Polymer Topology and the Depletion Interaction

Abstract

Using a theoretical model we show that ideal ring polymers are stronger depletants than ideal linear polymers of equal radii of gyration, but not of equal hydrodynamic radii. The difference in the depletion-induced force profile is largely controlled by the thickness of the depletion layer. Theory suggests that this thickness is equal to the average extent of a polymer along the direction perpendicular to the surfaces of the colloids. Within the limits of finite-size effects, Molecular Dynamics simulations support this conclusion.

Figures (4)

• Figure 1: Pictorial representation of the model. Two colloidal plates (gray) of area $A$ and polymers (beads) of radius of gyration $R_g^2\ll A$ are immerse a chamber of volume $V$. A thermostat and a polymer reservoir ensure that the temperature of the chamber is $T$, and the chemical potential of the polymers is equal to $\mu$.
• Figure 2: Confinement free energy and depletion interaction between colloidal plates. In all panels, theoretical results for linear polymers, ring polymers, and spherical micelles are shown as continuous red lines, dashed blue lines, and dotted black lines, respectively. Results from MD simulations are reported as gray dots, orange circles, and light-blue squares for spherical micelles, linear polymers, and ring polymers, respectively. (a) Ratio of the partition function of the polymer under confinement divided by the partition function for the polymer in bulk. The left inset highlights the results for $R_g<h<2R_g$. The inset on the right shows the discrepancy between theory and simulations, $\delta$, as a function of the length of the simulated polymer, where $\delta=\sum_{k=1}^M|y_{MD,i}-y_{Theory}(x_i)|/M$, with $y$ being the ratio of partition functions. The black line indicates a power law fit against all the data using SciPy. SciPy Panels (b) and (c) show the depletion force and the energy caused by various macromolecules. The force is scaled by the osmotic force, $\Pi A$, and the energy is measured in units of osmotic force multiplied by the radius of gyration, $\Pi A R_g$. In the insets of panels (b) and (c) the results are reported with the distance between the plates scaled by the hydrodynamic radius, $R_H$. In the inset of panel (c), the energy is reported relative to $\Pi A R_H$.
• Figure 3: Scaling by the thickness of the depletion layer, $\xi$. The panels and colors are the same as in Fig. \ref{['Fig:Theory+MD']} with the distance between plates scaled by the depletant-specific $\xi$.
• Figure 4: Polymer extension in one direction, $\Delta$. (a) Pictorial representation of $\Delta$. The first bead of the polymer chain ($x_1$) is marked in black, the one closest to the left wall in red ($x_{min}$), and the one nearest to the right wall in blue ($x_{max}$). The distance in the direction perpendicular to the wall between these last two beads is $\Delta$. One can rigidly move the chain by $h-\Delta$ without clashing with the walls. (b) The orange circles and light-blue squares indicate results for linear and ring polymers, respectively. The inset in the middle shows the mean of the distribution as a function of the length of the polymers. The red continuous line and the blue dashed line indicate the theoretical results for linear and ring polymers, respectively. The inset on the right shows the relative error between the theoretical and computational effective radius, $\delta=|\langle \Delta \rangle-2\xi|/(2\xi)$, as a function of polymer length. The black was fit using SciPy SciPy to a power law.