###### Abstract

The global agricultural industry requires improved efficacy of sprays being
applied to weeds and crops to increase financial returns and reduce
environmental impact. Enhancing foliar penetration is one way to improve
efficacy. Within the plant leaf, the cuticle is the most significant barrier
to agrochemical diffusion. It has been noted that a comprehensive set of
mechanisms for ionic active ingredient (AI) penetration through plant leaves
with surfactants is not well defined, and oils that enhance penetration have
been given little attention. The importance of a mechanistic mathematical
model has been noted previously in the literature. Two mechanistic
mathematical models have been previously developed by the authors, focusing on
plant cuticle penetration of calcium chloride through tomato fruit cuticles.
The models included ion binding and evaporation with hygroscopic water
absorption, along with the ability to vary the AI concentration and type,
relative humidity, and plant species. Here, we further develop these models to
include lipophilic adjuvant effects, as well as the adsorption and desorption,
of compounds on the cuticle surface with a novel Adaptive Competitive Langmuir
model. These modifications to a penetration model provide a novel addition to
the literature. We validate our theoretical model results against appropriate
experimental data, discuss key sensitivities, and relate theoretical
predictions to physical mechanisms. The results indicate the addition of the
desorption mechanism may be one way to predict increased penetration at late
times, and the sensitivity of model parameters compares well to those present
in the literature.

###### keywords:

plant cuticle; hydrophilic ionic active ingredient; porous diffusion;
adsorption; desorption; lipophilic; mathematical model; aqueous pores;
surfactant; competitive Langmuir; ion transport

labelinglabel xx 1 5 Received: date; Accepted: date; Published: date
Mathematical Modelling of Hydrophilic Ionic Fertiliser Diffusion in Plant
Cuticles: Lipophilic Surfactant Effects Eloise C. Tredenick 1,*, Troy W.
Farrell 1,2,* and W. Alison Forster 3 Eloise C. Tredenick, Troy W. Farrell and
W. Alison Forster Correspondence<EMAIL_ADDRESS>(E.C.T.);
<EMAIL_ADDRESS>(T.W.F.)

##  1 Introduction

The global agricultural industry requires improved efficacy of sprays applied
to crops and weeds (Shaner and Beckie, 2014). While spray application of
agrochemicals is known to be effective, it is often inefficient (Knoche,
1994). Enhancing the efficacy of agrochemicals has many benefits (Balneaves et
al., 1993; Schönherr, 2006; McKenna et al., 2013). For all foliar-applied
agrochemicals, inefficiencies arise during deposition (sprays are not reaching
the target) and retention (sprays are reaching the target but are not
retained). In the case of systemic pesticides and foliar fertilisers,
additional inefficiencies arise during penetration (not all active ingredient
(AI) penetrates the plant) and translocation (not all AI within the target
plant is transported to the site of biological activity).

The plant cuticle is considered the rate-limiting barrier to foliar
penetration of agrochemicals (Schönherr and Riederer, 1989). Aqueous pores are
dynamic nanopores within the cuticle that form only in the presence of water
(Schönherr, 2006; Riederer and Schreiber, 2001; Mays, 2007). Hydrophilic ionic
agrochemicals (ionic AIs) and water penetrate the plant cuticle through
aqueous pores via diffusion (Baur, 1999; Schönherr, 2006; Schreiber, 2006).
Ionic AI penetration has major practical importance to the agricultural
industry (Schreiber, 2005).

The mechanisms governing penetration of ionic AIs through plant cuticles has
been reviewed elsewhere (Tredenick et al., 2017, 2018; Fernández et al., 2017;
Schönherr, 2006; Schreiber, 2005). Significant factors include plant species
variations, ion binding to the cuticle surface, relative humidity, droplet
evaporation, and point of deliquescence with hygroscopic water absorption
Jeffree (2008); Schreiber et al. (2006); Santier and Chamel (1992); Yamada et
al. (1964); Tang et al. (1997); OxyChem (2014); Hunsche and Noga (2012).

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.
(2018), when considering the experimental data (Kraemer et al., 2009) for
penetration of ionic CaCl2 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.

Rapeseed oil is an ethoxylated natural oil that exhibits non-ionic surfactant
properties and is a versatile emulsifier (Aqn, 2016). The
hydrophilic–lipophilic balance (HLB) of RSO 5 is 5.2 (Kraemer et al., 2009),
indicating that it is lipophilic, or water-insoluble. A comprehensive set of
transport mechanisms for ionic AI penetration through plant leaves, in the
presence of surfactants, is not well defined (Hess and Foy, 2000; Wang and
Liu, 2007; Stock and Holloway, 1993). The mode of action of oils enhancing
ionic AI penetration through plant leaves has been given little attention
(Wang and Liu, 2007). The concentration of penetrated adjuvant is rarely
measured alongside ionic AI penetration in plant studies, RSO penetration has
not been measured to date and the transport of RSO through plant leaves is not
fully understood. However, some mechanisms have been identified. Surfactants
are known to decrease an ionic AI’s point of deliquescence (Chen and Lee,
2001), allowing further penetration to occur at lower relative humidity and
can change the contact area, contact angle, and surface tension of applied
droplets (Gaskin et al., 2005). Adjuvants, including surfactants, adsorb to
various surfaces (Zhang et al., 2017; Ahmadi and Shadizadeh, 2015; Bilaowas et
al., 2003; Peirce et al., 2016; Zhang et al., 2006). Aqueous surfactants
Triton X-100, SDS (sodium dodecylsulfate), and DTAB (dodecyltrimethylammonium
bromide) adsorb to different amounts on adaxial and abaxial wheat leaf
(Triticum aestivum) surfaces (Zhang et al., 2017). These results were
confirmed (Zhang et al., 2006) on cabbage (Brasslca oleracea) and wheat
(Triticum aestivum) leaves. They note that adsorption of non-ionic surfactants
on hydrophobic surfaces, such as plant cuticles, may lead to a complete
replacement of the surface by the adsorbed surfactant as a monolayer,
affecting the surface mechanisms. We note the possibility that the adsorbed
surfactant could play a role in changing the cuticle surface chemistry,
potentially influencing the adsorption of ionic AI, and further research is
required.

As RSO 5 has a HLB that is lipophilic, we will assume its transport can
appropriately be described as a lipophilic compound. Lipophilic non-
electrolytes are compounds that can traverse the plant cuticle exclusively via
the lipophilic pathway (Hess and Foy, 2000; Schönherr, 2006; Schreiber et al.,
2001). The lipophilic pathway is made of cutin and wax and is governed by
different mechanisms to transport in aqueous pores. Penetration of lipophilic
compounds through plant cuticles is a three step process: sorption into the
cuticular lipids, diffusion across the cuticular membrane and desorption into
the apoplast of the epidermal cells (Buchholz, 2006; Kirkwood, 1999; Schönherr
et al., 1999). The term sorption is commonly used as this term is nonspecific
and does not imply the location or nature of the interactions of the
lipophilic compound within the cuticle membrane (Bukovac and Petracek, 1993).
Sorption process can be described by an adsorption isotherm (Schreiber and
Schönherr, 1992). Lipophilic compounds diffuse through the cuticle by jumping
into voids or defects that arise due to molecular motion by the polymer
segments or chains (Schreiber et al., 2001; Schönherr, 2006). Schönherr and
Riederer (1988) studied the desorption of a lipophilic compound in various
isolated cuticles. They found that significant desorption, between 68$\%$ to
90$\%$ of the applied amount, occurs from the inner cuticle surface to the
water bath. Desorption also occurs from the outer cuticle surface. We note the
mechanism where lipophilic compounds desorb from the cuticle surface will play
an important role in developing the surface chemistry within this work.

Lipophilic compound penetration via the lipophilic pathway and hydrophilic
ionic compound penetration via the aqueous pathway are governed by distinct
mechanisms and each compound uses its pathway exclusively. In addition to the
mechanisms discussed earlier, penetration of hydrophilic ionic AIs through
plant cuticles is independent of temperature and plasticisers (accelerators),
only weakly affected by wax extraction and less molecular size-selective
compared to the lipophilic pathway Schreiber (2005); Schönherr (2000, 2001).
It has been noted that for lipophilic compounds, when the temperature
increases, voids appear and disappear more frequently, which leads to
diffusion rates greatly increasing Schönherr (2006); Buchholz (2006).
Lipophilic compounds mobility is significantly influenced by lipophilic
accelerators, which dissolve in the cutin and wax domains and act as
plasticisers Schreiber (2005); Riederer et al. (1995); Buchholz and Schönherr
(2000); Buchholz (2006).

