# Biomimetic Groundwork for Thermal Exchange Structures Inspired by Plant Leaf Design

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## Abstract

**:**

## 1. Introduction

#### 1.1. Thermal Design Innovation and Biomimetics

^{®}[13]), vascular cooling design for injection molds [14] or solar panels [15], responsive architectural façades [16], and evaporation-driven micropumps for drug delivery [17].

#### 1.2. Plant Structures, Leaf Exchange, and Thermodynamics

## 2. Materials and Methods

- Literature review and identification of botanical case studies pointing to a relation between leaf thermal function and morphology patterns—listed as leaf role models;
- Definition of leaf morphotypes and shape features of interest involved in such relations, presumed relevant for leaf exchange and plant thermal management, based on botany and transfer physics literature;
- Identification of thermal design features and hypotheses abstracted from the leaf literature review—listed as leaf-inspired design principles;
- Biology research addressing one instance of the reviewed case studies, that of sun–shade leaf dimorphism in oak trees. The experimental approach involved shape analysis of oak leaves with single-parameter metrics. This tested the suitability of basic geometrical parameters for differentiating sun and shade leaves (Section 2.1);
- Translation of a subset of design principles into a family of two-dimensional abstract geometries, reflecting leaf “morphotypes” and some results of the shape analysis of oak leaves. Paper models were used as leaf analogs in “proof-of-concept” evaporation tests, to observe and compare the evaporative transfer of these leaf-inspired geometries (Section 2.3).

#### 2.1. Biology Research: Shape Analysis of Oak Leaves

^{2}:surface area (P

^{2}/A), normalizing the margin extension to the leaf area. Roundness, calculated as 4πA/H

^{2}, is insensitive to leaf border irregularity (i.e., dissection) and tests the circularity of a shape’s overall spread, taking a maximum value of 1 for circles. The maximum inscribed circle diameter presumably gives the leaf effective width and characteristic dimension [32], an interpretation which is discussed in Section 4.3. Alternatively, a mathematical approach suggested in boundary layer theory applied to leaves was used for characterizing abstract geometries [33]:

_{eff}is the effective dimension, W is the maximum shape width in the airflow x-axis direction, Y(x) is the variable distance from one edge to the opposite in the y-axis direction, and n is an empirically determined parameter depending on boundary layer laminarity and flow conditions (typically, n = 0.5 or n = 0.75 for a free convection regime) [33]. Otherwise, $\sqrt{\mathrm{A}}$ was used for real leaves as characteristic leaf length, for tests on parameter size-dependence. While leaf aspect ratio was based on leaf maximum to minimum Feret diameter ratio, elongation e was calculated from the minimum bounding box side lengths (e = 1 − short side: long side). ShapeFilter estimation of the fractal dimension is based on a box-counting algorithm [31]. For each measured parameter, one-way analysis of variance tested if shape quantifiers significantly differentiated sun from shade leaves among all or within specific oak species (two-sample Student’s t-test assuming unequal variances, JMP statistics software). We used a significance level of α = 0.05 for all statistical tests. A total number of 206 leaves (96 sun, 110 shade) were analyzed.

#### 2.2. Biomimetic Design Principles: Abstracted Leaf-Inspired Geometries

_{eff}values little differed for n = 0.5).

#### 2.3. Proof-of-Concept Evaporation Study with Bio-Inspired Paper Models

^{2}; D = 10 cm, A = 79 cm

^{2}), representative of smaller/younger and larger/older leaves. Models’ dry and wet weight were satisfactorily constant among the different replicas and shapes (within 5% error), thus local disparities in the filter paper’s porosity were considered negligible. Paper models were coupled with equally shaped thin plastic plates keeping the paper flat and restraining evaporation to one side (analogous to most terrestrial plant leaves, which have transpiring pores only on the lower surface). Models were wetted with purified water via water level rising for uniform wetting, as described in a previous work [34], then covered with sealing lids cut to size, and transferred to an analytical scale (Denver Instrument M-220, class I accuracy ±0.1 mg) inside an environmental chamber (Associated Environmental Systems, model BHD-408) for air temperature and relative humidity control, set to T

_{air}= 35 °C and RH

_{air}= 35%. Lids were kept for 1 or 2 h to hinder evaporation while models thermally adjusted to the chamber, and then removed to release the water vapor. Thin supports held the models in place horizontally oriented, elevated from the balance plate (8.5 cm high), for proper heat and mass exchange with environmental air.

_{air}< 0.1 m/s, measured with a hot-wire anemometer). The top was kept open, to ventilate water vapor and assure constant air temperature and humidity inside the enclosure, as maintained by the chamber. During evaporation recording, a humidity-temperature sensor inside the analytical scale accompanied slight temperature (±0.5 °C) and relative humidity variations (±7%) along chamber regulation cycles, consistent among all tests. A thermal camera (FLIR T430 series, 320 × 240 pixels) was also mounted inside the chamber to capture time-lapse thermograms (1/min) of the evaporative cooling effect for two of the shapes.

