Non-Equilibrium Structural and Dynamic Behaviors of Polar Active Polymer Controlled by Head Activity

The most direct method to investigate the conformation of polymers is the calculation of the gyration radius $R_{g}$:

$$\centering R_{g}=\frac{1}{N}\sqrt{\sum_{i=1}^{N}\sum_{j=i}^{N}\langle\left(\mathbf{r}_{i}-\mathbf{r}_{j}\right)^{2}\rangle},\@add@centering$$

(7)

where $N$ is the number of monomers, $\mathbf{r}_{i.j}$ denotes the position of the monomers. FIG. 1(b) shows the probability distribution of radius of gyration $P(R_{g})$ for active polymers with various head activity under fixed ${\rm Pe}$ and $N$, where $P(R_{g})$ for passive polymers is also shown for comparison. In the absence of head activity ($\kappa=0$), we find strong activity-induced collapse of the polymer chain, consistent with previous studies Bianco et al. (2018); Jain and Thakur (2022). However, with increasing head activity, active polymer gradually swells and becomes even more expanded than the passive polymer.

In FIG. 2(a), we plot the average radius of gyration $\langle R_{g}\rangle$ as a function of ${\rm Pe}$ under different $\kappa$, where $\langle R_{g,p}\rangle$ for passive polymers is drawn as a dashed line for comparison. For small head activity $\kappa<0.5$, we find that $\langle R_{g}\rangle$ shows monotonic decrease with increasing ${\rm Pe}$. Nevertheless, for large head activity, an unusual non-monotonic behaviour of $\langle R_{g}\rangle$ is observed with a first-stage shrinkage when ${\rm Pe}\lesssim 10$ and late-state expand at large ${\rm Pe}$. Additionally, the Flory exponent $\nu$ is obtained through plotting $\langle R_{g}\rangle$ as a function of $N$ (See Supplemental FIG. S2 for more data). In FIG. 2(b), we show $\nu$ as a function of ${\rm Pe}$ under different $\kappa$ where the reference values of the random-walk chain ($\nu=0.5$) and self-avoid walks chain ($\nu=0.588$) are drawn as dashed lines. We find that under large head activity, $\nu$ also exhibits a non-monotonic variation with increasing ${\rm Pe}$, which indicates a fundamental change of the fractal dimension of polymer configurations by head activity.

To explore these phenomena deeper, we calculate the bond-vector correlation function:

$$\centering C_{b}(n)=\langle\frac{\mathbf{b}_{i}\cdot\mathbf{b}_{i+n}}{\mathbf{b}_{i}^{2}}\rangle,\@add@centering$$

(8)

where $\mathbf{b}_{i}=\mathbf{e}_{i,i+1}$. For typical semi-flexible polymers, $C_{b}(n)$ decays exponentially. The characteristic correlation length $n^{*}$ at which $C_{b}(n^{*})$ decays to $e^{-1}$ defines the persistence length $L_{p}=n^{*}b$. In FIG. 2(c), we plot $C_{b}(n)$ for the passive polymer and active polymers with different $\kappa$ under fixed ${\rm Pe}$. We find that compared with the passive polymer, $C_{b}(n)$ for active polymers with small $\kappa$ show a longer $L_{p}$, but with an abnormal negative correlation at intermediate length scale. This negative correlation indicates that the conformation of the active polymer has a strong inward or involution tendency which is associated with the collapsed configuration. Nevertheless, this negative correlation is absent for passive polymers in bad solvent (Supplemental FIG. S3), which suggests a fundamental difference between these two systems, despite their apparent similarity. As $\kappa$ grows, the persistence length increases significantly, with the negative correlation strength $h$ decrease rapidly to zero. Here, $h$ is defined as the area ratio between negative and positive regions in correlation function. Thus, active polymers with large head activity behave similarly to that of passive semi-flexible chains. This emerging “dynamic rigidity” is a pure non-equilibrium effect, since the chain still remains mechanically flexible. Unlike semi-flexible passive chains, this dynamic rigidity alone can not predict the conformation of polymers. For example, FIG. 2(a) shows that even $L_{p}$ is much larger than that of reference passive polymer, $R_{g}$ is still smaller than $R_{g,p}$ due to a large negative correlation in $C_{b}(n)$.

We summarize the behaviors of persistence length $L_{p}$ and $h$ in FIG. 2(d) and 2(e), respectively. One can see that with increasing ${\rm Pe}$, $L_{p}$ remains around the low value and even decreases for polymer with small $\kappa$, but rises significantly for polymer with large $\kappa$, while $h$ increases monotonically at small $\kappa$ but exhibit strong non-monotonic behavior at large $\kappa$. Therefore, the activity of single head monomer determines the direction of overall active forces acting on the chain: for small head activity, the totally active driving is “inward”, corresponding to a large $h$ (small $L_{p}$) and a collapsed polymer state, while for large head activity, the totally active forces is “outward”, corresponding to a large $L_{p}$ (small $h$) and an extended polymer state.

