2.1. Non-dimensionalisation

Throughout this analysis, we concentrate on droplets that are thin, which may be expressed herein as

(2.14) \begin{equation} 0\lt \delta =\frac { \tilde V_{\textit{init}}}{ {\tilde R}^3 }\ll 1. \end{equation}

Under the thin assumption, we may linearise the mixed-boundary problem for the vapour concentration (2.10)–(2.12) onto $\tilde z = 0$ , retrieving the classical problem for the electrostatic potential outside a charged disk of radius $\tilde R$ , whose solution has been long known (see e.g. Weber (Reference Weber1873) and Copson (Reference Copson1947) for details): the relevant information for the present analysis is the evaporative flux, which is determined by the appropriate linearised version of (2.13) to be

(2.15) \begin{equation} \tilde J= \tilde {M}_\ell \tilde D_{v}\big(\tilde {c}_{\textit{sat}}-\tilde {c}_\infty \big)\frac {2}{\pi }\frac {1}{\sqrt {\tilde {R}^2-\tilde {r}^2}}. \end{equation}

Since we assume that the only way fluid is lost from the system is through evaporation, the liquid volume $\tilde V(\tilde t)$ satisfies

(2.16) \begin{equation} \frac {\mathrm{d} \tilde V}{\mathrm{d} \tilde t}= -\frac {4\tilde M_\ell \tilde D_{v}\big(\tilde {c}_{\textit{sat}}-\tilde {c}_\infty \big) \tilde R}{\tilde \rho } \quad \implies \quad \tilde V=\tilde V_{\textit{init}}-\frac {4\tilde M_\ell \tilde D_{v}\big(\tilde {c}_{\textit{sat}}-\tilde {c}_\infty \big) \tilde R \tilde t}{\tilde \rho }, \end{equation}

where the right-hand side of the differential equation is the linearised net evaporation rate, determined by integrating (2.15) over the droplet. Hence the lifetime of the droplet $\tilde t_{\!f}$ is

(2.17) \begin{equation} \tilde {t}_{\!f}=\frac {\tilde \rho \tilde V_{\textit{init}}}{4\tilde R\,\tilde M_\ell \tilde D_{v}\big(\tilde {c}_{\textit{sat}}-\tilde {c}_\infty \big)}. \end{equation}

Following previous studies (e.g. Moore et al. Reference Moore, Vella and Oliver2021; D’Ambrosio et al. Reference D’Ambrosio, Wray and Wilson2025b ), we non-dimensionalise (2.1)–(2.9), (2.15)–(2.17) by introducing the scalings

(2.18) \begin{align} \tilde t=\tilde {t}_{\!f} t, \quad (\tilde r, \tilde z,\tilde h)& = \tilde R( r, \delta z, \delta h), \quad (\tilde {u},\tilde w)=\tilde {U}(u,\delta w), \quad \tilde p =\tilde {p}_{{\textit{atm}}}+\frac {\tilde \mu \tilde {U}}{\delta ^2\tilde R}p, \nonumber\\ \tilde J & = \delta \rho \tilde {U}\!{J}, \quad \tilde \phi =\tilde {\phi }_{\textit{init}}\phi , \quad \tilde {m}_s = \delta \tilde R \tilde {\phi }_{\textit{init}}\tilde {\rho }_{{s}}m_s, \end{align}

where the velocity scale $\tilde {U} = {\tilde R}^2 \tilde M_\ell \tilde D_{v}(\tilde {c}_{\textit{sat}}-\tilde {c}_\infty )/(\tilde \rho \tilde V_{\textit{init}})$ and time scale have been selected as those driven by evaporation, and the pressure scale is chosen to balance the viscous terms in the radial momentum equation.

Under these scalings and the thin droplet assumption, it is straightforward to show in the usual manner (see e.g. Hocking (Reference Hocking1983), Deegan et al. (Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten2000) and Oliver et al. (Reference Oliver, Whiteley, Saxton, Vella, Zubkov and King2015), for details) that the gravity-free lubrication equations pertain:

(2.19) \begin{equation} \bar {u} = -\frac {h^2}{3}\frac {\partial p}{\partial r}, \quad p = -\frac {1}{{\textit{Ca}}}\frac {1}{r}\frac {\partial }{\partial r}\left (\!r\frac {\partial h}{\partial r}\!\right )\!, \quad\! 4\frac {\partial h}{\partial t}+\frac {1}{r}\frac {\partial }{\partial r}\left (\textit{rh} \bar {u} \right )+J= 0 \quad \mbox{for} \quad 0\lt r,\;t\lt 1, \end{equation}

where the depth-averaged radial velocity $\bar {u}=\bar {u}(r,t)$ is given by

(2.20) \begin{equation} \bar {u} = \frac {1}{h}\int _0^h u{\,\mathrm{d}} z \end{equation}

and the capillary number ${\textit{Ca}}$ is

(2.21) \begin{equation} {\textit{Ca}} = \frac {\tilde \mu \tilde U}{\delta ^3\tilde \sigma }. \end{equation}

As discussed in detail in § 2.2 and in previous studies (e.g. Moore et al. Reference Moore, Vella and Oliver2021), for a wide variety of real-world applications, the pertinent limit is that in which ${\textit{Ca}}\ll 1$ . Hence, to leading order in an asymptotic expansion as ${\textit{Ca}}\rightarrow 0$ , we find that

(2.22) \begin{equation} h(r,t) = \frac {\theta (t)}{2}(1-r^2), \quad \bar {u}(r,t) = \frac {4}{\pi \theta (t) r}\left (\frac {1}{\sqrt {1-r^2}}-(1-r^2)\right )\!, \end{equation}

where $\theta (t) = \theta _0(1-t)$ is the contact angle of the droplet in the thin-film limit, $\theta _0=\theta (0)=4/\pi$ is the initial contact angle of the droplet and the corresponding capillary pressure is $p(t) = 2\theta (t)/{\textit{Ca}}$ . Note that $\theta (1) = 0$ corresponds to the droplet lifetime. One notable property of (2.22) is that time is separable in both the free surface and depth-averaged velocity, the latter of which means that pathlines coincide with streamlines. Further, we see that time enters purely through the contact angle $\theta (t)$ : sometimes it is useful to exploit this fact and use $\theta$ as our effective time variable. We state in the following where we use this property.

With the flow and droplet shape to hand, the dimensionless advection–diffusion equation (2.5) becomes

(2.23) \begin{equation} 4\frac {\partial \phi }{\partial t}+u\frac {\partial \phi }{\partial r}+w\frac {\partial \phi }{\partial z}=\frac {1}{{\textit{Pe}}}\left [ \frac {1}{r}\frac {\partial }{\partial r}\left (r\frac {\partial \phi }{\partial r}\right ) +\frac {1}{\delta ^2}\frac {\partial ^2\phi }{\partial z^2}\right ] \quad \mbox{for} \quad 0\lt r,\ t\lt 1,\ 0\lt z\lt h, \end{equation}

where the Péclet number that represents the relative importance of advective flux and diffusive flux of particles is defined by

(2.24) \begin{equation} {\textit{Pe}}=\frac {\tilde R \tilde U}{\tilde D_{{s}}}. \end{equation}

In this study, we concentrate on the regime in which ${\textit{Pe}}\gtrsim 1$ , which is consistent with a wide range of different applications (see § 2.2 and Moore et al. (Reference Moore, Vella and Oliver2021)). In particular, this means that on the droplet scale, the colloidal diffusive flux is small compared with the advective flux, except in a small region near the pinned contact line, as we discuss shortly.

The dimensionless no-penetration condition on the free surface (2.6) is given by

(2.25) \begin{equation} \left (1+\delta ^2 \left (\dfrac {\partial h}{\partial r}\right )^2\right )^{-1/2}\left (\frac {\partial \phi }{\partial z}-\delta ^2 \frac {\partial h}{\partial r} \frac {\partial \phi }{\partial r}\right )=\delta ^2 {\textit{Pe}}\, \phi J \quad \mbox{on} \quad z = h, \end{equation}

while the substrate condition (2.7) becomes

(2.26) \begin{equation} \frac {\partial \phi }{\partial z}=\delta ^2 {\textit{Da}} \, \phi \quad \mbox{on} \quad z = 0, \end{equation}

where the importance of substrate adsorption is encoded in the Damköhler number defined by

(2.27) \begin{equation} {\textit{Da}}=\frac {\tilde R \tilde {k}_{{a}}}{\delta \tilde {D}_{{s}} }, \end{equation}

with ${\textit{Da}}\ll 1$ ( ${\textit{Da}}\gg 1$ ) indicating that substrate adsorption is negligible (dominant) compared with diffusive colloid transport. Finally, the initial condition is given by

(2.28) \begin{equation} \phi (r,z,0) = 1. \end{equation}

We seek an asymptotic expansion of the transport problem of the form

(2.29) \begin{equation} \phi =\phi _0+\delta ^2 \phi _1+\cdots \quad \mbox{as} \quad \delta \rightarrow 0. \end{equation}

Assuming that ${\textit{Pe}}, {\textit{Da}} \ll \delta ^{-2}$ , at leading order we find

(2.30) \begin{equation} \frac {\partial ^2\phi _0}{\partial z^2}=0 \quad \mbox{for} \quad 0\lt r,\; t\lt 1, \; 0\lt z\lt h, \end{equation}

subject to homogeneous Neumann boundary conditions on $z = 0, h(r,t)$ . Thus, to leading order, the volume fraction is independent of depth, so that $\phi _0=\phi _0(r,t)$ .

Treating both the Péclet and Damköhler numbers as formally $O(1)$ as $\delta \rightarrow 0$ , the $O(\delta ^2)$ -correction to the volume fraction satisfies

(2.31) \begin{equation} 4\frac {\partial \phi _0}{\partial t}+u\frac {\partial \phi _0}{\partial r}=\frac {1}{{\textit{Pe}}}\left [ \frac {1}{r}\frac {\partial }{\partial r}\left (r\frac {\partial \phi _0}{\partial r} \right )+\frac {\partial ^2\phi _1}{\partial z^2} \right ] \quad \mbox{for} \quad 0\lt r,\; t\lt 1, \; 0\lt z\lt h, \end{equation}

subject to

(2.32) \begin{equation} \frac {\partial \phi _1}{\partial z} = {\textit{Da}}\, \phi _0 \quad \mbox{on} \quad z = 0; \quad \frac {\partial \phi _1}{\partial z} = \frac {\partial h}{\partial r}\frac {\partial \phi _0}{\partial r}+{\textit{Pe}}\, \phi _0 J \quad \mbox{on} \quad z = h(r,t). \end{equation}

Whence, upon integrating (2.31) across the droplet and employing the lubrication equations (2.19), we deduce

(2.33) \begin{equation} \frac {\partial }{\partial t}(h\phi _0)+\frac {1}{4r}\frac {\partial }{\partial r}\left [r\left (h\phi _0\bar {u}-\frac {1}{{\textit{Pe}}}h\frac {\partial \phi _0}{\partial r}\right )\right ]+\phi _0\frac {{\textit{Da}}}{4{\textit{Pe}}}=0 \end{equation}

for $0\lt r,\ t\lt 1$ . This must be solved subject to the condition of initial uniform distribution of solute:

(2.34) \begin{equation} \phi _0(r,0) = 1, \end{equation}

as well as relevant boundary conditions in the radial direction. The pertinent choice on physical grounds is to specify no-flux boundary conditions through the axis of symmetry and the contact line, so that

(2.35) \begin{equation} r\left (h\phi _0\bar {u}-\frac {1}{{\textit{Pe}}}h\frac {\partial \phi _0}{\partial r}\right ) = 0 \quad \mbox{at} \quad r = 0,1. \end{equation}

After non-dimensionalisation, the corresponding growth in the mass of particles on the substrate is given by $m_s = m_{s,0} + o(1)$ as $\delta \rightarrow 0$ , where

(2.36) \begin{equation} \frac {\partial m_{s,0}}{\partial t} = \frac {{\textit{Da}}}{4{\textit{Pe}}} \phi _0, \quad m_{s,0}(r,0) = 0. \end{equation}

Finally, integrating (2.33) and (2.36) over the footprint of the droplet, applying the conditions (2.34)–(2.35) and summing yields global conservation of the particle mass:

(2.37) \begin{equation} 1 = 2\pi \int _{0}^{1} \textit{rh}\phi _0 \,{\mathrm{d}} r + 2\pi \int _{0}^{1} {\textit{rm}}_{s,0}\,{\mathrm{d}} r, \end{equation}

where the term on the left-hand side gives the total initial particle mass and the integrals on the right-hand side give, respectively, the total mass of particles remaining in suspension and adsorbed onto the substrate.

Henceforth, we drop the ‘ $0$ ’ subscript notation denoting the leading-order variables for brevity.

