# Optimal Vein Density in Artificial and Real Leaves

Original Entry by Ryan Truby

AP 225 - Introduction to Soft Matter

September 16, 2012

## Contents

## Reference Information

**Authors:** X. Noblin, L. Mahadevan, I. A. Coomaraswamy, D. A. Weitz, N. M. Holbrook, M. A. Zwienieki

**Citation:** X. Noblin, et al. Optimal Vein Density in Artificial and Real Leaves. *PNAS*. **2008**, 105, 9140-9144.

**Keywords (from Ref.):** water transport, microfluidics, biomimetics, leaf hydraulic properties, evaporative pump

**Related Course Keywords: ** diffusion, osmotic pressure

## Background and Introduction

Stoma are pore-like structures found in the leaves of plants that are vital to two important mechanisms in plant physiology: photosynthesis and transpiration. Photosynthesis is the reaction by which plants use water and the energy of incident photons emitted from the Sun to convert carbon dioxide to glucose, a sugar the plant then uses as a source of chemical energy. When stoma are open, carbon dioxide diffuses into the leaves of plants and on into organelles called chloroplasts where photosynthesis takes place, while the oxygen produced from photosynthesis is released from the plant as it diffuses through these open pores. Consequently, precious water can diffuse from the plant too as water vapor while these stoma are open. This loss of water from the plant is called transpiration, and plants have evolved to use transpiration to their benefit. By careful control of these stoma, plants use transpiration as a means for regulating the osmotic pressures of their cells and their internal temperatures, as well for driving water upward against gravity by capillary action from the soil to their leaves.

The authors of this article employed multi-channeled microfluidic devices (shown in Fig. 1) as artificial leaves in order to model the transpiration-mediated evaporation of water from real leaves and better understand leaves' role in water transport in plants. Through various experimental trials and the development of a numerical model derived from both Fick's law and Darcy's law, the authors were able to elucidate a scaling relation correlating the artificial leaves' ideal channel density with their thicknesses. Interestingly, this scaling relation agreed with an analysis of the distances separating higher-order veins from both each other and the leaf's surface in leaves from 32 different plant species, revealing that leaves vascularize themselves with a network characterized by an ideal vein density.

## Summary

The artificial leaves designed by the authors were microfluidic devices fabricated from poly(dimethylsiloxane) (PDMS) and fixed to glass substrates (see Fig. 1). The devices possessed *n* parallel channels (of length = *L* = 35 mm, width = *ω*, height = *h*) separated by a distance *d* (that varied between 30µm and 1500µm); the channels were fixed between the glass substrate and a layer of PDMS with thickness *δ* (that varied between 45µm and 750µm). The width of the device was W = 30 mm. Water was pumped into the microfluidic devices, while air with a controlled relative percent humidity (RH, ranged from 20% to 100%) was flowed over them at 1 L*min^-1. Plexiglass was kept 1 mm above the devices to ensure a constant RH above the device. The microfluidic device was then connected to a beaker of water 30 cm below that rested on a computer-monitored mass balance. As water evaporated from the channels, water from the beaker would flow upward and into the device. Measurements of the volumetric flow rate of water vapor diffusing from the channels ranged from 0.1 nL*s^-1 to 10 mL*s^-1.

The authors studied the dependence of water vapor flow rate from the microfluidic devices on RH, the channel density in the device, and the thickness of the PDMS layer above the channels. Despite the channel density of the device, a decrease in water vapor flow rate was observed for increasing RHs (see Fig. 2). As shown in Fig. 3, increases in channel density resulted in increases in water vapor flow rates up to some critical density at which the flow rate would begin logarithmically increasing and eventually reach some maximum flow rate. For channel densities between 0 and 30 mm^-1, this maximum flow rate decreased as the PDMS layer above the channels increased in thickness (also illustrated by Fig. 4).

Following Kirchhoff's rule for flow, the total volumetric flow rate of water vapor from the microfluidic device equaled the sum of the flow rates of water vapor from all channels. To model the total volumetric flow rate, the author's used both Darcy's law and Fick's law: Darcy's law was needed to model the transport of water through the PDMS layer, and Fick's law described the diffusion of water vapor from the PDMS to the surrounding atmosphere. Darcy's law is given by

<math>J = \frac{-k}{\mu}\nabla P</math>,

where J is the flux, <math>\nabla P</math> is the pressure gradient [Pa/m], k is the permeability of the PDMS [m^2], and µ is the viscosity of the water diffusing through the PDMS [Pa*s]. Fick's law is the relationship

<math>J = -D\nabla c</math>,

where J is the flux of the water vapor from the channels, D is the diffusion coefficient of water vapor from the channels into air [m^2*s^-1], and c is the concentration of water vapor.

Using these transport equations, the authors were able to simply their model so that the total flux of water from the microfluidic devices could be expressed as

<math>J_T = \frac{2D(c_o-c_1)L}{\ln (\frac{(\delta + h)}{a})}N \arctan \frac{W}{2( \delta + h)N}</math>

When the distance between the microfluidic channels were much larger than the thickness of PDMS above them, the relation simplifies to

<math>J_T = \frac{\pi D(c_o-c_1)L}{\ln (\frac{(\delta + h)}{a})}N, d\gg\delta</math>

On the other hand, when the thickness of the PDMS layer is much greater than the separation distance,

<math>J_T = \frac{D(c_o-c_1)LW}{\ln (\frac{(\delta + h)}{a})(\delta + h)}N, d\ll\delta</math>

Fig. 3 demonstrates that there exists some ideal channel density, which the authors found by setting these last two equations equal to one another:

<math>d_c \simeq \delta</math>

Interestingly, this scaling relationship agreed with higher-order vein densities found in leaves from 32 different plant species, as seen in Fig. 5.

## Discussion and Relevance to Soft Matter

By employing multi-channeled microfluidic devices as "artificial leaves" and analyzing the diffusion of water vapor from them, the authors were able to derive and verify a mathematical model that reveals the optimal vein density for real leaves. The authors' model suggested that for maximum efficiency and optimal expenditure of metabolic resources and allocation of space within the leaf's body, higher-order veins will be separated from each other by distances equal to that between the veins and the leaf's surface. The model was verified by an analysis of vein placement and distribution from leaf samples collected from 32 different species.

## References

[1] X. Noblin, et al. Optimal Vein Density in Artificial and Real Leaves. *PNAS*. **2008**, 105, 9140-9144.