# The Influence of Shape on Parallel Self-Assembly

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## Abstract

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## 1. Introduction

## 2. The Experimental Self-Assembly Platform

## 3. Interaction Mechanisms and Measures of the System

**Figure 1.**The experimental setup (unit: mm). a) 3 different tiles. b) Agitated water tank. Stirring the water generated random, fluctuating forces, providing the system with the energy necessary for assembly.

#### 3.1. Magnetic potential energy

**r**from the magnetic moment ${\mathit{m}}_{j}$ is given by

**r**.

#### 3.2. Clustering coefficients in a self-assembly system

**Figure 2.**Example of clustering coefficients where ${X}_{comp}=7$. a) configuration with two clusters where ${x}_{1}=3$ (left) and ${x}_{2}=1$ (right). b) configuration with $x={X}_{comp}=7$. The two local clustering coefficients are ${c}_{1}=\frac{3}{7}=0.43$ (left) and ${c}_{2}=\frac{1}{7}=0.14$ (right) in (a), while $C=c=\frac{7}{7}=1$ in (b).

#### 3.3. Entropy and Degree of parallelism (DOP)

**Figure 3.**Example of the proposed degree of parallelism H where ${X}_{comp}=7$. It shows the tendency that the more assembly proceeds in parallel, the larger the value becomes.

## 4. The Experimental Results

#### 4.1. Assembly completion time

#### Comparison between different shapes

**Figure 4.**Comparison of assembly completion times. a) more than $80\%$ completion. b) more than $90\%$ completion. ($\mathcal{S}$: square, $\mathcal{C}$: circle, $\mathcal{R}$: rounded-square, $\mathcal{M}$: mixed). The boxes on the left side of each column show the assembly completion times for single magnet tiles, and on the right side show the times for double magnet tiles.

#### Comparison between differently magnetized tiles

**Table 1.**The lower quartile Q1, median, upper quartile Q3, and the averages of $80\%$ and $90\%$ completion level over all of the trials (unit: sec).

${\mathcal{S}}_{\text{s}}$ | ${\mathcal{S}}_{\text{d}}$ | |||||

80% | 90% | increase rate | 80% | 90% | increase rate | |

Q1 | 26 | 43 | 19 | 69 | ||

median | 45 | 56 | 26 | 172 | ||

Q3 | 83 | 129 | 38 | 300 | ||

average* | 56.5 | 76.6 | 136% | >68.5 | >184.2 | >269% |

${\mathcal{C}}_{\text{s}}$ | ${\mathcal{C}}_{\text{d}}$ | |||||

80% | 90% | increase rate | 80% | 90% | increase rate | |

Q1 | 10 | 26 | 6 | 13 | ||

median | 18 | 59 | 10 | 41 | ||

Q3 | 29 | 105 | 22 | 80 | ||

average | 31.2 | 71.0 | 228% | 20.9 | 54.2 | 259% |

${\mathcal{R}}_{\text{s}}$ | ${\mathcal{R}}_{\text{d}}$ | |||||

80% | 90% | increase rate | 80% | 90% | increase rate | |

Q1 | 12 | 33 | 5 | 14 | ||

median | 15 | 51 | 9 | 37 | ||

Q3 | 26 | 76 | 19 | 117 | ||

average* | 21.5 | 67.8 | 315% | 18.3 | >71.0 | >394% |

${\mathcal{M}}_{\text{s}}$ | ${\mathcal{M}}_{\text{d}}$ | |||||

80% | 90% | increase rate | 80% | 90% | increase rate | |

Q1 | 12 | 17 | 7 | 25 | ||

median | 15 | 26 | 13 | 42 | ||

Q3 | 18 | 77 | 33 | 96 | ||

average* | 21.5 | 42.7 | 199% | 23.8 | >71.4 | >300% |

#### 4.2. The formed structures

**Figure 5.**The formed structures of each assembly of the four different combinations. In each combination, the upper row depicts the case with one magnet and the lower row depicts the double magnet case. The trials are sorted with respect to the increase in assembly completion time. Tiles trapped by the magnetic shielding effect are marked with dotted circles.

**Figure 6.**Comparison of DOP against the assembly completion time. Considering the wide time range that square tiles took to complete, we displayed the other combinations in small windows, whose corresponding area is shown as a dotted square.

**Figure 7.**Image processing for the measurement of surface areas and perimeters of clusters. Note here the surface area means the entire area inside the cluster including the gaps.

**Figure 8.**Comparison of surface areas against perimeters. Both variables are normalized and represented as percentages from mean values. A large negative correlation was observed in $\mathcal{C}$ (−0.573), and a moderate negative correlation was observed in $\mathcal{R}$ (−0:322). Linearly fitted curves of $\mathcal{C}$ and $\mathcal{R}$ are shown in the figures.

