2.1. Profile of the liquid–vapour interface

An analytical solution of the drop shape has been derived by Carroll (Reference Carroll1976), but the elliptical integral involved in this solution would limit the analytical derivation of the present problem. Thus, we choose to describe the drop profile with an ansatz fitting Carroll's solution

(2.1)\begin{equation} h(z,t) = h_0(t) \left( 1 - \frac{z^2}{L^2} \right)^2, \end{equation}

where $h_0(t)$ is the drop height at the apex $z=0$ and $L$ is the distance from the apex to the contact line (figure 1a). We consider the regime for which contact lines are pinned, so the length $L$ remains constant. The contact angle is defined by Young's law and depends only on the liquid/solid properties. For an axisymmetric drop on a fibre, Carroll (Reference Carroll1976) demonstrated that the relation between the drop shape and the contact angle is not straightforward and, in particular, the contact angle cannot be read directly from the slope between the interface and the surface of the fibre as it is for sessile drops. Typically, for contact angles between 0$^\circ$ and 30$^\circ$, the slope of the interface vanishes in the vicinity of the contact line as shown in figure 1(b), where we compare drop profiles obtained from Carroll (Reference Carroll1976) at $L/a = 6.5$ to (2.1) for $\theta = 0^\circ, 5^\circ, 10^\circ \ \textrm {and}\ 15^\circ$. To describe the drop profile, we fit Carroll's profile with (2.1), where $L$ and $h_0$ are the two adjustable parameters. To keep the wetted-length constant over the dynamics, we choose to set $L$ as the fitted length obtained for $\theta = 0^\circ$, $L/a \approx 6.7$. Then, we use the values of $h_0$ obtained by the fit to calculate drop profile with (2.1) (see dashed lines in figure 1b). As shown in figure 1(b), the proposed description of the drop profile is reasonably close to the analytical solution and describes its main features. We tested the validity of the ansatz for various drop profiles and we found a good agreement for drop with small contact angles, typically lower than 30$^\circ$ and dimensionless volume $\varOmega /a^3 < 1000$.

The liquid volume is defined as

(2.2)\begin{equation} \varOmega(t) = \int_a^{a+h(z,t)} \int_0^{2{\rm \pi}} \int_{{-}L}^L r \, {\rm d}r \,{\rm d}\theta \,{\rm d}z = \tfrac{32}{315} {\rm \pi}L h_0(t) \left(21 a + 8 h_0(t)\right). \end{equation}

With (2.1) and (2.2), we can estimate the volume of the drop and compare it to the volume calculated by Carroll (Reference Carroll1976) from his analytical description of the drop profile. The ansatz gives a good approximation of the drop volume with a relative error of approximately $10\,\%$ for the largest contact angle tested, $\theta = 15^\circ$, for which the difference between the fit and the analytical profile is the largest.

2.2. Evaporative flux

The evaporation process is assumed to be limited by the diffusion of vapour in air (Cazabat & Guéna Reference Cazabat and Guéna2010), described by the vapour diffusion coefficient ${\mathcal {D}}_{v}$. The steady-state regime is reached on a time scale $L^2/{\mathcal {D}}_{v}$, which is, in most situations, short compared to the observation time scale. Thus, the vapour mass concentration field $c(r,z)$ is the solution of the Laplace equation ${\rm \Delta} c = 0$. The boundary conditions are $c=c_{sat}$ at the liquid–vapour interface, $c=c_{\infty }$ far from the interface and a no flux condition $\boldsymbol {n} \boldsymbol {{\cdot }} \boldsymbol {\nabla } c = 0$ on the solid–gas interface characterised by its normal $\boldsymbol {n}$ (see figure 1a). Once the mass concentration field is obtained, the evaporation velocity at the liquid–vapour interface is defined as $v_{e}(z) = ({{\mathcal {D}}_{ v}}/{\rho }) \boldsymbol {n} \boldsymbol {{\cdot }} \boldsymbol {\nabla } c|_{h(z,t)}$, where the derivative is calculated at the liquid–vapour interface and $\rho$ is the density of the liquid.

Corpart et al. (Reference Corpart, Dervaux, Poulard, Restagno and Boulogne2022) have recently performed numerical simulations using the finite element method to determine the local evaporative flux of a drop on a fibre. In particular, this study revealed that the evaporation velocity differs from the well-known sessile drop in two aspects. First, the divergence is localised in the close vicinity of the contact line and second, the contact angle has a weak effect on the evaporation velocity. A typical result of these computations is presented in figure 2. The localisation of the divergence near the contact line implies that the evaporation velocity cannot be written as a power law (Corpart et al. Reference Corpart, Dervaux, Poulard, Restagno and Boulogne2022). Instead, we propose to fit the results of the numerical simulations by

(2.3)\begin{equation} v_{e}(z) = v_{e}^0 \left[\beta\left( 1 - \frac{z^2}{L^2} \right)^{-\alpha} + 1 - \beta\right], \end{equation}

where $\alpha$ and $\beta$ are constants, independent of the contact angle, for $\theta < 20^\circ$ (Corpart et al. Reference Corpart, Dervaux, Poulard, Restagno and Boulogne2022). The prefactor $v_{e}^0=v_{e}(z=0)$ is the single parameter that depends on the environmental conditions and is proportional to ${\mathcal {D}}_{v} (c_{sat} - c_\infty ) / \rho$. The total evaporative flux writes $Q_{e}(t) = \int v_{e}\,{\rm d} S = 2{\rm \pi} \int _{-L}^{L} v_{e}(z) (a + h(z,t) )\,{\rm d} z$, which leads after integration to