Surfactants have the ability to increase penetration of ionic AIs, especially
at late times. Kraemer et al. (2009) measures penetration of ionic CaCl2,
which has been formulated with the lipophilic surfactant RSO 5, through
isolated astomatous tomato fruit (Solanum lycopersicum L., cultivar ‘Panovy’)
cuticles. If we focus on the penetration at late times, between 24 and 48 h,
when CaCl2 is formulated with RSO 5, penetration increases by 5.7$\%$ more
than without RSO 5. This increase in penetration at late times is confirmed
(Gauvrit et al., 2007), with glyphosate penetration and ethoxylated RSO, along
with other adjuvants with a range of HLBs, through barley (Hordeum vulgare L.,
cv. Plaisant) and ryegrass (Lolium multiflorum Lam.) leaves. They found a
similar penetration profile, where penetration of glyphosate significantly
increased to 72 h, and in some cases, a $40\%$ increase was obtained between
24 and 72 h. Coret and Chamel (1993) also confirmed this trend with glyphosate
and non-ionic adjuvant penetration through isolated tomato fruit cuticles,
where a mean of $20\%$ additional penetration was obtained between 20 and 100
h.

The mathematical models for ionic penetration in plant leaves and cuticles
have been reviewed elsewhere (Tredenick et al., 2017, 2018; Forster et al.,
2004; Trapp, 2004). The authors Tredenick et al. (2017) and Tredenick et al.
(2018) previously introduced two mechanistic models to simulate ionic
penetration in plant cuticles and incorporated pore swelling, relative
humidity, droplet evaporation, and hygroscopic water absorption that is
influenced by the point of deliquescence.

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. (2018), 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 (Butler and Ockrent, 1929; Markham and
Benton, 1931) and an adsorption–desorption model for chemicals in soil
(Cameron and Klute, 1977). However, the models in (Butler and Ockrent, 1929;
Markham and Benton, 1931; Cameron and Klute, 1977) do not consider desorption
with a moving droplet radius that is undergoing evaporation and Tredenick et
al. (2018) does not include desorption. We wish to consider this case for the
adsorption and desorption of agrochemicals, which includes the moving droplet
radius due to evaporation on the cuticle surface and create a novel
mechanistic model.

There are several models on penetration of lipophilic compounds through plant
tissues present in the literature. Previous modelling reviews are published
elsewhere (Forster et al., 2004; Trapp, 2004). Some models incorporate
diffusion (Keymeulen et al., 1995; Trapp, 2007; Riederer et al., 2002; Brazee
et al., 2004), while others employ empirical expressions (Schönherr and Baur,
1994). Whole plant leaf (Keymeulen et al., 1995; Trapp, 2007; Riederer et al.,
2002) and isolated cuticle models (Brazee et al., 2004; Schönherr and Baur,
1994) have been studied. The empirical model (Schönherr and Baur, 1994)
assumes equilibrium has been reached and relies on many experimental
parameters, such as partitioning coefficients. Penetration in plant leaves is
related to the physiochemical properties of the lipophilic compound,
especially molecular size and lipophilicity, but penetration cannot be
predicted by either property (Wang and Liu, 2007). Therefore, a mechanistic
model would be an advantage over an empirical model that relies solely on such
parameters.

Keymeulen et al. (1995) employed a model commonly applied in biological
applications, based on Fick’s first law of diffusion and Trapp and co-workers
have modified this formulation to account for whole plant transport in several
publications (Trapp, 2007; Legind et al., 2011). These models rely on many
experimental parameters and cannot be applied to the experimental setup
considered in Kraemer et al. (2009). Three experimental systems have been
developed (Riederer et al., 2002) to simplify plant penetration. These
experimental systems are a novel way to test the governing mechanisms as they
simplify the complex process of plant leaf penetration. The systems were
simulated using finite-element techniques with Fick’s first law of diffusion
with convection of vapour, incorporating wind. However, they focused on leaf
and stomatal transport and applied different experimental techniques to those
in Kraemer et al. (2009), so the models have limited applicability here.

We wished to consider the transport of lipophilic adjuvants by creating a
novel mechanistic model that accounts for adsorption, desorption, and
diffusion mechanisms and allows for any plant species, lipophilic adjuvant and
concentration, temperature, and relative humidity to be used. We aimed to
simulate the complex governing mechanisms involved in surfactant enhanced
hydrophilic ionic AI penetration through isolated astomatous plant cuticles by
utilising a predictive mathematical model. We will expand upon the models in
previous works (Tredenick et al., 2017, 2018) by including adsorption and
desorption of the ionic AI and lipophilic surfactant from the cuticle surface.
In this work, we will investigate the mechanism, where the lipophilic adjuvant
causes the ionic AI to desorb from the cuticle surface, and this may increase
penetration at late times. Without lipophilic adjuvant, desorption of ionic AI
is not possible. We are motivated to describe the increased penetration seen
in the Kraemer et al. (2009) experimental data at late times, and this was the
key objective of this work. The modelling results were validated against
experimental data (Kraemer et al., 2009) and a sensitivity analysis was
conducted.

##  2 Model Framework

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 (Tredenick et al., 2017, 2018) and we refer the reader to
Tredenick et al. (2018) for a full description of the auxiliary equations
governing evaporation in Appendix A. We account for three components,
hydrophilic ionic active ingredient (AI), water (H2O), and lipophilic adjuvant
(ADJ). The ionic AI can adsorb and desorb from the cuticle surface and the
free ionic AI in the droplet solution can diffuse through the aqueous pores.
The lipophilic adjuvant can adsorb and desorb from the cuticle surface, then
the adsorbed adjuvant can diffuse through the cuticle via the lipophilic
pathway. Water can diffuse through both the aqueous pores and the lipophilic
pathway (Schreiber et al., 2001; Schönherr, 2006) within the cuticle. The
authors previous model in Tredenick et al. (2018) 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. (2017) and Tredenick et al. (2018), and this will be the subject of
future works. All three components change primarily along the cuticle membrane
thickness, $x$ ($0\leq x\leq b$). Initially, a droplet with known contact
angle, volume, radius, concentration of ionic AI, and adjuvant is placed on
the outer cuticle surface, at $x=0$. A well stirred water bath exists at the
inner cuticle surface, at $x=b$. The model, including the variables and
parameters, as described in Table 1, governing partial differential equations,
initial conditions (ICs), boundary conditions (BCs), and parameters, is as
follows:

$\displaystyle\frac{\partial c_{\text{AI}}}{\partial t}$
$\displaystyle=D_{\text{AI}}\ \partialderivative[2]{c_{\text{AI}}}{x},$
$\displaystyle 0<x<b,\ t>0,$ (1) $\displaystyle\frac{\partial
c_{\scriptscriptstyle\text{H${}_{2}$O}}}{\partial t}$
$\displaystyle=\left(D_{\scriptscriptstyle\text{H${}_{2}$O,AqP}}\ +\
D_{\scriptscriptstyle\text{H${}_{2}$O,L}}\right)\
\partialderivative[2]{c_{\scriptscriptstyle\text{H${}_{2}$O}}}{x},$
$\displaystyle 0<x<b,\ t>0,$ (2)
$\displaystyle\frac{\partial\Gamma^{\text{ads}}_{\text{ADJ}}}{\partial t}$
$\displaystyle=D_{\text{ADJ}}\
\partialderivative[2]{\Gamma^{\text{ads}}_{\text{ADJ}}}{x},$ $\displaystyle
0<x<b,\ t>0,$ (3) $\displaystyle\text{ICs}:\quad c_{\text{AI}}(x,0)$
$\displaystyle=0,$ $\displaystyle 0<x\leq b,$ (4) $\displaystyle
c_{\text{AI}}(0,0)$ $\displaystyle=c_{\text{AI,0}}^{\text{drop}},$ (5)
$\displaystyle\Gamma^{\text{ads}}_{\text{AI}}(0)$ $\displaystyle=0,$ (6)
$\displaystyle c_{\scriptscriptstyle\text{H${}_{2}$O}}(x,0)$
$\displaystyle=c^{\text{pure}}_{\scriptscriptstyle\text{H${}_{2}$O}},$
$\displaystyle 0<x\leq b,$ (7) $\displaystyle
c_{\scriptscriptstyle\text{H${}_{2}$O}}(0,0)$
$\displaystyle=\frac{1-\bar{v}_{\text{AI}}\
c_{\text{AI}}(0,0)-\bar{v}_{\text{ADJ}}\
c_{\text{ADJ}}(0,0)}{\bar{v}_{\scriptscriptstyle\text{H${}_{2}$O}}},$ (8)
$\displaystyle\Gamma^{\text{ads}}_{\text{ADJ}}(x,0)$ $\displaystyle=0,$
$\displaystyle 0\leq x\leq b,$ (9)
$\displaystyle\qquad\qquad\qquad\qquad\qquad c_{\text{ADJ}}(0)$
$\displaystyle=c_{\text{ADJ,0}}^{\text{drop}},$ (10) $\displaystyle\text{BC -
AI (bath)}:\qquad\quad c_{\text{AI}}(b,t)$ $\displaystyle=0,$ $\displaystyle
t>0,$ (11) $\displaystyle\text{BC - H${}_{2}$O (drop)}:\qquad
c_{\scriptscriptstyle\text{H${}_{2}$O}}(0,t)$
$\displaystyle=\frac{1-\bar{v}_{\text{AI}}\
c_{\text{AI}}(0,t)-\bar{v}_{\text{ADJ}}\
c_{\text{ADJ}}(0,t)}{\bar{v}_{\scriptscriptstyle\text{H${}_{2}$O}}},\quad$
$\displaystyle t>0,$ (12) $\displaystyle\text{BC - H${}_{2}$O (bath)}:\qquad
c_{\scriptscriptstyle\text{H${}_{2}$O}}(b,t)$
$\displaystyle=c^{\text{pure}}_{\scriptscriptstyle\text{H${}_{2}$O}},$
$\displaystyle t>0,$ (13) $\displaystyle\text{BC - ADJ
(bath)}:\qquad\Gamma^{\text{ads}}_{\text{ADJ}}(b,t)$ $\displaystyle=0,$
$\displaystyle t>0,$ (14)
$\displaystyle\text{Parameters}:\qquad\varepsilon_{\text{AqP}}$
$\displaystyle=\pi\left(\dfrac{r_{\text{p}}^{\text{max}}}{L}\left(\mathchoice{{\hbox{$\displaystyle\sqrt{n_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\textstyle\sqrt{n_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=4.30554pt,depth=-3.44446pt}}}{{\hbox{$\scriptstyle\sqrt{n_{0}\,}$}\lower
0.4pt\hbox{\vrule
height=3.01389pt,depth=-2.41113pt}}}{{\hbox{$\scriptscriptstyle\sqrt{n_{0}\,}$}\lower
0.4pt\hbox{\vrule height=2.15277pt,depth=-1.72223pt}}}+1\right)\right)^{2},$
(15) $\displaystyle D_{\text{AI}}$ $\displaystyle=D_{\text{AI}}^{\text{bulk}}\
\varepsilon_{\text{AqP}}\
^{\left(\frac{F_{\text{s}}}{2-F_{\text{s}}}\right)},$ (16) $\displaystyle
D_{\scriptscriptstyle\text{H${}_{2}$O,L}}$
$\displaystyle=D_{\scriptscriptstyle\text{H${}_{2}$O}}^{\text{bulk}}\
\varepsilon_{\text{L}}\ ^{\left(\frac{F_{\text{s}}}{2-F_{\text{s}}}\right)},$
(17) $\displaystyle D_{\scriptscriptstyle\text{H${}_{2}$O,AqP}}$
$\displaystyle=D_{\scriptscriptstyle\text{H${}_{2}$O}}^{\text{bulk}}\
\varepsilon_{\text{AqP}}\
^{\left(\frac{F_{\text{s}}}{2-F_{\text{s}}}\right)},$ (18) $\displaystyle
D_{\text{ADJ}}$ $\displaystyle=D_{\text{ADJ}}^{\text{bulk}}\
\varepsilon_{\text{L}}\ ^{\left(\frac{F_{\text{s}}}{2-F_{\text{s}}}\right)},$
(19) $\displaystyle\text{CCR mode}:\quad\quad\diff{\theta}$
$\displaystyle=-\dfrac{\Lambda\ \left(1+\cos(\theta)\right)^{2}\
f(\theta)}{r_{\text{drop,0}}^{2}},$ $\displaystyle 0<t\leq t_{\text{rec}},$
(20)
$\displaystyle\diff{V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}}{t}$
$\displaystyle=-\pi\ \Lambda\ r_{\text{drop,0}}\ f(\theta),$ $\displaystyle
0<t\leq t_{\text{rec}}.$ (21) $\text{CCA
mode}:\quad\diff{V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}}{t}=-\pi\
\Lambda\ f(\theta_{\text{rec}})\left(\dfrac{3\ g(\theta_{\text{rec}})\
V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}}{\pi}\right)^{\frac{1}{3}}\\\
\times\left[\dfrac{\hbox to0.0pt{\hskip
0.62845pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$\chi$}}}{V_{0}}\
V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}\
\left(\dfrac{V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}}{V_{\text{Del}}}-1\right)\right]\
\left(1-\dfrac{c_{\text{AI}}(0,t)}{c_{\text{POD}}}\right)\\\
+\left.\dfrac{\left(\eta_{\text{pore}}\ A_{\Pi}\ \varepsilon_{\text{AqP}}\
D_{\scriptscriptstyle\text{H${}_{2}$O,AqP}}\ +\ \varepsilon_{\text{L}}\
D_{\scriptscriptstyle\text{H${}_{2}$O,L}}\right)\
M_{\scriptscriptstyle\text{w,H${}_{2}$O}}\
A_{\text{drop}}}{\rho_{\scriptscriptstyle\text{H${}_{2}$O}}}\ \frac{\partial
c_{\scriptscriptstyle\text{H${}_{2}$O}}}{\partial x}\right|_{x=0},\
t>t_{\text{rec}},$ (25) $\displaystyle A_{\text{drop}}(t)$
$\displaystyle=\pi^{\frac{1}{3}}\left(3\ g(\theta)\
V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}(t)\right)^{\frac{2}{3}},$
(26) $\displaystyle m(t)$ $\displaystyle=-\left.\ 10^{6}\ M_{\text{w,AI}}\
\eta_{\text{pore}}\ A_{\Pi}\ n_{\text{drops}}\ \varepsilon_{\text{AqP}}\
D_{\text{AI}}\ \int\limits_{0}^{t_{\text{final}}}\ A_{\text{drop}}\
\frac{\partial c_{\text{AI}}}{\partial x}\ dt\right|_{x=b},$ (27)
$\displaystyle\%\text{ Ca penetration}(t)$ $\displaystyle=\dfrac{m(t)\
100\%}{10^{6}\ 10^{3}\ c^{drop}_{AI,0}\ n_{\text{drops}}\ V_{0}}.$ (28)

Table 1: Model parameters.

Parameter |  Definition |  Value and Units |  Comments
---|---|---|---
$A_{\Pi}$ |  Control volume area |  m2 |
$A_{\text{drop}}(t)$ |  Drop surface contact area |  m2 |  Surface contact area of drop on cuticle surface
$A_{\text{drop,0}}$ |  Initial drop surface contact area |  m2 |  Surface contact area of drop on cuticle surface, (Erbil et al., 2002)
AI |  Active ingredient |  |
$b$ |  Thickness of cuticle |  $1.87\times 10^{-5}$ m |  (Chamel et al., 1991)
$c^{\text{drop}}_{\text{AI,0}}$ |  Concentration of AI in drop at $t=0$ |  mol$/$m3 |  (Kraemer et al., 2009)
$c^{\text{drop}}_{\text{ADJ,0}}$ |  Concentration of adjuvant in drop at $t=0$ |  1 mol$/$m3 (1 g$/$L) |
$c^{\text{pure}}_{\scriptscriptstyle\text{H${}_{2}$O}}$ |  Pure water concentration at $20\,^{\circ}\mathrm{C}$ and $t=0$ |  55,409.78 mol$/$m3 |  calculated
$c_{\text{POD}}$ |  Point of deliquescence concentration |  mol$/$m3 |  (Tredenick et al., 2018)
CCR |  Constant contact radius evaporation mode |  |
CCA |  Constant contact angle evaporation mode |  |
$c_{i}(x,t)$ |  Concentration of component $i$ |  mol$/$m3 |
$D_{\text{AI}}^{\text{bulk}}$ |  Self/bulk diffusion coefficient of AI |  7.93$\times 10^{-10}$ m${}^{2}/$s |  For CaCl2, Ca2+ diffuses the slowest, so Ca2+ value is used, (Yuan-Hui and Gregory, 1974)
$D_{\scriptscriptstyle\text{H${}_{2}$O}}^{\text{bulk}}$ |  Self/bulk diffusion coefficient of water |  2.299 $\times 10^{-9}$ m${}^{2}/$s |  (Holz et al., 2000)
$D_{\text{ADJ}}^{\text{bulk}}$ |  Self/bulk diffusion coefficient of adjuvant |  7.93 $\times 10^{-12}$ m${}^{2}/$s |  (fitted)
$D_{\text{evap}}$ |  Diffusivity of water in air |  2.4 $\times 10^{-5}$ m${}^{2}/$s |  (Semenov et al., 2013)
$D_{i}(x,t)$ |  Diffusivity of component $i$ |  m${}^{2}/$s |  (Liu and Nie, 2001)
$F_{\text{s}}$ |  Fractal scaling dimension |  1.15 (-) |  $1<F_{s}<2$ (fitted)
$f(\theta)$ |  Functional variation of $\theta$ |  |  (Popov, 2005)
$g(\theta)$ |  Functional of $\theta$ |  |  (Popov, 2005; Dash and Garimella, 2013)
$H$ |  Relative humidity |  0.7 (70%) |  (Kraemer et al., 2009)
HLB |  Hydrophilic Lipophilic Balance |  |
$i$ |  Component AI (CaCl2), H2O or ADJ (RSO 5) |  |
$k_{1}$ |  Adsorption rate constant ionic AI |  4.2 $\times 10^{-6}$ m${}^{3}/$(s mol) |  (fitted)
$k_{2}$ |  Desorption rate constant ionic AI |  6.5 $\times 10^{-8}$ m${}^{3}/$(s mol) |  (fitted)
$k_{3}$ |  Adsorption rate constant adjuvant |  2 $\times 10^{-8}$ m${}^{3}/$(s mol) |  (fitted)
$k_{4}$ |  Desorption rate constant adjuvant |  5 $\times 10^{-6}$ m${}^{3}/$(s mol) |  (fitted)
$k_{5}$ |  Kinetic rate constant ionic AI |  5.9 $\times 10^{-10}$ m3 |  (fitted)
$k_{6}$ |  Kinetic rate constant adjuvant |  8 $\times 10^{-9}$ m3 |  (fitted)
$\hbox to0.0pt{\hskip 0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{5}$ |  Kinetic rate constant ionic AI |  mol |  calculated
$\hbox to0.0pt{\hskip 0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{6}$ |  Kinetic rate constant adjuvant |  mol |  calculated
$L$ |  Control volume length |  1 m |
$M_{\text{w,AI}}$ |  Molecular weight of CaCl2 |  110.98 g$/$mol |
$M_{\text{w,ADJ}}$ |  Molecular weight of RSO 5 |  992 g$/$mol |  (Anastopoulos et al., 2009)
$M_{\scriptscriptstyle\text{w,H${}_{2}$O}}$ |  Molecular weight H2O |  $18.015$ g$/$mol |
$m_{\infty}$ |  Equilibrium mass of water absorbed per CaCl2 applied |  g${}_{\scriptscriptstyle\text{H${}_{2}$O}}/$g${}_{\text{AI}}$ |  (Tredenick et al., 2018)
$N_{\text{A}}$ |  Avogadro constant |  $6.02214\times 10^{23}$ mol-1 |
$n_{0}$ |  Number of aqueous pores on area $L^{2}$ (1 m2) of cuticle |  (-) |  $n_{0}=\eta_{\text{pore}}\ L^{2}$
$P_{\text{v}}$ |  Saturated water vapour pressure in air at $20\,^{\circ}\mathrm{C}$ |  2338.8 Pa |  (Lide, 2004)
|  $32\%$RH for CaCl2 (Kolthoff et al., 1969) and $27\%$RH for CaCl2 with RSO 5 |  |
R |  Gas constant |  8.3145 Pa$\cdot$m${}^{3}/$K$/$mol |
RSO |  Rapeseed oil surfactant |  |
$r_{\text{drop,0}}$ |  Initial drop contact radius |  m |  Contact radius of drop on cuticle surface (Erbil et al., 2002)
$r_{\scriptscriptstyle\text{H${}_{2}$O}}$ |  Van der Waals radius of a water molecule |  $1.5\times 10^{-10}$ m |  (Schreiber et al., 2006)
$r_{\text{p}}^{\text{max}}$ |  Maximum radius of aqueous pores |  $2.12\times 10^{-9}$ m |  For tomato fruit cuticle, ((Schreiber and Schönherr, 2009) p. 87)
$RH$ |  Relative humidity in text |  |
$t$ |  Time |  s |
$T$ |  Temperature |  293.15 K ($20\,^{\circ}\mathrm{C}$) |  (Kraemer et al., 2009)
$u$ |  Bound integration variable |  |