_{air}, model average surface temperature, t

_{model}, relative humidity, H, and saturated vapor densities, n

_{s}, at the model surface (air assumed saturated) and in the free stream:

_{V}[n

_{s}(t

_{model}) − H n

_{s}(t

_{air})]/δ,

_{V}is the diffusion coefficient of water vapor in air and δ is the boundary layer thickness. Assuming a laminar boundary layer, a first approximation may be given by δ ≈ 4.91$\sqrt{{\mathsf{\nu}\mathrm{L}}_{\mathrm{eff}}/{\mathrm{v}}_{\mathrm{air}}}$ [33], with ν kinematic viscosity of air. Evaporation rates (weight loss rate) were taken as the slope values extracted from linear regression fits to the time series plot of models’ weight (Microsoft Excel 2013). Weight loss within the initial drying phase was confirmed to be linear with our setup (R

^{2}> 0.999). Measured evaporation fluxes were compared via one-way ANOVA with JMP software.

## 3. Results

#### 3.1. Biology Research Findings

#### 3.1.1. Leaf Role Models for Evaporative Thermal Design

#### 3.1.2. Quantifying Oak Leaf Dimorphism

^{2}= 0.0003), convexity (R

^{2}< 0.006), and solidity (R

^{2}< 0.07) did not significantly correlate with the characteristic leaf length ($\sqrt{\mathrm{A}}$), suggesting that oak leaf shape is independent of size. Sun leaves’ significantly higher LDI and lower convexity reflect the extension of the border perimeter within a limited area, which is possible via shape dissection with more numerous and/or proportionally larger lobes (Q. falcata) or marginal teeth (e.g., Q. macrocarpa crown, Q. bicolor). These first results support the general observation that oak sun leaves are more deeply lobed and point to the possibly important role of the leaf border in exchange performance. They also show that simple parameters can quantify sun–shade dimorphism, and such parametrization may guide the design of dissipative exchangers inspired by sun leaves.

_{sun}= 10, N

_{shade}= 10), whose sun–shade subtle differences were statistically insignificant for the used parameters, trends for individual oak species were similar to the cross-species sample, with sun leaves significantly more dissected. Moreover, aspect ratio (p = 0.007) and elongation (p = 0.03) were significantly lower for Q. ellipsoidalis sun leaves, whose relatively longer lobes reduced the ellipticity of the overall leaf shape. Solidity was an effective parameter to differentiate lobation from toothiness. Both cases extend the leaf border (perimeter), but relatively small marginal teeth do not expand the leaf convex hull area as much as a few, relatively large lobes do, even when both yield similar LDI values. For instance, Q. macrocarpa and Q. bicolor toothed morphologies were identified as more “solid” than more dramatically lobed ones (Q. ellipsoidalis). The tendency of sun leaves to morph towards higher LDI and lower solidity is represented in Figure 3. Pairwise tests gave the following negative correlations between solidity and LDI (sun and shade leaves): Q. alba r = −0.90 (p < 0.0001); Q. ellipsoidalis r = −0.86 (p < 0.0001); Q. macrocarpa r = −0.58 (p = 0.0028); and Q. bicolor r = −0.15 (p = 0.53). These correlations link leaf shape dissection in oak species to border extension.

#### 3.2. Leaf-Inspired Design Principles

#### 3.2.1. Geometrical Abstraction from Leaf Role Models

#### 3.2.2. Leaf-Inspired Morphotypical Geometries

_{eff}(Equation (1)), and thus was expected to result in higher dissipation rates. However, the evaporation results presented in the following section reveal that none of these shape parameters completely dictate dissipative performance, demonstrating how difficult it is to theorize geometry–transfer relations, especially for mass transfer under low airflow conditions.

#### 3.3. Proof-Of-Concept Evaporation Tests

^{2}= 1.3), a less thermally challenging regime for real leaves, as will be discussed in Section 4.2.4.

_{eff}to relate to boundary layer thickness through Equation (2), were overall underestimated for the larger models (D = 10 cm) and overestimated for shape E.

## 4. Discussion

#### 4.1. Mass Transfer Enhanced by Geometry Dissection

#### 4.2. Biological and Biomimetic Significance

#### 4.2.1. Heterophyllous Leaf Dissection

#### 4.2.2. Sun–Shade Leaf Dimorphism

#### 4.2.3. Marginal Teeth

#### 4.2.4. Leaves from Temperate to Tropical Climates as Exchangers

#### 4.2.5. The Multifactorial and Multifunctional Realm of Leaf Morphology

#### 4.3. Applicability of Shape Parameters

#### 4.3.1. Sun–Shade Leaf Morphology

#### 4.3.2. Shape Parameters Relation to Evaporative Performance

_{eff}, defined by a weighted mean (Equation (1)) is proposed in botany literature [33,81] as a geometrical proxy for boundary layer thickness, which accounts for shape characteristics beyond scale.

_{eff}, was not a good predictor of evaporative performance for the tested paper models, as it was markedly overestimated for the elongated shape (E) and overall underestimated for the larger models (D = 10 cm). The computation of this theoretical parameter relies on the assumption that toothed edges do not interrupt flow–surface contact—an unlikely scenario for geometries with large-scale teeth—and is a good approximation only in either forced or free convection-dominant regimes, not mixed [81]. In our experiment, the relation between boundary layer thickness and L

_{eff}was likely compromised by the mixed flow conditions, especially for shape E, which was more resistant to transfer than estimated. For the larger models, whose evaporative fluxes were overall underestimated, results might also have diverged due to unaccounted evaporation-driven turbulence introduced in the boundary layer, causing it to be thinner than predicted. Nonetheless, because shapes L and T had similar L

_{eff}and evaporation performances, L

_{eff}is possibly more accurate than the inscribed circle diameter approach [32] as a geometrical predictor of boundary layer thickness over complex leaf geometries (e.g., lobed leaves). For simpler, non-dissected shapes, and flow regimes more largely governed by forced convection, the minimum Feret diameter gives an edge-to-edge length dimension that is easier to measure than the inscribed circle diameter and likely suffices in predicting transfer performance. In field research, leaf width strongly correlated with leaf–air temperature differences (indicative of boundary layer thickness) within a wind speed range up to 1 m/s [82].