(2.4)\begin{equation} Q_{e}(t) = 2{\rm \pi} v_{e}^0 L \left( A_1 a + A_2 h_0(t) \right), \end{equation}

with

(2.5a)$$\begin{gather} A_1=\sqrt{\rm \pi} \beta \frac{ \varGamma(1-\alpha) }{ \varGamma\left(\dfrac{3}{2} - \alpha\right)} + 2 (1 - \beta), \end{gather}$$

(2.5b)$$\begin{gather}A_2=\sqrt{\rm \pi} \beta \frac{ \varGamma(3-\alpha) }{ \varGamma\left(\dfrac{7}{2}- \alpha\right)} + \frac{16}{15} (1-\beta), \end{gather}$$

where $\varGamma$ is the Gamma-function (Abramowitz & Stegun Reference Abramowitz and Stegun1972).

Figure 2. Local dimensionless evaporation velocity $v_{e}(z)/v_{e}^0$ as a function of $1-z^2 / L^2$, where $v_{e}^0$ is the evaporation velocity at the drop apex. The points are obtained from numerical simulations using finite element methods presented by Corpart et al. (Reference Corpart, Dervaux, Poulard, Restagno and Boulogne2022) for $a=125\,\mathrm {\mu }$m, $\theta = 10^\circ$, $\varOmega = 1\,\mathrm {\mu }$L, which corresponds to $L/a \approx 6.7$. The dashed line is (2.3) with the fitted parameters $\alpha = 0.7$ and $\beta = 0.1$.

2.3. Hydrodynamics

2.3.1. Lubrication approximation

Now that the geometry and the evaporation dynamics are described, we analyse the flow in the liquid phase. By considerations of symmetries, the velocity field can be written in the cylindrical coordinates $(v_r(r, z), v_z(r, z))$. The continuity equation is

(2.6)\begin{equation} \frac{1}{r} \frac{\partial (r v_r)}{\partial r} + \frac{\partial v_z}{\partial z} = 0, \end{equation}

and the Navier–Stokes equations in the stationary regime are

(2.7a)$$\begin{gather} - \frac{\partial p}{\partial r} + \eta \left[\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial v_r}{\partial r}\right) + \frac{\partial^2 v_r}{\partial z^2}-\frac{v_r}{r^2} \right] = 0, \end{gather}$$

(2.7b)$$\begin{gather}- \frac{\partial p}{\partial z} + \eta \left[\frac{1}{r}\frac{\partial}{\partial r} \left(r \frac{\partial v_z}{\partial r}\right) + \frac{\partial^2 v_z}{\partial z^2}\right] = 0, \end{gather}$$

where $p(r,z)$ is the liquid pressure. The boundary conditions are no fluid slippage on the fibre and no stress at the liquid–vapour interface, i.e.

(2.8a)$$\begin{gather} v_z(r,z) = 0,\quad \mbox{on}\ r=a \quad \mbox{and}\quad |z| \leq L, \end{gather}$$

(2.8b)$$\begin{gather}\frac{\partial v_z}{\partial r} = 0,\quad \mbox{on}\ r=a+h(z, t) \quad \mbox{and} \quad |z| \leq L. \end{gather}$$

Equation (2.8b) means that tracers are assumed to have no surfactant effect, due to their size. We apply the lubrication approximation to (2.7a), (2.7b), which is valid for $({h_0}/{L})({\rho h(z,t) v_z}/{\eta }) \ll 1$ (Batchelor Reference Batchelor2000) and we get

(2.9)\begin{equation} \frac{{\rm d} p}{{\rm d} z} = \eta \left[\frac{1}{r}\frac{\partial}{\partial r} \left(r \frac{\partial v_z}{\partial r}\right) \right]. \end{equation}

Based on the differential equation (2.9) and the boundary conditions (2.8a) and (2.8b), we can calculate the velocity field $v_z$:

(2.10)\begin{equation} v_z(r, z, t) ={-} \frac{1}{\eta} \frac{{\rm d} p}{{\rm d} z} \left[\frac{ \left(a + h(z,t) \right)^2}{2} \ln\left(\frac{r}{r_0}\right) - \frac{r^2}{4} \right], \end{equation}

with $r_0 = a \exp ( - a^2 / ( 2 (a+h(z,t))^2 ) )$. We remark that the radial dependence of $v_z$ is not a quadratic function in contrast to the sessile drop that will be recalled thereafter in Appendix A.

2.3.2. Mass conservation

To fully obtain the fluid velocity $v_z$, the pressure gradient ${\rm d} p / {\rm d} z$ must be determined. To do so, we write the mass conservation over a slice between $z$ and $z + {\rm d} z$,

(2.11)\begin{equation} \frac{\partial h}{\partial t} + \frac{1}{2(a+h)} \frac{\partial }{\partial z} ((2ah + h^2) \bar{v}_z) + v_{e}(z) =0, \end{equation}

where $\bar {v}_z(z,t)$ is the velocity $v_z(r,z,t)$ averaged over a cross-section perpendicular to the fibre,

(2.12)\begin{equation} \bar{v}_z(z,t) = \frac{2}{2ah+h^2} \int_a^{a+ h(z,t)} v_z(r,z,t)\, r \, {\rm d}r. \end{equation}

