinel from fine particles of alumina and magnesia is undoubtedly the key factor influencing the properties of the material, where variables, such as the reaction rate and morphology of the spinel, can all significantly alter the performance of the resulting material. Therefore, a comprehensive understanding of the reaction mechanism of spinel formation from alumina and magnesia will certainly result in improved material design of the in situ spinel-containing refractories. The formation of spinel from alumina and magnesia is accompanied by expansion. C. Wagner was one of the first to quantify the expansion ratio, the results of which were far higher than the initial proposed expansion ratio of 8.1% calculated from the densities of MgO (3.58 g/ cm3), Al2O3 (3.99 g/cm3) and MgAl2O4 (3.58 g/cm3) [31]. In light of this finding, a charge neutrality model was subsequently proposed to explain the reaction mechanism between magnesia and alumina, which importantly noted that 3 mol of Mg2+ ions must be transferred from the magnesia side to the alumina side to neutralize the 2 mol of Al3+ ions

Corresponding author. Corresponding author. School of Iron and Steel, Soochow University, Suzhou, 215021, China. E-mail addresses: [email protected] (L. Hong), [email protected] (D. Hou).

∗∗

https://doi.org/10.1016/j.ceramint.2019.09.278 Received 27 February 2017; Received in revised form 27 September 2019; Accepted 28 September 2019 0272-8842/ © 2019 Elsevier Ltd and Techna Group S.r.l. All rights reserved.

Please cite this article as: Lan Hong, Weipeng Chen and Dong Hou, Ceramics International, https://doi.org/10.1016/j.ceramint.2019.09.278

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moving from the alumina side to the magnesia side. Taking charge neutrality into consideration, on the alumina side, 3 molar equivalents of spinel form, but only 1 mol equivalent of spinel forms on the magnesia side, resulting in the creation of pores inside magnesia particles. From this improved model, the calculated value of the volumetric expansion ratio was 16.7%, which is an increase over the traditional model based only on densities. However, this calculated value remains lower than experimental data, signifying the presence of additional factors influencing spinel formation and expansion. To further increase the accuracy of our theoretical understanding of spinel formation, Nakagawa introduced a new parameter, R, which is the ratio of spinel thickness formed at the alumina side to the spinel thickness formed at the magnesia side [32], where Wagner's neutrality charge model is reproduced when this R takes a value of 3. On the other hand, for the extreme instance when the diffusion of one type of ions is much faster than the diffusion of the other types of ions so that the Kirkendall effect may be apparent, the value of R can increase to infinitely large levels, resulting in the volumetric expansion ratio reaching up to 56%. The formation of Kirkendall pores in Al2O3–MgO refractory castables has been confirmed by E.Y. Sako et al. [33], supporting the validity of this model. However, as the pores generated during the reaction may have some impact on the reaction mechanism, more experimental results are required for further confirmation. Experimental work studying the reaction mechanism of spinel formation from magnesia and alumina has been carried out mainly by measuring the thickness of the spinel layer generated between the closely contacted magnesia and alumina particles, either single crystal or polycrystalline, under various temperatures and pressures [34–43], where the general conclusion obtained was that the solid-state reaction between magnesia and alumina is controlled by the inter-diffusion of Mg2+ and Al3+ ions. Despite the popularity of this experimental approach, these methodologies importantly ignore the effects of the generated pores, which would affect the validity of expansion evaluation. Moreover, monitoring the expansion ratio instead of the thickness of spinel would allow the generated pores within the reaction system to be taken into consideration. Furthermore, this methodology would be coherent with the notion that the expansion properties of a refractory contribute to a more fundamental characteristic of the material. Certainly, in the context of spinel, its formation is accompanied by dynamic volume expansion, and consequently, its study may provide key detailed insights into reaction kinetics in conjunction with a new perspective of theoretical development. The expansion ratio of in situ spinel-containing refractories has been studied in detail [44] in the presence of other raw materials, such as cement and silica, which interfere in the reaction between alumina and magnesia. Therefore, difficulties arise when using these data to clarify the kinetics of spinel formation. The current study utilizes powder compaction of fused magnesia and reactive alumina as samples for a direct measurement of the expansion ratio resulting from the formation of spinel. The expansion ratio results fit well in a kinetic model employing electrochemical potential as the driving force of the reaction.

