2.1. The evaporative problem

According to the well-known diffusion-limited model (see, for example, Picknett & Bexon Reference Picknett and Bexon1977; Deegan et al. Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten1997; Popov Reference Popov2005; Dunn et al. Reference Dunn, Wilson, Duffy, David and Sefiane2009; Wilson & Duffy Reference Wilson and Duffy2022; Wilson & D’Ambrosio Reference Wilson and D’Ambrosio2023), the concentration of vapour in the atmosphere, denoted by $\hat {c}=\hat {c}(\hat {r},\hat {z},\hat {t})$ , satisfies Laplace’s equation subject to conditions of complete saturation at the free surface, no flux of vapour through the substrate and an appropriate ambient value in the far field.

For a thin droplet ( $\hat {\theta }_0\ll 1$ ) we non-dimensionalise and scale the variables appropriately according to

(2.2) \begin{equation} (\hat {r},\hat {z})=\hat {R}_0 (r,\hat {\theta }_0z), \quad \hat {R}=\hat {R}_0R, \quad \hat {\theta }=\hat {\theta }_0\theta , \quad \hat {h}=\hat {\theta }_0\hat {R}_0h, \quad \hat {V}=\hat {\theta }_0\hat {R}_0^3V \end{equation}

for the droplet, and by

(2.3) \begin{align} (\hat {r},\hat {z})=\hat {R}_0 (r,z_{{a}}), \quad \hat {c}=\hat {c}_{\infty }+(\hat {c}_{\textit{sat}}-\hat {c}_{\infty })c, \quad \hat {J}=\dfrac {\hat {D}(\hat {c}_{\textit{sat}}-\hat {c}_{\infty })}{\hat {R}_0}J, \quad \hat {F}=\hat {D}(\hat {c}_{\textit{sat}}-\hat {c}_\infty )\hat {R}_0 F \end{align}

for the atmosphere, where $\hat {c}_{\textit{sat}}$ is the constant saturation concentration, $\hat {c}_\infty$ is the constant ambient concentration, $\hat {D}$ is the constant diffusion coefficient of vapour in the atmosphere, and $\hat {J}=\hat {J}(\hat {r},\hat {t})$ and $\hat {F}=\hat {F}(\hat {t})$ are the local and total evaporative mass fluxes from the droplet, respectively. Hence, in particular, the initial contact angle and initial contact radius of the droplet are both scaled to unity. Due to the different characteristic vertical length scalings in the thin droplet and in the atmosphere, we distinguish the vertical coordinates in the droplet and the atmosphere by denoting them by $z$ and $z_{{a}}$ , respectively.

Note that for the basic diffusion-limited model described here, the evaporative problem is decoupled from the thermal problem. This basic model has been extended by a number of authors to include a variety of additional physical effects, including coupling the thermal and evaporative problems by, for example, accounting for the temperature dependence of the saturation concentration (see, for example, Dunn et al. Reference Dunn, Wilson, Duffy, David and Sefiane2009; Ait Saada, Chikh & Tadrist Reference Ait Saada, Chikh and Tadrist2013) or surface tension (see, for example, Hu & Larson Reference Hu and Larson2005; Xu, Luo & Guo Reference Xu, Luo and Guo2012), but, for simplicity and clarity, in the present work we restrict our attention to the basic model.

The exact solution of the basic diffusion-limited model for a thin droplet is well known (see, for example, Wilson & Duffy Reference Wilson and Duffy2022) and leads to the familiar expression for the local evaporative flux $J$ , namely

(2.4) \begin{equation} J=\dfrac {2}{\pi \sqrt {R^2-r^2}}, \end{equation}

which is singular (but integrable) at the contact line of the droplet at $r=R$ . Integrating $J$ over the free surface of the droplet gives the total evaporative flux $F=F(t)$ , namely

(2.5) \begin{equation} F=2\pi \int _0^{R}{J\,r\;\text {d}r}=4R. \end{equation}

The droplet evolves according to the global mass-conservation condition

(2.6) \begin{equation} \dfrac {\text {d}V}{\text {d}t}=-F= -4R, \end{equation}

in which time has been non-dimensionalised by

(2.7) \begin{equation} \hat {t}=\dfrac {\hat {\rho }\hat {\theta }_{0}\hat {R}_0^2}{\hat {D}(\hat {c}_{\textit{sat}}-\hat {c}_{\infty })}t. \end{equation}

We note that the local and total evaporative fluxes, $J$ and $F$ , both depend on $t$ via their dependence on $R=R(t)$ .

2.2. The hydrodynamic problem

The pressure and velocity within the droplet, denoted by $\hat {p}=\hat {p}(\hat {r},\hat {z},\hat {t})$ and $\hat {\mathbf{u}}=(\hat {u}(\hat {r},\hat {z},\hat {t}),0,\hat {w}(\hat {r},\hat {z},\hat {t}))$ , satisfy the usual mass-conservation and Stokes equations subject to the usual boundary conditions.

We non-dimensionalise and scale the variables appropriately for the droplet according to

(2.8a,b) \begin{equation} (\hat {u},\hat {w})=\hat {U} \kern-1pt \big(u,\hat {\theta }_0w\big ), \quad \hat {p}-\hat {p}_{{a}}=\dfrac {\hat {\sigma }\hat {\theta }_0} {\hat {R}_0}p, \end{equation}

where $\hat {p}_{{a}}$ is the constant atmospheric pressure and $\hat {U}$ is the appropriate radial velocity scale which, for diffusion-limited evaporation, may be given by

(2.9) \begin{equation} \hat {U}=\dfrac {\hat {D}\!\left (\hat {c}_{\textit{sat}}-\hat {c}_\infty \right )}{\hat {\rho }\hat {\theta }_0\hat {R}_0} \end{equation}

(see, for example, Wray et al. Reference Wray, Wray, Duffy and Wilson2021).

We seek an asymptotic solution to the hydrodynamic problem in the form

(2.10) \begin{equation} (p,u,w)=\big (p^{(0)},u^{(0)},w^{(0)}\big )+ \textit{Ca} \big (p^{(1)},u^{(1)},w^{(1)}\big )+O\big (\hat {\theta }_0^2,\textit{Bo}, \textit{Ca}^2\big ). \end{equation}

Note that, as we shall show, the pressure $p$ is the only quantity required beyond leading order to describe the flow within the droplet and so, for clarity, henceforth we omit the ‘(0)’ superscript on all other leading-order quantities.

At leading order, the Stokes equations yield $\partial p^{(0)}/\partial r=\partial p^{(0)}/\partial z=0$ , and hence the leading-order pressure is independent of $r$ and $z$ , i.e. $p^{(0)}=p^{(0)}(t)$ , and is given by the normal stress condition at the free surface, namely

(2.11) \begin{equation} p^{(0)}=-\dfrac {1}{r}\dfrac {\partial }{\partial r}\left (r\dfrac {\partial h}{\partial r}\right ). \end{equation}

Since $p^{(0)}$ is independent of $r$ , the mean curvature of the free surface of the droplet is therefore spatially constant and the free-surface profile $h$ satisfies

(2.12) \begin{equation} \dfrac {\partial }{\partial r}\left (\dfrac {1}{r}\dfrac {\partial }{\partial r}\left ( r\dfrac {\partial h}{\partial r}\right )\right )=0 \end{equation}

subject to $h(R,t)=0$ and $\partial h/\partial r=-\theta$ at $r=R$ , and an appropriate regularity condition at $r=0$ , and is therefore given by the familiar paraboloidal form

(2.13) \begin{equation} h=\dfrac {\theta \big(R^2-r^2\big)}{2R}. \end{equation}

Evaluating (2.11) using (2.13) yields $p^{(0)}=2\theta /R$ . The volume $V=V(t)$ of the droplet is given by

(2.14) \begin{equation} V=2\pi \int _0^{R}{h\,r\;\text {d}r}=\dfrac {\pi \theta R^3}{4}. \end{equation}

At first order, the Stokes equations reduce to

(2.15a,b) \begin{equation} \dfrac {\partial ^2u}{\partial z^2}=\dfrac {\partial p^{(1)}}{\partial r}, \quad \dfrac {\partial p^{(1)}}{\partial z}=0 \end{equation}

(see, for example, Wray et al. Reference Wray, Wray, Duffy and Wilson2021). In particular, (2.15b ) shows that the first-order pressure $p^{(1)}$ is independent of $z$ , i.e. $p^{(1)}=p^{(1)}(r,t)$ . The leading-order velocities satisfy the usual mass-conservation equation

(2.16) \begin{equation} \dfrac {1}{r}\dfrac {\partial (ru)}{\partial r}+\dfrac {\partial w}{\partial z}=0 ,\end{equation}

subject to no-slip and no-penetration conditions on the substrate

(2.17) \begin{equation} u=w=0 \quad \text{on} \quad z=0, \end{equation}

and the leading-order tangential stress condition on the free surface of the droplet

(2.18) \begin{equation} \dfrac {\partial u}{\partial z}=0 \quad \text{at} \quad z=h. \end{equation}

Solving (2.15) and (2.16) subject to (2.17) and (2.18) leads to

(2.19a,b) \begin{align} u=\dfrac {1}{2}\dfrac {\partial p^{(1)}}{\partial r}\big (z^2-2hz\big ), \quad w=\dfrac {z^2}{6r}\left [\dfrac {\partial p^{(1)}}{\partial r}\left (3r\dfrac {\partial h}{\partial r}+3h-z\right ) +r\dfrac {\partial ^2p^{(1)}}{\partial r^2}\left (3h-z\right )\right ] \end{align}

(see, for example, Boulogne et al. Reference Boulogne, Ingremeau and Stone2017; D’Ambrosio et al. Reference D’Ambrosio, Wilson, Wray and Duffy2023).

The kinematic condition can be expressed as

(2.20) \begin{equation} \dfrac {\partial h}{\partial t}+\dfrac {1}{r}\dfrac {\partial (rQ)}{\partial r}=-J, \end{equation}

where $Q=Q(r,t)$ is the local radial volume flux of fluid, defined by

(2.21) \begin{equation} Q=\int _0^h{u\;\text {d}z}=h\bar {u}, \end{equation}

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

(2.22) \begin{equation} \bar {u}=\dfrac {1}{h}\int _0^h{u\;\text {d}z}. \end{equation}

Substituting the expression for $u$ given by (2.19a ) into (2.21) yields

(2.23) \begin{equation} Q=-\dfrac {h^3}{3}\frac {\partial p^{(1)}}{\partial r}. \end{equation}

We note that it will be convenient to rearrange and integrate (2.20) to express $Q$ as

(2.24) \begin{equation} Q=-\dfrac {1}{r}\int _0^r{\left (J+\dfrac {\partial h}{\partial t}\right )\,\tilde {r}\;\text {d}\tilde {r}}. \end{equation}

It is also convenient to eliminate $p^{(1)}$ between (2.19) and (2.23) to give

(2.25a,b) \begin{align} u=\dfrac {3Q}{2h^3}\big (2hz-z^2\big ), \quad w=\dfrac {z^2}{2h^4r} \left [3Qr\dfrac {\partial h}{\partial r}\left (2h-z\right )-\left (Qh+rh\dfrac {\partial Q}{\partial r}\right )\left (3h-z\right )\right ]\!. \end{align}

Specifically, $u$ and $w$ can be determined by substituting known expressions for $h$ and $Q$ given by (2.13) and (2.24) into (2.25).

2.3. The particle-transport problem

2.3.1. The concentration of particles

The motion of particles within the droplet is due to a combination of advection by the fluid flow and diffusion, and so the concentration of particles within the droplet, denoted by $\phi =\phi (r,z,t)$ (non-dimensionalised by the volume-averaged initial concentration of particles within the droplet, $\bar {\phi }_0$ ), satisfies the advection–diffusion equation subject to conditions of no flux of particles through either the substrate or the free surface of the droplet.

We consider situations in which the diffusion of particles is weak relative to advection on the scale of the droplet. More specifically, for a thin droplet we assume that the reduced particle Péclet number, $\hat {\theta }_0^2{\textit{Pe}}$ , satisfies

(2.26) \begin{equation} \hat {\theta }_0^2\ll \hat {\theta }_0^2{\textit{Pe}}\ll 1, \end{equation}

where $ \textit{Pe}$ is the appropriately defined particle Péclet number, namely

(2.27) \begin{equation} \textit{Pe}=\dfrac {\hat {R}_0\hat {U}}{\hat {D}_{{p}}}=\dfrac {\hat {D}\big(\hat {c}_{\textit{sat}}-\hat {c}_\infty \big)}{\hat {\rho }\hat {\theta }_0\hat {D}_{{p}}}, \end{equation}

in which $\hat {D}_{{p}}$ is the constant diffusivity of particles in the fluid. In particular, (2.26) means that we are in a regime in which advection dominates radial particle transport. Previous studies have shown that the Péclet number is typically larger than unity in a wide range of situations involving different fluids, particle materials and particle sizes (see, for example, Moore, Vella & Oliver Reference Moore, Vella and Oliver2021), and so the condition (2.26) is physically realisable provided that the Péclet number is not too large (specifically, provided that $ \textit{Pe} \ll \hat {\theta }_0^{-2}$ ) and that the droplet is sufficiently thin.

As is well known (see, for example, Jensen & Grotberg Reference Jensen and Grotberg1993; Wray et al. Reference Wray, Papageorgiou, Craster, Sefiane and Matar2014; D’Ambrosio et al. Reference D’Ambrosio, Wilson, Wray and Duffy2023), when the particle Péclet number satisfies (2.26), the leading-order concentration of particles, $\phi =\phi (r,t)$ , is independent of $z$ and satisfies

(2.28) \begin{equation} \dfrac {\partial \phi }{\partial t} +\bar {u}\dfrac {\partial \phi }{\partial r}=\dfrac {\phi J}{h}, \end{equation}

where the depth-averaged velocity $\bar {u}$ is given by (2.22).

Equation (2.28) may be solved via the method of characteristics, i.e.