Now, we can establish the relation between $\bar {v}_z$ and $v_z$. First, we integrate (2.12) with (2.10) to express $\bar {v}_z$ as a function of ${\rm d} p / {\rm d} z$. Equation (2.12) is written

(2.13)\begin{equation} \bar{v}_z(z,t) ={-} \frac{1}{\eta}\frac{{\rm d} p}{{\rm d} z} \mathcal{G}(a,h(z, t)), \end{equation}

where the geometrical function $\mathcal {G}(a,h(z, t))$ is given by

(2.14)\begin{align} \mathcal{G}(a, h(z,t)) =\frac{ 1 }{4(2ah+h^2)} \left( \frac{ a^4 }{2} + (a+h)^4 \left[ \frac{a^2}{(a+h)^2} \left( \frac{1}{2} - \ln \frac{a}{r_0} \right) + \ln \frac{a+h}{r_0} -1 \right]\right). \end{align}

Substituting the pressure derivative from (2.10) in (2.13) yields

(2.15)\begin{equation} v_z(r, z, t) = \frac{ h \bar{v}_z}{\mathcal{G}(a, h(z,t))} \left[ \frac{ \left(a + h(z,t) \right)^2}{2} \ln\left(\frac{r}{r_0} \right) - \frac{r^2}{4} \right]. \end{equation}

In the next subsections, we derive the time evolution of the drop profile $\partial h / \partial t$, which will be used along the evaporation velocity $v_{e}(z)$ in the mass conservation (2.11) to obtain the average velocity $\bar {v}_z$ as a function of the evaporation dynamics.

2.3.3. Liquid evaporation

First, we calculate the time derivative of the liquid profile, $\partial h / \partial t$, which appears in (2.11). The time derivative of the drop profile defined in (2.1) considering a constant drop length $L$ gives

(2.16)\begin{equation} \frac{\partial h(z,t)}{\partial t} = \frac{{\rm d} h_0 }{ {\rm d} t } \left( 1 - \frac{z^2}{L^2} \right)^2. \end{equation}

In addition, the loss of liquid by evaporation compensates the total evaporative flux as ${\rm d} \varOmega / {\rm d} t= - Q_{e}(t)$. Substituting the volume defined in (2.2), we have

(2.17)\begin{equation} \frac{{\rm d} h_0}{{\rm d} t} ={-} \frac{315}{32 {\rm \pi}} \frac{Q_{e}(t)}{ L (21 a + 16h_0(t)) }, \end{equation}

which fully defines the time derivative of the liquid profile.

Integrating (2.17) from 0 to $t$ and $h_{i}$ to $h_0$ leads to

(2.18)\begin{equation} \frac{16}{A_2} ( h_{i} - h_0(t)) + \frac{a( 21 A_2 - 16 A_1)}{A_2^2} \ln{ \left( \frac{A_1 a + A_2 h_{i}}{A_1 a + A_2 h_0(t)} \right)} = \frac{315}{16} v_{e}^0 t. \end{equation}

At the first leading order in $(h_{i} - h_0)/h_{i}$, we obtain the time variation of the height of the apex:

(2.19)\begin{equation} h_0(t) \approx h_{i} - \frac{315}{16}\frac{A_1 a + A_2 h_{i}}{21a + 16h_{i}} v_{e}^0 t. \end{equation}

2.3.4. Mean liquid velocity

Now, we derive the mean liquid velocity $\bar {v}_z$. By substituting (2.16) and (2.17) in the mass conservation (2.11), we have

(2.20)\begin{equation} \frac{1}{2(a+h)} \frac{\partial }{\partial z} ((2ah + h^2) \bar{v}_z) = \frac{315}{ 32 {\rm \pi}} \frac{Q_{e}(t)}{L(21a + 16 h_0)} \left( 1 - \frac{z^2}{L^2} \right)^2 - v_{e}(z). \end{equation}

The integration of this differential equation from $0$ to $z$ yields

(2.21)\begin{align} (2ah+h^2) \bar{v}_z = \frac{315}{16{\rm \pi}} \frac{Q_{e}(t)}{ (21a + 16h_0)} \frac{z}{L} \left[ a {\mathcal{P}}_1\left( \frac{z}{L} \right) + h_0 {\mathcal{P}}_2\left( \frac{z}{L} \right) \right] -2 \int_0^z (a+h) v_{e}\,{\rm d} z, \end{align}

where ${\mathcal {P}}_1(x) = 1 - \frac {2}{3} x^2 + \frac {1}{5}x^4$ and ${\mathcal {P}}_2(x) = 1 - \frac {4}{3}x^2 + \frac {6}{5}x^4 - \frac {4}{7}x^6 + \frac {1}{9}x^8$.

The plane perpendicular to the fibre at $z=0$ is a plane of symmetry, thus $\bar {v}_z(z=0) = 0$.

