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Electromagnetic behaviour of bulk MgB2 determined by numerical modelling using regional supercurrent properties
Journal of Alloys and Compounds 693 (2017) 1109e1115
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Electromagnetic behaviour of bulk MgB2 determined by numerical modelling using regional supercurrent properties K. Ozturk a, b, *, C.E.J. Dancer b a
Department of Physics, Faculty of Science, Karadeniz Technical University, 61080, Trabzon, Turkey International Institute for Nanocomposites Manufacturing (IINM), Warwick Manufacturing Group, University of Warwick, Coventry, CV4 7AL, United Kingdom
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 April 2016 Received in revised form 20 September 2016 Accepted 22 September 2016 Available online 23 September 2016
Previous studies in the literature have shown experimentally that the critical current density in bulk MgB2 depends on the position at which it is measured within the sample, and that the trapped magnetic field saturates above large values of sample diameter. To better understand this behaviour, we have carried out a detailed study of the trapped magnetic field and local critical current distribution in bulk MgB2 using numerical modelling with H-formulation. The properties of bulk MgB2 samples with radiusindependent (i.e. uniform) and radius-dependent critical current were modelled, based on experimentally-determined parameters obtained from the literature. Radius-independent samples took properties originally derived from either the central or edge region of the original samples. In our study it was determined that the peak trapped magnetic field value of the sample with the central region properties was higher by 22.4% and 9.20% than the samples with edge region properties and that with radius dependent parameter, respectively. Additionally, to understand why the trapped magnetic field saturates as the diameter of sample increases, the trapped magnetic field was modelled using the radius dependent critical current density Jc (B,r) as input data rather than using a constant bulk critical current density. The peak trapped magnetic field values increase by 8.6% and 6.2% respectively for the central region radius-independent and radius-dependent property bulk MgB2 samples as the bulk diameter increases from 25 mm to 35 mm. This tendency to saturation in the peak trapped field indicates that increasing the bulk diameter alone does not have a significant effect on the value of the trapped magnetic field, unless the bulk superconducting current density is improved uniformly throughout the MgB2 bulk. These results enable us to understand how we can experimentally enhance the trapped magnetic field in bulk MgB2 by producing samples with uniform high critical current density distribution using fabrication methods such as graded doping. © 2016 Elsevier B.V. All rights reserved.
Keywords: Bulk MgB2 Trapped field Regional critical current Numerical modelling
1. Introduction High magnetic field trapping properties and stable levitation of bulk high-temperature superconductors (HTS) are crucial for applications such as superconducting motors [1], flywheel energy storage [2] and Maglev systems [3,4]. In recent years it has been demonstrated that, like (RE)BCO (RE; rare earth), bulk magnesium diboride (MgB2) has high magnetic field trapping capacity and it can generate a stable levitation force [5]. Although at 39 K the superconducting transition temperature of MgB2 is lower than (RE)
* Corresponding author. Department of Physics, Faculty of Science, Karadeniz Technical University, 61080, Trabzon, Turkey. E-mail address: [email protected] (K. Ozturk). http://dx.doi.org/10.1016/j.jallcom.2016.09.239 0925-8388/© 2016 Elsevier B.V. All rights reserved.
BCO materials, bulk MgB2 superconductor may be preferable to RE (BCO) for many applications due to the cheap raw materials, light weight, and shorter processing time [6,7]. The other advantages of bulk MgB2 for superconducting magnet and levitation systems over conventional (RE)BCO bulks are uniformity of the superconducting properties, potential for superior mechanical properties as fracture strength (e.g. for samples fabricated by hot isostatic pressing [8]), and greater flexibility in sample shape and geometry due to the polycrystalline microstructure [5,9]. Studies on MgB2 indicate that some properties such as trapped magnetic field, levitation force, and critical current density can be improved considerably by introducing nanometer-scale nonsuperconducting regions which act as vortex pinning centres [10,11] and by improving material processing techniques [12,13]. In
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recent studies it was also shown that the critical current and trapped magnetic field values could be improved by decreasing porosity [14,15] and improving the connectivity between the grains of the bulk MgB2 with different methods as milling the starting powder [11,16], chemical doping [17] and high pressure pelletization [18]. Hot-pressing of nanocrystalline ball-milled powder produced MgB2 bulk samples of 20 mm diameter with high trapped magnetic fields of 5.4 T at 12 K and 3.2 T at 15 K [19]. These high trapped field values were attributed to the high critical current densities due to strong pinning, provided by nanocrystalline MgB2 grains. Another method to improve the trapped field is to use different practical trapped field techniques as field cooling (FC), zero field cooling (ZFC) and pulsed field magnetization (PFM), having advantages and disadvantages each other [20]. Trapped magnetic field properties of the bulk superconductors primarily depend on the magnetic field and temperature dependence of the critical current density Jc (B,T) [21]. In a bulk superconductor, the local Jc characteristics are usually calculated from M H hysteresis curves, while the macroscopic Jc characteristics are calculated from the experimental trapped field Btrap data [22,23]. In previously performed studies examining how the trapped magnetic field increases for samples produced using different methods such as ex situ uniaxial hot pressing (Durrell et al. [13]) and in situ capsule method (Naito et al. [23]), bulk MgB2 samples were observed to have local critical current density values which vary systematically within the sample geometry [13,23]. In these studies, the critical current density values taken at the centre section of the sample were generally higher than at edge section (the self-field critical current densities were determined locally to be 7 108 A m2 at the edge and 8 108 A m2 at centre of the sample for the sample of Durrell et al. [13]). More recently, studies by Zou et al. [20] and Fujishiro et al. [21] on the influence of geometric dimensions of bulk MgB2 on the measured trapped magnetic field have reported that the trapped field increases initially with increasing sample diameter but it saturates for large diameters. Besides the experimental researches in a few studies it was showed that the numerical simulations provide a fast, accurate and powerful tool to examine the trapped field and critical current profile of bulk superconductors, under any specific operating conditions [20,24]. Recently, several analytical and numerical methods have been used to calculate the levitation force or trapped field of HTS superconductors [25,26] and analytical methods mostly have been developed for simple geometries while the numerical methods have been developed for more complicated coupling effects and geometries [21,27]. Therefore, in literature the changing of the critical current density depends on region in bulk superconductor and the saturating of the trapped field after the large value of sample diameter point out the requirement of the additional detailed studies on trapped magnetic field and local Jc. Although there are a small number of papers published in the literature on the modelling of trapped field using the magnetic field and temperature dependent macro critical current density Jc (B,T) [20,21], there are no previous numerical investigations of the effect of regionally varying superconducting properties on bulk behaviour of MgB2. In this study, a numerical modelling is carried out by calculating and simulating the position dependent Jc (B,r) critical current density and Bz trapped magnetic field distributions to clarify the electromagnetic influence of the local critical current density and related parameter on the trapped field and flux pinning properties of the bulk MgB2. Understanding this will enable us to understand how to fabricate samples with microstructural variations which maximise the trapped magnetic field in bulk MgB2. The numerical modelling was carried out by using exponential relation for the dependence of Jc (B,r), the power-law for the E-J relation and the H-formulation in
Fig. 1. The re-plotted experimental Jc(B) characteristics of small specimens taken from the central and edge regions of each MgB2 bulk sample given in Durrell et al. [13] (denoted S1 here) and Naito et al. [23] (S2). “25” and “30” indicate the sample diameter in millimetres for S1 and S2 respectively. The inset graph shows the estimated curves following the data fitting that are used as input data for the numerical model.
the FEM package COMSOL Multiphysics 3.5a [17,25].
