# Non-Equilibrium Living Polymers

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## Abstract

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## 1. Introduction

#### 1.1. Review of Equilibrium Living Polymers

`++`code by the author and shared in a git repository (see acknowledgement section); briefly, one should simulate the diffusion of a particle deposited at random along a 1D interval with exponentially distributed length (and mean ${L}_{0}$). The diffusion depends on the instantaneous length as ${D}_{0}/L$ and breakages or fusions of an l-long segment can occur at rate $\kappa $ per unit time and unit length and at rate $\kappa {e}^{-l/{L}_{0}}$ per unit time and per each end, respectively (see also Figure 2A). The survival function $\mu \left(t\right)$ is computed as the probability of a particle to have not reached one of the two ends of the interval by time t. Examples of this function for different values of $\xi $ are given in Figure 2B. The adimensional viscosity is the suitably normalised integral of $\mu \left(t\right)$ and is plotted in Figure 2C as a function of $\xi $ for fixed ${L}_{0}=1$—thus confirming Equation (7)—and in Figure 2D as a function of the mean length ${L}_{0}$ for unbreakable chains and living polymers thereby confirming Equation (8).

#### 1.2. Non-Equilibrium Living Polymers

#### 1.2.1. Breakage Only

#### 1.2.2. Fusion Only

#### 1.3. Non-Equilibrium DNA Digestion and Ligation

## 2. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**A**) Worm-like micelles. In inset, the disposition of amphiphilic molecules to shield the hydrophobic part from the aqueous solution. (

**B**) Relaxation of a tube segment by reptation. (

**C**) Relaxation of two worm-like micelles via a reversible architectural reconnection.

**Figure 2.**

**Equilibrium living polymers.**(

**A**) Sketch of the numerical algorithm. From top to bottom: (i) diffusion step (showing the explicit dependence on instantaneous length) (ii) breakage event and (iii) fusion event with respective rates. Absorbing conditions are set at the boundaries of the interval. (

**B**) Survival function $\mu \left(t\right)$ for different values of the adimensional parameter $\xi ={\tau}_{b}/{T}_{d}$. Dotted and solid lines are stretched and normal exponential, respectively and are drawn as a guide for the eye. (

**C**) Corresponding zero-shear viscosity showing the dependence ${\eta}_{0}\sim {\xi}^{1/2}$ for $\xi \lesssim $ 1 (black line). (

**D**) Scaling of the adimensional viscosity with mean length ${L}_{0}$ showing that is, as expected, $\sim {L}_{0}$ for living polymers and $\sim {L}_{0}^{3}$ for unbreakable ones. Curves in (

**B**) and data points in (

**C**), (

**D**) are obtained from 1D Brownian simulations using the numerical algorithm described in (

**A**) with parameters ${L}_{0}$ = 1, $dL=0.01$, $dt=0.01$ and ${D}_{0}=0.01$ dL${}^{2}$/dt.

**Figure 3.**

**Non-equilibrium living polymers—breakage only.**Zero-shear viscosity as a function of the ageing time ${T}_{a}/{T}_{d}$ and for different values of $\chi ={T}_{d}/{\tau}_{b}$. Dashed lines are obtained from Equation (10) multiplied by a numerical pre-factor. The data points are from Brownian simulations as described in Figure 2A with no fusion and with parameters ${D}_{0}=0.01$ dL${}^{2}$/dt, $dL=0.01$, $dt=0.01$ and ${L}_{0}=1$.

**Figure 4.**

**Non-equilibrium living polymers—fusion only.**Zero-shear viscosity as a function of the ageing time ${T}_{a}/{T}_{d}$ and for different values of $\varphi ={T}_{d}/{\tau}_{f}$. In (

**A**) the case of a test chain within a system of non fusing chains is shown. Dashed lines are obtained from Equation (14) multiplied by a numerical pre-factor. In (

**B**), the self-consistent case—in which all chains in the system grow at the same rate—is shown. Dashed lines are obtained from Equation (17) multiplied by a numerical pre-factor. In this case I am also showing error bars as the dispersion is larger than the other cases. The data points are from Brownian simulations as described in Figure 2A with no breakage and with parameters ${D}_{0}=0.1$ dL${}^{2}$/dt, $dL=0.01$, $dt=0.01$ and ${L}_{0}=1$.

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Michieletto, D. Non-Equilibrium Living Polymers. *Entropy* **2020**, *22*, 1130.
https://doi.org/10.3390/e22101130

**AMA Style**

Michieletto D. Non-Equilibrium Living Polymers. *Entropy*. 2020; 22(10):1130.
https://doi.org/10.3390/e22101130

**Chicago/Turabian Style**

Michieletto, Davide. 2020. "Non-Equilibrium Living Polymers" *Entropy* 22, no. 10: 1130.
https://doi.org/10.3390/e22101130