From (2.3), we can calculate the remaining integral of (2.21):

(2.22)\begin{align} 2 \int_0^z (a+h) v_{e}\, {\rm d}z &= 2 v_{e}^0 z \left[ a \left( \beta {}_2F_1\left(\frac{1}{2},\alpha; \frac{3}{2};\frac{z^2}{L^2}\right) + (1-\beta) \right) \right. \nonumber\\ &\quad + \left. h_0 \left(\beta\, {}_2F_1\left(\frac{1}{2},\alpha-2;\frac{3}{2};\frac{z^2}{L^2}\right) + (1-\beta) {\mathcal{P}}_1\left( \frac{z}{L} \right)\right)\right], \end{align}

where the hypergeometric function ${}_2F_1(a,b;c;z)$ writes (Abramowitz & Stegun Reference Abramowitz and Stegun1972)

(2.23)\begin{equation} {}_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!} = 1 + \frac{ab}{c}\frac{z}{1!} + \frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^2}{2!} + \cdots. \end{equation}

The combination of (2.21) and (2.22) provides the mean velocity $\bar {v}_z(z,t)$. The velocity profile $v_z(r,z,t)$ is therefore obtained by a straightforward substitution in (2.15). This description of the fluid flow permits analysis of the induced particle transport, which will be discussed in § 3.

3.1. Methodology

To be able to compare the two geometries, a choice on the initial parameters has to be made. The possible parameters are $\theta _{i}$ the initial contact angle, $\mathcal {L}$ the wetted length, $\varOmega (t=0)$ the initial volume and $Q_{e}(t=0)$ the initial evaporation rate. A common practice for comparison is to consider the same liquid–solid system such that the initial contact angle $\theta _{i}$ is fixed. In addition, a similar evaporation rate $Q_{e}$ for both systems brings the advantage of a comparable driving force for the particle transport. With (2.4), we can calculate the evaporation rate of a drop on a fibre whose geometry is given by (2.1) at $L/a \approx 6.7$ and $\theta = \theta _{i} = 15^\circ$ (see figure 1b). Numerical simulation in a previous study by Corpart et al. (Reference Corpart, Dervaux, Poulard, Restagno and Boulogne2022) provides $v_{e}^0$ for a water droplet in the initial geometry described here and evaporating in dry air ($c_\infty = 0$) at 20 $^\circ$C ($c_{sat} = 1.72 \times 10^{-2}$ kg s$^{-1}$ and ${\mathcal {D}}_{v} = 2.36 \times 10^{-5}$ m$^2$ s$^{-1}$) (Lide Reference Lide2008). We obtained $v_{e}^0 \approx 4 \times 10^{-7}$ m s$^{-1}$. From (A4), we can calculate the wetted length of the sessile droplet having the same evaporation rate for the same ambient conditions. We find a similar length scale $R \approx L$. For instance, the barrel-shaped drop on a fibre of radius $a = 125\,\mathrm {\mu }$m has, initially, a wetted-length $L \approx 8.4 \times 10^{-4}$ m and a sessile drop evaporating at the same initial rate has a wetted length $R \approx 8.9 \times 10^{-4}$ m. We thus choose to make comparisons at the same wetted length $L/a \approx 6.7$ which is given by fitting Carroll's profile with (2.1), as explained in § 2 (see figure 1b). For the same wetted length and the same initial contact angle, the initial volume of the droplet is different in the two geometries. For example, for $L = R \approx 8.4\times 10^{-4}$ m and $\theta _{i} = 15^\circ$, the initial volume is $\varOmega (t = 0) \approx 0.1\,\mathrm {\mu }$L for the sessile case and $\varOmega (t = 0) \approx 0.6\,\mathrm {\mu }$L for the drop on a fibre. The sessile drop has a smaller initial volume and thus a shorter lifetime than the drop on a fibre even if the initial evaporation rate is the same in the two geometries. Additionally, since the particle transport depends on the particle size, we consider a particle diameter $2b=1\,\mathrm {\mu }$m motivated by the large number of studies on the coffee-stain using micrometer-sized particles.

To summarise, we choose to compare drops of same wetting length and same initial contact angle. This corresponds to drops having different initial volumes but the same initial evaporation rate for the two different geometries when evaporating in the same ambient conditions. In this framework, we can compare the mean velocities toward the contact line for the two systems. The competition between the particle transport and the Brownian motion will be rationalised by a Péclet number.

3.2. Time evolution of the drop shapes

We analyse the evolution of the drop shape in both configurations. The temporal evolution of drop heights, plotted in figure 3(c), are given by (2.18) (blue solid line) approximated by (2.19) (dashed line) for the drop on a fibre and (A2) (black line) for the sessile drop.

Figure 3. (a,b) Evolution of the drop shape at different contact angles in (a) the sessile case (A1) and (b) the fibre configuration for which the drop profile is described by elliptic integrals given by Carroll (Reference Carroll1976). (c) Time evolution of the drop height at the apex $h_0$ normalised by the initial height $h_{i}$ for the two configurations. The black line corresponds to the sessile drop (A2). The blue lines are for an axisymmetric drop on a fibre, the solid line is (2.18) and the dashed line is the approximation given by (2.19). (d) Dynamics of the contact angle $\theta$ for a sessile drop. (c,d) Circles represent (c) the apex heights and (d) contact angles of the profiles of drops on fibres obtained with our ansatz (2.1) and plotted in figure 1(b). Here, the comparisons are performed for the same wetted length ${\mathcal {L}}/a \approx 6.7$ and an initial contact angle $\theta _{i} = 15^\circ$. This corresponds to water on a glass substrate of initial volume $\varOmega (t=0) \approx 0.6\,\mathrm {\mu }$L for the fibre configuration ($a = 125\,\mathrm {\mu }$m) and $\varOmega (t=0) \approx 0.1\,\mathrm {\mu }$L for the sessile case. The initial evaporation rate $Q_{e}(t=0)$ is nearly the same for the two configurations. For both geometries, the ambient conditions are taken to be those of water evaporating in dry air at 20 $^\circ$C, $c_{sat} = 1.72 \times 10^{-2}$ kg m$^{-3}$, $c_\infty = 0$ and ${\mathcal {D}}_{v} = 2.36 \times 10^{-5}$ m$^2$ s$^{-1}$ (Lide Reference Lide2008). In these conditions, for a drop on a fibre ($\varOmega = 1\,\mathrm {\mu }$L, $\theta = 10^\circ$ and $a = 125\,\mathrm {\mu }$m), we obtained by numerical simulations $v_{e}^0 \approx 4 \times 10^{-7}$ m s$^{-1}$, $\alpha = 0.7$ and $\beta = 0.1$ (figure 2).