2.2. Parameter estimation

The remainder of our analysis focuses on solutions to (2.33)–(2.37). Naturally, this encompasses a wide parameter space spanned by the Péclet and Damköhler numbers. To that end, we first discuss realistic ranges for these numbers and focus our attention on these regimes. We also revisit some previous assumptions made herein, namely that  $\delta , \; {\textit{Ca}}, \; {{Bo}}\ll 1$ .

A limiting factor in this endeavour is available data for the substrate adsorption velocity  $\tilde {k}_{{a}}$ . Adsorption kinetics is a complex topic in its own right and may depend on factors such as electrostatic interactions; particle geometry and properties; and intermolecular forces. As a result, reliable experimental data can be difficult to obtain. In this section, we discuss some of the pertinent literature in the context of the present mathematical model.

First, we discuss how we may estimate the substrate adsorption velocity from experimental data in the literature. Perhaps the simplest theory available for estimating the adsorption rate of particles from fluid flow onto a ‘collector’ (i.e. a substrate) is the Smoluchowski–Levich approximation which assumes that, in the absence of colloidal and hydrodynamic interactions, deposition is controlled by convective–diffusive transport. The substrate adsorption velocity may then be expressed as

(2.38) \begin{equation} \tilde {k}_{{a}}=\dfrac {\tilde {D}_{{s}}}{\tilde {\delta }_D} \end{equation}

(see e.g. Adamczyk Reference Adamczyk2019), where $\tilde {\delta }_D$ is the thickness of the diffusive boundary layer close to the collector. In general, $\tilde {\delta }_D$ depends on the collector/flow configuration and is inversely proportional to the dimensionless mass transfer Sherwood number which has been estimated for a variety of ideal collectors (i.e. assuming the collector is a perfect sink) in different flow configurations (see e.g. Adamczyk et al. Reference Adamczyk, Dabros, Czarnecki and Van De Ven1983; Elimelech Reference Elimelech1994). For the present analysis of an evaporating droplet we assume that the collector/flow configuration is that of a spherical collector in uniform flow, and so the substrate adsorption velocity may then be estimated by

(2.39) \begin{equation} \tilde {k}_{{a}}\approx \dfrac {(\delta \tilde {U})^{1/3}\tilde {D}_{{s}}^{2/3}}{\tilde {R}^{2/3}} \end{equation}

(see e.g. Adamczyk et al. Reference Adamczyk, Dabros, Czarnecki and Van De Ven1983; Adamczyk Reference Adamczyk2002), where $\tilde {U}$ is the characteristic radial velocity within the droplet from (2.18) and $\delta \tilde {U}$ is therefore the approach velocity of a particle towards the substrate.

However, as previously mentioned, the Smoluchowski–Levich approximation neglects several factors, including particle–substrate repulsion forces. When there exists an energy barrier between the particle and the substrate an alternative theory may be used to estimate $\tilde {k}_{{a}}$ , namely interaction-force boundary-layer theory (see e.g. Spielman & Friedlander Reference Spielman and Friedlander1974; Adamczyk Reference Adamczyk2002; Adamczyk et al. Reference Adamczyk, Siwek, Weroński and Jaszczółt2003), sometimes also called surface boundary-layer theory. Specifically, assuming that substrate adsorption is a linear and irreversible process, the constant rate of adsorption between a spherical particle and a planar substrate may be approximated by the expression

(2.40) \begin{equation} \tilde {k}_{{a}}=\tilde {D}_{{s}}\left (\int _{\tilde \delta _{{m}}}^{\infty }{\left (e^{\tilde {V}_{{ t}}(\tilde d)/(\tilde k_{{B}}\tilde T)}-1\right )f(\tilde d)\,\text{d}\tilde d}\right )^{-1}\!, \end{equation}

where $\tilde V_{{t}}(\tilde d)$ , $\tilde k_{{B}}$ , $\tilde T$ , $f(\tilde d)$ , $\tilde d$ and $\tilde \delta _{{m}}$ denote the total interaction energy potential between the particle and the substrate, the Boltzmann constant, temperature, the hydrodynamic correction function, the distance between the particle and the substrate and the location of the primary minimum of the net interaction energy potential, respectively. In particular, the hydrodynamic correction function may be approximated by $f(\tilde d)=1+\tilde r_{{p}}/\tilde d$ (Adamczyk Reference Adamczyk2002), where $\tilde r_{{p}}$ is the radius of the particle, and which describes the resistance that arises due to the proximity of a particle to the substrate and therefore acts to reduce the particle diffusion coefficient relative to the bulk diffusivity of particles in the fluid, $\tilde {D}_{{s}}$ . The net interaction energy potential $\tilde V_{{t}}$ between a particle and the substrate may depend on a variety of long- and short-range interactions that balance attractive and repulsive forces. Here we follow Zigelman & Manor (Reference Zigelman and Manor2018a ) and Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010) by adopting Derjaguin–Landau–Verwey–Overbeek (DLVO) theory in which we assume that the dominant forces at play are electrostatic and van der Waals (attractive) forces. The net interaction energy potential may then be expressed as $\tilde V_{{t}}=\tilde V_{{el}}+\tilde V_{{v{\rm d}W}}$ with

(2.41) \begin{align} \tilde V_{{el}}=64\pi \tilde \varepsilon _0\tilde \varepsilon \left (\dfrac {\tilde k_{{B}} \tilde T}{\tilde e}\right )^2\tanh {\left (\dfrac {\tilde {\mathrm{e}} \tilde \psi _{{p}}}{4\tilde k_{{B}} \tilde T}\right )}\tanh {\left (\dfrac {\tilde {\mathrm{e}}\tilde \psi _{{s}}}{4\tilde k_{{B}}\tilde T}\right )}e^{-\tilde \kappa \tilde d} \quad \text{and} \quad \tilde V_{{v{\rm d}W}}=-\dfrac {\tilde A\tilde r_{{p}}}{6 \tilde d}\\[-20pt]\nonumber \end{align}

(see e.g. Zigelman & Manor Reference Zigelman and Manor2018a ; Bridonneau et al. Reference Bridonneau, Zhao, Battaglini, Mattana, Thévenet, Noël, Roché, Zrig and Carn2020), where $\tilde \varepsilon _0\tilde \varepsilon$ , $\tilde \kappa$ , $\tilde e$ , $\tilde \psi _p$ , $\tilde \psi _s$ and $\tilde A$ are the total permittivity of the liquid, the reciprocal of the Debye length, the elementary charge, the surface potential of the particle, the surface potential of the substrate and the Hamaker constant, respectively. With the Smoluchowski–Levich approximation and the interaction-force boundary-layer theory descriptions in hand, we are ready to estimate the substrate adsorption velocity from experimental data in the literature.

First, we consider the experimental data of Bridonneau et al. (Reference Bridonneau, Zhao, Battaglini, Mattana, Thévenet, Noël, Roché, Zrig and Carn2020) in which the deposition of silica nanoparticles from water droplets evaporating on gold-coated silicon wafer substrates is studied. In particular, the initial particle concentration is varied across four orders of magnitude, and the charges of the particles and the substrate are altered through different coatings and/or treatments prior to evaporation to study the effect of particle–substrate interaction on the deposit. Specifically, they found that the deposit shape remained the same regardless of the particle–substrate interaction in the studied range of DLVO forces, indicating that the aforementioned Smoluchowski–Levich approximation is appropriate to determine the substrate adsorption velocity for this study.

We look to determine our model parameters for two different experiments, specifically for droplets containing $1\,\text{g}\,\text{l}^{-1}$ of either negatively charged untreated silica nanoparticles or positively charged aluminium-coated silica nanoparticles evaporating on an oppositely charged gold substrate, respectively. These cases have been chosen since, whilst Bridonneau et al. (Reference Bridonneau, Zhao, Battaglini, Mattana, Thévenet, Noël, Roché, Zrig and Carn2020) found that the qualitative shape of the deposit remained the same regardless of the particle–substrate interaction, oppositely charged particles and substrates are the ideal configuration to use the Smoluchowski–Levich approximation since repulsion forces exist at small separations when the particles and the substrate have the same charge. For both experiments, the droplet has an initial volume $\tilde {V}_{\textit{init}}\approx 0.5$ μl. The initial contact angles of the droplets are ${\approx} 60.1^\circ$ and $54.4^\circ$ , from which we may estimate the initial contact radii to be $\tilde {R}\approx 0.79$ and $0.83\,\text{mm}$ , for droplets containing untreated and aluminium-coated silica nanoparticles, respectively. Hence, the aspect ratios of the droplets are $\delta \approx 1.01$ and $0.88$ . While these values of $\delta$ push the realms of the thin-film assumption initially, due to the contact line pinning, the thin droplet assumption becomes stronger as the droplets evaporate and the contact angles decrease. Furthermore, Larsson & Kumar (Reference Larsson and Kumar2022) demonstrate that the lubrication approximation is often a reasonable assumption in regimes where it strictly may not be asymptotically justified. The characteristic radial velocities are estimated to be $\tilde {U}\approx 4.14\times 10^{-7}$ and $4.55\times 10^{-7}\,\text{m}\,\text{s}^{-1}$ . The capillary and Bond numbers are therefore ${\textit{Ca}}\approx 5.12\times 10^{-9}$ and $5.62\times 10^{-9}$ and ${{Bo}}\approx 0.085$ and $0.093$ , respectively, which justifies our assumptions that surface tension dominates viscosity in the liquid flow and that gravity is negligible. The untreated and aluminium-coated silica nanoparticles have radii $\tilde {r}_{{p}}=37.5$ and $45\,\text{nm}$ and so the diffusivity of the particles in water may be determined from the Stokes–Einstein relation to be $\tilde {D}_{{s}}=6.55\times 10^{-12}$ and $5.46\times 10^{-12}\,\text{m}^2\,\text{s}^{-1}$ , respectively, so that the Péclet numbers are then ${\textit{Pe}}\approx 50.0$ and $69.1$ . Finally, the Smoluchowski–Levich approximation described above for the substrate adsorption velocity yields $\tilde {k}_{{a}}\approx 3.06\times 10^{-8}$ and $2.59\times 10^{-8}\,\text{m}\,\text{s}^{-1}$ , and so the Damköhler numbers are ${\textit{Da}}\approx 3.66$ and $4.47$ .

Next, we consider experimental data of Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010) in which the deposition of titania particles from water droplets of different pH values evaporating on glass substrates is studied. In particular, as outlined by Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010), both the electrostatic and van der Waals forces (or the DLVO force) between a particle and the substrate vary with the acidity of the base fluid, and a switch from particle–substrate attraction to repulsion is observed as the pH value of the fluid increases.

We look to determine our model parameters from values provided for two different experiments, namely when $\text{pH}=2.8$ and $\text{pH}=11.7$ . Specifically, when $\text{pH}=2.8$ the particles and the substrate are oppositely charged and the DLVO force is attractive, and when $\text{pH}=11.7$ the particles and the substrate are both negatively charged and the DLVO force is repulsive. The droplets have initial volume $\tilde {V}_{\textit{init}}\approx 5\,\text{nl}$ . The initial contact radii of the droplets are $\tilde {R}\approx 345$ and $410$ μm, from which we may estimate the initial contact angles to be ${\approx} 8.85^\circ$ and $5.29^\circ$ , for the attractive and repulsive cases, respectively. Hence, the aspect ratios of the droplets are $\delta \approx 0.12$ and $0.07$ , which are within our $0\lt \delta \ll 1$ assumption. The characteristic radial velocity scales are estimated to be $\tilde {U} \approx 7.22\times 10^{-6}$ and $1.02\times 10^{-5}\,\text{m}\,\text{s}^{-1}$ . The capillary and Bond numbers for this example are therefore ${\textit{Ca}}\approx 8.93\times 10^{-8}$ and $1.26\times 10^{-7}$ and ${{Bo}}\approx 0.016$ and $0.023$ , respectively. The titania particles have radii $\tilde {r}_{{p}}=12.5\,\text{nm}$ and so the diffusivity of the particles in water may be determined from the Stokes–Einstein relation to be $\tilde {D}_{{s}} \approx 1.96\times 10^{-11}\,\text{m}\,\text{s}^{-1}$ , so that the Péclet numbers are ${\textit{Pe}}\approx 127$ and $213$ . Finally, for the experiment in which $\text{pH}=2.8$ , the DLVO force is attractive and we may use the Smoluchowski–Levich approximation to estimate the substrate adsorption velocity to be $\tilde {k}_{{a}}\approx 1.42\times 10^{-7}\,\text{m}\,\text{s}^{-1}$ , which gives ${\textit{Da}}\approx 20.5$ . For the experiment in which $\text{pH}=11.7$ , the DLVO force is repulsive and we may use interaction-force boundary-layer theory to estimate $\tilde {k}_{{a}}\approx 2.86\times 10^{-8}\,\text{m}\,\text{s}^{-1}$ and hence ${\textit{Da}}\approx 8.23$ .