2. Numerical modelling methodology 2.1. Origin of experimentally derived data for superconducting properties In this paper, a numerical investigation is performed by calculating the radius dependent Jc (B,r) critical current density and Bz trapped magnetic field distributions in uniform external field to elucidate the electromagnetic influence of the local critical current density on the trapped field (for different diameter samples) and bulk critical current properties of the MgB2. As initial input data for the numerical calculation the Jc(B) experimental characteristic of small specimens, taken from different regions of the samples, have been taken from the previous studies in the literature by Durrell et al. [13] and Naito et al. [23] described above. The bulk MgB2 sample characteristics in Durrell et al. [13] are 25 mm diameter, 5.4 mm thickness and 3 T measured maximum trapped field (for two disk-shaped MgB2 bulks) at 17.5 K, while those in Naito et al. [23] are 30 mm diameter, 9 mm thickness and 1.33 T measured maximum trapped field at 20 K (and 1.50 T at 16.4 K). In this paper these samples are named S1 and S2, respectively. Fig. 1 shows the re-plotted experimental Jc(B) characteristics of small specimens taken from the centre and edge region of S1 and S2 MgB2 bulk samples, as taken from Durrell et al. [13] for S1 and Naito et al. [23] for S2. Values of critical current originally taken from the centre of the sample are denoted “Centr” while those from the edge are “Edg”. The diameter is indicated as the final number giving the final sample name. So, for example, S1-Edg-25 denotes the measurement from sample S1 (taken from Durrell et al. [13]), measured near the edge on a 25 mm diameter sample. The magnetic field (B) dependence of critical current density Jc(B) for each bulk small specimen are fitted as shown in the inset in Fig. 1. These parameters were determined using the exponential relation in Equation (1) [17]:
K. Ozturk, C.E.J. Dancer / Journal of Alloys and Compounds 693 (2017) 1109e1115
"
Jc ðBÞ ¼ Jc0 exp
B B0
b # (1)
Table 1 Data fitting parameters for the Jc(B) characteristics of each bulk MgB2 sample based on a small specimen taken from the local areas of S1 and S2 samples in Refs. [13,23] (see text) and related linear law for r-dependent Jc (B,r) characteristics. Jc0 (A/m2)
Sample
where Jc0 is the magnitude of critical current density when the local field is zero, and B0 and b are the material-dependent fitting parameters used in modelling as input data for S1-Centr-25, S1-Edg25 and S2-Centr-30, S2-Edg-30 samples. “Centr” and “Edg” samples are therefore taken to have uniform critical current densities throughout the sample, i.e. radius-independent character. The schematic depiction of the critical current density and their positions used in the numerical modelling is shown in Fig. 2 where r is 5.5 mm and 6 mm respectively for S1 and S2 samples. Also, to clarify the radius dependent electromagnetic behaviour of the bulk MgB2, in numerical modelling the radius dependent Jc0(r) and B0(r) characteristics are taken as input data by using the modified exponential law given in Equation (2),
"
b # B Jc ðB; rÞ ¼ Jc0 ðrÞexp B0 ðrÞ
(2)
1111
S1-Centr-25 S1-Edg-25 S1-rdep-25
8
8.33 10 6.77 108 Jc0(r) ¼ aþb r
B0 (T)
b, in equation (1, 2)
1.68 1.31 B0(r) ¼ c þ d r
2.10 1.39 2.10
a ¼ 8.33 108 A/m2, b ¼ 1.42 1010 A/m3 c ¼ 1.68 A/m2, d ¼ 34.4 A/m3 S2-Centr-30 S2-Edg-30 S2-rdep-30
1.08 109 7.14 108 Jc0(r) ¼ aþb r
0.90 0.87 B0(r) ¼ c þ d r
1.61 1.44 1.61
a ¼ 1.08 109 A/m2, b ¼ 3.02 1010 A/m3 c ¼ 0.90 A/m2, d ¼ 3.05 A/m3
sufficient for this model. In Table 1 and following figures S1-rdep-25 and S2-rdep-30 represent attained radius dependent modelling data respectively for S1 and S2, based on the combined using of the experimental data fitting parameters, linear law and Equation (2) in numerical modelling. 2.2. Numerical modelling procedure
where r denotes the distance from the sample centre. Table 1 shows the data fitting parameters for the Jc (B) characteristics used in numerical modelling of each bulk MgB2 sample based on a small specimen taken from the local areas of S1 and S2 samples in Refs. [13,23]. The r-dependent Jc (B,r) characteristics are calculated using the equations Jc0(r) ¼ aþb r and B0(r) ¼ c þ d r. The motivation behind these relationships is to use a simple mathematical formula that can easily describe the radius dependence of the critical current density Jc (B,r) in Equation (2). The critical current density value, Jc0 in the middle region of the bulk are obtained by taking the arithmetic average of the Jc0-center and Jc0-edge data fitting values, due to lacking data for different regions in literature (the same process is performed for middle region B0 value and also see Table 1). As shown in Fig. 1, the middle region critical current density is determined about as 2.9 108 Am2 at 1.5 T magnetic field by taking the average of the experimental Jc(B) values in the centre and the edge region. The bulk transport critical current density value of the S1 sample [13] was determined as 3.04 108 Am2 at 17.5 K using experimental maximum Bz trapped magnetic field value and analytical equation together, based on BioteSavart law including Bz trapped field, Jc bulk critical current density and size of the sample (reported the maximum the Bz value was about 3 T between the two disk-shaped MgB2 bulks). The critical current density values are almost identical, indicating that the average value approximation used for the middle region of the sample is
The numerical method, used in this study to implement the modelling on the electromagnetic behaviour of a bulk MgB2 depending on local superconducting parameters, was based on solving the magnetoquasistatic Maxwell equations using the Hformulation in the FEM package COMSOL Multiphysics 3.5a in two dimensions. The numerical formulation was firstly published by Hong et al. [25] to analyse the electromagnetic behaviour of the systems including high-temperature superconductors (HTSs) and the details can be found in Ref. [27]. Basically, the similar partial differential equations adapted for cylindrical symmetry were used in order to consistent with MgB2 cylindrical sample. The numerical procedure involves solving two PDEs using the general form PDE mode for the dependent variables Hr and Hz, the components of the magnetic fields in the r- and z-directions, respectively, in subdomains as the superconducting and dielectric region (air). The current density Jf and induced electrical field Ef in superconducting region flow in azimuthal direction as perpendicularly to the r-z plane. The PDEs in numerical calculation are derived using Faraday's and Ampere's law and these equations have the following forms in cylindrical symmetry [27], respectively as
b r
vEf 1 v rEf vHr vHz ¼ m0 mr b þb z þb z r r vr vz vt vt
Jf ¼
vHr vHz vz vr
(3)
(4)
As known the electrical behaviour of the superconductor material is nonlinear character and it can be modelled by E-J power law [28], given as;
Ef ¼ E0
Fig. 2. Schematic depiction of the positions within the bulk sample of the critical current density values used in the numerical modelling. r is 5.5 mm and 6 mm respectively for S1 and S2 samples.
