The agricultural industry, world-wide, requires improved efficacy of sprays applied to crops and weeds (Shaner and Beckie, 2014). Spray application of agrochemicals is known to be effective yet often inefficient (Knoche, 1994). There are many benefits from enhancing the efficacy of agrochemicals. Discovering attributes that increase the uptake of systemic agrochemicals can enhance efficacy and reduce the amount of active ingredient (AI) required (Balneaves et al., 1993), lower the spray water volume required, lead to more cost effective chemicals (Gaskin et al., 2013) and help minimize residues, maximize crop yield, crop quality and returns to growers (Schönherr, 2006; McKenna et al., 2013). The plant cuticle is considered the rate-limiting barrier in foliar uptake of agrochemicals (Schönherr and Riederer, 1989). Extensive research has been performed to investigate the factors involved in the mass transport of chemical compounds across the plant cuticle. However, many of the mechanisms influencing uptake are still unknown (Shaner and Beckie, 2014); the most perplexing being the differences in permeability among various plant species (Schreiber et al., 2006; Kerstiens, 2010; Forster and Kimberley, 2015). A reliable mathematical model to simulate uptake of agrochemicals would be of enormous benefit compared to performing uptake experiments, as AI uptake is specific to each AI, formulation and plant species combination, as well as environmental factors. Creating mathematical models to describe uptake should improve our understanding of the mechanisms governing the uptake of agrochemicals in plant cuticles. Zabkiewicz (2007) has noted agrochemical efficacy progress will not be made until appropriate models are created to simulate the multiple complex processes involved, creating a comprehensive agrochemical efficacy system.

Agrochemicals such as pesticides and fertilizers, along with other xenobiotics, can be either hydrophilic (ionic or uncharged) or lipophilic compounds. These two categories of AIs have opposite hydrophilicity. Lipophilic and hydrophilic compounds are governed by very different mass transport processes, which has given rise to the theory that the two types of compounds traverse the cuticle via two distinct routes (Schönherr, 2006). Hydrophilic compounds cross the cuticle via a diffusion process through aqueous pores (Baur, 1999; Schönherr, 2006). Two forms of hydrophilic compounds exist; ionic and uncharged. Ionic compounds are charged molecules that can diffuse across cuticles as they carry hydration shells (Schreiber, 2005). When ionic compounds enter the aqueous pores, they split into positive and negative ions. These ions must penetrate in equal numbers to maintain electroneutrality (Schönherr, 2006). Many plant hormones, growth regulators, plant nutrients [e.g., calcium chloride (CaCl₂)] and pesticides, such as bentazon and glyphosate, are ionic (Schönherr and Schreiber, 2004).

Water, which is a small, uncharged but polar molecule, can use both the lipophilic and aqueous pathway within the cuticle (Schreiber et al., 2006). Lipophilic compounds penetrate the cuticle via a dissolution-diffusion process along the lipophilic pathway. Here we will focus on the uptake of hydrophilic ionic compounds (ionic AIs) across aqueous pores as this has been said to have major practical importance in the agricultural industry and significantly less is known about the permeability of the cuticle to ionic compounds than lipophilic compounds (Schreiber, 2005).

Aqueous or polar pores form preferential sites in plant cuticles for ionic compound uptake. The aqueous pathway is made up of pores of molecular dimensions filled with water (Riederer and Schreiber, 2001). Aqueous pores are nanostructures that temporarily form only when water is present (Schönherr, 2006). Cutin is a major constituent of the polymer matrix within the cuticle, which contains polar polymers. These polar polymers sorb water and swell, giving rise to aqueous pores that traverse the cuticle (Kerstiens, 2006). Estimates of average aqueous pore radii from indirect measurements are 0.3 nm in Hedera helix L. (Popp et al., 2005), 0.45 nm in citrus (Schönherr and Schmidt, 1979), 2.0 nm in C. arabica (Eichert and Goldbach, 2008), and 2.12 nm in tomato fruit cuticle membranes (CMs) (Beyer et al., 2005; Schreiber and Schönherr, 2009, p. 87).