The evolution of the drop shape for different contact angles for a sessile drop is illustrated in figure 3(a). Due to the particle accumulation at the contact line that increases the pinning force (Joanny & de Gennes Reference Joanny and de Gennes1984; di Meglio Reference di Meglio1992; Boulogne et al. Reference Boulogne, Ingremeau, Limat and Stone2016), we consider that the depinning nearly occurs at a zero contact angle, which corresponds to a vanishing drop volume. Now, considering the drop on a fibre, we plot in figure 3(b) the drop shape on a fibre having the same wetted length and the same contact angles as the sessile drops in figure 3(a). By fitting the profiles of figure 3(b) with (2.1), we get the apex heights and the corresponding times by inserting them in (2.18). The results are represented by the circles in figure 3(c). This figure shows that the temporal evolution of the apex height of a drop on a fibre is well described by the approximated equation (2.19) (the dashed line in figure 3c) during the pinned-regime.

From figure 3(a,b), we also get the height of the apex at the depinning $h_0^{dep} = h_0(\theta = 0)$. In the example studied here, we get $h_0^{dep} = 0$ for the sessile drop and $h_0^{dep} = 2.4a$ (corresponding to $h_0^{dep} = 0.8 h_{i}$) for the drop on a fibre. From that, we get the duration of the pinned regime $\tau _{dep}$ of the drop on a fibre by inserting $h_0^{dep}$ into (2.18). In the example studied here, we find $\tau _{dep} \approx 110$ s. In the case of the sessile drop, the volume tends to zero at the end of the pinned regime, meaning that $\tau _{dep} = \tau _{e}$ (A5), the lifetime of a sessile drop evaporating entirely at constant contact radius. In the example studied here, we find $\tau _{e} \approx 90$ s. To establish the relationship between time and contact angle, we plot in figure 3(d) (circles) the temporal evolution of contact angle obtained from the drop profiles represented in figure 3(b) for which we know $h_0$ and $\theta$. By the solid black line, we represent $\theta (t) = \theta _{i} (1 - {t}/{\tau _{dep}})$ valid for a sessile drop (A2) which also describes very well the temporal evolution of the contact angle of a drop on a fibre. We thus find for both geometries that $h_0 \propto \theta \propto t$.

Moreover, as observed in figure 3(b,c), the relative variation of the drop height $h_0$ (and drop volume) during the pinned regime is small for the drop on a fibre. During the pinned regime, a small amount of the initial volume has evaporated, which means that the duration of the pinned regime is short compared to the drop lifetime $\tau _{dep} \ll \tau _{e}$. The remarkable difference with the sessile drop is the significant liquid volume remaining at a zero contact angle, the contact angle at which the depinning is supposed to occur. We can estimate the remaining volume at the end of the pinned regime from (2.2). For the geometry represented in figure 3(b), we find that only $35\,\%$ of the initial volume has evaporated before the contact line depinning.

From this comparison, we can state that the geometry imposes constraints on the dynamics of the drop profile. In particular, in contrast to the sessile drop, only a small fraction of the liquid volume can evaporate in the constant wetted length regime. Also, on a fibre, a drop has two contact lines such that we expect that only one of the two lines depins. At this time, the lower quantity of evaporated liquid implies that only a fraction of the particles are accumulated at the contact lines and that the complementary fraction of particles is still dispersed in the liquid phase. However, to compute the number of accumulated particles at the contact line, the velocity field inside the drop needs to be described, which is the object of the next section.

3.3. Transport of particles towards the contact line

3.3.1. Mean flow velocity towards the contact line

In figure 4(a), we plot $\bar {v}_x$ for both geometries, $x$ being either $z$ or $r$ according to the geometry. In the centre, $x=0$, the fluid velocity is equal to zero by symmetry. We observe that at the beginning of evaporation, i.e. for $\theta = 15^\circ$, the flow towards the contact line is one order of magnitude higher in the sessile drop at most of the radial positions. Near the contact line, the mean liquid velocity diverges due to the vanishing liquid thickness for both geometries. As the liquid evaporates, i.e. $\theta$ decreases, the so-called rush-hour effect (Hamamoto et al. Reference Hamamoto, Christy and Sefiane2011; Marin et al. Reference Marin, Gelderblom, Lohse and Snoeijer2011a) occurs in the sessile drop, i.e. $\bar {v}_r$ increases in time. However, $\bar {v}_z$ remains nearly constant for a drop on a fibre. This difference is due to the curvature of the substrate that enables the existence of the peculiar axisymmetric barrel morphology, for which the variation of the drop profile remains limited when $\theta$ decreases (see figure 3b) unlike the case of the spherical cap on a flat substrate (figure 3a).