Jf Jc ðB; rÞ
n ;
(5)
where Jc (B,r) (or radius independent Jc(B)) given as Equations (1) and (2) is the critical current density depending on magnetic field and radius r while E0 ¼ 1 104 V m1 and n ¼ 21 are generally typical values for type-II superconductors. The EeJ behaviour of the non-superconducting region is modelled depending on linear ohm law asEf ¼ rJf , where r is the resistivity of air. Also, for numerical calculation, in theqfield-dependent Equations (1) and (2), B is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi defined asB ¼ m0 mr H2r þ H 2z , where m0 and mr is the permeability
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of free space and relative permeability (taken as mr ¼ 1 here) of the other domains, respectively. To find dependent variables Hr and Hz components of the magnetic fields with suitable boundary conditions [25,27], the Equations (1), (2), (4) and (5) are inserted into (3) and finally two coupled PDEs expressed as:
v rE0
v rE0
vHz vr
r vH vz Jc ðB; rÞ
vHz vr
!n !,
r vH vz Jc ðB; rÞ
vz ¼ m0 mr r
vHr vt
!n !, vr ¼ m0 mr r
vHz vt
(6)
(7)
3. Results and discussion By defining suitable boundary conditions and different initial values for the PDEs on the interior boundary and the outer boundary, different kinds of magnetic field conditions can be constituted; such as a superconductor in a uniform field or different magnetization process as field-cooling, zero-field cooling and pulsed field magnetization. In this paper for numerical calculation, Dirichlet boundary condition was used for the outer boundary and r ¼ 0 boundary. In modelling a uniform external field was used to magnetize the superconducting domain, taking Hr ¼ 0 and Hz ¼ (B0/ m0) sinut as outer boundary condition, and the details can be find in Refs. [25,27,29]. Half a cycle of a sine wave is a reasonable approximation to magnetize bulk MgB2 for trapped magnetic field modelling in uniform field and the time with 0.1s was half a period of a used sinusoidal field (f ¼ 5s1) with an amplitude of 3 T. To focus the effect of the local critical current density and the related parameter on the electromagnetic behaviour of bulk MgB2, the temperature is taken as constant, so the heating or viscous forces were not considered in the magnetization modelling. Additionally, to avoid the long computation times and numerical instability, which are a common problem in the electromagnetic modelling of HTS (for large n values in Equation (5)), the magnetization time was taken small value as 0.1s. In order to avoid significant deviation from the experimental values given in Durrell et al. [13] and Naito et al. [23] and to easily compare the modelling data with each other, in trapped field modelling the sample sizes of S1-Centr-25, S1-rdep-25 and S1-Edg25 were taken as having diameter of 25 mm and thickness of 5 mm (temperature of 17.5 K). Besides, the sizes of S2-Centr-30, S2-rdep30 and S2-Edg-30 were taken as having diameter of 30 mm and thickness of 5 mm (temperature of 20 K). Fig. 3 shows the current density distribution and magnetic field of S1 sample with a 25 mm diameter at 17.5 K, exposed to a uniform external field of H(t) ¼ (3.0/m0) sin (10pt) in axially symmetric geometry, the magnetization modelling stage for ut ¼ 0.05p (a) and ut ¼ p (b). Surface plot shows induced superconducting currents in bulk MgB2, while streamline and arrows show the total magnetic field. In modelling the exponential relation was used for Jc(B) and Jc (B,r) as radius independent and dependent parameters, respectively and in figure the 2D rectangular sample plots represent a half of the sample. As shown in Figure (a) and (b), the modelling of the radius independent (for S1-Centr-25 and S1-Edg-25) and radius dependent critical current (for S1-rdep-25) shows different behaviour in the current density and magnetic field distribution of the bulk MgB2 sample. In the beginning stage of the magnetization for ut ¼ 0.05p in Fig. 3(a), as can be clearly seen from the magnetic field line orientation and the increasing of the current density (the increasing of colour darkness), the S1-Centr-25 sample with the central region character withstands the penetration of flux lines into sample more
than the S1-rdep-25 (with radius dependent character) and S1-Edg25 samples (with the edge region character). End of the half a cycle (ut ¼ p) in Fig. 3(b) the bulk superconductor also resists the flux lines motion while the external field are going to zero and so the superconductor has trapped magnetic field and remains magnetized. Fig. 4 shows current density taken from the magnetization modelling of S1 samples with a 25 mm diameter, exposed to an uniform external field of H(t) ¼ (3.0/m0) sin (10pt) as a function of radius (r) for a plane cut through the middle of the bulk (z ¼ 0) for time stage ut ¼ p. As known, magnetization capability of the HTS superconductor is related to the current density Jf, consisted in superconductor. The decreasing of the current density by going the outer sections of the sample can be seen in Figs. 3(b) and 4, when the external magnetic field becomes zero (for ut ¼ p). This circumstance, due to a decreasing in measured experimental Jc towards the outer region, points out that the radius dependent Jc (B,r) model (the modelling of S1-rdep-25 sample) has a more realistic explanations on bulk electromagnetic properties of MgB2. The peak trapped magnetic flux density on the sample axis of a MgB2 superconductor, Btrap, due to an induced persistent supercurrent, can be given in a simple form as Btrap ¼ Jckm0R, where k is a geometrical constant, Jc is the critical current density of the superconducting material and R is the sample diameter [20]. In some investigation it was reported that the enhancing of the flux pinning properties and consequently the improving of the magnetic and temperature dependent critical current density Jc (B,T) of the bulk MgB2 is a possible way to increase the Btrap value [20,21]. Fig. 5 shows z-component of the trapped magnetic field modelling, Bz, of samples with different region character, at z ¼ 0 and 0.5 mm above the top MgB2 surface, by using uniform external field of H(t) ¼ (3/m0)sin (10pt) in axially symmetric geometry for ut ¼ p. In this figure it was determined that the peak trapped magnetic field value of the S1-Centr-25 sample with central region parameter was higher by 22.4% and 9.20% than the S1-Edg-25 and the S1-rdep-25 samples with edge region parameter and radius dependent parameter, respectively. Also, similar situation can be clearly seen in figure for S2 samples. Another inferring of this figure is that the peak trapped magnetic field values of S1 samples are much higher than the S2 samples. The higher magnetic field trapping capability of the S1 samples than the S2 samples is due to the more robust critical current Jc(B) profile of S1 samples as seen in Fig. 1. In literature it has been stated that, dTc pinning, caused by the spatial variation of the Ginzburg-Landau coefficient linked to disorder in the transition temperature Tc, is dominant in pure MgB2 samples, while dl pinning, caused by the variations in the chargecarrier mean free path (l) near the lattice defects, is dominant in carbon doped MgB2 bulks [17,30]. Based on Collective pinning theory and experimental study it was reported that the transformation from transition temperature fluctuation, induced dTc pinning, to mean free path fluctuation, induced dl pinning, can be carried out [17] by adjusting the doping level of graphene oxide (GO) doped MgB2. Also, in our earlier study it was determined that, pinning character of the MgB2 superconductor shifted from surface pinning to normal point pinning character when the milling time of the powder was increased from 24 to 72 h [11]. In order to obtain a clear insight into the vortex pinning properties of MgB2 with different regional characters, an analysis on vortex pinning force (Fp) was performed, based on relation given as Fp ¼ JcB, where the data of Jc and B were obtained from the magnetization modelling by using uniform external field of H(t) ¼ (3/m0)sin (10pt) for quarter cycle of ut ¼ p/2. Fig. 6 shows the magnetic field dependence of the pinning force density, Fp (B), for the S1 and S2 samples with different region character. As shown in Fig. 6, the Fp values of the S1-Edg-25 and S2-Edg-30 samples with edge region character are lower than the S1-Centr-25 and S2-Centr-30 with central region
K. Ozturk, C.E.J. Dancer / Journal of Alloys and Compounds 693 (2017) 1109e1115
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Fig. 3. Current density distribution and magnetic field of S1 sample with a 25 mm diameter, exposed to a uniform external field of H(t) ¼ (3.0/m0) sin (10pt) in axially symmetric geometry, the magnetization modelling stage for ut ¼ 0.05p (a) and ut ¼ p (b). Surface plot shows induced superconducting currents in bulk MgB2 while streamline and arrows show the magnetic field.
