# Nanoscale Correlations of Ice Adhesion Strength and Water Contact Angle

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Aspects

## 3. Methods

#### 3.1. Atomistic Modelling

#### 3.2. Simulation Details

^{−1}nm

^{−2}to ensure ice detachment for the highest values of interaction energies.

## 4. Results

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Illustration of the most general water contact angle $\theta $. The surface energies $\gamma $ correspond to the parameters of Young’s equation in Equation (1), where ${\gamma}_{w,s}$, ${\gamma}_{s}$ and ${\gamma}_{w}$ refer to the specific energies of the solid–water, solid–vapour and water–vapour interfaces, respectively.

**Figure 2.**Two forms of the general relation between ice adhesion strength τ and water contact angle θ on a generic surface with properties C

_{0}; (

**a**) Normalised ice adhesion strength as function of the cosine of the contact angle (1 + cos θ); (

**b**) Normalised ice adhesion strength as function of the cosine of the contact angle (1 + cos θ). The relation is described by Equation (7).

**Figure 3.**Experimental results on the relation between receding contact angle and ice adhesion strength for a collection of surfaces; (

**a**) Results from Meuler et al. [22], with 22 samples. Adapted with permission [22], 2010 American Chemical Society; (

**b**) Results from He et al. [18], with 24 samples; (

**c**) Results from Golovin et al. [38], with 108 samples. Ice adhesion is average value; (

**d**) Illustration of receding contact angle compared to the equilibrium contact angle in Figure 1. The relation from Equation (7) and Figure 2b is clearest for (

**a**), and can be seen for higher ice adhesion strengths in (

**b**). The relation cannot be seen in (

**c**). The definition of the receding contact angle is shown in (

**d**).

**Figure 4.**Illustrations of the simulation system in this study of water molecules on a graphene surface, here represented by System A. This system had interaction potential ε

_{0}= 2.9288 × 10

^{−1}kJmol

^{−1}nm

^{−2}. (

**a**) Water drop at T = 275 K; (

**b**) Ice cube at T = 180 K.

**Figure 5.**Water contact angle $\theta $ as function of the interaction potential $\epsilon $ for the System A. The values of $\epsilon $ can be found in Table 1.

**Figure 6.**Ice adhesion strength $\tau $ as function of interaction potential $\epsilon $ for System A. The values of $\epsilon $ can be found in Table 1.

**Figure 7.**Ice adhesion strength as function of contact angle, with the relation from Equation (7) fitted to the data. It can be seen that the fit of the general relation shows a very high level of significance.

**Figure 8.**Contact angle as function of interaction energy $\epsilon $ for all system sizes. Separate figures can be seen in Figure S8.

**Figure 9.**Ice adhesion strength as function of interaction energy $\epsilon $ for all system sizes. Separate figures can be seen in Figure S9.

**Figure 10.**Overview of the correlation with Equation (7) for the four systems investigated in this study. The ice adhesion strength has been normalised with respect to the mean value from the interaction energy ${\epsilon}_{0}$. Data from the four individual systems is given in Figure S10.

**Table 1.**Overview of the energy well depths, or interaction energies, $\epsilon $, applied in the seven different simulation systems to change the water contact angle. All systems were simulated five times to obtain averages.

$\mathit{\epsilon}$ | Value [${\mathbf{kJ}\mathbf{mol}}^{-1}{\mathbf{nm}}^{-2}$] |
---|---|

$0.05{\epsilon}_{0}$ | $1.4644\times {10}^{-2}$ |

$0.1{\epsilon}_{0}$ | $2.9288\times {10}^{-2}$ |

$0.5{\epsilon}_{0}$ | $1.4644\times {10}^{-1}$ |

${\epsilon}_{0}$ | $2.9288\times {10}^{-1}$ |

$1.5{\epsilon}_{0}$ | $4.3932\times {10}^{-1}$ |

$2{\epsilon}_{0}$ | $5.8576\times {10}^{-1}$ |

$2.5{\epsilon}_{0}$ | $7.3220\times {10}^{-1}$ |

**Table 2.**Overview of the differently sized simulation systems. The different interaction energies $\epsilon $ in Table 1 were applied to all four systems to investigate water contact angles.

System | Number of Atoms | Area of Graphene Sheet | Area of Ice–Solid Contact |
---|---|---|---|

A | $\mathrm{28,376}$ | $19.9\mathrm{nm}\times 20.3\mathrm{nm}$ | $8.0\mathrm{nm}\times 7.6\mathrm{nm}$ |

B | 7336 | $8.0\mathrm{nm}\times 8.1\mathrm{nm}$ | $4.9\mathrm{nm}\times 4.6\mathrm{nm}$ |

C | $\mathrm{58,184}$ | $19.9\mathrm{nm}\times 19.9\mathrm{nm}$ | $14.7\mathrm{nm}\times 13.8\mathrm{nm}$ |

D | $\mathrm{102,568}$ | $26.3\mathrm{nm}\times 26.1\mathrm{nm}$ | $19.6\mathrm{nm}\times 18.5\mathrm{nm}$ |

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**MDPI and ACS Style**

Rønneberg, S.; Xiao, S.; He, J.; Zhang, Z.
Nanoscale Correlations of Ice Adhesion Strength and Water Contact Angle. *Coatings* **2020**, *10*, 379.
https://doi.org/10.3390/coatings10040379

**AMA Style**

Rønneberg S, Xiao S, He J, Zhang Z.
Nanoscale Correlations of Ice Adhesion Strength and Water Contact Angle. *Coatings*. 2020; 10(4):379.
https://doi.org/10.3390/coatings10040379

**Chicago/Turabian Style**

Rønneberg, Sigrid, Senbo Xiao, Jianying He, and Zhiliang Zhang.
2020. "Nanoscale Correlations of Ice Adhesion Strength and Water Contact Angle" *Coatings* 10, no. 4: 379.
https://doi.org/10.3390/coatings10040379