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Chu, K.; Zhu, Y.; Miljkovic, N.; Nam, Y.; Enright, R.; Wang, E.N. Enhanced boiling heat transfer with copper oxide hierarchical surfaces. In Proceedings of the 2013 Transducers Eurosensors XXVII, 17th International Conference on Solid-State Sensors, Actuators and Microsystems, Barcelona, Spain, 16–20 June 2013; pp. 2272–2275, ISBN 978-1-4673-5983-2. [Google Scholar] [CrossRef]
- Sharma, A.; Tyagi, V.V.; Chen, C.R.; Buddhi, D. Review on thermal energy storage with phase change materials and applications. Renew. Sustain. Energy Rev.
**2009**, 13, 318–345. [Google Scholar] [CrossRef] - Wang, Q.; Chen, G.; Chen, Q.; Zeng, M. Review of Improvements on Shell-and-Tube Heat Exchangers with Helical Baffles. Heat Transf. Eng.
**2010**, 31, 836–853. [Google Scholar] [CrossRef] - Incropera, F.P. Fundamentals of Heat and Mass Transfer; John Wiley: Hoboken, NJ, USA, 2007; pp. 574–576. [Google Scholar]
- Jaluria, Y. Computational Heat Transfer; Routledge: Abingdon, UK, 2017; pp. 203–222. [Google Scholar]
- Shang, J.S. Three decades of accomplishments in computational fluid dynamics. Prog. Aerosp. Sci.
**2004**, 40, 173–197. [Google Scholar] [CrossRef] - Shah, R.K.; Sekulic, D.P. Fundamentals of Heat Exchanger Design; John Wiley & Sons: Hoboken, NJ, USA, 2003; pp. 78–80. [Google Scholar]
- Angilletta, M.J., Jr.; Angilletta, M.J. Thermal Adaptation: A Theoretical and Empirical Synthesis; OUP Oxford: Oxford, UK, 2009; p. 126. [Google Scholar]
- Cafsia, L. Plyskin; insulating façade panel based on Biomimicry of the polar bear skin and fur. In Proceedings of the IABSE Conference, Bath, UK, 19–20 April 2017; IABSE Symposium Report. Volume 108, pp. 58–59. [Google Scholar] [CrossRef]
- Dongjin, C. Sea Otter Fur Structure Imitated Waterproof and Warm-Keeping Fabric and Preparation Method Thereof. CN Patent CN108330703A, 27 July 2018. [Google Scholar]
- Transport of Heat and Mass in Natural Porous Materials with Graded Structure. Available online: https://www.trr141.de/index.php/research-areas-2/research-areas/ (accessed on 20 May 2019).
- Turner, J.S.; Soar, R.C. Beyond biomimicry: What termites can tell us about realizing the living building. In Proceedings of the First International Conference on Industrialized, Intelligent Construction (I3CON), Loughborough University, Loughborough, UK, 14–16 May 2008. [Google Scholar]
- Kerbel, M.; Kulyk, R. Method and Apparatus for Managing an Energy Consuming Load. U.S. Patent 8,527,109B2, 3 September 2013. [Google Scholar]
- Smith, C.A.; Bernett, A.; Hanson, E.; Garvin, C. Tapping into Nature; Terrapin Bright Green LLC: New York, NY, USA, 2015; p. 27. [Google Scholar]
- Fraunhofer, I.S.E. FracTherm® Technology in Roll-Bond Solar Absorbers; Fraunhofer Institute for Solar Energy Systems ISE Press Release: Freiburg, Germany, 2013. [Google Scholar]
- Reichert, S.; Menges, A.; Correa, D. Meteorosensitive architecture: Biomimetic building skins based on materially embedded and hygroscopically enabled responsiveness. Comput.-Aided Des.
**2015**, 60, 50–69. [Google Scholar] [CrossRef] - Kim, H.; Kim, K.; Lee, S.J. Compact and Thermosensitive Nature-inspired Micropump. Sci. Rep.
**2016**, 6, 36085. [Google Scholar] [CrossRef] [PubMed] - Lütz, C. Plants in Alpine Regions: Cell Physiology of Adaption and Survival Strategies; Springer Science & Business Media: Berlin, Germany, 2011; pp. 62–63. [Google Scholar]
- Gates, D.M. Biophysical Ecology; Springer Inc.: New York, NY, USA, 1980; pp. 25–28. [Google Scholar]
- Sinha, R.K. Modern Plant Physiology; CRC Press: Boca Raton, FL, USA, 2004; p. 84. [Google Scholar]
- Mahan, J.R.; Upchurch, D.R. Maintenance of constant leaf temperature by plants—I. Hypothesis-limited homeothermy. Environ. Exp. Bot.