Much work has been done to characterize the important mechanisms involved in uptake of ionic agrochemicals. Many factors, both external and internal to the cuticle, affect uptake. Yamada et al. (1964) found the adaxial and abaxial surfaces of enzymatically isolated cuticles of several plant species to be highly directionally dependent in regards to permeability to various anions and cations. When calcium ion (Ca²⁺) penetration was measured through tomato fruit cuticles, the outside to inside direction produced about 3.5 times more penetration than the inside to outside direction after 40 h. They conclude this effect is due to differences in kind, size and polarity of ions and the binding capacity of the adaxial and abaxial surfaces of cuticles. Clearly ion binding effects on cuticle surfaces have a significant impact on uptake.

Water can travel through the cuticle as free molecules in aqueous pores lined with dipoles and/or fixed ionic charges and attach to these dipoles and/or fixed charges as a monolayer (Luque et al., 1995; Kerstiens, 2006). The water content of the cuticle increases with increasing relative humidity, causing the cuticle to swell (Schönherr, 2006). The timescale for this swelling is unknown. Permeability of cuticles to ionic compounds is highly affected by relative humidity; high penetration generally occurs at high relative humidity. Relative humidity impacts the rate of surface spray droplet evaporation, which in turn affects the uptake (Ramsey et al., 2005). High relative humidity increases the number and radius of aqueous pores, which in turn facilitates the transport of ionic compounds (Middleton and Sanderson, 1965; Schönherr and Schmidt, 1979; Schönherr, 2000, 2002). Schönherr (2002) found ionic glyphosate salts with surfactants penetrated 5–10 times faster when relative humidity was increased from 70 to 100%. The same work also shows a log-linear relationship when the log of the cumulative uptake is plotted against time. This relationship represents the process whereby the aqueous pores initially take up progressively more AI until such time that the concentration of AI in the pores approaches a maximum cumulative uptake, causing the rate of uptake to slow. However, deviations from a log-linear uptake relationship were observed by Schönherr (2000) on studies using CaCl₂. No general conclusions were made.

In summary, it has been shown experimentally that significant factors affecting the diffusion of ionic compounds include relative humidity, adjuvant, plant species, concentration of AI and ion binding capacities of cuticle surfaces (Yamada et al., 1964; Schönherr, 2000; Buchholz, 2006).

Several mathematical models on uptake through plant tissues are present in the literature. Comprehensive model reviews have been published in Forster et al. (2004) and Trapp (2004). Some models incorporate diffusion (Hsu, 1983; Satchivi et al., 2000b; Veraverbeke et al., 2003; Mercer, 2007; Pecha et al., 2012), while others employ empirical expressions (Forster et al., 2006; Schreiber and Schönherr, 2009, p. 132). Empirical models are limited to their specific measurement conditions. The formulation presented in Schreiber and Schönherr (2009) requires a single rate constant to be measured specific to each different plant species, hydrophilic AI, adjuvant and environmental conditions. Mercer (2007) has developed a diffusion model to simulate hydrophilic AI uptake through plant leaves with surfactants. They utilized a linear, cylindrical diffusion model where phloem translocation (tissue that carries nutrients to all other parts of the plant) is incorporated with a source term and no-flux boundary conditions were applied. Mercer focused on the size of the droplets on the surface and spread area effects with surfactant. No validation with experimental results was presented. Pecha et al. (2012) have developed a model for the uptake of ionic biostimulant through whole leaves by application through immersion as opposed to spraying. Their 1-D model consists of an evaporation formulation for the surface solution coupled to linear diffusion within the leaf proper and a no-flux, inner, symmetric boundary condition. The model does not incorporate porosity within the plant leaf aqueous pores. They apply their model to investigate the effect that the evaporation rate of the surface solution has on uptake. They have utilized a thin film model for evaporation, but we would be interested in droplet evaporation, which is governed by different mechanisms (Tang et al., 1997). No validation of their model predictions with experimental data is included. Satchivi et al. have produced a model in several parts (Satchivi et al., 2000a,b, 2001, 2006). They formulate a dynamic, compartment-type, nonlinear model for whole plant uptake from foliage-applied agrochemicals. The whole plant model accounts for uptake within the cuticle, leaf, stem and root in 19 compartments. Forster et al. (2004) noted that the number of experimental inputs was so vast that obtaining realistic values may not be possible.