(2.29a,b) \begin{equation} \dfrac {\text {d}\phi }{\text {d}t}=\dfrac {J\phi }{h} \quad \text{on the characteristics determined by} \quad \dfrac {\text {d}r}{\text {d}t}=\bar {u}, \end{equation}

subject to a prescribed initial condition $\phi (r,0)=\phi _0(r)$ . For simplicity, in the remainder of the present work we assume that the initial concentration of particles within the droplet is spatially uniform, and so, as a consequence of our choice of scalings, $\phi _0(r)\equiv 1$ .

2.3.2. The mass of particles

As the droplet evaporates, particles are advected towards the contact line by the flow within the droplet, resulting in a mass flux of particles into the contact line (see, for example, Deegan et al. Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten2000; Boulogne et al. Reference Boulogne, Ingremeau and Stone2017; D’Ambrosio et al. Reference D’Ambrosio, Wilson, Wray and Duffy2023). Hence, if a particle is advected by the flow into the contact line before the end of the evaporation then it is deposited onto the substrate and so, in general, the mass of particles within the bulk of the droplet will be a decreasing function of time. In the absence of particle–substrate adsorption (or any other similar physical mechanism) deposition onto the substrate via the contact line is the only way by which particles can leave the bulk of the droplet, and so the nature of the resulting deposit depends on the motion of the contact line. Specifically, during a CR phase particles that are advected into the contact line form a ring deposit at the fixed position of the pinned contact line, while during a CA phase they form a distributed deposit on the substrate in the region swept by the moving contact line. Figure 1(a)–(c) shows examples of ring deposits, while figure 1(d)–(j) shows examples of distributed deposits. As we shall show in the present analysis, depending upon the mode in which the droplet is evaporating, for diffusion–limited evaporation the final deposit may consist of a single ring deposit (see, for example, figure 1 a), multiple concentric-ring deposits (see, for example, figure 1 b), distributed deposits of near-uniform shape (see, for example, figure 1 j), or a combination thereof.

The mass of particles per unit area within the footprint of the droplet is $\phi h$ , and so the mass of particles in the bulk of the droplet at time $t$ , denoted by $M_{\textit{drop}}=M_{\textit{drop}}(t)$ (non-dimensionalised by $\hat {\theta }_0\hat {R}_0^3\bar {\phi }_0$ ), is given by

(2.30) \begin{equation} M_{\textit{drop}}=2\pi \int _0^{R(t)}{\phi (r,t)\,h(r,t)\,r\;\text {d}r}. \end{equation}

In particular, the initial mass of particles in the bulk of the droplet, denoted by $M_0=M_{\textit{drop}}(0)$ , is given by

(2.31) \begin{equation} M_0=2\pi \int _0^{1}{\phi _0(r)\,h(r,0)\,r\;\text {d}r}=\dfrac {\pi }{4}. \end{equation}

During a CR phase starting at $t=t_{{p}}$ in which the droplet has a pinned contact line located at $r=R(t_{{p}}) \, (\leqslant 1)$ , the mass flux of particles from the bulk of the droplet into the fixed contact line is

(2.32) \begin{equation} \lim _{r\to {R(t_{{p}})^{-}}}{2\pi \phi Qr}, \end{equation}

and therefore the mass of particles in the ring deposit located at $r=R(t_{{p}})$ at time $t \, (\geqslant t_{{p}})$ , denoted by $M_{\textit{ring}}=M_{\textit{ring}}(t)$ (also non-dimensionalised by $\hat {\theta }_0\hat {R}_0^3\bar {\phi }_0$ ), is given by

(2.33) \begin{equation} M_{\textit{ring}}=2\pi \int _{t_{{p}}}^{t}{\lim _{r\to {{R(t_{{p}})}^{-}}}{\phi (r,\tilde {t})\,Q(r,\tilde {t})\,r}\;\text {d}\tilde {t}}. \end{equation}

Note that $M_{\textit{drop}}$ given by (2.30) can be rewritten as

(2.34) \begin{equation} M_{\textit{drop}}=2\pi \int _0^{r_0(R(t_{{p}}),t)}{\phi (r,t_{{p}})\,h(r,t_{{p}})\,r\;\text {d}r} \end{equation}

and $M_{\textit{ring}}$ given by (2.33) can be rewritten as

(2.35) \begin{equation} M_{\textit{ring}}=2\pi \int _{r_0(R(t_{{p}}),t)}^{R(t_{{p}})}{\phi (r,t_{{p}})\,h(r,t_{{p}})\,r\;\text {d}r}, \end{equation}

where $r_0=r_0(R(t_{{p}}),t)$ denotes the initial radial position of particles at the beginning of the CR phase, i.e. at time $t=t_{{p}}$ , that reach the contact line at time $t$ (i.e. whose positions satisfy $r(t)=R(t_{{p}})$ ), and is determined by solving (2.29b ). Note that in the special case $t_{{p}}=0$ and $R(t_{{p}})=1$ the expression for $M_{\textit{ring}}$ given by (2.35) is equivalent to the corresponding expression obtained by Deegan et al. (Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten2000) for a droplet evaporating in the CR mode.

During a CA phase starting at $t=t_{\textit{d}}$ in which the droplet has a moving contact line located at $r=R(t) \, (\leqslant R(t_{\textit{d}}) \leqslant 1)$ , the mass flux of particles from the bulk of the droplet into the moving contact line is

(2.36) \begin{equation} \lim _{r\to {R(t)^{-}}} 2\pi \phi hr\left (\bar {u}-\frac {\text {d} R}{\text {d} t}\right )\!, \end{equation}

and therefore the mass of particles in the distributed deposit at time $t \, (\geqslant t_{\textit{d}})$ , denoted by $M_{\textit{dist}}=M_{\textit{dist}}(t)$ (also non-dimensionalised by $\hat {\theta }_0\hat {R}_0^3\bar {\phi }_0$ ), is given by

(2.37) \begin{equation} M_{\textit{dist}}=2\pi \int _{t_{\textit{d}}}^{t}{\lim _{r\to {R(\tilde {t})}^{-}} \phi (r,\tilde {t})\,h(r,\tilde {t})\,r\left (\bar {u}(r,\tilde {t})-\dfrac {\text {d}R(\tilde {t})}{\text {d}\tilde {t}}\right ) \;\text {d}\tilde {t}}. \end{equation}

Note that $M_{\textit{drop}}$ given by (2.30) can be rewritten as

(2.38) \begin{equation} M_{\textit{drop}}=2\pi \int _0^{r_0(R(t),t)}{\phi (r,t_{\textit{d}})\,h(r,t_{\textit{d}})\,r\;\text {d}r} \end{equation}

and $M_{\textit{dist}}$ given by (2.37) can be rewritten as

(2.39) \begin{equation} M_{\textit{dist}}=2\pi \int _{r_0(R(t),t)}^{R(t_{\textit{d}})}{\phi (r,t_{\textit{d}})\,h(r,t_{\textit{d}})\,r\;\text {d}r}, \end{equation}

where $r_0=r_0(R(t),t)$ denotes the initial radial position of particles at the beginning of the CA phase, i.e. at time $t=t_{\textit{d}}$ , that reach the contact line at time $t$ (i.e. whose positions satisfy $r(t)=R(t)$ ), and is determined by solving (2.29b ).

The expression for $M_{\textit{dist}}$ given by (2.39) can also be written as

(2.40) \begin{equation} M_{\textit{dist}}=2\pi \int _{R(t)}^{R(t_{\textit{d}})}{\phi _{\textit{dist}}(r)\,r\;\text {d}r}, \end{equation}

where $\phi _{\textit{dist}}=\phi _{\textit{dist}}(r)$ (non-dimensionalised by $\hat {\theta }_0\hat {R}_0\bar {\phi }_0$ ) is the density of the distributed deposit on the substrate in the annular region $R(t) \leqslant r \leqslant R(t_{\textit{d}})$ . An expression for $\phi _{\textit{dist}}$ can be obtained by differentiating (2.39) and (2.40) with respect to $R$ and equating the two expressions to give

(2.41) \begin{equation} \phi _{\textit{dist}}(R)=\phi (r_0(R,t),t_{\textit{d}})\,h(r_0(R,t),t_{\textit{d}})\dfrac {r_0(R,t)}{R}\dfrac {\partial \left (r_0(R,t)\right )}{\partial R}. \end{equation}

Note that in the special case $t_{\textit{d}}=0$ and $R(t_{\textit{d}})=1$ the expressions $M_{\textit{dist}}$ and $\phi _{\textit{dist}}$ given by (2.39) and (2.41), respectively, are equivalent to the corresponding expressions obtained by Freed-Brown (Reference Freed-Brown2014) for a droplet evaporating in the CA mode.

In the following four sections we analyse the evolution of, the flow within, and the deposition from, a droplet evaporating in four different modes, namely the CR, CA, SS and SJ modes, respectively.

3.1. The evolution of the droplet

For a droplet evaporating in the CR mode, i.e. with $R\equiv 1$ , $\theta =\theta (t)$ , (2.6) becomes

(3.1) \begin{equation} \dfrac {\text {d}V}{\text {d}t}=\dfrac {\pi }{4}\dfrac {\text {d}\theta }{\text {d}t}=-4. \end{equation}

Solving (3.1) yields the well-known simple explicit evolution of the droplet, namely

(3.2) \begin{equation} \theta =1-\dfrac {16}{\pi }t, \end{equation}

with $h$ and $V$ given by (2.13) and (2.14), respectively, and hence the lifetime of the droplet $t_{\textit{CR}}$ obtained by setting $\theta (t_{\textit{CR}})=0$ is

(3.3) \begin{equation} t_{\textit{CR}}=\dfrac {\pi }{16} \end{equation}

(see, for example, Wilson & Duffy Reference Wilson and Duffy2022).

3.2. The flow within the droplet

Substituting the expressions for $J$ and $h$ given by (2.4) and (2.13), respectively, into (2.24) and evaluating the integral gives the following solution for $Q$ and hence the depth-averaged velocity $\bar {u}$ in the CR mode:

(3.4a,b) \begin{equation} Q=\dfrac {2}{\pi r}\Big [\sqrt {1-r^2}-\big (1-r^2\big )^2\Big ], \quad \bar {u}=\dfrac {4}{\pi \theta r}\Big [\big (1-r^2\big )^{-1/2} -\big (1-r^2\big )\Big ] \end{equation}

(see, for example, equations 11 and 12 in Gelderblom et al. Reference Gelderblom, Diddens and Marin2022). Note that $\bar {u}$ depends on time only through its dependence on $\theta$ and so we will sometimes find it convenient to parametrise by $\theta$ rather than $t$ during a CR phase. This will be particularly useful when describing the deposition of particles from the droplet.

Figure 3 shows plots of $Q$ and $\bar {u}$ given by (3.4) at different times for a droplet evaporating in the CR mode. In particular, figure 3 illustrates that $Q$ is independent of $t$ in the CR mode and that the depth-averaged radial velocity is outwards towards the contact line throughout the evaporation, i.e. $Q\geqslant 0$ and $\bar {u}\geqslant 0$ for all $0\leqslant t\leqslant t_{\textit{CR}}$ . The solutions for the velocities $u$ and $w$ can be obtained by substituting the expression for $Q$ given by (3.4) into (2.25). The resulting expressions are given by, for example, Boulogne et al. (Reference Boulogne, Ingremeau and Stone2017), but are omitted here for brevity.

Figure 3.

Plot of (a) the radial volume flux $Q$ and (b) the depth-averaged radial velocity $\bar {u}$ given by (3.4) at times $t=(0$ , $1/10$ , …, $9/10)\times t_{\textit{CR}}$ for a droplet evaporating in the CR mode. The arrow in (b) indicates the direction of increasing $t$ .

3.3. The deposition of particles

For a droplet evaporating in the CR mode, the characteristic equations (2.29) may be used to write

(3.5a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}\theta }=\dfrac {\text {d}r/\text {d}t}{\text {d}\theta /\text {d}t}= \dfrac {\bar {u}}{\text {d}\theta /\text {d}t}, \quad \dfrac {\text {d}\phi }{\text {d}\theta }=\dfrac {\text {d}\phi /\text {d}t}{\text {d}\theta /\text {d}t} =\dfrac { J\phi }{h\, \text {d}\theta /\text {d}t}, \end{equation}

subject to the initial condition $\phi _0(r)=1$ . Substituting the expressions for $J$ , $h$ , $\text {d}\theta /\text {d}t$ and $\bar {u}$ from (2.4), (2.13), (3.1) and (3.4b ) into (3.5) gives

(3.6a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}\theta }=-\dfrac {1}{4\theta r}\Big [\big (1-r^2\big )^{-1/2}-\big (1-r^2\big )\Big ], \quad \dfrac {\text {d}\phi }{\text {d}\theta }=-\dfrac {\phi }{4\theta }\big (1-r^2\big )^{-3/2}. \end{equation}

Note that (3.6a ) is separable and using separation of variables gives

(3.7) \begin{equation} r_0(r,t)=\sqrt {1-\theta ^{1/2}\left (\theta ^{-3/4}-1+\big (1-r^2\big )^{3/2}\right )^{2/3}}, \end{equation}

where $r_0=r_0(r,t)$ denotes the initial radial position of particles at the beginning of evaporation, i.e. at time $t=0$ , that are at radial position $r$ at time $t$ . In particular, the time it takes for particles at initial position $r_0$ to travel to the pinned contact line, i.e. to $r=1$ , where they are then deposited into the ring deposit, defined by $t=t_{\textit{ring}}$ , is given implicitly by $r_0(1,t_{\textit{ring}})$ , which is determined by setting $r=1$ in (3.7) to yield

(3.8) \begin{equation} r_0(1,t_{\textit{ring}})=\sqrt {1-\left (1-\theta ^{3/4}\right )^{2/3}}, \end{equation}

where $\theta$ is given by (3.2) in the CR mode. Substituting the expression for $r$ given implicitly by (3.7) into (3.6b ), it may be shown that $\phi$ is given parametrically by

(3.9) \begin{equation} \phi =\left [\dfrac {\left (1-r_0^2\right )^{3/2}}{\theta ^{3/4}-1+\left (1-r_0^2\right )^{3/2}}\right ]^{1/3}. \end{equation}

Finally, eliminating $r_0$ between (3.7) and (3.9) yields an explicit expression for the concentration of particles within the droplet evaporating in the CR mode

(3.10) \begin{equation} \phi =\left [1+\big (1-r^2\big )^{-3/2}\big (\theta ^{-3/4}-1\big )\right ]^{1/3} \end{equation}

(see, for example, Zheng Reference Zheng2009; D’Ambrosio et al. Reference D’Ambrosio, Wilson, Wray and Duffy2023). The concentration of particles $\phi$ is a monotonically increasing function of $r$ that takes its minimum value at $r=0$ and is singular at the pinned contact line throughout the evaporation, i.e. $\phi =O ((1-r)^{-1/2} )\to \infty$ as $r\to 1^{-}$ for all $0\lt t\lt t_{\textit{CR}}$ . Further discussion regarding the behaviour of $\phi$ and $\phi h$ in the CR mode is given by D’Ambrosio et al. (Reference D’Ambrosio, Wilson, Wray and Duffy2023).