Fig. 4. Current density taken from the magnetization modelling of S1 samples with a 25 mm diameter, exposed to an uniform external field of H(t) ¼ (3.0/m0) sin (10pt) as a function of radius (r) for a plane cut through the middle of the bulk (z ¼ 0) for time stage ut ¼ p.
character. It can be said that, the value of trapped field and the levitation force, depending on the critical current density and the radius of a superconducting current loop (circulating inside bulk superconductor sample), can be increased considerable by improving of the electromagnetic and the structural properties of superconductor. As structural properties, which are capable of
Fig. 5. The z-component of the trapped magnetic field modelling, Bz, of samples with different region character, at z ¼ 0 and 0.5 mm above the top MgB2 surface, by using uniform external field of H(t) ¼ (3/m0)sin (10pt) in axially symmetric geometry for ut ¼ p .
improvement the electromagnetic behaviour direct or indirect, grain boundary coupling, bulk density of sample and quantity and size of the vortex pinning centre can be thought [11,20]. As mentioned before, the trapped field characteristics of the superconducting bulks can be vary widely depending on the magnetic field dependence of the critical current density Jc (B), because of the
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Fig. 6. The magnetic field dependence of the pinning force density, Fp (B), for the S1 and S2 samples with different region parameter by using the magnetization modelling.
changing of the density of the vortex pinning centres in bulk MgB2 sample. The reason of the lower Fp value in Fig. 6 going from the centre to edge section (for edge samples) can be negative intrinsic dl pinning mechanism, due to the spatial variation of the structural properties by increasing the number and size of the voids and/or decreasing of the connectivity between the grains boundary, depending on radius. To understand this more fully, analysis of the microstructural variations with position within bulk MgB2 samples are being studied and this will be the subject of a future publication. Additionally, it can be said that, the weakness of the vortex pinning properties, and so lower Fp value, going from the centre to edge section in bulk MgB2 induces lower Bz values of the samples with edge region and radius-dependent parameters (see Fig. 5). Although it is expected that the maximum trapped field value of the bulk superconductor improves with the increasing of the sample size, some investigation reported that peak trapped field value increases initially with increasing sample diameter but it saturates after the large diameter and thickness [21]. To understand the reason of the trapped magnetic field saturation, in modelling, as input data the radius dependent critical current Jc (B,r) densities were used instead of constant bulk critical current parameters. Fig. 7 shows the z-component of the trapped magnetic field modelling, Bz, of the S1 samples with (a) Central parameter, (b) Radius parameter for the different diameter as 25, 30 and 35 mm. The data were obtained by using uniform external field of H(t) ¼ (3/ m0)sin (10pt) in axially symmetric geometry for ut ¼ p, at z ¼ 0 and 0.5 mm above the top sample surface. As shown in Fig. 7, peak trapped magnetic field values increased from 1.840 and 1.685 to 1.998 and 1.790 T when the diameter increase from 25 mm to 35 mm, respectively for the central region radius-independent (S1Centr) and radius-dependent S1 (S1-rdep) samples. Also, it is determined that, the peak trapped magnetic field values increase by 8.6% and 6.2%, respectively for the central region radiusindependent S1-Centr and radius-dependent S1-rdep samples, while the bulk diameter increase from the 25 mm to 35 mm. The saturation tendency, by going of increment value from 8.6% to 6.2%, indicates that the increasing of the bulk diameter alone doesn't have a significant effect on the value of the trapped magnetic field, unless the bulk superconducting current is enhanced uniformly throughout the MgB2 bulk. These results show that both the trapped magnetic field and bulk critical current behaviour depends on
Fig. 7. The z-component of the trapped magnetic field modelling, Bz, of the S1 samples with (a) Central parameter and (b) Radius-dependent parameter for the different diameters as 25, 30 and 35 mm (with constant thickness of 5 mm).
the local superconducting parameters as well as applied external field. This is more critical for samples with large diameter, because the decreasing non-uniform critical current distribution when the radius increase will limit the maximum trapped field and bulk critical current performance of the MgB2 sample. Consequently, further work will now be carried out experimentally, such as using the graded doping of MgB2 to provide a uniform current density distribution. From this work, producing such samples is expected to result in superior flux trapping performance of bulk MgB2 samples. 4. Conclusions In order to clarify the electromagnetic influence of the experimentally observed regional critical current density variation and related parameters on the trapped field and flux pinning properties of MgB2, we have carried out a numerical modelling by calculating and simulating the position dependent Jc (B,r) critical current density and Bz trapped magnetic field distributions. It was determined that the peak trapped magnetic field value of the sample with central region properties was higher by 22.4% and 9.20% compared to the samples with edge region properties and the sample with radius dependent parameter, respectively.
K. Ozturk, C.E.J. Dancer / Journal of Alloys and Compounds 693 (2017) 1109e1115
To understand the reason of the trapped magnetic field saturation while the diameter of sample increases, in modelling, as input data the radius dependent critical current densities Jc (B,r) were used instead of constant bulk critical current parameters. It is determined that, the peak trapped magnetic field values increase by 8.6% and 6.2%, respectively for the central region radiusindependent S1-Centr samples and radius-dependent S1-rdep samples, while the bulk diameter increase from the 25 mm to 35 mm. The saturation tendency in the peak trapped field points out that, the increasing of the bulk diameter alone doesn't have a significant effect on the value of the trapped magnetic field, unless the bulk superconducting current is improved uniformly throughout the MgB2 bulk. Our results show that the trapped magnetic field and bulk critical current behaviour depends on the local superconducting parameters as well as applied external field. This becomes critically important as the sample diameter increases to larger sizes because the non-uniform current distribution in the bulk will then limit the maximum trapped magnetic field which can be attained. Following on from this study, future work will focus on fabrication methods such as graded doping in MgB2 which may produce a uniform current density distribution and therefore further increase the radius independent bulk critical current density, leading to superior trapped magnetic field performance. Acknowledgements This work was supported by the Scientific and Technological Research Council of Turkey (TÜBITAK), with program code 2219 (with grant number of 1059B191500307). References [1] J.R. Hull, M. Strasik, Supercond. Sci. Technol. 23 (2010), 124005 (7pp). [2] F.N. Werfel, U. Floegel-Delor, R. Rothfeld, T. Riedel, B. Goebel, D. Wippich, P. Schirrmeister, Supercond. Sci. Technol. 25 (2012), 014007 (16pp). [3] C. Beyer, O. de Haas, L. Kuehn, L. Schultz, IEEE Trans. Appl. Supercond. 17 (2007) 2129e2132. [4] K. Ozturk, E. Sahin, M. Abdioglu, M. Kabaer, S. Celik, E. Yanmaz, T. Kucukomeroglu, J. Alloys Comp 643 (2015) 201e206. [5] A. Patel, S.C. Hopkins, G. Giunchi, A.F. Albisetti, Y. Shi, R. Palka, D.A. Cardwell,
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B.A. Glowacki, IEEE Trans. Appl. Supercond. 23 (2013) 6800604. [6] M. Muralidhar, K. Inoue, M.R. Koblischka, M. Tomita, M. Murakami, J. Alloys Comp 608 (2014) 102e109. [7] A. Yamamoto, J. Shimoyama, S. Ueda, Y. Katsura, S. Horii, K. Kishio, Supercond. Sci. Technol. 18 (2005) 116e121. [8] A. Murakami, H. Teshima, T. Naito, H. Fujishiro, T. Kudo, A. Iwamoto, Phys. Procedia 67 (2015) 982e986. [9] E. Perini, G. Giunchi, Supercond. Sci. Technol. 22 (2009), 045021 (6pp). [10] T. Naito, T. Yoshida, H. Fujishiro, Supercond. Sci. Technol. 28 (2015), 095009 (8pp). [11] E. Yanmaz, K. Ozturk, C.E.J. Dancer, M. Basoglu, S. Celik, C.R.M. Grovenor, J. Alloys Comp 492 (2010) 48e51. [12] A.G. Bhagurkar, A. Yamamoto, L. Anguilano, A.R. Dennis, J.H. Durrell, N. Hari Babu, D.A. Cardwel, Supercond. Sci. Technol. 29 (2016), 035008 (8pp). [13] J.H. Durrell, C.E.J. Dancer, A. Dennis, Y. Shi, Z. Xu, A.M. Campbell, N. Hari Babu, R.I. Todd, C.R.M. Grovenor, D.A. Cardwell, Supercond. Sci. Technol. 25 (2012), 112002 (5pp). [14] C.E.J. Dancer, P. Mikheenko, A. Bevan, J.S. Abell, R.I. Todd, C.R.M. Grovenor, J. Eur. Ceram. Soc. 29 (2009) 1817e1824. [15] C.E.J. Dancer, D. Prabhakaran, M. Basoglu, E. Yanmaz, H. Yan, M. Reece, R.I. Todd, C.R.M. Grovenor, Supercond. Sci. Technol. 22 (2009), 095003 (7pp). [16] C.E.J. Dancer, D. Prabhakaran, A. Crossley, R.I. Todd, C.R.M. Grovenor, Supercond. Sci. Technol. 23 (2010), 065015 (10pp). [17] F.X. Xiang, X.L. Wang, X. Xun, K.S.B. De Silva, Y.X. Wang, S.X. Dou, Appl. Phys. Lett. 102 (2013), 152601 (5pp). [18] T.A. Prikhna, et al., Phys. C 402 (2004) 223e233. ۧler, K. Nenkov, J. Scheiter, O. Perner, A. Handstein, T. Kanai, [19] G. Fuchs, W. Ha L. Schultz, B. Holzapfel, Supercond. Sci. Technol. 26 (2013), 122002 (5pp). [20] J. Zou, M.D. Ainslie, H. Fujishiro, A.G. Bhagurkar, T. Naito, N. Hari Babu, J.F. Fagnard, P. Vanderbemden, A. Yamamoto, Supercond. Sci. Technol. 28 (2015), 075009 (9pp). [21] H. Fujishiro, T. Naito, T. Yoshida, Supercond. Sci. Technol. 27 (2014), 065019 (7pp). [22] K. Ozturk, S. Akbulut, S. Kutuk, S. Bolat, S. Celik, M. Basoglu, J. Alloys Comp 516 (2012) 167e171. [23] T. Naito, T. Sasaki, H. Fujishiro, Supercond. Sci. Technol. 25 (2012), 095012 (6pp). [24] A. Patel, S.C. Hopkins, A. Baskys, V. Kalitka, A. Molodyk, B.A. Glowacki, Supercond. Sci. Technol. 28 (2015), 115007 (10pp). [25] Z. Hong, A.M. Campbell, T.A. Coombs, Supercond. Sci. Technol. 19 (2006) 1246e1252. [26] K. Ozturk, S. Celik, A. Cansiz, Phys. Status Solidi A 206 (11) (2009) 2569e2575. [27] K. Ozturk, A. Patel, B.A. Glowacki, Phys. Status Solidi A 209 (12) (2012) 2558e2564. [28] J. Rhyner, Phys. C 212 (1993) 292e300. [29] A. Patel, R. Palka, B.A. Glowacki, Supercond. Sci. Technol. 24 (2011), 015009 (8pp). [30] J.L. Wang, R. Zeng, J.H. Kim, L. Lu, S.X. Dou, Phys. Rev. B 77 (2008) 174501.