Figure 4. (a) Mean velocity of the flow towards the contact line $\bar {v}_x$ as a function of the dimensionless coordinate $x/{\mathcal {L}}$ along the solid surface. The comparison is performed for water drops in both configurations with the same dimensionless wetted length $\mathcal {L}/a \approx 6.7$ and various contact angles (see caption). Dashed lines correspond to $\bar {v}_r$ (A6), the average fluid velocity towards the contact line in sessile drops. Solid lines are $\bar {v}_z$, the average fluid velocity in a drop on a fibre, given by (2.21). The dependence of $\bar {v}_z$ in $\theta$ is implicit and contained in the dimensions of the drop which are obtained by fitting Carroll's drop profile with (2.1) for each contact angle, as done in figure 1(b). (b) Péclet number $\overline {Pe}$ deduced from the mean velocity and drop profile (2.1) and (A1) as a function of $x/{\mathcal {L}}$. Solid lines correspond to a drop on a fibre and dashed lines to sessile drops having the same wetted length. The solid red line corresponds to $\overline {Pe}=1$. For both geometries, the ambient conditions are taken to be those of water evaporating in dry air at 20 $^\circ$C, described in the § 3.1 and in figure 3.

3.4. Number of particles accumulating at the contact line

3.4.1. Advection of a slice of fluid

We consider a fluid layer at a position $x_0$ of an infinitesimal width ${\rm d}\kern 0.06em x$, which is advected towards the contact line at a velocity $\bar {v}_x(x_0, t)$. The advection of a fluid layer is described by its position $x_0(t)$ that satisfies (Deegan Reference Deegan2000; Popov Reference Popov2005; Monteux & Lequeux Reference Monteux and Lequeux2011; Berteloot et al. Reference Berteloot, Hoang, Daerr, Kavehpour, Lequeux and Limat2012)

(3.1)\begin{equation} \frac{{\rm d}\kern 0.06em x_0}{{\rm d} t} = \bar{v}_x(x_0, t). \end{equation}

For the sessile drop, the solution is recalled in Appendix A (A9) and is represented in figure 5 as a solid black line. For the fibre, the complexity of (2.21) makes us unable to find an analytical solution of the differential equation (3.1). Instead, we proceed to a numerical integration with odeint from scipy (Jones, Oliphant & Peterson Reference Jones, Oliphant and Peterson2001). The solution is plotted as a solid blue line in figure 5.

Figure 5. Temporal evolution of the dimensionless positions of the advected fluid layer $x_0/{\mathcal {L}}$ (solid lines) and of $x^\star /{\mathcal {L}}$ (dashed lines), the positions beyond which the Péclet number is greater than unity. Blue lines are used for axisymmetric drops on fibres and black lines for sessile drops. Here, the comparison between the two geometries is performed in the conditions described in § 3.1 and in figure 3.

We define $x^\star$ as the position along the solid surface for which $\overline {Pe}(x^\star ) = 1$. For $x > x^\star$, we have $\overline {Pe} > 1$ such that the particles advection by the flow overcome their Brownian diffusion. The particles in the volume of liquid enclosed between $x = x^\star$ and $x = {\mathcal {L}}$ are advected by the flow and transported to the triple line.

In figure 5, we plot by dashed lines $x^\star /{\mathcal {L}}$ as a function of the dimensionless time for the two geometries. At the beginning of the drying, $x^\star /{\mathcal {L}}$ is close to unity for both geometries. This means that the region over which the particles are transported by the liquid flow, corresponding to the region between $x =x^\star$ and $x= {\mathcal {L}}$, is localised in the close vicinity of the contact line at the beginning of evaporation. We even note that initially, this region is smaller for a sessile drop than for a drop on a fibre $r^\star /R < z^\star /L$. Thus, at the beginning of evaporation, the transport of the particles towards the contact line is more efficient for the drop on a fibre. However, as the sessile drop evaporates, the region boundary position $r^\star /R$ continuously decreases to reach zero at the end of the pinned regime ($t = \tau _{dep}$). On the fibre, $z^\star /L$ remains nearly constant, close to unity. In other words, the width of the attraction zone in a sessile drop grows as the liquid evaporates. Progressively, this zone occupies the entire drop, which leads to transport of the majority of the suspended particles towards the contact line. This is not the case for the drop on a fibre for which the zones in which the particles are effectively transported by the flow to the contact lines remain small and located in the close vicinity of the triple line.

The curves presented in figure 5 provide a comparison of the relative positions of $x^\star$ (dashed lines) and $x_0$ (solid lines). During most of the pinned contact line regime, the area bounded by the position of the advected fluid layer $x_0$ includes small and large Péclet numbers domains ($x_0 < x^\star <\mathcal {L}$). Exceptions are noticed at short time scales after evaporation starts and at the end of the pinned regime for the sessile drop.

