**1.5 Sintering kinetics of MgB2 superconductors at low temperature**

Analysis of the kinetics of the sintering process can be performed based on the DTA data using different computational methods. Yan *et al.* [18] calculated the activation energy of the solid-solid reaction at low sintering temperature using the Ozawa–Flynn–Wall and Kissinger as 58.2 and 72.8 kJ⋅mol-1, respectively. The value of the pre-exponential factor calculated using the Kissinger method is 2.0 × 1015 s−1. They also report that the activation energy increases parabolic as the reaction progresses [18]. However, the study by Shi *et al.* [19] draws different conclusions to those of Yan et al. These authors use a new kinetic analysis (based on a variant on the Flynn-Wall-Ozawa method) under non-isothermal conditions and suggest that the solid-solid reaction between Mg and B powders follows an instantaneous nuclei growth (Avrami–Erofeev equation, *n* = 2) mechanism. The values of *E*  decrease from 175.4 to 160.4 kJ·mol-1 with the increase of the conversion degrees (α) from 0.1 to 0.8 in this model. However, the activation energy (*E*) increases to 222.7 kJ·mol-1 [19] again as the conversion degree reaches 0.9.

On the other hand, the solid-solid reaction between Mg and B exothermal peak is partly overlapped with the Mg melting endothermic peak in the DTA curves. This phenomenon makes it difficult to calculate the kinetics parameters exactly from the thermal analysis data and also could be the reason why the previous results are different from different research groups. It is necessary to further investigate the phase formation mechanism of MgB2 during the low-temperature sintering combined with advanced test methods.

In our recent work [38], in-situ X-ray diffraction technique is used to measure the degree of reaction between Mg and B as a function of time at several certain temperatures below Mg melting, respectively. Based on these isothermal data, the kinetics analysis of MgB2 phase formation during the low-temperature sintering is carried out.

Bulk samples of MgB2 were prepared by a solid-state sintering method using amorphous boron powder (99% purity, 25μm in size), magnesium powder (99.5% purity, 100 μm in size). Several reaction temperatures in the range of 550~600 oC, below the melting point of Mg, were chosen as the isothermal holding temperatures. Then the samples were fast-heated to the chosen temperature with a rate of 50 oC/min in order to prevent significant reaction between Mg and B before arriving at the isothermal annealing temperature. The x-ray diffraction measurement started as soon as the sample temperature reached the certain isothermal temperature and it will detect the sample every 15 min till the reaction is over. The weight fraction of synthesized MgB2 which corresponds to the degree of reaction was calculated from the XRD data of sample obtained after different soaking time according to the External Standard Method.

Fig. 6 illustrates the typical X-ray diffraction patterns of the Mg-B sample isothermally annealed at 575 oC for different periods. One can see that no organized MgB2 peak can be observed when the temperature just reaches 575 oC. It implies that the reaction between Mg and B did not occur during the rapid heating to the final isothermal holding temperature. As the holding time prolonging, the MgB2 phase appears and increases gradually while the Mg phase decreases. However, the increase in the intensity of MgB2 peaks becomes very slow and even stops when it reaches a certain degree despite of longer holding time (ie. longer than 480 min). Similar behavior is also found in the isothermal annealing experiments at 550 oC and 600 oC (the XRD patterns in not shown here).

Sintering Process and Its Mechanism of MgB2 Superconductors 477

and 600 oC, respectively. At 550 oC and 575 oC, the reaction seems stop even though the degree of reaction is still below 100% and there is residual Mg and B in the sample (see Fig. 6). It means that at the final stage of isothermal heating in present work, the reaction rate is so slow that it is difficult to observe the increase in the degree of reaction. At each isothermal annealing temperature, it can be also found that the reaction rate becomes slower and slower with the

Based on these isothermal data, kinetics analysis of the MgB2 phase formation during the low-temperature sintering is carried out. Early kinetics studies employed the currently-

()() *<sup>d</sup> kT f dt*

Where *t* represents time, *α* is the degree of reaction, *T* is the temperature, k*(T)* is the temperature-dependent rate constant and *f(α)* is a function that represents the reaction

( ) exp( ) *<sup>E</sup> kT A*

Where *A* is the pre-exponential factor, *E* is the activation energy and *R* is the gas constant.

