Wall slip of complex fluids: Interfacial friction versus slip length

Using a dynamic surface force apparatus (SFA), we demonstrate that the notion of slip length used to describe the boundary flow of simple liquids is not appropriate for viscoelastic liquids. Rather, the appropriate description lies in the original Navier's partial slip boundary condition, formulated in terms of an interfacial friction coefficient. We establish an exact analytical expression to extract the interfacial friction coefficient from oscillatory drainage forces between a sphere and a plane, suitable for dynamic SFA or atomic force microscopy noncontact measurements. We use this model to investigate the boundary friction of viscoelastic polymer solutions over 5 decades of film thicknesses and 1 decade in frequency. The proper use of the original Navier's condition describes accurately the complex hydrodynamic force up to scales of tens of micrometers, with a simple Newtonian-like friction coefficient that is not frequency dependent and does reflect closely the dynamics of an interfacial depletion layer at the solution-solid interface.

Figure 1

The dynamic surface force apparatus. Left: Large view of the sphere-plane contact. Center: Detailed view of the contact. Right: Schematic of the flow.

Figure 2

Left: inverse of the components of the dynamic force response $\tilde{Z}$ measured in an HPAM solution at a frequency of 220 Hz, as a function of the sphere-plane nominal distance $D$. Blue, $1/Z_R$; red, $1/Z_I$. The top left inset shows the quasistatic interaction force $F_{\mathrm{stat}}$, whose jump-to-contact defines the origin of distances. The bottom right inset is an enlargement at the submicrometric scale. Right: Slip length defined from the extrapolation length of $1/Z_I$ (intercept with the $x$ axis of the black dashed line of the left plot), as a function of the concentration of the solution, for various frequencies.

Figure 3

Left: schematic of the sphere-plane geometry and of the flow profile. Right: elastic component ($1/\tilde{Z}{)}_R$ (blue dots) and dissipative component ($1/\tilde{Z}{)}_I$ (red dots) of the inverse of the force response measured in an HPAM solution at a frequency of 220 Hz. The black continuous lines are the components of the theoretical expression (8) fitted with a real-valued boundary friction coefficient $\tilde{\lambda }={\lambda }_R$. Inset: enlargement below the micrometric scale. The red dashed line is the extrapolation of the far-field behavior of ${(1/\tilde{Z})}_I$.

Figure 4

Left: components ${\eta }_R$ ($\blacksquare$) and ${\eta }_I$ ($\square$) of the viscoelastic modulus $\tilde{\eta }$ of the solutions as a function of polymer concentration, at 30 Hz (black), 220 Hz (red), and 248 Hz (blue). Right: interfacial friction coefficient $\tilde{\lambda }={\lambda }_R$ at the solution-solid boundary as a function of the HPAM concentration, at 30 Hz (black), 220 Hz (red), and 248 Hz (blue). The dashed lines are a guide for the eye.

Figure 5

Log-log plot of the components of $1/\tilde{Z}$ (blue dots, real; red dots, imaginary) in a solution of 1.4 g/L at 220 Hz, showing the macro-micro transition. The dashed red (resp. green) line corresponds to a Newtonian fluid (resp. water) with a no-slip b.c. at the solid wall. The black continuous lines are the components of the theoretical expression (8) taking into account the bulk viscoelasticity and a Navier's b.c. with a real-valued friction coefficient ${\lambda }_R$. The dashed continuous lines are Eq. (8) with the boundary condition applied inside the liquid, at a distance $e_s/2={\eta }_{\mathrm{water}}/2{\lambda }_R=14$ nm of each solid surface.