# Self-Propulsion Enhances Polymerization

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## Abstract

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## 1. Introduction

## 2. Self-Propulsion, Self-Assembly and Self-Organization

## 3. Model Definition

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Parameter Values Used in the Numerical Simulations

**Table A1.**Parameters used in the numerical simulations. The molecules start at non overlapping random positions within the simulation cell.

Active Colloids Self-Assembly Model | ||
---|---|---|

N | Number of colloids | 512 |

${N}_{s}$ | Number of spherical subunits per colloid | 20 |

T | Temperature | $[0.1,0.75]$ |

${\sigma}_{s}$ | Spherical subunit radius | 0.25 |

${\gamma}_{r}$ | Translational Friction | 250 |

${D}_{r}$ | Translational Diffusion Coefficient | ${k}_{B}T/{\gamma}_{r}$ |

${\gamma}_{t}$ | Rotational Friction | ${D}_{r}/{k}_{b}T$ |

${D}_{t}$ | Rotational Diffusion Coefficient | $3{D}_{r}/{\sigma}_{s}^{2}$ |

${F}_{\mathrm{sp}}$ | Self-propulsion force | $[0,20]$ |

$\Delta t$ | Integration step | ${10}^{-3}$ |

${t}_{\infty}$ | Number of time steps | ${10}^{7}$ |

R | Channel size ratio | $[0.25,1]$ |

l | Size of the simulation cell | 135 |

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**Figure 1.**(

**a**) Schematic representation of the basic rigid molecule. The sides of the molecule are discretized into spherical subunits with positions ${\mathbf{e}}_{i}^{\alpha}$ which exert attractive or repulsive forces on the spherical subunits of other molecules. The entire molecule is characterized by the position ${\mathbf{r}}_{i}$ of the center of mass, its velocity ${\mathbf{v}}_{i}$ and orientation ${\widehat{\mathbf{u}}}_{i}$. Note that the velocity and orientation are not necessarily parallel. The self-propulsion force always acts in the direction of the orientation vector ${\widehat{\mathbf{u}}}_{i}$. The edges of the molecule are colored according to the repulsive or attractive forces they exert on other molecules: if two molecules collide through edges of the same color, the force is repulsive as in (

**b**), whereas the force is attractive only between the green and red edges, as in (

**c**). The dynamics take place on a 2D-space.

**Figure 2.**Snapshots of the system with fully periodic boundary conditions (the bulk) showing the effect of self-propulsion on the assembly of molecules. Different snapshots correspond to different values of the self-propulsion force: (

**a**) ${F}_{\mathrm{sp}=0}$, (

**b**) ${F}_{\mathrm{sp}=2.5}$, (

**c**) ${F}_{\mathrm{sp}=10}$, and (

**d**) ${F}_{\mathrm{sp}=15}$. In all cases the temperature is $T=0.1$ and the number of particles is $N=512$. Note that with no self-propulsion (panel (

**a**), ${F}_{\mathrm{sp}}=0$), only very small chains are formed. This case would correspond to spontaneous self-assembly. As the self-propulsion force ${F}_{\mathrm{sp}}$ increases, (panels (

**b**–

**d**)), longer chains appear. It is apparent that self-propulsion considerably improves the assembly of molecules as compared to spontaneous self-assembly. The color gradient represents the length of the chain. All the monomers ($L=1$) are colored in gray.

**Figure 3.**Effect of self-propulsion on the assembly of molecules within semi-periodic boundary conditions (the channel), where the longest sides are rigid boundaries and the shorter ones are periodic. All the snapshots correspond to temperature $T=0.1$, channel aspect ratio $R={l}_{1}/{l}_{2}=0.25$ and $N=512$ particles. (

**a**) For ${F}_{\mathrm{sp}}=0$, i.e., when the motion of the molecules is driven just by thermal noise, only small polymers are formed. The next three panels correspond to non-zero values of the self-propulsion force: (

**b**) ${F}_{\mathrm{sp}}=5$, (

**c**) ${F}_{\mathrm{sp}}=7.5$, and (

**d**) ${F}_{\mathrm{sp}}=10$. Note that as the value of ${F}_{\mathrm{sp}}$ increases the length of the assembled polymers also increases. Note also that in this case the polymers aggregate in the repulsive boundaries, allowing them to form even longer chains. The color gradient represents the length of the chain. All the monomers ($L=1$) are colored in gray.

