Controllable rectification on the thermal conductivity of porous YBa₂Cu₃O_7−x superconductors from 3D-printing

The fabrication of YBCO bulks with larger thermal conductivity is based on the DIW 3D-printing process, which is shown in figure 1. To fabricate simple YBCO without doping, the Y₂O₃, BaCO₃ and CuO powders are needed, and their diameters are all less than 1 µm. We choose Al₂O₃ nanoparticles as doping material because it has higher thermal conductivity. Since Al₂O₃ nanoparticles do not react with Y₂O₃, BaCO₃ and CuO at the sintering temperature, the Al₂O₃ nanoparticles were added to the YBCO precursor powder mixture. The mechanism of Al₂O₃ particles enhanced YBCO with higher thermal conductivity is that Al₂O₃ and YBCO combined and became a composite material. Meanwhile, with Al₂O₃ doping into YBCO, the density of YBCO can be enhanced, which causes higher thermal conductivity of a composite material. To make the precursor powder mixture into the slurry required for 3D printing, it is first necessary to prepare a binder solution. Generally, sodium carboxymethyl cellulose (RnOCH₂COONa) and deionized water are mixed in a weight ratio (1:15). In a glass beaker, a mechanical stirring process was adopted at room temperature for at least 72 h to obtain a uniform water-based precursor. Then, the ground powder is fully dispersed into the water-based precursor in an agate mortar. After fully stirring, a two-phase system printing paste is obtained. For the three-phase paste slurry, soybean oil epoxide (C₅₇H₉₈O₁₂) was dissolved in the binder solution, and the oil–water–powder three-phase mixed suspension was obtained by stirring. To obtain better paste uniformity, a roller mill can also be used to homogenize the mixture between steel rollers. Finally, the printing paste with uniform elasticity and good performance is obtained. If the printed YBCO sample is dried directly, a large number of irregular circular cavities will be generated in the sample. However, when the 3D printed structure is processed in an ice-cold environment (−60 °C), as the water in the structure freezes, cellulose fibers that absorb numerous particles are gradually excluded from the growing ice crystals. Therefore, the distance between the fibers is reduced, forcing them to form interconnected reinforced scaffolds together, and after freeze-drying, because many mechanically reinforced scaffolds are formed in the structure, they can be strong enough to withstand crystallization migration at the interface of annealing liquid phase without large deformation. Therefore, the samples were freeze-dried at −50 °C for 48 h. At the same time, during the freeze-drying process, the extrusion of ice crystals improved the close contact between different particles and formed a good intracellular structure. In addition, this method can prevent the internal crack propagation of 3D-printed YBCO bulk caused by initial temperature difference and spatial anisotropy. After the freeze-drying, the YBCO samples were annealed at 920 °C in an oxygen atmosphere to enhance their thermal conductivity. After sintering, to make up for the lack of oxygen in YBCO bulk materials during the preparation process, the YBCO samples were annealed at 480 °C in 0.4 MPa oxygen pressure atmosphere, and finally, the fully prepared YBCO bulk materials can be obtained.

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Fundamentally, the printability of ceramic precursor paste is determined by the rheological and thixotropic properties. Meantime, they also affect the structural stability and geometric accuracy of the 3D-printed samples [32]. Researchers have proposed that the viscosity of 2000 Pa·s is suitable for fabricating stable constructions of the YBCO [33]. In our work, the weight ratio of epoxidized soybean oil and water has been changed prepared thermal conductivity precursor paste to make sure it can be well used to fabricate stable structures. Zhang et al [30] confirmed the best anneal temperature of samples is 920 °C which makes samples have higher crystallinity and uniform grain size. Therefore, our work also chooses 920 °C as anneal temperature to keep Al₂O₃-doped YBCO has the same properties as pure YBCO.

The chemical composition and structure of Al₂O₃-doped 3D-printed YBCO bulk with high thermal conductivity were further characterized by x-ray diffraction (XRD), as shown in figure 2. The XRD pattern of pure YBCO, pure Al₂O₃ and Al₂O₃-doped YBCO with high thermal conductivity all show specific sharp peaks, which indicated the compositions of YBCO and Al₂O₃ nanoparticles. As shown in figures 3(a)–(c), the sintered Al₂O₃-doped YBCO with high thermal conductivity has a porous structure. According to figure 3(d), it is clear that Al₂O₃ nanoparticles had been doped in the YBCO structure homogeneously. Though the radius of Al₂O₃ nanoparticles was controlled at about 30 nm, the aggregation of the nanoparticles also appeared during the preparation of precursor paste and the freeze-drying process, which can be seen in the inset of figure 3(d). As shown in figures 3(e)–(j), corresponding to the synthesized elements, Y, Ba, Cu, O, and Al, respectively. According to these five elements' distribution's color mapping, Al₂O₃-doped YBCO with high thermal conductivity was confirmed to be fabricated successfully as an YBCO composite.