Substituting $t_{{p}}\equiv 0$ , $R(t_{{p}})\equiv 1$ , and the expression for $r_0(1,t_{\textit{ring}})$ given by (3.8) into (2.34) and evaluating the integral yields an explicit expression for the mass of particles in the bulk of the droplet, namely

(3.11) \begin{equation} M_{\textit{drop}}= 2\pi \int _0^{r_0(1,t_{\textit{ring}})}{h(r,0)\,r\;\text {d}r}=M_0\left [1-\big (1-\theta ^{3/4}\big )^{4/3}\right ]\!, \end{equation}

where the initial mass of particles in the droplet is given by (2.31) to be $M_0=\pi /4$ . The mass of particles in the ring deposit from (2.35) is given by

(3.12) \begin{equation} M_{\textit{ring}}=2\pi \int _{r_0(1,t_{\textit{ring}})}^1{h(r,0)\,r\;\text {d}r}= M_0\big (1-\theta ^{3/4}\big )^{4/3} \end{equation}

(see, for example, Deegan et al. Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten2000; Boulogne et al. Reference Boulogne, Ingremeau and Stone2017). For a droplet evaporating in the CR mode, the contact line never de-pins and therefore $M_{\textit{dist}}\equiv 0$ . Adding together the expressions for $M_{\textit{drop}}$ and $M_{\textit{ring}}$ given by (3.11) and (3.12), respectively, and recalling that $M_{\textit{dist}}\equiv 0$ confirms that the total mass of particles is conserved as the droplet evaporates, i.e. that $M_{\textit{drop}}+M_{\textit{ring}}+M_{\textit{dist}}=M_0$ .

Figure 4 shows a plot of the evolutions of $M_{\textit{drop}}/M_0$ and $M_{\textit{ring}}/M_0$ given by (3.11) and (3.12), respectively, as functions of $t/t_{\textit{CR}}$ for a droplet evaporating in the CR mode. In particular, as figure 4 shows, all of the particles are eventually deposited into a single ring deposit at $r=1$ in the CR mode, i.e. $M_{\textit{drop}}(t_{\textit{CR}})=0$ and $M_{\textit{ring}}(t_{\textit{CR}})=M_0=\pi /4$ .

Figure 4.

Plot of the evolutions of $M_{\textit{drop}}/M_0$ (solid line) and $M_{\textit{ring}}/M_0$ (dashed line) given by (3.11) and (3.12), respectively, as functions of $t/t_{\textit{CR}}$ for a droplet evaporating in the CR mode.

4.1. The evolution of the droplet

For a droplet evaporating in the CA mode, i.e. with $R=R(t)$ , $\theta \equiv 1$ , (2.6) becomes

(4.1) \begin{equation} \dfrac {\text {d}V}{\text {d}t}=\dfrac {3\pi R^2}{4}\dfrac {\text {d}R}{\text {d}t}=-4R. \end{equation}

In particular, from (4.1), the rate at which the contact line is receding is then given by

(4.2) \begin{equation} \dfrac {\text {d}R}{\text {d}t}=-\dfrac {16}{3\pi R}. \end{equation}

Solving (4.2) yields the well-known explicit evolution of the droplet, namely

(4.3) \begin{equation} R=\sqrt {1-\dfrac {32}{3\pi }t}, \end{equation}

with $h$ and $V$ given by (2.13) and (2.14), respectively, and hence the lifetime of the droplet $t_{\textit{CA}}$ obtained by setting $R(t_{\textit{CA}})=0$ is

(4.4) \begin{equation} t_{\textit{CA}}=\dfrac {3\pi }{32} \end{equation}

(see, for example, Wilson & Duffy Reference Wilson and Duffy2022).

4.2. The flow within the droplet

Substituting the expressions for $J$ and $h$ given by (2.4) and (2.13), respectively, into (2.24) and evaluating the integral gives the following solution for $Q$ in the CA mode:

(4.5) \begin{equation} Q=\dfrac {2R}{\pi r}\left [\sqrt {1-\dfrac {r^2}{R^2}}-\left (1+\dfrac {r^2}{3R^2}\right ) \left (1-\dfrac {r^2}{R^2}\right )\right ]\!, \end{equation}

and hence the corresponding depth-averaged velocity $\bar {u}$ is

(4.6) \begin{equation} \bar {u}=\dfrac {4}{\pi r}\left [\left (1-\dfrac {r^2}{R^2}\right )^{-1/2} -\left (1+\dfrac {r^2}{3R^2}\right )\right ] \end{equation}

(see, for example, equations 18 and 19 in Gelderblom et al. Reference Gelderblom, Diddens and Marin2022). Note that $\bar {u}$ depends on time only through its dependence on $R$ and so we will sometimes find it convenient to parametrise by $R$ rather than $t$ during a CA phase. As in the CR mode, this will be particularly useful when describing the deposition of particles from the droplet.

Figure 5.

Plot of (a) the radial volume flux $Q$ given by (4.5) and (b) the depth-averaged radial velocity $\bar {u}$ given by (4.6) at times $t=(0$ , $1/10$ , …, $9/10)\times t_{\textit{CA}}$ for a droplet evaporating in the CA mode. The arrows indicate the direction of increasing $t$ .

Figure 5 shows plots of $Q$ given by (4.5) and $\bar {u}$ given by (4.6) at different times for a droplet evaporating in the CA mode. In particular, figure 5 shows that the depth-averaged radial velocity is outwards towards the contact line throughout the evaporation, i.e. $Q\geqslant 0$ and $\bar {u}\geqslant 0$ for all $0\leqslant t\leqslant t_{\textit{CA}}$ . In addition, while $Q$ approaches zero at $r=0$ and $r=R$ , $\bar {u}$ approaches zero at $r=0$ but is square-root singular at the receding contact line throughout the evaporation. As discussed by Gelderblom et al. (Reference Gelderblom, Diddens and Marin2022), this behaviour for the depth-averaged radial flow is qualitatively similar to that for a droplet evaporating in the CR mode given in § 3.2. Moreover, the prediction of depth-averaged outward radial flow differs from the corresponding prediction of the spatially uniform evaporation model, for which the flow is inwards, as discussed in § 1. A comparison between the resulting transport of particles will be discussed in § 4.4.

Substituting the expressions for $h$ and $Q$ from (2.13) and (4.5) into (2.25) yields the solutions for the velocities $u$ and $w$ , namely

(4.7) \begin{equation} u=\dfrac {24z}{\pi r R}\left (1-\dfrac {r^2}{R^2}\right )^{-2}\left [\left (1-\dfrac {r^2}{R^2}\right )^{-1/2} -\left (1+\dfrac {r^2}{3R^2}\right )\right ]\left (1-\dfrac {r^2}{R^2}-\dfrac {z}{R}\right ) \end{equation}

and

(4.8) \begin{align} w&=\dfrac {8z^2}{\pi R^3}\left (1-\dfrac {r^2}{R^2}\right )^{-3}\left [4\left (1-\dfrac {r^2}{R^2}\right )-\dfrac {9}{2}\sqrt {1-\dfrac {r^2}{R^2}}-\dfrac {2z}{3R}\left (7+\dfrac {r^2}{R^2}\right )\right . \nonumber\\&\qquad \qquad \qquad \qquad \qquad \quad \left . - \dfrac {5z}{R}\left (1-\dfrac {r^2}{R^2}\right )^{-1/2}\right ]\!. \end{align}

Figure 6(a) shows instantaneous streamlines of the flow within a droplet evaporating in the CA mode for diffusion-limited evaporation calculated from (4.7) and (4.8) at $t=t_{\textit{CA}}/2=3\pi /64$ . As figure 6(a) illustrates, the flow is outwards and downwards towards the receding contact line everywhere within the droplet.

Figure 6.

The instantaneous streamlines of the flow within a droplet evaporating in the CA mode for (a) diffusion-limited evaporation calculated from (4.7) and (4.8) and (b) spatially uniform evaporation at $t=t_{\textit{CA}}/2=3\pi /64$ .

4.3. The deposition of particles

For a droplet evaporating in the CA mode, the characteristic equations (2.29) may be used to write

(4.9a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}R}=\dfrac {\text {d}r/\text {d}t}{\text {d}R/\text {d}t}= \dfrac {\bar {u}}{\text {d}R/\text {d}t}, \quad \dfrac {\text {d}\phi }{\text {d}R}=\dfrac {\text {d}\phi /\text {d}t}{\text {d}R/\text {d}t} =\dfrac { J\phi }{h\, \text {d}R/\text {d}t}, \end{equation}

subject to the initial condition $\phi _0(r)=1$ . Substituting the expressions for $J$ , $h$ , $\text {d}R/\text {d}t$ and $\bar {u}$ from (2.4), (2.13), (4.2) and (4.6) into (4.9) gives

(4.10a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}R}=\dfrac {3R}{4r}\left [\left (1+\dfrac {r^2}{3R^2}\right )- \left (1-\dfrac {r^2}{R^2}\right )^{-1/2} \right ]\!, \quad \dfrac {\text {d}\phi }{\text {d}R}=-\dfrac {3\phi }{4R}\left (1-\dfrac {r^2}{R^2}\right )^{-3/2}. \end{equation}

Using the substitution $\zeta =r/R$ , (4.10a ) becomes separable in $R$ and $\zeta$ and may be rewritten as

(4.11) \begin{equation} \int _{1}^{R}{\dfrac {1}{\hat {R}}\,\text {d}\hat {R}}=\int _{\zeta _0}^{\zeta }{\dfrac {4\hat {\zeta }\sqrt {1-\hat {\zeta }^2}} {3\left [\big (1-\hat {\zeta }^2\big )^{3/2}-1\right ]} \,\text {d}\hat {\zeta }}, \end{equation}

where $\zeta _0\equiv r_0(r,t)$ denotes the initial radial position of particles at the beginning of evaporation, i.e. at time $t=0$ , that are at radial position $r$ at time $t$ . Performing the integral in (4.11) and eliminating $\zeta$ gives

(4.12) \begin{equation} r_0(r,t)=\sqrt {1-R^{3/2}\left (R^{-9/4}-1+\left (1-\dfrac {r^2}{R^2}\right )^{3/2}\right )^{2/3}}. \end{equation}

The time it takes for particles at initial position $r_0$ to travel to the receding contact line, i.e. to $r=R$ , where they are then deposited in the distributed deposit, defined by $t=t_{\textit{dist}}$ , is given implicitly by $r_0(R,t_{\textit{dist}})$ , namely

(4.13) \begin{equation} r_0(R,t_{\textit{dist}})=\sqrt {1-\left (1-R^{9/4}\right )^{2/3}}. \end{equation}

Substituting the expression for $r$ given implicitly by (4.12) into (4.10b ) the equation becomes separable and may be rewritten as

(4.14) \begin{equation} \int _{1}^{\phi }{\dfrac {1}{\hat {\phi }}\,\text {d}\hat {\phi }}=-\int _{1}^{R}{\dfrac {3\hat {R}^{5/4}} {4\left [\hat {R}^{9/4}-1+\left (1-r_0^2\right )^{3/2}\right ]}\text {d}\hat {R}}, \end{equation}

and performing the integral shows that $\phi$ is given parametrically by

(4.15) \begin{equation} \phi =\left [\dfrac {\left (1-r_0^2\right )^{3/2}}{R^{9/4}-1+\left (1-r_0^2\right )^{3/2}}\right ]^{1/3}. \end{equation}

Figure 7.

Plot of (a) the concentration of particles $\phi$ and (b) the mass per unit area $\phi h$ within the droplet as functions of $r$ at $t=(0$ , $1/10$ , …, $9/10)\times t_{\textit{CA}}$ for a droplet evaporating in the CA mode. The dashed lines correspond to the initial values of (a) the concentration of particles given by $\phi _0(r)=1$ and (b) the mass of particles given by $\phi _0(r)h(r,0)=(1-r^2)/2$ , and the arrows indicate the direction of increasing $t$ .

Finally, eliminating $r_0$ between (4.12) and (4.15) yields an explicit expression for the concentration of particles within the droplet, namely

(4.16) \begin{equation} \phi = \left [1+\left (1-\dfrac {r^2}{R^2}\right )^{-3/2}\big (R^{-9/4}-1\big )\right ]^{1/3}. \end{equation}

Figure 7 shows $\phi$ given by (4.16) and $\phi h$ plotted as functions of $r$ at different times for a droplet evaporating in the CA mode. In particular, as figure 7(a) shows, $\phi$ is a monotonically increasing function of $r$ which takes its minimum value at $r=0$ and is singular at the receding contact line throughout the evaporation, i.e. $\phi =O ( (R-r )^{-1/2} )\to \infty$ as $r\to R^{-}$ for all $0\lt t\lt t_{\textit{CA}}$ . This singular behaviour of $\phi$ at the contact line is qualitatively similar to that for a droplet evaporating in the CR mode, as described in § 3.3. As figure 7(b) shows, $\phi h$ decreases everywhere within the droplet as it evaporates and is zero at the receding contact line throughout the evaporation. This behaviour is due to the fact that the advection of particles towards the contact line is balanced by the flux of particles into the receding contact line, as is the case for the CR mode (D’Ambrosio et al. Reference D’Ambrosio, Wilson, Wray and Duffy2023).