2.1. Experimental To apply the current research results to refractory formulation, commercially utilized calcined alumina and fused magnesia powders of high purity were employed as the raw materials for quick reaction rates. Calcined alumina has a purity of 99%, specific gravity of 3.89 g/cm3 and d50 of 1.55 μm, with the crystalline phase identified as corundum. However, fused magnesia has a purity of 98%, specific gravity of 3.58 g/cm3 and d50 of 80 μm, with the crystalline phase identified as periclase. Large magnesia particles were selected to mitigate the hydration of magnesia particles as strategized in industry, and small alumina particles were used for quick reaction rate. Magnesia and alumina powders were mixed homogeneously at the stoichiometric ratio (molar ratio of magnesia to alumina of 1:1) for spinel formation. To eliminate the influence of weight of the materials on the expansion, a very small amount (~0.7 g) of the mixture was pressed under various pressures of 125 MPa, 375 MPa and 750 MPa using a die of 10 mm in diameter to obtain pellets of various porosities. The initial weight, thickness and diameter of each pellet were measured before the pellet was set in the cold zone of an electric furnace in air. Then, the temperature was increased to a preset value ranging between 1573 K and 1773 K before the samples were pushed to the hot zone of the furnace and kept for a certain time interval varying from 0 to 25 h. After the experiments, the samples were immediately returned to the cold zone again. The heated pellets were measured for weight, thickness and diameter. The phase composition and microstructure were analyzed using X-ray diffraction (XRD) and scanning electron microscopy-energy dispersive spectrometry (SEM-EDS). 2.2. Treatment of data The apparent linear expansion ratio resulting from the formation of spinel through the solid-state reaction between magnesia and alumina at high temperatures, h (%), was calculated using the sample thickness before and after the experiments, as shown below. h

=

hfinal

hinitial

hinitial

× 100%

(1)

where hinitial (cm) and hfinal (cm) represent the sample thickness before and after experiments, respectively. The results of the apparent expansion ratio are graphed against time in Fig. 1, comparing samples of various pressing pressures consistently heated at 1573 K in Fig. 1a and comparing samples heated at different temperatures consistently prepared with a pressing pressure of 125 MPa in Fig. 1b. The apparent linear expansion ratio increases with holding time at 1573 K. Such an increase in the expansion ratio can primarily be attributed to more spinel being generated when the holding time is increased, which can be evidenced from the XRD results shown in Fig. 2 where the typical peaks corresponding to spinel intensify, whereas those peaks corresponding to alumina and magnesia weaken, with holding time. As the apparent linear expansion ratio of the powder compacts is directly correlated to the total thickness of the spinel layers generated, thus it is rational to use the apparent linear expansion ratio as a means of monitoring the reaction progress. The apparent linear expansion ratio increases with increasing pressing pressure (Fig. 1a), which can be attributed to the lower initial height of samples prepared under higher pressure, since the amount of spinel generated is independent of the pressing pressure as shown in Fig. 3, where the intensity of spinel, alumina and magnesia peaks is not influenced by the pressing pressure. For the same holding time and pressing pressure, the apparent linear expansion ratio also increases with temperature up to 1723 K, at which point the apparent linear expansion ratio decreases as the applied temperature further increases to 1773 K (Fig. 1b), although the sample after being heated at 1773 K for 5.5 h consists primarily of

2. Experimental design Most previous studies conducted experiments by placing large magnesia and alumina particles of single or poly-crystals [34–43] into contact for the convenient measurement of the thickness of the formed spinel layer. However, complications arise when utilizing this setup to monitor the expansion ratio, which is a key quality control parameter monitored in industry. Thus, to assess the expansion ratio of spinel formation in a more accurate and straightforward manner, the current research employs powder compacts of magnesia and alumina instead of large particles to evaluate the expansion ratio for mechanistic analysis. Another advantage of such a setup is short turnaround for each test. This kind of experimental setup has been successfully employed in investigating spinel formation from alumina and magnesia [45,46]. 2

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Fig. 1. Changes of linear expansion ratio with holding time.