We now turn to examples in the literature for which the substrate adsorption velocity is explicitly estimated. The present model of substrate adsorption as represented by (2.7) is based on pioneering modelling and experiments on the kinetics of bovine serum albumin (BSA) adsorption and the values for $\tilde {k}_{{a}}$ determined therein are used in the examples studied experimentally and numerically in Siregar (Reference Siregar2012) and Siregar et al. (Reference Siregar, Kuerten and Van der Geld2013). We take the value of $\tilde {k}_{{a}} \approx 2\times 10^{-8}$ m s $^{-1}$ for BSA in a HEPES buffer from Kurrat, Prenosil & Ramsden (Reference Kurrat, Prenosil and Ramsden1997) for a reference value of the substrate adsorption velocity. The diffusivity of BSA is given in Kurrat et al. (Reference Kurrat, Prenosil and Ramsden1997) to be $\tilde {D}_{{s}} \approx 3.26\times 10^{-11}$ m $^2$ s $^{-1}$ .

The first example in Siregar et al. (Reference Siregar, Kuerten and Van der Geld2013) considers a droplet based on data from Hu & Larson (Reference Hu and Larson2002), for an evaporating water droplet with an initial volume $\tilde {V}_{\textit{init}} \approx 0.60$ μl and an initial contact radius $\tilde {R} \approx 1$ mm. The aspect ratio parameter is thus $\delta \approx 0.597$ . The evaporation velocity scale is estimated to be $\tilde {U}\approx 2.19\times 10^{-7}$ m s $^{-1}$ . Hence, the capillary and Bond numbers for this example are ${\textit{Ca}}\approx 1.25\times 10^{-8}$ and ${{Bo}}\approx 0.13$ , respectively, which again justifies our assumptions that surface tension dominates viscosity in the liquid flow and that gravity is negligible. The Péclet number is given by ${\textit{Pe}}\approx 6.72$ , while the Damköhler number is ${\textit{Da}} \approx 1.03$ .

The second example in Siregar et al. (Reference Siregar, Kuerten and Van der Geld2013) is based on evaporation data from Golovko, Butt & Bonaccurso (Reference Golovko, Butt and Bonaccurso2009) and considers a $\tilde {V}_{\textit{init}} = 66.2$ pl water droplet of initial contact radius $\tilde {R} = 37.3$ μm. The evaporation velocity scale is estimated to be $\tilde {U}\approx 7.04\times{} 10^{-6}$ m s $^{-1}$ . The capillary and Bond numbers are therefore ${\textit{Ca}}\approx 4.1\times 10^{-8}$ and ${{Bo}} \approx 1.89\times 10^{-4}$ , justifying the supposition that surface tension dominates the droplet dynamics. On the other hand, the aspect ratio $\delta \approx 1.28$ for this example, which is due to the large initial contact angle ${\approx} 70^\circ$ . The above caveats about the role of $\delta$ pertain once again. The Péclet number is ${\textit{Pe}} \approx 8.05$ , while the Damköhler number is ${\textit{Da}}\approx 1.79\times 10^{-2}$ .

As a final example, we turn to a different colloid, namely polystyrene latex particles, whose adsorption onto mica is studied experimentally in Adamczyk et al. (Reference Adamczyk, Siwek, Weroński and Jaszczółt2003). The authors compare heterogeneous and homogeneous surfaces, with adsorption velocities in the range $\tilde {k}_{{a}} \approx (8.9{-}9.7)\times 10^{-8}$ m s $^{-1}$ , while the diffusion coefficient is estimated from figure 2(b) and equation (2) therein to be $\tilde {D}_{{s}} \approx 1.02\times 10^{-12}$ m $^2$ s $^{-1}$ . For reference, we consider the water droplet from the first example based on the data in Hu & Larson (Reference Hu and Larson2002) containing such particles, leading to ${\textit{Pe}} \approx 219$ and ${\textit{Da}} \approx 154$ .

From the above examples, it is of interest to consider a range of possible Péclet and Damköhler numbers. However, given that we are primarily concerned with the influence of substrate adsorption on coffee-ring formation, we focus our asymptotic analysis on the regime for which ${\textit{Pe}} \gg 1$ , since it is well known that it is the dominance of advection in the droplet bulk and its competition with diffusion near the contact line that leads to the nascent coffee ring (Moore et al. Reference Moore, Vella and Oliver2021) (see § 3). While this assumption is consistent with the majority of the studies referenced previously, there are examples for which the Péclet number approaches order unity, so we return to such cases when considering our numerical results (see § 4).

In terms of the Damköhler number, we note that while this can vary from large to small in the examples above, in each case we have ${\textit{Da}}\lesssim {\textit{Pe}}$ . To this end, we may introduce the parameter $\mathcal{V}$ defined by

(2.42) \begin{equation} \frac {2\mathcal{V}}{\pi } := \frac {{\textit{Da}}}{{\textit{Pe}}} = \frac {\tilde {k}_{{a}}}{\delta \tilde {U}}, \end{equation}

where the coefficient $2/\pi$ has been introduced for notational convenience in the following. This is the ratio between the substrate adsorption velocity $\tilde {k}_{{a}}$ and the vertical velocity across the droplet. Hence, if $\mathcal{V}\lesssim 1$ , we are still in a regime where evaporation is a dominant effect, while if $\mathcal{V}\gg 1$ , substrate adsorption dominates evaporation. Since the latter regime is beyond the scope of the present paper, where our primary interest is in ring formation, we focus on the regime for which ${\textit{Da}}\lesssim {\textit{Pe}}$ henceforth.

In what follows, we perform a matched asymptotic analysis of (2.33)–(2.37) in the advection-dominated regime in § 3 to both gain physical insight and act as a validation tool for our numerical simulations; the numerical methodology is outlined in Appendix A. In the interests of completeness, we briefly discuss the regime in which advection is weaker in § 4.

3.1. Outer problem

In the droplet bulk, the two physical effects influencing the volume fraction distribution are advection due to evaporation and settling due to substrate adsorption. To leading order, the volume fraction satisfies the first-order linear equation

(3.1) \begin{equation} \frac {\partial }{\partial t}(h\phi )+\frac {1}{4r}\frac {\partial }{\partial r}\left (r h\phi \bar {u}\right )+\phi \frac {\mathcal{V}}{2\pi }=0 \quad \mbox{for} \quad 0\lt r,\; t\lt 1, \end{equation}

which can be simplified using the lubrication equation and by switching our time variable to the more convenient contact angle using the transformation $\theta = \theta _0(1-t)$ , finding

(3.2) \begin{equation} \frac {\partial \phi }{\partial \theta } - \frac {\bar {u}}{4\theta _0}\frac {\partial \phi }{\partial r} = -\frac {\phi }{2\pi \theta _0 h}\left (\frac {1}{\sqrt {1-r^2}}-\mathcal{V}\right ) \quad \mbox{for} \quad 0\lt r\lt 1,\; 0\lt \theta \lt \theta _0. \end{equation}

This must be solved subject to the initial condition (2.34). This problem is exactly that studied in D’Ambrosio et al. (Reference D’Ambrosio, Wray and Wilson2025b ) and may be solved using the method of characteristics, yielding

(3.3) \begin{equation} \phi =\left (\frac {\theta }{\theta _0}\right )^{-1/4-\mathcal{V}/8}\frac {\xi }{\xi _0}\left (\frac {1-\xi }{1-\xi _0}\right )^{\mathcal{V}/2}\mbox{exp}\left [\frac {\mathcal{V}}{\sqrt {3}}\mbox{arctan}\left (\frac {\sqrt {3}(\xi _0-\xi )}{2\xi \xi _0+\xi +\xi _0+2}\right )\right ]\!, \end{equation}

where

(3.4) \begin{equation} \xi (r,\theta ) = \left (1 - \left (\frac {\theta }{\theta _0}\right )^{3/4}\big(1-\xi _0^3\big)\right )^{1/3}, \quad \xi _0 (r)= \sqrt {1-r^2}. \end{equation}

In particular, we note that

(3.5) \begin{equation} \phi (r,\theta )\sim \dfrac {\gamma (\theta )}{\sqrt {1-r}} \quad \text{as} \quad 1-r\to 0^{+} , \end{equation}

where

(3.6) \begin{equation} \gamma (\theta ) = \left (\!\frac {\theta }{\theta _0}\!\right )^{-1/4-\mathcal{V}/8}\frac {\zeta (\theta )(1-\zeta (\theta ))^{\mathcal{V}/2}}{\sqrt {2}}\mbox{exp}\!\left (\!-\frac {\mathcal{V}}{\sqrt {3}}\mbox{arctan}\!\left (\!\frac {\sqrt {3}\zeta (\theta )}{\zeta (\theta )+2}\!\right )\!\right )\!, \!\quad\! \zeta (\theta ) = \xi (1,\theta ). \end{equation}

Hence, the leading-order outer volume fraction becomes unbounded local to the contact line, necessitating a reconsideration of particle diffusion in a boundary layer.

We can use the expression for the behaviour of the volume fraction as $1-r\rightarrow 0^+$ , from (3.5) and (3.6), to calculate the accumulated mass flux into the contact line, $\mathcal{M}(\theta )$ , as given by

(3.7) \begin{equation} \mathcal{M}(\theta ) = -\frac {1}{4\theta _0}\int _{\theta _0}^{\theta } \left .(rh\bar {u}\phi )\right |_{r=1^{-}}{\,\mathrm{d}} \bar {\psi } = \frac {\chi }{2\theta _0}\int _{\theta }^{\theta _0} \gamma (\psi )\,{\mathrm{d}}\psi , \end{equation}

where $\chi = \sqrt {2}/\pi$ is the coefficient of the inverse square-root singularity in the diffusive evaporative flux, which arises from the local velocity expansion $\bar {u} \sim 2\chi /\theta \sqrt {1-r}$ as $1-r\rightarrow 0^+$ . We have introduced this notation to be consistent with previous analyses of the nascent coffee ring (Moore, Vella & Oliver Reference Moore, Vella and Oliver2022; Moore & Wray Reference Moore and Wray2023), in which the equivalent accumulated mass flux into the contact line in the absence of substrate adsorption is

(3.8) \begin{equation} \mathcal{M}_{\textit{MVO}}(\theta ) = \frac {\sqrt {2}\chi }{4}\left (1-\left (\frac {\theta }{\theta _{0}}\right )^{3/4}\right )^{4/3}\!. \end{equation}

We compare (3.7) and (3.8) in figure 2, where we see that the presence of substrate adsorption decreases $\mathcal{M}(\theta )$ relative to $\mathcal{M}_{\textit{MVO}}(\theta )$ , with mass now lost to the system via deposition. As the relative importance of substrate adsorption increases, this effect is amplified. We also show on the graph the small-time asymptotic form of $\mathcal{M}(\theta )$ for $\mathcal{V}\gt 0$ , namely

(3.9) \begin{align} \mathcal{M}(\theta ) \sim \frac {3^{4/3}\chi }{2^{25/6}\theta _0^{4/3}}(\theta _0-\theta )^{4/3}\left [1 - \frac {3^{1/3}2^{4/3}\mathcal{V}}{5\theta _0^{1/3}}(\theta _0-\theta )^{1/3} + \frac {\mathcal{V}^2}{3^{1/3}2^{4/3}\theta _0^{2/3}}(\theta _0-\theta )^{2/3}\right ]\\[-20pt]\nonumber \end{align}

as $\theta _0-\theta \rightarrow 0^+$ and the late-time expansion

(3.10) \begin{equation} \mathcal{M}(\theta ) \sim \frac {\chi }{2\theta _0}\left [\int _{0}^{\theta _0}\gamma (\psi )\,{\mathrm{d}} \psi - \frac {2\sqrt {2}e^{-\mathcal{V}\pi /6\sqrt {3}}}{3^{\mathcal{V}/2}(\mathcal{V}+6 )}\left (\frac {\theta }{\theta _0}\right )^{(\mathcal{V}+6)/8}\right ]\quad \mbox{as} \quad \theta \rightarrow 0^+. \end{equation}

Figure 2. Accumulated mass flux into the contact line $\mathcal{M}(\theta )$ for $\mathcal{V} = 0.25, 1, 4, 8$ (greyscales) and the equivalent coefficient in the no-substrate adsorption problem $\mathcal{M}_{\textit{MVO}}(\theta )$ (blue). The dashed red curves represent the small-time and large-time limits (3.9) and (3.10).