Substituting $t_{\textit{d}}\equiv 0$ , $R(t_{\textit{d}})\equiv 1$ , and the expression for $r_0(R,t_{\textit{dist}})$ given by (4.13) into (2.38) and evaluating the integral yields the explicit expression for the mass of particles in the bulk of the droplet, namely

(4.17) \begin{equation} M_{\textit{drop}}= 2\pi \int _0^{r_0\left (R,t_{\textit{dist}}\right )}{h(r,0)\,r\;\text {d}r}=M_0\left [ 1-\big (1-R^{9/4}\big )^{4/3}\right ]\!, \end{equation}

where, again, the initial mass of particles in the droplet is given by (2.31) to be $M_0=\pi /4$ . The mass of particles in the distributed deposit from (2.39) is given by

(4.18) \begin{equation} M_{\textit{dist}}=2\pi \int _{r_0\left (R,t_{\textit{dist}}\right )}^1{h(r,0)\,r\;\text {d}r}= M_0\big (1-R^{9/4}\big )^{4/3}. \end{equation}

For a droplet evaporating in the CA mode, the contact line never pins and therefore $M_{\textit{ring}}\equiv 0$ . Adding the expressions for $M_{\textit{drop}}$ and $M_{\textit{dist}}$ given by (4.17) and (4.18), respectively, and recalling that $M_{\textit{ring}}\equiv 0$ confirms that the total mass of particles is conserved as the droplet evaporates, i.e. that $M_{\textit{drop}}+M_{\textit{ring}}+M_{\textit{dist}}=M_0$ .

Substituting the expression for $r_0(R,t_{\textit{dist}})$ given by (4.13) into (2.41) and recalling that $t_{\textit{d}}=0$ yields the explicit expression for the density of the distributed deposit on the substrate, namely

(4.19) \begin{equation} \phi _{\textit{dist}}=\dfrac {3r^{1/4}}{8}\big (1-r^{9/4}\big )^{1/3} \end{equation}

for $0\leqslant r\leqslant 1$ . Specifically, $\phi _{\textit{dist}}$ approaches zero at both the centre of the droplet and at the contact line, i.e. $\phi _{\textit{dist}}=O (r^{1/4} )\to 0^{+}$ as $r\to 0^+$ and $\phi _{\textit{dist}}=O ((1-r)^{1/3} )\to 0^{+}$ as $r\to 1^{-}$ . The former is a consequence of the outward radial flow that transports particles away from the centre of the droplet throughout the evaporation. As we shall discuss in § 4.4, this behaviour is qualitatively different from the results obtained for spatially uniform evaporation.

Figure 8.

Plot of (a) the evolutions of $M_{\textit{drop}}/M_0$ (solid line) and $M_{\textit{dist}}/M_0$ (dashed line) given by (4.17) and (4.18), respectively, as functions of $t/t_{\textit{CA}}$ and (b) the density of the distributed deposit $\phi _{\textit{dist}}$ given by (4.19) as a function of $r$ for a droplet undergoing diffusion-limited evaporation in the CA mode.

Figure 8 shows a plot of the evolutions of $M_{\textit{drop}}/M_0$ and $M_{\textit{dist}}/M_0$ given by (4.17) and (4.18), respectively, as functions of $t/t_{\textit{CA}}$ and a plot of $\phi _{\textit{dist}}$ given by (4.19) as a function of $r$ for a droplet evaporating in the CA mode. In particular, as figure 8(a) shows, $M_{\textit{drop}}/M_0$ decreases monotonically from $M_{\textit{drop}}/M_0=1$ at $t=0$ to $M_{\textit{drop}}/M_0=0$ at $t=t_{\textit{CA}}$ and $M_{\textit{dist}}/M_0$ increases monotonically in $t/t_{\textit{CA}}$ , confirming that all of the particles are eventually deposited in a distributed deposit on the substrate in the region $0\leqslant r\leqslant 1$ in the CA mode. Specifically, particles are advected towards the receding contact line by the outward radial flow and are therefore continually deposited on the substrate in the region swept by the moving contact line throughout the evaporation. This results in a near-uniform deposit, as illustrated by the density profile of the mass of particles on the substrate in figure 8(b).

As discussed previously, the behaviour of the flow and the concentration of particles within the droplet in the CA mode is qualitatively similar to that predicted for a droplet evaporating in the CR mode. However, the presence of the receding contact line results in a qualitatively different deposit type, i.e. a switch from a single ring deposit in the CR mode to a near-uniform deposit in the CA mode.

4.4. Comparison with previous theoretical and experimental studies

As discussed in § 1, the spatially uniform evaporation model is sometimes used to approximate the diffusion-limited model. This is because it is simpler to use and has also been shown to predict qualitatively similar behaviour to that for diffusion-limited evaporation for the flow within and deposition from a droplet evaporating in the case where the contact line is pinned (see, for example, Boulogne et al. Reference Boulogne, Ingremeau and Stone2017; D’Ambrosio et al. Reference D’Ambrosio, Wilson, Wray and Duffy2023; Mills Reference Mills2023). Here we give a brief summary of the flow within, and the deposition from, a droplet undergoing spatially uniform evaporation in the CA mode and compare these predictions with the present results given in § 4.3 for diffusion-limited evaporation. In addition, we will validate the present theoretical approach by comparing both predictions with experimental results available in the literature.

4.4.1. Spatially uniform evaporation

For a spatially uniform evaporative flux of the form $J=4/(\pi R)$ , the total evaporative flux $F$ from (2.5) is the same as that for diffusion-limited evaporation, namely $F=4R$ . Therefore, the evolution of the droplet undergoing spatially uniform evaporation in the CA mode is given by the results for the diffusion-limited model summarised in § 4.1. However, as we shall see, the flow within and the deposition of particles from the droplet are not the same.

In particular, it may be shown that the fluid flux $Q$ from (2.24) and the depth-averaged radial velocity $\bar {u}$ from (2.22) are

(4.20a,b) \begin{equation} Q=-\dfrac {2r}{3\pi R}\left (1-\dfrac {r^2}{R^2}\right )\!, \quad \bar {u}=-\dfrac {4r}{3\pi R^2} \end{equation}

(see, for example, Freed-Brown Reference Freed-Brown2014). The expressions for the velocities $u$ and $w$ may be determined in the same manner as described in § 4.2, but are omitted here for brevity. Figure 6(b) shows instantaneous streamlines of the flow within a droplet evaporating in the CA mode for spatially uniform evaporation calculated at $t=t_{\textit{CA}}/2=3\pi /64$ . In particular, as figure 6(b) shows, the flow is inwards and upwards away from the contact line. Therefore, in contrast to the case of diffusion-limited evaporation, for spatially uniform evaporation the depth-averaged radial velocity is inwards towards the centre of the droplet throughout the evaporation, i.e. $\bar {u}\leqslant 0$ for all $0\leqslant t\leqslant t_{\textit{CA}}$ , and hence particles are advected towards the centre of the droplet. However, the contact line recedes at a faster rate than the maximum inward fluid velocity, which occurs at the contact line, i.e.

(4.21) \begin{equation} \left \lvert \dfrac {\text {d}R}{\text {d}t}\right \rvert =4\left \lvert \bar {u}(R,t)\right \rvert \gt \left \lvert \bar {u}(R,t)\right \rvert \geqslant \left \lvert \max _r\bar {u}(r,t)\right \rvert \end{equation}

for $0\lt R\lt 1$ . Therefore, particles are captured by the receding contact line as the droplet evaporates and are continually deposited on the substrate in the region swept by the moving contact line throughout the evaporation. In particular, for spatially uniform evaporation it may be shown from (2.29) that $r_0(r,t)=r R^{-1/4}$ , and therefore that the evolution of the mass of particles in the bulk of the droplet and in the distributed deposit, given by (2.38) and (2.39), are

(4.22a,b) \begin{equation} M_{\textit{drop}}=M_0\big (2R^{3/2}-R^3\big ), \quad M_{\textit{dist}}=M_0\big (1-R^{3/2}\big )^2=M_0-M_{\textit{drop}}, \end{equation}

respectively. The density profile of the distributed deposit from (2.41) is then

(4.23) \begin{equation} \phi _{\textit{dist}}=\dfrac {3}{8\sqrt {r}}\big (1-r^{3/2}\big ) \end{equation}

(see Freed-Brown Reference Freed-Brown2014).

Figure 9.

Plot of (a) the evolutions of $M_{\textit{drop}}/M_0$ (solid line) and $M_{\textit{dist}}/M_0$ (dashed line) given by (4.22) as functions of $t/t_{\textit{CA}}$ and (b) the density of the distributed deposit $\phi _{\textit{dist}}$ given by (4.23) as a function of $r$ for a droplet undergoing spatially uniform evaporation in the CA mode.

Figure 9 shows a plot of the evolutions of $M_{\textit{drop}}/M_0$ and $M_{\textit{dist}}/M_0$ given by (4.22) as functions of $t/t_{\textit{CA}}$ and a plot of $\phi _{\textit{dist}}$ given by (4.23) as a function of $r$ for a droplet undergoing spatially uniform evaporation in the CA mode. In particular, as figure 9(a) shows, despite the qualitatively different behaviour of the flow within the droplet, the evolutions of $M_{\textit{drop}}$ and $M_{\textit{dist}}$ are qualitatively the same as those for a droplet undergoing diffusion-limited evaporation in the CA mode shown in figure 8(a), i.e. all of the particles are eventually deposited in the distributed deposit. However, the density of the distributed deposit is qualitatively different. Specifically, $\phi _{\textit{dist}}$ given by (4.23) is a monotonically decreasing function of $r$ which takes its minimum value of zero at $r=1$ and is square-root singular at $r=0$ , as shown in figure 9(b). The qualitative change in behaviour of the depth-averaged radial flow from outwards for diffusion-limited evaporation to inwards for spatially uniform evaporation results in a qualitatively different deposit type, i.e. a switch from a near-uniform deposit for diffusion-limited evaporation to a peak deposit for spatially uniform evaporation. We note that Freed-Brown (Reference Freed-Brown2015) observed this difference in the resulting deposit types when comparing their theoretical predictions for the deposit density profile for a droplet undergoing spatially uniform evaporation with numerical calculations for diffusion-limited evaporation. In summary, whereas the spatially uniform evaporation model provides a good approximation to the diffusion-limited model for the flow within and the deposition from a droplet evaporating with a pinned contact line (see, for example, Boulogne et al. Reference Boulogne, Ingremeau and Stone2017; D’Ambrosio et al. Reference D’Ambrosio, Wilson, Wray and Duffy2023; Mills Reference Mills2023), it is not appropriate to use it when the contact line is receding.

4.4.2. Validation against previous experimental studies

The contrasting predictions for the deposit of the diffusion-limited and spatially uniform evaporation models raises the question as to which model most accurately predicts experimental observations. As discussed in § 1, previous experimental studies have observed uniform, near-uniform and doughnut deposits from a thin droplet evaporating in the CA mode (see, for example, Kajiya et al. Reference Kajiya, Monteux, Narita, Lequeux and Doi2009; Thokchom et al. Reference Thokchom, Zhou, Kim, Ha and Kim2017; Kim et al. Reference Kim, Rokoni, Kaneelil, Cui, Han and Sun2019; Lee & Kim Reference Lee and Kim2021; Matavž et al. Reference Matavž, Uršič, Močivnik, Richter, Humar, Čopar, Malič and Bobnar2022), indicating that the diffusion-limited model provides a qualitatively better prediction than spatially uniform evaporation for these experiments. In order to confirm this we will carry out a quantitative comparison with the experimental studies by Kajiya et al. (Reference Kajiya, Monteux, Narita, Lequeux and Doi2009), Kim et al. (Reference Kim, Rokoni, Kaneelil, Cui, Han and Sun2019) and Matavž et al. (Reference Matavž, Uršič, Močivnik, Richter, Humar, Čopar, Malič and Bobnar2022). Specifically, these experiments concern the deposition of polymers from water droplets on glass substrates (Kajiya et al. Reference Kajiya, Monteux, Narita, Lequeux and Doi2009; Kim et al. Reference Kim, Rokoni, Kaneelil, Cui, Han and Sun2019) and from ethylene glycol droplets on coated silicon wafers (Matavž et al. Reference Matavž, Uršič, Močivnik, Richter, Humar, Čopar, Malič and Bobnar2022), and report profiles of the final deposit (figure 6a in Kajiya et al. Reference Kajiya, Monteux, Narita, Lequeux and Doi2009; figure 9a in Kim et al. Reference Kim, Rokoni, Kaneelil, Cui, Han and Sun2019; figure 4d in Matavž et al. Reference Matavž, Uršič, Močivnik, Richter, Humar, Čopar, Malič and Bobnar2022), which enables comparisons with the theory.

Figure 10.

Plot of the density profile of the mass of particles on the substrate $\phi _{\textit{dist}}$ as a function of $r$ for a droplet undergoing diffusion-limited (solid black line) and spatially uniform (solid grey line) evaporation in the CA mode given by (4.19) and (4.23), respectively, as well as experimental results of Kajiya et al. (Reference Kajiya, Monteux, Narita, Lequeux and Doi2009) (dotted line), Kim et al. (Reference Kim, Rokoni, Kaneelil, Cui, Han and Sun2019) (dashed line) and Matavž et al. (Reference Matavž, Uršič, Močivnik, Richter, Humar, Čopar, Malič and Bobnar2022) (dot-dashed line).

