Shear rate threshold for the boundary slip in dense polymer films

The shear rate dependence of the slip length in thin polymer films confined between atomically flat surfaces is investigated by molecular dynamics simulations. The polymer melt is described by the bead-spring model of linear flexible chains. We found that at low shear rates the velocity profiles acquire a pronounced curvature near the wall and the absolute value of the negative slip length is approximately equal to the thickness of the viscous interfacial layer. At higher shear rates, the velocity profiles become linear and the slip length increases rapidly as a function of shear rate. The gradual transition from no-slip to steady-state slip flow is associated with faster relaxation of the polymer chains near the wall evaluated from decay of the time autocorrelation function of the first normal mode. We also show that at high melt densities the friction coefficient at the interface between the polymer melt and the solid wall follows a power-law decay as a function of the slip velocity. At large slip velocities the friction coefficient is determined by the product of the surface-induced peak in the structure factor, the temperature, and the contact density of the first fluid layer near the solid wall.

Figure 1

(Color online) A schematic of a steady-state shear flow over a flat solid wall. (a) The slip velocity and the slope of the linear velocity profile are related via $V_s=\dot{\gamma }{L}_s$, where $L_s$ is the slip length. (b) Apparent slip is associated with the lower viscosity boundary layer. (c) The velocity profile with a downward curvature due to higher viscosity boundary layer is described by the negative slip length. (d) A combination of the “true” slip at the liquid/solid interface and the curvature of the velocity profile.

Figure 2

(Color online) A snapshot of fluid monomers (open blue circles) and wall atoms (filled gray circles) positions projected on the $xz$ plane. The bottom wall is at rest and the top wall is moving with a constant velocity $U$ in the $\widehat{x}$ direction. Each fluid monomer belongs to a polymer chain. Five polymer chains are marked by solid lines. The fluid monomer density is $\rho =1.08{\sigma }^{-3}$ and the top wall speed is $U=0.1\sigma /\tau$.

Figure 3

(Color online) Averaged monomer density profiles near the stationary lower wall with ${\epsilon }_{\text{wf}}/\epsilon =0.9$. The uniform fluid densities away from the walls are (a) $\rho =1.04{\sigma }^{-3}$ and (b) $\rho =1.08{\sigma }^{-3}$. The upper wall velocities $U$ are tabulated in the inset. The left vertical axis indicates the location of the fcc lattice plane at $z=-15.24\sigma$. The dashed line at $z=-14.74\sigma$ denotes the reference plane for computing the slip length.

Figure 4

(Color online) Average normalized velocity profiles for the indicated values of the upper wall speed and the fluid density $\rho =1.04{\sigma }^{-3}$. The vertical axes coincide with the position of the fcc lattice planes at $z=-15.24\sigma$ and $z=8.77\sigma$. The dashed lines denote liquid/solid interfaces at $z=-14.74\sigma$ and $z=8.27\sigma$. The inset shows an enlarged view of the velocity profiles near the stationary lower wall.

Figure 5

(Color online) Averaged velocity profiles for the indicated upper wall speeds $U$ (in units of $\sigma /\tau$) and the fluid density $\rho =1.08{\sigma }^{-3}$. The vertical axes indicate the location of the fcc lattice planes at $z=-15.24\sigma$ and $z=7.86\sigma$. The dashed lines denote the position of the liquid/solid interfaces $0.5\sigma$ away from the fcc planes. The region near the lower wall is enlarged in the inset.

Figure 6

(Color online) Average normalized velocity profiles for the upper wall speed $U=0.01\sigma /\tau$ (see text for description). The vertical dashed lines indicate liquid/solid interfaces at $z=-14.74\sigma$ and $z=6.81\sigma$. The inset shows averaged velocity profiles for the upper wall speeds $U=0.125\sigma /\tau$ (dashed violet curve), $U=0.3\sigma /\tau$ (continuous blue curve), and $U=1.0\sigma /\tau$ (dashed-dotted green line).

Figure 7

(Color online) Shear-rate dependence of the viscosity in the bulk region $\mu /\epsilon \tau {\sigma }^{-3}$ for the indicated values of the polymer density. The dashed line with a slope $-0.37$ is shown for reference. Solid curves are a guide for the eye.

Figure 8

(Color online) Slip length $L_s/\sigma$ as a function of shear rate for the indicated values of the polymer melt density. At low shear rates and $\rho =1.11{\sigma }^{-3}$, the slip length is multivalued. Solid curves are drawn to guide the eye.

Figure 9

(Color online) Shear stress ${\sigma }_{xz}$ (in units of $\epsilon {\sigma }^{-3}$) averaged in the bulk region as a function of the slip velocity of the first fluid layer for the indicated values of the polymer melt density. Solid curves are a guide for the eye.

Figure 10

(Color online) Log-log plot of the friction coefficient $k={\sigma }_{xz}/V_1$ (in units of $\epsilon \tau {\sigma }^{-4}$) as a function of the slip velocity [computed from Eq. (7)] for the tabulated values of the polymer density. The dashed line with a slope of $-0.7$ is plotted for reference. The same data as in Fig. 9. Solid curves are drawn as a guide for the eye.

Figure 11

(Color online) Structure factor $S\left({\mathbf{G}}_1\right)$ evaluated at the first reciprocal lattice vector ${\mathbf{G}}_1=\left(7.23{\sigma }^{-1},0\right)$ in the shear flow direction (a), contact density near the stationary lower wall (b), and temperature (c) of the first fluid layer as a function of the slip velocity $V_1$ (in units of $\sigma /\tau$). The solid blue line $y=6.93–7.73\text{ln}\left(x\right)$ represents the best fit to the data. Solid curves are a guide for the eye.

Figure 12

(Color online) Log-log plot of the inverse friction coefficient $k^{-1}=V_1/{\sigma }_{xz}$ (in units of ${\sigma }^4/\epsilon \tau$) as a function of $T_1{k}_B/\epsilon {\left[S\left({\mathbf{G}}_1\right){\rho }_c{\sigma }^3\right]}^{-1}$ evaluated in the first fluid layer for the indicated polymer melt densities. The solid line with a slope of 0.9 is plotted as a reference.

Figure 13

(Color online) Log-log plot of the inverse friction coefficient $k^{-1}=V_1/{\sigma }_{xz}$ (in units of ${\sigma }^4/\epsilon \tau$) as a function of the variable $S\left(0\right)/\left[S\left({\mathbf{G}}_1\right){\rho }_c{\sigma }^3\right]$ computed in the first fluid layer. The fluid densities are listed in the inset. The dashed line with a slope of 1.15 is shown for reference.

Figure 14

(Color online) Normalized time autocorrelation function of the first normal mode at equilibrium for polymer chains (a) in the bulk region and (b) near the walls for the indicated values of the fluid density.

Figure 15

(Color online) Time autocorrelation function of the first normal mode for polymer chains (a) in the bulk region and (b) near the walls and the fluid density $\rho =1.08{\sigma }^{-3}$. The upper wall speed from right to left is $U=0$, 0.025, 0.1, 0.2, 0.5, 1.0, 2.0, 3.0, and 4.0 (in units of $\sigma /\tau$).

Figure 16

(Color online) Inverse relaxation time of the polymer chains near the walls (top) and the slip length (bottom) as a function of shear stress ${\sigma }_{xz}$ (in units of $\epsilon {\sigma }^{-3}$) for the indicated values of the melt density. The inset shows the stretched exponential coefficient for the same data.