#### 4.3. Time evolution

- ${\mathcal{S}}_{\text{d}}$: After the spacer was removed, the tiles moved randomly by changing their relative positions (1–3). The increase in potential energy and the clustering coefficients can be seen in the right figure. Once two tiles were attached (often adjusting their relative positions by sliding), the relatively strong connection force kept the connection tight (2, 7). Note that this results in a large value of the potential energy. This caused the tiles to stay in the same configuration, that is to say, reconfiguration was made more difficult (e.g., 8). As a result, the system produced an irregular shape (8). In this transition, the tiles first formed two small clusters (3–6) and subsequently they bonded together (7). It is worth noting that this large scale docking did not cause a big stored energy jump as expected (reflected in the right figure). This suggests that a major dominance of the energy is induced by locally connecting two tiles but among tiles that are apart. In addition to that, we observed that a white tile highlighted with a dotted circle was assisted to attach to the cluster by a red tile in the transformation from (3) to (5) (magnetic shielding effect, see Section 5).
- ${\mathcal{C}}_{\text{d}}$: In the beginning, several small groups were formed (1–2). The speed of aggregation was fast, whereas connections between two tiles were relatively weak and the tiles changed their relative positions smoothly (3–4 and 6–8). In particular, the transformation highlighted with a dotted circle that can be seen from (7) to (8) is supposed to be rarely observed in the square tile combinations (see Section 5). The increase in potential energy is lower than in the case of square tiles, especially since the closest distance between two magnets is greater (recall that we set the surface area of the tiles to be the same). The transition took 22s for 90% aggregation, and took 79s for the further global configuration (7–8).
- ${\mathcal{R}}_{\text{d}}$: The characteristic of these tiles was that they frequently rotated and changed directions according to the landscape of potential energy (2–4). These tiles possessed positive characteristics of both square and circle tiles, namely, a flexible reconfigurability and a stable lattice formation.
**Figure 9.**Representative aggregations of 4 combinations in which more than 95 % of tiles aggregated. For each case of raw data, we present the time sequence of trials listed from the left top to the right bottom (with an illustration of the most discriminative movement of the set on the left side). On the right side, we show the transitions of the magnetic potential energy and the clustering coefficients. - ${\mathcal{M}}_{\text{d}}$: This was the only heterogeneous combination in terms of shape. Rotation was also observed. The circle tiles acted as a “hinge”, carrying a connected square tile to another position (2–3, 6). Structured lattice regions were stabilized by square tiles fixing the relative positions (8), while due to the flexibility of such combinations, the system often produced branching shapes, which were characterized by lowest clustering coefficients (see Figure 5 ${\mathcal{M}}_{\text{d}}$).

**Figure 10.**Comparisons of the 4 transitions of clustering coefficients plotted against (a) potential energy, and (b) normalized potential energy.

## 5. Discussion

#### 5.1. Shape parameter consistency problem

#### 5.2. Magnetic shielding effect and the influence of shape on self-assembly

**Figure 12.**Magnetic shielding effect. a) The positive energy from two red tiles (B, C) acts as a shield, preventing the red tile (A) from connecting to the white tile. b) The small attractive region is displayed. The tile A is repelled from the cluster in the grey region. c) If the first two attachments are made with an “L” configuration instead of a straight one, a wider attractive region is kept open for the third red tile. d) An additional white tile can expand the attractive region even farther (“O” configuration).

**Figure 13.**Two examples of square tile assembly. a) Starting with two identical sets of opponent tiles. b) Starting with one red tile and three white tiles.

**Figure 14.**The DOP transitions of square tiles (Figure 5 ${\mathcal{S}}_{\text{d}}$) vs. the change in global clustering coefficients. a) Transitions whose assembly completion times ($90\%$) are less than one minute (Figure 5 ${\mathcal{S}}_{\text{d}}$ 11 - 14). b) Transitions which took more than two minutes (Figure 5 ${\mathcal{S}}_{\text{d}}$ 15 - 20). In the case of rapid transitions, the tiles tended to form two clusters and subsequently aggregated and configured a cluster. This tendency can be observed as large DOP reductions in the second half stages of their transitions, where the global clustering coefficients are between 0.4 and 0.8 (highlighted with a gray colored background).

**Figure 15.**Normalized potential energy ($-{U}_{total}^{\prime}/\left|{U}_{total}^{\prime}{|}_{\theta =90}\right|$) vs. the rotational angle θ. In the case of square tiles, a potential barrier has to be overcome to arrive at a stable position (B), whereas in the case of circle tiles, the tile can roll down to the position D without any assistance.

