Surface ageing at drop scale

Abstract

Surface ageing of a micro-drop populated by surfactants below the critical micellar concentration and subject to evaporation is considered, motivated by our interest in the transport of biomolecules in digital microfluidics. The classical approach based on diffusion–sorption processes is reviewed in order to address a finite-sized system of digital microfluidics. Short-time and long-time asymptotic approximations for a diffusion-limited regime, as well as full analytical expressions for adsorption- and evaporation-limited regimes, are constructed, which help to validate numerical calculations of full coupling between all these kinetics. The impact of the small drop size and the continuous lack of equilibrium induced by evaporation are described by introducing specific dimensionless numbers. By taking into account evaporative mass transfer in the mass balance, the boundary condition for surfactant transport to and from the interface is modified. Furthermore, the small size of the geometry suggests allowing a novel non-dimensionalisation for surface concentration, following from the mass balance of surfactant molecules in thermo-dynamic equilibrium. This order of magnitude quantifies the equilibrium surface concentration for small-sized geometries.

Appendix: Instantaneous diffusion within a sessile drop of arbitrary contact angle

The present model is extended to the case of a sessile drop with an arbitrary contact angle. A regime with (quasi) instantaneous diffusion is considered. The geometry of a (small enough) sessile drop corresponds to a spherical cap of radius R, with a contact angle 0 < θ ≤ π. The diffusion kinetics is considered much faster than the adsorption and evaporation kinetics (τ_diff ≪ τ_ads, τ_evap), such that the transport equation (2) does not need to be considered. Instead, the relation between surface and volume concentration, $\Upgamma$ and C, respectively, may be expressed by the non-dimensional equation for mass balance:

$$ C=\frac{1}{R^3}-\frac{3F}{\varepsilon_0 R}\Upgamma, $$

(18)

where the form factor, $F=2(1-\cos\theta)/(2-3\cos\theta+\cos^3\theta),$ is introduced, translating the dependences of the drop volume and drop surface on the contact angle, θ. Hence, the Henry law (1) may be reformulated to express the surfactant transport at the surface:

$$ j_{{\rm ads}} = \eta_{{\rm ea}} \left[\frac{1}{R^3} - \left(\frac{3F}{\varepsilon_0 R}+1\right)\Upgamma\right]. $$

(19)

Furthermore, by reconsidering the mass balance for a spherical cap, the impact of the variable radius on surface concentration may be expressed:

$$ j_{{\rm evap}}= -\frac{{\rm d} R}{{\rm d} t} \left(\frac{\varepsilon_0}{3FR^3} + \frac{\Upgamma}{R}\right). $$

(20)

Combining Eqs. (19) and (20) and writing,

$$ \frac{\hbox{d}\Upgamma}{\hbox{d}t} = j_{{\rm ads}}+ j_{{\rm evap}}, $$

an ordinary differential equation for the surface concentration can be derived as

$$ \frac{\hbox{d}\Upgamma}{\hbox{d}t} = P(t)+ Q(t)\Upgamma, $$

(21)

with the time-dependent coefficients,

$$ P(t)= \frac{1}{R^3}\left(\eta_{{\rm ea}} -\frac{\varepsilon_0}{3F}\frac{{\rm d} R}{{\rm d} t}\right), $$

and

$$ Q(t)=-\left(\frac{1}{R}\frac{{\rm d} R}{{\rm d} t} +\frac{3F}{\varepsilon_0 R}+1\right). $$

The evaporation law (9) also has to be modified for arbitrary contact angles:

$$ R(t)^2 = 1-\hat F t /\tau_{{\rm evap}}, $$

with $\hat F = 2/[(1-\cos\theta)(2+\cos\theta)]$ (McHale et al. 1998).

From this last expression for the time-dependent radius of the spherical cap, the differential equation (21) can be solved by making use of the integrating factor, $\exp(\int P(t)\;\hbox{d}t).$

Also, the surface concentration equilibrium value for microscopic systems, $\Upgamma_{{\rm e}}^\mu,$ depends on the contact angle. It may be calculated in diffusion–adsorption equilibrium $(\hbox{d}\Upgamma/\hbox{d}t =0)$ without evaporation:

$$ \Upgamma_{{\rm e}}^\mu = \frac{\Upgamma_{{\rm e}}}{1-\frac{3F}{\varepsilon_0}}. $$

Figure 5 represents the dimensionless curves for different contact angles ranging from π/6 to π, for the non-dimensional numbers, $\varepsilon_0=0.1$ and η_ea = 1. It is remarkable that in the hydrophobic range, π/2 ≤ θ ≤ π, surface ageing is hardly affected by a change in the contact angle, whereas in the hydrophilic range, π/6 ≤ θ ≤ π/2, the impact of the contact angle becomes significant.

Fig. 5

Impact of the contact angle, θ, upon surface ageing $(\varepsilon_0 = 0.1, \eta_{{\rm ea}} =1).$ Hydrophilic, hydrophobic and superhydrophobic contact angles are considered. The curves for θ = π/2 and θ = π yield consistently the same result as the one issued from the generalized formulation based on a spherical geometry (see Fig. 2a for comparison). Time is measured in units of τ_evap. The insert shows the same curves during a longer time (up to 0.5 τ_evap)