The above abnormal conformation behaviors have a deep connection with the dynamic motion of active polymer. Since the active propelling forces are along the chain’s contour, the active polymer adopts active reptation motion, which is also referred as “railway motion” Isele-Holder et al. (2015) (see Supplemental Movie S1-S5). To quantify such motion, we define a correlation function of head-tail position:

$$\centering\Delta r_{ht}^{2}(\Delta t)=\langle\left[{\mathbf{r}_{1}(t)-\mathbf{r}_{N}(t+\Delta t)}\right]^{2}\rangle,\@add@centering$$

(9)

where the angle bracket represents the time and ensemble average. For perfect railway motion, $\Delta r_{ht}^{2}$ decays to zero at time $\tau_{ht}\simeq Nb/v_{0}=N\tau_{0}$ and bounces back, which is the time for the tail to reach the original position of the head. For pure diffusive dynamics, $\Delta r_{ht}^{2}$ will be a flat line determined by polymer’s mean squared end-to-end distance $R_{e}^{2}\equiv\langle(\mathbf{r}_{N}-\mathbf{r}_{1})^{2}\rangle$. In FIG. 3(a), we plot $\Delta r_{ht}^{2}$ for active polymers with different $\kappa$ under fixed ${\rm Pe}$, where we find $\Delta r_{ht}^{2}$ shows pronounce dips at the same $\tau_{ht}$ independent of head activity. We further extract the minimal values of the correlation function of head-tail position, $\Delta r_{ht\_min}^{2}={\rm min}(\Delta r_{ht}^{2}(\Delta t))$, to quantify the deviation from the perfect railway motion, and plot them as a function of ${\rm Pe}$ in FIG. 3(b). We find that $\Delta r_{ht\_min}^{2}$ decreases with increasing ${\rm Pe}$ and $\kappa$, which indicates that polymers with a higher head activity show relatively stronger railway motion.

For perfect railway motion chain, the direction of the head monomer is the only degree of freedom that determines the chain conformation. This direction can be represented by the bond angle $\theta$ between the first three monomers. In FIG. 3(c), we give the angle distribution $P(\theta)$ for active polymers with different $\kappa$ along with the passive polymer under varying polymer lengths. We find that for the active polymer without head activity, the peak of $P(\theta)$ locates at a small value around $0.5\pi$, indicating the averaged bent configuration of polymer head. With increasing $\kappa$, the polymer head takes a more straight configuration. These results are independent of polymer length, indicating a similar way of the head activity affecting the polymer contour. Simple mechanical analysis in the inset of FIG. 3(c) shows that when the propulsion force of the head monomer is smaller than the second monomer, the pushing force from back will induce a “torque” that makes the bond $\mathbf{r}_{1,2}$ bending inward. One the contrary, when the propulsion force of the head monomer is large enough, the head monomer will pull the three monomers straight. For railway motion chain, $P(\theta)$ determines the bond angle distribution of the whole chain. Thus, an increasing average $\theta$ corresponds to an increasing persistence length $L_{p}$ and a decreasing involution degree $h$ of polymer. In FIG. 3(d), we summarized the average value of $\theta$, which is qualitatively agreed with the behaviour of $L_{p}$ and $h$ in FIG. 2(d) and 2(e).

To further explore the impact of head activity on the dynamics of polar active polymer, we calculate the end-to-end vector correlation function

$$\centering C_{ee}(t)=\frac{\mathbf{R_{e}}(0)\cdot\mathbf{R_{e}}(t)}{\mathbf{R_{e}}(0)^{2}},\@add@centering$$

(10)

and compare $C_{ee}(t)$ for polymers with different head activity $\kappa$ and polymer length $N$ in FIG. 4(a). A surprising finding is that all $C_{ee}(t)$ exhibit linear decay rather than exponential decay as for the passive polymer (Supplemental FIG. S4) Azuma and Takayama (1999); Descas et al. (2004). Moreover, we find $C_{ee}(t)$ is insensitive to the head activity $\kappa$ and overall activity ${\rm Pe}$ but only depends on the chain length $N$. In FIG. 4(c), we plot the characteristic decay time $\tau_{tr}$ of $C_{ee}(t)$ (for active polymers $C_{ee}(\tau_{tr})=0$ while for passive polymers $C_{ee}(\tau_{tr})=1/e$) as a function of $N$, we find all data from active polymer falling around the line of $\tau=N\tau_{0}$. This behavior is also distinct from the passive polymer for which $\tau_{tr}\propto N^{2}$.