**1988**, 28, 351–357. [Google Scholar] [CrossRef] - Michaletz, S.T.; Weiser, M.D.; Zhou, J.; Kaspari, M.; Helliker, B.R.; Enquist, B.J. Plant Thermoregulation: Energetics, Trait—Environment Interactions, and Carbon Economics. Trends Ecol. Evol.
**2015**, 30, 714–724. [Google Scholar] [CrossRef] [PubMed] - Vincent, J.F.V.; Bogatyreva, O.A.; Bogatyrev, N.R.; Bowyer, A.; Pahl, A.-K. Biomimetics: Its practice and theory. J. R. Soc. Interface
**2006**, 3, 471–482. [Google Scholar] [CrossRef] - Gu, Y.; Zhang, W.; Mou, J.; Zheng, S.; Jiang, L.; Sun, Z.; Wang, E. Research progress of biomimetic superhydrophobic surface characteristics, fabrication, and application. Adv. Mech. Eng.
**2017**, 9, 1687814017746859. [Google Scholar] [CrossRef] - Miguel, S.; Hehn, A.; Bourgaud, F. Nepenthes: State of the art of an inspiring plant for biotechnologists. J. Biotechnol.
**2018**, 265, 109–115. [Google Scholar] [CrossRef] - Zheng, Y.; Zhou, X.; Xing, Z.; Tu, T. Exploring the underwater air-retaining ability and thermal insulating effect of terry fabrics inspired by Salvinia molesta. Text. Res. J.
**2018**. [Google Scholar] [CrossRef] - Nicotra, A.B.; Leigh, A.; Boyce, C.K.; Jones, C.S.; Niklas, K.J.; Royer, D.L.; Tsukaya, H. The evolution and functional significance of leaf shape in the angiosperms. Funct. Plant Biol.
**2011**, 38, 535–552. [Google Scholar] [CrossRef] - Helms, M.; Vattam, S.S.; Goel, A.K. Biologically inspired design: Process and products. Des. Stud.
**2009**, 30, 606–622. [Google Scholar] [CrossRef] - Gratani, L. Plant Phenotypic Plasticity in Response to Environmental Factors. Adv. Bot.
**2014**, 208747. [Google Scholar] [CrossRef] - Kottek, M.; Grieser, J.; Beck, C.; Rudolf, B.; Rubel, F. World map of the Köppen-Geiger climate classification updated. Meteorol. Z.
**2016**, 15, 259–263. [Google Scholar] [CrossRef] - Wagner, T.; Lipinski, H.-G. IJBlob: An ImageJ Library for Connected Component Analysis and Shape Analysis. J. Open Res. Softw.
**2013**, 1, e6. [Google Scholar] - Leigh, A.; Sevanto, S.; Close, J.D.; Nicotra, A.B. The influence of leaf size and shape on leaf thermal dynamics: Does theory hold up under natural conditions? Plant Cell Environ.
**2017**, 40, 237–248. [Google Scholar] [CrossRef] - Schuepp, P. Tansley review no. 59 leaf boundary layers. New Phytol.
**1993**, 125, 477–507. [Google Scholar] [CrossRef] - Gruber, P.; Rupp, A.I.K.S. Investigation of leaf shape and edge design for faster evaporation in biomimetic heat dissipation systems. In Bioinspiration, Biomimetics, and Bioreplication VIII, Proceedings of the SPIE Smart Structures and Materials + Nondestructive Evaluation and Health Monitoring, Denver, CO, USA, 5–7 March 2018; SPIE-International Society for Optics and Photonics: Bellingham, WA, USA, 2018; Volume 10593, p. 105930H. [Google Scholar]
- Royer, D.L.; Wilf, P.; Janesko, D.A.; Kowalski, E.A.; Dilcher, D.L. Correlations of climate and plant ecology to leaf size and shape: Potential proxies for the fossil record. Am. J. Bot.
**2005**, 92, 1141–1151. [Google Scholar] [CrossRef] - Lee, S.J.; Kim, H.; Ahn, S. Water transport in porous hydrogel structures analogous to leaf mesophyll cells. Microfluid. Nanofluidics
**2015**, 18, 775–784. [Google Scholar] [CrossRef] - Shahraeeni, E.; Lehmann, P.; Or, D. Coupling of evaporative fluxes from drying porous surfaces with air boundary layer: Characteristics of evaporation from discrete pores. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef] - Winn, A.A. The Functional Significance and Fitness Consequences of Heterophylly. Int. J. Plant Sci.
**1999**, 160, S113–S121. [Google Scholar] [CrossRef] [PubMed] - Schmerler, S.B.; Clement, W.L.; Beaulieu, J.M.; Chatelet, D.S.; Sack, L.; Donoghue, M.J.; Edwards, E.J. Evolution of leaf form correlates with tropical-temperate transitions in Viburnum (Adoxaceae). Proc. Biol. Sci.
**2012**, 279, 3905–3913. [Google Scholar] [CrossRef] [PubMed] - Peppe, D.J.; Royer, D.L.; Cariglino, B.; Oliver, S.Y.; Newman, S.; Leight, E.; Enikolopov, G.; Fernandez-Burgos, M.; Herrera, F.; Adams, J.M.; et al. Sensitivity of leaf size and shape to climate: Global patterns and paleoclimatic applications. New Phytol.
**2011**, 190, 724–739. [Google Scholar] [CrossRef] - Talbert, C.M.; Holch, A.E. A study of the lobing of sun and shade leaves. Ecology
**1957**, 38, 655–658. [Google Scholar] [CrossRef] - Sack, L.; Melcher, P.J.; Liu, W.H.; Middleton, E.; Pardee, T. How strong is intracanopy leaf plasticity in temperate deciduous trees? Am. J. Bot.
**2006**, 93, 829–839. [Google Scholar] [CrossRef] - Billings, F.H. Precursory Leaf Serrations of Ulmus. Bot. Gaz.
**1905**, 40, 224–225. [Google Scholar] [CrossRef] - Macmillan, R.J. Aspects of heterophylly in Morus alba. Ph.D. Thesis, City University of New York, New York, NY, USA, 2002. [Google Scholar]
- Lewis, M.C. Genecological differentiation of leaf morphology in Geranium sanguineum L. New Phytol.