Table 2: Cont.

Parameter |  Definition |  Value and Units |  Comments
---|---|---|---
$V_{0}$ |  Volume of droplet at $t=0$ |  1 $\times 10^{-9}$ m3 |  (Kraemer et al., 2009)
$V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}(t)$ |  Volume of water in droplet at time $t$ |  |
$V_{\text{Del}}(t)$ |  Deliquescent droplet volume |  m3 |
$\bar{v}_{\text{AI}}$ |  Partial molar volume CaCl2 |  $1.6\times 10^{-5}$ m${}^{3}/$mol |  (Oakes et al., 1995)
$\bar{v}_{\text{ADJ}}$ |  Partial molar volume adjuvant |  $4.97\times 10^{-7}$ m${}^{3}/$mol |
$\bar{v}_{\scriptscriptstyle\text{H${}_{2}$O}}$ |  Partial molar volume water |  $1.8047\times 10^{-5}$ m${}^{3}/$mol |  (Zen, 1957)
$x$ |  Length |  m |
$\varepsilon_{\text{AqP}}$ |  Aqueous pore pathway porosity |  |
$\varepsilon_{\text{L}}$ |  Lipophilic pathway porosity |  0.03 |  (fitted)
$\Gamma^{\text{ads}}_{\text{AI}}(t)$ |  Concentration adsorbed ionic AI per droplet area on cuticle surface |  mol$/$m2 |
$\Gamma^{\text{ads}}_{\text{ADJ}}(t)$ |  Concentration adsorbed adjuvant per droplet area on cuticle surface |  mol$/$m2 |
$\Gamma_{S}$ |  Saturated adsorbed molecules per droplet area |  400 mol$/$m2 |  (fitted)
$\eta_{\text{pore}}$ |  Density of aqueous pores in cuticle |  $2.18\times 10^{15}$ m-2 |  (Tredenick et al., 2018)
$\Lambda$ |  Evaporation constants as a function of relative humidity |  m${}^{2}/$s |
$\psi$ |  Saturated water vapour concentration as a function of relative humidity |  g$/$m3 |  (Erbil, 2012)
$\Phi$ |  Point of deliquescence humidity shifting factor for choosing the humidity in $m_{\infty}$ and $c_{\text{POD}}$ |  |
$\rho_{\scriptscriptstyle\text{H${}_{2}$O}}$ |  Liquid density H2O at $20\,^{\circ}\mathrm{C}$ |  $9.98207\times 10^{5}$ g$/$m3 |  (Weast and Lide, 1989)
$\rho_{\scriptscriptstyle\text{AI}}$ |  Liquid density of CaCl2 at $20\,^{\circ}\mathrm{C}$ |  $2.16\times 10^{6}$ g$/$m3 |  (Dow Chemical Company, 2003)
$\theta(t)$ |  Contact angle of drop on cuticle surface that changes with time |  rads |
$\theta_{0}$ |  Contact angle of drop on cuticle surface at $t=0$ |  rads |  See Table 1 in Tredenick et al. (2018)
$\theta_{\text{rec}}$ |  Receding contact angle of drop on cuticle surface |  rads |  See Table 1 in Tredenick et al. (2018)
$\chi$ |  Logistic decay evaporation constant |  0.0428 L${}^{2}/$g2 |  (Tredenick et al., 2018)
$\overline{\hbox{}}$$\chi$ |  Logistic decay evaporation term (a constant) as a function of initial concentration of ionic AI |  |  calculated
$\xi$ |  Point of deliquescence humidity shifting factor to incorporate with the addition of adjuvants |  $5\%$RH |  (Tredenick et al., 2018)

AI can travel through plant cuticle aqueous pores via Fickian diffusion and
this is described by Equation (1). Water diffuses through aqueous pores and
the lipophilic pathway (Equation (2)). Therefore, two diffusion coefficients
are required, one for the diffusion of water through aqueous pores,
$D_{\scriptscriptstyle\text{H${}_{2}$O,AqP}}$ (Equation (18)) and the
lipophilic pathway, $D_{\scriptscriptstyle\text{H${}_{2}$O,L}}$ (Equation
(17)). The adsorbed lipophilic adjuvant can diffuse through the cuticle via
the lipophilic pathway (Equation (3)). We note the adsorbed adjuvant diffuses
in terms of $\Gamma^{\text{ads}}_{\text{ADJ}}$ in mol$/$m2, unlike ionic AI
and water, which diffuse in terms of a concentration, $c$, in mol$/$m3.

Initially, the lipophilic adjuvant has not been adsorbed to the cuticle
surface or through the cuticle (Equation (9)). An initially applied
concentration of adjuvant is present on the cuticle surface (Equation (6)). We
note the parameters described in Equation (15)–(19) are constants, as the
porosities are constant. The diffusion coefficient for water in the lipophilic
pathway, $D_{\scriptscriptstyle\text{H${}_{2}$O,L}}$ (Equation (17)), is the
same formulation as the diffusion coefficient for water in the aqueous
pathway, $D_{\scriptscriptstyle\text{H${}_{2}$O,AqP}}$ (Equation (18)), but is
based on the porosity of the lipophilic pathway, $\varepsilon_{\text{L}}$.
Following from this, the diffusion coefficient for adjuvant in the lipophilic
pathway, $D_{\scriptscriptstyle\text{ADJ}}$ (Equation (19)), is also a similar
formulation to Equation (18). Droplet evaporation is governed by Equation
(20)–(25) in constant contact radius mode (CCR) then constant contact angle
mode (CCA), which includes hygroscopic water absorption, and further details
are available in Tredenick et al. (2018). Water travels along the two pathways
so the two diffusivities ($D_{\scriptscriptstyle\text{H${}_{2}$O,AqP}}$ and
$D_{\scriptscriptstyle\text{H${}_{2}$O,L}}$) are included in the term for
evaporation on the third line of Equation (25), which accounts for the
possibility of water diffusing towards the droplet from the cuticle.

We introduce a mechanism for the droplet boundary condition on the cuticle
surface, as depicted in Figure 1, where ionic AI and lipophilic adjuvant can
be adsorbed and desorbed. Focusing on ionic AI, it first adsorbs, then the
adsorbed ionic AI can be desorbed by the adjuvant that is adsorbing onto the
cuticle surface. More adjuvant adsorbs as the concentration of adjuvant in the
droplet increases due to evaporation. The same process occurs for adjuvant and
ionic AI. The process is competitive, where ionic AI and adjuvant compete for
the same sites on the cuticle surface. The adsorbed ionic AI can only be
desorbed by adjuvant in the droplet solution, not ionic AI in solution. The
most important feature of this model is that the adjuvant will desorb ionic AI
that is adsorbed, and this desorbed ionic AI can then diffuse into the cuticle
at later times. The adsorbed ionic AI would otherwise not diffuse through the
cuticle.