Figure 10 shows a plot of $\phi _{\textit{dist}}$ as a function of $r$ for a droplet undergoing diffusion-limited and spatially uniform evaporation in the CA mode given by (4.19) and (4.23), respectively, as well as the experimental results of Kajiya et al. (Reference Kajiya, Monteux, Narita, Lequeux and Doi2009), Kim et al. (Reference Kim, Rokoni, Kaneelil, Cui, Han and Sun2019) and Matavž et al. (Reference Matavž, Uršič, Močivnik, Richter, Humar, Čopar, Malič and Bobnar2022). Figure 10 shows that, despite the simplicity of the present model described in § 2, the prediction for $\phi _{\textit{dist}}$ for diffusion-limited evaporation given by (4.19) compares well with the results obtained from physical experiments, with particularly good agreement with the studies by Kim et al. (Reference Kim, Rokoni, Kaneelil, Cui, Han and Sun2019) and Matavž et al. (Reference Matavž, Uršič, Močivnik, Richter, Humar, Čopar, Malič and Bobnar2022). By contrast, the theoretical prediction for spatially uniform evaporation is qualitatively different from the experimental results shown, as previously discussed.

5.1. The evolution of the droplet

In the SS mode the evolution of a droplet consists of a CR phase, i.e. with $R\equiv 1$ and $\theta =\theta (t)$ , until a time $t=t^*$ at which the critical receding contact angle $\theta =\theta ^*$ $(0\leqslant \theta ^*\leqslant 1)$ is reached, followed by a CA phase, i.e. with $R=R(t)$ and $\theta \equiv \theta ^*$ .

For a droplet evaporating in the SS mode (2.6) becomes

(5.1) \begin{equation} \dfrac {\text {d}V}{\text {d}t}=\dfrac {\pi }{4}\dfrac {\text {d}\theta }{\text {d}t}=-4 \end{equation}

for $0\leqslant t\leqslant t^*$ , and

(5.2) \begin{equation} \dfrac {\text {d}V}{\text {d}t}=\dfrac {3\pi \theta ^*R^2}{4}\dfrac {\text {d}R}{\text {d}t}=-4R \end{equation}

for $t^*\leqslant t\leqslant t_{\textit{SS}}$ . Solving (5.1) and (5.2) yields the well-known explicit evolution of the droplet, namely

(5.3) \begin{align}&\qquad\,\,\, R\equiv 1, \quad \theta =1-\dfrac {16}{\pi }t \quad \text{for} \quad 0\leqslant t\leqslant t^*, \end{align}

(5.4) \begin{align}& \theta \equiv 1, \quad R=\sqrt {1-\dfrac {32}{3\pi \theta ^*}\left (t-t^*\right )} \quad \text{for} \quad t^*\leqslant t\leqslant t_{\textit{SS}} , \end{align}

with $h$ and $V$ given by (2.13) and (2.14), respectively, the critical time $t^*$ obtained by setting $\theta (t^*)=\theta ^*$ is

(5.5) \begin{equation} t^*=\dfrac {\pi \left (1-\theta ^*\right )}{16}, \end{equation}

and the lifetime of the droplet $t_{\textit{SS}}$ obtained by setting $R(t_{\textit{SS}})=0$ is

(5.6) \begin{equation} t_{\textit{SS}}=\dfrac {\pi \left (2+\theta ^*\right )}{32} \end{equation}

(see, for example, Wilson & Duffy Reference Wilson and Duffy2022).

5.2. The flow within the droplet

Substituting the expressions for $J$ and $h$ given by (2.4) and (2.13), respectively, into (2.24) and evaluating the integral gives the following solution for $Q$ in the SS mode:

(5.7) \begin{equation} Q=\dfrac {2}{\pi r} {\begin{cases} \sqrt {1-r^2}-\left (1-r^2\right )^2 \quad &\text{for} \quad 0\leqslant t\leqslant t^*,\\[8pt] R\left [\sqrt {1-\dfrac {r^2}{R^2}}-\left (1+\dfrac {r^2}{3R^2}\right )\left (1-\dfrac {r^2}{R^2}\right )\right ] \quad &\text{for} \quad t^*\leqslant t\leqslant t_{\textit{SS}}, \end{cases}} \end{equation}

and hence the depth-averaged velocity $\bar {u}$ is

(5.8) \begin{equation} \bar {u}=\dfrac {4}{\pi r} {\begin{cases} \dfrac {1}{\theta }\Big [\!\left(1-r^2\right)^{-1/2}-\left(1-r^2\right)\!\Big ] \quad &\text{for} \quad 0\leqslant t\leqslant t^*,\\[8pt] \dfrac {1}{\theta ^* }\left [\left (1-\dfrac {r^2}{R^2}\right )^{-1/2}-\left (1+\dfrac {r^2}{3R^2}\right )\right ] \quad &\text{for} \quad t^*\leqslant t\leqslant t_{\textit{SS}}. \end{cases}} \end{equation}

The behaviour of $Q$ and $\bar {u}$ given by (5.7) and (5.8), respectively, in the CR and CA phases corresponds to the behaviour for a droplet evaporating in the CR and CA modes, respectively, and so are not plotted here for brevity. In particular, the depth-averaged radial velocity is outwards towards the contact line in both phases, i.e. $Q\geqslant 0$ and $\bar {u}\geqslant 0$ for all $0\leqslant t\leqslant t_{\textit{SS}}$ , $Q$ is independent of $t$ during the CR phase, and $\bar {u}$ is singular at the contact line in both phases. Note that (5.7) and (5.8) recover the solutions for a droplet evaporating in the CR and CA modes described in § 3.2 and § 4.2 when $\theta ^*=0$ and $\theta ^*=1$ , respectively.

5.3. The deposition of particles

For a droplet evaporating in the SS mode, the characteristic equations (2.29) may be expressed as

(5.9a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}\theta }=\dfrac {\text {d}r/\text {d}t}{\text {d}\theta /\text {d}t}= \dfrac {\bar {u}}{\text {d}\theta /\text {d}t}, \quad \dfrac {\text {d}\phi }{\text {d}\theta }=\dfrac {\text {d}\phi /\text {d}t}{\text {d}\theta / \text {d}t} =\dfrac { J\phi }{h\,\text {d}\theta /\text {d}t}, \end{equation}

subject to the initial condition $\phi _0(r)=1$ , for $0\leqslant t\leqslant t^*$ , and as

(5.10a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}R}=\dfrac {\text {d}r/\text {d}t}{\text {d}R/\text {d}t}= \dfrac {\bar {u}}{\text {d}R/\text {d}t}, \quad \dfrac {\text {d}\phi }{\text {d}R}=\dfrac {\text {d}\phi /\text {d}t}{\text {d}R/\text {d}t} =\dfrac { J\phi }{h\,\text {d}R/\text {d}t}, \end{equation}

subject to the initial condition $\phi (r,t^*)=\phi ^*(r)$ , for $t^*\leqslant t\leqslant t_{\textit{SS}}$ .

For $0\leqslant t\leqslant t^*$ , substituting the expressions for $J$ , $h$ , $\text {d}\theta /\text {d}t$ and $\bar {u}$ from (2.4), (2.13), (5.1) and (5.8) into (5.9) gives

(5.11a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}\theta }=-\dfrac {1}{4\theta r}\left [\big (1-r^2\big )^{-1/2}-\big (1-r^2\big )\right ]\!, \quad \dfrac {\text {d}\phi }{\text {d}\theta }=-\dfrac {\phi }{4\theta }\big (1-r^2\big )^{-3/2}. \end{equation}

The characteristic equations given by (5.11) are exactly those given by (3.6) for a droplet evaporating in the CR mode with the same prescribed initial condition and so may be solved in the same manner as described in § 3.3. In particular, the explicit expressions for $r_0(r,t)$ , $r_0(1,t_{\textit{ring}})$ and $\phi$ are given by (3.7), (3.8) and (3.10), respectively, during the CR phase.

For $t^*\leqslant t\leqslant t_{\textit{SS}}$ , substituting the expressions for $J$ , $h$ , $\text {d}R/\text {d}t$ and $\bar {u}$ from (2.4), (2.13), (5.2) and (5.8) into (5.10) gives

(5.12a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}R}=\dfrac {3R}{4r}\left [\left (1+\dfrac {r^2}{3R^2}\right )- \left (1-\dfrac {r^2}{R^2}\right )^{-1/2} \right ]\!, \quad \dfrac {\text {d}\phi }{\text {d}R}=-\dfrac {3\phi }{4R}\left (1-\dfrac {r^2}{R^2}\right )^{-3/2}. \end{equation}

The characteristic equations given by (5.12) are exactly those given by (4.10) for a droplet evaporating in the CA mode with the only difference being the initial condition. Therefore, (5.12a ) may be solved in the same manner as described in § 4 to yield $r_0(r,t)$ and $r_0(R,t_{\textit{dist}})$ given by (4.12) and (4.13), respectively. Substituting the expression for $r$ given implicitly by (4.12) into (5.12b ) the equation becomes separable and may be rewritten as

(5.13) \begin{equation} \int _{\phi ^*(r_0)}^{\phi }{\dfrac {1}{\hat {\phi }}\,\text {d}\hat {\phi }}=- \int _{1}^{R}{\dfrac {3\hat {R}^{5/4}} {4\left [\hat {R}^{9/4}-1+\left (1-r_0^2\right )^{3/2}\right ]}\text {d}\hat {R}}, \end{equation}

where $\phi ^*(r)=\phi (r,t^*)$ denotes the concentration of particles within the droplet at $t=t^*$ and may be determined from (3.10). Performing the integral in (5.13) shows that $\phi$ is given parametrically by

(5.14) \begin{equation} \phi =\left [\dfrac {{\theta ^*}^{-3/4}-1+\left (1-r_0^2\right )^{3/2}} {R^{9/4}-1+\left (1-r_0^2\right )^{3/2}}\right ]^{1/3}. \end{equation}

Finally, eliminating $r_0$ between (4.12) and (5.14) yields an explicit expression for the concentration of particles within the droplet during the CA phase, namely

(5.15) \begin{equation} \phi =\left [1+\left (1-\dfrac {r^2}{R^2}\right )^{-3/2}\big ({\theta ^*}^{-3/4}R^{-9/4}-1\big )\right ]^{1/3}. \end{equation}

Note that (3.10) holds for the entire lifetime of the droplet when $\theta ^*=0$ and (5.15) recovers the solution for a droplet evaporating in the CA mode given by (4.16) when $\theta ^*=1$ . Figure 11 shows $\phi$ given by (3.10) and (5.15) and $\phi h$ plotted as functions of $r$ at different times for a droplet evaporating in the SS mode when $\theta ^*=1/2$ . The behaviour of $\phi$ in the CR and CA phases corresponds to the behaviour for a droplet evaporating in the CR and CA modes, respectively. In particular, as figure 11(a) shows, $\phi$ is a monotonically increasing function of $r$ and is singular at the contact line throughout the evaporation in both phases.

Figure 11.

Plot of (a) the concentration of particles $\phi$ and (b) the mass per unit area $\phi h$ within the droplet as functions of $r$ at $t=0$ , $t^*/3$ , $2t^*/3$ , $t^*$ , $t^*+(t_{\textit{SS}}-t^*)/6$ , $t^*+(t_{\textit{SS}}-t^*)/3$ , …, $t^*+5(t_{\textit{SS}}-t^*)/6$ for a droplet evaporating in the SS mode when $\theta ^*=1/2$ . The dashed lines correspond to the initial values of (a) the concentration of particles given by $\phi _0(r)=1$ and (b) the mass of particles given by $\phi _0(r)h(r,0)=(1-r^2)/2$ , the dot-dashed lines correspond to the values at $t=t^*$ , and the arrows indicate the direction of increasing $t$ .

Substituting $t_{{p}}=0$ , $t_{\textit{d}}=t^*$ , $R_{{p}}=R_{\textit{d}}=1$ and the expressions for $r_0(1,t_{\textit{ring}})$ and $r_0(R,t_{\textit{dist}})$ given by (3.8) and (4.13), respectively, into (2.34) and (2.38) and evaluating the integrals yields the explicit expressions for the mass of particles in the bulk of the droplet, namely

(5.16) \begin{align} M_{\textit{drop}}&=2\pi {\begin{cases} {\displaystyle\int}_{\!\! \kern-2pt 0}^{r_0(1,t_{\textit{ring}})}{h(r,0)\,r\;\text {d}r} \quad &\text{for} \quad 0\leqslant t\leqslant t^*,\\[6pt] {\displaystyle\int}_{\!\! \kern-2pt 0}^{r_0\left (R,t_{\textit{dist}}\right )}{\phi ^*(r)\, h(r,t^*)\,r\;\text {d}r} \quad &\text{for} \quad t^*\leqslant t\leqslant t_{\textit{SS}}, \end{cases}} \nonumber \\ &=M_0 {\begin{cases} 1-\left (1-\theta ^{3/4}\right )^{4/3} \quad &\text{for} \quad 0\leqslant t\leqslant t^*,\\[6pt] 1-\big (1-{\theta ^*}^{3/4}R^{9/4}\big )^{4/3} \quad &\text{for} \quad t^*\leqslant t\leqslant t_{\textit{SS}}, \end{cases}} \end{align}

where, again, the initial mass of particles in the droplet is given by (2.31) to be $M_0=\pi /4$ . For $0\leqslant t\leqslant t^*$ , the mass of particles in the ring deposit from (2.35) is given by

(5.17) \begin{equation} M_{\textit{ring}}= 2\pi \int _{r_0(1,t_{\textit{ring}})}^1{h(r,0)\,r\;\text {d}r}= M_0\big (1-\theta ^{3/4}\big )^{4/3}. \end{equation}

For $t^*\leqslant t\leqslant t_{\textit{SS}}$ , the mass of particles in the distributed deposit from (2.39) can be calculated explicitly as

(5.18) \begin{align} M_{\textit{dist}}= 2\pi \int _{r_0\left (R,t_{\textit{dist}}\right )}^1{\phi ^*(r)\, h(r,t^*)\,r\;\text {d}r} =M_0\left [\big (1-{\theta ^*}^{3/4}R^{9/4}\big )^{4/3}-\big (1-{\theta ^*}^{3/4}\big )^{4/3}\right ]\!. \end{align}

Figure 12.