**1969**, 68, 481–503. [Google Scholar] [CrossRef] - Royer, D.L.; McElwain, J.C.; Adams, J.M.; Wilf, P. Sensitivity of leaf size and shape to climate within Acer rubrum and Quercus kelloggii. New Phytol.
**2008**, 179, 808–817. [Google Scholar] [CrossRef] - Royer, D.L.; Wilf, P. Why do Toothed Leaves Correlate with Cold Climates? Gas Exchange at Leaf Margins Provides New Insights into a Classic Paleotemperature Proxy. Int. J. Plant Sci.
**2006**, 167, 11–18. [Google Scholar] [CrossRef][Green Version] - Vogel, S. Leaves in the lowest and highest winds: Temperature, force and shape. New Phytol.
**2009**, 183, 13–26. [Google Scholar] [CrossRef] [PubMed] - Vogel, S. ‘Sun Leaves’ and ‘Shade Leaves’: Differences in Convective Heat Dissipation. Ecology
**1968**, 49, 1203–1204. [Google Scholar] [CrossRef] - Zwieniecki, M.A.; Boyce, C.K.; Holbrook, N.M. Hydraulic limitations imposed by crown placement determine final size and shape of Quercus rubra L. leaves. Plant Cell Environ.
**2005**, 27, 357–365. [Google Scholar] [CrossRef] - Jebb, M.H. A revision of the genus Trevesia (Araliaceae). Glasra
**1998**, 3, 85–113. [Google Scholar] - Gray, E.; Gray, R.E. Leaf Lobation Patterns in Mulberry. Castanea
**1987**, 52, 216–224. [Google Scholar] - Gottschlich, D.E.; Smith, A.P. Convective heat transfer characteristics of toothed leaves. Oecologia
**1982**, 53, 418–420. [Google Scholar] [CrossRef] [PubMed] - Bergles, A.E. Some Perspectives on Enhanced Heat Transfer—Second-Generation Heat Transfer Technology. J. Heat Transf.
**1988**, 110, 1082–1096. [Google Scholar] [CrossRef] - Llano-Sánchez, L.E.; Domínguez-Cajeli, D.M.; Ruiz-Cárdenas, L.C. Thermal transfer analysis of tubes with extended surface with fractal design. Fac. Ing.
**2018**, 27, 31–37. [Google Scholar] [CrossRef][Green Version] - Kuddusi, L.; Eğrican, N. A critical review of constructal theory. Energy Convers. Manag.
**2008**, 49, 1283–1294. [Google Scholar] [CrossRef] - Xu, P.; Yu, B.; Yun, M.; Zou, M. Heat conduction in fractal tree-like branched networks. Int. J. Heat Mass Transf.
**2006**, 49, 3746–3751. [Google Scholar] [CrossRef] - Huang, Z.; Hwang, Y.; Aute, V.; Radermacher, R. Review of Fractal Heat Exchangers. In Proceedings of the 16th International Refrigeration and Air Conditioning Conference, Purdue, IN, USA, 11–14 July 2016. [Google Scholar]
- Herr, M. Design Criteria for Low-Noise Trailing-Edges. In Proceedings of the 13th AIAA/CEAS Aeroacoustics Conference, Rome, Italy, 21–23 May 2007; American Institute of Aeronautics and Astronautics: Reston, VA, USA, 2007. [Google Scholar] [CrossRef]
- Wells, C.L.; Pigliucci, M. Adaptive phenotypic plasticity: The case of heterophylly in aquatic plants. Perspect. Plant Ecol. Evol. Syst.
**2000**, 3, 1–18. [Google Scholar] [CrossRef] - Rice, S.K.; Schuepp, P.H. On the ecological and evolutionary significance of branch and leaf morphology in aquatic Sphagnum (Sphagnaceae). Am. J. Bot.
**1995**, 82, 833–846. [Google Scholar] [CrossRef] - Potter, K.; Davidowitz, G.; Woods, H.A. Insect eggs protected from high temperatures by limited homeothermy of plant leaves. J. Exp. Biol.
**2009**, 212, 3448–3454. [Google Scholar] [CrossRef] [PubMed][Green Version] - Slot, M.; Krause, G.H.; Krause, B.; Hernández, G.G.; Winter, K. Photosynthetic heat tolerance of shade and sun leaves of three tropical tree species. Photosynth. Res.
**2018**, 141, 119–130. [Google Scholar] [CrossRef] [PubMed] - Terashima, I.; Miyazawa, S.-I.; Hanba, Y.T. Why are sun leaves thicker than shade leaves?—Consideration based on analyses of CO
_{2}diffusion in the leaf. J. Plant Res.**2001**, 114, 93–105. [Google Scholar] [CrossRef] - Schymanski, S.J.; Or, D.; Zwieniecki, M. Stomatal Control and Leaf Thermal and Hydraulic Capacitances under Rapid Environmental Fluctuations. PLoS ONE