In this work we develop a mathematical model and numerical simulation for the uptake of ionic agrochemicals through aqueous pores in plant cuticles. The model accounts for important biological and chemical mechanisms involved in uptake through aqueous pores, not previously incorporated in the modeling of agrochemical uptake. Specifically, we account for the formation and swelling of aqueous pores as a result of water uptake in the cuticle, ion binding effects and the evaporation of the spray droplet on the cuticle surface. We seek to validate the predictions of our model against available experimental results from the literature. We then discuss the key sensitivities within the model and relate this behavior to appropriate physical conditions.

To formulate a mathematical model for uptake we need to consider the experimental setup used. The solution to the model will be verified against well-established data collected from such an experimental setup. To this end, we consider the experimental setup of Kraemer et al. (2009), who study the uptake of CaCl₂ from droplets, containing no surfactant, applied to the surface of astomatous, isolated tomato fruit (Solanum lycopersicum L., cultivar “Panovy”) cuticles. Experimentally, droplets of a solution of known concentration and volume were applied to the adaxial surface of the isolated CM. In close contact with the abaxial surface of the CM was a water bath, acting as a receiver for the penetrated AI, which was analyzed for CaCl₂ at regular time intervals after application. Kraemer et al. (2009) found that as both the initial applied concentration and time increases, then so does the penetrated amount of calcium.

The model takes the form of a nonlinear porous two-component diffusion model. A schematic diagram of the model domain considered here, in relation to the Kraemer et al. (2009) experimental setup, is shown in Figure 1, where both water and AI are diffusing. Figure 1 shows the initial conditions on the left and a short time later on the right. We consider a quasi-one-dimensional spatial domain, where all variables change primarily along the cuticular membrane thickness, x (0 ≤ x ≤ b). A water droplet, having initial contact angle, θ₀, radius, r_drop, and containing a known concentration of AI (CaCl₂) sits on the adaxial surface of the cuticle, at x = 0. A well stirred water bath exists at the abaxial surface, at x = b. Water adsorbs to the surface of the aqueous pores and can be seen in Figure 1 as dark blue circles. For simplicity, in Figure 1, a single pore is depicted crossing the cuticle. The change of AI concentration within the droplet, at x = 0, and the flux of AI into the water bath, at x = b, take into account that the droplet will cover many pores. However, we are assuming that all pores can be modeled by considering what happens in a single pore and then scaling by the appropriate pore density. Further discussion of this is given around Figure 2.

Fickian diffusion is the transport mechanism ionic AI uses to traverse the plant CM aqueous pores (Schreiber and Schönherr, 2009). Diffusion of ionic AI is known to depend on the swelling of the aqueous pores (Kerstiens, 2006). Aqueous pores change in size based on adsorbed water, therefore porosity of the cuticle must be modeled. The porosity of the aqueous pores acts as a limiter to diffusion, both on the surface and through the CM.

Water facilitates the diffusion of AI. Water enters the CM and opens up the aqueous pores by attaching to the pore walls, forming an adsorption monolayer. As the pores open, water can also diffuse through the pores and act as a solvent for the mass transport for AI. Water molecules that are adsorbed onto aqueous pore walls are not available for diffusion.

The model makes the following assumptions: relative humidity and room temperature (T) are constant with values equal to those given experimentally by Kraemer et al. (2009); only water evaporates from the drop; droplet evaporation does not involve convection in liquid or gas phases; the water bath is well stirred. The AI solution is assumed to be homogeneously distributed over the surface of the CM. It is necessary to model the porosity of the cuticle as the change in porosity could be significant, therefore the aqueous pore radius needs to be modeled at every point in space and time. Pores may be very tortuous, traversing the thin cuticle laterally in lamellate like structures. The model accounts for this via the introduction of diffusivity functions that depend on porosity and tortuosity.