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Al₂O₃ is a kind of nanoparticle with high thermal conductivity. In this paper, we use Al₂O₃nanoparticles to improve the thermal conductivity of YBCO bulk materials for 3D-printing. Like 3D printed YBCO superconducting bulk materials, we tested the superconducting properties of bulk materials doped with alumina nanoparticles and measured the temperature dependence and magnetic field intensity of magnetization of 3D printed YBCO doped with Al₂O₃ nanoparticles by using a magnetic measuring system.

Critical current density (J_c, A cm⁻²) is an important evaluation index of superconductivity, which is calculated from the M–Ba curve by using Bean-model's formula [34]:

$$\begin{equation}{J_{\text{c}}} = \frac{{20\Delta M}}{{b(1 - {\raise0.7ex\hbox{$b$} \!/ \!\lower0.7ex\hbox{${3a}$}})}}\end{equation} \tag{ 1 }$$

where ΔM is the width of the magnetized hysteresis loop (emu cm⁻³), a and b (a > b) are the cross-sectional dimensions of the sample perpendicular to the magnetic field, respectively.

Figures 4(a)–(c) show graphs of magnetization versus magnetic field (from −90 kOe to 90 kOe) of 3D printed samples doped with different Al₂O₃ ratios with temperatures ranging from 2.5 K to 55 K. With different Al₂O₃ doping ratios, the YBCO superconducting properties show giant differences, which indicated that the sample doped with 15% Al₂O₃ has the best superconducting properties. Therefore, further calculations are given based on 15% Al₂O₃-doped YBCO. A larger magnetic moment can be obtained at a lower temperature when the external magnetic field strength is the same. The maximum magnetization of 3D printed samples at 2.5, 4, 15, 35 and 55 K are 5.68, 5.4, 4.19, 1.57 and 0.9 emu g⁻¹, respectively. Compared with the traditional cold-pressed sintered YBCO, 3D-printed YBCO shows considerable improvement in the magnetic moment and hysteresis loops at different temperatures. In figure 4(d), from which YBCO 3D-printing bulk with improved thermal conductivity doped with Al₂O₃nanoparticles also has a transition temperature of 92 K. As is shown in figures 4(e) and (f), at different temperatures (2.5, 4, 15, 35, and 55 K), J_c (H) decreases exponentially with the increase of the magnetic field. The double logarithmic plot of J_c (H) shows an approximately linear relationship to the magnetic field (1000–90 000 Gs) without the so-called fishtail effect. In addition, the power-law relationship between J_c and t is obtained, that is, J_c decreases with the temperature increasing from 0 K to 55 K. After doping with Al₂O₃ nanoparticles, the prepared enhanced 3D printed YBCO superconducting bulk material will have changes in thermal conductivity based on retaining the original superconductivity characteristics.

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Since Al₂O₃ nanoparticles have high thermal conductivity, 3D printed YBCO was doped with 5%, 10% and 15% Al₂O₃ nanoparticles respectively. YBCO high-temperature superconductors are always being used at about 77 K, so the thermal conductivity of undoped and doped YBCO with different proportions was measured from 0 K to 300 K. As shown in figure 5(a), the thermal conductivity of Al₂O_3-doped YBCO doped was improved, and the dependence of thermal conductivity on temperature was a process of first increasing, then decreasing and finally tending to be linear. The change rule of thermal conductivity before and after doping was consistent. Among them, the thermal conductivity of undoped YBCO, YBCO doped with 10% and 15% mass ratio reached the peak value of 16.98, 23.19 and 25.87 W Km⁻¹ respectively when the temperature reached 42 K. While the thermal conductivity of samples doped with 10% mass ratio reached the peak value of 22.78 W Km⁻¹, when the temperature reached 62 K. It can be seen that after doping, the thermal conductivity of 3D printed YBCO is improved regardless of the mass ratio of doping. At the same peak period, 3D printed YBCO doped with 10% Al₂O₃nanoparticles has the highest thermal conductivity, whose thermal conductivity is 1.52 times higher than pure YBCO, which has a great influence on the thermal conductivity of YBCO high-temperature superconducting materials. It can also be seen from the figure that after the peak position of the thermal conductivity of the sample, that is, the temperature is 42 K, the improvement of the overall thermal conductivity brought by the doped Al₂O₃nanoparticles with a mass ratio of 5% is the most significant. The improvement of the thermal conductivity of the doped 3D printed YBCO remains 1.5–2 times higher than pure YBCO after 42 K, which can even reach 2.1 times at 300 K. Therefore, the doping mass ratio of Al₂O₃nanoparticles and the improvement of thermal conductivity are not linearly related. In practical engineering applications, it is necessary to select different doping mass fractions of alumina nanoparticles according to the actual working temperature of YBCO high-temperature superconducting materials to achieve the optimal effect. Given the changing trend of thermal conductivity of YBCO high-temperature superconducting materials with temperature increasing first and then decreasing, many researchers have made detailed research and analysis on the mechanism. The existing mechanism explanation is to analyze the influence of phonon heat transfer, electron heat transfer and their coupling effect on thermal conductivity in YBCO at the microscopic level, and then give theoretical formulas for fitting and comparative analysis of experimental data. There is almost no macroscopic model to explain the special change of thermal conductivity of YBCO high-temperature superconducting materials with the temperature rising first and then falling.

