Van der Waals Interactions between YAG:Ce Particles in Tellurite Glass

The van der Waals interaction for two YAG:Ce particles in tellurite glass melt at 923 K can be approximately calculated by the following equation:^{41 (link)–43} $W_{\mathrm{vdW}}\left(D\right)=-\frac{A\left(a^2+RD\right)}{12D^2}$ $A=\frac{3}{4}{\mathrm{k}}_{\mathrm{B}}T{\left(\frac{{\epsilon }_{\mathrm{G}}-{\epsilon }_{\mathrm{Y}}}{{\epsilon }_{\mathrm{G}}+{\epsilon }_{\mathrm{Y}}}\right)}^2+\frac{3\mathrm{h}{v}_{\mathrm{e}}}{16\sqrt{2}}\frac{{\left(n_{\mathrm{G}}^2-n_{\mathrm{Y}}^2\right)}^2}{\left(n_{\mathrm{G}}^2+n_{\mathrm{Y}}^2\right)\sqrt{n_{\mathrm{G}}^2+n_{\mathrm{Y}}^2}}$ where D is the distance between two particles in nanometers. A is the Hamaker constants for the van der Waals interaction. a is the radius of the particle contact surface (a is negligible relative to the micron size R of the YAG:Ce particles). R is the radii of YAG:Ce particles. k_B is the Boltzmann constant; T is the ambient temperature; ${\epsilon }_{\mathrm{G}}$ and ${\epsilon }_{\mathrm{Y}}$ are the relative permittivity of tellurite glass and YAG:Ce, respectively. h is Planck constant. v_e is the main electron absorption frequency in ultraviolet light. n_G and n_Y are the refractive indices of tellurite glass and YAG:Ce, respectively. According to the parameters in Supplementary Table 1, it can be calculated that A is approximately equal to 2.15 zJ. Therefore, the van der Waals interaction between two similar YAG:Ce particles in tellurite glass melt can be further derived as follows: $W_{\mathrm{vdW}}\left(D\right)=-\frac{2.15R}{12D}$
Equation (3) is valid only if the contact distance D between two YAG:Ce particles in tellurite glass melt is about more than two atomic layers (about 0.4 nm)^{44 (link),45 (link)}. Thus, when D = 0.4 nm, the maximum attraction W_{vdW (min)} between two YAG:Ce particles in tellurite glass melt is calculated to be −2.42 × 10³ zJ. The van der Waals potential energy easily induces YAG:Ce particles to aggregate when the other repulsive force cannot be generated.