3.4.2. Particle accumulation dynamics in the large Péclet domain – $x^\star < x_0$

If $x^\star < x_0$, i.e. $\overline {Pe} > 1$ between $x_0$ and $\mathcal {L}$, the particles in this layer are transported towards the contact line such that their number is conserved. On the fibre, from the initial concentration (number of particles per unit volume) $c_{i}$ and the initial liquid profile $h(z, t = 0)$, the number of particles $N_{{CL}}$ accumulated at each contact line can be written as

(3.2)\begin{equation} N^{fibre}_{{CL}}(t) = c_{i} \int_a^{a+h(z, t = 0)} \int_0^{2{\rm \pi}} \int_{z_0(t)}^L r \,{\rm d} r \, {\rm d} \theta \, {\rm d} z, \end{equation}

which gives

(3.3)\begin{equation} N^{fibre}_{CL}(t) = {\rm \pi}c_{i} h_{i} L \left(h_{i} \mathcal{P}_3\left(\frac{z_0}{L}\right) + 2a \mathcal{P}_4\left(\frac{z_0}{L}\right) \right), \end{equation}

with $\mathcal {P}_3(x) = - ({x^{9}}/{9}) + {4 x^{7}}/{7} - {6 x^{5}}/{5} + {4 x^{3}}/{3} - x + \frac {128}{315}$ and $\mathcal {P}_4(x) =- ({x^{5}}/{5}) + {2 x^{3}}/{3} - x + \frac {8}{15}$. The above equation is shown as a dashed blue line in figure 6. The equivalent calculation for the sessile drop is recalled in Appendix A.1 (A11) and is represented by the dashed grey line in figure 6.

Figure 6. Time evolution of the dimensionless number of particles at the contact line $N_{CL}/N_{tot}$. Blue lines represent axisymmetric drops on fibres and black lines sessile drops. Solid lines are obtained from (3.3) (fibre) or (A11) (sessile) when $x_0 \geq x^\star$ and from (3.5) (fibre) or (A13) (sessile) when $x_0 \leq x^\star$. The dashed lines represent the results obtained under the assumption that all particles contained between $x_0$ and $\mathcal {L}$ are transported to the contact line and are plotted from (A11) for a sessile drop (grey dashed line) and (3.3) for a drop on a fibre (blue dashed line). Here, the comparison between the two geometries is performed in the conditions described in § 3.1 and in figure 3. The inset shows the temporal evolution of the number of particles $N_{CL}$ plotted in solid lines in the main figure for an initial particle concentration $c_{i} = 1 \times 10^8$ particles mL$^{-1}$.

3.4.3. Particle accumulation dynamics over small and large Péclet domains – $x^\star \geq x_0$

Now, if positions $x^\star$ and $x_0$ are swapped, the number of accumulated particles is decomposed from two contributions. The first contribution, from $x^\star$ to ${\mathcal {L}}$, is equivalent to (3.2). In this large Péclet domain, all the particles are advected with an increasing particle concentration as evaporation proceeds. Between $x_0$ and $x^\star$, the small Péclet number indicates that Brownian motion maintains a uniform particle concentration. Once the fluid layer reaches $x^\star$, particles are advected and concentrated as it is between $x^\star$ and ${\mathcal {L}}$. Therefore, we evaluate the number of particles in the fluid layer at the position $x^\star$ with a particle concentration $c_{i}$. The sum of these two contributions gives

(3.4)\begin{equation} N^{fibre}_{{CL}}(t) = c_{i} \int_a^{a+h(z, t = 0)} \int_0^{2{\rm \pi}} \int_{z^\star}^L r {\rm d} r \, {\rm d} \theta \, {\rm d} z + c_{i} \int_a^{a+h(z^\star, t=0)} \int_0^{2{\rm \pi}} \int_{z_0(t)}^{z^\star} r \,{\rm d} r \, {\rm d} \theta \, {\rm d} z. \end{equation}

After integration, we obtain

(3.5)\begin{align} N^{fibre}_{CL}(t) &= {\rm \pi}c_{i} h_{i} L \left[h_{i}\left(\mathcal{P}_3\left(\frac{z^\star}{L}\right) + \left(1 - \left(\frac{z^\star}{L}\right)^2\right)^4 \left(\frac{z^\star}{L} - \frac{z_0}{L}\right) \right) \right. \nonumber\\ &\left. + 2a\left(\mathcal{P}_4\left(\frac{z^\star}{L}\right) + \left(1 - \left(\frac{z^\star}{L}\right)^2\right)^2 \left(\frac{z^\star}{L} - \frac{z_0}{L}\right) \right) \right]. \end{align}

The same calculation is performed in Appendix A.1 for the sessile drop.

Using the appropriate conditions according to the relative positions of $x_0$ and $x^\star$ shown in figure 5, we plot by solid lines in figure 6 the dimensionless number of particles accumulated at the contact line. In the inset of figure 6, we plot the number of particles accumulated at the triple line over time for an initial concentration of $c_{i} = 1 \times 10^8$ particles mL$^{-1}$.

Figure 6 (grey dashed line) shows that the classical calculation (Deegan et al. Reference Deegan, Bakajin, Dupont, Huber, Nagel and Witten2000; Popov Reference Popov2005; Monteux & Lequeux Reference Monteux and Lequeux2011; Berteloot et al. Reference Berteloot, Hoang, Daerr, Kavehpour, Lequeux and Limat2012; Boulogne et al. Reference Boulogne, Ingremeau and Stone2017), made in the literature for a sessile drop, valid if all the particles contained between $x_0$ and $\mathcal {L}$ are transported by the flow, leads to overestimate the number of particles accumulated at the triple line during the first part of the pinned regime. However, at the end of the drying, the liquid height tends towards 0 which leads to $\overline {Pe} > 1$ in almost the entire drop (cf. figure 5) i.e. $r^\star \leq r_0$. The result is that almost all the particles are transported and deposited at the initial position of the contact line during drying, which corresponds to a typical density of $N_{tot} / 2{\rm \pi} R \approx 2$ particles $\mathrm {\mu }$m$^{-1}$ in the final deposit. In practice, we must note also that the threshold value $\overline {Pe} = 1$ is arbitrary and must be adjusted for a fine quantitative description.