**2013**, 8, e54231. [Google Scholar] [CrossRef] - Givnish, T.J.; Kriebel, R. Causes of ecological gradients in leaf margin entirety: Evaluating the roles of biomechanics, hydraulics, vein geometry, and bud packing. Am. J. Bot.
**2017**, 104, 354–366. [Google Scholar] [CrossRef][Green Version] - Royer, D.L. Climate reconstruction from leaf size and shape: New developments and challenges. Paleontol. Soc. Pap.
**2012**, 18, 195–212. [Google Scholar] [CrossRef][Green Version] - Seddon, G. Xerophytes, xeromorphs and sclerophylls: The history of some concepts in ecology. Biol. J. Linn. Soc.
**1974**, 6, 65–87. [Google Scholar] [CrossRef] - Wickens, G.E. Ecophysiology of Economic Plants in Arid and Semi-Arid Lands; Springer Science & Business Media: Berlin, Germany, 2013; pp. 155–160. [Google Scholar]
- Wright, I.J.; Dong, N.; Maire, V.; Prentice, I.C.; Westoby, M.; Díaz, S.; Gallagher, R.V.; Jacobs, B.F.; Kooyman, R.; Law, E.A.; et al. Global climatic drivers of leaf size. Science
**2017**, 357, 917–921. [Google Scholar] [CrossRef][Green Version] - Parkhurst, D.F.; Loucks, O. Optimal leaf size in relation to environment. J. Ecol.
**1972**, 60, 505–537. [Google Scholar] [CrossRef] - Bradshaw, A.D. Evolutionary Significance of Phenotypic Plasticity in Plants. Adv. Genet.
**1965**, 13, 115–155. [Google Scholar] - Chitwood, D.H.; Rundell, S.M.; Li, D.Y.; Woodford, Q.L.; Yu, T.T.; Lopez, J.R.; Greenblatt, D.; Kang, J.; Londo, J.P. Climate and Developmental Plasticity: Interannual Variability in Grapevine Leaf Morphology. Plant Physiol.
**2016**, 170, 1480–1491. [Google Scholar] [CrossRef][Green Version] - Vaz Monteiro, M.; Blanuša, T.; Verhoef, A.; Hadley, P.; Cameron, R.W. Relative importance of transpiration rate and leaf morphological traits for the regulation of leaf temperature. Aust. J. Bot.
**2016**, 64, 32–44. [Google Scholar] [CrossRef] - Grace, J.; Wilson, J. The Boundary Layer over a Populus Leaf. J. Exp. Bot.
**1976**, 27, 231–241. [Google Scholar] [CrossRef] - Yamazaki, K. Gone with the wind: Trembling leaves may deter herbivory. Biol. J. Linn. Soc.
**2011**, 104, 738–747. [Google Scholar] [CrossRef][Green Version] - Collatz, G.J.; Ball, J.T.; Grivet, C.; Berry, J.A. Physiological and environmental regulation of stomatal conductance, photosynthesis and transpiration: A model that includes a laminar boundary layer. Agric. Meteorol.
**1991**, 54, 107–136. [Google Scholar] [CrossRef] - Rohsenow, W.M.; Hartnett, J.P.; Ganic, E.N. Handbook of Heat Transfer Applications; McGraw-Hill Book Co: New York, NY, USA, 1985; pp. 1–7. [Google Scholar]
- McLellan, T.; Endler, J.A. The Relative Success of Some Methods for Measuring and Describing the Shape of Complex Objects. Syst. Biol.
**1998**, 47, 264–281. [Google Scholar] [CrossRef][Green Version] - Biot, E.; Cortizo, M.; Burguet, J.; Kiss, A.; Oughou, M.; Maugarny-Calès, A.; Gonçalves, B.; Adroher, B.; Andrey, P.; Boudaoud, A.; et al. Multiscale quantification of morphodynamics: MorphoLeaf software for 2D shape analysis. Development
**2016**, 143, 3417–3428. [Google Scholar] [CrossRef][Green Version] - Parkhurst, D.F.; Duncan, P.R.; Gates, D.M.; Kreith, F. Wind-tunnel modelling of convection of heat between air and broad leaves of plants. Agric. Meteorol.
**1968**, 5, 33–47. [Google Scholar] [CrossRef] - Lusk, C.H.; Clearwater, M.J.; Laughlin, D.C.; Harrison, S.P.; Prentice, I.C.; Nordenstahl, M.; Smith, B. Frost and leaf-size gradients in forests: Global patterns and experimental evidence. New Phytol.
**2018**, 219, 565–573. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Plant thermal exchange budget and leaf heteromorphism, illustrated with sun and shade leaves from different oak species.