spinel, with almost no alumina and magnesia detected (Fig. 4, left). This phenomenon is attributed to the fine particulate nature of the raw materials being easily sintered with each other at high temperatures so that densification occurs. This densification can be evidenced through SEM analysis of the sample heated at 1773 K for 5.5 h, as shown in Fig. 4 (right), in which highly crystalline spinel particles clearly displaying crystal orientation are very dense. To clarify the dependence of densification on temperature, compactions of fused magnesia powders were heated for 5.5 h at 1673 K and 1723 K, respectively, after being pressed at 375 MPa. The shrinkage is less than 0.1% at 1673 K whereas the shrinkage is approximately 5.8%

at 1723 K. Therefore, only the results obtained below 1673 K are used for kinetic analysis to eliminate the influence of densification on the expansion ratio. For the solid-state reaction between alumina and magnesia, it has been demonstrated by previous work that the kinetics of spinel formation are dominated by the diffusion of ions through the generated spinel layer(s), and therefore, the thickness of the spinel layer increases proportionally with the square root of time [34–43]. Fig. 5 shows the graphs of h2 against holding time (t), where a very good linear relation between h2 and t can be observed for various pressing pressures (Fig. 5a) and temperatures below 1673 K (Fig. 5b). This relation strongly

Fig. 2. XRD patterns of samples heated at 1573 K for various times. 3

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At the MgO(1 =

3 1

x

x)

Al2 O3(1 + x ) Al2 O3 interface:

MgO(1

x)

4 + 2x Al2 O3 + 3Mg2 + 1 x

Al2 O3(1 + x ) + 2Al3 +

(3)

The overall chemical reaction between MgO and Al2O3 is formulated as:

4 + 2x 3 Al2 O3 + 4MgO = MgO(1 1 x 1 x

+ MgO Al2 O3

x)

(4)

There is a concentration profile for both Mg2+ and Al3+ ions marked in the graph with an assumption of linearity for change of concentrations. Significant uncertainty should not arise from the assumption of linearity based on results from previous work [42,43]. Then, the concentration of Mg2+ and Al3+ ions at location l can be expressed as

CMg 2 + =

x+

x l l0

m V MgO ·Al2 O3

2(1 + x )

CAl3 + = Fig. 3. Influence of pressing pressure on spinel generation.

1

m VMgO ·Al2 O3

(5) 2x l l0

(6)

where is the molar volume of spinel, set to a constant value against x in the present instance, since the temperature range was between 1573 K and 1673 K corresponding to the x value distribution within a narrow range from 0.07 to 0.14 [42]. At the interface of 2 1 MgO–MgO·Al2O3, we have CMg 2 + = V m and CAl3 + = V m , m V MgO ·Al2 O3

suggests that similar to the previous studies, the diffusion of ions through formed spinel layer(s) controls spinel formation in the powder compacts as well.

whereas at 1 CMg 2 + = V m

2.3. Flux equations of the ions

MgO·Al2 O3

x

the MgO(1-x)·Al2O3(1+x)-Al2O3 2(1 + x ) and CAl3 + = V m .

interface,

MgO·Al2 O3

we

have

MgO·Al2 O3

MgO·Al2 O3

According to the previous studies, the diffusion of oxygen within the spinel layers is negligible to the overall reaction kinetics [38]. Therefore, only the diffusion of Mg2+ and Al3+ ions within the spinel layers is taken into consideration in the flux equations. Fig. 6 is a schematic diagram showing the Al2O3–MgO·Al2O3–MgO reaction system with a spinel layer thickness of l 0 . The value of l 0 increases with holding time and temperature, as observed above. At the interface between magnesia and spinel, stoichiometric spinel forms but at the interface between spinel and alumina, the nonstoichiometric phase MgO(1 x) ·Al2 O3(1+x) is generated due to the solubility of alumina in spinel. The value of x is a function of temperature as indicated by the MgO–Al2O3 phase diagram. The reaction at each interface is thus expressed as follows.

As shown in Fig. 6, Mg2+ ions diffuse from magnesia to alumina with a flux of JMg 2 +, and concurrently, Al3+ ions transfer from alumina to magnesia with a flux of JAl3 +. The driving force for the diffusion of ions is the electrochemical potential, which is expressed as

At the MgO–MgO·Al2O3 interface: 4MgO + 2Al3+ = MgO·Al2O3 + 3Mg2+ (2)

JMg 2 + =

i

=

i

(7)

+ zi F

where i is the electrochemical potential, i is the chemical potential, z i is the valence of the species i, F is the Faraday constant, and is the electrical potential. The fluxes of Mg2+ and Al3+ ions can be written as follows.