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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Electromagnetic behaviour of bulk MgB2 determined by numerical modelling using regional supercurrent properties K. Ozturk a, b, *, C.E.J. Dancer b a
Department of Physics, Faculty of Science, Karadeniz Technical University, 61080, Trabzon, Turkey International Institute for Nanocomposites Manufacturing (IINM), Warwick Manufacturing Group, University of Warwick, Coventry, CV4 7AL, United Kingdom
b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 April 2016 Received in revised form 20 September 2016 Accepted 22 September 2016 Available online 23 September 2016
Previous studies in the literature have shown experimentally that the critical current density in bulk MgB2 depends on the position at which it is measured within the sample, and that the trapped magnetic field saturates above large values of sample diameter. To better understand this behaviour, we have carried out a detailed study of the trapped magnetic field and local critical current distribution in bulk MgB2 using numerical modelling with H-formulation. The properties of bulk MgB2 samples with radiusindependent (i.e. uniform) and radius-dependent critical current were modelled, based on experimentally-determined parameters obtained from the literature. Radius-independent samples took properties originally derived from either the central or edge region of the original samples. In our study it was determined that the peak trapped magnetic field value of the sample with the central region properties was higher by 22.4% and 9.20% than the samples with edge region properties and that with radius dependent parameter, respectively. Additionally, to understand why the trapped magnetic field saturates as the diameter of sample increases, the trapped magnetic field was modelled using the radius dependent critical current density Jc (B,r) as input data rather than using a constant bulk critical current density. The peak trapped magnetic field values increase by 8.6% and 6.2% respectively for the central region radius-independent and radius-dependent property bulk MgB2 samples as the bulk diameter increases from 25 mm to 35 mm. This tendency to saturation in the peak trapped field indicates that increasing the bulk diameter alone does not have a significant effect on the value of the trapped magnetic field, unless the bulk superconducting current density is improved uniformly throughout the MgB2 bulk. These results enable us to understand how we can experimentally enhance the trapped magnetic field in bulk MgB2 by producing samples with uniform high critical current density distribution using fabrication methods such as graded doping. © 2016 Elsevier B.V. All rights reserved.
Keywords: Bulk MgB2 Trapped field Regional critical current Numerical modelling
1. Introduction High magnetic field trapping properties and stable levitation of bulk high-temperature superconductors (HTS) are crucial for applications such as superconducting motors [1], flywheel energy storage [2] and Maglev systems [3,4]. In recent years it has been demonstrated that, like (RE)BCO (RE; rare earth), bulk magnesium diboride (MgB2) has high magnetic field trapping capacity and it can generate a stable levitation force [5]. Although at 39 K the superconducting transition temperature of MgB2 is lower than (RE)
* Corresponding author. Department of Physics, Faculty of Science, Karadeniz Technical University, 61080, Trabzon, Turkey. E-mail address: [email protected] (K. Ozturk). http://dx.doi.org/10.1016/j.jallcom.2016.09.239 0925-8388/© 2016 Elsevier B.V. All rights reserved.
BCO materials, bulk MgB2 superconductor may be preferable to RE (BCO) for many applications due to the cheap raw materials, light weight, and shorter processing time [6,7]. The other advantages of bulk MgB2 for superconducting magnet and levitation systems over conventional (RE)BCO bulks are uniformity of the superconducting properties, potential for superior mechanical properties as fracture strength (e.g. for samples fabricated by hot isostatic pressing [8]), and greater flexibility in sample shape and geometry due to the polycrystalline microstructure [5,9]. Studies on MgB2 indicate that some properties such as trapped magnetic field, levitation force, and critical current density can be improved considerably by introducing nanometer-scale nonsuperconducting regions which act as vortex pinning centres [10,11] and by improving material processing techniques [12,13]. In
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recent studies it was also shown that the critical current and trapped magnetic field values could be improved by decreasing porosity [14,15] and improving the connectivity between the grains of the bulk MgB2 with different methods as milling the starting powder [11,16], chemical doping [17] and high pressure pelletization [18]. Hot-pressing of nanocrystalline ball-milled powder produced MgB2 bulk samples of 20 mm diameter with high trapped magnetic fields of 5.4 T at 12 K and 3.2 T at 15 K [19]. These high trapped field values were attributed to the high critical current densities due to strong pinning, provided by nanocrystalline MgB2 grains. Another method to improve the trapped field is to use different practical trapped field techniques as field cooling (FC), zero field cooling (ZFC) and pulsed field magnetization (PFM), having advantages and disadvantages each other [20]. Trapped magnetic field properties of the bulk superconductors primarily depend on the magnetic field and temperature dependence of the critical current density Jc (B,T) [21]. In a bulk superconductor, the local Jc characteristics are usually calculated from M H hysteresis curves, while the macroscopic Jc characteristics are calculated from the experimental trapped field Btrap data [22,23]. In previously performed studies examining how the trapped magnetic field increases for samples produced using different methods such as ex situ uniaxial hot pressing (Durrell et al. [13]) and in situ capsule method (Naito et al. [23]), bulk MgB2 samples were observed to have local critical current density values which vary systematically within the sample geometry [13,23]. In these studies, the critical current density values taken at the centre section of the sample were generally higher than at edge section (the self-field critical current densities were determined locally to be 7 108 A m2 at the edge and 8 108 A m2 at centre of the sample for the sample of Durrell et al. [13]). More recently, studies by Zou et al. [20] and Fujishiro et al. [21] on the influence of geometric dimensions of bulk MgB2 on the measured trapped magnetic field have reported that the trapped field increases initially with increasing sample diameter but it saturates for large diameters. Besides the experimental researches in a few studies it was showed that the numerical simulations provide a fast, accurate and powerful tool to examine the trapped field and critical current profile of bulk superconductors, under any specific operating conditions [20,24]. Recently, several analytical and numerical methods have been used to calculate the levitation force or trapped field of HTS superconductors [25,26] and analytical methods mostly have been developed for simple geometries while the numerical methods have been developed for more complicated coupling effects and geometries [21,27]. Therefore, in literature the changing of the critical current density depends on region in bulk superconductor and the saturating of the trapped field after the large value of sample diameter point out the requirement of the additional detailed studies on trapped magnetic field and local Jc. Although there are a small number of papers published in the literature on the modelling of trapped field using the magnetic field and temperature dependent macro critical current density Jc (B,T) [20,21], there are no previous numerical investigations of the effect of regionally varying superconducting properties on bulk behaviour of MgB2. In this study, a numerical modelling is carried out by calculating and simulating the position dependent Jc (B,r) critical current density and Bz trapped magnetic field distributions to clarify the electromagnetic influence of the local critical current density and related parameter on the trapped field and flux pinning properties of the bulk MgB2. Understanding this will enable us to understand how to fabricate samples with microstructural variations which maximise the trapped magnetic field in bulk MgB2. The numerical modelling was carried out by using exponential relation for the dependence of Jc (B,r), the power-law for the E-J relation and the H-formulation in
Fig. 1. The re-plotted experimental Jc(B) characteristics of small specimens taken from the central and edge regions of each MgB2 bulk sample given in Durrell et al. [13] (denoted S1 here) and Naito et al. [23] (S2). “25” and “30” indicate the sample diameter in millimetres for S1 and S2 respectively. The inset graph shows the estimated curves following the data fitting that are used as input data for the numerical model.