The variables and parameters with their associated units and references (where possible) are described in Table 1. The model, including the governing partial differential equations (PDEs), initial conditions (ICs), boundary conditions (BCs), and auxiliary functions is as follows:

PDEs:

Functions:

ICs:

BCs - AI (drop):

Functions:

The dimensionless, nonlinear plant cuticle diffusion model as described in Equations (A-2–A-19) is solved numerically. Figure 3 shows a comparison of the numerical solution of the model compared to the experimental data from Kraemer et al. (2009). All parameters used here are shown in Table 1 and are the same for all five plots, with the exception that five different applied initial concentrations of AI are used, namely cAI,0drop = 1 (A), 5 (B), 10 (C), 15 (D), 30 g/L (E) (corresponding to a mass of 5, 25, 50, 75, 150 μg). The formulation of the mass Ca penetrated is described previously in Equation (27). The mass Ca penetrated is converted to percent Ca uptake at 48 h as shown in Figure 3, as described in Section 3.1.

Overall the numerical solutions and the experimental data in Figure 3 compare well, considering the complex mechanisms involved. We do note, however, that the most significant discrepancy between our model and the Kraemer et al. (2009) data occurs in Figure 3B. When we calculate the total uptake in each of the (Figures 3A,C–E) as a percentage of the initial applied mass of Ca in the droplets, we find that the mean percentage uptake is 42% with a coefficient of variation of 14%. However, in the experimental data of Figure 3B, this uptake is only 18.6%. This may be an outlier. No explanation is provided by Kraemer et al. (2009) in regards to this.

We can consider the error associated with Figure 3 when the numerical solution is compared to experimental data. We include the error bars and exclude the data in Figure 3B and find an an R² value of 83.2%. This value is reasonable, given the complex mechanisms involved in the model. We do note that the original error bars in Kraemer et al. (2009) for Figure 3A were not discernible, so have been excluded.

In Figure 3, as cAI,0drop increases, the penetrated amount of Ca increases. From the figure, we can see that over the first 2–10 h, uptake occurs rapidly, then levels out and approaches a maximum value. This rapid increase in uptake initially matches the trend in the experimental data. The difference between the concentration of AI in the drop and the cuticle rapidly increases due to droplet evaporation, which in turn causes rapid initial uptake. Uptake levels out after approximately 10 h. Uptake levels out due to the concentration of AI in the drop reaching zero. The maximum AI uptake value is governed by ion binding. Ions are bound to the cuticle surface and therefore lost to diffusion. A high percentage of ions are lost to ion binding. The mean percentage uptake of the Kraemer et al. (2009) data at 48 h is 37%. This may then indicate that 63% of ions are lost to ion binding. The numerical uptake at 48 h agrees well with the Kraemer et al. (2009) data at 48 h.

Figures 4A,B show the results of AI diffusion, and water diffusion with adsorption in the cuticle aqueous pores over 48 h. The output in the figures is from a single applied AI concentration of 10 g/L. In both figures, the drop is located at x = 0 m and the water bath is located at x = 1.87e−5 m.

In Figure 4A, we see that the concentration of AI at x = 0 m increases to approximately 5,000 mol/m³ at 1 h as the surface solution becomes more concentrated due to evaporation. Once the concentration of AI reaches its solubility concentration (discussed in Section 2.4), it ceases to increase and then decreases as the concentration of AI is transported via diffusion into the cuticle. At the final time of 48 h, the concentration of AI at x = 0 m has reached zero as all the AI available for diffusion has been either transported into the cuticle via diffusion or lost to ion binding on the cuticle exterior. Within the cuticle, AI is transported from regions of high concentration to low concentration via diffusion, with the most rapid change in AI concentration occurring before 4 h. This matches the timescale evident in the data from Kraemer et al. (2009). At late times, the concentration of AI has reached equilibrium and is zero everywhere in the cuticle pore.

