Capillary bridge instabilities during evaporation fall within a multiscale problem. At the microscale, i.e. the grain scale, they result from the combined effects of capillarity and evaporation, and can therefore be described in terms of Laplace and Kelvin equilibria, respectively (Everett and Haynes, 1972). When the pore diameter is constant or decreases ahead of the meniscus (in the defending liquid phase), both equilibria are stable; otherwise, they are unstable. The onset of Haines jumps (Haines, 1930) during evaporation thus occurs when the pore diameter goes from decreasing to increasing, i.e. at pore throats. A necessary condition for this to happen is that the system be “soft” (Sun and Santamarina, 2019); for instance, in a porous system, when the grains are deformable or when the meniscii are interacting. The transient state of the bridge depends on various factors, such as the fluid viscosity, the surface tension, the humidity, the temperature, which will influence the jumps intensity. Upon neglecting the thermal effects, the latter has been shown to be inversely proportional to the Ohnesorge number O h = η w / ρ w γ l s (Zacharoudiou and Boek, 2016), ratio of the viscous effect over the inertial and capillary effects, where η _w denotes the viscosity of the wetting fluid, ρ _w its density, γ the surface tension, and l _s a characteristic length scale of the system. When these instabilities occur serially and percolate through the mesoscale, i.e. at the multi-grain scale, one can speak of air entry (see Figure 1). To reduce the intensity of air entry, one can thus increase the Ohnesorge number by increasing the viscosity of the defending fluid, or by decreasing the surface tension. In turn, air entry is a precursor of desiccation cracks at the macroscale (Shin and Santamarina, 2011; Hueckel et al., 2014). Desiccation cracks deteriorate the mechanical strength of soils and increase their hydraulic conductivity, which can be particularly detrimental for earth embankment (Khandelwal, 2011), slope stability (Stirling, 2014) and geobarriers integrity, such as in nuclear waste disposals (Dixon et al., 2002) and landfills (Omidi et al., 1996). It is therefore paramount to better comprehend air entry and the associated Haines instabilities. In this contribution, we will focus on the temporal evolution of capillary bridges during evaporation, before and just after their rupture, by using digital image correlation and varying the separation between the grains, their material and their geometry. We consider three-grain clusters, which is the minimal number of grains to allow pore throat passage, representing air entry in the early stages of drying of geomaterials (see Figure 1). Two-grain bridges, which represent the terminal stages of drying, were already experimentally studied by Mielniczuk et al. (2014); it was found that the rupture of the bridges is accompanied by an abrupt decrease of capillary forces and cohesiveness of the granular assembly. We will show here that this abruptness can be more precisely characterized as stemming from tertiary creep, in the form of a Voight instability. Voight’s relation (Voight, 1988; Voight, 1989), Ω ̈ = A Ω ̇ α , (1) was empirically introduced as a general description of rate-dependent material failure under constant loading conditions; in Eq. 1, Ω is an observable quantity (here the meniscus displacement d), A > 0 and α > 0 are parameters, and the superposed dot denotes the differentiation with respect to time. This relation was first introduced by Fukuzono (1985) in the case of landslides. Following its generalization by Voight (1988), it was experimentally verified to hold for a wide range of materials, including geomaterials, metal, and ice (Voight, 1989). It proved particularly useful in predicting volcanic eruptions and landslides (see Veveakis et al. (2007) and Intrieri et al. (2019) for a recent review). Noticeably, the parameter α is found to be close to 2 in general (see Voight (1989) and Chang and Wang (2021) for more recent data).

However, Voight’s relation has not yet been explored at the microscale, where macroscopic instabilities originate, especially in the case of geomaterials (see Kawamoto et al. (2018), Rattez et al. (2018a), Rattez et al. (2018b), Lesueur et al. (2020), Guével et al. (2020), Guével et al. (2022), e.g.). We suggest here that this relation, in addition to describing field-scale instabilities, may also underpin their microscopic origin. In the context of capillary instabilities, Voight’s relation is to be envisioned through the creep failure of the capillary bridge upon the constant thermal loading imposed by evaporation, due to constant environmental temperature and humidity, analogously to the constant mechanical loading due to gravity in landslides. Whereas the instability onset is due to thermal runaway in landslides (Veveakis et al., 2007), it is due here to the geometrical configuration (pore throat passage).

