Confined drying of a binary liquid mixture droplet: A quantitative interferometric study under humidity control

Abstract

We present a methodology that combines Mach-Zehnder interferometry, a custom relative humidity (RH) controlled chamber, and a confined two-dimensional droplet geometry to enable precise investigations of drying of complex fluids and the associated transport mechanisms. This approach is applied to a model binary mixture, water-glycerol, the concentration-dependent thermodynamic and transport properties of which are relatively well documented. High-resolution interferometric imaging (6 $μ$m pixel$^{-1}$, 1 frame s$^{-1}$) allows simultaneous measurement of drying kinetics and internal concentration fields with $\pm 0.5\%$ accuracy, characterized here over a wide range of RH (25-95%), and thus Péclet numbers. The experimental results closely match a quasisteady, isothermal model of vapor-diffusion-controlled evaporation coupled to diffusion within the droplet. These data enable extraction of both the concentration-dependent mutual diffusion coefficient $D(\varphi)$ and the water chemical activity $a_w(\varphi)$ over almost the entire range of glycerol volume fraction $\varphi$, even from a single low-RH experiment. While $a_w(\varphi)$ agrees well with literature values, our measurements yield a consistent fit for $D(\varphi)$. Complementary experiments with fluorescence microscopy confirm that buoyancy-driven convection, although present, remains negligible, so that mass diffusion dominates solute transport in this confined geometry. The overall agreement validates the methodology, demonstrating its robustness as a quantitative framework for probing drying dynamics and transport in complex fluids, with broad applicability to controlled evaporation studies.

Figures (10)

• Figure 1: 2D confined drying droplet configuration. (a) A liquid droplet of radius $R(t)$ dries while confined between two circular polydimethylsiloxane (PDMS)-coated glass wafers of radius $R_\text{W}$. The wafers are separated by a fixed gap of height $h$, maintained by spacers, forming a quasi-2D geometry with $h \ll R_\text{W}$. (b) Schematic cross-sectional view of the confined droplet and the main physical processes involved during its drying. The two color maps represent the two concentration gradients, in the droplet and in the gas phase.
• Figure 2: Physical properties of the water-glycerol mixture at $T=22^{\circ}$C. (a) Density $\rho$, (b) viscosity $\eta$, and (c) water chemical activity $a_w$ as a function of the glycerol volume fraction $\varphi$. Panel (a) corresponds to Eq. \ref{['equ:density']}, panel (b) is the empirical correlation of Ref. cheng_formula_2008, and panel (c) corresponds to Eq. \ref{['equ:bouchaudy_activity']}.
• Figure 3: Experimental setup for combined measurements of drying kinetics and concentration fields during drying. (a) Mach-Zehnder interferometer coupled with the 2D confined droplet cell inside a humidity-controlled chamber. (b) Typical raw interferogram of a confined drying droplet of a water-glycerol mixture recorded during an experiment. (c) Corresponding concentration field inside the droplet obtained after postprocessing of the interference fringes.
• Figure 4: Refractive index of the water-glycerol mixture as a function of the glycerol volume fraction at $\lambda=632.8$ nm and $T=22^{\circ}$C. Light and dark blue symbols represent autocalibration curves from interferometry experiments at two RH values, while black symbols correspond to independent measurements performed using a refractometer. Data are well described by the second-degree polynomial fit provided in Eq. \ref{['eq:refr_index']}.
• Figure 5: Drying kinetics of a confined 2D pure water droplet under different $\textrm{RH}$ levels. (a) Typical sequence of BF images for $R_0=1.56\textrm{ mm}$ and $\textrm{RH}=67\%$ (see also movie M1.avi in Ref. Supp_Mat). (b) Evolution of the normalized area $\alpha= A(t)/A_0$ over time for various $\textrm{RH}$ values. (c) Evolutions of $\alpha$ plotted against the normalized time $t/\tau_f$. Datasets collapse onto a single master curve in agreement with the analytical solution provided in Eq. \ref{['equ:analytical']}.