Figure 4B shows the diffusion and adsorption of water in the aqueous pores over time. Initially the concentration of water is a constant everywhere, except at x = 0 m, where it is slightly lower due to the presence of AI. At x = 0 m, the concentration of water decreases due to evaporation. The water concentration in the cuticle decreases up to 1 h, then increases after 1 h as water diffuses back toward the drop from the water bath. At late times, the concentration of water returns to the concentration of pure water throughout the whole cuticle.

Swelling of the aqueous pores (not shown) occurs whilst uptake takes place. In the context of the Kraemer et al. (2009) experimental setup, where cuticles are rehydrated initially and a water bath is located on the lower cuticle surface, the pore swelling is not significant. We leave pore swelling in the model with the view to investigating this in future works.

The values for F_S, k, and ρ₀ are described in Table 1. Here we will investigate these parameters. A low value of F_S is obtained from the fitting exercise. A low F_S would describe a pore at the low tortuosity end of the range. If a higher value was used, the uptake would be more gradual, as the pores would then be more tortuous. If a certain plant species cuticle is known to have slow uptake, a low diffusivity, very tortuous pores or a high lamellate structure, a higher F_S can be chosen appropriately. Fitting F_s is reasonable as this facilitates the calculation of the diffusion path length. The diffusion path length through plant cuticles cannot currently be established by a physical measurement (Riederer and Schreiber, 1995). Yuan and Sundén (2014) have provided values of 1.1 ≤ F_s ≤ 1.3 in porous structures to use as reference points. Therefore, our fitted value of F_s = 1.1 is reasonable. The value for ρ₀ in Table 1 closely agrees with values for ρ₀ found elsewhere ranging from 5.1e13 m⁻² in Populus x canescens (Aiton) Sm. leaves (Remus-Emsermann et al., 2011) to 2e15 m⁻² in Citrus aurantium cuticles (Schreiber and Schönherr, 2009, p. 85). Therefore, our fitted value for ρ₀ would seem to be reasonable.

This model can be theoretically applied to the uptake of most ionic hydrophilic AI without adjuvants. It cannot apply to uncharged hydrophilic compounds, ionic compounds which would dehydrate pores such as Fe chelates (Schönherr et al., 2005; Schlegel et al., 2006) or lipophilic compounds. The model can also theoretically be applied to any isolated astomatous plant leaf or fruit species cuticle, where the aqueous pores are sufficiently large to allow AI to be transported by Fickian diffusion. If ionic AI penetrates through a certain plant species at a very slow rate, transport is theorized to be a mechanism alternate to Fickian diffusion and this situation would require further investigation before the model could be applied.

Overall the validation results of our plant cuticle model with the Kraemer et al. (2009) data in Figure 3 shows good agreement with the experimental data, confirming that our model can be appropriately applied to uptake experiments, simulating the important governing mechanisms. This model provides a good basis for future work.

A nonlinear, porous diffusion model has been developed here to simulate diffusion of hydrophilic ionic AI and diffusion with adsorption of water through a plant cuticle. This model makes novel additions to a simple diffusion model by incorporating the swelling of aqueous pores with water, climatic conditions such as relative humidity that affect the evaporation of water in the applied droplet, parameters that account for differences between plant species, porosity and tortuosity of the aqueous pores, cuticle surface ion binding and a diffusivity function that changes through the cuticle over time. The nonlinear model has been solved numerically, producing results that show good agreement with experimental data. Major factors influencing our model's uptake of ionic AI through plant cuticles have been found to be cuticle structure, including tortuosity and density of the aqueous pores, and to a lesser extent (while still high), humidity and cuticle surface ion binding effects through the sensitivity analysis.

ET is responsible for article writing, model creation and adaptation, computational code creation, results, analysis, article editing and revision. TF is responsible for model creation and adaptation, results, analysis, article editing and revision. WF is responsible for motivating the project, the biological and agrochemical consultation, article editing and revision. SP is responsible for computational code editing, article editing and revision. All authors agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.