[ ]

 αα−

( ) ln( ) ln *<sup>E</sup> <sup>g</sup> <sup>t</sup> RT A* α

It needs different reaction time to reach certain degree of reaction at different isothermal holding temperatures. According to Eq. (4), at certain degree of reaction, one can plot the *lnt*~*1/T* and then do a linear fit. Following this way, the activation energy (*E*) can be obtained without referring to the reaction modes. Table 1 illustrates the reaction time and activation energy at different degree of reaction of the Mg-B sample. It is recognized that the activation energy (*E*) firstly decreases when *α* changes from 0.20 to 0.40 and then increases again when *α* changes from 0.50 to 0.80. It is indicated that the reaction between Mg and B at low temperature is not controlled by only one kinetics reaction model. At different stage of

In order to determine the kinetic modes of the reaction between Mg and B during the lowtemperature sintering, Model fitting method is performed. Following this method, The determination of the *g(α)* term is achieved by fitting various reaction models to experimental data. As described below in Eq. (5), the relationship between *g(α)* and *t* should be linear.

= +

α

α

1 <sup>0</sup> *g f* () () () *d kTt*

α

(1)

(4)

*RT* = − (2)

≡ = (3)

α=

reaction time increasing and the degree of reaction seems unchanged at last.

accepted kinetic equation:

model. *k(T)* can be described as:

Integrating Eq. (1), it comes:

Where *g(α)* is the integral form of *f(α).* 

reaction, the kinetics model is varied.

After substitution for *k(T)* and rearranging, it yields:

Fig. 6. Typical X-ray diffraction patterns of the Mg-B powder specimen isothermally annealed at 575 oC for different periods [38].

According to the XRD data, the weight fraction of the synthesized MgB2 phase, which is considered as the degree of reaction, is calculated using the External Standard Method. Plots of degree of reaction vs. time are given in Fig. 7 for isothermal sintering at 550 oC, 575 oC

Fig. 7. Plots of degree of reaction vs. time isothermally-annealed Mg-B powder specimens at 550 oC, 575 oC and 600 oC, respectively [38].

and 600 oC, respectively. At 550 oC and 575 oC, the reaction seems stop even though the degree of reaction is still below 100% and there is residual Mg and B in the sample (see Fig. 6). It means that at the final stage of isothermal heating in present work, the reaction rate is so slow that it is difficult to observe the increase in the degree of reaction. At each isothermal annealing temperature, it can be also found that the reaction rate becomes slower and slower with the reaction time increasing and the degree of reaction seems unchanged at last.

Based on these isothermal data, kinetics analysis of the MgB2 phase formation during the low-temperature sintering is carried out. Early kinetics studies employed the currentlyaccepted kinetic equation:

$$\frac{d\alpha}{dt} = k(T)f(\alpha) \tag{1}$$

Where *t* represents time, *α* is the degree of reaction, *T* is the temperature, k*(T)* is the temperature-dependent rate constant and *f(α)* is a function that represents the reaction model. *k(T)* can be described as:

$$k(T) = A \exp(-\frac{E}{RT})\tag{2}$$

Where *A* is the pre-exponential factor, *E* is the activation energy and *R* is the gas constant.

Integrating Eq. (1), it comes:

476 Sintering of Ceramics – New Emerging Techniques

Fig. 6. Typical X-ray diffraction patterns of the Mg-B powder specimen isothermally

According to the XRD data, the weight fraction of the synthesized MgB2 phase, which is considered as the degree of reaction, is calculated using the External Standard Method. Plots of degree of reaction vs. time are given in Fig. 7 for isothermal sintering at 550 oC, 575 oC

Fig. 7. Plots of degree of reaction vs. time isothermally-annealed Mg-B powder specimens at

annealed at 575 oC for different periods [38].