The ionic AI and adjuvant are contained within a droplet that evaporates and
the radius changes with time. This evaporation causes the droplet area to
become smaller over time. Only the adsorbed ionic AI or adjuvant that is under
the droplet can desorb into the droplet solution. Therefore, a portion of
ionic AI and adjuvant that are adsorbed onto the cuticle surface are left
behind by the moving droplet radius and are not available for diffusion. As
discussed in Tredenick et al. (2018); Schönherr (2001) and Schönherr and Luber
(2001), hygroscopic water absorption in the context of the Kraemer et al.
(2009) experimental setup, where relative humidity is $70\%$ and the point of
deliquescence of CaCl2 is $32\%$, will cause the total evaporation and
penetration time to increase. If ionic AI is present (as free molecules) in
the droplet, the droplet will not completely evaporate. We have assumed that
adsorbed ions (that are not free) do not contribute to the hygroscopic water
absorption, both inside (adsorbed under the droplet) and outside the droplet
(left behind by the moving droplet radius).

The adjuvant adsorbs as a monolayer (Zhang et al., 2006), as does the ionic
AI. It is known that adjuvants, including the surfactant RSO 5, adsorb to
various surfaces (Zhang et al., 2017; Ahmadi and Shadizadeh, 2015; Bilaowas et
al., 2003; Peirce et al., 2016; Zhang et al., 2006), lipophilic compounds
adsorb and desorb on the cuticle surface (Schönherr and Riederer, 1988), and
ionic AIs adsorb to the cuticle surface (Yamada et al., 1964). However, we
assume that ionic AIs can desorb and the process is competitive.

We modify the well-known competitive Langmuir formulation (Butler and Ockrent,
1929; Markham and Benton, 1931), to describe a model for ionic AI and adjuvant
adsorption (ads) and desorption on the cuticle surface. The well-known
competitive Langmuir formulation is described elsewhere for comparison
(Tredenick (2019) p. 188). This novel model is known henceforth as the
“Adaptive Competitive Langmuir model" and adapts to a moving droplet radius
that is undergoing evaporation, which influences the amount of compound
adsorbed and desorbed.

Figure 1: The Adaptive Competitive Langmuir model that includes adsorption and
desorption. The initial condition is at $t=0$ and $t>0$ is a short time later
after some droplet evaporation has occurred. Two generic molecules A (orange
circles) and B (purple circles) sit on the cuticle surface. Initially, A and B
are in a droplet solution. Then as time progresses, A and B adsorb and desorb
in a competitive process. As the droplet has moved due to evaporation, some of
the molecules have been left behind. We note A in this case is ionic AI and B
is lipophilic adjuvant. This figure is a simplification and describes the
Adaptive Competitive Langmuir model and moving droplet radius, and we note
other processes are involved in droplet evaporation, as described in text (not
to scale).

In Figure 1, we see the scenario of the Adaptive Competitive Langmuir model. A
droplet sits on the cuticle surface that contains ionic AI (A—orange circles)
and lipophilic adjuvant (B—purple circles), initially, at $t=0$ and a short
time later, at $t>0$. Initially, ionic AI and adjuvant are in a droplet
solution. Then after some time, the droplet has partially evaporated and some
adsorption of ionic AI and adjuvant has occurred. Ionic AI and adjuvant can
desorb each other, competing for the same sites, shown with the arrows. As the
droplet has moved due to evaporation, some of the molecules have been left
behind. We note that Figure 1 demonstrates the Adaptive Competitive Langmuir
model and moving droplet radius, and is an oversimplification, as the ionic AI
and adjuvant molecules may not be homogeneously distributed in the droplet
solution (Hunsche and Noga, 2012; Zhang et al., 2017; Bilaowas et al., 2003;
Zhang et al., 2006).

Our novel Adaptive Competitive Langmuir model for ionic AI is as follows:

$\displaystyle\diff{\Omega_{\text{AI}}(t)}{t}$ $\displaystyle=\
\underbrace{k_{1}\
\overbrace{\left(1-\Omega_{\text{AI}}(t)\right)}^{\text{Sites unoccupied by
AI}}\ c_{\text{AI}}(0,t)}_{\text{AI
adsorption}}-\underbrace{k_{2}\overbrace{\Omega_{\text{AI}}(t)}^{\text{Sites
occupied by AI}}c_{\text{ADJ}}(t)}_{\text{AI desorption}},$ (104)

where $\Omega_{\text{AI}}$ is the fraction of sites occupied by ionic AI on
the cuticle surface, $k_{1}$ is the rate constant for adsorption for ionic AI,
$c_{\text{AI}}$ is the concentration of ionic AI in the drop, $k_{2}$ is the
rate constant for desorption of AI and $c_{\text{ADJ}}$ is the concentration
of adjuvant in the drop. The term $(1-\Omega_{\text{AI}})$ is the fraction of
sites that are either vacant or unoccupied by ionic AI on the cuticle surface.
Therefore the first term in Equation (104) describes the rate of adsorption,
based on the sites available (vacant or no ionic AI) and the concentration of
ionic AI in the drop, $c_{\text{AI}}$. The second term in Equation (104)
describes the rate of desorption, based on the fraction of sites occupied by
ionic AI on the cuticle surface, $\Omega_{\text{AI}}$, and the concentration
of adjuvant on the cuticle surface, $c_{\text{ADJ}}$. The ionic AI desorption
term of Equation (104) only allows adsorbed ionic AI to be desorbed by
adjuvant in the droplet ($c_{\text{ADJ}}$), not ionic AI in the droplet.

We then define the fraction of sites occupied by ionic AI under the droplet on
the cuticle surface, $\Omega_{\text{AI}}$, which evolves in time with the
changing droplet area during evaporation, as:

$\Omega_{\text{AI}}(t)=\dfrac{A_{\text{drop,0}}\
\Gamma^{\text{ads}}_{\text{AI}}(t)}{\Gamma_{S}\ A_{\text{drop}}(t)},$ (105)

where $A_{\text{drop,0}}$ is the initial droplet contact area,
$\Gamma^{\text{ads}}_{\text{AI}}$ is the concentration of adsorbed ionic AI
per unit area of cuticle surface under the droplet,
$\Gamma^{\text{ads}}_{\text{ADJ}}$ is the concentration of adsorbed adjuvant
per unit area of cuticle surface under the droplet, $\Gamma_{S}$ is the
saturated adsorbed molecules per droplet area and $A_{\text{drop}}$ is the
droplet area on the cuticle surface.

The equation describing the concentration of adsorbed ionic AI per unit area
of cuticle surface under the droplet, $\Gamma^{\text{ads}}_{\text{AI}}$, is
found by substituting Equation (105) into Equation (104) and simplifying,
producing Equation (108) and an analogous equation for
$\Gamma^{\text{ads}}_{\text{ADJ}}$ in Equation (112).

$\diff{\Gamma^{\text{ads}}_{\text{AI}}(t)}{t}=\underbrace{k_{1}\
c_{\text{AI}}(0,t)\ \left[\dfrac{\Gamma_{S}}{A_{\text{drop,0}}}\
A_{\text{drop}}(t)-\Gamma^{\text{ads}}_{\text{AI}}(t)\right]}_{\text{AI
adsorption}}\\\ -\ \underbrace{k_{2}\ c_{\text{ADJ}}(t)\
\Gamma^{\text{ads}}_{\text{AI}}(t)}_{\text{AI desorption}}+\
\underbrace{\dfrac{\Gamma^{\text{ads}}_{\text{AI}}(t)}{A_{\text{drop}}(t)}\
\diff{A_{\text{drop}}(t)}{t},}_{\text{Change in droplet area}}$ (108)
$\diff{\Gamma^{\text{ads}}_{\text{ADJ}}(0,t)}{t}=\underbrace{k_{3}\
c_{\text{ADJ}}(t)\ \left[\dfrac{\Gamma_{S}}{A_{\text{drop,0}}}\
A_{\text{drop}}(t)-\Gamma^{\text{ads}}_{\text{ADJ}}(0,t)\right]}_{\text{ADJ
adsorption}}\\\ -\ \underbrace{k_{4}\ c_{\text{AI}}(0,t)\
\Gamma^{\text{ads}}_{\text{ADJ}}(0,t)}_{\text{ADJ desorption}}+\
\underbrace{\dfrac{\Gamma^{\text{ads}}_{\text{ADJ}}(0,t)}{A_{\text{drop}}(t)}\
\diff{A_{\text{drop}}(t)}{t}}_{\text{Change in droplet area}}\\\
+\underbrace{\left.\dfrac{\varepsilon_{\text{L}}\ A_{\text{drop}}(t)\
D_{\text{ADJ}}}{V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}(t)}\
\frac{\partial\Gamma^{\text{ads}}_{\text{ADJ}}(x,t)}{\partial
x}\right|_{x=0}}_{\text{Spreads into cuticle}},$ (112)