For a drop on a fibre, however, $z^\star$ is almost constant and therefore during most of the pinned regime, $z^\star > z_0$ which means that the classical calculation of (3.3), represented by blue dashed line in figure 6, overestimates the number of particles accumulated at the contact line. Indeed, taking into account the fact that Brownian diffusion dominates in the zone between $z_0$ and $z^\star$, we obtain a number of particles accumulated at the edge of the drop that is ten times lower than the one obtained by considering that all the particles between $z_0$ and $z^\star$ are advected by the flow.

Figure 6 also shows that the particles contained in the drop on a fibre are transported towards the contact lines. At the end of the pinned regime, the number of particles accumulated at the initial positions of the triple lines of a drop on a fibre corresponds to $N_{CL}(\tau _{dep}) / 2 {\rm \pi}a = 0.2$ particles $\mathrm {\mu }$m$^{-1}$, which is approximately 10 times lower than the sessile drop. As shown in the inset, the duration of the pinned regime is approximately the same in both geometries $\tau _{dep}^{fibre} \approx 110$ s and $\tau _{dep}^{sessile} \approx 90$ s, but the rate of accumulation of particles at the contact line is lower in the drop on fibre than in the sessile drop. This lower rate can be attributed to the overall lower fluid velocity and the narrow size of the advection-dominated domain in the fibre geometry.

Next we want to compare the results of the calculations with what is observed experimentally.

4.2. Observations

An example of the temporal evolution of the system is shown in figure 7(a) for a drop on a fibre and 7(b) for a sessile drop. The Worthington number associated to the drop on the fibre is ${Wo}\approx 0.2$, validating the small effect of gravity on the drop shape as observed in figure 7(a).

Figure 7. Experimental observations of particle transport induced by evaporation. (a,b) Temporal evolution of a drop on a fibre (side view) and on a flat surface (bottom view), respectively. The initial wetted length is $R = L \approx 0.75$ mm. The times framed in red correspond to the moment when the depinning of one of the contact lines occurs. (c,d) Photographs of the corresponding final deposit. The direction of gravity is shown in the pictures. The scale bars represent 0.2 mm and the yellow arrows indicate the initial position of the triple lines. Supplementary movies are available at https://doi.org/10.1017/jfm.2023.59.

One of the main differences between the two geometries is that there is only one contact line in a sessile drop, whereas two independent contact lines exist in a drop on a fibre. In both cases, the contact line depinning from its initial position is indicated by red frames in figures 7(a) and 7(b). The exact depinning time $\tau _{dep}$ of one of the two triple lines of a drop on a fibre is difficult to determine experimentally due to the curvature of the substrate and the interface. However, it is observed that, on a fibre, the time during which the wetted length is constant is small compared to the total lifetime of the drop and represents approximately 1 %–10 % of the lifetime. This is not observed for a sessile drop where the triple line remains pinned for the majority of the drying time, i.e. 80 %–90 % of the lifetime. The lifetime of the sessile drop is shorter than the lifetime of the drop on a fibre because its initial volume is lower. Corpart et al. (Reference Corpart, Dervaux, Poulard, Restagno and Boulogne2022) have shown that the evaporative flux of a barrel-shaped droplet on a fibre is correctly approximated by the one of a spherical droplet in the same condition, such that the lifetime of a drop on a fibre can be estimated as $\rho L^2 / (2 {\mathcal {D}}_{v} [c_{sat} - c_\infty ])$. Comparing this result to the lifetime of a sessile drop defined in (A5) for the experimental condition tested here, we obtain $\tau _{e}^{fibre}/\tau _{e}^{sessile} \approx 8 /( {\rm \pi}\theta _{i}) \approx 10$ for $\theta _{i} = 15^\circ$, which is in good agreement with what is measured experimentally, as we get $\tau _{e}^{fibre}/\tau _{e}^{sessile} \approx 11$.

The final deposition is shown in figures 7(c) and 7(d) for these two geometries. During the pinned regime of a drop on a fibre, it is observed that the areas over which particles are transported to the triple lines are small and remain localised in the vicinity of the contact line, so that few particles are deposited at the contact lines. Conversely, in a sessile drop evaporating at constant wetted length, we observe that the zone of attraction of the triple line progressively increases over time to finally invade the entire liquid. The particles are therefore mainly transported and deposited at the initial position of the contact line.

When one of the two contact lines unpins from the fibre, which is beyond the proposed model, the particles continue to accumulate at the stationary contact line, while the movement of the other triple line generates a complex flow in the drop. This contact line can then anchor again and a complex alternation of the movement of the left and right lines can be noticed, leading to the typical final morphology of the deposit observed in figure 7(c).

4.3. Comparison with the proposed model

There is a qualitative agreement between the observations and the predictions of the calculation. Indeed, for the drop on a fibre, we observe experimentally that the zone over which the particles are transported towards the triple line is small and remains localised at the edge of the drops during the pinned regime and that there are few particles accumulated at the contact line during the pinned regime. The duration of the pinned regime is also short compared to the lifetime of the drop and when the depinning occurs, there is a significant volume of liquid remaining on the fibre.

However, we were unable to obtain a quantitative agreement between the theory and the experiments because of the difficulty of tracking the particles due to the curvature of the substrate and the liquid–vapour interface. In addition, the area where we can measure the velocity is located at the edge of the drop where the velocity diverges, so we cannot see any difference between the two geometries.