3.1. Basic assumptions

The theoretical approach is based on the following premises.

We consider a sessile droplet composed of water and PG and evaporating into ambient air at a relative humidity $R\!H$. The problem is axisymmetric (although this assumption will partly be overridden in § 5 by considering a gradient of $R\!H$).

As the contact angles and film slopes are here not large (${<}20^\circ$), we rely upon the lubrication approximation in the liquid phase (inside the droplet). The composition variation across the droplet is neglected.

The experimental results (figure 3) permit us to conjecture that, shape-wise, the solutal Marangoni action is immediately essential just in a small vicinity of the contact line (foot region), setting off the apparent contact angle for the classical static shape (2.1a) in the core of the droplet. Accordingly, assuming such a separation of scales, we here advance a local approach/model intended to be valid in the foot region and match (2.1a).

We further conjecture (to be verified a posteriori) that, for such a strong localization in the foot region, there must be a good solutal-Marangoni mixing inside the droplet, which will smooth out concentration gradients in the core of the droplet and (to a lesser extent) in the foot region. Accordingly, we assume that the water mass fraction $c(r)$ deviates everywhere only slightly from its (spatially quasi-constant) value $c_m$ in the core of the droplet:

(3.1)\begin{equation} c=c_m+c'(r) \quad (c'\ll 1). \end{equation}

The mixing will here be treated by means of the Taylor dispersion (Taylor Reference Taylor1953, Reference Taylor1954) in the framework of the lubrication approximation.

Only quasi-steady regimes are considered. In particular, an initial fast-spreading stage (upon the droplet deposition) is assumed to be over. The strong mixing and localization also favour fast attaining a quasi-steady state from the viewpoint of concentration distribution, so that the composition $c_m$ has not yet changed significantly due to evaporation, as compared with the initial one upon the droplet deposition. (In fact, one can refer to figure 1(c) of a recent preprint by Ramírez-Soto & Karpitschka (Reference Ramírez-Soto and Karpitschka2021) to appreciate how fast the quasi-steadiness is really reached (in less than $1\,\mathrm {s}$).) Thus, $c_m$ is here treated as an input parameter, identified with the initial composition. The other input parameters (apart from the material properties) are the droplet size (represented by either the droplet contact radius $R$, or the droplet volume $V$) and the ambient relative humidity $R\!H$ (expressed either in fraction terms or in per cent as evident from the context).

A direct contribution of evaporation into the flow inside the droplet is neglected with respect to the solutal Marangoni flow. The apparent contact angles are assumed to be entirely associated with the latter.

The material properties of the liquid such as the surface tension $\gamma (c)$, $({\textrm {d}\gamma }/{\textrm {d} c})(c)$, diffusion coefficient $D_l(c)$ and dynamic viscosity $\eta (c)$ are all functions of the composition. These dependencies are fully accounted for in the present paper using results from the literature (cf. Appendix A). However, by virtue of the ansatz (3.1), they are here evaluated at $c=c_m$ and otherwise treated as constants.

In view of close densities of water and PG, we shall adopt an approximate constant value of the liquid density throughout, $\rho _l\approx 1015\,\mathrm {kg}\,\mathrm {m}^{-3}$, also implying no distinction between the mass and volume fractions. The PG volume fraction (used in figure 4) is then $(1-c_m)\times 100\,\%$. Neglecting the natural convection inside the droplet is, notwithstanding, a separate assumption, adopted in view of small contact angles (cf. Diddens, Li & Lohse Reference Diddens, Li and Lohse2021).

The volatility of PG is neglected. As for water, Raoult's law is assumed at any liquid composition, which use is supported by data from Verlinde, Verbeeck & Thun (Reference Verlinde, Verbeeck and Thun2010) and Karpitschka et al. (Reference Karpitschka, Liebig and Riegler2017, figure S2).