where $k_{3}$ is the adsorption rate constant for the adjuvant and $k_{4}$ is
the desorption rate constant for the adjuvant. The first term in Equation
(108) accounts for the adsorption of the ionic AI to the cuticle surface based
on the concentration of ionic AI in the droplet, $c_{\text{AI}}$, and the
sites that are unoccupied by ionic AI under the droplet,
$(\Gamma_{S}A_{\text{drop}}/A_{\text{drop,0}}-\Gamma^{\text{ads}}_{\text{AI}})$.
The second term accounts for the desorption of ionic AI from the cuticle
surface based on the concentration of adjuvant on the droplet surface,
$c_{\text{ADJ}}$, and the amount of ionic AI adsorbed per area,
$\Gamma^{\text{ads}}_{\text{AI}}$. The third term accounts for the changing
droplet area, $A_{\text{drop}}$. Equation (112) is similar to Equation (108),
except the last term is added to account for the adsorbed adjuvant,
$\Gamma^{\text{ads}}_{\text{ADJ}}$, being transported away from the cuticle
surface and into the lipophilic pathway of the cuticle, based on the porosity
of the lipophilic pathway, $\varepsilon_{\text{L}}$, the changing area of the
droplet, $A_{\text{drop}}$, the diffusion coefficient for the adjuvant,
$D_{\text{ADJ}}$, and the flux of the adsorbed adjuvant at the droplet
boundary.

The equations for the concentration of ionic AI and adjuvant in the drop,
$c_{\text{AI}}$ and $c_{\text{ADJ}}$, are as follows:

$\diff{}{t}\left(V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}(t)\
c_{\text{AI}}(0,t)\right)=\dfrac{\hbox to0.0pt{\hskip
0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{5}\
A_{\text{drop,0}}}{\Gamma_{S}\ A_{\text{drop}}(t)}\
\left[\dfrac{\Gamma^{\text{ads}}_{\text{AI}}(t)}{A_{\text{drop}}(t)}\
\diff{A_{\text{drop}}}{t}\ -\
\diff{\Gamma^{\text{ads}}_{\text{AI}}}{t}\right]\\\
+\left.\varepsilon_{\text{AqP}}\ \eta_{\text{pore}}\ A_{\Pi}\
A_{\text{drop}}(t)\ D_{\text{AI}}\ \frac{\partial c_{\text{AI}}}{\partial
x}\right|_{x=0},$ (115)
$\diff{}{t}\left(V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}(t)\
c_{\text{ADJ}}(t)\right)=\dfrac{\hbox to0.0pt{\hskip
0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{6}\
A_{\text{drop,0}}}{\Gamma_{S}\ A_{\text{drop}}(t)}\
\left[\dfrac{\Gamma^{\text{ads}}_{\text{ADJ}}(0,t)}{A_{\text{drop}}(t)}\
\diff{A_{\text{drop}}}{t}\ -\
\diff{\Gamma^{\text{ads}}_{\text{ADJ}}}{t}\right],$ (116)

where $\hbox to0.0pt{\hskip
0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{5}$
is the kinetic rate constant for the ionic AI and $\hbox to0.0pt{\hskip
0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{6}$
is the kinetic rate constant for the adjuvant in mol. In Equation (115), the
droplet becomes more concentrated due to evaporation, governed by the volume
of water in the drop, $V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}$,
and the first term is formulated to incorporate the changing droplet area,
$A_{\text{drop}}$. The first term in Equation (115) is found by substituting
Equation (105) into $\hbox to0.0pt{\hskip
0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{5}\
\text{d}\Omega_{\text{AI}}/\text{d}t$ and simplifying. The second term in
Equation (115) describes how ionic AI is transported into the cuticle via
diffusion, based on the porosity of the aqueous pores,
$\varepsilon_{\text{AqP}}$, the density of aqueous pores in cuticle,
$\eta_{\text{pore}}$, the control volume area, $A_{\Pi}$, the changing area of
the droplet, $A_{\text{drop}}$, the diffusion coefficient for ionic AI,
$D_{\text{AI}}$, and the flux of the ionic AI at the droplet boundary.
Equation (116) does not include the flux term for the concentration of
adjuvant, $c_{\text{ADJ}}$, transported into the cuticle, as the adsorbed
adjuvant, $\Gamma^{\text{ads}}_{\text{ADJ}}$, is transported into the cuticle
in Equation (112).

The $\hbox to0.0pt{\hskip
0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{5}$
and $\hbox to0.0pt{\hskip
0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{6}$
rate constants in Equation (115)–(116) are as follows:

$\displaystyle\hbox to0.0pt{\hskip
0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{5}$
$\displaystyle=k_{5}\ c^{\text{drop}}_{\text{AI,0}},$ (117)
$\displaystyle\hbox to0.0pt{\hskip
0.2604pt\leavevmode\hbox{\set@color$\overline{\hbox{}}$}\hss}{\leavevmode\hbox{\set@color$k$}}_{6}$
$\displaystyle=k_{6}\ c^{\text{drop}}_{\text{ADJ,0}},$ (118)

where $k_{5}$ and $k_{6}$ are the kinetic rate constants in m3 and
$c^{\text{drop}}_{\text{AI,0}}$ and $c^{\text{drop}}_{\text{ADJ,0}}$ are the
initial concentration of ionic AI and adjuvant in the drop in mol$/$m3. The
parameters $k_{5}$ and $k_{6}$ were found to be a function of the initial
concentrations, as changing the initial concentration alters the initial
droplet contract angle $\theta_{0}$ (see Tredenick et al. (2018)).

In summary, the novel Adaptive Competitive Langmuir formulation consists of
Equation (108),(118). The two points of difference from the well-known
competitive Langmuir model and the adaptive model described here, are that
ionic AI desorbs adjuvant (and vice-versa) and the droplet contact area
changes in time, causing some of the adsorbed ionic AI and adjuvant to be left
behind by the moving droplet radius and are not available for diffusion. The
Adaptive Competitive Langmuir model assumes the following: ionic AI and
lipophilic adjuvant adsorb as a monolayer, that the process is competitive,
all sites are equivalent, there are no interactions between the adsorbed ionic
AI and adjuvant molecules on the cuticle surface and the adsorption–desorption
process is not in equilibrium. As this model assumes a competitive process,
when one compound is adsorbing and desorbing, then the other must also, so
$k_{5}$ or $k_{6}$ can only be zero or non-zero pairwise.

###  2.1 Numerical Solution Procedure

The model as described in Equation (1)–(118), is solved numerically. This is
achieved using a finite volume method and discretising the model’s partial
differential equations with second order central differences to approximate
the spatial derivatives. The resulting system of ordinary differential
equations is solved using ‘ode15i’ (Shampine, 2002) within MATLAB® (MATLAB,
2017). The fitted parameter $F_{s}$ in Table 1 was found by using the value in
Tredenick et al. (2018), then the parameter was adjusted by focusing on
penetration for only the initial applied mass of 25 $\upmu$g (5 g$/$L) of
calcium using trial-and-error. The other fitted parameters in Table 1 were
found using trial-and-error for only the initial applied mass of 25 $\upmu$g
of calcium. The experimental data (Kraemer et al., 2009) includes penetration
of five different initial applied masses of Ca of 5, 25, 50, 75, and 150
$\upmu$g. All the fitted parameters were then kept constant and used to solve
the penetration of the other four initial applied masses.

##  3 Results and Discussion

Figure 2: Numerical solution results as percentage penetration with time of
the model (purple line) compared to experimental data (orange crosses) for
CaCl2 (applied at 5 g$/$L or 25 $\upmu$g) with RSO 5 (1 g$/$L) and the
numerical solution of the model in Tredenick et al. (2018) (black line).
Parameters are described in Table 1.

The plant cuticle model, as described in Equation (1)–(118), is solved
numerically, with parameters described in Table 1. In Figure 2, we can see the
numerical solution results as percentage of ionic AI penetration with time
(purple line) compared to experimental data (orange crosses—(Kraemer et al.,
2009)) for CaCl2 at 5 g$/$L (25 $\upmu$g) with RSO 5 (1 g$/$L). The results of
the model compare well to the experimental data, especially at late times,
between 4 and 48 h. Penetration increases rapidly over the first 4 h, then
begins to level out and the penetration can then continue to increase between
4 and 48 h by $15\%$, due to ionic AI adsorption with desorption. Penetration
at 48 h has not yet reached equilibrium and can continue to increase at much
later times, beyond 48 h.

In Figure 2, the fit of the model in the current work to the experimental data
is reasonable; however, the fit is reduced at 24 h. We surmise this could be
due to the Kraemer et al. (2009) experimental setup, where each data point
represents 10 repeats. Other authors (Schönherr, 2001; Schönherr and Luber,
2001; Schönherr, 2000) have noted they conducted 50–100 repeats as variability
among individual cuticles can be large (Baur, 1997).