The reduction in the flux of particles to the contact line due to the mass adsorbed onto the substrate is given by integrating (2.36) to leading order in the bulk, yielding

(3.11) \begin{equation} m_s(r,\theta ) = \frac {\mathcal{V}}{2\pi \theta _0}\int _{\theta }^{\theta _0} \phi (r,\psi ) \,{\mathrm{d}} \psi . \end{equation}

Notably, as we approach the contact line, this is unbounded, with

(3.12) \begin{equation} m_s(r,\theta ) \sim \left (\frac {\mathcal{V}}{2\pi \theta _0}\int _{\theta }^{\theta _0} \gamma (\psi ) \,{\mathrm{d}} \psi \right )\frac {1}{\sqrt {1-r}} = \frac {\mathcal{V}\mathcal{M}(\theta )}{\pi \chi \sqrt {1-r}} \quad \mbox{as} \quad 1-r\rightarrow 0^+. \end{equation}

With reference to figure 2, it is worth stressing that the transition from $\mathcal{V} = 0$ to $\mathcal{V}\gt 0$ appears to be regular in terms of the accumulated mass flux: indeed, an asymptotic expansion of the outer solution in the limit $\mathcal{V} \rightarrow 0$ retrieves the equivalent outer solution without substrate adsorption (see e.g. Deegan et al. Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten2000; Moore et al. Reference Moore, Vella and Oliver2021). Formally, perturbations due to substrate adsorption are dominant over colloidal diffusion in the bulk provided that $\mathcal{V} \gg {\textit{Pe}}^{-1}$ as ${\textit{Pe}}\rightarrow \infty$ . If $\mathcal{V}\ll {\textit{Pe}}^{-1}$ , perturbations due to diffusion dominate those due to substrate adsorption in the droplet bulk, which is beyond the scope of the present analysis.

3.2. Inner region

We now turn to the boundary layer near the contact line. Note that we continue to use the contact angle as our time variable in this section. Following Moore et al. (Reference Moore, Vella and Oliver2021), we introduce local variables

(3.13) \begin{equation} r = 1 - \frac {R}{{\textit{Pe}}^2}, \quad \bar {u} = {\textit{Pe}} \bar {U}, \quad h = \frac {H}{{\textit{Pe}}^2}, \quad \phi = {\textit{Pe}}^b \varPhi , \quad m_s = {\textit{Pe}}^b M_s, \end{equation}

where $b\gt 0$ is to be determined from conservation of mass considerations shortly. Then, to leading order as ${\textit{Pe}}\rightarrow \infty$ , the dominant terms in (2.33) are advection and diffusion, so that

(3.14) \begin{equation} \frac {\partial }{\partial R} \left (\bar {U}\!H\kern-0.5pt\varPhi + H\frac {\partial \varPhi }{\partial R}\right ) = 0 \quad \mbox{for} \quad R\gt 0, \; 0\lt \theta \lt \theta _0, \end{equation}

subject to the leading-order no-flux condition at the contact line,

(3.15) \begin{equation} \bar {U}\!H\kern-0.5pt\varPhi + H\frac {\partial \varPhi }{\partial R} = 0 \quad \mbox{on} \quad R=0, \, 0\lt \theta \lt \theta _0, \end{equation}

together with the leading-order forms of the free surface and velocity as found from (2.22).

The leading-order boundary-layer solution is thus

(3.16) \begin{equation} \varPhi (R,\theta ) = A(\theta )\mbox{exp}\left (-\frac {4\chi }{\theta }\sqrt {R}\right )\!, \end{equation}

where $A(\theta )$ must be determined by matching. The exponential form of the solution is exactly the same as that presented in Moore et al. (Reference Moore, Vella and Oliver2022) in the no-substrate adsorption regime: the difference between the two solutions will solely come from the coefficient $A(\theta )$ , as we discuss further shortly.

Having determined the leading-order inner volume fraction, the leading-order adsorbed mass is then given by

(3.17) \begin{equation} M_s(R,\theta ) = \frac {\mathcal{V}}{2\pi \theta _0}\int _{\theta }^{\theta _0} A(\psi )\mbox{exp}\left (-\frac {4\chi }{\psi }\sqrt {R}\right )\,{\mathrm{d}} \psi . \end{equation}

3.2.1. Matching using the conservation of particle mass

In order to match between the leading-order outer and inner solutions, we must now turn to conservation of particle mass (2.37). Each of the integrals on the right-hand side of (2.37) may be split into contributions from the outer and inner solutions by introducing a parameter $0\lt {\textit{Pe}}^{-2}\ll \varepsilon \ll 1$ . For example, the suspension mass may be written as

(3.18) \begin{align} \int _{0}^{1} (rh\phi ) (r,\theta )\,{\mathrm{d}} r = \int _{0}^{1-\varepsilon } (rh\phi )_{\textit{outer}}(r,\theta ) \,{\mathrm{d}} r + {\textit{Pe}}^{b-4} \int _{0}^{\varepsilon {\textit{Pe}}^2} \!\!\left (\!1-\frac {R}{{\textit{Pe}}^2}\!\right )\! (H\kern-0.5pt\varPhi)(R,\theta )\,{\mathrm{d}} R,\\[-20pt]\nonumber \end{align}

where $\phi _{\textit{outer}}$ is given by (3.3). Then, taking the limit $\varepsilon {\textit{Pe}}^2\rightarrow \infty$ and ${\textit{Pe}}\rightarrow \infty$ , we find to leading order

(3.19) \begin{align} \int _{0}^{1} (rh\phi )(r,\theta )\,{\mathrm{d}} r = \int _{0}^{1} (rh\phi )_{\textit{outer}}(r,\theta ) \,{\mathrm{d}} r + {\textit{Pe}}^{b-4} \int _{0}^{\infty } (H\kern-0.5pt\varPhi) (R,\theta )\,{\mathrm{d}} R.\\[-20pt]\nonumber \end{align}

A similar argument for the adsorbed mass yields

(3.20) \begin{equation} \int _{0}^{1} {\textit{rm}}_s(r,\theta )\,{\mathrm{d}} r = \int _{0}^{1} ({\textit{rm}}_s)_{\textit{outer}}(r,\theta ) \,{\mathrm{d}} r + {\textit{Pe}}^{b-2} \int _{0}^{\infty } M_s(R,\theta )\,{\mathrm{d}} R, \end{equation}

where $m_{s,{outer}}$ is given by (3.11).

Upon substituting these into (2.37), we have

(3.21) \begin{align} \displaystyle \frac {1}{2\pi } = \displaystyle \int _{0}^{1} \!(rh\phi + {\textit{rm}}_s)_{\textit{outer}}(r,\theta ) \,{\mathrm{d}} r + {\textit{Pe}}^{b-4} \!\int _{0}^{\infty } \!(\!H\kern-0.5pt\varPhi \!) (\!R,\theta )\,{\mathrm{d}} R + {\textit{Pe}}^{b-2} \!\int _{0}^{\infty }\! M_s(\!R,\theta )\,{\mathrm{d}} R.\\[-20pt]\nonumber \end{align}

Now, since some mass is advected to the contact line (cf. (3.7)), for conservation of mass to be satisfied to leading order as ${\textit{Pe}}\rightarrow \infty$ , we cannot only consider the outer region and must include one of the inner-region terms. Here, the correct dominant balance is to choose $b = 2$ , so that leading-order conservation of mass gives

(3.22) \begin{equation} \frac {1}{2\pi } = \int _{0}^{1} (rh\phi + {\textit{rm}}_s)_{\textit{outer}}(r,\theta ) \,{\mathrm{d}} r + \int _{0}^{\infty } M_s(R,\theta )\,{\mathrm{d}} R. \end{equation}

This is different from the equivalent formulation without substrate adsorption as discussed in Moore et al. (Reference Moore, Vella and Oliver2021, Reference Moore, Vella and Oliver2022), where we must take $b = 4$ . Substrate adsorption thus acts to reduce the size of the volume fraction in the boundary layer in the present regime.

If we differentiate (3.22) with respect to $\theta$ , we find that

(3.23) \begin{equation} 0 = -\frac {1}{4}\int _{0}^{1} \frac {\partial }{\partial r} \left ((rh\phi \bar {u})_{\textit{outer}}\right ) \,{\mathrm{d}} r + \frac {\mathcal{V}}{2\pi } \int _{0}^{\infty } \varPhi (R,\theta )\,{\mathrm{d}} R, \end{equation}

where we have used the governing advection equation for the outer solution and the inner adsorption equation. Whence, utilising the no-flux condition at the droplet centre and the known solution for $\varPhi (R,\theta )$ given in (3.16), we deduce that

(3.24) \begin{equation} A(\theta ) = -\frac {16\pi }{\mathcal{V}}\frac {\theta _0\chi ^2}{\theta ^2}\frac {{\,\mathrm{d}}\mathcal{M}}{{\,\mathrm{d}}\theta } = \frac {8\pi \chi ^3\gamma (\theta )}{\mathcal{V}\theta ^2}. \end{equation}

The expression (3.24) has been presented in a slightly unusual form to highlight where physical quantities in the system such as $\mathcal{M}(\theta )$ , $\chi$ and $\theta$ naturally occur. This is for ease of interpreting the results and for comparison with other regimes where substrate adsorption may be more or less important, or indeed, comparisons with more general geometries.

We display the coefficient $A(\theta )$ in figure 3, where we also include for reference the result for the zero-substrate adsorption regime in Moore et al. (Reference Moore, Vella and Oliver2021, Reference Moore, Vella and Oliver2022), which is

(3.25) \begin{equation} A_{\textit{MVO}}(\theta ) = \frac {64\chi ^4\mathcal{M}_{\textit{MVO}}(\theta )}{3\theta ^5}, \end{equation}

where $\mathcal{M}_{\textit{MVO}}(\theta )$ is given by (3.8). We note there are substantial differences for this coefficient in the presence of substrate adsorption: indeed, there is an increase in $A(\theta )$ relative to $A_{\textit{MVO}}(\theta )$ at the beginning of evaporation. This effect is reduced as $\mathcal{V}$ increases. As the droplet evaporates, the reduction of accumulated mass flux into the inner region from the droplet bulk leads to a substantial reduction (by several orders of magnitude at the later stages of evaporation) in $A(\theta )$ compared to (3.25).

Figure 3. The coefficient $A(\theta )$ for $\mathcal{V} = 0.25, 1, 4, 8$ (greyscales) and the equivalent coefficient in the no-substrate adsorption problem $A_{\textit{MVO}}(\theta )$ (blue). The dashed red curves represent the small-time and large-time limits (3.26) and (3.27).

These effects can be seen more clearly in the early- and late-time behaviour of the coefficient. In the early-time limit, we have

(3.26) \begin{align} A(\theta ) \sim \frac {8\pi \chi ^3}{\mathcal{V}\theta _0^{7/3}}\frac {3^{1/3}}{2^{7/6}}(\theta _0-\theta )^{1/3}\left [1-\frac {3^{1/3}\mathcal{V}}{2^{2/3}\theta _0^{1/3}}(\theta _0-\theta )^{1/3} + \frac {3^{2/3}V^2}{2^{7/3}\theta _0^{2/3}}(\theta _0-\theta )^{2/3}\right ]\\[-20pt]\nonumber \end{align}

as $\theta _0-\theta \rightarrow 0^+$ . In particular, we see that $A(\theta )\rightarrow 0$ as $\theta _0-\theta \rightarrow 0^+$ , retrieving the physically anticipated solution that the initial mass of colloid in the boundary layer is identically zero. Meanwhile, in the late-time limit, we find that

(3.27) \begin{equation} A(\theta ) \sim \frac {8\pi \chi ^3}{\mathcal{V}\theta _0^2}\left (\frac {e^{-\mathcal{V}\pi /6\sqrt {3}}}{3^{\mathcal{V}/2}\sqrt {2}}\right )\left (\frac {\theta }{\theta _0}\right )^{(\mathcal{V}-9)/4}\left [1 + \frac {(2\mathcal{V}-3)}{9}\left (\frac {\theta }{\theta _0}\right )^{3/4}\right ] \quad \mbox{as} \quad \theta \rightarrow 0^+. \end{equation}

These asymptotes are both indicated alongside the full solution for $A(\theta )$ in figure 3.

It is worth noting that the transition from $\mathcal{V} = 0$ to $\mathcal{V}\gt 0$ is not continuous in the coefficient $A(\theta )$ . Indeed, when $\mathcal{V} = O({\textit{Pe}}^{-1})$ as ${\textit{Pe}}\rightarrow \infty$ , the dominant balance in the matching procedure using global conservation of particle mass is different from that described above and we revisit this in § 3.6.

We note here that we can apply similar conservation of particle mass arguments for different evaporation models – specifically the one-sided model – as well, with similar findings in the reduction of the size of the nascent coffee ring. We discuss this in Appendix B.

3.3. Composite profiles

To view the particle distribution beyond the growing coffee ring, we must consider the droplet as a whole. However, we are unable to directly compute a leading-order composite profile for the volume fraction $\phi$ or the adsorbed mass $m_s$ due to their inverse square-root singularities at the contact line, which may not be matched to the exponentially decaying solutions in the boundary layer.

