Modelling diffusion: around particles and into plants
Keywords: leaf
cuticle
pesticide
PDE
electrochemistry
modelling
diffusion
simulation
particle
mass transport
plant

This thesis investigates diffusional mass transport about particles on a planar surface through numerical simulation. The results of these simulations are applied to physically significant systems involving particles at electrode and bio-membrane interfaces, with the bio-membrane referring specifically to plant cuticles and foliar uptake across them.
Chapter 1 introduces diffusion as a mass transport process and justifies this thesis’ focus upon it. Chapter 1 also introduces the electrochemical concepts that constitute necessary background for the work presented in Chapters 4-6. Chapter 2 introduces plant biology underpinning the models and simulations of agrochemical foliar uptake that are the subject of discussion in Chapters 5 and 6, including leaf structure and function, the epidermal plant cuticle, and practicalities of foliar uptake for plant chemical treatment.
Chapter 3 introduces the numerical methods used in this thesis. The finite difference solution framework is presented, including its explicit and implicit formulations. The problem is linearised to a matrix multiplication. Three solution methods are presented: the Thomas algorithm, the alternating direction implicit method, and fully implicit methods. Dimension reduction, grid construction, and the simulation validation is discussed.
Chapter 4 represents this thesis’ first “Results & Discussion” chapter. Simulations results are presented for diffusion at an electroactive particle on a supporting inert planar surface under chronoamperometry. Such simulations are pertinent to particle-modified electrodes and nano-impact electrochemistry. This chapter also serves as an introduction to the sphere-on-a-surface geometry for later chapters.
Chapter 4 introduces the diffusion indicator: a tool for quantifying the linearity/convergence of the diffusion regime behaviour. Total and local flux profiles at the particle surface are presented and discussed for a range of contact angles. Local diffusional stagnation is a consequence of the geometry and the diffusion indicator is used to characterise it. The simulation results are well-approximated by a mapping we propose to an isolated sphere. We introduce a novel, local variant of the diffusion indicator and discuss its utility. The gradient of the local diffusion indicator is shown to provide additional information about mass transport perpendicular to the electrode surface and general diffusional convergence.
Chapter 5 focusses on foliar uptake across the plant cuticle from a particulate under kinetic chemical release. Significant attention is given to establishing the model for the simulation of the uptake of lipophilic material from particulate. The uptake’s dependence on the particle’s contact angle, release kinetics, and the relative aqueous and cuticular solubilities and diffusion coefficients are demonstrated, including 4 limiting regimes. A direct and an indirect route are demonstrated for uptake: either via particle-cuticle contact or via dissolution into a surrounding medium. A factor for approximating the relative contributions of these two pathways is presented. A leaching effect, hitherto unrecognised for agrochemicals, is demonstrated. The model is finally compared against other models in the literature.
In Chapter 6, experimental results are presented for the study of foliar uptake into living maize plants of a propriety fungicide. We present data for the transfer of fungicidal material between leaf compartments over 24 hours and images from reflected light and cryogenic scanning-electron microscopy of the leaf surface with analysis. Alongside insights from Chapter 5, we establish a simplest viable model for this system. Dimensionless simulation results are presented and compared with experimental results: validating the model and providing an inference for the cuticular solubility and diffusion coefficient.