Molecular dynamics simulations of polymeric fluids in narrow channels: Methods to enhance mixing

Mixing of shear thinning polymeric fluids in long channels with patterned boundary conditions is studied through molecular dynamics simulations. Patterned wettability was shown to induce spatially varying slip lengths at the channel walls which in turn induce mixing in the fluid. To quantify the amount of mixing for different wave lengths of patterns, transverse velocity profiles were evaluated. The transverse velocity profiles from the molecular dynamics simulations were then compared with predictions from continuum modeling and good quantitative agreement was found. Offsetting the pattern was shown to produce better mixing in the center of the channel. Transverse flow is found to increase when the radius of gyration of the chains is smaller than the pattern length. We also implement an oscillating (time dependent) body force and find that the transverse flow increases significantly. However, we do not find an increase in transverse flow with frequency of the oscillation as predicted from continuum modeling and we postulate reasons for this behavior.

Figure 1

(Color online) $xy$-cross section of the simulation setup is shown here. The volume between the two parallel plates, placed at $\mp w$, was filled with polymeric fluid.

Figure 2

(Color online) Longitudinal velocity profiles obtained for different upper wall velocities $U$. The velocity profiles are linear except for slight curvature up to a distance of $\sigma$ away from both inner wall surfaces. The magnitude of the velocity profiles is linear in $U$.

Figure 3

The slip length is calculated for different shear rates. It increases almost linearly before starting to decrease.

Figure 4

(Color online) Log-log plot between shear viscosity $\eta$ and shear rate $\dot{\gamma }$. The viscosity of the fluid is given by $\eta =m{\left(\dot{\gamma }\right)}^{n-1}$. At high shear rates the plot is linear and can be seen in the embedded figure. A linear fit at high shear rates using the above equation yields a value of $n=0.60763\pm 0.013871$.

Figure 5

(Color online) Average longitudinal velocity profiles from simulations for different values of ${\epsilon }_{wf}$ for $y\ge 0$. Note, the longitudinal velocity profile is symmetric about $y=0$.

Figure 6

Slip length $\delta$ as a function of the interaction parameter ${\epsilon }_{wf}$.

Figure 7

(Color online) Schematic diagram representing the pattern slip boundary condition.

Figure 8

(Color online) Contour plots of the transverse velocity profile $u_y\left(x,y\right)$ for various $kw$. Alternative high positive or negative transverse velocity regions (dark) were observed in both upper and lower halves of the channel.

Figure 9

(Color online) The vorticity calculated for the same simulation as Fig. 8a. Note, maximum vorticity occurs in the region between maximum magnitude of transverse velocity.

Figure 10

(Color online) Snapshots of polymer conformations at different times. Note, on average chains near the middle of the channel are more elongated in the $x$ direction compared to chains near the boundaries. In addition, chains near the boundary are more compact.

Figure 11

(Color online) Contour plot of transverse velocity profile $u_y\left(x,y\right)$ for $kw=\pi /2$ with offset patterning.

Figure 12

The function $R$ as a function of $y/w$ solved using Eq. (15) with parameters selected to agree with molecular dynamics simulation (see text).

Figure 13

(Color online) The body force (circles) and maximum longitudinal velocity response (squares) as a function of time. The scale on the left-hand side represents body force and the one on the right-hand side represents the magnitude of $u_{x_{\text{max}}}$.

Figure 14

(Color online) Transverse velocity profile $u_y\left(x,y\right)$ for the time dependent body force case for the frequency $\omega =2\pi /10$.

Figure 15

(Color online) Transverse velocity profile $u_y\left(x,y\right)$ for constant body force case with 5000 averages.