Such abnormal dynamics can be explained theoretically based on railway motion of the polymer chain. As shown in FIG. 4(b), assuming the initial and final configurations of the polymer on the same “railway” have an overlap part with end-to-end vector $\mathbf{R}_{2}$, thus we have ${\mathbf{R}_{e}}(0)=\mathbf{R}_{1}+\mathbf{R}_{2}$ and ${\mathbf{R}_{e}}(t)=\mathbf{R}_{2}+\mathbf{R}_{3}$, where $\mathbf{R}_{1}$ and $\mathbf{R}_{3}$ are end-to-end vectors of the two non-overlapped parts. Since $\mathbf{R}_{1}$, $\mathbf{R}_{2}$, $\mathbf{R}_{3}$ are statistically independent, we have $\left\langle\mathbf{R}_{i}\cdot\mathbf{R}_{j}\right\rangle=0$ for $i\neq j$. Combining the nearly linear dependence of $\mathbf{R}_{e}^{2}$ on $N$ shown in FIG. 5(b), one can get

$$\centering C_{ee}(t)\approx\frac{\mathbf{R}_{2}^{2}}{\mathbf{R_{e}}(0)^{2}}\approx\frac{Nb-v_{0}t}{Nb}=1-\frac{t}{N\tau_{0}},\quad t<N\tau_{0},\@add@centering$$

(11)

which suggests $\tau_{tr}=N\tau_{0}$. Note that this result is independent of the shape of “railway” controlled by head activity.

The railway motion also results in interesting polymer diffusion dynamics. Starting from the railway motion hypothesis in FIG. 4(b), one can get the expression of the center-of-mass of the polymer as

$$\centering\mathbf{r}_{cm}(0)=\frac{1}{N}\sum_{i=1}^{N}\mathbf{r}_{i},\quad\mathbf{r}_{cm}(t)=\frac{1}{N}\sum_{i=t}^{N+t}\mathbf{r}_{i}.\@add@centering$$

(12)

When $t<N\tau_{0}$, the displacement of the center-of-mass could be rewritten as:

$$\centering\Delta\mathbf{r}_{cm}(t)=\frac{1}{N}\left(\sum_{i=t}^{N+t}\mathbf{r}_{i}-\sum_{i=1}^{N}\mathbf{r}_{i}\right)=\frac{1}{N}\left(\mathbf{r}_{N+i}-\mathbf{r}_{i}\right).\@add@centering$$

(13)

Using an approximation of $\mathbf{r}_{N+i}-\mathbf{r}_{i}\approx\mathbf{R_{e}}(i)$ and considering continuous motion, we get

$$\centering\Delta\mathbf{r}_{cm}(t)=\frac{1}{N}\sum_{i=1}^{t}\mathbf{R_{e}}(i)=\frac{1}{N\tau_{0}}\int_{0}^{t}\mathbf{R_{e}}(t){\rm d}t.\@add@centering$$

(14)

Thus, the mean square displacement (MSD) of polymer center-of-mass has the formation of

	$\displaystyle{\rm MSD}(t)$	$\displaystyle\equiv\Delta\mathbf{r}_{cm}^{2}(t)$
		$\displaystyle=\frac{1}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\int_{0}^{t}\mathbf{R_{e}}(t_{1})\mathbf{R_{e}}(t_{2}){\rm d}t_{1}{\rm d}t_{2}$		(15)

when $\left|t_{2}-t_{1}\right|\leq N\tau_{0}$. Combining the linear decay of $C_{ee}$ (Eq. (11)), we get

$$\centering{\rm MSD}(t)=\frac{1}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\int_{0}^{t}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\left(1-\frac{\left|t_{2}-t_{1}\right|}{N\tau_{0}}\right){\rm d}t_{1}{\rm d}t_{2}.\@add@centering$$

(16)

Considering the long-time limit $t\gg N\tau_{0}$, the internal integral can be completely computed and has symmetric properties. Hence,

	$\displaystyle{\rm MSD}(t)$	$\displaystyle=\frac{2}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\int_{t_{1}}^{t_{1}+N\tau_{0}}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\left(1-\frac{t_{2}-t_{1}}{N\tau_{0}}\right){\rm d}t_{1}{\rm d}t_{2}$
		$\displaystyle=\frac{2}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\int_{0}^{N\tau_{0}}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\left(1-\frac{t_{3}}{N\tau_{0}}\right){\rm d}t_{1}{\rm d}t_{3}$
		$\displaystyle=\frac{2}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\frac{N\tau_{0}}{2}{\rm d}t_{1},$		(17)

where $t_{3}=t_{2}-t_{1}$. During the whole time $\mathbf{R}_{\mathbf{e}}^{2}(t_{1})$ is randomly fluctuated around the average end-to-end distance $R_{e}^{2}$, hence the integral could be directly calculated as

$${\rm MSD}(t)=\frac{R_{e}^{2}}{N\tau_{0}}t,\quad t\gg N\tau_{0}.$$

(18)