the FEM package COMSOL Multiphysics 3.5a [17,25].
2. Numerical modelling methodology 2.1. Origin of experimentally derived data for superconducting properties In this paper, a numerical investigation is performed by calculating the radius dependent Jc (B,r) critical current density and Bz trapped magnetic field distributions in uniform external field to elucidate the electromagnetic influence of the local critical current density on the trapped field (for different diameter samples) and bulk critical current properties of the MgB2. As initial input data for the numerical calculation the Jc(B) experimental characteristic of small specimens, taken from different regions of the samples, have been taken from the previous studies in the literature by Durrell et al. [13] and Naito et al. [23] described above. The bulk MgB2 sample characteristics in Durrell et al. [13] are 25 mm diameter, 5.4 mm thickness and 3 T measured maximum trapped field (for two disk-shaped MgB2 bulks) at 17.5 K, while those in Naito et al. [23] are 30 mm diameter, 9 mm thickness and 1.33 T measured maximum trapped field at 20 K (and 1.50 T at 16.4 K). In this paper these samples are named S1 and S2, respectively. Fig. 1 shows the re-plotted experimental Jc(B) characteristics of small specimens taken from the centre and edge region of S1 and S2 MgB2 bulk samples, as taken from Durrell et al. [13] for S1 and Naito et al. [23] for S2. Values of critical current originally taken from the centre of the sample are denoted “Centr” while those from the edge are “Edg”. The diameter is indicated as the final number giving the final sample name. So, for example, S1-Edg-25 denotes the measurement from sample S1 (taken from Durrell et al. [13]), measured near the edge on a 25 mm diameter sample. The magnetic field (B) dependence of critical current density Jc(B) for each bulk small specimen are fitted as shown in the inset in Fig. 1. These parameters were determined using the exponential relation in Equation (1) [17]:
K. Ozturk, C.E.J. Dancer / Journal of Alloys and Compounds 693 (2017) 1109e1115
"
Jc ðBÞ ¼ Jc0 exp
B B0
b # (1)
Table 1 Data fitting parameters for the Jc(B) characteristics of each bulk MgB2 sample based on a small specimen taken from the local areas of S1 and S2 samples in Refs. [13,23] (see text) and related linear law for r-dependent Jc (B,r) characteristics. Jc0 (A/m2)
Sample
where Jc0 is the magnitude of critical current density when the local field is zero, and B0 and b are the material-dependent fitting parameters used in modelling as input data for S1-Centr-25, S1-Edg25 and S2-Centr-30, S2-Edg-30 samples. “Centr” and “Edg” samples are therefore taken to have uniform critical current densities throughout the sample, i.e. radius-independent character. The schematic depiction of the critical current density and their positions used in the numerical modelling is shown in Fig. 2 where r is 5.5 mm and 6 mm respectively for S1 and S2 samples. Also, to clarify the radius dependent electromagnetic behaviour of the bulk MgB2, in numerical modelling the radius dependent Jc0(r) and B0(r) characteristics are taken as input data by using the modified exponential law given in Equation (2),
"
b # B Jc ðB; rÞ ¼ Jc0 ðrÞexp B0 ðrÞ
(2)
1111
S1-Centr-25 S1-Edg-25 S1-rdep-25
8
8.33 10 6.77 108 Jc0(r) ¼ aþb r
B0 (T)
b, in equation (1, 2)
1.68 1.31 B0(r) ¼ c þ d r
2.10 1.39 2.10
a ¼ 8.33 108 A/m2, b ¼ 1.42 1010 A/m3 c ¼ 1.68 A/m2, d ¼ 34.4 A/m3 S2-Centr-30 S2-Edg-30 S2-rdep-30
1.08 109 7.14 108 Jc0(r) ¼ aþb r
0.90 0.87 B0(r) ¼ c þ d r
1.61 1.44 1.61
a ¼ 1.08 109 A/m2, b ¼ 3.02 1010 A/m3 c ¼ 0.90 A/m2, d ¼ 3.05 A/m3
sufficient for this model. In Table 1 and following figures S1-rdep-25 and S2-rdep-30 represent attained radius dependent modelling data respectively for S1 and S2, based on the combined using of the experimental data fitting parameters, linear law and Equation (2) in numerical modelling. 2.2. Numerical modelling procedure
where r denotes the distance from the sample centre. Table 1 shows the data fitting parameters for the Jc (B) characteristics used in numerical modelling of each bulk MgB2 sample based on a small specimen taken from the local areas of S1 and S2 samples in Refs. [13,23]. The r-dependent Jc (B,r) characteristics are calculated using the equations Jc0(r) ¼ aþb r and B0(r) ¼ c þ d r. The motivation behind these relationships is to use a simple mathematical formula that can easily describe the radius dependence of the critical current density Jc (B,r) in Equation (2). The critical current density value, Jc0 in the middle region of the bulk are obtained by taking the arithmetic average of the Jc0-center and Jc0-edge data fitting values, due to lacking data for different regions in literature (the same process is performed for middle region B0 value and also see Table 1). As shown in Fig. 1, the middle region critical current density is determined about as 2.9 108 Am2 at 1.5 T magnetic field by taking the average of the experimental Jc(B) values in the centre and the edge region. The bulk transport critical current density value of the S1 sample [13] was determined as 3.04 108 Am2 at 17.5 K using experimental maximum Bz trapped magnetic field value and analytical equation together, based on BioteSavart law including Bz trapped field, Jc bulk critical current density and size of the sample (reported the maximum the Bz value was about 3 T between the two disk-shaped MgB2 bulks). The critical current density values are almost identical, indicating that the average value approximation used for the middle region of the sample is
The numerical method, used in this study to implement the modelling on the electromagnetic behaviour of a bulk MgB2 depending on local superconducting parameters, was based on solving the magnetoquasistatic Maxwell equations using the Hformulation in the FEM package COMSOL Multiphysics 3.5a in two dimensions. The numerical formulation was firstly published by Hong et al. [25] to analyse the electromagnetic behaviour of the systems including high-temperature superconductors (HTSs) and the details can be found in Ref. [27]. Basically, the similar partial differential equations adapted for cylindrical symmetry were used in order to consistent with MgB2 cylindrical sample. The numerical procedure involves solving two PDEs using the general form PDE mode for the dependent variables Hr and Hz, the components of the magnetic fields in the r- and z-directions, respectively, in subdomains as the superconducting and dielectric region (air). The current density Jf and induced electrical field Ef in superconducting region flow in azimuthal direction as perpendicularly to the r-z plane. The PDEs in numerical calculation are derived using Faraday's and Ampere's law and these equations have the following forms in cylindrical symmetry [27], respectively as
b r
vEf 1 v rEf vHr vHz ¼ m0 mr b þb z þb z r r vr vz vt vt
Jf ¼
vHr vHz vz vr
(3)
(4)
As known the electrical behaviour of the superconductor material is nonlinear character and it can be modelled by E-J power law [28], given as;
Ef ¼ E0
Fig. 2. Schematic depiction of the positions within the bulk sample of the critical current density values used in the numerical modelling. r is 5.5 mm and 6 mm respectively for S1 and S2 samples.