**Figure 2.**Leaf shape study based on qualitative features (e.g., leaf lobes, sinuses, marginal teeth) and quantitative parameters (e.g., perimeter, area, convex hull, inscribed circle).

**Figure 3.**Sensitivity of shape parameters to sun leaf dissection in Q. bicolor (left, green dots), Q. alba (left, purple dots), Q. macrocarpa (right, red dots), and Q. ellipsoidalis (right, yellow dots). The species’ datapoints are grouped and emphasized by merely qualitative colored sets. Hollow dots represent the other oak species’ leaf datapoints from the collected sample.

**Figure 4.**Mean evaporation flux for filter paper models with leaf-inspired morphotypes, for two characteristic dimensions (control circle D = 5 and 10 cm). Error bars represent ±1 SD. Orange markers represent theoretical estimates based on L

_{eff}(Equation (2)), for t

_{model}= 28 °C and v

_{air}= 0.015 m/s.

**Figure 5.**Thermal time-lapse of evaporating models of lobed and toothed shapes (characteristic dimension 10 cm): (

**a**) Shortly after cap removal (evaporation release); (

**b**) 1 min after; and (

**c**) 20 min after.

Leaf Shape Variation | Plant Case Studies, Reported Observations | References |
---|---|---|

Temporal variation:SEASONAL • summer vs. winter heterophylly | Southern coastal violet (Viola septemloba): Developed more lobed leaves in summer, which maintained lower leaf temperatures | [38] |

Spatial variation:GLOBAL • geographical trends • plant local adaptation | Geranium sanguineum: Leaves develop more elongated lobes in drier, continental habitats. | [45] |

Viburnum (Adoxaceae) clade: Leaves in warmer climates are more elongated and entire, overall. | [39] | |

LEAF • developing leaves • margin transpiration | Maple (Acer genus): Greater transpiration at the margins of young leaves; leaves grown in colder environments become more dissected, develop more numerous and larger marginal teeth. | [46,47] |

Elm (Ulmus genus): Leaf tissues with evaporative role develop prematurely at the margins of young leaves. | [43] | |

PLANT CANOPY • sun vs. shade leaf dimorphism | Oak (Quercus genus): Sun leaves have deeper lobes and greater transpiring capacity; transpiring sun leaves reach colder temperatures than shade leaves; sun leaf models in low wind convect heat better, more independently of orientation thanks to sinuses. | [40,48,49,50] |

Combined variationtemporal and spatial:PLANT LIFETIME • heteroblasty • young vs. mature plants | Snowflake Aralia (Trevesia palmata): Sun leaves are more dissected; young plants have palmately lobed leaves, mature plants have pseudo-compound leaves. | [51] |

Mulberry (Morus genus): Lobes develop preferentially on leaves’ outer side; many-lobed leaves retain lower temperatures; young plants have more lobed leaves. | [44,52] |

**Table 2.**Sun vs. shade leaf shape parameters across multiple oak species (two-sample t test)

^{1}(abbreviations: SD = standard deviation, Dif. = difference, Std Err = standard error, df = degrees of freedom, CL = confidence level).

Shape Parameter | Leaf Type | Mean | SD | Dif. | t ratio | Std Err Dif. | df | p | 95% CL Dif. | |
---|---|---|---|---|---|---|---|---|---|---|

Lower | Upper | |||||||||

Perimeter (cm) | SUN | 59 | 24 | 14 | 4.72 | 3.06 | 182.81 | <0.0001 | 8 | 20 |

SHADE | 45 | 19 | ||||||||

LDI (normalized perimeter) | SUN | 89 | 47 | 26 | 4.30 | 6.02 | 184.07 | <0.0001 | 14 | 38 |

SHADE | 63 | 38 | ||||||||

Convexity | SUN | 0.55 | 0.15 | −0.08 | −3.75 | 0.02 | 203.46 | 0.0002 | −0.13 | −0.04 |

SHADE | 0.64 | 0.17 | ||||||||

Roundness | SUN | 0.60 | 0.08 | −0.04 | −3.28 | 0.01 | 186.64 | 0.001 | −0.07 | −0.02 |

SHADE | 0.64 | 0.10 |

^{1}N

_{shade}= 110, N

_{sun}= 96; p-values for μ

_{shade}≠ μ

_{sun}(assuming unequal variances).

**Table 3.**Sun vs. shade leaf circularity and solidity within three oak species (two-sample t tests)

^{1}(abbreviations: SD = standard deviation, Dif. = difference, Std Err = standard error, df = degrees of freedom, CL = confidence level).

Oak Species (Sample Size) | Parameter | Leaf Type | Mean | SD | Dif. | t Ratio | Std Err Dif. | df | p | 95% CL Dif. | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Lower | Upper | |||||||||||

Q. alba^{2} | ||||||||||||

N_{sun} = 21 | LDI | SUN | 81 | 33 | 38 | 4.89 | 7.71 | 27.14 | <0.0001 | 22 | 54 | |

SHADE | 43 | 15 | ||||||||||

N_{sha} = 26 | Solidity | SUN | 0.68 | 0.12 | −0.13 | −4.53 | 0.03 | 28.45 | <0.0001 | −0.19 | −0.07 | |

SHADE | 0.82 | 0.06 | ||||||||||

Q. macrocarpa^{2} | ||||||||||||

N_{sun} = 11 | LDI | SUN | 127 | 43 | 55 | 3.61 | 15.19 | 16.80 | 0.0022 | 23 | 87 | |

