Finite Element Simulation of Temperature and Current Distribution in a Superconductor, and a Cell Model for Flux Flow Resistivity—Interim Results

Abstract

A critical problem arises when current distribution in a high-temperature superconductor and its stability against quench shall be predicted: is it correct to assume homogeneous temperature distribution in superconductors, in general or only in LHe-cooled devices? The finite element analysis presented in this paper shows that during the very first instants following a disturbance, like single Dirac or periodic heat pulses, or large fault currents, temperature distribution in a BSCCO 2223 conductor is highly inhomogeneous. This is because disturbances, of transient or continuous, isolated or extended types in conductor volumes, create hot spots of comparatively long life cycle. As a consequence, separation between Ohmic and flux flow current limiter types, or decisions on the mechanism that initialises current sharing, cannot be made definitely. A semi-empirical cell model is presented in this paper to estimate flux flow resistivity in multi-filamentary superconductors in a successive approximation approach. Weak links are modelled, as nano- and microscopic surface irregularities and corresponding resistances, in analogy to thermal transport. Though the model requests input of a large amount of data (dimensions, porosities, field-dependent quantities) that still have to be verified experimentally, it is by its flexibility superior to ideas relying on, for example, imagination of separate, non-interacting chains of strong and weak links switched in parallel. In particular, and in contrast to the standard expression to calculate flux flow resistivity, the cell model suggests to replace solid conduction by an effective resistivity, a method that is more appropriate for multi-filamentary conductors. The paper also discusses integration time steps in numerical simulations that have to be selected in conformity with several characteristic times of current and thermal transport.

Appendices

Appendix A1 Material Composition of Weak Links

Microporous materials of any kind inevitably contain weak links. They constitute obstacles, at least against the following:

• (i) Electrical transport (Ohmic resistance to flow of electron charges)

• (ii) Magnetic transport (movement of vortices under flux flow)

• (iii) Thermal transport (propagation of lattice excitations), and even against fluid transport, if any is involved

Electrical resistance calculations to items (i) and (iii) are straightforward; a very large number of experimental data exists. The proper problem concerns item (ii) in case it is not a homogeneous solid but a micro-porous superconductor the flux flow resistivity of which has to be determined.

In the finite element calculations, as described in the text, this problem is tentatively circumvented by calculation of an effective resistivity, ρ _eff, by application of a cell model. Weak links between grains and domains built up from crystallites and grains are electrically conducting or insulating, which means they can be composed of superconductor and of normal conductor components or of contaminations.

Because of the small coherence length, weak link geometry in high temperature superconductors ranges from Josephson junctions with nanoscopic dimensions to solid bridges of finite, but tiny volume; all of these form a network of resistances to current flow and most of them are related to materials manufacture. Different crystalline and amorphous structures, variations between low to large angle tilt grain boundaries, twist boundaries and different textures, may contribute to electrical resistances within grains and contacts between grains; among these, low-angle tilt grain boundaries may be considered as (relatively) “strong” links (essentially, these are the superconductor grains). Results for critical current density reported in [36] may be the consequence of “current sharing”, here not between superconductor grains and matrix material, but between said strong links and current paths via tiny spatial or interfacial or contact weak links.

Besides spatial (contact) inhomogeneity of weak links, there are also variations of material composition, like in grains. Variations of materials properties in grains comprise mixtures of residual 2212 and 2223 phases of the BSCCO family (the 2223 phase emanates during solid-state reaction from the 2212 phase). This means relative contributions from different phases depend on temperature, length of sintering steps and their repetitions, and on phase stabilising measures like Pb doping to achieve high stability of the 2223 phase (and to reduce synthesis temperature). There is also strong anisotropy of flux flow not only in grains but also in grain boundaries [39], much larger than the anisotropy of resistance to standard current transport.

More variations of grain materials properties result from different oxygen contents, different additions like Sr, as well as from presence of the BSCCO 2201 phase. The 2223 phase is surrounded by a manifold of undesired Bi-, Pb- and Cu-containing phases; their relative contribution by volume is about 10 % [40]. It is hardly possible to obtain a one-phase 2223 in PIT industrial conductor manufacture. But under the superconducting secondary phases, the 2212 is dominating, at low enough temperature.

Bi-containing phases other than BSCCO 2223, like 2201, 2212 and 2234, have T _Crit= 13, 94 and 104 K, respectively. Since it is difficult to synthesise the 2234 phase, it will hardly appear, neither as a grain nor as a possible weak link component, which means, at T≥ 77 K, probably the only competitor to 2223 precipitation in grain and weak link materials is the phase 2212. But at given working core temperature, T > 94 K, still below the T _Crit= 108 K of the 2223 phase, the T _Crit of the 2212 phase is exceeded, and it will enter into its normal conducting state. Contributions from this phase by flux flow to resistivity ρ _FF accordingly disappear (the same considerations apply to corresponding critical current density and critical magnetic fields). This applies to all non-superconductor components of the weak links if their critical temperature is below the T _Crit of the BSCCO 2223 phase. Their contributions to flux flow resistivity are zero at all temperatures.

However, all materials other than BSCCO 2223 may contribute to Ohmic resistance between grains and domains and corresponding losses in the bridges, at all temperatures. These materials bridges accordingly have to be considered separately in the finite element calculations. A rough estimate assumes that at T > 94 K, normal conducting contributions by the 2212 phase (and from foreign constituents) to the resistivity of weak links, all independent of magnetic field, are in the order of 1 to 10 %, at the most, in good quality 1G conductors.