**Figure 4.**Average length $\langle L\rangle $ of polymer chains as a function of time t for different values of the aspect ratio R, temperature T and self-propulsion force ${F}_{\mathrm{sp}}$. Panels (

**a**–

**c**) correspond to temperature $T=0.1$ and panels (

**d**–

**f**) to $T=0.25$. Panels (a,d) show the evolution of $\langle L\left(t\right)\rangle $ in the case of periodic boundary conditions, whereas the remaining panels correspond to the channel geometry for different aspect ratios: in (

**b**,

**e**) $R=0.75$ and in (

**c**,

**f**) $R=0.25$. It can be observed that during the computing time $\langle L\rangle $ does not reach a stationary value. However, it is evident in all cases that increasing the self-propulsion force ${F}_{\mathrm{sp}}$ speeds up the formation of polymers. The maximum computing time $t=10,000$ corresponds to ${10}^{7}$ time steps in the molecular dynamics simulation. For each curve, the ensemble average $\langle L\rangle $ was computed over 20 different realizations.

**Figure 5.**Results for the system with periodic boundary conditions. (

**a**) Probability $P\left(L\right)$ that a polymer of length L is created after ${10}^{7}$ time steps. Different colors correspond to different values of the self-propulsion force ${F}_{\mathrm{sp}}$. All the simulations in this panel were computed at constant temperature $T=0.1$. The inset shows a magnification of the tail of the distribution $P\left(L\right)$ in order to better appreciate the existence of long polymers. (

**b**) Average chain length $\langle L\rangle $ as a function of the self-propelled force ${F}_{\mathrm{sp}}$ for different values of the temperature T. Note that for lower temperatures, there is an optimal value of ${F}_{\mathrm{sp}}^{\star}$ of the self-propulsion force at which $\langle L\rangle $ exhibits a maximum. This value is plotted in panel (

**c**) as a function of the temperature T. (

**d**) Temporal evolution of the average polymer length, $\langle L\left(t\right)\rangle $, in the high-temperature regime for different values of ${F}_{\mathrm{sp}}$. Note that in this case $\langle L\left(t\right)\rangle $ does reach a stationary value which increases with ${F}_{\mathrm{sp}}$. The inset shows the fraction of monomers ($L=1$) and dimers $(L=2)$ in the system in this regime. (Polymers with $L>2$ appear in such low quantities that cannot be appreciated on the histogram).

**Figure 6.**Results for the channel with semi-periodic boundary conditions. All the results presented in this figure were computed at constant temperature $T=0.1$. (

**a**) Probability $P\left(L\right)$ that a chain of length L is formed after ${10}^{7}$ time steps, for different values of the self-propulsion force ${F}_{\mathrm{sp}}$ in a channel with an aspect-ratio $R=0.75$. The inset shows a magnification of the tail of the distribution $P\left(L\right)$. Note that in this case, even longer polymers are formed with respect to the case with fully periodic boundary conditions depicted in Figure 5. Panels (

**b**,

**c**) show the average chain length $\langle L\rangle $ as a function of the self-propulsion force ${F}_{\mathrm{sp}}$ for different values of the channel aspect-ratio R, and temperatures $T=0.1$ and $T=0.25$, respectively. For low temperature ($T=0.1$), the average chain length $\langle L\rangle $ first increases and then saturates as a function of ${F}_{\mathrm{sp}}$, whereas at a higher temperature ($T=0.25$) it reaches a maximum at an optimal value ${F}_{\mathrm{sp}}^{\star}$. (

**d**) Optimal value ${F}_{\mathrm{sp}}^{\star}$ as a function of the channel aspect ratio R for $T=0.25$.

**Figure 7.**Average maximum polymer length $\langle {L}_{\mathrm{max}}^{C}\rangle $ for the channel against the average maximum polymer length $\langle {L}_{\mathrm{max}}^{B}\rangle $ for the bulk. Each point corresponds to a specific value of ${F}_{\mathrm{sp}}$, whereas the different symbols correspond to different values of the channel aspect ratio R. Panel (

**a**) shows data for $T=0.1$ whereas panel (

**b**) shows similar data at a higher temperature $T=0.25$. Note that almost all the points (except for ${F}_{\mathrm{sp}}=0$) fall above the identity line, which means that the polymers formed in the channel are considerably longer than the ones observed in the bulk. The ensemble averages $\langle \xb7\rangle $ were computed over 20 different realizations for each condition.

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Aldana, M.; Fuentes-Cabrera, M.; Zumaya, M.
Self-Propulsion Enhances Polymerization. *Entropy* **2020**, *22*, 251.
https://doi.org/10.3390/e22020251

**AMA Style**

Aldana M, Fuentes-Cabrera M, Zumaya M.
Self-Propulsion Enhances Polymerization. *Entropy*. 2020; 22(2):251.
https://doi.org/10.3390/e22020251

**Chicago/Turabian Style**

Aldana, Maximino, Miguel Fuentes-Cabrera, and Martín Zumaya.
2020. "Self-Propulsion Enhances Polymerization" *Entropy* 22, no. 2: 251.
https://doi.org/10.3390/e22020251