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In the present work, the dual phase lag heat conduction in functionally graded hollow spheres is investigated under spherically symmetric and axisymmetric thermal loading. The heat conduction equation is given based on the dual phase lag theory to consider the details of energy transport in the material in comparison with the non-Fourier hyperbolic heat conduction. All the material properties of the sphere are taken to vary continuously along the radial direction following a power-law with arbitrary non-homogeneity indices except the phase lags which are assumed to be constant for simplicity. The specified spherically symmetric and axisymmetric boundary conditions of the sphere lead to a 1D and 2D heat conduction problem, respectively. Employing the Laplace transform to eliminate the time dependency of the problem, analytical solutions are obtained for the temperature and heat flux. The final results in the time domain are obtained by a numerical Laplace inversion method. The speed of thermal wave in the functionally graded sphere based on the dual phase lag is compared with that of the hyperbolic heat conduction. Furthermore, the numerical results are shown to clarify the effects of phase lags and non-homogeneity indices on the thermal response. The current results are verified with those reported in the literature.

Using NiCl${}_2\cdot 6{\text{H}}_2$O and Y(NO)₃ solution as raw material and urea as the precipitating agent, Yttrium-doped NiO hollow spheres (Y-NiOHS) were prepared through homogeneous precipitation method with melamine–formaldehyde polymer microspheres (MF) as templates. The electrochemical performances of Y-NiOHS with difference Y-doped mole ratios (Y:$\mathrm{\text{Ni}}=0.3$%, 0.5%, 1%Y-NiOHS) were characterized by cyclic voltammetry (CV), constant current charge–discharge test (GCD) and AC impedance test (EIS). The results of CV show that 0.5%Y-NiOHS in 5mV/s scan rate can achieve the maximum capacity of 320.92F/g. The results of GCD show that the capacity of 0.5%Y-NiOHS is 226.2F/g at 0.5A/g discharge current density and the 1000 charge–discharge cycles after the capacitance retention rate is 85%. The electrochemical performance of 0.5%Y-NiOHS are better than NiOHS and NiO powder.

Fe₃O₄ hollow microspheres with good dispersibility and high saturation magnetization were synthesized through a facile one-step solvothermal method. The formation mechanism of the hollow structure was studied by taking time-dependent experiments. Porous $\alpha$-FeOOH and $\alpha$-Fe₂O₃ nanosheets were firstly fabricated. Fe₃O₄ solid spheres aggregated by small particles were obtained from the transition of $\alpha$-FeOOH and $\alpha$-Fe₂O₃. Finally, the solid sphere is transferred to hollow sphere through Ostwald ripening. The maximum saturation magnetization of the hollow spheres is $115.4\pm 0.1$emu/g, which is higher than some results reported in references. The Fe₃O₄ hollow spheres show potential applications in microwave absorption and photocatalysis.

In this paper, we report an interesting approach for efficient synthesis of uniform sub-micrometer carbon supported Fe₃O₄ hollow spheres. Fe₃O₄ precursor was first coated on the surface of sulfonated polystyrene hollow microspheres. Then, the precursor and sulfonated polystyrene hollow microspheres were converted into Fe₃O₄ and carbon hollow spheres when heated at 550°C in N₂ atmosphere. The obtained Fe₃O₄ @ carbon hollow microspheres exhibit enhanced lithium storage properties compared with Fe₂O₃ hollow spheres as anode materials, delivering a reversible capacity of 612 mA hg⁻¹ after 50 cycles at a high current density of 400 mA g⁻¹.