Entropic Torque

Quantitative predictions are presented of a depletion-induced torque and force acting on a single colloidal hard rod immersed in a solvent of hard spheres close to a planar hard wall. This torque and force, which are entirely of entropic origin, may play an important role for the key-lock principle, according to which a biological macromolecule (the key) is functional only in a particular orientation with respect to a cavity (the lock).

Figure 1

A spherocylinder of length $L$ and diameter $\sigma$ at angle $\theta$ and distance $z$ relative to a planar hard wall at $z=0$. The minimal value of $z$ is $z_{\mathrm{m}\mathrm{i}\mathrm{n}}(\theta )=(\sigma +L|\cos \theta |)/2$. The rod is immersed in a solvent of hard spheres of diameter ${\sigma }_s$ and number density ${\rho }_s$ far from the wall.

Figure 2

Depletion potential $\beta W(z,\theta )$ for the spherocylinder shown in Fig. 1 for $L/\sigma =10$, $\sigma /{\sigma }_s=1$, and a packing fraction ${\eta }_s={\rho }_s\pi {\sigma }_s^3/6=0.2239$ of the hard-sphere solvent. The dashed and dotted lines represent the positions of local minima and maxima, respectively.

Figure 3

Different cuts through the depletion potential $\beta W[z_{\mathrm{m}\mathrm{i}\mathrm{n}}(\theta )+j\sigma /4,\theta ]$ for $j=0,1,2,3$ as calculated within DFT (full line) and the corresponding quantity within the AOa (dotted line) for the system described in Fig. 2. The correlation effects in the solvent cause quantitative and qualitative differences. Note that for reasons of clarity the curves for $j=2$ and $j=3$ have been shifted by 2 and 4, respectively.

Figure 4

The torque $\beta M(z,\theta )$ as function of $\theta$ for various values of $z$. For reasons of clarity the curves for $z=4.5,3.4,\dots$ are shifted downward by $-5,-10,\dots$, respectively. The horizontal lines $M=0$ are indicated partially. The parameters of the system are the same as in Fig. 2. Because of the hard wall at $z=0$ the torque is defined only for $\theta \ge \arccos (2z/L-\sigma /L)$ if $z<(\sigma +L)/2$. $M>0$ $(<0)$ corresponds to a force which tends to align the rod parallel (normal) to the wall. The lines denote our DFT results which are in excellent agreement with our simulations (symbols). The error bars of the simulations are of the order of the symbol size.