Droplet evaporation is controlled by vapour diffusion in the gas. The vapour is dilute (hence no Stefan flow), given that the saturation pressure of water is much smaller than the atmospheric pressure at the temperatures considered (${\sim }20\,^\circ$C). In view of the small slopes, the droplet is approximated by a flat disk from the viewpoint of the vapour diffusion problem in the gas. By virtue of the ansatz (3.1), the distribution of the evaporation flux density $j(r)$ (in $\mathrm {kg}\,\mathrm {m}^\mathrm {-2}\,\mathrm {s}^\mathrm {-1}$) along the droplet diameter is assumed to be the one for a droplet of a uniform composition $c=c_m$, neglecting a feedback from $c'(r)$. This gives rise to a well known expression (e.g. Popov Reference Popov2005)

(3.2)\begin{equation} j(r)=\frac{2 D_g\rho_{sat} (\chi_m-R\!H)}{{\rm \pi} \sqrt{R^2-r^2}} \end{equation}

with $D_g=2.55\times 10^{-5}\,\textrm {m}^2\,\textrm {s}^{-1}$ the water vapour diffusivity and $\rho _{sat}=0.017\,\textrm {kg}\,\textrm {m}^{-3}$ (at 20 $^{\circ }$C) the saturation density of water vapour, except that it is here modified by a prefactor $(\chi _m-R\!H)$ accounting for Raoult's law and the ambient humidity. The molar fraction of water in the liquid $\chi$ is related to $c$ by

(3.3a,b)\begin{equation} \chi=\frac{M_{pg} c}{M_{pg} c+ M_w (1-c)},\quad \chi_m=\frac{M_{pg} c_m}{M_{pg} c_m+ M_w (1-c_m)} \end{equation}

(likewise for $\chi _m$ versus $c_m$), where $M_w=18\,\textrm {g}\,\textrm {mol}^{-1}$ and $M_{pg}=76\,\textrm {g}\,\textrm {mol}^{-1}$ are the molar masses of water and PG, respectively. The prefactor and the evaporation flux (3.2) both vanish at equilibrium. The equilibrium relative humidity $R\!H_{eq}$ for a given liquid composition $\chi _m$ is then obviously $R\!H_{eq}=\chi _m$. Inversely, for a given $R\!H$ the equilibrium composition $\chi _{m,{eq}}$ corresponds to $\chi _{m,{eq}}=R\!H$. In the present paper, we study the case $\chi _m>R\!H$ (in other words, $\chi _m>\chi _{m,{eq}}$, or $R\!H< R\!H_{eq}$), when, as argued in § 1, the evaporation and the solutal Marangoni stresses due to the ensuing composition inhomogeneity make for a contraction regime of the droplet. Note in (3.2) a well known integrable divergence of the evaporation flux at the contact line (as $r\to R$).

In the narrow foot region near the contact line, for which our local model will be developed, the evaporation flux (density) distribution can be established as the edge behaviour ($x\equiv R-r$, $x\ll R$) within (3.2):

(3.4)\begin{equation} j(x)=\frac{\sqrt{2}D_g\rho_{sat}}{\rm \pi}\frac{1}{\sqrt{R x}} (\chi_m-R\!H). \end{equation}

The local model based upon (3.4) and thus neglecting a feedback between $j(x)$ and $c'(x)$ (cf. above (3.2)) will be referred to as the ‘non-rectified’ local model. In contrast, the ‘rectified’ local model will be the one fully accounting for the mentioned feedback (‘rectification’) and yielding a substitute to (3.4) to be considered later on. Regardless, even in the latter case, it will still be assumed that the ‘non-rectified’ expression (3.2) is valid, although just in the core of the droplet (excluding the foot region). This is justified by an expected higher degree of mixing and composition uniformity in the core of the droplet and a higher degree of water depletion (larger values of $-c'$) in the foot region. Thus, the rectification will thereby just stay confined to the local problem in the foot region.

The rationale behind considering the non-rectified local model (despite the eventual availability of a more general, rectified one here) is its incomparable simplicity, which will permit us to probe the essence of the phenomenon in a more subtle and clear way. Furthermore, the non-rectified approach can already prove sufficient at least in certain cases. For instance, one can expect it to work for droplets with a large water content, when close to the pure-water case. At the same time, it will prepare ground for passing to the rectified local model should the need arise. In the remainder of the present section, the developments of §§ 3.2, 3.3 and 3.6 equally apply to both local models. The consideration in §§ 3.4, 3.5, 3.7 and 3.8 is based on the non-rectified approach, although the discussion in §§ 3.5 and 3.7 is deemed to be pertinent regardless of the model type. Finally, the rectified local model is tackled in § 3.9.