So now we have a strict expression for ${\rm MSD}$ on large time-scale. When the situation is opposite, $t\ll N\tau_{0}$, the internal integral of Eq. (16) cannot be completely computed. Considering the symmetry of $t_{1}>t_{2}$ and $t_{1}<t_{2}$, we have

	$\displaystyle{\rm MSD}(t)$	$\displaystyle=\frac{2}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\int_{0}^{t_{1}}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\left(1-\frac{t_{1}-t_{2}}{N\tau_{0}}\right){\rm d}t_{1}{\rm d}t_{2}$
		$\displaystyle=\frac{2}{N^{2}\tau_{0}^{2}}\int_{0}^{t}\mathbf{R}_{\mathbf{e}}^{2}(t_{1})\left(t_{1}-\frac{t_{1}^{2}}{2N\tau_{0}}\right){\rm d}t_{1}.$		(19)

When $t\ll N\tau_{0}$, $\mathbf{R}_{\mathbf{e}}^{2}(t_{1})$ has little fluctuation and nearly maintains the initial value with averages of $R_{e}^{2}$, hence MSD has an approximate value of

$${\rm MSD}(t)\approx\frac{2R_{e}^{2}}{N^{2}\tau_{0}^{2}}\left(\frac{t^{2}}{2}-\frac{t^{3}}{6N\tau_{0}}\right),\quad t\ll N\tau_{0}.$$

(20)

In summary, the expression of the mean squared displacement of center-of-mass of an active polymer has the formation of

$${\rm MSD}(t)\simeq\left\{\begin{aligned} &\frac{R_{e}^{2}}{N^{2}\tau_{0}^{2}}{t^{2}},~{}~{}~{}~{}~{}\quad&t\ll N\tau_{0},\\ &\frac{R_{e}^{2}}{N\tau_{0}}t,~{}~{}~{}~{}~{}\quad&t\gg N\tau_{0}.\end{aligned}\right.$$

(21)

Eq. (21) predicts a ballistic dynamics regime (${\rm MSD}\sim t^{2}$) and a diffusion dynamics regime (${\rm MSD}\sim t^{1}$) separated by the characteristic time $\tau_{R}=N\tau_{0}$ and characteristic ${\rm MSD}^{*}(\tau_{R})\simeq{R_{e}^{2}}$. In FIG. 5(a) and Supplemental FIG. S5, we plot the $\rm MSD(t)$ for active polymers with different $\kappa$, where the reference $\rm MSD(t)$ of the passive polymer is shown in gray color. We find that the simulation results agree well with our theoretical prediction. Especially $\rm MSD^{*}(\tau_{R})$ match with $R_{e}^{2}$ excellently when varying the head activity and polymer length (see FIG. 5(b)). Furthermore, based on Stokes-Einstein relation ${\rm MSD}=6Dt$ and Eq. (21), we can obtain the diffusion coefficient

$$D=\frac{R_{e}^{2}}{6N\tau_{0}}.$$

(22)

Since $R_{e}^{2}\sim N$, the diffusion coefficient is independent of chain length, which is in contrast with the behavior of classical Rouse chain $D\sim\frac{1}{N}$ Tejedor and Ramírez (2019) and consistent with previous work Bianco et al. (2018). These predictions are also confirmed by the simulation results shown in FIG. 5(c) and Supplemental FIG. S6, at which $D$ is independent of $N$ and matches with $R_{e}^{2}/6N\tau_{0}$ very well under different $\kappa$. Moreover, from FIG. 5(c), one can see that the conformation and dynamics are coupled: the diffusion coefficient of an extended active polymer with large head activity can be an order of magnitude larger than the collapsed active polymer with small head activity. Finally, we emphasize another scaling relation hidden in Eq. (22), $D\sim\tau_{0}^{-1}$. Since $\tau_{0}$ is defined as $\tau_{0}=b/v_{0}$ with $v_{0}=F_{act}/\gamma$, we have $D\sim F_{act}$, which provids a connection between dynamics and activity strength. This correlation has been a subject of continuous debate and contention in literatures Anand and Singh (2018); Bianco et al. (2018); Mokhtari and Zippelius (2019). The dynamic theory developed here is based on railway motion, hence it is valid for all polar active polymers exhibiting railway motion, regardless of the model (Supplemental FIG. S7). This provides us an efficient way to control the conformation and dynamics of active polymer, which may have application values in drug delivery and biomedical engineering Sasaki et al. (2014); Vuijk et al. (2021); Isele-Holder et al. (2016). Recently, Philipps et al. submitted an analytical study on polar active polymers, which reports dynamic equations similar to Eq. (22) Philipps et al. (2022).