# Seaweed and Dendritic Growth in Unsaturated Fatty Acid Monolayers

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Pockels-Langmuir Trough and Isotherms

^{2}. The compression speed can be varied. The experiments were performed in ambient air. The trough temperature was kept constant ±0.1 °C with a thermostat (DC-30 Thermo-Haake, Haake Technik, Karlsruhe, Germany). The fatty acid was dissolved in chloroform solution (c = 0.1 mM). The solution was spread with a 100 μL syringe (model 1710, Hamilton, Bonaduz, Switzerland) and the chloroform was allowed to dissipate for a few minutes. Then, the monolayer was compressed with a predetermined compression speed and the isotherm was recorded.

#### 2.3. Brewster Angle Microscopy (BAM)

^{2}(using Scheimpflug’s principle), corresponding to 1360 pixels × 1024 pixels and a spatial resolution of 2 µm. Due to the implemented Scheimpflug optics, it is possible to generate an overall focused image. However, the obtained images are distorted. The rectification and the background correction are performed by Accurion_Image (Accurion, Göttingen, Germany, version 1.2.3.).

#### 2.4. Image Processing

#### 2.4.1. Contrast Enhancement

#### 2.4.2. Determination of the Fractal Dimension

## 3. Results and Discussion

#### 3.1. Isotherms of Erucic Acid Monolayers at Different Compression Velocities

^{2}[16].

#### 3.2. Domain Growth Visualized with Brewster Angle Microscopy (BAM) Videos

#### 3.3. Parameters Characterizing Domain Growth

#### 3.3.1. Influence of the Compression Velocity ${v}_{\mathrm{C}}$ on Fractal Dimension ${D}_{F}$

#### 3.3.2. The Growth Speed ${v}_{R}$ of the Domains

#### 3.3.3. Dependence of Growth Speed ${v}_{R}$ on Compression Velocity and Supersaturation

#### 3.3.4. The Influence of Excess Lateral Pressure $\mathsf{\Delta}\pi $ and Supersaturation $\mathsf{\Delta}c$ on the Tip Radius $r$

#### 3.3.5. Side Branch Separation λ for Seaweed Domains and Dendrites

#### 3.3.6. Influence of the Compression Velocity on the Flow in the LE Phase

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Scheme A1.**The correlation between molecular area A and surface concentration c in the coexistence region of monolayers is indicated in an exemplary isotherm.

**Table A1.**The measured molecular area $A$ of erucic acid in the coexistence regime (cf. Figure 1), $\mathsf{\Delta}A=A-{A}_{\infty}$, the supersaturation $\mathsf{\Delta}c$ (calculated with Equation (A2)) and the normalized supersaturation $\mathsf{\Delta}c/{c}_{\infty}$ (calculated from Equation (A6)), for an excess lateral pressure $\mathsf{\Delta}\pi =\pi -{\pi}_{\infty}$.

$\Delta \mathit{\pi}[\mathbf{mN}/\mathbf{m}]$ | $\mathit{A}$$\left[{\u212b}^{2}\right]$ | $\Delta \mathit{A}$$\left[{\u212b}^{2}\right]$ | $\Delta \mathit{c}$$\left[{\u212b}^{-2}\right]$ | $\Delta \mathit{c}$$/{\mathit{c}}_{\mathit{\infty}}$ |
---|---|---|---|---|

0 | 27.50 | 0.00 | 0.0000 | 0.000 |

0.4 | 27.22 | 0.28 | 0.0004 | 0.010 |

1 | 26.80 | 0.70 | 0.0010 | 0.026 |

2 | 26.10 | 1.40 | 0.0020 | 0.054 |

3 | 25.40 | 2.10 | 0.0030 | 0.083 |

4 | 24.70 | 2.81 | 0.0041 | 0.114 |

5 | 24.00 | 3.51 | 0.0053 | 0.146 |

6 | 23.30 | 4.21 | 0.0066 | 0.181 |

**Figure A1.**Supersaturation $\mathsf{\Delta}c$ in dependence of $\mathsf{\Delta}\pi =\pi -{\pi}_{c}$, the lateral pressure above the LE/LC phase transition at equilibrium conditions, of erucic acid of Figure 1.

## Appendix B

#### Determination of the Fractal Dimension

**Figure A2.**

**Left**: The contrast-enhanced black and white picture of a typical seaweed structure at $t=49.9\mathrm{s}$. The scale bar is $100\mathsf{\mu}\mathrm{m}$.