CMg 2 + DMg 2 + d RT

Mg 2 +

dl

+ zMg 2 + F

Fig. 4. XRD (left)and SEM (right) results for the sample heated at 1773 K for 5.5 h. 4

d dl

(8)

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Fig. 5. Relationship between

2 h and

holding time.

2CMg 2 + DMg 2 + d

Fd =

Mg 2 +

+ 3CAl3 + d

Al3 +

4CMg 2 + DMg 2 + + 9CAl3 + DAl3 +

Therefore, the flux of

Al3 +

ions can be obtained as below.

C

D C D Mg 2 + Mg 2 + Al3 + Al3 +

JAl3 + =

RT 4C

Mg 2 +

D

Mg 2 +

4

d

+ 9C 3 + D 3 + Al Al

C

C Ð Mg 2 + Al3 +

=

(14)

m RTVMgO ·Al2 O3 2CMg 2 + + 3CAl3 +

2

2

Al3 +

dl

6

d

d Al2 O3 dl

Mg 2 +

dl

6

d MgO dl

(15)

Inside the generated spinel layer, the following relationship among the chemical potentials of magnesia, alumina and spinel can be validated according to the overall reaction (4). Fig. 6. Schematic diagram of Al2O3–MgO·Al2O3–MgO reaction system.

JAl3 + =

CAl3 + DAl3 + d Al3 + d + z Al3 + F RT dl dl

4

Ð = RT =

NMg 2 +

d

NAl3 +

4d

Al3 +

dl

m NAl3 + = V MgO ·Al2 O3 CAl3 +

JAl3 + = NMg 2 + JAl3 +)

=

3 1

x

MgO(1 x ) Al2 O3(1 + x )

+

(16)

MgOAl2 O3

=

4 + 2x d 1 x

(17)

Al2 O3

CMg 2 + CAl3 + Ð 8 + x d Al2 O3 m 21 2 + 3 + RTVMgO (2 C + 3 C ) x dl ·Al2 O3 Mg Al

(18)

Ð 8 + x [(1 + x ) l0 xl][(1 x ) l 0 + xl] d Al2 O3 m RTVMgO x 8[(2 + x ) l 0 xl)]2 dl ·Al2 O3 1 (19)

(10)

Separating the variables and integrating the above equation from the interface between Al2O3 and MgO(1-x)·Al2O3(1+x) to the interface between MgO and MgO·Al2O3 across the spinel layer that has been generated results in the following.

(11)

l0

JAl3 +

(12)

0

8[(2 + x ) l 0 xl)]2 1 x dl = 8 + x [(1 + x ) l 0 xl][(1 x ) l 0 + xl] MgO MgO·Al2 O3 Al2 O3

When charge neutrality is applied to the system, the following equation also becomes valid.

2JMg 2 + + 3JAl3 + = 0

Al2 O3

Introducing the expressions for CMg 2 + and CAl3 + , the above equation becomes

NAl3 + JMg 2 +)

where NMg 2 + and NAl3 + are the molar fractions of Mg2+ and Al3+ ions. The following relationships are self-explanatory. m NMg 2 + = VMgO ·Al2 O3 CMg 2 +

MgO

JAl3 + =

dl d

4 + 2x 1 x

leading to the following Eq. (18).

Mg 2 +

m V MgO ·Al2 O3 (NAl3 + JMg 2 +

+

At a certain temperature, the right side of Eq. (16) is constant, in which instance

(9)

where CMg 2 + and CAl3 + are molar concentrations of Mg2+, Al3+ ions; DMg 2 + and DAl3 + are the diffusion coefficients of Mg2+ and Al3+ ions, respectively; l is the diffusion distance and T is the temperature. J.P. Stark [47] demonstrated the invariance between the movement of one component with respect to the other in a binary diffusion system, based on which a parameter called the inter-diffusion coefficient, Ð, was introduced and expressed by the following equation. m V MgO ·Al2 O3 (NMg 2 + JAl3 +

