Particle adhesion to rough surfaces

While particle adhesion to smooth surfaces is well understood, real surfaces are not perfectly smooth, and the effects of surface roughness on adhesion are not easily characterized. We develop a theory for the effects of surface roughness on the strength of particle adhesion due to van der Waals forces, in the Derjaguin-Muller-Toporov (DMT)-type adhesion regime. We first address a well-defined rough surface created by embedding spheres in a smooth substrate, which had been previously examined experimentally. We derive an analytic expression for the adhesive force of particles to this well-defined surface, with the key distinction from the previous work being the inclusion of interactions from surface asperities not in direct contact with the particle. We show that our theory is in good agreement with experimental results in the DMT regime. Within appropriate limits, we extend our theory to general rough surfaces and verify the theory by comparing to the exact numerical results. We show that the interactions from surface asperities not in direct contact with the particle are the dominant contribution to the adhesive force under some conditions, and our theory predicts the experimental and numerical adhesion forces very accurately.

Figure 1

Using the Rumpf and Rabinovich models, the total van der Waals force between a sphere and a rough surface can be approximated by the sum of two parts: the interaction of the particle with the asperity it is atop, represented as a sphere (blue dashed circle), and the interaction of the particle with a smooth surface at, in the Rumpf model, the center of the spherical asperity (green dashed line), and in the Rabinovich model, the average height of the rough surface (red dashed line).

Figure 2

(a) At large λ the particle can sit anywhere on the surface, but at small λ the particle must sit atop the embedded spheres. (b) Definition of parameters describing geometry of particle and surface.

Figure 3

Our modeled interaction between a spherical particle and the well-defined surface of Ramakrishna et al. [20, 39]. In Eq. (15), the second term accounts for contact interaction with the embedded sphere below the particle (blue), and the first and third term account for the noncontact interactions with the substrate (green) and embedded spheres (red), respectively.

Figure 4

(a) The total force [red solid line; Eq. (15)], the force from the substrate [blue dashed line; first term of Eq. (15)], and the force from the embedded spheres [black dashed line; last two terms of Eq. (15)] predicted by our proposed theory. Note there are no adjustable parameters in our theory. (b) Experimental results of Ramakrishna et al. [20, 39] (black squares) compared with the Rabinovich model [green; Eq. (16)], the Katainen model [blue; Eq. (17)], and our proposed theory [red; Eq. (15)]. The demarcation of the DMT and JKR regimes is approximate, and based on the experimental results of Ramakrishna et al. [39] that showed that the DMT regime applied at $245\mu {\mathrm{m}}^{-2}$ while the JKR regime applied at $450\mu {\mathrm{m}}^{-2}$.

Figure 5

Our modeled interaction between a sinusoidal surface and sphere. In Eq. (22), the first term accounts for the contact force contribution from the asperity directly under the center of the sphere (in blue) and second term accounts for the noncontact force contributions of all other asperities (in red).

Figure 6

$F_V/F_V^S$ as a function of $R$ for a particle interacting with a sinusoidal surface with $A=50\delta$ and $\lambda =100\delta$ and 300δ, separated by a distance of 1δ. Squares: explicit sum [Eq. (23)] with $r_{\mathrm{cut}}=8\delta$; Solid line: our proposed theory [Eq. (22)]. Dashed line: Rabinovich model [Eq. (4)].