Relative importance of electrostatic and van der Waals forces in particle adhesion to rough conducting surfaces

It is commonly assumed that van der Waals forces dominate adhesion in dry systems and electrostatic forces are of second order importance and can be safely neglected. This is unambiguously the case for particles interacting with flat surfaces. However, all surfaces have some degree of roughness. Here we calculate the electrostatic and van der Waals contributions to adhesion for a polarizable particle contacting a rough conducting surface. For van der Waals forces, surface roughness can diminish the force by several orders of magnitude. In contrast, for electrostatic forces, surface roughness affects the force only slightly, and in some regimes it actually increases the force. Since van der Waals forces decrease far more strongly with surface roughness than electrostatic forces, surface roughness acts to increase the relative importance of electrostatic forces to adhesion. We find that for a particle contacting a rough conducting surface, electrostatic forces can be dominant for particle sizes as small as ∼1–10 μm.

Figure 1

Schematic of the two systems considered in this study: (a) a flat dielectric slab interacting with a rough conducting and grounded surface and (b) a dielectric sphere interacting with a rough conducting and grounded surface. For presentation purposes we depict the system in 2D, but the simulations were carried out in 3D.

Figure 2

(a) The electrostatic force and (b) the fractional error as a function of r/R for a particle with $R=0.1\mu \mathrm{m}$ and $\varepsilon =1$ adhering to a rough surface with $A=0.1\mu \mathrm{m}$, and $\lambda =1\mu \mathrm{m}$. The fractional error is that between the force calculated using the approximation shown in Fig. 1, where the radius of the rough surface region is $r$, and the force calculated when the entire surface in the cell is simulated as a rough surface.

Figure 3

The enhancement of the electrostatic adhesion force, represented by $F_E/F_E^S-1$, due to roughness on the conducting surface as a function of roughness amplitude, $A$, and wavelength, λ. Results are for a dielectric flat slab with $\varepsilon =20$; the dielectric slab has a surface charge, but the ratio $F_E/F_E^S$ is independent of the value of the surface charge.

Figure 4

The ratio of electrostatic to van der Waals forces, $F_E/F_V$, for a charged flat dielectric surface with $\varepsilon =20$ and ${\sigma }^2/H={10}^9{\mathrm{C}}^2{\mathrm{m}}^{–4}{\mathrm{J}}^{–1}$ in contact with a sinusoidal conducting surface as a function of roughness amplitude, $A$, for three wavelengths of roughness: $\lambda =1\mu \mathrm{m}$ (black), $\lambda =5\mu \mathrm{m}$ (red), $\lambda =25\mu \mathrm{m}$ (blue), and for a smooth conducting surface (black-dashed).

Figure 5

The effect of roughness on the electrostatic adhesion force, represented by $F_E/F_E^S$, on an insulating sphere with $\varepsilon =1$ as a function of $R/\lambda$ for rough surfaces with $A/\lambda =0.05$ (green), 0.1 (red), and 0.3 (blue). Three different wavelengths of roughness are calculated for each $A/\lambda , \lambda =200\mathrm{nm}$ (squares), 500 nm (circles), and 1000 nm (triangles). Due to computational limitations, we could not carry out calculations for indefinitely large particle sizes.

Figure 6

The effect of surface roughness on the electrostatic adhesion force, represented by $F_E/F_E^S$, on an dielectric sphere with $\varepsilon =20$ as a function of $R/\lambda$ for rough surfaces with (a) $A/\lambda =0.05$, (b) $A/\lambda =0.1$, and (c) $A/\lambda =0.3$. Three different wavelengths of substrate roughness are calculated for each $A/\lambda , \lambda =200\mathrm{nm}$ (black squares), 500 nm (red circles), and 1000 nm (blue triangles). Due to computational limitations, we could not carry out calculations for indefinitely large particle sizes.

Figure 7

The effect of surface roughness on the electrostatic adhesion force, represented by $F_E/F_E^S$, on spheres with $\varepsilon =1$ (black), 2 (blue), 5 (red), and 20 (green) as a function of $R/\lambda$ for a rough surface with $A=100\mathrm{nm}$ and $\lambda =1000\mathrm{nm}$.

Figure 8

The ratio of electrostatic to van der Waals adhesion force, $F_E/F_V$, for a dielectric particle with (a) $\varepsilon =1$ and (b) $\varepsilon =20$. The results are obtained with ${\sigma }^2/H={10}^9{\mathrm{C}}^2{\mathrm{m}}^{–4}{\mathrm{J}}^{–1}$. Three rough substrates were considered: $A=60\mathrm{nm}$ and $\lambda =200\mathrm{nm}$ (black); $A=150\mathrm{nm}$ and $\lambda =500\mathrm{nm}$ (red); and $A=300\mathrm{nm}$ and $\lambda =1000\mathrm{nm}$ (blue). The electrostatic forces were calculated using the numerical methodology (solid lines) and analytic approximations (dotted lines). The analytic approximations of $F_E/F_V$ were calculated by the ratio of (a) Eqs. (2) and (8), and (b) the empirical fit from Matsuyama and Yamamoto [18] for the electrostatic force between a polarizable sphere and a flat surface and Eq. (8). The rough substrate $F_E/F_V$ is compared with that of a smooth substrate represented by the black-dashed line.