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As shown in figure 5(b), the 3D printed YBCO samples have a regular porous structure by SEM characterization at room temperature, and through careful observation found that the inner pore structures are mostly regular and irregular hexagons, which are the dominant shape. And the characteristic structure of YBCO is the basis of establishing a macroscopic theoretic model to explain the time dependence of YBCO thermal conductivity. To simplify the model calculation, porous structures inside samples were considered as the same shape at the same temperature. And we assumed these structures as truss structures made up of several hinged bars with the same thickness. In 2008, analytical and finite element methods were used by Bezazi et al [35] to analyze the load-bearing and heat-conducting multi-functional applications of a new type of multi-concave honeycomb structure [36]. On this basis, in 2009, Inocenti et al [37] established a transient thermal analysis model considering both the thermal radiation and the thermal conduction of solid materials, furthermore, the thermal protection performance of the new honeycomb structure was described [38]. In 2014 [39], based on the Bernoulli–Euler beam model, the Poisson's ratio of honeycomb structure was calculated by using the strain energy principle under pure bending load, on this basis, the influence of the expansion effect of negative Poisson's ratio on the thermal conductivity of the structure is further analyzed [40]. As early as 1935, Bruggeman [41] proposed a new structural model in which two phases are connected and each phase is continuous. Subsequently, Landauer [42] derived the effective thermal conductivity equation of the structure in 1952. Since most of the voids in 3D printed YBCO were hexagonal at room temperature, the temperature of the YBCO high-temperature Superconductor classification increased from 0 K to 300 K, therefore, it is deduced that the inner structure of the 3D printed sample is hexagonal at room temperature because of the thermal expansion and deformation of the inner structure caused by temperature. The internal structure is a continuous process in which the top-to-top perovskite (structure) of two triangles gradually deforms into a hexagon as the temperature expands. Because the alumina-doped 3D-printing YBCO heat transfer enhancement material is formed by a 3D-printing paste nozzle, the precursor paste is the internal skeleton component of the sintered product, which is itself a porous material. As shown in figure 5(e), the finite element method was used for simulating the structural deformation inside the sample with temperature changing. When the temperature decreases, its structure shrinks, and gradually changes from the initial hexagonal structure to a rectangular structure. Because the rectangular structure itself is unstable, which continues to shrink in the process of continuous temperature decrease, and finally deforms into a special structure with two triangles placed on top. Therefore, our model is based on the porous material as a framework to support the formation of a larger scale of porous materials. Using the heat transfer coefficient formula given by the above researchers, the heat transfer coefficient of 3D printed YBCO light porous material doped with Al₂O₃particles is calculated and the numerical simulation is carried out.

The model calculation is based on the following equations that combine the honeycomb's structural thermal conductivity with the effective thermal conductivity of cellular composite material [36–39]. In the theoretical calculation of the microscopic model, a modified formula has been used as follows to calculate the thermal conductivity of our designed structure with making up of a high thermal conductivity microstructure,

$$\begin{equation}{k_s} = \rho k\end{equation} \tag{ 2 }$$

where ${k_s}$ represents the sample's conductivity, $\rho$ is a dimensionless value that represents the relative structural density of the model structure, and $k$ denotes the thermal conductivity of the composite material. Next, we give the calculation formula of $\rho$ and $k$, respectively. Similar to the presentation in the literature [37], we supply one structural influencing factor $\alpha$, which represents the structural deformation, as shown in the following formula:

$$\begin{align}\rho = \frac{{m(n + 2)}}{{2\alpha }}\frac{1}{{n\sin \theta + \sin \theta \cdot\cos \theta }},\nonumber\\[11pt] \left(\theta = - \frac{\pi }{{120}}T + a\pi ,m = {\raise0.7ex\hbox{$t$} \!/ \!\lower0.7ex\hbox{$l$}},n = {\raise0.7ex\hbox{$s$} \!/ \!\lower0.7ex\hbox{$l$}}\right)\end{align} \tag{ 3 }$$

where $t$ represents the thickness of model, $s$ and $l$ represent the length of cross rod and sway rod of the model, respectively. $T$ represents the temperature, $a$ is a constant, and $\theta$ represents the angle of the model's concave angle which is determined as a function of temperature.