3.2. Formulation

We now focus on the liquid film in the foot region. Within the lubrication approximation, the problem is described by two dependent variables, which are the height $h(x)$ and concentration $c'(x)$ distributions along the film (Cartesian coordinate $x>0$ with $x=0$ chosen at the contact line). Therefore, two equations need to be formulated (momentum and species conservation), which are considered in the following two paragraphs.

With no slip at the substrate ($z=0$) and the solutal Marangoni stress at the free surface $z=h$, the solution of the momentum equation $0=-\partial _x p+\eta \partial _{zz} u$ for the velocity field component parallel to the substrate in the lubrication approximation is

(3.5)\begin{equation} u=\eta^{{-}1} (\partial_x p) \left(\frac{1}{2} z^2-h z\right)+\eta^{{-}1} \frac{\textrm{d}\gamma}{\textrm{d} c} (\partial_x c') z, \end{equation}

a sum of the (half-) Poiseuille and linear shear flows. Here $z$ is the Cartesian coordinate across the film, and $p=-\gamma \partial _{xx} h$ the Laplace pressure (gravity being negligible in this small region). With the evaporation-related flow neglected as stated in § 3.1, a quasi-steady lubrication equation merely reduces to $\int _0^h u\,\textrm {d} z=0$, and hence

(3.6)\begin{equation} \gamma h \partial_{xxx} h +\frac{3}{2}\frac{\textrm{d}\gamma}{\textrm{d} c}\partial_x c'=0. \end{equation}

According to (3.6), the film is shaped by the solutal Marangoni effect moderated by the capillary pressure.

We next turn to an equation for $c'(x)$. We recall (cf. § 3.1) that the dependence of $c'$ on $z$ is neglected to leading approximation assuming a good levelling by molecular diffusion in our thin droplet, hence $c'=c'(x)$. The water species balance in any element ${\textrm {d}\kern0.06em x}$ of the film consists of a diffusion influx $h\rho _l D_{eff}\partial _x c'\vert _x^{x+{\textrm {d}\kern0.06em x}}$, convective suction influx $j c_m\,{\textrm {d}\kern0.06em x}$ (here neglecting $c'$ against $c_m$, cf. § 3.1) due to the overall evaporation flux $j(x)$ (in $\textrm {kg}\,\textrm {m}^{-2}\,\textrm {s}^{-1}$), and losses $(-j_w\,\textrm {d}\kern0.06em x)$ due to water evaporation flux $j_w(x)$ (in $\textrm {kg}\,\textrm {m}^{-2}\,\textrm {s}^{-1}$). In a quasi-steady state, the sum of the three must vanish. As water is considered as the only volatile component in a water–PG mixture, and hence $j_w=j$, we arrive at the following equation:

(3.7)\begin{equation} \partial_x (h D_{eff} \partial_x c')- j(1-c_m)/\rho_l=0, \end{equation}

according to which the concentration profile is a result of the competition between the evaporation and diffusion. A more general outlook on (3.6) and (3.7) can be found in Appendix B. Here $D_{eff}$ is the effective diffusion coefficient resulting from the Taylor dispersion (Taylor Reference Taylor1953, Reference Taylor1954; Van den Broeck Reference Van den Broeck1990). The idea behind this is that the combination of molecular diffusion $D_l$ along $z$ and a flow $u(z)$ with $\int _0^h u\,\textrm {d} z=0$ (as here) results in an additional diffusion-like smearing of the profile $c'(x)$. Hence the form $D_{eff}=D_l (1+B P\!e^2)$, where $B$ is a numerical coefficient and $P\!e$ a Péclet number. Proceeding in a standard way, one can obtain $BP\!e^2=(D_l^2 h)^{-1} \int _0^h (\int _0^z u(z')\,\textrm {d} z' )^2 \,\textrm {d} z$. Using $u(z)$ from (3.5) and using (3.6) as a constraint, one finally arrives at

(3.8)\begin{equation} D_{eff}= D_l \left[1+\frac{1}{1680} \left(\frac{1}{D_l\eta} \frac{\textrm{d}\gamma}{\textrm{d} c} h^2 \partial_x c' \right)^2 \right]. \end{equation}

For a closure, one still requires $j(x)$, which is given by (3.4) for the non-rectified local model and will be considered in § 3.9 for the rectified one.