Possible contributions to the (proper) flux flow resistivity, ρ _FF, from phases other than 2223 (because of their lower T _Crit) thus depend on working temperature, not only on their volume fractions. In principal, (3b) has to be considered separately for each superconductor component that grains and weak links (if they contain superconductor phases) might consist of: the higher the working temperature, the less the contribution to ρ _eff from phases other than 2223.

Flux flow resistivity, ρ _FF, not the normal resistivity, ρ _NC, of the same grain or weak link material, in the present simulations (98 ≤T≤ 108 K) accordingly can be expected as being solely determined by the BSCCO 2223 phase.

Voids arise during manufacture and handling, like pores and longitudinal or transversal cracks (compare, for example, Fig. 5.30 in [40]). All these can be taken into account in the resistivity, ρ _eff, though in rough approximations only. For this purpose, the resistivity of the field-independent part of the solid material (contaminations) is increased in calculation of ρ _eff by at the most 5 or 10 % of grains, domains and weak links, respectively.

Accordingly, like in grains, inhomogeneity of weak link properties probably resembles the same broad spectrum of material composition. But structural and physical/chemical properties of weak link materials are not identical with, and may not even be close to, properties of the proper superconductor solids.

For step 1 (compare text), the conductivity k _Shell has to be inserted into (3b). We roughly have assumed $k_{\text {Shell}}\approx 1/100\, k_{{\exp }}$. The factor 1/100 results from measurements [41] of the anisotropy of thermal conductivity in c-axis vs. ab-plane directions. Measurements were performed using the 3 ω method on the bare superconductor core, after removal of the Ag-jacket, of a BSCCO/Ag PIT conductor prepared by ABB Corporate Research. Principal porosity was π_Solid= 0.11. The 3 ω method is explained in [41], and the measured anisotropy, r(T), is shown in Fig. 8. In an approximation, we assume that the ratio of electrical conductivity between c-axis and ab-plane is about the same as the r(T) of thermal conductivity. The conductivity of the weak link materials with porosity π_{Weak link} then is estimated as $k_{\text {Shell}}=(k_{{\exp }}/r)/({\Pi }_{{\text {Weak}}_{\text {link}}}/{\Pi }_{\text {Solid}})$.

Fig. 8

Measured anisotropy ratio, r(T)=λ _ab(T)/ λ _c(T), of thermal conductivity, λ(T), of the BSCCO 2223 PIT bare core (after removal of the Ag-jacket) in ab-plane vs. c-axis direction. The conductor was provided by ABB Corporate Research, Heidelberg (Germany). Results are reported in [41]; the applied 3 ω method is explained in [42]

Step 4 assigns another porosity to domains and grains to simulate separation of field dependency of the weak link components again by application of the Russell cell model. We applied π₄=π_Split= 0.75 and 10 ⁻⁴, for domain and grain weak links, respectively. The field-dependent part of the grain weak links to resistivity thus is rather close to the total value that results from step 3 while only 1/4 of the total weak link resistivity from domain weak links is assumed to depend on magnetic field. The latter estimate is uncertain but at least qualitatively reflects the poor concentration of BSCCO 2223 phase in domain weak link material.

Appendix A2:: Weak Links and Solid Point Contacts in Particle Beds

The contact radius, a, between two crossed cylinders, each of radius, r, according to [27] reads as follows:

$$ a=C r [(1 - \mu^{2}) p / Y (1 - {\Pi} )^{2}]^{\mathrm{1/3}} $$

(4)

with C= 1.55 a constant. Equation 4 is applicable to elastic deformations; below their fracture load, elastic deformation applies to most ceramics. Because of their platelet geometry, (4) is a better approximation to the contact radius between grains and domains, touching one superimposed upon the other in plane-parallel or transversally, than the contact radius of two contacting spheres.

Young’s modulus for BSCCO 2223 prepared in the PIT process is in the order of 100 GPa [42], as estimated by the authors, and the real intrinsic modulus may be higher. In the present calculations, Y≈ 240 GPa and μ≈ 0.2 are used; both values are characteristic for hard ceramics of very small porosity (compare Fig. 2.9 in [32]). The impact of the Ag-jacket on size of the contact radii, though is inner surface is indented and interleaved with the ceramic components, is neglected since its modulus of elasticity is much smaller, below 10 GPa (the stress vs. strain curve of the filaments thus is expected to depend only on the modulus of the ceramic superconductor). Pressure load, p, in y-direction, applied during manufacturing is in the order of 10 ⁸ Pa if a critical current density larger than 10 ⁸ A/m ² shall be achieved with PIT ([41], Fig. 12 in Sect. A). But in x- and z-directions, an effective pressure load applies that is experienced by the superconductor material; it is much smaller, probably by at least two orders of magnitude.

Under these conditions, contact radii, a, of weak links in z-direction, in core and shell cross sections of Fig. 6a, between either domains or grains, ranges from 2.2 to 6.1 μm and from 13 to 288 nm, respectively. This procedure can be applied to BSCCO 2223 only if the bed of superconductor platelets is highly densified, under strong pressure load in y-directions during manufacture, yielding ab-planes oriented parallel to the x,z-plane.

These estimates urgently need experimental verification with superconductor particles. A statistically contact model formulated to PIT conductors, like the cell model described in [43], might be helpful for development steps to obtain the desired flux flow resistivity.