**Right**: The deduced boxplot leading to a fractal dimension of ${D}_{F}=1.62\pm 0.06$.

**Figure A3.**

**Left**: The contrast-enhanced black and white picture of a typical dendrite at $t=8.3\mathrm{s}$. The scale bar is 100 μm.

**Right**: The deduced double logarithmic boxplot leading to a fractal dimension of ${D}_{F}=1.87\pm 0.15$.

## References

- Oliveira, O.N., Jr.; Caseli, L.; Ariga, K. The past and the future of Langmuir and Langmuir–Blodgett films. Chem. Rev.
**2022**, 122, 6459–6513. [Google Scholar] [CrossRef] [PubMed] - Kaganer, V.M.; Möhwald, H.; Dutta, P. Structure and phase transitions in Langmuir monolayers. Rev. Mod. Phys.
**1999**, 71, 779–819. [Google Scholar] [CrossRef][Green Version] - Blume, A. Lipids at the air–water interface. ChemTexts
**2018**, 4, 1–25. [Google Scholar] [CrossRef] - Stefaniu, C.; Brezesinski, G.; Möhwald, H. Langmuir monolayers as models to study processes at membrane surfaces. Adv. Colloid Interface Sci.
**2014**, 208, 197–213. [Google Scholar] [CrossRef] - Miller, A.; Helm, C.A.; Möhwald, H. The colloidal nature of phospholipid monolayers. J. Phys.
**1987**, 48, 693–701. [Google Scholar] [CrossRef] - Miller, A.; Möhwald, H. Diffusion limited growth of crystalline domains in phospholipid monolayers. J. Chem. Phys.
**1987**, 86, 4258–4265. [Google Scholar] [CrossRef] - Gutierrez-Campos, A.; Diaz-Leines, G.; Castillo, R. Domain growth, pattern formation, and morphology transitions in Langmuir monolayers. A new growth instability. J. Phys. Chem. B
**2010**, 114, 5034–5046. [Google Scholar] [CrossRef][Green Version] - Flores, A.; Corvera-Poire, E.; Garza, C.; Castillo, R. Pattern formation and morphology evolution in Langmuir monolayers. J. Phys. Chem. B
**2006**, 110, 4824–4835. [Google Scholar] [CrossRef] [PubMed][Green Version] - Akamatsu, S.; Bouloussa, O.; To, K.; Rondelez, F. Two-dimensional dendritic growth in Langmuir monolayers of D-myristoyl alanine. Phys. Rev. A
**1992**, 46, R4504. [Google Scholar] [CrossRef] - Yoon, D.K.; Zhu, C.; Kim, Y.H.; Shen, Y.; Jung, H.-T.; Clark, N.A. Dendritic growth in a two-dimensional smectic E freely suspended film. Mol. Syst. Des. Eng.
**2020**, 5, 815–819. [Google Scholar] [CrossRef] - Mullins, W.W.; Sekerka, R.F. Morphological stability of a particle growing by diffusion or heat flow. J. Appl. Phys.
**1963**, 34, 323–329. [Google Scholar] [CrossRef] - Mullins, W.W.; Sekerka, R. Stability of a planar interface during solidification of a dilute binary alloy. J. Appl. Phys.
**1964**, 35, 444–451. [Google Scholar] [CrossRef] - Bruinsma, R.; Rondelez, F.; Levine, A. Flow-controlled growth in Langmuir monolayers. Eur. Phys. J. E
**2001**, 6, 191–200. [Google Scholar] [CrossRef] - Luviano, A.S.; Campos-Terán, J.; Langevin, D.; Castillo, R.; Espinosa, G. Mechanical properties of DPPC–POPE mixed langmuir monolayers. Langmuir
**2019**, 35, 16734–16744. [Google Scholar] [CrossRef] - Espinosa, G.; López-Montero, I.; Monroy, F.; Langevin, D. Shear rheology of lipid monolayers and insights on membrane fluidity. Proc. Natl. Acad. Sci. USA
**2011**, 108, 6008–6013. [Google Scholar] [CrossRef][Green Version] - Vollhardt, D. Effect of unsaturation in fatty acids on the main characteristics of Langmuir monolayers. J. Phys. Chem. C
**2007**, 111, 6805–6812. [Google Scholar] [CrossRef] - Moisy, F. Boxcount (MATLAB Central File Exchange). 2008. Available online: https://ww2.mathworks.cn/matlabcentral/fileexchange/13063-boxcount (accessed on 18 June 2022).
- Vysotsky, Y.B.; Belyaeva, E.; Fainerman, V.; Vollhardt, D.; Aksenenko, E.; Miller, R. Thermodynamics of the clusterization process of cis isomers of unsaturated fatty acids at the air/water interface. J. Phys. Chem. B
**2009**, 113, 4347–4359. [Google Scholar] [CrossRef] - Miller, A.; Knoll, W.; Möhwald, H. Fractal growth of crystalline phospholipid domains in monomolecular layers. Phys. Rev. Lett.
**1986**, 56, 2633. [Google Scholar] [CrossRef] - Li, J.; Miller, R.; Möhwald, H. Characterisation of phospholipid layers at liquid interfaces 2. Comparison of isotherms of insoluble and soluble films of phospholipids at different fluid/water interfaces. Colloids Surf. A Physicochem. Eng. Asp.
**1996**, 114, 123–130. [Google Scholar] [CrossRef] - Helm, C.A.; Moehwald, H. Equilibrium and nonequilibrium features determining superlattices in phospholipid monolayers. J. Phys. Chem.
**1988**, 92, 1262–1266. [Google Scholar] [CrossRef] - Mandelbrot, B.B. The Fractal Geometry of Nature; WH Freeman and Company: New York, NY, USA, 1982; Volume 1. [Google Scholar]
- Ivantsov, G. Temperature Field Around a Spherical, Cylindrical, and Needle-Shaped Crystal, Growing in a Pre-Cooled Melt. In Temperature Field Around a Spherical; Akademiya Nauk SSR: Kiev, Ukraine, 1985; Volume 58, pp. 567–569. [Google Scholar]
- Ihle, T.; Müller-Krumbhaar, H. Fractal and compact growth morphologies in phase transitions with diffusion transport. Phys. Rev. E
**1994**, 49, 2972. [Google Scholar] [CrossRef] [PubMed][Green Version] - McFadden, S.; Browne, D.J. A generalised version of an Ivantsov-based dendrite growth model incorporating a facility for solute measurement ahead of the tip. Comput. Mater. Sci.
**2012**, 55, 245–254. [Google Scholar] [CrossRef] - Dhar, P.; Eck, E.; Israelachvili, J.N.; Lee, D.W.; Min, Y.; Ramachandran, A.; Waring, A.J.; Zasadzinski, J.A. Lipid-protein interactions alter line tensions and domain size distributions in lung surfactant monolayers. Biophys. J.
**2012**, 102, 56–65. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bouda, M.; Caplan, J.S.; Saiers, J.E. Box-counting dimension revisited: Presenting an efficient method of minimizing quantization error and an assessment of the self-similarity of structural root systems. Front. Plant Sci.
**2016**, 7, 149. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dehkhoda, P.; Tavakoli, A. Crown-Sierpinski microstrip antenna: Further reduction of the size of a crown square fractal. In Proceedings of the 2005 IEEE Antennas and Propagation Society International Symposium, Washington, DC, USA, 3–8 July 2005; pp. 247–250. [Google Scholar]