550 oC, 575 oC and 600 oC, respectively [38].

$$\mathcal{S}\_{\mathcal{S}}(\alpha) \equiv \int\_0^{\alpha} \left[ f(\alpha) \right]^{-1} d\alpha = k(T)t \tag{3}$$

Where *g(α)* is the integral form of *f(α).* 

After substitution for *k(T)* and rearranging, it yields:

$$\ln(t) = \frac{E}{RT} + \ln\left[\frac{\mathcal{g}(\alpha)}{A}\right] \tag{4}$$

It needs different reaction time to reach certain degree of reaction at different isothermal holding temperatures. According to Eq. (4), at certain degree of reaction, one can plot the *lnt*~*1/T* and then do a linear fit. Following this way, the activation energy (*E*) can be obtained without referring to the reaction modes. Table 1 illustrates the reaction time and activation energy at different degree of reaction of the Mg-B sample. It is recognized that the activation energy (*E*) firstly decreases when *α* changes from 0.20 to 0.40 and then increases again when *α* changes from 0.50 to 0.80. It is indicated that the reaction between Mg and B at low temperature is not controlled by only one kinetics reaction model. At different stage of reaction, the kinetics model is varied.

In order to determine the kinetic modes of the reaction between Mg and B during the lowtemperature sintering, Model fitting method is performed. Following this method, The determination of the *g(α)* term is achieved by fitting various reaction models to experimental data. As described below in Eq. (5), the relationship between *g(α)* and *t* should be linear.

Sintering Process and Its Mechanism of MgB2 Superconductors 479

linear fitting. At certain degree of reaction, if the *g(a)* is given, the value of *A* can be

calculated value of *k*(*T*) should be consistent with the linear fitted slope of plot *g*(*a*)*~t* if the

Following this method, we verify these two models discussed above at 575oC. At the initial

2.09×10-5, which is comparable to the corresponding value of slope given in Table 2. In the

much smaller than the corresponding value of the slope presented in Table 2. At the middle stage ((*α*=0.50) of reaction, the calculated value of *k*(*T*) from the contracting cylinder model is 2.18×10-5, which is still comparable to the corresponding value of slope. For the onedimensional diffusion model, the calculated *k*(*T*) is 2.64×10-5, which is smaller than the corresponding value given in Table 2. However, at the final stage (*α*=0.80), the calculated *k*(*T*) from the one-dimensional diffusion model is 3.62×10-5, which is more consistent with the corresponding value presented in Table 2 than that in the case of the contracting

Hence, one can say that the reaction between Mg and B during low-temperature sintering is firstly mainly controlled by the contacting sphere model, which is a kind of the phase boundary reaction mechanism. As the reaction prolongs, the one-dimensional diffusion

In our previous study [39], we have investigated the MgB2 phase formation process during the sintering based on the phase identification and microstructure observation. It is found that the sintering 'necks' between individual Mg and B particles occurs at the first stage of the sintering. Then the solution active regions form at the sintering necks. With the isothermal heating prolonging, a few Mg and B atoms in the solution active regions can be activated and react with each other. The activated atoms are limited at this initial stage and thus the reaction rate is slow and mainly determined by the phase boundary reaction mechanism. Meanwhile, an MgB2 phase layer is gradually formed at the sintering necks between Mg and B particles and the Mg atoms have to diffuse through the whole MgB2 phase layer to reach the reaction interface, , as shown in Fig. 8. As the reaction prolonging, the synthesized MgB2 layer becomes thicker and thicker, and it should be more and more difficult for Mg atoms to reach the reaction interface through diffusion. Finally, the reaction rate is controlled by the diffusion-limited mechanism. As a result, the reaction rate becomes slower and slower and it takes a very long time to be totally completed due to the slow diffusion rate of Mg. The corresponding activation energy is also decreased firstly and then increased again during the whole reaction process, as discussed above. It is also the reason

Based on above analysis and compared with previous studies [18, 19] in which they propose the reaction is controlled by single mechanism, the varied mechanisms are more valid and more consistent with the actual sintering process. The value of activation energy in present

model, one kind of diffusion-limited mechanism, gradually becomes dominant.

why the residual Mg is still present even after holding for 10 h at 575 oC.

