Elucidating fluctuating diffusivity in center-of-mass motion of polymer models with time-averaged mean-square-displacement tensor

There have been increasing reports that the diffusion coefficient of macromolecules depends on time and fluctuates randomly. Here a method is developed to elucidate this fluctuating diffusivity from trajectory data. Time-averaged mean-square displacement (MSD), a common tool in single-particle-tracking (SPT) experiments, is generalized to a second-order tensor with which both magnitude and orientation fluctuations of the diffusivity can be clearly detected. This method is used to analyze the center-of-mass motion of four fundamental polymer models: the Rouse model, the Zimm model, a reptation model, and a rigid rodlike polymer. It is found that these models exhibit distinctly different types of magnitude and orientation fluctuations of diffusivity. This is an advantage of the present method over previous ones, such as the ergodicity-breaking parameter and a non-Gaussian parameter, because with either of these parameters it is difficult to distinguish the dynamics of the four polymer models. Also, the present method of a time-averaged MSD tensor could be used to analyze trajectory data obtained in SPT experiments.

Figure 1

Correlation functions ${\widehat{\mathrm{\Phi }}}_i\left(t\right)(i=1,2)$ of the TMSD tensor calculated from trajectory data ${\mathit{R}}_G\left(t\right)$ of the Rouse model (circles and triangles). The COM trajectories ${\mathit{R}}_G\left(t\right)$ are generated through numerical simulations of the Rouse model. Time is measured in units of ${\tau }_0:=b^2/D$, where $b$ is the bond length and $D$ is the diffusion constant of the beads. The number $N$ of beads and the lag time $\mathrm{\Delta }$ are set as $N=50$ and $\mathrm{\Delta }=0.01{\tau }_0$, respectively. The solid lines are the theoretical predictions given by Eqs. (45) and (46). There are no fitting parameters (the same is true of Figs. 2–4).

Figure 2

Correlation functions ${\widehat{\mathrm{\Phi }}}_i\left(t\right)(i=1,2)$ of the TMSD tensor calculated from trajectory data ${\mathit{R}}_G\left(t\right)$ of the Zimm model (circles and triangles). The COM trajectories ${\mathit{R}}_G\left(t\right)$ are generated through numerical simulations of the Zimm model (see Appendix pp3). Distance is measured in units of the bond length $b$ and time in units of ${\tau }_0:=b^2/D$, where $D$ is the diffusion coefficient of each bead. Results for three different values of the bead radius $a$ are presented: (a) $a=0.1b$, (b) $a=0.15b$, and (c) $a=0.2b$. The number $N$ of beads and the lag time $\mathrm{\Delta }$ are set as $N=50$ and $\mathrm{\Delta }=0.01{\tau }_0$. The longest relaxation time ${\tau }_1$ is estimated from Eq. (60) as (a) ${\tau }_1=61.0{\tau }_0$, (b) ${\tau }_1=40.7{\tau }_0$, and (c) ${\tau }_1=30.5{\tau }_0$. The dotted lines are the theoretical predictions for ${\widehat{\mathrm{\Phi }}}_1^{\mathrm{id}}(\mathrm{\Delta },t)$ and ${\widehat{\mathrm{\Phi }}}_1^{\mathrm{ex}}\left(t\right)$ given by Eqs. (89) and (90). The dashed lines are the theoretical predictions for ${\widehat{\mathrm{\Phi }}}_2^{\mathrm{id}}(\mathrm{\Delta },t)$ and ${\widehat{\mathrm{\Phi }}}_2^{\mathrm{ex}}\left(t\right)$ given by Eqs. (91) and (92). The solid lines are the sums ${\widehat{\mathrm{\Phi }}}_i(\mathrm{\Delta },t)(i=1,2)$ of the ideal and excess parts [Eqs. (24) and (34)].

Figure 3

Correlation functions ${\widehat{\mathrm{\Phi }}}_i\left(t\right)(i=1,2)$ of the TMSD tensor calculated from trajectory data ${\mathit{R}}_G\left(t\right)$ of the reptation model (circles and triangles). The COM trajectories ${\mathit{R}}_G\left(t\right)$ are generated through numerical simulations of the reptation model. Distance is measured in units of the bond length $b$ between the tube segments, and time in units of ${\tau }_0:=b^2\zeta /k_{B}T$, where $\zeta$ is a friction coefficient of the tube segment. The number $N$ of tube segments and the lag time $\mathrm{\Delta }$ are set as $N=85$ and $\mathrm{\Delta }=250{\tau }_0$, respectively. For details of the simulation setup, see Ref. [47]. The longest relaxation time (a disengagement time) ${\tau }_d=N^3{\tau }_0/{\pi }^2$ is estimated as ${\tau }_d\approx 6.2\times {10}^4{\tau }_0$. The dotted lines are the theoretical predictions for ${\widehat{\mathrm{\Phi }}}_1^{\mathrm{id}}(\mathrm{\Delta },t)$ and ${\widehat{\mathrm{\Phi }}}_1^{\mathrm{ex}}\left(t\right)$ given by Eqs. (100) and (101). The dashed lines are the theoretical predictions for ${\widehat{\mathrm{\Phi }}}_2^{\mathrm{id}}(\mathrm{\Delta },t)$ and ${\widehat{\mathrm{\Phi }}}_2^{\mathrm{ex}}\left(t\right)$ given by Eqs. (102) and (103). The solid lines are the sums ${\widehat{\mathrm{\Phi }}}_i(\mathrm{\Delta },t)(i=1,2)$ of the ideal and excess parts [Eqs. (24) and (34)].

Figure 4

Correlation functions ${\widehat{\mathrm{\Phi }}}_i\left(t\right)(i=1,2)$ of the TMSD tensor calculated from trajectory data ${\mathit{R}}_G\left(t\right)$ of the rigid rodlike polymer (circles and triangles). The COM trajectories ${\mathit{R}}_G\left(t\right)$ are generated through numerical simulations of the rigid rodlike polymer [Eqs. (2), (106), and (107)]. Distance is measured in units of the rod's diameter $b$ and time in units of ${\tau }_0:=b^3\eta /k_{B}T$. The length $L$ of the rod is set as $L=30b$, and $\mathrm{\Delta }$ is $0.5{\tau }_0$. The values of the three diffusion coefficients ($D_{\perp },D_{\parallel },D_r$) are given by Eqs. (109) and (110). In particular, the rotational relaxation time $1/D_r$ is estimated from Eq. (110) as $1/D_r\approx 8.3\times {10}^3{\tau }_0$. The dashed lines are the theoretical predictions for ${\widehat{\mathrm{\Phi }}}_2^{\mathrm{id}}(\mathrm{\Delta },t)$ and ${\widehat{\mathrm{\Phi }}}_2^{\mathrm{ex}}\left(t\right)$ given by Eqs. (118) and (116). The solid lines are the sums ${\widehat{\mathrm{\Phi }}}_i(\mathrm{\Delta },t)(i=1,2)$ of the ideal and excess parts [Eqs. (24) and (34)].