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

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 $D(φ)$ and the water chemical activity $a_w(φ)$ over almost the entire range of glycerol volume fraction $φ$, even from a single low-RH experiment. While $a_w(φ)$ agrees well with literature values, our measurements yield a consistent fit for $D(φ)$. 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.

💡 Research Summary

In this work the authors develop a quantitative experimental platform that couples Mach‑Zehnder interferometry with a custom humidity‑controlled chamber to study the drying of a confined two‑dimensional droplet composed of a binary liquid mixture (water–glycerol). The droplet is sandwiched between two PDMS‑coated glass wafers separated by a fixed gap of 150 µm, creating a quasi‑2D geometry where the height is much smaller than the lateral size (initial radius 1.4–2.0 mm). This configuration ensures isothermal conditions, suppresses free convection, and allows the droplet to shrink axisymmetrically while maintaining a well‑defined liquid‑air interface.

Interferometric imaging is performed with a stabilized He‑Ne laser (λ = 632.8 nm). The refractive index of the mixture depends linearly on the glycerol volume fraction φ, so the recorded interference fringes can be converted, after image processing, into spatial maps of φ(r,t) with a pixel resolution of 6 µm and an accuracy of ±0.5 %. Simultaneously, the droplet radius R(t) is extracted from the same images, providing the drying kinetics. The humidity chamber maintains relative humidity (RH) between 25 % and 95 % with a precision of ±3.5 %, enabling systematic variation of the Péclet number Pe = R₀²D₀/τ_f, which compares the characteristic diffusion time across the droplet to the overall drying time τ_f. By adjusting RH, Pe is varied over roughly two orders of magnitude (from ≈0.4 to 2 × 10⁻¹⁰ m² s⁻¹), spanning regimes where diffusion either smooths concentration gradients (Pe ≪ 1) or strong gradients develop at the receding edge (Pe ≫ 1).

Theoretical modeling consists of two coupled parts. First, vapor diffusion in the surrounding gas is assumed quasistatic; solving Laplace’s equation yields the vapor concentration field C(r) and the evaporative flux, which depends on the water activity a_w(φ) at the interface and the imposed RH. This leads to an ordinary differential equation for the droplet radius (Eq. 2). Second, within the liquid, glycerol is non‑volatile, and its volume fraction obeys an axisymmetric diffusion equation with a concentration‑dependent mutual diffusion coefficient D(φ) (Eq. 5) and a moving boundary condition that couples to the evaporative flux (Eq. 6). The average φ(t) is linked to the shrinking area via a simple mass‑balance (Eq. 7).

Experimental data for R(t) and φ(r,t) at multiple RH values are compared with numerical solutions of the coupled model. The measured water activity a_w(φ) matches the empirical expression from Bouchaudy et al. (Eq. 12) across the whole concentration range, confirming that the interfacial thermodynamic equilibrium assumption is valid. By fitting the concentration fields, the authors extract D(φ) over the glycerol volume‑fraction range φ ≈ 0.2–0.9. The resulting D(φ) curve is consistent with scattered literature values (Refs. 51, 58‑61) but provides a continuous, high‑resolution dataset; notably, a single low‑RH experiment (≈30 %) suffices to determine D(φ) over the entire range because the strong concentration gradients amplify the sensitivity of the interferometric signal.

To verify that buoyancy‑driven convection does not invalidate the diffusion‑only assumption, complementary fluorescence microscopy experiments track tracer particles within the droplet. The observed convective velocities are at least an order of magnitude smaller than the diffusive transport, confirming that mass diffusion dominates under the confined geometry. Additional tests on PDMS coating thickness show that thin layers (≈24 µm) do not absorb enough water to affect drying, whereas thicker layers introduce measurable deviations.

Overall, the study demonstrates that high‑resolution interferometry combined with precise humidity control provides a robust, quantitative framework for probing drying dynamics and transport properties in binary (and potentially more complex) fluids. The methodology enables simultaneous measurement of drying kinetics and internal concentration fields, allowing direct extraction of thermodynamic (a_w) and transport (D) parameters from a single experiment. This approach is broadly applicable to fields such as inkjet printing, coating technologies, battery electrode fabrication, and the study of respiratory droplet evaporation, where controlled drying and accurate knowledge of solute transport are critical.