**Figure 1.**Isotherms of erucic acid monolayers at different compression velocities ${v}_{C}$ at pH 3,

**T**= 10 °C. The inset shows the shift of the phase transition lateral pressure ${\pi}_{t}-{\pi}_{\infty}$ in dependence on the shift in the molecular area, ${A}_{\infty}-{A}_{t}$, while ${v}_{C}$ is increased. A blue arrow marks the transition pressure ${\pi}_{\infty}$ for the isotherm measured at the lowest compression velocity, which is approximated as the equilibrium isotherm.

**Figure 2.**Domain growth in the LE/LC coexistence region of slowly (left, ${v}_{C}$ = 1.2 ${\u212b}^{2}/\left(\mathrm{molecule}\xb7\mathrm{min}\right)$ ) and quickly compressed erucic acid monolayers (right, ${v}_{C}$ = 2.3 ${\u212b}^{2}/\left(\mathrm{molecule}\xb7\mathrm{min}\right)$ ). The images were obtained with Brewster angle microscopy; all scale bars are 100 μm long. Experimental conditions as in Figure 1.

**Figure 3.**Fractal dimension ${D}_{F}$ of the domains in the coexistence region of erucic acid monolayers (experimental conditions as in Figure 1) at different compression velocities ${v}_{C}$ as a function of time $t$. The fractal dimension was determined as described in Appendix B. Exemplary error bars are included.

**Scheme 1.**The parameters characterizing a domain with a fractal dimension are branch length $l$, tip radius $r$, side branch separation $\lambda $, and branch width $w$.