MgO

Al2 O3 MgO(1 x )·Al2 O3(1 + x ) Al2 O3

(13)

Ð d m RTVMgO ·Al2 O3

Al2 O3

(20) At the interface of Al2O3 and MgO(1-x)·Al2O3(1+x), the activity of Al O MgO(1 x )·Al2 O3(1 + x ) = 1; the activities of both alumina is unity, aAl22O33

Then, the term Fd can be solved by substituting the expressions of fluxes into the above equation. 5

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magnesia

and

spinel

are

also

unity

MgO aMgO

MgO·Al2 O3

=1

results in the formation of

and

= 1 at the interface of MgO and MgO·Al2O3, where alumina is in equilibrium with magnesia according to the following reaction 48

MgO MgO·Al2 O3 aMgO ·Al2 O3

0 MgO + Al 2O3 = MgO Al2 O3 GMgO Al2 O3 =

35530

JMgO·Al2 O3

Therefore,

K=

MgO MgO·Al2 O3 aMgO ·Al2 O3 MgO MgO·Al2 O3 MgO MgO·Al2 O3 aMgO aAl2 O3

RT

l 0 dl 0 =

0 GMgO ·Al2 O3

= exp

RT

=

i

0 Al2 O3

l02 =

JAl3 + 0

0 + GMgO ·Al2 O3

k =

0 Al2 O3

Ð d m RTVMgO ·Al2 O3

Al2 O3

9 ln(1 2x

x)

1 =

Ð 0 GMgO ·Al2 O3 m RTV MgO ·Al2 O3

h

0 GMgO ·Al2 O3

8+x m RTV MgO x ) ln(1 + x ) ·Al2 O3 4(1

x 9 ln(1

x)

2x

2 h

(28)

Table 1

x (8 + x )(4 (1

x )2 [2 ln(1 + x )

0 GMgO ·Al2 O3 , RT

x) 9 ln(1

- [48]

x )]

,-

RT

4(1

x (4 x )(8 + x ) x ) 2 [ln(1 + x ) 9 ln(1

x)

2x ]

t = kt

Ð

0 GMgO ·Al2 O3

RT

4(1

x (8 + x )(4 x ) x )2 [ln(1 + x ) 9 ln(1

x)

(34)

2x ]

(35)

= l0

=

2l 2 0

=

2Ð

0 GMgO ·Al2 O3

RT

(1

x (8 + x )(4 x ) 2 [2 ln(1 + x )

x) 9 ln(1

x )]

t=

x) 9 ln(1

x )]

and

0 GMgO ·Al2 O3 RT

2

(36)

The values of x and MgO·Al2 O3 at various temperatures are listed in RT the following Table 1 according to Refs. [42,48]. The inter-diffusion coefficient of Al3+ and Mg2+ ions in the formed spinel layer, Ð, obeys the Arrhenius law.

(29)

where JMgO·Al2 O3 is the flux of the spinel. From the reaction equation between magnesia and alumina, we know that the flux of 2 mol of Al3+

x (8 + x )(4

0 GMgO ·Al2 O3

G0

dl 0 m = JMgO·Al2 O3 V MgO ·Al2 O3 dt

x )2 [2 ln(1 + x )

Ð

Here, k is the reaction rate constant in terms of expansion ratio and is dependent on the pressing pressure and the three factors influencing k . The value of k can be obtained by taking the slope of h2 against t from Fig. 5.

The following equation represents the growth rate for spinel formation from magnesia and alumina.

x value, - [42]

dt

k t = kt

2.4. Kinetics of spinel formation

Temperature, K

2x ]

where is a constant influenced only by the pressing pressure. Then, h and t should exhibit a relationship as given below by re-arranging Eq. (33) and Eq. (35).

Hence,

(1

x)

The changes in the apparent linear expansion ratio, h , are primarily if not solely dependent on the solid-state reaction between magnesia and alumina and are proportional to the thickness of the spinel layer, l 0 .

(27)

Calculated value of

8(1

3. Discussion (26)

Ð

RT

The value of k is the slope of the linear relationship between and t , and it is dependent on three factors, which are the diffusion of the reactant ions, the chemical reaction between magnesia and alumina, and the dissolution of alumina into spinel.