Plot of the evolutions of (a) $M_{\textit{drop}}/M_0$ given by (5.16), (b) $M_{\textit{ring}}/M_0$ given by (5.17), and (c) $M_{\textit{dist}}/M_0$ given by (5.18) as functions of $t/t_{\textit{SS}}$ for $\theta ^*=1/6$ , $2/6$ , …, $5/6$ , and (d) $M_{\textit{ring}}(t_{\textit{SS}})/M_0$ (solid line) and $M_{\textit{dist}}(t_{\textit{SS}})/M_0$ (dot-dashed line) given by (5.19) as functions of $\theta ^*$ for a droplet evaporating in the SS mode. The squares in (a) correspond to $t=t^*$ , the dots in (b) and (c) correspond to the values of $M_{\textit{ring}}/M_0$ and $M_{\textit{dist}}/M_0$ at $t=t_{\textit{SS}}$ , respectively, and the dotted and dashed lines correspond to the solutions when $\theta ^*=0$ (CR mode) and $\theta ^*=1$ (CA mode), respectively. The arrows indicate the direction of increasing $\theta ^*$ .

The mass of particles on the substrate at the end of the evaporation is therefore split between the ring deposit and the distributed deposit with the final mass in each given by evaluating $M_{\textit{ring}}$ given by (5.17) and $M_{\textit{dist}}$ given by (5.18) at $t=t^*$ and $t=t_{\textit{SS}}$ , respectively, to yield

(5.19a,b) \begin{align} M_{\textit{ring}}(t^*)=M_{\textit{ring}}(t_{\textit{SS}})= M_0\big (1-{\theta ^*}^{3/4}\big )^{4/3}, \,\, M_{\textit{dist}}(t_{\textit{SS}})= M_0\left [1-\big (1-{\theta ^*}^{3/4}\big )^{4/3}\right ]\!. \end{align}

Note that (5.16), (5.17) and (5.18) recover the solutions for a droplet evaporating in the CR mode, given by (3.11) and (3.12), and the CA mode, given by (4.17) and (4.18), when $\theta ^*=0$ and $\theta ^*=1$ , respectively.

Figure 12 shows plots of the evolutions of $M_{\textit{drop}}/M_0$ given by (5.16), $M_{\textit{ring}}/M_0$ given by (5.17), and $M_{\textit{dist}}/M_0$ given by (5.18) as functions of $t/t_{\textit{SS}}$ for a range of values of $\theta ^*$ , and of $M_{\textit{ring}}(t_{\textit{SS}})/M_0$ and $M_{\textit{dist}}(t_{\textit{SS}})/M_0$ given by (5.19) as functions of $\theta ^*$ for a droplet evaporating in the SS mode. In particular, the deposit is a combination of a single ring deposit at $r=1$ and a near-uniform deposit in the region $0\leqslant r\leqslant 1$ in the SS mode. The final masses in the ring and the distributed deposits depend upon the value of the receding contact angle, and are monotonically decreasing and increasing functions of $\theta ^*$ , respectively, as illustrated in figure 12(d). Moreover, there is a critical receding contact angle, $\theta ^*\simeq 0.300$ , at which the final masses in the ring deposit and in the distributed deposit are the same, and above which more mass is deposited onto the substrate in the distributed deposit than in the ring deposit.

Substituting the expression for $r_0(R,t_{\textit{dist}})$ given by (4.13) into (2.41) and recalling that $t_{\textit{d}}=t^*$ yields the explicit expression for the density of the distributed deposit on the substrate, namely

(5.20) \begin{equation} \phi _{\textit{dist}}=\dfrac {3{\theta ^*}^{3/4}r^{1/4}}{8}\big (1-{\theta ^*}^{3/4}r^{9/4}\big )^{1/3}. \end{equation}

Specifically, $\phi _{\textit{dist}}$ approaches zero at the centre of the droplet, i.e. $\phi _{\textit{dist}}=O (r^{1/4} )\to 0^{+}$ as $r\to 0^+$ , and is finite and non-zero for all $0\lt \theta ^*\lt 1$ at the contact line, i.e. $\phi _{\textit{dist}}(1)=3{\theta ^*}^{3/4} (1-{\theta ^*}^{3/4})^{1/3}$ . Note that (5.20) recovers the solution for a droplet evaporating in the CA mode given by (4.19) when $\theta ^*=1$ . Since $\phi _{\textit{dist}}$ describes the density of the distributed deposit, i.e. the mass of particles on the substrate deposited via the receding contact line, it does not include the mass in the ring deposit that forms during the CR phase of the SS mode.

Figure 13 shows a plot of $\phi _{\textit{dist}}$ given by (5.20) as a function of $r$ for a range of values of $\theta ^*$ for a droplet evaporating in the SS mode. In particular, as figure 13 shows, $\phi _{\textit{dist}}$ is nearly uniform and non-zero at the contact line for all $0\lt \theta ^*\lt 1$ . In contrast to the CA mode when $\theta ^*=1$ , the non-zero density of the distributed deposit at the contact line is due to the presence of particles at the contact line when the droplet initially de-pins at $t_{\textit{d}}=t^*$ .

Figure 13.

Plot of the density profile of the mass of particles on the substrate $\phi _{\textit{dist}}$ given by (5.20) as a function of $r$ when $\theta ^*=1/6$ , $2/6$ , …, $5/6$ for a droplet evaporating in the SS mode. The dashed line corresponds to the solution when $\theta ^*=1$ (CA mode), and the arrow indicates the direction of increasing $\theta ^*$ .

5.4. Validation against previous experimental studies

The behaviour that the mass left behind on the substrate is split between a ring deposit and some distributed deposit within the original footprint of the droplet when the droplet evaporates in the SS mode has been reported in the literature on several occasions (see, for example, Kim et al. Reference Kim, Ahn, Kim and Zin2007; Chhasatia & Sun Reference Chhasatia and Sun2011; Yunker et al. Reference Yunker, Still, Lohr and Yodh2011; Li et al. Reference Li, Lv, Li, Quéré and Zheng2015; Iqbal et al. Reference Iqbal, Majhy, Shen and Sen2018; Kim et al. Reference Kim, Pack, Rokoni, Kaneelil and Sun2018; Patil, Bhardwaj & Sharma Reference Patil, Bhardwaj and Sharma2018; Yu et al. Reference Yu, Wang, Zhu, Zhou and Zhou2019), which is consistent with the predictions of the present model. In order to make a quantitative comparison with these experimental observations, we compare the final mass in the ring deposit and the distributed deposit with the experimental measurements by Yunker et al. (Reference Yunker, Still, Lohr and Yodh2011), Li et al. (Reference Li, Lv, Li, Quéré and Zheng2015) and Kim et al. (Reference Kim, Pack, Rokoni, Kaneelil and Sun2018). Specifically, these experiments concern the deposition of spherical polystyrene particles from water droplets evaporating on glass substrates and either report values for the mass within the ring deposit (figure 1b in Li et al. Reference Li, Lv, Li, Quéré and Zheng2015) or provide deposit profiles for which this value may be extracted (figure 1d in Yunker et al. Reference Yunker, Still, Lohr and Yodh2011; figure 6a in Kim et al. Reference Kim, Pack, Rokoni, Kaneelil and Sun2018), as well as values for the time at which the contact line de-pins, which enables comparisons with the theory.

Figure 14 shows a plot of $M_{\textit{ring}}(t_{\textit{SS}})/M_0$ and $M_{\textit{dist}}(t_{\textit{SS}})/M_0$ for a droplet undergoing diffusion-limited evaporation in the SS mode given by (5.19), as well as the experimental results of Yunker et al. (Reference Yunker, Still, Lohr and Yodh2011), Li et al. (Reference Li, Lv, Li, Quéré and Zheng2015) and Kim et al. (Reference Kim, Pack, Rokoni, Kaneelil and Sun2018). In particular, as figure 14 shows, despite the simplicity of the present model, the theoretical predictions agree very well with the experimental results for a range of de-pinning angles showing that the present model provides an accurate prediction for the mass of particles in the ring deposit after evaporation.

Figure 14.

Comparison between the theoretical predictions for the final mass deposited on the substrate in the ring deposit $M_{\textit{ring}}(t_{\textit{SS}})/M_0$ (solid line) and the distributed deposit $M_{\textit{dist}}(t_{\textit{SS}})/M_0$ (dot-dashed line) given by (5.19) for a droplet evaporating in the SS mode with experimental data from Li et al. (Reference Li, Lv, Li, Quéré and Zheng2015) (squares $\blacksquare$ ), Yunker et al. (Reference Yunker, Still, Lohr and Yodh2011) (diamonds ) and Kim et al. (Reference Kim, Pack, Rokoni, Kaneelil and Sun2018) (triangles $\blacktriangle$ ).

6.1. The evolution of the droplet

In the SJ mode the evolution of a droplet consists of an infinite series of CR phases separated by an infinite series of jump phases in which the contact angle jumps instantaneously from $\theta _{\textit{min}}$ to $\theta _{\textit{max}}$ , where $0\leqslant \theta _{\textit{min}}\leqslant \theta _{\textit{max}}\leqslant 1$ , with a corresponding jump decrease in the contact radius. We denote the constant value of the pinned contact radius during the $n{\text{th}}$ CR phase by $R_n$ , which lasts from $t=t_{n-1}$ to $t=t_n$ .

We assume that mass is conserved during the $n^{\text{}}$ th jump, i.e. $V(R_n,\theta _{\textit{min}})=V(R_{n+1},\theta _{\textit{max}})$ at $t=t_n$ . Therefore, from (2.14), the contact radius $R=R_n$ satisfies

(6.1) \begin{equation} R_{n}=\left (\dfrac {\theta _{\textit{min}}}{\theta _{\textit{max}}}\right )^{1/3}R_{n-1}=\left (\dfrac {\theta _{\textit{min}}}{\theta _{\textit{max}}}\right )^{(n-1)/3}. \end{equation}

During the $n{\text{th}}$ CR phase, lasting from $t=t_{n-1}$ to $t=t_n$ , (2.6) becomes

(6.2) \begin{equation} \dfrac {\text {d}V}{\text {d}t}=\dfrac {\pi R_n^3}{4}\dfrac {\text {d}\theta }{\text {d}t}=-4R_n. \end{equation}

Solving (6.2) yields the well-known explicit evolution of the droplet, namely

(6.3) \begin{align}&\qquad\qquad\qquad\qquad\quad R_1\equiv 1, \quad \theta =1-\dfrac {16}{\pi }t \quad \text{for} \quad 0\leqslant t\leqslant t_1, \end{align}

(6.4) \begin{align}& R_n\equiv \left (\dfrac {\theta _{\textit{min}}}{\theta _{\textit{max}}}\right )^{(n-1)/3}\!, \!\!\quad \theta =\theta _{\textit{max}}-\dfrac {16}{\pi R_n^2}\left (t-t_{n-1}\right ) \quad\!\! \text{for} \quad t_{n-1}\leqslant t\leqslant t_n,\, n=2,3,4,\ldots , \end{align}

with $h$ and $V$ given by (2.13) and (2.14), respectively, and the critical times $t_1$ and $t_n$ at which the droplet jumps satisfy $\theta (t_i)=\theta _{\textit{min}}$ are therefore given by

(6.5) \begin{equation} t_1=\dfrac {\pi }{16}\left (1-\theta _{\textit{min}}\right ) \end{equation}

and

(6.6) \begin{equation} t_n= \dfrac {\pi }{16}\left [1-\theta _{\textit{max}}+\left (\theta _{\textit{max}}-\theta _{\textit{min}}\right ) \left (\dfrac {1-\left (\dfrac {\theta _{\textit{min}}}{\theta _{\textit{max}}}\right )^{2n/3}}{1-\left (\dfrac {\theta _{\textit{min}}}{\theta _{\textit{max}}}\right )^{2/3}} \right )\right ] \end{equation}

(see, for example, Wilson & Duffy Reference Wilson and Duffy2022). The lifetime of the droplet $t_{\textit{SJ}}$ is found by taking the limit of (6.6) as $n\to \infty$ , which gives

(6.7) \begin{equation} t_{\textit{SJ}}= \dfrac {\pi }{16}\left [1-\theta _{\textit{max}}+\left (\theta _{\textit{max}}-\theta _{\textit{min}}\right ) \dfrac {\theta _{\textit{max}}^{2/3}}{\theta _{\textit{max}}^{2/3}-\theta _{\textit{min}}^{2/3}}\right ]\!. \end{equation}

6.2. The flow within the droplet

Substituting the expressions for $J$ and $h$ given by (2.4) and (2.13), respectively, into (2.24) and evaluating the integral gives the following solution for $Q$ in the SJ mode:

(6.8) \begin{equation} Q=\dfrac {2R_n}{\pi r}\left [\sqrt {1-\dfrac {r^2}{R_n^2}}-\left (1-\dfrac {r^2}{R_n^2}\right )^2\right ]\!, \end{equation}

and hence the depth-averaged velocity $\bar {u}$ is

(6.9) \begin{equation} \bar {u}=\dfrac {4}{\pi \theta r}\left [\left (1-\dfrac {r^2}{R_n^2}\right )^{-1/2} -\left (1-\dfrac {r^2}{R_n^2}\right )\right ]\!. \end{equation}

The behaviour of $Q$ and $\bar {u}$ given by (6.8) and (6.9), respectively, corresponds to the behaviour for a droplet evaporating in the CR mode, and so are not plotted here for brevity. In particular, in each CR phase, the depth-averaged radial velocity is outwards towards the pinned contact line $R_n$ , i.e. $Q\geqslant 0$ and $\bar {u}\geqslant 0$ for all $0\leqslant t\leqslant t_{\textit{SJ}}$ , $Q$ is independent of $t$ , and $\bar {u}$ is singular at the pinned contact line $R_n$ .