SHADE | 72 | 28 | ||||||||||

N_{sha} = 13 | Solidity | SUN | 0.75 | 0.04 | −0.06 | −4.04 | 0.01 | 16.02 | 0.0009 | −0.09 | −0.02 | |

SHADE | 0.80 | 0.02 | ||||||||||

Q. ellipsoidalis^{2} | ||||||||||||

N_{sun} = 7 | LDI | SUN | 121 | 33 | 29 | 2.16 | 13.65 | 8.59 | 0.0609 | −2 | 61 | |

SHADE | 91 | 23 | ||||||||||

N_{sha} = 16 | Solidity | SUN | 0.46 | 0.06 | −0.12 | −4.17 | 0.03 | 11.79 | 0.0014 | −0.19 | −0.06 | |

SHADE | 0.58 | 0.07 |

^{1}p-values for μ

_{shade}≠ μ

_{sun}(assuming unequal variances);

^{2}leaf contours not to scale.

BIOLOGY Plant Role Model | ABSTRACTION Shape Features, Transfer Hypotheses | APPLICATION Transfer Regime |
---|---|---|

Viola septemloba Geranium sanguineum | • Elliptic lobation • Lobe elongationElliptic lobes in two-dimensional exchangers enhance convection. Elliptic lobes of higher aspect ratio (elongated) enhance convection. | SENSIBLE HEAT convective cooling |

Viburnum genus | • Leaf blade elongationShapes of high aspect ratio enhance convection. | |

Acer rubrum | • Toothed edges • Teeth proportions and shape Toothed edges, especially with proportionally large teeth, enhance vapor dissipation. | LATENT HEAT evaporation |

Ulmus genus | • Hierarchically toothed edges Hierarchical teeth enhance vapor dissipation. | |

Quercus genus | • Sinus profile in lobed shapesSinuses of lobed shapes enhance orientation-independence of transfer in free convection, and inclination-independence under strong airflows. | HEAT and MASS TRANSFER evaporative cooling |

Trevesia palmata | • Compoundness • FenestrationA large surface dissected into semi-distinct, space-filling parts enhances transfer. | |

Morus genus | • Circular lobation • Convex teethA hierarchical design of major obovate lobes and marginal curved teeth enhances transfer. |

Geometry (i.d. and Visual) | Abstract Design Features | Relative ^{1} Perimeter | LDI | Relative ^{1} Max. Inscribed Circle Diameter | Relative ^{1} L_{eff} ^{2} (Effective Dimension) | Protrusion to Core Ratio | Convexity | Solidity | |
---|---|---|---|---|---|---|---|---|---|

C | circular, control | 1 | 13 | 1 | 1 | 0 | 1 | 1 | |

E | ellipse, aspect ratio and elongation | 1.1 | 15 | 0.70 | 0.70 ^{2} | 0 | 1 | 1 | |

L | “lobes”, few and large protrusions | 1.5 | 28 | 0.78 | 0.79 | 0.4 | 0.87 | 0.67 | |

T | marginal teeth, many small protrusions | 1.8 | 41 | 0.9 | 0.82 | 0.2 | 0.61 | 0.83 | |

^{1}relative parameters are in relation to the control shape C;

^{2}L

_{eff}computed for airflow perpendicular to the ellipse.

**Table 6.**Mean evaporation flux of leaf-inspired shape models (one-way ANOVA) (abbreviations: SD = standard deviation, Std Err = standard error, df = degrees of freedom).

Characteristic Dimension | Shape | Evap. Flux Mean ^{1} (mg/min/cm^{2}) | SD | Std Err | Relative^{2} Enhancement | df | F Ratio | p | |
---|---|---|---|---|---|---|---|---|---|

D = 5 cm | C | 0.67 | 0.03 | 0.01 | - | 3 | 19.08 | <0.0001 | |

E | 0.69 | 0.02 | 0.01 | +4% | |||||

L | 0.76 | 0.02 | 0.01 | +14% | |||||

T | 0.76 | 0.01 | 0.01 | +14% | |||||

D = 10 cm | C | 0.59 | 0.01 | 0.005 | - | 3 | 28.89 | <0.0001 | |

E | 0.59 | 0.01 | 0.005 | −1% | |||||

L | 0.64 | 0.01 | 0.005 | +7% | |||||

T | 0.63 | 0.01 | 0.005 | +6% |

^{1}N = 4 for each mean;

^{2}enhancement in relation to control model C.

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rupp, A.I.K.S.; Gruber, P. Biomimetic Groundwork for Thermal Exchange Structures Inspired by Plant Leaf Design. *Biomimetics* **2019**, *4*, 75.
https://doi.org/10.3390/biomimetics4040075

**AMA Style**

Rupp AIKS, Gruber P. Biomimetic Groundwork for Thermal Exchange Structures Inspired by Plant Leaf Design. *Biomimetics*. 2019; 4(4):75.
https://doi.org/10.3390/biomimetics4040075

**Chicago/Turabian Style**

Rupp, Ariana I. K. S., and Petra Gruber. 2019. "Biomimetic Groundwork for Thermal Exchange Structures Inspired by Plant Leaf Design" *Biomimetics* 4, no. 4: 75.
https://doi.org/10.3390/biomimetics4040075