Translational and rotational diffusion of a single nanorod in unentangled polymer melts

Polymer nanocomposites have been an issue of both academic and industrial interest due to promising electrical, mechanical, optical, and magnetic properties. The dynamics of nanoparticles in polymer nanocomposites is a key to understanding those properties of polymer nanocomposites and is important for applications such as self-healing nanocomposites. In this article we investigate the translational and the rotational dynamics of a single nanorod in unentangled polymer melts by employing extensive molecular dynamics simulations. A nanorod and polymers are modeled as semiflexible tangent chains of spherical beads. The stiffness of a nanorod is tuned by changing the bending potential between chemical bonds. When polymers are sufficiently long and the nanorod is stiff, the nanorod translates in an anisotropic fashion along the nanorod axis within time scales of translational relaxation times even in unentangled polymer melts. The rotational diffusion is suppressed more significantly than the translational diffusion as the polymer chain length is increased, thus the translational and rotational diffusion of the nanorod are decoupled. We also estimate the winding numbers of polymers, i.e., how many times a polymer winds the nanorod. The winding number increases with longer polymers but is relatively insensitive to the nanorod stiffness.

Figure 1

(a) A simulation snapshot for $N=32$ and $K_b=20$. A yellow molecule is the nanorod. Red polymers are ones that wind the nanorod with the winding number $W>0.6$. Grey polymers do not wrap but interact with the nanorod via nonbonding intermolecular interactions. (b) A schematic for the estimation of the winding number $W$. $\mathrm{\Delta }{\varphi }_{ij}$ is the angle between two vectors projected onto the surface perpendicular to the chemical bond of the nanorod.

Figure 2

Six schematic examples for how a polymer (red) may wind a nanorod (yellow) and corresponding winding numbers $W$. Monomers of the polymer and the nanorod bead of interest are projected onto the plane normal to the chemical bond for the nanorod bead.

Figure 3

Simulation results for (a) the persistence length ($l_p$) and (b) the mean-square end-to-end distance ($R_E$) of a single nanorod as a function of $K_b$. In this case, $N=32$ and $\rho =0.8$.

Figure 4

(a) The site-site intermolecular correlation functions [$g_{NP}\left(r\right)$] between the nanorod and polymers and (b) site-site intermolecular correlation functions [$g_{PP}\left(r\right)$] between polymers.

Figure 5

(a) $\langle {\left(\mathrm{\Delta }r\right)}^2\left(t\right)\rangle$ and (b) $U\left(t\right)$ of the nanorod in polymer melts for $K_b=20$ and different values of N.

Figure 6

(a) The translational diffusion coefficient $D_T$ and (b) relative relaxation times (${\tau }_T/{\tau }_{T,N=1}$ and ${\tau }_R/{\tau }_{R,N=1}$) of the nanorod in polymer melts for $K_b=20$ and different values of $N$. ${\tau }_{T,N=1}$ and ${\tau }_{R,N=1}$ are the translational and rotational relaxation times of the nanorod for $N=1$.

Figure 7

(a) $\langle {\left(\mathrm{\Delta }r\right)}^2\left(t\right)\rangle$ and (b) $U\left(t\right)$ of the nanorod in polymer melts for $N=32$ and different values of $K_b$.

Figure 8

The probability distribution functions [$P\left(W\right)$] of the winding number $W$ for (a) $K_b=20$ and (b) $N=64$.

Figure 9

(a) The probability distribution functions $P\left(T\right)$ of the total winding number $T$ and (b) the probability distribution functions $P\left(S\right)$ of the effective winding number $S$. $K_b=20$ and $N$ ranges from 2 to 64.

Figure 10

A simulation snapshot for a single nanorod (yellow) in polymers melts of $N=64$. Red, orange, green, pink, purple, and ice-blue neighbor polymers wind the nanorod at the same time with winding numbers $W=0.93$, 0.56, 0.94, 0.71, 0.81, and 0.56, respectively. All other polymers are colored in grey.

Figure 11

$A\left(t\right)$ of the nanorod for (a) $K_b=20$ and different values of $N$ and (b) $N=32$ and different values of $K_b$.

Figure 12

(a) The orientational correlation coefficient ($S_2$) between two end-to-end vectors of both the nanorod and the local segment of polymers as a function of the shortest distance ($r_{\min }$) between the local segment and the nanorod. (b) The time-dependent diffusion coefficient $\langle D(t;r)\rangle$ as a function of time $t$ for different initial shortest distance $r$ between the monomer and the nanorod for $N=64$ and $K_b=20$. (c) $\langle D(t;r)\rangle$ of monomers during time $t=96$ as a function of the initial shortest distance $r$.