Investigation of surface/bulk stresses of nanoparticles with diffusive interfaces using the phase field crystal model

We investigate the surface stress of solid-liquid diffusive interfaces at equilibrium using the phase field crystal model in two dimensions. To analytically study the surface energy dependence on elastic strains, we employ the amplitude equation formalism and recast the free energy functional in terms of strains and amplitudes of density waves to examine the intricate coupling between them. For planar interfaces, the surface stress and its anisotropy are explored using the phase field crystal model and its amplitude equations. The anisotropy of surface stress is shown to be closely related to the anisotropic density waves across the interface, and a stronger anisotropy is observed in the surface stress than that in the surface energy. In addition, to investigate the curvature effect, we examine resultant strain fields at equilibrium within nanoparticles of various sizes subjected to their surface stresses. The measured strains are compared with the classical sharp interface model. The discrepancy in strain fields arises as the size of nanoparticles becomes smaller which suggests a curvature dependent surface stress for diffusive interfaces. We construct the effective surface stress using the measured strain fields and classical linear elasticity theory. The overall magnitude of the effective surface stress decreases with the radius of the nanoparticle, and the effective surface stress is shown to be more isotropic for small nanoparticles.

Figure 1

The surface shearing and the substrate shearing share the same deformation effect on the surface which is indicated in a blue line. The solid line represents the deformed state and the dashed line represents the reference state.

Figure 2

The profiles of (a) the magnitude squared of effective RLVs, $|K_j^{\mathrm{eff}}{|}^2$, as well as (b) the normal components $u_{n,n}$ and ${\phi }_{,n}$ across the interface for the close packed orientation of which the normal vector of the interface is $\widehat{n}=(1,0)$. The interface is located at $z=0$ and the solid (liquid) is in the negative (positive) regime of $z$.

Figure 3

The profiles of the transverse stress ${\sigma }_{tt}$ for two orientations across the interface.

Figure 4

The snapshots of the solid-liquid interfaces at equilibrium in PFC simulation for two orientations, namely, (a) the close-packed orientation and (b) the zigzag orientation.

Figure 5

The excess interfacial energy as function of strains for $\epsilon =0.1$. The PFC simulation results for $0^{\circ }$ (close-packed) and ${30}^{\circ }$ (zigzag) orientations are shown in the blue circles and red squares, respectively. The solid lines are the fitted quadratic curves.

Figure 6

The profiles of amplitudes for two orientations, (a) the close-packed and (b) the zigzag orientations of which the surface normal $\widehat{n}$ are $(1,0)$ and $(\frac{\sqrt{3}}{2},\frac{1}{2})$, respectively. The real and imaginary parts of the amplitudes are shown in the solid and dashed lines, respectively. Note that $A_1=A_2$ for the close packed orientation and the real parts of $A_1$ and $A_3$ are identical for the zigzag orientation.

Figure 7

(a) The surface energy and (b) the surface stress of different orientations for $\epsilon =0.1$ are obtained from the PFC model (black squares) and the AE model (red circles).

Figure 8

The $\epsilon$ dependence of the surface stress for $0^{\circ }$ orientations. The result of PFC simulation is shown in squares and the AE result is shown in the solid line.

Figure 9

The $\epsilon$ dependence of anisotropy parameters $\alpha$ of $\tau$ (black) and $\gamma$ (red) is plotted for PFC (squares) and AE (lines) simulations.

Figure 10

The density shifts are measured in both phases and compared with the prediction of sharp interface model. The density variation for solid (liquid) is shown in black (blue). The simulation results from the PFC and the AE are depicted in squares and circles, respectively. The theoretical prediction in Eq. (54) is shown for PFC and AE in solid lines and dashed lines, respectively.

Figure 11

The anisotropy of the equilibrium shape of various crystal size is plotted for PFC (squares) and AE (circles) simulations. The theoretical estimation of the shape anisotropy for large nanoparticles, ${\alpha }_R\cong {\alpha }_{\gamma }$, is shown in dashed lines.

Figure 12

The strains plot of (a) the radial and (b) the angular components for $R_0=991$ measured in PFC, and (c) the radial and (d) the angular components for $R_0=935$ calculated in AE. The dashed circle represents the position of interface. The strains are only plotted for the solid regime where the elastic moduli remain constant.

Figure 13

The strain profiles of (a) the radial and (b) the angular components measured in PFC $\left(R_0=991\right)$, and (c) the radial and (d) the angular components calculated in AE $\left(R_0=935\right)$. The results are shown in black dots for the closed packed orientation $\left(0^{\circ }\right)$ and blue dots for the zigzag orientation $\left({30}^{\circ }\right)$. The sharp interface prediction in Eqs. (13) and (14) is plotted in red lines for $0^{\circ }$ and green lines for ${30}^{\circ }$.

Figure 14

The bulk strain ${\varepsilon }_{\alpha \beta }^{\left(0\right)}$ of the PFC (black) and the AE (blue) simulations is compared with the prediction by the sharp interface theory in Eq. (10) which are plotted in dashed lines.

Figure 15

The profiles of the first harmonics of (a) the radial and (b) the angular strain components measured from PFC simulations, and (c) the radial and (d) the angular components calculated in AE method. The green line is the prediction by sharp interface theory in Eqs. (13) and (14).

Figure 16

(a) The average value and (b) the first harmonics of effective surface stress in seed simulation is compared with the sharp interface theory. The simulation result is shown in black squares for PFC and red circles for AE, and the prediction is shown in a dashed line. The mean curvature ${\kappa }_0$ is defined as ${\kappa }_0\equiv 1/R_0$.