To circumvent the algebraic complexity of proceeding to higher order, we may follow Moore et al. (Reference Moore, Vella and Oliver2022) and introduce an intermediate region using the scalings

(3.28) \begin{equation} r = 1 - \frac {\mathring {r}}{{\textit{Pe}}^{k}}, \quad h = \frac {\mathring {H}}{{\textit{Pe}}^{k}}, \quad \bar {u} = {\textit{Pe}}^{k/2}\mathring {\bar {u}}, \quad \phi = {\textit{Pe}}\mathring {\phi }, \quad m_s = {\textit{Pe}}\mathring {m}_s \end{equation}

in (2.33)–(2.36), where $k\in (1,2)$ is chosen to be smaller than the contact line region, but large enough to enable the neglect of the time derivative terms in the advection–diffusion equation (2.33).

Under these scalings, retaining both advection and diffusion terms in (2.33), we find that

(3.29) \begin{equation} \frac {\partial }{\partial \mathring {r}}\left [\frac {2\chi \sqrt {\mathring {r}}}{{\textit{Pe}}^{k/2-1}\theta }\mathring {\phi } + \mathring {r}\frac {\partial \mathring {\phi }}{\partial \mathring {r}}\right ] = 0, \end{equation}

which yields the solution

(3.30) \begin{equation} \mathring {\phi }(\mathring {r},\theta ) = \mbox{e}^{-4\chi \sqrt {\mathring {r}}{\textit{Pe}}^{1-k/2}/\theta }\left [{\textit{Pe}} A(\theta ) + \frac {4\chi \gamma (\theta )}{\theta }\mbox{Ei}\left (\frac {4\chi \sqrt {\mathring {r}}{\textit{Pe}}^{1-k/2}}{\theta }\right )\right ]\!, \end{equation}

where $\mathrm{Ei}(x)$ is the exponential integral and the constants of integration have been chosen to match to the inner solution (3.16) as $\mathring {r}\rightarrow 0$ and the local expansion of the outer solution (3.5) as $\mathring {r}\rightarrow \infty$ . The corresponding solution for the intermediate adsorbed mass is given by

(3.31) \begin{equation} \mathring {m}_s(\mathring {r},t) = \frac {\mathcal{V}}{2\pi \theta _0}\int ^{\theta _0}_{\theta }\mathring {\phi }(\mathring {r},\psi )\,{\mathrm{d}} \psi . \end{equation}

We can construct a leading-order additive composite solution for the volume fraction using van Dyke’s rule:

(3.32) \begin{align} \displaystyle \phi _{\textit{comp}}(r,\theta ) & = \displaystyle \phi (r,\theta ) + {\textit{Pe}}\mathring {\phi }(r,\theta ) + {\textit{Pe}}^{2}\varPhi (r,\theta ) - \frac {\gamma (\theta )}{\sqrt {1-r}} \displaystyle - \mbox{exp}\left (-\frac {4\chi {\textit{Pe}}\sqrt {(1-r)}}{\theta }\right ) \nonumber\\ &\quad\times \left [{\textit{Pe}}^{2}A(\theta ) + \frac {4\chi \gamma (\theta )}{\theta }\log \left (\frac {4\chi {\textit{Pe}}\sqrt {(1-r)}}{\theta }\right )\right ]\!, \end{align}

where $\phi$ , $\mathring {\phi }$ and $\varPhi$ are the leading-order outer, intermediate and inner solutions given by (3.3)–(3.4), (3.30) and (3.16), respectively; the final term on the first line is the overlap contribution between the outer and intermediate regions; and the terms on the second line account for the overlap between the intermediate and inner regions, and the logarithmic singularity in the exponential integral, respectively.

Similarly, a leading-order additive composite solution for the adsorbed mass is given by

(3.33) \begin{align}& \displaystyle m_{s,\textit{comp}}(r,\theta ) = \displaystyle m_{s}(r,\theta ) + {\textit{Pe}}\mathring {m}_{s}(r,\theta ) + {\textit{Pe}}^{2}M_{s}(r,\theta ) - \frac {\mathcal{V}\mathcal{M}(\theta )}{\pi \chi \sqrt {1-r}} \displaystyle - {\textit{Pe}}\frac {\mathcal{V}}{2\pi \theta _0} \nonumber\\&\quad \times \int _{\theta }^{\theta _0} \mbox{exp}\left (\!-\frac {4\chi {\textit{Pe}}\sqrt {(1-r)}}{\psi }\right )\!\left [\!{\textit{Pe}} A(\psi ) + \frac {4\chi \gamma (\psi )}{\psi }\log \!\left (\!\frac {4\chi {\textit{Pe}}\sqrt {(1-r)}}{\psi }\right )\!\right ]{\mathrm{d}}\psi , \end{align}

where $m_{s}$ , $\mathring {m}_{s}$ and $M_{s}$ are the leading-order outer, intermediate and inner solutions given by (3.11), (3.31) and (3.17), respectively.

In figure 4, we display the composite mass profiles $m_{\textit{comp}} = h\phi _{\textit{comp}}$ , $m_{s,{comp}}$ alongside numerical results from the full advection–diffusion model (2.33)–(2.37) for ${\textit{Pe}} = 219,$ ${\textit{Da}} = 154$ ( $\mathcal{V} \approx 1.1$ ) based on parameters taken from Hu & Larson (Reference Hu and Larson2002) and Adamczyk et al. (Reference Adamczyk, Siwek, Weroński and Jaszczółt2003) as discussed in § 2.2. As the evaporation time increases, we see that the suspension mass in both the droplet bulk and the contact-line region decreases, as more and more particles are adsorbed onto the substrate, leading to an increase in the deposited mass throughout the droplet. Nevertheless, we see that the nascent coffee-ring profile persists in the suspension mass throughout the process, albeit with a gradually decreasing peak with advancing peak location, as illustrated in figure 4(b). This movement of the peak results in notable kinks in the deposited mass profile curves near the contact line, as illustrated in figure 4(d).

Figure 4. The leading-order composite suspension (a,b) and deposited (c,d) mass as a function of time for ${\textit{Pe}} = 219$ , ${\textit{Da}} = 154$ ( $\mathcal{V} \approx 1.1$ ). The initial profile mirrors the droplet shape and is shown as the bold blue line, while profiles at $15\,\%$ intervals of the drying time are shown in increasingly lighter shades of grey. A close-up of the nascent coffee-ring profile is shown in (b,d), whereby we see the characteristic formation of the peak just inside the contact line. The red dashed curves indicate the corresponding numerical solution of (2.33)–(2.37), while the arrows indicate increasing evaporation time.

The deposited mass is relatively uniform in the bulk of the droplet at all shown drying times, only substantially rising as we approach the contact line. The profile is monotonically increasing in $r$ : this mirrors the volume fraction, which is also monotonically increasing. Note that this is an inherent feature of the model herein: given that the flow velocity is increasing with $r$ , the depth-averaged thin-film model necessarily produces a monotonically increasing $\phi$ in the limit ${\textit{Pe}}\rightarrow \infty$ . This may not be the case if finite thickness effects are fully accounted for – indeed the analysis of Widjaja & Harris (Reference Widjaja and Harris2008) suggests both monotonic and non-monotonic profiles for the deposited mass – but this is beyond the scope of the present analysis.

We note that we see excellent agreement between the leading-order composite profiles and the numerical solutions throughout the evaporation time, which acts as a check on the veracity of our asymptotic results. Given that it is challenging to obtain good accuracy in the narrow boundary-layer region numerically, this validation gives us reassurance in using the asymptotic solution in this region.

In figure 4, we can clearly see that the dominant contributions to the leading-order distribution of particulate mass come from the outer suspension mass and the outer and inner deposited mass, with the size of the inner suspension mass significantly smaller. This is a vindication of the matching procedure used to obtain $A(\theta )$ and illustrates the departure from the previous theory in the absence of substrate adsorption (Moore et al. Reference Moore, Vella and Oliver2021).

3.4. Distribution of the mass

We may use our analysis to map the distribution of particle mass in the system throughout the drying process. A first metric to illustrate where the mass is at a given time is to calculate the incomplete integral mass functions, denoted by $G(r,\theta )$ and $H(r,\theta )$ for the suspension and deposited mass, respectively, where

(3.34) \begin{equation} G(r,\theta ) = 2\pi \int _{0}^{r} \bar {r}(h\phi )(\bar {r},\theta ) \,{\mathrm{d}}\bar {r}, \quad H(r,\theta ) = 2\pi \int _{0}^{r} \bar {r}m_{s}(\bar {r},\theta ) \,{\mathrm{d}}\bar {r}. \end{equation}

By construction, we initially have $G(r,\theta _0) = 1/2\pi$ , $H(r,\theta _0) = 0$ , while for $\theta \lt \theta _0$ , the mass redistributes itself according to the competing transport mechanisms of advection, diffusion and substrate adsorption. Here, we use the asymptotic results to approximate the integral masses to leading order as ${\textit{Pe}}\rightarrow \infty$ by using the composite solutions (3.32) and (3.33) in (3.34).

In figure 5, we display results for two specific cases: ${\textit{Pe}} = 219,\ {\textit{Da}} = 154$ ( $\mathcal{V} \approx 1.1$ ) and ${\textit{Pe}} = 40,\ {\textit{Da}} = 60$ ( $\mathcal{V} \approx 2.4$ ). The curves in the figures are coded by colour: the suspension mass is in orange/yellow, while the deposited mass is in blue. The curves are shown at $30\,\%$ , $60\,\%$ and $90\,\%$ of the drying time in each case. We see from the figure that when the relative effect of substrate adsorption is stronger, that is, with higher $\mathcal{V}$ , the proportion of particle mass deposited on the substrate grows more rapidly. Indeed, at $90\,\%$ of the drying time, there is ${\approx} 84\,\%$ of the total mass in the deposited layer for $\mathcal{V} \approx 1.1$ , while for $\mathcal{V}\approx 2.4$ , this rises to ${\approx} 97\,\%$ of the total mass.

Figure 5. The cumulative mass distribution in the suspension (orange-scale curves) and deposited onto the substrate (blue-scale curves) for (a) ${\textit{Pe}} = 219,\ {\textit{Da}} = 154$ ( $\mathcal{V} \approx 1.1$ ) and (b) ${\textit{Pe}} = 40,\ {\textit{Da}} = 60$ ( $\mathcal{V} \approx 2.4$ ). Results are displayed at $30\,\%,\ 60\,\%$ and $90\,\%$ of the drying time.

It is noticeable that for each case and across each time snapshot displayed, the majority of the integral suspension mass is congregated in the bulk of the droplet, rather than the boundary layer. As an example, for $\mathcal{V} \approx 1.1$ at $60\,\%$ of the drying time, the cumulative suspension mass at $1-r = 10^{-1}$ is ${\approx} 40\,\%$ of the total mass, while there is only ${\approx} 2\,\%$ more of the total mass between $1-r = 10^{-1}$ and $1-r = 10^{-6}$ . This is indicative of the relatively small amount of particle mass in the inner region: the presence of substrate adsorption reducing this from an $O(1)$ amount as described in Moore et al. (Reference Moore, Vella and Oliver2021) to an $O({\textit{Pe}}^{-2})$ amount here.

In contrast, as anticipated by the matching procedure in the asymptotics, the percentage of the total mass in the adsorbed layer close to the contact line is substantial: for the same example at the same dryout time, the cumulative deposited mass at $1-r = 10^{-1}$ is ${\approx} 28\,\%$ of the total mass, while between $1-r = 10^{-1}$ and $1-r = 10^{-6}$ , there is ${\approx} 29\,\%$ of the total mass. We are clearly embedded in a regime where the mass is primarily distributed between the suspension in the bulk, the deposited mass in the bulk and the deposited mass in the boundary layer. Notably, this changes over time, with the percentage left in suspension reducing as we approach dryout since the mass is necessarily deposited on the substrate according to the model.

We can investigate the mass distribution in the droplet bulk and local to the contact line as the droplet dries in more detail by considering each contribution separately. To do so, we choose $r = 1-1/{\textit{Pe}}$ as the boundary between the bulk and the contact line; this value is of course flexible, but we have selected it as it sits in the overlap between the outer and inner regions of the asymptotics. We then evaluate four integrals,

(3.35) \begin{align} \mathcal{G}_{{b}}(\theta ) &= 2\pi \!\!\int _{0}^{1-1/{\textit{Pe}}} \!r(h\phi _{\textit{comp}})(r,\theta )\, {\mathrm{d}} r, \quad\! \mathcal{G}_{{cl}}(\theta ) = 2\pi \!\!\int _{1-1\!/\!{\textit{Pe}}}^1 \!r(h\phi _{\textit{comp}})(r,\theta )\, {\mathrm{d}} r, \end{align}

(3.36) \begin{align} \mathcal{H}_{{b}}(\theta ) &= 2\pi \!\!\int _{0}^{1-1/{\textit{Pe}}} \!{\textit{rm}}_{s,{comp}}(r,\theta )\, {\mathrm{d}} r, \quad\! \mathcal{H}_{{cl}}(\theta ) = 2\pi \!\!\int _{1-1\!/\!{\textit{Pe}}}^1 \!{\textit{rm}}_{s,{comp}}(r,\theta )\, {\mathrm{d}} r, \end{align}

representing the suspension mass in the bulk, the suspension mass in the local contact-line region, the deposited mass in the bulk and the deposited mass in the contact-line region, respectively. We then demonstrate the role of substrate adsorption by fixing ${\textit{Pe}} = 40$ and varying ${\textit{Da}}$ so that we span from small $\mathcal{V}$ to large and show the results in figure 6.