The dynamics of the capillary bridges can be decomposed into two distinct regimes. Initially, the water volume decreases quasi-statically following the evaporation rate. Then, upon reaching the pore throat, corresponding to the onset of unstable Laplace and Kelvin equilibria, the meniscus abruptly accelerates until reaching a new stable equilibrium position in the form of pendular bridges, corresponding to air entry. We observe, for three-grain clusters, three modes of air entry, depending on the separation of the grains: 1) thin-film entry for the lowest separations, 2) asymmetric finger entry for intermediate separations, and 3) symmetric finger entry for the largest separations (see Figure 3 in the case of the glass cylinders and Mielniczuk et al. (2021) in the case of five-grain clusters). We note that in the third mode, symmetric air entry was not always guaranteed as symmetry breaking could sometimes occur, owing to the sensitivity of the experiments to external perturbations. As a result, the separation threshold between the second and third modes could not be precisely identified but lies, for instance, between S = 500 µm and S = 1,000 µm for the glass cylinders. That said, far enough below this threshold (and above the threshold with the first mode), asymmetric air entry was always obtained. The bridge finally reaches a post-instability pendular configuration, which analysis is already touched upon in Mielniczuk et al. (2014) and Mielniczuk et al. (2015), and will be further studied in coming works. Fitting to the focus of this contribution, we consider the time of air entry as the final time t _f , or failure time, which is used to normalize all temporal quantities (see Table 1), so that the normalized time is t* = t/t _f ∈ [0, 1]. The duration to reach air entry is called the lifetime of the capillary bridge.

In a given assembly of grains (see Figure 1 e.g.), air entry will occur through the largest pores, where less energy is demanded to displace the meniscus, as per Laplace’s law. Therefore, we may restrict our attention to configurations of largest grain separation, approximately S ≥ 500 μm, leading to finger-type instabilities. This assertion is verified in Figure 4 showing that the failure time t _f is a decreasing function of the separation S, which may be fitted with a power law.

We now proceed to characterizing air entry in time. Using the least-square method, we find that the evolution in time of the meniscus position during evaporation is well described by Voight’s relation Eq. 1 for α = 2 (see Eq. 5 in Voight (1988), with Ω₀ = 0 and t ₀ = 0), regardless of the separations and type of material: d * = − ln 1 − t * , (2) where the distance traveled by the meniscus d is normalized for each configuration by the fitting parameter A: d i * = d i / A i , where i denotes one the 16 configurations (see Table 1). Eq. 2 can also be expressed in the more familiar form d ̇ * = 1 / ( 1 − t * ) used in landslides prediction. The collapse of the self-similar curves onto Voight’s relation are shown in Figure 5.

While different types of Voight’s relation are possible, depending on the value of α in Eq. 1, we found that the best fit is provided by α = 2. In particular, the case α = 1 (exponential function) yields a similar overall fitting accuracy but does not capture the asymptote for t*→ 1.

We have proposed a temporal characterization of Haines jumps, through showing experimentally that they can be described as tertiary creep instabilities with Voight’s relation. In doing so, we adopted an engineering perspective, that encapsulates the complex mechanisms underlying capillary instabilities into a “black box” representation. A detailed physics-based analysis of these mechanisms will be pursued in upcoming works. Voight’s relation, popularized in the study of geohazards, is here extended to microscopic instabilities, suggesting surface asperities as a common origin. The multiscale validity of Voight’s law, at least for granular materials, indicates a clear coupling between the microscopic and macroscopic scales, which should be harvested to better predict and prevent material failure.