Jf Jc ðB; rÞ
n ;
(5)
where Jc (B,r) (or radius independent Jc(B)) given as Equations (1) and (2) is the critical current density depending on magnetic field and radius r while E0 ¼ 1 104 V m1 and n ¼ 21 are generally typical values for type-II superconductors. The EeJ behaviour of the non-superconducting region is modelled depending on linear ohm law asEf ¼ rJf , where r is the resistivity of air. Also, for numerical calculation, in theqfield-dependent Equations (1) and (2), B is ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi defined asB ¼ m0 mr H2r þ H 2z , where m0 and mr is the permeability
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of free space and relative permeability (taken as mr ¼ 1 here) of the other domains, respectively. To find dependent variables Hr and Hz components of the magnetic fields with suitable boundary conditions [25,27], the Equations (1), (2), (4) and (5) are inserted into (3) and finally two coupled PDEs expressed as:
v rE0
v rE0
vHz vr
r vH vz Jc ðB; rÞ
vHz vr
!n !,
r vH vz Jc ðB; rÞ
vz ¼ m0 mr r
vHr vt
!n !, vr ¼ m0 mr r
vHz vt
(6)
(7)
3. Results and discussion By defining suitable boundary conditions and different initial values for the PDEs on the interior boundary and the outer boundary, different kinds of magnetic field conditions can be constituted; such as a superconductor in a uniform field or different magnetization process as field-cooling, zero-field cooling and pulsed field magnetization. In this paper for numerical calculation, Dirichlet boundary condition was used for the outer boundary and r ¼ 0 boundary. In modelling a uniform external field was used to magnetize the superconducting domain, taking Hr ¼ 0 and Hz ¼ (B0/ m0) sinut as outer boundary condition, and the details can be find in Refs. [25,27,29]. Half a cycle of a sine wave is a reasonable approximation to magnetize bulk MgB2 for trapped magnetic field modelling in uniform field and the time with 0.1s was half a period of a used sinusoidal field (f ¼ 5s1) with an amplitude of 3 T. To focus the effect of the local critical current density and the related parameter on the electromagnetic behaviour of bulk MgB2, the temperature is taken as constant, so the heating or viscous forces were not considered in the magnetization modelling. Additionally, to avoid the long computation times and numerical instability, which are a common problem in the electromagnetic modelling of HTS (for large n values in Equation (5)), the magnetization time was taken small value as 0.1s. In order to avoid significant deviation from the experimental values given in Durrell et al. [13] and Naito et al. [23] and to easily compare the modelling data with each other, in trapped field modelling the sample sizes of S1-Centr-25, S1-rdep-25 and S1-Edg25 were taken as having diameter of 25 mm and thickness of 5 mm (temperature of 17.5 K). Besides, the sizes of S2-Centr-30, S2-rdep30 and S2-Edg-30 were taken as having diameter of 30 mm and thickness of 5 mm (temperature of 20 K). Fig. 3 shows the current density distribution and magnetic field of S1 sample with a 25 mm diameter at 17.5 K, exposed to a uniform external field of H(t) ¼ (3.0/m0) sin (10pt) in axially symmetric geometry, the magnetization modelling stage for ut ¼ 0.05p (a) and ut ¼ p (b). Surface plot shows induced superconducting currents in bulk MgB2, while streamline and arrows show the total magnetic field. In modelling the exponential relation was used for Jc(B) and Jc (B,r) as radius independent and dependent parameters, respectively and in figure the 2D rectangular sample plots represent a half of the sample. As shown in Figure (a) and (b), the modelling of the radius independent (for S1-Centr-25 and S1-Edg-25) and radius dependent critical current (for S1-rdep-25) shows different behaviour in the current density and magnetic field distribution of the bulk MgB2 sample. In the beginning stage of the magnetization for ut ¼ 0.05p in Fig. 3(a), as can be clearly seen from the magnetic field line orientation and the increasing of the current density (the increasing of colour darkness), the S1-Centr-25 sample with the central region character withstands the penetration of flux lines into sample more
than the S1-rdep-25 (with radius dependent character) and S1-Edg25 samples (with the edge region character). End of the half a cycle (ut ¼ p) in Fig. 3(b) the bulk superconductor also resists the flux lines motion while the external field are going to zero and so the superconductor has trapped magnetic field and remains magnetized. Fig. 4 shows current density taken from the magnetization modelling of S1 samples with a 25 mm diameter, exposed to an uniform external field of H(t) ¼ (3.0/m0) sin (10pt) as a function of radius (r) for a plane cut through the middle of the bulk (z ¼ 0) for time stage ut ¼ p. As known, magnetization capability of the HTS superconductor is related to the current density Jf, consisted in superconductor. The decreasing of the current density by going the outer sections of the sample can be seen in Figs. 3(b) and 4, when the external magnetic field becomes zero (for ut ¼ p). This circumstance, due to a decreasing in measured experimental Jc towards the outer region, points out that the radius dependent Jc (B,r) model (the modelling of S1-rdep-25 sample) has a more realistic explanations on bulk electromagnetic properties of MgB2. The peak trapped magnetic flux density on the sample axis of a MgB2 superconductor, Btrap, due to an induced persistent supercurrent, can be given in a simple form as Btrap ¼ Jckm0R, where k is a geometrical constant, Jc is the critical current density of the superconducting material and R is the sample diameter [20]. In some investigation it was reported that the enhancing of the flux pinning properties and consequently the improving of the magnetic and temperature dependent critical current density Jc (B,T) of the bulk MgB2 is a possible way to increase the Btrap value [20,21]. Fig. 5 shows z-component of the trapped magnetic field modelling, Bz, of samples with different region character, at z ¼ 0 and 0.5 mm above the top MgB2 surface, by using uniform external field of H(t) ¼ (3/m0)sin (10pt) in axially symmetric geometry for ut ¼ p. In this figure it was determined that the peak trapped magnetic field value of the S1-Centr-25 sample with central region parameter was higher by 22.4% and 9.20% than the S1-Edg-25 and the S1-rdep-25 samples with edge region parameter and radius dependent parameter, respectively. Also, similar situation can be clearly seen in figure for S2 samples. Another inferring of this figure is that the peak trapped magnetic field values of S1 samples are much higher than the S2 samples. The higher magnetic field trapping capability of the S1 samples than the S2 samples is due to the more robust critical current Jc(B) profile of S1 samples as seen in Fig. 1. In literature it has been stated that, dTc pinning, caused by the spatial variation of the Ginzburg-Landau coefficient linked to disorder in the transition temperature Tc, is dominant in pure MgB2 samples, while dl pinning, caused by the variations in the chargecarrier mean free path (l) near the lattice defects, is dominant in carbon doped MgB2 bulks [17,30]. Based on Collective pinning theory and experimental study it was reported that the transformation from transition temperature fluctuation, induced dTc pinning, to mean free path fluctuation, induced dl pinning, can be carried out [17] by adjusting the doping level of graphene oxide (GO) doped MgB2. Also, in our earlier study it was determined that, pinning character of the MgB2 superconductor shifted from surface pinning to normal point pinning character when the milling time of the powder was increased from 24 to 72 h [11]. In order to obtain a clear insight into the vortex pinning properties of MgB2 with different regional characters, an analysis on vortex pinning force (Fp) was performed, based on relation given as Fp ¼ JcB, where the data of Jc and B were obtained from the magnetization modelling by using uniform external field of H(t) ¼ (3/m0)sin (10pt) for quarter cycle of ut ¼ p/2. Fig. 6 shows the magnetic field dependence of the pinning force density, Fp (B), for the S1 and S2 samples with different region character. As shown in Fig. 6, the Fp values of the S1-Edg-25 and S2-Edg-30 samples with edge region character are lower than the S1-Centr-25 and S2-Centr-30 with central region
K. Ozturk, C.E.J. Dancer / Journal of Alloys and Compounds 693 (2017) 1109e1115
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Fig. 3. Current density distribution and magnetic field of S1 sample with a 25 mm diameter, exposed to a uniform external field of H(t) ¼ (3.0/m0) sin (10pt) in axially symmetric geometry, the magnetization modelling stage for ut ¼ 0.05p (a) and ut ¼ p (b). Surface plot shows induced superconducting currents in bulk MgB2 while streamline and arrows show the magnetic field.
