Shear deformation of low-density polymer brushes in a good solvent

Self-consistent field approach is used to model a single end-tethered polymer chain on a substrate subject to various forces in three dimensions. Starting from a continuous Gaussian chain model, the following perturbations are considered: (i) hydrodynamic interaction with an externally imposed shear flow for which an alternative theoretical framework is formulated; (ii) excluded volume effect in a good solvent, treated in a mean field approximation; (iii) monomer-substrate repulsion. While the chain stretches along the flow, the change of the density profile perpendicular to the substrate is negligible for any reasonable simulation parameters. This null effect is in agreement with multiple neutron scattering studies.

Figure 1

Left panel: experimental reflectivity data on a logarithmic scale of base $e$ for a protonated PS brush $M_n=250\mathrm{kg}{\mathrm{mol}}^{-1}$ in air (black diamonds), in d-toluene static (blue circles) and in d-toluene at a shear rate of 500 ${\mathrm{s}}^{-1}$ (red squares). Right panel: fitted SLD profiles, where the dashed line is the corresponding monomer density of the brush, shown on the right axis.

Figure 2

Simulation summary. The semi-transparent cloud in the center shows the mushroom density profile (orange is denser). The streamlines of the solvent flow are visualized with cone glyphs, seeded at three different heights: green near the substrate, yellow in the middle of the brush, red at the top. The velocity field deviates from laminar, flowing around the core of the mushroom. As a result, there is upwards drag on the right side of the mushroom but also an equal and opposite downwards pull on the left side. The net vertical force is close to zero, resulting in zero change of the density profile perpendicular to the substrate. The simplified equations around the picture show key steps of the algorithm, described in detail in the main text.

Figure 3

Comparison between the modified diffusion equation and the diffusion-advection equation, both models of a uniform flow field. Parameters used in this example are $u=3, g=9/4$ (see Appendix).

Figure 4

The effect of the excluded volume on free [(a), (c), chain centered at (100,100,100) nm] and grafted [(b), (d), chain grafted at (100,100,0) nm] chains in $\theta$ (a), (b) and good (c), (d) solvents. The color code shows the 2D cross sections of the density profile, $\rho (x,100,z)$. The silver circle is the radius of gyration in each case. The thick yellow curve is the integrated density along the $x$-axis: $\rho \left(x\right)=\int dzdy\rho (x,y,z)$, displayed as an overlapping inset.

Figure 5

The lateral shift of the chain center of mass $\langle x\rangle =\int x\rho dydz$, as a function of the applied shear rate normalized to the Rouse relaxation time $\dot{\gamma }{\tau }_r$. The shift is maximum for a free draining chain and decreases with decreasing liquid screening length $\xi /R_g$.

Figure 6

The polymer density profile along the flow $x$ (blue and red curves), and perpendicular to flow $z$ (yellow and purple curves). We compare a free draining polymer (dotted lines, liquid penetration length $\xi /R_g\approx 1$) against one which strongly screens the flow (solid lines, $\xi /R_g\ll 1$), under the same applied shear rate. The effect of flow is entirely along the $x$-axis, where the density profile stretches and the center of mass moves away from the grafting point $x_0=0.5\mu \mathrm{m}$.