钛酸钡纳米颗粒铁电性临界尺寸的理论分析

从铁电体的Eular-Lagrange方程出发, 取贝塞尔方程级数解的形式, 得到了钛酸钡陶瓷颗粒的总极化强度表达式, 分析了各系数对总极化强度的影响. 根据总极化强度表达式, 采用MATLAB软件对尺寸在100 nm以下的钛酸钡纳米颗粒的铁电性进行了仿真分析. 结合实际数据探讨了尺寸效应对陶瓷颗粒铁电性的影响, 获得了与实验数据相符的数值解和极小值, 从而预测了钛酸钡纳米颗粒铁电性存在的临界尺寸为6 nm.

Theoretical analysis on ferroelectricity critical dimension of BaTiO3 nanoparticles

The expression of the total polarization intensity of BaTiO₃ nanoparticles based on Euler-Lagrange equation for ferroelectric particles and the form of the series solution of Bessel function are obtained, the influences of coefficient on the total polarization intensity are analyzed. According to the expression for the total polarization intensity, the ferroelectricity of BaTiO₃ nanoparticles with different size below 100 nm is simulated and analyzed by MATLAB. Based on experimental data, the effect of grain size on the ferroelectricity is discussed by searching for the numerical value of the solution, and the critical grain size of BaTiO₃ nanoparticles with ferroelectricity is predicted to be 6 nm subsequently; or, more specifically, based on the Euler-Lagrange equation of ferroelectric particle's total free energy, and according to the boundary condition, the equation is given in spherical coordinates, and the transformed equation has the form and characteristics of the Bessel equation, Therefore, it can be discussed according to the characteri-stics of the Bessel equation. It is considered that it has the series solution, and according to the form of deduced series solution, at the same time, under reasonable conditions, the change of polarization is simulated. By combining with the boundary conditions, the total expression of polarization of nanoparticles may be obtained. It contains some factors, first of all, it is analyzed on the whole, and the effect the factors have on the total expression of polarization of the nanoparticles is analyzed. The factors that directly affect, or indirectly affect the polarization intensity, and thus influence the trend of polarization intensity change is analyzed and identified. Then, the effect of each factor on the dielectric constant is simulated by adjusting the factors, and the numerical solution obtained is consistent with the experimental data, so the predictive value can be obtained.