Figure 6. Percentages of the total mass in suspension (black) and deposited (red) in the droplet bulk and in suspension (blue) and deposited (magenta) near the contact line as a function of drying time. In each case, ${\textit{Pe}} = 40$ and we take ${\textit{Da}} =$ (a) $10$ ( $\mathcal{V} \approx 0.39$ ), (b) $20$ ( $\mathcal{V} \approx 0.79$ ), (c) $60$ ( $\mathcal{V} \approx 2.36$ ) and (d) $100$ ( $\mathcal{V} \approx 3.92$ ).

The results largely corroborate our previous findings, particularly that the percentage of the total mass in the local region in suspension is substantially smaller than the other contributions, even for the relatively small $\mathcal{V} \approx 0.39$ example shown in figure 6(a). Of note is that the relative dominance of the deposited mass in the bulk and in the contact-line region changes with $\mathcal{V}$ : a stronger $\mathcal{V}$ favours deposition in the bulk as substrate adsorption begins to dominate advection. Even for $\mathcal{V}\approx 0.79 \lt 1$ , we find that at early stages the bulk deposited mass accounts for a greater percentage of the total mass than that in the contact-line region, although this switches after about $45\,\%$ of the drying time. In addition, we find that the evolutions of the deposited mass both in the bulk and in the contact-line region are concave up when substrate adsorption is relatively weak, as illustrated in figure 6(a,b), and are concave down when substrate adsorption dominates, as illustrated in figure 6(c,d). This indicates that more particles are deposited onto the substrate at early stages of evaporation when substrate adsorption is sufficiently strong, as expected.

We can further visualise the effect of $\mathcal{V}$ on the final distribution of mass by considering the residue densities – that is, adsorbed mass per unit area – in the droplet bulk and contact-line regions, defined by

(3.37) \begin{equation} \mathcal{D}_{{b}} = \frac {\mathcal{H}_{{b}}(0)}{\pi \big(1-{\textit{Pe}}^{-1}\big)^2}, \quad \mathcal{D}_{{cl}} = \frac {\mathcal{H}_{{cl}}(0)}{\pi \big(1-\big(1-{\textit{Pe}}^{-1}\big)^2\big)}, \end{equation}

respectively. The variation of these densities with $\mathcal{V}$ for different Péclet numbers is displayed in figure 7. Clearly, for smaller $\mathcal{V}$ , the dominance of advection in particle transport leaves a substantially larger density of material in the contact-line region: for example, for $\mathcal{V} = 1,\ {\textit{Pe}} = 40$ , we have $\mathcal{D}_{{cl}}/\mathcal{D}_{{b}} \approx 16.1$ . However, as the strength of substrate adsorption is increased through $\mathcal{V}$ , we see that there is a crossover value, beyond which the substantial deposition in the droplet bulk leads to a final density that is higher in the interior than at the contact line. In this sense, substrate adsorption is reducing the coffee-ring effect. For example, for ${\textit{Pe}} = 40$ this crossover point is $\mathcal{V} \approx 5.65$ . The crossover value clearly increases with ${\textit{Pe}}$ , as larger ${\textit{Pe}}$ correlates to stronger advection. Notably, across the entire range of Péclet numbers shown here, the density in the interior of the droplet remains largely unchanged: the main effect of the Péclet number is to increase the density in the contact-line region, again connected to the stronger coffee-ring effect for stronger advection.

Figure 7. The adsorbed mass per unit area in the droplet bulk $\mathcal{D}_{{b}}$ (solid curves) and contact-line region $\mathcal{D}_{{cl}}$ (dashed curves) for ${\textit{Pe}} = 20$ (black), $40, 60, 80, 100$ (light grey).

It is important to stress that the above comments referencing the effect of substrate adsorption on reducing the coffee-ring effect do not mean that there is less pointwise mass deposited near the contact line than in the bulk: indeed, since the volume fraction is always monotonically increasing in $r$ to leading order as ${\textit{Pe}}\rightarrow \infty$ , the same is true for the deposited mass, so that there is still an effective strengthening of the deposit at the contact line. This assumption will break down in regimes in which $\mathcal{V} \gg 1$ , since the dominant time scale is no longer that of the evaporation-driven flow, rather that dictated by substrate adsorption. This regime is beyond the scope of the present study.

3.4.1. The nascent coffee ring

To strengthen the above point, here it is pertinent to investigate how substrate adsorption influences the growth of the coffee ring. Since we are only interested in contributions near the contact line, we can forgo the composite profile in favour of the leading-order inner solutions. In previous studies in the absence of substrate adsorption, the nascent coffee ring has been identified, to leading order as ${\textit{Pe}}\rightarrow \infty$ , with the suspension mass in the inner region (Moore et al. Reference Moore, Vella and Oliver2021, Reference Moore, Vella and Oliver2022), which here is given by

(3.38) \begin{equation} m_{\textit{susp}}(R,\theta ) = H\kern-0.5pt\varPhi = \theta A\kern-0.5pt(\theta ) R\exp\left (-\frac {4\chi }{\theta }\sqrt {R}\right )\!. \end{equation}

However, in the presence of substrate adsorption, we must also consider the particle mass that has deposited on the substrate, which to leading order is simply

(3.39) \begin{equation} m_{\textit{dep}}(R,\theta ) = {\textit{Pe}}^2 M_{s} = {\textit{Pe}}^2 \frac {\mathcal{V}}{2\pi \theta _0}\int _{\theta }^{\theta _0} A(\psi )\exp\left (-\frac {4\chi }{\psi }\sqrt {R}\right )\,{\mathrm{d}} \psi . \end{equation}

If we interpret the coffee ring as the increase of particle mass due to evaporative transport to the contact line, we must consider the behaviour of both of these contributions.

Figure 8. The leading-order suspension (left) and deposited (right) mass in the inner region as a function of $R = {\textit{Pe}}^2(1-r)$ for $(a)$ $\mathcal{V}=0.25$ , $(b)$ $\mathcal{V}=1$ , $(c)$ $\mathcal{V}=4$ and $(d)$ $\mathcal{V}=8$ . In each case, we show profiles at $25\,\%$ $(\theta = 0.75\theta _0)$ , $50\,\%$ $(\theta = 0.5\theta _0)$ and $75\,\%$ $(\theta = 0.25\theta _0)$ of drying time: drying time is indicated by the arrow in each figure.

We display $m_{\textit{susp}}$ and $m_{\textit{dep}}$ as functions of $R$ for ${\textit{Pe}} = 10$ and $\mathcal{V} = 0.25,\ 1,\ 4,\ 8$ in figure 8. The curves are shown at $25\,\%$ , $50\,\%$ and $75\,\%$ of drying time in each case. In the figure, we see that the suspension mass retains its characteristic gamma distribution profile as discussed in detail in Moore et al. (Reference Moore, Vella and Oliver2021, Reference Moore, Vella and Oliver2022). However, not only is the size of this profile substantially reduced as compared with the zero-substrate adsorption regime, it also reduces with evaporation time, due to the increasing amount of particle mass lost to substrate adsorption. Notably, there is a simultaneous reduction of adsorbed mass in the inner region, which is due to the decrease in the coefficient $A(\theta )$ as $\mathcal{V}$ increases. Of course, these simultaneous local reductions happen concurrently with a rise in the particle mass deposited in the droplet bulk. This is consistent with a transition from advection-dominated particle transport to substrate adsorption-dominated particle transport as $\mathcal{V}\rightarrow \infty$ .

3.5. Breakdown of the model

While there are several limitations of the model we have presented, there are two particular potential points to investigate in terms of the breakdown of the theory, namely finite particle size effects in the suspension – sometimes referred to as jamming – and the growth of the deposited layer on the solid substrate. We use our asymptotic solution to discuss each of these in turn.

3.5.1. Finite particle size effects and jamming

Throughout, we have assumed that the particles are dilute in the fluid so that the flow and transport problems decouple. As we have seen, the largest volume fractions in the resulting flow are in the inner region in which ${\textit{Pe}}^2 \varPhi$ is monotonically increasing, with maximum value

(3.40) \begin{equation} \varPhi _{\textit{max}}(\theta ) = \varPhi (0,\theta )= A(\theta ), \end{equation}

where $A(\theta )$ is given by (3.24). Now, there will be a critical volume fraction $\tilde {\phi }_c$ at which the particles should no longer be considered as dilute, so that we expect to see a breakdown of the model at $\theta = \theta _{\textit{cb}}(\mathcal{V},{\textit{Pe}})$ such that

(3.41) \begin{equation} {\textit{Pe}}^2\varPhi _{\textit{max}}(\theta _{\textit{cb}})=\dfrac {\tilde {\phi }_c}{\tilde {\phi }_{\textit{init}}} \quad \implies \quad \frac {\gamma (\theta _{\textit{cb}})}{\theta _{\textit{cb}}^2} = \frac {\mathcal{V}\tilde {\phi }_c}{8\pi \chi ^3\tilde {\phi }_{\textit{init}}{\textit{Pe}}^2}. \end{equation}

We plot the curves of $\theta _{\textit{cb}}(\mathcal{V},{\textit{Pe}})$ as a percentage of the total drying time for varying $\mathcal{V}$ and ${\textit{Pe}} = 5,\ 10,\ 20,\ 40$ in figure 9 as the bluescale curves. We take the critical volume fraction to be $\tilde {\phi }_c = 0.64$ and for reference we take $\tilde {\phi }_{\textit{init}} = 0.05$ , based on the volume fraction for the data in table 4 in Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010), which, as discussed in § 2.2, looks at titania particles in water (calculated from mass fraction $X = 0.2$ in table 4, density of titania $\tilde {\rho }_p = 4230$ kg m $^{-3}$ , density of water $10^3$ kg m $^{-3}$ ).

Figure 9. The onset of jamming (bluescale curves, left-hand axis) and deposited layer effects (redscale curves, right-hand axis) as a percentage of the total drying time given by the roots of (3.41) and (3.46), respectively. For each case, we vary the Péclet number, ${\textit{Pe}} = 5,\ 10,\ 20,\ 40$ , while for $\theta _{{ct}}$ , we take $R = 1$ as an illustrative value.

For small substrate adsorption velocities, we observe similar behaviour to Moore et al. (Reference Moore, Vella and Oliver2021) in the sense that diffusive evaporation leads to rapid breakdown near the contact line. For example, for $\mathcal{V} = 0.4$ and ${\textit{Pe}} = 10$ , breakdown is predicted to occur after $0.002\,\%$ of the drying time for the values in the Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010) experiments. However, it is worth again stressing that the presence of substrate adsorption here does inhibit finite particle size effects slightly (since volume fraction scales like ${\textit{Pe}}^2$ with substrate adsorption rather than ${\textit{Pe}}^4$ without).

When substrate adsorption is stronger, the onset of finite particle size effects is substantially delayed due to the particle mass lost to the substrate. For ${\textit{Pe}} = 10$ and $\mathcal{V} = 4$ , the suspension of particles may still be modelled as dilute for $88\,\%$ of the drying time. As can be seen, if we further increase $\mathcal{V}$ , there is a rapid decrease of the critical contact angle $\theta _{\textit{cb}}$ : studying the profiles of $A(\theta )$ in figure 3, a local maximum develops in $A$ near $\theta _0$ and for sufficiently large $\mathcal{V}$ , this maximum is below the jamming threshold, so that we only see jamming in the late stages when $A$ is singular as $\theta \rightarrow 0$ . This latter behaviour holds for $\mathcal{V} \lt 9$ (cf. (3.27)). For larger substrate adsorption, $A$ is bounded at $\theta = 0$ , so that jamming does not occur. However, as noted previously, for $\mathcal{V}\gg 1$ , we are no longer in the advection-dominated regime discussed herein.

3.5.2. Growth of the deposited layer

The second method of breakdown we discuss here is the invalidation of the assumption that we may neglect the deposited layer on the substrate in our analysis. This assumption is realised in the application of the no-slip, no-flux boundary conditions and the substrate adsorption condition on $z = 0$ , as well as the linear nature of the substrate adsorption condition. While this linearisation is reasonable as a first approximation initially, it may be called into question as the deposited layer grows. Not only may the size of the deposited layer play a role, but it may also cause a blocking effect, reducing the substrate adsorption velocity (Adamczyk Reference Adamczyk2002). Since the adsorbed mass is monotonically increasing in $r$ , this breakdown will necessarily occur first in the contact-line region, so that we may again concentrate our investigations using the solution in the inner region.