Integrating both sides of the above equation, assuming div JAl3 + is 0, gives

JAl3 + l 0 =

0 GMgO ·Al2 O3

l02

8[(2 + x ) l 0 xl)]2 1 x dl = 8 + x [(1 + x ) l 0 xl][(1 x ) l 0 + xl]

8(1 x ) 1 l0 ln(1 + x ) 8+x 2x

Ð

Hence, the reaction rate constant in terms of the thickness of the spinel layer, k , is thus obtained as

(25)

0 0 Al2 O3 + GMgO·Al2 O3

JAl3 +

x (4 x )(8 + x ) x )2 [ln(1 + x ) 9 ln(1

(31)

(33)

Then, the above integration becomes l0

2x ]

2.5. Integrating the above equation gives

Then, the chemical potential of Al2O3 at the interface between MgO MgO·Al2 O3 , can be calculated as Al2O3 and MgO(1-x)·Al2O3(1+x), Al 2 O3 below.

=

x)

(32)

(24)

+ RTlnai

MgO MgO·Al2 O3 Al2 O3

x (4 x )(8 + x ) x )2 [ln(1 + x ) 9 ln(1

(23)

where K is the equilibrium constant of the spinel formation reaction 0 from MgO and Al2O3, and GMgO ·Al2 O3 is the standard free energy change of formation of spinel MgO·Al2O3. Since the potential of species i , by definition, can be written as 0 i

(30)

Separating the variables,

(22)

Hence, MgO MgO·Al2 O3 aAl 2 O3

moles of spinel. Therefore,

x J 3+ x ) Al

0 Ð GMgO·Al2 O3 RT 8(1 l0

dl 0 = dt

0 GMgO ·Al2 O3

= exp

x x

By replacing JMgO·Al2 O3 with its relationship with JAl3 +, the following is obtained.

(21)

2.097J/mol

4 = 2(1

4 1

at various temperatures.

1573

1598

1623

1648

1673

0.07 3.26

0.08 3.31

0.09 3.37

0.12 3.55

0.14 3.67

−2.96818

−2.92567

−2.88448

−2.84454

−2.80579

6

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present study. This small discrepancy can be attributed to the raw materials used in this study being of commercial grade, and the presence of minor impurities would thus influence the diffusion of Mg2+ ions. This influence has been pointed out previously by H.-P. Liermann et al. [52]; the activation energy for Mg2+ ion diffusion decreased to 202±8 kJ/mol in the presence of iron. When the ion mobility of one component noticeably exceeds the ion mobility of the other, the Kirkendall effect may be instigated within the reaction system, where numerous (Kirkendall) pores are generated at the sites occupied originally by the ions of better mobility. Such an occurrence is observed in Fig. 8, which shows the phase compositions and morphology of the sample soaked at 1573 K for 3.5 h. The XRD results clearly show that some spinel has formed while a large amount of unreacted Al2O3 and MgO remain in this sample. In the SEM image of this sample, in addition to the spinel formed on the surface of alumina particles, magnesia particles display high porosity in comparison to the very smooth surface of fused magnesia particles before reaction, clearly demonstrating the formation of Kirkendall pores originating from a much faster transfer of Mg2+ ions than Al3+ ions or the Kirkendall effect.

Fig. 7. Activation energy of Al3+ and Mg2+ ion inter-diffusion through spinel layer.

Ð = Ð0 exp

E RT

(37)

3.1. Validation of current kinetic model of the spinel formation process

where Ð0 is the pre-exponential constant, and E is the inter-diffusion activation energy. Since 2 is correlated only to the pressing pressures, for the same pressing pressure, the slope of lnÐ against 1 is equivalent to the slope