3.3. Rescaling

Rescaling in accordance with

(3.9a–f)\begin{gather} x^*=\frac{x}{\delta},\quad h^*=\frac{h}{\epsilon\delta},\quad \theta_{app}^*=\frac{\theta_{app}}{\epsilon},\quad \theta_{max}^*=\frac{\theta_{max}}{\epsilon},\quad c'^*=\frac{c'}{\zeta},\quad j^*=\frac{j}{\psi}, \end{gather}

(3.10)\begin{gather}\epsilon = \left( \frac{3^{7/4} 35^{1/4} (1-c_m) D_g \dfrac{\textrm{d} \gamma}{\textrm{d} c} \eta^{1/2} \rho_{sat} (\chi_m -R\!H)}{{\rm \pi} D_l^{1/2} R^{1/2} \gamma^{3/2} \rho_l}\right)^{1/5}, \end{gather}

(3.11a–c)\begin{gather}\zeta = \frac{2\gamma}{3 {\rm d}\gamma/{\rm d}c} \epsilon^2,\quad \delta=6\sqrt{105}\frac{D_l \eta}{\gamma} \epsilon^{{-}4},\quad \psi=\frac{\sqrt{2}D_g\rho_{sat}}{\rm \pi}\frac{1}{\sqrt{R \delta}} (\chi_m-R\!H) \end{gather}

enables us to get rid of all the prefactors in the formulated system of equations. Here $\delta$ is the length scale defining the longitudinal extent of the foot region, $\epsilon$ is the scale of film slopes and apparent contact angles (in radians), $\zeta$ is the scale of mass fraction variations and $\psi$ is the scale of the evaporation flux density (3.4) in the foot region. For the typical values (apart from already listed) $\gamma \sim 50\ \textrm{mN\ m}^{-1}$, ${\textrm {d}\gamma }/{\textrm {d} c}\sim 20\,\textrm {mN}\,\textrm {m}^{-1}$, $\eta \sim 0.01\,\textrm {Pa}\,\textrm {s}$, $D_l\sim 0.3\,\textrm {mm}^2\,\textrm {s}^{-1}$, $R\!H\sim 0.3$, $c_m\sim 0.5$ and $R\sim 1\,\mathrm {mm}$, one obtains $\epsilon \sim 0.18\ll 1$ ($\epsilon \sim 10^\circ$), $\zeta \sim 0.05\ll c_m$ and $\delta \sim 3\,\mathrm {\mu }\mathrm {m}\ll R$. These are quite in agreement with the assumptions made, such as small slopes (lubrication approximation), small composition non-uniformity (3.1) and length scale separation (narrowness of the foot region), respectively (cf. § 3.1).

3.4. Solution within the non-rectified local model

The system of ODEs thereby becomes

(3.12)\begin{gather} h^* \partial_{x^*x^*x^*} h^*+\partial_{x^*} c'^*=0, \end{gather}

(3.13)\begin{gather}\partial_{x^*}\left[(1+h^{*4}(\partial_{x^*} c'^*)^2) h^* \partial_{x^*} c'^*\right]=j^*, \end{gather}

(3.14)\begin{gather}j^*=\frac{1}{\sqrt{x^*}}, \end{gather}

which is defined in the interval $0< x^*<+\infty$ (the infinity formally representing the core of the droplet in the framework of our local approach). At the contact line ($x^*=0$), we impose the boundary conditions

(3.15a–c)\begin{equation} h^*=0,\quad \partial_{x^*} h^*=0,\quad c'^*<\infty \quad \text{at}\ x^*=0, \end{equation}

where for definiteness we have chosen $\theta _{mic}=0$, for its value is anyway unknown and besides only weakly affects the sought final result (cf. Appendix C). At the opposite end of the interval, in accordance with the present local approach, we wish to satisfy the boundary conditions

(3.16a,b)\begin{equation} \partial_{x^* x^*} h^*=0,\quad c'^*=0 \quad \text{as}\ x^*\to +\infty \end{equation}

(small, negligible curvature and ‘full mixing’ in the core of the droplet as compared with the foot region). The system of ODEs (3.12) and (3.13) is of the fifth order and there are formally five boundary conditions in (3.15) and (3.16). However, both $x^*=0$ and $x^*=+\infty$ are singular points. Therefore, an additional exploration by means of coordinate expansions can be helpful, which we carry out following the next paragraph.