α

*A* α

α

), the calculated *k*(*T*) is 1.19×10-5, which is

*RT* = − can also be determined at certain temperatures. The

), can be obtained from the

), the calculated *k*(*T*) is

According to Eq. (4), the intercept of plot *lnt ~1/T* ( ( ) ln( ) *<sup>g</sup>*

stage (*α*=0.20) of reaction, for contracting sphere model ( 1/3 1 (1 ) − −

calculated. Then ( ) exp( ) *<sup>E</sup> kT A*

given *g*(*a*) is the valid reaction model.

case of one-dimensional diffusion model ( <sup>2</sup>

cylinder model (the calculated value is 2.33×10-5).

$$\mathbf{g}(\alpha) = A \exp(-\frac{E}{RT})\mathbf{t} \tag{5}$$

Set of alternate reaction models is linearly-fitted to the experimental data obtained from the is-situ X-ray diffraction results at 575 oC and then obtained results are collected in Table 2. The coefficient *r* of the contracting cylinder, contracting sphere and one-dimensional diffusion models are the better ones. According the Eq. (5), the intercept *t* during the linear fitting should be zero. But the intercept of contracting cylinder model is 0.0821, which is too high compared to 0. And the corresponding value of *r* is also not as good as the case in the Contracting sphere model. Hence, the contracting cylinder model is ignored here. On the other hand, the coefficient *r* of the contracting sphere is better than that of the onedimensional diffusion model. But the intercept of one-dimensional diffusion model is more near to 0 than that of the contracting sphere model.


Table 1. The reaction time (*t*) and activation energy (*E*) at different degree of reaction (α) during the low-temperature sintering of the Mg-B powders [38].


Table 2. Linear fitting results of the experimental data obtained from the in-situ X-ray diffraction results at 575 oC by adopting alternate reaction models [38].

( ) exp( ) *<sup>E</sup> g A t*

Set of alternate reaction models is linearly-fitted to the experimental data obtained from the is-situ X-ray diffraction results at 575 oC and then obtained results are collected in Table 2. The coefficient *r* of the contracting cylinder, contracting sphere and one-dimensional diffusion models are the better ones. According the Eq. (5), the intercept *t* during the linear fitting should be zero. But the intercept of contracting cylinder model is 0.0821, which is too high compared to 0. And the corresponding value of *r* is also not as good as the case in the Contracting sphere model. Hence, the contracting cylinder model is ignored here. On the other hand, the coefficient *r* of the contracting sphere is better than that of the onedimensional diffusion model. But the intercept of one-dimensional diffusion model is more

> Reaction time (*t*) at 575 oC (s)

Table 1. The reaction time (*t*) and activation energy (*E*) at different degree of reaction (

α

α

α

α

α

α

α

α

Table 2. Linear fitting results of the experimental data obtained from the in-situ X-ray

α

α

α

α

diffraction results at 575 oC by adopting alternate reaction models [38].

Reaction model *G(α)* Slope (*b*) *r*-square Intercept (*a*)

during the low-temperature sintering of the Mg-B powders [38].

0.20 5200 1500 520 275.39 0.30 6700 3600 2250 130.58 0.40 11700 5700 4500 114.75 0.50 21000 8100 5100 169.71 0.60 36000 11700 6150 181.71 0.70 42300 14100 7800 202.63 0.80 46800 18900 10260 211.74

α

near to 0 than that of the contracting sphere model.

Reaction time (*t*) at 550 oC (s)

One-dimensional diffusion <sup>2</sup>

Two-dimensional diffusion 1/2 1/2 [1 (1 ) ] − −

Mampel (first order) − − ln(1 )

Avrami-Erofeev 1/2 [ ln(1 )] − −

Avrami-Erofeev 1/3 [ ln(1 )] − −

Avrami-Erofeev 1/4 [ ln(1 )] − −

Contracting cylinder 1/2 1 (1 ) − −

Contracting sphere 1/3 1 (1 ) − −

Power law 3/2

Power law 1/2

Power law 1/3

Power law 1/4

Degree of reaction (*α*) *RT*

= − (5)