While, of course, the neglect of such a deposited layer will always be invalidated very close to the contact line, since the droplet free surface forms a wedge profile in the inner region, the modelling assumption has been that these local effects are small and do not propagate back towards the droplet bulk. However, clearly if the deposited layer becomes appreciable throughout the inner region, our analysis must be reconsidered.

We can estimate when this occurs by considering the leading-order deposited mass in the inner region. The dimensional mass deposited onto a region of radial extent $\tilde {\varDelta }$ about $\tilde {r} = \tilde {R}(1-R/{\textit{Pe}}^2)$ is given by

(3.42) \begin{equation} \tilde {m}_{s} \approx 2\pi \delta \tilde {R}^2 \tilde {\rho }_{{s}} \tilde {\phi }_{\textit{init}} {\textit{Pe}}^2 M_{s}(R,\theta ) \tilde {\varDelta }. \end{equation}

Now, if we suppose that the particles form a layer of thickness $\tilde {\epsilon }_{\textit{layer}}$ with packing fraction $0.64$ , we must have

(3.43) \begin{equation} \tilde {m}_{\textit{layer}} \approx 2\pi \tilde {R} \times 0.64\tilde {\rho }_{{s}} \tilde {\varDelta } \tilde {\epsilon }_{\textit{layer}} \approx 2\pi \delta \tilde {R}^2 \tilde {\rho }_{{s}}\tilde {\phi }_{\textit{init}} {\textit{Pe}}^2 M_{s}(R,\theta ) \tilde {\varDelta }. \end{equation}

The layer thickness can thus be estimated by

(3.44) \begin{equation} \tilde {\epsilon }_{\textit{layer}} \approx \frac {\delta \tilde {R} \tilde {\phi }_{\textit{init}}{\textit{Pe}}^2}{0.64} M_{s}(R,\theta ). \end{equation}

Meanwhile, the thickness of the droplet in this region is given by

(3.45) \begin{equation} \tilde {h} \approx \delta \tilde {R} {\textit{Pe}}^{-2}\theta R. \end{equation}

For the sake of comparison, let us suppose that our theory is invalidated if the deposited layer is 10 $\,\%$ of the droplet thickness: this occurs at the critical time $\theta = \theta _{{ct}}(\mathcal{V},{\textit{Pe}},R)$ , which is the root of

(3.46) \begin{equation} \frac {\mathcal{V}}{2\pi \theta _0}\int _{\theta _{{ct}}}^{\theta _0} A(\theta )\mbox{exp}\left (-\frac {4\chi }{\theta }\sqrt {R}\right )\,{\mathrm{d}} \theta = \frac {0.064}{\tilde {\phi }_{\textit{init}}{\textit{Pe}}^4} \theta _{{ct}} R. \end{equation}

Clearly, and as expected, the critical time for finite layer effects additionally depends on the distance from the contact line in the inner variables, $R$ . For illustration, we display in figure 9 as the redscale curves the breakdown time at $R = 1$ for ${\textit{Pe}} = 5,\ 10,\ 20,\ 40$ and varying $\mathcal{V}$ with $\tilde {\phi }_{\textit{init}} = 0.05$ taken from the experiments in Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010). Perhaps unsurprisingly, the factor of ${\textit{Pe}}^4$ in the denominator of (3.46) means that breakdown due to the growth of the deposited layer occurs rapidly as ${\textit{Pe}}\rightarrow \infty$ for all $\mathcal{V}$ , although since the initial adsorbed mass is zero, there is a timeframe of validity.

In addition, we see that the breakdown due to the finite layer thickness depends strongly on $\mathcal{V}$ . When we increase $\mathcal{V}$ , we increase the importance of substrate adsorption throughout the droplet, which reduces the amount adsorbed onto the substrate in the inner region, reducing the layer growth there and increasing the range of validity. For example, for ${\textit{Pe}} = 10$ , when $\mathcal{V} = 1$ , we see breakdown after ${\approx} 2.51\,\%$ of the total drying time, while for $\mathcal{V} = 10$ , this increases to ${\approx} 50.01\,\%$ .

It is also clear from figure 9 that for a fixed Péclet number and radial location $R$ , there is a transition from breakdown due to finite particle size effects to breakdown due to the deposited layer thickness as we increase $\mathcal{V}$ as the increasing significance of substrate adsorption results in more particles being found in the deposited layer rather than in suspension. In particular, for small $\mathcal{V}$ , the first breakdown is due to finite particle size effects. However, as we increase $\mathcal{V}$ , the reduction in the coefficient $A(\theta )$ coupled with the rapid growth of the deposited layer eventually leads to breakdown due to the latter. For example, if ${\textit{Pe}} = 10$ and $R = 1$ , this transition occurs at $\mathcal{V} \approx 1.52$ . With $R$ fixed, this transition value increases with ${\textit{Pe}}$ . Similarly, with ${\textit{Pe}}$ fixed, the transition value increases with $R$ .

However, for very small $\mathcal{V}$ , we note here an important caveat regarding the above results and discussion. As we noticed previously, the coefficient $A(\theta )$ and the matching procedure outlined in § 3.2.1 are no longer valid as $\mathcal{V}\rightarrow 0$ , due to the singular behaviour of the asymptotic expansion. We now present the resolution of this singularity.

3.6. The limit $\mathcal{V}\rightarrow 0$

When $\mathcal{V}$ is small, the asymptotic analysis presented previously breaks down when considering conservation of particle mass. For completeness, we briefly summarise the asymptotic results for $\mathcal{V} = o(1)$ as ${\textit{Pe}}\rightarrow \infty$ .

3.6.1. Outer region

In the outer region, substrate adsorption now plays a lower-order role in particulate transport so that to leading order as ${\textit{Pe}}\rightarrow \infty$ , the solution is precisely that presented in the no-substrate adsorption problem as given in Moore et al. (Reference Moore, Vella and Oliver2021, Reference Moore, Vella and Oliver2022):

(3.47) \begin{equation} \phi (r,\theta ) = \frac {1}{\sqrt {1-r^2}}\left (\left (\frac {\theta _0}{\theta }\right )^{3/4}-(1-(1-r^2)^{3/2})\right )^{1/3}\!. \end{equation}

As previously stated, this is regular in the sense that this is the leading-order solution if we take the limit of (3.3)–(3.4) as $\mathcal{V}\rightarrow 0$ . Note that the volume fraction is still square-root singular at the contact line: that is, $\phi \sim \gamma (\theta )/\sqrt {1-r}$ as $1-r\rightarrow 0^+$ , where

(3.48) \begin{equation} \gamma (\theta ) = \frac {1}{\sqrt {2}} \left (\left (\frac {\theta _0}{\theta }\right )^{3/4}-1\right )^{1/3} \end{equation}

is the leading-order term of (3.6) as $\mathcal{V}\rightarrow 0$ .

The corresponding leading-order solution for the deposited mass is given by $m_{s} \sim \mathcal{V}m_{s,0}$ , where

(3.49) \begin{align} m_{s,0}(r,\theta ) &= \frac {1}{2\pi \sqrt {1-r^2}(1-(1-r^2)^{3/2})}\nonumber\\&\quad\times\left [\left (1-\left (\frac {\theta }{\theta _0}\right )^{3/4}(1-(1-r^2)^{3/2})\right )^{4/3}- (1-r^2)^2\right ]\!. \end{align}

As we approach the contact line, we therefore have

(3.50) \begin{equation} m_{s} \sim \frac {\mathcal{V}\mathcal{M}_{\textit{MVO}}(\theta )}{\pi \chi \sqrt {1-r}} \quad \mbox{as} \quad 1-r\rightarrow 0^+, \end{equation}

as previously.

3.6.2. Inner region

In the inner region, the scalings (3.13) remain the same in the regime in which $\mathcal{V}\rightarrow 0$ , with the exception that we now have

(3.51) \begin{equation} \phi = \varphi \varPhi , \quad m_{s} = \mathcal{V}\varphi M_{s}, \end{equation}

where $\varphi$ is to be determined shortly. The solution then follows identically, with the leading-order inner particle volume fraction given by (3.16) and the leading-order inner deposited mass given by (3.17) with $\mathcal{V} =1$ .

3.6.3. Matching using conservation of particle mass

It is in the matching procedure that things differ most substantially as $\mathcal{V}\rightarrow 0$ . With the updated scales for the deposited mass in each region, (3.21) now becomes

(3.52) \begin{align} \displaystyle \frac {1}{2\pi } & = \displaystyle \int _{0}^{1} (rh\phi )_{\textit{outer}}(r,\theta ) \,{\mathrm{d}} r + \varphi {\textit{Pe}}^{-4} \int _{0}^{\infty }\varPhi\!H(R,\theta )\,{\mathrm{d}} R \nonumber\\ &\quad +\mathcal{V}\displaystyle \int _{0}^{1} ({\textit{rm}}_{s,0})(r,\theta ) \,{\mathrm{d}} r + \mathcal{V}\varphi {\textit{Pe}}^{-2} \int _{0}^{\infty } M_s(R,\theta )\,{\mathrm{d}} R. \end{align}

Clearly, in this regime, the leading-order deposited mass in the outer region – represented by the first integral in the second line – no longer enters the leading-order balance in the conservation law. Nevertheless, we must still balance the mass lost out of the outer region to leading order due to advection. There are three distinct regimes.

For weak substrate adsorption when ${\textit{Pe}}^{-2}\ll \mathcal{V}\ll 1$ , the contribution due to substrate adsorption in the inner region dominates the contribution from the suspension mass in the inner region, so that we choose

(3.53) \begin{equation} \varphi = \mathcal{V}^{-1}{\textit{Pe}}^2. \end{equation}

Proceeding, we find that the coefficient $A(\theta )$ is then given by (3.24) with $\mathcal{V} = 1$ . In this weak substrate adsorption regime, substrate adsorption only affects the suspension mass to leading order in the inner region. Notably, the coefficient $A(\theta )$ is different from that of the zero-substrate adsorption problem, due to the different dominant terms in the conservation of mass equation. Furthermore, we note that substrate adsorption reduces the scaling for the suspension mass from that in Moore et al. (Reference Moore, Vella and Oliver2021, Reference Moore, Vella and Oliver2022). This is expected to reduce the role of finite particle size effects in the suspension, as discussed above for the case $\mathcal{V} = O(1)$ .

For the border case whereby $\mathcal{V} = \mathcal{W}{\textit{Pe}}^{-2}$ where $\mathcal{W} = O(1)$ as ${\textit{Pe}}\rightarrow \infty$ , the leading-order contributions from both the suspension mass and deposited mass in the inner region are of the same order of magnitude, so we take

(3.54) \begin{equation} \varphi = {\textit{Pe}}^4. \end{equation}

We may then derive the following integral equation for $A(\theta )$ :

(3.55) \begin{equation} \mathcal{M}_{\textit{MVO}}(\theta ) = \frac {3\theta ^5}{64\chi ^4}A(\theta ) + \frac {\mathcal{W}}{16\pi \theta _0\chi ^2}\int _{\theta }^{\theta _0} \psi ^2 A(\psi )\,{\mathrm{d}}\psi . \end{equation}

This has solution

(3.56) \begin{eqnarray} A(\theta ) = -\frac {64\chi ^4}{3\theta ^5}\int _{\theta }^{\theta _0} {\mathcal{M}^\prime}_{\textit{MVO}}(\psi )\,\mbox{exp}\left (\frac {2\mathcal{W}\chi ^2}{3\pi \theta _0}\left (\frac {1}{\theta ^2} - \frac {1}{\psi ^2}\right )\right )\,{\mathrm{d}} \psi . \end{eqnarray}

In this border regime, substrate adsorption has a leading-order effect on the suspension mass in the inner region only, through the exponential in the integral of (3.56). Moreover, the scaling for the size of the suspension mass is the same as that in the zero-substrate adsorption model of Moore et al. (Reference Moore, Vella and Oliver2021).

Finally, when substrate adsorption is very weak so that $\mathcal{V} = o({\textit{Pe}}^{-2})$ , each leading-order deposition term in (3.52) is small, so that we take

(3.57) \begin{equation} \varphi = {\textit{Pe}}^{4}, \end{equation}

and retrieve the solution of Moore et al. (Reference Moore, Vella and Oliver2021, Reference Moore, Vella and Oliver2022) with $A = A_{\textit{MVO}}(\theta )$ as given by (3.25). Clearly, in this regime, substrate adsorption has a lower-order effect on the advection–diffusion competition that drives the growth of the nascent coffee ring. In principle, this will manifest as a temporal boundary layer close to lifetime, during which the increase in concentration results in substrate adsorption becoming significant, although in practice the model would likely break down prior to this point as a result of finite particle size effects.

3.6.4. Validation and comparisons

As previously, one may find composite profiles for the volume fraction $\phi$ and deposited mass $m_{s}$ in each of the three regimes discussed above.