On account of (3.14), (3.13) can be integrated once to yield

(3.17)\begin{equation} (1+h^{*4}(\partial_{x^*} c'^*)^2) h^* \partial_{x^*} c'^*=2 \sqrt{x^*}, \end{equation}

where the integration constant was set equal to zero on the premise of no diffusion flux ($h^* \partial _{x^*} c'^*$ in dimensionless terms) at the contact line (as $x^*\to 0$), or alternatively, in order to ensure compatibility with the boundary conditions (3.15).

The solution of (3.12) and (3.17) with (3.15) can be found to start off as

(3.18)\begin{gather} h^*\sim 6\left({\tfrac{2}{35}}\right)^{1/3} x^{*7/6} + K x^{*\alpha}+\cdots\quad\text{as}\ x^*\to 0, \end{gather}

(3.19)\begin{gather}c'^*\sim c'^*_0+\left(\tfrac{35}{2}\right)^{1/3} x^{*1/3}+\cdots\quad\text{as}\ x^*\to 0, \end{gather}

where $\alpha =2.134$ is the largest root of the cubic equation $\alpha (\alpha -1)(\alpha -2)=\frac {35}{108}$. Here $K$ and $c'^*_0$ are free parameters that can be used to shoot for the conditions (3.16). The latter conditions can be seen to be compatible with (3.12) and (3.17), implying the behaviour

(3.20)\begin{gather} h^*\sim \theta_{app}^* x^*-\frac{8\times 2^{1/3}}{3\theta_{app}^{*8/3}} x^{*1/2} -\frac{32\times 2^{2/3}}{9 \theta_{app}^{*19/3}} \ln x^*+C +\cdots\quad \text{as}\ x^*\to +\infty, \end{gather}

(3.21)\begin{gather}c'^*\sim{-} \frac{2^{4/3}}{\theta_{app}^{*5/3}} x^{*-1/2}+\cdots \quad \text{as}\ x^*\to +\infty. \end{gather}

Here the value of $\theta _{app}^*$ (a free parameter of the expansion) is to be found together with the overall solution. It is interpreted as a rescaled apparent contact angle as seen from a larger scale (core of the droplet), the true (non-rescaled) one then being $\theta _{app}= \theta _{app}^* \epsilon$, cf. (3.9c) and (3.10). Here $C$ is another free parameter. We note that the order of the system of ODEs (3.12) and (3.17) is four, and there are accordingly just four free parameters $K$, $c'^*_0$, $\theta _{app}^*$ and $C$ in the coordinate expansions (3.18), (3.19), (3.20) and (3.21) . We also note that (3.12) can be integrated once, on account of (3.18) and (3.19), to yield $h^*\partial _{x^* x^*}h^*-\frac {1}{2} (\partial _{x^*}h^*)^2+c'^*-c'^*_0=0$. Using (3.20) and (3.21) in here, one can deduce

(3.22)\begin{equation} c'^*_0={-}\tfrac{1}{2} \theta_{app}^{*2} \end{equation}

for the rescaled water mass fraction deficit at the contact line, the true (non-rescaled) one then being $c'_0= c'^*_0 \zeta$, cf. (3.9e), (3.10) and (3.11a).

The dimensionless problem is parameter free and only needs to be solved once and for all. The computation yields a unique solution, satisfying all boundary conditions, with the profiles shown in figure 5 as well as the values of the free parameters of the coordinate expansions such as

(3.23)\begin{equation} \theta_{app}^*=2.40\quad \Longrightarrow \quad \theta_{app}=2.4 \epsilon \end{equation}

and $c'^*_0=-2.87$ (marked by a point in figure 5 and compatible with (3.22) and (3.23)). This wraps up the local solution in the foot region. To return to the original, dimensional variables, one just needs to use (3.9)–(3.11).

Figure 5.