To obtain the thermal conductivity of Al₂O₃-doped YBCO material, we use the formula (4) which has been given in the literature [38, 39] to calculate the thermal conductivity of the composite material with two components,

$$\begin{align}k &= \frac{1}{4}\left\{ (3{v_2} - 1){k_2} + (3{v_1} - 1){v_1} \right.\nonumber\\[12pt] &\quad\left. + \sqrt {{{[(2{v_2} - 1){k_2} + (3{v_1} - 1){k_1}]}^2}} + 8{k_1}{k_2}\right\} \end{align} \tag{ 4 }$$

where ${v_1},{v_2}$ represent the volume ratio of two main materials, and ${k_1},{k_2}$ represent two main materials' thermal conductivity. In the end, we obtained the thermal conductivity of the nanoparticle-doped YBCO by the 3D-printing process, which is also fit for other porous materials suitable for 3D-printing,

$$\begin{equation}{k_s} = \frac{{m(n + 2)}}{{8\alpha }}\frac{{\left\{ {(3{v_2} - 1){k_2} + (3{v_1} - 1){v_1} + \sqrt {{{[(2{v_2} - 1){k_2} + (3{v_1} - 1){k_1}]}^2}} + 8{k_1}{k_2}} \right\}}}{{n\sin ( - \frac{\pi }{{120}}T + a\pi ) + \sin ( - \frac{\pi }{{120}}T + a\pi )\cdot\cos ( - \frac{\pi }{{120}}T + a\pi )}}.\end{equation} \tag{ 5 }$$

As shown in figure 5(c), the inside microstructure of the sample changed with different $\Phi$ values since the structure we put forward has changed with the temperature rising (As shown in figure S2, supporting information). It is clear that when the $\Phi$ is about 60°, the sample's thermal conductivity reached its peak value. As shown in figure 5(d), the green lines represent the numerically calculated results of the temperature dependence of the thermal conductivity of the 10% Al₂O₃-doped YBCO sample on the macroscopic scale model we have established, and the red asterisks represent 3D printed YBCO samples doped with 10% Al₂O₃ nanoparticles, the black squares represent the undoped 3D printed YBCO samples. It can be seen that our macroscopic theoretic model based on thermal expansion-induced structural deformation can well explain the unique temperature dependence of the thermal conductivity of 3D printed YBCO superconductor classification in doped or undoped conditions. Since we assumed structures inside samples are all the same shape at the same temperature and assumed the density of samples is 100%, there is a difference between the experimental results and theoretical calculation, which is acceptable.

As mentioned above, a macroscopic theoretic model has been established based on the thermal expansion deformation of the frame structure to calculate the thermal conductivity of the macroscopic structure. When the thermal conductivity reaches its peak, the inner structure is shown as a concave hexagon. And through our calculation of the thermal conductivity of the model, it can be concluded that when the thermal conductivity of the sample reaches its peak, the concave angle of the structure is 120°. Therefore, with the help of the results of calculation and analysis, the DIW 3D-printing YBCO technology has been used to directly design and print the YBCO structure with the largest thermal conductivity in the range of 0–300 K. As shown in figure 6(a), for the YBCO bulk with high thermal conductivity fabricated by 3D-printing technique, the YBCO exhibited a regular periodic structure, these structures are formed by the deposition of cylindrical slurry from a nozzle with an inner diameter of about 200 μm, which is set up according to the procedure of the printing model. Because the prepared ceramic printing paste has excellent rheological properties, it can be successfully extruded without powder agglomeration, which allows the geometry to be accurately copied from the structure so that 3D-printed samples can be made accurately. As shown in figures 6(c) and (d). The rod-like structure of 3D printed YBCO is also porous, so the YBCO bulk with high thermal conductivity retained its ultra-light property. As shown in figure 7, we tested the thermal conductivity of both YBCO bulks with vertical and horizontal orthogonal deposition printed by Zhang et al [30], and YBCO bulks with designed structure in the normal temperature section. It has been found that the thermal conductivity of three YBCO bulks with optimized structures was significantly improved compared with that of YBCO blocks with a square grid, and all of the samples are undoped with Al₂O₃. At temperature of 360 K, the maximum thermal conductivity of the designed structure can reach 1.47 times that of the square grid structure. Therefore, a structural design method of controllable thermal conductivity based on 3D-printing technology has been confirmed from the experimental point of view, and the functional YBCO 3D-printed structural composite material has been successfully prepared, which provides a high reference value for the later practical application.

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