## 6. Conclusions

## Acknowledgements

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## Appendix

${\mathcal{S}}_{\text{s}}$ | ${\mathcal{S}}_{\text{d}}$ | ${\mathcal{C}}_{\text{s}}$ | ${\mathcal{C}}_{\text{d}}$ | |||||

trials | 80% | 90% | 80% | 90% | 80% | 90% | 80% | 90% |

1 | 12 | 34 | 19 | 136 | 9 | 26 | 16 | 56 |

2 | 45 | 45 | 142 | 162 | 8 | 27 | 6 | 10 |

3 | 51 | 56 | 33 | 46 | 19 | 54 | 13 | 29 |

4 | 19 | 129 | 19 | 20 | 18 | 109 | 22 | 24 |

5 | 26 | 53 | 17 | 172 | 29 | 59 | 7 | 13 |

6 | 11 | 24 | 38 | 266 | 18 | 26 | 56 | 119 |

7 | 46 | 55 | 27 | 27 | 10 | 23 | 26 | 117 |

8 | 58 | 58 | 26 | >300 | 23 | 166 | 6 | 11 |

9 | 90 | 90 | >300 | >300 | 29 | 97 | 11 | 80 |

10 | 83 | 157 | 26 | 26 | 182 | 192 | 11 | 46 |

11 | 36 | 67 | 20 | 300 | 12 | 13 | 10 | 41 |

12 | 138 | 138 | 21 | 135 | 25 | 105 | 7 | 11 |

13 | 164 | 164 | 29 | 273 | 14 | 59 | 6 | 17 |

14 | 26 | 36 | 11 | 300 | 65 | 65 | 5 | 73 |

15 | 43 | 43 | >300 | >300 | 16 | 44 | 111 | 166 |

average* | 56.5 | 76.6 | >68.5 | >184.2 | 31.2 | 71.0 | 20.9 | 54.2 |

${\mathcal{R}}_{\text{s}}$ | ${\mathcal{R}}_{\text{d}}$ | ${\mathcal{M}}_{\text{s}}$ | ${\mathcal{M}}_{\text{d}}$ | |||||

trials | 80% | 90% | 80% | 90% | 80% | 90% | 80% | 90% |

1 | 14 | 183 | 82 | 139 | 10 | 17 | 20 | 38 |

2 | 18 | 138 | 8 | 14 | 13 | 77 | 7 | 20 |

3 | 13 | 43 | 10 | 28 | 12 | 17 | 36 | 96 |

4 | 26 | 65 | 10 | 174 | 12 | 36 | 5 | 81 |

5 | 8 | 20 | 19 | 39 | 18 | 84 | 124 | 124 |

6 | 12 | 46 | 5 | 11 | 17 | 26 | 14 | 25 |

7 | 24 | 32 | 9 | 11 | 14 | 26 | 10 | 106 |

8 | 27 | 76 | 5 | 8 | 26 | 48 | 13 | 42 |

9 | 15 | 45 | 8 | 17 | 20 | 23 | 34 | 50 |

10 | 13 | 130 | 22 | 37 | 15 | 81 | 15 | 34 |

11 | 12 | 51 | 5 | 37 | 17 | 17 | 8 | 12 |

12 | 15 | 33 | 9 | >300 | 10 | 17 | 33 | 37 |

13 | 62 | 62 | 43 | 104 | 12 | 21 | 19 | >300 |

14 | 12 | 33 | 11 | 117 | 112 | 132 | 13 | 21 |

15 | 52 | 60 | 29 | 29 | 15 | 19 | 6 | 65 |

average* | 21.5 | 67.8 | 18.3 | >71.0 | 21.5 | 42.7 | 23.8 | >71.4 |

© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Miyashita, S.; Nagy, Z.; Nelson, B.J.; Pfeifer, R.
The Influence of Shape on Parallel Self-Assembly. *Entropy* **2009**, *11*, 643-666.
https://doi.org/10.3390/e11040643

**AMA Style**

Miyashita S, Nagy Z, Nelson BJ, Pfeifer R.
The Influence of Shape on Parallel Self-Assembly. *Entropy*. 2009; 11(4):643-666.
https://doi.org/10.3390/e11040643

**Chicago/Turabian Style**

Miyashita, Shuhei, Zoltán Nagy, Bradley J. Nelson, and Rolf Pfeifer.
2009. "The Influence of Shape on Parallel Self-Assembly" *Entropy* 11, no. 4: 643-666.
https://doi.org/10.3390/e11040643