Dimensionless/rescaled height $h^*$, slope $\partial _{x^*} h^*$ and water mass fraction deviation $c'^*$ profiles (solid lines) and apparent contact angle $\theta _{app}^*=\partial _{x^*} h^*\vert _{x^*\to +\infty }$ (dashed line) within the non-rectified local model (3.12)–(3.16) in the foot region. The scales for the quantities with an asterisk are defined in (3.9), (3.10) and (3.11).

3.6. Calculation of $\theta _{max}$ by composite expansions

The local solution is only valid in the foot region, whereas the classical static shape just in the core of the droplet. One can build therefrom a uniformly valid ‘composite’ solution for the overall droplet profile by summing up the local solution $h(x)$ (as in figure 5) and the classical static shape $h(r)$, given by (2.1a) with $r=R-x$, and then subtracting their ‘common’ part $\theta _{app} x$. This will also permit us to obtain $\theta _{max}$ from the maximum slope of the composite profile, thus drawing a finer distinction between $\theta _{app}$ and $\theta _{max}$ as announced following (2.2). In the present paper, however, we shall limit ourselves to a lighter and even more approximate version of this procedure. Namely, we just represent the composite profile in a vicinity of the maximum slope point (which is also the inflection point, cf. figure 1) by the first two terms of the asymptotics (3.20) plus a curvature contribution $-\frac {1}{2} \kappa ^* x^{*2}$, where $\kappa =-\partial _{rr} h\vert _{r=R}>0$ is the first curvature of the classical static shape (2.1a) at the contact line and $\kappa ^*=({\delta }/{\epsilon }) \kappa$. One can then readily deduce

(3.24a,b)\begin{gather} \theta_{max}^*\approx\theta_{app}^* - \frac{2^{8/9} 3^{1/3} \kappa^{*1/3}}{\theta_{app}^{*16/9}},\quad \theta_{max}=\theta_{max}^* \epsilon, \end{gather}

(3.25)\begin{gather}\kappa^* = \theta_{app}^* \frac{\delta}{R} \left(\frac{R}{l_c} \frac{I_0\left(\dfrac{R}{l_c}\right)}{I_1\left(\dfrac{R}{l_c}\right)}-1\right), \end{gather}

where note that $\theta _{max}$ is slightly smaller than $\theta _{app}$, their difference being small to the extent the curvature in the core of the droplet is small on the scale of the foot region (i.e. to the extent $\kappa ^*\ll 1$). Note also that (3.25) yields $\kappa ^*=\theta _{app}^*({\delta }/{R})$ in the spherical cap limit ($R\ll l_c$). At the same time with (3.24a), one obtains the location of the maximum slope (inflection) point at $x^*=2^{8/9}/(3^{2/3}\theta _{app}^{*16/9} \kappa ^{*2/3})$. This corresponds to $x^*=O({R}/{\delta })^{2/3}$, and thus we see that the maximum slope point actually belongs to an intermediate zone between the foot region $x^*=O(1)$ and the core of the droplet $x^*=O({R}/{\delta })$ The interest of the result (3.24) lies in the fact that our experimental measurements correspond exactly to $\theta _{max}$, and not to any other possible version of the apparent contact angle, the same being deemed true as far as the reflectometric measurements by Cira et al. (Reference Cira, Benusiglio and Prakash2015) and Benusiglio et al. (Reference Benusiglio, Cira and Prakash2018) are concerned.

3.7. Verification by a non-rectified global model

To cross-check the various points raised in §§ 3.5 and 3.6 on the basis of the local approach in the foot region, we also carry out some test computations formulated for the entire droplet (‘global model’, in contrast with the local model used elsewhere in § 3). In doing so, non-rectified (‘Popov’) expressions for the evaporation flux $j$ are still used. The computation details are relegated to Appendix D. Some key results are illustrated here in figure 6, where note the additional definitions

