Translational and rotational coupling in Brownian rods near a solid surface

An anisotropic macromolecule confined between two surfaces displays Brownian motion predominantly in the plane parallel to these surfaces. It can be expected that both the rotational and translational diffusion coefficients are strongly affected by hydrodynamic interactions with the walls. This work studies the more extreme case in which a rodlike particle comes into contact with a wall or in very close proximity (order of 100 nm). Experimental data have been gathered and analyzed demonstrating the rod tethering on a surface. This is compared with numerical simulations to allow estimates of proximity to the surface. The experimental data show that particle tethered motion is subject to varied degrees of constraining which imply subtle deviations in the Brownian dynamical behavior. The key finding is that a rotational-translational coupling occurs which is markedly different from the translational and rotational movements normally predicted for anisotropic macromolecules.

Figure 1

(Color online) Description of a Brownian rod (state A) in close proximity to a surface which may attach directly (state A to C) or via an intermediate tethered mode (state A to B to C).

Figure 2

(Color online) (A) Image (cropped) of a nanofiber, annotated with the axes used in the subsequent analysis. (B) A shish-kebab rod consisting of ten consecutive spheres in a slit of height $L_z$, as used for the computational model. The longest rod axis is aligned in the $yz$ plane and is (almost) parallel to the confining walls. The distance $h$ indicates the closest distance between wall and rod, and $l^{\prime }$ is the pivot to centroid length, as explained in the text.

Figure 3

(Color online) Computed translational diffusion coefficients as a function of normalized closest distance $h/L_z$ for an untethered shish-kebab rod of $6\mu \text{m}$ length and $0.6\mu \text{m}$ diameter, lying almost parallel to the walls of a slit of height $L_z=4.5\mu \text{m}$ filled with water $\left(\eta ={10}^{-3}\text{Pa}\text{s}\right)$. Different directions are indicated by different symbols: translations along the longest rod axis ($\sim y$; circles), translations perpendicular to the rod but along the wall ($\sim x$; squares), and translations perpendicular to the rod and perpendicular to the wall ($\sim z$; diamonds). Open (closed) symbols are for an angle of $80°$ $\left(90°\right)$ between wall normal and rod axis. Dashed lines indicate the rod diffusion coefficients $D_a$ and $D_b$ in bulk.

Figure 4

(Color online) Computed rotational diffusion coefficients for rotations around the sphere located at the rod end which is closest to the wall [see Fig. 2B] as a function of normalized closest distance $h/L_z$. Rotations around the $x$ axis are indicated by circles, and rotations around the $z$ axis are indicated by squares. Open (closed) symbols are for an angle of $80°$ $\left(90°\right)$ between wall normal and rod axis.

Figure 5

(Color) (A) The plot of the motion of the centroid of a rod undergoing the transition from free random movement (blue) to tethered arc like movement (red); (B) the associated angular values which complete the data set.

Figure 6

(Color online) The diffusion coefficient calculated by segmenting the full data set into windows consisting of 8000 sequential images, with the window being moved in steps of 100 images, hence giving plots of (A) $D_x$, (B) $D_a$, (C) $D_b$, and (D) $D_{\theta }$ against time. In (B) the different regimes are shown: region I indicates a free rod, region II is a transitional stage which comes about due to the length of the window, and region III is a tethered rod.

Figure 7

(Color) Spatial trajectories of the centroid (arcs) and tethered ends (spots) for carbon nanofibers measuring (A) $5.56\times 0.87$ $\left(\text{length}\times \text{diameter}\right)\mu {\text{m}}^2$ and (B) $7.50\times 1.04\mu {\text{m}}^2$. Color coding has been used to split the data into time segments; in the case of (A) the segment lengths are chosen to highlight the movement of the tethered end, while in (B) the segments are equal in length. While both nanofibers are end tethered, they exhibit subtle differences which have important ramifications for the Brownian dynamics at play.

Figure 8

(Color online) The rotational mean square difference parameter calculated for two carbon nanofibers, $5.56\times 0.87$ $\left(\text{length}\times \text{diameter}\right)\mu {\text{m}}^2$ in blue (upper) line [case (A)] and $7.50\times 1.04\mu {\text{m}}^2$ in green (lower) line [case (B)], both of which exhibit end tethered behavior. For case (A) the data are linear, while for case (B) a loss of linearity occurs at larger time intervals.

Figure 9

(Color online) (A) Time evolution of the angular position $\theta$ (degrees) and (B) the calculated radius of rotation $l$ $\left(\mu \text{m}\right)$ for the case of the carbon nanofiber measuring $5.56\times 0.87$ $\left(\text{length}\times \text{diameter}\right)$.