Mathematical Modeling of Diffusion of a Hydrophilic Ionic Fertilizer in Plant Cuticles: Surfactant and Hygroscopic Effects

Abstract

The agricultural industry requires improved efficacy of sprays being applied to crops and weeds to reduce their environmental impact and increase financial returns. One way to improve efficacy is by enhancing foliar penetration. The plant leaf cuticle is the most significant barrier to agrochemical diffusion within the leaf. The importance of a mechanistic mathematical model has been noted previously in the literature, as each penetration experiment is dictated by its specific parameters, namely plant species, environmental conditions such as relative humidity and spray formulation including adjuvant addition. A mechanistic mathematical model has been previously developed by the authors, focusing on plant cuticle diffusion of calcium chloride through tomato fruit cuticles including pore swelling, ion binding and evaporation, along with the ability to vary the active ingredient concentration and type, relative humidity and plant species. Here we further develop this model to include adjuvant effects as well as the hygroscopic nature of deliquescent ionic solutions with evaporation on the cuticle surface. These modifications to a penetration and evaporation model provide a novel addition to the literature and allow the model to be applied to many types of evaporating ionic hygroscopic solutions on many types of substrates, not just plant cuticles. We validate our theoretical model results against appropriate experimental data, discuss key sensitivities and relate theoretical predictions to physical mechanisms. The important governing mechanisms influencing surfactant enhanced penetration of ionic active through plant cuticles were found to be aqueous pore radius, pore density, cuticle thickness and initial contact angle of the applied droplet; ion binding, relative humidity and evaporation including hygroscopic water absorption parameters for point of deliquescence. The sensitivity analysis indicated surfactants increase penetration by changing the point of deliquescence of a solution, which alters the water absorption and the initial contact angle, which alters the number of pores under the droplet. The results of the validation and sensitivity analysis imply that this model accounts for many of the mechanisms governing penetration in plant cuticles.

Mathematical Modelling of Hydrophilic Ionic Fertiliser Diffusion in Plant Cuticles: Lipophilic Surfactant Effects

“…The model takes the form of a quasi-one-dimensional, diffusion model. We will briefly describe the modelling formulation, that is based on the authors previous works [12,13] and we refer the reader to Tredenick et al [13] for a full description of the auxiliary equations governing evaporation in Appendix A. We account for three components, hydrophilic ionic active ingredient (AI), water (H 2 O), and lipophilic adjuvant (ADJ).…”

“…Water can diffuse through both the aqueous pores and the lipophilic pathway [4,34] within the cuticle. The authors previous model in Tredenick et al [13] is simplified here by assuming the porosity does not significantly change in time, to focus and expand on the adsorption mechanism. Pore swelling may be influential under other environmental, initial and boundary conditions, as discussed Tredenick et al [12] and Tredenick et al [13], and this will be the subject of future works.…”

“…The authors previous model in Tredenick et al [13] is simplified here by assuming the porosity does not significantly change in time, to focus and expand on the adsorption mechanism. Pore swelling may be influential under other environmental, initial and boundary conditions, as discussed Tredenick et al [12] and Tredenick et al [13], and this will be the subject of future works. All three components change primarily along the cuticle membrane thickness, x (0≤x≤b).…”

“…The focus of this paper was to further analyse the mechanisms involved in lipophilic surfactant addition to a hydrophilic ionic AI spray solution that penetrates through isolated astomatous plant cuticles. In Tredenick et al [13], when considering the experimental data [22] for penetration of ionic CaCl 2 formulated with the surfactant rapeseed oil (RSO 5), the model was able to predict the overall trends in the experimental data. However, the model was unable to replicate the data trends, where penetration increased at late times, between 24 and 48 h.…”

“…Mechanistic models for adsorption and desorption, with a moving droplet radius due to evaporation, are limited in the well-established literature. We previously presented a model in Tredenick et al [13], where ions bind to the leaf surface under the moving droplet radius, but do not desorb. Several adsorption–desorption models are available in the well-established literature, such as the competitive Langmuir model [49,50] and an adsorption–desorption model for chemicals in soil [51].…”