Reaction time (*t*) at 600 oC (s)

3.49×10-5 0.9851 -0.0165

2.14×10-5 0.9568 0.3411

9.20×10-5 0.9720 -0.0060

4.53×10-5 0.9830 0.4432

3.06×10-5 0.9730 0.6088

2.32×10-5 0.9648 0.6977

2.54×10-5 0.9855 0.0821

2.05×10-5 0.9869 0.0417

3.39×10-5 0.9779 0.0711

2.09×10-5 0.9042 0.4934

1.57×10-5 0.8821 0.6276

1.25×10-5 0.8699 0.7061

Activation energy (*E*) (kJ/mol)

> α)

According to Eq. (4), the intercept of plot *lnt ~1/T* ( ( ) ln( ) *<sup>g</sup> A* α ), can be obtained from the linear fitting. At certain degree of reaction, if the *g(a)* is given, the value of *A* can be calculated. Then ( ) exp( ) *<sup>E</sup> kT A RT* = − can also be determined at certain temperatures. The calculated value of *k*(*T*) should be consistent with the linear fitted slope of plot *g*(*a*)*~t* if the given *g*(*a*) is the valid reaction model.

Following this method, we verify these two models discussed above at 575oC. At the initial stage (*α*=0.20) of reaction, for contracting sphere model ( 1/3 1 (1 ) − −α ), the calculated *k*(*T*) is 2.09×10-5, which is comparable to the corresponding value of slope given in Table 2. In the case of one-dimensional diffusion model ( <sup>2</sup> α ), the calculated *k*(*T*) is 1.19×10-5, which is much smaller than the corresponding value of the slope presented in Table 2. At the middle stage ((*α*=0.50) of reaction, the calculated value of *k*(*T*) from the contracting cylinder model is 2.18×10-5, which is still comparable to the corresponding value of slope. For the onedimensional diffusion model, the calculated *k*(*T*) is 2.64×10-5, which is smaller than the corresponding value given in Table 2. However, at the final stage (*α*=0.80), the calculated *k*(*T*) from the one-dimensional diffusion model is 3.62×10-5, which is more consistent with the corresponding value presented in Table 2 than that in the case of the contracting cylinder model (the calculated value is 2.33×10-5).

Hence, one can say that the reaction between Mg and B during low-temperature sintering is firstly mainly controlled by the contacting sphere model, which is a kind of the phase boundary reaction mechanism. As the reaction prolongs, the one-dimensional diffusion model, one kind of diffusion-limited mechanism, gradually becomes dominant.

In our previous study [39], we have investigated the MgB2 phase formation process during the sintering based on the phase identification and microstructure observation. It is found that the sintering 'necks' between individual Mg and B particles occurs at the first stage of the sintering. Then the solution active regions form at the sintering necks. With the isothermal heating prolonging, a few Mg and B atoms in the solution active regions can be activated and react with each other. The activated atoms are limited at this initial stage and thus the reaction rate is slow and mainly determined by the phase boundary reaction mechanism. Meanwhile, an MgB2 phase layer is gradually formed at the sintering necks between Mg and B particles and the Mg atoms have to diffuse through the whole MgB2 phase layer to reach the reaction interface, , as shown in Fig. 8. As the reaction prolonging, the synthesized MgB2 layer becomes thicker and thicker, and it should be more and more difficult for Mg atoms to reach the reaction interface through diffusion. Finally, the reaction rate is controlled by the diffusion-limited mechanism. As a result, the reaction rate becomes slower and slower and it takes a very long time to be totally completed due to the slow diffusion rate of Mg. The corresponding activation energy is also decreased firstly and then increased again during the whole reaction process, as discussed above. It is also the reason why the residual Mg is still present even after holding for 10 h at 575 oC.