(3.26a–c)\begin{equation} r^*=\frac{r}{\delta},\quad R^*=\frac{R}{\delta},\quad l_c^*=\frac{l_c}{\delta} \end{equation}

and the fact that the problem now depends on two dimensionless parameters $R^*$ and $R/l_c$ (unlike a parameter-free local formulation). We see that disregarding the Taylor dispersion would have noticeable negative consequences upon the results along several lines. For instance, the localization in the foot region would be smeared (mathematically, it would cease to exist as a distinguished region, along the arguments advanced in § 3.5), and it would be impossible to capture the sharp experimental slope profiles shown in figure 3. Larger $\theta _{max}$ values would be predicted, which would generally deteriorate the agreement with experiment as one will be able to appreciate later on (and a similar point was recently indicated by Ramírez-Soto & Karpitschka Reference Ramírez-Soto and Karpitschka2021). The mass fraction variation inside the droplet would have a tendency to be much greater, including a significant variation that would take place not only near the foot, but also in the core of the droplet. We note that the calculations in the absence of the Taylor dispersion in figure 6 were undertaken just for tendency illustration purposes, for a number of assumptions used here such as the ansatz (3.1) or quasi-steadiness are more likely to break down for such a case. As for other aspects manifest in figure 6, we notice a significant multifold enhancement of the effective diffusion relative to the molecular one, quite in agreement with the tendency earlier established with the help of the local model and also in agreement with the corresponding recent calculation by Ramírez-Soto & Karpitschka (Reference Ramírez-Soto and Karpitschka2021). Finally, the composite-expansion formula (3.24) for $\theta _{max}$ (short solid lines) has thereby been put to test and confirmed to work well (within $\sim$1 % in the studied $R^*$ range).

Figure 6.

Dimensionless/rescaled slope $-\partial _{r^*}h^*$, water mass fraction deviation $c'^*$ and effective diffusivity $D_{eff}$ profiles for sessile droplets of $R^*=10, 30\ \text {and}\ 50$ at $R\ll l_c$ (a,c,e) and $R/l_c=3$ (b,d,f) within the non-rectified global model, cf. § 3.7. Solid curves are results with Taylor dispersion (Marangoni mixing) as considered in the present paper. Dot–dashed curves are results ignoring the Taylor dispersion, for comparison. Short solid lines are $\theta _{max}^*$ levels in accordance with the composite-expansion formula (3.24a) with (3.23) and (3.25) for corresponding $R^*=R/\delta$ and $R/l_c$. Dashed line is $\theta _{app}^*$ level (3.23) given by the non-rectified local model. The scales for the quantities with an asterisk are defined in (3.9)–(3.11) and (3.26).

3.8. Non-rectified local model: parametric study of contact angles

To recapitulate, the result for $\theta _{app}$ within the non-rectified local model is given (in radians) by (3.23) on account of (3.10), while the result for $\theta _{max}$ by (3.24) on account of (3.10), (3.23) and (3.25). In the present subsection, we carry out a parametric study by plotting these results in the same format and for the same parameters as the experimental data shown in figure 4. However, to reduce figure cluttering, they are plotted in a separate figure 7 and the intermediate humidity case is omitted. Note that the above mentioned formulae are explicit in the droplet contact radius $R$, but not in the droplet volume $V$, whereas it is $V$ that is presumed fixed in figure 4 (and hence in figure 7). Therefore, an additional algebraic resolution using (2.1b) is herewith implied.

Figure 7.

Contact angle modelling results for the same parameters and in the same format as in figure 4 (excluding the intermediate humidity for space reasons), and the solid lines are also the same. Dotted and dot–dashed lines are predictions for $\theta _{app}$ and $\theta _{max}$, respectively, using the non-rectified local model, cf. § 3.8. Dashed and solid lines are similar predictions using the rectified local model, cf. § 3.9.

The results for $\theta _{app}$ and $\theta _{max}$ are shown in figure 7 by the dotted and dot–dashed curves, respectively. We see that the difference between $\theta _{app}$ and $\theta _{max}$ is expectedly subtle yet not unnoticeable, justifying a finer distinction (3.24) as compared with merely (2.2). The dashed and solid curves will be explained in § 3.9 later on and can be ignored in substance for the moment. However, in form, the solid curves can be used in figure 7 as a guide for the eye, representing fairly well the experimental results in figure 4.