Figure 3: Numerical solution (purple line) of the model compared to
experimental data (orange circles) for CaCl2 with RSO 5, using an initial
applied mass of Ca of 5 $\upmu$g (A), 25 $\upmu$g (B), 50 $\upmu$g (C), 75
$\upmu$g (D), 150 $\upmu$g (E) with RSO 5 (1 g$/$L) over 48 h. Percentage Ca
penetration is compared to time in hours, calculated according to Equation
(28). The numerical solution can be seen as the continuous purple line and the
experimental data as orange circles with error bars. The final percent Ca
penetration is shown on each subfigure at 48 h. Figure 4: Numerical solution
(purple line) of the model compared to experimental data (orange circles) for
CaCl2 with RSO 5, using an initial applied mass of Ca of 5 $\upmu$g (A), 25
$\upmu$g (B), 50 $\upmu$g (C), 75 $\upmu$g (D), 150 $\upmu$g (E) with RSO 5 (1
g$/$L) over 48 h. The mass of Ca penetrated in $\upmu$g is compared to time in
hours, calculated according to Equation (27). The numerical solution can be
seen as the continuous purple line and the experimental data as orange circles
with error bars. The final percent Ca penetration is shown on each subfigure
at 48 h.

In Figure 3,4, we can see the validation results of the model, compared with
experimental data for CaCl2 with RSO 5, across 5 initial applied
concentrations of CaCl2. Percentage Ca penetration is compared to time in
hours in Figure 3 and as a mass in $\upmu$g in Figure 4. The parameters, as
described in Table 1, are the same across all 5 graphs, except the initial
applied concentration of CaCl2; 5 $\upmu$g (A), 25 $\upmu$g (B), 50 $\upmu$g
(C), 75 $\upmu$g (D), 150 $\upmu$g (E). Figure 3 is calculated according to
Equation (28) and Figure 4 according to Equation (27). We have included these
two figures as Figure 4 shows that increasing the initial concentration of
ionic AI increases the mass penetration, while Figure 3 shows the penetration
trend clearly.

The results in Figure 3,4 show a reasonable fit across all five plots,
considering the complex mechanisms involved, producing an R-squared value of
97.95$\%$, including the error bars. This indicates the model can predict
changes in the initial applied concentration. In the figure, the desired trend
of increased penetration at late times, after penetration has begun to level
out, can be seen in all five graphs.

In Figure 2–4, we can see that the penetration timescale of the current work
is significantly extended to more than 48 h. This extended timescale is caused
by two mechanisms. The first mechanism is the desorption of ionic AI. At late
times, ionic AI is continuously released back into the solution. The second
mechanism is the hygroscopic water absorption. As the concentration of ionic
AI increases in the drop, due to desorption, hygroscopic water absorption is
increased. This in turn extends the evaporation timescale and the penetration
timescale. Both the ionic AI desorption and the hygroscopic water absorption
promote penetration at late times and both mechanisms must be included in a
plant cuticle model for extended penetration to occur at late times.

The increase in penetration in Figure 2 in the current work, between 4 and 48
h, confirms the addition of the desorption mechanism may be one way to achieve
increased penetration at late times. It has been well established that
surfactants including RSO increase the penetration of ionic AI, especially at
late times between 24 and 100 h, to a significant degree Kraemer et al.
(2009); Gauvrit et al. (2007); Coret and Chamel (1993). Earlier work by
Tredenick et al. (2018), which focused on penetration and droplet evaporation,
found that penetration ceased after 9 h, as shown by the black line in Figure
2. Penetration ceased after 9 h as the ionic AI was either adsorbed on the
cuticle surface to a significant degree or transported through the cuticle.
Desorption was not included in the previous work. The current work produces a
penetration profile that is closer to the experimental data.

In future works, the model presented here can be adapted to consider a range
of constant temperatures. Equation (17)–(19) for the diffusivities for
adjuvant and water in the lipophilic pathway could be modified to be a
function of temperature by utilising an Arrhenius equation (Laidler, 1984). It
has been noted that for lipophilic compounds, when the temperature increases,
voids appear and disappear more frequently, which leads to diffusion rates
greatly increasing (Schönherr, 2006; Buchholz, 2006). Likewise, an Arrhenius
equation could be used to formulate the $k_{i}$ terms if temperature varied.
For lipophilic compounds, it has been found that desorption hysteresis occurs
(Schönherr and Riederer, 1988) at the inner water bath boundary (in real
leaves the boundary condition would represent the apoplast of the epidermal
cells), where the adsorbed lipophilic compound may desorb into the bath at a
different rate and a portion of compound may be retained within the cuticle.
This can be incorporated into the model by utilising a time varying function
for $\Gamma^{\text{ads}}_{\text{ADJ}}$ at the bath boundary condition in
Equation (14).

The inner cuticle boundary condition for ionic AI is based on a well stirred
water bath, so the concentration of ionic AI is instantaneously removed and is
zero in Equation (11). In real plants, the inner cuticle surface would be
adjacent to the apoplast of the epidermal cells, and transport through the
apoplast is thought to be governed by diffusion Sattelmacher (2001); Liu and
Gaskin (2004). A flux condition based on diffusion could replace Equation (11)
for the ionic AI, and a similar equation for the lipophilic adjuvant in
Equation (14).

In this work, we have added the mechanism, where the lipophilic adjuvant
causes the ionic AI to desorb from the cuticle surface, and this increases
penetration at late times. Without lipophilic adjuvant, desorption of ionic AI
is not included, and the model can be applied when no adjuvant is included in
the formulation by setting the initial applied concentration of adjuvant in
Equation (10) to zero.

The fitted model parameters are shown in Table 1. Some of these parameters
have physiological significance, such as the tortuosity or fractal scaling
dimension, $F_{s}$. A low value for $F_{s}$ of 1.15 was found with the fitting
exercise. A low value for $F_{s}$ indicates the aqueous pores in tomato fruit
cuticles are in the lower range of tortuosity, tending to straight.
Penetration is quite rapid, even without adjuvant, so this value is logical.
Fitting $F_{s}$ is reasonable as it facilitates the diffusion path length
calculation, which currently cannot be found by a physical measurement
(Riederer and Schreiber, 1995). The value for the partial molar volume of the
adjuvant, $\bar{v}_{\text{ADJ}}$, was not available in the literature and was
found by fitting with trial-and-error. Likewise, the value for the bulk
diffusion coefficient of the adjuvant, $D_{\text{ADJ}}^{\text{bulk}}$, was not
available in the literature and was found by fitting with trial-and-error. The
value for the porosity of the lipophilic pathway, $\varepsilon_{\text{L}}$,
was found to be similar to the value for the porosity of the aqueous pathway,
$\varepsilon_{\text{AqP}}$. The values in Tredenick et al. (2018) for the
aqueous pore density, $\eta_{\text{pore}}$, and logistic decay evaporation
constant, $\chi$, were utilised. We note the model has many fitted parameters.
However, the penetration of ionic AI showed little sensitivity to
$D_{\text{ADJ}}^{\text{bulk}}$ and $\varepsilon_{\text{L}}$, when fitting with
trial-and-error. The parameters $D_{\text{ADJ}}^{\text{bulk}}$,
$\bar{v}_{\text{ADJ}}$ and $\Gamma_{\text{S}}$ may be found with further
experimental works, reducing the number of fitted parameters. The parameters
$D_{\text{ADJ}}^{\text{bulk}}$ and $\bar{v}_{\text{ADJ}}$ for RSO 5 can then
be used in any material, not just plant cuticles.

###  3.1 Sensitivity Analysis

Figure 5: Percentage of calcium (Ca) penetration sensitivity to $k_{1}$ and
$k_{2}$ (A), $k_{3}$ and $k_{4}$ (B), $k_{5}$ (C) and $k_{6}$ (D), with
parameters described in Table 1 at an initial concentration of CaCl2 of 5
g$/$L and RSO 5 of 1 g$/$L. The kinetic rate constants for the ionic AI are
$k_{1}$ and $k_{2}$ (A) and $k_{5}$ (C). The kinetic rate constants for the
lipophilic adjuvant are $k_{3}$ and $k_{4}$ (B) and $k_{6}$ (D).

A sensitivity analysis was performed with the results from the model, using
parameters in Table 1 at an initial concentration of CaCl2 of 5 g$/$L (or 25
$\upmu$g) with RSO 5 (1 g$/$L). We have investigated the adsorption–desorption
parameters, with the method described in Tredenick et al. (2017) and Tredenick
et al. (2018). The results of the sensitivity analysis are seen in Figure
5A–D, where dimensionless percentage penetration of ionic AI or percent
calcium (Ca) penetration is investigated over a timescale of 48 h. A
representative range of $k_{i}$ values are investigated, where $i$ is between
1 and 6, along with dimensionless penetration.