Based on above analysis and compared with previous studies [18, 19] in which they propose the reaction is controlled by single mechanism, the varied mechanisms are more valid and more consistent with the actual sintering process. The value of activation energy in present

Sintering Process and Its Mechanism of MgB2 Superconductors 481

Shimoyama *et al.* [40] found that a small amount of silver addition decreases dramatically the reaction temperature of magnesium and boron in the formation of bulk MgB2 without degradation of either the critical temperature or the critical current density. Although the added silver forms an Ag-Mg alloy after the heat treatment, these impurity particles exist mainly at the edge of voids in the sample microstructure and therefore do not provide a significant additional restriction to the effective current path. Accordingly MgB2 bulks with excellent *J*c properties have been fabricated at a temperature as low as 550 °C with the 3 at.% Ag doping. The sintering time of doped samples is also reduced significantly compared to that required for undoped samples fabricated by low temperature sintering. This effectively widens the processing window for the development of practical, low-cost MgB2

Hishinuma *et al.* [41] synthesized Mg2Cu-doped MgB2 wires with improved *J*c by sintering at low temperature for 10 h. They found that the formation of the MgB2 phase is improved due directly to the lower melting point of Mg2Cu (568 °C) than Mg (650 °C), which can promote the diffusion of Mg in the partial liquid (the MgB2 phase forms by the diffusion reaction between released Mg from Mg2Cu and amorphous B powder [41]). The *J*c of sample prepared in this way can be improved further in Mg2Cu-doped MgB2 wires by sintering at lower temperature (450 °C) for longer time (more than 100 h). The maximum core *J*<sup>c</sup> value of these samples was found to be over 100, 000 A cm-2 at 4.2 K in a magnetic field of 5 T for a tape sintered for 200 h. Bulk MgB2 has been fabricated successfully in other studies by Cudoping and sintering at 575 oC for only 5 h [42]. Thermal analysis indicates that the Mg-Cu liquid forms through the eutectic reaction between Mg and Cu at about 485 °C, which leads to the accelerated formation of MgB2 phase at low temperature. The SEM images of the sintered Cu-doped samples are shown in Fig. 9. It is observed that the undoped sample is porous and consists of small irregular MgB2 particles and large regular Mg particles which are in poor connection with each other (see Fig. 9a). On the other hand, the MgB2 particles of the doped sample become larger and more regular accompanying with the increasing amount of Cu addition. The doped samples also become denser with the amount of Cu addition increasing for the reason that the MgB2 particles are in better connection with each other and give birth to less voids (see Figs. 9b-9d). Especially, as shown in the Fig. 9d, the MgB2 grains in the (Mg1.1B2)0.9Cu0.1 sample exhibit platy structure with a typical hexagonal shape [42]. The high *J*c in MgB2 samples doped with Cu is attributed mainly to the grain boundary pinning mechanism that results from the formation of small MgB2 grains during low temperature sintering. As with the Ag-doped samples, the concentration of Mg-Cu alloy in these samples tends to form at the edge of voids in the microstructure and does not degrade significantly the connectivity between MgB2 grains, which contributes directly to enhanced *J*c. Recently, the addition of Sn to the precursor powder has also been observed to assist the formation of the MgB2 phase during low temperature sintering, and bulk Sn-

superconductors by reaction at low temperature [40].

doped MgB2 prepared at 600 ºC for 5 h exhibit good values of *J*c [43].

Interestingly, although Ag and Sn addition can form a local eutectic liquid with the Mg precursor at lower temperature than the addition of Cu, the Cu has been found to play a more effective role in the improvement of MgB2 phase formation than Ag and Sn at low temperature. The Cu-doped samples take significantly less time to form the primary MgB2 phase than those containing similar concentrations of Ag or Sn at a similar sintering

work is also comparable to the calculated value in Shi et al.'s study [19] whereas the models are different from theirs. But in their study, the activation energy is calculated using a model-free method, just as in our work.

Fig. 8. Schematic illustration of solid-solid reaction between Mg and B particles based on the inter-diffusion mechanism [39].

Based on above discussion, it is concluded that the reaction between Mg and B during the low-temperature sintering is controlled by varied mechanisms. At initial stage, the reaction rate is mainly determined by the phase boundary reaction mechanism. As the reaction prolonging and the synthesized MgB2 layer increasing, the diffusion-limited mechanism gradually becomes dominant. The corresponding activation energy is also decreased firstly and then increased again.