6.3. The deposition of particles

For a droplet evaporating in the SJ mode, the characteristic equations (2.29) may be expressed as

(6.10a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}\theta }=\dfrac {\text {d}r/\text {d}t}{\text {d}\theta /\text {d}t}= \dfrac {\bar {u}}{\text {d}\theta /\text {d}t}, \quad \dfrac {\text {d}\phi }{\text {d}\theta }=\dfrac {\text {d}\phi /\text {d}t}{\text {d}\theta /\text {d}t}= \dfrac {J\,\phi }{h\,\text {d}\theta /\text {d}t}, \end{equation}

subject to the initial condition $\phi _0(r)=1$ for $0\leqslant t\leqslant t_1$ , and $\phi (r,t_{n-1})=\phi _{n,0}(r)$ for $t_{n-1}\leqslant t\leqslant t_n$ and $n=2,3,4,\ldots$ .

For $0\leqslant t\leqslant t_1$ , substituting the expressions for $J$ , $h$ , $\text {d}\theta /\text {d}t$ and $\bar {u}$ from (2.4), (2.13), (6.2) and (6.9) into (6.10) gives

(6.11a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}\theta }=-\dfrac {1}{4\theta r}\Big [\big (1-r^2\big )^{-1/2}-\big (1-r^2\big )\Big ], \quad \dfrac {\text {d}\phi }{\text {d}\theta }=-\dfrac {\phi }{4\theta }\big (1-r^2\big )^{-3/2}. \end{equation}

The characteristic equations given by (6.11) are exactly those given by (3.5) for a droplet evaporating in the CR mode with the same prescribed initial condition and so may solved in the same manner as detailed in § 3.3. In particular, the explicit expressions for $r_0(r,t)$ , $r_0(1,t_{\textit{ring}})$ and $\phi$ are given by (3.7), (3.8) and (3.10), respectively, during the first CR phase.

For $t_{n-1}\leqslant t\leqslant t_n$ and $n=2,3,4,\ldots$ , substituting the expressions for $J$ , $h$ , $\text {d}\theta /\text {d}t$ and $\bar {u}$ from (2.4), (2.13), (6.2) and (6.9) into (6.10) gives

(6.12a,b) \begin{equation} \dfrac {\text {d}r}{\text {d}\theta }=-\dfrac {R_n^2}{4\theta r}\left [\left (1-\dfrac {r^2} {R_n^2}\right )^{-1/2} -\left (1-\dfrac {r^2}{R_n^2}\right )\right ]\!, \quad \dfrac {\text {d}\phi }{\text {d}\theta }=-\dfrac {\phi }{4\theta }\left (1-\dfrac {r^2}{R_n^2}\right )^{-3/2}. \end{equation}

Note that (6.12a ) is separable and using separation of variables gives

(6.13) \begin{equation} r_{0}(r,t)=R_n\sqrt {1-\left (\dfrac {\theta }{\theta _{\textit{max}}}\right )^{1/2} \left [\left (\dfrac {\theta }{\theta _{\textit{max}}}\right )^{-3/4}-1+\left (1- \dfrac {r^2}{R_n^2}\right )^{3/2}\right ]^{2/3}}, \end{equation}

where $r_{0}= r_{0}(r,t)$ denotes the initial radial position of particles at the beginning of the $n{\text{th}}$ CR phase, i.e. at $t=t_{n-1}$ , that are at radial position $r$ at time $t$ . In particular, the time it takes for particles to travel to the pinned contact line, i.e. to $r=R_n$ , where they are then deposited into the $n^{\text{}}$ th ring deposit, defined by $t=t_{\textit{ring}_n}$ , is given implicitly by $r_{0} (R_n,t_{\textit{ring}_n} )$ , namely

(6.14) \begin{equation} \quad r_{0}(R_n,t_{\textit{ring}_n})=R_n\sqrt {1-\left [1-\left (\dfrac {\theta }{\theta _{\textit{max}}}\right )^{3/4}\right ]^{2/3}}. \end{equation}

Substituting the expression for $r$ given implicitly by (6.13) into (6.12b ), the equation becomes separable and may be rewritten as

(6.15) \begin{equation} \int _{\phi _{n,0}(r_0)}^{\phi }{\dfrac {1}{\hat {\phi }}\,\text {d}\hat {\phi }}= -\int _{\theta _{\textit{max}}}^{\theta }{\dfrac {1}{4\hat {\theta }^{1/4}\theta _{\textit{max}}^{3/4}\left [ \left (\dfrac {\hat {\theta }}{\theta _{\textit{max}}}\right )^{3/4}-1+\left (1-\dfrac {r_{0}^2}{R_n^2}\right )^{3/2} \right ]}\,\text {d}\hat {\theta }}, \end{equation}

where $\phi _{n,0}(r)=\phi (r,t_{n-1})$ denotes the concentration of particles within the droplet at the beginning of the $n{\text{th}}$ CR phase, i.e. immediately after the ${(n-1)}{\text{th}}$ jump, which is yet to be determined. Performing the integral in (6.15) shows that $\phi$ is given parametrically by

(6.16) \begin{equation} \phi =\phi _{n,0}(r_0)\left [\dfrac {\left (1-\dfrac {r_0^2}{R_n^2}\right )^{3/2}} {\left (\dfrac {\theta }{\theta _{\textit{max}}}\right )^{3/4}-1+\left (1-\dfrac {r_0^2}{R_n^2}\right )^{3/2}}\right ]^{1/3}. \end{equation}

In order to determine an explicit solution for $\phi$ , we have to make an assumption regarding what happens to the particles within the droplet during the ${(n-1)}{\text{th}}$ jump so that we can determine $\phi _{n,0}(r)$ . Since the jump is theoretically instantaneous we assume conservation of the mass of particles within the droplet across the jump phase. It follows that $\phi _{n,0}(r)$ must satisfy

(6.17) \begin{equation} \int _0^{r}{\phi _{n,0}(\tilde {r})\,h_{n,0}(\tilde {r})\,\tilde {r}\;\text {d}\tilde {r}}= \int _0^{r^*}{\phi _{n-1,\textit{end}}(\tilde {r})\,h_{n-1,\textit{end}}(\tilde {r})\,\tilde {r}\;\text {d}\tilde {r}}, \end{equation}

where

(6.18) \begin{equation} r^*=\dfrac {R_{n-1}}{R_n}r, \end{equation}

$0\leqslant r\leqslant R_n$ and $0\leqslant r^*\leqslant R_{n-1}$ , $h_{n,0}(r)=h(r,t_{n-1})$ denotes the profile of the droplet at the beginning of the $n{\text{th}}$ CR phase, i.e. immediately after the ${(n-1)}{\text{th}}$ jump, and $\phi _{n-1,\textit{end}}(r)=\phi (r,t_{n-1})$ and $h_{n-1,\textit{end}}(r)=h(r,t_{n-1})$ denote the concentration of particles within the droplet and the profile of the droplet, respectively, at the end of the ${(n-1)}{\text{th}}$ CR phase, i.e. immediately before the ${(n-1)}{\text{th}}$ jump. Differentiating (6.17) with respect to $r$ and rearranging yields

(6.19) \begin{equation} \phi _{n,0}(r) =\phi _{n-1,\textit{end}}\left (r^*\right )\dfrac {h_{n-1,\textit{end}}(r^*)}{h_{n,0}(r)}\dfrac {r^*}{r}\dfrac {\text {d}r^*}{\text {d}r}, \end{equation}

which, using (6.18), simplifies to

(6.20) \begin{equation} \phi _{n,0}(r) =\phi _{n-1,\textit{end}}\left (\dfrac {R_{n-1}}{R_n}r\right )\!, \end{equation}

since, from (2.13),

(6.21) \begin{equation} \dfrac {h_{n-1,\textit{end}}\left ( \dfrac {R_{n-1}}{R_n}r\right )}{h_{n,0}(r)}\left (\dfrac {R_{n-1}}{R_n}\right )^2=1. \end{equation}

Substituting the expression for $\phi _{n,0}(r)$ given by (6.20) into (6.16) gives

(6.22) \begin{equation} \phi =\phi _{n-1,\textit{end}}\left (\dfrac {R_{n-1}}{R_n}r_0\right ) \left [\dfrac {\left (1-\dfrac {r_0^2}{R_n^2}\right )^{3/2}} {\left (\dfrac {\theta }{\theta _{\textit{max}}}\right )^{3/4}-1+\left (1-\dfrac {r_0^2}{R_n^2}\right )^{3/2}}\right ]^{1/3}. \end{equation}

Finally, eliminating $r_0$ between (6.13) and (6.22) and recalling that the solution for $\phi$ during the first CR phase is given explicitly by (3.10), it may be shown by induction that the concentration of particles within the droplet during the $n^{\text{}}$ th CR phase is given explicitly by

(6.23) \begin{equation} \phi = \left [1+\left (1-\dfrac {r^2}{R_n^2}\right )^{-3/2}\big (\theta ^{-3/4}R_n^{-9/4}-1\big )\right ]^{1/3}\!. \end{equation}

Details of the proof by induction of (6.23) may be found in Appendix A. Note that (3.10) holds for the entire lifetime of the droplet when $\theta _{\textit{min}}=0$ , and that (6.23) recovers the solution for a droplet evaporating in the CA mode given by (4.16) and in the SS mode given by (5.15) when $R_n=R$ , $\theta _{\textit{min}}=\theta _{\textit{max}}=1$ , and when $R_n=R$ , $\theta _{\textit{max}}\to \theta _{\textit{min}}=\theta ^*$ , respectively. Figure 15 shows $\phi$ given by (6.23) and $\phi h$ plotted as functions of $r$ at different times for a droplet evaporating in the SJ mode for $\theta _{\textit{min}}=1/2$ , $\theta _{\textit{max}}=1$ , and the first $6$ jump phases. The behaviour of $\phi$ in each CR phase corresponds to the behaviour for a droplet evaporating in the CR mode. Specifically, $\phi$ is a monotonically increasing function of $r$ and is singular at the pinned contact line $R_n$ in each CR phase, as illustrated in figure 15(a).

Figure 15.

Plot of (a) the concentration of particles $\phi$ and (b) the mass per unit area $\phi h$ within the droplet as functions of $r$ at $t=t_{n-1},t_{n-1}+(t_n-t_{n-1})/3, t_{n-1}+2(t_n-t_{n-1})/3,t_n$ for a droplet evaporating in the SJ mode for $\theta _{\textit{min}}=1/2$ , $\theta _{\textit{max}}=1$ , and the first $6$ jump phases. The dashed and dotted lines correspond to (a) $\phi _{n,0}(r)$ and $\phi _{n-1,\textit{end}}(r)$ and (b) $\phi _{n,0}(r)h_{n,0}(r)$ and $\phi _{n-1,\textit{end}}(r)h_{n-1,\textit{end}}(r)$ , respectively, and the arrow in (a) indicates the direction of increasing $t$ .

Substituting $t_{{p}}=0$ , $R_{{p}}=1$ , and the expression for $r_0(1,t_{\textit{ring}})$ given by (3.8) for $n=1$ and $t_{{p}}=t_{n-1}$ , $R_{{p}}=R_n$ , and the expression for $r_0(R_n,t_{\textit{ring}_n})$ given by (6.13) for $n=2,3,4\ldots$ into (2.34) and evaluating the integrals yields the explicit expressions for the mass of particles in the bulk of the droplet, namely

(6.24) \begin{align} M_{\textit{drop}}&=2\pi {\begin{cases} {\displaystyle\int}_{\!\! \kern-2pt 0}^{r_0(1,t_{\textit{ring}})}{ h(r,0)\,r\;\text {d}r} \quad &\text{for} \quad 0\leqslant t\leqslant t_1,\\[5pt] {\displaystyle\int}_{\!\! \kern-2pt 0}^{r_0(R_n,t_{\textit{ring}_n})}{ \phi _{n,0}(r)\,h_{n,0}(r)\,r\;\text {d}r} \quad &\text{for} \quad t_{n-1}\leqslant t\leqslant t_n,\,n=2,3,4,\ldots , \end{cases}} \nonumber \\ &= M_0 {\begin{cases} 1-\left (1-\theta ^{3/4}\right )^{4/3} \quad &\text{for} \quad 0\leqslant t\leqslant t_1,\\ 1-\big (1-\theta ^{3/4}R_n^{9/4}\big )^{4/3} \quad &\text{for} \quad t_{n-1}\leqslant t\leqslant t_n,\,n=2,3,4,\ldots , \end{cases}} \end{align}

where, again, the initial mass of particles in the droplet is given by (2.31) to be $M_0=\pi /4$ . The mass of particles in the $n^{\text{}}$ th ring deposit at time $t$ , defined by ${M_{\textit{ring}}}_n={M_{\textit{ring}}}_n(t)$ and located at $r=R_n$ , from (2.35) is given by

(6.25) \begin{align} {M_{\textit{ring}}}_n&= 2\pi {\begin{cases} {\displaystyle\int}_{\!\! \kern-2pt r_0(1,t_{\textit{ring}})}^1{ h(r,0)\,r\;\text {d}r} \quad &\text{for} \quad n=1,\\[5pt] {\displaystyle\int}_{\!\! \kern-2pt r_0(R_n,t_{\textit{ring}_n})}^{R_n}{ \phi _{n,{max}}(r)\,h_{n,{max}}(r)\,r\;\text {d}r} \quad &\text{for} \quad n=2,3,4,\ldots , \end{cases}}\nonumber \\ &=M_0 {\begin{cases} \left (1-\theta ^{3/4}\right )^{4/3} \quad &\text{for} \quad n=1,\\ \big (1-\theta ^{3/4}R_n^{9/4}\big )^{4/3}- \big (1-\theta _{\textit{max}}^{3/4}R_n^{9/4}\big )^{4/3} \quad &\text{for} \quad n=2,3,4,\ldots . \end{cases}} \end{align}

The final mass in the $n^{\text{}}$ th ring deposit at the end of the evaporation is determined by evaluating $M_{\textit{ring}_n}$ given by (6.25) at $t=t_n$ to yield

(6.26) \begin{align} {M_{\textit{ring}}}_n(t_n)= M_0 {\begin{cases} \left (1-\theta _{\textit{min}}^{3/4}\right )^{4/3}\quad &\text{for} \quad n=1,\\[5pt] \left (1-\theta _{\textit{min}}^{3/4}R_n^{9/4}\right )^{4/3}- \left (1-\theta _{\textit{max}}^{3/4}R_n^{9/4}\right )^{4/3} \quad &\text{for} \quad n=2,3,4,\ldots . \end{cases}} \end{align}

Figure 16.