**Figure 4.**Bottom: Determining the length of a main branch at different times from BAM images of an erucic acid monolayer compressed at ${v}_{C}=2.5{\u212b}^{2}/\left(\mathrm{molecule}\xb7\mathrm{min}\right)$) (experimental conditions as in Figure 1). Top: The length of the main branch of different domains in dependence on the time t. Three monolayers were analyzed, each with a different compression speed ${v}_{C}$ as indicated. The lines are linear fits, whose slopes correspond to the constant growth velocity ${v}_{R}$ as indicated. Exemplary error bars are included. For each monolayer, $t=0\mathrm{s}$ refers to the first observation of a domain. For a selected domain, the lowest value of $t$ corresponds to the first observation of this domain, when it is still very small.

**Figure 5.**The growth speed ${v}_{R}$ as a function of the excess lateral pressure $\mathsf{\Delta}\pi =\pi -{\pi}_{\infty}$ for three monolayers compressed with three different compression velocity ${v}_{C}$ as indicated. Additionally, shown is the supersaturation $\mathsf{\Delta}c$ in the LE phase. The data were derived from Figure 4.

**Figure 6.**Tip radius $r$ of domains from three different monolayers characterized by different compression velocity ${v}_{C}$ (indicated) as a function of the excess lateral pressure $\mathsf{\Delta}\pi =\pi -{\pi}_{\infty}$, the lateral pressure above the LE/LC phase transition of erucic acid (cf. Figure 1) at equilibrium conditions. The tip radius is shown at the lateral pressure where it could be first unambiguously resolved. With further compression of the monolayer, the tip radius did not change. As long as $\mathsf{\Delta}\pi $ is small, it is proportional to the supersaturation $\mathsf{\Delta}c=1/A-1/{A}_{\infty}$ in the LE phase, which is also shown.

**Figure 7.**Side branch separation $\lambda $ versus the compression velocity ${v}_{C}$ for six different monolayers of erucic acid (cf. Figure 1). For each monolayer, about 20 different domains were analyzed. The domains nucleated at different lateral excess pressure $\mathsf{\Delta}\pi $.

**Table 1.**Comparison of velocities influencing domain growth. ${v}_{C}$ is the compression velocity calculated from the dimensions of the Langmuir trough and the barrier velocity ${v}_{Barrier}$. ${v}_{R}$ is the growth velocity of the main branch and ${v}_{\infty}$ the flow velocity in the LE phase of molecules far away from the domain, yet flowing already toward the domain.

${\mathit{v}}_{\mathit{C}}$ $[{\u212b}^{2}/\left(\mathrm{molecule}\xb7\mathrm{min}\right)]$ | ${\mathit{v}}_{\mathit{B}\mathit{a}\mathit{r}\mathit{r}\mathit{i}\mathit{e}\mathit{r}}$ $[\mathrm{cm}/\mathrm{min}]$ | ${\mathit{v}}_{\mathit{B}\mathit{a}\mathit{r}\mathit{r}\mathit{i}\mathit{e}\mathit{r}}$ $[\mathsf{\mu}\mathrm{m}/\mathrm{s}]$ | ${\mathit{v}}_{\mathit{R}}$ $[\mathsf{\mu}\mathrm{m}/\mathrm{s}]$ | ${\mathit{v}}_{\mathit{\infty}}$ $[\mathsf{\mu}\mathrm{m}/\mathrm{s}]$ |
---|---|---|---|---|

1.2 | 0.46 | 76.67 | 2.1–4.9 | 0.8–1.2 |

2.3 | 1.08 | 180.00 | 6.0–24.8 | 3.8–5.5 |

2.5 | 1.20 | 200.00 | 11.3–16.9 | 4.3–6.3 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gellert, F.; Ahrens, H.; Wulff, H.; Helm, C.A. Seaweed and Dendritic Growth in Unsaturated Fatty Acid Monolayers. *Membranes* **2022**, *12*, 698.
https://doi.org/10.3390/membranes12070698

**AMA Style**

Gellert F, Ahrens H, Wulff H, Helm CA. Seaweed and Dendritic Growth in Unsaturated Fatty Acid Monolayers. *Membranes*. 2022; 12(7):698.
https://doi.org/10.3390/membranes12070698

**Chicago/Turabian Style**

Gellert, Florian, Heiko Ahrens, Harm Wulff, and Christiane A. Helm. 2022. "Seaweed and Dendritic Growth in Unsaturated Fatty Acid Monolayers" *Membranes* 12, no. 7: 698.
https://doi.org/10.3390/membranes12070698