Here, we investigate the veracity of the asymptotic analysis by considering a specific example for which ${\textit{Pe}} = 40$ and ${\textit{Da}} = 5$ , so that $\mathcal{V} \approx 0.20$ and we may reasonably say that $\mathcal{V} = O({\textit{Pe}}^{-1/2})$ , so that we are in the weak substrate adsorption regime. The results are shown in figure 10, where the suspension mass (figure 10 a,b) and deposited mass (figure 10 c,d) are shown at $15\,\%$ intervals of the drying time using the same colour coding as in figure 4. We see good agreement between the leading-order composite profiles and the numerical simulations in each case, with expected magnitude of errors given the relatively small Péclet number and large $\mathcal{V}$ chosen for this example. The behaviour of the nascent coffee ring is of particular interest here, with the peak initially rising for around $30\,\%$ of the drying time (cf. figure 10 b) due to the early-stage influence of the competition between advection and diffusion as reported in previous analyses (Moore et al. Reference Moore, Vella and Oliver2021). However, after this, the peak of the suspension mass profile begins to fall as the effect of substrate adsorption increases, reducing the mass left in the suspension at later times.

Figure 10. The leading-order composite suspension (a,b) and deposited (c,d) mass as a function of time for ${\textit{Pe}} = 40$ , ${\textit{Da}} = 5$ ( $\mathcal{V} \approx 0.2$ ). The initial profile mirrors the droplet shape and is shown as the bold blue line, while profiles at $15\,\%$ intervals of the drying time are shown in increasingly lighter shades of grey. A close-up of the nascent coffee-ring profile is shown in (b,d), whereby we see the characteristic formation of the peak just inside the contact line. The red dashed curves indicate the corresponding numerical solution of (2.33)–(2.37), while the arrows indicate increasing evaporation time.

Similar to figure 4, we see that the deposited mass is relatively uniform in the droplet bulk, although at later times we begin to see notable variation in $r$ away from the contact line. If we interpret the deposited mass as representative of the stain left after evaporation, this may suggest a slightly smoother transition from the ring to the internal deposit.

The figure also further illustrates that the particulate mass is predominantly contained in the outer suspension mass and the inner deposited mass in this regime, which justifies the balance chosen in the matching above. In particular, even at $90\,\%$ of the drying time, the outer deposited mass is still roughly $40\,\%$ smaller than the outer suspension mass.

It is worth noting that, while the composite solution is valid to leading order throughout the droplet, the intermediate solution for the volume fraction has a correction that is $O(1)$ as ${\textit{Pe}}\rightarrow \infty$ . Hence, for relatively small ${\textit{Pe}}$ or large $\mathcal{V}$ , it can often be more beneficial to directly use the leading-order outer/inner solutions for investigating behaviours in the droplet interior or at the contact line, respectively.

5.1. Comparison with previous numerical studies

First, we consider the simulations in Widjaja & Harris (Reference Widjaja and Harris2008), where the fully coupled diffusive evaporation, Stokes flow and particle transport problems are solved numerically. The majority of the examples studied therein are in the regime that ${\textit{Da}} \gg {\textit{Pe}}$ , and thus fall outside the scope of this paper. However, an example that falls into our scope is that in their figure 9(a), for which ${\textit{Pe}} = {\textit{Da}} = 12.5$ (where we have accounted for the slightly different definition of the Péclet and Damköhler numbers herein). Note that the droplet under consideration has a relatively large aspect ratio, $\delta = 0.8$ , which pushes the limits of applicability of the depth-averaged model we use.

Figure 13. A comparison between simulation data from figure 9(a) in Widjaja & Harris (Reference Widjaja and Harris2008) (circles), the leading-order composite asymptotic adsorbed mass given by (3.33) (solid grey curve) and the numerical solution of (2.33)–(2.37) (dashed red curve). In the language of the present paper, the results are presented for ${\textit{Da}} = {\textit{Pe}} = 12.5$ $(\mathcal{V} = \pi /2)$ .

In figure 13, we compare data extracted from figure 9(a) in Widjaja & Harris (Reference Widjaja and Harris2008) with the leading-order composite solution (3.33) and the numerical solution of (2.33)–(2.37). We see that, despite the relatively large initial droplet aspect ratio, the model presented herein does a good job of qualitatively capturing the monotonic increasing nature of the adsorbed mass profile. The agreement is particularly good in the droplet bulk, although our model slightly under-predicts the profile near the contact line. It is anticipated that this is a combination of the large aspect ratio and that the Péclet number is quite small, so that the assumption that $\delta ^2{\textit{Pe}} \ll 1$ made in deriving our depth-averaged model is not met well by this example. Nevertheless, despite being at the limits of the validity of the model, these comparisons are encouraging.

5.2. Comparison with previous experimental studies

Let us now turn to experiments. Generally, the qualitative behaviour that more mass is adsorbed onto the substrate in the bulk when there is an increase in substrate adsorption, e.g. when particle–substrate interaction is attractive rather than repulsive, is a common observation in experimental studies (see e.g. Yan et al. Reference Yan, Gao, Sharma, Chiang and Wong2008; Bhardwaj et al. Reference Bhardwaj, Fang, Somasundaran and Attinger2010; Anyfantakis & Baigl Reference Anyfantakis and Baigl2015; Devineau et al. Reference Devineau, Anyfantakis, Marichal, Kiger, Morel, Rudiuk and Baigl2016; Molchanov et al. Reference Molchanov, Roldughin, Chernova-Kharaeva and Senchikhin2019; Kumar et al. Reference Kumar, Basavaraj and Satapathy2023). However, as discussed previously in § 2.2, it is challenging to get experimental data that accurately estimates all the required parameters in the model to make a quantitative comparison, with the substrate adsorption velocity $\tilde {k}_{{a}}$ proving the most troublesome. In addition, due to the importance of finite particle size effects such as jamming or particle packing in the vicinity of the coffee ring and in the deposited layer, the model presented herein is unlikely to give good quantitative comparisons with the final deposited profile. However, given that for the vast majority of problems, the dilute regime is the natural starting point – and that we further demonstrated that substrate adsorption can extend the validity of this regime – we are more likely to have success when comparing quantities such as the mass in the ring and the central deposit. Here we carry out a quantitative comparison with two experimental studies by Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010) and Bridonneau et al. (Reference Bridonneau, Zhao, Battaglini, Mattana, Thévenet, Noël, Roché, Zrig and Carn2020), in which water droplets containing silica and titania nanoparticles evaporate on gold-coated silicon wafer and glass substrates, respectively, for which we were able to determine approximations for $\tilde {k}_{{a}}$ as outlined in § 2.2. These studies also provide profiles of the final deposit from which we may determine the mass in the ring and the central deposit.

Before making our comparisons it is important to clarify that the experiments considered here study droplets evaporating with a pinned contact line for most of their lifetime. That is, the contact line will eventually de-pin towards the end of evaporation. In particular, for the experiments by Bridonneau et al. (Reference Bridonneau, Zhao, Battaglini, Mattana, Thévenet, Noël, Roché, Zrig and Carn2020) the de-pinning angle is approximately $ 6^\circ$ , indicating that the contact line is pinned for only $86\,\%$ and $84\,\%$ of lifetime for the two experiments of untreated and aluminium-coated silica nanoparticles, respectively, outlined in § 2.2. In addition, close inspection of supplementary videos 1 and 2 in Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010) indicate that the droplet is pinned for ${\approx} 93\,\%$ of lifetime when $\text{pH}=11.7$ but only for ${\approx} 70\,\%$ of lifetime when $\text{pH}=2.8$ . Therefore, in order to test the validity of the present model we compare quantities for the mass in the ring when the contact line de-pins, i.e. at $\theta =\theta ^*$ , rather than at $\theta =0$ , assuming that after the contact line recedes no more mass can be deposited into the ring.

Figure 14 shows a comparison between the leading-order deposited mass in the contact-line region at the time of contact-line de-pinning $\mathcal{H}_{{cl}}(\theta ^*)$ with the experimental data for the final mass in the ring from Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010) and Bridonneau et al. (Reference Bridonneau, Zhao, Battaglini, Mattana, Thévenet, Noël, Roché, Zrig and Carn2020). Specifically, as figure 14 shows, despite the number of simplifying assumptions made herein, the asymptotic solution does a good job of predicting the final mass in the ring deposit, slightly over-predicting the experimental values of Bridonneau et al. (Reference Bridonneau, Zhao, Battaglini, Mattana, Thévenet, Noël, Roché, Zrig and Carn2020) by approximately $6\,\%$ and $9\,\%$ , but with particularly good agreement when comparing with the experiments of Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010). The latter is perhaps unsurprising given that this example is comfortably in the regime in which ${\textit{Pe}}\gg 1$ and $\delta \ll 1$ .

Figure 14. Comparison between the leading-order deposited mass in the contact-line region at the time of contact-line de-pinning $\mathcal{H}_{{cl}}(\theta ^*)$ given by (3.36) for (a) ${\textit{Pe}}\approx 50.0$ , ${\textit{Da}}\approx 3.66$ and $\mathcal{V}\approx 0.115$ (black line) and ${\textit{Pe}}\approx 69.1$ , ${\textit{Da}}\approx 4.47$ and $\mathcal{V}\approx 0.102$ (grey line) and the experimental data for the final mass in the ring from supplementary figures S8 and S9 in Bridonneau et al. (Reference Bridonneau, Zhao, Battaglini, Mattana, Thévenet, Noël, Roché, Zrig and Carn2020) for untreated (star) and aluminium-coated (diamond) particles, and for (b) ${\textit{Pe}}\approx 127$ , ${\textit{Da}}\approx 20.5$ and $\mathcal{V}\approx 0.253$ (black line) and ${\textit{Pe}}\approx 213$ , ${\textit{Da}}\approx 8.23$ and $\mathcal{V}\approx 0.061$ (grey line) and experimental data for the final mass in the ring from figures 6(a) and 6(b) in Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010) for $\text{pH}=11.7$ (circle) and $\text{pH}=2.8$ (square). The dashed lines correspond to the predicted mass in the contact-line region when the droplet evaporates with a pinned contact line throughout evaporation $\mathcal{H}_{{cl}}(0)$ .

In all the cases considered, the predicted mass in the contact-line region at $\theta =0$ is substantially larger than the true value from experiments, as well as those predicted by the model at the time of contact-line de-pinning, highlighting the importance of contact-line motion in the formation of the final deposit. For example, for the experiments by Bridonneau et al. (Reference Bridonneau, Zhao, Battaglini, Mattana, Thévenet, Noël, Roché, Zrig and Carn2020) in which the droplets are pinned for $84\,\%$ – $86\,\%$ of lifetime, the final predicted adsorbed mass in the contact-line region after contact-line de-pinning and in the bulk after complete evaporation is approximately $74\,\%$ and $8\,\%$ – $9\,\%$ of the total mass, respectively. This indicates that approximately $17\,\%$ – $18\,\%$ of the mass is left unaccounted for and therefore must be deposited in the bulk due to late contact-line de-pinning. This proportion of final mass is nearly double the predicted mass deposited in the bulk due to substrate adsorption effects. We note that whilst the majority of previous studies on droplet evolution and particle deposition often ignore late contact-line de-pinning, a recent study by D’Ambrosio et al. (Reference D’Ambrosio, Wray and Wilson2025a ) gives theoretical predictions for the final mass of particles deposited on the substrate split between the contact-line region and the bulk for a variety of de-pinning angles, $\theta ^*$ , for a thin droplet of dilute suspension in the absence of substrate adsorption. Specifically, their model predicts that for the experiments by Bridonneau et al. (Reference Bridonneau, Zhao, Battaglini, Mattana, Thévenet, Noël, Roché, Zrig and Carn2020) approximately $23\,\%$ – $25\,\%$ of the total mass will be deposited in the bulk due to contact-line de-pinning, which are similar values to those outlined here although slightly higher presumably due to the absence of substrate adsorption in their model.

Moreover, it is clear from the experiments by Bhardwaj et al. (Reference Bhardwaj, Fang, Somasundaran and Attinger2010) that substrate adsorption has a strong effect on contact-line pinning. Specifically, for the experiment when $\text{pH}=11.7$ , the DLVO force is repulsive, substrate adsorption is weak and particles are advected to the contact line which remains pinned for ${\approx} 93\,\%$ of the evaporation. The predicted de-pinning angle of ${\approx} 0.24^\circ$ is significantly lower than the value predicted for clean water on glass by Hu & Larson (Reference Hu and Larson2002), namely $2^\circ$ – $4^\circ$ , showing that the strong coffee-ring effect enhances the pinning in this case. However, when $\text{pH}=2.8$ , the DLVO force is attractive and fewer particles are advected to the contact line due to substrate adsorption, i.e. the coffee-ring effect is reduced. The predicted de-pinning angle of ${\approx} 1.96^\circ$ is much closer to the value for clean water and the earlier de-pinning time further reduces the coffee-ring effect in this case as more particles are deposited in the bulk as the contact line recedes.