Plot of the evolutions of (a) $M_{\textit{drop}}/M_0$ given by (6.24) and (b) $M_{\textit{rings}}/M_0$ given by (6.27) as functions of $t/t_{\textit{SJ}}$ for a droplet evaporating in the SJ mode for $\theta _{\textit{min}}=1/2$ and $\theta _{\textit{max}}=1$ . The dots correspond to (a) $M_{\textit{drop}}(t_n)$ and (b) $M_{\textit{ring}_n}(t_n)$ , the dotted lines correspond to the solutions for $M_{\textit{drop}}$ and $M_{\textit{ring}}$ when $\theta _{\textit{min}}=0$ (CR mode), and the (barely distinguishable) dashed lines correspond to the solutions for $M_{\textit{drop}}$ and $M_{\textit{dist}}$ when $\theta _{\textit{min}}=\theta _{\textit{max}}=1$ (CA mode).

The total mass of particles that have left the droplet to be deposited into the multiple ring deposits during the $n{\text{th}}$ CR phase, denoted by $M_{\textit{rings}}=M_{\textit{rings}}(t)$ , is then

(6.27) \begin{equation} M_{\textit{rings}}=M_{\textit{ring}_n}+\sum _{k=1}^{n-1}M_{\textit{ring}_k}(t_k). \end{equation}

The density of the $n{\text{th}}$ ring deposit at the end of the evaporation, defined by $\phi _{\textit{ring}_n}=M_{\textit{ring}_n}(t_n)/(2\pi R_n)$ , is then

(6.28) \begin{align} \phi _{\textit{ring}_n} =\dfrac {1}{8} {\begin{cases} \left (1-\theta _{\textit{min}}^{3/4}\right )^{4/3} \quad &\text{for} \quad n=1,\\[5pt] \left (R_n^{-3/4}-\theta _{\textit{min}}^{3/4}R_n^{3/2}\right )^{4/3}- \left (R_n^{-3/4}-\theta _{\textit{max}}^{3/4}R_n^{3/2}\right )^{4/3} \quad &\text{for} \quad n=2,3,4,\ldots . \end{cases}} \end{align}

Figure 16 shows plots of the evolutions of $M_{\textit{drop}}/M_0$ given by (6.24) and $M_{\textit{rings}}/M_0$ given by (6.27) as functions of $t/t_{\textit{SJ}}$ for a droplet evaporating in the SJ mode for $\theta _{\textit{min}}=1/2$ and $\theta _{\textit{max}}=1$ . In particular, as figure 16 shows, the deposit is a combination of multiple (theoretically infinitely many) ring deposits at radial positions $r=R_n$ in the SJ mode. Moreover, the behaviour reduces to that for a droplet evaporating in the CR mode, i.e. all of the particles eventually being transferred to a single ring deposit at $r=1$ , and in the CA mode, i.e. all of the particles being transferred to a distributed deposit in the region $0\leqslant r\leqslant 1$ , when $\theta _{\textit{min}}=0$ and $\theta _{\textit{max}}\to \theta _{\textit{min}}=\theta ^*$ , respectively. Figure 17 shows a plot of $M_{\textit{ring}_n}(t_n)/M_0$ given by (6.26) and $\phi _{\textit{ring}_n}$ given by (6.28) as functions of $\theta _{\textit{min}}$ for $\theta _{\textit{max}}=1$ and the first $20$ jump phases for a droplet evaporating in the SJ mode. In particular, the final mass within, and density of, the ring deposits decrease monotonically with $n$ for sufficiently small $\theta _{\textit{min}}$ , but are non-monotonic in $n$ above a critical value as illustrated in figure 17. Therefore, there is a region of values of $\theta _{\textit{min}}$ and $\theta _{\textit{max}}$ for which $M_{\textit{ring}_n}(t_n)$ and $\phi _{\textit{ring}_n}$ are non-monotonic in $n$ , i.e. the outermost ring does not contain the most particles and is therefore not the most dense ring, which we will now discuss.

Figure 17.

Plot of (a) $M_{\textit{ring}_n}/M_0$ given by (6.26) and (b) $\phi _{\textit{ring}_n}$ given by (6.28) as functions of $\theta _{\textit{min}}$ for $\theta _{\textit{max}}=1$ and the first $20$ jump phases for a droplet evaporating in the SJ mode. The dotted lines correspond to the solutions for the first CR phase, i.e. for $n=1$ , and the dashed lines correspond to the critical values of (a) $\theta _{\textit{min}}\simeq 0.600$ and (b) $\theta _{\textit{min}}\simeq 0.418$ that satisfy $M_{\textit{ring}_1}(t_1)=M_{\textit{ring}_2}(t_2)$ and $\phi _{\textit{ring}_1}=\phi _{\textit{ring}_2}$ , respectively. The arrows indicate the direction of increasing $n$ .

The critical values of the minimum and maximum contact angles at which the final masses within the first and second ring deposits are equal, denoted by $\theta _{\textit{min}}^{\textit{crit},\textit{mass}}$ and $\theta _{\textit{max}}^{\textit{crit},\textit{mass}}$ , are determined by solving $M_{\textit{ring}_1}(t_1)=M_{\textit{ring}_2}(t_2)$ from (6.26), and are therefore given implicitly by

(6.29) \begin{equation} \theta _{\textit{max}}^{\textit{crit,mass}}=\dfrac {{\theta _{\textit{min}}^{\textit{crit,mass}}}^{2}}{\left [1-2^{3/4}\left (1-{\theta _{\textit{min}}^{\textit{crit,mass}}}^{3/4}\right ) \right ]^{4/3}}. \end{equation}

Similarly, the critical angles at which the densities of the first and second ring deposits are equal, denoted by $\theta _{\textit{min}}^{\textit{crit,dens}}$ and $\theta _{\textit{max}}^{\textit{crit,dens}}$ , are determined by solving $\phi _{\textit{ring}_1}(t_1)=\phi _{\textit{ring}_2}(t_2)$ from (6.28), and are therefore given implicitly by

(6.30) \begin{equation} \theta _{\textit{max}}^{\textit{crit,dens}}=\dfrac {{\theta _{\textit{min}}^{\textit{crit,dens}}}^{2}}{\left [1-\left \{1+\left (\dfrac {\theta _{\textit{min}}^{\textit{crit,dens}}}{\theta _{\textit{max}}^{\textit{crit,dens}}}\right )^{1/3}\right \}^{3/4}\left (1-{\theta _{\textit{min}}^{\textit{crit,dens}}}^{3/4}\right ) \right ]^{4/3}}. \end{equation}

Hence the region of values for which $M_{\textit{ring}_n}(t_n)$ and $\phi _{\textit{ring}_n}$ are non-monotonic in $n$ are

(6.31) \begin{equation} 0.600\simeq \left (2^{3/4}-1\right )^{4/3}\leqslant \theta _{\textit{min}}\leqslant 1, \quad \theta _{\textit{max}}^{\textit{crit,mass}}\left (\theta _{\textit{min}}^{\textit{crit,mass}}\right )\leqslant \theta _{\textit{max}}\leqslant 1 \end{equation}

and

(6.32) \begin{equation} 0.418\lesssim \theta _{\textit{min}}\leqslant 1, \quad \theta _{\textit{max}}^{\textit{crit,dens}}\left (\theta _{\textit{min}}^{\textit{crit,dens}}\right )\leqslant \theta _{\textit{max}}\leqslant 1, \end{equation}

with $\theta _{\textit{max}}^{\textit{crit},\textit{mass}} (\theta _{\textit{min}}^{\textit{crit},\textit{mass}} )$ and $\theta _{\textit{max}}^{\textit{crit,dens}} (\theta _{\textit{min}}^{\textit{crit,dens}} )$ given implicitly by (6.29) and (6.30), respectively. Figure 18 shows a plot of $\theta _{\textit{max}}^{\textit{crit},\textit{mass}}$ given by (6.29) and $\theta _{\textit{max}}^{\textit{crit,dens}}$ given by (6.30) as functions of $\theta _{\textit{min}}^{\textit{crit},\textit{mass}}$ and $\theta _{\textit{min}}^{\textit{crit,dens}}$ . In particular, the two shaded regions indicate the region of values of $\theta _{\textit{min}}$ and $\theta _{\textit{max}}$ for which the final mass within (the darker shaded region) and the density of (the lighter and darker shaded regions) the ring deposits are non-monotonic in $n$ , given by (6.31) and (6.32). In particular, as figure 18 shows, the region of parameter values for which $M_{\textit{ring}_n}(t_n)$ is non-monotonic in $n$ is much smaller than, and contained within, the region for which $\phi _{\textit{ring}_n}$ is non-monotonic in $n$ .

Figure 18.

Plot of $\theta _{\textit{max}}^{\textit{crit},\textit{mass}}$ (solid black line) given by (6.29) and $\theta _{\textit{max}}^{\textit{crit,dens}}$ (dashed black line) given by (6.30) as functions of $\theta _{\textit{min}}^{\textit{crit},\textit{mass}}$ and $\theta _{\textit{min}}^{\textit{crit,dens}}$ . The dotted line corresponds to $\theta _{\textit{max}}=1$ . The shaded regions indicate the regions of values in which $M_{\textit{ring}_n}(t_n)$ and $\phi _{\textit{ring}_n}$ are non-monotonic in $n$ .

6.4. Validation against previous experimental results

The general behaviour that multiple ring deposits are left behind on the substrate after a droplet evaporates in the SJ mode is a common observation in experiments (see, for example, Orejon et al. Reference Orejon, Sefiane and Shanahan2011; Askounis et al. Reference Askounis, Sefiane, Koutsos and Shanahan2014; Yang et al. Reference Yang, Li and Sun2014; Kim et al. Reference Kim, Pack, Rokoni, Kaneelil and Sun2018). However, in practice, due to surface roughness in physical experiments the contact line may not necessarily jump inwards symmetrically and the critical minimum and maximum contact angles may vary throughout the evaporation. These behaviours cannot be captured by the present model and so, in general, comparing theoretical predictions with physical experiments that evaporate in the SJ mode is a difficult task. We will compare the predictions of the present model for the final mass in the ring deposits with the experimental observations by Kim et al. (Reference Kim, Pack, Rokoni, Kaneelil and Sun2018). Specifically, as discussed in § 5.4, this study concerns the deposition of spherical polystyrene particles from water droplets evaporating on glass. When the particles are less hydrophilic the droplet evolves in six distinct SJ phases followed by a CA phase for the remainder of evaporation. The experiments report a deposit profile (their figure 6b) for which the mass within the six rings may be determined, a value for $\theta _{\textit{min}}$ , and a plot of the evolution of $R$ (their figure 5a) for which we may obtain a best fit for the value of $\theta _{\textit{max}}$ . Note that the droplet in this study therefore actually evaporates in what we may term a ‘stick–jump–slide’ (SJS) mode of evaporation, for which we will denote the lifetime of the droplet as $t_{\textit{SJS}}$ , however, here we consider only the mass within the ring deposits due to the SJ phases.

Figure 19.

Comparison between the theoretical predictions (circles, solid lines) for the evolutions of (a) $R$ and (b) $M_{\textit{rings}}/M_0$ given by (6.27) for $0\leqslant t\leqslant t^*$ as functions of $t/t_{\textit{SJS}}$ , as well as (c) the locations of, and the final masses of particles within, the $n{\text{th}}$ ring deposits $M_{\textit{ring}_n}(t_n)/M_0$ given by (6.26) when $\theta _{\textit{min}}\simeq 0.667$ , $\theta _{\textit{max}}\simeq 0.853$ , and $n=1$ , $2$ , $\ldots$ , $6$ for a droplet evaporating in the SJS mode with the experimental data from Kim et al. (Reference Kim, Pack, Rokoni, Kaneelil and Sun2018) (squares, dashed line).

Figure 19 shows a quantitative comparison between the theoretical predictions for the evolutions of $R$ and $M_{\textit{rings}}/M_0$ given by (6.27) for $0\leqslant t\leqslant t^*$ as functions of $t/t_{\textit{SJS}}$ , as well as the locations of, and the final masses of particles within, the $n{\text{th}}$ ring deposits $M_{\textit{ring}_n}/M_0$ given by (6.26) when $\theta _{\textit{min}}\simeq 0.667$ , $\theta _{\textit{max}}\simeq 0.853$ and $n=1$ , $2$ , $\ldots$ , $6$ for a droplet evaporating in the SJS mode with the experimental data from Kim et al. (Reference Kim, Pack, Rokoni, Kaneelil and Sun2018). In particular, as figures 19(b) and 19(c) show, the theoretical predictions of the present model compare well with the results of the experiments for the total mass of particles in the multiple ring deposits, although the present model does not predict that the second most outer ring contains more mass than the first (i.e. the outer) ring, as is observed in experiments. Given that the present model captures parameter regions in which the mass within the ring deposits is non-monotonic in $n$ , as discussed in detail in § 6.3, this disagreement could be due to variation in the critical contact angles throughout the evaporation in the experiments.