Fig. 4. Current density taken from the magnetization modelling of S1 samples with a 25 mm diameter, exposed to an uniform external field of H(t) ¼ (3.0/m0) sin (10pt) as a function of radius (r) for a plane cut through the middle of the bulk (z ¼ 0) for time stage ut ¼ p.
character. It can be said that, the value of trapped field and the levitation force, depending on the critical current density and the radius of a superconducting current loop (circulating inside bulk superconductor sample), can be increased considerable by improving of the electromagnetic and the structural properties of superconductor. As structural properties, which are capable of
Fig. 5. The z-component of the trapped magnetic field modelling, Bz, of samples with different region character, at z ¼ 0 and 0.5 mm above the top MgB2 surface, by using uniform external field of H(t) ¼ (3/m0)sin (10pt) in axially symmetric geometry for ut ¼ p .
improvement the electromagnetic behaviour direct or indirect, grain boundary coupling, bulk density of sample and quantity and size of the vortex pinning centre can be thought [11,20]. As mentioned before, the trapped field characteristics of the superconducting bulks can be vary widely depending on the magnetic field dependence of the critical current density Jc (B), because of the
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Fig. 6. The magnetic field dependence of the pinning force density, Fp (B), for the S1 and S2 samples with different region parameter by using the magnetization modelling.
changing of the density of the vortex pinning centres in bulk MgB2 sample. The reason of the lower Fp value in Fig. 6 going from the centre to edge section (for edge samples) can be negative intrinsic dl pinning mechanism, due to the spatial variation of the structural properties by increasing the number and size of the voids and/or decreasing of the connectivity between the grains boundary, depending on radius. To understand this more fully, analysis of the microstructural variations with position within bulk MgB2 samples are being studied and this will be the subject of a future publication. Additionally, it can be said that, the weakness of the vortex pinning properties, and so lower Fp value, going from the centre to edge section in bulk MgB2 induces lower Bz values of the samples with edge region and radius-dependent parameters (see Fig. 5). Although it is expected that the maximum trapped field value of the bulk superconductor improves with the increasing of the sample size, some investigation reported that peak trapped field value increases initially with increasing sample diameter but it saturates after the large diameter and thickness [21]. To understand the reason of the trapped magnetic field saturation, in modelling, as input data the radius dependent critical current Jc (B,r) densities were used instead of constant bulk critical current parameters. Fig. 7 shows the z-component of the trapped magnetic field modelling, Bz, of the S1 samples with (a) Central parameter, (b) Radius parameter for the different diameter as 25, 30 and 35 mm. The data were obtained by using uniform external field of H(t) ¼ (3/ m0)sin (10pt) in axially symmetric geometry for ut ¼ p, at z ¼ 0 and 0.5 mm above the top sample surface. As shown in Fig. 7, peak trapped magnetic field values increased from 1.840 and 1.685 to 1.998 and 1.790 T when the diameter increase from 25 mm to 35 mm, respectively for the central region radius-independent (S1Centr) and radius-dependent S1 (S1-rdep) samples. Also, it is determined that, the peak trapped magnetic field values increase by 8.6% and 6.2%, respectively for the central region radiusindependent S1-Centr and radius-dependent S1-rdep samples, while the bulk diameter increase from the 25 mm to 35 mm. The saturation tendency, by going of increment value from 8.6% to 6.2%, indicates that the increasing of the bulk diameter alone doesn't have a significant effect on the value of the trapped magnetic field, unless the bulk superconducting current is enhanced uniformly throughout the MgB2 bulk. These results show that both the trapped magnetic field and bulk critical current behaviour depends on
Fig. 7. The z-component of the trapped magnetic field modelling, Bz, of the S1 samples with (a) Central parameter and (b) Radius-dependent parameter for the different diameters as 25, 30 and 35 mm (with constant thickness of 5 mm).
the local superconducting parameters as well as applied external field. This is more critical for samples with large diameter, because the decreasing non-uniform critical current distribution when the radius increase will limit the maximum trapped field and bulk critical current performance of the MgB2 sample. Consequently, further work will now be carried out experimentally, such as using the graded doping of MgB2 to provide a uniform current density distribution. From this work, producing such samples is expected to result in superior flux trapping performance of bulk MgB2 samples. 4. Conclusions In order to clarify the electromagnetic influence of the experimentally observed regional critical current density variation and related parameters on the trapped field and flux pinning properties of MgB2, we have carried out a numerical modelling by calculating and simulating the position dependent Jc (B,r) critical current density and Bz trapped magnetic field distributions. It was determined that the peak trapped magnetic field value of the sample with central region properties was higher by 22.4% and 9.20% compared to the samples with edge region properties and the sample with radius dependent parameter, respectively.
K. Ozturk, C.E.J. Dancer / Journal of Alloys and Compounds 693 (2017) 1109e1115
To understand the reason of the trapped magnetic field saturation while the diameter of sample increases, in modelling, as input data the radius dependent critical current densities Jc (B,r) were used instead of constant bulk critical current parameters. It is determined that, the peak trapped magnetic field values increase by 8.6% and 6.2%, respectively for the central region radiusindependent S1-Centr samples and radius-dependent S1-rdep samples, while the bulk diameter increase from the 25 mm to 35 mm. The saturation tendency in the peak trapped field points out that, the increasing of the bulk diameter alone doesn't have a significant effect on the value of the trapped magnetic field, unless the bulk superconducting current is improved uniformly throughout the MgB2 bulk. Our results show that the trapped magnetic field and bulk critical current behaviour depends on the local superconducting parameters as well as applied external field. This becomes critically important as the sample diameter increases to larger sizes because the non-uniform current distribution in the bulk will then limit the maximum trapped magnetic field which can be attained. Following on from this study, future work will focus on fabrication methods such as graded doping in MgB2 which may produce a uniform current density distribution and therefore further increase the radius independent bulk critical current density, leading to superior trapped magnetic field performance. Acknowledgements This work was supported by the Scientific and Technological Research Council of Turkey (TÜBITAK), with program code 2219 (with grant number of 1059B191500307). References [1] J.R. Hull, M. Strasik, Supercond. Sci. Technol. 23 (2010), 124005 (7pp). [2] F.N. Werfel, U. Floegel-Delor, R. Rothfeld, T. Riedel, B. Goebel, D. Wippich, P. Schirrmeister, Supercond. Sci. Technol. 25 (2012), 014007 (16pp). [3] C. Beyer, O. de Haas, L. Kuehn, L. Schultz, IEEE Trans. Appl. Supercond. 17 (2007) 2129e2132. [4] K. Ozturk, E. Sahin, M. Abdioglu, M. Kabaer, S. Celik, E. Yanmaz, T. Kucukomeroglu, J. Alloys Comp 643 (2015) 201e206. [5] A. Patel, S.C. Hopkins, G. Giunchi, A.F. Albisetti, Y. Shi, R. Palka, D.A. Cardwell,
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