In this way, we see that for large water contents (small PG contents, roughly ${\lesssim }20\,\%$), the agreement with experiment (i.e. between the dot–dashed curves and roughly the corresponding solid curves) appears to be rather good. In this case, the use of the present simple, non-rectified approach proves to be well justified, as hypothesized in § 3.1. However, an appreciable overestimation is observed for larger PG contents. Clearly, a key reason can be traced back to using the expression (3.4) throughout the foot region. It is exactly valid for constant concentration in the liquid and hence disregards possible local rectification of the evaporation flux due to a relatively significant local water depletion ($c'<0$) in the foot region. The assumption of merely $c'\ll c_m$ or $\zeta \ll 1$ (implied throughout the present paper) generally proves to be insufficient to neglect the rectification, as described in the following subsection.

3.9. Rectified local model

The details of such a rectified local model are provided in Appendix E (still without any fitting parameters), and a good fit of the corresponding numerical solution is given by

(3.27a,b)\begin{gather} \theta_{app}^*=2.4 (1+1.93 \beta)^{{-}0.182},\quad \theta_{app}=\theta_{app}^* \epsilon, \end{gather}

(3.28)\begin{gather}\beta= \frac{\rm \pi}{\sqrt{2}} \frac{\dfrac{M_w}{M_{pg}}}{\left[c_m + \dfrac{M_w}{M_{pg}} (1-c_m)\right]^2} \sqrt{\frac{R}{\delta}} \frac{\zeta}{\chi_m - R\!H}, \end{gather}

where the previous, non-rectified result (3.23) is recovered at $\beta \ll 1$. The latter is indeed seen to be a stronger inequality than merely $c'\ll c_m$ (or $\zeta \ll 1$ for that matter) principally due to the presence of another small parameter, $\sqrt {\delta /R}\ll 1$, by which $\zeta$ is actually divided in (3.28). This originates from the typical scale of local vapour concentration variation in the gas around the foot region, which is just $O(\sqrt {\delta /R})$ of the absolute value of vapour concentration (cf. Appendix E). For the rectification to be negligible, $\zeta$ need actually be small with respect to $\sqrt {\delta /R}$, and not just unity. The typical values of the parameter $\beta$ defined in (3.28) are represented in figure 8, confirming indeed the importance of the rectification for smaller water (larger PG) contents. The rectified evaporation flux profiles at various $\beta$ are shown in figure 9. One can notice a progressive reduction of the evaporation rate spike near the contact line as $\beta$ is increased; given that $\beta$ generally increases with the PG content (cf. figure 8), this result is qualitatively similar to the one obtained by Karpitschka et al. (Reference Karpitschka, Liebig and Riegler2017). It is also worthwhile to note that the result (3.22) holds irrespective of the rectification, and hence $c'^*_0$ within the rectified local model can be calculated therefrom upon the substitution of (3.27a).

Figure 8.

The rectification parameter $\beta$ defined in (3.28) as a function of the liquid composition and relative humidity for a droplet with $R=1\,\mathrm {mm}$ at $20\,^\circ$C. The evaporation-flux rectification can be ignored (and the non-rectified local model applied) inasmuch as $\beta \ll 1$.

Figure 9.

Dimensionless evaporation flux density profiles $j^*(x^*)$ in the foot region within the rectified local model for a number of non-vanishing values of the rectification parameter $\beta$ defined in (3.28) (solid lines, for all of which $j^*=0$ at $x^*=0$ – partial regularization – even if not always well discernible in the graph). The corresponding non-rectified profile (3.14) used within the non-rectified local model (dashed line, formally $\beta =0$), which is asymptotically attained by all profiles at large $x^*$. The scales for the quantities with an asterisk are defined in (3.9)–(3.11).

Using now the rectified result (3.27) with (3.28) instead of the non-rectified one (3.23) when plotting the contact angle $\theta _{max}$ in the way described in the beginning of § 3.8, we obtain the solid lines shown both in figure 4 and in figure 7. The agreement with experiment is seen to drastically improve with respect to the non-rectified approach, confirming the importance of the rectification where required (i.e. at larger PG contents). For comparison, instead of $\theta _{max}$ as above, we have also plotted just directly $\theta _{app}$, giving rise to the dashed lines in figure 7. The dashed lines are seen to agree with experiment slightly less well than the solid ones (cf. figures 4 and 7), which seems reasonable given that it is $\theta _{max}$ and not any other possible $\theta _{app}$ that is represented by the measurements.