In Figure 5A, we investigate the sensitivity of the models results for
percentage penetration with the adsorption and desorption rate constants for
ionic AI; $k_{1}$ and $k_{2}$. In the figure, the ionic AI adsorption rate
constant, $k_{1}$, controls the percent penetration where the penetration
starts to level out and is inversely proportional to penetration. A larger
value for $k_{1}$ indicates more ions are being adsorbed so penetration is
reduced. The ionic AI desorption rate constant, $k_{2}$, is directly
proportional to penetration and influences the amount of desorption of ionic
AI. If $k_{2}$ is zero, penetration levels out earlier and no desorption
occurs. As $k_{2}$ increases, more ionic AI is released at later times, which
increases penetration and the penetration timescale is extended. Only when no
adsorption occurs ($k_{i}=0$) can penetration reach 100$\%$. The sensitivity
in Figure 5A aligns to a similar adsorption–desorption model in the well-
established literature (Cameron and Klute, 1977).

In Figure 5B, we investigate the sensitivity of the model’s results for ionic
AI percentage penetration with the adjuvant adsorption and desorption rate
constants; $k_{3}$ and $k_{4}$. In the figure, the adjuvant adsorption rate
constant, $k_{3}$, has minimal influence over AI penetration and is inversely
proportional to penetration. As $k_{3}$ increases, ionic AI penetration
decreases as more adjuvant is adsorbed, which decreases the concentration of
adjuvant in the drop and this causes less ionic AI to desorb and penetration
ceases earlier. The adjuvant desorption rate constant, $k_{4}$, has minimal
influence over ionic AI penetration and is directly proportional to
penetration. As $k_{4}$ increases, ionic AI penetration increases at late
times as more adjuvant is being desorbed, which increases the concentration of
adjuvant in the drop and increases ionic AI desorption.

Figure 5C shows the sensitivity of the model to the kinetic rate constant for
ionic AI, $k_{5}$, which controls how influential the adsorption and
desorption of ionic AI is, when included into the droplet boundary condition
in Equation (115). Penetration of ionic AI is very sensitive to $k_{5}$ and
penetration is inversely proportional to $k_{5}$. As the value of $k_{5}$
increases, both adsorption and desorption increase. As adsorption increases,
penetration levels out at a lower value. Increasing $k_{5}$ also increases the
amount of desorption, which can be seen as the gradient for $k_{5}$ at
$1.1\text{\times}{10}^{-9}$ is larger than at $2.2\text{\times}{10}^{-10}$. A
balance needs to be obtained between adsorption and desorption to explain the
experimental data trends.

Figure 5D shows the sensitivity of the model results to the kinetic rate
constant for adjuvant, $k_{6}$, which controls the level of adsorption and
desorption of adjuvant. Penetration is inversely proportional to $k_{6}$. When
$k_{6}$ is small, this reduces both the adsorption and desorption of adjuvant,
resulting in the concentration of adjuvant in the droplet increasing. This
causes an increased concentration of adjuvant in the droplet solution that
increases ionic AI desorption, which increases ionic AI penetration at late
times. Conversely, when $k_{6}$ is large, both the adsorption and desorption
of adjuvant are increased and the increase of adjuvant adsorption causes the
concentration of adjuvant in the drop to significantly reduce and thus
decreases ionic AI desorption. Therefore, a balance needs to be maintained.

The parameters discussed in the sensitivity analysis show appropriate
sensitivities within the model and it is clear that a balance needs to be
maintained between adsorption and desorption of ionic AI. We note that there
are few experimental studies that measure the adsorption and desorption
kinetic rate constants for adjuvants. Future experimental works, similar to
Yamada et al. (1964), could be conducted to directly measure the $k_{i}$ rate
constants for RSO 5 and identify the mechanisms that are involved in adjuvant
transport through plant leaves and cuticles.

The Adaptive Competitive Langmuir model applies to sets of two substances,
which are adsorbing and desorbing in a competitive manner, under a dynamic
droplet contact area. The plant cuticle diffusion model can apply to the
penetration of most hydrophilic ionic AIs including lipophilic adjuvants in
the formulation. It can apply to isolated astomatous plant leaf or fruit
cuticle, where the aqueous pores are sufficiently large to allow ionic AI to
be transported through the cuticle via Fickian diffusion. The model for
lipophilic transport of adjuvant through the cuticle can be applied to both
lipophilic adjuvants and lipophilic AIs. Adaptations can be made for
hydrophilic adjuvants as transport is thought to be similar to hydrophilic
ionic AIs. The model cannot be applied, in its current form, to whole leaf
penetration, hydrophilic uncharged compounds, nor compounds, such as Fe
chelates (Schönherr et al., 2005; Schlegel et al., 2006), that dehydrate
aqueous pores. Adaptations could be made to this model to account for these
compounds and this is the subject of future work. We note that if the type of
hydrophilic ionic AI, lipophilic adjuvant, or plant species was changed, the
model and mechanisms would not change. However, new parameter values would
need to be found, and this may present a challenge depending on the extent of
the available experimental literature.

##  4 Conclusions

We created a novel Adaptive Competitive Langmuir model that allows a dynamic
droplet contact area, which influences the amount of adsorption and desorption
of molecules. A mechanistic model has, therefore, been developed to simulate
diffusion of hydrophilic ionic AIs, including lipophilic adjuvants through
plant cuticles. This model makes novel additions to a simple diffusion model
by incorporating the important governing mechanisms of the adsorption and
desorption of ionic AI and adjuvant at the plant cuticle surface, along with
the adsorption and subsequent diffusion of the adsorbed lipophilic adjuvant
through the cuticle. The model has been solved numerically, producing results
that show reasonable agreement with the experimental data. The addition of the
desorption mechanism at the cuticle surface enables penetration of AI to
increase at later times and further explains the experimental data trends. The
sensitivity analysis indicated the parameters that controlled the adsorption
and desorption had a significant impact on penetration. The results indicate
the chemistry within the droplet on the cuticle surface is significant.
Overall, the results of the validation and sensitivity analysis imply the
model has included many important governing mechanisms simulating ionic AI
penetration through plant cuticles.

conceptualisation, E.C.T., T.W.F., W.A.F.; methodology, E.C.T., T.W.F.,
W.A.F.; computational code creation, E.C.T., T.W.F.; model creation and
adaptation, E.C.T., T.W.F.; validation, E.C.T., T.W.F.; formal analysis,
E.C.T., T.W.F.; writing—original draft preparation, E.C.T.; writing—review and
editing, E.C.T., T.W.F., W.A.F.

This research received no external funding.

###### Acknowledgements.

We thank S.T.P. Psaltis for his contribution to the partial development of the
numerical scheme. The authors declare no conflict of interest. yes

##  Appendix A Additional Droplet Evaporation Equations

The following describe the additional droplet evaporation equations and
further details can be found in Tredenick et al. (2018):

$\displaystyle A_{\scriptscriptstyle\text{drop,0}}$
$\displaystyle=\pi^{\frac{1}{3}}\left(3\ g(\theta_{0})\
V_{0}\right)^{\frac{2}{3}},$ $\displaystyle c_{\text{mass}\%}$
$\displaystyle=-0.8307\ \mathrm{e}^{3.618\ \Phi}+55.44\ \mathrm{e}^{-0.612\
\Phi},$ $\displaystyle c_{\text{POD}}$
$\displaystyle=\dfrac{c_{\text{mass}\%}\
\rho_{\scriptscriptstyle\text{AI}}}{100\%\ M_{\text{w,AI}}},$ $\displaystyle
g(\theta)$ $\displaystyle=\dfrac{\sin^{3}(\theta)}{(1-\cos(\theta))^{2}\
(2+\cos(\theta))},$ $\displaystyle m_{\infty}$ $\displaystyle=0.307\
\mathrm{e}^{2.763\ \Phi}+1.218\times 10^{-9}\ \mathrm{e}^{24\ \Phi},$
$\displaystyle r_{\text{drop,0}}$ $\displaystyle=\left(\dfrac{3\
g(\theta_{0})\ V_{0}}{\pi}\right)^{\frac{1}{3}},$ $\displaystyle
V_{\text{Del}}$ $\displaystyle=\dfrac{m_{\infty}\ M_{\text{w,AI}}\
c_{\text{AI}}(0,t)\
V^{\text{drop}}_{\scriptscriptstyle\text{H${}_{2}$O}}}{\rho_{\scriptscriptstyle\text{H${}_{2}$O}}},$
$\overline{\hbox{}}$$\chi$ $\displaystyle=\chi\ c^{\text{drop}\
2}_{\text{AI,0}},$ $\displaystyle\Lambda$
$\displaystyle=\dfrac{D_{\text{evap}}\
\psi}{\rho_{\scriptscriptstyle\text{H${}_{2}$O}}},$ $\displaystyle\psi$
$\displaystyle=\dfrac{M_{\scriptscriptstyle\text{w,H${}_{2}$O}}\
P_{\text{v}}(1-H)}{R\ T},$ $\displaystyle f(\theta)$
$\displaystyle=\tan\left(\dfrac{\theta}{2}\right)+8\int\limits_{0}^{\infty}\cosh^{2}(\theta
u)\ \text{csch}(2\pi u)\ \tanh\left[\left(\pi-\theta\right)u\right]\ du,$
$\displaystyle t_{\text{rec}}$
$\displaystyle=\int\limits_{\theta_{\text{rec}}}^{\theta_{0}}\dfrac{r_{\text{drop,0}}^{2}}{\Lambda\
\left(1+\cos(\theta)\right)^{2}\ f(\theta)}\ d\theta.$

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