H.M. Rietveld [53] suggested a structure refinement method for XRD profiles obtained from step-scanning measurement of powder samples, by which the area and FMHM of peaks can be calculated. The peak area of a crystal is dependent on its fraction in the sample, and the FMHM value corresponds to its crystalline size. Fig. 2 indicates that the intensity of the spinel peaks increases, but the intensity of both magnesia and alumina peaks decreases, with increasing holding time. The FMHM of peak (113) of the spinel phase corresponds to its crystalline size. Similarly, the most intense peaks of magnesia (200) and alumina (104) are selected as their typical peaks, respectively. The change in the intensity of these three peaks with holding time is graphed in Fig. 9. All the peaks in Fig. 9 shift slightly to right with increasing holding time, implying that the index spacing or lattice constant of the crystals decreases with the holding time (Fig. 10a). However, the crystalline size, calculated from FMHM values, of both MgO and Al2O3 after heating is larger than the crystalline size of their respective raw material but shows a very limited increase with holding time (Fig. 10b). The fractions of MgAl2O4, MgO and Al2O3 in the samples calculated from the areas covered by their respective typical peaks are graphed in

T

of lnÐ + ln 2 against 1 , from which the activation energy E of interT diffusion can be calculated as shown in Fig. 7, where,

lnÐ + ln

2

= lnk

ln

0 GMgO ·Al2 O3 RT

ln

x (8 + x )(4 (1

x )2 [2 ln(1 + x ) 3+

x) 9 ln(1

x )]

.

2+

The average value of E for inter-diffusion of Al and Mg ions is 310.6 kJ/mol, which is independent of the pressing pressure. According to the simulation results using atomic scale computer simulations at temperatures ranging from 500 K to 1500 K, Mg2+ ions are more mobile than Al3+ ions in magnesium aluminate spinel [49]. Similar results were obtained by testing the mobility of cations in magnesium aluminate spinel in an electric field at 1273 K [50]. The conclusions from both studies suggest that the kinetics of the interdiffusion of Al3+ and Mg2+ ions should be analogous to the kinetics of the inter-diffusion of Mg2+ ion diffusion. The activation energy of Mg2+ self-diffusion in spinel was determined to be 343±25 kJ/mol [51], which is slightly larger than the result obtained for spinel in the

Fig. 8. XRD (left) and SEM (right) results for the sample soaked at 1573 K for 3.5 h. 7

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Fig. 9. Intensity change of typical XRD peaks with holding time for MgAl2O4 (113), MgO (200) and Al2O3 (104).

Fig. 10. Changes of lattice constant (a) and crystalline size (b) with holding time.

from the XRD results is compared to the change in the MgAl2O4 fraction with holding time predicted by the kinetic model established in this research (Fig. 12). Very good consistency can be found, showing the validation of the current kinetic model in elucidating the formation process of spinel from magnesia and alumina. 4. Conclusions The kinetics of spinel formation from reactive alumina and fused magnesia powder compactions were investigated by monitoring the expansion ratio in place of the thickness of the spinel layer that was generated. Homogeneous samples with the stoichiometric mixing ratio (MgO:Al2O3 in moles) of 1:1 were pressed under pressures of 125, 375 and 750 MPa. Experiments were performed in the temperature range from 1573 K to 1773 K. The expansion ratio increased with increasing pressing pressure, soaking time and temperature between 1573 and 1673 K, whereas above 1673 K, sintering occurred among the fine particles to cause shrinkage. A kinetic model was established by taking the electrochemical potential as the reaction driving force. The reaction kinetics are influenced by the thermodynamics of spinel formation, dissolution of Al2O3 and inter-diffusion of Al3+ and Mg2+ ions in spinel. The apparent activation energy of the inter-diffusion of Al3+ and Mg2+ in spinel is 310.6 kJ/mol, which is very close to the apparent activation energy of the inter-diffusion of Mg2+ ion self-diffusion. Using the proposed model, data derived experimentally are reproduced with good parity. The validation of the current kinetic model is also confirmed by XRD results.

Fig. 11. Fraction change of components MgAl2O4, MgO and Al2O3 in samples with holding time.

Fig. 11. MgO is always in excess relative to Al2O3 after reaction due to the solubility of Al2O3 in spinel when the initial molar mixing ratio of MgO and Al2O3 is unity. In the beginning, the fraction of spinel increases, and the fractions of MgO and Al2O3 decrease sharply, with increasing holding time. The fraction change of all three compounds slows when time is further increased, which is in good agreement with the results assessed by using a linear expansion ratio. The change in the MgAl2O4 fraction with holding time obtained

Declaration of competing interest The authors declare no competing financial interest. 8

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Fig. 12. Consistency of calculated spinel fraction from XRD results with the kinetic model.

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